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[
"NONCONVEX REGULARIZATION FOR SPARSE NEURAL NETWORKS",
"NONCONVEX REGULARIZATION FOR SPARSE NEURAL NETWORKS"
] |
[
"Konstantin Pieper ",
"Armenak Petrosyan "
] |
[] |
[] |
Convex 1 regularization using an infinite dictionary of neurons has been suggested for constructing neural networks with desired approximation guarantees, but can be affected by an arbitrary amount of over-parametrization. This can lead to a loss of sparsity and result in networks with too many active neurons for the given data, in particular if the number of data samples is large. As a remedy, in this paper, a nonconvex regularization method is investigated in the context of shallow ReLU networks: We prove that in contrast to the convex approach, any resulting (locally optimal) network is finite even in the presence of infinite data (i.e., if the data distribution is known and the limiting case of infinite samples is considered). Moreover, we show that approximation guarantees and existing bounds on the network size for finite data are maintained.
|
10.1016/j.acha.2022.05.003
|
[
"https://arxiv.org/pdf/2004.11515v2.pdf"
] | 249,210,069 |
2004.11515
|
2f4978fc41ffae4994c0dbeceae732d23888fa3c
|
NONCONVEX REGULARIZATION FOR SPARSE NEURAL NETWORKS
Konstantin Pieper
Armenak Petrosyan
NONCONVEX REGULARIZATION FOR SPARSE NEURAL NETWORKS
Convex 1 regularization using an infinite dictionary of neurons has been suggested for constructing neural networks with desired approximation guarantees, but can be affected by an arbitrary amount of over-parametrization. This can lead to a loss of sparsity and result in networks with too many active neurons for the given data, in particular if the number of data samples is large. As a remedy, in this paper, a nonconvex regularization method is investigated in the context of shallow ReLU networks: We prove that in contrast to the convex approach, any resulting (locally optimal) network is finite even in the presence of infinite data (i.e., if the data distribution is known and the limiting case of infinite samples is considered). Moreover, we show that approximation guarantees and existing bounds on the network size for finite data are maintained.
Introduction
Many currently employed neural network training procedures are based on minimizing nonconvex functionals, and employ random initialization and stochastic optimization methods to avoid nonoptimal local minima. An alternative way to address these problems is by convex reformulation of the objective. Here, the network architecture can be adapted during training by gradually adding neurons, while the output layer weights can be penalized by a convex sparsity-promoting functional (or regularization term). The latter approach has the potential to set redundant network connections to zero during the training, which can be subsequently removed from the network. Apart from reducing the network size, which improves computational efficiency of network evaluation, this also serves the purpose of reducing the generalization error in the model due to weight shrinkage. For early works on convex neural networks and their theoretical analysis based on infinite feature spaces and optimization problems on spaces of measures, we refer to, e.g., [4,40,2]. Additional characterizations of the associated optimal networks have recently been provided in [41,34,14,35,33,15].
In this paper, we demonstrate that convex sparsity promoting penalties such as the canonical 1 norm employed in the previous works do not always effectively eliminate redundancy in the presence of large amounts of data, and the associated training procedures may still be affected by a certain amount of unnecessary over-parametrization. As a remedy, we develop a corresponding framework that incorporates nonconvex penalties but keeps most of the aforementioned advantages of convex neural networks, including the measure space interpretation. In order to do that, we focus on shallow networks with one hidden layer, which are of fundamental importance and -compared to deep networks -are relatively well understood theoretically. Moreover, we focus attention on the activation function given as the popular ReLU function σ(z) = max{ z, 0 }; however, most of our theory applies to a much larger class of activation functions and also other kernel based methods. We define a shallow neural network with N neurons to be a function N : R d → R of the form
(1) N ω,c (x) = N n=1
c n σ(a n · x + b n ),
where σ(a n · x + b n ) are the single neurons and N is also called the width of the network. Additionally, by ω n = (a n , b n ) ∈ R d+1 we denote the nodes consisting of inner weights a n ∈ R d and b n ∈ R. The numbers c n ∈ R are called the outer weights (output layer weights). Let f be a target function defined on some set D ⊆ R d (it can be an image, solution to a PDE, a specific parameter associated with a model, etc.). The aim is to find a neural network N with as few as possible neurons N such that N fits the training points {(x k , y k )} k=1,...,K . Here, x k ∈ D are the input data and y k the output data, which we assume to be given as y k = f (x k ) + ε k . Additionally, ε k represents error, which could be random in nature (e.g., measurement error) or have a deterministic origin (e.g., modeling or computational truncation error). The network training considered here will be based on the following minimization problem:
(2) min N ∈N, {an,bn,cn} N n=1 , an 2 +|bn| 2 ≤1 l (N ω,c ; y) + α N n=1 φ(|c n |).
For simplicity, we focus our attention on the least squares loss function (3) l (N ω,c ; y) = l K,x (N ω,c ; y) :
= 1 2K K k=1 |N ω,c (x k ) − y k | 2 ,
although most of our results can be transferred to a much more general class of data fidelity terms. We emphasize that in problem (2) we do not fix the width N of the network, which will be chosen together with the corresponding coefficients to minimize the objective. To achieve a compromise between a good fit and a simple network (representing a hopefully regular function with few neurons N ) we employ a sparsity promoting sublinear cost term involving the scalar penalty function φ for the outer weights with hyperparameter α > 0. Note that N ω,c is linear in c and that neurons with c n = 0 can be dropped from the network. For the inner weights, we impose bound constraints. This does not limit the set of functions that can be written as (1), owing to the positive homogeneity of the ReLU activation function, but prevents the inner weights from growing arbitrarily large. Concerning the penalty φ : R + → R + , we always impose the following assumption: (A1 φ ) The function φ is a concave, nondecreasing, and differentiable with φ(0) = 0, φ (0) = 1, and φ(z) → +∞ for z → +∞; Moreover, φ is γ-convex, i.e., there exists a γ ≥ 0 such that the derivative φ fulfills −γ(z 2 − z 1 ) ≤ φ (z 2 ) − φ (z 1 ) ≤ 0 for all 0 ≤ z 1 ≤ z 2 . These assumptions imply that φ fulfills φ(z) ≤ z for all z ≥ 0 and that φ is subadditive, i.e. φ(z 1 + z 2 ) ≤ φ(z 1 ) + φ(z 2 ). This property enhances the sparsity of the solution, and in the case that the inequality is strict (referred to as strongly subadditive), it actively promotes it. We will discuss this in more detail below. Moreover, positive homogeneity of the ReLU activation function and the monotonicity of φ also imply that (a n , b n ) = 1 will always be fulfilled for an optimal solution of the problem (2). Consequently, (2) is equivalent to the following problem where S d = {(a, b) ∈ R d+1 : (a, b) 2 = 1} is the unit sphere in R d+1 .
Note that the convex 1 sparsity promoting penalty (i.e. φ(z) = z) is still included in the above assumptions. It is known that the problem for φ(z) = z can be understood as a convex problem on the space of measures, and always admits a global solution with N ≤ K. Moreover, this solution can be efficiently approximated with (generalized) conditional gradient methods (see, e.g., [2,9,8,39]). While the 1 problem with φ(z) = z has many favorable theoretical properties, it does not completely solve the issue of over-parametrization, especially in the case where K is large. In Figure 1, we visualize this with a simple one-dimensional example, where we can interpret N ω,c as a piecewise linear spline with knot points x n = −b n /a n . We see that a nonconvex penalty term is able to substantially reduce the number of neurons without affecting the quality of the approximation. , y k = f (x k ) + ε k perturbed by white noise with std. dev. 0.05. Top: outer weights c n over the knot points x n = −b n /a n . Bottom: Noisy data y k = f (x k ) + ε k (blue), locally optimal network N ω,c (black), and knot points of the corresponding linear spline (orange).
To shed more light on this effect, we consider the limit of l K,x where the number of data points K grows infinitely large. If we interpret the data points x k to be random samples from a probability distribution ν on the domain D, the loss function l K,x can be understood as the empirical estimation of the loss
l ν (N ω,c ; y) = 1 2 D (N ω,c (x) − y(x)) 2 dν(x) = 1 2 N ω,c − y 2 L 2 (D,ν) , where y(x) = f (x) + ε(x)
, and ε(x) is an error term. Consequently, we consider the problem (2) with l = l ν , which can be seen as approximating the function in the whole domain D instead of a finite number of points.
Since N is maintained as a free optimization variable in (P φ ), one may expect that in this case the problem may not have minimizers (local or global): a network with larger and larger number of neurons could decrease the value of the functional. That is indeed the case for 1 regularization as our numerical experiments (performed for large K) indicate. The solutions of problem (P φ ) with the 1 -penalty (φ(z) = z) tend to form clusters of nodes (a n , b n ) with very small coefficients c n (this effect is more severe for a larger number of training data K). The main disadvantage of the 1 cost functional is that it is not encouraging nodes (a n , b n ) that are very close to merge into one. This is related to the additivity of the absolute value on the positive real axis: if we replace a node (a, b) with coefficient c by two nodes (a 1 , b 1 ) and (a 2 , b 2 ) very closely placed to (a, b), with corresponding coefficients c 1 and c 2 (of the same sign as c) with c = c 1 + c 2 we obtain a network with the same 1 -norm, while potentially decreasing the fidelity term l. However, switching to a nonconvex, strongly subadditive penalty, this effect is remedied; as illustrated in Figure 1b.
The above discussion highlights why the 1 regularized problem
(P 1 ) min N ∈N, {cn}∈R N , {ωn=(an,bn)}∈(S d ) N l (N ω,c ; y) + α N n=1 |c n |
is not the best choice for sparsity promoting regularization of neural networks. Certainly, this also affects formulations where we replace the penalty term in (P 1 ) with a constraint c 1 ≤ M , or the fidelity term with a constraint N ω,c − f ≤ δ, since they essentially lead to the same solution manifolds (parametrized by different hyperparameters α, M , and δ). Moreover, we point out that the global solutions of the problem (P 1 ) are also global solutions of the popular formulation below, employing 2 regularization:
(P 2 , 2 ) min N ∈N, {(an,bn,cn)}∈(R d ×R×R) N l (N ω,c ; y) + α 2 N n=1 (a n , b n ) 2 + |c n | 2 .
This equivalence relies on the positive homogeneity of the ReLU activation function; see, e.g., [32]. Thus, (P 2 , 2 ) is surprisingly already equivalent to a sparsity regularized problem in this setting, and is also affected by the same issues as (P 1 ). More generally, if we replace the cost term in (P 2 , 2 ) with R(a, b, c) = (1/p) n (a n , b n ) p + |c n | p , we obtain the problem formulation (P φ ) with φ(z) = (2/p) z p/2 (see Appendix F). For 0 < p < 2, the choice of φ(z) = (2/p) z p/2 is concave, monotonous, and subadditive, and would be appropriate for some parts of this paper. However, it does not fulfill the other requirements (A1 φ ) imposed above, since it has an unbounded derivative at zero and can not be normalized to fulfill φ (0) = 1. Since this assumption is crucial for this paper, we instead consider a strongly subadditive function φ : R + → R + fulfilling (A1 φ ) and the following additional assumption: (A2 φ ) There exists a γ > 0 and z > 0 such that
φ (z 2 ) − φ (z 1 ) ≤ − γ(z 2 − z 1 ) for all 0 ≤ z 1 ≤ z 2 ≤ z.
It can be observed that any φ possessing the properties (A1 φ ) and (A2 φ ) also satisfies the inequalities
z − (γ/2)z 2 ≤ φ(z) ≤ z − ( γ/2)z 2 for z ∈ [0, z]. The function (4) φ log,γ (z) = 1 γ log(1 + γz) = z 0 1 1 + γζ dζ
for γ > 0, which is a scaled version of the log-penalty function (considered in, e.g., [29]), and its convex combination with the 1 norm, will be the main function of choice for us in the numerical examples. Another option is the MCP function [46],
MCP γ (z) = z 0 max{ 0, 1 − γζ } dζ = z − (γ/2)z 2 for z < 1/γ, 1/(2γ) else,
which fulfills (A2 φ ) for γ > 0, but lacks the property φ(z) → +∞ for z → +∞. However, a proper convex combination of MCP with 1 , e.g. φ(z) = (1/2)(z + MCP 2γ (z)), fulfills both (A1 φ ) and (A2 φ ). We discuss the former choices in the context of the general assumptions in Appendix A. Another penalty function, the SCAD penalty function, given by
SCAD γ,λ (z) = z 0 min {1, max{0, 1 − γ(ζ − λ)}} dζ = z for z < λ z − (γ/2)(z − λ) 2 for λ ≤ z < λ + 1/γ, λ + 1/(2γ) else,
where λ > 0 and γ > 0, does not fulfill (A2 φ ) due to being linear close to z ≈ 0. We refer to Figure 2 Figure 2. Comparison of different penalty functions φ and their derivatives φ . The 1 and log-penalty fulfill (A1 φ ), MCP and SCAD fulfill (A1 φ ) aside from φ(z) → +∞ for z → ∞. The log-penalty and MCP also fulfill (A2 φ ).
for a visualization of different penalty functions.
1.1. Contribution. For φ satisfying the conditions (A1 φ ) and (A2 φ ) we show that the problem (P φ ) has global and local minimizers (with finite N ) for finite and infinite data (see Theorem 3). Since the existence of finite minimizers of the convex problem relies on the finite data, and this is not the case for the nonconvex problem, this is a rather unexpected result.
Since the regularization term in (P φ ) is nonconvex, finding the global solution of the problem may not be feasible. In fact, for a nonconvex optimization problem of similar structure, finding global minima with very high precision is shown to be an NP-hard problem; see [12]. On the other hand, it is observed in practice and confirmed in theory (see, e.g., [27]) that local minima (or stationary points, in general) of nonconvex regularized problems tend to be well behaved. We will develop a similar theory for the problem (P φ ), where N is a free optimization variable. First, we will define a concept of local minimality, which is based on the notion of locality in the space of shallow neural networks of the form (1) defined in terms of the associated measure
(5) µ = N n=1 c n δ (an,bn) ,
where δ (an,bn) is the Dirac measure at (a n , b n ). We will refer to these solutions as "local solutions in the sense of measures", which will be further explained in the context of integral neural networks discussed in Section 2. We only mention that a local solution in our setting will be any measure (5), where the outer weights c n are minimal on in a suitable neighborhood, and where adding to N ω,c any additional node (a, b) ∈ S d with suitably small outer weight c will also increase the training objective; see Theorem 4. For these local solutions in the sense of measures of (P φ ) (which also include the global solutions), we show that:
• They are always finitely supported; see Theorem 3. • In the case K < ∞ it holds N ≤ K; see Theorem 6.
• The approximation error can be estimated by
l(N ω,c ; f ) ≤ 2 α f W(D) + l(y; f )(6)
for any f ∈ W(D), as introduced in Section 2.2; see Theorem 5.
The last point quantitively affirms the assertion that the hyperparameter α can be treated as a trade-off between the network sparsity and reconstruction accuracy (at least for a well-behaved subclass of functions f ). In particular, it shows that local solutions in the sense of measures of (P φ ) can reduce the fitting term to a similar level as the level of noise (or bias) in the data by an appropriate choice of α, i.e., α ≤ C l(y; f ).
Finally, we propose a method to algorithmically approximate local solutions in the sense of measures of (P φ ). This is based upon an extension of the methods developed for 1 regularization [9,8,2,39,19] to the setting of nonconvex penalties φ. Here, we combine adaptive node insertion and deletion with local minimization of the outer weights (or, optionally, full gradient-based training of all the weights (a, b, c)). Again, we rely on the property φ (0) = 1 and the corresponding optimality conditions to guide the node insertion and deletion steps.
1.2.
Limitations. The deterministic error estimate is one of main points that justify the presented solution concept. Let us briefly mention several shortcomings of the presented approach and discuss possible extensions, which are, however, outside of the scope of this paper.
• The bound (6) is only applicable to well-behaved (regular) functions in the space W(D), which is endowed with a norm that is sometimes referred to as the variation of f = N ω,c ; cf., e.g., [2,24], the references therein, and Section 3 for a detailed discussion. Since the variation is defined here in the canonical way (independent of the functional φ), we can apply known results to extend the bound to less regular functions g (e.g., Lipschitz-continuous). We first approximate g by a function g ε ∈ W(D) with l(g; g ε ) ≤ ε 2 ; cf., e.g., [2,Section 4.3].
Then we can use (6) with f = g ε and apply the triangle inequality for an estimate with g. • The error of the fit in (6) is defined in terms of the functional used for training. In practice, the exact distribution ν is usually not available or computationally accessible, and training is performed on a sample, using l K,x for noisy data y. However, bounds for the error N ω,c −f are desired for every x ∈ D. This problem of generalization can be addressed by combining (6) with estimation bounds for l ν (f ; y) − l K,x (f, y) over all f W(D) ≤ δ; see, e.g., [2, Section 5]. • The property φ (0) = 1, which is fundamentally required for (6), could be replaced by φ (0) < +∞ with minor modifications. However, as mentioned before, Assumption (A1 φ ) excludes the "q-norms" φ(z) = (1/q) z q with q < 1 since here φ (0) = +∞. In this case, inserting additional neurons with sufficiently small outer weights c will always increase the objective (due to φ (0) = +∞) and thus the zero measure is always a local solution in the sense of measures. Thus, no approximation guarantees can be given for arbitrary local solutions. Certainly, there can be local solutions that fit the data properly, but finding them algorithmically with gradient based optimization has to rely on an appropriate initialization. In particular, the node insertion strategy of the algorithm presented in Section 4 relies on φ (0) = 1 and can not be directly adapted to φ(z) = (1/q) z q .
1.3. Related work. Sparsity has been widely employed with the dual purpose of removing noninformative connections from neural networks [17,21] and also to guide adaptive architecture search [4,2,13]; in our case it is the adaptive choice of the network width. Training procedures with nonconvex penalties have been employed in order to eliminate certain weights from the network. In [28] a nonconvex penalty with φ(z) = (β + 1)z/(β + z) is proposed. Note, that a rescaled version of this penalty fulfills all the requirements of our analysis. Similarly, in [45] a different nonconvex regularization strategy is proposed that is based on the ratio of 1 and 2 norms and does not have separable form as we consider in (P φ ). However, unlike discussed here, these works apply the penalty to all the weights in the network with a fixed architecture and do not consider N to be variable. Moreover, the nonconvex regularization approaches proposed above rely on random initialization to find a "good" local minimum by local optimization and do not lead to any deterministic approximation guarantees, which is a general problem known to negatively affect training procedures based on random initialization and local all-weight training (see, e.g., [1]).
Nonconvex penalties for sparse regularization have been considered in the statistics literature. The functions like SCAD, the MCP and capped 1 are popular choices [18,47,46,44]. We remark that the MCP penalty fulfills most of the conditions of our analysis; cf. Figure 2. However, the dictionary and the data set can be infinite in our work, and the existing results do not directly apply to the neural network model we consider here.
Nonconvex functionals on spaces of measures also appear in the study of gradient regularization, such as problems involving functions of bounded variation (TV-norm of the gradient). In fact, such problems initially prompted the characterization of lower-semicontinuity of nonconvex functionals of measures [5,6,7], which is also used for the existence theory provided here. However, the gradient of a function can only have atomic parts (Dirac delta functions) in one spatial dimension, and not in higher spatial dimensions. Therefore, nonconvex regularizers of the gradient face additional challenges [22], and are usually implemented after discretizing the problem. We note that the aforementioned works do not provide the results from Section 2 on finitely supported minima of optimal solutions under the additional condition (A2 φ ).
Integral neural networks have been around for a while and there is a large volume of work in this direction; see, e.g., [2,33]. Integral neural networks have, in particular, been used for demonstrating approximation capabilities of shallow neural networks [3,26,16]. Under the integral representation assumption, it is possible to prove certain convergence rates for greedy type algorithms. Other related work includes integral neural network representation results like those in [23,25] and the ridgelet transform [11,30,42].
Organization.
In Section 2, we develop the main theoretical framework as outlined above. We introduce integral neural networks, and extend problems (P 1 ) and (P φ ) to this framework. Moreover we state the main contributions concerning various properties of local solutions in the sense of measures, their existence, necessary conditions, finiteness, and good fidelity. Section 3 is about the least total variational norm solution of the exact representation constrained problem. This problem is strongly connected to the fidelity estimate in Theorem 5. In Section 4 we propose an algorithmic solution method, which is an adaptation of the generalized conditional gradient method to the nonconvex setting. Finally, in Section 5, we illustrate the results of this paper with concrete examples in one to three dimensions. The paper ends with an Appendix, where we provide proofs of the theorems from Section 2.2.
General theory
Problem (P φ ) is a particular case of a more general framework that will be discussed in this section. We will make use of the concept of integral neural networks which is a generalization of the neural network (1) where the sum in N ω,c (x) is replaced with an integral. We will introduce the extensions of problems (P φ ) and (P 1 ) for integral neural networks, and obtain various analytic results concerning their local solutions. Most of the results here apply to a larger class of activation functions than just the ReLU so the theory will be developed in this setting.
Integral neural networks.
Let Ω be a compact subset of R d+1 . Denote by M (Ω) the space of real-valued Borel measures on Ω of bounded total variation, and by µ M (Ω) the total variation norm of the measure µ ∈ M (Ω). Consider a set D ⊂ R d and a function σ ∈ C(D × Ω) such that for some Λ > 0 it holds
|σ(x; ω 1 ) − σ(x; ω 2 )| ≤ Λ (1 + x ) ω 1 − ω 2 for all x ∈ D, ω 1 , ω 2 ∈ Ω,(7)
and |σ(x; ω 0 )| ≤ Λ(1 + x ) for some ω 0 ∈ Ω and all x ∈ D. Note that with the ReLU activation function, i.e. σ(x; ω) = max{ a·x+b, 0 }, this condition is satisfied with Λ = 1 for ω = (a, b) ∈ Ω = S d .
An integral neural network is a function of the form
[N µ](x) = Ω σ(x; ω) dµ(ω) where µ ∈ M (Ω). Finally, by ϕ, µ = Ω ϕ(ω) dµ(ω),
we denote the canonical duality pairing of ϕ ∈ C(Ω) and µ ∈ M (Ω) = C(Ω) * .
To see that the above integral network is an extension of the network in (1)
, let Ω = S d , σ(x; ω) = σ(a · x + b) for ω = (a, b) ∈ S d , and define the discrete measure (8) µ = N n=1 c n δ ωn ∈ M (Ω).
Then it can be observed that
(9) [N µ](x) = N n=1 c n σ(a n · x + b n ) = N ω,c (x).
Additionally, it holds that
µ M (Ω) = N n=1 |c n | = c 1 , ϕ, µ = N n=1 ϕ(ω n )c n = (ϕ(ω), c) R N ,
which relates the total variation norm of µ to the 1 norm of c and the duality pairing to an Euclidean inner product of the vector (ϕ(ω n )) n ∈ R N with c. Now, we turn to the loss function. Let ν be a probability measure supported on the set D = supp(ν) ⊆ R d with finite first and second moments; i.e. D x 2 dν(x) = x 2 L 2 (D,ν) < ∞. Associated to this, we define the Hilbert space L 2 (D, ν) of square integrable functions with respect to ν. The last property ensures that N is bounded as an operator from M (Ω) to L 2 (D, ν), where we let N = max ω∈Ω σ(·; ω) L 2 (D,ν) < ∞ be its operator norm. Note that σ(·; ω) L 2 (D,ν) ≤ σ(·; ω) − σ(·; ω 0 ) L 2 (D,ν) + σ(·; ω 0 ) L 2 (D,ν) , and both terms can be bounded using using (7) by CΛ(1 + x L 2 (D,ν) ), where C only depends on the compact set Ω. Moreover, let f ∈ L 2 (D, ν) be the target function we aim to approximate with integral neural networks. From the observation above, it can be seen that a consistent extension of the 1 -regularized problem (P 1 ) from finitely supported to arbitrary measures is given as
(P L 1 ) min µ∈M (Ω) L(µ) + α µ M (Ω) where L(µ) = l(N µ; y) = 1 2 N µ − y 2 L 2 (D,ν) .
Here, y = f + ε ∈ L 2 (D, ν) is a potentially biased or noisy version of function f to be approximated. We remark that the empirical functional l K from (3) is still included in this formulation by the choice
ν = ν K = (1/K) k δ x k and identifying y ∈ L 2 (D, ν K ) with the vector y k = y(x k ) = f (x k )+ε(x k ) = f (x k ) + ε k .
To extend the problem (P φ ) for integral neural networks, we need to define the analog of the penalty term in (P φ ) for arbitrary measures µ ∈ M (Ω). To do this, we first recall that any finite measure µ can be uniquely decomposed into an atomic part, which is a (potentially infinite) sum of Dirac-delta measures, and the remaining continuous part. Denote by atom µ = { ω n } n the atoms of µ ∈ M (Ω), of which there are either a finite number or countably infinitely many. Then we define
(10) Φ(µ) = |µ| (Ω \ atom µ) + n φ(|µ|({ ω n })),
where |µ| is the total variation measure of µ. Defining the decomposition into atomic and continuous part µ = µ atom + µ cont , where µ atom = n c n δ ωn with c n = µ({ ω n }) and µ cont = µ| Ω\atom µ , we can equivalently write
Φ(µ) = µ cont M (Ω) + n φ(|c n |).
We note that this functional is weakly lower semicontinuous with respect to weak- * convergence (see [5,Theorem 3.3]), which will be important in the following. Here, the assumption φ (0) = 1 is essential, otherwise the first term in the definition of Φ would need to be multiplied by φ (0). Moreover, Φ is identical to the weakly- * lower semicontinuous envelope of Φ atom (µ) = ω∈atom µ φ(|µ|({ ω })); see [7,Theorem 3.2]. Clearly, Φ(µ) = µ M (Ω) for a measure with no atoms and in general the following inequality holds
φ( µ M (Ω) ) ≤ Φ(µ) ≤ µ M (Ω)
, using the sub-additivity of φ; see Lemma 1 in Appendix A. As the canonical generalization of the problem (P φ ), we then consider the following problem
(P Φ ) min µ∈M (Ω) L(µ) + α Φ(µ).
One advantage we gain from expanding the definition of finite width neural networks to infinite width neural networks is that measures come equipped with the total variation norm topology that allows us to define a concept of locality in a straightforward way.
2.2.
Local solutions of the φ-regularized problem. Finding the global solution of the nonconvex problem (P Φ ) may not be realistic, so instead, we will investigate its local minima, and show that they possess desirable properties. The first result is to show that minima of the functional (11) J(µ) = L(µ) + α Φ(µ) in fact exist. To this purpose, we introduce an appropriate notion of a local minimum.
Definition 1.μ ∈ M (Ω) is a local minimum if there exists an > 0, such that J(µ) ≥ J(μ) for all µ ∈ M (Ω) with µ −μ M (Ω) ≤ .
The next theorem establishes the existence of minimizers under the minimal assumptions (A1 φ ), which also cover the 1 -penalty function.
Theorem 1. If φ satisfies conditions (A1 φ ), then (11) admits at least one global minimizer in M (Ω) which is bounded in terms of the data:
φ µ M (Ω) ≤ l(0; y) α .
The proof is given in Appendix A. Note, that the global minimum is also a local minimum; however, the optimization algorithm that we employ in practice can only approximate a local minimum. Moreover, the local and global solutions of (P Φ ) only correspond to solutions of (P φ ) in a generalized sense; thus far, there is no guarantee that the solutions are discrete, i.e. have a representation as in (8). To analyze this, we first derive first-order conditions for the local solutions.
To characterize local solutions of (P Φ ) with a first-order necessary condition, we require some additional notation. Denote by ∇L(µ) the gradient of the loss function L, which is defined as
∇L(µ), u = lim τ →0 (1/τ ) [L(µ + τ u) − L(µ)] , ∀u ∈ M (Ω). It holds that ∇L(µ) = N * ∇l(N µ; y) = N * (N µ − y) ∈ C(Ω), where N * : L 2 (D, ν) → C(Ω), [N * g](ω) = D σ(x; ω)g(x) dν(x) ∀g ∈ L 2 (D, ν)
is the (pre-)adjoint of N . The gradient p = N * (N µ − y) ∈ C(Ω) will be called the dual variable in the following. We note that the dual variable gives the inner product of the residual N µ − y with σ(·; ω), i.e.p
(ω) = D σ(x; ω)[N µ − y](x) dν(x) = (σ(·; ω), N µ − y) L 2 (D,ν) .
It serves to characterize the local solutions of (P Φ ) as follows.
Theorem 2. Let φ fulfill the conditions (A1 φ ). Ifμ is a local minimum of functional J, then the optimal dual variablep = ∇L(μ) = N * (Nμ − y) ∈ C(Ω) has the following properties:
|p(ω)| ≤ α for ω ∈ Ω, p(ω) = −α φ (|μ({ ω })|) sign(μ)(ω) forμ-almost all ω ∈ suppμ.
Here, sign(μ) : suppμ → {−1, 1} denotes the signum ofμ, definedμ-a.e. uniquely for ω ∈ suppμ (by the Hahn decomposition).
We refer to Appendix B for the proof of this result. We note that this only gives a necessary condition for optimality, and that the interpretation of the second condition requires abstract tools from measure theory.
In the previous results, we still include the case φ(z) = z corresponding to (P L 1 ). In this situation, we can not guarantee that a solution of the form (8) exists (which is also evidenced by our numerical experiments in Section 5). However, if (A2 φ ) holds, we derive that the local solutions to (P Φ ) are finitely supported and thus of the form (8).
Theorem 3. Suppose φ satisfies conditions (A1 φ ) and (A2 φ ). Ifμ is a local solution of (P Φ ), then there exists N < ∞,ω n and corresponding coefficientsc n = 0, n = 1, . . . , N , withμ = N n=1c n δω n . The proof of this result is given in Appendix C. In the case of an atomic local minimum, the necessary optimality conditions from Theorem 2 can be further simplified.
Remark 1. Let φ fulfill the conditions (A1 φ ). Letμ be a finitely supported local minimum of J (as in Theorem 3) withc n = 0. Then, the second condition of Theorem 2 reads as (12)p(ω n ) = −α φ (|c n |) signc n , n = 1, . . . , N,
where the dual variable isp = N * (Nω ,c − y). For the optimalω define also the functional
Jω(c) = l(Nω ,c ; y) + α N n=1 φ(|c n |).
It can be seen that condition (12) is the first order necessary condition of local optimality ofc being a local minimizer of Jω. This follows from 0 = ∂ cn Jω(c n ) =p(ω n ) + α φ (|c n |) signc n employing straightforward calculations.
We note that with (A2 φ ), due to φ (z) ≤ 1 − γz, these conditions imply that |p(ω n )| < α for all n = 1, . . . , N , in addition to |p| ≤ α, which holds uniformly on Ω. However, for a nonconvex problem, the necessary conditions above are not sufficient for optimality. The next theorem provides a slightly stronger condition that turns out to be sufficient for local optimality. ii) For all ω ∈ Ω \ {ω n } n=1,...,N it holds |p(ω)| < α, wherep = N * (Nω ,c − y) is the associated dual variable. Then,μ is a local minimum of J (i.e., a local solution of (P Φ )).
Proof. First, sincec is a local minimum of Jω, there exists an > 0, such that Jω(c) ≥ Jω(c) for all c ∈ R N with c −c 1 ≤ . Due to Remark 1 and (A2 φ ), |p(ω n )| = αφ (|c n |) < αφ (0) = α, and thus there exists a δ > 0, such that sup ω∈Ω |p(ω)| ≤ (1 − δ)α. Without restriction, assume in the following that ≤ δα/( N 2 + (αγ/2)), where γ is from (A1 φ ). To verify local optimality ofμ, let µ ∈ M (Ω) be arbitrary with µ −μ M (Ω) ≤ . By decomposing µ into atomic and non-atomic part, we can write
µ = µ 0 + µ, with µ 0 = N n=1 c n δω n and µ = n c n δ ωn + µ cont ,
where { ω n } = atom µ \ atomμ, c n = µ({ω n }), c n = µ({ ω n }), and µ cont is the continuous part of µ.
Therefore, it follows N µ = N µ 0 + N µ. Moreover, c −c 1 + c 1 + µ cont M (Ω) = µ −μ M (Ω) ≤ .
By the quadratic form of the loss, we obtain
1 2 N µ − y 2 L 2 (D,ν) = 1 2 N µ 0 − y 2 L 2 (D,ν) + (N µ 0 − y, N µ) L 2 (D,ν) + 1 2 N µ 2 L 2 (D,ν) ≥ 1 2 Nω ,c − y 2 L 2 (D,ν) + (N [µ 0 −μ], N µ) L 2 (D,ν) + (Nμ − y, N µ) L 2 (D,ν) ≥ 1 2 Nω ,c − y 2 L 2 (D,ν) − N 2 µ M (Ω) + p, µ , using that |(N [µ 0 −μ], N µ) L 2 (D,ν) | ≤ N 2 c −c 1 µ M (Ω) ≤ N 2 µ M (Ω) .
Moreover, for the penalty it holds
Φ(µ) = Φ(µ 0 ) + Φ( µ) = N n=1 φ(|c n |) + n φ(| c n |) + Ω d|µ cont |.
Combining this, we obtain
J(µ) ≥ Jω(c) − N 2 µ M (Ω) + Ω [α − |p|] d|µ cont | + n [αφ(| c n |) − |p( ω n )|| c n |] .
By the optimality ofc, it follows that Jω(c) ≥ Jω(c) = J(μ). For the third term, we use that
Ω [α − |p|] d|µ cont | ≥ δα Ω d|µ cont |.
For the fourth term we use (
A1 φ ) for φ(| c n |) = φ (0)| c n | + | cn| 0 φ (z) − φ (0) dz ≥ | c n | − (γ/2)| c n | 2 to obtain n [αφ(| c n |) − |p( ω n )|| c n |] ≥ n (α − |p( ω n )|) | c n | − (αγ/2)| c n | 2 ≥ δα c 1 − (αγ/2) c 2 2 ≥ δα c 1 − (αγ/2) c 2 1 .
Combining these estimates, we consequently obtain
J(µ) ≥ J(μ) + δα Ω d|µ cont | + c 1 − (αγ/2) c 2 1 − N 2 µ M (Ω) ≥ J(μ) + δα − (αγ/2) + N 2 µ M (Ω) ≥ J(μ).
We can interpret the conditions from the previous result in terms of the original problem (P φ ). Condition i) simply requires the outer weights c in (P φ ) to be chosen as a local minimum. Condition ii) can be read as follows: Adding any number of additional nodes ω with small outer weights to N will increase the training objective of (P φ ).
The next theorem proves that any local minimizer is not only finitely supported, guaranteed by Theorem 3, but also fits the data properly. To state the theorem we need to introduce the following notation; cf. [2,33]. Let W(D) be the space of functions f on D that satisfy f (x) = [N µ](x) for every x ∈ D and some µ ∈ M (Ω). A characterization of this space for the ReLU activation function will be provided below in Section 3. For f ∈ W(D), denote (13) f W(D) = min
µ∈M (Ω) µ M (Ω) subject to f (x) = [N µ](x) for every x ∈ D.
Note that, for any f ∈ W(D), there exists a minimizer
µ f with f = N µ f such that f W(D) = µ f M (Ω)
. The existence of the optimal measure in (13) follows from the direct method of variational calculus by showing that a minimizing sequence has a convergent subsequence in the weak- * sense (cf. Appendix A).
Theorem 5. If φ satisfies conditions (A1 φ ), and f ∈ W(D) then, for any local solutionμ of (P Φ ),
Nμ − f 2 L 2 (D,ν) ≤ 2 α f W(D) + y − f 2 L 2 (D,ν) . Proof. Let µ f be such that f (x) = [N µ f ](x) for all x ∈ D and µ f M (Ω) = f W(D)
. Then, for any local minimizerμ of (P Φ ), we have
Nμ − f 2 L 2 (D,ν) = (Nμ − y, N (μ − µ f )) L 2 (D,ν) + (y − f, Nμ − f ) L 2 (D,ν) ≤ N * (Nμ − y),μ − µ f + 1 2 y − f 2 L 2 (D,ν) + 1 2 Nμ − f 2 L 2 (D,ν) ,
using Young's inequality. Withp = N * (Nμ − y) and bringing the last term to the left-hand side, we arrive at
1 2 Nμ − f 2 L 2 (D,ν) ≤ p,μ − p, µ f + 1 2 y − f 2 L 2 (D,ν) .
Now, we can estimate the first term by zero, due to
p,μ = −α Ω φ (|μ({ ω })|) sign(μ)(ω) dμ(ω) = −α Ω φ (|μ({ ω })|) d|μ|(ω) ≤ 0,
using the optimality conditions forμ from Theorem 2. Finally, the second term is estimated as − p, µ f ≤ α µ f M (Ω) using p C(Ω) ≤ α, resulting in the desired estimate.
The condition f ∈ W(D) may appear restrictive, but it turns out that a large class of functions is included. In particular, when σ is given by the ReLU function, all sufficiently smooth functions are contained in W(D). Moreover, if additionally D = R d , the quantity f W(R d ) can be explicitly computed. We will discuss this topic further in Section 3.
Finally, we give an additional upper bound in the case of finite data. The representer theorem for the 1 minimization (see, e.g., [2, Section 2.2]) claims that the problem (P L 1 ) has a global solution with at most K nodes. A similar but stronger version of the representer theorem holds for the problem (P Φ ), which is proved in Appendix D.
Theorem 6. If φ satisfies conditions (A1 φ ) and (A2 φ ), and supp(ν) = { x k } k=1,...,K is a finite set with K points then any local solution of (P Φ ) has at most K nodes.
Exact representation with integral ReLU neural networks
We return to the space W(D), more specifically to the case when the activation function is the ReLU function σ(x; ω) = max{ 0, a · x + b} and (a, b) = ω ∈ Ω = S d . Here, it can be easily seen that any f ∈ W(D) is Lipschitz continuous on D. First, we consider D = R d , where ν is any probability measure supported on the whole R d . We can now characterize the kernel of N : It is given by M − (S d ), the set of all odd measures µ, i.e. dµ(−a, −b) = − dµ(a, b), that satisfy the conditions
a µ := S d a dµ(a, b) = 0 and b µ := S d b dµ(a, b) = 0.
It can be seen that N µ ≡ 0, if and only if µ ∈ M − (S d ). Indeed, by noting that
max{0; a · x + b} = 1 2 (a · x + b − |a · x + b|),
for any measure µ it holds
N µ = 1 2 S d [a · x + b] dµ + S d |a · x + b| dµ = 1 2 a µ · x + b µ + S d |a · x + b| dµ .(14)
Note, that the first two terms only depend on the odd part of µ, and the last term only on the even part. Thus, if µ ∈ M − (S d ), then N µ = 0. The converse follows from [33, Lemma 10] which, by a change of variables, claims that for every f = N µ there exists a unique linear function a · x + b and a unique even measure µ f with f = a · x + b + N µ f . Thus, for N µ = 0 the unique even measure will be the zero measure and (a, b) = 0. Taking into account (14), we must therefore have that µ is an odd measure with a µ = 0 and b µ = 0. Next, we derive an explicit formula for the even measure µ f that represents exactly a given smooth function f .
c f (a, b) = (−1) (d+1) /2 2(2π) d−1 1 a d+2 ∂ d+1 ∂b d+1 R[f ](a, b)
if d is odd;
(−1) d /2 2(2π) d−1 1 a d+2 ∂ d+1 ∂b d+1 H R[f ](a, ·) (b)
if d is even.
Here, R[f ] is the Radon transform of f and H[g] is the Hilbert transform of a function g : R → R.
Then f (x) = [N µ f ](x) where dµ f (a, b) = c f (a, b) d(a, b) is the measure with density c f (a, b) with respect to the d-dimensional Hausdorff measure d(a, b) on S d . Moreover, f (x) = [N µ](x) for µ ∈ M (S d ) if and only if µ = µ f + µ − , where µ − ∈ M − (S d ), and f W(R d ) = µ f M (Ω) = c f L 1 (Ω) , where f W(R d ) is defined in (13).∈ C c (R d ) with F | D = f , we have f W(D) ≤ F W(R d ) = µ F M (S d ) .
Proof. It easily follows from the observation that
{µ ∈ M (S d ) : f (x) = [N µ](x), ∀x ∈ R d } ⊂ {µ ∈ M (S d ) : F (x) = [N µ](x), ∀x ∈ D}.
Additionally, in the setting where D is bounded, we point out that the W norm can be estimated by what is know as the Barron constant in the literature (going back to [3] and [10]). In particular, it can be shown that, for any continuous function f : R d → R with Fourier transform f and
C f = R d ω 2 | f (ω)| dω < ∞,
the restriction of f to the bounded set D is in W(D) with norm bounded by a constant factor of C f . We refer to the introduction of [24] (cf. also [16]), where we note that the variation of the function f introduced there is equivalent to the definition (13) given above due to density of finite linear combinations of Dirac-delta functions in the space of measures (in the weak- * sense) and the equivalence in R d+1 of the 1-norm to the Euclidean norm. This characterizes a different subset of W(D) (different from C d+1 ).
We conclude this section with the following observation: A smooth function is optimally represented by a smooth density c(a, b) as opposed to a sparse discrete measure (8). Thus, we can not expect the solution of (P L 1 ) (which can be considered an approximate version of (13)) to provide such a discrete measure in the infinite data case. Indeed, in the numerical experiments of Section 5, we will observe that the optimal solution of (P L 1 ) will tend to approximate the continous density rather than a maximally discrete one, in contrast to the nonconvex (P Φ ).
The optimization algorithm
To numerically solve the problem (P φ ), we deploy the following algorithm which consists of three phases that are executed consecutively. The first phase adds new neurons in a greedy fashion, the second optimizes the weights, and redundant neurons are pruned in phase three as detailed below. The proposed method can be considered an accelerated version of the conditional gradient method [2,39] and is also closely related to the gradient boosting [20] and the CoSaMP [31] algorithms. Sample N trial random nodes ω ∈ Ω.
4:
Optimize (in parallel) the function (15) starting from the initial nodes.
5:
Select nodes with |p t (ω)| > α and add them to the network (of width N (t + 1/2)).
6:
Perform local training based on (16).
7:
Remove nodes with outer weight zero (resulting in width N (t + 1)). N (t) ] be the lists of network inner and outer weights in the t-th iteration and N (t) be the corresponding number of neurons. Network initialization is arbitrary: one can start from any network, including the empty network, then add and extract neurons to derive an optimal network. Phase 1. To determine new points to insert, in the greedy insertion step, we compute the nodes ω ∈ Ω for which the correlation of σ(·; ω) with the residual g t = N ω (t) ,c (t) − y is largest. Thus, we maximize (15) Ω
ω → |p t (ω)| = 1 K K k=1 σ(x k ; ω)g t (x k )
where we have assumed that K is finite. Note that p t (ω) is exactly the dual variable as defined in Section 2.2. Finding a global maximum of (15) (in a high dimensional space) is a challenging problem, and its reliable determination up to a guaranteed tolerance for the specific problem here is subject of ongoing research; cf. [2]. As an ersatz, we use the following heuristic which is commonly employed in practice: we test all local maxima of (15) that are found by a gradient maximization, initialized at N trial random points on Ω. This corresponds to solving N trial simple constrained optimization problems (in parallel); cf., e.g., [8]. Moreover, the constraint (a n , b n ) = ω n ∈ S d can be removed by parametrization of the sphere, using, e.g., a stereographic projection; see Appendix E. Of these points, we insert all that violate the constraint |p t (ω)| ≤ α (after removing possible duplicates). Phase 2. Let N (t + 1/2) ≤ N (t) + N trial denote the number of neurons in the resulting network. Next, we compute an approximate local solution to the objective function of (P 1 ), for N = N (t + 1/2), given by
(16) (Ω × R) N (t+1/2) (ω, c) → l (N ω,c ; y) + α N (t+1/2) n=1 φ(|c n |).
Here, it is sufficient to optimize with respect to the outer weights only according to Theorem 4. The resulting nonsmooth optimization problem can be solved by standard training methods based on (proximal) gradient descent, using the old values as initialization for the weights from the previous iteration. In practice, we employ more efficient second order semismooth Newton methods; see Appendix E. Alternatively, we can optimize in terms of all weights, using an appropriate nonsmooth optimization algorithm. Phase 3. Finally, we note that the objective (16) contains a sparsity promoting term for the outer weights, and we can expect several of the outer weights to be zero after having solved the problem up to a specified tolerance. Thus, we can eliminate these outer weights (together with their corresponding inner weights) from the network without changing the underlying function. This results in a new network of final width N (t + 1).
The resulting procedure is outlined in Algorithm 1. In Appendix E, we provide a more detailed explanation of the implementation of single steps of the method. The code for the numerical experiments is provided at [38].
Numerical examples
In this section, we supply numerical examples in one and two dimensions. We demonstrate the difference between 1 and φ regularization and the effect of the parameter γ on the optimal number of neurons. Here, we take φ as (17) φ γ (z) = 1 2 (z + φ log,2γ (z)) , which fulfills (A1 φ ) and (A2 φ ) with γ = γ/4 and z = 1/(2γ), where φ log,2γ (z) is defined in (4).
5.1.
One dimensional example. In Figure 3, we consider the function 1]. The data set consists of 1000 uniformly arranged points x k and y k = f (x k ) (without noise). We fix α = 10 −5 and the number of iterations of Algorithm 1 is T = 15.
f (x) = exp −x 2 /2 sin 7 1 + x 2 on the interval D = [−1,
During each iteration, up to N trial = 50 nodes can be added to the network. In Figure 3, we plot the results for (P 1 ) and (P φ ) with γ = 1. We observe that the 1 optimal network, while providing a sparse approximation of the function in some areas, contains dense "clusters" of inner weights on the sphere, corresponding to a large cluster of knot points in the corresponding linear spline. These are located in areas of the sphere where the dual variable assumes the extreme values ±α. For the φ minimal network the knot points appear more sparsely spaced and the dual variable is reduced below the extreme values. To further investigate the influence of the hyperparameter γ, we also solve (P φ ) for γ ∈ { 10 −3 , 10 −2 , 10 −1 }, and give their optimal number of neurons and approximation quality in Table 1a. We observe that increasing γ leads to an increased reduction in the number of neurons while keeping the approximation quality essentially constant. We note that γ = 0 is the 1 solution. Table 1. Optimal N and fidelity for the solutions of (P φ ) with (17) and different γ.
Two-dimensional example.
For the two dimensional experiment, we uniformly sample
(18) f (x 1 , x 2 ) = exp − x 2 1 + x 2 2 2 cos(10 x 1 x 2 ).
on a 31 × 31 grid in D = [−1, 1] 2 and solve (P 1 ) and (P φ ) with φ γ from (17) and α = 10 −5 and γ = 5. For both methods, the algorithm is iterated T = 10 times, with up to N trial = 50 nodes added in each iteration. We give the results in Figure 4. As before, we observe a reduction in the number of nodes for the nonconvex regularization with essentially the same approximation quality.
In contrast to the convex regularized solution, the nonconvex regularization is not affected by the clustering of nodes in a small area of the sphere. (17) for (18). Left: Location of optimal inner weights (color indicates sign, size outer weight magnitude) under stereographic projection of S 2 from the pole (a, b) = (0, −1). Right: Data points and optimal neural network.
To illustrate the effect of increased γ on removal of "clusters" in more detail we consider a function with analytically known representing measure µ f given by (19) f
(x) = x −x = (x 1 −x 1 ) 2 + (x 2 −x 2 ) 2 , wherex = (0.1, 0.1),
which is uniformly sampled on a 21 × 21 grid in D = [−1, 1] 2 . The results are provided in Table 1b. Concerning the number of neurons, we observe a steady reduction from N = 105 to N = 14 with increasing γ. In this example the approximation quality is slightly reduced; however, it is still consistently small following the estimate N ω,c − f 2 ≤ α f W(D) in Theorem 5.
For this example, we can give the integral representation exactly (cf. Theorem 7). One can show that f = N µ f for a measure µ f supported on the great circle Sx = { (a, b) ∈ S 2 | a ·x = b }:
f (x) = Sx max{ a · x + b, 0 } 1 2 a dS(a, b) = 1 2 S 0 max{ a · y, 0 } dS(a, b) y=x−x .
Here, dS is the one-dimensional line integral on Sx (resp. S 0 ). Therefore, µ f is given by dµ f = 1/(2 a ) dS(a, b)| Sx . In Figure 5 we investigate more closely the properties of the solutions. We observe that the nodes of the solution of (P 1 ) densely cluster on the great circle Sx, up to gaps arising due to the symmetry of the nodes with respect to (a, b) ∼ (−a, −b) (cf. Theorem 7). This is remarkable since the support of the regularized solution -representing a compromise between a fit of the data and a small regularization term -is essentially the same as the exact measure µ f that achieves the perfect fit. In contrast, for large γ, the points are sparsely placed on the circle and regularly spaced up to the symmetry (a, b) ∼ (−a, −b). By reducing the dual variable below the bound |p| ≤ α in the existing nodes, no additional point with a small weight can be inserted without increasing the regularized objective. on a random set of 1000 uniformly distributed points in D = [−1, 1] 3 and solve (P 1 ) and (P φ ) with φ γ from (17) and γ = 5. To evaluate the influence of α, we consider α = 10 −2 · (1/4) j for j = 0, 1, . . . , 3. For both functionals, the algorithm is initialized with the previous solution (for larger α, if available), iterated T = 10 times with up to N trial = 50 nodes added in each iteration. In Figure 6 we investigate more closely the properties of the solutions for the smallest α, which again illustrates the improved sparsity of the nonconvex approach. Moreover, Table 2 confirms that this reduction holds independently of α, that the fidelity is comparable between the convex and nonconvex solution, and that α determines the quality of the fit. We note that the error is reduced by a factor ∼ 1/2 when α is reduced by a factor 1/4, which is in accordance with Theorem 5. Table 2. Optimal N and fidelity for the solutions of (P 1 ) and (P φ ) with (17) for γ = 5 and different α using f from (20).
Conclusions
In this work, we introduce a nonconvex regularization method for finding sparse neural networks. Even though it is challenging to solve the problem globally, we provide theoretical confirmation of local minimizers satisfying the desirable sparsity and approximation properties. We numerically solve the problem using an adaptive method that gradually adds and removes nodes from the network.
We assume that the target function outputs scalar values. This restriction can be easily lifted by considering the outer weights c n to be vectors, and by replacing the penalty term φ(|c n |) with φ( c n ) in the problem formulation. Then, the generalized problem with measures will be modified by switching from signed measures to vector-valued measures. We can expect all the results of this paper to hold true with small modifications.
Throughout the paper, we largely limited ourselves to the ReLU activation function, although the optimization problem is not specific to this choice and many activation functions that fulfill the Lipschitz continuity requirement could be employed instead. However, the ReLU allowed us to restrict the inner weights from R d+1 to S d . For other activation functions that are not positively homogeneous (in particular smoother ones), the restriction from the whole space to a bounded set can impose restrictions on the approximation properties of the resulting architecture, and it may be necessary to consider an unbounded dictionary in R d+1 .
A more thorough investigation of the numerical optimization algorithm, algorithmic choices, and implementation details may warrant more attention. Additionally, we only consider nonconvex regularization and adaptive training for shallow neural networks. For the case of deep neural networks, a direct reformulation in terms of appropriate integral representations remains a challenging problem.
which fulfills the conditions on the second derivative with, e.g., γ = γ/4 and z = 1/γ, and in the latter case we have
MCP γ (z) = −γ for z ≤ 1/γ 0 else,
where we can take γ = γ and z = 1/γ. We also note that for any φ γ that fulfills the conditions on the second derivative with γ = γ 1 γ and z = z 1 /γ, the convex combination φ(z) = τ φ γ/τ (z) + (1 − τ )z, 0 < τ < 1, fulfills the conditions (A1 φ ) and (A2 φ ) with the same γ, γ and z = τ z 1 /γ using that φ(z) ≥ (1 − τ )z → ∞ for τ → ∞.
Lemma 1. Assume (A1 φ ) is fulfilled. For any µ ∈ M (Ω) it holds φ( µ M (Ω) ) ≤ Φ(µ) ≤ µ M (Ω) .
Proof. The second inequality follows directly from φ(|z|) ≤ |z|. Concerning the first, we let atom µ = { ω n } n be the atoms of µ and estimate
Φ(µ) ≥ φ (|µ| (Ω \ atom µ)) + n φ (|µ|({ω n })) ≥ φ |µ| (Ω \ atom µ) + n |µ|({ω n }) = φ( µ M (Ω) ),
where we used first the subadditivity of φ and second the σ-additivity of |µ|.
For a sequence of measures µ (k) ∈ M (Ω), k = 1, 2, . . . , and a measure µ ∈ M (Ω), denote µ (k) * µ if µ (k) ∈ M (Ω) converges to µ in weak- * sense as functionals on C(Ω), i.e. for any ϕ ∈ C(Ω), lim n→∞ ϕ, µ (k) = ϕ, µ .
The next lemma can be derived from Theorem 3.3 in [5].
Lemma 2. Assume (A1 φ ) is fulfilled. Φ is weak- * lower semicontinuous on M (Ω): if µ (k) * µ then (21) lim inf n→∞ Φ(µ (k) ) ≥ Φ(µ).
Proof. The functional Φ can be written as
Φ(µ) = Ω\atom µ d|µ| + ω∈atom µ φ(|µ({ ω })|),
which is an instance of the functional discussed in [5] (where we choose f (z) = |z|). We verify the assumptions of [5,Theorem 3.3], which gives the desired result: The assumptions (H 1 )-(H 3 ) are trivially fulfilled. Continuity and subadditivity of z → φ(|z|), i.e., φ(|z
1 + z 2 |) ≤ φ(|z 1 | + |z 2 |) ≤ φ(|z 1 |) + φ(|z 2 |), imply assumptions (H 4 )-(H 5 )
, assumption (H 6 ) follows from φ(|z|) ≤ |z|, and assumption (H 7 ) simplifies to φ (0)|z| = |z| for all c ∈ R, i.e. φ (0) = 1.
With these lemmas we can give the basic existence result from section 2.2.
Proof of Theorem 1. We employ the direct method of variational calculus. Since J ≥ 0, we can select a minimizing sequence µ (k) ∈ M (Ω). Now, with
αφ( µ (k) M (Ω) ) ≤ αΦ(µ (k) ) ≤ J(µ (k) ) → inf µ∈M (Ω) J(µ)
and the fact that φ(z) → +∞ for z → ∞, µ (k) M (Ω) is bounded and we can select a subsequence that converges toμ ∈ M (Ω) in the weak- * sense. By weak- * lower semicontinuity of Φ and continuity of the loss function L(µ), we conclude that the minimum is attained atμ. The bound on the global solution follows with αφ( μ M (Ω) ) ≤ αΦ(μ) ≤ J(μ) ≤ J(0) = l(0; y).
Appendix B. Proof of Theorem 2
In the following, we letμ be a local solution of (P Φ ), i.e. a local minimum of J (given in (11)). To derive the conditions for the given local solution, we consider
J(μ + τ u) − J(μ) = L(μ + τ u) − L(μ) + αΦ(μ + τ u) − αΦ(μ) ≥ 0
with arbitrary u ∈ M (Ω) and 0 < τ < / u M (Ω) where is the radius from Definition 1. Dividing by τ > 0 and letting τ → 0, from local optimality, it follows
− ∇L(μ), u = − p, u ≤ α lim τ →0 + (1/τ ) [Φ(μ + τ u) − Φ(μ)] ,
as long as the limit on the right exists. We consider different values of u in the following:
For u = ±δ ω for ω ∈ atomμ, i.e.μ({ω}) = 0, we obtain
∓p(ω) = − p, u ≤ α lim τ →0 + (1/τ )φ(τ ) = α lim τ →0 (1/τ ) [φ(τ ) − φ(0)] = α φ (0) = α.
Hence, |p(ω)| ≤ α for ω ∈ atomμ. Now, take u = ±δ ω for ω ∈ atomμ, i.e.μ({ω}) = c = 0. Here, we obtain
∓p(ω) = − p, u ≤ α lim τ →0 + (1/τ ) [φ(|c ± τ |) − φ(|c|)] = ±α φ (|c|) sign c.
Hence, it followsp
(ω) = −α φ (|c|) sign c.
Using the fact that φ (0) = 1, φ is increasing and φ is decreasing we have that φ (ω) ∈ [0, 1] for ω ≥ 0. Hence |p(ω)| ≤ α for ω ∈ atomμ also proving the first estimate in the lemma: |p(ω)| ≤ α for all ω ∈ Ω. Next, we take u = ±μ cont = ±μ| Ω\atomμ to be the continuous part ofμ and deduce
∓ p,μ cont ≤ α lim(1/τ )[(1 ± τ ) μ cont M (Ω) − μ cont M (Ω) ] = ±α μ cont M (Ω)
and thus − p,μ cont = α μ cont M (Ω) , which together with |p(ω)| ≤ α for ω ∈ Ω implies that p = −α signμ cont forμ cont almost all ω ∈ Ω. Combined with the atomic case above we get the second part of the theorem.
Appendix C. Proof of Theorem 3 First, let us show thatμ is atomic. Assume otherwise and letμ cont =μ| Ω\atomμ = 0 be the continuous part ofμ. Then there exist a pointω ∈ suppμ cont \ atomμ, i.e., such thatμ({ω}) = 0 and for any δ > 0, |μ|(B δ (ω)) > 0 where B δ (ω) is the open ball of radius δ in Ω aroundω. Without restriction, letω ∈ suppμ cont,+ whereμ cont,+ is the positive part ofμ cont . Let D + ⊂ Ω \ atomμ be a set withμ cont,+ =μ| D + (given by the Hahn decomposition theorem) and D δ = D + ∩ B δ (ω). Now, we define
µ δ =μ −μ| D δ + C δ δω, where C δ =μ(D δ ) > 0
which replacesμ on D δ for any δ > 0 by a single Dirac measure of the same total variation norm. We note that by construction C δ is positive and converges to zero for δ → 0. We will show that µ δ , for δ small enough, improves the function value of J in contradiction to optimality ofμ.
Using the Lipschitz continuity assumption on σ, i.e. |σ(x;ω) − σ(x; ω)| ≤ Λ(1 + x ) ω −ω , one readily obtains that
|[N (µ δ −μ)](x)| = |[N (μ| D δ − C δ δω)](x)| ≤ D δ σ(x; ω) dμ(ω) − C δ σ(x;ω) = D δ [σ(x; ω) − σ(x;ω)] dμ(ω) ≤ δΛ(1 + x )|μ|(D δ ) = δΛ(1 + x )μ(D δ ) = δΛ(1 + x )C δ
for any x ∈ R d . Thus, it also follows
N (µ δ −μ) L 2 (D,ν) ≤ δΛC ν C δ = δΛ 1 C δ .
Here, we define C 2 ν = D (1 + x ) 2 dν(x) and Λ 1 = ΛC ν . Consequently, by the quadratic form of L and, using the fact thatp = N * (Nμ − y), we have
L(µ δ ) = L(μ) + (Nμ − y, N (µ δ −μ)) L 2 (D,ν) + 1 2 N (µ δ −μ) 2 L 2 (D,ν) ≤ L(μ) + p, µ δ −μ + 1 2 δ 2 Λ 2 1 C 2 δ .
By the optimality condition and continuity ofp, we havep(ω) = −α for all ω ∈ D δ ∩ suppμ ⊂ suppμ cont,+ (note that alsoω ∈ D δ ∩ suppμ ⊂ suppμ cont,+ ) and therefore the term p, µ δ −μ = p, −μ| D δ + C δ δω vanishes. Hence,
L(µ δ ) ≤ L(μ) + 1 2 δ 2 Λ 2 1 C 2 δ . Concerning the cost term, we estimate Φ(µ δ ) = Φ(μ) −μ(D δ ) + φ(C δ ) = Φ(μ) + φ(C δ ) − C δ = Φ(μ) + C δ 0 (φ (ξ) − φ (0)) dξ ≤ Φ(μ) − γ C δ 0 ξ dξ = Φ(μ) − γ 2 C 2 δ ,
for δ small enough, using φ (0) = 1, the definition of Φ and (A2 φ ). Combining both estimates, we obtain
J(µ δ ) = L(µ δ ) + αΦ(µ δ ) ≤ J(μ) + 1 2 δ 2 Λ 2 1 − α γ C 2 δ . Therefore J(µ δ ) ≤ J(μ) − 1 2 α γ − δ 2 Λ 2 1 C 2 δ < J(μ)
, for δ < α γ/Λ 1 , contradicting the optimality ofμ. Now let us show that the number of atoms inμ is finite. Assume otherwise, then there exists a subsequence of distinct atomsω n converging to someω due to compactness of Ω. Without restriction, assume that c n =μ({ω n }) > 0 for all n. By optimality ofμ and continuity ofp, from Theorem 2, it holds αφ (c n ) = −p(ω n ) → −p(ω) for n → ∞. Since φ (c n ) → 1 due to c n → 0, it follows thatp(ω) = −α. Hence, from Theorem 2,ω cannot be an atom ofμ. Define now
µ N =μ − ∞ n=N c n δω n + C N δω where C N = ∞ n=N c n > 0,
replacing an infinite number of atoms by a single one. Set δ N = max n≥N |ω n −ω|. As before, we obtain
L(µ N ) ≤ L(μ) + p, µ N −μ + 1 2 δ 2 N C 2 N Λ 2 1 .
Here, the second term is given as
p, µ N −μ = C Np (ω) − ∞ n=N c np (ω n ) = −αC N + α ∞ n=N c n φ (c n ),
using the optimality conditions. Concerning Φ, there holds
Φ(µ N ) = Φ(μ) − ∞ n=N φ(c n ) + φ(C N ).
Combining these estimates, we obtain
J(µ N ) = L(µ N ) + αΦ(µ N ) ≤ J(μ) − α ∞ n=N φ(c n ) − φ (c n )c n − α[C N − φ(C N )] + 1 2 δ 2 N C 2 N Λ 2 1 . Now, we use concavity of φ for φ(c n ) − φ (c n )c n ≥ φ(c n − c n ) = φ(0) = 0 and uniform concavity of φ on [0, z], using (A2 φ ), for φ(C N ) = φ(0) + φ (0)C N + C N 0 [φ (ξ) − φ (0)] dξ ≤ C N − ( γ/2)C 2 N when N is large enough, to obtain J(µ N ) ≤ J(μ) − 1 2 α γ − δ 2 N Λ 2 1 C 2 N .
Similar to the previous case, we choose now N such that δ N < α γ/Λ 1 , which again results in a contradiction to the optimality ofμ. This, together with the optimality conditions obtained in Theorem 2, concludes the proof.
Appendix D. Proof of Theorem 6 For the total variation norm (i.e. φ(z) = z), the Carathéodory lemma implies an upper bound on the number of atoms for some optimal solutionμ in some cases. In particular, we consider the special case of finitely supported ν, which is given by a sum of K Dirac delta measures. In this case, the space L 2 (D, ν) is finite dimensional, i.e. dim L 2 (D, ν) = K. To prove the Theorem 6 we need to show that any local solution of (P Φ ) is atomic and its support consists of at most K points.
By the previous result we know that any locally optimal solution is representable as µ = N n=1c n δω n , |c n | > 0,ω n ∈ Ω, N ∈ N.
Clearly,
Nμ = N n=1c n N (δω n ) = N n=1c n σ(·;ω n ).
Assume that N > K. Then there exists a nontrivial vector λ ∈ R N such that N n=1 λ n c n σ(x k ;ω n ) = 0 for all k = 1, . . . , K or, equivalently, N n=1 λ n c n σ(·;ω n ) = 0 in L 2 (D, ν). For any τ ∈ R we definē
µ τ = N n=1
(1 + τ λ n )c n δω n .
Note that Nμ τ = Nμ + τ N n=1 λ n c n σ(·;ω n ) = Nμ in L 2 (D, ν) for any τ . Now, we assume also that τ is small enough such that 1 + λ n τ ≥ 0 for all n and turn our attention to the objective functional of (P Φ ). Taking into account the previous argument, we have, for any τ = 0, that
J(μ τ ) − J(μ) = αΦ(μ τ ) − αΦ(μ) = α N n=1 [φ ((1 + τ λ n )|c n |) − φ(|c n |)] < τ α N n=1 φ (|c n |) λ n |c n | = τ α Φ (μ; δµ),
where Φ (μ; δµ) is the directional derivative of Φ atμ in direction δµ =μ 1 −μ, taking into account that L(μ τ ) = L(μ), the restrictions on τ and the strict concavity of φ. Depending on the sign of Φ (μ; δµ), we choose τ > 0 or τ < 0 sufficiently small such that J(μ τ ) − J(μ) < 0, contradicting the local optimality ofμ.
Appendix E. The optimization algorithm In the following, we give more details on the concrete implementation of the steps summarized in Algorithm 1.
Initialization: Let ω (t) = [ω (t) 1 , . . . , ω (t) N (t) ], c (t) = [c (t) 1 , . . . , c (t) N (t) ]
be the lists of network inner and outer weights in the t-th iteration and N (t) be corresponding the number of neurons. In addition, denote by N ω (t) ,c (t) (x) the corresponding network. Network initialization is arbitrary: one can start from any network, including the empty network, then add and extract nodes to derive an optimal network. Phase 1. To determine new points to insert, in the greedy insertion step, we compute the nodes ω ∈ Ω for which the correlation of σ(ω, x) with the residual g t (x) = N ω (t) ,c (t) (x) − y(x) is largest. Thus, we maximize the absolute value of
p t (ω) = D σ(ω, x) g t (x) dν(x) = 1 K K k=1 σ(ω, x k )g (t) k
where we assume that K is finite and g (t) k = g t (x k ). Note that p t (ω) is exactly the dual variable as defined in Subsection 2.2. Finding a global maximum in a high dimensional space is a challenging problem which requires expensive computations, and its reliable determination up to a guaranteed tolerance for the specific problem here is subject of ongoing research; cf. [2]. As an ersatz, we use the following heuristic which is commonly employed in practice: we test all local maxima of (22) Ω ω → |p t (ω)|, which are found by a gradient maximization, initialized at N trial random points on Ω. This corresponds to solving N trial simple unconstrained optimization problems (in parallel); cf., e.g., [8]. Of these points, we insert all that violate the constraint |p t (ω)| ≤ α (after removing possible duplicates). Here, we rely on the random initialization of the N trial problems in order to have a chance to identify the global maximum with some probability. Moreover, more than one identified local maximum can be added to the network in each iteration to identify a potentially wide network faster. According to Theorem 4, at any node ω ∈ Ω where |p t (ω)| < α, it is not possible to decrease the objective by inserting the corresponding node with small non-zero weight. Conversely, the nodes ω ∈ Ω where |p t (ω)| > α, represent locations where local decrease can still be achieved. All the corresponding outer weights at the trial nodes are initialized to zero, and are going to be optimized in the next phase of the algorithm.
Before we address the next phase, we point out that in the case when we consider Ω to be the sphere and σ be the ReLU activation function, we opt to parametrize ω by its stereographic projection ω = (a, b) = (2z, 1 − z 2 )/(1 + z 2 ), where z ∈ R d . This is done to avoid dealing with the algebraic constraint ω = 1. Here, we use the southern pole as the projection point, which corresponds to (a, b) = (0, −1). The corresponding neuron represents the zero function and removing it from Ω does not affect the approximation capability of the network. Phase 2. Let N (t + 1/2) ≤ N (t) + N trial denote the number of nodes in the resulting network. Next, we compute an approximate local solution to the following problem (23) (Ω × R) N (t+1/2) (ω, c) → l (N ω,c ; y) + α N (t+1/2) n=1 φ(|c n |)
in terms of all weights, using the old values as initialization for the weights from the previous iteration. The resulting nonsmooth optimization problem can be approximately solved by standard training methods based on (proximal) gradient descent. In particular, we can eliminate the constraint for the inner weights by stereographic projection and use gradient descent, and treat the outer weights with proximal gradient descent (see, e.g., [36]). We note that the proximal map for the cost term φ can still be determined uniquely for small stepsize due to γ-convexity. For instance, for the function φ γ (z) = log(1 + γz)/γ from the introduction (4), it is given as
Prox λφγ (q) = arg min c∈R 1 2 |c − q| 2 + λφ γ (c) = sign q 2γ (γ|q| − 1) + (γ|q| − 1) 2 + 4γ (|q| − λ), for |q| > λ, 0 else.
In the case that λ < 1/γ, this proximal mapping is uniquely determined. We note that this has the potential to set a number of outer weights to zero because of the proximal descent step. We do not specify the details here (in particular, the optimal choices of the stopping criteria for the different nonlinear and nonsmooth optimization routines that we employ), and leave a detailed analysis to future work. Phase 3. Finally, we note that the objective (16) contains a sparsity promoting term for the outer weights, and we can expect several of the outer weights to be zero after having solved the problem up to a specified tolerance. Thus, we can eliminate these outer weights (together with their corresponding inner weights) from the network without changing the underlying function. This results in a new network of final width N (t + 1).
E.1. Second-order methods for outer weights. The aforementioned gradient-based methods in phase two of Algorithm 1 are efficient in that they only require derivatives of the objective function in terms of inner and outer weights, which are readily available in modern computational toolboxes via automatic differentiation. However, these methods suffer from slow convergence once we are close to the minimum, and can be slow to eliminate redundant nodes ω (as already observed in the context of a convex sparse problem with measures; cf. [39]). In order to provide accurate results (that are not influenced by the choice of the solver), in our numerical experiments we employ a second order semi-smooth Newton method for the outer weights. In an additional step after step 6. of Algorithm 1 this solves the problem in terms of outer weights up to machine precision for the fixed inner weights ω (t+1/2) , which in particular serves to reliably eliminate redundant nodes. (24) l N ω (t+1/2) ,c ; y + α is a continuously differentiable function of c = [c 1 , . . . , c N (t+1/2) ] with Lipschitz continuous derivative due to conditions (A1 φ ) we imposed on φ. For twice continuously differentiable φ, such as the log-penalty function, the same property holds for F . Note that if c is a local minimizer of (24) then (25) − ∇F (c) ∈ α∂ c 1
where ∂ c 1 is the subdifferential of the 1 norm. For λ > 0 and q = [q 1 , . . . , q N (t+1/2) ], let Prox λ (q) be the proximal operator of the 1 norm (also known as the soft-thresholding operator) which modifies each entry of q according to the formula Prox λ (q) n = sign q n max{|q n | − λ, 0}.
Using a reformulation of the optimality condition in terms of Robinson's normal map, c satisfies (25) if and only if c = Prox λ (q) for some q and (26) ∇F (Prox λ (q)) + α λ (q − Prox λ (q)) = 0;
see, e.g., [37,Prop. 3.5]. The first advantage of this reformulation is that the inclusion condition (25) is replaced with an equation. This nonsmooth equation can then be solved using a semi-smooth Newton method (see, e.g., [43]), which exhibits locally superlinear convergence. In particular, once the optimal sparsity pattern is identified, it converges at the quadratic rate of the classical Newton method. The second important advantage is that at each iteration, the soft-thresholding operator outputs a sparse c: we use this feature to drop the zero entries and reduce the number of nodes in the network.
Appendix F. Equivalences of outer-and all-weights regularization Here, we consider networks with the ReLU activation function σ(ω, x) = max{ a · x + b, 0 }, ω = (a, b) ∈ R d+1 , and prove the equivalence of certain cost terms. l(N ω,c ; y) + α N n=1 1 p |c n /τ n | p + 1 q τ q n .
Moreover, since the first term does not depend on τ , we can compute τ n as the minimum of τ → (1/p)|c n | p τ −p + (1/q)τ q .
Differentiating with respect to τ , we obtain 0 = −|c n | p τ −p−1 n + τ q−1 n ↔ τ q+p n = |c n | p ↔ τ n = |c n | p/(q+p) . Inserting the analytical solution for τ n into the cost term above, we obtain (1/p)|c n | p−p 2 /(q+p) + (1/q)|c n | qp/(q+p) = (1/p + 1/q)|c n | qp/(q+p) , which completes the proof.
As a corollary, by taking r(ω) = ω p for ω ∈ R d+1 , we get that solving the problem where S d p = { (a, b) ∈ R d+1 | a p p + |b| p = 1 } is the unit p-sphere in R d+1 , is equivalent to solving the problem min N ∈N, (an,bn,cn)∈R d ×R×R l (N ω,c ; y) + α p N n=1 a n p p + |b n | p + |c n | p .
(
P φ ) min N ∈N, {cn}∈R N , {ωn=(an,bn)}∈(S d ) N l (N ω,c ; y) + α N n=1 φ(|c n |),
Global 1 -solution (with φ(z) = z): N = 36 neurons and N ω,c − f = 0.044. Local solution for φ(z) = log(1 + z): N = 10 neurons and N ω,c − f = 0.026.
Figure 1 .
1Comparison of solutions of (P φ ) for f (x) = cos(10(10 −3 + x 2 ) 1/8 ) with different convex and noncovex φ: We choose α = 10 −4 , x k by 5000 uniformly distributed points on the interval [−1, 1]
Theorem 4 .
4Let φ fulfill the conditions (A1 φ ) and (A2 φ ). Letμ = N n=1c n δω n be a finitely supported measure such that: i)c ∈ (R \ {0}) N is a local minimum of Jω.
Theorem 7 .
7For a compactly supported function f ∈ C d+1 c (R d ), we define the coefficient function
Proof.
The proof of Theorem 7 follows by combining [15, Theorem 1] and [33, Lemma 10]. Concerning approximation on a bounded domain, for f ∈ C d+1 (D), we have C d+1 (D) ⊂ W(D), and Theorem 7 provides an upper bound for f W(R d ) . Here we say that f ∈ C d+1 (D) if there exists an extension F ∈ C d+1 c (R d ) such that F | D = f .
Corollary 1 .
1If f ∈ C d+1 (D), then f ∈ W(D), and for any extension F
Algorithm 1
1Iterative node insertion and optimization 1: Initial network ω (0) , c (0) of width N (0) 2: while t < T do 3:
Local φ γ -regularized solution for γ = 1.
Figure 3 .
3Comparison between solutions of (P 1 ) and (P φ ), for φ γ (z) in (17) in one dimension. Top: True function samples (dashed blue) and neural network approximation (black, orange crosses at the knot points x n = −b n /a n ). Middle: Optimal sparse measure in angular coordinates. Bottom: Optimal dual variable in angular coordinates.
Optimal measure (P 1 ): γ = 0, N = 197.
Optimal network and data (black circles) for (P 1 ):N ω,c − f = 9.4 · 10 −3 . The plot for (P φ ) with γ = 5 is indistinguishable: N ω,c − f = 8.7 · 10 −3 .
Figure 4 .
4Comparison of (P 1 ) and (P φ ) with
5. 3 .
3Three-dimensional example. Here, we consider for d = 3 the function(20) f (x 1 , x 2 , x 3 ) = (x 2 2 + x 2 3 )/(x 1 + 1.05) + 2(x 1 + 1.05)
Dual variable for γ = 10.
Figure 5 .
5Comparison of (P 1 ) and (P φ ) with(17) for(19). Left: Location of inner weights (dot color indicates sign, dot size outer weight magnitude) under stereographic projection of S 2 from the pole (a, b) = (0, −1). Right: Dual variable and nodes under stereographic projection.
Figure 6 .
6Comparison of solutions of (P 1 ) and (P φ ) with φ γ from (17) for f from(20): Location of inner weights (dot color indicates sign, dot size outer weight magnitude) under stereographic projection of S 3 from the pole (a, b) = (0, 0, −1).
F
(c) = l N ω (t+1/2) ,c ; y + α N (t+1/2) n=1 [φ(|c n |) − |c n |]
Proposition 3 .NN
3Let p ≥ 1 and q ≥ 1 and r(ω) be a 1-homogeneous functional: r(τ ω) = |τ | r(ω) for any ω = (a, b) ∈ R d+1 . ∈N, (an,bn,cn)∈R d ×R×R l(N ω,c ; y) ∈N, N ∈N, {cn}∈R N , {ωn=(an,bn): r(ωn)≤1} l(N ω,c ; y) + 2α s N n=1 |c n | s/2 , where s = 2pq/(q + p) = 2/(1/p + 1/q) is the harmonic mean of p and q, are equivalent; i.e. their minimum values are the same and the solutions of one problem solve the other up to a re-normalization. Proof. Note that due to positive homogeneity of the ReLU activation function σ, we have N ω,c = N ωτ ,cτ where ω τ = τ ω, c τ = c/τ. It is easy to see that the problem (27) is equivalent to min N ∈N, {cn}∈R N , {ωn=(an,bn): r(ωn)≤1}, τn≥0
N
∈N, {cn}∈R N , {ωn=(an,bn)}∈(S d p ) N l (N ω,c ; y)
Appendix A. Properties of Φ and the problem (P Φ ) First, we discuss some simple consequences of the requirements (A1 φ ) and (A2 φ ).Proof. By integrating the first condition from 0 ≤ z 1 ≤ z 2 we obtain the desired conditionwhich implies that φ is concave and γ-convex, which are all conditions from (A1 φ ) except for the unboundedness at infinity. Plugging in z 1 = 0, using φ (0) = 1 and φ(0) = 0 and integrating over z 2 from 0 to z yields the second inequality. Similarly, integrating the second inequality once yields the condition in (A2 φ ) and integrating twice yields the second inequality.Proof. For any z 1 ≥ 0, z 2 ≥ 0 there holdsWith this, it is easy to verify that the functions φ log,γ (z) and MCP γ mentioned in section 1 fulfill the conditions. In the former case, we have φ log,γ (z) = − γ (1 + γz) 2 ,
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|
[] |
[
"Closing the Gap between Single-User and Multi-User VoiceFilter-Lite",
"Closing the Gap between Single-User and Multi-User VoiceFilter-Lite"
] |
[
"Rajeev Rikhye [email protected] \nGoogle LLC\nUSA\n",
"Quan Wang [email protected] \nGoogle LLC\nUSA\n",
"Qiao Liang \nGoogle LLC\nUSA\n",
"Yanzhang He [email protected] \nGoogle LLC\nUSA\n",
"Ian Mcgraw [email protected] \nGoogle LLC\nUSA\n"
] |
[
"Google LLC\nUSA",
"Google LLC\nUSA",
"Google LLC\nUSA",
"Google LLC\nUSA",
"Google LLC\nUSA"
] |
[] |
VoiceFilter-Lite is a speaker-conditioned voice separation model that plays a crucial role in improving speech recognition and speaker verification by suppressing overlapping speech from non-target speakers. However, one limitation of VoiceFilter-Lite, and other speaker-conditioned speech models in general, is that these models are usually limited to a single target speaker. This is undesirable as most smart home devices now support multiple enrolled users. In order to extend the benefits of personalization to multiple users, we previously developed an attention-based speaker selection mechanism and applied it to VoiceFilter-Lite. However, the original multi-user VoiceFilter-Lite model suffers from significant performance degradation compared with single-user models. In this paper, we devised a series of experiments to improve the multi-user VoiceFilter-Lite model. By incorporating a dual learning rate schedule and by using feature-wise linear modulation (FiLM) to condition the model with the attended speaker embedding, we successfully closed the performance gap between multi-user and single-user VoiceFilter-Lite models on single-speaker evaluations. At the same time, the new model can also be easily extended to support any number of users, and significantly outperforms our previously published model on multi-speaker evaluations.
|
10.21437/odyssey.2022-41
|
[
"https://arxiv.org/pdf/2202.12169v2.pdf"
] | 247,084,072 |
2202.12169
|
5953bf2278262d7c2226d1aab56feaab81f305a0
|
Closing the Gap between Single-User and Multi-User VoiceFilter-Lite
Rajeev Rikhye [email protected]
Google LLC
USA
Quan Wang [email protected]
Google LLC
USA
Qiao Liang
Google LLC
USA
Yanzhang He [email protected]
Google LLC
USA
Ian Mcgraw [email protected]
Google LLC
USA
Closing the Gap between Single-User and Multi-User VoiceFilter-Lite
VoiceFilter-Lite is a speaker-conditioned voice separation model that plays a crucial role in improving speech recognition and speaker verification by suppressing overlapping speech from non-target speakers. However, one limitation of VoiceFilter-Lite, and other speaker-conditioned speech models in general, is that these models are usually limited to a single target speaker. This is undesirable as most smart home devices now support multiple enrolled users. In order to extend the benefits of personalization to multiple users, we previously developed an attention-based speaker selection mechanism and applied it to VoiceFilter-Lite. However, the original multi-user VoiceFilter-Lite model suffers from significant performance degradation compared with single-user models. In this paper, we devised a series of experiments to improve the multi-user VoiceFilter-Lite model. By incorporating a dual learning rate schedule and by using feature-wise linear modulation (FiLM) to condition the model with the attended speaker embedding, we successfully closed the performance gap between multi-user and single-user VoiceFilter-Lite models on single-speaker evaluations. At the same time, the new model can also be easily extended to support any number of users, and significantly outperforms our previously published model on multi-speaker evaluations.
Introduction
Speaker-conditioned speech models are a class of speech models that are conditioned on a target speaker embedding, allowing the model to produce personalized outputs. For example, in personalized speaker separation, prior knowledge of a target speaker's voice profile is used to suppress overlapping speech from non-target speakers [1,2,3,4,5,6,7]. In personalized Automatic Speech Recognition (ASR), a speaker's voice profile is used to improve the overall recognition accuracy [8,9,10,11]. Additionally, in personalized Voice Activity Detection (VAD), the target speaker profile is used to determine when the target speaker begins or stops talking, which in turn improves the accuracy of downstream components such as ASR [12].
While beneficial, speaker-conditioned speech models are often only limited to a single enrolled user. This makes them incompatible with many devices, such as smart displays and smart speakers, which currently support multiple users [13]. One naive approach to mitigate this would be to have multiple passes of the same model -one pass for each enrolled user. This approach is, however, computationally expensive and unacceptable for on-device applications. As such, extending speaker-conditioned models to support multiple users remains an open and relevant problem [14,15] To overcome this limitation, we previously described an attention-based [16] speaker selection mechanism, and ex-tended VoiceFilter-Lite to support an arbitrary number of enrolled users [17] (see Fig. 1). This multi-user VoiceFilter-Lite model significantly reduces speech recognition Word Error Rate (WER) and speaker verification Equal Error Rate (EER) when the input audio contains overlapping speech. We also demonstrated how this multi-user VoiceFilter-Lite model is critical to a personalized keyphrase detection system on shared devices [18,17] by reducing the false rejection rate caused by speaker mis-identification. Although the performance is promising, this original multi-user model suffered from two issues. First, on single-user evaluations, the multi-user model had worse performance than a single-user model on both speech recognition and speaker verification tasks. This is undesirable as we do not wish to degrade performance on devices with a single enrolled user. Second, the original multi-user model seems to overfit the training data and failed to generalize well to unseen combinations of enrolled users. These limitations raise severe concerns regarding the deployment of the multi-user VoiceFilter-Lite model in production environments.
In this paper, we focus on addressing these limitations by exploring variations of each component of the multi-user VoiceFilter-Lite model, and developed a new version of the model that closes the performance gap on single-user evaluations. It is important to note that in this paper, we focus only on improving speaker identification in the multi-talker setting. In summary, the original contributions of this paper include:
1. We introduce a dual learning rate scheduler where the At-tentionNet is independently trained with a learning rate that is an order of magnitude smaller than the Voice-FilterNet. Experiments in Section 4.2 show that the dual learning rate scheduler prevents the AttentionNet from overfitting and significantly improves model quality.
2. We introduce feature-wise linear modulation (FiLM) [19,20,21] as an efficient way to condition the Voice-FilterNet on the attended embedding. Doing so reduces model size from 3.47 MB to 3.23 MB, and significantly improves performance of the model on a speaker verification task, as shown in Section 4.3.
3. As a complement to the original multi-user VoiceFilter-Lite paper [17], we carefully compared different implementations of aggregating multiple enrolled speaker embeddings into a single embedding, and confirmed that the attention mechanism is critical to the performance, as shown in Section 4.1.
4. Using a combination of the best practices from above, the new multi-user VoiceFilter-Lite model performs identically to the single-user model when there is only one enrolled user, and at the same time significantly reduces speaker verification EER when there are multiple enrolled users. The resulting model meets the quality bar for deployment to production environments.
Review of VoiceFilter-Lite
VoiceFilter-Lite is a targeted voice separation model for streaming, on-device automatic speech recognition (ASR) [2], as well as text-independent speaker verification (TI-SV) [18]. It assumes that the target speaker has completed an offline enrollment process [22,23], which uses a speaker recognition model to produce an aggregated embedding vector e that presents the voice characteristics of this speaker. In this work, we use the dvector embedding [24] trained with the generalized end-to-end extended-set softmax loss [25] as the speaker embedding. Let x (t) be the input feature frame at time t from the speech to be processed (e.g. stacked log Mel-filterbank energies). This feature is first frame-wise concatenated with the d-vector e, then fed into an LSTM network [26] followed by a fully connected neural network to produce a mask y (t) , which has the same dimensionality as x (t) :
y (t) = FC • LSTM(Concat(x (t) , e)).(1)
At runtime, the mask y (t) is element-wise multiplied to the input x (t) to produce the final enhanced features. Separately, we also use another LSTM-based neural network followed by a fully connected layer to estimate the noise type (either overlapping or non-overlapping speech) from the input x (t) . This noise type prediction is then used during inference to deactivate the VoiceFilter-Lite model when the input frame contains no overlapping speech. For more details, we refer the reader to [2].
We have previously demonstrated that the VoiceFilter-Lite model is an important component of text-independent speaker verification [18]. In particular, in the presence of overlapping background speech (the multitalker scenario), speaker verification tends to fail. Adding VoiceFilter-Lite to the feature frontend of speaker verification helps to suppress overlapping speech, which in turn improves the accuracy of target speaker verification. This in turn helps to reduce the false rejection rate of personalized keyphrases. For more details, we refer the reader to [18].
Review of multi-user VoiceFilter-Lite
To extend the VoiceFilter-Lite model to support multiple enrolled users, we added an AttentionNet to the VoiceFilter-Lite model, as illustrated in Fig. 1. This AttentionNet uses an attention mechanism to compute the most relevant speaker embedding given an input frame over an inventory of multiple speaker embeddings: e1, e2, · · · , eN .
The AttentionNet comprises two parts -the PreNet and the ScorerNet. The PreNet is a stack of three LSTM layers that computes, for each frame, a compressed representation of features in the stacked log Mel-filterbank energies, referred to as the key vector k (t) :
k (t) = PreNet(x (t) ).(2)
This compressed representation is then individually combined with each of the N enrolled speaker embeddings in the ScorerNet to generate a score for each enrolled speaker. The attention weights α (t) i > 0 are the softmax over these scores: Figure 1: Overall architecture of the multi-user VoiceFilter-Lite model proposed in [17]. This model comprises two partsan AttentionNet which computes the most relevant speaker from a noisy frame, and a VoiceFilterNet, which is identical to the single-use VoiceFilter-Lite model [2].
s (t) i = ScorerNet(Concat(k (t) , ei)),(3)α (t) i = exp (s (t) i ) N j=1 exp (s (t) j ) .(4)
Finally, the attended embedding is the dot product of these attention weights and the matrix of the N enrolled speaker emebddings. In this way, the ScorerNet selects one of the N enrolled speaker embeddings that is most relevant to the compressed representation, and therefore the most probable speaker in that frame:
e (t) att = N i=1 α (t) i · ei.(5)
This attended embedding is used as a conditioning input in the VoiceFilterNet, which is identical to the original VoiceFilter-Lite as described previously:
y (t) = FC • LSTM(Concat(x (t) , e (t) att )).(6)
Both the AttentionNet and VoiceFilterNet in the multi-user VoiceFilter-Lite model are jointly trained with an Adam optimizer [27] using a weighted linear combination of the following three loss functions:
1. Lasym: an asymmetric L2 loss for signal reconstruction; 2. Lnoise: a noise type prediction loss for adaptive suppression at runtime;
3. Latt: an attention loss that measures how well the attention weights predict the target speaker:
Latt = t CrossEntropy(α (t) , wgt) + λ||α (t) ||∞ (7)
where wgt is the ground truth attention weights and λ is the weight of the L∞ regularization term. The ground truth attention weights are the one-hot encoding of the position of the target speaker embedding. For example, if N = 4 and the target speaker is the second enrolled speaker, then wgt = [0, 1, 0, 0]. This loss ensures that the attention weights of the non-target speakers tend towards 0, while the target speaker weight tends to 1.
For more details on the performance on this model on a variety of tasks, we refer the reader to [17].
Closing the performance gap between the single-user and multi-user models
As previously discussed in Section 1, the original multi-user VoiceFilter-Lite described in Section 2.2 suffers from performance degradation on single-user evaluations when compared with single-user VoiceFilter-Lite models, which prevents us from deploying such models in production environments. However, we observed an interesting fact -the loss functions of the multi-user VoiceFilter-Lite model look reasonable during training. This implies the attention mechanism in the original multi-user VoiceFilter-Lite model is likely overfitting the training data, and specifically, the combinations of enrolled speakers in the training data.
To address this overfitting issue, we use a dual learning rate schedule, where the AttentionNet is trained independently and with a smaller learning rate than the VoiceFilterNet. Doing so ensures smaller weight updates for the AttentionNet, allowing the optimizer to more effectively minimize the loss function to produce an optimal solution. We found that this approach prevents the AttentionNet from memorizing the training data, which in turn allows it to generalize better to unseen examples.
To further improve the VoiceFilter-Lite model for speaker verification, we also replace the frame-wise concatenation operation between the attended embedding e (t) att and the input features x (t) with a feature-wise linear modulation (FiLM). In FiLM, the input features are modulated by the embedding via the following affine transformation:
x (t) trans = FC1(e (t) att ) x (t) + FC2(e (t) att ),(8)y (t) = FC • LSTM(x (t) trans ),(9)
where FC1 and FC2 are two different fully connected neural networks, and denotes the element-wise product. We used two-layer FC networks, where the final layer projects the attended embedding to the same dimension as the input features with a tanh activation function. Unlike concatenation, FiLM learns to influence each input frame in an element-wise fashion by applying an affine transformation. As a result, the attended embedding is able to scale features in the input frame up or down, or negate them or even selectively threshold them allowing a more fine grained control than simple concatenation. Furthermore, FiLM only requires two parameters (FC1 and FC2) per input frame, making it a computationally more efficient conditioning method. Numerous studies have described the benefit of using FiLM in ASR [28,29] and speech enhancement [21,20], demonstrating FiLM's broad relevance for speaker-conditioned speech models.
Experimental Setup
Experimental design
Although the multi-user VoiceFilter-Lite model supports an arbitrary number of enrolled speaker embeddings as side input, there are additional constraints to consider when implementing this model in TFLite [30,31]. Since TFLite does not support inputs with an unknown dimension, we had to pre-define a maximal number of enrolled speaker embeddings, i.e. N in our implementation. Then, at runtime, if the actual number of enrolled speakers is smaller than N , we use an all-zero vector as the embedding of any missing speaker. Thus in the experiments to be shown in Section 4, for simplicity, we first assume the maximal number of speakers is N = 2 in our studies. Then in Section 4.4, we demonstrate that the observations from N = 2 experiments are also valid when we extend it to N = 4. Furthermore, in this paper, we focus only on addressing the multi-talker speaker verification challenge, especially for the multi-user multi-talker case. For example, when both speaker A and speaker B enrolled their voices on the device (multi-user), and at runtime, speaker A and speaker C speak at the same time (multi-talker), we expect the speaker verification system to accept the input, because it contains speech from one of the enrolled users (i.e. speaker A).
For consistency, the acoustic feature frontend and the speaker verification model we used in our experiments are exactly the same as the ones used in [17].
Model topology
For all the models in our experiments, the VoiceFilterNet has 3 LSTM layers, each with 256 nodes, and a fully connected layer with sigmoid activation function. The noise type prediction network has 2 LSTM layers, each with 128 nodes, and a fully connected layer with 64 nodes. In the multi-user setup, the PreNet has 3 LSTM layers, each with 128 nodes; the ScorerNet has two feedforward layers, each with 64 nodes.
Training and evaluation data
All the VoiceFilter-Lite models in our experiments are trained on a combination of: (1) The LibriSpeech training set [32]; and (2) a vendor-collected dataset of English speech queries with a grand total of 2,902,102 utterances from 18,851 unique speakers (see Table 1). To generate the noisy inputs, we augment these training utterances with different noise sources (speech and non-speech) and with different room configurations [33,34,35], using a signal-to-noise ratio (SNR) drawn from a uniform distribution between 1dB and 10dB. In the multiuser setup, each training utterance is attached with both the target speaker embedding, and randomly sampled speaker embeddings from other speakers. For example, for a 4-enrolled user model, we randomly sample 3 speaker embeddings from other speakers. And to ensure that we train on all possible speaker combinations (e.g. 1, 2, and 3 enrolled users), we use a dropout probability of 25% to randomly replace each non-target speaker embedding with an all-zero vector.
For evaluation, we use a vendor-provided English speech query dataset. The enrollment list comprises 8,069 utterances from 1,434 speakers, while the test list comprises 194,890 utterances from 1,241 speakers. Each speaker verification task is evaluated on 193k positive trials and 200k negative trials based on the enrollment and test sets. The interfering speech are drawn from a separate English dev-set consisting of 220,092 utterances from 958 speakers. More details on each subset of the training and evaluation data are provided in Table 1. During evaluation, we apply different noise sources and room configurations to the data. We use "Clean" to denote the original nonnoisified data, although they could be quite noisy already. The non-speech noise source consists of ambient noises recorded in cafes, vehicles, and quiet environments, as well as audio clips of music and sound effects downloaded from Getty Images [36]. The speech noise source is a distinct development set without overlapping speakers from the testing set. We evaluate on reverberating room conditions, which consists of 3 million convolutional room impulse responses generated by a room simulator [35] with three SNR values: −5dB, 0dB, and 5dB.
Experimental Results
Experiment 1 -Attention is required for accurate voice separation
The aim of our first experiment is to determine whether the At-tentionNet is required or not. There are two naive alternative approaches one can feed the N enrolled speaker embeddings to the VoiceFilter-Lite model without the AttentionNet:
• Averaging Model: The attended embedding is the average (arithmetic mean) of all enrolled speaker embeddings.
• Concat Model: The attended embedding is an unordered concatenation of all enrolled speakers embeddings. To preserve the size of the attended embeddig, we linearly project this concatenated vector down to the size of a single speaker embedding.
We also explore two variations of the attention-based model, where we generate the attended embedding by:
• Weighted Sum Model: We take the dot product (see Eq. 5) between the attention weights and the N speaker embeddings, to compute a weighted sum of all the enrolled speaker embeddings. This is different from the Averaging Model, which uses identical weight 1 N for each enrolled speaker.
• Concat Top-K Model: Out of N enrolled speaker embeddings, we pick the K embeddings with the largest attention weights, and concatenate them by the order of the corresponding attention weights, and project this vector down to the size of a single speaker embedding. It is important to note that this is different from the naive Concat Model, because here the concatenated speaker embeddings are ordered by their attention weights.
The evaluation results are shown in Table 2. From this table, we make three key observations. First, compared to "No Voice-Filter", adding VoiceFilter-Lite model (rel. −70.8% for singleuser and −71.4% for the best multi-user model for speech noise at SNR 0 dB) to the feature frontend of the text-independent speaker verification system significantly reduces the equal error rate for speaker verification, confirming our previous results [18,17]. Since VoiceFilter-Lite is disabled when there is no overlapping speech, we observe no difference in the EER for the non-speech noise cases for all models.
Second, amongst the multi-user VoiceFilter-Lite models, we see that neither the Averaging Model (relative 10.1% increase in EER for two enrolled users) nor the Concat Model (rel. 12.1% increase in EER) perform as well as the attention-based multi-user VoiceFilter-Lite models. This result suggests that the AttentionNet, which finds the most relevant target speaker, is required for good performance. Importantly, the AttentionNet is also able to find the target speaker from evaluation data, which it has not seen during training. Furthermore, since the multiuser VoiceFilter-Lite models with AttentionNet have single user EERs that closely match the single-user VoiceFilter-Lite model, we can confidently say that the attention mechanism is indeed able to generalize to unseen examples and is able to correctly identify the target speaker.
Third, between the two AttentionNet models, we find that the Weighted Sum Model outperforms the Concat Top-K Model for the two-enrolled speaker case (rel. 6.2% increase in EER). Similarly, we notice that the Averaging Model also performs better than the Concat Model for the same two-enrolled speaker case. This suggest that concatenating the two speaker embeddings, with or without ordering, and then projecting it to 256 dimensions (i.e. the size of a single speaker embedding) does not contain sufficient information for the VoiceFilterNet to identify and enhance speech features of the target speaker in the input data. Rather, using a weighted sum of the speaker embeddings is a much better predictor of the target speaker embedding. The difference in single-user EER between the Averaging Model and the Weighted Sum model further reinforces the fact that the AttentionNet is selecting the correct speaker.
Taken together, the results of our first experiment indicate that the AttentionNet with weighted sum is critical to the multi-user VoiceFilter-Lite model. The simpler, non-attentionbased strategies are insufficient for such tasks. In all multi-user VoiceFilter-Lite models in subsequent sections, we will use the AttentionNet + Weighted Sum Model configuration.
Experiment 2 -Dual learning rate schedule helps to avoid AttentionNet overfitting
One observation we made in our previous multi-user VoiceFilter-Lite study [17] is that the attention mechanism tends to overfit and memorize training data. Our next experiment is aimed at addressing this limitation by tuning the learning rate of the model.
Evaluation results are shown in Table 3. Since changing the learning rate or model architecture does not affect performance on non-speech background noise (see Table 2), we omit the nonspeech noise results from the next two tables.
First, for the single-user VoiceFilter-Lite model, we notice that using a smaller learning rate of 10 −6 results in a signifi- cantly worse model with a much higher EER across all SNR values compared to the model trained with a higher learning rate 10 −5 (rel. 56.6% increase at SNR 0 dB). Secondly, for the multi-user VoiceFilter-Lite model, we observe a regression in the EER with two-enrolled users with the higher learning rate (rel. 29.6% increase at SNR 0 dB). This suggests that with a higher learning rate the AttentionNet tends to overfit on training data and fails to generalize to the evaluation data. Therefore, we implemented a dual learning rate scheduler where the Atten-tionNet is trained with a smaller learning rate of 10 −6 , while the VoiceFilterNet is trained with a larger learning rate of 10 −5 . As shown in Table 3, this significantly improves both the singleand two-user (28.7% reduction relative to the original learning rate at SNR 0 dB) performance of the model.
Experiment 3 -FiLM-based speaker conditioning improves model performance
So far, we have shown that having an AttentionNet and training it with a smaller learning rate than the VoiceFilterNet is necessary for good performance in reducing EER when the multiuser VoiceFilter-Lite model is present in the text-independent speaker verification frontend. Another aspect of the model that can be further optimized is how the attended embedding is used There are several ways in which the attended embedding can be used to condition the VoiceFilterNet:
• Concat-Conditioned Model: The attended embedding is concatenated with each input frame before being fed into the VoiceFilterNet LSTM stack. This increases the dimensions of the input frame by the size of the attended embedding (256 dimensions).
• FiLM-Conditioned Model: An affine transformation, shown in Eq. 8, is applied to each input frame. This affine transformation allows the attended embedding to modulate the input frame in a feature-wise manner. This does not change the dimensions of the input frame.
Evaluation results for these different models are shown in Table 4. In these experiments, we keep the AttentionNet architecture the same (Weighted Sum Model) and use a dual learning rate schedule as we have described in the preceeding sections. We observed that the multi-user VoiceFilter-Lite model that uses FiLM to condition the input frames with the attended emebedding performs significantly better than the model that uses concatenation. Specifically, relative to the concatconditioned multi-user model, we find a 5.6% reduction in EER for one enrolled user and a 26.7% reduction (at SNR 0dB) for two enrolled users. We also observe that relative to the singleuser VFL, the FiLM-conditioned multi-user model has comparable performance with a single enrolled user (2.42 vs. 2.37 at SNR 0dB). Therefore, using FiLM to condition the model on the speaker embedding is far more robust method than concatenation. Table 5. Compared to our previously published model [17], the new four-user model, which uses FiLM and dual learning rates (see Fig. 2) results in a significantly lower EER for all speaker combinations. Interestingly, we observe a regression in EER between the best two-user VoiceFilter-Lite model and the fouruser model for the 1-speaker (2.37 vs. 2.54, 7.1% increase at SNR 0dB) and 2-speaker (3.55 vs. 4.22, 18.9% increase at SNR 0dB) evaluations. One reason for this could be that there are fewer 1-speaker and 2-speaker examples during training the four-user model (P1−user = 0.25 3 ) than the two-user model (P1−user = 0.25) due to the way we process our training data (see Section 3.3). In fact, for the four-user model, only about 15.6% of the training data contains one or two enrolled users. As a result, during evaluation, the model does not generalize very well on 1-speaker or 2-speaker evaluations compared with 3-speaker and 4-speaker evaluations. To address this issue, one of our future work directions is to balance our training data according to the realistic distributions of the number of users on shared devices, as well as to make the model more robust to unbalanced data.
Conclusions
In this paper, we devised a series of experiments to evaluate the impact of various design choices in the multi-user VoiceFilter-Lite model. We confirmed that an attention mechanism is critical for the multi-user model to function well, which cannot be replaced by naive aggregation logic such as either averaging or concatenating all enrolled speaker embeddings. We found that training the attention mechanism with a learning rate that is an order of magnitude smaller than the rest of the model addresses the overfitting issue, and is critical to close the performance gap between single-user and multi-user models on single-user evaluations. Additionally, the performance of the model could be further improved by using FiLM to modulate the attended speaker embedding.
Although all experiments in this paper are carried out for multi-user VoiceFilter-Lite, it is important to note that the pro-posed attention-based speaker selection mechanism is a generic solution that can be applied to any speaker-conditioned speech models. This is crucial as most smart home devices, such as smart displays and smart speakers, usually support multiple enrolled users. Thus as our future work, we would like to adopt the best practices from the multi-user VoiceFilter-Lite to other speaker-conditioned speech models, including personalized ASR or personal VAD.
Figure 2 :
2Anatomy of the proposed new AttentionNet with FiLM-based speaker modulation.
Table 1 :
1Number of utterances and speakers in each subset of training and evaluation data.Training data for VoiceFilter-Lite
Num. of utts Num. of spks
LibriSpeech
281,241
2,338
Vendor-collected
2,620,867
16,513
Evaluation data for speaker verification
Num. of utts Num. of spks
Enrollment set
8,069
1,434
Test set
194,890
1,241
Interference speech
220,092
958
Table 2 :
2Equal Error Rate (EER) of text-independent speaker verification with different VoiceFilter-Lite (VF) models in the frontend and different number of enrolled users. The multi-user VoiceFilter-Lite models all use dual learning rates. Bold green text indicates best model.Model Name
Num. of
enrolled users
Clean
Non-speech Noise
Speech Noise
-5dB 0dB 5dB -5dB 0dB 5dB
No VoiceFilter
-
0.71
5.04 2.23 1.50 12.40 8.29 5.13
Single-user VoiceFilter
1
0.71
5.01 2.19 1.48
3.97
2.42 1.65
Multi-user VoiceFilter
Averaging Model
1
0.71
5.01 2.20 1.48
4.75
2.66 1.72
2
0.71
5.02 2.21 1.48
7.12
3.91 2.29
Concat Model
1
0.71
5.01 2.20 1.48
4.57
2.58 1.72
2
0.71
5.02 2.21 1.48
7.41
3.98 2.29
AttentionNet
+ Weighted Sum Model
1
0.71
5.01 2.22 1.49
3.94
2.37 1.64
2
0.72
5.03 2.21 1.47
7.11
3.55 1.99
AttentionNet
+ Concat Top-K Model
1
0.71
5.01 2.20 1.48
3.92
2.39 1.62
2
0.72
5.02 2.22 1.49
7.23
3.77 2.13
Table 3 :
3EER of text-independent speaker verification with dif-
ferent VoiceFilter-Lite (VFL) models. Here, we vary the learn-
ing rate (LR). Each model is trained for 25 million steps. All
models use the weighted sum attention mechanism. "Num.
Spk" is the number of enrolled speakers during evaluation. For
the "Dual LR" setup, we use a LR of 10 −5 for VoiceFilterNet
and a LR of 10 −6 for AttentionNet.
Model Name
Num.
users
Clean
Speech Noise
-5dB 0dB 5dB
No VFL
-
0.71
12.40 8.29 5.13
Single-user VFL
LR: 10 −5
1
0.71
3.97
2.42 1.65
Single-user VFL
LR: 10 −6
1
0.71
6.67
3.79 2.24
Multi-user VFL
LR: 10 −5
1
0.71
4.13
2.50 1.68
2
0.71
10.39 6.79 4.31
Multi-user VFL
LR: 10 −6
1
0.71
6.97
3.88 2.25
2
0.71
9.52
5.24 2.81
Multi-user VFL
Dual LR
1
0.71
4.02
2.51 1.73
2
0.71
8.46
4.84 3.43
Table 4 :
4EER of text-independent speaker verification with dif-
ferent VoiceFilter-Lite (VFL) models. Here, the attended em-
bedding conditioning mechanism is changed. All multi-user
VFL models use Weighted Sum and Dual Learning Rate sched-
ule. Bold green text indicates best model.
Model Name
Num.
users
Clean
Speech Noise
-5dB 0dB 5dB
No VFL
-
0.71
12.40 8.29 5.13
Single-user VFL
1
0.71
3.97
2.42 1.65
Multi-user VFL
+ Concat Cond.
1
0.71
4.02
2.51 1.73
2
0.71
8.46
4.84 3.49
Multi-user VFL
+ FiLM Cond.
1
0.71
3.94
2.37 1.64
2
0.71
7.11
3.55 1.99
by the VoiceFilterNet.
Table 5 :
5EER of text-independent speaker verification with different VoiceFilter-Lite (VFL) models. Here, the best 2-enrolled user and best 4-enrolled user are compared with the previously published model.Finally, we demonstrate that the best two-user model can be easily extended to support four enrolled users. In the following experiments, we trained a four-user model with the same model architecture. Evaluation results for this model is shown inModel Name
Num. of
enrolled users
Clean
Non-speech Noise
Speech Noise
-5dB 0dB 5dB -5dB 0dB 5dB
No VFL
-
0.71
5.04 2.23 1.50 12.40 8.29 5.13
Single-user VFL
1
0.71
5.01 2.19 1.48
3.97
2.42 1.65
Best
Two-user VFL
1
0.71
5.01 2.22 1.49
3.94
2.37 1.64
2
0.72
5.03 2.21 1.47
7.11
3.55 1.99
Previously Published
Four-user VFL [17]
1
0.71
5.03 2.21 1.47
7.32
3.90 2.19
2
0.72
5.04 2.21 1.49
9.34
5.18 2.78
3
0.72
5.01 2.22 1.49 10.36 5.73 3.01
4
0.72
5.05 2.21 1.49 10.99 6.10 3.14
New
Four-user VFL
1
0.71
5.03 2.21 1.47
4.32
2.54 1.71
2
0.71
5.03 2.21 1.47
7.59
4.21 2.50
3
0.72
5.03 2.22 1.49
8.05
5.14 2.85
4
0.72
5.03 2.21 1.50
9.78
5.38 2.99
4.4. Experiment 4 -Same observations hold for four en-
rolled users
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|
[] |
[
"The emergence of dark matter-deficient ultra-diffuse galaxies driven by scatter in the stellar mass-halo mass relation and feedback from globular clusters",
"The emergence of dark matter-deficient ultra-diffuse galaxies driven by scatter in the stellar mass-halo mass relation and feedback from globular clusters"
] |
[
"Sebastian Trujillo-Gomez \nZentrum für Astronomie\nAstronomisches Rechen-Institut\nUniversität Heidelberg\nMonchhofstraße 12-14D-69120HeidelbergGermany\n",
"J M Diederik Kruijssen \nZentrum für Astronomie\nAstronomisches Rechen-Institut\nUniversität Heidelberg\nMonchhofstraße 12-14D-69120HeidelbergGermany\n",
"Marta Reina-Campos \nZentrum für Astronomie\nAstronomisches Rechen-Institut\nUniversität Heidelberg\nMonchhofstraße 12-14D-69120HeidelbergGermany\n\nDepartment of Physics & Astronomy\nMcMaster University\n1280 Main Street WestL8S 4M1HamiltonCanada\n\nCanadian Institute for Theoretical Astrophysics (CITA)\nUniversity of Toronto\n60 St George StM5S 3H8TorontoCanada\n"
] |
[
"Zentrum für Astronomie\nAstronomisches Rechen-Institut\nUniversität Heidelberg\nMonchhofstraße 12-14D-69120HeidelbergGermany",
"Zentrum für Astronomie\nAstronomisches Rechen-Institut\nUniversität Heidelberg\nMonchhofstraße 12-14D-69120HeidelbergGermany",
"Zentrum für Astronomie\nAstronomisches Rechen-Institut\nUniversität Heidelberg\nMonchhofstraße 12-14D-69120HeidelbergGermany",
"Department of Physics & Astronomy\nMcMaster University\n1280 Main Street WestL8S 4M1HamiltonCanada",
"Canadian Institute for Theoretical Astrophysics (CITA)\nUniversity of Toronto\n60 St George StM5S 3H8TorontoCanada"
] |
[
"MNRAS"
] |
In addition to their low stellar densities, ultra-diffuse galaxies (UDGs) have a broad variety of dynamical mass-to-light ratios, ranging from dark matter (DM) dominated systems to objects nearly devoid of DM. To investigate the origin of this diversity, we develop a simple, semiempirical model that predicts the structural evolution of galaxies, driven by feedback from massive star clusters, as a function of their departure from the mean SMHM relation. The model predicts that a galaxy located 0.5 dex above the mean relation at halo = 10 10 M will host a factor of ∼ 10−100 larger globular cluster (GC) populations, and that feedback from these GCs drives a significant expansion of the stellar component and loss of DM compared to galaxies on the SMHM relation. This effect is stronger in haloes that collapse earlier and have enhanced star formation rates at > ∼ 2, which leads to increased gas pressures, stellar clustering, and mean cluster masses, and significantly enhances the energy loading of galactic winds and its impact on the DM and stellar orbits. The impact on galaxy size and DM content can be large enough to explain observed galaxies that contain nearly the universal baryon fraction, as well as NGC1052-DF2 and DF4 and other isolated UDGs that contain almost no DM. The trend of increasing galaxy size with GC specific frequency observed in galaxy clusters also emerges naturally in the model. Our predictions can be tested with large and deep surveys of the stellar and GC populations in dwarfs and UDGs. Because stellar clustering drives the efficiency of galactic winds, it may be a dominant factor in the structural evolution of galaxies and should be included as an essential ingredient in galaxy formation models.
|
10.1093/mnras/stab3401
|
[
"https://arxiv.org/pdf/2103.08610v2.pdf"
] | 232,240,674 |
2103.08610
|
fe82a50e2e2433a5aadeec807e2956bedeff885e
|
The emergence of dark matter-deficient ultra-diffuse galaxies driven by scatter in the stellar mass-halo mass relation and feedback from globular clusters
2021
Sebastian Trujillo-Gomez
Zentrum für Astronomie
Astronomisches Rechen-Institut
Universität Heidelberg
Monchhofstraße 12-14D-69120HeidelbergGermany
J M Diederik Kruijssen
Zentrum für Astronomie
Astronomisches Rechen-Institut
Universität Heidelberg
Monchhofstraße 12-14D-69120HeidelbergGermany
Marta Reina-Campos
Zentrum für Astronomie
Astronomisches Rechen-Institut
Universität Heidelberg
Monchhofstraße 12-14D-69120HeidelbergGermany
Department of Physics & Astronomy
McMaster University
1280 Main Street WestL8S 4M1HamiltonCanada
Canadian Institute for Theoretical Astrophysics (CITA)
University of Toronto
60 St George StM5S 3H8TorontoCanada
The emergence of dark matter-deficient ultra-diffuse galaxies driven by scatter in the stellar mass-halo mass relation and feedback from globular clusters
MNRAS
0002021Accepted 2021 November 5. Received 2021 November 4; in original form 2021 March 12Preprint 4 February 2022 Compiled using MNRAS L A T E X style file v3.0Galaxies -galaxies: evolution -galaxies: formation -galaxies: structure - galaxies: haloes -galaxies: star clusters: general
In addition to their low stellar densities, ultra-diffuse galaxies (UDGs) have a broad variety of dynamical mass-to-light ratios, ranging from dark matter (DM) dominated systems to objects nearly devoid of DM. To investigate the origin of this diversity, we develop a simple, semiempirical model that predicts the structural evolution of galaxies, driven by feedback from massive star clusters, as a function of their departure from the mean SMHM relation. The model predicts that a galaxy located 0.5 dex above the mean relation at halo = 10 10 M will host a factor of ∼ 10−100 larger globular cluster (GC) populations, and that feedback from these GCs drives a significant expansion of the stellar component and loss of DM compared to galaxies on the SMHM relation. This effect is stronger in haloes that collapse earlier and have enhanced star formation rates at > ∼ 2, which leads to increased gas pressures, stellar clustering, and mean cluster masses, and significantly enhances the energy loading of galactic winds and its impact on the DM and stellar orbits. The impact on galaxy size and DM content can be large enough to explain observed galaxies that contain nearly the universal baryon fraction, as well as NGC1052-DF2 and DF4 and other isolated UDGs that contain almost no DM. The trend of increasing galaxy size with GC specific frequency observed in galaxy clusters also emerges naturally in the model. Our predictions can be tested with large and deep surveys of the stellar and GC populations in dwarfs and UDGs. Because stellar clustering drives the efficiency of galactic winds, it may be a dominant factor in the structural evolution of galaxies and should be included as an essential ingredient in galaxy formation models.
INTRODUCTION
Over the last few years, interest in the study of so-called ultradiffuse galaxies (UDGs) has rapidly increased. UDGs are typically defined to be galaxies with the stellar mass of a dwarf (10 7 < ∼ * / M < ∼ 3 × 10 8 ), and very large extent, e > 1.5 kpc, similar to the sizes of * galaxies (van Dokkum et al. 2015). Due to their very low surface brightness > 24 mag arcsec −2 , these galaxies were mostly absent in large surveys like SDSS 1 . When going to lower brightness limits, UDGs have been found in large numbers in galaxy clusters (Caldwell 2006;van Dokkum et al. 2015;Koda et al. 2015;Mihos et al. 2015;van der Burg et al. 2016), but also in groups (Merritt et al. 2016;Trujillo et al. 2017;Román & Trujillo 2017b), and in the field (Leisman et al. 2017), which ★ E-mail: [email protected] 1 Sloan Digital Sky Survey, https://www.sdss.org suggest that they may represent a substantial fraction of the galaxy population (Jones et al. 2018).
Much of the interest in these objects is focused on establishing the mechanisms that lead to their formation, as well as how they fit, alongside normal galaxies, into the broader theory of galaxy formation in the context of the cold dark matter (ΛCDM) cosmology. Many studies of their properties find that most UDGs populate dark matter (DM) haloes comparable to those hosting dwarf galaxies ( halo < ∼ 10 11 M ; Beasley & Trujillo 2016;Peng & Lim 2016;Román & Trujillo 2017a;Leisman et al. 2017;Amorisco 2018;Amorisco et al. 2018;Chilingarian et al. 2019). Among the current theoretical and numerical efforts to understand their formation, isolated UDGs have been found to naturally arise in some cosmological hydrodynamical simulations as a result of feedback-driven winds that expand the dark matter and stellar orbits Jackson et al. 2021b). Other models predict that UDGs form in the high-spin tail of the distribution of DM haloes obtained from cosmological simulations (Amorisco & Loeb 2016). However, not all of the observed UDGs can be explained by current models.
Recent studies have uncovered UDGs with surprising properties. van Dokkum et al. (2018b) and van Dokkum et al. (2019a) found two UDGs in a nearby galaxy group, NGC1052-DF2 and DF4, hosting populations of ∼ 10 unusually massive GCs ( GC ≈ 9×10 5 M ). Both galaxies have stellar masses ∼ 10 8 M and seem to contain very little DM, with stars providing nearly the entire dynamical mass in the central ∼ 8 kpc (van Dokkum et al. 2018a(van Dokkum et al. , 2019a. For an NFW DM halo density profile (Navarro et al. 1997), this would imply a baryon fraction bar ≡ bar / dyn > ∼ 50 per cent, in contradiction with the < ∼ 3 per cent expected in galaxies of the same mass in ΛCDM (Papastergis et al. 2012). Many theoretical studies have focused on understanding how DM-deficient galaxies could form in a Universe dominated by DM, pointing to the need for intense ram pressure and tidal stripping of satellites in highly eccentric orbits to preferentially remove the DM (Ogiya 2018;Carleton et al. 2019;Jiang et al. 2019;Nusser 2020;Sales et al. 2020;Macciò et al. 2020;Jackson et al. 2021a). These results seem are in general very sensitive to the infall times and orbits of the galaxies.
UDGs found in isolation seem to present a greater challenge to current models. These are usually H -rich and star-forming, are commonly found in blind H surveys (Leisman et al. 2017), and have lower gas rotation rates than dwarfs of similar mass (Jones et al. 2018). Mancera Piña et al. (2019Piña et al. ( , 2020 studied the H rotation curves of six of these objects and found that they imply very high baryon fractions of > 47 per cent (or conversely, very low DM content) within the extent of their gas discs, ∼ 8 − 10 kpc. These values are well above the budget defined by the cosmic mean, bar ≈ 15 per cent (Planck Collaboration et al. 2016). In the most extreme cases, the baryonic mass alone is enough to account for the rotation velocity of the gas. Since environmental processes cannot account for the large sizes and low DM content of these isolated systems, an internal mechanism is necessary. However, none of the mechanisms suggested so far for forming field UDGs, such as high spin DM haloes (Amorisco & Loeb 2016) or feedback-driven expansion Jiang et al. 2019;Jackson et al. 2021b), are able to explain the extreme DM deficiency of these of objects.
The relation between the stellar mass of a galaxy and that of its host DM halo is at the heart of the problem of galaxy formation in the context of the current cosmological paradigm. The stellar mass-halo mass (SMHM) relation has been studied extensively in observations using many techniques (e.g. Leauthaud et al. 2012;Behroozi et al. 2019). The halo abundance matching (HAM) technique combined with observed galaxy clustering (Conroy et al. 2006;Trujillo-Gomez et al. 2011;Reddick et al. 2013;Campbell et al. 2018)provide useful constraints on the mean and scatter of the relation for massive galaxies (Tasitsiomi et al. 2004;Kravtsov et al. 2004;Conroy et al. 2006;Moster et al. 2010;Behroozi et al. 2010;Rodríguez-Puebla et al. 2012;Moster et al. 2018). However, at the scales of dwarf galaxies ( halo < ∼ 10 11 M ), independent statistical constraints on the scatter in the SMHM relation, such as galaxy clustering or weak lensing, are not yet available. Dynamical modelling of the stellar and gaseous components of dwarf galaxies provide estimates of the DM halo mass, but these are often subject to significant systematics, including extrapolation from the galaxy extent out to the virial radius (Read et al. 2016b;Campbell et al. 2017;Oman et al. 2019). Taken at face value, these estimates show broad agreement with the mean HAM expectation, but with considerably larger scatter in stellar mass at fixed halo mass (Katz et al. 2017;Read et al. 2017;Schneider et al. 2017;Forbes et al. 2018;Li et al. 2020).
In addition to observational evidence, larger scatter is also expected on theoretical grounds at the scales of dwarf galaxies due to their shallower potential wells, and the strong role of feedback and reionisation in the stochasticity of their star formation. These considerations raise the interesting possibility that dwarf galaxies could scatter around the mean SMHM relation by more than a decade in stellar mass at fixed halo mass, and that this scatter could have important consequences for the evolution of galaxies. In this work we develop a semi-empirical model to investigate the effect of this scatter on the structure, evolution, and star cluster populations of present day low-mass galaxies. The model relates the collapse epoch of a DM halo with its excursion from the mean SMHM relation, and predicts how this determines its star and cluster formation rate at early times, and how SN feedback from the enhanced stellar clustering modifies the structure of the galaxy and its DM halo at present. This paper is organised as follows. Section 2 describes the semi-empirical model. The predictions for galaxy and DM halo structure and GC populations at = 0 are presented in Section 3, including the implications for the origin of UDGs, and how DMdeficient galaxies may be explained as a result of upward scatter in the SMHM relation. Section 5 shows the effect of structural evolution on the inferred halo masses of low-mass galaxies, Section 6 discusses the implications of the model, and Section 7 summarises our findings. The analysis below assumes the Planck cosmological parameters (Planck Collaboration et al. 2016): Ω m = 0.308, Ω bar = 0.049, 8 = 0.816, and 0 = 67.8 km s −1 Mpc −1 .
MODELLING THE IMPACT OF SCATTER IN THE SMHM RELATION ON GALAXY EVOLUTION AND STRUCTURE
In this section we describe a semi-empirical model that relates the departure of a galaxy from the SMHM relation to the evolution of its structure. In the model, both the integrated properties and the evolution of the galaxy are defined by its position in the 2-dimensional plane defined by * and halo . The model is summarised as follows:
(i) For a given DM halo mass halo , the mean stellar mass at = 0 is determined by assuming an empirical mean SMHM relation.
(ii) The scatter in * at fixed halo is assumed to correlate with the scatter in maximum circular velocity max at fixed halo , such that more concentrated galaxies (i.e. with deeper potential wells and higher max ) have larger * at = 0.
(iii) The DM halo concentration (now a function of both * and halo ) is used to obtain the halo collapse time through the direct relation found in cosmological simulations. This implies that galaxies with an excess of * at a fixed halo will inhabit denser DM haloes with earlier formation times relative to those on the mean SMHM relation.
(iv) Galaxies are assumed to have formed a fixed fraction of their present stellar mass during the collapse epoch of their DM halo. Gas accretion rates are by definition higher during halo collapse, leading to elevated gas pressures and star formation rates. Hence, this will correspond to the dominant epoch of GC formation in the galaxy.
(v) At the collapse time, sizes are assigned to the stellar component of galaxies based on an empirical relation between the effective radius e and halo at that epoch. Using the size and stellar mass at the collapse time, the time-averaged SFR surface density within e is obtained.
(vi) The gas surface density Σ gas is calculated from the SFR surface density Σ SFR at the time of collapse using the Kennicutt-Schmidt (Kennicutt 1998) relation. The galactic rotation frequency Ω is obtained using the DM, stellar, and gas masses enclosed within e . A model describing the dependence of the clustering of star formation on the galactic environment (specified by Σ gas , Ω, and Toomre ) is used to estimate the fraction of star formation occurring in bound clusters. A complementary model is applied to predict the environmentally-dependent initial cluster mass function (ICMF). The stellar cluster populations are fully determined by the bound fraction and the ICMF, including the number of GCs and the mean cluster mass.
(vii) Using the results of detailed numerical simulations, the effect of supernovae (SN) clustering on the energy loading of galactic winds is estimated. The SN clustering is obtained from the SFR, the bound fraction of star formation, and the ICMF.
(viii) To model the process of DM core formation due to feedback-driven outflows, a fixed fraction of the wind energy is assumed to couple to the DM halo potential energy at = 0. The model then predicts the amount of expansion of the collisionless components (DM and old stars) due to the increase in halo potential energy from galactic winds. This is done using a parametrized feedback-modified cored DM density profile calibrated using hydrodynamic simulations.
(ix) The masses and sizes of the DM and stellar components at the collapse time, and at = 0 derived above fully specify the structural evolution of the galaxy and define the dynamical mass of each component at all radii.
The model outlined above thus connects the position of a galaxy in the * − halo plane to its mean star formation rate density, cold gas content, massive star cluster populations, galactic winds, and feedback-driven structural evolution of its DM and stellar components. We have deliberately chosen to neglect the effect of individual mergers because these are difficult to model analytically and because we are primarily interested in the evolution of the galaxy and its GCs at > ∼ 2. At this epoch, smooth gas accretion from the cosmic web is the dominant mechanism for the production of massive clusters in the progenitors of * galaxies . Furthermore, an individual massive merger can be thought of as a temporary enhancement in the gas, stellar, and DM mass accretion rate that boosts the conversion of gas into stars, effectively scattering the remnant galaxy upwards from the mean SMHM relation. In a broad statistical sense, using mean halo mass accretion histories, our model then accounts for the effects of mergers. Furthermore, we do not account for GCs accreted from mergers, but these are expected to be subdominant in galaxies with halo < 10 11 M (Choksi & Gnedin 2019). In the following subsections we describe each component of the model in detail.
The SMHM relation
The starting point is the assumption of a mean SMHM relation at = 0. We employ the SMHM relation as derived by Behroozi et al. (2013) using the halo abundance matching technique (HAM) and considering the stellar mass function, specific star formation rates, and the cosmic history of star formation as constraints.
HAM generally consists of fitting the function * ( halo ) by statistically matching the number density of observed galaxies with the DM halo mass function predicted by ΛCDM. The low mass end of the SMHM relation, halo < ∼ 10 11 M , is subject to significant systematics due to uncertainties in the stellar mass function, and the lack of constraints from galaxy clustering at low masses (e.g. see figure 34 in Behroozi et al. 2019). Furthermore, the relation extends down to halo 10 10 M , and extrapolation is required below this mass. Assuming a different mean SMHM relation would modify the quantitative predictions of our model, but not the overall behaviour of galaxies that scatter above (and below) compared to those on the mean relation. Figure 1 shows the mean SMHM relation assumed in this work and compares it to a comprehensive set of dynamical models of nearby low-mass galaxies from (Read et al. 2017), Forbes et al. (2018, and the SPARC database (Li et al. 2020). With the exception of a few Local Group members, most objects are isolated late-type galaxies 2 . The mass models use a variety of different techniques, but all of them account for the possibility of a cored DM halo using coreNFW DM density profile (Read et al. 2016b). For comparison, the figure also includes simple dynamical estimates based on H observations of gas-rich UDGs by Mancera Piña et al. (2020), baryon-dominated galaxies (Guo et al. 2020), and velocity dispersion measurements for NGC1052-DF2 and DF4 (van Dokkum et al. 2018a(van Dokkum et al. , 2019a. We refer the reader to Appendix A for details on each of these datasets and mass models. In general, dynamical measurements have significant uncertainties and may suffer from systematics which are difficult to estimate. However, across several datasets, these dynamical models suggest the possibility of large amounts of scatter, in excess of the uncertainties, around the mean SMHM relation obtained from abundance matching 3 . The cosmic baryon fraction bar ≡ Ω bar /Ω m ≈ 0.15 (Planck Collaboration et al. 2016) places a strict upper limit on the galaxy stellar (and total baryonic) mass for a given halo mass. However, many galaxies appear to exceed it. This is another way of stating that these galaxies appear to be "DM-deficient", as this feature is commonly described. In fact, except for NGC1052-DF2 and DF4 and a few early-types (brown points), most of the objects in these samples are H -rich (with median gas fractions in the range ∼ 2 − 5 for * < 10 9 M ), and therefore their total baryon content is even closer to the cosmic mean (see Mancera Piña et al. 2020;Guo et al. 2020, for extreme examples).
In our model, galaxies are initialised on a uniform grid in the * − halo plane at = 0. In the following analysis we focus on the region of the plane extending down to 1 dex below the mean SMHM, and up to the maximum stellar mass allowed by the cosmic baryon fraction.
Relating stellar mass and DM halo formation time
A key step in the model is the link between galaxy stellar mass and host DM halo formation time. We assume a simple deterministic linear relation between the scatter in * at fixed halo , log( * / * ), and the scatter in halo concentration at that mass, log − log . field UDGs (MP20) Figure 1. Stellar mass of galaxies as a function of their DM halo mass at = 0. The solid line corresponds to the mean relation obtained from HAM and other observational constraints by Behroozi et al. (2013). The points show results of dynamical modelling of predominantly isolated nearby galaxies from Forbes et al. (2018), Li et al. (2020), and the triangles show the baryon-dominated galaxies from Guo et al. (2020). The Forbes et al. (2018) data is coloured according to morphology (blue: late types; brown: early types). The pentagons are simple mass models based on H rotation curves from Mancera Piña et al. (2020), and the stars are dynamical inferences for DF2 (van Dokkum et al. 2018a) andDF4 (van Dokkum et al. 2019a). For all the UDGs, the data probe out to ∼ 8 − 10 kpc, where DM is expected to dominate. The dotted line shows the upper limit set by the cosmological baryon fraction, and the dash-dotted line marks the region where * ≥ halo . Error bars on the Li et al. (2020) data are omitted for clarity. Taken at face value, the dynamical measurements indicate a large scatter around the mean relation. One of the key assumptions of our model is that the scatter at fixed halo mass is only limited by the cosmological baryon fraction. Note that most of the low-mass galaxies shown here (except DF2 and DF4 and the early types from Forbes et al. (2018)) contain significant amounts of H , which would shift their total baryon mass closer to or even above the cosmic baryon budget.
The structure of dark matter haloes in cosmological simulations is well described by the two-parameter NFW density profile (Navarro et al. 1997),
( ) = 4 s / s (1 + / s ) 2 ,(1)where s ≡ 200 /(2)
is the scale radius that marks the transition in the logarithmic slope from −1 to −3, s is the density at the scale radius, and is the halo concentration. The halo virial radius is
200 ≡ 3 halo 4 (200 crit ) ,(3)
where crit is the critical density of the Universe. The density at the scale radius is related to the concentration parameter through the equation
s = 200 crit 12 3 [ln(1 + ) − /(1 + )] .(4)
Cosmological DM-only simulations find an anti-correlation between halo mass and concentration ,
log ( halo ) = 0.905 − 0.101 log halo 10 12 ℎ −1 M ,(5)
such that low-mass haloes have higher concentrations (and therefore higher central densities) than higher mass haloes. The scatter in concentration at fixed halo is found to have a constant logarithmic width log = 0.11 . In the ΛCDM cosmology, the concentration is set by the formation time of the DM halo (Wechsler et al. 2002),
= 1 coll ,(6)
where coll = (1 + coll ) −1 is the scale factor at the time when the logarithmic mass accretion rate d log /d log drops below 2, and 1 = 4.1. This implies that low-mass haloes formed earlier than higher mass haloes, and that at a fixed mass, haloes with higher central densities (i.e. higher concentration) formed earlier than lower density haloes. The maximum circular velocity,
max ≡ √︂ G (< ) max ,(7)
is a proxy for the central gravitational potential, and increases with both halo mass and concentration, such that haloes with higher concentration have higher max at fixed halo . In HAM models, observed galaxy clustering is best reproduced when assuming a (positive) correlation between galaxy stellar mass and halo maximum circular velocity max (Campbell et al. 2018). Based on physical arguments, max , which directly traces the central gravitational potential, is expected to be the halo property that sets the gas accretion rate and therefore the baryonic budget of galaxies.
This implies a correlation between halo assembly time and * at fixed halo , which is also observed in cosmological hydrodynamical simulations tuned to reproduce the overall properties of galaxies at = 0. In the EAGLE simulations (Schaye et al. 2015;Crain et al. 2015), galaxies that scatter upwards from the mean SMHM relation are hosted by earlier-forming DM haloes because these were able to retain a larger fraction of their baryons (due to their deeper potential) and had a longer time to grow their stellar mass (Matthee et al. 2017;Kulier et al. 2019). The same correlation between halo concentration and = 0 stellar mass is also found in the Illustris (Artale et al. 2018) and IllustrisTNG simulations (Bose et al. 2019), and in the EMERGE semi-empirical model (Moster et al. 2020). Following this argument, we assume a simple linear scaling between concentration and stellar mass at fixed halo mass,
log = log + log log( * / * ),(8)
where the free parameter = 0.5 controls the slope of the relation. The relation implies that galaxies that inhabit haloes with concentrations 1 away from the mean (at a given halo ) should have stellar masses that deviate from the mean SMHM by 2 orders of magnitude. This large scatter in * assumes that galaxies with such a large deviation in stellar masses are relatively common in haloes with halo < ∼ 10 10 M . This precise form of this relation is difficult to deduce from observations because of the effect of SN feedback on the halo mass profiles. The value of is observationally unknown in low mass galaxies, and the value we use here is arbitrary and corresponds to a shallow correlation between concentration and stellar mass (at fixed halo ). This choice produces good agreement with UDG mass profiles (see Section 4), and its value does not impact the predictions for galaxy structure significantly (as shown in Appendix B). As shown in Figure 1, this is consistent with what is found in inferences from dynamical modelling of dwarf galaxies with * < ∼ 10 8 M (Ferrero et al. 2012;Schneider et al. 2017;Forbes et al. 2018;Li et al. 2020). The sensitivity of the predictions to the value of is explored in Appendix B.
The left and right panels of Figure 2 show the DM halo concentration and collapse redshift, respectively, on the plane defined by * and halo . Galaxies with halo ∼ 10 10 M that lie on the mean SMHM relation have concentrations ∼ 13, while galaxies that scatter maximally above the relation, with * = 1.5×10 9 M , have concentrations as large as ∼ 17. At this halo mass, this means that the collapse redshift of a galaxy on the SMHM relation is coll ∼ 2, while it can reach coll 3 above it.
To summarise, at a fixed halo mass, galaxies above the mean SMHM relation have higher concentrations, and earlier collapse times.
Star-forming environment at the collapse epoch
Next, we need to determine the star-forming environment at the collapse epoch of the halo. First, in order to assign sizes to galaxies based on their halo mass, we use the empirical relation from Kravtsov (2013),
e = 0.015 200 ,(9)
which relates the effective radius e of the galaxy to the virial radius 200 of its DM halo at = 0. This relation is found to hold over 8 orders of magnitude in stellar mass and across all morphological types (Kravtsov 2013). Given that the relation was calibrated using halo abundance matching, we assume that galaxy sizes depend only on halo through equations (3) and (9). Since the Kravtsov (2013) relation only evolves by ∼ 50 per cent between = 3 and = 0 (Mowla et al. 2019), we make the simplifying assumption that it is constant in time. The effective radius can be obtained at any epoch using the predicted halo mass growth histories implemented in the software package (Correa et al. 2015a,b,c), coll e = 0.015 coll 200 .
As discussed above, the aim of this model is to investigate the effect of feedback from stellar clusters on the structure of their host galaxies. Models that follow the formation and evolution of GCs in cosmological simulations find that earlier-forming DM haloes host a larger number of GCs at = 0 (Kruijssen et al. 2019, Table B1). This suggests that the high gas accretion rates during halo collapse drive the formation of a significant fraction of GCs. We therefore assume the peak GC formation epoch is the collapse epoch of the halo (also see e.g. Reina-Campos et al. 2019). We estimate the galaxy SFR at this epoch using the time-averaged SFR, * , at > coll ,
SFR = coll * coll ,(11)
where coll is the halo collapse time (i.e. the time since the Big Bang at coll ), and coll * = * *
is the stellar mass at = coll . The parameter * sets the fraction of the total stellar mass formed before coll , and is set to * = 0.2 in the fiducial model. This value is broadly consistent with the star formation histories of low-mass galaxies in semi-empirical models (e.g. Moster et al. 2013;Behroozi et al. 2013). In Appendix B we examine the sensitivity of the results to this assumption. The SFR surface density then follows from the SFR and the effective radius of the galaxy,
Σ SFR = 0.5 SFR ( coll ) 2 ,(13)
where the factor 0.5 accounts for the assumption that the SFR follows the distribution of the stellar mass, where half is within e . We use the Kennicutt-Schmidt relation measured at ∼ 1−3 by Genzel et al. (2010) to obtain the mean gas surface density within the effective radius from Σ SFR . This relation is slightly shallower than the original Kennicutt (1998) fit, but also provides a good fit to the = 0 data. After calculating the mean gas surface density within e , we then assume (following Di Cintio & Lelli 2016) that the cold gas distribution is described by an exponential profile with a scale length ∼ 2 times larger than for the stellar component, coll e,gas = 2 coll e .
This allows us to calculate the total gas mass, coll gas = 2 ( coll e ) 2 Σ gas ,
Lastly, the DM density profile, together with the stellar and gas mass profiles at the collapse epoch can be used to obtain the total mass enclosed within the disc effective radius, and hence the rotation frequency of the disc, Figure 3 shows the predictions for the SFR, e , Σ SFR , Σ gas , gas , and Ω at the collapse time as a function of * and halo . At a fixed halo mass of halo = 10 10 M , the SFR increases from ∼ 2 × 10 −3 M yr −1 at the mean * , to ∼ 0.2 M yr −1 for the maximum upward scatter. Disc effective radii (at the collapse time) decrease only slightly from ∼ 0.09 kpc on the mean relation to ∼ 0.06 kpc above it. As expected from the trend in the SFR and galaxy sizes, the SFR surface density can increase by a factor of ∼ 200 between the mean and the maximum stellar mass at this halo mass. The gas surface density also increases by a factor of 100. The gas fraction decreases by about 10 per cent, and the angular frequency at e , which is a proxy for the shear, increases substantially by up to a factor of ∼ 15. Upwards scatter from the mean SMHM relation therefore results in galaxies with larger stellar and gas surface densities, and hence larger gas pressures. As we will show in the next section, these conditions lead to increased efficiency of star cluster formation.
Forming stellar cluster populations
Stellar cluster populations can be described at birth in terms of the fraction of star formation occurring in bound clusters, and their initial mass distribution (i.e. the ICMF).
Under the hypothesis of a differentially-rotating disc in hydrostatic equilibrium, these quantities can be determined from the gas surface density, the angular rotation frequency, and the Toomre parameter (Kruijssen 2012;Trujillo-Gomez et al. 2019). We assume the value = 0.5, which is typically observed at = e in ∼ 2 star-forming galaxies (Genzel et al. 2014). We calculate the bound fraction bound with the model presented in Kruijssen (2012), and assume that the ICMF is well described by a double Schechter function (Schechter 1976) with low and high-mass truncation masses min and max , respectively (Trujillo-Gomez et al. 2019). In the ICMF model, the low-mass truncation is set by the mass scale within the hierarchical cloud structure that can remain bound after gas expulsion by stellar feedback. The high-mass truncation is set by the fraction of the largest shear-limited gas cloud that is able to collapse and form stars before feedback disrupts it.
These models generally predict a steep increase of the bound fraction and both the minimum and maximum cluster masses with increasing gas surface density. An additional, shallower dependence on Ω reduces max as the degree of shear support increases (for details, see Reina-Campos & Kruijssen 2017;Trujillo-Gomez et al. 2019). These environmental models reproduce the observed populations of young clusters in the nearby Universe, finding good agree-ment with the observed cluster formation efficiency (e.g. Ginsburg & Kruijssen 2018;Adamo et al. 2020a;Adamo et al. 2020b), and cluster mass functions (Messa et al. 2018;Trujillo-Gomez et al. 2019) across a wide range of galactic environments (also see Pfeffer et al. 2019). The bound fraction and maximum cluster mass models have been implemented in cosmological galaxy formation simulations, and are found to produce GC populations with mass and metallicity distributions, and specific frequencies in good agreement with observations (Pfeffer et al. 2018;Kruijssen et al. 2019).
Together, the stellar mass of the galaxy, the bound fraction, and the ICMF completely determine the number and masses of the star clusters formed at ∼ coll . The ICMF is given by
d d = Φ norm −2 exp − min exp − max ,(18)
where the normalisation factor Φ norm is obtained by requiring that the total mass under the ICMF equal the fraction of the galaxy stellar mass in bound clusters, coll cl = bound coll * .
To compute the number of star clusters, we neglect the effects of random sampling and simply integrate the ICMF,
coll cl = ∫ ∞ 0 d d d .(20)
We define GCs as star clusters of initial mass > 10 5 M , and obtain their total number by integration of the ICMF:
coll GC = ∫ ∞ GC,min d d d ,(21)
where GC,min = 10 5 M . This mass is near the observed median GC mass across most galaxies (Jordan et al. 2007), and also corresponds to the minimum mass of clusters that are expected to survive disruption for a Hubble time in low-mass galaxies (Kruijssen 2015).
We explore the effect of varying this threshold mass in Section 3. Lastly, the specific frequency is a measure of the number of GCs per unit galaxy stellar mass, and it is defined as coll = coll GC * /10 9 M .
(22) Figure 4 shows the effect that the departure from the mean SMHM relation has on the cluster populations formed at the collapse epoch of the halo. Driven by the steep increase in gas pressure (traced by Σ gas ), bound can increase from ∼ 0.3 to 1.0 for galaxies with halo = 10 10 M as they scatter upwards from . Dependence of galaxy properties at the collapse redshift on the present position in the * − halo plane. First row: time-averaged star formation rate and galaxy effective radius at = coll . Second row: SFR surface density (left) and gas surface density (right). Last row: gas fraction (left) and disc rotation frequency (right). The solid line is the mean SMHM relation from Behroozi et al. (2013), and the dotted line indicates the cosmic baryon fraction. At a fixed halo mass, galaxies that scatter above the mean SMHM relation had increasingly higher SFRs, and smaller sizes. This resulted in higher mean SFR surface densities, mean gas surface densities, global gas fractions, and rotation frequencies at their formation epoch, = coll . the relation 4 . The minimum cluster mass follows a similar trend. For halo = 10 10 M , galaxies on the relation form clusters with min ∼ 10 2 M , which increases to 10 3 M for an upward scatter of ∼ 1 dex, and then steeply increases to 10 6 M near the cosmic baryon fraction. The maximum truncation mass also increases as galaxies scatter upwards from the relation 5 , with a secondary trend of increase towards more massive DM haloes due to their larger discs producing weaker shear (see Figure 3). Together, these trends result in the steep growth of the population of massive clusters with increasing * at fixed halo . The mean cluster mass 4 These values are relatively high compared to nearby galaxies, and originate from the high values of Σ gas at coll . The corresponding gas surface densities are comparable to observed massive star-forming galaxies at > 1 (Tacconi et al. 2013) 5 The model predicts no massive clusters in dwarfs similar to Fornax, which lies near the mean SMHM relation with halo ≈ 10 10 M , and contains 5 GCs with ∼ 10 5 M . However, at this mass, galaxies hosting such large GC populations seem to be rare (Shao et al. 2020). A single gas-rich major merger (as opposed to many minor mergers) could in principle account for the majority of Fornax's GCs ). Appendix C shows that assuming a larger value of Toomre = 2.0 can account for the GC population of Fornax without significantly altering the model predictions for UDGs. also increases steeply with upwards departure from the SMHM relation, reaching > ∼ 10 6 M in galaxies with stellar masses near the cosmic baryon fraction. The increase in bound and coll cl for galaxies above the SMHM relation implies that they produce a larger fraction of stars in clusters, and that these clusters are significantly more massive than for galaxies on the relation. Section 3 shows how this impacts GC populations, and in the next subsection, we explore how these GC populations impact the evolution of galaxy structure.
Feedback-driven expansion
Having defined the gas, stellar, and star cluster content of the galaxy at the collapse epoch of the halo, we can now estimate the effect of supernova feedback from both clusters and field stars on the structural evolution of the galaxy from = coll until = 0. For this, we assume that massive gas outflows can generate large enough fluctuations in the gravitational potential to irreversibly heat the orbits of DM particles and reduce their central density, forming a shallow core in the halo density profile. This assumption is supported by a vast body of literature (e.g., Navarro et al. 1996;Read & Gilmore 2005;Mashchenko et al. 2008;Pontzen & Governato 2012;Teyssier et al. 2013;Di Cintio et al. 2014;Oñorbe et al. 2015;Read et al. 2016a;Freundlich et al. 2020;Burger & Zavala 2021). Two elements are necessary to predict the effect of feedback from clustered star First row: gravitationally bound fraction of star formation (left), and minimum cluster mass (right). Second row: maximum cluster mass (left), and mean cluster mass (right). The solid line is the mean SMHM relation from Behroozi et al. (2013), and the dotted line indicates the cosmic baryon fraction. At a fixed halo mass, galaxies that scatter above the mean SMHM relation show a steep increase in the fraction of star formed in bound clusters, accompanied by a rapid increase in max and a slower increase min . Together, the increase in both min and max drives a steep increase in the mean cluster mass. Galaxies near the cosmic baryon fraction will form a higher fraction of stars in clusters, and more massive cluster populations at = coll compared to galaxies on the SMHM relation. formation in our model: the impact of SN clustering on the energy loading of galactic winds, and the coupling of the wind energy to the gravitational potential energy of the DM halo. The first element is the fraction of total SN energy that drives galactic winds, and its dependence on the degree of spatial and temporal clustering of star formation. This has been studied extensively in analytical models and controlled numerical experiments where SNe are detonated uniformly or in a clustered (in space and time) fashion within a patch of the ISM (e.g., Sharma et al. 2014;Keller et al. 2014;Kim & Ostriker 2015;Walch & Naab 2015;Girichidis et al. 2016;Gentry et al. 2017;Fielding et al. 2018;Gentry et al. 2019) or in entire galaxies Keller et al. 2021). Fielding et al. (2017) investigated this effect on simulated galactic discs as a function of the number of overlapping SNe, the gas surface density, and the gas scale-height. They found that the resulting wind energies are consistent with those expected from superbubbles that break out of the gas disc when their cooling radius is large enough to exceed the scale-height. The wind energy loading (i.e. the fraction of SN energy that goes into driving a wind) in the simulations can be parametrized using the expression 6
w (ℎ, Σ gas , cl,SN ) = 10 −2 ℎ 0.03 kpc −1.5 Σ gas 10 M pc −2 −1 1.05 cl,SN ,(23)
where ℎ is the scale-height of the gas, and cl,SN is a measure 6 To obtain this expression we adjusted equation (2) of Fielding et al. (2017) to account for the power-law index of the dependence on cl,SN shown in the bottom panel of their figure 4. Since their equation does not include a normalisation, we also obtain it using the fits in their figure 4.
of the degree of SN clustering. It corresponds to the number of supernovae progenitors formed per 100 M of stars, relative to the case with spatially uniform star formation. We use this expression to calculate the wind energy loading. The scale-height is obtained using eqs. (33) and (34) from Krumholz & McKee (2005),
ℎ = Σ gas 2 gas ,(24)
which assume a disc in hydrostatic equilibrium and a flat rotation curve. The gas mid-plane density is
gas = Σ 2 gas 2 2 ,(25)
where is the gas velocity dispersion. The dispersion can be estimated using eq. (35) in Krumholz & McKee (2005),
dyn = Σ gas √ 2Ω ,(26)
where the factor = 3 accounts for the gravity due to the stars, and = 0.5 as before. This dynamical estimate of the gas velocity dispersion is corrected for thermal support using the weighted mean of the atomic and molecular components,
= √︃ (1 − H2 ) 2 HI + H2 2 H2 ,(27)
where the velocity dispersion of gas phase is
2 = 2 dyn + 2 s, .(28)
The sound speeds of the two components are set to s,HI = 5.0 km s −1 (i.e. HI ≈ 3000 K), and s,H2 = 0.3 km s −1 (Krumholz & McKee 2005). Lastly, the molecular gas fraction is given by eq. (73) in Krumholz & McKee (2005),
H2 = 1 + 0.025 Σ gas 10 2 M pc −2 −2 −1 .(29)
For the high-redshift conditions of interest here, H2 ≈ 1 such that the contribution of thermal pressure to the velocity dispersion is very small. At the collapse epoch, a galaxy with halo = 10 10 M and the maximum allowed stellar mass (i.e. the cosmic baryon fraction) is predicted to have a scale-height a factor of ∼ 2 smaller than a galaxy on the SMHM relation.
We define cl,SN as the average number of SNe per cluster in the bound component of star formation,
cl,SN = coll cl SN ,(30)
where coll cl is the mean cluster mass, and SN = 99 M is the total mass of stars in a simple stellar population that contains at least one SN progenitor (assuming a progenitor mass of 8 M for a Chabrier 2003 IMF). At a fixed SFR, a higher cl,SN implies an increase in both the spatial and in the temporal clustering of SNe because a larger fraction of SNe detonate simultaneously in massive star clusters. The total wind energy loading of the galaxy is then the weighted sum of the bound (i.e. star cluster) and unbound (i.e. field star) components,
gal = (1.0 − bound ) w (ℎ, Σ gas , 1.0) + bound w (ℎ, Σ gas , cl,SN ),(31)
where the value cl,SN = 1.0 for the unbound component reflects the normalization of the SN rate surface density employed by Fielding et al. (2017) in the case of uniformly distributed SNe. Figure 5 shows the impact of the degree of spatial and temporal clustering of star formation on the generation of galactic winds by SNe. The average number of overlapping SNe across the galaxy cl,SN systematically increases with the upward scatter in stellar mass at fixed halo mass (left panel), with a maximum near cl,SN ∼ 10 3 − 10 5.5 in galaxies near the cosmic baryon fraction. This is an increase of up to ∼ 3 − 5 orders of magnitude relative to lowmass galaxies near the SMHM relation with cl,SN = 1 − 10. The right panel of Figure 5 shows the impact of stellar clustering on the energy loading of galactic winds at the collapse epoch. SNe become increasingly more efficient at powering galactic winds as their host galaxies scatter upwards from the SMHM relation. The fraction of SN energy driving the winds rises from gal < ∼ 0.1 to a saturation at gal = 1 as the scatter above the relation increases by 1 dex. This effect is driven by both the increase in SN clustering ( cl,SN ), and the change in the underlying structure of the ISM (i.e. the decrease of the gas disc scale-height, see equation 23). Note that the increase in cl,SN and gal are both independent of the increase in the SFR of galaxies above the SMHM relation, and depend only on the structure of the cold ISM. In Section 3.3 we show that the increased clustering of SNe in galaxies above the SMHM relation is the dominant factor driving the rise in the efficiency of galactic wind generation.
Once the wind energy coupling is obtained, the second ingredient is the fraction of wind energy that produces potential fluctuations large enough to expand the DM distribution and reduce the central density of the halo. Unfortunately, there are no estimates of this value in the literature. However, values for the fraction of SN energy that couples to the DM cover a broad range studied the time evolution of this coupling in cosmological simulations of dwarf galaxies, finding values ∼ 0.01−0.1. In simulations with very different subgrid physics, Chan et al. (2015) obtain total coupling fractions ∼ 0.005 − 0.1. For simplicity, we assume that a constant fraction of the wind energy is transferred into the halo gravitational potential,
DM = DM w ,(32)
and set the coupling to the upper bound obtained by Madau et al. (2014), DM = 0.1. In Appendix B we examine the dependence of the model predictions on this assumption. The complexity of the core formation process has prompted many numerical studies. There is general agreement that impulsive energy injection due to stellar feedback results in an overall flattening of the initially steep NFW density profile in the central few kiloparsecs (see Pontzen & Governato 2014, for a review). Read et al. (2016a) provide a convenient parametrization of the modified profile given by the coreNFW model,
cNFW (< ) = NFW (< ) ,(33)where = tanh core ,(34)
and and core control the slope and size of the core, respectively. Their study shows that the DM haloes of isolated galaxy simulations are well fit when these parameters are given by
= tanh 0.04 SF dyn ,(35)
and core = 1.75 e ,
where SF is the total duration of star formation in the galaxy, and dyn is the circular orbit period at the scale radius of the NFW profile. This allows the DM core size core to be inferred from the observed effective radius of the stars e . Because at the collapse redshift most galaxies have already been forming stars for several dynamical times, we assume = 1 throughout. However, in our model we cannot obtain core by directly modelling the effect of stellar feedback on the galaxy effective radius. Instead, we use the DM coupling efficiency DM to obtain the size of the DM core. This is done by solving for the value of core that increases the gravitational potential energy by an amount equal to the total energy available from winds. This requires numerically solving the equation
Δ ( core ) = DM SN SN ,(37)
where SN is the total number of SNe at > coll , SN = 10 51 erg is the energy output of a single supernova, and
Δ ( core ) = − 1 2 ∫ ∞ 0 [ 2 fin ( core ) − 2 init ] 2 d(38)
is the difference in potential energy between the final coreNFW profile (and its embedded stellar disc), and the initial pristine NFW halo (and its expanded stellar disc) for a given value of core . The initial and final total mass profiles are 7 Behroozi et al. (2013), and the dotted line indicates the cosmic baryon fraction. The steep increase in the bound fraction of star formation and mean cluster mass as galaxies scatter upwards from the SMHM relation results in an increase of cl,SN of ∼ 3 − 5 orders of magnitude. As stellar mass increases at a fixed halo mass, SN energy is more efficiently injected into the ISM to power galactic winds. This results in larger amounts of energy becoming available to dynamically heat the DM orbits and grow the DM core. Galaxies above the SMHM relation not only have larger SFR and therefore larger feedback budget, but are also more efficient at injecting feedback energy into galactic winds. and fin ( core ,
init ( ) = NFW ( ) + * ( coll e , ),(39)) = cNFW ( core , ) + * ( fin e , ),(40)
where * ( e , ) is the stellar mass profile assuming an exponential stellar disc with effective radius e . Lastly, we estimate the effect of expansion on the stellar disc. We denote as 'old' the stellar disc already present at the collapse epoch that expands alongside the DM halo. The final = 0 effective radii of the 'old' stellar disc is (Eq. 36), old e = fin e = core 1.75
.
After the main episode of GC formation at = coll , we assume that the galaxy continues to form stars and grows in stellar mass, and that its feedback-driven winds have a gradually diminishing effect on the galaxy structure. The effective radius and stellar mass of the 'young' disc that forms at < coll are then given by (Eqs. 10 and 12) young e = 0.015 200 ,
and young * = (1 − * ) * ,(42)
where the virial radius 200 and stellar mass * correspond to the values at = 0. The stellar mass distribution at = 0 is then determined by a smooth transition in the scale-length of the exponential disc between the old and the young populations,
e = * old e + (1 − * ) young e .(44)
This relation implicitly assumes a smooth evolution in time starting from the strong expansion of the old stellar component to the progressively weaker effect of expansion on increasingly younger populations. It also explicitly enforces the observed = 0 mass-size relation from Kravtsov (2013) for galaxies on the mean SMHM relation (see Section 3.2). The quantities core and e are then completely specified by the = 0 position of a galaxy in the * − halo plane, and define the mass distribution of the DM, and stellar components at present. For simplicity, we avoid the complication of predicting the gas mass and structure at = 0. Figure 6 provides a summary of the general trends predicted by the model.
" SFR and " P ISM < l a t e x i t s h a 1 _ b a s e 6 4 = " j N l e h / U l w 3 6 M g q j I 9 1 0 F d d 5
e K F M = " > A A A C F H i c b V D L S g M x F M 3 U V 6 2 v q k s 3 w Y 4 g C G W m G 1 0 W B d G F U K 1 9 Q G c o m T T T h i a Z I c k o Z e h H u P F X 3 L h Q x K 0 L d / 6 N 6 W N R W w 9 c O J x z L / f e E 8 S M K u 0 4 P 1 Z m a X l l d S 2 7 n t v Y 3 N r e y e / u 1 V W U S E x q O G K R b A Z I E U Y F q W m q G W n G k i A e M N I I + h c j v / F A p K K R u N e D m P g c d Q U N K U bQ g T h R f j p + a g i P j N K B Y S R N C Q 3 H 6 u x E i r h S A x 6 Y T o 5 0 T 8 1 7 I / E / r 5 X o 8 M x P q Y g T T Q S e L A o T B n U E R w n B D p U E a z Y w B G F J z a 0 Q 9 5 B E W J s c c y Y E d / 7 l R V I v F V 2 n 6 N 6 W C u X z a R x Z c A A O w T F w w S k o g y t Q A T W A w R N 4
A W / g 3 X q 2 X q 0 P 6 3 P S m r G m M / v g D 6 y v X 3 4 U n J A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " j N l e h / U l w 3 6 M g q j I 9 1 0 F d d 5
e K F M = " > A A A C F H i c b V D L S g M x F M 3 U V 6 2 v q k s 3 w Y 4 g C G W m G 1 0 W B d G F U K 1 9 Q G c o m T T T h i a Z I c k o Z e h H u P F X 3 L h Q x K 0 L d / 6 N 6 W N R W w 9 c O J x z L / f e E 8 S M K u 0 4 P 1 Z m a X l l d S 2 7 n t v Y 3 N r e y e / u 1 V W U S E x q O G K R b A Z I E U Y F q W m q G W n G k i A e M N I I + h c j v / F A p K K R u N e D m P g c d Q U N K U bQ g T h R f j p + a g i P j N K B Y S R N C Q 3 H 6 u x E i r h S A x 6 Y T o 5 0 T 8 1 7 I / E / r 5 X o 8 M x P q Y g T T Q S e L A o T B n U E R w n B D p U E a z Y w B G F J z a 0 Q 9 5 B E W J s c c y Y E d / 7 l R V I v F V 2 n 6 N 6 W C u X z a R x Z c A A O w T F w w S k o g y t Q A T W A w R N 4
A W / g 3 X q 2 X q 0 P 6 3 P S m r G m M / v g D 6 y v X 3 4 U n J A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " j N l e h / U l w 3 6 M g q j I 9 1 0 F d d 5
e K F M = " > A A A C F H i c b V D L S g M x F M 3 U V 6 2 v q k s 3 w Y 4 g C G W m G 1 0 W B d G F U K 1 9 Q G c o m T T T h i a Z I c k o Z e h H u P F X 3 L h Q x K 0 L d / 6 N 6 W N R W w 9 c O J x z L / f e E 8 S M K u 0 4 P 1 Z m a X l l d S 2 7 n t v Y 3 N r e y e / u 1 V W U S E x q O G K R b A Z I E U Y F q W m q G W n G k i A e M N I I + h c j v / F A p K K R u N e D m P g c d Q U N K U bQ g T h R f j p + a g i P j N K B Y S R N C Q 3 H 6 u x E i r h S A x 6 Y T o 5 0 T 8 1 7 I / E / r 5 X o 8 M x P q Y g T T Q S e L A o T B n U E R w n B D p U E a z Y w B G F J z a 0 Q 9 5 B E W J s c c y Y E d / 7 l R V I v F V 2 n 6 N 6 W C u X z a R x Z c A A O w T F w w S k o g y t Q A T W A w R N 4
A W / g 3 X q 2 X q 0 P 6 3 P S m r G m M / v g D 6 y v X 3 4 U n J A = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " j N l e h / U l w 3 6 M g q j I 9 1 0 F d d 5
e K F M = " > A A A C F H i c b V D L S g M x F M 3 U V 6 2 v q k s 3 w Y 4 g C G W m G 1 0 W B d G F U K 1 9 Q G c o m T T T h i a Z I c k o Z e h H u P F X 3 L h Q x K 0 L d / 6 N 6 W N R W w 9 c O J x z L / f e E 8 S M K u 0 4 P 1 Z m a X l l d S 2 7 n t v Y 3 N r e y e / u 1 V W U S E x q O G K R b A Z I E U Y F q W m q G W n G k i A e M N I I + h c j v / F A p K K R u N e D m P g c d Q U N K U bQ g T h R f j p + a g i P j N K B Y S R N C Q 3 H 6 u x E i r h S A x 6 Y T o 5 0 T 8 1 7 I / E / r 5 X o 8 M x P q Y g T T Q S e L A o T B n U E R w n B D p U E a z Y w B G F J z a 0 Q 9 5 B E W J s c c y Y E d / 7 l R V I v F V 2 n 6 N 6 W C u X z a R x Z c A A O w T F w w S k o g y t Q A T W A w R N 4
A W / g 3 X q 2 X q 0 P 6 3 P S m r G m M / v g D 6 y v X 3 4 U n J A = < / l a t e x i t > " c and " z coll of DM halo < l a t e x i t s h a 1 _ b a s e 6 4 = " p c / v 9 o f 8 3 s 9 2 U 0 w 8 8 m A 4 7 1 P o 8 y " M ⇤ at fixed M halo < l a t e x i t s h a 1 _ b a s e 6 4 = " t D g K P S L R e W k 5 1 r a I B / z I y c + z z
A = " > A A A C I H i c b V D L S g N B E J z 1 G e M r 6 t H L Y F b w F H Z z i c e g H r w I E c w D k h B 6 J 7 P J k N m Z Z W Z W i U s + x Y u / 4 s W D I n r T r 3 H y O M T E g o a i q p v u r i D m T B v P + 3 Z W V t f W N z Y z W 9 n t n d 2 9 / d z B Y U 3 L R B F a J Z J L 1 Q h A U 8 4 E r R p m O G 3 E i k I U c F o P B p d j v 3 5 P l W Z S 3 J l h T N s R 9 A Q L G Q F j p U 6 u 5 L a S G J S S D y 5 2 i Y t B d P G 8 9 N h J W y r C d h E f u V i G + O o G 9 4 H L T i 7 v F b w J 8 D L x Z y S P Z q h 0 c l + t r i R J R I U h H L R u + l 5 s 2 i k o w w i n o 2 w r 0 T Q G M o A e b V o q I K K 6 n U 4 e H O F T q 3 R x K J U t Y f B E n Z 9 I I d J 6 G A W 2 M w L T 1 4 v e W P z P a y Y m P G + n T M S J o Y J M F 4 U J x 0 b i c V q 4 y x Q l h g 8 t A a K Y v R W T P i g g x m a a t S H 4 i y 8 v k 1 q x 4 H s F / 7 a Y L 1 / M 4 s i g Y 3 S C z p C P S q i M r l E F V R F B T + g F v a F 3 5 9 l 5 d T 6 c z 2 n r i j O b O U J / 4 P z 8 A i v / o R 0 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " p c / v 9 o f 8 3 s 9 2 U 0 w 8 8 m A 4 7 1 P o 8 y A = " > A A A C I H i c b V D L S g N B E J z 1 G e M r 6 t H L Y F b w F H Z z i c e g H r w I E c w D k h B 6 J 7 P J k N m Z Z W Z W i U s + x Y u / 4 s W D I n r T r 3 H y O M T E g o a i q p v u r i D m T B v P + 3 Z W V t f W N z Y z W 9 n t n d 2 9 / d z B Y U 3 L R B F a J Z J L 1 Q h A U 8 4 E r R p m O G 3 E i k I U c F o P B p d j v 3 5 P l W Z S 3 J l h T N s R 9 A Q L G Q F j p U 6 u 5 L a S G J S S D y 5 2 i Y t B d P G 8 9 N h J W y r C d h E f u V i G + O o G 9 4 H L T i 7 v F b w J 8 D L x Z y S P Z q h 0 c l + t r i R J R I U h H L R u + l 5 s 2 i k o w w i n o 2 w r 0 T Q G M o A e b V o q I K K 6 n U 4 e H O F T q 3 R x K J U t Y f B E n Z 9 I I d J 6 G A W 2 M w L T 1 4 v e W P z P a y Y m P G + n T M S J o Y J M F 4 U J x 0 b i c V q 4 y x Q l h g 8 t A a K Y v R W T P i g g xi D m T B v P + 3 Z W V t f W N z Y z W 9 n t n d 2 9 / d z B Y U 3 L R B F a J Z J L 1 Q h A U 8 4 E r R p m O G 3 E i k I U c F o P B p d j v 3 5 P l W Z S 3 J l h T N s R 9 A Q L G Q F j p U 6 u 5 L a S G J S S D y 5 2 i Y t B d P G 8 9 N h J W y r C d h E f u V i G + O o G 9 4 H L T i 7 v F b w J 8 D L x Z y S P Z q h 0 c l + t r i R J R I U h H L R u + l 5 s 2 i k o w w i n o 2 w r 0 T Q G M o A e b V o q I K K 6 n U 4 e H O F T q 3 R x K J U t Y f B E n Z 9 I I d J 6 G A W 2 M w L T 1 4 v e W P z P a y Y m P G + n T M S J o Y J M F 4 U J x 0 b i c V q 4 y x Q l h g 8 t A a K Y v R W T P i g g xi D m T B v P + 3 Z W V t f W N z Y z W 9 n t n d 2 9 / d z B Y U 3 L R B F a J Z J L 1 Q h A U 8 4 E r R p m O G 3 E i k I U c F o P B p d j v 3 5 P l W Z S 3 J l h T N s R 9 A Q L G Q F j p U 6 u 5 L a S G J S S D y 5 2 i Y t B d P G 8 9 N h J W y r C d h E f u V i G + O o G 9 4 H L T i 7 v F b w J 8 D L x Z y S P Z q h 0 c l + t r i R J R I U h H L R u + l 5 s 2 i k o w w i n o 2 w r 0 T Q G M o A e b V o q I K K 6 n U 4 e H O F T q 3 R x K J U t Y f B E n Z 9 I I d J 6 G A W 2 M w L T 1 4 v e W P z P a y Y m P G + n T M S J o Y J M F 4 U J x 0 b i c V q 4 y x Q l h g 8 t A a K Y v R W T P i g g xv 0 = " > A A A C E X i c b V C 7 T s M w F H X K q 5 R X g Z H F o k G q G K q k C 4 w V L C x I R a I P q Y 2 q G 9 d p r T p x Z D t A F f U X W P g V F g Y Q Y m V j 4 2 9 w 2 g 7 Q c q Q r H Z 9 z r 3 z v 8 W P O l H a c b y u 3 s r q 2 v p H f L G x t 7 + z u F f c P m k o k k t A G E V z I t g + K c h b R h m a a 0 3 Y s K Y Q + p y 1 / d J n 5 r T s q F R P R r R 7 H 1 A t h E L G A E d B G 6 h X L d j e J Q U p x b 2 P 7 u n d q Y 9 A 4 Y A + 0 n z 3 T r g z x E L i Y 2 L 1 i y a k 4 U + B l 4 s 5 J C c 1 R 7 x W / u n 1 B k p B G m n B Q q u M 6 s f Z S k J o R T i e F b q J o D G Q E A 9 o x N I K Q K i + d X j T B J 0 b p 4 0 B I U 5 H G U / X 3 R A q h U u P Q N 5 0 h 6 K F a 9 D L x P 6 + T 6 O D c S 1 k U J 5 p G Z P Z R k H C s B c 7 i w X 0 m K d F 8 b A g Q y c y u m A x B A t E m x I I J w V 0 8 e Z k 0 q x X X q b g 3 1 V L t Y h 5 H H h 2 h Y 1 R G L j p D N X S F 6 q i B C H p E z + g V v V l P 1 o v 1 b n 3 M W n P W f O Y Q / Y H 1 + Q M z Z Z t b < / l a t e x i t >
< l a t e x i t s h a 1 _ b a s e 6 4 = " t D g K P S L R e W k 5 1 r a I B / z I y c + z z
v 0 = " > A A A C E X i c b V C 7 T s M w F H X K q 5 R X g Z H F o k G q G K q k C 4 w V L C x I R a I P q Y 2 q G 9 d p r T p x Z D t A F f U X W P g V F g Y Q Y m V j 4 2 9 w 2 g 7 Q c q Q r H Z 9 z r 3 z v 8 W P O l H a c b y u 3 s r q 2 v p H f L G x t 7 + z u F f c P m k o k k t A G E V z I t g + K c h b R h m a a 0 3 Y s K Y Q + p y 1 / d J n 5 r T s q F R P R r R 7 H 1 A t h E L G A E d B G 6 h X L d j e J Q U p x b 2 P 7 u n d q Y 9 A 4 Y A + 0 n z 3 T r g z x E L i Y 2 L 1 i y a k 4 U + B l 4 s 5 J C c 1 R 7 x W / u n 1 B k p B G m n B Q q u M 6 s f Z S k J o R T i e F b q J o D G Q E A 9 o x N I K Q K i + d X j T B J 0 b p 4 0 B I U 5 H G U / X 3 R A q h U u P Q N 5 0 h 6 K F a 9 D L x P 6 + T 6 O D c S 1 k U J 5 p G Z P Z R k H C s B c 7 i w X 0 m K d F 8 b A g Q y c y u m A x B A t E m x I I J w V 0 8 e Z k 0 q x X X q b g 3 1 V L t Y h 5 H H h 2 h Y 1 R G L j p D N X S F 6 q i B C H p E z + g V v V l P 1 o v 1 b n 3 M W n P W f O Y Q / Y H 1 + Q M z Z Z t b < / l a t e x i t >
< l a t e x i t s h a 1 _ b a s e 6 4 = " t D g K P S L R e W k 5 1 r a I B / z I y c + z z
v 0 = " > A A A C E X i c b V C 7 T s M w F H X K q 5 R X g Z H F o k G q G K q k C 4 w V L C x I R a I P q Y 2 q G 9 d p r T p x Z D t A F f U X W P g V F g Y Q Y m V j 4 2 9 w 2 g 7 Q c q Q r H Z 9 z r 3 z v 8 W P O l H a c b y u 3 s r q 2 v p H f L G x t 7 + z u F f c P m k o k k t A G E V z I t g + K c h b R h m a a 0 3 Y s K Y Q + p y 1 / d J n 5 r T s q F R P R r R 7 H 1 A t h E L G A E d B G 6 h X L d j e J Q U p x b 2 P 7 u n d q Y 9 A 4 Y A + 0 n z 3 T r g z x E L i Y 2 L 1 i y a k 4 U + B l 4 s 5 J C c 1 R 7 x W / u n 1 B k p B G m n B Q q u M 6 s f Z S k J o R T i e F b q J o D G Q E A 9 o x N I K Q K i + d X j T B J 0 b p 4 0 B I U 5 H G U / X 3 R A q h U u P Q N 5 0 h 6 K F a 9 D L x P 6 + T 6 O D c S 1 k U J 5 p G Z P Z R k H C s B c 7 i w X 0 m K d F 8 b A g Q y c y u m A x B A t E m x I I J w V 0 8 e Z k 0 q x X X q b g 3 1 V L t Y h 5 H H h 2 h Y 1 R G L j p D N X S F 6 q i B C H p E z + g V v V l P 1 o v 1 b n 3 M W n P W f O Y Q / Y H 1 + Q M z Z Z t b < / l a t e x i t >
< l a t e x i t s h a 1 _ b a s e 6 4 = " t D g K P S L R e W k 5 1 r a I B / z I y c + z z
v 0 = " > A A A C E X i c b V C 7 T s M w F H X K q 5 R X g Z H F o k G q G K q k C 4 w V L C x I R a I P q Y 2 q G 9 d p r T p x Z D t A F f U X W P g V F g Y Q Y m V j 4 2 9 w 2 g 7 Q c q Q r H Z 9 z r 3 z v 8 W P O l H a c b y u 3 s r q 2 v p H f L G x t 7 + z u F f c P m k o k k t A G E V z I t g + K c h b R h m a a 0 3 Y s K Y Q + p y 1 / d J n 5 r T s q F R P R r R 7 H 1 A t h E L G A E d B G 6 h X L d j e J Q U p x b 2 P 7 u n d q Y 9 A 4 Y A + 0 n z 3 T r g z x E L i Y 2 L 1 i y a k 4 U + B l 4 s 5 J C c 1 R 7 x W / u n 1 B k p B G m n B Q q u M 6 s f Z S k J o R T i e F b q J o D G Q E A 9 o x N I K Q K i + d X j T B J 0 b p 4 0 B I U 5 H G U / X 3 R A q h U u P Q N 5 0 h 6 K F a 9 D L x P 6 + T 6 O D c S 1 k U J 5 p G Z P Z R k H C s B c 7 i w X 0 m K d F 8 b A g Q y c y u m A x B A t E m x I I J w V 0 8 e Z k 0 q x X X q b g 3 1 V L t Y h 5 H H h 2 h Y 1 R G L j p D N X S F 6 q i B C H p E z + g V v V l P 1 o v 1 b n 3 M W n P W f O Y Q / Y H 1 + Q M z Z Z t b < / l a t e x i t >
" r core , " r e , " N z coll GC , and # M DM (< 8 kpc)
< l a t e x i t s h a 1 _ b a s e 6 4 = " U X / c L u I V w 1 N 7 b L O j j a 7 E W 5 O V H 6 o = " > A A A C b 3 i c b V H J S g M x G M 6 M W 6 1 b r Q c P F Q l 2 B A U p M 7 3 Y g 4 d i B b 1 U F K w t t L V k 0 l R D M 8 m Q Z C x 1 G I 8 + o D f f w Y t v Y L q B W n 8 I f H x L l i 9 + y K j S r v t h 2 Q u L S 8 s r q d X 0 2 v r G 5 l Z m O 3 u v R C Q x q W H B h G z 4 S B F G O a l p q h l p h J K g w G e k 7 v c r I 7 3 + T K S i g t / p Y U j a A X r k t E c x 0 o b q Z N 6 c V h Q i K c X A g Y 7 s x C 0 Z Q C w k S Z w T + I 8 0 x 1 9 P + M t K 8 h C / z O K M J S M f 4 l 3 j 7 Y o B n 7 m r E 8 d F N T k 6 K 7 2 O c T / E y b H T y e T d g j s e O A + 8 K c i D 6 d x 0 M u 9 m Y x w F h G v M k F J N z w 1 1 O 0 Z S U 8 x I k m 5 F i o Q I 9 9 E j a R r I U U B U O x 7 3 l c B D w 3 R h T 0 i z u I Z j 9 m c i R o F S w 8 A 3 z g D p J / V X G 5 H / a c 1 I 9 0 r t m P I w 0 o T j y U G 9 i E E t 4 K h 8 2 K W S Y M 2 G B i A s q b k r x E 9 I I q z N F 6 V N C d 7 f J 8 + D + 2 L B c w v e b T F f P p / W k Q I 5 c A C O g A d O Q R l c g R t Q A x h 8 W l k r Z + 1 Z X / a u v W / D i d W 2 p p k d 8 G v s 4 2 / M 0 L j W < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " U X / c L u I V w 1 N 7 b L O j j a 7 E W 5 O V H 6 o = " > A A A C b 3 i c b V H J S g M x G M 6 M W 6 1 b r Q c P F Q l 2 B A U p M 7 3 Y g 4 d i B b 1 U F K w t t L V k 0 l R D M 8 m Q Z C x 1 G I 8 + o D f f w Y t v Y L q B W n 8 I f H x L l i 9 + y K j S r v t h 2 Q u L S 8 s r q d X 0 2 v r G 5 l Z m O 3 u v R C Q x q W H B h G z 4 S B F G O a l p q h l p h J K g w G e k 7 v c r I 7 3 + T K S i g t / p Y U j a A X r k t E c x 0 o b q Z N 6 c V h Q i K c X A g Y 7 s x C 0 Z Q C w k S Z w T + I 8 0 x 1 9 P + M t K 8 h C / z O K M J S M f 4 l 3 j 7 Y o B n 7 m r E 8 d F N T k 6 K 7 2 O c T / E y b H T y e T d g j s e O A + 8 K c i D 6 d x 0 M u 9 m Y x w F h G v M k F J N z w 1 1 O 0 Z S U 8 x I k m 5 F i o Q I 9 9 E j a R r I U U B U O x 7 3 l c B D w 3 R h T 0 i z u I Z j 9 m c i R o F S w 8 A 3 z g D p J / V X G 5 H / a c 1 I 9 0 r t m P I w 0 o T j y U G 9 i E E t 4 K h 8 2 K W S Y M 2 G B i A s q b k r x E 9 I I q z N F 6 V N C d 7 f J 8 + D + 2 L B c w v e b T F f P p / W k Q I 5 c A C O g A d O Q R l c g R t Q A x h 8 W l k r Z + 1 Z X / a u v W / D i d W 2 p p k d 8 G v s 4 2 / M 0 L j W < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " U X / c L u I V w 1 N 7 b L O j j a 7 E W 5 O V H 6 o = " > A A A C b 3 i c b V H J S g M x G M 6 M W 6 1 b r Q c P F Q l 2 B A U p M 7 3 Y g 4 d i B b 1 U F K w t t L V k 0 l R D M 8 m Q Z C x 1 G I 8 + o D f f w Y t v Y L q B W n 8 I f H x L l i 9 + y K j S r v t h 2 Q u L S 8 s r q d X 0 2 v r G 5 l Z m O 3 u v R C Q x q W H B h G z 4 S B F G O a l p q h l p h J K g w G e k 7 v c r I 7 3 + T K S i g t / p Y U j a A X r k t E c x 0 o b q Z N 6 c V h Q i K c X A g Y 7 s x C 0 Z Q C w k S Z w T + I 8 0 x 1 9 P + M t K 8 h C / z O K M J S M f 4 l 3 j 7 Y o B n 7 m r E 8 d F N T k 6 K 7 2 O c T / E y b H T y e T d g j s e O A + 8 K c i D 6 d x 0 M u 9 m Y x w F h G v M k F J N z w 1 1 O 0 Z S U 8 x I k m 5 F i o Q I 9 9 E j a R r I U U B U O x 7 3 l c B D w 3 R h T 0 i z u I Z j 9 m c i R o F S w 8 A 3 z g D p J / V X G 5 H / a c 1 I 9 0 r t m P I w 0 o T j y U G 9 i E E t 4 K h 8 2 K W S Y M 2 G B i A s q b k r x E 9 I I q z N F 6 V N C d 7 f J 8 + D + 2 L B c w v e b T F f P p / W k Q I 5 c A C O g A d O Q R l c g R t Q A x h 8 W l k r Z + 1 Z X / a u v W / D i d W 2 p p k d 8 G v s 4 2 / M 0 L j W < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " U X / c L u I V w 1 N 7 b L O j j a 7 E W 5 O V H 6 o = " > A A A C b 3 i c b V H J S g M x G M 6 M W 6 1 b r Q c P F Q l 2 B A U p M 7 3 Y g 4 d i B b 1 U F K w t t L V k 0 l R D M 8 m Q Z C x 1 G I 8 + o D f f w Y t v Y L q B W n 8 I f H x L l i 9 + y K j S r v t h 2 Q u L S 8 s r q d X 0 2 v r G 5 l Z m O 3 u v R C Q x q W H B h G z 4 S B F G O a l p q h l p h J K g w G e k 7 v c r I 7 3 + T K S i g t / p Y U j a A X r k t E c x 0 o b q Z N 6 c V h Q i K c X A g Y 7 s x C 0 Z Q C w k S Z w T + I 8 0 x 1 9 P + M t K 8 h C / z O K M J S M f 4 l 3 j 7 Y o B n 7 m r E 8 d F N T k 6 K 7 2 O c T / E y b H T y e T d g j s e O A + 8 K c i D 6 d x 0 M u 9 m Y x w F h G v M k F J N z w 1 1 O 0 Z S U 8 x I k m 5 F i o Q I 9 9 E j a R r I U U B U O x 7 3 l c B D w 3 R h T 0 i z u I Z j 9 m c i R o F S w 8 A 3 z g D p J / V X G 5 H / a c 1 I 9 0 r t m P I w 0 o T j y U G 9 i E E t 4 K h 8 2 K W S Y M 2 G B i A s q b k r x E 9 I I q z N F 6 V N C d 7 f J 8 + D + 2 L B c w v e b T F f P p / W k Q I 5 c A C O g A d O Q R l c g R t Q A x h 8 W l k r Z + 1 Z X / a u v W / D i d W 2 p p k d 8 G v s 4 2 / M 0 L j W < / l a t e x i t >
" f cl,SN , " ⌘ w , and " ⌘ gal < l a t e x i t s h a 1 _ b a s e 6 4 = " D 1 l a 2 z a 1 q z C / R Y 4 v j 9 8 q z K U Q j c Y = " > A A A C P 3 i c d Z A 7 T 8 M w F I U d n q W 8 C o w s F g 0 S Q 1 U l X W C s Y G F C R d C H 1 F T V j e u 0 V h 0 n s h 2 q K u o / Y + E v s L G y M I A Q K x v u Y y g t X M n S 0 f n u 1 f U 9 f s y Z 0 o 7 z Y q 2 s r q 1 v b G a 2 s t s 7 u 3 v 7 u Y P D m o o S S W i V R D y S D R 8 U 5 U z Q q m a a 0 0 Y s K Y Q + p 3 W / f z X m 9 Q c q F Y v E v R 7 G t B V C V 7 C A E d D G a u d q t p f E I G U 0 s L E d t F N P h p j w w t 3 N y C 7 g e e Z R D V M 8 G C M Q n X 9 w F / j I b u f y T t G Z F F 4 W 7 k z k 0 a w q 7 d y z 1 4 l I E l K h C Q e l m q 4 T 6 1 Y K U j P C 6 S j r J Y r G Q P r Q p U 0 j B Y R U t d L J / S N 8 a p w O D i J p n t B 4 4 s 5 P p B A q N Q x 9 0 x m C 7 q l F N j b / Y s 1 E B x e t l I k 4 0 V S Q 6 a I g 4 V h H e B w m 7 j B J i e Z D I 4 B I Z v 6 K S Q 8 k E G 0 i z 5 o Q 3 M W T l 0 W t V H S d o n t b y p c v Z 3 F k 0 D E 6 Q W f I R e e o j K 5 R B V U R Q Y / o F b 2 j D + v J e r M + r a 9 p 6 4 o 1 m z l C v 8 r 6 / g E W 6 a 0 z < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " D 1 l a 2 z a 1 q z C / R Y 4 v j 9 8 q z K U Q j c Y = " > A A A C P 3 i c d Z A 7 T 8 M w F I U d n q W 8 C o w s F g 0 S Q 1 U l X W C s Y G F C R d C H 1 F T V j e u 0 V h 0 n s h 2 q K u o / Y + E v s L G y M I A Q K x v u Y y g t X M n S 0 f n u 1 f U 9 f s y Z 0 o 7 z Y q 2 s r q 1 v b G a 2 s t s 7 u 3 v 7 u Y P D m o o S S W i V R D y S D R 8 U 5 U z Q q m a a 0 0 Y s K Y Q + p 3 W / f z X m 9 Q c q F Y v E v R 7 G t B V C V 7 C A E d D G a u d q t p f E I G U 0 s L E d t F N P h p j w w t 3 N y C 7 g e e Z R D V M 8 G C M Q n X 9 w F / j I b u f y T t G Z F F 4 W 7 k z k 0 a w q 7 d y z 1 4 l I E l K h C Q e l m q 4 T 6 1 Y K U j P C 6 S j r J Y r G Q P r Q p U 0 j B Y R U t d L J / S N 8 a p w O D i J p n t B 4 4 s 5 P p B A q N Q x 9 0 x m C 7 q l F N j b / Y s 1 E B x e t l I k 4 0 V S Q 6 a I g 4 V h H e B w m 7 j B J i e Z D I 4 B I Z v 6 K S Q 8 k E G 0 i z 5 o Q 3 M W T l 0 W t V H S d o n t b y p c v Z 3 F k 0 D E 6 Q W f I R e e o j K 5 R B V U R Q Y / o F b 2 j D + v J e r M + r a 9 p 6 4 o 1 m z l C v 8 r 6 / g E W 6 a 0 z < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " D 1 l a 2 z a 1 q z C / R Y 4 v j 9 8 q z K U Q j c Y = " > A A A C P 3 i c d Z A 7 T 8 M w F I U d n q W 8 C o w s F g 0 S Q 1 U l X W C s Y G F C R d C H 1 F T V j e u 0 V h 0 n s h 2 q K u o / Y + E v s L G y M I A Q K x v u Y y g t X M n S 0 f n u 1 f U 9 f s y Z 0 o 7 z Y q 2 s r q 1 v b G a 2 s t s 7 u 3 v 7 u Y P D m o o S S W i V R D y S D R 8 U 5 U z Q q m a a 0 0 Y s K Y Q + p 3 W / f z X m 9 Q c q F Y v E v R 7 G t B V C V 7 C A E d D G a u d q t p f E I G U 0 s L E d t F N P h p j w w t 3 N y C 7 g e e Z R D V M 8 G C M Q n X 9 w F / j I b u f y T t G Z F F 4 W 7 k z k 0 a w q 7 d y z 1 4 l I E l K h C Q e l m q 4 T 6 1 Y K U j P C 6 S j r J Y r G Q P r Q p U 0 j B Y R U t d L J / S N 8 a p w O D i J p n t B 4 4 s 5 P p B A q N Q x 9 0 x m C 7 q l F N j b / Y s 1 E B x e t l I k 4 0 V S Q 6 a I g 4 V h H e B w m 7 j B J i e Z D I 4 B I Z v 6 K S Q 8 k E G 0 i z 5 o Q 3 M W T l 0 W t V H S d o n t b y p c v Z 3 F k 0 D E 6 Q W f I R e e o j K 5 R B V U R Q Y / o F b 2 j D + v J e r M + r a 9 p 6 4 o 1 m z l C v 8 r 6 / g E W 6 a 0 z < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " D 1 l a 2 z a 1 q z C / R Y 4 v j 9 8 q z K U Q j c Y = " > A A A C P 3 i c d Z A 7 T 8 M w F I U d n q W 8 C o w s F g 0 S Q 1 U l X W C s Y G F C R d C H 1 F T V j e u 0 V h 0 n s h 2 q K u o / Y + E v s L G y M I A Q K x v u Y y g t X M n S 0 f n u 1 f U 9 f s y Z 0 o 7 z Y q 2 s r q 1 v b G a 2 s t s 7 u 3 v 7 u Y P D m o o S S W i V R D y S D R 8 U 5 U z Q q m a a 0 0 Y s K Y Q + p 3 W / f z X m 9 Q c q F Y v E v R 7 G t B V C V 7 C A E d D G a u d q t p f E I G U 0 s L E d t F N P h p j w w t 3 N y C 7 g e e Z R D V M 8 G C M Q n X 9 w F / j I b u f y T t G Z F F 4 W 7 k z k 0 a w q 7 d y z 1 4 l I E l K h C Q e l m q 4 T 6 1 Y K U j P C 6 S j r J Y r G Q P r Q p U 0 j B Y R U t d L J / S N 8 a p w O D i J p n t B 4 4 s 5 P p B A q N Q x 9 0 x m C 7 q l F N j b / Y s 1 E B x e t l I k 4 0 V S Q 6 a I g 4 V h H e B w m 7 j B J i e Z D I 4 B I Z v 6 K S Q 8 k E G 0 i z 5 o Q 3 M W T l 0 W t V H S d o n t b y p c v Z 3 F k 0 D E 6 Q W f I R e e o j K 5 R B V U R Q Y / o F b 2 j D + v J e r M + r a 9 p 6 4 o 1 m z l C v 8 r 6 / g E W 6 a 0 z < / l a t e x i t > Figure 6. Schematic of the trends predicted by the model for a galaxy that scatters above the mean SMHM relation at = 0. The arrows indicate the direction of causal relations.
EFFECTS OF DEPARTURE FROM THE MEAN SMHM RELATION ON GALAXY EVOLUTION
In the previous section we introduced a semi-empirical model to describe the influence of clustered stellar feedback on galaxy structure. In this section we explore the demographics of the GC populations predicted by the model across the plane defined by * and halo , and how these affect the evolution of the structure of their galaxies.
GC populations
Figures 7, 8, and 9 show the predictions of the model for the effect of scatter around the mean SMHM relation on the structure and GC populations of galaxies at = 0. The total numbers of GCs (i.e. massive stellar clusters) with masses > 10 4 , > 10 5 , and > 10 6 M , are shown in Figure 7. Upward scatter from the SMHM relation has a strong impact on the massive cluster populations of low-mass galaxies. This is caused by the earlier collapse times, the larger SFR surface densities (especially at early times), as well as the higher gas pressures of objects that lie above the relation. These conditions lead to higher fractions of bound clusters and larger mean cluster masses (see Section 2.4 and Fig. 4). The result is a steep increase in the number of massive GCs with increasing * at a fixed halo . In addition, there is also an increase in the number of GCs coll GC with increasing halo mass due to the larger max in more massive haloes. In the model, galaxies with halo < ∼ 10 10 M only form GCs when they scatter above the mean SMHM assumed here (see discussion in Section 2.4). This is consistent with the increasingly dominant role of mergers in the formation of GCs in low-mass galaxies in cosmological simulations . The downturn in the number of GCs near the cosmic baryon fraction in the left and middle panels is due to the minimum cluster mass increasing past 10 4 M , which shifts the ICMF towards more massive, but less numerous objects (see Figure 4). The last panel of Figure 7 shows that galaxies with * 10 9 M located near the cosmic baryon fraction can form significant populations of very massive ( > 10 6 * ) GCs.
For comparison, Figure 7 also shows the number of GCs hosted by each galaxy in the Forbes et al. (2018) sample of nearby dwarfs. These data are included in both the left and middle panels because the sample does not have a consistent minimum GC mass, and dwarf galaxies tend to contain more low-mass GCs than massive galaxies. Despite the limited sample size, the data suggest a trend of increasing number of GCs with stellar mass at fixed halo mass albeit with significant scatter. Since we do not model cluster dynamical mass loss, a quantitative comparison with the observations is not possible. However, the model qualitatively reproduces the general trend suggested by the data, predicting increasing GC numbers in galaxies with larger * (at a fixed halo ), although with a steeper dependence on * . Interestingly, observed dwarfs which do not host any GCs are located near the mean SMHM relation for halo < ∼ 10 10 M . This matches the region where no GCs are predicted to form. The right panel shows the good agreement between the number of massive GCs with > 10 6 M found in DF2 by van Dokkum et al. (2018b) and the model prediction. To place that galaxy in the *halo plane, we assume the best-fit halo mass obtained in Section 4.1, halo = 10 9.1−9.5 M (with the range indicated by the error bar). Large samples of low-mass galaxies with stellar and halo mass estimates, measured effective radii, and GC counts will be essential for testing the predicted relation between these quantities in our model.
In simulated * galaxies from the E-MOSAICS project (Pfeffer et al. 2018;Kruijssen et al. 2019) the number of GCs GC is found to correlate with the formation time of the DM halo, such that earlier-forming haloes host larger GC populations (Kruijssen et al. 2019, Table B1). This trend is naturally reproduced in our model via the assumption that the time-averaged SFR at > ∼ 2 was higher in haloes that collapsed earlier. For a fixed galaxy size e , the increase in coll GC directly follows from the observed increase in both the cluster formation efficiency (i.e. the fraction of stellar mass in bound clusters), and the ICMF truncation mass max , with Σ SFR (Adamo et al. 2020a).
To quantify the abundance of GCs relative to field stars, Figure 8 shows the GC specific frequency (Eq. 22) for GCs with > 10 4 M . The general trend of increasing specific frequency with decreasing stellar mass (for lines parallel to the cosmic baryon fraction) predicted by the model matches qualitatively the dominant observed trend in the Forbes et al. (2018) sample (see also Peng et al. 2008;Georgiev et al. 2010). The model overpredicts the present-day observed coll in the most massive haloes, but this is not a problem since observations only account for the surviving clusters. In fact, Kruijssen (2015) predicts that disruption is a key driver of the observed relation between coll and galaxy mass, and can suppress it by a factor of 10 − 100 in * galaxies (see also Bastian et al. 2020). The specific frequency in the model is relatively constant for galaxies on the mean SMHM relation, and increases steeply (and then decreases) with stellar mass at fixed halo , reaching the highest values at halo = 10 7 − 10 8 M . This occurs because the high gas pressure environments of galaxies with larger * at fixed halo generally produce a larger fraction of their stars in massive bound clusters compared to galaxies near the SMHM relation. This trend may also explain the increased specific frequencies of low-mass galaxies in the central region of the Virgo cluster, which Peng et al. (2008) argue is a predictable result of their relatively early formation times compared to galaxies in the outskirts. Our model also predicts a drop in the specific frequency coll for galaxies near the cosmic baryon fraction due to the rapid increase in the minimum cluster mass (see Figure 7). However, mass loss in clusters with < 10 5 M may modify this trend.
Galactic structure
The predicted structural properties of galaxies at = 0 are shown in Figure 9. From left to right, the panels show the size of the DM core, the galaxy effective radius, and the increase in the effective radius relative to the case with no feedback-driven expansion. Galaxies on or below the mean SMHM relation do not expand significantly relative to the case where SN feedback is not included, and reproduce the observed relation between halo mass and galaxy size from Kravtsov (2013). On the contrary, galaxies located more than ∼ 0.5 dex above the relation inject a larger fraction of their SN energy into galactic winds, increasing their effective radius by a factor of 10 as the stellar mass increases at fixed halo . The transition into the expansion regime occurs closer to the mean SMHM relation in lower mass dwarfs. Galaxies in which SN feedback is most efficient at expanding the DM halo and stars, are generally those with the largest GC abundance per unit stellar mass, coll .
The region inhabited by UDGs according to the common observational definition, e > 1.5 kpc for * < ∼ 5 × 10 8 M , is indicated by the white dashed line in Figure 9. The model predicts that isolated UDGs will form from galaxies hosted by haloes with halo > ∼ 10 7.5 M that scatter upwards from the SMHM relation by > ∼ 0.5 dex. Together with the elevated specific frequencies in the same region of the SMHM plane (see Figure 8), this provides a natural explanation for the higher GC abundance and specific frequency observed in low surface brightness galaxies compared to normal dwarfs at fixed stellar mass ( Behroozi et al. (2013), and the dotted line indicates the cosmic baryon fraction. The model is in broad agreement with the trend found in the observations, in which galaxies with higher * at fixed halo host more GCs. Note that the Forbes et al. (2018) sample does not use a consistent lower mass limit to define a GC, and that low-mass dwarfs tend to host GCs of lower mass compared to massive galaxies.
The role of the degree of clustering of star formation in galaxy evolution
Scatter above the mean SMHM relation influences the structure of model galaxies in two main ways. First, galaxies that scatter above the relation have larger stellar masses at fixed halo mass, and therefore more total SN energy available for winds to expand the DM and stellar components. Second, these galaxies formed in earlier collapsing haloes with larger SFR surface densities at = coll due to higher gas pressures. This results in an increase in both their bound fraction and mean cluster mass, which in turn increase the spatial and temporal clustering of SNe and the energy loading of galactic winds. To understand which of these two effects is the dominant driver of the expansion of the galaxy and halo, we show in Figure 10 the wind energy loading gal , the DM core size core , and the galaxy effective radius e , of the fiducial model compared to a model with no SN clustering (setting cl,SN = 1 in Equation 23).
It is evident from Figure 10 that the spatial and temporal clustering of SNe in massive clusters is the dominant mechanism that determines galactic wind energies and galaxy structure in those galaxies above the SMHM relation which exhibit the largest amounts of feedback-induced expansion, halo ≈ 10 7.5 − 10 10 M . Including the degree of clustering of stellar feedback sources in the model has a large impact on galactic outflows: for galaxies that scatter > ∼ 0.5 dex above the SMHM relation it increases wind energies by a factor of up to ≈ 1000. For these galaxies, the middle and right panels of Figure 10 show that the enhanced energy loading of galactic outflows due to clustered feedback strongly influences galaxy evolution: it increases the DM core sizes by up to a factor of ∼ 1000, and galaxy effective radii by up to a factor of ∼ 30 compared to the case with no SN clustering. This implies that a treatment of the clustering of feedback sources could be essential for future galaxy formation models aiming to reproduce the full diversity of the low-mass galaxy population.
IMPLICATIONS FOR THE FORMATION OF UDGS
A large fraction of the region above the mean SMHM relation appears to fall within the regime of observed UDGs (see Fig. 9). The model predicts that isolated UDGs form from dwarf galaxies that scatter > ∼ 1 dex above the mean SMHM relation as a result of the relatively early assembly of their host DM haloes, and its effect on stellar clustering and feedback-driven expansion (see Section 3.3). The last panel of Figure 9 shows that UDGs in the model have a broad range of dynamical mass-to-stellar mass ratios, dyn / * ≈ 1−100, when measured well outside the extent of their stellar component at = 8 kpc.
In this section we compare the predictions of the model to the detailed properties of some of the most extreme outliers from the SMHM relation in Figure 1, NGC1052-DF2 and DF4, and six isolated gas-rich UDGs with extremely high baryon fractions bar ≡ bar / dyn ∼ 1. Because their mass distribution is known at very large galactocentric radii ( ∼ 200 ), their halo masses can be measured directly. The low DM content of these objects (consistent with zero in some cases) therefore presents the greatest challenge to galaxy formation in the ΛCDM context, in which DM is always dominant at scales > ∼ 200 .
The case of NGC1052-DF2 and DF4
The predictions for a galaxy with the stellar mass of DF2/DF4, * ≈ 2 × 10 8 M , are presented in Figure 11. The top row shows how the formation of the core impacts the DM density profiles of galaxies when measured at a galactocentric radius = 8 kpc, which approximately corresponds to the outermost dynamical constraints for DF2 and DF4. The left panel shows the DM mass deficit, i.e. the fraction of DM mass that was removed by feedback-driven expansion within 8 kpc. Interestingly, the DM mass deficit is below 1 per cent for galaxies near or below the mean SMHM relation, and 10 8 10 9
10 10 10 11 M halo [M ] The GC specific frequency quantifies the abundance of GCs relative to stars, coll ≡ coll GC (10 9 M / * ), and is calculated here for clusters with > 10 4 M . The observational data for nearby low-mass galaxies from Forbes et al. (2018) is shown using diamonds coloured by coll . Empty symbols correspond to galaxies that do not host any GCs. The solid line is the mean SMHM relation from Behroozi et al. (2013), and the dotted line indicates the cosmic baryon fraction. As seen in the observations, dwarf galaxies with lower stellar mass have more dominant GC populations relative to field stars. The model predicts an additional trend of increasing specific frequency with stellar mass at fixed halo mass which is also present in the Forbes et al. (2018) data. The drop in coll near the cosmic baryon fraction is due to a shift in the low-mass truncation of the ICMF. The predicted higher coll in galaxies that assembled earlier resembles what is observed in the Virgo cluster (Peng et al. 2008). increases steeply to 10 per cent above it. Galaxies with masses halo ∼ 10 9 − 10 10 M that lie more than ∼ 1dex above the mean SMHM relation reach the largest DM deficiencies, nearly 100 per cent, due to their large massive cluster populations (see Figure 7), and the resulting high clustering of SN events (see Figure 5). The right panel shows the DM mass enclosed within 8 kpc, and compares it to the possible locations of DF2 and DF4 on the * − halo plane, given that their total halo mass is not known (shaded band). To provide an upper limit on halo , we assume no DM has been stripped. The red line indicates the upper limit on the enclosed DM mass from van Dokkum et al. (2018a). The small region left of the red line that overlaps with the shaded band represents the part of the SMHM plane where DF2 and DF4 could have formed. It corresponds to a DM halo with halo < ∼ 2 × 10 9 M which contains nearly its entire cosmic baryon fraction in stars, and lost > ∼ 90 per cent of its central DM mass due to feedback-driven expansion. To explore the effects of the variation in the * / halo ratio, the bottom row of Figure 11 shows the circular velocity profiles corresponding to the positions of the three star symbols in the upper right panel. The halo and stellar masses, and the enclosed baryon fraction within 8 kpc are indicated in each panel. Dynamical constraints for DF2 (van Dokkum et al. 2018a;Emsellem et al. 2019), andDF4 (van Dokkum et al. 2019a) are shown for comparison. Evidently, the model at the mean SMHM relation (right panel) fails to reproduce the dynamical constraints due to its massive DM halo with a relatively small core. The two models that scatter by more than 1 dex above the SMHM relation (left and middle panels) satisfy the dynamical constraints due to a combination of two factors: a lower mass host DM halo with a lower circular velocity circ , and stronger winds due to increased stellar clustering, which more easily carve a very large DM core in a halo of lower central density and binding energy. The models that reproduce the mass profiles predict effective radii e ∼ 5 kpc, larger than the observed values for DF2 and DF4, ∼ 2 kpc. Due to the many simplifying assumptions in the model, we do not expect this to be a major shortcoming of the model.
Isolated gas-rich UDGs
As Figure 1 clearly shows, NGC1052-DF2 and DF4 are not unique in their apparent DM deficiency. Many galaxies from several samples including the nearby dwarfs from (Forbes et al. 2018), the SPARC database Li et al. 2020), and many other isolated low-mass galaxies in the Local Volume and beyond (e.g. Geha et al. 2006;Ferrero et al. 2012;Oman et al. 2016;Klypin et al. 2015;Papastergis et al. 2015;Schneider et al. 2017;Trujillo-Gomez et al. 2018) seem to inhabit very low mass DM haloes relative to the expectation from the mean SMHM relation. Similar examples of this phenomenon are also found among massive early type galaxies (Padmanabhan et al. 2004), and massive disc galaxies (Posti et al. 2019).
When the dynamical masses of DF2 and DF4 were first estimated, they appeared to be the most extreme examples of DMdeficient galaxies, with bar ∼ 1 within a radius of ∼ 8 kpc (see Fig. 1). However, in a recent study, Mancera Piña et al. (2019Piña et al. ( , 2020 found six gas-rich isolated UDGs for which the rotation velocity of the H gas is very slow compared to non-UDGs with the same baryonic mass. As a result, the stellar and cold gas content of these galaxies accounts for most (and sometimes all) of their dynamical mass within the extent of their H disc, ∼ 8 − 11 kpc. Through a careful analysis of the uncertainties, Mancera Piña et al. (2020) rule out systematics in the derivation of the rotation curves as a source of this effect. We build simple mass models of these UDGs and show the inferred halo masses in Figure 1 8 , highlighting the striking similarity with DF2 and DF4 in terms of their DM deficiency. Four out of the six isolated UDGs in this sample have baryon fractions above the cosmic value, and three of these (indicated by upper limits on their halo masses) do not require any DM to explain their dynamics at large radii.
By construction, our model accounts for galaxies with baryon fractions as high as the cosmic mean. However, objects that lie above it and have extremely low DM content can potentially be explained by the effect of feedback-driven DM expansion due to enhanced stellar clustering in galaxies above the mean SMHM. As in the case of DF2 and DF4, this could increase their baryon fractions to bar ≡ bar / dyn ∼ 1. Figure 12 shows the predictions for galaxies with properties in the range spanned by the isolated UDGs from Mancera Piña et al. (2020), with baryonic masses 1.1 × 10 9 M < bar < 2.3 × 10 9 M and baryon fractions 0.47 < bar ( < ∼ 8 kpc) < 1. The top panel shows the DM mass of galaxies enclosed within = 8 kpc in the * − halo parameter space. Since the model does not predict the cold gas properties at = 0, we simply assume an exponential gas density profile with a scale-length equal to the mean disc scalelength of the sample in order to facilitate comparison with our predictions. The blue shading indicates the range of stellar mass of the observed UDGs. The bottom left and middle panels show the predicted circular velocity profile for an object in this region with halo = 1 − 3 × 10 9 M and the mean stellar mass of the isolated UDG sample, * = 1.6 × 10 8 M . Although these model galaxies should contain significant amounts of DM in the central ∼ 10 kpc . Impact of scatter in the SMHM relation on galaxy and DM halo structural properties at = 0. From left to right the panels show the size of the feedback-induced DM profile core, the galaxy effective radius, the increase in the effective radius relative to the case where there is no feedback-induced halo expansion, and the dynamical mass-to-stellar mass ratio within a radius of 8 kpc, respectively. The region defined by the white dashed line corresponds to the commonly adopted observational definition of UDGs, e > 1.5 kpc for * < ∼ 5 × 10 8 M . The solid line is the mean SMHM relation from Behroozi et al. (2013), and the dotted line indicates the cosmic baryon fraction. Galaxies with the largest upward scatter in * at fixed halo can form DM cores with core 10 kpc, and expand their stellar components dramatically, reaching effective radii up to e ∼ 10 kpc. The model predicts that UDGs with a broad range of dynamical mass-to-light ratios form when the upwards scatter in * at fixed halo is 1 dex. e Figure 10. Impact of stellar clustering at the peak GC formation epoch on DM halo and galaxy structural evolution. The panels show the wind energy loading (left), the DM core size (middle), and the galaxy effective radius (right) at = 0 for the fiducial model relative to a model where clustering of SNe does not influence the driving of galactic winds (i.e. cl,SN = 1). The solid line is the mean SMHM relation from Behroozi et al. (2013), and the dotted line indicates the cosmic baryon fraction. The enhanced coupling of SN energy to galactic winds due to overlapping SNe in star clusters is the dominant effect driving the expansion of the DM haloes and stellar discs of low-mass galaxies that scatter above the SMHM relation. As a result, our model predicts that feedback from massive star clusters is a necessary ingredient for explaining the origin of isolated ultra-diffuse galaxies.
(the 'no feedback' case shown with a dashed line), the increased clustering of star formation and wind driving efficiency have carved a very large core core > 25 kpc in their DM profiles (solid grey line). The corresponding reduction in the inner DM density, together with the increased baryon mass due to the early collapse of the halo, lead to the large enclosed baryon fraction bar > 0.73 within 8 kpc.
Also note that the model predicts very large disc effective radii e = 3.2 − 5.2 kpc, in agreement with the observed range of the UDGs, ∼ 3 − 7 kpc. For comparison, the bottom right panel shows the profile of a galaxy with the same stellar mass but with a halo mass determined by the mean SMHM relation, halo = 5 × 10 10 M . In this case the baryonic circular velocity profile fits the observed galaxies, but the total circular velocity is significantly overpredicted, and the central baryon fraction is only 0.15 due to the large halo mass and relative inefficacy of SN feedback. H -bearing UDGs are commonly found in the field. They represent about 6 per cent of all low-mass galaxies with 8.5 < log HI / M < 9.5, and have a cosmic abundance similar to that of cluster and group UDGs (Jones et al. 2018). Leisman et al. (2017), Jones et al. (2018), and Guo et al. (2020) find that these galaxies generally rotate slower than normal galaxies of similar baryonic mass, indicating that DM-deficiency may be a general property of field UDGs. This raises the intriguing possibility that a single formation channel, like the one proposed here, could account for the structural properties of both field and cluster/group UDG populations. This mechanism may also remove the need for intense tidal stripping for explaining UDGs in dense environments.
INFERRED HALO MASSES AND BARYON FRACTIONS
The modification of the mass distribution by stellar feedback for galaxies above the mean SMHM relation could have important consequences for dynamical models aimed at inferring the mass of the host DM halo from the stellar or gas kinematics. Figure 13 shows the effect of fitting two commonly assumed DM halo density profiles to observations that determine the circular velocity at a radius of = 8 kpc. This radius is representative of the most extended dwarf galaxy and field UDG gas rotation curves Mancera Piña et al. 2020), as well as the outermost GC dynamical measurements in DF2 andDF4 (van Dokkum et al. 2018a, 2019a). The figure shows that galaxies which scatter upwards from the mean SMHM relation and become very expanded due to the enhanced clustering of SN feedback, have such large DM cores that naive fits to the kinematics at < ∼ 8 kpc assuming the NFW profile result in a large underestimation of the true cosmological halo mass (i.e. the halo mass without expansion). For galaxies near the cosmic baryon fraction, this effect can result in an underestimation of the true halo mass by more than an order of magnitude. Interestingly, Mancera Piña et al. (2020) find that the sparsely-sampled rotation curves of their gas-rich UDGs (also reproduced in Figure 13) can only be fit with an NFW profile of extremely low concentration compared to the mean expected in ΛCDM haloes. When fit using NFW profiles (which do not account for stellar feedback), the effect of enhanced DM expansion in galaxies that scatter above the SMHM relation could explain both the apparent position of these galaxies in the * − halo plane, and their seemingly low concentrations. Figure 13 also suggests that the apparently large scatter in * at fixed halo in the dwarf galaxy samples in Figure 1 could be enhanced in large part by underestimation of the true halo mass. This would push their apparent positions above the cosmic baryon fraction, with * > halo in the most extreme cases, and in contradiction with ΛCDM expectations. We therefore expect that the real scatter in * ( halo ) may be considerably smaller 9 . Detailed modelling of observed rotation curves in large galaxy samples is needed to constrain the model and to provide an estimate the true scatter in * / halo .
DISCUSSION
Our model predicts that low-mass galaxies that scatter significantly above the SMHM relation formed earlier, and experience more clustered star formation and more energetic outflows, causing them to expand enough to become UDGs. This intrinsic channel for UDG formation has also been suggested by numerical studies Jiang et al. 2019;Jackson et al. 2021b). A novel result of our analysis is that the growth of the DM core in UDGs can be so pronounced for objects with the highest * / halo ratios (near the cosmic baryon fraction), that the DM mass enclosed within ∼ 10 kpc is extremely reduced. In addition, the model simultaneously explains how the early formation of these galaxies leads to an excess in the number of GCs. These predictions are strikingly similar to the puzzling observations of NGC1052-DF2 and DF4, and of six isolated gas-rich UDGs, which all seem to have very low DM content ( < ∼ 50 per cent of the total mass) at large radii, where DM is expected to dominate. The model therefore accounts for the formation of galaxies with low DM content in the field, as well as in groups and clusters, without the need for highly eccentric orbits. Trujillo-Gomez et al. (2020) used the GC mass function of the 11 unusually massive GCs in DF2 to reconstruct its galactic environment ∼ 9 Gyr ago. Due to the implications of the peculiar GC mass function for the size of the star-forming region, they conclude that the massive GCs likely formed during a major merger. Our model does not explicitly include the effect of mergers. However, we expect the overall expansion of the galaxy to be mostly insensitive to the details of how the GCs were produced (i.e. by rapid gas accretion or by a major merger), such that the results of the merger model are complementary to the ones presented here.
We find that the observed large scatter in the SMHM relation of low-mass galaxies can originate in part from the combined effects of early collapse and increased stellar clustering on the central mass distribution. When mass models based on DM-only simulations are fitted to the dynamics of galaxies with the largest upward scatter, the strong expansion of the DM results in an underestimation of the true (pre-expanded) halo mass. This could help explain the objects that lie above the cosmic baryon fraction, and especially those with * > halo (see Figure 13). The model also qualitatively reproduces the higher GC specific frequency in galaxies with lower surface brightness, found across dwarfs, UDGs, and low surface brightness galaxies in galaxy clusters (Lim et al. 2018;Prole et al. 2019a;Lim et al. 2020). Because UDGs are more commonly found near the cluster core, and galaxies in this region formed earlier (Peng et al. 2008), our model predicts that they should have scattered above the mean SMHM relation, causing them to form more GCs and expand. The existence of DM-deficient isolated UDGs, as well as many other field galaxies with low DM content (i.e. large baryon fractions; see Section 4.2) indicates that the lack of DM is not exclusively caused by stripping in a group environment as proposed by many numerical studies (e.g. . Impact of feedback-driven expansion on the inferred halo mass of observed galaxies. Circles show the true halo mass of model galaxies with varying amounts of scatter from the mean SMHM relation at various values of halo . Lines connect the true halo to the value obtained by fitting an NFW (triangles) or a coreNFW (diamonds) profile to the DM circular velocity measured at a large radius, = 8 kpc. The dotted and dash-dotted lines indicate the cosmic baryon fraction and the * = halo line, respectively. The symbols are coloured to indicate the galaxy effective radius. The grey symbols with error bars reproduce the dynamical constraints for DF2, DF4, and gas-rich isolated UDGs from Figure 1. The solid line is the mean SMHM relation from Behroozi et al. (2013), the dotted line shows the cosmic baryon fraction, and the dash-dotted line indicates * = halo . Assuming an NFW profile to fit the enclosed dynamical mass of strongly expanded galaxies severely underestimates their true halo , and shifts their apparent position towards the left, making them appear very DM-deficient, with * > halo in some cases. The low inferred halo masses of DF2/DF4 and some field UDGs could be explained by this effect.
Instead, intrinsic mechanisms acting on isolated galaxies, such as enhanced outflows due to stellar clustering as proposed here, are needed to explain this phenomenon. Most field UDGs are blue and star-forming (Prole et al. 2019b). Furthermore, gas-rich field UDGs represent a substantial fraction of the galaxy population, and tend to rotate slower than normal galaxies of the same baryonic mass (Jones et al. 2018). This suggests that DM-deficiency might be generic to all field UDGs that scatter well above the SMHM relation, and highlights the need to understand these objects. Furthermore, the model predicts that galaxies with * ∼ 10 5 −10 7 M could constitute a large fraction of the UDG population that is yet to be discovered. Silk (2019) proposed a mechanism for removing DM and forming massive GCs in high-speed collisions between dwarf galaxies in gas-rich proto-groups. These predictions were confirmed by Lee et al. (2021) using hydrodynamical simulations, showing that the collision results in the formation of a gas-poor compact merger remnant and several massive GCs. While DF2 and DF4 could be consistent with forming in high speed collisions, the six gas-rich DM-deficient UDGs studied by Mancera Piña et al. (2020) cannot be explained by this mechanism due to their high gas content, ongoing star formation, and isolation. Shen et al. (2021) recently estimated the relative distance between DF2 and DF4 to be 2.1 ± 0.5 Mpc. This large separation implies that at least one of the two galaxies is isolated, and likely rules out environmental formation mechanisms, including high-speed collisions.
While our model does not account for environmental effects, it predicts an large intrinsic diversity (at formation) in the sizes and mass-to-light ratios of field galaxies (Fig. 9). This intrinsic diversity would only be enhanced by the effects of ram pressure stripping of gas and tidal stripping of stars and DM on the broad distribution of satellite infall times and trajectories. Detailed predictions for these effects should be made using numerical simulations with initial conditions provided by the model presented here. Another element that the model does not include is the evolution of the cold gas content. While we assume here that isolated galaxies retain their cold gas until = 0, it is possible that in certain cases a strong clustered feedback episode could remove all the gas permanently, effectively quenching the galaxy. Suggestive evidence of this process may be found in the first study of the spatially-resolved stellar populations in a UDG. Villaume et al. (2021) studied DF44, a UDG in the outskirts of the Coma cluster, and determined that the galaxy has a very narrow SFH and a flat radial stellar metallicity and age gradient. They conclude that this is consistent with DF44 having formed entirely, along with its large population of ∼ 100 GCs 10 (van Dokkum et al. 2016), in a single burst of star formation ∼ 10 Gyr ago. Since it is likely that DF44 was quenched before falling into Coma, this raises the possibility that the same SF burst that produced the large population of GCs and expanded the stars also quenched the galaxy ∼ 10 Gyr ago. In this scenario, DF44 formed near the mean SMHM relation (van Dokkum et al. 2019b) and had an early strong burst of SF that expanded the stellar component to e = 4.6 kpc, flattening the age and metallicity gradients, and quenching the galaxy. Early self-quenching prevented subsequent generations of star formation (which are centrally concentrated due to the infall of fresh gas) from producing a young disc with a smaller effective radius (see Eq. 44). The lack of a young stellar component could therefore preserve the flat gradients caused by expansion.
Our results also show that due to their higher GC occupation, UDGs may be offset from the linear GC − halo relation observed in normal galaxies. This suggests that using this relation to infer the DM halo mass of UDGs may be seriously biased (also see the discussion in van Dokkum et al. 2018b). The high values of coll GC in cluster UDGs would therefore place them in the more massive DM haloes of normal galaxies with the same GC occupation, creating the illusion of UDGs that scatter downwards from the mean SMHM relation. Interestingly, this is the result that Prole et al. (2019a) obtain for Fornax UDGs, assuming that they follow the GC − halo relation.
CONCLUSIONS
Through a simple semi-empirical model, we have explored the effect of significant scatter in the low-mass SMHM relation -as suggested by dynamical probes -on the star cluster populations, feedbackdriven winds, and structural evolution of galaxies and their host DM haloes. To minimise the number of free parameters, the model uses empirical galaxy scaling relations, clustering constraints on the galaxy-halo connection, as well as the results of large hydrodynamic cosmological simulations, and detailed numerical models of the generation of galactic winds by clustered SNe. For simplicity, it accounts for mergers as part of the smooth matter accretion component. Due to the lack of constraints on some elements of the model, several assumptions must be made. For reasonable parameter choices, the results show that DM haloes that collapse earlier and therefore achieve larger SFRs at > ∼ 2, reach the high pressure ISM conditions necessary to produce a larger fraction of stars in bound clusters that are on average more massive. In general, the larger the departure of a galaxy upwards from the mean SMHM relation, the more clustered its star formation and therefore its SN feedback will be at the collapse epoch. This in turn increases the energy loading of galactic winds, and the energy injected into the DM orbits by potential fluctuations due to massive outflows. Assuming that a small constant fraction of the wind energy couples to the potential of the halo, we estimate that the DM core size increases steeply with the upwards scatter, with galaxies near the cosmic baryon fraction forming cores with sizes core 10 kpc.
The stars that formed before the epoch of halo collapse expand along with the DM, and this has a strong effect on galaxy sizes and surface brightness at = 0. It allows galaxies with * < 10 9 M that scatter significantly ( 0.5−1.0 dex) above the SMHM relation to expand enough to become UDGs, and suggests that all UDGs could in principle be formed through this mechanism.
Our conclusions are summarised as follows:
• A simple semi-empirical model predicts that scatter around the mean SMHM relation has important consequences for the evolution of the structure and star cluster populations of galaxies and their host DM haloes.
• The model assumes that galaxies with larger than average * / halo ratios are hosted by DM haloes that collapse earlier, allowing for larger SFR at early epochs ( Figure 2). Their higher SFR surface densities (Figure 3) indicate higher ISM pressures and therefore increased stellar clustering and larger GC populations compared to galaxies on the mean relation ( Figure 4). The predicted trend of increasing GC numbers and specific frequencies with scatter from the SMHM relation qualitatively matches the observed trend in nearby dwarfs, and may explain why UDGs in clusters have more GCs than normal galaxies of the same stellar mass (Figs. 7 and 8).
• The degree of clustering of star formation in space and time has a strong impact on galaxy evolution. For galaxies that scatter upwards from the SMHM relation by > ∼ 0.5 − 1 dex, a steep increase in stellar clustering and in the mean cluster mass drives a significant increase in the wind energy loading due to clustered SNe. This boost increases to a factor of > 100 for galaxies with the largest * / halo (Figure 10). The higher SFR at the collapse epoch in these galaxies also increases the total SN energy available to power galactic winds. This suggests that the spatial and temporal clustering of star formation and feedback should be included as an essential ingredient in galaxy formation models.
• Outflows in galaxies above the SMHM relation are more energetic and can potentially expand the orbits of the DM and old stars by a factor of > 10 at = 0 compared to those on the mean relation. UDGs are formed naturally in progenitors with halo > ∼ 10 7.5 M which scatter upwards by > ∼ 1 dex (Figure 9). The model predicts that a large fraction of low surface brightness field galaxies with * ∼ 10 5 − 10 7 M could still lurk in the dark, comprising the low-mass end of the UDG population that is yet to be discovered.
• Enhanced feedback due to SN clustering drives the formation of a large core in the DM density profile, with the maximum effect at halo ∼ 10 9 − 10 10 M and * ∼ 10 8 M (Figure 9). Cores with sizes core > 10 kpc significantly reduce the DM content, and boost the baryon fraction within ∼ 10 kpc. Galaxies with the largest upwards scatter at halo = 10 9 − 10 10 M fit the dynamical constraints of DM-deficient galaxies like DF2/DF4 and several isolated UDGs, providing a natural pathway for the formation of isolated galaxies lacking DM. This mechanism accounts for the formation of DM-deficient galaxies in the field, and in groups/clusters, without the need for highly eccentric orbits to remove the DM within ∼ 10 kpc (Figs. 11 and 12). Feedback from a large population of GCs formed in a single early burst of star formation could potentially account for the old ages and flat metallicity and age gradients observed in DF44 (Section 6).
• The model naturally explains the large diversity in the dynamical mass-to-light ratios measured well beyond the extent of the stellar disc of observed UDGs, with a predicted range of dyn / * (< 8 kpc) ≈ 1 − 100 (Fig. 9). This diversity is driven by the increase in the stellar mass and its effect on stellar (and SN) clustering and the expansion of the inner DM halo with growing scatter above the mean SMHM relation. While environmental effects such as DM stripping are not needed to reproduce the properties of UDGs in the model, they are expected to increase the diversity of UDGs in group and cluster environments.
• Fitting the mass profile of galaxies with large DM cores with commonly used density profiles (such as the NFW profile) results in a significant underestimation of the halo mass by more than an order of magnitude in objects with the largest * / halo . This provides an explanation for galaxies that appear to exceed the cosmic baryon fraction, and in the most extreme cases where * > ∼ halo (Fig. 13). It also means that the intrinsic scatter around the SMHM relation is smaller than observed.
These predictions can be compared to recent cosmological simulations of galaxy formation which resolve dwarf galaxies in representative volumes. Recently, Jackson et al. (2021b) used the NewHorizon simulation to identify low surface brightness galaxies and find that these form preferentially in high density environments which lead to earlier formation times and stronger feedback-driven expansion. The dominant uncertainty in our model lies in the parameters * , DM , and ( * , halo ), which control the fraction of stars formed during the halo collapse epoch, the fraction of galactic wind energy that couples to the DM, and the relation between stellar mass and halo concentration, respectively (see Appendix B). These parameters can be constrained by high-resolution cosmological simulations that model cluster formation and disruption, and include its effect on the driving of galactic winds. Including the effects of stellar clustering on feedback and galaxy structure will be key to understanding how galaxies form and evolve in the ΛCDM paradigm. velocity max as derived from the H linewidth, which has been shown to be a good approximation (Trujillo-Gomez et al. 2018). GC occupation data for the isolated galaxies was obtained from Georgiev et al. (2010), while for the Local Group dwarfs it was obtained from a variety of sources (see Forbes et al. 2018, for details). Li et al. (2020) fitted mass models to the SPARC (Spitzer Photometry and Accurate Rotation Curves) database ) of high quality rotation curves and 3.6 m Spitzer data for 175 late-type galaxies ranging in mass from dwarfs to giant spirals. The models used here assume the coreNFW profile (Read et al. 2016a) and uniform priors on the halo concentration in the range 0 < < 1000. To avoid systematics due to large inclination corrections, we follow Lelli et al. (2016) and limit the sample to inclinations > 30 deg, and remove objects with poor quality rotation curves (i.e. flag = 3). The stellar mass is calculated from the 3.6 m luminosity assuming a mass-to-light ratio Υ * = 0.5. This introduces a small but negligible inconsistency with the mass models, which fit the stellar mass distribution using log-normal priors for the mass-tolight ratio of the disc and bulge components centred at Υ disc = 0.5 and Υ bulge = 0.7, with a standard deviation of 0.1 dex.
Mancera Piña et al. (2020) analysed low-resolution H rotation curves of 6 gas-rich isolated UDGs found in the ALFALFA survey. They modelled the full 3D data cubes (accounting for the effect of beam-smearing) to obtain the inclination, and provide measurements of stellar and H mass, effective radius, and circular velocity out at the outermost point of the rotation curve, out ∼ 8 − 10 kpc. The circular velocity of the DM component is obtained by removing the contribution of stars and gas, 2 DM = 2 circ − 2 bar , where 2 bar = ( * + 1.33 HI )/ out , and with the factor 1.33 accounting for the contribution from Helium. Since the rotation curves are too coarse to properly fit mass models, we performed a simple fit to only the outermost rotation velocity using the coreNFW density profile from Read et al. (2016a) which includes the effect of feedback-induced DM cores. Since the density profile depends on halo mass and concentration, an additional constraint is needed. For this, we assume the cosmological concentration-mass relation and its 1 scatter from to obtain an estimate of the uncertainty in halo . For three of the objects, the baryonic mass exceeds the dynamical mass at out , and the halo mass is consistent with zero. To obtain the upper limits on halo shown in Figure 1 for these objects, we use the minimum integer multiple of the 1uncertainty in circ which results in a non-zero halo mass. Guo et al. (2020) searched the ALFALFA (Haynes et al. 2011) catalogue for galaxies with luminosities > −18 where the baryonic mass accounts for more than 50 per cent of the dynamical mass within the H radius. For the 19 objects they find, they fit NFW mass models to the inclination-corrected rotation velocity obtained from the 20 per cent velocity width of the H line profile. To avoid underestimation of halo due to the presence of a DM core, here we use only those galaxies with HI > 2 e .
The stellar, dynamical, and halo masses for NGC1052-DF2 were obtained directly from van Dokkum et al. (2018a). The stellar and dynamical masses for NGC1052-DF4 were obtained from van Dokkum et al. (2019a), and the limit on the halo mass was calculated by fitting a coreNFW profile (Read et al. 2016a) of average concentration to the 2 upper limit on the DM mass within 7 kpc. Compared to a pure NFW profile, the inclusion of a DM core has a negligible effect on the halo mass obtained.
APPENDIX B: IMPACT OF PARAMETER CHOICES
Here we examine the dependence of the predictions of the model on the choice of parameters. For this, we select the parameters that are least constrained by observations and that contribute the largest uncertainties. These are the fraction of wind energy that couples to the DM, DM , the fraction of stellar mass in place at = coll , * , and the concentration -stellar mass relation, ( * , halo ). Figure B1 shows the predictions for the GC populations and galaxy and halo structure for a model with the fiducial parameters and a 5 times lower value of the coupling fraction of wind energy to the DM, DM = 0.02 (set to DM = 0.1 in the fiducial model). As expected, the GC populations are unchanged, but the amount of expansion due to feedback is reduced by about a factor of ∼ 3, while the maximum effect shifts to lower mass haloes with halo ∼ 10 8.5 M . Figure B2 shows the predictions for a model in which only 5 per cent of the present stellar mass of the galaxy formed by the collapse redshift = coll , * = 0.05, a factor of 4 lower than the fiducial value ( * = 0.2). In this case, the number of massive clusters in galaxies near the mean SMHM relation is reduced by a factor of ∼ 4, and the effect of expansion of the galaxy and halo shifts to lower halo masses halo < ∼ 10 9 M compared to the fiducial case. Figure B3 shows the predictions for a model in which the relation between DM halo concentration and offset from the mean SMHM relation has a steeper slope by a factor of 2, = 1.0. This results in galaxies which scatter upwards having larger DM halo concentrations and earlier collapse times relative to the fiducial model (where = 0.5). The main consequence of the earlier collapse times is a slight reduction in the size of the GC populations of galaxies near the mean SMHM relation, and an earlier saturation for galaxies near the cosmic baryon fraction. The impact on galaxy and halo structure is minimal relative to the fiducial model. Figure C1 shows the predictions for the fiducial model assuming a larger value of the Toomre parameter, = 2, corresponding to the critical value for gravitational instability in a thin disc and gas = 0.5. Larger values of increase the largest unstable cloud scale and hence the maximum cluster mass, allowing for larger populations of massive GCs. Using this assumption, dwarf galaxies with very high specific frequencies like Fornax (located near the mean SMHM at halo ≈ 10 10 M , and hosting 5 GCs) become typical in the model, with a prediction of coll GC ( > 10 4 M ) ≈ 14. The model predictions for DF2/DF4 and the isolated gas-rich UDGs (see Section 4) are able to fit the observational constraints as well as those with = 0.5. This paper has been typeset from a T E X/L A T E X file prepared by the author. Figure B2. Predictions for GC populations and galaxy structure assuming a factor of 4 reduction in the fraction of stars formed at = coll , * = 0.05, relative to the fiducial value ( * = 0.2). Due to the lower SFR at the collapse epoch, massive cluster populations are reduced in number, and the total SN energy is lower, leading to reduced feedback-driven expansion compared to the fiducial case (see Figs. 7,8,and 9). Figure C1. Predictions for GC populations and galaxy structure assuming a larger value of the Toomre parameter, = 2 in the fiducial model. A larger results in slightly larger GC populations for the most massive dwarf galaxies near the mean SMHM relation (see Fig. 7). It also slightly reduces the region of UDG formation in the SMHM plane (see Fig. 9).
APPENDIX C: INFLUENCE OF THE TOOMRE PARAMETER
Figure 2 .
2DM (< e ) + coll * (< e ) + coll gas (< e )] Predicted host DM halo properties as a function of the position of the galaxy in the * − halo plane. Left: DM halo concentration at = 0. Right: collapse redshift. The dotted line shows the cosmic baryon fraction. At a fixed halo mass, galaxies above the mean SMHM relation are hosted by haloes with higher concentrations and earlier collapse times.
Figure 3
3Figure 3. Dependence of galaxy properties at the collapse redshift on the present position in the * − halo plane. First row: time-averaged star formation rate and galaxy effective radius at = coll . Second row: SFR surface density (left) and gas surface density (right). Last row: gas fraction (left) and disc rotation frequency (right). The solid line is the mean SMHM relation from Behroozi et al. (2013), and the dotted line indicates the cosmic baryon fraction. At a fixed halo mass, galaxies that scatter above the mean SMHM relation had increasingly higher SFRs, and smaller sizes. This resulted in higher mean SFR surface densities, mean gas surface densities, global gas fractions, and rotation frequencies at their formation epoch, = coll .
Figure 4 .
4Demographics of star cluster populations formed at the collapse epoch as a function of the present position of the galaxy in the * − halo plane.
∼ 0.002−0.4 (Peñarrubia et al. 2012; Read et al. 2016a). Madau et al. (2014)
Figure 5 .
5Impact of scatter away from the mean SMHM relation on the efficiency of galactic wind driving by supernovae. Left: Average SN clustering factor cl,SN . Right: energy loading factor gal of galactic winds due to SNe. The solid line is the mean SMHM relation from
van Dokkum et al. 2017; Lim et al. 2018; Prole et al. 2019a; Lim et al. 2020).
Figure 7 .
7Demographics of the GC populations formed at = coll as a function of the position of their host galaxy in the * − halo plane. The panels show the total number of > 10 4 M GCs (left), > 10 5 M GCs (middle), and > 10 6 M GCs (right). The sample of nearby dwarfs from Forbes et al. (2018) is shown as diamonds in the left and middle panels, with the colour of each symbol indicating the observed number of GCs hosted by each galaxy, and galaxies with no GCs indicated by empty symbols. The position of the galaxy NGC1052-DF2 is indicated by the circle with error bars, coloured according to the observed number GCs with > 10 6 M (7 out of 11 in van Dokkum et al. 2018b) (the error bars indicate the uncertainty in halo from the range of best-fit halo masses obtained in Section 4.1). The solid line is the mean SMHM relation from
Figure 8 .
8Abundance of galactic GC populations relative to stars at = 0 as a function of the position of their host galaxy in the * − halo plane.
Figure 9
9Figure 9. Impact of scatter in the SMHM relation on galaxy and DM halo structural properties at = 0. From left to right the panels show the size of the feedback-induced DM profile core, the galaxy effective radius, the increase in the effective radius relative to the case where there is no feedback-induced halo expansion, and the dynamical mass-to-stellar mass ratio within a radius of 8 kpc, respectively. The region defined by the white dashed line corresponds to the commonly adopted observational definition of UDGs, e > 1.5 kpc for * < ∼ 5 × 10 8 M . The solid line is the mean SMHM relation from Behroozi et al. (2013), and the dotted line indicates the cosmic baryon fraction. Galaxies with the largest upward scatter in * at fixed halo can form DM cores with
Figure 11 .
11Impact of feedback-driven expansion on the density profile of galaxies as a function of their position in the * − halo plane. First row: amount of DM mass lost from the inner 8 kpc relative to the case with no feedback (left), and DM mass within 8 kpc (right). The range of stellar mass for DF2 and DF4 including uncertainties is indicated by the horizontal shaded band, and the observational upper limit on the enclosed DM mass, DM (< 8 kpc) < 1.5 × 10 8 M (van Dokkum et al. 2018a), is shown as a red line. The star symbols indicate the three halo masses considered in the bottom row. The solid line is the mean SMHM relation fromBehroozi et al. (2013). Second row: predicted circular velocity curves for a galaxy with the stellar mass of DF2/DF4 ( * ≈ 1−2×10 8 M ) and three different possible values of halo mass (from left to right) corresponding to the star symbols in the upper right panel. Points show the dynamical constraints for DF2 (green symbols) and DF4 (blue square). The halo and stellar masses, and the baryon fraction within 8 kpc are indicated in each panel. While the model on the mean SMHM relation (bottom right) is DM-dominated and well above the constraints, models with halo masses up to halo ∼ 3 × 10 9 M satisfy both the DM mass constraints and the circular velocity curve at all radii.
Figure 12 .
12Predicted DM content and mass profiles at large radii compared to observations of gas-rich isolated UDGs. Top: enclosed DM mass within the central 8 kpc as a function of position in the * − halo plane. The blue shaded band indicates the range of stellar masses of the isolated UDG sample from ManceraPiña et al. (2020). The star symbols indicate the location of the three galaxy models shown in the bottom row. The solid line is the mean SMHM relation fromBehroozi et al. (2013). Bottom left and middle: mass profiles of two model galaxies that fit the gas-rich UDG mass profile constraints. Bottom right: mass profile of a galaxy with the mean observed stellar mass of the UDG sample and the mean halo mass from the SMHM relation. Each of the bottom panels lists the corresponding halo mass, stellar mass, and baryonic fraction within 8 kpc bar , and shows the baryonic and total observed circular velocities from ManceraPiña et al. (2020) as points with error bars. The ∼ 2 dex upwards departure from the mean SMHM relation (bottom left panel) produces more efficient feedback-driven expansion of the DM and stars, resulting in dynamical dominance of the baryonic mass within the inner 8 − 11 kpc. A galaxy with the mean halo mass (bottom right panel) fits the baryonic rotation velocity but overpredicts the total circ . Since the model does not predict gas properties, we assumed exponential gas density profiles with a scale-length set by the mean stellar scale-length of the UDG sample.
Figure 13
13Ogiya 2018; Carleton et al. 2019; Nusser 2020; Macciò et al. 2020; Jackson et al. 2021a).
Figure B1 .
B1Predictions for GC populations and galaxy structure assuming a factor of 5 lower coupling of the wind energy to the DM, DM = 0.02, compared to the fiducial value ( DM = 0.1). Feedback-driven expansion of the DM halo and the galaxy is reduced due to the lower amount of energy available relative to the fiducial model (seeFigs. 7, 8, and 9).
Figure B3 .
B3Global predictions for GC populations and galaxy structure assuming a steeper relation between halo concentration and offset from the mean SMHM relation (with = 1.0), relative to the fiducial model ( = 0.5). Compared to the fiducial model (seeFigs. 7, 8, and 9), there is a slight reduction in the feedback-induced expansion of the galaxy and DM halo.
S.Trujillo-Gomez et al.
Galaxies in high density environments could have their DM halo masses reduced through tidal stripping, artificially increasing the intrinsic scatter in the SMHM relation. However, while 18 of theForbes et al. (2018) late-types are not isolated, their high H content indicates a lack of significant stripping of their DM haloes. 3 Large scatter in * at fixed halo would bias the estimation of the mean halo ( * ) relation due to the steeply increasing number of galaxies at low masses that scatter upwards in stellar mass. However, in our analysis we do not estimate population statistics using observations, and therefore do not need to include this effect.MNRAS 000, 1-21 (2021) 4 S.Trujillo-Gomez et al.
MNRAS 000, 1-21(2021)
Here we neglect the potential from the gas component because it is dissipative and couples directly to the SN energy.MNRAS 000, 1-21(2021)
See Appendix A for a description of the mass models.MNRAS 000, 1-21(2021)
By extension, we also expect the true scatter on the relation between the number of GCs and the DM halo mass to be smaller than observed (see e.g. Forbes et al. 2018; Bastian et al. 2020).MNRAS 000, 1-21(2021)
This estimate has been challenged by Saifollahi et al. (2021) and Bogdán (2020). Saifollahi et al. (2021) estimate a much lower number, ≈ 20 GCs, in DF44.MNRAS 000, 1-21(2021)
ACKNOWLEDGEMENTSThe authors are grateful to the anonymous referee for a constructive review, and to Alexa Villaume, Aaron Romanowsky, and Ben Keller for insightful discussions and feedback. STG, JMDK, and MRC gratefully acknowledge funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme via the ERC Starting Grant MUSTANG (grant agreement number 714907). JMDK gratefully acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through an Emmy Noether Research Group (grant number KR4801/1-1) and the DFG Sachbeihilfe (grant number KR4801/2-1). MRC gratefully acknowledges the Canadian Institute for Theoretical Astrophysics (CITA) National Fellowship for partial support. This work made use of the software packages(Oliphant 2006),(Virtanen et al. 2019),(Hunter 2007)and(Diemer 2018).DATA AVAILABILITYThe data underlying this article are available within the article and in the references to published sources.APPENDIX A: OBSERVATIONAL DATA AND MASS MODELSThis section describes the observational data and corresponding mass models for the galaxies shown inFigure 1.Forbes et al. (2018)provide mass models of nearby Local Group and isolated dwarf galaxies. These include analysis of high quality H and H rotation curves of galaxies without GCs fromRead et al. (2016b), and a compilation of published rotation curves for galaxies with GCs from other sources in addition to velocity dispersion measurements at e for a few dSphs. For some dIrr galaxies without published rotation curves, the halo mass is obtained from a fit to the maximum circular
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|
[] |
[
"Reduction of Interaction Delays in Networks",
"Reduction of Interaction Delays in Networks"
] |
[
"Leonhard Lücken \nInstitute of Mathematics\nHumboldt-University of Berlin\nUnter den Linden 610099Berlin Germany\n",
"Jan Philipp Pade \nInstitute of Mathematics\nHumboldt-University of Berlin\nUnter den Linden 610099Berlin Germany\n",
"Serhiy Yanchuk \nInstitute of Mathematics\nHumboldt-University of Berlin\nUnter den Linden 610099Berlin Germany\n"
] |
[
"Institute of Mathematics\nHumboldt-University of Berlin\nUnter den Linden 610099Berlin Germany",
"Institute of Mathematics\nHumboldt-University of Berlin\nUnter den Linden 610099Berlin Germany",
"Institute of Mathematics\nHumboldt-University of Berlin\nUnter den Linden 610099Berlin Germany"
] |
[] |
Delayed interactions are a common property of coupled natural systems. For instance, signals in neural or laser networks propagate at finite speed giving rise to delayed connections. Such systems are often modeled by delay differential equations with discrete delays. We show that by a selective timeshift transformation it is often possible to reduce the number of different delays. We identify dynamic invariants of this transformation, determine its capabilities to reduce the number of delays and interpret these findings in terms of the topology of the underlying graph. In particular, we show that networks with identical generalized round-trip times along the fundamental semicycles are dynamically equivalent.
|
10.1209/0295-5075/103/10006
|
[
"https://arxiv.org/pdf/1206.1170v2.pdf"
] | 55,984,025 |
1206.1170
|
72f4eb21bc7ea42ee468d90bdd804ec1ccad4249
|
Reduction of Interaction Delays in Networks
6 Jun 2012
Leonhard Lücken
Institute of Mathematics
Humboldt-University of Berlin
Unter den Linden 610099Berlin Germany
Jan Philipp Pade
Institute of Mathematics
Humboldt-University of Berlin
Unter den Linden 610099Berlin Germany
Serhiy Yanchuk
Institute of Mathematics
Humboldt-University of Berlin
Unter den Linden 610099Berlin Germany
Reduction of Interaction Delays in Networks
6 Jun 2012
Delayed interactions are a common property of coupled natural systems. For instance, signals in neural or laser networks propagate at finite speed giving rise to delayed connections. Such systems are often modeled by delay differential equations with discrete delays. We show that by a selective timeshift transformation it is often possible to reduce the number of different delays. We identify dynamic invariants of this transformation, determine its capabilities to reduce the number of delays and interpret these findings in terms of the topology of the underlying graph. In particular, we show that networks with identical generalized round-trip times along the fundamental semicycles are dynamically equivalent.
Time delays play an important role in networks of coupled dynamical systems. For instance, in neuronal networks [1][2][3][4][5] delays occur due to the finite propagation times along the axons or to reaction times at chemical synapses. Similarly, the propagation of light between interacting lasers [6][7][8] may cause significant time delays. Depending on the physiological properties of axons and synapses, the delays of signals between different cells in the network differ and in practice there are as many different delays in the network as there are links between single neurons. Such a diversity of delays was shown to enable the design of robust pattern generating systems [9,10] and to play an important role in neural processing of temporal information [11]. Hence, to study such natural systems, one would need to study models involving numerous different delays. Independently of the physical origin, dynamics in networks of interacting systems can often be described by the equationṡ
x j (t) = f j x j (t), (x k (t − τ jk )) k∈P j ,(1)
where x j (t), j = 1, . . . , N, denotes the dynamics of a node j and P j is the set of all other nodes of the network connected to the node j. For instance, x j may be considered as a dynamical variable describing the dynamics of one neuron in the network. Taking into account finite time delays τ jk , which occur due to the signal propagation time from a node k to the node j, the coupling terms contain all variables x k which are connected to x j with corresponding time delays τ jk . Here we consider directed networks, thus one may generally have τ jk = τ kj or even k ∈ P j while j / ∈ P k . We assume that the network is connected and hence, the number L of links is larger than N − 1.
For a network of N systems, up to N 2 different time delays may occur in (1). By allowing several connections from one system to another, the number of delays can increase even more.
This creates immense challenges for the analysis of the system, since every single delay may alter the properties and dynamics significantly [12][13][14][15][16]. The interaction of several delays is even more complicated [8-10, 17, 18] and not fully understood so far.
In this Letter we show how the number of different delays can be reduced. It appears that any connected network possesses a characteristic number of delays which is essential for describing the dynamics. This number of essential delays equals the cycle space dimension C = L − (N − 1) of the underlying graph and is usually smaller than the number of distinct τ jk . We show that the essential delays correspond to generalized round-trip times along fundamental semicycles in the network. Networks which have the same local dynamics and the same set of essential delays can be considered as equivalent from the dynamical point of view.
As a simple illustrative example, let us consider a ring of unidirectionally coupled systemṡ
x j (t) = f (x j (t) , x j+1 (t − τ j ))(2)
with inhomogeneous coupling delays τ j . It is known [19,20]
y j (t) = x j (t + η j ) , 1 ≤ j ≤ N.(3)
Indeed, the transformed variables y j (t) fulfill
y j (t) = f (y j (t) , y j+1 (t − τ j + η j − η j+1 )) ,
and if the shifts η j are chosen appropriately, all hitherto distinct delays become the same while their sum along the ring, the round-trip time Nτ , is preserved. This homogenization of delays was applied to simplify the analysis of the solutions in systems (2) with inhomogeneous delays [20,21]. Conversely, the same transformation was applied to design pattern generators [9,10] starting from the system with homogeneous delays. Our aim is to use the timeshift transformation (3) to reduce the number of different delays in networks with arbitrary topology.
Applying (3) to (1), we obtain the new systeṁ
y j (t) = f j y j (t) , (y k (t −τ jk )) k∈P j(4)
withτ jk = τ jk + η k − η j . We proceed by exploring possible delay reductions in basic network motifs, which consist of exactly one semicycle (a cycle in the underlying undirected graph) and are depicted in Fig. 1.
Motif I. Dynamics on network I is described by the system of equationṡ
x 1 (t) = f 1 (x 1 (t)) , x 2 (t) = f 2 (x 2 (t) , x 1 (t − τ 1 ) , x 1 (t − τ 2 )) .
The transformation (3) leads to the same system for y j but with different delaysτ j = we obtainτ 1 = 0 andτ 2 = τ 2 − τ 1 > 0. Thus, the transformed system contains only one delayed connection -the other one becomes instantaneous. In case τ 1 = τ 2 we obtaiñ τ 1 =τ 2 = 0, which means that both delays can be eliminated. Thus, the dynamics in motif I for τ 1 = τ 2 can be described by a system with just one delay, and for τ 1 = τ 2 the system reduces to an ordinary differential equation. Note that the quantityτ 2 −τ 1 = τ 2 − τ 1 remains invariant.
τ j + η 1 −η 2 , j = 1, 2. Without loss of generality we assume τ 1 < τ 2 . By choosing η 2 −η 1 = τ 1 , τ 1 τ 2 (I) (II) (III) τ 1 τ 3 τ 2 τ 4 1 2 τ 1 τ 3 τ
Motif II. After applying transformation (3) to motif II, the delays change toτ 1 = τ 1 +
η 1 − η 2 ,τ 2 = τ 2 + η 1 − η 3 ,τ 3 = τ 3 + η 2 − η 3 .
It is easy to check that by appropriate choice of η j one obtains a system of equations with a single delay. For instance, it can be done by requiring that all new delays are identicalτ 1 =τ 2 =τ 3 ≡ τ . In this case the delay time is τ = τ 1 − τ 2 + τ 3 . The corresponding time shifts can be chosen as η 1 = µ, η 2 = µ + τ 2 − τ 3 , η 3 = µ + 2τ 2 − τ 1 − τ 3 with an arbitrary constant µ. Simple calculations give five different reductions to a single delay as shown in Table I. Note that some of the reductions lead to negative delays in the resulting network, which should be avoided since the dynamics in such systems are unstable [22]. For example, in the case τ = τ 1 − τ 2 + τ 3 < 0 there is only one suitable reduction (τ 1 =τ 3 = 0 andτ 2 = −τ ). Summarizing, the motif II always admits a reduction to one delay T = |τ 1 − τ 2 + τ 3 | = |τ 1 −τ 2 +τ 3 |. This invariant quantity can be considered as a generalized round-trip time, taking into account the directionality of the connections. In the case τ = 0 the delays can be eliminated.
Motif III. Similarly, for the motif III [ Fig. 1-III], one can reduce the four different delays τ 1 ,τ 2 , τ 3 , and τ 4 to a single delay τ . Formally, ten different combinations are possible, which are listed in Table I(c). As in the case of motif I a reduction to uniform delays at each link is impossible for this case. But, as in the other examples, a reduction to a system with only one delayed connection can be achieved. Further, the generalized round-trip T = |τ 1 − τ 2 − τ 3 + τ 4 | is preserved for any reduction. When this round-trip is zero, all
(a): motif Ĩ τ 1 0 τ τ 2 −τ 0 (b): motif IĨ τ 1 0 0 τ τ /2 τ τ 2 0 −τ 0 0 τ τ 3 τ 0 0 τ /2 τ (c): motif IIĨ τ 1 0 0 0 0 0 τ τ /2 τ τ −τ τ 2 0 0 −τ −τ /2 −τ 0 0 0 τ −τ τ 3 0 −τ 0 −τ /2 −τ 0 0 τ 0 −τ τ 4 τ 0 0 0 −τ 0 τ /2 τ τ 0τ = τ 1 − τ 2 (a), τ = τ 1 − τ 2 + τ 3 (b), τ = τ 1 − τ 2 − τ 3 + τ 4 (c)
. In each case, the generalized round-trip is given by T = |τ |.
delays can be eliminated. Γ j τ (ℓ j ) .
Here τ (ℓ j ) is the time delay along the link ℓ j and Γ j is either +1 or −1 depending on the direction of the link ℓ j with respect to an arbitrary but fixed orientation of the semicycle.
Note that the orientation can be chosen in two different ways, which corresponds to opposite signs of Γ j . The obtained value for T (c) is independent of the choice of orientation.
It is possible to prove that there is a choice of timeshifts η j and a set S = {ℓ 1 , ..., ℓ N −1 } of N − 1 links, such that the delaysτ jk = τ jk − η j + η k in the transformed equation ( equals the round-trip delay of this fundamental semicycle [ Fig. 2(a)]. To prove this claim one can inductively construct timeshifts and the corresponding spanning tree, which contains instantaneous links. Details will be presented elsewhere [23]. Consequently, the number of delays in the transformed system can be reduced to at most
C = L − (N − 1) .(6)
The number C has a well known meaning for network graphs [24]: It is the cycle space dimension which is defined as the maximal number of independent cycles. Here, a set of cycles is called independent if neither of the cycles is obtained as a sequence of the other ones in the set. For instance, in the network given in Fig. 2(a) the cycle space dimension is C = 5 and in the one depicted in (b) it is C = 3.
We call C the essential number of delays since it is the minimal number of different delays to which the number of delays in a network can be reduced generically. Here, generically means that the conditions which allow for further reduction of delays form a nullset in the parameter space of delays R L ≥0 = {τ jk | τ jk ≥ 0} of the original system (1) [23]. Dynamical invariants. There is a natural correspondence between solutions of the original system (1) and the transformed system (4): Any solution x (t) = (x j (t)) 1≤j≤N of (1) can be identified with its transformation, y (t) = (x j (t + η j )) 1≤j≤N , which is a solution of (4), see Fig. 3. As a result, many important properties of solutions like stability, frequency of oscillation, amplitude, etc., remain the same within a class of equivalent systems. They depend only on a set of generalized round-trip time delays, but not on the specific delays τ jk . As a consequence it is possible (and favorable) to perform bifurcation analysis in the reduced parameter space. Practically, one considers round-trips on a set of fundamental semicycles instead of single delays as parameters of the system.
The following example is intended to provide an idea of possible applications and merits of the results presented in this article. Let us consider the network of Mackey-Glass systems [7,25,26]ẋ
j (t) = −γx j (t) + β c x k (t − τ jk ) 1 + (c x k (t − τ jk )) 10 ,(7)
where the summation is performed over all coupled nodes k ∈ P j . The coupling topology is shown in Fig. 2(b). The parameters are fixed to γ = 0.1, c = 0.525 and β = 0.2. Although the total number of delays is 8, one can parametrize all of them by the three round-trips T (c j ), j = 1, 2, 3, on the fundamental semicycles [ Fig. 2(b)]. For further investigation we fix T (c 3 ) = 7 and T (c 2 ) = 10 wherever only T (c 1 ) is varied.
Firstly, the delays in the network were set up randomly in such a way that the round-trips were preserved with T (c 1 ) = 5. For each realization of delays, the system was integrated numerically with constant initial conditions [see caption to Fig. 4]. The empirical distribution of the individual delays and the observed attractors is shown in Fig. 4(b). One can observe two attractors, one equilibrium and one periodic solution, which exist independently of the choice of single delays. They can be identified by their componentwise supremum norm |x| = N j=1 sup t |x j (t)| 2 which is invariant under (3). Starting from those solutions, we obtain the bifurcation diagram with respect to T (c 1 ) [ Fig. 4(a)]. For T (c 1 ) = 0, two stable equilibria are the only attractors for the system. The lower equilibrium looses stability in a Hopf bifurcation (H) and the emergent periodic solution undergoes a period doubling bifurcation (P1) which connects to a reverse period doubling at (P2). Snaking of the main branch accompanied by period doubling bridges and divergent periods indicates the presence of a homoclinic saddle-focus [27]. A step further can be done by a two-parametric bifurcation diagram [ Fig. 4(c)]. There, the Hopf bifurcation (H) was followed numerically in the region 0 ≤ T (c 1 ) ≤ 500 and 0 ≤ T (c 2 ) ≤ 500. Allthough the same diagram might have been obtained accidentally by studying variations of the delays τ 5 and τ 7 , its proper meaning as a diagram of codimension one is first made accessible by the presented results.
Discussion. We have presented a universal method for the reduction of discrete interaction-delays in networks of dynamical systems. It turned out that the cycle space dimension of the network equals the number of essential delays which determine the dynamics. The networks whose generalized round-trip times along the fundamental semicycles coincide, form a class of dynamically equivalent systems. This is an important step towards the systematization and classification of dynamics on networks with many different delays.
Note that all results which are presented in this paper can be extended straightforwardly to networks with multiple links from one cell to another (see, for instance, motif I in Fig. 1) and to other types of local dynamics like time-discrete systems or evolution equations.
We thank K. Knauer and M. Zaks for useful discussions and the DFG for financial support in the framework of the Collaborative Research Center SFB910.
Figure 1 :
1Simple motifs consisting of one semicycle.
General networks. The cases considered above suggest that the generalized round-trip time along a semicycle in the network remains invariant under the time shift transformation (3). Indeed, this holds for arbitrary networks. Let us first define a semicycle c = (ℓ 1 , ..., ℓ k ) of a directed graph as a closed undirected path, which is represented as an ordered set of its constituting links. Then we define the generalized round-trip time T (c) along a given semicycle c = (ℓ 1 , ..., ℓ k ) as follows T (c) := k j=1
Figure 2 :
2all links contained in S and non-negative on all other links. In fact, the set S is a spanning tree: a set of N − 1 links which does not contain any semicycle. The addition of any link ℓ / ∈ S results in a set S ′ = S ∪ {ℓ} which contains exactly one semicycle. This is the fundamental semicycle corresponding to ℓ. The transformed delayτ (ℓ) on the link ℓ (a) A network with cycle space dimension C = 5. A spanning tree S [solid links] and one of its fundamental cycles, c (ℓ) [blue dotted path], are displayed. The ±-signs indicate the value of the corresponding coefficients Γ j in the the round-trip T (c (ℓ)), see (5). (b) A network of N = 6 cells with cycle space dimension C = 3. We depict a spanning tree [solid links] and three corresponding fundamental cycles which are indicated by differently dashed paths. The numbers refer to an indexing of the links which is used in the text. According to our results, any admissible dynamics in this network involves at most three different interaction delays.
Figure 3 :
3Equivalent stable solutions of unidirectional rings of four Mackey-Glass oscillators (7) with distinct single delays but equal round-trip. In each row, the time evolution of the j-th cell is indicated by the coloring. In (a), the delays are homogeneous, τ j,j+1 = 25. Chart (b) shows the corresponding solution of the transformed system with delaysτ j,j+1 = τ j,j+1 − η j + η j+1 and η = (−5.8, 0, −5.6, −18.4). Other parameters: β = 0.2, γ = 0.1, c = 4.
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Figure 4 :
4Dynamical features of six-node network in Fig. 2(b) of Mackey-Glass systems (7). (a): Bifurcation diagram for T (c 1 ) ∈ [0, 23] and T (c 2 ) = 10. Black lines correspond to stable solutions and gray to unstable. Solid branches are equilibria, dashed are periodic solutions. Bifurcations are denoted by 'H' (Hopf), 'P[n]' (period doubling), and 'F[n]' (fold). The ordinate measures the norm |x| and the inset shows the period along the main branch. Red crosses indicate the observed solutions from chart (b). (b): 50 random realizations of single delays which keep constant T (c 1 ) = 5, T (c 2 ) = 10, and T (c 3 ) = 7. Each dot corresponds to a realization of τ j . Mean and standard deviation are indicated by squares and bars. For each choice of delays, constant initial functions were drawn from a uniform distribution on [0, 1.5]. The inset bars indicate the number of observed solutions x (t), t ∈ [10 5 , 1.1 × 10 5 ], with the norm |x| in their base interval. Chart (c) shows the trace and reappearances of the Hopf bifurcations in the T (c 1 )-T (c 2 )-plane.
Table I :
IPossible reductions to a single delay in motifs I, II, and III [see Fig. 1]. The values of the delay τ are as follows:
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|
[] |
[
"Dynamical analysis of nonminimal coupled theories",
"Dynamical analysis of nonminimal coupled theories"
] |
[
"Rafael Ribeiro \nDepartamento de Física\nDepartamento de Física e Astronomia and Centro de Física do Porto\nFaculdade de Ciências da Universidade do Porto\nInstituto Superior Técnico\nUniversidade de Lisboa\nAv. Rovisco Pais 1, Rua do Campo Alegre 6871049-001, 4169-007Lisboa, PortoPortugal, Portugal\n",
"Jorge Páramos \nDepartamento de Física\nDepartamento de Física e Astronomia and Centro de Física do Porto\nFaculdade de Ciências da Universidade do Porto\nInstituto Superior Técnico\nUniversidade de Lisboa\nAv. Rovisco Pais 1, Rua do Campo Alegre 6871049-001, 4169-007Lisboa, PortoPortugal, Portugal\n"
] |
[
"Departamento de Física\nDepartamento de Física e Astronomia and Centro de Física do Porto\nFaculdade de Ciências da Universidade do Porto\nInstituto Superior Técnico\nUniversidade de Lisboa\nAv. Rovisco Pais 1, Rua do Campo Alegre 6871049-001, 4169-007Lisboa, PortoPortugal, Portugal",
"Departamento de Física\nDepartamento de Física e Astronomia and Centro de Física do Porto\nFaculdade de Ciências da Universidade do Porto\nInstituto Superior Técnico\nUniversidade de Lisboa\nAv. Rovisco Pais 1, Rua do Campo Alegre 6871049-001, 4169-007Lisboa, PortoPortugal, Portugal"
] |
[] |
In this work a dynamical system approach to nonminimal coupled f (R) theories is made. The solutions of three distinct models are obtained and their stability and physical interpretation are studied to ascertain their viability as candidates for dark energy. Comparison is drawn with previous works in the context of f (R) and nonminimally coupled models.
|
10.1103/physrevd.90.124065
|
[
"https://arxiv.org/pdf/1409.3046v2.pdf"
] | 119,239,177 |
1409.3046
|
01150dd692c6361a7d05182fcd72d7c8e32f9329
|
Dynamical analysis of nonminimal coupled theories
(Dated: September 11, 2014)
Rafael Ribeiro
Departamento de Física
Departamento de Física e Astronomia and Centro de Física do Porto
Faculdade de Ciências da Universidade do Porto
Instituto Superior Técnico
Universidade de Lisboa
Av. Rovisco Pais 1, Rua do Campo Alegre 6871049-001, 4169-007Lisboa, PortoPortugal, Portugal
Jorge Páramos
Departamento de Física
Departamento de Física e Astronomia and Centro de Física do Porto
Faculdade de Ciências da Universidade do Porto
Instituto Superior Técnico
Universidade de Lisboa
Av. Rovisco Pais 1, Rua do Campo Alegre 6871049-001, 4169-007Lisboa, PortoPortugal, Portugal
Dynamical analysis of nonminimal coupled theories
(Dated: September 11, 2014)PACS numbers: 04.20.Fy, 04.50.Kd, 98.80.Jk
In this work a dynamical system approach to nonminimal coupled f (R) theories is made. The solutions of three distinct models are obtained and their stability and physical interpretation are studied to ascertain their viability as candidates for dark energy. Comparison is drawn with previous works in the context of f (R) and nonminimally coupled models.
I. INTRODUCTION
Although its experimental success [1,2], it is known that General Relativity (GR) does not exhibit the most general form to couple matter with curvature. In fact, these can be coupled in a nonminimal way [3] (for early proposals see Ref. [4][5][6][7]) that has already been shown to be able to mimic dark matter [8,9], dark energy [10,11] and explain post-inflationary preheating [12,13] and cosmological structure formation [14].
This nonminimal coupling (NMC) can give rise to several implications, from Solar System [15] and stellar dynamics [16][17][18][19] to close like-time curves [20], wormholes [21] and modifications to the well-known energy conditions [22] (see Ref. [23] for a thorough review).
From a fundamental standpoint, a NMC can arise from one-loop vacuum-polarization effects in the formulation of Quantum Electrodynamics in a curved spacetime [24], as well as in the context of matter scalar fields [25,26]. In the framework of Riemann-Cartan geometry, a NMC was considered in an earlier proposal [27] and another study showed that it clearly affects the features of the ground state [28]. Phenomenologically, it can be viewed as a natural continuation of so-called f (R) theories [29], where the standard Einstein-Hilbert action is replaced by a non-linear function f (R) of the scalar curvaturean extension of General Relativity that has garnered a strong interest in the past decade.
The purpose of this work is to make a dynamical system approach on NMC theories in a cosmological setting and derive the solutions for some models; for a very recent and similar study, albeit less general, see Ref. [30]. This work is organized as follows: the nonminimal gravitational model is discussed in section II; the formulation of the equivalent dynamical system is presented in section III; a confirmation of the dynamical system obtained in f (R) theories is shown in section IV; the discussions of the results obtained for two pure NMC models and for a power law correction model are presented in sections V and VI, respectively. Finally, the conclusions are presented in section VII.
II. THE MODEL
Following the generalization of the Einstein-Hilbert action put forward in f (R) theories [29], a NMC model is embodied in the action [3],
S = d 4 x √ −g [κf 1 (R) + f 2 (R)L] , κ = c 4 16πG ,(1)
where f i (R) are arbitrary functions of the scalar curvature R, g is the metric determinant and L is the matter Lagrangian density. The standard Einstein-Hilbert action is obtained by taking f 1 (R) = R−2Λ and f 2 (R) = 1. The field equations are obtained by imposing a null variation of the action with respect to the metric,
F G µν = 1 2 f 2 T µν + µν F + 1 2 g µν κf 1 − 1 2 g µν RF,(2)
where F = κf 1 + f 2 L, the prime denotes derivation with respect to the scalar curvature (omitted), µν ≡ µ ν −g µν , and the matter energy-momentum tensor is defined as
T µν = − 2 √ −g δ ( √ −gL) δg µν .(3)
The generalized Bianchi identities imply the noncovariant conservation law Since there is an equivalence between this model and a two-scalar field model, this non-conservation may be interpreted as an energy exchange between matter and those scalar fields [26].
µ T µν = f 2 f 2 (g µν L − T µν ) µ R.(4
To study the recent accelerated expansion of our universe, a flat universe is considered with the line element
ds 2 = −dt 2 + a 2 (t)dV 2 ,(5)
where a(t) is the scale factor and dV is the volume element with comoving coordinates, and matter is assumed to behave as a perfect fluid, with an energy-momentum tensor
T µν = (ρ + P ) u µ u ν + P g µν ,(6)
derived from the Lagrangean density L = −ρ (see Ref.
[31] for a discussion), where ρ and P are the energy density and pressure of the perfect fluid, respectively, and u µ is its four-velocity.
One can see that the energy-momentum tensor is again conserved, just like in GR or f (R) theories, since Eq. (4) yields the continuity equatioṅ
ρ + 3H(1 + w)ρ = 0,(7)
where H =ȧ/a is the Hubble parameter and w = P/ρ is the equation of state (EOS) parameter. Inserting the metric in the field equations (2), one obtains the modified field equations
H 2 = 1 3F 1 2 f 2 ρ − 3HF Ṙ − 9H 2 (1 + w)f 2 ρ − (8) − 1 2 κf 1 + RF 2 , 2Ḣ + 3H 2 = − 1 2F f 2 wρ + 2F + κf 1 − RF + 4HḞ .(9)
III. DYNAMICAL SYSTEM
One way to obtain the solutions of the field equations is via the study of the ensuing dynamical system, in which the field equations are replaced by a equivalent dynamical system of dimensionless variables
x = − F Ṙ F H , y = R 6H 2 , z = − κf 1 6F H 2 ,(10)Ω 1 = f 2 ρ 6F H 2 , Ω 2 = − 3(1 + w)f 2 ρ F
with F ≡ κf 1 − f 2 ρ (note that the latter is the partial derivative of F with respect to the scalar curvature R). The modified Friedmann equation (8) becomes
1 = x + y + z + Ω 1 + Ω 2 ,(11)
acting as a restriction to the phase space. Since ρ is indirectly dependent on R, one finds that for a constant F , (as studied in Ref. [32]),
− d log F dN = − F Ṙ F H + f 2ρ F H = x + Ω 2 = 0.(12)
where N = ln a is the number of e-folds. The introduction of this coupling increases the number of variables of the problem -for f (R) theories, only four variables were required [33].
Calculating the variation of this variables with respect to N , one obtains the following autonomous system, equivalent to the field equations (8),
dx dN = x x − y + Ω 2 1 + α 2 α − 1 − y − 3z + 3wΩ 1 + Ω 2 [3 (1 + w) − y] dy dN = y [−x/α + 2 (2 − y)] dz dN = z x 1 − α 1 α + Ω 2 + 2 (2 − y) dΩ 1 dN = Ω 2 xy 3α (1 + w) + Ω 1 [1 − 3w + x + Ω 2 − 2y] dΩ 2 dN = Ω 2 x 1 − α 2 α − 3 (1 + w) + Ω 2 (13) with the dimensionless parameters, α = F R F , α 1 = f 1 R f 1 , α 2 = f 2 R f 2 .(14)
These will depend on the choice of the functions f 1 (R) and f 2 (R) and must be computed as a function of the variables for each particular model (they are analogous to the Υ parameter defined in Ref. [33]). It is possible to show that
α 1 = y z Ω 2 3(1 + w) − 1 ,(15)
so that, as expected from Eq. (11),
0 = dx dN + dy dN + dz dN + dΩ 1 dN + dΩ 2 dN .(16)
Other useful relations are
α = f 1 R f 1 1 − Ω 2 3(1 + w) + α 2 Ω 2 3(1 + w) ,(17)
and
f 2 R = − Ω 2 y 3(1 + w)Ω 1 f 2 .(18)
The determination of the fixed points of any dynamical system analysis depends crucially on the choice of the variables. The number of dynamical variables for the pure NMC case is the same as in Ref. [30]. Since in Ref. [30], f 1 (R) = R and f 2 (R) remains unspecified, the fixed points appear as a function of y (in the present notation). Different values of y correspond to distinct cosmological eras, since this parameter is related to the decelerated parameter q = 1 − y (defined in the following section) and to w ef f = (2q − 1)/3. Furthermore, it also assumed w = 0, thus limiting its scope to a Universe filled with pressureless dust.
A. Physical Quantities
With the adopted metric (5), the Ricci scalar reads R = 6(2H 2 +Ḣ).
One important parameter used in cosmology is the deceleration parameter
q ≡ −ä ȧ a 2 = 1 − y,(20)
so that the scalar curvature may be written as
R = 6H 2 (1 − q).(21)
Since our universe appears to be expanding at an accelerated rate, one is searching for a model with q < 0 → y > 1. In GR this parameter yields
q = 1 2 (1 + 3w).(22)
which would require an exotic fluid with negative pressure, w < −1/3. After determining the fixed points of the dynamical system for each particular choice of functions f 1 (R) and f 2 (R), one may straightforwardly determine the scale factor for each fixed point. From Eq. (19), one can see thaṫ H = (y − 2)H 2 , and it is possible to determine the scale factor with this relation. Note that this quantity also depends on y, with the distinct evolutions,
a(t) = t t0 1 2−y , y = 2 e H0t , y = 2(23)
For the first case, the scale factor evolves as a power of time, while in the second result the Hubble parameter will be constant and thus the scale factor will rise exponentially, i.e. a De Sitter phase.
Other important physical quantity is the energy density: one can determine its evolution for each fixed point from the continuity equation (7). The general solution for this equation is the familiar result
ρ(t) = ρ 0 a(t) −3(1+w) .(24)
Considering the definition of the variable Ω 2 from equation (10), one can see that
ρ = κf 1 Ω 2 f 2 [Ω 2 − 3(1 + w)] ,(25)
so, for a particular fixed point, it should be possible to determine the energy density from this relation. Note that for Ω 2 = 3(1 + w) there appears to be a divergence in the density: physically, a fixed point with this value of Ω 2 will correspond to a regime where f 2 ρ κf 1 .
IV. f (R) THEORIES
Let us now consider the case of f (R) theories, in order to confirm the results obtained in Ref. [33]. In this case,
f 1 (R) = f (R), f 2 (R) = 1 ⇒ F = κf , α 1 = −y/z,(26)
the variable Ω 2 vanishes trivially, α will only depend on the derivatives of our arbitrary function f (R) and α 2 is not well determined, but does not appear in the equations. The dynamical system can be simplified to
dx dN = x(x − y) − y − 3z + 3wΩ 1 − 1 dy dN = y 4 − x α − 2y dz dN = z[x + 4 − 2y] + xy α dΩ 1 dN = Ω 1 (1 − 3w + x − 2y)(27)
which is equivalent to the system presented in Ref. [33], as expected (for an extensive discussion of a dynamical system approach on f (R) theories see also Ref. [34]).
V. PURE NONMINIMAL COUPLING CASE
To study the influence of the NMC in cosmology, a simple case where f 1 (R) = R and f 2 (R) = f (R) is considered. One can see that
F = κ − f ρ, F = −f ρ(28)
and, from relation (15), Ω 2 can be written as
Ω 2 = 3(1 + w) 1 + z y .(29)
Also, equation (17) implies that α = α 2 (1 + z/y), and thus the dynamical system can be written as
dx dN = 3 (1 + w) [3 (1 + w) + x] z y − y (4 + 3w + x) + +8 + 9w 2 + 6x + x 2 − 6z + 3w (6 + 2x − z + Ω 1 ) dy dN = y 4 − xy (y + z) α 2 − 2y dz dN = z 4 + x − 2y + 3 (1 + w) (y + z) y − xy (y + z) α 2 dΩ 1 dN = xy α 2 + Ω 1 4 + x − 2y + 3(1 + w)z y(30)
restricted by the modified Friedmann equation
1 = x + y + z + Ω 1 + 3(1 + w) 1 + z y .(31)
A. Power law Nonminimal Coupling
Let us consider a simple function
f 2 (R) = C + R 12M 2 n ,(32)
already studied in Ref. [10] with C = 1. The parameters C and M are both constant and the latter is related to the energy scale of the theory. For this model, α 2 = n−1, independently of C. For C = 0, the exponent n should be close to zero so as to introduce a small deviation from f 2 (R) = 1; conversely, for C = 1, n may take any value. The fixed points obtained for both cases are the same, but the evolution of the physical quantities will differ.
The fixed points of this system are obtained imposing a null variation of the dynamic variables. Their values associated with the solutions are shown in Table I.
Comparing these fixed points with the solutions obtained in the article Ref. [10], one can verify that the second fixed point (with w = 0) corresponds to the f 2 ρ κ regime; the third fixed point has some similar features to the f 2 ρ κ regime, although there is not an exact equivalence, as discussed in the corresponding paragraph. Also there appears to be an extra solution not mentioned in Ref. [10], corresponding to a De Sitter phase of exponential expansion of the Universe.
To have a point that can replicate the effects of dark energy, one requires it to be stable with q < 0. Also, since the model is a power law of R, one requires that n < 0, so that it dominates only for late times, when the scalar curvature sufficiently small. In the regime f 2 ρ κ, one can see that
(x, y, z, Ω 1 ,Ω 2 ) a(t) ρ(t) q 1 (0, 2, 0, −4 − 3w, 3(1 + w)) e H 0 t ρ → ∞ −1 2 4 − 2n(4 + 3w) 2n − 1 , n(−2 + 4n + 3w) 1 − 3n + 2n 2 , 0, 2 − 4n − 3w 1 − 3n + 2n 2 , 3(1 + w) t t 0 1−3n+2n 2 2−n(4+3w) , n = 2 4 + 3w ρ → ∞ −1 + 2 − n(4 + 3w) 1 − 3n + 2n 2 3 6n(1 + w) 1 − 4n − 3w , − 1 − 4n − 3w 2(n − 1) , 1 − 2n − 3w 2(n − 1) , 1 1 − n , − 6n(1 + w) 1 − 4n − 3w t t 0 2(1−n) 3(1+w) t t 0 2n−2 −1 + 3(1 + w) 2(1 − n)Ω 1 ∼ − f 2 6f 2 H 2 = − y n 1 + C 12M 2 R n ,(33)
This relation will be used to further explore the physical significance of relevant fixed points.
First fixed point
This point corresponds to a De Sitter solution in the regime f 2 ρ κ, since Ω 2 = 3(1 + w) and considering Eq. (25). For C = 0, the above yields y = −nΩ 1 , which leads to the restriction n = 2/(4 + 3w). For C = 1, the relation (33) yields
H 0 = M n 2 + 3w 2 − 1 −1/(2n) ,(34)
for n > 2/(4 + 3w). The stability of the point is shown in Fig. 1.
Notice that the NMC exponent is positive, n > 0, so that its effect should be dominant at early times, when the curvature is high, R M 2 . Furthermore, Fig. 1 shows that the fixed point is never an atractor for any pair (w, n), but unstable or a saddle point. Thus, it is not a viable candidate for dark energy, but could have some bearing on inflation.
Second fixed point
Since Ω 2 = 3(1+w), this point is in the regime f 2 ρ κ again due to Eq. (25). If n = 2/(4 + 3w), it is equal to the first fixed point, so a De Sitter phase is attained. The stability of the point is shown in Fig. 2.
Notice that one has y = −Ω 1 /n: from Eq. (33), this is only physical when C = 0 or, if C = 1, when (R/12M 2 ) n 1 -so that the NMC should be dominant in the latter case. This is in accordance with the corresponding regime f 2 ρ κ studied in Ref. [10], and confirmed by the value for the deceleration parameter when w = 0, q = (1 + n)/(1 − n), and requires a negative value for the exponent n for the NMC to dominate at late times.
In GR, for the era of matter dominance, w = 0 → q = 1/2 and for the radiation era, w = 1/3 → q = 1. In table II it is shown that it is possible to have quite different values of q with respect to the latter: for example, it is possible to have an evolution characteristic of the radiation era in GR, i.e. q = 1, even when w = 0.
Third fixed point
Notice that this point has x = −Ω 2 so, from Eq. (12), F is constant and its value can be determined
z = − κR 6F H 2 ⇒ F = − κy z = κ 1 − 4n − 3w 1 − 2n − 3w .(35)
For this case, one has
ρ(t) = 24κM 2 1 − 2n − 3w 3 2 (1 + w) 2 M 2 (1 − n)(1 − 4n − 3w) t 2 n−1 ,(36)
independent of C. For w = 0, this reads
ρ 0 = 8 3 3 4 n (1−5n+4n 2 ) −n (1 − n)(1 − 4n) (1 − 2n) t 0 t 2 2n κ t 2 0 ,(37)
with t 2 ≡ 1/(2 √ 3M ). One can see that the result obtained here is different from the one attained in the f 2 ρ ρ regime studied in Ref. [10]: in the latter, f 2 ρ = 0 was effectively assumed, and thus F = κ. Conversely, Eq. (35) with w = 0 (a universe filled with pressure-less dust) reads
F = κ 1 − 4n 1 − 2n = κ → f 2 ρ κ = 2n 1 − 2n .(38)
and indeed one finds that f 2 ρ/κ can be of the order unity or larger. Nevertheless, both the third fixed point here obtained (for w = 0) and the regime f 2 ρ κ studied in Ref. [10] predict the same evolution for the scale factor, typified by a deceleration factor q = −1 + 3/[2(1 − n)].
Also, one can see that
Ω 1 = 1 1 − n 1 + C 12M 2 R n → C 12M 2 R n = 0,(39)
and a consistent solution is obtained when C = 0, or alternatively if C = 1 and (R/12M 2 ) n 1: the latter implies that the NMC must be dominant, again requiring a negative value for the exponent n in order to replicate late time dark energy.
The stability of the point is shown in Fig. 3. The third point is not a viable candidate for dark energy, since the stable region corresponds to q > 0. In table II the values of n are shown for typical values of q and w. As expected, when n = 0, these coincides exactly with the results of GR. Again, this fixed point allows one type of matter to mimic another (e.g. NMC dust leads to a behaviour typical of radiation in GR), as depicted on Table II.
B. Exponential Nonminimal Coupling
The study of an exponential model,
f 2 (R) = exp R R 0 ,(40)
might be of interest because when the scalar curvature tends to zero, the NMC vanishes asymptotically. One can see that the effects of an exponential function in f (R) theories, Ref. [35], is richer than in NMC theories. For this model, one can determine α 2 using equation (18), since for this particular case α 2 = f 2 R/f 2 = f 2 R/f 2 . Also, the relation α = α 2 (1 + z/y) is still valid, and thus the dynamical system is well determined. The fixed points obtained are shown in Table III.
Coordinates (x, y, z, Ω 1 ,Ω 2 )
1 − 3 2 (1 + w), 2, −1, 0, 3 2 (1 + w) 2 (0, 2, 0, −4 − 3w, 3(1 + w)) 3 (−4 − 3w, 2, 0, 0, 3(1 + w))
First fixed point
This is a saddle point with no physical meaning. First, it presents the unusual case where y = 2, so that the scalar curvature is constant, but x = 0. One can see that x = −Ω 2 , which implies that F is constant, from Eq. (12). From Eq. (25), one can see that ρ = −κ/f 2 . Also, considering the definitions presented in Eq. (10), in order to have x = 0 with a constant curvature, one needs H = 0 or F = 0. Note that,
F = κ − f 2 ρ = 0 ⇒ ρ = κ/f 2 ,(41)
which disagrees with the previous result unless R 0 → ∞ and GR is recovered. For H = 0,
Ω 1 = f 2 ρ 6F H 2 → ∞ = 0,(42)
unless ρ = 0, but it will also disagree with the previous result -thus, proving the inconsistency of this point.
Second fixed point
This is a saddle point in the regime f 2 ρ κ, since Ω 2 = 3(1 + w), with
H 2 0 = R 0 6(4 + 3w)
.
This is the only consistent fixed point for this model. For y = 2, R = 12H 2 0 is constant, implying x = 0. Also,
z ∼ κR 6f 2 ρH 2 0 ∼ κ f 2 ρ → 0.(44)
Third fixed point
The third point is a stable point in the regime f 2 ρ κ, since Ω 2 = 3(1 + w). This appears to be another case where y = 2, R constant with x = 0 and it is inconsistent. The definition of x from Eq. (10) and f 2 ρ κ implies that
x ∼ −Ṙ R 0 H .(45)
SinceṘ = 0, to obtain x = 0, it is necessary that H = 0, to induce an indetermination. When considering the definition of Ω 1 ,
Ω 1 ∼ − R 0 6H 2 → ∞ = 0,(46)
which makes this an inconsistent point. Note that this was the only point of this model with a stable region: although all the fixed points correspond to a De Sitter phase, none of them can be used to describe dark energy.
VI. POWER LAW NMC AND CURVATURE TERM
Let us now consider the model
f 1 (R) = R + 12M 2 1 R 12M 2 1 n1 ,
(47)
f 2 (R) = 1 + R 12M 2 2 n2 ,
where M i are characteristic energy scales. One can see that α 2 = n 2 − 1 and relation (15) is still valid; to determine α, one resorts to equation (17) and writes
f 1 R f 1 = n 1 1 − 1 α 1 (48)
The dynamical system is obtained by replacing all this parameters in the initial system (13), but it is too extensive to be presented here. The fixed points obtained from this dynamic system are shown in Table VI and the corresponding solutions in Table VII. Note that for n 1 = 1 → f 1 (R) = 2R, the fixed points coincide with the ones presented in the subsection V A, considering the restriction for Ω 2 , Eq. (29). For the GR case (n 1 = n 2 = 0), the fixed points obtained will collapse to only two: an unstable matter dominance and a stable cosmological constant dominance.
First, second and third fixed point
These are points of little interest with an evolution similar to the radiation era. Since Ω 2 = 0, one might expect these points to be related to pure f (R) solutions.
In fact, they appear in Ref. [33], that studies the f (R) = R + R 0 (R/R 0 ) n model. The first is a saddle point, the stability of the second fixed point is shown in Fig. 4 and the third is a saddle point when 0 < w < 2/3, and for w > 2/3 the stability depends on both parameters n 1 and n 2 .
Forth fixed point
In this point the energy density is null so the NMC is neglected, f 2 ρ κf 1 . This corresponds to a point based on a pure f (R) theory that was studied in Ref. [33]. The stability of the point is shown in Fig. 5.
It is easy to see that a De Sitter phase is obtained when n 1 = 2, which corresponds to x = 0, y = 2 and z = −1 -the same as the previous fixed point, but their origin is completely different. The Starobinsky inflation model, Ref. [13], corresponds to n 1 = 2 and n 2 = 0. Since the NMC is neglected, this point corresponds to that solution.
The scale factor and deceleration parameter are independent of w, but the stability has some dependency, as shown in Fig. 5. One can see that when
n 1 = (7 ± √ 73)/12 → q = 1/2,(49)
and n 1 = 0, 5/4 → q = 1. (50)
Fifth fixed point
The scale factor only depends on n 1 , reflecting a stronger influence of κf 1 . This is also visible, since Ω 1 = 0 and Ω 2 = 0. The regime f 2 ρ κf 1 is verified, but with non-null energy density. This fixed point also appears in Ref. [33], and is a point based on pure f (R) theory. The stability of the point is shown in Fig. 6.
A De Sitter solution is only obtained when w = −1, which is similar to the use of a cosmological constant. Table IV shows the values of n 1 needed to obtain the usual values of q, when w = 0, 1/3. For n 1 = 1, the result q = (1 + 3w)/2, typical of GR, is recovered; interestingly, this does not depend on n 2 .
Sixth fixed point
This is stable when 1 < n 1 < 2 and a saddle point in rest of the region. It corresponds to a De Sitter phase, with no matter. The Hubble parameter has to satisfy the condition
H 2 0 = M 2 1 1 n 1 − 2 1/(n1−1) ,(51)
for n 1 = 1 and n 1 = 2. This point also appears in Ref. [33].
Seventh, eighth and ninth fixed point
These points correspond to the first, second and third point of the power law of pure NMC, presented in section V A, respectively. The stability of these points is shown in Fig. 1,7 and 8, respectively. Despite this correspondence between the points of both models, the stability of the eighth and ninth points is altered by the NMC.
Tenth fixed point
This point appears to be similar to the fifth, but with no dominant regime and it is the only point that depends explicitly on both functions. The stability of the point is shown in Fig. 9 and 10, for w = 1/3 and w = 0, respectively. This point is divergent when n 1 = n 2 .
A De Sitter solution is only obtained when w = −1, which is similar to the use of a cosmological constant. Table V shows the relation between n 1 and n 2 needed to obtain the usual values of q, when w = 0, 1/3. The normal results for GR are obtained when n 1 − n 2 = 1.
Point
Coordinates (x, y, z,
Ω 1 , Ω 2 ) 1 (−4, 0, 5, 0, 0) 2 (1, 0, 0, 0, 0) 3 (−1 + 3w, 0, 0, 2 − 3w, 0) 4 − 2(n 1 − 2) 2n 1 − 1 , n 1 (−5 + 4n 1 ) 1 − 3n 1 + 2n 2 1 , 5 − 4n 1 1 − 3n 1 + 2n 2 1 , 0, 0 5 3(−1 + n 1 )(1 + w) n 1 , − 3 − 4n 1 + 3w 2n 1 , 3 − 4n 1 + 3w 2n 2 1 , −3(1 + w) −8 4 − 2n 2 (4 + 3w) −1 + 2n 2 , n 2 (−2 + 4n 2 + 3w) 1 − 3n 2 + 2n 2 2 , 0, 2 − 4n2 − 3w 1 − 3n2 + 2n2 2 , 3(1 + w) 9 − 6n 2 (1 + w) −1 + 4n 2 + 3w , 1 − 4n 2 − 3w 2 − 2n 2 , 1 − 2n 2 − 3w 2(−1 + n 2 ) , 1 1 − n 2 , 6n 2 (1 + w) −1 + 4n 2 + 3w 10 − 3(1 + w) (−1 + n 2 )(3 + 4n 2 + 3w) + n 1 7 − 2n 2 2 + 3w − 9n 2 (1 + w) + n 2 1 [−4 + n 2 (8 + 6w)] (n 1 − n 2 )[4n 1 − 4n 2 − 3(1 + w)] , − 3 − 4n 1 + 4n 2 + 3w 2n 1 − 2n 2 , 3 − 5n 2 − 2n 2 2 + 3w − 9n 2 w + n 1 [−4 + n 2 (8 + 6w)] 2(n 1 − n 2 ) 2 , −4n 2 − 3(1 + w) − 2n 2 1 (4 + 3w) + n 1 (13 + 2n 2 + 9w) 2(n 1 − n 2 ) 2 , 3n 2 (1 + w)[3 + 4n 2 + 3w + n 2 1 (8 + 6w) − n 1 (13 + 2n 2 + 9w)] (n 1 − n 2 )[4n 1 − 4n 2 − 3(1 + w)]
A. Modified Friedmann equation
In this section we attempt a comparison between the results obtained above and those of Ref. [32], which is based upon the phenomenological study of modifications of the Friedmann equation, of the form
H 2 ∼ ρ 1+β .(52)
As shown in that study, the above relation can be obtained in the regime F = const. when f 1 (R) ∼ R and f 2 (R) ∼ R β/(1+β) . From Eq. (24), the scale factor is given by and the deceleration parameter is
a(t) ∼ t 2 3(1+w)(1+β) ,(53)Point a(t) ρ(t) q 1 t/t0 0 1 2 t/t0 0 1 3 t/t0 t t 0 −3(1+w)/2 1 4 e H 0 t , n1 = 2 t t 0 −1+3n 1 −2n 2 1 n 1 −2 , n1 = 2 0 −1 + 2 − n1 1 − 3n1 + 2n 2 1 5 t t 0 2n1 3(1 + w) t t 0 −2n 1 −1 + 3(1 + w) 2n1 6 e H 0 t 0 −1 7 e H 0 t ρ → ∞ −1 8 t t 0 1−3n 2 +2n 2 2 2−4n 2 −3n 2 w , n2 = 2 4 + 3w ρ → ∞ −1 + 2 − n2(4 + 3w) 1 − 3n2 + 2n 2 2 9 t t 0 2(1−n 2 ) 3(1+w) t t 0 2(n 2 −1) −1 + 3(1 + w) 2(1 − n2) 10 t t 0 2(n 1 −n 2 ) 3(1+w) t t 0 2(n 2 −n 1 ) −1 + 3(1 + w) 2(n1 − n2)q = −1 + 3(1 + β)(1 + w) 2 .(54)
From the above, a comparison between this solutions and the solutions obtained from the fixed points is possible. Considering n 1 = 1 and n 2 = β/(1 + β), one can see The remaining region corresponds to saddle points. Large, medium and short dash indicate q = 0, q = 1/2 and q = 1, respectively. Black traces correspond to w = 1/3 and grey traces to w = 0. β = −1, which corresponds to n 2 → ∞. The ninth and tenth fixed points have a deceleration parameter exactly like Eq. (54). The eighth point has F = const. and satisfies Eq. (54) when β = (1−3w)/(1+3w). The fifth point also satisfies Eq. 54 when β = 0, which corresponds to GR case. Thus, one concludes that the modifications to Friedmann equation due to a NMC are indeed obtainable from a dynamical system's approach, as correctly argued in Ref. [32].
B. Linear NMC
Note that, for n 2 = 1 divergences appear in some points and a more detail study by direct substitution is required, which is done in Ref. [30]. A direct comparison between the points obtained above and the ones attained in Ref. [30] is not done due to the different choice of the variables: in particular, note that that study resorts to a variable proportional to ρ 2 .
Nevertheless, one can compare the deceleration parameter obtained in both works (n 1 = n), which clearly marks the physical significance of the underlying fixed points: one finds that the tenth point presented here has the same deceleration parameter, q = (5−2n)/(2(n−1)), of the forth point of the mentioned article. Also, there is a fixed point in Ref. [30] with q = −1, that can be related to the fixed points obtained here with the same FIG. 7: The two lightest grey region correspond to the stable regions of the eighth fixed point when w = 1/3 and w = 0, from lightest to darkest, respectively. The two darkest grey regions are overlapped but correspond to an unstable region, where from the lightest to the darkest corresponds to w = 1/3 and w = 0. Large, medium and short dash indicate q = 0, q = 1/2 and q = 1, respectively. The continuous line corresponds to q = −1. The black traces corresponds to w = 1/3 and the grey to w = 0.
value. These comparisons are only valid when the power law term dominates over R, since our choice of model was Eq. (47).
VII. CONCLUSION
In this work, a dynamical system approach was made on NMC theories. The dynamical system for the most general case with two arbitrary functions was obtained. Also, the solutions and their stability for three different models were obtained and compared with previous works.
As expected, the NMC dynamical system can be particularized to a pure f (R) theory when f 2 (R) = 1, yielding the same results obtained in Ref. [33]. One can see that the variable Ω 2 introduced by the NMC, is the key to determine whether F is constant or if the NMC dominates over the usual f (R) theory.
In the pure NMC case described by a power law, the solutions obtained are in agreement with the ones presented in Ref. [10]. In addition, the obtained result for the energy density for the third fixed point is different from the one in Ref. [10], due to the assumption of the latter that ρ = 0 → F = κ, which differs from the result here obtained, Eq. (35).
Furthermore, the pure NMC exponential case appears to have less diversity of solutions than the usual expo- nential f (R) model, as seen in Ref. [35]. The last model considered of power law corrections to GR yields the solutions for the pure f (R) case and the pure NMC case, as if it considered the regimes for which function dominates over the other, and also a solution that depends simultaneously on both models. Furthermore, it was determined which fixed points correspond to the general solution of H 2 ∼ ρ 1+β , presented in Ref. [32].
This method is a good way to determine the solutions of a particular model, since it does not assume solutions a priori. Note that, in the pure NMC power law case, there is a solution obtained by this method not considered in Ref. [10].
However, and despite its success in determining a variety of solutions, this method depends on the chosen variables. In addition, the existence of fixed points with desired stability does not imply that there is a trajectory in the phase space (i.e. a history for the Universe) that connects these points. Even if that can be guaranteed, only a more profound mathematical study of the field equations (including their numerical solution) can ensure that the universe may evolve from a matter dominated to an accelerated expansion era, in a way compatible with the experimental observations and with reasonable physical parameters, n and M .
FIG. 1 :
1The dark grey region corresponds to the unstable region of the first fixed point. There is no stable region and the remaining phase space corresponds to a saddle point.FIG. 2: Stability region of the second fixed point. The light grey region corresponds to the a stable fixed point, the dark grey region to an unstable fixed point and the remaining to a saddle point. Large, medium and short dash indicate q = 0, q = 1/2 and q = 1, respectively. The continuous line corresponds to q = −1.
FIG. 3 :
3Stability region of the third fixed point. The light grey region corresponds to a stable fixed point, the dark grey region to an unstable fixed point and the remaining to a saddle point. Large, medium and short dash indicate q = 0, q = 1/2 and q = 1, respectively.
VII: Solutions associated with the fixed points of the model, Eq. (47). FIG. 4: Stability region for the second fixed point. There is no stable region for w = 1/3 and w = 0. The unstable region for w = 0 (dark grey) is overlapped with the w = 1 (light grey). The remaining regions corresponds to saddle points.
FIG. 5 :
5The two lightest grey region correspond to the stable regions of the forth fixed point when w = 1/3 and w = 0, from lightest to darkest respectively. For n1 > 2, the regions overlap and are both stable. The two darkest grey regions are overlapped but correspond to an unstable region for w = 1/3 and w = 0. Large, medium and short dash indicate q = 0, q = 1/2 and q = 1, respectively. The continuous line corresponds to q = −1.that the fifth, sixth, ninth and tenth fixed points correspond to a constant F . The sixth fixed point corresponds to a De Sitter phase and q is only equal to Eq. (54) whenFIG. 6: The two overlapped regions correspond to the stable regions of the fifth fixed point when w = 1/3, light grey, and w = 0, dark grey. There is no unstable region for both cases.
FIG. 8 :
8Stability region for the ninth fixed point. There is no stable region for w = 1/3 and w = 0. The dark grey region corresponds to the unstable region when w = 0 and lightest when w = 1/3. The remaining regions corresponds to saddle points. Large, medium and short dash indicate q = 0, q = 1/2 and q = 1, respectively. The black traces corresponds to w = 1/3 and the grey to w = 0.FIG. 9: Stability region of the tenth point when w = 1/3. Light grey corresponds to the stable region and the dark grey to the unstable. The remaining corresponds to a saddle point. Large, medium and short dash indicate q = 0, q = 1/2 and q = 1, respectively.
FIG. 10 :
10Stability region of the tenth point when w = 0. Light grey corresponds to the stable region and the dark grey to the unstable. The remaining corresponds to a saddle point. Large, medium and short dash indicate q = 0, q = 1/2 and q = 1, respectively.
TABLE I :
IFixed points and respective solutions of the model, Eq. (32).
TABLE II :
IIValues of n needed to obtain the usual deceleration values for different w.
TABLE III :
IIIFixed points of the model, Eq. (40).
TABLE IV :
IVValues of n needed to obtain the usual deceleration values for different w for the fifth fixed point.
TABLE V :
VValues of n needed to obtain the usual deceleration values for different w for the sixth fixed point.
TABLE VI :
VIFixed points of the model, Eq. (47).
TABLE
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|
[] |
[
"GRAM SCHMIDT BASED GREEDY HYBRID PRECODING FOR FREQUENCY SELECTIVE MILLIMETER WAVE MIMO SYSTEMS",
"GRAM SCHMIDT BASED GREEDY HYBRID PRECODING FOR FREQUENCY SELECTIVE MILLIMETER WAVE MIMO SYSTEMS"
] |
[
"Ahmed Alkhateeb [email protected] \nThe University of Texas at Austin\nTXUSA\n",
"Robert W Heath [email protected] \nThe University of Texas at Austin\nTXUSA\n"
] |
[
"The University of Texas at Austin\nTXUSA",
"The University of Texas at Austin\nTXUSA"
] |
[] |
Hybrid analog/digital precoding allows millimeter wave MIMO systems to leverage large antenna array gains while permitting low cost and power consumption hardware. Most prior work has focused on hybrid precoding for narrow-band mmWave systems. MmWave systems, however, will likely operate on wideband channels with frequency selectivity. Therefore, this paper considers frequency selective hybrid precoding with RF beamforming vectors taken from a quantized codebook. For this system, a low-complexity yet nearoptimal greedy algorithm is developed for the design of the hybrid analog/digital precoders. The proposed algorithm greedily selects the RF beamforming vectors using Gram-Schmidt orthogonalization. Simulation results show that the developed precoding design algorithm achieves very good performance compared with the unconstrained solutions while requiring less complexity.Index Terms-Millimeter wave communications, frequency selective, Gram-Schmidt, hybrid precoding.
|
10.1109/icassp.2016.7472307
|
[
"https://arxiv.org/pdf/1601.07223v1.pdf"
] | 15,364,784 |
1601.07223
|
be42857c34d90001f884985e82a32fe1249f127c
|
GRAM SCHMIDT BASED GREEDY HYBRID PRECODING FOR FREQUENCY SELECTIVE MILLIMETER WAVE MIMO SYSTEMS
Ahmed Alkhateeb [email protected]
The University of Texas at Austin
TXUSA
Robert W Heath [email protected]
The University of Texas at Austin
TXUSA
GRAM SCHMIDT BASED GREEDY HYBRID PRECODING FOR FREQUENCY SELECTIVE MILLIMETER WAVE MIMO SYSTEMS
Hybrid analog/digital precoding allows millimeter wave MIMO systems to leverage large antenna array gains while permitting low cost and power consumption hardware. Most prior work has focused on hybrid precoding for narrow-band mmWave systems. MmWave systems, however, will likely operate on wideband channels with frequency selectivity. Therefore, this paper considers frequency selective hybrid precoding with RF beamforming vectors taken from a quantized codebook. For this system, a low-complexity yet nearoptimal greedy algorithm is developed for the design of the hybrid analog/digital precoders. The proposed algorithm greedily selects the RF beamforming vectors using Gram-Schmidt orthogonalization. Simulation results show that the developed precoding design algorithm achieves very good performance compared with the unconstrained solutions while requiring less complexity.Index Terms-Millimeter wave communications, frequency selective, Gram-Schmidt, hybrid precoding.
INTRODUCTION
Millimeter wave (mmWave) communication can leverage the large bandwidth potentially available at the high frequency bands to provide high data rates [1][2][3][4][5]. To guarantee sufficient received signal power at these high frequencies, though, large antenna arrays need to be deployed at both the transmitter and receiver [2,3,6]. Designing precoding and combining matrices for these mmWave wideband large MIMO systems differs from lower-frequency solutions. This is mainly due to the different hardware constraints on the mixed signal components because of their high cost and power consumption [7]. Therefore, developing precoding schemes for wideband mmWave systems is important for building these systems.
For the sake of low power consumption, hybrid analog/digital precoding solutions, that divide the precoding between analog and digital domains, and hence requiring smaller number of RF chains, were proposed in [8][9][10][11][12][13][14][15][16]. For general MIMO systems, hybrid precoding design with diversity and spatial multiplexing objectives were investigated in [8,9]. In [10], the sparse nature of mmWave channels was exploited, and low-complexity iterative algorithms based on matching pursuit were devised, assuming perfect channel knowledge at the transmitter. Extensions to the case when only partial channel knowledge is required was considered in [11,12]. Other heuristic algorithms that do not rely on orthogonal matching pursuit were also proposed in [13][14][15] for the hybrid precoding design with perfect channel knowledge at the transmitter. The work in [10][11][12][13][14] assumed a narrow-band mmWave channel, with perfect or partial This work is supported in part by the National Science Foundation under Grant No. 1319556, and by a gift from Nokia. channel knowledge at the transmitter. In [16], hybrid beamforming with only a single-stream transmission over MIMO-OFDM system was considered. The solution in [16] though relied on the joint exhaustive search over both RF and baseband codebooks which results in high-complexity. As mmWave communication is expected to employ broadband channels, developing spatial multiplexing hybrid precoding algorithms for wideband mmWave systems is important.
In this paper, we investigate the frequency selective hybrid precoding design to maximize the achievable mutual information given that the RF precoders are taken from a quantized codebook. We first derive the optimal baseband precoders as functions of the RF precoders. Then, we design a greedy hybrid precoding algorithm based on Gram-Schmidt orthogonalization. Despite its low-complexity, the proposed algorithm is illustrated to achieve a similar performance compared with the optimal hybrid precoding design that requires an exhaustive search over the RF codebooks.
SYSTEM MODEL
Consider the OFDM based system model in Fig. 1 where a basestation (BS) with NBS antennas and NRF RF chains is assumed to communicate with a single mobile station (MS) with NMS antennas and NRF RF chains. The BS and MS communicate via NS length-K data symbol blocks, such that NS ≤ NRF ≤ NBS and NS ≤ NRF ≤ NMS.
At the transmitter, the NS data symbols s k at each subcarrier k = 1, ..., K are first precoded using an NRF × NS digital precoding matrix F[k], and the symbol blocks are transformed to the timedomain using NRF K-point IFFT's. Note that our model assumes that all subcarriers are used and, therefore, the data block length is equal to the number of subcarriers. A cyclic prefix of length D is then added to the symbol blocks before applying the NBS × NRF RF precoding FRF. It is important to emphasize here that the RF precoding matrix FRF is the same for all subcarriers. This means that the RF precoder is assumed to be frequency flat while the baseband precoders can be different for each subcarrier. The discretetime transmitted complex baseband signal at subcarrier k can therefore be written as
y[k] = FRFF[k]s[k],(1)they meet FRFF[k] ∈ UN BS ×N S , with the set of semi-unitary matrices UN BS ×N S = U ∈ C N BS ×N S |U * U = I .
At the MS, assuming perfect carrier and frequency offset synchronization, the received signal is first combined in the RF domain using the NMS×NRF combining matrix WRF. Then, the cyclic prefix is removed, and the symbols are returned back to the frequency domain where the symbols at each subcarrier k are combined using the NRF × NS digital combining matrix W[k]. Denoting the NMS × NBS channel matrix at subcarrier k as H[k], the received signal at subcarrier k after processing can be then expressed as
y[k] = W[k] * W * RF H[k]FRFF[k]s[k] + W[k] * W * RF n[k], (2) where n[k] ∼ N (0, σ 2 N I) is a Gaussian noise vector.
PROBLEM FORMULATION
The paper objective is to develop a low-complexity hybrid precoding design to maximize the achievable system spectral efficiency. Given the system model in Section 2. For simplicity of exposition, we will assume that the receiver can perform optimal nearest neighbor decoding based on the NMS-dimensional received signal with fully digital hardware. This allows decoupling the transceiver design problem, and focusing on the hybrid precoders design to maximize the mutual information of the system [10], defined as
I FRF, {F[k]} K k=1 = 1 K K k=1 log 2 IN MS + ρ NS H[k]FRFF[k]F[k] * F * RF H[k] * ,(3)
where ρ = P Kσ 2 is the SNR. As combining with fully digital hardware is not a practical mmWave solution, the hybrid combining design problem needs also to be considered. The design ideas that will be given in this paper for the hybrid precoders, however, provide direct tools for constructing the hybrid combining matrices, WRF,
{W[k]} K k=1
, and is therefore omitted due to space limitations. If the RF beamforming vectors are taken from a codebook FRF that captures the RF hardware constraints, then the maximum mutual information under the given hybrid precoding model is
I HP = max F RF ,{F[k]} K k=1 I FRF, {F[k]} K
One challenge of the hybrid precoding design to solve the optimization problem in (4) is the coupling between baseband and RF precoders that arises in the power constraint (the second constraint of (4)). In the following proposition, we show that the baseband precoders can be written optimally as a function of the RF precoders.
Proposition 1 Define the SVD decompositions of the matrices
H[k] = U[k]Σ[k]V[k] * and Σ[k]V[k] * FRF (F * RF FRF) − 1 2 = U[k]Σ[k]V[k] * , then the baseband precoders {F[k]} K k=1 that solve (4) are given by F[k] = (F * RF FRF) − 1 2 V[k] :,1:N S , k = 1, 2, ..., K.(5)
Proof: The proof follows using change of variables. It is omitted due to space limitation, but available in the journal version [17]. 2 Given proposition 1, the optimal hybrid precoding based mutual information can be given by making an exhaustive search over only the RF precoding codebook. To avoid this search. we propose efficient greedy hybrid precoding algorithms in the following sections.
GREEDY HYBRID PRECODING
A natural greedy approach to construct the hybrid precoder is to iteratively select the NRF RF beamforming vectors from the codebook FRF to maximize the mutual information. In this paper, we call this the direct greedy hybrid precoding (DG-HP) algorithm. Let the NBS × (i − 1) matrix F (i−1) RF denote the RF precoding matrix at the end of the (i − 1)th iteration. Then by leveraging the optimal baseband precoder structure in (5), the objective of the ith iteration is to select f RF n ∈ FRF that solves
I (i) HP = max f RF n ∈F RF 1 K K k=1 i =1 log 2 1 + ρ NRF × λ H [k]F (i,n) RF F (i,n) * RFF (i,n) RF −1F (i,n) * RF H [k] * ,(6)withF (i,n) RF = F (i−1) RF , f RF n .
The best vector f RF n will be then added to the RF precoding matrix to form F . The main limitation of this algorithm is that it still requires an exhaustive search over FRF and eigenvalues calculation in each iteration. In the next section, we will make a first step towards developing a low-complexity algorithm that has a similar (or very close) performance to this DG-HP algorithm.
GRAM-SCHMIDT GREEDY HYBRID PRECODING
In hybrid analog/digital precoding architectures, the effective channel seen at the baseband is through the RF precoders lens. This gives the intuition that it is better for the RF beamforming vectors to be orthogonal (or close to orthogonal), as this physically means that the effective channel will have a better coverage over the dominant subspaces belonging to the actual channel matrix. This intuition is also confirmed by the structure of the optimal baseband precoder discussed in Proposition 1, as the overall matrix FRF (F * RF FRF) − 1 2 has a semi-unitary structure. This note means that in each iteration i of the greedy hybrid precoding algorithm in (6) with a selected codeword f RF n , the additional mutual information gain over the previous iterations is due to the contribution of the component of f RF n that is orthogonal on the existing RF precoding matrix F
(i−1)
RF . This is similar to the greedy user scheduling in MIMO broadcast channels based on the orthogonal channel components [18], but in a different context. Based on that, we modify the DG-HP algorithm by adding a Gram-Schmidt orthogonalization step in each iteration i to project the candidate beamforming codewords on the orthogonal complement of the subspace spanned by the selected codewords in F (i−1)
RF . This can be simply done by multiplying the candidate vectors by the projection
matrix P (i−1) ⊥ = Ii − F (i−1) RF F (i−1) * RF F (i−1) RF −1 F (i−1) * RF .
Given the optimal precoder design in (5), the mutual information at the ith iteration of the modified Gram-Schmidt hybrid precoding (GS-HP) algorithm can be written as
I (i) HP = max f RF n ∈F RF 1 K K k=1 i =1 log 2 1 + ρ NRF λ H [k] F (i,n) RF × F (i,n) RF * F (i,n) RF −1 F (i,n) RF * H [k] * ,(7)(a) = max f RF n ∈F RF 1 K K k=1 i =1 log 2 1 + ρ NRF λ T (i−1) +H[k]P (i−1) ⊥ f RF n f RF n * P (i−1) ⊥ * H * [k] ,(8)
with
T (i−1) =H [k] F (i−1) RF F (i−1) RF * F (i−1) RF −1 F (i−1) RF * H [k] * , and F (i,n) RF = F (i−1) RF , P (i−1) ⊥ f RF n .
Note that T (i−1) is a constant matrix at iteration i, and (a) follows from the Gram-Schmidt orthogonalization which allows the matrix F
(i,n) RF F (i,n) RF * F (i,n) RF − 1 2 to be written as F (i−1) RF F (i−1) RF * F (i−1) RF − 1 2 , P (i−1) ⊥ f RF n .
Hence, the eigenvalues calculation in (8) can be calculated as a rank-1 update of the previous iteration eigenvalues, which reduces the overall complexity [19]. The best vector f RF n will be then added to the RF precoding matrix to form F . In the following proposition, we prove that this Gram-Schmidt hybrid precoding algorithm is exactly equivalent to the DG-HP algorithm.
Proposition 2 The achieved mutual information of the direct greedy hybrid precoding algorithm in (6) and the Gram-Schmidt hybrid precoding algorithm in (7)
APPROXIMATE GRAM-SCHMIDT BASED GREEDY HYBRID PRECODING
The main advantage of the Gram-Schmidt hybrid precoding design in Section 5 is that it leads to a near-optimal low-complexity design of the frequency selective hybrid precoding as will be discussed in this section. Given the optimal baseband precoding solution in (5), the mutual information at the ith iteration in (7) can be written as
I (i) HP = max f RF n ∈F RF 1 K K k=1 i =1 log 2 1 + ρ NRF λ H [k] F (i,n) RF × F (i,n) RF * F (i,n) RF −1 F (i,n) RF * H [k] * ,(9)c) F (i) RF = F (i−1) RF f RF n d) Π = Π Ii − F (i) RF F (i) * RF F (i) RF −1 F (i) * RF Digital Precoder Design 3) F[k] = F (N RF ) RF F (N RF ) * RF F (N RF ) RF − 1 2 V[k] :,1:N S , k = 1, ..., K, with V[k] defined in (5). (a) ≥ max f RF n ∈F RF 1 K K k=1 i =1 log 2 1 + ρ NS λ Σ [k]Ṽ * [k] × F (i,n) RF F (i,n) RF * F (i,n) RF −1 F (i,n) RF * Ṽ [k]Σ * [k] ,(10)(b) ≈ 1 K K k=1 log 2 I + ρ NSΣ [k] 2 − tr Σ [k] + max f RF n ∈F RF 1 K K k=1 Σ [k]Ṽ[k] * F (i,n) RF F (i,n) RF * F (i,n) RF − 1 2 2 F(11)
where follows from using the large mmWave MIMO approximations used in [10]. The objective of the ith iteration is then to solve
f RF n = arg max f RF n ∈F RF 1 K K k=1 Σ [k]Ṽ[k] * F (i,n) RF F (i,n) RF * F (i,n) RF − 1 2 2 F(12)
= arg max
f RF n ∈F RF Σ HṼ * H F (i−1) RF F (i−1) * RF F (i−1) RF − 1 2 2 F ,(13)(a) = arg max f RF n ∈F RF Σ HṼ * H P (i−1) ⊥ f RF n 2 2 ,(14)
whereΣ The problem in (14) is simple to solve with just a maximum projection step. We call this algorithm the approximate Gram-Schmidt hybrid precoding (Approximate GS-HP) algorithm. As shown in Algorithm 1, the developed algorithm sequentially build the RF and baseband precoding matrices in two separate stages. First, the RF beamforming vectors are iteratively selected to solve (14). Then, the baseband precoder is optimally designed according to (5). Despite its sequential design of the RF and baseband precoders, which reduces the complexity when compared with prior solutions that mostly depend on the joint design of the baseband and RF precoding matrices [10,12], Algorithm 1 achieves a significant gain over prior solutions, and gives a very close performance to the optimal solution given by exhaustive search, as will be shown in Section 7.
SIMULATION RESULTS
In this section, we evaluate the performance of the proposed algorithm using numerical simulations. We adopt a wideband mmWave channel model that consists of L = 6 clusters. The center AoAs/AoDs of the L clusters θ , φ are assumed to be uniformly distributed in [0, 2π). Each cluster has R = 5 rays with Laplacian distributed AoAs/AoDs [10,20], and angle spread of 10 o . The number of system subcarriers K equals 512, and the cyclic prefix length is D = 128, which is similar to 802.11ad [21]. The paths delay is uniformly distributed in [0, DTs]. Both the BS and MS have ULAs with NRF = 3.
In Fig. 2, we validate the result in Proposition 2, in addition to evaluating the approximate Gram-Schmidt based hybrid precoding algorithm. The spectral efficiencies achieved by these greedy algorithms are compared with the optimal hybrid precoding design given by the exhaustive search over the RF codebooks. The rates are also compared with the prior solution in [10]. For a fair comparison, we assume that each RF beamforming vector is selected from a beamsteering codebook with a size NCB = 64. First, Fig. 2 shows that the direct greedy and Gram-Schmidt based hybrid precoding algorithms achieve exactly the same performance which verifies Proposition 2. Despite its low-complexity, the developed approximate Gram-Schmidt hybrid precoding design in Algorithm 1 achieves very close performance to the exhaustive-search based optimal solution. We emphasize here that any hybrid precoding design can not perform better that the shown optimal hybrid precoding solution with the considered RF codebook, which confirms the nearoptimal result of the proposed algorithm. This is also clear from the considerable gain obtained by the proposed algorithm compared with the prior solution in [10]. Also, it is worth mentioning that the developed hybrid precoding algorithms in this paper can be applied to any large MIMO system (not specifically mmWave systems).
CONCLUSION
In this paper, we investigated hybrid precoding design for wideband mmWave systems. First, we derived the optimal hybrid precoding design that maximizes the achievable mutual information for any given RF codebook, and showed that the optimal baseband structure can be decomposed into an RF precoder dependent matrix and a unitary matrix. Second, we developed a novel greedy hybrid precoding algorithm based on Gram-Schmidt orthogonalization. Thanks to this Gram-Schmidt orthogonalization, we showed that only sequential design of the RF and baseband precoders is required to achieve the same performance of more sophisticated algorithms that requires a joint design of the RF and baseband precoders in each step. Simulation results illustrated that the proposed precoding algorithms improve over prior work and stay within a small gap from the unconstrained perfect channel knowledge solutions.
PROOF OF PROPOSITION 2
Proof: To prove that I GS−HP
k=1 s.t. [FRF] :,r ∈ FRF, r = 1, ..., NRF FRFF[k] ∈ UN BS ×N RF , k = 1, 2, ..., K.
are exactly equal, i.e., I DG−HP HP = I GS−HP HP . Proof: See Section 9.
Algorithm 1
1Approximate Gram-Schmidt Greedy Hybrid Precoding Initialization 1) Construct Π =Σ HṼH , withΣ H = Σ 1, ...,ΣK and V H = Ṽ 1, ...,ṼK . Set FRF = Empty Matrix. Set ACB = f RF 1 , ..., f RF N v CB , where f RF n , n = 1, ..., N v CB are the codewords in FRF. RF Precoder Design 2) For i, i = 1, ..., NRF a) Ψ = Π * ACB b) n = arg max n=1,2,..N v CB [Ψ] :,n 2 .
(a) is by considering only the first NS dominant singular values of H[k],Σ[k] = [Σ[k]] :,1:N S ,Ṽ[k] = [V[k]] :,1:N S , and (b)
H = Σ [1], ...,Σ[K] ,Ṽ H = Ṽ [1], ...,Ṽ[K] , and (a) is a result of the Gram-Schmidt processing as described in Section 5.
Fig. 2 .
2The performance of the approximate Gram-Schmidt hybrid precoding design in Algorithm 1 compared with the optimal hybrid precoding solution, the unconstrained SVD solution, and the prior work in[10]. The system has NBS = 32 antennas, NMS = 16 antennas, and NS = NRF = 3.
IN S , and P is the average total transmit power. Since FRF is implemented using analog phase shifters, its entries are of constant modulus. To reflect that, we normalize the entries [FRF] m,n = 1. Further, we assume that the angles of the analog phase shifters are quantized and have a finite set of possible values. With these assumptions, [FRF] m,n = e jφm,n , where φm.n is a quantized angle. The hybrid precoders are assumed to have a unitary power constraint,i.e., arXiv:1601.07223v1 [cs.IT] 26 Jan 2016Fig. 1. A block diagram of the OFDM based BS-MS transceiver that employs hybrid analog/digital precoding.where s[k] is the NS × 1 transmitted vector at subcarrier k,
such that E [s[k]s[k] * ] = P
KN S 2
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+
+
F RF
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HP= I DG−HP HP , it is sufficient to prove that F (N RF ) RF of the GS-HP and DG-HP algorithms are equal. To do that, we will show that both the algorithms choose the same RF beamforming vector in each iteration, i.e., F (i) RF is equal for i = 1, ..., NRF. This can be proved using mathematical induction as follows. At the first iteration, the two algorithms do the exhaustive search over the same codebook FRF, and consequently select the same beamforming vectors. Now, suppose that the two algorithms reach the same RF precoding matrix Fwe need to prove that they both select the same RF beamforming vector at iteration i, i.e., we need to prove that both(6)and(7)choose beamforming vectors with the same index. To prove that, it is enough to show that the contributions of the nth beamforming vector f RF n from F v RF in(6)and(7)are equal.Given the optimal baseband precoder in(5), and denoting the SVD ofF , equation(6)can be written asEquation(7)can be similarly written, but withÛ. Hence, we need to prove thatÛRF . As f n is a result of successive Gram-Schmidt operations, we can write fn = f n + F , as EC is an i × i full-rank matrix.2
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[] |
[
"Small-Gain Theorem for Safety Verification of Interconnected Systems",
"Small-Gain Theorem for Safety Verification of Interconnected Systems"
] |
[
"Ziliang Lyu ",
"Xiangru Xu ",
"Yiguang Hong "
] |
[] |
[] |
A small-gain theorem in the formulation of barrier function is developed in this work for safety verification of interconnected systems. This result is helpful to verify inputto-state safety (ISSf) of the overall system from the safety information encoded in the subsystem's ISSf-barrier function. Also, it can be used to obtain a safety set in a higher dimensional space from the safety sets in two lower dimensional spaces.
|
10.1016/j.automatica.2022.110178
|
[
"https://arxiv.org/pdf/2011.09173v1.pdf"
] | 227,012,746 |
2011.09173
|
450c7976e2b2e21edcf02a865fe835ae8a7fd8e0
|
Small-Gain Theorem for Safety Verification of Interconnected Systems
Nov 2020 1
Ziliang Lyu
Xiangru Xu
Yiguang Hong
Small-Gain Theorem for Safety Verification of Interconnected Systems
Nov 2020 1Input-to-state safety (ISSf)set invariancebarrier functioninterconnected systemssmall-gain theorem
A small-gain theorem in the formulation of barrier function is developed in this work for safety verification of interconnected systems. This result is helpful to verify inputto-state safety (ISSf) of the overall system from the safety information encoded in the subsystem's ISSf-barrier function. Also, it can be used to obtain a safety set in a higher dimensional space from the safety sets in two lower dimensional spaces.
I. Introduction
Safety is crucial in the design of control systems, which requires any trajectories of the control systems initialized in a prescribed safety set to be kept out of the unsafe region. Applications concerned with safety are ubiquitous in our daily life, ranging from the cars on the ground to the aeroplanes in the air. For example, autonomous vehicles are equipped with lane keeping modules [1], a robot team is designed to avoid the collision between robots [2], and aircrafts must satisfy the safety requirement during takeoff and landing [3].
For safety-critical systems, a rigorous verification of safety is the first step towards other control objectives (e.g., stabilization and regulation). In general, the techniques for safety verification can be classified into two sorts: model checking [4] and deductive verification [5]. Compared with the former one, deductive verification provides a safety certificate by mathematical inferences rather than exhaustively checking all of the possible system behaviors. The barrier function gives a promising deductive verification approach, with a set of Lyapunov-like criterions for safety verification; e.g., see [6]- [13] for more details. In [6], barrier functions were used to formulate the verification tasks as convex programming problems. A framework by combining control Lyapunov functions and barrier functions was proposed in [7] to balance the objective of stabilization and safety verification. In [8], a novel barrier function, called the zeroing barrier function, was proposed for establishing safety and analyzing the robustness of safe sets. Later, [10] redefined a new notion of input-to-state safety (ISSf) and provided a sufficient condition in the sense of barrier function to check ISSf, which could be regarded as the counterpart of Sontag's input-to-state stability (ISS) [14] for safety verification. However, it should be mentioned that most of these results were not given for interconnected systems.
The small gain technique is effective for interconnected systems to keep good properties, which completes the verification tasks of a large-scale system to be completed by analyzing its less complicated subsystems. In the stability analysis, this technique has been investigated [15]- [19]. For example, the Lyapunov-based small-gain theorem developed in [16] has been found useful to establish ISS for interconnected systems. This result has been extended to switched systems [17] and hybrid systems [19] in recent years.
Considering that the barrier function provides a Lyapunovlike verification approach, it is natural to ask a question: whether there exists a small-gain condition such that one can establish ISSf for the overall system from the safety information encoded in the subsystem's ISSf-barrier function. This question is not trivial. First, the existing small-gain theorems for stability rely on some nonnegative assumptions, which cannot be met in safety verification. Therefore, we cannot directly employ the tools developed in [15]- [19] to analyze the small-gain condition for safety verification. Second, the ISS plays important roles on the small-gain conditions, but there are no small-gain results of ISSf.
The objective of this paper is to develop a small-gain theorem based on barrier functions for safety verification of interconnected system, in order to establish a higher dimensional safety set from two lower dimensional safety sets. Based on the proposed ISSf-barrier function, we prove that, if the absolute value of the composition of ISSf gains is smaller than that of the identity function, then the overall system is ISSf. This condition is different from traditional small-gain conditions for stability.
Notations and Terminologies. Throughout this paper, '•' denotes the composition operator, i.e., f • g(s) = f (g(s)); 'T' denotes the transpose operator; |·| denotes the Euclidean norm; R and R ≥0 denote the set of real numbers and nonnegative real numbers, respectively. For any measurable function u : R ≥0 → R m , u = sup{|u(t)|, t ≥ 0}. In the following, we review the comparison functions, whose details can be founded in [8] and [20]. A continuous function γ: R ≥0 → R ≥0 with γ(0) = 0 is of class K, if it is strictly increasing. Moreover, a class K function γ is of class K ∞ if it is unbounded. A continuous function γ : R → R with γ(0) = 0 is of extended class K if it is strictly increasing. In particular, an extended class K function γ is of extended class K ∞ if it is unbounded.
II. Preliminaries
Consider the systeṁ
x = f (x, u), x(t 0 ) = x 0(1)
where x ∈ R n is the state, u ∈ R m is the external input, and the vector field f : R n × R m → R n is locally Lipschitz continuous with f (0, 0) = 0. We first review two definitions related to safety.
Definition 1 (Forward Robust Invariance [21]). A set S is forward robustly invariant if for every x 0 ∈ S, there exists an external input u such that the solution x(t) to (1) satisfies x(t) ∈ S for all t ≥ t 0 .
Definition 2 (Input-to-State Safety [10]). The system (1) is input-to-state safe (ISSf) on the set
S γ( u ) = {x ∈ R n : h(x) + γ( u ) ≥ 0}(2)
with respect to the external input u, if S γ( u ) is forward robustly invariant. Herein, h : R n → R is a continuously differentiable function, and γ is a function of class K ∞ . In particular, we say that system (1) is safe if there is no external input (i.e.,
u = 0), in which case S γ( u ) = {x ∈ R n : h(x) ≥ 0}.
Because in the proof of our small-gain theorem we need to handle nonsmooth functions, here we introduce the Dini derivative and corresponding results.
Definition 3 (Dini Derivative [22]). Consider a function h : R → R. The upper and the lower Dini derivatives of h at r ∈ R are, respectively, defined by
D + h(r) = lim sup s→0 + h(r + s) − h(r) s , D + h(r) = lim inf s→0 + h(r + s) − h(r) s .
Given a continuous function h : R → R, it is known that h is not increasing at r ∈ R if and only if the Dini derivative of h at r is nonpositive, and is not decreasing if and only if the Dini derivative of h at r is nonnegative.
Lemma 1. For i = 1, 2, suppose that h i : R n i → R is locally Lipschitz and let h(x) = min{h 1 (x 1 ), h 2 (x 2 )}. Then h(x)
is locally Lipschitz as well, and
D + h(x) =ḣ 1 (x 1 ), if h 1 (x 1 ) < h 2 (x 2 ); D + h(x) =ḣ 2 (x 2 ), if h 1 (x 1 ) > h 2 (x 2 ); D + h(x) = min{ḣ 1 (x 1 ),ḣ 2 (x 2 )}, if h 1 (x 1 ) = h 2 (x 2 ). Let f i (x) = −h i (x) and f (x) = −h(x). Then Lemma 1 follows directly from [23, Theorem 2.1] due to D + f (x) = −D + h(x) and max{ḟ 1 (x 1 ),ḟ 2 (x 2 )} = − min{ḣ 1 (x 1 ),ḣ 2 (x 2 )}.
Lemma 2 (Comparison Principle [24]). Consideṙ
x = f (x), x(t 0 ) = x 0 where f : R → R is locally Lipschitz. Let [t 0 , T ) be the maximal interval of existence of the solution x(t). Suppose that y(t) is a continuous function with D + y(t) ≥ f (y(t)), y(t 0 ) ≥ x 0 .(3)
Then
y(t) ≥ x(t) for all t ∈ [t 0 , T ).
III. ISSf-Barrier Function
In this section, we present a new notion of ISSf-barrier function, which is suitable to characterize the ISSf gains of our small-gain condition.
Definition 4 (ISSf-Barrier Function). A continuously differentiable function h : R n → R is called an ISSf-barrier function for system (1), if there exist functions γ of class K and α of extended class K such that for each u ∈ R m ,
h(x) ≤ −γ(|u|) ⇒ ∇h(x) f (x, u) ≥ −α(h(x)).(4)
Herein, γ is referred to as the ISSf gain.
Note that [10] has presented another definition of ISSfbarrier functions; referring to (5) below. The relation between the ISSf-barrier function defined in (4) and the one of [10] is shown by the following result. Proposition 1. A continuously differentiable function h : R n → R is an ISSf-barrier function for system (1) with α of extended class K ∞ if and only if there exist functions χ of extended class K ∞ and φ of class K ∞ such that
∇h(x) f (x, u) ≥ −χ(h(x)) − φ(|u|).(5)
Proof. Select a constant c ∈ (0, 1). Then it follows from (5) that
h(x) ≤ −φ(|u|) ⇒ ∇h(x) f (x, u) ≥ −(1 − c)χ(h(x)) whereφ(r) = −χ −1 (−φ(r)/c).
The remainder is to verify that φ is of class K. To see this, note thatχ(r) = −χ −1 (−r/c) is continuous and strictly increasing on the interval [0, ∞), which in turn, implies thatφ(r) =χ • φ(r) is of class K.
Then we show that the converse is also true. Assume that (4) holds with some α of extended class K ∞ and γ of class K. According to (4), we have the following two cases.
Case 1. h(x) ≤ −γ(|u|). In this case, there exists a function φ of class K ∞ such that,
∇h(x) f (x, u) ≥ −α(h(x)) ≥ −α(h(x)) − φ(|u|).(6)Case 2. h(x) > −γ(|u|). Since α is of extended class K ∞ , ∇h(x) f (x, u) + α(h(x)) ≥ ∇h(x) f (x, u) + α(−γ(|u|)) ≥ inf h(x)≥−γ(|u|) ∇h(x) f (x, u) + α(−γ(|u|))(7)Letφ(r) = max{0,φ(r)} witĥ φ(r) = − inf h(x)≥−γ(|r|) ∇h(x) f (x, r) + α(−γ(|r|)).
This implies thatφ is continuous andφ(0) = 0 (recalling f (0, 0) = 0). Note thatφ(r) ≥ 0 for each r ≥ 0. By combining this with (7), we can select a class K ∞ function φ with φ(r) ≥φ(r) for all r ≥ 0 such that
∇h(x) f (x, u) + α(h(x)) ≥ −φ(|u|).
By combining the two cases above, we conclude that the converse is true as well.
Then we are ready to present the main result of this section, which can be used as a tool to establish ISSf. Theorem 1. Consider the system (1) and the set S γ( u ) defined in (2). Suppose that h : R n → R is an ISSf-barrier function satisfying (4). Then, (i) S γ( u ) is forward robustly invariant, and system (1) is ISSf on S γ( u ) ;
(ii) the safety set S γ( u ) is ISS.
Proof. We show the conclusion of (i) by contradiction. If this is not true, then there exist some t ≥ t 0 and some ǫ > 0 such that h(
x(t)) < −γ( u ) − ǫ. Let τ = inf{t ≥ t 0 : h(x(t)) ≤ −γ( u ) − ǫ}. Therefore, h(x(τ)) ≤ −γ( u ) ≤ −γ(|u(τ)|), which in turn, implies that d dt t=τ h(x(t)) = ∇h(x(τ)) f (x(τ), u(τ)) ≥ −α(h(x(τ))) > 0.(8)
Thus, h(x(t)) ≤ h(x(τ)) for some t in (t 0 , τ). This contradicts the minimality of τ and consequently, x(t) ∈ S γ( u ) for all t ≥ t 0 .
Then we consider (ii). Construct a Lyapunov-like function as in [8]:
V(x) = 0, if x ∈ S γ( u ) −h(x), if x ∈ R n \ S γ( u ) With (4), we have V(x) ≥ γ(|u|) ⇒ ∇V(x) f (x, u) ≤ α(−V(x)), ∀x ∈ R n \ S γ( u ) .
Moreover, with [14, Lemma 2.14] and [8, Proposition 4], we conclude that the set S γ( u ) is ISS.
IV. Small-Gain Theorem for Safety Verification Consider the interconnected systeṁ
x 1 = f 1 (x 1 , x 2 , u 1 ) (9a) x 2 = f 2 (x 1 , x 2 , u 2 ) (9b)
where, for i = 1, 2, x i ∈ R n i , u i ∈ R m i and f i :
R n 1 ×R n 2 ×R m i → R n i is locally Lipschitz. Let x = (x T 1 , x T 2 ) T , u = (u T 1 , u T 2 ) T and f (x, u) = ( f 1 (x 1 , x 2 , u 1 ) T , f 2 (x 1 , x 2 , u 2 ) T ) T
The small-gain theorem for safety verification of the interconnected system (9) is summarized as follows.
Theorem 2. Consider the interconnected system given in (9). Assume that there exist functions α i of extended class K, φ i of extended class K ∞ and γ i of class K such that
h 1 (x 1 ) ≤ − max{φ 1 (|h 2 (x 2 )|), γ 1 (|u 1 |)} ⇒ ∇h 1 (x 1 ) f 1 (x 1 , x 2 , u 1 ) ≥ −α 1 (h 1 (x 1 )),(10)h 2 (x 2 ) ≤ − max{φ 2 (|h 1 (x 1 )|), γ 2 (|u 2 |)} ⇒ ∇h 2 (x 2 ) f 2 (x 1 , x 2 , u 2 ) ≥ −α 2 (h 2 (x 2 )).(11)
Then there exist an ISS-barrier function h : R n 1 × R n 2 → R, which is dependent on h 1 and h 2 , and a class K ∞ function γ : R ≥0 → R ≥0 , which is dependent on γ 1 and γ 2 , such that the interconnected system (9) is ISSf on the set
S γ( u ) = {x ∈ R n : h(x) + γ( u ) ≥ 0}(12)
with respect to the external input u, if
φ 1 • φ 2 (r) > r, ∀r < 0; φ 1 • φ 2 (r) < r, ∀r > 0.(13)
Remark 1. Clearly, h i (x i ) is an ISSf-barrier function for x isubsystem. According to Theorem 1, the barrier-type conditions given in (10) and (11) imply that the sets
S 1 = x 1 : h 1 (x 1 ) + max{φ 1 ( h 2 (x 2 ) ), γ 1 ( u 1 )} ≥ 0 , S 2 = x 2 : h 2 (x 2 ) + max{φ 2 ( h 1 (x 1 ) ), γ 2 ( u 2 )} ≥ 0
are forward robustly invariant. Intuitively, Theorem 2 can be regarded as a tool to establish a robustly invariant set S γ( u ) in a higher dimensional space from the robustly invariant sets S 1 and S 2 in lower dimensional spaces. This tool is particularly useful for safety verification of large scale systems, since it allows the safety verification to be completed by analyzing its less complicated subsystems rather than analyzing the overall complex system.
The small-gain condition (13) is different from the traditional small-gain condition of [15], whose analysis tools cannot be directly employed in our result. To prove Theorem 2, we first introduce two lemmas, which are extended from those given in [15,Appendix].
Lemma 3. Let ρ 0 : R → R be a continuous function with ρ 0 (r) < 0 for all r < 0 and ρ 0 (r) > 0 for all r > 0. Then there exists a continuous function ρ : R → R such that • ρ 0 (r) < ρ(r) < 0 for all r < 0, and 0 < ρ(r) < ρ 0 (r) for all r > 0; • ρ is continuously differentiable on R, and ρ ′ (r) < 1 2 for all r ∈ R.
Proof. See Appendix. Lemma 4. Let φ i be of extended class K ∞ and satisfy (13). Then there exists an extended class K ∞ function φ such that • φ −1 1 (r) < φ(r) < φ 2 (r) for all r < 0, and φ 2 (r) < φ(r) < φ −1 1 (r) for all r > 0; • φ(r) is continuously differentiable on R\0, and φ ′ (r) > 0 for all r ∈ R\0.
Proof. See Appendix.
Then it is time to prove Theorem 2.
Proof of Theorem 2. By applying Lemma 4 to φ 1 and φ 2 , one can select an extended class K ∞ function φ, which is continuously differentiable on R\0, such that
φ −1 1 (r) < φ(r) < φ 2 (r), ∀r < 0; (14a) φ 2 (r) < φ(r) < φ −1 1 (r), ∀r > 0. (14b) Define a function h(x) = min{φ(h 1 (x 1 )), h 2 (x 2 )}.(15)
Let
S φ 1 ( h 2 ) = {x 1 : h 1 (x 1 ) + φ 1 ( h 2 ) ≥ 0}, S γ 1 ( u 1 ) = {x 1 : h 1 (x 1 ) + γ 1 ( u 1 ) ≥ 0}, S φ 2 ( h 1 ) = {x 2 : h 2 (x 2 ) + φ 2 ( h 1 ) ≥ 0}, S γ 2 ( u 2 ) = {x 2 : h 2 (x 2 ) + γ 2 ( u 2 ) ≥ 0}.
Clearly, S 1 = S φ 1 ( h 2 ) ∪ S γ 1 ( u 1 ) and S 2 = S φ 2 ( h 1 ) ∪ S γ 2 ( u 2 ) . For every (x 1 , x 2 ) ∈ ∂S φ 1 ( h 2 ) × ∂S φ 2 ( h 1 ) , we have the following three cases. Case 1. φ(h 1 (x 1 )) < h 2 (x 2 ). Note that h i (x i ) ≤ 0 when (x 1 , x 2 ) ∈ ∂S φ 1 ( h 2 ) × ∂S φ 2 ( h 1 ) . According to (14), φ −1 1 (h 1 (x 1 )) < φ(h 1 (x 1 )) < h 2 (x 2 ) and hence, it follows immediately from (10) that
D + h(x) = φ ′ (h 1 (x 1 ))∇h 1 (x 1 ) f 1 (x 1 , x 2 , u 1 ) ≥ −φ ′ (h 1 (x 1 ))α 1 (h 1 (x 1 ))(16)
when h(x) ≤ −γ 1 (|u|) withγ 1 (r) = −φ(−γ 1 (r)). The derivation of (16) has employed the fact that φ ′ (h 1 (x 1 )) > 0. Case 2. φ(h 1 (x 1 )) > h 2 (x 2 ). According to (14),
h 2 (x 2 ) < φ(h 1 (x 1 )) < φ 1 (h 1 (x 1 )) which implies D + h(x) = ∇h 2 (x 2 ) f 2 (x 1 , x 2 , u 2 ) ≥ −α 2 (h 2 (x 2 ))(17)
when h(x) ≤ −γ 2 (|u 2 |). Case 3. φ(h 1 (x 1 )) = h 2 (x 2 ). With Lemma 1, it follows from (16) and (17) that
D + h(x) ≥ − max{φ ′ (h 1 (x 1 ))α 1 (h 1 (x 1 )), α 2 (h 2 (x 2 ))} (18) when h(x) ≤ −γ(|u|) with γ(r) =γ 1 (r) + γ 2 (r).
Let α(r) := max{φ ′ (r)α 1 (r), α 2 (r)}. Since φ ′ (r) > 0 for each r ∈ R, α is an extended class K ∞ function. By summarizing the three cases above, we have
h(x) ≤ −γ(|u|)⇒D + h(x) ≥ −α(h(x)).(19)
Without loss of generality, we can assume that α is smooth. In the following, we use the smoothing technique given in [25,Remark 4.1] to show that such an assumption is always possible.
Pick an extended class K ∞ functionα, which is smooth on R, such thatα (s) ≥ −sα(s), ∀s ∈ [−1, 0); α(s) ≤ sα(s), ∀s ∈ (0, 1]. This is possible since α(r) is positive definite when r > 0 and negative definite when r < 0. Then let η : R → R be an extended class K ∞ function, which is smooth in R, such that • η(r) ≤ r for all r ∈ [−1, 0); • η(r) < −α(r)/α(r) for all r < −1. Define
β(r) = s 0 η(s), ∀r ≥ 0; 0 s η(s), ∀r < 0.(20)
Note that β is of extended class K ∞ . Let W(h(x)) = β(h(x)) and then,
D + W(h(x)) = β ′ (h(x))D + h(x) ≥ η(h(x))α(h(x))(21)
when h(x) ≤ −γ(|u|). In the following, we show that D + W(h(x)) is lower bounded by −α(h(x)). Since h(x) ≤ −γ(|u|) ≤ 0, we have the following two cases.
• If h(x) ∈ [−1, 0], η(h(x)) ≤ h(x) ≤ −α (h(x)) α(h(x)) .(22)
• If h(x) ∈ (−∞, −1), with the definition of η on this interval, we have
η(h(x)) ≤ −α (h(x)) α(h(x)) .(23)
Therefore,
h(x) ≤ −γ(|u|) ⇒ D + W(h(x)) ≥ −α(h(x)).(24)
By combining the proof of Theorem 1 and Lemma 2, we can conclude from (24) that the set S γ( u ) = {x ∈ R n : h(x) + γ( u ) ≥ 0} is forward robustly invariant. In other words, the interconnected system (9) is ISSf on the set S γ( u ) .
Before the end of this section, we provide a simple example for illustration. Example 1. Consider the interconnected systeṁ
x 1 = −x 1 − 0.24x 2 sin(x 1 − x 2 ) − 0.5u 3 1 , x 2 = −x 2 − 0.16x 1 sin(x 2 − x 1 ) − 0.5u 2 .
Suppose that x 1 -subsystem and x 2 -subsystem are ISSf on the safety sets S 1 and S 2 , respectively, where
S 1 = x 1 : x 1 + max{0.96 x 2 , 2 u 1 3 } ≥ 0 , S 2 = x 2 : x 2 + max{0.64 x 1 , 2 u 2 } ≥ 0 .
The objective of this example is to verify that the overall system is ISSf. To achieve the control objective of Example 1, we choose h 1 (x 1 ) = x 1 and h 2 (x 2 ) = x 2 , and then it follows that h 1 (x 1 ) ≤ − max{0.96|h 2 |, 2|u 1 | 3 } ⇒ḣ 1 (x 1 ) ≥ −0.5h 1 (x 1 ), h 2 (x 2 ) ≤ − max{0.64|h 1 |, 2|u 2 |} ⇒ḣ 2 (x 2 ) ≥ −0.5h 2 (x 2 ).
Thus, the barrier-type conditions given in (10) and (11) are satisfied. Moreover, we can specify that φ 1 (r) = 0.96r, φ 2 (r) = 0.64r, γ 1 (r) = 2r 3 and γ 2 (r) = 2r. Then we obtain φ 1 • φ 2 (r) = 0.6144r, which implies that the small-gain condition (13) is satisfied. By combining this with Lemma 4, we can take φ(r) = r. With taking h(x) = min{x 1 , x 2 } and recalling (19), we have
h(x) ≤ −2|u| 3 − 2|u|⇒D + h(x) ≥ −0.5h(x),
which implies the set
S = {x : h(x) + 2 u 3 + 2 u ≥ 0}
is forward robustly invariant. Clearly, we can see that S is a safety set obtained from the safety sets S 1 and S 2 in the lower dimensional spaces.
V. Conclusion
We have developed a small-gain theorem based on ISSfbarrier functions for safety verification. It is shown that an interconnected system, which consists of two ISSf subsystems, is ISSf if the absolute value of the composition of ISSf gains of two subsystems is smaller than that of the identity function.
Appendix
A. Proof of Lemma 3
Suppose − 1 2 ≤ ρ 0 (r) < 0 for all r < 0 and 0 < ρ 0 (r) ≤ 1 2 for all r > 0. Otherwise, we use
max{− 1 2 , ρ 0 (r)}, if r < 0 0, if r = 0 min{ 1 2 , ρ 0 (r)}, if r > 0
to replace ρ 0 (r). Let ρ 1 (0) = 0 and
ρ 1 (r) = ρ − 1 (r), if r < 0 ρ + 1 (r), if r > 0 (25) with ρ − 1 (r) = max s∈[−2,r] ρ 0 (s) if − 1 ≤ r < 0; max s∈[r−1,−1] ρ 0 (s) if r < −1; ρ + 1 (r) = min s∈[r,2] ρ 0 (s) if 0 < r ≤ 1 min s∈[1,r+1] ρ 0 (s) if r > 1.
Since ρ 0 (r) < 0 for r ∈ [−2, 0),
lim r→0 − ρ 1 (r) = max s∈[−2,0] ρ 0 (s) = 0;
and analogously, lim r→0 + ρ 1 (r) = min s∈ [r,2] ρ 0 (s) = 0.
Thus, ρ 1 is continuous at zero. Clearly, ρ 1 is continuous at r = −1 and at r = 1. Thus, ρ 1 is continuous on R.
To get a desired function ρ satisfying the properties given in Lemma 3, we take
ρ(r) = r+1 r ρ 1 (s)ds, if r < −1; 0 r ρ 1 (s)ds, if − 1 ≤ r < 0; r 0 ρ 1 (s)ds, if 0 ≤ r ≤ 1; r r−1 ρ 1 (s)ds if r > 1.(26)
Since ρ 1 is a continuous function, we can verify that ρ is continuously differentiable on R. Note that ρ 0 (r) ≤ ρ 1 (r) < 0 for r < 0, and 0 < ρ 1 (r) ≤ ρ 0 (r) for r > 0. As a result, it is easy to see that ρ ′ (r) ≤ |ρ 1 (r)| ≤ 1 2 for all r ∈ R. The second requirement given in Lemma 3 is met.
Next, we show that ρ also meets the first requirement of Lemma 3. Note that ρ 1 is not increasing on (−∞, −1) ∪ (1, ∞), and not decreasing on (−1, 1). Also note that ρ 1 (r + 1) ≥ ρ 0 (r) for r ≤ −1, and ρ 1 (−1) ≤ ρ 0 (r) for all r ≥ 1. Then, with (26), it can be observed that ρ(r) ≥ −rρ 1 (r) ≥ ρ 0 (r), ∀r ∈ [−1, 0); ρ(r) ≥ ρ 1 (−1) ≥ ρ 0 (r), ∀r ∈ (−2, −1); ρ(r) ≥ ρ 1 (r + 1) ≥ ρ 0 (r), ∀r ∈ (−∞, −2].
The first and the third inequalities are easy to verify by noting that ρ 1 is not decreasing on (−1, 0) and not increasing on (−∞, −2]. For the second inequality, note that ρ 1 (−1) ≤ ρ 1 (r) < 0 for all r ∈ (−2, −1). Similarly, ρ(r) ≤ rρ 1 (r) ≤ ρ 0 (r), ∀r ∈ (0, 1]; ρ(r) ≤ ρ 1 (1) ≤ ρ 0 (r), ∀r ∈ (1, 2); ρ(r) ≤ ρ 1 (r − 1) ≤ ρ 0 (r), ∀r ∈ [2, ∞).
Thus, the conclusion follows.
B. Proof of Lemma 4
Let
ρ 0 (r) = 1 2 [r − φ 1 • φ 2 (r)].(27)
According to (13),
φ 1 • φ 2 (r) > r − ρ 0 (r), if r < 0; φ 1 • φ 2 (r) < r − ρ 0 (r), if r > 0;
and hence, φ 2 (r) > φ −1 1 (r − ρ 0 (r)), ∀r < 0; φ 2 (r) < φ −1 1 (r − ρ 0 (r)), ∀r > 0. By Lemma 3, there exists a continuously differentiable function ρ : R → R with ρ ′ (r) ≤ 1 2 such that ρ 0 (r) < ρ(r) < 0 for each r < 0, and 0 < ρ(r) < ρ 0 (r) for each r > 0. Without loss of generality, we assume that |ρ(r)| < |r|. Now we let φ(0) = 0 and φ(r) = 1 ρ(r) r r−ρ(r) φ −1 1 (s)ds, ∀r 0.
With such a function, it yields immediately that φ −1 1 (r) < φ(r) < φ −1 1 (r − ρ(r)) < φ 2 (r) for all r < 0, and φ 2 (r) < φ −1 1 (r − ρ(r)) < φ(r) < φ −1 1 (r) for all r > 0. Recall that φ −1 1 (0) = 0 and φ −1 1 is continuous on R. It means that lim r→0 + φ(r) = lim r→0 − φ(r) = 0, which further implies that φ(r) is continuous at zero, and consequently, φ(r) is continuous on R as well.
Since ρ is continuously differentiable on R, φ is continuously differentiable on R\0, and φ ′ (r) = − ρ ′ (r) ρ 2 (r) Then we show that φ ′ (r) > 0 for all r < 0. According to ρ ′ (r), we have the following two cases.
• If ρ ′ (r) ≤ 0, then − ρ ′ (r) ρ(r) r r−ρ(r) φ −1 1 (s)ds + ρ ′ (r)φ −1 1 (r − ρ(r)) ≤ −ρ ′ (r)φ −1 1 (r − ρ(r)) + ρ ′ (r)φ −1 1 (r − ρ(r)) ≤ 0 (30) and hence, by substituting this into (29), one obtains φ ′ (r) > 0.
• If ρ ′ (r) > 0, then − ρ ′ (r) ρ(r) r r−ρ(r) φ −1 1 (s)ds + ρ ′ (r)φ −1 1 (r − ρ(r)) ≤ −ρ ′ (r)φ −1 1 (r) + ρ ′ (r)φ −1 1 (r − ρ(r))
and hence,
φ ′ (r) ≥ 1 ρ(r) [1 − ρ ′ (r)][φ −1 1 (r) − φ −1 1 (r − ρ(r))] ≥ 1 2ρ(r) [φ −1 1 (r) − φ −1 1 (r − ρ(r))] > 0.(32)
By combining the two cases above, one can conclude that φ ′ (r) > 0 for all r < 0. Moreover, we can obtain from [15, Lemma A.1] that φ ′ (r) > 0 for all r > 0.
s)ds + ρ ′ (r)φ −1 1 (r − ρ(r)) . (29)
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|
[] |
[
"Conceptual and practical bases for the high accuracy of machine learning interatomic potentials",
"Conceptual and practical bases for the high accuracy of machine learning interatomic potentials"
] |
[
"Akira Takahashi \nDepartment of Materials Science and Engineering\nKyoto University\n606-8501KyotoJapan\n",
"Atsuto Seko \nDepartment of Materials Science and Engineering\nKyoto University\n606-8501KyotoJapan\n\nCenter for Elements Strategy Initiative for Structure Materials (ESISM)\nKyoto University\n606-8501KyotoJapan\n\nCenter for Materials Research by Information Integration\nNational Institute for Materials Science\n305-0047TsukubaJapan\n\nJST, PRESTO\n332-0012KawaguchiJapan\n",
"Isao Tanaka \nDepartment of Materials Science and Engineering\nKyoto University\n606-8501KyotoJapan\n\nCenter for Elements Strategy Initiative for Structure Materials (ESISM)\nKyoto University\n606-8501KyotoJapan\n\nCenter for Materials Research by Information Integration\nNational Institute for Materials Science\n305-0047TsukubaJapan\n\nFine Ceramics Center\nNanostructures Research Laboratory\n456-8587NagoyaJapan, Japan\n"
] |
[
"Department of Materials Science and Engineering\nKyoto University\n606-8501KyotoJapan",
"Department of Materials Science and Engineering\nKyoto University\n606-8501KyotoJapan",
"Center for Elements Strategy Initiative for Structure Materials (ESISM)\nKyoto University\n606-8501KyotoJapan",
"Center for Materials Research by Information Integration\nNational Institute for Materials Science\n305-0047TsukubaJapan",
"JST, PRESTO\n332-0012KawaguchiJapan",
"Department of Materials Science and Engineering\nKyoto University\n606-8501KyotoJapan",
"Center for Elements Strategy Initiative for Structure Materials (ESISM)\nKyoto University\n606-8501KyotoJapan",
"Center for Materials Research by Information Integration\nNational Institute for Materials Science\n305-0047TsukubaJapan",
"Fine Ceramics Center\nNanostructures Research Laboratory\n456-8587NagoyaJapan, Japan"
] |
[] |
Machine learning interatomic potentials (MLIPs) based on a large dataset obtained by density functional theory (DFT) calculation have been developed recently. This study gives both conceptual and practical bases for the high accuracy of MLIPs, although MLIPs have been considered to be simply an accurate black-box description of atomic energy. We also construct the most accurate MLIP of the elemental Ti ever reported using a linearized MLIP framework and many angulardependent descriptors, which also corresponds to a generalization of the modified embedded atom method (MEAM) potential.PACS numbers: 31.50. Bc,71.15.Pd
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10.1103/physrevmaterials.1.063801
|
[
"https://arxiv.org/pdf/1708.02741v1.pdf"
] | 119,352,636 |
1708.02741
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d80dbb028620041e0a82518b425896eb494b8ec2
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Conceptual and practical bases for the high accuracy of machine learning interatomic potentials
9 Aug 2017
Akira Takahashi
Department of Materials Science and Engineering
Kyoto University
606-8501KyotoJapan
Atsuto Seko
Department of Materials Science and Engineering
Kyoto University
606-8501KyotoJapan
Center for Elements Strategy Initiative for Structure Materials (ESISM)
Kyoto University
606-8501KyotoJapan
Center for Materials Research by Information Integration
National Institute for Materials Science
305-0047TsukubaJapan
JST, PRESTO
332-0012KawaguchiJapan
Isao Tanaka
Department of Materials Science and Engineering
Kyoto University
606-8501KyotoJapan
Center for Elements Strategy Initiative for Structure Materials (ESISM)
Kyoto University
606-8501KyotoJapan
Center for Materials Research by Information Integration
National Institute for Materials Science
305-0047TsukubaJapan
Fine Ceramics Center
Nanostructures Research Laboratory
456-8587NagoyaJapan, Japan
Conceptual and practical bases for the high accuracy of machine learning interatomic potentials
9 Aug 2017(Dated: July 30, 2018)
Machine learning interatomic potentials (MLIPs) based on a large dataset obtained by density functional theory (DFT) calculation have been developed recently. This study gives both conceptual and practical bases for the high accuracy of MLIPs, although MLIPs have been considered to be simply an accurate black-box description of atomic energy. We also construct the most accurate MLIP of the elemental Ti ever reported using a linearized MLIP framework and many angulardependent descriptors, which also corresponds to a generalization of the modified embedded atom method (MEAM) potential.PACS numbers: 31.50. Bc,71.15.Pd
Interatomic potentials (IPs) have played a central role in performing atomistic simulations, such as molecular dynamics simulation. A wide variety of conventional IPs have been developed by considering the nature of chemical bonding in specific systems of interest, such as Lennard-Jones [1], embedded atom method (EAM) [2][3][4], modified EAM (MEAM) [5,6], and Tersoff [7][8][9] potentials. However, the accuracy and transferability of conventional IPs are often lacking owing to the simplicity of their potential forms. As an example, the phonon dispersion relationships of hexagonal close-packed (HCP) Ti computed from several EAM and MEAM potentials are shown in Fig. 1, along with that computed on the basis of the density functional theory (DFT). The overall phonon dispersions of EAM and MEAM potentials are scattered and markedly deviate from that obtained by DFT calculation.
On the other hand, the machine learning IP (MLIP) based on a large dataset obtained by DFT calculation has great potential for improving its accuracy and transferability effectively. Once the MLIP is established, it does not increase the order of computational cost as compared with conventional IPs. The MLIP has also been increasingly applied to a wide range of materials regardless of their type of chemical bonding. Its frameworks applicable to periodic systems have recently been proposed [21][22][23].
Although the MLIP can provide an accurate energy description, its physical interpretation or relationship with the existing IPs is still lacking. In this study, we introduce an interpretation of the MLIP on the basis of the framework of EAM and MEAM potentials. The interpretation provides a conceptual basis for the high accuracy of the MLIP. Secondly, we develop the most accurate MLIP of the elemental Ti ever reported using a linearized MLIP framework. As shown later, the high accuracy of
10 Γ A H K Γ Γ A H K Γ Γ A H K Γ FIG. 1.
Phonon dispersion curves of elemental HCP Ti calculated using conventional EAM [10][11][12][13] and MEAM [14][15][16][17] potentials. Some of these curves are obtained from the interatomic potential repository project [18] and KIM project [19]. Black broken lines indicate the phonon dispersion curves obtained by DFT calculation. Force constants are calculated using the lammps [20] code.
the linearized MLIP implies that the high accuracy and transferability of MLIPs are based mainly on the use of a large number of relevant descriptors, although it has been considered that the use of flexible black-box functions, such as neural network and Gaussian process models, is essential for modeling atomic energy. The framework of EAM potentials is based on the concept of the embedding energy of an atom into a host described by electron density [24]. The embedding energy of atom i is defined as a functional of the host electron density ρ(r) expressed as
E (i) = F (i) [ρ(r)] ,(1)
where F (i) denotes the embedding energy functional for atom i. Although the application of this concept is not exclusive to metallic systems, the framework of EAM potentials is compatible only with metallic systems owing to the introduction of some approximations. A main approximation is the uniform density approximation (UDA), in which the embedding energy is assumed to be a function of the scalar local electron density, written as
E (i) = F (ρ(r i )) ,(2)
where r i denotes the position of atom i. Another one is a pairwise approximation in which the local electron density is assumed to be equal to the sum of contributions from neighboring atoms expressed by a single pairwise function. Adding a short-range pairwise interaction, the EAM atomic energy is expressed as
E (i) = F j p(r ij ) + 1 2 j φ(r ij ),(3)
where p(r ij ) and φ(r ij ) denote the pairwise contribution of the neighbor atom j to the local electron density and short-range pairwise interaction including repulsive energy, respectively [2]. In an extended manner, the MEAM atomic energy is given by
E (i) = F (ρ(r i )) + 1 2 j φ(r ij ),(4)ρ(r i ) = j p(r ij ) + j,k f (r ij )f (r ik )g(cos γ jik ),(5)
where the local electron density is described by a threebody function g in addition to the pairwise contribution.
Since the function forms of p, f , and g have not been established, a wide range of approximated forms have been proposed in the literature. In addition, polynomials and spline models have been simply used as function F . On the other hand, all MLIPs with pairwise descriptors are formulated as
E (i) = F b (i) 10 , b (i) 20 , . . . , b (i) nmax0 ,(6)
where b
(i)
n0 denotes a pairwise descriptor expressed as
b (i) n0 = j f n (r ij ).(7)
A large number of pairwise descriptors are generally used for formulating MLIPs, and neural network models, Gaussian process models, and polynomials have been used as functions F . This formulation is obviously a generalization of the EAM atomic energy. Similarly, most MLIPs with angular-dependent descriptors are formulated as
E (i) = F (b (i) 10 , b (i) 20 , . . . , b (i) 11 , b (i) 21 , . . . , b (i) nmaxlmax ),(8)
where b
(i)
nl denotes an angular-dependent descriptor. Most angular-dependent descriptors specified by number l belong to the class of angular Fourier series, which corresponds to a set of rotationally invariant descriptors derived from spherical harmonics [25]. The angular Fourier series is given by
b (i) nl = j,k f n (r ij )f n (r ik ) cos l (γ jik ) (l ≥ 1),(9)
where γ jik denotes the bond angle between atoms j−i−k.
From the comparison between Eqns. (4) and (8), the formulation of the MLIP with angular-dependent descriptors is clearly a generalization of the MEAM potential.
We have demonstrated that the MLIP formulations can be regarded as the generalizations of the EAM and MEAM potentials by comparing their equations for atomic energy. We will show that the MLIP formulations can also be derived from the concept of embedding energy using a higher-order approximation beyond the UDA. This derivation interprets MLIPs. Using a higherorder approximation for the embedding energy functional (Eqn.(1)), atomic energy may be described by a function of local electron density and its derivatives as
E (i) =F (i) [ρ(r)] =F ρ(r i ), ∂ρ ∂x (r i ), ∂ρ ∂y (r i ), ∂ρ ∂z (r i ), . . . .(10)
Then, the local electron density is assumed to be described by direction-dependent contributions from neighbor atoms, ρ(r i ) = j p(r ij ). Eqn.(10) is rewritten as
E (i) = F j p(r ij ), j ∂ ∂x p(r ij ), j ∂ ∂y p(r ij ), j ∂ ∂z p(r ij ), . . . .(11)
Expanding the electron density contribution p using a basis set {f n (r ij )} n=1,2,...,nmax as
p(r ij ) = nmax n=1 c n f n (r ij ),(12)
embedding atomic energy is written as
E (i) =F j f 1 (r ij ), . . . , j f nmax (r ij ) ,(13)
where another symbolF for the embedding energy function is derived from both function F and expansion coefficients {c n } n=1,2,...,nmax . Replacing the vector r ij with the pair distance r ij , Eqn.(13) becomes the pairwise MLIP formulation. Generally, the basis set is not necessarily pairwise. When functions based on spherical harmonics are used as a basis set and functionF satisfying the rotational invariance, the angular-dependent MLIP (Eqn. (8)) is derived. Thus, MLIP formulations are derived from the concept of embedding energy using an approximation beyond the UDA. This implies that the lack of accuracy and transferability of the EAM and MEAM potentials can be ascribed to their poor representation for embedding energy due to the limitation of the UDA [26].
On the basis of the relationship between MLIPs and EAM potentials, we construct two MLIPs for the elemental Ti in this study. The first one is constructed by a third-order polynomial approximation of Eqn.(6) expressed as
E (i) = w 0 + n w n b (i) n0 + n,n ′ w n,n ′ b (i) n0 b (i) n ′ 0 + n,n ′ ,n ′′ w n,n ′ ,n ′′ b (i) n0 b (i) n ′ 0 b (i) n ′′ 0 ,(14)
where w 0 , w n , w n,n ′ , and w n,n ′ ,n ′′ denote regression coefficients. The second one is constructed by a secondorder polynomial approximation of Eqn.(8) with angular Fourier series descriptors expressed as
E (i) = w 0 + n,l w n,l b (i) nl + n,l,n ′ ,l ′ w n,l,n ′ ,l ′ b (i) nl b (i) n ′ l ′ .(15)
Here, we fixed l max to ten. We used pairwise Gaussiantype functions as radial functions f n (r) expressed as
f n (r) = f c (r) exp −p(r − q n ) 2 ,(16)
where f c (r) denotes a cosine-type cutoff function. p and q n are given parameters, and we used a single p value and a set of q n values given by an arithmetic sequence. Also in the EAM and MEAM potentials, Gaussian functions have sometimes been used for expressing the pairwise electron density contribution. In addition, a polynomial approximation for the embedding energy function F has been used for EAM and MEAM potentials. Therefore, the only difference between the MLIP and EAM (MEAM) potentials is in the number of descriptors being used in the formulation of atomic energy. Eqns. (14) and (15) are also a generalization of our previous linearized model where only the power of b n is considered [23,27]. Training and test datasets were generated by DFT calculation for 2700 and 300 atomic configurations, respectively. We firstly optimized the atomic positions and lattice constants of face-centered cubic (FCC), bodycentered cubic (BCC), HCP, simple cubic (SC), ω, and β-Sn structures, and supercells were then developed by the 2 × 2 × 2, 3 × 3 × 3, 3 × 3 × 3, 4 × 4 × 4, 3 × 3 × 3, and 2 × 2 × 2 expansions of their conventional unit cells, respectively. Atomic configurations were generated by isotropic expansion, random expansions, random distortions, and random displacements. Both the energy and forces acting on each atom were used for training. Therefore, the total number of training data was 430650. We adopted linear ridge regression to estimate MLIPs involving the minimization of a function defined by the energy and forces acting on atoms. The function is defined elsewhere [27]. DFT calculation was performed using the plane-wave basis projector augmented wave (PAW) method [28,29] within the Perdew-Burke-Ernzerhof exchange-correlation functional [30] as implemented in the vasp code [31,32]. The cutoff energy was set to 400 eV. The total energies converged to less than 10 −3 meV/supercell. The lattice constants of the ideal structures were optimized until the residual forces became less than 10 −3 eV/Å.
We will show the accuracy of MLIPs for the elemental Ti. We regard the root mean square error (RMSE) for the energy of the test dataset as a measure of prediction error. Figure 2 (a) shows the dependence of prediction error on the number of regression coefficients. The number of regression coefficients was controlled using only the number of radial functions f n for both pairwise and angular-dependent MLIPs. By examining the convergence of RMSE with respect to the number of regression coefficients, we obtained an optimized pairwise MLIP with a prediction error of 3.8 meV/atom (2925 coefficients). Similarly, we obtained an optimized angular-dependent MLIP with a prediction error of 0.5 meV/atom (35245 coefficients), which means that it is very important to consider angular-dependent descriptors for expressing the interatomic interactions of the elemental Ti. Figure 2 (b) also shows the distribution of the absolute energy difference between DFT and MLIPs for the test dataset. The distribution for the angulardependent MLIP is much narrower than that for the pairwise MLIP, which is consistent with the degree of prediction error. For the angular-dependent MLIP, more than a hundred structures show the absolute energy difference within only 0.1 meV/atom. In addition, some outliers can be found in the distribution for the pairwise MLIP. A structure shows the maximum absolute energy difference of 23.0 meV/atom of the pairwise MLIP, whereas the absolute energy difference of the angular-dependent MLIP does not exceed 2.8 meV/atom. We then compare the distribution of the energy difference between DFT and IPs for the test data, elastic constants and phonon dispersion relationships obtained from EAM [10] and MEAM [17] potentials, the pairwise MLIP and the angular-dependent MLIP along with a reference of the DFT calculation. Figure 3 shows the comparison of the distribution of energy difference between DFT and IPs for the test dataset. EAM and MEAM potentials show very large energy differences for almost the entire test dataset, while both the MLIPs show very small energy differences. hand, the pairwise MLIP is worst for predicting most of the elastic constants and bulk moduli of both HCP and BCC structures, despite its small prediction error. Including angular-dependent terms, the prediction of elastic constants and bulk moduli is much improved. This is consistent with the fact that the angular-dependent descriptors are essential for predicting the mechanical behavior of the elemental Ti. The phonon dispersion curves were also calculated using the supercell approach [33] for HCP and BCC structures with the DFT equilibrium lattice constant. To evaluate a dynamical matrix, each symmetrically independent atomic position was displaced by 0.01Å. The forces acting on atoms were then computed. Supercells were fabricated by the 4 × 4 × 4 expansion of conventional unit cells for both HCP and BCC structures. Phonon calculations were performed using the phonopy code [34]. Figure 5 shows the phonon dispersion curves of (a) HCP and (b) BCC structures computed from EAM and MEAM potentials, and the MLIPs. As shown in Fig. 5, the phonon dispersion curves from EAM and MEAM potentials differ largely from that obtained by DFT calculation. Imaginary phonon modes are observed in the DFT phonon dispersion for the BCC structure, but not in the EAM and MEAM phonon dispersions. Although the pairwise MLIP reproduces the DFT phonon dispersion better than the EAM and MEAM potentials, phonon frequencies tend to be overestimated. The angular-dependent MLIP significantly improves the inconsistency of phonon frequency.
In summary, this study provides both conceptual and practical bases for the high accuracy of MLIPs. We have shown that MLIPs can be regarded as a description of embedding energy beyond the UDA, which is a funda- mental approximation of both EAM and MEAM potentials. In other words, the high accuracy of MLIPs is based on the use of higher-order approximation of embedding energy. We have then applied a linearized MLIP approach to the elemental Ti, which is also a generalization of the MEAM potential. An angular-dependent linearized MLIP predicts the energetics and phonon frequencies much more accurately than the existing MEAM potentials. The only difference between the MEAM potentials and linearized MLIP is in the number of descriptors being used. This indicates that the use of a systematic set of numerous descriptors is the most important practical feature for building MLIPs with high accuracy.
enough to express some functions such as j f1(rij )
Zhou et al.) EAM (Mendelev et al. Type 1) EAM (Mendelev et al. Type 2) EAM (Mendelev et al. Type 3) EAM (Ackland) EAM (Zope and Mishin, for TiAl) MEAM (Ko et al., for NiTi ) MEAM (Gibson et al.) MEAM (Hennig et al.
FIG. 2 .
2(a) Dependence of RMSE of MLIPs on number of terms for elemental Ti. (b) Distribution of absolute energy difference between DFT values and MLIPs.
FIG. 3 .
3Distribution of energy difference between DFT and IPs.
Figure 4 FIG. 4 .
44shows the elastic constants and bulk moduli of (a) HCP-Ti and (b) BCC-Ti obtained from EAM and MEAM potentials and the MLIPs. The elastic constants of EAM and MEAM potentials are close to those of DFT calculation, except for the C 33 of HCP and the C 44 of BCC obtained from the EAM potential. On the other Elastic constant C and bulk modulus B values of (a) HCP-Ti and (b) BCC-Ti calculated on the basis of DFT and IPs.
FIG. 5 .
5Phonon dispersion curves of (a) HCP-Ti and (b) BCC-Ti calculated from IPs. Broken black lines indicate the phonon dispersion curves calculated on the basis of DFT.
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[
"THE CYLINDRICAL CONTACT HOMOLOGY OF UNIVERSALLY TIGHT SUTURED CONTACT SOLID TORI",
"THE CYLINDRICAL CONTACT HOMOLOGY OF UNIVERSALLY TIGHT SUTURED CONTACT SOLID TORI"
] |
[
"Roman Golovko "
] |
[] |
[] |
We calculate the sutured version of cylindrical contact homology of a sutured contact solid torus (S 1 × D 2 , Γ, ξ), where Γ is arbitrary and ξ is a universally tight contact structure.2000 Mathematics Subject Classification. Primary 53D42; Secondary 57M50, 53D10.
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10.2140/pjm.2015.274.73
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[
"https://arxiv.org/pdf/1006.4073v8.pdf"
] | 119,717,610 |
1006.4073
|
31dcd9c1fb8dc6a23d60557d39368dc1c0435cd3
|
THE CYLINDRICAL CONTACT HOMOLOGY OF UNIVERSALLY TIGHT SUTURED CONTACT SOLID TORI
24 Jan 2012
Roman Golovko
THE CYLINDRICAL CONTACT HOMOLOGY OF UNIVERSALLY TIGHT SUTURED CONTACT SOLID TORI
24 Jan 2012
We calculate the sutured version of cylindrical contact homology of a sutured contact solid torus (S 1 × D 2 , Γ, ξ), where Γ is arbitrary and ξ is a universally tight contact structure.2000 Mathematics Subject Classification. Primary 53D42; Secondary 57M50, 53D10.
INTRODUCTION
The cylindrical contact homology of a (closed) contact manifold was introduced by Eliashberg and Hofer and is the simplest version of the symplectic field theory of Eliashberg, Givental and Hofer [6]. It is the homology of a differential graded module whose differential counts genus zero holomorphic curves in the symplectization with one positive puncture and one negative puncture.
A natural condition to impose on a compact, oriented contact (2m + 1)-manifold (M, ξ) with boundary is to require that ∂M be convex, i.e., there is a contact vector field X transverse to ∂M. To a transverse contact vector field X we can associate the dividing set Γ = Γ X ⊂ ∂M, namely the set of points x ∈ ∂M such that X(x) ∈ ξ(x). By the contact condition, (Γ, ξ ∩ T Γ) is a (2m − 1)dimensional contact submanifold of (M, ξ); the isotopy class of (Γ, ξ ∩ T Γ) is independent of the choice of X. We will denote by (M, Γ, ξ) the contact manifold (M, ξ) with convex boundary and dividing set Γ = Γ X ⊂ ∂M with respect to some transverse contact vector field X. Note that the actual boundary condition we need is slightly different and is called a sutured boundary condition. (In the early 1980's, Gabai developed the theory of sutured manifolds [7], which became a powerful tool in studying 3-manifolds with boundary.) For the moment we write (M, Γ, ξ) to indicate either the convex boundary condition or the sutured boundary condition.
It turns out that there is a way to generalize cylindrical contact homology to sutured manifolds. This is possible by imposing a certain convexity condition on the contact form. This construction is described in the paper of Colin, Ghiggini, Honda and Hutchings [3] and will be summarized in Section 2.
In this paper, we construct a sutured contact solid torus with 2n parallel sutures of slope k l using the gluing method of Colin, Ghiggini, Honda and Hutchings [3], and calculate the sutured cylindrical contact homology of it. Here n ∈ N, (k, l) = 1 and |k| > l > 0. In order to define the slope, we choose an oriented identification ∂(S 1 × D 2 ) ≃ T 2 = (R/Z) 2 as follows: map {pt} × ∂D 2 (the meridian) to (1, 0) (slope is 0) and S 1 × {pt} (a longitude) to (0, 1). This calculation, together with the calculation of the sutured cylindrical contact homology of the sutured contact solid torus with 2n parallel longitudinal sutures, where n ≥ 2, that has been done in [8], finishes the calculation of the cylindrical contact homology of (S 1 × D 2 , Γ, ξ), where Γ is arbitrary and ξ is a universally tight contact structure.
Our goal is to prove the following theorem: Theorem 1.1. Let (S 1 ×D 2 , Γ) be a sutured manifold, where Γ is a set of 2n parallel closed curves of slope k l , where (k, l) = 1, |k| > l > 0 and n ∈ N. Then there is a contact form α which makes (S 1 × D 2 , Γ, α) a sutured contact manifold with a universally tight contact structrure ξ = ker α, HC cyl (S 1 ×D 2 , Γ, α) is defined, is independent of the contact form α for ξ = ker α and the almost complex structure J and
HC cyl,h (S 1 × D 2 , Γ, ξ) ≃ Q, for k ∤ h > 0; Q n−1 , for k | h > 0; 0,
otherwise.
Here h corresponds to the homological grading.
ACKNOWLEDGEMENTS
The author is deeply grateful to Ko Honda for his guidance, help and support. He also thanks Dmytro Chebotarov, Oliver Fabert, Paolo Ghiggini, Jian He, Michael Hutchings and Mark McLean for helpful suggestions and interest in his work. In addition, the author is extremely grateful to Andrew Cotton-Clay for his critical comments on the first version of the paper. Finally, the author thanks the Mathematical Sciences Research Institute and the organizers of the "Symplectic and Contact Geometry and Topology" program for their hospitality.
BACKGROUND
The goal of this section is to review definitions of sutured contact manifold and the relative version of cylindrical contact homology. This section can be considered as a summary of [3].
2.1. Review of sutured contact manifolds. In this section we recall some definitions and describe some constructions from [3]. We first start with the notion of a Liouville manifold. Definition 2.1. A Liouville manifold (often also called a Liouville domain) is a pair (W, β) consisting of a compact, oriented 2n-dimensional manifold W with boundary and a 1-form β on W , where ω = dβ is a positive symplectic form on W and the Liouville vector field Y given by i Y (ω) = β is positively transverse to ∂W . It follows that the 1-form β 0 = β| ∂W (this notation means β pulled back to ∂W ) is a positive contact form with kernel ζ.
We now recall the definition of a sutured contact manifold.
Definition 2.2.
A compact oriented 2n + 1-dimensional manifold M with boundary and corners is a sutured contact manifold if it comes with an oriented, not necessarily connected submanifold 1], such that the following holds:
Γ ⊂ ∂M of dimension 2n − 1 (called the suture), together with a neighborhood U(Γ) = [−1, 0] × [−1, 1] × Γ of Γ = {0} × {0} × Γ in M, with coordinates (τ, t) ∈ [−1, 0] × [−1,(1) U ∩ ∂M = ({0} × [−1, 1] × Γ) ∪ ([−1, 0] × {−1} × Γ) ∪ ([−1, 0] × {1} × Γ); (2) ∂M \ ({0} × (−1, 1) × Γ) = R − (Γ) ⊔ R + (Γ),
where the orientation of ∂M agrees with that of R + (Γ) and is opposite that of R − (Γ) and the orientation of Γ agrees with the boundary orientation of R + (Γ);
(3) the corners of M are precisely {0} × {±1} × Γ.
In addition, M is equipped with a contact structure ξ, which is given by the kernel of a positive contact 1-form α such that:
(i) (R ± (Γ), β ± = α| R ± (Γ) ) is a Liouville manifold; (ii) α = Cdt + β inside U(Γ),
where C > 0 and β is independent of t and does not have a dt-term;
(iii) ∂ τ = Y ± , where Y ± is a Liouville vector field for β ± .
Such a contact form α is said to be adapted to (M, Γ, U(Γ)).
Here we briefly describe the way to glue sutured contact manifolds. This procedure was first described by Colin and Honda in [4] and then generalized by Colin, Ghiggini, Honda and Hutchings in [3].
Let (M ′ , Γ ′ , U(Γ ′ ), ξ ′ ) be a sutured contact 3-manifold with an adapted contact form α ′ . We denote by π the projection along ∂ t defined on U(Γ ′ ).
Take 2-dimensional submanifolds P ± ⊂ R ± (Γ ′ ) such that ∂P ± is the union of (∂P ± ) ∂ ⊂ ∂R ± (Γ ′ ), (∂P ± ) int ⊂ int(R ± (Γ ′ )
) and ∂P ± is positively transversal to the Liouville vector field Y ′ ± on R ± (Γ ′ ). Whenever we refer to (∂P ± ) int and (∂P ± ) ∂ , we assume that closures are taken as appropriate. Moreover we make the assumption that π((∂P − ) ∂ ) ∩ π(∂P + ) ∂ ) = ∅.
Let ϕ be a diffeomorphism which sends (P + , β ′ + | P + ) to (P − , β ′ − | P − ) and takes (∂P + ) int to (∂P − ) ∂ and (∂P + ) ∂ to (∂P − ) int . Note that, since dim M = 3, we only need β ′ + | P + and ϕ * (β ′ − | P − ) to match up on ∂P + , since we can linearly interpolate between primitives of positive area forms on a surface.
Topologically, we construct the sutured manifold (M, Γ) from (M ′ , Γ ′ ) and the gluing data (P + , P − , ϕ) as follows:
Let M = M ′ / ∼, where • x ∼ ϕ(x) for all x ∈ P + ; • x ∼ x ′ if x, x ′ ∈ π −1 (Γ ′ ) and π(x) = π(x ′ ) ∈ Γ ′ .
Then
R ± (Γ) = R ± (Γ ′ ) \ P ± (∂P ± ) int ∼ π ± ((∂P ∓ ) ∂ ) and Γ = Γ ′ \ π(∂P + ⊔ ∂P − ) π((∂P + ) int ∩ (∂P + ) ∂ ) ∼ π((∂P − ) int ∩ (∂P − ) ∂ ).
For the detailed description of the gluing procedure we refer to [3]. Finally, we describe the way to complete sutured contact manifold (M, α) to a noncompact contact manifold (M * , α * ). This construction was first described in [3].
Let (M, Γ, U(Γ), ξ) be a sutured contact manifold with an adapted contact form α. The form α is then given by
Cdt+β ± on [1−ε, 1]×R + (Γ) and [−1, −1+ε]×R − (Γ) of R + (Γ) = {1}×R + (Γ) and R − (Γ) = {−1} × R − (Γ), where t ∈ [−1, −1 + ε] ∪ [1 − ε, 1] extends the t-coordinate on U.
On U, α = Cdt + β, β = β + = β − and ∂ τ is a Liouville vector field Y for β. We first extend α to [1, ∞) × R + (Γ) and (−∞, −1] × R − (Γ) by taking Cdt + β ± as appropriate. The boundary of this new manifold is {0} × R × Γ. Notice that since ∂ τ = Y , the form dβ| [−1,0]×{t}×Γ is the symplectization of β| {0}×{t}×Γ in the positive τ -direction. We glue [0,
∞) × R × Γ with the form Cdt + e τ β 0 , where β 0 is the pullback of β to {0} × {t} × Γ.
We denote by M * the noncompact extension of M described above and by α * the extension of α to M * .
2.2. Review of cylindrical contact homology. Let (M, Γ, U(Γ), ξ) be a sutured contact manifold with an adapted contact form α and (M * , α * ) be its completion.
The Reeb vector field R α * that is associated to a contact form α * is given by dα * (R α * , ·) = 0 and α * (R α * ) = 1.
We assume that R α * is nondegenerate, i.e., the first return map along each (not necessarily simple) periodic orbit does not have 1 as an eigenvalue. Observe that nondegeneracy can always be achieved by a small perturbation. Remark 2.3. Note that every periodic orbit of R α * lies in M. Hence, the set of periodic Reeb orbits of R α * coincides with the set of periodic Reeb orbits of R α .
A Reeb orbit γ is called elliptic or positive (respectively negative) hyperbolic if the eigenvalues of P γ are on the unit circle or the positive (resp. negative) real line respectively.
If τ is a trivialization of ξ over γ, we can then define the Conley-Zehnder index. In 3-dimensional situation, we can explicitly describe the Conley-Zehnder index and its behavior under multiple covers as follows: 10]). If γ is elliptic, then there is an irrational number φ ∈ R such that P γ is conjugate in SL 2 (R) to a rotation by angle 2πφ and
Proposition 2.4 ([µ τ (γ k ) = 2⌊kφ⌋ + 1,
where 2πφ is the total rotation angle with respect to τ of the linearized flow around the orbit.
If γ is positive (respectively negative) hyperbolic, then there is an even (respectively odd) integer r such that the linearized flow around the orbit rotates the eigenspaces of P γ by angle πr with respect to τ and µ τ (γ k ) = kr.
A closed orbit of R α * is said to be good if it does not cover a simple orbit γ an even number of times, where the first return map ξ γ 0 → ξ γ T has an odd number of eigenvalues in the interval (−1, 0). Here T is the period of the orbit γ. An orbit that is not good is called bad.
We now recall the notion of an almost complex structure on R × M * that is tailored to (M * , α * ). Let (W, β) be a Liouville manifold and ζ be the contact structure given on ∂W by ker(β 0 ), where β 0 = β| ∂W . In addition, let ( W , β) be the completion of (W, β), i.e.,
W = W ∪ ([0, ∞) × ∂W ) and β| [0,∞)×∂W = e τ β 0 , where τ is the [0, ∞)-coordinate. An almost complex structure J 0 on W is β-adapted if J 0 is β 0 -adapted on [0, ∞) × ∂W ; and dβ(v, J 0 v) > 0 for all nonzero tangent vectors v on W .
Definition 2.5. Let (M, Γ, U(Γ), ξ) be a sutured contact manifold, α be an adapted contact form and (M * , α * ) be its completion. We say that an almost complex structure J on R × M * is tailored to (M * , α * ) if the following hold:
(1) J is α * -adapted, i.e, J is R-invariant, J(ξ) = ξ, dα(v, Jv) > 0 for nonzero v ∈ ξ and J(∂ s ) = R α * , where s denotes the R-coordinate;
(2) J is ∂ t -invariant in a neighborhood of M * \ int(M);
(3) The projection of J to T R ± (Γ) is a β ± -adapted almost complex structure J 0 on the com-
pletion ( R + (Γ), β + ) ( R − (Γ), β − ) of the Liouville manifold (R + (Γ), β + ) (R − (Γ), β − ). Moreover, the flow of ∂ t identifies J 0 | R + (Γ)\R + (Γ) and J 0 | R − (Γ)\R − (Γ) .
Given a sutured contact manifold (M, Γ, U(Γ), α) and an α * -adapted almost complex structure J, we define the sutured cylindrical contact homology group HC cyl (M, Γ, α, J) to be the cylindrical contact homology of (M * , α * , J). The cylindrical contact homology chain complex C(α, J) is a Q-module freely generated by all good Reeb orbits, where the grading | · | and the boundary map ∂ are defined as in [1] with respect to the α * -adapted almost complex structure J. The homology of C(α, J) is the sutured cylindrical contact homology group HC cyl (M, Γ, α, J).
For our calculations we need the following construction of a "global" symplectic trivialization described in [1]. Assume that all the Reeb orbits of R α are good. Let us now choose trivializations τ (γ) consistently for all Reeb orbits γ. Assume that H 1 (M; Z) is a free module. We pick representatives C 1 , . . . , C s in H 1 (M; Z) for a basis of H 1 (M; Z), together with a trivialization of ξ along each representative C i , i = 1, . . . , s. Now for a Reeb orbit γ, we distinguish the following cases:
(1) [γ] = 0 ∈ H 1 (M; Z). Choose a spanning surface S γ and use it to trivialize ξ along γ.
(2) 0 = [γ] ∈ H 1 (M; Z).
We choose a surface S γ realizing a homology between γ and a linear combination of the representatives C i , i = 1, . . . , s. We then use S γ to extend the chosen trivializations of ξ along the C i , i = 1, . . . , s to γ. We denote the obtained trivialization by τ . To a J-holomorphic curve in M J (γ; γ ′ ), we can glue the chosen surfaces S γ and S γ ′ and obtain a closed surface in M. Let A ∈ H 2 (M; Z) be its homology class; we can use it to decorate the corresponding connected component M J A (γ; γ ′ ) of the moduli space. Using τ we can write
ind(u) = |γ| − |γ ′ | + 2 c 1 (ξ), A (2.2.1) for u ∈ M J A (γ; γ ′ ), where |γ| is the Conley-Zehnder grading of γ defined by |γ| := µ τ (γ) − 1. (2.2.2)
We will use Formulas 2.2.1 and 2.2.2 for our calculations.
In addition, we will need the following fact, which is a consequence of Lemma 5.4 in [2]: Fact 2.6. Let (M, α) be a closed, oriented contact manifold with nondegenerate Reeb orbits and
u = (a, f ) : (Ṡ, j) → (R × M, J) be a J-holomorphic curve in M J (γ; γ ′ )
, where γ and γ ′ are good Reeb orbits, J is an α-adapted almost complex structure on R × M and M J (γ; γ ′ ) is a moduli space of J-holomorphic curves that we consider in cylindrical contact homology. Then the following inequality holds:
A(γ) def = γ α ≥ γ ′ α def = A(γ ′ )
with equality if and only if γ = γ ′ and in this case the moduli space consists of a single element R × γ.
Now we recall the following theorem: When M is closed and R × M is 4-dimensional, the following transversality result has been proven by Momin, see Proposition 2.10 in [11]:
Theorem 2.8 ([11]). Let u ∈ M J (γ; γ ′ ) be such that ind(u) = 1.
Then the linearization of the Cauchy-Riemann operator is surjective at u. Remark 2.9. Observe that Theorem 2.8 does not require J to be generic. Remark 2.13. Observe that Theorem 2.11 and Remark 2.12 rely on the assumption that the machinery, needed to prove the analogous properties for contact homology and cylindrical contact homology in the closed case, works.
CONSTRUCTION
3.1. Gluing map. First we construct H ∈ C ∞ (R 2 ). The time-1 flow of the Hamiltonian vector field associated to H composed with an appropriate rotation will play a role of the gluing map when we will apply the gluing construction described in Section 2.1 to the sutured contact solid cylinder constructed in Section 3.2.
We fix p ∈ R 2 and consider H sing : R 2 → R given by H sing = µr 2 cos(n|k|θ) in polar coordinates about p, where µ > 0, n ≥ 1 and k ∈ Z \ {−1, 0, 1}. Note that H sing is singular only at p. We obtain H ∈ C ∞ (R 2 ) from H sing by perturbing H sing on a disk D(r sing ) about p in such a way that H has n|k| equally spaced saddle points, critical point at p and interpolates with no critical points with H sing on D(r sing ). In other words, H = H sing on R 2 \ D(r sing ). For the level sets of H sing and H in the case n = 1, |k| = 3 we refer to Figure 1.
The construction of H is a modification of the construction described in [5]. We proceed in four steps. (1) We consider
H 1 = H sing + f (r, θ) = H sing + f exp (r, θ) + g(r, θ) = µr 2 cos(n|k|θ) − Ae −mr 2 + g(r, θ),
where A and m are positive constants, and g(r, θ) is a smooth function to be chosen later. We are interested in the critical points of H 1 away from the origin. We calculate
∂H 1 ∂r = 2µr cos(n|k|θ) + 2mrAe −mr 2 + ∂g ∂r , ∂H 1 ∂θ = −n|k|µr 2 sin(n|k|θ).
Thus, at the critical points of H 1 we must have sin(n|k|θ) = 0. In this case, cos(n|k|θ) = ±1. If cos(n|k|θ) = 1, then ∂H 1 ∂r − ∂g ∂r cannot be zero. When cos(n|k|θ) = −1, ∂H 1 ∂r − ∂g ∂r = −2µr + 2mrAe −mr 2 . For r > 0, ∂H 1 ∂r − ∂g ∂r = 0 when e mr 2 = mA µ , i.e., when r = r c := 1 m ln( mA µ ). We impose the restriction that mA > µ. Note that by making m large, we can make r c arbitrarily small. When cos(n|k|θ) = −1, H 1 − g(r, θ) = − µ m (ln( mA µ ) + 1). Let g(r) be equal to µ m (ln( mA µ ) + 1) on the annular neighborhood of r = r c . For such g, H 1 is 0 at the critical points, i.e., at the points (r c , θ), where cos(n|k|θ) = −1.
In summary, we get critical points at one value of r at the values of θ when cos(n|k|θ) = −1, that is, for n|k| values of θ. These are our n|k| saddle points (it's not hard to see they are saddle points; alternatively, we can deduce that they must be for index reasons).
(2) Keeping f exp solely a function of r and keeping g constant, we cut off f exp smoothly starting at some point past r c to give a Hamiltonian H 2 which agrees with H sing + g outside a ball. As long as ∂fexp ∂r < 2µr, there are no new critical points. Note that f exp (r c ) = − µ m . Keeping ∂fexp ∂r near µr c (which, using e.g. A = eµ m , is 1 √ m ), we can bring f exp to zero in a radial distance of a constant times 1 √ m ; i.e. for m large we can make H 2 agree with H sing + g outside an arbitrarily small ball.
For A = eµ m , g = 2 µ m . Then keeping g solely a function of r, we cut off g(r, θ) smoothly starting at some point past the point where H 2 = H sing + g to give Hamiltonian H 3 . As long as ∂g ∂r > −2µr, there are no new critical points. We can make it in such a way that H 3 agrees with H sing outside a small ball.
(3) Recall that H 3 = H sing + f exp + g near the origin and g(r, θ) = 2 µ m > 0. Note that g(r, θ) is small for large m. Now keeping g constant we modify H sing + f exp + g near the origin to give us H 4 which is Br 2 − C near the origin (for B > 0), which corresponds to the Hamiltonian flow rotating at a constant angular rate. Since ∂H 3 ∂r = ∂(H sing +fexp) ∂r > 0 for r < r c , we can patch together Br 2 − C near the origin with H 2 outside a small ball of radius less than r c in a radially symmetric manner to get H 4 such that ∂H 4 ∂r > 0 for r < r c (we do this by choosing C sufficiently large). Note that H 4 has a critical point at the origin. Let p 1 , . . . , p n|k| denote the equally spaced saddle points of H ordered counterclockwise, i.e., R n|k| (p i ) = p i+1 , where R n|k| corresponds to the 2π n|k| -rotation around the center of D(r sing ).
Remark 3.1. We first note that H(p s ) = 0 for s = 1, . . . , n|k|. Hence, by Morse lemma (arguing the same way as in Lemma 3.2 in [8]) we get that there is a neighborhood U s of p s such that H = axy on U s , where s = 1, . . . , n|k| and a > 0. In addition, observe that H is 2π n|k| -symmetric with respect to θ. Therefore, U s 's together with coordinates (x, y) are 2π n|k| -symmetric with respect to θ, i.e., R n|k| (U s ) = U s+1 and coordinates on U s maps to the coordinate on U s+1 , where R n|k| denotes 2π n|k| -rotation with respect to θ. Finally, note that H =Br 2 −C on a neighborhood of the center of D(r sing ), which we call U, whereC > 0 andB is a small positive number and hence Hamiltonian flow rotates at a constant rate near the origin.
3.2. Sutured contact solid tori. In this section, we construct the sutured contact solid torus with 2n sutures of slope k l , where n ∈ N, (k, l) = 1, |k| > l > 0. Let γ p,ps be an embedded curve in R 2 which starts at p and ends at p s for s = 1, . . . , n|k|. For the time being, we can think about γ p,ps as about the segment connecting p and p s .
Lemma 3.2.
There exists a 1-form β on R 2 satisfying the following:
(1) dβ > 0;
(2) its singular foliation given by ker β has isolated singularities and no closed orbits;
(3) β = εc 2 r 2 dθ on U with respect to the polar coordinates whose origin is at the center of D(r sing ); β = εsym 2 (xdy − ydx) on U s with respect to the coordinates from Remark 3.1, where s ∈ {1, . . . , n|k|}; β = 1 2 r 2 dθ on R 2 \ D(r sing ) with respect to the polar coordinates whose origin is at the center of D(r sing ); here 0 < ε c ≪ ε sym ≪ 1; (4) the set of hyperbolic points of the singular foliation of β is given by {q s } n|k| s=1 such that q s lies on γ p,ps outside of U s and U;
(5) β is 2π n|k| -symmetric, i.e., R * n|k| (β) = β, where R n|k| : R 2 → R 2 is a 2π
n|k| -rotation with respect to the center of D(r sing ).
Proof. Consider a singular foliation F on R 2 which satisfies the following:
(1) F is Morse-Smale and has no closed orbits.
(2) The singular set of F consists of elliptic points and hyperbolic points. The elliptic points are the equally spaced saddle points of H and the center of D(r sing ). The set of hyperbolic points of the singular foliation of β is given by {q s } n|k| s=1 such that q s lies on γ p,ps outside of U s and U.
(3) F is oriented and for one choice of orientation the flow is transverse to and exits from ∂D(r sing ). (4) F is 2π n|k| -symmetric with respect to θ. Next, we modify F near each of the singular points so that F is given by β 0 = 1 2 (xdy − ydx) on U s with respect to the coordinates from Remark 3.1 and β 0 = 2xdy + ydx near a hyperbolic point. On R 2 \ D(r sing ), β 0 = 1 2 r 2 dθ with respect to the polar coordinates whose origin is at the center of D(r sing ). In addition, on U, β 0 = 1 2 r 2 dθ with respect to the polar coordinates whose origin is at the center of D(r sing ). From Remark 3.1 it follows that we can do it in such a way that the modification of F is still 2π n|k| -symmetric. Finally, we get F given by β 0 , which satisfies dβ 0 > 0 near the singular points and on R 2 \ D(r sing ). Now let β = gβ 0 , where g is a positive function with dg(X) ≫ 0 outside of U ∪ (∪ n|k| s=1 U s ) ∪ (R 2 \ D(r sing )), g| ∪ n|k| s=1 Us = ε sym , g| U = ε c , g| R 2 \D(r sing ) = 1 and X is an oriented vector field for F (nonzero away from the singular points).
Here 0 < ε c ≪ ε sym ≪ 1. Since dβ = dg ∧ β 0 + g ∧ dβ 0 , dg(X) ≫ 0 guarantees that dβ > 0.i X H (dβ) = 2B ε c ∂ ∂θ (ε c rdr ∧ dθ) = −2Brdr = −dH, and β(X H ) − H = ε c 2 r 2 dθ 2B ε c ∂ ∂θ −Br 2 +C =C.
Next, we work on U s , where s = 1, . . . , n|k|. From Remark 3.3 it follows that β = εsym 2 (xdy − ydx) and H = axy on U s . Let X H be a Hamiltonian vector field defined by i X H dβ = −dH.
We show that
X H = − ax ε sym ∂ ∂x + ay ε sym ∂ ∂y
is a solution of the equation
β(X H ) = H (3.2.1) on U s . We calculate i X H (dβ) = − ax ε sym ∂ ∂x + ay ε sym ∂ ∂y (ε sym dx ∧ dy) = −axdy − aydx = −dH and β(X H ) = ε sym 2 (xdy − ydx) − ax ε sym ∂ ∂x + ay ε sym ∂ ∂y = axy = H.
Finally, Remark 3.3 says that β = 1 2 r 2 dθ and H = µr 2 cos(n|k|θ) on R 2 \ D(r sing ). As in the previous case, we show that
X H = n|k|µr sin(n|k|θ) ∂ ∂r + 2µ cos(n|k|θ) ∂ ∂θ is a solution of Equation (3.2.1) on R 2 \ D(r sing ).
We calculate i X H (dβ) = (n|k|µr sin(n|k|θ)∂ r + 2µ cos(n|k|θ)∂ θ ) (rdr ∧ dθ)
= −2µr cos(n|k|θ)dr + n|k|µr 2 sin(n|k|θ)dθ = −dH, and β(X H ) = 1 2 r 2 dθ n|k|µr sin(n|k|θ) ∂ ∂r + 2µ cos(n|k|θ) ∂ ∂θ = µr 2 cos(n|k|θ) = H.
Let X H be the Hamiltonian vector field of H with respect to dβ and ϕ s X H be the time-s flow of X H . Now we introduce the following notations:
S := {x ∈ R 2 \ D(r sing ) | ϕ s X H (x) ∈ R 2 \ D(r sing ) ∀s ∈ [0, 1]}, V := {x ∈ U | ϕ s X H (x) ∈ U ∀s ∈ [−1, 1]}, and V i := {x ∈ U i | ϕ s X H (x) ∈ U i ∀s ∈ [−1, 1]}.
For simplicity, let us denote ϕ X H := ϕ 1 X H . Remark 3.5. Using the form of X H on U i , where i = 1, . . . , n|k|, we may assume that the curves γ p,p i 's in Lemma3.2 satisfy the following list of properties:
(1) γ p,p i is an embedded curve which starts at p and ends at p i ;
(2) γ p,p i is a part of one of the curves of the singular foliation given by ker β;
(3) γ p,p i coincides with one of the level sets of H on V i and near p i can be presented as
W s (ϕ X H , p i ) = {x | (ϕ X H ) n (x) → p as n → ∞}.
Recall that the following claim was proven in [8]:
(ϕ t X H ) * β − β = df t , where f t = t 0 (−H + β(X H )) • ϕ s X H ds.
Remark 3.7. Observe that from Lemma 3.4 and Claim 3.6 it follows that ϕ *
X H (β) − β = dh, where h := f 1 = 0 on S ∪ (∪ n|k| i=1 V i ) and h =C > 0 on V . Hence, we get ϕ * X H (β) = β on S ∪ V ∪ (∪ n|k| i=1 V i ). Now we define ϕ k l := R − k l • ϕ X H , where R − k l : R 2 → R 2 is a − 2πl
k -rotation around the center of D(r sing ).
Remark 3.8. Since R * n|k| (β) = β, we get R * − k l (β) = β and hence ϕ * k l (β) = (R − k l • ϕ X H ) * (β) = ϕ * X H (R * − k l (β)) = ϕ * X H (β).
Fix R * ≫ r sing such that there is an annular neighborhood
V R * of ∂D(R * ) in R 2 with V R * ⊂ S. Consider D(R * ) with β 0 := β| D(R * ) and β 1 := ϕ * X H (β)| D(R * ) (= ϕ * k l (β)| D(R * ) ). Note that dβ 1 = d(ϕ * X H (β)| D(R * ) ) = ϕ * X H (dβ)| D(R * ) = (dβ)| D(R * ) = dβ 0 > 0. (3.2.2)
In addition, from the definitions of V (R * ) and D(R * ) it follows that (2) α = dt + εβ 1 in a neighborhood of {1} × D 2 ;
β 0 = β 1 on V R * ∩ D(R * ).(3) R α is collinear to ∂ ∂t on [−1, 1] × D 2 ; (4) R α = ∂ ∂t in a neighborhood of [−1, 1] × ∂D 2 .
Here ε is a small positive number.
In addition, recall that
α = (1 + εχ 1 (t)h)dt + ε((1 − χ 0 (t))β 0 + χ 0 (t)β 1 ), (3.2.4) where h ∈ C ∞ (D 2 ) such that β 1 − β 0 = dh; χ 0 : [−1, 1] → [0, 1] is a smooth map for which χ 0 (t) = 0 for −1 ≤ t ≤ −1 + ε χ 0 , χ 0 (t) = 1 for 1 − ε χ 0 ≤ t ≤ 1, χ ′ 0 (t) ≥ 0 for t ∈ [−1, 1]
and ε χ 0 is a small positive number; χ 1 (t) := χ ′ 0 (t); ε is a sufficiently small positive number. Remark 3.10. Note that dα = εω, where α is a 1-form given by Formula 3.2.4 and ω = dβ 0 = dβ 1 > 0 on D 2 .
Observe that from Formulas 3.2.2 and 3.2.3 it follows that β 0 and β 1 described above satisfy the conditions of Lemma 3.9. We now take [−1, 1] × D(R * ) equipped with the contact 1-form α given by Formula 3.2.4. For simplicity, let us denote β − := εβ 0 and β + := εβ 1 , where ε is a constant from Lemma 3.9 which makes α contact.
Then we construct P + , P − and D in the way described in [8]. Recall that are surfaces with boundary which satisfy the following properties:
P + , P − , D ⊂ D(R * ) ⊂ R 2
(1) P ± ⊂ D;
(2) (∂P ± ) ∂ ⊂ ∂D and (∂P ± ) int ⊂ int(D);
(3) ϕ X H maps P + to P − in such a way that ϕ X H ((∂P + ) int ) = (∂P − ) ∂ and ϕ X H ((∂P
+ ) ∂ ) = (∂P − ) int ; (4) (∂P − ) ∂ ∩ (∂P + ) ∂ = ∅.
Note that in our case
• ∂P + = (∪ n|k|−1 s=0 a + s ) ∪ (∪ n|k|−1 s=0 b + s ), • ∂P − = (∪ n|k|−1 s=0 a − s ) ∪ (∪ n|k|−1 s=0 b − s ), • ∂D = (∪ n|k|−1 s=0 a + s ) ∪ (∪ n|k|−1 s=0 b − s ) ∪ (∪ n|k|−1 s=0 c + s ) ∪ (∪ n|k|−1 s=0 c − s )
, which is slightly different from the case described in [8] when Figure 3 for the schematic visualization of P + (bounded by the bold line), P − and D. For more details of this construction we refer to [8].
• ∂P + = (∪ n−1 s=0 a + s ) ∪ (∪ n−1 s=0 b + s ), • ∂P − = (∪ n−1 s=0 a − s ) ∪ (∪ n−1 s=0 b − s ), • ∂D = (∪ n−1 s=0 a + s ) ∪ (∪ n−1 s=0 b − s ) ∪ (∪ n−1 s=0 c + s ) ∪ (∪ n−1 s=0 c − s ). See
Remark 3.11. Note that a ± i 's, b ± i 's and c ± i 's are constructed in such a way that
a ± i , b ± i , c ± i ⊂ D(R * ) ∩ S for i = 0, . . . , n|k| − 1. Hence, we see that ∂P + , ∂P − , ∂D ⊂ D(R * ) ∩ S. In addition, R n|k| (a ± i ) = a ± i+1 and R n|k| (b ± i ) = b ± i+1 ,
where R n|k| is a 2π n|k| -rotation around p and i, i+1 are considered modulo n|k|. Proof. First note that α| R− = β − and α| R + = β + . Let us check that (R − , β − ) and (R + , β + ) are Liouville manifolds. From the construction of β ± it follows that d(β − ) = d(β + ) > 0. Since
β − = β + on D ∩ S and by Formula 3.2.4, α = dt + β − on U(Γ). Recall that β − = β + = ε 2 r 2 dθ on D ∩ S. Hence, α| U (Γ) = dt + ε 2 r 2 dθ. The calculation i Y ± | R ± ∩U (Γ) (dβ ± ) = 1 2 r∂ r (εrdr ∧ dθ) = ε 2 r 2 dθ = β ±
implies that the Liouville vector fields Y ± | R ± ∩U (Γ) are equal to 1 2 r∂ r . From the construction of D it follows that Y ± is positively transverse to ∂R ± . Therefore, (R − , εβ 0 ) and (R + , εβ 1 ) are Liouville manifolds. As we already mentioned, α = dt + β − on U(Γ). Finally, if we take τ such that ∂ τ = 1 2 r∂ r , then ([−1, 1] × D, Γ, U(Γ), ξ) becomes a sutured contact manifold with an adapted contact form α.
Then we use ϕ k l for the gluing construction. Note that ϕ X H maps a + s to a − s and b + s to b − s . Hence, using Remark 3.11, we see that ϕ k l maps a + s to a − s−nl and b + s to b − s−nl for k > 0, and ϕ k l maps a + s to a − s+nl and b + s to b − s+nl for k < 0. Then we follow the gluing procedure briefly described in Section 2.1 and completely written in [3]. Finally, we get a sutured contact solid torus (S 1 × D 2 ,Γ, U(Γ)) with a contact formα δ , whereΓ is a set of 2n parallel closed curves of slope k l , where n ∈ N, (k, l) = 1, |k| > l > 0 and δ is the rotation angle of the map ϕ X H near p. Remark 3.13. We have constructed (S 1 × D 2 ,Γ, U(Γ)) using the gluing construction for sutured manifolds. However, since there is a close connection between sutured contact manifolds and contact manifolds with convex boundary, we observe that the gluing construction we used for the sutured contact solid cylinder corresponds to the gluing construction for the contact 3-ball with convex boundary and one dividing curve on the boundary. The corresponding gluing construction for the contact 3-ball with convex boundary corresponds (is inverse) to the convex decomposition of the contact solid torus S 1 × D 2 with convex boundary with respect to the convex meridional disk {pt} × D 2 with ∂-parallel dividing curves. Hence, the constructed sutured contact solid tori are universally tight sutured contact manifolds by the gluing/classification result from Section 2 in [9] (more precisely, Corollary 2.3, Theorem 2.5 and Corollary 2.6).
CALCULATION
4.1. Reeb orbits. Note that ϕ k l | P + has n orbits of period |k| obtained from the equally spaced saddle points of H. Lemma 3.9 and the gluing procedure briefly described in Section 2.1 imply that these orbits correspond to the Reeb orbits, which we call γ 1 , . . . , γ n such that
[γ s ] = [γ t ] = |k| ∈ H 1 (S 1 × D 2 ; Z)
for s, t = 1, . . . , n. In addition, ϕ k l | P + has a periodic point of period 1, which is p. It corresponds to the Reeb orbit, which we call γ, such that [γ] = 1 ∈ H 1 (S 1 × D 2 ; Z). Lemma 4.1. γsα δ = γtα δ and |k| γα δ > γsα δ , where s, t = 1, . . . , n. −N)).
Proof. Let
M (0) = (([−1, 1] × D) ∪ (R + (Γ) × [1; ∞)) ∪ (R + (Γ) × (−∞; −1])) andM = M (0) \ ((P + × (N, ∞) ∪ (P − × (−∞,
In addition, let αM denote the contact form onM and let ξM denote the contact structure defined by αM .
Consider For simplicity, from now on we assume that l = 1. The calculation for the case when l > 1 is completely analogous.
Lemma 4.2.
All closed orbits of Rα δ are nondegenerate. Moreover, γ is an elliptic orbit and γ i is a hyperbolic orbit such that γ t and γ s i are good orbits for i = 1, . . . n; s, t ∈ N. There exists a symplectic trivialization τ of ξ along γ and γ i 's constructed in the consistent way as described in Section 2.2, and N δ ∈ N such that µ τ (γ s i ) = 2s, for k < −1; −2s, for k > 1, and µ τ (γ t ) = 2m − 1, for k < −1 and (m − 1)|k| ≤ t < m|k|; −2m + 1, for k > 1 and (m − 1)|k| < t ≤ m|k|
for i = 1, . . . , n; t ≤ N δ , s ≤ N δ |k| .
Proof. Fix i = 1, . . . , n. We first observe that H| V i = axy, where a > 0 and hence
ϕ X H | V i = λ 0 0 λ −1 ,
where λ = e a = 1. Let the symplectic trivialization of ξM along [−N, N] × {p i } be given by the
framing (λ −N−t 2N ∂ x , λ t+N 2N ∂ y ), where i = 1, .
. . , n|k| and (x, y) are coordinates on V i which coincide with the coordinates on U i from Remark 3.1. Since Remark 3.1 implies that R n|k| maps coordinates on V i to the coordinate on V i+1 , where i, i + 1 are considered modulo n|k| and R n|k| denotes 2π n|k|rotation around p, we conclude that the symplectic trivializations of ξM along [−N, N]×{p i+nm }'s for m = 0, . . . , |k| −1 and fixed i = 1, . . . , n give rise to the symplectic trivialization τ γ i ofξ along γ i . It is easy to see that the linearized return map P γ i with respect to this trivialization is given by
P γ i = λ |k| 0 0 λ −|k| .
Since the eigenvalues of P γ i are positive real numbers different from 1, γ i is a positive hyperbolic orbit. In addition, P γ s i = P s γ i . Therefore, the eigenvalues of P γ s i are different from 1. Hence, γ s i is a nondegenerate orbit for s ∈ N and i = 1, . . . , n. We now observe that the linearized Reeb flow around γ i (with respect to τ γ i ) rotates the eigenspaces of P γ i by angle 2π for k < −1 and −2π for k > 1. Hence, we get
µ τγ i (γ s i ) =
2s, for k < −1; −2s, for k > 1
(cos(θ δ,k,N (t))∂ x + sin(θ δ,k,N (t))∂ y , − sin(θ δ,k,N (t))∂ x + cos(θ δ,k,N (t))∂ y ), where θ δ,k,N (t) = π(1−δk)(t+N ) N k and t ∈ [−N, N]. Observe that R −k • ϕ X H | V is a rotation by angle 2π(− 1 k + δ), where R −k is a − 2π
k -rotation about p and δ is a small positive irrational number. It is easy to see that with respect to this framing P γ is a rotation by 2π(− 1 k + δ). Hence, since δ is irrational, we see that γ is an elliptic orbit and γ t is nondegenerate for t ∈ N. Let
N δ := max{m ∈ N | mδ < 1 |k| }.
Note that we get µ τγ (γ t ) = 2m − 1, for k < −1 and (m − 1)|k| ≤ t < m|k|; −2m + 1, for k > 1 and (m − 1)|k| < t ≤ m|k| (4.1.3) for t ≤ N δ . Formulas 4.1.2 and 4.1.3 and the fact that δ is irrational imply that the parity of µ τγ i (γ s i ) is independent of s for given i and the parity of µ τγ (γ t ) is independent of t. Hence, we conclude that γ s i 's and γ t 's are good Reeb orbits for i = 1, . . . , n and s, t ∈ N. We now show that the symplectic trivializations τ γ i 's and τ γ are consistent in the sense of Section 2.2. We first consider {−N}×P − ⊂M . Note that Remark 3.5 implies that for i = 1, . . . , n|k| there is an embedded curve γ p,p i with the following properties:
(1) γ p,p i is an embedded curve which starts at p and ends at p i ;
(2) γ p,p i coincides with one of the level sets of H on V i and near p i can be presented as
W s (ϕ X H , p i ) = {x | (ϕ X H ) n (x) → p as n → ∞}; (3) γ p,p i isv i ) = v i+1 , where R n|k| denotes 2π
n|k| -rotation around p and i, i + 1 are considered modulo n|k|.
Let
v i,t := −ε β t ϕ −N−t 2N X H * (v i ) 1 + εhχ 1,ext (t) ∂ t + ϕ −N−t 2N X H * (v i ), where β t = (1 − χ 0,ext (t))β 0 + χ 0,ext (t)β 1 ,αM (v i,t ) = ((1 + εχ 1,ext (t)h)dt + εβ t ) −ε β t ϕ −N−t 2N X H * (v i ) 1 + εhχ 1,ext (t) ∂ t + ϕ −N−t 2N X H * (v i ) = 0 along ϕ (−N −t)/2N (γ p,p i ), where t ∈ [−N, N] and i = 1, . . . , n|k|. Hence, we conclude that v i,t ∈ ξM along ϕ (−N −t)/2N (γ p,p i ) for i = 1, . . . , n|k|; t ∈ [−N, N]. In addition, observe that (ϕ (−N −t)/2N X H ) * (v i ) ∈ T ϕ (−N −t)/2N (γ p,p i ) does not vanish, since v i does not vanish. Thus, v i,t does
not vanish as well. We now note that Properties (2) and (3) imply that ((ϕ
(−N −t)/2N X H ) * v i )| V ∪V i ∈ ξM . Hence, we conclude that β t ((ϕ (−N −t)/2N X H ) * (v i )| V ∪V i ) = 0 for t ∈ [−N, N]. Thus, v i,t = (ϕ (−N −t)/2N X H ) * (v i ) on [−N, N] × (V ∪ V i ).
In addition, Property (2), Property (3) and the fact that p and p i are fixed points of ϕ
(−N −t)/2N X H for t ∈ [−N, N] imply that ((ϕ (−N −t)/2N X H ) * (v i ))(p i ) = h(t)v(p i ), (4.1.4)
where h(t) does not vanish for t ∈ [−N, N]. Besides that, observe that Property (2) implies that v(p i ) is a multiple of a coordinate vector on V i . Then we take (v i,t , w i,t ) as a symplectic basis of ξM along (ϕ
(−N −t)/2N X H )(γ p,p i ),
where w i,t is a symplectic complement of v i,t with respect to dαM . We now observe that Remark 3.10 and the fact that αM for i = 1, . . . , n; t ≤ N δ , s ≤ N δ |k| . 4.2. Sutured cylindrical contact homology. In this section, we calculate the sutured version of cylindrical contact homology of the sutured contact solid torus that we have constructed in Section 3.2 (in the case when l = 1).
| [1,N ]×(V ∪V i ) = dt + β + , αM | [−N,−1]×(V ∪V i ) = dt + β − imply that dαM | [−N,N ]×(V ∪V i ) = εω. Hence, dαM is invariant under (ϕ
We first recall that Lemma 4.1 implies that all closed orbits of Rα δ are nondegenerate.
Remark 4.4. Note that there are no contractible Reeb orbits. Hence, from Theorem 2.7, Remark 2.12, and the fact that
π 1 (S 1 × D 2 ; Z) ≃ H 1 (S 1 × D 2 ; Z) ≃ Z it follows that for all h ∈ H 1 (S 1 × D 2 ; Z) HC cyl,h * (S 1 × D 2 ,Γ,α δ , J)
is defined, i.e., ∂ 2 = 0, and is independent of contact formα δ for the given contact structureξ and the almost complex structure J. for i = 1, . . . , n, s ≤ N δ |k| and t ≤ N δ . Hence, we get
(4.2.3) C h m (α δ , J) = Q γ h , for h > 0 and m = 2⌊h(− 1 k + δ)⌋; Q γ h/|k| 1 , . . . , γ h/|k| n , for k | h > 0 and m = − 2h k − 1; 0, otherwise for h ≤ N δ . Now since by Lemma 4.1 A(γ |k| ) > A(γ i ) for i = 1, .
. . , n, we can use Fact 2.6 and Remark 2.12 and conclude that ∂(γ s i ) = 0 for i = 1, . . . , n and s > 0. Then we prove that ∂(γ t ) = 0 for k ∤ t ≤ N δ . Since [γ i ] = |k|[γ] in H 1 (S 1 × D 2 ; Z) ∼ = Z, the cylindrical contact homology differential at γ t counts only cylinders with negative end at γ t . Then, similarly to the previous case, Fact 2.6 and Remark 2.12 imply that ∂(γ t ) = 0 for k ∤ t ≤ N δ .
We now consider the case when k | t and will show that ∂(γ t ) = 0 for k | t ≤ N δ . Is this situation, by arguing in the same way as in the case when k ∤ t, we get that ∂(γ t ) counts only cylinders with negative end at γ t/|k| i . Now we note that
ind(u) = |γ t | − |γ t/|k| i | (4.2.4)
for any pseudoholomorphic curve u in the moduli space M J (γ t ; γ t/|k| i ), where k | t ≤ N δ and J is an almost complex structure tailored to ((R × S 1 × D 2 ) * ,α * δ ). The index formula can be written in this way, since H 2 (S 1 × D 2 ; Z) = 0 and hence < c 1 (ξ), A >= 0 for all A ∈ H 2 (S 1 × D 2 , Z). We now use Equations 4.2.1 and 4.2.2 in the case when k < −1 and get
|γ t | − |γ t/|k| i | = 2m − 2 t |k| + 1 = 2(m − t |k| ) + 1,
and m = t |k| for i = 1, . . . , n; t ≤ N δ . Hence, we can rewrite Equation 4.2.4 as
ind(u) = |γ t | − |γ t/|k| i | = 2( t |k| − t |k| ) + 1 = 1 (4.2.5)
for i = 1, . . . , n and t ≤ N δ . Therefore, Theorem 2.8 and Remark 2.12 imply that for every u ∈ M(γ t , γ t/|k| i ) the linearization of the Cauchy-Riemann operator is surjective at u; here k | t ≤ N δ , J is any almost complex structure tailored to ((S 1 × D 2 ) * ,α * δ ) and i = 1, . . . , n. Similarly, we use Equations 4.2.1 and 4.2.2 in the case when k > 1 and get for i = 1, . . . , n and t ≤ N δ . Therefore, Theorem 2.8 and Remark 2.12 imply that for every u ∈ M(γ t , γ t/|k| i ) the linearization of the Cauchy-Riemann operator is surjective at u; here k | t ≤ N δ , J is any almost complex structure tailored to ((S 1 × D 2 ) * ,α * δ ) and i = 1, . . . , n. Let (S 1 × D 2 , Γ long , U(Γ long ), α long δ ) be a sutured contact solid torus obtained from ([−1, 1] × D, Γ, U(Γ), α) by using ϕ X H as a gluing map. Recall that we get (S 1 × D 2 ,Γ, U(Γ),α δ ) from ([−1, 1] × D, Γ, U(Γ), α) by using ϕ k = R −k • ϕ X H as a gluing map. Here R −k is a − 2π k -rotation around the center of D. We now note that (S 1 × D 2 , Γ long , U(Γ long ), α long δ ) is a universally tight sutured contact solid torus with 2n|k| parallel longitudinal sutures, |k| > 1. This follows from the gluing/classification result for universally tight contact structures on a sutured solid torus, see Section 2 in [9] (more precisely, Corollary 2.3, Theorem 2.5 and Corollary 2.6). The cylindrical contact homology of this sutured contact manifold is computed in [8] and is given buy HC cyl,h (S 1 × D 2 , Γ long , ξ long ) ≃ Q n|k|−1 , for h ≥ 1; 0, otherwise. Here ξ long = ker α long .
|γ t | − |γ t/|k| i | = −2m − (−2 t |k| − 1) = 2( t |k| − m) + 1,
Note that (S 1 × D 2 , Γ long , U(Γ long ), α long δ ) has n|k| hyperbolic orbits γ long for i, j = 1, . . . , n|k|. Hence, Theorem 2.7, Remark 2.12 together with Fact 2.6, and Formulas 4.2.7, 4.2.8 and 4.2.9 imply that ∂γ long = 0; otherwise we come to contradiction to Formula 4.2.7 (∂γ long = 0 implies that the exponent of Q in Formula 4.2.7 must be n|k| + 1).
We now take almost complex structure J long tailored to ((S 1 × D 2 ) * , (α long δ ) * ) such that as a map ξ long → ξ long it is obtained from some fixed J cyl : ξ → ξ which is defined on ([−1, 1] × D, Γ, U(Γ), α) and satisfies the following properties:
(1) (J cyl ) 2 = −I, dα(J cyl ·, J cyl ·) = dα(·, ·), dα(·, J cyl ·) > 0;
(2) J cyl is 2π n|k| -symmetric, i.e., it is invariant under 2π n|k| -rotation with respect to the center of D.
Here ξ long = ker α long and ξ = ker α. By saying that J long is obtained from J cyl we simply mean that the gluing procedure with ϕ X H applied to ([−1, 1] × D, Γ, U(Γ), α) transforms J cyl to J long . Since ξ is 2π n|k| -symmetric on ([−1, 1]×D, Γ, U(Γ), α), we claim that J cyl , which satisfies Properties (1) and (2), exists and that Property (2) is not a serious restriction on J cyl . The symmetry of ξ follows from the symmetry of β and X H , and from the construction of α. Now we takeJ on (S 1 × D 2 ,Γ, U(Γ),α δ ), which is obtained from the same J cyl defined on ([−1, 1] × D, Γ, U(Γ), α) by applying the gluing procedure with ϕ k = R −k • ϕ X H to ([−1, 1] × D, Γ, U(Γ), α), and possibly modify it near the boundary of (S 1 × D 2 ,Γ, U(Γ),α δ ) (far from the Reeb orbits) so that it becomes tailored to ((S 1 × D 2 ) * , (α δ ) * ). From the symmetry of J cyl and the form of the gluing maps for (S 1 × D 2 ,Γ, U(Γ),α δ ) and (S 1 × D 2 , Γ long , U(Γ long ), α long δ ) we see that ∂γ long = 0 for the count with respect to J long implies that ∂γ t = 0 for k | t ≤ N δ for the count with respect toJ. Observe that this choice of almost complex structures is possible, since Theorem 2.8 and Remark 2.12 imply that we do not need to require almost complex structures to be generic. Finally, from We now note thatξ = kerα δ is independent of δ. This follows from the gluing/classification result for universally tight contact structures on a sutured solid torus, see Section 2 in [9] (more precisely, Corollary 2.3, Theorem 2.5 and Corollary 2.6). Hence, from Theorem 2.7 and Remark 2.12 it follows that HC cyl,h (S 1 × D 2 ,Γ,ξ) = HC cyl,h (S 1 × D 2 ,Γ,α δ )
.
Let (M, α) be a closed, oriented contact manifold with nondegenerate Reeb orbits. Let C h m (M, α) be the cylindrical contact homology complex, where h is a homotopy class of Reeb orbits and m corresponds to the Conley-Zehnder grading. If C 0 k (M, α) = 0 for k = −1, 0, 1, then for every free homotopy class h (1) ∂ 2 = 0; (2) H(C h * (M, α), ∂) is independent of the contact form α for ξ, the almost complex structure J and the choice of perturbation for the moduli spaces.
Remark 2.10. Note that Theorem 2.8 can be considered as a consequence of the automatic transversality result of Wendl, see Theorem 0.1 in[12].Finally, we recall the following result of Colin, Ghiggini, Honda and Hutchings from [3]: Theorem 2.11 ([3]). Let (M, Γ, U(Γ), ξ) be a sutured contact 3-manifold with an adapted contact form α, (M * , α * ) be its completion and J be an almost complex structure on R × M * which is tailored to (M * , α * ). Then the contact homology algebra HC(M, Γ, ξ) is defined and independent of the choice of contact 1-form α with ker(α) = ξ, adapted almost complex structure J, and abstract perturbation. Remark 2.12. Fact 2.6, Theorems 2.7 and 2.8, Formulas 2.2.1 and 2.2.2 hold for J-holomorphic curves in the symplectization of the completion of a sutured contact manifold, provided that we choose the almost complex structure J on R × M * to be tailored to (M * , α * ).
FIGURE 1 .
1The level sets of H sing (left) and the level sets of H (right) in the case n = 1, |k| = 3
( 4 )
4Finally, to ensure no fixed points of the time-1 flow of the Hamiltonian vector field of H, we let H be H 4 multiplied by a radially symmetric function which is ǫ for r < R (for ǫ sufficiently small that the only fixed points of the time-1 flow inside radius R are the critical points and for R large enough that H 4 agrees with H sing for r > R) and 1 for r > 2R. This creates no new fixed points in the region R < r < 2R because H 4 and ∂H4 ∂r have the same sign there. Now there are no fixed points of the time-1 flow of the Hamiltonian vector field of H, except for the n|k|+1 critical points of H because outside radius R there are no compact flow lines.
Remark 3. 3 .
3From the previous lemma we get β defined on R 2 with the following properties: (i) dβ > 0 on R 2 ; (ii) β = εc 2 r 2 dθ and H =Br 2 −C on U, whereC > 0; here ε c is a small positive real number; (iii) β = εsym 2 (xdy − ydx) and H = axy on U s for s = 1, . . . , n|k|; here 0 < ε c ≪ ε sym ≪ 1; (iv) β = 1 2 r 2 dθ and H = µr 2 cos(nθ) on R 2 \ D(r sing ). For the comparison of the level sets of H with the singular foliation of β in the case n = 1, |k| = 3 we refer toFigure 2.
Lemma 3. 4 .
4Let β be a 1-form from Lemma 3.2. The Hamiltonian vector field X H of H with respect to the area form dβ satisfies β(X H ) = H on (∪ n|k| s=1 U s ) ∪ (R 2 \ D(r sing )). In addition, the Hamiltonian vector field X H of H with respect to the area form dβ satisfies β(X H ) − H =C on U.
FIGURE 2 .
2The level sets of H (left) and the characteristic foliation of β (right) in the case n = 1, |k| = 3 Proof. First, Remark 3.3 implies that β = εc 2 r 2 dθ, H =Br 2 −C on U and ε c is a small positive number. Now we show that X H = 2B εc ∂ ∂θ is a solution of β(X H ) − H =C on U. We calculate
Claim 3. 6
6([8]). If (M, ω) is an exact symplectic manifold, i.e., ω = dβ, then the flow ϕ t X H of a Hamiltonian vector field X H consists of exact symplectic maps, i.e.,
recall Lemma 3.10 from[8], which provides the construction of the contact 1-form on[−1, 1] × D 2 .
Lemma 3.9 ([8]). Let β 0 and β 1 be two 1-forms on D 2 such that β 0 = β 1 in a neighborhood of ∂D 2 and dβ 0 = dβ 1 = ω > 0. Then there exists a contact 1-form α and a Reeb vector field R α on [−1, 1] × D 2 with coordinates (t, x), where t is a coordinate on [−1, 1] and x is a coordinate on D 2 , with the following properties: (1) α = dt + εβ 0 in a neighborhood of {−1} × D 2 ;
FIGURE 3 .
3Construction of P + , P − and D in the case n = 1, |k| = 3
We take [−1, 1] × D with a contact form α := α| [−1,1]×D . Let Γ = {0} × ∂D in [−1, 1] × D and U(Γ) := [0, 1] × [−1, 1] × Γ be a neighborhood of Γ with coordinates (τ, t) ∈ [0, 1] × [−1, 1],where t is a usual t-coordinate on [−1, 1] ×D. From the definition of S and Remark 3.11 it follows that we may assume that U(Γ) ⊂ [−1, 1] × (S ∩ D).
Lemma 3 .
312. ([−1, 1] × D, Γ, U(Γ), ξ) is a sutured contact manifold and α is an adapted contact form.
[− 1 , 1 ]
11× D ⊂M . From the construction of α it follows that β + = β − on V s and α| [−1,1]×Vs = dt+β − for s = 1, . . . , n|k|. Hence, since the contact structure on [1, ∞)×P + is given by dt+β + and the contact structure on (−∞, −1]×P − is given by dt+β − , αM | [−N,N ]×Vs = dt+β − on [−N, N] × V s ⊂M for s = 1, . . . , n|k|. Therefore, we get = 1, . . . , n|k|. From the gluing construction and Equation (4.1.1) it follows that γsα δ = 2N|k| for s = 1, . . . , n. Note that γsα δ does not depend on s. Hence, γsα δ = γtα δ for s, t = 1, . . . , n. Now from the fact that α = (1 + εχ 1 (t)h)dt + β − on [−1, 1] × V , where h > 0 and χ 1 (t) × V . Hence, from the gluing construction we obtain |k| γα δ > 2N|k|. Thus,
∈ N and i = 1, . . . , n. Now let the symplectic trivialization of ξM along [−N, N] × {p} be given by the framing
a part of one of the curves of the characteristic foliation of αM on {−N}×P − ⊂M . Now consider γ p,p i with a vector field v i ∈ T γ p,p i such that v i does not vanish and (R n|k| ) * (
for t ∈ [1, N]; χ 0 (t), for t ∈ [−1, 1]; 0, for t ∈ [−N, −1],and χ 1,ext (t) := χ ′ 0,ext (t) on [−N, N]. It is easy to see that
for k < −1 and t = (m − 1)|k|; −2ml + 1,for k > 1 and t = m|k|.
t | = 2m − 2, for k < −1 and (m − 1) ≤ t |k| < m; −2m, for k > 1 and (m − 1) < t |k| ≤ m (4.2.2)
and m = t |k| for i = 1, . . . , n; t ≤ N δ . Thus, we can rewrite Equation 4.2.4 as ind(u) = |γ t | − |γ
1 ,
1. . . , γ long n|k| and one elliptic orbit γ long . Here γ long i 's correspond to the equally spaced saddle points of H and γ long corresponds to the critical point of H at the center of D(r sing ). In addition, observe that [γ long i ] = [γ long ] = 1 ∈ H 1 (S 1 × D 2 ; Z).
CONTACT HOMOLOGY OF UNIVERSALLY TIGHT SUTURED CONTACT SOLID TORI 21Finally, note that from Lemma 4.1 and from the construction of γ long and γ long 1 , . . . , γ long n|k| it follows that A(γ long ) > A(
h > 0 and m = 2⌊h(− 1 k + δ)⌋; Q n−1 , for k | h > 0 and m = − 2h k − 1; 0, otherwise for h ≤ N δ .
Therefore, using the fact that v i,t = (ϕwhere w i := w i,t | γp,p i and t ∈ [−N, N]. Now we take the symplectic trivialization given by the framing. . , n|k|. In addition, for a fixed i = 1, . . . , n we consider the sequence of framingsFinally, note that sincewe see thatwhere i, i − n are considered modulo n|k|. Similarly, since β − and hence dαM = dβ − is invariant under 2π n|k| -rotaion around p,Hence, we getwhere i, i − n are considered modulo n|k|. Formulas 4.1.7 and 4.1.8 imply that the framings given by Formula 4.1.5 and Formula 4.1.6 give rise to the symplectic trivializations τ (γ i ) along γ i and τ (γ |k| ) along γ |k| which are consistent in terms of Section 2.2 along the surfaces obtained from (ϕby gluing them with ϕ k . In addition, note that from the property described by Equation 4.1.4 and the fact thatwhere q(t) = 0, i = 1, . . . , n|k| and t ∈ [−N, N]. Thus, τ (γ i ) and τ γ i are in the same homotopy class of symplectic trivializations of the 2-plane bundle γ * i ξ over S 1 . In addition, it easy to see that the symplectic trivialization τ (γ |k| ) and τ γ |k| obtained by pull-back from γ are in the same homotopy class of symplectic trivializations of the 2-plane bundle (γ |k| ) * ξ over S 1 . Thus, we see that the symplectic trivializations τ γ i 's and τ γ are consistent in the sense of Section 2.2. From now on, for simplicity we denote the consistent trivialization by τ . Now observe that N δ → ∞ when δ → 0. In addition, we note that for fixed n, k and two small positive irrational numbers δ 1 = δ 2 , the sets of closed orbits of Rα δ 1 and Rα δ 2 are the same, and the corresponding orbits with the same first homology class h ≤ min{N δ 1 , N δ 2 } have the same Conley-Zehnder gradings in the corresponding complexes. Therefore, for every 0 < h ∈ Z = H 1 (S 1 × D 2 ; Z), there exists δ such thatwhere 0 < δ k ≪ 1 |k| . Finally, Formula 4.2.10 implies thatotherwise.(4.2.11)This completes the proof of Theorem 1.1 when l = 1.Remark 4.5. For l > 1, one can use the same observations as in the case when l = 1 and show that the only non-zero part of the cylindrical contact homology differential is given by < ∂γ t , γ t/|k| i > = 0 for |k| | t ≤ N δ . This will lead to Formula 4.2.11 for all l such that (k, l) = 1, |k| > l > 0.Remark 4.6. Theorem 1.3 from[8]and Theorem 1.1 provide the formula for the sutured version of cylindrical contact homology of (S 1 × D 2 , Γ, ξ), where Γ is arbitrary and ξ is a universally tight contact structure. This follows from the gluing/classification result for universally tight contact structures on a sutured solid torus, see Section 2 (more precisely, Corollary 2.3, Theorem 2.5 and Corollary 2.6) in[9].
A survey of Contact Homology, lectures at Yashafest. F Bourgeois, F Bourgeois, A survey of Contact Homology, lectures at Yashafest, 2007.
F Bourgeois, H Eliashberg, K Hofer, E Wysocki, Zehnder, Compactness results in symplectic field theory, Geom. and Top. 7F Bourgeois, Y Eliashberg, H Hofer, K Wysocki and E Zehnder, Compactness results in symplectic field theory, Geom. and Top. 7 (2003), 799-888.
. V Colin, K Ghiggini, M Honda, Hutchings, Sutures and contact homology I, Geom. Topol. 15V Colin, P Ghiggini, K Honda and M Hutchings, Sutures and contact homology I, Geom. Topol. 15 (2011), 1749-1842.
Constructions contrôlées de champs de Reeb et applications. V Colin, Honda, Geom. Topol. 9V Colin and K Honda, Constructions contrôlées de champs de Reeb et applications, Geom. Topol. 9 (2005), 2193-2226.
Symplectic Floer homology of area-preserving surface diffeomorphisms. A Cotton-Clay, Geom. Topol. 13A Cotton-Clay, Symplectic Floer homology of area-preserving surface diffeomorphisms, Geom. Topol. 13 (2009), 2619-2674.
Introduction to symplectic field theory. Y Eliashberg, A Givental, H Hofer, Geom. Funct. Anal. Special. 10Y Eliashberg, A Givental and H Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. Special Volume 10 (2000), 560-673.
Foliations and the topology of 3-manifolds. D Gabai, J. Diff. Geom. 18D Gabai, Foliations and the topology of 3-manifolds, J. Diff. Geom 18 (1983), 445-503.
The embedded contact homology of sutured solid tori. R Golovko, Alg. Geom. Topol. 11R Golovko, The embedded contact homology of sutured solid tori, Alg. Geom. Topol. 11 (2011), 1001-1031.
Gluing tight contact structures. K Honda, Duke Math. J. 115K Honda, Gluing tight contact structures, Duke Math. J. 115 (2002), 435-478.
An index inequality for embedded pseudoholomorphic curves in symplectizations. M Hutchings, J. Eur. Math. Soc. 4M Hutchings, An index inequality for embedded pseudoholomorphic curves in symplectizations, J. Eur. Math. Soc. 4 (2002), 313-361.
Contact Homology of orbit complements and implied existence. A Momin, arXiv:1012.1386preprint 2010A Momin, Contact Homology of orbit complements and implied existence, preprint 2010, arXiv:1012.1386.
Automatic transversality and orbifolds of punctured holomorphic curves in dimension four. C Wendl, Comment. Math. Helv. 852C Wendl, Automatic transversality and orbifolds of punctured holomorphic curves in dimension four, Comment. Math. Helv. 85 (2010), no. 2, 347-407.
. Département De Mathématiques Et De Statistique, Université De Montréal, Montréal , Q C H3t 1j4, Canada , DÉPARTEMENT DE MATHÉMATIQUES ET DE STATISTIQUE, UNIVERSITÉ DE MONTRÉAL, MONTRÉAL, QC H3T 1J4, CANADA
. Université Département De Mathématiques, Du Québecà, Montréal Montréal, Q C H2x 3y7, Canada , [email protected] URL. DÉPARTEMENT DE MATHÉMATIQUES, UNIVERSITÉ DU QUÉBECÀ MONTRÉAL, MONTRÉAL, QC H2X 3Y7, CANADA E-mail address: [email protected] URL: http://www.dms.umontreal.ca/˜rgolovko
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[] |
[
"EXACT LAGRANGIANS IN PLUMBINGS",
"EXACT LAGRANGIANS IN PLUMBINGS"
] |
[
"Mohammed Abouzaid ",
"Ivan Smith "
] |
[] |
[] |
Consider a Stein manifold M obtained by plumbing cotangent bundles of manifolds of dimension greater than or equal to 3 at points. We prove that the Fukaya category of closed exact Spin Lagrangians with vanishing Maslov class in M is generated by the compact cores of the plumbing. As applications, we classify exact Lagrangian spheres in A 2 -Milnor fibres of arbitrary dimension, derive constraints on exact Lagrangian fillings of Legendrian unknots in disk cotangent bundles, and prove that the categorical equivalence given by the spherical twist in a homology sphere is typically not realised by any compactly supported symplectomorphism.
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10.1007/s00039-012-0162-y
|
[
"https://arxiv.org/pdf/1107.0129v3.pdf"
] | 119,556,249 |
1107.0129
|
fa1bfb00db9aa2c246ffbc9b74077d97f52cc0ec
|
EXACT LAGRANGIANS IN PLUMBINGS
Mohammed Abouzaid
Ivan Smith
EXACT LAGRANGIANS IN PLUMBINGS
Consider a Stein manifold M obtained by plumbing cotangent bundles of manifolds of dimension greater than or equal to 3 at points. We prove that the Fukaya category of closed exact Spin Lagrangians with vanishing Maslov class in M is generated by the compact cores of the plumbing. As applications, we classify exact Lagrangian spheres in A 2 -Milnor fibres of arbitrary dimension, derive constraints on exact Lagrangian fillings of Legendrian unknots in disk cotangent bundles, and prove that the categorical equivalence given by the spherical twist in a homology sphere is typically not realised by any compactly supported symplectomorphism.
Introduction
1.1. Context. A classical problem in symplectic topology is to understand the topology of exact Lagrangian submanifolds of Stein manifolds. The case which has received most attention is Arnold's "nearby Lagrangian submanifold" conjecture, asserting that an exact L ⊂ T * M should be Hamiltonian isotopic (and in particular diffeomorphic) to M . A homotopy-theoretic version of this, namely that the inclusion L ⊂ T * M is a homotopy equivalence whenever the Maslov class µ L = 0, has been proved in [7]. Moving away from cotangent bundles, much less is known. On the one hand, Dehn twists provide constructions of infinite families of distinct Lagrangian submanifolds in fixed homology or even smooth isotopy classes, for instance in the A n -surfaces arising as Milnor fibres of the singularities C 2 /Z n+1 with n ≥ 2 [31]. On the other, there are a few uniqueness or non-existence theorems:
• Hind [16] proved there is a unique Lagrangian sphere up to isotopy in the A 1 -space T * S 2 ; • Ishii, Ueda and Uehara [17,18] classified Lagrangian spheres in the A n -surfaces up to isomorphism in the Fukaya category, in particular showing that they formed a single orbit under the natural action of the braid group; • Ritter [28] proved that the A n -surfaces contain no exact Lagrangian tori. Hind's work relied on existence results for holomorphic foliations which seem special to four dimensions, Ishii et al appealed to sheaf theory computations valid only on complex surfaces, whilst Ritter used non-exact deformations of the symplectic form relying on positivity of the second Betti number.
On the other hand, the systematic study of the symplectic topology of Stein manifolds has undergone new developments, with progress on at least 3 distinct fronts. First, the work of Oancea [26], Seidel [33], McLean [21] and others on symplectic homology (as defined by Floer-Hofer and Viterbo) showed the importance of this invariant for distinguishing examples; second, the work of Bourgeois, Eliashberg and Ekholm [10] on Legendrian surgery rendered symplectic homology and its cousins amenable to inductive computation; and finally, there has been progress on foundational and structural results for Fukaya categories [2,4,30], and their connection with symplectic cohomology.
The purpose of this note is to illustrate some of these developments, and their effectiveness in constraining the topology of Lagrangian embeddings in Stein manifolds.
Convention. We work throughout over a field k. If k is not of characteristic 2, then all Lagrangian submanifolds should be equipped with (relative) Spin structures: note that the zero-section of a cotangent bundle T * Q is always relatively spin, relative to the background class w 2 (Q). As a standing convention throughout the paper, Lagrangian submanifolds are orientable and connected (unless explicitly stated otherwise). A small number of arguments in the paper rely on the freedom to choose the characteristic of k coprime to the index of a covering space; in such situations we restrict to working with Spin Lagrangians, and then the relevant Fukaya categories are defined over Z.
1.2.
Results. Let Q 0 and Q 1 be smooth Riemannian manifolds. Recall the plumbing M = D * Q 0 #D * Q 1 of two cotangent bundles T * Q i is given by choosing balls B i in Q i , and identifying their disc cotangent bundles D * B i ⊂ D * Q i by a symplectomorphism which interchanges the zero-section and fibre directions (see Definition 2.1 and Section 4 for more details). The plumbing M is naturally a Liouville domain, which admits an exact symplectic form for which the obvious inclusions Q i → M are exact Lagrangian submanifolds, and can be completed to a Liouville manifold we denote T * Q 0 #T * Q 1 . There is a trivialisation of the bicanonical bundle K ⊗2 M , well-defined up to homotopy, which restricts on neighbourhoods of Q i ⊂ M to the trivialisations arising from complexified volume forms on the Lagrangians Q i . Remark 1.2. The proof generalises to plumbings of many components, at least when the plumbing graph is a tree. It seems likely that Theorem 1.1 holds in more generality (for self-plumbings; for plumbings along submanifolds of codimension ≥ 3), but our argument employs a geometric trick using Lefschetz fibrations (Section 3.1) which is more restrictive. Theorem 1.1 is a generation, rather than split-generation, result for Fukaya categories, generalising the fact that the zero-section generates the Fukaya category of closed exact Lagrangians in a cotangent bundle. Twisted complexes, rather than summands thereof, are more readily manipulated algebraically, whence the possibility of new applications. Whilst these applications are not exhaustive, they should serve to illustrate the reach of Theorem 1.1. Statements in the main body of the text are sometimes sharper.
The first application generalises several of the four-dimensional results described in Section 1.1 to arbitrary dimensions, in the first non-trivial case. Let A n 2 denote the complex n-dimensional A 2 -Milnor fibre obtained by plumbing two copies of T * S n , which can also be described as the affine variety A n 2 = x 2 1 + · · · + x 2 n + t 3 = 1 ⊂ C n+1 equipped with the restriction of the standard symplectic form ω C n+1 = i 2π n+1 j=1 dz j ∧ dz j . Corollary 1.9. The image of the natural map ρ : π 0 (Symp ct (T * Σ)) → Aut(F(M Σ )/ [1] intersects the Z-subgroup defined by the spherical twist in Σ only in the identity.
Our results imply the image of ρ is torsion-free, and presumably the (wrapped) Fukaya category of the cotangent bundle cannot be used to detect torsion classes in π 0 Symp ct (T * Σ) (should such exist).
Remark 1. 10. In [20], Kontsevich introduced a notion of stabilisation (by dimension) of a symplectic manifold, and asked whether the map from the group of components of the symplectomorphism group to the autoequivalences of the Fukaya category becomes an isomorphism after sufficiently many stabilisations. In Section 2.2 we shall explain that (the argument underlying) Corollary 1.9 implies that the answer is negative, assuming that the Fukaya categories of stabilisations -which have not been defined or studied explicitly in the literature -behave in the expected manner under passing to covers, cf. Conjecture 2.27.
Spherical twists are one of the few known sources of interesting autoequivalences of Fukaya categories. The underlying geometric Dehn twist in a Lagrangian sphere exists because the sphere admits a metric with periodic geodesic flow. Similar metrics exist on other compact rank one symmetric spaces (complex projective spaces, the Cayley plane, etc), and for such a manifold Q, the category F(M Q ) again has infinite-order autoequivalences that are nontrivial modulo shift. A classical result of Bott [9] states that if Q admits a metric with periodic geodesic flow, then it has, as cohomology algebra, a truncated polynomial ring H * (Q) ∼ = Z[u]/ u k . It seems likely that there is a Fukaya-categorical version of Bott's theorem.
Conjecture 1.11. Let Q be simply connected. If the map ρ of Corollary 1.9 is non-zero then H * (Q) is a truncated polynomial ring. For any Q, the image of ρ is trivial or Z.
There is a more natural formulation of the Conjecture, bypassing the auxiliary manifold M Q but invoking a version of the Fukaya category of the cotangent bundle which incoporates certain non-closed Lagrangian submanifolds Legendrian at infinity, for instance the Nadler-Zaslow category [25]. The version above is tailored to our technical machinery. We prove two partial results. We will say that the representation ρ is categorically trivial if any autoequivalence in the image of ρ acts trivially on objects of F(M Q ). Proposition 1.12. If Q is a product of spheres, or if Q has homology concentrated in 3 degrees and the middle Betti number is greater than 1, then ρ is categorically trivial.
In particular, if Q is a simply-connected 4-manifold and M Q admits a non-trivial autoequivalence, then Q is homotopy equivalent to one of S 4 and CP 2 (this is essentially sharp, in that each of these manifolds does admit a non-trivial Dehn twist). Wendl [36] used delicate constructions of planar holomorphic foliations to prove that Symp ct (T * (S 1 × S 1 )) is connected, and the first case of Proposition 1.12 can be viewed as a partial generalisation of that result.
Acknowledgements. The authors are grateful to Tobias Ekholm, Gabriel Paternain and Paul Seidel for helpful conversations, and to the anonymous referee, whose detailed comments located and corrected several small errors.
Proof of the applications
In this section, we derive the consequences of Theorem 1.1 which are mentioned in the introduction. We shall consider, throughout, a minimal model H * F(M ) for the Z-graded ordinary Fukaya category of a Liouville domain M . This is a strictly unital A ∞ category, whose objects are closed Lagrangian branes, i.e. closed connected exact Lagrangians, equipped with a Spin structure and a grading in the sense of [31], and whose morphism spaces are Floer cohomology groups. We shall use the notation
(2.1) Tw(H * (F(M )))
for the category of twisted complexes, an object of which consists of a sequence of Lagrangian branes
{L i } D i=1 , carrying finite-dimensional local systems {V i } D i=1
-equivalently modules for the group ring Z[π 1 (L i )] -concentrated in a single degree, and degree 1 maps
(2.2) δ i,j ∈ HF * (V i , V j )
satisfying r≥1 µ r (δ, . . . , δ) = 0, with µ r the A ∞ -operations of the additive enlargement of HF(M ) and δ = (δ ij ) the upper triangular matrix of differentials. Whenever the local systems are trivial, we may alternatively write δ i,j as a map
(2.3) δ i,j : V i → HF * (L i , L j ) ⊗ V j .
This category admits a shift functor
(2.4) X → X[1]
which decreases the grading of each local system V i by 1. For background on the category of twisted complexes, see [30, Chapter 1, Section 3]. Whenever Σ ⊂ M is an exact Lagrangian submanifold of a Liouville manifold M , there is an associated algebraic twist functor
T Σ ∈ NuFun(HF * (M ))
defined as usual by the cone of evaluation
(2.5) T Σ = Cone HF (Σ, •) ⊗ Σ ev −→ • .
If H * (Σ) ∼ = H * (S n ), the twist functor is invertible, hence defines an autoequivalence of H * (F(M )). Explicitly, the inverse functor is given by the cone on the dual of evaluation
(2.6) T −1 Σ = Cone • ev ∨ −→ HF (•, Σ) ∨ ⊗ Σ ,
compare to [30,Remark 5.12].
In the body of the paper, most of our Liouville manifolds M will arise as plumbings. Define the model plumbing to be the open neighbourhood
(2.7) {|x||y| ≤ 1} = R ∞ ⊂ C n = R n × √ −1R n
of the union of transverse linear Lagrangian planes R n ∪ √ −1R n = {y = 0} ∪ {x = 0}. Now given base-points q i ∈ Q i , choose Riemannian metrics g i locally flat near q i , and pick isometric embeddings η i : (B(1/2), 0) → (Q i , q i ) of the ball of radius 1/2 in R n to Q i . There are then natural symplectic inclusions
Dη i : D * B(1/2) → D * Q i .
On the other hand, inclusions ι 0 : B(1/2) ⊂ R n and ι 1 :
B(1/2) ⊂ √ −1R n induce symplectic inclusions Dι j : D * B(1/2) → R 1 ⊂ C n .
Definition 2.1. The plumbing M = T * Q 0 #T * Q 1 is the Liouville manifold obtained by completing a Weinstein domain defined by identifying points in the image of Dη 0 with points in the image of Dη 1 whenever their Dι j -images in R ∞ ⊂ C n co-incide.
Essentially, we are identifying the two copies of D * B(1/2) ∼ = B n × B n via the map (x, y) → (y, −x), which preserves the symplectic form dx∧dy (the result of the identification inherits a natural Liouville structure).
2.1. Exact Lagrangians in A 2 Milnor spaces. Throughout this section, fix a dimension n ≥ 3. We work over an arbitrary coefficient field k.
Let A n 2 denote the n-dimensional A 2 -Milnor fibre obtained by plumbing two copies of T * S n , which can also be described as the affine variety
A n 2 = x 2 1 + · · · + x 2 n + t 3 = 1 ⊂ C n+1 equipped with the restriction of the standard symplectic form ω C n+1 = i 2π n+1 j=1 dz j ∧ dz j .
This section will classify exact Lagrangian submanifolds of Maslov class zero inside A n 2 up to quasi-equivalence in the Fukaya category. The main technical step applies in slightly more generality, so we begin with a Stein manifold (M, ω) which is the plumbing of two cotangent bundles T * Q i for Z-homology n-spheres Q 0 and Q 1 . We fix gradings up to a global shift by requiring that
(2.8) HF (Q 0 , Q 1 ) ∼ = k is concentrated in degree 1.
and consider Q i as objects of H * F(M ). The category H * F(M ) admits an action of the braid group Br alg 3 on three strings, obtained from the twist functors T i in the spherical objects Q i . An easy consequence of the definition of the algebraic Dehn twist is the following result from [30,Lemma 5.11].
Lemma 2.2. T i (Q i ) is quasi-equivalent, in Tw(H * F(M )), to Q i [1 − n].
Proposition 2.3. Suppose π 1 (Q i ) admits no non-trivial finite-dimensional representation, for i = 0, 1. Up to quasi-equivalence and shift in Tw(H * F(M )), every closed Lagrangian brane lies in the braid group orbit of Q 0 or Q 1 .
Remark 2.4. In fact Q 1 and Q 0 lie in the same orbit of the braid group action.
We will break up the proof into stages. Given a Lagrangian brane L, generation of F(M ) by the cores (Theorem 1.1) implies L, as an object of Tw(H * (F(M ))) is equivalent to a twisted complex built from local systems on the Lagrangians Q i . Lemma 2.5. There is a twisted complex which is quasi-isomorphic to L, built from the objects Q i , and in which none of the arrows are identity morphisms.
Proof. The hypothesis on the fundamental groups implies that the modules V i appearing in this complex are trivial, i.e. just finite-dimensional k-vector spaces. Moreover, the compact cores Q i define rank one simple modules over the wrapped Floer cohomology algebra of the cotangent fibres T * qi Q i . Proposition 4.2 guarantees that this algebra is non-positively graded, and Lemma 4.7 asserts that in a twisted complex of (ordinary) modules over an A ∞ -algebra graded in non-positive degree, all arrows arise from morphisms of degree at least 2 in the A ∞ -algebra.
We shall use throughout this section the fact that, in the subcategory of H * (F(M )) with objects Q i , all higher products vanish for degree considerations (recall that our minimal model is assumed strictly unital; the vanishing result is in fact true whenever n ≥ 2, but n > 2 is the easy case). Proposition 2.3 is a consequence of the following purely algebraic result: Lemma 2.6. Under the same hypotheses as in Proposition 2.3, let C • be a twisted complex over Q 0 and Q 1 whose endomorphism algebra is supported in non-negative degrees. Then C • is in the braid group orbit of a direct sum of copies of Q j supported in a single degree, for either j = 0 or j = 1.
By applying a suitable shift, we assume that the twisted complex C • is concentrated in degrees 0 ≤ i ≤ N for some N ≥ 0. The degree i part of the complex is
U i ⊗ Q 0 ⊕ V i ⊗ Q 1 where U i , V i are k-vector spaces concentrated in degree i; the internal differential (of degree +1) in the twisted complex is comprised of arrows U i → V i , V j → U j−n+2 , U i → U i−n+1 and V i → V i−n+1
, labelled by the appropriate generators of HF (Q i , Q j ). Diagrammatically (taking n = 4):
(2.9) U 0 U 1 U 2 U 3 z z · · · z z V 0 V 1 V 2 h h V 3 h h d d · · · d d
with vertical arrows labelled by HF (Q 0 , Q 1 ), dotted arrows by HF (Q 1 , Q 0 ) and the curved arrows by the fundamental classes of the Q i (note that Equation 2.8 implies that the total differential is indeed degree 1).
Remark 2.7. The curved arrows emanate only from U j and V j with j ≥ n − 1 ≥ 2.
The key ingredient is a measure of complexity of C • defined as follows.
Definition 2.8. Let |U i | = 2i + 1, |V j | = 2j and define cx(C • ) = max max Ui =0 |U i |, max Vi =0 |V i | − min min Ui =0 |U i |, min Vi =0 |V i | .
In other words, the complexity is the longest zig-zag path ↓ ↓ · · · that can be drawn in C • . Note that we are not assuming that all the arrows in the zig-zag are represented by non-trivial differentials in the complex (the here is a map V i → U i−1 whose degree would not be compatible with a differential), just that the complex stretches out over the requisite number of degrees. We will prove by induction on cx(
C • ) that C • σ(Q 0 ) for some σ ∈ Br alg 3 = T 0 , T 1 . Lemma 2.9. Assume U 0 ⊕ V 0 = 0. Let N = max i {U i ⊕ V i = 0}. Then V 0 = 0 ⇔ U N = 0.
Proof. Suppose for contradiction that V 0 = 0, but U N = 0. Consider the component of
End HF (M ) (C • , C • ) containing Hom(V N , V 0 ) ⊗ e Q1 .
Since the twisted complex has no arrows out of V 0 , and no arrows into V N by the hypothesis U N = 0, this morphism must be closed. The fact that µ k = 0 for k ≥ 3 and that there are no identity morphisms e Q1 in the differentials of the twisted complex furthermore implies that Hom(V N , V 0 ) ⊗ e Q1 cannot be exact: if µ 2 (p, q) = e Q1 for differentials p, q then p = e Q1 = q. However, this morphism has negative degree, which is a contradiction to our assumption that
End Tw(HF (M )) (C • , C • )
is concentrated in non-negative degrees. Therefore, V 0 = 0 ⇒ U N = 0. The reverse implication is analogous, looking instead at Hom(U N , U 0 ) ⊗ e Q0 and recalling that e Q0 does not appear as a differential.
Remark 2.10. The proof of Lemma 2.9 actually yields a slightly stronger conclusion:
(1) If U N = 0 then V j = 0 for j < n − 2; (2) If V 0 = 0 then U N −j = 0 for j < n − 2.
This is because, say in the first case, the proof U N = 0 ⇒ V 0 = 0 only used the fact that no arrows come out of V 0 , which also holds true for V j with j < n − 2.
Lemma 2.11. In the situation of Lemma 2.9, suppose V 0 = 0. Then either V N −i = 0 for i < n − 1 or U j = 0 for j < n − 1. In the first case T −1 0 (C • ) has lower complexity than C • and in the second T 1 (C • ) has lower complexity than C • .
Proof. Suppose first that N ≥ n − 1. We will prove that one can reduce the complexity by applying one of the moves indicated. Iteratively, one can then simplify the twisted complex by a braid until N ≤ n − 2. At this point, there are no curved arrows; in this special case, a minor modification of the arguments given below allows one to further reduce complexity directly.
The start of the analysis applies to both cases. We claim that
• U i → V i is injective for i = 0, 1, . . . , n − 2; • V i+n−2 → U i vanishes for i = 0, 1, . . . , n − 2.
For the first statement, suppose the vector space morphism U i → V i has kernel, with i < n − 1. Pick a non-trivial element γ ∈ Hom(U N , U i ) with image in that kernel, noting that U N = 0 by Lemma 2.9. Then γ ⊗ e Q0 is closed (since no other arrows come out of U i ; this follows from Remark 2.7) but not exact, being a multiple of an identity element. It survives to the cohomology of endomorphisms of C • , but has negative degree, a contradiction. Vanishing of V i+n−2 → U i when i < n − 1 now follows from the fact that d C • • d C • = 0 in the twisted complex C • . Indeed, the Hom(V i−n+2 , V i )-term in the Maurer-Cartan equation for d C • is a tensor product Hom(V i−n+2 , U i ) ⊗ Hom(U i , V i ). Analogously, we claim that
• U N −i V N −i is surjective for i = 0, 1, . . . , n − 2; • V N −i → U N −i−n+2 is zero for i = 0, 1, . . . , n − 2.
The first assertion follows since if the map had cokernel, one would split V N −i ∼ = im(U N −i )⊕ V N −i and consider an element of Hom(V N −i , V 0 ) ⊗ e Q1 to obtain a cohomologically essential endomorphism of negative degree. Closedness again follows from Remark 2.7, and the fact that i < n − 1. The second assertion then follows from d 2 = 0 for the twisted complex C • , as previously; the Hom(U N −i , U N −i−n+2 )-component in the Maurer-Cartan equation comes from morphisms factoring through V N −i .
We now apply the autoequivalence T −1
0 to C • , cf. (2.6). Since U N −i → V N −i is onto, we can split the map to write U N −i ∼ = U N −i ⊕ V N −i . By Lemma 2.2, applying T −1 0
has the effect of shifting the U N −i -term of the complex leftwards by n − 1 places, and replacing the V N −i ⊗ (Q 0 ⊕ Q 1 ) term by V N −i ⊗ Q 1 only. In particular, after applying T −1 0 , the top right n − 2 places in the complex T −1 0 (C • ) are zero. One can apply a similar analysis at the left hand side of the original complex C • . Using the fact that U i ⊂ V i if i < n − 1, and writing
V i = V i ⊕ im(U i ), the autoequivalence T −1 0 has the effect of replacing U i ⊗ (Q 0 ⊕ Q 1 ) by U i ⊗ Q 1 and V i ⊗ Q 1 by V i ⊗ Q 1 → V i ⊗ Q 0 [2 − n].
The upshot is that one has a twisted complex of the shape (for the diagram, n = 4 again, and †, * denote vector spaces which can vary from instance to instance, or depend on the value of N ):
(2.10) V 0 V 1 † ⊕ V 2 * y y * y y · · · 0 0 0 0 V 0 h h V 1 i i V 2 h h · · · e e V N −1 V N h h
We now consider the two cases of the statement of the Lemma.
(1) If V N −i = 0 for every i = 0, 1 . . . , n − 2, then although the complex now extends an additional n − 2 columns to the left, the overall complexity has been reduced from 2N + 1 to ≤ 2N , hence by at least 1 (essentially because of the gap at the bottom left of the diagram above).
(2) If some V N −j = 0 with 0 ≤ j ≤ n − 2, then we claim that the diagonal arrows V k V k are injective for 0 ≤ k ≤ n − 2.
Otherwise, in the usual fashion, we can build a negative degree cohomologically essential morphism by taking an element of Hom(V N −j , V k ) ⊗ e Q1 . An additional step is needed in case k = n − 2, since the vector space occuring at that point in the top row of (2.10) is †⊕V n−2 which may be larger than V n−2 . Namely, we must observe that there is no non-trivial component V n−2 → †: indeed, † = ker(U n−1 → V n−1 ). Since V k is the quotient of V k by U k , we therefore conclude that U k = 0 for 0 ≤ k ≤ n − 2. In other words, our initial twisted complex is of the shape
(2.11) 0 0 0 U 3 U 4 · · · U N −1 U N V 0 V 1 V 2 V 3 d d V 4 d d · · · d d V N −1 V N i i
We now claim that the vertical arrows U N −j → V N −j must be isomorphisms for 0 ≤ j ≤ n − 2 at the top right of the diagram (we already know they are surjective). If this was not true, then take an element in the kernel of one of these maps, and consider Hom(U N −j , V 0 ) ⊗ p to obtain a closed and non-exact endomorphism of negative degree, where p ∈ HF (Q 0 , Q 1 ) is the unique intersection point. Finally, apply the autoequivalence T 1 to the resulting complex. The terms V j for j < n − 1 are shifted rightwards, so the leftmost n − 2 columns now vanish; whilst the isomorphisms U N −j → V N −j in the rightmost columns are replaced by terms U N −j on the upper row, and V N −j−n+1 on the lower row. Thus, we see
(2.12) 0 0 0 U 3 U 4 · · · U N −1 U N 0 0 0 V 0 V 1 · · · V N −3 V N −2 i i
The complexity has again been reduced by at least one, which completes the proof of the second case.
Lemma 2.12. In the situation of Lemma 2.9, suppose V 0 = 0. Then either V i = 0 for i = 0, 1, . . . , n−2 or U N −i = 0 for i = 0, 1, . . . , n−2; the twist T −1 1 (C • ) has lower complexity than C • in the first case, and T 0 (C • ) has lower complexity in the second case.
Proof. The proof is exactly analogous to that of Lemma 2.11, and indeed reduces to that Lemma by suitable relabelling. More precisely, we replace the ordered pair (Q 0 , Q 1 ) by the pair (Q 1 [n − 2], Q 0 ), noting that in both cases the unique morphism from the first brane to the second is concentrated in degree 1. The effect on the complex (2.9) is to shift the lower row to the left, so the dotted arrows become vertical, and the previously vertical arrows move n − 2 places to the left. Finally, since we have assumed throughout that U N ⊕ V N = 0, and the hypothesis V 0 = 0 implies that U N = 0 by Lemma 2.9, we know that V N = 0. Up to reversing the roles of the U i and V i , this brings us back into exactly the situation of Lemma 2.11. Moreover, Remark 2.10 implies that the relabelling operation does not itself increase complexity. The conclusion of Lemma 2.11, given the relabelling, is therefore to say that one can reduce complexity either by applying the inverse twist in Q 1 [n − 2] or the twist in Q 0 . The twist functor in a shifted object is exactly the same as the twist functor in the unshifted object, which completes the proof.
Applying Lemmata 2.11, 2.12, we immediately conclude Lemma 2.6.
Proof of Proposition 2.3. Since every exact Lagrangian has endomorphism algebra supported in degree [0, n], Lemma 2.6 implies that such a Lagrangian is in the braid group orbit of a direct sum of copies of one of the components of the skeleton. (A choice of component is immaterial, by Remark 2.4.) Whenever this direct sum has more than one factor, the endomorphism algebra has dimension > 1 in degree 0. We conclude that a connected exact Lagrangian must in fact be in the braid group orbit of (either of) the two components of the skeleton.
As an immediate Corollary, one has: Corollary 2.13. Let M be the plumbing of two Z-homology spheres Q i for which the groups π 1 (Q i ) admit no non-trivial finite-dimensional representation. A Maslov zero exact Lagrangian submanifold of M is a Z-homology sphere, which moreover realises a primitive homology class.
Remark 2.14. If Q 0 ∼ = S n ∼ = Q 1 , then the autoequivalences T i are realised by the Dehn twists in the Q i . Later we will prove that if π 1 (Q i ) = {1}, then T i is not induced by any compactly supported symplectomorphism, and the images of the Q i by general braids are not represented by embedded Lagrangians.
Remark 2.15. If the Q i are k-homology spheres for some field k, with the same hypotheses on π 1 (Q i ), then the conclusion of Corollary 2.13 holds in the Fukaya category linear over k; in particular, every exact L ⊂ M of Maslov class zero is then a k-homology sphere.
One can vary the hypotheses on the π 1 (Q i ) somewhat. In general, the modules V i appearing in the twisted complex C • may be non-trivial local systems. Morphisms between such are still supported in non-negative degrees. The proof relies on triviality of the V i only for this statement, and for the fact that the horizontal arrows in the complex C • are all obtained from the fundamental classes of the Q i . For non-trivial V i , there may in general be additional classes in HF
(V i ⊗ Q i , V j ⊗ Q j ) in intermediate degree.
However, if such classes vanish under pullback to a covering space, one can use the methods of Section 2.2, and in particular Theorem 2.21, to eliminate the contributions of these classes. As a special case: Corollary 2.16. If each Q i either admits no non-trivial finite-dimensional local systems, or has contractible universal cover, then the conclusions of Corollary 2.13 hold.
We will leave the interested reader to fill in the details.
Remark 2.17. Note that the proof of Lemma 2.6 did not use at any stage the fact that the vector spaces V i and U i are finite dimensional. In particular, if one knows that every local system (of arbitrary rank) on a Lagrangian lies in the category generated by (possibly infinite) direct sums of the components of the skeleton, the same argument would show that any such object in the Fukaya category of a plumbing would be equivalent to an object in the braid group orbit of direct sums of Q 0 . Quasi-isomorphism in the wrapped category implies quasi-isomorphism in the usual category of dg-local systems, by results in [7,Appendix B]. In particular, we would conclude that every representation of π 1 (L) would split as a direct sum of trivial ones. As we note in Remark 4.8, the results of [7] imply such a generation statement, from which we conclude the simple connectivity statement asserted in the introduction, Remark 1.5.
2.2.
Spherical twists in homology spheres. The invertibility of the spherical twist functor T Σ relies only on the fact that H * (Σ) ∼ = H * (S n ). If Σ ∼ = S n ⊂ M is diffeomorphic to the sphere, then there is also a geometric Dehn twist τ Σ . This admits a canonical lift to a graded symplectomorphism, hence acts on F(M ), and Seidel (following Kontsevich) proved that [30,Corollary 17.17] T Σ ∼ = τ Σ are quasi-isomorphic in the category of A ∞ -endofunctors of F(M ). By contrast, for Σ ∼ = S n , there is no candidate geometric Dehn twist, since Σ will not in general admit a metric with periodic geodesic flow. Informally, one would like the non-existence of a Dehn twist to be reflected in the non-surjectivity of a map
π 0 Symp ct (T * Σ) −→ Auteq(HF(T * Σ))/ [1] .
As formulated, this is slightly problematic, since the category F(T * Σ) has too few objects (all closed objects being isomorphic up to shift). One solution is to consider an enlarged category associated to a cotangent bundle which incorporates suitable non-compact Lagrangian submanifolds, for instance those co-inciding outside a compact subset with a fixed cotangent fibre. Such a category was invoked by Seidel in [34], and related but more general versions have been considered in the work of Nadler and Zaslow [25]. Thus, in the Nadler-Zaslow category N Z(T * S n ), the Dehn twist and the shift functor are indeed distinct autoequivalences, in contrast to their actions on F(T * S n ).
To keep technical details to a minimum, we take a slightly more circumspect approach. Let M Σ = T * Σ#T * S n denote the Weinstein manifold obtained by plumbing T * Σ and T * S n . Equivalently, M Σ is obtained by performing a Weinstein surgery along the Legendrian unknot which is the boundary of a cotangent fibre in T * Σ. Then M Σ contains two obvious closed exact Lagrangians, the components Σ and S n of its compact core. Compactly supported symplectomorphisms of T * Σ act on the Fukaya category F(M Σ ) provided we quotient out by the shift [1] (alternatively one could consider graded symplectomorphisms).
Remark 2.18. The wrapped Fukaya category W(T * Q) is invariant under arbitrary exact symplectomorphisms of T * Q, whilst the Nadler-Zaslow category is invariant only under compactly supported symplectomorphisms of Q. The Fukaya category of the plumbing F(T * Σ#T * S n ) mediates between the two, being invariant under symplectomorphisms which co-incide with the identity near the boundary of a single cotangent fibre. It is therefore more sensitive to "compactly supported" phenomena than the full wrapped category W(T * Q), even if a priori less so than N Z(T * Q).
The following is straightforward.
Lemma 2.19. The twist functor T Σ ∈ Auteq(HF(M Σ ))/ [1] has infinite order, in particular this quotient group is non-trivial.
Proof. Given our cohomological assumptions, this follows from the usual proof of injectivity of the braid group Br alg 3 into the autoequivalences of the Fukaya category of the A n 2 -Milnor fibre. Explicitly, writing T i Σ for the i-fold self-composition of T Σ , the Floer cohomology groups HF (S n , T i Σ (S n )) are unbounded in rank as i → ∞. By contrast, the autoequivalence T Σ cannot arise from a symplectomorphism unless Σ is homeomorphic to S n . Proposition 2.20. If π 1 (Σ) = 1, the cyclic subgroup Z generated by T Σ meets the image of the natural map ρ : π 0 Symp ct (T * Σ) −→ Auteq(HF(M Σ ))/ [1] only in the identity.
The proof will rely on passing to the universal cover, so we let M be any Liouville manifold with universal cover π :M → M . The following is a summary of the results proved in [7, Section 6]:
Theorem 2.21.
There is a wrapped Fukaya category W(M ; π) which comes with a pullback functor π * : W(M ) −→ W(M ; π) which acts on objects L of W(M ) by taking the total inverse image π −1 (L) ⊂M and such that the map on morphisms
(2.13) HF * (L, L) → HF * (π −1 (L), π −1 (L))
agrees with the classical pullback on cohomology whenever L ⊂ M is closed. Moreover, deck transformations of π act by autoequivalences of W(M ; π).
Remark 2.22. For infinite coverings, the category W(M ; π) is not in general the same as the wrapped category of the underlying Stein manifold as defined without reference to the covering π. For instance, if π : T * R → T * S 1 is the universal covering, then W(T * R; π) is not empty -the fibre and zero-section are non-trivial -whereas the usual wrapped category W(C) vanishes.
We proceed to the proof of Proposition 2.20. Suppose Σ is a homology sphere with nontrivial fundamental group. Let ι :Σ → Σ denote the universal covering. We fix a coefficient field k of characteristic dividing the index of the covering ι, interpreted as meaning that k is arbitrary if this covering is infinite. Suppose for contradiction that T Σ is geometric, induced by a compactly supported symplectomorphism τ . Let M Σ be the Weinstein manifold obtained by capping the Legendrian unknot in D * Σ. Considering τ • τ , or its inverse, and recalling the definition of the twist functor in Equation 2.5, we infer that there is an exact Lagrangian submanifold L ⊂ M Σ representing the twisted complex
(2.14) Σ[n − 1] ← Σ ← S n
with the leftmost arrow given by multiplication by the fundamental class [Σ], and where S n is the core component coming from capping a fibre of D * Σ. (Applying shifts, we have assumed that S n is placed in degree zero and that the morphism Σ ← S n is in degree 1.)
Let π :M Σ → M Σ denote the (universal) covering induced by ι :Σ → Σ. The inverse image of this complex in W(M Σ ; π) is represented bỹ Σ[n − 1] ←Σ ← π −1 (S n )
in which the left differential is obtained by pullback from the fundamental class of Σ, hence vanishes (trivially if ι is infinite; by our choice of characteristic for the coefficient field k if ι is finite). By the long exact triangle for the Dehn twist, the complex Σ ← S n is the image of Σ under the twist in S n . Now, π −1 (Σ) is connected, so the corresponding object in the category W(M Σ ; π) is indecomposable (considering the rank of its HW 0 ). This in turn implies thatΣ[n − 1] ← π −1 (S n ) does not admit any non-trivial summand. This follows immediately from taking Floer cohomology with a cotangent fibre to one of the components of π −1 (S n ): the Floer group with have rank 0 in the first case, and 1 in the other.
Proof of Proposition 2.20. Being the preimage of a connected Lagrangian submanifold L under a covering, all the components of π −1 (L) are diffeomorphic and moreover related by deck transformations. These lift to autoequivalences of W(M Σ ; π) by Theorem 2.21. The analysis above showed that, up to shifts of the pieces,
(2.15) π −1 (L) ∼ =Σ ⊕ Σ ← π −1 (S n )
in the category W(M Σ ). Each connected componentL α ⊂ π −1 (L) is itself an indecomposable object of the category, since HW 0 (L α ,L α ) has rank 1. For the same reason,Σ is indecomposable, and Lemma 2.23 implies thatΣ ← π −1 (S n ) is not isomorphic to a deck transformation image ofΣ. It follows that the RHS of (2.15) is not quasi-isomorphic to a direct sum of indecomposables lying in a single orbit of the covering group, which is a contradiction (recall that working over a field, the decomposition of an object into a sum of indecomposables is unique). The conclusion is that the twisted complex of (2.14) cannot represent an exact Lagrangian L, hence τ did not exist.
This argument is actually a bit more general. Suppose Σ is a homology sphere to which the analysis underlying Proposition 2.3 applies, for instance π 1 (Σ) has no non-trivial finitedimensional representations (or the universal cover of Σ is acyclic). Let φ : T * Σ → T * Σ be a compactly supported symplectomorphism, and write Φ for the associated element of Auteq(HF(M Σ )), defined with respect to an arbitrary choice of grading of φ. By the classification of exact Lagrangian submanifolds of Maslov class zero in T * Σ, the symplectomorphism φ preserves the object Σ up to shift. The image L of the core component S n ⊂ M Σ under φ co-incides with S n outside a compact subset of T * Σ, hence (2.16) HF (L, T * x S n ) ∼ = k has rank 1. Theorem 1.1 implies that L is a twisted complex on S n and Σ, and (2.16) (together with the fact that multiplication by [S n ] on HF (S n , T * x S n ) is trivial) implies that this complex contains a unique copy of S n . Lemma 2.9, and its analogue with the roles of Q 0 and Q 1 reversed, then implies that L may be represented by a complex of the shape
(2.17) C • ← S n or S n ← C • with C • ∈ T w(Σ)
the twisted complex Σ → · · · → Σ of arbitrary length and with all differentials the fundamental class [Σ] (more explicitly, Lemma 2.9 implies that V i =0 = 0 and that V 0 has dimension 1). The two cases of (2.17) are again analogous, and we treat only the first. Place S n in degree zero, and consider the inverse image in W(M Σ ; π), represented by a complex
(2.18)C • ← π −1 (S n )
where all differentials inC • vanish, being obtained by pullback from the fundamental class of Σ. This preimage is again quasi-isomorphic to a direct sum
(2.19) π −1 (L) Σ [k(n − 1)] ⊕ · · · ⊕Σ[2(n − 1)] ⊕ Σ [n − 1] ← π −1 (S n )
Lemma 2.23 yields a contradiction unless (2.20) L (Σ ← S n ) or L S n by following the same steps as in the proof of Proposition 2.20. In the first case, replacing φ by an iterate if necessary, we would obtain a new Lagrangian submanifold representing the complex Σ ← Σ ← S n , which contradicts (2.20). Therefore L S n , and moreover one can easily check that φ acts trivially on the Floer cohomology algebra generated by Σ and S n . In other words, we conclude:
Corollary 2.24. Let Σ be a homology sphere with π 1 (Σ) = 1. Suppose π 1 (Σ) admits no non-trivial finite-dimensional representation. Any φ ∈ Symp ct (T * Σ) acts trivially modulo shift on H * (F(M Σ )).
The symplectomorphism φ plays a rather minor role above, which leads to a slightly more general formulation of the conclusion. Inside T * Σ, consider an exact Lagrangian submanifold K which is a filling of the Legendrian unknot Λ. That is, inside the unit disk cotangent bundle, we consider a properly embedded Lagrangian submanifold with Λ as Legendrian boundary.
Corollary 2.25. If Σ is a homology sphere with π 1 (Σ) = 1, and if π 1 (Σ) admits no nontrivial finite-dimensional representations, then any exact Maslov zero Lagrangian filling K of Λ is a homology ball.
Proof. Adding a Weinstein handle along the Legendrian unknot yields an associated closed Lagrangian submanifoldL ⊂ M Σ , with M Σ = T * Σ#T * S n the plumbing considered previously. The analysis surrounding Equation 2.20 implies that
K S n orK (Σ ← S n ) orK (S n ← Σ) ∈ HF(M Σ ).
These three objects are quasi-represented by the core component S n and its two possible Lagrange surgeries with the other component Σ, respectively. It follows that K is Floer cohomologically indistiguishable from at least one of the fibre T *
x Σ or one of its surgeries with the 0-section, in particular has the same cohomology groups as one of these spaces. Since the complement of a disk in a homology sphere is a homology ball, the result follows.
Remark 2.26. Some care should be exercised in extending the results here from Z-homology spheres to k-homology spheres. Although the basic classification of twisted complexes carries over, cf. Remark 2.15, there is potentially conflict between the characteristic of k for which one obtains a homology sphere and the characteristic required in the vanishing argument for differentials in the twisted complex in the universal cover. For instance, RP 3 is a homology sphere except in characteristic 2; however, the vanishing argument after Equation 2.18 would precisely require characteristic 2, i.e. a divisor of |π 1 (RP 3 )|, and T * RP 3 does admit a non-trivial autoequivalence (Dehn twist).
Stablisations and symplectomorphisms.
Consider the operation which assigns to a Liouville manifold M its product with the unit disc D 2 together with the function
W : M × D 2 → D 2 (2.21) (m, z) → z 2 . (2.22)
Using the techniques developed in [30], one may construct a well defined Fukaya category F(M × D 2 , W ) with objects being exact Lagrangian branes with boundary contained in M × {1}. The study of these categories is not fully developed, but one expects the following conjecture to hold:
Conjecture 2.27. For each coverM of M there is a commutative diagram (2.23) F(M ) / / F(M × D 2 , W ) F(M ) / / O O F(M × D 2 , W ). O O
where the horizontal arrows are equivalences, and the vertical arrows are pullback functors which assign to a Lagrangian its inverse image.
Assuming this conjecture, we observe that whenever M is the plumbing of T * Σ and T * S n , with Σ a homology sphere with non-trivial fundamental group, there can be no geometric automorphism ofM × D 2 realising the Dehn twist in F(M × D 2 , W ), cf. Remark 1.10. Indeed, most arguments given in the proof of Proposition 2.20 use only algebra, and hence would apply just as well after stabilisation. The only geometric part of the argument appears in our analysis of Equation (2.15), where we use the fact that the inverse images of a Lagrangian under a covering map are disjoint, hence Floer cohomologically orthogonal, and related by deck transformations. While we are not proposing a precise definition of the class of symplectomorphisms ofM × D 2 which give rise to automorphisms of the category F(M × D 2 , W ), any sufficiently natural definition will still have the property that the deck transformations act, and that is all that was needed.
2.3. Truncated polynomial rings. The construction of the geometric Dehn twist τ L in a Lagrangian sphere L ⊂ M relies on the existence of a metric on S n with periodic geodesic flow. Similar metrics exist on all the real, complex and quaternionic projective spaces, and the Cayley projective plane. In general, if a manifold Z admits a metric for which all geodesics are closed, they are necessarily of the same length, and the geodesic flow is periodic. Such manifolds are called Zoll manifolds, and have been the subject of much classical investigation. Bott [9] proved, using Morse theoretic analysis of spaces of geodesics, that if a manifold Z admits a Zoll metric then H * (Z) is a truncated polynomial ring (and π 1 (Z) is either trivial or of order 2). is non-trivial, then H * (Q; Z) is a truncated polynomial ring.
A cleaner formulation, avoiding the auxiliary manifold M Q , invokes the Nadler-Zaslow category N Z(T * Q) in place of F(M Q ), cf. Remark 2.18. We note that classical work of Adem gives strong constraints on the manifolds whose cohomology is a truncated polynomial algebra; writing H * (Q) ∼ = Z[x]/ x d , necessarily the degree |x| of x is a power of 2, and if |x| > 4 then x 3 = 0. The only known simply-connected manifolds with truncated polynomial cohomology are homeomorphic to the compact rank one symmetric spaces. The rest of this section is devoted to providing evidence for Conjecture 2.28, proving a partial analogue in two non-trivial cases.
Proposition 2.29. In the situation of Conjecture 2.28, suppose moreover that all higher products
µ p : H i1 (Q; k) ⊗ · · · ⊗ H ip (Q; k) → H i1+···+ip+2−p (Q; k) with p ≥ 2 and 0 < i j < i j + 2 − p < n = dim R (Q)
vanish, i.e. vanishing holds whenever the output lies in cohomological degree < n. If for φ ∈ Symp ct (T * Q) the autoequivalence Φ = ρ(φ) acts non-trivially on objects of F(M Q ), then 0<i<n rk k H i (Q; k) ∈ {0, 1}.
Note that the hypothesis applies to an (n − 1)-connected 2n-manifold Q, implying that H n (Q; k) has rank ≤ 1. The hypothesis is also satisfied if Q has formal cohomology ring, and if the only non-trivial cup-products are those forced by Poincaré duality (i.e. those into the top cohomological degree); for instance, if Q is a product of spheres, or a connect sum of such. Thus, Proposition 2.29 implies Proposition 1.12 from the Introduction.
We establish some preliminary lemmas relevant for Proposition 2.29. Suppose an exotic autoequivalence Φ arising from a compactly supported symplectomorphism φ does exist, and apply it to the core component S n . By Theorem 1.1, the image is a twisted complex on S n and Q. Compact support of φ implies that Φ(Q) Q[j] is quasi-isomorphic to some shift of Q, by uniqueness of exact Lagrangian submanifolds of Maslov class zero in simply-connected cotangent bundles [14,24]. In particular, the group
(2.24) HF (Φ(S n ), Q) ∼ = HF (S n , Q[j]) ∼ = k
has rank 1 over the field. Equation (2.24) implies that the twisted complex representing Φ(S n ) contains a unique copy of S n . Although Q is not a sphere, the conclusion of Lemma 2.9 still applies to twisted complexes over Q and S n , since that argument only used the fact that no identity morphisms appeared in the differentials of the twisted complex. As in the previous subsection, (the analogue of) Lemma 2.9 then implies that Φ(S n ) is equivalent to a twisted complex of the form
(C • ← S n ) or (S n ← C • )
with C • only involving the component Q. Since we are assuming that Φ acts non-trivially on objects modulo shift, and since Φ fixes Q modulo shift, the complex C • is not empty. Replacing φ by φ −1 and Φ by Φ −1 if necessary, we will henceforth concentrate on the case Φ(S n ) (C • ← S n ).
Lemma 2.30. HF (C • , Q) has rank 2.
Proof. C • lies in an exact triangle with S n and Φ(S n ), which have Floer homology of rank 1 with Q, in the latter case by (2.24). Exactness implies that HF (C • , Q) has rank 0 or 2.
However, since C • is a twisted complex on Q, it can only be orthogonal to Q if it is identically zero, which we have excluded by the hypothesis that Φ acts non-trivially on objects.
Let V i denote the vector spaces underlying the twisted complex C • , so
Φ(S n ) ∼ = V 0 ⊗ Q ← V 1 ⊗ Q ← · · · ← V N ⊗ Q ← S n
where by assumption V 0 = 0 and V N = 0.
Lemma 2.31. Either Q is a k-homology sphere, or N > 0.
Proof
. If C • = V ⊗ Q, then HF (C • , Q) ∼ = V ∨ ⊗ H * (Q)
. This can only have rank 2 if V ∼ = k and Q is a homology sphere.
Since we know that homology spheres admit categorical Dehn twists, we will henceforth exclude this case, and suppose that N > 0. Proof. Any element of Hom(V 0 , H n (Q)) defines a morphism C • → Q as follows:
(2.27) V 0 ⊗ Q [Q] · · · o o V N ⊗ Q o o Q
which is closed, since V 0 is terminal in C • , but cannot be exact, from the usual assumption that the twisted complex C • contains no differentials represented by e Q (and using strict unitality to exclude higher-order A ∞ -products). Analogously, Hom(V N , H 0 (Q)) embeds into HF (C • , Q): the morphism
(2.28) V 0 ⊗ Q · · · o o V N ⊗ Q o o e Q Q
is exact, since the V N -copy of Q is initial, and cannot be closed since the differential d C • contains no e Q -terms. It is moreover easy to see that distinct morphisms arising in this way cannot be cohomologous if the corresponding elements of V 0 respectively V N are not linearly dependent, so Hom(V 0 , H n (Q)) ⊕ Hom(V N , H 0 (Q)) ⊂ HF (C • , Q).
Hom(⊕ N −1 i=0 V, H 0 (Q)) / / 4 4 , , Hom(⊕ N −1 i=1 V i , H 0<i<n (Q)) / / Hom(⊕ N i=1 V, H n (Q)) Hom(V N , H 0<i<n (Q)) 4 4
with an additional differential from the first to the last term, as indicated. Acyclicity implies that the differential surjects onto the top term in the middle column. Let U 0 denote the kernel of the composition of the differential with the projection onto this summand; this is a proper subspace of Hom(⊕ N −1 i=0 V, H 0 (Q)) whose rank is equal to dim(V )−1− 0<i<n b i (Q), with b i (Q) = rk k H i (Q; k) the i-th Betti number. Dually, let U N denote the quotient of Hom(⊕ N i=1 V, H n (Q)) by the image of the restriction of the differential to the last term in the middle column; this vector space also has rank dim(V ) − 1 − 0<i<n b i (Q).
Acyclicity of Equation (2.29) is now equivalent to the fact that the complex
(2.30) U 0 / / * * Hom(⊕ N −1 i=1 V i , H 0<i<n (Q)) / / U N is exact. Exactness in the middle now implies that (dim(V ) − 2) · 0<i<n b i (Q) ≤ 2 dim(V ) − 1 − 0<i<n b i (Q) (2.31) dim(V ) · 0<i<n b i (Q) − 2 ≤ −2 (2.32)
This is impossible whenever 0<i<n b i (Q) > 1, because the left hand side is then nonnegative, which implies the required conclusion.
It seems reasonable to call a compactly supported symplectomorphism of T * Q yielding an autoequivalence of T * Q which acts non-trivially on objects of F(M Q ) a "categorical Dehn twist".
Corollary 2.33. Let Q be a simply-connected closed 4-manifold. If T * Q admits a categorical Dehn twist then Q is homotopy equivalent to S 4 or CP 2 .
Proof. Proposition 2.29 implies that H 2 (Q; k) is trivial or of rank 1. Since this holds for any co-efficient field k, the result is true integrally, and then the universal coefficient theorem implies that H * (Q; Z) is torsion-free. Q is therefore a homotopy sphere or a homotopy complex projective plane, by (Milnor's refinement of) Whitehead's theorem [23].
Fibres generate the wrapped Fukaya category
Let Q 0 and Q 1 be two smooth closed oriented manifolds of the same dimension n. From [6], we know that a cotangent fibre generates the wrapped Fukaya category of each of T * Q 0 and T * Q 1 . The goal of this section is to prove the analogous result for the plumbing of the two cotangent bundles. To state the result, note that one can always arrange the plumbing parameters so that the plumbing region is disjoint from a pair of cotangent fibres T * qi Q i . In particular, these give rise to exact Lagrangians in T * Q 0 #T * Q 1 which we denote L qi , and hence to objects of its wrapped Fukaya category.
Theorem 3.1. Any pair of cotangent fibres L q0 and L q1 which are disjoint from the plumbing region, generate the subcategory of the wrapped Fukaya category of T * Q 0 #T * Q 1 consisting of closed Lagrangians.
The weaker statement that these Lagrangians split-generate the wrapped Fukaya category should follow from work of Bourgeois, Ekholm, and Eliashberg [10], together with the generation criterion of [4]. Unfortunately, the bridge between the Symplectic Field theoretic and Hamiltonian approaches to studying the Fukaya category is not fully in place, and the technicalities of building such a bridge would go beyond the scope of this paper.
Instead, we give an alternative approach in two parts. First, we consider an additional Lagrangian in T * Q 0 #T * Q 1 , coming from the diagonal Lagrangian in a copy of C n = R n ⊕ √ −1R n containing the model plumbing region. Using Seidel's double covering trick [30,Section 18], and the Viterbo restriction functor [2], we show that this Lagrangian, together with the two cotangent fibres appearing in the statement of Theorem 3.1, generates the Fukaya category. Then we use an explicit geometric argument, together with the restriction functor again, to show that the diagonal can be removed from our generating collection.
3.1.
Embedding the plumbing in a Lefschetz fibration. The starting point is to consider the manifold Q obtained by removing a small ball from each of Q 0 and Q 1 , and gluing the resulting boundary spheres via their natural identifications with S n−1 . Clearly Q is diffeomorphic to the usual connect sum of the Q i , and contains a separating sphere V .
By work of [14, Section 3], we know that any Morse function on Q extends to a Lefschetz fibration π 0 on an exact manifold with corners E 0 , over the disc of radius 1/2, containing Q as an exact Lagrangian. We shall choose a Morse function which maps the region coming from Q 0 to the negative reals, the region coming from Q 1 to the positive reals, and which has no critical values near the origin. The inverse image of 0 is therefore the separating n−1 dimensional sphere V ⊂ Q, which gives rise to a Lagrangian sphere in the fibre of π 0 . Note that this is a framed Lagrangian in the sense of Seidel, carrying a canonical identification with the sphere up to the action of O(n).
By perturbing the Lefschetz fibration, we may arrange that there are no critical points along the y-axis, that the fibration is trivial in a neighbourhood thereof, and that the only critical points along the x-axis lie on Q (see Figure 1). We may construct a new exact manifold E equipped with a Lefschetz fibration π over the disc of radius 1, with the same fibre as π 0 , by adding exactly one new critical point to π 0 , whose value lies on the segment between (0, 1/2) and (0, 1), and whose vanishing cycle agrees with V by parallel transport There are exact Lagrangian embeddings of Q 0 and Q 1 in E which intersect transversely at one point, whose image is contained in a neighbourhood of Q ∪ T , and which agree with Q away from T .
Proof. We construct Q 0 as the union of two pieces: First, consider the Lagrangian obtained by taking the intersection of Q with the inverse image of (−1/2, − ]. This is a Lagrangian diffeomorphic to the complement of a ball in Q 0 , with the boundary sphere mapping to − . Next, choose any smooth path contained in the neighbourhood of the y-axis connecting (− , 0) to the new critical value of π, which only intersects the y-axis at this critical value and agrees with a horizontal line in a neighbourhood thereof (see Figure 1). Having assumed that the fibration π 0 is trivial in a neighbourhood of the y-axis, this is also a thimble with the correct vanishing cycle (and hence topologically a ball). The union of these two Lagrangians is obviously homeomorphic to Q 0 , but it is actually diffeomorphic because the framing on V was obtained by using its identification with the unit sphere in the tangent space of a point on Q 0 . We construct Q 1 symmetrically so that it lies in the positive x half-plane. Note that these two Lagrangians only meet at the critical point of π which lies on the y-axis, proving the first part of the Lemma. The remaining claims are obvious from the construction.
Any neighbourhood of a pair of exact Lagrangians which meet transversely at one point is symplectomorphic to a neighbourhood in their plumbing. Choosing this neighbourhood to be small enough, we may assume that the only critical points of the Lefschetz fibration that it includes are those which lie on either Q 0 or Q 1 .
3.2.
General results. The results of this section apply beyond the special case we are considering. Recall that a thimble in the total space of a Lefschetz fibration is a Lagrangian ball obtained by transporting a vanishing cycle from a distinguished boundary point (say (0, 1) in our case) to a critical point along an embedded arc in the disc. A basis of thimbles is a collection of such balls, one for each critical point, which are obtained by arcs which do not intersect in the interior. We shall fix such a basis ∆ 1 , . . . , ∆ m .
The double covering trick of Seidel embeds E as a Liouville submanifold of a manifoldẼ, such that there are Lagrangian spheres∆ 1 , . . . ,∆ m inẼ whose intersections with E agree with ∆ 1 , . . . , ∆ m . The construction is arranged in such a way that the composition of the Dehn twists along the spheres∆ i is Hamiltonian isotopic to a symplectomorphism which maps every closed Lagrangian in E to a region which is disjoint from E.
Using the correspondence between algebraic and geometric twists, Seidel concludes: Lemma 3.3 (Lemma 18.15 [30]). Let L be a closed Lagrangian brane in E. There exists (i) an object C in the subcategory of F(Ẽ) generated by∆ i (ii) a morphism from C to L and (iii) a Lagrangian L − which is disjoint from E, such that we have a quasi-isomorphism
(3.1) Cone(C → L) ∼ = L −
in the derived Fukaya category ofẼ.
Remark 3.4. The reader familiar with Seidel's book may benefit from being reminded that this result, because it appeals to the Fukaya category ofẼ rather than to that of a Lefschetz fibration, does not require the imposition of any assumption on the characteristic of the ground field.
In order to derive a conclusion about wrapped Fukaya categories, we shall invoke a general result about such categories. Let W be an exact symplectic manifold whose Liouville form λ defines a contact structure on the boundary, and assume that W in ⊂ W is a codimension 0 submanifold which is also Liouville with respect to the restriction of λ. The wrapped Fukaya category W(W ) of W has, as objects, exact Lagrangians whose primitive is locally constant near the boundary, equipped with additional data to obtain integral gradings and oriented moduli spaces of holomorphic discs. Consider a subcategory L(W ) of W(W ) consisting of Lagrangians L satisfying the following additional property
(3.2)
L admits a primitive for λ which is constant on ∂W in ∩ L.
The following result was proved in Section 5 of [2]: Proof. Let L be a closed exact Lagrangian brane in E. We shall apply the restriction functor, Proposition 3.5, from the subcategory of F(Ẽ) whose objects are∆ 1 , . . . ,∆ m , L and L − to the wrapped Fukaya category of E. The main subtlety is that such a functor is only defined under the assumption that each Lagrangian is exact and admits a primitive which is constant on its intersection with ∂E. This is obvious for L and L − , and follows for ∆ i because the boundary is connected and Legendrian for an appropriate choice of contact form on the boundary of E.
At the level of objects, the restriction functor assigns to each Lagrangian its intersection with E, so we obtain that L − goes to 0, L to itself, and∆ i to ∆ i . In particular, Lemma 3.3 implies that the image of Cone(C → L) vanishes in the wrapped Fukaya category of E. Since the image of C lies in the category generated by the basis of thimbles, we conclude that so does L.
3.3.
Generators for the wrapped Fukaya category of the plumbing. We now return to the specific Lefschetz fibration we were studying, taking the origin as basepoint. We choose thimbles with the following property:
(3.3)
for j = 0, 1, either ∆ i is disjoint from Q j , or it intersects Q j transversely at one point. The thimbles disjoint from Q 0 ∪Q 1 correspond to critical points which do not lie on Q 0 ∪Q 1 . These may essentially be discarded, by the discussion at the end of Section 3.1 (they may be connected to the origin via arcs which follow the boundary of the disk of Figure 1 into the lower half of that picture, and which therefore do not intersect the projections of Q 0 and Q 1 ). The other thimbles are those arising from critical points which lie on Q 0 and Q 1 , so we cannot avoid the existence of an intersection. In this second situation, there are three different possibilities: First, we have the critical points which lie on Q 0 . As the associated thimbles intersect Q 0 in one point, we may find a Weinstein neighbourhood of Q 0 which intersects the thimble in a (disc) cotangent fibre. Next, there is the symmetric case replacing Q 0 by Q 1 . Finally, there is the distinguished critical point which lies on the y-axis. After translation in the base, we consider a local model Lefschetz fibration
C n → C (3.4) (z 1 , . . . , z n ) → z 2 i . (3.5)
Recall that we have arranged for Q 0 and Q 1 to project to paths which are horizontal near the critical point on the y-axis, so that after translation they agree with the x-axis. In particular, in this local model, they must come from the Lagrangians R n and √ −1R n . Moreover, we may choose the thimble associated to this critical point to project to the positive y-axis in this model, which then forces it to agree with (1 + √ −1)R n . Using the fact that a Weinstein neighbourhood of the union of Q 0 and Q 1 is modelled after the plumbing, we conclude: Proof. Apply the restriction functor from E to D * Q 0 #D * Q 1 , together with Proposition 3.6.
3.4.
Expressing the diagonal in terms of cotangent fibres. In order to prove Theorem 3.1, we shall show that the additional generator given in Corollary 3.8 is in fact redundant. The proof will use a geometric argument, together with the restriction functor a few more times.
We shall start by considering a point q ∈ R n and the Lagrangians q + √ −1R n and √ −1q + R n . Assuming that the model region in Equation (2.7) is sufficiently large so as to contain the point q + √ −1q, we respectively obtain Lagrangians L q and L p in the plumbing, which intersect. Using Polterovich's surgery construction [27], we may construct two other Lagrangians which agree with L q ∪ L p away from the intersection point. We recall a convenient model for these Lagrange surgeries. Working in a suitable Darboux ball, it is sufficient to define the surgeries at the origin of the union R n ∪ √ −1R n ⊂ C n . Let γ : R → C be a smooth embedded path satisfying the following conditions:
(1) for t 0, γ is a linear parametrization of the negative imaginary axis; (2) for t 0, γ is a linear parametrization of the positive real axis; (3) im(γ) ⊂ C lies in the upper left quadrant.
Associated to such a path γ we have the Lagrangian submanifold
(3.6) V γ = t∈im(γ) t.S n−1 ⊂ C n
where S n−1 ⊂ R n ⊂ C n is the unit sphere; this co-incides outside a compact set with R n ∪ √ −1R n . Any two such submanifolds are Lagrangian isotopic via a Lagrangian isotopy with compact support. V γ defines one of the two surgeries. The other surgery is obtained similarly, but starting with a path γ with the same asymptotic conditions, but now passing below the origin, so im(γ ) is contained in the union of the three quadrants in C whose interiors do not meet γ. Cone(L p → L q ) and Cone(L q → L p ).
Proof. Consider the Liouville domain obtained by attaching two Lagrangian handles to the plumbing along ∂L p and ∂L q . The Lagrangian discs L p and L q extend to Lagrangian spheres which we denote V p and V q . Let τ Vp (V q ) and τ Vq (V p ) denote the Lagrangian spheres obtained by applying the indicated Dehn twists. Seidel's long exact sequence [32] asserts that we have quasi-isomorphisms
(3.8) τ Vp (V q ) ∼ = Cone(V p → V q ) and τ Vq (V p ) ∼ = Cone(V q → V p ).
Applying the restriction functor, we see that V p and V q respectively map to L p and L q , while the two twists map to the two possible surgeries.
Remark 3.10. For the purposes of this paper, we do not need to analyse the morphism appearing in Equation (3.7). However, it follows easily from the construction of [2] that in both cases, the morphism corresponds to the generator of Floer cohomology arising from the intersection point in the local model. By increasing the norm of q, we may isotope the Lagrangians L p and L q to Lagrangian balls which are disjoint. By the analysis in Section 4.2, there is a "short" Reeb chord (one entirely contained in the local model) between these Lagrangians in only one direction, cf. Figure 4. Using a continuation homomorphism in Floer cohomology, we conclude that the connecting homomorphism in the cone defined by L p → L q in fact vanishes. From that, we conclude that the surgery is equivalent in the wrapped Fukaya category to the disjoint union of L p and L q , hence one obtains an easy example of pairs of Lagrangians which are isomorphic non-zero objects in the Fukaya category but not Lagrangian isotopic, indeed which have different topology. More generally, one can start with any pair of exact Lagrangians which do not intersect, and consider the shortest Reeb chord between them (there is a unique one generically). There is a corresponding Hamiltonian isotopy which makes the pair intersect at exactly one point.
Using the surgery formula of [13], one can prove again that one of the two surgeries is equivalent to the disjoint union in the wrapped Fukaya category, so this phenomenon is rather general. We shall now show that the cone of L q → L p is equivalent, in the wrapped Fukaya category of the plumbing, to the diagonal Lagrangian. The proof will use, yet again, the restriction functor, as well as the invariance of the quasi-isomorphism classes of objects under Hamiltonian isotopies. The key geometric observation is the following: (1) γ(0) = √ n(−1 + √ −1); (2) the tangent line to γ at γ(0) is R (1 + √ −1) ⊂ C; (3) the imaginary part im(γ(t)) is non-decreasing and monotonically increasing near 0. Consider the associated submanifold V γ ⊂ C n . It is straightforward to check that the first and third conditions imply
V γ ∩ (−1, . . . , −1) + √ −1R n ∪ ( √ −1, . . . , √ −1) + R n = {(−1 + √ −1, . . . , −1 + √ −1)}
Hence this intersection is exactly zγ(0), where z = (1, . . . , 1)/ √ n ∈ S n−1 . Moreover, this isolated intersection point is transverse, and the tangent space to V γ at the intersection point agrees with the translated diagonal (by our prescription of the tangent space to γ at γ(0)). We now define ∆ −1, √ −1 by linearising V γ in a small neighbourhood of this point. At this stage, we are ready to complete the proof of the main result of this section:
Proof of Theorem 3.1. Comparing the statement of the Theorem with that of Corollary 3.8, we see that it suffices to prove that the diagonal gives rise to a Lagrangian in the plumbing which is quasi-isomorphic to an iterated cone on cotangent fibres of Q 0 and Q 1 . In fact, we shall show that it is isomorphic to a single cone L q → L p . In order to prove this, we first choose q so that the compactly supported isotopy in Lemma 3.11 is contained in the model region, and abusively write ∆ p,q for the resulting Lagrangian in the plumbing. Then we choose a smaller model region which intersects ∆ p,q in the diagonal, and attach this region to disc cotangent bundles of small radius to obtain a Liouville subdomain
(3.9) D * ≤ Q 0 #D * ≤ Q 1 ⊂ D * Q 0 #D * Q 1 .
We now apply the restriction functor to the subcategory of W(D * Q 0 #D * Q 1 ) with objects L p , L q , and ∆ p,q . Since ∆ p,q is quasi-isomorphic to the cone L q → L p , the restriction of ∆ p,q is also quasi-isomorphic to such a cone. We conclude that ∆ = ∆ p,q ∩ D * ≤ Q 0 #D * ≤ Q 1 lies in the category generated by L q = L q ∩ D * ≤ Q 0 #D * ≤ Q 1 and L p = L p ∩ D * ≤ Q 0 #D * ≤ Q 1 . By construction, ∆ is the diagonal Lagrangian, while L q and L p are respectively cotangent fibres to Q 0 and Q 1 . We complete the proof by noting that the wrapped Fukaya category of the plumbing is independent of because an appropriate Liouville flow rescales
D * Q 0 #D * Q 1 into D * ≤ Q 0 #D * ≤ Q 1 .
Generation by compact cores
Let M be the Liouville manifold obtained by plumbing M = T * Q 0 #T * Q 1 . From the previous section, we know that the Fukaya category of closed exact Lagrangian submanifolds of Maslov class zero embeds in the category generated by the cotangent fibres. In this section, we shall prove is supported in non-positive degree. Moreover:
(1) if n > 3, A 0 = ⊕ i HW 0 (L qi , L qi );
(2) if n = 3, the subspace A 0 0,1 = HW 0 (L q0 , L q1 ) is an ideal on which all higher products vanish, whilst A 0 1,0 = HW 0 (L q1 , L q0 ) = 0. The quotient A * /A 0 0,1 is the direct sum of ⊕ i HW 0 (L qi , L qi ). For our application, we shall need to compute the degree 0 part of the wrapped Floer cohomology of a cotangent fibre, and relate its category of (ordinary) modules to the category of representations of the fundamental group. Recall that every local system on a Lagrangian gives rise to an object of the Fukaya category. Since Q i intersects L qi at one point, a module E i over the group ring of π 1 (Q i , q i ) gives rise to a module over HW 0 (L qi , L qi ) by considering HW * (L qi , E i ); since Q i is disjoint from L qj if i = j, we obtain a representation of A * by projecting to HW 0 (L qi , L qi ).
Proposition 4.4. If n ≥ 3, there is an isomorphism from HW 0 (L qi , L qi ) to the group ring of π 1 (Q i , q i ) such that the pullback of E i is isomorphic to HW * (L qi , E i ). [19]), which implies that any A ∞ algebra A is equivalent to one for which the differential vanishes. Recall that an A ∞ module is a graded vector space P together with a collection of maps
(4.2) µ 1|d : P ⊗ A d → P
of degree 1 − d satisfying a homotopy associativity equation; Homological Perturbation similarly implies any such is equivalent to one with µ 1|d = 0 only for d > 0. We shall call such algebras and modules minimal.
The main structure result for a minimal A ∞ algebra A supported in non-positive degrees is the following, cf. related ideas in [11]:
Lemma 4.6. The ascending degree filtration on any module is a filtration by A ∞ -submodules. The subquotients, which are supported in a single degree, are determined up to A ∞ equivalence by the corresponding representation of A 0 .
Proof. Under the assumption of minimality, all the A ∞ -operations µ k have degree 2−k ≤ 0, and only the product has degree 0. Similarly, all the µ 1|d have degree < 0 except for the multiplication µ 1|1 . The rest is straightforward.
Proof of Proposition 4.1. By Theorem 3.1, it suffices to prove that every module over A * which is supported in finitely many cohomological degrees has a filtration by such modules induced by local systems on Q 0 and Q 1 . It follows from the discussion above that every such module has a filtration by modules supported in a single cohomological degree, and that such a module is completely determined by the representation of A 0 = ⊕ i,j HW 0 (L qi , L qj ). If n > 3, this group is a isomorphic to the sum of group rings ⊕ i π 1 (Q i , q i ), and the module action is the obvious one, by Proposition 4.4. Hence, every such module arises from a local system on Q i . When n = 3 there is an additional step. Recall that A 0 0,1 = HW 0 (L q0 , L q1 ) is an ideal in A 0 . Given any module P over A 0 , A 0 0,1 · P is a submodule on which A 0 0,1 acts trivially because the product vanishes on it. Since A 0 0,1 also acts trivially on P/ A 0 0,1 · P , we may write P as an extension (4.3) P/ A 0 0,1 · P → A 0 0,1 · P of two modules on which A 0 0,1 acts trivially. Each of these is determined by its structure as a module over ⊕ i HW * (L qi , L qi ). By the same argument as above, such a module corresponds to a local system on Q i . We continue to assume that A * is graded in non-positive degrees. The following gives some control on the morphisms in a twisted complex representing a suitable shift of a closed exact Lagrangian, and was used in the discussion after the statement of Proposition 2.3.
Lemma 4.7. Any finite rank module over A * is equivalent to a twisted complex C • = (⊕ i Z[i]⊗P −i , δ ij ), with each P −i an ordinary A 0 -module and Z[i] a free graded abelian group concentrated in degree −i. The morphisms δ ij are given by elements of Hom(P −i , P −j ) of degree ≥ 2.
Proof. Applying the homological perturbation lemma, we assume that such a module P is minimal, and define P k to be its component in degree k. The ascending degree filtration
(4.4) P (l) = k≤l P k
is a filtration by submodules. Note that we have an exact triangle
(4.5) P (l−1) / / P (l) y y Z[−l] ⊗ P l e e
In particular, P (l) is quasi-isomorphic to the cone of a morphism (4.6) Z[−l] ⊗ P l → P (l−1) . Since P is of finite rank, the degree filtration terminates, and we obtain, by induction, a description of P as a twisted complex. Note that, in our convention for a twisted complex, the differential δ i,j is non-vanishing only if i < j. We therefore define the i th part of this complex to be Z[i] ⊗ P −i . Since the morphisms δ ij have total degree 1, and the homological shift is strictly negative, the corresponding morphism in Hom(P −i , P −j ) has degree strictly greater than 1.
Remark 4.8. Lemma 4.6 was implicitly concerned with finite rank modules, in particular finite rank local systems over exact Lagrangians, but there is a version which holds for arbitary rank systems. The relevant result is essentially [7,Proposition 3.3]. We fix a Liouville manifold M , a collection of (not necessarily closed) exact Lagrangian submanifolds {L i }, and a further collection of closed exact Lagrangians {Q j }, satisfying:
(1) the L i split-generate the Fukaya category;
(2) the algebra A * = ⊕ i,j HW * (L i , L j ) is supported in non-positive degrees;
(3) every indecomposable A 0 -module is isomorphic to that defined by some (not necessarily finite rank) local system V j → Q j over one of the Q j . Then [7,Proposition 3.3] asserts that every (arbitrary rank) local system over a closed exact Lagrangian submanifold of M can be written as a finite twisted complex over (arbitrary rank) local systems on the Q j . It is this stronger result which is needed to prove that exact Lagrangians in the A n 2 -spaces are actually simply connected. 4.1. Neck stretching. The final three sections are devoted to the proofs of Proposition 4.2 and Proposition 4.4. For a single manifold Q, the fact that HW * (T * q Q, T * q Q) is supported in non-positive degree arises from the relation between the Maslov index of binormal chords in a cotangent bundle, and the Morse index of the corresponding critical points of the path space action functional. This relation reduces to the fact that a curve in the Lagrangian Grassmannian of T * Q defined by parallel transport of the vertical distribution along a curve in T * Q of the form (q(t),q(t)) meets the Maslov cycle always with the same co-orientation [22]. To obtain an analogue for plumbings, we begin by equipping Q i with metrics g i which can be written as (4.7) f (|x|)g eucl with f a smooth function which is identically 1 near the origin, in some charts U i centered at q i , whose domains are balls of radius e 2 in R n . Moreover, we require that (4.8) f (|x|) = 1 |x| 2 whenever 1 < |x|. This condition on f shall be used to facilitate "stretching the neck:" Lemma 4.9. The complement of the origin in R n , equipped with the metric
(4.9) g eucl |x| 2 .
is isometric to the product metric on R × S n−1 .
Proof. Using radial coordinates, we may write every point in the complement of the origin as rθ, with θ ∈ S n−1 and r ∈ (0, +∞). The map (4.10) (s, θ) → e s θ provides the desired isometry between the product metric and Equation (4.9). With this in mind, we find that the Riemannian manifold (Q i , g i ) admits an isometric embedding of [0, 2] × S n−1 . We define a family of metrics g T i which agrees with g i away from this region, and which, in this region is stretched, so that it admits an isometric embedding of [0, T + 2] × S n−1 . More precisely, let us fix the coordinates
(2 − log(|x|), θ) on the region {1 < |x| < e 2 } ⊂ U i .
Note that in these co-ordinates, {0} × S n−1 ⊂ [0, 2] × S n−1 corresponds to the sphere of radius e 2 around q i , whilst {2} × S n−1 corresponds to the sphere of radius 1. In particular, a neighbourhood of 0 in the collar [0, 2] × S n−1 extends the complement of U i , at the outer boundary of the chart centred on q i . In these coordinates, the metrics g T i are defined as
(4.11) dχ T ds g [0,2] ⊕ g S n−1
where χ T : [0, 2] → R is a family of strictly monotone functions satisfying the following conditions (see Figure 3):
(1) χ T ≡ s in a neighbourhood of 0. This condition ensures that g i extends this metric smoothly to the complement of U i . (2) χ T ≡ T + s on a neighbourhood of [1,2]. This condition implies that a path in the radial direction between the two boundary spheres has length T + 2, and that g i extends this metric smoothly to the rest of U i . (3) χ −1 S (0, S/2) = χ −1 T (0, S/2) whenever S < T . Let Q T i denote the union of χ −1 T (0, T /2) with the complement of U i . From this condition, we conclude that we have an inclusion Q S i ⊂ Q T i whenever S < T , and that the restriction of g T i and g S i to Q S i agree. (4) If s < 1, lim T →+∞ χ T (s) is well defined and finite. Let Q op i denote the complement of the set of points in U i which are the images of points with norm |x| ≤ e. This final condition implies (4.12) ∪ T Q T i = Q op i . We now consider the family of Liouville domains associated to these metrics. First, we set the "plumbing region"
(4.13) R 0 = {|x| 2 |y| 2 ≤ 1, |x| < e, |y| < e} ⊂ R n × √ −1R n ,
and we define (4.14)
M = D * Q op 0 ∪ R 0 ∪ D * Q op 1
where the unit cotangent bundle is taken with respect to the metric g i . The reader may check that this definition makes sense, by computing that the unit disc cotangent bundle of the product [0, 1] × S n−1 , isometrically embedded in Q i with respect to the metric g i , is symplectomorphic to the set of points in R 0 for which the norm of the x variable lies in the interval [1, e].
Since ∂M is a smooth contact manifold, we have an associated complete Liouville man-ifoldM = M ∪ ∂M × [1, +∞). Note that, by construction, we have (disjoint) embeddings T * Q op i ⊂M . With this in mind, we may construct a Liouville subdomain (4.15) M T ⊂M by taking the union
(4.16) M T = D * T Q op 0 ∪ R 0 ∪ D * T Q op 1
where D * T Q op i is the unit cotangent bundle with respect to the metric g T i . Note thatM is also symplectomorphic to the completion of M T . In particular, all Floer homological invariants are independent of T .
There is a slightly different description of the manifolds M T which is related to the idea of stretching the neck. For T ∈ R >0 , we may define a plumbing region
(4.17) R T = {|x| 2 |y| 2 ≤ 1, |x| < e T +1 , |y| < e T +1 } ⊂ R n × √ −1R n .
The set of points whose x coordinate lies in the interval [1, e T +1 ] can be identified with the unit disc cotangent bundle of the product [0, T + 1] × S n−1 , which, by construction, isometrically embeds in Q i with respect to the metric g T i . In particular, M T has an alternative description as Figure 5 for a picture in the lowest-dimensional case. Since it is hard to make these Lagrangian foliations completely explicit in dimension > 2, we will reduce the computation of indices to a problem involving holomorphic disks. By applying a suitable conformal dilation to the symplectic form, we now work with the Liouville domain M which contains a standard plumbing region R ∞ ∩ B 0 (2) ⊂ C n . Fix a Reeb chord γ from L q0 to L q1 inside the boundary of M , where by standard convention these cotangent fibres are disjoint from R ∞ ⊂ M . We will decompose γ into a sequence of chords γ = γ 1 * · · · * γ n where each γ i is • either a segment of a geodesic inside the (vertical) boundary of the open region D * Q op i for i = 0, 1; • or a Reeb chord for the Hamiltonian flow on ∂R ∞ for the model region described above. In particular, once γ is given, one can choose a level set |x| = 1 + ε of the radial function in the local model which γ crosses transversely. In particular, we may find 0 < δ < ε such that γ crosses every level set between 1 + ε and 1 + δ transversely. Under this assumption, every time the geodesic crosses the level set |x| = 1 + ε, it reaches |x| = 1 + δ, and vice versa depending on whether γ is leaving or entering D * Q 0 . Following [3], we fix a Lagrangian foliation L ⊂ T M , depending on γ, with the following properties:
(4.18) D * Q T 0 ∪ R T ∪ D * Q T 1 .
• L co-incides with the vertical foliations near S * Q i except in the collar region |x| ∈ (1 + δ, 1 + ε); • L interpolates smoothly between the vertical foliations in the collar. L will be tangent to the boundary at some points inside the collar, cf. Figure 5. All the foliations (for different chords or slicings) can be taken to be homotopic, hence all define the same Maslov indices. For any choice of gradings of the cotangent fibres L qi , the foliation L defines a canonical Maslov index µ(γ) ∈ Z. Moreover, each γ i is a Reeb chord between Legendrian boundaries of disk cotangent fibres, hence again has a uniquely defined Maslov index. The first claim is that the index of the whole chord is governed by the indices of the pieces (this is presumably well-known, compare to [29, Theorem 2.3] for a corresponding local statement).
Lemma 4.11. µ(γ) = j µ(γ j ).
Proof. Fix a Hamiltonian chordγ : [0, 1] → ∂M between Legendrian submanifolds Λ i γ(i). There is a unique trivialisation ofγ * T M which takes the Lagrangian foliation L to the constant real subbundleγ×R n ⊂γ×C n . The Maslov index of this Reeb chord is given by the index of a ∂-operator on the half-plane (disk with one boundary puncture), with Lagrangian boundary condition given by the image under parallel transport by the Hamiltonian flow of the tangent plane to Λ 0 . Now consider a pair of Reeb chordsγ + andγ − with the property thatγ + (1) =γ − (0) and with all 3 endpoints of the chords lying on Legendrian submanifolds whose tangent spaces are contained in the Lagrangian foliation L. Letγ • denote the obvious concatenation ofγ ± . The usual additivity of the index for ∂-operators implies that
µ(γ • ) − (µ(γ + ) + µ(γ − )) = index(P)
where P is the operator given by counting holomorphic sections over a three-punctured disk with boundary conditions given by the end-points of the three chords appearing on the left side of the equation. Since all these boundary conditions lie in the foliation L, in the given trivialisation the Lagrangian boundary condition on this pair of pants is constant, hence it has index 0.
To prove Proposition 4.2, it suffices to show that the indices of the γ j are all non-positive when dim(Q i ) ≥ 2. By our choice of foliation and slicing of γ, for those Reeb chords contained in the D * Q op i this follows from the usual non-positivity (in our convention) of indices for geodesic chords in any Riemannian metric.
The remaining pieces γ i of chord are those which traverse the plumbing region; in other words, which crosses the region separated by the hypersurfaces |x| = 1 + ε and |x| = 1 + δ. Choosing and δ sufficiently close, these chords correspond to short geodesics on the product of an interval with the sphere, which start and end on different boundary components.
A Hamiltonian chord between L 0 and L 1 for the time-1 flow φ 1 H is also an intersection point of L 0 and φ 1 H (L 1 ), so the problem of determining the index of a chord is equivalent to the problem of determining the index of a Lagrangian intersection. In the wrapped category, indices for such intersection points are invariant under arbitrary (not necessarily compactly supported) Lagrangian isotopies which do not change the configuration of intersection points of the two Lagrangians. In particular, we can deform the cotangent fibres near the boundary of the plumbing inwards towards the origin until they intersect, and by continuity the index of the unique Reeb chord will then be given by the index of the unique Lagrangian intersection point of the deformed fibres. Take these for definiteness to be the fibres through Proof. Projecting to the co-ordinate axes C ⊂ C n , the boundary conditions define the four distinct lines R, √ −1R, R + √ −1/2, √ −1R + 1/2 ⊂ C in the first factor, and the real and imaginary axes (each repeated) in the remaining n−1 factors. Any holomorphic quadrilateral with these boundary conditions is therefore constant under projection to the co-ordinates z 2 , . . . , z n , whilst there is a unique rigid quadrilateral under projection to the z 1 -plane. This fixes the modulus of the quadrilateral, and the corners are all convex, so the curve is isolated and transversely cut out.
We now grade the compact cores and cotangent fibres of the plumbing as follows:
HF (Q i , L qi ) ∼ = k[0]; HF (Q 0 , Q 1 ) ∼ = k[n/2] n even k[(n − 1)/2] n odd
The other gradings are determined by Poincaré duality, so
HF (L qi , Q i ) ∼ = k[n]; HF (Q 1 , Q 0 ) ∼ = k[µ 3 : HF (L q1 , Q 1 ) ⊗ HF (L q0 , L q1 ) ⊗ HF (Q 0 , L q0 ) −→ HF (Q 0 , Q 1 )[−1]
which in turn forces the group HF (L q0 , L q1 ) to be concentrated in degree (4.19) HF (L q0 , L q1 ) ∼ = k[(2 − n)/2] n even k[(1 − n)/2] n odd If n > 3, it follows that for this (maximally symmetric) choice of grading, the chord from L q0 to L q1 , or the shortest chord in the reverse direction as appropriate, cf. Remark 4.10, has strictly negative degree. If n = 3, one chord has negative degree whilst the other has degree 0, whilst if n = 2 the chords both have degree 0. At this stage, we conclude that the algebra A * = ⊕ i,j HW * (L qi , L qj ) is graded in degrees ≤ 0 whenever n ≥ 2, and its degree 0 part is a quotient of ⊕ i HW 0 (L qi , L qi ) whenever n ≥ 4. There is a final delicacy when n = 3, which is that there may still be additional chords of degree 0 not coming from loops in any component; however, (4.19) implies these chords can arise only in one direction, i.e. from Q 1 to Q 0 but not vice-versa. Therefore A 0 differs from a quotient of ⊕ i HW 0 (L qi , L qi ) by a nilpotent ideal, as required. (An alternative would be to restore symmetry to the grading of A * in (4.19) by taking gradings in (1/2)Z, in which case the argument when n = 3 becomes formally identical to the higher-dimensional case.) 4.3. Computation of the degree 0 part of wrapped Floer cohomology. We shall prove Proposition 4.4 in this Section. The main point is to adapt to our setting the results of Abbondandolo and Schwarz [1] and of [5], proving an isomorphism between wrapped Floer cohomology of cotangent fibres and the homology of the based loop space. Note that u(0, 0) = u(1, 0) = q, since this is the only point lying both on Q and L. In particular, the path t → u(t, 0) is a based loop on Q. In this special case, Lemma 4.5 [5], asserts that the evaluation map Now, let E be a local system on Q, with fibre E q at q. Every local system defines an object of the Fukaya category (see [7] for a detailed description), and, in this situation, the fact that L and Q intersect at one point implies that we have a natural isomorphism therefore makes E q into a module over HW * (L); for degree reasons only HW 0 (L) acts nontrivially. This product counts maps (4.21) which are rigid. Therefore, given a generator x ∈ CW 0 (L), the associated map (4.27) E q → E q is obtained by taking the sum of the monodromy of the local system E over all based loops on Q associated to elements of H(x). Note that these monodromy maps make E q into a module over H − * (Ω q Q); again, for degree reasons, only H 0 (Ω q Q), which is the group ring of π 1 (Q, q), acts non-trivially. We conclude that the HW 0 (L)-module structure on E q is obtained by pulling back the Z[π 1 (Q, q)]-module structure by the map induced by Equation
∼ = Z[π 1 (Q, q)] ⊗ E q / / E q .
Returning to the manifolds of interest, note that L qi and Q i do intersect at one point. In order to prove Proposition 4.4, it remains to show that the map induced by Equation (4.24) yields an isomorphism (4.29) HW 0 (L qi ) ∼ = H 0 (Ω qi Q i ) ∼ = Z[π 1 (Q i , q i )].
We shall prove this by exhibiting an inverse.
4.3.2.
Action functionals on based paths. Let Ω qi (Q i ) denote the space of loops on Q i , based at a point q i , of class W 1,2 . Abbondandolo and Schwarz have related the Morse homology of the action functional on the spaces of such based loops to the wrapped Floer homology of cotangent fibres. In the discussion below, we shall essentially follow [1, Section 2], modifying certain points to account for the fact that we need to compute the Floer homology in the plumbing. We define the action of a path to be the integral Lemma 4.14. If S < T , the set of based loops of action equal to S 2 is independent of T .
Proof. The Cauchy-Schwartz inequality implies that (4.31) Length(γ) 2 ≤ E T (γ), hence a loop of action bounded by S 2 has length bounded by S. Since the basepoint lies in the complement of the cylindrical region, every path which has length less than S cannot escape a cylinder of length S/2. From our choice of metric in Section 4.1, this implies that such a loop must be contained in Q S i , where the metrics g T i are independent of T whenever S < T .
Let us write Ω S qi (Q i ; g T i ) for the set of based loops of action bounded by S 2 with respect to the metric g T i . By the above Lemma, we have an inclusion (4.32) Ω S qi (Q i ; g S i ) ⊂ Ω T qi (Q i ; g T i ) whenever S < T . In particular, we can take the increasing union (4.33) ∪ T Ω T qi (Q i ; g T i ) ⊂ Ω qi (Q i ). Note that all the loops in the above increasing union lie in Q op i . Moreover, every loop in Q op i has uniformly bounded length with respect to the metrics g T i by Equation (4.12). We conclude that CM S * (E T ) ⊂ CM * (E T ) generated by geodesics whose action is bounded by S 2 .
Lemma 4.14, together with the fact that the negative gradient flow of E T preserves Ω S qi (Q i ; g T i ), imply that we have a canonical identification such that the projection of u(t, 0) to Q is an element of σ, and u converges, at −∞ to x. If y is a generator of CM * (E), and W u (y) is the descending manifold with respect to the gradient flow of E, Abbondandolo and Schwarz prove that the count of rigid elements of M(x; W u (y)) defines a chain map (4.40) CM − * (E) → CW * (T * q Q). For the Hamiltonian H = |p| 2 , the non-constant Hamiltonian chords with endpoints on T * q Q are in bijective correspondence with Reeb chords with endpoints on the unit conormal, and with non-constant geodesics based at q. Theorem 3.1 of [1] states that Equation (4.40) is a chain equivalence, which respects the action filtration; on the right hand side, the action of a chord x is
(4.41) A(x) = x * (λ) − H • x dt
where λ is the Liouville form. More precisely, [1] asserts that, after passing to the associated graded with respect to the action filtration, the map from Morse homology to Floer homology assigns to each geodesic the corresponding Reeb chord.
Let us now consider the situation of a plumbing: From Equation (4.18), we conclude that we have a codimension 0 embedding (4.42)
T * Q T i ⊂M . Let us write L qi for the cotangent fibre at q i , considered as a Lagrangian inM . Since all based geodesics for g T i of action less than T 2 are contained in Q T i (see Lemma 4.14), we may use the previous machinery to define a moduli space M(x; W u (y)) whenever x is a chord in M with endpoints on L i , and y a geodesic for g T i of action less than T . Note that, even in the special case when x lies in T * Q T i , elements of this moduli space do not, a priori, have image contained in T * Q T i . From this moduli space, we obtain a map . Note that this function extends the squared norm (with respect to g T i ) of a cotangent vector in T * Q T i . Let CW * T (L qi ; r 2 T ) denote the subcomplex of the right hand side generated by chords whose action is bounded by T 2 . HM T 0 (E T ) → HW 0 T (L qi ; r 2 T ). is surjective.
Proof. From the discussion above, the natural bases of CM T 0 (E T ) and CW 0 T (L qi ; r 2 T ) can be identified, since they both correspond to geodesics in Q T i with vanishing Jacobi index. The exactness ofM and Equation (3.1) of [2] imply that Equation (4.43) preserves the action filtration: in the natural bases above, the corresponding matrix is therefore upper triangular. Moreover, as in the case of cotangent bundles, the diagonal entries of this matrix are ±1, corresponding to stationary solutions of the pseudo-holomorphic curve equation satisfied by elements of M(x; W u (y)).
We conclude that the map (4.45) CM T 0 (E T ) → CW 0 T (L qi ; r 2 T ). is an isomorphism. Since the Morse complex is supported in non-negative degrees, whilst the Floer complex is supported in non-positive degrees, passing to cohomology in degree 0 is obtained by taking the quotients of the two sides in Equation (4.45) by the image of degree 1 elements on the left, and degree −1 elements on the right. We conclude that the induced map on cohomology is surjective.
Since wrapped Floer homology is a symplectic invariant, the homology of the right hand side of Equation (4.43) is independent of T . In particular, we obtain a map
Theorem 1 . 1 .
11Let M be a plumbing of two cotangent bundles. The Fukaya category F(M ) of closed exact Lagrangians fully faithfully embeds in the subcategory of the wrapped Fukaya category W(M ) generated by cotangent fibres. If the real dimension of M is greater than 4, then every closed exact Lagrangian with vanishing Maslov class is equivalent to a twisted complex over the components of the compact core equipped with local systems.
Lemma 2 . 23 .
223In the category W(M Σ ; π),Σ[n − 1] andΣ ← π −1 (S n ) do not lie in the same orbit under the action by deck transformations.
Conjecture 2. 28 .
28Given a simply-connected n-manifold Q, let M Q denote the plumbing T * Q#T * S n given by adding a Weinstein handle to the Legendrian unknot in a fibre of T * Q. If the map ρ : π 0 Symp ct (T * Q) −→ Auteq(HF(M Q ))/[1]
Lemma 2 . 32 .
232The outermost vector spaces in C • are each of rank 1, i.e.
( 2 .
225) dim(V 0 ) = 1 = dim(V N ), and we have an isomorphism(2.26) Hom(V N , H 0 (Q)) ⊕ Hom(V 0 , H n (Q)) ∼ = HF (C • , Q)induced by the inclusion of the left hand side in the chain complex of morphisms from C • to Q.
Lemma 2 .
230 now implies that dim(V 0 ) + dim(V N ) ≤ 2, which in turn implies(2.25).Proof of Proposition 2.29. Let V = ⊕ i V i denote the sum of the vector spaces in the twisted complex defining C • . Lemma 2.30 implies that under the given assumptions on Q, namely vanishing of non-trivial cup-products and higher products into H <n (Q; k), the complex computing HF (C • , Q) includes an acylic complex(2.29) Hom(V 0 , H 0<i<n (Q))
Figure 1 .
1along the y-axis from this new critical point to the origin. Let T denote the Lagrangian disc (thimble) obtained by parallel transport of V . Note that Q ∩ T = V .
Proposition 3. 5 (
5Viterbo restriction functor). There exists an A ∞ functor L(W ) → W(W in ) which assigns to each Lagrangian its intersection with W in . This result has a fairly straightforward consequence which forms the starting point of investigations of the wrapped Fukaya categories of Lefschetz fibrations undertaken by the first author with Seidel: Proposition 3.6. The Fukaya category of closed Lagrangians in E lies in the subcategory of W(E) generated by ∆ 1 , . . . , ∆ m .
Lemma 3 . 7 .
37There exists an exact Liouville embedding of D * Q 0 #D * Q 1 into E which intersects all thimbles either in a cotangent fibre to Q i or in the diagonal in the local model of Equation 2.7.
Corollary 3 . 8 .
38The Fukaya category of closed Lagrangians in T * Q 0 #T * Q 1 lies in the subcategory of the wrapped category which is generated by a collection of three objects: A cotangent fibre to each Lagrangian Q i , and the intersection of the diagonal with the local model of the plumbing region.
Lemma 3 . 9 .
39In the wrapped Fukaya category of the plumbing, the surgeries are quasiisomorphic to cones (3.7)
Figure 2 .
2Figure 2.
Lemma 3 . 11 .
311There exists a compactly supported isotopy of Lagrangians in C n between the Polterovich surgery of −(1, . . . , 1) + √ −1R n and ( √ −1, . . . , √ −1) + R n , and a Lagrangian ∆ −1, √ −1 which agrees with the diagonal in a neighbourhood of R n ∪ √ −1R n . Proof. Translating co-ordinates, it suffices to prove that there is a Lagrangian submanifold, Lagrangian isotopic via a compactly supported isotopy, to a Lagrange surgery of R n ∪ √ −1R n , and agreeing in some open neighbourhood of −(1, . . . , 1) + √ −1R n ∪ ( √ −1, . . . , √ −1) + R n with the linear Lagrangian submanifold obtained by translating the diagonal in R n ⊕ √ −1R n by (−1 + √ −1, . . . , −1 + √ −1). Recall the model of Equation 3.6 for the surgery. We choose the path γ, in the upper left quadrant, to satisfy the conditions:
agrees with the antidiagonal near the origin, and of (−1 + √ −1R) ∪ ( √ −1 + R) which agrees with the diagonal near the origin, cf.
Figure 2 .
2This accounts for the choice of translated axes above. (For any n, the space of linear Lagrangians transverse to both R n and √ −1R n has two components, distinguished by the Maslov index.)
Proposition 4 . 1 .
41Let L ⊂ M be a closed exact Lagrangian of Maslov class zero. Every finite rank local system over L lies in the category generated by finite rank local systems over Q 0 and Q 1 . The proof of Proposition 4.1, relies on two computations of wrapped Floer homology, respectively given in Sections 4.2 and 4.3
Proposition 4 . 2 .
42If n ≥ 3, there is a choice of gradings on the cotangent fibres L qi such that the wrapped Floer cohomology algebra
* = ⊕ i,j HW * (L qi , L qj )
Example 4. 3 .
3Proposition 4.2 is false when dim(Q i ) = 1. Let M be the plumbing of two copies of T * S 1 , so equivalently M = T 2 \{pt} is a punctured torus. The Reeb flow on ∂M ∼ = S 1 is periodic, and there is a spectral sequence converging to SH * (M ) for which the columns of the E 1 -page are given by one copy of H * (M ) and infinitely many copies of H * (∂M ), in suitably shifted degrees. From here Seidel computed [33, Example 3.3] that SH * (M ) is non-trivial in infinitely many positive degrees. Analogously, one can directly compute the indices of Reeb chords to see that in this case A * is also non-trivial in infinitely many positive degrees.
Remark 4 . 5 .
45In the case where one of the Lagrangians Q i is a sphere, this Proposition can be derived from results of Bourgeois-Ekholm-Eliashberg[10].Proposition 4.1 follows from Propositions 4.2 and 4.4 by application of the general theory of modules over non-positively graded A ∞ algebras, which we review briefly. The starting point is the Homological Perturbation Lemma of Kadeishvili (see
Figure 3 .
3Graphs of the functions χ T and χ S
Figure 4 .
4Indeed, the moon has a dark side.
Remark 4 . 10 .
410The presentation R ∞ = {|x| 2 |y| 2 ≤ 1} for the plumbing region suggests a misleading symmetry exchanging x and y. However, the symplectic form dx ∧ dy is negated under this symmetry, and the corresponding Hamiltonian flow is therefore reversed to give negative geodesic flow if one views Q 0 as the local zero-section. For instance, consider two cotangent fibres L qi . We claim there is a Hamiltonian chord contained in ∂R ∞ between the boundary Legendrians of these fibres in at most one of the two possible directions. The result is particularly vivid in the flat case, n = 2, illustrated inFigure 4; from the viewpoint of the Q 0 -plane, one Legendrian boundary projects to a point q 0 , whilst the other projects to a sphere {|q| = 1/|q 1 |}, and the possible chords are constrained by the associated tangent data.4.2.Non-positivity of Maslov indices in plumbings. In the plumbing, the Maslov index measures twisting along a Hamiltonian chord of a Lagrangian foliation which interpolates between the usual vertical foliations in the two pieces, see
Figure 5 .
5Figure 5.
0, . . . , 0) in the local model R ∞ .
Lemma 4 . 12 .
412There is a rigid holomorphic quadrilateral with boundary edges on the linear Lagrangian subspaces R n , √ −1R n , R n + ( √ −1/2, 0, . . . , 0), √ −1R n + (1/2, 0, . . . , 0).
4. 3 . 1 .
31From Floer theory to the homology of the based loop space. Let M be a Liouville manifold, and Q ⊂ M a closed exact Lagrangian, which is an object of its Fukaya category. The main result of [5] (see the discussion preceding Corollary 1.5) is that there exists a functor from the Fukaya category of M to the category of modules over the chains on the based loop space of Q. Let us consider the special case when a Lagrangian L meets Q at one point q. Given a time-1 Hamiltonian chord x with endpoints on L, define
the space of pseudo-holomorphic maps(4.21) u : [0, 1] × [0, +∞) → M which asymptotically converge to x as the second coordinate goes to +∞, and with boundary conditions (4.22) u(0, s) ∈ L, u(t, 0) ∈ Q, and u(1, s) ∈ L.
( 4 .
423) ev : H(x) → Ω q (Q) defines, at the level of complexes, a chain map(4.24) CW * (L) → C − * (Ω q Q).
( 4 . 25 )
425HW * (L, E) ∼ = E q .The product in the Fukaya category(4.26) HW * (L, E) ⊗ HW * (L) → HW * (L, E)
( 4 .
424) on homology. More precisely: Lemma 4.13. The product in Floer theory, the isomorphism in Equation (4.25), and the map in Equation (4.24) fit into a commutative diagram:
( 4 . 28 )
428HW 0 (L) ⊗ HW 0 (L, E) / / HW 0 (L, E)
( 4 .
434) ∪ T Ω T qi (Q i ; g T i ) = Ω qi (Q op i ) ⊂ Ω qi (Q i ).4.3.3. Morse homology of based paths. The critical points of E T are based geodesics, and the gradient flow of E is known to satisfy the Palais-Smale condition (see [1, Proposition 2.5]) and the Morse homology of E T computes the homology of the space of loops based at q i . More precisely, the homology of Ω S qi (Q i ; g T i ) is computed by the homology of the subcomplex (4.35)
( 4 . 36 )
436CM S * (E T ) = CM S * (E T ) whenever T and T are both greater than S. By Equation (4.34), the direct limit of the homology groups HM T * (E T ) is the homology of the based loops inQ i − U i , with U i the chart of (* (E T ) = H * (Ω qi (Q i − U i )) .4.3.4. Surjection in degree 0. Let Q be a compact smooth manifold, H a Hamiltonian on T * Q, and J an almost complex structure on T * Q. Given a chain σ of loops on Q based at q ∈ Q, and a time-1 Hamiltonian chord x of H with both endpoints on T * q Q, define (4.38) M(x; σ) to be the moduli space of pseudo-holomorphic half-cylinders (4.39) u : [0, 1] × (−∞, 0] → T * Q
( 4 .
443) CM T − * (E T ) → CW * (L qi ; r 2 T ). Here r 2 T :M → [0 + ∞) isthe square of the cylindrical coordinate with respect to the decompositionM = M T ∪ ∂M T × [1, +∞)
Lemma 4 . 15 .
415The induced map on homology
T
HM T − * (E T ) → lim T HW * T (E T ) ≡ HW * (L qi ).
n/2] n even k[(n + 1)/2] n odd Via construction, cf. Definition 2.1, open subsets of these cores and fibres are identified with open disks in the linear Lagrangian subspaces of Lemma 4.12. Rigidity of the quadrilateral described in Lemma 4.12 means that it contributes to a non-trivial higher multiplication
M. ABOUZAID, I. SMITH
Lemma 4.18. Equations (4.29)and (4.47) are isomorphisms.Proof. Since the compositions of Equations (4.29) and (4.47) is the identity, the first map must be surjective, and the second injective. However, we already know that Equation (4.47) is a surjection, so it is therefore an isomorphism. Since the inverse to an isomorphism is necessarily injective, we conclude that Equations (4.29) is an isomorphism as well.Corollary 4.16. Equation (4.46) is surjective in degree 0.From the fact that CM T * (E T ) computes the ordinary homology of the space of based loops on Q T i of bounded energy, and Equation (4.34), we compute that the homology of the limit in Equation (4.46) is the homology of the space of based loops in Q op i . Note that, since the dimension of Q i is greater than 1, every loop in Q i is homotopic to a loop lying in Q op i , and, since the dimension is greater than 2, every homotopy between loops in Q i can be assumed to lie in Q op i . We conclude Corollary 4.17. There is a surjective homomorphism The proof is a straightforward cobordism argument, along the same lines as the case of cotangent bundles, considered in Section 5 of[5]. The key idea is that one can glue the domains of the half-cylinders in Equations (4.39) and(4.21), to obtain a pseudo-holomorphic map whose source is an annulus of very large modular parameter. By considering the moduli space of pseudo-holomorphic maps whose source is an annulus of arbitrary modular parameter, one obtains a homotopy between the composition of (4.43) with (4.24) and the natural map) which assigns to every critical point of E T the chain represented by a triangulation of its descending manifold.Passing to the direct limit over T , and to homology, we conclude that the compositionis homotopic to the map (4.50) H − * (Ω qi (Q op i )) → H − * (Ω qi (Q i )) induced by the inclusion Q op i ⊂ Ω qi (Q i ). By our assumption on the dimension of Q i , this map is an isomorphism in degree 0. We conclude:
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|
[] |
[
"Search for the field-induced magnetic instability around the upper critical field of superconductivity in H c in CeCoIn 5",
"Search for the field-induced magnetic instability around the upper critical field of superconductivity in H c in CeCoIn 5"
] |
[
"Takanori Taniguchi [email protected] \nDepartment of Physics\nKyoto University\n606-8502KyotoJapan\n",
"Shunsaku Kitagawa \nDepartment of Physics\nKyoto University\n606-8502KyotoJapan\n",
"Masahiro Manago \nDepartment of Physics\nKyoto University\n606-8502KyotoJapan\n",
"Genki Nakamine \nDepartment of Physics\nKyoto University\n606-8502KyotoJapan\n",
"Kenji Ishida \nDepartment of Physics\nKyoto University\n606-8502KyotoJapan\n",
"Hiroaki Shishido \nDepartment of Physics and Electronics\nOsaka Prefecture University\n980-8577OsakaJapan\n"
] |
[
"Department of Physics\nKyoto University\n606-8502KyotoJapan",
"Department of Physics\nKyoto University\n606-8502KyotoJapan",
"Department of Physics\nKyoto University\n606-8502KyotoJapan",
"Department of Physics\nKyoto University\n606-8502KyotoJapan",
"Department of Physics\nKyoto University\n606-8502KyotoJapan",
"Department of Physics and Electronics\nOsaka Prefecture University\n980-8577OsakaJapan"
] |
[] |
We present nuclear spin-lattice relaxation rate (1/T 1 ) at the Co site and ac-susceptibility results in the normal and superconducting (SC) states of CeCoIn 5 for H c near the SC upper critical field H c2 above 0.1 K. At 4.2 T, 1/T 1 rapidly decreases below the SC transition temperature, consistent with the previous reports. Although the field dependence of 1/T 1 T at 0.1 K shows a peak at 5.2 T above H c2 , the temperature dependence of 1/T 1 T at 5.2 T is independent of temperature below 0.2 K, showing a Fermi-liquid behavior. In addition, we found no NMR-spectrum broadening by the appearance of internal fields around H c2 at 0.1 K. We could not detect any field-induced magnetic instability around H c2 down to 0.1 K although the remarkable non-Fermi-liquid behavior towards H c2 was observed in various physical quantities.
|
10.7566/jpscp.30.011107
|
[
"https://arxiv.org/pdf/1911.01572v1.pdf"
] | 207,780,151 |
1911.01572
|
35d7d14d7030e9899a642a06e173a1b80b68eca3
|
Search for the field-induced magnetic instability around the upper critical field of superconductivity in H c in CeCoIn 5
Takanori Taniguchi [email protected]
Department of Physics
Kyoto University
606-8502KyotoJapan
Shunsaku Kitagawa
Department of Physics
Kyoto University
606-8502KyotoJapan
Masahiro Manago
Department of Physics
Kyoto University
606-8502KyotoJapan
Genki Nakamine
Department of Physics
Kyoto University
606-8502KyotoJapan
Kenji Ishida
Department of Physics
Kyoto University
606-8502KyotoJapan
Hiroaki Shishido
Department of Physics and Electronics
Osaka Prefecture University
980-8577OsakaJapan
Search for the field-induced magnetic instability around the upper critical field of superconductivity in H c in CeCoIn 5
(Received February 21, 2019)field-induced criticalityNMRac susceptibilityheavy fermion
We present nuclear spin-lattice relaxation rate (1/T 1 ) at the Co site and ac-susceptibility results in the normal and superconducting (SC) states of CeCoIn 5 for H c near the SC upper critical field H c2 above 0.1 K. At 4.2 T, 1/T 1 rapidly decreases below the SC transition temperature, consistent with the previous reports. Although the field dependence of 1/T 1 T at 0.1 K shows a peak at 5.2 T above H c2 , the temperature dependence of 1/T 1 T at 5.2 T is independent of temperature below 0.2 K, showing a Fermi-liquid behavior. In addition, we found no NMR-spectrum broadening by the appearance of internal fields around H c2 at 0.1 K. We could not detect any field-induced magnetic instability around H c2 down to 0.1 K although the remarkable non-Fermi-liquid behavior towards H c2 was observed in various physical quantities.
Introduction
Heavy fermion systems have many examples of a quantum critical point (QCP), where magnetic transition temperature is suppressed to zero by applied a pressure or magnetic field, and unconventional superconductivity has been discovered in many cases. This fact provides strong evidence for magnetically mediated Cooper pairing in these superconductors [1][2][3]. In particular, CeT In 5 (T = Co, Ir, Rh) series, called 1-1-5 family, have attracted much attention for QCP studies because of an emergence of various phenomena originating from 4 f electrons such as antiferromagnetic (AFM) order, heavy fermion (FL) state, and superconducting (SC) state under the pressure or magnetic field [4][5][6].
In this paper, we focus on CeCoIn 5 . This compound is located in the vicinity of the AFM phase and shows superconductivity at a transition temperature T c = 2.3 K at zero-field and ambient pressure [7]. The Knight shift decreased rapidly on entering the SC phase for H c, indicating a singlet Cooper pair forming in CeCoIn 5 [8,9]. Nuclear spin-lattice relaxation rate (1/T 1 ) was proportional to ∼ T 3 far below T c , suggesting that CeCoIn 5 is a line-nodal superconductor [9]. In addition, the d x 2 −y 2 symmetry was suggested from the several measurements such as the field-angle resolved thermalconductivity [10] and heat-capacity [11] measurements and the spectroscopic imaging scanning tunneling microscopy [12,13] measurement.
For H ab, coexistence of Flude-Ferrell-Larkin-Ovchinnikov (FFLO) state and incommensurate spin-density-wave order (called Q-phase) was observed near H c2 [14,15]. The NMR spectrum at the In(2) site abruptly shifts at T c with the first order character determined with specific-heat and magnetization measurements near H c2 , and splits in the Q-phase inside the SC state [15][16][17][18]. It is noted that the magnetic Q-phase can exist inside the SC state. On the other hand, the presence of pure FFLO state in H c was suggested from the various measurements near H c2 [18,19], but the crucial evidence of the FFLO state in H c has not been obtained. In addition, the presence of the fieldinduced magnetic instability similar to the Q phase is another interesting issue for H c near H c2 , since a remarkable non Fermi liquid behavior towards H c2 was observed in various measurements. Sakai et al. reported that 1/T 1 T measured at 5 T increased down to 0.2 K with decreasing temperature due to the development of the quasi-2D spin fluctuations [5], but 1/T 1 T below 0.2 K was constant (Korringa-law), indicating the FL state. This was consistent with the dHvA [21], resistibility [22], and thermal conductivity measurements [23]. In the NMR study by Sakai et al., the presence of the maximum of 1/T 1 T near H c2 was shown and suggested the possibility of the coincidence with the field-induced magnetic critical point and H c2 in H c, but the detailed search of the magnetic phase has not been performed [5]. In this paper, we report the detailed temperature and field dependence of 59 Co-NMR and ac susceptibility results for H c near H c2 . No superconductivity was observed from the ac susceptibility measurement above 5.0 T and 0.2 K, but 1/T 1 T at 0.1 K shows a maximum at 5.2 T above H c2 . However, 1/T 1 T measured at 5.2 T stayed constant below 0.2 K, indicative of the absence of field-induced magnetic instability around H c2 .
Experimental Procedures
Single crystals of CeCoIn 5 were synthesized by the In-flux method [20]. We selected the thinplate single crystal (1.5 mm × 2.0 mm × 0.2 mm) with the c axis normal to the plate, and performed the 59 Co-NMR measurement on the single crystal. The 59 Co-NMR spectra were obtained by summing the Fourier transform spectra from the spin-echo signal obtained at equally spaced rf frequencies at a fixed magnetic field. Since 59 Co nuclear spin is 7/2 (I = 7/2), the 59 Co-NMR spectrum consists of seven peaks. The sharpness of the resonance line indicates the high quality of the crystal. 1/T 1 at the Co site was obtained from a central line arising from the |−1/2 ⇐⇒ |+1/2 transition. The crystal alignment with respect to the magnetic field was precisely performed with a single-axis rotator in the horizontal field generated by a split-magnet, and the misalignment is within 0.5 • .
Results
Figure 1(a) shows the crystal structure of CeCoIn 5 . The crystal structure of CeCoIn 5 is the tetragonal HoCoGa 5 type structure and the space group P4/mmm [7]. The Ce and Co sites, occupying the 1a and 1b Wyckoff positions with the both point symmetries 4/mmm, are at the same fourfold axis along the c axis. There are two crystallographycally inequivalent In sites, In(1) and In (2), occupying the 1c and 4i Wyckoff positions with the point symmetries 4/mmm and 2mm., respectively. Figure 1(b) shows the temperature dependence of ac susceptibility χ ac for various magnetic fields parallel to the c axis. Below 4.8 T, χ ac decreases below T c (H), caused by the diamagnetic shielding. On the other hand, above 5.0 T, a clear anomaly is not observed down to 0.2 K, which is consistent with the previous reports (µ 0 H c2 = 5.0 T) [7]. Figure 2(a) shows the field dependence of 1/T 1 T and χ ac at 0.1 K near H c2 for H c. At 4.8 T, the 59 Co-NMR spectrum was splitting below T c due to the first order transition, and we follow the normal-state component. 1/T 1 T showed a peak at 5.2 T, suggesting that the field-induced peak of 1/T 1 T is above H c2 . Figure 2(b) shows the temperature dependence of 1/T 1 T in various magnetic fields parallel to c axis. Above T c , 1/T 1 T increases with decreasing temperature due to the development of AFM fluctuations. In smaller fields than H c2 (2.0 and 4.2 T), 1/T 1 T abruptly decreases without a coherent peak just below T c , supporting the d-wave superconductivity. At 5.2 T, 1/T 1 T is independent of temperature below 0.2 K, showing the Korringa behavior even at the critical field where 1/T 1 T shows a maximum. This indicates that a field-induced magnetic instability is absent in CeCoIn 5 near H c2 in H c.
Ce
In (1) Co T near H c2 . The shape of lines are insensitive to the field, and any linewidth broadening of the 59 Co-NMR spectra was not observed at 5.2 T, as shown the Fig. 3(b), also indicative of the absence of due to the appearance of static internal field a field-induced magnetic instability around H c2 as well as the result of 1/T 1 T .
In(2) a b c ! " ! # $ % $ ! " & ' ( $ ) # $ $ " * + ' $ , - . / # 0 ' 1 0 ' 2 2 ' 3 2 ' 4 2 ' ( 2 ' 1 " $ " 5 # $ ( ' & 4 $ ) $ ( ' 6 2 $ ) $ ( ' 3 2 $ ) $ & ' 2 2 $ ) $ & ' 1 2 $ ) " ! (a) (b)
Many anomalous states due to the development of spin-fluctuations, such as a non-Fermi liquid behavior, have been observed near QCPs in various heavy-fermion compounds. A typical example is YbRh 2 Si 2 , in which 1/T 1 T of 29 Si in YbRh 2 Si 2 continues to increase at the field-induced critical point with decreasing temperature [24]. However, our result shows that 1/T 1 T is constant below 0.2 K, indicating that the FL state holds even at the critical field where 1/T 1 T shows the maximum. The 1/T 1 T behavior observed in the critical region of CeCoIn 5 is reminiscent of the criticality observed in CeRu 2 Si 2 showing the meta-magnetic criticality. Sakakibara et al. reported that the FL state holds at low temperatures even at the critical region, and suggested that the origin of this criticality is related with the change of the f electron character from itinerant to localized [25]. 1/T 1 T of CeRu 2 Si 2 at the critical field increased down to T M and stayed constant below T M [26], as if CeRu 2 Si 2 possesses a finite crossover temperature at T M [25]. We suggest that the critical behavior observed in CeCoIn 5 near H c2 might originate from the field continuous change of the Fermi-surfaces, in which the large Fermi-surfaces still remain above H c2 as shown in [21]. The 1/T 1 T behaviors against H and T show the crossover from the non-FL to FL occurring at around 5.2 T below 0.2 K, in good agreement with the recent thermal expansion measurement [27], but inconsistent with the thermal conductivity results showing the non-FL behaviors above H c2 [23].
Recently, the field-induced phase transition was reported at around 8 T and 15 mK from the dHvA measurement, but the order parameter is unknown yet [28]. From the present 59 Co-NMR measurement, it seems that the field-induced phase transition is not ascribed to magnetic in origin but would -! .
-" .
-
$ # - ! ' ( / , " 0 1 ' ( 0 2 3 - ' 4 5 6 7 , ' ' 8 ' ! - % ' + ! " # ! " # # ! $ % & ' ( ) ( * + # , ! " + ! " + # % & ) * % ! , # # % - % " , ! ! % - % . , # # % - % .
, originate from the Fermi-surface instability probed by the transport measurement, since AFM magnetic fluctuations are suppressed at the higher fields [5]. However it is noted that the phase-transition was observed up to 9.5 T and seems to exist above 10 T in H c, which is anticipated to the relation with the magnetic Q phase observed in H ⊥ c. To understand the origin of the field-induced transition, further measurements, particularly angle dependence of the NMR measurements in the ac plane down to 10 mK would give a clue to clarify the origin.
! # % - / (a) (b)
Conclusion
We have performed the single crystal 59 Co-NMR measurements for H c near H c2 in CeCoIn 5 to investigate whether the field induced magnetic phase like the Q phase is observed or not. Although low temperature 1/T 1 T shows a peak at 5.2 T above H c2 , 1/T 1 T at 5.2 T is almost constant below 0.2 K, indicative of an absence of field-induced magnetic instability down to 0.1 K.
Fig. 1 .
1(a) The crystal structure of CeCoIn 5 . The blue semitransparent sheets present the Ce and In(1) plane. (b) The temperature dependence of the χ ac near H c2 . For the sake of clarity, the plots in (b) are vertically shifted consecutively. T c is determined by liner fitting of the diamagnetic signals around the turning point, like the red lines at 4.56 T, and is indicated the black arrows.
Figure 3 (
3a) shows the 59 Co-NMR spectra measured at 0.1 K in 5.0, 5.2, and 5.4
Fig. 2 .
2(a) The field dependence of 1/T 1 T and ac magnetic susceptibility at 0.1 K. (b) The temperature dependence of 1/T 1 T at several fields. The black arrow shows the T c s at 4.22 T determined by the ac susceptibility measurement.
Fig. 3 .
3(a) 59 Co-NMR spectra measured at 0.1 K in 5.0, 5.2 and 5.4 T. (b) The temperature dependence of 59 Co-NMR spectra at 5.2 T. Panels (a) and (b) show the Co-NMR center line arising from |−1/2 ⇐⇒ |+1/2 transition.
AcknowledgmentWe would like to thank Y. Maeno and S. Yonezawa for stimulating discussions. This
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|
[] |
[
"Research on Anonymization and De-anonymization in the Bitcoin System",
"Research on Anonymization and De-anonymization in the Bitcoin System"
] |
[
"Qingchun Shentu ",
"Jianping Yu \nATR Defense Science & Technology Lab\nShenzhen University\nShenzhenChina\n",
"\nBitbank Research Labs *\n\n"
] |
[
"ATR Defense Science & Technology Lab\nShenzhen University\nShenzhenChina",
"Bitbank Research Labs *\n"
] |
[] |
The Bitcoin system is an anonymous, decentralized crypto-currency. There are some deanonymizating techniques to cluster Bitcoin addresses and to map them to users' identifications in the two research directions of Analysis of Transaction Chain (ATC) and Analysis of Bitcoin Protocol and Network (ABPN). Nowadays, there are also some anonymization methods such as coin-mixing and transaction remote release (TRR) to cover the relationship between Bitcoin address and the user. This paper studies anonymization and de-anonymization technologies and proposes some directions for further research.
| null |
[
"https://arxiv.org/pdf/1510.07782v1.pdf"
] | 14,958,959 |
1510.07782
|
ff93ed1741e2a19c5702085c95e610b55d9324ce
|
Research on Anonymization and De-anonymization in the Bitcoin System
Qingchun Shentu
Jianping Yu
ATR Defense Science & Technology Lab
Shenzhen University
ShenzhenChina
Bitbank Research Labs *
Research on Anonymization and De-anonymization in the Bitcoin System
1
The Bitcoin system is an anonymous, decentralized crypto-currency. There are some deanonymizating techniques to cluster Bitcoin addresses and to map them to users' identifications in the two research directions of Analysis of Transaction Chain (ATC) and Analysis of Bitcoin Protocol and Network (ABPN). Nowadays, there are also some anonymization methods such as coin-mixing and transaction remote release (TRR) to cover the relationship between Bitcoin address and the user. This paper studies anonymization and de-anonymization technologies and proposes some directions for further research.
2
The transaction and user networks Taint analysis is a service provided by Blockchain.info, which is used to calculate the percentage of Bitcoins in an address from another address. Obviously, taint analysis is premised on that these two addresses are related to each other in a transaction chain. As shown in Fig.2 (b), the degree of taint of C and E, D and E are 2/3, 1/3 respectively.
Amount analysis is another method to deduce the relationship between the inputs and the outputs within a CoinJoin transaction, according to the characteristic that the input amount of a Bitcoin transaction equals to the output amount plus transaction fee. There is a CoinJoin transaction shown in Fig.4, known condition that coin-mixing fee is 1%, we could deduce that the 0 th input corresponds to the 0 th output, and the 1 st input corresponds to the 1 st output, and the 2 nd and 3 rd inputs correspond to the 2 nd output, and then the 3 rd output is coin-mixing fee.
Timing sequence: after the coin-mixing service provider (mixer) completes the coin-mixing operation he should return Bitcoins to users within a specific time. Hence, the attacker could estimate that the transaction returning Bitcoins were within a block range, and interacting with amount analysis so as to reduce users' possible output addresses to a lesser extent.
Reid [3] analyzed the transaction chain in Bitcoin system and cluster addresses using the transaction and user networks mentioned above in 2011. They studied the event that 25,000 BTC belonging to the Slush Pool was stolen in 2011, and demonstrated how to trace transaction through transaction chain and external information, and they gained some useful clues. Ober [5] and Ron [6] analyzed the Bitcoin blockchain, and studied the amount change of Bitcoins in all Bitcoin addresses, and studied the procedure how users acquired and spent their Bitcoins, and how they transferred Bitcoins within multiple addresses 5 in order to protect privacy. Ron [6] also traced 364 transactions which amount is larger than 50,000 BTC, and found that they were related to a transaction with 90,000 BTC in 2010.
Androulaki [7] simulated Bitcoin payment scenarios in a university in 2012 and estimated the anonymity degree of Bitcoin quantitatively through two heuristic rules, gathering some or all inputs in a transaction and change address. They found that even the receiver generates a new address each time when receiving Bitcoins, 40% of users could be found the real identify. Meiklejohn [8] clustered addresses within the Bitcoin blockchain based on the same heuristic rules and identified 1.9 million addresses, which accounted for 16% of all addresses at that time. He also connected these addresses with addresses of same Bitcoin service provider and found 500,000 addresses that Mt.Gox used before and 250,000 addresses that
Silk Road used before.
The analysis methods mentioned above are not always able to find users' identities. It is partly because of being short of the corresponding relationship between real or virtual identity and Bitcoin addresses; another reason is that the heuristic rules would bring about some mistakes such as coin-mixing inputs making the rule of gathering inputs get wrong inputs, identifying wrong change address, other users' addresses including in a transaction chain closure. In spite of shortages, the above analysis methods can provide a variety of valuable clues, if it is relative to users' identities of Bitcoin service providers, and then there is a high probability to find the true identity of a specific Bitcoin address.
Coin-Mixing
For the ATC attacks, people obfuscate the transaction chain, and separate the corresponding relationship between the input and output of a Bitcoin transaction and even hide the amount of transaction, which denoted as coin-mixing.
Current study regarding coin-mixing focuses on three directions. The first is the use of centralized coin-mixing, such as DarkWallet [14], BitcoinFog.com, BitLaundry.com, and Blockchain.info, where
CoinJoin [15], Mixcoin [16], and coin-mixing algorithms based on blind signature such as BlindCoin [17], RSA Coin-Mixing [18] and Blind-Mixing [19] are representative studies. The second regards decentralized protocols such as CoinSwap [20], Fair Exchange [21], XIM [22], CoinShuffle [23] and CoinParty [24], which are compatible with the Bitcoin protocol, where Bitcoins are mixed through decentralized protocols, and no trust is required among users with no possibility of currency loss. The last direction regards novel coin-mixing technologies such as blind signature transaction [25], Zerocoin [26], Zerocash [27], Pinocchio [28], CryptoNote [29] and SideChain [30], which are not compatible with the Bitcoin Protocol, and must be applied in new blockchains. 6 2.3.1 Centralized coin-mixing: research regarding Bitcoin coin-mixing originated from the CoinJoin anonymization method proposed by Gmaxwell [15]. General Bitcoin transactions have 1 to 2 inputs and 1 to 2 outputs, so it is easy to analyze the transferring path of Bitcoins. The CoinJoin transction combines many inputs and outputs and put them into a single transaction so that an input in the CoinJoin transaction is difficult to be corresponded to an output. Suppose that the numbers of inputs and outputs within a transaction are all N, and each input corresponds to a output respectively, and given event A is that corresponds a certain input to a certain output. When N = 10, then the possibility of A P(A) = 0.1; after 5 coin-mixing activities, P(A)=10 -5 , it seems CoinJoin transaction is the most simple and most effective anonymization method.
According to above principle, centralized coin-mixing providers receive Bitcoins from users through website and perform coin-mixing activities. They have some countermeasures against amount analysis: ① each user inputs same amount; ② random coin-mixing fee, attackers cannot estimate output amount accurately; ③ extend coin-mixing path and increase the frequency of coin-mixing, so that attackers are not able to perform amount analysis within a single CoinJoin transaction; ④ reduce the amount of coinmixing, such as 0.1 to 5 BTC each time; ⑤ receive Bitcoins from multiple addresses.
Methods against the timing sequence: ① buy more time to mix coins, with higher probability to be obfuscated with other normal transactions ; ② make the time when the mixers send Bitcoins back to users random within a long range, such as from 2 days to 7 days, attackers cannot estimate the time accurately.
Method against the user and transaction networks is make use of shared wallet, that is, the mixers receive Bitcoins through address A, however, send Bitcoins to users through address B. Thus, the input and output addresses of a certain user are no relation in the transaction chain.
The mixers hide users' IP addresses through TOR or I2P, and receive Bitcoins from users, and then perform coin-mixing, therefore develop the anonymity of the Bitcoin system. However, there are two serious weaknesses of centralized coin-mixing: ① mixers must know a user's input address and output addresses, and hence cannot provide true anonymity for users; ② users must trust mixers and send Bitcoins to mixers in advance, so users face with the risk of currency loss.
A blind digital signature represents a condition where a signer signs the digest of a message while the content of the message is unknown to the signer. In 1982, Chaum [31] first proposed to implement an anonymous e-Cash system based on the use of blind signatures, which was intended to protect the anonymity of a sender unconditionally. In 2015, BlindCoin [17] was proposed based on bilinear groups to make centralized coin-mixing more anonymous; however, it may be deanonymized because that it uses a public log to reach accountability, which may reveal the deposit time and the withdrawal time. In July 2015, Wu [18] also proposed a blind coin-mixing algorithm based on RSA (RSA Coin-Mixing), however, a user's Bitcoins may be falsely claimed by another. In 2015, Shentu et al [19] adopted elliptic curve cryptography (ECC) and blind signature, and propose a Blind-Mixing scheme, thus improve the performance of ECC blind signature. The scheme also uses the private key from the user' input address to sign the deposit voucher, thus the Bitcoins sent to the mixer cannot be falsely claimed by another.
As for the users' risk of currency loss, the present studies such as Mixcoin and BlindCoin intend to make centralized coin-mixing reach accountability. However, they could not prevent Bitcoins loss when the mixers were gone. Therefore, it is also an important subject.
Decentralized coin-mixing protocols:
because of the risk of currency loss and the application of the secure multi-party computation protocol (SMC) [37] in the Bitcoin [38,39,40], there are some decentralized coin-mixing protocols, and no trust is required among users with no possibility of currency loss.
Fair Exchange [21] was proposed by Barber, which is a two-party Bitcoin exchange protocol, and the two parties exchange their Bitcoins without any trust between each other using Bitcoin script and three
Bitcoin transactions including commit transaction, refund transaction and claim transaction.
XIM [22] is a two-party Bitcoin exchange protocol resisting Sybil attacks and DOS attacks, no third party is required. Bissias invents a method finding coin-mixing partners in the Bitcoin blockchain, and adopts Fair Exchange as exchange protocol, and prevent Sybil attacks and DOS attacks through transaction fee.
CoinShuffle [23] is a decentralized multi-party coin-mixing protocol, and communicates with other partners using anonymous group communication protocol Dissent [41] to ensure anonymity. When coinmixing, all output addresses from all users would be changed the order randomly, and no one could know the corresponding relationship between users and their output addresses. At last, a CoinJoin transaction with multi-input and multi-output will be generated and released.
CoinParty [24] is also as multi-party coin-mixing protocol based on SMC, and there is a centralized server acting as a communication platform to gathering coin-mixing requirements and to transfer messages among users. When coin-mixing, an escrow address is generated and controlled by all coin-mixing partners, and all users send Bitcoins to this address, and then all output addresses are obfuscated through the similar method as CoinShuffle does, and at last a CoinJoin transaction, which input address is the escrow address, is generated and released. 8
Decentralized coin-mixing Protocols in blockchain:
are not compatible with the Bitcoin Protocol, and must be applied in new blockchains. Ladd [25] introduced blind signature to Bitcoin transaction the first time, and proposed the idea of blind signature transaction, which means that the sender intents to send
Bitcoins to a group of receivers, and then he signs and releases the transaction, however he cannot connect receivers with their addresses. Blind signature transaction is a good theoretical exploration, fusing the blind signature, zero knowledge proof and Paillier public-key system.
Amount analysis is an import ATC method, however, the SideChain [30] developed by BlockStream has a kind of secret transaction using Pederson Commitment to replace amount, where the commitment stands for an Bitcoin amount within a certain range such as from 0 to 2 BTC but anyone don't know exact number except the private owner. Even so, the commitment could be verified by the third party and ensure the owner not to spend more money. Thus, Pederson Commitment renders the amount analysis ineffective.
Ring signature was first introduced in CryptoNote [29], here are features: ① the signer selects any user's public keys to take part in signing, no notify required; ② unforgeablity, attackers don't know any member's private key, so they cannot forge signature; ③ unconditional anonymity, in case attackers gain all possible private keys, however, the possibility of the signer being recognized is not more than 1/n, where n indicates the number of possible signers. Some crypto-currencies bases on CryptoNote such as Bytecoin [42] and DarkNetSpace [43] used ring signature to hide the sender, equally performed a coinmixing activity.
Stealth address was also first introduced by CryptoNote, which is originated from the Diffie-Hellman encryption key exchange protocol based on Elliptic Curve (ECDH). The receiver makes a special address public, where the address is called Stealth address, the sender transfers Bitcoins to the address with a one-time public key, and attackers are unable to find any transactions according to the address.
However, the receiver could calculate the correct receiving address and the corresponding private key, and gains the Bitcoins transferred by the sender. DarkWallet [14] and BitShares [44] are also using this technology.
Zerocoin [26] is a decentralized coin-mixing protocol built on new blockchain, including Mint and
Spent these two kinds of operations. Mint operation transforms Bitcoins to Zerocoins, and Spent operation transfers Zerocoins in the Zerocoin blockchain or exchanges Zerocoins to Bitcoins. Zerocoin hides the addresses of the sender and the receiver, however, some weaknesses exist: ① the man who defines the initiation data can acquire all Zerocoins; ② the performance is really not good, the size of zero knowledge in each transaction is larger than 45KB, and it needs 450 ms time to verify; ③ Can't split the amount, and cannot be used to make payment.
Zerocash [27] is the improved version, here are features: ① the initiation data being defined by users; ② the size of zero knowledge being decreased 97.7%; ③ the verification time being reduced by 98.6%; ④ it could be used to pay; ⑤ input amount could be changed or hidden. Another improved version Pinocchio [28] reduced the size of zero knowledge to 288 bytes, and the verification time is less than 10ms.
2.3.4
Quantifying the anonymity: in order to estimate the anonymity degree of coin-mixing transaction and coin-mixing system, we need to quantify the anonymity degree. Suppose a coin-mixing system with N members. When one of members sends a message, ideally, the probability of identifying him is 1/N.
However, if attackers learn some knowledge through participating in and observing this system, the probability of some members being identified will increase. Diaz [45] used the following quantifying mode to calculate the anonymity degree.
(1)
where H(X) indicates the entropy of attacked system, pi indicates the probability of the ith user being the sender, this probability is decided by attackers. HM indicates the maximum entropy of the system. Formation (1) indicates that the anonymity degree equals to the ratio of the system entropy with attackers' knowledge to the maximum system entropy.
Moser [46] adopted above quantifying method and ATC, and estimated the coin-mixing effect of mixers such as BitcoinFog.com, BitLaundry.com and Blockchain.info, and then found that he could find some clues about coin-mixing paths of the mixers except blockchain.info.
Quantifying the anonymity degree is a new method to estimate the coin-mixing effects, however, the present studies are too few and more attention and further research are required.
The Bitcoin Protocol and Sybil Attacks
The Bitcoin protocol
Bitcoin nodes communicate with each other via unencrypted TCP connections using port 8333 [47].
A Bitcoin wallet which does not accept incoming connections is known as the Client while others are called Bitcoin Nodes. Both the client and node save the copy of the IP addresses and ports of other clients
2 1 2 * ( ) ( ) ( ) N i i i M p log p H X d H log N
and nodes. By default, they always keep 8 outgoing connections. If the number of outgoing connections falls below 8 they will reconnect until then number returns back to 8 entry nodes.
Both the client and node keep a record of other client's and node's penalty points. Penalty points are used as the basis of a disconnecting mechanism to avoid denial of service (DOS) attacks. When illegal blocks and transactions occur, the originating node will incur penalty points. Then, when the points total reaches 100, all connections from it will be rejected for 24 hours as a punishment.
When the client generates a new transaction, the command 'inv' is sent to the entry node. The entry node checks the transaction id in its own transaction database, if it exists, the id is disregarded, if not, it will send the command 'getdata' to request the contents of this transaction. The client replies with the command 'tx' as well as the transaction data or replies 'notfound' otherwise. Then the entry node verifies the transaction. If the transaction is not correct, it will return 'reject', if it is correct, the transaction will be transmitted to its entry nodes.
Anonymous Network
TOR [32] and I2P [33] are anonymous networks, which hide the real IP address for users and encrypt the transferred data, preventing original data from being exposing.
The key technologies are: ① route selection, the client decides the route randomly. As for TOR, Each node knows the identity of its previous node and next node, but it does not know the source IP and destination IP. The last node decrypts data and accesses the Internet in plain text. As for I2P, the original data was divided into several packages, and each package was transferred in a different route; ② data encryption, the original data was encrypted when across through the relay nodes, and then decrypted at the last node, and the relay nodes don't know the original data.
TOR and I2P are general anonymous networks, not specially serving for the Bitcoin network. In 2015, FBI closed 400 illegal websites on the TOR network, and it seems the security of TOR under serious threat.
Sybil Attcack
In active or passive ABPN, there are several attacks in which the new transaction can be linked to an IP address so that the attacker could find the relationship between the Bitcoin address and IP address.
Bitcoin Protocol Sniffer:
In the Bitcoin protocol, the data is not encrypted so a well-formed sniffer could monitor all outgoing 'inv' commands to check whether the transaction id is likely to be a new transaction. Hence, we could obtain the relationship between the Bitcoin address and the IP address, and get the real identity with the help of the telecom operator's IP records.
Sybil Attack: Kaminsky [10] proposes the Sybil attack using a Bitcoin client to connect to all nodes in the Bitcoin network. The first source IP address of a new transaction is owned by the original sender.
Koshy's [11] experiments, which had gathered 2,500,000 pairs of address and IP within 5 months, show that this method works, but there are three problems remaining: (1) Bitcoin via TOR hides the true IP address, (2) a large number of clients cannot be connected directly, (3) the same client owns different sessions, different IP addresses and different networks (anonymous and not anonymous), so it is difficult to link the transactions and IP addresses.
Sybil attack plus entry nodes: Biryukov [12] implemented a method that makes all Bitcoin nodes deny connections from TOR exit nodes in 2014. Meanwhile, they succeeded in detecting the entry nodes of a specified client. With these two tricks, they solved the first two problems of the Sybil attack. Those suspicious IP addresses collected in a Sybil attack include the IP address of the sender, IP addresses of entry nodes, and IP addresses of non-entry nodes. Through the delivery time of every mentioned IP addresses, we could probably then find the source IP address of a new transaction. Their results on a Bitcoin test network show that there is a 60% chance of identifying the source IP of a new transaction successfully using this method.
Fake nodes attack: Alex Biryukov [13] developed a TOR middle-man attack and 'Address cookies'
to solve the third problem in 2015 via what could be called a fake Bitcoin nodes attack. This works by firstly, establishing a sufficient number of fake TOR exit nodes (the amount should reach 3% of all exit nodes in TOR network) and fake Bitcoin nodes (1,000 to 1,500). These fake nodes behave like normal nodes but they run code from the attackers. Then, address cookies aim to identify a certain client even if it uses different IP addresses, different sessions, and different networks.
Transaction remote release (TRR)
In the Bitcoin protocol, the only way that the attackers can connect the Bitcoin address with an IP address is in the process of spreading a new transaction. If we encrypt the new transaction and obfuscate the source IP of the sender then the attacker may not succeed. Shentu etc. [34] proposed a new anonymization technology for Bitcoin, which denoted as Transaction Remote Release (TRR). A client encrypts a new transaction layer by layer, using the public key from different TRR nodes (Bitcoin nodes supporting TRR protocol). Then it establishes an independent connection to TRR nodes, one by one. When a TRR node receives data, it decrypts the data using its private key and then transmits the remaining data to the next node. The last TRR node releases the transaction to the Bitcoin network. Each node knows the identity of its previous node and next node, but it does not have access to the transaction content. Only the client and the last node know the content of the transaction, but the last node does not know the IP address of the client. TRR doesn't require TOR or I2P, using public key to encrypt data, avoiding entry nodes of the Bitcoin nodes, and rendering the ABPN ineffective, thus provides strong anonymity for Bitcoin. However, TRR would require to modify the Bitcoin protocol, thus cannot be applied at once. Setting up a new blockchain and establishing a Transaction Delivery Network (TDN) are two possible avenues for further research.
DarkNetspace [42] is an independent blockchain of crypto-currency, based on TRR technology to enhance the anonymity of currency transactions. TDN is an independent network based on TRR technology to distribute new transactions from any blockchain anonymously, supporting multi-currencies and multiblockchains.
Research directions and Prospects
We think that there are several directions worth further research.
(1) There are some representative studies on ATC but not yet reaching the practical stage. A large number of stolen Bitcoins failed to be identified its owner, so it is necessary to set up a global identify database including Bitcoin addresses and users identifies. In addition, some stolen
Bitcoins is difficult to be traced through ATC because they were mixed by mixers.
(2) Research on the anonymity of Bitcoin network is very limited, when the TOR denying and TOR middle-man were conducted successfully, the anonymity of the Bitcoin network was seriously under threat. TRR is a useful exploration, and more research work is expected.
(3) The security and practicability of the decentralized coin-mixing protocols have not estimated adequately. More deanonymization research is expected to attack decentralized coin-mixing protocols.
(4) Expecting that more cryptography algorithms would be applied on the Bitcoin system and would strength the security of privacy of Bitcoin, which are group signature, group blind signature, privacy sharing, homomorphic encryption, lattice cryptography and other algorithms. (6) Getting rid of the risk of currency loss for users is also an urgent subject for centralized coinmixing.
( 5 )
5The research on the anonymity of Bitcoin is always qualitative. Quantifying research method and mathematical model are badly needed.
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|
[] |
[
"X-ray Linear Dichroism in cubic compounds: the case of Cr 3+ in MgAl 2 O 4",
"X-ray Linear Dichroism in cubic compounds: the case of Cr 3+ in MgAl 2 O 4"
] |
[
"Amélie Juhin \nInstitut de Minéralogie et de Physique\nUMR CNRS 7590\nMilieux Condensés (IMPMC)\nUniversité Pierre et Marie Curie\nParis 6 & Paris 7\n\nIPGP\n4 place Jussieu75052, Cedex 05ParisFrance\n",
"Christian Brouder \nInstitut de Minéralogie et de Physique\nUMR CNRS 7590\nMilieux Condensés (IMPMC)\nUniversité Pierre et Marie Curie\nParis 6 & Paris 7\n\nIPGP\n4 place Jussieu75052, Cedex 05ParisFrance\n",
"Marie-Anne Arrio \nInstitut de Minéralogie et de Physique\nUMR CNRS 7590\nMilieux Condensés (IMPMC)\nUniversité Pierre et Marie Curie\nParis 6 & Paris 7\n\nIPGP\n4 place Jussieu75052, Cedex 05ParisFrance\n",
"Delphine Cabaret \nInstitut de Minéralogie et de Physique\nUMR CNRS 7590\nMilieux Condensés (IMPMC)\nUniversité Pierre et Marie Curie\nParis 6 & Paris 7\n\nIPGP\n4 place Jussieu75052, Cedex 05ParisFrance\n",
"Philippe Sainctavit \nInstitut de Minéralogie et de Physique\nUMR CNRS 7590\nMilieux Condensés (IMPMC)\nUniversité Pierre et Marie Curie\nParis 6 & Paris 7\n\nIPGP\n4 place Jussieu75052, Cedex 05ParisFrance\n",
"Etienne Balan \nInstitut de Minéralogie et de Physique\nUMR CNRS 7590\nMilieux Condensés (IMPMC)\nUniversité Pierre et Marie Curie\nParis 6 & Paris 7\n\nIPGP\n4 place Jussieu75052, Cedex 05ParisFrance\n\nInstitut de Recherche pour le Développement (IRD)\n213 rue La FayetteUR T058, 75480, Cedex 10ParisFrance\n",
"Amélie Bordage \nInstitut de Minéralogie et de Physique\nUMR CNRS 7590\nMilieux Condensés (IMPMC)\nUniversité Pierre et Marie Curie\nParis 6 & Paris 7\n\nIPGP\n4 place Jussieu75052, Cedex 05ParisFrance\n",
"Georges Calas \nInstitut de Minéralogie et de Physique\nUMR CNRS 7590\nMilieux Condensés (IMPMC)\nUniversité Pierre et Marie Curie\nParis 6 & Paris 7\n\nIPGP\n4 place Jussieu75052, Cedex 05ParisFrance\n",
"Sigrid G Eeckhout \nEuropean Synchrotron Radiation Facility\n6 rue Jules Horowitz, BP 22038043Grenoble CedexFrance\n",
"Pieter Glatzel \nEuropean Synchrotron Radiation Facility\n6 rue Jules Horowitz, BP 22038043Grenoble CedexFrance\n"
] |
[
"Institut de Minéralogie et de Physique\nUMR CNRS 7590\nMilieux Condensés (IMPMC)\nUniversité Pierre et Marie Curie\nParis 6 & Paris 7",
"IPGP\n4 place Jussieu75052, Cedex 05ParisFrance",
"Institut de Minéralogie et de Physique\nUMR CNRS 7590\nMilieux Condensés (IMPMC)\nUniversité Pierre et Marie Curie\nParis 6 & Paris 7",
"IPGP\n4 place Jussieu75052, Cedex 05ParisFrance",
"Institut de Minéralogie et de Physique\nUMR CNRS 7590\nMilieux Condensés (IMPMC)\nUniversité Pierre et Marie Curie\nParis 6 & Paris 7",
"IPGP\n4 place Jussieu75052, Cedex 05ParisFrance",
"Institut de Minéralogie et de Physique\nUMR CNRS 7590\nMilieux Condensés (IMPMC)\nUniversité Pierre et Marie Curie\nParis 6 & Paris 7",
"IPGP\n4 place Jussieu75052, Cedex 05ParisFrance",
"Institut de Minéralogie et de Physique\nUMR CNRS 7590\nMilieux Condensés (IMPMC)\nUniversité Pierre et Marie Curie\nParis 6 & Paris 7",
"IPGP\n4 place Jussieu75052, Cedex 05ParisFrance",
"Institut de Minéralogie et de Physique\nUMR CNRS 7590\nMilieux Condensés (IMPMC)\nUniversité Pierre et Marie Curie\nParis 6 & Paris 7",
"IPGP\n4 place Jussieu75052, Cedex 05ParisFrance",
"Institut de Recherche pour le Développement (IRD)\n213 rue La FayetteUR T058, 75480, Cedex 10ParisFrance",
"Institut de Minéralogie et de Physique\nUMR CNRS 7590\nMilieux Condensés (IMPMC)\nUniversité Pierre et Marie Curie\nParis 6 & Paris 7",
"IPGP\n4 place Jussieu75052, Cedex 05ParisFrance",
"Institut de Minéralogie et de Physique\nUMR CNRS 7590\nMilieux Condensés (IMPMC)\nUniversité Pierre et Marie Curie\nParis 6 & Paris 7",
"IPGP\n4 place Jussieu75052, Cedex 05ParisFrance",
"European Synchrotron Radiation Facility\n6 rue Jules Horowitz, BP 22038043Grenoble CedexFrance",
"European Synchrotron Radiation Facility\n6 rue Jules Horowitz, BP 22038043Grenoble CedexFrance"
] |
[] |
The angular dependence (x-ray linear dichroism) of the Cr K pre-edge in MgAl2O4:Cr 3+ spinel is measured by means of x-ray absorption near edge structure spectroscopy (XANES) and compared to calculations based on density functional theory (DFT) and ligand field multiplet theory (LFM). We also present an efficient method, based on symmetry considerations, to compute the dichroism of the cubic crystal starting from the dichroism of a single substitutional site. DFT shows that the electric dipole transitions do not contribute to the features visible in the pre-edge and provides a clear vision of the assignment of the 1s→3d transitions. However, DFT is unable to reproduce quantitatively the angular dependence of the pre-edge, which is, on the other side, well reproduced by LFM calculations. The most relevant factors determining the dichroism of Cr K pre-edge are identified as the site distortion and 3d -3d electronic repulsion. From this combined DFT, LFM approach is concluded that when the pre-edge features are more intense than 4 % of the edge jump, pure quadrupole transitions cannot explain alone the origin of the pre-edge. Finally, the shape of the dichroic signal is more sensitive than the isotropic spectrum to the trigonal distortion of the substitutional site. This suggests the possibility to obtain quantitative information on site distortion from the x-ray linear dichroism by performing angular dependent measurements on single crystals.
|
10.1103/physrevb.78.195103
|
[
"https://arxiv.org/pdf/0806.1586v1.pdf"
] | 39,618,975 |
0806.1586
|
889fb4094f7b1372ddf82de510e5851cd9fd3a3d
|
X-ray Linear Dichroism in cubic compounds: the case of Cr 3+ in MgAl 2 O 4
10 Jun 2008
Amélie Juhin
Institut de Minéralogie et de Physique
UMR CNRS 7590
Milieux Condensés (IMPMC)
Université Pierre et Marie Curie
Paris 6 & Paris 7
IPGP
4 place Jussieu75052, Cedex 05ParisFrance
Christian Brouder
Institut de Minéralogie et de Physique
UMR CNRS 7590
Milieux Condensés (IMPMC)
Université Pierre et Marie Curie
Paris 6 & Paris 7
IPGP
4 place Jussieu75052, Cedex 05ParisFrance
Marie-Anne Arrio
Institut de Minéralogie et de Physique
UMR CNRS 7590
Milieux Condensés (IMPMC)
Université Pierre et Marie Curie
Paris 6 & Paris 7
IPGP
4 place Jussieu75052, Cedex 05ParisFrance
Delphine Cabaret
Institut de Minéralogie et de Physique
UMR CNRS 7590
Milieux Condensés (IMPMC)
Université Pierre et Marie Curie
Paris 6 & Paris 7
IPGP
4 place Jussieu75052, Cedex 05ParisFrance
Philippe Sainctavit
Institut de Minéralogie et de Physique
UMR CNRS 7590
Milieux Condensés (IMPMC)
Université Pierre et Marie Curie
Paris 6 & Paris 7
IPGP
4 place Jussieu75052, Cedex 05ParisFrance
Etienne Balan
Institut de Minéralogie et de Physique
UMR CNRS 7590
Milieux Condensés (IMPMC)
Université Pierre et Marie Curie
Paris 6 & Paris 7
IPGP
4 place Jussieu75052, Cedex 05ParisFrance
Institut de Recherche pour le Développement (IRD)
213 rue La FayetteUR T058, 75480, Cedex 10ParisFrance
Amélie Bordage
Institut de Minéralogie et de Physique
UMR CNRS 7590
Milieux Condensés (IMPMC)
Université Pierre et Marie Curie
Paris 6 & Paris 7
IPGP
4 place Jussieu75052, Cedex 05ParisFrance
Georges Calas
Institut de Minéralogie et de Physique
UMR CNRS 7590
Milieux Condensés (IMPMC)
Université Pierre et Marie Curie
Paris 6 & Paris 7
IPGP
4 place Jussieu75052, Cedex 05ParisFrance
Sigrid G Eeckhout
European Synchrotron Radiation Facility
6 rue Jules Horowitz, BP 22038043Grenoble CedexFrance
Pieter Glatzel
European Synchrotron Radiation Facility
6 rue Jules Horowitz, BP 22038043Grenoble CedexFrance
X-ray Linear Dichroism in cubic compounds: the case of Cr 3+ in MgAl 2 O 4
10 Jun 2008numbers: 6172Bb7870Dm7115Mb
The angular dependence (x-ray linear dichroism) of the Cr K pre-edge in MgAl2O4:Cr 3+ spinel is measured by means of x-ray absorption near edge structure spectroscopy (XANES) and compared to calculations based on density functional theory (DFT) and ligand field multiplet theory (LFM). We also present an efficient method, based on symmetry considerations, to compute the dichroism of the cubic crystal starting from the dichroism of a single substitutional site. DFT shows that the electric dipole transitions do not contribute to the features visible in the pre-edge and provides a clear vision of the assignment of the 1s→3d transitions. However, DFT is unable to reproduce quantitatively the angular dependence of the pre-edge, which is, on the other side, well reproduced by LFM calculations. The most relevant factors determining the dichroism of Cr K pre-edge are identified as the site distortion and 3d -3d electronic repulsion. From this combined DFT, LFM approach is concluded that when the pre-edge features are more intense than 4 % of the edge jump, pure quadrupole transitions cannot explain alone the origin of the pre-edge. Finally, the shape of the dichroic signal is more sensitive than the isotropic spectrum to the trigonal distortion of the substitutional site. This suggests the possibility to obtain quantitative information on site distortion from the x-ray linear dichroism by performing angular dependent measurements on single crystals.
I. INTRODUCTION
Transition metal elements play an essential role in physics (magnetic materials, superconductors...), coordination chemistry (catalysis, metalloproteins) or geophysics (3d elements are major constituents of the Earth and planets). To understand the properties that transition elements impart to the materials they are inserted in, X-ray Absorption Near Edge Structure (XANES) spectroscopy has been widely used, since it provides unique information on their local surrounding and electronic structure. In particular, the position and intensity of the localized transitions observed at the K pre-edge (1s→3d transitions) are sensitive to the cation oxidation state, the geometry of its environment (coordination number and symmetry), and the degree of admixture between p and d orbitals. For example, the shape and area of the pre-edge are commonly used to quantify the redox states of transition elements in crystals, glasses and coordination complexes, by comparison to those recorded on reference compounds. [1][2][3] However, it is not straightforward to obtain this kind of information on single crystals. Indeed, it is well known that the XANES spectra of non-cubic crystals show an angular dependence, when the polarization and the direction of the incident x-ray beam (here, designated as unit vectors,ε andk, respectively) are varied. For cubic crystals, the problem may seem at first sight more simple. Electric dipole transitions (e.g., 1s→p transitions) are isotropic. They contribute mainly to the edge, but also to the pre-edge if one of the three following situations is encountered : (i) there is p-d intrasite hybridization (e.g., the crystallographic site does not show an inversion center), (ii) the thermally activated vibrations remove the inversion center, (iii) there is p-d intersite hybridization (in samples highly concentrated in the investigated element). Electric quadrupole transitions are anisotropic and the cubic crystal thus shows an angular dependence. The information carried by the pre-edge features can be derived for cubic crystals by taking advantage of this angular dependence. In particular, the respective proportion of electric dipole and quadrupole transitions in the pre-edge can be derived, by measuring XANES spectra for various known orientations (ε,k) of the incident beam. 4 In addition, the symmetry of the crystallographic sites, that host the investigated element, is often a subgroup of the cubic group. The number of equivalent sites is given by the ratio of the multiplicity of the space group and the multiplicity of the point group. The XANES spectrum of the cubic crystal is thus the average over the equivalent sites of the individual site spectra. Hence, the derivation of structural and electronic information for a single site is not straightforward, which makes the use of group theory and theoretical computations mandatory.
Among cubic oxides, spinels have attracted considerable interest for their optical, electronic, mechanical and magnetic properties. [5][6][7] In the Earth's interior, the formation of silicate spinels has major geophysical implications. 8 More specifically, MgAl 2 O 4 spinels are used in a broad range of applications, including optically transparent materials, catalyst supports, nuclear Table I. Site distortion has been slightly exagerated for clarity. The sample is cut along the (110) plane (red) and rotated along the [110] direction, whileε andk are fixed in the laboratory system. waste management and cement castables. [10][11][12] Cr 3+ often substitutes for Al 3+ in MgAl 2 O 4 , which causes a red color and makes natural Cr-spinels valuable gemstones. Cr 3+ is intentionally added to high-temperature refractory materials to improve their thermal and mechanical properties. 13 In MgAl 2 O 4 spinel (Fd3 m space group symmetry), Al 3+ cations occur at octahedral sites, which exhibit D 3d (or3 m) symmetry and build chains aligned along the six twofold axis of the cubic structure. 14 The number of equivalent octahedral sites in the unit cell is four, denoted hereafter as sites A, B, C and D, depending on their direction of distortion, either [111], [111], [111] or [111], respectively ( Fig. 1 and Tab. I). During the Al to Cr substitution, the local D 3d symmetry is retained and the relaxed Cr-site remains centrosymmetric, which indicates the absence of Cr 3d -4p mixing. 15 Hence, the K pre-edge features arise from pure electric quadrupole transitions (1s→3d ) but an experimental evidence of this is still lacking. As the Cr-site remains distorted in the same direction as for the Al-site, four equivalent relaxed sites are available for Cr. Hence, the electric dipole and electric quadrupole absorption cross-sections, for a given single crystal configuration, are expected to be different for a Cr impurity located at site A, B, C or D, since their orientations with respect to the incident beam are different.
In this paper, we compare the experimental angular dependence of the Cr K pre-edge MgAl 2 O 4 :Cr 3+ to those obtained by theoretical calculations, combining a mono-electronic approach based on density functional theory (DFT) and multielectronic methods based on the ligand field multiplet theory (LFM). The monoelectronic approach is usually dedicated to the study of delocalized final states (e.g., the calculation of K-edge spectra) but has also provided satisfactory results for the study of Ti K pre-edge in TiO 2 and SrTiO 3 , [16][17][18][19] and also for the study of Fe K pre-edge in FeS 2 . 20 The multielectronic approach, usually dedicated to the study of localized final states (e.g., K pre-edge, L 2,3 edges of 3d transition elements) has been succesfully applied to the case of K preedge in several systems. 21,22 Our aim is to determine the factors (site distortion, electronic interactions) prevailing at the angular dependence of Cr K pre-edge in spinel and to provide a comparison between the monoelectronic and multielectronic approaches. We also present a powerful method, based on symmetry considerations, to reduce the number of calculations needed to reconstruct the angular dependence of the cubic crystal from that of a single site. The paper is organized as follows. Section II is dedicated to the experimental work, including the sample description, the X-ray absorption measurements and analysis. Section III is devoted to the computational work, including the theoretical framework (Sec. III A), the details of DFT calculations (Sec. III B) and of the multiplet calculations (Sec. III C). Results are presented in Sec. IV and discussed in Sec. V.
II. EXPERIMENTS
A natural gem-quality red spinel single crystal from Mogok (Burma), with composition (Mg 0.95 Fe 0.01 ) 0.96 (Al 2.02 Cr 0.01 ) 2.03 O 4 , was investigated (for details, see Ref. 15). The single crystal was cut along the (110) plane (plotted in red on Fig.1) and orientated according to the Laue method. Cr K-edge (5989 eV) XAS spectra were collected at room temperature at beamline ID26 of the European Synchrotron Radiation Facility (Grenoble, France). 23 The energy of the incident radiation was selected using a pair of He-cooled Si crystals with (111) orientation. The spot size on the sample was approximatively 250×50 µm 2 . The orientated sample was placed on a rotating holder at 45 • with respect to the incident beam, and turned around the [110] direction from a rotation angle α rot . The starting configuration (α rot = 0 • ) corresponds toε = [010] andk = [100] (see Fig.1). The (α rot = 90 • ) configuration corresponds toε = [ 1 2 , 1 2 , 1
√ 2 ] andk = [-1 2 ,-1 2 , 1 √ 2 ]
. For this sample cut and this experimental setup, the maximum variation effect is obtained by substracting the absorption recorded for α rot = 0 • from that recorded for α rot = 90 • . One spectrum was recorded every 15 • from α rot = 0 • to α rot = 360 • , which enables to reconstruct the complete angular dependence of the crystal. The absorption was measured by a photodiode fluorescence detector. For
( 1 4 , 1 4 , 1 2 ) ( 3 4 , 3 4 , 1 2 ) , ( 1 4 , 3 4 ,0) , ( 3 4 , 1 4 ,0) C [111] ( 1 2 , 1 2 , 1 2 ) (0,0, 1 2 ), (0, 1 2 ,0) , ( 1 2 ,0,0) D [111] ( 1 4 , 1 2 , 1 4 ) ( 3 4 , 1 2 , 3 4 ) , ( 1 4 ,0, 3 4 ) , ( 3 4 ,0, 1 4 )
each α rot angle, ten pre-edge spectra ranging from 5987 to 5998 eV were recorded with an energy step of 0.05 eV and averaged. Two additional scans were recorded between 5985 and 6035 eV by step of 0.2 eV, in order to merge the pre-edge on the XANES spectrum, and two more spectra were recorded between 5950 and 6350 eV by step of 0.5 eV, in order to normalize the XANES to the K-edge jump far from the edge. Self-absorption effects are negligible, because of the low Cr-content of the sample.
III. THEORY
In this section, we recall the general expressions of the electric dipole and quadrupole absorption cross-sections for a cubic crystal and for a site with D 3d symmetry (Subsec. A). Then, we use the general method described in Ref. 24 to calculate the angular dependence of the cubic crystal from that of a single site. This framework is illustrated in the particular case of spinel. Finally, we report the details of the monoelectronic and multielectronic calculations performed for substitutional Cr in spinel (Subsec. B and C).
A. Theoretical framework
Absorption cross-sections for a cubic crystal
The total absorption cross-section for a crystal (cubic or non-cubic), σ, is expressed as:
σ(ε,k) = σ D (ε) + σ Q (ε,k)(1)
where σ D is the electric dipole cross-section and σ Q is the electric quadrupole cross-section. The expression given above is valid in the absence of coupling between the electric dipole and the electric quadrupole terms: this condition is fulfilled if the system is either centrosymmetric or if, at the same time, the system is nonmagnetic (no net magnetic moment on the absorbing ion) and one uses exclusively linear polarization. For Cr in MgAl 2 O 4 , the two types of conditions are satisfied. The dipole and quadrupole cross-sections can be expressed in function of spherical tensor components, respectively (σ D (0,0), σ D (2,m)) and (σ Q (0,0), σ Q (2,m), σ Q (4,m)), which transform under rotation like the corresponding spherical harmonics (Y 0 0 , Y m 2 and Y m 4 ). 25 The tensor components are functions of ω, omitted for clarity in this paper. The symmetries of the crystal restrict the possible values of σ D (2,m) and σ Q (4,m), as will be precised hereafter for the cubic case.
The electric dipole cross-section for a cubic crystal, σ D cub , is isotropic (e.g., it does not depend on the direction of the polarization vector) and is equal to σ D cub (0,0): 25
σ D cub (ε) = σ D cub (0, 0).(2)
The electric quadrupole cross-section for a cubic crystal, σ Q cub , is expressed, according to group theory (Appendix A), as:
σ Q cub (ε,k) = σ Q cub (0, 0)+ 20 √ 14 (ε 2 x k 2 x +ε 2 y k 2 y +ε 2 z k 2 z − 1 5 )σ Q cub (4, 0),(3)
where σ Q cub (0, 0) is the isotropic electric quadrupole crosssection, and σ Q cub (4, 0) is a purely anisotropic electric quadrupole term. The polarization unit vectorε and the wave unit vectork have their coordinates expressed in the Cartesian reference frame of the cube.
Absorption cross-sections for a site with D 3d symmetry
For a site with D 3d symmetry, the reference frame is chosen consistently with the symmetry operations of the point group, i.e. with the z -axis parallel to the C 3 axis of the D 3d group. 26 The polarization and the wave unit vectors are expressed as:ε = sin θ cos φ sin θ sin φ cos θ andk = cos θ cos φ cos ψ − sin φ sin ψ cos θ sin φ cos ψ + cos φ sin ψ − sin θ cos ψ Hence, θ, which appears in the expression ofε and k, is the angle betweenε and the C 3 axis. The electric dipole absorption cross-section in D 3d is given by: 25
σ D D 3d (ε) = σ D D 3d (0, 0) − 1 √ 2 (3 cos 2 θ − 1)σ D D 3d (2, 0).(4)
In order to determine σ D D 3d (ε) for any experimental configuration (ε), one needs first to determine σ D D 3d (0, 0) and σ D D 3d (2, 0), for example by performing calculations for at least two independent orientations ofε. The isotropic term, σ D D 3d (0, 0), can be calculated directly by choosing θ = arccos 1 √ 3 .
The electric quadrupole absorption cross-section in D 3d is given by: 25
σ Q D 3d (ε,k) = σ Q D 3d (0, 0) + 5 14 (3 sin 2 θ sin 2 ψ − 1) σ Q D 3d (2, 0) + 1 √ 14 (35 sin 2 θ cos 2 θ cos 2 ψ + 5 sin 2 θ sin 2 ψ − 4) σ Q D 3d (4, 0) − √ 10 sin θ[(2 cos 2 θ cos 2 ψ − 1) cos θ cos 3φ − (3 cos 2 θ − 1) sin ψ cos ψ sin 3φ] σ Q D 3d (4, 3).(5)
To determine σ Q D 3d (ε,k) for any experimental configuration (ε,k), one needs first to determine σ Q D 3d (0, 0), σ Q D 3d (2, 0), σ Q D 3d (4, 0) and σ Q D 3d (4, 3), for example by performing calculations for at least four independent orientations (ε,k).
From a single site D 3d to the cubic crystal
In order to reconstruct the angular dependence of the cubic crystal from that of a single site with D 3d symmetry, the tensor components have to be averaged over the equivalent sites of the cubic cell. For the electric dipole cross-section, we need a relation between (σ D D 3d (2,0), σ D D 3d (0, 0)) and σ D cub (0,0), and for the electric quadrupole cross-section, we need a relation between (σ Q D 3d (0,0), σ Q D 3d (2,0), σ Q D 3d (4,0), σ Q D 3d (4,3)) and (σ Q cub (0,0), σ Q cub (4,0)). To do so, we have used the formulas given in Ref. 24, which have been obtained from a spherical tensor analysis. This general method uses the symmetry operations of the crystal, which exchange the equivalent sites of the cubic cell, and is here illustrated in the case of spinel.
The averages over the four equivalent sites are given by: 24
σ D cub (0, 0) = σ D D 3d (0, 0),(6)
σ D cub (2, 0) = 0
Similarly, we have: 24
σ Q cub (0, 0) = σ Q D 3d (0, 0),(8)σ Q cub (4, 0) = − 1 18 (7σ Q D 3d (4, 0) + 2 √ 70 σ Q D 3d (4, 3)) (9)
4. Calculation of the absorption cross-sections for the experimental orientations
The electric dipole isotropic cross-section of the cubic crystal, σ D cub (ε), does not depend on the direction of the incident polarization vectorε. Hence, it will be the same for every experimental configuration:
σ D cub (α rot ) = σ D cub (0, 0).(10)
For the sample cut and the experimental setup used in this study, the expression of the electric quadrupole crosssection of the cubic crystal, is given in function of the rotation angle α rot by:
σ Q cub (α rot ) = σ Q cub (0, 0) + 1 16 √ 14 [−19 − 60 cos(2α rot ) +15 cos(4α rot )]σ Q cub (4, 0).(11)
The connection between Eqs. 11 and 3 is made following the definition ofε andk as functions of α rot (Appendix B). Eq. 11 shows that the total angular dependence of the cubic crystal is a π-periodic function. The fact that the rotation axis might not be perfectly aligned with the x-ray beam or that the sample might not be perfectly homogeneous, could have introduced an additional 2π periodic component. This component would be removed from the signal, using a filtering algorithm, based on the angular dependence recorded from 0 • to 360 • . 20 In our experiments, this 2π periodic-component was measured to be very small, and no filtering was applied.
For the present sample cut and experimental setup, the maximum variation of the electric quadrupole crosssection is expected between α rot = 0 • and α rot = 90 • .
• For α rot = 0 • :
σ Q cub (α rot = 0 • ) = σ Q 0 − 4 √ 14 σ Q cub (4, 0).(12)
• For α rot = 90 • :
σ Q cub (α rot = 90 • ) = σ Q 0 + 7 2 √ 14 σ Q cub (4, 0).(13)
• The isotropic cross-section is :
σ Q iso = 1 15 (8 σ Q cub (α rot = 90 • ) + 7 σ Q cub (α rot = 0 • )) = σ Q cub (0, 0).(14)
• The dichroic term is :
σ Q dichro = σ Q cub (α rot = 90 • ) − σ Q cub (α rot = 0 • ) = 15 2 √ 14 σ Q cub (4, 0).(15)
B. Computational details
Density Functional Theory Calculations
The computation of the electric dipole and electric quadrupole absorption cross-sections were done using a first-principles total energy code based on DFT in the Local Density Approximation with spin-polarization (LSDA). 27 We used periodic boundary conditions, plane wave basis set and norm conserving pseudopotentials 28 in the Kleiman Bylander form. 29 The parameters for the pseudopotential generation are given in Ref. 15.
We started from a host structure of MgAl 2 O 4 , which is obtained by an ab initio energy minimization calculation. In this calculation, the lattice parameter was fixed to its experimental value, 30 while the atomic positions were allowed to vary to minimize the total energy and the interatomic forces. We then relaxed a 2×2×2 rhomboedral supercell containing one Cr atom in substitution for Al (i.e., 1 Cr, 31 Al, 16 Mg and 64 O), with the basis vectors expressed in a cubic frame. The supercell was large enough to avoid interactions between neighboring Cr atoms. As the Cr impurity is in its high-spin state, the spin multiplet S z = 3 2 is imposed for the supercell. The atomic positions in the supercell were allowed to vary, in order to minimize the total energy and the interatomic forces. We used a 90 Ry energy cutoff and a single kpoint sampling in the Brillouin zone. The Cr-site, after relaxation, still exhibits a D 3d symmetry, with an inversion center, one C 3 axis and three C 2 axis.
The Cr K-edge absorption cross-section was computed using the method described in Refs. 31,32. First, we calculated self-consistently the charge density of the system, with a 1s core-hole on the substitutional Cr atom. Then, the all-electron wave functions were reconstructed within the projector augmented wave framework. 33 The absorption cross-section was computed as a continued fraction, using a Lanczos basis constructed recursively. 34, 35 We used a 70 Ry energy cutoff for the plane-wave expansion, one k point for the self-consistent spin-polarized charge density calculation, and a Monkhorst-Pack grid of 3×3×3 k -points in the Brillouin zone for the absorption cross-section calculation. For the convolution of the continued fraction, we used an energy-dependent broadening parameter γ, which takes into account the main photoelectron damping modes (core-hole lifetime and imaginary part of the photoelectron self-energy). The energy-dependent γ used in this study is that of Ref. 36. The calculated spectrum was then shifted in energy to the experimental one: the maximum of absorption is set at 6008.5 eV. The absorption edge jump is set to 1, so that experimental and calculated spectra for all figures are normalized absorption. In such a way, the calculated pre-edge could be compared directly to the experimental one. As mentioned previously, the four substitutional sites will exhibit different spectra for the electric dipole
(ε = [ 1 2 , 1 2 , 1 √ 2 ],k = [-1 2 ,-1 2 , 1 √ 2 ]).
and quadrupole operators, since their orientations are different with respect to the incident beam absorption of x-rays with given (ε,k). The general method to obtain the angular dependence measured for the cubic crystal is to compute the electric dipole and electric quadrupole absorption cross-sections for a Cr impurity lying in each of the four trigonally distorted sites A, B, C and D, and then to take the average. However, this heavy brute force method requires the calculation of four monoelectronic potentials with core-hole (after previous associated structural relaxation). The number of calculations can be drastically reduced if we take advantage of the symmetry properties of the crystal, which enables to perform the calculations for only one substitutional site (site A, with coordinates of (0, 1 4 , 3 4 ) and direction of distortion [111]). This method is detailed in Appendix C. Figure 2 presents the normalized electric quadrupole cross-section calculated for the four equivalent sites, for α rot = 0 • and α rot = 90 • . The spectra calculated for α rot = 0 • (orange line) are equal for the four sites. For α rot = 90 • , sites B and C give the same spectra (black solid line), as well as sites A and D (black dashed line). We observe a slight difference in intensity for the peak at 5993.2 eV: this is indeed a consequence of the fact that sites (A, D) and (B, C) have different orientations with respect to the incident beam, and that their symmetry differs from O h . Because the trigonal distortion of the octahedra is small in spinel, the anisotropic behaviour of the sites is limited for the investigated configurations. However, the effect of the trigonal environment can have drastic consequences when the distortion is more pronounced.
Ligand Field Multiplet Calculations
In order to extract quantitative information from the angular dependence of the pre-edge, we have performed LFM calculations using the method developed by T. Thole in the framework established by Cowan and Butler. 26,37,38 In this approach, Cr 3+ is considered as an isolated ion embedded in a crystal field potential. The band structure of the solid is not taken into account, which prevents to calculate transitions to delocalized (i.e., non-atomic) levels. In other words, the LFM approach can be used to calculate K pre-edge spectra, but the edge region cannot be computed. Since the Cr-site is centrosymmetric, no hybridization is allowed between the 3d -orbitals and the 4p-orbitals of Cr. Hence, the preedge is described by the transitions from the initial state 1s 2 3d 3 to the final state 1s 1 3d 4 .
We expose briefly the principles of multiplet calculations but details can be found in other references (see for example Ref. 39). This approach takes into account all the 3d -3d and 1s-3d electronic Coulomb interactions, as well as the spin-orbit coupling on every open shell of the absorbing atom, and treats its geometrical environment through a crystal field potential. In the electric quadrupole approximation, the spectrum is calculated as the sum of all possible transitions for an electron jumping from the 1s level toward one 3d level according to:
σ Q (ε,k) = π 2 k 2 α ω I,F 1 d I | F |ε·rk·r|I | 2 δ(E F −E I − ω),(16)
where |I and |F are the multielectronic initial and final states, of respective energies E I , E F , and d I the degeneracy of the initial state. 40 .
Once the |I and |F states have been calculated, the absolute intensities of the pre-edge spectra are calculated inÅ 2 at T = 300 K. The population of the ground-state levels |I is given by a Boltzmann law. The spectra are convoluted by a Lorentzian (with HWHM = 0.54 eV) and a Gaussian (with FWHM = 0.85 eV), which respectively take into account the lifetime of the 1s core-hole for Cr and the instrumental resolution. Finally, the transitions are normalized by the edge jump at the Cr K edge, calculated for a Cr atom from Ref. 41 as 4.48 10 −4Å2 . Hence, the calculated spectra can be directly compared to the normalized experimental ones.
The electric quadrupole absorption cross-section was calculated for a Cr 3+ ion lying in D 3d symmetry, according to the method described above. The crystal-field parameters used in the calculation are those derived from optical absorption spectroscopy (D q = 0.226 eV, D σ = -0.036 eV, D τ = 0.089 eV). 42 We used the scaling factor of the Slater integrals (κ= 0.7), related to B and C Racah parameters, given in the same reference. The only adjustable parameter is the absolute position in energy.
As mentioned in Sec. III A 2 (Eq. 5), one needs first to determine σ Q D 3d (0, 0), σ Q D 3d (2, 0), σ Q D 3d (4, 0) and σ Q D 3d (4,3), in order to determine σ Q cub (0, 0) and σ Q cub (4, 0) using Eqs 8 and 9. This is done by performing four multiplet calculations, which provide four independent values of the electric quadrupole cross-section (Appendix D). Once this first step has been performed, we used Eq. 8 and 9 to derive σ Q cub (0, 0) and σ Q cub (4, 0). The electric quadrupole cross-section of the cubic crystal can then be calculated for any experimental configuration using Eq. 11. The electric quadrupole cross-section for α rot = 0 • and α rot = 90 • , the dichroic and the isotropic spectra were determined respectively according to Eqs 12-15.
IV. RESULTS
A. DFT calculations
Comparison with experiment
The XANES spectrum, calculated for the cubic crystal by first-principles calculations (solid line), is shown in Fig. 3 and compared to the experimental spectrum (dotted line). As we mentioned above, the main absorption edge is due to electric dipole transitions. Hence, the XANES spectrum does not show any angular dependence, except in the pre-edge region. The agreement between the experimental and theoretical spectra is good, since all the features are reproduced by the calculation. A more detailed discussion is reported in Ref. 15. The inset of Fig. 3 shows the theoretical isotropic XANES spectrum in the pre-edge region (black solid line). This spectrum is the sum of the isotropic
toε = [ 1 2 , 1 2 , 1 √ 2 ],k = [-1 2 ,-1 2 , 1 √ 2 ] (αrot = 90 • ).
The dichroic signal (green lines) is the difference between the black and red lines, for the experimental (dotted) and the calculated spectra (solid), respectively. electric dipole (orange dashed line) and the electric quadrupole contributions. Our calculations show that electric dipole transitions do not contribute to the pre-edge, except by a background, which is actually the tail of the absorption edge (1s→p transitions). This is a clear confirmation that Cr K pre-edge features are due to a pure electric quadrupole contribution. In the pre-edge region, the calculated isotropic spectrum is in satisfactory agreement with experiment, since the two features visible in the pre-edge are reproduced. Similar calculations have been successfully performed to calculate the K pre-edge for substitutional Cr 3+ in corundum and beryl, 36,43 with a good agreement between the experimental and theoretical data. This shows that a monoelectronic approach can reproduce pre-edge features, as can be measured on powder spectra. However, the position of the theoretical spectrum is shifted by about 0.9 eV relative to experiment. This shift, which has been already observed in several systems, 17,20,36,43 is due the limitation of DFT-LSDA in the modelling of electron-hole interaction. In the calculation, the effect of the core-hole is to shift the 3d levels to lower energy, with respect to the main edge. Unfortunately, this effect is not sufficient to reproduce the experimental data, because the core-hole seems to be partly screened. 16 This could be improved by taking into account the self-energy of the photoelectron. 44 The experimental and calculated pre-edge spectra for the two configurations, which give the maximum dichroic signal for the sample cut, are shown in Fig. 4. The number of peaks is well reproduced in both cases by the calculation. For both spectra, the intensity of the first peak at about 5990.7 eV is close to 4 % of the absorption edge on the experimental data, but underestimated by 25 % in the calculation. The relative intensity of the peak at about 5992.7 eV is overestimated in the 90 • configuration. Additionally, the energy splitting between the two peaks is underestimated by the calculation (1.6 eV vs 2.0 eV experimentally). The small energy shift of the first peak between the two configurations, observed as positive in the experimental data, is calculated as negative. As a consequence of those several discrepancies, the theoretical dichroic signal is not in good agreement with the experimental one. Compared to the Ti K pre-edge calculations in rutile and SrTiO 3 , using a similar monoelectronic approach and reported in several studies, [16][17][18][19] the significant discrepancy observed for Cr in spinel may seem at first sight unexpected. However, we underline the fact that interlectronic repulsions become crucial for localized final states (e.g., for 1s→3d transitions), and that Ti has no d electrons in the systems studied. This shows that the electronic interactions on the Cr atom are too significant to reproduce quantitatively the angular dependence of Cr-spinel in a monoelectronic approach, although the average description (i.e., the isotropic spectrum) is satisfactory.
Nevertheless, the monoelectronic calculation is able to reproduce the correct number of peaks. Since this monoelectronic approach does not take into account spin-orbit coupling and does not fully describe the 3d -3d electronic repulsion, a monoelectronic chemical vision of an isolated Cr 3+ ion can be applied for the interpretation of the calculated features.
Assignment of the calculated monoelectronic transitions within an atomic picture
In the following, we shall concentrate on the spectra associated to the local symmetry (i.e., calculated for site A). In the monoelectronic calculations, the spin multiplet S z = 3 2 is imposed for the supercell, since the Cr impurity is in its high-spin state. Cr 3+ has an initial electronic configuration (t ↑ 2g ) 3 (e g ) 0 , which means that Cr 3+ is a fully magnetized paramagnetic ion in the calculation, while it is paramagnetic in the experiment. Indeed, it is not possible to impose the fourfold degenerate S= 3 2 ground-state in the DFT calculation, that requires nondegenerate ground states. In order to assign the transitions visible in the experimental spectra, we would need to calculate the average of the spectra for S z = 3 2 , S z =-3 2 , S z = 1 2 and S z =-1 2 . However, it is not possible to do the calculation for S z =± 1 2 in the Kohn-Sham formalism, since they are linear combinations of three Slater determinants. 45 Nevertheless, the spin-polarized computation of the XANES spectrum for S z = 3 2 enables to understand the origin of the pre-edge features: the contri- bution of the two spins (↑ and ↓) can be indeed separated, which means that we can deduce whether the 3d -orbitals have been reached by a 1s electron with spin ↑ or ↓, and this for different expressions of the electric quadrupole operator.
As shown in Fig. 1, although the distortion of the octahedra has been slighlty exagerated, the oxygen ligands are located approximately along the fourfold axis of the cube for all the equivalent sites A, B, C and D. Thus, the analysis made for site A provides an assignment, which is also valid (mutatis mutandis) for the equivalent sites. For a given configuration (ε,k), we can easily deduce from the expression of the electric quadrupole operator which 3d -orbital has been probed in this transition. The interpretation of the features is possible through group theory in the monoelectronic approach, using the branching rules of O h ⊃ D 3d (Appendix E). The d orbitals belong to the t 2g (O h ) and e g (O h ) irreducible representations within octahedral symmetry. When lowering the symmetry to D 3d , the t 2g (O h ) irreducible representation is split into the e g (D 3d ) and a 1g (D 3d ) irreducible representations. To indicate that they come from t 2g (O h ), they will be written as e g (t 2g ) and a 1g (t 2g ). The e g (O h ) irreducible representation becomes the e g (D 3d ) irreducible representation, designed hereafter as e g (e g ).
The normalized electric quadrupole cross-sections calculated for site A are shown in Fig. 5 for three different configurations (ε,k). For a better understanding of the structures, the electric quadrupole transitions to both occupied and empty states are represented. For (Fig. 5a), the electric quadrupole operator is expressed asÔ a = 1 2 (x 2 − y 2 ), which enables to probe the 3d electronic density in the x 2 -y 2 direction, i.e., along Cr-O bonds: the orbitals probed are the e g (e g ), which are empty for spin ↑ and ↓, since they are coming from the e g (O h ) levels. Fig. 5a shows that, indeed, two peaks are obtained at 5991.6 eV and 5993.2 eV, above the Fermi level. Below the Fermi level at 5990 eV, a broad structure is observed between 5982 eV and 5990 eV, which corresponds to e g states hybridized with the p-orbitals of the oxygens. For α rot = 0 • (ε = [010],k=[100]), one single peak is obtained in the empty states at 5991.6 eV for spin ↓ (see Fig. 5b, black line). For this orientation, the electric quadrupole operator, expressed asÔ b = xy, enables to probe the d electronic density, projected on Cr, in the xy direction, i.e., between the Cr-O bonds. The e g (t 2g ) and a 1g (t 2g ) orbitals having a component along xy, as indicated by their expressions in Appendix D (Eq. 44), they are probed in the transition. As these states coming from the t ↑ 2g (O h ) are fully occupied, they can be reached only by a photoelectron with spin ↓. This is indeed consistent with our results. The splitting between e ↓ g and a ↓ 1g is not visible, which is an indication of a small trigonal distortion for the Cr-site in spinel. Below the Fermi level, a broad structure with an intense peak at 5988.7 eV is observed. The intense peak corresponds to the occupied e ↑ g (t 2g ) levels. To interpret the origin of the broad structure, we have to remind that the e g (t 2g ) states can hybridize with the e g (e g ) levels, since they belong to the same irreducible representation in D 3d . As mentioned previously, the hybridization of the mixed e g states with the p-orbitals of the oxygens gives rise to the structures visible below 5988 eV. (Fig. 5c), two peaks are obtained above the Fermi level. For this orientation, the electric quadrupole operator is expressed asÔ c = 3z 2 −r 2 4 − xy 2 , which enables to probe the 3d electronic density both in the xy and 3z 2 − r 2 directions. For the 3z 2 − r 2 component, the levels probed are the e g (e g ), as for the first orientation studied (Fig. 5a). For the xy component, the levels probed are the e g (t 2g ) and a 1g (t 2g ), as for the second orientation (Fig. 5b). Fig. 5c shows that the spectrum is a close combination of the transitions visible on the two previous spectra (Fig. 5a and b), and the assignment of the structures is made clear from the two previous cases. The position of the t ↓ 2g (e g ) peak is close to that of the e ↑ g (e g ) at 5991.6 eV. The energy difference (1.6 eV) between t ↓ 2g (e g ) and e ↓ g (e g ) gives an idea of the t ↓ 2g (O h )-e ↓ g (O h ) splitting due to the crystal field. This can be compared to the experimental crystal-field splitting (2.26 eV), derived from optical absorption spectroscopy, but one should keep in mind that the crystal field splitting in the monoelectronic picture is associated with spin ↑ levels. For the configurationÔ c , the 3z 2 −r 2 component enables to probe the e g (O h ) states, as x 2 −y 2 . Considering the normalization factors in the expression of the d orbitals, the magnitude of the transition operator along 3z 2 − r 2 is √ 3 times bigger than the magnitude of the transition operator along x 2 −y 2 , which is 2 times bigger than the magnitude of the transition operator along xy.Ô c thus appears as a linear combination of the two operatorŝ O a andÔ b , with respective weights of √ 3 2 and 1 2 . If no coupling occurs betweenÔ a andÔ b when calculating the square matrix elements | f |ε · rk · r|i | 2 , the third cross-section (c) should be the linear combination of the two cross-sections (a) and (b) obtained forÔ a and O b , with respective weights of 3 4 and 1 4 . However, the linear combination of the two cross-sections (not shown in Fig. 5) and the cross-section obtained for the linear combination of the transition operators are slightly different, which indicates a small interference between the xy and 3z 2 − r 2 (or x 2 − y 2 ) components. The interference is a clear evidence of the e ↓ g (e g ) and e ↓ g (t 2g ) hybridization due to the D 3d local symmetry.
(ε = [ 1 √ 2 , 1 √ 2 ,0],k = [ 1 √ 2 ,-1 √ 2 ,0])For α rot = 90 • (ε = [ 1 2 , 1 2 , 1 √ 2 ],k = [-1 2 ,-1 2 , 1 √ 2 ])
B. LFM calculations
Comparison with experiment
For the two experimental configurations, Fig. 6 presents the experimental Cr K pre-edge spectra (dotted line), the theoretical spectra obtained by LFM calculations (solid line) and the corresponding dichroic signals. The calculated pre-edges have been obtained for the cubic crystal from a calculation performed for a single site with D 3d symmetry. For each configuration, the shape of the spectrum is well reproduced by the calculation. In the experimental data, the position of the first peak is shifted by approximately +0.15 eV for α rot = 0 • , compared to that in the α rot = 90 • configuration. This relative shift is also well reproduced in the calculated spectra. For α rot = 90 • , the relative intensity of the two peaks is in good agreement with the experimental data. The shape of the dichroic signal is well reproduced by the calculation: in fact, the x-ray linear dichroism of the crystal is well described in the multiplet approach, suggesting that the calculation includes the necessary multielectronic interactions on the Cr atom. We recall that the crystal-field parameters used in the calculation are those obtained from optical absorption spectroscopy (see Appendix F & G for the correspondance between the experimental crystal-field parameters and the parameters used in the multiplet calculations). However, the intensity of the dichroic signal is overestimated by 20 % in the calculation. The first reason for this overestimation is that the calculated spectra have been normalized by the edge jump at the Cr K edge, which was calculated for an isolated Cr without considering the influence of the crystal structure according to Ref. 41. This can account for a few percent in the discrepancy. Another few percent possibly lie in the normalization of the experimental data, since we used the average of two spectra, which were recorded between 5950 eV and 6350 eV with a rather large energy step (0.5 eV). This can introduce limited noise and thus uncertainty on the normalization. A third source of error is that the crystal-field parameters used in the calculation might be slightly different in the excited state than in the ground-state, because of the influence of the core-hole: for example, if D q is increased by 2 % in the excited state, the intensity of the first peak in the dichroic signal decreases by 14 %. The shape and intensity of the calculated dichroism are quite sensitive to the crystal-field parameters used in the excited state.
Nevertheless, despite this slight intensity mismatch with the experimental data, the angular dependence of the crystal is well reproduced by the calculation, which means that the multielectronic approach takes into account the necessary interactions. Since isotropic and dichroic calculated spectra fit well with experiment, the analysis of the calculation is very likely to yield valuable insight into the origin of the experimental transitions in the pre-edge region. In the following, we shall investigate the influence of the different terms in the Hamiltonian taken into account in the LFM approach (trigonal distortion, fourfold degeneracy of the ground state trigonal S= 3 2 ( 4 A 2g ), spin-orbit coupling on the 3d levels, 3d -3d or 1s-3d Coulomb repulsion) on the angular dependence.
Influence of trigonal distortion on dichroism
In this paragraph, we investigate the influence of the trigonal distortion on the angular dependence. The isotropic and dichroic spectra in O h symmetry have been obtained by setting the trigonal distortion of the crystal field to zero. They are compared to those calculated in D 3d symmetry using the distortion parameters given by optical absorption spectroscopy (D σ = -0.036 eV, D τ = 0.089 eV). As shown in Fig. 7, the difference between the calculations performed in O h and D 3d symmetries (orange solid line and black dotted line, respectively) is weak, since the isotropic and dichroic signals have similar shape and intensity. This result is consistent with the small values of the parameters D σ and D τ , which quantify the trigonal distortion of the Cr-site in spinel. This means that, provided that trigonal distortion is limited, the calculation of pre-edge spectra could have been performed for a single site with O h symmetry (see Appendix A2 for simplified formula). This is also in line with the monoelectronic calculation, for which the splitting between e ↓ g (t 2g ) and a ↓ 1g (t 2g ) could not be resolved in the calculated spectra.
We have investigated the effect of the intensity of the trigonal distortion on the calculated spectra by choosing two other sets of the distortion parameters. In Fig. 7, the spectra labeled D 3d -i (i = 2,3) are calculated with the set of distortion parameters (i × D σ , i × D τ ), for (D σ = -0.036 eV, D τ = 0.089 eV). The crystal-field parameters used are the same in the ground-and excitedstate. As seen in Fig. 7, the intensity of the isotropic spectra (D 3d , D 3d -2 and D 3d -3 ) are almost identical, indicating that the isotropic signal is not sensitive to site distortion. The intensity of the maximum at 5990.75 eV remains close to 2.5 % of the electric dipole edge jump. On the contrary, the shape and intensity of the linear dichroic signal is highly sensitive to the trigonal distortion. The intensity of the first feature at 5990.25 eV is lowered when the distortion is increased: one observes a 20 % decrease when the distortion parameters are doubled (signal labeled D 3d -2 in Fig.7), and a 50 % decrease when the distortion parameters are tripled (signal labeled D 3d -3 ). It should be noticed that, for our parameter sets, the increase of site distortion is accompanied by a rather counter-intuitive decrease of the intensity of the linear dichroic signal, thus indicating the relevance of the theoretical developments performed within this paper. This means that site distortion has to be carefully taken into account when calculations are performed to mimic the angular dependence of the pre-edge. In that case, the calculation for a single site with D 3d symmetry should follow the method described in Sec. III.
Influence of ground-state degeneracy, spin-orbit coupling and interelectronic repulsion on dichroism
Beyond the site symmetry distortion, the other ingredients of the calculation are the fourfold degeneracy of the S= 3 2 ground state, the 3d -3d Coulomb repulsion, the 1s-3d Coulomb repulsion and the 3d spin-orbit coupling. We shall check the influence of these different parameters. We have performed multiplet calculations restricting the ground-state to the non-degenerate S z = 3 2 state of the S= 3 2 multiplet. The calculated electric quadrupole transitions are almost identical to those with the fourfold ground state. Differences are below 0.1% of the maximum intensity of the isotropic electric quadrupole spectrum. This clearly indicates that the procedure followed in monolectronic calculations to take into account the spin degeneracy is sound and appropriate. The radial integrals for Coulomb interaction and spin-orbit coupling are calculated by relativistic Hartree-Fock atomic calculations. One finds: the 1s-3d exchange Slater integral G 1 1s,3d = 0.052 eV, the direct 3d -3d Slater integrals F 2 3d,3d = 10.78 eV and F 4 3d,3d = 6.75 eV, and the 3d spin-orbit coupling ζ 3d = 0.035 eV. G 1 1s,3d is small compared to F 2 3d,3d and F 4 3d,3d . Using G 1 1s,3d = 0 in the multiplet calculation, we found almost no difference with the isotropic and dichroic signal calculated with the ab initio atomic value of G 1 1s,3d . By calculating the dichroic signal with ζ 3d = 0, we found a small difference concerning the intensity of isotropic and dichroic signals, when compared to the associated spectra with ζ 3d = 0.035 eV. The maximum relative difference is less than a few percent (2 %) of the feature intensity. The observed small dependence of the pre-edge features with G 1 1s,3d and ζ 3d is in line with results obtained at the Fe K pre-edge. 21 We also performed calculations setting F 2 3d,3d and F 4 3d,3d
to zero, and we observed that the isotropic and dichroic calculated spectra (not shown) were in complete disagreement with experimental data. This clearly indicates that the direct Slater integrals on the 3d shell, and thus the multielectronic 3d -3d Coulomb interactions, are the essential ingredients governing the shape of the isotropic as well as the dichroic signals.
From the preceding analysis, we can unambiguously determine the parameters governing the shape and intensities of the pre-edge features. Spin-orbit coupling on the 3d orbitals, ground-state degeneracy and 1s-3d Coulomb repulsions have only limited impact on the calculated LFM isotropic spectra. This explains the reasonable agreement between calculation and experiment for isotropic pre-edge in the DFT formulation, where the two first previous ingredients are missing, and where the 1s-3d Coulomb repulsion is taken into account in an approximate way. The D 3d distortion has almost no influence on the isotropic pre-edge but can have a large one on the dichroic signal. In the case of Cr in spinel, the trigonal distortion is such small that it does not provide detectable features on the dichroic signal. The major ingredient for the interpretation of the Cr pre-edge features is 3d -3d Coulomb repulsion. This effect is highly multielectronic and complicates the simple interpretation provided by the monoelectronic scheme. This ingredient is mandatory to get correct intensities and energies for both isotropic and dichroic signals.
V. DISCUSSION AND CONCLUSION
On the one hand, monoelectronic calculations allow to make contact between electric dipole and electric quadrupole calculations. They show that electric dipole transitions do not contribute to the features visible in the pre-edge and they provide a clear vision of the assignment of the 1s-3d transitions occuring in the pre-edge. However, they are unable to reproduce quantitatively the linear dichroism in cubic crystals, since the interelectronic repulsion on the 3d levels of the Cr ion cannot fully be described in the LSDA framework. On the other hand, multielectronic calculations well reproduce the angular dependence of the pre-edge in cubic crystals, as well as the isotropic spectrum, with no adjusted parameters. However, in this approach, the main absorption edge, associated to electric dipole transitions, cannot be reproduced since the band structure (or at least the electronic structure of a large enough cluster around the absorbing atom) is not taken into account. The agreement between experiment and multiplet calculations indicates that the assignment of the transitions is no more straightforward, as could have been expected from a more simple atomic monoelectronic picture. Hence, the two approaches are highly complementary.
From this monoelectronic-multielectronic combined approach, our first finding is that the 3d -3d electronic repulsions and the crystal field are the main interactions prevailing at the K pre-edge of Cr in spinel. The multiplet approach seems mandatory to describe quantitatively the K pre-edge of 3d transition ions, and more generally the K-edge spectra of elements for which electronic correlations are significant. The effect of the 3d spin-orbit coupling and of the 1s-3d Coulomb repulsion are very weak: this explains that the monoelectronic approach (which does not fully take into account these interactions) can provide a satisfactory simulation of the isotropic spectrum.
Our second finding concerns the maximum proportion of electric quadrupole transitions in the Cr K pre-edge that can be estimated, with respect to the edge jump (here, normalized to 1): the intensity of the largest peak (5990.75 eV) on the pre-edge isotropic spectrum is less than 2.5 % of the edge jump. From the monoelectronic calculations, we estimate that the non-structured slope from electric dipole origin contributes to about 0.9 % of the edge jump at 5990.75 eV. Thus, the total intensity of the largest pre-edge feature does not exceed 3.5 %.
We can conclude that, if the pre-edge features are more intense than 4 % of the edge jump, pure quadrupole transitions alone cannot explain the origin of the structures. It gives a strong limitation to the often encountered idea that electric quadrupole transitions could explain large pre-edge features. This result, which is consistent with previous studies on Fe 2+ and Fe 3+ in minerals and glasses, 21,22,46 can probably be extended to the other 3d transition ions.
Our final finding concerns the relation between the spectral feature of the pre-edge with the local site distortion of the absorbing ion. The effect of the trigonal distortion does not affect significantly the pre-edge isotropic spectrum. This is a general trend already observed for other related spectra (electric dipole transitions for K-edge and L 2,3 edges of 3d elements). On the contrary, the dichroic signal is much more sensitive. This indicates the possibility to obtain quantitative information on site distortion from the linear dichroic dependence of the pre-edge feature. This can only be recorded if angular dependent measurements on single crystals are performed to yield the full dependence of the absorption signal.
The connection between site distortion and linear dichroism is then made by simulations in the LFM method within the geometrical analysis developed throughout this paper.
VI. APPENDIX
A. Proof of Equation 3
1. General expression of the electric quadrupole cross-section for a cubic crystal
We start from the defining formula of the electric quadrupole cross-section for linearly polarized x-rays:
σ Q (ε,k) = π 2 k 2 α ω f | f |(ε · rk · r|g | 2 δ(E g + ω − E f ), = ijlm ε i ε j k l k m σ iljm ,(17)
where :
σ ijlm = π 2 k 2 α ω f g|r i r l |f f |r j r m |g δ(E i + ω −E f ),
ε andk are the polarization and wave unit vectors, respectively.
To calculate the form of this sum when the sample has a symmetry group G, we use the fact that the absorption cross-section is invariant by any symmetry operation that acts on both the sample variables σ ijlm and the x-ray variablesε,k. Therefore, the cross-section is left invariant by the crystal symmetries applied to the x-ray variables. In other words, for any operation S of the symmetry group G of the sample, we have σ(ε,k) = σ(S(ε), S(k)). Therefore, if G is the symmetry group of the sample, or a subgroup of it, we can write
σ Q (ε,k) = 1 |G| S σ(S(ε), S(k)),(18)
where |G| denotes the number of elements of G.
We rewrite Eq. 17 as
σ Q (ε,k) = i =j ε 2 i k 2 j σ ijij + i =j ε i ε j k i k j σ iijj + i =j ε i ε j k j k i σ ijji + i ε 2 i k 2 i σ iiii + R.(19)
Eq. 19 defines term R. The term R is the sum of the terms that are not (i = j and l = m) or (l = i and m = j) or (m = i and l = j). So R is a sum of 4 terms of the type (i = j and i = l and i = m) plus the three cyclic permutations of (i, j, l, m) and 4 terms of the type (i = j and j = l and l = i) plus the three cyclic permutations of (i, j, l, m). We want to prove Eq. 3 that is valid for a cubic crystal in a reference frame such that x, y, z axis are taken along the fourfold symmetry of the cubic crystal. We first show that if the sample has three perpendicular mirror planes, the term R is zero. Consider the term (i = j and i = l and i = m) and take the symmetry (S(ε i ) = −ε i ) the other variables j, k, l are different from i, so the symmetry leaves them invariant and changes only one sign. Therefore, using Eq. 18, this term disappears from R. The same is true for the three cyclic permutations. Consider now the term (i = j and j = l and l = i). Since the values of the indices is 1, 2 or 3, one of the three indices i, j, k is different from the other two and from m. So one index is again different from the other ones and the same reasoning can be applied to show that the corresponding term vanishes. This holds also for the three cyclic permutations and we have shown that, when there are three perpendicular symmetry planes, the absorption cross-section is
σ Q (ε,k) = i =j ε 2 i k 2 j σ ijij + i =j ε i ε j k i k j σ iijj + i =j ε i ε j k j k i σ ijji + i ε 2 i k 2 i σ iiii .(20)
The group O h has a subgroup made by the six permutations of (x, y, z). An average over this subgroup gives the following result
σ Q cub (ε,k) = i =j ε 2 i k 2 j A + i =j ε i ε j k i k j B + i ε 2 i k 2 i C,(21)
where
A = i =j σ ijij 6 , B = i =j σ iijj + σ ijji 3 , C = i σ iiii 3 .
To complete the proof, we use the fact that i =j
ε 2 i k 2 j =ε ·εk ·k − i ε 2 i k 2 i , i =j ε i ε j k i k j = (ε ·k) 2 − i ε 2 i k 2 i ,
and the identitiesε ·ε = 1,k ·k = 1 andε ·k = 0 to get
σ Q cub (ε,k) = A + i ε 2 i k 2 i (C − A − B).(22)
To compare this result with the expansion over spherical tensors, we need to determine the isotropic contribution σ Q 0 , which is obtained as the average of σ Q cub (ε,k) over angles. We write σ(ε,k) in terms of θ, φ, ψ as in Sec. III A 2.
Thus,
σ Q cub (ε,k) = A + C−A−B 4
sin 2 θ 7 cos 2 θ cos 2 ψ + cos 4φ cos 2 θ cos 2 ψ) + 2 sin 2 2φ sin 2 ψ − cos θ sin 4φ sin 2ψ , and the average over all directions is
σ Q 0 = 1 8π 2 π 0 sin θdθ 2π 0 dφ 2π 0 dψσ Q cub (ε,k) = A + C−A−B 5 . Therefore, σ Q cub (ε,k) = σ Q 0 + i ε 2 i k 2 i − 1 5 (C − A − B) and σ Q cub (ε,k) = σ Q 0 + (ε 2 x k 2 x + ε 2 y k 2 y + ε 2 z k 2 z − 1 5 )σ Q 1 ,(23)
where σ Q 1 = (C − A − B) and k x , k y , k z are the coordinates ofk.
This expression is valid for any cubic crystal, providing that the reference frame is such that the x, y, z axis are taken along the fourfold symmetry axis of the cubic crystal.
Absorption cross-sections in O h symmetry
For a site with O h symmetry, the orthonormal reference frame chosen is that of the cubic crystal. The z -axis of the reference frame is parallel to the fourfold axis of the cube. The angle θ is thus the angle between the polarization vector and the z -axis of the cube.
The cross-section calculated for a single site in O h is equal to the cross-section of the cubic crystal, since a perfect octahedron and the cube have the same symmetry operations:
σ Q O h (ε,k) = σ Q cub (ε,k) and σ D O h (ε) = σ D cub (ε).
For a single site with O h symmetry, the expression of the electric dipole cross-section is very simple:
σ D O h (ε) = σ D (0, 0),(24)
where σ D (0, 0) is the isotropic electric dipole crosssection.
For the cubic crystal, one obtains:
σ D cub (ε) = σ D O h (ε) = σ D (0, 0).(25)
The electric quadrupole absorption cross-section for a site with O h symmetry is given by:
σ Q O h (ε,k) = σ Q O h (0, 0) + 1 √ 14
[35 sin 2 θ cos 2 θ cos 2 ψ + 5 sin 2 θ sin 2 ψ − 4 + 5 sin 2 θ(cos 2 θ cos 2 ψ cos 4 φ − sin 2 ψ cos 4φ) − 2 cos θ sin ψ cos ψ sin 4φ ] σ Q O h (4, 0), (26) where σ Q cub (0, 0) is the isotropic electric quadrupole cross-section, and σ Q cub (4, 0) a purely anisotropic electric quadrupole term.
Using σ Q O h (ε,k) = σ Q cub (ε,k) and (θ = π 2 , φ = π 2 , ψ = π 2 ), Eq. 26 is equivalent to Eq. 23 with:
σ Q 0 = σ Q O h (0, 0) = σ Q cub (0, 0),(27)σ Q 1 = 20 √ 14 σ Q O h (4, 0) = 20 √ 14 σ Q cub (4, 0).(28)
Eq. 23 can be rewritten as:
σ Q cub (ε,k) = σ Q cub (0, 0) + 20 √ 14 (ε 2 x k 2 x + ε 2 y k 2 y + ε 2 z k 2 z − 1 5 )σ Q cub (4, 0).(29)
This is the proof of Eq. 3.
B. Expression ofε(αrot) andk (αrot)
For the sample cut and the experimental setup used in this study, we have:
ε(α rot ) = 1−cos αrot 2 1+cos αrot 2 sin αrot √ 2 , k(α rot ) = −1−cos αrot 2 −1+cos αrot 2 sin αrot √ 2 .
C. Symmetry adapted method used in monoelectronic calculations
Firstly, the absorption cross-section was calculated in the electric dipole approximation, in order to derive σ D cub (0, 0), the isotropic electric dipole cross-section. This term can be obtained from a single calculation of the electric dipole absorption cross-section performed for site A. The expression of the electric dipole cross-section in D 3d , σ D A , is given by Eq. 4, where θ is the angle between the polarization vector and the C 3 axis. The program we used for the ab initio calculations calculates the average of σ D A (ε) forε along the x -, y-and z -axis of the cubic frame. The angle θ between the [111] direction (parallel to the C 3 axis of the site) and each of these three directions, is arccos 1 √ 3 . This implies that, for a polarization vectorε taken along the x -, y-or z -axis,
σ D cub (ε) = σ D A (ε) = σ D D 3d (0, 0) = σ D cub (0, 0).(30)
This means that the average value calculated by the program is directly equal to σ D cub (0, 0). Hence, σ D cub (0, 0) was obtained from a single calculation performed at site A. Secondly, the calculation was performed in the electric quadrupole approximation, in order to derive σ Q cub (0, 0) and σ Q cub (4, 0). Once these two terms are determined, we will be able to calculate the electric quadrupole absorption cross-section for the cubic crystal, according to Eq. 3, for any (ε,k) configuration. We used a symmetry adapted method to determine σ Q cub (0, 0) and σ Q cub (4, 0) in order to reduce the number of calculations: this way, it is possible to consider one single substitutional site (site A) and take advantage of the symmetry properties of the crystal. This method allows to save significant computational time, since we perform only two self-consistent calculations (instead of eight with the brute force method): one calculation to do the structural relaxation of the system (substituted at site A), and a second one to calculate the charge density with a core-hole on Cr. As mentioned in Ref. 24, assuming that we have calculated the spectrum for a given site X, it is possible to obtain the spectrum for any site Y equivalent to X by calculating the spectrum of site X for a rotated x-ray beam. More precisely, in the case of electric quadrupole transitions, if site Y is the image of site X by a rotation R, the spectrum of site Y for a configuration (ε,k) is equal to the spectrum of site X for the rotated configuration (R −1 (ε),R −1 (k)). As in Ref. 24, let us consider the site with reduced coordinates (0, 1/4, 3/4), with is a representative of site A. By applying the three rotations about the z -axis of the crystal through angles of π/2, π, 3π/2, we obtain the positions of three sites, which are respectively representative of sites B, D and C. The three rotations will denoted as R π/2 , R π and R 3π/2 . This implies that
σ Q B (ε,k) = σ Q A (R −1 π/2 (ε), R −1 π/2 (k)),(31)σ Q C (ε,k) = σ Q A (R −1 3π/2 (ε), R −1 3π/2 (k)),(32)σ Q D (ε,k) = σ Q A (R −1 π (ε), R −1 π (k)).(33)
We shall now apply the previous equations to the case of the two experimental configurations, α rot = 0 • and α rot = 90 • , in order to see if additional simplifications can be found. To do so, we need to consider the expression of the electric quadrupole transition operatorQ:
Q = i 2ε · rk · r.(34)
Note that, in text, the assignment of the calculated monoelectronic transitions is discussed usingÔ =ε · rk · r.
We have to consider the absolute value ofÔ (orQ) because the cross-section does not depend on the sign of O.
For α rot = 0 • (ε = [010],k = [100]):
|ε · rk · r| = |R −1 π/2 (ε) · rR −1 π/2 (k) · r| = |R −1 π (ε) · rR −1 π (k) · r| = |R −1 3π/2 (ε) · rR −1 π/2 (k) · r| = xy. Expression of the electric quadrupole cross-section calculated in D 3d group for the four independent orientations used to derive σ Q D 3d (0, 0), σ Q D 3d (2, 0), σ Q D 3d (4, 0) and σ Q D 3d (4, 3) (see text). The coordinates ofε andk are given in the reference frame chosen for D 3d , with the z -direction parallel to the C3 axis.
label θ φ ψεk σ Q D 3d (ε,k) s1 arccos( 1 √ 3 ) − 2π 3 π 4 (-1 √ 6 , -1 √ 2 , 1 √ 3 ) ( 1 √ 6 ,-1 √ 2 ,-1 √ 3 ) σ Q D 3d (0,0)+ √ 14 9 σ Q D 3d (4, 0)+ 4 √ 5 9 σ Q D 3d (4,3) s2 arccos( 1 √ 3 + 1 √ 6 ) 0 0 ( −1+ √ 2 √ 6 ,0, 1+ √ 2 √ 6 ) ( 1+ √ 2 √ 6 ,0, 1− √ 2 √ 6 ) σ Q D 3d (0,0)- q 5 14 σ Q D 3d (2, 0)-109 36 √ 14 σ Q D 3d (4,0)-2 √ 5 9 σ Q D 3d (4,3) s3 π 2 π 2 π 2 (0,1,0) (-1,0,0) σ Q D 3d (0,0)+ q 10 7 σ Q D 3d (2, 0)+ 1 √ 14 σ Q D 3d (4,0) s4 3π 4 π 2 π (0, 1 √ 2 ,-1 √ 2 ) (0, 1 √ 2 , 1 √ 2 ) σ Q D 3d (0,0)- q 5 14 σ Q D 3d (2, 0)+ 19 4 √ 14 σ Q D 3d (4,0)
Hence, we have:
σ Q A (ε,k) = σ Q A (R −1 π/2 (ε), R −1 π/2 (k)) = σ Q A (R −1 π (ε), R −1 π (k)) = σ Q A (R −1 3π/2 (ε), R −1 3π/2 (k)).
This means that:
σ Q A (ε,k) = σ Q B (ε,k) = σ Q C (ε,k) = σ Q D (ε,k).(35)
For α rot = 0 • , we thus need to perform one single calculation of the electric quadrupole cross-section, since
σ Q cub (ε,k) = σ Q A (ε,k).(36)
For α rot = 90
• (ε = [ 1 2 , 1 2 , 1 √ 2 ] andk = [-1 2 ,-1 2 , 1 √ 2 ]):
|ε · rk · r| = |R −1 π (ε) · rR −1 π (k) · r| = |z 2 /2 − (x − y) 2 /4| and |R −1 π/2 (ε) · rR −1 π/2 (k) · r| = |R −1 3π/2 (ε) · rR −1 π/2 (k) · r| = |z 2 /2 − (x + y) 2 /4|. This means that:
σ Q A (ε,k) = σ Q D (ε,k),(37)σ Q B (ε,k) = σ Q C (ε,k).(38)
For α rot = 90 • , we thus need to perform two calculations of the electric quadrupole cross-sections since
σ Q cub (ε,k) = σ Q A (ε,k) + σ Q C (ε,k) 2(39)
As mentioned previously, instead of doing the calculation for the two sites A and C, it is more convenient to compute the spectrum for site A, for the two orientations (ε,k) and (R −1 As mentioned in Sec. III A 2 (Eq. 5), one needs first to determine σ Q D 3d (0, 0), σ Q D 3d (2, 0), σ Q D 3d (4, 0) and σ Q D 3d (4,3), in order to determine σ Q cub (0, 0) and σ Q cub (4, 0) using Eqs 8 and 9. This is done by performing four multiplet calculations, which provide four independent values of the electric quadrupole cross-section, s1, s2, s3 and s4, whereε andk are defined in Table II. The components σ Q D 3d (0, 0), σ Q D 3d (2, 0), σ Q D 3d (4, 0) and σ Q D 3d (4, 3) are obtained by a combination of s1, s2, s3 and s4, according to:
−3π 2 (ε),R −1 −3π 2 (k)σ Q D 3d (0, 0) = 1 5 s1 + 2 5 s2 + 4 15 s3 + 2 15 s4, σ Q D 3d (2, 0) = − 1 √ 70 s1 − 2 35 s2 + 5 14 s3 − 2 35 s4, σ Q D 3d (4, 0) = − 2 √ 14 35 s1 − 4 √ 14 35
s2
+ 2 √ 14 105 s3 + 16 √ 14 105
s4,
σ Q D 3d (4, 3) = 2 √ 5 s1 − 1 2 √ 5 s2 − 2 3 √ 5 s3 − √ 5 6 s4.(40)
These equations have been obtained by inversing the system of equations, which give the expressions of s1, s2, s3 and s4, in function of σ Q D 3d (0, 0), σ Q D 3d (2, 0), σ Q D 3d (4, 0) and σ Q D 3d (4, 3) (Table II). Once this first step has been performed, Eq. 8 and 9 are used to derive σ Q cub (0, 0) and σ Q cub (4, 0). The electric quadrupole cross-section of the cubic crystal can then be calculated for any experimental configuration using Eq. 11.
E. Expression of the d -eigenstates in D 3d
The d -eigenstates in D 3d point group are determined by the branching rules of the irreducible representation 2 + (O 3 ) in D 3d . In order to get the complete eigenstates, we must consider the O 3 ⊃ O h ⊃ D 3d ⊃ C 3i subduction. To simplify the notation, we will make no use of
SO3 → O → D3 → C3 |k(SO3)ρ(O)σ(D3)λ(C3) 2 →1 → 1 → 1 |e+(t2) 2 →1 → 1 → −1 |e−(t2) 2 →1 → 0 → 0 |a1(t2) 2 → 2 → 1 → 1 |e+(e) 2 → 2 → 1 → −1 |e−(e)
the parity (± or g/u) in the rest of the appendix. Hence, we will use the subduction SO 3 ⊃ O ⊃ D 3 ⊃ C 3 . In the following, the irreducible representations are labeled according to Ref. 26. For example, in O group, the irreducible representation1 designates T 2 in Schönflies notation, while 2 designates E. The complete eigenstates are written as |k(SO 3 )ρ(O)σ(D 3 )λ(C 3 ) 3 , where λ is the irreducible representation of C 3 subgroup, coming from the k irreducible representation of SO 3 , that becomes ρ in O, σ in D 3 and λ in C 3 . The branching rules for k= 2 are given in Tab. III.
We recall that in D 3 (or C 3 ), the reference frame chosen is not the one used in O.
Therefore, we need to express the |k(
|2211 3 = − 1 √ 2 |2200 4 + i 1 √ 2 |2222 4 |221−1 3 = − 1 √ 2 |2200 4 − i 1 √ 2 |2222 4 |2100 3 = (1 + i) 1 √ 6 |2111 4 + (1 − i) 1 √ 6 |211−1 4 − i √ 3 |2122 4 |2111 3 = (1 + i)( 1 2 √ 2 − 1 2 √ 6 )|2111 4 + (−1 + i)( 1 2 √ 2 + 1 2 √ 6 )|211−1 4 − i √ 3 |2122 4 |211−1 3 = −(1 + i)( 1 2 √ 2 + 1 2 √ 6 )|2111 4 + (1 − i)( 1 2 √ 2 − 1 2 √ 6 )|211−1 4 − i √ 3 |2122 4(41)
If we now express the |k(SO 3 )ρ(O)σ(D 4 )λ(C 4 ) 4 as |JM partners (Ref. 26 p. 527), we obtain:
|2200 4 = −|20 |2222 4 = − 1 √ 2 |22 − 1 √ 2 |2−2 |2111 4 = −|21 |211−1 4 = |2−1 |2122 4 = 1 √ 2 |22 − 1 √ 2 |2−2(42)
In O, the d orbitals are expressed as:
d xy = i √ 2 (|2−2 − |22 ) d yz = i √ 2 (|2−1 + |21 ) d xz = 1 √ 2 (|2−1 − |21 ) d 3z 2 −r 2 = |20 d x 2 −y 2 = 1 √ 2 (|22 + |2−2 )(43)3 + i 2 )d xz + ( 1 2 √ 3 + i 2 )d yz e + (t 2 ) = |211−1 3 = 1 √ 3 d xy − ( 1 2 √ 3 + i 2 )d xz + ( 1 2 √ 3 − i 2 )d yz(44)
In D 3 , (e + (t 2 ), e − (t 2 )) is basis of the irreducible representation e(t 2 ). Similarly, (e + (e) , e + (e)) is a basis of the irreducible representation e(e). In D 3 , a mixing is thus possible between the functions belonging to the two e irreducible representations, originating from the e and t 2 levels in O.
F. Definition of the crystal-field parameters used in the LFM calculations
In SO 3 symmetry, the crystal-field Hamiltonian can be written as a combination of the q components of unit tensors U (k) with rank k. Each tensor U (k) is associated to the k irreducible representation of SO 3 . For the Cr 3+
SO3 → O → D3 → C3 X k(SO 3 )ρ(O)σ(D 3 )λ(C 3 ) 4 → 0 → 0 → 0 X 4000 4 →1 → 0 → 0 X 4100 2 →1 → 0 → 0 X 2100
ion, k = 0, 2 or 4, the term k = 0 contributing only to the average energy of the configuration. Again, we will not make use of the parity (± or g/u). To study the Cr 3+ ion in trigonal symmetry D 3 , we need to consider the subduction SO 3 ⊃ O ⊃ D 3 ⊃ C 3 . The crystal-field Hamiltonian is expressed as:
H cc = k=2,4 X k(SO3)ρ(O)σ(D3)λ(C3) U k(SO3)ρ(O)σ(D3)λ(C3) .
The unit tensor U k(SO3)ρ(O)σ(D3)λ(C3) is related to the λ irreducible representation of C 3 subgroup, coming from the k irreducible representation of SO 3 , that becomes ρ in O, σ in D 3 and λ in C 3 . The terms X k(SO3)ρ(O)σ(D3)λ(C3) are the crystal-field parameters used in the LFM code. Their definition, in function of (D σ , D τ , D q ) or (ν, ν', D q '), are given in Appendix G. The branching rules which give 0 as irreducible representation in D 3 are summarized in Tab. IV. This implies that :
H cc = X 4000 U 4000 + X 4100 U 4100 + X 2100 U 2100 . (45) Using the expression of the d -eigenstates in C 3 given in value of X 4100 and X 2100 were set to zero, while the value of X 4000 was fixed to the value used in D 3d symmetry. Things can be simplified by defining D ′ q , so that it contains also the contribution of D τ to the cubic field. This leads to the definition of two other distortion parameters, ν and ν ′ .
e ± (t 2 )|H ′ trig |e ± (t 2 ) = − 1 3 ν, a 1 (t 2 )|H ′ trig |a 1 (t 2 ) = 2 3
ν, e ± (t 2 )|H ′ trig |e ± (e) = ν ′ .
FIG. 1 :
1(Color online) Cubic cell of the spinel structure and experimental setup. The four equivalent octahedral sites are labeled according to the coordinates given in
The figure corresponds to the experimental setup taken as starting point (αrot = 0 • ). For this configuration, the [001] axis of the cube is in the vertical plane, perpendicular tô ε = [010] andk = [100].
FIG. 2 :
2(Color online) Pre-edge spectra calculated for site A, B, C and D in the electric quadrupole approximation. The orange lines present the spectra calculated for αrot = 0 • (ε = [010],k = [100]), while the black lines are the spectra calculated for αrot = 90 •
FIG. 3 :
3(Color online) Comparison between experimental (dotted line) and calculated (solid line) isotropic XANES spectra at the Cr K-edge in spinel. The calculation was performed using DFT-LSDA (see Sec. III B1). The inset presents the spectra in the pre-edge region. The dashed orange line is the calculated electric dipole contribution.
FIG. 4 :
4(Color online) Comparison between experimental (dotted line) and calculated (solid line) Cr K-pre edge spectra in spinel, using DFT-LSDA. The orange lines correspond tô ε =[010],k = [100] (αrot=0 • ). The black lines correspond
FIG. 5 :
5(Color online) Electric quadrupole transitions calculated for site A, using spin-polarization. The upper panel (a) presents the spectrum calculated for (ε = [ ). The middle panel (b) presents the spectrum calculated for αrot = 0 • (ε = [010],k = [100]). The lower panel (c) presents the spectrum calculated for αrot = 90 • (ε = [ 1 2 , 1 2 , 1 √ 2 ],k = [-1 2 ,-1 2 , 1 √ 2 ]). For each configuration, the respective contributions of the two spins are plotted (black solid line for spin down, orange solid line for spin up), as well as the sum (dashed line). The Fermi level is located approximately around 5990 eV.
FIG. 6 :2
6(Color online) Comparison between experimental (dotted line) and calculated (solid line) Cr K pre-edge spectra in spinel, using Ligand Field Multiplet theory in D 3d symmetry. The orange lines correspond toε = [010],k = [100] (αrot = 0 • ). The black lines correspond toε = [ ] (αrot = 90 • ). The dichroic signal is the difference between the black and orange lines.
FIG. 7 :
7(Color online) Comparison between calculated Cr K pre-edge spectra in spinel in D 3d symmetry for increasing trigonal distortion. The orange solid line, labeled O h corresponds to a perfect O h symmetry (Dσ = 0, Dτ = 0). The black dotted line,labeled D 3d corresponds to (Dσ = -0.036 eV, Dτ = 0.089 eV), which are the distortion parameters obtained from Optical Absorption Spectroscopy. 42 The black solid line, labeled D 3d -2, was obtained for a doubled distortion parameters (Dσ = -0.072 eV, Dτ = 0.178 eV), and the orange dotted line, labeled D 3d -3 for tripled distortion parameters (Dσ = -0.108 eV, Dτ = 0.267 eV)
SO 3 )ρ(O)σ(D 3 )λ(C 3 ) 3 in function of |k(SO 3 )ρ(O)σ(D 4 )λ(C 4 ) 4 , where |k(SO 3 )ρ(O)σ(D 4 )λ(C 4 ) 4 are determined using the SO 3 ⊃ O ⊃ D 4 ⊃ C 4 subduction.To do so, we use the relations given for k = 2 in Ref.26 (p. 549):
Combining
Eqs 41, 42 and 43, we obtain the expression of the d -functions, which are basis of the irreducible representations in C 3 , in function of the d -orbitals in O: e + (e) = |2211 3
TABLE I :
ICoordinates of Cr atom and direction of site distortion for the four equivalent substitutional sites belonging to the rhombohedral unit cell. We also give the coordinates of the twelve other sites, obtained from the previous by the three translations of the fcc lattice (see text andFig.1).site identification
direction of site distortion
Cr-coordinates in rhombohedral unit cell
Cr-coordinates in cubic cell
A
[111]
( 1
2 , 1
4 , 1
4 )
( 1
2 , 3
4 , 3
4 ) , (0, 3
4 , 1
4 ) , (0, 1
4 , 3
4 )
B
[111]
TABLE II :
II
). This corresponds to (ε= [ 1
2 , 1
2 , 1
√
2 ],k = [-1
2 ,-1
2 , 1
√
2 ]) and (ε = [-1
2 , 1
2 , 1
√
2 ],k =
[ 1
2 ,-1
2 , 1
√
2 ]), respectively.
D. Method used to perform the multiplet
calculations
TABLE III :
IIIBranching rules for k = 2 using the SO3 ⊃ O ⊃ D3 ⊃ C3 subduction
TABLE IV :
IVBranching rules giving 0 as irreducible representation in D3
2. (ν, ν ′ , D ′ q )parameter set The crystal-field Hamiltoninan H cc is now defined as H cc = H ′ cub +H ′ trig , where H ′ cub contains the contribution of D τ . According to Ref. 47 (Eq. 3.90), we have:
AcknowledgmentsThe authors are very grateful to M. Sikora (ID26 beamline) for help during experiment, and to F. Mauri, M. Calandra, M. Lazzeri and C. Gougoussis for fruitful discussions. The theoretical part of this work was supported by the French CNRS computational Institut of Orsay (Institut du Développement et de Recherche en Informatique Scientifique) under projects 62015 and 72015. This is IPGP Contribution n • XXXX.We obtain the following equations:G. Relations between the crystal-field parameters used in LFM calculations and those derived from optical absorption spectroscopyAs mentioned in Ref.47, two equivalent parameter sets are available in optical absorption spectrocopy to describe the crystal-field in trigonal symmetry: (D σ , D τ , D q ) and (ν, ν', D ′ q ). In the following, we make connexion between the two sets.If the cubic term H cub is added, we have:Combining Eqs 46 and 48, we obtain:X 4000 contains only the cubic part of the crystal field, although D τ appears in its expression. This means that the trigonal distortion, via D τ , contributes to the e − t 2 splitting of the d -orbitals. On the contrary, X 4100 and X 2100 are entirely due to the trigonal distortion. Hence, when we investigated the effect of the trigonal distortion on the XANES spectra in the LFM calculations, the If we add the cubic term H ′ cub , which is here defined so that the center of gravity is maintained for the trigonally split t 2g orbitals, the following equations are obtained:Combining Eqs 46 and 51, we have:Relations between the two parameter setsCombining Eqs 49 and 52, we obtain the relations given in Ref.47 (Eq. 3.91):
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|
[] |
[
"Adaptive transmission for radar arrays using Weiss-Weinstein bounds",
"Adaptive transmission for radar arrays using Weiss-Weinstein bounds"
] |
[
"Christian Greiff ",
"David Mateos-Núñez ",
"María A González-Huici ",
"Stefan Brüggenwirth "
] |
[] |
[] |
We present an algorithm for adaptive selection of pulse repetition frequency or antenna activations for Doppler and DoA estimation. The adaptation is performed sequentially using a Bayesian filter, responsible for updating the belief on parameters, and a controller, responsible for selecting transmission variables for the next measurement by optimizing a prediction of the estimation error. This selection optimizes the Weiss-Weinstein bound for a multi-dimensional frequency estimation model based on array measurements of a narrow-band far-field source. A particle filter implements the update of the posterior distribution after each new measurement is taken, and this posterior is further approximated by a Gaussian or a uniform distribution for which computationally fast expressions of the Weiss-Weinstein bound are analytically derived. We characterize the controller's optimal choices in terms of SNR and variance of the current belief, discussing their properties in terms of the ambiguity function and comparing them with optimal choices of other Weiss-Weinstein bound constructions in the literature. The resulting algorithms are analyzed in simulations where we showcase a practically feasible real-time evaluation based on look-up tables or small neural networks trained off-line.
|
10.1049/iet-rsn.2018.5253
|
[
"https://arxiv.org/pdf/1806.06641v2.pdf"
] | 116,253,182 |
1806.06641
|
4cccdcb006197b4d6db77b8b38f60b1f88f8207f
|
Adaptive transmission for radar arrays using Weiss-Weinstein bounds
Christian Greiff
David Mateos-Núñez
María A González-Huici
Stefan Brüggenwirth
Adaptive transmission for radar arrays using Weiss-Weinstein bounds
1 This paper is a preprint of a paper accepted by IET Radar, Sonar & Navigation and is subject to Institution of Engineering and Technology Copyright. When the final version is published, the copy of record will be available at the IET Digital Library
We present an algorithm for adaptive selection of pulse repetition frequency or antenna activations for Doppler and DoA estimation. The adaptation is performed sequentially using a Bayesian filter, responsible for updating the belief on parameters, and a controller, responsible for selecting transmission variables for the next measurement by optimizing a prediction of the estimation error. This selection optimizes the Weiss-Weinstein bound for a multi-dimensional frequency estimation model based on array measurements of a narrow-band far-field source. A particle filter implements the update of the posterior distribution after each new measurement is taken, and this posterior is further approximated by a Gaussian or a uniform distribution for which computationally fast expressions of the Weiss-Weinstein bound are analytically derived. We characterize the controller's optimal choices in terms of SNR and variance of the current belief, discussing their properties in terms of the ambiguity function and comparing them with optimal choices of other Weiss-Weinstein bound constructions in the literature. The resulting algorithms are analyzed in simulations where we showcase a practically feasible real-time evaluation based on look-up tables or small neural networks trained off-line.
I. INTRODUCTION
Software-defined radar systems offer degrees of freedom such as waveform diversity, beam-steering, or antenna selection. The concept of Cognitive Radar [1] describes a dynamic systems approach for the control of these degrees of freedom in a real-time, closed-loop fashion, motivating research on waveform design [2]- [4], matched illumination [5], [6], radar resource management [7], [8], or spectral coexistence [9], [10]. Application areas include multi-functional active electronically scanned array (AESA) systems in airborne or maritime scenarios [11], [12], automotive multiple-input multiple-output (MIMO)-radars [13]- [15] or distributed sensor networks [16].
The perception-action cycle [17], [18] describes a sequential process of extracting information from a scene and using that knowledge for adapting the transmission and processing of subsequent measurements. These are tasks of estimation and control that can be formulated in a Bayesian estimation framework [19], [1]. Sequential estimation is performed by updating a belief distribution over the parameter of interest according to motion and measurement models, which is suitable for tracking and Track-Before-Detect approaches. Practical implementations such as Particle Filters [20] or Cubature Kalman Filters [1] can handle nonlinear models crucial for radar systems. On the control side, the estimation performance can be predicted and optimized using Bayesian
The authors are with the Department of Cognitive Radar, Fraunhofer FHR, Wachtberg, Germany. Email: [email protected] bounds [19] that provide a lower bound on the expectation over the current belief of the covariance matrix of the error of any estimator. Classical examples include the Bayesian Cramer-Rao bound (BCRB), and other members of the family of Weiss-Weinstein bounds (WWB) [21], [22], including the Bobrovsky-Zakaï bound (BZB). The latter, together with the Ziv-Zakai bound (ZZB) [23], take into account estimation errors in nonlinear estimation problems that occur below a certain signal-to-noise ratio (SNR) due to sidelobes in the ambiguity function [24]. These are underestimated by the Cramér-Rao bound (CRB), which is related to the mainlobewidth and therefore measures the accuracy and resolution limit when the SNR is large. Instances of such nonlinear estimation problems include frequency estimation in radar systems such as the estimation of Doppler frequency and direction of arrival (DoA). These scenarios have been the context for adaptive strategies for the selection of pulse repetition frequency [25] using the BCRB, and for antenna selection using the BZB [14], [15] and Weiss-Weinstein bound (WWB) [26].
The optimization metrics prescribed by the WWB are themselves an optimization over so-called test points of an expression that contains integrals of the measurement and prior distributions. These probability models affect the characterization of optimal sensing parameters, and also the existence of analytical expressions for the integrals. In the context of sequential estimation of a dynamic Markovian parameter, these metrics can be constructed to bound the Bayesian mean squared error (BMSE) given all previous measurements. In this case, the motion and measurement updates can be embedded in a sequential computation of the WWB under an ample class of dynamics [27], resulting in explicit formulas for some families of prior and measurement distributions [28]. Alternatively, the works [13], [14], [29] show that under exact posteriors, the Bayesian bound becomes too a conditional bound given previous measurements, and suggest approximating these posteriors with a particle filter or the Metropolis-Hastings algorithm. In this work we use a particle filter to update the posterior, that is further approximated by a combination of Gaussian and uniform distributions for the selection of sensing parameters using the WWB. For this purpose and those priors, we have derived the WWB for frequency estimation using array measurement models for a single source with known SNR but random initial phase, with a rigorous derivation of testpoint domains. This model can be particularized to azimuth and elevation estimation, azimuth and Doppler estimation in time-division multiplexing (TDM) MIMO. Modeling-wise, in the case of DoA or Doppler estimation, we characterize the WWB-based optimal selection of a scaling parameter that models the carrier frequency or the pulse repetition interval (PRI), in terms of the field of view (FoV) or variance of the prior density, and compare the benefits of the proposed model with alternative WWB constructions called conditional, which we refer to as known-phase, and unconditional signal models [30], demonstrating the influence of modeling the initial phase as random while regarding the SNR as deterministic and known. Computationally, this formulation has the advantage of fast, vectorized evaluation over test-points thanks to explicit formulas without needing to evaluate the inverse of a matrix. We then show in simulations the closed-loop performance of the particle filter combined with the above criteria using feedback on the variance of the posterior for adaptation of pulse repetition frequency (PRF) or array scaling, and for antenna selection.
Organization: The paper is organized as follows: Section II describes the adaptive sensing algorithm for sequential Bayesian estimation. Section III presents a derivation of the WWB for general array processing tasks for a single source of known SNR under a spatio-temporal sampling scheme with a random initial phase associated to the coherent processing interval. Section IV includes an analysis of the controller choices for several WWB models and analyzes the consequences of assuming knowledge of the initial phase or lack thereof. Section V applies the general framework to the problem of adaptive PRF or array scaling for Doppler or DoA estimation, and to the problem of channel selection for DoA estimation. The resulting adaptive policies are compared in simulations, exemplifying the practical implementation of our strategies with the use of look-up tables and neural nets. Section VI discusses our conclusions and ideas for future work, and we include Appendices with auxiliary results.
Notational conventions: R N and C N denote the Ndimensional real and complex vector spaces, respectively. The real part of a complex number z ∈ C is Re{z}, while |z| stands for the absolute value. Likewise, the Euclidean volume for sets Θ ⊂ R q is denoted by |Θ| = χ Θ (θ)dθ, where χ Θ is the indicator function. For a symmetric real or Hermitian complex matrix A, the induced norm is
x A := √
x H Ax where x H is the conjugated transpose of vector x. The weighted trace is defined as trace ρ (A) := i ρ i A ii . We denote by 1 N ∈ R N the vector of ones and I N ∈ R N ×N the identity matrix. For functions, p, f : R q → R, we define the expectation of f with respect to the density p as E p(θ) [f ] = f (θ)p(θ)dθ. Bracketed integer superscripts serve as abbreviation for a collection of variables, e.g. x (k) = {x 1 , . . . , x k }. We adhere to the convention that lower case letters denote scalars (e.g. γ ∈ R), lower case boldface letter denote vectors (e.g. d ∈ R N ) and upper case boldface letters denote matrices (e.g. H ∈ R q×M ). We frequently use a Matlab inspired notation for vector evaluations of a scalar function, e.g. for a vector d = (d n ) N n=1 ∈ R N , the expression e id = (e idn ) N n=1 ∈ C N also denotes a vector.
II. BAYESIAN ADAPTIVE SENSING FOR SEQUENTIAL
ESTIMATION
Here we describe a framework for sequential adaptive sensing using as feedback a belief distribution of the parameter of interest. The control system comprises a processor, which uses a Bayesian filter to incorporate information from measurements about a parameter of interest into the belief distribution, and a controller, which is a rule for selecting the transmission variables for the next measurement using feedback on the current knowledge. Next we describe the processor and the controller.
A. Processor
The processor is in charge of incorporating information from the latest measurement into the belief distribution of the parameter of interest. Consider a parameter vector that at time k is modeled by the random vector θ k ∈ R N . To relate the measurement x k at step k with the parameter θ k , we need a measurement model, p(x k | θ k , g k ), that depends on the sensing parameters g k used in that measurement. In Section III, we substantiate this model focusing on multidimensional frequency estimation of a single complex sinusoid with additive Gaussian white noise where the sensing parameters are sampling schemes in time and space. We keep this section general for suitable distributions p(x k | θ k , g k ).
To model the evolution between measurement steps of the parameter being estimated, we assume a Markovian transition (or state evolution) model of the form p(θ k | θ k−1 , g k ). Note that in general there can be a dependence on the sensing parameter, e.g., if the latter specifies the time of the measurement at step k. This transition probability is assumed known.
A Bayesian filter proceeds in two steps: the motion update (or prediction) and the measurement update (or filtering). The initial belief distribution for the parameter, denoted by p(θ 0 ) = p 0 (θ 0 ) is a modeling choice.
Motion update: The motion update predicts the state of the parameter at the time of the next measurement using the model for state evolution. Suppose that after measurement step k −1, we have a belief given by p + k−1 (θ k−1 ). Then the motion update of the belief distribution is given by the Chapman-Kolmogorov equation [20],
p − k (θ k ) := p(θ k | θ k−1 , g k ) p + k−1 (θ k−1 )dθ k−1 . (1)
This probability is employed as predicted prior belief at step k. Measurement update: The measurement update filters the prediction using the likelihood of the measurement,
p + k (θ k ) := c p(x k | θ k , g k ) p − k (θ k )(2)
with c chosen so that p + k is a probability density over values of θ k , and p + 0 (θ 0 ) := p 0 (θ 0 ). The recurrences (1) and (2) have properties that can depend on the policy for sensing parameters at the Controller. [18], [26], where we consider in this work the optional choice of training a neural network for WWB ranking of candidate sensing parameters.
B. Controller
The controller is in charge of selecting sensing parameters for the next measurement using as input the current belief distribution. That is, at step k it takes as input the posterior of the last step p + k−1 (θ k−1 ) before the motion update, or approximation thereof, and returns g k . Any criterion to make this selection should employ the state evolution model p(θ k | θ k−1 , g k ) and the observation model according to candidate sensing parameters p(x k | θ k , g k ). The criterion used in this paper is a tight lower bound on the BMSE. For the data model p(x, θ), the Bayesian Mean Squared Error (BMSE) of an estimatorθ ≡θ(x) of θ is defined as the Bayesian covariance matrix of the error e :=θ(x)−θ, i.e.,
BMSE(θ; p(x, θ)) := E p(x,θ) (θ(x) − θ)(θ(x) − θ) T .(3)
The BMSE in (3) can be used as optimization metric for adaptive sensing [31], but is expensive to compute because it involves Monte Carlo integrals over the parameter space and over realizations of the observation. Instead, we follow the common practice of replacing the BMSE by one of its lower bounds, e.g. the Weiss-Weinstein bound (WWB). The WWB provides a lower bound on the BMSE of any estimator and thus gives an indication of the achievable estimation performance. Formally, the bound is obtained from a covariance inequality in the sense of the Loewner order on positive semi-definite matrices [19, p. 333] as
WWB(H; p(x, θ)) BMSE(θ; p(x, θ)),(4)
where WWB(H; p(x, θ)) ∈ R q×q is a member of the family of WWBs parametrized by the test point matrix H for a data model p(x, θ) as described in the Appendix A. Depending on the estimation task, we are interested in the contribution to the BMSE of a subset of coordinates, and thus we define the following objective function for candidate sensing parameters,
C k (g) := sup H trace ρ (WWB(H; p(x k | θ k , g)p − k (θ k ))),(5)
where the predicted prior p − k (θ k ) ≡ p − k (θ k ; g) depends on g and is obtained from the input p + k−1 (θ k−1 ) through the motion update (1). The weighting vector ρ ∈ R q ≥0 can be used to balance units or weight the components of θ. Optimization over test points H is performed to obtain the tightest bound within the parametric family of bounds. The sensing parameters are then found as
g k = arg min g C k (g).(6)
This selection requires a double optimization procedure, first over test point matrix H, to evaluate the prediction of the BMSE, and then over sensing parameters. The former is non-convex and we use a global optimization algorithm (e.g. simulated annealing [32]). A visualization of the closed-loop between the processor and the controller is depicted in Fig. 1. Algorithm 1 summarizes the steps.
Algorithm 1 (Adaptive selection of sensing parameters).
Input: Initial belief distribution p + 0 (θ); Measurement likelihood model p(x | θ, g); State evolution model p(θ k |θ k−1 )
Output: Belief distribution p + k (θ k |g k ) Procedure: Set k = 1. 1. Motion update of belief distribution via (1) to obtain p − k (θ k )
2. The controller finds "optimal" sensing parameters g k by minimizing the cost function g k = arg min g C k (g)
3. Measurement is performed, yielding observation x k 4. The processor performs the measurement update of belief distribution via (2) to obtain the posterior p + k (θ k ) 5. Start next cycle with p + k (θ k ) as new initial belief by increasing k ← k + 1 and repeating from step 1 Remark 1 (Dependence of motion model on sensing parameter ). Note that if the motion update depends on the sensing parameter (e.g., if it refers to the time of measurement), then the controller has to perform the motion update in (1) for each evaluation of the cost function C k (g). For the special case of a g-independent state evolution model, the motion update in (1) needs to be performed only once before passing the resulting prediction to the controller.
C. Legitimation of the closed-loop
The closed-loop formed by the recurrences (1) and (2) and the selection of sensing parameters (6) governs the evolution of the belief distribution of the parameter. Under this evolution, the cost function is a lower bound of the BMSE conditional to previous measurements x (k) := {x 1 , . . . , x k } [13], [14], [29].
Proposition 1 (Properties of the closed-loop). The following relations hold for the closed-loop system formed by the Processor updates (1) and (2), and the Controller selection (6).
(i) The motion and measurement updates satisfy
p − k (θ k ) = p(θ k | x (k−1) , g (k) ) (7a) p + k (θ k ) = p(θ k | x (k) , g (k) ) (7b) c = 1 p(x k | x (k−1) , g (k) ) ,(7c)
i.e., p + k (θ k ) is the posterior belief at step k conditioned to all measurements x (k) and sensing parameters g (k) .
(ii) The WWB in the cost function (5) satisfies the inequality
WWB(H; p(x k | θ k , g)p − k (θ k )) BMSE(θ k , p(x k , θ k | x (k−1) , g (k−1) , g)),
whereθ k is any estimator based on x (k) , g (k−1) , g. Similarly for the corresponding inequality taking trace ρ on both sides.
This result is proved in Appendix D. Relations (7) are what we would expect without selection of sensing parameters using previous measurements. Next we derive the WWB for a frequency estimation model based on array measurements.
III. STATISTICAL MODEL FOR MULTI-DIMENSIONAL FREQUENCY ESTIMATION FOR RANDOM INITIAL PHASE
In this section we derive the statistical performance bound based on the WWB for a family of array processing models under Gaussian and independent uniform priors. This metric can be used both for adaptation of transmission variables and for optimal design of constrained sparse arrays and sampling schemes (cf. [33]).
A. Observation model for spatio-temporal sampling
Here we present an observation model for array processing tasks that include direction of arrival (DoA) and Doppler estimation of a single source, and also MIMO schemes such as TDM MIMO.
Consider the following data model for an observation x according to a sensing scheme depicted in Fig. 2,
x = a(θ) √ γ + n ∈ C N ,(8)
where θ := (u 1 , ..., u q−1 , ϕ) T ∈ R q is the vector of unknown parameters, n ∼ N C (0, I N ) is standard complex Gaussian noise, and γ is the (single element) SNR, which is assumed known or estimated beforehand. Furthermore a(θ) denotes the spatio-temporal steering vector for one source with frequencies u j and initial phase ϕ, defined as
a(θ) := e iDθ = e i j dj uj e iϕ ∈ C N ,(9)
which depends on the q − 1 sampling vectors d j ∈ R N , which we combine, for convenience, to form the sampling matrix,
D := (d 1 , ..., d q−1 , 1 N ) ∈ R N ×q .(10)
The sampling matrix D ≡ D(g) can be parametrized by a sensing parameter g that can be designed or adapted, and which we omit in this section. We refer to the generic parameter u = (u 1 , ..., u q−1 ) as frequency vector, whereas ϕ is called initial phase or phase.
This data model can be applied to the estimation of several quantities for one source, including DoA or Doppler estimation, where d 1 refers to antenna positions or pulse times; joint azimuth-elevation estimation with 3-dimensional arrays [30, eq. (38)], where d 1 , d 2 , d 3 , are the coordinates of the antenna locations in some basis and θ are the electronic angles; and range-Doppler-azimuth estimation [34] in automotive applications after a Fourier transform in the fast-time domain for each Tx and Rx pair and each pulse.
Next we show an example of application to the problem of TDM MIMO array processing for joint DoA-Doppler estimation [35]. This is a general template that we use in Section V-D for adaptive selection of antenna elements in the scenario of DoA estimation.
B. TDM MIMO DoA-Doppler estimation
Adaptive sampling for DoA-Doppler estimation can involve the activation of receivers and transmitter activation sequences. Consider N Rx receivers in a linear array at positions d Rx ∈ R NRx that collect the echoes from a total of N P pulses transmitted by a subset of a total of N Tx transmitters available, allowing repetitions, located at positions d Tx ∈ R NTx . The pulses are sent one after the other at time instances t P ∈ R NP . The transmission variables to be optimized are (i) the specific subset and order of transmitter activations, codified by the
matrix G Tx ∈ {0, 1} NP×NTx (where 1 NP = G Tx 1 NTx );
(ii) the subset of N R receivers whose signals are processed, codified by, G Rx ∈ {0, 1} NR×NRx ; and (iii) possibly the carrier frequency. A model for the observation of a single target with DoA u, Doppler frequency ω = 4πv r /λ, and complex amplitude s = |s|e iϕ , is given by [35, eq. (4),(5)]
x = b(u, v r , ϕ)|s| + n,(11)
where the spatio-temporal steering vector for TDM MIMO,
b(u, v r , ϕ) := e i 1 λ (d V u+t V vr) e iϕ ,
can be written as in (9) in terms of the positions of the active virtual elements, i.e., pairs Tx, Rx, and the pulse times,
d V := G Tx d Tx ⊗ 1 NR + 1 NP ⊗ G Rx d Rx t V := t P ⊗ 1 NR , (units of 1 2π for d V and 1 4π for t V ) yielding the sampling matrix D(g) = 1 λ (d V , t V , λ1 NPNR ) ∈ R NPNR×3 .
This fits into our general model (8) by identifying θ = (u, v r , ϕ) as parameters to be estimated, γ = |s| 2 σ 2 as SNR,
d 1 = d V
λ as virtual array positions, and d 2 = t V λ as virtual pulse times.
In the next section we describe the construction of the WWB for the model (9), (10) that includes the above scenarios. Fig. 2. Design of sensing parameters in spatial and temporal domains during a coherent processing interval (CPI), or frame, with random initial phase. The antenna positions and pulse timing can be nonuniform and sparse and can be selected or scaled between frames using the proposed Bayesian adaptive framework.
C. Random-phase WWB for array processing
Here we derive the WWB for the data model introduced in (8) both for Gaussian and independent uniform belief distributions on the frequency parameters. The calculation is similar to the general formulation in [30], but employs a different class of test points. Performing the calculations outlined in the Appendix A for
the choice of test point h = (h u1 , ..., h uq−1 , h ϕ ) T ∈ R q×1 , we obtain WWB(h) = 1 Q hh T ∈ C q×q (12a) Q = 2 η(h, h) − η(h, −h) η(h, 0) 2 (12b) η(v,ṽ) =ή(v,ṽ)ξ(v,ṽ) (12c)
where the integral over observationsή ≡ή θ according to (27) iś
η(v,ṽ) = exp(− γ 2 (N − Re{1 T N e iD(ṽ−v) })),(13)
and the following integral over parameters remains to be determined after the choice of prior distribution,
ξ(v,ṽ) := Θ p(θ) p(θ + v)p(θ +ṽ) p(θ) 2 dθ.(14)
Analytic expressions for this integral are given in the Appendix B in the cases of Gaussian (33) and independent uniform (29) priors. Remark 3 (Factorization of integrals). Note that the factorization of η =ή · ξ in (12c) (cf. equation (26) in the Appendix) into the product of an integral over observations and an integral over parameters is due to the fact that, for this choice of model and parameters, the functionή is independent of θ. Unfortunately, this is not true if e.g. the SNR γ is included into the set of random parameters θ or in the case of several targets. The consequence of the latter is an increase in computation time.
Combining the expressions in (12), the optimization problem (5) can be written as
C = sup h∈HΘ trace ρ WWB(h)(15)
for h in the domain (cf. (25) in the Appendix)
H Θ := {h ∈ R q×1 : Θ ∩ (Θ + h) = ∅}.(16)
Note that this set depends on the prior belief through its support, Θ := supp(p(θ)) = {θ ∈ R : p(θ) > 0}.
Remark 4 (Symmetry of WWB and H Θ ). A sufficient condition for the symmetry WWB(h) = WWB(−h) results from the corresponding symmetries (v,ṽ) → (−v, −ṽ) foŕ η, which holds in general in view of (13), and also for ξ, which depends on the specific prior distribution. Since (16), this symmetry implies that we can neglect those test points of H Θ which are on one side of an arbitrarily chosen hyperplane through 0 ∈ H Θ when solving the optimization problem (15).
h ∈ H Θ ⇐⇒ −h ∈ H Θ by definition of H Θ in
We thus choose to restrict the optimization to test points with positive phase component h ϕ ≥ 0.
For both uniform and Gaussian priors, the integral over priors ξ and thus the corresponding WWB (31), (34) depend only on the variance of the prior, and is independent of its mean. The dependence on the prior only through the variance makes it convenient to numerically characterize the controller based on the corresponding WWB cost function for observation models in low-dimensional estimation and for low-dimensional sensing parameters. We study the controller output based on inputs given by distributions with this property for an array scaling task in the following section.
IV. ANALYSIS OF RANDOM-PHASE WWB FOR SCALING OF SAMPLING PERIOD
Here we analyze the WWB constructed in Section III for a model of frequency estimation in one dimension plus an extra dimension for the initial phase. We compare them with the corresponding bounds assuming a known initial phase in the decision problem of selecting the optimal scaling of the array. In particular, we characterize the controller choices numerically using a look-up table that can be used for realtime computation and also interpret the choices, specially the added robustness of considering the initial phase unknown.
A. Observation model for array scaling
We investigate the following estimation problem as a special case of model (8): Consider a sampling vector d ∈ R N that can be scaled by a factor g k ∈ R + , that we wish to adapt at each measurement step k = 1, 2, . . . , according to the data model
x k = e ig k du e iϕ k √ γ + n k .(17)
We recall that the Gaussian noise realizations n k ∼ N C (0, I N ) and the uniform random initial phases ϕ k ∼ U([−π, π]) are assumed independent from each other and between time steps. Regarding applications, the sampling vector and the frequency parameter in model (17) can have various interpretations. For instance, the scaling parameter can be the (inverse) wavelength λ of the carrier frequency in narrow-band DoA estimation, or the PRI for a train of pulses in Doppler estimation (after the range bin is determined). For adaptation steps on a time scale exceeding the coherency interval of the radar, the prescribed randomization of the initial phase is necessary. It is worth noting that even though these applications can be cast into the mathematical shape of model (17), it is necessary to reflect on how changing the scaling affects other radar properties in the context of the broader estimation task (e.g. changing the PRF affects range ambiguities). On the other hand, the more general array processing model (8) allows to consider these broader scenarios.
B. Characterization of controller for uniform array scaling
Here we analyze the controller choices based on optimization of the WWB cost function (15) for uniform and Gaussian priors, respectively given by expressions (34) and (31) in Section III-C. To this end, we visualize the dependence of the optimal scaling on prior variance and SNR value and compare with the results obtained from related statistical performance bounds. (31) and (34), as g
(RP ) opt ≡ g (RP ) opt (σ 2 u )
. Those depend only on the variance σ 2 u of the Gaussian or uniform belief distribution, which indicates the certainty we have on the parameter u. For comparison, we discuss also the optimal scaling choices according to the WWBs for two slightly different models: the scalings referred to as g We focus on two aspects of the scaling selection, (i) the dependence on variance and SNR, and (ii) the sensitivity of the WWB with respect to variations of the optimal scaling. a) Dependencies of RP and KP models: Regarding the dependence on variance (or Field-of-View length), consider Fig. 3 that shows the scaling selections for the RP model with uniform and Gaussian priors (red). As we expect, the more uncertainty we have about u (i.e. σ 2 u large), the smaller the optimal array scaling g to avoid aliasing, and vice-versa, the more certainty about u (i.e. σ 2 u small), the larger the scaling, to trade off ambiguity suppression with accuracy. The difference between Gaussian and uniform priors of same variance is comparatively insignificant. We observe that in this high SNR scenario with γ = 0 dB, the choices based on the KP WWB (black) are roughly twice the value as compared to the random-phase model, g
(KP ) opt ≈ 2g (RP )
opt . Further inspection (not shown here), reveals that this relationship holds for SNR values approximately above γ = −1.5 dB. For lower SNR values, we find that the optimal scalings according to the random and known-phase models coincide g
(KP ) opt ≈ g (RP )
opt .
Remark 5 (Dependence of KP WWB on coordinate origin).
The known-phase WWB depends on the coordinate origin chosen to define the sampling vector d, which in these visualizations is taken as the array center of mass. This makes it harder to analyze this WWB model and also argues against using it. This dependence also occurs for the Cramér-Rao bound (CRB) where the Fisher information matrix satisfies FIM = 2γg 2 d 2 for the known-phase model, while for the random-phase model FIM = 2γg 2 d −d1 N 2 .
b) Dependencies of UC model: The unconditional model considers a signal with random amplitude s ∼ N C (0, γ ) (cf. Appendix C) yielding a notion of SNR γ with a different interpretation. The deterministic SNR notion γ of the other models obey an exponential distribution γ ∼ Exp(γ ) under this model, such that, even though E[γ] = γ , a given value of γ emphasizes low SNR values according to the previous notion. The optimal scaling g (U C) opt (blue) in Fig. 3 (for γ = 0 dB) is thus more conservative compared to the other models for high SNR. The unsteady behavior of the curve for the unconditional model can be understood in view of Fig. 4, which shows the cost function plotted over scalings. We observe that the unconditional model exhibits an almost flat 'basin' of low values with only slight oscillation in which the optimal scaling lies for this model. It is thus likely that the scaling in the optimization grid with the lowest cost function value is found in a neighboring peak of the analytical optimum. At low SNR, the respective scalings of the other two models (not shown here) are smaller than g (U C) opt . In summary, the optimal scaling does not depend very strongly on the SNR γ for the unconditional model and is generally more conservative.
c) Sensitivity of RP model: With regards to the sensitivity of the RP WWB with respect to scaling, consider Figure 6 depicting the optimal choices for the case of uniform priors at both high and low SNR, including the value of the RP WWB Fig. 6. Color code: random-phase (RP) WWB with uniform prior for each pair (∆u, g). The red curve shows the optimal scaling for each ∆u (i.e. minimum of RP WWB along each vertical line) at the given SNR. The black curve shows the corresponding scaling choice according to the known-phase WWB. For high SNR (left), we observe the latter yields consistently larger scalings. For low SNR (right), the choices of the known-phase WWB, which depend on the convention for the choice of coordinate origin, are the same as for the random-phase WWB.
cost (in color code) for each pair (∆u, g). It is noteworthy, from a sensitivity perspective, the change of the RP WWB with respect to scaling choice. We observe that for high SNR (cf. left plot in Fig. 6), the optimal scaling according to the RP WWB g (RP ) opt (red), is only slightly smaller than scalings which abruptly exhibit significantly higher cost values, whereas for low SNR (cf. right plot of Fig. 6), the optimal scalings are not so close to such a threshold. This is relevant because for high SNR the optimal choices for the alternative metric of knownphase WWB g (KP ) opt (black) are slightly bigger and thus in the region of higher cost from the perspective of the RP WWB. This phenomenon is studied in the next section, where we show using the array factor that the RP WWB captures the notion of aliasing differently than the KP WWB.
C. Ambiguity function for optimal choices of known-phase and random-phase WWB for array scaling
Here we interpret the behavior of the RP WWB and the KP WWB calculated in the previous section in terms of the array factor for the same model of frequency estimation (17) as a function of array scaling. The array factor or ambiguity function with respect to the sampling vector (or array) d is given by
B(u, u + h) := e igdu , e igd(u+h) e igdu e igd(u+h) = 1 N , e igdh N = 1 N N n=1 e igdnh ≡ B(h).(18)
This quantity appears in the WWB through the functionή in (13). It can be interpreted in several ways: (i) Cosine distance or ambiguity function between signals e igdu ; (ii) The discrete-time Fourier transform (DTFT), or projection, of a signal e igdu into the frequency-shifted signal e igd(u+h) . Figs. 7 and 8 depict the array factor of the optimal arrays for the RP and KP models, respectively, for a specific Field-of-View length ∆u for two SNR values, γ = 0, regarded here as high, and γ = −10, regarded as low.
We make the following observations:
(i) For high SNR, the random-phase WWB favors the largest scaling which places the first grating lobe (i.e. smallest h 1 > 0 with |B(h)| = 1) right outside of ∆u as measured from the main lobe (c.f. Fig. 7). This choice maximizes the accuracy (since larger apertures correspond to thinner mainlobes) while avoiding aliasing even for the extreme case of the parameter value being in the extremes of the prior belief distribution. This choice explains the sensitivity phenomenon displayed in Figs. 4 and 6 (left) where the cost function shows a dramatic increase for bigger scalings. A lesson from this regarding the application of the RP WWB for adaptive scaling is the following: if the prior is given by a uniform approximation of the filter's empirical density output, then the support length ∆u needs to be chosen conservatively, at least in the case of high SNR.
(ii) For low SNR, the behavior is governed by the sidelobes (c.f. Fig. 8). In the context of Fig. 6 (right), we note that the optimal scaling is in a region of relatively small slope or change of the RP WWB. It is noteworthy that for low SNR below approximately −1.5 dB the KP WWB and RP WWB yield the same choice.
(iii) The KP WWB can identify only aliasing problems for test points that satisfy Re{B(h)} = 1. Their location depends on the convention for the array's center of massd. For a uniform array, the array factor is B(h) = 1 N e ighd sin(N πgh 2 )/ sin( πgh 2 ). Choosing the array's center of mass as coordinate origin (i.e.d = 0), we find that every other of the grating lobes at h k = 2k g is a minimum for Re{B(h)} if the number of observations N is even, and not a maximum as for |B(h)|. This is especially the case for the first grating lobe h 1 and thus the controller based on the known-phase WWB chooses, for fixed ∆u, a scaling twice as large than it should. (In the case of the uniform array, we could choose an offset d = ± π 2 to detect the grating lobe of Re{B(h)}, but in general such an offset depends on the array.)
V. ADAPTIVE ARRAY SCALING AND CHANNEL SELECTION FOR FREQUENCY ESTIMATION
Here we apply the adaptive sensing framework of Section II using the WWB metric derived in Section III to the problem of frequency estimation in two scenarios: (i) adaptation of array scaling, and (ii) antenna selection. In the first case, the parameter optimized is 1-dimensional and we can use the numerical characterization in Section IV. In the second case, the parameter optimized is discrete, with as many elements as groups of antennas that can be active, and we use a neural network to fit the optimal test point evaluation of the WWB. First we define the Bayesian updates and their particle filter implementation, and then we simulate the closed-loop between the filter and the controller in both scenarios. Fig. 7. Array factor (18) of optimal array according to the WWB for randomphase and known-phase models for high SNR. The alternating symmetry between the absolute value and the real part only appears when the coordinate origin is the center of mass. The optimal scaling according to RP WWB seems to depend on B(h) through the absolute value (which is coordinate origin invariant), in contrast with the optimal scaling for the KP WWB that depends on the real part and thus depends on the coordinate origin. Fig. 8. Array factor (18) of optimal array according to the WWB for randomphase and known-phase models. At low SNR, the optimal scaling is the same for both models.
A. Bayesian measurement and motion updates
A characterization of the Bayesian filter requires to define the measurement and the motion updates. The likelihood function, p(x | θ, g) for θ = (u, ϕ) T ∈ R 2 , required in the measurement update for model (17), obeys the general Gaussian model in (8) and is straightforward. The transition model between measurement steps is as follows. The frequency parameter of interest u is assumed, in this example, constant (u k = u, ∀k) along the execution of the algorithm k = 1, 2, . . . , and the initial belief distribution is assumed uniform in the interval [a, b]. Naturally, other motion models can be implemented by the particle filter. The initial phase ϕ however, undergoes a transition that is crucial for our observation model: after each measurement step it is reinitialized at random, to capture the fact that no information is available due to incoherent measurements. Formally, the initial belief p + 0 (θ 0 ) is uniform on the Cartesian product [a, b] × [−π, π], and the state evolution is modeled by
θ k = u k−1 0 + 0 m ϕ with m ϕ ∼ U([−π, π]).(19)
This yields transition probabilities independent of the scaling g k , i.e.
p(θ k |θ k−1 ) = δ u k−1 (u k ) 1 2π χ [−π,π] (ϕ k ).
The measurement and motion updates are implemented using a particle filter, described next.
B. Particle filter implementation
We employ a particle filter {p θ , w} (see e.g. [20]), comprising N P particles p θ = {θ i = (u i , ϕ i ) T } N P i=1 ∈ R q×N P and weights w ∈ R N P ×1 to represent, at each step k ≥ 1, the belief distribution of θ k = (u k , ϕ k ) T ∈ R 2 . The particle filter is initialized with particles drawn from p + 0 (θ 0 ) := p 0 (θ 0 ), i.e. uniformly at random from [a, b] × [−π, π], and with equal weights w = 1 N P 1 N P . The motion update required to obtain p − k can be realized with the particle filter by applying the target dynamics (19) independently to each particle. The measurement update is performed by resampling all particles at each step according to the weights given by the likelihood function w i = p(x | θ i , g) with the residual resampling method [36]; after the resampling the weights are reset to w = 1 N P 1 N P .
C. Simulation of adaptive array scaling
Here we compare in simulations the performance of the closed-loop between the Bayesian filter and the controllers described in Section IV-B in the sequential frequency estimation task (17). The resulting adaptive strategies are compared with a fixed scaling, a linearly increasing scaling, and a random scaling.
The metric used to evaluate the estimation quality of the policies is the average of squared-errors over N T independent trials or executions of the algorithm at a given step k, where the ground truth u 0 is drawn randomly according to the initial belief p 0 (θ 0 ) at the beginning of each trial,
MSE(û k ) := 1 N T N T n=1 ((û k ) n − (u k ) n ) 2 ,(20)
andû k is the conditional mean estimator at the k-th step, u k =p u k = p u k w. Further simulation specifications are the following. The target SNR is fixed to γ = −5 dB and assumed known by the controller. To relax this condition a further dimension can be added to the particle filter and then the conditional mean estimate, or a conservative guess, can be used to evaluate the WWB. The sampling vector d ∈ R N consists of N = 12 uniformly spaced elements. We employ a particle filter with N P = 10 4 particles to represent the joint belief distribution of the frequency parameter u and phase ϕ. The functional dependence of the optimal scaling g = g(∆u, γ) for the WWB policies has been computed beforehand on a sufficiently fine grid (Fig. 5). The decision time of the controller is thus made negligible and it is suited for real-time applications. This computation speed is particularly beneficial to analyze the performance in simulations because we find that on the order of 10 4 trials are required to obtain reproducible results for the empirical mean squared error (MSE) defined in (20). For the adaptive array scaling estimation task (17), using as objective function (15), in principle the prior can be approximated by the empirical density of the particles. However, computing the parameter integral via (14) for an arbitrary empirical density is expensive due to the large number of particles required for a good representation. For this reason, we approximate the belief distribution represented by the particles (p θ , w) by a uniform or Gaussian distribution of judiciously chosen variance, e.g., in terms of the empirical variancê
σ 2 u = (p u −p u 1 N P ) T diag(w)(p u −p u 1 N P ).(21)
We have observed that the estimation quality of the adaptive sensing policies based on Gaussian and uniform approximations of the empirical density given by the particles can benefit from choosing a larger (i.e., more conservative) variance for the controller input, σ 2 u = δ ·σ 2 u , i.e., multiplying by a factor the variance of the particles in (21). The choice of δ ≥ 1 that works well seems to depend on the SNR: For high SNR, the resulting policies benefit from bigger (more conservative) values. This can be explained based on the abrupt increase of cost reported in Fig. 6 (left), which reflects the fact that at high SNR there is a possibility of abruptly introducing aliasing in the field of view. For low SNR, as in the simulation with γ = −5 dB, choosing equal variances for the empirical and approximate distribution (i.e. δ = 1) worked fine. This again can be due to the smaller sensitivity of the scaling with respect to the variance at low SNR as shown in Fig. 6 (right) wherein the cost is dominated by sidelobes and not by grating lobes. Fig. 9 shows one realization of scaling choices for each of the strategies, while Fig. 10 shows the empirical MSE for 10 5 trials of each of the strategies, confirming the benefit of adaptation strategies over ad hoc policies without feedback. For the adaptive policies, we observe the influence of approximating the empirical density of the particles by a Gaussian or uniform prior, which can have a bigger impact than the SNR modeling choice that distinguishes the RP and UC WWB. Fig. 9. Array scaling over measurement steps for one trial. Top: fixed choice and linearly increasing scaling. Bottom: our adaptive algorithms based on the RP WWB where the priors are given by the output of the particle filter approximated using uniform or Gaussian densities. Fig. 10. Comparison of MSE at each step over 10 5 independent trials of each policy for SNR γ = −5 dB. Note that a high number of trials is necessary for this metric to converge because each choice of sensing parameters depends on the filtered belief distribution from the previous step and thus on the unique history of previous choices.
We find it interesting to compare, in addition to the average MSE(û k ), also the histogram of the squared errors at each step, cf. Fig. 11. It can be seen that the linear scaling strategy often produces estimates equally exact as the adaptive strategy, but is more prone to outliers. Conversely, the fixed scaling, which is more conservative, is equally well suited to avoid outliers as the adaptive strategy, but in the prevailing part of trials its estimates are less accurate. Fig. 11. Histogram of errors at given steps of the trial of the algorithm for each policy. The vertical lines represent the MSE over all trials at the given step. Note that at the beginning of each trial the ground truth is sampled randomly and therefore this metric resembles the empirical BMSE.
D. Simulation of adaptive channel selection
Here we simulate the controller performance for DoA estimation in TDM MIMO (11) assuming that Doppler is known and equal to 0. The controller needs to determine at each step the subset of transmitters and receivers that are active [15] [26].
In contrast with the case of array or sampling scaling, where the sensing parameter is one-dimensional and can be computed off-line, stored in a look-up table and interpreted visually, the adaptation of antenna selection presents a number of discrete choices that grows exponentially with the number of available antennas. This has motivated us to train a neural network to predict the values of the evaluation of the tightest WWB over test points in (5), cf. Fig. 1.
We have trained a fully connected neural network to approximate the KP WWB used in our previous work [26]. The concept is similar for the newly presented RP WWB. The input data comprises the antenna choices and the variance of the prior distribution, and the output is the optimal KP WWB (5). The choice of antennas is formatted using 1-hot encoding of the virtual array elements that are active for a scenario where the available Tx and Rx elements are placed in a uniform grid 0.9{1, . . . , 8} in units of half-wavelength, and Tx 1 and Rx 1 are fixed. That is, at each step the controller chooses one transmitter and one receiver out of 7 available.
For training, we have used the Tensorflow library for Python. For this small problem, the neural network is allowed to overfit the training data because we have computed the WWB in a sufficiently fine grid of variance values. The problem remains for the future to show the application of the closedloop Bayesian adaptive framework in Fig. 1 to scenarios where the neural network learns to abstract relevant array properties based on limited training data. Fig. 12. Channel selection at each measurement step for one trial of the policies defined by a fixed choice, the KP WWB, and a neural network that approximates the KP WWB. The prior distribution is assumed Gaussian with variance equal to the variance of the distribution given by the particle filter. (Overlapping virtual elements are represented with concentric circles.) Fig. 13. Comparison of MSE for policies using the KP WWB [26], [30], the associated neural network approximation, and the uniform MIMO array with Tx {1, 3} and Rx {1, 2}. The MSE is obtained at each step averaging over 300 realizations of the measurement.
In Fig. 12 we show that the antenna choices in a typical execution of the neural network resemble the ones of the exact WWB, and Fig. 13 shows that the performance is similar. From a computation standpoint, this example shows the practical side of the adaptive framework in Fig. 1 based on previous work of the authors [26].
VI. CONCLUSIONS
We have studied frequency estimation tasks for radar arrays in the context of a Bayesian setting where adaptation of sampling vectors based on the WWB prediction of estimation error is shown to be feasible for real-time implementations, at least at the software level, and provides a significant improvement of accuracy. From the ambiguity function standpoint, we have discussed the impact of incorporating knowledge of the phase or lack thereof in the model of the WWB, as this modeling choice affects the characterization of aliasing.
We have derived the Weiss-Weinstein bound for a generic multi-dimensional frequency estimation model for a single source with random initial phase, which can be efficiently implemented for uniform and Gaussian priors, and stated the optimization problem rigorously to obtain optimal sensing parameters. We have shown the applicability in two scenarios of 1D sequential frequency estimation, adapting, respectively, the scaling of sampling vector or PRF, e.g., for Doppler estimation, and antenna selection for DoA estimation. In the first case we have characterized the optimal controller choices of scaling parameter in terms of prior variance and SNR. By storing the values in a look-up table, we can achieve real-time computation. Analogously, in the case of antenna selection for DoA estimation, we have shown that a neural net trained offline can over-fit the predictions of the WWB for a given SNR, suggesting that the evaluation of the optimal WWB is feasible for real-time implementations.
Future work needs to address the bottleneck of the computational cost of the WWB for empirical densities, e.g., given by a particle filter. This may be overcome with neural networks trained off-line using as input not the variance but a higherdetail representation of the densities. Through this means, one might obtain well adjusted sensing choices for a larger class of belief distributions than currently possible with Gaussian or uniform approximations of the empirical densities. We also envision applications of this framework to scenarios like channel selection in TDM MIMO for joint DoA and Doppler estimation, and PRF adaptation for ground moving target indication (GMTI) with colored noise. Quantitative guarantees on the benefits of adaptation are also an open problem, particularly to complement the need of numerical analysis that requires a large number of Monte Carlo realizations in multidimensional problems to extract conclusions about the average behavior of the closed-loop.
VII. ACKNOWLEDGMENTS APPENDIX
A. Background on the Weiss-Weinstein bound
For convenience of the reader, we include here the general expression of the WWB for Gaussian observations following [19] and [30]. These are the expressions that we explicitly evaluate in Section III-C for our array processing models with random initial phase in the case of uniform and Gaussian priors. The parametric family of Weiss-Weinstein bounds WWB(H) ∈ R q×q for a data model comprising observations x ∈ Ω ⊆ C N and random parameter vector θ ∈ R q depends on their joint probability distribution p(x, θ) and is defined by
WWB(H) := HQ −1 H T ,(22)
where the elements of the matrix Q ∈ R M ×M are given by [30] Q k,l :=
η(h k , h l ) + η(−h k , −h l ) − η(h k , −h l ) − η(−h k , h l ) η(h k , 0)η(0, h l ) .(23)
The real-valued function η is the expectation of scaled "likelihood ratios" l(x;θ, θ) := p(x,θ) p(x,θ) given by
η(v,ṽ) = E p(x,θ) [l 1 2 (x; θ + v, θ)l 1 2 (x; θ +ṽ, θ)].(24)
It is related to the Bayesian Bhattacharyya coefficient [28] and quantifies the overlap between the shifted densities on the support of the unshifted density.
H Θ := {H ∈ R q×M : Θ ∩ (Θ + h m ) = ∅, ∀m}(25)
where Θ := supp(p(θ)) = {θ ∈ R : p(θ) > 0} denotes the support of the prior. Note that if the intersection of supports in (25) was empty for one i, then η(h i , 0) = 0 and therefore Q −1 cannot be computed since the i-th row/column of Q is not defined. In practice, the joint probability distribution is decomposed as p(x, θ) = p(x|θ)p(θ), because the likelihood function p(x|θ), denoting the probability of the observation x given the parameter vector θ, and the prior probability distribution p(θ), can be modeled more naturally. With regard to equation (24), we find
η(v,ṽ) = Θή θ (v,ṽ)p 1 2 (θ + v)p 1 2 (θ +ṽ)dθ,(26)whereή θ (v,ṽ) = Ω p 1 2 (x|θ + v)p 1 2 (x|θ +ṽ)dx.
Up to this point the formulation of the WWB applies to general probability distributions of vector parameters and vector observations. Now we consider likelihood functions corresponding to Gaussian observation models parametrized by the mean, as for the conditional model described in [30], where x ∼ N C (a(θ), R) with a known noise covariance matrix R. The authors of [30, eq. (15)] offer the following analytic expression for the integration of likelihoods over observation space
logή θ (v,ṽ) = − 1 4 R −1/2 (a(θ + v) − a(θ +ṽ)) 2 (27)
which is obtained after using the parallelogram law to the terms that remain after a null addition trick to complete the Gaussian integral.
B. Integral ξ for uniform and Gaussian priors
Here we give explicit formulas for the WWB in (12) for Gaussian and uniform priors providing expressions for (14). These priors can be applied to design problems, such as array design, where the parameter of interest is assumed in a given interval or Field-of-View [33]. In this work, we use them in our Bayesian adaptive algorithm to approximate the outcome of the particle filter and accelerate the computations of the controller.
1) Uniform belief distribution: Consider a uniform belief distribution with support Θ ⊂ R q for the parameter of interest, θ ∼ U(Θ), i.e., p(θ) = 1 |Θ| χ Θ (θ). We restrict our analysis to independent parameters. This implies a rectangular support
Θ = (µ u + × j [− ∆u j 2 , ∆u j 2 ]) × [−π, π] =: [α, β](28)
of volume |Θ| = 2π j ∆u j with edge lengths ∆u ∈ R q−1 and covariance Σ u = diag([ ∆u 2 12 ]). Using (28), the integral over priors ξ in (14) takes the form
ξ(v,ṽ) = |Θ(v,ṽ)| |Θ| ,(29)
where the volume |Θ(v,ṽ)| of the shifted-support intersectioñ
Θ(v,ṽ) := Θ ∩ (Θ − v) ∩ (Θ −ṽ) = (max(α, α − v, α −ṽ), min(β, β − v, β −ṽ))
can be readily seen to be
|Θ(v,ṽ)| = j r([β − α + min(0, −v, −ṽ) − max(0, −v, −ṽ)] j ) = j r([β − α − 1 2 (|v −ṽ| + |v| + |ṽ|)] j )(30)
where r(x) := max(0, x) is the ramp function, i.e. the product |Θ(v,ṽ)| must be set to zero if one of the factors is negative. We thus find the expression
WWB(h) = hh T 2|Θ|ή (h, 0) 2 |Θ(h)| 2 |Θ(h)| −ή(h, −h)|Θ(h, −h)| (31) withή (h, 0) = exp(− γ 2 (N − 1 T N e iDh )) (32a) η(h, −h) = exp(− γ 2 (N − 1 T N e i2Dh )) (32b) |Θ(h)| = (2π − |h ϕ |) j (∆u j − |h uj |) |Θ(h, −h)| = max(0, 2π − 2|h ϕ |) j max(0, ∆u j − 2|h uj |).
Depending on the shape of the support Θ, the function ξ can exhibit certain symmetries. For the case of our rectangular domain (28), we easily observe ξ(v,ṽ) = ξ(−v, −ṽ) from the representation in (30). As noticed in Remark 4, the optimization in (15) can be performed for h ∈ ( × j [−∆u j , ∆u j ]) × [0, 2π].
2) Gaussian belief distribution: Consider a Gaussian belief distribution for the frequency parameter u ∼ N R (µ u , Σ u ), which is independent of the uniformly distributed phase ϕ, i.e.
p(θ) = p(u)p(ϕ) p(u) = 1 (2π) q−1 det(Σ u ) exp(− 1 2 u − µ u Σ −1 u ) p(ϕ) = 1 2π χ [−π,π] (ϕ).
Denoting Φ := [−π, π], the integral over priors ξ in (14) becomes
ξ(v,ṽ) = BC(v u ,ṽ u ) · 1 2π |Φ(v ϕ ,ṽ ϕ )| (33) BC(v u ,ṽ u ) = exp(− 1 8 v u −ṽ u 2 Σ −1 u ) |Φ(v ϕ ,ṽ ϕ )| = |Φ ∩ (Φ − v ϕ ) ∩ (Φ −ṽ ϕ )| = max(0, 2π − 1 2 (|v ϕ −ṽ ϕ | + |ṽ ϕ | + |v ϕ |))
where we used the expression for the Bhattacharyya coefficient for the case of two Gaussians with same variance but different means [37, eq. (61)]). (The derivation follows along the same lines as the derivation of the expression for the complex Gaussian likelihood integral in (27).) We thus find the expression The optimization (15) is for h ∈ R q−1 × [0, 2π] since the symmetry ξ(v,ṽ) = ξ(−v, −ṽ) noticed in Remark 4 is evident from (33).
WWB(h) =(
C. Background on the unconditional WWB
For convenience of the reader we include the unconditional WWB [30, eq. (56)]. It is based on the model x = a(u)s + n ∈ C N , where n is standard complex Gaussian noise, the steering vector is a(u) = e idu and the complex amplitude s ∼ N C (0, γ ) is also a Gaussian random variable, i.e. |s| 2 ∼ γ 2 χ 2 2 = Exp(γ ) has an exponential distribution. Note that the notion of SNR according to the KP and RP models, denoted as γ, is related to γ by E[|s| 2 ] = E[γ] = γ . When the belief distribution on u is a uniform prior of length ∆u, the corresponding WWB reads
UWWB(h) = h 2 2∆u ×(35)
(∆u − |h|) 2 (1 + κ 4 (N 2 − |1 T N e idh | 2 )) −2 (∆u − |h|) − max(0, ∆u − 2|h|)(1 + κ 4 (N 2 − |1 T N e i2dh | 2 )) −1 where κ = γ 2 N γ +1 . Inspiration for this formula came from eq. [30, eq. (56)], which is the special case for ∆u = 2. The ramp function r(x) = max(0, x) is required when optimization is performed over H Θ = [−∆u, ∆u]. Due to the symmetry h → −h, the optimization can be restricted to [0, ∆u].
D. Bound on BMSE conditioned to previous history
Here we prove Proposition 1. We restate part i) in the following Lemma where we spell out the assumed probability dependencies that hold for our observation and transition models.
Lemma 1 (Motion and measurement updates under sequence of sensing parameters). Let the following assumptions be satisfied (i) State independence of previous measurements, i.e.
p(θ k | θ k−1 , g k ) = p(θ k | θ k−1 , x (k−1) , g (k) )
(ii) Conditional independence to next sensing parameter, i.e.
p(θ k−1 | x (k−1) , g (k−1) ) = p(θ k−1 | x (k−1) , g (k) ).
(iii) x k is independent of g (k−1) , x (k−1) given θ k and g k , i.e.,
p(x k | θ k , g k ) = p(x k | θ k , x (k−1) , g (k−1) , g k ). (37) Then the recurrences for the motion and measurement updates (1) and (2) satisfy (7).
Proof. We carry out the proof by complete induction. We can see the assertions to hold for k = 0 by definition of p + 0 . Now suppose it is true for k − 1. To show (7a), we note that p(θ k | θ k−1 , g k ) p + k−1 (θ k−1 )dθ k−1 = p(θ k | θ k−1 , x (k−1) , g (k) ) p(θ k−1 | x (k−1) , g (k−1) )dθ k−1 = p(θ k , θ k−1 | x (k−1) , g (k) )dθ k−1 = p(θ k | x (k−1) , g (k) ).
In the first step we use (i) and the hypotheses of induction for p + k−1 (θ k−1 ); afterwards we use assumption (ii) and standard properties of probabilities. To show (7b), we note that
p + k (θ k ) = c p(x k | θ k , g k )p(θ k | x (k−1) , g (k) ) = c p(x k | θ k , x (k−1) , g (k) )p(θ k | x (k−1) , g (k) ) =c p(θ k | x k , x (k−1) , g (k) )
where in the 2nd step we have used assumption (iii), and in the last step we have used Bayes rule applied to the probabilityp(x k |θ k ) := p(x k | θ k , x (k−1) , g (k) ), resulting iñ c = c · p(x k | x (k−1) , g (k) ). Since the normalizing constant c is chosen so that p + k is a probability density, it isc = 1, and we obtain the value for c in (7c).
Next we observe that the sensing parameters optimized according to (5) satisfy condition (ii) in Lemma 1.
Remark 6.
Consider the selection of sensing parameters according to (5) and (6). Then, for k ≥ 1, it holds that p(θ k−1 | x (k−1) , g (k−1) ) = p(θ k−1 | x (k−1) , g (k) ), where for k = 1 it us understood that p(θ 0 ) = p(θ 0 | g 1 ).
This follows from the fact that g k is computed in a deterministic manner in (6) from the previous observations x (k−1) and sensing parameters g (k−1) (requiring in addition only the initial belief p(θ 0 ) = p 0 (θ 0 ) and the transition and measurement models to carry out the recurrences (1) and (2)). As such, g k is independent of every other random variable conditioned to x (k−1) and g (k−1) , and is in particular independent of θ k−1 .
Using the previous results we can provide the proof of Proposition 1. Proof of i) follows from Lemma 1.
Proof of ii) in Proposition 1. Follows using the identity in (7a) for p − k (θ k ) (see Lemma 1) in inequality (4). Note that the assumptions (i) and (iii) of Lemma 1 are satisfied for the observation and transition models considered, and condition (ii) is verified in Remark 6.
Fig. 1 .
1Diagram of adaptive sensing based on the WWB. This Bayesian framework is analogous to the ones in
Fig. 3 .
3Optimal scaling versus variance σ 2 u of Gaussian and uniform priors. The optimal choices according to the Random-Phase WWB are depicted in red, the choices for the Known-Phase WWB [30, Eq. (57)] in black, and for the unconditional WWB [30, Eq. (56)] in blue.
Fig. 4 .
4Cost function C(g) for all three models. The unconditional model optimal scaling is harder to determine precisely for the unconditional model due to the slight variations in the cost function 'basin' and the optimal value is smaller and thus more conservative.The scaling choices are computed for a uniform array of N = 12 elements with positions d = π(1, . . . , N ) − N +1 2 (naturally, other arrays are possible). The array's center of mass is at the origin, i.e.d = 1 N n d n = 0. We denote the optimal scaling choice according to the presented randomphase (RP) WWB in
(
KP ) opt are found from a known-phase (KP) model which assumes the phase ϕ as known (based on [30, Eq. (57)]), while the scalings referred to as g (U C) opt correspond to an unconditional (UC) model with Gaussian amplitude based on [30, eq. (56)]. Details on the latter are provided in the Appendix C.
Fig. 5 .
5Optimal scaling according to random-phase WWB versus SNR γ and support length ∆u of uniform prior.
2 (BC(h u , 0)|Φ(h ϕ )|) 2 |Φ(h ϕ )| −ή(h, −h) BC(h u , −h u )|Φ(h ϕ , −h ϕ )|withή as in (32a), (32b), and|Φ(h ϕ )| = 2π − |h ϕ | |Φ(h ϕ , −h ϕ )| = max(0, 2π − 2|h ϕ |) BC(h u , 0)
Remark 2 (Choice of test point matrix). The computation of the WWB (22) requires to select a test point matrix H. In[19, sec. 4.4.1.4], it is suggested to use test point matrices with at least as many columns M as rows q (i.e. number of random parameters in the model). While still being valid lower bounds, the WWB matrices that arise from test points with M < q are by construction rank-deficient and therefore suboptimally suited to produce tight bounds to a presumably full-rank BMSE matrix. For the sake of simplicity in terms of derivation and computational time, we nonetheless select the test point matrix to comprise only one column and perform global optimization to find the tightest bound in this class.
The matrix of test points has the shapeH = h 1 , ..., h M ∈ R q×M for some M ≥ 1, although M ≥ q is recommended in [19, seq. 4.4.1.4].The domain of valid test points H Θ is restricted for practical purposes at least to matrices H satisfying
The authors would like to thank the German Ministry of Defense, particularly the WTD 81, for supporting this work. We also thank Prof. Joseph Tabrikian for helpful discussions on the topic during the German-Israeli exchange program TA44 and the anonymous reviewers that helped us improve the quality of the paper.
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[] |
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"Impact of Service Sector Loads on Renewable Resource Integration",
"Impact of Service Sector Loads on Renewable Resource Integration"
] |
[
"Nina Voulis \nSection Systems Engineering\nFaculty of Technology, Policy and Management\nDelft University of Technology\nJaffalaan 52628 BXDelftthe Netherlands\n",
"Martijn Warnier \nSection Systems Engineering\nFaculty of Technology, Policy and Management\nDelft University of Technology\nJaffalaan 52628 BXDelftthe Netherlands\n",
"Frances M T Brazier \nSection Systems Engineering\nFaculty of Technology, Policy and Management\nDelft University of Technology\nJaffalaan 52628 BXDelftthe Netherlands\n"
] |
[
"Section Systems Engineering\nFaculty of Technology, Policy and Management\nDelft University of Technology\nJaffalaan 52628 BXDelftthe Netherlands",
"Section Systems Engineering\nFaculty of Technology, Policy and Management\nDelft University of Technology\nJaffalaan 52628 BXDelftthe Netherlands",
"Section Systems Engineering\nFaculty of Technology, Policy and Management\nDelft University of Technology\nJaffalaan 52628 BXDelftthe Netherlands"
] |
[] |
HighlightsThe importance of the service sector for renewable resource integration is shown Extensive number of sources is combined to acquire detailed service sector demand Load mix with service sector uses more renewable energy than households only Differences are significant for a broad range of renewables penetration scenariosAbstractUrban areas consist of a mix of households and services, such as offices, shops, schools, etc. Yet power systems research usually only considers households. Omitting the service sector can lead to incorrect assessments of reality, in particular of measures needed to safeguard reliable supply as power generation transitions to intermittent renewable resources. This paper is the first fundamental study which explicitly addresses the impact of service sector loads on renewable resource integration. First, a broad range of renewable resource penetration scenarios is explored. Second, for a single scenario, different times of the day, days of the week, and weather conditions are compared. Third, the impact of load flexibility and storage availability is assessed. In each case, results for a realistic mix of household and service sector loads are compared to those for household loads only. Numerical results are based on a simulation model which uses the Netherlands as a case study. Results show that mixed loads can absorb significantly more renewable energy directly in a broad range of scenarios. For the single scenario, the biggest differences between the two load types are found on weekdays, with differences of up to 21% of peak load. These results indicate that service sector loads need to be explicitly considered when assessing renewable resource integration.
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10.1016/j.apenergy.2017.07.134
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[
"https://arxiv.org/pdf/1605.09667v3.pdf"
] | 2,967,723 |
1605.09667
|
6fd19d29d498e821c31cc5237f1e6814656f0b53
|
Impact of Service Sector Loads on Renewable Resource Integration
Nina Voulis
Section Systems Engineering
Faculty of Technology, Policy and Management
Delft University of Technology
Jaffalaan 52628 BXDelftthe Netherlands
Martijn Warnier
Section Systems Engineering
Faculty of Technology, Policy and Management
Delft University of Technology
Jaffalaan 52628 BXDelftthe Netherlands
Frances M T Brazier
Section Systems Engineering
Faculty of Technology, Policy and Management
Delft University of Technology
Jaffalaan 52628 BXDelftthe Netherlands
Impact of Service Sector Loads on Renewable Resource Integration
service sector loadsurban power systemsrenewable energyenergy demand profile
HighlightsThe importance of the service sector for renewable resource integration is shown Extensive number of sources is combined to acquire detailed service sector demand Load mix with service sector uses more renewable energy than households only Differences are significant for a broad range of renewables penetration scenariosAbstractUrban areas consist of a mix of households and services, such as offices, shops, schools, etc. Yet power systems research usually only considers households. Omitting the service sector can lead to incorrect assessments of reality, in particular of measures needed to safeguard reliable supply as power generation transitions to intermittent renewable resources. This paper is the first fundamental study which explicitly addresses the impact of service sector loads on renewable resource integration. First, a broad range of renewable resource penetration scenarios is explored. Second, for a single scenario, different times of the day, days of the week, and weather conditions are compared. Third, the impact of load flexibility and storage availability is assessed. In each case, results for a realistic mix of household and service sector loads are compared to those for household loads only. Numerical results are based on a simulation model which uses the Netherlands as a case study. Results show that mixed loads can absorb significantly more renewable energy directly in a broad range of scenarios. For the single scenario, the biggest differences between the two load types are found on weekdays, with differences of up to 21% of peak load. These results indicate that service sector loads need to be explicitly considered when assessing renewable resource integration.
Introduction
Urban areas are major energy consumers [1]. In the coming decades, urban areas will need to transition to renewable energy resources to satisfy their demand in a sustainable manner [2,3]. Urban demand broadly falls in three categories: industrial, residential and service sector. This paper considers only the latter two since industrial consumers are typically located on city edges and have case-specific requirements to transition to renewable supply [4,5]. Considerable work has already been done on characterising and modelling residential demand [6,7]. The service sector has received much less attention. Yet service sector loads have markedly different demand profiles as compared to households [8,9]. Disregarding their presence in urban areas is misleading when assessing the effects of renewable resource integration.
The importance of service sector consumers for power balancing in future urban grids is increasingly acknowledged in literature [10,11,12,13,14]. However, most modelling studies are based on residential loads only, with some notable exceptions, such as [15], who take some service sector loads into account. Studies on service sector demand primarily focus on subsector energy use [8,9,16,17,18] (see further). Overall, little literature is available on the effects of service sector loads on the integration of renewable resources. This paper addresses this hiatus by explicitly modelling service sector loads as an inherent part of urban demand. Different renewable resource integration metrics are compared for two load types: (1) mixed residential and service sector loads in realistic proportions, and (2) residential loads only. This is the first fundamental study of the impact of service sector loads on renewable resource integration.
Service Sector Demand
The service sector, also termed commercial, business or tertiary sector, comprises a highly heterogeneous group of power consumers. Sector definitions differ, but mostly include nonmanufacturing commercial activities and exclude agriculture and transportation [9,18,19]. This paper defines the service sector as the collection of non-manufacturing commercial and governmental activities, excluding agriculture, transportation, power sector, street lighting and waterworks.
The service sector power demand in developed countries currently accounts for one quarter to one third of the total national power demand, and is thus on par with residential demand [20,21,22]. In 2050, the demand shares of the service and the residential sectors are projected to increase to 40% each, at the expense of the industry demand, which will account for only 20% of the total national demand [21,23,24].
Although the service sector demand is increasingly important, its load characteristics are poorly understood. The sector's heterogeneity and its historically modest share in the total national power demand are seen as the main causes for the existing lack of knowledge [18,19]. Some efforts are undertaken to remedy the situation. Recent studies seek to characterise power consumption per subsector and end-use. These studies are based on cumulative annual values [8,9,16,17,18], and mostly aim to estimate the effects of energy efficiency [8,17] and demand response [16] programs. Studies using hourly data are rare, with [9] a noteworthy exception. Yet such detailed (e.g., hourly) data are needed to study the interrelations between variably demand and variable non-dispatchable renewable supply, and thus to prepare the power system for the transition to renewable resources.
Importance of Detailed Demand Characteristics
The current power system is built on the paradigm of dispatchable generation which follows a variable immutable load [25]. The main load characteristics considered, are cumulative annual load and maximal load peak [26,27,28]. Power systems with a high share of intermittent renewables require the consideration of more detailed load characteristics. In such systems supporting measures, such as demand response and storage, are needed to safeguard the balance between supply and demand. To design such supporting measures appropriately, it is key to understand the extent and timing of the imbalances between renewable generation and demand. These imbalances depend on (1) the type of load, (2) the time (e.g., time of the day, day of the week), and (3) the weather (which governs renewable generation). The latter two can only be understood if detailed (e.g., hourly) data are available. A novel time and weather dependency classification system is presented, showing the potential of detailed demand data. This paper relies on modelled hourly service sector demand data provided by the United States Department of Energy (DOE) [29] as sufficiently detailed measured open source demand data are unavailable.
Contributions
This paper reports on three experiments. First, a broad range of solar and wind resource penetration scenarios is explored. Second, for a single scenario of installed solar and wind capacity, this paper zooms in on the differences between different times of the day, days of the week, and weather conditions. Third, the impact of load flexibility and storage availability is assessed for the same scenario as used in the second experiment. The main contributions of this paper are:
1. Extensive data combination and analysis yielding a quantitative model for urban demand, consisting of a realistic mix of service sector and residential loads.
2. Development of a novel time and weather dependency classification system.
3. Quantitative analysis of a broad range of renewable resource penetration scenarios with respect to the importance of the service sector for renewable resource integration.
4. Quantitative analysis of different times of the day, days of the week and weather conditions with respect to the importance of the service sector for renewable resource integration.
5. Quantitative analysis of the impact of load flexibility and storage availability with respect to the importance of the service sector for renewable resource integration.
The remainder of this paper is structured as follows. Section 2 presents the theoretical rationale for explicit consideration of service sector loads. Section 3 outlines the methods used for data collection and profile calculation. Section 4 provides more details on the three modelling experiments. Section 5 presents the results of these experiments, which are further discussed in Section 6. A final conclusion is given in Section 7.
Rationale
Urban areas are typically a mix of residential and service sector loads. Residential load profiles are more readily available, making them an attractive proxy for urban areas as a whole. However, residential and service sector load profiles differ considerably. Fig. 1 illustrates the difference in load profiles between (1) residential loads only and (2) mixed residential and service sector loads. This paper hypothesises that realistic load profile differences are large enough to lead to significant differences in renewable resource integration metrics. In this section a theoretical intuition supporting this hypothesis is developed from two perspectives: mismatch between renewable generation and demand, and renewable energy utilisation. These form the basis for the metrics used in the remainder of this paper.
Load Type Comparison
Two load types are compared, (1) residential loads only, denoted by L r , and (2) mixed residential and service sector urban loads, denoted by L m . Let h(t) and s(t) represent respectively household and service sector load over time. Then,
L r (t) = φ · h(t) (1) L m (t) = h(t) + s(t)(2)
Note that L r is scaled by a factor φ to ensure that 8760 t=1 L r = 8760 t=1 L m for hourly steps of t. Figure 1: Comparison of load profiles of residential loads only (orange) and mixed residential and service sector loads (blue) on an average weekday. The shaded area represents the cumulative load difference. Blue shaded area shows energy consumption underestimation by residential loads only as compared to mixed urban loads. Orange shaded area represents the contrary case. Note that for the mixed load the proportions of residential and service sector are chosen to be representative for the Netherlands.
Mismatch
Generation and load must be in perfect balance for the proper operation of the power system. Let mismatch M be the difference between generation G and coinciding load L, thus M = G − L. For a given generation G, the difference in mismatch between calculations considering residential and mixed load equals:
∆M = M r − M m = s(t) − (φ − 1) · h(t)(3)
From Eq. 3 follows that the difference in mismatch ∆M only depends on the loads h(t) and s(t), not on the generation G. Fig. 1 shows that ∆M > 0 during the day and ∆M < 0 in the evening. This suggests that in urban areas with a mixed demand, power imbalance calculations based on residential load only will lead to underestimations of the mismatch between supply and demand during the day and overestimations of this mismatch during the evening. Numerical values and statistical significance of these errors are shown in the following experimental sections.
Renewable Energy Utilisation
A similar analysis can be carried out for renewable energy utilisation, denoted by R. Assuming no storage or load flexibility, R is computed as:
R = G if G ≤ L L if G > L(4)
Given a generation G, the difference in renewable energy utilisation between residential loads only and mixed loads equals:
∆R = R r − R m(5)
Expanding Eq. 5 yields ∆R as a function of both renewable generation and load. As both are time and weather dependent, assessment of renewable energy utilisation requires an analysis of time and weather interactions, in addition to correct load profile estimation. In this paper a model is presented which allows to study the influence of load type, both on an annual basis, and in specific time and weather conditions. This model is described in the next section.
Simulation Model
This paper uses an extensive data collection and simulation approach to assess the impact of service sector loads on renewable resource integration. Measured detailed service sector load data are scarce, therefore a large number of data sources is combined to create a detailed realistic data series. To obtain numerical results, the Netherlands is chosen as a case study. The influence of load type, and of time and weather on renewable resource integration metrics is studied using a novel simulation model (developed in Matlab [30]). Load type effects are assessed by comparing two load cases: residential loads only, and mixed residential and service sector loads. Time and weather effects are studied using a novel time interval classification system. The approach is conceptually shown in Fig. 2. It consists of three steps: (1) data collection, (2) profile modelling, and (3) simulation experiments. The first two steps, the core of the simulation model, are outlined in the next section. The experiments carried out are described in Section 4.
Data Collection and Profile Modelling
Synthetic load and generation profiles are calculated based on a large number of data sources, detailed below. To ensure spatial and temporal consistency, all calculations are done for the same area (Amsterdam, the Netherlands) and the same period (2014), taking into account official Dutch holidays and daylight saving times. The network is assumed to be a "copper plate". All resulting profiles have an hourly granularity.
Load Types
Two types of loads are defined: residential load (comprised of household loads only) and mixed load (a mix of household and service sector loads). For both load types, household load is represented by a single average Dutch household profile. For the mixed load type, service sector load is calculated as a weighted sum of thirteen subsector load profiles (described below in more detail). To ensure that comparison between the two load types is fair, an equal annual cumulative consumption (718 GWh/year) is used for both load types (see also 1). To achieve this, the residential load is weighted by a factor φ (for the Netherlands, φ = 2.05, see also Eq. 1).
3.1.1.1. Household Load Profile. Household demand data are obtained from [31]. The average yearly household consumption is assumed to be 3500 kWh [32]. The selected profile describes the average Dutch residential load. The use of this single profile is deemed representative at the scale used in the simulations in this paper (100 000 or 205049 households, depending on load type).
Service Sector Load Profiles.
Sufficiently detailed open source service sector consumption data are only available from the United States Department of Energy (DOE) [29]. These data are used in this paper. The DOE provides data for 16 types of commercial consumers. Of these, 13 types are considered in this paper (as listed in Table 1). The remaining three (Midrise Apartment, Outpatient Health Care and Strip Mall) are not used due to their poor matching within the Dutch context.
The available consumer types are assumed to represent subsectors of the service sector. To create a single realistic service sector profile for the Netherlands, the profiles of the 13 consumer types are combined. Each consumer type reference building, for instance a hospital, is scaled to represent the whole subsector, in this case healthcare. These scaling factors are expressed as the number of U.S. reference buildings necessary to realistically account for a given Dutch subsector load (e.g., 4 hospitals to account for healthcare for a small city of 100 000 households). Due to a lack of consistent data, the scaling factors are based on different data, such as floor area, number of employees, number of students, etc. An overview of the service sector consumer types and corresponding data is given in Table 1.
The service sector profiles themselves are obtained using the DOE EnergyPlus modelling software [33]. This software builds demand profiles based on the building age, climate data, and the building location. As the simulations assume a future situation, new construction (post-2004) standard is used. To create profiles representative for the Netherlands, Amsterdam climate data are used [34]. Finally, the location match in terms of climate zone is based on both the ASHRAE climate classification [35] and the available U.S. locations for the reference models, yielding Seattle as the closest match for Amsterdam.
Generation Profiles
Solar and wind power generation are modelled using weather data from the Royal Netherlands Meteorological Institute (KNMI) [51]. Table 1: Service sector consumer types considered in this paper. A single representative service sector profile is constructed through the combination of the profiles of the different consumer types. Each consumer type reference building is scaled to represent a subsector. The number of reference buildings necessary to model the entire subsector is estimated based on different data sources, as consistent data are not available. The number of reference buildings is based on a small city with 100 000 households. The roof area is the area available per building for solar panels. References for subsector scaling factors are listed in the last column.
Service Sector
Number [52]. The technical specifications are based on Solarex MSX-60 panels [53]. This paper assumes that solar panels can be placed on roofs of residential and service sector buildings. The roof area is used as a constraint to the number of solar panels which can be used. For the service sector, the maximal available roof area is calculated as the ratio between the total floor area and the number of storeys [29]. For households, an average roof area of 33 m 2 is used [54]. All roofs are assumed to be flat and allowing optimal positioning of solar panels.
Wind Power Generation.
Wind power generation is modelled as described in [55]. The technical turbine specifications are obtained from 500 kW EWT DIRECTWIND 52/54-500kW wind turbines [56].
Experiments Description
Three simulation experiments are carried out to study the impact of service sector loads on renewable resource integration. This impact is quantified using four metrics. The next paragraph outlines the metrics used, the subsequent paragraphs provide details for the three experiments.
Metrics
The following four renewable resource integration metrics are used in this paper:
Positive Mismatch. Positive mismatch accounts for generation excess. It is calculated as the difference between generation and load when generation exceeds load.
Negative Mismatch. Negative mismatch accounts for generation shortage. It is calculated as the difference between generation and load when load exceeds generation.
Renewable Energy Utilisation. Renewable energy utilisation is the amount of renewable energy which can be used by the coinciding load. It is assumed that whenever renewable energy is available, it is utilised first. Only if no renewable energy is available, other (non-modelled) sources are used.
Self-Consumption. Self-consumption is the ratio of renewable energy utilised by the coinciding loads to the total renewable energy generated.
Experiment 1: Renewable Resource Penetration Scenarios
Renewable resources considered in this paper are solar photovoltaic (PV) panels and wind turbines. For both solar PV and wind turbines, the installed generation capacity is varied between 0 MW and 530 MW with steps of 53 MW (121 scenarios in total). In case of residential load, 530 MW represents 300% of peak load (177 MW). In case of mixed load, 530 MW is 368% of peak load (144 MW), as mixed load has a flatter profile (see Fig. 1). The considered capacities are comparable to [21], where renewable resource capacity of up to 341% of peak load is considered for 2050.
In each scenario, the corresponding generation profile is calculated. This generation profile is combined with, on one hand, the demand profile of residential loads only, and, on the other hand, with the demand profile of mixed loads. For each scenario and for each load type, a year-long hourly simulation is run. From the results, annual metrics are calculated and reported.
Experiment 2: Time and Weather Dependency
For a single scenario of solar PV and wind turbine penetration, this paper zooms in on the role time and weather conditions play on the importance to consider the service sector in systems with high renewable resource penetration. A novel time and weather dependency classification system is introduced to study the impact of different days of the week, times of the day, and weather conditions. The single scenario of solar PV and wind turbine penetration is obtained as a result of an area-constrained optimisation.
Time and Weather Dependency Classification System
In a power system with a high penetration of renewables, not only load variations, which mainly depend on the time of the day and day of the week determine the system state, but also weather variations which govern renewable generation. To account for the future system dependency on both time and weather, this paper proposes a novel time and weather classification system. In this system, each hour of the year is classified according to four parameters: (1) day of the week, (2) time of day, (3) solar irradiance and (4) wind speed. Two categories are distinguished for the day of the week: weekday and weekend. Three categories are distinguished for the time of the day: night (00:00 -08:00), day (08:00 -16:00) and evening (16:00 -00:00). Five categories are distinguished for both solar power generation and wind power generation. In both cases, the categories are based on quantiles. In total, 150 time and weather dependent categories are defined. Their frequency of occurrence is summarised in Table 2.
Area-Constrained Renewable Mix Optimisation
The single renewable resource penetration scenario is based on an area-constrained optimisation. The optimisation problem is formulated as a constrained multi-objective non-linear problem with design variables x the number of solar PV panels and wind turbines:
x = [x P V , x turbine ]: minimize x f (x) = p pos * M + (x) + p neg * M − (x) + p ren * R(x) subject to 0 ≤ x P V ≤ α · A roof 0 ≤ x turbine ≤ (α − 1) · A roof x P V + x turbine ≤ α · A roof(6)
where:
p pos : weighting factor of positive mismatch (p pos > 0, in this paper p pos = 1) p neg : weighting factor of negative mismatch (p neg > 0, in this paper p neg = 1) p ren : weighting factor of renewable energy utilisation (p ren < 0, in this paper p ren = −5)
A roof : roof area available α : factor accounting for additional area available (φ ≥ 1, in this paper φ = 3)
Positive mismatch M + (x), negative mismatch M − (x) and renewable energy utilisation R(x) are all function of the decision variables x P V and x turbine through their dependency on generation G. Generation G is calculated as: G = x P V · g P V + x turbine · g turbine , with g P V and g turbine respectively the generation profiles of 1 m 2 PV and one 500 kW wind turbine. This paper assumes that the total area available for renewable power generation is three times the size of the cumulative roof area of all the buildings considered (i.e. α = 3). The roof area itself is only available for solar power generation, while at most twice the roof area is available for wind power generation (α − 1 in Eq. 6). The footprint of a wind turbine is assumed to be 0.345 km 2 /MW [57]. The optimisation problem is solved using the genetic algorithm in Matlab.
Experiment 3: Flexibility
In future power systems additional flexibility measures, such as demand response and storage, are expected to be implemented. The effects of these measures on renewable power integration metrics are assessed by assuming that part of the load is flexible and limited storage is available. The results are compared to those from experiment 2.
Demand Response
This paper assumes that a fixed percentage of the load can be shifted. Based on literature review it is estimated that on average about 20% of both household and service sector demand is shiftable for 2 hours, both to an earlier and to a later consumption time [9,10]. Within these constraints, it is assumed that loads are shifted as to mitigate the largest positive and negative mismatches first, i.e. to reduce residual peak supply and peak demand.
Storage
This paper assumes that some of the excess energy can be stored in batteries. The total storage is assumed to be 600 MWh (i.e. sufficiently large to supply the average demand for 7.3 hours). Grid-to-battery efficiency is assumed to be 80%, battery-to-grid efficiency 90%. When demand response and storage are combined, it is assumed that demand response capabilities are used first as they can mitigate shorter-term fluctuations, while storage is used after all demand response capabilities are exhausted.
Statistical Analysis
Metric differences between residential loads only and mixed loads are analysed for statistical significance using the two-sample t-test. Since multiple scenarios or categories are compared at once, the significance level is corrected using the Holm-Bonferroni correction to control the familywise error rate at 5%. For the first experiment, the correction is made for 121 comparisons. For second and third experiments, the correction is made for 150 comparisons.
Results
This section presents the results of the three experiments conducted. The overarching aim is to study what the influence of service sector loads is on renewable resource integration. The first experiment addresses this question over a broad range of renewable resource penetration scenarios. The second experiment zooms in on the differences between different days of the week, times of the day, and weather conditions for a single scenario. The third experiment addresses the influence of load flexibility and storage availability, further adhering the same assumptions as used in the second experiment. : Annual average differences between residential loads only and mixed residential and service sector loads. The XY plane represents scenarios of wind and solar penetration (expressed as percentage of peak load assuming mixed loads). Statistically significant differences are shown as red areas. Simulations for residential loads only assume 205 049 households. Simulations for mixed loads assume 100 000 households and the corresponding number of service sector consumers as shown in Table 1. Note that the cumulative annual demand in both load types is equal (718 GWh/year). Figure 3 shows annual average differences between residential loads only and mixed residential and service sector loads for four renewable resource integration metrics across a broad range of renewable resource penetration scenarios. Scenarios with solar and wind generation capacity of up to 530 MW are considered, i.e. 300% of peak load for the residential loads only and 368% of peak load for the mixed loads (mixed loads have a flatter profile, see Figure 1). Figures 3a and 3b show respectively the annual average positive and negative mismatch differences between residential loads only and mixed loads.
Experiment 1: Renewable Resource Penetration Scenarios
Mismatch
Positive mismatch represents renewable generation excess, i.e. renewable energy which cannot be used by the local loads. Positive mismatch differences indicate to what extent renewable generation excess is larger for the case of residential loads only, as compared to mixed loads. The larger and more positive the differences are, the more excess for residential loads. When solar and wind penetration equals zero, the positive mismatch difference is zero, since no renewable power is generated. For all other penetration scenarios, differences increase with increasing solar penetration, while the variation as a function of wind is limited. Statistically significant positive mismatch differences are found for solar penetration levels above 74% of peak load and for wind penetration scenarios below 74% of peak load. It should be noted that these cut-off values are based on the scenario step granularity of 37% of peak load.
Negative mismatch represents generation shortage, i.e. additional energy to be supplied by non-renewable resources. Negative mismatch differences indicate to what extent generation shortage is larger for the case of residential loads only, as compared to mixed loads (more negative differences indicate larger shortage for residential loads only). Negative mismatch difference is zero when solar and wind penetration equal zero as no renewable generation is available for either load type. Negative mismatch is larger for residential loads only than for mixed loads, leading to negative mismatch differences below zero across all remaining scenarios. Negative mismatch differences are statistically significant for scenarios with solar penetration above 147% of peak load (at wind capacity above 37% of peak load). Figure 3c shows renewable energy utilisation differences between residential loads only and mixed loads. Renewable energy utilisation is the amount of renewable energy that can be used by the coinciding demand. Renewable energy utilisation differences indicate to what extent less renewable energy is used directly by the residential loads only than by the mixed loads (more negative differences indicate less utilisation by residential loads only). Renewable energy utilisation differences follow the same pattern as negative mismatch differences. For all scenarios, renewable energy utilisation is higher for the mixed loads than for the residential loads only, thus the renewable energy utilisation difference is negative. Statistically significant differences are found for scenarios with solar capacity at or exceeding 147% of peak load, at any wind penetration. Figure 3d shows self-consumption differences between residential loads only and mixed loads. Self-consumption is the ratio of renewable energy utilised by the coinciding demand relative to the total renewable energy generated. Self-consumption differences indicate to what extent less of the generated renewable energy can be used by the residential loads only, as compared to mixed loads (more negative differences indicate less self-consuption by residential loads only). Self-consumption is highest at very low penetration of renewable technologies and undefined for zero penetration. If only a small amount of renewable power is generated, there is a high probability that coinciding load will be sufficiently high to use it entirely, irrespective of the load profile. Self-consumption differences have a similar pattern as renewable energy utilisation differences, although differences at low wind penetration scenarios are more pronounced. Statistically significant differences are found for low solar capacity scenarios above 74% of peak load and for wind penetration of at most 147% of peak load.
Renewable Energy Utilisation
Self-Consumption
Summary
Differences in renewable resource integration metrics between residential loads only and mixed loads are found across a broad range of scenarios. Statistical significance differs between metrics, yet is found in all scenarios except low solar, very high wind. All metrics are dependent on the presence of renewable generation and thus tend to zero at low solar and low wind penetration scenarios, leading to non-significant results. Overall, this experiment shows a significant difference in annual average metrics between residential loads only and mixed loads. It demonstrates that (1) metrics differ significantly between residential and mixed areas with the same annual cumulative load, and that (2) in mixed load urban areas assessment of renewable power integration metrics based on residential load only leads to significant errors.
The relative magnitude of these average annual differences is relatively small, up to approximately 5% of the total annual load. However, the differences between the metrics for residential loads only and mixed loads vary throughout the year, depending on both time and weather conditions. These variations are assessed in the next experiment.
Experiment 2: Time and Weather Dependency
A power system with a high penetration of renewables is highly dependent on both time and weather. To study this dependency, all hours of the reference year (2014) are classified using the time and weather classification system introduced in this paper. Each category has four parameters: day of the week, time of the day, solar generation, and wind generation. In total, 150 time and weather dependent categories are analysed. An example of a category is: all weekday night (0:00 -8:00) hours with solar generation between 0% and 3% of the installed capacity and wind generation between 0% and 5% of the installed capacity (this category is indicated on The results shown are obtained assuming an optimal renewable mix for the mixed loads: 399 MW solar PV and 30 MW wind turbines (i.e. total renewable capacity amounting to 243% of peak load in case of residential loads only and 298% of peak load in case of mixed loads). Figure 4 shows mismatch dependency on time and weather and compares residential loads only and mixed loads. Positive mismatch indicates renewable generation excess. Negative mismatch indicates renewable generation shortage. Mismatches closer to zero are better.
Mismatch
During weekdays and on weekend nights (Figure 4a-d), the mismatch is more positive for the residential loads only than for the mixed loads. In the weekends during the day and in the evening (Figure 4e-f), the mismatch is more positive for the mixed loads, although the differences are relatively small compared to the weekday categories. The largest differences occur on sunny weekdays at daytime (Figure 4b-c), and amount to up to 21% of the peak load (for mixed loads).
On an annual basis, 62% of positive mismatches occurs during weekdays at daytime when solar generation exceeds 40% of installed capacity, which corresponds to 7% of the time. Most negative mismatches (46%) occur during weekdays in the evening with solar generation below 3% of installed capacity, which corresponds to 20% of the time.
Statistical significance is not shown in the graph, yet is calculated as described in Section 3. Significant differences between mismatch results for the two load types are found for all data points on weekdays during the day (Figure 4b), as well as weekday and weekend evenings (Figure 4c and f) for low solar (generation below 3% of installed capacity). In other periods, statistically significant differences occur for some categories. The disparity in statistical significance between periods can be attributed to two reasons: the number of data points and the relative difference between residential loads only and mixed loads for a given period. First, since weather patterns are not dependent on the day of the week, weekdays have on average 2.5 times more data points per weather category than weekends. Second, during weekends and during night periods, the difference between residential loads only and mixed loads is smaller than during other periods since most service sector activities are shut down. Each category thus has four parameters: day of the week, time of the day, solar generation, and wind generation. For example, the category indicated by the red arrow represents all weekday night (0:00 -8:00) hours of 2014 with solar generation between 0% and 3% of installed capacity and wind generation between 0% and 5% of installed capacity. The average mismatch in these hours is -41 MW for both load types. The values on the x-and y-axes are quantiles. Note that some high solar categories are missing because they do not occur in the modelled reference year. The results shown assume 399 MW solar PV and 30 MW wind turbines as installed capacity. Figure 5 shows renewable energy utilisation dependency on time and weather and compares residential loads only and mixed loads. Renewable energy utilisation is the amount of generated renewable energy which can be used by the coinciding loads directly. Higher renewable energy utilisation is better.
Renewable Energy Utilisation
At low solar and wind power generation levels, the differences between residential loads only and mixed loads are small, both on weekdays and in weekends and during all times of the day. Differences increase as solar generation increases. Wind generation has limited effects as it represents only a small portion of the total renewable generation due to area constraints (see optimisation problem definition in Section 3). At higher solar generation levels, renewable energy utilisation is higher for the mixed loads than for the residential loads only. The service sector consumption profile coincides better with the solar power generation profile as both peak during the day. During the weekend at day and evening times (Figure 5e-f), the renewable energy utilisation at high solar irradiance levels is higher for residential loads only since many service sector loads do not operate during the weekend. The largest differences occur on sunny weekdays at daytime and in the evening (Figure 5b-c), and amount to up to 13% of the peak load. Statistically significant differences are found during weekdays at high solar generation levels for all periods (Figure 5a-c). Most renewable energy utilisation (26%) occurs during weekdays at daytime with high solar generation levels (above 40% of installed capacity), these categories correspond to 7% of the time. Further, 7% of the renewable energy is consumed during night and evening periods with lowest sun and highest wind (occurring 10% of the time). Figure 6 shows self-consumption dependency on time and weather and compares residential loads only and mixed loads. Self-consumption is the amount of renewable energy utilised relative to the amount generated.
Self-Consumption
Similarly to the mismatch and renewable energy utilisation metrics, during weekdays and weekend nights the mixed loads performs better than the residential loads only (Figure 6a-d).
During weekend days and evenings the opposite is the case, although differences are again small (Figure 6e-f). Similarly to other metrics, the largest differences are found on sunny weekdays at daytime and in the evening (Figure 6b-c), mixed loads yield a self-consumption up to 11% higher than residential loads. Statistically significant differences occur for similar categories as for mismatch. At low solar generation levels and at all wind generation levels, the self-consumption is 100%, meaning that all renewable power generated can be used by the modelled loads. As solar generation increases, selfconsumption decreases. During weekdays the differences between the two load types are biggest (Figure 6b). In these periods the self-consumption decreases faster for the residential loads only than for the mixed loads. This result illustrates that modelling only households underestimates the self-consumption of realistic mixed urban areas.
Result Dependency on Load Assumptions in the Optimisation Step
The results presented above rely on a renewable resource generation mix obtained by solving an optimisation problem assuming mixed loads. In this paper, the optimisation is constrained by area (see Section 4.3.2). This is the binding constraint for the number of wind turbines, regardless of the load type assumed. However, the optimal solar generation capacity changes with the load type. It is 15% lower if residential loads only instead of mixed loads are assumed. The general trends for time and weather dependency as shown in Figures 4 -6 remain similar if residential loads only instead of mixed loads are assumed. However, overall mismatches become more negative, renewable energy utilisation decreases and self-consumption increases.
Summary
Renewable power integration metrics vary as a function of both time and weather. The results shown rely on the proposed time and weather classification system. Pronounced solar generation dependency is found for all metrics due to the high share of solar PV in the generation mix. Relative metric performance of residential loads only and mixed loads differs per period. Overall, on weekdays (subplots a-c on Figures 4, 5 and 6) mixed loads leads to lower mismatches and higher renewable energy utilisation, on the weekends (subplots d-f on Figures 4, 5 and 6) the contrary is the case. This difference can be attributed to service sector operation hours. Statistically significant differences between residential loads only and mixed loads are primarily found on weekdays due to a larger number of datapoints per category and a larger difference between the two load type profiles. Overall, these results show complex interdependencies between time, weather and load type.
Experiment 3: Flexibility
New technologies, such as storage (e.g., electrical vehicles) and flexible loads capable of participating in demand response (e.g., smart appliances) are expected to penetrate the power system, alongside with renewable power generation. Their integration in the grid is expected to influence the utilisation of renewable energy. Here, the influence is explored as a function of time and weather dependency. The approach used in this experiment is identical to experiment 2, the results should be compared with those obtained in the previous experiment.
For this experiment it is assumed that 20% of both household and service sector demand is shiftable for 2 hours, both to an earlier and to a later consumption time. The total storage is assumed to be 600 MWh (i.e. sufficiently large to supply the average demand for 7.3 hours). Figure 7 shows the mismatch between supply and demand after demand response and storage, comparing residential loads only and mixed loads. This figure can be compared to Figure 4. Similarly to the case with no demand response and storage, on weekdays and weekend nights (Figure 7a-d) the mismatches remain more positive for the residential loads only. The contrary remains the case during other periods (Figure 7e-f).
Mismatch
Overall, the absolute value of mismatches decreases as compared to the case without storage and demand response. The largest decrease in positive mismatch is observed during high sun weekday and weekend nights (by up to 133 MW or 100%) and high sun weekday daytimes (by up to 108 MW or 65%). Negative mismatches decrease only slightly during weekday and weekend daytimes, indicating that the available flexibility does not suffice to supply the demand during daytimes with little renewable generation. Notably, negative mismatches do decrease on weekday and especially on weekend evenings (by respectively 36% and 60%). The largest remaining differences in mismatch are found on sunny weekday evenings (Figure 7c), and amount to up to 24% of peak load.
The number of statistically significant categories decreases, although statistically significant differences remain across all periods. Most negative mismatches (43%) occur during weekday evenings with low solar power generation (corresponding to 20% of the time). Due to demand flexibility, positive mismatches (i.e. generation excess) decrease considerably (from 164 GWh/year to 36 GWh/year). The remaining mismatches occur primarily during high solar weekdays and weekends at daytime (accounting for respectively 56% and 33% of the annual positive mismatch). Figure 8 shows renewable energy utilisation after demand response and storage, comparing residential loads only and mixed loads. This figure can be compared to Figure 5. A first difference between the two figures is the shape of the surfaces. With flexible load and storage, renewable energy utilisation does not level off when renewable generation is high. During weekday daytimes ( Figure 5 and 8b) with high solar power generation, renewable energy utilisation doubles. In similar categories during weekend days ( Figure 5 and 8e) renewable energy utilisation increases with 150%. Renewable energy utilisation remains low during low sun nights and daytimes, but increases fifteen fold in low sun, low wind evenings (from around 3 MW to 44 MW). A second difference between Figures 5 and 8 is the increased similarity between the two load type surfaces due to demand flexibility. Yet, some differences remain. The largest remaining differences in renewable energy utilisation are found on sunny weekday evenings (Figure 8c), and amount to up to 24% of peak load.
Renewable Energy Utilisation
Statistical differences between the two load types disappear in all but a few categories (high sun weekday daytimes and low sun evenings). Most renewable power is utilised during high sun weekdays at daytime (24%). Overall, 45% of all renewable power is used on weekdays between 08:00 and 16:00. The availability of local storage enables delayed renewable power consumption, for instance, 10% of all renewable power is utilised on weekday evenings during low sun periods. Figure 9 shows self-consumption after demand response and storage, comparing residential loads only and mixed loads. This figure can be compared to Figure 6. With flexible load and storage, self-consumption increases from a minimum of 33% to a minimum of 54%. Renewable energy utilisation of 100% is achieved in more categories than without system flexibility. During high sun weekday and weekend daytimes, renewable energy utilisation is above 75% (Figures 9b and 9e), compared to 33% without system flexibility (Figures 6b and 9e). Least self-consumption improvements are on weekday and weekend evenings (improvements of about 10%). The largest remaining differences in self-consumption are found on sunny weekday evenings (Figure 9c), selfconsumption is up to 20% higher for mixed loads than for residential loads only. Since overall the differences between load types diminish, the number of statistically significant differences decreases.
Self-Consumption
Summary
Demand flexibility and storage improve the three metrics considered: mismatches decrease and both renewable energy utilisation and self-consumption increase. Since generation capacity does not change, the metric trends remain primarily dependent on solar power generation levels. The differences between the two load types diminish due to demand flexibility. On an annual basis, self-consumption increases from 91% to 98%, corresponding to a rise from 324 GWh/year to 533 GWh/year. Negative mismatch (i.e. energy needed from other sources) improves from 394 GWh/year to 298 GWh/year. Positive mismatch (i.e. generation excess) decreases from 164 GWh/year to 18 GWh/year. Due to system flexibility, with the same renewable generation capacity, 74% instead of 45% of demand can be supplied by local renewable resources.
Discussion
In the coming decades, power systems are expected to transition to renewable generation. The European Union for instance envisions that up to 97% of its power will be generated by renewables by 2050 [3]. As urban areas are major power consumers [1], their power grid will need to undergo considerable transformations. A thorough understanding of system characteristics, for both demand and generation, is needed to facilitate such transition. This paper focuses on the demand side, in particular on the role service sector loads play in renewable resource integration in urban areas.
The service sector has thus far been mostly omitted in power system research. This paper is the first fundamental study of the impact of service sector loads on renewable resource integration. It compares renewable resource integration metrics for two types of loads: residential loads only and mixed residential and service sector loads.
Currently service sector demand data are not readily available. This paper proposes demand profile models based on available data. This section first discusses the modelling choices and validation. Next, the influence of service sector loads on renewable resource integration is discussed. Finally, the time and weather classification system, introduced in this paper to study time and weather dependency of renewable resource integration metrics, is evaluated.
Service Sector Load Profile Modelling and Validation
This paper relies on modelled service sector load profiles, since sufficiently detailed measured open service sector demand data are not available. The Netherlands is chosen as case study. The modelling choices and model validation are discussed next, followed by an assessment of possible generalisation of the obtained results to other countries.
The Netherlands as a Case Study
In the Netherlands, only cumulative average annual service sector electricity consumption data are openly available ( [32,58,59]). Therefore this study relies on U.S. DOE commercial reference building models. These models are used to generate realistic Dutch service sector profiles. The share of each subsector in the final mix is estimated using several cross-checked sources (see references in Table 1). Further, the best climate match possible between the U.S. models and the Netherlands is assured to ensure comparable heating and cooling requirements (see Section 3).
It is nevertheless difficult to estimate whether specific U.S. assumptions underlying the DOE reference building profiles cause deviations from real Dutch profiles. Perez-Lombard et al. [60] compared office energy end-use between U.S., Spain and UK. End-use differences exist between the three countries. The differences between U.S. and the two European countries are however not larger than between the two European countries themselves. This suggests that using U.S. data for the Netherlands does not lead to larger errors than using data from another European country. Although undesirable, the practice of using data from other countries is currently common due to limited service sector data availability [8].
Further comparison of the values obtained in this paper with those found in the literature is not straightforward. Data available from literature are cumulative annual values (e.g., [8,17]) or average daily profiles (e.g., [9]). Comparison with these data is hard due to both their lack of detail and the inconsistencies in service sector definitions, an issue also raised by others [8,18,60]. Therefore, obtained results are compared with cumulative annual Dutch service sector load data. The modelled service sector loads account for 70% of the cumulative Dutch service sector power consumption [32,58,59]. The remainder can be attributed to unaccounted for subsectors, inaccuracies in subsector share estimations and load profile deviations.
Given the lack of service sector data, model and result validation now relies on source combination and cross-referencing. To improve service sector modelling and model validation, three issues in this research field need to be addressed: (1) inconsistent service sector definitions, (2) lack of open data in general, and (3) lack of detailed (e.g., hourly) load profiles in particular.
Generalisation
The numerical results in this paper are based on the service sector consumer mix in the Netherlands (see Table 1). The composition of the service sector and its share in the total national demand differ between countries [9,17,18]. Yet the shape of the service sector demand profile, with a peak during the day, is similar across developed countries [8,29,60,61] and differs from the shape of household demand profiles, which typically peak in the evening [31,62]. The results obtained based on the Dutch service sector can most likely be generalised for other developed countries.
Based on the results presented in this paper, can be expected that the more important solar generation is in a country's renewable resource mix, the bigger the impact of service sector loads is. Solar power generation peaks during the day, matching better with the service sector demand peak than with the household demand peak.
To numerically validate the generalisation expectations, the presented model needs to be adapted for other countries and the simulations need to be repeated.
Impact of the Service Sector on Renewable Resource Integration
Despite the limitations posed by data availability, this paper is able to show the impact of service sector loads on renewable resource integration metrics. It thus provides a proof-ofconcept for realistic load modelling in urban areas, allowing an improved and more accurate assessment of renewable resource integration in urban power systems. The purpose is two-fold: (1) quantifying the errors made by estimating metrics for a mixed load area based solely on more readily available residential load profiles, and (2) assessing renewable resource integration metrics in areas with similar cumulative annual load, but with different load profiles differences arising from differences in load types.
The influence of load type is analysed by comparing residential loads only and mixed residential and service sector loads. Since the future generation mix is uncertain [63], first the influence of renewable energy penetration scenarios is addressed. Next, for a single generation mix scenario, the influence of time and weather is studied. Finally, the impact of load flexibility and storage is explored.
Experiment 1: Renewable Resource Penetration Scenarios
The impact of service sector loads on renewable resource integration is studied over a broad range of renewable resource penetration scenarios. Statistically significant differences between renewable resource integration metrics for residential loads only and mixed loads are found in all renewable resource penetration scenarios, except in those with high installed wind turbine capacity and low installed solar PV capacity, and those with few renewable resources (left upper side on each subplot in Figure 3). In most cases, mixed loads lead to less renewable energy excess, less energy requirements from other non-renewable resources, and thus to a higher renewable energy utilisation and higher self-consumption. Thus, given a similar annual load, mixed urban areas perform better in terms of renewable resource integration than what is expected based on residential loads only.
This paper assumes renewable resource penetration scenarios of up to 368% of peak load. This assumption is high, yet it is comparable to renewable generation capacities in other studies (e.g.up to 340% of peak load in [21]). Currently the Netherlands produces ten times more renewable power from wind (5300 GWh/year) than from solar (504 GWh/year) [64]. However, such relative proportions are unlikely at very high renewable resource penetrations. Wind turbines with an installed capacity of 368% of peak load of mixed residential and service sector in the entire Netherlands would cover approximately 30% of the land area. On the other hand, solar PV with an installed capacity of 368% of mixed loads for the entire Netherlands would cover only 1% of the land area. Thus, in the Netherlands, very high wind and low solar penetration scenarios are unlikely due to area constraints.
These considerations show that within the realistic range of renewable resource penetration scenarios in the Netherlands, service sector loads significantly influence renewable resource integration metrics. They should be taken into account when assessing the impact of renewable resource integration in urban power systems.
Experiment 2: Time and Weather Dependency
A renewable power system is highly dependent both on time and weather. Time governs diurnal, weekly and seasonal patterns in demand, and diurnal and seasonal patterns in solar generation patterns. Weather governs both solar and wind power generation, as well as some portion of the demand. The dependencies of renewable resource integration metrics on the interplay between time and weather are analysed using the time and weather dependency classification system introduced in this paper. This system itself is discussed further, here the focus is on the comparison between the metric trends obtained for residential loads only and for the mixed loads.
An initial inspection of the results presented in Figures 4 -6 shows that general trends are similar between residential loads only and mixed loads. All metrics depend on time and primarily on solar generation level (since solar power accounts for 93% of the installed renewable capacity due to area constraints).
Further statistical analysis reveals significant differences between the two load types in a number of time and weather dependent categories, in particular on weekdays during daytime, and for some metrics during the evenings and nights (depending on solar generation levels). Metric differences of up to 21% of peak load are found. These results thus supports the hypothesis that load profile differences between residential loads only and mixed loads lead to significant differences in corresponding renewable resource integration metrics.
Experiment 3: Flexibility
In the future, loads are expected to become partly flexible. In this study this is taken into account by assuming that 20% of the loads are shiftable up to two hours. Furthermore, storage capacity of 600 MWh is assumed, which corresponds to, for instance, 2000 households having an electric car with a 30 kWh battery (that is 1-2% of the households, depending on the load type). Load flexibility considerably improves all metrics. Renewable power consumption increases by 65% and covers 74% of total demand (up from 45%). Only 2% of the renewable energy cannot be consumed by the modelled loads. Most statistical differences between residential loads only and mixed loads disappear due to increased system flexibility. However, it should be noted that the model assumes that both the households and the service sector participate in demand response. Yet, existing demand response programs primarily target household consumers [5,6,65]. In literature only a limited number of studies exist on service sector demand response [9,16] and very few, if any, programs are implemented. If service sector loads remain untargeted by demand response programs, a considerable share of load flexibility will remain unharnessed leading to lower renewable energy utilisation and higher mismatches between generation and demand.
Time and Weather Dependency Classification System
The power system dependency on the weather increases generation becomes increasingly renewable. Although load also partly depends on the weather, currently power system metrics are assessed mainly from a time perspective [66]. This paper proposes a novel time and weather dependency classification system which takes both time and weather into account.
The time and weather dependency classification system is flexible and can be readily applied to a wide range of dataseries. In this paper the system application is shown for a reference case with time intervals of one hour, and full-year data. Classification parameter choices yielded 150 time and weather dependent categories. Time interval, dataseries size and number of categories can be varied, depending on the data and modelling purposes. For instance, the time and weather dependency classification system can be used with statistical data from multiple years to identify critical combinations of time and weather, to plan and dispatch system operations accordingly.
In this proof-of-concept example such critical combinations are reported for the reference year 2014 (see Section 5). The ability to identify such critical values as a function of time and weather and to assess their likelihood of occurrence is key for designing supporting measures (such as storage, load flexibility and dispatchable generation) for power systems with a high share of renewable resources.
Conclusions and Future Work
This paper seeks to contribute to the understanding of the impact of the service sector, an inherent part of urban areas, on the transition of urban power systems to renewable generation. So far, the service sector has been largely disregarded in power systems research. This is the first fundamental study addressing the impact of the service sector on renewable resource integration. The main contributions of this paper are:
1. The extensive data collection, combination and analysis which enables a realistic simulation of mixed service sector and residential demand. Currently, measured service sector load profiles are scarce. Therefore this paper relies on modelled profiles provided by U.S. DOE, which are adapted for the Netherlands, the chosen case study region.
2. This paper proposes a novel time and weather dependency classification system which allows metric analysis taking both time and weather into account. The existing approach primarily considers metric dependency on time, while power systems with a high penetration of renewables depend on both time and weather.
3. The impact of the service sector is shown in a broad range of solar and wind resource penetration scenarios. Statistically significant differences in renewable resource integration metrics between residential loads only and mixed loads are reported in all realistic scenarios.
4. Metric differences between residential loads only and mixed loads are analysed using the novel time and weather classification system. Largest differences between both load types occur on sunny weekdays at daytimes and in the evenings (i.e. in more than 1200 hours per year). Differences of up to 21% of peak load are found.
5. Renewable resource integration metric improvements are shown when part of the load is flexible and some storage is available. Load flexibility and storage availability diminish metric differences between the two load types. The largest remaining differences, up to 24% of peak load, are found on sunny weekday evenings. The simulated improvements in renewable resource integration metrics can only fully be realised in practice in mixed urban areas if dedicated demand response programs for service sector consumers are developed.
This paper is a first step towards a better understanding of the importance of the service sector in the transition of urban areas to sustainable power generation. Considerable work remains to be done. Similar analyses need to be carried out both on a smaller scales, i.e. for specific urban neighbourhoods as load characteristics vary locally, as on a larger scales, i.e. for other countries, as both generation and load conditions vary between geographical regions.
Current lack of open, detailed, measured service sector demand data thwarts its inclusion in power systems research. Therefore, detailed open service sector data and consensus building on definitions are indispensable for future research.
Figure 2 :
2Load and generation profiles calculation. This flow diagram shows how different data sources are combined. The resulting profiles are used in three subsequent experiments.
Figure 3
3Figure 3: Annual average differences between residential loads only and mixed residential and service sector loads. The XY plane represents scenarios of wind and solar penetration (expressed as percentage of peak load assuming mixed loads). Statistically significant differences are shown as red areas. Simulations for residential loads only assume 205 049 households. Simulations for mixed loads assume 100 000 households and the corresponding number of service sector consumers as shown in Table 1. Note that the cumulative annual demand in both load types is equal (718 GWh/year).
Figures 4, 5 and 6 with a red arrow). For each category, average metrics over all hours within that category are calculated and reported. Results are shown in Figures 4, 5 and 6. In each figure, the upper row represents weekdays, the lower row -weekends. The columns represent three different times of the day: night, day and evening. Within each subfigure, 25 weather-dependent categories are shown. The three figures show the same metrics as considered for the renewable energy penetration scenarios, with positive and negative mismatches shown on one figure.
Figure 4 :
4Mismatch dependency on time and weather: comparison between residential loads only and mixed loads. Surface plots depict the average mismatches per time and weather category. The six subfigures represent different days of the week and times of the day. Within each subfigure, 25 weatherdependent categories are shown.
Figure 5 :
5Renewable energy utilisation dependency on time and weather: comparison between residential loads only and mixed loads. Surface plots depict the average renewable energy utilisation per time and weather category. The six subfigures represent different days of the week and times of the day. Within each subfigure, 25 weather-dependent categories are shown. The values on the x-and y-axes are quantiles. The arrow indicates an example category introduced in Figure 4.
Figure 6 :
6Self-consumption dependency on time and weather: comparison between residential loads only and mixed loads. Surface plots depict the average self-consumption per time and weather category. The six subfigures represent different days of the week and times of the day. Within each subfigure, 25 weather-dependent categories are shown. The values on the x-and y-axes are quantiles. The arrow indicates an example category introduced in Figure 4.
Figure 7 :
7Mismatch after demand response and storage, comparison between residential loads only and mixed loads. Surface plots depict the average mismatch per time and weather category. The six subfigures represent different days of the week and times of the day. Within each subfigure, 25 weatherdependent categories are shown. The values on the x-and y-axes are quantiles. The arrow indicates an example category introduced in Figure 4.
Figure 8 :
8Renewable energy utilisation after demand response and storage, comparison between residential loads only and mixed loads. Surface plots depict the average renewable energy utilisation per time and weather category. The six subfigures represent different days of the week and times of the day. Within each subfigure, 25 weather-dependent categories are shown. The values on the x-and y-axes are quantiles. The arrow indicates an example category introduced inFigure 4.
Figure 9 :
9Self-consumption after demand response and storage, comparison between residential loads only and mixed loads. Surface plots depict the average self-consumption per time and weather category. The six subfigures represent different days of the week and times of the day. Within each subfigure, 25 weather-dependent categories are shown. The values on the x-and y-axes are quantiles. The arrow indicates an example category introduced in Figure 4.
Table 2 :
2Hours in 2014 classified according to the proposed time and weather dependency classification system. Numbers represent the hours per year in each category. The table distinguishes between three parameters: time of the day, solar generation, and wind generation. The latter two are categorised using quantiles 1 and expressed as percentage of installed capacity. Note that for the sake of conciseness, this table makes no distinction between weekdays and weekends. The number of weekday and weekend hours in each category can be inferred statically: 5/7 th of all hours fall on weekdays and 2/7 th on weekends. Note 1. During the night and in the evening more than 1752 hours fall in the first solar generation bracket (0-3%) because solar generation quantiles are calculated based on daylight hours.Time of
Wind Generation 0-5% 5-13% 13-26% 26-71% 71-100%
Total
the Day
Solar Generation
(hours)
0-3%
627
419
426
409
414
2295
3-9%
92
53
40
29
29
243
Night
9-21%
73
59
36
25
17
210
21-40%
35
48
37
13
9
142
40-100%
8
12
4
5
1
30
0-3%
31
26
48
68
99
272
3-9%
31
58
93
128
198
508
Day
9-21%
50
78
106
157
173
564
21-40%
90
120
128
166
186
690
40-100%
132
226
177
196
155
886
0-3%
540
526
532
450
419
2467
3-9%
21
58
45
39
18
181
Evening
9-21%
15
42
47
31
23
158
21-40%
7
24
28
30
10
99
40-100%
0
3
5
6
1
15
Total (hours)
1752
1752
1752
1752
1752
8760
AcknowledgementsThis work was supported by the Netherlands Organisation for Scientific Research (NWO) [grant number 408-13-012].
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|
[] |
[
"Radical parallelism on projective lines and non-linear models of affine spaces",
"Radical parallelism on projective lines and non-linear models of affine spaces"
] |
[
"Andrea Blunck ",
"Hans Havlicek "
] |
[] |
[] |
We introduce and investigate an equivalence relation called radical parallelism on the projective line over a ring. It is closely related with the Jacobson radical of the underlying ring. As an application, we present a rather general approach to non-linear models of affine spaces and discuss some particular examples.MSC 2000 : 51C05, 51B05, 51N10.
| null |
[
"https://arxiv.org/pdf/1304.0181v1.pdf"
] | 14,414,379 |
1304.0181
|
ff28fd0f98288bde36c3433d757b3d7ddb385829
|
Radical parallelism on projective lines and non-linear models of affine spaces
31 Mar 2013
Andrea Blunck
Hans Havlicek
Radical parallelism on projective lines and non-linear models of affine spaces
31 Mar 2013arXiv:1304.0181v1 [math.AG]Projective line over a ringJacobson radicalCremona transformationaffine space
We introduce and investigate an equivalence relation called radical parallelism on the projective line over a ring. It is closely related with the Jacobson radical of the underlying ring. As an application, we present a rather general approach to non-linear models of affine spaces and discuss some particular examples.MSC 2000 : 51C05, 51B05, 51N10.
Introduction
If two points of the projective line over a ring are non-distant then they are also said to be parallel. This terminology goes back to the projective line over the real dual numbers, where parallel points represent parallel spears of the Euclidean plane [1, 2.4]. In general, this parallelism of points is not an equivalence relation. In the present article we shall introduce another concept of "parallelism" on the projective line over a ring. In order to avoid ambiguity we call this the radical parallelism, since it reflects the Jacobson radical of a ring R in terms of the projective line over R. The two kinds of parallelism coincide exactly for local rings. The radical parallelism is defined and discussed in Section 2. There are several results on the projective line over a local ring which can be generalized to an arbitrary ring R as follows: Consider radically parallel points instead of parallel points and the Jacobson radical of R instead of the only maximal ideal of a local ring. For example, the radical parallelism is always an equivalence relation. It is the equality relation if, and only if, rad R = {0}. Next, in Section 3, we consider a K-algebra R and the associated affine chain geometry. Its automorphism group contains bijective transformations R → R (without "exceptional points") which are not affine transformations, provided that rad R = {0} and K = GF (2). In particular, when dim K R is finite, then these mappings are birational, i.e., they are Cremona transformations. We may regard R as an affine space over K and fix one of the non-affine transformations from the above. Then R together with the images of the lines under this transformation yields a non-linear model of the affine space R over K. Two particular cases of such models are investigated in detail. The first example arises from the ring of dual numbers over K. It yields, for K = R, the well-known parabola model of the real affine plane; cf. [12, p. 67]. For an arbitrary ground field K, a similar parabola model is described. However, some properties of the parabola model of the real affine plane do not hold any more if K has characteristic 2. This is due to the fact that in this case all tangent lines of a parabola are parallel. The second example is based upon the three-dimensional K-algebra of upper 2 × 2-matrices over K. As before, we obtain a kind of "parabola model" which can be easily described in terms of the associated chain geometry. Throughout this paper we shall only consider associative rings with a unit element 1, which is inherited by subrings and acts unitally on modules. The trivial case 1 = 0 is excluded. The group of invertible elements of a ring R will be denoted by R * . Let us recall the definition of the projective line over a ring R: Consider the free left Rmodule R 2 . Its automorphism group is the group GL 2 (R) of invertible 2 × 2-matrices with entries in R. A pair (a, b) ∈ R 2 is called admissible, if there exists a matrix in GL 2 (R) with (a, b) being its first row. Following [8, p. 785], the projective line over R is the orbit of the free cyclic submodule R(1, 0) under the action of GL 2 (R). So
P(R) := R(1, 0) GL 2 (R)
or, in other words, P(R) is the set of all p ⊂ R 2 such that p = R(a, b) for an admissible pair (a, b) ∈ R 2 . Two such pairs represent the same point exactly if they are left-proportional by a unit in R. We adopt the convention that points are represented by admissible pairs only. (Cf. [5,Proposition 2.1] for the possibility to represent points also by non-admissible pairs.) The point set P(R) is endowed with the symmetric and anti-reflexive relation distant (△) defined by △ := (R(1, 0), R(0, 1)) GL 2 (R) . Letting p = R(a, b) and q = R(c, d) gives then
p △ q ⇔ a b c d ∈ GL 2 (R).
The vertices of the distant graph on P(R) are the points of P(R), two vertices of this graph are joined by an edge if, and only if, they are distant. Given a point p ∈ P(R) let
P(R) p := {x ∈ P(R) | x △ p}
be the neighbourhood of p in the distant graph. We shall no longer use the term "parallel points" in the present paper, but we speak instead of "non-distant points" ( △). The sign will be used for the radical parallelism which is defined below. The Jacobson radical of a ring R, denoted by rad R, is the intersection of all the maximal left (or right) ideals of R. It is a two sided ideal of R and its elements can be characterized as follows:
b ∈ rad R ⇔ 1 − ab ∈ R * for all a ∈ R ⇔ 1 − ba ∈ R * for all a ∈ R;
see [10, pp. 53-54].
Radical parallelism
A point p ∈ P(R) is called radically parallel to a point q ∈ P(R) if
x △ p ⇒ x △ q(2)
holds for all x ∈ P(R). In this case we write p q. Clearly, the relation is reflexive and transitive; we shall see in due course that is in fact an equivalence relation. Each matrix γ ∈ GL 2 (R) determines an automorphism P(R) → P(R) : p → p γ of the distant graph. Hence, by definition,
p q ⇔ p γ q γ(3)
holds for all p, q ∈ P(R) and all γ ∈ GL 2 (R). The connection between the radical parallelism on P(R) and the Jacobson radical of R is as follows:
Theorem 2.1 The point R(1, 0) is radically parallel to q ∈ P(R) exactly if there is an element b in the Jacobson radical rad R such that q = R(1, b).
Proof: We start with a characterization of rad R in terms of matrices. For all a, b ∈ R we have
1 b 0 1 1 − ba 0 a 1 = 1 b a 1 .(4)
So, by (1), we get
b ∈ rad R ⇔ 1 b a 1 ∈ GL 2 (R) for all a ∈ R.(5)
Clearly, we have
P(R) R(1,0) = {x ∈ P(R) | x △ R(1, 0)} = {R(a, 1) | a ∈ R}.(6)
So (5) shows immediately that R(1, 0) is radically parallel to every point (6) and R(1, 0) q imply that the right hand side of (5) is fulfilled, whence b ∈ rad R.
q = R(1, b) with b ∈ rad R. Conversely, suppose that R(1, 0) q. Then R(0, 1) △ R(1, 0) implies R(0, 1) △ q. So we may set q = R(1, b) with b ∈ R. Now
In order to obtain an alternative description of the radical parallelism we consider the factor ring R/rad R =: R and the canonical epimorphism R → R : a → a + rad R =: a. It has the crucial property a ∈ R * ⇔ a ∈ R *
for all a ∈ R; cf. [10,Proposition 4.8]. Now we turn to the corresponding projective lines. The mapping
P(R) → P(R) : p = R(a, b) → R(a, b) =: p(8)
is well defined and surjective [5,Proposition 3.5]. Furthermore, as a geometric counterpart of (7) we have
p △ q ⇔ p △ q (9)
for all p, q ∈ P(R), where we use the same symbol to denote the distant relations on P(R) and on P(R), respectively. See [5, Propositions 3.1, 3.2].
Theorem 2.2 The mapping given by (8) has the property
p q ⇔ p = q(10)
for all p, q ∈ P(R).
Proof: Let q = R(c, d) ∈ P(R)
. Then (7) shows that R(1, 0) = q if, and only if, c ∈ R * and d ∈ rad R or, equivalently,
q = R(1, b) with b := c −1 d ∈ rad R. So from Theorem 2.1 we get R(1, 0) q ⇔ R(1, 0) = q(11)
for all q ∈ P(R). Now consider arbitrary points p, q ∈ P(R). There is a matrix γ ∈ GL 2 (R) with p γ = R(1, 0). We have
r γ = r γ(12)
for all r ∈ P(R), where γ ∈ GL 2 (R) is obtained by applying the canonical epimorphism to the entries of the matrix γ ∈ GL 2 (R); cf. [5, Proposition 3.1]. With (3), (11), and (12) we conclude
p q ⇔ p γ = R(1, 0) q γ ⇔ p γ = R(1, 0) = q γ ⇔ p γ = q γ ⇔ p = q.(13)
This completes the proof.
As an immediate consequence of Theorem 2.2 we obtain:
Corollary 2.3
The radical parallelism on the projective line over a ring is an equivalence relation.
In particular, is a symmetric relation despite its (seemingly asymmetric) definition in formula (2). Since p q means that the neighbourhood of p in the distant graph is a subset of the neighbourhood of q, we get, by virtue of this symmetry:
Corollary 2.4
The neighbourhood of a point p ∈ P(R) in the distant graph cannot be a proper subset of the neighbourhood of a point q ∈ P(R).
Furthermore, we have #{x ∈ P(R) | x p} = #rad R(14)
for all p ∈ P(R); in fact, Theorem 2.1 implies that (14) holds for p = R(1, 0), whence the assertion follows from the transitive action of GL 2 (R) on P(R) and (3). Thus the "size" of rad R can be recovered from the distant graph on P(R) as the cardinality of an (arbitrarily chosen) class of radically parallel points. In particular, is the equality relation if, and only if, rad R = {0}.
Another easy consequence of (9) and Theorem 2.2 is
p q ⇔ p = q ⇒ p △ q ⇔ p △ q(15)
for all p, q ∈ P(R). Note that here our assumption 1 = 0 is essential, since it guarantees that △ is an antireflexive relation. (The only point of the projective line over the zero-ring is distant to itself.) In general, however, the converse of (15) is not true:
Theorem 2.5 The relations "radically parallel" and "non-distant" on P(R) coincide if, and only if, R is a local ring.
Proof: Since GL 2 (R) acts transitively on P(R) and leaves △ and invariant, it suffices to characterize those rings where
{x ∈ P(R) | R(1, 0) △ x} = {x ∈ P(R) | R(1, 0) x}.(16)
Furthermore, we recall the following property: If a pair (a, b) ∈ R 2 is admissible and so the first row of an invertible matrix, then the first column of the inverse matrix shows that
(a, b) is unimodular, i.e., there are a ′ , b ′ ∈ R such that aa ′ + bb ′ = 1. Now let R be local. Then R \ R * = rad R,(a, b) ∈ R 2 is a unit. From this we get that a point x ∈ P(R) satisfies R(1, 0) △ x if, and only if, x = R(1, b) with b ∈ rad R. But this is equivalent to R(1, 0) x by Theorem 2.1. Conversely, suppose that (16) holds. Choose any non-unit b ∈ R. Then R(1, b) is a point and R(1, 0) △ R(1, b) implies b ∈ rad R by Theorem 2.1. Hence R \ R * ⊂ rad R and, since rad R ⊂ R \ R * holds trivially, R \ R * = rad R
Non-linear models of affine spaces
In this section R is a ring and K = R is a field contained in the centre of R. So R is an algebra over K with finite or infinite dimension. The point set of the chain geometry Σ(K, R) is the projective line over R, the chains are the K-sublines of P(R); cf. [8, p. 790]. We fix the point R(1, 0) =: ∞. By (6), the mapping
ι : R → P(R) ∞ : z → R(z, 1)(17)
is a bijection. We consider R as an affine space over K.
For each subset S ⊂ P(R) let (S ∩ P(R) ∞ ) ι −1 be the affine trace of S. The affine traces of the chains through ∞ are precisely the so-called regular lines Ku + v (u ∈ R * , v ∈ R); cf. [8,Proposition 3.5.3]. By reversing the order of the coordinates in Theorem 2.1, we obtain
(rad R) ι = {x ∈ P(R) | x R(0, 1)},
whence rad R is the affine trace of {x ∈ P(R) | x R(0, 1)}. One can easily check that the affine trace of {x ∈ P(R) | x △ R(0, 1)} equals R \ R * . In general, however, (R \ R * ) ι is not equal to {x ∈ P(R) | x △ R(0, 1)}. Let, for example, R = R + Rj be the ring of real anormalcomplex numbers, where j 2 = 1 and j / ∈ R; cf. [1, p. 44]. Then R(1 − j, 1 + j) △ R(0, 1), but R(1 − j, 1 + j) / ∈ R ι , because 1 + j is not invertible. Each matrix γ ∈ GL 2 (R) defines an automorphism of the chain geometry Σ(K, R). Let us write D γ for the set of all points z ∈ R such that z ιγ ∈ P(R) ∞ . Then the mapping
γ ′ : D γ → R : z → z ιγι −1(18)
is injective, but in general the domain and the image of γ ′ will be proper subsets of R. (19) is satisfied then the corresponding mapping γ ′ : R → R is an affine transformation for b = 0, and a non-affine bijective transformation for b ∈ rad R \ {0} and K = GF(2).
(b) If(19)
Proof: (a) Let γ ∈ GL 2 (R) and suppose that γ ′ is defined for all points of R. So we obtain
(P(R) ∞ ) γ ⊂ P(R) ∞ .(20)
By definition, the distant relation △ is invariant under GL 2 (R). Therefore (P(R) ∞ ) γ = P(R) ∞ γ , so that (20) is equivalent to R(a, b) = ∞ γ ∞. Thus a ∈ R * and b ∈ rad R by Theorem 2.2. Furthermore, R(c, d) △ R(a, b) ∞ yields R(c, d) △ ∞, so that d ∈ R * . Conversely, we infer from (19) that a − bd −1 c ∈ R * . Hence
−1 0 0 d 1 −b 0 1 −a + bd −1 c 0 d −1 c 1 = a b c d(21)
shows that γ ∈ GL 2 (R). It follows from d ∈ R * , b ∈ rad R, and (1) that zb + d ∈ R * for all z ∈ R. Therefore γ yields the mapping
γ ′ : R → R : z → (zb + d) −1 (za + c)(22)
with domain D γ = R. If ∞ = ∞ γ then b ∈ rad R \ {0}. We observe, as above, that the first and the third matrix on the left hand side of (21) both yield affine transformations. Hence we may confine our attention to the transformation
β ′ : R → R : z → (1 − zb) −1 z(23)
arising from the second matrix in (21). Let now K = GF (2). Then there is an element k ∈ K \ {0, 1}. The image of the line K under β ′ carries the points 0 β ′ = 0, 1 β ′ = (1 − b) −1 , and k β ′ = (1 − kb) −1 k. These points are non-collinear, since b ∈ rad R \ {0} implies that b / ∈ K, whence 1 − b and 1 − kb are linearly independent over K. Thus β ′ cannot be an affine transformation.
The following example shows that we cannot drop the assumption K = GF(2) in Theorem 3.1 (b). Let R = GF(2) + GF(2)ε be the ring of dual numbers over GF (2), where ε 2 = 0 and ε / ∈ GF(2). The invertible matrix δ := 1 ε 0 1 + ε yields a transformation on P(R) that interchanges ∞ with R(1, ε), but fixes the remaining four points of P(R). Hence δ ′ = id R is an affine transformation, even though ∞ δ = ∞. Suppose that γ ′ is a non-affine bijection according to Theorem 3.1 (b). We obtain a nonlinear model of the affine space on the K-vector space R by applying the bijection γ ′ to the points and lines of this affine space. So we get a "new" space which has the same point set, but the β ′ -images of the "old" lines will be the lines in the "new" sense. In view of Theorem 3.1 (b), such non-linear models of affine spaces are possible whenever the radical of R is non-zero and K = GF (2). It would be interesting to describe explicitly the "new lines" in a purely geometric way. However, this is beyond the scope of this article. Below we just give two examples, one of it generalizes the well-known parabola model of the real affine plane [12, p. 67]. We have several distinguished subgroups of GL 2 (R) which, by Theorem 3.1, fix P(R) ∞ as a set. The commutative group
B := 1 −b 0 1 | b ∈ rad R(24)
acts regularly on the set of points that are radically parallel to ∞; cf. Theorem 2.1. For each β ∈ B the induced mapping β ′ : R → R is given by (23). Next, there is the commutative group T :
= 1 0 c 1 | c ∈ R .(25)
Each τ ∈ T fixes ∞ and, by Theorem 3.1, it yields the translation τ ′ : R → R : z → z + c. Every translation of R arises in this way. A transformation τ ∈ T need not fix every point p ∞. In fact, if τ is the matrix in formula (25) then p = R (1, b), with b ∈ rad R, remains fixed if, and only if, bcb = 0.
For a subset S ⊂ R let ann(S) := {a ∈ R | aS = Sa = 0} denote the annihilator of S in R. So, for example, c ∈ ann(rad R) implies that (26) is fulfilled for all b ∈ rad R. Finally, a straightforward calculation shows that N := 1 + n 1 0
n 2 1 | n 1 , n 2 ∈ ann(rad R) ∩ rad R(27)
is a commutative subgroup of GL 2 (R). (Observe that (1 + n 1 )(1 − n 1 ) = 1.) Each ν ∈ N stabilizes ∞ and, by Theorem 3.1, it determines an affinity ν ′ : R → R : z → z(1 + n 1 ) + n 2 . The groups N and B have the property that νβ = βν for all ν ∈ N and all β ∈ B.
Every point p ∞ remains fixed under every transformation ν ∈ N. For, clearly, ∞ ν = ∞ and, since p can be written as ∞ β with β ∈ B, we obtain p ν = ∞ βν = ∞ νβ = p from (28). We adopt the notation B ′ := {β ′ | β ∈ B}; T ′ and N ′ are defined similarly.
In the remainder of this section, we suppose that m := dim K R is finite. Then the so-called cone of singularity R \ R * is an algebraic set; see [8,Remark 3.5.4]. Also, the affine trace of a chain is an affine normal rational curve of degree ≤ m, provided that it has at least two points in common with P(R) ∞ ; see [8,Theorem 3.6.5]. According to [8, p. 804], the mappings given in (18) are Cremona transformations; cf. also [2] and [3, p. 129]. In particular, the mappings described in Theorem 3.1 (b) are bijective Cremona transformations R → R. Let s be the dimension of the Jacobson radical of R. Then 1 / ∈ rad R implies s ≤ m − 1. All elements of rad R are nilpotent; see [10,Proposition 4.18]. Thus y s+1 = 0 for all y ∈ rad R. So, for each β ∈ B, formula (23) can be written in polynomial form as
β ′ : R → R : z → (1 + zb + · · · + (zb) s ) z.(29)
The final part of this section is devoted to the investigation of two particular examples, where we are able to describe explicitly the images of the lines under a fixed non-identical transformation β ′ ∈ B ′ . It will be easy to show that non-regular lines go over to non-regular lines and that the images of the regular lines are "certain" parabolas. Our main objective is to make more precise this last statement. We rule out, however, the field with two elements from our discussion, because in an affine space over GF(2) a parabola has only two points, and it would take rather complicated formulations to include this case.
Example 3.2 Let R = K + Kε be the ring of dual numbers over K, where K = GF(2). This is a local commutative ring, and its radical is Kε. The lines parallel to Kε are called vertical. In formula (29) we may put z = z 1 + z 2 ε and b = tε with z 1 , z 2 , t ∈ K. Thus we get
β ′ : R → R : (z 1 + z 2 ε) → z 1 + (tz 2 1 + z 2 )ε.(30)
We assume that t = 0. All non-regular lines or, said differently, all vertical lines are invariant under β ′ . In order to describe the images of the regular lines, we consider the group N ′ . Its transformations are obtained from (27) by substituting n 1 = l 1 ε and n 2 = l 2 ε, where l 1 , l 2 ∈ K, and this gives
ν ′ : R → R : z 1 + z 2 ε → z 1 + (z 1 l 1 + z 2 + l 2 )ε.(31)
Then, either l 1 = 0, whence ν ′ is a non-trivial shear with the vertical axis z 1 = −l 2 /l 1 , or l 1 = 0, whence ν ′ is a vertical translation, i.e. a translation along the vertical line Kε. Altogether, since l 1 and l 2 can be chosen arbitrarily in K, the transformations in N ′ are all the shears with a vertical axis and all the vertical translations. From a projective point of view this is the group of all elations whose centre is the point at infinity of all the vertical lines.
If L is a regular line then there is a ν ′ ∈ N ′ such that L ν ′ = K; for if L and K are parallel then ν ′ can be chosen as a vertical translation, and otherwise as a non-trivial shear whose vertical axis contains the point K ∩ L. Hence the group N ′ acts transitively and, by the commutativity of N ′ , even regularly on the set of regular lines. It is clear that the image under β ′ of the regular line K is a parabola C, say, with an equation z 2 = tz 2 1 . By the transitivity of N ′ on the set of regular lines and by (28), the set of β ′ -images of the regular lines is the orbit of the parabola C under the action of the group N ′ , i.e. the set of all parabolas with an equation
z 2 = tz 2 1 + l 1 z 1 + l 2 with l 1 , l 2 ∈ K.(32)
In projective terms this is a net of conics mutually osculating at the point at infinity of all vertical lines. For each translation τ ′ : z → z + c, c ∈ R, the point ∞ β = R(1, tε) is fixed under τ , because (tε)c(tε) = 0; cf. (26). But C is the affine trace of a chain through ∞ β ; so C τ ′ is the affine trace of a chain through ∞ βτ = ∞ β , whence, by the above,
C τ ′ ∈ C N ′ . Therefore C T ′ ⊂ C N ′ .
There are two cases: If char K = 2 then C T ′ = C N ′ , since in this case for every parabola in C T ′ all its tangent lines are parallel to the line K, whereas there is a non-trivial shear ν ′ ∈ N ′ which maps C to a parabola whose mutually parallel tangent lines are not parallel to K. Suppose now that char K = 2. Then equation (32) can be written in the form
z 2 + l 2 1 4t − l 2 = t z 1 + l 1 2t 2 .
Hence for each ν ′ ∈ N there exists a translation τ ′ ∈ T ′ with C ν ′ = C τ ′ . This is illustrated (with ν ′ = τ ′ ) in Figure 1. (We just proved an affine version of a theorem on osculating conics; see [6, 2.5.4] or [11,Satz 2].) Thus C T ′ = C N ′ . So the mapping (30) leads us in a natural way to the aforementioned parabola model of the affine plane over K, char K = 2.
The point set of this model is the ring R; its line set consists of all vertical lines together with all translates of the parabola C.
z 1 z 2 C C ν ′ = C τ ′ Figure 1.
Observe that there is also a parabola model for char K = 2. However, since C T ′ = C N ′ , we have to use all the vertical lines and the orbit of C under the group N ′ (rather than the translation group) in order to obtain its line set.
The paper [16] gives, for the real dual numbers, an explicit description and some applications of the transformations described in Theorem 3.1 (b). There are two maximal ideals in R, namely Kj 1 + Kε and Kj 2 + Kε, and their union is the cone of singularity. The radical is Kε. A line or plane is said to be vertical if it is parallel to Kε. In formula (29) we may put z = z 1 j 1 + z 2 j 2 + z 3 ε and b = tε with z 1 , z 2 , z 3 , t ∈ K. Thus we get
β ′ : R → R : z 1 j 1 + z 2 j 2 + z 3 ε → z 1 j 1 + z 2 j 2 + (z 3 + tz 1 z 2 )ε.(33)
We assume that t = 0. All vertical lines are invariant under β ′ . Each point on the cone of singularity remains fixed. Consider a plane which is parallel to one of the planes of the cone of singularity. The restriction of β ′ to such a plane is a planar shear, fixing the intersection of the plane with the cone of singularity. Hence all non-regular lines go over to non-regular lines.
The group N ′ is obtained from (27) by putting n 1 = l 1 ε and n 2 = l 2 ε, where l 1 , l 2 ∈ K. So we get ν ′ : R → R :
z 1 j 1 + z 2 j 2 + z 3 ε → z 1 j 1 + z 2 j 2 + (z 1 l 1 + z 3 + l 2 )ε.(34)
If l 1 = 0 then ν ′ is a non-trivial admissible shear, i.e. a shear in the direction of Kε with an axis parallel to the plane z 1 = 0. In fact, the axis of ν ′ is the vertical plane z 1 = −l 2 /l 1 . If l 1 = 0 then ν ′ is a vertical translation, i.e. a translation along the vertical line Kε. Altogether, the transformations in N ′ are all the admissible shears and all the vertical translations. In projective terms this is the group of all elations whose centre is the point at infinity of all the vertical lines and whose axis is a plane through the line at infinity of all the planes z 1 = const. Let P be the plane with equation z 3 = 0. It is clear that P β ′ is a hyperbolic paraboloid H with equation z 3 = tz 1 z 2 , and that the set of images of the regular lines in P is the set H of all the parabolas contained in H. (Figure 2 shows, for K = R, the cone of singularity and the plane P . In Figure 3 their images under β ′ are displayed.) As in Example 3.2, the group N ′ operates regularly on the set of non-vertical lines in a vertical plane which is non-parallel to z 1 = 0: If L is a regular line then there is a unique vertical plane V L through L, and this plane is not parallel to z 1 = 0. Thus there is a unique mapping ν ∈ N such that L ν ′ = V L ∩ P . So, by this action of N ′ and by (28), the set of β ′ -images of the regular lines is the union of all orbits C N ′ with C ∈ H. An alternative description is possible using the translation group T ′ . (The straightforward calculations leading to the following results are left to the reader.) Fix a parabola C ∈ H lying in the vertical plane V , say. Then V := {V τ ′ ∩ H | τ ′ ∈ T ′ } is a set of parabolas. It follows that each parabola in V is a translate of a parabola in C N ′ and vice versa. There are two cases. If char K = 2 then no parabola in V \ {C} is a translate of C. If char K = 2 then all parabolas in V are translates of C. (This is well known for K = R.) Irrespective of char K, the β ′ -images of the regular lines are-up to translations-precisely the parabolas in H. Furthermore, if char K = 2 then this result remains true if H is replaced by H 0 := {C ∈ H | 0 ∈ C}. Also we obtain the following parabola model of the affine 3-space over K. The point set of this model is the ring R; its line set consists of all non-regular lines together with all translates of the parabolas in H (for arbitrary characteristic of K) or in H 0 (for char K = 2 only). If K = R then R is isomorphic to the ring of real ternions. A detailed investigation of the chain geometry over the real ternions can be found in [2].
Theorem 3. 1
1Let γ = a b c d be a 2 × 2-matrix over R. Then the following hold:(a) The matrix γ is invertible and the corresponding mapping γ ′ , given by(18), is defined for all points of R if, and only if, a, d ∈ R * , and b ∈ rad R.
(b) We deduce from (a) that (20) is satisfied. By Corollary 2.4, applied to the points ∞ and ∞ γ , it follows that (P(R) ∞ ) γ = P(R) ∞ γ = P(R) ∞ , whence the injective mapping γ ′ is bijective. There are two cases: If ∞ = ∞ γ then b = 0. This implies that γ ′ : R → R : z → d −1 (za + c) is an affine transformation; see also[8, Lemma 3.5.7].
Example 3. 3
3Let R be the ring of upper triangular 2 × 2-matrices over K, where K = GF(2). So, R has a
Figure 2 .
2Figure 2.
Figure 3 .
3Figure 3.
since this is the only maximal left ideal in R; see [10, Theorem 19.1]. The previous remark on unimodularity, applied to the local ring R, shows that at least one entry of each admissible pair
is a two-sided ideal. But this means that R is a local ring.Corollary 2.3, Corollary 2.4, and (14) generalize well-known results on the projective line over a local ring. Cf. [8, Proposition 2.4.1].
The mappings discussed in Example 3.2 are closely related with the geometry of the isotropic (or: Galilean) plane. Likewise, Example 3.3 leads to a three-dimensional Cayley-Klein geometry, namely the geometry of the pseudo-isotropic space. We refer, among others, to[13],[17], [7, p. 136], and [14, p. 24]. The parabola model of the real affine plane is the starting point of the theory of shift planes. See [15, p. 420]. Such a plane arises, for example, from the real affine plane if the vertical lines and the translates of a curve which is in a certain sense "close to a parabola" are defined to be the "new lines". Similarly, it seems plausible that "in the neighbourhood" of our parabola model of the real affine 3-space there could exist so called R 3 -spaces (in the sense of[4]) other than the real affine 3-space. The reader should consult[9]for results and a lot of references on this interesting class of topological geometries.
Vorlesungenüber Geometrie der Algebren. W Benz, SpringerBerlinBENZ, W.: Vorlesungenüber Geometrie der Algebren, Springer, Berlin, 1973.
Über eine Cremonasche Raumgeometrie. W Benz, Math. Nachr. 80BENZ, W.:Über eine Cremonasche Raumgeometrie, Math. Nachr. 80 (1977), 225-243.
Cross ratios and a unifying treatment of von Staudt's notion of reeller Zug. W Benz, H.-J Samaga, Schaeffer , H , Geometry -von Staudt's Point of View. Plaumann, P. and Strambach, K.Reidel, DordrechtBENZ, W., SAMAGA, H.-J., and SCHAEFFER, H.: Cross ratios and a unifying treatment of von Staudt's notion of reeller Zug, In Plaumann, P. and Strambach, K., editors, Geometry -von Staudt's Point of View, pages 127-150, Reidel, Dordrecht, 1981.
Topologische Geometrien auf 3-Mannigfaltigkeiten. D Betten, Simon Stevin. 55BETTEN, D.: Topologische Geometrien auf 3-Mannigfaltigkeiten, Simon Stevin 55 (1981), 221-235.
Projective representations I. Projective lines over rings. A Blunck, H Havlicek, Abh. Math. Sem. Univ. Hamburg. 70BLUNCK, A. and HAVLICEK, H.: Projective representations I. Projective lines over rings, Abh. Math. Sem. Univ. Hamburg 70 (2000), 287-299.
Geometrie projektiver Räume I, BI-Wissenschaftsverlag. H Brauner, MannheimBRAUNER, H.: Geometrie projektiver Räume I, BI-Wissenschaftsverlag, Mannheim, 1976.
Vorlesungenüber höhere Geometrie. O Giering, Vieweg, Braunschweig WiesbadenGIERING, O.: Vorlesungenüber höhere Geometrie, Vieweg, Braunschweig Wiesbaden, 1982.
Handbook of Incidence Geometry. A Herzer, Buekenhout, F.ElsevierAmsterdamChain geometriesHERZER, A.: Chain geometries, In Buekenhout, F., editor, Handbook of Incidence Geometry, pages 781-842, Elsevier, Amsterdam, 1995.
Collineations of V -spaces. H Klein, Geom. Dedicata. 83KLEIN, H.: Collineations of V -spaces, Geom. Dedicata 83 (2000), 313-318.
A First Course in Noncommutative Rings. T Y Lam, SpringerNew YorkLAM, T.Y.: A First Course in Noncommutative Rings, Springer, New York, 1991.
H P Paukowitsch, Über oskulierende Quadriken und oskulierende quadratische Kegel im reellen m-dimensionalen projektiven Raum. 188PAUKOWITSCH, H.P.:Über oskulierende Quadriken und oskulierende quadratische Kegel im reellen m-dimensionalen projektiven Raum, Sb.österr. Akad. Wiss, Abt. II 188 (1979), 429-450.
B Polster, G Steinke, Geometry on Surfaces. CambridgeCambridge University PressPOLSTER, B. and STEINKE, G.: Geometry on Surfaces, Cambridge University Press, Cambridge, 2001.
. H Sachs, Ebene isotrope Geometrie. ViewegSACHS, H.: Ebene isotrope Geometrie, Vieweg, Braunschweig Wiesbaden, 1987.
H Sachs, Isotrope Geometrie des Raumes. Braunschweig WiesbadenViewegSACHS, H.: Isotrope Geometrie des Raumes, Vieweg, Braunschweig Wiesbaden, 1990.
H Salzmann, D Betten, T Grundhöfer, H Hähl, R Löwen, Stroppel , M , Compact Projective Planes. Berlinde GruyterSALZMANN, H., BETTEN, D., GRUNDHÖFER, T., HÄHL, H., LÖWEN, R., and STROPPEL, M.: Compact Projective Planes, de Gruyter, Berlin, 1996.
Über die auf der affinen Ebene operierenden Laguerre-Abbildungen. H Schaal, Sb. osterr. Akad. Wiss. Abt. II. 191SCHAAL, H.:Über die auf der affinen Ebene operierenden Laguerre-Abbildungen, Sb. osterr. Akad. Wiss. Abt. II 191 (1982), 213-231.
Handbook of Incidence Geometry. E M Schröder, Buekenhout, F.ElsevierAmsterdamMetric geometrySCHRÖDER, E.M.: Metric geometry, In Buekenhout, F., editor, Handbook of Inci- dence Geometry, pages 945-1013, Elsevier, Amsterdam, 1995.
. Andrea Blunck, Fachbereich Mathematik, Bundesstraße. 5520146Universität HamburgAndrea Blunck, Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, D-20146
. Germany Hamburg, Email, Hamburg, Germany Email: [email protected]
Wiedner Hauptstraße 8-10, A-1040 Wien, Austria Email: [email protected]. Hans Havlicek, Institut für Geometrie, Technische UniversitätatHans Havlicek, Institut für Geometrie, Technische Universität, Wiedner Hauptstraße 8-10, A-1040 Wien, Austria Email: [email protected]
|
[] |
[
"Time-resolved spectroscopy of exciton states in single crystals, single crystalline films and powders of YAlO",
"Time-resolved spectroscopy of exciton states in single crystals, single crystalline films and powders of YAlO"
] |
[
"Yalo ",
": Ce \nInstitute for Problems of Material Science NAS Ukraine\nKrjijanovskogo 303142KievUkraine\n",
"V Babin \nInstitute of Physics\nUniversity of Tartu\nRiia 14251014TartuEstonia\n\nInstitute for Problems of Material Science NAS Ukraine\nKrjijanovskogo 303142KievUkraine\n",
"L Grigorjeva \nInstitute of Solid State Physics\nUniversity of Latvia\nKengaraga 8LV-1063RigaLatvia\n",
"I Kondakova \nInstitute for Problems of Material Science NAS Ukraine\nKrjijanovskogo 303142KievUkraine\n",
"T Kärner \nInstitute of Physics\nUniversity of Tartu\nRiia 14251014TartuEstonia\n",
"V V Laguta \nInstitute for Problems of Material Science NAS Ukraine\nKrjijanovskogo 303142KievUkraine\n\nInstitute of Physics AS CR\nCukrovarnicka 10162 53PragueCzech Republic\n",
"M Nikl \nInstitute of Physics AS CR\nCukrovarnicka 10162 53PragueCzech Republic\n",
"K Smits \nInstitute of Solid State Physics\nUniversity of Latvia\nKengaraga 8LV-1063RigaLatvia\n",
"S Zazubovich \nInstitute of Physics\nUniversity of Tartu\nRiia 14251014TartuEstonia\n",
"Yu Zorenko \nIvan Franko National University of Lviv\nGen. Tarnavsky 10779017LvivUkraine\n"
] |
[
"Institute for Problems of Material Science NAS Ukraine\nKrjijanovskogo 303142KievUkraine",
"Institute of Physics\nUniversity of Tartu\nRiia 14251014TartuEstonia",
"Institute for Problems of Material Science NAS Ukraine\nKrjijanovskogo 303142KievUkraine",
"Institute of Solid State Physics\nUniversity of Latvia\nKengaraga 8LV-1063RigaLatvia",
"Institute for Problems of Material Science NAS Ukraine\nKrjijanovskogo 303142KievUkraine",
"Institute of Physics\nUniversity of Tartu\nRiia 14251014TartuEstonia",
"Institute for Problems of Material Science NAS Ukraine\nKrjijanovskogo 303142KievUkraine",
"Institute of Physics AS CR\nCukrovarnicka 10162 53PragueCzech Republic",
"Institute of Physics AS CR\nCukrovarnicka 10162 53PragueCzech Republic",
"Institute of Solid State Physics\nUniversity of Latvia\nKengaraga 8LV-1063RigaLatvia",
"Institute of Physics\nUniversity of Tartu\nRiia 14251014TartuEstonia",
"Ivan Franko National University of Lviv\nGen. Tarnavsky 10779017LvivUkraine"
] |
[] |
Luminescence characteristics of single crystals (SC), single crystalline films (SCF), powders and ceramics of YAlO 3 and YAlO 3 :Ce have been studied at 4.2-300 K under photoexcitation in the 4-20 eV energy range and X-ray excitation. The origin and structure of defects responsible for various exciton-related emission and excitation bands have been identified. The ~5.6 eV emission of YAlO 3 SCF is ascribed to the self-trapped excitons. In YAlO 3 SC, the dominating 5.63 eV and 4.12 eV emissions are ascribed to the excitons localized at the isolated antisite defect Y 3+ Al and at the Y 3+ Al defect associated with the nearest-neighbouring oxygen vacancy, respectively. Thermally stimulated release of electrons, trapped at these defects, takes place at 200 K and 280 K, respectively. The formation energies of various Y 3+ Al -related defects are calculated. The presence of Y Al antisite-related defects is confirmed by NMR measurements. The influence of various intrinsic and impurity defects on the luminescence characteristics of Ce 3+ centers is clarified.
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10.1088/0022-3727/44/31/315402
|
[
"https://arxiv.org/pdf/1101.0524v1.pdf"
] | 94,159,262 |
1101.0524
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44a236d9fca19f50137dc7ebed86410a6aa24a1e
|
Time-resolved spectroscopy of exciton states in single crystals, single crystalline films and powders of YAlO
Yalo
: Ce
Institute for Problems of Material Science NAS Ukraine
Krjijanovskogo 303142KievUkraine
V Babin
Institute of Physics
University of Tartu
Riia 14251014TartuEstonia
Institute for Problems of Material Science NAS Ukraine
Krjijanovskogo 303142KievUkraine
L Grigorjeva
Institute of Solid State Physics
University of Latvia
Kengaraga 8LV-1063RigaLatvia
I Kondakova
Institute for Problems of Material Science NAS Ukraine
Krjijanovskogo 303142KievUkraine
T Kärner
Institute of Physics
University of Tartu
Riia 14251014TartuEstonia
V V Laguta
Institute for Problems of Material Science NAS Ukraine
Krjijanovskogo 303142KievUkraine
Institute of Physics AS CR
Cukrovarnicka 10162 53PragueCzech Republic
M Nikl
Institute of Physics AS CR
Cukrovarnicka 10162 53PragueCzech Republic
K Smits
Institute of Solid State Physics
University of Latvia
Kengaraga 8LV-1063RigaLatvia
S Zazubovich
Institute of Physics
University of Tartu
Riia 14251014TartuEstonia
Yu Zorenko
Ivan Franko National University of Lviv
Gen. Tarnavsky 10779017LvivUkraine
Time-resolved spectroscopy of exciton states in single crystals, single crystalline films and powders of YAlO
Luminescence characteristics of single crystals (SC), single crystalline films (SCF), powders and ceramics of YAlO 3 and YAlO 3 :Ce have been studied at 4.2-300 K under photoexcitation in the 4-20 eV energy range and X-ray excitation. The origin and structure of defects responsible for various exciton-related emission and excitation bands have been identified. The ~5.6 eV emission of YAlO 3 SCF is ascribed to the self-trapped excitons. In YAlO 3 SC, the dominating 5.63 eV and 4.12 eV emissions are ascribed to the excitons localized at the isolated antisite defect Y 3+ Al and at the Y 3+ Al defect associated with the nearest-neighbouring oxygen vacancy, respectively. Thermally stimulated release of electrons, trapped at these defects, takes place at 200 K and 280 K, respectively. The formation energies of various Y 3+ Al -related defects are calculated. The presence of Y Al antisite-related defects is confirmed by NMR measurements. The influence of various intrinsic and impurity defects on the luminescence characteristics of Ce 3+ centers is clarified.
Introduction
Single crystals (SC) of Ce 3+ -doped yttrium-aluminum perovskite YAlO 3 :Ce (YAP:Ce) were found to have favorable scintillation characteristics especially for medical applications [1,2] which stimulated intense studies of this material. Luminescence characteristics of Ce 3+ centers, as well as energy transfer and defects creation processes in YAP:Ce SC were studied in [1][2][3][4][5][6]. The crystals grown at high temperatures from the melt contain antisite (e.g., Y 3+ Al -, Ce 3+ Al -type) and vacancy-related defects which can negatively influence the scintillation characteristics of YAP:Ce SC. The concentration of these defects is strongly suppressed in YAP single crystalline films (SCF) prepared by the lowtemperature Liquid Phase Epitaxy (LPE) method [7]. However, at the preparation of the SCF with the use of the PbO-containing flux, lead ions are introduced into the SCF. As a result, various lead-related centers appear. Their luminescence characteristics were studied in [8][9][10].
The luminescence of undoped YAP SC was studied for more than thirty years (see, e.g., [7][8][9][11][12][13][14][15][16][17] and references therein). Two emission bands observed at 5.9 eV (FWHM=0.7 eV) and 4.2 eV (FWHM=0.9 eV) under excitation in the exciton absorption region were ascribed to the self-trapped excitons (STE) and recombination processes, respectively [12][13][14][15][16]. In [7][8][9]17], the role of antisite defects of the type of Y 3+ Al in the formation of intrinsic luminescence centers was considered. After that, the highest-energy intrinsic emission of YAP SC was concluded to arise from the Y 3+ Al defect itself [7], or the exciton localized near the Y 3+ Al defect [8][9], or the STE [17], while the emission bands located in the 3.0-4.5 eV range, from the excitons localized near various vacancy-related defects [17].
Thus, in SC and SCF of YAP, several types of intrinsic defects can exist. This reduces the efficiency of energy transfer from the host lattice to a luminescence center. Photo-thermally stimulated disintegration of various exciton states into electron and hole centers and their subsequent tunneling recombination can result in the appearance of slow luminescence decay components. The overlap of intrinsic emission bands with the absorption bands of Ce 3+ can allow an effective energy transfer from the defect-related centers to Ce 3+ ions which negatively influences the scintillation characteristics of Ce 3+ -doped YAP.
However, an origin of the defect-related states as well as the processes responsible for the luminescence of YAP are not still clear. In different papers, different data are reported on the positions of the exciton, defect-related and impurity-related excitation bands in the exciton absorption region (see, e.g., Table 1). Therefore, their detailed comparison is not always possible. In the present work, the characteristics of various emission bands measured for the same undoped and Ce 3+ -doped YAP SC and YAP SCF samples at the same experimental conditions are compared. The aim of the work was to clarify the origin and structure of defects responsible for the unrelaxed (denoted further as perturbed) and the relaxed (localized) exciton states in the SC and SCF of YAP and to obtain an information on the processes of energy transfer from the defect-related centers to the luminescence Ce 3+ centers.
Experimental procedure
Single crystals of undoped and Ce 3+ -doped YAP were grown in CRYTUR Ltd., Turnov, Czech Republic, by the Czochralski method in a molybdenum crucible and studied in [3,6]. Single crystalline films of undoped YAP and YAP:Ce were prepared in Lviv University, Ukraine, by the LPE method from the PbO-B 2 O 3 flux in a platinum crucible and investigated in [10]. The nanopowders of YAP and YAP:Ce with the size of particles of about 27 nm were synthesized by the coprepitation method. The ceramics were obtained by the heating of the mentioned powders at 1000 o C.
Luminescence characteristics were studied at 10 K under excitation by the synchrotron radiation at SUPERLUMI station (HASYLAB at DESY, Hamburg, Germany) with the time windows 1-9 ns and 96-149 ns. The excitation spectra of different emissions were measured in the 4-20 eV energy range at exactly the same conditions and corrected for the spectral distribution of the excitation light.
The emission spectra were measured with the use of the monochromator ARC 0.3 m Spectra Pro300i, detected with the photomultiplier Hamamatsu R6358P and not corrected. The experiments at low temperatures were carried out with the use of a close-circle refrigerator. The X-ray luminescence spectra were measured at 80 K (40 kV, 10 mA) using Andor Sharock B-303i spectrograph with CCD camera (Andor DU-401A-BV).
Thermally stimulated luminescence (TSL) glow curves were measured with the heating rate 5 K/min after X-ray irradiation of a sample at 70 K for 30 min (40 kV, 15 mA). The sample located in an Oxford Instruments OptistatCF (Continuous Flow) helium cryostat was cooled and heated by He exchange gas, the heating rate was controlled with a ITC5205 temperature controller. The luminescence was registered using a Hamamatsu H8259-01 photon counting head.
Results and discussion
In YAP SCF, all the intrinsic and defect-or impurity-related emissions are most effectively excited in the exciton absorption region. The emissions of YAP SC are effectively excited in the band-to-band transitions region as well. The positions of the excitation bands in the exciton region observed for various exciton emission bands as well as for the emission of Ce 3+ centers in YAP SC and YAP SCF, and single and dimer Pb 2+ -based centers in YAP SCF are shown in Table 1. It is evident that they are characteristic for each emission band studied.
Self-trapped and localized exciton luminescence in YAP single crystalline films
As it was mentioned above, the number of the antisite and vacancy-related defects is strongly suppressed in SCF as compared with SC. Therefore, in the emission spectrum of YAP SCF, the STE band should be the dominating one. In the uncorrected emission spectrum of YAP SCF, measured at 10 K under excitation in the 8.40-7.95 eV range, the most intense band peaking at ~5.6 eV is observed Table 1). The band gap energy in YAP is E g =8.8 eV [18,19].
Besides the STE emission, a complex band is observed in YAP SCF, consisting of at least three components with the maxima at 4.16 eV, 3.63 eV, and 3.15 eV (Fig. 1a, curve 4). Two later bands arise from the single and dimer Pb 2+ -based centers [10]. A wide 4.16 eV emission, observed in [8] at 4.19 eV and ascribed to an exciton localized near a Pb 2+ ion, is excited around 7.6 eV (Fig. 2b). The origin of the analogous (4.3 eV) emission in LuAG SCF was studied in [20]. By analogy with [20], we suggest that the 4.16 eV emission arises from an exciton localized near a Pt 4+ ion whose presence is caused by the use of a platinum crucible at the SCF preparation. In the excitation spectrum of the 3.63 eV emission of the single Pb 2+ -based centers, the 7.66 eV band is observed (Fig. 3a). In the excitation spectrum of the 3.15 eV emission of the dimer {Pb 2+ -V O -Pb 2+ } centers, the band located at 7.74 eV is present (Fig. 3b). These bands can arise from the excitons perturbed by the single and the dimer Pb 2+ - (Table 1) confirm our conclusion that the 4.16 eV emission is not connected with the Pb 2+ -related centers.
Intrinsic luminescence of YAP single crystals
In the uncorrected emission spectrum of YAP SC, the 5.63 eV and 4.12 eV emissions are observed ( Fig. 1b). As it was mentioned in the Introduction, different interpretations were given in [7][8][9][13][14][15][16][17] for the highest-energy intrinsic emission band. As in SC, containing a lot of antisite and vacancyrelated defects, the STE emission should be strongly suppressed, the 5.63 eV emission is surely connected with the antisite defect. In [17], a weak 5.28 eV emission was ascribed to ex 0 Y 3+ Al . However, this emission is practically absent under excitation in the exciton absorption region (Fig. 1b). This is evident also from the complete coincidence of the excitation spectra shown in Figs. 4a and 4b. The relative intensity of the 5.28 eV emission is much larger under X-ray excitation. By analogy with the ~4.3 eV emission of LuAG [20], we suggest that the 5.28 eV emission arises from recombination processes. The 4.2 eV emission found in [12] has been ascribed to the exciton-based center [15,16,21].
This emission was observed also in the thermally stimulated luminescence (TSL) spectra of undoped YAP SC [13,16,22] and ascribed in [22] to an exciton localized around a single oxygen vacancy.
However, according to [23], oxygen vacancies in YAP SC are mainly associated with the antisite Y 3+ Al defects.
The results obtained indicate that the characteristics of the 5.63 eV and 4.12 eV emissions in YAP SC differ from the characteristics of the STE emission in YAP SCF. In the YAP SCF, these emissions are absent [8]. We conclude that both the 5.63 eV and the 4.12 eV emission arise from the excitons localized near Y 3+ Al -related defects. As the positions of their lowest-energy excitation bands are different: the 5.63 eV emission is excited at ~7.9 eV (Fig. 5a), while the 4.12 eV emission, at 7.13, 7.7 eV (Fig. 5b), it means that the defects responsible for these emissions are different as well. One can suggest that these emissions arise from an exciton localized near a single antisite defect: ex 0 Y 3+ Al and from an exciton localized near the Y 3+ Al defect associated with an oxygen vacancy V O : ex 0 {Y 3+ Al -V O }. Indeed, the defects of the type of {Y 2+ Al -V O } were recently detected by the ESR method in the irradiated YAP crystals [23]. Their emission band is located at 2.45 eV [24]. The origin of this emission is confirmed by the comparison of the X-ray excited luminescence spectra of YAP SC with those of YAP nanopowder and ceramics prepared at the temperatures below 1000 C. In the latter materials, unlike YAP SC, the emission bands in the 4.0-6.0 eV energy range are completely absent decreases twice at about 220 K [3,12]. The reduction of the 5.63 eV emission intensity is accompanied with the enhancement of the 4.12 eV emission [3]. In YAP:Ce SC, the TSL peaks are observed at the temperatures (~200 K and ~280 K, Fig. 7) corresponding to the thermal quenching of the 5.63 eV and 4.12 eV emissions, respectively. However, these peaks are absent in the abovementioned YAP ceramics and nanopowders (Fig. 7). Therefore, we conclude that they arise from the recombination with the hole Ce 4+ centers of electrons, thermally released from the Y 3+ Al and {Y 3+ Al -V O } traps, respectively.
The assumption about effective formation of the {Y 3+ Al -V O }-type defects in YAP SC is supported also by the comparison of the Y 3+ Al and {Y 3+ Al -V O } defects formation energies.
Calculation of the Y 3+ Al , {Y 3+ Al -V O }, and {Y 3+ Al -Ce 3+ } defects formation energies
Ab-initio computations allow determination the total energies as a function of ionic coordinates and types of ions, and thus provide independent information about the microscopic structure of a defect and clarify some models of defects.
The density functional theory (DFT) calculations were performed by us within the local spin-density approximation (LSDA). We have used the all-electron full-potential local-orbital (FPLO)_ code _version 7.00-28 [25]. In our scalar relativistic calculations, the exchange and correlation potential of Perdew and Wang [26] was employed. The k-mesh gives the subdivision of the Brillouin zone along the three axes from which the k-space integration mesh is constructed. We have used the default value 12x12x12 for pure YAlO 3 and 6x6x6 for supercells with defects. The defect energy was calculated using 20-atom P nma -symmetry cell , where the lattice parameters were taken from x-ray diffraction data [27].
Using the supercell calculations we obtained the formation energy of antisite defect by comparing the ground state total energy of perfect cell with that of cells with single Y-Al interchanges. The structural parameters used are summarized in Table 2. As can be seen from the The calculated energies are summarised in Table 3 We have also calculated the total energy for supercell in which Ce 3+ ion replaced Y ion and found that the antisite defect formation is facilitated when Ce 3+ occupies the nearest Y 3+ site (see Table 3).
Our calculations have shown also that the ground state energy of the Ce 3+ -doped YAP lattice with the oxygen vacancy near Ce 3+ is lower on 0.25 eV than the energy of the lattice in which the oxygen vacancy are located far from the Ce 3+ ion.
In the case when the Y 3+ substitutes for the Al nearby the oxygen vacancy, the Y 3+ ion must be shifted towards the vacancy because Y 3+ ion is much larger than Al 3+ ion. The total energy dependence on the displacement of Y 3+ antisite ion is presented in Fig. 8. One can see that the calculation predicts the Y 3+ ion displacement from the Al position in the direction towards the oxygen vacancy by the distance about 0.34 Å.
The excitons perturbed by Ce 3+ -related centers in SC and SCF of YAP:Ce
At 4.2 K, the emission spectrum of Ce 3+ centers in YAP:Ce consists of two components peaking at 3.30 eV and 3.55 eV (see, e.g., [3,6]). In SC and SCF of YAP:Ce, the excitation spectra of the Ce 3+ emission in the exciton absorption region are considerably different (see also [7,9]). The main exciton band is located at ~7.7 eV in SC (Fig. 9a), but at about 7.9 eV, in SCF (Fig. 9b). These bands are most probably arising from an exciton perturbed by a Ce 3+ -related defect. As the main difference between SCF and SC is in the absence of antisite defects in SCF, we assume that the 7.9 eV band arises from an exciton perturbed by a single Ce 3+ ion, while the band around 7.7 eV, from an exciton perturbed by a Ce 3+ ion associated with the antisite defect Y 3+ Al (see Table 1). This suggestion is confirmed by the fact, that in LuYAP:Ce, where the content of antisite defects should be larger [29], the abovementioned difference in the excitation spectra in SC and SCF is also larger [9]. The presence of nonequivalent Ce 3+ centers in YAP:Ce and LuYAP:Ce has been noticed also in [6,30].
The comparison of the excitation spectra of the slow and fast decay components of the Ce 3+ emission in the exciton region ( Fig. 9) indicates that in YAP SC, unlike YAP SCF, a competition exists between the processes responsible for these components (see also [7][8][9]). Indeed, the excitation spectrum maximum of the fast component coincides with the excitation spectrum minimum of the slow component. In [6], a similar competition was observed in YAP:Ce SC between the processes of defects creation and the excitation of the Ce 3+ emission in the exciton absorption region. As the absorption/excitation bands of Ce 3+ are strongly overlapped with the intrinsic emission bands of YAP SC and SCF (see Fig. 1), one can assume that the appearance of the slow components in the Ce 3+ emission decay is caused by the energy transfer from the corresponding defects to Ce 3+ ions. However, the excitation bands of the defect-related emissions (Figs. 2b-5) do not coincide with the excitation spectrum of the slow component of the Ce 3+ -related emission in the exciton absorption region (Fig. 9).
Only in the 7.6-7.7 eV range, the slow component of the Ce 3+ emission in YAP SC can arise from the above-mentioned energy transfer processes. Therefore, one can assume that the slow component of the Ce 3+ emission appears around 8.0 eV mainly due to the tunneling recombinations between electron centers and hole Ce 4+ centers produced as a result of the photostimulated disintegration of the regular exciton. The tunnelling recombination processes in YAP were considered also in [22,31]. In the energy range <7.5 eV, the slow component can appear in YAP SC due to the tunneling recombinations in the pairs consisting of an antisite-defect-related electron center and hole Ce 4+ center, both produced as a result of the photoionization of the higher-energy excited states of Ce 3+ or the photostimulated electron transfer processes (see also [6]). The presence of {Y 3+ Al -defect} centers in YAP SC was confirmed by the ESR studies [23].
The comparison of the lowest-energy exciton bands in the excitation spectra of the lead-related emission (Fig. 3) and the Ce 3+ emission (Fig. 9b) indicates that there is no energy transfer from the lead-induced centers to Ce 3+ -related centers. However, the energy transfer from Ce 3+ to dimer lead centers can take place (see also [10]).
89 Y and 27 Al NMR study
We have also performed studies of 89 Y, and 27 Al NMR in YAP:Ce grinded single crystals and powders including also nanopowders and nanocrystalline ceramics with the aim to prove existence of Y Al and Al Y antisite defects and to determine possible secondary phases in these materials. Such defects are believed to act as effective traps for charge carriers. NMR provides a unique approach to quantitatively measure the site occupancies and other imperfections in crystalline and even amorphous or liquid materials. NMR spectra of powders also give structural information. However, in contrast to single crystal measurements such spectra need extensive simulations. In particular, we found that 89 Y NMR is very sensitive to the presence of antisite-related defects while 27 Al NMR is more suitable for detection of the unwanted secondary phases. 89 Y NMR spectra measured in grinded single crystal and commercial powder are shown in Fig. 10. NMR spectrum in grinded single crystal is well fitted by orthorhombic symmetry chemical shift tensor with δ = -121 ppm and asymmetry parameter η = 0.12.
In commercial powder NMR spectrum is complex. It contains two components. One component is completely similar to that measured in grinded single crystal and the second one is described by the [23]. 27 Al NMR spectra were measured in grinded single crystals, microcrystalline powder and nanocrystalline ceramics (Fig. 11). One can see that the spectra in grinded single crystal and microcrystalline powder are practically similar while the spectrum in the nanocrystalline ceramics is much broader. It contains several components and only one component is related to the perovskite phase of YAP. Calculation shows that the nanocrystalline ceramics studied by us contained about 80% of the secondary non-perovskite phase. Therefore the synthesis method of YAP nanopowder has to be essentially improved.
Conclusions
The detailed study of luminescence characteristics of undoped and Ce 3+ -doped SC and SCF of YAP and the analysis of published data has allowed us to identify the origin and structure of the defects responsible for various localized exciton emission bands. In YAP SCF, the relatively weak ~5.6 eV emission of the STE is the dominating one. In YAP SC, the STE emission is practically absent. The
(
Fig. 1a, curves 1-3) which should arise from the STE. The excitation band of this emission is located at about 7.83 eV(Fig. 2a)(see
related centers, respectively. The appearance of the inner-centre 3.63 eV and 3.15 eV emissions under excitation in the exciton 7.66 eV and 7.74 eV bands means that the excitation energy is effectively transferred to the corresponding lead centers. Different positions of the excitation bands of the 4.16 eV emission and the 3.63 eV, 3.15 eV emissions
(
Fig. 6) indicating to the absence of antisite defects. In ceramics and nanopowders, the 2.45 eV emission is absent as well. Unlike YAP SC, in these materials the centers with the 2.45 eV photoluminescence are not created by the X-ray irradiation. Thermal stability of the ex 0 {Y 3+ Al -V O } excitons should be higher as compared with the ex 0 Y 3+ Al excitons. Under 8.3-8.4 eV excitation, the 5.63 eV emission intensity is constant up to 150 K and then
. We found the antisite defect formation energies in the presence of the nearest-neighbouring oxygen vacancy as well as in the presence of Ce 3+ impurity ion. The formation of antisite defect is preferable in cases where there is an oxygen vacancy near, which settles close to the direction Al -V O -Al. The formation energy of such type defect is by 25 % lower as compared with the energy of the antisite ion with vacancy V O in more far environment.
following parameters: δ = -35 ppm and η = 0.62. The second component is probably related to both Y Al antisite-related defects and resonances from near-surface regions of crystallites. Calculation shown that the total concentration of such defective Y sites in commercial YAP powder is up to 15-20%. On other hand, in commercial YAP crystals only Y Al antisite defects were detected with the concentration one order lower, i.e. about 2-3%
5.63 eV emission arises from ex 0 Y 3+ Al , while the 4.12 eV emission, from ex 0 {Y 3+ Al -V O }. The antisite defects of the type of Y 3+ Al and Y 3+ Al -V O are absent in the YAP SCF, nanopowders and ceramics studied. In X-irradiated YAP:Ce SC, the thermally stimulated destruction of the corresponding electron centers around 200 K and 280 K, respectively, and recombination of the released electrons with Ce 4+ centers is accompanied with the Ce 3+ emission. The possibility of an effective formation of the {Y 3+ Al -V O }-type defects in YAP single crystals is confirmed by the theoretical calculations. In the excitation spectra of various intrinsic, defect-related and impurity-related emission bands, the regular and various perturbed exciton bands are identified. The comparison of their positions allowed to clarify the influence of various defects on the appearance of undesirable slow components in the luminescence decay kinetics of Ce 3+ centers which negatively influence the scintillation characteristics of YAP:Ce. The conclusion is made that the slow decay is mainly caused by the tunnelling recombination of electron and hole centers created at the disintegration of the regular exciton states and the photoionization of Ce 3+ ions. The contribution of the energy transfer processes between defect and Ce 3+ states is found to be smaller. 27. N. L. Ross, J.Zhao, R. J. Angel, J. Solid State Chem. 177, A. Vedda, M. Fasoli, M. Nikl, V.V. Laguta, E. Mihokova, J. Peichal, A. Yoshikawa, M.Zhuravleva, Phys. Rev. B 80, 045113 (2009).
Figure captions
Figure captions
Fig. 1 .
1(a) Emission spectra of YAP SCF measured under 8.4 eV (curve 1), 8.15 eV (curve 2), 7.95 eV (curve 3) and 7.6 eV (curve 4) excitations. (b) Time-resolved emission spectra of YAP SC measured under E exc =7.95 eV. Excitation spectrum of the fast component of the Ce 3+ emission in YAP:Ce SC (dotted line). T=10 K.
Fig. 2 .
2Time-resolved excitation spectra of YAP SCF (shown in the 6-9 eV range) measured at 10 K for (a) the STE emission and (b) the 4.16 eV emission of the localized exciton.
Fig. 3 .
3Time-resolved excitation spectra of YAP SCF (shown in the 4-10 eV range) measured at 10 K for (a) the 3.63 eV emission of single Pb 2+ -based centers and (b) the 3.15 eV emission of dimer Pb 2+ centers.
Fig. 4 .
4Time-resolved excitation spectra of YAP SC measured at 10 K for (a) E em =5.7 eV and (b) E em =5.2 eV.
Fig. 5 .
5Time-resolved excitation spectra of YAP SC (shown in the 6-9 eV energy range) measured for: (a) E em =5.7 eV and (b) E em =4.1 eV. T=10 K.
Fig. 6 .
6(a) X-ray excited emission spectra of YAP single crystal, ceramics, and nanopowder measured at 80 K at the same conditions.
Fig. 7 .
7TSL glow curves measured with the heating rate 5 K/min for different YAP:Ce samples after their X-ray irradiation at the same conditions (70 K, 30 min, 40 kV, 15 mA): (a) single crystal (7566/1) (solid line) and powder QM58/N-S1 (Phosphor Technology UK) (dashed line); (b) ceramics (2 wt% Ce) (solid line) and nanopowder (2 wt% Ce) (dashed line).
Fig. 8 .
8The total energy dependence on the displacement of Y 3+ ion which replaces Al 3+ obtained for the case when the oxygen vacancy is close by.
Fig. 9 .
9Time-resolved excitation spectra of the Ce 3+ emission measured at 10 K for (a) YAP:Ce SC and (b) YAP:Ce SCF (shown only in the 6-9 eV range).
Fig. 10 .
1089 Y NMR spectra measured at Larmor frequency 19.607 MHz in (a) grinded single crystal and (b) commercial powder of YAP. Points are measured spectra and solid lines are computer simulated spectra.
Fig. 11 .
1127 Al NMR spectra measured in grinded single crystal, microcrystalline powder and nanocrystalline ceramics.
Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .
123456(a) Emission spectra of YAP SCF measured under 8.4 eV (curve 1), 8.15 eV (curve 2), 7.95 eV (curve 3) and 7.6 eV (curve 4) excitations. (b) Time-resolved emission spectra of YAP SC measured under E exc =7.95 eV. Excitation spectrum of the fast component of the Ce 3+ emission in YAP:Ce SC (dotted line). T=10 K. Time-resolved excitation spectra of YAP SCF (shown in the 6-9 eV range) measured at 10 K for (a) the STE emission and (b) the 4.19 eV emission of the localized exciton. Time-resolved excitation spectra of YAP SCF (shown in the 4-10 eV range) measured at 10 K for (a) the 3.63 eV emission of single Pb 2+ -based centers and (b) the 3.15 eV emission of dimer Pb 2+ centers. Time-resolved excitation spectra of YAP SC measured at 10 K for (a) E em =5.7 eV and (b) E em =5.2 eV. Time-resolved excitation spectra of YAP SC (shown in the 6-9 eV energy range) measured for: (a) E em =5.7 eV and (b) E em =4.1 eV. T=10 K. (a) Emission spectra of YAP single crystal, ceramics and nanopowder measured at the same conditions at 80 K under the X-ray excitation.
Fig. 7 .
7TSL glow curves measured with the heating rate 5 K/min for different YAP:Ce samples after their X-ray irradiation at the same conditions (70 K, 30 min, 40 kV, 15 mA): (a) single crystal (7566/1) (solid line) and powder QM58/N-S1 (Phosphor Technology UK) (dashed line); (b) ceramics (2 wt% Ce) (solid line) and nanopowder (2 wt% Ce) (dashed line).
Fig. 8 .Fig. 9 .
89The total energy dependence on the displacement of Y 3+ ion which replaces Al 3+ obtained for the case when the oxygen vacancy is close by. Time-resolved excitation spectra of the Ce 3+ emission measured at 10 K for (a) YAP:Ce SC and (b) YAP:Ce SCF (shown only in the 6-9 eV range).
Fig. 10 .Fig
1089 Y NMR spectra measured at Larmor frequency 19.607 MHz in (a) grinded single crystal and (b) commercial powder of YAP. Points are measured spectra and solid lines are computer simulated spectra. . 11. 27 Al NMR spectra measured in grinded single crystal, microcrystalline powder and nanocrystalline ceramics.
Table 1 .
1Luminescence characteristics of perturbed and localized excitons and defects in YAP SC and YAP SCF at 10 K. The maxima positions obtained in the present paper at the same conditions are shown in bold. Table 2. Structural parametres of YAlO 3 . Space group Pmna (53) and lattice parametres a = 5.334 Å, b = 7.375 Å, c = 5,180 Å were taken from the x-ray diffraction data [27]. The fractional atomic coordinates, Y (x,1/4, z), Al (0,0,1/2) O1 (x,1/4,z), O2 (x, y, z) are obtained by total energy minimization and compared with x-ray refinement (experiment). There are four formula units per primitive cell.Sample Exciton
Exciton emission,
eV
Defect emission,
eV
Lowest-energy exciton band,
eV
YAP SC ex 0 Y 3+
Al
5.9 [12]; 5.69 [8];
5.63
-
7.88 [8]; 7.95 [16]; 7.9
ex 0 {Y 3+
Al -V O } 4.12
2.45 [29]
7.7; 7.13
ex 0 V O
4.2 [24];
-
7.67 [17]; 7.6 [13]; 7.52 [8]
ex 0 Ce 3+
No
Ce 3+
7.80 [9]; 7.9
ex 0 {Ce 3+ -Y 3+
Al } No
~Ce 3+
7.75 [9]; ~7.7
YAP SCF STE
~5.6
-
7.91 [17]; 7.83
ex 0 (Pt 4+ )
4.16
-
~7.6
ex 0 (Pb 2+ )
4.19 [8]
-
7.59 [8]
ex 0 (Pb 2+ )
No
3.63 [8]
7.66
ex 0 (Pb 2+ -V O -
Pb 2+ }
No
3.15 [10]
7.74
ex 0 Ce 3+
No
Ce 3+
7.80 [9]; 7.9
Other code [28]
FPLO
Experiment [27]
Y x
0.0562
0.0540
0.0526
Y z
0.9868
0.9881
0.9896
O1 x
0.4756
0.4756
0.475
O1 z
0.0884
0.0884
0.086
O2 x
0.2949
0.2949
0.293
O2 y
0.0465
0.0465
0.044
O2 z
0.7045
0.7045
0.703
Table 3 .
3Calculated LDA total and defect energies of YAlO 3 based on 20-atom supercells.Type of defects
Energy of
antisite pair
defect (eV)
Total energy
(Hartree) 1Hr=27.2
eV
YAlO 3
-15394.773028
Al Y -Y Al (antisite pair)
10.8
-15394.370739
Al Y -Y Al (V O is nearby)
8.1
-15319.478418
Al Y -Y Al (V O is far)
10.6
-15319.384259
Al Y Y Al -Ce 3+
Y
10.2
-20871.244242
Y Al (V O is far)
-18459.58375
Y Al (V O is nearby)
-18459.66158
Ce 3+
Y (V O is nearby)
-20796.646814
Ce 3+
Y (V O is far)
-20796.637682
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|
[] |
[
"Alpha Conjugate Neck Structures in the Collisions of 35 MeV/nucleon 40 Ca with 40 Ca",
"Alpha Conjugate Neck Structures in the Collisions of 35 MeV/nucleon 40 Ca with 40 Ca"
] |
[
"K Schmidt \nCyclotron Institute\nTexas A&M University\n77843College Station, Texas\n\nInstitute of Physics\nUniversity of Silesia\nKatowicePoland\n",
"X Cao \nCyclotron Institute\nTexas A&M University\n77843College Station, Texas\n\nShanghai Institute of Applied Physics\nChinese Academy of Sciences\n201800ShanghaiChina\n",
"E J Kim \nCyclotron Institute\nTexas A&M University\n77843College Station, Texas\n\nDivision of Science Education\nChonbuk National University\n561-756JeonjuKorea\n",
"K Hagel \nCyclotron Institute\nTexas A&M University\n77843College Station, Texas\n",
"M Barbui \nCyclotron Institute\nTexas A&M University\n77843College Station, Texas\n",
"J Gauthier \nCyclotron Institute\nTexas A&M University\n77843College Station, Texas\n",
"S Wuenschel \nCyclotron Institute\nTexas A&M University\n77843College Station, Texas\n",
"G Giuliani \nCyclotron Institute\nTexas A&M University\n77843College Station, Texas\n\nLaboratori Nazionali del Sud\nINFN\nvia Santa Sofia, 6295123CataniaItaly\n",
"M R D Rodrigues \nCyclotron Institute\nTexas A&M University\n77843College Station, Texas\n\nInstituto de Física\nUniversidade de São Paulo\nCaixa Postal\n66318, 05389-970São PauloCEP, SPBrazil\n",
"H Zheng \nCyclotron Institute\nTexas A&M University\n77843College Station, Texas\n\nLaboratori Nazionali del Sud\nINFN\nvia Santa Sofia, 6295123CataniaItaly\n",
"M Huang \nCyclotron Institute\nTexas A&M University\n77843College Station, Texas\n\nCollege of Physics and Electronics information\nInner Mongolia University for Nationalities\n028000TongliaoChina\n",
"N Blando \nCyclotron Institute\nTexas A&M University\n77843College Station, Texas\n",
"A Bonasera \nCyclotron Institute\nTexas A&M University\n77843College Station, Texas\n\nLaboratori Nazionali del Sud\nINFN\nvia Santa Sofia, 6295123CataniaItaly\n",
"R Wada \nCyclotron Institute\nTexas A&M University\n77843College Station, Texas\n",
"C Botosso \nCyclotron Institute\nTexas A&M University\n77843College Station, Texas\n",
"G Liu \nShanghai Institute of Applied Physics\nChinese Academy of Sciences\n201800ShanghaiChina\n",
"G Viesti \nDipartamento di Fisica dell'Università di Padova and INFN Sezione di Padova\nItaly\n",
"S Moretto \nDipartamento di Fisica dell'Università di Padova and INFN Sezione di Padova\nItaly\n",
"G Prete \nINFN Laboratori Nazionali di Legnaro\nItaly\n",
"S Pesente \nDipartamento di Fisica dell'Università di Padova and INFN Sezione di Padova\nItaly\n",
"D Fabris \nDipartamento di Fisica dell'Università di Padova and INFN Sezione di Padova\nItaly\n",
"Y El Masri \nUniversite Catholique de Louvain\nLouvain-la-NeuveBelgium\n",
"T Keutgen \nUniversite Catholique de Louvain\nLouvain-la-NeuveBelgium\n",
"S Kowalski \nInstitute of Physics\nUniversity of Silesia\nKatowicePoland\n",
"A Kumar \nDepartment of Physics\nNuclear Physics Laboratory\nBanaras Hindu University\nVaranasiIndia\n",
"G Zhang \nCyclotron Institute\nTexas A&M University\n77843College Station, Texas\n\nShanghai Institute of Applied Physics\nChinese Academy of Sciences\n201800ShanghaiChina\n",
"J B Natowitz \nCyclotron Institute\nTexas A&M University\n77843College Station, Texas\n"
] |
[
"Cyclotron Institute\nTexas A&M University\n77843College Station, Texas",
"Institute of Physics\nUniversity of Silesia\nKatowicePoland",
"Cyclotron Institute\nTexas A&M University\n77843College Station, Texas",
"Shanghai Institute of Applied Physics\nChinese Academy of Sciences\n201800ShanghaiChina",
"Cyclotron Institute\nTexas A&M University\n77843College Station, Texas",
"Division of Science Education\nChonbuk National University\n561-756JeonjuKorea",
"Cyclotron Institute\nTexas A&M University\n77843College Station, Texas",
"Cyclotron Institute\nTexas A&M University\n77843College Station, Texas",
"Cyclotron Institute\nTexas A&M University\n77843College Station, Texas",
"Cyclotron Institute\nTexas A&M University\n77843College Station, Texas",
"Cyclotron Institute\nTexas A&M University\n77843College Station, Texas",
"Laboratori Nazionali del Sud\nINFN\nvia Santa Sofia, 6295123CataniaItaly",
"Cyclotron Institute\nTexas A&M University\n77843College Station, Texas",
"Instituto de Física\nUniversidade de São Paulo\nCaixa Postal\n66318, 05389-970São PauloCEP, SPBrazil",
"Cyclotron Institute\nTexas A&M University\n77843College Station, Texas",
"Laboratori Nazionali del Sud\nINFN\nvia Santa Sofia, 6295123CataniaItaly",
"Cyclotron Institute\nTexas A&M University\n77843College Station, Texas",
"College of Physics and Electronics information\nInner Mongolia University for Nationalities\n028000TongliaoChina",
"Cyclotron Institute\nTexas A&M University\n77843College Station, Texas",
"Cyclotron Institute\nTexas A&M University\n77843College Station, Texas",
"Laboratori Nazionali del Sud\nINFN\nvia Santa Sofia, 6295123CataniaItaly",
"Cyclotron Institute\nTexas A&M University\n77843College Station, Texas",
"Cyclotron Institute\nTexas A&M University\n77843College Station, Texas",
"Shanghai Institute of Applied Physics\nChinese Academy of Sciences\n201800ShanghaiChina",
"Dipartamento di Fisica dell'Università di Padova and INFN Sezione di Padova\nItaly",
"Dipartamento di Fisica dell'Università di Padova and INFN Sezione di Padova\nItaly",
"INFN Laboratori Nazionali di Legnaro\nItaly",
"Dipartamento di Fisica dell'Università di Padova and INFN Sezione di Padova\nItaly",
"Dipartamento di Fisica dell'Università di Padova and INFN Sezione di Padova\nItaly",
"Universite Catholique de Louvain\nLouvain-la-NeuveBelgium",
"Universite Catholique de Louvain\nLouvain-la-NeuveBelgium",
"Institute of Physics\nUniversity of Silesia\nKatowicePoland",
"Department of Physics\nNuclear Physics Laboratory\nBanaras Hindu University\nVaranasiIndia",
"Cyclotron Institute\nTexas A&M University\n77843College Station, Texas",
"Shanghai Institute of Applied Physics\nChinese Academy of Sciences\n201800ShanghaiChina",
"Cyclotron Institute\nTexas A&M University\n77843College Station, Texas"
] |
[] |
The de-excitation of alpha-conjugate nuclei produced in reactions of 35 MeV/nucleon 40 Ca with 40 Ca has been investigated. Particular emphasis is placed on examining the dynamics of collisions leading to projectile-like fragment exit channels. A general exploration of the reaction systematics reveals the binary dissipative character of the collisions and a hierarchy effect similar to that seen for heavier systems. Investigation of the subset of events characterized by a total α-conjugate mass (α particles plus α-conjugate fragments) equal to 40 and atomic number equal to 20 reveal a dominance of α-conjugate exit channels. The hierarchy effect for these channels leads to the production of αclustered neck structures with potentially exotic geometries and properties.
|
10.1103/physrevc.95.054618
|
[
"https://arxiv.org/pdf/1612.00267v2.pdf"
] | 119,350,877 |
1612.00267
|
c0fd5749d3f9d5c198c36cc7040bb0515a9d9e4c
|
Alpha Conjugate Neck Structures in the Collisions of 35 MeV/nucleon 40 Ca with 40 Ca
K Schmidt
Cyclotron Institute
Texas A&M University
77843College Station, Texas
Institute of Physics
University of Silesia
KatowicePoland
X Cao
Cyclotron Institute
Texas A&M University
77843College Station, Texas
Shanghai Institute of Applied Physics
Chinese Academy of Sciences
201800ShanghaiChina
E J Kim
Cyclotron Institute
Texas A&M University
77843College Station, Texas
Division of Science Education
Chonbuk National University
561-756JeonjuKorea
K Hagel
Cyclotron Institute
Texas A&M University
77843College Station, Texas
M Barbui
Cyclotron Institute
Texas A&M University
77843College Station, Texas
J Gauthier
Cyclotron Institute
Texas A&M University
77843College Station, Texas
S Wuenschel
Cyclotron Institute
Texas A&M University
77843College Station, Texas
G Giuliani
Cyclotron Institute
Texas A&M University
77843College Station, Texas
Laboratori Nazionali del Sud
INFN
via Santa Sofia, 6295123CataniaItaly
M R D Rodrigues
Cyclotron Institute
Texas A&M University
77843College Station, Texas
Instituto de Física
Universidade de São Paulo
Caixa Postal
66318, 05389-970São PauloCEP, SPBrazil
H Zheng
Cyclotron Institute
Texas A&M University
77843College Station, Texas
Laboratori Nazionali del Sud
INFN
via Santa Sofia, 6295123CataniaItaly
M Huang
Cyclotron Institute
Texas A&M University
77843College Station, Texas
College of Physics and Electronics information
Inner Mongolia University for Nationalities
028000TongliaoChina
N Blando
Cyclotron Institute
Texas A&M University
77843College Station, Texas
A Bonasera
Cyclotron Institute
Texas A&M University
77843College Station, Texas
Laboratori Nazionali del Sud
INFN
via Santa Sofia, 6295123CataniaItaly
R Wada
Cyclotron Institute
Texas A&M University
77843College Station, Texas
C Botosso
Cyclotron Institute
Texas A&M University
77843College Station, Texas
G Liu
Shanghai Institute of Applied Physics
Chinese Academy of Sciences
201800ShanghaiChina
G Viesti
Dipartamento di Fisica dell'Università di Padova and INFN Sezione di Padova
Italy
S Moretto
Dipartamento di Fisica dell'Università di Padova and INFN Sezione di Padova
Italy
G Prete
INFN Laboratori Nazionali di Legnaro
Italy
S Pesente
Dipartamento di Fisica dell'Università di Padova and INFN Sezione di Padova
Italy
D Fabris
Dipartamento di Fisica dell'Università di Padova and INFN Sezione di Padova
Italy
Y El Masri
Universite Catholique de Louvain
Louvain-la-NeuveBelgium
T Keutgen
Universite Catholique de Louvain
Louvain-la-NeuveBelgium
S Kowalski
Institute of Physics
University of Silesia
KatowicePoland
A Kumar
Department of Physics
Nuclear Physics Laboratory
Banaras Hindu University
VaranasiIndia
G Zhang
Cyclotron Institute
Texas A&M University
77843College Station, Texas
Shanghai Institute of Applied Physics
Chinese Academy of Sciences
201800ShanghaiChina
J B Natowitz
Cyclotron Institute
Texas A&M University
77843College Station, Texas
Alpha Conjugate Neck Structures in the Collisions of 35 MeV/nucleon 40 Ca with 40 Ca
(Dated: October 22, 2018)APS/123-QEDPACS numbers: 2570Mn, 2570Pg
The de-excitation of alpha-conjugate nuclei produced in reactions of 35 MeV/nucleon 40 Ca with 40 Ca has been investigated. Particular emphasis is placed on examining the dynamics of collisions leading to projectile-like fragment exit channels. A general exploration of the reaction systematics reveals the binary dissipative character of the collisions and a hierarchy effect similar to that seen for heavier systems. Investigation of the subset of events characterized by a total α-conjugate mass (α particles plus α-conjugate fragments) equal to 40 and atomic number equal to 20 reveal a dominance of α-conjugate exit channels. The hierarchy effect for these channels leads to the production of αclustered neck structures with potentially exotic geometries and properties.
I. INTRODUCTION
Nuclei are normally treated as consisting of fermions. However, in medium correlations and the strong binding of the α particle can lead to situations in which an α cluster picture can be employed to understand nuclear structure and decay properties [1][2][3][4]. Both theoretical calculations and experimental observations provide strong support for the α clustered nature of light α -conjugate (even-even N=Z) nuclei [5][6][7]. Loosely bound states with excitation energies near the alpha emission thresholds states may be a manifestation of the tendency of low density low temperature nuclear matter to undergo Bose condensation [8][9][10][11][12]. For example, the 7.65 MeV Hoyle-state in 12 C , important for the solar 3α capture process [13] is known to possess a large radius [14], which could allow the α particles to retain their quasi-free characteristics.
The role of α clusters in reaction dynamics is itself an interesting topic. Cluster effects are often seen in transfer reactions involving light nuclei [15]. Studies of more violent collisions of α conjugate nuclei might reveal impor- * Electronic address: [email protected] tant effects of these correlations on the collision dynamics and in determination of the reaction exit channels. Given that near Fermi energy nuclear collisions can drastically modify the temperatures, densities and cluster properties of nucleonic matter, the possibility that short-lived Bose Condensates might be fleetingly produced in such collisions is an intriguing idea. Recently the emission of three α from the Hoyle state has been characterized for 12 C produced in several different reactions [16][17][18][19]. Results for the ratio of simultaneous to sequential de-excitation differ and the influence of medium or proximity effects on the de-excitation modes of that state in complex reactions remains an open question. The authors of reference [20] have argued that enhanced α emission occurs during the thermal expansion of 16 O , 20 Ne and 24 Mg projectilelike fragments produced in 25 MeV/nucleon, 40 Ca + 12 C collisions, reflecting the α-conjugate nature of the parent fragments. Signatures of and possible evidence for Bose Einstein condensation and Fermi quenching in the decay of hot nuclei produced in 35MeV/nucleon 40 Ca + 40 Ca collisions have been discussed in references [21][22][23]. Evidence of cluster effects in the dynamics at much higher energies were reported in reference [24].
To pursue the question of the effects of α-like correlations and clustering in collisions between α-conjugate nuclei we have embarked on a program of experimental studies of such collisions at and below the Fermi energy using the NIMROD-ISiS array at TAMU [25]. A dominating α clustered nature of the colliding matter could manifest itself in the kinematic properties and yields of the α conjugate products. While the granularity of our detection system is not sufficient for high resolution fragment and particle correlations, we are able to explore certain features of the reactions which lead to large cross sections for α conjugate reaction products.
In this paper we report results for a study of α clusterization effects in mid-peripheral collisions of 40 Ca + 40 Ca at 35 MeV/nucleon. We first present some global observations and then focus on collisions in which excited projectile-like fragments disassemble into α-conjugate products.
II. EXPERIMENTAL DETAILS
The experiment was performed at Texas A&M University Cyclotron Institute. 40 Ca beams produced by the K500 superconducting cyclotron impinged on 40 Ca targets at the energy of 35 MeV/nucleon. The reaction products were measured using a 4π array, NIMROD-ISiS (Neutron Ion Multidetector for Reaction Oriented Dynamics with the Indiana Silicon Sphere) [25] which consisted of 14 concentric rings covering from 3.6 • to 167 • in the laboratory frame. In the forward rings with θ lab ≤ 45 • , two special modules were set having two Si detectors (150 and 500 µm) in front of a CsI(Tl) detector (3 − 10 cm), referred to as super-telescopes. The other modules (called telescopes) in the forward and backward rings had one Si detector (one of 150, 300 or 500 µm) followed by a CsI(Tl) detector. The pulse shape discrimination method was employed to identify the light charged particles with Z ≤ 3 in the CsI(Tl) detectors. Intermediate mass fragments (IMFs), were identified with the telescopes and super-telescopes using the "∆E − E" method. In the forward rings an isotopic resolution up to Z = 12 and an elemental identification up to Z = 20 were achieved. In the backward rings only Z = 1-2 particles were identified, because of the detector energy thresholds. In addition, the Neutron Ball surrounding the NIMROD-ISiS charged particle array provided information on average neutron multiplicities for different selected event classes. Further details on the detection system, energy calibration, and neutron ball efficiency can be found in [25][26][27].
It is important to note that, for symmetric collisions in this energy range, the increasing thresholds with increasing laboratory angle lead to a condition in which the efficiencies strongly favor detection of projectile-like fragments from mid-peripheral events. The modeling of these collisions using an Antisymmetrized Molecular Dynamics (AMD) code [28,29] coupled with the statistical code GEMINI [30] as an afterburner, and applying the experimental filter demonstrates that this is primarily an effect of energy thresholds.
III. GENERAL CHARACTERIZATION OF THE REACTIONS
Our previous study of the 40 Ca + 40 Ca at 35 MeV/nucleon focused on the multi-fragment exit channels and led to the conclusion that even the most violent and most central collisions were binary in nature [31]. Similar conclusions on the dominant binary nature of reactions with 35 MeV/nucleon 24 Mg projectiles were reported by Larochelle et al. [32].
We initiated the present analysis of the new data by reconstructing the "initial apparent excitation energy", E * , of the projectile-like fragments through calorimetry. E * was defined as the sum, for accepted particles, of the particle kinetic energies in the frame of the total projectile-like nucleus (determined by reconstruction of the mass and velocity of the primary excited nucleus from its de-excitation products), minus the reaction Qvalue. See equation (1).
E * = M i=1 K cp (i) + M n K n − Q.(1)
Here M is the total charged particle multiplicity, K cp (i) is the source frame kinetic energy of charged particle i, M n is the average neutron multiplicity, K n is the average neutron kinetic energy and Q is the disassembly Q value. For this purpose the average kinetic energy of the neutrons was taken to be equal to the average proton kinetic energy with a correction for the Coulomb barrier energy. Average neutron multiplicities were determined by applying efficiency corrections to the average neutron multiplicities observed with the neutron ball [27]. For a compound nucleus this initial apparent excitation energy would correspond to the energy available for statistical decay of the primary nucleus. We caution that given the binary nature of the collisions studied, the deexciting projectile-like nucleus is not necessarily a fully equilibrated nucleus. Nevertheless, this measure of energy deposition into the systems studied can serve as a useful sorting parameter. For the initial event selection we included all particles and fragments detected in an event. As will be seen, this event selection is revised in subsequent sections where we employ a more restrictive filtering to derive excitation energies. In Fig. 1 the mass numbers, A, of the three heaviest fragments in each event are plotted against their laboratory-frame parallel velocities for 1 MeV increments in E * /A. The favored detection of projectile-like species for all windows is clearly seen in this figure. Most of the fragments have velocities above the center of mass velocity, 4.0[cm/ns]. Increasing excitation energy corresponds, at least qualitatively, to decreasing impact parameter and increased collision violence. This is manifested in the figure by the decrease in yields of the heaviest mass products and increasing yields of lighter mass products as excitation increases. At low excitation energies the majority of the heavier products have parallel velocities near the beam velocity of 8.0[cm/ns]. The similar mean lab velocities suggest that the lighter fragments are produced in by statistical de-excitation of the initial projectile-like fragment. As the excitation energy increases, a clear correlation between parallel velocity and fragment mass is observed. For these excitations, corresponding to the region of mid-peripheral collisions, the parallel velocity decreases as the fragment mass decreases. This trend could reflect a greater degree of energy dissipation with decreasing impact parameter and/or the onset of neck emission [33][34][35]. We shall return to this question. Fig. 2 shows the results of AMD-GEMINI calculations for this 35 MeV/nucleon 40 Ca + 40 Ca system filtered using our experimental geometries and thresholds. The AMD calculation [28,29] followed the reaction until 300 fm/c after the collision. The code GEMINI [30] was employed as an afterburner to de-excite the primary fragments. We note that the plots in Fig. 2 look qualitatively similar to those in Fig. 1. However at the lower excitation energies the AMD exhibit narrower velocity distributions and different yield distributions. This may be a manifestation of more transparency in the AMD collision than in the experiment [36].
IV. SELECTION OF A = 40, Z = 20 PLF
Our previous analyses of near Fermi energy collisions [26,27] indicate that significant proton emission occurs in the earliest stages of the collision as the nucleon momentum distributions are thermalizing, not in the later stage disassembly. To better characterize the source of the light particles in the selected events we explored the Z = 1 and Z = 2 light particle emission by carrying out both 2-source and 3-source fits assuming that the observed light charged particle emission can be attributed to primary sources moving in the laboratory frame, a projectile-like source (PLF), a target-like source (TLF) and a (virtual) intermediate velocity source (IV) moving at a velocity ∼ 1 /2 the projectile velocity [37]. This latter source reflects nucleon-nucleon collisions occurring early in the process. In each source frame the emission was assumed to have a Maxwellian distribution and each of the sources is described by a source velocity, temperature, Coulomb barrier and particle multiplicity [26]. A comparison of the total yields with those obtained from the source fits to the proton energy spectra indicates the proton emission is low, with average multiplicities ∼2 and is dominated by emission from an intermediate velocity source having an apparent velocity of ∼ 1 /2 that of the projectile rather than from later statistical de-excitation. For this light symmetric system we expect the same to be true for the neutrons. While the neutron kinetic energies are not accessible in this experiment, the efficiency corrected neutron multiplicities obtained using the neutron ball are similar to the proton multiplicities. For d and t emission the average multiplicities are much lower and about half the particles are emitted from the IV source. The 3 He emission was too low to allow reasonable fits. To pursue our analysis we focus on events for which A = 40 and Z = 20. However in this selection we have neglected both protons and neutrons.
V. TESTS OF STATISTICAL BEHAVIOR
Horn and co-workers suggested that the ratio of average excitation energy to the average exit channel separation energy could be used as a test for statistical emission from highly excited lighter nuclei [32-35, 37, 38]. For their model assumptions regarding a Fermi gas level density, negligible emission barriers and a linear increase of available exit channels with increasing excitation energy, they concluded that the ratio should be constant with a value near 2. They also concluded that the statistical variance of this ratio would be small enough to enable this ratio to be used as an identifier of statistical de-excitation on an event by event basis [38]. Experimental observations of constant values of the ratio have been cited as evidence for strong dominance of statistical deexcitation of projectile-like fragments [32-35, 37, 38]. In Fig. 3 we present, for all observed PLF exit channels with 10 or more events having A = 40 and Z = 20 (not including n or p as discussed above), a plot of average excitation energy, E * , vs exit channel separation energy, -Q. In general these data are similar to previous results [32,38,39]. A linear fit to these data leads to a slope parameter of 2.39. This result, well above 2, is close to that extracted in reference [32]. Based upon comparisons with statistical model results the authors of reference [32] concluded that there are important dynamic effects in mid peripheral and central reactions at 35 MeV/nucleon and above. A closer investigation of Fig 3 indicates that some prominent channels, particularly at lower separation energies, have ratios well above the average values. This observed deviation suggests that these reactions warrant additional exploration. We return to these results in the section VII.
VI. ALPHA-CONJUGATE EXIT CHANNELS
The main purpose of the present study was to explore exit channels composed of α particles or α-conjugate nuclei. To focus on such channels the event by event data were sorted as a function of the total detected "α-like mass", A L , i.e., the sum of the masses of the detected products that are either α particles or α-conjugate nuclei. Fig. 4 of the entrance channel mass is seen, but with very low statistics. The shoulder in the A L ∼40 region and rapid decrease beyond that reflects the detector selectivity for projectile-like fragments from mid-peripheral events.
VII. ALPHA-CONJUGATE AL = 40 EXIT CHANNELS
For the analyses which follow we have chosen to focus on those events for which A L = 40 and compare the properties of the 19 possible exit channels for the disassembly of the 40 Ca nucleus into α particles or α-conjugate nuclei. The 19 possible combinations of α -conjugate nuclei which satisfy this total α-conjugate mass = 40 criterion are schematically indicated in Fig. 5. This depiction is similar to that of the Ikeda diagram which is commonly invoked in discussions of the cluster structure of light nuclei [40]. The events selected typically have a few Z = 1 particles (and neutrons) and, in rare cases, a heavier nonα-conjugate fragment, associated with them. To further refine our event selection we exclude the fraction of the A L = 40 events (11%) with non-α-conjugate fragments from the analysis. In our selection we have allowed Z = 1 particles and neutrons but we have re-determined the excitation energies by excluding the Z = 1 particles and neutrons as they are primarily pre-equilibrium particles, representing energy dissipation but not energy deposition into the PLF [41]. This leads to slightly smaller excitation energies and introduces a small uncertainty. As invariant velocity plots for the α particles indicate that a small fraction of the α particles may result from pre-equilibrium emission or from the target like source, α particles with PLF source frame energies greater than 40 MeV were also excluded to remove those contributions. The excitation energy distributions derived for all A = 40, Z = 20 PLF events defined in this manner are presented in Fig. 6. The A L = 40 events account for 61% of the A = 40, Z = 20 PLF events detected. Detected events with α-conjugate mass = 40 account for to 0.23% of the total experimental events collected. Filtered AMD calculations predict about half that amount, 0.11%. In Fig. 7 the excitation functions for the different A L = 40 exit channels detected in this reaction are presented. The distribution of yields in the different exit channels are presented in Fig. 8 as percentages of the total A L = 40 yields. Both the experimental results and those from the filtered AMD-GEMINI calculation are presented. They both suggest that the most probable decay modes are those with one heavy α-like mass fragment and several α particles in the exit channel. While the two distributions are similar, there are some significant differences between the experimental and calculated results. In Fig. 9 channel of the decay of the selected A = 40, Z = 20 nuclei, the fractional yield vs the ratio of average excitation energy to exit channel separation energy (see Fig. 3). Results are presented for both the experimental data (top) and AMD simulation (bottom). Each exit channel is represented by a solid circle. We have further identified the A L = 40 exit channels using open diamonds. We see that, in both frames of Fig. 9, these channels are those with the largest values of E * /-Q from the systematics. Their ratios are well above the values for the other channels with similar separation energies and in general their yields are quite high. An exploration of other high yield channels reveals that these are generally channels in which the deviations from A L = 40 reflect the existence of deuterons or 6 Li nuclei in the exit channel. These are exit channels such as ( 30 P,2α,d),( 26 Al,3α,d), ( 22 Na,4α,d) and ( 22 Na, 6 Li,3α) for example. We identify such channels with additional open circles around the solid circles. These channels might well be those in which an initial breakup into α particles and/or α-conjugate fragments is followed by a secondary emission or break-up. If so, the fraction of initial α-conjugate break-ups of A = 40, Z = 20 nuclei is much larger than the 61% observed in suggest that it is the dynamic evolution which favors the extension of these excitation functions to higher energies and shifts the ratios higher. The degree to which this large fraction of α-conjugate de-excitations reflects the initial α-conjugate nature of 40 Ca or the dynamic evolution of the excitation and density warrants further investigation.
VIII. COLLISION DYNAMICS FOR AL = 40
To more explicitly probe the dynamics of the A L = 40 events we have constructed momentum space representations of the correlations among exit channel products using sphericity and co-planarity to characterize the event shapes [42,43]. Sphericity, S, and Coplanarity, C, are defined as:
S = 3 2 λ 1 + λ 2 λ 1 + λ 2 + λ 3 ,(2)C = √ 3 2 λ 2 − λ 1 λ 1 + λ 2 + λ 3 .(3)
Where the λ are the eigenvalues of the flow tensor in the c.m. of the source system and are ordered so that λ 1 < λ 2 < λ 3 . A combined plot of S and C reveals the dominant shape in momentum space.
The upper left panel of Fig. 10 provides a schematic representation of the interpretation of momentum space distributions using these coordinates. Events at 0.0, 0.0 are rod-like. Those at 0.75, 0.43 are disk like. Events along the line between these points are co-planar. The events at 1.0, 0.0 are spheres. Oblate and prolate shapes will appear in the regions between these extremes. It is important to note that the shapes in the sphericitycoplanarity plots do not reflect the actual geometric shape of the decaying nuclei, but they represent the shape of the momentum flow during the decay. In the rest of Fig. 10 we present the experimental sphericity -coplanarity plots for the A L = 40 exit channels. We do not include the channels with only two α-conjugate fragments which would necessarily appear at 0.0 in the sphericitycoplanarity plane. Most of exit channel event distributions fall closer to the co-planar region of the rod to disk axis than that of the sphere and only the larger multiplicity events approach the latter. In some previous work similar observations have been attributed to multiplicity effects [43]. While it is obvious that fluctuations will be important and that two and three fragment events will necessarily be co-planar in this representation, in general, the distribution will reflect the initial momentum distribution resulting from the collision as well as the mode and sequence of subsequent de-excitations and momentum conservation in that sequence rather than the multiplicity, per se. The generally prolate nature of the sphericity co-planarity plots of Fig. 10 suggest that the exit channels with large numbers of α particles result from processes in which an initial breakup into larger excited fragments is followed by α particle de-excitation. Under very specific circumstances of simultaneous fragmentation, the observed momentum space shape should be more directly related to the initial geometric configuration of the de-exciting system [44]. To understand these A L = 40 events in more detail, we have constructed invariant velocity distributions for the single fragment (xα) exit channels, Fig. 11, and the two fragment exit channels, Fig. 12. The products of the different decay channels are transformed into the rest frame of the reconstructed α-like mass 40 nucleus. The decay channels are indicated in the various panels. The vertical lines indicate the rest frame parallel velocity of the reconstructed emitting source. In these figures the right-hand panels show the invariant velocity distributions for the heaviest fragment in the event and the left-hand panels show the invariant velocity distribution for the α particles or other remnants of the de-excitation. In Fig. 11 we note that the velocity spectra of the heaviest fragment is peaked at a parallel velocity above the reconstructed source velocity while the α particle velocities are centered at lower parallel velocities than the reconstructed source velocity. We also note that the α particle velocity distributions become more symmetric about the source velocity as the multiplicity of α particles increases. In Fig. 12 we show the decay channels of α-like mass 40 nuclei which consist of pairs of heavier α-like mass fragments. Except for the symmetric two 20 Ne channel, we observe a similar behavior -the heavier fragment velocities are centered at velocities larger than the velocity of the decaying nucleus while the velocity distributions for the lighter fragments peak at parallel velocities smaller than the parallel velocity of the decaying nucleus.
To emphasize the generality of this observation for the A L = 40 exit channels, we show distributions of observed mass vs parallel velocity for all the different decay channels in Fig. 13. We note again in this figure that the heaviest fragment in the different decay channels always tends to be observed at velocities larger than that of the neck region and that light particles tend to be observed as originating from the velocity region between the source velocity and the COM velocity, i.e., from a neck region. To verify that the effect is real and not the result of some biasing by the experimental acceptance of NIMROD, we have done statistical model calculations using the statistical de-excitation code GEMINI. When these events were filtered through our experimental acceptance the resultant parallel velocity distributions remained symmetric about the source velocity. In the present case these necks exhibit important α clustering effects. The manifestation of this neck can be either a single α-conjugate fragment or one or more α particles either independently formed or derived from the de-excitation of an excited α-conjugate precursor. The observed emission patterns, in which the lighter fragments trail the heavier fragments, are strongly reminiscent of the "hierarchy" effect reported for other systems in a similar energy range [33,34]. It reflects a dynamics in which mass and velocity are correlated such that, for fragments emitted forward in the center of mass, the heaviest fragments are emitted at forward angles and are on average the fastest ones, the second heaviest fragment is the second fastest one, and so on. Such behavior is inconsistent with production of a fully equilibrated compound nucleus. Rather it signals a binary nature of the reaction with neck formation between the quasiprojectile and the quasi-target [34,35]. The breakup of this neck is fast enough that memory of the neck geometry is retained. Of course these emissions from the neck region are subject to possible modification by proximity effects [45][46][47][48][49]. The results in Figs. 11 -13 suggest that the α particles in xα events observed in the left panels of Fig. 11 could originate from the same process as the fragments seen in the left-hand side of Fig. 12. As previously noted, the 40 Ca + 12 C reaction at 25 MeV/nucleon populates excited states of 12 C nuclei which decay by 3α emission, primarily in a sequential manner [16]. It is reasonable to expect that similar excited α de-exciting states are produced in the present reaction. Indeed, we have already noted that the sphericity co-planarity plots of Fig. 10 suggest that the exit channels with large numbers of α particles result from processes in which an ini- tial breakup into larger excited fragments is followed by α particle de-excitation. Further evidence for such precursors is found in our data in the large numbers of 8 Be nuclei emitted. The granularity of the detector in our experiment is such that most of these are observed as two α particles simultaneously striking a single detector and identified by their combined ∆E, E signal. The granularity of our detector is not well suited to measuring the 8 Be correlation function so we do not pursue this question further.
IX. SUMMARY AND CONCLUSIONS
Reactions of 35 MeV/nucleon 40 Ca with 40 Ca have been investigated with an emphasis on peripheral and mid-peripheral collisions leading to excited projectilelike fragments. A global analysis of the de-excitation channels of A = 40 PLF fragments agrees with previous studies that total equilibration of all degrees of freedom is not achieved in the mid-peripheral collisions. A hierarchy effect is observed in the collision dynamics. The selection of the subset of A = 40 projectile-like fragment exit channels characterized by a total α-conjugate mass (α particles plus α-conjugate fragments) equal to 40 indicates that these projectile-like exit channels generally have important dynamic contributions. Most of the α particles observed in such events trail larger α-conjugate leading fragments and originate from α-conjugate neck structures formed during the collisions. The manifestation of this neck can be a single α-conjugate fragment or one or more α particles either independently formed or derived from the de-excitation of an excited α-conjugate precursor. This mechanism significantly increases the difficulty of isolating clean projectile decay samples [49] Transport model calculations typically indicate that the neck structures formed in mid-peripheral collisions have densities lower than normal density [34]. Lowering of the density is expected to favor α clustering. Using a constrained HFB approach, Girod and Schuck have explored the nuclear equation of state for self-conjugate N=Z nuclei and concluded that those nuclei will cluster into a metastable phase of α particles (or in some cases α-conjugate light clusters) at excitations above 3 MeV/nucleon and densities below 0.33 normal density [50]. We believe that the reaction dynamics observed in this paper can provide a natural entry point to study the disassembly of α clustered systems with potentially exotic geometries and properties. and by The Robert A. Welch Foundation under Grant #A0330. We appreciate useful conversations with S. Shlomo. S. Umar and A. Ono. We also greatly appreci-ate the continued excellent work of the staff of the TAMU Cyclotron Institute.
online.) Yields of the three heaviest fragments in the event as a function of the fragment parallel velocity in different windows of initial apparent excitation energy E * /A. The projectile velocity is 8.0[cm/ns]. The c.m. velocity is 4.0[cm/ns].These two velocities are indicated by vertical lines in each panel.
online.) Filtered AMD-GEMINI results, similar as Fig. 1.
depicts the resultant event yields. For a given total α-like mass, several different decay channels are often possible. Events for which all of the detected α conjugate mass is in α particles are indicated by the large open circles inFig. 4. A total α-like mass as large as 85% online.) Average excitation energy vs exit channel separation energy for the de-excitation channels of A = 40, Z = 20 nuclei selected as described in the text. Data are represented by small filled dots. The linear least squares fit to the data is represented by the solid line.
online.) Detected number of events yielding α particles or α-conjugate nuclei in the collision of 40 Ca with40 Ca at 35A MeV, plotted against total detected mass of α-conjugate nuclei. Small filled circles represent total yields.Open circles represent yields for events in which only α particles contribute to the AL.
online.) Excitation energy distributions for A = 40, Z = 20 derived as indicated in the text, The blue area represents the data for all such events. The hatched area represents the data for AL = 40 events.
FIG. 7 :
7Excitation functions for the AL = 40 events discussed in the text.
FIG. 8 :
8Percentages of AL = 40 events appearing in the possible exit channels. The experimental results are represented by solid dark grey bars. The filtered AMD-GEMINI results are represented by solid silver bars.
Fig. 6 .
6The excitation energy evolution of Figs. online.) The fraction of exit channel events as a function of the ratio of the average excitation energy to the separation energy. Top -Experimental data, Bottom -AMD Calculation. Each solid circle represents an exit channel, The AL = 40 channels are identified using large open diamonds. Events identified by open circles may be AL = 40 channels which have undergone secondary decays with d and 6 Li emissions (See text).
online.) Sphericity Co-planarity plots for the AL = 40 exit channels. Two-body exit channels are excluded. See text.
online.) Source frame invariant velocity plots for the α-like exit channels containing α particles. Vertical lines at 0 are to aid the eye in comparisons of these distributions.
online.) Source frame invariant velocity plots for the two fragment α-like exit channels. Vertical lines at 0 are to aid the eye in comparisons of these distributions.
FIG. 13: (Colour online.) Parallel velocity distributions for the α-conjugate exit channels. In each panel distributions are color coded for different products. Solid red diamonds: heaviest fragment in the event, black lines: second heaviest fragment, open blue circles: α particles.10 -2
10 -1
10 0
+ 36 Ar
2 + 32 S
3 + 28 Si
4 + 24 Mg
5 + 20 Ne
10 -2
10 -1
10 0 6 + 16 O
7 + 12 C
10
20 Ne+ 20 Ne 12 C+ 28 Si
10 -2
10 -1
10 0 16 O+ 24 Mg 2 12 C+ 16 O
2 +2 16 O
+ 12 C+ 24 Mg
2 4 6 8 10
+3 12 C
2 4 6 8 10
10 -2
10 -1
10 0 + 16 O+ 20 Ne
2 4 6 8 10
4 +2 12 C
2 4 6 8 10
3 + 12 C+ 16 O
2 4 6 8 10
2 + 12 C+ 20 Ne
m 1 > m 2 > m
m 1
m 2
m
V [cm/ns]
Number of Events
AcknowledgmentsThis work was supported by the United States Department of Energy under Grant #DE-FG03-93ER40773
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|
[] |
[
"Prefactor in the dynamically assisted Sauter-Schwinger effect",
"Prefactor in the dynamically assisted Sauter-Schwinger effect"
] |
[
"Christian Schneider \nFakultät für Physik\nUniversität Duisburg-Essen\nLotharstr. 147057DuisburgGermany\n",
"Ralf Schützhold \nFakultät für Physik\nUniversität Duisburg-Essen\nLotharstr. 147057DuisburgGermany\n"
] |
[
"Fakultät für Physik\nUniversität Duisburg-Essen\nLotharstr. 147057DuisburgGermany",
"Fakultät für Physik\nUniversität Duisburg-Essen\nLotharstr. 147057DuisburgGermany"
] |
[] |
The probability of creating an electron-positron pair out of the quantum vacuum by a strong electric field can be enhanced tremendously via an additional weaker time-dependent field. This dynamically assisted Sauter-Schwinger effect has already been studied in several works. It has been found that the enhancement mechanism depends on the shape of the weaker field. For example, a Sauter pulse 1/ cosh(ωt) 2 and a Gaussian profile exp(−ω 2 t 2 ) exhibit significant, qualitative differences. However, so far most of the analytical studies were focused on the exponent entering the pair-creation probability. Here, we study the subleading prefactor in front of the exponential using the worldline instanton method. We find that the main features of the dynamically assisted Sauter-Schwinger effect, including the dependence on the shape of the weaker field, are basically unaffected by the prefactor. To test the validity of the instanton approximation, we compare the number of produced pairs to a numerical integration of the full Riccati equation.
|
10.1103/physrevd.94.085015
|
[
"https://arxiv.org/pdf/1603.00864v3.pdf"
] | 118,407,703 |
1603.00864
|
9830400e7e626d18e4490fa2b41a0f579fbdce18
|
Prefactor in the dynamically assisted Sauter-Schwinger effect
Christian Schneider
Fakultät für Physik
Universität Duisburg-Essen
Lotharstr. 147057DuisburgGermany
Ralf Schützhold
Fakultät für Physik
Universität Duisburg-Essen
Lotharstr. 147057DuisburgGermany
Prefactor in the dynamically assisted Sauter-Schwinger effect
(Dated: 20 October 2016)PACS numbers: 1220-m, 1115Tk
The probability of creating an electron-positron pair out of the quantum vacuum by a strong electric field can be enhanced tremendously via an additional weaker time-dependent field. This dynamically assisted Sauter-Schwinger effect has already been studied in several works. It has been found that the enhancement mechanism depends on the shape of the weaker field. For example, a Sauter pulse 1/ cosh(ωt) 2 and a Gaussian profile exp(−ω 2 t 2 ) exhibit significant, qualitative differences. However, so far most of the analytical studies were focused on the exponent entering the pair-creation probability. Here, we study the subleading prefactor in front of the exponential using the worldline instanton method. We find that the main features of the dynamically assisted Sauter-Schwinger effect, including the dependence on the shape of the weaker field, are basically unaffected by the prefactor. To test the validity of the instanton approximation, we compare the number of produced pairs to a numerical integration of the full Riccati equation.
I. INTRODUCTION
The Sauter-Schwinger effect is a striking phenomenon predicted by Quantum Electrodynamics (QED), that describes nonperturbative pair creation from the QED vacuum by a strong electric field [1][2][3][4]. Intuitively, one can visualize this process as an electron tunnelling from the Dirac sea to the positive continuum. So far, direct experimental verification has not been possible, due to the extremely high critical field strength E S = m 2 c 3 /(hq) ≈ 1.3 × 10 18 V/m (corresponding to an intensity of 4.6 × 10 29 W/cm 2 ) where pair production is expected for a uniform, static electric field.
An extension that can significantly lower this threshold is dynamical assistance [5], where an additional weak, time dependent field with the frequency scalehω 2mc 2 is superimposed onto a static, or slowly varying field. In [6] the impact of different pulse shapes on the dynamically assisted Sauter-Schwinger mechanism has been compared, by calculating the exponent of the pair production rate, neglecting the fluctuation prefactor.
In the following, we will apply the worldline instanton method [7][8][9][10][11][12][13] to calculate the full pair production rate in the dynamically assisted Sauter-Schwinger effect for different shapes of time dependent pulses. Section II will give a summary of the method, which is then used in section III to yield both numerical results without any further approximations and analytical estimates in certain parameter regions. In section IV we will present the numerical methods used to solve the Riccati equation which gives the exact number of produced pairs (up to numerical accuracy).
We will work in 1+1 spacetime dimensions throughout, a choice that will be explained in section IV.
This is sufficient to represent the fields considered in this work, as we only need one spatial dimension (in * [email protected] which the electric field is oriented) and the temporal dimension (to study time dependent fields) [14].
II. WORLDLINE INSTANTON METHOD
Let us first briefly review the worldline instanton method (for a detailed derivation see, e.g. [11,12]), and in particular how the prefactor differs in 1 + 1 and 3 + 1 spacetime dimensions.
We start out with the vacuum persistence amplitude, i.e., the probability amplitude that an initial vacuum state |0 in remains vacuum |0 out , which can be expressed using the effective action Γ M ,
0 out |0 in = e iΓM .(1)
We use the subscript M to indicate Minkowskian quantities, in contrast to the Euclidean versions we will mostly be concerned with in the following. If the effective action were to gain an imaginary part, the absolute value of the vacuum persistence would deviate from one, which can be interpreted as the probability amplitude for pair production:
P e + e − = 1 − | 0 out |0 in | 2 ≈ 2 Γ M .(2)
After analytic continuation, the Euclidean effective action can be expressed using the worldline path integral
Γ[A µ ] = ∞ 0 dT T e −m 2 T d 2 x (0) x(T )=x(0)=x (0) Dx × exp − T 0 dτ ẋ 2 4 + iqA ·ẋ(3)
over closed loops x µ (τ ) in Euclidean spacetime, where µ only takes on the values 1 and 2, soẋ 2 =ẋ 2 1 +ẋ 2 2 and A ·ẋ = A 1ẋ1 + A 2ẋ2 . In this case x 1 denotes the spatial component (e.g. z) and x 2 imaginary time.
arXiv:1603.00864v3 [hep-th] 27 Oct 2016
Note that we are considering scalar QED here, i.e. a complex scalar field coupled to the electromagnetic potential. Compared to QED this lacks the spin degree of freedom, which for the class of fields studied in the following can be shown to result in a trivial factor of 2 only [11].
The worldline instanton approach is a semiclassical approximation to (3), by evaluating both the path integral and the integral over T using the saddle point method.
For the path integral, we need to find a path x µ (τ ) with x µ (T ) = x µ (0) that extremizes
A[x µ ](T ) = T 0 dτ ẋ 2 4 + iqA ·ẋ(4)
for a given T . The Euler-Lagrange equations for this action functional give the equations of motion
x µ = iqF µνẋν .(5)
A solution to (5) that satisfies the periodicity conditions is called a world line instanton, in analogy to the instantons in nonrelativistic quantum tunneling [15]. The saddle point approximation includes an additional prefactor, arising from the fluctuations around the extremum. For the path integral, this amounts to the determinant of a second order differential operator [16]. Remarkably, this determinant can be found using a finite determinant comprised of solutions to a certain initial value problem (for a derivation using methods of complex analysis, see [17]).
Including the fluctuation prefactor, the saddle point approximation of the path integral in (3) is given by
x(T )=x(0)=x (0) Dx exp − T 0 dτ ẋ 2 4 + iqA ·ẋ ≈ e iθ 4πT | det[η (ν) µ,free (T )]| | det[η (ν) µ (T )]| exp −A[x cl µ ](T ) . (6)
The η µ are solutions to the fluctuation equations of motion and θ is the Morse index [18] of the fluctuation operator (omitted here for brevity). In contrast to the (3+1)dimensional case, the finite determinants are 2 × 2 and we get a factor of (4πT ) −1 instead of (4πT ) −2 .
Let us now restrict ourselves to time dependent, homogeneous electric fields of constant direction. In particular, we use the Euclidean four-potential
iA 1 = E ω f (ωx 2 ),(7)
leading to the electric field
E = Ef (iωt)e z .(8)
In this case, both the instanton action and the fluctuation determinant can be found, up to quadrature, in terms of the function f . The remaining T -integral can be done explicitly using the saddle point approximation as well. The imaginary part of the Minkowski effective action is then completely determined by the single function
g(γ) = 4 π χ * 0 dχ 1 − 1 γ 2 f (γχ) 2 ,(9)
where γ = mω/(qE) is the Keldysh parameter and the turning point χ * is determined by the implicit equation
γ = f (γχ * ).(10)
Using this function, we find our final expression for the imaginary part of the effective action, and thus the pair production rate:
Γ M [A µ ] ≈ L m −1 E E S √ 2 8πγΦ(γ) exp −π E S E g(γ) ,(11)
with the function
Φ(γ) = − d 2 d(γ 2 ) 2 (γ 2 g(γ)).(12)
The length L is given by the spatial extent of the electric field, for example the focal spot of a laser beam.
To check the validity of the semiclassical approximation, we turn to a single Sauter pulse, where an exact treatment is possible. In this case (for details see [12])
g(γ) = 2 1 + 1 + γ 2 ,(13)Φ(γ) = 1 √ 2 1 + γ 2 −3/4 ,(14)
and thus
Γ M ≈ L m −1 1 4π E E S 1 + γ 2 3/4 γ × exp − E S E 2 1 + 1 + γ 2 .(15)
In Fig. 1 we compare (15) to the known exact solution (see, e.g., [19]). As expected, the semiclassical approximation breaks down for large E/E S , but even for E = E S /5 the agreement is excellent, only for E = E S /3 the results start to deviate visibly.
In 3 + 1 space-time dimensions, the prefactor differs from (11) by additional factors [12] Γ 3+1
M Γ 1+1 M = V 2 m −2 E E S 1 4π 2 d d(γ 2 ) (γ 2 g(γ))
. Apart from the obvious two-volume/area V 2 and an additional power of E, this also includes nontrivial scaling with γ. For the single Sauter pulse, this amounts to
(16) E /E S = 1/2 1/3 1/5 1/7 1.0 1.5 2.0 2.5 3.0 γ -8 -6 -4 -2 log 10 ( pp )1 d d(γ 2 ) (γ 2 g(γ)) = 1 + γ 2 ,(17)
which does not modify the qualitative behavior.
III. DYNAMICALLY ASSISTED SAUTER-SCHWINGER EFFECT
We now apply (9) and (11) to the dynamically assisted Sauter-Schwinger effect.
We choose the function
f (χ) = 1 ρ tan(ρχ) + εh(χ), ε 1, ρ 1,(18)
representing the sum of a strong, slow field and a weak,
faster profile with E weak /E = ε, Ω/ω = ρ, E = E(cosh −2 (Ωt) + εh (iωt))e z .(19)
Note that in this case, the combined Keldysh parameter γ = mω/(qE) compares the frequency scale ω of the weak pulse with the field strength E of the slow, strong field. Note that we do not approximate the slow field by a static one (as in [6]), as the electric field has to vanish for large times for the numerical integration of the Riccati equation to work and the limit ε → 0 would pose problems in the instanton method.
In the following, we will choose the following profiles for the fast pulses, see also [6] • Cosine cos(ωt), h cos (χ) = sinh χ
• Gaussian exp(−ω 2 t 2 ), h Gauss (χ) = √ π 2 erfi χ • Sauter cosh −2 (ωt), h Sauter (χ) = tan χ • Lorentzian (1 + ω 2 t 2 ) −1 , h Lorentz (χ) = artanh χ
First, we will numerically calculate g(γ) and Φ(γ) (and thus Γ M ) for these pulse shapes h(χ). In section III B we will then present analytical approximations for the pair production rate.
A. Numerical evaluation
The effective action (11) can be evaluated straightforwardly using numerical methods. The only challenge is solving the implicit equation (10). For the cosine and Gaussian profiles, h is smooth and a simple root finding algorithm converges using basically any choice of starting point. For the other two profiles however, h diverges (at χ = π/2 or χ = 1 respectively), so for small ε, the starting points for the root finding method have to be chosen with some care.
As soon as χ * is found, a standard numerical integration routine can be used to evaluate (9) and (12), yielding the effective action (11). Figure 2 shows the results of this procedure. In all cases there is a region of relatively weak dependence on the frequency of the weak field and a region of strong enhancement, as soon as the Keldysh parameter crosses a threshold value γ crit . For the cosine and Gauss profiles, this threshold depends on ε, while it is approximately constant for the Sauter and Lorentz pulses. This is the same behavior as seen in [6] considering the exponent only, so we can now conclude that the prefactor does not change this qualitatively. Now that we have an approximation for the full pair production rate using the worldline instanton method, the question that remains is if this approximation actually works well for these field configurations. Thus we compare it to a solution of the full Riccati equation as outlined in section IV. In Fig. 3, we can see that both methods agree perfectly below threshold or for large ε. Only for small ε and large γ the results deviate visibly. This is very interesting, because naïvely one might expect the quality of the approximation to depend mainly on the pair production probability (as it appears to do in Fig. 1), while in this case the interplay between multiple scales leads to a different behavior.
Furthermore, for small ε the instanton method predicts a highly unintuitive "dip" just below the critical value of γ where the pair production actually decreases with increasing Keldysh parameter (visible in Fig. 3 at the lower left edge of the displayed surfaces, more obvious in Fig. 4). The numerical solution of the Riccati equation however does not show this anomaly, so we can conclude it to be an artifact of the semiclassical approximation, again stressing that its validity has to be carefully examined in each situation.
Regardless, the Riccati equation predicts the same qualitative result of dynamical assistance above a threshold value of γ, which is roughly independent of ε.
B. Analytical approximations
To find analytical expressions for g(γ) and thus Φ(γ) and Γ M , we can use different approaches for γ < γ crit and γ > γ crit .
Below threshold, we can Taylor expand g(γ) in ε and ρ:
g(γ) = 4 π χ * 0 dχ 1 − tan(ργχ) ργ + ε h(γχ) γ 2 ≈ 4 π χ * ε=0 0 dχ 1 − tan(ργχ) ργ 2 − ε 1 0 dχ χ 1 − χ 2 h(γχ) γ = 2 1 + 1 + ρ 2 γ 2 − 4ε πγ 1 0 dξ h(γ 1 − ξ 2 ) = 2 1 + 1 + ρ 2 γ 2 − 4ε πγ G(γ).(20)
Note that this approximation works only for subcritical γ, because otherwise h(γχ) grows large, invalidating the expansion. The ε-independent term is just the result for a single Sauter pulse with Keldysh parameter ργ (see (13)). Substituting this expression for g(γ) in (12) we get
Φ(γ) ≈ ρ 2 2 (1 + ρ 2 γ 2 ) 3/2 + ε πγ 3 (γ 2 G + γG − G). (21)
Here, it is evident why we kept the slow pulse explicit, instead of approximating it as static. If the strong field were independent of time, we would have Φ → 0 as ε → 0, leading to Γ M → ∞. This is expected, because then for ε → 0, the instanton is not confined in the time direction anymore, giving rise to a zero mode. Using a slow Sauter pulse for the strong field solves this problem. Now all that is left is to calculate G(γ) for the different pulse profiles:
Cosine:
G cos (γ) = π 2 I 1 (γ) (22) Φ cos (γ) = ρ 2 2 (1 + ρ 2 γ 2 ) 3/2 + ε 2 I 1 (γ) γ(23)
where I ν denotes the modified Bessel functions of the first kind.
Gauss: Φ Gauss (γ) =
G Gauss (γ) = πγ 4 e γ 2 /2 I 0 (γ 2 /2) − I 1 (γ 2 /2)(24)ρ 2 2 (1 + ρ 2 γ 2 ) 3/2 + ε 2 e γ 2 2 I 0 γ 2 2 + I 1 γ 2 2 (25)
Lorentz:
G Lorentz (γ) = π 2 1 − 1 − γ 2 γ (26) Φ Lorentz (γ) = ρ 2 2 (1 + ρ 2 γ 2 ) 3/2 + ε 2 1 (1 − γ 2 ) 3/2(27)
For the Sauter profile, it is unfortunately not possible to find a closed form expression for the integral in G(γ), so we can not give an analytic expression for its subcritical behavior.
For γ > γ crit however, the function g(γ) can be approximated in the limit ε 1 for the Sauter and Lorentzian pulses by geometric considerations [5,20], leading to
g Sauter (γ > π/2) = 2 π arcsin π 2γ + γ 2 − π 2 2 γ 2 .(28)
The same method applies to the Lorentzian pulse, the only difference being the different value of the critical Keldysh parameter:
g Lorentz (γ > 1) = 2 π arcsin 1 γ + γ 2 − 1 γ 2 .
(29) Figure 4 shows the numerical calculation of Γ Lorentz for different values of ε, the approximations (26) and (27) for γ < 1 and (29) for γ > 1. Far above threshold, the pair production rate converges to the approximation (29), independent of ε. Closer to the threshold, the geometric approach breaks down for larger values of ε, although for ε = 10 −2 the numerical values agree with the approximation very well. As expected, the approximation (26) below threshold works better, the smaller the expansion parameter ε gets.
For ε = 10 −2 the instanton method again predicts the anomalous decrease in pair production at γ ≈ γ crit mentioned before.
IV. NUMERICAL SOLUTION OF THE RICCATI EQUATION
To test the instanton approximations we compared the results to a numerical evaluation of the Riccati equation, this chapter explains how we obtained these results. A brief derivation of the Riccati formalism can be found in [6] or in some more detail in [21]. For a time dependent field pointing in the z-direction represented by a vector potential A 3 (t), we may employ a Fourier transformation in order to account for the spatial dependence of our mode functions. After that, the time-dependence of the instantaneous Bogoliubov coefficients α k and β k is governed by the Riccati equatioṅ
R k = Ξ k (t) e +2iφ k (t) + R 2 k (t)e −2iφ k (t)(30)
with
R k = β k /α k and Ξ k (t) = qȦ 3 (t) m 2 + k 2 ⊥ 2Ω k (t) 2 , φ k (t) = t −∞ dt Ω k (t )
,
Ω k (t) = m 2 + k 2 ⊥ + (k 3 + qA 3 (t)) 2 ,(31)
and the initial condition R k (−∞) = 0. Here, k labels the different momentum modes, where k 3 denotes momentum parallel to the electric field and k ⊥ = (k 1 , k 2 ) the perpendicular momenta. To arrive at the number of produced pairs per volume, we need to integrate over all modes:
N pp = d 3 k (2π) 3 |R k (∞)| 2 ,(32)
where the factor of (2π) −3 stems from the choice of normalization in the mode decomposition. We will, as mentioned in the introduction, work in 1+1 spacetime dimensions, which amounts to setting k ⊥ = 0. This is due to the fact that we need to find R k (∞) for sufficiently many values of k to approximate the integral in (32), which is computationally intensive. The payoff however is small: While the longitudinal momentum spectrum includes important physical effects like interference patterns (see, e.g. [21,22]), the perpendicular momenta only amount to a rescaling of the electron mass, as is evident in (31). This leads to further exponential suppression for k 2 ⊥ > 0 so they hardly contribute to (32) and especially do not modify the qualitative response of the pair production rate to the field profile.
To numerically integrate (30), we need to introduce dimensionless quantities. First, we choose the vector potential to be
A 3 (t) = E ω f (ωt),(33)
with a dimensionless shape function f and scaling frequency ω. We then introduce the quantities
p = k 3 m , γ = mω qE , τ = tqE m , E = E E S = qE m 2 ,(34)
leading to the dimensionless Riccati equation, which can be treated numerically:
R p (τ ) = Ef (γτ ) 2 1 + (p + f (γτ )/γ) 2 × e i2ϕp(τ )/E + R 2 p (τ )e −i2ϕp(τ )/E , (35a) ϕ p (τ ) = 1 + (p + f (γτ )/γ) 2 .(35b)
While ϕ p (τ ) can of course be obtained immediately in terms of an integral, this can usually not be done analytically. Now, since we would like to use a variable step width integration algorithm to solve (35), we do not know in advance for which values of τ we need ϕ p (τ ), so we cannot efficiently precompute the integral. Thus, it is most convenient to directly solve the system of equations (35) in lockstep. Of course, we cannot perform infinitely many integration steps to find R(∞) from R(−∞) = 0, but have to choose a sufficiently long time range τ ∈ [−T , T ], so that E(|τ | > T ) is negligible compared to E(|τ | < T ). Then, using the initial condition R p (−T ) = 0, ϕ p (−T ) = 0 we arrive at the number of produced pairs per Compton length
N pp = 1 m −1 dp 2π |R p (T )| 2 .(36)
Unfortunately, actually obtaining R p (T ) for large T and small values of E is far from trivial. Since the exponentials in the Riccati equation oscillate wildly with a frequency of order E −1 , the step width ∆τ has to be sufficiently small, which means many steps have to be taken to reach T . Using a standard ODE integration algorithm [23], for fields weaker than ≈ E S /10 machine precision is not sufficient anymore and rounding errors begin to dominate.
Instead, we use the software package TIDES [24] which is based on the Taylor series method and supports the GNU MPFR [25] library for multiple precision arithmetic, allowing us to integrate the Riccati equation using as many significant digits in the calculation as needed to give accurate results. As a benchmark, we calculated R p=0 (T ) for a single Sauter pulse, which we can again compare to the known analytical result. Indeed, using very little computational resources [26], it is possible to reproduce the analytic solution for p = 0 and various values of γ with a relative error of less than 10 −14 for a field strength of E = E S /100, see Fig. 5.
V. SUMMARY AND CONCLUSION
Using the worldline instanton method, we have been able to numerically calculate and find analytical approximations for the pair production rate in the dynamically assisted Sauter-Schwinger effect. Building on [6], this now includes the quantum mechanical fluctuation prefactor, which is shown not to counteract the mechanism of dynamical assistance.
Comparing the different pulse shapes considered, they all exhibit a similar qualitative behavior. This includes a region of negligible dependence on the time dependent field up to a threshold value of the Keldysh parameter γ, beyond which the pair production rate is exponentially enhanced.
This threshold γ crit is independent of ε for the Sauter and Lorentzian pulses, in contrast to the sinusoidally varying field and Gaussian pulse which is caused by the different analytic structure of the field profiles.
Furthermore, using a numerical integration of the Riccati equation, we have shown that these results are not an artifact of the semiclassical approximation, but are present in a full numerical simulation as well.
FIG. 1 .
1Predicted number density of produced pairs by a single Sauter pulse with Keldysh parameter γ. The points depict the worldline instanton approximation (15), the lines show the exact analytical result. The field strength increases from bottom to top.
FIG. 2 .
2Imaginary part of the effective action(11) in the assisted Sauter Schwinger effect for different pulse shapes h(χ), Keldysh parameters γ and relative field strengths ε. The color scale spans from 10 −40 (blue, bottom left corners) to 10 −16 (yellow, top right corners). Note the dependence of the threshold γ crit on ε for the cosine and Gauss profiles, while γ crit ≈ const. for the Sauter and Lorentz profiles.
FIG. 3 .
3Density of produced pairs from a strong, slow and another weak, fast Sauter pulse. The strong field strength is E/ES = 0.033, corresponding to an intensity of ≈ 5 × 10 26 W/cm 2 . The blue surface (lying above for large γ and small ε) is the instanton result (11), the orange surface shows the numerical integration of the full Riccati equation.
FIG. 4 .
4Number density of produced pairs for a Lorentzian pulse. The dots represent the numerical results for different values of ε, the lines represent the approximations (26) and (27) for γ < 1 and (29) for γ > 1. ε increases from bottom to top.
FIG. 5 .
5Comparison of the analytic solution (line) and numerical results (points) of the Riccati equation for a single Sauter pulse with E = ES/100 and k3 = 0. The top plot shows the relative deviation from the analytic result.
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About one minute per point on a typical desktop PC. About one minute per point on a typical desktop PC.
|
[] |
[
"Transience in Countable MDPs",
"Transience in Countable MDPs"
] |
[
"Stefan Kiefer \nDepartment of Computer Science\nSchool of Informatics\nUniversity of Oxford\nUK\n",
"Richard Mayr \nUniversity of Edinburgh\nUK\n",
"Mahsa Shirmohammadi \nDepartment of Computer Science\nUniversité de Paris\nCNRS\n75013ParisIRIF, FFrance\n",
"Patrick Totzke \nUniversity of Liverpool\nUK\n"
] |
[
"Department of Computer Science\nSchool of Informatics\nUniversity of Oxford\nUK",
"University of Edinburgh\nUK",
"Department of Computer Science\nUniversité de Paris\nCNRS\n75013ParisIRIF, FFrance",
"University of Liverpool\nUK"
] |
[] |
The Transience objective is not to visit any state infinitely often. While this is not possible in any finite Markov Decision Process (MDP), it can be satisfied in countably infinite ones, e.g., if the transition graph is acyclic.We prove the following fundamental properties of Transience in countably infinite MDPs. 1. There exist uniformly ε-optimal MD strategies (memoryless deterministic) for Transience, even in infinitely branching MDPs. 2. Optimal strategies for Transience need not exist, even if the MDP is finitely branching. However, if an optimal strategy exists then there is also an optimal MD strategy.3.If an MDP is universally transient (i.e., almost surely transient under all strategies) then many other objectives have a lower strategy complexity than in general MDPs. E.g., ε-optimal strategies for Safety and co-Büchi and optimal strategies for {0, 1, 2}-Parity (where they exist) can be chosen MD, even if the MDP is infinitely branching.ACM Subject Classification Theory of computation → Random walks and Markov chains;Mathematics of computing → Probability and statistics
|
10.4230/lipics.concur.2021.11
|
[
"https://arxiv.org/pdf/2012.13739v3.pdf"
] | 229,679,975 |
2012.13739
|
b4e35652862a0b9e3dbef5b1b7deb25c119d5bed
|
Transience in Countable MDPs
Stefan Kiefer
Department of Computer Science
School of Informatics
University of Oxford
UK
Richard Mayr
University of Edinburgh
UK
Mahsa Shirmohammadi
Department of Computer Science
Université de Paris
CNRS
75013ParisIRIF, FFrance
Patrick Totzke
University of Liverpool
UK
Transience in Countable MDPs
Related Version This is the full version of a CONCUR 2021 paper [13].and phrases Markov decision processesParityTransience
The Transience objective is not to visit any state infinitely often. While this is not possible in any finite Markov Decision Process (MDP), it can be satisfied in countably infinite ones, e.g., if the transition graph is acyclic.We prove the following fundamental properties of Transience in countably infinite MDPs. 1. There exist uniformly ε-optimal MD strategies (memoryless deterministic) for Transience, even in infinitely branching MDPs. 2. Optimal strategies for Transience need not exist, even if the MDP is finitely branching. However, if an optimal strategy exists then there is also an optimal MD strategy.3.If an MDP is universally transient (i.e., almost surely transient under all strategies) then many other objectives have a lower strategy complexity than in general MDPs. E.g., ε-optimal strategies for Safety and co-Büchi and optimal strategies for {0, 1, 2}-Parity (where they exist) can be chosen MD, even if the MDP is infinitely branching.ACM Subject Classification Theory of computation → Random walks and Markov chains;Mathematics of computing → Probability and statistics
Introduction
Those who cannot remember the past are condemned to repeat it.
George Santayana (1905) [22] The famous aphorism above has often been cited (with small variations), e.g., by Winston Churchill in a 1948 speech to the House of Commons, and carved into several monuments all over the world [22].
We prove that the aphorism is false. In fact, even those who cannot remember anything at all are not condemned to repeat the past. With the right strategy they can avoid repeating the past equally well as everyone else. More formally, playing for Transience does not require any memory. We show that there always exist ε-optimal memoryless deterministic strategies for Transience, and if optimal strategies exist then there also exist optimal memoryless deterministic strategies. 1 finitely branching. If they do exist then there are also MD optimal strategies. More generally, there exists a single MD strategy that is optimal from every state that allows optimal strategies for Transience. 3. If an MDP is universally transient (i.e., almost surely transient under all strategies) then many other objectives have a lower strategy complexity than in general MDPs, e.g., ε-optimal strategies for Safety and co-Büchi and optimal strategies for {0, 1, 2}-Parity (where they exist) can be chosen MD, even if the MDP is infinitely branching. For our proofs we develop some technical results that are of independent interest. We generalize Ornstein's plastering construction [20] from reachability to tail objectives and thus obtain a general tool to infer uniformly ε-optimal MD strategies from non-uniform ones (cf. Theorem 7). Secondly, in Section 6 we develop the notion of the conditioned MDP (cf. [17]). For tail objectives, this allows to obtain uniformly ε-optimal MD strategies wrt. multiplicative errors from those with merely additive errors. base its decision also on a memory mode m ∈ {0, 1}. Formally, a 1-bit strategy σ is given as a tuple (u, m 0 ) where m 0 ∈ {0, 1} is the initial memory mode and u : {0, 1} × S → D({0, 1} × S) is an update function such that for all controlled states s ∈ S , the distribution u ((m, s)) is over {0, 1} × {s ′ | s−→s ′ }. for all random states s ∈ S , we have that m ′ ∈{0,1} u((m, s))(m ′ , s ′ ) = P (s)(s ′ ). Note that this definition allows for updating the memory mode upon visiting random states. We write σ[m 0 ] for the strategy obtained from σ by setting the initial memory mode to m 0 .
MD strategies are both memoryless and deterministic; and deterministic 1-bit strategies are both deterministic and 1-bit.
Objectives. The objective of the controller is determined by a predicate on infinite runs. We assume familiarity with the syntax and semantics of the temporal logic LTL [9]. Formulas are interpreted on the underlying structure (S, −→) of the MDP M. We use φ M,s ⊆ sS ω to denote the set of runs starting from s that satisfy the LTL formula φ, which is a measurable set [27]. We also write φ M for s∈S φ M,s . Where it does not cause confusion we will identify φ and φ and just write P M,s,σ (φ) instead of P M,s,σ ( φ M,s ).
Given a set T ⊆ S of states, the reachability objective Reach(T ) def = FT is the set of runs that visit T at least once. The safety objective Safety(T ) def = G¬T is the set of runs that never visit T .
Let C ⊆ N be a finite set of colors. A color function C ol : S → C assigns to each state s its color C ol(s). The parity objective, written as Parity(C ol), is the set of infinite runs such that the largest color that occurs infinitely often along the run is even. To define this formally, let even(C) = {i ∈ C | i ≡ 0 mod 2}. For ∈ {<, ≤, =, ≥, >}, n ∈ N, and Q ⊆ S, let [Q] C ol n def = {s ∈ Q| C ol(s) n} be the set of states in Q with color n. Then
Parity(C ol) def = i∈even(C) GF[S] C ol=i ∧ FG[S] C ol≤i .
We write C-Parity for the parity objectives with the set of colors C ⊆ N. The classical Büchi and co-Büchi objectives correspond to {1, 2}-Parity and {0, 1}-Parity, respectively.
An objective φ is called a tail objective (in M) iff for every run ρ ′ ρ with some finite prefix ρ ′ we have ρ ′ ρ ∈ φ ⇔ ρ ∈ φ. For every coloring C ol, Parity(C ol) is tail. Reachability objectives are not always tail but in MDPs where the target set T is a sink Reach(T ) is tail.
Optimal and ε-optimal Strategies. Given an objective φ, the value of state s in an MDP M, denoted by val M,φ (s), is the supremum probability of achieving φ. Formally, we have val M,φ (s) def = sup σ∈Σ P M,s,σ (φ) where Σ is the set of all strategies. For ε ≥ 0 and state s ∈ S, we say that a strategy is ε-optimal from s iff P M,s,σ (φ) ≥ val M,φ (s) − ε. A 0-optimal strategy is called optimal. An optimal strategy is almost-surely winning iff val M,φ (s) = 1.
Considering an MD strategy as a function σ : S → S and ε ≥ 0, σ is uniformly ε-optimal (resp. uniformly optimal) if it is ε-optimal (resp. optimal) from every s ∈ S.
Throughout the paper, we may drop the subscripts and superscripts from notations, if it is understood from the context. The missing proofs can be found in the appendix.
Transience and Universally Transient MDPs
In this section we define the transience property for MDPs, a natural generalization of the well-understood concept of transient Markov chains. We enumerate crucial characteristics of this objective and define the notion of universally transient MDPs.
w 0 w 1 w 2 w 3 w 4 · · · 1 p p p p 1 − p 1 − p 1 − p 1 − p 1 − p Figure 1
Gambler's Ruin with restart: The state wi illustrates that the controller's wealth is i, and the coin tosses are in the controller's favor with probability p. For all i, Pw i (Transience) = 0 if p ≤ 1 2 ; and Pw i (Transience) = 1 otherwise.
Fix a countable MDP M = (S, S , S , −→, P ). Define the transience objective, denoted by Transience, to be the set of runs that do not visit any state of M infinitely often, i.e.,
Transience def = s∈S FG ¬s.
The Transience objective is tail, as it is closed under removing finite prefixes of runs. Also note that Transience cannot be encoded in a parity objective.
We call M universally transient iff for all states s 0 , for all strategies σ, the Transience property holds almost-surely from s 0 , i.e., ∀s 0 ∈ S ∀σ ∈ Σ P M,s0,σ (Transience) = 1.
The MDP in Figure 1 models the classical Gambler's Ruin Problem with restart; see [10,Chapter 14]. It is well-known that if the controller starts with wealth i and if p ≤ 1 2 , the probability of ruin (visiting the state w 0 ) is P wi (F w 0 ) = 1. Consequently, the probability of re-visiting w 0 infinitely often is 1, implying that P wi (Transience) = 0. In contrast, for the case with p > 1 2 , for all states w i , the probability of re-visiting w i is strictly below 1. Hence, the Transience property holds almost-surely. This example indicates that the transience property depends on the probability values of the transitions and not just on the underlying transition graph, and thus may require arithmetic reasoning. In particular, the MDP in Figure 1 is universally transient iff p > 1 2 . In general, optimal strategies for Transience need not exist:
▶ Lemma 1. There exists a finitely branching countable MDP with initial state s 0 such that val Transience (s) = 1 for all controlled states s, there does not exist any optimal strategy σ such that P s0,σ (Transience) = 1. Figure 2. For all i ≥ 1 the state x i+1 is the unique successor of x i so that (x i ) i≥1 form an acyclic ladder; the value of Transience is 1 for all x i . The state ⊥ is sink, and its value is 0. The states (r i ) i≥1 are all random, and r i
Proof. Consider a countable MDP
M with set S = {ℓ i , ℓ ′ i , r i , x i | i ≥ 1} ∪ {ℓ 0 , ⊥} of states; see1−2 −i − −−− → x i and r i 2 −i − − → ⊥. Observe that the value of Transience is 1 − 2 −i for the r i .
The states (ℓ i ) i∈N are controlled whereas the states (ℓ ′ i ) i≥1 are random. By interleaving of these states, we construct a "recurrent ladder" of decisions: ℓ 0 → ℓ 1 and for all i ≥ 1, state ℓ i has two successors ℓ ′ i and r i . In random states ℓ ′ i , as in Gambler's Ruin with a fair coin, the successors are ℓ i−1 or ℓ i+1 , each with equal probability. In each state (ℓ i ) i≥1 , the controller decides to either stay on the ladder by going to ℓ ′ i or leaves the ladder to r i . As in Figure 1, if the controller stays on the ladder forever, the probability of Transience is 0.
Starting in ℓ 0 , for all i > 0, strategy σ i that stays on the ladder until visiting ℓ i (which happens eventually almost surely) and then leaves the ladder to r i achieves Transience with probability 1 − 2 i . Hence, val Transience (ℓ 0 ) = 1.
Recall that transience cannot be achieved with a positive probability by staying on the acyclic ladder forever. But any strategy that leaves the ladder with a positive probability ℓ 0 ℓ 1 ℓ ′
Figure 2
A partial illustration of the MDP in Lemma 1, in which there is no optimal strategy for Transience, starting from states ℓi. For readability, we have three copies of the state ⊥. We call the ladder consisting of the interleaved controlled states ℓi and random states ℓ ′ i a "recurrent ladder": if the controller stays on this ladder forever, it faithfully simulates a Gambler's Ruin with a fair coin, and the probability of Transience will be 0.
comes with a positive probability of falling into ⊥, thus is not optimal either. Thus there is no optimal strategy for Transience. ◀ Reduction to Finitely Branching MDPs. In our main results, we will prove that for the Transience property there always exist ε-optimal MD strategies in finitely branching countable MDPs; and if an optimal strategy exists, there will exist an optimal MD strategy. We generalize these results to infinitely branching countable MDPs by the following reduction:
▶ Lemma 2. Given an infinitely branching countable MDP M with an initial state s 0 , there exists a finitely branching countable M ′ with a set S ′ of states such that s 0 ∈ S ′ and 1. each strategy α 1 in M is mapped to a unique strategy β 1 in M ′ where P s0,α1 (Transience) = P s0,β1 (Transience), 2. and conversely, every MD strategy β 2 in M ′ is mapped to an MD strategy α 2 in M where P s0,α2 (Transience) ≥ P s0,β2 (Transience).
Proof sketch. See Appendix B for the complete construction. In order to construct M ′ from M, for each controlled state s ∈ S in M that has infinitely many successors (s i ) i≥1 , a "recurrent ladder" is introduced; see Figure 3. Since the probability of Transience is 0 for all those runs that eventually stay forever on a recurrent ladder, the controller should exit such ladders to play optimally for Transience. Infinitely branching random states can be dealt with in an easier way. ◀
Properties of Universally Transient MDPs.
Notice that acyclicity implies universal transience, but not vice-versa. Proof. Towards (1) ⇒ (2), consider an arbitrary strategy σ from the initial state s 0 and some state s. By (1) we have ∀σ.P M,s0,σ (Transience) = 1 and thus 0 = P M,s0,σ (¬Transience) = P M,s0,σ ( s ′ ∈S GF(s ′ )) ≥ P M,s0,σ (GF(s)) which implies (2). Towards (2) ⇒ (1), consider an arbitrary strategy σ from the initial state s 0 . By (2) we have 0 = s∈S P M,s0,σ (GF(s)) ≥ P M,s0,σ ( s∈S GF(s)) = P M,s0,σ (¬Transience) and thus P M,s0,σ (Transience) = 1.
· · · s i−1 s i · · · MDP M ⇓ reduction MDP M ′ s ℓ 0 ℓ 1 ℓ ′ 1 · · · ℓ i−1 ℓ ′ i−1 ℓ i ℓ ′ i · · · s 1 · · · s i−1 s i · · ·
We now show the implications (2) ⇒ (3) ⇒ (4) ⇒ (5) ⇒ (2). Towards ¬(3) ⇒ ¬(2), ¬(3) implies ∃s.Re(s) = 1 and thus ∀ε > 0.∃σ ε P M,s,σε (XF(s)) ≥ 1 − ε. Let ε i def = 2 −(i+1) . We define the strategy σ to play like σ εi between the i-th and (i + 1)th visit to s. Since (2), where s 0 = s. Towards (3) ⇒ (4), regardless of s 0 and the chosen strategy, the expected number of visits to s is upper-bounded by B(s) def = ∞ n=0 (n + 1) · (Re(s)) n < ∞. The implication (4) ⇒ (5) holds trivially. Towards ¬(2) ⇒ ¬(5), by ¬(2) there exist states s 0 , s and a strategy σ such that P M,s0,σ (GF(s)) > 0. Thus the expected number of visits to s is infinite, which implies ¬(5). ◀
∞ i=1 ε i < ∞, we have ∞ i=1 (1 − ε i ) > 0. Therefore P M,s,σ (GF(s)) ≥ ∞ i=1 (1 − ε i ) > 0, which implies ¬
We remark that if an MDP is not universally transient (unlike in Lemma 3(5)), for a strategy σ, the expected number of visits to some state can be infinite, even if σ attains Transience almost surely.
Consider the MDP M with controlled states {s 0 , s 1 , . . . }, initial state s 0 and transitions s 0 → s 0 and s k → s k+1 for every k ≥ 0. We define a strategy σ that, while in state s 0 , proceeds in rounds i = 1, 2, . . . . In the i-th round it tosses a fair coin. If Heads then it goes to s 1 . If Tails then it loops around s 0 exactly 2 i times and then goes to round i + 1. In every round the probability of going to s 1 is 1/2 and therefore the probability of staying in s 0 forever is (1/2) ∞ = 0. Thus P M,s0,σ (Transience) = 1. However, the expected number of visits to s 0 is ≥
∞ i=1 1 2 i · 2 i = ∞.
MD Strategies for Transience
We show that there exist uniformly ε-optimal MD strategies for Transience and that optimal strategies, where they exist, can also be chosen MD. First we show that there exist ε-optimal deterministic 1-bit strategies for Transience (in Corollary 5) and then we show how to dispense with the 1-bit memory (in Lemma 6).
It was shown in [14] that there exist ε-optimal deterministic 1-bit strategies for Büchi objectives in acyclic countable MDPs (though not in general MDPs). These 1-bit strategies will be similar to the 1-bit strategies for Transience that we aim for in (not necessarily acyclic) countable MDPs. In Lemma 4 below we first strengthen the result from [14] and construct ε-optimal deterministic 1-bit strategies for objectives Büchi(F ) ∩ Transience. From this we obtain deterministic 1-bit strategies for Transience (Corollary 5).
▶ Lemma 4. Let M be a countable MDP, I a finite set of initial states, F a set of states and ε > 0. Then there exists a deterministic 1-bit strategy for Büchi(F ) ∩ Transience that is ε-optimal from every s ∈ I.
Proof sketch. The full proof can be found in Appendix C. It follows the proof of [14, Theorem 5], which considers Büchi(F ) conditions for acyclic (and hence universally transient) MDPs. The only part of that proof that requires modification is [14, Lemma 10], which is replaced here by Lemma 18 to deal with general MDPs.
In short, from every s ∈ I there exists an ε-optimal strategy σ s for φ def = Büchi(F ) ∩ Transience. We observe the behavior of the finitely many σ s for s ∈ I on an infinite, increasing sequence of finite subsets of S. Based on Lemma 18, we can define a second stronger objective φ ′ ⊆ φ and show ∀ s∈I P M,s,σs (φ ′ ) ≥ val M,φ (s) − 2ε. We then construct a deterministic 1-bit strategy σ ′ that is optimal for φ ′ from all s ∈ I and thus 2ε-optimal for φ. Since ε can be chosen arbitrarily small, the result follows. ◀ Unlike for the Transience objective alone (see below), the 1-bit memory is strictly necessary for the Büchi(F ) ∩ Transience objective in Lemma 4. The 1-bit lower bound for Büchi(F ) objectives in [14] holds even for acyclic MDPs where Transience is trivially true.
▶ Corollary 5. Let M be a countable MDP, I a finite set of initial states, F a set of states and ε > 0. 1. If ∀s ∈ I val M,Büchi(F ) (s) = val M,Büchi(F )∩Transience (s) then there exists a deterministic 1-bit strategy for Büchi(F ) that is ε-optimal from every s ∈ I.
If M is universally transient then there exists a deterministic 1-bit strategy for Büchi(F )
that is ε-optimal from every s ∈ I. 3. There exists a deterministic 1-bit strategy for Transience that is ε-optimal from every s ∈ I.
Proof. Towards (1), since ∀s ∈ I val M,Büchi(F ) (s) = val M,Büchi(F )∩Transience (s), strategies that are ε-optimal for Büchi(F ) ∩ Transience are also ε-optimal for Büchi(F ). Thus the result follows from Lemma 4. Item (2) follows directly from (1), since the precondition always holds in universally transient MDPs.
Towards (3), let F def = S. Then we have Büchi(F ) ∩ Transience = Transience and we obtain from Lemma 4 that there exists a deterministic 1-bit strategy for Transience that is ε-optimal from every s ∈ I. ◀ Note that every acyclic MDP is universally transient and thus Corollary 5(2) implies the upper bound on the strategy complexity of Büchi(F ) from [14] (but not vice-versa).
In the next step we show how to dispense with the 1-bit memory and obtain non-uniform ε-optimal MD strategies for Transience. ▶ Lemma 6. Let M = (S, S , S , −→, P ) be a countable MDP with initial state s 0 , and ε > 0. There exists an MD strategy σ that is ε-optimal for Transience from s 0 , i.e., P M,s0,σ (Transience) ≥ val M,Transience (s 0 ) − ε.
Proof. By Lemma 2 it suffices to prove the property for finitely branching MDPs. Thus without restriction in the rest of the proof we assume that M is finitely branching.
Let ε ′ def = ε/2. We instantiate Corollary 5(3) with I def = {s 0 } and obtain that there exists an ε ′ -optimal deterministic 1-bit strategyσ for Transience from s 0 .
We now construct a slightly modified MDP M ′ as follows. Let S bad ⊆ S be the subset of states whereσ attains zero for Transience in both memory modes, i.e., S bad
def = {s ∈ S | P M,s,σ[0] (Transience) = P M,s,σ[1] (Transience) = 0}. Let S good def = S \ S bad .
We obtain M ′ from M by making all states in S bad losing sinks (for Transience), by deleting all outgoing edges and adding a self-loop instead. It follows that
P M,s0,σ (Transience) = P M ′ ,s0,σ (Transience) (1) ∀σ. P M,s0,σ (Transience) ≥ P M ′ ,s0,σ (Transience)(2)
In the following we show that it is possible to play in such a way that, for every s ∈ S good , the expected number of visits to s is finite. We obtain the deterministic 1-bit strategy σ ′ in M ′ by modifyingσ as follows. In every state s and memory mode x ∈ {0, 1} whereσ[x] attains 0 for Transience andσ[1 − x] attains > 0 the strategy σ ′ sets the memory bit to 1 − x. (Note that only states s ∈ S good can be affected by this change.) It follows that
∀s ∈ S. P M ′ ,s,σ ′ (Transience) ≥ P M ′ ,s,σ (Transience)(3)
Moreover, from all states in S good in M ′ the strategy σ ′ attains a strictly positive probability of Transience in both memory modes, i.e., for all s ∈ S good we have
t(s, σ ′ ) def = min x∈{0,1} P M ′ ,s,σ ′ [i] (Transience) > 0.
Let r(s, σ ′ , x) be the probability, when playing σ ′ [x] from state s, of reaching s again in the same memory mode x. For every s ∈ S good we have r(s, σ ′ , x) < 1, since t(s, σ ′ ) > 0.
Let R(s) be the expected number of visits to state s when playing σ ′ from s 0 in M ′ , and R x (s) the expected number of visits to s in memory mode x ∈ {0, 1}. For all s ∈ S good we have that
R(s) = R 0 (s) + R 1 (s) ≤ ∞ n=1 n · r(s, σ ′ , 0) n−1 + ∞ n=1 n · r(s, σ ′ , 1) n−1 < ∞(4)
where the first equality holds by linearity of expectations. Thus the expected number of visits to s is finite. Now we upper-bound the probability of visiting S bad . We have P M ′ ,s0,σ ′ (Transience) ≥ P M ′ ,s0,σ (Transience) = P M,s0,σ (Transience) ≥ val M,Transience (s 0 ) − ε ′ by (3), (1) and the ε ′ -optimality ofσ. Since states in S bad are losing sinks in M ′ , it follows that
P M ′ ,s0,σ ′ (FS bad ) ≤ 1 − P M ′ ,s0,σ ′ (Transience) ≤ 1 − val M,Transience (s 0 ) + ε ′(5)
We now augment the MDP M ′ by assigning costs to transitions as follows. Let i : S → N be an enumeration of the state space, i.e., a bijection. Let S ′ good def = {s ∈ S good | R(s) > 0} be the subset of states in S good that are visited with non-zero probability when playing σ ′ from s 0 . Each transition s ′ → s is assigned a cost:
If s ′ ∈ S bad then s ∈ S bad by def. of M ′ . We assign cost 0. If s ′ ∈ S good and s ∈ S bad we assign cost
K/(1 − val M,Transience (s 0 ) + ε ′ ) for K def = (1 + ε ′ )/ε ′ .
If s ′ ∈ S good and s ∈ S ′ good we assign cost 2 −i(s) /R(s). This is well defined, since R(s) > 0. s ′ ∈ S good and s ∈ S good \ S ′ good we assign cost 1. Note that all transitions leading to states in S good are assigned a non-zero cost, since R(s) is finite by (4).
When playing σ ′ from s 0 in M ′ , the expected total cost is upper-bounded by
P M ′ ,s0,σ ′ (FS bad ) · K/(1 − val M,Transience (s 0 ) + ε ′ ) + s∈S ′ good R(s) · 2 −i(s) /R(s)
The first part is ≤ K by (5) and the second part is ≤ 1, since R(s) < ∞ by (4). Therefore the expected total cost is ≤ K + 1, i.e., σ ′ witnesses that it is possible to attain a finite expected cost that is upper-bounded by K + 1. Now we define our MD strategy σ. Let σ be an optimal MD strategy on M ′ (from s 0 ) that minimizes the expected cost. It exists, as a finite expected cost is attainable and M ′ is finitely branching; see [21,Theorem 7.3.6].
We now show that σ attains Transience with high probability in M ′ (and in M). Since σ is cost-optimal, its attained cost from s 0 is upper-bounded by that of σ ′ , i.e.,
≤ K + 1. Since the cost of entering S bad is K/(1 − val M,Transience (s 0 ) + ε ′ ), we have P M ′ ,s0,σ (FS bad ) · K/(1 − val M,Transience (s 0 ) + ε ′ ) ≤ K + 1 and thus P M ′ ,s0,σ (FS bad ) ≤ K + 1 K (1 − val M,Transience (s 0 ) + ε ′ )(6)
For every state s ∈ S good , all transitions into s have the same fixed non-zero cost. Thus every run that visits some state s ∈ S good infinitely often has infinite cost. Since the expected cost of playing σ from s 0 is ≤ K + 1, such runs must be a null-set, i.e.,
P M ′ ,s0,σ (¬Transience ∧ GS good ) = 0(7)
Thus P M,s0,σ (Transience)
≥ P M ′ ,s0,σ (Transience) by (2) = 1 − P M ′ ,s0,σ (FS bad ) by (7) ≥ 1 − K + 1 K (1 − val M,Transience (s 0 ) + ε ′ ) by (6) = val M,Transience (s 0 ) − ε ′ − (1/K)(1 − val M,Transience (s 0 ) + ε ′ ) ≥ val M,Transience (s 0 ) − ε ′ − (1/K)(1 + ε ′ ) = val M,Transience (s 0 ) − 2ε ′ def. of K = val M,Transience (s 0 ) − ε def. of ε ′ ◀
Now we lift the result of Lemma 6 from non-uniform to uniform strategies (and to optimal strategies) and obtain the following theorem. The proof is a generalization of a "plastering" construction by Ornstein [20] (see also [16]) from reachability to tail objectives, which works by fixing MD strategies on ever expanding subsets of the state space. ▶ Theorem 7. Let M = (S, S , S , −→, P ) be a countable MDP, and let φ be an objective that is tail in M. Suppose for every s ∈ S there exist ε-optimal MD strategies for φ. Then: 1. There exist uniform ε-optimal MD strategies for φ. 2. There exists a single MD strategy that is optimal from every state that has an optimal strategy.
▶ Theorem 8. In every countable MDP there exist uniform ε-optimal MD strategies for Transience. Moreover, there exists a single MD strategy that is optimal for Transience from every state that has an optimal strategy.
Proof. Immediate from Lemma 6 and Theorem 7, since Transience is a tail objective. ◀
Strategy Complexity in Universally Transient MDPs
The These results (Theorems 10 and 11) ultimately rely on the existence of uniformly ε-optimal strategies for safety objectives. While such strategies always exist for finitely branching MDPs -simply pick a value-maximal successor -this is not the case for infinitely branching MDPs [17]. However, we show that universal transience implies the existence of uniformly ε-optimal strategies for safety objectives even for infinitely branching MDPs.
▶ Theorem 9. For every universally transient countable MDP, safety objective and ε > 0 there exists a uniformly ϵ-optimal MD strategy.
Proof. Let M = (S, S , S , −→, P ) be a universally transient MDP and ε > 0. Assume w.l.o.g. that the target T ⊆ S of the objective φ = Safety(T ) is a (losing) sink and let ι : S → N be an enumeration of the state space S.
By Lemma 3(3), for every state s we have Re(s) def = sup σ P M,s,σ (XF(s)) < 1 and thus
R(s) def = ∞ i=0 Re(s) i < ∞.
This means that, independent of the chosen strategy, Re(s) upper-bounds the chance to return to s, and R(s) bounds the expected number of visits to s.
Suppose that σ is an MD strategy which, at any state s ∈ S , picks a successor s ′ with
val(s ′ ) ≥ val(s) − ε 2 ι(s)+1 · R(s) .
This is possible even if M is infinitely branching, by the definition of value and the fact that R(s) < ∞. We show that P M,s0,σ (Safety(T )) ≥ val(s 0 ) − ε holds for every initial state s 0 , which implies the claim of the theorem.
Towards this, we define a function cost that labels each transition in the MDP with a realvalued cost: For every controlled transition s−→s ′ let cost((s, s ′ )) def = val(s) − val(s ′ ) ≥ 0. Random transitions have cost zero. We will argue that when playing σ from any start state s 0 , its attainment w.r.t. the objective Safety(T ) equals the value of s 0 minus the expected total cost, and that this cost is bounded by ε.
For any i ∈ N let us write s i for the random variable denoting the state just after step i, and Cost(i) def = cost(s i , s i+1 ) for the cost of step i in a random run. We observe that under σ the expected total cost is bounded in the limit, i.e.,
lim n→∞ E n−1 i=0 Cost(i) ≤ ε.(8)
We moreover note that for every n,
E(val(s n )) = E(val(s 0 )) − E n−1 i=0 Cost(i) .(9)
Full proofs of the above two equations can be found in Appendix E. Together they imply
lim inf n→∞ E(val(s n )) = val(s 0 ) − lim n→∞ E n−1 i=0 cost(i) ≥ val(s 0 ) − ε.(10)
Finally, to show the claim let [s n / ∈ T ] : S ω → {0, 1} be the random variable that indicates that the n-th state is not in the target set T . Note that [s n / ∈ T ] ≥ val(s n ) because target states have value 0. We have:
P M,s0,σ (Safety(T )) = P M,s0,σ ∞ i=0 X i ¬T semantics of Safety(T ) = G¬T = lim n→∞ P M,s0,σ n i=0 X i ¬T continuity of measures = lim n→∞ P M,s0,σ (X n ¬T ) T is a sink = lim n→∞ E([s n / ∈ T ]) definition of [s n / ∈ T ] ≥ lim inf n→∞ E(val(s n )) as [s n / ∈ T ] ≥ val(s n ) ≥ val(s 0 ) − ε Equation (10). ◀
We can now combine Theorem 9 with the results from [15] to show the existence of MD strategies assuming universal transience. Formally, let M be a universally transient MDP with states S, C ol : S → {0, 1, 2}, and φ = Parity(C ol). There exists an MD strategy σ ′ that is optimal for all states s that have an optimal strategy: [15,Def. 19] for a precise definition). By Lemma 17, M + is still a universally transient MDP and therefore by Theorem 9, there exist uniformly ε-optimal MD strategies for every safety objective and every ε > 0.
∃σ ∈ Σ. P M,s,σ (φ) = val M (s) =⇒ P M,s,σ ′ (φ) = val M (s). Proof. Let M + be the conditioned version of M w.r.t. φ (see(φ) ≥ val M (s) − ε.
Proof. This directly follows from Theorem 9 and [15, Theorem 25]. ◀
The Conditioned MDP
Given an MDP M and an objective φ that is tail in M, a construction of a conditioned MDP M + was provided in [17, Lemma 6] that, very loosely speaking, "scales up" the probability of φ so that any strategy σ is optimal in M if it is almost surely winning in M + . For certain tail objectives, this construction was used in [17] to reduce the sufficiency of MD strategies for optimal strategies to the sufficiency of MD strategies for almost surely winning strategies, which is a special case that may be easier to handle. However, the construction was restricted to states that have an optimal strategy. In fact, states in M that do not have an optimal strategy do not appear in M + . In the following, we lift this restriction by constructing a more general version of the conditioned MDP, called M * . The MDP M * will contain all states from M that have a positive value w.r.t. φ in M. Moreover, all these states will have value 1 in M * . It will then follow from Lemma 13 (3) below that an ε-optimal strategy in M * is εval M (s 0 )-optimal in M. This allows us to reduce the sufficiency of MD strategies for ε-optimal strategies to the sufficiency of MD strategies for ε-optimal strategies for states with value 1. In fact, it also follows that if an
MD strategy σ is uniform ε-optimal in M * , it is multiplicatively uniform ε-optimal in M, i.e., P M,s,σ (φ) ≥ (1 − ε) · val M (s) holds for all states s. ▶ Definition 12. For an MDP M = (S, S , S , −→, P ) and an objective φ that is tail in M, define the conditioned version of M w.r.t. φ to be the MDP M * = (S * , S * , S * , −→ * , P * ) with S * = {s ∈ S | val M (s) > 0} S * = {s ∈ S | val M (s) > 0} ∪ {s ⊥ } ∪ {(s, t) ∈ −→ | s ∈ S , val M (s) > 0} −→ * = {(s, (s, t)) ∈ (S × −→) | val M (s) > 0, s−→t} ∪ {(s, t) ∈ S × S | val M (s) > 0, val M (t) > 0} ∪ {((s, t), t) ∈ (−→ × S) | val M (s) > 0, val M (t) > 0} ∪ {((s, t), s ⊥ ) ∈ (−→ × {s ⊥ }) | val M (s) > val M (t)} ∪ {(s ⊥ , s ⊥ )} P * (s, t) = P (s, t) · val M (t) val M (s) P * ((s, t), t) = val M (t) val M (s) P * ((s, t), s ⊥ ) = 1 − val M (t) val M (s) P * (s ⊥ , s ⊥ ) = 1 for a fresh state s ⊥ . The conditioned MDP is well-defined. Indeed, as φ is tail in M, for any s ∈ S we have val M (s) = s−→t P (s, t)val M (t), and so if val M (s) > 0 then s−→t P * (s, t) = 1. ▶ Lemma 13. Let M = (S, S , S , −→, P ) be an MDP, and let φ be an objective that is tail in M. Let M * = (S * , S * , S * , −→ * , P * ) be the conditioned version of M w.r.t. φ. Let s 0 ∈ S * ∩ S.
Let σ ∈ Σ M * , and note that σ can be transformed to a strategy in M in a natural way. Then: 1. For all n ≥ 0 and all partial runs s 0 s 1 · · · s n ∈ s 0 S * * in M * with s n ∈ S:
val M (s 0 ) · P M * ,s0,σ (s 0 s 1 · · · s n S ω * ) = P M,s0,σ (s 0 s 1 · · · s n S ω ) · val M (s n ) ,
where w for a partial run w in M * refers to its natural contraction to a partial run in M;
i.e., w is obtained from w by deleting all states of the form (s, t).
For all measurable
R ⊆ s 0 (S * \ {s ⊥ }) ω we have P M,s0,σ (R) ≥ val M (s 0 ) · P M * ,s0,σ (R) ≥ P M,s0,σ (R ∩ φ s0 ) ,
where R is obtained from R by deleting, in all runs, all states of the form (s, t).
3.
We have val M (s 0 ) · P M * ,s0,σ (φ) = P M,s0,σ (φ). In particular, val M * (s 0 ) = 1, and, for any ε ≥ 0, strategy σ is ε-optimal in M * if and only if it is εval M (s 0 )-optimal in M. Lemma 13.3 provides a way of proving the existence of MD strategies that attain, for each state s, a fixed fraction (arbitrarily close to 1) of the value of s:
▶ Theorem 14. Let M = (S, S , S , −→, P ) be an MDP, and let φ be an objective that is tail in M. Let M * = (S * , S * , S * , −→ * , P * ) be the conditioned version of M w.r.t. φ. Let ε ≥ 0. Any MD strategy σ that is uniformly ε-optimal in M * (i.e., P M * ,s,σ (φ) ≥ val M * (s) − ε holds for all s ∈ S * ) is multiplicatively ε-optimal in M (i.e., P M,s,σ (φ) ≥ (1 − ε)val M (s) holds for all s ∈ S).
Proof. Immediate from Lemma 13.3. ◀
As an application of Theorem 14, we can strengthen the first statement of Theorem 8 towards multiplicatively (see Theorem 14) uniform ε-optimal MD strategies for Transience.
▶ Corollary 15. In every countable MDP there exist multiplicatively uniform ε-optimal MD strategies for Transience.
Proof. Let M be a countable MDP, and M * its conditioned version w.r.t. Transience. Let ε > 0. By Theorem 8, there is a uniform ε-optimal MD strategy σ for Transience in M * . By Theorem 14, strategy σ is multiplicatively uniform ε-optimal in M. ◀
The following lemma, stating that universal transience is closed under "conditioning", is needed for the proof of Lemma 17 below.
▶ Lemma 16. Let M = (S, S , S , −→, P ) be an MDP, and let φ be an objective that is tail
in M. Let M * = (S * , S * , S * , −→ * , P * ) be the conditioned version of M w.r.t. φ, where s ⊥ is replaced by an infinite chain s 1 ⊥ −→s 2 ⊥ −→ · · · . If M is universally transient, then so is M * .
In [17, Lemma 6] a variant, say M + , of the conditioned MDP M * from Definition 12 was proposed. This variant M + differs from M * in that M + has only those states s from M that have an optimal strategy, i.e., a strategy σ with P M,s,σ (φ) = val M (s). Further, for any transition s−→t in M + where s is a controlled state, we have val M (s) = val M (t), i.e., M + does not have value-decreasing transitions emanating from controlled states. The following lemma was used in the proof of Theorem 10:
Conclusion
The Transience objective admits ε-optimal (resp. optimal) MD strategies even in infinitely branching MDPs. This is unusual, since ε-optimal strategies for most other objectives require infinite memory if the MDP is infinitely branching (in particular all objectives generalizing Safety [17]).
Transience encodes a notion of continuous progress, which can be used as a tool to reason about the strategy complexity of other objectives in countable MDPs. E.g., our result on Transience is used in [18] as a building block to show upper bounds on the strategy complexity of certain threshold objectives w.r.t. mean payoff, total payoff and point payoff.
A Strategy Classes
We formalize the amount of memory needed to implement strategies. Let M be a countable set of memory modes. An update function is a function u :
B Missing Proofs from Section 3
In this section, we prove Lemma 2 from the main body.
▶ Lemma 2. Given an infinitely branching countable MDP M with an initial state s 0 , there exists a finitely branching countable M ′ with a set S ′ of states such that s 0 ∈ S ′ and 1. each strategy α 1 in M is mapped to a unique strategy β 1 in M ′ where P s0,α1 (Transience) = P s0,β1 (Transience), 2. and conversely, every MD strategy β 2 in M ′ is mapped to an MD strategy α 2 in M where P s0,α2 (Transience) ≥ P s0,β2 (Transience).
Proof.
Given an infinitely branching MDP M = (S, S , S , −→, P ) with set S of states and an initial state s 0 ∈ S, we construct a finitely branching M ′ with set S ′ of states such that s 0 ∈ S ′ . The reduction uses the concept of "recurrent ladders"; see Figure 2.
− → z 1 , z i 1−p ′ i − −− → z i+1 , z i p ′ i − → s i for all i ≥ 1,
with fresh random states z i and suitably adjusted probabilities p ′ i to ensure that the gadget is left at state s i with exact
probability p i , i.e., p ′ i = p i /( i−1 j=1 (1 − p ′ j )).
See Figure 3 for a partial illustration. Given ρ = q 0 q 1 · · · q n ∈ S + denote by last(ρ) = q n the last state of ρ.
u(m, q)(m ′ , q ′ ) = P (q)(q ′ ) if q ∈ S ; α 1 (ρq)(q ′ ) if q ∈ S is finitely branching in M;
for all m = (ρ, ⊥) and m ′ = (ρq, j) with j ≥ 1, and u(m, q)(m ′ , q ′ ) = 0 otherwise.
u(m, q)(m ′ , q ′ ) = α 1 (ρq)(q j ) if q ∈ S is infinitely branching in M with q → q i for all i ≥ 1, and q ′ = ℓ q,0 ; for all m, m ′ = (ρ, j) with j, u(m, q)(m, q ′ ) = 1 if s = last(ρ)
The strategy β 1 consists of the above update function u and initial memory m 0 = (ϵ, ⊥) where ϵ is the empty run. Intuitively speaking, in every step β 1 considers the memory (ρ, x) and the current state q to simulate what α 1 would have played in M. The memory (ρ, x) is such that ρ invariantly demonstrates the history of run projected into the state space S of M (omitting the introduced states due to the reduction). The second component x in the memory is ⊥ if the current state is in S, and otherwise it is a natural number j ≥ 1. Such a natural number j indicates that the controller is currently on a recurrent ladder and must leaves the ladder at the j-th controlled state on the ladder. Subsequently, β 1 starts with memory (ϵ, ⊥) and q = s 0 , when q is a random state in M, β 1 only append q to ρ to keep track of the history; when q is a finitely branching state in M, β 1 plays as α 1 (ρq) and append q to ρ; when q is an infinitely branching state in M with successors (q j ) j≥1 , for every j ≥ 1, the strategy β 1 chooses the first state ℓ q,0 of the recurrent ladder for q while flipping the memory from (ρ, ⊥) to (ρ, j) with probability σ(ρq)(q j ). This requires the ladder to be traversed to state ℓ q,j and left from there to q j , the j-th successor of q in M. Furthermore, β 1 append q to ρ;
when q is ℓ s,i and memory is (ρ, j) with last(ρ) = s, if i ≤ j then β 1 continues to stay on the recurrent ladder by picking ℓ ′ s,i ; when q is ℓ s,j and memory is (ρ, j) with last(ρ) = s, β 1 leaves the ladder from ℓ s,j to q j which is the j-th successor of state s in M;. In addition, the memory (ρ, j) is flipped back to (ρ, ⊥). By the construction of M ′ and β 1 , it follows that β 1 in M ′ faithfully simulates α 1 in M and thus P M,s0,α1 (Transience) = P M ′ ,s0,β1 (Transience).
For the second item, let β 2 : S ′ → S ′ be an MD strategy in M ′ where S ′ ⊆ S is the set of controlled states in M ′ . We define an MD strategy α 2 : S → S in M as follows. For all controlled states s ∈ S ′ ,
α 2 (s) = β 2 (q) if s ∈ S is finitely branching in M; s j if s ∈ S is infinitely branching in M with the successors (s i ) i≥1 ,
and if there exists j ∈ N such that β 2 (s) = ℓ s,0 , β 2 (ℓ s,0 ) = ℓ s,1 and β 2 (ℓ s,i ) = ℓ ′ s,i for all 0 < i < j, and β 2 (ℓ s,j ) = s j ; s 1 if s ∈ S is infinitely branching in M with the successors (s i ) i≥1 , and if β 2 (s) = ℓ s,0 , β 2 (ℓ s,0 ) = ℓ s,1 and β 2 (ℓ s,i ) = ℓ ′ s,i for all i > 0.
Note that the above strategy is well-defined, as in every recurrent ladder in M ′ , either there exists some j such that β 2 exits the ladder at its j-th controller state, or β 2 choose to stay on the ladder forever. In the latter case, by a Gambler's Ruin argument, the probability of Transience for those runs staying on the ladder forever is 0. By the construction of M ′ , α 2 faithfully simulates β 2 unless when β 2 stays on a ladder forever and the prospect of Transience becomes 0. In those cases, α 2 continues playing what β 2 would have played if it exited the s-ladder at ℓ s,1 . It follows that P M,s0,α2 (Transience) ≥ P M ′ ,s0,β2 (Transience). ◀
C 1-Bit Strategy for Büchi(F ) ∩ Transience
▶ Lemma 4. Let M be a countable MDP, I a finite set of initial states, F a set of states and ε > 0. Then there exists a deterministic 1-bit strategy for Büchi(F ) ∩ Transience that is ε-optimal from every s ∈ I.
Proof. We prove the claim for finitely branching M first and transfer the result to general MDPs at the end. Let M = (S, S , S , −→, P ) be a finitely branching countable MDP, I ⊆ S a finite set of initial states and F ⊆ S a set of goal states and φ def = Büchi(F ) ∩ Transience the objective. For every ε > 0 and every s ∈ I there exists an ε-optimal strategy σ s such that
P M,s,σs (φ) ≥ val M,φ (s) − ε.(11)
However, the strategies σ s might differ from each other and might use randomization and a large (or even infinite) amount of memory. We will construct a single deterministic strategy σ ′ that uses only 1 bit of memory such that ∀ s∈I P M,s,σ ′ (φ) ≥ val M,φ (s) − 2ε. This proves the claim as ε can be chosen arbitrarily small. In order to construct σ ′ , we first observe the behavior of the finitely many σ s for s ∈ I on an infinite, increasing sequence of finite subsets of S. Based on this, we define a second
I · · · · · · K 1 L 1 K 2 L 2 K 3 L i−1 K i L i Figure 4
To show the bubble construction. The green region in K1 is F1, and for all i ≥ 2, the green region in Ki \ Li−1 is Fi.
stronger objective φ ′ with φ ′ ⊆ φ,(12)
and show that all σ s attain at least val M,φ (s) − 2ε w.r.t. φ ′ , i.e.,
∀ s∈I P M,s,σs (φ ′ ) ≥ val M,φ (s) − 2ε.(13)
We construct σ ′ as a deterministic 1-bit optimal strategy w.r.t. φ ′ from all s ∈ I and obtain P M,s,σ ′ (φ) ≥ P M,s,σ ′ (φ ′ ) by Equation (12) ≥ P M,s,σs (φ ′ ) by optimality of σ ′ for φ ′ ≥ val M,φ (s) − 2ε by Equation (13).
Behavior of σ, objective φ ′ and properties Equation (12) and Equation (13).
We start with some notation. Let bubble k (X) be the set of states that can be reached from some state in the set X within at most k steps. Since M is finitely branching, bubble k (X) is finite if X is finite. Let F ≤k (X) def = {ρ ∈ S ω | ∃t ≤ k. ρ(t) ∈ X} and F ≥k (X) def = {ρ ∈ S ω | ∃t ≥ k. ρ(t) ∈ X} denote the property of visiting the set X (at least once) within at most (resp. at least) k steps. Moreover, let ε i def = ε · 2 −(i+1) .
▶ Lemma 18. Assume the setup of Lemma 4, φ def = Büchi(F ) ∩ Transience and a strategy σ s from each s ∈ I. Let X ⊆ S be a finite set of states and ε ′ > 0. 1. There is k ∈ N such that ∀ s∈I P M,s,σs (φ ∩ ¬(F ≤k (F \ X))) ≤ ε ′ . 2. There is l ∈ N such that ∀ s∈I P M,s,σs (φ ∩ F ≥l (X)) ≤ ε ′ .
Proof. It suffices to show the properties for a single s, σ s since one can take the maximal k, l over the finitely many s ∈ I.
We observe that φ ⊆ Transience = s∈S FG¬(s) ⊆ s∈X FG¬(s) = FG¬(X), where the last equivalence is due to the finiteness of X.
Towards 1, we have φ = GFF ∩Transience ⊆ GFF ∩FG¬(X) ⊆ GF(F \X) ⊆ F(F \X) = k∈N F ≤k (F \X) and therefore that φ∩ k∈N ¬(F ≤k (F \X)) = ∅. It follows from the continuity of measures that lim k→∞ P M,s,σs (φ ∩ ¬(F ≤k (F \ X))) = 0.
Towards 2, we have φ ∩ ∩ l F ≥l (X) ⊆ FG¬(X) ∩ ∩ l F ≥l (X) = ∅. By continuity of measures we obtain lim l→∞ P M,s,σs (φ ∩ F ≥l (X)) = 0. ◀ In the following, let us write X to denote the complement of a set X ⊆ S ω of runs.
By Lemma 18(1) there is a k 1 such that for K 1 def = bubble k1 (I) and F 1 def = F ∩ K 1 we have ∀ s∈I P M,s,σs (φ ∩ K * 1 F 1 S ω ) ≤ ε 1 . We define the pattern
R 1 def = (K 1 \ F 1 ) * F 1
and obtain ∀ s∈I P M,s,σs (φ ∩ R 1 S ω ) ≤ ε 1 . By Lemma 18 (2) there is an l 1 > k 1 such that ∀ s∈I P M,s,σs (F ≥l1 (K 1 )) ≤ ε 1 . Define L 1 def = bubble l1 (I). By Lemma 18(1) there is a k 2 > l 1 such that for K 2 def = bubble k2 (I) and F 2 def = F ∩K 2 \L 1 we have ∀ s∈I P M,s,σs (φ∩K * 2 F 2 S ω ) ≤ ε 2 . We define the pattern
R 2 def = (K 2 \ F 2 ) * F 2
and obtain ∀ s∈I P M,s,σs (φ∩R 2 S ω ) ≤ ε 2 and, via a union bound, ∀ s∈I P M,s,σs (φ∩R 2 (S \ K 1 ) ω ) ≤ ε 1 + ε 2 . By another union bound it follows that ∀ s∈I P M,s,σs (φ ∩ R 1 R 2 (S \ K 1 ) ω ) ≤ 2ε 1 + ε 2 .
Proceed inductively for i = 2, 3, . . . as follows (see Figure 4 for an illustration). By Lemma 18 (2)
there is an l i > k i such that ∀ s∈I P M,s,σs (F ≥li (K i )) ≤ ε i . Define L i def = bubble li (I). By Lemma 18(1) there is k i+1 > l i such that for K i+1 def = bubble ki+1 (I) and F i+1 def = F ∩ K i+1 \ L i we have ∀ s∈I P M,s,σs (φ ∩ (K i+1 \ F i+1 ) * F i+1 S ω ) ≤ ε i+1 . By a union bound, ∀ s∈I P M,s,σs (φ ∩ (K i+1 \ F i+1 ) * F i+1 (S \ K i ) ω ) ≤ ε i + ε i+1 . By an induction hypothesis we have ∀ s∈I P M,s,σs (φ ∩ R 1 R 2 . . . R i (S \ K i−1 ) ω ) ≤ 2ε 1 + · · · + 2ε i−1 + ε i . We define the pattern R i+1 def = (K i+1 \ (F i+1 ∪ K i−1 )) * F i+1 . Using that (K i+1 \ F i+1 ) * F i+1 (S \ K i ) ω ∩ R 1 R 2 . . . R i (S \ K i−1 ) ω ⊆ R 1 R 2 . . . R i+1 (S \ K i ) ω , we get ∀ s∈I P M,s,σs (φ ∩ R 1 R 2 . . . R i+1 (S \ K i ) ω ) ≤ 2ε 1 + · · · + 2ε i + ε i+1 ≤ ε.(14)
We now define the Borel objectives R ≤i
def = R 1 R 2 . . . R i S ω and φ ′ def = i∈N R ≤i .
Since F i ∩ F k = ∅ for i ̸ = k and φ ′ implies a visit to the set F i for all i ∈ N, we have φ ′ ⊆ Büchi(F ). Now we show that φ ′ ⊆ Transience. Let s be an arbitrary state and ρ a run from some state in I that satisfies φ ′ . If s is not reachable from I then ρ never visits s. Otherwise, there exists some minimal j such that s ∈ K j . The run ρ must eventually visit F j+1 and after visiting F j+1 it cannot visit K j (and thus s) any more. Therefore ρ visits s only finitely often. Thus φ ′ ⊆ Transience. Together we have φ ′ ⊆ Büchi(F ) ∩ Transience = φ and obtain Equation (12).
Moreover, R ≤1 ⊇ R ≤2 ⊇ R ≤3 .
. . is an infinite decreasing sequence of Borel objectives.
For every s ∈ I we have
P M,s,σs (φ ′ ) = P M,s,σs (∩ ∞ i=1 R ≤i ) by def. of φ ′ = lim i→∞ P M,s,σs (R ≤i ) by cont. of measures = lim i→∞ 1 − P M,s,σs (R ≤i ) by duality = lim i→∞ 1 − (P M,s,σs (R ≤i ∩ φ) + P M,s,σs (R ≤i ∩ φ)) case split ≥ lim i→∞ 1 − (ε + P M,s,σs (R ≤i ∩ φ))
by Equation (14) ≥ lim (12) ≥ val M,φ (s) − 2ε by Equation (11) Thus we obtain property Equation (13).
i→∞ 1 − (ε + P M,s,σs (φ ′ ∩ φ)) since φ ′ ⊆ R ≤i = 1 − (ε + 1 − P M,s,σs (φ ′ ∪ φ)) by duality = P M,s,σs (φ) − ε by Equation
Definition of the 1-bit strategy σ ′ . We now define our deterministic 1-bit strategy σ ′ that is optimal for objective φ ′ from every s ∈ I. First we define certain "suffix" objectives of
φ ′ . Recall that R i = (K i \(F i ∪K i−2 )) * F i . Let R i,j def = R i R i+1 . . . R j S ω and R ≥i def = j≥i R i,j .
In particular, this means that φ ′ = R ≥1 . Every run w from some state s ∈ I that satisfies φ ′ can be split into parts before and after the first visit to set F i , i.e., w = w 1 s ′ w 2 where w 1 s ′ ∈ R ≤i , s ′ ∈ F i and s ′ w 2 ∈ R ≥i+1 . (Note also that w 2 cannot visit any states in K i−1 .) Thus it will be useful to consider the objectives R ≥i+1 for runs that start in states s ′ ∈ F i . For every state s ′ ∈ F i we consider its value w.r.t. the objective R ≥i+1 , i.e., val M,R ≥i+1 (s ′ ) def = supσ P M,s ′ ,σ (R ≥i+1 ). For every i ≥ 1 we consider the finite subspace K i \ K i−2 . In particular, it contains the sets F i−1 and F i . (For completeness let K 0 def = F 0 def = I and K −1 def = ∅.) It is not enough to maximize the probability of reaching the set F i in each K i individually. One also needs to maximize the potential of visiting further sets F i+1 , F i+2 , . . . in the indefinite future. Thus we define the bounded total reward objective B i for runs starting in F i−1 as follows. Runs that exit the subspace (either by leaving K i or by visiting K i−2 ) before visiting F i get reward 0. When some run reaches the set F i for the first time in some state s ′ then this run gets the reward of val M,R ≥i+1 (s ′ ). We can consider an induced finite MDPM with state space K i \ K i−2 , plus a sink state (with reward 0) that is reached immediately after visiting any state in F i and whenever one exits the set K i \ K i−2 . InM one gets a reward of val M,R ≥i+1 (s ′ ) for visiting s ′ ∈ F i as above. By [21, Theorem 7.1.9], there exists a uniform optimal MD strategy σ i for this bounded total reward objective on the induced finite MDP M, which can be directly applied for objective B i on the subspace K i \ K i−2 in M. (The strategy σ i is not necessarily unique, but our results hold regardless of which of them is picked.)
We now define σ ′ by combining different MD strategies σ i , depending on the current state and on the value of the 1-bit memory. The intuition is that the strategy σ ′ has two modes: normal-mode and next-mode. In a state s ′ ∈ K i \ K i−1 , if the memory is i (mod 2) then the strategy is in normal-mode and plays towards reaching F i . Otherwise, the strategy is in next-mode and plays towards reaching F i+1 (normally this happens because F i has already been seen).
Initially σ ′ starts in a state s ∈ I with the 1-bit memory set to 1. We define the behavior of σ ′ in a state s ′ ∈ K i \ K i−1 for every i ≥ 1. Figure 5 Memory updates along runs π1, π2, π3, drawn in blue while the memory-bit is one and in red while the bit is zero. Both π1 and π3 violate φ ′ and are drawn as dotted lines once they do.
· · · K 1 L 1 K 2 L 2 K 3 π 3 π 2 π 1 I
If the 1-bit memory is i (mod 2) and s ′ / ∈ F i then σ ′ plays like σ i . (Intuitively, one plays towards F i , since one has not yet visited it.) If the 1-bit memory is i (mod 2) and s ′ ∈ F i then the 1-bit memory is set to (i + 1) (mod 2), and σ ′ plays like σ i+1 . (Intuitively, one records the fact that one has already seen F i and then targets the next set F i+1 .) If the 1-bit memory is (i + 1) (mod 2) then σ ′ plays like σ i+1 . (Intuitively, one plays towards F i+1 , since one has already visited F i .) Observe that if a run according to σ ′ exits some set K i (and thus enters K i+1 \ K i ) with the bit still set to i (mod 2) (normal-mode) then this run has not visited F i and thus does not satisfy the objective φ ′ . (Or the same has happened earlier for some j < i, in which case also the objective φ ′ is violated.) An example is the run π 1 in Figure 5.
However, if a run according to σ ′ exits some set K i (and thus enters K i+1 \ K i ) with the bit set to (i + 1) (mod 2) (thus σ i+1 in next-mode) then in the new set K i ′ \ K i ′ −1 with i ′ = i + 1 the bit is set to i ′ (mod 2) and σ ′ continues to play like σ i+1 in normal-mode. Even if this run returns (temporarily) to K i (but not to K i−1 ) the strategy σ ′ continues to play like σ i+1 in next-mode. An example is the run π 2 in Figure 5.
Finally, if a run returns to K i−1 after having visited F i then it fails the objective φ ′ . An example is the run π 3 in Figure 5.
The 1-bit strategy σ ′ is optimal for φ ′ from every s ∈ I. In the following let s ∈ I be an arbitrary initial state in I. For any run from s, let firstin(F i ) be the first state s ′ in F i that is visited (if any). We define a bounded reward objective B ′ i for runs starting at s as follows. Every run that does not satisfy the objective R ≤i gets assigned reward 0. Otherwise, consider a run from s that satisfies R ≤i . When this run reaches the set F i for the first time in some state s ′ then this run gets a reward of val M,R ≥i+1 (s ′ ). Note that this reward is ≤ 1.
We show that for all
i ∈ N val M,φ ′ (s) = val M,B ′ i (s)(15)
Towards the ≥ inequality, letσ be anε-optimal strategy for B ′ i from s. We define the strategŷ σ ′ to play likeσ until a state s ′ ∈ F i is reached and then to switch to someε-optimal strategy for objective R ≥i+1 from s ′ . Every run from s that satisfies φ ′ can be split into parts, before and after the first visit to the set F i , i.e.,
φ ′ = {w 1 s ′ w 2 | w 1 s ′ ∈ R ≤i , s ′ ∈ F i , s ′ w 2 ∈ R ≥i+1 }. Therefore we obtain that P M,s,σ ′ (φ ′ ) ≥ E M,s,σ (B ′ i ) −ε ≥ val M,B ′ i (s) − 2ε.
Since this holds for everyε > 0, we obtain val M,φ ′ (s) ≥ val M,B ′ i (s). Towards the ≤ inequality, letσ be any strategy for φ ′ from s. We have P M,s,σ (φ ′ ) ≤
s ′ ∈Fi P M,s,σ (R ≤i ∩ firstin(F i ) = s ′ ) · val M,R ≥i+1 (s ′ ) = E M,s,σ (B ′ i ). Thus val M,φ ′ (s) ≤ val M,B ′ i (s).
Together we obtain Equation (15). For all i ∈ N and every state s ′ ∈ F i we show that
val M,R ≥i+1 (s ′ ) = val M,Bi+1 (s ′ )(16)
Towards the ≥ inequality, letσ be anε-optimal strategy for B i+1 from s ′ ∈ F i . We define the strategyσ ′ to play likeσ until a state s ′′ ∈ F i+1 is reached and then to switch to someε-optimal strategy for objective R ≥i+2 from s ′′ . We have that
P M,s ′ ,σ ′ (R ≥i+1 ) ≥ E M,s ′ ,σ (B i+1 ) −ε ≥ val M,Bi+1 (s) − 2ε. Since this holds for everyε > 0, we obtain val M,R ≥i+1 (s ′ ) ≥ val M,Bi+1 (s ′ ).
Towards the ≤ inequality, letσ be any strategy for R ≥i+1 from s ′ ∈ F i . We have
P M,s ′ ,σ (R ≥i+1 ) ≤ s ′′ ∈Fi+1 P M,s ′ ,σ (R i+1 S ω ∩ firstin(F i+1 ) = s ′′ ) · val M,R ≥i+2 (s ′′ ) = E M,s ′ ,σ (B i+1 ). Thus val M,R ≥i+1 (s ′ ) ≤ val M,Bi+1 (s ′ ).
Together we obtain Equation (16). We show, by induction on i, that σ ′ is optimal for B ′ i for all i ∈ N from start state s, i.e.,
E M,s,σ ′ (B ′ i ) = val M,B ′ i (s)(17)
In the base case of i = 1 we have that B ′ 1 = B 1 . The strategy σ ′ plays σ 1 until reaching F 1 , which is optimal for objective B 1 and thus optimal for B ′ 1 . For the induction step we assume (IH) that σ ′ is optimal for B ′ i .
val M,B ′ i+1 (s) = val M,B ′ i (s) by Equation (15) = E M,s,σ ′ (B ′ i ) by (IH) = s ′ ∈Fi P M,s,σ ′ (R ≤i ∩ firstin(F i ) = s ′ ) · val M,R ≥i+1 (s ′ ) by def. of B ′ i = s ′ ∈Fi P M,s,σ ′ (R ≤i ∩ firstin(F i ) = s ′ ) · val M,Bi+1 (s ′ ) by Equation (16) = s ′ ∈Fi P M,s,σ ′ (R ≤i ∩ firstin(F i ) = s ′ ) · E M,s ′ ,σi+1 (B i+1 ) opt. of σ i+1 for B i+1 = E M,s,σ ′ (B ′ i+1 ) by def. of σ ′ and B ′ i+1
So σ ′ attains the value val M,B ′ i+1 (s) of the objective B ′ i+1 from s and is optimal. Thus Equation (17). Now we show that σ ′ performs well on the objectives R ≤i for all i ∈ N.
P M,s,σ ′ (R ≤i ) ≥ val M,φ ′ (s)(18)
We have
P M,s,σ ′ (R ≤i ) ≥ E M,s,σ ′ (B ′ i ) since B ′ i
gives rewards 0 for runs / ∈ R ≤i and ≤ 1 otherwise = val M,B ′ i (s) by Equation (17) = val M,φ ′ (s) by Equation (15) So we get Equation (18). Now we are ready to prove the optimality of σ ′ for φ ′ from s.
P M,s,σ ′ (φ ′ ) = P M,s,σ ′ (∩ i∈N R ≤i ) by def. of φ ′ = lim i→∞ P M,s,σ ′ (R ≤i ) by continuity of measures from above ≥ lim i→∞ val M,φ ′ (s) by Equation (18) = val M,φ ′ (s)
This concludes the proof that σ ′ is optimal for φ ′ and hence 2ε-optimal for φ for every initial state s ∈ I.
From finitely to infinitely branching MDPs. Let M be an infinitely branching MDP with a finite set of initial states I and ε > 0. We derive a finitely branching MDP M ′ with sufficiently similar behavior wrt. our objective φ = Büchi(F ) ∩ Transience. Every controlled state x with infinite branching x → y i for all i ∈ N is replaced by a gadget x → z 1 , z i → z i+1 , z i → y i for all i ∈ N with fresh controlled states z i . Infinitely branching random states with x pi − → y i for all i ∈ N are replaced by a gadget x
1 − → z 1 , z i 1−p ′ i − −− → z i+1 , z i p ′ i − → y i for all i ∈ N,
with fresh random states z i and suitably adjusted probabilities p ′ i to ensure that the gadget is left at state y i with probability p i , i.e.,
p ′ i = p i /( i−1 j=1 (1 − p ′ j )
). We apply the above result for finitely branching MDPs to M ′ and obtain a 1-bit deterministic ε-optimal strategy σ ′ for our objective φ = Büchi(F ) ∩ Transience from all states s ∈ I. We construct a 1-bit deterministic ε-optimal strategy σ ′′ for M as follows. Consider some state x that is infinitely branching in M and its associated gadget in M ′ . Whenever a run in M ′ according to σ ′ reaches x with some memory value α ∈ {0, 1} there exist values p i for the probability that the gadget is left at state y i . Let p def = 1 − i∈N p i be the probability that the gadget is never left. (If x is controlled then only one p i (or p) is nonzero, since σ ′ is deterministic. If x is random then p = 0.) Since σ ′ is deterministic, the memory updates are deterministic, and thus there are values α ′ i ∈ {0, 1} such that whenever the gadget is left at state y i the memory will be α ′ i . We now define the behavior of the 1-bit deterministic strategy σ ′′ at state x with memory α in M.
If x is controlled and p ̸ = 1 then σ ′′ picks the successor state y i where p i = 1 and sets the memory to α ′ i . If p = 1 then any run according to σ ′ that enters the gadget does not satisfy the objective φ = Büchi(F ) ∩ Transience, since the states in the gadget are disjoint from F . I.e., every run that eventually stays in some gadget forever does not even satisfy Büchi(F ), and thus does not satisfy φ. Thus σ ′′ performs at least as well in M regardless of its choice, e.g., pick successor y 1 and α ′ = α.
If x is random then p = 0 and the successor is chosen according to the defined distribution (which is the same in M and M ′ ) and σ ′′ can only update its memory. Whenever the successor y i is chosen, σ ′′ updates the memory to α ′ i . In states that are not infinitely branching in M, σ ′′ does exactly the same in M as σ ′ in M ′ .
Since the gadgets do not intersect F , σ ′′ performs at least as well in M as σ ′ in M ′ and is thus ε-optimal from every s ∈ I. ◀ ▶ Remark 19. Note that the last step in the proof of Lemma 4, lifting the result from finitely branching MDPs to infinitely branching MDPs, does require this particular construction. It cannot be shown by applying Lemma 2. The construction used for Lemma 2 (i.e., Figure 3) can only lift MD strategies, but not deterministic 1-bit strategies. The problem is that the construction in Figure 3 introduces extra randomness and multiple paths to the same exit from the ladder. While an MD strategy on the finitely branching MDP M ′ induces a corresponding MD strategy on the infinitely branching MDP M, the same does not hold for deterministic 1-bit strategies. In contrast, the different construction in the last part of the proof of Lemma 4 preserves deterministic 1-bit strategies, but works only for the Büchi(F ) ∩ Transience objective, not for Transience alone.
D Missing Proofs from Section 4
We prove Theorem 7 from the main body: ▶ Theorem 7. Let M = (S, S , S , −→, P ) be a countable MDP, and let φ be an objective that is tail in M. Suppose for every s ∈ S there exist ε-optimal MD strategies for φ. Then: 1. There exist uniform ε-optimal MD strategies for φ.
2.
There exists a single MD strategy that is optimal from every state that has an optimal strategy.
D.1 Proof of Item 1 of Theorem 7
Proof. We follow Ornstein's proof [20] as presented in [16]. Recall that an MD strategy σ can be viewed as a function σ : S → S such that for all s ∈ S , the state σ(s) is a successor state of s. Starting from the original MDP M we successively fix more and more controlled states, by which we mean select an outgoing transition and remove all others. While this is in general an infinite (but countable) process, it defines an MD strategy in the limit. Visually, we "plaster" the whole state space by the fixings. Put the states in some order, i.e., s 1 , s 2 , . . . with S = {s 1 , s 2 , . . .}. The plastering proceeds in rounds, one round for every state. Let M i be the MDP obtained from M after the fixings of the first i − 1 rounds (with M 1 = M). In round i we fix controlled states in such a way that (A) the probability, starting from s i , of φ using only random and fixed controlled states is not much less than the value val Mi (s i ); and (B) for all states s, the value val Mi+1 (s) is almost as high as val Mi (s). The purpose of goal (A) is to guarantee good progress towards φ when starting from s i . The purpose of goal (B) is to avoid fixings that would cause damage to the values of other states. Now we describe round i. Consider the MDP M i after the fixings from the first i−1 rounds, and let ε i > 0. Recall that we wish to fix a part of the state space so that s i has a high probability of φ using only random and fixed controlled states. By assumption there is an MD strategy σ such that P Mi,si,σ (φ) ≥ val Mi (s i ) − ε 2 i . Fixing σ everywhere would accomplish goal (A), but potentially compromise goal (B). So instead we are going to fix σ only for states where σ does well: define
G def = {s ∈ S | P Mi,s,σ (φ) ≥ val Mi (s) − ε i }
and obtain M i+1 from M i by fixing σ on G. (Note that σ does not "contradict" earlier fixings, because in the MDP M i the previously fixed states have only one outgoing transition left.)
We have to check that with this fixing we accomplish the two goals above. Indeed, we accomplish goal (A): by its definition strategy σ is ε 2 i -optimal from s i , so the probability of ever entering S \ G (where σ is less than ε i -optimal) cannot be large:
P Mi,si,σ (Reach(S \ G)) ≤ ε i(19)
In slightly more detail, this inequality holds because the probability that the ε 2 i -optimal strategy σ enters a state whose value is underachieved by σ by at least ε i can be at most ε i . We give a detailed proof of (19) in Lemma 20 below. It follows from the ε 2 i -optimality of σ and from (19)
that we have P Mi,si,σ (φ ∧ ¬Reach(S \ G)) ≥ val Mi (s i ) − ε i − ε 2 i . So in M i+1 we obtain for all strategies σ ′ : P Mi+1,si,σ ′ (φ) ≥ val Mi (s i ) − ε i − ε 2 i(20)
We also accomplish goal (B): the difference between M i and M i+1 is that σ is fixed on G, but σ performs well from G on. So we obtain for all states s:
val Mi+1 (s) ≥ val Mi (s) − ε i(21)
In slightly more detail, this inequality holds because any strategy in M i can be transformed into a strategy in M i+1 , with the difference that once the newly fixed part G is entered, the strategy switches to the strategy σ, which (by the definition of M i+1 ) is consistent with the fixing and (by the definition of G) is ε i -optimal from there. We give a detailed proof of (21) in Lemma 21 below. This completes the description of round i. Let ε ∈ (0, 1), and for all i ≥ 1, choose ε i def = ε 2 · 2 −i . Let σ be an arbitrary MD strategy that is compatible with all fixings. (This strategy σ is actually unique.) It follows that σ is playable in all M i . We have for all i ≥ 1:
P M,si,σ (φ) ≥ val Mi (s i ) − ε i − ε 2 i by (20) ≥ val Mi (s i ) − 2ε i as ε i < 1 ≥ val Mi (s i ) − ε 2 choice of ε i ≥ val M (s i ) − i−1 j=1 ε j − ε 2 by (21) ≥ val M (s i ) − ε choice of ε j
Thus, the MD strategy σ is ε-optimal for all states. ◀ ▶ Lemma 20. Equation (19) holds.
Proof. For a state s ∈ S \ G, define the event L s as the set of runs that leave G such that s is the first visited state in S \ G. Then we have:
P Mi,si,σ (Reach(S \ G)) = s∈S\G P Mi,si,σ (L s )
Since φ is tail and using the Markov property:
P Mi,si,σ (φ) = P Mi,si,σ (¬Reach(S \ G) ∧ φ) + s∈S\G P Mi,si,σ (L s ) · P Mi,s,σ (φ)
By the definition of G it follows:
P Mi,si,σ (φ) ≤ P Mi,si,σ (¬Reach(S \ G) ∧ φ) + s∈S\G P Mi,si,σ (L s ) · (val Mi (s) − ε i )(22)
On the other hand, σ is ε 2 i -optimal for s i , hence:
P Mi,si,σ (φ) ≥ −ε 2 i + val Mi (s) ≥ −ε 2 i + P Mi,si,σ (φ ∧ ¬Reach(S \ G)) + s∈S\G P Mi,si,σ (L s ) · val Mi (s)(23)
By combining (22) and (23) we obtain:
ε 2 i ≥ ε i · s∈S\G P Mi,si,σ (L s ) = ε i · P Mi,si,σ (Reach(S \ G)) ◀ ▶ Lemma 21. Equation (21) holds.
Proof. For a state s ′ ∈ G, define the event E s ′ as the set of runs that enter G such that s ′ is the first visited state in G. Fix any state s ∈ S and any strategy σ i in M i . We transform σ i into a strategy σ i+1 in M i+1 such that σ i+1 behaves like σ i until G is entered, at which point σ i+1 switches to the MD strategy σ, which we recall is compatible with M i+1 and is ε i -optimal from G in M i . To show (21) it suffices to show that P Mi+1,s,σi+1 (φ) ≥ P Mi,s,σi (φ) − ε i . We have:
P Mi+1,s,σi+1 (φ) = P Mi+1,s,σi+1 (¬Reach(G) ∧ φ) + φ is tail s ′ ∈G P Mi+1,s,σi+1 (E s ′ ) · P Mi+1,s ′ ,σi+1 (φ) Markov property = P Mi,s,σi (¬Reach(G) ∧ φ) + using def. of σ i+1 s ′ ∈G P Mi,s,σi (E s ′ ) · P Mi,s ′ ,σ (φ)
Further we have for all s ′ ∈ G:
P Mi,s ′ ,σ (φ) ≥ val Mi (s ′ ) − ε i as s ′ ∈ G ≥ P Mi,s ′ ,σi (φ) − ε i
Plugging this in above, we obtain:
P Mi+1,s,σi+1 (φ) ≥ P Mi,s,σi (¬Reach(G) ∧ φ) + s ′ ∈G P Mi,s,σi (E s ′ ) · (P Mi,s ′ ,σi (φ) − ε i ) ≥ P Mi,s,σi (¬Reach(G) ∧ φ) + s ′ ∈G P Mi,s,σi (E s ′ ) · P Mi,s ′ ,σi (φ) − ε i = P Mi,s,σi (φ) − ε i ◀
D.2 Proof of Item 2 of Theorem 7
Proof. As discussed in Section 6, in [17, Lemma 6] there is a construction of a certain conditioned version of M (similar to M * from Definition 12), say M + . The construction is such that φ is tail also in M + . By [17, Lemma 6, item 2] it suffices to exhibit a single MD strategy in M + that is almost surely winning from all states that have an almost surely winning strategy.
Obtain from M + an MDP M ′ by restricting the state space to those states that have an almost surely winning strategy, and eliminating all transitions leaving these states. In M ′ all states have an almost surely winning strategy, as an almost surely winning strategy may never enter a state that does not have an almost surely winning strategy (using the fact that φ is tail). Let σ be a uniform 1 2 -optimal MD strategy (in M ′ ), which exists by item 1. It suffices to show that σ is (in M ′ ) almost surely winning from all states that have an almost surely winning strategy.
We follow the argument from [16, Theorem 6]. We have P M ′ ,s,σ (φ) ≥ 1 2 for all states s. Thus, for any run s 0 s 1 · · · in M ′ we have P M ′ ,si,σ (¬φ) ≤ 1 2 for all i; in particular, the sequence (P M ′ ,si,σ (¬φ)) i does not converge to 1. As a consequence of Lévy's zero-one law, since ¬φ is tail, the events ¬φ and {s 0 s 1 · · · | lim i→∞ P M ′ ,si,σ (¬φ) = 1} are equal up to a null set. Thus, for all states s we have P M ′ ,s,σ (¬φ) = 0; hence, P M ′ ,s,σ (φ) = 1. Suppose that σ is an MD strategy which, at any state s ∈ S , picks a successor s ′ with
val(s ′ ) ≥ val(s) − ε 2 ι(s)+1 · R(s) .
This is possible even if M is infinitely branching, by the definition of value and the fact that R(s) < ∞. We show that P M,s0,σ (Safety(T )) ≥ val(s 0 ) − ε holds for every initial state s 0 , which implies the claim of the theorem. Towards this, we define a function cost that labels each transition in the MDP with a realvalued cost: For every controlled transition s−→s ′ let cost((s, s ′ )) def = val(s) − val(s ′ ) ≥ 0. Random transitions have cost zero. We will argue that when playing σ from any start state s 0 , its attainment w.r.t. the objective Safety(T ) equals the value of s 0 minus the expected total cost, and that this cost is bounded by ε.
For any i ∈ N let us write s i for the random variable denoting the state just after step i, and Cost(i) def = cost(s i , s i+1 ) for the cost of step i in a random run. We now show that under σ the expected total cost is bounded in the limit, i.e.,
lim n→∞ E n−1 i=0 Cost(i) ≤ ε.(24)(i) ≤ ∞ i=0 Re(s) i ε 2 ι(s)+1 ·R(s) = ε 2 ι(s)+1 , which in turn im- plies Eq. (24) as then lim n→∞ E n−1 i=0 Cost(i) = s∈S lim n→∞ E n−1 i=0 Cost s (i) ≤ s ε 2 ι(s)+1 = ε.
Next, we show that for every n,
E(val(s n )) = E(val(s 0 )) − E n−1 i=0 Cost(i) .(25)
By induction on n where the base case n = 0 trivially holds. For the induction step,
E(val(s n+1 )) = E(val(s n ) + val(s n+1 ) − val(s n )) = E(val(s n )) + E(val(s n+1 ) − val(s n )) = E(val(s n )) + P(s n ∈ S )E(val(s n+1 ) − val(s n ) | s n ∈ S ) + P(s n ∈ S )E(val(s n+1 ) − val(s n ) | s n ∈ S ) = E(val(s n )) + 0 − P(s n ∈ S )E(Cost(n) | s n ∈ S ) = E(val(s n )) − P(s n ∈ S )E(Cost(n) | s n ∈ S ) − P(s n ∈ S )E(Cost(n) | s n ∈ S ) = E(val(s n )) − E(Cost(n)) = E(val(s 0 )) − n−1 i=0 E(Cost(i)) − E(Cost(n)) = E(val(s 0 )) − E n i=0
Cost(i) . (24) and (25) we get
From Equations
lim inf n→∞ E(val(s n )) = val(s 0 ) − lim n→∞ E n−1 i=0 cost(i) ≥ val(s 0 ) − ε.(26)
Finally, to show the claim let [s n / ∈ T ] : S ω → {0, 1} be the random variable that indicates that the n-th state is not in the target set T . Note that [s n / ∈ T ] ≥ val(s n ) because target states have value 0. We have: (26). ◀
P M,s0,σ (Safety(T )) = P M,s0,σ ∞ i=0 X i ¬T semantics of Safety(T ) = G¬T = lim n→∞ P M,s0,σ n i=0 X i ¬T continuity of measures = lim n→∞ P M,s0,σ (X n ¬T ) T is a sink = lim n→∞ E([s n / ∈ T ]) definition of [s n / ∈ T ] ≥ lim inf n→∞ E(val(s n )) as [s n / ∈ T ] ≥ val(s n ) ≥ val(s 0 ) − ε Equation
F Missing Proofs from Section 6
We prove Lemma 13 from the main body:
▶ Lemma 13. Let M = (S, S , S , −→, P ) be an MDP, and let φ be an objective that is tail in M. Let M * = (S * , S * , S * , −→ * , P * ) be the conditioned version of M w.r.t. φ. Let s 0 ∈ S * ∩ S. Let σ ∈ Σ M * , and note that σ can be transformed to a strategy in M in a natural way. Then: 1. For all n ≥ 0 and all partial runs s 0 s 1 · · · s n ∈ s 0 S * * in M * with s n ∈ S: val M (s 0 ) · P M * ,s0,σ (s 0 s 1 · · · s n S ω * ) = P M,s0,σ (s 0 s 1 · · · s n S ω ) · val M (s n ) , where w for a partial run w in M * refers to its natural contraction to a partial run in M; i.e., w is obtained from w by deleting all states of the form (s, t).
For all measurable
R ⊆ s 0 (S * \ {s ⊥ }) ω we have P M,s0,σ (R) ≥ val M (s 0 ) · P M * ,s0,σ (R) ≥ P M,s0,σ (R ∩ φ s0 ) ,
where R is obtained from R by deleting, in all runs, all states of the form (s, t). 3. We have val M (s 0 ) · P M * ,s0,σ (φ) = P M,s0,σ (φ). In particular, val M * (s 0 ) = 1, and, for any ε ≥ 0, strategy σ is ε-optimal in M * if and only if it is εval M (s 0 )-optimal in M.
Proof.
We prove the equality in item 1 by induction on n. For n = 0 it is trivial. For the step, suppose the equality holds for some n. Let s 0 s 1 · · · s n ∈ s 0 S * * be a partial run in M * with s n ∈ S.
Let s n ∈ S and s n+1 ∈ S * ∩ S. We have:
val M (s 0 ) · P M * ,s0,σ (s 0 s 1 · · · s n (s n , s n+1 )s n+1 S ω * )
= val M (s 0 ) · P M * ,s0,σ (s 0 s 1 · · · s n S ω * ) · σ(s 0 s 1 . . . s n )((s n , s n+1 )) · val M (s n+1 ) val M (s n ) def. of P * = P M,s0,σ (s 0 s 1 · · · s n S ω ) · σ(s 0 s 1 . . . s n )((s n , s n+1 )) · val M (s n+1 ) ind. hyp.
= P M,s0,σ (s 0 s 1 · · · s n S ω ) · σ(s 0 s 1 . . . s n )(s n+1 ) · val M (s n+1 ) σ in M = P M,s0,σ (s 0 s 1 · · · s n (s n s n+1 )s n+1 S ω ) · val M (s n+1 ) Let s n ∈ S and s n+1 ∈ S * ∩ S. We have:
val M (s 0 ) · P M * ,s0,σ (s 0 s 1 · · · s n s n+1 S ω * ) = val M (s 0 ) · P M * ,s0,σ (s 0 s 1 · · · s n S ω * ) · P * (s n )(s n+1 ) = P M,s0,σ (s 0 s 1 · · · s n S ω ) · P * (s n )(s n+1 ) · val M (s n ) ind. hyp.
= P M,s0,σ (s 0 s 1 · · · s n S ω ) · P (s n )(s n+1 ) · val M (s n+1 ) def. of P * = P M,s0,σ (s 0 s 1 · · · s n s n+1 S ω ) · val M (s n+1 )
This completes the inductive step, and we have proved item 1. Towards item 2, define an MDP M ′ * = (S ′ * , S ′ * , S ′ * , −→ ′ * , P ′ * ) with "intermediate" states like (s, t) in M * , but with transition probabilities as in M; more precisely: , t), t) ∈ (−→ × S) | s ∈ S } P ′ * (s, t) = P (s, t) P ′ * ((s, t), t) = 1
S ′ * = S S ′ * = S ∪ {(s, t) ∈ −→ | s ∈ S } −→ ′ * = {(s, (s, t)) ∈ (S × −→) | s−→t} ∪ (S × S) ∪ {((s
Then we have P M ′ * ,s0,σ (R) = P M,so,σ (R) for all measurable R ⊆ s 0 (S ′ * ) ω .
Let s 0 s 1 · · · s n ∈ s 0 (S ′ * ) * . If s 0 s 1 · · · s n is a partial run in M * , then we have: P M ′ * ,s0,σ (s 0 s 1 · · · s n (S ′ * ) ω ) = P M,s0,σ (s 0 s 1 · · · s n S ω ) Equation (27) ≥ P M,s0,σ (s 0 s 1 · · · s n S ω ) · val M (s n )
= val M (s 0 ) · P M * ,s0,σ (s 0 s 1 · · · s n S ω * ) item 1 = val M (s 0 ) · P M * ,s0,σ (s 0 s 1 · · · s n (S ′ * ) ω ∩ S ω * )
Otherwise (i.e., s 0 s 1 · · · s n is not a partial run in M * ), the same inequality holds trivially. Invoking Lemma 22 below with S := S ′ * and s := s 0 and µ(R) := P M * ,s0,σ (R ∩ S ω * ) and µ ′ (R) := P M ′ * ,s0,σ (R) and x := val M (s 0 ) yields P M ′ * ,s0,σ (R) ≥ val M (s 0 ) · P M * ,s0,σ (R ∩ S ω * ) for all measurable R ⊆ s 0 (S ′ * ) ω .
By Equation (27), the first inequality of item 2 follows. Towards the second inequality of item 2, define φ s0 − def = {ρ ∈ s 0 (S ′ * ) ω | ρ ∈ φ s0 }. If P M ′ * ,s0,σ (s 0 s 1 · · · s n (S ′ * ) ω ∩ φ s0 − ) > 0, then s 0 s 1 · · · s n is a partial run in M * and we have:
val M (s 0 ) · P M * ,s0,σ (s 0 s 1 · · · s n (S ′ * ) ω ∩ S ω * ) = val M (s 0 ) · P M * ,s0,σ (s 0 s 1 · · · s n S ω * ) = P M,s0,σ (s 0 s 1 · · · s n S ω ) · val M (s n ) item 1 ≥ P M,s0,σ (s 0 s 1 · · · s n S ω ) · P M,s0,σ ( φ s0 | s 0 s 1 · · · s n S ω ) φ is tail = P M,s0,σ (s 0 s 1 · · · s n S ω ∩ φ s0 ) = P M ′ * ,s0,σ (s 0 s 1 · · · s n (S ′ * ) ω ∩ φ s0 − ) Equation (27) Otherwise (i.e., P M ′ * ,s0,σ (s 0 s 1 · · · s n (S ′ * ) ω ∩ φ s0 − ) = 0), the same inequality holds trivially. Invoking Lemma 22 with S := S ′ * and s := s 0 and µ(R) := P M ′ * ,s0,σ (R ∩ φ s0 − ) and µ ′ (R) := P M * ,s0,σ (R ∩ S ω * ) and x := 1/val M (s 0 ) yields val M (s 0 ) · P M * ,s0,σ (R ∩ S ω * ) ≥ P M ′ * ,s0,σ (R ∩ φ s0 − ) for all measurable R ⊆ s 0 (S ′ * ) ω .
By Equation (27), the second inequality of item 2 follows. Item 3 follows from item 2, with R = φ s0 . ◀
The following lemma was used in the preceding proof.
▶ Lemma 22. Let S be countable and s ∈ S. Call a set of the form swS ω for w ∈ S * a cylinder. Let µ, µ ′ be measures on sS ω defined in the standard way, i.e., first on cylinders and then extended to all measurable sets R ⊆ sS ω . Suppose there is x ≥ 0 such that x · µ(C) ≤ µ ′ (C) for all cylinders C. Then x · µ(R) ≤ µ ′ (R) holds for all measurable R ⊆ sS ω .
Proof. Let C = {C ⊆ sS ω | C cylinder} denote the class of cylinders. This class generates an algebra C * ⊇ C, which is the closure of C under finite union and complement. The classes C and C * generate the same σ-algebra σ(C). The class C * is a set of countable disjoint unions of cylinders [4, Section 2]. Hence x · µ(R) ≤ µ ′ (R) for all R ∈ C * . Define
Q = {R ∈ σ(C) | x · µ(R) ≤ µ ′ (R)} .
We have C ⊆ C * ⊆ Q ⊆ σ(C). We show that Q is a monotone class, i.e., if R 1 , R 2 , . . . ∈ Q, then R 1 ⊆ R 2 ⊆ · · · implies i R i ∈ Q, and R 1 ⊇ R 2 ⊇ · · · implies i R i ∈ Q. Suppose R 1 , R 2 , . . . ∈ Q and R 1 ⊆ R 2 ⊆ · · · . Then:
x · µ i R i = sup i x · µ(R i ) measures are continuous from below ≤ sup i µ ′ (R i ) definition of Q = µ ′ i R i measures are continuous from below So i R i ∈ Q.
Using the fact that measures are continuous from above, one can similarly show that if R 1 , R 2 , . . . ∈ Q and R 1 ⊇ R 2 ⊇ · · · then i R i ∈ Q. Hence Q is a monotone class. Now the monotone class theorem (see, e.g., [4, Theorem 3.4]) implies that σ(C) ⊆ Q, thus Q = σ(C). Hence x · µ(R) ≤ µ ′ (R) for all R ∈ σ(C). Proof. For any state s 0 ∈ S * ∩ S, let R s0 def = {s 0 s 1 · · · ∈ s 0 S ω * | ∃ i ≥ 1 : s 0 = s i } denote the event of returning to s 0 . Suppose M * is not universally transient. By Lemma 3(3) there exists s 0 ∈ S * ∩ S such that val M * ,Rs 0 (s 0 ) = 1. We show that, in M, for any C > 0 there exists a strategy under which the expected number of returns to s 0 is at least C. By Lemma 3(4) this implies that M is not universally transient.
Let C > 0. Let R be the event, in M * , starting in s 0 , of returning to s 0 at least 2C/val M,φ (s 0 ) times, and denote by X the random variable counting the number of returns to s 0 . Since val M * ,Rs 0 (s 0 ) = 1, we also have val M * ,R (s 0 ) = 1, and so there exists a strategy σ with P M * ,s0,σ (R) ≥ 1 2 . By the first inequality of Lemma 13.2 we have P M,s0,σ (R) ≥ val M,φ (s 0 ) · 1 2 . It follows: E M,s0,σ (X) ≥ P M,s0,σ (R) · 2C/val M,φ (s 0 ) ≥ C ◀
In [17, Lemma 6] a variant, say M + , of the conditioned MDP M * from Definition 12 was proposed. This variant M + differs from M * in that M + has only those states s from M that have an optimal strategy, i.e., a strategy σ with P M,s,σ (φ) = val M (s). Further, for any transition s−→t in M + where s is a controlled state, we have val M (s) = val M (t), i.e., M + does not have value-decreasing transitions emanating from controlled states.
As a consequence, in contrast to M * , in M + there is no need for intermediate states of the form (s, t): Since val M (s) = val M (t), an intermediate state (s, t) would transition to t with probability 1. Therefore, such intermediate states do not appear in M + . Instead, in M + there is a direct transition from s to t like in the original MDP M. As a further consequence, the state s ⊥ does not appear in M + (it would not be reachable).
Any strategy σ in M + can be naturally applied also in M * : whenever σ moves from a controlled state s to a state t (hence s and t have the same value), in M * strategy σ moves instead to the random state (s, t) (from which M * transitions to t with probability 1).
This correspondence is exploited in the proof of the following lemma from the main body: Proof. Suppose M is universally transient. We show that M + is universally transient. Indeed, let s 0 be any state in M + , and let σ be any strategy in M + . Write R for the event of returning to s 0 in M * , and R for the event of returning to s 0 in M + . We have P M+,s0,σ (R) = P M * ,s0,σ (R). Since M * is universally transient by Lemma 16, by Lemma 3(3) this probability is less than 1. Applying Lemma 3(3) again, it follows that M + is universally transient. ◀
▶
Theorem 10. For universally transient MDPs optimal strategies for {0, 1, 2}-Parity, where they exist, can be chosen uniformly MD.
▶
Lemma 17. Let M be an MDP, and let φ be an objective that is tail in M. Let M + be the conditioned version w.r.t. φ in the sense of [17, Lemma 6]. If M is universally transient, then so is M + .
M × S → D(M × S) that meets the following two conditions, for all modes m ∈ M:for all controlled states s ∈ S , the distribution u((m, s)) is over M × {s ′ | s−→s ′ }.for all random states s ∈ S , we have that m ′ ∈M u((m, s))(m ′ , s ′ ) = P (s)(s ′ ).An update function u together with an initial memory m 0 induce a strategy u[m 0 ] : S * S → D(S) as follows. Consider the Markov chain with states set M × S, transition relation (M × S) 2 and probability function u.Any partial run ρ = s 0 · · · s i in M gives rise to a set H(ρ) = {(m 0 , s 0 ) · · · (m i , s i ) | m 0 , . . . , m i ∈ M} ofpartial runs in this Markov chain. Each ρs ∈ s 0 S * S induces a probability distribution µ ρs ∈ D(M), the probability µ ρs (m) is the probability of being in state (m, s) conditioned on having taken some partial run from H(ρs). We define u[m 0 ] such that u[m 0 ](ρs)(s ′ ) def = m,m ′ ∈M µ ρs (m)u((m, s))(m ′ , s ′ ) for all ρs ∈ S * S and s ′ ∈ S.We say that a strategy σ can be implemented with memory M (and initial memory m 0 ) if there exists an update function u such that σ = u[m 0 ]. In this case we may also write σ[m 0 ] to explicitly specify the initial memory mode m 0 . Based on this, we can define several classes of strategies:A strategy σ is memoryless (M) (also called positional) if it can be implemented with a memory of size 1. We may view M-strategies as functions σ : S → D(S). A strategy σ is finite memory (F) if there exists a finite memory M implementing σ. More specifically, a strategy is 1-bit if it can be implemented with a memory of size 2. Such a strategy is then determined by a function u : {0, 1} × S → D({0, 1} × S). Deterministic 1-bit strategies are are both deterministic and 1-bit.
For the first item, let α 1
1be a general strategy α 1 : S * S → D(S) in M. We define β 1 in M ′ with the use of memory M = S * × {⊥} ∪ {i ∈ N | i ≤ 1}) and an update function u; see Appendix A. The definition of u : M × S ′ → D(M × S ′ ) is as follows. For all q, q ′ ∈ S ′ and ρ ∈ S * , for all m = (ρ, ⊥) and m ′ = (ρq, ⊥),
was infinitely branching in M, and if q = ℓ s,i , q ′ = ℓ ′ s,i and i < j; for all m = (ρ, j) and m ′ = (ρ, ⊥) with j ≥ 1, u(m, q)(m, q ′ ) = 1 if s = last(ρ) was infinitely branching in M with s → s i for all i ≥ 1, and if q = ℓ s,j and q ′ = s j ;
▶
Theorem 9. For every universally transient countable MDP, safety objective and ε > 0 there exists a uniformly ϵ-optimal MD strategy. Proof. Let M = (S, S , S , −→, P ) be a universally transient MDP and ε > 0. Assume w.l.o.g. that the target T ⊆ S of the objective φ = Safety(T ) is a (losing) sink and let ι : S → N be an enumeration of the state space S. By Lemma 3(3), for every state s we have Re(s) def = sup σ P M,s,σ (XF(s)) < 1 and thus R(s) def = ∞ i=0 Re(s) i < ∞. This means that, independent of the chosen strategy, Re(s) upper-bounds the chance to return to s, and R(s) bounds the expected number of visits to s.
▶
Lemma 16. Let M = (S, S , S , −→, P ) be an MDP, and let φ be an objective that is tail in M. Let M * = (S * , S * , S * , −→ * , P * ) be the conditioned version of M w.r.t. φ, where s ⊥ is replaced by an infinite chain s 1 ⊥ −→s 2 ⊥ −→ · · · . If M is universally transient, then so is M * .
For every state s the value of the objective to re-visit s is strictly below 1, i.e.▶ Lemma 3. For every countable MDP M = (S, S , S , −→, P ), the following conditions
are equivalent.
1. M is universally transient, i.e., ∀s 0 , ∀σ. P M,s0,σ (Transience) = 1.
2. For every initial state s 0 and state s, the objective of re-visiting s infinitely often has
value zero, i.e., ∀s 0 , s sup σ P M,s0,σ (GF(s)) = 0.
3. ,
Re(s)
def
= sup σ P M,s,σ (XF(s)) < 1.
s
s 1
For all states s 0 , s, under every strategy σ from s 0 the expected number of visits to s is finite.1
2
1
2
1
2
1
2
1
2
1
2
1
2
Figure 3 A partial illustration of the reduction in Lemma 2.
4. For every state s there exists a finite bound B(s) such that for every state s 0 and strategy
σ from s 0 the expected number of visits to s is ≤ B(s).
5.
strategy complexity of parity objectives in general MDPs is known [15]. Here we show that some parity objectives have a lower strategy complexity in universally transient MDPs. It is known [14] that there are acyclic (and hence universally transient) MDPs where ε-optimal strategies for {1, 2}-Parity (and optimal strategies for {1, 2, 3}-Parity, resp.) require 1 bit. We show that, for all simpler parity objectives in the Mostowski hierarchy [19], universally transient MDPs admit uniformly (ε-)optimal MD strategies (unlike general MDPs [15]).
The claim now follows from[15, Theorem 22]. ◀ ▶ Theorem 11. For every universally transient countable MDP M, co-Büchi objective and ε > 0 there exists a uniformly ε-optimal MD strategy.Formally, let M be a universally transient countable MDP with states S, C ol : S → {0, 1} be a coloring, φ = Parity(C ol) and ε > 0.There exists an MD strategy σ ′ s.t. for every state s, P M,s,σ ′
23 Manfred Schäl. Markov decision processes in finance and dynamic options. In Handbook of Markov Decision Processes. Springer, 2002. 24 Olivier Sigaud and Olivier Buffet. Markov Decision Processes in Artificial Intelligence. John Wiley & Sons, 2013. 25 William D. Sudderth. Optimal Markov strategies. Decisions in Economics and Finance, 2020. 26 R.S. Sutton and A.G Barto. Reinforcement Learning: An Introduction. Adaptive Computation and Machine Learning. MIT Press, 2018. 27 Moshe Y. Vardi. Automatic verification of probabilistic concurrent finite-state programs. In Annual Symposium on Foundations of Computer Science. IEEE Computer Society, 1985. doi:10.1109/SFCS.1985.12.
For all random states s in M with infinite branching s pi − → s i for all i ≥ 1, we use a gadget sThe reduction is as follows.
For all controlled state s in M with infinite branching s → s i for all i ≥ 1, we intro-
duce a recurrent ladder in M ′ , consisting the controlled states (ℓ s,i ) i∈N and random
states (ℓ ′
s,i ) i≥1 . The set of transitions includes s → ℓ s,0 and ℓ s,0 → ℓ s,1 , and for all i ≥ 1
two transitions ℓ s,i → ℓ ′
s,i , and ℓ s,i → s i . Moreover, ℓ ′
s,i
1
2
− → ℓ s,i+1 and ℓ ′
s,i
1
2
− → ℓ s,i−1 . Here,
all states of the recurrent ladder are fresh states.
1
To show this, let us decompose the cost function as cost = s cost s where cost s is a local cost function for state s that assigns val(s) − val(s ′ ) to all controlled transitions s−→s ′ starting in s and zero otherwise. Similarly, we let Cost(n) def = s Cost s (n), where Cost s (n) is the random variable denoting the cost incurred on in step n from s. We thus haveCost s (i)where the last equality holds by convergence of monotone series.We now show an upper bound on lim n→∞ E i=0 Cost s (i) for some fixed state s. Costs are only incurred at state s, and each time they are upper-bounded by ε 2 ι(s)+1 ·R(s) . Moreover, the probability of returning from s to s is upper-bounded by Re(s). This means that lim n→∞ Elim
n→∞
E
n−1
i=0
Cost(i) = lim
n→∞
E
s∈S
n−1
i=0
Cost s (i) =
s∈S
lim
n→∞
E
n−1
i=0
n−1
n−1
i=0 Cost s
▶ Lemma 17. Let M be an MDP, and let φ be an objective that is tail in M. Let M + be the conditioned version w.r.t. φ in the sense of[17, Lemma 6]. If M is universally transient, then so is M + .
|
[] |
[
"A multi-protocol framework for the development of collaborative virtual environments",
"A multi-protocol framework for the development of collaborative virtual environments"
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[
"Luciano Argento [email protected] \nUniversity of Calabria P. Bucci\n41C I-87036RendeCSItaly\n",
"Angelo Furfaro [email protected] \nUniversity of Calabria P. Bucci\n41C I-87036RendeCSItaly\n",
"D I M E S \nUniversity of Calabria P. Bucci\n41C I-87036RendeCSItaly\n"
] |
[
"University of Calabria P. Bucci\n41C I-87036RendeCSItaly",
"University of Calabria P. Bucci\n41C I-87036RendeCSItaly",
"University of Calabria P. Bucci\n41C I-87036RendeCSItaly"
] |
[] |
Collaborative virtual environments (CVEs) are used for collaboration and interaction of possibly many participants that may be spread over large distances. Both commercial and freely available CVEs exist today. Currently, CVEs are used already in a variety of different fields: gaming, business, education, social communication, and cooperative development. In this paper, a general framework is proposed for the development of a cooperative environment which is able to exploit a multi protocol network infrastructure. The framework offers support to concerns such as communication security and inter-protocol interoperability and let software engineers to focus on the specific business of the CVE under development. To show the framework effectiveness we consider, as a case of study, the design of a reusable software layer for the development of distributed card games built on top of it. This layer is, in turn, used for the implementation of a specific card game.
|
10.1109/cscwd.2015.7231001
|
[
"https://arxiv.org/pdf/1412.3260v2.pdf"
] | 513,126 |
1412.3260
|
d5b4b86d535c3a175cc78a4ea927ebfa2b14bc74
|
A multi-protocol framework for the development of collaborative virtual environments
Luciano Argento [email protected]
University of Calabria P. Bucci
41C I-87036RendeCSItaly
Angelo Furfaro [email protected]
University of Calabria P. Bucci
41C I-87036RendeCSItaly
D I M E S
University of Calabria P. Bucci
41C I-87036RendeCSItaly
A multi-protocol framework for the development of collaborative virtual environments
Collaborative virtual environments (CVEs) are used for collaboration and interaction of possibly many participants that may be spread over large distances. Both commercial and freely available CVEs exist today. Currently, CVEs are used already in a variety of different fields: gaming, business, education, social communication, and cooperative development. In this paper, a general framework is proposed for the development of a cooperative environment which is able to exploit a multi protocol network infrastructure. The framework offers support to concerns such as communication security and inter-protocol interoperability and let software engineers to focus on the specific business of the CVE under development. To show the framework effectiveness we consider, as a case of study, the design of a reusable software layer for the development of distributed card games built on top of it. This layer is, in turn, used for the implementation of a specific card game.
I. INTRODUCTION
A collaborative virtual environment (CVE) is a space where several people, often spread over different locations, interact with each other. The aim of these people is to share ideas and experiences in a cooperative setting -hence the name [1]. Recently, the term "Web 3.0" has been introduced to refer to the future aspects of the Web; some groups think that the Semantic Web will play the role of the main new technology in this context, while others consider CVEs as the most important advancement of the Web [2].
Both commercial and freely available CVEs exist today. These systems are used in a variety of different fields such as gaming, business, education and cooperative development. A number of companies have started exploiting CVEs, like the prominent Second Life [3], for business purposes. Second Life has also been used in the educational field; on-line lectures have been given and pedagogical usefulness of this novel medium has been investigated by a considerable number of colleges and educational researchers [4]. One of the most famous application area for 3D CVEs is gaming. Even older games like Doom [5] can be considered as an early form of CVE.
This paper proposes a general framework for the development of cooperative virtual environments which exploits a multi protocol network infrastructure. The framework offers support to concerns such as communication security and interprotocol interoperability and let software engineers to focus on the specific business of the CVE under development. The framework was designed to build applications independent from the communication technology, which means that it is possible to seamlessly add new communication protocols, as new features, without affecting the remaining code. For instance, an existing application, featuring only Bluetooth technology, can be integrated with Wi-Fi technology in a transparent way. Moreover, as stated early, users can join the environment by using different technologies. To show the framework effectiveness we consider, as a case of study, the design of reusable software layer for the development of distributed card games. This layer is, in turn, used for the implementation of a specific card game, i.e. the Italian card game named Tressette [6].
A deep study was conducted to guarantee a high degree of reusability, therefore a great number of card games can be developed, such as Poker, Blackjack, Spades, Uno, etc., and the same card game can be easily deployed on different platforms.
The rest of the paper is organized as follows. Section II briefly surveys the related work. Section III presents the framework design. Section IV describes the case study. Section V concludes the paper and outlines some possible future work.
II. RELATED WORK
Collaborative virtual environments, also referred to as collaborative workspaces (CW) or networked virtual environments (NVE or NVE-VE), have been around since the late 80s. The key areas and methodologies regarding the most pioneering CVE systems from 1987 to 2003 are discussed in [7]. There exists a variety of application domains for CVE systems such as gaming, business and education.
A virtual environment enabling physics experiments to be cooperatively accessible via Internet, named iLabs, has been proposed in [8]. The objective of iLabs is to let students and educators work together in a collaborative way by using a three dimensional CVE, in the field of physics education. Virtual reality (VR) technologies have been exploited in [9] to achieve a Collaborative Learning Environment with Virtual Reality (CLEV-R) as a sort of VR campus where students go to learn, socialize and collaborate on-line. This research also investigates the use of mobile systems for e-learning. The use of CVEs for implementing a novel business modeling approach is described in [10] where the authors provide a complete and usable environment for collaborative (re-)design of business processes by extending a number of 3-D tools.
A study on whether virtual worlds such as Second Life and World of Warcraft (a well-known MMORPG developed by Blizzard Entertainment) can offer a basis for trade (B2C and C2C e-commerce) and whether credible studies of ecollaboration behavior and related outcomes can be performed on them has been conducted in [11].
As stated in the previous section, to test the effectiveness of the framework proposed in this paper, a reusable software layer for the development of distributed card games was realized and a specific card game was developed on top of it. Many open source projects regarding the development of a framework to build card games can be found on the web. AJAX Cards [12] is a framework for the development of single-player card games on the web by using HTML, CSS and JavaScript. Future plans involves the inclusion of AJAX support for multiplayer games. The Lua card game framework [13] provides a web page in which it is possible to develop multiplayer card games by uploading images for cards and scripts, written in the Lua programming language [14], to describe the rules of the game. None of the above referenced frameworks takes contemporaneously into account multi platform deployment, heterogeneous communication technologies and security, which are instead the benefits inherited by using our framework for CVE development.
III. FRAMEWORK DESIGN
This section explains in details the framework designed to provide basic tools to build collaborative virtual environment. The basic building block of an environment is the Room entity, which represents the cooperation space on the side of each participant. Such entities interact among them by means of communication channels achieved by exploiting specific abstractions that hide modules which handle the needed communication technologies. A typical scenario, the framework is able to support, is shown in Fig. 1 The design of the framework has been carried out by taking into account the following aspects:
• the evaluation of technologies suitable to achieve a portable software; • the definition of a software architecture whose modules are as decoupled as possible in order to guarantee a high degree of reusability, and which is able to support the deployment of domain-specific collaborative applications in a distributed environment; • the achievement of a multiprotocol network infrastructure able to shield clients from details about communication technologies; • the choice and analysis of a domain-specific application as a case of study in order to assess the framework effectiveness.
For the last point, we realized a reusable software layer, for the development of distributed card games, built on top of the framework. Many card games were analyzed to identify the most significant and generic entities and operations. Subsequently, proper abstractions were defined to build a common basis for the development of concrete card games.
The resulting architecture, shown in Fig. 2, is composed of 3-layers:
• application clients • domain-specific API • cooperative multi-protocol framework. The next subsections explains the study performed for this work, the solutions adopted in order to build the mentioned layers and the way they were decoupled.
A. Communication
One of the main goals of this work was to design a framework which is able to provide a set of features suitable for the development of a multi-protocol network infrastructure.
The framework supports multiple architectural patterns for distributed communication such as peer to peer (P2P) and client/server. In this paper, the devices which host the component in charge of the data elaboration will be referred to, by abuse of terminology, as servers while the others as clients.
Two distinct hierarchies were designed to handle all the aspects regarding the communication between devices. The highest level abstractions of the mentioned hierarchies are represented by the following two interfaces: ServerCommunicator and Communicator.
The first interface provides services thought to create a communication channel while the second one deals with the handling of the connection. Only these two interfaces are accessible from client code, the implementations of the concrete classes cannot be seen. The design pattern Factory [15] has been used in order to create concrete instances. This allows the entities which need to communicate with a remote object to transparently use different communication protocols just by resorting to the suitable concrete factory object.
1) Devices scanning: To join a virtual environment, advertising and discovering operations are needed. Two interfaces, Discoverer and Advertiser, were respectively devised in order to let a player search for an existing environment or create a new one.
To build a multi-protocol network infrastructure, different implementations of these two interfaces can be exploited. For instance, if an application provides Bluetooth and Wi-Fi connection, a user could advertise the cooperation environment, he previously created, by using both Wi-Fi-based and Bluetooth-based implementations of the Advertiser interface, simultaneously. On the other hand, a user that wants to join an environment will use only the needed implementation of Discoverer. During the registration phase the server creates a channel with each client based on the chosen communication technology. These details are hidden by the ServerCommunicator and Communicator interfaces so there is no need to concern with these aspects during the development of the domain-specific cooperation logic.
B. The environment
This section describes the modules designed to build the environment in which the cooperation takes place. Specifically, the Room entity has been devised in order to achieve such a goal. It is in charge of handling basic operations such as participants' registration, opening and closing of the cooperative sessions and provides services which let a participant to join the environment, send and receive data. To account of the participant role, the following specific interfaces were defined: ServerRoom and ClientRoom. Room data are hosted on the side of the participant playing the server role, so it is necessary to make them remotely accessible to the other participants. The Remote Proxy design pattern offers a solution to this problem by hiding communication details and making data to appear as they were locally available. Actually, the goal of the Remote Proxy pattern is to give a local representation for an object that lives in a different address space [15].
The proxy entity implements the ClientRoom interface on the client device while on the server a skeleton entity is created. The hierarchies previously introduced are exploited to obtain a modular separation that leads to a system which can be naturally evolvable, i.e. it is possible to easily integrate new communication technologies without requiring modification of the existing code.
C. The client applications
One of the framework requirements is the capability of binding any client application to the cooperative environment built on top of it, regardless of the application nature. This decoupling has been obtained by resorting to the Observer pattern [15]. The object representing the cooperation room, maintains the reference to a list of observers whose aim is to receive notification data about the status of the environment and update the respective client application. For instance let us suppose there are four participants A, B, C and D. With reference to Fig. 1, A uses an Android app on a mobile phone connected, via bluetooth, to B, which also uses an Android app and is in turn connected, via Wi-Fi to a LAN, C is a desktop client connected to same LAN and D a web application which runs over the Internet and it is reachable from the LAN through a router. Each of these applications has to register itself as observer of the room and be able to properly handle data received from the environment.
D. Session handling
While a cooperation session is active, it may happen that a user disconnects, e.g. because a connection problem or because he needs to temporarily leave. During the registration phase, the host, into which the room resides, generates a token for each participant. This token is used to handle the session accordingly to an application specific policy.
For example in a game environment, these disconnection events could last for few seconds, thereby if they are handled properly it is possible to avoid ending the game in unexpected way. When a player leaves temporarily a game, he can rejoin the environment by using the token previously obtained. The main benefit of the use of the tokens is that only players which originally joined the environment can rejoin after a disconnection. Obviously a timeout can be set for the session by the server, so if a user fails to rejoin before the time runs out, the session, will be terminated. Tokens can also carry identification information, e.g. digitally signed messages, allowing users' authentication.
IV. CASE STUDY
As a case of study, the design of a reusable software layer for the development of distributed card games was built on top of the framework. In order to build such a layer it was necessary to study in depth the main characteristics of existing card games and figure out what elements can be considered as a basis on which to build a new card game. The following aspects were analyzed, during the study:
• the possible set of values associated with the cards; • the set of seeds associated with the cards; • order relationship among cards; • number of players and possible partnerships; • actions available for a player during his turn; • actions available for a player during the turn of another one; • the interaction among players; • score computation; • card picking; • card distribution; • how the next turn is chosen; • the conditions by which a player or a group of players is declared as the winner. The fundamental entities and relations of a generic card game were identified, as a result of the study, and are shown in the class diagram depicted in Fig. 4.
A. Architecture for a generic card game
A certain number of abstraction were designed in order to capture the essential elements which characterize a generic The operations needed to handle the hand of a player, the set of cards distributed to every player, are generic enough to suit a generic card game well, so the concrete class Hand has been purposely devised.
There are two more entities that were identified during the mentioned study: the player and the game coordinator. The first serves the purpose of letting a user interact with the game performing actions like checking the hand, playing one or more cards and so on. The second has different tasks to accomplish, such as elaborating and communicating information about the current status of the game and coordinating the actions of all players. The player and the game coordinator will not necessarily be located on the same machine. The framework has been thought to build a distributed system in which each device can either adopt the role of a server or the role of a client, so the entity player will be assigned to every user while the entity game coordinator only to the user whose device act as a server. It is also possible to implement a centralized coordination approach where the coordinator is not tied to any player and reside on a different machine.
The game coordinator knows the players who participate in the game so a communication between objects on different machines will take place. The Player interface and the abstract class called AbstractPlayer are exploited to hide the true nature of a player, so the game coordinator will never know whether it is communicating with a remote object, whether it is interacting with the local user or whether it is interacting with and intelligent artifact.
The Proxy pattern [15] was exploited so that a transparent communication channel can be created between the player Fig. 5). The ProxyPlayer class implements the Player interface and forwards all the requests, received by the game coordinator, to its associated client. On the client machine the SkeletonPlayer module, another class that implements the Player interface and knows the real player, receives the forwarded requests.
The communication between a server and a client is bidirectional, so the player needs a way to communicate to the game coordinator. The game coordinator carries out a number of operations; the whole set of operations can be split up in subsets, each of them deals with a specific aspect of the system. In the light of the previous consideration, three interfaces were designed:
• GameCoordinatorElaboration; • GameCoordinatorCommunication; • GameCoordinatorRegistration.
The first interface defines operations regarding the server, while the second and the third ones were thought for the client. The Remote Proxy pattern was used once again to let a player communicate to the game coordinator. This time the proxy and the correspondent skeleton implement the GameCoordinator-Communication and GameCoordinatorRegistration interfaces. To gather the operations of the two interfaces mentioned previously and to hide the implementation details of the proxy and the skeleton, the following two interfaces were created:
• ProxyGameCoordinator; • SkeletonGameCoordinator.
It was also created the GameCoordinator abstract class which implements the GameCoordinatorElaboration, GameCoordi-natorCommunication and GameCoordinatorRegistration interfaces, to provide a default implementation for a portion of the whole set of operations and a module to handle properly the entity game coordinator on the server. The player and the game coordinator, just like the room entity, use the services provided by the ServerCommunicator and Communicator interfaces (see Fig. 3).
B. Security aspects
During the design phase the existence of fake client application was taken into account. Therefore the system was designed so that the server checks the validity of the data received by each client. Whenever a fake client tries to play an unexpected card the system would be able to recognize this anomaly. If an anomaly is detected a notification will be sent to each client to communicate the presence of a fake client and subsequently the game would be ended.
C. Tressette: a specific card game
Tressette, an Italian card game, was built on top of the framework. The application was thought for Android and personal computer devices; the Java programming language was used in order to guarantee the portability of the software and the Bluetooth and the Socket technologies were chosen as communication technologies.
Tressette is played with a standard Italian 40-card deck and the cards are ranked as follows from highest to lowest: 3-2-Ace-King-Knight-Knave and then all the remaining cards in numerical order from 7 down to 4. The game may be played with four players playing in two partnerships, or in headsup play. In either case, ten cards are dealt to each player. In the developed application the players play only in two partnerships. The object of the game is to score as many points as possible until a score of 21 is achieved. Players must follow suit unless that suit does not remain in their hand, and players must show the card they pick up off the card pile to their opponent. More information about the Tressette card game can be found in [6].
The entities that have been introduced for implementing the Tressette game are reported in the class diagram depicted in Fig. 6. There was no need to override any methods of the Hand class, its services resulted generic enough for the application. TressetteCard is a concrete class which extends Card, it introduces an operation to mark a card as playable or not: in this game a card can be played only when specific conditions hold, see [6] for more details. The Enum Shape represents the concept of seed, so it implements the Seed interface, while the concrete class called TressetteValue realises the Value interface. TressetteTeam extends Team, it keeps track of the game score. Analogously TressetteDeck extends Deck with gamespecific deck handling aspects. In the method elaboration of the TressetteGameCoordinator class lies the core of the logic of the game, here the turns and scorers are updated and the winners are proclaimed. TressettePlayer provides specific operations for Tressette, like the TressetteCard class does. It was necessary to define two more interfaces to handle specific aspects of the game: TressetteCode and TressetteMessage. The first interface serves the purpose of exchanging messages between the room and a client, while the second one defines all the statements available for a user.
V. CONCLUSIONS AND FUTURE WORK
We presented a modular framework for the development of domain specific CVEs. One of the main feature of our proposal is the support for a transparent and seamless multi protocol interaction among participants. The framework is not tied to any particular architectural distributed communication pattern and can be exploited to support any of them. The effectiveness of the framework has been experimented by implementing on top of it a reusable software layer for the development of distributed card games. Future research directions include others practical experimentation of the framework in more heterogeneous distributed settings, e.g. involving web-services [16] and cloud based applications [17]. The proposed framework can be used for building collaborative environments in many fields. For example, existing e-learning platforms based on CVE (e.g. [18]) could be extended in order to gain the multiprotocol features we have introduced in the framework.
Fig. 1. A multi-protocol cooperative scenario
Fig. 2 .
2Framework architecture devices interact among them by using different communication media.
Fig. 3 .
3Framework entities
Fig. 4 .
4Card game entities card game. The abstract class Card can be used to model cards of different games, it implements the Comparable interface so that a comparison is possible among the cards. Seed and Value interfaces are abstractions which describe the corresponding two features of the card concept. The entities deck and team are respectively represented by the abstract classes Deck and Team. These classes provide some services which are common to every card game. Other operations, i.e. the way the cards are distributed, depend on the nature of the game: in Poker hands are of five cards, in Tressette with four players, see section IV-C, ten cards are given, at game begin, to each player.
Fig. 5 .
5Player entities and the game coordinator (see
Fig. 6 .
6Tressette game entities
where different types ofClient Applications
Card Game API
Multiprotocol CVE Framework
Bluetooth
Internet
Infrared
Room
Proxy
Skeleton
Player
Coordinator
Team
Card
Poker
Blackjack
Spades Uno Tressette
Mobile App
Web App
Desktop App
TV set App
...
...
...
...
...
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E Gamma, R Helm, R Johnson, J Vlissides, Design Patterns: Elements of Reusable Object-oriented Software. Boston, MA, USAAddison-Wesley Longman Publishing Co., IncE. Gamma, R. Helm, R. Johnson, and J. Vlissides, Design Patterns: Elements of Reusable Object-oriented Software. Boston, MA, USA: Addison-Wesley Longman Publishing Co., Inc., 1995.
An approach to web service compatibility in cooperative processes. V De Antonellis, M Melchiori, P Plebani, Applications and the Internet Workshops. Proceedings. 2003 Symposium onV. De Antonellis, M. Melchiori, and P. Plebani, "An approach to web service compatibility in cooperative processes," in Applications and the Internet Workshops, 2003. Proceedings. 2003 Symposium on, Jan 2003, pp. 95-100.
Cooperative cloud computing in research and academic environment using virtual cloud. S Malik, F Huet, D Caromel, 2012 International Conference on. in Emerging Technologies (ICETS. Malik, F. Huet, and D. Caromel, "Cooperative cloud computing in research and academic environment using virtual cloud," in Emerging Technologies (ICET), 2012 International Conference on, Oct 2012, pp. 1-7.
Architectures supporting e-learning through collaborative virtual environments: the case of invite. C Bouras, G Hornig, V Triantafillou, T Tsiatsos, Proceedings. IEEE International Conference on. IEEE International Conference onAdvanced Learning TechnologiesC. Bouras, G. Hornig, V. Triantafillou, and T. Tsiatsos, "Architectures supporting e-learning through collaborative virtual environments: the case of invite," in Advanced Learning Technologies, 2001. Proceedings. IEEE International Conference on, 2001, pp. 13-16.
|
[] |
[
"Nucleosynthesis Calculations from Core-Collapse Supernovae",
"Nucleosynthesis Calculations from Core-Collapse Supernovae"
] |
[
"Christopher L Fryer [email protected] ",
"Patrick Young ",
"Michael Bennett ",
"Steven Diehl ",
"Falk Herwig ",
"Raphael Hirschi ",
"Aimee Hungerford ",
"Marco Pignatari ",
"Georgios Magkotsios ",
"Gabriel Rockefeller ",
"Francis X Timmes ",
"\nThe NuGrid Collaboration b Computational Methods (CCS\n\n",
"\nLos Alamos National Laboratory\n87544Los AlamosNMUSA\n",
"\nSchool of Earth and Space Exploration\nArizona State University\n85287TempeAZUSA\n",
"\nAstrophysics Group\nKeele University\nST5 5BGUK\n",
"\nTheoretical Astrophysics Group (T-6)\nLos Alamos National Laboratory\n87544Los AlamosNMUSA\n",
"\nDept. of Physics & Astronomy\nV8W 3P6VictoriaBCCanada\n",
"\nJoint Institute for Nuclear Astrophysics\nUniversity of Notre Dame\n46556INUSA\n",
"\nMackinac Island\nMichiganUSA\n"
] |
[
"The NuGrid Collaboration b Computational Methods (CCS\n",
"Los Alamos National Laboratory\n87544Los AlamosNMUSA",
"School of Earth and Space Exploration\nArizona State University\n85287TempeAZUSA",
"Astrophysics Group\nKeele University\nST5 5BGUK",
"Theoretical Astrophysics Group (T-6)\nLos Alamos National Laboratory\n87544Los AlamosNMUSA",
"Dept. of Physics & Astronomy\nV8W 3P6VictoriaBCCanada",
"Joint Institute for Nuclear Astrophysics\nUniversity of Notre Dame\n46556INUSA",
"Mackinac Island\nMichiganUSA"
] |
[] |
We review some of the uncertainties in calculating nucleosynthetic yields, focusing on the explosion mechanism. Current yield calculations tend to either use a piston, energy injection, or enhancement of neutrino opacities to drive an explosion. We show that the energy injection, or more accurately, an entropy injection mechanism is best-suited to mimic our current understanding of the convection-enhanced supernova engine. The enhanced neutrino-opacity technique is in qualitative disagreement with simulations of core-collapse supernovae and will likely produce errors in the yields. But piston-driven explosions are the most discrepant. Piston-driven explosion severely underestimate the amount of fallback, leading to order-of-magnitude errors in the yields of heavy elements. To obtain yields accurate to the factor of a few level, we must use entropy or energy injection and this has become the NuGrid collaboration approach.10th Symposium on Nuclei in the Cosmos
| null |
[
"https://arxiv.org/pdf/0811.4648v1.pdf"
] | 14,362,025 |
0811.4648
|
a61b3d36ffe3c11a402b3c18a079892d3cb3ef88
|
Nucleosynthesis Calculations from Core-Collapse Supernovae
28 Nov 2008 July 27 -August 1 2008
Christopher L Fryer [email protected]
Patrick Young
Michael Bennett
Steven Diehl
Falk Herwig
Raphael Hirschi
Aimee Hungerford
Marco Pignatari
Georgios Magkotsios
Gabriel Rockefeller
Francis X Timmes
The NuGrid Collaboration b Computational Methods (CCS
Los Alamos National Laboratory
87544Los AlamosNMUSA
School of Earth and Space Exploration
Arizona State University
85287TempeAZUSA
Astrophysics Group
Keele University
ST5 5BGUK
Theoretical Astrophysics Group (T-6)
Los Alamos National Laboratory
87544Los AlamosNMUSA
Dept. of Physics & Astronomy
V8W 3P6VictoriaBCCanada
Joint Institute for Nuclear Astrophysics
University of Notre Dame
46556INUSA
Mackinac Island
MichiganUSA
Nucleosynthesis Calculations from Core-Collapse Supernovae
28 Nov 2008 July 27 -August 1 2008* Speaker.
We review some of the uncertainties in calculating nucleosynthetic yields, focusing on the explosion mechanism. Current yield calculations tend to either use a piston, energy injection, or enhancement of neutrino opacities to drive an explosion. We show that the energy injection, or more accurately, an entropy injection mechanism is best-suited to mimic our current understanding of the convection-enhanced supernova engine. The enhanced neutrino-opacity technique is in qualitative disagreement with simulations of core-collapse supernovae and will likely produce errors in the yields. But piston-driven explosions are the most discrepant. Piston-driven explosion severely underestimate the amount of fallback, leading to order-of-magnitude errors in the yields of heavy elements. To obtain yields accurate to the factor of a few level, we must use entropy or energy injection and this has become the NuGrid collaboration approach.10th Symposium on Nuclei in the Cosmos
Downflow
Shock from Infalling Stellar Material PNS Upflow Figure 1: A slice of the x-y plane of a 3-dimensional supernova explosion calculation modeling the collapse of a 23 M ⊙ star [6]. The proto-neutron star (PNS) and outer edge of the convective region defined at the position where the infalling stellar material shocks against the convection are labeled. Note that this outer edge moves outward with time. Energy is injected into this convective region rather uniformly, striving to produce a constant entropy profile as the region expands.
Nucleosynthesis and Understanding Supernova Explosions
The first step in producing a yield for core-collapse supernovae is to introduce a realistic explosion. Although scientists are still working hard to determine the exact physics behind corecollapse supernova explosions, there is growing support for the convection-driven mechanism [1,2,3,4]. Even so, a lot of work remains (both in the progenitor evolution and the explosion mechanism itself) if we want to accurately predict the explosion energy for a given stellar mass. For the foreseeable future, we will have to artificially induce explosions and explore a range of explosion energies (producing error bars) for nuclear yields. However, a qualitative understanding of the explosion mechanism can help us better induce these explosions so that our range of answers will actually bracket the true answer. As we shall see, some mechanisms used to induce explosions will not produce results consistent with the convection-enhanced neutrino driven mechanism.
Our current understanding of the explosion mechanism behind core-collapse supernovae involves a series of phases (for a review, see [5]). When the mass of a core becomes so large that electron capture and iron dissociation can occur, the core collapses. The core collapses to nuclear densities and bounces, sending a shock through the star. Most of the energy in the shock is thermal and, when neutrinos can escape the shock, they sap the bounce shock's energy and itself. After the bounce shock stalls, the region between the edge of the proto-neutron star and the shock front of the infalling material is unstable to a number of convective instabilities. It is in this convective region that neutrino energy leaking out of the core is converted into kinetic energy that eventually pushes out the infalling star and drives an explosion. Figure 1 shows annotated plots of a convection-driven explosion at 2 different times. As the energy in the convective region grows (and the infall rate decreases), the outer edge of the convective region moves out. Ultimately, the convective region has enough energy to drive an explosion (although in this case, the explosion is so weak that most of the star will fall back onto the proto-neutron star, forming a black hole).
What can we learn from simulations of this convection-enhanced explosion mechanism? First, energy is deposited in a region covering a few tenths of a solar mass. Convection strives to flatten the entropy gradient, so energy is deposited fairly uniformly across the convective region. As long as the shock (outer edge of the convective region) is moving out slowly (slow enough that convection can redistribute the energy), energy is deposited throughout the convective region. In mass coordinates, the region does not change dramatically with time. Finally, no energy is deposited beyond the convective region.
With this understanding, let's compare the different mechanisms currently used to drive explosions for nucleosynthesis: piston-driven explosions, energy-driven explosions, and enhanced neutrino-opacity driven explosions. Piston-driven explosions have been used extensively in the past and much of the comprehensive yields in the literature are based on these explosions. Pistondriven explosions work by placing a hard surface at the inner boundary (generally assumed to be at the edge of the iron core, but it would be more realistic to use the outer edge of the convective region). This hard surface is then pushed outward, accelerating the star and driving an explosion. Such an approximation keeps accelerating the inner material in the ejecta, not allowing it to slow down and ultimately fall back on the star, severely underestimating the fallback and overestimating the amount of heavy elements (such as 56 Ni) for a given explosion energy. This has been discussed at some detail [7] and it is now generally accepted in the explosion community that piston-driven yields are not accurate.
Two alternate options are being used in the literature. A straight energy deposition in the inner few tenths of a solar mass. This method is designed to incorporate the energy increase in the convective region. The energy is limited to a few tenths of a solar mass (the rough mass size of the convective region throughout most calculations). And the energy is injected uniformly (in specific internal energy) across this region (as we would expect from convection). More realistic might be to uniformly increase the entropy throughout this region.
The second method is to artificially increase the neutrino opacity in this region. The argument for this method is that it includes neutrino changes to the electron fraction (albeit at an exaggerated level). The disadvantages are many. First, it injects energy even beyond the convective region. Although it is true that neutrinos in a realistic engine will do this, the opacity is lower, so we are over-estimating this energy injection. Second, the energy deposition is highly peaked toward the dense material and not distributed across the convective engine as we would expect in a real convection-enhanced supernova. Although these artifacts are probably small when compared to the piston/energy deposition differences, they all point a direction opposite from what we would expect from the convection mechanism. This method will be less like the convective engine than a simple direct energy deposition.
Comparing Nuclear Yields
We have now discussed in detail the differences between the methods used to simulate a explosions for supernova nucleosynthesis. Table 1 shows the yields for models in the 20-25 M ⊙ range by 3 different groups [6,8,9]. The stars are all evolved with an initial metallicity at solar, but pre- [7], the models all use piston explosions. This is is evident from the small remnant masses for a given explosion (this can not be reproduced in a real explosion calculation). Note that for a given explosion energy and remnant mass, we get considerable scatter in the yield (more than an order of magnitude). Most of the difference is caused by those results using piston explosions and those using the more realistic models (which include fallback).
scriptions for winds vary somewhat and each group uses its own method to drive explosions. The first difference between the models can be seen in the remnant masses. Note that the WW models both predict remnant masses below 1.5 M ⊙ for explosion energies of roughly 2 × 10 51 erg. The CL remnants are also small. But at the same energy, the 23e-series produces a 2.6M ⊙ remnant.
The difference between these remnant masses is entirely an artifact of our method of artificially induced explosions. Different methods produce very different amounts of fallback. To better understand the fallback, let's briefly review its history. The idea of fallback was first brought up by Colgate [10] to overcome nucleosynthesis issues arising from the supernova ejection of neutron rich material produced in stellar cores [11,12]. Colgate argued that the inner layers of the ejected material would deposit its energy to the stellar material above it, ultimately reducing its energy below that needed to escape the neutron star, and it would fall back onto the neutron star. In such a scenario, one would expect the inner material to fall back quickly (within the first few to ten seconds). It was argued that this material (the neutron rich material from the initial explosion) would accrete onto the neutron star, alleviating any nucleosynthesis issues.
Piston models for explosions misled many scientists on the issue of fallback and the supernova field in general. By artificially preventing fallback, piston modelers became concerned with the ejecta of neutron rich material (recall, this is why Colgate first thought about fallback in the first place). Supernova modelers have worked extensively to try to reset the electron fraction and nucleosynthesis modelers put in knobs to reset the electron fraction and move out the mass cut. The Colgate idea of fallback was all but forgotten. With more modern, energy-injected explosion models, fallback occurs (renewing Colgate's original idea) and removes issues with neutron rich ejecta. This makes it easier for explosion models to match compact remnant mass measurements [14] and may even explain the r-process [15]. Table 1 also shows the yields for some key elements from these models. There is a lot of scatter in these models, so it is difficult to pick out any specific trend, but note that some elements (e.g. 45 Sc) are overproduced by piston models by more than a factor of 10. Also, the ratio of 44 Ti to 56 Ni can be an order of magnitude higher in some energy-driven explosions (making it easier to explain the supernova that produced Cassiopeia A). Not until we model a full suite of models will we truly understand the extent of the errors introduced by piston-driven models.
For our in-progress NuGrid calculations, we use a constant entropy injection process. This is the closest match to the convection-enhanced explosion mechanism. When the shock moves beyond 1000 km, we stop the energy injection (which due to the entropy increase process starts to decrease as the density lowers anyway). This still leaves 2 parameters: total energy injection and rate at which the energy is injected. The rate has been studied at some level [7] and it can lead to order of magnitude differences in the yield. Fortunately, for a given explosion energy, we can constrain the delay time [13], so we believe we can fix this parameter somewhat, limiting its errors.
Inside the supernova: A powerful convective engine. M Herant, W Benz, W R Hix, C L Fryer, S A Colgate, ApJ. 435339M. Herant, W. Benz, W.R. Hix, C.L. Fryer, S. A. Colgate 1994, Inside the supernova: A powerful convective engine, ApJ, 435, 339
Modeling Core-Collapse Supernovae in Three Dimensions. C L Fryer, M S Warren, ApJ. 57465C.L. Fryer, Warren, M. S. 2002, Modeling Core-Collapse Supernovae in Three Dimensions, ApJ, 574, L65
Two-dimensional hydrodynamic core-collapse supernova simulations with spectral neutrino transport. I. Numerical method and results for a 15 Mo star. R Buras, M Rampp, H Th, K Janka, Kifonidis, A&A. 4471049R. Buras, M. Rampp, H.-Th. Janka,K. Kifonidis, 2006, Two-dimensional hydrodynamic core-collapse supernova simulations with spectral neutrino transport. I. Numerical method and results for a 15 Mo star, A&A, 447, 1049
A New Mechanism for Core-Collapse Supernova Explosions. A Burrows, E Livne, L Dessart, C Ott, J Murphy, ApJ. 640878A. Burrows, E. Livne, L. Dessart, C. Ott, J. Murphy 2006, A New Mechanism for Core-Collapse Supernova Explosions, ApJ, 640, 878
. C L Fryer, Stellar Collapse, IJMPD. 121795C.L. Fryer 2003, Stellar Collapse, IJMPD, 12, 1795
Late-Time Convection in the Collapse of a 23M ⊙ Star. C L Fryer, P A Young, ApJ. 6591438C.L. Fryer, P.A. Young 2007, Late-Time Convection in the Collapse of a 23M ⊙ Star, ApJ, 659, 1438
Uncertainties in Supernova Yields. I. One-Dimensional Explosions. P A Young, C L Fryer, ApJ. 6641033P.A. Young, C.L. Fryer 2007, Uncertainties in Supernova Yields. I. One-Dimensional Explosions, ApJ, 664, 1033
The Evolution and Explosion of Massive Stars. II. Explosive Hydrodynamics and Nucleosynthesis. S E Woosley, T A Weaver, ApJS. 101181S.E. Woosley, T.A. Weaver 1995, The Evolution and Explosion of Massive Stars. II. Explosive Hydrodynamics and Nucleosynthesis, ApJS, 101, 181
A Chieffi, M Limongi, Explosive Yields of Massive Stars from Z=0 to Z=Z ⊙. 608405A. Chieffi, M. Limongi 2004, Explosive Yields of Massive Stars from Z=0 to Z=Z ⊙ , ApJ, 608, 405
Neutron Star Formation, Thermonuclear Supernovae, and Heavy-Element Reimplosion. S A Colgate, ApJ. 163221S.A. Colgate 1971, Neutron Star Formation, Thermonuclear Supernovae, and Heavy-Element Reimplosion, ApJ, 163, 221
W D Arnett, Supernova Light Curves and Presupernova Models ApJ. 16311W.D. Arnett 1971, Supernova Light Curves and Presupernova Models ApJ, 163, 11
. P A Young, C L Fryer, A Hungerford, D Arnett, G Rockefeller, F X Timmes, B Voit, C Meakin, K A Eriksen, Constraints on the Progenitor of Cassiopeia A. 640891ApJP.A. Young, C.L. Fryer, A. Hungerford, D. Arnett, G. Rockefeller, F.X. Timmes, B. Voit, C. Meakin, K.A. Eriksen 2006, Constraints on the Progenitor of Cassiopeia A, ApJ, 640, 891
C L Fryer, Fallback in Stellar Collapse. 50492C.L. Fryer 2006, Fallback in Stellar Collapse, New Astronomy, 50, 492
C L Fryer, V Kalogera, Theoretical Black Hole MAss Distributions. 554548C.L. Fryer,V. Kalogera 2001, Theoretical Black Hole MAss Distributions, ApJ, 554, 548
C L Fryer, F Herwig, A L Hungerford, F X Timmes, Supernova Fallback: A Possible Site for the r-Process. 646131C.L. Fryer, F. Herwig, A.L. Hungerford, F.X. Timmes, 2006, Supernova Fallback: A Possible Site for the r-Process, ApJ, 646, L131
|
[] |
[
"OPTIMALLY CONVERGENT HDG METHOD FOR THIRD-ORDER KORTEWEG-DE VRIES TYPE EQUATIONS",
"OPTIMALLY CONVERGENT HDG METHOD FOR THIRD-ORDER KORTEWEG-DE VRIES TYPE EQUATIONS"
] |
[
"B O Dong "
] |
[] |
[] |
We develop and analyze a new hybridizable discontinuous Galerkin (HDG) method for solving third-order Korteweg-de Vries type equations. The approximate solutions are defined by a discrete version of a characterization of the exact solution in terms of the solutions to local problems on each element which are patched together through transmission conditions on element interfaces. We prove that the semi-discrete scheme is stable with proper choices of stabilization function in the numerical traces. For the linearized equation, we carry out error analysis and show that the approximations to the exact solution and its derivatives have optimal convergence rates. In numerical experiments, we use an implicit scheme for time discretization and the Newton-Raphson method for solving systems of nonlinear equations, and observe optimal convergence rates for both the linear and the nonlinear third-order equations. arXiv:1610.06968v3 [math.NA] 24 Apr 2017 recurrence of initial states. Phys. Rev. Lett., 15:240-243, 1965.
|
10.1007/s10915-017-0437-4
|
[
"https://arxiv.org/pdf/1610.06968v3.pdf"
] | 35,169,304 |
1610.06968
|
a01b4b59d0da2313834194ee2dac5051e50d7738
|
OPTIMALLY CONVERGENT HDG METHOD FOR THIRD-ORDER KORTEWEG-DE VRIES TYPE EQUATIONS
B O Dong
OPTIMALLY CONVERGENT HDG METHOD FOR THIRD-ORDER KORTEWEG-DE VRIES TYPE EQUATIONS
We develop and analyze a new hybridizable discontinuous Galerkin (HDG) method for solving third-order Korteweg-de Vries type equations. The approximate solutions are defined by a discrete version of a characterization of the exact solution in terms of the solutions to local problems on each element which are patched together through transmission conditions on element interfaces. We prove that the semi-discrete scheme is stable with proper choices of stabilization function in the numerical traces. For the linearized equation, we carry out error analysis and show that the approximations to the exact solution and its derivatives have optimal convergence rates. In numerical experiments, we use an implicit scheme for time discretization and the Newton-Raphson method for solving systems of nonlinear equations, and observe optimal convergence rates for both the linear and the nonlinear third-order equations. arXiv:1610.06968v3 [math.NA] 24 Apr 2017 recurrence of initial states. Phys. Rev. Lett., 15:240-243, 1965.
Introduction
In this paper, we develop and analyze a new hybridizable discontinuous Galerkin (HDG) method for the following initial-boundary value problem of the Korteweg-de Vries (KdV) type equation on a finite domain (1.1)
Here f ∈ L 2 (Ω) and F (u) = βu m , where β is a constant and m ≥ 0 an integer. The well-posedness of the problem (1.1) and properties of the solution have been theoretically and numerically studied; see [4,18,3,5,17,30] and references therein. KdV type equations play an important role in applications, such as fluid mechanics [26,7,25], nonlinear optics [1,19], acoustics [28,33], plasma physics [6,37,32,29], and Bose-Einstein condensates [31,21] among other fields. They also have an enormous impact on the development of nonlinear mathematical science and theoretical physics. Many modern areas were opened up as a consequence of the basic research on KdV equations. Due to their importance in applications and theoretical studies, there has been a lot of interest in developing accurate and efficient numerical methods for KdV equations. In particular, an ongoing effort on developing discontinuous Galerkin (DG) methods for KdV type equations has been made in the last decade. The first DG method, the local discontinuous Galerkin c XXXX American Mathematical Society [36] and further studied for the linear case in [23,34,35,20]. In [10], a DG method for the KdV equation was devised by using repeated integration by parts. Recently, several conservative DG methods [2,9,22] were developed for KdV type equations to preserve quantities such as the mass and the L 2 -norm of the solutions. When solving KdV equations, one can use these DG methods for spatial discretization together with explicit schemes for time-marching if the coefficient before the thirdorder derivative is very small. However, when such coefficient is of order one, for example, implicit time-marching methods might be the methods of choice.
Traditional DG methods, despite their prominent features such as hp-adaptivity and local conservativity, were criticized for having larger number of degrees of freedom than continuous finite element methods when solving steady-state problems or problems that require implicit-in-time solvers. Here, we develop an HDG method which is very suitable for solving KdV equations when implicit time-marching is used. HDG methods [13,11,15,14] were first introduced for diffusion problems and they provide optimal approximations to both the potential and the flux. Due to the feature that the global coupled degrees of freedom only live on element interfaces, they are significantly advantageous for solving steady-state problems or time-dependent problems that require implicit time-marching. In [8], we introduced the first family of HDG methods for stationary third-order linear equations, which allow the approximations to the exact solution u and its derivatives u x and u xx to have different polynomial degrees. We proved superconvergence properties of these methods on projection of errors and numerical traces, and numerical results indicate that the HDG method using the same polynomial degree k for all three variables is quite robust with respect to the choice of the stabilization function and provides a converging postprocessed solution with order 2k + 1 with the least amount of degrees of freedom. This suggests that the HDG method using the same polynomial degrees for all variables is the method of choice for solving onedimensional third-order problems. Therefore, in this paper we extend this HDG method to time-dependent third-order KdV type equations.
To construct the HDG method for KdV equations, we follow the approach used in [8] for stationary third-order equations. That is, given any mesh of the domain, we show that the exact solution can be obtained by solving the equation on each element with provided boundary data that are determined by transmission conditions. Then we define HDG methods by a discrete version of this characterization, which ensures that the only globally-coupled degrees of freedom are those associated to the numerical traces on element interfaces. In [8], it was shown that HDG methods derived by providing boundary data to local problems in different ways are indeed equivalent to each other when the stabilization function is finite and nonzero. So here we just need to consider the one that takes the numerical trace of u at both ends of the interval and the numerical trace of u xx at the right end as boundary data for the local problems. Our method is different from the HDG method in [27], which was designed from implementation point of view. That HDG method involves two sets of numerical traces for u x , and there is no error analysis for the method.
Our way of devising HDG methods from the characterization of the exact solution allows us to carry out stability and error analysis. We first apply an energy argument to find conditions on the stabilization function in the numerical traces, under which the HDG method has a unique solution for KdV type equations. Then by deriving four energy identities and combining them together, we prove that the method has optimal approximations to u as well as its derivatives u x and u xx for linear equations; this technique is similar to that in [35]. In implementation, implicit time-marching schemes such as BDF or DIRK methods can be used, and at each time step a stationary third-order equation is solved by the HDG method together with the Newton-Raphson method (see Appendix A). Due to the one-dimensional setting of the KdV equations, the global matrix of the HDG method that needs to be numerically inverted at each time step is independent of the polynomial degree of the approximations, its size is only 2N + 1, where N is the number of intervals of the mesh, and its condition number is of the order of h −2 , where h denotes the size of the intervals of the mesh.
The paper is organized as follows. In Section 2, we define the HDG method for third-order KdV type equations and state and discuss our main results. The details of all the proofs are given in Section 3. We show numerical results in Section 4 and some concluding remarks in Section 5. The details on implementation of the method are in Appendix A.
Main Results
In this section, we state and discuss our main results. We begin by describing the characterizations of the exact solution that the HDG method is a discrete version of. We then introduce our HDG method for KdV type equations, and state our stability result and optimal a priori error estimate.
Characterizations of the exact solution.
To display the characterizations of the exact solution we are going to work with, let us first rewrite our third-order model equation as the following first-order system:
q − u x = 0, p − q x = 0, u t + p x + F (u) x = f for x ∈ Ω, t ∈ (0, T ], (2.1a)
with the initial and boundary conditions
u = u 0
in Ω for t = 0, (2.1b)
u = u D on ∂Ω, (2.1c) q = q N on ∂Ω N . (2.1d)
We partition the domain Ω as
T h = {I i := (x i−1 , x i ) : a = x 0 < x 1 < · · · < x N −1 < x N = b},
and introduce the set of the boundaries of its elements,
∂T h := {∂I i : i = 1, . . . , N }. We also set E h := {x i } N i=0 , h i = x i − x i−1 and h := max N i=1 h i . We know that, when f is smooth enough, if we provide the values { u i } N i=0 and { p i } N i=1
and, for each i = 1, . . . , N , solve the local problem
Q − U x = 0, P − Q x = 0, U t + P x + F (U ) x = f in I i , U = u 0 for t = 0, U (x + i−1 ) = u i−1 , U (x − i ) = u i , P (x − i ) = p i ,thenU = u D on ∂Ω, Q = q N on ∂Ω N
are satisfied. There are other possible characterizations of the exact solution corresponding to different choices of boundary data for the local problem; see [8].
Note that for these characterizations, the boundary data of the local problems are the unknowns of a global problem obtained from the transmission conditions and boundary conditions, and the system of equations for the global unknowns is square.
2.2. HDG method. To define our HDG method, we first introduce the finite element spaces to be used. We let the approximations (
u h , q h , p h , u h , q h , p h ) to (u| Ω , q| Ω , p| Ω , u| E h , q| E h , p| E h ) be in the space W k h × W k h × W k h × L 2 (E h ) × L 2 (∂T h ) × L 2 (∂T h ) where W k h = {w ∈ L 2 (T h ) : w| Ii ∈ P k (I i ) ∀ i = 1, · · · , N }. Here P k (I i )
is the space of polynomials of degree at most k on the domain I i . For any function ζ lying in L 2 (∂T h ), we denote its values on
∂I i := {x + i−1 , x − i } by ζ(x + i−1 ) (or simply ζ + i−1 ) and ζ(x − i ) (or simply ζ − i ). Note that ζ(x + i ) is not necessarily equal to ζ(x − i ). In contrast, for any η in the space L 2 (E h ), its value at x i , η(x i ) (or simply η i ) is uniquely defined; in this case, η(x − i ) or η(x + i ) mean nothing but η(x i ).
To obtain the HDG formulation, we use a discrete version of the characterization of the exact solution. Assuming that the values { u hi } N i=0 and { p − hi } N i=1 are given, for each i = 1, . . . , N , we solve a local problem on the element I i by using a Galerkin method. To describe it, let us introduce the following notation. By (ϕ, v) Ii , we denote the integral of ϕ times v on the interval I i , and by ϕ, vn ∂Ii we simply mean the expression ϕ(
x − i )v(x − i )n(x − i ) + ϕ(x + i−1 )v(x + i−1 )n(x + i−1 ).
Here n denotes the outward unit normal to I i : n(x + i−1 ) := −1 and n(x − i ) := 1. On the element I i = (x i−1 , x i ), we give f and the boundary data u h i−1 , u h i and p − h i and take the HDG approximate solutions (p h , q h , u h ) ∈ P k (I i ) × P k (I i ) × P k (I i ) to be the solution of the equations
(q h , v) Ii + (u h , v x ) Ii − u h , vn ∂Ii = 0, (p h , z) Ii + (q h , z x ) Ii − q h , zn ∂Ii = 0, (u ht , w) Ii − (p h + F (u h ), w x ) Ii + p h + F h , wn ∂Ii = (f, w) Ii , for all (v, z, w) ∈ P k (I i )×P k (I i )×P k (I i ),
where the remaining undefined numerical traces are given by
p h = p h + τ pu ( u h i−1 − u h ) n at x + i−1 , q h = q h + τ qu ( u h i−1 − u h ) n at x + i−1 , q h = q h + τ qu ( u h i − u h ) n + τ qp ( p − h i − p h ) n at x − i , F h = F ( u h ) − τ F ( u h , u h )( u h − u h ) n at x + i−1 and x − i .
The functions τ qu , τ pu , τ qp , and τ F ( u h , u h ) are defined on ∂T h and are called the components of the stabilization function; they have to be properly chosen to ensure that the above problem has a unique solution. In particular, due to the nonlinearity of F , the function τ F (·, ·) : ∂T h → R can be nonlinear in terms of u h and u h . In the case of F = 0, we simply take τ F = 0.
It remains to impose the transmission conditions
[[ q h ]](x i ) = 0 and [[ p h + F h ]](x i ) = 0 for all i = 1, . . . , N − 1,
and the boundary conditions
u h = u D on ∂Ω and q h = q N on ∂Ω N . Here, [[ζ]](x i ) := ζ(x − i ) − ζ(x + i ).
This completes the definition of the HDG methods using the characterization of the exact solution. Note that this way of defining the HDG methods immediately provides a way to implement them.
On the other hand, the above presentation of the HDG methods is not very well suited for their analysis. Thus, we now rewrite it in a more compact form using the notation
(ϕ, v) := N i=1 (φ, v) Ii , ϕ, vn := N i=1 ϕ, vn ∂Ii . Let M h (g) := {ζ ∈ L 2 (E h ) : ζ| ∂Ω = g},M h := L 2 (E h \ {a}).
The approximation provided by the HDG method,
(u h , q h , p h , u h , p − h ), is the ele- ment of W k h × W k h × W k h × M h (u D ) ×M h which solves the equations (q h , v) + (u h , v x ) − u h , vn = 0, (2.2a) (p h , z) + (q h , z x ) − q h , zn = 0, (2.2b) (u ht , w) − (p h + F (u h ), w x ) + p h + F h , wn = (f, w), (2.2c) and q h , µn = q N , µn ∂Ω N , p h + F h , χn =0 (2.2d) for all (v, z, w, µ, χ) ∈ W k h × W k h × W k h ×M h × M h (0), where, on ∂T h , we have (2.2e) p + h = p + h + τ + pu ( u h − u + h ) n + , q + h = q + h + τ + qu ( u h − u + h ) n + , q − h = q − h + τ − qu ( u h − u − h ) n − + τ − qp ( p − h − p − h ) n − , F h = F ( u h ) − τ F ( u h , u h ) ( u h − u h ) n.
It is not difficult to define HDG methods that are associated to other characterizations of the exact solution, but these methods are actually the same, provided that the corresponding stabilization function allows for the transition from one characterization to the other; see [16,8]. In fact, the choice of characterization to use is more relevant for the actual implementation of the HDG method rather than for its actual definition. The implementation of the HDG method (2.2) is discussed in the Appendix.
When above scheme is discretized in time, we can choose the initial approxi-
mation (u 0 h , q 0 h , p 0 h , u 0 h , p 0 h ) to be the HDG approximate solutions of the stationary equation v + v xxx + F (v) x = g, where g = u 0 + (u 0 ) xxx + F (u 0 )
x and u 0 is the initial data of the time-dependent problem (1.1); see [8] for HDG methods on stationary third-order equations. The initial approximation
(u 0 h , q 0 h , p 0 h , u 0 h , p 0 h ), is the element of W k h × W k h × W k h × M h (u D ) ×M h which solves the equations (q 0 h , v) + (u 0 h , v x ) − u 0 h , vn = 0, (p 0 h , z) + (q 0 h , z x ) − q 0 h , zn = 0, (u 0 h , w) − (p 0 h + F (u 0 h ), w x ) + p 0 h + F 0 h , wn = (g, w), q 0 h , µn = q N , µn ∂Ω N , p 0 h + F 0 h , χn = 0 for all (v, z, w, µ, χ) ∈ W k h × W k h × W k h ×M h × M h (0), where q 0 h , p 0
h , and F 0 h are defined in the same ways as q h , p h , and F h in (2.2e). Note that the equations above are almost the same as those in (2.2) except the third one. This way of choosing initial data for time-dependent problems by solving corresponding stationary problems has been used in [12,9].
Next, we present our stability result and a priori error estimate of the HDG method under some conditions on the stabilization function.
Stability.
To discuss the L 2 -stability of the HDG method, we let
τ (u h , u h ) := 1 (u h − u h ) 2 u h u h (F (s) − F ( u h ))n ds.
We have the following stability result.
Theorem 2.1. Assume that u D = q N = 0. If the stabilization function satisfies (τ + F −τ + ) − τ + pu − 1 2 (τ + qu ) 2 ≥ 0, and (τ − F −τ − ) + 1 2 (τ − qu ) 2 ≥ 0, (τ − F −τ − )(τ − qp ) 2 + τ − qu τ − qp − 1 2 ≥ 0, (2.3)
then for the HDG method (2.2), we have
d dt u h 2 ≤ 2(f, u h ).
Note that if the nonlinear term F = 0, then we have τ F =τ = 0 and the condition (2.3) in the Theorem above can be simplified as
−τ + pu − 1 2 (τ + qu ) 2 ≥ 0 and τ − qu τ − qp − 1 2 ≥ 0. (2.4)
If F (u) = 0, we just need to have τ F ≥τ and take τ ± qu , τ + pu and τ − qp to satisfy (2.4). Sinceτ
= 1 (u h − u h ) 2 u h u h F (ξ)(s − u h )n ds ≤ 1 2 sup s∈J(u h , u h ) |F (s)|, where J(u h , u h ) = [min{u h , u h }, max{u h , u h }], the stabilization function τ F satis- fies the condition τ F ≥τ if τ F ≥ 1 2 sup s∈J(u h , u h ) |F (s)|.
For other choices of τ F which satisfies the condition τ F ≥τ , see [24].
2.4.
A priori error estimate for linear equations. Now we consider the convergence properties of our HDG method for linear equations in which F = 0. We proceed as follows. We first define an auxiliary projection and state its optimal approximation property. Then, we provide an estimate for the L 2 -norm of the projections of the errors in the primary and auxiliary variables. Let us introduce a key auxiliary projection that is tailored to the numerical traces. The projection of the function (u, q,
p) ∈ H 1 (T h ) × H 1 (T h ) × H 1 (T h ), Π(u, q, p) := (Πu, Πq, Πp), is defined as follows. On an element I i = (x i−1 , x i ), the projection is the element of P k (I i ) × P k (I i ) × P k (I i )
which solves the following equations:
(δ u , v) Ii = 0 ∀ v ∈ P k−1 (I i ), (2.5a) (δ q , z) Ii = 0 ∀ z ∈ P k−1 (I i ), (2.5b) (δ p , w) Ii = 0 ∀ w ∈ P k−1 (I i ), (2.5c) δ p − τ + pu δ u n = 0 on x + i−1 , (2.5d) δ q − τ + qu δ u n = 0 on x + i−1 , (2.5e) δ q − τ − qu δ u n − τ − qp δ p n = 0 on x − i , (2.5f)
where we use the notation δ ω := ω − Πω for ω = u, q, and p. Note that the last three equations have exactly the same structure as the numerical traces of the HDG method in (2.2e).
The following result for the optimal approximation properties of the projection Π was shown in [8]. To state it, we use the following notation. The H s (D)-norm is denoted by · s,D . We drop the first subindex if s = 0, and the second one if D = Ω or D = T h .
Lemma 2.2. Suppose that (2.6) τ + qu + τ − qu − τ + pu τ − qp = 0.
Then the projection Π in (2.5) is well defined on any interval I i . In addition, if τ + qu , τ − qu , τ + pu and τ − qp are constants, we have that, for ω = u, q and p, there is a constant C such that
ω − Πω Ii ≤ C h s+1 for s ∈ [1, k],
provided ω ∈ H s+1 (I i ).
Next, we provide estimates for the L 2 -norm of the projection of the errors
u := Πu − u h , q := Πq − q h , p := Πp − p h ,τ + qu ∈ [0, 1], τ + pu ∈ [−1 − 1 − τ + qu 2 , − 1 2 − 1 2 τ + qu 2 ],(2.
7)
then for k > 0 and h small enough, we have
u (t) + q (t) + p (t) + ut (t) ≤ Ch k+1 for 0 ≤ t ≤ T, where C is independent of h.
It is easy to see that if the stabilization function satisfies the condition (2.7), then it also satisfies the conditions (2.4) and (2.6). Using Lemma 2.2, Theorem 2.3 and the triangle inequality, we immediately get the following L 2 error estimate for the actual errors.
e u (t) + e q (t) + e p (t) + e ut (t) ≤ Ch k+1 for 0 ≤ t ≤ T, where C is independent of h.
Proofs
In this section, we provide detailed proofs of our main results. We first prove Theorem 2.1 on the L 2 -stability of the HDG method for general KdV type equations. Then we combine several energy identities to prove the error estimate in Theorem 2.3 for linear third-order equations.
3.1. L 2 -stability. Now let us prove Theorem 2.1 on the stability of the HDG method for the KdV equation. We treat the nonlinear term in a way similar to that in [24].
Proof. Taking ω = u h , v = −p h and z = q h in (2.2a)-(2.2c) and adding the three equations together, we get
(f, u h ) =(u ht , u h ) − (p h + F (u h ), u hx ) + p h + F h , u h n − (q h , p h ) − (u h , p hx ) + u h , p h n + (p h , q h ) + (q h , q hx ) − q h , q h n .
Using integration by parts and (2.2d), we have
(f, u h ) = 1 2 d dt u h 2 − (F (u h ), u hx ) − p h + F h − p h , ( u h − u h )n + 1 2 ( q h − q h ) 2 , n + 1 2 q 2 h (0). (3.1)
Let G(s) be such that dG(s)/ds = F (s). It is easy to see that
−(F (u h ), u hx ) = −( d dx G(u h ), 1) = − G(u h ), n = − u h u h F (s)ds, n .
Using it for the second term on the right hand side of (3.1), we get that
(f, u h ) = 1 2 d dt u h 2 + Φ + 1 2 q h (0) 2 , where Φ = − u h u h (F (s) − F ( u h ))ds, n − F h − F ( u h ), ( u h − u h )n − p h − p h , ( u h − u h )n + 1 2 ( q h − q h ) 2 , n .
Next, we just need to show that Φ ≥ 0. Let
τ := 1 ( u h − u h ) 2 u h u h (F (s) − F ( u h ))nds.
Using the definition of F h in (2.2e), we have
Φ = τ F −τ , ( u h − u h ) 2 − p h − p h , ( u h − u h )n + 1 2 ( q h − q h ) 2 , n .
By the definition of p h and q h in (2.2e), we get
Φ + := Φ| ∂T + h = τ + F −τ + − τ + pu − 1 2 (τ + qu ) 2 , ( u h − u h ) 2 ∂T + h , Φ − := Φ| ∂T − h = τ − F −τ − + 1 2 (τ − qu ) 2 , ( u h − u h ) 2 ∂T − h + 1 2 (τ − qp ) 2 , ( p h − p h ) 2 ∂T − h + τ − qu τ − qp − 1, ( p h − p h )( u h − u h )n ∂T − h .
It is easy to check that if the stabilization function satisfies the condition (2.3), then we get Φ + ≥ 0 and Φ − ≥ 0. This shows that 1 2
d dt u h 2 ≤ (f, u h ).
Error analysis.
In this section, we prove the optimal error estimate for the projections of the errors in Theorem 2.3 for linear equations with F = 0. First, we obtain the equations for the projection of the errors.
for all (v, z, w) ∈ W k h × W k h × W k h ,
where e ω = ω − ω h for ω = u, q, and p. From (2.2e) and (2.2d), it is easy to see that
e + p = e + p + τ + pu ( e u − e + u ) n + , e + q = e + q + τ + qu ( e u − e + u ) n + , e − q = e − q + τ − qu ( e u − e − u ) n − + τ − qp ( e − p − e − p ) n − , + p = + p + τ + pu ( u − + u ) n + , + q = + q + τ + qu ( u − + h ) n + , − q = − q + τ − qu ( u − − u ) n − + τ − qp ( − p − − p ) n − .
Using the equations (2.5d)-(2.5f), after some simple algebra manipulations we get that + p = e + p and ± q = e ± q . Therefore, by the definition of the projection Π, (2.5a)-(2.5c), we easily obtain the following equations for the projections of errors
( q , v) + (δ q , v) + ( u , v x ) − u , vn = 0, (3.3a) ( p , z) + (δ p , z) + ( q , z x ) − q , zn = 0, (3.3b) ( ut , w) + (δ ut , w) − ( p , w x ) + p , wn = 0, (3.3c) q , µ n = 0, p , χ n = 0 (3.3d) for all (v, z, w, µ, χ) ∈ W k h × W k h × W k h ×M h × M h (0).
Energy identities.
To prove the L 2 -error estimate in Theorem 2.3, we begin by establishing a key identity involving the quantity 2 := u 2 + q 2 + p 2 + ut 2 by energy arguments.
Lemma 3.1. We have that 1 2
d dt 2 + S + Ψ = 0, where S =(δ ut , u ) − (δ q , p ) + (δ p , q ) + (δ q t , q ) + (δ p , ut ) − (δ ut , p ) + (δ p t , p ) − (δ ut , q t ) + (δ q t , ut ) + (δ utt , ut ) − (δ q t , p t ) + (δ p t , q t ), Ψ = − p − p , ( u − u )n + 1 2 ( q − q ) 2 , n + q − q , ( ut − ut )n + 1 2 ( p − p ) 2 , n + qt − qt , ( p − p )n + 1 2 ( ut − ut ) 2 , n − pt − pt , ( ut − ut )n + 1 2 ( qt − qt ) 2 , n + 1 2 2 q (x 0 ) + 1 2 ( p + qt ) 2 (x 0 ) − 1 2 2 p (x N ).
Proof. Differentiating the error equations (3.3a)-(3.3c) with respect to t, we get
( q t , v) + (δ q t , v) + ( ut , v x ) − ut , vn = 0, (3.4a) ( p t , z) + (δ p t , z) + ( q t , z x ) − qt , zn = 0, (3.4b) ( utt , w) + (δ utt , w) − ( p t , w x ) + pt , wn = 0. (3.4c)
Next, we use (3.3) and (3.4) to get four energy identities.
(i) Taking w = u , v = − p , and z = q in (3.3) and adding the three equations together, we have 0 =( ut , u ) + (δ ut , u ) − ( p , ux ) + p , u n − ( q , p ) − (δ q , p ) − ( u , p x ) + u , p n + ( p , q ) + (δ p , q ) + ( q , q x ) − q , q n .
Using integration by parts, (3.3d), and the fact that
u | ∂Ω = e u | ∂Ω = 0, q | ∂Ω N = e q | ∂Ω N = 0, we get 0 = 1 2 d dt u 2 + (δ ut , u ) − (δ q , p ) + (δ p , q ) − p − p , ( u − u )n + 1 2 ( q − q ) 2 , n + 1 2 2 q (x 0 ).
(3.5)
(ii) Similar to (i), taking v = q in (3.4a), z = ut in (3.3b), and w = − p in (3.3c) and adding the three equations together, we get (3.4b), and w = − q t in (3.3c) and adding the equations together, we get
0 = 1 2 d dt q 2 + (δ q t , q ) + (δ p , ut ) − (δ ut , p ) + q − q , ( ut − ut )n + 1 2 ( p − p ) 2 , n − 1 2 2 p (x N ) + 1 2 2 p (x 0 ). (3.6) (iii) Taking v = ut in (3.4a), z = p in0 = 1 2 d dt p 2 + (δ p t , p ) + (δ q t , ut ) − (δ ut , q t ) + qt − qt , ( p − p )n + 1 2 ( ut − ut ) 2 , n + qt p (x 0 ). (3.7)
(iv) Taking v = − p t , z = q t , and w = ut in (3.4a)-(3.4c) and adding the equations together, we get
0 = 1 2 d dt ut 2 + (δ utt , ut ) − (δ q t , p t ) + (δ p t , q t ) − pt − pt , ( ut − ut )n + 1 2 ( qt − qt ) 2 , n + 1 2 2 qt (x 0 ). (3.8)
The proof is completed by adding the four equations (3.5)-(3.8) together.
3.2.3.
Proof of the L 2 -error estimate. Using Lemma 3.1, we first get the following result.
Lemma 3.2. If the stabilization function satisfies the condition (2.7), then we have
(t) 2 ≤ (0) 2 + Θ(0) + t 0 2 p (x N ) dt + 2 | t 0 S dt| for 0 ≤ t ≤ T, , where Θ = τ + qu − τ + pu τ + qu , ( u − u ) 2 ∂T + h + 1, τ − qu ( u − u ) 2 + τ − qp ( p − p ) 2 ∂T − h ,
and S is the same as in Lemma 3.1.
Proof. Using the definition of + p and q in (3.2), for the Ψ term in Lemma 3.1, we have
Ψ = Ψ + + Ψ − , where Ψ + = − τ + pu , ( u − u ) 2 ∂T + h − 1 2 (τ + qu ) 2 , ( u − u ) 2 ∂T + h + τ + qu , ( u − u )( u − u ) t ∂T + h − 1 2 (τ + pu ) 2 , ( u − u ) 2 ∂T + h − τ + pu τ + qu , ( u − u )( u − u ) t ∂T + h − 1 2 1, ( ut − ut ) 2 ∂T + h − τ + pu , ( ut − ut ) 2 ∂T + h − 1 2 (τ + qu ) 2 , ( ut − ut ) 2 ∂T + h + 1 2 2 q (x 0 ) + 1 2 ( p + qt ) 2 (x 0 ) and Ψ − = − p − p , u − u ∂T − h + 1 2 1, (τ − qu ( u − u ) + τ − qp ( p − p )) 2 ∂T − h + τ − qu ( u − u ) + τ − qp ( p − p ), ( u − u ) t ∂T − h + 1 2 1, ( p − p ) 2 ∂T − h + τ − qu ( u − u ) t + τ − qp ( p − p ) t , p − p ∂T − h + 1 2 1, ( ut − ut ) 2 ∂T − h − pt − pt , ut − ut ∂T − h + 1 2 1, (τ − qu ( ut − ut ) + τ − qp ( pt − pt )) 2 ∂T − h − 1 2 2 p (x N ).
We can rewrite the term Ψ + as
Ψ + = Γ 1 + 1 2 d dt Θ 1 , where Γ 1 = −τ + pu − 1 2 (τ + qu ) 2 − 1 2 (τ + pu ) 2 , ( u − u ) 2 ∂T + h + − 1 2 − τ + pu − 1 2 (τ + qu ) 2 , ( ut − ut ) 2 ∂T + h + 1 2 2 q (x 0 ) + 1 2 ( p + qt ) 2 (x 0 ), Θ 1 = τ + qu − τ + pu τ + qu , ( u − u ) 2 ∂T + h .
Similarly, if we assume that τ − qu τ − qp = 1, after some calculations we get
Ψ − = Γ 2 + 1 2 d dt Θ 2 − 1 2 2 p (x N ), where Γ 2 = ( 1 2 τ − qu ) 2 , ( u − u ) 2 ∂T − h + 1 2 , τ − qp ( p − p ) + ( u − u ) t ) 2 ∂T − h + ( 1 2 τ − qp ) 2 , ( pt − pt ) 2 ∂T − h + 1 2 , ( p − p ) + τ − qu ( u − u ) t 2 ∂T − h , Θ 2 = 1, τ − qu ( u − u ) 2 + τ − qp ( p − p ) 2 ∂T − h .
So from Lemma 3.1 we get
(3.9) 1 2 d dt ( 2 + Θ 1 + Θ 2 ) + Γ 1 + Γ 2 = 1 2 2 p (x N ) − S.
Now we integrate the equation (3.9) with respect to t and get
1 2 (t) 2 + Θ 1 (t) + Θ 2 (t) + t 0 (Γ 1 + Γ 2 )dt = 1 2 (0) 2 + Θ 1 (0) + Θ 2 (0) + 1 2 t 0 2 p (x N )dt − t 0 S dt.
It is easy to check that if τ ± qu , τ + pu and τ − qp satisfy the condition (2.7), we have
Θ 1 ≥ 0, Θ 2 ≥ 0, Γ 1 ≥ 0, Γ 2 ≥ 0 for any t ∈ [0, T ].
Therefore,
(t) 2 ≤ (0) 2 + Θ(0) + t 0 2 p (x N ) dt + 2 | t 0 S dt|, where Θ = Θ 1 + Θ 2 .
To prove Theorem 2.3, we also need the following Lemma for error estimates of the initial approximations at t = 0 (See Theorem 2.2 and Theorem 2.3 in [8]). Lemma 3.3. If τ ± qu , τ + pu , τ − qp satisfy the condition (2.6), then for k > 0,
u (0) + q (0) + p (0) ≤ Ch k+2 , e u (0) E h + e q (0) E h + e p (0) E h ≤ Ch 2k+1 .
In addition, let us get an estimate for ut at t = 0.
Lemma 3.4. If τ ± qu , τ + pu , τ − qp satisfy the condition (2.6), then for k > 0 ut (0) ≤ Ch k+1 . Proof. Taking t = 0 and w = ut (0) in the error equation (3.3c), we have ( ut (0), ut (0)) + (δ ut (0), ut (0)) − ( p (0), utx (0)) + p (0), ut (0)n = 0.
By Cauchy inequality, trace inequality and inverse inequality, we get Proof. We first estimate the term
ut (0) 2 ≤ C δ ut (0) 2 + Ch −2 p (0) 2 + Ch −1 p (0) 2 E h .t 0 2 p (x N )dt. Taking ω to be ω 1 := x−x0 x N −x0 in (3.3c), we get p (x N ) = −( ut , ω 1 ) − (δ ut , ω 1 ) + ( p , 1 x N − x 0 )
by the fact that ω 1 (x 0 ) = 0 and ω 1 (x N ) = 1. Using Cauchy inequality, we have
| p (x N )| ≤ |( ut , ω 1 )| + |(δ ut , ω 1 )| + |( p , 1 x N − x 0 )| ≤ C( ut + δ ut + p ).
Then by the approximation property of the projection Π in Lemma 2.2, we obtain where S 1 =(δ ut , u ) − (δ q , p ) + (δ p , q ) + (δ q t , q ) + (δ p , ut ) − (δ ut , p ) + (δ p t , p ) + (δ q t , ut ) + (δ utt , ut ),
S 2 = − (δ ut , q t ) − (δ q t , p t ) + (δ p t , q t ).
Using Cauchy inequality and the approximation property of the projection Π in Lemma (2.2), we get
t 0 |S 1 |dt ≤ Ch k+1 t 0 dt.
Integrating S 2 with respect to t, we have
t 0 S 2 dt = − (δ ut , q )| t 0 + t 0 (δ utt , q )dt − (δ q t , p )| t 0 + t 0 (δ q tt , p )dt + (δ p t , q )| t 0 − t 0 (δ p tt , q )dt.
By the approximation property of the projection Π in Lemma 2.2,
t 0 S 2 dt ≤Ch 2k+2 + C (0) 2 + 1 4 (t) 2 + Ch k+1 t 0 dt. So we get t 0 Sdt ≤ t 0 |S 1 |dt + | t 0 S 2 dt| ≤ Ch 2k+2 + C (0) 2 + 1 4 (t) 2 + Ch k+1 t 0 dt. (3.11)
Applying (3.10) and (3.11) to Lemma 3.2, we have
(t) 2 ≤C (0) 2 + CΘ(0) + Ch 2k+2 + C t 0 (t) 2 dt. Since Θ(0) ≤ C( u (0) 2 E h + p (0) 2 E h ) + Ch −1 ( u (0) 2 + p (0) 2 ) by
Numerical Results
In this section, we carry out several numerical experiments to study the accuracy and capability of our HDG method. In the first and the second numerical experiments, we examine the orders of convergence of the method for linear and nonlinear third-order problems. In the third and the fourth experiments, we apply the method to solve some well-known dispersive wave problems. For all the experiments, we use the following second-order midpoint rule [2,9] for time discretization. Let 0 = t 0 < t 1 < · · · < t J = T be a partition of the interval [0, T ] and ∆t j = t j+1 − t j . For j = 0, · · · , J − 1 and ω ∈ {u h , q h , p h }, let ω j+1 ∈ W k h be defined as ω j+1 = 2ω j,1 − ω j , where ω j,1 is the solution of the equation
ω j,1 − ω j 1 2 ∆t j + (ω j,1 ) xxx + F (ω j,1 ) x = 0.
The components of the stabilization function, (τ + qu , τ + pu , τ − qu , τ − qp ) are taken to be (0, −1, 1, 1) in all the following numerical tests.
Numerical experiment 1: In this test, we use the HDG method to solve the time-dependent third-order linear problem
u t + u xxx = f, where f is chosen so that the exact solution is u(x, t) = sin(x + t) on the domain (x, t) ∈ [0, 1] × [0, 0.1].
The initial condition is u 0 = sin(x) and the boundary conditions are u(0, t) = sin(t), u(1, t) = sin(1 + t) and u x (1, t) = cos(1 + t). We take h = 2 −n for n = 1, · · · , 5. The step size for time discretization is ∆t = 0.1 * h 2 for k = 0, 1, and ∆t = 0.1 * h 3 for k = 2, 3 so that the temporal errors are very small. We compute the orders of convergence of u h , q h , p h at the final time T = 0.1, and the orders we observe in the numerical experiments are listed in Table 1.
Our numerical results indicate that the orders of convergence of (e u , e q , e p ) are optimal as predicted by the error estimate in Theorem 2.4 for any k > 0. For k = 0, although our error analysis is inclusive, we observe that the method converges optimally in the numerical experiment.
Numerical experiment 2: Now we use the HDG method to solve the nonlinear third-order equation u t + u xxx + (3u 2 ) x = f. The function f , the initial condition and the boundary conditions are chosen so that the exact solution is u(x, t) = sin(2x+t) in the domain (x, t) ∈ [0, π]×[0, 0.1]. Here, we take the stabilization function τ F = 3, given that F (u) = 3u 2 and 1 2 |F (u)| = 3|u| ≤ 3 for the solution u. The mesh size for the HDG method is h = 2 −n for n = 3, · · · , 7. The step size for time discretization is ∆t = 0.1 * h 2 for k = 0, 1 and ∆t = 0.1 * h 3 for k = 2, 3 so that the temporal errors are much smaller than the spatial errors. The orders of convergence of u h , q h , p h at the final time T = 0.1 are displayed in Table 2. Our numerical results show that the orders of convergence of (e u , e q , e p ) are also optimal for any k ≥ 0 for the nonlinear problem.
In the previous two tests, we have observed optimal convergence rates of the HDG method for both linear and nonlinear third-order problems. In the next two tests, we apply the method to solve the KdV equation (4.1) u t + u xxx + (3u 2 ) x = 0. 2] with the initial condition u 0 = 2 sech 2 (x − 4) and the boundary conditions u(−10, t) = 2 sech 2 (−10 − 4t + 4), u(0, t) = 2 sech 2 (−4t + 4), u x (0, t) = −4 sech 2 (−4t + 4) tanh(−4t + 4). The exact solution to this initialboundary value problem is the classical solitary-wave solution [2,27] u(x, t) = 2 sech 2 (x − 4t + 4).
In the computation, we use 100 elements, piecewise cubic polynomials, and timestep size ∆t = 10 −3 , and take τ F = (F ( u)) 2 + 1 4 so that τ F > 1 2 |F ( u h )|. The spacetime graphs of the computed solution (u h , q h , p h ) as well as the exact solutions (u, q, p) at the final time T = 2 are displayed in Figure 1. We observe a good match between the approximate solutions and the exact solutions.
Numerical experiment 4: In this test, we simulate the interaction of two solitary waves with different propagation speeds using our HDG method. We consider the KdV equation ( Table 2. The error (e u , e q , e p ) and their convergence orders for the nonlinear problem in the numerical experiment 2.
and boundary data u(−20, t), u(0, t), u x (0, t), which admits the solution (see [27])
u(x, t) = 5 4.5 csch 2 [1.5(x − 9t + 14.5)] + 2 sech 2 (x − 4t + 12)
{3 coth[1.5(x − 9t + 14.5)] − 2tanh(x − 4t + 12)} 2 .
In our computation, we use 50 elements, piecewise cubic polynomials, and the time-step size ∆t = 10 −4 . The stabilization function τ F is taken in the same way as in the previous test. The space-time graphs of the HDG approximate solutions and the exact solutions are displayed in Figure 2. From the side-by-side comparison, we see that the HDG solutions are good approximations to the exact solutions. They show that the two waves are moving toward the same direction. The faster soliton catches up with the slower one and they overlap around t = 0.5. Afterwards, the faster soliton continues to propagate and the slower one falls behind.
Concluding remarks
In this paper, we develop a new HDG method for time-dependent third-order equations in one space dimension based on the characterization of the exact solution as the solutions to local problems that are "glued" together by transmission conditions. We find conditions on the stabilization function under which the method is L 2 stable for KdV type equations. We also obtain optimal error estimates for the linear third-order equation. Numerical results from computation verify the theoretical error analysis and show that the method is able to accurately simulate solitary wave solutions of the KdV equation. Our future work is to develop and analyze HDG methods for fifth-order KdV equations and third-order equations in multiple dimensions and complex systems.
for any (v, z, w, µ, χ)
∈ W k − h , χn ∂T − h − p h n + τ pu (¯ u h −ū h ), χ ∂T + h − F (¯ u h ) −τ F (¯ u h −ū h ), χ .
Here we have used the notationτ F := τ F (¯ u h ,ū h ), and ∂ 1τF (respectively, ∂ 2τF ) denotes the first-order partial derivative of τ F with respect to the first argument (respectively, second argument) evaluated at (¯ u h ,ū h ).
The discretization of the system above gives rise to matrix equations of the form (A.1)
0 A 1 B 1 A 2 −B T 1 C B 2 0 A 3 δp h δq h δu h + D 1 0 D 2 E 2 D 3 E 3 δ u h δ p − h = R 1 R 2 R 3 , and (A.2) G 1 G 2 G 3 G 4 0 G 5 δp h δq h δu h + D 4 E 4 D 5 E 5 δ u h δ p − h = R 4 R 5 .
From (A.1), we get
(A.3) δp h δq h δu h = 0 A 1 B 1 A 2 −B T 1 C B 2 0 A 3 −1 R 1 R 2 R 3 − D 1 0 D 2 E 2 D 3 E 3 δ u h δ p − h
We emphasize that the above inverse can be computed on each element independently of each other since the matrices A 1 , A 2 , A 3 , B 1 , B 2 and C are block-diagonal owing to the discontinuous nature of the approximation spaces. Applying (A.3) to (A.2), we get the global linear system
K δ u h δ p − h = F, where K = D 4 E 4 D 5 E 5 − G 1 G 2 G 3 G 4 0 G 5 0 A 1 B 1 A 2 −B T 1 C B 2 0 A 3 −1 D 1 0 D 2 E 2 D 3 E 3 and F = R 4 R 5 − G 1 G 2 G 3 G 4 0 G 5 0 A 1 B 1 A 2 −B T 1 C B 2 0 A 3 −1 R 1 R 2 R 3 .
Therefore, the only globally coupled degrees of freedom are those associated with δ u h and δ p − h , which live only on element interfaces. Due to the one-dimensional setting of the KdV equation, the size and the bandwidth of the global linear system are independent of the degrees of polynomials used; it only depends on the number of subintervals in the mesh. Once δ u h and δ p − h are obtained, (δp h , δq h , δu h ) can be locally computed by using (A.3).
u t + u xxx + F (u) x = f for x ∈ Ω := (a, b), t ∈ (0, T ], u = u 0 in Ω for t = 0, u = u D on ∂Ω := {a, b}, u x = q N on ∂Ω N := {b}.
(LDG) method, for the KdV equation was introduced in 2002 by Yan and Shu in 2000 Mathematics Subject Classification. Primary 65M60, 65N30.
(P, Q, U ) coincides with the solution (p, q, u) of (2.1) if and only if the transmission conditions Q(x − i ) = Q(x + i ), P (x − i ) = P (x + i ), i = 1, . . . , N − 1 and the boundary conditions
and deduce from them the estimates for the L 2 -norm of the errors e u := u − u h , e q := q − q h , e p := p − p h .
Theorem 2. 3 .
3Suppose that F (u) = 0 in the problem (2.1) and the exact solution(u, q, p) ∈ W 2,∞ ((0, T ]; H k+1 (T h ))×W 1,∞ ((0, T ]; H k+1 (T h ))×W 1,∞ ((0, T ]; H k+1 (T h )).If the stabilization function of the HDG method (2.2) satisfies the condition τ − qu > 0, τ − qu τ − qp = 1, and
Theorem 2. 4 .
4Suppose that the hypotheses of Theorem 2.3 are satisfied. Then we have
3. 2 . 1 .
21The error equations. From the equations defining the HDG method, (2.2a)-(2.2c), and the fact that the exact solution also satisfy these equations, we obtain the following error equations (e q , v) + (e u , v x ) − e u , vn = 0, (e p , z) + (e q , z x ) − e q , zn = 0, (e ut , w) − (e p , w x ) + e p , wn = 0,
Then the conclusion follows by using Lemma 2.2 and Lemma 3.3. Now let us finish the proof of Theorem 2.3 by estimating the right hand side of the inequality in Lemma 3.2 and using Lemma 3.3 and Lemma 3.4.
p
(x N ) dt ≤ Ch 2k+2 + Let S = S 1 + S 2 ,
3.4. Now we use Grönwall's inequality and get(t) 2 ≤ Ch 2k+2 ,where C depends on t but not on h. This completes the proof of Theorem 2.3.
Figure 1 .
1Space-time graphs of one soliton in the domain (x, t) ∈ [−10, 0] × [0, 2]. Evolution of the HDG approximate solution (left) and the exact solution (right) of (A): u, (B): q, and (C): p.
Figure 2 .
2Space-time graphs of the interaction of two solitary waves in the domain (x, t) ∈ [−20, 0] × [0, 2]. Evolution of the HDG approximate solution (left) and the exact solution (right) of (A): u, (B): q, and (C): p.
Table 1. The error (e u , e q , e p ) and their convergence orders for the linear problem in the numerical experiment 1.Numerical experiment 3: In this test, we consider the KdV equation (4.1) in the domain (x, t) ∈ [−10, 0] × [0,k
eu
Order
eq
Order
ep
Order
0
1.27e-01
-
1.07e-01
-
1.94e-01
-
6.87e-02
0.89
6.26e-02
0.77
1.13e-01
0.78
3.83e-02
0.84
3.52e-02
0.83
6.10e-02
0.89
2.08e-02
0.88
1.94e-02
0.86
3.31e-02
0.88
1.07e-02
0.96
1.03e-02
0.92
1.85e-02
0.84
1
1.13e-02
-
1.22e-02
-
6.83e-03
-
3.28e-03
1.79
3.08e-03
1.99
1.90e-03
1.85
8.62e-04
1.93
7.69e-04
2.00
4.87e-04
1.97
2.17e-04
1.99
1.92e-04
2.00
1.22e-04
1.99
5.44e-05
2.00
4.80e-05
2.00
3.06e-05
2.00
2
3.66e-04
-
3.27e-04
-
7.41e-04
-
4.59e-05
2.99
4.33e-05
2.92
6.99e-05
3.41
5.71e-06
3.01
5.50e-06
2.98
1.12e-05
2.64
7.10e-07
3.01
6.94e-07
2.99
1.49e-06
2.91
8.86e-08
3.00
8.73e-08
2.99
1.90e-07
2.97
3
1.97e-05
-
5.43e-05
-
7.32e-04
-
1.05e-06
4.23
2.24e-06
4.60
8.53e-05
3.10
6.50e-08
4.01
7.77e-08
4.85
4.19e-06
4.35
4.07e-09
4.00
3.88e-09
4.32
1.86e-07
4.49
2.55e-10
4.00
2.32e-10
4.06
5.68e-09
5.03
4.1) in the domain (x, t) ∈ [−20, 0] × [0, 2] with the initial conditionu 0 (x) = 5
4.5 csch 2 [1.5(x + 14.5)] + 2 sech 2 (x + 12)
{3 coth[1.5(x + 14.5)] − 2 tanh(x + 12)} 2
k
eu
Order
eq
Order
ep
Order
0
6.63e-01
-
1.34e-00
-
2.63e-00
-
4.08e-01
0.70
8.58e-01
0.64
1.79e-00
0.56
2.37e-01
0.78
5.17e-01
0.73
1.16e-00
0.64
1.32e-01
0.84
2.94e-01
0.82
6.78e-01
0.76
7.11e-02
0.90
1.59e-01
0.89
3.71e-01
0.87
1
5.35e-02
-
9.60e-02
-
2.31e-01
-
1.29e-02
2.05
2.36e-02
2.03
5.29e-02
2.12
3.18e-03
2.02
5.86e-03
2.01
1.28e-02
2.05
7.92e-04
2.01
1.47e-03
2.00
3.17e-03
2.01
1.98e-04
2.00
3.67e-04
2.00
7.92e-04
2.00
2
3.31e-03
-
5.81e-03
-
1.25e-02
-
4.01e-04
3.05
7.32e-04
2.99
1.61e-03
2.96
4.97e-05
3.01
9.20e-05
2.99
1.99e-04
3.01
6.20e-06
3.00
1.15e-05
3.00
2.48e-05
3.00
7.74e-07
3.00
1.44e-06
3.00
3.10e-06
3.00
3
1.54e-04
-
2.81e-04
-
6.52e-04
-
9.57e-06
4.01
1.77e-05
3.99
3.82e-05
4.09
5.97e-07
4.00
1.17e-06
3.99
2.39e-06
4.00
3.73e-08
4.00
6.97e-08
4.00
1.49e-07
4.00
2.33e-09
4.00
4.36e-09
4.00
1.03e-08
3.86
Acknowledgements The author would like to acknowledge the support of National Science Foundation grant DMS-1419029.Appendix A. ImplementationTo implement the HDG method (2.2), we use an implicit scheme for the discretization of the time derivative. One may use high order BDF or an implicit Runge-Kutta method for time discretization. Here, for simplicity we consider the backward Euler method with time-step ∆t. At time-level t j , inserting the definition of the numerical traces (2.2e) into (2.2a)-(2.2e), we obtain the equationsfrom which (u h , q h , p h ) can be locally solved in terms of f , u h and p − h , and the equationswhich determine the globally coupled unknowns ( u h , p − h ). Next, we apply the Newton-Raphson method to solve the above nonlinear system. Denoting the approximations at the current iteration byand g 1 (δp h , µ) + g 2 (δq h , µ) + g 3 (δu h , µ) + d 4 (δ u h , µ) + e 4 (δ p − h , µ) = r 4 (µ),
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[
"A Geometrical Formulation of the Renormalization Group Method for Global Analysis II: Partial Differential Equations",
"A Geometrical Formulation of the Renormalization Group Method for Global Analysis II: Partial Differential Equations"
] |
[
"Teiji Kunihiro \nFaculty of Science and Technology\nRyukoku University\nOhtsu-city520-21SetaJapan\n"
] |
[
"Faculty of Science and Technology\nRyukoku University\nOhtsu-city520-21SetaJapan"
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It is shown that the renormalization group (RG) method for global analysis can be formulated in the context of the classical theory of envelopes: Several examples from partial differential equations are analyzed. The amplitude equations which are usually derived by the reductive perturbation theory are shown to be naturally derived as the equations describing the envelopes of the local solutions obtained in the perturbation theory.
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10.1143/ptp.94.503
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"https://arxiv.org/pdf/patt-sol/9508001v2.pdf"
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hep-th/9505166
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0734c2aea8c6a8f4fdce6da7e637d5c7ae251994
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A Geometrical Formulation of the Renormalization Group Method for Global Analysis II: Partial Differential Equations
24 Aug 1995 August, 1995
Teiji Kunihiro
Faculty of Science and Technology
Ryukoku University
Ohtsu-city520-21SetaJapan
A Geometrical Formulation of the Renormalization Group Method for Global Analysis II: Partial Differential Equations
24 Aug 1995 August, 1995arXiv:patt-sol/9508001v2
It is shown that the renormalization group (RG) method for global analysis can be formulated in the context of the classical theory of envelopes: Several examples from partial differential equations are analyzed. The amplitude equations which are usually derived by the reductive perturbation theory are shown to be naturally derived as the equations describing the envelopes of the local solutions obtained in the perturbation theory.
Introduction
Renormalization group (RG) equations [1] appear in various fields of science. In quantum field theory [2], the RG equation improves results obtained in the perturbation theory. In recent years, the improvement of the effective potential [2,3,4] has acquired a renewed interest [5]. The RG equation has also a remarkable success in statistical physics especially in the critical phenomena [2]. One may also note that there is another successful theory of the critical phenomena called coherent anomaly method (CAM) [6]; CAM utilizes envelopes of susceptibilities and other thermodynamical quantities as a function of temperature. It is well known that Feigenbaum [7] applied RG equation to deduce a universality of some chaotic phenomena.
Recently, Illinois group [8,9] have shown that the RG equation can be also used for nonquantum mechanical problems: They proposed to use the RG equation to get an asymptotic behavior of solutions of differential equations including ones of singular and reductive perturbation problems in a unified way. Mathematically, the RG equation is used to improve the global behavior of the local solutions which were obtained in the perturbation theory. This fact suggests that the RG method may be formulated in a purely mathematical way without recourse to the notion of the RG. In the previous paper [11], we showed that the RG method can be formulated in the context of the classical theory of envelopes [12]: We pointed out that the RG equation is nothing but the envelope equation, and gave a proof why the RG equation can give a globally improved solution to ordinary differential equations(ODE's). In fact, if a family of the curves {C τ } τ in the x-y plane is represented by the equation F (x, y, τ ) = 0, the equation G(x, y) = 0 representing the envelope E is given by eliminating the parameter τ from the equation
∂F (x, y, τ ) ∂τ τ =x = 0. (1.1)
Here we have chosen the parameter to be the x-coordinate of the point of the tangency of a curve C τ and the envelope E. The relevance of the envelope equation Eq.(1.1) and the (Gell-Mann-Low type) RG equations is evident; the parameter τ corresponds to the renormalization point. Thus one would also readily recognize that improving the effective potential in quantum field theory is constructing the envelope of the effective potentials with the renormalization point varied. [11] The purpose of the present paper is to show that the formulation can be naturally extended to a class of partial differential equations (PDE's); the PDE's dealt in the present work include a dissipative nonlinear hyperbolic equation, one-and two-dimensional Swift-Hohenberg equations, damped Kuramoto-Shivashinsky equation [14] and Barlenblatt equation [13]. Most of these examples were examined by the Illinois group [8,9,10]. Therefore there will be some overlaps in the exposition with theirs. It is inevitable, however, because our purpose is to give a purely mathematical formulation of their method without recourse to the notion of the renormalization group. We will show that the amplitude equations usually obtained by the reductive perturbation methods are given naturally as envelope equations. In the course of the formulation, relevance of the characteristic manifold as the branch stripmanifold will be indicated to constructing the global solutions. In Appendix A, we give a short summary of the classical theory of envelope surfaces; for PDE's, envelope surfaces may be more relevant in some cases than envelope curves discussed in [11]. In Appendix B, we examine Barenblatt's equation in our approach; for this equation, only the anomalous exponent of the long-time behavior of the solution is given.
Dissipative nonlinear hyperbolic equation
We first consider the following slightly dissipative nonlinear hyperbolic equation [9] to apply the envelope theory 1 to construct a global solution:
∂u ∂t + λ(u) ∂u ∂x = η ∂ 2 u ∂x 2 ,(2.1)
where λ(u) is a sufficiently smooth function of u, and η is a positive constant. Following [9], we consider a small amplitude wave in the background of constant solution u 0 ;
u = u 0 + ǫu 1 + ǫ 2 u 2 + · · · . (2.2)
Here ǫ denotes the amplitude of the wave. λ(u) is expanded as
λ(u) = λ 0 + ǫλ ′ (u 0 )u 1 + · · · ,(2.3)
where λ 0 ≡ λ(u 0 ). We further assume that the dissipation is weak in the sense η ∼ ǫ; we write as η = µǫ, where µ = O(1). Then equating the coefficients of ǫ n (n = 0, 1, 2, ...), we have
∂ t u 1 + λ 0 ∂ x u 1 = 0, ∂ t u 2 + λ 0 ∂ x u 2 = −λ ′ 0 u 1 ∂ x u 1 + µ∂ 2 x u 1 ,(2.4)
and so on. Here,
λ ′ 0 = λ ′ (u 0 ).
It is now convenient to introduce the new variable ξ ≡ x−λ 0 t, the characteristic direction of the unperturbed equation; we describe the solutions in terms of (ξ, t). Then
∂ t u 1 = 0, ∂ t u 2 = −λ ′ 0 u 1 ∂ ξ u 1 + µ∂ 2 ξ u 1 ,(2.5)
and so on. One readily gets
u 1 (ξ, t) = F 0 (ξ), u 2 (ξ, t) = (t − t 0 )(−λ ′ 0 F 0 (ξ)∂ ξ F 0 (ξ) + µ∂ 2 ξ F 0 (ξ)),(2.6)
where F 0 (ξ) is an arbitrary function of ξ. We note that u 2 is a secular term. 2 We have thus an approximate solution to Eq. (2.1)
u(ξ, t; t 0 ) = u 0 + ǫF 0 (ξ) + ǫ 2 (t − t 0 )(−λ ′ 0 F 0 (ξ)∂ ξ F 0 (ξ) + µ∂ 2 ξ F 0 (ξ)) + O(ǫ 3 ),(2.7)
where we have made it explicit that u is dependent on an arbitrary time t 0 . We stress that this (approximate) solution is only valid for t around t 0 . Now, geometrically speaking, we have a family of surfaces S t 0 represented by u(ξ, t; t 0 ) with a parameter t 0 . Let us obtain the envelope surface E of this family of surfaces, following the classical theory of envelopes given in Appendix A: we represent E by u E (ξ, t).
We first note that F 0 (ξ) may be functionally dependent on the arbitrary initial time t 0 :
u(ξ, t; t 0 ) = u 0 + ǫF 0 (ξ, t 0 ) + ǫ 2 (t − t 0 )(−λ ′ 0 F 0 (ξ, t 0 )∂ ξ F 0 (ξ, t 0 ) + µ∂ 2 ξ F 0 (ξ, t 0 )) + O(ǫ 3 ). (2.8)
It is natural to set the tangent curve C t 0 of E and S t 0 to lie along t = t 0 -line, which is parallel to the characteristic direction (ξ-direction):
C t 0 : t = t 0 , u = u(ξ, t 0 , t 0 ), ξ = ξ. (2.9)
Then the envelope equation reads ∂u ∂t 0 = 0, and t 0 = t, (2.10) which leads to
∂ t F (ξ, t) + ǫλ ′ (u 0 )F ∂ ξ F = η∂ 2 ξ F. (2.11)
This is Burgers equation which is usually obtained in the reductive perturbation theory. Then
u E (ξ, t) = u(ξ, t; t) = u 0 + ǫF (ξ, t). (2.12)
Now one may wonder if u E (ξ, t) satisfies Eq. (2.1), although u(ξ, t; t 0 ) does up to O(ǫ 3 ) for any t 0 . We remark that the question is not trivial because u E (ξ, t) ≡ u(ξ, t; t). We shall show that the answer is yes. In fact, the time derivative of u E (ξ, t) at t = ∀t 0 coincides with ∂u(ξ, t; t 0 )/∂t at t = t 0 ;
∂u E (ξ, t) ∂t t=t 0 = ∂u(ξ, t; t 0 ) ∂t t=t 0 + ∂u(ξ, t 0 ; t ′ 0 ) ∂t ′ 0 t ′ 0 =t 0 , = ∂u(ξ, t; t 0 ) ∂t t=t 0 , (2.13)
where Eq. (2.10) has been used. Furthermore, needless to say, u E (ξ, t 0 ) = u(ξ, t 0 , t 0 ). Thus, at t = ∀t 0 ,
∂u E ∂t + λ(u E ) ∂u E ∂x = η ∂ 2 u E ∂x 2 + O(ǫ 3 ).
(2.14)
Thus one sees that our envelope function u E satisfies the Eq.(2.1) uniformly for t in the global range.
A comment is in order here: one may say that to get the approximate but global solution u E from the local solution Eq.(2.8), we have utilized the fact that the uniqness of the solution of the Cauchy problem is violated when the initial data are given along the characteristic manifold (curve); see Appendix A for characteristic manifolds(curves).
One-dimensional Swift-Hohenberg equation
In this section, we deal with the one-dimensional Swift-Hohenberg equation [15,14]:
L 1 u = ǫu − u 3 ,L 1 ≡ ∂ t + (∂ 2 x + k 2 ) 2 , (3.1)
where ǫ is a small parameter. We shall show that the envelope of local solutions of Eq. (3.1) satisfies the time-dependent Ginzburg-Landau equation.
Following [8], we scale u as
u = √ ǫφ. (3.2)
Then φ satisfiesL
1 φ = ǫ(φ − φ 3 ). (3.3)
We solve this equation in the perturbation theory, expanding φ as φ = φ 0 + ǫφ 1 + ...; we havê
L 1 φ 0 = 0,L 1 φ 1 = φ 0 − φ 3 0 ,(3.4)
and so on. As the 0-th order solution, we take
φ 0 (x, t) = Ae ikx + c.c., (3.5)
where c.c. denotes the complex conjugate. Then the 1st-order equation readŝ
L 1 φ 1 = Ae ikx − A 3 e 3ikx + c.c., (3.6) where A ≡ A(1 − 3|A| 2 ). (3.7)
A special solution to Eq. (3.6) is found to be[9]
φ 1 = A{µ 1 (t − t 0 ) − µ 2 8k 2 (x 2 − x 2 0 )}e ikx + A 3 64k 3 e 3ikx + c.c.,(3.8)
with µ 1 + µ 2 = 1. We note that the secular terms have appeared in φ 1 because of the first term in r.h.s. of Eq. (3.6).
Now one may say that we have a family of surfaces S t 0 x 0 represented by (3.9) parametrized by t 0 and x 0 . Let us obtain the envelope of the family of the surfaces in two steps, by assuming that the amplitude A is dependent on x 0 and t 0 . First, fixing x 0 , we obtain the envelope E 1 of the surfaces with t 0 being the parameter. The resultant envelope E x 0 has the parameter x 0 ; then we obtain the envelope of the family of the surfaces E 1 .
φ(x, t; x 0 , t 0 ) = [Ae ikx + ǫA{µ 1 (t − t 0 ) − µ 2 8k 2 (x 2 − x 2 0 )}e ikx + ǫ A 3 64k 3 e 3ikx ] +c.c.,
The first step is achieved by setting
∂φ ∂t 0 = 0, with t 0 = t. (3.10)
This is the condition to get the envelope that has the common tangent curve with S x 0 t 0 along x-direction. Thus we have the equation for A(x 0 , t);
∂ t A = µ 1 ǫA + O(ǫ 2 ). (3.11)
Then the envelope E 1 is represented by
φ E 1 (x, t; x 0 ) ≡ φ(x, t; x 0 , t 0 = t), = [A(x 0 , t)e ikx − ǫA(x 0 , t) µ 2 8k 2 (x 2 − x 2 0 )e ikx + ǫ A(x 0 , t) 3 64k 3 e 3ikx ] +c.c.. (3.12)
The envelope E of the family of the surfaces given by Eq. (3.12) is obtained as follows;
∂φ E 1 ∂x 0 = 0, with x 0 = x, (3.13) which leads to ∂ x A(x, t) = −µ 2 ǫA x 4k 2 + O(ǫ 2 ). (3.14)
Here we have utilized the fact that ∂ x A is O(ǫ). Differentiating this equation with respect to x, we get
∂ 2 x A(x, t) = −ǫ µ 2 4k 2 A + O(ǫ 2 ). (3.15)
Here we note again that ∂ x A is O(ǫ).
Now combining Eq.'s (3.10) and (3.14), we have
∂ t A = 4k 2 ∂ 2 x A + ǫA(1 − 3|A| 2 ), (3.16) up to O(ǫ 2 ). The envelope is represented by φ E (x, t) ≡ φ E 1 (x, t; x 0 = x), = A(x, t)e ikx + ǫ A(x, t) 3 64k 4 e 3ikx + c.c.. (3.17)
A comment is in order here: To derive Eq.(3.15), we started from the first-order differential equation Eq.(3.13) with respect to x 0 . One could start from the second-order differential equation as is done in [9];
∂ 2 φ ∂x 2 0 x 0 =x = 0,(3.∂φ E ∂t t=t 0 = ∂φ(t, t 0 ) ∂t t=t 0 + ∂φ(t, t 0 ) ∂t 0 t=t 0 = ∂φ(t, t 0 ) ∂t t=t 0 ,(3.19)
on account of Eq.(3.10). Similary, for ∀x = x 0 , one can easily verify that
∂ 2 φ E ∂x 2 x=x 0 = ∂ 2 φ ∂x 2 x=x 0 +O(ǫ 2 ),(3.20)
on account of Eq.'s (3.12), (3.13) and (3.18). For instance,
∂ 2 φ E /∂x∂x 0 | x=x 0 = O(ǫ 2 ).
Damped Kuramoto-Shivashinsky equation
In this section, we deal with the (one-dimensional) damped Kuramoto-Shivashinsky equation, which is given byL
1 u = ǫu − u∂ x u,(4.1)
whereL 1 is defined in Eq.(3.1). This equation is not examined by the Illinois group.
By scaling u as
u = √ ǫφ,(4.2)
we have 3L
1 φ = −ǫ 1/2 φ∂ x φ + ǫφ. (4.3)
We first try to solve this equation by the perturbation theory with the expansion
φ = φ 0 + ǫ 1/2 φ 1 + ǫφ 1 + .... (4.4)
The equations for φ 0 , φ 1 , φ 2 ... are found to bê
L 1 φ 0 = 0,L 1 φ 1 = −φ 0 ∂ x φ 0 ,L 1 φ 2 = φ 0 − (φ 0 ∂ x φ 1 + φ 1 ∂ x φ 0 ),(4.5)
and so on.
As the 0-th order solution, we take
φ 0 = Ae ikx + c.c,(4.6)
where A is a constant. Then φ 1 is found to be
φ 1 = − i 9k 2 A 2 e 2ikx + c.c.. (4.7)
Thus the equation for φ 2 readŝ
L 1 φ 2 = A(1 − |A| 2 9k 2 )e ikx − A 3 3k 3 e 3ikx + c.c.,(4.8)
which is in a similar form with Eq.(3.6). One easily gets for a special solution to this equation
φ 2 = A(1 − |A| 2 9k 2 ){µ 1 (t − t 0 ) − µ 2 8k 2 (x 2 − x 2 0 )}e ikx − A 3 192k 6 e 3ikx + c.c.. (4.9)
Thus we reach the solution in the perturbation theory up to O(ǫ 3/2 ),
φ(x, t; x 0 , t 0 ) = Ae ikx − iǫ 1/2 A 2 9k 3 e 2ikx + ǫA(1 − |A| 2 9k 2 ){µ 1 (t − t 0 ) − µ 2 8k 2 (x 2 − x 2 0 )}e ikx −ǫ A 3 192k 6 e 3ikx + c.c..
(4.10)
Now we have a family of surfaces S x 0 t 0 in x-t-φ plane represented by φ(x, t; x 0 , t 0 ) with x 0 and t 0 being the parameters. We repeat the procedure of the previous section to obtain the envelope E of the family of the surfaces: We first note that A may depend on x 0 and t 0 , i.e., A = A(x 0 , t 0 ). Then fixing x 0 , we first obtain the envelope of S x 0 t 0 with t 0 being the parameters. The envelope E 1 is obtained by
∂φ ∂t 0 = 0, with t 0 = t,(4.11)
where we have assumed that the tangent curve (the characteristic curve) is along x-direction by setting t 0 = t. The above equation leads to
∂ t A(x 0 , t) = ǫµ 1 A(1 − |A| 2 9k 2 ) + O(ǫ 3/2 ). (4.12)
With A satisfying this equation, the envelope E 1 is represented by
φ E 1 (x, t; x 0 ) ≡ φ(x 0 , t 0 ; x 0 , t 0 = t) = Ae ikx − iǫ 1/2 A 2 9k 3 e 2ikx − ǫA(1 − |A| 2 9k 2 ) µ 2 8k 2 (x 2 − x 2 0 )e ikx −ǫ A 3 192k 6 e 3ikx + c.c.. (4.13)
This function can be regarded as representing a family of surfaces with x 0 being the parameter. The envelope E of this family of surfaces are obtained by setting
∂φ E 1 ∂x 0 = 0, with x 0 = x,(4.14)
which leads to
∂ x A(x, t) = −ǫ µ 2 4k 2 A(1 − |A| 2 9k 2 )x + O(ǫ 3/2 ). (4.15)
Here we have utilized the fact that ∂ x A ∼ O(ǫ). Further differentiating with respect to x, one has
∂ 2 x A = −ǫ µ 2 4k 2 A(1 − |A| 2 9k 2 ) + O(ǫ 3/2 ). (4.16)
Combining Eq.'s (4.12) and (4.16), one sees that the amplitude satisfies equation
(∂ t − 4k 2 ∂ 2 x )A = ǫA(1 − |A| 2 9k 2 ), (4.17) up to O(ǫ 2 ).
With this amplitude, the envelope E is given by
φ E (x, t) ≡ φ(x, t; x 0 = x), = Ae ikx − iǫ 1/2 A 2 9k 2 e 2ikx − ǫ A 3 192k 6 e 3ikx + c.c.. (4.18)
A couple of comments are in order here:
(1) Eq.(4.16) could be obtained as an exact relation by imposing the condition that
∂ 2 φ ∂x 2 0 x 0 =x = 0. (4.19)
We must note, however, that the geometrical meaning of this condition is not clear. In contrast, in our formulation, the second derivative is evaluated to be
∂ 2 φ ∂x 2 0 x 0 =x = O(ǫ 2 ).
(4.20)
(2) As was done in the preceding section, one can easily show that φ E (x, t) satisfies the original equation Eq.(4.1) up to O(ǫ 2 ) but uniformly in the global domain.
Two-dimensional Swift-Hohenberg equation
In this section, we deal with the two-dimensional Swift-Hohenberg equation given bŷ
L 2 u = ǫu − u 3 ,L 2 = ∂ t + (∂ 2 x + ∂ 2 y + k 2 ) 2 . (5.1) Scaling u as u = √ ǫφ, (5.2) we haveL 2 φ = ǫ(φ − φ 3 ). (5.3)
We first solve this equation in the naive perturbation theory expanding φ = φ 0 +ǫφ 1 +ǫ 2 φ 2 +...;
L 2 φ 0 = 0,L 2 φ n = φ n−1 − φ 3 n−1 , (n = 1, 2, ...). (5.4)
If we assume the roll solution along y-axis for the zero-th order equation,
φ 0 = Ae ikx + c.c.,(5.5)
then the solution up to O(ǫ 2 ) is found to be [10] φ(x, t;
x 0 , t 0 ) = A + ǫ{µ 1 (t − t 0 ) − µ 2 x 2 − x 2 0 8k 2 + µ 3 xy 2 − x 0 y 2 0 8ik + µ 4 y 4 − y 4 0 4! }A e ikx +c.c., with A ≡ A(1 − 3|A| 2 ), (5.6)
where i=1∼4 µ i = 1 and x 0 , y 0 and t 0 are arbitrary constants. Here we have omitted the terms that do not give rise to secular terms.
Now one may regard that we have a family of "surfaces" in the x-y-t-φ space with x 0 , y 0 and t 0 being the parameters. Let us obtain the envelope of this "surfaces" in the three steps by noting that A may be dependent on x 0 , y 0 and t 0 : First we regard that only t 0 is the parameter of the "surfaces" with both x 0 and y 0 fixed. The condition reads
∂φ ∂t 0 = 0, t 0 = t. (5.7)
This condition may be also regarded as the one for the envelope curve of a family of curves in t-φ plane with x and y being fixed. The condition leads to
∂ t A(x 0 , y 0 , t) = ǫµ 1 A + O(ǫ 2 ). (5.8)
Inserting this solution, we have the envelope
φ E 1 (x, y, t; x 0 , y 0 ) ≡ φ(x, y, t; x 0 , y 0 , t 0 = t) = A(x 0 , y 0 , t) + ǫ{−µ 2 x 2 − x 2 0 8k 2 + µ 3 xy 2 − x 0 y 2 0 8ik + µ 4 y 4 − y 4 0 4! }A e ikx
+c.c., (5.9) which we may regard as a family of curves in x-φ plane with x 0 being the parameter where y and y 0 are fixed. The envelope E 2 for this family of curves is given by setting
∂φ E 1 ∂x 0 = 0, with x 0 = x, (5.10) which leads to ∂ x A(x, y 0 , t) = −ǫ( µ 2 4k 2 x − µ 3 8ik y 2 0 )A + (ǫ 2 ). (5.11)
Accordingly, the envelope is represented by (5.12) which is regarded as representing a family of curves in y-φ plane. The envelope of φ E 2 is obtained as usual by setting ∂φ E 2 ∂y 0 = 0, with y 0 = y, (5.13) which leads to
φ E 2 (x, y, t; y 0 ) ≡ φ E 1 (x, y, t; x 0 = x, y 0 ) = A(x, y 0 , t)e ikx + ǫ{µ 3 x y 2 − y 2 0 8ik + µ 4 y 4 − y 4 0 4! }Ae ikx + c.c.,∂ y A(x, y, t) = ǫ( µ 3 4ik xy + µ 4 3! y 3 )A + O(ǫ 2 ). (5.14)
Differentiating Eq.'s (5.11) and (5.14), we have
∂ 2 x A(x, y, t) = −ǫ µ 2 4k 2 A + O(ǫ 2 ), (5.15) ∂ x ∂ 2 y A(x, y, t) = −ǫ µ 3 4ik A + O(ǫ 2 ), (5.16) ∂ 4 y A(x, y, t) = ǫµ 4 A + O(ǫ 2 ), (5.17)
where we have utilized the fact that
∂ x A ∼ O(ǫ), ∂ y A ∼ O(ǫ). (5.18)
Combining these equation together with Eq.(5.8), we finally reach
∂ t A − 4k 2 ∂ 2 x A + 4ik∂ x ∂ 2 y A + ∂ 4 y A = ǫA(1 − 3|A| 2 ), (5.19) up to O(ǫ 2 ).
With this amplitude, the envelope is given by
φ E (x, y, t) = φ E 1 (x, y, t; y 0 = y), = A(x, y, t)e ikx + c.c.. (5.20)
A few comments are in order here:
(1) By imposing that the any order of the differentiation of φ with respect to x 0 and y 0 should vanish, one could get Eq.'s (5.15-17) as exact relations [10], although the geometrical meaning of these conditions are unclear. In our formulation, instead, the following are derived as approximate relations
∂ 2 φ E1 ∂ 2 x 0 x 0 =x = O(ǫ 2 ), ∂ 3 φ E2 ∂x∂ 2 y 0 y 0 =y = O(ǫ 2 ), ∂ 4 φ E2 ∂ 4 y 0 y 0 =y = O(ǫ 2 ). (5.21)
(2) It can be shown that φ E satisfies Eq.(5.1) up to O(ǫ 2 ) but uniformly in the global domain owing to the above relations together with Eq. (5.8).
A brief summary and concluding remarks
We have shown in the present paper that the RG method for global analysis can be formulated for partial differential equations, too. An interesting equation which is treated by the Illinois group [8] but left untouched in the text is Barenblatt's equation. A complete global solution to this equation is not given in the RG method [8] but only the anomalous exponent for the long-time behavior is obtained. For completeness, we show that the same result for the anomalous exponent can be obatined in our envelope theory in Appendix B. Thus together with our previous paper [11] where ordinary differential equations are discussed, we have shown that almost all the classes of problems treated in the RG method by the Illinois group [8,9,10] are nicely formulated on the basis of the classical theory of envelopes without recourse to the notion of the renormalization group. Actually, this is natural because the resulting equations indeed describe the amplitudes of the nonlinear waves as given by the solutions of the nonlinear equations.
It is already indicated [8,9] that the renormalizability is equivalent to the solvability of equations [16]. It would be interesting to apply the theory developed here to systems of equations 4 and see possible geometrical meaning of the solvability condition.
In deriving the whole envelopes of local solutions, we have taken a multi-step approach in sections 3, 4 and 5. It is interesting that a kind of multi-step approach is also proposed for improving the effective potentials in quantum field theories with multi-scales; see the paper by Ford [5]. It may imply that our approach that identifies the RG equation with the envelope equation naturally leads to the effective potentials as given by Ford when applied to quantum field theories. [18] Appendix A In this appendix, we give a short review of the classical theory of envelopes [12]. We shall first consider envelopes in three-dimensional space. i.e., envelope surfaces. Then the extension to higher dimensional cases will be briefly described. 5 Let {S τ } τ be a family of surfaces parametrized by τ in the x-y-u space; here S τ is represented by the equation Φ(x, y, u; τ ) = 0.
(A.1)
We suppose that {S τ } τ has the envelope E, which is represented by the equation
Ψ(x, y, u) = 0. (A.2)
The problem is to obtain Ψ(x, y, u) from Φ(x, y, u; τ ).
Let E and S τ has a common tangent plane on a common curve C τ ; with τ being varied, {C τ } τ forms E. C τ is called a characteristic curve. The necessary condition for S τ to have an envelope E is given as follows. Let C τ be represented by x = x(σ, τ ), y = y(σ, τ ), u = u(σ, τ ), with σ being a parameter. We assume that rank
x σ y σ u σ x τ y τ u τ = 2, (A. 3) in order for {C τ } τ to give a non-degenerate surface E. Now, since C τ is on S τ , Φ(x(σ, τ ), y(σ, τ ), u(σ, τ ); τ ) = 0. Differentiating this equation with respect to τ , one has
Φ x x τ + Φ y y τ + Φ u u τ + Φ τ = 0, (A.4)
where Φ x ≡ ∂Φ/∂x and so on. On the other hand, since (x τ , y τ , u τ ) ≡ t τ is a tangent vector both of E and S τ , and (Φ x , Φ y , Φ u ) ≡ n τ is a normal vector of S τ , n τ · t τ = Φ x x τ + Φ y y τ + Φ u u τ = 0. Accordingly, the characteristic curve is given by the conditions;
Ψ(x, y, u) = 0 and τ (x, y, u) = const. (A.8) 5 In [11], a review is given on how to construct envelope curves.
Note that the second equation gives a constraint on x, y and u.
We remark that the sufficient conditions for the existence of an envelope is supplemented by [12] Φ τ τ = 0.
(A.9)
When S τ is given by u = ϕ(x, y; τ ), the condition Eq. (A.6) is reduced to ∂ϕ(x, y; τ ) ∂τ = 0, (A.10)
which gives τ as a function of x and y. Thus we get for the envelope
u = ϕ E (x, y) = ϕ(x, y; τ (x, y)). (A.11)
The characteristic curve is accordingly given by The characteristic curve C τ is given by
u = ϕ E (x, y) τ (x,y)=const.x = τ = constant ≡ x 0 , and u = exp(x 0 y) + exp(−x 0 ). (A.16)
We remark that ∂ 2 ϕ/∂τ 2 = 0 provided that y = 0.
The theory can be extended to the envelope of a family of hyper-surfaces {S τ } τ in the (n+ 1)-dimensional space R n+1 . Let {S τ } τ be represented by the equation Φ(x 1 , x 2 , · · · , x n , u; τ ) = 0 with a parameter set τ = (τ 1 , τ 2 , · · · , τ n−1 ). Then the function representing the envelope is given by eliminating parameters τ i (i = 1, 2, · · · , n − 1) from Φ(x 1 , x 2 , · · · , x n , u; τ ) = 0, ∂Φ ∂τ i = 0, (i = 1, 2, · · · , n − 1). (A.17)
A few comments are in order here:
(i) The envelope of a family of surfaces has usually an improved global nature compared with the surfaces in the family. So it is natural that the theory of envelopes may have some power for global analysis.
(ii) It should be stressed here that Eq.(A.6), which we call the envelope equation, has the same form as renormalization group (RG) equations. Rather, it should be said conversely; RG equations in general are envelope equations [11]. This is the reason why RG equations "improve" things; one should note that the improvement by an RG equation usually means that a function with a better global nature is constructed from functions with a local nature only valid around the renormalization point. from which τ is given as a function of x and y, namely, τ = τ (x, y). Then one sees that the envelope function u = ϕ E (x, y; [A]) = ϕ(x, y, A(τ (x, y)), τ (x, y)) is the general solution to F (x, y, u, p, q) = 0.
where
G(x, t) = 1 √ 2πt e − x 2 2t (B.4)
is the Green's function. 6 We expand u as u(x, t) = u 0 (x, t) + ǫu 1 (x, t) + · · · . (B.5)
As an initial condition, we take
u(x, 0) = u 0 (x, 0) = m 0 √ 2πl 2 e − x 2 2l 2 , (B.6)
where m 0 and l are parameters. Then one finds [8] that
u(x, t) = m 0 √ 2πt e − x 2 2t {1 − ǫ √ 2πe ln t l 2 } + O(ǫ 2 ) + O(l 2 /t, ǫ), (B.7)
where we have retained in u 1 only the term which may contribute to the anomalous dimension.
We renormalize m 0 at t = t 0 as follows; Here we have made the t 0 dependence of u explicit.
We now leave the line of argument of Goldenfeld et al [8]. We have a family of surfaces represented by u(x, t; t 0 ); the surfaces are parametrized by t 0 . Let us obtain the envelope E of this family of surfaces. We can suppose that the tangent curve C t 0 (the characteristic curve) is along t = t 0 : t = t 0 , u = u(x, t 0 ; t 0 ).
(B.11) 6 If the Green's function is defined as usual by
(∂ t − 1 2 ∂ 2 x )G(x, t) = δ(x)δ(t),
G(x, t) = θ(t)G(x, t),
where θ(t) is the step function. hence the anomalous exponent reads α = ǫ/ √ 2πe, which of course coincides with the result of Goldenfeld et al. 7
necessary condition for {S τ } to have an envelope E. Thus, solving Eq. (A.6) for τ = τ (x, y, u), one finally gets Ψ(x, y, u) = Φ(x, y, u; τ (x, y, u)) = 0.(A.7)
u
= ϕ(x, y, τ ) ≡ exp(τ y)(1 + y(x − τ )) + exp(−x). (A.13) Then the equation ∂ϕ/∂τ = 0 gives τ = x, (A.14) except at y = 0. Thus the envelope is given by u = exp(xy) + exp(−x). (A.15)
(
iii) Let u = ϕ(x, y; σ, τ ) be a complete solution of a PDEF (x, y, u, p, q) = 0, (p ≡ ∂u/∂x, q ≡ ∂u/∂y), (A.18)where σ and τ are constant. Assuming a functional dependence of σ on τ , i.e., σ = A(τ ), one can obtain the envelope as follows;
+ O(ǫ 2 ) + O(l 2 /t, ǫ). (B.10)
a constant number. Thus the envelope is given byu E (x, t) = u(x, t; t) =mt −(1/2+α) F (x 2 /t), (B.15)
18 )
18which gives Eq.(3.15) as an exact relation without the remainder of O(ǫ 2 ). We remark that the geometrical meaning of Eq.(3.18) is not clear in contrast to Eq.(3.13). In our formulation, Eq.(3.18) is derived as an approximate relation, as is shown below. Now one may ask the question as to the relation of φ E (x, t) given in Eq. (3.17) and the original equation Eq.(3.1): The answer is that φ E (x, t) satisfies Eq.(3.1) up to O(ǫ 2 ) uniformly in the global domain due to the very envelope conditions Eq.(3.10) and Eq.(3.13). In fact, for ∀t = t 0 ,
See Appendix A and[11] for classical theory of envelopes.
One might have added another arbitrary function G 0 (ξ)of ξ to u 2 . However, the effect of G 0 could be renormalized away to F 0 .
If one scales u as u = ǫφ, one will get a different equation from that given below.
The RG method or the envelope theory developed in[11] and here is of course applicable to systems of ODE's[17].
It should be mentioned that u E (x, t) thus obtained does not satisfy the Barlenblatt's equation even in the order of ǫ; this is because the Gaussian form for F (x 2 /t) is not correct. To obtain the precise form of F (x 2 /t), one may insert u E (x, t) into the Barlenblatt's equation and solve F (ξ).
AcknowledgementsThe author acknowledges G. C. Paquette, who gave lectures on the RG method at Ryukoku University in June 1994. He is also grateful to R. Kobayashi, who organized the seminar where these lectures were given. He thanks Y. Morita, the conversation with whom motivated the author to think about the mathematical structure of the RG method seriously. The author is grateful to M. Yamaguti for his interest in this work, encouragement and useful comments on Cauchy's problem and Burgers equation. He also acknowledges Y. Oono for his correspondence and sending the preprint[10]prior to publication. The author thanks J. Matsukidaira for discussions on applications of the envelope theory to systems of ODE's. A part of the present work was completed while the author stayed at Brookhaven Natinal Laboratory(BNL) as a summer-program visitor in August 3-14, 1995. He thanks BNL and especially R. Pisarski for their hospitality.Appendix BIn this Appendix, we examine a non-linear partial differential equation which has an anomalous exponent in the long-time behavior of its solutions;This is called Barlenblatt's equation.[13,8]This equation is a nonlinear PDE with the step function in r.h.s. Without this term (ǫ = 0), the equation is a simple diffusion equation and has a self-similar form after a long time; u(x, t) ∼ t −1/2 f (x 2 /t). However, with ǫ = 0, the long-time behavior is found to beThe anomalous exponent α can be interpreted as an anomalous dimension[8]. The appearance of the anomalous dimension is due to the fact that the scale l with a spatial dimension which characterizes the initial distribution of u(x, 0) can not be neglected even in the longtime limit t → ∞ in contrast to the case of the simple diffusion equation[13]. We are interested in determining the anomalous dimension α in the perturbation theory.To solve the problem, we shall follow[8]for a while. First we convert the equation to an integral equation;u(x, t) =
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[
"SUBCANONICAL POINTS ON ALGEBRAIC CURVES",
"SUBCANONICAL POINTS ON ALGEBRAIC CURVES"
] |
[
"Evan Merrill Bullock "
] |
[] |
[] |
If C is a smooth, complete algebraic curve of genus g ≥ 2 over the complex numbers, a point p of C is subcanonical if K C ∼ = O C (2g − 2)p . We study the locus G g ⊆ M g,1 of pointed curves (C, p) where p is a subcanonical point of C. Subcanonical points are Weierstrass points, and we study their associated Weierstrass gap sequences. In particular, we find the Weierstrass gap sequence at a general point of each component of G g and construct subcanonical points with other gap sequences as ramification points of certain cyclic covers and describe all possible gap sequences for g ≤ 6.
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10.1090/s0002-9947-2012-05506-1
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[
"https://arxiv.org/pdf/1002.2984v2.pdf"
] | 85,561,740 |
1002.2984
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90c56c5eec0d565fc586af62322e81a73c461d16
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SUBCANONICAL POINTS ON ALGEBRAIC CURVES
Evan Merrill Bullock
SUBCANONICAL POINTS ON ALGEBRAIC CURVES
If C is a smooth, complete algebraic curve of genus g ≥ 2 over the complex numbers, a point p of C is subcanonical if K C ∼ = O C (2g − 2)p . We study the locus G g ⊆ M g,1 of pointed curves (C, p) where p is a subcanonical point of C. Subcanonical points are Weierstrass points, and we study their associated Weierstrass gap sequences. In particular, we find the Weierstrass gap sequence at a general point of each component of G g and construct subcanonical points with other gap sequences as ramification points of certain cyclic covers and describe all possible gap sequences for g ≤ 6.
Definition. If C is a smooth curve of genus g ≥ 2, we say that a point p of C is a subcanonical point if K C ∼ = O C (2g − 2)p .
Equivalently, p is subcanonical if there exists a holomorphic differential on C which vanishes at p to order 2g − 2 and nowhere else. Let G g ⊆ M g,1 be the locus of pointed curves (C, p) where p is a subcanonical point of C. It follows from results in [KZ03] that in genus g ≥ 4, G g has three components G hyp g , G odd g and G even g , corresponding to hyperelliptic curves and to the cases where the associated theta characteristic O C (g − 1)p is odd or even. In Section 2, we determine the set of Weierstrass gaps associated to the general point of each of these components:
Theorem. Let g ≥ 4, then
(1) a general point of G hyp g has Weierstrass gaps {1, 3, 5, . . . , 2g − 5, 2g − 3, 2g − 1}, (2) a general point of G odd g has Weierstrass gaps {1, 2, 3, . . . , g − 2, g − 1, 2g − 1}, and (3) a general point of G even g has Weierstrass gaps {1, 2, 3, . . . , g − 2, g, 2g − 1}.
In Section 4, we show that for g ≤ 5, these are the only possible Weierstrass gap sequences for subcanonical points, and in Section 3 we construct for g ≥ 6 loci within G odd g and G even g consisting of subcanonical points with various other gap sequences as branch points of cyclic covers.
1.2. Weierstrass points. Let C be a smooth curve of genus g. If p is a point of C, the vanishing sequence (of the complete canonical series |K C |) at p is the sequence 0 = a K C 0 (p) < a K C 1 (p) < . . . < a K C g−1 (p) ≤ 2g − 2 of orders of vanishing of the holomorphic differentials at p, so that a K C k (p) = v p (ω) : ω ∈ H 0 (C, K C ) . Equivalently, we may consider the sequence 0 = α K C 0 (p) ≤ α K C 1 (p) ≤ . . . ≤ α K C g−1 (p) ≤ g− 1 defined by α K C k (p) = a K C k (p) − k, which we call the ramification sequence (of the canonical series) at p.
Historically, Weierstrass points were defined using the Weierstrass gap sequence at p. This consists of, in increasing order, the positive integers n for which there does not exist a meromorphic function on C with pole divisor np. One can check, using the Riemann-Roch theorem, that the set of Weierstrass gaps is just {a K C i (p) + 1}. (One direction is easy: if there were a meromorphic function f with pole divisor (a i + 1)p and a holomorphic differential ω with v p (ω) = a i , then the product f ω would be a meromorphic differential with only a single simple pole at p.)
One advantage of thinking in terms of the gap sequence is that the set of nongaps N − {a K C i (p) + 1}, the set of positive numbers n for which there does exist a meromorphic function on C with pole divisor np, forms a semigroup under addition (multiplying functions adds pole orders). This provides a necessary (but not sufficient-for example, see [EH87]) condition for a sequence of numbers 0 = a 0 < a 1 < . . . < a g−1 ≤ 2g − 2 to be the vanishing sequence for the canonical series at some point of some curve.
At a general point of our curve C, there is no meromorphic function having a pole only at p of order less than g + 1, or equivalently the vanishing sequence at p is the smallest possible vanishing sequence, 0, 1, . . . , g − 1, and correspondingly the ramification sequence is just 0, 0, . . . , 0. We call a point p with a larger ramification sequence (α K C k (p)) a Weierstrass point of weight w(α) = g−1 k=0 α k . A subcanonical point is then a Weierstrass point with α g−1 = g − 1. For example, a hyperelliptic curve C of genus g has a degree 2 map π : C → P 1 . Each of the 2g + 2 ramification points of π is in fact a Weierstrass point of C with the maximal possible vanishing sequence 0, 2, 4, . . . , 2g −2 and ramification sequence 0, 1, 2, . . . , g − 1, and is thus an example of a subcanonical point. Our main goal will be to try to determine what ramification sequences a subcanonical point can have and to study the stratification of G g by these sequences.
Every smooth curve has only finitely many Weierstrass points, the sum of whose weights is g(g 2 − 1). On a general curve, there are g(g 2 − 1) distinct Weierstrass points, each of weight one (cf. [ACGH85] I.E). For a given ramification sequence α = (α k ), every component of the locally closed subset C α = {(C, p)| α K C (p) = α} ⊆ M g,1 has codimension at most the weight of α. We say that (C, p) is a dimensionally proper Weierstrass point if C α(p) has codimension exactly w(p) in a neighborhood of (C, p). It is known, for example (cf. [EH87] for this and some slightly stronger results), that if α is any such sequence 0 = α 0 ≤ α 1 ≤ . . . ≤ α g−1 ≤ g − 1 with weight at most g/2, then C α contains dimensionally proper points.
1.3. Theta characteristics. A theta characteristic or spin structure on a smooth curve C of genus g is a line bundle L of degree g − 1 on C satisfying L ⊗ L ∼ = K C . In [Cor89], Cornalba constructs a compactified moduli space of curves with theta characteristics. The parity of the number h 0 (C, L) is constant in families of curves with theta characteristics, and a theta characteristic is defined to be odd or even as this number is.
Any smooth curve has exactly 2 g−1 (2 g −1) odd theta characteristics and 2 g−1 (2 g +1) even theta characteristics. Since a subcanonical point p ∈ C is a point for which O C ((2g − 2)p) ∼ = K C , the line bundle L = O C ((g − 1)p) is a theta characteristic, and we call the subcanonical point odd or even as this associated theta characteristic is.
The parity of a subcanonical point p ∈ C is in fact determined by its vanishing sequence: if p is subcanonical, then h 0 C, O C ((g − 1)p) = h 0 C, K C (−(g − 1)p) is the dimension of the space of holomorphic differentials vanishing to order at least g − 1 at p, which is the number of a K C i (p) which are at least g − 1. This allows us to compute, for example that if p is a ramification point of a hyperelliptic curve C, then h 0 (C, L) = g+1 2 , so p is an odd subcanonical point if g ≡ 1, 2 (mod 4) and even if g ≡ 0, 3 (mod 4).
1.4. Previous results. In [KZ03], Kontsevich and Zorich studied the moduli spaces H g (k 1 , k 2 , . . . , k n ) of pairs (C, ω) where C is a compact Riemann surface of genus g and ω is a holomorphic differential on C with exactly n zeroes, with multiplicities k 1 , . . . , k n , where k 1 + . . . + k n = 2g − 2. They showed that these spaces are smooth (as orbifolds) of dimension 2g + n − 1 and classified their connected components. The central case they considered was the case of H g (2g − 2), where the holomorphic differential ω has only a single zero of order 2g − 2 at a point p which is then a subcanonical point of C. They proved that for g ≥ 4, the space H g (2g − 2) has three disjoint components, each of dimension 2g, namely the locus where C is hyperelliptic, and the loci where C is non-hyperelliptic and p is even and odd. It follows immediately that G g ⊆ M g,1 has exactly three irreducible components, each of dimension 2g − 1, namely the hyperelliptic and the non-hyperelliptic even and odd subcanonical points. (The dimension is one lower than that in [KZ03] because of the freedom to multiply ω by a non-zero scalar without changing the point.)
Acknowledgements
This paper is based on my Ph.D. dissertation at Harvard University. I would like to thank my advisor, Joe Harris, for the very great deal of help he provided on this project. This work was partially supported by an NSF Graduate Fellowship.
General subcanonical points
It follows from the results of [KZ03] that for g ≥ 4, the locus G g ⊆ M g,1 consisting of pairs (C, p) where p is a subcanonical point of C, has three disjoint irreducible components:
(1) the locus G hyp g of pairs (C, p) where C is a hyperelliptic curve of genus g and p is a ramification point of the hyperelliptic double cover, (2) the locus G odd g of pairs (C, p) where C is a non-hyperelliptic curve of genus g and p is a subcanonical point such that (g − 1)p is an odd theta characteristic, and (3) the locus G even g of pairs (C, p) where C is a non-hyperelliptic curve of genus g and p is a subcanonical point such that (g −1)p is an even theta characteristic. In this section, we will prove the following theorem, which essentially states that a general subcanonical point (C, p) in each component of G g has the smallest ramification sequence α K C (p) possible.
Theorem 2.1. Let g ≥ 4, then
(1) a general point of G hyp g has ramification sequence 0, 1, 2, . . . , g − 3, g − 2, g − 1, (2) a general point of G odd g has ramification sequence 0, 0, 0, . . . , 0, 0, g − 1, and (3) a general point of G even g has ramification sequence 0, 0, 0, . . . , 0, 1, g − 1.
In the hyperelliptic case, every ramification point of the hyperelliptic double cover has this ramification sequence and these 2g + 2 ramification points are all the Weierstrass points on the curve. This is a standard result (see Chapter I of [ACGH85]) whose statement we include here for completeness. It will also follow as a special case from our work on cyclic covers in Section 3.
As for the odd and even cases, the above ramification sequences are as small as possible: if p ∈ C is any subcanonical point on a curve of genus g, then certainly
α K C (p) ≥ (0, . . . , 0, 0, g − 1)
and if p is an even subcanonical point, then we must have
h 0 (C, O((g − 1)p)) ≥ 2; it follows that a K C g−2 (p) ≥ g − 1, so that α K C (p) ≥ (0, . . . , 0, 1, g − 1).
This means, by the upper semi-continuity of the ramification sequence, that in order to prove Theorem 2.1 we need only show that there exist points of G odd g and G even g having the desired ramification sequences. The basic approach of our construction will be that used in [EH87] to construct Weierstrass points of low weight having prescribed ramification sequences: using limit linear series (cf. [EH86]), we will begin by describing possible limit "subcanonical points" with the desired ramification sequences on certain reducible curves, and then show that they smooth to points with the same ramification sequences on nearby smooth curves.
We recall briefly the definition of a limit linear series in the case we will need. Suppose that X = C ∪ E is the union of a smooth curve C of genus g − h and a smooth curve E of genus h, meeting at a single node q. Then a crude limit g r d on X consists of a g r d on C, i.e. a pair L C = (L C , V C ) where L C is a line bundle of degree d on C and V C is a dimension r + 1 subspace of H 0 (C, L C ), together with a g r d on E, L E = (L E , V E ), satisfying the compatibility condition that for 0 ≤ i ≤ r,
a L C i (q) + a L E r−i (q) ≥ d.
Given a one-parameter family of curves whose special fiber is X and whose general fiber is a smooth curve of genus g, along with a family of g r d 's on the smooth fibers, there is a well-defined crude limit g r d on X. Strict inequalities in the compatibility condition arise when ramification points of the g r d 's approach the node of X; this complication can be avoided by blowing up the family at the node and base-changing. A (refined) limit g r d on X is a crude limit g r d in which the above inequalities are all equalities.
The following lemma provides a partial description of the crude and refined limit canonical series on X that have a "subcanonical point" on E (i.e. a point p with a L E g−1 (p) = 2g − 2) in the case where E has genus 1. Lemma 2.2. Let X = C ∪ E be the union of a smooth curve C of genus g − 1 and a smooth curve E of genus 1, meeting at a single node q. Assume g ≥ 3.
(1) Let L = {(L C , V C ), (L E , V E )} be a crude limit g g−1 2g−2 on X. (a) Suppose that there is a point p = q of E such that α L E g−1 (p) = g − 1.
Then
O E (p − q) ∈ Pic 0 (E) is (2g − 2)-torsion, q is a subcanonical point of C (i.e. α K C g−2 (q) = g − 2), and α L E 0 (p) = 0. (b) If moreover O E (p − q) has order exactly 2g − 2 in Pic 0 (E), then α K C i−1 (q) ≥ α L E i (p) for all 1 ≤ i ≤ g − 2. (c) If instead O E (p − q) has order exactly g − 1, then α K C i−1 (q) ≥ α L E i (p)
for all but possibly one value of i satisfying 1 ≤ i ≤ g − 2, for which it must be the case that
g − 1 = a L E i (p) = α L E i (p) + i and α K C i−1 (q) = α L E i (p) − 1.
(2) Suppose conversely that q is a subcanonical point of C with ramification sequence
α = (α 0 , α 1 , . . . , α g−3 , g − 2)
and that p is a point on E such that O E (p − q) has order 2g − 2 or g − 1.
Then on X there is a unique (refined) limit
g g−1 2g−2 , L = {(L C , V C ), (L E , V E )} satisfying the ramification condition α L E (p) ≥ (0, α 0 , α 1 , . . . , α g−3 , g − 1) at p. Then α L E i+1 (p) = α i for 0 ≤ i ≤ g − 3,
except in the special case where O E (p−q) has order g−1, α i = g−3−i, and either i = g−3 or α i+1 > α i ; in this case, which may only occur for at most one value of i, instead α L E i+1 (p) = α i +1.
Proof.
We begin by noting that if L = {(L C , V C ), (L E , V E )} is a crude limit g g−1
2g−2 on X, then by "Clifford's Theorem" (Theorem 4.1 of [EH86]) we know that L C ∼ = K C (2q) and that L E ∼ = K E (2(g − 1)q) ∼ = O E ((2g − 2)q). This completely determines the C-aspect of L: by Riemann-Roch we know that h 0 (C, K C (2q)) = g, so that V C = H 0 (C, K C (2q)), of which H 0 (C, K C ) is a codimension one subspace. We see then that a L C 0 (q) = 0 and a L C i (q) = a K C i−1 (q) + 2 for 1 ≤ i ≤ g − 1, or equivalently that α L C 0 (q) = 0 and α L C i (q) = α K C i−1 (q) + 1 for 1 ≤ i ≤ g − 1. We thus need only consider the E-aspects of these linear series. Now, to prove the first part of the lemma, we see that
α L E g−1 (p) = g − 1 means a L E g−1 (p) = 2g − 2, so there exists some σ ∈ V E ⊆ H 0 (E, L E ) vanishing at p to order 2g − 2, but L E ∼ = O E ((2g − 2)q), so (2g − 2)p ∼ (2g − 2)q on E and O E (p − q) is (2g − 2)
-torsion as desired. Moreover, as σ does not vanish at q, we have a L E 0 (q) = 0, whence by the basic inequality for crude limit series
a L E 0 (q) + a L C g−1 (q) ≥ 2g − 2,
we conclude that a K C g−2 (q) = a L C g−1 (q) − 2 ≥ 2g − 2 − 2 = 2(g − 1) − 2 and that q is a subcanonical point of C.
Likewise, if α L E 0 (p) > 0, then every σ ∈ V E would vanish at p so that a L E g−1 (q) < 2g − 2. This, however, would imply that a L C 0 (q) > 0 by the basic inequality, which we know can not be the case. Hence we must have α L E 0 (p) = 0.
To show that if moreover O E (p − q) has order exactly 2g − 2 then α K C i−1 (q) ≥ α L E i (p) for 1 ≤ i ≤ g − 2, we use a simplified version of the proof of Proposition 5.2 of [EH87]. We would like to show that
a L E g−1−i (q) + a L E i (p) ≤ 2g − 3 for 1 ≤ i ≤ g − 2 since then by the basic inequality a L E g−1−i (q) + a L C i (q) ≥ 2g − 2 we would know that a L C i (q) − 1 ≥ a L E i (p) and hence that α K C i−1 (q) = α L C i (q) − 1 ≥ α L E i (p) for 1 ≤ i ≤ g − 2 as desired. To prove our claim, suppose that a L E g−1−i (q) + a L E i (p) ≥ 2g − 2. Let W 1 , W 2 ⊆ V E be the subspaces of V E defined by W 1 = {σ ∈ V E : v q (σ) ≥ a L E g−1−i (q)} and W 2 = {σ ∈ V E : v p (σ) ≥ a L E i (p)}. Then dim W 1 = g − (g − 1 − i) = i + 1 and dim W 2 = g − i,
and as these are subspaces of a g-dimensional vector space, we must have dim W 1 ∩ W 2 ≥ 1. Let σ ∈ W 1 ∩ W 2 be a nonzero element of the intersection. We know that σ has 2g − 2 zeroes, but since
a L E g−1−i (q) + a L E i (p) ≥ 2g − 2 we have accounted for all of these, i.e. we must have that a L E g−1−i (q) + a L E i (p) = 2g − 2 and (2g − 2)q ∼ (σ) = a L E g−1−i (q)q + a L E i (p)p so that −(2g − 2)q + a L E g−1−i (q)q + a L E i (p)p = a L E i (p)(p − q) ∼ 0 and since O E (p − q) has order 2g − 2, we must have a L E i (p) = 0 or a L E i (p) = 2g − 2, so that i = 0 or i = g − 1. This shows that a L E g−1−i (q) + a L E i (p) ≤ 2g − 3 for 1 ≤ i ≤ g − 2 as desired. The case where O E (p − q) has order exactly g − 1 is entirely analogous, except that now (2g − 2)p ∼ (g − 1)p + (g − 1)q, so it is possible that a L E g−1−i (q) + a L E i (p) = 2g − 2 when a L E g−1−i (q) = a L E i (p) = g − 1,
in which case our use of the basic inequality as above only yields
α K C i−1 (q) ≥ α L E i (p) − 1.
Since the a i are increasing, this case may only occur for at most one value of i. This completes the proof of the first part of the lemma.
The second part of the lemma follows more or less directly from Proposition 5.2 of [EH87], which states in this case that given points p and q on an elliptic curve E and given ramification sequences β and γ with
(*) g − 1 ≥ β g−1−j + γ j ≥ g − 2 for j = 0, . . . , g − 1,
there exists at most one g g−1 2g−2 , L = (O E ((2g − 2)q), V ) on E with α L (q) = β and α L (p) = γ, and one exists if and only if
β g−1−j + γ j = g − 1 =⇒ b g−1−j q + c j p ∼ (2g − 2)q, and b g−1−j q + (c j + 1)p ∼ (2g − 2)q =⇒ γ j+1 = γ j , (**)
for each j, where b j = β j + j and c j = γ j + j are the associated vanishing sequences. Now, the ramification sequence at q is completely determined by the refined limit linear series condition and the known ramification sequence of the C-aspect at q: we must have
β = (0, g − 2 − α g−3 , g − 2 − α g−2 , . . . , g − 2 − α 0 , g − 1) so that β j = g − 2 − α g−2−j for j = 1, . . . , g − 2.
On the other hand, we must consider all ramification sequences γ at p satisfying
γ ≥ γ = (0, α 0 , α 1 , . . . , α g−3 , g − 1)
and show that for only one of them does there actually exist a g g−1 2g−2 , L = (O E ((2g − 2)q), V ) with α L (q) = β and α L (p) = γ; we will show that such a γ must satisfy the hypothesis (*) of Proposition 5.2 of [EH87], so uniqueness will then be automatic.
In the case where O E (p − q) has order 2g − 2, we must have γ = γ , since if γ i > γ i = α i−1 were bigger then the first part of the lemma would imply that
α i−1 = α K C i−1 (q) ≥ α L E i (p) = γ i > α i−1 .
In this case, condition (**) is clearly satisfied since bq + cp ∼ (2g − 2)q is impossible except when b and c are 0 and 2g − 2.
In the case where instead O E (p − q) has order g − 1, however, there might be a g g−1 2g−2 with α L (q) = β and α L (p) = γ ≥ γ where γ i > γ i for some i. However, the same argument using the first part of the lemma shows that this can happen for at most one index i and that we must have γ i = γ i + 1 for that index. It also shows that i is completely determined by the α j : it must satisfy
g − 1 = b g−1−i = c i = c i + 1 = α i−1 + i + 1.
When no such i exists, our only possible ramification sequence at p is γ, and Proposition 5.2 of [EH87] tells us that there is a unique g g−1 2g−2 on E with the given ramification sequences at p and q, since the second condition of (**) that
b g−1−i q + (c i + 1)p ∼ (2g − 2)q =⇒ γ i+1 = γ i is vacuous as there is no i which makes b g−1−i = c i + 1 = g − 1.
On the other hand, when there is such an i, we may have two different sequences of numbers, γ and γ , at p which each satisfy the hypothesis (*) of the proposition together with the sequence β at q, where again γ and γ are equal except that γ i = γ i + 1. Of course, in order for γ to actually be a ramification sequence as well, we must have that
γ i + 1 = γ i ≤ γ i+1 = γ i+1
If this is not the case, then γ i+1 = γ i , and hypothesis (**) is satisfied for β and γ.
We are left with the case where there is such an i but γ i+1 ≥ γ i + 1. In this case, β and γ do not satisfy hypothesis (**) since γ i+1 = γ i , but β and γ do satisfy hypothesis (**) since now b g−1−j q + c j p ∼ (2g − 2)q. We have seen in each case then that there is a unique limit g g−1 2g−2 on X satisfying the given ramification condition at p, which completes the proof of the lemma.
2.1. Odd case. We can now prove these exists a point of G odd g which has ramification sequence 0, . . . , 0, g − 1. As in [EH87], the proof is by induction on the genus. We show in Section 4.2 that the result is true in genus 3, which provides the base case for our induction.
Suppose then that the result is known to be true in genus g − 1, so that there exists a subcanonical point q ∈ C on a curve of genus g − 1 with ramification sequence 0, . . . , 0, g − 2. As a Weierstrass point, q is dimensionally proper: we know from
[KZ03] that G odd g−1 has dimension 2(g − 1) − 1 = 3(g − 1) − 2 − (g − 2)
. As in [EH87], we consider the curve of compact type X = C ∪ E consisting of the curve C meeting some elliptic curve E in a node at q. We pick a point p ∈ E so that O E (p − q) ∈ Pic 0 (E) has order 2g − 2. Now, by Lemma 2.2, there is a unique
limit g g−1 2g−2 , L = ((L C , V C ), (L E , V E )
) on X with ramification sequence 0, . . . , 0, g − 1 at p. Moreover, there is no such limit g g−1 2g−2 satisfying the ramification condition at a general point p ∈ E, since by the first part of Lemma 2.
2, O E (p − q) must be (2g − 2)-torsion.
Thus, as in Proposition 5.1 of [EH87] we may apply Corollary 3.7 of [EH86] to conclude that L may be smoothed, maintaining the ramification condition α ≥ (0, . . . , 0, g − 1) near p. We briefly sketch the Eisenbud-Harris argument:
Let ( X → B, Bp − → X) be a miniversal deformation space of the pointed curve (X, p), with discriminant hypersurface ∆ ⊂ B. Then dim B = 3g − 2. By Theorem 3.3 of [EH86], there is a scheme G = G g−1
2g−2 X/ B; (p, (0, 0, . . . , 0, g − 1)) −→B whose fiber over each point z ∈B parametrizes the (refined) limit g g−1 2g−2 's L z on X z satisfying the ramification condition
α Lz (p(z)) ≥ (0, 0, . . . , 0, g − 1)
at the marked pointp(z) of X z . Moreover, every component of G has dimension at least the expected dimension:
dim G ≥ dim B + ρ(g, g − 1, 2g − 2; (0, 0, . . . , 0, g − 1)) = dim B + (g − 1 + 1)(2g − 2 − (g − 1)) − (g − 1)g − (g − 1) = (3g − 2) − (g − 1) = 2g − 1.
Suppose that the component of G containing our given limit g g−1 2g−2 L on (X, p) were to lie entirely over ∆ (i.e. suppose that L doesn't smooth). Then by the first part of Lemma 2.2, G must in fact lie over the locus D ⊂ ∆ parameterizing pointed nodal curves C z ∪q (z) E z in whichq(z) is a subcanonical point of C z and O Ez (p(z) −q(z)) is (2g − 2)-torsion. Since, by the induction hypothesis, we know that the locus of genus g − 1 curves (C z ,q(z)) with a marked subcanonical point has dimension (3(g − 1) − 2) − ((g − 1) − 1) = 2g − 3, and since the locus of elliptic curves with a marked point of order (2g − 2) has dimension 1, we find that dim D = 2g − 2.
But we know by the second part of Lemma 2.2 that the curves over D each have a unique g g−1 2g−2 satisfying the ramification condition, i.e. that the fibers of G over D each consist of a single point. Thus the dimension of this component of G must be 2g − 2, contradicting the known bound dim G ≥ 2g − 1. We conclude that G does not lie entirely over ∆.
This means there are nearby smooth pointed curves with ramification sequence (of their canonical series, the only g g−1 2g−2 on a smooth curve) at least 0, . . . , 0, g − 1. However, L itself has exactly this ramification sequence at p, so by upper semi-continuity of ramification sequences, the general smooth curve in the smoothing family must also have ramification sequence exactly 0, . . . , 0, g−1, as desired. This completes our proof by induction that a general point of G odd g has ramification sequence 0, . . . , 0, g − 1.
2.2. Even case. We turn now to the case of G even g . While we know from [KZ03] that G even g has the same dimension, 2g − 1, as G odd g , the smallest possible ramification sequence 0, . . . , 0, 1, g − 1 has a weight which is one greater. This means that a general point of G even g cannot possibly be dimensionally proper, which is a problem: the inductive framework of [EH87] only applies to dimensionally proper Weierstrass points.
In order to deal with this issue, we note that while these are not dimensionally proper Weirstrass points, if we think of G even g as being the subcanonical points whose associated theta-characteristic is even, then it does have the "expected dimension" in the sense that while the ramification condition α K C g−1 (p) ≥ g − 1 ought give a codimension of g −1, we might expect that the additional condition that O C (g −1)p be an even theta-characteristic would not increase the codimension further since the parity of a theta-characteristic is constant in families.
For the even case, we consider the same nodal curve X = C ∪ E as in the odd case, where an elliptic curve E meets a curve C of genus g − 1 at an odd subcanonical point q of C with ramification sequence 0, . . . , 0, g − 2. Now, we pick a distinguished point p of E such that O E (p − q) has order exactly g − 1, instead of 2g − 2 as in the even case. We will see shortly that this corresponds to picking an odd theta characteristic on E rather than an even one.
As in the odd case, the point of attachment is a dimensionally proper Weierstrass point and Lemma 2.2 again shows that there is a unique limit
g g−1 2g−2 , L = ((L C , V C ), (L E , V E )) on X satisfying the same ramification condition α L E (p) ≥ (0, . . . , 0, g − 1)
at p and that there is no such limit g g−1 2g−2 satisfying the ramification condition at a general point of E. Thus, the proofs of Proposition 5.1 of [EH87] and Corollary 3.7 of [EH86] again may be applied to show that L can be smoothed, preserving the ramification condition α ≥ (0, . . . , 0, g − 1) near p.
More precisely, there is a family of stable curves π : X → ∆, over a smooth, one-dimensional base, with smooth fibers X t = π −1 (t) away from the special fiber X = X 0 = C ∪ E, together with a sectionp: ∆ → X such thatp(0) = p and α K X t (p(t)) ≥ 0, . . . , 0, 0, g − 1 for t = 0, as again the complete series K Xt is the only g g−1 2g−2 on the smooth curve X t . We thus know that the pointsp(t) are subcanonical points of the smooth curves X t for t = 0, and we know by Lemma 2.2 that α L E (p) = 0, . . . , 0, 1, g − 1. By upper semicontinuity, this leaves only two possibilities for the ramification sequence atp(t) for a general t, namely 0, . . . , 0, 0, g − 1 and 0, . . . , 0, 1, g − 1. In order to show that it is in fact the latter, we will need to apply a result about the limits of theta characteristics on smooth curves approaching a curve of compact type.
In [Cor89], Cornalba constructs a compactified moduli space of curves with thetacharacteristics by describing objects associated to a stable curve which correspond to limits of theta-characteristics on nearby smooth curves. In this compactificaton, the odd and even loci remain disjoint irreducible components (cf. section 6 of [Cor89]).
In the case of curves of compact type, the answer is especially simple: a "thetacharacteristic" on a curve of compact type should consist of a choice of a thetacharacteristic on each of its components of positive genus, and its parity should be the sum of the parities on those components. The special case that we require is the following:
Lemma 2.3. Let π: X → ∆ be a family of stable curves of genus g over a smooth, one-dimensional base, such that for some 0 ∈ ∆, the family is smooth away from 0 and the special fiber π −1 (0) = C ∪ C consists of a curve C of genus i and a curve C of genus g − i meeting at a single node.
Let L 0 → (X − π −1 (0)) be a family of theta-characteristics on the smooth fibers of X. Let L be the unique extension of L 0 to all of X which has degree i − 1 on C and let L be the unique extension of L 0 to all of X which has degree g − i − 1 on C .
Then L| C is a theta-characteristic on C, L | C is a theta-characteristic on C , and the parity of the theta characteristic L 0 | π −1 (λ) for λ = 0 is equal to the sum of the parities of L| C and L | C .
In the case of our family X → ∆, on each smooth curve X t , the associated theta-characteristic is O Xt ((g − 1)p(t)). Thus one extension of this family of thetacharacteristics to a line bundle on all of X is simply O X ((g − 1)p). The theta characteristics on C and E associated to this family are thus simply the twists of this line bundle of degrees g − 2 on C and degree 0 on E, respectively. These are O C ((g − 2)q) and O E ((g − 1)p − (g − 1)q). Now, we know that q is an odd subcanonical point of C (since it has ramification sequence 0, . . . , 0, g − 2) so O C ((g − 2)q) is an odd thetacharacteristic. On the other hand O E ((g − 1)p − (g − 1)q) ∼ = O E is effective, since p was chosen so that O E (p − q) has order g − 1 in Pic 0 (E), so the theta-characteristic on the elliptic curve E is odd.
By Lemma 2.3, O Xt ((g − 1)p(t)) is an even theta-characteristic for t = 0. This implies that the ramification sequence of the subcanonical pointp(t) for general t is 0, . . . , 0, 1, g − 1 rather than 0, . . . , 0, 0, g − 1, completing the proof that the general point of G even g has ramification sequence 0, . . . , 0, 1, g − 1.
Remark 1. A slightly closer examination of the proof of Theorem 2.1, in particular of the use of Corollary 3.7 of [EH86], also provides a new proof, without using the methods of [KZ03], that G odd g and G even g are non-empty for g ≥ 4 and that each has some component of the correct dimension 2g−1. It seems likely that this proof, unlike that in [KZ03], might extend to the case of characteristic p, at least when p g, using the theory of limit linear series in characteristic p developed in [Oss06].
There does not, however, seem to be any easy way to show the irreducibility of G odd g and G even g using these techniques.
Remark 2. While the proof of Theorem 2.1 would generalize using any other dimensionally proper subcanonical point as a base case for the induction, in fact no such points can exist. The corollary to Theorem 2 of [EH87] shows that a dimensionally proper point p ∈ C must be primitive, which means that all smaller sequences α ≤ α K C (p) must satisfy the semigroup condition. One can check directly that the sequence 0, 0, . . . , 0, 1, g − 2 fails to satisfy the semigroup condition and that thus 0, 0, . . . , 0, g − 1 is the only possible ramification sequence for a dimensionally proper subcanonical point.
Cyclic covers
Perhaps the simplest construction of subcanonical points is as ramification points of the hyperelliptic double cover on a hyperelliptic curve. In this section, we show how non-hyperelliptic subcanonical points can be constructed as ramification points of certain cyclic covers, and describe in some cases how to compute the vanishing sequences of those subcanonical points.
Let d > 1 be a fixed natural number, and B be a given smooth curve of genus h. The preimage of this section in the total space on L is the desired d-sheeted cyclic cover π: C → B, totally ramified over D: it is easily checked that it has a local analytic equation of the form z → z n at the ramification points, and the cyclic automorphism group is giving simply by the group of dth roots of unity acting on the total space of L by multiplication on each fiber. Let g be the genus of C.
We now give a more algebraic description of this construction (see [Har77] II.5, IV.3). Consider the locally free sheaf of rank d
F = O B ⊕ L −1 ⊕ L ⊗(−2) ⊕ · · · ⊕ L ⊗(−(d−1)) on B. The maps L ⊗(−i) ⊗ L ⊗(−j) ∼ = L ⊗(−i−j) and L ⊗(−i) ⊗ L ⊗(−j) ∼ = L ⊗(−i−j) ∼ = L ⊗(−i−j+d) (−D) → L ⊗(−i−j+d)
give F the structure of a sheaf of Z/dZ-graded algebras over O B . The desired cyclic cover is then the global spec π: C = Spec F → B, and here the action of a dth root of unity ζ is induced by multiplication by ζ i on the ith graded piece. This tells us in particular that
π * O C ∼ = O B ⊕ L −1 ⊕ L ⊗(−2) ⊕ · · · ⊕ L ⊗(−(d−1)) ,
and moreover, looking in local analytic coordinates near a ramification point p ∈ C with π(p) = q so that π is given near p by z → z n , we see that the L ⊗(−i) component in this direct sum decomposition corresponds to functions which are multiplied by ζ i when z is replaced by ζz. Equivalently, a holomorphic section of L −i near q corresponds in this decomposition to a holomorphic function on a z-disc near p in whose Taylor expansion z j may only appear when j ≡ i (mod d). This shows in particular that a section of L ⊗(−i) vanishing to order m at q corresponds to a holomorphic function on C near p which vanishes to order dm + i, and likewise for meromorphic sections and pole orders in the case where m is negative.
Thus if p is a ramification point of π and π(p) = q, then
π * O C (kp) ∼ = O B k d q ⊕L −1 k+1 d q ⊕L ⊗(−2) k+2 d q ⊕· · ·⊕L ⊗(−(d−1)) k+d−1 d q .
We would like to determine when p is subcanonical and calculate its vanishing sequence when it is subcanonical, or in other words we would like to compute the quantity
h 0 C, K C (−np) = h 0 B, π * K C (−np) ,
especially in the case where n = 2g − 2, where it will be one if p is subcanonical and zero otherwise. To do this, we note that by Riemann-Hurwitz, π * K B ∼ = K C (−R), where R is the ramification divisor of π, consisting of each ramification point with multiplicity d − 1, so that π(R) = (d − 1)D. We may then compute
π * K C (−np) ∼ = π * (O C (R − np) ⊗ π * K B ) ∼ = π * (O C (R − np)) ⊗ K B ∼ = K B d−1−n d q ⊕ K B ⊗ L −1 D + −n d q ⊕ K B ⊗ L ⊗(−2) D + 1−n d q ⊕ · · · ⊕ K B ⊗ L ⊗(−(d−1)) D + d−2−n d q ∼ = K B d−1−n d q ⊕ K B ⊗ L ⊗(d−1) −n d q ⊕ K B ⊗ L ⊗(d−2) 1−n d q ⊕ · · · ⊕ K B ⊗ L ⊗2 d−3−n d q ⊕ K B ⊗ L d−2−n d q ∼ = d−1 i=0 L ⊗i ⊗ K B d−1−i−n d q , since L ⊗d ∼ = O B (D). By Riemann-Hurwitz, 2g − 2 = d(2h − 2) + (d − 1) deg D = d((2h − 2) + (d − 1) deg L)
and in particular 2g − 2 is a multiple of d. When we set n = 2g − 2, only one of the line bundles in the above direct sum has non-negative degree, namely
L ⊗(d−1) ⊗ K B − 2g−2 d q , which has degree (d − 1) 2g−2−d(2h−2) d(d−1) + (2h − 2) − 2g−2 d = 0. We see then that p is a subcanonical point of C ⇐⇒ L ⊗(d−1) ⊗ K B ∼ = O B 2g−2 d q ⇐⇒ K B (D) ∼ = L 2g−2 d q ⇐⇒ L ∼ = K B D − 2g−2 d q .
Thus, given q and B, we would like to determine whether there is some effective divisor D on B, consisting of distinct points, one of which is q, so that
K B D − 2g−2 d q ⊗d ∼ = O B (D),
or equivalently, so that
(d − 1)D ∼ (2g − 2)q − dK B .
Now if we let D be some divisor (not necessarily effective) such that (d − 1)D ∼ (2g − 2)q − dK B , we would like to find an effective divisor in |D − q| which consists of deg D − 1 distinct points other than q. By Bertini's theorem, this is possible as long as |D − q| is base-point-free, i.e. if
0 < h 0 (B, O B (D − q − r)) < h 0 (B, O B (D − q))
for every r ∈ B. By Riemann-Roch, this is guaranteed to be the case if deg D − 2 ≥ 2h − 1, i.e. if the number of distinct branch points of the cover we are constructing is at least 2h + 1.
We would now like to find the vanishing sequence of the subcanonical point p ∈ C that we have constructed, for which we must compute h 0 C, K C (−np) = h 0 B, π * K C (−np) for 0 ≤ n ≤ 2g − 2. However, in our direct sum decomposition for π * K C (−np), without additional hypotheses we only have control over the terms
d − m − 1 ∈ {a K B j (q)}.
3.1. Double covers. The other terms may depend on tensor powers of K B in ways that are not completely determined by the vanishing sequence of K B itself at q, so if we are to compute the entire vanishing sequence we will need to make an additional assumption. One additional assumption we may make is that d = 2, in which case the two terms we can control are the only terms. This gives us the following:
Theorem 3.1. Let B be a curve of genus h and q be a point of B. Let g ≥ 3h. Then there exists a double cover π: C → B, where C has genus g, such that q is a branch point of π, and the point p ∈ C with π(p) = q is a subcanonical point with vanishing sequence as follows:
2m + 1 ∈ {a K C j (p)} ⇐⇒ m ∈ {a K B j (q)} 2m ∈ {a K C j (p)} ⇐⇒ g − 2 − m ∈ {a K B j (q)} and 0 ≤ m ≤ g − 1.
Conversely, every subcanonical point on a curve of genus g which is a branch point of a double cover to a curve of genus h has vanishing sequence as above.
The converse here follows from the fact that in the special case of d = 2, every double cover can be constructed in the way we have described. When d > 2 it is no longer the case that every d-sheeted cyclic cover arises in this way. Applying Theorem 3.1 in the case h = 0 recovers the standard computation of the vanishing sequence of a hyperelliptic Weierstrass point:
Corollary 3.2. A branch point of the hyperelliptic double cover from a hyperelliptic curve C of genus g is a subcanonical point with vanishing sequence 0, 2, 4, . . . , 2g − 4, 2g − 2 and ramification sequence 0, 1, 2, . . . , g − 2, g − 1.
Applying Theorem 3.1 in the case h = 1 recovers the computation of the vanishing sequence of a subcanonical point which is a ramification point on a bielliptic curve. This is proven in [BDC99] in the course of classifying Weierstrass points on bielliptic curves. Their results imply also that a subcanonical point of a bielliptic curve π : C → E which is not a ramification point of π must have ramification sequence 0, . . . , 0, 0, g − 1 or 0, . . . , 0, 1, g − 1.
Corollary 3.3. For g ≥ 3, there exists a subcanonical point on a bielliptic curve C of genus g which is a branch point of the bielliptic double cover C → E. Any subcanonical point on a bielliptic curve which is the branch point of the bielliptic cover has vanishing sequence 0, 1, 2, 4, 6, . . . , 2g − 8, 2g − 6, 2g − 2 and ramification sequence 0, 0, 0, 1, 2 . . . , g − 5, g − 4, g − 1.
In the case h = 2, there are two cases for q: the point can be a general point or a Weierstrass point with ramification sequence 0, 2. Applying Theorem 3.1 to those two cases we get:
Corollary 3.4. For g ≥ 6, there exist subcanonical points on curves of genus g with the ramification sequence 0, 0, 0, 0, 0, 1, 2 . . . , g − 7, g − 6, g − 1.
For g = 6, there exist subcanonical points with ramification sequence 0, 0, 0, 2, 2, 5, for g = 7, there exist subcanonical points with ramification sequence 0, 0, 0, 1, 1, 3, 6, and for g ≥ 8, there exist subcanonical points with ramification sequence 0, 0, 0, 1, 1, 1, 2, . . . , g − 8, g − 7, g − 4, g − 1.
In the case h = 3, there are four different possible vanishing sequences for a point on a genus 3 curve, namely 0, 1, 2; 0, 1, 3; 0, 1, 4; and 0, 2, 4. Applying Theorem 3.1 to those four cases we get:
Corollary 3.5. For g ≥ 14, there exist subcanonical points on curves of genus g that have each of the following ramification sequences: 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, . . . , g − 8, g − 1, 0, 0, 0, 0, 0, 1, 1, 1, 2, 3, 4, . . . , g − 9, g − 6, g − 1, 0, 0, 0, 0, 0, 1, 2, 2, 2, 3, 4, . . . , g − 10, g − 7, g − 6, g − 1, and 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 4, . . . , g − 10, g − 7, g − 4, g − 1.
0,
While this the proof of the corollary works for all g ≥ 9, writing down the sequences it yields for 9 ≤ g ≤ 13 requires some care, as in the g = 6, 7 cases of Corollary 3.4.
We could also let h vary; for example, setting h = g 3 , the biggest h can be in Theorem 3.1 and taking q to be a general point on a curve of genus h, we get:
Corollary 3.6. For g ≥ 6, let g ≡ i (mod 3), with i ∈ {0, 1, 2}. Then there exist subcanonical points on curves of genus g with ramification sequence 0, . . . , 0, 1, 2, 3, . . . , g 3 − 2 + i, g − 1. 3.2. Cyclic covers of higher degree. Alternatively, instead of requiring d = 2, we can allow d > 2, but control the L ⊗(d−1−i) ⊗ K B i−n d terms by assuming that (2h − 2)q ∼ K B ; this is automatic in the cases where h = 0 or h = 1 and equivalent to saying that q is itself a subcanonical point of B when h ≥ 2. This simplifies the condition (d − 1)D ∼ (2g − 2)q − dK B to (d − 1)D ∼ (2g − 2 − d(2h − 2))q, and allows us to pick D with
D ∼ 2g−2−d(2h−2) d−1 q and L ∼ = O B 2g−2−d(2h−2) d(d−1) q .
For the rest of the section, we will assume that D, and thus also L, is as above.
Letting = deg L = 2g−2−d(2h−2) d(d−1)
, this yields
L ⊗(d−1−i) ⊗ K B i−n d ∼ = K B i−n d + (d − 1 − i) q ,
and we see that
n = dm + i ∈ {a K C j (p)} if and only if either (d − 1 − i) − m ≤ 0 with m − (d − 1 − i) ∈ {a K B j (q)}, or d − 1 − i − m ≥ 2,
since meromorphic 1-forms exist with pole locus kq for k ≥ 2 but not for k = 1. Solving for g in terms of , this gives us:
Theorem 3.7. Let B be a curve of genus h and let q be a point of B such that K B ∼ = O B (2h − 2)q . Let d ≥ 2 be an integer and be a positive integer satisfying
≥ 2h+1 d . Then if g = d(d−1) 2 + d(h − 1)+1
is at least 2, there is a curve C of genus g, which is a cyclic d-sheeted cover π: C → B of B such that there is a totally ramified p ∈ C with π(p) = q, where p is a subcanonical point of C whose vanishing sequence is determined by the following, for i = 0, 1, . . . , d − 1:
dm + i ∈ {a K C j (p)} ⇐⇒ either 0 ≤ m ≤ (d − 1 − i) − 2 or m − (d − 1 − i) ∈ {a K B j (q)}.
Taking h = 0 or h = 1, with g = 3 − 2 or g = 3 + 1, respectively, we get the following:
Corollary 3.8. Let g = 3k + 1 ≥ 7. Then there exist subcanonical points on curves of genus g with each of the following ramification sequences: 0, 0, 1, 1, 2, 2, . . . , k − 1, k − 1, k, k + 2, k + 4, . . . , g − 5, g − 3, g − 1, and 0, 0, 0, 0, 0, 1, 1, 2, 2, . . . , k − 3, k − 3, k − 2, k, k, k + 1, k + 3, . . . , g − 8, g − 6, g − 1.
Low genus examples
The following table summarizes the possible ramification sequences of subcanonical points on curves of genus g ≤ 6, which we will be describing for the rest of the section.
genus a K C (p) α K C (p) parity weight codim M g,1 C α 2 0, 2 0, 1 odd 1 1 3 0, 1, 4 0, 0, 2 odd 2 2 0, 2, 4 0, 1, 2 even 3 2 4 0, 1, 2, 6 0, 0, 0, 3 odd 3 3 0, 1, 3, 6 0, 0, 1, 3 even 4 3 0, 2, 4, 6 0, 1, 2, 3 even 6 3 5 0, 1, 2, 3, 8 0, 0, 0, 0, 4 odd 4 4 0, 1, 2, 4, 8 0, 0, 0, 1, 4 even 5 4 0, 2, 4, 6, 8 0, 1, 2, 3, 4 odd 10 4 6 0, 1, 2, 3, 4, 10 0, 0, 0, 0, 0, 5 odd 5 5 0, 1, 2, 3, 5, 10 0, 0, 0, 0, 1, 5 even 6 5 0, 1, 2, 4, 6, 10 0, 0, 0, 1, 2, 5 even 8 7 0, 1, 2, 5, 6, 10 0, 0, 0, 2, 2, 5 odd 9 6 0, 2, 4, 6, 8, 10 0, 1, 2, 3, 4, 5 odd 15 5 4.1. Genus 2. Every genus 2 curve is hyperelliptic, so every curve has exactly 2g + 2 = 6 subcanonical points, the ramification points of the hyperelliptic double cover. These subcanonical points all have ramification sequence 0, 1, and the associated theta characteristics are odd. Any non-hyperelliptic curve of genus 3 can be embedded by the canonical series as a smooth quartic in P 2 , and conversely every smooth plane quartic is a non-hyperelliptic curve of genus 3. Given such a smooth plane quartic curve C ⊂ P 2 , the canonical series is cut out on C by the lines in P 2 .
Thus a subcanonical point on C is a point p ∈ C such that for some line L ⊂ P 2 we have that C ∩ L = 4p, or in other words (C.L) p = 4. In this case, any other The existence of such plane quartics is a simple application of Bertini's Theorem: if we fix a point and line p ∈ L ⊂ P 2 , and consider the linear system of plane quartics C satisfying the linear conditions (C.L) p ≥ 4, then this linear system has p as its only basepoint (consider unions of four lines through p), so Bertini's Theorem shows that a general such C is smooth away from p. On the other hand, a general such C is smooth at p (for example, consider C = L ∪ C with p ∈ C ), so we see that a general plane quartic satisfying (C.L) p ≥ 4 is smooth everywhere.
The locus of such plane quartics has dimension 6 2 − 1 − 4 = 10, and the locus of choices of p ∈ L has dimension 3. Two non-hyperelliptic curves of a given genus are isomorphic if and only if their canonical models are projectively equivalent. Since a genus 3 curve has only finitely many automorphisms and dim PGL(3) = 8, we see that the locus in M 3 of non-hyperelliptic genus 3 curves which possess a subcanonical point has dimension 10 + 3 − 8 = 5, or codimension (3 · 3 − 3) − 5 = 1 in M 3 as expected.
We might also have expected codimension one in this case since a general smooth plane quartic C does have ordinary flexes, that is points p whose tangent line L satisfies (C.L) p = 3; possessing a hyperflex should be one additional condition. Alternatively, note that a general smooth plane quartic has bitangents (in fact, it has 28 bitangents, corresponding to the 28 odd theta-characteristics), so it should be one additional condition for the two points of tangency to come together.
Genus 4.
A non-hyperelliptic curve of genus 4 canonically embeds as C ⊆ P 3 , a degree 6 curve which is the complete intersection of an irreducible quadric Q and a cubic (cf. [ACGH85] ch. III).
If Q is smooth, then C may be regarded as a smooth curve of bidegree (3, 3) on the surface Q ∼ = P 1 × P 1 . The canonical series K C is then cut on C by hyperplanes in P 3 , and thus by curves of bidegree (1, 1) on Q. A curve of bidegree (3, 3) meets a curve of bidegree (1, 1) in 6 points, counting multiplicities. Suppose that p is a subcanonical point of C, so that C meets some bidegree (1, 1) curve H at p with multiplicity 6 and nowhere else. We claim that if H is any other (1, 1) curve, then (C.H ) p ≤ 2; for otherwise (H.H ) ≥ 3, but two (1, 1) curves meet in at most 2 points unless they share a common component, and certainly H must be irreducible, for if it consisted of two lines, they could not each meet C at p with multiplicity 3. We see then that the vanishing sequence of K C at p must then be 0, 1, 2, 6, with corresponding ramification sequence 0, 0, 0, 3.
To check that such curves do in fact exist, we note given a smooth quadric Q ⊆ P 3 , so that Q ∼ = P 1 × P 1 , and a smooth curve H ⊂ Q of bidegree (1, 1), we have that H is a hyperplane section of Q, i.e. a smooth plane conic, so that H ∼ = P 1 . It may then be checked by direct computation that O H (3) ∼ = O P 1 (6) and that the restriction maps 1 (3, 3)) → H 0 (P 1 , O P 1 (6)) are surjective. Thus, given a point p ∈ H, there do exist curves of bidegree (3, 3) meeting H at p with multiplicity exactly 6. Moreover, meeting H at p with multiplicity at least 6 imposes exactly 6 linear conditions on the space of bidegree (3, 3) curves on Q. By a Bertini's theorem argument, there exist smooth curves with this property. By the above surjectivity, these curves arise as the complete intersection of a smooth quadric surface and a cubic surface, and one can check (e.g. using the adjunction formula) that such a curve does in fact have genus 4. We may estimate the dimension of the locus in M 4,1 arising from these curves as follows:
H 0 (P 3 , O P 3 (3)) → H 0 (Q, O Q (3)) ∼ = H 0 (P 1 × P 1 , O P 1 ×Pchoice of Q 5 3 − 1 + choice of H 2 · 2 − 1 + p 1 + C bidegree (3, 3) 4 · 4 − 1 − (C.H)p ≥ 6 6 − PGL(4) 15 = 7
As expected, this estimate is equal to dim G odd 4 = 2 · 4 − 1. A more rigorous version of this dimension count would show irreducibility as well.
In order to find non-hyperelliptic curves of genus 4 with even subcanonical points, we will need to look at the case where Q is not smooth, but rather a cone over a smooth conic. Here, we take a smooth curve C on Q, defined by the intersection of Q with a cubic hypersurface in P 3 , so that it meets a point p of a line L of the ruling with multiplicity 3. Then the tangent plane H to Q at p will intersect Q in the double line L and H will intersect C with multiplicity 6, meaning p is a subcanonical point of C. Here, however, if H is another hyperplane containing L, then H meets C with multiplicity 3. We may certainly find hyperplanes not meeting C at p at all or meeting C at p with multiplicity 1, so we see that the vanishing sequence for K C at p is 0, 1, 3, 6, with corresponding ramification sequence 0, 0, 1, 3.
1 To prove that such curves actually exist, we would first note that homogeneous cubic polynomials on P 3 restrict to L to give all the homogeneous polynomials of degree 3 on L, including those vanishing to order exactly 3 at p. We would then apply Bertini's theorem to the curves cut out on Q by cubics that vanish to order at least 3 along L. Note that such curves containing the line L certainly vanish to order at least 3 along L and can easily be made smooth at p.
The dimension estimate of the corresponding locus in M 4,1 is as follows:
choice of cone Q 5 3 − 1 − 1 + L 1 + p 1 + cubic C restricted to Q 6 3 − 4 3 − 1 − (C.L)p ≥ 3 3 − PGL(4) 15 = 7.
As in the odd case, we see that this agrees with dim G even 4 = 2 · 4 − 1 = 7.
4.4. Genus 5. In genus 5, the only possible ramification sequences for a non-hyperelliptic subcanonical point are still just 0, 0, 0, 0, 4 and 0, 0, 0, 1, 4; to show this, one can simply check that no other ramification sequences correspond to non-gap sequences that satisfy the semigroup condition. We will describe a general genus 5 curve with each of these ramification sequences.
First of all, any non-hyperelliptic, non-trigonal curveC of genus 5 is the normalization of a plane sextic C with 5 ordinary double points, q 1 , . . . , q 5 . The canonical series on C is cut out by the plane cubics through the 5 double points. A point p on C is then subcanonical if there is some plane cubic E, passing through q 1 , . . . , q 5 , which otherwise meets C only at p, with multiplicity 8. Suppose that E is a smooth E C p q q q q q Figure 4. genus 5 non-hyperelliptic, non-trigonal case elliptic curve. Then the points q 1 , . . . , q 5 , p ∈ E are not arbitrary: since C cuts out the divisor 2q 1 + . . . 2q 5 + 8p on E, they must satisfy the relation 2q 1 + 2q 2 + . . . + 2q 5 + 8p ∼ 6H on E, where the divisor H is cut out on E by a line. Now suppose that E = E is another cubic passing through q 1 , . . . , q 5 which satisfies (E .C) p ≥ 4. Then (E.E ) p ≥ 4, so by Bézout, we must have (E .C) p = (E.E ) p = 4, and in this case it must moreover be true that q 1 + q 2 + . . . + q 5 + 4p ∼ 3H on E. Conversely, if q 1 + q 2 + . . . + q 5 + 4p ∼ 3H, then there is a section of O E (3) ∼ = O E (q 1 + q 2 + . . . + q 5 + 4p) with zero divisor q 1 + q 2 + . . . + q 5 + 4p, and since the map
H 0 (P 2 , O P 2 (3)) → H 0 (P 2 , O E (3)) is surjective (consider the short exact sequence 0 → I E (3) → O P 2 (3) → O E (3) → 0)
there is some cubic in P 3 which cuts out q 1 + q 2 + . . . + q 5 + 4p on E.
We thus see that if q 1 + q 2 + . . . + q 5 + 4p ∼ 3H on E, then p has vanishing sequence 0, 1, 2, 4, 8 and ramification sequence 0, 0, 0, 1, 4, whereas if 2q 1 + 2q 2 + . . . + 2q 5 + 8p ∼ 6H but q 1 + q 2 + . . . + q 5 + 4p ∼ 3H, then p has vanishing sequence 0, 1, 2, 3, 8 and ramification sequence 0, 0, 0, 0, 4.
Of course, again we should check that such curves actually exist and calculate the dimensions of the corresponding loci in M 5,1 . We begin by fixing a smooth cubic E ⊂ P 2 and points q 1 , . . . , q 5 , p ∈ E satisfying 2q 1 + 2q 2 + . . . + 2q 5 + 8p ∼ 6H on E. We would like to compute how many linear conditions we impose on H 0 (P 2 , O P 2 (6)) by requiring that a sextic curve C have at least double points at q 1 , . . . , q 5 and satisfy (C.E) p ≥ 8. We will then show by Bertini's theorem that there exist such curves which are smooth away from the q i and have simple nodes at the q i . Now, since as above the map H 0 (P 2 , O P 2 (6)) → H 0 (E, O E (6)) is surjective, and since we have chosen p and the q i so that 2q 1 + 2q 2 + . . . + 2q 5 + 8p ∼ 6H, we find that there exist sextics C ⊂ P 2 which cut out exactly the divisor 2q 1 + 2q 2 + . . . + 2q 5 + 8p on E. Moreover, we see that requiring that a sextic vanish along E to order at least 2 at each of the q i and to order 8 at p imposes exactly h 0 (E, O E (6)) − 1 = (18 − 1 + 1) − 1 = 17 linear conditions on H 0 (P 2 , O P 2 (6)). Now, for C to have at least a double point at each point q i is 5 additional linear conditions (that at each q i a derivative in a direction away from E be zero as well); in fact, these 5 linear conditions are independent, as we may consider sextics of the form E ∪ F i , where F i is a cubic vanishing at the four q j with j = i but not at q i . Now, let X be the projective space (of dimension 8 2 − 1 − 17 − 5) of sextics which have double points (or worse) at the q i and which intersect E at p with multiplicity at least 8. A general C ∈ X cuts out the divisor 2q 1 + 2q 2 + . . . + 2q 5 + 8p on E. We note first that the set of base points of X is just {q 1 , . . . , q 5 , p}, since for other points of E we already know a general C ∈ X does not contain them, and if x ∈ P 2 is not in E, we may find some plane cubic E which contains q 1 , . . . , q 5 but not x, and then E ∪ E ∈ X would not contain x. This shows, by Bertini's theorem, that a general C ∈ X is smooth away from {q 1 , . . . , q 5 , p}.
In fact, a general C ∈ X must also be smooth at p, since if C ⊃ E but C is not smooth at p, then we may find some plane cubic E which contains q 1 , . . . , q 5 but not p, and then C + EE still satisfies the vanishing conditions and is smooth at p. (Here, in an abuse of notation, we are writing C, E, E for both the curves and their defining homogeneous polynomials.) Likewise, to control the singularity of C at q i , we may find a conic Z containing q j for j = i but not containing q i , and then for a general choice of a line L through q i , the curves C + EZL would still satisfy the vanishing conditions but have a simple node at q i whose branches have arbitrary tangent directions (aside from not being tangent to E, since then the curve would contain E). This implies (e.g. by applying Bertini's theorem on the blowup of P 2 at the q i ) that a general C ∈ X is smooth away from the q i with nodes at the q i , as desired.
To find the dimension of the corresponding loci (our parameter space will have two components depending on whether or not q 1 + q 2 + . . . + q 5 + 4p ∼ 3H), we must note that the map from an abstract smooth curve to the plane to give a degree 6 singular curve is not unique, but rather there is a 2-parameter family of such maps (essentially, the plane curves we are dealing with are projections of the canonical curve of degree 8 in P 4 from two general points on the curve). We thus calculate the dimensions of This again agrees with the known dimension: dim G 5 = 2 · 5 − 1 = 9.
4.5. Genus 6. In genus 6, there are more possible ramification sequences. A curvẽ C of genus 6 which is not hyperelliptic, trigonal, bielliptic, or isomorphic to a smooth plane quintic, is the normalization of a plane sextic C having 4 double points (cf.
[ACGH85] V.A). In this case, the situation is analogous to that in genus 5: if p ∈ C is the subcanonical point and q 1 , . . . , q 4 are the double points and E ⊂ P 2 is a smooth elliptic curve through the q i cutting out the divisor 10p on C, then 2q 1 + . . . + 2q 4 + 10p ∼ 6H on E, where H is a hyperplane section, and p has vanishing sequence 0, 1, 2, 3, 5, 10 and ramification sequence 0, 0, 0, 0, 1, 5 if q 1 + . . . + q 4 + 5p ∼ 3H and otherwise has vanishing sequence 0, 1, 2, 3, 4, 10 and ramification sequence 0, 0, 0, 0, 0, 5. As in the genus 5 case, we calculate the dimension as for the corresponding loci in M 6,1 , which again agrees with the known dimension of 2 · 6 − 1 for G odd 6 and G even 6 . By Corollary 3.3, there exist subcanonical points on bielliptic curves of genus 6 that have vanishing sequence 0, 1, 2, 4, 6, 10 and ramification sequence 0, 0, 0, 1, 2, 5.
To study the corresponding locus in M 6,1 , we recall from Section 3 that given an elliptic curve E and distinct points q 1 , q 2 , . . . , q 10 ∈ E, there exists a bielliptic double cover of E of genus 6 with branch locus {q 1 , . . . , q 10 } having a subcanonical point that maps to q 1 if and only if q 1 + q 2 + . . . + q 10 ∼ 10q 1 on E, and in this case the bielliptic double cover is unique. We can show then that the corresponding locus in M 6,1 is irreducible of dimension dim M 1,10 − 1 = 9.
A general subcanonical point p on a smooth plane quintic C has vanishing sequence 0, 1, 2, 3, 4, 10. This is because the canonical series on C is cut by plane conics, and if there is a smooth quadric Q meeting C only at p with (C.Q) p = 10, then we can find quadrics meeting C at p with multiplicities 0, 1, 2, 3, 4 simply by taking unions of lines (to get (C, Q ) p = 3, for example, take the union of the tangent line to C at p with some other line through p).
There however exist smooth plane quintics which possess a 5-fold flex, that is, there exists a smooth plane quintic curve C and a line L ⊂ P 2 meeting C at a single point p with (C.L) p = 5. Then p is a subcanonical point of C with vanishing sequence . genus 6, plane quintic case 0, 1, 2, 5, 6, 10 and ramification sequence 0, 0, 0, 2, 2, 5. To show this we can again construct sections with those vanishing orders by simply taking unions of lines.
The proof that there exist smooth plane smooth quintics which possess a 5-fold flex is entirely analogous to the proof in genus 3 of the existence of smooth plane quartics with a hyperflex: we fix a point p and a line L containing it, and show by Bertini's theorem that there exists a smooth quintic C satisfying (C.L) p = 5. Since a smooth plane quintic can have only one embedding into P 2 (cf. [ACGH85] ch. V), we compute the dimension of the corresponding locus in M 6,1 as: A more detailed proof of this dimension count would also show the irreducibility of the corresponding locus.
1. 1 .
1Main results. A point p on a smooth curve C is a Weierstrass point if its associated set of Weierstrass gaps{n ∈ Z ≥0 : h 0 (C, O C (np)) = h 0 (C, O C ((n − 1)p))}is not equal to {1, 2, . . . , g}. A point is a subcanonical if and only if 2g − 1 is a gap, so subcanonical points are Weierstrass points.
Let D be an effective divisor on B consisting of distinct points, with deg D divisible by d. Let L be a line bundle on B satisfying L ⊗d ∼ = O B (D). Then we may construct a d-sheeted cyclic cover of B, totally ramified over each point of D, as follows: the isomorphism L ⊗d ∼ = O B (D) determines a dth power map from the total space of the line bundle L to the total space of the line bundle O B (D). The total space of the line bundle O B (D) has a distinguished section, the constant section 1, which as a section of O B (D) vanishes on D.
d
determine only the portion of the vanishing sequence where n ≡ 0, d − 1 (mod d); for n is in the vanishing sequence for K C at p if and only if h 0 C, K C (−np) = h 0 C, K C (−(n + 1)p) , and only if n ≡ i (mod d), so whether a number n ≡ i (mod d) appears in the vanishing sequence depends entirely on the L ⊗(d−1−i) ⊗ K B i−n d term. We see then that n = dm + (d − 1) is in the vanishing sequence for K C at p if and only if m is in the vanishing sequence for K B at q. Likewise, if n = dm, then n ∈ {a K C j (p)} if and only if h 0 B, − (m + 1) q , or in other words, if and only if 2g−2 d − m is a Weierstrass non-gap of p, or if and only if 0 ≤ m ≤ 2g−2 d and 2g−2
4. 2 .
2Genus 3. In genus 3, we expect each component of G 3 to have codimension 3 − 1 = 2 in M 3,1 , so each component of the locus of curves which have a subcanonical point should have codimension 1 in M 3 . This is of course true of the locus of hyperelliptic curves of genus 3, which have subcanonical points with ramification sequence 0, 1, 2.
Figure 1 .
1genus 3 non-hyperelliptic case line L ⊂ P 2 either does not contain p, in which case (C.L ) p = 0, or does contain p but is not tangent to C at p, so that (C.L ) p = 1. (If instead (C.L ) p ≥ 2, then (L.L ) p ≥ min{(C.L ) p , (C.L) p } ≥ 2, and L = L by Bézout.) Thus the vanishing sequence for the canonical series of C at p is 0, 1, 4, and the ramification sequence is 0, 0, 2.
Figure 2 .
2genus 4 odd case, vanishing sequence 0, 1, 2, 6
Figure 3 .
3genus 4 even case, vanishing sequence 0, 1, 3, 6
Figure 5
5Figure 5. genus 6, plane quintic case
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"Service d'Astrophysique\nCNRS URA 2052\nCEA Saclay\nF-91191Gif-sur-YvetteFrance",
"Université Paris VII, fédération APC\n2 place JussieuF-75005Paris CedexFrance",
"Service d'Astrophysique\nCNRS URA 2052\nCEA Saclay\nF-91191Gif-sur-YvetteFrance",
"Department of Physics and Astronomy\nX-Ray Astronomy Group\nLeicester University\nLE1 7RHU.K",
"Service d'Astrophysique\nCNRS URA 2052\nCEA Saclay\nF-91191Gif-sur-YvetteFrance",
"Service d'Astrophysique\nCNRS URA 2052\nCEA Saclay\nF-91191Gif-sur-YvetteFrance"
] |
[] |
We report the results of an XMM-Newton observation of the persistent X-ray source 1E 1743.1-2843, located in the Galactic Center (GC) direction. We determine the position of the source at αJ2000=17 h 46 m 21.0 s , δJ2000=−28 • 43 ′ 44 ′′ (with an uncertainty of 1.5 ′′ ), which is the most accurate to date, and will enable crossidentifications at other wavelengths. The source was bright during this observation (L 2−10 keV ∼ 2.7 × 10 36 d 2 10kpc erg s −1 for a power-law continuum), with no significant variability. We propose that 1E 1743.1-2843 may be explained in terms of a black hole candidate in a low/hard state. There is an indication that the source exhibits different states from a comparison of our results with previous observations (e.g., ART-P, BeppoSAX). However, the present spectral analysis does not rule out the hypothesis of a neutron star low-mass X-ray binary as suggested previously. If 1E 1743.1-2843 is actually located in the GC region, we might expect to observe significant 6.4 keV fluorescent iron line emission from nearby molecular clouds (e.g., GCM+0.25+0.01).
|
10.1051/0004-6361:20030770
|
[
"https://arxiv.org/pdf/astro-ph/0305489v1.pdf"
] | 12,021,474 |
astro-ph/0305489
|
4d00ffcea520b5512e064814328e3011b00fb3a3
|
XMM-Newton study of the persistent X-ray source 1E 1743.1-2843 located in the Galactic Center direction
May 2003 October 15, 2018
D Porquet
Service d'Astrophysique
CNRS URA 2052
CEA Saclay
F-91191Gif-sur-YvetteFrance
J Rodriguez
Service d'Astrophysique
CNRS URA 2052
CEA Saclay
F-91191Gif-sur-YvetteFrance
Integral Science Data Center
16 CH-1290Chemin d'Ecogia, VersoixSwitzerland
S Corbel
Service d'Astrophysique
CNRS URA 2052
CEA Saclay
F-91191Gif-sur-YvetteFrance
Université Paris VII, fédération APC
2 place JussieuF-75005Paris CedexFrance
P Goldoni
Service d'Astrophysique
CNRS URA 2052
CEA Saclay
F-91191Gif-sur-YvetteFrance
R S Warwick
Department of Physics and Astronomy
X-Ray Astronomy Group
Leicester University
LE1 7RHU.K
A Goldwurm
Service d'Astrophysique
CNRS URA 2052
CEA Saclay
F-91191Gif-sur-YvetteFrance
A Decourchelle
Service d'Astrophysique
CNRS URA 2052
CEA Saclay
F-91191Gif-sur-YvetteFrance
XMM-Newton study of the persistent X-ray source 1E 1743.1-2843 located in the Galactic Center direction
May 2003 October 15, 2018Received ... / accepted ...arXiv:astro-ph/0305489v1 26 Astronomy & Astrophysics manuscript no. ms3545 (DOI: will be inserted by hand later)X-Rays: general -Individual: 1E 17431-2843 -Stars: neutron -Black hole physics -Binaries: general
We report the results of an XMM-Newton observation of the persistent X-ray source 1E 1743.1-2843, located in the Galactic Center (GC) direction. We determine the position of the source at αJ2000=17 h 46 m 21.0 s , δJ2000=−28 • 43 ′ 44 ′′ (with an uncertainty of 1.5 ′′ ), which is the most accurate to date, and will enable crossidentifications at other wavelengths. The source was bright during this observation (L 2−10 keV ∼ 2.7 × 10 36 d 2 10kpc erg s −1 for a power-law continuum), with no significant variability. We propose that 1E 1743.1-2843 may be explained in terms of a black hole candidate in a low/hard state. There is an indication that the source exhibits different states from a comparison of our results with previous observations (e.g., ART-P, BeppoSAX). However, the present spectral analysis does not rule out the hypothesis of a neutron star low-mass X-ray binary as suggested previously. If 1E 1743.1-2843 is actually located in the GC region, we might expect to observe significant 6.4 keV fluorescent iron line emission from nearby molecular clouds (e.g., GCM+0.25+0.01).
Introduction
The source 1E 1743.1-2843 was discovered during the first X-ray imaging observations of the Galactic Center (GC) region performed with the Einstein Observatory (Watson et al. 1981). Its column density (> 10 23 cm −2 ) is one of the highest observed in the bright X-ray sources found in this region of sky, suggesting a distance similar to, or greater, than the GC (d=7.9±0.3 kpc, McNamara et al. 2000). 1E 1743.1-2843 has been detected by all Xray satellites with X-ray imaging capability above 2 keV (Watson et al. 1981, Kawai et al. 1988, Sunyaev et al. 1991, Pavlinsky et al. 1994, Lu et al. 1996, Cremonesi et al. 1999, whereas in soft X-rays (e.g. ROSAT) the source is not detectable due to the high column density along the line-of-sight (see Predehl & Trümper 1994). Kawai et al. (1988) suggested that part of the measured absorption could be intrinsic, as is the case for Vela X-1 and GX 301-2. The inferred X-ray 2-10 keV luminosity of this bright persistent source is about 2×10 36 ×d 2 10kpc erg s −1 ruling out models involving coronal or wind emission from normal stars but, conversely, strongly favoring the presence of an accreting compact object. Cremonesi et al. (1999) Send offprint requests to: Delphine Porquet, e-mail: [email protected] (current address) suggested that the absence of periodic pulsations (and/or eclipses) and the relatively soft X-ray spectrum favor a low mass X-ray binary (LMXB) containing a neutron star. LMXBs are systems in which a compact object (either a neutron star or a black hole) accretes matter from a low-mass (< 1 M ⊙ ) companion star. Most LMXBs containing neutron stars are characterized by the occurrence of Type I X-ray bursts produced by the thermonuclear flashes of the accreted material on the surface of the neutron star. However, no bursts have been observed from 1E 1743.1-2843 in extensive observations over the last 20 years. If 1E 1743.1-2843 is a neutron star LMXB, the lack of bursts is noteworthy because its X-ray luminosity (10 36 -10 37 erg s −1 ) is in the range which is typical of Xray bursters. The lack of bursting activity in neutron star LMXB of higher luminosity is generally ascribed to the stable burning of H and He in sources operating close to the Eddington limit (Fujimoto et al. 1981). Such a high luminosity would require 1E 1743.1-2843 to be at a distance greater than several tens of kiloparsecs. Type-I bursts are also suppressed in pulsars due to the higher surface magnetic fields (e.g., Lewin et al. 1995). Cremonesi al. (1999) did not rule out other interpretations such as an extragalactic source seen through the Galactic plane.
Here we present the results of the first observation of 1E 1743.1-2843 with XMM-Newton. Section 2 details the observation and data reduction procedures. Section 3 presents the determination of the accurate X-ray position of this object. Sections 4 and 5 describe, respectively, the timing and spectral analysis. The analysis results are discussed in the last section.
Observations and data analysis
1E 1743.1-2843 was observed by XMM-Newton on September 19, 2000 about 5.5 ′ off-axis from the center of the pointing. The EPIC-MOS cameras were operated in the standard full-frame mode (time resolution: 2.6 s), and the EPIC-PN camera in the extended full frame mode (time resolution: 200 ms), with the medium filter used in both cases. The effective exposure times were ∼29.1 ksec and ∼22 ksec for the MOS and PN cameras, respectively. The data were reprocessed using version 5.3.3 of the Science Analysis Software (SAS) and further filtered using xmmselect. The datasets were screened by rejecting periods of high background arising from marked increases in the incident flux of soft protons. After this data cleaning, the useful observing times are respectively for MOS1 and MOS2 about 22.2 ksec and 23 ksec, and 18.4 ksec for PN. Unfortunately, in the MOS1 CCD, 1E 1743.1-2843 is located on a bright pixel column which makes the data difficult to process. Our present analysis is based largely on the PN data for which we use single events (corresponding to pattern 0) to avoid any possibility of pileup effects, although the MOS2 data (event patterns 0-12) did provide a valuable check of the results. The inferred PN flux in the 2-10 keV energy range is then about 2×10 −10 erg cm −2 s −1 , thus implying a negligible pile-up fraction (<2%, see Fig. 98 in the XMM-Newton Users' Handbook 1 ). Counts from 1E 1743.1-2843 were extracted within a radius of 26.4 ′′ , thereby avoiding a gap between the PN CCDs. We extracted a source-free local background from an annulus centered on 1E 1743.1-2843 with inner and outer radii of 2 ′ and 5 ′ respectively.
X-ray position and multi-wavelength counterpart
In the field of view of our pointing, a bright Xray point source is associated to an optical foreground Tycho-2 source (HD 316297). This very accurate position allow to refine the astrometry and we find for 1E 1743.1-2843 the following position: α J2000 =17h 46m 21.0s, δ J2000 =−28 • 43 ′ 44 ′′ , with a final accuracy limited by the systematic residual uncertainties of 1.5 ′′ (Kirsch 2002). The position inferred from XMM-Newton data is by far much more accurate than those determined from earlier observations, i.e about 60". According to the relation between the visual extinction (A v ) and the hydrogen column density along the line-ofsight (N H ) reported in Predehl & Schmitt (1995), we find for 1E 1743-2843 (N H ∼ 2×10 23 cm −2 , see §5) a value of A v of about 110 magnitudes. The source is too absorbed to be detected by 2MASS (Two Micron All Sky Survey; http://www.ipac.caltech.edu/2mass/). An examination of the maps of the NVSS, NRAO VLA Sky Survey (Condon et al. 1998) did not reveal any possible radio counterpart at the position of 1E 1743.1-2843 with flux at 20 cm greater than ∼50 mJy (Cremonesi et al. 1999
Timing analysis
The 2-10 keV background subtracted light curves of 1E 1743.1-2843 obtained during the XMM-Newton observation are presented in Figure 1 for two different time binnings (100 s and 500 s). The light curve shows some variation around the mean value of 2.42±0.10 cts s −1 (Fig.1).
Fitting the light curve binned at 500 s with a constant count rate gives a moderate fit with χ 2 /d.o.f=104/49. In order to determine whether or not this variability was significant, we produced a power spectrum using POWSPEC v1.0, between 2.4 mHz and 2.5 Hz. The resultant power spectrum is flat, and the (leahy) normalized power density spectrum is well fit with a constant value of 1.99±0.01 (χ 2 = 25 for 28 d.o.f.). This value is compatible with the expected value of 2 for a purely poissonian noise (white noise). The data mode employed in the present observation allows timing studies only up to a frequency of 2.5 Hz, which is quite limited, since many XRBs present quasi periodic variations above that value. No pulsations or quasi periodic oscillations (QPO) are detected in the 2.4 mHz-2.5 Hz frequency range, down to a relatively low level. Indeed using the relation
N σ = 0.5 × S 2 S + B r 2 T ∆ν 1 2 ,
with S the source net count rate, B the background count rate, T the exposure time, r the fractional amplitude, we can estimate a 3 sigma upper limit for a given pulsation (whose width is equal to the frequency resolution of the power spectrum). In our case, this leads to a 3σ upper limit of ∼ 2.4% for a periodic pulsation and higher for any QPO (which by definition posseses a natural width higher than the frequency resolution). As already pointed out by Cremonesi et al. (1999), a typical Type I X-ray burst with a peak luminosity close to the Eddington limit would have produced in 1E 1743.1-2843 a very large count rate increase. For EPIC-PN assuming that 1E 1743.1-2843 is located at the Galactic Center (d∼8 kpc), a factor of 50 increase would be observed between the "quiescent state" and the "burst" count rates. Then such bursts would be readily seen even in a light curve with a time binning as low as 1 s, which is very short compared to the typical duration of a few tens of seconds in LMXB. In addition the non-detection of eclipses in the X-ray light curve implies that the orbital inclination of the system is smaller than 70
• (Cowley et al. 1983). We do not see any indication of an orbital variation on long time scale as suggested by Cremonesi et al. (1999) but this is consistent with our shorter observation duration. Analysis of the Power Density Spectrum indicates that the fractional variability of 1E 1743.1-2843 in the frequency range 10 −4 -2.5 Hz is less than 18% rms (3 sigma upper limit). This is not a strong constraint, regarding the state of the source (if it is a black hole for example), as only a limited frequency domain has been explored, and such a limit is compatible with either a soft or hard state (Nowak 1995).
Spectral analysis
For spectral fitting, the data were rebinned with a minimum of 25 counts per bin to allow use of the χ 2 statistic. xspec (v11.1.0) is used for the spectral fitting. The re- sponse matrice (.rmf) and ancillary (.arf) files were computed using the SAS package. The spectrum of 1E 1743.1-2843 was fitted between 2 and 12 keV. The photoelectric absorption cross-sections of Wilms et al. (2000) are used throughout this paper with abundances taken from Anders & Grevesse (1989). All errors are quoted at 90% confidence.
We fit the background-subtracted source spectrum with various single-component spectral models as follows: black-body (bb), power-law (pow), and multi-color disk black-body (MCD, diskbb). In all cases the hydrogen column density (N H ) was included as a free parameter. The results of this analysis are presented in Table 1. All of the simple models noted above provided a statistical good fit to the observed spectrum. For the single bb and diskbb models, the fits are good but the inferred temperatures are higher (kT∼1.9 keV and kT∼3.4 keV, respectively) than those found in general for neutron star LMXBs, i.e., 0.5-1.5 keV (e.g., Barret 2001). The parameters found here for the bb model are consistent, within the error bars, with those found with BeppoSAX in April 1998, i.e. N H =1.3±0.1×10 23 cm −2 , and kT=1.8±0.1 keV. The unabsorbed 2-10 keV flux found in the present data appears slightly lower than the one found in April 1998, i.e. 1.65×10 −10 erg cm −2 s −1 .
Recently, absorption features associated to H-like and/or He-like K α resonance lines of Fe, Ca, Ne, O, and N have been observed in several neutron star LMXBs (e.g, GX 13+1: Ueda et al. 2001;EXO0748-676: Cottam et al. 2001;MXB1659-298: Sidoli et al. 2001X1624-490: Parmar et al. 2002. Such features are not statistically required in the present data, with equivalent width (EW) upper limits (at 90% confidence) of about 10 eV, 1 eV, and 15 eV, respectively for Fe xxvi (∼ 7 keV), Fe xxv (∼ 6.7 keV), and Ca xx (∼ 4.1 keV). The others lines of Ne, O, and N, below 2 keV, are not accessible due to the very large absorption in the line-of-sight.
The excellent power-law fit (χ 2 red =0.975) contrasts with the corresponding results from Cremonesi et al. (1999) where a power-law model gave χ 2 red =1.49. Moreover, fixing the parameters at the values found by Cremonesi et al. (i.e., N H =2×10 23 cm −2 , and Γ=2.2), we also obtain a bad fit for the power-law model (χ 2 red =1.53). Figure 2 shows the PN spectrum of 1E 1743.1-2843 and the residuals of the best-fitting power-law model to the present data. This implies that 1E 1743.1-2843 could be in the present observation a black hole candidate (BHC) in its low/hard state or, conceivably, an Active Galactic Nucleus (AGN) observed through the obscuration of the Galactic Plane. The inferred photon index for 1E 1743.1-2843 is about 1.8 which is within the range found in both type of objects (e.g., Wu et al. 2001, Malizia et al. 1999. BHC in our Galaxy are usually associated with weak (few mJy) radio counterpart (e.g., Fender & Hendry 2000, Corbel et al. 2000. For example, the BHC 1E 1740.7-2942 located in the GC region, has a radio flux at 20 cm of about 1.4 mJy (Gray et al. 1992), and an unabsorbed 2-10 keV flux of about 5×10 −10 erg cm −2 s −1 , which corresponds to a luminosity of about 5.7 × 10 36 d 2 10kpc erg s −1 . Similarly, Seyfert galaxies are also rather weak radio sources (Nagar et al. 2000). It follows that both the BHC and AGN hypotheses cannot be ruled out by the radio flux limits for 1E 1743.1-2843 quoted earlier.
We found that the presence of an iron K α emission line at 6.4 keV (from neutral to moderatly ionized iron, i.e. < Fe xvii), is not statistically required by our data with ∆χ 2 <1 for one additional parameter. We found an upper limit (at 90% confidence) for the EW of 12 eV. This value is compatible with the known properties of LMXBs (i.e., less than 10 to 170 eV; Asai et al. 2000), and with extremely high luminosity Radio-Quiet quasars (George et al. 2000), but rather weak for a typical Seyfert galaxy (EW∼100-150 eV, Nandra & Pounds 1994).
Although the single power-law component model provides a rather good fit to the PN spectrum, it is worth investigating whether constrained 2-component models add any further information. We fitted a model combining a power-law with a bb (or a MCD). We let all the parameters free in the fitting procedure. We found a very good representation of the present data, and we found a ∆χ 2 of about 32 for only two additional parameters compared to the one-component power-law model. The results are shown in Table 2. The low value of the temperature together with the hard spectral index found are compatible with a BH in a low hard state, as already observed for example in GX 339-4 (kT∼0.12 keV, Wilms et al. 1999). Due to the very high absorption below 2 keV, we cannot obtain a strong constraint on the normalization factor of the disk component (which is related to the inner radius of the disk), and hence no reliable constraints on the inner radius of a 0.1 keV accretion disk.
A possible explanation of the differences between our spectral fits and those of Cremonesi et al. (1999) may be due to a different state of the object. We checked for possible spectral variations compared to previous observations obtained from the X-ray coded mask telescope ART-P in the 4-20 keV band (Pavlinsky et al. 1994). On the basis of the XMM-Newton measurements (specifically the powerlaw model) we obtain 1.31 +0.13 −0.12 ×10 −2 photon cm −2 s −1 , in the 4-20 keV band which is smaller that the average flux determined using ART-P during Fall 1990 (1.77±0.08 ×10 −2 photon cm −2 s −1 ; Table 1 in Pavlinsky et al. 1994). In fact 1E 1743.1-2843 is known to be variable in hard X-rays from ART-P observations carried out from Spring 1990 to Winter 1992. Figure 3 compares the photon spectrum inferred from the XMM-Newton observation with the average photon spectrum measured by ART-P during Fall 1990. In the more recent observation, the intensity is significantly lower at energies below 6 keV. For example the flux at 3.5 keV is about 7 times lower than during fall 1990. The difference could be due to a change in absorption between the two observations. Above 6 keV the spectral slope appears flatter. This is at least weak evidence for the fact that the source changes of state from time to time.
The observation of 1E 1743.1-2843 at higher energies (E> 30 keV) would certainly help to determine the nature of the object, indeed up to 12 keV a very absorbed blackbody model with kT about 1.8 keV and a very absorbed power-law model have similar shapes. Unfortunately no significant detection above 30 keV of 1E 1743.1-2843 has ever been obtained (see e.g. Goldwurm et al. 1994). This has previously been justified in terms of the source having a soft thermal spectrum; however the XMM-Newton ob- Diamonds: average spectrum obtained with ART-P during Fall 1990 (Pavlinsky et al. 1994). servation indicates that the source can enter states where the spectral form is relatively hard.
Summary and discussion
We report on XMM-Newton observation of the bright X-ray source 1E 1743-2843. During the observation the source flux remained relatively steady at a level corresponding to a 2-10 keV luminosity of about 2.7 × 10 36 d 2 10kpc erg s −1 (assuming a power-law continuum). As in previous X-ray observations, there was no evidence for either X-ray bursts, strong chaotic variability or significant pulsations.
The XMM-Newton spectrum of 1E 1743.1-2843 can be well fitted with simple very absorbed (N H =13-20 × 10 22 cm −2 ), featureless, one-component emission models, such as power-law continuum. We found that the present data are compatible with a black hole candidate in a low/hard state. The fact that the measured spectrum is slightly harder than measured earlier by BeppoSAX and ART-P observations is a tentative indication to a change of state but our spectral coverage is too limited to draw a firm conclusion. The hypothesis of a neutron star LMXB, as proposed earlier by Cremonesi et al. (1999), is not ruled out by our spectral analysis. It could be also an extragalactic source seen through the Galactic Plane. Unfortunately, the data mode employed in the present observation is not suitable for advanced timing analysis, in particular we are unable to investigate whether the source exhibits millisecond X-ray pulsations, which may be detectable from neutron star LMXBs.
It is noteworthy that the X-ray source 1E 1743.1-2843 is located within 20 ′ of the Galactic Center and, in projec-tion, lies on the periphery of SNR G0.33+0.04 where the SNR emission is the brightest at 90 cm (Kassim & Frail 1996). Also 1E 1743.1-2843 lies very close, again in projection, to a giant molecular cloud core GCM+0.25+0.01 (α J2000 =17h 46m 10.1s, δ J2000 =-28 • 42 ′ 48.4 ′′ ), which appears to be located at the GC region, and to contain embedded low-mass star formation (Lis et al. 1994). If 1E 1743.1-2843 is located in the GC region, and behind this cloud, this could explain the high absorption of its soft X-ray flux. In such circumstances we might expect to observe significant 6.4 keV fluorescent line emission in this cloud due to the high X-ray illumination from 1E 1743.1-2843. Indeed, Lis et al. (1994) have already remarked that this cloud might be a further example of an X-ray irradiated reflection nebula. If no significant Fe K α line at 6.4 keV (neutral or moderatly ionized iron) is detected, then one can infer that 1E 1743.1-2843 is far from the Galactic Center region, i.e. at a distance greater than 8 kpc, and has a X-ray luminosity higher than 10 36 erg s −1 . A mosaic with a higher S/N of the Galactic Center region may answer this point (Decourchelle et al. 2003, in preparation). According to the formula of Sunyaev & Churazov (1998) and assuming a minimum projected distance, we obtained a line flux of about 10 −5 erg s −1 , i.e. approximatively 6 times lower than the very bright line emission observed from the giant molecular cloud Sgr B2 (5.6×10 −5 erg s −1 , Murakami et al. 2001). Then 1E 1743.1-2843 could also be a contributor in a smaller part as shown above to the 6.4 keV, iron line observed in Sgr B2 (1E 1743.1-2843 which lies 63 pc away, in terms of the minimum projected distance, Murakami et al. 2000). Long-term monitoring of 1E 1743.1-2843 coupled with a more detailed investigation into its interaction with GC molecular clouds (if any) could therefore be useful for the development of a more complete view of GC activity.
The greatly improved positional constraints from XMM-Newton should, in the future, help in the search for a possible counterpart which, in term, would contribute greatly to our understanding of its true nature. Observations at higher energies are needed to determine the source spectral state, but up to now the location of the source in the crowded Galactic Center region and the limited sensitivity and spatial resolution of the available instrumentation have hampered such an investigation. Observations with the IBIS instrument onboard the INTEGRAL mission, will bring information about the hard X-ray (i.e. > 30 keV) and possible gamma-ray emission of 1E 1743.1-2843 without any spatial confusion, which should help in the determination of its nature. In particular, significant hard X-ray and gamma-ray emissions are expected from BHC in low/hard states.
Fig. 2 .
2The 2-12 keV PN spectrum of 1E 1743.1-2843 (binning at 15σ) and the best-fit power-law continuum model. The lower panel shows the residuals of the data to the model.
Fig. 3 .
3Photon index spectra of 1E 1743.1-2843. Filled circles: present XMM-Newton observation (power-law model).
).0
5
10
15
20
25
TIME (ksec)
1.5
1.7
1.9
2.1
2.3
2.5
2.7
2.9
3.1
3.3
3.5
Counts/s
0
5
10
15
20
25
TIME (ksec)
1.5
1.7
1.9
2.1
2.3
2.5
2.7
2.9
3.1
3.3
3.5
Counts/s
Fig. 1. The 2-10 keV light curve of 1E 1743-2843 measured
with XMM-Newton PN detector after background subtraction.
Top Panel: time binning of 500 s. Bottom Panel: time binning
of 100 s.
Table 1 .
1The results of fitting different continuum models to the PN data in the 2-12 keV energy range. The fluxes correspond to the 2-10 keV band and are corrected for absorption. The units are 10 −10 erg cm −2 s −1 .Models
NH
kT (keV)
χ 2 /d.o.f.
Flux
(10 23 cm −2 )
or Γ
(2-10 keV)
pow
2.02±0.04 1.83±0.05 1111/1141 2.40±0.23
bb
1.31±0.03 1.93±0.03 1206/1141 1.52±0.02
diskbb
1.69±0.03
3.4±0.1
1099/1141 1.89±0.21
Table 2 .
2The results of fitting to the PN data with twocomponent models, in the 2-12 keV energy range.parameters
bb+pow
diskbb+pow
NH
2.10 +0.07
−0.06
2.11 +0.06
−0.05
(10 23 cm −2 )
kT (keV)
0.15 +0.05
−0.04
0.16 +0.04
−0.05
Γ
1.89±0.05
1.89±0.05
χ 2
ν /d.o.f.
1078.7/1139
1078.6/1139
Acknowledgements. This work is based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and the USA (NASA). We would like to thank the anonymous referee for a careful reading of the manuscript. D.
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|
[] |
[
"Parameterization of Fullerenes Via Singularity- Combinations",
"Parameterization of Fullerenes Via Singularity- Combinations"
] |
[
"Shaoqing Li \nAnhui Jianzhu University\nHefeiAnhuiChina\n"
] |
[
"Anhui Jianzhu University\nHefeiAnhuiChina"
] |
[] |
Fullerene can be embedded in a piecewise linear 2-manifold with singularities corresponding to the nonhexagonal faces. Adjacent two or three singularities can be combined as a whole cut out from a parent cone led by a parent singularity. With proper combinations, fullerene can be simplified to a parent structure easily to be parameterized. In this paper, singularity-combinations of fullerenes were studied, and the symmetries of the combining singularities on the parent cone were revealed. Then, fullerenes with different shapes were studied for parameterization with singularity-combinations, and parameters characterizing its shape were obtained for each type. These parameters are practical for fullerenes, especially those far from icosahedrons.
| null |
[
"https://arxiv.org/pdf/2202.06475v1.pdf"
] | 246,822,870 |
2202.06475
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69cc590c419ad3b642a2a86f1e0066f15de88b1c
|
Parameterization of Fullerenes Via Singularity- Combinations
Shaoqing Li
Anhui Jianzhu University
HefeiAnhuiChina
Parameterization of Fullerenes Via Singularity- Combinations
1fullereneparameterizationsingularity-combinationmanifoldparent structure
Fullerene can be embedded in a piecewise linear 2-manifold with singularities corresponding to the nonhexagonal faces. Adjacent two or three singularities can be combined as a whole cut out from a parent cone led by a parent singularity. With proper combinations, fullerene can be simplified to a parent structure easily to be parameterized. In this paper, singularity-combinations of fullerenes were studied, and the symmetries of the combining singularities on the parent cone were revealed. Then, fullerenes with different shapes were studied for parameterization with singularity-combinations, and parameters characterizing its shape were obtained for each type. These parameters are practical for fullerenes, especially those far from icosahedrons.
Introduction
A fullerene graph (fullerene for short) is a polyhedral graph with only pentagonal and hexagonal faces, modeling an allotrope of carbon. It can be embedded in a piecewise linear manifold, which can be obtained through cut-and-glue from a graphene sheet, with singularities corresponding to the non-hexagonal faces [1][2][3]. The latticed manifold is called fullerene manifold, or fullerene for short. As easy to measure, the fullerene manifold is the default model of fullerenes for symmetries and parameterization in literature. Researchers have focused on symmetries and parameterization for decades, trying to establish a systematic theory for the structure of fullerene [4,5].
A fullerene manifold can be treated as the surface of a master polyhedron. With icosahedron (not necessarily regular) as the master polyhedron, the present author has studied the intrinsic symmetries of fullerene and accomplished a general parameterization. However, in some cases, such as carbon nanotubes, the icosahedron model is not perfect as it may be distorted, and the value of some parameters may be negative [1].
In a fullerene manifold, the area around each singularity is a cone. The area around a cluster of singularities is part of a master cone (or cylinder in special). Adjacent two or three singularities can be combined as a cluster cut out from the master cone. If so, the master cone is called the parent cone, and the virtual apex of the master cone is called the parent singularity of the cluster. With a proper combination of singularities, a fullerene may have a simple parent structure. If the parent can be parameterized, the fullerene can also be parameterized.
In 1998, William P. Thurston had proposed that fullerenes can be simplified and parameterized with singularity-combinations [4]. However, the proposal has not attracted enough attention and the related exploration has been delayed for decades.
In this paper, combinations of singularities were studied and parameterizations of different types of fullerenes were explored in light of the combinations. The remainder of the paper is as follows. In section 2 and section 3, the combination of two singularities and the combination of three singularities were studied respectively, and intrinsic symmetries of the combining singularities on their parent cones were revealed. In section 4, in light of the combinations, the caps of tube-like fullerenes are classified and parameterized separately for parameterization of the fullerenes. In Section 5, the octahedral, tetrahedral, and D3 fullerenes were parameterized with the combinations. In section 6, the conclusion and outlook were presented.
This article is a sequel of literature [1]. Readers are recommended to read literature [1] first as many results of [1] are quoted in this paper.
Combination of two singularities
Geometric analysis
Let S, T be two adjacent singularities. Cones S and T can be unfolded together as Figure 1a. First, unfold cone T along a generatrix, and then unfold cone S along line segment ST. In the figure, T1, and T2 represent the different positions of singularity T after unfolding. As the angular defects of the singularities are all 60°, the angle between the two positions of the generatrix is 120° after twice unfolding. The parent cone can be built up by gluing along the generatrix and filling the opening as the surface extension. Let P be the parent singularity of S&T as Figure 1b. Since T1, T2 are coincident after gluing, so, PT1=PT2, ∠T1PT2 =120°, As triangle ST1T2 is equilateral according to unfolding, point P is its center. Then in Figure 1c,
∠SPT1=∠SPT2=120°.
This means that two combining singularities are symmetrically distributed on the parent cone. Their intrinsic field angle from the parent singularity is half of the cone angle, i.e. 120°.
The singularities S and T can be trimmed out from the parent cone P: flatten the surface of the cone P along lines PS, PT, trim off the apex of the cone along the connecting line segment of ST, and seal the remaining part of the cone. Then singularities, each with an angular defect of 60°, will appear at points S and T.
Coxeter coordinates for the parent singularity
Because of the symmetry, the Coxeter coordinates from point P to S and T are identical. Let the Coxeter coordinates be (m, n), and the Coxeter coordinates between S and T be (s, t) as Figure1c, their relationship can be derived as follows with the Eisenstein plane [2].
The hexagonal lattice is congruent to an Eisenstein plane characterized by a triangular lattice. The Eisenstein plane is a complex plan with unit vectors in the six directions denote with 1, ω, ω 2 , ω 3 , ω 4 , ω 5 , where ω = e i2π/6 is the complex root of equation ω 6 = 1. As ω 3 = −1, ω 2 = ω -1, each vector in the Eisenstein plane can be express as (a+bω), with (a, b) known as Eisenstein integers. In Figure 2c,
1 1 ST SP PT =+ so 5 ( ) ( ) = (1 ) = ( 2 ) ( ) s t n m m n n m m n n m n m + = + + + + − + + + + − This means 2 s n m t n m =+ =− (1)
From (1) we can get:
( ) / 3 ( 2 ) / 3 m s t n s t =− =+ (2)
According to formula (2), when s ≡ t mod (3), the Coxeter coordinates m, n are all integers. In this case, the parent singularity corresponds to the center of a square, i.e. a singularity with 4 degrees.
When S and T are not congruent with module 3, then m and n are not integers. In this case, the parent singularity corresponds to a carbon atom. Such parent singularity is called the first kind of unconventional singularity, to distinguish from another kind in section 3.
Combination of three singularities
Geometric analysis
Let A, B, and C be three adjacent singularities. Cones A, B, and C can be unfolded together as Figure 2a. First, unfold cone C along a generatrix, and then unfold cones A, B along lines segments CA, CB respectively. As the angular defects of the singularities are all 60°, the angle between two positions of the generatrix is 180° after triple unfolding. The parent cone can be built up by gluing along the generatrix and filling the opening as the surface extension. Let P be the parent singularity ( Figure 2b). As Points C1 and C2 are coincident after gluing, then, PC1=PC2, ∠C1PC2 =180°. This means point P (marked with ⊕ in Figure 2) is just at the center point of line segment C1C2, and points C1 and C2 are symmetrical about point P. The point C1&C2 after gluing indicates the location of singularity C on the parent cone. Because of this symmetry, only two of the three locations of A, B, and C on the parent cone are independent. As triangles BC1C3 and AC3C2 in Figure 2b are equilateral according to unfolding, with the location of points C1 and B, we can obtain the position for points C2, C3, and then for point A.
The quadrilateral ABC1C2 represents the extension area of the parent cone. If ∠ACB = 60°, then, ∠C1C3B +∠AC3B +∠AC3C2 = 180°. In this case, point
C3 is online C1C2. Similarly, if ∠ ACB > 60°, point C3 is inside the quadrilateral; if ∠ACB < 60°, point C3 is outside the quadrilateral.
The singularities A, B, and C can be truncated out from the parent cone P: take the cone manifold as the surface of triangular pyramid P-ABC, truncate off the apex through points A, B, and C, and extend the lattice to section ABC. Then singularities, each with an angular defect of 60°, will appear at points A, B, and C.
Coxeter coordinates for the parent singularity
Let a, b, c, and Δ be the four parameters for the triangle ABC as literature [1], the Coxeter coordinates from P to A, B, C be (ma, ma), (mb, nb), (mc, nc), separately. Then the Coxeter coordinates between C1 and C2 in Figure 2c is (2mc, 2nc). Just as section 2, with the help of the Eisenstein plane, the following relationship can be obtained:
( ) / 2 ( ) / 2 c c m c a n c b = + + =+ (3)
The expressions of ma, na, mb, and nb can also be obtained with the same method. All these expressions can be written in a formula with matrix： 1 / 2 1 / 2 0 1 1 / 2 1 / 2 0 0 0 1 / 2 1 / 2 1
0 1 / 2 1 / 2 0 1 / 2 0 1 / 2 1 1 / 2 0 1 / 2 0 a b b c c a m n a m b n c m n = (4)
According to formula (4), if a, b, and c have the same parity, the Coxeter coordinates of point A, B, C to point P are all integers. In this case, the parent singularity corresponds to the center of a triangle, i.e. a singularity with three degrees.
If a, b, and c don't have the same parity, the Coxeter coordinates of point A, B, C to point P are not all integers. In this case, the parent singularity does not correspond to the center of a triangle, just as Figure 2d. The parent cone can also be obtained by cut-and-glue from a graphene sheet, with vertex corresponding to the center of a carbon bond. Such parent singularity is called the second kind of unconventional singularity.
Parameterization of caps
Just as carbon nanotubes, most fullerenes have a tube-like structure. The twelve singularities of a tube-like fullerene cluster into two sextuples, each forming a cap of the tube. According to the number of singularities adjacent to the tube, from more to less, the caps can be classified into five types: hexagon, pentagonal pyramid, trimmed square pyramid, truncated triangular pyramid, and trimmed shrunk pyramid. The two caps of one fullerene are not necessarily of the same type.
Cap of pentagonal pyramid If there are five singularities adjacent to the tube, the cap is a pentagonal pyramid. As shown in Figure 3a, five pairs of Coxeter coordinates (mi, ni) for the bottom edges can be selected as the independent parameters for the cap. These parameters determine the relative position of the bottom vertices in the unfolding. As the triangle AB1B2 is equilateral according to the unfolding, the position for apex A is determined, and the detail for the cap can be obtained. Figure 3b, the tube connecting two caps has four dimensions, (m, n) for the feature of the tube and (k, l) for the relative position of the caps. To match the tube, the parameters for the cap should satisfy the following constraint equations
As shown in
Cap of trimmed square pyramid
If there are four singularities adjacent to the tube, the cap can be trimmed out from a square pyramid. Just as figure 4a, four singularities A, B, C, and D are adjacent to the tube. They and the parent singularity P of E&F form a square pyramid P-ABCD.
Similar to Figure 3a, four pairs of Coxeter coordinates of the bottom edges can be selected as the independent parameters for the pyramid. According to section 2, PE = PF,∠EPF=120°. A pair of Coxeter coordinates are needed to locate the singularities E and F. These 10 parameters in total are adequate for the cap of trimmed pyramid.
With the locations E and F, we can draw an equilateral triangle EFG as Figure 4a. The unfolding of the pyramid falling in the triangle is to be trimmed off. Then we get the detailed unfolding of the cap.
As P may be the first kind of unconventional singularity, to avoid non-integer values, the Coxeter coordinates for E/F can refer to the base vertices (point A for example) of the pyramid instead of the apex P.
(b) (a)
Cap of truncated triangular pyramid
If there are three singularities adjacent to the tube, the cap can be truncated out from a triangular pyramid. Just as Figure 4b, three singularities A, B, C of the cap are adjacent to the tube. They and the parent singularity P of D, E, and F form a triangular pyramid P-ABC.
Similar to Figure 3a, the three pairs of Coxeter coordinates of its bottom edges can be selected as its independent parameters. According to section 3. points D1 and D2 are symmetrical about point P, triangles D1D3E and D2D3F are equilateral. Two pairs of Coxeter coordinates are needed for two of three locations of singularities D, E, and F. These 10 parameters in total are adequate for a truncated triangular pyramid.
As P may be the second kind of unconventional singularity. To avoid non-integer values, the Coxeter coordinates for D, E, and F can refer to the base vertices of the pyramid point instead of the apex P (point A for D, point B for E, for example).
Cap of trimmed shrunk pyramid
If there are only two singularities adjacent to the tube, the cap can be transformed into a shrunk pyramid with combinations. The shrunk pyramid has only two lateral faces. Just as Figure 4c, singularities A and B are adjacent to the tube; G is the parent singularity of C and D; H is the parent singularity of E and F. The virtual singularities G and H can also be combined to get a parent singularity P, which is the apex of the shrunk pyramid P-AB. The angular defect of singularity P is 240° as those of G and H are all 120°. Just as the combination of two singularities with angular defects of 60° in section 2, G and H are symmetrically distributed on the parent cone P, and their intrinsic field angle from P is half of the cone angle. Therefore, in the figure, ∠GPH=60°, and the triangle GHP is equilateral.
On the parent cone P, only three locations of four points C, D, E, and F are independent. For example, if points C and D are known, point G can be obtained according section 2, and then point H can also be obtained according to the symmetry mentioned above. If point E is also known, point F can be obtained. With three pairs of Coxeter coordinates for these locations and two pairs of Coxeter coordinates for the base edges of the shrunk pyramid, the detailed unfolding of the cap can be obtained. These five pairs of Coxeter coordinates can be selected as independent parameters.
Cap of hexagon
If six singularities are all adjacent to the tube, the cap is a hexagon as Figure 4d. As a closedloop of edges, its shape can be determined by five of the six pairs of Coxeter coordinates for the edges. These five pairs of Coxeter coordinates can be selected as the independent parameters. With the help of the Eisenstein plane, the Coxeter coordinates for the sixth side can be obtained as:
Independent Parameters for a tube-like fullerene
With an inner point selected as a virtual apex, the hexagon cap can be treated as a hexagonal pyramid. Therefore, each tube-like fullerene has a parent structure as a tube with two caps of pyramids. Just as Figure 3b, to match the tube, the parameters for each pyramid should satisfy two constraint equations similar to formula (5), with different ranges of variable i according to the number of its base edges. For a tube-like fullerene, two pairs of constraint equations can be simplified to two constraint equations without parameters m, n.
Each cap has ten independent parameters. Only 18 of the 20 parameters for the two capes of a tube-like fullerene are independent because of the two constrain equations mentioned above. Together with another two parameters for the relative position of the two caps, as (k, l) in Figure 3, they make up twenty independent parameters for the fullerene.
Parameterization of octahedral/tetrahedral/D3 fullerenes
Except for clustering into two sextuples as a tube-like fullerene, the 12 singularities of a fullerene may also evenly cluster into three quads, four triples, or six pairs. In these cases, they have simple polyhedral parent structures.
Octahedral fullerenes A fullerene whose singularities cluster into six pairs is called an octahedral fullerene, or Ofullerene for short, as its parent structure has an octahedral master polyhedron.
If the parent singularities of the six pairs are all conventional 4-degree singularities, the octahedral fullerene is a trimmed (4,6)-fullerene. As each pair of singularities have two independent parameters according to section 2, a trimmed (4,6)-fullerene has 12 such parameters in total. Together with the 8 independent parameters for the parent (4,6)-fullerene selected in literature [1], they make up 20 independent parameters for the trimmed (4,6)-fullerene.
Just as Figure 5a, if the lattice is refined with one-third-sized hexagons, the first kind of unconventional singularity will become a conventional 4-degree singularity. Therefore, other octahedral fullerenes can be parameterized with the same method as trimmed (4,6)-fullerenes, and the difference is that the value of the parameters may not be integers.
Tetrahedral fullerenes A fullerene whose singularities cluster into four triples is called a tetrahedral fullerene [5], or T-fullerene for short [6], as its parent structure has a tetrahedral master polyhedron.
If the parent singularities for each cluster are all conventional 3-degree singularities, the tetrahedral fullerene is a truncated (3,6)-fullerene. As each triple of singularities has four independent parameters according to section 3, a truncated (3,6)-fullerene has 16 such parameters in total. Together with the 4 independent parameters for the parent (3,6)-fullerene selected in literature [1], they make up 20 independent parameters for the truncated (3,6)-fullerene. Just as Figure 5b, if the lattice is refined with half-sized hexagons, the second kind of unconventional singularities will become conventional 3-degree singularities. Therefore, other tetrahedral fullerenes can also be parameterized as truncated (3,6)-fullerenes, and the difference is that the value of the parameters may not be integers.
The Tetrahedral fullerene mentioned in literature [5] is a truncated (3,6)-fullerene. Its parent fullerene is a Goldberg polyhedron with two independent parameters. The truncations for its four cones are all regular truncations with the same pair of parameters. Therefore, the tetrahedral fullerene has only four independent parameters.
D3-fullerenes
A fullerene whose singularities cluster into three quads is called D3-fullerene [6], as its parent has a master polyhedron as dihedron D3. Just as singularities C, D, E, and F in Figure 4c, each quad has six independent parameters. Therefore, a D3-fullerene has 18 such parameters in total. Its parent structure, the triangular dihedron, has only two independent parameters [4]. Together, they make up 20 independent parameters for the D3-fullerene.
Conclusion and outlook
Fullerenes come in different shapes depending on the distribution of their singularities. Their parent structures which represent their major shape as frameworks can be obtained by singularitycombinations. Fullerenes can be parameterized with the parameters of their parent structures If the lattice is refined with one-third-sized hexagons, the first kind of unconventional singularity will be transformed to a conventional 4-degree singularity. (b) If the lattice is refined with half-sized hexagons, the second kind of unconventional singularity will be transformed to a conventional 3-degree singularity. together with those of the location of the combining singularities. Characterizing the shapes of the fullerenes, these parameters may vary from type to type, but they are practical, especially for those fullerenes far from icosahedrons.
A pair of Coxeter coordinates from a four-degree singularity has two possible trimming results. Two pairs of Coxeter coordinates from a three-degree singularity can correspond to up to six possible truncations. Therefore, it is essential to specify the Coxeter coordinates for parameters (illustrate in the unfolding graph for example) or eliminate ambiguity with further classification.
Fig. 1 .
1combination of two singularities. (a) Together unfolding of two adjacent cones S&T. (b) Position of the parent singularity P. (c) The relationship of S, T, and P. (d) Unconventional singularity.
Fig. 2 .
2combination of three singularities. (a) Together unfolding of three adjacent cones A, B, and C. (b) Position of the parent singularity P. (c) The relationship of A, B, C and P. (d) Unconventional singularity.
4
Parameterization of tube-like fullerenes
Fig. 3 .
3The cap of pentagonal pyramid. (a) Independent parameters for the cap of pentagonal pyramid. (b) Unfolding of fullerene with two caps of pentagonal pyramids.
Fig. 4 .
4Other types of caps. (a) Cap of trimmed square pyramid. (b) Cap of truncated triangular pyramid. (c) Cap of trimmed shrunk pyramid. (d) Cap of hexagon.
Fig. 5 .
5Transformation of unconventional singularities. (a)
On the intrinsic symmetries and parameterization of fullerenes. S Li, Comput. Theor. Chem. 1200S. Li, On the intrinsic symmetries and parameterization of fullerenes, Comput. Theor. Chem. 1200 (2021).
The topology of fullerenes. P Schwerdtfeger, L N Wirz, J Avery, Wiley Interdiscip. Rev. Comput. Mol. Sci. 5P. Schwerdtfeger, L.N. Wirz, J. Avery, The topology of fullerenes, Wiley Interdiscip. Rev. Comput. Mol. Sci. 5 (2015) 96-145.
The Clar structure of fullerenes. E J Hartung, 112Syracuse UniversityE.J. Hartung, The Clar structure of fullerenes, Syracuse University, 2012, pp. 112.
Shapes of polyhedra and triangulations of the sphere. W P Thurston, Geom. Top. Monographs. 1W.P. Thurston, Shapes of polyhedra and triangulations of the sphere, Geom. Top. Monographs, 1 (1998) 511- 549.
P W Fowler, D E Manolopoulos, An atlas of fullerenes. Mineola, NYDover Publications Inc2nd ed.P.W. Fowler, D.E. Manolopoulos, An atlas of fullerenes, 2nd ed. Mineola, NY: Dover Publications Inc.2006.
Structure and properties of the non-face-spiral fullerenes T-C380, D3-C384, D3-C440 and D3-C672 and their halma and leapfrog transforms. L N Wirz, J E Avery, P Schwerdtfeger, J Chem Inf Model. 54Wirz LN, Avery JE, Schwerdtfeger P. Structure and properties of the non-face-spiral fullerenes T-C380, D3-C384, D3-C440 and D3-C672 and their halma and leapfrog transforms. J Chem Inf Model 2014, 54: 121-130.
|
[] |
[
"Multiscaling in the sequence of areas enclosed by coalescing random walkers",
"Multiscaling in the sequence of areas enclosed by coalescing random walkers"
] |
[
"Peter Welinder ",
"Gunnar Pruessner [email protected] \nMathematics Institute\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK\n",
"Kim Christensen ",
"\nCondensed Matter Theory Group, Blackett Laboratory\nImperial College London\nPrince Consort RdSW7 2BWLondonUK\n"
] |
[
"Mathematics Institute\nUniversity of Warwick\nGibbet Hill RoadCV4 7ALCoventryUK",
"Condensed Matter Theory Group, Blackett Laboratory\nImperial College London\nPrince Consort RdSW7 2BWLondonUK"
] |
[] |
We address the question whether the sequence of areas between coalescing random walkers displays multiscaling and in the process calculate the second moment as well as the two point correlation function exactly. The scaling of higher order correlation functions is estimated numerically, indicating a logarithmic dependence on the system size. Together with the analytical results, this confirms the presence of multiscaling.
|
10.1088/1367-2630/9/5/149
|
[
"https://arxiv.org/pdf/cond-mat/0702684v1.pdf"
] | 55,678,628 |
cond-mat/0702684
|
386d7ecb8b8c298e26493a952ae5ae551a448ae1
|
Multiscaling in the sequence of areas enclosed by coalescing random walkers
28 Feb 2007
Peter Welinder
Gunnar Pruessner [email protected]
Mathematics Institute
University of Warwick
Gibbet Hill RoadCV4 7ALCoventryUK
Kim Christensen
Condensed Matter Theory Group, Blackett Laboratory
Imperial College London
Prince Consort RdSW7 2BWLondonUK
Multiscaling in the sequence of areas enclosed by coalescing random walkers
28 Feb 2007Submitted to: NJPnumbers: 0540Fb0250-r0540-a
We address the question whether the sequence of areas between coalescing random walkers displays multiscaling and in the process calculate the second moment as well as the two point correlation function exactly. The scaling of higher order correlation functions is estimated numerically, indicating a logarithmic dependence on the system size. Together with the analytical results, this confirms the presence of multiscaling.
Introduction
Gap scaling is found frequently in the context of scale invariance, such as equilibrium statistical mechanics of phase transitions [1], growth phenomena [2], reaction diffusion processes [3] or self organised criticality [4,5]. In the present context, gap scaling [1] means that the nth moment of the observable s, denoted as s n , in leading order scales like t ∆(1+n−τ ) in some large parameter t. The "gap" refers to the constant difference ∆ between two exponents for n and n + 1. In a stochastic process recently introduced as the "totally asymmetric Oslo model" [6,7], the moments of the area under a random walk along an absorbing wall was found to obey gap scaling with τ = 4/3 and ∆ = 3/2.
If the moments scale asymptotically as power laws (possibly with logarithmic corrections) but the exponents do not increase linearly with n, they are said to display multiscaling. The presence of multiscaling is sometimes confused with the absence of scaling altogether and very few examples of simple dynamical processes are known, which can be shown by exact calculation to display multiscaling. It is therefore highly desirable to find a simple stochastic process, such as Brownian motion, for which multiscaling can be derived analytically. Motivated by a recent study of cluster aggregation [8,9], we study the scaling of "sequential moments" or, more accurately, correlation functions of the area size between the trajectories of coalescing random walkers, illustrated in Figure 1. We hope that future research will make the link between our results and the results recently obtained by Munasinghe et al [10] on the scaling of the n-point correlation function of coalescing random walkers.
Recent years have seen a considerable interest in the distribution of the area size under a random walker trajectory. It has been investigated in the context of "directed rice piles" [7,11], as well as in the context of extreme value statistics [12,13]. In both cases, the object of interest is the area between trajectories of coalescing random walkers with initial spacing x 0 , or, equivalently, the area under the trajectory of a random walker, which starts at time t ′ = 0 at distance r(t ′ = 0) = x 0 away from an absorbing wall. In the next section, the model will be introduced in detail.
Addressing problems of extreme value statistics, Majumdar and Comtet [13] constrain the ensemble to trajectories with r(t ′ = 0) = x 0 and, in addition, with r(t ′ = t) = x 0 at a particular "termination time" t ′ = t in the limit x 0 → 0. In this limit, they derive very elegantly the exact distribution function of the area sizes. Due to the lack of additional scales, and, correspondingly, due to the lack of a dimensionless parameter, in the case x 0 → 0 the scaling of all observables is readily derived from dimensional analysis, i.e. the exponents are immediately known and gap-scaling is a necessity for all existing moments. In contrast, we will be concerned with finite x 0 (so that a priori neither exponents nor type of scaling are known) and an ensemble not constrained by the termination time in the form r(t ′ = t) = x 0 . This problem has been addressed earlier [11] using some of the results in [13] for a calculation of the distribution function of the areas in leading order of x 0 .
Unfortunately, only the first moment, which is trivial, is known exactly. Very little is
x 0 t ′ 0 t r 0 (t) r 1 (t) s 1 s 2 s 3 s 4 s 5 s 6 s 7 s 8 s 9 Figure 1. A sequence of trajectories r 0 (t ′ ), r 1 (t ′ ), .
. . starting at t ′ = 0 with initial spacing x 0 and demarcating areas s 1 , s 2 , . . . between them. The dashed lines indicate the "walls" at t ′ = 0 and the terminating time t ′ = t. Whenever random walkers meet, they coalesce and perform the rest of the walk together.
known about higher moments beyond leading order. We employ analytical calculations, based on the formalism introduced in [7] as well as extensive numerics to overcome this problem. In the course, we determine the exact second moment of the distribution, as well as the two point correlation function, which displays anti-correlations, sometimes interpreted as a hallmark of multiscaling [10,14]. To our knowledge, these are the first non-trivial exact results for the area size distribution under a random walker with x 0 > 0.
At least three correlation functions are required to decide about the presence of multiscaling. So far, we have been unable to perform the calculation of the third moment analytically. We have therefore resorted to Monte Carlo techniques, which are, unfortunately, hampered by slow convergence due to rare events. Nevertheless, the numerics strongly suggests multiscaling.
In the next section, we will introduce the model and its observables. The third section contains most of the analytical and numerical results, which are briefly summarised in the last section. Figure 1 shows a lattice realisation (see below) of a sequence of random walkers i = 0, 1, 2, . . . with diffusion constant D 0 and trajectories r i (t ′ ) which start at t ′ = 0 from r i (t ′ = 0) = ix 0 . Whenever two random walkers meet, they coalesce and the resulting, single walker continues its walk. The walkers are indistinguishable and it is therefore irrelevant whether walkers actually run simultaneously or sequentially, in which latter case they are thought to stop as soon as they intersect the trajectory of another walker. This is the picture we will adopt henceforth.
Model
For the sake of definiteness, all trajectories are considered to run from t ′ = 0 to t ′ = t. If two walker coalesce, their trajectories coincide for the remaining time. The area s i between two walkers i − 1 and i with trajectories r i−1 (t ′ ) and r i (t ′ ) respectively,
s i = t 0 dt ′ (r i (t ′ ) − r i−1 (t ′ )) ≥ 0 (1)
is a random variable, the moments of which will be denoted by s n , where is the expectation value over all configurations.
Given an individual trajectory, it is impossible to tell whether it is produced by merging walkers or not. Producing the trajectories sequentially, the only trajectory relevant to the fate of walker i is the trajectory i − 1. There is no transient (other than producing the very first trajectory), i.e. stationarity is reached immediately. That means, for example, that the expectation value s 1 s 2 is translation invariant,
s 1 s 2 = s i s i+1 for all i ≥ 1.
If the random walkers are unbiased, any sequence of areas (s i , s i+1 , . . . , s i+n ) is, by construction, as likely as the sequence (s i+n , s i+n−1 , . . . , s i ). This can be seen by mirroring the entire sequence which, in case of an unbiased walk, leaves the probabilities of individual trajectories unchanged. In the presence of a bias this is no longer case and so the joint probability of a sequence changes under inversion. Thus, in the presence of a bias, one expects, for example, s 2 1 s 2 = s 2 2 s 1 , while the two averages would be equal in the absence of a bias. Of course, the observable itself might be inversion invariant, for example s 1 s 2 = s 2 s 1 regardless of a bias.
As long as the two paths r i and r i−1 have not merged, the variance of the difference r i −r i−1 corresponds to the variance of a random walker with twice the diffusion constant D = 2D 0 , walking along an absorbing wall. However, while the individual trajectories r i might be biased, i.e. in the presence of a drift, the difference does not have such a bias. It is therefore clear that whatever is said about the (single) area size distribution between random walkers with initial spacing x 0 , biased or not, equally applies to the (single) area size distribution under an unbiased random walker with twice the diffusion constant starting at distance r(t = 0) = x 0 from an absorbing wall. However, the notion of a sequence is missing in the latter case.
The main aim of this work is to calculate the correlation functions s 1 s 2 , s 1 s 2 s 3 , and s 1 s 2 s 3 s 4 and determine their asymptotes, that means the exponent γ n and possibly the polynomial P so that lim t→∞ s 1 s 2 . . . s n t γn P(ln t) = c n > 0 (2) suspecting that γ n is not linear in n and/or that P is non-trivial.
As it turns out, s 1 s 2 and indeed s 1 s n+1 can be derived from s 2 , the calculation of which is detailed in the first part of the result section. The other two moments could only be tackled numerically, presented in the second part of the result section.
Most asymptotes will be determined in the limit of large t. It is important to realise the rôle x 0 plays in this context. The problem is fully parametrised by t, D and x 0 . With the area having the same dimension as t √ Dt, in case of vanishing x 0 the scaling of the expectation value of any product of these areas, say s 1 s 2 3 , is fixed to be a corresponding power of t √ Dt, i.e. s 1 s 2 3 ∝ t 9/2 . In fact, there cannot even be any corrections to this behaviour. However, if x 0 = 0, there is a dimensionless quantity Dt/x 2 0 , which can and, indeed, does change the exponents of the asymptotes and gives rise to corrections to scaling [15].
Results
In this section, we will first discuss in detail the analytical results, starting with a brief review of the method which is used to calculate the second moment s 2 in closed form. After deriving its expansion in powers of t, some shortcomings of the perturbative approach of [7] are discussed. Using these results, the correlation function s 1 s n+1 is derived, which includes the original problem s 1 s 2 as a special case. This part finishes with a discussion on the effect of anti-correlations and a brief section on higher moments.
The second part of the result section is concerned with our numerical results. First, the lattice effects are estimated by a comparison of the numerical estimate for s 1 s 2 with the analytical result. After discussing some technical problems, the results for s 1 s 2 s 3 and s 1 s 2 s 3 s 4 are shown, indicating the presence of logarithmic corrections.
Exact calculations
When calculating correlation functions of the form s 1 s 2 , one of the central observations is that they can be obtained by considering "local moments" of the form s 2 in a system with twice the initial spacing, ‡ which will be discussed in detail below. We therefore start the section with the calculation of the second moment and its implications for earlier results, in particular the approximation scheme used in [7]. Knowing s 2 then allows us to calculate s 1 s n+1 .
All analytical calculations are based on the hierarchy of differential equations introduced in [7],
∂ t ψ n (t, x; x 0 ) = D∂ 2 x ψ n (t, x; x 0 ) + xnψ n−1 (t, x; x 0 )(3)
with boundary conditions
lim t→0 ψ n (t, x; x 0 ) = δ n,0 δ(x − x 0 ) (4) ψ n (t, 0; x 0 ) = 0(5)
Starting with n = 0, ψ 0 (x, t; x 0 ) is the probability to find at x a fair random walker along an absorbing wall with diffusion constant D = 2D 0 that started at x 0 . For n > 0, ψ n is the local contribution to the n-th moment of the area under the trajectory, see [7]. For n = 0 one finds, consistently, s 0 = 1, for n > 0 one has
s n (t; x 0 ) = t 0 dt ′ ∞ 0 dx ′ x ′ nψ n−1 (t ′ , x ′ ; x 0 ) .(6)
‡ We thank Alan Bray for sharing this insight with us.
To ease notation, the dimensionless form of ψ n is introduced as
ψ n (x, t; x 0 ) = 1 x 0 x 3 0 D n ψ n (y, τ )(7)
with y = x/x 0 and τ = t/(x 2 0 /D), so that the differential equation reads
∂ τ ψ n (τ, y) = ∂ 2 y ψ n (τ, y) + yn ψ n−1 (τ, y)(8)
with δ(x−x 0 ) in boundary condition (4) replaced by δ(y−1). The propagator G(y, τ ; y 0 ) of Equation (3) can be constructed easily,
G(y, τ ; y 0 ) ≡ 1 √ 4τ π e − (y−y 0 ) 2 4τ − e − (y+y 0 ) 2 4τ ,(9)
which at y 0 = 1 coincides with ψ 0 (y, τ ), i.e.
ψ 0 (y, τ ) = G(y, τ ; 1) = 1 √ τ π e − y 2 +1 4τ sinh y 2τ .(10)
The formal solution is the hierarchy
ψ n (y, τ ) = τ 0 dτ ′ ∞ 0 dy ′ ny ′ ψ n−1 (y ′ , τ ′ )G(y, τ − τ ′ ; y ′ )(11)
which is, as it turns out, not easily calculated in closed form. Only the first moment is calculated immediately
s (t; x 0 ) = t 0 dt ′ ∞ 0 dx ′ x ′ ψ 0 (t ′ , x ′ ; x 0 ) = x 0 t .(12)
It has been shown [7,11] that the moments follow gap scaling,
lim t→∞ s n (t; x 0 ) t ∆(1+n−τ ) = x 0 D (n−1)/2 C n(13)
with τ = 4/3 and ∆ = 3/2 and C n dimensionless. Using the hierarchy (11) and the definition of the moments (6), the second moment is
s 2 (t; x 0 ) = 2 x 3 0 D 2 T 0 dτ ∞ 0 dyy τ 0 dτ ′ ∞ 0 dy ′ y ′ × 1 4(τ − τ ′ )π e − (y−y ′ ) 2 4(τ −τ ′ ) − e − (y+y ′ ) 2 4(τ −τ ′ ) 1 √ 4τ ′ π e − (y ′ −1) 2 4τ ′ − e − (y ′ +1) 2 4τ ′(14)
where T = T (t; x 0 ) = t x 2 0 /D . The integration of (14) would be fairly straight forward, was it not for the factors y and y ′ .
There is no reason to expect convergence problems from the improper integrals, because the Gaussians (and even more so differences of Gaussians) effectively cut them off. We therefore feel confident when changing the integration order, so that the improper integrals are done first and in reverse order. The integration over y can be done immediately and produces only a factor of the form y ′ 4(τ − τ ′ )π. The integral over y ′ is slightly more complicated, as it is of the form dy ′ y ′2 G(y ′ , τ ′ ; 1), and it produces an exponential and an error function, all multiplied by powers of τ ′ . Remarkably, the resulting integrand for the integration over τ ′ and τ is independent of τ , so that the integral over τ ′ can be replaced by the same expression evaluated at τ with a prefactor T − τ . Due to the presence of pre-factors involving various powers of τ , the final integration over τ produces many different terms:
s 2 (t; x 0 ) = 1 180 x 3 0 D 2 (1 + 28T + 132T 2 ) 2 √ π √ T e − 1 4T − (1 + 30T ) + (1 + 30T + 180T 2 + 120T 3 )E 1 √ 4T (15) = 1 180 x 3 0 D 2 1 + 28 tD x 2 0 + 132 tD x 2 0 2 2 √ π √ tD x 0 e − x 2 0 4tD − 1 + 30 tD x 2 0 + 1 + 30 tD x 2 0 + 180 tD x 2 0 2 + 120 tD x 2 0 3 E x 0 √ 4tD (16) where E is the error function E(u) = (2/ √ π) u 0 dxe −x 2 .
To our knowledge this is the first non-trivial moment of the area distribution under a random walker that has been calculated exactly.
It is very instructive to expand the result in powers of T = T (t;
x 0 ) = t/(x 2 0 /D): s 2 (t; x 0 ) (x 3 0 /D) 2 = 32 15 √ π T 5/2 + 8 9 √ π T 3/2 − 1 6 T + 1 15 √ π T 1/2 − 1 180 + 1 1260 √ π T −1/2 − 1 181440 √ π T −3/2 + 1 13305600 √ π T −5/2 + O T −7/2 .(17)
with the terms beyond leading order to be considered corrections to scaling [15]. There are two terms without a factor π −1/2 , namely − 1 6 T and − 1 180 , which are present in the same form, i.e. without any further factor, already in the exact expression (15). What makes these terms very different from all others is that they are not an odd-half power of T . In fact, to any order, all terms apart from −(1 + 30T )/180, are odd-half powers of T . In [7] only the odd-half powers had been anticipated, suggesting that no deviation from this form is possible, but that turns out to be incorrect now.
It is instructive to understand why the systematic, perturbative expansion in [7] misses the two terms. It relies on a perturbative expansion of ψ 0 ( √ µy, µτ ), (10), for large µ at fixed, finite y and τ . The scaling in µ is then inherited by ψ 1 ( √ µy, µτ ) through the integral (11) and by s 2 (µt; x 0 ) through (6). This relation gives rise to the scaling of s 2 (t; x 0 ) by parameterising t in convenient units, t = µt 0 . Along the lines of a renormalisation group calculation [16], µ can be considered the "flow parameter" and t 0 = x 2 0 /D as the normalisation point. Using this procedure for all orders in µ produces the prediction of only odd-half powers of µ.
However, when evaluating the perturbative expansion of ψ 0 ( √ µy, µτ ) in powers of µ in the integrals producing ψ 1 and s 2 , one violates the premise that y and τ are finite, as in the integral (11) τ runs from 0 to a finite value and the integral over y is to be evaluated at divergent upper bound. As a result, one finds integrals of the form
T 0 dτ τ 0 dτ ′ τ ′1/2−i for i = 0, 1, 2, .
. .. For i = 0, 1 the integrals are convergent, producing terms of order T 5/2 and T 3/2 respectively, however for i ≥ 2 one (i = 2) or both (i > 2) integrals diverge in the lower limit. Ignoring this divergence, i.e. only keeping the upper limit, produces the correct amplitudes nevertheless. We have tested the integrals for i = 0, 1, . . . , 5 by direct evaluation and comparison to the exact result in the form of the Taylor expansion (17), of course, with the terms of order T 1 and T 0 missing. In general, the integrals for the perturbative expansion give the following odd-half powers:
s 2 (T ; x 0 ) + x 3 0 D 2 1 + 30T 180 = 1 2 1 √ π x 3 0 D 2 ∞ i=0 T 5/2−i i j=0 4 2+j−i 3 2 − i 5 2 − i (j + 1)!(−) i−j (2j + 1)!(i − j)! .(18)
The terms 1/(3/2 − i) and 1/(5/2 − i) are reminiscent of the integration T 0 dτ τ 0 dτ ′ τ ′1/2−i . As long as none of the coefficients or the error term is divergent, the series expansion can be considered a simple Taylor series in µ and therefore in particular the leading order terms are reliable, as in an asymptotic expansion.
Similarly, one can use the perturbative expansion scheme to determine the amplitude of higher orders of s 3 . We can confirm agreement with [11] of the amplitude of the leading order T 4 , expecting the next to leading order T 3 to be correctly predicted in the perturbative approach as well. Following the integration for s 2 , now three integrals over τ might diverge and one might speculate that the first term missed in the perturbative expansion is T 5/2 and the (actually divergent) amplitude for T 2 to be correct, a T 3/2 to be missed, T 1 to be correct, T 1/2 missed again, and terms of order T 0 and lower to be correct again.
We now turn back to the expansion of the exact result, Equation (15). When it comes to the large distance behaviour of the correlation function, the second moment s 2 (t; x 0 ) will be evaluated for small T . Expanding Equation (15) for small T gives
s 2 (t; x 0 )(19)= x 3 0 D 2 T 2 + 2 3 T 3 + e − 1 4T 1 √ π −512T 13/2 + 28672T 15/2 − O T 17/2 .
Again, the leading, polynomial orders, which together with the pre-factor give just s 2 (t; x 0 ) = x 2 0 t 2 + (2/3)Dt 3 + . . ., are solely due to the two unusual terms with integer powers identified in (15). However, it is consistent with the method in [7] being a large T expansion, that they would have been missed there. All other terms in (19) are exponentially suppressed with a remarkably high leading power in the polynomial. In fact, the first six terms, starting with T 1/2 , all turn out to have vanishing amplitude.
and more generally with S = s 2 + s 3 + . . . + s n−2 for n ≥ 3 (where S = 0 at n = 3) Figure 2. Pruning the network of trajectories, such as the one shown in figure 1, by leaving only every nth trajectory produces an ensemble of walkers with initial spacing nx 0 (trajectories removed are shown dotted). Pruning is identical to merging the enclosed areas, in this example n = 3 so that s ′ 1 = s 1 + s 2 + s 3 , s ′ 2 = s 4 + s 5 + s 6 ,. . . . Such a joining/pruning scheme, mapping an ensemble with initial spacing x 0 onto an ensemble with initial spacing nx 0 is exact, even on the lattice.
(s 1 + S) 2 (t; x 0 ) − 2 (s 1 + S + s n−1 ) 2 (t; x 0 ) + (s 1 + S + s n−1 + s n ) 2 (t; x 0 ) = 2 s 1 s n (t; x 0 ) ≡ 2c n−1 (t; x 0 ) (21) t ′ 0 t 3x 0 s ′ 1 s ′ 2 s ′ 3
for n ≥ 0. Given Equation (15), the second moment u n (t; x 0 ) is easily calculated because u n (t; x 0 ) = u 1 (t; nx 0 ) = s 2 (t; nx 0 ). This is illustrated in Figure 2: The network of trajectories is pruned by joining n consecutive areas s 1 , . . . , s n to one large area. Thereby a sequence of larger areas is constructed, as if produced by random walkers with initial spacing nx 0 . The exact two point correlation function therefore is
s 1 s n+1 (t; x 0 ) = 1 2 s 2 (t; |n − 1|x 0 ) − 2 s 2 (t; |n|x 0 ) + s 2 (t; |n + 1|x 0 )(23)
for n ∈ Z, with the exact expression given in (15). The modulus guarantees the validity even for n ≤ 0, although the above results for s 2 (t; x 0 ) have been derived using x 0 > 0. In the following, the asymptotes of s 1 s n+1 are examined. We first consider the behaviour of s 1 s n+1 (t; x 0 ) in the limit of large t at fixed n. By dimensional analysis, s 2 (t; x 0 ) consists of a pre-factor (x 3 0 /D) 2 and a dimensionless function in T = Dt/x 2 0 . According to the Taylor expansion (17), a term of order T 5/2 gives rise to a contribution in total linear in x 0 , and lower order terms in T correspond to larger powers in x 0 . Under the provision of large T according to Equation (17) the powers are x 0 , x 3 0 , x 4 0 , x 5 0 , x 6 0 , x 7 0 , x 9 0 , . . .. When taking the lattice Laplacian according to (23), Equation (17) is evaluated term by term for x 0 → nx 0 and x 0 → (n ± 1)x 0 , which enters in the pre-factor (x 3 0 /D) 2 as well as in T (t; x 0 ) = tD/x 2 0 . Since (n + 1) − 2n + (n − 1) = 0, the first power of x 0 to contribute therefore is x 3 0 , so that in the limit of fixed n > 0 and large t s 1 s n+1 (t; x 0 )
= x 3 0 D 2 8n 3 √ π T 3/2 (t; x 0 ) − 1 + 6n 2 6 T (t; x 0 ) + n + 2n 3 3 √ π T 1/2 (t; x 0 ) + O T 0 ,(24)
again in terms of T = T (t; x 0 ) = tD/x 2 0 . § As discussed below, s 1 s n+1 − s 2 < 0 for all T greater than some threshold and n = 0.
Of course, in the limit of large n at fixed t, one expects asymptotic independence,
lim n→∞ s 1 s n+1 (t; x 0 ) = s 2 (t; x 0 ) = (tx 0 ) 2 = x 3 0 D 2 T 2 (t; x 0 ) .(25)
This large n behaviour can be obtained from the small T expansion, Equation (19). The polynomial part T 2 together with the pre-factor (x 3 0 /D) 2 contributes the expected constant (tx 0 ) 2 , while T 3 (x 3 0 /D) 2 is independent of x 0 and its contribution vanishes therefore after operating with the lattice Laplacian. The exponential prefactor exp(−1/(4T (t; nx 0 ))) = exp(−n 2 /(4T (t; x 0 ))) renders the contribution from n−1 exponentially more relevant than from n or n + 1:
s 1 s n+1 (t; x 0 ) = (tx 0 ) 2 + x 3 0 D 2 256 √ π T 13/2 e −(n−1) 2 1 4T × −n −7 − 7n −8 + 28(2T − 1)n −9 + O n −10 + O e −n 2 1 4T(26)
in the limit of large n at fixed t.
Anti-correlations
The Laplacian structure (22) gives rise to rather peculiar behaviour of the error of the numerical estimate of s . In a naïve implementation one probably takes a series of N samples, s 1 , s 2 , . . . , s N and estimates s by
s = 1 N N i=1 s i ,(27)
which is indeed an unbiased estimator, s = s [17], independent of N. In the following, we will use an over-bar to indicate numerical estimates, for example s for the estimate of s . Considering its error, for large N one arrives at a picture as cartooned in Figure 3:
The walkers start from r 0 (t ′ = 0) = 0, r 1 (0) = x 0 , r 2 (0) = 2x 0 , . . . , r N (0) = Nx 0 , so that for sufficiently large N the very first one, r 0 , is extremely unlikely to interact with the very last one, r N . The total area between r 0 and r N is exactly the sum over all areas,
Ns = Ntx 0 + t 0 dt ′ [(r N (t ′ ) − Nx 0 ) − r 0 (t ′ )] .(28)
Assuming that the first and the last are in fact independent, suggests that the total area is well modelled by a Langevin type approach r 0,N (t) − r 0,N (t = 0) = t 0 dt ′ η 0,N (t ′ ) § To avoid confusion henceforth the short hand T (without arguments) is used whenever it is to be evaluated at t and x 0 , i.e. where any n dependence has been removed, with two independent noise sources η 0,N (t) with vanishing mean and correlator η 0,
T (t; nx 0 ) = n −2 T (t; x 0 ) = n −2 T . 0 tN (t 1 )η 0,N (t 2 ) = 2Γ 2 δ(t 1 − t 2 )
. This correlator has variance r 2 0,N (t) = 2Γ 2 t = 2D 0 t, so that Γ 2 = D 0 = D/2. Assuming r 0 and r N are independent, the variance of the numerical estimate is 2Dt 3 /(3N 2 ), which means that the error in the estimate from the correlated sequence s 1 , . . . , s N vanishes faster than in the fully uncorrelated case, where the variance decays like σ 2 (s)/N = 32D 1/2 t 5/2 x 0 /(15N √ π)+. . ., with σ 2 (s) ≡ s 2 − s 2 .
This surprising behaviour is explained by anti-correlations. To see this more clearly, we point out the "sum rule"
s 2 = 1 N N i=1 s i 2 (t; x 0 ) = 2 N 2 N −1 i=0 (N − i) s 1 s i+1 (t; x 0 ) − 1 N s 2 (t; x 0 ) = 1 N 2 (s 1 + . . . + s N ) 2 = 1 N 2 s 2 (t; Nx 0 )(29)
where the equality of the first and second line is reminiscent of the Laplacian property (22). When considering large N, comparing to (19) confirms,
s 2 − s 2 = 2 3N 2 Dt 3 + . . .(30)
and more importantly sharply from
σ 2 (s)(t; x 0 ) = s 2 (t; x 0 )− s 2 (t; x 0 ) = x 3 0 D 2 32 15 √ π T 5/2 − T 2 + 8 9 √ π T 3/2 + . . .(32)
at n = 0, Equation (17), to
s 1 s 1+n (t; x 0 ) − s 2 (t; x 0 ) = x 3 0 D 2 −T 2 + 8 3 √ π nT 3/2 + . . .(33)
at fixed n > 0, see (24). Of course, in the limit of large n asymptotic independence prevails and only exponentially decaying correlations are left
s 1 s 1+n (t; x 0 ) − s 2 (t; x 0 ) (34) = x 3 0 D 2 256 √ π T 13/2 e −(n−1) 2 1 4T − 1 n 7 − 7 n 8 + 28(2T − 1) n 9 + O n −10 + O e −n 2 1
4T see (26). Figure 4 shows s 1 s 1+n together with its numerical estimate.
Other moments
In the pursuit to determine whether the correlation functions s 1 , s 1 s 2 , s 1 s 2 s 3 , . . . display gap scaling or multiscaling in T , any third correlation function, other than s ∝ T , (12), or s 1 s 2 ∝ T 3/2 , (24), is to be determined. Along the same lines as above and using translational invariance, one finds
s 1 s 2 s 3 = 1 6 (s 1 + s 2 + s 3 ) 3 − 2 (s 1 + s 2 ) 3 − (s 1 + s 3 ) 3 + 3 s 3 (35) = 1 6 (s 1 + s 2 + s 3 ) 3 − 3 s 3 − 6 s 2 1 s 2 − 6 s 1 s 2 2 − 3 s 2 1 s 3 − 3 s 1 s 2 3(36)
with the leading order of s 3 ∝ T 4 known from [7,11]. In case of an unbiased random walk, the second line simplifies further, for example s 2 1 s 2 = s 1 s 2 2 . Unfortunately, the above expression does not suffice to determine the leading order of s 1 s 2 s 3 : On the right hand side of Equation (35) (s 1 + s 3 ) 3 is unknown, on the right hand side of Equation (36) effectively the same term is unknown, namely s 2 1 s 3 + s 1 s 2 3 . The remaining terms can be determined using the form suggested above,
s 3 (t; x 0 ) = x 3 0 D 3 C 1 T 4 + C 2 T 3 + . . . ,(37)
where only the leading fourth order has so far been proven to exist with C 1 = 15/8. In particular,
s 2 1 s 2 + s 1 s 2 2 = 1 3 (s 1 + s 2 ) 3 − 2 s 3 = x 3 0 D 3 2C 2 T 3 + . . . (38)
and similarly for the leading order of
6 s 1 s 2 s 3 + (s 1 + s 3 ) 3 = (s 1 + s 2 + s 3 ) 3 − 2 (s 1 + s 2 ) 3 + 3 s 3 (39) = 2C 1 T 4 + 14C 2 T 3 + . . . ,
which is (35) with all unknown terms on the left hand side and all known terms on the right hand side. So, if s 1 s 2 s 3 has leading order higher or equal to (s 1 + s 3 ) 3 , it must be T 4 , given that the leading order always has positive amplitude. If it is of lower order, it might not appear on the right hand side at all, since it might be cancelled by a negative amplitude from (s 1 + s 3 ) 3 . This is potentially a very useful numerical criterion, because if s 1 s 2 s 3 / (s 1 + s 3 ) 3 does not converge to 0, it means that s 1 s 2 s 3 has leading order T 4 ; from numerics, however, it seems that the ratio asymptotically vanishes.
Numerics
All numerical simulations have been done on the lattice, all analytical calculations in the continuum. On the lattice, in every time step the random walker takes one step in the up or down direction with probabilities p and q ≡ 1 − p respectively. The resulting diffusion constant is calculated from the variance after n steps, 4npq = 2D 0 t with t = n∆t and units chosen so that ∆t = 1 and D = 2D 0 = 4pq. In most of our simulations, the walker was unbiased, p = q = 1/2, implying D = 1, and the initial spacing is chosen to be minimal, x 0 = 2. ¶ This choice could, in principle, introduce discretisation effects, which are discussed below.
The random number generator used throughout this study is the "Mersenne Twister" [18]. ¶ Note that x i (t ′ ) is even at even times and odd at odd times.
The central objective of the numerical approach was to determine the large T behaviour of s 1 s 2 s 3 and s 1 s 2 s 3 s 4 . As mentioned earlier, the most naïve implementation estimates s 1 s 2 . . . s n+1 by
s 1 s 2 . . . s n+1 = 1 N − n N −n i=1 s i s i+1 . . . s i+n(40)
which, however, introduces correlations. To avoid that, one can produce independent realisations at a significantly higher cost: While N trajectories produce N −n correlated samples, they produce only N/(n + 1) uncorrelated samples, using a new set of n + 1 trajectories for every realisation of s 1 , s 2 , . . . , s n . Obviously, a correlated sample, for instance s 1 s 2 , s 2 s 3 , . . . , s 7 s 8 (N = 9, n = 2), contains multiple uncorrelated sub-samples, for example s 1 s 2 , s 4 s 5 , s 7 s 8 but also s 2 s 3 , s 5 s 6 etc. The correlated sample therefore converges at least as fast as its uncorrelated sub-samples, however, possibly including anti-correlations. It is very important to produce large samples, because of "exponentially rare but important events" [19]. For example, estimating s 1 s 2 s 3 we found an exceptionally large event where s 1 s 2 s 3 was almost 10 12 times bigger than the average s 1 s 2 s 3 up to that point, estimated from about 3.4·10 11 sequential samples, for x 0 = 2 and t = 2 17 . At the end of the simulations, the inclusion of the extreme event still increased the average s 1 s 2 s 3 by a factor of 3.7.
In addition to the statistical error, there are errors from the discretisation and, closely related, the size of x 0 . Ideally, in a numerical simulation of a continuous random walk, x 0 and t were chosen as large as possible, which is, however, computationally very costly. To estimate the influence of lattice effects due to small x 0 , we compared s 1 s 2 (t; x 0 ) to the exact result (15) for different x 0 = 2, 4, 8, 32. The result is shown in Figure 5 in the form s 1 s 2 / s 1 s 2 versus t. As expected, it suggests large x 0 to avoid lattice effects, which clashes with the demand for large T = tD/x 2 0 to determine large time asymptotes in T . Comparing the product of the square of the statistical error and the CPU-time spent on the results for different x 0 indicates that a sequential, i.e. correlated, see (40), simulation of x 0 = 2 is most efficient.
For technical reasons, the error bars shown in the graphs are estimated from the variance of the estimator for small subsamples. For example, the independent samples were created from 1800 individual runs, each consisting of about 7.2 · 10 6 to 61 · 10 6 independent samples totalling to 4.1 · 10 10 samples, produced by fair random walks as described above. However, while s 1 s 2 suggests satisfactory agreement between discrete numerical and continuous analytical results, as mentioned above, rare events made it very hard to estimate s 1 s 2 s 3 or higher correlation functions. The most obvious way to overcome this problem is to use importance sampling [20] which, however, requires a method to produce these rare, important events and a control over their frequency. We decided to introduce a repulsive potential between the walkers which therefore run simultaneously rather than sequentially. Having no analytical reference other than s and s 1 s 2 , it is difficult to assess the quality of this approach. Given the strong impact Sfrag replacements Figure 5. The ratio of the numerical estimate s 1 s 2 and the theoretical value s 1 s 2 . The number of samples varies between 4 · 10 10 and 1.16 · 10 12 . The symbols are shifted relatively to each other to reduce clutter, the error bars span two standard deviations in total. The dashed line marks perfect agreement between numerics and theory. Data marked (i) is from using independent samples, data marked with (s) comes from sequential, correlated runs, Equation (40). The data points marked with arrows have been produced by importance sampling, (r). of rare events on s 1 s 2 s 3 , the quality of the numerical estimate s 1 s 2 compared to s 1 s 2 has only very limited indicative power for the quality of s 1 s 2 s 3 . It is remarkable how large the error of the importance sampling scheme is for s 1 s 2 compared to the naïve methods, see Figure 5, and how small it is for s 1 s 2 s 3 , see Figure 6. Moreover, the numerically estimated error bar of any estimator is expected to suffer from the same problems as the estimator itself, i.e. too small or wrongly biased samples.
x 0 = 2 (r) x 0 = 2 (i) x 0 = 2 (s) x 0 = 4 (s) x 0 = 8 (s) x 0 = 32 (s) t s 1 s 2 / s 1 s 2
The key result for s 1 s 2 s 3 is shown in Figure 6 in the form s 1 s 2 s 3 /[(x 3 0 /D) 3 T 3/2 ], where we divide by (x 3 0 /D) 3 to render the result dimensionless and by T 3/2 to remove the suspected leading order. It remains somewhat inconclusive whether the fit to a logarithm in the intermediate region breaks down for the sequential runs at large T only because of a lack of statistics. The latter is, however, consistent with the observation that underestimation of s 1 s 2 s 3 is more than twice as frequent as overestimation even when averaging over about 4·10 8 samples. The estimates produced from the importance sampling method seem to confirm the presence of the logarithm even for the largest t = 131072.
From the present result, we can quite confidently rule out gap scaling, which would PSfrag replacements Figure 6. The numerical estimate s 1 s 2 s 3 in the form s 1 s 2 s 3 /[(x 3 0 /D) 3 T 3/2 ] in a loglin plot. Symbols corresponding to different initial spacings x 0 are shifted relatively to each other to reduce clutter. Data marked (i) is from independent samples, (s) from sequential samples, (r) from importance sampling with repulsive potential. The straight line is a fit to a logarithm a ln(T /b) to guide the eye. The importance sampling result (see arrows) fully agrees with the logarithmic fit in the intermediate region of the naïve scheme. It remains unclear why for large T the results from independent samples deviate so significantly from the logarithmic behaviour. require s 1 s 2 s 3 /t 2 to converge in the limit of large t to a non-vanishing value. A direct measure for the validity of gap scaling is a moment ratio of the form [6]
x 0 = 2 (r) x 0 = 2 (i) x 0 = 2 (s) x 0 = 4 (s) x 0 = 8 (s) x 0 = 32 (s) T s 1 s 2 s 3 /[(x 3 0 /D) 3 T 3/2 ]
which converges to a non-vanishing value if the observables obey gap scaling, irrespective of the value of the exponents. The numerical estimate of g 3 is shown in Figure 7 (using the analytical results for s and s 1 s 2 ). Its decay to 0 indicates once more that s 1 s 2 s 3 does not diverge fast enough. Along the same lines, Figure 8 shows s 1 s 2 s 3 s 4 /[(x 3 0 /D) 4 T 3/2 ] together with a fit to a parabola in ln(T ). While the fit is not perfect and the data is, apart from the result from importance sampling, slightly inconsistent for large T , the suggested parabola seems plausible. The data is certainly not consistent with gap scaling.
Summary
Motivated by the question whether the sequence of the areas between trajectories of coalescing random walkers exhibits multiscaling, we have calculated the second moment and subsequently the two point correlation function exactly. The correlation function displays anti-correlations, as was discovered in the analysis of the variance of the numerical estimator of the first moment. Anti-correlations are sometimes regarded as indicating multiscaling [10,14].
Three moments are necessary to expose the absence of gap scaling, which necessitated a numerical calculation. Due to the presence of rare but important events, convergence is slow, yet an importance sampling scheme produces results consistent with logarithmic scaling, which is often observed in conjunction with multiscaling [10,14].
While the analytical proof for multiscaling in the present system is still lacking, the numerical results indicate it convincingly. Two possible avenues may be pursued in the future: An exact solution along the lines discussed in section 3.1.3, or an exact calculation based on the work by Munasinghe et al [10,14].
3. 1 . 1 .
11Two point correlation function Based on s 2 one can derive the two point correlation function by noting that (s 1 + s 2 ) 2 − s 2 1 − s 2 2 = 2 s 1 s 2
s 1 + S)s n−1 = (S + s n−1 )s n . With u n (t; x 0 ) = (s 1 + . . . + s n ) 2 (t; x 0 ) and the convention u −1 = u 1 and u 0 = 0, the correlation function is just the lattice Laplacian of the second moment u n (t; x 0 ), 2 s 1 s n+1 (t; x 0 ) = u n+1 (t; x 0 ) − 2u n (t; x 0 ) + u n−1 (t; x 0 ) = ∇ 2 n u n
Figure 3 .
3Estimating the average area s by considering a sequence s 1 , . . . , s N is equivalent to considering the entire area enclosed by the first and the last trajectory, r 0 and r N (shown in bold) and dividing by the total number of walkers N .
s j − s 2 = −Nσ 2 (s)(t; x 0 ) + s 2 (t; Nx 0 ) − N 2 s 2 (t; x 0 ) = − Nσ 2 (s) + 2 3 Dt 3 + . . .(31)In other words: In the variance of the estimator(27), the anti-correlations fall short of cancelling the contribution of the variance of the area σ 2 (s)/N by only a small correction 2Dt 3 /(3N 2 ), see (30). Thus, in the limit of large T , the correlator s 1 s n+1 − s 2 drops
Figure 4 .
4The numerical estimate of the correlation function s 1 s 1+n for n = 1, 2, . . . , 130 for x 0 = 2, t = 4096 and D = 1, shown in the form s 1 s 1+n / s 2 where s = x 0 t. The symbols represent numerical estimates (no error bar is shown), the straight line is the theoretical result (23). The second moment, n = 0, is not shown because it is orders of magnitude greater than s 2 .
Figure 7 .
7The moment ratio g 3 , Equation (41), would converge to a nonzero value in the case of gap scaling. As in previous figures, data are shifted to avoid clutter. They show different initial spacings x 0 and different simulation methods [(i) independent, (s) sequential, (r) repulsive].
Figure 8 .
8The numerical estimate s 1 s 2 s 3 s 4 in the form s 1 s 2 s 3 s 4 /[(x 3 0 /D) 4 T 3/2 ] in a log-lin plot, similar to Figure 6. Data is shifted to avoid clutter, and shows different initial spacings x 0 and different simulation techniques. The full line is a fit to a parabola in the logarithm of T , i.e. a ln(T /b) + c ln(T /b) 2 .
AcknowledgmentsThe authors would like to thank A. Thomas, M. J. Morris, D. Moore, N. Emarporats and R. Toumi for technical support for the SCAN computing facility. GP would like to thank A. Bray and C. Connaughton for very useful discussions, and O. Peters, R. Ecke, S. Revell and P. Watson for hospitality. PW would like to thank the Nuffield foundation for generous support (Grant No. URB/33147).
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"ON THE ANDRÉ MOTIVE OF CERTAIN IRREDUCIBLE SYMPLECTIC VARIETIES",
"ON THE ANDRÉ MOTIVE OF CERTAIN IRREDUCIBLE SYMPLECTIC VARIETIES"
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[
"Ulrich Schlickewei "
] |
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[] |
We show that if Y is an algebraic deformation of the Hilbert square of a K3 surface, then the André motive of Y is an object of the category generated be the motive of Y truncated in degree 2. arXiv:0909.1889v1 [math.AG]
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10.1007/s10711-011-9594-z
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[
"https://arxiv.org/pdf/0909.1889v1.pdf"
] | 17,621,493 |
0909.1889
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2f740ac8195f7f56cd8ea2b86bd68f17a5e22b09
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ON THE ANDRÉ MOTIVE OF CERTAIN IRREDUCIBLE SYMPLECTIC VARIETIES
Ulrich Schlickewei
ON THE ANDRÉ MOTIVE OF CERTAIN IRREDUCIBLE SYMPLECTIC VARIETIES
We show that if Y is an algebraic deformation of the Hilbert square of a K3 surface, then the André motive of Y is an object of the category generated be the motive of Y truncated in degree 2. arXiv:0909.1889v1 [math.AG]
An irreducible symplectic variety Y is a smooth, projective variety over C which is simply connected and which admits a nowhere degenerate, holomorphic two-form σ ∈ H 0 (Y, Ω 2 Y ). A general principle says that most of the geometry of Y is encoded by the cohomology group H 2 (Y, Z) together with the Hodge decomposition and the Beauville-Bogomolov quadratic form. Beauville [B] found two series of examples of irreducible symplectic varieties. Apart from these only two exceptional examples have been discovered by O'Grady [O'G2], [O'G3].
In this note, we study the André motive of irreducible symplectic varieties which are deformation equivalent to the Hilbert scheme of points on a K3 surface or equivalently which are deformations of a smooth, compact moduli space of stable sheaves on a K3 surface. In all even dimensions, there is one family of such deformations, these families build one of Beauville's series of examples. Let Y be such a variety. Denote by h 2 (Y ) the André motive of Y truncated in degree 2. We use results of Markman on the monodromy of moduli spaces of sheaves on K3 surfaces and André's deformation principle to derive Theorem. a)The motive of Y is an object of h 2 (Y ) , the category generated by h 2 (Y ).
b) The motive of Y is an object of the category generated by motives of Abelian varieties. c) All Hodge classes on Y are motivated, hence absolute in the sense of Deligne.
Item a) can be seen as a motivic manifestation of the above-mentioned principle. Items b) and c) are consequences of a) and of André's results [An2] on degree 2 motives of Hyperkähler varieties.
It has been proved by Arapura [Ar] that the motive of a moduli space Y parametrizing sheaves on a K3 surface S is an object of the category generated by the motive of S. Since the motive of S is also an object of h 2 (Y ) , our result can be seen as a generalization of Arapura's result. After quickly reviewing André's motives in Section 1, we collect in Section 2 some of Markman's results [M1], [M2] on the cohomology of moduli spaces of sheaves on K3 surfaces and on their monodromy groups. The proof of our result is given in Section 3.
André motives
The idea of the category of motives is to provide the target of a universal cohomology functor for smooth, projective varieties. Grothendieck dreamed of a category which should be Tannakian and semisimple. However, Jannsen [J] proved that Grothendieck's category of homological motives can only be semisimple if homological and numerical equivalence of algebraic cycles coincide on all varieties. This is one of Grothendieck's standard conjectures which are widely open.
In order to circumvent the standard conjectures and to obtain nonetheless a Tannakian and semisimple category of motives, André introduced a category in which he formally inverted the dual Lefschetz operator. The basic ingredient in the theory is the notion of a motivated cohomology class. Let X be a smooth, projective variety over C. A cohomology class α ∈ H * (X, Q) is motivated if there exist a smooth, projective variety Y and algebraic cycles
Z 1 , Z 2 on X × Y such that α = p 1, * ([Z 1 ] ∪ Λ H×G [Z 2 ]) .
Here, p 1 : X × Y → X and p 2 : X × Y → Y are the projections, and Λ H×G is the dual Lefschetz operator with respect to some product polarization p * 1 H + p * 2 G. Clearly, algebraic cohomology classes are motivated. Vice versa, Grothendieck's standard conjectures would imply that motivated cohomology classes are algebraic.
André [An1] proves that the category of motives defined in terms of motivated correspondences is Tannakian and semisimple over Q. The Betti realization which maps a motive to the underlying singular cohomology group is a conservative fibre functor. (Recall that a functor F : C → C is conservative if a morphism f in C is an isomorphism if and only if so is F (f ).)
One of the big advantages of motivated cohomology classes is that examples are rather easy to produce. This is mainly due to the following result which gives a positive answer to Grothendieck's invariant cycle conjecture in the motivated world.
Theorem 1.1 (André, see [An1], 5.1). Let f : X → S be a smooth projective morphism where S is a smooth, connected, algebraic variety. Let s ∈ S be a closed point, m, n ∈ N and let
α ∈ H * (X s , Q) ⊗n ⊗ H * (X s , Q) ∨ ⊗m
be a motivated class which is invariant under a subgroup of finite index of π 1 (S, s) (acting on H * (S, Q) via the monodromy representation). Then any translate of α under parallel transport to
H * (X t , Q) ⊗n ⊗ (H * (X t , Q) ∨ ) ⊗m for t ∈ S is motivated on X t .
Markman's results
Let S be a projective K3 surface, polarized by an ample divisor H. The Todd genus of S is td(S) = 1 + 2[x] where x is an arbitrary point of S. A square root is given by td(S) = 1 + [x].
For a coherent sheaf E on S define the Mukai vector by
v(E) = ch(E) td(S) ∈ H * (S, Z).
We associate with S a rational weight two Hodge structure
H(S, Q) := H * (S, Q), H 2,0 (S) = H 2,0 (S), H 1,1 (S) = H 0 (S) ⊕ H 1,1 (S) ⊕ H 4 (S).
There is a natural duality operator
D S : H(S, Q) → H(S, Q)
acting as (−1) i id on H 2i (S, Q). Since the Künneth components of the diagonal in S × S are algebraic, D S is given by an algebraic class.
The Mukai pairing on H(S, Q) is given by
α, β = − S D S (α) ∪ β.
This is a non-degenerate, symmetric bilinear form of signature (4+, 20−).
Let now v ∈ H * (S, Z) be a primitive and effective (cf. [M2, Def. 1.1]) vector. Then by results of Mukai, Huybrechts, O'Grady and Yoshioka, there exist a polarization H on S and a non-empty, smooth, projective variety X := M H (v) which parametrizes H-stable sheaves with Mukai vector v on S. Moreover, X is an irreducible symplectic variety of dimension d = v, v + 2 which is deformation equivalent to Hilb d 2 (S). We assume that d > 2 and for simplicity we assume that X is a fine moduli space.
Let E be a universal sheaf on S × X. Then E is uniquely determined up to the twist by the pull-back of a line bundle from X. Denote by p : S × X → S and by q : S × X → X the projections, let π H 2 : H * (X, Q) → H 2 (X, Q) be the projection in degree 2. Define
ϕ 1 : H(S, Q) → H 2 (X, Q), α → π H 2 q * ch(E) ∪ p * td(S) ∪ p * D S (α) .
According to a result of O'Grady [O'G1], the restriction of ϕ 1 to v ⊥ is an isomorphism of Hodge structures (even over Z).
We normalize the correspondence ch(E)p * td(S) following [M2, Lemma 3.1]: let η :=
ϕ 1 (v) v, v −1 ∈ H 2 (X, Q). Put (1) u := ch(E) ∪ p * td(S) ∪ q * exp(η).
Then u is independent of the universal sheaf E and we define
ϕ 1 : H(S, Q) → H 2 (X, Q), α → π H 2 q * (u ∪ p * D S (α)) .
Note that
ϕ 1 (α) = ϕ 1 (α) − α, v v, v ϕ 1 (v).
This implies that ϕ 1 (v) = 0 and that ϕ 1|v ⊥ = ϕ 1|v ⊥ . Next, we note that ϕ 1 is an algebraic correspondence. This is, because u and D S are so and because the projection H * (X, Q) → H 2 (X, Q) is algebraic (cf. [Ar] where the conjecture B is shown for X).
Since the standard conjecture B holds for S as well, there is an algebraic right inverse ψ : H 2 (X, Q) → H(S, Q) (see [K,Cor. 3.14]). Since the (Mukai-)orthogonal projection H(S, Q) → Qv is given on S ×S by the class −( v, v −1 D S (v))⊗v, the orthogonal projection H(S, Q) → v ⊥ is algebraic. Thus we may assume that ψ induces an isomorphism
(2) ψ : H 2 (X, Q) ∼ → v ⊥ ⊂ H(S, Q)
which is inverse to ϕ 1|v ⊥ .
Let G v be the fix group of v in Aut( H(S, Z), , ). Markman defines two representations of G v on H * (X, Z). We will now describe both of them.
1.) Let p ij be the projection from X × S × X to the (i, j)-th factor. For g ∈ G v set
γ g := (p 13 ) * p * 12 D X×S (id ⊗ g)( t u) ∪ p * 23 u −1 ∈ H * (X × X, Q),
where D X×S is the duality operator acting by (−1) i on H 2i (X × S, Q) and the class u was introduced in (1). Let l : H * (X, Q) → H * (X, Q) be the universal polynomial map which takes the Chern character (r + a 1 + a 2 + . . .) of a coherent sheaf to its total Chern class (1 + a 1 + ( a 2 1 2 − a 2 ) + . . .). Then by definition γ g := degree d part of l(γ g ) = c d (γ g ).
Theorem 2.1 (Markman, [M2], Thm. 3.10 and Cor. 3.14). i) For g ∈ G v the correspondence γ g acts as a (degree-preserving) automorphism on H * (X, Q).
ii) The map γ : G v → Aut(H * (X, Q)), g → γ g is a faithful representation of G v .
iii) The class u ∈ H(S, Q) ⊗ H * (X, Q) is invariant under the product representation of G v , where G v acts on the first factor via the natural representation.
The theorem implies that the algebraic maps
(3) ϕ i : H(S, Q) → H 2i (X, Q), α → π H 2i q * (u ∪ p * D S (α)) are G v -equivariant.
2.) To define the second representation of G v recall that the space H(S, Q) has four positive directions. Given any two positive four-spaces F and F , orientations of these spaces can be compared using orthogonal projections. An isometry g ∈ Aut( H(S, Q), , ) is called orientation preserving if for an oriented, positive four-space F the space g(F ) has the same orientation. This induces the covariance or orientation character (4) cov : G → Z/2Z sending g to 0 or 1 according to whether it preserves orientations or not.
Then Markman defines the representation
γ mon : G v → Aut(H * (X, Q)), g → (D X ) cov(g) • γ g
where again D X is the duality operator of X, acting by (−1) i id on H 2i (X). The subscript is justified by the following result of Markman. An element g ∈ Aut(H * (X, Q)) is called a monodromy operator if there exist a family X → B of Hyperkähler manifolds with fibre X b = X for some b ∈ B and a g ∈ π 1 (B, b) such that g is the image of g under the monodromy representation π 1 (B, b) → Aut(H * (X, Q)). Let Mon(X) be the subgroup of Aut(H * (X, Q)) generated by monodromy operators.
Theorem 2.2 (Markman, [M2], Thm. 1.6). The image of the representation γ mon : G v → Aut(H * (X, Q)) is a normal subgroup of finite index in Mon(X).
In particular, since γ and γ mon coincide on the kernel N of the orientation character, its image N := γ(N ) in Aut(H * (X, Q)) is a subgroup of finite index in Mon(X).
Following an idea of Beauville, Markman had proved in previous work that the class of the diagonal in X × X can be expressed in terms of the Chern classes of the universal sheaf E. This implies Summary of results used in the sequel. We have seen that there are homomorphisms ϕ i : H(S, Q) → H 2i (X, Q) and ψ : H 2 (X, Q) → v ⊥ with the following properties:
i) The ϕ i and ψ are induced by algebraic cycles on S × X resp. on X × S.
ii) The homomorphism ϕ 1 induces an isomorphism v ⊥ → H 2 (X, Q) whose inverse is ψ.
iii) There is a subgroup of finite index N ⊂ Mon(X) such that the compositions
η i,0 := ϕ i • ψ : H 2 (X, Q) → H 2i (X, Q)
are N -equivariant. This follows from Theorem 2.1 and 2.2. iv) For i ≥ 2, the classes ϕ i (v) ∈ H 2i (X, Q) are N -invariant. Again, this is implied by Theorem 2.1.
v) The sum of H 0 (X, Q), of the image of ⊕ i≥1 η i,0 and of the ϕ i (v) generate the cohomology ring H * (X, Q) as a Q-algebra. This is a consequence of Theorem 2.3.
3. Proof of the Theorem a) Let X = M H (v) be as in the previous section, let Y be a fixed algebraic deformation of X. By this we mean that there exists a smooth, projective morphism of connected, smooth, complex algebraic varieties X → B which admits X and Y as fibers.
We have to prove that h(Y ) is an object of h 2 (Y ) . Recall that by definition, this is the smallest full subcategory of the category M of André motives which contains h 2 (Y ) and the unit object 1 = h(Spec(C)), which is stable under ⊗ and under duals and which contains all subobjects resp. quotients in M of objects in h 2 (Y ) .
The idea is to identify H(S, Q) with G(X) = H 2 (X, Q) ⊕ Q and to use Markman's results to define a surjection of a sum of products of G(X) to H * (X, Q) which is monodromy invariant. By André's deformation principle, this will induce a surjection of a motive m(Y ) to h(Y ) where m(Y ) is an object of h 2 (Y ) . Let's make this precise now.
For any fibre V of X → B, let g 0 (V ) := h 0 (V ) 1 and for i = 1, . . . , d = dim(X) define
g i (V ) := h 2 (V ) ⊕ 1(−1) (−i + 1). (For V = X = M H (v), the motive g i (X) plays the role of h(S)(−i + 1) = h 0 (S)(−1) ⊕ h 2 (S) ⊕ h 4 (S)(1) (−i + 1).) Next, we put m(V ) := (i 1 ,...,i d )∈{0,...,d} d g i 1 (V ) ⊗ . . . ⊗ g i d (V ) .
Note that m(V ) is an object of h 2 (V ) and that m(V ) can be seen as a submotive of the motive of a variety Z(V ) which is a disjoint union of products of V and P 1 .
We fix an isomorphism η 0 : 1 → h 0 (X). For i = 1, . . . , d we will define below morphisms of
motives η i : g i (X) → h 2i (X)
with the following properties:
1) there exists a subgroup N of finite index in Mon(X) such that η i is N -invariant.
2) if we define the morphism η : m(X) → h(X)
as the composition of the morphism
(η i 1 ⊗ . . . ⊗ η i d ) : m(V ) → (i 1 ...,i d )∈{0,...,d} d h 2i 1 (X) ⊗ . . . ⊗ h 2i d (X)
with the cup-product morphism
h 2i 1 (X) ⊗ . . . ⊗ h 2i d (X) → h(X),
then η is surjective.
Assume for one moment, that the η i are defined. Consider the family Z → B which is constructed by letting vary the Z(V ) over B. The morphism η corresponds to a motivated cohomology class on Z(X)×X. Property 1) implies that this class is invariant under a subgroup of finite index of the monodromy group of the family Z × B X → B. By André's Theorem 1.1 we get a surjection m(Y ) → h(Y ). Since m(Y ) is an object of h 2 (Y ) and since this category is closed under quotients in M, the proof is reduced to the construction of the η i .
Let η i,0 : h 2 (X)(−i + 1) → h 2i (X) be the morphism of motives corresponding to the algebraic homomorphism η i,0 in item iii) in the summary at the end of the last section. Next, we define η i,1 : 1(−i) → h 2i (X) as the motivated cohomlogy class η i,1 = ϕ i (v) ∈ H 2i (X, Q). Finally we define η i := η i,0 ⊕ η i,1 : g i (X) → h 2i (X).
Property 1) has been checked in items iii) and iv) at the end of the preceding section.
Property 2) is a direct consequence of item v). There we have seen that the Betti realization of η is surjective. But the Betti realization is a conservative functor. Thus, η is surjective in M. This proves a) b) is a direct consequence of [An2,Thm. 1.5.1]. This theorem says that the motive h 2 (Y ) of an irreducible symplectic variety Y is an object of M(Ab), the smallest Tannakian subcategory of the category of André motives which contains the motives of Abelian varieties. The proof of this theorem relies on the Kuga-Satake correspondence. André shows that the Kuga-Satake homomorphism P 2 (Y ) → H 2 (A × A, Q) is motivated where P 2 (Y ) is the primitive part of H 2 (Y, Q) with respect to some polarization and A is a Kuga-Satake variety for P 2 (Y ). Thus, p 2 (Y ), the motive corresponding to P 2 (Y ), and hence also h 2 (Y ) are objects of M(Ab). c) follows from b) and from [An1,Thm. 0.6.2], which says that all Hodge classes on Abelian varieties are motivated. The proof of this theorem uses the deformation principle to reduce first to Abelian varieties with CM, then to Weil classes and finally to products of elliptic curves.
Theorem 2 . 3 (
23Markman,[M1], Cor.2). The Künneth factors of the Chern classes of the universal sheaf E generate H * (X, Q).
This paper is a part of my Ph.D. thesis prepared at the University of Bonn. I would like to thank my advisor Daniel Huybrechts for his continuous encouragement. Moreover, I am grateful to Eyal Markman for valuable comments on a previous version of this note. This work was supported by the SFB/TR 45 'Periods, Moduli Spaces and Arithmetic of Algebraic Varieties' of the DFG (German Research Foundation) and by the Bonn International Graduate School in Mathematics (BIGS).
Pour une théorie inconditionnelle des motifs. Y André, Publ. Math. I.H.E.S. 83Y. André, Pour une théorie inconditionnelle des motifs, Publ. Math. I.H.E.S. 83 (1996), 5-49.
On the Shafarevich and Tate conjectures for hyperkähler varieties. Y André, Math. Ann. 305Y. André, On the Shafarevich and Tate conjectures for hyperkähler varieties, Math. Ann. 305 (1996), 205-248.
Motivation for Hodge cycles. D Arapura, Adv. Math. 2072D. Arapura, Motivation for Hodge cycles, Adv. Math. 207, no. 2 (2006), 762-781.
Variétés Kählériennes dont la première classe de Chern est nulle. A Beauville, J. Diff. Geom. 18A. Beauville, Variétés Kählériennes dont la première classe de Chern est nulle, J. Diff. Geom. 18 (1983), 755-782.
Motives, numerical equivalence, and semi-simplicity. U Jannsen, Invent. Math. 1073U. Jannsen, Motives, numerical equivalence, and semi-simplicity, Invent. Math. 107, no. 3 (1992), 447- 452.
Algebraic cycles and the Weil conjectures. S Kleiman, Dix exposés sur la cohomologie des schémas. North-Holland; AmsterdamS. Kleiman, Algebraic cycles and the Weil conjectures, in: Dix exposés sur la cohomologie des schémas, North-Holland, Amsterdam (1968), 359-386.
Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces. E Markman, J. reine angew. Math. 544E. Markman, Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces, J. reine angew. Math. 544 (2002), 61-82.
On the monodromy of moduli spaces of sheaves on K3 surfaces. E Markman, J. Alg. Geom. 17E. Markman, On the monodromy of moduli spaces of sheaves on K3 surfaces, J. Alg. Geom. 17 (2008), 29-99.
The weight-two Hodge structure of moduli spaces of sheaves on a K3 surface. K O'grady, J. Alg. Geom. 6K. O'Grady, The weight-two Hodge structure of moduli spaces of sheaves on a K3 surface, J. Alg. Geom. 6 (1997), 599-644.
Desingularized moduli spaces of sheaves on a K3. K O'grady, J. Reine Angew. Math. 512K. O'Grady, Desingularized moduli spaces of sheaves on a K3, J. Reine Angew. Math. 512 (1999), 49-117.
A new six-dimensional irreducible symplectic variety. K O'grady, J. Alg. Geom. 123K. O'Grady, A new six-dimensional irreducible symplectic variety, J. Alg. Geom. 12, no. 3 (2003), 435-505.
|
[] |
[
"A Compact Fermion To Qubit Mapping",
"A Compact Fermion To Qubit Mapping"
] |
[
"Charles Derby \nDepartment of Computer Science\nPhasecraft Ltd\nUniversity College London\nPhasecraft Ltd\n\n",
"Joel Klassen \nDepartment of Computer Science\nPhasecraft Ltd\nUniversity College London\nPhasecraft Ltd\n\n"
] |
[
"Department of Computer Science\nPhasecraft Ltd\nUniversity College London\nPhasecraft Ltd\n",
"Department of Computer Science\nPhasecraft Ltd\nUniversity College London\nPhasecraft Ltd\n"
] |
[] |
Mappings between fermions and qubits are valuable constructions in physics. To date only a handful exist. In addition to revealing dualities between fermionic and spin systems, such mappings are indispensable in any quantum simulation of fermionic physics on quantum computers. The number of qubits required per fermionic mode, and the locality of mapped fermionic operators strongly impact the cost of such simulations. We present a novel fermion to qubit mapping which outperforms all previous local mappings in both the qubit to mode ratio, and the locality of mapped operators.
|
10.1103/physrevb.104.035118
|
[
"https://arxiv.org/pdf/2003.06939v2.pdf"
] | 227,072,953 |
2003.06939
|
e5cf0e5390712ed45d48c3a9c6d20ed86bd7e364
|
A Compact Fermion To Qubit Mapping
Charles Derby
Department of Computer Science
Phasecraft Ltd
University College London
Phasecraft Ltd
Joel Klassen
Department of Computer Science
Phasecraft Ltd
University College London
Phasecraft Ltd
A Compact Fermion To Qubit Mapping
Mappings between fermions and qubits are valuable constructions in physics. To date only a handful exist. In addition to revealing dualities between fermionic and spin systems, such mappings are indispensable in any quantum simulation of fermionic physics on quantum computers. The number of qubits required per fermionic mode, and the locality of mapped fermionic operators strongly impact the cost of such simulations. We present a novel fermion to qubit mapping which outperforms all previous local mappings in both the qubit to mode ratio, and the locality of mapped operators.
One of the most striking features of fermions is the nonlocality of their state space. This non-locality is necessitated by their anti-symmetric exchange statistics -the phase of the wavefunction yielded by a fermion tracing a path past an even number of its counterparts differs from that yielded by a path past an odd number. However causality is preserved by parity superselection [1], which forbids superpositions of even and odd fermion number, preventing the direct measurement of these phase differences.
A consequence of this non-locality is that any representation of fermionic systems on collections of local quantum systems, such as qubits or distinguishable spins, must introduce non-local structure [2]. This is most readily seen in the Jordan-Wigner (JW) transform [3], which maps fermionic creation (a † i ) and annihilation (a i ) operators, which create and annihilate a fermion at mode i and satisfy the canonical anti-commutation relations
{a † i , a j } = δ ij , {a † i , a † j } = 0 , {a i , a j } = 0,(1)
to string-like Pauli operators
a † i → 1 2 Z 1 ...Z i−1 (X i − iY i ) .(2)
Under this mapping even local observables conserving fermion parity, such as lattice hopping terms (a † i a j + a † j a i ), are mapped to strings of Pauli operators which may be as large as the size of the system. The JW transform is an example of a mapping between fermions and qubits. Such mappings describe a correspondence between states of fermions and states of qubits, or, equivalently, between fermionic operators and multi-qubit operators. They are restricted to fermionic systems with a discrete set of modes, since qubits posses finite dimensional Hilbert spaces, and are typically applied to fermionic lattice models. Many mappings are tailored to specific lattices.
A potential application where fermion to qubit mappings would be indispensable is the simulation of fermions by quantum computers. The accurate simulation of fermions has long posed a fundamental challenge to classical computers. Since the conception of quantum computers it has been understood that one of their primary applications would be in addressing this challenge, with substantial potential impact on a broad range of scientific disciplines.
Using a fermion to qubit mapping, a fermionic Hamiltonian may be mapped to a qubit Hamiltonian H = i H i , with the terms H i constituting tensor products of Pauli operators. A quantum computer can perform an effective simulation of the fermionic Hamiltonian by simulating time dynamics under H. The primary strategy to do this is via a Trotter expansion, which consists of dividing the time evolution unitary into a product of short evolutions generated by the terms H i [4][5][6]. This is followed by a further decomposition of these short evolutions into sequences of quantum gates. However the greater the number of qubits on which these individual Hamiltonian terms H i act -ie the Pauli weight -the more costly the circuit decomposition [7,8]. Similar considerations also inform the performance of other quantum algorithms, such as VQE [9]. Thus there has emerged a practical need to design fermion to qubit mappings which minimize the Pauli weight of commonplace fermionic operators. In particular those fermionic interactions which couple nearby fermionic modes -ie geometrically local operators -which feature prominently in physically realistic systems.
The JW transform performs poorly in this respect because all of the requisite fermionic non-locality is manifest in the observables, as opposed to the states -the fermionic Fock states map directly to seperable binary states. One may instead design a mapping which encodes the non-locality in the states, by mapping fermionic states into a highly entangled subspace of the multi-qubit system. In this way one can retrieve low weight qubit representations of geometrically local fermionic operators. We refer to such mappings as local mappings.
There currently exist a handful of local mappings [10][11][12][13][14][15]. A comparison of these mappings is given in Table I. Two terms which are ubiquitous in fermionic Hamiltonians are lattice hopping, and Coulomb interactions. The minimum upper bounds on the Pauli weights of these terms under any of these local mappings is 4 and 2 respectively. Furthermore, all of these local mappings employ approximately 2 or more qubits per fermionic mode. In this letter we present a new local mapping that, when applied to square lattices, not only outperforms all existing local mappings in terms of Pauli weight, yielding for instance max weight 3 hopping terms, but also employs fewer than 1.5 qubits per mode. We expect these features to find significant use in near term quantum computing applications, where resources are limited. The mapping can be thought of as a modified toric code [16,17] that condenses local pairs of particle excitations, yielding a low energy subspace which corresponds to a fermionic Hilbert space. For clarity we focus in this work on the square lattice, however the design scheme of the mapping may also be applied to other interaction graphs, yielding similar cost benefits.
All local mappings encode fermionic states in a subspace of a multi-qubit Hilbert space via the formalism of stabilizer codes, by defining a set of mutually commuting Pauli operators (stabilizers) for which the subspace constitutes a common +1 eigenspace. There are two design strategies that all existing mappings employ. One strategy, employed by mappings presented in refs. [10][11][12], leverages the Jordan-Wigner transform, and defines stabilizers that "cancel out" sections of the long strings, discarding a portion of the fermionic modes in the process.
The second strategy, employed by the mappings presented in refs. [13][14][15], as well as the mapping presented in this work, focuses instead on finding a set of low weight Pauli operators which reproduce all of the local (anti)commutation relations of the fermionic "edge" (E jk ) and "vertex" (V j ) operators -which are most concisely defined in terms of Majorana operators γ j := a j + a † j and
γ j := aj −a † j i : E jk := −iγ j γ k , V j =: −iγ jγj .(3)
These operators are hermitian, traceless, self inverse and E jk = −E kj . The local (anti)-commutation relation which fermion to qubit mappings of this second kind aim to reproduce is that pairs of operators anti-commute if and only if they share a vertex. More precisely:
{E jk , V j } = 0, {E ij , E jk } = 0,(4)
and for all i = j = m = n:
[V i , V j ] = 0, [E ij , V m ] = 0, [E ij , E mn ] = 0.(5)
All even fermionic operators (ie even products of creation and annihilation operators and sums thereof) can be expressed in terms of edge and vertex operators [13]. Furthermore, all parity preserving operators are even fermionic operators and so in accordance with parity superselection all physical fermionic observables are even fermionic operators. Associating Pauli operators to each edge and vertex operator such that the above conditions are satisfied almost completely defines a mapping from the even fermionic operators to qubit operators. However there exists an additional non-local relation: the product of any loop of edge operators must equal the identity. More precisely
i (|p|−1) (|p|−1) i=1 E pipi+1 = 1 (6)
for a cyclic sequence of sites p = {p 1 , p 2 , ..}. Given a mapping from edge operators to Pauli operators, the expression on the left hand side of Eq. 6 also in general maps to a Pauli operator. However by restricting the mapped fermionic states to the common +1 eigenspace of these Pauli operators (ie taking them to be the stabilizers) Eq. 6 becomes satisfied in that subspace.
For all prior mappings which have employed this second strategy, the stabilizers additionally generate the total fermion parity operator i V i , fixing its value to +1, so that the mappings only admit representations of even fermionic states. The mapping given in this work is the first employing this design strategy that can avoid this side-effect. In this case one may represent states violating parity superselection, and so the full fermionic algebra admits a representation. This additional structure is completely specified by mapping a single Majorana operator to a qubit operator, satisfying appropriate (anti)commutation relations.
We now proceed with a complete description of the new mapping. It suffices to define mappings from edge and vertex operators to qubit operators. Consider a square lattice of fermionic sites at the vertices. Label the faces of the lattice even and odd in a checker-board pattern. For each fermionic site j associate a "vertex" qubit indexed by j. Associate a "face" qubit to the odd faces. Assign an orientation to the edges of the lattice so that they circulate around the even faces clockwise or counterclockwise, alternating on every row of faces. This is illustrated in Fig. 1.
Here and throughout we denote mapped operators with a tilde overscript. Let f (i, j) index the unique odd face adjacent to edge (i, j). For every edge (i, j), with i pointing to j, define the following mapped edge operators: For those edges on the boundary which are not adjacent to an odd face, omit the Pauli operator that would otherwise be acting on a face qubit. For every vertex j define the mapped vertex operators
E ij := X i Y j X f (i,j) (i, j) oriented downwards −X i Y j X f (i,j) (i, j) oriented upwards [18] X i Y j Y f (i,j) (i, j) horizontal(7)E ji := −Ẽ ij .(8)(L − 1) 2L 2 2L(L − 1) 2L 2 − L 3L 2 1.5L 2 − L 1.5L 2 − L − 1 1.5L 2 − L + 1 Qubit to Mode Ratio 2 − 2 L 2 2 − 2 L 2 − 1 L 3 1.5 − 2 L 1.5 − 2 L − 1 2L 2 1.5 − 2 L + 1 2L 2V j := Z j(9)
This specifies all mapped vertex and edge operators and is illustrated in Figure 1. This mapping satisfies the local (anti)-commutation relations 4 and 5. The intuition is that a directed edge has an X on the tail and a Y on the head. Whenever the head of one edge touches the tail of another the edge operators anti-commute, while if two edges touch head to head or tail to tail they commute. By adding a qubit at some faces, and choosing an appropriate orientation for the edges, one can enforce the additional necessary anti-commutation relations at the face qubits.
Given a lattice with M fermionic modes, this mapping uses fewer than 1.5M qubits. Furthermore this construction yields hopping and Coulomb terms with Pauli weight at most 3. The reason that the Pauli weights and qubit numbers are so low is that the face qubits are used extremely efficiently, each one enforcing anti-commutation relations at four bounding corners.
In order to satisfy Equation 6 one must project into the common +1 eigenspace of all loops of edge operators. In the case of a planar graph the loops around faces form a minimal generating set. The stabilizers associated with even faces are non-trivial and illustrated in Figure 2, while the stabilizers associated with odd faces are equal to 1. Therefore the number of independent constraints, dividing the Hilbert space in two, is given by the number of even faces. By a simple counting argument the dimension of the subspace to which fermionic states are mapped is thus given by:
subspace dimension = 2 M+OF−EF(10)
where M is the number of fermionic modes, OF is the number of odd faces, and EF is the number of even faces. There are three distinct cases for which the subspace dimension differs:
(I) There are an even number of faces, and so an equal number of even and odd faces.
(II) There is one more even face than odd face.
(III) There is one more odd face than even face.
The reader may wish to examine sublattices in Figure 1 to develop an intuition for these cases.
In case (I) the mapping represents the full fermionic Fock space F, with dimension 2 M . Therefore single Majorana operators also admit a representation. It suffices to specify one Majorana, and all others may be constructed using edge and vertex operators. A Majorana operator γ j must anti-commute with all edges incident on site j as well as the vertex operator V j . Any corner vertex j which bounds an odd face must either have arrows pointing into it or pointing away from it. If the arrows point into (away from) the corner, then the mapped Majorana isγ j = X j (Y j ). There are two possible choices of corners. The choice is arbitrary, but once a corner is chosen then the equivalent operator at the remaining corner corresponds to a Majorana hole operator h i := γ i j V j .
In case (II) the dimension of the subspace is 2 M −1 , which is half of the full fermionic Fock space. Furthermore, up to multiplication by stabilizers, iṼ i = 1. Thus this mapping represents the even fermionic fock space. This is evidenced by the fact that there are no corner vertices bounding odd faces on which to define a Majorana operator.
Finally, in case (III) the dimension of the subspace is 2 M +1 , ie the full fermionic Fock space plus an additional "logical" qubit degree of freedom: F ⊗ C 2 . In this case there are four corner vertices on which to define a Majorana operator. These four Majoranas can be thought of as four distinct species A i , B i , C i and D i , each introduced at a different corner and translated by edge operators to site i. Pairwise annihilation of differing species of Majorana yield three distinct vacua: 1 , 2 , 3 . Identifying one species of Majorana (in this case A) to act as the canonical Majorana operator on the fermionic system and identity on the logical qubit, the other three species become identified with majorana hole operators multiplied by logical Pauli operators on the logical qubit.
A i =γ i ⊗ I,(11)B i =h i ⊗X, C i =h i ⊗Ỹ , D i =h i ⊗Z.(12)
The vacua are thus identified with applying logical Pauli operators on the logical qubit:
1 =X, 2 =Ỹ , 3 =Z.(13)
These logical Pauli operators are mapped to multiqubit Pauli operators which span the length of the system [19], thus the logical qubit is topologically protected. This is highly suggestive of a connection to the toric code.
Indeed the stabilizers of the mapping presented here are tensor products of toric code star (Π S ) and plaquette (Π P ) operators on the face qubits (up to local rotations) and four qubit Z parity checks on the vertex qubits. This is illustrated in Figure 3.
Z Z Z Z Z Y Y X X Z Z Z Z Z Y Y X X FIG. 3.
The toric code (dotted purple) embedded in the compact mapping. Each stabilizer is a tensor product of either a plaquette Πp (red) or star Πs (blue) operator, with a four qubit Z parity operator (black)
The toric code Hamiltonian:
H toric = − Π S − Π P(14)
has localised electric (e) and magnetic (m) excitations, corresponding to energy contributions from Π S and Π P terms. These excitations exhibit fermionic mutual statistics and bosonic self statistics. Consequently a composite of e and m excitations exhibits fermionic self statistics. The addition of the four qubit Z parity operators yields a modified Hamiltonian:
H map = − Π S ⊗ Z s − Π p ⊗ Z p(15)
for which specific localized pairings of e and m particles no longer cause an energy penalty. More concretely, any path of edge operators E ij = −iγ i γ j corresponds to the creation of a pair of Majoranas at either end of the path. Each of these Majoranas give rise to bound pairs of e and m particles in H toric which have no energy penalty in H map and exhibit fermionic exchange statistics. This is illustrated in Figure 4. In this sense the mapping presented here leverages the topological order of the toric code to generate non-local exchange statistics. Similar connections appear in other fermionic mappings [10,13,15], but not as explicitly.
The fermion to qubit mapping presented here constitutes a significant improvement on both the mode to qubit ratio and the Pauli weights of local operators. For near term quantum computing hardware, such gains are essential. See Ref. [8] for an example how this specific mapping can yield improvements on quantum simulation techniques. Additionally, this mapping has error mitigating properties which we discuss in concurrent work [20]. The design principles outlined for this encoding can also be applied to other lattice types. In the supplementary material we include a similar mapping for a hexagonal lattice.
We would like to thank Johannes Bausch, Toby Cubitt, Laura Clinton, Raul Santos and Tom Scruby for their helpful discussions and proofreading.
A Compact Fermion To Qubit Mapping: Supplementary Material
DETAILS OF LOGICAL QUBIT OPERATORS IN CASE III
As discussed in the main text, in case (III) the encoded subspace consists of the full fermionic Fock space plus an additional logical qubit degree of freedom (F ⊗ C 2 ) and the square lattice has four corners from each of which either all incident edges point into or away. Applying an X Pauli operator at a corner where edges point away (and Y where edges pointing into) must correspond to applying an encoded Majorana or hole operator (h i := γ i k V k ) -as these are the only fermionic operators satisfying the resulting (anti-)commutation relations at that site -along with potentially some additional action on the logical qubit space. We define the four operators A i , B i , C i and D i , each corresponding to a distinct corner, as the application of an X (or Y ) at that corner followed by the application of a sequence of edge operators from the corner to site i. These operators satisfy the following (anti-)commutation relations ∀i = j and ∀M = M ∈ {A, B, C, D}:
{M i , M j } = [M i , M j ] = {M i , M i } = 0(1)
An assignment of Majorana, hole and Pauli operators to these M which satisfy these commutation relations is:
A i =γ i ⊗ I,(2)B i =h i ⊗X, C i =h i ⊗Ỹ , D i =h i ⊗Z.(3)
One can retrieve the isolated logical Pauli operators by taking the product of select pairs:
C i D i = iI ⊗X(4)D i B i = iI ⊗Ỹ(5)B i C i = iI ⊗Z(6)
Such products correspond to string-like operations extending the length of the lattice, from one corner to another. Examples of these operators are illustrated in Figure 1. Note that if one treats one of these operators as a stabilizer, then one restricts to the full fermionic code space without an extra logical qubit. In this case, as one should expect, there are only two corners in which to inject a Majorana, since injecting a Majorana at either of the other two corners would anti-commute with the chosen stabilizer. These two corners correspond to the Majorana and its hole counterpart, as in case (I) where there are only two odd corners.
HEXAGONAL LATTICE MAPPING
Hexagonal lattices admit a similar construction to that described in the main text for square lattices. Again we follow the scheme of orienting the edges, introducing ansatz edge operators with an X at the tail and a Y at the head, and then introducing interactions on face qubits to satisfy the anti-commutation relations.
In the hexagonal lattice every edge, except for the bottom edge of every face, is oriented clockwise on even columns of faces and counterclockwise on odd columns, as illustrated in Figure 2. This ensures that heads touch tails for all edges except for the bottom edge of every hexagon. A qubit is introduced at every face. For every bottom edge of every face f , the edge operator acts on the face qubit of f with Y f . The two edges adjacent to this bottom edge and also adjacent to f act on the face qubit of f with X f . In this way all anti-commutation relations are satisfied. Just as in the square lattice case, the vertex operators are Z operators on the vertex qubit. We illustrate this construction in Figure 2.
X Y X Y X Y Y X X Y X X X Y Y X Y X
Once again, the stabilizers of this mapping are cycles. However in this case there are no trivial cycles, so there is a stabilizer generator for every face. This implies that arXiv:2003.06939v2 [quant-ph] 19 Nov 2020 the code space is the full fermionic space. One again, single fermion operators may be injected into the code at those vertices from which edges are either uniformly pointing towards or away. This mapping yields hopping and coulomb terms with Pauli weight at most 3. With M modes, this mapping uses fewer than 1.5M qubits.
Similar generalizations may be applied to other lattices.
PROOF OF CORRECTNESS OF SQUARE LATTICE FERMIONIC MAPPINGS
For the sake of completeness we include here a more explicit proof of our claims. The arguments presented here should be relatively familiar to those readers acquainted with fermionic mappings. Definition 1. A mapping from some Hilbert space H 1 to another Hilbert space H 2 is an isometry [1] J : H 1 → H 2 , as defined in [2,Sec. 5].
Let f m = C m correspond to an m mode single fermion Hilbert space. The n fermion Fock space on m modes is defined as Proof. The mapping employs m+n F /2 qubits. The number of non-trivial stabilizers is n F /2. Let L ⊂ H = (C 2 ) m+n F /2 be the joint +1 eigenspace of these nontrivial stabilizers, then dim(L) = 2 m = dim(F m ).
∧ n f m = span 1 n! π∈Sn sgn(π) n i=1 ψ π(i) : ψ j ∈ f m ,
If edge operators E jk and vertex operators V k satisfy the anti-commutation relations
{E jk , V j } = 0, {E ij , E jk } = 0,(7)
and for all i = j = m = n:
[V i , V j ] = 0, [E ij , V m ] = 0, [E ij , E mn ] = 0.(8)
and loop condition
i (|p|−1) (|p|−1) i=1 E pipi+1 = 1(9)
then they generate the even fermionic algebra L(F E m ). In order to generate the entire fermionic algebra L(F m ), we need an additional generator; for instance a single Majorana operator γ i , which together with a vertex operator V i γ i ∝γ i generates the entire algebra L(F m ).
By inspection the Pauli operatorsẼ jk andṼ k satisfy relations 8 and 7. Furthermore, since the number of faces is even there are always exactly two corner faces of the lattice which are odd according to the checker-board labelling. Choose a corner vertex c, associated with an odd corner face, and define an encoded Majorana operator γ c = X c arrows pointing into corner, or Y c otherwise.
Then {γ c ,Ẽ jk } = {γ c ,Ṽ k } = 0. By inspection we see thatẼ jk ,Ṽ k and the additional Majoranaγ c commute with the stabilizers of the mapping; as such, restriction to the joint +1 eigenspace L of the stabilizers-denoted ·| L -retains all their algebraic properties. Furthermore, the operatorsẼ jk | L satisfy Equation 9. Thus by the same argument as in [2,Sec. 8], the identification
E jk −→Ẽ jk | L V k −→Ṽ j | L γ c −→γ c | L extends to a * -homomorphism µ : L(F m ) −→ L(L).
Since the CAR algebra is unique up to an isomorphism [3], and dim(L) = dim(F m ), the mapping is an isomorphism. The claim follows by 1.
Theorem 2.
A square (L 1 ×L 2 ) lattice mapping with an odd number of faces n F = (L 1 − 1)(L 2 − 1), and an extra even face (case II) describes an mapping J : F E m → L, where m = L 1 L 2 and L ⊂ (C 2 ) m+ n F /2 .
Proof. The mapping employs m + n F /2 qubits. The number of non-trivial stabilizers is n F /2 . Let L ⊂ (C 2 ) m+ n F /2 be the +1 eigenspace of the non-trivial stabilizers, then dim(L) = 2 m+ n F /2 − n F /2 = 2 (m−1) . Since n F is odd, it must be the case that L 1 − 1 and L 2 − 1 are odd, and so m is even. Thus dim(F E m ) = 2 m−1 = dim(L).
L(F E m ) ⊕ L(F O m ) is generated by E ij and V i . However under the algebraic constraint that i V i = I, the algebra only generates L(F E m ). Thus we have a mapping µ : L(F E m ) → L(L) by µ(E ij ) =Ẽ ij | L and µ(V i ) =Ṽ i | L . By a similar argument to the previous theorem, µ is a *-isomorphism, and therefore specifies an mapping J : F E m → L.
Theorem 3. A square (L 1 ×L 2 ) lattice mapping with an odd number of faces n F = (L 1 − 1)(L 2 − 1), and an extra odd face (case III), is a fermionic mapping J : C 2 ⊗F m → L, where m = L 1 L 2 and L ⊂ (C 2 ) m+ n F /2 .
Proof. The mapping employs m + n F /2 qubits. The number of non-trivial stabilizers is n F /2 . Let L ⊂ (C 2 ) m+ n F /2 be the +1 eigenspace of the non-trivial stabilizers, then dim(L) = 2 (m+1) = dim(C 2 ⊗ F E m ). There are four corner faces of the lattice which are odd according to the checker-board labelling. One may choose a corner vertex c and define an encoded operator acting trivially on the logical qubit, and as a Majorana on the logical fermionic space:Ĩ ⊗γ c = X c or Y c depending on if the arrows point into or respectively away from that corner. The other three corners, which we label c x , c y , c z , may then be defined as logical Pauli operators combined with logical hole operators:
X ⊗h cx = X cx or Y cx Y ⊗h cy = X cy or Y cỹ Z ⊗h cz = X cz or Y cz here the choice of which corners to associate with which logical Paulis is a matter of convention, and the choice of X or Y will again depend on if the arrows point into or respectively away from that corner.
By Proposition 2, the operators I ⊗ γ j , X ⊗ h j and Y ⊗ h j , along with the edge and vertex operators, generate L(C 2 ⊗F m ). Define the mapping µ : L(C 2 ⊗F m ) → L(L) by µ(I ⊗ γ c ) =Ĩ ⊗γ c | L etc. By a similar argument to the previous theorems, µ is a *-isomorphism, and therefore specifies an mapping J : C 2 ⊗ F m → L.
FIG. 2 .
2Non-trivial stabilizer of the encoding.
FIG. 4 .
4A pair of Majorana particles, generated by a string of edge operators (black) in the fermionic mapping correspond to localized pairs of e (blue) and m (red) particles in the toric code.
FIG. 1 .
1The logical X and Y operators on the extra logical qubit in case (III)
FIG. 2 .
2Edge orientation, qubit placement and edge operators for the hexagonal lattice mapping.
and the full fermionic Fock space of m modes is F m = m n=0 (∧ n f m ). The dimension of F m is 2 m . Let F E m = m n∈even (∧ n f m ) and F O m = m n∈odd (∧ n f m ). If m is even the dimension of F E m is 2 m−1 . In the following we denote with L(S) to be the Linear operators on a space S, which form a C * -algebra in the case where S is a complex Hilbert space. Definition 2 (Fermionic Mapping). A fermionic mapping is an isometry J : F m −→ H, where H is a qubit Hilbert space, i.e. J satisfies the property J † J = I Fm Proposition 1 ([2]). A * -isomorphism µ : L(H 1 ) → L(H 2 ) induces an mapping J : H 1 −→ H 2 which is unique up to a global phase. An operator X ∈ L(H 1 ) is represented via the map µ(X) = JXJ † . Proposition 2 ([2]). The algebra L(F E m )⊕L(F O m ) is generated by the edge and vertex operators E ij and V i . The algebra L(F m ) is generated by a single Majorana operator γ and the edge and vertex operators E ik and V i .
Theorem 1 .
1A square (L 1 × L 2 ) lattice mapping with an even number of faces n F = (L 1 − 1)(L 2 − 1) (case I) is a fermionic mapping J : F m → L, where m = L 1 L 2 and L ⊂ (C 2 ) m+n F /2 .
TABLE I. A comparison of existing local fermion to qubit mappings on an L × L lattice of fermionic modes. The mapping presented in this work is given in the three rightmost columns. Max weight Coulomb and max weight hopping denote the maximum Pauli weights of the mapped Coulomb (a † i aia † j aj) and nearest neighbour hopping (a † i aj + a † j ai) terms respectively. Encoded fermionic space denotes whether the full or even fermionic fock space is represented. Graph geometry denotes the hopping interaction geometry which the mapping is tailored to.FIG. 1. Qubit assignment, edge orientation, and examples of mapped edge and vertex operators for a 4 × 5 square lattice. Vertices are numbered in snaking order, left to right, top to bottom.Max Weight
Hopping
6
4
4
5
4
3
3
3
Max Weight
Coulomb
8
2
6
6
6
2
2
2
Encoded
Fermionic
Space
Even
Full
Even
Full
Even
Full
Even
Full
Plus Qubit
Graph
Geometry
General General
Square
Lattice
Square
Lattice
General
Square
Lattice
Square
Lattice
Square
Lattice
Z
V1
E4,3
Y
X
Y
E12,17
Y
X
X
E19,20
X
Y
1
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|
[] |
[
"Checking account activity and credit default risk of enterprises: An application of statistical learning methods",
"Checking account activity and credit default risk of enterprises: An application of statistical learning methods"
] |
[
"Jinglun Yao ",
"Maxime Levy-Chapira ",
"Mamikon Margaryan "
] |
[] |
[] |
The existence of asymmetric information has always been a major concern for financial institutions. Financial intermediaries such as commercial banks need to study the quality of potential borrowers in order to make their decision on corporate loans. Classical methods model the default probability by financial ratios using the logistic regression. As one of the major commercial banks in France, we have access to the the account activities of corporate clients. We show that this transactional data outperforms classical financial ratios in predicting the default event. As the new data reflects the real time status of cash flow, this result confirms our intuition that liquidity plays an important role in the phenomenon of default. Moreover, the two data sets are supplementary to each other to a certain extent: the merged data has a better prediction power than each individual data. We have adopted some advanced machine learning methods and analyzed their characteristics. The correct use of these methods helps us to acquire a deeper understanding of the role of central factors in the phenomenon of default, such as credit line violations and cash inflows.RésuméL'existence de l'asymétrie de l'information est une problématique majeure pour les institutions financières. Les intermédiaires financiers, telles que banques commerciales, doivent étudier la qualité des emprunteurs potentiels afin de prendre leurs décisions sur les prêts commerciaux. Les méthodes classiques modélisent la probabilité de défaut par les ratios financiers en utilisant la régression logistique. Au sein d'une principale banque commerciale en France, nous avons accès aux informations sur les activités du compte des clients commerciaux. Nous montrons que les données transactionnelles surperforment les ratios financiers sur la prédiction du défaut. Comme ces nouvelles données reflètent le flux de trésorerie en temps réel, ce résultat confirme notre intuition que la liquidité joue un rôle essentiel dans les phénomènes de défault. En outre, les deux bases de données sont complémentaires l'une à l'autre d'une certaine mesure : la base fusionnée a une meilleure performance de prédiction que chaque base individuelle. Nous avons adopté plusieurs méthodes avancées de l'apprentissage statistique et analysé leurs caractéristiques. L'utilisation appropriée de ces méthodes nous aide à acquérir une compréhension profonde du rôle des facteurs centraux dans la prédiction du défaut, tel que la violation de l'autorisation du découvert et les flux de trésorerie.
| null |
[
"https://arxiv.org/pdf/1707.00757v1.pdf"
] | 158,045,344 |
1707.00757
|
5eeff0e5955cbcb34894ee05cb2061e592974cb3
|
Checking account activity and credit default risk of enterprises: An application of statistical learning methods
July 5, 2017
Jinglun Yao
Maxime Levy-Chapira
Mamikon Margaryan
Checking account activity and credit default risk of enterprises: An application of statistical learning methods
July 5, 2017* Student at Ecole Polytechnique † Quantitative Risk Project Manager at Société Générale ‡ Head of Credit Risk Modeling at Société Générale
The existence of asymmetric information has always been a major concern for financial institutions. Financial intermediaries such as commercial banks need to study the quality of potential borrowers in order to make their decision on corporate loans. Classical methods model the default probability by financial ratios using the logistic regression. As one of the major commercial banks in France, we have access to the the account activities of corporate clients. We show that this transactional data outperforms classical financial ratios in predicting the default event. As the new data reflects the real time status of cash flow, this result confirms our intuition that liquidity plays an important role in the phenomenon of default. Moreover, the two data sets are supplementary to each other to a certain extent: the merged data has a better prediction power than each individual data. We have adopted some advanced machine learning methods and analyzed their characteristics. The correct use of these methods helps us to acquire a deeper understanding of the role of central factors in the phenomenon of default, such as credit line violations and cash inflows.RésuméL'existence de l'asymétrie de l'information est une problématique majeure pour les institutions financières. Les intermédiaires financiers, telles que banques commerciales, doivent étudier la qualité des emprunteurs potentiels afin de prendre leurs décisions sur les prêts commerciaux. Les méthodes classiques modélisent la probabilité de défaut par les ratios financiers en utilisant la régression logistique. Au sein d'une principale banque commerciale en France, nous avons accès aux informations sur les activités du compte des clients commerciaux. Nous montrons que les données transactionnelles surperforment les ratios financiers sur la prédiction du défaut. Comme ces nouvelles données reflètent le flux de trésorerie en temps réel, ce résultat confirme notre intuition que la liquidité joue un rôle essentiel dans les phénomènes de défault. En outre, les deux bases de données sont complémentaires l'une à l'autre d'une certaine mesure : la base fusionnée a une meilleure performance de prédiction que chaque base individuelle. Nous avons adopté plusieurs méthodes avancées de l'apprentissage statistique et analysé leurs caractéristiques. L'utilisation appropriée de ces méthodes nous aide à acquérir une compréhension profonde du rôle des facteurs centraux dans la prédiction du défaut, tel que la violation de l'autorisation du découvert et les flux de trésorerie.
Introduction
As Mishkin and Eakins (2006) point out, asymmetric information is one of the core issues in the existence of financial institutions. Financial intermediaries, such as commercial banks, play an important role in the financial system because they reduce transaction costs, share risk, and solve problems raised by asymmetric information. One of the most important channels of achieving this role is the effective analysis of the quality of potential corporate borrowers. Banks need to distinguish reliable borrowers from unreliable ones in order to make their decisions on corporate loans. From the banks' point of view, this reduces the losses associated with corporate defaults, while it is also beneficial for the whole economy because resources are efficiently attributed to prominent projects.
Altman (1968), Beaver (1966) and Ohlson (1980) are pioneers of using statistical models in the prediction of default. They have used financial ratios which are calculated from the balance sheet and the income statement. Their inspiring work has been widely recognized, which is proved by the fact that the method has become the standard of credit risk modeling for many financial institutions. One might doubt, however, if the phenomenon of default can be "explained" by the financial ratios. Intuitively, default takes place when the cash flows of a firm are no longer sustainable. The financial structure of a firm might well be the result of an upcoming default instead of being the cause of it because the firm might be obliged to sell some of its assets when it is short of cash flows. Leland (2006) distinguishes two kinds of credit risk models: structural models and statistical models (or reduced form models). According to him, the statistical model above is not directly based on firm's cash flows or values, but empirically estimates a "jump rate" to default. What's more, reduced form models do not allow an integrated analysis of a firm's decision to default or its optimal financial structure decisions. On the other hand, structural models, such as those proposed by Black and Scholes (1973), Merton (1974) and Longstaff et al. (2005), associate default with the values of corporate securities, as the valuation of corporate securities depends on their future cash flows, which in turn are contingent upon the firm's operational cash flows. The diffusion models of market values of securities allow us to investigate the evolution of cash flows, and thus the default probabilities. This suggestion is insightful, but does not provide a practical approach for commercial banks vis-avis their corporate clients. Most small and medium-sized enterprises do not sell marketed securities. For these firms, using structural models based on corporate securities is simply impossible. Fortunately, however, commercial banks possess the information on cash flows in another way. Corporate clients not only borrow from banks but also open checking accounts in these banks. Norden and Weber (2010) demonstrate that credit line usage, limit violations, and cash inflows exhibit abnormal patterns approximately 12 months before default events. Measures of account activity substantially improve default predictions and are especially helpful for monitoring small businesses and individuals. This is another good example of economies of scale in which a bank shares information within itself to achieve better global performance.
Instead of using a structural model, we choose to use some statistical learning methods which improve considerably the prediction performance compared with classical logistic regression. This choice is due to the fact that it is difficult to construct a structural model at the first stage which gives a general image and a good prediction at the same time. There is limited literature which explains the default by using checking account information. By using statistical learning methods, we can empirically tell which variables are the most important in default prediction. This can help us in the next stage construct a structural model. On the other hand, if we are only interested in prediction, a reduced form model is sufficient for our concern.
However, We should underline the fact that application of machine learning methods does not eliminate the necessity of economic understanding. As we will show, the construction of meaningful economic variables is an essential preliminary step for machine learning. What's more, the "important variables" given by machine learning should be taken with a grain of salt. Strobl et al. (1993) resume that variable selection in CART (classification and regression trees) is affected by characteristics other than information content, e.g. variables with more categories are preferred. To solve the problem, Strobl et al. (2007) propose an unbiased split selection based on exact distribution hypothesis. As with all exact procedures, this method is computationally too intensive. Hothorn et al. (2006) propose a more parsimonious algorithm, conditional classification tree (ctree), which is based on the framework of permutation test developed by Strasser and Weber (1999). What's more, unbiased random forest (conditional random forest, or cforest) is constructed based on ctree. But cforest is still too heavy to be executed for our data. Besides, it is not clear whether the unbiasedness in the sense of random forest is still valuable for other machine learning methods. That is to say, it is disputable to find an universally valuable subset of variables which contain the same level of information in any statistical method. Instead of using these computationally expensive methods, we will compare the variables selected by boosting, stepwise selection and lasso. An thorough understanding of these machine learning methods is efficient to shed light on the interpretation of model selections.
We begin by introducing basic random forest and boosting, as well as some important modifications to accommodate characteristics in our data. Section 3 compares three approaches of treating checking account data, illustrates the importance of economically meaningful variables and shows some particularities of machine learning methods. Section 4 compares the performance of financial ratios and questionnaires with that of account data, combines the two data to achieve better prediction performance. Section 5 does three model selections, respectively based on AIC, lasso and boosting. We use the logistic regression to interpret the marginal effect of these most important variables. Section 6 concludes the article.
Introduction to random forest and boosting
Classification tree
For random forest and boosting, the most commonly used basic classifier is the classification tree. Suppose we want to classify a binary variable Y by using two explicative variables X 1 and X 2 . An example of the classification tree is given in Figure 1 1 . The two graphical representations are equivalent. And the tree can be represented by the form
f (X) = 5 m=1 c m I{(X 1 , X 2 ) ∈ R m } where c m ∈ {0, 1}, I is the indicator function.
(1)
Figure 1 -A simple example of classification tree
To grow a tree, the central idea is to choose a loss function and to minimize the loss function with respect to the tree. and James et al. (2013) give a full introduction to the most important loss criteria in the context of classification trees. We use the Gini index as the loss function in our research. It should be underlined, however, that it is computationally too expensive to find a global optimal solution. Instead, in practice one uses the "greedy algorithm" which admits the part already constructed and searches the optimal solution based on this part. A tree grown in this way is called a CART (classification and regression tree), which was proposed by Breiman et al. (1984) and has become the most popular tree algorithm in machine learning.
The advantage of tree is obvious: it is intuitive and easy to be interpreted. Nonetheless, it generally has poor predictive power on training set and test set if the model is mildly fitted. Conversely, an overfitting with training set (or overly reduced bias) is generally not expected in machine learning. Ensemble methods, such as Random Forest and Boosting, are conceived to solve this dilemma.
Random Forest
Random Forest aims at reducing model variance and thus increasing prediction power on test set. Instead of growing one single tree, we plant a forest. A general description of the algorithm is given in Figure 2 2 . In practice, the optimal value of m is around √ p for classification problem, where p is the total number of variables. We can of course, use cross-validation to optimise the value of this parameter. This small value of m looks strange at first sight, but it is in fact the key of random forest. In fact, for B identically distributed random variables, each with variance σ 2 and positive pairwise correlation ρ, the variance of their average is
ρσ 2 + 1 − ρ B σ 2(2)
Even with large B (the number of trees in the case of random forest), we still need to decrease ρ to reduce the variance of average. The role of a small m is to reduce the correlation ρ across trees, However, the basic random forest works poorly for our data because it is imbalanced (fewer than 6% observations defaulted). Several remedies exist for this characteristic, including weights adjustment (Ting (2002)) and stratified sampling (Chen et al. (2004)). We have adopted the stratification method which is easy to be implemented and yields satisfying results. Instead of sampling uniformly default and non-default observations for each tree in step 1.(a) (eg. sampling 2/3 observations uniformly), we take 2/3 default observations and an equal number of non-default observations. This apparently small modification leads to tremendous amelioration in confusion matrices. For a given checking account data with 30 variables, the comparison is shown in Table 2. The test AUCs are respectively 78.72% and 79.87%.
Boosting
The most commonly used version of boosting is AdaBoost (Freund et al. (1996)). Contrary to random forest which plants decision trees in parallel, AdaBoost cultivates a series of trees. If an observation is wrongly classified in previous trees, its weight will be accentuated in latter trees until it is correctly classified. The central idea is intuitive, yet it had been purely an algorithmic notion until Friedman et al. (2000), who pointed out the inherent relationship between AdaBoost and additive logistic regression model: where additive logistic regression model is defined as having the following form for a two-class problem:
log P (y = 1|x) P (y = 0|x) = M m=1 f m (x)(3)
In the case of boosting trees, f m are individual trees adjusted by weights. According to Result 1, boosting is by its nature an optimisation process. This insight paves the way for xgboost (Extreme Gradient Boosting by Friedman (2001)), which searches the gradient of objective function and implements efficiently the basic idea of boosting. Moreover, the intimate relationship between boosting and logistic regression leads to some interesting results on which we will discuss later on.
Overfitting in machine learning
A model is overfitted if it suits well the training set but poorly the test set. In our research, the model performance criterion is AUC (Area Under the ROC Curve), which measures the discrimination power of a given model. It should be noticed that AUC is immune to imbalance in data.
Some methods, like the random forest, aim at reducing the model variance, i.e., by decorrelating the training data and the model, we obtain a model which is less sensitive to data change. For example, using 30 checking account variables to explain default, we get AU C = 79.45% for training set and AU C = 79.85% for test set in balanced random forest. Boosting had also been considered to work in this way. But Friedman et al. (2000) point out that boosting seems mainly a bias reducing procedure. This conclusion is coherent with our experiment. Using the same variables, we get AU C = 87.45% for training set and AU C = 79.8% for test set. Boosting has necessarily overfitted the model, but this feature does not undermine its ability of predicting the test set.
Additional remarks should be made on parameters in machine learning methods. While it is not the major concern of this article, it is nonetheless crucial to let the machine run correctly. One important parameter is related to the complexity of model, for example, the number of candidate variables for each node splitting in random forest, the number of learning steps in boosting. Cross-validation is adopted to ensure the appropriate level of complexity and to avoid over-fitting. Appendix C gives an exhaustive explanation on the most important parameters in our models.
Organising checking account data: Three approaches
In current literature, treating checking account data does not have mature approaches as we can find for financial structure data. In the latter case, corporate finance suggests some particularly useful ratios such as working capital/total assets, retained earnings/total assets, market capitalization/total debt etc (Ross et al. (2008)). Defining new features based on checking account data becomes a central issue in our study. We have tried three approaches detailed below. They will be combined with three different statistical methods (logistic regression, random forest and boosting).
Variable Definition 1 (Continuous Variables Based on Economic Intuition)
This definition is inspired by Norden and Weber (2010). At the end of each year, which we note time t, we define the explained variable, default, as the binary variable of going bankrupt in the next year. The explicative variables are created based on monthly account variables in the last two years. These 30 variables are listed in Appendix A and can be classed mathematically into four categories: the difference of a characteristic (eg. balance, monthly cumulative credits) between the begin and the end of a period (one or two years); the value of this characteristic at time t; the standard deviation of this characteristic during a certain period; attributes of the firm (annual sales, sector). The basic idea is to use stock and flow variables for a complete but also concise description of a certain characteristic. Moreover, the standard deviation of, e.g. monthly cumulative credits, allows us to quantify the risk associated with unstable income.
The size of firms may influence considerably the model in an undesirable way. A firm might have a higher balance than another one only because it is larger: this larger balance does not "reflect" a smaller probability of default. Norden and Weber (2010) have used the line of credit as the normalisation variable for the corporate clients of a German universal bank. However, this variable is not available in our research. We thus need to figure out another appropriate normalisation variable.
One suggestion is to use information on the balance sheet or the income statement, such as total sales. But larger firm may open accounts in several different commercial banks, reflecting only a fragment of cash flow information in each account. There exists thus a discrepancy between the size of the account and the size of the firm. In order to capture the account size, we need a variable within the account itself which reflects the account's normal level of vitality. The average monthly cumulative credits in the last two years, responds to the defined criteria and is used to normalize the variables proportional to account size. Intuitively, monthly cumulative credits is the equivalent of total sales in the context of checking account in the sense of total resources.
Variable Definition 2 (Automatically built variables)
As well as in Definition 1, we still use account information in the past year to predict the default in the coming year. But the explanatory variables used in statistical methods are built in a much more "computer science" way. Instead of using economic intuitions above to organise raw information, we rely on automatic methods to build the model inputs. 50 variables are firstly resumed from raw monthly information, and then interact with each other using the four basic arithmetic operations. Together with some raw variables, the data set contains around 5000 variables in total. It should be noticed that these combinations are usually not intuitively interpretable. While it might be possible to give some far-fetched explanation for "average monthly balance/cumulative number of intended violations", it is far more difficult to interpret other variables.
One might argue that the simple arithmetic interactions are not capable of exhausting possible meaningful combinations of raw information, making this approach not representative. However, it should first of all be noticed that boosting with 5000 variables is already computationally expensive for an ordinary computer. In practice, we launch the boosting for each kind of arithmetic interaction and select the most important ones according to their contributions to the Gini index. These variables are then used to run a final and lighter boosting with around 200 variables. Secondly, it is simply computationally impossible to exhaust most meaningful combinations. Suppose we want to create automatically the 30 variables in definition 1. These variables are based on more than 10 basic monthly variables (e.g. TCREDIT, monthly number of violations), i.e. more than 120 variables if we take the month into consideration. Var16 is the difference of TCREDIT between time t and t-12 (substraction of 2 variables), while var9 is the sum of monthly number of violations during one year (sum of 12 variables). This simple example shows that for a new variable, there is no limit a priori on the number of participating raw monthly variables. That is to say, any variable among the 120 variables might be included in or excluded from the combination. The number of possible forms of combination is astronomical: 2 120 , even if we allow only one arithmetic operation, for example the addition. Let alone other forms of operations. Thirdly, there is no reason to delimitate a priori a set of reasonable operations. The use of standard deviation for TCREDIT (var24), for example, is based on the intuition of the stability of revenue. It is not reasonable, however, to include a priori this operation, which is more complicated than sinus, cosinus or other simple functions, in the set of reasonable operations, if we investigate the question in a purely mathematical way.
Variable Definition 3 (Discrete Variables Based on Economic Intuition)
Similar to Definition 1, this definition is also economically interpretable. In contrast, we create 5 variables which are highly discretized. Four of them are binary, the fifth has three categories. These variables are listed in Appendix B.
Comparison of Performance
The performance, measured by test AUC, is given in Table 3. We have selected the 20 best variables in Group 1 and Group 3 respectively by AIC and by variable importance in boosting. The 5 best variables in Group 2 are chosen according to variable importance in boosting. Despite the difference in variable selection methods, all the variables in Group 2 are included in Group 1. Among the 20 variables in Group 3, three variables are not available for most of the observations (> 50%) and are eliminated for random forest and for logistic regression.
We can remark that balanced random forest and boosting always outperform logistic regression, except for Group 4. (The failure of imbalanced random forest vis-a-vis balanced random forest justifies the use of stratification for imbalanced data.) This result clearly favors the application of machine learning methods in the default prediction for our data. But why does boosting exhibit the same level of performance as logit for Group 4? For this group, the logit even outforms the balanced random forest. It seems to us that discretization is the reason for this. While it is a common approach to discretize continuous variables for logistic regression because this can create a certain kind of non-linearity of a given explanatory variable within the linear framework, this will nonetheless reduce the information contained in it. The discretization is especially detrimental for imbalanced random forest. AU C = 46.81% suggests a worse performance than randomly distributed classes and should be considered as a pathology. Even the balanced random forest performs worse than logistic regression. In fact, the individual trees grown in a random forest are usually very deep (depth > 1000 with default settings for our data). The capacity of classification is thus intimately related to the number of potential splits permitted by the variables. The split of a specific node in a classification tree can be seen as an automatic discretization process. It would better to let the tree choose the splitting point by itself according to the optimisation criterion rather than fixing it a priori. As for boosting, the trees are usually shallow (depth = 5 in our setting). As suggest, experiences so far indicate that 4 <= depth <= 8 works well in the context of boosting, with results being fairly insensitive to particular choices in this range. In any case, it is unlikely that depth > 10 will be required. This probably suggests that boosting relies much less heavily on the variables' ability of offering potential splits, making it less sensitive to discrete variables.
In fact, using stumps (depth = 2) is sufficiently efficient for yielding good prediction. Using all the 30 variables in Definition 1, the AUCs are respectively 79.47% for depth = 2 and 79.82% for depth = 5. (It should be remarked, however, that the optimal number of rounds validated by cross-validation is higher in the case depth = 2. They are 2811 and 997 respectively for depth = 2 and depth = 5 with other parameters fixed according to Appendix C.) In the case of M stumps, the additive logistic regression model becomes:
log P (y = 1|x) P (y = 0|x) = M m=1 α m 1 xm<sm where x m ∈ {x 1 , x 2 , ..
., x p }, p is the number of explicative variables (4) Remark that M is usually much larger than p. In the case above, the ratio M/p is about 93, which means that on average 93 dummy variables are created for each continuous explanatory variable. The linear combination of these dummies can be very good approximation of any ordinary nonlinear function. With depth > 2, the approximation is extended to multivariate functions. So the advantage of boosting over logistic regression seems to be the capacity of the former to take nonlinearity into consideration. This clearly explains why boosting is mainly a bias-reducing method, as mentioned by .
Does the out-performance of boosting and balanced random forest also imply their superiority of identifying rich data set? Comparing Group 1 and Group 3, we can remark that logit AUC is higher in Group 3, while boosting AUC is lower. If we trust in logistic regression in the case of prediction, we should conclude that Group 3 contains more information than Group 1 and that machine learning methods such as boosting are not reliable for distinguishing a rich data set from a poorer one. However, looking at Group 2, we can easily reverse this conclusion. The logit AUC in Group 2 is nearly the same as that in Group 3, while Group 2 contains apparently less information than Group 1 because all the variables in Group 2 are included in Group 1. Instead, a plausible explanation for the low logit AUC in Group 1 should be the multicollinearity between explanatory variables (James et al. (2013)). Boosting and random forest, in contrast, split each node by individual variable and should not be impacted by the haunting multicollinearity. With less variables (Group 2), the prediction accuracy is higher in logit. This phenomenon probably suggests that logit can not well "digest" rich information because of its restrictive linear form. It is thus more reliable to use AUCs of machine learning methods as a measure of information contained in the data set.
The close relationship between boosting and logistic regression explains some results which may seem strange at first sight. The higher logit AUC in Group 3 compared with Group 1 should be interpreted by the model selection method: "good variables" in the sense of boosting should generally be "good" in the sense of logit. It is thus not surprising to find that 20 variables selected from about 5000 variables works better in logit than 20 variables selected from 30. On the other hand, the same variables have lower AUC in balanced random forest than in boosting (76.67% vs 78.13%). Does this mean that random forest is worse than boosting for predicting? If we look at Table 4, balanced random forest and boosting generally have the same prediction power. The difference between them for Group 3, as well as for Group 2, should be explained by the model selection method. These variables are selected by their contribution to boosting and naturally fit less well the random forest. These phenomena raise the question of the existence of universally valuable selection method, in the sense that the "good" variables are equally good for any machine learning method, not just for one or several methods which is identical or are close to the method used in the selection process. While this question is difficult to answer, we can at least conclude that variables should be defined a priori (Definition 1) based on economic intuition, instead of by a pure "computer science" way (Definition 2) and being selected by machine learning methods at a latter stage. Besides the bias in variable selection, automatically created variables in Group 3 also contain less information than Group 1 (AUCs of both balanced random forest and boosting are lower in Group 3). At least for our data, an economically meaningful construction of variables allows us not only to interpret marginal effects of explanatory variables but also possess information in a more concise way. The discussion can be extended to controversial epistemological discussions which are out of the scope of the article. But at least it seems to us that, as Williamson (2009) points out, while someone hope that machine learning can "close the inductive loop"-i.e., automate the whole cyclic procedure of data collection, hypothesis generation, further data collection, hypothesis reformulation...-the present reality is that machine successes are achieved in combination with human expertise.
Combination of checking account information, financial ratios and managerial information
Traditional reduced form methods for default prediction mainly focused on financial structure of enterprises(Altman (1968), Beaver (1966) and Ohlson (1980)), as financial structure does reflect to a large extent the solvability of enterprises, and is relatively more available than real time account information. What's more, in a reduced form method, as we merely try to match a pattern to the data (Fayyad et al. (1996)) without worrying much about causality, the problem of endogeneity is not a primary concern. But once we want to get some causal interpretation, financial structure data may suffer from endogeneity and should be carefully interpreted as a "cause" of credit default.
On the other hand, we should remark the difference between book value and market value, and the accounting principle associated with this difference (Ross et al. (2008)). For small and mediumsized enterprises, their market values are simply not available because they usually don't sell any marketed securities, while their book values are historic and subjected to accounting manipulations.
Commercial banks have both a necessity and an advantage in the analysis of credit default. The possession of corporate account information helps them to acquire a more direct and "frank" image of the firms' account. Not only the information may be more reliable, but also more real-time. Balance sheet and income statement are resumed once a year by firms, while checking account information can be theoretically daily. In practice, we use monthly variables as raw variables for the purpose of simplification. This allows commercial banks to supervise the solvability of corporate borrowers on a more frequent basis. Given the advantage of checking account information, we should expect a better performance of prediction based on checking account data. This is represented in the first two columns of Table 4. The AUCs based on account data in balanced random forest and boosting are significantly larger than those based on financial and management data. (One might argue that this superiority is simply due to more explicative variables. In fact, with the same number of variables (11), the boosting AUC of account data is 79.19%, which is nearly the same as that of 20 account variables (79.66%) and significantly larger than that of financial and managerial variables (76.17%)).
But this does not suggest the inutility of financial statements in default evaluation. In fact, if we combine Group 1 and Group 5 to yield a data with all the three components(financial ratios, questionnaires on firms and checking account data), the test AUC is the highest ever (84.24%). (Once again, one might argue that the higher AUC is simply due to more variables, instead of orthogonal nature between different sources of information. This argument is refuted by our experiment: Using all the 30 variables in Definition 1, the test AUC of boosting is only 79.8%, which is nearly the same as that of Group 1 and significantly less than that of Group 6.) We can thus conclude that the three sources of information are complementary, which corresponds to our intuition on the real functioning of enterprises. First, The checking account information is a reflection of a firm's cash flow, which is most directly related to a firm's solvability. Second, financial ratios illustrate the firm's financial structure and its ability to earn profits. We should remark that the financial ratios we used are primarily concerned with the firm's profitability and expenses (Interest expenses, earnings before interest and taxes etc.) and are more tightly related to cash flow, which is also the case for Atiya (2001). Third, other non-financial reasons should be taken into consideration, for example, the managerial expertise of cadres.
Of course, these is not a complete list of all the factors which are related to credit default. Some macroeconomic factors, for example, can be additionally taken into account. We have observed a decreasing quarterly default rate during 2013-2014, which might be explained by decreasing interest rate in Europe during the same period. If we use data from 2009 to 2012 as training set, and that from 2013 to 2014 as test set, the statistical pattern works less well for defaults at the end of 2013 and at the beginning of 2014.
Selection and interpretation of the most important variables in checking account data
Because of the multicollinearity problem between 30 variables in Definition 1, a variable selection process is needed in order to obtain and interpret the marginal effect of each prominent variable by logistic regression. The list of important variables in Definition 1 is shown in Figure 3. We can see that according to boosting, the most important variables are especially related to number of violations (var9, var11, var13) and current status (var27, var32, var33, var34). Intuitive as it be, this variable importance in the sense of boosting should be taken with a grain of salt. For example, does it mean that var10 (number of intended violations during the period [t − 23, t − 12], ranked 29 in the importance list) is much less useful than var24 (which reflects the stability of credits during the period [t − 11, t], ranked 7 in the importance list)? In fact, if we draw two conditional distributions (conditioned on default) of each variable and calculate their individual AUCs which reflects their individual discriminality, the AUC of var10 (68.68%) is much higher than that of var24 (56.61%). This seeming paradox comes from the mechanism by which boosting attributes the variable importance. In fact, at each split in each tree, the improvement in the split criterion is the importance measure attributed to the splitting variable, and is accumulated over all trees in the boosting separately for each variable. var9 and var10 refer to the same kind of information, except that var9 concerns more recent information (period [t − 11, t]) and naturally has a better discriminatory power than var10 (AU C = 72.68% vs AU C = 68.68%). For each node splitting, both of the two variables are candidates. As var10 has no advantage over var9, it is rarely used for splitting and is thus considered as a "bad" variable. In this case, it is better to be a mediocre but irreplaceable variable than a brilliant but replaceable one. But it also shows the advantage of boosting in recognizing redundant information and eliminating them.
Figure 3 -Variable importance of 30 variables in Definition 1 according to boosting
In order to be more rigorous on variable selection, we have tried out two other different methods which are based on logistic regression: stepwise selection and lasso. For stepwise selection, AIC was used as the criterion. Forward and backward selection have generated the same 8 variables marked in Table 5. In order to compare between different model selection methods, we have adjusted λ in lasso so as to yield exactly 8 non-zero coefficients. These 8 variables are also resumed in Table 5. Remark that there are 7 among 8 variables which are identical to those selected by AIC. Thus for our data, there is no apparent difference between stepwise and lasso in model selection. In contrast, 4 among 8 variables selected by boosting differ from that of lasso and stepwise selection! These variables all belong to the variables of current status.
It is difficult to confirm by experiment the reason for this difference. Our intuition centers on the multicollinearity between the 4 favored variables in boosting. Table 6 shows the Spearman correlation between these variables 3 . It seems to us that because of its restrictive linear form, logit is incapable of disentangling the interweaving information contained in these variables. On the contrary, boosting seems to be able to digest this intricate information. This hypothesis is loosely confirmed by the regression results in Table 7 and 8. All the coefficients of the variables selected by AIC are significantly different from zero at the 0.1% level. The 4 variables favored by boosting (var27, var32, var33, var34), on the other hand, are less significant: var27 and var33 are significant at the 5% level, while var32 and var34 are not significant. We should remark, however, that all the signs of these 4 variables correspond to our intuition and that var32 and var34 are not far from being significant (P values=20.89% and 11.23% respectively). This is a common syndrome of multicollinearity because it increases the variances of related estimated coefficients and renders the coefficients insignificantly different from zero.
Returning on the regression in Table 7, several insights can be gained from the marginal effect of these variables. First, var9, var11 and var13 are always the best variables in any method that we have used (also valid for balanced random forest, of which we haven't presented the variable selection). These variables concern intended or rejected violations of credit line. The negative sign for var11 (number of rejected violations) should not be regarded as counter-intuitive, because of the presence of var9 (number of intended violations) and its positive coefficient which is larger than that of var11 in absolute value. This suggests that larger number of violations, whether rejected or not, indicates a higher probability of default. We have used the amount of violations instead of the number of violations to construct var13, in order to capture more precisely the confidence on each client given by the bank advisor. This variable seems to work particularly well, in the sense that the same variable on the previous year, var14, is also included by stepwise selection and by lasso. This suggests that front line staff have acquired some important experiences and intuitions in distinguishing solvable clients from insolvable ones. These experiences may be hard to be formally formulated, but are truly valuable and should be paid attention to. Second, the risk of default is intimately related to the risk of income. As var24 (standard deviation of cumulative monthly credits) shows, the more the income is unstable, the more the firm is likely to default. Var31 (cumulative monthly credits at month t) is also related to credit and decreases the default probability by having more income. Credits, rather than debits, may be considered more seriously as the source of default. Norden and Weber (2010) point out that there exists a very strong correlation between debits and credits and that the latter should be considered as the constraint of the former. Shih (2001). See Appendix D for the details of the theorem.This theorem allows us to transform a categorical variable into a discrete numeric variable for classification trees. The corresponding numeric values of sectors are shown in Table 9.
Higher values are associated with higher average default rate. This is also validated by the logistic regression in Table 7. Fourth, larger firms are less likely to default. They are more mature than startups. Commercial banks have reason to be unwilling to lend money to startups, who in some cases might need to search investment from venture capitals or angel investors.
conclusion
We have investigated the relationship between corporate checking account and credit default and shown that account information outperforms traditionally used financial ratios in predicting the default for our data sets. This result aligns with our understanding of default as a phenomenon of liquidity. Checking account information reflects a more direct and real-time status of the firm's cash flow and is a privilege of commercial banks when the firm's market value is not available. Banks can exploit economies of scale and use information on the firms' checking account to make reasonable decisions on corporate loans. Despite the importance of this subject, there is currently little literature except Norden and Weber (2010), Mester et al. (2007) and Jiménez et al. (2009). Inspired by their work, we have investigated a broader range of explicative variables and systematically compared the performance of different data sets by statistical learning methods. We have shown that these methods, together with the AUC criterion, are more accurate and reliable approaches to measure the information contained in data sets than logistic regression. While the latter often suffers from multicollinearity, machine learning methods such as random forest and boosting separately make use of these variables and are capable of disentangling intricate information. By using random forest and boosting, we have significantly increased the prediction accuracy. Tree-based methods have other advantages such as being immune to extreme values.
We should remark particularly, however, that successful statistical learning process is achieved with human expertise. Meaningful economic variables must be first of all created based on raw checking account information, just as pioneers on corporate finance have created financial ratios based on balance sheet and income statement. We also need to normalise these variable so as to eliminate the effect of account size. As we have shown, it is technically not possible (and epistemologically unacceptable for some) to create explicative variables which contain the same level of concise information simply by automated program. The 30 variables created by Definition 1 need to be perfected by eliminating about one half less useful variables and adding other potential important indicators. But even at this early stage, the importance of human expertise in financial study is illustrated.
Financial ratios and managerial questionnaire are nonetheless still important in predicting credit default. By combining them with checking account data, the model has the best prediction performance and outperforms any other model with only one single data. This suggests a certain kind of orthogonality between the information of different data sets: the financial structure, profitability, and managerial experience should be considered in parallel with checking account information in a reduced form model.
By careful approaches of model selection, we have shown some particularities of boosting in selecting important variables. We have used the 8 most important variables given by stepwise selection to gain intuitions on the mechanism of default. Violations of credit line, whether rejected or not, are particularly good indicators of upcoming default. Moreover, front line advisors seem to have notable experience in distinguishing acceptable violations, which is reflected in the percentage of permitted amount of violations. While the default is at first sight due to excessive expenses, Norden and Weber (2010) and us have focused on the importance of credits. Low level of income, as well as instability of income, increases significantly the default rate.
A Variable Definition 1
For simplicity, variables are abbreviated according to Operation Meaning X t Value of X at month t ∆X t X t − X t−11 ∆∆X t X t − X t−23 mean t (X) Mean of X during the period [t-11, t] sd t (X) Standard Deviation of X during the period [t-11, t] One might wonder why the variables are not nominated from 1 to 30. This is purely a historical problem: we have done a first version of 30 variables before modifying them to get the second version that we see right now.
B Variable Definition 3
The 5 discrete variables are defined in Table 13
C Parameters in Random Forest and Boosting
In our research, the parameters of random forest and boosting are set in the following way:
• Random Forest (R package RandomForest) -The number of candidate variables for node splitting (mtry): For a classification problem, √ p is the "standard" choice. We can also use cross-validation for determining the value.
-Balanced random forest: use the parameter "sampsize" for stratified sampling. If the forest is well balanced, the minimal number of observations in each node ("nodesize") should not greatly influence the prediction power.
• Boosting (R package xgboost)
-Choose the number of rounds ("nrounds") by the cross-validation.
-Choose a sufficiently small number for the shrinkage parameter ("eta"). For our data, the performance is stable when eta < 0.1. eta = 0.01 is used in our program.
-Maximum depth of each tree ("max_depth"): between 4 and 8. We have used max d epth = 5.
-The proportion of observations used for each tree ("subsample") does not influence greatly the prediction performance. We have used subsample = 0.5.
Figure 2 -
2Algorithm of random forest thus decrease the model variance.
Theorem 1
1The real AdaBoost algorithm fits an additive logistic regression model by stagewise and approximate optimization of J(F ) = E[e −yF (x) ].
Table 1 -
1Add captionTraining set
Test set
Error rate
Error rate
Imbalanced Balanced
Imbalanced Balanced
True value
0
0.057% 25.829%
True value
0
0.055% 25.588%
1
98.861% 28.599%
1
99.108% 27.340%
Global
3.830% 25.930%
Global
3.940% 25.660%
Table 2 -Error rates of imbalanced and balanced random forest. False negative rates are extremely
high for both the training set and the test set using imbalanced random forest.In contrast, the errors
rates using balanced random forest are much more reasonable.
Table 3 -
3Test AUCs of four groups of account data (3 definitions) in logit, random forest and
boosting. The 20 variables in Group 1 and Group 3 are selected respectively by AIC and by variable
importance in boosting. The 5 best variables in Group 2 are chosen according to variable importance
in boosting. All the variables in Group 2 are included in Group 1. Among the 20 variables in Group
3, three variables are not available for most of the observations (> 50%) and are eliminated for
random forest and for logistic regression.
Table 5 -
5Variable selection among 30 variables of Definition 1 according to boosting, stepwise
selection and lasso. Stepwise selection with AIC criterion (both forward and backward) has yielded
8 variables. For the purpose of comparison, we have chosen the 8 best variables in boosting. For
lasso, we have adjusted the parameter λ so as to yield exactly 8 non-zero coefficients.
Table 9 -
9Transformation of sector variable to numeric variable according to average defaultIncrease in expenses might be direct reason for default, but income decrease or instability may be more fundamental. Third, different economic sectors clearly have different default rate. We have constructed var29 (sector) by using a theorem in
Table 10 .
10of intended violations from the beginning of the year REJ_CNVIOL cumulative number of rejected violations from the beginning of the year INT_CAVIOL cumulative amount of intended violations from the beginning of the year REJ_CAVIOL cumulative amount of rejected violations from the beginning of the year M EAN _T CREDIT t mean TCREDIT during the period [t-23, t], used for nomalisationTable 10 -Variable AbbreviationsThe 30 variables defined in Definition 1 are built by applying the operations inTable 11. Their definition formulas are shown inTable 12. 4Abbreviation
Explanation
MIN_BAL
monthly min account balance
MAX_BAL
monthly max account balance
MEAN_BAL
monthly average account balance
MEAN_CRBAL
monthly average credit balance
MEAN_DBBAL
monthly average debit balance
TCREDIT
monthly total credits
TDEBIT
monthly total debits
INT_CNVIOL
cumulative number
Table 11 -
11Operations for creating variables4
.M EAN _CRBAL t /M EAN _CRBAL t−1>= c yearsTable 13 -5 variables defined in Definition 3. The exact values of a, b and c are not presented because of confidential agreement.Variable before discretization
Discrete classes
sum of MEAN_CRBAL during [t-2, t]
<= a€
> a€
sum of monthly intended number of violations S1,
and of monthly rejected number of violations S2,
during [t-2, t]
S1 = 0
S1 > 0&S2 = 0
S1 > 0&S2 > 0
existence of unpayed loan during [t-11, t]
NO
YES
< b
>= b
history of relationship
with the bank (years)
< c years
D Theorem for transforming sector variable to discrete numeric variable Theorem 2 Suppose there are two classes, class 1 and class 2. Let X be a categorical variable taking values on {1, 2, ..., L} where the categories are in increasing p(1|X = i) values. If φ is a concave function, then one of the L − 1 splits, X ∈ {1, 2, ..., l} where 1 <= l < L, minimizes p Lef t φ(p 1Lef t ) + p Right φ(p 1Right ).
Extracted fromJames et al. (2013)
Extracted from
It should be more appropriate to calculate the Pearson correlation because we are interested in linear correlation in the case of logistic regression. However, this correlation is not stable with respect to manipulations such as elimination of missing values or extreme values. Spearman correlation, on the other hand, seems to be quite stable with data manipulations, which shows the advantage of tree-based methods. Tree-based methods depend on ordinal properties of variables instead of cardinal ones.
Our research have adopted rigorous statistical methods to obtain a well-performed prediction model based on checking account and to identify key indicators in this data by an inductive methodology. We had a thorough discussion on the mechanisms of these methods which have significant implications on the results. This has enriched the scarce literature on this topic and can provide suggestions to banks on their decision of corporate loans. Further research may try to identify other key factors in checking account information or construct a structural model for credit default of small and medium sized enterprises.
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|
[] |
[
"Weakly nonlinear analysis of pattern formation in active suspensions",
"Weakly nonlinear analysis of pattern formation in active suspensions"
] |
[
"Laurel Ohm \nDepartment of Mathematics\nPrinceton University\n08540PrincetonNJ\n",
"† ",
"Michael J Shelley \nCourant Institute of Mathematical Sciences\nNew York University\n10012New YorkNY\n\nCenter for Computational Biology\nFlatiron Institute\nSimons Foundation\n10010New YorkNY\n"
] |
[
"Department of Mathematics\nPrinceton University\n08540PrincetonNJ",
"Courant Institute of Mathematical Sciences\nNew York University\n10012New YorkNY",
"Center for Computational Biology\nFlatiron Institute\nSimons Foundation\n10010New YorkNY"
] |
[] |
We consider the Saintillan-Shelley kinetic model of active rodlike particles in Stokes flow (Saintillan & Shelley 2008a,b), for which the uniform, isotropic suspension of pusher particles is known to be unstable in certain settings. Through weakly nonlinear analysis accompanied by numerical simulations, we determine exactly how the isotropic steady state loses stability in different parameter regimes. We study each of the various types of bifurcations admitted by the system, including both subcritical and supercritical Hopf and pitchfork bifurcations. Elucidating this system's behavior near these bifurcations provides a theoretical means of comparing this model with other physical systems which transition to turbulence, and makes predictions about the nature of bifurcations in active suspensions that can be explored experimentally.
|
10.1017/jfm.2022.392
|
[
"https://arxiv.org/pdf/2205.04942v1.pdf"
] | 248,665,524 |
2205.04942
|
9cf75875299917bbd026f2608a974d550c36cab8
|
Weakly nonlinear analysis of pattern formation in active suspensions
Laurel Ohm
Department of Mathematics
Princeton University
08540PrincetonNJ
†
Michael J Shelley
Courant Institute of Mathematical Sciences
New York University
10012New YorkNY
Center for Computational Biology
Flatiron Institute
Simons Foundation
10010New YorkNY
Weakly nonlinear analysis of pattern formation in active suspensions
(Received xx; revised xx; accepted xx)Under consideration for publication in J. Fluid Mech. 1 Banner appropriate to article type will appear here in typeset article
We consider the Saintillan-Shelley kinetic model of active rodlike particles in Stokes flow (Saintillan & Shelley 2008a,b), for which the uniform, isotropic suspension of pusher particles is known to be unstable in certain settings. Through weakly nonlinear analysis accompanied by numerical simulations, we determine exactly how the isotropic steady state loses stability in different parameter regimes. We study each of the various types of bifurcations admitted by the system, including both subcritical and supercritical Hopf and pitchfork bifurcations. Elucidating this system's behavior near these bifurcations provides a theoretical means of comparing this model with other physical systems which transition to turbulence, and makes predictions about the nature of bifurcations in active suspensions that can be explored experimentally.
Introduction
Inherently nonequilibrium suspensions of active particles abound in biological and experimental settings (Gompper et al. 2020;Marchetti et al. 2013). For example, motile bacteria such as E. coli and Bacillus subtilis propel themselves through their surrounding fluid environment, interacting through their induced flow fields (Lauga & Powers 2009;Lushi et al. 2014;Mendelson et al. 1999;Pedley & Kessler 1992), while likewise immersed microtubule bundles slide and extend, driven by ATP-driven molecular motors (DeCamp et al. 2015;Henkin et al. 2014;Needleman & Dogic 2017;Opathalage et al. 2019;Sanchez et al. 2012). These active suspensions are remarkable because, despite the near lack of inertial effects relative to viscous ones, the activity of the particles can produce large-scale coherent flows and even so-called active or bacterial turbulence, characterized by chaotic fluctuations in particle concentration and fluid velocity (Dombrowski et al. 2004;Doostmohammadi et al. 2017;Dunkel et al. 2013;Gachelin et al. 2014;Koch & Subramanian 2011;Nishiguchi et al. 2017;Peng et al. 2021;Simha & Ramaswamy 2002;Sokolov & Aranson 2012;2 Sokolov et al. 2007;Stenhammar et al. 2017;Thampi et al. 2014;Zhang et al. 2010).
Here we consider the kinetic model for a dilute suspension of active elongated particles developed by Saintillan and Shelley (Saintillan & Shelley 2008a,b). We note that a similar model was developed independently by Subramanian & Koch (2009), and that both models share many similarities with the Doi-Edwards model for passive polymers (Doi 1981;Doi & Edwards 1986). In the dilute limit, particles only interact with each other hydrodynamically by exerting an 'active stress' on the surrounding fluid. Even in this setting, changes in particle density and activity are known to cause the suspension to transition from a uniform, isotropic state to more complex states -many of which are observed experimentally -involving large-scale patterns in particle alignment and concentration. The kinetic model (Saintillan & Shelley 2008a,b) thus presents an opportunity to elucidate some of the fundamental mechanisms behind the transition to collective dynamics and active turbulence.
To understand this transition, we perform a multiple timescales expansion to determine exactly how the uniform, isotropic steady state in the 2D kinetic model loses stability in different parameter regimes. The variety of predicted behaviors near the onset of instability, which we verify through numerical simulations, is surprisingly complex. Linear stability analysis alone shows that, depending on the (fixed) ratio of particle diffusivity to concentration, the uniform isotropic state can lose stability through either a pitchfork or Hopf bifurcation. Here the bifurcation parameter is a ratio of the particle swimming speed to the particle concentration and magnitude of active stress the particles exert. Our weakly nonlinear analysis shows that both the pitchfork and Hopf parameter regions can be further subdivided into subcritical and supercritical regions, again depending on the ratio of particle diffusivity to concentration.
Numerically, we find hysteresis in the subcritical Hopf region, where far-from-isotropic quasiperiodic patterns of particle alignment are bistable with the uniform isotropic state. The patterns in this region are perhaps precursors to active turbulence. However, the dimensionality of the initial perturbation to the isotropic steady state makes a difference. If the initial perturbation is one dimensional, i.e. purely in the or direction, then only a supercritical Hopf bifurcation can occur, and we numerically locate the stable limit cycle that arises. An example of a 2D limit cycle is also located numerically within the region in which both 1D and 2D perturbations give rise to a supercritical Hopf bifurcation. In the supercritical pitchfork setting, which includes immotile (but active) particles, we identify the stable steady states emerging just beyond the bifurcation. These steady states resemble the steady vortex found in Wioland et al. (2013), although we consider periodic boundary conditions rather than confinement.
A key takeaway is that even this simple kinetic model is capable of capturing many different types of transitions to collective behavior in an active suspension. The different bifurcations analyzed here can be compared with systematic numerical studies of phase transitions in other active suspension models (Forest et al. 2004a,b;Giomi et al. 2011Giomi et al. , 2012Xiao-Gang et al. 2014;Yang et al. 2014) and help explain the 1D and 2D patterns -including limit cycles and other attractors -located numerically in Forest et al. (2014Forest et al. ( , 2013Forest et al. ( , 2015 for a similar version of the kinetic model. The weakly nonlinear analysis performed here also facilitates comparison with the normal forms arising in more classical pattern formation processes in fluid mechanics, especially thermal convection (Crawford & Knobloch 1991;Cross & Hohenberg 1993;Knobloch 1986;Pomeau 1986;Schöpf & Zimmermann 1993;Swift & Hohenberg 1977;Swinney & Gollub 1981), but also other phenomena arising in complex fluids such as electrohydrodynamic convection in nematic liquid crystals (Bodenschatz et al. 1988) and the transition from sub-to supercritical instability in viscoelastic pipe flow (Wan et al. 2021).
The paper begins by introducing the kinetic model (section 2.1) and recapping the wellstudied linear stability analysis (sections 2.2, 2.3) and role of rotational diffusion (section 2.4). Readers familiar with the model may wish to skip directly to the outline of results in section 2.5, where the types of bifurcations are mapped out in greater detail. Here ∇ · denotes the divergence on the unit sphere. The translational and rotational fluxes are given by
Background
Kinetic model of an active suspension
= 0 + − ∇(log Ψ) (2.2) = ( − T )(∇ ) − ∇ (log Ψ). (2.3)
The translational velocity consists of a particle swimming term with speed 0 in direction , particle advection by the surrounding fluid flow, and translational diffusion. For simplicity we take the translational diffusion to be isotropic. The rotational velocity depends on a Jeffery term for the rotation of an elongated particle in Stokes flow (Jeffery 1922), written here in the infinitely slender limit, along with rotational diffusion.
Finally, the surrounding fluid medium satisfies the Stokes equations with active forcing:
− Δ + ∇ = ∇ · , ∇ · = 0 (2.4) = 0 ∫ 1 Ψ( , , ) ( T − 1 2 )
.
(2.5)
Here ( , ) and ( , ) are the fluid velocity and pressure, is the fluid viscosity, and ( , ) is the trace-free active stress exerted by the particles on the fluid. The active stress is the orientational average of the force dipoles exerted by the particles on the fluid, and the sign of the coefficient 0 corresponds to the sign of the dipoles: 0 > 0 for puller particles, while 0 < 0 for pushers. Note that vanishes when the particles are in complete nematic alignment.
Nondimensionalization and quantities of interest
We choose to nondimensionalize equations (2.1)-(2.5) over slightly different characteristic velocity, length, and time scales from those commonly used in the literature (Ezhilan et al. 2013;Hohenegger & Shelley 2010;Saintillan & Shelley 2008b,a). In particular, letting denote the number of particles in the system and denote the length of the periodic box in which the particles are suspended, we nondimensionalize the model (2.1)-(2.5) according to
Ψ = Ψ / 2 , = /2 , = | 0 | 2 , = 2 | 0 | , 4
which results in the following system of equations:
Ψ = −div ( Ψ ) − ∇ ( Ψ ) = + − ∇ (log Ψ ) = (I − T )(∇ ) − ∇ (log Ψ ) −Δ + ∇ = ± ∫ 1 ( T − 1 2 I)∇ Ψ ( , , ) , div = 0.
(2.6)
Here we note the presence of three dimensionless parameters: the diffusion coefficients
= 4 2 | 0 | and = 2 | 0 | ,
and a nondimensional 'swimming speed'
= 2 0 | 0 | . (2.7)
We choose this nondimensionalization in order to easily incorporate the immotile state ( 0 = 0) into the analysis. Without swimming, equations (2.1)-(2.5) describe a suspension of shakers (Ezhilan et al. 2013;Stenhammar et al. 2017), particles which do not swim but still exert an active stress on the surrounding fluid. We also fix the domain to be the 2-dimensional torus T 2 := R 2 /(2 Z) 2 .
The parameter contains more information than just the particle swimming speed: it is really the ratio of swimming speed to the active stress magnitude and particle concentration. Note that may be related to the more familiar nondimensionalization used in Hohenegger & Shelley (2010); Saintillan & Shelley (2008b) via = (2 ℓ)/(| | ), where ℓ is the length of typical swimmer, = 0 /( 0 ℓ) is a dimensionless signed active stress coefficient ( > 0 for pullers and < 0 for pushers), and = ℓ 2 / 2 is the relative volume concentration of swimmers.
In (2.6), the active stress coefficient in the Stokes equations is scaled to unit magnitude but retains the sign of the force dipole exerted by the particles on the fluid: +1 for puller particles and −1 for pusher particles.
Dropping the prime notation, the system (2.6) may be written more succinctly as
Ψ = − · ∇Ψ − · ∇Ψ − div (I − T ) (∇ )Ψ + ΔΨ + Δ Ψ −Δ + ∇ = ± ∫ 1 ( T − 1 2 I)∇Ψ( , , ) , div = 0.
(2.8)
One way to measure deviations of the swimmer density Ψ from the uniform, isotropic steady state Ψ 0 = 1/(2 ) is to consider the relative entropy
S( ) = ∫ T 2 ∫ 1 Ψ Ψ 0 log Ψ Ψ 0 .
(2.9)
Using (2.8), the entropy can be shown to evolve according to 2008b)). The first term in (2.10) arises 5 from the viscous dissipation of the active stress exerted by the particles and is negative for pullers and positive for pushers. The two diffusive terms are negative; hence we expect puller suspensions to always relax to isotropy over time. For pushers, however, we may expect to see some more interesting behaviors: indeed, simulations show that patterns and fluctuations in particle alignment and concentration arise in certain parameter regimes (Hohenegger & Shelley 2010;Saintillan & Shelley 2008b,a, 2013, 2015. We aim to study the onset of pattern formation in these active pusher suspensions.
S( ) = ∓ 4 Ψ 0 ∫ T 2 |E | 2 − ∫ T 2 ∫ 1 |∇(log Ψ)| 2 + ∇ (log(Ψ)) 2 Ψ Ψ 0 (2.10) where E = 1 2 (∇ + (∇ ) T ) (see (Saintillan & Shelley
It will be useful to first define some system quantities that can be measured numerically and used to verify analytical predictions. One such quantity is the active power input, defined for pusher particles by
P ( ) = ∫ T 2 ∫ 1 T E ( , ) Ψ( , , )
.
(2.11)
The sign is opposite for puller particles. This quantity can be understood as the perturbative power input due to the interaction of the active particles with the flow (as opposed to the power input of each individual particle). Using the Stokes equations in (2.8), the active power balances the rate of viscous dissipation in the fluid:
P ( ) = ∫ T 2 2 |E ( , )| 2 .
(2.12)
Note that P ( ) = 0 for the uniform, isotropic steady state, and tracking the growth of P ( ) (or, equivalently, the rate of viscous dissipation) serves as a measure of the instability of the uniform state (Saintillan & Shelley 2015).
We also define the concentration field
( , ) = ∫ 1 Ψ( , , ) (2.13)
and nematic order tensor
Q( , ) = 1 ( , ) ∫ 1 ( T − I/2)Ψ( , , )
.
(2.14)
We can then define the scalar-valued nematic order parameter as
N ( , ) = maximum eigenvalue of Q( , ). (2.15)
The nematic order parameter measures the local degree of nematic alignment: N ( , ) = 0 when the particle orientations are isotropic and N ( , ) = 1 when all surrounding particles are exactly aligned.
2.3. Linear stability: the eigenvalue problem From here, we restrict our attention to the pusher case in two spatial dimensions. We begin by recalling the results of the eigenvalue problem resulting from a linear stability analysis about the uniform, isotropic steady state Ψ 0 = 1/(2 ) and = 0 (
Ψ = − · ∇Ψ + 2 T ∇ + ΔΨ + Δ Ψ −Δ + ∇ = − 1 2 ∫ 1 ( T − 1 2 I)∇Ψ( , , ) , div = 0.
(2.16) 6 (a) (b) Figure 1: (a) Real part (shifted by 2 ) and (b) imaginary part of the growth rate ( ) versus | |. Figure (a) can be read as follows: fix a value of 0 < 2 < 0.25. Look at the corresponding horizontal green line. As is lowered, the green line intersects the blue curve. This is where the eigenvalue ( ) becomes unstable. Different types of bifurcations are possible depending on the value of 2 : for example, point B corresponds to a purely real eigenvalue crossing, while the presence of nonzero Im( ) signals a Hopf bifurcation at points D and E. At point C, indicated by a red dot, two real eigenvalues meet and become complex. In the case of shakers ( = 0), we may consider the real eigenvalue crossing at point A along the -axis.
We insert the plane wave ansatz Ψ = ( , )e i · + , ∈ C, into (2.16), and choose coordinates such that the wavevector = and the particle orientation = cos + sin , 0 < 2 . Defining := + 2 , we obtain an eigenvalue problem for in the form of an integrodifferential equation on 1 :
( , ) = −i cos ( , ) + 1 cos sin ∫ 2 0 ( , ) sin cos + ( , ).
(2.17) We note that while it may be more suggestive to write in terms of sin(2 ) rather than sin cos , we find that the details of the weakly nonlinear calculations in the following sections are slightly simpler in terms of cos = · and sin = · only.
In the absence of rotational diffusion ( = 0), we may solve (2.17) for ( , ) as
( , ) = cos sin + i cos ,
where is such that
[ ] := ∫ 2 0 ( , ) cos sin = .
(2.18)
In particular, satisfies the implicit dispersion relation 2 2 + 2 3 − 2 2 √︃ 2 2 + 2 4 4 = 1.
(2.19)
Recalling that = + 2 , we may then numerically solve for the relationship between and (see figure 1).
A similar calculation for = shows that the eigenvalues ( ) plotted in figure 1 also correspond to a -direction eigenfunction whose eigenmodes are a 90-degree rotation (in ) 7 of the -direction eigenmodes. The solutions of (2.16) arising from eigenfunctions of the linearized operator are thus given by (2.20) and all scalar multiples of (2.20), where
Ψ( , , ) = ( , )e i + + ( , )e i + , 2 + 2 = 1,( , ) = cos sin + 2 + i cos , ( , ) = cos sin + 2 + i sin .
In particular, besides point C, each eigenvalue ( ) has a 4-dimensional eigenspace over C, spanned by the -and -direction components with ± . When | | < √ 3/9, there are two distinct real-valued eigenvalues ( ) which satisfy the required condition ∫ 2 0 cos sin = ∫ 2 0 cos sin = , both of which then have a 4-dimensional eigenspace over R.
Note that the dispersion relation (2.19) is exact only in the absence of rotational diffusion, but for the purposes of this paper, we will consider 0 < 1. This alters figure 1, but only slightly (see next section 2.4 for details). In return, we do not have to contend with the continuous spectrum present in the Numerical simulations indicate that if the point ( 2 , | |) lies to the right of the blue contour in figure 1a for each , then the uniform, isotropic steady state is stable to small perturbations (see sections 4 and 5). In this region, particle diffusion and swimming -processes which tend to decrease order among the particles -are large relative to both the active stress magnitude and the particle concentration, which favor local alignment. As is decreased, Re( ( )) crosses the green line corresponding to the (fixed) value of 2 and the uniform, isotropic state becomes unstable.
Since we are considering the system (2.6) on a periodic domain and have nondimensionalized according to the length of the domain, the | | = 1 mode is the lowest nontrivial mode in the system and hence the first to lose stability as decreases. As noted in Koch & Subramanian (2011), "the finiteness of the domain may act to stabilize the system." Indeed, the stabilizing effects of mixing on T ( = 2, 3) due to swimming are studied in Albritton & Ohm (2022) and play a role in the patterns observed here.
We aim to understand the onset of pattern formation in (2.6) by exploring the many different ways the | | = 1 mode can lose stability as is decreased. From figure 1, we can see that depending on , the type of bifurcation that we expect to see for the | | = 1 mode will change. We aim to characterize all of the types of initial bifurcations admitted by the system through a weakly nonlinear analysis. According to the dispersion relation, if 1/9 < < 1/4, as decreases, the purely real eigenvalue will change sign from positive to negative across the top branch of the blue curve, between points A and C. For 0 < < 1/9, however, we expect to see a Hopf bifurcation as crosses the blue curve roughly between points C and E, since for these values of , has nonzero imaginary part.
As noted, figure 1 does not quite give the exact locations of the bifurcations that we will consider here, since we still need to consider the effects of (small) > 0. This will be the subject of the next section.
Role of rotational diffusion
The dispersion relation (2.19) was obtained in the absence of rotational diffusion; however, studying pattern formation near the isotropic steady state will require > 0. When = 0, the system (2.8) has infinitely many steady states; in particular, any spatially uniform swimmer distribution Ψ = Ψ( ) is a steady state. The continuum of nearby steady states obscures the mechanism by which the isotropic state Ψ = 1/(2 ) loses stability; indeed, any function Ψ( ) belongs to the kernel of the linearized operator. When we add in > 0 (along with the assumption that the total number of swimmers is conserved), this kernel is eliminated. Thus we aim to determine when the effect of small > 0 can be considered as a perturbation of the dispersion relation (2.19) and figure 1.
In particular, given a wavenumber , for small > 0, we wish to determine when an expansion (in ) of the form
= 0 + 1 + ( 2 ), = 0 + 1 + ( 2 ) (2.21)
is valid for some ( 1 , 1 ).
Plugging this expansion into the eigenvalue problem (2.17) and separating scales, at (1) we obtain the = 0 eigenfunctions and eigenvalue relation
0 ( , ) = cos sin 0 + 2 + i cos
where 0 is such that
[ 0 ] = ∫ 2 0 0 cos sin = .
(2.22)
At ( ) we obtain the expression
1 = 1 0 ∫ 2 0 1 cos sin − 1 0 0 + 2 + i cos + 2 0 0 + 2 + i cos .
Taking [·] of both sides and using (2.22) then yields an integral expression for 1 :
1 ∫ 2 0 cos 2 sin 2 ( 0 + 2 + i cos ) 2 = ∫ 2 0 2 0 cos sin 0 + 2 + i cos .
As long as Re( 0 + 2 ) ≠ 0 (which holds for the | | = 1 modes as long as ≠ 0), we obtain an expression for 1 : which is finite as long as 2 2 ≠ 3( 2 + Re( 0 )) 2 . The line
1 = 5 6 2 2 2 2 − 3( 2 + 0 ) 2 + 1 2 2 2 2 2 + ( 2 + 0 ) 2 + 5( 2 + 0 ) 3 ( 2 2 − 3( 2 + 0 ) 2 ) √︁ 2 2 + ( 2 + 0 ) 2 − 7 3 , 9 (a) (b)0 + 2 = | | √ 3 (2.23)
is plotted along with the real part of the = 0 dispersion relation (2.19) in figure 2a. Away from this line, for small > 0, we may consider the solutions of the eigenvalue problem (2.17) as perturbations of the = 0 eigenvalues and eigenfunctions. As we can see, this line (2.23) passes through the point C from figure 1 where the two real eigenvalues meet and become two complex conjugate eigenvalues. A very precise choice of and should correspond to a codimension 2 bifurcation at point C; however, a different scaling than (2.21) with respect to is likely needed to study this point in detail. We will not attempt to study point C in detail here, and will instead focus on the more generic bifurcations between points A and C and between points C and E.
In figures 2a and 2b we plot the dispersion relation for 0 + 1 on top of 0 using = 0.001. Away from the crossing with the line (2.23), the 0 + 1 curve aligns very closely with the 0 curve. Hereafter, for most numerical purposes, we will take = 0.001 -small enough to use figure 1 as our roadmap for determining roughly where in the ( , ) parameter space to look to see different system behaviors. We note that, while the expansion in is not rigorous, it is backed later on by the close agreement of the predicted behavior near bifurcation points (from amplitude equations) with the observed behavior from numerical simulations. In particular, it appears that, away from point C, the small > 0 picture is largely captured by this expansion.
Outline of results
The remainder of the paper is devoted to a weakly nonlinear analysis of the different possible bifurcations apparent in figure 1, which we map out in greater detail in figure 3. We begin in section 3 by considering the immotile case = 0. We examine the real eigenvalue crossing at point A for all values of for which a bifurcation occurs and show that the resulting pitchfork bifurcation is always supercritical. In section 4 we assume is very small and Figure 3: Diagram of the various types of bifurcations through which the uniform isotropic steady state loses stability, depending on the location of the bifurcation value .
Here the subscript is used to reflect that the value of depends on the translational diffusion through the dispersion relation plotted in figure 1. Note that the letters A -E correspond to the positions in figure 1a, which we also repeat here for clarity. The upper line labeled 2D corresponds to the evolution of initial perturbations to the uniform isotropic state with both components in both the and direction ( , ≠ 0 in (2.20)), while the lower line labeled 1D corresponds to perturbations with or component only
( = 0 or = 0 in (2.20)).
analyze the Hopf bifurcation occuring along the curve between points C and E. We show that for initial perturbations to the uniform isotropic state with both and components (i.e. both , ≠ 0 in (2.20); see line labeled 2D in figure 3), the bifurcation transitions from supercritical to subcritical at roughly point D, but is always supercritical for initial perturbations with either = 0 or = 0 (labeled 1D in figure 3). In section 5, we again consider very small and study real eigenvalue crossing occuring along the curve between points A and C. The pitchfork bifurcation also transitions from supercritical to subcritical in both the 2D and 1D cases, with the change occuring roughly at point B in the case of an initial perturbation in both and , and just before point C in the case of an -only or -only initial perturbation.
Each section is accompanied by numerical simulations verifying the predicted behaviors near the different bifurcations and illustrating the various states that arise. The numerics are a pseudo-spectral implementation of equations (2.6) with time-stepping via a second order implicit-explicit backward differentiation scheme.
Immotile particles: supercritical pitchfork bifurcation
The simplest scenario for studying the onset of pattern formation in the model (2.8) is in the case = 0; i.e. the particles are immotile (or shakers (Ezhilan et al. 2013;Stenhammar et al. 2017)) but still exert a dipolar force on the surrounding fluid. The uniform, isotropic steady state in a suspension of immotile particles undergoes a bifurcation to an alignment instability, indicated by point A in figure 1, which we study in greater detail. We first show via weakly nonlinear analysis that the resulting pitchfork bifurcation is always supercritical, and we then numerically explore examples of the emerging nontrivial steady state.
Weakly nonlinear analysis
In the case of immotile particles, we can explicitly calculate the eigenvalues and eigenfunctions of the linearized operator when > 0. The (purely real) eigenvalues ( ) and eigenmodes ( , ), ( , ) are given by
( ) = 1 4 − 2 − 4 , ( , ) = ( , ) = cos sin . (3.1)
From the immotile dispersion relation (3.1), when > 1 4 − 4 , all eigenvalues ( ) are negative, but as is decreased, the | | = 1 modes are the first to change sign as = * = 1 4 − 4 is crossed. We study the nature of this bifurcation for different via the method of multiple scales. For 0 < 1, we fix = 1 4 − 4 − 2 , so the | | = 1 modes are just barely growing, and define the slow timescale = 2 . We then assume the following expansions in :
Ψ = 1 2 (1 + Ψ 1 + 2 Ψ 2 + 3 Ψ 3 + · · · ), = 1 + 2 2 + 3 3 + · · · . (3.2)
Plugging each of these expansions into equation (2.8) with = 0 and separating by orders of , at ( ) we obtain the = 0 version of the eigenvalue equation (2.17), evaluated at the critical value * = 1 4 − 4 where (1) = 0:
L [Ψ 1 ] := −2 T ∇ 1 − 1 4 − 4 ΔΨ 1 − Δ Ψ 1 = 0. (3.3)
This equation is satisfied by the | | = 1 modes of the eigenfunctions (2.20), where we recall that the eigenmodes in the immotile case are given by (3.1). Thus Ψ 1 and 1 have the form
Ψ 1 = cos sin ( )e i + ( )e i + c.c., 1 = − i 8 ( )e i + ( )e i + c.c. (3.4)
Here we have inserted the complex amplitudes ( ), ( ) which depend solely on the slow timescale , and for which we aim to find an equation. Throughout, we use c.c. to denote the complex conjugate of each of the preceding terms.
At ( 2 ) we obtain the equation
L [Ψ 2 ] = − 1 · ∇Ψ 1 − div (I − T ) (∇ 1 )Ψ 1 ) (3.5)
where the operator L is as defined in (3.3). Using the expressions (3.4) for Ψ 1 and 1 , the right hand side of equation (3.5) can be calculated explicitly (see equation (A 1) in Appendix A). Noting that the right hand side expression contains only exponential terms of the form e ±2i , e ±2i , and e ±i( ± ) , with no terms proportional to e ±i or e ±i , equation (3.5) is solvable without any additional conditions on the coefficients and . In particular, due to the form of the right hand side expression, we look for Ψ 2 and corresponding 2 of the form Ψ 2 = 2,1 e i( + ) + 2,2 e i( − ) + 2,3 2 e 2i + 2,4 2 e 2i + 2,5 | | 2 + 2,6 2 + c.c.,
2 = − i 8 ∫ 2 0 2,1 e i( + ) ( − ) + 2,2 e i( − ) ( + ) (cos 2 − sin 2 ) − i 4 ∫ 2 0
2,3 e 2i 2 + 2,4 e 2i 2 sin cos + c.c.,
(3.6) where each 2, = 2, ( ). Plugging these expressions (3.6) into the left hand side of (3.5) (see Appendix A, equation (A 2) for the full expression), after matching exponents with the right hand side, we can explicitly solve for each 2, : (3.7)
2,1 = − 1 + 16 (sin 4 + cos 4 ) − 2(1 − 8 ) sin cos + 3 4(1 + 16 ) 2,2 = − 1 + 16 (sin 4 + cos 4 ) + 2(1 − 8 ) sin cos + 3 4(1 + 16 ) 2,3 = 2 4 − 2 cos 4 + 3(1 − 16 ) 2(1 − 12 ) cos 2 + 3 1 − 12 2,4 = 2 4 − 2 sin 4 + 3(1 − 16 ) 2(1 − 12 ) sin 2 + 3 1 − 12 2,5 = −
Inserting each of the coefficients (3.7) in the expression (3.6) for 2 , we have that each -integral vanishes and therefore 2 = 0. At ( 3 ) we thus obtain the following equation for
Ψ 3 : L [Ψ 3 ] = − Ψ 1 − 1 · ∇Ψ 2 − div (I − T ) (∇ 1 )Ψ 2 − ΔΨ 1 .
(3.8) Letting R ( , , ) denote the right hand side of (3.8), we have that R may be calculated explicitly using (3.4) and (3.6); in particular, R is of the form
R ( , , ) = ( , )e i + ( , )e i + 2 + ( , )e i(2 + ) + 2 + ( , )e i( +2 ) + 2 − ( , )e i(2 − ) + 2 − ( , )e i(− +2 ) + 3 ( , )e i3 + 3 ( , )e i3 + c.c.
(3.9) By the Fredholm alternative, equation (3.8) admits a solution Ψ 3 as long as
∫ 2 0 ∫ T 2 R ( , , )Φ( , ) = 0 for all Φ( , ) such that L * [Φ] = 0. (3.10)
Since the operator L defined in (3.3) is self-adjoint in the immotile case, we have that any such Φ has the form Φ = cos sin ( e i + e i ) + c.c. for any 2 + 2 = 1.
Thus (3.10) is automatically satisfied for each term of R except for ( , )e i + ( , )e i + c.c. The exact form of and is given in Appendix A, equation (A 3).
Since the ratio / is arbitrary, we need that both
∫ 2 0 ( , ) = 0 and ∫ 2 0 ( , ) = 0.
These two conditions together lead to a coupled system of ODEs for the amplitudes , :
= + ( 2 1 | | 2 + 2 2 2 ) = + ( 2 1 2 + 2 2 | | 2 ) ) (3.11) where 1 = − 3(3 − 28 − 32 2 ) 1024 (1 − 12 ) , 2 = 7 − 136 − 2432 2 1024 (1 − 8 ) (1 + 16 )
.
(3.12) 13 Figure 4: In the immotile setting, the coefficients 1 + 2 (3.13) and 1 (3.12) are both negative for all values of for which a bifurcation occurs (0 < < 1/16), indicating that a supercritical pitchfork bifurcation occurs for both 2D ( and ) and 1D ( -only) initial perturbations to the uniform isotropic state.
If 1 + 2 < 0, the system (3.11) has real, nonzero steady states of the form
= ± 1 √︁ −( 1 + 2 ) , = ± 1 √︁ −( 1 + 2 )
.
We have that
1 + 2 = −1 − 104 + 560 2 + 9600 3 − 6144 4 512 (1 − 8 ) (1 − 12 ) (1 + 16 ) , (3.13)
which, as we can see from figure 4, is indeed negative for all values of for which a bifurcation occurs. Thus for any relevant level of rotational diffusion, the uniform, isotropic steady state loses stability through a supercritical pitchfork bifurcation and nontrivial stable steady states emerge.
To leading order in = √︁ * − = √︃ 1 4 − 4 − , the stable steady states which bifurcate from the uniform isotropic state are of the form
Ψ = 1 2 ± 2 √︄ 512 (1 − 8 )(1 − 12 ) (1 + 16 ) 1 + 104
− 560 2 − 9600 3 + 6144 4 cos sin e i ± e i + c.c.
(3.14) Note that as long as the initial perturbation coefficients and are both nonzero, the form of (3.14) does not depend on or . If either = 0 or = 0 in (2.20), the bifurcating stable steady states take on a different form. Without loss of generality, we consider = 0. In this case, the coupled system (3.11) reduces to the single amplitude equation
= + 1 | | 2 (3.15)
where 1 is as in (3.12). As shown in figure 4, 1 is negative for all 0 < < 1/16, and therefore the uniform steady state still loses stability through a supercritical pitchfork bifurcation for all meaningful choices of . To leading order in , the stable steady states which emerge are of the form
Ψ( , ) = 1 2 1 ± √︄ 1024 (1 − 12 ) 3(3 − 28 − 32 2 ) cos sin e i + c.c. (3.16)
Numerical evidence of supercriticality along with simulated examples of the emerging 14 steady states (3.14) and (3.16) are presented in the following section.
Numerics
To study the immotile bifurcation numerically, we begin by checking for supercriticality. We first fix = 0.0125, so that by (3.1), * = 0.2 is the bifurcation value. Taking our initial condition to be a random, small-magnitude perturbation to the uniform isotropic state in both and , we begin by running the simulation with = 0.1 until = 500. Then the value of is increased by 0.02 every 100 until = 0.3. The bifurcation value * = 0.2 is reached at = 1000.
Over the course of the simulation we keep track of the 2 norm of the velocity field
(·, ) 2 2 (T 2 ) = ∫ T 2 | ( , )| 2 ,
which is plotted continuously over time in figure 5a. We also keep track of the time-averaged rate of viscous dissipation in the fluid for each value of , which we recall from (2.12) balances the active power input P. Given a constant value of over the time interval ( 1 , 2 ), we measure the value of
P = 1 2 − 1 ∫ 2 1 P ( ) = 1 2 − 1 ∫ 2 1 ∫ T 2 2 |E ( , )| 2 ,(3.17)
which we consider as a function of . In our case, 2 − 1 = 100 for each different value of . We plot P in figure 5b over the course of the simulation for the various values of . As expected for a supercritical bifurcation, we see that P smoothly decays to zero as increases toward the bifurcation value * = 0.02, and remains zero after the bifurcation is reached. This smooth transition from nontrivial dynamics to the uniform, isotropic steady state as is slowly varied from a very unstable value through the bifurcation value and beyond may be contrasted with the hysteresis seen later in the subcritical Hopf region for motile particles (Section 4.2).
We next consider what the emerging stable steady states actually look like. Given , we choose such that 2 = * − = 0.02. We initialize the simulation by perturbing the uniform, isotropic state with a small random-magnitude perturbation to a random assortment of the five lowest spatial modes in both and with random orientation , and run until the dynamics settle into a steady state. The resulting nematic order parameter and direction of preferred local nematic alignment are plotted in figures 6a and 6b for ( , ) = (0.0025, 0.22) ( * = 0.24) and ( , ) = (0.03125, 0.105) ( * = 0.125), respectively. We see that the higher spatial modes decay over time, and the resulting steady state consists only of the unstable | | = 1 mode. For ( , ) = (0.03125, 0.105) we also plot the fluid vorticity field ( , ) = ∇ × ( , ) (3.18) and a sampling of the velocity field ( , ) throughout the domain.
The supplementary video Movie1 shows the system approaching an example of the steady state shown in figure 6 as the bifurcation is approached from below. In this case = 0.0125 so * = 0.2. The steady state is reached after approaching = 0.18 from below.
We also consider the same parameter combinations as in figure 6 for an -only initial perturbation, and plot the results in figure 7. Given the form of the calculated steady state (3.14) for a 2D ( and ) perturbation and (3.16) for a 1D ( -only) perturbation, we expect the particles to display a slight preference for the (nematic) orientations = 4 (≡ 5 4 ) and = 3 4 (≡ 7 4 ) where ± cos sin is maximized. This preference is clearly visible in figures 6a and 6b as well as in figures 7b and 7a.
In addition, in the 2D case, figure 4 suggests that given there should be a ∈ (0, 1 16 ) which maximizes the magnitude of the deviation from isotropy in the emerging stable steady state (3.14), and that as → 0 the correlation in particle alignment should disappear. We can see numerical evidence of this dependence in the difference in the magnitude of Similarly, in the case = 0, the dependence in (3.16) suggests that as increases, the deviation from isotropy at a distance 2 from the bifurcation continues to increase. This increase is shown in figures 7b and 7a.
Finally, we make a quantitative comparison between the expression (3.14) and the numerical solution to the full system (2.8) with = 0 as 2 = * − is varied. We fix = 0.03125, so * = 0.125, and consider five different values of 2 . For each , we allow the system to reach a steady state, and in figure 8, we plot the difference between the predicted steady state Ψ p ( , , ) (3.14) and the computed steady state Ψ c ( , , ) over a 1D slice of -values for fixed and . As is decreased, the pointwise difference between the predicted and computed steady states decreases like 2 , as expected. This is further corroborated by table 1, which displays the maximum difference max 0 , , 2 Ψ p ( , , ) − Ψ c ( , , ) . Table 1: Maximum difference between the predicted steady state Ψ p (3.14) and the computed steady state for (2.8) with = 0 for five different values of . The difference scales with 2 , as expected.
Motile particles: Hopf bifurcation
When > 0, the system (2.8) can experience a much richer catalogue of bifurcations, as evidenced by figure 1. We focus first on the Hopf region (roughly between C and E on figure 1), where the eigenvalue corresponding to the | | = 1 mode is complex-valued. Fixing a very small level of rotational diffusion (e.g. = 0.001), we choose a value of translational diffusion < 1 9 such that for some value of = , we have (±1) = ±i ; i.e. the real part of (±1) vanishes and a Hopf bifurcation occurs. Here the subscript is used to denote that the values of and depend on the choice of through the implicit expression (2.19).
We perform a weakly nonlinear analysis of the Hopf bifurcation for the | | = 1 mode in Section 4.1. We show that for relatively large , the bifurcation at = is supercritical and a stable limit cycle arises just beyond the bifurcation. However, for small , the bifurcation is subcritical for general initial perturbations in both and , but supercritical for -only perturbations. We find numerical evidence of hysteresis in this subcritical region, which we explore in Section 4.2. In the supercritical region, we numerically locate an example of the emerging stable limit cycle (see Section 4.4).
Weakly nonlinear analysis
As in the immotile case, we consider = − 2 for 1, so the | | = 1 modes are very slightly unstable. We again consider a slow timescale = 2 and expand Ψ and in as (3.2). Plugging these expressions into (2.8), at ( ) we obtain the equation
L [Ψ 1 ] := Ψ 1 + · ∇Ψ 1 − 2 T ∇ 1 − ΔΨ 1 − Δ Ψ 1 = 0. (4.1)
Note that when > 0 we no longer have an explicit expression for the eigenvalues and eigenfunctions of the linearized operator (2.16) when > 0, and therefore the following analysis must be performed in the limit of very small rotational diffusion, which we treat perturbatively using Section 2.4.
Using the form (2.20) of the eigenvalues and eigenfunctions of the linearized operator when = 0, and using the perturbative expression for > 0 from Section 2.4, we have that Ψ 1 and 1 are given up to ( ) by At ( 2 ) we obtain the equation
L [Ψ 2 ] = − 1 · ∇Ψ 1 − div (I − T (∇ 1 )Ψ 1 ) ,(4.3)
where the operator L is as defined in (4.1). Using (4.2), we can explicitly calculate the expression on the right hand side of (4.3) up to terms of ( ) (see Appendix B, equation (B 1) for the full expression). As in the immotile case, using the form of the right hand side as a guide, we look for (Ψ 2 , 2 ) of the form Ψ 2 = 2,1 e i( + )+2i + 2,2 e i( − ) + 2,3 2 e 2i +2i + 2,4 2 e 2i +2i + 2,5 | | 2 + 2,6 2 + c.c., (4.4) where 2, = 2, ( ). Inserting the ansatz (4.4) into the left hand side of (4.3), we then match exponents with the right hand side and solve for each of the coefficients 2, . Further details are contained in Appendix B, equations (B 3) and (B 4), but we obtain that both 2,5 and 2,6 are ( −1 ) for small , while each of the other coefficients are (1) in as → 0. Thus for sufficiently small , the coefficients 2,5 and 2,6 dominate the behavior of Ψ 2 . We may solve for 2,5 and 2,6 explicitly up to terms of (1) in : 2,5 = 2 1 cos + 2 (2 cos 2 − 1) + 3 cos 3 + 4 log( + i + i cos ) − 1 2 ∫ 2 0 log( + i + i cos ) + ( ) 2,6 = 2 1 sin + 2 (2 sin 2 − 1) + 3 sin 3
+ 4 log( + i + i sin ) − 1 2 ∫ 2 0 log( + i + i sin ) + ( ) (4.5) where 1 = − i( + i ) 2 2 3 , 2 = − + i 8 2 , 3 = i 6 , 4 = ( + i ) 3 2 4 .
(4.6)
Note that we must enforce ∫ 2 0 2,5 = ∫ 2 0 2,6 = 0 in order to have ∫ T 2 ∫ 2 0 Ψ 2 = 0, i.e. to enforce that the total concentration of particles in the system is preserved at 1/(2 ).
We may thus rewrite (4.4) as
Ψ 2 = 2,5 ( ) | | 2 + 2,6 ( ) 2 + ( 0 ), 2 = ( 0 ).
At ( 3 ) we obtain the equation
L [Ψ 3 ] = − Ψ 1 + · ∇Ψ 1 − 1 · ∇Ψ 2 − 2 · ∇Ψ 1 − div (I − T )(∇ 2 )Ψ 1 ) − div (I − T ) (∇ 1 )Ψ 2 .
(4.7)
Using (4.2) and (4.4), we may calculate the form of the right hand side. First, we note that
Ψ 1 = ,1 ( )( )e i +i + ,1 ( ) ( )e i +i + c.c.
· ∇Ψ 1 = i cos ,1 ( ) e i +i + i sin ,1 ( ) e i +i + c.c., (4.8)
where ,1 and ,1 are as in (4.2). The remaining terms on the right hand side of (4.7) may be written
− 1 · ∇Ψ 2 − 2 · ∇Ψ 1 − div (I − T )(∇ 2 )Ψ 1 ) − div (I − T ) (∇ 1 )Ψ 2 = R 1 ( , , , ) + R 3 ( , , , ) + c.c., R 1 ( , , , ) = 1 ( ) 3 | | 2 e i +i + ( ) 2 2 e i +i + ( ) 2 | | 2 e i +i + ( ) 3 2 e i +i R 3 ( , , , ) = 2 + ( , )e i(2 + )+i3 + 2 + ( , )e i( +2 )+i3 + 2 − ( , )e i(2 − )+i + 2 − ( , )e i(− +2 )+i + 3 ( , )e i3 +i3 + 3 ( , )e i3 +i3 .
(4.9) The expressions for , ( ), ( ), and ( ) are written up to ( ) in Appendix B, equation (B 5). As in the immotile setting, the exact form of the coefficients in R 3 ( , , , ) will not be important due to the solvability condition for equation (4.7).
In particular, in order for the ( 3 ) equation (4.7) to have a solution Ψ 3 , the right hand side of (4.7) must satisfy the same Fredholm condition (3.10) as in the immotile case, except now the operator L defined in (4.1) is no longer self-adjoint. The adjoint L * is given by
L * [Φ] = − Φ − · ∇Φ − 2 T ∇ − ΔΦ − Δ Φ,
and, by the same calculation as in Sections 2.3 and 2.4, any Φ satisfying L * [Φ] = 0 has the form Φ = ,1 ( )e i +i + ,1 ( )e +i + c.c.,
2 + 2 = 1 ,1 = cos sin − i − i cos + ( ), ,1 = cos sin − i − i sin + ( ).
(4.10) Due to the form of R 3 (4.9), we have that ∫
T 2 R 3 ( , , , )Φ( , , ) = 0
for Φ as in (4.10), so the Fredholm condition (3.10) is automatically satisfied. Thus it remains to ensure that
∫ 2 0 ∫ T 2 − Ψ 1 + · ∇Ψ 1 + R 1 Φ = 0
for any Φ as in (4.10). Using the forms of (4.8), (4.9), and (4.10), this leads to the following 20 coupled system of equations for the amplitudes and :
0 ( ) = 3 + 1 ( 2 1 | | 2 + 2 2 2 ) 0 ( ) = 3 + 1 ( 2 1 2 + 2 2 | | 2 ) ) (4.11)
where the complex constants 0 , 1 , 2 , and 3 are given by
0 = ∫ 2 0 2 ,1 ( ) = ∫ 2 0 2 ,1 ( ) 1 = ∫ 2 0 ( ) ,1 ( ) = ∫ 2 0 ( ) ,1 ( ) 2 = ∫ 2 0 ( ) ,1 ( ) = ∫ 2 0 ( ) ,1 ( ) 3 = i ∫ 2 0 2 ,1 ( ) cos = i ∫ 2 0 2 ,1 ( ) sin .
(4.12)
Explicit expressions for each depending on , , and are given in Appendix (B), equation (B 7). Note that although ,1 ( ) ≠ ,1 ( ), etc., the coefficients for the and directions are equal.
In the Hopf setting, each coefficient is now complex-valued. Writing ( ) = ( )e i ( ) and ( ) = ( )e i ( ) , we may separate (4.11) into equations for the magnitudes , and the phases , . We then look for conditions on the coefficients such that (4.11) admits a nontrivial limit cycle satisfying = = 0.
We find that if 0 ≠ 0, Re ( 1 + 2 )/ 0 ≠ 0, and In particular, if (4.13) holds, the stable limit cycle arising just after the bifurcation, to leading order in = √ − , is given by
Ψ = 1 2 1 ± √︁ − Re 3 0
Re 1 + 2 0 ,1 ( )e i +i( + 2 ) ± ,1 ( )e i +i( + 2 ) .
(4.16) If the condition (4.13) does not hold, the bifurcation at = is subcritical and, while the system behavior for < is less predictable, we may expect to see hysteresis if is then increased beyond . In particular, the system may remain in a stable, nontrivial state well after the uniform isotropic state has also become stable.
As in the immotile setting, we also consider the effects of an initial perturbation in the -direction only. When = 0, we obtain the single equation
0 ( ) = 3 + 1 | | 2 (4.17)
for the -direction amplitude . In this case a stable limit cycle emerges beyond the bifurcation if 0 ≠ 0, Re 1 / 0 ≠ 0, and and, to leading order in , the emerging solution after the bifurcation has the form
Ψ = 1 2 1 ± √︁ − Re 3 0 Re 1 0 ,1 ( )e i +i( + 2 ) (4.20)
In figure 9, we plot each of 0 , Re 3 / 0 , Re ( 1 + 2 )/ 0 , and Re 1 / 0 using the perturbed dispersion relation (2.21) (see figure 2) with = 0.001 for ∈ [0.2, 0.7]. This range of essentially covers all for which a Hopf bifurcation exists and for which the perturbative expression in is valid.
From figure 9a, we see that 0 ≠ 0 for all values of in the region of interest, so division by 0 always makes sense. Furthermore, from figure 9b, we see that Re 3 / 0 is always positive. Therefore Re ( 1 + 2 )/ 0 and Re 1 / 0 will determine the sign of the quantities of interest in conditions (4.13) and (4.18), respectively. Interestingly, figure 9c indicates that Re ( 1 + 2 )/ 0 changes sign for some ∈ [0.2, 0.7]. Note that since 1 and 2 are calculated only up to ( ), the precise location where Re ( 1 + 2 )/ 0 = 0 cannot be determined since the location may depend on lower order terms in . However, for a sufficiently small bifurcation value (determined by choosing sufficiently large), we should see a stable limit cycle emerge beyond the bifurcation, while for sufficiently large ( sufficiently small), the bifurcation should be subcritical.
In contrast, Re 1 / 0 < 0 for all ∈ [0.2, 0.7] (see figure 9d), indicating that in the Hopf region, for initial perturbations in the -direction only, a stable limit cycle should always arise immediately beyond the bifurcation.
In the following sections we explore these predictions numerically.
Subcritical region and bistability
We first consider the subcritical region for initial perturbations in both and , where we find strong numerical evidence of hysteresis. For the following simulations, we fix = 0.001 Figure 9: The relevant relationships among the coefficients 0 , 1 , 2 , and 3 of the amplitude equations (4.11) are plotted over the Hopf bifurcation range ∈ [0.2, 0.7]. In particular, since the curves for (a) 0 and (b) Re( 3 / 0 ) are both strictly positive for all such , the type of Hopf bifurcation is determined by (c) Re ( 1 + 2 )/ 0 (for 2D initial perturbations) or (d) Re 1 / 0 (for 1D initial perturbations). In the 2D case, there is a transition from supercritical to subcritical at some value of ∈ [0.2, 0.7], indicated by the vertical dotted line in (c). and = 0.02, so the bifurcation occurs at roughly ≈ 0.63. According to figure 9c, this value of lies well within the subcritical region for generic initial perturbations with both and components.
22 (a) (b) (c) (d)
Similar to the immotile bifurcation study in figure 5, we take our initial condition to be a small random perturbation in both and to the uniform isotropic state, and begin by running the simulation with = 0.48 for 500 , allowing the system to move away from the isotropic state. We then increase by 0.03 every 100 until = 0.93, well beyond the bifurcation value of ≈ 0.63. We then decrease by 0.05 every 100 until we reach = 0.53, again passing through the bifurcation point. The bifurcation value ≈ 0.63 is reached from below at = 1000 and again from above at = 2600.
Again we keep track of the time-averaged rate of viscous dissipation P (3.17), except now the average is taken over each constant value of throughout the simulation. We plot P versus in figure 10a. In contrast to the supercritical bifurcation seen in the immotile setting (figure 5), here we can see clear hysteresis: as increases past the bifurcation at ≈ 0.63, the system remains in a nontrivial state -away from the uniform, isotropic state -well beyond the bifurcation point (up to about = 0.81) before finally dropping down to the uniform, isotropic state (P = 0). Then as is decreased, the system remains in the uniform isotropic steady state until the uniform state loses stability at = 0.63, after which Hysteresis is also evident in the plot of 2 2 (T 2 ) over the entire simulation described above. The bifurcation value of ≈ 0.63 is reached from below at = 1000 and again from above at = 2600. After the bifurcation is passed from below, the system remains in a nontrivial state well beyond the bifurcation value. (c) An almost periodic structure appears after the bifurcation value is passed, which persists until = 0.84 at = 1700. Here we plot 2 2 (T 2 ) from = 1300 to 1500, where = 0.72 ( = 1300 to 1400) and = 0.75 ( = 1400 to 1500).
the system transitions to a nontrivial state again. This apparent region of bistability between 0.63 < < 0.81 is characteristic of a subcritical bifurcation.
We also plot the 2 norm of the velocity field 2 2 (T 2 ) over the entire course of the simulation in figure 10b. Note that fluctuations in the velocity field persist well past the bifurcation point, which is reached at = 1000 ( = = 0.63). The fluctuations remain until = 1700 (where is increased to 0.84), when they quickly begin to decay. After reaches a maximum of = 0.93 from = 2000 to 2100, the velocity field stays motionless as is decreased again. After = 0.63 is reached, 2 2 (T 2 ) begins to increase again. We note that this hysteretic behavior is replicated even if we start the above procedure much closer to the bifurcation point -e.g. at = 0.6. This indicates that the bistability we are seeing directly relates to the subcritical nature of the bifurcation.
Looking more closely at the region immediately following the bifurcation at = 1000 (see figure 10c), the system develops a very regular, nearly periodic structure, especially from = 1300 to 1500 ( = 0.72 and 0.75). This structure persists until is increased to 0.84 at = 1700. Snapshots of the nematic order parameter N ( , ) and particle concentration field ( , ) along this upper solution branch at = 0.75 are displayed in figure 15. In addition, the supplementary videos Movie2 and Movie3 show the quasiperiodic nature of the dynamics along this upper branch.
The unexpectedly regular structure of the temporal dynamics in figures 10c and 11 prompts a closer look at the nontrivial hysteretic state at = 0.75. We simulate this state over a long time and, among other values, record the velocity ( , ) evaluated at the center point of the computational domain. The value of the -coordinate over 1500 time units is plotted in figure 12a; the -coordinate behaves similarly. The near-perfect periodicity here is striking. We plot the power spectrum of ( ) in figure 12b and note that, remarkably, ( ) decomposes into essentially just two temporal modes: a large mode at frequency The overlap is nearly exact. The simplicity of the signal in Figure 12 is surprising given that these dynamics occur in a region beyond the predictive scope of the preceding weakly nonlinear analysis, where we do not necessarily expect such a regular structure.
Supercriticality for 1D initial perturbations
While an initial perturbation in both and gives rise to a subcritical bifurcation at = 0.63, as predicted by figure 9, an initial perturbation in only the -direction should result in a supercritical bifurcation. Indeed, if we perform the same numerical test as in figure 10 but with an initial perturbation in only the -direction instead, the resulting relationship between the average viscous dissipation P and (figure 13a) is characteristic of a supercritical bifurcation. In particular, P smoothly decreases to zero as the bifurcation is approached from below, similar to the behavior seen in the immotile bifurcation ( figure 5). Furthermore, we can numerically locate the limit cycle which emerges just below the bifurcation value of = 0.63. Snapshots from a single period of this limit cycle are shown Figure (b) is the power spectrum of over the time interval plotted in (a). The signal decomposes into just two temporal modes. In figure (c), we plot ( ) along with the simple signal ( ) (4.21) composed of the two modes in (b). The agreement is nearly perfect.
in figure 13, and a few periods of the cycle are documented in the supplementary video Movie4. The alignment among particles is very weak, but they display a clear preferred direction which oscillates over time. Note that the period of the limit cycle corresponds to every other peak of the velocity 2 norm (figure 13b); roughly 15-15.5 . This may be compared with the predicted period of 2 / = 2 /0.43 ≈ 14.6 .
Supercritical region (2D)
While initial perturbations in only the -direction give rise to a supercritical bifurcation for all ∈ [0.2, 0.7], generic 2D perturbations in both and should transition from subcritical to supercritical near = 0.5. We fix = 0.001 and = 0.075 so that ≈ 0.40, which according to figure 9c lies within the supercritical region for 2D perturbations. Setting = 0.38, we simulate the longtime system dynamics starting with a 2D initial perturbation. After an initial period of slow growth, a stable limit cycle develops, as shown in figure 14).
Although the peaks in 2 2 (T 2 ) over time are not perfectly equal in height, they occur at regular intervals (∼ 6 ), and give rise to a regular, repeated pattern in the particle alignment and generated velocity field, which can be viewed in the supplementary videos Movie5 (nematic order parameter) and Movie6 (vorticity field and velocity direction). Snapshots of a single period of the cycle are plotted in figure 15.
From figure 15 we can see that the period of the limit cycle corresponds to every 4 peaks in 2 2 (T 2 ) , so every 24 . This can be compared with the predicted imaginary part of the growth rate ≈ 0.24, which yields a period of 2 / ≈ 26 .
The periodic behavior displayed in figure 15 may be compared to the noisy quasiperiodic behavior along the upper solution branch in the bistable region for the subcritical Hopf bifurcation in figure 11. The dynamics in the supercritical case are much more regular over time. Figure 15: Snapshots of (a) the nematic order parameter N ( , ) and the direction of local nematic alignment and (b) the fluid vorticity and velocity fields over one period of the limit cycle. The snapshots alternate between peaks and valleys in 2 2 (T 2 ) in figure 14, starting with a peak. As in figure 6, the extensile flow produced by the aligned dipoles is clear.
Motile particles: pitchfork bifurcation
Keeping fixed and small, we now fix translational diffusion within the range 1/9 < < 1/4 such that one of the (now purely real) eigenvalues corresponding to the | | = 1 modes crosses zero for some ∈ (0, √ 3/9). Again, the subscript is used to denote that the bifurcation value of depends on the choice of translational diffusion .
In this parameter regime, the same weakly nonlinear calculations as in the Hopf setting may be performed (see Section 4.1), except now = 0. We thus arrive at the same form of amplitude equations as (4.11): for -only perturbations. Then, to leading order in = √ − , a stable steady state emerges after the real eigenvalue crossing of the form
0 ( ) = 3 + 1 ( 2 1 | | 2 + 2 2 2 ) 0 ( ) = 3 + 1 ( 2 1 2 + 2 2 | | 2 ) ),(5.Ψ = 1 2 1 ± √︁ √︂ − 3 1 + 2 ,1 ( )e i ± ,1 ( )e i + c.c. (5.6)
for initial perturbations in both and and
Ψ = 1 2 1 ± √︁ √︂ − 3 1 ,1 ( )e i + c.c. (5.7)
for initial perturbations in the -direction only. If the conditions (5.2) and (5.3) do not hold for 2D and 1D perturbations, respectively, then the bifurcation is subcritical and the system behavior beyond the bifurcation value is less predictable.
As in the Hopf setting, using the expressions from Appendix B, equation (B 7), we plot 0 , 3 , 1 + 2 , and 1 over the desired range of ( , ) in figure 16. Here we again use the perturbed dispersion relation (2.21) of Section 2.4 (see figure 2) with = 0.001. We find that both 0 and 3 are always positive within our region of interest, although 0 appears to approach 0 as → √ 3/9 ≈ 0.192 while 3 → 0 as → 0. Thus, as in the Hopf case, the existence of a new stable steady state following the real eigenvalue crossing is determined by the sign of 1 + 2 (for 2D initial perturbations in and ) or the sign of 1 (for -only perturbations). From figures 16b and 16c, we see that both 1 + 2 and 1 are negative for small , indicating the emergence of a nontrivial stable steady state after the bifurcation, but both 1 + 2 and 1 are positive for larger , indicating that the bifurcation type switches to subcritical somewhere in the interval 0 < < √ 3/9.
We again explore the supercritical and subcritical regions numerically in the following sections.
Supercritical region
When is small, we expect dynamics near the bifurcation to look very similar to the immotile case, where the isotropic steady state loses stability to a supercritical pitchfork bifurcation. We fix = 0.001 and = 0.23, so the bifurcation occurs at roughly ≈ 0.09. According to figure 16, this value of lies within the supercritical region for 29 (a) (b) (c) Figure 16: Plots of the coefficients (a) 0 and 3 , (b) 1 + 2 , and (c) 1 given by the expressions (4.12) in the region 0 < < √ 3/9 (or 1/9 < < 1/4), where each is real-valued. The vertical dotted lines in (b) and (c) indicate the value of where the pitchfork bifurcation transitions from supercritical to subcritical for 2D and 1D initial perturbations, respectively.
(a) (b) (c) Figure 17: (a) Plot of P versus using = 0.001, = 0.23 (so ≈ 0.09), and a 2D ( and ) initial perturbation. The bifurcation is supercritical, and we plot the nematic order parameter N ( , ) and preferred local alignment direction for the stable state which emerges just beyond the bifurcation in the case of (b) a 2D initial perturbation and (c) an -only initial perturbation.
both 2D ( and ) and 1D ( -only) initial perturbations.
We begin with a small random perturbation to the uniform, isotropic state in both and . We initialize the simulation with = 0 until = 1500 and then increase by 0.02 every 100 until = 0.2. We plot the average viscous dissipation P (3.17) versus each different value of in figure 17a. Again, the relationship between P and supports the expectation of supercriticality. We note that the viscous dissipation P behaves qualitatively the same as figure 17a using an -only initial perturbation.
The emergent stable steady state for < is plotted in figure 17b for a 2D initial perturbation and in figure 17c for an initial -only perturbation. Not surprisingly, in both cases the steady state essentially looks like the stable states which arise following the immotile bifurcation ( figure 6). The calculation in the immotile case helps to explain the very weak alignment seen in the emerging steady state, since the rotational diffusion is very small in this setting. 5.2. Subcritical region We next fix = 0.001 and = 0.13, so ≈ 0.19 is the bifurcation value. According to figure 16, both 2D ( and ) and 1D ( -only) initial perturbations to the isotropic state give rise to a subcritical bifurcation at this value of .
Using a small, random 2D perturbation to the uniform, isotropic state as our initial condition, we simulate the model dynamics until = 3000 using = 0.17, just on the unstable side of . As predicted by the amplitude equation coefficients in figure 16, and in contrast to the supercritical setting, the system does not settle into a stable nontrivial steady state. Rather, the system undergoes a relatively quick spike in activity before settling into what appears to be a stable limit cycle, as shown in figure 18 as well as the supplementary video Movie7).
The fast initial spike in activity may be contrasted with the supercritical Hopf bifurcation, where the oscillations grow slowly in amplitude over a much longer time before reaching the near-periodic dynamics shown in figure 14. Since the isotropic state is predicted to lose stability through a subcritical pitchfork bifurcation for this parameter combination, we may be seeing the results of a second bifurcation. Unlike the subcritical Hopf setting, this behavior does not persist beyond the bifurcation value -the system appears to quickly return (a) (b) (c) Figure 19: An initial -only perturbation also results in a limit cycle following the subcritical pitchfork bifurcation. Figure to the isotropic state when is increased above .
Similar behavior is observed for -only initial perturbations, as shown in figure 19. In particular, using the same parameter values as in the 2D setting, we see a quick initial spike in activity which then settles into what appears to nearly be a limit cycle (the amplitude here is very slightly decreasing over time). Again, this near-periodic behavior may be the result of a second bifurcation beyond the subcritical pitchfork. In both the 2D ( and ) and 1D ( -only) cases, however, we find no evidence of any type of hysteresis like what is seen in the subcritical Hopf setting.
Discussion
We have determined exactly how the uniform isotropic steady state loses stability in the Saintillan-Shelley kinetic model of a dilute suspension of active rodlike particles. Our weakly nonlinear analysis reveals a surprisingly complex array of possible bifurcations that the system may undergo as the nondimensional swimming speed (ratio of swimming speed to concentration and active stress magnitude) is decreased below the instability threshold. The type of bifurcation depends on the particle diffusivity, and passes from supercritical pitchfork (relatively high diffusion or immotile particles) to subcritical pitchfork to supercritical Hopf to subcritical Hopf as the diffusivity is decreased. The analysis is supported by numerical examples of each different bifurcation in the model. The numerical examples include uncovering surprising structures in the hysteretic solution that is bistable with the uniform isotropic state in the subcritical Hopf region, as well as locating different 1D and 2D limit cycles and steady states which emerge following the supercritical Hopf and pitchfork bifurcations.
The bifurcation analysis presented here provides mathematical insight into the onset of collective particle motion and active turbulence. It would be interesting to see if the full range of different transitions to collective motion can be realized experimentally. The patterns observed here may depend strongly on the aspect ratio of the periodic domain, as evidenced especially by the difference in behavior for 1D versus 2D initial perturbations in the Hopf region (section 4.2). In particular, a long, thin 2D domain may give rise to patterns more similar to the 1D case. Furthermore, while a similar linear stability picture to section 2.3 holds in 3D (Hohenegger & Shelley 2010), the effects of an additional spatial and orientational dimension on sub-versus supercriticality remain unclear. These questions both warrant further study.
From a modeling perspective, it would be useful to perform a similar bifurcation analysis for different closure models derived from the kinetic theory to see to what extent different closures can capture the complexity of the transition to collective behavior seen in the full kinetic model. It also may be interesting to analyze how the inclusion of steric interactions among particles (Gao et al. 2017) affects the nature of the possible types of bifurcations to collective motion. From a mathematical perspective, it would be interesting to prove that the uniform isotropic steady state is indeed stable above the bifurcation at = and that this stability boundary is sharp. Our numerical tests indicate that the eigenvalues calculated in Section 2.3 provide a fairly full picture of the stability boundaries, but it would be reassuring to verify this rigorously, particularly in the absence of translational or rotational diffusion. See forthcoming work (Albritton & Ohm 2022) addressing the stabilizing effects of the swimming term in the Saintillan-Shelley model.
Appendix B. Hopf bifurcation
As in the immotile case, we may use the expressions (4.2) for Ψ 1 and 1 to compute the right hand side of equation (4.3) up to ( ):
− 1 · ∇Ψ 1 − div (I − T )(∇ 1 )Ψ 1 ) = 1 ( )e i( + )+2i + 2 ( )e i( − ) + 3 ( )(e 2i +2i 2 + | | 2 ) + 4 ( ) (e 2i +2i 2 + 2 ) + c.c., (B 1) where 1 = − 2 −3 cos 2 sin 2 + sin 4 + cos sin + i + i cos − i sin 4 cos ( + i + i cos ) 2 + −3 cos 2 sin 2 + cos 4 + cos sin + i + i sin − i cos 4 sin ( + i + i sin ) 2 + ( ) 2 = − 2 −3 cos 2 sin 2 + sin 4 − cos sin + i + i cos − i sin 4 cos ( + i + i cos ) 2 + −3 cos 2 sin 2 + cos 4 − cos sin − i − i sin + i cos 4 sin ( − i − i sin ) 2 + ( ) 3 = − 2 2 −3 cos 2 sin 2 + cos 4 + i + i cos − i − cos 3 sin 2 ( + i + i cos ) 2 + ( ) 4 = − 2 2 −3 cos 2 sin 2 + sin 4 + i + i sin − i − sin 3 cos 2 ( + i + i sin ) 2 + ( ).
Again, as in the immotile case, the form of the right hand side expression (B 1) leads us to consider the ansatz (4.4) for (Ψ 2 , 2 ). Using this ansatz, the left hand side of the ( 2 ) equation (4.3) takes the following form:
L [Ψ 2 ] = 2i
2,1 e i( + )+2i + 2,3 e 2i +2i 2 + 2,4 e 2i +2i 2 + i (cos + sin ) 2,1 e i( + )+2i + (cos − sin ) 2,2 e i( − ) + 2 cos 2,3 e 2i +2i 2 + 2 sin 2,4 e 2i +2i 2 − 1 4 (cos 2 − sin 2 ) ∫ 2 0 2,1 e i( + )+2i + 2,2 e i( − ) (cos 2 − sin 2 ) − 1 cos sin ∫ 2 0 2,3 e 2i +2i 2 + 2,4 e 2i +2i 2 sin cos + 2 2,1 e i( + )+2i + 2 2,2 e i( − ) + 4 2,3 e 2i +2i 2 + 4 2,4 e 2i +2i 2 − ( 2,1 )e i( + )+2i + ( 2,2 )e i( − ) + ( 2,3 )e 2i +2i 2 + ( 2,4 )e 2i +2i 2 + ( 2,5 ) | | 2 + ( 2,6 ) 2 + c.c.
(B 2)
Equating exponents from the right hand side (B 1) with the left hand side (B 2), we obtain six independent integrodifferential ODEs for the coefficients 2, ( ).
For each of 2, , = 1, 2, 3, 4, it can be shown that (small) > 0 results in a small 36 perturbation of the = 0 solution to the following set of equations: 2 + 2i + i (cos + sin ) 2,1 − cos 2 − sin 2 4 ∫ 2 0 2,1 (cos 2 − sin 2 ) − 2,1 = − 2 −3 cos 2 sin 2 + sin 4 + cos sin + i + i cos − i sin 4 cos ( + i + i cos ) 2 + −3 cos 2 sin 2 + cos 4 + cos sin + i + i sin − i cos 4 sin ( + i + i sin ) 2 + ( ), 2 + i (cos − sin ) 2,2 − cos 2 − sin 2 4 ∫ 2 0 2,2 (cos 2 − sin 2 ) − −3 cos 2 sin 2 + sin 4 + i + i sin − i − sin 3 cos 2 ( + i + i sin ) 2 + ( ).
(B 3) In particular, following a similar approach to Section 2.4, it may be shown that the solutions to each of the above four equations is bounded independent of as → 0 for all values of the triple ( , , ) of interest.
In contrast, the remaining two coefficients 2,5 and 2,6 blow up like 1/ as → 0: 2,5 = 2 2 −3 cos 2 sin 2 + cos 4 + i + i cos + i cos 3 sin 2 ( + i + i cos ) 2 + ( ) 2,6 = 2 2 −3 cos 2 sin 2 + sin 4 + i + i sin + i sin 3 cos 2 ( + i + i sin ) 2 + ( ) . (B 4) In particular, for sufficiently small > 0, the behavior of 2,5 and 2,6 dominate over each of the other terms which are quadratic in the amplitudes and . We may then integrate the equations (B 4) in to obtain the expressions for 2,5 and 2,6 given in equation (4.5). the following expressions for 0 , 1 , 2 , 3 : 0 = − 4 6( + i ) 2 + 2 − (6( + i ) 2 + 4 2 ) ( + i ) √︃ 2 + ( + i ) 2 1 = 96 8 5 6 + 232 2 (
+ i ) 4 + 512( + i ) 6 − 28 4 ( + i ) 2 − 8( + i ) 3 √︃ 2 + ( + i ) 2 61 2 ( + i ) 2 + 64( + i ) 4 + 4 4 + ( + i ) 3 2 4 1 ( , , ) + ( ) 2 = ( + i ) 3 4 8 √︃ 2 + ( + i ) 2 4( + i ) 2 + 2 + 2 2 ( + i ) 2 2 + 2( + i ) 2 − 4( + i ) 3 − 4 2 ( + i ) + ( + i ) 3 2 4 2 ( , , ) + ( ) 3 = 2 ( + i ) 5 4 + i ) 2 + 2 − (4( + i ) 2 + 3 2 ( + i ) √︃ 2 + ( + i ) 2 ,(B 7
In the kinetic model (Saintillan & Shelley 2008a,b), a suspension of elongated particles in a 2-dimensional periodic box of length is modeled as a number density Ψ( , , ) of particles with center-of-mass position and orientation ∈ 1 at time . Due to conservation of the number of particles, the density Ψ evolves according to a Smoluchowski equation: Ψ = −∇ · ( Ψ) − ∇ · ( Ψ); ∇ := ( − T ) . (2.1)
= 0 spectral analysis of Subramanian et al. (2011). Furthermore, when > 0, the = 0 mode is always linearly stable, as it satisfies the heat equation in : if Ψ = Ψ( , ) then Ψ = Δ Ψ. The eigenvalue relation (2.19) ceases to be valid for Re( ) 0; i.e. when Re( ( )) − 2 (Hohenegger & Shelley 2010; Subramanian et al. 2011). For fixed 0 < < 1/4, however, the eigenvalue analysis does capture a sign change in the real part of the growth rate ( ) as | | is varied. For each , this sign change occurs where the blue curve in figure 1 (corresponding to ) intersects the green line corresponding to the fixed value of 2 .
Figure 2 :
2The (a) real part and (b) imaginary part of the perturbed dispersion relation 0 + 1 (dotted black curve) is plotted for = 0.001 on top of the unperturbed relation with = 0 (solid light blue curve). The perturbative expression fails to be valid along the gray line 0 + 2 = | | / √ 3 plotted in (a). The inset in figure (a) shows in greater detail the behavior of the perturbative expression 0 + 1 near the point ( √ 3/9, 1/9) where the gray line intersects the unperturbed expression. In particular, the perturbed expression blows up as | | approaches √ 3/9.
Figure 5 :Figure 6 :
56Numerical evidence of supercriticality in the pitchfork bifurcation for immotile particles ( = 0). Here = 0.0125 is fixed and the bifurcation occurs at * = 0.2. The simulation is initiated with = 0.1 until = 500; then every 100 the value of is increased by 0.02, so the bifurcation value is reached at = 1000.Figure(a) shows the 2 norm of the fluid velocity field over time. Figure (b) shows the time-averaged viscous dissipation P (3.17) for the different values of . The apparently smooth decrease to zero as → * indicates that the uniform isotropic steady state loses stability through a supercritical pitchfork bifurcation. Plots of the nematic order parameter N ( , ) and the direction of local nematic alignment for (a) ( , ) = (0.0025, 0.22) ( * = 0.24) and (b) ( , ) = (0.03125, 0.105) ( * = 0.125) demonstrate the dependence of the emerging immotile steady state on , as predicted by the form of (3.14). In both cases, 2 = * − = 0.02. Figure (c) shows the vorticity field ( , ) (colors) and velocity field ( , ) (arrows) for the same steady state pictured in figure (b). Note the clear extensile flow produced by the aligned dipoles in the top right and bottom left of the domain.
Figure 7 :Figure 8 :
78In the case of an -only initial perturbation, a similar dependence of the emerging steady state on can be seen in the two plots of the nematic order parameter N ( , ) and the direction of local nematic alignment for (a) ( , ) = (0.0025, 0.22) ( * = 0.24) and (b) ( , ) = (0.03125, 0.105) ( * = 0.125). Again, in both cases, 2 = * − = 0.02. Figure (c) shows the vorticity field ( , ) (colors) and velocity field ( , ) (arrows) for the -only steady state with ( , ) = (0.03125, 0.105).Difference between the predicted steady state Ψ p (3.14) and the computed steady state for (2.8) with = 0 for a fixed value of = and two different fixed values of : (a) = and (b) = 23 16 . As expected, the difference between Ψ p − Ψ c is ( 2 ). N ( , ) between figures 6b and 6a.
+ i + i cos + ( ), ,1 () = cos sin + i + i sin + ( ) 1 = − i 2 e i +i + e i +i + c.c. + ( ).
and ( ) are complex-valued amplitudes which depend only on the slow timescale and for which we wish to obtain an equation. Here again c.c. denotes the complex conjugate of the preceding terms.
conditions are satisfied, the magnitude and phase of the emerging limit cycle are
Figure 10 :
10(a) The plot of the time-averaged viscous dissipation P versus displays bistability in the system above the subcritical Hopf bifurcation at ≈ 0.63. (b)
Figure 11 :
11Snapshots of (a) the nematic order parameter N ( , ) and the direction of local nematic alignment, and (b) the concentration field ( , ) along the nontrivial upper solution branch which emerges above the subcritical Hopf bifurcation at = 0.63. Here = 0.75, and figures are taken at successive local peaks and valleys in the velocity 2 norm (every 5 to 5.5 ), starting with a peak. 0.096 = 1/10.4 and a small mode at 0.62 = 1/16.2. In figure 12c we plot the signal of ( ) for 200 , where the value 2065 was chosen to qualitatively match with .
Figure 12 :
12The surprisingly regular temporal dynamics in the nontrivial hysteretic state at = 0.75.Figure(a) displays the near-periodic dynamics of the -coordinate of the velocity field ( , ) evaluated at the center point of the computational domain.
Figure 13 :Figure 14 :
1314(a) Plot of P versus when = 0.001 and = 0.02 (so ≈ 0.63) for initial perturbations in the -direction only. The behavior here can be contrasted with figure 10 for 2D ( and ) initial perturbations. (b) Plot of 2 2 (T 2 ) over time and (c) plot of the -component (blue) and -component (red) of the velocity field evaluated at the center point of the computational domain. Here = 0.6 is fixed. Both plots show that a stable limit cycle develops following an -only initial perturbation. (d) Snapshots of N ( , ) over the period of one limit cycle. The particles are very weakly aligned here, but their preferred direction oscillates. The stable limit cycle which develops just below the Hopf bifurcation at = 0.40. Figure (a) shows the -component (blue) and -component (red) of the velocity evaluated at the center point of the computational domain. Figure (b) shows the 2 norm of the velocity
0 , 1 , 2 , and 3 given in Appendix B, equation (B 7) are all real-valued for each ( , ) in the region of interest. Since the coefficients are real-valued, similar to the immotile setting, we may look for conditions under which (5.1) admits a nontrivial steady state solution = = 0. If the coefficients satisfy 0 ≠ 0, 1 + 2
Figure 18 :
18For an initial perturbation in and , after a quick spike in activity, the system settles into a limit cycle following the subcritical pitchfork bifurcation.Figure(a) shows 2 2 (T 2 ) over time. Figure (b) shows the -component (blue) and -component (red) of evaluated at the center of the domain over time. Figure (c) displays snapshots of the nematic order parameter N ( , ) and direction of local nematic alignment at successive peaks and valleys in 2 2 (T 2 ) over one period of the cycle, starting with a peak.
T 2 ) over time.Figure(b) shows the -component (blue) and -component (red) of evaluated at the center of the domain over time.Figure (c)shows the nematic order parameter N ( , ) and direction of local nematic alignment at the 2 norm peak and valley, respectively. The system oscillates slowly between the two states pictured.
the stabilizing effect of swimming in an active suspension. arXiv preprint . electrically driven patternforming instabilities in planar nematics. Journal de Physique 49 (11)and symmetry-breaking bifurcations in fluid dynamics. Annual Review of Fluid Mechanics 23 (1formation outside of equilibrium. Reviews of modern physics 65 (3order of motile defects in active nematics. Nature materials 14 (11), 1110-1115. D , M 1981 Molecular dynamics and rheological properties of concentrated solutions of rodlike polymers in isotropic and liquid crystalline phases. J. Polym. Sci. B 19 (2), 229-243. D , M & E , S F 1986 The Theory of Polymer Dynamics, , vol. 73. Oxford University Press. Self-concentration and large-scale coherence in bacterial dynamics. Physical review letters 93 (9), 098103. of meso-scale turbulence in active nematics. Nature communications 8 (1), 1-7.
D
Hohenegger & Shelley 2010; Saintillan & Shelley 2008b; Subramanian et al. 2011). Linearizing (2.8) about this state, we obtain
† Email address for correspondence: [email protected]
(T 2 ) over time.
Moving on to the ( 3 ) equation (4.7), we note that out of all the cubic-in-terms in the right hand side expression (4.9), each term is (1) in except for −div (I − T ) (∇ 1 )Ψ 2 , which is ( −1 ). For sufficiently small , this term determines the behavior of the cubic-in-terms and gives rise to the following expressions for ( ),
Acknowledgements.We thank Scott Weady for supplying a base code for numerics. Declaration of interests. The authors report no conflict of interest.Appendix A. Immotile bifurcationUsing the expressions for Ψ 1 and 1 (3.4), we can calculate the right hand side of equation (3.5), given by − 1 · ∇Ψ 1 − div (I − T )(∇ 1 )Ψ 1 ) = 1 ( )e i( + ) + 2 ( )e i( − ) + 3 ( )(e 2i 2 + | | 2 ) + 4 ( ) (e 2i 2 + 2 ) + c.c.,33The form of (A 1) leads us to consider the ansatz (3.6) for Ψ 2 and 2 . Plugging this ansatz into the operator L, the left hand side of the ( 2 ) equation (3.5) takes the form L [Ψ 2 ] = − 1 4 (cos 2 − sin 2 ) ∫ 2 0 2,1 e i( + ) + 2,2 e i( − ) (cos 2 − sin 2 ) − 1 cos sin ∫ 2 0 2,3 e 2i 2 + 2,4 e 2i 2 sin cos + 1 4 − 4 2 2,1 e i( + ) + 2 2,2 e i( − ) + 4 2,3 e 2i 2 + 4 2,4 e 2i 2(A 2) Matching the exponents in (A 1) to (A 2), we obtain a series of six independent integrodifferential ODEs which may each be solved to yield the coefficients 2, listed in (3.7).We also need to keep track of the components of ( , )e i and ( , )e i appearing in the expression R (3.9) for the right hand side of the ( 3 ) equation (3.8).Each term of R contributes the following to ( , )e i and ( , )e i :Similarly, the contribution of each term in (A 3) to [ ] is given by
|
[] |
[
"A comparison of Fourier and POD mode decomposition methods for high-speed Hall thruster video",
"A comparison of Fourier and POD mode decomposition methods for high-speed Hall thruster video"
] |
[
"J W Brooks [email protected]&[email protected] \nNaval Research Laboratory\nWashingtonDC\n",
"M S Mcdonald \nNaval Research Laboratory\nWashingtonDC\n",
"A A Kaptanoglu \nUniversity of Washington\nSeattleWA\n"
] |
[
"Naval Research Laboratory\nWashingtonDC",
"Naval Research Laboratory\nWashingtonDC",
"University of Washington\nSeattleWA"
] |
[] |
Hall thrusters are susceptible to large-amplitude plasma oscillations that impact thruster performance and lifetime and are also difficult to model. High-speed cameras are a popular tool to study these dynamics due to their spatial resolution and are a popular, nonintrusive complement to in-situ probes. Highspeed video of thruster oscillations can be isolated (decomposed) into coherent structures (modes) with algorithms that help us better understand the evolution and interactions of each. This work provides an introduction, comparison, and step-by-step tutorial on established Fourier and newer Proper Orthogonal Decomposition (POD) algorithms as applied to high-speed video of the unshielded H6 6-kW laboratory model Hall thruster. From this dataset, both sets of algorithms identify and characterize m = 0 and m > 0 modes in the discharge channel and cathode regions of the thruster plume, as well as mode hopping between the m = 3 and m = 4 rotating spokes in the channel. The Fourier methods are ideal for characterizing linear modal structures and also provide intuitive dispersion relationships. By contrast, the POD method tailors a basis set using energy minimization techniques that better captures the nonlinear nature of these structures and with a simpler implementation. Together, the Fourier and POD methods provide a more complete toolkit for studying Hall thruster plasma instabilities and mode dynamics. Specifically, we recommend first applying POD first to quickly identify the nature and location of global dynamics and then using Fourier methods to isolate dispersion plots and other wave-based physics.
| null |
[
"https://arxiv.org/pdf/2205.14207v1.pdf"
] | 249,191,422 |
2205.14207
|
e6bfd12d211bc22eed0fdb3e59fe865c2570f510
|
A comparison of Fourier and POD mode decomposition methods for high-speed Hall thruster video
27 May 2022
J W Brooks [email protected]&[email protected]
Naval Research Laboratory
WashingtonDC
M S Mcdonald
Naval Research Laboratory
WashingtonDC
A A Kaptanoglu
University of Washington
SeattleWA
A comparison of Fourier and POD mode decomposition methods for high-speed Hall thruster video
27 May 2022Hall thrusterhigh-speed videomode decompositionsingular value decomposition
Hall thrusters are susceptible to large-amplitude plasma oscillations that impact thruster performance and lifetime and are also difficult to model. High-speed cameras are a popular tool to study these dynamics due to their spatial resolution and are a popular, nonintrusive complement to in-situ probes. Highspeed video of thruster oscillations can be isolated (decomposed) into coherent structures (modes) with algorithms that help us better understand the evolution and interactions of each. This work provides an introduction, comparison, and step-by-step tutorial on established Fourier and newer Proper Orthogonal Decomposition (POD) algorithms as applied to high-speed video of the unshielded H6 6-kW laboratory model Hall thruster. From this dataset, both sets of algorithms identify and characterize m = 0 and m > 0 modes in the discharge channel and cathode regions of the thruster plume, as well as mode hopping between the m = 3 and m = 4 rotating spokes in the channel. The Fourier methods are ideal for characterizing linear modal structures and also provide intuitive dispersion relationships. By contrast, the POD method tailors a basis set using energy minimization techniques that better captures the nonlinear nature of these structures and with a simpler implementation. Together, the Fourier and POD methods provide a more complete toolkit for studying Hall thruster plasma instabilities and mode dynamics. Specifically, we recommend first applying POD first to quickly identify the nature and location of global dynamics and then using Fourier methods to isolate dispersion plots and other wave-based physics.
Introduction
The Hall thruster (HT) is a standard spacecraft electric propulsion system that uses crossed electric and magnetic fields (E × B) to ionize and accelerate propellant [1,2]. Their high specific impulse and technological maturity make them ideal for long duration satellite station keeping and time-insensitive missions, such as orbit raising, with thousands currently in orbit and more planned. However, HTs experience anomalous electron transport across their magnetic field that is not sufficiently understood to permit fully predictive thruster models, motivating continued improvement of diagnostics for model validation.
This lack of validation is especially important for increased qualification by simulation for new thruster designs at ever-higher power, such as the N30 or X3 nested Hall thrusters, where full life qualification in a ground test facility would be prohibitively expensive for 10s-100s kW thrusters [3,4].
Anomalous electron transport has been strongly linked to plasma oscillations in HTs [5] in both the thruster discharge channel [6,7] and cathode regions [8]. Long-wavelength azimuthal oscillations, the focus of this work, are typically characterized in a Fourier representation with integer mode numbers, m, including the azimuthally uniform m = 0 "breathing" modes in HT channels [9,10] and cathodes [11,12], m > 0 azimuthally rotating "spoke" modes in the channel, and a m = 1 counter rotating "antidrift" modes around the cathode [13]. The rotating spoke channel modes, first observed in an early Hall accelerator by Janes and Lowder [14], are now known to be ubiquitous in unshielded thrusters [15] though they may be less common in magnetically shielded thrusters [13,16]. Cathode m = 1 anti-drift modes have been previously observed in both shielded and unshielded versions of the H6 Hall thruster [13] (the unshielded H6 is studied in this work). All of these modes are historically characterized by one or more in-situ diagnostics, but the localized nature of these diagnostics both perturb the plasma and make it difficult to relate individual measurements to global mode structures.
High-speed imaging (HSI) has become a popular complementary diagnostic to in situ probes and has proven itself well suited to characterizing mode dynamics [6,17,13]. Its popularity is due to its ease of use, nonintrusive nature, high speed (100s of kHz), and fine spatial resolution of order millimeters per pixel. HSI was first used on HTs to relate image brightness to plasma density oscillations measured by electrical probes [18]. This and later work has led to the use of pixel light intensity as a proxy for discharge current [19].
To isolate and characterize individual mode dynamics from HSI, various post-processing algorithms have been developed. Early work subtracted the timeaverage from each pixel to reveal multiple m > 1 modes in the H6 Hall thruster channel [15]. Later, Fourier techniques were used with azimuthal binning to isolate the frequency spectrum associated with individual channel modes [20]. Fourier methods have also been used to provide azimuthal dispersion plots within the CHT (cylindrical Hall thruster) [6] and H6 [17] thruster channels. A phase-based Fourier analysis was developed to isolate the spatial structure of the m = 1 cathode mode on the H6 thruster [13] and m > 0 channel and cathode modes on the HERMeS thruster [16]. An alternative Fourier-based visualization method, Cross-Spectral-Density (CSD), was developed on the CHT thruster [21].
The Fourier methods remain the most established method for isolating and analyzing Hall thruster plasma oscillations, but they have several disadvantages. First, their linear sine/cosine bases are not ideal for nonlinear features. Second, Fourier methods require several preprocessing steps for high-speed video; this added complexity requires additional computational overhead and makes it more difficult to study multi-dimensional dynamics.
An alternative to Fourier methods are Singular Value Decomposition (SVD) based algorithms [22]. The chief advantage of SVD is that is tailors custom bases for each dataset based on the energy contribution of the coherent dynamics instead of using a presumed basis (e.g. Fourier's sines and cosines). This facilitates improved characterization and reconstruction of nonlinear dynamics without making any physical assumptions of the system. SVD also requires minimal preprocessing compared with Fourier methods.
The most notable and likely simplest SVD algorithm is Proper Orthogonal Decomposition (POD) which is extensively used in fluid mechanics [23,24,22]. A classic POD application is to decompose fluid vortex shedding around a body (e.g. a cylinder) into discrete modes [25,26]. POD has also been used in plasma Distribution A: Approved for public release; Distribution unlimited physics to characterize plasma oscillations [27,28,29,30,31] but is often referred to as Biorthogonal Decomposition (BD). Very recently, POD has been used to characterize axial and azimuthal modes in Hall thruster high-speed video [32] and azimuthal modes in a hollow cathode plume [33]. More sophisticated SVD algorithms exist, most notably DMD (Dynamic Mode Decomposition) [34,35] and have applications in active control, linear dynamics, and plasma physics [36,37,38].
The goal of this work is to compare Fourier and POD techniques as applied to Hall thruster highspeed imaging in a tutorial format. To this end, this work analyzes a high-speed video recording of the unshielded H6 Hall thruster plume and provides a stepby-step explanation of algorithm's implementation and results. This work starts by introducing the H6 thruster, high-speed video dataset, and video preprocessing in Section 2.
Section 3 discusses several established Fourier mode analysis methods and, when applied to the H6 dataset, identifies simultaneous cathode and channel modes in addition to mode hopping within the channel. Section 4 introduces the SVD algorithm and its most common implementation: Proper Orthogonal Decomposition (POD). When applied to the H6 dataset, POD identifies the same mode behavior and provide several improvements over the Fourier methods. When used together, POD and Fourier methods provide a more complete toolkit for identifying and the isolating global mode dynamics. The code and dataset used for this work are available online [39].
Experimental setup
This section covers the experimental hardware (thruster, facility, high-speed camera), the HSI dataset, and common HSI prepossessing techniques.
Hardware and dataset
This work focuses on a single illustrative operating condition for the unshielded H6 Hall thruster. The H6 is a laboratory model 6-kW thruster (see Figure 1) with a design operating point of 300 V and 20 mg/s (where 1 mg/s ≈ 1 A) on xenon. In this work, we focus on a 600 V and 10 mg/s condition that exhibits several simultaneous modes, most prominently an m = 1 cathode spoke, a shared m = 0 mode between the channel and cathode regions, and mode-hopping between m = 3 and m = 4 in the channel.
Recordings took place in the University of Michigan Plasmadynamics and Electric Propulsion Laboratory's Large Vacuum Test Facility (LVTF) circa 2011. The LVTF is a cylindrical chamber 9 m long and 6 m in diameter and at the time was maintained at high vacuum by seven TM-1200 cryopumps with a combined pumping speed of 210,000 L/s on xenon.
A Photron SA5 FASTCAM high-speed camera placed 6.5 m axially downstream of the thruster exhaust imaged the plume through a quartz window with 152 x 192 resolution in monochrome (B&W) at a framerate of 175 kHz. The fastest mode observed was a 80 kHz cathode spoke, just below the 87.5 kHz camera Nyquist frequency. The camera used a Nikon ED AF Nikkor 80-200 mm lens at its maximum aperture of f/2.8. The bright, central cathode saturates several pixels, and the algorithms largely ignore these pixels.
In preparation for video processing, any highspeed video dataset should be thought of as a 3D matrix of pixel measurements, p(t, x, y), with dimensions of time, t, and Cartesian space, x and y, and with lengths N t , N x , and N y , respectively [17].
Video preprocessing
Before applying mode analysis algorithms, several preprocessing steps should first be considered. This section outlines two prominent steps: 1) spatial centering and normalization and 2) amplitude normalization. Other preprocessing steps, not covered here, include masking and filtering which are useful in isolating specific spatial regions or dynamics, respectively. Please note that all of these steps are optional for POD.
Spatial identification and scaling
This step identifies and the scales the spatial geometries in preparation for converting to polar coordinates, a requirement for Fourier analysis. This step is not required for POD. To identify the thruster channel origin and radius, we fit an annular Gaussian function G(x, y; x 0 , y 0 , r 0 , a, w, G 0 ) = a exp − 1 2
r(x, y; x 0 , y 0 ) − r 0 w 2 to the time-average,p(x, y), of the high-speed video. In this equation, r 0 is the radius of the channel, the radius at each pixel is
+ G 0(1)r(x, y; x 0 , y 0 ) = (x − x 0 ) 2 + (y − y 0 ) 2 ,(2)
x 0 and y 0 are the center (origin) of the channel, a is the amplitude, w is related to the channel width, and G 0 is an offset. With these fit parameters solved, we next center and normalize the spatial coordinates to the channel radius (i.e. x norm = (x − x 0 )/r 0 and y norm = (y − y 0 )/r 0 ). These normalized coordinates can then be optionally multiplied by the dimensioned channel radius to provide actual units to x and y. For this work, we remain with normalized coordinates. Figure 2 shows the results of this step as applied to the time-averaged video,p(x, y), of our H6 dataset. In this figure, the two regions with the most plasma dynamics, the annular channel and center cathode, are clearly visible. The pixel intensity has raw integer units based on the camera's 12 bit depth. The pixels associated with the cathode's center were allowed to saturate to better capture dynamics in the channel, and the pixels adjacent to the saturated pixels still capture the cathode's dynamics. The channel radius, r 0 , as identified by the Gaussian fit, is indicated with a black dashed line. The channel edges, indicated with white dashed lines, are identified as approximately at (r 0 ± w)/r 0 . The x and y coordinates have been centered and normalized to the channel radius as described above.
Alternatives to the annular Gaussian fit have been used in previous Hall thruster work. McDonald [17] discussed both the Kasa [41] and Taubin [42] methods and recommended Taubin for cases where the entire channel annulus is not visible.
Another option previously used [21] is a circle detection algorithm called a Hough transform. While no one method is obviously superior over the others, we recommend using 1) the Gaussian fit presented here because the solved parameters are directly relatable to physical dimensions or 2) the Hough transform as it is available prewritten in many programming languages.
Video amplitude scaling
This step scales the video's arbitrary intensity measurements to a more meaningful range. Common practice is to assume that oscillations in a video's amplitude is roughly linear with the plasma density oscillations [18] and that camera measurements are linear with light emission [43]. As we cannot scale the data to a definitive physical value, this section instead discusses several data normalization methods to provide better physical intuition of the oscillations. This step is optional for both Fourier and POD methods.
Before normalizing, the first step is to subtract the time-averaged image (also known as AC coupling) from each frame of the raw video dataset to better isolate the oscillations.
Next, we normalize the video amplitude in one of two ways: pixel-wise normalization or channel-average normalization. The first method divides each pixel by its standard deviation in time, and this has the advantage of making the mode dynamics easier to visualize after mode decomposition. Unfortunately, this method artificially amplifies the oscillations at different spatial locations and makes quantifying global mode amplitudes untenable. As an alternative, the second option divides each pixel by the average brightness within the channel, which allows the modes to be scaled as a percentage of the average channel brightness. While both methods are used in this work, channel-average normalization is default. Figure 3 shows the results of AC coupling and channel normalization at two separate instances in time. At t = 22.9 ms (Figure 3a), the normalized video snapshot reveals a dominant 3-lobed azimuthal wave (m = 3 mode) in the channel. At t = 24.8 ms (Figure 3b), a 4-lobed azimuthal wave (m = 4 mode) is dominant. The amplitudes of both modes are around 10% of the average channel brightness. Both snapshots also show an m = 1 azimuthal wave around the cathode.
Fourier-based methods
Fourier based algorithms are the most established methods for mode decomposition and identification in HT high-speed video [44,17,13,21]. This is because Fourier series' bases are periodic sines and cosines and are therefore ideal for characterizing wave-like oscillations, such as plasma waves. In this section, we use Fourier analysis to characterize simultaneous
Detecting modes
Mode dynamics in Hall thruster plasmas are typically characterized by oscillating waves with slowly evolving amplitudes and frequencies. The easiest way to detect these modes is to apply FFT and Welch-averaged FFT [45] algorithms to diagnostic measurements of signal pixels of the high-speed video. Figure 4 shows an example of this applied to three signals from our dataset: the discharge current flowing from the anode to cathode, a high-speed pixel centered in the channel, and a high-speed pixel adjacent to the cathode. Figure 4a shows a 1 ms time window of the three signals and that they have similar oscillatory behavior. Each signal has been AC coupled and normalized by its standard deviation for ease of comparison. Figure 4b shows the Welch-averaged FFT of each signal over 100 ms, and the resulting power spectrum reveals prominent frequency peaks with each. Most notable is a broad spectrum peak at 20 kHz that is common to all three signals. Additional peaks at 7.5 kHz and 80 kHz are unique to the channel and cathode measurements, respectively; this uniqueness suggests that their dynamics are isolated to their respective regions. The broader width of the 20 kHz peak suggests that its frequency is more erratic than the two narrower peaks. The remainder of this paper will identify the modes associated with these three peaks and few less pronounced peaks.
Azimuthal and radial binning
Waves around the channel and cathode of a Hall thruster are primarily azimuthal [5] with the approximate Fourier form
p(t, θ) ∼ e i(mθ−ωt)(3)
where m is the integer azimuthal mode number, θ is the azimuthal angle, and ω = 2πf is the angular frequency. After scaling the video's coordinates and amplitudes (Section 2.2), the next step is convert the 3D video, p(t, x, y), in Cartesian coordinates to a 2D video, p(t, θ), in polar coordinates so that it matches Eq. 3. To do this, we first isolate the channel region (i.e. 0.9 < r/r 0 < 1.1) with radial masking and discard the rest. The radial dependence within this narrow region is assumed constant, and the radial coordinate is therefore dropped. Next, we convert the unmasked x and y coordinates within the channel to the azimuthal coordinate with a four-quadrant inverse tangent function, θ(x, y) = arctan2(y, x). The resulting data is azimuthally binned and averaged to provide a result with uniform azimuthal spacing. Figure 5 illustrates the azimuthal and radial binning for a single time snapshot (t = 22.9 ms). Figure 5a shows the radial mask isolating a narrow region within the channel and also the edges of N θ = 100 azimuthal bins. Figure 5b shows the unbinned values (black) overlayed by the bin-averaged result (red). This shows a clear m = 3 structure and also a few non-ideal features: a non-uniform spacing between peaks, a steepened waveform, and a weak fourth peak at 3π/4. This process is repeated for each time step within p(t, x, y), and the result is the 2D dataset, p(t, θ), within the channel.
In this example, we applied the binning process to the channel region. However, this same procedure could be applied to other radial regions, including the region around the cathode [13] or at radial slices between the cathode and channel. Optionally, multiple radial slices could be made to provide a 3D dataset, p(t, θ, r).
Azimuthal mode identification
With p(t, θ) solved and matching the assumed Fourier form (Eq. 3), we can next identify the azimuthal modes within the channel. To do this, we apply a 2D FFT, F t,θ , in both time and θ to identify the mode numbers and their characteristic frequencies. This provides the 2D complex matrix,
P (ω, m) = F t,θ p(t, θ) ,(4)
with dimensions in angular frequency, ω, and azimuthal mode number, m, and with dimensional lengths, N t and N θ , respectively. Note that the coordinates ω = 2πf and m range between their negative and positive Nyquist frequencies (-87.5 kHz ≤ f < 87.5 kHz and −N θ /2 ≤ m < N θ /2). Due to symmetry, the negative frequencies can be truncated. The absolute value of this result, P (ω, m) , is the azimuthal dispersion relationship within the channel and is shown in Figure 6a. It reveals a series of evenly-spaced discrete modes (2 ≤ m ≤ 5) and what appears to be a continuous wave. Both the continuous wave and the discrete modes are propagating clockwise (negative θ) and therefore have negative wavenumbers. By convention, we present the azimuthal modes and wave numbers as positive. Figure 6b shows several slices of the dispersion relationship at m = 0, 2, 3, and 4 and more clearly identifies the peaks originally observed in Figure 4b. From this plot, the m = 0 mode is the broad-spectrum peak at 20 kHz, the m = 3 mode is the narrow peak at 7.5 kHz, and the m = 4 mode is the peak at 12.5 kHz. In addition, a weak m = 2 mode is observed at 2.5 kHz. Figure 6b shows that all three of the indicated m > 0 modes have a roughly uniform spacing of 5 kHz. Figure 6a also identifies an m = 6 mode at roughly double the m = 3 frequency which makes it a harmonic of the m = 3 mode [46]. To observe the 80 kHz peak in Figure 4b, the above analysis could be applied to the region around the cathode instead of inside the channel.
Mode evolution
To capture the time evolution of each mode, we apply 1D FFT, F θ , to each time step in the 2D dataset, p(t, θ), along the θ dimension. The result is the 2D The power spectrum of select mode numbers helps to identify their characteristic frequency.
complex matrix,
P (t, m) = F θ p(t, θ)(5)
with dimensions in time, t, and azimuthal mode number, m. Depending on the FFT algorithm, P (t, m) then needs to be multiplied by a constant to return the correct amplitude, typically 2/N θ . Figure 7 plots the real (cosine) component, the imaginary (sine) component, and the amplitude of P (t, m) at m = 0, 3, and 4 and shows the time evolution of each. Figure 7a reveals the m = 0 breathing mode to have a mostly consistent amplitude around 10% to 20% of the average channel brightness. Figures 7b and 7c shows the m = 3 and m = 4 modes, respectively, with amplitudes between 3% to 5%. These figures also clearly identify repeated mode hopping between these two modes. In the m = 3 and 4 figures, the real component leads the real component by roughly 90 degrees, indicating that the modes are rotating clockwise. The m = 0 mode is not rotating, and therefore its imaginary component is zero.
The modes' spatial structures
To isolate the spatial structure associated with each mode, we first apply 1D Fourier analysis,
P (f, x, y) = F t p(t, x, y) ,(6)
to each pixel in the 3D video, p(t, x, y), with respect to time. Next, we index the resulting matrix, P (f, x, y), at the frequencies associated with each mode as identified in Figure 6. Figure 8 shows the resulting real (cosine) and imaginary (sine) component of each mode, which reveals the spatial extent and phase of each. The number of peaks and troughs of each structure accurately corresponds to its mode number. This approach is a computationally efficient alternative to bandpass-filtering each pixel as done previously [17]. These plots are also nearly identical to plots in previous works [13,16] with the difference being that the past works plot the phase, atan(imag./real), of each mode instead of the real and imaginary components. For better visualization, each pixel in p(t, x, y) can optionally be normalized by its standard deviation before applying Eq. 4. Figure 9 shows an example of this for the m = 3 mode. This figure reveals improved spatial detail within the channel and also suggests mode structure outside the channel as well.
Fourier methods conclusion
In this section, we applied the Fourier mode decomposition and identification methods to our H6 high-speed video dataset. First, the FFT of individual pixels was first able to identify the presence of coherent mode dynamics in the channel and cathode regions. After radially and azimuthally binning the data within the channel, a 2D FFT provided an azimuthal dispersion plot and helped relate mode numbers to their characteristic frequency. A 1D FFT was then used to isolate the temporal evolution of each mode and most notably identified mode hopping between the m = 3 and m = 4 modes in the channel. Finally, a 1D FFT was applied to the original video, and plotting the frequencies associated with each provided the mode's spatial structure. While these methods were only applied to the channel region in this work, this process can be applied to any radial region including around the cathode.
SVD-based methods
Matrix factorization-based methods, based on the SVD (Singular Value Decomposition) algorithm [47], are an alternative approach to decomposing and identifying modes from high-speed video and provide several advantages over Fourier methods. Below, we first detail the SVD algorithm, discuss the simplest SVDbased mode analysis method (the POD algorithm), and apply it to the present dataset.
The SVD algorithm
The SVD algorithm is the foundation for POD and for more advanced algorithms. This section discusses the SVD algorithm and its implementation. The most notable feature of SVD is that it does not presume spatial or temporal bases (e.g. Fourier sines and cosines) when decomposing coherent structures.
Instead, SVD provides a tailored, orthonormal basis for a particular video dataset by minimizing the L 2 norm (error) between the original video and the video reconstructed from the new SVD bases [22]. SVD orders these bases from the highest to lowest contribution to the reconstruction instead of by increasing wavenumber or frequency. A downside to this is that the bases often need to be associated with their Fourier counterparts through post-processing (manually or algorithmically). An advantage of SVD is that it does not require any preprocessing, most notably conversion to a particular coordinate system (e.g. converting the Cartesian video to polar). More details on the SVD algorithm can be found in the extensive literature on the subject [47,48,49,22].
In order to apply the SVD algorithm to a video dataset, p(t, x, y), we first convert it from 3D to 2D. We do this by stacking the data associated with the two spatial dimensions into a single spatial dimension, z = stack(x, y) to get the 2D p(t, z) dataset with dimensions in time and space and with sizes N t and N z = N x N y , respectively. This process is the 3D equivalent of stacking columns in a 2D matrix to get a 1D array. The Python data structure xarray has builtin functions (stack and unstack ) that make stacking and unstacking very convenient. By convention, we also transpose p(t, z) to p(z, t) so that the spatial dimension is ordered before the temporal dimension. For our high-speed video dataset, N z > N t , so p(z, t) is a "tall-skinny" matrix.
When applied to p(z, t), the SVD algorithm outputs three real matrices: U (z, n), Σ(n, n), and V (t, n) T . Here, we are using n to represent the SVD basis numbers to distinguish them from the Fourier mode numbers, m. First, U (z, n) is a non-square, 2D matrix associated with the spatial bases (called the topos, i.e., "shapes") with dimensions of space, z, and mode number, n, and lengths N z and N n = N t , respectively. Σ(n, n) is a square 2D diagonal matrix associated with the basis energies (these diagonal values are referred to as the singular values) with dimensions n by n each with length N n . Finally, the transposed matrix, V (t, n) T , is a square 2D matrix that contains the temporal evolution of the bases (called the chronos, i.e., "times") with dimensions of n and t, each with length, N n = N t . The exact dimension and format of these three matrices may vary depending on the particular SVD algorithm being used and shape of p(z, t). For example, many SVD algorithms return Σ(n, n) as 1D array of its diagonal elements.
Multiplying these three matrices together, imaginary (sine) Figure 9.
U (z, n)Σ(n, n)V (t, n) T = | | | u 1 u 2 . . . u N | | | σ 1 . . . σ N | | | v 1 v 2 . . . v N | | | T = σ 1 u 1 v T 1 + σ 2 u 2 v T 2 + . . . + σ N u N v T N ≈ p(z, t),(7)
Pre-normalizing each pixel in p(t, x, y) by its standard deviation instead of by the average channel brightness provides better visualization of the mode structure. Amplitude is spectral density (au). These plots are directly comparable to the m = 3 subplots in Figure 8 where the channel normalization is used. Figure 10. The SVD algorithm orders its bases from highest to lowest energy (i.e. their total contribution to the reconstructed video), and therefore the lower numbered bases are more likely to contain the most prominent dynamics.
reconstructs p(z, t) with remarkable accuracy (owing to the orthonormality of the u n and v n ). Related to this, another major advantage of SVD over Fourier is its ability to reconstruct data (like this high-speed video) with fewer bases. In this equation, u n , σ n , and v n , are the topo, singular value (energy), and chrono associated with the n th basis, which are ordered from highest contribution (energy) to lowest. In Eq. 7, the multiplication of u n v T n is an outer product. As a final note, we refer to σ n as the energy because our measurement is light intensity. However, it is common in the literature to measure the velocity or magnetic field and therefore refer to σ 2 n as the energy.
Proper Orthogonal Decomposition (POD)
The POD method, also referred to as Biorthogonal Decomposition (BD), has been used extensively for plasma physics datasets [27] but has only been very recently introduced to Hall thruster and hollow cathode high-speed video [32,33]. The POD algorithm is nearly identical to the SVD algorithm with the main distinction being that the U , Σ, and V T matrices are often trimmed to retain a smaller set of bases (e.g. n¡30). After applying the POD algorithm (Section 4.1) to our dataset, we inspect each of the U , Σ, and V T matrices to identify and interpret the dynamics associated with each mode. First, the energy for each basis, σ n , is plotted in Figure 10. This highlights that the lower numbered bases capture the majority of the dynamics of the video and are therefore the focus of this section.
To visualize the spatial bases (topos), we unstack the z dimension in U (z, n) to get U (x, y, n). Figure 11 shows the first 10 topos, u n , and each reveals an m = 0 or m > 0 azimuthal structure in the channel or around the cathode. The bases associated with m > 0 rotating modes each have a near-duplicate (but rotated) topo that represents an effective sinecosine pairing. For example, the n = 2 and 3 bases Table 1 are effectively the Fourier sine and cosine (real and imaginary) components of the m = 3 rotating mode in the channel and therefore well matches the modes in Figure 8. Note that the POD bases associated with the m = 2 Fourier mode is not shown in Figure 11 despite being present (as shown by Figure 8). This is because its energy contribution is lower than the first 10 POD bases. Each of these topos has an associated time evolution, i.e. chronos, associated with it. Several select chronos, σ n v T n , are plotted in Figure 12. Referring back to Figure 11, we see that their corresponding topos are roughly equivalent to the m = 0, 3, and 4 Fourier modes in the channel. Therefore, it is not a surprise that their chronos in Figure 12 are nearly identical to the Fourier-solved mode evolution shown in Figure 7. Finally, the Welch-averaged FFT of select POD chronos are shown in Figure 13. Note that only a single basis from a sine-cosine pair is presented as their power spectrums of each are nearly identical. Many of the peaks in Figure 13 are the same peaks as identified in the raw data ( Figure 4b) and the Fourier mode analysis (Figure 6b). A new structure is revealed by the n = 4 basis, related to the Fourier m = 1 cathode mode, which has two peaks around 38 and 80 kHz. From this analysis alone, it is not clear if the m = 1 cathode mode truly has two characteristic frequencies, if this basis is the combination of two Fourier modes, or if there is another explanation.
Comparing the results of the POD bases (Figures 11-13) to the Fourier modes (Figures 6-8) reveals a near one-to-one mapping as is indicated in Table 1. This is not too surprising as there are already established scenarios in which POD modes reduce to Fourier modes [50].
SVD methods conclusion
The primary advantage of the SVD-based methods is that they do not presume a universal basis, Fourier or otherwise, and instead create tailored bases which optimally capture particular features of the dynamics. In addition, these methods require less preconditioning of the video, e.g. radial binning and converting the pixel coordinates to polar, and therefore they can provide detailed images of the spatial structures of each basis or mode ( Figure 11). The most basic SVD method, POD, is easier to implement than the Fourier methods, does not require a linear assumption, and produces very similar results. More advanced SVD methods, such as DMD, have potential for Hall thruster modal analysis but require more development.
Conclusions
This work provided a survey of existing Fourier and new SVD-based mode decomposition techniques for high-speed Hall thruster video data. To highlight the implementation of these methods, they were each applied to the same H6 dataset which contained simultaneous modes associated with the channel and cathode. Both Fourier and POD methods were able to characterize the spatial and temporal evolution of each mode, including mode hopping between the m = 3 and m = 4 channel modes. Both methods also had their various strengths and weaknesses.
Fourier methods are well suited for characterizing linear wave dynamics and therefore excel at identifying both discrete mode and continuous wave dynamics in the various regions of the Hall thruster plasma. A notable feature of the Fourier methods are their ability to naturally provide dispersion relationships. However, Fourier methods require preprocessing, e.g.
converting the video to polar coordinates, which adds computational complexity.
The main advantage of the SVD methods (notably POD) is that they produce a tailored set of orthogonal bases for each dataset that are ordered by their relative amplitudes (high to low), and these bases do not presume any form (e.g. sines and cosines in Fourier). The POD method is relatively simple to implement as preprocessing, most notably coordinate conversion, is optional. In addition, the POD method can naturally locate mode structures at different spatial locations (e.g. cathode and anode) unlike Fourier which requires the dataset to be preprocessed with a particular spatial location in mind.
Together, Fourier and POD methods provide for a more complete toolkit for studying global Hall thruster dynamics as captured by high-speed imaging. Moving forward, we recommend first applying POD to a dataset in order to identify its major dynamics and the spatial locations of each. With this information, Fourier methods can then be intelligently targeted to extract dispersion relationships or other select modal information. The code and data used in this work are made are available online [39] to assist future researchers.
Figure 1 .
1a) The unshielded H6 Hall thruster. b) End-onview of the thruster and the plasma in its channel (annular ring) and the cathode (center dot). Figure reproduced here with permission from the author[40].
Figure 2 .
2An annular Gaussian function is fit to the timeaveraged video and identifies the channel origin and radius. The video's coordinates are then centered at the origin and normalized by the channel radius.
Figure 3 .
3Two time snapshots, that have been AC coupled and normalized by the average channel brightness, reveal prominent mode structures. a) At t = 22.9 ms, an m = 3 mode is dominant in the channel. b) At t = 24.8 ms, an m = 4 mode is dominant in the channel. Both snapshots show an m = 1 mode around the cathode. m = 0 and m > 0 modes associated with the thruster discharge channel and cathode in addition to mode hopping between the m = 3 and m = 4 modes in the channel.
Figure 4 .
4Three signals are analyzed (a pixel in the thruster channel (r/r 0 = 1), a pixel adjacent to the thruster cathode (r/r 0 = 0.1), and the discharge current), and their power spectrum reveals the existence of several prominent modes (peaks). a) The three time-series signals over 1 ms. Each has been subtracted by their mean and divided by their standard deviation. b) Their power spectrums, calculated over 100 ms.
Figure 5 .
5The azimuthal binning process for a single instant in time is shown. a) A radial mask isolates a narrow radial region within the channel, and the N θ =100 azimuthal bin boundaries are overlayed. b) The average of each azimuthal bin (red) overlays the raw (black) data.
Figure 6 .
6a) The azimuthal dispersion plot within the channel. A continuous wave and series of azimuthal mode numbers are identified and travel counter-clockwise (i.e. are negative). b)
Figure 7 .
7Time evolution of the m = 0, 3 and 4 modes as captured by the 1D FFT in θ. Repeated mode hopping is observed between the m = 3 and m = 4 modes. The signals are normalized by the average channel brightness.
Figure 8 .
8Fourier analysis (Eq. 4) reveals the spatial structure of several dominant modes in the channel. The amplitude is spectral density (au) with each subfigure having a different scaling. m=3 real (cosine) m=3
Figure 11 .
11The POD topos for the first 10 bases show m = 0 or m > 0 azimuthal structures in the channel or around the cathode. Amplitude units are arbitrary. By inspection, topos correspond to Fourier sine/cosine pairs as shown in
Figure 12 .
12The POD chronos for the n = 1, 2, 3, 5 and 6 bases. These bases are roughly equivalent to the time evolutions of the m = 0, 3, and 4 Fourier modes inFigure 7.
Figure 13 .
13The Welch-averaged FFT of select POD chronos helps in identifying the presence of a coherent mode structure and their characteristic frequencies. This figure is directly comparable to the Fourier results inFigure 6b.
Table 1 .
1The POD bases and Fourier modes show a near oneto-one mapping.POD basis, n Fourier mode, m Location1
0
Channel
2, 3
3, sine and cosine
Channel
4
0
Cathode
4, 6
4, sine and cosine
Channel
7, 8
1, sine and cosine
Cathode
AcknowledgmentsThis analysis was performed while JWB held an NRC Research Associateship award at the Naval Research Laboratory (NRL), with JWB and MM supported by the NRL Base Program. The imaging data was collected at the University of Michigan by MM under advisor Alec Gallimore in the Plasmadynamics and Electric Propulsion Laboratory, where the use of the Photron SA5 FASTCAM was made possible via grant FA9550-09-1-0695 from the Air Force of Scientific Research (AFOSR). AAK acknowledges funding from the Army Research Office (ARO W911NF-19-1-0045) and the Air Force Office of Scientific Research (AFOSR FA9550-18-1-0200).
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[] |
[
"Constraint from D-D Mixing in Left-Right Symmetric Models",
"Constraint from D-D Mixing in Left-Right Symmetric Models"
] |
[
"Bhaskar Dutta \nDepartment of Physics\nTexas A&M University\n77843-4242College StationTXUSA\n",
"Yukihiro Mimura \nDepartment of Physics\nTexas A&M University\n77843-4242College StationTXUSA\n"
] |
[
"Department of Physics\nTexas A&M University\n77843-4242College StationTXUSA",
"Department of Physics\nTexas A&M University\n77843-4242College StationTXUSA"
] |
[] |
We study the constraint arising from the recently observed D-D mixing in the context of supersymmetric models with left-right symmetry. In these models, the supersymmetric contributions in the mixing amplitudes of D-D, K-K and B-B are all correlated. We compare the constraint from the D-D mixing with the K-K mixing and find that the D-D mixing constrains the maximal supersymmetric contribution to the Bs-Bs mixing amplitude. The maximal supersymmetric contribution can allow a large CP phase of Bs-Bs mixing which agrees with the recent measurement of the CP asymmetry of Bs → J/ψφ decay. PACS numbers: 12.10.-g, 12.15.Ff Recently, BaBar and Belle have observed signals of D 0 -D 0 mixing [1]. The HFAG [2] interpretation of the current data gives us x D = 8.7 +3.0 −3.4 × 10 −3 , y D = (6.6 ± 2.1) × 10 −3 , (1) where x D = ∆M D /Γ D and y D = ∆Γ D /(2Γ D ), Γ D is the average decay width of two neutral D meson mass eigenstates. The mass difference of D 0 -D 0 is obtained as
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10.1103/physrevd.77.051701
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[
"https://arxiv.org/pdf/0708.3080v1.pdf"
] | 118,312,264 |
0708.3080
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17ebaa24dfa74384e5a1ed2bcbf486475f504f7b
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Constraint from D-D Mixing in Left-Right Symmetric Models
22 Aug 2007 (Dated: August 22, 2007)
Bhaskar Dutta
Department of Physics
Texas A&M University
77843-4242College StationTXUSA
Yukihiro Mimura
Department of Physics
Texas A&M University
77843-4242College StationTXUSA
Constraint from D-D Mixing in Left-Right Symmetric Models
22 Aug 2007 (Dated: August 22, 2007)
We study the constraint arising from the recently observed D-D mixing in the context of supersymmetric models with left-right symmetry. In these models, the supersymmetric contributions in the mixing amplitudes of D-D, K-K and B-B are all correlated. We compare the constraint from the D-D mixing with the K-K mixing and find that the D-D mixing constrains the maximal supersymmetric contribution to the Bs-Bs mixing amplitude. The maximal supersymmetric contribution can allow a large CP phase of Bs-Bs mixing which agrees with the recent measurement of the CP asymmetry of Bs → J/ψφ decay. PACS numbers: 12.10.-g, 12.15.Ff Recently, BaBar and Belle have observed signals of D 0 -D 0 mixing [1]. The HFAG [2] interpretation of the current data gives us x D = 8.7 +3.0 −3.4 × 10 −3 , y D = (6.6 ± 2.1) × 10 −3 , (1) where x D = ∆M D /Γ D and y D = ∆Γ D /(2Γ D ), Γ D is the average decay width of two neutral D meson mass eigenstates. The mass difference of D 0 -D 0 is obtained as
∆M D = (1.4 ± 0.5) × 10 −11 MeV.(2)
This new data can constrain new physics such as supersymmetry (SUSY) in the similar way as the traditional constraint from the K-K mixing data [3]. In SUSY models, the flavor degeneracy is often assumed in squark and slepton mass matrices to suppress flavor changing neutral currents (FCNC) [4]. The flavor violation effects in the sfermion mass matrices can only come from the evolution of renormalization group equations (RGE). If this is the case, the flavor violation highly depends on the unification scenario of quarks and leptons. In the minimal extension of SUSY standard model (MSSM), the induced FCNCs from RGE effects are not large in the quark sector, but sizable effects can be generated in the lepton sector since the neutrino mixings are large [5]. In quark-lepton unified models, the loop effects due to the large neutrino mixings can induce sizable effects also in the quark sector. Therefore, it is important to investigate the FCNC effects to obtain a footprint of the unification models.
Left-right symmetric model construction is an interesting candidate to unify matter (including righthanded neutrino) in the gauge group SU (3) c × SU (2) L × SU (2) R × U (1) B−L [6]. The left-right parity is broken spontaneously, and the hypercharge arises from a linear combination of U (1) B−L and the U (1) subgroup of SU (2) R . This left-right symmetric branch can be easily unified in SO(10) grand unified models. In the SUSY version of left-right symmetric models, the box diagrams for meson mixing (K-K, B-B and D-D) can be enhanced by gluino contribution. Therefore, the newly observed D-D mixing can be an important probe for left-right symmetric models. Further, in such models, D-D, K-K, B d -B d and B s -B s mixing amplitudes get correlated. This creates an interesting opportunity for cross-checking these models, since the most interesting observation from the sizable SUSY contribution will be the phase of B s -B s mixing, which can be measured by B s → J/ψφ decay.
The mass difference of B s -B s (the absolute value of the mixing amplitude M 12 ) has been measured [7], and the measurement is consistent with the Standard Model (SM) prediction. Therefore, if there is a sizable SUSY contribution, the phase of B s -B s mixing (argument of the amplitude) must be large. The CP asymmetry of the B s decay is being measured and the current result is [8] 2β s = −0.70 +0.47 −0.39 (rad),
while the SM prediction is ∼ 0.03 − 0.04. If this result holds in future, then it will indicate an existence of new physics.
The SUSY contribution to the B s -B s mixing is related to the 23 off-diagonal elements of the squark mass matrices, which may be large since it can be related to the large atmospheric mixing. On the other hand, it is hard to predict the amount of the SUSY contribution to the K-K and D-D mixings due to cancellation. However, we can show that cancellations for both K-K and D-D mixings are not allowed simultaneously when the non-universal terms in squark mass matrices originate from left-right symmetric models. Consequently, the recent observation of the D-D mass difference restricts the amount of SUSY contribution, and thus, it also restricts the phase of B s -B s mixing. In this Letter, we will show how to obtain the constraint of SUSY contribution from the D-D mixing, and study the correlation of the constrained phase of B s -B s mixing to other measurements, e.g., phase of B d -B d mixing, in left-right symmetric models.
In left-right symmetric models, the Lagrangian is invariant under the exchange Q L ↔ Q c R , where Q L is SU (2) L doublet and Q c R is SU (2) R doublet which contains conjugate of right-handed up-and down-type quarks U c , D c . As a result, Yukawa couplings are given by symmetric matrices. The squark matrices are given at unification scale as [9]
M 2 F = m 2 0 1 − κ U F k 1 k 2 1 U † F ,(4)
where U F is a unitary matrix and F denotes Q, U c and D c . In the original basis where we respect the left-right symmetry, U F is common for Q, U c and D c . We note that the squark mass matrices are given in the notation:
(M 2 Q ) ijQiQ † j + (M 2 U ) ijŨ c iŨ c † j + (M 2 D ) ijD c iD c † j .
The nonuniversal part of squark mass matrices can be generically parameterized by the κ term. We consider that the κ term is generated from a loop diagram in the
form ∝ f f † . Here, f is the quark Majorana coupling f (Q L Q L ∆ qq + Q c R Q c R ∆ c qq )
, which can be unified into the neutrino Majorana coupling in a SO(10) model. In general, U F is parameterized as U F = P U q where P is a diagonal phase matrix and U q includes 3 mixing angles (θ q ij ) and 1 phase (δ q ). We parameterize U q in the basis where the down-type quark Yukawa matrix is diagonal. The mixing angles and a phase are parameterized in the same convention as the CKM matrix. If we consider the type II seesaw scenario [10] in a SO(10) model, k 2 corresponds to the ratio of neutrino mass squared and U F is the neutrino mixing matrix in the basis where the charged-lepton mass matrix is diagonal, and thus θ q ij correspond to neutrino mixing angles when both chargedlepton and down-type quark mass matrices are simultaneously diagonalized. In general, θ q ij are not necessarily exactly same as the neutrino mixings. The Yukawa matrices for up-and down-type quarks (Y u and Y d ) are given as
Y u = V T CKM Y diag u P u V CKM , Y d = Y diag d P d ,(5)
where P u,d are diagonal phase matrices. We can calculate the off-diagonal elements of the squark mass matrices δ ij ≡ (M 2 F ) ij /m 2 0 in the above notation.
|δ d 12 | ≃ κ 1 2 k 2 sin 2θ q 12 cos θ q 23 + e iδ q sin θ q 13 sin θ q 23 , (6) |δ d 13 | ≃ κ 1 2 k 2 sin 2θ q 12 sin θ q 23 − e iδ q sin θ q 13 cos θ q 23 , (7) |δ d 23 | ≃ 1 2 κ sin 2θ q 23 ,(8)
where superscript d stands for that it is given in the basis where the down-type Yukawa matrix is diagonal. These quantities enter into the calculation of K-K, B d -B d and B s -B s mixing amplitudes. When a flavor degeneracy is assumed at the unification scale and only the MSSM RGE is considered, the chargino diagram contribution dominates the SUSY contribution. However, if the flavor violation is induced by a loop diagram at the unification scale (as discussed before), the gluino diagram can generate the dominant contribution. This contribution to the mixing amplitude Mg 12 can be written in the following mass insertion form
Mg 12 M SM 12 ≃ a [(δd LL ) 2 ji + (δd RR ) 2 ji ] − b (δd LL ) ji (δd RR ) ji ,(9)
(ji = 21, 31, 32 for K-K, B d -B d and B s -B s , respectively) where a and b depend on squark and gluino masses, and δd LL,
RR = (M 2 d ) LL,RR /m 2 (m is an averaged squark mass). The matrix M 2 d is a down-type squark mass matrix (Q,D c † )M 2 d (Q † ,D c ) T
in the basis where the down-type quark mass matrix is real (positive) diagonal. When squark and gluino masses are less than 1 TeV, a ∼ O(1) and b ∼ O(100). We also have contributions from δ d LR , but we neglect them since they are suppressed by (m b /m SUSY ) 2 . It is worth noting that the left-right symmetric boundary conditions give much larger SUSY contribution since both off-diagonal elements for LL and RR are large and b ≫ a in the mass insertion formula.
When LL-RR contributions are dominant, the phases in P are cancelled due to (M 2 d ) LL = M 2 F and (M 2 d ) RR = (M 2 F ) T (when we neglect the RGE effects). The phase of the mixing amplitude is generated from the phases in P d . Since there is no constraint for the phases in P d , the phase of the mixing amplitude is free. However, there are only two physical phases in P d and therefore the phases of the SUSY contributions for K-K, B d -B d and B s -B s are correlated. We will show the impact of this correlation later.
The gluino contribution for the D-D mixing is obtained when we changed toũ, but it needs to be written in the basis where the up-type quark Yukawa matrix is diagonal. The important quantity for the D-D mixing is δ u 12 (in Y u diagonal basis), which can be written as
[V * CKM (δ d )V T CKM ] 12 ∼ δ d 12 + V us δ d 22 ,(10)
up to the P u phase (P u phase gives just an overall phase of δ u 12 and it is not important for the cancellation since the short-distance SM contribution of D-D is small.), and δ d 22 ≃ κ sin 2 θ q 23 . Therefore, when κ sin 2 θ q 23 is large, both K-K (δ d 12 ) and D-D (δ u 12 ) SUSY contribution cannot be cancelled away simultaneously.
In Fig.1, we show the maximal value for κ allowed by the experimental results for K-K and D-D mixings as a function of sin θ q 13 . We use sin 2 θ q 23 = 1/2, tan 2 θ q 12 = 0.4, k 1 = 0 and k 2 = 0.05. The SUSY parameters are chosen to be m 0 = 1 TeV, m 1/2 = 300 GeV (gaugino mass), A 0 = 0 (trilinear scalar coupling) and tan β H = 10 (ratio of Higgs vacuum expectation values). The phase δ q and the other phases are chosen to make the κ value maximal. In the usual convention, sin θ q 13 is positive since its negative value can be redefined by rephasing δ q . But, in order to show the figure simply, we also use negative sin θ q 13 as a convention. The K-K (δ d 12 ) is cancelled at sin θ q 13 ∼ − 1 2 k 2 sin 2θ q 12 cot θ q 23 and D-D (δ u 12 ) is cancelled at sin θ q 13 ∼ ± sin θ q 23 V us . Due to the fact that phases in P are free, D-D (δ u 12 ) can be cancelled for both positive and negative θ 13 .
For most of Fig.1, the D-D constraint, using the recent experimental result, is weaker than the K-K constraint. However, the D-D mixing is important at the K-K cancellation region (δ d 12 → 0). As a result, the newly observed D-D mixing can restrict the maximal SUSY contribution to the B-B mixing. From Fig.1, we see that the maximal SUSY contribution is obtained at the K-K cancellation region after satisfying the D-D constraint.
We can classify the solution for fitting the of K-K mixing amplitude in the following three cases, which are illustrated in Fig.2. The K-K mixing amplitude is given as
M 12 = M SM 12 + M SUSY 12 .
The mass difference is given as ∆M K = 2|M 12 |, and the CP violation parameter |ǫ K | = ImM 12 /( √ 2∆M K ). The SM predication for M 12 is in the fourth quadrant of the M 12 -complex plain. The experimental measurement for M 12 is more accurate rather than the illustration in the Fig.2 should be 0 or π in solution C. The solutions A, B, C which provide maximal value of κ are shown in the Fig.1. In solutions B and C, the amount of cancellation of δ d 12 is larger than in solution A for a given κ ∼ 0.2. As noted, the phases of SUSY contributions for K-K, B d -B d , B s -B s mixing amplitudes are related since the phases of δ d ij are cancelled (up to small RGE modification) and only two physical phases in P d remain. Since all solutions A,B,C of the K-K mixing provide restriction to the phases of the SUSY contributions, phases of the SUSY contribution for B d,s -B d,s mixings are restricted for large SUSY contribution. We draw Fig.3 to show the correlation of the phases. We choose CKM parameters as sin 2β SM ≃ 0.77 and sin 2β SM s ≃ 0.04. We use the same parameters for θ q 23 , θ q 12 , k 2 and SUSY mass parameters as we have used to draw Fig.1. We choose κ = 0.2 in each solution. As described, the phase of the SUSY contribution M K-K 12 is almost π in solution A. We choose the SUSY phases to be π/2 and π for solutions B and C, respectively. In the plot, we choose the absolute values of SUSY contributions to be same for both solutions B and C. Since the B s -B s SUSY contribution is determined by δ d 23 (eq. (8)), the maximal values of |β s | are almost same in all three solutions. On the other hand, the B d -B d SUSY contribution depends on θ q 13 and it is different for solutions A and B,C. One finds that sin 2β eff is smaller than the SM value when β eff s becomes positive in solutions A and C. Solution B gives us opposite result. This correlation is a consequence of the fact that there are only two physical phases for three different mixing amplitudes.
The global fit of the experimental data [11,12] shows that sin 2β arising from the V ub measurement has a 2σ discrepancy from the sin 2β measurement from B d → J/ψK [11], sin 2β = 0.678 ± 0.026 [2]. Thus a negative SUSY contribution is favored for sin 2β eff . The present data for β s , eq.(3), favors negative value. As a result, solution A is disfavored by the experimental result. For solutions B and C, the phases of SUSY contributions have ambiguity. In order to produce negative contribution for sin 2β and generate negative sin 2β s , a phase of magnitude from 0 to π/2 is favored for the K-K SUSY contribution.
We have chosen θ q 23 = π/4, which generates the maximal SUSY contribution to B s -B s for a given κ. It is important that the D-D mixing constrains κ sin 2 θ q 23 at the K-K cancellation region, and thus, the D-D mixing data constrains the B s -B s mixing for a given θ q 23 . Naively, the SUSY contribution of B s -B s is proportional to (κ sin 2θ q 23 ) 2 . Thus, when the SUSY contribution saturates the observed D-D mixing, the maximal value of |β s | becomes larger for smaller θ q 23 . Such a direction also decreases sin 2β eff , which is favored by the experimental result. So we see that the meson mixings are all related in left-right symmetric models. More accurate measurement of B s -B s phase will impose interesting constraint on the model.
If we consider a SO(10) model, the SUSY contribution of B s -B s also gets correlated to the τ → µγ decay amplitude [13], which is more important compared to the D-D constraint for small m 0 and large tan β H . In the case of large m 0 , however, the D-D constraint can be stronger than τ → µγ. The µ → eγ decay amplitude is small due to the same cancellation condition for δ d 12 (which reduces K-K mixing amplitude) and δ l 12 . We assume that the left-right symmetry under the exchange of Q L ↔ Q c R . We can also consider the exchange Q L ↔ (Q c R ) * . In this case, the Yukawa matrices are Hermitian instead of symmetric matrices [14]. The phase matrices P d and P d become just signature matrices. However, the squark mass matrices satisfy M 2 U = M 2 D = (M 2 Q ) * and therefore, the phases in P are not cancelled in the meson mixings. As a result, in the Hermitian Yukawa case, the phases of meson mixings are also correlated at the K-K cancellation region as in the symmetric Yukawa case.
In conclusion, we have studied the importance of D-D mixing in left-right symmetric models. We showed that the D-D mixing data constrains the phase of B-B mixing for given parameters in left-right symmetric models, and studied the correlation of the meson mixings. If we consider unified models without left-right symmetry such as SU (5), where only right-handed squark mixings can be large naively, the SUSY contribution is not very enhanced. Besides, since right-handed squark mixings are unknown, both K-K and D-D can be cancelled away separately, and therefore there is no constraint. Therefore, left-right symmetric models are very interesting candidates to investigate correlation among measurements especially when the SUSY contribution is maximal and the B s -B s phase is large. The improved result of this mixing phase will further shed light on the correlation of meson mixings and left-right models.
This work was supported in part by the DOE grant DE-FG02-95ER40917.
FIG. 1 :
1Maximal values for κ are shown as a function of the angle θ q 13 from the constraints of K-K and D-D mixing separately. Cancellation happens for the SUSY contribution when the maximal κ is large. The points A, B, C correspond to the solutions illustrated inFig.2.
FIG. 2 :
2. However, the numerical value can have ambiguity from bag parameters and the charm quark mass. The experiment measures only the absolute values of real and imaginary parts of M 12 . So the possible solutions to satisfy the experiment are the four separate regions as shown in the Fig.2. Solution A is given as M SUSY 12 ∼ −2M SM 12 . In this solution, M 12 is in the second quadrant. Since ImM 12 ≪ ReM 12 , M 12 lying in the third quadrant is almost same as solution A. In solution A, the M SUSY 12 phase is almost π. In solution B, M 12 is in the first quadrant, and the M SUSY 12phase is about π/2. In solution C, M 12 is in the fourth quadrant. When |M SUSY 12 | ≪ ImM SM 12 , the SUSY contribution is negligible in the K system, and phase of M Illustration of the experimentally allowed solutions (shaded area) for amplitude M12 for K-K mixing. Details are described in the text.can be arbitrary. When |M SUSY 12 | ∼ ImM SM 12 , the phase of M SUSY 12
FIG. 3 :
3Correlations for the phases of B d -B d (β eff ) and Bs-Bs (β eff s ) mixings. A, B, C correspond to the solutions illustrated in Fig.2.
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"Heuristic model of teaching A simple physical model differentiates effective from ineffective teaching and identifies elements that make teaching productive, with specific implications concerning training of teachers",
"Heuristic model of teaching A simple physical model differentiates effective from ineffective teaching and identifies elements that make teaching productive, with specific implications concerning training of teachers"
] |
[
"Stanis Law \nInstitute of Theoretical Physics\nUniversity of Warsaw\nPoland\n",
"D G Lazek \nInstitute of Theoretical Physics\nUniversity of Warsaw\nPoland\n"
] |
[
"Institute of Theoretical Physics\nUniversity of Warsaw\nPoland",
"Institute of Theoretical Physics\nUniversity of Warsaw\nPoland"
] |
[] |
A simple physical model differentiates effective from ineffective teaching and identifies elements that make teaching productive, with specific implications concerning training of teachers. 01.40.Ha, 12.90.+b, 01.40.Fk I.
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"https://arxiv.org/pdf/0804.4796v2.pdf"
] | 119,273,127 |
0804.4796
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be8c7ed12f97b72b450184eee905ef68b34d92db
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Heuristic model of teaching A simple physical model differentiates effective from ineffective teaching and identifies elements that make teaching productive, with specific implications concerning training of teachers
5 Sep 2008 (Dated: 4 September 2008)
Stanis Law
Institute of Theoretical Physics
University of Warsaw
Poland
D G Lazek
Institute of Theoretical Physics
University of Warsaw
Poland
Heuristic model of teaching A simple physical model differentiates effective from ineffective teaching and identifies elements that make teaching productive, with specific implications concerning training of teachers
5 Sep 2008 (Dated: 4 September 2008)arXiv:0804.4796v2 [physics.ed-ph]numbers: 0140-d0140Ha0140J-1290+b0140Fk
A simple physical model differentiates effective from ineffective teaching and identifies elements that make teaching productive, with specific implications concerning training of teachers. 01.40.Ha, 12.90.+b, 01.40.Fk I.
INTRODUCTION
It is reasonable to attempt to describe the process of teaching in terms of a model because it is known that models can lead to improvement in our understanding of natural phenomena and the understanding may allow us to distinguish useful from useless action. A physical model proposed in this article provides a picture in which effective teaching is clearly separated from ineffective teaching. The model picture also allows for identification of elements which lead to teaching that may be called productive. These insights have implications for the training of teachers.
Section II discusses the reasons why a model of teaching is needed and sets the stage for the next two sections, in which the model proposed in this article is described. Section III describes a kinematical part of the model that is sufficient to identify the difference between effective and ineffective teaching. Section IV describes a dynamical part of the model that leads to the concept of productive teaching. Conclusions are in Section V.
II. THE NEED FOR A MODEL
Building models is a time-honored practice to organize thinking about concepts in physics. For example, the concept of the atom has been discussed since ancient times. About a century ago, Bohr was equipped with enough data on atomic phenomena to conceive and describe a concrete image of an atom [1]. Bohr's model allowed physicists to focus their attention on the dynamical issues in the physics of atoms and gradually replace the model with quantum mechanics. The latter provided a mathematical basis for the progress that followed [2].
Thus, one can say that physical models may be useful in two ways. One way is that a model provides a context in which a concept can be spoken about in concrete terms. The concreteness of the context helps eliminate confusion and allows researchers to focus on the dynamics of the observed phenomena. Another way is that a model can be wrong in the sense that it disagrees with results of experiments or observations. Such findings provide the basis for seeking a better next-generation model. The same methodology may be applied to teaching. There is no reason to limit this methodology to physics teaching; it applies also to teaching in other disciplines. A simple physical model of teaching (MOT, or just "model") described in this paper incorporates the concepts of a teacher and a student through the concept of observers. Two observers can communicate with each other about phenomena they observe and one can help the other in understanding what happens. In a preliminary way, the model also allows for incorporation of the concept of forces that drive this process. Recognition of the existence of relevant forces, and how they manifest themselves in a teacher-student relationship, leads to a description of the concept of productive teaching. Since the same model structure appears valid in all contexts in which the process of teaching may occur, the scope of the model is not limited to the case of a teacher teaching a student in a classroom at a school or university.
The claim that a physical MOT may be formulated in a simple way requires explanation because it is known from physics education research [3,4,5] that the process of teaching is not simple. Procedures used by teachers to specifically check what students actually learn in a physics course involve complex physical notions and as such are not simple [6]. One may expect that a process of teaching is even more complex than in physics when the subject matter involves phenomena that cannot be explained simply in terms of physics.
From a psychological point of view, the process of teaching cannot be separated from the process of learning and the latter is not commonly defined in simple terms [7,8,9,10,11,12,13]. From a neuroscientific point of view, the process of learning by the brain is a subject of intense study, and complex phenomenological and theoretical models are needed to describe how individual neurons and networks of neurons work [14,15,16,17,18,19]. Besides the brain, the process of learning involves a student's body and the body is also complex. For example, the involvement of hands in the process of learning is not simple [20]. The role that hands play in learning is related to the complex process of evolution of species [21], including the ability to learn as a means of increasing the chances of survival and reproduction. Highly-evolved learning abilities in humans emerged from processes that correlate the behavior of individuals with the environment they live in, and the latter influences individuals in very complex ways (for a popular explanation of how advanced human behaviors, such as altruism of teachers, could emerge in the evolution, see [22]; cf. [23,24]). At the level of contemporary society, the processes of teaching and learning in a large system of education can be seen as exceedingly complex [25,26] and may fascinate physicists [27,28,29,30]. The need to corral the complexity of the process of teaching provides perhaps the strongest motivation for building a suitable physical model. Without a MOT, the purpose of educational reform is ambiguous.
A physical MOT can be simple only in the sense that the basic elements of the process of teaching can be identified using physical concepts such as an observer and a frame of reference. The complexity of teaching comes from the complexity of the events that observers communicate about and from the complexity of the observers themselves, including the forces that drive their learning and ultimately determine what teaching may accomplish. Given this starting point, a simple model will be obtained in the next two sections.
III. KINEMATICAL PART: EFFECTIVE TEACHING
In order to kinematically describe what happens in the process of teaching, one first postulates that every student has a mental "dictionary" that associates specific word definitions with specific concepts, such as the sequences of words associated with "velocity" or "acceleration" [31]. A growing body of research on misconceptions in physics shows how bizarre the word entries for these two concepts can be in a student's mental dictionary [32,33]. The source of this variety of entries is that every student's dictionary results from his or her prior learning history and every student has a different history.
By definition, in the process of teaching a teacher communicates with a student and, as a result, the student's dictionary grows and improves in accuracy. However, one should remember that students who take a course have different mental dictionaries concerning concepts to be taught, such as in physics, and faculty members would only deceive themselves if they believed that every student has the same interpretation for the language that they use, especially when the students are only beginning to learn this language.
It is postulated that the words and concepts learned by a student are stored in his or her brain in a coded form that ultimately amounts to sequences of coordinates (numbers) in a multidimensional space. For brevity, this space will be called the student's space of knowledge. Nothing more needs to be said about this space except that the model will require an additional structure to provide a representation of what a word or a concept means to the student. This additional structure will be introduced soon.
In support of the concept of coordinates in the student's space of knowledge one might recall mathematical considerations which imply that statements in any logical system can be represented in terms of numbers [34]. Sim-ilarly, neural science suggests that learning and memory storage processes can be understood in terms of numbers that represent in specific units how networks of neurons work, such as voltages, currents, densities of ions, etc. [18,19]. Psychological studies also have a history of attempts to describe the meaning of words and concepts to individuals in terms of numbers [35]. Attempts to develop artificial intelligence presume that the human brain can be understood as an information-processing machine that ultimately operates only with numbers [36,37,38]. However, the MOT described here postulates the student's space of knowledge not for the purpose of explaining, but for the purpose of circumventing, the issues of complexity and lack of understanding of the human brain, its development and function. The goal is to model concepts of teaching in such basic terms that a person interested in these concepts can form suitable entries for them in his or her own dictionary without the necessity of first engaging in life-long studies of what is currently known about the human brain and its psychology.
Thus, the changes in a student's mental dictionary are represented in the model by transformations of the coordinates of words and concepts in the student's space of knowledge. Teaching a subject matter to a student is meant to cause the coordinates of words and concepts pertaining to the subject matter in the student's space of knowledge to form the intended relations among each other, thereby forming an accurate dictionary.
It follows that a teacher and a student need to communicate well with each other in order for the teaching process to stay on track. The communication must be good in both directions (not only from the teacher to the student but also from the student to the teacher) because the only person who knows what the student's knowledge consists of after interpretation using the student's dictionary, is the student.
The internal mental processes in the student's brain enable the student to produce conversations or questions using his or her own dictionary. These internal processes involve the coordinates of the words and concepts in the student's brain but the student is not conscious of what the coordinates are and what their numerical values are. What the student can be conscious of regarding his or her mental dictionary is, for example, a sense of confusion about what the teacher is saying, or a visible disagreement between a prediction the student makes using his or her words and concepts about a possible result of an observation or experiment and a result he or she actually obtains.
When the student undertakes a conscious effort to make a change in his or her space of knowledge and asks questions, he or she needs to get useful answers from the teacher, which means answers that suggest to him or her what needs to be changed and how. In order to be able to help the student, the teacher must know what the student thinks. Using the model, one can say that the task of the teacher is to establish (as best as one can) the coordinates of relevant words and concepts in the student's brain, identify the transformation that will reduce or remove the confusion that stems from the improper relations among these coordinates, and then use the words and concepts already known to the student to provide him or her with information about what needs to be reconsidered in order to deal with the confusion effectively, the effect being the required change in the student's dictionary.
A student may learn without constant help from a teacher, or people who play this role, directly or through textbooks and other means. However, by definition, the model presented here pertains to a process of teaching that involves a teacher who helps a student [39].
Further description of the model concerns a representation of the meaning of words and concepts and how the meaning is conveyed from person to person. Both aspects are incorporated into the model using the concept of observers and how they communicate about events.
A. Students and teachers as observers
Consider a student to be an observer whose brain registers events and stores information about them in memory. Every event is registered at some place and time and has some content. It is reasonable to postulate that the content of an event registered in a brain ultimately amounts to a sequence of parameters. (A concrete way of imagining these parameters is described in Appendix A.) Essentially, the parameters describe activity in all parts of the brain involved in the process of registration. The parameters are treated in the model as coordinates in the space of events. If the content of every event registered in a brain is assumed equivalent to a sequence of values of n parameters, the number of dimensions in the space of events is N = 4 + n; 4 for place and time and n for content. This space of events is used below to model the process of transfer of information about the meaning of words and concepts between a teacher and a student.
Firstly, it is postulated that the meaning of a word (or concept) is represented in the student's brain in the form of memory about a set of events that are associated in the brain with the word (or concept). The model is not meant to explain the neurobiological nature of how the memory functions so that the association can be formed. The association is taken for granted in order to represent the fact that concrete examples allow people to quickly identify the intended meaning of words and concepts.
Secondly, it is postulated that a student creates and changes his or her own dictionary of words and concepts on the basis of observation of correlations between two sets of events: one set that the student associates with a word definition and another set that the student associates with a concept. The identification of these correlations does not have to be fully conscious. But it is not necessary to determine the extent to which the identification of correlations is conscious or not conscious for the model to provide a picture of what happens in the process of teaching, including the transfer of meaning of words and concepts.
In the process of teaching, a teacher introduces a set of events considered relevant and provides students with information about them, using words (defined to a various degree of precision) to describe the events and related concepts. The relevant events are, for example, experimental demonstrations of phenomena. According to the model, a student's dictionary changes as a result of the student's attempts to correlate demonstrated events with their description by the teacher. The teacher is another observer who helps the student understand the observed events using the introduced concepts. The student tries to improve his or her own dictionary by making it, from the student's point of view, resemble the one of the teacher. More generally, teachers and students become familiar with many events in their lives and teaching often refers to and certainly draws on this experience instead of using direct experimental demonstrations.
B. Communication between observers
Once the kinematics of teaching is reduced to communication between observers about events, further description of the model concerns the process of communication.
In physics, effective communication between two observers (meaningful, precise, and fast) uses coordinates in their respective frames of reference. The use of frames of reference is a hallmark of physics [40]. Each observer builds her or his own frame of reference according to universal rules. Two observers can effectively exchange information using coordinates of events as soon as they establish and know (understand and remember) how their frames are related. The relationship is found in physics using the following procedure [40,41].
The observers make use of a set of physically welldefined events that each of them can unambiguously recognize (in the sense of classical physics). For example, an event in which two initially distinct material points meet each other is a physically well-defined event. The coordinates of the two points coincide in all frames of reference when the points coincide. Having introduced a suitable set of well-defined events, both observers can find the coordinates of these events in their respective frames. Knowing the coordinates in both frames, the observers can figure out the rules of correspondence between their coordinates for all events in the set. Subsequently, they can extrapolate this correspondence to a general relationship between their frames of reference that is supposed to connect their coordinates for all events. They inspect many cases and, if they gain confidence in the extrapolated relationship they use, they consider it valid.
For example, if the correspondence in the set of welldefined events is figured out to be a linear one in an N -dimensional vector space, a set of only N well-defined and linearly independent events (in terms of their coordinates) is sufficient to find all parameters of a general linear relationship between the two frames of reference. If the required relationship is not linear, a more complex, case-specific reasoning is required and a simple, generally valid relationship is not guaranteed. Note that Appendix A suggests significant topological complexity of the space of events registered in a student's brain. The same concerns the teacher's space of events. A linear relationship between the frames of reference in the student's and teacher's spaces of events is unlikely.
In the case of teaching introductory physics, the set of well-defined events may take the form of tabletop experiments that students carry out with their own hands and whose results they first analyze each from her or his own point of view and in a small group. They then discuss their observations, confusions, discoveries, and conclusions with an instructor. Such discussions create and change the dictionaries students have for thinking and communicating about physical phenomena [42,43].
Since the process of teaching is postulated to involve the space of knowledge and the space of events, it follows that the process of communication between brains requires the establishment of frames of reference in two spaces per brain. Indeed, the model associates such two frames with every brain. Thus every word, concept, and event becomes equivalent in a brain to some sequence of co-ordinates, and discussions about events using all kinds of languages are reduced to a transfer of sequences of co-ordinates from one brain to another. Since it is obvious from the point of view of physics that a meaningful exchange of sequences of co-ordinates between observers must be preceded by establishment of a relationship between their frames of reference, it is now also clear that the process of establishing a dictionary must involve the processes of building frames of reference and finding relationships between them.
However, the process of communication between two brains cannot proceed in as simple and direct a way as the process of communication between two observers in physics. The postulated reason is that we do not have access through thinking to the values of coordinates that our brains assign to events they register and to words and concepts we use in communication about the events. A teacher may imagine the coordinates and relationships among them in a student's brain for the purpose of teaching. But neither the teacher nor the student knows the coordinates with which the student's brain registers events or stores words and concepts. Therefore, one has to accept that the process of associating words and concepts with events must exhibit a large degree of ambiguity, which cannot be eliminated by precise matching and changing of the numerical values of coordinates of words and concepts on the basis of directly measuring and comparing them.
Instead of the direct measurement and comparison of coordinates, the model postulates that the creation and development of a dictionary in the student's space of knowledge is essentially based on a process of trial and error. A word or concept is unconsciously assigned its initial coordinates in the space of knowledge. These coordinates are associated with the coordinates of concrete events stored in memory. The initial association can happen quickly but it is not certain and requires testing before it is promoted to a more stable memory. One may postulate that the initial association depends on the unconscious structural and functional features of the brain. One is also free to postulate that the association involves feedback from complex processes that include both shortand long-term memory and conscious thinking. But the nature of the association does not matter for the model.
Of importance is the postulate that the initial association between coordinates in both spaces is modified and stabilized in the brain via the process of registration of new events and communication about them. The growth and change in the brain tissue and its functioning that result from this process are accepted in the model as an underlying neurobiological realization of learning. However, precise knowledge of the neurobiological realization is not needed to understand the model. For example, the meaning of a word is postulated to become eventually established in the following way.
The coordinates initially assigned to a word in the space of knowledge are unconsciously correlated in the brain with coordinates of some events stored in memory. The initial association is not entirely random, neither is it precise. Then the word is used in thinking, absorbing information, and communication about events. This process uncovers defects in the initial association through identification of misunderstandings [44]. The association is changed to a new one that appears to reduce the confusion and is used in further thinking, experimenting, observation, and communication. Eventually, subsequent uses of the word cease to lead to confusions that require modification of the association and the meaning of the word is obtained in the form of a stable association between the coordinates of the word in the space of knowledge and the coordinates of corresponding events in memory. A similar, multistage process is postulated for concepts. Identification of the meaning of concepts in terms of events precedes the introduction of their names in the dictionary.
Ultimately, the rank and order in the space of knowledge is postulated in the model to be brought about by observing events, and communicating about them with other observers. Thus, a language is built and learned, and the words in it and concepts required to understand and predict events are tuned with increasing accuracy.
Whether one communicates about current events or events that are only remembered, or merely imagined on the basis of memory of the actual events (such as the motion of a point along a straight line, Einstein's gedanken experiment [45,46,47], or an event described in a book or shown in a film), is of secondary importance. The same scheme is expected to apply to teaching in all disciplines, not just physics.
C. Effective teaching
What follows is the model description of the process referred to as effective teaching. For the purpose of differentiating between effective teaching and another process that may be referred to as ineffective teaching, the latter will be presented first. This order of presentation brings out the conceptual contrast between the two processes, providing an example of the utility of the model.
Consider a teacher, for brevity called T , and a student, for brevity called S. T is supposed to teach S.
In the case of ineffective teaching, T tries to elicit from S a reproduction of a sequence t of words (or other symbols) that T presents to S. Eventually, S reproduces t, from memory, but without having much of an idea why and what T wanted to convey by t. The sequence t may be similar to some sequences that S already knows, but whether these sequences correspond to what T meant by t and what was the goal of talking about t remains unclear to S. S will soon forget the sequence t because there is no reason to remember it.
Such ineffective teaching does not include the process of establishing the meanings of words and concepts that T uses and S is supposed to learn about. In the model picture of ineffective teaching, what T does is, in essence, supplying S with a sequence of co-ordinates valid in the frame of reference of T without trying to find out how the frame of reference of S is related to the frame of reference of T . Then T expects to obtain the same sequence back, but has no way to establish what this sequence means in the S's frame of reference. Thus, the model makes it obvious that the process of ineffective teaching, based on transferring information only one way, is fundamentally inadequate. At the same time it becomes clear that ineffective teaching is characterized by arbitrary assumptions about how S's brain associates meaning with words and concepts in terms of events. It is assumed that S does it in the same way as T so that the dictionaries of T and S are the same. T 's dictionary is assumed to be well-calibrated with respect to the world, including S as a part of the world. It is also assumed that S can be considered well-taught if he or she repeats the statements that T wants him or her to repeat. This is what is ordinarily described as rote memorization.
The model thus predicts that ineffective teaching leads to a lack of communication between T and S. It generates confusion and correlates word definitions with names of concepts in the mind of S in a disordered fashion, devoid of meaning. On the surface, things appear not that bad but only because the ambiguities of thinking and language hide the imperfections of the communication.
Further, the model implies that when T interacts ineffectively with a whole group of students instead of just one, most of them are confused and each of them in a different way. Having no chance to understand what is going on, the students lose interest. T is not able to regain the students' attention and disciplinary measures are likely to dominate other aspects of teaching. An edu-cational system may be unable to identify dysfunctional situations of ineffective teaching and students may be required to pass tests. They are then judged on the basis of the numerical values of their scores. The problem is that these values reflect not whether the students learn but how the system functions.
In the case of effective teaching the situation is remarkably different. According to the model, the initial task of T is to establish how the frame of reference of S is oriented with respect to the frame of reference of T . This means that T first tries to find out what statements S already knows regarding the subject of study and what S means by them, i.e., what events correspond to these statements according to S. T uses these events to estimate how the dictionaries of T and S are related to each other. Further tuning of communication between T and S involves hands-on activity during which concrete events are associated with concrete words and concepts.
When key parts of the dictionaries of T and S are already tuned, which is described in the model by saying that the relation between T 's and S's frames of reference is approximately known to T and T knows how to use the dictionaries, T becomes able to try to understand what to do in order to convey information contained in the sequence t to S in terms of sequences that S already understands and can think and talk about with confidence. Eventually, T finds the sequence s that conveys to S the same information about the world that the sequence t corresponds to in the dictionary of T . If s represents some important insight, S will use it and not forget. This experience paves the way for further steps in building S's dictionary with T 's help. T can always arrange a dialogue around issues that are comprehensible to S, and the dialog helps T identify issues that still confuse S.
In the case of a group of students, a discussion is led by T in order to identify a set of issues that all members of the group find unclear. In terms of the model, this discussion allows T to approximately understand orientation of the students' frames of reference and co-ordinates of entries in their dictionaries. This understanding allows T to help the students in beginning a coherent study of the identified issues. Thanks to the meaningful communication, T is welcomed by students to play the role of an advisor and as such can transparently influence the course of study so that it may reach its stated goals. Teaching of this kind usually leads students to discoveries of new aspects of the subject matter. Students may subsequently consider the new aspects important and learn more about them. Students may also realize the value of the process of clear communication about the subject matter, which T enables them to practice.
D.
Training in effective teaching
Teachers can incorporate effective teaching into their work with students if they are trained to do so. Both pre-service and in-service teachers need training. The training must itself be an example of effective teaching so that the trainees can experience the process first from the standpoint of students. The resulting appreciation of the great value of effective teaching in comparison with ineffective teaching, the value explained here in physical terms using the MOT, motivates teachers to incorporate effective teaching into their work with students.
McDermott and the Physics Education Group at the University of Washington have developed a program for training science teachers [5,42,43]. The model concept of effective teaching suggests the possibility of similar elements also being useful in training in other disciplines. Indeed, one can compare effective teaching as defined by the MOT with the teaching of actors that was described by Stanislavski in 1936 [48] (his three-volume textbook has had more than 40 editions since then). Such a comparison may initially appear pointless to a reader focused on physics education. But, when equipped with the model, the reader will find that the same elements are present in both cases, even though they concern different subject matters and different dictionaries of words and concepts. Another example, in which the same elements of teaching can be identified by a trainee equipped with the model, is provided by the program of Clay [49,50,51]. Clay's program concerns teaching young children who have extreme difficulty learning to read and write (it is now in use in thousands of schools).
The point of these comparisons is that the MOT allows trainees to notice common aspects of teaching in all disciplines. This in turn allows them to properly identify the elements that they could otherwise misinterpret (associate exclusively with their specialty) if they thought that such elements occur only in their discipline. MOT implies that teaching in all disciplines requires the establishment of relationship between the frames of reference of T and S before a meaningful communication between them can begin. In all disciplines, T must understand S's initial vocabulary before T becomes able to help S in building a proper dictionary. Thus, MOT predicts that the same basic elements must be present in training of teachers in all disciplines.
A chief example of a universal feature is that training requires time. It must extend over the period that the trainees need to work out hands-on examples which enable them to discover the meanings of relevant words and concepts. The required amount of time can only be decided by research. Training that does not provide the required time cannot accomplish the mission [52]. MOT makes these predictions physically obvious.
For illustration, consider the case of in-service science teachers. The minimal period of training required for acquisition of physical concepts of mass, force, energy, electric current, and magnetic field, can be estimated as not shorter than about two months of study every day of the week [53,54]. At least a year of further study should follow in which trainees attempt to implement effective teaching in their work at school. The implementation will encounter difficulties. For example, one can now predict that it will be impossible for a science trainee to achieve success with students if the time allocated in a school program for teaching the words and concepts of physics is much shorter than the time required by a typical human brain for learning these words and concepts in an effective way. Note, however, that this and other [55] inconsistencies between training in effective teaching and known practices of schooling are not specific to the subject matter and stem instead from the way the educational system is organized. This brings the development of MOT offered here to the key point where it becomes clear that the process of training of teachers cannot be considered complete unless it addresses the issue of success with students and what is meant by the success.
The issue of a trainee's success or failure in implementing what he or she learns in the process of training requires an extended discussion. The discussion begins here and extends into Section IV, where it proceeds to the issues of dynamics of teaching and leads to the concept of productive teaching.
When a teacher steps in front of a class of students and begins to teach, the concept of effective teaching in the teacher's brain is soon confronted with challenging, unavoidable questions: Do the students want to learn what they are taught? What if they don't? Should they be forced? What will the students actually learn by going through an obligatory course if they are not interested in the subject and all they want is a passing grade required for graduation? Will they ever appreciate the subject if they are not interested in it and do not want to learn? How to find out what they really think? What is the teacher to do? The training of teachers is incomplete if it does not address these questions. These questions pertain to teaching in all disciplines and are not specific to any one subject matter.
A trainer of teachers is confronted with similar questions. This is exemplified by replacing the words: teacher, student, and passing grade required for graduation, with the words: trainer, trainee, and career incentive, respectively. The resulting questions lead to a new one: How to design a training program, including selection of candidates to be trained and trainers to train them, so that a large percentage of trainees will afterwards achieve success in working with their students? The question is important because teachers cannot consciously engage in a long-term process of improvement of teaching in schools unless they are successful with students. Such success confirms for them that they know what to do and how to do it, and the clarity of purpose enables them to continue their work every day. Without success with students, they burn out.
That a MOT should address the questions stated above is also suggested by results of a longitudinal study of the same program that was used earlier to estimate the minimal period of training for science teachers [53]. In addition to other findings of interest to educators, the study found that a year after completion of the training in basic physics almost three quarters of the program participants eagerly responded to mail surveys concerning their subsequent experiences in teaching. However, only about one third responded after four years. It was not found why the number dropped so much and the drop could have occurred for multiple reasons. On the other hand, teachers' commitment to enhance the quality of their work can be sustained and reinforced for many years if they achieve success with students. One may not exclude that the drop in the number of responses occurred because the majority of trainees were not sufficiently motivated by the results of their work with students to stay in touch with and provide data to the program that trained them.
Since the concept of effective teaching does not answer the questions of practice with students, it is postulated that the process of training teachers of all kinds and levels misses something essential if it is limited to the concept of effective teaching. A MOT would be most useful if it could help in the identification and inclusion of the missing elements. A heuristic point of view toward this goal is discussed in the next section.
IV. DYNAMICAL PART: PRODUCTIVE TEACHING
Section III describes the process of teaching as the sequence of changes in a student's dictionary that a teacher helps the student make in order to enable the student to interpret events that belong in the subject area, and communicate about them in terms of a language suitable for the subject. Effective teaching corresponds to the well-defined sequence that is described kinematically in Section III. It includes establishment of how the frames of reference of T and S in their spaces of knowledge and events are related to each other, establishing what are the co-ordinates of relevant words, concepts, and events, which are used in communication, and establishing new entries in the relevant dictionaries on the basis of observation, experiment, and exchange of information.
In the context of training of teachers, Section III D shows that the kinematical concept of effective teaching is not sufficient to address the issue of teachers' success in working with students. Students respond to teaching in various ways and a model of teaching must be able to incorporate the dynamics of the teacher-student relationship. This section discusses the dynamics. The discussion addresses the issue of success with students and leads to the concept of productive teaching.
A.
Extension of the dictionary
The dictionary of words and concepts in the space of knowledge (of a student or a teacher) introduced in Section III is limited to the subject matter. For example, in the case of physics, the dictionary includes words such as "velocity" and "acceleration" or concepts such as "mass" and "force." But Section III D shows that the concept of teaching involves words and concepts such as "success with students" or "wanting to learn," and these suggest a host of other related words and concepts that are all absent in the dictionaries for specific subjects such as physics. Therefore, a complete MOT must include an extended dictionary that by definition includes specific dictionaries for all disciplines and the additional words and concepts that pertain to the process of teaching irrespective of the discipline. This extended dictionary will be called the meta-dictionary of teaching. The metadictionary is required to discuss the dynamics of teaching.
The following examples illustrate what kind of words and concepts must be included in the meta-dictionary (again, S means a student and T a teacher).
When T is to teach a subject matter that is important for S to know but S does not understand why he or she needs to know the subject, is not interested in learning, and does not want to do what T tells S to do, it is clear that there exists a barrier in communication between T and S through which T must break. The key step is to explain to S how the subject to be taught is related to S's future (this will be clarified within the model picture in Sections IV D and IV F). The dictionary of the subject itself is of marginal importance in this step because S does not know it yet. T must instead refer to the concepts and words that S already knows, understands, and appreciates regarding S's own future. This is why the meta-dictionary needs to contain the concept of starting from where S is in a broader sense than in effective teaching, where the corresponding concept is limited to starting where S is only with respect to the dictionary of a particular subject matter. The broader concept means that T must assess and utilize a variety of dictionaries that already exist in S's space of knowledge. Unless T starts from where S is in this broader sense and explains why the subject to be taught is important for S, it is likely that S will not be interested in learning the subject.
A primary example of obligatory courses that students may be not interested in is mathematics. The lack of appreciation is reflected in their performance. The percentage of students performing at required levels drops down from grade to grade (from above 60% in 4th grade to below 50% in 12th grade), being lowest at the time of graduation from high school [56]. The drop occurs despite the fact that results of math tests count toward admission to college. Recent publicly-discussed data on students' performance in mathematics in grades 3 to 8 [57,58] reflect the same trend. The percentage of students who score at or above proficiency levels drops from about 80% in third grade to below 60% in eighth grade. If the drop is interpreted as a consequence of students losing interest in what math is about and what it is needed for, one has to conclude that teaching of math is inadequate [59]. The inadequacy is not describable in terms of the language of mathematics alone. One needs an extended dictionary that can be used in discussing the relationship between mathematics and students, students and their future in society, and thus also mathematics and society.
When S thinks that teaching is not adequate, he or she should be able to say so. For example, it may be unclear to S why knowledge of the mathematical concept of integration is required. T must respond to S's concern using the dictionary that already exists in S's space of knowledge. Note, however, that an entirely new concept emerges. S may be afraid to ask about the purpose of lessons on integration due to a fear of consequences if the question is interpreted as unintelligent or violating the school order. Such fear prevents S from learning what it means to speak up on matters that concern S while the way the teaching process is conducted influences the development of S's brain and determines what S knows and understands about the meaning and significance of the taught subject. Thus, the way S is treated in the process of teaching determines S's competence and selfimage as a member of society. The apparently narrow problem with teaching math is ultimately related to the concept of educating a citizen, which includes experience of social control, democracy, freedom, and critical thinking [60,61,62,63,64,65].
These aspects are not specific to any area of study and play an important role in the dynamics of teaching (see next sections). They require a dictionary of words and concepts that are fundamental to the well-being of society. The corresponding meta-dictionary of teaching is much richer than the dictionary-to-be-taught concerning just mathematics, or any other discipline taken out of its relevant context. In the case of mathematics, T may help the class understand not only why integration is taught but also why S's question about the purpose of lessons and the way T answers this question are all important. Subsequently, the obligatory lesson of mathematics and the role of T will appear justified instead of arbitrary from the point of view of S and the class.
Of course, students will not voice doubts about the adequacy or purpose of the process of teaching unless they have a feeling of safety with T . Otherwise, the fear of negative judgment by T and hence uncertainty of the future will silence students. But to gain the trust of students and achieve openness in communication with them so that real issues on their minds have a chance to get resolved, T needs to think in terms far broader than the subject matter alone. The corresponding metadictionary of teaching must contain concepts of safety and trust in communication between people, in addition to the words and concepts limited to the dictionaries of a specific subject and judgment of students' progress in learning the specific subject.
Safety and trust are important for asking questions about the subject matter. Asking a question that discloses confusion may imply that S does not know or does not understand something in a situation where others appear to already know and understand. Such exposure could put S at a disadvantage in an already stressful situ-ation. Without feelings of safety and trust, S may refrain from asking questions.
Teaching does not have to become inadequate and misguided in purpose if T starts from where the students are and takes advantage of their curiosity. Curiosity is a central concept in the meta-dictionary of teaching.
A curious S eagerly confronts a problem to be solved. Finding it difficult, S is glad to receive help from T , and appreciates T 's contribution. In order to contribute to the self-motivated study, T must be interested in and capable of helping S in the learning process. The selfmotivated learning is hard for S because it involves unlearning, a change in brain structure that was in place, and learning anew, creating a new structure. This requires that S overcomes the stress that is associated with the changes and performs the work that is needed to make the changes happen. Curiosity can take S through the difficulties. So, T should sustain S's curiosity, a condition valid irrespective of the subject of study.
B. Inclusion of internal events
Effective teaching concerns a dictionary of a concrete subject matter and uses events observed by S and T as the vehicle that conveys between them the meaning of words and concepts in the dictionary. For example, S and T discuss well-defined events on a laboratory table.
A complete MOT contains the meta-dictionary described in Section IV A. This dictionary extends beyond the subject matter itself and includes "new" words and concepts such as success with students, wanting to learn, starting from where the student is, educating a citizen, safety, trust, and curiosity. One can also talk about attitude, attention, engagement, etc. According to Section III, in order to identify the meanings of these words and concepts, one should associate them with concrete events. However, the required events are definitely not of the kind that happen on a laboratory table.
In the case of new words and concepts, the events that matter happen inside S or inside T (see below), instead of outside of them, as is the case with the events on a laboratory table. Therefore, the space of events that is needed in a complete MOT must include events inside an observer. Such events will be called internal. Their description in the model is provided in a few detailed steps below. The same details will be useful later in the discussion of the concept of productive teaching. More precisely, they will be used in the model to show that the concept of productive teaching is not merely an addition to the concept of effective teaching but underlies the latter as a foundation of the whole process of teaching.
Firstly, consider T teaching S some motor skill, such as riding a bike, catching or throwing a ball, or pressing strings on the fingerboard of a violin; or a mental skill, such as the ability to focus, be patient, contain emotion, or perform a gedanken experiment. The process actively involves S and T as performers of the skill and thus in-volves phenomena that happen inside S or T in ways not reducible to learning a dictionary for description of events outside observers. Namely, S must focus on what happens within the body and mind of S when performing a skill. T judges S's performance from outside but focuses on what happens within the body and mind of S. Moreover, the analysis of S's performance that T makes is based on what T knows is happening in the body and mind of T when performing the same skill. It should be clarified that learning a skill includes a buildup of a dictionary concerning the skill. One cannot fully comprehend concepts and words used in communicating about the skill unless one is able to perform the skill in a way corresponding to the meanings that count.
Secondly, consider the model's definition of the space of events registered in a brain that is provided in Appendix A. The definition makes it clear that the events registered in a brain are determined not only by physical events outside an observer but also by a multitude of processes that happen concurrently in the observer's brain. These processes depend on the neurophysiological history and state of the brain as a central organ of a living person, which includes how the brain functions using the senses, memory, and thinking, both subconsciously and consciously. It is clear that many physical events outside an observer are not registered in the observer's brain because the input they provide is below the threshold for altering the concurrent activity in the brain.
Thirdly, while registration of events is associated with changes in an observer's brain, a change in the functioning of the observer's brain is not necessarily expressed in the observer's overt behavior. If it is not overtly and clearly expressed, the change cannot be registered and unambiguously interpreted by another observer [66].
In essence, using the model of Appendix A, one can say that events registered in a brain can be approximately divided into three classes according to their dominant coordinates. Some events have dominant coordinates associated with what is happening outside of an observer. These will be called external events. Other events have dominant coordinates associated with what is happening inside of the observer. These will be called internal events. The third class contains events that have their dominant coordinates associated both with what happens inside and outside of the observer. For brevity, these will be called engaging events.
In terms of the model nomenclature, one can say that the concept of effective teaching involves external events, such as an event on a laboratory table; engaging events, such as those that cause changes in S's dictionary concerning what happens on a laboratory table; and internal events, such as thinking about an event on a laboratory table, experiencing confusion regarding the subject, or formulating questions about it. The concept of effective teaching ignores and obscures all internal events that are not directly related to the improvement of S's dictionary for the taught subject.
The ignored internal events include, for example, wan-dering of thoughts away from the taught subject toward issues of main interest to S, or S's plain feeling of aversion toward learning the subject. The possibility that S sees the subject as having no relevance to S's way of life and future is also overlooked. Phenomena such as boredom, a sense of violation by enforcement of classroom lessons or homework, feeling insecure in the presence of T , or not trusting T enough to share important thoughts or information with T , are not included in the model picture of effective teaching. All internal events that contribute to (positive) attitudes and manifest themselves in S's attention or engagement, are considered given, or stimulated and reinforced by effective teaching, although no reason is identified in the kinematical part of the model for the necessity of their occurrence. Internal events that contribute to the phenomenon of curiosity are not explicitly considered. Instead, curiosity for the subject matter is assumed to be always in place, as it by definition is assumed to be in place in the case of T , who is also assumed to be interested in teaching the subject to the students. In summary, the kinematical concept of effective teaching in the model does not explicitly include a host of feelings, thoughts, attitudes, wants, and expectations that are important for the course and outcome of the process of teaching.
The above examples make it clear that the concept of effective teaching described in Section III omits the internal events that provide meaning to the meta-dictionary described in Section III. Of course, ignoring the internal events does not change the fact that the behavior of S hinges on them and the process of teaching cannot be successful if it is incompatible with the internal events. A realistic MOT must account for their significance.
C. Communication about internal events
The issues that require the meta-dictionary of Section IV A, cannot be discussed between T and S without communication about internal events. But the communication cannot proceed according to the scheme of effective teaching described in Section III B.
In effective teaching, the dictionary of words and concepts concerning a subject of study is built, used, and changed with the help of a process of communication based on a set of well-defined events; both T and S observe and discuss these events as two different observers who use two different frames of reference to describe the same phenomena. Such a set of well-defined events does not exist for communication between observers about events that happen inside one of them. The difficulty is that, by definition, only one observer has access to what happens in an internal event.
Since the standard methodology adopted in effective teaching does not apply, the question arises how T and S may proceed. The model asserts that the situation is not hopeless because T and S may rely on their own internal events in concrete situations and each can try to find similarities in what the other says regarding these examples.
The model's description of what may happen in a dialog between T and S when they attempt to establish orientation of their frames of reference with respect to each other, including internal events [67], is provided below in two parts. The first part describes only what may happen and does not address the issue of why the dialog may proceed in the described way. The second part, which contains a heuristic point of view toward the origin of forces that can keep the dialog on track, is provided in the next subsection.
Suppose the observer of an internal event is S (an event happens inside S). T can infer what happens but only from the overt, nonverbal behavior of S, and from the verbal information provided by S in terms of S's dictionary. In order to decode such information, T must hypothesize about what happens inside S and about the meaning of the words S uses. T must conduct the dialog using and expressing the hypotheses so that the dialog can lead to improvement of the hypotheses, or to making new ones. However, it is up to S whether the required dialog with S will proceed.
In particular, the dialog's prospects depend on how S perceives T . S will judge how T comes across according to the rules chosen by S, not by T . The judgment will include the utility of the relationship between S and T for the purpose of dealing with the internal events in S by S (see next subsection). No quick fix or superficial verbal assurance can change the judgment that S builds over time. Instead, overt behavior of T in response to overt behavior of S is registered in the brain of S and this extended process contributes to S's judgment of T .
If S judges T as helpful, the dialog will proceed because S expects a benefit. If T is judged as not helpful, S will not be motivated to cooperate. These regularities in behavior are observed in countless examples [13].
The key to S's willingness to cooperate with T is that T comes across as somebody who wants and is able to help S unfold the potential that S believes himself or herself to have; potential that is not reducible to any subject matter or dictionary. Inclusion of this regularity in communication between T and S about internal events is a challenge for every MOT. It will be addressed in the present MOT in the next sections (including relevant references). The goal is to capture in the MOT that T 's authentic respect for the potential of S to develop as a learning human being and the interest of T in S having a chance to realize his or her true potential are both seen and appreciated by S in the process of building up their interpersonal relationship. Eo ipso the cooperation happens voluntarily on both their parts. Only a conscious process supported from both sides allows T and S to build the required meta-dictionary. This mutually supported process is identified in the model as the way around the basic difficulty of an absence of well-defined events: wanting allows the participants to continue despite misunderstandings. This is also why the signifi-cance of safety and trust was stressed in Section IV A, even though every normal human being is able to identify the significance of safety and trust in their experience.
Since the task of establishing meaningful communication between two brains is at the discretion of the brains that are unique as individuals, and since this task depends on the interpersonal relationship between them and the context of their communication that are not fully predictable, it cannot be carried out in steps enforced by a predetermined curriculum according to a rigid schedule. The magnitude of complexity involved in communication between two brains about internal events boggles the mind and requires study. In fact, science is yet to discover a precise methodology [68] to deal with this complexity because the standard procedure based on welldefined events is not available. The fact that the present MOT eventually produces such clear conclusion in disagreement with most common educational practices, is an example of the power of models that is needed in theory of teaching according to Section II.
This means that teaching involves an art of communicating despite ambiguities that cannot be systematically eliminated by any simple procedure. Performance of this art requires extensive preparation and continuous training such as performance in other disciplines [69]. The issue of required teacher selection and training is taken up in Section IV F.
Even if the buildup of the meta-dictionary depends on the internal events, which depend in turn on the interpersonal relationship between T and S and complex contexts of their interactions, all of these elements being unpredictable to such an extent that teaching cannot be reduced to simple procedures, the model must now provide a dynamical idea for what drives the process of teaching in the right direction. The next subsection incorporates the required concepts of "dynamics" and "right direction" into the model.
D.
A heuristic view toward the origin of forces
Challenges for a MOT stated in the previous sections must be approached with the realization that the advanced processes that go on in the brain cannot be ultimately explained today in terms of their physics, strictly speaking, because they are too complex and too little is known and understood about them. Approaches of a far less-precise nature than physics, such as Freud's attempts [70] to understand psychology and Hebb's idea [71] of explaining the same in terms of neurophysiology, are still far away from reaching conclusion [72]. Moreover, all these approaches concern just one brain. The process of a teacher teaching students is of a much finer kind because it involves at least two brains in interaction in a complex environment and it causes complex changes in them in complex ways.
But the current lack of fundamental understanding of the forces relevant to the MOT does not imply that such forces do not exist, that one may succeed in teaching by acting against them, or that one has a chance to seriously tackle problems of education being ignorant of their role and strength. The situation is like one with the force of gravity that causes stones to roll only down a slope irrespective of whether one understands the law of gravity or not.
This section employs the model picture of teaching developed in the previous sections to describe a heuristic point of view toward the origin of the relevant forces. The model view suggests that the same basic forces always determine the course and outcome of the process of teaching and that attempts to teach in ways that act against these forces cannot lead to success. As a result of recognizing the depth of the relevant forces' origin, and thus also their strength and significance, the next subsection will introduce the concept of productive teaching.
To begin with, the concept of "force" as a dynamical agent in the process of teaching should be distinguished from other known concepts that are associated with the word "force" in teaching. For example, there exist forces that act in the daily practice of teaching and do not directly refer to the buildup of dictionaries through interactions between brains. This concept of forces includes systemic rules mandated by law or economic and social pressures that act on teachers, students, and other stake-holders in the system. Such forces qualify as constraints or influences on what happens in schools (and other places where teaching occurs) rather than causes of the process of teaching that the model is about.
Another important concept is that the force of teaching is the teacher. In a trivial way, this may be interpreted to mean that the teacher is the person who forces students to learn, no matter how it is done. An opposite meaning associated with the concept of teacher as a force of teaching is that the teacher is the identifier, facilitator, creator, and guardian of contexts in which students learn [73,74]. A vast difference between these two interpretations demonstrates the magnitude of ambiguity that needs to be eliminated. A more subtle ambiguity is contained in popular statements that students must realize that "only they can learn" and that "it depends on them how much they learn," which seem to imply that the teacher plays only a secondary role. On the other hand, it is also said that teachers need to "create a desire to learn" in students. The latter statement can be confronted with the fact that most children are eager to learn. Evidently, there is a need for clarification of the concept of forces in the process of teaching.
Consider first a few examples which show that the action of forces relevant in teaching manifests itself in recognizable psychological phenomena. These phenomena may lead to unproductive outcomes of teaching in major ways even though the subject has been learned by S sufficiently well to pass some test. Consider that S has questions concerning his or her self-image in the context of learning a dictionary. For example, if S feels judged as "not gifted enough to learn physics" and never talks about it with T , S's entire life may later be tainted by efforts to overcome the feeling of inferiority. The other highly unproductive outcome would be that S resigns to the feeling of inferiority and never becomes a fully developed person because of the internal feeling of failure. An opposite case of concern is an overconfident S who will have to deal with consequences of serious errors resulting from the illusion of competence. A common psychological effect of school or college is that alumni avoid contact with subjects that were forced upon them during their academic training. This effect negatively influences the perception of arts and sciences in society.
To be more specific, it is known that test anxiety in students has major implications for their learning and for measurements of what they learn [75,76,77]. In an environment that stimulates test anxiety, many individuals cannot learn effectively and their actual knowledge cannot be reliably assessed. Of course, the anxiety is not invented by testers: some natural inner forces lead to anxiety in some conditions. Although the forces that cause anxiety are not precisely known, it is known that anxiety has devastating consequences for teaching. If, instead of being ignored and blindly caused to backfire, the same underlying dynamics were better understood and taken advantage of in the process of teaching [78,79], it would be possible to create and maintain conditions that do not stimulate the destructive anxiety in students who are prone to it, and do not induce a whole spectrum of unproductive consequences of such anxiety in their future behavior. One could employ the dynamics to instead stimulate the same students' curiosity about the world and useful dictionaries, with an entirely different spectrum of consequences in their future behavior.
In order to locate forces in a potentially complete physical MOT (even if the forces are not fully understood), one needs to delineate a plausible way in which known physical processes might in principle be responsible for the existence of these forces. Once room is made for these processes in the model structure, and thus the basic (i.e., following from physical laws) origin of the dynamics of teaching is incorporated (even if not directly because of the absence of required details), a model definition of the concept of productive teaching will be offered.
In the model, T is considered an observer who communicates with S as another observer of the same world. There is no postulate in the model of any fundamental difference between observers. However, T is a more experienced observer than S and has two special features. One feature is that T wants to possess expertise about the world's mechanisms and share it with S. Another feature is that T knows what to do in order to share this expertise with S. Among the mechanisms of the world, T understands and appreciates the role of personal growth in human life. On the other hand, S needs and wants T 's help in learning about the world because S is convinced that T possesses the expertise S wants to gain and S sees that T wants and knows how to share it with S.
A heuristic point of view toward the physical origin of these features of T and S involves several observations put together. These observations concern an individual and a large number of generations of many individuals.
There is nothing new about these observations except for their inclusion into the description of a physical MOT. The purpose is to make the heuristic point of view as concrete as possible within as small an amount of text as possible. From a vast relevant literature, only a few references are selected that seem to most accurately put the observations in the model's context. The first observation is that humans learn eagerly from the moment of birth. Children are curious, explore, and ask questions of adults. That adults are motivated to explore is seen in their tendency to travel, especially after they retire and are free to make choices about how they spend time. Apparently, humans need to explore to feel well because this is the only way available to them for discovering regularities of the environment in which they live. They need to know these regularities to successfully operate in the environment. Similarly, all anomalies (news) catch their attention because humans need to figure out the new elements in order to know how to deal with them. A brain must feel pleasure when exploring, uncovering regularities, and explaining surprises, in order to develop strategies for survival and apply them in an ever-changing and unpredictable environment. As suggested already by Hebb [80], the feeling of pleasure coming from exploration, observing regularities, and solving puzzles, can be associated with "directed growth or development in cerebral organization."
The second observation is that a human being is the product of evolution [21]. This means that the "growth or development in cerebral organization" in a human brain and the associated feeling of pleasure are a result of a chain of physical events that took many millions of years. A huge amount of information accumulated in this chain of events is encoded in the capabilities of the genetic material of individuals [81]. In the course of the life of a single individual, the individual's genetic code is expressed in varying conditions in different ways to various degrees [82,83]. This process results in the learning brain. Interactions between different brains are critical to the continuation of the chain because they are involved in the transfer and survival of the genetic material. Ultimately, the formation of genetic codes and their expression in processes that produce functioning human brains happen according to the dynamical laws of physics [84]. But there is no simple path that connects the basic laws of physics to a single living human brain. Instead, a human brain is built and functions in the ways that result from and are informed by an unimaginably large number of intermediate structural and functional steps that involve simpler structures involved in simpler functions. As a result, many layers of complexity are superimposed in the brain dynamics [85,86]. An ultimate understanding of the brain is a fascinating prospect and one of, if not the greatest, challenges to science.
The second observation means that the forces which determine how brains function and interact result from and are tested by such a long "battle of life" [21] that they will not yield to any artificial concept of teaching. Instead, teaching must be adjusted to them. For example, there is no point in artificially suppressing the role of feelings because of the assumption that they only "jumble" rigorous thinking, or in artificially suppressing rigorous thinking as "secondary" to the essence of humanity [87]. In fact, feelings and thinking are coupled into an inseparable whole. One cannot argue that logic is more important than emotion or that emotion is more important than logic. The only form of the human brain that passes the test of evolution involves both. Attempts at artificial separation will encounter resistance since only a whole brain can function as a learner and a teacher. The third observation is that teaching (such as parents' teaching of children, or teaching of less experienced group members by more experienced members) helps individuals develop values in behavior and knowledge, including the development of language [21]. This is why individuals can grow far beyond what any one of them could ever accomplish from scratch or separately. The ultimate recursion is that the increasing capability for teaching using increasingly advanced language allows us to describe the world we live in to new generations with increasing precision. The new generations are thus equipped to achieve greater scope, accuracy of knowledge and understanding than previous ones, achieving greater capability to take action.
In the context of dynamics of the model of teaching developed here, these three observations are interpreted as inescapably suggesting that the origin of internal events which manifest themselves as wanting to learn, wanting to teach, and wanting to take action, is of the same depth as the origin of man. Therefore, it is postulated that the dominant internal events in the process of teaching are related to who S and T are and may become as human beings. Each has a strategy for this purpose. The evolutionary perspective makes it obvious that disclosure of such a strategy to a stranger is associated in a brain with danger. On the other hand, the disclosure may lead to success thanks to cooperation. Both S and T judge and choose the depth of communication that suits them.
In the process of teaching, when S and T encounter difficulties in communication due to ambiguities in their respective dictionaries and when they need to share information about internal events in order to explain what happens, they operate near the limits of communication that they set previously as useful and safe. Pushing these limits, with some necessary elements of curiosity, pleasure, and risk [88], is what breaks new ground in the process of teaching according to the present model. The concepts of a "right direction" and "dynamics" of this process can now be included in the model.
The process of teaching is essentially "blind" [89] in the same sense, in which the underlying laws of physics and the resulting process of evolution are blind, and it is driven by the same forces that drive evolution. But since it is the highest existing form of transfer of information from generation to generation (actually, from a brain to a brain) and as such includes the transfer of values that allow humans to operate at the level we have currently reached, many characteristics of teaching appear to have a purpose. This is viewed as a misconception analogous to suggesting that cows have milk so that people may milk them. Instead, the model postulates that the concept of "right direction" can be associated with the increase of options that individuals in principle have for choosing their own steps into the future as their understanding of the world increases. (Examples from modern history that illustrate the complexity of the phenomena that contribute to this process can be found in Refs. [90,91,92].) But "in principle" does not imply that every individual is equally aware of the possibilities and limitations, has equal access to the resources and equally contributes to their replenishment, can benefit from the development in the same way as others and provides others with equal services, etc. The concept of "right direction" that T may adopt in teaching S is defined in the model by saying that T chooses steps that lead S to greater awareness of possibilities and limitations, more efficient access to resources and participation in their replenishment, wiser consumption of benefits and more competent service to others, etc. Most succinctly, T and S share an understanding of their world and values.
The concept of "dynamics" which causes T and S to make concrete choices in interacting with each other is reduced in the model to the statement that the phenomenon of teaching is shaped through evolution. It takes forms corresponding to the qualities and knowledge of its participants and how they communicate. Most succinctly, T and S cooperate with each other in agreement with their needs and understanding of the world and values.
As a result of gathering enough experience and insight, T becomes consciously aware of the process of creation and transfer of knowledge and values, and realizes that this process includes T 's own communication with students. From that moment on, T can consciously work on better understanding how to communicate with students and transfer to them foundations of the meaning of what T has comprehended, so that the values and knowledge the students learn in the process help them move into their futures.
Self-consistency of the model dynamics requires that the process of buildup and transfer of the meta-dictionary of teaching is perpetual. This means that the transfer happens in agreement with the existing values and knowledge and is carried on and evolves as these imply, from generation to generation. So, through participation in the process, T 's students become familiar with the mission T serves. Appreciation of T 's work by S signifies that T 's mission is accomplished in the case of S.
That the model dynamics may resemble what currently happens only in a minority of schools (some examples are described in Refs. [94,95,96]) is a separate issue of great practical significance. An in-depth discussion of this issue is beyond the scope of this article. But if the process of teaching in every school and the model picture of teaching were similar today, it would mean that we have already entered the era of consciously taking advantage of the mechanisms of our origin in designing conditions of our further development. In fact, as of yet we have not.
E.
Productive teaching
Teaching may be called productive when it includes all the elements of effective teaching found in Section III C, uses the meta-dictionary of teaching to solve the problems presented in Section III D (as described in Sections IV A-C), and proceeds as a result of action of the forces discussed in Section IV D.
In other words, the process of productive teaching is not blindly enforced by T but stems and branches from the will of S with the help of T . T helps S understand what is relevant and motivates S to participate in making the teaching process happen in terms of engaging events. In particular, S feels free to exchange information about internal events with T (and other members of the group when more people are involved). This freedom results from and contributes to the buildup of an interpersonal relationship between S and T and becomes palpable along the establishment of the meta-dictionary that they use to solve problems they encounter.
While effective teaching discussed in Section III is a concept that may be tied to some subject of study (and measured solely in terms of knowledge and understanding of the specific subject), productive teaching produces a person who finds pleasure in learning and wants and knows how to learn more, where the word "more" includes future subjects that are not known and not predictable by T during the process of teaching. No matter what subjects are covered in the process of productive teaching, S learns to use the inner forces of learning that manifest themselves in S's curiosity, wanting to learn, and joy, even if it gets hard to make progress. The open state of mind of S that results from productive teaching is not measurable in terms of knowledge or understanding of the subject matter that S is taught. One may say that productive teaching concerns teaching a person, not a subject matter. A person thus educated knows the forces that drive learning and how to use them.
In particular, the new insights, and observation of many instances of T 's helpful behavior in the interpersonal relationship with S (and with others as witnessed by S), allow S to realize that the approach of productive teaching, adopted by T , is driven by the respect that T has for S as another human being in need of learning. This instructive interpersonal relationship, created by T and perceived by S as a fruitful one, shows S the necessity and value of T 's work [93].
Appreciation of T 's work by S is what completes the definition of productive teaching according to the model. The outcome of productive teaching is not only the mastery of subjects and skills but also appreciation by S of the value of understanding among people, how it happens and bears fruits. Through this appreciation and understanding, productive teaching prepares S for further learning and making choices for action. The concept of productive teaching is closely related to the concept of context of productive learning that was introduced by Sarason [73,74] but so far has not been discussed in terms of a model. Using the model, one can now say that productive teaching proceeds through employment of contexts of productive learning. The insight provided by the physical MOT is threefold: 1) effective communication between T and S requires comprehension of the relationship between the frames of reference in their spaces of knowledge and events, which implies that T and S have to start from finding out what this relationship is and learning how to use it, 2) the internal events determine the course and outcome of the teaching process and T and S need to communicate about them in order to keep the teaching process on track, and 3) teaching becomes productive when it is driven by the forces that characterize T and S as human beings in pursuit of their goals. One can now also say that the context of productive learning is a context in which issues of importance to S are the center stage and subject of effective communication with T so that the three elements identified in the model are present and S can learn with the help of T . This means that teaching begins with and proceeds in concrete, real-life contexts that S and T identify together and in which T facilitates S's learning, creating opportunities for S to discover new concepts and guarding S's process of learning from derailment. In the process of teaching, T also learns: constantly studying how students learn, building a model of how to teach, and seeking validation for the model, aiming at improvement.
Documented examples that illustrate the concept of a context of productive learning, including its long-term consequences [94,95,96], also help illustrate in practical terms what kind of teaching is described by the physical MOT offered here. In particular, they indicate what may happen in the dialog between S and T (see Refs. [97,98]). Examples that illustrate the same concepts and can be seen in popular cinema, are described in Refs. [13,99]. An example from a program of teaching science [53] based on Ref. [42], can be found in Ref. [100].
A few examples given below further illustrate what typically happens in the context of productive learning and should occur in practice of productive teaching according to the model. S thinks it is best to say, "I do not know" when he or she does not know how to find an answer to a question or solve a problem. S is confident that this is the best way to respond when missing a point because S knows that the shortest path to learning is to take advantage of communication with T and other people in the process of filling gaps and seeking connections.
Learning is impeded by hiding gaps in knowledge and pretending to understand. Similarly, if a personal image and opinion are more important than actual learning, disclosure of shortcomings in knowledge or understanding is out of the picture.
In the process of productive teaching, S has no reason to be afraid to say, "I do not know," because S knows that T understands what such a statement means, listens to S carefully, and helps. So, S does not hesitate to say, "Please explain this and that because I do not understand" when he or she is confused, or, "I do not remember" when memory fails, or, "Why do you say such and such?" if what T says does not sound clear to S, etc.
In order to appreciate how difficult it is to create and sustain contexts of productive learning (especially by the unprepared for the unprepared), these apparently simple examples can be compared not only with what happens today in many classrooms, but also with how people communicate in other contexts. Consider the contexts of discussion between a teacher and a principal, a principal and a superintendent, an employee and a boss, or a citizen and a government official. In these and other contexts, the subject matter can suddenly become secondary in importance to the issue of dominance of one person's position in the system. At this point, communication about the subject matter breaks down and there is no room for communication about internal events. Instead, the only course of action for the person in a weaker position in the system is to adjust to the decision of the person in a stronger position in the system. In different circumstances, the hierarchy of their positions may change and a reaction based on memory takes place.
In the process of productive teaching, there is no need for blind measures of discipline and judgment. Instead, the values transferred in the process bond S to the idea of participation. Testing of S by T is essentially replaced by gathering of information by T about what S does while learning. This information is the basis for advice that T gives to S so that S can improve the process and reach the intended results [101] (see Section IV F).
In summary, one may say that productive teaching differs from effective teaching by the dynamical condition that the relevant sequence of changes in S's dictionary occurs as a result of the action of the natural forces of learning in S. Therefore, S learns what the word "learn" means and becomes a learner for the rest of his or her life.
Learning in agreement with the natural forces that S is equipped with as a human being, allows S to discover the meaning of words and concepts such as human rights, law, democracy, achievement, and other entries of fundamental value in S's dictionary. For example, since the rules of communication between T and S are not decided solely by T , the issues of motivation and power in the process of teaching get discussed between S and T . Decisions are made in ways that do not subject S to the will of T . Instead, S is engaged in making decisions in the contexts familiar to S. If T must decide because S does not have the required knowledge, T must also explain to S why T is responsible for making the decision, and why T considers a particular decision to be the right one.
The model described here helps in identifying several meanings of the word "productive" in connection with teaching. For example, many correct entries are produced in the multiple dictionaries concerning subject matters that belong to S's space of knowledge. The large number may be achieved because the process of teaching takes advantage of the forces that drive learning and they accelerate it to large speeds. Since the entries are correct and understood, they are useful in thinking about new problems and become stable (are not forgotten, as useless information typically is). When teaching takes advantage of self-motivated learning, it produces entries in S's dictionary that are basic to the well-being of S as an individual, such as inquiry, discovery, and study. By learning how to use inner forces, a person may become creative. A rich meta-dictionary is produced in S for communication about internal events. This dictionary contains entries for the values that are fundamental to S's future as a member of a society, indispensable for S becoming a conscious citizen. The teaching produces a person who carries these concepts and values on in relation to other people. The overarching meaning is that the teaching is productive when it contributes to the student's future as a fully developing person. This short statement is itself a sequence of words whose meaning cannot be grasped without a proper dictionary.
F. Training in productive teaching
Training in productive teaching is more difficult and time consuming than training in effective teaching (see Section III D) because productive teaching requires the buildup of many words and concepts in the metadictionary of teaching on top of a dictionary of specific discipline(s). However, in a system in which basic competence of trainees in a subject matter is itself obtained as a result of productive teaching, the difficulty and time consumption can be predicted considerably smaller than in a system in which a basic familiarity with the same subject matter is obtained by the trainees without the first-hand experience of productive teaching (first as students, and only then as teachers). Let the productive system be called P , and the questionable one Q.
In the case of P , teachers are trained in the subject matter in ways that create entries in the meta-dictionary in parallel to the creation of entries in the dictionary for a discipline. In the case of Q, the dictionary for a discipline remains uncorrelated with the meta-dictionary. Moreover, the dictionary for a discipline is plagued with wrong entries, such as the illusion that science results from pure logic, the false impression that art results only from emotion, the mistaken belief that interpersonal relationships are irrelevant to the development of science, and the invalid assumption that the context of discov-eries and creations is not important for how they occur. Q compounds the difficulties of training in productive teaching because the trainees first have to unlearn what was inculcated in them during their training in the subject matter. Then they can learn anew. But it is much harder to change an existing structure in a brain than to build a new one from scratch. Unfortunately, the Q-like systems greatly outnumber P -like systems (see below).
In a system of the dominant type (Q), almost all practical suggestions for training in productive teaching originate one way or another in a small number of examples that have already been mentioned (some only through references) during construction of the model in previous sections. A few new sources are included in this section as particularly relevant to the training of teachers. One can say that this section illustrates that a model, which organizes thinking about teaching, leads also to a selection of suggestions for training.
The model principle of beginning from establishment of relationship between frames of reference and starting where the trainees are implies that the events suitable for beginning a buildup of the meta-dictionary of teaching, irrespective of the discipline, should involve a subject matter that is familiar to the trainees. The more familiar the subject matter the easier it is for a trainee to focus on the issues of teaching rather than on the issues of the subject matter [102]. For example, consider a physicist to be trained in productive teaching (a professional in another discipline could just as easily be considered instead of a physicist). Let the subject matter be reading. Instead of reading Shakespeare or Goethe, however, consider the case of teaching young children who have difficulty learning to read. Imagine a visit by the physicist (as a trainee in productive teaching) to a training session for teachers of reading in a program called Reading Recovery [51,103]. The physicist witnesses how the trainees observe what happens between a child learning to read and a veteran Reading Recovery teacher who helps the child overcome the difficulties the child encounters. The required setup includes a sound-proof room with a microphone and a one-way mirror so that the trainees (and the physicist) can hear from a loudspeaker and observe through the mirror what happens in the room between the child and teacher without being seen or heard by the child.
Since the physicist and all trainees already know and understand what it means to read, they are able to focus on what the teacher does in response to the difficulties that the child encounters. The whole group observes how the teacher works with the child and they discuss what they see, among themselves and with an instructor who observes the reading room with them and explains what happens when the trainees have questions or confusions arise in their discussion. Using the model, one can say that the observed events allow the trainees to build entries in their meta-dictionaries of teaching without unnecessarily focusing their attention on the subject matter, which would impede the process of building the right entries.
The physicist sees that the teacher behind the mirror is focused on finding out what blocks the child in reading a story composed of a few simple words. The teacher discusses with the child the meaning of the individual words and the whole story, illustrated with a picture, and helps the child understand how to self-correct mistakes the child makes [104]. It becomes clear to the physicist that what the teacher does is not focused on some abstract concepts of reading but on discovering how to help a unique child solve a concrete problem. Once the child finds a solution, the teacher uses it as an example and helps the child understand how to seek solutions to similar problems with concrete letters, words, or groups of words, explaining through practice the role of comprehension of the text and the methods of self-correction. This is a complex interpersonal process, immersed in a context in which the child wants to learn to read.
The physicist also observes the behavior of the teacher trainees and their instructor as they follow the lesson through the mirror and loudspeaker. It becomes clear to the physicist that the instructor and trainees do not talk about meanings of the read words or sentences. These are understood well and do not require special attention. They talk about concepts that matter in helping the child succeed in becoming a learner. This is a very complex issue, incomparably more complex than the simple sentence the child tries to read. The other clear lesson is that in order to grasp the meaning of words and concepts in the meta-dictionary of teaching one must observe events such as those behind and in front of the one-way mirror.
The next suggestion about training in productive teaching concerns reading documents that illustrate what productive teaching can accomplish. Consider the experiment carried out by Schaefer-Simmern in the nineteen forties [105]. He demonstrated in a number of cases ranging from mental defectives to people in business and the professions that one can teach a person to see art as experience [106]. His students unfolded their creativity in visual art. Schaefer-Simmern concluded from his experiment with mental defectives that [107]: the experiment "seems to confirm the fact that creative activity in the visual arts can be unfolded and developed in mentally defective persons to a degree analogous to that of their mental potentialities. Real difficulties appeared usually only with individuals with higher IQ's who had previously received art instruction based on copying. With such a background, feeble-minded persons cling to a technique and slavish imitation which are in no way related to their stage of visual comprehension. Nevertheless, the imitative, memorized picture seems to give them a certain security. Any attempt to lead such persons back to their own stage of visual conceiving is usually resented vigorously because of their anxiety over losing that sterile mental possession and being thrown back into a state of uncertainty." The above is only one of many lessons from Schaefer-Simmern's experiments. Dewey stressed in his foreword to Ref. [105] that Schaefer-Simmern's ex-periments provided an "effective demonstration of what is sound and alive in theoretical philosophies of art and of education." Trainees in productive teaching may be asked if they see parallels between Schaefer-Simmern's experiments and what they think is possible to achieve in their disciplines.
Other examples of literature that reports on the practice of productive teaching in schools are Refs. [94,95] (see Section IV E). Trainees in productive teaching can observe exemplary ongoing programs in action and learn from individuals working in these programs. Experienced individuals can play the role of mentors to the trainees. The trainees may learn from their mentors how the latter achieved understanding of productive teaching with the help of their own mentors of a previous generation. What were the breakthrough events that helped the mentors get on their paths to productive teaching? How long did it take and how did it happen that as a result of these events they learned how to create and sustain contexts of productive learning for their students? Such contexts are certainly not limited to the classroom in the cases described in Refs. [94,95]. A challenge to training programs is identifying people who can be mentors, and the schools that are suitable for study and accessible to the trainees [108]. Currently, the concept of apprenticeship in productive teaching as a common path to professionalism in education can only be considered a concept of the future [109]. What the physical MOT provides here is a logical structure which explains why the above examples of recommendations for training are not merely items in another trade book in education but a necessity for building a culture of a meaningful communication between teachers and students as learning observers of the world.
In the current circumstances, one of the primary objectives of training is that the trainees learn the difference between the classroom regularities in systems of type Q and P . In P , students may study the neighborhood of the school and spend a lot of their time outside the school building, and in Q they may almost always stay in the building. But the trainees need to know and understand the concept of regularities in the process of teaching. Some regularities will agree with the MOT, and some will not.
An example of a study of classroom regularities that qualifies for incorporation in a training program in productive teaching is Susskind's study of question asking [110,111]. Susskind found that teachers often think that they ask approximately the same number of questions as students do and that they expect that only a slightly larger number of students' questions would be better. In fact, teachers ask about 25 times more questions per period than all students in a classroom. They receive less than one fifth the rate of students' questions that they estimate as occurring and as desirable. The bottom line is that on average a student asks one question per month in all his or her social studies classes combined (assuming four 45-minute social studies lessons per week and realistic estimates of time available for asking questions), whereas teachers ask about one question per minute. The main reasons for students to not ask questions are identified by Susskind to be yelling by teachers and laughing by peers. Students appear to assume and adjust to a rule that only teachers are to ask questions, while the duty of students is to provide answers.
Susskind reported on rates of asking low-level questions (related to memorization, i.e., of the type: Who, what, where?) and high-level questions (such as concerning causes of wars) by teachers. He observed that these two rates are about the same on average. But the students of teachers who ask less low-level and more highlevel questions ask more questions than the students of teachers who ask more low-level and less high-level questions. Susskind's study also includes a category of competitive questions that have a particularly negative impact on students' interest in asking about anything.
In addition, Susskind showed that a measurable change in the classroom frequencies of question-asking can occur as a result of the following sequence of events (see Susskind's work for important procedural details, such as anonymity of records): 1) teachers make predictions of the numbers of questions that will be found in their classrooms, 2) researchers measure what happens and demonstrate the results to the teachers, 3) the results are discussed at a few seminar meetings, at which the researchers and teachers exchange ideas about the origin of differences between expectations and facts. Such discussions stimulate teachers to try different approaches and improve the situation. After the seminars, teachers are observed again. It is found that on average the number and types of questions teachers ask before and after the sequence of events 1, 2, 3 do not differ much, although some differences are discernible (for example, the average number of questions asked by teachers dropped by about 10%, while the average percentage of high-level questions increased from about 50 by about 2 and the average percentage of competitive questions decreased from about 8 by about 2, with all these changes being comparable with the magnitudes of corresponding standard deviations but correlated in ways that nevertheless allowed Susskind to draw his conclusions). On the other hand, the observed change in the behavior of students of the teachers who went through the sequence 1, 2, 3, is dramatic: the rate of their students' question-asking doubles. Susskind's work warns readers that there may be many reasons for why the seminars could help teachers to change their classroom behaviors and thus also change the students' behavior (see also two paragraphs below). The MOT suggests that when the human brain sees a conflict between an assumption or prediction it makes, especially about itself, and actual results of observation and experiment, it attempts to make changes in its space of knowledge in order to become better prepared to deal with new events (see Sections III A and IV D). It tries out new ideas. This is precisely what Susskind says the teachers did as a result of participation in the seminars.
Although no simple explanation is offered by Susskind, the correlations he observed in the study (see the original articles) still allow him to make a few recommendations.
Susskind's recommendations for teachers are: to reduce the number of questions teachers ask, to ask questions of greater complexity, and to write the best questions asked by students on the blackboard. Furthermore, if teachers ask questions about students' personal experiences, students feel encouraged to ask questions concerning issues that truly interest them.
Susskind also recommends videotaping of classrooms as a means of creating a reliable source of information for teachers about how they and their students behave (the same recommendation is made by many authors [112]). Unfortunately, studies of classroom regularities are rare (on the order of 10 per 50 years before Susskind's [110], cf. [109]). Today's technology provides many means for recording (including self-recording) and discussing classroom regularities by the trainers and trainees that are significantly better than the ones that existed at the time of Susskind's studies. The virtue of a film record is that a trainee can see how his or her action appeared from outside while the trainee also knows what internal events in the trainee accompanied what happened.
An example of observations that trainees need to learn to make is that Susskind's study describes only correlations among measured numbers of various types of questions asked by teachers and students (statistically analyzed assuming that systematic errors cannot be large if various observers agree in their counting and judgment of the type of the questions). There is no claim of observing or understanding causal relationships or dynamical origins of the observed correlations, even though researchers are tempted to think that they see some causality [113].
A description of what is observed must be distinguished by trainees from an understanding of the dynamical origin of what is observed. Using the analogy of Bohr's model (see the beginning of Section II), one may say that the description of the observed atomic spectra by Bohr's model was an essential step on the way to the discovery of the underlying dynamics of the atom, but by itself did not provide an understanding of this dynamics. Moreover, trainees need to realize that understanding the dynamics of behavior of teachers and students in interaction is incomparably more difficult than just observing and describing their behavior. This realization can motivate the trainees to seek a better understanding of the dynamics of their own interactions with students.
Studies by Csikszentmihalyi et al. [114] used modern remote communication technology (such as beepers) to monitor students' behavior during various activities around the clock (the technique is called experience sampling method). Results of these studies agree with the model assumption stated in Section IV A and elaborated on in Section IV D that students are strongly motivated by the visions they have of their own future [115]. On the other hand, the personal desire for and enjoyment of participation in activities is associated with the concept that Dewey described (in the context of artistic expression) as follows [116]: "Because of this wholeness of artistic activity, because the entire personality comes into play, artistic activity which is art itself is not an indulgence but is refreshing and restorative, as is always the wholeness that is health. There is no inherent difference between fullness of activity and artistic activity; the latter is one with being alive." Csikszentmihalyi calls such ultimate involvement in an activity "flow" [118], since one can say that a person feels flowing fully immersed in the activity and ceases to sense the flow of time. The same phenomenon is known to exist in all disciplines, including science [119], art, sport, and teaching [120]. The training of teachers should include the experience of flow in the process of teaching. Productive teaching is a source of immense joy for a teacher who sees how a student moves on in the process of learning, and crosses otherwise insurmountable barriers. This feature is heuristically included in the dynamical part of MOT discussed in Section IV D.
The training of teachers must include basic issues of communication and cooperation with parents. Numerous examples illustrate that teachers and parents must communicate [121]. Parents are major players in the student's life [122]. They can help a teacher understand where the student is coming from and how to help a student in getting on the path of learning, which are indispensable elements in productive teaching according to the MOT. On the other hand, the model suggests that communication between teachers and parents as observers of the same reality from different frames of reference encounters the same type of difficulties as communication between teachers and students. Of course, the case of teachers and parents involves significant internal events (most parents deeply care about their children) and needs an extended meta-dictionary (for parents, the subject matter is of secondary importance to how their children fare). The MOT predicts and explains why a clear communication in these circumstances must be difficult and requires training, see Section IV C.
The degree of success in training of a pool of teachers in productive teaching will depend on the selection of candidates [123]. Ideally, a selection process should be based on strengths that the candidates exhibit in their records and during "auditions" in the role of a teacher. As in the case of good schools of music, where it is not enough to just know how to hold an instrument and play a sequence of notes, so in the case of teaching it is not enough to know something about the subject matter and spell it out in front of the class (e.g., see the MOT predictions concerning difficulties in communication about internal events described in Section IV C). The analogy with music makes it clear that people who start learning how to teach early, for example, in scouts, can develop their skills far beyond average before they come to the "audition" appropriate for their admission to the school of productive teaching as defined by the MOT.
If someday training in productive teaching becomes a regularity of the educational system, one can imagine that promising students will be immersed in the contexts of productive learning by giving them responsible teaching and administrative roles in their schools and they will be helped in practicing their skills as part of their work. The help they may provide in return in teaching younger students and running the school could in principle reduce the burdens on teachers and provide the teachers with more time than they have now for their own professional development [124]. The MOT provides physical arguments for the necessity of increase in the amount of time made available to teachers for training.
It is clear at this point that the heuristic concept of productive teaching defined by the MOT implies a long list of suggestions for training of teachers and a corresponding list of questions that require answers. With current capabilities for research on teaching and learning it should be possible to begin systematic studies on the applicability of the concepts of productive teaching and training in practice.
It is expected that a trainee teacher who learns the concept of productive teaching will voluntarily express gratitude to a trainer for providing the lesson. The reason is that if the trainer works on the lesson and the trainee understands what the result is supposed to be, it will be clear to the trainee familiar with the MOT that confirmation of the result is what the trainer needs to obtain in order to know the degree to which the task is accomplished. Observing how the trainee behaves in contact with students under supervision (or aware of being observed or filmed) is not sufficient because a trainee may behave like an actor, who knows how to follow instructions given by a director and perfectly fakes a character while not actually becoming one. Since the concept of productive teaching is learned inside the trainee, only the trainee can provide the information that the lesson is learned. There is no point in asking the trainee if he or she has understood the concept before it is observed as emitted by the trainee. When it occurs, the information is provided by the trainee in a voluntary expression of appreciation of the concept (the concept includes the appreciation of the work of the trainer and understanding by the trainee that the voluntary expression of this appreciation is the way by which the trainer knows the result). Honest, self-motivated discussion of the concept cannot be faked since it can only be based on concrete events in which the trainee behaves in a sincere way and thus learns from them. If that is not the case, the trainee only pretends comprehension of the concept. Attempts to hide confusion and maintain superficial claims to understanding are not difficult to identify. When the trainer points out a problem related to the trainee's conception of productive teaching, the trainee's reaction is itself a measure of the depth of understanding of the concept. So, it is relatively easy to know whether a trainee is still in the woods. But it is not possible to tell that the training is completed until the trainee voluntarily provides the evidence.
Everyone who works in the capacity of a teacher does so in large part as a result of prior training. If that training was not based on the principles of communication between brains that are identified in the model, the principles may remain unknown to the person. It is then not the person's fault that she or he is not able to create a productive teaching environment for learning by others. Such cases can be gradually eliminated. According to the MOT, the elimination requires an extension of the concept of teaching from effective to productive. Those who are to train teachers would have to demonstrate understanding of the difference before becoming trainers.
V. CONCLUSION
The MOT proposed here provides a description of the concepts of effective and productive teaching that are not at all new in their meanings. Authors of great influence on our thinking have discussed these meanings extensively. The original aspect of the model is its simplicity. Namely, it sketches a picture of these concepts in simple physical terms. This is useful for seeking systemic solutions to problems in Education since one has to define the goals of the system before one attempts to figure out how to create systemic conditions in which these goals can be achieved. It is clear that the system must be designed in such a way that the forces pushing toward the goal are not counterbalanced by other forces that might also act in a system and win if the system is designed without the necessary understanding. If the goal is productive teaching defined in the MOT, the system forces must be arranged in such a configuration that the basic forces that drive productive teaching can function and perform their work.
However, the major problem of figuring out a suitable arrangement is not solved here [125]. For example, the question of how to train people who are to legislate, administer, and judge the work of teachers is not resolved. This problem can be called the system government problem [126]. It is clear that it depends on the competence of the system government if teachers (appropriately trained) can perform in agreement with their training in the conditions created for their work (see Section III D). The system government problem also means that the search for systemic solutions cannot be limited to the educational system alone. The bedrock assumption is that in all circumstances where teaching occurs in a system, both teacher and student are learning and they learn from each other in identical ways as observers who exchange information about the same reality.
The insights the model offers regarding effective teaching (Section III) and productive teaching (Section IV) help in comprehending Sarason's early prediction of the failure of educational reforms [127]. Sarason's prediction is based on his concept of a context of productive learning. Until now, there existed no simple description of what this concept means. According to the model, a context of productive learning exists for a student in the process of productive teaching (see Section IV E). A reform effort cannot be successful in creating contexts of productive learning for students unless the processes of productive teaching are center stage in the system. Thus, if an expert looks at a reform plan, including plans to monitor and document the performance of the system in the long run, and sees that nowhere in it the key processes are paid due attention, the prediction of failure may be made long before the reform is implemented. According to Sarason, Bensman [94] and Levine [95] provided credible descriptions of reform efforts that come the closest to incorporating contexts of productive learning in school practice.
The model picture also sheds some light on the belief that a rigorous scheme for training teachers in science (and through them their students) can by itself stimulate interest in learning. The model suggests instead that the kinematical image of the process of science must result from its proper dynamics (whatever it may be). When the image is artificially created through special constraints and incentives, neither teachers nor students can learn what the relevant forces are (whatever they may be). A less arbitrary alternative for the design of reforms is to secure facilitation of action of the natural forces of productive teaching: strengths of teachers and students. For this purpose, the knowledge of teachers needs to include the meta-dictionary of teaching. The model provides a preliminary definition of the meta-dictionary, showing that a dictionary that is limited to subject matter alone is insufficient to discuss educational reform.
When teachers and students are seen as observers of the world from different frames of reference and the kinematics and dynamics of their communication about what they observe are analyzed in simple terms of a physical model, it becomes clear that the interpersonal relationship between a teacher and a student is the key to productive teaching defined in the MOT. According to the model, productive teaching is not only effective in terms of mastery of the subject of study. It also helps a student to understand the role the teacher plays in the process, a facilitator of learning rooted in the individual's strengths. Through appreciation of this role, a student becomes a person who understands the principles and virtues of communication between people. This is why the model concept of productive teaching appears to provide a picture of what systemic solutions need to support in order to make a transition to a society in which words and ideas can help people in carrying out the work that is necessary for their well-being [128]. However, the model also says that the number of dimensions in the space of events registered in a brain is large (see Appendix A). Therefore, the corresponding relationships between frames of reference of different brains are complex, nonlinear, and topologically nontrivial. MOT predicts through these circumstances that an exchange of sequences of coordinates without paying much attention to their proper interpretation in different frames of reference may be easily mistaken for the con-cept of communication between brains. But if teaching is equated to handing down to a student a sequence of coordinates t formed using a language known in fact only to the teacher, and then checking if the sequence t is correctly handed back, not only is the teaching ineffective and students unfairly judged, but also a new phenomenon is created. Every sequence t can be handed around without clarity or need for clarity of its meaning in terms of real events. In such case, everybody can "teach" because there is no need to know either the content or how to transfer the content. The model predicts in simple terms that teaching cannot be improved by streamlining such a mindless process of handing down statements without content. The prediction holds no matter how much money may be spent on such a process because the process has nothing to do with productive teaching defined in the model. Thus, the model supports a claim that this process will not lead to transition toward new forms of society such as the one envisioned by Drucker [128].
According to the MOT, productive teaching demands preparation of teachers who can communicate with students in many more dimensions than only those of the subject matter (Section IV F). Such advanced preparation requires an apprenticeship system in which novices are taken care of by mentors over extended periods of time and may stay in touch with the mentors during their subsequent professional activities. Certainly, such preparation of teachers resembles processes of induction or mediated entry in professional disciplines and as such implies an open-ended self-correcting evolutionary process of change in the discipline of teaching based on ongoing research and development of the highest standards. By comparison with contemporary systems of education, such an evolutionary process of change demands new organization because the existing organization is focused too much on handing down sequences of words without paying due attention to their meanings. This illustrates how potentially far-reaching predictive power physical models of teaching may have.
Understood as a process based on interaction between different brains, teaching cannot be described by models that are limited to learning by only one brain and do not involve the transfer and processing of information in interaction between different brains. However, models of how a single brain learns are very important for teaching despite their limitations [14,15,129,130]. For example, consider that the process of gene expression in the formation of an individual's memory [83,131] (and perhaps also in other Hebbian processes) depends on factors such as attention [82]. When combined with the MOT, this consideration suggests that an educational system harms the development of students' brains if productive teaching is rare and students are regularly prevented from focusing their attention on productive learning. Given the model of teaching, such an educational system can no longer be considered legitimate. In order to become legitimate, it has to engage in a long-term research and development process of the highest quality on an appro-priate scale, concerning productive teaching and its implementation on a regular basis.
The model described here may be seen as an attempt to incorporate a number of ideas that originate in various arts and sciences into a physically-motivated structure. Such attempts are in need of building bridges between different disciplines; for example, between psychology and physics in terms of neural science. It is encouraging that physicists and neural scientists are already involved in discussions [132,133,134], and that neural scientists appear to be in communication with psychologists [17]. Even though a physical MOT may in principle offer a framework in which all required elements have a chance to eventually find their place, it is predicted that much more precise physical models of teaching will be needed than the very preliminary one described here. Nevertheless, because this preliminary MOT has a simple physical structure with many concrete implications regarding otherwise complex issues of teaching, it can be predicted on general grounds [135] that physical MOTs will be useful in guiding reform of educational systems. not based solely on the momentary input from the visual system. It depends also on the input from all other senses and from a large number of body organs. Ultimately, the processes that occur in the brain itself determine how events are registered. Factors such as attention and interest (or lack thereof for various, complex reasons) play significant roles. Also, since the brain develops over time, its activity at every moment depends on its biophysical history and the current structure this history has produced, including memory. It is clear that the brain's reaction to every momentary input it receives is not a simple function of this input. Thus, the concept of space of events registered in a brain requires a definition.
It is postulated that at every moment in time a brain located at any place in space consists of particles and fields in some state that in principle may be described using a very large number of physical parameters. A considerably smaller number of parameters describes the biophysical state of the brain, which corresponds to the underlying state of particles and fields. These parameters change over time. A change in this set of parameters over a short period will be called a raw representation of an event in the brain. The length of the suitable period is of secondary importance. For the MOT, periods on the order of one hundredth of a second are perhaps appropriate since a human eye cannot discern images that appear with frequency larger than on the order of one hundred per second, ears cannot discern words spoken faster than about ten per second, typical muscular reaction time is on the order of one fifth of a second, etc.
The raw representations of events are somehow processed in a brain so that the results can be stored in memory, recalled, and recognized. These results are called events registered in a brain and, by definition, they are elements of the space of events registered in the brain.
The space of events registered in a brain is certainly not isomorphic with the space of physical events outside the volume swept in space-time by the brain because the events registered in the brain depend on processes that involve physical events inside this volume. The latter are specific to the brain in which they happen. It is clear that events registered in different brains cannot be compared in any simple way.
Studies of the nervous systems of simple organisms [72,82,83,131] indicate how the raw representations of events may be processed and how the results of this processing may be registered in an organism's brain. The human brain is so complex that attempts to fully explain how it works are unlikely to succeed in the near future [15]. Nevertheless, one may imagine the space of events registered in a human brain using the concept of coordinates. This coordinate picture is referred to in Sections III and IV and helps in appreciation of the magnitude of dimensionality of the events that underlie the process of productive teaching discussed in Section IV.
In order to introduce the relevant coordinates, one may start from the postulate that the human brain is built from units that are connected by links. The smallest unit to think about would be a neuron and the smallest link would be a synapse. A more suitable level of analysis is that neurons work in groups and these groups may be considered units that are connected by complex links. A candidate to consider for a unit would be Hebb's assembly [14]. Fortunately for the purposes of the model, one may postulate the existence of units and links without specifying precisely their nature, except that there are many units and many more links among them. The physical arrangement of units and links in the brain, such as their apparent spatial extent or overlap, is irrelevant.
In order to quickly imagine the coordinates of events, it is best to consider first only two units connected by just one link. It is postulated that the magnitude of activity in the link at any moment and any position of the brain can be described by one number. This number is treated as a coordinate of content of the event registered in the brain. Note that the coordinate most likely describes a compact dimension since the link ceases to function properly and is turned off when the activity increases above a certain threshold. Thus, a coordinate above threshold does not exist and the threshold value is considered equivalent to zero [136].
Since the brain is considered to be built from many units and the number of just pair-wise links between m units, denoted by n, is much larger than m, n = m(m − 1)/2, the content of an event registered in the brain can be identified with an element in an ndimensional space with a large n. It is postulated that n is fixed and the same for all brains. More complex alternatives do not need to be discussed here. Including the time and place of registration, the number of dimensions in the space of events registered in a brain is N = 4+n, a much larger number than 4 that applies only to space-time. The magnitude of N suggests greater difficulties with precise communication between observers about events their brains register than the observers typically encounter already in communication about pointlike events in four-dimensional spaces.
Acknowledgments The author would like to thank Seymour B. Sarason and Kenneth G. Wilson for many discussions.APPENDIX A: SPACE OF EVENTS REGISTERED IN A BRAINThe word "event" is associated in physics with what happens in a set of points in space-time. But a learning human brain obtains information about events only in the form of input it can receive. A heuristic idea of parameterization of the information about events registered in a brain is used below to define and visualize the concept of space of events that is used in Sections III and IV.Consider how a snapshot of a learning individual's environment forms a record of an event and how such a record is received as input in the individual's brain. The input comes from the visual system and causes changes in the brain. Complete description of the changes undoubtedly requires a large set of parameters. But even without knowing the appropriate set of parameters and their values one may postulate that the changes ultimately result from the interplay of three elements: 1) The light actually received in the eyes, 2) performance of the system of vision that provides input to the brain, and 3) the activity in the brain at the moment of reception of the input from the visual system.The first two elements contribute to the process of registration of events by the individual's brain in familiar ways. For example, a student may be observing what happens on a laboratory table and, perhaps, must be wearing corrective lenses in order to see with sufficient precision. The third element, i.e., the ongoing brain activity whose change constitutes registration of an event, is least well-understood. Obviously, such brain activity is
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The Ego and the Id. S Freud, NortonS. Freud, The Ego and the Id (Norton, 1962).
E R Kandel, Nobel Lectures in Physiology or Medicine. H. JörnvallWorld ScientificE. R. Kandel, in Nobel Lectures in Physiology or Medicine 1996-2000, ed. H. Jörnvall (World Scientific, 2003).
. See Ref, and trace back to Ref. [74], where the school culture is identified as one of the obstacles to change in the classroom; then see Refs. [12] and [13See Ref. [25], Ch. 21, and trace back to Ref. [74], where the school culture is identified as one of the obstacles to change in the classroom; then see Refs. [12] and [13].
The school culture and process of change. S B Sarason, The Psycho-Educational Clinic. F. Kaplan and S. B. Sarason4University of MarylandDept. of Mental HealthS. B. Sarason, The school culture and process of change, Brechbill Lecture, University of Maryland, January 10, 1966; published in The Psycho-Educational Clinic, Vol. 4 (1969), Eds. F. Kaplan and S. B. Sarason, Dept. of Mental Health, The Commonwealth of Massachusetts;
Library of Congress Catalog Card No. 74-625106. See also two other articles by Sarason in the same volume. Library of Congress Catalog Card No. 74-625106. See also two other articles by Sarason in the same volume.
. S B Sarason, Anxiety in Elementary School Children. WileyS. B. Sarason et al., Anxiety in Elementary School Chil- dren (Wiley, 1960).
A Longitudinal Study of the Relation of Test Anxiety to Performance on Intelligence and Achievement Tests, Monographs of the Society for Research in Child Development. S B Sarason, K T Hill, P G Zimbardo, Serial No. 98. 29S. B. Sarason, K. T. Hill, P. G. Zimbardo, A Longitu- dinal Study of the Relation of Test Anxiety to Perfor- mance on Intelligence and Achievement Tests, Mono- graphs of the Society for Research in Child Develop- ment, Serial No. 98, Vol. 29, No. 7 (1964).
The Relation of Test Anxiety and Defensiveness to Test and School Performance over the Elementary-School Years, a further longitudinal study. K T Hill, S B Sarason, 31K. T. Hill, S. B. Sarason, The Relation of Test Anxi- ety and Defensiveness to Test and School Performance over the Elementary-School Years, a further longitudi- nal study, ibid., Serial No. 104, Vol. 31, No. 2 (1966).
The process of teaching cannot be understood here as limited to what T does because T works in a system and the system may dictate norms that generate anxiety. E.g., see Ref. 79The process of teaching cannot be understood here as limited to what T does because T works in a system and the system may dictate norms that generate anxiety. E.g., see Ref. [79].
. R L Cutts, An Empire of Schools. R. L. Cutts, An Empire of Schools (Sharpe, 1997).
See Ref. [22] for a popular exposition of this point. See Ref. [22] for a popular exposition of this point.
. C G Kentros, Neuron. 42283C. G. Kentros at al., Neuron 42, 283 (2004).
. H Udo, Neuron. 45887H. Udo et al., Neuron 45 887 (2005).
Physical laws lead to the formation of the genetic code and its expression in ways that may be not understandable using classical concepts, because these concepts do not apply in description of quantum phenomena. Physical laws lead to the formation of the genetic code and its expression in ways that may be not understand- able using classical concepts, because these concepts do not apply in description of quantum phenomena.
For example, in terms of renormalization group [86], suitably generalized, a long chain of steps connects the complex dynamics of complex effective degrees of freedom, such as entire brains, with the simple dynamics of basic initial degrees of freedom. such as particles and fieldsFor example, in terms of renormalization group [86], suitably generalized, a long chain of steps connects the complex dynamics of complex effective degrees of free- dom, such as entire brains, with the simple dynamics of basic initial degrees of freedom, such as particles and fields.
K G Wilson, Nobel Lectures. World ScientificK. G. Wilson, in Nobel Lectures (World Scientific, 1993).
C P Cf, Snow, The Two Cultures. Cambridge University PressCf. C. P. Snow, The Two Cultures (Cambridge Univer- sity Press, 1986).
The word "blind" is borrowed from R. Dawkins, The Blind Watchmaker. NortonThe word "blind" is borrowed from R. Dawkins, The Blind Watchmaker (Norton, 1996).
How the West Grew Rich. N Rosenberg, L E BirdzellJr, Basic BooksN. Rosenberg, L. E. Birdzell, Jr., How the West Grew Rich (Basic Books, 1986).
. T P Hughes, American Genesis. T. P. Hughes, American Genesis (Penguin, 1989).
When Dreams Came True (Brassey's. M J Bennett, M. J. Bennett, When Dreams Came True (Brassey's, 1996).
using information from others almost entirely without direct personal contact (e.g., only reading books), and the teaching they receive this way may be effective in the sense of learning a specialty. Productive teaching includes personal contacts in order to avoid unproductive consequences for the learner of learning almost entirely without personal contacts. In rare cases, individuals are able to learnIn rare cases, individuals are able to learn using informa- tion from others almost entirely without direct personal contact (e.g., only reading books), and the teaching they receive this way may be effective in the sense of learning a specialty. Productive teaching includes personal con- tacts in order to avoid unproductive consequences for the learner of learning almost entirely without personal contacts.
D Bensman, Central Park East and its graduates. Teachers College PressD. Bensman, Central Park East and its graduates (Teachers College Press, 2000).
One kid at a time: Big lessons from a small school. E Levine, Teachers College PressE. Levine, One kid at a time: Big lessons from a small school (Teachers College Press, 2001).
Three cups of Tea (Penguine. G Mortenson, D O Relin, G. Mortenson and D. O. Relin, Three cups of Tea (Pen- guine, 2007).
Wieman's observation that teaching may be not as productive for students as a teacher thinks [30], appears related to a possibility that it is difficult for a person to pay attention to issues of teaching when the person needs to focus on the subject matter. Wieman's observation that teaching may be not as pro- ductive for students as a teacher thinks [30], appears related to a possibility that it is difficult for a person to pay attention to issues of teaching when the person needs to focus on the subject matter.
The author observed training sessions for teachers and teacher leaders in the Reading Recovery program led by G. S. Pinnell at The Ohio State University. The author observed training sessions for teachers and teacher leaders in the Reading Recovery program led by G. S. Pinnell at The Ohio State University, in 1996.
The same need for self-correction based on comprehension is evident in students of physics who mechanically transform equations without realizing what they are doing and cannot see and correct errors they make. The same need for self-correction based on comprehen- sion is evident in students of physics who mechanically transform equations without realizing what they are do- ing and cannot see and correct errors they make.
H Schaefer-Simmern, The Unfolding of Artistic Activity. University of California PressH. Schaefer-Simmern, The Unfolding of Artistic Activity (University of California Press, 1961).
Art as Experience (Penguin, 1980). J Dewey, See also RefJ. Dewey, Art as Experience (Penguin, 1980). See also Ref.[69], Chapter 7.
For example, could the teachers in the private Shady Hill School, Cambridge, Massachusetts (www.shs.org), be able and willing to provide mentoring to the trainees in the Boston area, nationally. or interantionallyFor example, could the teachers in the private Shady Hill School, Cambridge, Massachusetts (www.shs.org), be able and willing to provide mentoring to the trainees in the Boston area, nationally, or interantionally?
The school teacher. D C Lortie, University of Chicago PressD. C. Lortie, The school teacher (University of Chicago Press, 2002).
Susskind, Questioning and curiosity in the elementary school classroom. E C Susskind, F. Kaplan and S. B. Sarason4Yale UniversityCommunity Mental Health Monograph of Psycho-Educational Clinic. Ph.D. ThesisE. C. Susskind, in Community Mental Health Mono- graph of Psycho-Educational Clinic, Eds. F. Kaplan and S. B. Sarason, Vol. 4 (1969), pp. 130-151; see also E. C. Susskind, Questioning and curiosity in the elementary school classroom, Ph.D. Thesis, Yale University, 1969.
. E Susskind, J. of Clinical Child Psychology. E. Susskind, J. of Clinical Child Psychology, Summer 1979, pp. 101-106.
94] can be studied with the help of a documentary film High School II by F. Wiseman, photography by R. Leiterman; see C. James, Film Review; 25 Years Later. New York Times. Wiseman Goes Back to SchoolRef. [94] can be studied with the help of a documentary film High School II by F. Wiseman, photography by R. Leiterman; see C. James, Film Review; 25 Years Later, Wiseman Goes Back to School, New York Times, July 6, 1994.
This reservation is made explicitly in Refs. 110, 111This reservation is made explicitly in Refs. [110, 111].
M Csikszentmihalyi, K Rathunde, S Whalen, M Wong, Talented teenagers. Cambridge University PressM. Csikszentmihalyi, K. Rathunde, S. Whalen, M. Wong, Talented teenagers (Cambridge University Press, 1997).
See Dewey's foreword to Schaefer-Simmern's work in Ref. 105See Dewey's foreword to Schaefer-Simmern's work in Ref. [105].
. M Csikszentmihalyi, Flow. HarperCollinsM. Csikszentmihalyi, Flow (HarperCollins,1991).
Hadamard reports on similar phenomenon in mathematics, see Ref. 47Hadamard reports on similar phenomenon in mathe- matics, see Ref. [47].
The chapter is focused on teachers providing conditions for flow to students and suggests that teachers can do it on the basis of their own experience of flow. Ref. [114], Chap. 9. The chapter is focused on teachers providing conditions for flow to students and suggests that teachers can do it on the basis of their own expe- rience of flow.
. E G See Ref, 67Ref. [95], pp. 97, 137; Ref. [13], pp. 64E.g., see Ref. [94], p. 103; Ref. [95], pp. 97, 137; Ref. [13], pp. 64, 67.
System dynamics occurs at the end of the chain. 85System dynamics occurs at the end of the chain [85].
Educational Reform: a self-scrutinizing memoir. S B Sarason, Teachers College PressS. B. Sarason, Educational Reform: a self-scrutinizing memoir (Teachers College Press, 2002).
. P F Drucker, Harper BusinessP. F. Drucker, Post-Capitalist Society (Harper Business, 1993); pp. 6-9, 201-204, and Chap. 12.
J J Hopfield, ibid. 81Proc. Nat. Acad. Sci. USA. Nat. Acad. Sci. USA793088J. J. Hopfield, Proc. Nat. Acad. Sci. USA 79, 2554 (1982), ibid. 81, 3088 (1984).
. T J Sejnowski, O Paulsen, J. Neurosci. 261673T. J. Sejnowski, O. Paulsen, J. Neurosci. 26, 1673 (2006).
. C H Bailey, E R Kandel, K Si, Neuron. 4449C. H. Bailey, E. R. Kandel, K. Si, Neuron 44, 49 (2004).
. L Borg-Graham, Nature Neurosci. 31191L. Borg-Graham, Nature Neurosci. 3, 1191 (2000).
. J J Hopfield, Nature Neurosci. 31204J. J. Hopfield, Nature Neurosci. 3, 1204 (2000).
. J Knight, Nature. 419244J. Knight, Nature 419, 244 (2002).
E M Rogers, Diffusion of Innovations. Free Press257E. M. Rogers, Diffusion of Innovations (Free Press, 2003); e.g., see p. 257.
On a circle, as example of a compact dimension, an angle can be chosen as a co-ordinate: it ranges from 0 to 2π, and the point with co-ordinate 2π is the same as the one with. co-ordinate 0On a circle, as example of a compact dimension, an angle can be chosen as a co-ordinate: it ranges from 0 to 2π, and the point with co-ordinate 2π is the same as the one with co-ordinate 0.
|
[] |
[
"On \"Electromagnetic Potential Vectors and Spontaneous Symmetry Breaking\"",
"On \"Electromagnetic Potential Vectors and Spontaneous Symmetry Breaking\""
] |
[
"V V Dvoeglazov [email protected] \nUAF\nUniversidad Autónoma de Zacatecas Apartado Postal 636\n98061Zacatecas, ZacMéxico\n"
] |
[
"UAF\nUniversidad Autónoma de Zacatecas Apartado Postal 636\n98061Zacatecas, ZacMéxico"
] |
[] |
The appearance of terms, which are analogous to ones required for symmetry breaking, in Lagrangian of Ref.[1] is shown to be caused by gauge invariance of quantum electrodynamics (QED) and by inaccuracy of the cited author in the choice of canonical variables. These terms do not have physical significance within modern quantum electrodynamics.
| null |
[
"https://arxiv.org/pdf/hep-ph/9305295v2.pdf"
] | 119,426,820 |
hep-ph/9305295
|
cbe3c6ad96bdec4a7215b2e6ef677290eff47d9b
|
On "Electromagnetic Potential Vectors and Spontaneous Symmetry Breaking"
arXiv:hep-ph/9305295v2 9 Oct 2018
V V Dvoeglazov [email protected]
UAF
Universidad Autónoma de Zacatecas Apartado Postal 636
98061Zacatecas, ZacMéxico
On "Electromagnetic Potential Vectors and Spontaneous Symmetry Breaking"
arXiv:hep-ph/9305295v2 9 Oct 2018.De, 04.20.Cv, 04.20.Fy, 11.10.Ef .
The appearance of terms, which are analogous to ones required for symmetry breaking, in Lagrangian of Ref.[1] is shown to be caused by gauge invariance of quantum electrodynamics (QED) and by inaccuracy of the cited author in the choice of canonical variables. These terms do not have physical significance within modern quantum electrodynamics.
In Ref. [1,2] the following k-space Lagrangian for electromagnetic field, interacting with the current j and with the charge density ρ, has been obtained:
Λ( k) = iΨ + ( k)Ψ( k) + A * ( k) − k j( k) − mΨ( k)Ψ( k)+ + i k C * ( k)ρ( k) k 2 − | ρ( k) | 2 2 k 2 + 1 2 | C ⊥ ( k) | 2 − k 2 | A ⊥ ( k) | 2 .(1)
The approach was used, in which A, the vector potential, and C =˙ A = ∂ A ∂t are supposed to be independent to each other. The author of cited paper considers (1) as the Lagrangian with the spontaneous-symmetry-breaking terms (fourth and fifth in the above formula).
Let us mark, the approach using the additional vector variable (it is designated as C in Ref. [1]), which is different from field variables, and it is considered as independent, is not an innovation. This is just the well-known Hamiltonian canonical formalism (see e.g. [3]- [6]). In Ref. [5] the canonical-conjugated variable to A µ is defined identically with the quantity C in [1], if we don't take into account the inessential coefficient 1 4π :
π i = 1 4π ∂L ∂(∂ t A i ) = 1 4π ∂ A ∂t .(2)
This canonical-conjugated quantities are due to use of the following Lagrangian:
L = − 1 8π A µ,ν A µ,ν = − 1 16π F µν F µν − 1 8π ∂A µ ∂x ν ∂A ν ∂x µ(3)
But, in the case of the x-space Lagrangian
L = − 1 4 F µν F µν(4)
the quantities A and C =˙ A are not the canonical-conjugated quantities, as opposed to the case of classical mechanics where x, the coordinate, and˙ x, the velocity, are, in fact, the canonical quantities. It is not clear, what quantzation procedure are implied by the author of Ref [1]. In the case of canonical quantization the Lagrangian ∼ F µν F µν does not give us π 0 , which is equal to zero. In the case of Lagrange quantization it is not clear, what commutation rules should be implemented, e.g., for [ A( x), C( x ′ )]. Moreover, in the Lagrange approach the field and the momenta of field are not considered as two independent quantities. It is also not obvious, how P µ , the energy-momentum operator, is expressed by A and C in the quantum case. Let us not forget, under quantization of electromagnetic field it is impossible to use the Lorentz condition ab initio. According to Fermi [7] it exists as the condition for the state vectors only. It is necessary to choose the definite Lagrangian and the definite quantization approach. and we are able to use the Lorentz condition in a weaker form only after setting up the commutation relations:
∂A (−) ∂x Φ = 0.(5)
In the case of the Lagrangian (3) we are able to quantizate the electromagnetic field canonically, using the variable π =˙ A as independent. Following the techniques of [1], we then have the additional terms to the k-space Lagrangian, which do contract one of the term in (1):
L add = − i k 2 ρ( k) k C * ( k) + 1 k 2 k C( k) k C * ( k) − k A( k) k A * ( k) .(6)
The total Lagrangian does not contain the symmetry breaking terms. As a result of gauge invariance of QED it is possible to use the other Lagrangians differing from (4) by the supplementary term 1 λ ∂A ∂x 2 , which, on the first view, brings nothing in (18) of cited paper. However, in this case the canonical quantities are π 0 = 1 λ ( ∂A µ ∂x µ ) and π i = F i0 and the expressions (10, 11) between the canonical quantities in Ref. [1] are no longer valid.
Moreover, it is well-known that the Lagrangian can be defined up to the total derivative only. If we implement the function ∂ µ f µ = ∂ µ (g · h) µ it is easy to select g and h in such kind that both of the symmetry-breaking terms in (18) are contracted out.
In the end, it is not clear, why the Coulomb gauge ( k · A = 0 and k · C = 0) was used by the author of [1] in the formula (18), the Lagrangian, but it was not used before, e.g. in (10) and (15).
In conclusion, the appearance of interaction terms of the form a · | Ψ | 2 +b · | Ψ | 4 in the QED Lagrangian is caused by gauge invariance of electrodynamics, implementing the Lorentz condition ab initio, inaccuracy of the author in the choice of the canonicalconjugated quantities. These terms are nonphysical and can be eliminated as a result of using the appropriate gauge. Consequently, they have no any physical meaning in quantum theory. I would still like to mention that investigations of interaction electromagnetic field with currents deserves serious elaboration. In Ref. [8] the equation was presented:
(−iγ µ ∂ µ − m)Ψ( x) = = e 2 γ µ Ψ( x) d yD( x − y)Ψ( y)γ µ Ψ( y) + eγ µ Ψ( x)A in µ ( x)(7)
(where D( x − y) is the Green's function for electromagnetic field, A in µ ( x) is the solution of Maxwell's equations), which should be resolved (see also [6]).
Acknowledgements. I am very grateful to Prof. A. M. Cetto, Head of the Departamento de Física Teórica, IFUNAM, for creation of excellent conditions for research. The technical help of A. Wong is greatly acknowledged. This work has been financially supported by the CONACYT (México) under the contract No. 920193.
. J V Shebalin, Nuovo Cim, 10899J. V. Shebalin, Nuovo Cim., 108 B, 99 (1993).
. J V Shebalin, Physica D. 66381J. V. Shebalin, Physica D, 66, 381 (1993);
. Phys. Lett. A. 2261Phys. Lett. A. 226, 1 (1997).
A Visconti, Quantum Field Theory. Pergamon PressI254A. Visconti, Quantum Field Theory. Vol. I (Pergamon Press, 1969), p. 254.
S N Gupta, Quantum Electrodynamics. Gordon and Breach Sci.Publ59S. N. Gupta, Quantum Electrodynamics (Gordon and Breach Sci.Publ., 1977), p. 59.
A A Sokolov, Quantum Electrodynamics. EnglishMoscow, RussiaMir Publisher107Russian editionA. A. Sokolov et al., Quantum Electrodynamics (Mir Publisher, Moscow, Russia, 1988), in English, p. 107. Revised from the 1983 Russian edition.
G Källén, Quantum Electrodynamics. Springer-Verlag48G. Källén, Quantum Electrodynamics (Springer-Verlag, 1972), p. 48.
. E Fermi, Rev. Mod. Phys. 487E. Fermi, Rev. Mod. Phys., 4, 87 (1932).
Foundations of Radiation Theory and Quantum Electrodynamics. A. O. BarutPlenum Press169Foundations of Radiation Theory and Quantum Electrodynamics. Ed. A. O. Barut (Plenum Press, 1980), p. 169.
|
[] |
[
"Galaxy clusters in presence of dark energy: a kinetic approach",
"Galaxy clusters in presence of dark energy: a kinetic approach"
] |
[
"M Merafina [email protected] \nDepartment of Physics\nUniversity of Rome \"La Sapienza\"\nPiazzale Aldo Moro 2I-00185RomeItaly\n",
"G S Bisnovatyi-Kogan \nSpace Research Institute\nRussian Academy of Sciences\nMoscowRussia\n\nNational Research Nuclear University MEPhI\nMoscowRussia\n",
"M Donnari \nDepartment of Physics\nUniversity of Rome \"La Sapienza\"\nPiazzale Aldo Moro 2I-00185RomeItaly\n\nDepartment of Physics\nUniversity of Rome \"Tor Vergata\"\nVia della Ricerca Scientifica 1I-00133RomeItaly\n"
] |
[
"Department of Physics\nUniversity of Rome \"La Sapienza\"\nPiazzale Aldo Moro 2I-00185RomeItaly",
"Space Research Institute\nRussian Academy of Sciences\nMoscowRussia",
"National Research Nuclear University MEPhI\nMoscowRussia",
"Department of Physics\nUniversity of Rome \"La Sapienza\"\nPiazzale Aldo Moro 2I-00185RomeItaly",
"Department of Physics\nUniversity of Rome \"Tor Vergata\"\nVia della Ricerca Scientifica 1I-00133RomeItaly"
] |
[] |
Context. The external regions of galaxy clusters may be under strong influence of the dark energy, discovered by observations of the SN Ia at redshift z < 1. Aims. The presence of the dark energy in the gravitational equilibrium equation, with the Einstein Λ term, contrasts the gravity, making the equilibrium configuration more extended in radius. Methods. In this paper we derive and solve the kinetic equation for an equilibrium configuration in presence of the dark energy, by considering the Newtonian regime, being the observed velocities of the galaxies inside a cluster largely smaller than the light velocity.Results. The presence of the dark energy in the gravitational equilibrium equation leads to wide regions in the W 0 -ρ Λ diagram where the equilibrium solutions are not permitted, due to the prevalence of the effects of the dark energy on the gravity.
|
10.1051/0004-6361/201424062
|
[
"https://arxiv.org/pdf/1404.7744v1.pdf"
] | 56,370,286 |
1404.7744
|
179ca37ce4f5f4bcc9a1c2e324daa4458c6a06ad
|
Galaxy clusters in presence of dark energy: a kinetic approach
May 1, 2014 May 1, 2014
M Merafina [email protected]
Department of Physics
University of Rome "La Sapienza"
Piazzale Aldo Moro 2I-00185RomeItaly
G S Bisnovatyi-Kogan
Space Research Institute
Russian Academy of Sciences
MoscowRussia
National Research Nuclear University MEPhI
MoscowRussia
M Donnari
Department of Physics
University of Rome "La Sapienza"
Piazzale Aldo Moro 2I-00185RomeItaly
Department of Physics
University of Rome "Tor Vergata"
Via della Ricerca Scientifica 1I-00133RomeItaly
Galaxy clusters in presence of dark energy: a kinetic approach
May 1, 2014 May 1, 2014arXiv:1404.7744v1 [astro-ph.GA] 30 Apr 2014 Astronomy & Astrophysics manuscript no. aajournal c ESO 2014 Preprint online version:galaxies: clusters: general -(cosmology:) dark energy -hydrodynamics
Context. The external regions of galaxy clusters may be under strong influence of the dark energy, discovered by observations of the SN Ia at redshift z < 1. Aims. The presence of the dark energy in the gravitational equilibrium equation, with the Einstein Λ term, contrasts the gravity, making the equilibrium configuration more extended in radius. Methods. In this paper we derive and solve the kinetic equation for an equilibrium configuration in presence of the dark energy, by considering the Newtonian regime, being the observed velocities of the galaxies inside a cluster largely smaller than the light velocity.Results. The presence of the dark energy in the gravitational equilibrium equation leads to wide regions in the W 0 -ρ Λ diagram where the equilibrium solutions are not permitted, due to the prevalence of the effects of the dark energy on the gravity.
Introduction
It was shown by Chernin [2001Chernin [ , 2008 that outer parts of galaxy clusters (GC) may be under strong influence of the dark energy (DE), discovered by observations of SN Ia at redshift z ≤ 1 [Riess et al., 1998, Perlmutter et al., 1999, and in the spectrum of fluctuations of CMB radiation [see e.g. Spergel et al., 2003, Tegmark et al., 2004. Equilibrium solutions for polytropic configurations in presence of DE have been obtained in papers of Balaguera-Antolínez et al. [2006, 2007 and Merafina et al. [2012]. Here we derive a Boltzmann-Vlasov kinetic equation in presence of DE, in Newtonian gravity, and obtain its solutions. These solutions generalize the ones, obtained by Bisnovatyi-Kogan et al. [1993 for the kinetic equation without DE. Here we consider the problem in Newtonian approximation, because the observed chaotic velocities of galaxies inside a cluster are much less than the light velocity v gc ≪ c.
The general relativistic solution in presence of DE could be applied for equilibrium configurations of point masses from some exotic particles, interacting only gravitationally. On early stages of the universe expansion, before and during the inflation stage, these particles may form gravitationally bound configurations which collapse during the inflation, when antigravity decreases. As a result of this collapse such hypothetical objects may be transformed into primordial black holes appearing after the end of inflation. The relativistic kinetic equation, and its solutions in presence of DE will be considered elsewhere.
Newtonian approximation in description of DE
The substance, which is called now as DE was first introduced by Einstein (1918) for a stationary universe, in the form of cosmological constant Λ, during his unsuccessful attempts to construct a solution for a stationary universe. Soon after de Sitter (1917) had shown, that in presence of Λ the solution for an empty space describes an exponential expansion. Friedmann [1922Friedmann [ , 1924 was the first, who had obtained exact solutions for the expanding universe containing matter, in presence of the cosmological constant Λ. Another exact solution for the metric in presence of Λ, around the gravitating point mass, was obtained by Carter [1973]. This solution is a direct generalization of the Schwarzschild solution for a black hole (BH) in vacuum with a metric ds 2 = g 00 c 2 dt 2 − g 11 dr 2 − r 2 (dθ 2 + sin 2 θdϕ 2 ) ,
has a form
g 00 = 1 g 11 = 1 − 2GM c 2 r − Λr 2 3 = 1 − 2GM c 2 r − 8πGρ Λ r 2 3c 2 ,(2)
where the density of DE ρ Λ is connected with Λ as
ρ Λ = Λc 2 8πG .(3)
A transition to the Newtonian limit, where DE is described by the antigravity force in vacuum was done by Chernin [2008]. In the limit of a weak gravity (v 2 ≪ c 2 , GM/r ≪ c 2 ) the metric coefficients are connected with a gravitational potential Φ g as [Landau and Lifshitz, 1962] g 1/2
00 = 1 + Φ g c 2 .(4)
Then, using the Eq.(4) and the Eq.
(2) at Λ = 0, we obtain the expression for the Newtonian potential Φ g and the Newtonian gravity force acting on the unit mass F g
Φ g = − GM r , F g = − dΦ g dr = − GM r 2 .(5)
For the Schwarzschild-de Sitter metric (2) we have in the Newtonian limit
Φ = − GM r − 4πGρ Λ r 2 3 , F = F g + F Λ = − GM r 2 + 8πGρ Λ r 3 . (6)
So, the cosmological constant create in a vacuum a repulsive (antigravity) force between a BH and a test particle, which increases linearly with a distance between them. The normalization of the potential here is chosen so that Φ g = 0 at r = ∞, and Φ Λ = 0 at r = 0. Let us consider now the equilibrium of a self-gravitating object in presence of DE. In general relativity the equations describing the equilibrium in a spherically symmetric configuration in vacuum (without DE) had been derived by Oppenheimer and Volkoff [1939]
dP dr = − G(ρc 2 + P)(M r c 2 + 4πPr 3 ) r 2 c 4 − 2GM r rc 2 (7) dM r dr = 4πρr 2 .
Here ρ, P are the total density and total pressure of the matter, and M r is the total (gravitating) mass, including a gravitational binding energy, inside a radius r in the Schwarzschild-like metric
ds 2 = e ν c 2 dt 2 − e λ dr 2 − r 2 (dθ 2 + sin 2 θdϕ 2 ) ,(8)e λ = 1 − 2GM r rc 2 −1 ,(9)e ν = exp 2 ∞ r dP/dr P + ρc 2 dr .(10)
In presence of DE, ρ and P are represented as
ρ = ρ m + ρ Λ = ρ m + Λc 2 8πG , P = P m + P Λ = P m − Λc 4 8πG .(11)
Let us consider a Newtonian limit when
P m ≪ ρ m c 2 , r ≫ 2GM (m) r c 2 .(12)
Here we used a definition M (m) r = 4π r 0 ρ m r 2 dr. In the Newtonian limit we have from Eq.(7)
dP dr = − ρ m 3GM (m) r − Λc 2 r 3 r 2 3 − Λr 2 .(13)
Let us estimate the last term in the denominator. For existence of an equilibrium configuration with a finite radius we need a positive sign of the numerator, so using the conditions (12) we have
Λr 2 < 3GM (m) r rc 2 ≪ 1 .
Therefore, in the denominator we have Λr 2 ≪ 3 and we can neglet the term with Λ. So in the Newtonian approximation, in presence of DE, we obtain the following equilibrium equation
dP dr = −ρ m GM (m) r r 2 − Λc 2 r 3 = −ρ m GM (m) r r 2 − 8πGρ Λ r 3 ,(14)
with ρ Λ given by the definition (3), which was used without derivation by Merafina et al. [2012]. On the other hand, we can write the Poisson equation for the gravity of the matter together with the hydrostatic equilibrium equation
∇ 2 Φ g = 4πGρ m , ∇P ρ m = −∇Φ g − ∇Φ Λ ,(15)
and then, the potential created by DE in the vacuum, taking into account that P = P m + P Λ , satisfies the Poisson equation
∇ 2 Φ Λ = −8πGρ Λ , ρ Λ = Λc 2 8πG .(16)+ (v · ∇)v + ∇P ρ m = −∇Φ g − ∇Φ Λ .(17)
Kinetic equation for self-gravitating cluster in presence of DE
The kinetic Boltzmann-Vlasov equation for a distribution function f of non-collisional gravitating points of equal mass m, in spherical coordinates (r, θ, ϕ) is written as
∂ f ∂t + v r ∂ f ∂r + v θ r ∂ f ∂θ + v ϕ r sin θ ∂ f ∂ϕ + + v 2 θ + v 2 ϕ r − ∂Φ ∂r ∂ f ∂v r + − v r v θ r + cot θ v 2 ϕ r − 1 r ∂Φ ∂θ ∂ f ∂v θ + + − v r v ϕ r − cot θ v ϕ v θ r − 1 r sin θ ∂Φ ∂ϕ ∂ f ∂v ϕ = 0 ,(18)
where, in presence of DE, we have Φ = Φ g + Φ Λ . In a spherically symmetric stationary cluster, we have ∂Φ/∂t = 0 and Φ = Φ(r). Moreover, the kinetic equation (18) has four first integrals, written in Cartesian coordinates (x, y, z) as
E m = 1 2 (v 2 x + v 2 y + v 2 z ) + Φ , L x m = y v z − z v y , L y m = z v x − x v z , L z m = x v y − y v x .(19)
In spherical coordinates, these integrals can be expressed by
E m = 1 2 (v 2 r + v 2 θ + v 2 ϕ ) + Φ , L x m = −r v θ sin ϕ − r v ϕ cos θ cos ϕ , L y m = r v θ cos ϕ − r v ϕ sin ϕ cos θ , L z m = r v ϕ sin θ ,(20)
where E and L i (i = x, y, z) are the energy and the projection of the angular momentum on the corresponding axis, respectively. From the last three integrals follows the conservation of the absolute value of the angular momentum L, written in the form
L 2 m 2 = r 2 (v 2 θ + v 2 ϕ ) .(21)
Then, the solution of the kinetic equation (18) is an arbitrary function of the first integrals (20). We restrict ourselves to an isotropic distribution function f (E). For a uniform DE, a normalization of its energy at r = ∞ is not possible, therefore we choose Φ Λ = 0 at r = 0 as the most convenient one [Merafina et al., 2012]. Thus, from Eqs.(15), we have
Φ Λ = − 4πG 3 ρ Λ r 2 = − Λc 2 6 r 2 .(22)
Following Zel'dovich and Podurets [1965], Bisnovatyi-Kogan et al. [1993,1998], we consider a Maxwell-Boltzmann distribution function with a cut-off
f = Be −E/T for E ≤ E cut f = 0 for E > E cut ,(23)
where the cutoff energy E cut is given by
E cut = − αT 2(24)
and α is the so called cutoff parameter, while T is the temperature in energy units. The total energy is
E = mv 2 2 + mΦ = mv 2 2 + mΦ g − mΛc 2 r 2 6 ,(25)
where the total potential Φ and the velocity v are given by
Φ = Φ g + Φ Λ and v = (v 2 r + v 2 θ + v 2 ϕ ) 1/2 .(26)
The constant B in the first of the Eqs.(23) depends on the total potential Φ and therefore it is not the same for each model. In order to consider a unique distribution function for all the equilibrium configurations, following Merafina and Ruffini [1989], we must choose a different normalization by introducing a new constant A connected with B through the following relation 1
B = Ae mΦ R /T ,(27)
with Φ R the value of the total potential Φ at r = R. In this way, the expression of the distribution function (23) for E ≤ E cut becomes
f = A exp mΦ R T − mv 2 2T − m T Φ g − Λc 2 r 2 6 .(28)
The maximum kinetic energy ǫ c is connected with the potential Φ by the relation
ǫ c = m(Φ R − Φ) .(29)
Then the distribution function can be rewritten as
f = A e −(ǫ−ǫ c )/T ,(30)
where ǫ = mv 2 /2 is the kinetic energy of the single point mass. The Poisson equation (15) in a spherical symmetry referred to gravitational field is given by
1 r 2 d dr r 2 dΦ g dr = 4πGρ m ,(31)
with the boundary conditions Φ g (0) = Φ g0 and Φ ′ g (0) = 0. The matter density can be expressed as
ρ m = 4πm p max 0 f p 2 d p , with p = mv ,(32)
1 Eq. (27) is not arbitrary but justified by considerations of statistical mechanics, being E = constant along the motion of each single component of mass m and taking into account the presence of the chemical potential µ in the constant B, where µ + mΦ = constant along the radial coordinate r.
where the expression of the maximum momentum p max is given by
p max = 2m −mΦ − αT 2 = 2m −mΦ g + mΛc 2 r 2 6 − αT 2 .
The cluster with a finite radius is possible only when the following condition is satisfied
αT 2m < −Φ max , with Φ max < 0 .
Then, by using the form of the distribution given in Eq. (30), we can finally rewrite the matter density as
ρ m = 4 √ 2πAm 5/2 ǫ c 0 e −(ǫ−ǫ c )/T √ ǫ dǫ , where ǫ c = p 2 max 2m
. (33) Introducing dimensionless variables
W = p 2 max 2mT = ǫ c T and x = p 2 2mT = ǫ T ,(34)
we obtain W = m(Φ R −Φ)/T and the expression of matter density ρ m becomes
ρ m = 4 √ 2πAm 5/2 T 3/2 W 0 e W−x √ xdx ,(35)
where, as usual, at
W = 0 we have ρ m = 0, being Φ = Φ R .
From the Poisson equation (31) we can deduce the equation describing the structure of the Newtonian configurations in presence of DE by considering also the potential Φ Λ . In fact, inserting the expression of the gravitational potential Φ g = Φ − Φ Λ in Eq.(31) and using the Eq. (22), we obtain
1 r 2 d dr r 2 dΦ dr = 4πGρ m − Λc 2 ,(36)
where the potential Φ now includes all the contributions. Then, by considering the first relation in Eq.(11), the equilibrium equation becomes
1 r 2 d dr r 2 dΦ dr = 4πG(ρ m − 2ρ Λ ) .(37)
Now, we have to consider the boundary conditions for the potential Φ with respect to ones given for the potential Φ g in the Eq.(31). Starting from Eq. (22), we can write
Φ = Φ g − Λc 2 6 r 2 and Φ ′ = Φ ′ g − Λc 2 3 r(38)
and therefore, for
r = 0, we have Φ(0) = Φ g0 and Φ ′ (0) = Φ ′ g (0) = 0.
In order to write the dimensionless form of the equilibrium equation, we can express the radial coordinate as r = ηr and, using the definition W = m(Φ R − Φ)/T , the equilibrium equation can be rewritten as
1 r 2 d dr r 2 dW dr = − 4πGmη 2 T (ρ m − 2ρ Λ ) .(39)
In the same way, following Merafina and Ruffini [1989], we can introduce the expression of dimensionless densities by defining the following quantities
ρ m = σ 2 Gη 2ρ m and ρ Λ = σ 2 Gη 2ρ Λ ,(40)
where σ 2 = 2T/m. Thus, the dimensionless form of the equilibrium equation will be given by
1 r 2 d dr r 2 dW dr = −8π(ρ m − 2ρ Λ ) ,(41)
with the boundary conditions W(0) = W 0 and W ′ (0) = 0. Moreover, it is important to note that the relationρ m0 > 2ρ Λ must be satisfied at the center of the equilibrium configuration in order to obtain the condition of initial decreasing density W ′′ (0) < 0. However, this is a necessary but not sufficient condition for the existence of the equilibrium solution, because the presence of the DE can make possible to reach conditions of increasing density (W ′ > 0) at other values of the radial coordinate. It remains to define the expression of the dimensional quantity η. For getting the result, we can use the relations (35) and (11) for the densities ρ m and ρ Λ , respectively, and compare them with the definitions (40). We obtain
η = (Am 4 Gσ) −1/2 ,(42)
witĥ
ρ m = 2π W 0 e W−x √ xdx andρ Λ = Λη 2 c 2 8πσ 2 ,(43)
whereρ Λ is given by the value of Λ. The total mass M (m) at radius R is given by
M (m) = 4π R 0 ρ m r 2 dr = σ 2 η G R 0 4πρ mr 2 dr ,(44)
wherê
M (m) = R 0 4πρ mr 2 dr and M (m) = σ 2 η GM (m) .(45)
Finally, in order to make explicit the dependence of the dimensional quantities on the velocity σ, we can introduce the quantity
ζ = η σ 1/2 = (Am 4 G) −1/2(46)
and the dimensional quantities can be rewritten as
ρ m = σ 3 Gζ 2ρ m and ρ Λ = σ 3 Gζ 2ρ Λ(47)
and
M (m) = σ 3/2 ζ GM(m)
and
R = ζ σ 1/2R .(48)
Turning to the condition (24) on the energy E cut , we can express the cutoff parameter α by using the condition at the edge of the configuration
α 2 = − mΦ R T = − m T (Φ g + Φ Λ ) r=R .(49)
Thus, being Φ g (R) = −GM (m) /R and Φ Λ (R) = −Λc 2 R 2 /6, we obtain
α = 2GmM (m) RT + mΛc 2 R 2 3T(50)
and, finally, by using dimensionless quantities (47), (48) and the relation (3), we have
α = 4M (m) R 1 + 4πρ ΛR 3 3M (m) .(51)
For small values of the cutoff parameter α, maintaining a finite value of α T which corresponds to high values of the temperature T , the distribution function (23) may be taken as a constant [Bisnovatyi-Kogan et al., 1998]. Then, the solutions exist only for small values of W 0 andρ Λ , assuming a more simplified form which converges to a limiting sequence. In the limit of W → 0, the dimensionless density ρ m can be expressed aŝ
ρ m = 2π W 0 √ xdx = 4π 3 W 3/2 ,(52)
whereas the equilibrium equation (41) becomes
1 r 2 d dr r 2 dW dr = − 32π 2 3 W 3/2 + 16πρ Λ .(53)
Expressing in dimensional terms, the density can be written as
ρ m = 4π 3 σ 3 Gζ 2 W 3/2 = ρ p W 3/2 ,(54)
where
ρ p = 4π 3 σ 3 Gζ 2 .(55)
Moreover, by imposing a change of radial coordinate fromr to y for whicĥ
r = y 3 32π 2 1/2 ,(56)
the dimensionless equilibrium equation can be rewritten as
1 y 2 d dy y 2 dW dy = −W 3/2 + 3 2πρ Λ .(57)
We can also substitute the densityρ Λ by using Eqs. (47) and (55) and finally get
1 y 2 d dy y 2 dW dy = −W 3/2 + 2ρ Λ ρ p ,(58)
which corresponds, if we take ρ p = ρ m0 and W ≡ θ, exactly to the equilibrium equation for a polytropic configuration with index n = 3/2 in presence of DE introduced by Merafina et al. [2012] in accordance to the dimensionless Emden variables and the initial conditions W(0) = θ(0) = 1 and W ′ (0) = θ ′ (0) = 0. Therefore, the polytropic configurations calculated by Merafina et al. [2012] in hydrostatic approach, can be used for the description of clusters of gravitating point masses with a distribution function (28) with energy cutoff (29), at small values of Λ and large values of T .
Numerical results
The dimensionless equilibrium equation (41) depends on two parameters: the gravitational potential at the center of the configuration W 0 andρ Λ , which determines the intensity of DE through the value of the cosmological constant Λ. Different values of these parameters give a two-dimensional family of equilibrium solutions. The set of solutions forρ Λ = 0 at different values of W 0 was obtained by Bisnovatyi-Kogan et al. [1998]. We solved numerically the Poisson equation for gravitational equilibrium at different values of the two parameters (W 0 ,ρ Λ ) mentioned above. First of all, we focused our attention on the matter density profiles ρ m (r) of the equilibrium configurations; in detail, we investigated how they change for increasing values of the dimensionless DE densityρ Λ at fixed values of the dimensionless gravitational potential W 0 . We chose three values of W 0 and, respectively, four values ofρ Λ , founding the existence of pairs of parameters (W 0 ,ρ Λ ) which do not allow equilibrium solutions. This peculiarity is well represented in the matter density profiles shown in Figs. 1, 2 and 3. Hereafter in the figures, for the sake of compactness, we define the following quantities
ρ * = σ 3 Gζ 2 , M * = σ 3/2 ζ G , R * = ζ σ 1/2(59)
and, therefore, the dimensionless quantities introduced in Eqs. (47) and (48) can be rewritten aŝ
ρ m = ρ m ρ * ,ρ Λ = ρ Λ ρ * ,M (m) = M (m) M * ,R = R R * ,r = r R * .(60)
For each value of the central potential W 0 , it exists one value of the parameterρ Λ after which the matter density profile does not converge to zero, but oscillates indefinitely. If we assert that the radius R of an equilibrium configuration is defined as the value of the radial coordinate r at which the matter density ρ m (r) becomes zero, it is clear that everytime this does not happen we are unable to estimate the radial extension of the system. All the configurations with a given value of W 0 andρ Λ which correspond to oscillating density profiles cannot be considered in gravitational equilibrium. Moreover, the calculation of the total radius R of an equilibrium configuration is strictly connected to the one related to the total mass M (m) . Following Eq.(45) and expressing in terms of dimensionless quantities, the massM (m) r within the radiusr is given bŷ
M (m) r = r 0 4πρ m ξ 2 dξ .(61)
As a consequence, the non-equilibrium solutions for which the total radius R cannot be defined do not even allow the evaluation of the total mass M (m) of the system. Bisnovatyi-Kogan et al. [1998] found the set of solutions at Λ = 0 forM(ρ m0 ) andM(α) curves, in the Newtonian case. These curves are shown in Figs. 4 and 5 (continuous line) together with the ones given for different values ofρ Λ (Λ 0). Analyzing the former figure (Fig. 4), when the parameterρ Λ is different from zero, and for increasing values of this parameter, the curves are not longer continuous and the absolute maximum of the mass disappears. Within the interval 0.6 ≤ρ Λ ≤ 0.8, the curves present several branches (in the figure, only the branches forρ Λ = 0.8 are shown in order to keep clear the understanding of the different behaviors). Out of the interval 0.6 ≤ρ Λ ≤ 0.8 the branches reduce to a unique curve and, in particular, for ρ Λ > 0.8, the curve becomes gradually shorter at increasing values ofρ Λ , until reaching the critical valueρ Λ ≃ 1.38 in which the curve reduces to a unique point (we discuss this critical value in the following). It is clear that the unusual behavior of thê M(ρ m0 ) is related to the density profiles of the non-equilibrium solutions. In order to analyze the latter figure (Fig. 5), by considering Eq.(51), we can conclude that also the parameter α is connected to the values of the total radiusR and the masŝ M (m) . Consequently it is easy to show that for non-equilibrium solutions it is not possible to calculate the cutoff parameter α. Therefore, also in this case we can expect the existence of different branches of solutions. Moreover, the behavior of theM(α) curves at different values ofρ Λ extends the range of solutions at values of α larger than the critical value (α = 2.87) valid for Λ = 0 [Bisnovatyi-Kogan et al., 1998]. As previously underlined, it is possible to distinguish several branches of solutions with a limiting value of α changing in dependence of the value ofρ Λ . This limiting value, sistematically larger than 2.87, increases at increasing values ofρ Λ until the absolute limiting value α lim ≃ 3.42.
The DE background in which all the bodies of the Universe are embedded, produces the antigravity that changes their gravitational equilibrium, acting on the contrary of the matter gravity. In order to establish when we are in presence of configurations where the presence of the DE can change the gravitational equilibrium, following Bisnovatyi-Kogan and Chernin [2012], we introduce the so called zero gravity radius R Λ . This is a physical parameter which is defined as the distance from the center of the system where the matter gravity and DE antigravity balance each other exactly. Let us considering the total force acting on the unit mass
F = F g + F Λ = − GM (m) r r 2 + 8πGρ Λ 3 r ,(62)
where, differently from Eq. (6), this relation is also valid within the matter and not only in the vacuum. Then, the total force F defined in Eq.(62) and, consequently, the acceleration, are both zero at the distance
R Λ = 3M (m) R Λ 8πρ Λ 1/3 ,(63)
where the zero gravity radius depends on the total mass of the equilibrium configurations if R ≤ R Λ , while, if the condition F = 0 is satisfied inside the configuration, we have not equilibrium and the mass to consider is M (m) r with r = R Λ . Therefore, every cluster has its zero gravity radius. This definition allows us to identify a gravitationally bound system only if it is enclosed within the sphere of radius R Λ , namely only if its total radius is less than its zero gravity radius (R < R Λ ). Galaxies in the external regions where r ≥ R Λ can flow out from the center of the cluster under the action of the DE antigravity force.
In Fig. 6 we have represented the curve of the equilibrium configurations having R = R Λ , through the behavior of W 0 as a function ofρ Λ . In addition we have also shown the curves representing the families of equilibrium solutions at fixed values of α. By keeping constant the value of W 0 and increasing the value of ρ Λ there exists one limiting value, lying on the curve, after which it is no longer possible to obtain equilibrium solutions. Along this limiting curve, which separates two regions (solid line), the equilibrium configurations have the total radius exactly equal to the zero gravity radius and the matter density profiles vanishing with a minimum in correspondence of the total radius R = R Λ . Therefore, it is possible to define the region on the right side of the figure in which no gravitational equilibrium can establish and no curves at constant α can lie, corresponding to configurations with matter density profiles not converging to zero. On the contrary, in the region corresponding to the left side of the figure, we can assert that the force due to the presence of the DE, F Λ , is less strong than the one due to the gravity, F g , and the gravitational equilibrium can be achieved. In other words, speaking in terms of radial extension, the condition R ≤ R Λ is satisfied for each configuration belonging to this region and the matter density profiles are regular and converging to zero in correspondence of the total radius R.
Finally, from Eq.(63), we can see that the zero gravity radius R Λ is inversely proportional to the DE density ρ Λ . Therefore, by decreasing the value of the DE density, the zero gravity radius increases until the condition R Λ → ∞ for ρ Λ = 0. Actually, by considering the plane (W 0 -ρ Λ ) of Fig. 6, the gravitational equilibrium is achieved more easily, and for more values of W 0 , when theρ Λ parameter is small. In particular, forρ Λ = 0, the equilibrium solutions are possible for each value of W 0 , recovering the well know results of Bisnovatyi-Kogan et al. [1998].
Conclusions
We have calculated the equilibrium configurations of Newtonian clusters with a truncated Maxwellian distribution function, in presence of DE. All these clusters have a structural equilibrium, being satisfied the condition R < R Λ , and result dynamically stable. By considering the relaxation time, we obtain a value larger than the age of the Universe and, therefore, we can conclude that thermodynamical instabilities are not relevant in current evolution of galaxy clusters. On the other hand, the evaluation of parameters characterizing this kind of clusters suggests that these systems are collisionless. In any case, the critical point of the onset of thermodynamic instability lies far from the first maximum mass, at larger values of W 0 in the curve withρ Λ = 0 of Fig. 4 [Bisnovatyi-Kogan and Merafina, 2006], as well in curves witĥ ρ Λ 0, the most part of the equilibrium configurations results thermodynamically stable.
Presently the density distribution inside galaxy clusters is described by several phenomenological functions, some if which follow from numerical simulations [see Chernin et al., 2013]. Qualitatively the truncated Maxwellian distribution, considered here is similar to the non-singular density distribution suggested by Chernin et al. [2013]. It may be used for the more detailed study of the density and velocity distributions on the periphery of rich clusters, where the influence of DE is important, and their comparison with observations. The galaxies in the outer parts of the clusters are not numerous, and they have smaller masses and luminosities in presence of even weaker relaxation. Therefore, only largest telescopes should be used for a search of galaxies at the cluster peripheries. The most sensitive X-ray telescopes are needed for detection of the hot gas in the outer parts of the clusters, and its possible outflow in presence of DE, considered by Bisnovatyi-Kogan and .
Fig. 1 .
1Dimensionless matter density profiles for equilibrium configurations with W 0 = 4 andρ Λ = 0 (dashed line), 0.5 (dotted line), 0.9 (dash-dotted line), for which the matter density goes to zero, andρ Λ = 1.25 (solid line), chosen inside the region of non-equilibrium solutions.
Fig. 2 .
2Dimensionless matter density profiles for equilibrium configurations with W 0 = 8 andρ Λ = 0 (dashed line), 0.3 (dotted line), 0.5 (dash-dotted line), for which the matter density goes to zero, andρ Λ = 0.8 (solid line), chosen inside the region of non-equilibrium solutions.
Fig. 3 .
3Dimensionless matter density profiles for equilibrium configurations with W 0 = 12 andρ Λ = 0 (dashed line), 0.3 (dotted line), 0.5 (dash-dotted line), for which the matter density goes to zero, andρ Λ = 0.9 (solid line), chosen inside the region of non-equilibrium solutions.
Fig. 4 .
4Dimensionless mass as a function of the dimensionless central matter density forρ Λ = 0 (solid line), 0.5 (dashed line), 0.8 (dotted lines), 1.3 (dash-dotted line).
Fig. 5 .Fig. 6 .
56Dimensionless mass as a function of the cutoff parameter α forρ Λ = 0 (solid line), 0.5 (dashed line), 0.8 (dotted lines), 1.3 (dash-dotted line). Limiting curve of the equilibrium configurations with R = R Λ (solid line), expressed in terms of W 0 as a funtion ofρ Λ . Labelled curves at constant α (dashed lines) are also considered.
AcknowledgementsThe work of G.S. Bisnovatyi-Kogan was partly supported by RFBR grant 11-02-00602, the RAN Program Formation and evolution of stars and galaxies, and Russian Federation President Grant for Support of Leading Scientific Schools NSh-5440.2012.2 .
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Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant. A G Riess, A V Filippenko, P Challis, A Clocchiatti, A Diercks, P M Garnavich, R L Gilliland, C J Hogan, S Jha, R P Kirshner, B Leibundgut, M M Phillips, D Reiss, B P Schmidt, R A Schommer, R C Smith, J Spyromilio, C Stubbs, N B Suntzeff, J Tonry, AJ. 116A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P. M. Garnavich, R. L. Gilliland, C. J. Hogan, S. Jha, R. P. Kirshner, B. Leibundgut, M. M. Phillips, D. Reiss, B. P. Schmidt, R. A. Schommer, R. C. Smith, J. Spyromilio, C. Stubbs, N. B. Suntzeff, and J. Tonry. Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant. AJ, 116:1009-1038, September 1998.
First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters. D N Spergel, L Verde, H V Peiris, E Komatsu, M R Nolta, C L Bennett, M Halpern, G Hinshaw, N Jarosik, A Kogut, M Limon, S S Meyer, L Page, G S Tucker, J L Weiland, E Wollack, E L Wright, ApJS. 148D. N. Spergel, L. Verde, H. V. Peiris, E. Komatsu, M. R. Nolta, C. L. Bennett, M. Halpern, G. Hinshaw, N. Jarosik, A. Kogut, M. Limon, S. S. Meyer, L. Page, G. S. Tucker, J. L. Weiland, E. Wollack, and E. L. Wright. First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters. ApJS, 148:175-194, September 2003.
Cosmological parameters from SDSS and WMAP. M Tegmark, M A Strauss, M R Blanton, K Abazajian, S Dodelson, H Sandvik, X Wang, D H Weinberg, I Zehavi, N A Bahcall, F Hoyle, D Schlegel, R Scoccimarro, M S Vogeley, A Berlind, T Budavari, A Connolly, D J Eisenstein, D Finkbeiner, J A Frieman, J E Gunn, L Hui, B Jain, D Johnston, S Kent, H Lin, R Nakajima, R C Nichol, J P Ostriker, A Pope, R Scranton, U Seljak, R K Sheth, A Stebbins, A S Szalay, I Szapudi, Y Xu, J Annis, J Brinkmann, S Burles, F J Castander, I Csabai, J Loveday, M Doi, M Fukugita, B Gillespie, G Hennessy, D W Hogg, Ž Ivezić, G R Knapp, D Q Lamb, B C Lee, R H Lupton, T A Mckay, P Kunszt, J A Munn, L O'connell, J Peoples, J R Pier, M Richmond, C Rockosi, D P Schneider, C Stoughton, D L Tucker, D E Berk, B Yanny, D G York, Phys. Rev. D. 6910103501M. Tegmark, M. A. Strauss, M. R. Blanton, K. Abazajian, S. Dodelson, H. Sandvik, X. Wang, D. H. Weinberg, I. Zehavi, N. A. Bahcall, F. Hoyle, D. Schlegel, R. Scoccimarro, M. S. Vogeley, A. Berlind, T. Budavari, A. Connolly, D. J. Eisenstein, D. Finkbeiner, J. A. Frieman, J. E. Gunn, L. Hui, B. Jain, D. Johnston, S. Kent, H. Lin, R. Nakajima, R. C. Nichol, J. P. Ostriker, A. Pope, R. Scranton, U. Seljak, R. K. Sheth, A. Stebbins, A. S. Szalay, I. Szapudi, Y. Xu, J. Annis, J. Brinkmann, S. Burles, F. J. Castander, I. Csabai, J. Loveday, M. Doi, M. Fukugita, B. Gillespie, G. Hennessy, D. W. Hogg,Ž. Ivezić, G. R. Knapp, D. Q. Lamb, B. C. Lee, R. H. Lupton, T. A. McKay, P. Kunszt, J. A. Munn, L. O'Connell, J. Peoples, J. R. Pier, M. Richmond, C. Rockosi, D. P. Schneider, C. Stoughton, D. L. Tucker, D. E. vanden Berk, B. Yanny, and D. G. York. Cosmological parameters from SDSS and WMAP. Phys. Rev. D, 69(10):103501, May 2004.
The Evolution of a System of Gravitationally Interacting Point Masses. Y B , M A Podurets, AZh. 42963Y. B. Zel'dovich and M. A. Podurets. The Evolution of a System of Gravitationally Interacting Point Masses. AZh, 42:963, 1965.
|
[] |
[
"\"Light Sail\" Acceleration Revisited",
"\"Light Sail\" Acceleration Revisited",
"\"Light Sail\" Acceleration Revisited",
"\"Light Sail\" Acceleration Revisited"
] |
[
"Andrea Macchi \nCNR/INFM/polyLAB\nPisaItaly\n\nDipartimento di Fisica \"Enrico Fermi\"\nUniversità di Pisa\nLargo B. Pontecorvo 3I-56127PisaItaly\n",
"Silvia Veghini \nDipartimento di Fisica \"Enrico Fermi\"\nUniversità di Pisa\nLargo B. Pontecorvo 3I-56127PisaItaly\n",
"Francesco Pegoraro \nDipartimento di Fisica \"Enrico Fermi\"\nUniversità di Pisa\nLargo B. Pontecorvo 3I-56127PisaItaly\n",
"Andrea Macchi \nCNR/INFM/polyLAB\nPisaItaly\n\nDipartimento di Fisica \"Enrico Fermi\"\nUniversità di Pisa\nLargo B. Pontecorvo 3I-56127PisaItaly\n",
"Silvia Veghini \nDipartimento di Fisica \"Enrico Fermi\"\nUniversità di Pisa\nLargo B. Pontecorvo 3I-56127PisaItaly\n",
"Francesco Pegoraro \nDipartimento di Fisica \"Enrico Fermi\"\nUniversità di Pisa\nLargo B. Pontecorvo 3I-56127PisaItaly\n"
] |
[
"CNR/INFM/polyLAB\nPisaItaly",
"Dipartimento di Fisica \"Enrico Fermi\"\nUniversità di Pisa\nLargo B. Pontecorvo 3I-56127PisaItaly",
"Dipartimento di Fisica \"Enrico Fermi\"\nUniversità di Pisa\nLargo B. Pontecorvo 3I-56127PisaItaly",
"Dipartimento di Fisica \"Enrico Fermi\"\nUniversità di Pisa\nLargo B. Pontecorvo 3I-56127PisaItaly",
"CNR/INFM/polyLAB\nPisaItaly",
"Dipartimento di Fisica \"Enrico Fermi\"\nUniversità di Pisa\nLargo B. Pontecorvo 3I-56127PisaItaly",
"Dipartimento di Fisica \"Enrico Fermi\"\nUniversità di Pisa\nLargo B. Pontecorvo 3I-56127PisaItaly",
"Dipartimento di Fisica \"Enrico Fermi\"\nUniversità di Pisa\nLargo B. Pontecorvo 3I-56127PisaItaly"
] |
[] |
The dynamics of the acceleration of ultrathin foil targets by the radiation pressure of superintense, circularly polarized laser pulses is investigated by analytical modeling and particle-in-cell simulations. By addressing self-induced transparency and charge separation effects, it is shown that for "optimal" values of the foil thickness only a thin layer at the rear side is accelerated by radiation pressure. The simple "Light Sail" model gives a good estimate of the energy per nucleon, but overstimates the conversion efficiency of laser energy into monoenergetic ions.Radiation Pressure Acceleration (RPA) of ultrathin solid targets by superintense laser pulses has been proposed as a promising way to accelerate large numbers of ions up to "relativistic" energies, i.e. in the GeV/nucleon range[1,2,3,4,5,6,7,8,9]. The simplest model of this acceleration regime is that of a "perfect" (i.e. totally reflecting) plane mirror boosted by a light wave at perpendicular incidence[10], which is also known as the "Light Sail" (LS) model. The LS model predicts the efficiency η, defined as the ratio between the mechanical energy of the mirror over the electromagnetic energy of the light wave pulse, to be given bywhere V is the mirror velocity; hence, RPA becomes more and more efficient (η → 1) as β → 1. Heuristically, Eq.(1) can be explained by the conservation of the number of "photons" N of the light wave reflected by the moving mirror in a small time interval: each photon has energyhω, thus the total energy of the incident and reflected pulses are given by Nhω and Nhω r , where ω r = ω(1 − β)/(1 + β) due to the Doppler effect, and the energy transfered to the mirror is given by their differenceThe predictions of the LS model are very appealing for applications, but one may wonder to what extent this picture is appropriate to describe the acceleration of a solid target by a superintense laser pulse. In the present paper, we revisit the LS model with the help of simple modeling and particle-in-cell (PIC) simulations. We address issues outside the model itself, such as the effects of nonlinear reflectivity and charge depletion, and on this basis we explain a few features observed in simulations. Our main result is that the LS model is accurate in predicting the ion energy but overstimates the corresponding conversion efficiency, i.e. the fraction of the laser pulse energy transferred into quasi-monoenergetic ions, due to the fact that only a layer of the foil at its rear side is accelerated by RPA.Our analysis is confined to a one-dimensional (1D) approach for the sake of simplicity and because multidimensional simulations showed that a "quasi-1D" geometry has to be preserved in the acceleration stage (by using flat-top intensity profiles) to avoid early pulse transmission due to the expansion of the foil in the radial direction[11]. Circularly polarized pulses are used to reduce electron heating[12], an approach followed by several groups for efficient acceleration of thin foils[2,3,4,5,6,11]. We do not consider intensities high enough that ions become relativistic within the first laser cycle; this condition may affect the early stage of charge depletion (e.g. by narrowing the temporal scale separation between ions and electrons), and lead to different estimates[1,6].The LS model is based on the following equation of motion for the foil d dtwhere γ = 1/ 1 − β 2 , dX/dt = V , I is the light wave intensity, ρ and ℓ are the mass density and thickness of the foil, R(ω ′ ) is the reflectivity in the rest frame of the foil, and ω ′ = ω (1 − β)/(1 + β). For suitable expressions of R(ω ′ ), the final velocity β f can be obtained from Eq.(2) as a function of the pulse fluence F = Idt. For R = 1, one obtainsIn the last equality we wrote the fluence in dimensionless units as a 2 0 τ , where a 0 = I/m e c 3 n c is the dimensionless pulse amplitude and τ is the pulse duration in units of the laser period, and introduced the parameter ζ = π(n 0 /n c )(ℓ/λ) which characterizes the optical properties of a sub-wavelength plasma foil[13]. In these equations, n 0 is the initial electron density, n c = πm e c 2 /e 2 λ 2 is the cut-off density, and λ is the laser wavelength. In practical units, n c = 1.1 × 10 21 cm −3 [λ/µm] −2 and a 0 = (0.85/ √ 2)(Iλ 2 /10 18 W cm −2 µm 2 ) 1/2 for a circularly polarized laser pulse. Using Eq.(3) it is found that with a 1 ps = 10 −12 s, 1 PW = 10 12 W laser pulse and a 10 nm target of 1 g cm −3 density, ∼ 1 GeV per nucleon may be obtained. As the LS model assumes the target to be a perfect mirror (i.e. rigid and totally reflecting), it implies that all the ions are accelerated to the same velocity and the spectrum is perfectly monoenergetic.
|
10.1103/physrevlett.103.085003
|
[
"https://arxiv.org/pdf/0905.2068v2.pdf"
] | 46,378,808 |
0905.2068
|
152deebdc4d36c52ac944a0d0c37766ddbdebf70
|
"Light Sail" Acceleration Revisited
23 Jul 2009
Andrea Macchi
CNR/INFM/polyLAB
PisaItaly
Dipartimento di Fisica "Enrico Fermi"
Università di Pisa
Largo B. Pontecorvo 3I-56127PisaItaly
Silvia Veghini
Dipartimento di Fisica "Enrico Fermi"
Università di Pisa
Largo B. Pontecorvo 3I-56127PisaItaly
Francesco Pegoraro
Dipartimento di Fisica "Enrico Fermi"
Università di Pisa
Largo B. Pontecorvo 3I-56127PisaItaly
"Light Sail" Acceleration Revisited
23 Jul 2009(Dated: July 23, 2009)
The dynamics of the acceleration of ultrathin foil targets by the radiation pressure of superintense, circularly polarized laser pulses is investigated by analytical modeling and particle-in-cell simulations. By addressing self-induced transparency and charge separation effects, it is shown that for "optimal" values of the foil thickness only a thin layer at the rear side is accelerated by radiation pressure. The simple "Light Sail" model gives a good estimate of the energy per nucleon, but overstimates the conversion efficiency of laser energy into monoenergetic ions.Radiation Pressure Acceleration (RPA) of ultrathin solid targets by superintense laser pulses has been proposed as a promising way to accelerate large numbers of ions up to "relativistic" energies, i.e. in the GeV/nucleon range[1,2,3,4,5,6,7,8,9]. The simplest model of this acceleration regime is that of a "perfect" (i.e. totally reflecting) plane mirror boosted by a light wave at perpendicular incidence[10], which is also known as the "Light Sail" (LS) model. The LS model predicts the efficiency η, defined as the ratio between the mechanical energy of the mirror over the electromagnetic energy of the light wave pulse, to be given bywhere V is the mirror velocity; hence, RPA becomes more and more efficient (η → 1) as β → 1. Heuristically, Eq.(1) can be explained by the conservation of the number of "photons" N of the light wave reflected by the moving mirror in a small time interval: each photon has energyhω, thus the total energy of the incident and reflected pulses are given by Nhω and Nhω r , where ω r = ω(1 − β)/(1 + β) due to the Doppler effect, and the energy transfered to the mirror is given by their differenceThe predictions of the LS model are very appealing for applications, but one may wonder to what extent this picture is appropriate to describe the acceleration of a solid target by a superintense laser pulse. In the present paper, we revisit the LS model with the help of simple modeling and particle-in-cell (PIC) simulations. We address issues outside the model itself, such as the effects of nonlinear reflectivity and charge depletion, and on this basis we explain a few features observed in simulations. Our main result is that the LS model is accurate in predicting the ion energy but overstimates the corresponding conversion efficiency, i.e. the fraction of the laser pulse energy transferred into quasi-monoenergetic ions, due to the fact that only a layer of the foil at its rear side is accelerated by RPA.Our analysis is confined to a one-dimensional (1D) approach for the sake of simplicity and because multidimensional simulations showed that a "quasi-1D" geometry has to be preserved in the acceleration stage (by using flat-top intensity profiles) to avoid early pulse transmission due to the expansion of the foil in the radial direction[11]. Circularly polarized pulses are used to reduce electron heating[12], an approach followed by several groups for efficient acceleration of thin foils[2,3,4,5,6,11]. We do not consider intensities high enough that ions become relativistic within the first laser cycle; this condition may affect the early stage of charge depletion (e.g. by narrowing the temporal scale separation between ions and electrons), and lead to different estimates[1,6].The LS model is based on the following equation of motion for the foil d dtwhere γ = 1/ 1 − β 2 , dX/dt = V , I is the light wave intensity, ρ and ℓ are the mass density and thickness of the foil, R(ω ′ ) is the reflectivity in the rest frame of the foil, and ω ′ = ω (1 − β)/(1 + β). For suitable expressions of R(ω ′ ), the final velocity β f can be obtained from Eq.(2) as a function of the pulse fluence F = Idt. For R = 1, one obtainsIn the last equality we wrote the fluence in dimensionless units as a 2 0 τ , where a 0 = I/m e c 3 n c is the dimensionless pulse amplitude and τ is the pulse duration in units of the laser period, and introduced the parameter ζ = π(n 0 /n c )(ℓ/λ) which characterizes the optical properties of a sub-wavelength plasma foil[13]. In these equations, n 0 is the initial electron density, n c = πm e c 2 /e 2 λ 2 is the cut-off density, and λ is the laser wavelength. In practical units, n c = 1.1 × 10 21 cm −3 [λ/µm] −2 and a 0 = (0.85/ √ 2)(Iλ 2 /10 18 W cm −2 µm 2 ) 1/2 for a circularly polarized laser pulse. Using Eq.(3) it is found that with a 1 ps = 10 −12 s, 1 PW = 10 12 W laser pulse and a 10 nm target of 1 g cm −3 density, ∼ 1 GeV per nucleon may be obtained. As the LS model assumes the target to be a perfect mirror (i.e. rigid and totally reflecting), it implies that all the ions are accelerated to the same velocity and the spectrum is perfectly monoenergetic.
The dynamics of the acceleration of ultrathin foil targets by the radiation pressure of superintense, circularly polarized laser pulses is investigated by analytical modeling and particle-in-cell simulations. By addressing self-induced transparency and charge separation effects, it is shown that for "optimal" values of the foil thickness only a thin layer at the rear side is accelerated by radiation pressure. The simple "Light Sail" model gives a good estimate of the energy per nucleon, but overstimates the conversion efficiency of laser energy into monoenergetic ions.
Radiation Pressure Acceleration (RPA) of ultrathin solid targets by superintense laser pulses has been proposed as a promising way to accelerate large numbers of ions up to "relativistic" energies, i.e. in the GeV/nucleon range [1,2,3,4,5,6,7,8,9]. The simplest model of this acceleration regime is that of a "perfect" (i.e. totally reflecting) plane mirror boosted by a light wave at perpendicular incidence [10], which is also known as the "Light Sail" (LS) model. The LS model predicts the efficiency η, defined as the ratio between the mechanical energy of the mirror over the electromagnetic energy of the light wave pulse, to be given by
η = 2β/(1 + β), β = V /c,(1)
where V is the mirror velocity; hence, RPA becomes more and more efficient (η → 1) as β → 1. Heuristically, Eq.(1) can be explained by the conservation of the number of "photons" N of the light wave reflected by the moving mirror in a small time interval: each photon has energyhω, thus the total energy of the incident and reflected pulses are given by Nhω and Nhω r , where ω r = ω(1 − β)/(1 + β) due to the Doppler effect, and the energy transfered to the mirror is given by their difference
[2β/(1 + β)]Nhω.
The predictions of the LS model are very appealing for applications, but one may wonder to what extent this picture is appropriate to describe the acceleration of a solid target by a superintense laser pulse. In the present paper, we revisit the LS model with the help of simple modeling and particle-in-cell (PIC) simulations. We address issues outside the model itself, such as the effects of nonlinear reflectivity and charge depletion, and on this basis we explain a few features observed in simulations. Our main result is that the LS model is accurate in predicting the ion energy but overstimates the corresponding conversion efficiency, i.e. the fraction of the laser pulse energy transferred into quasi-monoenergetic ions, due to the fact that only a layer of the foil at its rear side is accelerated by RPA.
Our analysis is confined to a one-dimensional (1D) approach for the sake of simplicity and because multidimensional simulations showed that a "quasi-1D" geometry has to be preserved in the acceleration stage (by using flat-top intensity profiles) to avoid early pulse transmission due to the expansion of the foil in the radial direction [11]. Circularly polarized pulses are used to reduce electron heating [12], an approach followed by several groups for efficient acceleration of thin foils [2,3,4,5,6,11]. We do not consider intensities high enough that ions become relativistic within the first laser cycle; this condition may affect the early stage of charge depletion (e.g. by narrowing the temporal scale separation between ions and electrons), and lead to different estimates [1,6].
The LS model is based on the following equation of motion for the foil d dt
(βγ) = 2I(t − X/c) ρℓc 2 R(ω ′ ) 1 − β 1 + β ,(2)
where γ = 1/ 1 − β 2 , dX/dt = V , I is the light wave intensity, ρ and ℓ are the mass density and thickness of the foil, R(ω ′ ) is the reflectivity in the rest frame of the foil, and ω ′ = ω (1 − β)/(1 + β). For suitable expressions of R(ω ′ ), the final velocity β f can be obtained from Eq.(2) as a function of the pulse fluence F = Idt. For R = 1, one obtains
β f = (1 + E) 2 − 1 (1 + E) 2 + 1 , E = 2F ρℓc 2 = 2π Z A m e m p a 2 0 τ ζ .(3)
In the last equality we wrote the fluence in dimensionless units as a 2 0 τ , where a 0 = I/m e c 3 n c is the dimensionless pulse amplitude and τ is the pulse duration in units of the laser period, and introduced the parameter ζ = π(n 0 /n c )(ℓ/λ) which characterizes the optical properties of a sub-wavelength plasma foil [13]. In these equations, n 0 is the initial electron density, n c = πm e c 2 /e 2 λ 2 is the cut-off density, and λ is the laser wavelength. In practical units, n c = 1.1 × 10 21 cm −3 [λ/µm] −2 and a 0 = (0.85/ √ 2)(Iλ 2 /10 18 W cm −2 µm 2 ) 1/2 for a circularly polarized laser pulse. Using Eq.(3) it is found that with a 1 ps = 10 −12 s, 1 PW = 10 12 W laser pulse and a 10 nm target of 1 g cm −3 density, ∼ 1 GeV per nucleon may be obtained. As the LS model assumes the target to be a perfect mirror (i.e. rigid and totally reflecting), it implies that all the ions are accelerated to the same velocity and the spectrum is perfectly monoenergetic. 1 shows a parametric study of the ion spectrum vs. ℓ and a 0 from PIC simulations. For all runs, n 0 = 250n c , Z/A = 1/2 and the pulse has a flat-top envelope with 1 cycle rise and fall times and 8 cycles plateau. For each value of a 0 and for ℓ less than a threshold value ℓ opt we observe a narrow spectral peak, whose energy increases with decreasing ℓ and is in very good agreement with the predictions of the LS model, assuming R = 1. A typical lineout of the spectrum is shown in Fig.2 a). For ℓ > ℓ opt , the peak disappears and a thermal-like spectrum is observed. This is correlated with an almost complete expulsion of the electrons from the foil in the forward direction at the beginning of the interaction, leading to a Coulomb explosion of the ions.
The results of Fig.1 show that the LS model is useful for quantitative predictions of the ion energy, but also suggest several questions of interest both for the basic physics of RPA and its applications. How is ℓ opt determined? Does the reflectivity of the foil and relativistic effects on the latter play a role? As the radiation pressure tends to separate electrons from ions, does the foil remain neutral before and/or after the acceleration stage? Moreover, as shown shown in Fig.2 b), the "monoenergetic" peak contains just a fraction of the total number of ions, and such fraction depends on ℓ and a 0 . This is different from the assumption of the LS model, which assumes all the ions in the foil to move coherently with the foil, and may sound surprising, since the peak energy is in agreement with the LS formula where the whole mass of the foil, including low-energy ions out of spectral peak, is used. In the following we provide answers to the questions above by discussing effects not included in the simplest LS model, i.e. beyond the description of the foil as a perfect, rigid mirror.
First we discuss effects related to the reflectivity R of the plasma foil. For very high intensities, electrons oscillate with relativistic momenta in the laser field, leading Fig.1. b) Fraction of ions contained in the spectral peak vs. the target thickness ℓ for three values of a0 = 10 (black, crosses), 30 (blue, triangles) and 50 (red, squares). The dashed lines correspond to Eq.(8) for F . All other parameters for both a) and b) are the same as in Fig.1. c) Approximate profiles of ion (ni, green) and electron (ne, blue) densities and of the electrostatic field (Ex, red) in the early stage of the interaction, before ions move.
to a nonlinear dependence of R upon a 0 . An explicit expression can be found analytically by using the model of a delta-like "thin foil" [13], i.e. a plasma slab located at x = 0 with electron density n e (x) = n 0 ℓδ(x). The expression obtained for R in the rest frame of the foil is very well approximated by
R ≃ ζ 2 /(1 + ζ 2 ) (a 0 < 1 + ζ 2 ) ζ 2 /a 2 0 (a 0 > 1 + ζ 2 ) .(4)
A threshold for self-induced transparency of the foil may thus be defined as a 0 = 1 + ζ 2 ≃ ζ when ζ ≫ 1, i.e. in most cases of interest. According to Eq.(4), the total radiation pressure P rad on the target
P rad = 2RI/c = 2m e c 3 n c a 2 0 R(5)
becomes independent upon a 0 for a 0 > ζ. Thus, the maximum radiation pressure is obtained for a 0 < ∼ ζ, and in this condition typically R ≃ 1 for solid densities. This suggests that the optimal thickness ℓ opt is determined by the condition a 0 ≃ ζ, in good agreement with the simulation results in Fig.1 and as also found by other studies [8,15].
The nonlinear reflectivity of the thin foil is determined by the transverse motion of electrons (in the foil plane). However, for a thin but "real" target the radiation pressure tends to push electrons also in longitudinal direction, and may remove them from the foil. Let us compare P rad with the electrostatic pressure P es that would be generated if all electrons would be removed from the foil. The condition P rad ≥ P es = 2π(en 0 ℓ) 2 (6) corresponds to the threshold for the removal of all electrons from the foil. However, when Eq.(4) is used for a 0 ≤ 1 + ζ 2 , Eq.(6) reduces to a 0 ≥ 1 + ζ 2 , while for a 0 ≥ 1 + ζ 2 we find that P rad = P es holds. It is thus possible to produce a density distribution where all electrons pile at the rear surface of the foil. In fact, if a 0 < ∼ ζ and R ≃ 1, the laser pulse compresses the electron layer while keeping R constant since the product n e ℓ does not change during the compression; at the same time almost no electrons are ejected from the rear side because the ponderomotive force vanishes there if R ≃ 1, and so does the electrostatic field: the qualitative profiles of the electron density and of the electric field are shown in Fig.2 c). Since for a 0 close to ζ the equilibrium between the electrostatic and radiation pressures occurs only when the depth d of the region of charge depletion is close to ℓ, electrons are compressed in a very thin layer. The depletion depth d may be estimated from the equilibrium condition
P es = 2π(en 0 d) 2 ≃ 2I/c (7)
when R ≃ 1, which yields d ≃ ℓ(a 0 /ζ) < ∼ ℓ. It is worth to point out that these considerations are appropriate for a circularly polarized laser pulse; for linear polarization, all electrons may be expelled for a transient stage under the action of the J × B force whose peak value per unit surface exceeds 2RI/c due to its oscillating component. Complete expulsion of electrons for a 0 > ζ has been discussed in Ref. [14].
The snaphshots from a PIC simulation shown in Fig.3 for a case with a 0 = 30 and ζ = 31.4 confirm the scenario outlined above. The electron density n e reaches values (out of scale in Fig.3) up to tens of the initial density. A very high resolution ∆x = λ/2000 is used to resolve the density spike properly. For a laser pulse with flat-top envelope the density spiking at the rear side of the foil is particularly evident, but we verified that it occurs also for a "sin 2 " envelope. Similar features were observed also in Refs. [5,8], but not discussed in detail.
The electron compression in a thin layer during the initial "hole boring" stage has important consequences for the later acceleration stage. Let us refer to the approximate field profiles in the initial stage, sketched in Fig.2 c), which were the basis of the model presented in Ref. [12]. This model suggests that only the ions located initially in the electron compression layer (d < x < ℓ) will be bunched and undergo RPA (via a "cyclic" acceleration as discussed in Refs. [2,3,4]) because for these ions only the electrostatic pressure balances the radiation pressure, The ion density ni (green), the electron density ne (blue), the longitudinal electric field Ex (red, dashed) and the pulse field amplitude EL = E 2 y + E 2 z (red, dotted) are shown. The target left boundary is at x = 0 where the pulse impinges at t = 0. Times are normalized to the laser period T , fields to meωc/e, and densities of nc. The laser pulse has amplitude a0 = 30 and the foil thickness is ℓ = 0.04λ. All other parameters are the same of Fig.1. while the ions in the electron depletion layer (0 < x < d) will be accelerated via Coulomb explosion, i.e. by their own space-charge field. This is exactly what is observed in the PIC simulations, both in the density profiles (see Fig.3 at t = 2.2T ) and in the ion spectra. This effect also explains how RPA with circularly polarized pulses may work also in double layer targets [15], if the thickness of a thin layer on the rear side matches ℓ eff = ℓ − d. Fig.2 shows ion spectra for the same simulation of Fig.3 and for a simulation with the same parameters, but where ions in a surface layer of 0.01λ thickness have been replaced by protons. A fraction of heavier ions is also accelerated to the same energy per nucleon as the protons, a typical feature of RPA of a thin plasma foil.
As an additional consequence of the piling up of electrons at the rear surface, the portion of the foil which is boosted by the laser pulse is negatively charged due to the excess of electrons. However, the simulations show that when the laser pulse is over the excess electrons detach from the foil and move in the backward direction, so that the accelerated layer is eventually neutral. This is important to avoid a later Coulomb explosion of the layer and to preserve a monoenergetic spectrum. During the acceleration, the longitudinal field at the surface of the accelerated layer is almost constant implying that the charge there contained is also constant. It is thus possible to estimate the fraction F of accelerated ions from the initial equilibrium condition, Eq.(7), as
F ≃ ℓ eff /ℓ ≃ (1 − a 0 /ζ).(8)
The agreement with data in Fig.2 b) is qualitative, with large deviations as F becomes significantly smaller than one. As explained below, a lower bound on F is determined by energy conservation. A simple argument of force balance also explains why the energy of the spectral peak in Fig.2 is in very good agreement with the predictions of the LS model where the initial value ℓ of the foil thickness is used, while only a layer of thickness ℓ eff < ℓ is accelerated via RPA. Let us refer again to the profiles of Fig.2. The equilibrium condition for electrons implies
2I c . = ℓ d eE 0 ℓ − x ℓ − d n p dx = 1 2 en 0 E 0 ℓ.(9)
The electric field pushes ions in the compression layer d < x < ℓ, exerting a total pressure
P c = ℓ d eE 0 ℓ − x ℓ − d n i dx = 2I c ℓ − d ℓ ,(10)
where we used Eq.(9) and assumed R = 1. The equation of motion for the ion layer, in the early stage, can be thus written as
d dt [ρ(ℓ − d)γβ] = P c c = 2I c ℓ − d ℓ ,(11)
which is trivially equivalent to
d dt (ρℓγβ) = 2I c ,(12)
i.e. to the equation of motion one would write for the whole foil. The argument may be applied also when the layer is in motion leading to the same conclusion. Having the same β(t) as the whole foil implies that the energy per nucleon and the efficiency (1) will be also the same, but the total kinetic energy will be lower for the thin layer. The rest of the absorbed energy is stored in the electrostatic field and as kinetic energy of the ions in the x < X(t) region. Let us consider for example the energy stored in the electrostatic field. At the time t, the field E x between the initial (x = 0) and the actual (x = X(t)) positions of the front surface of the foil is given approximately by E x = E 0 x/X(t), where E 0 = 4πen 0 d, corresponding to an electrostatic energy per unit surface
U es = U es (t) = X(t) 0 E 2 x (x, t) 8π dx,(13)
which varies in time as
dU es dt = 1 8π E 2 x [X(t)] dX dt = 1 8π E 2 0 βc.(14)
Dividing (14) by the laser intensity we obtain the "conversion efficiency" into electrostatic energy η es
η es = 1 I dU es dt = 2β d ℓ 2 ζ a 0 2 .(15)
If ζ ≃ a 0 and thus d ≃ ℓ, we would obtain η es ≃ 2β > η that is unphysical. Thus, the energy stored in the electrostatic field also prevents the accelerated layer thickness to shrink to zero.
In conclusion, we have revisited the "Light Sail" model of Radiation Pressure Acceleration of a thin plasma foil. The nonlinear reflectivity of the foil determines the "optimal" condition ζ ≃ a 0 , for which the energy in the RPA spectral peak is highest and in good agreement with the LS model formula where the total thickness (or the total mass) of the foil enters as a parameter. However, not all the foil is accelerated, but only a thin layer at the rear side of thickness ℓ eff < ℓ; the apparent paradox is solved by observing that, to keep electrons in a mechanical quasi-equilibrium, the electrostatic pressure pushing ions in the accelerated layer is ℓ eff /ℓ times the radiation pressure on electrons, so that the equation of motion for the thin layer is the same as if the whole foil were accelerated. Finally, we showed that the energy stored in the electrostatic field is comparable to the kinetic energy and must be taken into account. For applications, the most relevant consequences and differences with respect to the simplest LS picture are that the number of "monoenergetic" ions is reduced, so that the actual efficiency may be quite lower than given by Eq. (1), and that also light ions in a thin layer at the rear surface (e.g., hydrogen impurities) may be accelerated by RPA. Support from CNR via a RSTL project and use of supercomputing facilities at CINECA (Bologna, Italy) sponsored by the CNR/INFM supercomputing initiative are acknowledged.
FIG. 1 :
1(Color online) Parametric study of the ion energy spectra vs. laser amplitude a0 and foil thickness ℓ. The contours of log 10 fi(E) are shown, with fi(E) the energy per nucleon distribution normalized to unity. For all runs, n0 = 250nc, Z/A = 1/2, τ = 9. The dashed line shows the prediction of the LS model for the ion energy. The dotted horizontal line marks ℓopt given by the ζ = a0 condition.
Fig.
Fig.1 shows a parametric study of the ion spectrum vs. ℓ and a 0 from PIC simulations. For all runs, n 0 = 250n c , Z/A = 1/2 and the pulse has a flat-top envelope with 1 cycle rise and fall times and 8 cycles plateau. For each value of a 0 and for ℓ less than a threshold value ℓ opt we observe a narrow spectral peak, whose energy increases with decreasing ℓ and is in very good agreement with the predictions of the LS model, assuming R = 1. A typical lineout of the spectrum is shown in Fig.2 a). For ℓ > ℓ opt , the peak disappears and a thermal-like spectrum is observed. This is correlated with an almost complete expulsion of the electrons from the foil in the forward direction at the beginning of the interaction, leading to a Coulomb explosion of the ions. The results of Fig.1 show that the LS model is useful for quantitative predictions of the ion energy, but also suggest several questions of interest both for the basic physics of RPA and its applications. How is ℓ opt determined? Does the reflectivity of the foil and relativistic effects on the latter play a role? As the radiation pressure tends to separate electrons from ions, does the foil remain neutral before and/or after the acceleration stage? Moreover, as shown shown in Fig.2 b), the "monoenergetic" peak contains just a fraction of the total number of ions, and such fraction depends on ℓ and a 0 . This is different from the assumption of the LS model, which assumes all the ions in the foil to move coherently with the foil, and may sound surprising, since the peak energy is in agreement with the LS formula where the whole mass of the foil, including low-energy ions out of spectral peak, is used. In the following we provide answers to the questions above by discussing effects not included in the simplest LS model, i.e. beyond the description of the foil as a perfect, rigid mirror. First we discuss effects related to the reflectivity R of the plasma foil. For very high intensities, electrons oscillate with relativistic momenta in the laser field, leading
FIG. 2 :
2(Color online) a): Ion energy spectra (in energy per nucleon) from a simulation with a0 = 30 and a ℓ = 0.04λ thick foil of a single ion species with Z/A = 1/2 (top) and one with the same parameters but where ions in a thin surface layer (0.01λ) at the rear side are replaced by protons (bottom).
FIG. 3 :
3(Color online) Snapshots from a 1D PIC simulation of the interaction of a laser pulse with a thin plasma slab.
* Electronic address: [email protected]. * Electronic address: [email protected]
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|
[] |
[
"ONE-DIMENSIONAL GAME-THEORETIC DIFFERENTIAL EQUATIONS",
"ONE-DIMENSIONAL GAME-THEORETIC DIFFERENTIAL EQUATIONS"
] |
[
"Rafa L M Lochowski ",
"ANDNicolas Perkowski ",
"David J Prömel "
] |
[] |
[] |
We provide a very brief introduction to typical paths and the corresponding Itô type integration. Relying on this robust Itô integration, we prove an existence and uniqueness result for one-dimensional differential equations driven by typical paths with non-Lipschitz continuous coefficients in the spirit of Yamada-Watanabe as well as an approximation result in the spirit of Doss-Sussmann.
|
10.1016/j.ijar.2021.03.003
|
[
"https://arxiv.org/pdf/2101.08041v1.pdf"
] | 231,648,100 |
2101.08041
|
d6b493b961131cd9acbe944565e6d74e496b2c43
|
ONE-DIMENSIONAL GAME-THEORETIC DIFFERENTIAL EQUATIONS
20 Jan 2021
Rafa L M Lochowski
ANDNicolas Perkowski
David J Prömel
ONE-DIMENSIONAL GAME-THEORETIC DIFFERENTIAL EQUATIONS
20 Jan 2021model-free financeVovk's outer measurepathwise stochastic calculusstochastic differential equationsYamada-Watanabe theoremDoss-Sussmann approximation Mathematics Subject Classification (2020): 91A4060H10
We provide a very brief introduction to typical paths and the corresponding Itô type integration. Relying on this robust Itô integration, we prove an existence and uniqueness result for one-dimensional differential equations driven by typical paths with non-Lipschitz continuous coefficients in the spirit of Yamada-Watanabe as well as an approximation result in the spirit of Doss-Sussmann.
Introduction
In the past decade, ideas from game-theoretic probability (see the books [SV01,SV19]) have led to various related notions of outer measures based on the concept of pathwise super-hedging coming from mathematical finance, see e.g. [Vov09,TKT09,Vov12,PP16]. These outer measures allow to use arbitrage considerations to examine which path properties are satisfied by "typical (price) paths", that is, which path properties hold except null sets with respect to these outer measures. For instance, V. Vovk proved that non-constant typical continuous paths have infinite p-variation for p < 2 and finite p-variation for p > 2, see [Vov08]. Additionally, due to the financial nature of these outer measures, they found many applications in mathematical finance under model uncertainty, like robust versions of the pricing-hedging duality [BCH + 17, BKPT19, BKN20, CKPS19], the role of measurability to avoid arbitrage [Vov17] and game-theoretic portfolio theory [SV19,Chapter 17].
Since typical paths are too irregular to apply classical calculus, a novel approach to work with typical paths is required. This motivated the development of a game-theoretic Itô calculus, which can be viewed as a robust version of the stochastic Itô calculus developed for (semi-)martingales like the Brownian motion, cf. for instance [KS88,RY99]. Game-theoretic ideas coming in particular from mathematical finance, indeed, allowed to set up a rather mature game-theoretic (or model-free) Itô integration for typical paths, see [PP16,Vov16,LPP18]. Since the stochastic Itô integration was originally motivated by stochastic differential equations, it rose the natural question whether the game-theoretic Itô integration is sufficiently powerful to treat differential equations driven by typical paths. A first affirmative answer to this question was given by [BKN19]. Assuming the coefficients of the differential equations are Lipschitz continuous, [BKN19] introduces an outer measure allowing to show existence and uniqueness results for differential equations driven by typical paths even in a Hilbert space setting. Based on an alternative outer measure, the work [CG LM18] provides also existence and uniqueness results for multi-dimensional differential equations with Lipschitz coefficients driven by typical paths.
The general interest to gain a deeper understanding of differential equations driven by typical paths stems from their benefits on a theoretical as well as on an applied level. To illustrate these benefits, let us recall that differential equations perturbed by random noises are frequently used as mathematical models for real-world evolutions. As commonly observed, mathematical modeling comes with the issue of model uncertainty. One approach in mathematical finance to treat such model uncertainty is to simultaneously work under a family of probability measures instead of under one fixed probability measure, cf. e.g. [STZ11]. The solution theory for differential equations driven by typical paths allows immediately to simultaneously work under sensible families of probability measures, see Proposition 2.2, and such represents in this direction a more general approach to (stochastic) differential equations than the classical Itô theory.
In the present work we study one-dimensional differential equations driven by typical paths. Similarly to the theory of stochastic differential equations, we shall see that the one-dimensional case is special in certain aspects. For instance, it allows to treat stochastic differential equations with non-Lipschitz continuous diffusion coefficients, like the Cox-Ingersoll-Ross process, cf. Example 4.4, which is a frequently used model for the evolution of interest rates and of the volatility on financial markets.
First, we show the existence of a unique solution to differential equations driven by typical paths with Hölder continuous diffusion coefficients of order 1/2, see Subsection 4.2. This corresponds to the famous Yamada-Watanabe theorem [YW71] from probability theory, which is only known to hold true in a basically one-dimensional setting. Like for the classical Yamada-Watanabe result, it is remarkable that differential equations driven by typical paths possess unique solutions under a non-Lipschitz assumption while usually uniqueness results for differential equations require the involved vector fields to be almost Lipschitz continuous. Intuitively, this demonstrates the regularizing effect of typical paths similar to the one of a Brownian motion.
Secondly, we show a stability result between ordinary differential equations and differential equations driven by typical paths, see Subsection (4.3), saying that suitable ordinary differential equations can approximate a differential equation driven by typical paths. This result can be viewed as a game-theoretic version of the Doss-Sussmann theorem [Dos77,Sus78] from probability theory. The stability theorem of Doss and Sussmann can be again motivated by questions regarding the suitable mathematical model for a real-world evolutions. While for example stochastic differential equations driven by a Brownian motion form convenient mathematical models, the underlying real-world evolutions might be actually more suitably represented by a differential equation perturbed by a non-Markov noise process of bounded variation. Hence, a stability result like the Doss-Sussmann theorem is of upmost importance from a modeling perspective.
Organization of the paper: Section 2 introduces the game-theoretic outer measure and the notion of typical paths. In Section 3 we present the essential results regarding game-theoretic integration and provide an Itô type formula. Section 4 treats one-dimensional differential equations driven by typical paths.
Game-theoretic probability and typical paths
We consider the sample space Ω := C([0, ∞); R), which is the space of all continuous functions ω : [0, ∞) → R with ω(0) = 0, and the coordinate process on Ω is denoted by S = (S t ) t∈[0,∞) where S t (ω) := ω(t). We equip the space Ω with the right-continuous filtration (F t ) t∈[0,∞) with F t := s>t σ(S u : u ≤ s) and set F := t∈[0,∞) F t . Stopping times τ and the associated σ-algebras F τ are defined as usual. A mapping F : Ω → R is called gametheoretic variable if F is F-measurable. The indicator function of a set A is denoted by 1 A , x ∧ y := min{x, y} for x, y ∈ R and the space R d is equipped with the Euclidean norm · .
A process H : Ω × [0, ∞) → R is called a simple strategy if it is of the form H t (ω) = ∞ n=0 F n (ω)1 (τn(ω),τ n+1 (ω)] (t), (ω, t) ∈ Ω × [0, ∞),
where F n : Ω → R are F τn -measurable bounded functions for n ∈ N and 0 = τ 0 (ω) ≤ τ 1 (ω) ≤ . . . are stopping times such that for every ω ∈ Ω one has lim n→∞ τ n (ω) = ∞ and for every interval [s, t] ⊂ [0, ∞) there are at most finitely many stopping times (τ n ) n=N,...,M satisfying τ n ∈ [s, t]. For such a simple strategy H the corresponding capital process
(H · S) t (ω) = ∞ n=0 F n (ω)(S τ n+1 (ω)∧t (ω) − S τn(ω)∧t (ω)) is well-defined for every ω ∈ Ω and every t ∈ [0, ∞). A simple strategy H is called λ-admissible for λ ≥ 0 if (H · S) t (ω) ≥ −λ for all t ∈ [0, ∞) and all ω ∈ Ω.
The previous definitions come all with natural interpretation from a game-theoretic and a financial perspective. The sample space Ω can be interpreted as the set of all possible price evolution on a financial market. In this context, a simple strategy H represents a trading strategy of an investor, who changes her position at preselected stopping times, and the process ((H · S) t ) t∈[0,∞) stands for the capital generated by trading according to H into the price process (S t ) t∈[0,∞) . The admissible condition can be understood as a maximal credit limit as it is commonly imposed in mathematical finance.
Like in classical mathematical finance, to consider trading only with respect to simple strategies is often not sufficient. Therefore, we need to work with all capital processes in the the liminf-closure of capital processes generated by simple strategies. This is comparable with the Itô integral, which is an operator on the L 2 -closure of simple integrands, except that we need to work in the present setting with a pointwise closure instead of a closure with respect to a probability measure. For this purpose, we introduce H λ for the set of λ-admissible simple strategies and the set of capital gain processes by V λ := C · = lim inf n→∞ (H n · S) · : (H n ) n∈N ⊂ H λ for λ ≥ 0. This allows us to introduce an outer measure and the notion of typical paths, as initiated by Vovk [Vov08]. While Vovk's original definition was based on a closure of simple strategies using countable convex combinations of them, we rely here on the set V λ . Of course, both ways to introduce an outer measure are naturally justified, see e.g. [PP16, Section 2.3]. The following definition presents the modified Vovk's outer measure as introduced by [PP15,PP16].
Definition 2.1. LetΩ ⊂ Ω be a non-empty set. The outer measure P (·;Ω) of the set A ⊆Ω is defined as the cheapest super-hedging price for 1 A , that is
P (A;Ω) := inf λ ≥ 0 : ∃C ∈ V λ s.t. ∀ω ∈Ω λ + lim inf t→∞ C t (ω) ≥ 1 A (ω) .
A set of paths A ⊆Ω is called a null set inΩ if P (A;Ω) = 0. A property (P) holds for typical paths inΩ if the set A where (P) is violated is a null set w.r.t. P (·;Ω).
Furthermore, we set P (A) := P (A; Ω) and say a property (P) holds for typical paths if it holds for typical paths in Ω.
Keeping the financial interpretation of the above definitions in mind, the outer measure P represents the cheapest super-hedging price, essentially as in the classical setting of mathematical finance but with one important difference: we require here super-hedging for all ω ∈ Ω and not just almost surely. An additional reason leading to the great interest of the outer measure P in mathematical finance under model uncertainty is that it dominates all local martingale measures on the space Ω. Recall, a probability measure Q is called a (local) martingale measure if the coordinate process (S t ) t∈[0,∞) is a (local) martingale w.r.t. Q. Local martingale measures appear in mathematical finance to characterize arbitrage-free market models and as "pricing" measures for financial derivatives.
Furthermore, a null set can essentially be viewed as a model-independent arbitrage opportunity of the first kind, cf. [PP15, Lemma 3.2]. Let us recall that, given a probability measure P on (Ω, F), we say that (S t ) t∈[0,∞) satisfies no arbitrage of the first kind (NA1) under P if the set
W ∞ 1 := 1 + ∞ 0 H u dS u : H ∈ H 1 is bounded in probability, that is if lim n→∞ sup X∈W ∞ 1 P(X ≥ n) = 0.
Here H 1 stands for the set of all integrable processes w.r.t. P.
The following proposition collects properties of the outer measure P and presents its link to mathematical finance, which we discussed vaguely in the previous paragraph.
Proposition 2.2 (Proposition 3.3 in [PP15]).
(i) P is an outer measure with P (Ω) = 1, i.e. P is non-decreasing, countably sub-additive, and P (∅) = 0. (ii) Let P be a probability measure on (Ω, F) such that the coordinate process (S t ) t∈[0,∞) is a local martingale under P, and let A ∈ F. Then P(A) ≤ P (A). (iii) Let A ∈ F be a null set, and let P be a probability measure on (Ω, F) such that the coordinate process (S t ) t∈[0,∞) satisfies (NA1) under P. Then P(A) = 0.
φ : [0, ∞) → [0, ∞) satisfying φ(0) = 0 is said to be a time-change.
The set of all time-changes will be denoted by G 0 and the group of all time-changes that are strictly increasing and unbounded will be denoted by
G. A subset A ⊂ Ω is called time-superinvariant if for each ω ∈ Ω for all φ ∈ G 0 it holds that (2.1) ω • φ ∈ A ⇒ ω ∈ A. A set A ⊂ Ω is called time-invariant if (2.1) holds true for all φ ∈ G.
For a comprehensive and intuitive explanation of time-superinvariance we refer the interested reader to [Vov12, Remark 3.3]. With these definitions at hand, we can present Vovk's game-theoretic Dambis-Dubins-Schwarz theorem.
Theorem 2.4 (Theorem 3.1 in [Vov12]). Each time-superinvariant set A ⊂ Ω satisfies P (A) = W(A).
Of course, since we consider here a slight modification of Vovk's original outer measure, one needs to verify that the game-theoretic Dambis-Dubins-Schwarz theorem due to Vovk still holds for P , see [BCH + 17, Theorem 2.6]. Keeping in mind Theorem 2.4, we observe the following regarding time-superinvariant sets: the definition of time-superinvariance ensures that all martingale measures on Ω assign a time-superinvariant set exactly the same probability. This also motivates the name "Dambis-Dubins-Schwarz theorem", which roughly says that every one-dimensional martingale is a time-changed Brownian motion.
If one considers only sets of nowhere constant and divergent paths, the notions of timesuperinvariance and time-invariance turn out to be equivalent.
Definition 2.5. A path ω ∈ Ω is said to be nowhere constant if there is no interval (s, t) ⊂ [0, ∞) such that ω is constant on (s, t) and ω is said to be divergent if there is no c ∈ R such that lim t→∞ ω(t) = c.
The set DS ⊂ Ω denotes the set of all ω ∈ Ω that are nowhere constant and divergent.
Lemma 2.6 (Lemma 3.5 in [Vov12]
). A set A ⊂ DS is time-superinvariant if and only if it is time-invariant.
Itô integration w.r.t. typical paths
In the spirit of stochastic Itô integration, it is possible to develop a game-theoretic integration theory for typical paths, see [PP16,Vov16,LPP18] and also [SV19,Chapter 14]. Like the classical Itô integration ([RY99, Chapter IV]), the game-theoretic integration requires as a fundamental ingredient the existence of quadratic variation for typical paths.
3.1. Quadratic variation. For a continuous path ω : [0, ∞) → R and n ∈ N we introduce the Lebesgue stopping times σ n 0 (ω) := 0 and σ n k (ω) := inf t ≥ σ n k−1 : ω(t) ∈ 2 −n Z and ω(t) = ω(σ n k−1 ) , (3.1) for k ∈ N and ω ∈ Ω. For n ∈ N the discrete quadratic variation of ω is given by
V n t (ω) := ∞ k=0 ω(σ n k+1 (ω) ∧ t) − ω(σ n k (ω) ∧ t) 2 , t ∈ [0, ∞).
To establish the convergence of the sequence (V n · (ω)) n∈N of discrete quadratic variations, we recall the concept of locally uniform convergence in
C([0, ∞); R). A sequence (f n ) n∈N ⊂ C([0, ∞); R) is said to converge locally uniformly to f ∈ C([0, ∞); R) if lim n→∞ sup x∈[0,T ] f n (x) − f (x) = 0, for every T > 0.
In this case, f is called the locally uniform limit of (f n ) n∈N .
Proposition 3.1. For typical paths ω ∈ Ω, the quadratic variation
S t (ω) := lim n→∞ V n t (ω), t ∈ [0, ∞),
exists as a locally uniform limit in C([0, ∞); R). Moreover, for typical paths ω ∈ Ω, the quadratic variation S (ω) : [0, ∞) → R is a non-negative and non-decreasing function.
Proposition 3.1 can be found, for instance, in [Vov12, Lemma 8.1] and was generalized to typical paths in the space of càdlàg functions with mildly restricted jumps (see [Vov15, Theorem 1]) and to typical paths in the space of càdlàg functions satisfying a mild restriction on the jumps directed downwards (see [ LPP18, Theorem 3.2]).
Remark 3.2. The existence of quadratic variation S provided in Proposition 3.1 ensures the existence of quadratic variation in the sense of Föllmer, see [Föl81]. Hence, typical paths can be used as integrators in the purely pathwise Itô calculus initiated by Föllmer [Föl81].
Similarly to the quadratic variation in probability theory, the existence of quadratic variation of a (typical) path is stable under time-changes.
Lemma 3.3. Let φ : [0, ∞) → R be a time-change in G 0 and ω ∈ Ω. If the quadratic variation S t (ω) := lim n→∞ V n t (ω), t ∈ [0, ∞),
exists as a locally uniform limit in C([0, ∞); R), then
S t (ω • φ) = lim n→∞ V n t (ω • φ), t ∈ [0, ∞),
exists as a locally uniform limit in C([0, ∞); R) and
S φ(t) (ω) = S t (ω • φ), t ∈ [0, ∞).
Proof. Keeping in mind the definition of the Lebesgue stopping times (σ n k ) in (3.1), we notice that
V n t (ω • φ) = ∞ k=0 ω • φ(σ n k+1 (ω • φ) ∧ t) − ω • φ(σ n k (ω • φ) ∧ t) 2 = ∞ k=0 ω(σ n k+1 (ω) ∧ φ(t)) − ω(σ n k (ω) ∧ φ(t)) 2 = V n φ(t) (ω),
for all t ∈ [0, ∞) and every n ∈ N. Hence, (V n · (ω)) n∈N converges locally uniformly if and only if (V n · (ω • φ)) n∈N converges locally uniformly, and for t ∈ [0, ∞) we get shall review here only the essential basics to treat differential equations driven by typical paths. To that end, we need to introduce some concepts concerning processes.
S t (ω • φ) = lim n→∞ V n t (ω • φ) = lim n→∞ V n φ(t) (ω) = S φ(t) (ωA process X : Ω × [0, ∞) → R is called adapted if the game-theoretic variable X t is F t - measurable for all t ∈ [0, ∞).
The process X is said to be continuous if the sample path t → X t (ω) is continuous for typical paths ω ∈ Ω. In order to work with "game-theoretic" processes, we recall the outer expectation E associated to P . For a non-negative gametheoretic variable F we define
E[F ] := inf λ ≥ 0 : ∃ C ∈ V λ s.t. ∀ω ∈ Ω λ + lim inf t→∞ C t (ω) ≥ F (ω) . Given two processes X, Y : Ω × [0, ∞) → R we introduce X(ω) − Y (ω) ∞;[0,T ] := sup t∈[0,T ] X t (ω) − Y t (ω) , ω ∈ Ω, for T > 0. Now we identify two processes X, Y if d T (X, Y ) := E X − Y ∞;[0,T ] ∧ 1 = 0 for all T ∈ [0, ∞).
The resulting space of equivalent classes of processes is denoted by L 0 loc ([0, ∞); R). In order to construct a game-theoretic Itô integration, we start to define the integral of step functions and then extend the construction to a more general class of integrands. A process F : Ω × [0, ∞) → R d is called a step function if there exist stopping times 0 = τ 0 ≤ τ 1 ≤ . . . , and F τn -measurable functions F n : Ω → R d , such that for every ω ∈ Ω we have τ n (ω) = ∞ for all but finitely many n, and such that
F t (ω) = ∞ n=0 F n (ω)1 [τn(ω),τ n+1 (ω)) (t), (ω, t) ∈ Ω × [0, ∞). The corresponding integral of F w.r.t. (S t ) t∈[0,∞) is given by (F · S) t := ∞ n=0 F τn S τ n+1 ∧t − S τn∧t , t ∈ [0, ∞),
which is well-defined for every ω ∈ Ω.
The following lemma (Lemma 3.4) and the next corollary (Corollary 3.5) are direct consequences of [PP16, Theorem 3.5 and Corollary 3.6].
Lemma 3.4 (Model-free Itô integration). Let X : Ω × [0, ∞) → R be an adapted and continuous process. Then, there exists a process X dS ∈ L 0 loc ([0, ∞); R) with the following continuity property: for every T > 0, if (X (n) ) n∈N is a sequence of simple functions and (c n ) n∈N ⊂ R is a sequence of real numbers such that X (n) (ω) − X(ω) ∞;[0,T ] ≤ c n for all ω ∈ Ω and all n ∈ N, then for typical paths ω ∈ Ω there exists a constant C(ω) > 0 such that
(X (n) · S)(ω) − X dS(ω) ∞;[0,T ] ≤ C(ω)c n log n for all n ∈ N.
The integral process X dS is continuous for typical paths, and there exists a representative X dS which is adapted, although it may take the values ±∞. We usually write t 0 X s dS s := X dS(t), and we call X dS the model-free Itô integral of X w.r.t. S.
One of fundamental properties of Itô integrals in probability theory is that they can be approximated by left-point Riemann sums if one considers sufficiently regular integrands and a sufficiently strong concept of convergence. An example of this property is formulated in the next corollary for the model-free Itô integral.
Corollary 3.5. Suppose we are in the setting of Lemma 3.4. If c n = o((log n) −1/2 ), then for typical paths ((X (n) · · S)) n∈N converges locally uniformly to X dS.
A more general integration theory for typical paths was developed in [PP16], [Vov16] and [ LPP18] providing, e.g., more sophisticated continuity estimates for the model-free Itô integral and integration for not necessarily continuous integrands, and not necessarily continuous typical paths as integrators.
3.3. Itô's formula. It is known that typical paths are as irregular as the sample paths of martingales. More precisely, typical paths have finite p-variation only for p > 2, see [Vov08,Theorem 1]. Therefore, the model-free Itô integral from Lemma 3.4 cannot satisfy the fundamental theorem of calculus but it does satisfy an Itô type formula, as we shall show.
Let A : Ω × [0, ∞) → R and B : Ω × [0, ∞) → R be adapted and continuous processes. We consider the integral process Y : Ω × [0, ∞) → R given by
(3.2) Y t := t 0 A u dS u + t 0 B u du, t ∈ [0, ∞),
where the first integral denotes the model-free Itô integral, as defined in Lemma 3.4, and the second integral a classical Riemann-Stieltjes integral. For this type of integral processes we can derive the following Itô type formula.
Proposition 3.6. If (Y t ) t∈[0,∞) has the representation (3.2) and f : R → R is a twice continuously differentiable function, then the Itô type formula
(3.3) f (Y t ) = f (Y 0 ) + t 0 f ′ (Y u )B u du + t 0 f ′ (Y u )A u dS u + 1 2 t 0 f ′′ (Y u )A 2 u d S u ,
for t ∈ [0, ∞), holds for typical paths.
Note, since S (ω) exists and is a non-decreasing and continuous function for typical paths ω ∈ Ω, the integral
= inf t ≥ ρ n k−1 : |A t − A ρ n k−1 | ≥ 2 −n or |B t − B ρ n k−1 | ≥ 2 −n , for k ∈ N. The corresponding approximation (Y (n) · ) n∈N of (Y t ) t∈[0,∞) is defined by Y (n) t := t 0 A (n) u dS u + t 0 B (n) u du, t ∈ [0, ∞).
By the continuity of the model-free Itô integration (use e.g. Corollary 3.5 with c n := 2 −n ), we have
lim n→∞ sup t∈[0,T ] t 0 A (n) u dS u − t 0 A u dS u = 0
for every T > 0, for typical paths. Furthermore, by the continuity of Riemann-Stieltjes integration (see e.g. [FV10, Proposition 2.7]), we know that
lim n→∞ sup t∈[0,T ] t 0 B (n) u (ω) du − t 0 B u (ω) du = 0
for every T > 0 and all ω ∈ Ω. Hence, (Y (n) · ) n∈N converges locally uniform to (Y t ) t∈[0,∞) for typical paths.
We also notice that, for typical paths ω ∈ Ω, by the definition of quadratic variation, by the Cauchy-Schwarz inequality and Proposition 3.1, the quadratic variation of (Y (n) t ) t∈[0,∞) exists and is given by
Y (n) t (ω) := lim n→∞ ∞ k=0 Y (n) σ n k+1 (ω)∧t (ω) − Y (n) σ n k (ω)∧t (ω) 2 = lim n→∞ ∞ k=0 σ n k+1 ∧t 0 A (n) u dS u (ω) − σ n k ∧t 0 A (n) u dS u (ω) 2 = t 0 (A (n) u (ω)) 2 d S u (ω), t ∈ [0, ∞),
where the convergence takes place locally uniformly and (σ n k ) k∈N denotes again the Lebesgue stopping times as defined in (3.1). Using Remark 3.2 and Föllmer's pathwise Itô formula ([Föl81, THÉORÈME]), we observe that
f (Y (n) t ) − f (Y (n) 0 ) = t 0 f ′ (Y (n) u ) dY (n) u + 1 2 t 0 f ′′ (Y (n) u ) d Y (n) u = t 0 f ′ (Y (n) u )B (n) u du + t 0 f ′ (Y (n) u )A (n) u dS u + 1 2 t 0 f ′′ (Y (n) u ) A (n) t 2 d S u .
(3.4)
where we used the definition of (Y (n) t ) t∈[0,∞) and the previous identity for ( Y (n) t ) t∈[0,∞) in the last line. Since (Y (n) · ) n∈N converges locally uniformly to (Y t ) t∈[0,∞) for typical paths, we can conclude the following as n → ∞: Due to these observations about the convergence behavior, the identify (3.4) reveals the assertion by sending n → ∞.
• f (Y (n) · ), f ′ (Y (n) · ), f ′′ (Y (n) · ) converge locally uniformly to f (Y · ), f ′ (Y · ), f ′′ (Y · ), respec- tively, since f is twice continuously differentiable; • · 0 f ′ (Y (n) u )B (n) u du and · 0 f ′′ (Y (n) u ) A (n) t 2 d S u converge locally uniformly to the limits · 0 f ′ (Y u )B u du and · 0 f ′′ (Y u ) A t 2 d S u ,
Game-theoretic differential equations
One of the main motivations to develop classical stochastic Itô integration was to set up a well-posedness theory for stochastic differential equations. In a related manner, gametheoretic integration can be used to treat differential equations driven by typical paths, cf.
[BKPT19] and [CG LM18].
Remark 4.1. The work [BKN19] provides existence and uniqueness results for differential equations on a finite time horizon [0, T ] driven by typical paths in a Hilbert space setting (the typical paths attain their values in some Hilbert space), assuming that the coefficients are Lipschitz continuous. This approach relies on an extended path space. As a result one obtains a smaller outer measure than the outer measure defined in Definition 2.1. By the extension of the path space we mean that together with the coordinate process (S t ) t∈[0,T ] the investor is allowed to buy or sell assets, whose prices at the moment t ∈ [0, T ] are equal to S 2 t − S t , where here · denotes the Hilbert space norm and ( S t ) t∈[0,T ] denotes the quadratic variation process of the coordinate process (S t ) t∈[0,T ] but defined in a different way than the usual tensor quadratic variation of a Hilbert space-valued semi-martingale, see [BKN19, Remark 2.7]. Additionally, the measure d S is supposed to be absolutely continuous with respect to the Lebesgue measure dt and the density d S /dt is supposed to be globally bounded).
The work [CG LM18] obtains existence and uniqueness results for multi-dimensional differential equations driven by typical paths on a finite time horizon [0, T ] under Lipschitz assumptions. In order to obtain a Burkholder-Davis-Gundy type inequality, [CG LM18] is based on a modified outer expectation which may be interpreted as the super-hedging cost of not only the terminal value of some process (Z t ) t∈[0,T ] , i.e. Z T , but of the value Z τ for any stopping time τ such that τ ∈ [0, T ]. As a result one obtains a possibly greater outer measure than the outer measure defined in Definition 2.1.
While the case of multi-dimensional differential equations driven by typical paths was already studied, we focus here on one-dimensional differential equations. The one-dimensional case is in various ways special and allows to obtain results which do not hold in general in a multi-dimensional setting. A famous example is the Yamada-Watanabe theorem providing the existence and uniqueness of a solution for differential equations with non-Lipschitz diffusion coefficients, see [YW71]. For examples and a more comprehensive discussion about the different necessary regularity assumptions on the coefficients in a one-and multi-dimensional setting, respectively, we refer to [WY71, Remark 2 and 3].
In this section we consider one-dimensional differential equations driven by typical paths of the form (4.1) (i) We say that X = (X t ) t∈[0,∞) is a solution to (4.1) inΩ if X : Ω × [0, ∞) → R is an adapted and continuous process and (4.1) holds for typical paths ω ∈Ω.
X t = x 0 + t 0 b(X u ) d S u + t 0 σ(X u ) dS u , t ∈ [0, ∞),
For Ω =Ω we usually omit "in Ω" and just call (X t ) t∈[0,∞) a solution to (4.1).
(ii) We say that X = (X t ) t∈[0,∞) is the unique solution to (4.1) inΩ if (X t ) t∈[0,∞) is a solution to (4.1) inΩ and, for every
solution Y = (Y t ) t∈[0,∞) inΩ we have X(ω) − Y (ω) [0,T ],∞ = 0 for all T ∈ [0, ∞), for typical paths ω ∈Ω.
For Ω =Ω we usually omit "in Ω" and just call (X t ) t∈[0,∞) the unique solution to (4.1).
We shall study the differential equation (4.1) assuming the following regularity assumptions on the coefficients.
|b(x) − b(y)| ≤ C b |x − y| and |σ(x) − σ(y)| ≤ C σ |x − y| 1/2 ,
for all x, y ∈ R, where C b and C σ are positive constants.
Example 4.4. In mathematical finance the Cox-Ingersoll-Ross (CIR) process serves as a frequently applied model for the evolution of interest rates or volatility on financial market. The CIR process (r t ) t∈[0,∞) can be described by the stochastic differential equation
dr t = a(b − r t ) dt + σ √ r t dW t , t ∈ [0, ∞),
where (W t ) t∈[0,∞) denotes a standard Wiener process and a, b, σ are constants. While the diffusion coefficient x → σ √ x is not Lipschitz continuous, it satisfies the regularity assumption of Assumption 4.3.
Existence theorem.
To prove the existence of a solution to the differential equation (4.1) driven by typical paths, we introduce the following Euler type approximation:
For n ∈ N we set X (n) 0 = x 0 and X (n) t := X (n) τ n k + b(X (n) τ n k )( S t − S τ n k ) + σ(X (n) τ n k )(S t − S τ n k ) for t ∈ (τ n k , τ n k+1 ]
where τ n 0 := 0 and, for k ∈ N, τ n k (ω) := inf t ≥ τ n k−1 : (S t ∈ 2 −n Z and S t = S τ n k−1 ) or ( S t ∈ 2 −n Z and S t = S τ n k−1 ) . Proposition 4.5. Suppose Assumption 4.3 holds true. For typical paths ω ∈ Ω, the limit
X t := lim n→∞ X (n 3 ) t , t ∈ [0, ∞),
exists as locally uniform limit and (X t ) t∈[0,∞) is a solution to the differential equation (4.1).
As a preparation for the proof of Proposition 4.5, we show that the assertion holds true under the Wiener measure W on (Ω, F).
Lemma 4.6. Suppose Assumption 4.3 holds true. For all T > 0 there exists a constant C which only depends on T, C b , C σ and, such that
(4.2) E sup t∈[0,T ] |X (n) t − X t | ≤ C n 1/2 , n ∈ N,
where E denotes the expectation operator with respect to the Wiener measure W on (Ω, F).
In particular, (X (n 3 ) · ) n∈N converges almost surely locally uniformly to (X t ) t∈[0,∞) , where X = (X t ) t∈[0,∞) denotes the strong solution to (4.1) under the Wiener measure W, that is (X t ) t∈[0,∞) is an adapted process on the filtered probability space (Ω, F,
(F t ) t∈[0,∞) , W) such that (X t ) t∈[0,∞) satisfies (4.1) almost surely and W( t 0 |b(X u )| d S u + t 0 σ 2 (X u ) dS u < ∞) = 1 for every t ∈ [0, ∞). Proof. We define κ n (t) := ∞ k=0 1 (τ n k ,τ n k+1 ] (t)τ n k , t ∈ [0, ∞).
Notice that
X (n) t = x 0 + t 0 b(X (n) κn(u) ) d S u + t 0 σ(X (n) κn(u) ) dS u , t ∈ [0, ∞),
and that by the definition of τ n k we have
|X (n) u − X (n) κn(u) | ≤ C b,σ 2 −n ,
where C b,σ ≥ 1 is a positive constant depending on the bounds of b and σ. Moreover, recall that there exists a strong solution (X t ) t∈[0,∞) to the stochastic differential equation
X t = x 0 + t 0 b(X u ) d S u + t 0 σ(X u ) dS u , t ∈ [0, ∞),
under the Wiener measure W by classical results from probability theory, see e.g. [KS88, Section 5.2 C]. For n ∈ N we introduce the process (Y
(n) t ) t∈[0,∞) with Y (n) t := X t − X (n) t .
Step 1: Analogously to [GR11, Proposition 2.2] we claim: there exists C > 0, depending on C b , C σ and C b,σ only, such that
|Y (n) t | ≤ 1 + C S t n + C b t 0 |Y (n) u | d S u + M (n) t , t ∈ [0, ∞),M (n) t ≤ 2C 2 σ C b,σ t 0 (|Y (n) u | + 2 −n ) d S u , t ∈ [0, ∞).
For δ > 1 and ε > 0 let Ψ ε,δ be the same function as in the proof of [GR11, Proposition 2.2], i.e. Ψ ε,δ (x) ≤ 2
x log(δ) and Ψ ε,δ is non-negative and supported on [ε/δ, ε], and R Ψ ε,δ (x) dx = 1. Let Φ ε,δ (x) = |x| 0 y 0 Ψ ε,δ (z) dz dy. Then, Φ ε,δ is twice continuously differentiable and we have |x| ≤ ε + Φ ε,δ (x), |Φ ′ ε,δ (x)| ≤ 1 and
(4.4) Φ ′′ ε,δ (x) = Ψ ε,δ (|x|) ≤ 2 |x| log(δ) 1 [ε/δ,ε] (|x|) ≤ 2δ ǫ log(δ)
.
For t ∈ [0, ∞), Itô's formula (cf. Proposition 3.6) yields
Φ ε,δ (Y (n) t ) = Φ ε,δ (Y (n) 0 ) + M (n) t + t 0 Φ ′ ε,δ (Y (n) u )(b(X u ) − b(X (n) κn(u) )) + 1 2 Φ ′′ ε,δ (Y (n) u )|σ(X u ) − σ(X (n) κn(u) )| 2 d S u with M (n) t := t 0 Φ ′ ε,δ (Y (n) u ) σ(X u ) − σ(X (n) κn(u) ) dS u .
Hence, by the properties of Φ ε,δ we get
|Y (n) t | ≤ ε + Φ ε,δ (Y (n) t ) = ε + t 0 Φ ′ ε,δ (Y (n) u )(b(X u ) − b(X (n) κn(u) )) + 1 2 Φ ′′ ε,δ (Y (n) u )|σ(X u ) − σ(X (n) κn(u) )| 2 d S u + M (n) t . The contribution from the drift is t 0 Φ ′ ε,δ (Y (n) u ) b(X u ) − b(X (n) κn(u) ) d S u = t 0 Φ ′ ε,δ (Y (n) u ) b(X u ) − b(X (n) u ) d S u + t 0 Φ ′ ε,δ (Y (n) u ) b(X (n) u ) − b(X (n) κn(u) ) d S u ≤ C b t 0 |Y (n) u | d S u + C b C b,σ 2 −n S t ,
where we used that |Φ ′ ε,δ | ≤ 1. The contribution from the quadratic variation is
t 0 1 2 Φ ′′ ε,δ (Y (n) u )|σ(X u ) − σ(X (n) κn(u) )| 2 d S u ≤ t 0 Φ ′′ ε,δ (Y (n) u )|σ(X u ) − σ(X (n) u )| 2 d S u + t 0 Φ ′′ ε,δ (Y (n) u )|σ(X (n) u ) − σ(X (n) κn(u) )| 2 d S u ≤ C 2 σ t 0 Φ ′′ ε,δ (Y (n) u )|X u − X (n) u | d S u + C 2 σ t 0 Φ ′′ ε,δ (Y (n) u )|X (n) u − X (n) κn(u) | d S u ≤ C 2 σ 2 log(δ) S t + C 2 σ 2δ ε log(δ) 2 −n S t ,
where we used Assumption 4.3 in the second last line and the estimate (4.4) is the last one. So overall
|Y (n) t | ≤ ε + C b t 0 |Y (n) u | d S u + C b C b,σ 2 −n S t + C 2 σ 2 log(δ) S t + C 2 σ 2δ ε log(δ) 2 −n S t + M (n) t .
Choosing ε = 1/n and δ = 2 n/2 this becomes
|Y (n) t | ≤ 1 n + C b t 0 |Y (n) u | d S u + C S t (2 − n 2 + n −1 ) + M (n) t ≤ 1 + C S t n + C b t 0 |Y (n) u | d S u + M (n) t ,
for some C > 0 which depends on C b , C σ and C b,σ only. The quadratic variation of (M
(n) t ) t∈[0,∞) is M (n) t = t 0 |Φ ′ ε,δ (Y (n) u )(σ(X u ) − σ(X (n) κn(u) ))| 2 d S u ≤ 2C 2 σ t 0 |Y (n) u | d S u + 2C 2 σ C b,σ t 0 2 −n d S u ,
for t ∈ [0, ∞), as claimed.
Step 2: Under the Wiener measure W we have S t = t. Using the estimate (4.3) and a standard localization argument gives
E[|Y (n) t |] ≤ 1 + Ct n + C b t 0 E[|Y (n) u |] du, t ∈ [0, ∞).
Hence, applying Gronwall's inequality leads to
(4.5) E[|Y (n) t |] ≤ 1 + Ct n , t ∈ [0, ∞),
where C > 0 depends on C b , C σ and C b,σ only, but may differ from the constant denoted by C in Step 1. The Burkholder-Davis-Gundy inequality together with the estimate (4.3) and (4.5) yields
E sup u∈[0,t] |Y (n) u | ≤ 1 + Ct n + C b t 0 E[|Y (n) u |] du + E sup u∈[0,t] |M (n) u | ≤ 1 + Ct n + C b t 0 E[|Y (n) u |] du + C BDG 2C 2 σ C b,σ t 0 (E[|Y (n) u |] + 2 −n ) du 1/2 ≤ 1 + Ct n + C t 0 E sup r∈[0,u] |Y (n) r | du + (C BDG 2C 2 σ ((1 + Ct) + t)) 1/2 n −1/2 ,
for a new constant C > 0 which depends on C b , C σ and C b,σ . So the claimed estimate (4.2) follows by applying again Gronwall's inequality.
Step 3: After having established (4.2) the almost sure locally uniform convergence of (X (n 3 ) · ) n∈N to (X t ) t∈[0,∞) follows by a routine argument: For any T > 0 we have ) n∈N converges almost surely uniformly on [0, T ] to (X t ) t∈[0,∞) . By choosing a countable sequence T m → ∞, we obtain, almost surely, the locally uniform convergence of (X
E ∞ n=1 sup t∈[0,T ] |X (n 3 ) t − X t | ≤ C 3 ∞ n=1 n −3/2 < ∞,(n 3 ) · ) n∈N to (X t ) t∈[0,∞) .
Proof of Proposition 4.5. Consider the event E 1 := ω ∈ Ω : X (n 3 ) (ω) converges locally uniformly and denote by E c 1 the complement of the set E 1 . We want to show that E c 1 is timesuperinvariant in the sense of Definition 2.3 in order to apply the pathwise Dambis-Dubins-Schwarz theorem (Theorem 2.4). For this purpose, it is sufficient to show that ω ∈ E 1 implies ω • φ ∈ E 1 for every φ ∈ G 0 .
Let ω ∈ E 1 and φ ∈ G 0 be a time-change. Thanks to Lemma 3.3 and the definition of the stopping times (τ n k ), for n, k ∈ N, we have
S τ n k (ω•φ) (ω • φ) = S τ n k (ω) (ω) and S τ n k (ω•φ) (ω • φ) = S τ n k (ω) (ω). Furthermore, we have X (n) 0 (ω) = x 0 = X (n) 0 (ω • φ). Suppose now that X (n) τ n k (ω) (ω) = X (n) τ n k (ω•φ) (ω • φ)
for some k ∈ N and let us apply an induction argument over k ∈ N. For t ∈ (τ n k (ω •φ), τ n k+1 (ω • φ)] we observe that
X (n) t (ω • φ) = X (n) τ n k (ω•φ) (ω • φ) + b(X (n) τ n k (ω•φ) (ω • φ)) S t (ω • φ) − S τ n k (ω•φ) (ω • φ) + σ(X (n) τ n k (ω•φ) (ω • φ)) S t (ω • φ) − S τ n k (ω•φ) (ω • φ) = X (n) τ n k (ω) (ω) + b(X (n) τ n k (ω) (ω)) S φ(t) (ω) − S τ n k (ω) (ω) + σ(X (n) τ n k (ω) (ω)) S φ(t) (ω) − S τ n k (ω) (ω) = (X (n) ) φ(t) (ω). This implies X (n) t (ω • φ) = X (n) φ(t) (ω) for all t ∈ [0, ∞). Hence, if (X (n 3 ) ·
(ω)) n∈N converges locally uniformly, then (X (n 3 ) · (ω • φ)) n∈N converges locally uniformly. This means ω ∈ E 1 implies ω • φ ∈ E 1 , that is E c 1 is time-superinvariant. Combining Theorem 2.4 and Lemma 4.6 ensure that P (E c 1 ) = 0. Hence, for typical paths, the process (X t ) t∈[0,∞) , defined by X t := lim n→∞ X (n 3 ) t , exists as locally uniform limit. Furthermore, by Lemma 3.4 and the continuity property of Riemann-Stieltjes integration, we deduce that (X t ) t∈[0,∞) is a solution to (4.1), which completes the proof.
4.2. Yamada-Watanabe theorem. In probability theory, a famous theorem of Yamada-Watanabe states that there exists a unique solution to one-dimensional stochastic differential equations driven by a Wiener process assuming that the diffusion coefficient is Hölder continuous of order 1/2, see [YW71,WY71]. This is a very deep insight as usually differential equations require basically Lipschitz continuous coefficients to ensure the uniqueness of solutions.
In this subsection we prove a Yamada-Watanabe type theorem (Theorem 4.7) for the differential equation (4.1), which is driven by typical paths. In order to recall the definition of the set DS, we refer to Definition 2.5.
Theorem 4.7. Suppose that Assumption 4.3 holds. Then, there exists a unique solution X = (X t ) t∈[0,∞) in DS to the differential equation (4.1).
Before we proceed to the proof of Theorem 4.7 we will prove some preparatory results. Let (X t ) t∈[0,∞) be the solution to the differential equation (4.1). From Corollary 3.5 we know that if the stopping times (ρ n k ) are given by ρ n 0 := 0 and ρ n k := inf{t > ρ n k−1 : |σ(X t ) − σ(X ρ n k−1 )| ≥ 2 −n or |S t − S ρ n k−1 | ≥ 2 −n } for n, k ∈ N, then for typical paths
(4.6) t 0 σ(X u ) dS u := lim n→∞ ∞ k=0 σ(X ρ n k ) S ρ n k+1 ∧t − S ρ n k ∧t , t ∈ [0, ∞),
and the convergence in (4.6) is locally uniform, for typical paths, since the stopping times (ρ n k ) ensure that the integrand σ(X u ) is uniformly approximated. From this approximation one can also derive the behaviour of the model-free Itô integral under time-changes. σ(X ρ n k ((ω)) (ω)) S ρ n k+1 (ω)∧t (ω) − S ρ n k (ω)∧t (ω) , t ∈ [0, ∞), exists as locally uniform limit, then
t 0 σ((X • φ) u ) d(S • φ) u (ω), t ∈ [0, ∞),
as defined in (4.6), exists as locally uniform limit and
φ(t) 0 σ(X u ) dS u (ω) = t 0 σ((X • φ) u ) d(S • φ) u (ω).
Furthermore, if ω ∈ Ω is such that the quadratic variation ( S t (ω)) t∈[0,∞) exists, then
φ(t) 0 b(X u (ω)) d S u (ω) = t 0 b(X u (ω • φ)) d S u (ω • φ), t ∈ [0, ∞).
Proof. By the definition of the stopping times (ρ n k ), for n ∈ N and t ∈ [0, ∞) we observe that
φ(t) 0 σ(X s ) dS s (ω) := lim n→∞ ∞ k=0 σ(X ρ n k (ω)) S ρ n k+1 ∧φ(t) (ω) − S ρ n k ∧φ(t) (ω) = lim n→∞ ∞ k=0 σ(Xτ n k (ω)) Sτ n k+1 ∧t (ω) − Sτ n k ∧t (ω)
where the last equality holds for the new stopping times (τ n k ) defined byτ n 0 := 0 and τ n k := inf{t >τ n k−1 :
|σ((X • φ) t ) − σ((X • φ)τ n k−1 )| ≥ 2 −n or |(S • φ) t − (S • φ)τ n k−1 ∧t | ≥ 2 −n } for n, k ∈ N. Hence, t 0 σ((X • φ) s ) d(S • φ) s (ω) = lim n→∞ ∞ k=0 σ(Xτ n k (ω)) Sτ n k+1 ∧t (ω) − Sτ n k ∧t (ω) = φ(t) 0 σ(X s ) dS s (ω),
which reveals the first assertion. The second assertion follows by Lemma 3.3
With this preparatory results at hand we are in a position to prove Theorem 4.7.
Proof of Theorem 4.7. Since the existence of a solution (X t ) t∈[0,∞) to (4.1) in DS follows by Proposition 4.5, it remains to show uniqueness in DS. Let us suppose that there are two continuous and adapted processes X (1) = (X (1) t ) t∈[0,∞) and X (2) = (X (2) t ) t∈[0,∞) solving the differential equation (4.1) driven by typical paths. Let us consider the event
E 2 := ω ∈ DS : sup t∈[0,∞) X (1) t (ω) − X (2) t (ω) > 0 .
We shall show that the event E 2 is time-superinvariant in the sense of Definition 2.3. Let φ ∈ G and ω ∈ DS. Without loss of generality we may assume that for ω the quadratic variation ( S t (ω)) t∈[0,∞) in the sense of Proposition 3.1 and (X (1) t (ω)) t∈[0,∞) and (X (2) t (ω)) t∈[0,∞) satisfy equation (4.1) together with (4.6). Due to Lemma 4.8, for i = 1, 2 we observe that 4.3. Doss-Sussmann approximation. While differential equations of the form (4.1) appear frequently in the mathematical modeling of random phenomena, the actual "noise" term (S t ) t∈[0,∞) in many applications is represented by a process of bounded variation. This led to the natural question how ordinary differential equations, perturbed by random noises of bounded variation, and stochastic differential equations are linked. A first answer to this fundamental question was given by Doss [Dos77] and Sussmann [Sus78], stating that a suitably chosen sequence of ordinary differential equations can approximate a differential equation driven by typical paths.
X (i) t (ω • φ) = x 0 + t 0 b(X (i) u (ω • φ)) d S u (ω • φ) + t 0 σ(X (i) u ) dS u (ω • φ) = x 0 + φ(t) 0 b(X (i) u (ω)) d S u (ω) + φ(t) 0 σ(X (i) u ) dS u (ω) = X (i) φ(t) (ω) for t ∈ [0, ∞). This reveals that ω • φ ∈ E 2 implies ω ∈ E 2 .
For n ∈ N let (S (n) t ) t∈[0,∞) be a process such that function t → S (n) t (ω) is continuous and of locally bounded variation for every ω ∈ Ω. Let us consider the differential equation
(4.7) X (n) t = x 0 + t 0 b(X (n) u ) d S u + t 0 σ(X (n) u ) dS (n) u , t ∈ [0, ∞),
where x 0 ∈ R, b : R → R and σ : R → R are Lipschitz functions and X (n) : Ω × [0, ∞) → R. Notice that the differential equation (4.7) possesses a unique solution (X (n) t (ω)) t∈[0,∞) for all ω ∈ Ω such that ( S t (ω)) t∈[0,∞) exists in the sense of Proposition 3.1, see [FV10,Corollary 3.9]. Under suitable assumptions, if the sequence (S (n) · ) n∈N uniformly approximates the coordinate process (S t ) t∈[0,∞) , it turns out that the corresponding sequence (X (n) · ) n∈N of solutions indeed converges to an adapted and continuous process (X t ) t∈[0,∞) solving a differential equation driven by typical paths.
Remark 4.10. A natural and most straightforward way to approximate uniformly the coordinate process (S t ) t∈[0,∞) with continuous processes (S (n) · ) n∈N of locally bounded variation is to choose (S (n) · ) n∈N as a piecewise linear approximation along the sequence of Lebesgue stopping times, cf. (3.1). While these approximations may be not adapted, one can use the concept of truncated variation, as in [ LM13], to obtain uniform, continuous and adapted approximations of (S t ) t∈[0,∞) , which have locally bounded variation.
The following theorem formulates precisely the indicated convergence result for typical paths and can be seen as a game-theoretic version of the Doss-Sussmann approximation result in probability theory.
Theorem 4.11. Suppose that b : R → R is Lipschitz continuous and σ : R → R is twice continuously differentiable with bounded first and second derivative. Let (S (n) · ) n∈N be a sequence of processes such that the function t → S (n) t (ω) is continuous and of locally bounded variation for every ω ∈ Ω, and denote by (X
X t = x 0 + t 0 b(X u ) + 1 2 σ(X u )σ ′ (X u ) d S u + t 0 σ(X u ) dS u , t ∈ [0, ∞),
for typical paths, which is unique for typical paths in DS.
The proof adapts the classical probabilistic arguments, cf. e.g. [KS88, Section 5.2 D].
Proof. Without loss of generality, we consider ω ∈ Ω such that ( S t (ω)) t∈[0,∞) exists in the sense of Proposition 3.1, recalling that the complement of this set has outer measure zero.
Step 1: Let g : R 2 → R be the solution to the ordinary differential equation ∂g ∂x = σ(g) and g(0, y) = y, y ∈ R.
Hence, for x, y ∈ R we get ∂ 2 g ∂x 2 = σ(g)σ ′ (g), ∂ 2 g ∂x∂y = σ ′ (g) ∂g ∂y and ∂ ∂y g(0, y) = 1, which gives that 1 ρ(x, y) := ∂ ∂y g(x, y) = exp x 0 σ ′ (g(z, y)) dz > 0.
As shown in the proof of [KS88, Chapter 5, Proposition 2.21], the continuous function f (x, y) := ρ(x, y)b(g(x, y)), x, y ∈ R, is locally Lipschitz continuous in y, bounded in x and has locally linear growth in y. Hence, due to [FV10, Corollary 3.9], there exists a unique solution (Y t (ω)) t∈[0,∞) to the ordinary differential equation
(4.9) Y t (ω) = x 0 + t 0 f (S u (ω), Y u (ω)) d S u (ω), t ∈ [0, ∞),
since the coefficient f is a locally Lipschitz continuous function of linear growth. This allows us to define the process X t := g(S t , Y t ) for t ∈ [0, ∞). Using Föllmer's pathwise Itô Corollary 3.9] that, for typical path ω ∈ Ω, there is a unique pathwise solution (Z t (ω)) t∈[0,∞) to the equation
Z t (ω) = t 0b
(g(Z s (ω) + S s (ω), x 0 )) d S s (ω), t ∈ [0, ∞).
Then, we obtain from Föllmer's pathwise Itô formula that (X t (ω)) t∈[0,∞) , given by X t (ω) := g(Z t (ω) + S t (ω), x 0 ), solves X t (ω) = g(0, x 0 ) + t 0 σ(u(Z s (ω) + S s (ω), x 0 ))b(g(Z s (ω) + S s (ω), x 0 )) d S s (ω) + t 0 σ(g(Z s (ω) + S s (ω), x 0 )) dS s (ω) + 1 2 t 0 σ ′ (g(Z s (ω) + S s (ω), x 0 ))σ(g(Z s (ω) + S s (ω), x 0 )) d S s (ω) = x 0 + t 0 σ(X s (ω))b(X s (ω)) + 1 2 σ ′ (X s (ω))σ(X s (ω)) d S s + t 0 σ(X s (ω)) dS s (ω).
Uniqueness of solutions and the approximation via the solutions (X (n) · ) n∈N of (4.7) follow exactly as in the previous proof.
See also the recent paper [KR16] for closely related results on pathwise solutions of onedimensional SDEs via Doss-Sussmann and Lamperti transforms in a semi-martingale context.
t 0 f
0′′ (Y u )A 2 u d S u in (3.3) can be defined as a Riemann-Stieltjes integral. Proof of Proposition 3.6. Since A : Ω × [0, ∞) → R and B : Ω × [0, ∞) → R are adapted and continuous processes, we can locally uniformly approximate them by A
u converges locally uniformly to
· 0 f
0′ (Y u )A u dS u by the continuity of the model-free Itô integration.
where x 0 ∈
0R, b : R → R and σ : R → R are continuous functions and X : Ω × [0, ∞) → R is supposed to be an adapted and continuous process. Thanks to Lemma 3.4, the model-free Itô integral t 0 σ(X u ) dS u exists in L 0 loc ([0, ∞); R). Definition 4.2. LetΩ ⊂ Ω be a set.
Assumption 4. 3 .
3Suppose that b : R → R and σ : R → R are continuous and bounded functions satisfying the conditions
t∈[0,∞) is some continuous local martingale with quadratic variation
and therefore almost surely ∞ n=1 sup t∈[0,T ] |X (n 3 ) t − X t | < ∞. Hence, (X (n 3 ) ·
Lemma 4 . 8 .
48Let φ : [0, ∞) → [0, ∞) be a time-change in G 0 and ω ∈ Ω.
Remark 4. 9 .
9The boundedness assumption on b and σ made in Assumption 4.3 is not needed for the uniqueness result provided in Theorem 4.7. However, it is required by Proposition 4.5 to provide the existence of a solution.
·
) n∈N the sequence of corresponding solutions to (4.7). If lim n→∞ S (n) − S ∞;[0,T ] = 0, then lim n→∞ X (n) − X ∞;[0,T ] = 0, for typical paths, where X = (X t ) t∈[0,∞) denotes the solution to
). 3.2. Game-theoretic integration. Based on the existence of quadratic variation for typical paths, one can derive Itô's isometry type estimates w.r.t. the outer measure P , see [PP16, Lemma 3.4] or [ LPP18, Lemma 4.5 or 4.8]. While this allows to develop a comprehensive integration theory for typical paths (see [PP16, Vov16, LPP18] and [SV19, Chapter 14]), we
Hence, E 2 is time-invariant and by Lemma 2.6 E 2 is also time-superinvariant. By Vovk's pathwise Dambis-Dubins-Schwarz theorem (Theorem 2.4 and [Vov12, Corollary 3.7]) and the classical uniqueness result of Yamada and Watanabe for stochastic differential equations driven by a Brownian motion ([KS88, Chapter 5.2, Proposition 2.13]), we obtain P (E 2 ; DS) = W(E 2 ) = 0.
formula[Föl81], we see that X t (ω) = g(S 0 (ω), Y 0 (ω)) + t 0 ∂ ∂x g(S u (ω), Y u (ω)) dS u (ω)is the unique solution to (4.8). Note that the uniqueness holds due to Remark 4.9.Step 2: Similar to Step 1, since (SMoreover, we get that X) is the unique solution to (4.7). Indeed, by classical calculus, for t ∈ [0, ∞), we haveStep 3: Recall that X · = g(S · , Y · ) and X). Hence, since g is a continuous function and (S (n) · (ω)) n∈N converges locally uniformly to (S t (ω)) t∈[0,∞) , it is sufficient to prove that (Y (n) · (ω)) t∈[0,∞) converges locally uniformly to (Y t (ω)) t∈[0,∞) in order to show that (X (n) · (ω)) n∈N converges locally uniformly to (X t (ω)) t∈[0,∞) . Furthermore, notice that (Y (n) t (ω)) t∈[0,∞) and (Y t (ω)) t∈[0,∞) are solutions to the ordinary differential equations (4.10) and (4.9), respectively, which are both driven by processes of locally bounded variation. Therefore, the locally uniform convergence of (S (n) · (ω)) n∈N to (S t (ω)) t∈[0,∞) implies the locally uniform convergence of (Y (n) · (ω)) t∈[0,∞) to (Y t (ω)) t∈[0,∞) , which is a classical stability result for ordinary differential equations, see for instance[FV10,Theorem 3.15].
Note that ∂ ∂x g(x, y) = σ(g(x, y)), and since σ is continuously differentiable, we get ∂ 2 ∂x 2 g(x, y) = σ ′ (g(x, y))σ(g(x, y)), so g(·, y) is twice continuously differentiable even if σ is only continuously differentiable. In particular, the mapb(g(·, x 0 )) is locally Lipschitz continuous as concatenation of locally Lipschitz continuous maps. If σ is bounded, thenb(g(·, x 0 )) is of linear growth as concatenation of maps of linear growth, while ifb is bounded. Mathias Beiglböck, M G Alexander, Martin Cox, Nicolas Huesmann, David J Perkowski, Prömel, Finance Stoch. 214ous functionb of linear growth, then it suffices to assume that σ is continuously differentiable and of linear growth, and that at least one of the two functionsb or σ is bounded. Indeed, in that case we can apply the Lamperti transform rather than the Doss-Sussmann transform: Let g be as in the previous proof. of course alsob(g(·, x 0 )) is bounded. Therefore, it follows again from [FV10, References [BCH + 17. Pathwise superreplication via Vovk's outer measureRemark 4.12. If the drift term is of the form b(x) = σ(x)b(x) for a locally Lipschitz continu- ous functionb of linear growth, then it suffices to assume that σ is continuously differentiable and of linear growth, and that at least one of the two functionsb or σ is bounded. Indeed, in that case we can apply the Lamperti transform rather than the Doss-Sussmann transform: Let g be as in the previous proof. Note that ∂ ∂x g(x, y) = σ(g(x, y)), and since σ is continu- ously differentiable, we get ∂ 2 ∂x 2 g(x, y) = σ ′ (g(x, y))σ(g(x, y)), so g(·, y) is twice continuously differentiable even if σ is only continuously differentiable. In particular, the mapb(g(·, x 0 )) is locally Lipschitz continuous as concatenation of locally Lipschitz continuous maps. If σ is bounded, thenb(g(·, x 0 )) is of linear growth as concatenation of maps of linear growth, while ifb is bounded, of course alsob(g(·, x 0 )) is bounded. Therefore, it follows again from [FV10, References [BCH + 17] Mathias Beiglböck, Alexander M. G. Cox, Martin Huesmann, Nicolas Perkowski, and David J. Prömel, Pathwise superreplication via Vovk's outer measure, Finance Stoch. 21 (2017), no. 4, 1141-1166.
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|
[] |
[
"An IRT-based Model for Omitted and Not-reached Items",
"An IRT-based Model for Omitted and Not-reached Items"
] |
[
"Jinxin Guo \nNortheast Normal University\n\n",
"Xin Xu \nNortheast Normal University\n\n"
] |
[
"Northeast Normal University\n",
"Northeast Normal University\n"
] |
[] |
Missingness is a common occurrence in educational assessment and psychological measurement. It could not be casually ignored as it may threaten the validity of the test if not handled properly. Considering the difference between omitted and not-reached items, we developed an IRT-based model to handle these missingness. In the proposed method, not-reached responses are captured by the cumulative missingness. Moreover, the nonignorability is attributed to the correlation between ability and person missing trait. We proved that its item parameters estimate under maximum marginal likelihood (MML) estimation is consistent. We further proposed a Bayesian estimation procedure using MCMC methods to estimate all the parameters. The simulation results indicate that the model parameters under the proposed method are better recovered than that under listwise deletion, and the nonignorable model fits the simulated nonignorable nonresponses better than ignorable model in terms of Bayesian model selection. Furthermore, the Program for International Student Assessment (PISA) data set was analyzed to further illustrate the usage of the proposed method.
| null |
[
"https://arxiv.org/pdf/1904.03767v1.pdf"
] | 102,351,025 |
1904.03767
|
1d44aa33a468aa5f719e3cb2ae5a7ae56949a303
|
An IRT-based Model for Omitted and Not-reached Items
Jinxin Guo
Northeast Normal University
Xin Xu
Northeast Normal University
An IRT-based Model for Omitted and Not-reached Items
missing data mechanismomittednot-reachednonig- norableignorablePISA
Missingness is a common occurrence in educational assessment and psychological measurement. It could not be casually ignored as it may threaten the validity of the test if not handled properly. Considering the difference between omitted and not-reached items, we developed an IRT-based model to handle these missingness. In the proposed method, not-reached responses are captured by the cumulative missingness. Moreover, the nonignorability is attributed to the correlation between ability and person missing trait. We proved that its item parameters estimate under maximum marginal likelihood (MML) estimation is consistent. We further proposed a Bayesian estimation procedure using MCMC methods to estimate all the parameters. The simulation results indicate that the model parameters under the proposed method are better recovered than that under listwise deletion, and the nonignorable model fits the simulated nonignorable nonresponses better than ignorable model in terms of Bayesian model selection. Furthermore, the Program for International Student Assessment (PISA) data set was analyzed to further illustrate the usage of the proposed method.
Introduction
Missing data is always unavoidable in many studies, including educational assessment and psychological measurement (Rose et al., 2017;Yuan et al., 2018). Recently modeling missing data mechanism has gained increasing prominence and been widely considered in order to get a more reliable 1 arXiv:1904.03767v1 [stat.ME] 7 Apr 2019 evaluation. Actually, missingness would occur under many conditions. For example, test takers may fail to reach some items due to time limits. Or sometimes they may tend to omit some items for individual reasons, such as their abilities and item preference. If these missing responses could not be dealt with properly, it would bring biased parameter estimation and further threaten the validity of tests (Pohl et al., 2014;Rose et al., 2015).
To better tackle the problem of missing data, Rubin (1976) and Little and Rubin (2014) have defined three kinds of missingness: "missing completely at random"(MCAR), "missing at random"(MAR), and "not missing at random"(NMAR).
Suppose that there are N examinees and J items in the test. Let Y ij denote the dichotomous response variable of examinee i ∈ {1, · · · , N } to item j ∈ {1, · · · , J}, where Y ij = 1 means that examinee i answer item j correctly while Y ij = 0 otherwise. Let Y ij = (Y obs ij , Y mis ij ), where Y obs ij is observed and Y mis ij is missing. And Y = {Y ij } N ×J is the complete response matrix which can be decomposed into an observed part Y obs and a missing part Y mis .
Denote by R = {R ij } N ×J the missing indicator matrix, where the missing variable R ij can be defined as
R ij = 0, if Y ij is observed, 1, if Y ij is missing.
Actually, the three categories of missingness can be characterized by the conditional distribution of R given Y , denoted by P (R|Y , Ω), where Ω is the unknown parameters set. The missingness is MCAR if the distribution of R does not depend on the response data Y , either observed or unobserved, which can be formulated as P (R|Y , Ω) = P (R|Ω) for all Y , Ω.
MAR is the situation in which missingness is independent of missing response given the observed ones, which can be written as P (R|Y , Ω) = P (R|Y obs , Ω) for all Y mis , Ω.
The missingness of MCAR or MAR is also called ignorable or uninformative (Schafer and Graham, 2002;Rose et al., 2015). However, NMAR is obviously distinct from the above two kinds of missingness. It is not independent of missing response given the observed responses, which is also called nonignorable or informative. For example, in the context of IRT, test takers may fail to answer some items because their abilities are too low to answer correctly.
This kind of missing responses can be viewed as NMAR.
In fact, there exist several methods to handle missing item responses.
One of the most direct and simplest approaches is listwise deletion which is also the default method dealing with missing data in some statistical softwares, such as SPSS and SAS. In this method, all cases with missing responses would be deleted. The method is very direct and effective when the missing data response rate is small. However, if the proportion of missing data is high, especially when the missingness is nonignorable, this method would cause bias and thus lead to errors in statistical inference (Rose et al., 2015(Rose et al., , 2017Wu et al., 2017).
So we paid more attention to modeling missing responses for nonignorable missingness. The general methods are typically based on the joint distribution of R and Y was constructed. Two commonly used joint models are selection models (SLM; Heckman, 1976) and pattern mixture models (PMM; Little, 1993). SLM is based on the following factorization:
P (R, Y |Ω) = P (R|Y , Ω)P (Y |Ω)(1)
And PMM can be written as:
P (R, Y |Ω) = P (Y |R, Ω)P (R|Ω)(2)
Based on these joint models, several methods have been proposed by researchers. For example, Holman and Glas (2005) and Glas et al. (2015) introduced an IRT model for omitted items based on PMM that could simultaneously estimate IRT item parameters and the parameter about the propensity of missing data. Rose et al. (2010) derived multidimensional IRT (MIRT) to handle nonignorable item nonresponses, which was believed to have originated from general SLM (Rose, 2013). However, these methods could only be applied to omitted responses. To differentiate the omitted and not-reached items, latent regression models (LRMs) were proposed to model omitted and not-reached items (Rose et al., 2017).
Specifically, not-reached items (also called "dropout") occurs when the test takers fail to reach some items at the end of test and omitted items (also called "intermittent") refer to the situation where they skip one or more
items and then answer the next one.
Motivated by the previous methods, this paper proposed an approach to model the omitted and not-reached items on the basis of SLM. In details, the effects of previous nonresponses on current item are modeled for notreached items. And the correlation between ability parameter and latent person missing parameter is employed to clarify whether the missingness is nonignorable or ignorable.
The remainder of this paper was organized as follows. In Section 2, we presented the proposed method to handle binary missing item responses.
The MML estimation of the item parameters and related consistency results were given in Section 3, followed by the Bayesian estimation using MCMC method. To evaluate parameter recovery and model selection, two simulation studies were conducted in Section 4. In Section 5, we carried out a detailed analysis of PISA data set to illustrate the usage of the proposed method.
Finally, some issues that need to be resolved were addressed and further 4 research directions were discussed in Section 6.
2 Handling omitted and not-reached items with IRT-based model 2.1 Two parameter IRT models
In the proposed method, two-parameter IRT models were employed to model the item response data. The probability of examinee i correctly answering item j can be expressed as
p ij = P (Y ij = 1 |θ i , a j , b j ) = F (a j (θ i − b j )) ,(3)
where θ i denotes the ability parameter for the ith individual; a j and b j are the discrimination and difficulty parameters of item j, respectively. And F (·)
is a cumulative distribution function (CDF) of standard normal or standard logistic distribution. In details, probit link could yield two-parameter normal ogive (2PNO) IRT model, that is, p ij = Φ (a j (θ i − b j )), where Φ(·) is the CDF of standard normal distribution. Similarly, when F (·) is the CDF of standard logistic distribution, that is, logit link is employed, it follows that
logit(p ij ) = a j (θ i − b j ). Therefore, p ij = 1/{1 + exp (−a j (θ i − b j ))}, which
is called two-parameter logistic (2PL) IRT model.
Modeling missing data mechanism
Considering the difference between omitted and not-reached items, different effect indexes were applied in the proposed method. Especially, notreached responses are characterized by the cumulative missingness. Motivated by the idea of IRT models, missingness is captured by latent missing trait from two perspectives: item and person. On the basis of this, the missing data process can be modeled as
π ij =P (R ij = 1 |τ i , ζ j , R i,j−1 , Y ij , γ ) =G (γ 0 − τ i + ζ j + g (R i,j−1 , γ 1 ) + l (Y ij , γ 2 )) ,(4)
5 where G (·) is similar to F (·) in Equation (3), specified by probit or logit link, γ 0 is the intercept parameter with constraint of γ 0 < 0 that could influence the baseline probability of missingness, τ i is the person missing parameter that measures the individual latent trait to nonresponses, and ζ j is the item missing parameter that represents the inclination of missingness caused by item. Moreover, for the ith examinee,
R i,j−1 = (R i1 , R i2 , ..., R i,j−1 )
is the previous missing vector before jth item and
Y ij = (Y i1 , Y i2 , ..., Y ij )
is responses vector of the first j items. Moreover, l (·) is a function of the responses vector that characterizes the effect from response variables. And we simply took l (Y ij , γ 2 ) = γ 2 Y ij , where γ 2 < 0. In addition, g (·) is a function of the missing indicators, which denotes the impact of previous nonresponses on current item. We set g (·) = 0 when j = 1 since there are no previous missing responses. Specially, in the proposed method, g (R i,j−1 , γ 1 ) = γ 1 In the proposed method, the ability parameter θ in IRT model and person missing parameter τ in missing part were assumed to be bivariate joint normally distributed with mean vector µ = µ θ µ τ and covariance matrix Σ = σ 2 θ σ θτ σ θτ σ 2 τ . A graphical representation of the proposed method is present in Figure 1.
j−1 h=2 R i,h with
To guarantee the model identification, µ θ and µ τ are set to 0, and σ 2 θ was fixed to 1 (Browne, 2006). Note that the generation of the nonignorable mechanism is attributed to the correlation between θ and τ . To be more specific, if the missingness is ignorable, τ is independent of θ, therefore, p (θ, τ |µ, Σ) = p(θ|µ θ , σ 2 θ )p(τ |µ τ , σ 2 τ ). If the missing data depends on both the latent ability θ and the person missing parameter τ , the missingness Item domain
µ a , σ 2 a , µ b , σ 2 b , µ ζ , σ 2 ζ Person µ θ , µ τ , Σ θτ a j , b j θ i ζ j τ i Y ij R ij R i1
. . . is nonignorable. So the proposed method could handle both ignorable and nonignorable missingness.
The likelihood function
Let θ = (θ 1 , · · · , θ N ), τ = (τ 1 , · · · , τ N ), a = (a 1 , · · · , a J ), b = (b 1 , · · · , b J ), ζ = (ζ 1 , · · · , ζ J ), and γ = (γ 0 , γ 1 , γ 2 ) . And the parameter set can be written as Ω = (θ, τ , a, b, ζ, γ, σ θτ , σ 2 τ ). Denote by y ij = (y i1 , · · · , y ij ) and r ij = (r i1 , · · · , r ij ) the observation vectors of Y ij = (Y i1 , · · · , Y ij ) and R ij = (R i1 , · · · , R ij ).
A sequence of one-dimensional conditional distributions modeling method proposed by Ibrahim et al. (1999) was employed to construct the conditional joint distribution of R i· given Y iJ , which can be written as
P (R iJ = r iJ |γ, τ i , ζ, Y iJ ) = J j=1 P (R ij = r ij |τ i , ζ j , Y ij , R i,j−1 ) = J j=1 π I(r ij =1) ij (1 − π ij ) I(r ij =0) ,(5)
where I(·) is the indicator function and π ij is given by Equation (4).
Based on Equation (1), the likelihood function of complete data could be given by
L(Ω|y, r) = N i=1 P (Y iJ = y iJ |a, b, θ i )P (R iJ = r iJ |γ, τ i , ζ, Y iJ ) = N i=1 J j=1 P (Y ij = y ij |θ i , a j , b j ) P (R ij = r ij |τ i , ζ j , Y ij , R i,j−1 ) = N i=1 J j=1 p y ij ij (1 − p ij ) 1−y ij π r ij ij (1 − π ij ) 1−r ij .(6)
where p ij is given by Equation (3). Note that if Y ij in Equation (6) is missing, it would be imputed from Bernoulli(q ij ), and q ij can be computed as
q ij = P (Y mis ij = 1|a j , b j , ζ j , θ i , τ i , γ) = p ij π 11 ij p ij π 11 ij + (1 − p ij )π 10 ij ,(7)
where
π 11 ij =P (R ij = 1|τ i , ζ j , γ, R i,j−1 , Y ij = 1), π 10 ij =P (R ij = 1|τ i , ζ j , γ, R i,j−1 , Y ij = 0).(8)
Integrating over all the missing item responses y mis finally yields
L(Ω|y obs , r) = y mis N i=1 J j=1 P Y ij = (y obs ij , y mis ij ) |θ i , a j , b j · P (R ij = r ij |τ i , ζ j , Y ij , R i,j−1 ) P (Y ij = y mis ij |a j , b j , ζ j , θ i , τ i , γ) = y mis N i=1 J j=1 p y obs ij ij (1 − p ij ) 1−y obs ij π r ij ij (1 − π ij ) 1−r ij q y mis ij ij (1 − q ij ) 1−y mis ij ,(9)
where y obs ij is the observation of Y obs ij and y mis ij is the imputation of Y mis ij .
3 Estimation of model parameters and consistency results
MML estimation and consistency results
We first presented the MML estimation of IRT item parameters and item missing parameter. By integrating over the ability and person missing parameters, the marginal likelihood function L(a, b, ζ|y obs , r) could by given
by y mis N i=1 J j=1 p y obs ij ij (1−p ij ) 1−y obs ij π r ij ij (1−π ij ) 1−r ij q y mis ij ij (1−q ij ) 1−y mis ij p(θ, τ |µ, Σ) dθ dτ,
where σ θτ in Σ is assumed to be known. The MML estimation of (a, b, ζ)
should satisfy:
(â,b,ζ) = arg max a,b,ζ l(a, b, ζ|y obs , r)
where l(a, b, ζ|y obs , r) = log L(a, b, ζ|y obs , r) is the log likelihood. In practice, the log likelihood equations for the item parameters could be derived using the NewtonRaphson algorithm, EM algorithm, or a combination of the two (Bock and Aitkin, 1981). Therefore, the MML estimation of (a, b, ζ)
could be easily obtained. Actually the MML estimation is consistent under some assumptions.
Assumption 1 For sufficiently small A 1 > 0 and for sufficiently large A 2 > 0, B > 0, C > 0, the following integrals are finite:
m 1 (A 1 , B) = − log F (A 1 (θ − B))p(θ, τ |µ, Σ) dθ dτ, m 2 (A 2 , B) = − log[1 − F (A 2 (θ + B))]p(θ, τ |µ, Σ) dθ dτ, m 3 (C) = − log G(c − τ − C)p(θ, τ |µ, Σ) dθ dτ, m 4 (C) = − log[1 − G(c − τ − C)]p(θ, τ |µ, Σ) dθ dτ,
where c is a known finite constant.
Assumption 2 Given known γ, (a, b, ζ) are identifiable.
Theorem 1 The MML estimation (â,b,ζ) are consistent under Assumption 1-2.
Theorem 2 If the estimation (â,b,ζ) are consistent, then they are asymptotically normal with mean centered at the true parameters and variance being the inverse Fisher information matrix.
9
The proof of Theorem 1-2 would be presented in Appendix A.
Bayesian estimation using MCMC method
Though the MML estimation of (a, b, ζ) are consistent, the other parameters could not be estimated by MML estimation. One natural idea is to estimate the parameters in the proposed model by Monte Carlo Markov Chain (MCMC) method. Actually, MCMC and MML estimation have been already compared in context of IRT (Kieftenbeld and Natesan, 2012;Hendrick, 2014). It was verified that there were little difference in item parameter recovery between the two methods with samples of 300 or more (Kieftenbeld and Natesan, 2012). So MCMC method was eventually employed to estimate the whole parameters in the proposed method.
We only take the proposed based on probit link as a demonstration, as 2PL IRT model can be very close to 2PNO IRT model through multiplying by a scaling constant 1.702 for the logistic item discrimination parameter (Baker and Kim, 2004). In details, Gibbs sampling was employed to estimate in unknown parameters in Equation (3) and Equation (4). And Metropol-isHastings algorithm was adopted to estimate σ θτ and σ 2 τ . In order to realize the Gibbs sampling for the 2PNO IRT model, the aug- (Albert, 1992;Albert and Chib, 1993). It was assumed that
mented Z ij was introduced for response variable Y ij , where Z ij |θ i , a j , b j , Y ij ∼ N (a j (θ i − b j ) , 1)Y ij = 1 if Z ij ≥ 0, 0 if Z ij < 0.(10)
Similarly, the independent random variables W ij were augmented for the missing data indicator R ij , which were assumed to follow the normal (4) could be reformulated as
distribution N (γ 0 − τ i + ζ j + g (R i,j−1 , γ 1 ) + l (Y ij , γ 2 ) , 1). So EquationR ij = 1 if W ij ≥ 0, 0 if W ij < 0.(11)
The detailed sampling process would be presented in Appendix B.
Simulation Studies
Two simulation studies were conducted to investigate the empirical performance of the proposed model, including parameter recovery and Bayesian model assessment.
Simulation I
Simulation I is used to compare the parameter recovery of the proposed method with that of listwise deletion.
Design
In the data generation, the number of examinees and the number of items were set to N = 500 and J = 20, respectively. The true values of model parameters were set as follows:
a j ∼ Uniform (0.5, 1.5) , b j ∼ N (0, 1) , ζ j ∼ N (0, 1) , j = 1, ...J. θ i τ i ∼ N 0 0 , 1 σ θτ σ θτ 1 , i = 1, ..., N.
As to covariance σ θτ , also regarded as the correlation between θ and τ (denoted as ρ ), were set as 0, 0.4, 0.8 to elucidate the effect of no, a little, large strength of nonignorability.
Equation (3) was used to simulate the dichotomous response data. And then Equation (4) was employed to generate the missing responses. Actually, the proportions of missing response is adjusted by γ = (γ 0 , γ 1 , γ 2 ). The true values of γ and corresponding average missing proportions (denoted by p)
were set as follows:
γ 0 γ 1 γ 2 p = -2.2 -1.6 -1.1 -0.65 -0.2 0.02 0.04 0.04 0.05 0.05 -0.2 -0.2 -0.2 -0.25 -0.25 0.096 0.178 0.266 0.371 0.472
100 datasets were simulated for 5 (γ) × 3 (σ θτ ) =15 conditions. The MCMC sampling procedure was iterated 20,000 and the first 1,5000 iterations were discarded as burn-in. And then the expected a posteriori (EAP) estimation of each parameter can be obtained from its Markov Chain.
Criteria
To assess the performance of parameter recovery, two criteria were applied: mean Bias and mean absolute error (MAE).
Simply speaking, let λ = (λ 1 , · · · , λ K ) be a vector of true parameter value. And denote by λ l = ( λ 1l , · · · , λ Kl ) be its EAP estimation, l = 1, · · · , L, where L is the number of replications. The mean Bias and MAE could be estimated by Table 1 presents the results of parameter recovery under both the proposed method and listwise deletion for the ignorable missingness (ρ = 0).
Bias(λ) = (KL) −1 L l=1 K k=1 ( λ kl − λ k ), M AE(λ) = (KL) −1 L l=1 K k=1 | λ kl − λ k |.
Results
The results show that the parameter recovery under the two methods are nearly similar for a, b and θ. That is, under the two methods, the item parameters a and b are well recovered, as their biases are close to 0 and MAEs are around 0.1. Relatively higher MAE for the person ability parameter θ is
τ -0.003 0.439 - - ζ 0.012 0.198 - - γ 0 -0.059 0.166 - - γ 1 0.024 0.025 - - γ 2 0.064 0.070 - - σ θτ 0.056 0.056 - - σ 2 τ -0.087 0.110 - - (-1.6,0.04,-0.2) 0.173 a 0.015 0.109 0.016 0.111 b -0.010 0.101 -0.028 0.105 θ 0.004 0.309 0.005 0.310 τ 0.001 0.370 - - ζ 0.036 0.208 - - γ 0 -0.095 0.192 - - γ 1 0.010 0.017 - - γ 2 0.103 0.103 - - σ θτ 0.059 0.059 - - σ 2 τ -0.033 0.081 - - (-1.1,0.04,-0.2) 0.264 a 0.009 0.116 0.009 0.116 b -0.017 0.112 -0.041 0.117 θ 0.010 0.331 0.011 0.333 τ 0.002 0.326 - - ζ 0.008 0.195 - - γ 0 -0.079 0.190 - - γ 1 0.008 0.015 - - γ 2 0.127 0.127 - - σ θτ 0.053 0.053 - - σ 2 τ -0.040 0.089 - - (-0.65,0.05,-0.25) 0.370 a 0.017 0.130 0.017 0.130 b -0.063 0.133 -0.091 0.145 θ 0.003 0.361 0.005 0.362 τ 0.004 0.297 - - ζ -0.001 0.191 - - γ 0 -0.104 0.193 - - γ 1 0.006 0.015 - - γ 2 0.188 0.188 - - σ θτ 0.050 0.050 - - σ 2 τ -0.046 0.091 - - (-0.2,0.05,-0.25) 0.463 a 0.018 0.146 0.020 0.146 b -0.074 0.148 -0.109 0.164 θ 0.007 0.398 0.009 0.399 τ -0.003 0.287 - - ζ 0.047 0.201 - - γ 0 -0.180 0.220 - - γ 1 0.006 0.012 - - γ 2 0.198 0.198 - - σ θτ 0.056 0.056 - - σ 2 τ -0.047 0.084 - -
Note. Bias and MAE refer to the mean Bias and MAE for each parameter in the third column, respectively.
13 14 present. And the bias of θ is very close to 0. For the other parameters in the proposed method, they are almost well recovered, except the person missingness parameter τ . One explanation might simply be that the dimensions of person parameters θ and τ are much higher than the item parameters. Table 2 and Table 3 show the results of parameters recovery under little (ρ = 0.4) and large (ρ = 0.8) nonignorable missingness, respectively. Generally speaking, the recovery of parameters under the proposed method is better than applying listwise deletion. At the same time, as the missing proportion is higher and nonignorability is stronger, the superiority is more obvious. In details, for the proposed method, the item parameters a and b
are always well recovered across all 5 (γ) × 2 (σ θτ ) =10 conditions as the bias is very close to 0 and the MAE is no more than 0.2. Note that for the fixed ρ, the MAE of τ decreases with the increasing of missing proportions.
That is, τ recovers better when the missing responses is more, which also confirms that the missingness could be attributed to latent person missing trait.
Simulation II
Besides evaluating the parameter recovery, we are also interested in assessing whether the missingness is ignorable or nonignorable. Simulation II was conducted to assess whether the missingness is ignorable or nonignorable.
Design
The settings of true model parameters are the same as in Simulation I, except parameter ρ. More specifically, ρ is only to confirm whether the missingness is nonignorable or ignorable in this simulation. That is, in this simulation, the nonignorable (ρ = 0) and ignorable (ρ = 0) models were employed to fit the data generate from the nonignorable model (ρ = 0.8).
Criteria
Within Bayesian framework, there are some commonly used criteria for model selection. In this section, these criteria were only applied to the distribution of R|Y , because we only focus on the missing data mechanism.
One criterion to evaluate the model fit is deviance information criterion (DIC; Spiegelhalter et al., 2002). This criterion takes into account the tradeoff relationship between the adequacy of model fitting and the number of model parameters. The model with a smaller DIC value fits the data better.
The other criterion to compare the two models in terms of fitting is the logarithm of the pseudo marginal likelihood (LPML; Geisser and Eddy, 1979;Ibrahim et al., 2001). The model with a larger LPML has a better fit of the data.
The detailed computational processes of the two criteria would be presented in Appendix C.
Results
The DIC and LPML difference between the nonignorable and ignorable model were calculated and presented by boxplots in Figure 2.
The boxes of DIC difference between nonignorable and ignorable model are always below 0, indicating that the nonignorable model fits the data better. Meanwhile, the LPML differences between nonignorable and ignorable model are always more than 0, which shows the same conclusion as DIC.
Furthermore, the DIC and LPML difference increase with the proportions of missing responses. So the nonignorable model has more distinct advantages in terms of model selection when the missing proportion is higher.
Analysis of the PISA data
The PISA is an international education assessment that measures students' skills and knowledge in science, mathematics, reading, and so on. Its data set is available and free on http://www.oecd.org/pisa/data/.
In this section, PISA data set was employed to illustrate the detailed use of the proposed method and further interpret the parameters.
Data set
In this study, the data set is chosen from a science subtest in the 2015 computer-based PISA in Dominican Republic. The invalid and not applicable samples are excluded from the data set. The valid sample size is 493, in which 173 individuals reached all 17 items. The overall missing proportion of the dataset is 22.9% (9.1% omitted items and 13.8% not-reached items). Only DS465Q01C is scored polytomously with 0 (no credit), 1 (partial credit), and 2 (full credit) scores. As the proposed method is focused on dichotomous responses, only full credit is treated as a correct response, while the other 18 two score categories are treated as incorrect responses.
Analysis
The nonignorable and ignorable models under the proposed method were used to fit the data. DIC and LPML of R|Y under the both models were computed for the purpose of model selection. (Brooks and Gelman, 1998) was computed to assess the convergences of all parameters in the proposed model. Convergence is evaluated by the value ofR. That is, if theR is less than 1.1, the parameter achieves convergence.
The values ofR can be obtained based on the "coda" R package (Plummer et al., 2006). The trace plots ofR for all parameters in the proposed method are presented in Figure 3. It suggests thatR is generally less than 1.1 after 5,000 iterations for each parameter, indicating the perfect convergence of all the model parameters.
Similar to Simulation I, listwise deletion was also employed to get the comparative analysis of parameter estimate. That is, 173 samples with complete responses were used to get the parameter estimates based on traditional IRT models. DS465Q01C and DS438Q03C, their EAP of ζ is much higher than the other items, which means the two items are more likely to be omitted. For the last three items, the proportions of missing are higher, despite their ζ are estimated to be negative. One possible explanation is that the missing data mechanism for the three items is mainly the effect that previous missing items bring on current item, so the missingness for these items may be largely notreached. In general, the items with higher ζ or at the end of test tend to miss.
Results
The results of item parameter based on listwise deletion are presented in Table 5. Obviously, for the same parameter, there is a big difference for the two methods. Specifically, the EAPs of item difficulty parameters in Table 4 are always higher that in Table 5. A likely cause is that the missing 20 21 responses often occur for the examinees with low ability and their responses are deleted in the listwise deletion leading to the underestimates of item difficulty parameters.
The posterior histograms of person parameters in the proposed method are shown in Figure 4, and the posterior histogram of ability parameters based on listwise deletion is presented in Figure 5. For the posterior histograms of ability parameters, there also exits a big difference between the two figures. The posterior histograms of ability parameters in Figure 4 is sharper than that in Figure 5, indicating the variance of the former is smaller.
The results of parameter estimation for other parameters in the proposed model are present in Table 6. σ θτ and σ 2 τ are estimated at 0.405 and 1.000, respectively. So the correlation between θ and τ is about 0.405, which is significantly more than 0. This also confirms that the missingness is nonignorable, which is consistent with the result of Bayesian model selection. And the EAP of γ 1 is 0.204 and more than 0 obviously, showing that the term (4) is reasonable and necessary. were demonstrated by using the 2015 PISA computer-based science subtest data as an example. The real data analysis indicated that the missingness is nonignorable, as the correlation between θ and τ is significantly greater than 0. And both the DIC and LPML also gave the same conclusion.
g (R i,j−1 , γ 1 ) = γ 1 j−1 h=2 R i,h−1 in Equation
The proposed method could handle both ignorable and nonignorable missingness by adjusting the correlation between person missing parameter and ability. If the correlation is more than 0, π ij in Equation (4) depends on the ability θ i , so that the missingness is nonignorable (NMAR). If the correlation is 0, π ij only depends on the observed response τ i and ζ j , as well as cumulative number of missing response. Therefore, the missingness is ignorable.
Despite such promising results, other issues should be investigated in the future. The first one is whether the proposed method could be applied to estimate more complicated IRT models, such as the partial credit model (Masters, 1982), the generalized partial credit model (Muraki, 1992). Second, if the last item at the end of test is missed and the penultimate one is observed, the last nonresponse may occur due to skipped item or the time limit. It is still unknown how to clarify whether the last nonresponse is omitted or not-reached. In this case, maybe response time could be taken into account to get further information. Finally, the missingness not only occurs in item response data but also response time data for computer based test. It is still a promising issue to extend the proposed method to the mixture model for response times and response accuracy (Wang and Xu, 2015). Let Y = (Y obs , Y mis ) and Y i be its item response for examinee i, for i = 1, · · · , N , where Y mis is the imputed value in the proposed method.
Acknowledgments
Define
G N (λ) = 1 N N i=1 log P (Y obs i , R i |λ), G(λ) = log P (Y obs , R|λ) dY obs dR, where P (Y obs , R|λ) = Y mis P ( Y , R|λ, θ, τ )P ( Y mis )p(θ, τ |µ, Σ) dθ dτ .
25 By Jensen's inequality,
log P (Y obs , R|λ) Y mis log P ( Y , R|λ, θ, τ ) P ( Y mis )p(θ, τ |µ, Σ) dθ dτ = Y mis P ( Y mis ) Y log F (a(θ − b)) dθ dτ + (1 − Y ) log 1 − F (a(θ − b)) dθ dτ + R log G(c − τ − C) dθ dτ + (1 − R) log 1 − G(c − τ − C) dθ dτ
Using the notations in Assumption 1, therefore,
E |log P (Y obs , R|λ)| ≤E Y mis P ( Y mis ) Y · m 1 (A 1 , B) + (1 − Y ) · m 2 (A 2 , B) + R · m 3 (C) + (1 − R) · m 4 (C) = m 1 (A 1 , B) − m 2 (A 2 , B) E Y obs + m 2 (A 2 , B) + m 3 (C) − m 4 (C) E R + m 4 (C)
Further, by Assumption 1,
m 1 (A 1 , B)−m 2 (A 2 , B) E Y obs +m 2 (A 2 , B) + m 3 (C)−m 4 (C) E R +m 4 (C) < ∞
Accordingly, the following uniform law of large numbers holds:
sup λ∈ D |G N (λ) − G(λ)| P − → 0,
where P − → denotes convergence in probability. Moreover, by Assumption 2 and Gibbs' inequality, G(λ) has a unique maximum at the true parameter λ 0 . Further, by the continuity of G(λ) with respect to λ, we have λ P − → λ 0 .
Proof of Theorem 2 Since the derivative of G N with respect to (a, b, ζ) at λ is 0 and then by Taylor's expansion, we have that
√ N ( λ − λ 0 ) = − √ N H −1 λ 0 ∂G N ∂a ∂G N ∂b ∂G N ∂ζ λ 0 + o(1),
where o(1) means the remainder converges to 0 in probability and
H = ∂ 2 G N ∂a 2 , ∂ 2 G N ∂a∂b , ∂ 2 G N ∂a∂ζ ∂ 2 G N ∂a∂b , ∂ 2 G N ∂b 2 , ∂ 2 G N ∂b∂ζ ∂ 2 G N ∂a∂ζ , ∂ 2 G N ∂b∂ζ , ∂ 2 G N ∂ζ 2 .
Then, by the multidimensional central limit theorem, we have
√ N ∂G N ∂a ∂G N ∂b ∂G N ∂ζ λ 0 d − → N(0, I(λ 0 )),
where d − → denote convergence in distribution and
I(λ 0 ) = E[ ∂p λ ∂a ] 2 , E[ ∂p λ ∂a ∂p λ ∂b ], E[ ∂p λ ∂a ∂p λ ∂ζ ] E[ ∂p λ ∂a ∂p λ ∂b ], E[ ∂p λ ∂b ] 2 , E[ ∂p λ ∂b ∂p λ ∂ζ ] E[ ∂p λ ∂a ∂p λ ∂ζ ], E[ ∂p λ ∂b ∂p λ ∂ζ ], E[ ∂p λ ∂ζ ] 2 λ 0
is the Fisher information matrix of p λ = log P (Y obs , R|λ) at λ 0 . At the same time, by the weak law of large numbers,
−H λ 0 P − → − E[ ∂ 2 p λ ∂a 2 ], E[ ∂ 2 p λ ∂a∂b ], E[ ∂ 2 p λ ∂a∂ζ ] E[ ∂ 2 p λ ∂a∂b ], E[ ∂ 2 p λ ∂b 2 ], E[ ∂ 2 p λ ∂b∂ζ ] E[ ∂ 2 p λ ∂a∂ζ ], E[ ∂ 2 p λ ∂b∂ζ ], E[ ∂ 2 p λ ∂ζ 2 ] λ 0 = I(λ 0 ).
Further, by the Continuous Mapping theorem and Slutsky's theorem, we
have √ N ( λ − λ 0 ) d − → N(0, I −1 (λ 0 )).
27
Appendix B: The detailed sampling procedure
The detailed sampling procedure using MCMC can be broken down into the following steps:
Step 1: Sample the augmented variable Z ij from the truncated normal distribution
Z ij |θ i , a j , b j , Y ij ∼ N (a j (θ i − b j ), 1)I(Z ij ≥ 0) if Y ij = 1, N (a j (θ i − b j ), 1)I(Z ij < 0) if Y ij = 0.
Step 2: Similarly, sample an additional augmented variable W ij as follow:
W ij |τ i , ζ j , γ, R ij , Y ij ∼ N (γ 0 − τ i + ζ j + g(R i,j−1 , γ 1 ) + l(Y ij , γ 2 ), 1)I(W ij ≥ 0) if R ij = 1, N (γ 0 − τ i + ζ j + g(R i,j−1 , γ 1 ) + l(Y ij , γ 2 ), 1)I(W ij < 0) if R ij = 0.
Step 3: Sample the latent ability parameter θ i for examinee i. As previously mentioned, θ i τ i ∼ N (µ, Σ). So the prior distribution of θ i is the conditional normal distribution
θ i |τ i , µ, Σ ∼ N µ θ i |τ , σ 2 θ|τ ,
where µ θ i |τ = µ θ + σ θτ σ −2 τ (τ i − µ τ ) and σ 2 θ|τ = σ 2 θ − σ 2 θτ σ −2 τ . And the full conditional posterior distribution of θ i is
θ i |τ i , µ, Σ, Z i· ∼ N Var θ × µ θ i |τ σ −2 θ|τ + J j=1 a j (Z ij + a j b j ) , Var θ , where Var θ = σ −2 θ|τ + J j=1 a 2 j −1 .
Step 4: Sample the discrimination parameter a j for item j. A prior for a j is a truncated normal distribution with mean µ a and variance σ 2 a . That is, a j ∼ N (µ a , σ 2 a ) I (a j > 0) . Therefore, the full conditional posterior distribution of a j is
a j |θ, b j , Z ·j ∼ N Var a j × µ a σ −2 a + N i=1 Z ij (θ i − b j ) , Var a j I (a j > 0) , where Var a j = σ −2 a + N i=1 (θ i − b j ) 2 −1 .
Step 5: Sample the difficulty parameter b j for item j. A prior for b j is a normal distribution with mean µ b and variance
σ 2 b . That is, b j ∼ N (µ b , σ 2 b ) . Therefore, the full conditional posterior distribution of b j is b j |θ, a j , Z ·j ∼ N Var b j × µ b σ −2 b + N i=1 a j (a j θ i − Z ij ) , Var b j , where Var b j = σ −2 b + N a 2 j −1 .
Step 6: Sample the missing response Y mis ij from Bernoulli(q ij ), where q ij was defined in Equation (7).
Step 7: Sample the person missing parameter τ i . Similar to θ i , the conditional prior distribution of τ i is
τ i |θ i , µ, Σ ∼ N µ τ i |θ , σ 2 τ |θ ,
where µ τ i |θ = µ τ + σ τ θ σ −2 θ (θ i − µ θ ) , and σ 2 τ |θ = σ 2 τ − σ 2 θτ σ −2 θ . The full conditional posterior distribution of τ i is a normal distribution
N Var τ × µ τ i |θ σ −2 τ |θ + J j=1 (γ 0 + ζ j + g(R i,j−1 , γ 1 ) + l(Y ij , γ 2 ) − W ij ) , Var τ , where Var τ = σ −2 τ |θ + J −1 .
Step 8: Sample the item missing parameter ζ j in the missing mechanism model. A prior for ζ j is a normal distribution N µ ζ , σ 2 ζ . Therefore, the full conditional posterior distribution of ζ j is a normal distribution
N Var ζ × µ ζ σ −2 ζ + N i=1 (−γ 0 + τ i − g(R ij−1 , γ 1 ) − l(Y ij , γ 2 ) + W ij ) , Var ζ ,
where Var ζ = σ −2 ζ + N −1 .
Step 9: Sample the intercept parameter γ 0 in the missing mechanism model. A prior for γ 0 is a truncated normal distribution with mean µ γ 0 and variance σ 2 γ 0 . That is, γ 0 ∼ N µ γ 0 , σ 2 γ 0 I (γ 0 < 0). Let
E = W 11 + τ 1 − ζ 1 − g (R 10 , γ 1 ) − l (Y 11 , γ 2 ) W 12 + τ 1 − ζ 2 − g (R 11 , γ 1 ) − l (Y 12 , γ 2 ) . . . W N J + τ N − ζ J − g (R N,J−1 , γ 1 ) − l (Y N J , γ 2 ) and e = e 11 e 12 . . . e N J .
where g (R i0 , γ 1 ) = 0. Therefore,
E = γ 0 1 + e,
where 1 is a N × J dimension vector with elements 1. So the full conditional
posterior distribution of γ 0 is γ 0 |τ , ζ, W , R, Y ∼ N Var γ 0 × µ γ 0 σ −2 γ 0 + 1 E , Var γ 0 I (γ 0 < 0) ,
where Var γ 0 = σ −2 γ 0 + 1 1 −1 .
Step 10: Sample γ 1 in the missing mechanism model. Similar to γ 0 , the prior for γ 1 is N µ γ 1 , σ 2
γ 1 I (γ 1 > 0) . Let H = W 11 − γ 0 + τ 1 − ζ 1 − l(Y 11 , γ 2 ) W 12 − γ 0 + τ 1 − ζ 2 − l(Y 12 , γ 2 ) . . . W N J − γ 0 + τ 1 − ζ J − l(Y N J , γ 2 ) and R = 0 R 11 . . . J−1 h=1 R N h .
Accordingly,
H = γ 1 R + e.
Therefore, the full conditional posterior distribution of γ 1 is
γ 1 |τ , ζ, W , R, Y ∼ N Var γ 1 × µ γ 1 σ −2 γ 1 + R H , Var γ 1 I (γ 1 > 0) , where Var γ 1 = σ −2 γ 1 + R R −1 .
Step 11: Sample γ 2 in the missing mechanism model. Similar to γ 0 and γ 1 , the prior for γ 2 is N µ γ 2 , σ 2 γ 2 I (γ 2 < 0) . Let
K = W 11 − γ 0 + τ 1 − ζ 1 − g (R 10 , γ 1 ) W 12 − γ 0 + τ 1 − ζ 2 − g (R 11 , γ 1 ) . . . W N J − γ 0 + τ 1 − ζ J − g (R N,J−1 , γ 1 ) and Y = Y 11 Y 12 . . . Y N J .
Equivalently,
K = γ 2 Y + e.
Therefore, the full conditional posterior distribution of γ 2 is γ 2 |τ , ζ, W , R, Y ∼ N Var γ 2 × µ γ 2 σ −2 γ 2 + Y K , Var γ 2 I (γ 2 < 0) ,
where Var γ 2 = σ −2 γ 2 + Y Y −1 .
Step 12: Sample the covariance σ θτ between θ and τ . We will use random-walk Metropolis sampling with a truncated normal proposal. At mth iteration, the proposal distribution is N σ (m−1) θτ , s 2 01 I (0 < σ * θτ < p 01 ),
where p 01 = σ 2,(m−1) τ and s 2 01 is the proposal variance. Therefore, the probability of acceptance α σ (m−1) θτ , σ * θτ can be written as
min 1, N i=1 p τ i θ (m) i , σ 2,(m−1) τ , σ * θτ p (σ * θτ ) Φ p 01 −σ (m−1) θτ s 01 − Φ −σ (m−1) θτ s 01 N i=1 p τ i θ (m) i , σ 2,(m−1) τ , σ (m−1) θτ p σ (m−1) θτ Φ p 01 −σ * θτ s 01 − Φ −σ * θτ s 01 ,
where p (τ i |θ i ) is the conditional density function of the person parameter in missing model, and p (·) is a density of uniform prior.
Step 13: Sample σ 2 τ . Similar to σ θτ , the random walk Metropolis sampler is also applied to σ 2 τ . The proposal distribution is the truncated normal distribution N σ where s 2 02 is the proposal variance, and p ·; n 2 , nα 2 is the density function of the scaled inverse Gamma distribution IG( n 2 , nα 2 ). What's more, the priors of parameters in above sampling procedure are set as follows: a j ∼ N (0, 1) I (a j > 0) , b j ∼ N (0, 1) , ζ j ∼ N (0, 1) , j = 1, ...J. 31 γ 0 ∼ N (0, 1) I (γ 0 < 0) , γ 1 ∼ N (0, 1) I (γ 1 > 0) , γ 2 ∼ N (0, 1) I (γ 2 < 0) µ θ = µ τ = 0, s 2 01 = s 2 02 = 0.01, σ θτ ∼ Uniform (0, 1) , σ 2 τ ∼ IG 0.0001 2 , 0.0001 2 Appendix C: The computational processes of DIC and LPML Let Ψ = (τ , ζ, γ) denote the vector of missing data model parameter in
Equation (4). Let Ψ ij = (τ i , ζ j , γ 0 , γ 1 , γ 2 ) and Ψ The log-likelihood function of R|y evaluated at Ψ (m) is given by
log f R = r Ψ (m) , Y = N i=1 J j=1 log f r ij Ψ (m) ij , Y ij = N i=1 J j=1 [r ij log(π (m) ij ) + (1 − r ij ) log(1 − π (m) ij )]
where π (m) ij = P R ij = 1 Ψ (12) avoids the situation that P D in traditional definition may be negative (Spiegelhalter et al., 2014).
To calculate the conditional LPML (Hanson et al., 2011), the conditional predictive ordinates (CPO; Hanson et al., 2011) index is needed. A Monte Carlo estimate of the CPO is given by
log (CPO ij ) = −U ij,max −log 1 M M m=1 exp − log f R ij Ψ (m) ij , Y ij − U ij,max ,(13)
where U ij,max = max
1≤m≤M − log f R ij Ψ (m)
ij , Y ij . Note that U ij,max plays an important role in numerical stabilization in computing Equation (13).
The LPML is the summary of log( CPO ij ), so
LPML (R|Ψ ) = N i=1 J j=1 log (CPO ij ).
γ 1
1> 0 was chosen to model the effect of previous missingness on the current item. Actually, the statistic j−1 h=2 R i,h exactly captures the missingness of notreached items, and reduces the number of nuisance parameters for modeling the missing data mechanism.
Figure 1 :
1The graphical representation of the proposed method.
Figure 2 :
2Boxplots of DIC and LPML difference between nonignorable and ignorable model.
19 Figure 3 :
1934 summarized the missing proportions of each item and the results of item parameters estimation based on the proposed method. For The trace plots ofR. Note. The dashed line is 1.1.
Figure 4 :
4proposed an IRT-based method to model omitted and notreached items based on SLM. It was proved that the item parameter estimate based on MML estimation is consistent. Further more, MCMC methods was given to estimate all the model parameters based on Probit link. Bayesian The posterior histograms of person parameters based on the proposed method.
Figure 5 :
5The posterior histograms of person parameters based on listwise deletion. model selection for nonignorable and ignorable model was explored using DIC and LPML criteria. The usage and the performance of the proposed method
Theorem 1
1Since the items are assumed to be independent in context of IRT, we only prove the consistency results for the fixed item. For notational convenience, the item subscript j was omitted in the following proof. Moreover, define λ = (a, b, ζ) as the item parameters vector for the item and λ 0 as its true value. Actually, the parameters space can be written as D = (0, ∞) × R × R. It is easy to verify that the MML estimation of λis not at the bound of D. So we only consider the consistency on its closed subset D = [A 1 , A 2 ] × [−B, B] × [−C, C], where A 1 > 0 is sufficient small and A 2 , B, C are sufficient large.
posterior distribution at time m for m = 1, ..., M .
, R ij−1 , Y ij and is modeled in Equation (4). And then the DIC can be calculated as follows:DIC (R|Y ) = D(Ψ) + 2P D = D(Ψ) + 2 D(Ψ) − D(Ψ) , f R Ψ (m) , Y and is the EAP of the deviance function D(Ψ) = −2 log f (R |Ψ ). And D(Ψ) = −2 max 1≤m≤M log f R Ψ (m) , Ywhich is an approximation of Dev(Ψ), when the prior is relatively noninformative and Ψ is the posterior mode. What's more, P D = D(Ψ) − D(Ψ) is the effective number of parameters. Based on the construction, the DIC (R|Ω ) given in Equation
Table 1 :
1The results of parameters recovery for ignorable nonresponses.(γ 0 , γ 1 , γ 2 )p
Parameter
The Proposed Method
Listwise Deletion Method
Bias
MAE
Bias
MAE
(-2.2,0.02,-0.2)
0.093
a
0.005
0.107
0.005
0.108
b
-0.002
0.094
-0.016
0.095
θ
0.000
0.292
0.001
0.292
Table 2 :
2The results of parameters recovery for nonignorable nonresponses (ρ = 0.4).Note. Bias and MAE refer to the mean Bias and MAE for each parameter in the third column, respectively.(γ 0 , γ 1 , γ 2 )p
Parameter
The Proposed Method
Listwise Deletion Method
Bias
MAE
Bias
MAE
(-2.2,0.02,-0.2)
0.098
a
0.013
0.106
0.004
0.106
b
0.003
0.098
-0.023
0.103
θ
0.008
0.289
0.007
0.295
τ
0.003
0.421
-
-
ζ
0.009
0.228
-
-
γ 0
-0.041
0.189
-
-
γ 1
0.020
0.021
-
-
γ 2
0.057
0.070
-
-
σ θτ
-0.014
0.048
-
-
σ 2
τ
-0.051
0.096
-
-
(-1.6,0.04,-0.2)
0.178
a
0.006
0.112
-0.005
0.114
b
-0.016
0.106
-0.055
0.118
θ
0.005
0.306
0.005
0.316
τ
0.008
0.357
-
-
ζ
0.024
0.204
-
-
γ 0
-0.067
0.199
-
-
γ 1
0.004
0.015
-
-
γ 2
0.103
0.103
-
-
σ θτ
0.010
0.044
-
-
σ 2
τ
0.012
0.100
-
-
(-1.1,0.04,-0.2)
0.266
a
0.010
0.127
-0.010
0.124
b
-0.034
0.117
-0.091
0.141
θ
0.004
0.325
0.006
0.339
τ
-0.007
0.316
-
-
ζ
-0.005
0.165
-
-
γ 0
-0.075
0.159
-
-
γ 1
0.005
0.014
-
-
γ 2
0.130
0.130
-
-
σ θτ
0.014
0.051
-
-
σ 2
τ
0.004
0.088
-
-
(-0.65,0.05,-0.25) 0.371
a
0.008
0.133
-0.008
0.135
b
-0.069
0.139
-0.151
0.189
θ
0.006
0.355
0.009
0.377
τ
0.002
0.296
-
-
ζ
-0.020
0.187
-
-
γ 0
-0.075
0.193
-
-
γ 1
0.002
0.014
-
-
γ 2
0.190
0.190
-
-
σ θτ
0.009
0.053
-
-
σ 2
τ
0.041
0.103
-
-
(-0.2,0.05,-0.25)
0.480
a
-0.003
0.153
-0.018
0.153
b
-0.123
0.177
-0.231
0.255
θ
0.003
0.386
0.009
0.419
τ
-0.002
0.286
-
-
ζ
-0.021
0.189
-
-
γ 0
-0.110
0.171
-
-
γ 1
0.006
0.014
-
-
γ 2
0.203
0.203
-
-
σ θτ
-0.011
0.048
-
-
σ 2
τ
0.004
0.093
-
-
Table 3 :
3The results of parameters recovery for nonignorable nonresponses (ρ = 0.8).Note. Bias and MAE refer to the mean Bias and MAE for each parameter in the third column, respectively.(γ 0 , γ 1 , γ 2 )p
Parameter
The Proposed Method
Listwise Deletion Method
Bias
MAE
Bias
MAE
(-2.2,0.02,-0.2)
0.097
a
0.011
0.105
-0.015
0.110
b
-0.009
0.104
-0.051
0.120
θ
0.001
0.278
-0.002
0.303
τ
0.001
0.373
-
-
ζ
0.026
0.206
-
-
γ 0
-0.053
0.163
-
-
γ 1
0.024
0.024
-
-
γ 2
0.038
0.079
-
-
σ θτ
-0.020
0.062
-
-
σ 2
τ
-0.050
0.122
-
-
(-1.6,0.04,-0.2)
0.180
a
0.006
0.116
-0.032
0.126
b
-0.018
0.110
-0.086
0.140
θ
0.004
0.283
0.005
0.323
τ
0.002
0.328
-
-
ζ
0.008
0.205
-
-
γ 0
-0.058
0.196
-
-
γ 1
0.008
0.017
-
-
γ 2
0.095
0.111
-
-
σ θτ
0.000
0.062
-
-
σ 2
τ
0.021
0.130
-
-
(-1.1,0.04,-0.2)
0.267
a
-0.002
0.122
-0.057
0.137
b
-0.033
0.124
-0.139
0.185
θ
0.003
0.298
0.006
0.361
τ
0.003
0.304
-
-
ζ
0.017
0.200
-
-
γ 0
-0.094
0.196
-
-
γ 1
0.008
0.017
-
-
γ 2
0.119
0.136
-
-
σ θτ
-0.003
0.063
-
-
σ 2
τ
0.019
0.120
-
-
(-0.65,0.05,-0.25) 0.365
a
-0.013
0.142
-0.078
0.161
b
-0.085
0.161
-0.247
0.276
θ
0.007
0.314
0.011
0.407
τ
0.005
0.284
-
-
ζ
0.022
0.195
-
-
γ 0
-0.117
0.204
-
-
γ 1
0.002
0.014
-
-
γ 2
0.192
0.193
-
-
σ θτ
0.025
0.053
-
-
σ 2
τ
0.077
0.128
-
-
(-0.2,0.05,-0.25)
0.464
a
-0.018
0.158
-0.082
0.181
b
-0.110
0.183
-0.324
0.347
θ
0.185
0.335
0.011
0.455
τ
0.004
0.281
-
-
ζ
0.052
0.207
-
-
γ 0
-0.180
0.216
-
-
γ 1
0.009
0.017
-
-
γ 2
0.202
0.202
-
-
σ θτ
0.003
0.064
-
-
σ 2
τ
0.043
0.132
-
-
Three Markov chains started at over dispersed starting values were used and each chain had 20000 iterations. The Gelman-Rubin convergence statis-The DICs under nonignorable
and ignorable model are 5504 and 5857, respectively. And the LPMLs under
nonignorable and ignorable model are -2895 and -3016, respectively. Judging
by the both criteria, nonignorable model was selected. So the following
analysis based on the proposed method refers to the nonignorable models.
ticR
Table
Table 4 :
4Item parameters estimation for the PISA subtest data based on the proposed method.Note. p refers to the missing proportion for each item. EAP =expected a posteriori, SE=standard error.Item
p
a
b
ζ
EAP
SE
EAP
SE
EAP
SE
DS465Q01C 0.296 1.078 0.003
2.302 0.004
0.784 0.001
CS465Q02S 0.142 0.329 0.001
0.743 0.004
-0.023 0.001
CS465Q04S 0.123 0.177 0.000
3.150 0.008
-0.225 0.001
DS131Q02C 0.203 0.730 0.002
2.615 0.005
0.215 0.001
DS131Q04C 0.237 1.099 0.003
1.720 0.003
0.333 0.001
CS428Q01S 0.097 0.867 0.002
1.182 0.002
-0.748 0.002
CS428Q03S 0.102 0.827 0.002
1.102 0.002
-0.741 0.002
DS428Q05C 0.275 1.218 0.004
2.243 0.004
0.418 0.001
DS514Q02C 0.247 0.781 0.002
0.019 0.002
0.191 0.001
DS514Q03C 0.169 0.784 0.002
0.584 0.002
-0.355 0.001
DS514Q04C 0.298 1.535 0.003
1.689 0.002
0.342 0.001
CS438Q01S 0.197 0.711 0.002
0.584 0.002
-0.355 0.001
CS438Q02S 0.239 0.558 0.001
0.872 0.003
-0.125 0.001
DS438Q03C 0.459 1.045 0.003
2.347 0.004
0.965 0.001
CS415Q07S 0.247 0.503 0.001
0.087 0.002
-0.277 0.001
CS415Q02S 0.279 0.709 0.002
0.243 0.002
-0.123 0.001
CS415Q08S 0.275 0.546 0.002
1.344 0.004
-0.228 0.005
Table 5 :
5Item parameters estimation for the PISA subtest data based on listwise deletion.Item
a
b
EAP
SE
EAP
SE
DS465Q01C
1.624 0.006
1.757 0.003
CS465Q02S
0.500 0.002
0.368 0.004
CS465Q04S
0.259 0.001
2.165 0.008
DS131Q02C
0.904 0.003
1.964 0.005
DS131Q04C
1.477 0.005
1.186 0.002
CS428Q01S
1.118 0.003
0.840 0.002
CS428Q03S
0.834 0.003
0.739 0.003
DS428Q05C
1.298 0.005
1.755 0.004
DS514Q02C
0.627 0.002
-0.570 0.003
DS514Q03C
0.801 0.003
1.763 0.005
DS514Q04C
1.629 0.005
1.203 0.002
CS438Q01S
0.729 0.002
-0.246 0.002
CS438Q02S
0.771 0.002
0.469 0.002
DS438Q03C
1.224 0.004
1.950 0.005
CS415Q07S
0.427 0.002
-0.526 0.004
CS415Q02S
0.694 0.002
-0.116 0.002
CS415Q08S
0.465 0.002
0.962 0.005
Note. EAP =expected a posteriori, SE=standard er-
ror.
Table 6 :
6The EAP and SE of the other parameters based on the proposed method.Parameter σ θτ
σ 2
τ
γ 0
γ 1
γ 2
EAP
0.405
1.000
-1.53
0.204
-0.034
SE
0.001
0.002
0.001
0.000
0.000
Note. EAP =expected a posteriori, SE=standard error.
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"Hyper-Hue and EMAP on Hyperspectral Images for Supervised Layer Decomposition of Old Master Drawings",
"Hyper-Hue and EMAP on Hyperspectral Images for Supervised Layer Decomposition of Old Master Drawings"
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"Amirabbas Davari [email protected] \nDepartment of Computer Science\nPattern Recognition Lab\nFriedrich-Alexander University Erlangen Nuremberg\nBibliotheca Hertziana -Max-Planck-Institute for Art HistoryErlangen, RomeGermany, Italy\n",
"Nikolaos Sakaltras \nDepartment of Computer Science\nPattern Recognition Lab\nFriedrich-Alexander University Erlangen Nuremberg\nBibliotheca Hertziana -Max-Planck-Institute for Art HistoryErlangen, RomeGermany, Italy\n",
"Armin Haeberle \nDepartment of Computer Science\nPattern Recognition Lab\nFriedrich-Alexander University Erlangen Nuremberg\nBibliotheca Hertziana -Max-Planck-Institute for Art HistoryErlangen, RomeGermany, Italy\n",
"Sulaiman Vesal \nDepartment of Computer Science\nPattern Recognition Lab\nFriedrich-Alexander University Erlangen Nuremberg\nBibliotheca Hertziana -Max-Planck-Institute for Art HistoryErlangen, RomeGermany, Italy\n",
"Vincent Christlein \nDepartment of Computer Science\nPattern Recognition Lab\nFriedrich-Alexander University Erlangen Nuremberg\nBibliotheca Hertziana -Max-Planck-Institute for Art HistoryErlangen, RomeGermany, Italy\n",
"Andreas Maier \nDepartment of Computer Science\nPattern Recognition Lab\nFriedrich-Alexander University Erlangen Nuremberg\nBibliotheca Hertziana -Max-Planck-Institute for Art HistoryErlangen, RomeGermany, Italy\n",
"Christian Riess \nDepartment of Computer Science\nPattern Recognition Lab\nFriedrich-Alexander University Erlangen Nuremberg\nBibliotheca Hertziana -Max-Planck-Institute for Art HistoryErlangen, RomeGermany, Italy\n"
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"Department of Computer Science\nPattern Recognition Lab\nFriedrich-Alexander University Erlangen Nuremberg\nBibliotheca Hertziana -Max-Planck-Institute for Art HistoryErlangen, RomeGermany, Italy",
"Department of Computer Science\nPattern Recognition Lab\nFriedrich-Alexander University Erlangen Nuremberg\nBibliotheca Hertziana -Max-Planck-Institute for Art HistoryErlangen, RomeGermany, Italy",
"Department of Computer Science\nPattern Recognition Lab\nFriedrich-Alexander University Erlangen Nuremberg\nBibliotheca Hertziana -Max-Planck-Institute for Art HistoryErlangen, RomeGermany, Italy",
"Department of Computer Science\nPattern Recognition Lab\nFriedrich-Alexander University Erlangen Nuremberg\nBibliotheca Hertziana -Max-Planck-Institute for Art HistoryErlangen, RomeGermany, Italy",
"Department of Computer Science\nPattern Recognition Lab\nFriedrich-Alexander University Erlangen Nuremberg\nBibliotheca Hertziana -Max-Planck-Institute for Art HistoryErlangen, RomeGermany, Italy",
"Department of Computer Science\nPattern Recognition Lab\nFriedrich-Alexander University Erlangen Nuremberg\nBibliotheca Hertziana -Max-Planck-Institute for Art HistoryErlangen, RomeGermany, Italy"
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Old master drawings were mostly created step by step in several layers using different materials. To art historians and restorers, examination of these layers brings various insights into the artistic work process and helps to answer questions about the object, its attribution and its authenticity. However, these layers typically overlap and are oftentimes difficult to differentiate with the unaided eye. For example, a common layer combination is red chalk under ink.In this work, we propose an image processing pipeline that operates on hyperspectral images to separate such layers. Using this pipeline, we show that hyperspectral images enable better layer separation than RGB images, and that spectral focus stacking aids the layer separation. In particular, we propose to use two descriptors in hyperspectral historical document analysis, namely hyper-hue and extended multiattribute profile (EMAP). Our comparative results with other features underline the efficacy of the three proposed improvements.
|
10.1109/icip.2018.8451768
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[
"https://arxiv.org/pdf/1801.09472v1.pdf"
] | 9,076,748 |
1801.09472
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223c231a1f6bbe5682433a5ea94e206154b43943
|
Hyper-Hue and EMAP on Hyperspectral Images for Supervised Layer Decomposition of Old Master Drawings
Amirabbas Davari [email protected]
Department of Computer Science
Pattern Recognition Lab
Friedrich-Alexander University Erlangen Nuremberg
Bibliotheca Hertziana -Max-Planck-Institute for Art HistoryErlangen, RomeGermany, Italy
Nikolaos Sakaltras
Department of Computer Science
Pattern Recognition Lab
Friedrich-Alexander University Erlangen Nuremberg
Bibliotheca Hertziana -Max-Planck-Institute for Art HistoryErlangen, RomeGermany, Italy
Armin Haeberle
Department of Computer Science
Pattern Recognition Lab
Friedrich-Alexander University Erlangen Nuremberg
Bibliotheca Hertziana -Max-Planck-Institute for Art HistoryErlangen, RomeGermany, Italy
Sulaiman Vesal
Department of Computer Science
Pattern Recognition Lab
Friedrich-Alexander University Erlangen Nuremberg
Bibliotheca Hertziana -Max-Planck-Institute for Art HistoryErlangen, RomeGermany, Italy
Vincent Christlein
Department of Computer Science
Pattern Recognition Lab
Friedrich-Alexander University Erlangen Nuremberg
Bibliotheca Hertziana -Max-Planck-Institute for Art HistoryErlangen, RomeGermany, Italy
Andreas Maier
Department of Computer Science
Pattern Recognition Lab
Friedrich-Alexander University Erlangen Nuremberg
Bibliotheca Hertziana -Max-Planck-Institute for Art HistoryErlangen, RomeGermany, Italy
Christian Riess
Department of Computer Science
Pattern Recognition Lab
Friedrich-Alexander University Erlangen Nuremberg
Bibliotheca Hertziana -Max-Planck-Institute for Art HistoryErlangen, RomeGermany, Italy
Hyper-Hue and EMAP on Hyperspectral Images for Supervised Layer Decomposition of Old Master Drawings
Old Master DrawingLayer SeparationHyper- HueEMAPSpectral Focus Stacking
Old master drawings were mostly created step by step in several layers using different materials. To art historians and restorers, examination of these layers brings various insights into the artistic work process and helps to answer questions about the object, its attribution and its authenticity. However, these layers typically overlap and are oftentimes difficult to differentiate with the unaided eye. For example, a common layer combination is red chalk under ink.In this work, we propose an image processing pipeline that operates on hyperspectral images to separate such layers. Using this pipeline, we show that hyperspectral images enable better layer separation than RGB images, and that spectral focus stacking aids the layer separation. In particular, we propose to use two descriptors in hyperspectral historical document analysis, namely hyper-hue and extended multiattribute profile (EMAP). Our comparative results with other features underline the efficacy of the three proposed improvements.
I. INTRODUCTION
Red chalk was a highly popular drawing material until the late nineteenth century [1], [2]. In the artistic work process, it has oftentimes been used for creating a first sketch, in order to later overdraw it with ink. For art historians today, these sketches provide insights into the creation process of the art work. In particular, differences between the underlying sketch and the overdrawn picture can indicate changes in the direction of the work.
In this work, we investigate the particular case where red chalk is overdrawn by ink. A widely used technique to visualize structures below a layer of ink is to image via infrared reflectography (IRR) the object in the infrared range, at wavelengths above 2000 nm. In this regime, ink becomes transparent. However, this approach is not applicable to make red chalk. Red chalk consists primarily of natural red clay containing iron oxide, and the reflectance of red chalk at wavelengths above 2000 nm is very similar to the image carrier (i.e., the paper or parchment). As a consequence, this range of wavelengths can not be used to visualize over-painted strata of red chalk [3], [4]. The difficulties of displaying and distinguishing the drawn strata by conventional IRR, or with remissionspectroscopy poses a significant challenge to recover the underlying substrate layers. This is also shown in the comparative sequence of images from the apocryphal Rembrandt drawings in Munich (visible spectrum versus infrared imaging), published by Burmester and Renger [3,.
In this work, we propose to close this diagnostic gap to visualize red chalk below ink by using hyperspectral imaging together with a pattern recognition pipeline. There are many works in the literature that used hyperspectral imaging for document analysis and proved its superiority to RGB imaging [5], [6], [7]. Our contributions are three-fold: We propose two descriptors for using in hyperspectral historical document analysis, namely hyperhue and extended multi-attribute profile (EMAP), and we address a common artifact in hyperspectral imaging called focus shifting, and propose spectral focus stacking as its solution. We evaluate the proposed approaches on drawings that are created to exactly mimic the original work process.
II. HYPERSPECTRAL DESCRIPTORS FOR SKETCH LAYER SEPARATION
A. Extended Multi-Attribute Profile (EMAP)
Attribute profiles are popular tools in remote sensing [8], [9]. The idea is to abstract morphological operators like opening or closing from specific shapes of structuring elements. The building blocks of attribute profiles are attribute filters that operate on connected components (CC) of lower or equal gray level intensities. On each CC in the image, an attribute A (e.g., the area, standard deviation, or diameter of the CC) is computed and compared to a threshold λ. If A(CC i ) ≥ λ, it is preserved. Otherwise, the i-th CC is merged with the closest neighboring CC. Analogously to classical morphological operators, attribute thickening (denoted as φ A λ (f )) is the process of merging the CCs of image f to neighboring CC with higher gray level. Attribute thinning (denoted as γ A λ (f )) is the process of merging the CCs of image f to neighboring CC with lower gray level.
The attribute thinning profile of an image f , denoted by Π(γ A λ )(f ), is generated by concatenating series of attribute thinning with an increasing criterion size λ:
Π(γ A λ )(f ) = {Π(γ A λ ) : Π(γ A λ ) = γ A λ (f ), ∀λ ∈ [0, ..., n]}(1)
Analogously, attribute thickening profile of an image f , denoted by Π(Φ A λ )(f ), is generated by concatenating series of attribute thickenings with an increasing criterion size λ:
Π(Φ A λ )(f ) = {Π(Φ T λ ) : Π(Φ T λ ) = Φ T λ (f ), ∀λ ∈ [0, ..., n]} (2)
The attribute profile (AP) is generated by concatenating series of attribute thickening and thinning profiles with an increasing criterion size λ:
AP (f ) = { ∀λ∈[λ1,...,λn] Π(Φ T λ )(f ) , f, ∀λ∈[λ1,...,λn] Π (Φ T λ (f )} (3) In the case of λ = 0, Π(γ T 0 ) = Π(Φ T 0 ) = f .
Therefore, attribute profile vector's size will be 2n + 1, i.e., n for attribute thinning, n for attribute closing and one for the original image.
By using more than one attribute and concatenating the generated APs, multi-attribute profiles (MAPs) are generated. Finally, stacking the computed MAPs over each spectral channel of a multi-/hyper-spectral image results in the extended multi-attribute profile (EMAP). EMAPs use both spatial and spectral signatures of a hyperspectral image and are capable of modeling and describing an image based on different attributes, e.g. area, standard deviation and moment of the CCs. In this work, we used the same attributes and threshold values as the work by Ghamisi et al. [10].
B. Hyper-Hue
In RGB color space, pixels are 3-dimensional. In this cube, (0, 0, 0) T corresponds to black color and (1, 1, 1) T represents the white color. The vector connecting these two points, the diagonal of the cube, is called achromatic axis. By projecting all points in the RGB cube on a plane which is perpendicular to the achromatic axis and includes the point (0, 0, 0) T , the so-called chromatic plane with regular hexagon shaped borders is formed.
For n-dimensional hyperspectral images, the same concept can be extended. Each point x i is represented by n values. Therefore, an n-dimensional hyperspectral image is defined as
f : D f ⊂ Z 2 → [0, 1] n .(4)
The vector connecting the n-dimensional points, let (0, . . . , 0) T denote the black in n dimensions, which we call HyperBlack, and let analogously denote (1, . . . , 1) T HyperWhite. Let furthermore a denote the achromatic hyper-axis, which is the normal vector of the hyperchromatic plane P. In order to mathematically define P, we derive its spanning unit vectors. In an n-dimensional space, P is spanned by n − 1 pairwise perpendicular ndimensional unit vectors, {u 1 , u 2 , . . . , u n } T . The vectors u i have the properties that (1) they start from the point HyperBlack, (2) they are pairwise perpendicular, (3) they are unit vectors and therefore their norm is 1, (4) the direction of u 1 points towards the chromatic hyper-space, (5) u i are orthogonal to a.
Suppose the first n − m elements of u i are 0 and the remaining m elements are non-zero. From these m elements, denote the first one as a and the remaining elements as b. As it is derived in [11], we obtain a basis for P by setting a =
(a) (b) (c) (d) (e)m−1 √ m(m−1) and b = −1 √ m(m−1)
. The projection of a hyperspectral point x j onto P is then
c j = (x j · u 1 )u 1 + (x j · u 2 )u 2 + · · · + (x j · u n )u n . (5)
Liu et al. [11] defined hyper-hue h, saturation S and intensity I of a hyperspectral point x via its projection c as
h = c c ,(6)S = c c max = max{x 1 , . . . , x n }−min{x 1 , . . . , x n } ,(7)I = 1 n (x 1 + · · · + x n ) .(8)
In this way, an extension of HSI color space is defined for hyperspectral images.
III. PROCESSING PIPELINE A. Sensitivity Normalization
Hyperspectral imaging setups suffer from various limitations and artifacts which need to be corrected. Fig. 1 (a)-(e) show five sample channels of a raw hyperspectral image (HSI), namely channel 20 (representing 407.31 nm wavelength), channel 40 (representing 455.41 nm wavelength), channel 70 (representing 528.32 nm wavelength), channel 130 (representing 676.93 nm wavelength), channel 230 (representing 932.82 nm wavelength). As it can be observed, the sensitivity of the HS camera sensor along the spectrum is not uniform. Using a white reference, the uneven sensitivity can be corrected. Fig. 2-(a) shows the normalized sensitivity diagram of the sensor measured from a white reference. The inverse of this diagram is used as the sensitivity normalization coefficient. Fig. 2
(b)-(f)
show the sensitivity-normalized version of the channels presented in Fig. 1 (a)-(e), respectively.
Every imaging setup needs good lighting for an acceptable acquisition and HS imaging is not an exception. In real world scenario, in a museum for instance, the subject may not be evenly illuminated. To simulate this situation,
B. Focus Stacking
Common HS cameras suffer from focus shifting, which is a well-known artifact in the field. It leads to the issue that not all of the channels are simultaneously in focus when making a multispectral acquisition. Fig. 4 shows this behavior for two hyperspectral images, namely H 1 and H 2 . For acquiring H 1 , the lens is focused with the blue range aimed to be in focus. H 2 , on the other hand, is captured by having the red spectrum in focus. Fig. 4 (a) shows channel 41, representing 458.82 nm wavelength, of H 1 . Fig. 4 (b) shows the same channel of H 2 . Especially on fine edges, we can observe that (a) is sharper and more in focus. Similarly, Fig. 4 (c) and (d) show the channel 200, representing the 854.97 nm wavelength, of H 1 and H 2 , respectively. This time the channel corresponding to H 2 is sharper than H 1 .
One contribution of this work lies in producing one hyperspectral image with all channels in focus via spectral focus stacking. To this end, we acquire two images with two different focus points, one in the blue spectrum and one in the red spectrum. The final all-in-focus image is generated from the in-focus channels of the two input images. In our work, we generate our final all-in-focus image by using the first 75 channels from H 1 and the remaining 183 channels from H 2 . We quantitatively compared our all-in-focus HSI with H 1 and H 2 . The results are presented in Table I and are discussed in Sec.IV-C.
(a) (b) (c) (d) (e) (f)
C. Classification
In a previous study on this application we followed an unsupervised approaches with k-means and GMM clustering algorithms [12], which performed weakly, especially for diluted red chalk. In this work, we assume that it is feasible to obtain a limited number of labeled pixels by a specialist, e.g., an art historian. This allows to use supervised learning to evaluate the proposed features and processing pipeline for layer separation. We consider the three classes red chalk, diluted red chalk and black ink. Classification is performed using a random forest (RF), with 10 trees. The number of variables for training the trees and bagging is set to the square root of the number of features, as proposed by Breiman [13]. We used 100 random samples per class for training and the rest for testing. We repeated this process 25 times and reported the average classification performance metrics and their standard deviation (SD). In our dataset, the number of pixels for these classes is 10791, 23528 and 85000, respectively. For training, we select 100 pixels from each class, which corresponds to 0.9%, 0.4% and 0.1% of each class, respectively.
IV. EVALUATION
A. Dataset 1) Phantom Data: We created a set of sketches with multiple layers of graphite, chalk, and different inks of the same chemical composition that were commonly used in old master drawings. After each layer was drawn, the picture was scanned with a book scanner (Zeutschel OS 12000, in RGB mode). This step-by-step documentation of the controlled creation process allows to compute ground truth drawing layers, by subtracting two subsequent scanned images. A sample sketch from this data is shown in Figure 5.
2) Hyperspectral Imaging: For imaging, we use a Specim PFD-CL-65-V10E hyperspectral camera equipped with a CMOS sensor, capable of capturing the spectrum in a wavelength range of 400 nm to 1000 nm. The imaging setup is shown in Fig. 6. We use a lens with 16 mm focal length and the distance between the subject and the camera is 68 cm. The document is illuminated with a 500 W tungsten lamp.
3) Simulated RGB: In order to compare the effectiveness of using hyperspectral images for layer separation with RGB images, we simulated RGB images from our hyperspectral images. The blue color in RGB domain is corresponding to the wavelengths between 415 nm and 495 nm (HSI channels 24 to 56). Similarly, the green color Fig. 1, (b) RGB image generated from the sensitivitynormalized HSI in Fig. 2, (c) RGB image generated from illumination-corrected HSI in Fig. 3.
corresponds to the wavelengths range of 495 nm-570 nm (HSI channels 57 to 87) and the red color lies between 620 nm-750 nm (HSI channels 108 to 156). We generated the red, green and blue channels by taking the average of HSI channels 108-156, 57-87 and 24-56, respectively. The reason that we do not use the RGB image acquired by the board scanner for comparison is that the board scanner has newer sensor, higher resolution, higher signal to noise ratio (SNR) and better lighting condition. Therefore, the comparison would not be fair. Fig.7 shows the simulated RGBs from the HSIs, before and after pre-processing.
B. Evaluation Protocol 1) Registration of HSI to the ground truth: Our ground truth, generated from Fig. 5, is acquired by a board scanner. The HSI images are acquired via a line scanner hyperspectral camera. Different modalities, resolutions, aspect ratios and the non-flat surface of the paper make the images from these modalities geometrically different. In order to compare the HSI analysis output, hyperspectral images need to be registered to the board scanner image. In a previous study [16], we concluded that a non-rigid registration using residual complexity similarity measure (RC) [17] suits our purpose well. Therefore, we use RC to register our HSI to the RGB image acquired by the board scanner. 2) Metrics: To evaluate the classification performances, we used overall accuracy (OA), average accuracy (AA) and Kappa coefficient metrics. OA is the number of correctly classified instances divided by the number of all samples, while AA is the mean class-based accuracies. The kappa statistic is a measure of how closely the classified samples matches the ground truth. By measuring the expected accuracy, it results in a statistic expressing the accuracy of a random classifier.
C. Results
1) Impact of Spectral Focus Stacking:
In order to study the effect of spectral focus stacking, we conducted two sets of experiments on simulated RGB images and HSI images. In the first experiment, we generated RGB images from H 1 , H 2 and all-in-focus HSI. In the second experiment, we carried out the classification on the illumination-corrected H 1 , H 2 and all-in-focus HSI. The results for these two experiments are presented in Table I. It can be seen that in both scenarios, spectral focus stacking yields better AA, OA and Kappa performance.
2) Layer Separation Performance of the Proposed Features: As spectral focus stacking results in better performance, the remaining computations are performed over all-in-focus images. To study the impact of illumination correction, hyper-hue, and EMAP, we generated the following features. 1) SimRGB: Simulated RGB image, generated from the illumination-uncorrected all-in-focus HSI, 2) SimRGB-IC: Simulated RGB image, generated from the illumination-corrected HSI, 3) SimRGB-IC-SI: SimRGB-IC, saturation (S in Eq. 7) and illumination (I in Eq. 8) concatenated together, 4) SimRGB-IC-EMAP: EMAP computed on SimRGB-IC. We used area as the only EMAP attribute with 20 thresholds λ. In order to choose the threshold values, we followed a similar approach to Ghamisi et al. [10], 5) HSI: Illumination-uncorrected all-in-focus HSI, 6) HSI-IC: Illumination-corrected all-in-focus HSI, 7) HSI-DR: HSI-IC projected to its PCA components such that 99.9% of its variance are preserved, 8) HSI-h: Hyper-hue computed from the illuminationcorrected HSI, 9) HSIhSI: HSI-IC, hyper-hue, saturation (S) and illumination (I) concatenated together, 10) HSIhSI-DR: Dimensionality reduced HSIhSI via PCA so that 99.9% of its variance is preserved. Table II. The first observation is that illumination correction always improves the results, both for SimRGB vs. SimRGB-IC and for HSI vs. HSI-IC. Furthermore, comparing the SimRGB-IC with HSI-IC indicates that an illuminationcorrected HSI performs better than an RGB image. In HSI-DR, applying PCA further improves the HSI performance.
Hyper-hue computed over HSI (HSI-h) results in a big jump in performance. Furthermore, the standard deviation in HSI-h is smallest among all the other features which indicates a high stability of this feature. Combining hyperhue, saturation and illumination on the HSI image (HSI-hSI) can not exceed the performance of hyper-hue alone. Also dimensionality reduction on this combination (HSI-hSI-DR) can not compete with HSI-h, and performs even worse than HSI-hSI. EMAP computed on the HSI (HSI-EMAP) results in a performance that is well comparable with HSI-h. Finally, computing EMAP over HSIhSI leads to a slight improvement and results in the overall best layer separation performance.
It is worth mentioning that the threshold values we choose for EMAP by following the proposed method in [10] are probably not optimal. Observing a competitive performance by EMAP in this work motivates us to study other attributes and threshold values in our future works.
In order to see the effect of a better sensor, SNR and lighting, we also classified on the down-sampled RGB image that is acquired by the RGB board scanner (see Table III). These results are superior to the HSI-based results, which was expected. Along with the results in Table II, we conclude that multi-/hyper-spectral imaging with suitable processing can outperform RGB imaging when operating on images with identical noise and photon statistics. While the photon statistics is typically bounded by the fact that historic documents may not be exposed to too much light, it will be interesting to investigate multispectral imaging with a DSLR camera due to the improved resolution, SNR, and dynamic range in future work.
The results are qualitatively compared in Fig. 8. In this figure, (a) represents the ground truth (GT). Red color in GT corresponds to the red chalk, green represents the red chalk that is overlaid by the black ink and blue color, is the black ink class. Black color in GT represents the background and is not considered during the classification. As it can be observed from this image, SimRGB label map contains high portion of misclassification, which is highly improved by HSI-h, HSI-EMAP and HSIhSI-EMAP.
V. CONCLUSION
In this work, we propose and evaluate a hyperspectral imaging pipeline for decomposing the layers of old Master drawings. Our particular focus lies on distinguishing the commonly used red chalk and black ink. We propose two descriptors to the field of hyperspectral historical document analysis, namely hyper-hue and extended multiattribute profile. We also address focus shifting, an artifact in hyperspectral imaging, by focus stacking.
Our comparative results confirm that hyperspectral images are at identical resolution and SNR more informative than RGB images and result in better layer separation performance. EMAP and hyper-hue both outperform the raw hyperspectral features, and focus stacking of hyperspectral images positively impacts the layer separation.
Figure 1 :
1Sample channels of the raw hyperspectral image. (a) channel 20, (b) channel 40, (c) channel 70, (d) channel 130, (e) channel 230. Comparing the channels, the uneven sensitivity of the camera sensor along the spectrum is obvious.
Figure 2 :
2Sensitivity-normalized sample channels of the raw hyperspectral image. (a) Normalized hyperspectral sensor sensitivity vs. wavelength (nm), (b) channel 20, (c) channel 40, (d) channel 70, (e) channel 130, (f) channel 230.we sidelit our scene. Using a white reference, we estimate the uneven lighting, as shown inFig. 3-(a).Fig. 3(b)-(f) show the illumination-corrected version of theFig. 2(b)-(f), respectively.
Figure 3 :Figure 4 :
34Illumination-corrected and registered sample channels of the raw hyperspectral image to the ground truth. (a) Normalized uneven illumination field, (b) channel 20, (c) channel 40, (d) channel 70, (e) channel 130, (f) channel 230. H 1 which is focused on the blue spectrum vs. H 2 which is focused on the red spectrum. (a) H 1 channel 41 (457.82 nm), (b) H 2 channel 41 (457.82 nm), (c) H 1 channel 200 (854.97 nm), (d) H 2 channel 200 (854.97 nm).
Figure 5 :
5Sample layers from the data of the creation process as basis of evaluation: (a) Step 1: first graphite sketch. (b) Step 2: underdrawings with red chalk. (c) Steps 3 and 4: drawing with pen and iron gall ink plus final wash with two dilutions of ink in "two bowl technique", as described by Armenini [14][pp.54-55]. and Meder [15][pp.54-55]. Delineation after: Stefano della Bella, "Mother with two children", Florence, Galleria degli Uffici, Gabinetto Disegni e stampe, Inv.-Nr. 5937S.
Figure 6 :Figure 7 :
67Imaging Simulated RGB image from the hyperspectral image. (a) RGB image generated from the raw HSI in
Figure 8 :
8Label maps. (a) Ground truth, (b) SimRGB, (c) HSI-h, (d) HSI-EMAP, (e) HSIhSI-EMAP.
Table I :
ISpectral focus stacking results.Feature
AA% (±SD)
OA% (±SD)
Kappa (±SD)
Simulated RGB image from HSI
H 1
70.63 (±1.41) 60.82 (±2.51) 0.3515 (±0.0227)
H 2
72.32 (±1.15) 63.62 (±3.09) 0.3777 (±0.0313)
Focus Stacking 73.72 (±1.10)
64.96 (±2.40)
0.3980 (±0.0257)
Illumination-corrected HSI
H 1
74.76 (±0.94) 64.67 (±1.45) 0.3998 (±0.0169)
H 2
76.12 (±0.96) 66.34 (±1.42) 0.4186 (±0.0165)
Focus Stacking 76.57 (±0.94)
67.21 (±3.56)
0.4304 (±0.0366)
Table II :
IIPerformances of the features 1 to 12. IC: EMAP's parameters are chosen similar to SimRGB-IC-EMAP, 12) HSIhSI-EMAP: EMAP computed on dimensionality reduced HSIhSI. EMAP parameters are chosen similar to SimRGB-IC-EMAP. The results for the features are presented inFeature
AA% (±SD) OA% (±SD) Kappa (±SD)
SimRGB
71.83 (±0.79) 62.05 (±1.90) 0.3632 (±0.0178)
SimRGB-IC
73.72 (±1.10) 64.96 (±2.40) 0.3980 (±0.0257)
SimRGB-IC-SI
74.29 (±0.61) 66.08 (±2.57) 0.4119 (±0.0261)
SimRGB-IC-EMAP 74.63 (±0.77) 67.25 (±1.84) 0.4251 (±0.0170)
HSI
75.43 (±1.05) 66.94 (±2.11) 0.4196 (±0.0217)
HSI-IC
76.57 (±0.94) 67.21 (±3.56) 0.4304 (±0.0366)
HSI-DR
80.35 (±0.66) 72.58 (±1.53) 0.5019 (±0.0183)
HSI-h
83.00 (±0.47) 77.39 (±1.28) 0.5731 (±0.0161)
HSIhSI
82.86 (±0.52) 77.16 (±1.53) 0.5701 (±0.0213)
HSIhSI-DR
79.58 (±0.86) 71.00 (±2.41) 0.4817 (±0.0273)
HSI-EMAP
82.61 (±1.11) 77.35 (±2.53) 0.5719 (±0.0350)
HSIhSI-EMAP
83.08 (±0.89) 77.70 (±1.18) 0.5766 (±0.0191)
11) HSI-EMAP: EMAP computed on dimensionality
reduced HSI-
Table III :
IIIDown-sampled board scanner-acquired RGB image performance. ±0.66) 83.68 (±1.31) 0.6773 (±0.0206)AA% (±SD)
OA% (±SD)
Kappa (±SD)
87.71 ((a)
(b)
(c)
(d)
(e)
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|
[] |
[
"Execution of Partial State Machine Models",
"Execution of Partial State Machine Models"
] |
[
"Mojtaba Bagherzadeh ",
"Nafiseh Kahani ",
"Karim Jahed ",
"Juergen Dingel "
] |
[] |
[] |
The iterative and incremental nature of software development using models typically makes a model of a system incomplete (i.e., partial) until a more advanced and complete stage of development is reached. Existing model execution approaches (interpretation of models or code generation) do not support the execution of partial models. Supporting the execution of partial models at early stages of software development allows early detection of defects, which can be fixed more easily and at lower cost. This paper proposes a conceptual framework for the execution of partial models, which consists of three steps: static analysis, automatic refinement, and input-driven execution. First, a static analysis that respects the execution semantics of models is applied to detect problematic elements of models that cause problems for the execution. Second, using model transformation techniques, the models are refined automatically, mainly by adding decision points where missing information can be supplied. Third, refined models are executed, and when the execution reaches the decision points, it uses inputs obtained either interactively or by a script that captures how to deal with partial elements. We created an execution engine called PMExec for the execution of partial models of UML-RT (i.e., a modeling language for the development of soft real-time systems) that embodies our proposed framework. We evaluated PMExec based on several use-cases that show that the static analysis, refinement, and application of user input can be carried out with reasonable performance, and that the overhead of approach, which is mostly due to the refinement and the increase in model complexity it causes, is manageable. We also discuss the properties of the refinement formally, and show how the refinement preserves the original behaviors of the model. Index Terms-MDD, Model-level Debugging, Partial Models, Incomplete Models, Model Execution ! 1 INTRODUCTION M Odel Driven Engineering (MDE) is a software development methodology that advocates the use of model for the description and development of systems [1].These models can capture relevant concepts on a level of abstraction higher than source code, and thus can facilitate communication, automation, and reuse. In general, the level and form of the use of models varies greatly, from the use of models only for documentation and communication to the use of models for code generation and as the main software development artifacts rather than source code [2], [3], [4],[5]. While modeling is used in a range of industries such as telecom, automotive, aerospace, business, and military, its use for the development of Real-time Embedded (RTE) systems appears to be one of the most prevalent [6], [7],[8].Over the last two decades, an impressive number of MDE tools and techniques for RTE systems have been introduced such as IBM RSARTE [9], ANSYS SCADE Suite [10], YAKINDU Statecharts [11], and AUTOFocus[12]. These tools provide a range of capabilities to simplify the development of RTE systems using models. E.g., the models can be executed, debugged, analyzed, visualized, and transformed. Our work concerns the execution of models, which is typically supported either by interpreting the models or by translating them into an existing programming language, often by code generation (translational execution)[13],[14].The ability to execute models in some ways is an important capability of MDE tools because it enables many Mojtaba Bagherzadeh and Nafiseh Kahani are with the school of EECS, University of Ottawa.
|
10.1109/tse.2020.3008850
|
[
"https://arxiv.org/pdf/2103.17194v1.pdf"
] | 226,691,599 |
2103.17194
|
3b0166c6bdcbb84d27d0611a098fd574384c3e30
|
Execution of Partial State Machine Models
Mojtaba Bagherzadeh
Nafiseh Kahani
Karim Jahed
Juergen Dingel
Execution of Partial State Machine Models
1Index Terms-MDDModel-level DebuggingPartial ModelsIncomplete ModelsModel Execution !
The iterative and incremental nature of software development using models typically makes a model of a system incomplete (i.e., partial) until a more advanced and complete stage of development is reached. Existing model execution approaches (interpretation of models or code generation) do not support the execution of partial models. Supporting the execution of partial models at early stages of software development allows early detection of defects, which can be fixed more easily and at lower cost. This paper proposes a conceptual framework for the execution of partial models, which consists of three steps: static analysis, automatic refinement, and input-driven execution. First, a static analysis that respects the execution semantics of models is applied to detect problematic elements of models that cause problems for the execution. Second, using model transformation techniques, the models are refined automatically, mainly by adding decision points where missing information can be supplied. Third, refined models are executed, and when the execution reaches the decision points, it uses inputs obtained either interactively or by a script that captures how to deal with partial elements. We created an execution engine called PMExec for the execution of partial models of UML-RT (i.e., a modeling language for the development of soft real-time systems) that embodies our proposed framework. We evaluated PMExec based on several use-cases that show that the static analysis, refinement, and application of user input can be carried out with reasonable performance, and that the overhead of approach, which is mostly due to the refinement and the increase in model complexity it causes, is manageable. We also discuss the properties of the refinement formally, and show how the refinement preserves the original behaviors of the model. Index Terms-MDD, Model-level Debugging, Partial Models, Incomplete Models, Model Execution ! 1 INTRODUCTION M Odel Driven Engineering (MDE) is a software development methodology that advocates the use of model for the description and development of systems [1].These models can capture relevant concepts on a level of abstraction higher than source code, and thus can facilitate communication, automation, and reuse. In general, the level and form of the use of models varies greatly, from the use of models only for documentation and communication to the use of models for code generation and as the main software development artifacts rather than source code [2], [3], [4],[5]. While modeling is used in a range of industries such as telecom, automotive, aerospace, business, and military, its use for the development of Real-time Embedded (RTE) systems appears to be one of the most prevalent [6], [7],[8].Over the last two decades, an impressive number of MDE tools and techniques for RTE systems have been introduced such as IBM RSARTE [9], ANSYS SCADE Suite [10], YAKINDU Statecharts [11], and AUTOFocus[12]. These tools provide a range of capabilities to simplify the development of RTE systems using models. E.g., the models can be executed, debugged, analyzed, visualized, and transformed. Our work concerns the execution of models, which is typically supported either by interpreting the models or by translating them into an existing programming language, often by code generation (translational execution)[13],[14].The ability to execute models in some ways is an important capability of MDE tools because it enables many Mojtaba Bagherzadeh and Nafiseh Kahani are with the school of EECS, University of Ottawa.
important quality assurance activities such as testing and debugging. However, while existing MDE tools offer good support for the execution of complete models, none of them make much effort to extend that support to models that are incomplete. For instance, a state machine model may be incomplete because the behaviour of a component or a composite state has not yet been specified, or the specification of a transition is missing or incomplete (due to, e.g., missing triggers or guards, or incomplete action code). Although these models might contain many executable parts and there might be great value in the ability to execute them, existing tools typically do not allow this. One reason is that code generation or build operations might fail. But, even if these steps succeed, the tools do not allow a 'best effort' treatment of partial models in which execution proceeds as far as possible and when it cannot proceed any further due missing information, then this missing information can be supplied manually or automatically to allow the execution to continue. Our work aims to enable this kind of best effort treatment.
Generally, the benefits of partial model execution can be classified as follows:
1) Facilitate design decisions. In very early stages of development, the ability to execute a partial model may help perform design space exploration, evaluate design alternatives and explore tradeoffs in a more efficient, focused fashion and without having to flesh out details that are irrelevant to the design decision, but required to achieve executability. The goal is to allow the model to be useful as early as possible and with a minimum of procedural or notational accidental complexity [15]. 2) Facilitate validation and improvement of models. A basic tenet in software engineering is that development should facilitate the early detection of bugs, because the cost of fixing a bug tends to increase with the amount of time that it goes unnoticed [16], [17], [18]. Agile development activities such as continuous testing are motivated by this observation. In the context of MDE, this means that developers should be able to carry out validation and debugging activities as early as possible and not only after additional effort has been invested to make the model complete. The ability to execute partial models is necessary to achieve this vision and a key prerequisite for making MDE more agile [15]. But, partial model execution can also help with the validation of large, complete models. In large MDE projects, the system can have hundreds of components [19], [20], much reducing the feasibility and practicality of system-wide validation activities, especially when execution requires code generation (e.g., as reported in [21] the code generation of large systems can take hours to complete). In these settings, unit testing of individual components is required. Each such component c is a partial model that typically is not executable in isolation. To achieve executability, the current state-of-the-art demands that c be completed by a harness that mocks or stubs out the parts of the system that c relies on. The creation of a suitable harness and connecting it to c can involve a significant amount of accidental complexity. Our approach facilitates unit testing, because the developer can focus on supplying appropriate information to c if and when c needs it. 3) Facilitate collaborative, heterogeneous development. MDE is often collaborative and different model parts can be owned by different, possibly geographically distributed, teams [22]. As a result, components may be out-of-date, incorrect, or unavailable which may affect the executability of any other components using them. Partial model execution can help protect the developers of a part of the system from these issues by allowing them to, e.g., perform validation without having to wait for a new version or attempt to make their part work with an old, out-dated version. But, MDE can also be part of a larger, heterogeneous development process in which, e.g., the code generated from a model is integrated with other code that has been developed by a different team using a different process or tool, or been purchased from a vendor [22], [23]. Partial model execution can help here, too, because the model can be validated without having to obtain this additional code and integrate it with the model. According to surveys, the integration of MDE into industrial development processes can be challenging, especially when distributed development and interoperability with existing code or tools is required [3], [22], [23], [24]. Partial model execution can increase the flexibility of MDE and thus may help mitigate these problems and improve industrial adoption of MDE.
Despite the importance and benefits of the execution of partial models, no work has addressed the execution of partial models, to the best of our knowledge. Existing work in this context deals with partial models at design time, which allows for specification, analysis, verification, and transformation of partial models (e.g., [25], [26], [27], [28]). As mentioned, a possible approach to executing partial models is the simulation of missing components by techniques such as mocking or stubbing [29]. These solutions are mainly designed for unit testing, and have several deficiencies when used for the debugging of models: (1) They are not fully automated, and developers still need to do extra work to, e.g., create stubs for the missing components. (2) Often, they are applicable only at the componentlevel to simulate a component fully, while for debugging purposes, developers may need to simulate only parts of a component. (3) More importantly, while in the code-base development context, there are several mocking frameworks (e.g., Mockito, EasyMock, JMock, Opmock, etc.) that can be used to simulate components of a system [30], there is a lack of facilities, guidelines, and frameworks in the context of MDE to help to create mockers [31].
This work advances the state of the art in model-level execution and utilizing partial models by providing support for the execution of partial models. We propose a conceptual framework for the execution of partial models, which consists of three steps: static analysis, automatic refinement, and input-driven execution. First, a static analysis that respects the execution semantics of models is applied. It detects problematic elements that prevent the execution from progressing or reaching certain states. Second, to make the partial models executable, model-to-model (M2M) transformations [32] are used to refine the models automatically by adding decision points where the elements are missing. The refined models preserve the original behavior of the userdefined models and its execution does not not get stuck and can reach all defined states in a finite number of execution steps, assuming proper inputs are provided. Third, during the execution, these decision points allow users to interactively (1) inspect and modify the system using debugging services, and (2) select one of the possible options to continue the execution. The interactive execution requires manual intervention, which can be repetitive, tedious, and time-consuming. To mitigate this problem, our approach includes a scripting language that captures user input as execution rules that can be applied automatically during execution, without stopping the execution and interacting with users. Also, the approach allows user input to be saved to the script of the execution rules or the design model, to avoid any unnecessary duplication by minimizing the effort required for writing the script and completion of the design model.
We extended our previous work on model-level debugging [13], [33] in the context of UML-RT (i.e., a language for modeling of soft real-time systems) [34], and created an execution engine of partial UML-RT models (PMExec [35]) that embodies the proposed framework. To maximize the impact of our work, our implementation is publicly available, and only uses open source tools, including the Papyrus-RT MDE tool, for modelling and code generation, and the Epsilon [36] tools for model transformation. We evaluated PMExec based on several use-cases that show that the static analysis, refinement, and handling of user input were performed with reasonable quality, and the overhead of approach, which is caused by the increase of complexity of models by the refinement, is manageable. The rest of this paper is organized as follows. In Section 2, we describe our formalization and a running example. Section 3 presents a conceptual framework for execution of partial models, and Section 4 discusses the application of the framework into UML-RT. We present our evaluation approach and results in Section 6 and discuss the limitations and issues of our solution in Section 5. We review related work in Section 7, and then conclude the paper with a discussion, summary, and directions for future research.
PRELIMINARIES
In this section, we define, exemplify, and discuss the formalization that we will use to specify and justify our refinement approach. To be able to illustrate our approach, we use the UML profile for Real-time systems (UML-RT). UML-RT [34], [37] is a language specifically designed for Real-Time Embedded (RTE) systems, with soft real-time constraints. Over the past two decades, it has been used successfully in industry to develop several large-scale industrial projects (e.g., [20]), and has a long, successful track record of application and tool support, via, e.g., IBM RSA-RTE [9], RTist [38], Eclipse eTrice [39] , and Papyrus-RT [40]. Our formalization is simplified, and focused on aspects that matter most to the execution of partial models. Interested readers can refer to [34], [37] for more in-depth information regarding UML-RT.
A Running Example
We use the control system of a simple traffic light (Traf-ficLight) as a running example throughout the paper. The structure of the system is shown in Figure 1, which consists of three components: UserConsole (UC), Controller (CTR), and StopLightDriver (SLD). The UC component collects user input, which it passes on to the CTR component, the component controlling the light. Using the corresponding messages, the CTR component sends the control actions to the SLD component, which transfers the messages through a hardware port to the traffic light.
Let us assume that we have a partial model of TrafficLight in which the behaviour of the component UC is missing, the behaviour of capsule SLD is complete, and the behaviour of capsule CTR is incomplete as shown in Figure 2 (e.g., no outgoing transition is defined from yellow). Let us discuss two exemplary situations where the execution of partial models can be helpful for early evaluation of design decisions and unit testing.
Scenario-1: (Early evaluation of design decisions)
We want to execute the model as is and test the current design of the component CTR. Existing MDE tools do not support the execution of the example out-of-the-box, due to two issues: (1) An appropriate stub for capsule UC needs to be created, which is a time-consuming task, (2) even if the stub is provided, the execution of CTR stops at state yellow, due to the missing specification, i.e., no outgoing transition is defined from state yellow. Currently, the only way to resolve the second issue is to postpone the evaluation until the missing specifications are provided.
Scenario-2: (Unit testing) We want to perform unit testing of the component CTR. To do that, all relevant components except CTR should be replaced by appropriate stubs. Note that even though the design of SLD is complete, it is not reasonable to use it for unit testing since it requires interaction with hardware which can be time-consuming and complicates the testing process.
In this work, we will present a systematic solution for automatically creating an executable version of a partial model that allows testing and debugging. For example, using our solution both of the above scenarios can be addressed without a need for the creation of stubs and completion of models.
Modelling of a Real-time Embedded (RTE) System
Using UML-RT Definition 1. (Read function (Projection)) Let tp be a tuple of attributes a 1 , . . . , a n where a 1 . . . a n refer to attributes names. We use tp.a i to read the value of attribute a i . E.g., to read the value of attribute name of tuple person = name, f amily we can use person.name.
Definition 2. (Interface)
Let us define an interface as a set of pairs (m, d), where m ∈ M u (i.e., a universal set of messages) is a message, and d ∈ {input, output} specifies whether a message is consumed (i.e., it is an input message) or produced (i.e., it is an output message). A message can have a payload, which is a set of values conveyed by the message. ( s is a state, sm is an HSM, t is a transition, q is a queue, E is mapping from variables to their values, sc is a composite state, a1 . . . an is a sequence of actions. ) Definition 3. (Component) Let us define a component as a tuple P, V, β , where P ⊆ P u (i.e., a universal set of ports) is a set of ports, V is a set of variables, and β refers to the specification of the component's behavior. A port is defined as a pair (t, conjugated), where t denotes the type of the port where the type of a port is an interface, and conjugated ∈ {true, false} specifies whether or not the port is conjugated. The direction of messages of conjugated ports is reversed. We will use conjugation to ensure that connected ports are compatible by requiring that (1) they have the same type, and (2) one of them is conjugated while the other one is not. Non conjugated ports are also called base ports.
Definition 4.
(Structure of an RTE system) Let us define the structure of an RTE system as a tuple C, I, con, in , where C is a set of components, I is a set of interfaces, con is a connectivity relationship ⊆ P u × P u , and in is an acyclic containment relation ⊆ C × C. Whenever two ports p1, p2 are connected by con (i.e., (p1, p2) ∈ con) then both have the same type (i.e., p1.t = p2.t) and exactly one of them must be conjugated. This condition ensures that connected ports are 'compatible'.
Often, MDE tools provide timing services that can be used to define timed behaviors. To support time in our formalization, we assume that an RTE system contains a timing interface {(startTimer, input), (timeout, output)} and a component called RTS with a port of type timing. Any component using timing services requires a connection with the RTS component.
Let us exemplify the above definition in the context of the running example (see Fig. 1). The CTR is connected to UC and SLD using two ports UCPort (base port) and SLDPort (conjugate port), which are typed by interfaces ControlP and StopLightP, respectively. ControlP has two messages (on() and off()) and StopLightP has five messages (red(), green(), yellow(), on(), and off()).
Definition 5. (Action language)
Action languages support primitive operations such as accessing/updating variables, arithmetic/logical expressions, control flow constructs, and sending messages. MDE tools provide action languages either by adapting a subset of well-known programming languages or by offering a specific, dedicated action language. E.g., Papyrus-RT uses a subset of C++ as the action language, UML assumes the use of the UML Alf action language [41], and YAKINDU [11] provides its own action language. In this work, we assume the existence of an action language with the standard capabilities, but not define a particular syntax for it. Definition 6. (Hierarchical State Machine (HSM)) We specify the behavior of a component c using a hierarchical state machine (HSM) that is defined as a tuple S, T , in . S = S b ∪ S c ∪ S p is a set of states, T is a set of transitions, and in ⊆ S c × (S ∪ T ) denotes an acyclic containment relation. States can be basic (S b ), composite (S c ), or pseudostates (S p ). Basic states are primitive states that the execution stays in until an outgoing transition is triggered. Composite states encapsulate a sub-state machine. Pseudo-states are transient control-flow states. There are six kinds of pseudostates, called initial, choice-point, history, junction-point, entry-point, and exit-point, (i.e., S p = S in ∪ S ch ∪ S h ∪ S j ∪ S en ∪ S ex ). Composite and basic states can have entry and exit actions that are coded using an action language. Definition 7. (Transition) Let inp(c) refer to the messages that can be received by component c. A transition t is a 5-tuple (src, guard, trig, act, des), where src, des ∈ S refer to non-empty source and destination of the transition respectively, guard is a logical expression coded using the action language, trig ⊆ inp(c) is a set of messages that trigger the transition, and act is the transition's action coded using the action language. Figure 2 shows an example of HSM, and the corresponding graphical notations. Table 1 lists the helper functions (along with samples in the context of the running example, if possible) that will be used in the rest of the paper. Note that we treat the root of an HSM as a composite state, which can be accessed using the root(HSM) function.
Definition 8. (Helper functions)
Definition 9.
(Well-formedness constraints of HSMs) Following [9], [37], [40], we define the well-formedness constraints of HSMs as follows:
• Only transitions that start from a choice-point can have a guard, and no transition that starts from a pseudostate can have a trigger. This constraint is defined to simplify the formalization, and our implementation addresses this case. • There are no AND-states (orthogonal regions), and UML concepts fork, join, shallow history, and final states are also not used. • Transitions cannot cross state boundaries, i.e., ∀t ∈ T : parent(t.src)=parent(t.des). Entry-point and exit-point states can be used to create transitions with different parents. • States do not have idle (do) actions. • There is no notation for history. Instead, any transition to a composite state is assumed to end in an implicit history state inside the composite state. • Triggers of transitions starting from the same basic or composite state must be disjoint, i.e., ∀ t1, t2 ∈ T : t1.src = t2.src ∧ t1.src ∈ S p =⇒ t1.trig ∩ t2.trig = ∅. • None of the pseudo-states except choice-points can have more than one outgoing transition. • Composite states and the root of the HSM cannot have more than one initial state.
Note that except the first constraint, these constraints are also enforced by existing UML-RT tools and none of them has been defined specifically for this study. Still, our approach can be extended to support AND-states and other concepts not offered in UML-RT.
Definition 10. (Configuration)
A configuration γ of component c is defined as a tuple σ, E, H where σ ∈ S refers to the current state of the configuration, E refers to a mapping from the component variables to values, and H is a partial mapping from composite states to their last visited substates.
Definition 11. (Execution of HSMs) We use Labeled Transition Systems (LTS) to define the execution semantics of an HSM of a component c. An LTS is a tuple Γ, A, γ 0 , Q, → , where Γ is a set of configurations, A is the set of actions (i.e., entry, exit, and transition actions defined in HSM), Q is a first-in, first-out (FIFO) queue that stores received messages, → is a transition relation (to avoid confusion with the syntax of HSM, we use the term 'execution step' instead of 'transition' in the rest of the paper), and γ 0 ∈ Γ is the initial configuration.
Definition 12. (Execution
Step) An execution step is defined as a tuple γ, a 1 . . . a n , γ that moves the execution from configuration γ = σ, E, H (source configuration) to configuration γ = σ , E , H (target configuration), while executing a possibly empty sequence of actions a 1 . . . a n with a i ∈ A for all 1 ≤ i ≤ n that may result in updating H and E, and producing outputs. We use the following notation to show an execution step.
σ, E, H E ←exec(E,a1···an) −−−−−−−−−−−−→ σ , E , H
Definition 13. (Stuck Configuration)
A stuck configuration is a configuration that no execution step can start from, i.e., the execution cannot progress anymore when it reaches a stuck configuration. We use notation γ s to show that configuration γ s is a stuck configuration.
Definition 14. (Initial Configuration)
The initial configuration of an HSM is is defined as γ 0 = initial, E 0 , ∅ , where initial refers to the initial state inside the root of the HSM (i.e., initial = S in ∩ child(root(HSM ))) and E 0 refers to default values of the variables. The execution of the HSM starts from its initial configuration and if the initial state of the HSM is not defined, the execution cannot start (missing initial state).
Definition 15.
(Execution Rules of an HSM) Let us assume that γ = σ, E, H refers to the current configuration. The rules in Figure 3 define the operational semantics [43] of HSMs. The presentation of the rules makes use of definitions from Table 1. The rules are adapted from the execution semantics of UML-RT, presented in [37], [42]. Rule-1, 2: These rules are applicable to configurations whose current state is one of the pseudo-states, except for history and choice-point. According to Rule-1, an execution step is taken if there is an outgoing transition from the current state that executes the related actions and moves the execution to a new configuration. Conversely (Rule-2), if there is no outgoing transition, the execution stops there, and the current configuration is considered stuck (issue 'broken chain' in Sec. 4.1). Rule-3, 4, 5: These rules are applicable to configurations whose current state is a basic state. If the current state is a deadlock state (Rule-3), the execution stops there, and the current configuration is considered stuck (issue deadlock state). Otherwise, if a message exists in the queue, one of the following rules is applied based on the result of the function next t(σ, head(Q)) (Ref. Table 1), a transition can be triggered, which results in an execution step that executes the related actions and moves the execution to a new configuration as shown in the bottom of the rule (Rule-4). Conversely (Rule-5), if a transition cannot be triggered (i.e., the incoming message is an unexpected message), an
σ ∈ S p \ (S h ∪ S ch ), t = out t(σ) σ, E, H E ←exec(E,act(t), entry(t.des)) − −−−−−−−−−−−−−−−−−−− → t.des, E , u h(t.des, H) (1) σ ∈ S p \ (S h ∪ S ch ), out t(σ) = ∅ σ, E, H (2) σ ∈ S b , deadlock(σ) σ, E, H (3) Q = ∅, σ ∈ S b , ¬deadlock(σ) , t = next t(σ, head(Q))
σ, E, H E ←exec(E,exit(t.src),exit(up s(γ.σ,t)),act(t), entry(t.des))
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ t.des, E , u h(t.des, H) (4) Q = ∅, σ ∈ S b , ¬deadlock(σ), next t(σ, head(Q)) = ∅ σ, E, H(5)
σ ∈ S c , s = next s(σ, H)
σ, E, H E ←exec(E,entry(s)) −−−−−−−−−−−−−→ s, E , u h(s, H) (6) σ ∈ S c , next s(σ, H) = ∅ σ, E, H(7)
σ ∈ S ch , t = out t(σ) ∧ eval(E, t.guard) Figure 3: Execution rules of an HSM adapted from [37], [42] execution step cannot be taken. We consider the configuration to be stuck. However, it is also possible to configure the RTS to throw away the unexpected messages. As a result, the execution can recover and continue. We argue that in the domain of RTE systems, in which most of the applications are safety-critical, it is not safe to throw away any message. Rule-6, 7: These rules are applicable to configurations whose current state is a composite state (implicit history state). If function next s(σ, H) (Ref. Table 1) returns a state, then an execution step is taken that applies the entry code of the related composite state and moves the execution to a new configuration, as shown in the bottom of the rule (Rule-6). Conversely, if the selection is unsuccessful, the execution cannot move, and the configuration is a stuck configuration (Rule-7). This can happen due to two reasons: (1) the current state has no child (issue childless composite state), (2) the current state has no initial state (issue missing initial state). Rule-8, 9: These rules are applicable to configurations whose current state is a choice-point. Guards of the outgoing transitions from the current state are evaluated, and the first transition whose guard evaluates to true is selected. This results in an execution step that executes the related action code and moves the execution to a new configuration, as shown in the bottom of the rule (Rule-8). Conversely, if none of the outgoing transitions' guards holds (issue nonexhaustive guards), the execution cannot move, and the configuration is a stuck configuration. Note that Rules 2, 3, 5, 7, and 9 (all of which are related to stuck configurations) can be merged into one rule. However, we use the different rules for the sake of clarity.
σ, E, H E ←exec(E,act(t), entry(t.des)) − −−−−−−−−−−−−−−−−−−− → t.des, E, u h(t.des, H) (8) σ ∈ S ch , ∃ t ∈ out t(σ)| eval(E, t.guard) σ, E, H(9)
Definition 16.
(Execution of an RTE system) The execution of an RTE system can be defined as a collection of its components' HSM executions, which interact with each other by passing messages. We do not describe the details of the composition here, and we assume that the RTE system execution is managed by a controller. The controller is responsible for scheduling and message-passing between components, and guarantees that an incoming message will be fully processed before the processing of the next message starts (run-to-completion semantics). Figure 4 shows a conceptual framework for executing partial models, which consists of three parts (Static Analysis, Automatic Refinement and Input-driven Execution). In the following, we discuss the setting and the three parts.
A CONCEPTUAL FRAMEWORK
Setting
As discussed in Section 1, a partial model can be executed for different purposes. In all cases, the completeness level of a model is often decided by different stakeholders, based on different constraints and goals. We do not have an automatic check to determine when a model is complete. Instead, we allow users to specify the completeness level of each component based on their need. Supported levels are clevels ={complete, partial, absent/ignored}. Complete components are assumed to be complete, and are not required to be analyzed and refined. By default, each component is assumed to be complete, unless its completeness level is explicitly set to something else. Partial components are assumed to be incomplete. Thus, their current specification (structure and behavior) is analyzed and refined. Absent/ignored components are assumed to have no behavior specification. However, their existence may be necessary for the execution of other (partial and complete) components, due to the dependency between them. Thus, the absent/ignored components are analyzed based on their structure (inputs and outputs) and possible dependencies of other components on them. Then they are given behaviour sufficient for simulation.
The setting allows the execution of the models for different purposes. E.g., in the context of the running example which is discussed in Subsection 2.1, we can set the completeness level of CTR to partial, that of SLD to complete, and that of UC to absent/ignored to execute the system for Scenario-1 (evaluation of design decision). For scenario-2 (unit testing), the level of CTR should be set to partial and the levels of the other components to absent/ignored.
Static Analysis
Input-driven Execution
Batch
Execution Rules
Interactive Define
Static Analysis
Assuming that the execution semantics of the language is defined, we perform static analysis with respect to the execution semantics to detect the problematic elements that can cause a problem for the execution. Depending on the semantics of the language different types of problems may be detected. In the context of state machines, the problems associated with executing partial models fall into two groups: lack of progress and lack of reachability. The former is related to situations in which the execution cannot progress anymore from a certain point. The latter concerns the execution being unable to reach certain, specific states. The static analysis is performed on the user-defined model and identifies problematic elements including: (1) missing elements, (2) existing elements with problematic specifications, and (3) missing and unhandled inputs.
Automatic Refinement
During the refinement phase, depending on the results of the static analysis and the setting, the user-defined model is refined automatically using model-to-model transformation techniques. The goal of the refinement is to fix the problematic elements or modify them in such a way that users can provide more information about them during execution. Depending on the modelling language, certain language constructs can be used to enable models to interact with users during the execution. E.g., for HSMs, we use choicepoints with certain actions and guards.
The refinement should meet the following constraints: (1) Refined models must preserve the original behavior of the user-defined models. (2) The execution of the refined model must not get stuck, assuming proper inputs are provided. (3) The execution of the refined model must be able to reach all defined states in a finite number of execution steps, assuming proper inputs are provided. (4) The execution of the refined model must allow users to select one of the possible options to fix the problematic element. The options must be exhaustive and include all possible situations which can be applied at design time without limiting the users to only a subset of them.
Input-driven Execution
The refined model can be executed via interpretation or code generation. During execution, the executed model provides an interface for reading user input either in interactive or batch mode. It is also crucial to provide debugging facilities. Thus, users can investigate the execution before providing inputs.
APPLICATION OF THE FRAMEWORK TO PARTIAL UML-RT MODELS
Generally, the discussed framework to allow execution of partial models can be applied in the context of different modeling languages. However, since the execution of models is a language-dependent concept, there is no way to provide a generic implementation of the framework using existing techniques and tools. In the rest of this section, we discuss the application of the framework for executing partial UML-RT models and demonstrate a tool that embodies the framework.
Static Analysis
In the following, we discuss the details of the static analysis of UML-RT models, with respect to the execution semantics of UML-RT as discussed in Def. 11. We categorize the problems based their effect on lack of progress or reachability. Note that a lack of progress entails a lack of reachability.
Lack of progress
Based on the execution rules of HSMs (see Def. 15), the execution of an HSM can be stopped due to several issues, which can be divided into two groups, as follows.
Missing/problematic elements: The execution of an HSM cannot start, or moves to a stuck configuration, due to the following issues. Except for P6, the elements with these issues can be described by queries on the structure of the HSM, as follows.
P1 ← {s c ∈ (root(HSM ) ∪ S c ) : S in ∩ child(s c ) = ∅} P2 ← {s c ∈ (root(HSM ) ∪ S c ) : child(s c ) = ∅} P3 ← {s ∈ S p \ (S h ∪ S ch ) : out t(s) = ∅} P4 ← {s ∈ S b : handled(s) = ∅} P5 ← {s ∈ S b : inp(c) \ handled(s) = ∅}
As for P6, we assume that all choice-points have this problem, (i.e., P6 ← {s ∈ S ch }). This is an overestimation, and covers all possible situations of P6. Checking the exhaustiveness of guards of a choice-point during design time is a difficult problem, and requires expensive computation. Thus, fixing this problem at design-time can increase the refinement time significantly, and even make it unsolvable. Also, the applicability of the existing techniques on partial models is not supported by default, and requires extra work and research.
Missing inputs: A prerequisite for taking an execution step from basic states is the reception of a new message (see Rule-4 of Fig. 3) that can enable an outgoing transition from the current state. The execution can be stuck, if the required messages for triggering possible transitions are not produced by the connected components. This can happen for two reasons: (1) the connected component lacks a behavior specification, i.e., components are set as absent/ignored (P7), (2) the behavior of a connected component is partial (P8). Detecting P7 is trivial and can be determined from the interface specification of components. Let I be a set of possible input messages of all partial and complete compo-nents, and let O be a set of possible output messages of absent/ignored components, then P7 ← I ∩ O. Detecting P8 at design time suffers from a similar problem as P6. Thus, we overestimate again and assume that all partial components have this problem, (i.e., P8 ← all partial components). As we will discuss later, we provide debugging commands for sending messages to the other components from partial components, during the execution. That way, users can fix this problem by manually injecting the related messages during the execution.
Lack of Reachability
Anything causing the lack of progress problem also causes a lack of reachability. There is no way for the execution to reach any state after being stopped. In addition, the following missing or problematic elements can cause a lack of reachability.
• P9: isolated states are states that do not have incoming transitions (with the exception of initial and history states). There is no way for the execution to reach these states. • P10: not-takeable transitions originate from a basic or composite state and have no trigger. • P11: as discussed, one of the main benefits of the execution of partial models is enabling early evaluation of different design decisions, the support of which requires all states to be reachable during the execution in a finite number of steps. Otherwise, some design decisions cannot be evaluated due to the lack of reachability issue. For example, in the context of the running example (see Fig. 2), assume that transition t22 is missing, and a user needs to evaluate the effect of action of t23. The evaluation is not possible without steering the execution to state s22. Thus, we define another condition that concerns steering the execution from configurations whose current state is a basic state to any configuration whose current state is any basic state. Except for P11, the elements with these issues can be queried from the structure of a component c with an HSM as follows.
P9 ← {s ∈ S \ (S in ∪ S h ) : in trans(s) = ∅} P10 ← {t ∈ T : src(t) ∈ S b ∪ S c ∧ trig(t) = ∅}
As for P11, this capability needs to be addressed in all basic stats, i.e., P11 ← S b .
Note that sets P 1 − P 11 are not disjoint and an element may have several issues. For instance, in the context of the running example (see Fig. 2), s 23 is a deadlock state and has a reachability problem, i.e., s 23 ∈ P 4 ∩ P 11.
Refinement of Partial UML-RT Models
In the following, we discuss the details of the refinement, applied to fix the problematic elements (P1-P11) extracted during the analysis phases. Main Loop of the Refinement. Algorithm 1 shows the main loop of the refinement of a UML-RT model. It takes a UML-RT model and a setting as inputs. A setting is a set of tuples c ∈ C, clevel ∈ clevels , where c is a component and clevel specifies the level of completeness of
t 2 ← add trans(s b , dec p) 15 t 2 .trig ← inp(c) \ handled(s b ) 16 forall s ∈ (child(s c ) \ (S in ∪ dec p ) do // Fix isolated states (P9) and step 2 of fix for P11 17 t 3 ←add trans(dec P, s) 18 forall t ∈ P 10 do // Fix not-takeable transition (P10) 19 t.src = dec p 20 forall s cc ∈ child(s c ) ∩ S c do // Step 3 of fix for P11 21 ex p ← ex p∪ add state(s cc , S ex ) 22
en p ← en p∪ add state(s cc , S en ) 23 to child ← add trans(dec p, en p ∩ child(s cc )) 24 f rom child ← add trans(ex p ∩ child(s cc ), dec p) 25 if parent(s c ) = ∅ then // Last step of fix for P11 26 to parent ←add trans(dec p, ex p ∩ child(s cc )) 27 f rom parent ← add trans(en p ∩ child(s cc ), dec p) the component. The algorithm first adds a new component to the model, called dbg agent with an empty HSM, creates a debugging interface, and adds debugging and timing ports into dbg agent. dbg agent is responsible for receiving from external applications and transferring dbg messages to partial and absent/ignored components. After setting up dbg agent, the algorithm tries to apply certain refinements based on the setting of the components, as follows.
1) For partial components, it adds a debugging port into the components and creates a connection between them and dbg agent. This allows them to receive the dbg message during the execution, which is essential for fixing elements in P7-P8. Then it calls the refineHSM function, which applies the behavioral refinement to fix the issues (line# 4-7).
2) The behavior of absent/ignored components are removed, which results in an empty HSM. Then, their empty HSM is refined, which results in an HSM that can receive and send all possible input and output messages of the component (line# [8][9][10][11][12]. 3) Finally, the HSM of dbg agent is also refined as a absent/ignored component that results in an HSM capable of processing debugging commands (line# 14).
Behavioral Refinement. Algorithm 2 presents function re-fineHSM, which refines the HSM of a component with respect to elements in P1-P11 except for elements in P7-P8. Before discussing the details, let us define add state (s c ∈ S c , ty ∈ S) → S as a function that adds a state of type ty inside the s c and add trans (src, trg ∈ S) → T as a function that adds a transition from state src to state trg.
The algorithm iterates over all composite states, and the root of the HSM, and refines them in 9 steps, as follows.
1) It creates a choice-point state called dec p. dec p is used as a decision point during the execution (line# 2). When a specification is missing, we refine the HSM so that the execution is directed to dec p. 2) Fix elements in P2 which have no child, by adding a new basic state inside the related state (line# [3][4]. Note that the added basic state has issues P4, P5, and P9, and requires the corresponding fixes. 3) Fix elements in P1 which miss initial states, by adding a new initial state inside the related state (line# 5-6). Note that the added initial state has issue P3, and requires the corresponding fixes. 4) The elements in P3 (Broken chain) are fixed by adding a transition from the problematic states to dec p (line# 7-8). This ensures that the execution will move to dec p instead of stopping at the problematic states, and thus users can steer the execution to other states from dec p. 5) Fix elements in P6 (Non-exhaustive guards) by adding a transition from each choice-point to dec p so that its guard is set to the negation of the disjunction of the outgoing transitions' guards (line# [9][10][11]. This ensures that the execution moves to the dec p if none of the guards of the outgoing transition holds, instead of stopping there.
Refined CTRSM
1 1 1 2 off ( ) 1 1 on ( ) 1 1 P5-P11 2 2 red ( ) 2 1 P5-P11 2 3 green ( ) 2 2 P4-P5-P11 yellow ( ) 2 3 (P10) 1 3 1 1 []/ 0 1 dec_p1 from_child (P11) dec_p2 P11 P11 2 1 to_parent (P11) 1 from_parent to_child (P11) P11 on[]/ 1 / 2 timeout[]/ 2 2 []/ 2 1 timeout[]/
Arguably, this solution is much cheaper than the design time analysis to detect and fix this issue. 6) During this step, a new transition is added from each basic state to dec p and its trigger is set to all un-handled messages in the state (line# [13][14][15]. This not only fixes the elements in P4 (deadlock state) by adding a transition from them to dec p, but also (1) allows all un-handled messages to be handled as the new transition's trigger (P5), and (2) allows the steering of the execution from any basic state to dec p which is the first step of the fix for elements in P11. 7) A transition is added from dec p to all basic states and isolated states (line#16-17). This fixes issue P9 (isolated states), and also allows the steering of the execution to any basic state from dec p which is the second part of the fix for P11. 8) To fix not-takeable transitions (P9), their source is changed to dec p (line# [18][19]. This allows them to be taken whenever the execution reaches dec p. Since each state has a transition to dec p which is added in step 6 with a trigger set to all un-handled messages, the nottakable transitions can be activated by any of the unhandled messages. Note that the dbg message, which is added using Algorithm 1 is assumed to be an un-handled message. 9) At the end, an exit-point (line# 21) and an entry-point (line# 22) states are added to each composite state to allow the execution to be steered from their sub-states to states in their parent state and vice-versa, which is the last part of the fix for elements in P10. Two transitions to parent and from parent allow the execution to be steered from their sub-states to states in their parent state, and transitions to substates and from upper allow the execution to be steered from states in their parent state to their sub-states (line# [20][21][22][23][24][25][26][27]. Note that the refinement algorithms do not contain details for the actions which are added to HSMs. E.g., (1) added actions to the HSM of dbg agent for processing and injecting the message dbg, (2) guards of outgoing transitions from decision points which are set in a way that allows users to select one of them, (3) actions of incoming transitions to decision points that call a function to read user input. Interested readers can refer to the source code of the refinement [44]. Figure 5 shows the result of running Algorithm 2 on the partial CTRSM with partial completeness level in which transitions and states are annotated with corresponding issues P1-P11. Let us review some examples of how the execution can be performed despite missing specifications: (1) No transition from yellow to red was possible in the original model. This is fixed by adding transitions from yellow and red to dec p and vice-versa. Thus, any of the input messages or dbg messages can move the execution to state dec p from state yellow where users can select one of the outgoing transitions (e.g., the transition from dec p to state red). (2) The transition t 13 is not-takeable in the original model and its action cannot be executed. In the refined model an off or dbg message can move the execution to dec p in which the transition t 13 is one of the possible transitions and can be selected and its action be executed.
Refinement Result on the Running Example
Execution of Refined Partial UML-RT Models
In this section, we discuss our method for the execution of incomplete UML-RT models. The discussion will emphasize key concepts over low-level implementation detail. The definitions below identify two such concepts.
Execution Flow of a Refined Partial Models
As discussed, the partial models are refined by adding decision points where elements are missing or are partial that allow users to execute the partial models and provide information about the missing or partial element during the execution. This requires a mechanism that (1) enables executed models to obtain user input either in interactive or batch mode, (2) provides debugging features to investigate and modify the execution of the model. Figure 6 shows the execution flow of a refined HSM in which a debugging probe is hooked into the execution of an HSM by adding relevant actions in the initial transition of the HSM. When an HSM starts executing, two threads main and probe are started but only one of them is active at each time of the execution. Thread main executes the models as specified until it reaches a decision point where it sends the relevant execution context to the probe and waits for the user input. Thread probe starts a debugging session (batch or interactive) that allows users to investigate the execution and provide input. At the end of the session, the user input is returned to thread main to continue the execution. The interaction between the threads is simply a function call from thread main to the probe that is implemented using an action in the transitions ending at the decision points. In the following subsections, we review two execution modes of partial models.
Interactive execution
With the interactive execution, users are allowed to issue debugging commands listed in interactiveStatement of Listing 1, e.g., view and modify variables. Most of debugging services are ported from MDebugger [33]. The new debugging commands to facilitate the execution of incomplete models are as follows: 1) viewCmd lists the possible options to continue the execution. 2) selectCmd allows users to select one of the possible options. Note that selectCmd is the last statement that is applied when the execution has been stopped, and any command after that in the body of the rule is ignored (similar to the return statement in many programming languages). Also, selectCmd can accept more than one option during the batch execution, and in that case the execution switches to interactive mode to capture the user input. 3) simpleStatement allows users to define new variables which can be accessed during the debugging session, and to access all attributes (i.e., HSM's variables and newly defined variables during the debugging session) and modify them using arithmetic expressions. 4) saveCmd allows users to save their decision during the interactive session. We discuss this command in detail in Section 4.3.3. 5) umlrtCmd consists of send, reply, and receipt commands.
Commands send and reply allow users to send/reply (inject) messages to other components. Command receipt accepts a message as an input and returns true if the message is the most recently received message by the component, and false otherwise.
Batch execution
Interactive execution stops and delays the execution which is not suitable in some situations, especially for the debugging of time-sensitive systems, to repeat a debugging scenario, or to test and explore a design decision. For these situations, a batch execution mode is supported that allows users to provide inputs using a script that consists of execution rules (see Def. 18). An execution rule prescribes how the execution of a refined partial model is to be continued when the current execution context matches the where of the rule and the when of the rule evaluates to true. Depending on the current execution context and defined rules, multiple rules may be applicable at each time, but only one rule can be applied at each time. The rule selection for a decision point of an execution context with component c and the current execution state s is performed by following the steps below:
1) Guards of rules whose where (i.e., component and state name) is equal to c and s are evaluated based on the order of their appearance in the script file of the rule. The first rule whose guard holds is selected and applied. 2) If (1) is unsuccessful, guards of rules whose state name is exactly equal to s and component name is equal to * or empty are evaluated based on the order of their appearance in the rules' script file. The first rule whose guard holds is selected and applied. 3) If (2) is unsuccessful, guards of rules whose component name is exactly equal to c and state name is equal to * or empty are evaluated based on the order of their appearance in the rules' script file. The first rule whose guard holds is selected and applied. 4) If (3) is unsuccessful, guards of rules whose state and component name is * are evaluated based on the order of their appearance in the script file of the rule. The first rule whose guard holds is selected and applied. 5) If none of the above holds, the execution stops and the user is asked to provide input to continue the execution.
Automation
In general, the partial models that we consider exhibit one of two kinds of partialness: un-intentional and intentional. The former is related to situations (e.g., early debugging) in which the model is still under development, and not yet complete, due to the iterative and incremental nature of the software development process. The latter relates to situations (e.g., unit testing and partial analysis) in which users intentionally execute a complete model as a partial model often by writing scripts for execution rules to mock the ignored or unavailable part of the model. This can, e.g., increase the efficiency of testing or help deal with the unavailability of external components that the model relies on.
With un-intentional partialness, the user has to provide input (in the form of interactive decisions at runtime or execution rules) to steer the execution of the refined partial model. It is possible that this input resolves the partialness in a generally satisfactory way and that the user then also wants to use it for the next incremental development step and apply it to the design model. However, as we discuss later, our approach is careful to avoid duplicate effort from the user by allowing the input to be used directly to update the design model, instead of requiring the user to provide it again when completing the design model. While this double effort may incur negligible costs for the debugging of only a small part of a partial model, its cost can be significant for the debugging of a large part of a partial model, due to the large number of inputs that may need to be provided. Note that in the case of intentional partialness, the model is already complete, and writing the scripts for execution rules is only for simulation and mocking of the existing components.
To minimize the overhead of writing execution rules, and to reduce the need for double efforts, we provide three automation features, including (1) generation of default execution rules, (2) saving of users' interactive decisions as execution rules, and (3) application of execution rules into the design model. As shown in Figure 7, the features are complementary and allow users to automatically create execution scripts and update the design model by application of the execution rules. In the following, we discuss the details of each feature.
Generation of Default Execution Rules
As discussed, at each decision point that has been added during the refinement (see Section 4.4), users are given a set of all possible options, one of which needs to be selected to continue the execution of the model. Considering and selecting one of the options can be time-consuming, specifically for the execution of large partial models. To minimize the efforts for making decisions, we generate default execution rules for the decision points that filter out options that are less likely to be selected by users. When the filtering results in a single option, the generated rule executes the model without user intervention, otherwise one of the remaining options needs to be selected by the user either interactively or by editing the generated rule. Note that the default execution rules are generated as a script and can be viewed, modified, or even ignored by users depending on the execution scenarios.
Algorithm 3 presents the method for the generation of default execution rules. The algorithm accepts as input the refined model and the partial elements which have been detected by the static analysis (P1-P11 as discussed in Section 4.1) and then it generates a set of execution rules.
For each decision point dec p, the algorithm iterates over all transitions that end at dec p (line# 4-5) and creates at least one rule for each transition (line# 4-12). The where part Let us assume that when applied to a transition t that may have been refined by Algorithm 2, the function org(t ) returns the original transition t before the refinement. Function genRuleBody applies the following three heuristics to filter out options less likely to be of interest to the user and generates the body of the rule. fix it compared to states that already have incoming transitions. 3) If the rule handles a broken chain, the algorithm tries items c) and d), in the same way as the previous heuristic (line# [30][31].
We note that the generated rules may require further manual modification, due to the fact that a rule contains more than one option for selection, or the user finds one of the heuristics unsuitable for their needs. However, the generated rules should provide a useful initial version even in these cases.
For illustration, Listing 2 shows the default execution rules that are generated for the refined CTRSM (see Fig. 5). Rules r1-r2 are examples for the second heuristic, and rules r3-6 filter out not-takable transitions t 13 by explicitly selecting transition t r1 . Note that rules r1-r2 filter out all options except one, and therefore users may not need to change them or provide inputs at runtime interactively. However, rules r3-r6 still select more than one option, and therefore the user still needs to update them or provide interactive input at runtime.
Save user decisions (inputs) as execution rules
As discussed, via interactive execution, users need to provide input to steer the execution at decision points. Depending on the goals of the execution, users may need to repeat the execution, and therefore providing the same input can be time-consuming and tedious. To deal with this issue, a feature is provided that allows users to save interactive decisions in the form of execution rules. Thus, the execution can be repeated based on the saved rules without having to provide the input again. The implementation of the feature is heavily dependent on appropriate support for recording and viewing the execution of a partial model together with the user decision (input) provided during the execution. In the following, we discuss a high-level overview of how this feature is realized. A central notion is that of an execution record.
Definition 19. (Execution Records)
Let us assume that the execution of a partial model is saved as a set of records c, d, o , where c denotes the execution context at the time of the decision, d refers to a sequence of debugging commands that have been issued by the user while the execution was stopped (i.e., from when the decision point was first reached and until the execution is resumed with a select command), and o refers to the runtime decision (input) that is taken by the user (i.e., the argument of the select command).
Function saveDecesionsAsRules in Algorithm 4 presents how the interactive decisions are saved as execution rules. It accepts a set of execution records and returns a set of execution rules. The function first checks that the decisions are consistent (i.e., unique decisions are taken for the same execution contexts) and then resolves inconsistencies by consulting with users, as will be discussed below. Then, it finds the rule that matches the execution context, and if no rule is matched, it creates an execution rule based on the execution context. Finally, it generates the body for the execution rule by using the issued debugging commands that modify the execution state (e.g., changing a variable value). Note that the interactive debugging commands (i.e., 'dbgCommands' and 'umlrtCmd' in Listing 1) are a subset of the statements used for writing the body of an execution rule and there is no mismatch that complicates the use of sequences of debugging commands as rule bodies.
Resolving the inconsistency. Users are allowed to view their previous decisions and save one or more decisions as rules. When a user wants to save more than one decision, it is possible that some of the decisions are not consistent with each other, such that different decisions are taken at the same decision point for the same execution context. Thus, saving these decisions can cause non-determinism and should be avoided. To resolve inconsistencies between decisions, we ask users to select one of them.
Application of execution rules into design model
To mitigate the issue of double effort, we allow users to apply the execution rules into the design model automatically. To do that, we use the information in the execution rules about how to resolve partialness at runtime to fix the partialness in the design model. This feature is useful when users are satisfied that the way to deal with partialness at runtime expressed in the rules is correct and final, and they want to fix and remove partialness in the design model.
Application of execution rule into the design model is addressed by function saveRuleToModel in Algorithm 4. It accepts the original state machine (sm) and an execution rule (r) as input and fixes the relevant partialness in sm according to the definition of r, when there is only one possible solution to fix the partialness that is addressed by r and r.where explicitly refers to a state, i.e., r.where does not contain * or does not refer to a component. The function first extracts the selected states and transitions based on the select statements from the rule's body, e.g., the selected states of rule r1 and r3 in Listing 3 are {off} and Let selT rans and selStates be the selected states and transitions by the rule extracted from the r.body 16 if selStates has one member ∧ r.h.where refers to a state in sm then 17 Let des be a state equal to the only member of selStates in sm and src be a state equal to r.h.where in sm 18 Let selT ran be the first member of selT rans // selT rans has maximum one member since selStates has one member and for a state, more than one transition can not be selected 1) It checks if the rule selects only one state. This check is necessary, because the application of a rule that selects more than one state can make the resulting model nondeterministic. E.g., rule r6 in Listing 3 selects more than one state (red, green, yellow, off ), and therefore saving this rule into the design model would require adding four transitions from state green into the mentioned states with the same trigger (off ) and would cause nondeterminism. The function also checks if the where part of the rule explicitly refers to a state because otherwise, it is not clear which element in the design model should be fixed (line# 16). 2) It creates a transition (t) whose source is set to the where part of the rule and whose destination is set to the state that is selected by the rule if the rule does not address a not-takeable transition (P 10 partialness). There is no need to add a new transition to fix a not-takeable transition (line# [17][18][19][20][21][22]. Note that if the source and destination states of t are not contained in the same composite state, function add trans adds required transitions and related entry-point and exitpoint states to assure the transition does not cross the boundary of its parent state (see Def. 9). 3) It then sets the action of t based on the body of the rule.
Note that since a not-takeable transition may already have actions. Therefore the action is appended to the body of the rule. The trigger of t is set based on the receipt statements in the when part of rules, and its guard is set based on the when part of the rule, excluding the receipt statements (line# 23-25). 4) Finally, if the rule handles the non-exhaustive guard of a choice point ch 1 , the guard of t is conjoined with the negation of the disjunction of the guards of all transitions leaving ch 1 , as calculated during the refinement (line# [26][27]. Note that when a rule is applied, it fixes the relevant partialness and there is no need to manually modify the model after its application, because the model is updated in a way that follows the semantics of the rules precisely, and respects all HSM well-formedness constraints (see Def. 9). Also, the algorithm only presents the application of an execution rule into the design model. Saving an interactive decision into the design model can be performed by saving it as an execution rule, which is then can be applied to the design model as discussed.
Tool Support (PMExec)
We have developed PMExec 1 that embodies our approach and supports execution of partial UML-RT models. We used [36] to implement the transformation rules required for refining the models into executable models. EOL supports a set of instructions to create, query, and modify models. The part for the execution of the refined models (debugging probe) is implemented using C++, ANTLR [45], and the Boost C++ Library [46].
PMExec Features
In the following, we discuss the features of PMExec 2 from the user point of view. When it is possible, the use of features is explained using the running example. Setup and run The PMExec is integrated into Papyrus-RT as an Eclipse plug-in and can be downloaded and installed from the PMExec repository. After installation, it can be used to run partial UML-RT models simply by defining a run configuration (i.e., an Eclipse run configuration) inside Papyrus-RT. The static analysis, transformation, code generation and build run automatically in the background without distracting the user. Upon successful execution, PMExec loads a UI as shown in Figure 8 as soon as the execution requires user input to continue the execution. The UI is split in two parts, a HSM view ( 1 of Figure 8) and a DBG console ( 2 of Figure 8). In the HSM view the user can see the HSM of the capsule where the current execution state is highlighted. In the DBG console the user can interactively issue commands to investigate and fix the execution problems. Some of the most important commands are discussed in the following in the context of the running example.
View/select options List the possible options to fix/continue the execution. E.g., the output of view options for the CTR when its execution is stuck in state yellow is shown in part 2 of Figure 8. Using the console, the execution can now be steered to any of the defined states inside the HSM. The command select allows users to select one of the possible options, e.g., 'select state red' steers the execution to state red.
Simple expressions Similar to scripting languages (e.g., Python's interactive console), PMExec allows the user to issue simple expressions (e.g., arithmetic expressions) and statements (to, e.g., define a new variable, or change/view variable values). This allows the user to investigate and modify the execution before deciding how to advance the execution. E.g., 'x=5+1' creates a new variable x and sets its value to 6. Defined variables can help the user record certain properties of the execution and define complex debugging and testing scenarios. Once defined, they can be used till the end of the execution.
Communication commands To allow the user to fix The command inject sends a signal to a capsule to start a debugging session, the command send sends messages on behalf of the capsule being debugged to the connected capsules, and the command reply sends an incoming message back on the same channel it has been received. E.g., in the context of running example, no behavior is defined for the component UC. Thus, the execution of the CTR will get stuck in state off and the overall execution of the system will be deadlocked. The user can fix the problem by using the following communication command (1) 'inject UC' to start a debugging session with capsule UC. Note that the refinement fixes the behavior of capsules even with no defined behavior. (2) 'send message on' to send message on to the CTR where it will trigger a transition to turn on the red light. Batch execution PMExec supports a batch execution mode which allows users to provide inputs using a script of execution rules. The Listing 3 is an example of an execution script in the context of the running example.
1) The rule r1 steers the execution to state red when a message timeout is received while in state yellow.
2) The rule r2 replies to any received message using a random message and then moves the execution to a random state. The rules with header '*' are only selected when no other rule matches in the current execution context. Note that having only one rule similar to r2 is enough for the random execution of any partial model using PMExec. Save command To save users' decisions which are provided interactively as execution rules, users can view the history of the execution using 'view exec'. The output shows all previous decisions (inputs), each of which is given a unique id. Then, the user can use save command ('save input id ') to save the input with the related id as an execution rule. Also, to save an execution rule into the design model, users can use 'save rule id ' that saves the rule with the related id into the design model. Note that in both cases (save input/rule), more than one input/rule can be processed by providing more than one id.
VALIDATION
This section explains the validation of our approach which consists of three parts: online survey, formal validation and empirical evaluation. The goal of the online survey was to collect the opinions of MDE researchers and practitioners w.r.t. whether or not (1) the execution of partial models is a necessary and useful technology in the context of MDE, and (2) our approach is helpful to address the execution of 14 Participants' Occupation Figure 9: Survey participants' occupation and experience with MDE partial models. The formal validation is concerned with the properties of the refinement approach and shows formally how the applied refinement does not change the behaviour of the original specification of the models but fixes the problems of lack of reachability and progress. The empirical evaluation is concerned with the applicability of the approach in practice. It applies our approach to several partial UML-RT models in different scenarios and evaluates the performance and overhead. In the following, we discuss each part in detail.
Online survey
Survey Design
The survey 3 consists of two steps: First, we ask participants who are MDE researchers or practitioners to view a short demonstration video (5 minutes) of PMExec to familiarize them with the execution of partial models. Note that the video does not demonstrate the automation features as discussed in Sec. 4.4, since the features were inspired by the suggestions of the participants. Second, we ask them to answer 15 questions classified into three groups: demographic (4 questions), general questions regarding the execution of partial models (5 questions), and specific questions concerning our proposed approach (6 questions). However, participants are allowed to provide other answers rather than the choices or scales presented to them. Note that, providing an email address is optional to allow participants to be anonymous in case they want to provide negative feedback.
Participants
We approached MDE practitioners and researchers in person at the MODELS (2019) conference, as well as by email and social media and asked them to participate in the survey. Our efforts resulted in 39 participants whose demographic data is shown in Figure 9. More than 81% of The execution of incomplete (partial) models is a necessary feature that MDD tools should support
The fact that existing MDD tools do not allow execution of partial models has a negative impact on when and how MDD developers debug and test models The demonstrated solution is a promising first step toward supporting the execution of partial models In general, the manual completion of models will be easier than the execution of models using the demonstrated solution The proposed process for the execution of partial models is easy to understand and use, assuming that its implementation and tool support has a sufficient degree of maturity Figure 11: Participants' opinion about which activities the execution of partial models can facilitate participants have more than two years of experience with MDE, 46% of them work in the industry, 13% of them are university professors, 10% are postdoctoral research fellows, and 31% of them are graduate students. Notably, all of the participants except one are currently dealing with MDE tools in their work, and around 36% of participants have more than 10 years of experience and can arguably be said to be internationally known experts in the field. Finally, only 13 (33%) of participants provided their email addresses, and 26 (67%) of them participated anonymously.
Results
The relevance of the execution of partial models. We have asked four questions from the participant to understand whether they think the execution of partial models is essential and how it can help developers. As shown in Fig. 10, except for one of the participants, all of them either strongly agree (51.3%) or agree (46.2%) that the execution of partial models is a necessary feature that needs to be supported by MDE tools. Also, as shown in Fig. 11, participants think that the execution of partial models can be helpful for several software development activities, mainly early and interactive debugging, unit testing and agile software developments. Also, more than 51% of participants think that the execution of partial models can improve the user experience of MDE tools. Overall, since the vast majority of the participants perceive the execution of partial models as a relevant technology, we can safely conclude that addressing of the execution of the partial models is a relevant problem and worth addressing.
The usefulness of our approach As shown in Figure 10, all of the participants except two of them perceive our approach as a first promising step toward addressing the execution of partial models. Also, 84.4% of the participants either strongly agree (15.4%) or agree that the current approach can be useful for MDD users, assuming that the approach has good tool support. 86.4% of the participants prefer to use both interactive and scripting methods for providing input, depending on the execution scenarios. 69.2% of participants are of the opinion that the execution of models by providing input through scripts is easier than through a manual completion of models. However, 15.4% of participants have the opposite view.
In addition, we received valuable and constructive responses to the open-ended question as discussed in the following.
1) As quoted in the following, one of the participants pointed out correctly that our approach only is applicable to modeling languages with step-based execution semantics. "This can likely work well with behavioral models such as statecharts or business process models, but I wonder about whether the value proposition also covers non-behavioural models such as goal models, where "execution" is not a sequence of events but a set of values or initial decisions"
2) As discussed, the video does not demonstrate the automation features as mentioned in Sec. 4.4, since the following suggestion inspired the features.
"The script option sounds like it does not much improve on manual fixing/completion of models. I suspect that it may prove more useful if: (a) the script is automatically generated (as a user option) during an interactive session and then applied (as a user option) on subsequent runs. (b) used as a means of automatically modifying the incomplete model -again, as a user option."
3) As quoted in the following, one of the participants suggested an interesting extension to the work by augmenting the semantics of modeling language to support the execution in the presence of holes that are specified with certain notations. We agree with the participant. However, we left addressing this extension to future work.
"The partiality of the model in the demo seems to pertain only to violated statically checkable constraints such as whether a state is exitable/reachable. This would certainly help modellers. Another type of partiality is the presence of "holes", i.e. properties not being filled in (cf. Scala's "???"), not resolved, etc. I think it's a good idea in general to augment semantics of a modelling language that they stay defined (but possibly defined in terms of a fault mode) in the presence of missing model parts. This is akin to interpret every .-operator as ?.-operator (Kotlin, TS, etc.) and propagate nulls/undefined in a as meaningful as possible manner." 4) Not surprisingly, as quoted in the following, two of the participants have constructive feedback concerning the tooling issues. However, in this work, our main focus is the creation of a prototype as proof of concept, and improving of tooling is left to future work.
"I found your presentation of the context to be rather technical. For example, at 2 minutes in the video you showed the output for the empty UserConsole. There you presented one option how I could continue. I think you might have a technical reason to call the states Init State 3 and New State 2 but to me as a user it is unclear what these numbers mean (any why a state that is not even created yet(?) would have a lower number than the initial state you already generated). But I really liked, that you highlighted the current state in the visual representation of the diagram."
Overall, based on the participants' opinions, we can conclude that while our proposed approach is a step in the right direction, it is not the final solution (i.e., it has limitations). Still, further research and development are required in this context, some of which will be discussed in Section 6.
Formal Validation
We use HSM to refer to the result of applying Algorithm 2 on an HSM and call it the RefinedHSM of the HSM. In the following, first, we define the simulation relationship between LTSs, and then discuss the properties of HSM .
Behavioural Preservation
Definition 20. (Simulation Relation) Let L 1 = Γ 1 , A 1 , γ 10 , Q 1 , → 1 and L 2 = Γ 2 , A 2 , γ 20 , Q 2 , → 2 refer to LTSs of two HSM, HSM 1 and HSM 2 respectively (LTSs are discussed in detail in Def. 11). We write L 1 L 2 and say L 2 simulates L 1 if there is a binary relation R ∈ Γ 1 × Γ 2 with the following two properties. Start property: γ 10 = ∅ =⇒ (γ 10 , γ 20 ) ∈ R. Note that, in the execution of HSMs only one initial state is allowed.
Step property: Let (γ 1 , ∈ Γ 1 , γ 2 ∈ Γ 2 ) ∈ R.
For all γ 1 ∈ Γ 1 and a i , whenever γ 1
a1···an −−−−→ 1 γ 1 there exist γ 2 ∈ Γ 2 such that γ 2 a1···an −−−−→ 2 γ 2 ∧ (γ 1 , γ 2 ) ∈ R.
The step property implies that when (γ 1 , γ 2 ) ∈ R, any execution step started from γ 1 can be matched by an execution step started from γ 2 such that they both execute the same actions and reach configurations that again are in relation R.
Simulation implies trace containment, i.e., every sequence of actions that is possible by the simulated LT S, is also possible by the simulating LT S [48] (i.e., the simulating LT S preserves the specification of the simulated LT S).
Definition 21. Let LTSs
L o = Γ o , A o , γ o0 , Q o , → o
and L r = Γ r , A r , γ r 0 , Q r , → r represent the execution semantics of HSM and HSM respectively, E refers to a mapping from newly introduced variables during the refinement to their values, and R ∈ Γ o × Γ r is a binary relation defined as follows.
R = {(γ o ∈ Γ o , γ r ∈ Γ r ) | γ o .σ = γ r .σ ∧ γ o .E = γ r .E \ E ∧ γ o .H = γ r .
H} Proposition 1. Assuming that L o and L r receive the same sequence of messages (Q r = Q o ) and users do not issue any debugging commands during the execution of HSM , the relation R (as defined above) is a simulation relation, i.e., execution of an HSM simulates the execution of HSM (L o L r ). Lemma 5.1. For message µ and basic state s, if function next t(s, µ)) (see Table 1) returns transition t in the context of HSM, then it returns the same transition (t) in the context of HSM .
Proof. (Lemma 5.1) According to Algorithm 2, the refinement applies the following changes to the basic states: (I) Add a transition from a basic state to dec p. The trigger of this transition is set to in(c) \ handled(s b ) in order to not affect the existing transitions. This ensures that if a transition of HSM can be triggered by message µ, it still can be triggered by the same message in HSM and next t in both cases returns the same transition. (II) The source of not-takeable transitions is changed to dec p. next t never returns a not-takeable transition, thus this change does not affect function next t. (III) A transition is added from dec p to isolated states. Function next t never returns an incoming transition to a state as the result. Thus, this change does not affect next t either. Based on (I), (II), and (III) the proof of this lemma is complete. Proof. (Lemma 5.2) No state or takeable transition is removed by the refinement. Thus if state s or one of its ancestors (parents(s)) has a takeable transition (t) that prevents s from being dead, the same transition also exists in HSM . This completes the proof of this lemma.
Proof. (Proposition 1) To prove that R is a simulation relationship, first, we need to show that the start property holds which includes the following two cases.
• The initial state of HSM is missing. This case is trivial, since without initial state, the execution of HSM cannot start (see Def. 11) and γ o0 = ∅. Thus, the start property holds for this case. • HSM contains the initial state (i.e., γ o0 = ∅). In this case we need show that (γ o0 , γ r 0 ) ∈ R where γ r 0 is the initial configuration of HSM . (I) According to (lines # 5-6 of Algorithm 2), when the original HSM contains the initial state in 0 , the refinement keeps the same initial state in the refined HSM (i.e., the initial states of HSM and HSM are equal to in 0 ). According to execution semantics of HSM (see Def. 11), the execution of HSM starts from the initial configuration where its current state is set to the initial state of the HSM. Thus, there is an initial configuration of γ r 0 ∈ Γ r , in which the current state is equal to in 0 i.e., (γ o0 .σ = γ r 0 .σ).
(II) γ o0 .H = γ r 0 .H because the history is set to empty for the initial configuration (see Def. 11). (III) Similarly, γ o0 .E = γ r 0 .E \ E , because the initial values of variables are set to default values and the refinement does not remove any existing variable. Based on (I), (II), and (III) we can conclude that (γ o0 , γ r 0 ) ∈ R and conclude that the start property of simulation holds for this case as well. Second, we have to show that the step property holds for any (γ o ∈ Γ o , γ r ∈ Γ r ) ∈ R which includes two main cases according to the execution semantics of HSM (see Def. 11).
• γ o is a stuck configuration, i.e., no execution step can originate from it. Thus, the step property holds for this case. • γ o is a not a stuck configuration. Based on the execution rules (see Def. 11), this case includes 4 sub-cases: (1) the current state (γ 0 .σ) is a pseudo-state of kind initial, entry-point, exit-point, or junction-point (Rule-1), (2) the current state is a basic state (Rule-4), (3) the current state is a history state (Rule-6), and (4) the current state is a choice-point (Rule-8). Proof of all sub-cases is similar and here we only prove sub-case (2). Let us assume that γ o .σ ∈ S b , (γ o ∈ Γ o , γ r ∈ Γ r ) ∈ R and an execution step
st o = (γ o → o γ o )
is taken. First, we have to show that an execution step st r = (γ r → r γ r ) can be started from γ r that executes the same actions as st o . To prove the existence of st r , we have to show that the following condition holds (see Rule-4, Def. 11). Based on the statements above and Lemma 5.1, we can conclude that the last part of the above formula (∃ t ∈ T | t = next t(γ r .σ, head(Q r ))) holds. Based on I, II, III, and IV we conclude that execution step st r exists.
Q r = ∅ ∧ γ r .σ ∈ S b ∧ ¬dead(γ r .σ) ∧ ∃ t ∈ T | t = next t(γ r .σ, head(Q r )) (I) Definition of R and assumption γ o .σ ∈ S b imply that γ r .σ ∈ S b . (II) Q o = ∅
Next The other sub-cases can be proven similarly, and we conclude that the relation R ∈ Γ o × Γ r is a simulation relation and by that execution of HSM simulates the execution of
HSM.
While HSM preserves the behavior of the original HSM, it also never gets stuck and provides useful features to steer the execution to relevant states and debug the execution of the HSM, assuming the required inputs are provided. In the rest of this section, we discuss the important properties of HSM .
Reachability of the execution
Proposition 2. (Reachability of States) Assume L r = Γ r , A r , γ r 0 , Q r , → r is the execution semantics of HSM . Let γ be the current configuration of L r , where γ.σ ∈ S b . By injecting a dbg message, the execution can be steered by a finite number of execution steps to any configuration γ in which the current state is any state except initial, choicepoints, and composite (implicit history) states, assuming that proper inputs are provided.
Proof. (Reachability of States) Let σ be the current state of configuration γ, and σ be the current state of configuration γ to which we want to steer the execution. Proving that γ is reachable by taking a finite number of execution steps, includes three cases. (I) Both states σ and σ have the same parent, i.e., parent(σ) = parent(σ ). In this case, based on the execution semantics of HSM s, injecting a dbg message starts an execution step that moves the execution to dec p with the same parent (line# 12-14 of Algorithm 2). dec p has outgoing transitions to all states that have same parent as γ (added by line# 16 of Algorithm 2). Thus the execution can be steered to γ by providing the required input. (II) parent(σ) ∈ parents(σ ) ∧ parent(σ) = parent(σ ). In this case, after the execution reaches the first dec p, it then can be moved using a series of to child and f rom parent transitions (line# 20-26 of Algorithm 2) until reaching γ whose current state is σ . (III) parent(σ ) ∈ parents(σ) ∧ parent(σ) = parent(σ ). In this case, after the execution reaches the first dec p, it can then be moved using a series of to parent and f rom child transitions (lines# 20-26 of Algorithm 2) until reaching γ whose current state is σ . Based on (I), (II), and (III), the proof of Proposition 2 is complete.
Proposition 3. (Reachability of Transitions) Let γ be the current configuration of L r , where γ.σ ∈ S b . By injecting dbg and related messages, a sequence of execution steps can be taken to execute the action of any transition, except for initial transitions and transitions starting from choicepoints.
Proof. (Reachability of transitions) Based on the execution semantics of HSMs a prerequisite for the execution of the action of transition t is to move the execution to a configuration γ whose current state is (1) the source state of transition t, or (2) a basic state inside the composite state which is the source of transition t (when a transition t start from a composite state). According to Proposition 2, this can be done by injecting a dbg message and providing proper inputs. Thus, we have to show that after reaching the source state of the transition t according to (1) or (2), an execution step can be taken to execute the action of the transition which includes five cases based on the source state of transition t: (1) pseudo-state except for choice-points and initial states, (2) basic state, (3) composite state, (4) choice-points, (5) initial state. (I) The proof for Case-1 is trivial. According to the Rule-1 (see Def. 11), the execution step is taken from these states if there is an outgoing transition originating from them. Thus, an execution step can be taken that executes the action of transition t. (II) The poof of case-2 and case-3 is similar to Case-1 assuming proper input messages are provided (trigger of transition t). As we discussed in Sec. 4.3.1, we provide a message injection feature that simplifies this. (III) Case-4 is not part of the proposition, because it is not possible to ensure that the transition t is executed due to its guard expression. Any buggy guard statement can prevent the execution of the transitions originating from a choicepoint.
(IV) Case-5 is not part of the proposition. There is no way for the execution to re-visit the transition starting from the initial state, except by restarting the execution. Based on (I) and (II), proof of Lemma 3 is complete. Proof. (Sketch) As we discussed in Sec. 4.4, the execution gets stuck due to two groups of issues: (1) Missing/problematic specification. (2) Missing input messages. All of the elements in group (1) are fixed by the refinement. Also, dbg and other relevant messages can be injected by users which prevents a component from getting stuck because of missing inputs. Here, we do not present a detailed proof, but it can be performed similar to the previous proofs.
Progress of the execution
Empirical Evaluation
This section details experiments we conducted to assess the performance and overhead of our approach. In the following, we describe use-cases, evaluation metrics, experiments, and results.
Use-cases
To perform experiments, several use-cases are used. As shown in Table 2, models have different complexities that range from simple models containing 11 states to models with 350 states. Simple models include the Car Door Central Lock system and the Digital Watch. The Car Door Central Lock system is a control system for locking and unlocking car doors. The Digital Watch is an implementation of classical digital watch, which is described in [49].
The Parcel Router [50], [51] is an automatic system where tagged parcels are routed through successive chutes and switchers to a corresponding bin. The system is timesensitive and jams can appear due to variations in the time required by a parcel to transit through the different chutes. It checks for potential parcel jams, and prevents parcels from being transferred from one chute to another until the next chute is empty. The simplified version ignores jams.
The Rover system model [52], [53] allows an autonomous robot to move in different directions. It is equipped with three wheels, driven by two engines. It can move forward, move backward, and rotate. Additionally, it is equipped with several sensors, such as temperature and humidity sensors, to collect data from the environment, and an ultrasonic detection sensor, to detect and avoid obstacles.
The FailOver system [54], [55] is an implementation of the fail-over mechanism. It involves a set of servers processing client requests. To meet high availability, the system supports two replication modes, passive and active [56]. In passive replication, one server component works as the master, handling all the client requests while backup servers are mainly idle, except for handshake operations. Whenever a malfunction occurs, resulting in a failure of the master server, a backup server is ranked up as the new master. In active replication, client requests are load-balanced between several servers. The Debuggable FailOver system is a debuggable version of the FailOver system, which is generated using MDebugger [13]. The complexity of this model is high, and allows us to check that the refinement and analysis time do not skyrocket when the model size grows exponentially.
Evaluation Metrics
We formulated the following metrics to assess the practicality of our approach.
Metric 1 (Performance of Analysis and Refinement).
We use model analysis and transformation to fix partial models for the execution. The analysis and refinement are the core of our approach, and their performance is a crucial metric for the practicality of our approach. Thus, this metric measures the time required for the analysis and transformation of models.
Metric 2 (Overhead of refinement). As discussed, the refinement adds certain elements to fix the execution of partial models. These new elements increase the complexity of the models in terms of the number of components, states, and transitions. This metric first measures how the complexity of refined models changes in comparison with the original ones. The refined model is created temporarily before execution, and is only used for code generation. Thus, this metric also measures the code generation time for original and refined models, in order to determine the side effects of the increased model size. Metric 3 (Performance of the debugging probe). When executing the partial models, the execution of HSM is passed to the debugging probe, to read and apply user input. In the interactive model, there is always a delay imposed by users in the loop, and the performance is not an important factor. However, in the batch mode, it is essential that the debugging probe efficiently selects and applies the execution rules. This metric measures the time required to load, select, and parse rules. Lock 5 11 15 418 925 250 2024 3704 782 Digital Watch 9 47 57 717 1535 322 5219 11126 2225 Parcel Router 8 14 25 418 1674 220 2877 5305 1279 Rover 6 16 21 604 925 313 3062 5001 1254 FailOver 7 31 43 739 2247 257 4523 10416 1685 Debuggable FailOver 8 350 620 1694 8454 347 13376 35500 1887 C: Component, S: State, T: Transition, Orig.: Original
Experiments
In the following, we discuss the experiments used to calculate the metrics.
Measuring the performance of static analysis and model transformation (EXP-1).
To effectively measure the performance of analysis and transformation, first we used Epsilon [36] to create nine versions of each model (partial versions), listed in Table 2 by removing 10%-90% of their elements, randomly. This results in 60 models (including the original ones). In the rest of this section, we refer to these versions by merely mentioning the model name appended with the percentage of removed elements (e.g., Rover%10 refers to a version of the model of the Rover system that has 10% of its elements removed randomly). Also, we use the percentage without a model name to refer to all models with the same level of missing elements (e.g., 10% refers to model versions of all use-cases that have 10% of their elements missing). Second, we ran the model analysis and refinements 20 times against the original and their partial versions, with a configuration in which all components are assumed to be partial. The rationale for the configuration is to measure the performance in the worst-case scenario. As discussed, the refinement and analysis of a partial components is much more expensive than the complete and absent/ignored components. No refinement is applied on complete components, and the behaviour of an absent/ignored component is replaced with a simple generic state machine. We recorded the time required for analysis and refinement, which is a reflection of their performance in the worst-case scenario. We also saved the partial and refined versions of the model that are used in EXP-2. Measuring the overhead of the refinement (EXP-2) First, we measure the complexity of the models, and their refined version resulting from EXP-1 in terms of the number of components, states, and transitions. Second, we generated code from them 20 times, and recorded the execution time of the code generation. This experiment reveals how the model complexity is increased when applying refinement, and what the effects of this increase are on the code generation. Measuring the performance of execution rule selection and application (EXP-2) To measure the loading/selection time of the execution rules, we generated 10,000 rules with 100 Lines of Code (LOC) in the context of the ABM system which is a controller of an Automatic Banking Machine (ABM) designed using UML-RT. We performed a test that loads the rules in four scenarios, in which 10, 100, 1000, 10,000 rules are used accordingly. We recorded the loading time in each scenario. Then, using a test program, we called the rule selection method for the random context based on the ABM system 1000 times, and measured the rule selection times.
To measure the time required to apply execution rules, we randomly generated four execution rule bodies, containing 1, 10, 100, 1000 lines in the context of the ABM system. We ran a test to measure the time required to parse the rule bodies, 20 times. We did not measure the execution time of the rules' body, since their execution time is dependent on their content, which is controlled by users. The debugging probe executes the body of the rules as they are.
Setting and Reproducibility of Experiments
We used a computer equipped with a 2.7 GHz Intel Core i5 and 8GB of memory, for all experiments, which is typical development PC. The experiments are automated using bash scripts. The scripts and models are publicly available at [44] and can be used to repeat our experiments. Note that we intentionally used a standard computer comparable to those used by developers, rather than more powerful hardware, because the debugging of partial models typically needs to be carried out daily.
Results
Metric 1 (Performance of analysis and refinement).
Based on the result of EXP-1, the Analysis Time and Transformation Time columns of Table 2 show the median, maximum and minimum time required to analyze and transform the ten versions of each use-case. For the largest model (Debuggable failover), the medians of analysis and transformation are less than two and 14 seconds, respectively. It is therefore safe to conclude that the performance of analysis and refinement is reasonable even when the configuration of all components is set to partial which is the worst-case configuration. Typically, the execution of partial models is focused on executing specific components, and the rest of the components are assumed to be complete or absent/ignored which is less expensive to analyze and refine. Figure 12 shows number of elements in P1-P11 except P10 for the different versions (Orig. and 10%-90%) of the sets are highest for original models. This is because of the overestimation for extracting these sets which is based on the numbers of elements in the HSM. For example, P11 includes all basic states of the HSM. Thus removing more elements (states and transitions) in the HSM causes a decrease in the number of elements in P11. However, the number of elements for P1, P2, P3, P4, P9 reaches its maximum between versions 30%-60%, because in these cases the removed elements cause a maximal amount of issues for the remaining elements of the HSM. This number then decreases in the subsequent versions when more and more of these remaining elements are also removed. Note that the number of elements in P10 is not included in the figure, because we only remove states and transitions from the original model. Thus, the number of elements in P10 does not change between versions.
Metric 2 (Overhead of the refinement).
Based on the results of EXP-2, Fig. 13 shows the percentage of added elements (states and transitions) to the original models and their partial versions, during the refinement (i.e., the number of the added element divided by the number of elements before refinement multiplied by 100). Not surprisingly, the number of added elements increases as the number of removed elements from models increases. Removing more elements introduces more problems for the execution, which in turn requires more elements to be added to fix these problems.
The percentage of added states is between 20% (the median of the percentage of added states for original versions of models) and 216% (the median of the percentage of added states for model versions with 90% removed elements). The percentage of added transitions is between 67% (median of the percentage of added transitions for the original model versions) and 300% (median of the percentage of added transitions for model versions with 90% removed elements). Note that the percentage of added transitions for the versions with 90% removed elements is almost fixed (300%) because almost all are removed and the refinement always adds almost the exact same elements to refine them similar to absent/ignored components. The percentage of added transitions is higher than the percentage of added states, since many of the execution problems are fixed by adding transitions. Also, the number of components increases only by one (i.e., the dbg agent component). We argue that these overheads are reasonable compared to the capabilities provided by the refined models, for the following reasons:
• In most of the cases, the refinement adds elements when there is a missing/problematic element and there is no other way to fix them using existing tools and techniques. Our approach simply automates the fix for problematic elements. Otherwise, users have to fix them manually, which is time-consuming and tedious. • The refined models are temporary models, which are shows that the code generation of the refined model is only 8% slower than the code generation for original models, which is calculated based on the median of the time for code generation from refined models, divided by the time for code generation from the original models. • As discussed, the experiments are performed using the worst-case configuration, and their results reflect the maximum costs of our approach. Otherwise, using realistic configurations, which focus on the execution of certain components, can even decrease the complexity of the refined models with respect to the original models. E.g., the refinement of the Debuggable Failover system by setting the completion level of the Client component to partial and the level of the other components to absent/ignored results in a refined model with 138 states and 326 transitions, which is almost 50% smaller than the original model!
Metric 3 (Performance of the debugging probe).
Based on the result of EXP-3, Table 3 shows the time required for loading and selecting rules by the debugging probe. The selection time is the median time of rule selection for 1,000 times. The loading of rules occurs only once the execution of the system starts. During the loading, the script of execution rules is loaded and parsed. The parsing in this phase only parses the rules' header, and saves their body as text. As shown in Table 3, the debugging probe can load 10,000 rules in less than a second, which is acceptable, because it happens only once. As discussed in Sec. 4.3.1, in batch mode execution, the debugging probe must select an applicable rule from the defined rules whenever the execution reaches the decision points. As shown in Table 3, the rule selection time is negligible (less than a millisecond), and it is, therefore, safe to conclude that rule selection performance is acceptable.
When an execution rule is selected, the debugging probe parses the rule's body and executes it. Thanks to ANTLR [45], the parsing time of the rule's body has reasonable performance. Rule bodies with 1-1000 LOC can be executed in less than a second (the median execution time for a rule body with 1000 LOC is 550 milliseconds).
According to the results mentioned above (i.e., acceptable performance of analysis, refinement, and debugging probe and reasonable overhead of the refinement), we conclude that our approach is a practical approach for the execution and debugging of partial models.
DISCUSSION
In the following we discuss issues with the input-driven execution of partial models, alternative solutions for the refinements of partial models, and threats to the validity of this work.
Issues with the input-driven execution
Since the execution rules are defined using a scripting language, it may contain bugs similar to any other scripting language. Generally, there is no solution to this problem, and to mitigate this issue partially, our refinement method guarantees behavioral preservation (see Sec. 5.2). Therefore, it cannot introduce new bugs into the completed part of the models. Also, to help users when the script is buggy, the execution engine switches back to interactive mode and allows users to provide a correct input. Finally, providing proper tooling such as a high-quality editor for writing and validating scripts can mitigate the challenge of correct script authoring.
Alternative refinement solutions
Note that the proposed refinement (see Sec. 4.4) approach is devised experimentally by experimenting with and evaluating possibly many different solutions. In Section 5, we discussed the correctness of our refinement approach concerning the relevant constraints (see Sec. 3). We also showed that our approach has reasonable performance and overhead for the refinement of partial models. However, at this stage, we do not claim that our refinement approach is the optimal solution, and alternative and even better refinement approaches can be proposed, especially if certain trade-offs or assumptions are made. For example, the refinement can only focus on fixing specific problems rather than addressing all of them depending on the users' needs, e.g., refining only elements that participate in the lack of progress can be simpler and faster than our current approach. Nevertheless, the proposed refinement is comprehensive and can be used as a reference method to devise more specialized methods targeting more specific problems.
Threats to the Validity
Internal threats. (1) To evaluate this work, we use generated models rather than using real partial models. Thus, the results of our evaluation may not be generalized to real partial models. To mitigate this issue, we tried to generate models with a wide range of partialness over several case studies to make sure they are representative of typical partial models. Also, we used a worst-case configuration for the evaluation.
(2) The implementation of our approach is not trivial, and therefore our implementation may have bugs. To mitigate this problem, we rely on well-known tools and frameworks, such as EMF and Epsilon. We also have performed a thorough test and validation of our implementation. (3) The online survey participants viewed only a short demonstration video to familiarise themselves with the execution of partial models and our proposed approach. We partially mitigate this issue by carefully designing the video and targeting the MDE experts as participants, most of them already aware of the problems surrounding the execution of partial models.
(4) Our study may be designed in a way that, inadvertently, steers participants towards specific answers. To mitigate this issue, the participants are allowed to provide other answers rather than what is presented to them. Also, providing an email address is not mandatory to enable participants to be anonymous in case they want to provide negative feedback.
External threats. We targeted MDE experts as our survey participants to make sure they can provide us relevant and high-quality feedback. However, this may also be a threat to the survey's results since some of the participants may be biased about how, e.g., the problem of executing partial models should be dealt with.
RELATED WORK
A large amount of related work exists, and only the most relevant can be discussed here. Existing work can be divided into three categories: (1) work on model-level debugging and execution, (2) work on partial models, which tries to address specification, analysis, and transformation of partial models, and (3) work on partial programs that deals with the parsing, analysis, and completion of partial programs in the context of different programming languages. Model-level execution and debugging. Existing techniques of model execution are based on either interpretation or translation. Interested readers can refer to [14] which provides a comprehensive survey of existing work in the context of the model execution.
Model-level debugging techniques can be classified into interactive debugging and debugging by tracing. Interactive debugging allows users to directly investigate and modify the execution of models during the execution, by providing features such as setting breakpoints and stepping over the execution. Interactive model-level debugging is supported by several MDD tools, e.g., Matlab StateFlow [57], AF3 [12], xtUML [58] and YAKINDU [11]. We also presented a new approach for supporting interactive model-level debugging in [13], [33] by using model transformation techniques.
In debugging by tracing, the model or the generated code is instrumented to generate useful execution traces. Then, the traces are collected and used for offline analysis and debugging. Hojaji et al. [59] surveys the existing work in the context of model execution tracing. Examples of existing work and MDD tools supporting trace analyses via code instrumentation include [60], [61], [62], [63], [64], [65]. For instance, Iyengar et al. [63], [64], [66] propose an optimized model-based debugging technique for RTE systems with limited memory. They use a monitor on the target platform to collect the generated traces and a debugger (executed on a host with sufficient memory) to analyze the traces offline, and to display results on the model elements. Das et al. [67] propose a configurable tracing tool based on LTTng. They rely on code instrumentation to produce tracepoints useful for LTTng.
To the best of our knowledge, none of the existing work in the context of model execution and debugging supports the execution and debugging of partial models.
Partial models
In the context of MDD, the partial models are mainly used to deal with uncertainties of type 'known unknown'. Existing research proposes mechanisms to define partial models using relaxed meta-models [68], model annotation [69], UML profiles [70], and graphical notations [71]. They leverage the partial models for analysis [69], [72], requirement management and analysis [73], testing [68], [74], and bi-directional transformation [75]. Also, some research addresses the refinement [76], [77], transformation [78] and completion [79] of partial models. E.g., the wok [69], [80] presents a rich formalism for partial models, which marks model elements with four special annotations (may, set, variable, and open) with well defined semantics. They show how the partial models can be concretized into possible design candidates. Sen et al. [68] present a semi-automated tool that supports the specification and completion of partial models, which are used for the testing of model transformations. They show that the testing of model transformations using partial models is as effective as using human-made models.
To the best of our knowledge, no work in the context of partial models addresses the execution and debugging of partial models. Our work does not require specification of partial elements explicitly by users, since it detects all of them automatically by static analysis. Automatic detection of partial elements allows users to execute the models with minimum effort. Note that the partiality that our approach detects only concerns the execution, and may not be suitable for managing uncertainties in requirements or design models.
Partial programs
An extensive body of work exists for dealing with and leveraging partial programs. The most important of them can be classified as follows.
(1) Parsing of partial programs Typically existing compilers can handle only complete programs. As a partial program is a subset of a complete program, many of its variables' types and library calls are unknown. Thus, parsing partial programs requires extra effort, mainly for the inference of missing types, and resolving unknown function calls. E.g., Zhong et al. [81] propose an approach that resolves unknown types and function calls for partial Java programs by analyzing the existing complete program versions. Melo et al. [82] present a technique to support the compilation of incomplete C code.
Koppler [83] presents a systematic approach to implement fuzzy parsers, which extract high-level structures out of incomplete or syntactically incorrect programs. Moonen [84] proposes a solution in the form of island grammars that partitions code into islands (recognizable constructs of interest) and water (remaining parts). Dagenais et al. [85] propose a framework that uses heuristics to recover the declared type of expressions and resolve ambiguities in partial Java programs. Note that since the models are saved in the form of an abstract syntax tree (AST), the need for this type of research is unnecessary in the context of MDD.
(2) Partial program analysis/verification to deal with poor scalability and missing components E.g., modular model checking, introduced in [86], verifies properties of system modules, under some assumptions about the environment. Colby et al. [87] present an approach for automatically closing an open concurrent reactive system (i.e., a system with missing components) by generating an environment that can provide any input at any time to the system. This result is a self-executable system, which can exhibit all the possible reactive behaviors of the original system and therefore can be used for the state space exploration that is required for verification and analysis purposes. Our refinement of absent/ignored components is similar to this work.
(3) Program synthesis techniques based on partial programs (synthesis by sketching). Instead of synthesizing a program from scratch, work in this category uses a partial program (i.e., a program with holes) along with a specification, test harness, or reference implementation, and tries to fill the holes using synthesis techniques. E.g., Solar-Lezama et al. [88] introduce the concept of programming with sketches and presents Stream Bit as a new programming approach based on sketching. Existing sketching techniques (e.g., [89]) translate the partial program into a propositional satisfiability problem, and leverage counter-exampleguided inductive synthesis to generate a program using existing SAT solvers. Hua et al. [90] introduce EdSketch that performs execution-driven sketching for synthesizing Java programs using a backtracking depth-first search.
CONCLUSION AND FUTURE WORK
In this paper, we have proposed a conceptual framework for the execution and debugging of partial models, which consists of static analysis, automatic refinement, and input-driven execution. Using static analysis, we extract the problematic elements that prevent execution. The problematic elements are automatically fixed by adding decision points and related specifications into the partial models. Finally, the refined models are executed with the help of user input, either interactively or via a script. We have created a debugger for the debugging of partial UML-RT models (PMExec) based on the proposed framework. We have applied PMExec to the debugging of several use-cases, and have evaluated its performance for analysis, refinement, and handling of users input. Despite being a prototype, the performance of PMExec is acceptable, which shows that our approach is a viable approach for the debugging of partial models.
We have made the implementation of PMExec publicly available. The modeling community can extend it and use it for more research on, e.g., (1) the exhaustive execution of partial models for testing or run-time verification, (2) the synthesis of models by sketching, (3) using the proposed framework to support partial execution and debugging of partial models expressed in other modeling languages, and (4) automatic completion of missing specifications, rather than taking inputs from users.
Figure 1 :
1The structure of TrafficLight
Figure 2 :
2Partial Bahaviour of component CTR (CTRHSM)
Figure 4 :
4A Conceptual Framework for Executing Partial Models
•
P1: missing initial state (see Def. 14), • P2: childless composite states (see Rule-7 of Fig. 3), • P3: broken chain (see Rule-2 of Fig. 3), • P4: deadlock state (see Rule-3 of Fig. 3) , • P5: unexpected messages (see Rule-5 of Fig. 3), • P6 : non-exhaustive guards of choice-points (seeRule-9 of Fig. 3).Algorithm 1: Refinement of a Partial UML-RT Model Input : A UML-RT model sys and a setting conf Output: A refined model 1 Add a debugging interface dbg int and a debugging agent dbg agent into sys 2 Add ports types with timing and dbg int into dbg agent forall c ∈ sys.C do // sys.Add port p of type dbg int into component c 10 Add a connection using port p with dbg agent 11 Delete all elements from HSM of c (c.β) 12 c.β ← refineHSM(c.β, 14 dbg agent.β ← refineHSM(dbg agent.β, c)
Algorithm 2 :
2Refinement of HSM (refineHSM) Input : An HSM sm and a component c Output: A refined HSM // The following loop refines states in order of their nesting level, with the least deeply nested state (root(HSM)) refined first. 1 forall s c ∈ root(sm) ∪ (sm.S ∈ S c ) do 2 dec p ← add state(s c , S ch ) // Add decision point 3 if s c ∈ P 2 then // Fix childless composite state (P2) 4 state p h ← add state(s c , S b ) 5 if s c ∈ P 1 then // Fix missing initial state (P2) 6 add state(s c , S in ) 7 forall s p ∈ (child(s c ) ∩ P 3 \ dec p) do 8 add trans(s p , dec p) // Fix broken chain (P3) 9 forall s ch ∈ child(s c ) \ dec p do // Fix non-exhaustive guards for choice-points (P6) 10 t 1 ← add trans(s ch , dec p) 11 t 1 .guard ← ¬ guard(out trans(s ch ) 12 forall s b ∈ child(s c ) ∩ P 11 do // Fix unexpected messages (P5), deadlock states (P4), and step 1 of fix for P11 13 if s b ∈ (P 4 ∪ P 5) then 14
Figure 5 :
5Refined version of CTRSM in Fig. 2 (added elements are coloured blue, and modified elements are coloured red)
Definition 17 .
17(Execution Context) Intuitively, an execution context captures the most relevant runtime information of an execution stopped at some decision point. Formally, an execution context is a tuple γ, dec p, m, O , where dec p refers to the decision point at which the execution is stopped, γ denotes the configuration (see Def. 10) right before the execution reached dec p, m denotes the last processed message by the HSM (the trigger of the most recently taken transition starting from γ.σ), and O is a list of possible options available to continue the execution (i.e., the transitions originating from dec p ). Definition 18. (Execution Rule) We define an execution rule as a tuple h, b , where h refers to the header and b refers to the body of the rule. A header is a tuple name, where, when , where name refers to the name of the execution rule, where refers either to the qualified name of a state (component.state), name of a component, *, or *.state as shown in Listing 1 (Line #7), and when refers to a boolean condition. A body is a sequence of statements as defined in Listing 1. The semantics and use of execution rules is discussed in Section 4.3.3.
Figure 6 :
6Execution flow of a refined Partial HSM
Figure 7 :
7Automated generation of executions rules and completion of the design (user-defined) model of each rule is set to the state the transition originated from, and the when part is set to a guard indicating the arrival of a message m triggering the transition (if any). Note that no rule is created for message dbg (i.e., the debugging message). Thus, when the default rules are used, the execution can still be steered to any specific state by injecting the debugging message. Finally, the algorithm calls the function genRule-Body (line# 13), which generates a body for the rule.
1 )
1If the rule handles a choice-point (i.e., the where part of the rule is a choice-point such as ch 1) with nonexhaustive guards, then the algorithm omits the transitions into states that are reachable via the transitions leaving ch 1 (line# 20-23) in the original model. The rationale is that the user has already decided when those states are to be reached from ch 1 by specifying the guards of the outgoing transitions from ch 1 in the original model. 2) If the rule handles an unexpected message (i.e., the guard in the when part of the rule indicates the arrival of an unexpected message) (line# 24-29), the algorithm performs one of the following actions. (a) It selects not-takeable transitions that end at an isolated state, if any (line #25); (b) otherwise, it selects not-takeable transitions, if any (line #26-27); (c) otherwise, it selects transitions that end at an isolated state (line #28-29); and (d) otherwise, it selects all possible transitions (line #33-34
1
Let R be an empty set 2 Let D be a set that contains all of dec p of sm which have been added during refinement 3 Let P e be exit-point/entry-point states that are added during the refinement 4 forall dec p ∈ D do 5 forall t ∈ {t : t.des = dec p} do6 if t.src ∈ P R 18 Function genRuleBody (Rule r, Decision point dec p)19 O ← {t : t.src = dec p} 20 if r.h.where ∈ P 6 then 21 T tmp ← {t : org(t).src = r.where} 22 S tmp ← {s : s = t.des ∧ t ∈ T tmp } 23 O ← O \ {t : t.src = dec p ∧ t.des ∈ S tmp } 24 else if r.h.when ∈ P 5 then 25 O tmp ← O ∩ {t : org(t) ∈ P 10 ∧ org(t).des ∈ P 4} 26 if O tmp = ∅ then 27 O tmp ← O ∩ {t : org(t) ∈ P 10} 28 if T tmp = ∅ then 29 O tmp ← O ∩ {t : org(t).src ∈ P 4} 30 else if r.h.where ∈ P 3 then 31 O tmp ← O ∩ {t : org(t).src ∈ P 4} 32 O ← O tmp 33 if O = ∅ then 34 O ← {t : t.src = dec p} 35 Generate body of r according to O
{r ∈ R : r.h.where = l.c.γ.σ ∧ r.h.when = receipt(l.c.of r based on the recorded debugging commands (l.d)12 return R 13 Function saveRuleToModel (HSM sm, execution rule r)14 Let P 1 − P 11 refer to the problematic elements of sm15
act ← (r.body without select statement) + t.act 24 Set t.trig based on the receipt statements from r.h.when 25 Set t.guard based on the r.h.when by excluding the receipt statements 26 if r.h.where ∈ P 6 then 27 t.guard ← t.guard ∧ ¬ t ∈out trans(t.src) guard(t ) {red, green, yellow, off} and the selected transitions are {t13} and {} respectively. It then takes the following steps.
Figure 8 :
8User interface of PMExec the Epsilon Object Language (EOL)
7 questions are 5-level Likert scale questions [47] (Strongly disagree, Disagree, Neutral, Agree, Strongly agree), 6 questions are multiple choice questions, one question is asking for the email address of the participant (email question), and one question is an open-ended question. All of the questions except the open-ended and email question are mandatory.
Lemma 5 . 2 .
52For basic state s, if function dead(s)) (seeTable 1) returns false in the context of HSM, then it also returns false for state s in the context of HSM .
because the execution step st o is not possible with an empty queue (see Rule-4, Def. 11). Also, Q o = Q r based on the assumption of the proposition. Thus, Q r = ∅. (III) ¬dead(γ o .σ) holds, because without that execution step st o is not possible. Thus, based on Lemma 5.2 ¬dead(γ r .σ) holds. (IV) First, ∃ t ∈ T | t = next t(γ o .σ, head(Q o )) holds, otherwise the execution step st o is not possible. Second, Q r = Q o implies that head(Q r ) = head(Q o ). Third, γ o .σ = γ r .σ.
, we have to show that st r and st o execute the same sequence of actions. According to Rule-4 (see Def. 11), definition of relation R, and Lemma 5.1, both st r and st o execute the same actions, i.e, exit(t.src), exit(up s(s, t)), act(t), entry(t.des)),where s = γ o .σ = γ r .σ, µ = head(Q o ) = head(Q r ) and t = next t(s, µ).Finally, we have to show that (γ o , γ r ) ∈ R. (I) We have already shown that next t returns the same result for both current states γ o .σ and γ r .σ. Thus γ o .σ = γ r .σ with t.des as active state where t is the result of next t (Rule-4). (II) Similarly, γ o .H = γ r .H which are set to u h(t.des, H). H refers to history of γ o and γ r which are equal (Def. 21). (III) Variables are only changed by the execution of actions. Since the same sequence of actions is executed by st 0 and st r , and γ o .E = γ r .E \ E , we have γ o .E = γ r .E \ E . Based on (I), (II), and (III) we can conclude that (γ o ∈ Γ o , γ r ∈ Γ r ) ∈ R. Thus, the proof for sub-case (2) is complete.
Proposition 4. (Progress of the Execution) The execution ofHSM never reaches a stuck configuration assuming proper inputs are provided.
Figure 12 :
12Number of elements (y axis) in sets P1-P11 for versions (Orig. and 10%-90%)
Figure 13 :
13Increased complexity of refined models (original and partial versions)
Table 1 :
1Helper functions returns triggers of outgoing transitions of s and parents(s). E.g., handled(s11)= {on}.returns true if state s and its parents do not handle any message (i.e., handled(s)=∅). E.g., deadlock(s23)= true and deadlock(s11)= false. evaluates guard g based on the values in map E and returns the result. exec(E, a1 . . . an) executes a sequence of actions a1 . . . an based on the values in map E and returns the updated E.Function
Description
inp(c)
returns possible input messages of component c. E.g., inp(UC)= {on, off}.
in t(s)
returns incoming transitions to state s. E.g., in t(en1)= {t12}.
out t(s)
returns outgoing transitions from state s. E.g., out t(en1)= {t21}.
handled(s)
root(sm)
returns root of the HSM sm.
child(s)
returns states inside state s. E.g., child(root(CTRSM))= {s11, s21, s22, s23, en1, in11, c11}.
parent(s)
returns the first-level container state of state s. E.g., parent(s21)= {c11}.
parents(s)
returns all container states of state s. E.g., parents(s21)= {c11, root(CTRSM)}.
deadlock(s)
u h(s, h)
if parent(s) = ∅ and s ∈ S b , updates the last visited state of parent(s) to s (i.e., entry of parent(s) in h) and
returns the updated h.
head(q)
reads, removes, and returns the first element in queue q.
next s(sc, H)
(1) returns the last visited state inside state sc from history H, (2) if (1) is unsuccessful (i.e., the composite
state is active for the first time), returns the default state (initial state) inside sc, and (3) if (1) and (2) are
unsuccessful, returns ∅.
next t(s, µ)
checks state s and its ancestors in bottom-up order, and returns the first (i.e., most deeply nested) outgoing
transition, which can be triggered by message µ. It returns ∅ if no transition can be triggered. E.g., next t(s21,
on)= ∅, next t(s21, timeout)= t22.
up s(s, t)
returns s and a subset of its parents in bottom-up order from state s to the state that t originated from. E.g.,
up s(s21, t13)= {s21, c11}.
eval(E, g)
when: 'when' '('guard')'; 7 where: component.state | component| * | * .state; 8 statement: scriptStatement | interactiveStatement ; 9 scriptStatement: simpleStatement | complexStatement; 10 interactiveStatement: dbgCmd | umlrtCmd ; 11 umlrtCmd: sendMsgCmd | replyMsgCmd | receipt; 12 dbgCmd: viewCmd | selectCmd | simpleStatement | visited | controlCmd | saveCmd;1 grammar ExecRules;
2 script: execRule+;
3 execRule: 'rule' rID=ID 'where' '(' where ')'
4 when?
5
'{' body=scriptStatement * '}';
6 Listing 1: High-level Grammar of the Interactive/Batch
Execution Commands
Execution of a Refined HSM
Decision
Point
Debugging
Probe
Main
Execution
Execution Context
User Decision
Debugging
Session
Data
Flow
Inactive
Thread
Active
Thread
Table 2 :
2Model Complexity of Use-cases, Worst Case Transformation and Analysis TimeModel
Orig. Model Complexity
Analysis Time (ms)
Transformation Time (ms)
C
S
T
Median Max. Min. Median
Max
Min
Car Door Central
Debuggable FailOver system. The number of elements in P5, P6, P7, P8, P11 decreases as the number of removed elements increases, i.e., the number of elements in these0
50
100
150
200
Org.
10%
20%
30%
40%
50%
60%
70%
80%
90%
P7
P8
P5 & P11
P6
0
10
20
30
40
50
Org.
10%
20%
30%
40%
50%
60%
70%
80%
90%
P1
P2
P3
P4
P9
Table 3 :
3Required time for loading and selection of execution rules only used for code generation. Thus, the overhead of added elements has no side effect except for the code generation. The result of the second part of EXP-2#
of
Rules
LOC of
Rules
Loading time
(ms)
Selection
time (ms)
10
100
3
0.006
100
100
13
0.006
1000
100
96
0.006
10,000 100
950
0.010
https://[email protected]/moji1/partialmodels.git
A demonstration video can be found at https://youtu.be/ BRKsselcMnc
https://tinyurl.com/yxv8embf
ACKNOWLEDGMENTThis work has been supported by Zeligsoft Ltd, Malina Software Corp., Cmind Inc., and the Natural Sciences and Engineering Research Council of Canada (NSERC).
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|
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"Measuring distances in Hilbert space by many-particle interference",
"Measuring distances in Hilbert space by many-particle interference"
] |
[
"Karol Bartkiewicz \nInstitute of Physics of Academy of Sciences of the Czech Republic\nJoint Laboratory of Optics of Palacký University\nRCPTM\n17. listopadu 12772 07OlomoucCzech Republic\n\nFaculty of Physics\nAdam Mickiewicz University\nPL-61-614PoznańPoland\n",
"Vojtěch Trávníček \nInstitute of Physics of Academy of Sciences of the Czech Republic\nJoint Laboratory of Optics of Palacký University\nRCPTM\n17. listopadu 12772 07OlomoucCzech Republic\n",
"Karel Lemr \nInstitute of Physics of Academy of Sciences of the Czech Republic\nJoint Laboratory of Optics of Palacký University\nRCPTM\n17. listopadu 12772 07OlomoucCzech Republic\n"
] |
[
"Institute of Physics of Academy of Sciences of the Czech Republic\nJoint Laboratory of Optics of Palacký University\nRCPTM\n17. listopadu 12772 07OlomoucCzech Republic",
"Faculty of Physics\nAdam Mickiewicz University\nPL-61-614PoznańPoland",
"Institute of Physics of Academy of Sciences of the Czech Republic\nJoint Laboratory of Optics of Palacký University\nRCPTM\n17. listopadu 12772 07OlomoucCzech Republic",
"Institute of Physics of Academy of Sciences of the Czech Republic\nJoint Laboratory of Optics of Palacký University\nRCPTM\n17. listopadu 12772 07OlomoucCzech Republic"
] |
[] |
The measures of distances between points in a Hilbert space are one of the basic theoretical concepts used to characterize properties of a quantum system with respect to some etalon state. These are not only used in studying fidelity of signal transmission and basic quantum phenomena but also applied in measuring quantum correlations, and also in quantum machine learning. The values of quantum distance measures are very difficult to determine without completely reconstructing the state. Here we demonstrate an interferometric approach to measuring distances between quantum states that in some cases can outperform quantum state tomography. We propose a direct experimental method to estimate such distance measures between two unknown two-qubit mixed states as Uhlmann-Jozsa fidelity (or the Bures distance), the Hilbert-Schmidt distance, and the trace distance. The fidelity is estimated via the measurement of the upper and lower bounds of the fidelity, which are referred to as the superfidelity and subfidelity, respectively. Our method is based on the multiparticle interactions (i.e., interference) between copies of the unknown pairs of qubits.
|
10.1103/physreva.99.032336
|
[
"https://arxiv.org/pdf/1812.07406v2.pdf"
] | 102,483,611 |
1812.07406
|
ec00179163e6cf8630e6a0cfe4c5d45b6904c24d
|
Measuring distances in Hilbert space by many-particle interference
Karol Bartkiewicz
Institute of Physics of Academy of Sciences of the Czech Republic
Joint Laboratory of Optics of Palacký University
RCPTM
17. listopadu 12772 07OlomoucCzech Republic
Faculty of Physics
Adam Mickiewicz University
PL-61-614PoznańPoland
Vojtěch Trávníček
Institute of Physics of Academy of Sciences of the Czech Republic
Joint Laboratory of Optics of Palacký University
RCPTM
17. listopadu 12772 07OlomoucCzech Republic
Karel Lemr
Institute of Physics of Academy of Sciences of the Czech Republic
Joint Laboratory of Optics of Palacký University
RCPTM
17. listopadu 12772 07OlomoucCzech Republic
Measuring distances in Hilbert space by many-particle interference
numbers: 0367Mn0365Ud4250Dv
The measures of distances between points in a Hilbert space are one of the basic theoretical concepts used to characterize properties of a quantum system with respect to some etalon state. These are not only used in studying fidelity of signal transmission and basic quantum phenomena but also applied in measuring quantum correlations, and also in quantum machine learning. The values of quantum distance measures are very difficult to determine without completely reconstructing the state. Here we demonstrate an interferometric approach to measuring distances between quantum states that in some cases can outperform quantum state tomography. We propose a direct experimental method to estimate such distance measures between two unknown two-qubit mixed states as Uhlmann-Jozsa fidelity (or the Bures distance), the Hilbert-Schmidt distance, and the trace distance. The fidelity is estimated via the measurement of the upper and lower bounds of the fidelity, which are referred to as the superfidelity and subfidelity, respectively. Our method is based on the multiparticle interactions (i.e., interference) between copies of the unknown pairs of qubits.
I. INTRODUCTION
In classical [1] as well as in quantum [2,3] communication theories the measures of distance between states quantify the accuracy of signal transmission through an imperfect communication channel. Here we focus on the problem of measuring or estimating three of the most popular distances by performing less measurements than required when applying full quantum state tomography. We discuss the problem on the example of a two-qubit states, which is of great significance to modern-day applications of quantum information processing. However, our method can be directly extended to be applicable to more complex systems or possibly other distance measures. We focus on the analysis of nonlinear properties of two-qubit states because they play an important role in quantum protocols exploiting quantum correlations. Thus, establishing methods of testing various properties of these states is well motivated. This is especially important for photonic qubits since photons are typical carriers of quantum information used in quantum communication protocols.
The most popular signal quality quantifier is Uhlmann-Jozsa fidelity, which is also referred to as the Uhlmann transition probability. It is commonly applied in quantum optics, quantum information, and condensed-matter physics. The fidelity of two mixed quantum states represented by density matrices ρ 1 and ρ 2 , which can represent input and output states of a transmission line, was de- * [email protected] † [email protected] ‡ [email protected] fined by Uhlmann [4] and Jozsa [5] as:
F (ρ 1 , ρ 2 ) ≡ Tr √ ρ 1 ρ 2 √ ρ 1 2 .(1)
This quantity is also referred to as the Uhlmann transition probability [4]. Note that the alternative definition given by Nielsen and Chuang [2] is denoted as √ F and is sometimes also called fidelity. Some of its important properties were studied, e.g., in Refs. [4][5][6][7]. Fidelity can be used to construct Bures metric [8], which defines distance between density matrices of quantum states. The Bures metric is equal to Fubini-Study metric [16] when considering only pure states. It is being used to quantify, e.g., a degree of quantum entanglement [9,10] (e.g., in quantum phase transitions), a degree of polarization [11][12][13][14], and nonclassicality [15]. The fidelity is related to square of Bures metric by
D 2 B (ρ 1 , ρ 2 ) = 2[1 − F (ρ 1 , ρ 2 )].
Another popular metric is the trace distance. It provides information about statistical distinguishability between two states. The trace distance is defined for Hermitian density matrices as
T (ρ 1 , ρ 2 ) = 1 2 Tr (ρ 1 − ρ 2 ) 2 = 1 2 i=1 |λ i |,(2)
where λ i are the eigenvalues of the Hermitian matrix (ρ 1 − ρ 2 ). This measure due to being Euclidean has intuitive geometric properties that can be utilized in depicting relations between quantum states (see, e.g., Ref. [17]) Trace distance of two mixed states is related to the fidelity by the following inequalities
1 − F (ρ 1 , ρ 2 ) ≤ T (ρ 1 , ρ 2 ) ≤ 1 − F (ρ 1 , ρ 2 ) 2 .(3)
When ρ 1 and ρ 2 are pure states the upper bound on T is saturated.
arXiv:1812.07406v2 [quant-ph] 20 Dec 2018
The last measure that we study in this paper is the Hilbert-Schmidt distance defined as
H(ρ 1 , ρ 2 ) = Tr ρ 1 − ρ 2 2 ,(4)
which is related to trace distance via Cauchy-Schwartz inequality, i.e, 0 ≤ H(ρ 1 , ρ 2 ) ≤ 2T (ρ 1 , ρ 2 ). The problem that we tackle in this paper is how to efficiently measure distances in Hilbert space. A natural solution is to perform complete quantum state tomography of ρ 1 and ρ 2 , then to calculate a given distance measure, including the ones considered above. As we demonstrate here, this solution cab be inefficient because in some cases it requires measuring redundant information and the number of measurements grows exponentially with the dimension of the Hilbert space. Full quantum tomography in some cases can also lead to negative density matrices which require further postprocessing in order to represent physical systems. This problem depends on the uncertainty of the collected data and on the errorrobustness of a specific tomographic protocol [17,18]. The direct calculation of fidelity via Eq. (1) for mixed states can be a challenging task. Examples of analytic formulas for calculating fidelity were be found, e.g., for single-qubit states [3] and multimode Gaussian fields [19]. Here, we propose an interferometric approach for direct and efficient measurement of the overlaps defined as
O n (ρ 1 , ρ 2 ) = Tr[(ρ 1 ρ 2 ) n ](5)
or O(ρ 1 , ρ 2 , ) = Tr(ρ 1 ρ 2 ) for n = 1, that can be used directly to express trace distance, Hilbert-Schmidt distance, and sub-and superfidelities. Which are, respectively, the lower and upper bounds on the fidelity F (ρ 1 , ρ 2 ) [7,20]. Note that the first-order overlap O(ρ 1 , ρ 2 ) becomes purity χ for ρ 1 = ρ 2 . These overlaps can be interpreted as interaction of two or four particles if the states ρ n (n = 1, 2) represent pairs of qubits. Quantum circuts for measuring overlaps can be also designed using the method of Ekert et al. [21] based on programmable quantum networks with the Fredkin (controlled-SWAP) gates [22][23][24][25][26][27]. Both our and the Ekert et al. methods can be applied to design setups for measuring linear and nonlinear functionals of arbitrary states. It was shown tehoretically by Miszczak et al. [7] that the network method enables measuring the first-and second-order overlaps between a pair of two-qubit states for the estimation of their fidelity bounds. In this work, we apply a purely algebraic method for estimating some second order overlaps of arbitraty two-qubit states and discuss an experimentally-friendly linear-optical implementation. As in Refs. [7,21] we assumed that we have access to copies of a given quantum state, which can be implemented either by producing two identical states simultaneously, or by storing the state produced earlier in order to measure it together with the second copy of the state available later. We note that some experimental demonstrations of direct measurements of fidelity of single qubits was already reported by Du et al. [28], Bovino et al. [29] and Adamson [30]. Moreover, fully-entangled fraction, which is directly related to maximum fidelity of two-qubit state with respect to a maximally entangled state, was measured in Ref. [31]. Theoretical works relevant to measuring purity and overlaps of two quantum states include,e.g., Refs. [32,33]. These results can be applied to measuring sub-and superfidelities, which can be also measured and as described by [7] by following the approaches of Ekert et al. [21] and Bovino et al. [29]. The method presened here for the experimental measurements of the first-and second-order overlaps are inspired by the method for the measurement of nonclassical correlations described in detail in Ref. [34], which can be also used for measuring, e.g., a degree of the CHSH inequality violation [31,35]. In contrast to previous proposals, our method is devised for easy experimental implementation on the platform of linear optics as it requires only trivial two-qubit manipulations implemented for instance by a simple beam splitter. It can also be quickly tested with hyper-entangled photons (see, e.g., Ref. [36]) because the required operations are then a mere deterministic single-photon projections.
This article is organized as follows: In Sec. II, we express the analyzed distance measures in terms of manyparticle overlaps. In Sec. III, we describe efficient methods for measuring the first-and second-order overlaps. We conclude in Sec. IV.
II. DISTANCE MEASURES IN TERMS OF MANY-PARTICLE INTERFERENCE
The density matrix of a two-qubit (quartit) system can be expressed in the Bloch representation using Einstein summation convention as
ρ = 1 4 R mn σ m ⊗ σ n .(6)
Here, R mn = Tr(ρσ m ⊗ σ n ) are the elements of correlation matrix and σ m , σ n the Pauli matrices with m, n = 0, ..., 3,, and σ 0 = I denotes the identity operator.
Note that a single-qubit density matrix can be obtained after tracing out the other qubit from the two-qubit density matrix, which results in
ρ a = 1 2 R m0 σ m , and ρ b = 1 2 R 0m σ m .(7)
To make our considerations less abstract let us assume that a qubit is encoded as polarization degree of freedom of a single, i.e., σ 3 = |H H| − |V V |, where H and V correspond to horizontal and vertical polarizations, respectively.
A. Fidelity, superfidelity, and subfidelity
One can express the fidelity of single-qubit density matrices ρ a and ρ b as
F (ρ a , ρ b ) = O(ρ a , ρ b ) + S L (ρ a )S L (ρ b ),(8)
where the S L (ρ) = 1 − χ(ρ) = 1 − O(ρ, ρ) is linear entropy (linear approximation to the von Neumann entropy), which can be directly measured by the method proposed in this article. For two-qubit and higher-dimensional density matrices the situation becomes quite complicated. However, to estimate F using a finite number of overlaps we can use its upper and lower bounds given by Miszczak et al.in Ref. [7]:
E(ρ 1 , ρ 2 ) ≤ F (ρ 1 , ρ 2 ) ≤ G(ρ 1 , ρ 2 ).(9)
The lower and upper bounds are referred to as the subfidelity and superfidelity and are defined as
E(ρ 1 , ρ 2 ) = O(ρ 1 , ρ 2 ) + 2[O 2 (ρ 1 , ρ 2 ) − O 2 (ρ 1 , ρ 2 )],(10a)G(ρ 1 , ρ 2 ) = O(ρ 1 , ρ 2 ) + S L (ρ 1 )S L (ρ 2 ). (10b)
To measure these bounds, the first-order overlap, the second-order overlap and the linear entropies (purities) have to be measured. If one or both states ρ 1 , ρ 2 are pure, the fidelity is equal to first-order overlap O(ρ 1 , ρ 2 ).
Measuring the first-order overlaps O(ρ 1 , ρ 2 ), χ(ρ 1 ) = O(ρ 1 , ρ 1 ), and χ(ρ 2 ) = O(ρ 2 , ρ 2 ) is enough for the determination of the superfidelity G(ρ 1 , ρ 2 ). If it is known than one of the states is pure, then we do not need to proceed with estimating the subfidelity because in this case we already have all the data needed for calculating the fidelity F (ρ 1 , ρ 2 ) = O(ρ 1 , ρ 2 ). In the simplest qubit case, the superfidelity and fidelity are equivalent, i.e., G(ρ 1 , ρ 2 ) = F (ρ 1 , ρ 2 ) and no further work is required for estimating the fidelity. However, in the case of quartits one also has to estimate the subfidelity E(ρ 1 , ρ 2 ) to know in what range is the fidelity F (ρ 1 , ρ 2 ). The only missing quantity needed for estimating the subfidelity E(ρ 1 , ρ 2 ) is the second-order overlap O 2 (ρ 1 , ρ 2 ), which depends on the Hilbert-space dimension of a given system (i.e., qubit or quartit) and requires from four to eight photons.
B. Trace distance
As stated in Eq. (2), the trace distance equals to the half of the sum of eigenvalues for the Hermitian matrix Λ = ρ 1 − ρ 2 . These eigenvalues can be calculated by using the Cayley-Hamilton theorem [37], thus solving the characteristic equation p(Λ) = 0, which for the 4 × 4 matrix reads
p(Λ) = Λ 4 − 1 2 Π 2 Λ 2 − 1 3 Π 3 Λ + I 4 det(Λ) = 0,(11)
where det(Λ) = 1 4 ( 1 2 Π 2 2 − Π 4 ) and Π n = Tr(Λ) n = Tr(ρ 1 − ρ 2 ) n . Then the roots of Eq. (11) are the eigenvalues λ [see Eq. (2)] of the matrix Λ. The moments Π n can be decomposed into measurable overlaps simply by expanding Π n , i.e.,
Π 1 = 0, (12a) Π 2 = O(ρ 1 , ρ 1 ) + O(ρ 2 , ρ 2 ) − 2O(ρ 1 , ρ 2 ), (12b) Π 3 = O(ρ 2 1 , ρ 1 ) − O(ρ 2 2 , ρ 2 ) +3[O(ρ 2 2 , ρ 1 ) − O(ρ 2 1 , ρ 2 )], (12c) Π 4 = O 2 (ρ 1 , ρ 1 ) + O 2 (ρ 2 , ρ 2 ) − 2O 2 (ρ 1 , ρ 2 ) +4O(ρ 2 1 , ρ 2 2 ) + 4[O(ρ 3 2 , ρ 1 ) − O(ρ 3 1 , ρ 2 )].
(12d) Thus, if we consider optical implementation it is necessary to work with four to eight photons.
C. Hilbert-Schmidt distance
It is now easy to see that the Hilbert-Schmidt distance within our framework can be expressed via first-order overlaps (i.e., two-particle interference) as
H(ρ 1 , ρ 2 ) = O(ρ 1 , ρ 1 ) + O(ρ 2 , ρ 2 ) − 2O(ρ 1 , ρ 2 ). (13)
This makes it the simplest quantity to measure of the three considered metrics as in the case of two-qubit states, it requires working only with four photons.
D. Two-particle overlap
Let us first recall (see Ref. [33]) how the first-order overlap O(ρ 1 , ρ 2 ) [or purities O(ρ 1 , ρ 1 ) and O(ρ 2 , ρ 2 )] can be observed directly if one possesses two copies of the system. For two-qubit states the first-order overlap (purity)can be expressed as
O(ρ 1 , ρ 2 ) = 1 16 R (1) mn R (2) kl Tr[(σ m σ k ) ⊗ (σ k σ l )] = 1 4 R (1) mn R (2) mn(14)
where
R (1) mn = Tr[(σ m ⊗ σ n )ρ 1 ],(15a)R (2) mn = Tr[(σ m ⊗ σ n )ρ 2 ].(15b)
To derive this relation we applied basic properties of Pauli algebra, i.e.,
σ a σ b = iε abc σ c + δ ab σ 0 , and Tr(σ a σ b ) = 2δ ab ,(16)
where i is imaginary unit, a, b, c = 0, 1, 2, 3, δ ab is the Kronecker delta. The Levi-Civita symbol ε abc is zero, if the at least two indexes are equal or abc = 0. By expressing a product traces as a trace of a tensor product we obtain
O(ρ 1 , ρ 2 ) = 1 4 Tr[(σ m ⊗ σ n ⊗ σ m ⊗ σ n )(ρ 1 ⊗ ρ 2 )] = 1 4 Tr[(σ m ⊗ σ m ) ⊗ (σ n ⊗ σ n )(ρ 1 ⊗ ρ 2 ) ] = 1 4 Tr[(V a1a2 ⊗ V b1b2 ) (ρ 1 ⊗ ρ 2 )],(17)
where V = σ m σ m = 2I − 4|Ψ − Ψ − |, |Ψ − is the singlet state, and (ρ 1 ⊗ ρ 2 ) = S a2b1 (ρ 1 ⊗ ρ 2 )S a2b1 , where S a2b1 = I ⊗ S ⊗ I is unitary matrix swapping modes b 1 and a 2 . Within this framework it is possible to introduce the Hermitian overlap operator O measured on ρ 1 ⊗ ρ 2 , i.e,
O = S a2b1 V a1a2 V b1b2 S a2b1 .(18)
Measuring the purity or first-order overlp can be performed by measuring a product of two V operators, which was shown in Ref. [34] can be experimentally implemented within the framework of linear optics. Alternatively, one can directly perform projections on the maximally entangled states by using wave-plates, a beam splitter and a pair of single-photon detectors (see, e.g., Refs. [17,18]).
E. Four-particle overlap
Here we describe the main result of our paper, i.e., the second order overlap for four-particle interference that are necessary to measure subfidelity E given in (10a) and trace distance T defined in (2). Note that, we can use the same reasoning as in this case to discuss lower-order interactions by preparing one or more of the particles in a completely mixed state. To develop to only multi-particle Hong-Ou-Mandel interference based method of measuring the the second order overlap we have applied the procedure described below. The calculations are nontrivial as they require utilizing a number of algebraic properties and because of the complexity of the problem we also used a computer algebra system [38]. Let us start with expressing a product of two density matrices as
ρ 1 ρ 2 = 1 16 R (1) mn R (2) kl (σ m σ k ) (1) ⊗ (σ n σ l ) (2) .(19)
Now, the second order overlap can be expressed as
Tr(ρ 1 ρ 2 ) 2 = Tr[S(ρ 1 ρ 2 ) (12) ⊗ (ρ 1 ρ 2 ) (34) ],(20)
where the shift operator reads
S = S 23 S 34 S 12 S 23 .(21)
Thus, using the cyclic property of trace we have (22) where S = S 34 S 12 , S 12 = 1 − 2P − 12 , S 23 = 1 − 2P − 23 , and P − = 1 4 (1 − σ i ⊗ σ i ). Further transforming the shift operator S results in
Tr(ρ 1 ρ 2 ) 2 = 2 −8 R (1) mn R (2) kl R (1) xy R (2) rs Tr[S (σ m σ k ) (1) ⊗(σ x σ r ) (2) ⊗ (σ n σ l ) (3) (σ r σ s ) (4) ],S = 1 4 (1−σ (1) i ⊗σ (2) i −σ (3) j ⊗σ (4) j +σ (1) i ⊗σ (2) i ⊗σ (3) j ⊗σ (4) j ).(23)
Hence, we can express the overlap as
Tr(ρ 1 ρ 2 ) 2 = 2 −10 R (1) mn R (2) kl R (1) xy R (2) rs Tr[(1 − σ (1) i ⊗ σ (2) i −σ (3) j ⊗ σ (4) j + σ (1) i ⊗ σ (2) i ⊗ σ (3) j ⊗ σ (4) j ) ×(σ m σ k ) (1) ⊗ (σ x σ r ) (2) ⊗ (σ n σ l ) (3) (σ r σ s ) (4) ](24)
or equivalently as
Tr(ρ 1 ρ 2 ) 2 = R (1) mn R (2) kl R (1) xy R (2) rs [A 1 − A 2 − A 3 + A 4 ] (25)
where the sate-independent tensors read as A 1 = 2 −10 δ mk δ xr δ nl δ ys , (26a)
A 2 = 2 −6 δ (3) rx δ (3) mk + δ (3) mx δ (3) rk − δ (3) mr δ (3) kx +δ (3) mr δ k0 δ x0 + δ (3) kx δ r0 δ m0 + δ (3) kr δ m0 δ x0 +δ (3) mk δ r0 δ x0 + δ (3) rx δ k0 δ m0 + δ k0 δ m0 δ x0 δ r0 , (26b) A 3 = 2 −6 δ (3) sy δ (3) nl + δ (3) ny δ (3) sl − δ (3) ns δ (3) ly +δ (3) ns δ l0 δ y0 + δ (3) ly δ s0 δ n0 + δ (3) ls δ n0 δ y0 +δ (3) nl δ s0 δ y0 + δ (3)
sy δ l0 δ n0 + δ l0 δ n0 δ y0 δ s0 , (26c)
A 4 = 2 −10 (δ (3) im δ k0 + δ (3) ik δ m0 + δ (3) mk δ i0 + δ i0 δ m0 δ k0 +i mki ) × (δ (3) ix δ r0 + δ (3) ir δ x0 + δ (3) xr δ i0 +δ i0 δ x0 δ r0 + i xri ) × (δ (3) jn δ l0 + δ (3) jl δ n0 + δ (3) nl δ j0 +δ j0 δ n0 δ l0 + i nlj ) × (δ (3) jy δ s0 + δ (3) js δ y0 + δ (3) sy δ j0 +δ j0 δ s0 δ y0 + i ysj ). (26d)
The resulting expressions are rather complex as they describe 2-, 3-, and 4-particle interactions. However, due to the properties of Pauli algebra, we can express the final outcome as a polynomial of 2-particle interactions that can be implemented in an optical setup by Hong-Ou-Mandel antibunching events. Remarkably, we can simplify the resulting expression for the second order overlap so that it is given by a small number of terms by observing that matrix multiplication of the form R
nk = Tr[(ρ 1 ⊗ ρ 2 ) σ (1a) m ⊗ σ (2b) k ⊗ (1 − 4P − 1b2a )] = Tr[(ρ 1 ⊗ ρ 2 ) σ (1a) m ⊗ σ (2b) k ] − 4Tr[(ρ 1 ⊗ ρ 2 ) σ (1a) m ⊗ σ (2b) k ⊗ P − 1b2a ]. This expression for R (1) mn R (2)
nk can be represented graphically as presented in Fig. 1. In similar manner (with help of computer algebra program [38]), by using Eq. (25) the expression for the four-particle overlaps can be written as
O(ρ 3 1 , ρ 1 ) = Tr(ρ 1 ρ 1 ρ 1 ρ 1 ) = θ (1111) · x,(27a)O(ρ 3 1 , ρ 2 ) = Tr(ρ 1 ρ 1 ρ 1 ρ 2 ) = θ (1112) · x,(27b)O(ρ 2 1 , ρ 2 2 ) = Tr(ρ 1 ρ 1 ρ 2 ρ 2 ) = θ (1122) · x,(27c)O 2 (ρ 1 , ρ 2 ) = Tr(ρ 1 ρ 2 ρ 1 ρ 2 ) = θ (1212) · x, (27d) O(ρ 1 , ρ 3 2 ) = Tr(ρ 1 ρ 2 ρ 2 ρ 2 ) = θ (1222) · x, (27e) O(ρ 2 , ρ 3 2 ) = Tr(ρ 2 ρ 2 ρ 2 ρ 2 ) = θ (2222) · x,(27f)
where θ (n) for n = 1111, 1112, 1122, 1212, 1222, 2222 are vectors of state-independent coefficients (for their explicit form Tab. III) and components of x are given in Tab. I. Note that in the second overlap used to estimate subfidelity is given by θ (1212) . Remarkably, the moments Π n can be also expressed as similar dot-products, i.e.,
Π 2 = Tr(ρ 1 − ρ 2 ) 2 = β (2) · x,(28a)Π 3 = Tr(ρ 1 − ρ 2 ) 3 = β (3) · x,(28b)Π 4 = Tr(ρ 1 − ρ 2 ) 4 = β (4) · x. (28c)
All three β (n) vectors can be found in Tab. II. To obtain these final expressions we have summed the equivalent graphs and factorized the remaining unique graphs (see elements of x in Tab. I and g n shown in Fig. 1). Note that the number of measurements g n needed to determine x is 63. The number of measurements (projections) required to perform full two-qubit tomography for two arbitrary two-qubit states (30 projections per state).
III. DESIGNING A MULTIPARTICLE INTERFEROMETER
A. Hilbert-Schmidt distance
Our results can be used in practice. In an experiment designed to measure Hilbert-Schmidt metrics we would use two copies of both states. The original state is encoded as two-photon polarization, while the copy is encoded into spatial degree of freedom. By using this approach we are able to implement all the measurements required for determining Π 2 , which correspond to g n for n = 2, 4,7,8,10,12,13,24,29. This is because x measurements corresponding to nonzero elements of β (2) can be expressed as products of other measurements making the number of the prime measurements as low as 9. For β (2) the nonzero elements are β
(2) 2 = 4, β (2) 3 = −2, β (2) 4 = 4, β (2) 5 = −2, β (2) 6 = −2, β (2) 7 = −2, β(2)13 = −8, β(2)28 = 4, β(2)
42 = 4. Note that it is sufficient to design an interferometer corresponding to the most complex graphs g 13 , g 24 , and g 29 as the less complex graphs are measured if the singlet projection is replaced by identity operation (i.e., intensity measurement). Hence, Π 2 can be measured in a 6-photon-pair interferometric configuration i n for n = 1 associated with the projective measurements g 13 , g 24 , g 29 , where 6 photon pairs are used and V operators are measured instead of singlet projections. At the same time, while using only 6 photon pairs it is possible to access only 6 out of 30 parameters of density matrices. This demonstrates the superior performance (in terms of the number of measured parameters) of the interferometric method with respect to the tomographic approach.
B. Fidelity, subfidelity, and superfidelity
We can apply our results to design an experiment aimed at measuring subfidelity G.
As it follows from Eq. (10b) in addition to measuring first-order overlap, the second order overlap O 2 (ρ 1 , ρ 2 ) needs to be measured. Within our framework of singlet projections, superfidelity can be measured as described in Ref. [33].
However, the method for measuring subfidelity presented in in Ref. [33], does not utilize simple antibunching events and may be problematic in implementation. Here, present an alternative solution which is free of this shortcoming. The set of measurements needed to estimate O 2 corresponds to antibunching events shown in Fig. 1 , where g n for n = 1, 2, 3, 4, 5, 6,7,8,9,10,11,12,13,15,16,18,19,20,21,22,24,25,26,27,29,30,31,32,33,34,35,37,38,39,41,42,43,44,45,46,47, This makes the total number of required projections equal to 41. However, after closer examination, we see that all the required quantities can be measured in in 4-photon-pair interferometric configurations i n for n = 1, 2, 3, 4, 5, 6, 7, 8, 9 associated with the projective measurements in the following way: i 1 = g 10 g 34 , i 2 = g 1 g 38 , i 3 = g 13 g 18 , i 4 = g 24 g 29 , i 5 = g 26 g 30 , i 6 = g 43 , i 7 = g 44 , i 8 = g 45 , i 9 = g 46 , i 10 = g 47 , where 20 photon pairs are used and V operators are measured instead of singlet projections. In case of full quantum state tomography, we would have Thus, measuring subfidelity can be performed more efficiently with interferometric method, without recourse to full quantum-state tomography.
C. Trace distance
For measuring trace distance T (ρ 1 , ρ 2 ) with our method we would measure Π n for n = 2, 3, 4, then calculate the eigenvalues of Λ as given by Eq. (11) by replacing Λ with with a variable for which we solve the characteristic equation. Finally, we would use definition (2). The projections required for measuring Π 3 , are g n for n = 1, 3,5,6,9,11,14,15,16,17,18,19,20,21,22,23,25,26,27,28,30,34,38,48,49, 50, 53, 54, 55 and additionally n = 2, 4,7,8,10,12,13,24,29,43,44,45,46,47,51,52,56,57,59, 60, 62, 63 for Π 4 . The total amount of projections required to determine T via moments is 51 (using 104 photon pairs). This number is larger than 60 protective measurements required for quantum state tomography. Thus, measuring T with interferometric method is more challenging than applying two-qubit tomography.
IV. CONCLUSIONS
In this paper we put forward a direct method for measuring the lower bound (i.e., subfidelity [7]) of the Uhlmann-Jozsa fidelity (equivalent to the Bures distance), Hilbert-Schmidt distance and trace distance for arbitrary unknown mixed two-qubit states. Our proposal of a experimentally-friendly method for direct measuring the second-order overlaps of two arbitrary two-qubit states enables the qualitative determination of their similarity. In particular, we demonstrated that while having access simultaneously to 12 photons, we could measure Hilbert-Schmidt distance directly. To this date, experiments utilizing interference of up to 16 photons in linear optical circuts have been reported, see, e.g., [41]. This lets us believe, that our method can be implemented experimentally. Distances between states can be utilized as cost functions for optimization problems solved in quantum machine learning [42,43]. Thus, we believe that our results can be useful in this context.
For Hilbert-Schmidt distance the number of required projective measurements is 10, which is smaller than 60 required for full quantum state tomography of both twoqubit states. In case of trace distance the number of the required projective measurements is much larger. As discussed in Sec. III, while analyzing both subfidelity and Hilbert-Schmidt distance, many of the measurements can be performed simultaneously. However, the example of trace distance shows that many-particle interference based method does not always outperform quantum state tomography. III. The nonzero components of vectors θ (n) for n = 1111, 2222, 1112, 1222, 1212, 1122 (used to calculate four-particle overlaps) corresponding to components of vector x described in Tab. I.
FIG. 1 .
1(color online) The complete set of 63 graphs representing the singlet projections needed for measuring all relevant four-particle interactions. The pairs of gray or white vertices denote the modes forming a copy of ρ1 or ρ2 state, respectively. The red (dark) lines mark the singlet projections (Hong-Ou-Mandel antibunching events).
of 60 required projective measurements (i.e., 20 of 30 required tensor products of Pauli operators) while using the same number of photon pairs.
TABLE I .
IComponents of vector x given in terms of 63 measurement outcomes gn shown inFig. 1.n
xn
n
xn
n
xn
n
xn
n
xn
n
xn
n
xn
n
xn
1
1
23
g21
45
g1g3g9
67
g38
89
g 2
18
111
g 3
1 g9
133
g12g13
155
g 2
24
2
g2
24
g22
46
g1g6g9
68
g39
90
g9g34
112
g 2
1 g18
134
g1g 3
9
156
g59
3
g4
25
g3g6
47
g1g 2
9
69
g9g21
91
g45
113
g1g9g26
135
g 2
9 g18
157
g 2
26
4
g7
26
g3g9
48
g1g18
70
g40
92
g46
114
g1g34
136
g1g9g30
158
g60
5
g8
27
g23
49
g1g27
71
g41
93
g47
115
g13g24
137
g9g38
159
g 2
10
6
g10
28
g24
50
g1g20
72
g10g24
94
g24g29
116
g51
138
g13g29
160
g10g12
7
g12
29
g4g10
51
g1g22
73
g12g24
95
g 2
9 g26
117
g18g26
139
g56
161
g 2
12
8
g1g9
30
g4g12
52
g1g30
74
g4g29
96
g26g30
118
g9g50
140
g18g30
162
g 2
9 g11
9
g1g6
31
g25
53
g 2
1 g11
75
g42
97
g1g5
119
g52
141
g1g55
163
g9g54
10
g1g11
32
g26
54
g2g13
76
g9g25
98
g48
120
g9g11
142
g57
164
g10g29
11
g1g3
33
g5g9
55
g7g13
77
g9g26
99
g2g4
121
g53
143
g 2
4
165
g12g29
12
g 2
1
34
g5g11
56
g31
78
g11g26
100
g4g7
122
g2g10
144
g4g8
166
g11g30
13
g13
35
g27
57
g32
79
g5g 2
9
101
g49
123
g7g10
145
g 2
8
167
g61
14
g14
36
g6g9
58
g33
80
g5g30
102
g2g8
124
g54
146
g 2
1 g5
168
g 4
9
15
g15
37
g28
59
g3g18
81
g8g29
103
g7g8
125
g2g12
147
g1g49
169
g 2
9 g30
16
g16
38
g 2
7
60
g6g18
82
g 2
1 g 2
9
104
g 3
1
126
g7g12
148
g4g24
170
g9g55
17
g17
39
g8g10
61
g9g18
83
g1g9g18
105
g1g26
127
g 3
9
149
g8g24
171
g 2
29
18
g 2
2
40
g8g12
62
g34
84
g1g38
106
g2g24
128
g9g30
150
g5g26
172
g62
19
g2g7
41
g 2
9
63
g35
85
g 2
1 g30
107
g7g24
129
g2g29
151
g58
173
g 2
30
20
g18
42
g29
64
g36
86
g 2
13
108
g4g13
130
g7g29
152
g 4
1
174
g63
21
g19
43
g30
65
g9g19
87
g43
109
g50
131
g10g13
153
g 2
1 g26
22
g20
44
g 2
1 g9
66
g37
88
g44
110
g8g13
132
g55
154
g1g50
TABLE II .
IIThe nonzero components of vectors β (n) for n = 2, 3, 4 (used for calculating moments Πn) corresponding to components of vector x described in Tab. I.
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|
[] |
[
"Anomalous optical saturation of low-energy Dirac states in graphene and its implication for nonlinear optics",
"Anomalous optical saturation of low-energy Dirac states in graphene and its implication for nonlinear optics"
] |
[
"Behrooz Semnani \nDepartment of Physics\nChalmers University of Technology\nGöteborgSweden\n\nWaterloo Institute for Nanotechnology\nUniversity of Waterloo\nWaterlooONCanada\n\nDepartment of Electrical & Computer Engineering\nUniversity of Waterloo\nWaterlooONCanada\n",
"Roland Jago \nDepartment of Physics\nChalmers University of Technology\nGöteborgSweden\n",
"Safieddin Safavi-Naeini \nWaterloo Institute for Nanotechnology\nUniversity of Waterloo\nWaterlooONCanada\n\nDepartment of Electrical & Computer Engineering\nUniversity of Waterloo\nWaterlooONCanada\n",
"Amir Hamed Majedi \nWaterloo Institute for Nanotechnology\nUniversity of Waterloo\nWaterlooONCanada\n\nDepartment of Electrical & Computer Engineering\nUniversity of Waterloo\nWaterlooONCanada\n\nPerimeter Institute for Theoretical Physics (PI)\nWaterlooONCanada\n",
"Ermin Malic \nDepartment of Physics\nChalmers University of Technology\nGöteborgSweden\n",
"Philippe Tassin \nDepartment of Physics\nChalmers University of Technology\nGöteborgSweden\n"
] |
[
"Department of Physics\nChalmers University of Technology\nGöteborgSweden",
"Waterloo Institute for Nanotechnology\nUniversity of Waterloo\nWaterlooONCanada",
"Department of Electrical & Computer Engineering\nUniversity of Waterloo\nWaterlooONCanada",
"Department of Physics\nChalmers University of Technology\nGöteborgSweden",
"Waterloo Institute for Nanotechnology\nUniversity of Waterloo\nWaterlooONCanada",
"Department of Electrical & Computer Engineering\nUniversity of Waterloo\nWaterlooONCanada",
"Waterloo Institute for Nanotechnology\nUniversity of Waterloo\nWaterlooONCanada",
"Department of Electrical & Computer Engineering\nUniversity of Waterloo\nWaterlooONCanada",
"Perimeter Institute for Theoretical Physics (PI)\nWaterlooONCanada",
"Department of Physics\nChalmers University of Technology\nGöteborgSweden",
"Department of Physics\nChalmers University of Technology\nGöteborgSweden"
] |
[] |
We reveal that optical saturation of the low-energy states takes place in graphene for arbitrarily weak electromagnetic fields. This effect originates from the diverging field-induced interband coupling at the Dirac point. Using semiconductor Bloch equations to model the electronic dynamics of graphene, we argue that the charge carriers undergo ultrafast Rabi oscillations leading to the anomalous saturation effect. The theory is complemented by a many-body study of the carrier relaxations dynamics in graphene. It will be demonstrated that the carrier relaxation dynamics is slow around the Dirac point, which in turn leads to a more pronounced saturation. The implications of this effect to the nonlinear optics of graphene is then discussed. Our analysis show that the conventional perturbative treatment of the nonlinear optics, i.e., expanding the polarization field in a Taylor series of the electric field, is problematic for graphene, in particular at small Fermi levels and large field amplitudes.Graphene is a two-dimensional material made of carbon atoms in a honeycomb structure. Its reduced dimensionality and the symmetries of its crystalline structure render graphene a gapless semiconductor [1]. Graphene exhibits a wealth of exceptional properties, including a remarkably high mobility at room temperature [2], Klein tunneling and Zitterbewegung[3,4], existence of a nonzero Berry phase, anomalous quantum Hall effect[5][6][7], quantum limited intrinsic conductivity [8], and a unique Landau level structure [9, 10]. Underlying these peculiar electronic properties are its pseudo-relativistic quasiparticles that obey the massless Dirac equation[1]. As a direct consequence of their massless nature, the Dirac fermions have definite chiralities[11,12]. Owing to the specific symmetries of the crystalline structure of graphene, the dynamics of the massless Dirac quasiparticles and their chiral character are topologically preserved-i.e., many-body induced band renormalizations as well as any moderate perturbations of the lattice will not open a gap in graphene's band structure[13]. A large number of the unusual properties of graphene are associated with the topologically protected band-crossing and the chiral dynamics of the charge carriers [3].One major consequence of the topologically protected chirality of the charge carriers is the anomalous structure of the interband coupling mediated by an electromagnetic field. Its dipole matrix element obtained in the length gauge [14] exhibits a singularity at the degeneracy points, in contrast to ordinary (and even other gapless) semiconductors[15,16]. This has raised some controversy regarding the treatment of the optical response of graphene[15,17]. Specifically, the perturbative treatment of the nonlinear optical response has been questioned[17,18]. The nonlinear optical coefficients of graphene obtained by means of perturbation theory suffer from a nonresolvable singularity[15,17]. Although substantial effort has been spent on developing comprehensive models for the nonlinear optical response of graphene [17-23], a self-consistent theoretical model that can resolve the above issue is still lacking. In addition, many experimental studies of the nonlinear optics of graphene have been reported [24-28]-some of these studies are difficult to reconcile with existing theoretical models.In this Letter, we show that the singular nature of the interband dipole coupling causes the charge carriers in the vicinity of the Dirac points to undergo ultrafast Rabi oscillations accompanied by slow relaxation dynamics, which, intriguingly, yields an anomalous saturation effect. This finding necessitates revisiting the perturbative treatment of the nonlinear optical response of graphene to account for the extreme nonlinear interactions around the Dirac points. These conclusions will be reached by describing the dynamics of the charge carriers with semiconductor Bloch equations (SBEs)[29,30].We consider a free-standing graphene monolayer (in the xy plane) illuminated by a normally incident electromagnetic field. The monochromatic and spatially uniform optical field at the graphene layer is described by E(t) = E 0 e iωt + c.c., where E 0 is parallel to the graphene plane. The light-matter interaction is considered semiclassically and the external field coupling is obtained in the length gauge[14]. For photon energies below approximately 2eV, the electronic dynamics of the quasiparticles in the absence of external radiation is adequately described by the massless Dirac equation yielding the relativistic energy-momentum dispersion E k = ± v F |k| [1], where k is the Bloch wave vector with respect to the Dirac point and v F is the Fermi velocity. Coupling to electromagnetic radiation alters the band structure of graphene by dressing the eigenstates and a bandgap can arXiv:1806.10123v1 [cond-mat.mes-hall]
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10.1088/2053-1583/ab1dea
|
[
"https://arxiv.org/pdf/1806.10123v1.pdf"
] | 119,226,438 |
1806.10123
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6fc151ec101c3d3fa6adc70467996d2dace3a7ef
|
Anomalous optical saturation of low-energy Dirac states in graphene and its implication for nonlinear optics
26 Jun 2018
Behrooz Semnani
Department of Physics
Chalmers University of Technology
GöteborgSweden
Waterloo Institute for Nanotechnology
University of Waterloo
WaterlooONCanada
Department of Electrical & Computer Engineering
University of Waterloo
WaterlooONCanada
Roland Jago
Department of Physics
Chalmers University of Technology
GöteborgSweden
Safieddin Safavi-Naeini
Waterloo Institute for Nanotechnology
University of Waterloo
WaterlooONCanada
Department of Electrical & Computer Engineering
University of Waterloo
WaterlooONCanada
Amir Hamed Majedi
Waterloo Institute for Nanotechnology
University of Waterloo
WaterlooONCanada
Department of Electrical & Computer Engineering
University of Waterloo
WaterlooONCanada
Perimeter Institute for Theoretical Physics (PI)
WaterlooONCanada
Ermin Malic
Department of Physics
Chalmers University of Technology
GöteborgSweden
Philippe Tassin
Department of Physics
Chalmers University of Technology
GöteborgSweden
Anomalous optical saturation of low-energy Dirac states in graphene and its implication for nonlinear optics
26 Jun 2018
We reveal that optical saturation of the low-energy states takes place in graphene for arbitrarily weak electromagnetic fields. This effect originates from the diverging field-induced interband coupling at the Dirac point. Using semiconductor Bloch equations to model the electronic dynamics of graphene, we argue that the charge carriers undergo ultrafast Rabi oscillations leading to the anomalous saturation effect. The theory is complemented by a many-body study of the carrier relaxations dynamics in graphene. It will be demonstrated that the carrier relaxation dynamics is slow around the Dirac point, which in turn leads to a more pronounced saturation. The implications of this effect to the nonlinear optics of graphene is then discussed. Our analysis show that the conventional perturbative treatment of the nonlinear optics, i.e., expanding the polarization field in a Taylor series of the electric field, is problematic for graphene, in particular at small Fermi levels and large field amplitudes.Graphene is a two-dimensional material made of carbon atoms in a honeycomb structure. Its reduced dimensionality and the symmetries of its crystalline structure render graphene a gapless semiconductor [1]. Graphene exhibits a wealth of exceptional properties, including a remarkably high mobility at room temperature [2], Klein tunneling and Zitterbewegung[3,4], existence of a nonzero Berry phase, anomalous quantum Hall effect[5][6][7], quantum limited intrinsic conductivity [8], and a unique Landau level structure [9, 10]. Underlying these peculiar electronic properties are its pseudo-relativistic quasiparticles that obey the massless Dirac equation[1]. As a direct consequence of their massless nature, the Dirac fermions have definite chiralities[11,12]. Owing to the specific symmetries of the crystalline structure of graphene, the dynamics of the massless Dirac quasiparticles and their chiral character are topologically preserved-i.e., many-body induced band renormalizations as well as any moderate perturbations of the lattice will not open a gap in graphene's band structure[13]. A large number of the unusual properties of graphene are associated with the topologically protected band-crossing and the chiral dynamics of the charge carriers [3].One major consequence of the topologically protected chirality of the charge carriers is the anomalous structure of the interband coupling mediated by an electromagnetic field. Its dipole matrix element obtained in the length gauge [14] exhibits a singularity at the degeneracy points, in contrast to ordinary (and even other gapless) semiconductors[15,16]. This has raised some controversy regarding the treatment of the optical response of graphene[15,17]. Specifically, the perturbative treatment of the nonlinear optical response has been questioned[17,18]. The nonlinear optical coefficients of graphene obtained by means of perturbation theory suffer from a nonresolvable singularity[15,17]. Although substantial effort has been spent on developing comprehensive models for the nonlinear optical response of graphene [17-23], a self-consistent theoretical model that can resolve the above issue is still lacking. In addition, many experimental studies of the nonlinear optics of graphene have been reported [24-28]-some of these studies are difficult to reconcile with existing theoretical models.In this Letter, we show that the singular nature of the interband dipole coupling causes the charge carriers in the vicinity of the Dirac points to undergo ultrafast Rabi oscillations accompanied by slow relaxation dynamics, which, intriguingly, yields an anomalous saturation effect. This finding necessitates revisiting the perturbative treatment of the nonlinear optical response of graphene to account for the extreme nonlinear interactions around the Dirac points. These conclusions will be reached by describing the dynamics of the charge carriers with semiconductor Bloch equations (SBEs)[29,30].We consider a free-standing graphene monolayer (in the xy plane) illuminated by a normally incident electromagnetic field. The monochromatic and spatially uniform optical field at the graphene layer is described by E(t) = E 0 e iωt + c.c., where E 0 is parallel to the graphene plane. The light-matter interaction is considered semiclassically and the external field coupling is obtained in the length gauge[14]. For photon energies below approximately 2eV, the electronic dynamics of the quasiparticles in the absence of external radiation is adequately described by the massless Dirac equation yielding the relativistic energy-momentum dispersion E k = ± v F |k| [1], where k is the Bloch wave vector with respect to the Dirac point and v F is the Fermi velocity. Coupling to electromagnetic radiation alters the band structure of graphene by dressing the eigenstates and a bandgap can arXiv:1806.10123v1 [cond-mat.mes-hall]
We reveal that optical saturation of the low-energy states takes place in graphene for arbitrarily weak electromagnetic fields. This effect originates from the diverging field-induced interband coupling at the Dirac point. Using semiconductor Bloch equations to model the electronic dynamics of graphene, we argue that the charge carriers undergo ultrafast Rabi oscillations leading to the anomalous saturation effect. The theory is complemented by a many-body study of the carrier relaxations dynamics in graphene. It will be demonstrated that the carrier relaxation dynamics is slow around the Dirac point, which in turn leads to a more pronounced saturation. The implications of this effect to the nonlinear optics of graphene is then discussed. Our analysis show that the conventional perturbative treatment of the nonlinear optics, i.e., expanding the polarization field in a Taylor series of the electric field, is problematic for graphene, in particular at small Fermi levels and large field amplitudes.
Graphene is a two-dimensional material made of carbon atoms in a honeycomb structure. Its reduced dimensionality and the symmetries of its crystalline structure render graphene a gapless semiconductor [1]. Graphene exhibits a wealth of exceptional properties, including a remarkably high mobility at room temperature [2], Klein tunneling and Zitterbewegung [3,4], existence of a nonzero Berry phase, anomalous quantum Hall effect [5][6][7], quantum limited intrinsic conductivity [8], and a unique Landau level structure [9,10]. Underlying these peculiar electronic properties are its pseudo-relativistic quasiparticles that obey the massless Dirac equation [1]. As a direct consequence of their massless nature, the Dirac fermions have definite chiralities [11,12]. Owing to the specific symmetries of the crystalline structure of graphene, the dynamics of the massless Dirac quasiparticles and their chiral character are topologically preserved-i.e., many-body induced band renormalizations as well as any moderate perturbations of the lattice will not open a gap in graphene's band structure [13]. A large number of the unusual properties of graphene are associated with the topologically protected band-crossing and the chiral dynamics of the charge carriers [3].
One major consequence of the topologically protected chirality of the charge carriers is the anomalous structure of the interband coupling mediated by an electromagnetic field. Its dipole matrix element obtained in the length gauge [14] exhibits a singularity at the degeneracy points, in contrast to ordinary (and even other gapless) semiconductors [15,16]. This has raised some controversy regarding the treatment of the optical response of graphene [15,17]. Specifically, the perturbative treatment of the nonlinear optical response has been questioned [17,18]. The nonlinear optical coefficients of graphene obtained by means of perturbation theory suffer from a nonresolvable singularity [15,17]. Although substantial effort has been spent on developing comprehensive models for the nonlinear optical response of graphene [17][18][19][20][21][22][23], a self-consistent theoretical model that can resolve the above issue is still lacking. In addition, many experimental studies of the nonlinear optics of graphene have been reported [24][25][26][27][28]-some of these studies are difficult to reconcile with existing theoretical models.
In this Letter, we show that the singular nature of the interband dipole coupling causes the charge carriers in the vicinity of the Dirac points to undergo ultrafast Rabi oscillations accompanied by slow relaxation dynamics, which, intriguingly, yields an anomalous saturation effect. This finding necessitates revisiting the perturbative treatment of the nonlinear optical response of graphene to account for the extreme nonlinear interactions around the Dirac points. These conclusions will be reached by describing the dynamics of the charge carriers with semiconductor Bloch equations (SBEs) [29,30].
We consider a free-standing graphene monolayer (in the xy plane) illuminated by a normally incident electromagnetic field. The monochromatic and spatially uniform optical field at the graphene layer is described by E(t) = E 0 e iωt + c.c., where E 0 is parallel to the graphene plane. The light-matter interaction is considered semiclassically and the external field coupling is obtained in the length gauge [14]. For photon energies below approximately 2eV, the electronic dynamics of the quasiparticles in the absence of external radiation is adequately described by the massless Dirac equation yielding the relativistic energy-momentum dispersion E k = ± v F |k| [1], where k is the Bloch wave vector with respect to the Dirac point and v F is the Fermi velocity. Coupling to electromagnetic radiation alters the band structure of graphene by dressing the eigenstates and a bandgap can arXiv:1806.10123v1 [cond-mat.mes-hall] 26 Jun 2018 be opened [31], which is reproduced by our model.
The SBEs describe the coupled dynamics of the population difference N (k, t) and the polarization (coherence) P(k, t) in the momentum state k. In the absence of electromagnetic radiation, the population difference relaxes to N eq
k = f ( v F k) − f (− v F k), where f (E)
is the Fermi-Dirac distribution. An electromagnetic field drives the system out of equilibrium via the coupled intraband and interband dynamics. In a moving frame {τ, k } = {t, k − δk(t)}, where δk obeys ∂δk ∂t + Γδk = − e E(t) (Γ is a phenomenological intraband relaxation coefficient), the dynamics of the charge carriers is governed by
∂N (k , τ ) ∂τ = −γ (1) k (N (k , τ ) − N eq k ) − 2Φ(k , τ )Im {P(k , τ )} , (1a) ∂P(k , τ ) ∂τ = −γ (2) k P(k , τ )+ i k P(k , τ ) + i 2 Φ(k , τ )N (k , τ ), (1b) where Φ(k, t) = eE·φ k k
is the matrix element of the external potential of the direct optical transition, and the unit vectorφ k is defined asφ k =ẑ × k/k. The frequency k = 2E k is the energy difference between the energy levels of the conduction and valence bands. γ (1) k and γ
(2) k are k-dependent relaxation coefficients stemming from many-body effects such as electron-electron and electron-phonon interactions. The detailed derivation of our theoretical model is provided in the Supplemental Material [32].
The light-graphene interaction as described by Eqs. (1a)-(1b) can be interpreted as an ensemble of inhomogeneously broadened two-level systems (one for each k). The last term in each of the two equations will lead to Rabi oscillations. Because of the singularity in Φ(k , τ ) for |k| → 0, we can expect ultrafast Rabi oscillations around the Dirac point, which are damped by manybody interactions. The decay terms drive the two-level systems towards an equilibrium state. Since the interband coupling is strong around the Dirac point (equivalent to highly intense illumination), the effective field leaves the two-level systems in a statistical mixture of the ground and excited states with equal weights and absorption quenching takes place. Thus, the states around the Dirac points undergo a saturation effect, even when illuminated by an arbitrarily weak electromagnetic field.
This saturation behavior can be further understood by studying the steady-state solution of the SBEs, which is
N st k = N eq k γ (1) k γ (1) k + γ (2) k Φ k 2 / γ (2) k + i∆ k 2 ,(2)
where Φ k = eE 0 ·φ k / k is the complex phasor associated with Φ(k, t).The function ∆ k = ω − k denotes the detuning of the two-level system at k with respect to the excitation.
Since | Φ k | is arbitrarily large for small-k states, in the vicinity of the Dirac point, the population N k cannot be expanded in a Taylor series of the fieldΦ. This implies that the nonlinear optics of graphene is in principle a nonperturbative problem. Indeed, due to the singularity of the interband coupling in graphene, there is always a region around the Dirac point where graphene is optically saturated. The saturation threshold E sat k is given by
eE sat k = k ∆ 2 k γ (1) k γ (2) k + γ (1) k γ (2) k .(3)
Saturation occurs of course in any two-level system at high field intensities. However, in graphene the saturation threshold field E sat k is zero at the Dirac point (k → 0) and, hence, there is always a region of k-space where E 0 > E sat k , even for arbitrarily weak intensities. The peculiar low-threshold saturation mechanism in graphene can be quantitatively resolved using a timedomain analysis of the graphene SBEs. For the sake of comparison, this analysis has been performed for two distinct continuous excitations with optical frequency of ω = 80meV (terahertz range) and ω = 800meV (infrared), respectively. In both cases, the electric field is linearly polarized along theŷ direction with magnitude E 0 = 10 6 V/m. Graphene is assumed to be undoped here and is initially held at room temperature. The relaxation coefficients γ (1) k and γ (2) k are determined using a microscopic theory, which encompasses carrier-carrier as well as carrier-phonon scattering channels and takes into account all relevant relaxation paths including interband and intraband and even inter-valley processes [33,34]. We refer to the Supplemental Material for the details of the many-body model [32] as well as the methodology used to extract the relaxation coefficients. The resulting relaxation coefficients are plotted in Fig. 1(b) and (c). We note in particular that γ (1) k tends to be zero around the Dirac point, which confirms the slow relaxation dynamics suggested above.
The relative change in the stationary component of the population difference due to the optical excitation as well as the amplitude of the oscillating induced polarization are shown in Fig. 1(d) and (e). To obtain the steadystate components, we performed a Fourier analysis within a time window where the transient response has died out. As expected, we observe a well-pronounced modified population difference around the Dirac point due to the spontaneous polarization effect (dark red region around the center). This effect is stronger for lower-frequency excitations-indeed, according to Eq. (3) a smaller detuning yields a weaker saturation threshold. The region in k-space where the spontaneous optical saturation is significant is well extended from the Dirac point. We note here that the size of the region depends on the applied field intensity-we will show below that this is the origin of the nonperturbative nature of the nonlinear optical response. In addition, there is of course the traditional optical saturation region for ∆ k ≈ 0 (indicated by the yellow dashed line).
Before we continue with the importance of this anomalous optical saturation for the nonlinear optics of graphene, let us briefly make a few remarks regarding the origin of this anomaly. First, the origin of the inverse dependence of the interband transition matrix element on the wave number can be linked to the distinctive mathematical structure of the current operator. It is straightforward to show that the interband coupling matrix element at wavenumber k isr cv ≈
i ev F J cv (k)/[E c (k)−E v (k)]
where J cv is the off-diagonal element of the current operator. In contrast to ordinary semiconductors, the offdiagonal components of the current operator in graphene and other chiral materials are strictly nonzero even at the band crossing points [35]. As a direct consequence of this property of massless Dirac quasiparticles, the interband part of the position operator carries a first-order singularity at the degeneracy point. Second, one may wonder why we have not used the velocity gauge, in which optical coupling is obtained by minimal substitution k → k + eA where E = −∂A/∂t. It is important to note that this approach is not gauge invariant in the "effective Hamiltonian" picture [36][37][38]. We show in the Supplemental Material that a modification of the velocity gauge is indeed required to yield a physically correct result [32]. This modification gives rise to the 1/k dependence of the interband coupling in the vicinity of the Dirac point [38].
We now turn to the nonlinear optics of graphene. Let us consider a nonlinear pump-probe experiment in which graphene is simultaneously subjected to the a pump (ω c ) and a weaker probe (ω p ) laser beam. The conductivity tensor of graphene in the presence of the pump field and "seen" by the probe field is calculated in the Supplementary Material [32]. Fig. 2 displays the change to the conductivity tensor due to the pump fields described earlier. The latter is related to the absorption coefficient of the probe beam: α ≈ Re{σ}/4ε 0 c, where c is the speed of light in vacuum. The relative change in absorption of the probe beam is also shown in Fig. 2. For the 80meV pump beam, there is strong saturation for a probe beam at the The change in the conductivity of graphene seen by the probe field (with frequency ωp) in a pump-probe experiment for the pump frequecies (a) ωc = 80meV and (b) ωc = 800meV. The conductivity is normalized to σ 0 = e 2 /4 . The corresponding changes of the absorption coefficient of graphene for different intensities of the pump field are shown to the right of the conductivity plots. FIG. 3. (a) In the calculation of the semiperturbative nonlinear coefficient, the saturated region (blue disc) is excluded, and only states in the unsaturated region (green region) play a role. (b-d) The Kerr-type nonlinearity of graphene obtained from the analytical approach (blue shaded regions) and the semiperturbative approach (i.e., σ (3) (ω, ω, −ω), black dotted curves) plotted for different field magnitude. The vertical axis shows different electric field magnitudes (i.e E 0 ). The two distinctive red shaded regions are the saturation regions in E 0 − ω plane due to, respectively, zero detuning and the strong interband coupling in the vicinity of the Dirac points. Results for different Fermi levels (i.e. E f ) are displayed in (b-d). Note the different scales in the insets, which depict the third-order nonlinear coefficient.
same frequency-this is because the pump beam has saturated the interband transition. For a probe beam with a much lower frequency (ω p ≈ ω c /100), there is also strong saturation-the latter must be due to the unconventional effect discussed above. For the 800meV pump, a weaker saturation is observed for a low-frequency probe. Indeed, for a higher-frequency pump the region of the lowthreshold saturation is smaller [observe in Eq. (3) that a larger detuning results in a larger saturation field]. Although the above example discusses the anomalous saturation effect for undoped graphene, the effect happens for a range of Fermi levels. Therefore, it will also be observed in samples where the charge-neutrality point fluctuates in space.
We now move on to discuss the applicability of perturbation theory in the analysis of the nonlinear response of graphene. As detailed in Ref. [15], the standard perturbative treatment of the optical response of graphene leads to a nonresolvable singularity in its higher-order nonlinear coefficients (beyond the linear response) originating from the small-k states. We now understand that these states should be excluded because they are saturated. However, since the saturation threshold depends on the field intensity, the nonlinear response coefficients calculated with standard perturbation theory become field dependent too. But let us investigate under what circumstances the standard nonlinear coefficients have meaning.
As mentioned earlier, neglecting the momentum of the absorbed photon, the optical transitions are vertical in k-space and, therefore, every point in the reciprocal space can be treated independently. It is argued in Refs. [15,17,21] that nonlinear frequency mixing in graphene can be decomposed into a number of additive contributions (see the Supplemental Material for a complete theoretical analysis [32]), i.e., the nonlinear conductivity tensor is σ (l) ≈ k,|k|>Ksat I (l) k , where K sat is the radius (with respect to the Dirac point) of the spontaneously saturated region in k-space and is obtained from Eq. (3):
v F K sat ≈ 1 2 ω − ( 1 2 ω) 2 − 2 v F eE 0 γ (2) γ (1) ,(4)
and I (l) k represents the contribution of the quasiparticles with Bloch index k to the l'th order nonlinear optical response. E 0 is the magnitude of the largest electric field component (most often a pump field) participating in the nonlinear process and ω is its frequency.
In order to gain insight into the intensity dependence of the nonlinear response coefficients obtained from the above-described semiperturbative approach, we compare in Fig. 3 the third-order nonlinear response defined as |σ (3) xxxx (ω, ω, −ω)| (where we have now used kindependent relaxation constants as usual) with the results of the full solution of SBEs (see the Supplemental Material [32]). The blue shaded regions display the nonperturbative solution and the black dotted curves are the results of semiperturbative approach. First, owing to the low-threshold saturation effect, there is a noticeable field dependence of the third-order nonlinear (Kerr-like) response for lower Fermi energies. When the field intensity becomes large enough to extend the saturation region to the excited k-states, the semiperturbative approach fails. As the Fermi energy becomes larger, the optically induced Pauli blocking becomes less important as the low-energy states are already Pauli blocked. It is worth pointing out that we have observed significant dependence of the results on K sat . Therefore, the exact exclusion of the saturated region is necessary to achieve accurate results.
In conclusion, we have demonstrated that the topologically protected singular interband coupling in graphene leads to ultrafast Rabi oscillations which excites the quasiparticles faster than they can relax back to the ground state. This leads to an anomalous optical saturation of the low-energy quasiparticles in graphene. Subsequently, we have shown that due to this effect the small-k states have to be excluded for the perturbative calculation of nonlinear optical coefficients of graphene. As a result the nonlinear coefficients obtained from perturbation theory exhibit a noticeable field dependence particularly for higher field intensities and small Fermi levels. We speculate that similar effects may be found in other Dirac materials and in Weyl semimetals.
Work at Waterloo has been supported by the Natural Science and Engineering Research Council of Canada (NSERC) and Quantum Quest Seed Fund (QQSF). Work at Chalmers has been supported by the Rune Bernhardsson Graphene fund, by Vetenskapsrådet under grant No. 2016-03603, and by the European Union's Horizon 2020 research and innovation programme under grant agreement No. 696656.
FIG. 1 .
1(a) The low energy band structure of graphene. The angleφ k and the magnitude of the Bloch wavenumber k are defined in polar coordinate system in reciprocal space. (b)& (c) The k-dependent population and coherence relaxation coefficients shown by γ The coefficients are calculated for ω = 80meV and 800meV electromagnetic excitations. In both cases the electric field magnitude is 10 6 V/m (d) & (e) The relative change in the population difference with respect to the equilibrium δN = N − Neq for ω = 80meV and 800meV, respectively. (f)& (g) The corresponding steady-state polarization.
FIG. 2 .
2FIG. 2.
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|
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"The quantum determinant of the elliptic algebra A q,p ( gl N )",
"The quantum determinant of the elliptic algebra A q,p ( gl N )",
"The quantum determinant of the elliptic algebra A q,p ( gl N )",
"The quantum determinant of the elliptic algebra A q,p ( gl N )"
] |
[
"L Frappat [email protected]@[email protected] ",
"D Issing ",
"E Ragoucy ",
"\nLaboratoire d'Annecy-le-Vieux de Physique Théorique LAPTh\n\n",
"\nUniv. Grenoble Alpes\nUSMB\nCNRS\n4 Address: BP 110 Annecy-le-VieuxF-74000, F-74941Annecy, Annecy CedexFrance\n",
"L Frappat [email protected]@[email protected] ",
"D Issing ",
"E Ragoucy ",
"\nLaboratoire d'Annecy-le-Vieux de Physique Théorique LAPTh\n\n",
"\nUniv. Grenoble Alpes\nUSMB\nCNRS\n4 Address: BP 110 Annecy-le-VieuxF-74000, F-74941Annecy, Annecy CedexFrance\n"
] |
[
"Laboratoire d'Annecy-le-Vieux de Physique Théorique LAPTh\n",
"Univ. Grenoble Alpes\nUSMB\nCNRS\n4 Address: BP 110 Annecy-le-VieuxF-74000, F-74941Annecy, Annecy CedexFrance",
"Laboratoire d'Annecy-le-Vieux de Physique Théorique LAPTh\n",
"Univ. Grenoble Alpes\nUSMB\nCNRS\n4 Address: BP 110 Annecy-le-VieuxF-74000, F-74941Annecy, Annecy CedexFrance"
] |
[] |
We introduce the quantum determinant for the elliptic quantum algebra A q,p ( gl N ) and prove that it generates the center of this algebra. We also show that it is group-like for the quasi-Hopf structure, which allows us to define the elliptic quantum algebra A q,p ( sl N ).1 It is well known that certain algebraic structures are characterized by solutions of the Yang-Baxter equation (YBE) with spectral parameter: for example, the rational and trigonometric solutions lead to the Yangian and quantum affine algebras respectively. In this framework, called the FRT formalism [1], the generators of the algebra under consideration are encapsulated into the Lax matrix L(z), whose intertwining properties are coded in the Rmatrix (RLL relations). Within this approach, Foda et al.[2,3]were able to define the elliptic quantum algebra A q,p ( gl 2 ) using the elliptic solution of the YBE (R-matrix of the 8-vertex model) discovered by Baxter [4]. This solution exhibits entries expressed in terms of the Jacobi theta function and depends on a deformation parameter q and an elliptic nome p.When properly normalized, Baxter's elliptic R-matrix satisfies R 12 (q −1 ) = A 12 where A 12 is the antisymmetrizer on (C 2 ) ⊗2 . This property implies a quantum determinant formula, such a quantum determinant lying in the center of the algebra. This allows to define the elliptic algebra A q,p ( sl 2 ) by imposing further a restriction on the quantum determinant, namelywhere c is the central charge of the algebra. The generalization to the gl N case builds on the elliptic Z N -symmetric R-matrix derived by Belavin [5]. This matrix reduces to Baxter's elliptic R-matrix when N = 2. Along the same lines as above, an elliptic quantum algebra A q,p ( gl N ) was defined in [6] by using the elliptic Z N -symmetric R-matrix. However, an explicit formula for the quantum determinant is still missing in the case of A q,p ( gl N ) when N > 2. The main result of this paper is the derivation of the quantum determinant formula for A q,p ( gl N ) and the proof that it generates the center of the algebra. Hence formula (1.1) can be extended to generic N.Let us remark that this derivation has a slightly different final step, compared to the usual approach used in the quantum affine or Yangian cases[1,7,8]. Indeed, the usual approach uses the fact that the antisymmetrizer on (C N ) ⊗N can be expressed as some product of Rmatrices. In the elliptic case, this does not seem to be true when N > 2. It does not prevent to apply the antisymmetrizer to an N-fold product of Lax matrices and to construct an analog of the quantum determinant that is still central. However, it remains to compute the value of the quantum determinant in the fundamental evaluation representation. Without the expression of the antisymmetrizer in term of R-matrices, it requires some more work with respect to the usual case.The plan of the paper is as follows. In section 2, we recall the definition and basic properties of the elliptic quantum algebra A q,p ( gl N ). Section 3 is a reminder devoted to the quantum affine algebra U q ( gl N ). Here, we note in particular that the limit p → 0 of the elliptic R-matrix is a twisted version of the affine R-matrix in the principal gradation. The main result of the paper, the formula for the quantum determinant of A q,p ( gl N ), is stated in section 4. Proofs are gathered in section 5.2 The elliptic quantum algebra A q,p ( gl N ) 2.1 Definition of A q,p ( gl N ) Let us briefly review the construction of the elliptic quantum algebra A q,p ( gl N ). We consider a free, associative, unital algebra generated by operators L i,j [n] (1 ≤ i, j ≤ N,
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10.1088/1751-8121/aae296
|
[
"https://arxiv.org/pdf/1803.00311v3.pdf"
] | 53,706,798 |
1803.00311
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12f3b9e456dfc03090d4c026437a29d1ac3aa240
|
The quantum determinant of the elliptic algebra A q,p ( gl N )
9 Oct 2018 March 2018
L Frappat [email protected]@[email protected]
D Issing
E Ragoucy
Laboratoire d'Annecy-le-Vieux de Physique Théorique LAPTh
Univ. Grenoble Alpes
USMB
CNRS
4 Address: BP 110 Annecy-le-VieuxF-74000, F-74941Annecy, Annecy CedexFrance
The quantum determinant of the elliptic algebra A q,p ( gl N )
9 Oct 2018 March 2018
We introduce the quantum determinant for the elliptic quantum algebra A q,p ( gl N ) and prove that it generates the center of this algebra. We also show that it is group-like for the quasi-Hopf structure, which allows us to define the elliptic quantum algebra A q,p ( sl N ).1 It is well known that certain algebraic structures are characterized by solutions of the Yang-Baxter equation (YBE) with spectral parameter: for example, the rational and trigonometric solutions lead to the Yangian and quantum affine algebras respectively. In this framework, called the FRT formalism [1], the generators of the algebra under consideration are encapsulated into the Lax matrix L(z), whose intertwining properties are coded in the Rmatrix (RLL relations). Within this approach, Foda et al.[2,3]were able to define the elliptic quantum algebra A q,p ( gl 2 ) using the elliptic solution of the YBE (R-matrix of the 8-vertex model) discovered by Baxter [4]. This solution exhibits entries expressed in terms of the Jacobi theta function and depends on a deformation parameter q and an elliptic nome p.When properly normalized, Baxter's elliptic R-matrix satisfies R 12 (q −1 ) = A 12 where A 12 is the antisymmetrizer on (C 2 ) ⊗2 . This property implies a quantum determinant formula, such a quantum determinant lying in the center of the algebra. This allows to define the elliptic algebra A q,p ( sl 2 ) by imposing further a restriction on the quantum determinant, namelywhere c is the central charge of the algebra. The generalization to the gl N case builds on the elliptic Z N -symmetric R-matrix derived by Belavin [5]. This matrix reduces to Baxter's elliptic R-matrix when N = 2. Along the same lines as above, an elliptic quantum algebra A q,p ( gl N ) was defined in [6] by using the elliptic Z N -symmetric R-matrix. However, an explicit formula for the quantum determinant is still missing in the case of A q,p ( gl N ) when N > 2. The main result of this paper is the derivation of the quantum determinant formula for A q,p ( gl N ) and the proof that it generates the center of the algebra. Hence formula (1.1) can be extended to generic N.Let us remark that this derivation has a slightly different final step, compared to the usual approach used in the quantum affine or Yangian cases[1,7,8]. Indeed, the usual approach uses the fact that the antisymmetrizer on (C N ) ⊗N can be expressed as some product of Rmatrices. In the elliptic case, this does not seem to be true when N > 2. It does not prevent to apply the antisymmetrizer to an N-fold product of Lax matrices and to construct an analog of the quantum determinant that is still central. However, it remains to compute the value of the quantum determinant in the fundamental evaluation representation. Without the expression of the antisymmetrizer in term of R-matrices, it requires some more work with respect to the usual case.The plan of the paper is as follows. In section 2, we recall the definition and basic properties of the elliptic quantum algebra A q,p ( gl N ). Section 3 is a reminder devoted to the quantum affine algebra U q ( gl N ). Here, we note in particular that the limit p → 0 of the elliptic R-matrix is a twisted version of the affine R-matrix in the principal gradation. The main result of the paper, the formula for the quantum determinant of A q,p ( gl N ), is stated in section 4. Proofs are gathered in section 5.2 The elliptic quantum algebra A q,p ( gl N ) 2.1 Definition of A q,p ( gl N ) Let us briefly review the construction of the elliptic quantum algebra A q,p ( gl N ). We consider a free, associative, unital algebra generated by operators L i,j [n] (1 ≤ i, j ≤ N,
1 Introduction n ∈ Z), compactly represented by the formal series of the spectral parameter z ∈ C L i,j (z) = n∈Z L i,j [n] z −n (2.1) and encapsulated into the so-called Lax matrix
L(z) = N i,j=1 L i,j (z) e i,j ,
where the matrices e i,j are the usual elementary matrices. We adjoin an invertible central element written as q c (q is the deformation parameter and c the central charge). The A q,p ( gl N ) algebra is defined by imposing a set of exchange relations (commonly refereed to as RLL or FRT relations [1]) on these generators:
R 12 ( z w )L 1 (z)L 2 (w) = L 2 (w)L 1 (z) R * 12 ( z w ) . (2.2)
The indices refer to the spaces in which the Lax operators and the matrix R(z) operate, with L 1 (z) = L(z) ⊗ I and L 2 (z) = I ⊗ L(z). In addition to its spectral parameter dependence, the R-matrix R(z) depends on two complex parameters q (deformation parameter) and p (elliptic nome). To lighten presentation, unless needed, we will not write explicitly the dependence in q and p. The second R-matrix appearing in (2.2) is related to the original one through R * 12 (z) = R 12 (z) p→p * =pq −2c . The A q,p ( gl N ) algebra is a quasi-Hopf algebra with coproduct ∆L(z; p) = 1⊗L(zq c (1) /2 ; p) · L(zq −c (2) /2 ; pq −2c (2) )⊗1 , (2.3)
∆c = c ⊗ 1 + 1 ⊗ c ≡ c (1) + c (2) (2.4)
where we have specified the p dependence in L(z), due to shifts occurring when we apply ∆ (for more details see [9,10]). One can easily check that ∆ is an algebra morphism. In order to define explicitly the matrix R(z) appearing in here, we first introduce a slightly modified matrix R(z) = R b,d a,c (z) e a,b ⊗ e c,d , whose non-vanishing entries obey a + c = b + d, the addition of indices being understood modulo N. For 1 ≤ a, b, c, d ≤ N, one defines
R b,d a,c (z) = η(z)S b a,c (z) ω (a+c−b−d)/2 δ (mod N )
a+c,b+d ,
(2.5) where S b a,c (z) = z 2(b−a) N q 2(c−b) N p (b−a)(c−b) N Θ p N (p N +c−a q 2 z 2 ) Θ p N (p N +c−b z 2 )Θ p N (p N +b−a q 2 )
. (2.6) Remark that since ω = e 2iπ/N , the factor ω (a+c−b−d)/2 equals ±1. This factor results from the gauge transformation done on the Belavin R-matrix that allows one to recover the original A q,p ( gl 2 ) matrix. Note that in (2.6) one can consider any index c ∈ Z because of the identity S b a,c+N (z) = S b a,c (z). The normalization coefficient η(z) is given by
η(z) = z 2 N κ N (z 2 ) (p N , p N ) 3 ∞ (p, p) 3 ∞ Θ p (q 2 )Θ p (pz 2 ) Θ p (q 2 z 2 ) (2.7) with 1 κ N (z 2 ) = (q 2N z −2 ; p, q 2N ) ∞ (q 2 z 2 ; p, q 2N ) ∞ (pz −2 ; p, q 2N ) ∞ (pq 2N −2 z 2 ; p, q 2N ) ∞ (q 2N z 2 ; p, q 2N ) ∞ (q 2 z −2 ; p, q 2N ) ∞ (pz 2 ; p, q 2N ) ∞ (pq 2N −2 z −2 ; p, q 2N ) ∞ . (2.8)
The matrix S is Z N -symmetric by construction, i.e. the coefficients satisfy S b+n a+n,c+n (z) = S b a,c (z), n = 1, ..., N, where again the addition of indices is modulo N.
Here, Θ p (z) denotes the Jacobi theta function defined by (p ∈ C such that |p| < 1)
Θ p (z) = (z; p) ∞ (pz −1 ; p) ∞ (p; p) ∞ (2.9)
and the infinite q-Pochhammer symbols are given by
(z; p 1 , . . . , p m ) ∞ = n i ≥0 (1 − zp n 1 1 . . . p nm m ) . (2.10)
It is easy to show that the Jacobi theta function enjoys the following properties:
Θ a (az) = Θ a (z −1 ) and Θ a (a n z) = (−1) n z n a n(n−1)/2 Θ a (z) , ∀ n ∈ Z, (2.11)
Θ a 2 (az) = Θ a 2 (az −1 ). (2.12)
The matrices R(z) and R(z) differ only by a suitable normalization, which reads
R(z) = τ N (q 1 2 z −1 )R(z) , (2.13) where τ N (z) = z 2 N −2 Θ q 2N (qz 2 ) Θ q 2N (qz −2 ) .
(2.14)
Example: the A q,p ( gl 2 ) algebra
Specifying N = 2, the R-matrix (2.6) reads explicitly as: 2.15) and the entries take the following form:
R(z) = 1 κ 2 (z 2 ) (p 2 ; p 2 ) ∞ (p; p) 2 ∞ a(z) 0 0 d(z) 0 b(z) c(z) 0 0 c(z) b(z) 0 d(z) 0 0 a(z) (a(z) = z −1 Θ p 2 (pz 2 )Θ p 2 (pq 2 ) Θ p 2 (pq 2 z 2 ) , b(z) = qz −1 Θ p 2 (z 2 )Θ p 2 (pq 2 ) Θ p 2 (q 2 z 2 ) , c(z) = Θ p 2 (pz 2 )Θ p 2 (q 2 ) Θ p 2 (q 2 z 2 ) , d(z) = − p 1 2 qz 2 Θ p 2 (z 2 )Θ p 2 (q 2 ) Θ p 2 (pq 2 z 2 ) .
(2.16)
The Z 2 -symmetry of the R-matrix corresponds to a symmetry w.r.t. the diagonal and w.r.t. the anti-diagonal. It implies in particular that R(z) t 1 t 2 = R(z). This is no longer the case for higher rank algebras.
Remark 2.1 It has been shown [9,11] that the elliptic quantum algebras are quasi-Hopf algebras, obtained by a twisting procedure on the quantum affine algebra U q ( gl N ). The explicit expression of the Drinfel'd twist and the proof that this twist indeed satisfies the shifted cocycle condition were given in [9], see also [12]. In the case of A q,p ( gl 2 ), the evaluated R-matrix (2.15) was recovered through this procedure. Note that for N > 2, the explicit expression of the twist in the N-dimensional evaluation representation is, to our knowledge, still unknown.
Properties of the R-matrix
Let g and h be the matrices of order N defined by g ij = ω i δ ij and h ij = δ i+1,j for 1 ≤ i, j ≤ N with ω = e 2iπ/N , the addition of indices being understood modulo N.
We recall the following properties of the R-matrix R(z) [13,14]:
• Yang-Baxter equation (also holds for R(z)):
R 12 ( z 1 z 2 ) R 13 ( z 1 z 3 ) R 23 ( z 2 z 3 ) = R 23 ( z 2 z 3 ) R 13 ( z 1 z 3 ) R 12 ( z 1 z 2 ) .
(2.17)
• Unitarity:
R 12 (z) R 21 (z −1 ) = I ,(2.
18)
• Regularity (P 12 is the permutation matrix):
R 12 (1) = P 12 ,(2.
19)
• Crossing-symmetry:
R 12 (z) t 2 R 21 (z −1 q −N ) t 2 = I ,(2.
20)
• Antisymmetry:
R 12 (−z) = ω (g −1 ⊗ I) R 12 (z) (g ⊗ I) ,(2.
21)
• Quasi-periodicity:
R 12 (−zp 1 2 ) = (g 1 2 hg 1 2 ⊗ I) −1 R 21 (z −1 ) −1 (g 1 2 hg 1 2 ⊗ I) , (2.22) • Invariance: (h ⊗ h) R 12 (z) = R 12 (z) (h ⊗ h) .
(2.23)
Remark 2.2
The crossing-symmetry and the unitarity properties of R 12 allow to exchange the inversion and the transposition when applied to the matrix R 12 (or to the matrix R 12 ). It provides a crossing-unitarity relation (also valid for R thanks to the q N -periodicity of the function τ N ):
R 12 (x) t 2 −1 = R 12 (q N x) −1 t 2 .
(2.24)
Note also that the unitarity property for R 12 reads
R 12 (z) R 21 (z −1 ) = τ N (q 1 2 z) τ N (q 1 2 z −1 ) ≡ U(z),(2.25)
where the function U(z) is defined as
U(z) = q 2 N −2 Θ q 2N (q 2 z 2 ) Θ q 2N (q 2 z −2 ) Θ q 2N (z 2 )Θ q 2N (z −2 ) . (2.26) 3
The quantum affine algebra U q ( gl N )
We recall in this section the different gradations that may be used in constructing the R-matrix defining the algebra U q ( gl N ) in the FRT formalism. As will be shown below, the R-matrix obtained as the non-elliptic limit of (2.13) appears to be a twisted version of the R-matrix in the principal gradation.
Homogeneous gradation.
In the FRT formalism, the quantum affine algebra U q ( gl N ) is described as an associative algebra defined by generators and relations. The generators L ± i,j [∓n], where n ∈ Z ≥0 , 1 ≤ i, j ≤ N and L + i,j [0] = L − j,i [0] = 0 for i > j, are coded in formal generating functions L ± i,j (z), themselves encapsulated into the Lax matrices L ± (z):
L ± (z) = N i,j=1 L ± i,j (z) e i,j and L ± i,j (z) = ∞ n=0 L ± i,j [∓n] z ±n . (3.1)
The relations are the well-known RLL relations
R 12 ( z ± w ± )L ± 1 (z)L ± 2 (w) = L ± 2 (w)L ± 1 (z)R 12 ( z ± w ± ) , (3.2) R 12 ( z ± w ∓ )L + 1 (z)L − 2 (w) = L − 2 (w)L + 1 (z)R 12 ( z ∓ w ± ) ,(3.3)
where z ± = zq ±c/2 , w ± = wq ±c/2 , c is the central charge and the matrix R 12 (z) is given by
R 12 (z) = ρ N (z) i e i,i ⊗ e i,i + q(1 − z) 1 − q 2 z i =j e i,i ⊗ e j,j + (1 − q 2 ) 1 − q 2 z i<j +z i>j e i,j ⊗ e j,i . (3.4)
The normalization factor ρ N (z) expressed as
ρ N (z) = q 1 N −1 (q 2 z; q 2N ) ∞ (q 2N −2 z; q 2N ) ∞ (z; q 2N ) ∞ (q 2N z; q 2N ) ∞ . (3.5) Let e i , f i (0 ≤ i ≤ N − 1) and h i (0 ≤ i ≤ N) denote the generators of U q ( gl N )
in the Serre-Chevalley basis and let R be the universal R-matrix of U q ( gl N ) (see e.g. [15]). The
R-matrix (3.4) is obtained from R by calculating its image R(z/w) = (π z ⊗ π w )R in the N-dimensional evaluation representation π z such that (1 ≤ i ≤ N) π z (e i ) = e i,i+1 , π z (f i ) = e i+1,i , π z (h i ) = e i,i , (3.6) π z (e 0 ) = ze N,1 , π z (f 0 ) = z −1 e 1,N , π z (h 0 ) = e N,N − e 1,1 . (3.7)
This defines the so-called homogeneous gradation. The quantum affine algebra U q ( gl N ) is endowed with the following coproduct structure:
∆ L ± i,j (z) = N k=1 L ± k,j (zq ∓c (2) /2 ) ⊗ L ± i,k (zq ±c (1) /2 ) , (3.8) where c (1) = c ⊗ 1 and c (2) = 1 ⊗ c.
The quantum determinant is given in the homogeneous gradation by
qdet L(z) = σ∈S N sgn(σ) q ℓ(σ) L + 1,σ(1) (z) L + 2,σ(2) (zq −2 ) . . . L + N,σ(N ) (zq 2−2N ) ,(3.9)
where ℓ(σ) denotes the length of the permutation σ and sgn(σ) = (−1) ℓ(σ) . Finally, thanks to the RLL relations, the action of the finite Cartan generators on the Lax matrices is given by (
1 ≤ i, j, k ≤ N) q h i L ± j,k (w) = L ± j,k (w) q h i +δ ik −δ ij . (3.10)
Principal gradation
Another possible choice is the principal gradation. In that case, the evaluation map π z is given by
π z (e i ) = z 2/N e i,i+1 , π z (f i ) = z −2/N e i+1,i , π z (h i ) = e i,i , (3.11) π z (e 0 ) = z 2/N e N,1 , π z (f 0 ) = z −2/N e 1,N , π z (h 0 ) = e N,N − e 1,1 . (3.12)
The R-matrix in the principal gradation reads:
R(z) = ρ N (z 2 ) i e i,i ⊗ e i,i + q(1 − z 2 ) 1 − q 2 z 2 e i,i ⊗ e j,j + z(1 − q 2 ) 1 − q 2 z 2 i<j z (2j−2i−N )/N + i>j z (2j−2i+N )/N e i,j ⊗ e j,i . (3.13)
The two matrices (3.4) and (3.13) are related by a gauge transformation
R(z/w) = V (z) ⊗ V (w) R(z 2 /w 2 ) (V (z) ⊗ V (w)) −1 (3.14) with V (z) = N i=1 z (N +1−2i)/N e i,i .
It follows that the Lax matrices L ± (z) and L ± (z) that define the quantum affine algebra in the homogeneous and principal gradations respectively are related by
L + (z) = V (zq c/2 ) L + (z 2 ) V (zq −c/2 ) −1 , (3.15) L − (z) = V (z) L − (z 2 ) V (z) −1 . (3.16)
Note that these relations ensure that equation (3.10) also holds for the Lax matrices L ± (z) in the principal gradation. The quantum determinant is then given in the principal gradation by
qdet L(z) = σ∈S N sgn(σ) q ℓ(σ)+ 2 N N i=1 i(σ(i)−i) L + 1,σ(1) (z) L + 2,σ(2) (zq −1 ) . . . L + N,σ(N ) (zq 1−N ) . (3.17)
Non-elliptic presentation
The limit p → 0 of the A q,p ( gl N ) algebra allows us to reveal still another presentation of the quantum affine algebra U q ( gl N ). Since this presentation is related to the non-elliptic limit of A q,p ( gl N ), we will call it the non-elliptic presentation. The R-matrix obtained in this limit reads:
R ′ (z) = ρ N (z 2 ) i e i,i ⊗ e i,i + q(1 − z 2 ) 1 − q 2 z 2 i<j q (2j−2i−N )/N + i>j q (2j−2i+N )/N e i,i ⊗ e j,j + z(1 − q 2 ) 1 − q 2 z 2 i<j z (2j−2i−N )/N + i>j z (2j−2i+N )/N e i,j ⊗ e j,i . (3.18)
When N > 2, this matrix differs from the previous one, essentially by some powers of q in the diagonal terms. Obviously, it is still Z N -symmetric. It can be obtained from (3.13) by a (constant non factorized) diagonal twist:
R ′ (z) = F 21 R(z) F −1 12 (3.19) where F 12 = N i=1 e i,i ⊗ e i,i + 1≤i =j≤N q α ij e i,i ⊗ e j,j (3.20)
with, for i < j, α ij = 1 2 + (i − j)/N and α ji = −α ij . We set by convention α ii = 0 for all i. Remark that for N = 2, α 12 = 0, so that the twist is I ⊗ I. The algebra is still defined by Eqs. (3.2)-(3.1) where the Lax matrices L ± (z) are now replaced by L ′± (z). At the universal level, the twisted R-matrix is given by
R F = F 21 R F −1 12 (3.21) with F 12 = q ij α ij h i ⊗h j . (3.22)
Here h i (i = 1, . . . , N) are the Cartan generators of the finite quantum algebra U q (gl N ) satisfying the following commutation relations (j = 1, . . . , N − 1):
[h i , e j ] = (δ ij − δ i,j+1 )e j , [h i , f j ] = −(δ ij − δ i,j+1 )f j , [e j , f j ] = q h j −h j+1 − q h j+1 −h j q − q −1 . (3.23)
The universal twist (3.22) satisfies the cocycle condition F 12 (∆ ⊗ id)F = F 23 (id ⊗ ∆)F , ensuring that the universal R-matrix R F satisfies the Yang-Baxter equation while the Rmatrix R does. The relation between the corresponding Lax matrices L ± and L ′± can be expressed as
L ′± (z) = (π z ⊗ id)F 21 L ± (z) (π z ⊗ id)F −1 12 . (3.24)
In the evaluation representation π z , one gets
(π z ⊗ id)F 12 = (π z ⊗ id)F −1 21 = N i=1 q N j=1 α ij h j e i,i . (3.25)
The twist being diagonal and depending only on the finite Cartan generators, the equation (3.10) also holds for the Lax matrices L ′± (z). The coproduct of the twisted algebra is given by ∆ F = F 12 ∆ F −1 12 . A direct calculation shows that
∆ F L ′± i,j (z) = N k=1 L ′± k,j (zq ∓c (2) /2 ) ⊗ L ′± i,k (zq ±c (1) /2 ) (3.26)
from which it follows that the twisted algebra gets the same coproduct structure as the original algebra.
Applying the twist to the expression (3.17), and using the correspondence (3.24), we get an expression for the quantum determinant in this new presentation. It is again expressed as a sum over permutations:
qdet L(z) = σ∈S N sgn(σ) q nσ L ′+ 1,σ(1) (z) L ′+ 2,σ(2) (zq −1 ) . . . L ′+ N,σ(N ) (zq 1−N ) , (3.27) where n σ = ℓ(σ) + 2 N N i=1 i(σ(i) − i) + 1≤i<j≤N (α σ(i),σ(j) − α ij )
. A detailed analysis of n σ , using the explicit expression of the coefficients α ij , shows that it vanishes identically. Then, the quantum determinant is given in the non-elliptic limit by
qdet L ′ (z) = σ∈S N sgn(σ) L ′+ 1,σ(1) (z) L ′+ 2,σ(2) (zq −1 ) . . . L ′+ N,σ(N ) (zq 1−N ) . (3.28)
Let us remark that the relation (3.28) is based on the (undeformed) antisymmetrizer, contrarily to the expressions found for the homogeneous and principal gradations that are based on q-deformed versions of it. When N = 2, R(z) and R ′ (z) coincide, and only the homogeneous gradation provides a deformed antisymmetrizer.
Main results
We are now in position to state the main result of this article. The next section is devoted to the proofs of this theorem. By generic values of the parameters p, q and of the central charge c, we mean that there is no functional relation among them, such as the ones used in [6,16] to define deformations of W N algebras, and that they do not obey any algebraic relation, such as c = −N or q being a root of unity, where it is known that the center is extended, see [17] for the former case and [18,19] for the latter.
Remark that for N = 2, it was already proven in [3] that the quantum determinant is central. To the best of our knowledge, the case N > 2 was not studied yet.
Taking the limit p → 0, we recover the formula (3.28), as expected.
Corollary 4.2 The quantum determinant is group-like:
∆qdetL(z) = qdetL(zq −c (2) /2 ; pq −2c (2) ) ⊗ qdetL(zq c (1) /2 ; p),
4)
where we have specified the p-dependence, as in (2.3).
Proof: We apply the coproduct (2.3) to the expression (4.1). For brevity and for this calculation, we note L(z) ≡ L(z; p) and L * (z) ≡ L(z; pq −2c (2) ). We get ∆qdetL(z) A (N )
N = ∆L 1 (z) ∆L 2 (zq −1 ) . . . ∆L N (zq 1−N ) A (N ) N = 1⊗L 1 (zq c (1) /2 ) L * 1 (zq −c (2) /2 )⊗1 . . . 1⊗L N (zq 1−N q c (1) /2 ) L * N (zq 1−N q −c (2) /2 )⊗1 A (N ) N = 1⊗L 1 (zq −c (1) /2 ) . . . L N (zq 1−N q −c (1) /2 ) L * 1 (zq c (2) /2 ) . . . L * N (zq 1−N q c (2) /2 )⊗1 A (N ) N = 1⊗L 1 (zq −c (1) /2 )L 2 (zq −1 q −c (1) /2 ) . . . L N (zq 1−N q −c (1) /2 )A (N ) N qdetL * (zq c (2) /2 )⊗1 = 1⊗qdetL(zq −c (1) /2 ) A (N ) N qdetL * (zq c (2) /2 )⊗1 = qdetL * (zq −c (2) /2 )⊗ qdetL(zq c (1) /2 ) A (N ) N
where in the last step we have used that the quantum determinant is central.
Theorem 4.1 and corollary 4.2 allow us to introduce the elliptic quantum algebra associated to sl N :
Definition 4.3 The elliptic quantum algebra A q,p ( sl N ) is the quasi-Hopf algebra defined by the coset A q,p ( sl N ) = A q,p ( gl N )/ qdet L(z) − q c/2 . (4.5)
Proof: Since the quantum determinant and c are central, the coset defines an algebra. Moreover, qdet L(z) and q c/2 are both group-like, so that the coset is a quasi-Hopf algebra.
Proofs
The proof of the main theorem relies on different lemmas and properties. Proof: It has been shown in [14] that R 12 (q)(1−P 12 ) = 0, which implies ker
Preliminary lemmas
R 12 (q) ⊃ im A (N ) 2 .
Then, it remains to show that these two spaces have same dimension. Since the entries of the R-matrix of A q,p ( gl N ) are products of (fractional) power of p and analytical functions of p, it is sufficient to consider the non-elliptic limit of the matrix, i.e. p → 0, leading to the R-matrix (3.18) of U q ( gl N ).
Denoting λ ′ = ρ N (q 2 ) −1 λ and Q = q 1+q 2 , one immediately gets that
det(R ′ (q) − λ Id) ρ N (q 2 ) N 2 = (1 − λ ′ ) N 1≤i<j≤N (Qq (2i−2j+N )/N − λ ′ )(Qq −(2i−2j+N )/N − λ ′ ) − Q 2 = (λ ′ ) N(N−1) 2 (1 − λ ′ ) N 1≤i<j≤N λ ′ − Q(q (2i−2j+N )/N + q −(2i−2j+N )/N ) (5.1)
from which it follows that for generic values of q, the eigenvalue 0 has multiplicity 1 2 N(N −1). It shows that ker R 12 (q) and im A (N ) 2 have same dimension, hence the result.
Lemma 5.2
In the A q,p ( gl N ) algebra, the following identity holds
L i,j (z)L k,l ( z q ) − L i,l (z)L k,j ( z q ) = L k,l (z)L i,j ( z q ) − L k,j (z)L i,l ( z q ) ∀ i, j, k, l = 1, . . . , N. (5.2)
In particular, we have
L i,j (z)L i,l ( z q ) = L i,l (z)L i,j ( z q ) ∀ i, j, l = 1, . . . , N. (5.3)
Proof: We consider the RLL relations (2.2) for w = z/q and project them onto an arbitrary element e i,j ⊗ e k,l . This leads to the following equation, valid for all i, j, k, l = 1, . . . , N:
N n,m=1 R n,m i,k (q)L n,j (z)L m,l ( z q ) = N n,m=1 R * j,l m,n (q)L k,n ( z q )L i,m (z), (5.4)
We will refer to this equation as X j,l i,k (z). Note that the lemma 5.1 implies the relations
R j,l i,k (q) = R l,j i,k (q) . (5.5)
Hence, looking at the difference X j,l i,k (z) − X l,j i,k (z), the R.H.S. is equal to Note that the indices j and l do not play any role in these relations, so if we can solve (5.7) for one pair j, l, we can do it for any. We thus consider the equations for fixed indices j and l, and omit them to ease the notation. Denoting
T n,m (z) ≡ T j,l n,m (z) = L n,j (z)L m,l ( z q ) − L n,l (z)L m,j ( z q ) + (n ↔ m),(5.8)
one gets for fixed i, j, k, l, using again the property (5.5), Since T (z) is in the symmetric part of C N ⊗ C N , lemma 5.1 implies that the only solution of the linear system (5.9) is T n,m (z) = 0, which is relation (5.2).
Explicit expression of the quantum determinant
The antisymmetrizer A (N ) N in (C N ) ⊗N is a rank 1 projector, the eigenvector corresponding to the eigenvalue 1 being given by w = σ∈S N sgn(σ)e σ(1) ⊗ · · · ⊗ e σ(N ) , (5.10) where {e i } denotes the standard vector basis of C N . A
(N )
N projects any given vector v on w:
A N v = w, v w ∀v ∈ (C N ) ⊗N ,(5.11)
where w, v = σ∈S N sgn(σ)v σ(1)...σ(N ) . Due to equality (5.11), to get an expression for the quantum determinant, it is enough to compute
L 1 (z) . . . L N (zq 1−N )w = N i 1 ,...,i N =1 σ∈S N sgn(σ)L i 1 ,σ(1) (z) . . . L i N ,σ(N ) (q 1−N z)(e i 1 ⊗ · · · ⊗ e i N ). (5.12)
We first prove that all the indices i 1 , . . . , i N in (5.12) must be different. For such a purpose, we prove that terms with identical indices vanish. The proof is done by recursion on the 'distance' between two identical indices.
Consider the terms with i k = i k+1 . Without loss of generality, we can check what happens for k = N −1, the reasoning naturally translates to all other possible pairs of adjacent indices. Focusing on the coefficient of e i 1 ⊗ · · · ⊗ e i N−2 ⊗ e i N ⊗ e i N only, we write (all indices arbitrary, but fixed)
σ∈S N sgn(σ)L i 1 ,σ(1) (z) . . . L i N ,σ(N −1) (q 2−N z)L i N ,σ(N ) (q 1−N z) = = σ ′ ∈S N sgn(σ ′ • s N,N −1 )L i 1 ,σ ′ (1) (z) . . . L i N ,σ ′ (N ) (q 2−N z)L i N ,σ ′ (N −1) (q 1−N z) = 1 2 σ ′ ∈S N sgn(σ ′ )L i 1 ,σ ′ (1) (z) . . . L i N−2 ,σ ′ (N −2) (q 3−N z)× × L i N ,σ ′ (N −1) (q 2−N z)L i N ,σ ′ (N ) (q 1−N z) − L i N ,σ ′ (N ) (q 2−N z)L i N ,σ ′ (N −1) (q 1−N z) = 0 (5.13)
where the last equality is done by virtue of (5.3). Suppose now that the terms where i k = i k+n with 1 ≤ n ≤ m have zero contribution and consider the term where i k = i k+m+1 :
σ∈S N sgn(σ)L i 1 ,σ(1) (z) . . . L i k ,σ(k) (q 1−k z) . . . L i k ,σ(k+m+1) (q −m−k z) . . . L i N ,σ(N ) (q 1−N z) = 1 2 σ ′ ∈S N sgn(σ ′ )L i 1 ,σ ′ (1) (z) . . . L i k−1 ,σ ′ (k−1) (q 2−k z) × L i k ,σ ′ (k) (q 1−k z)L i k+1 ,σ ′ (k+1) (q −k z) − L i k ,σ ′ (k+1) (q 1−k z)L i k+1 ,σ ′ (k) (q −k z) × L i k+2 ,σ ′ (k+2) (q 3−k z) . . . L i k ,σ ′ (k+m+1) (q −m−k z) . . . L i N ,σ ′ (N ) (q 1−N z) (5.14) = − 1 2 σ ′ ∈S N sgn(σ ′ )L i 1 ,σ ′ (1) (z) . . . L i k−1 ,σ ′ (k−1) (q 2−k z) × L i k+1 ,σ ′ (k) (q 1−k z)L i k ,σ ′ (k+1) (q −k z) − L i k+1 ,σ ′ (k+1) (q 1−k z)L i k ,σ ′ (k) (q −k z) × L i k+2 ,σ ′ (k+2) (q 3−k z) . . . L i k ,σ ′ (k+m+1) (q −m−k z) . . . L i N ,σ ′ (N ) (q 1−N z) (5.15) = − σ∈S N sgn(σ)L i ′ 1 ,σ(1) (z) . . . L i ′ k ,σ(k) (q 1−k z) . . . L i ′ k ,σ(k+m) (q −m−k z) . . . L i ′ N ,σ(N ) (q 1−N z) .
(5.16)
To get (5.14), we have used the same trick as in the calculation of (5.13). To go from (5.14) to (5.15) we have used the relation (5.2). In the last equality, we have introduced the indices
i ′ ℓ = i ℓ for ℓ / ∈ {k, k + 1} and i ′ k = i k+1 , i ′ k+1 = i k .
This last expression vanishes due to the recursion hypothesis.
Since all indices i r are different, we can replace the sum on i 1 , . . . , i N by a sum over permutations µ ∈ S N . We pick one such permutation and examine the coefficient of e µ(1) ⊗ · · · ⊗ e µ(N ) : But we can also look at a different permutation µ ′ = µ • s k . In this case, we find that
χ µ :=χ (µ•s k ) = 1 2 σ∈S N sgn(σ)L µ(1),σ(1) . . . L µ(k−1),σ(k−1) × L µ(k+1),σ(k) L µ(k),σ(k+1) − L µ(k+1),σ(k+1) L µ(k),σ(k) L µ(k+2),σ(k+2) . . . L µ(N ),σ(N ) . (5.19)
Once more, condition (5.2) shows that χ (µ•s k ) = −χ µ . This allows us to conclude that in fact, for any σ, µ ∈ S N , we have χ µ = sgn(σ)χ (µ•σ) . In particular, χ µ = sgn(µ)χ id , and we finally arrive at the following result:
L 1 (z) . . . L N (zq 1−N )w = 1 N! µ∈S N χ µ e µ(1) ⊗ · · · ⊗ e µ(N ) (5.20) = 1 N! µ∈S N sgn(µ) χ id e µ(1) ⊗ · · · ⊗ e µ(N ) = χ id w . (5.21)
From this, we directly infer that the quantum determinant is χ id , which proves the equality (4.3).
Remark that in this way we have proved that where π = π 1 ⊗ π 2 ⊗ ... ⊗ π N and M 0 (z) is a matrix M(z) (yet to be determined) acting on the space 0 only. One can show from the evaluation of the relation (4.3) that M(z) is a diagonal matrix:
L 1 (z) . . . L N (zq 1−N ) A (N ) 1...N = M(z) A (N ) 1...N ,(5.M(z) = N −1 j=0 η z q j N k=1 q 2k−N −1 σ∈SN sgn(σ) S σ(1) 1,k (z) . . . S σ(N ) N,k+ N−1 i=1 (i−σ(i)) z q N −1 e k,k ≡ N k=1 m k (z) e k,k ,(5.25)
where we set, for any indices a, b, c,
S b a,c (z) := Θ p N (p N +c−a q 2 z 2 ) Θ p N (p N +c−b z 2 )Θ p N (p N +b−a q 2 )
.
(5.26)
Using the expression of η(z), we get
m k (z) = − (p N ; p N ) ∞ (p; p) ∞ 3N q 2k−2N Θ p (q 2 ) N Θ p (z 2 ) Θ p (q 2 z 2 ) σ∈SN sgn(σ) N ℓ=1 S σ(ℓ) ℓ,k+ ℓ−1 i=1 (i−σ(i)) ( z q ℓ−1 ) .
(5.27) Now, from the invariance property (2.23) of the R-matrix, it is easy to show that Using property (2.11), it is easy to show that m(z; qp N/2 , p) = m(z; q, p), where we have indicated explicitly the dependence in the parameters q and p. In other words, since |p| < 1, we have m(z; q, p) = m(z; qp ℓN/2 , p) = lim ℓ→∞ m(z; qp ℓN/2 , p) = lim q ′ →0 m(z; q ′ , p), ∀ p, q. (5.29) This shows that m(z; q, p) does not depend on q. To compute it, we take the limit q → 1. To show that this limit is well-defined, we computed explicitly relation (5.23) in the limit p → 0 and generic values of q and N (it corresponds to the non-elliptic presentation of U q ( gl N ), see section 3.3). We got m(z; q, p)| p=0 = 1. Since p, q and c are generic, it shows that the limit (5.29) exists at least in a neighborhood of p = 0. Due to the term Θ p (q 2 ) N in (5.27), which vanishes in the limit q → 1, one sees that only the term σ = id contributes to m(z; q, p).
h 0 h 1 · · · h N π qdetL(z) A (N ) 1...N = π qdetL(z) h 0 h 1 · · · h N A (N ) 1...N = π qdetL(z) A (N ) 1...N h 0 h 1 · · · h N .
Then, a direct calculation shows that m(z; 1, p) = 1 for generic values of p and N.
We checked relation (5.23) for N = 2, 3 and generic values of p and q.
Centrality of the quantum determinant
We wish to show that, for z, w ∈ C
[qdetL(z), L 0 (w)] = 0 . where we used that L 0 (w) and the antisymmetrizer commute as they live in different spaces, applied the inverse of (5.23), and finally traced over the antisymmetrizer. As a consequence, the quantum determinant lies in the center of the algebra A q,p ( gl N ) as desired.
Theorem 4. 1
1Let A (N ) N be the antisymmetrizer of N spaces C N . For generic values of the parameters p, q and of the central charge c, one has the following identity L 1 (z) . . . L N (zq 1−N ) A qdetL(z), called the quantum determinant, is a scalar function that lies in the center of the A q,p ( gl N ) algebra. It can be rewritten as qdetL(z) = tr 1...N L 1 (z) . . . L N (zq 1−N )A . . . L N,σ(N ) (zq 1−N ) , (4.3)where S N is the set of permutations of N objects. Conversely, for generic values of the parameters p, q and of the central charge c, the quantum determinant generates the center of the A q,p ( gl N ) algebra.
words, X j,l i,k (z) − X l,j i,k (z) does not depend on p * . Finally, the L.H.S.
N sgn(σ)L µ(1),σ(1) . . . L µ(k−1),σ(k−1) × L µ(k),σ(k) L µ(k+1),σ(k+1) − L µ(k),σ(k+1) L µ(k+1),σ(k) L µ(k+2),σ(k+2) . . . L µ(N ),σ(N ) .(5.18)
22)where M(z) is scalar in the spaces 1,...,N and given by(4.3). It remains to prove that M(z) is central in A q,p ( gl N ): it is done in section 5.4.5.3 Value of q-detL(u) in the fundamental representationLemma 5.3 The R-matrix (2.6) for the A q,p ( gl N ) algebra obeys the following relation R 10 (z) . . . R N 0 zq 1−N A We apply the evaluation maps π j : L j (z) → R j0 (z), j = 1, ..., N to the equality (5.22): R 10 (z) . . . R N 0 zq 1−N A
the expression (5.24), it implies [h 0 , M 0 (z)] = 0 which can be recasted as m k (z) = m k+1 (z), that is to say M(z) = m(z) I.
be achieved by commuting L 0 (w) through the expression (4.2) for the quantum determinant: qdetL(z) L 0 (w) = tr 1...N L 1 (z) . . . L N (zq 1−N )L 0 (w)A (N ) 1...N = tr 1...N L 1 (z) . . . L N −1 (zq 2−N ) (w)L 1 (z) . . . L N (zq 1−N ) (w)L 1 (z) . . . L N (zq 1−N ) A used the RLL relations (2.2) and the fact that generators acting in different subspaces commute. The last equalities are due to lemma 5.3 and definition 4.1. Next, using the fact that the quantum determinant is a scalar in the spaces 0, 1, . . . , N, we get qdetL(z) L 0 (w) = tr 1...N R
be the antisymmetrizer on (C N ) 2 . Then for generic values of q, ker R 12 (q) = im ALemma 5.1 Let A
(N )
2
(N )
2 .
Center of the algebra A q,p ( gl N ). Center of the algebra A q,p ( gl N )
It is known that in U q ( gl N ) and for generic values of q and c, the quantum determinant. as defined in (3.9), generates the center of this quantum algebra [1It is known that in U q ( gl N ) and for generic values of q and c, the quantum determinant, as defined in (3.9), generates the center of this quantum algebra [1].
N ) is isomorphic to U q ( gl N ) as an algebra [9] and for generic values of p, q and c (in the sense explained in section 4), (3.9) also describes the full center of the algebra A q,p ( gl N ). The same is true for the other two expressions (3.17) and (3.28), that are just the same quantum determinant in different presentations. Moreover, we have shown that expression. Since, ), and q,p ( gl N ) to a generating set. defined by (3.28)) of the same center. Thus, it implies that (4.3) also defines a generating set of the center of A q,p ( gl N ) for genericSince, being a twist of it, A q,p ( gl N ) is isomorphic to U q ( gl N ) as an algebra [9] and for generic values of p, q and c (in the sense explained in section 4), (3.9) also describes the full center of the algebra A q,p ( gl N ). The same is true for the other two expressions (3.17) and (3.28), that are just the same quantum determinant in different presentations. Moreover, we have shown that expression (4.3) also lies in the center of A q,p ( gl N ), and q,p ( gl N ) to a generating set (defined by (3.28)) of the same center. Thus, it implies that (4.3) also defines a generating set of the center of A q,p ( gl N ) for generic values of p, q and c.
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|
[] |
[
"Pancreas Segmentation in Abdominal CT Scan: A Coarse-to-Fine Approach",
"Pancreas Segmentation in Abdominal CT Scan: A Coarse-to-Fine Approach"
] |
[
"Yuyin Zhou ",
"Lingxi Xie ",
"Wei Shen \nSchool of Communications and Information Engineering\nShanghai University\n\n",
"Elliot Fishman ",
"Alan Yuille \nCenter for Imaging Science\nJohns Hopkins University\n\n",
"\n1,2\n"
] |
[
"School of Communications and Information Engineering\nShanghai University\n",
"Center for Imaging Science\nJohns Hopkins University\n",
"1,2"
] |
[] |
Deep neural networks have been widely adopted for automatic organ segmentation from CT-scanned images. However, the segmentation accuracy on some small organs (e.g., the pancreas) is sometimes below satisfaction, arguably because deep networks are easily distracted by the complex and variable background region which occupies a large fraction of the input volume. In this paper, we propose a coarse-to-fine approach to deal with this problem. We train two deep neural networks using different regions of the input volume. The first one, the coarse-scaled model, takes the entire volume as its input. It is used for roughly locating the spatial position of the pancreas. The second one, the fine-scaled model, only sees a small input region covering the pancreas, thus eliminating the background noise and providing more accurate segmentation especially around the boundary areas. At the testing stage, we first use the coarse-scaled model to roughly locate the pancreas, then adopt the fine-scaled model to refine the initial segmentation in an iterative manner to obtain increasingly better segmentation. We evaluate our algorithm on the NIH pancreas segmentation dataset with 82 volumes, and outperform the state-of-theart [18] by more than 4%, measured by the Dice-Sørensen Coefficient (DSC). In addition, we report 62.43% DSC for our worst case, significantly better than 34.11% reported in[18].
| null |
[
"https://arxiv.org/pdf/1612.08230v1.pdf"
] | 2,953,632 |
1612.08230
|
788f341d02130e1807edf88c8c64a77e4096437e
|
Pancreas Segmentation in Abdominal CT Scan: A Coarse-to-Fine Approach
Yuyin Zhou
Lingxi Xie
Wei Shen
School of Communications and Information Engineering
Shanghai University
Elliot Fishman
Alan Yuille
Center for Imaging Science
Johns Hopkins University
1,2
Pancreas Segmentation in Abdominal CT Scan: A Coarse-to-Fine Approach
4 Radiology and Radiological Science, the Johns Hopkins Medical Institutions 1Pancreas Segmentation, A Coarse-to-Fine Approach
Deep neural networks have been widely adopted for automatic organ segmentation from CT-scanned images. However, the segmentation accuracy on some small organs (e.g., the pancreas) is sometimes below satisfaction, arguably because deep networks are easily distracted by the complex and variable background region which occupies a large fraction of the input volume. In this paper, we propose a coarse-to-fine approach to deal with this problem. We train two deep neural networks using different regions of the input volume. The first one, the coarse-scaled model, takes the entire volume as its input. It is used for roughly locating the spatial position of the pancreas. The second one, the fine-scaled model, only sees a small input region covering the pancreas, thus eliminating the background noise and providing more accurate segmentation especially around the boundary areas. At the testing stage, we first use the coarse-scaled model to roughly locate the pancreas, then adopt the fine-scaled model to refine the initial segmentation in an iterative manner to obtain increasingly better segmentation. We evaluate our algorithm on the NIH pancreas segmentation dataset with 82 volumes, and outperform the state-of-theart [18] by more than 4%, measured by the Dice-Sørensen Coefficient (DSC). In addition, we report 62.43% DSC for our worst case, significantly better than 34.11% reported in[18].
Introduction
In recent years, due to the fast development of deep neural networks, we have witnessed rapid progress in both medical image analysis and computer-aided diagnosis. This paper focuses on a specific task in this research field, namely to automatically segment small organs (e.g., the pancreas) from CT-scanned images. This is a difficult task, as the pancreas often suffers high variety in shape, size and location [19], while occupying only a very small fraction (e.g., < 0.5%) of the entire CT volume. In such cases, deep neural networks can be distracted by the background region, which occupies a large fraction of the input volume and may include complex and variable contents. Consequently, the segmentation result becomes less accurate especially around the boundary areas.
To deal with this problem, we propose a coarse-to-fine approach for pancreas segmentation. Starting with a baseline approach, i.e., a coarse-scaled model which takes the entire volume as the input, we train an additional fine-scaled model on a small region covering the pancreas, so as to alleviate the background noise. We find that the latter model works much better, especially in detecting details on the boundary areas. At the testing stage, we first use the coarse-scaled model to roughly detect the pancreas, and then apply the fine-scaled model to refine the initial segmentation in an iterative manner. On a modern GPU, we need around 3 minutes to process a CT volume during the testing stage. This is comparable to [18], but we report higher accuracy.
We train and evaluate our algorithm on the NIH pancreas segmentation dataset [19]. Following [19], we partition the 82 samples into 4 folds and evaluate by cross-validation. Compared to recently published work [18], our average segmentation accuracy, measured by the Dice-Sørensen Coefficient (DSC), increases from 78.01% to 82.37%. Meanwhile, we report 62.43% DSC on the worst case, which guarantees reasonable performance on the particularly challenging test samples. In comparison, [18] reports 34.11% DSC on the worse case and [19] reports 23.99%. Some previous methods [4] [26] report even lower numbers. We point out that, although our algorithm is only tested on a pancreas dataset, the approach is directly generalizable to other organ segmentation tasks, especially for those small organs such as the spleen or the duodenum.
The remainder of this paper is organized as follows. Section 2 briefly introduces related work, and Section 3 describes the algorithm. Experiments are shown in Section 4, and we conclude our work in Section 5.
Related Work
Contrast-enhanced computed tomography (CECT) is a popular way of producing detailed images of internal organs, bones, soft tissues and blood vessels. Based on this technology, computer-aided diagnosis (CAD) is widely studied as a tool to assist physicians. Automatic segmentation is an important prerequisite of a CAD system [25] [8]. The difficulty mainly comes from the high anatomical variability and/or the small volume of the target organs. Indeed researchers sometimes design a specific segmentation algorithm for each organ [1][19] [23].
Over the past few years, the rapid development of deep learning has led to a revolution to medical image analysis. The most successful methods are based on the deep convolutional neural network (CNN), a hierarchical model which is learning complicated data distributions. In the computer vision field, deep learning has been widely used for image classification [11][24] [20], object detection [7] and instance segmentation [13]. These neural network methods can be transferred to medical image analysis. But there is one challenge. CT-scanned data take the form of 3D volumes, and not 2D images. There are basically two types of solution, i.e., using a 2D-based network to process 3D data, or training a 3D-based network directly. A straightforward idea borrowed from computer vision is to cut the 3D volume into slices (2D images) and train a 2D model on these. These models can be trained in a patch-based manner [5] [19], or directly applied to the full images [18]. To consider and exploit the underlying relationship between neighboring slices, 3D-based networks are also widely studied. Due to their heavier computational overhead, 3D models are often trained in a patchbased manner [17][10] [14] [15]. A more efficient solution is to integrate 3D cues to 2D models, for example by using a tri-planar structure (three orthogonal planes intersecting at the point to be classified) [16] or interpreting the 3D volume as a sequence of 2D image slices and applying recurrent neural networks [22] [3]. A detailed discussion about 2D and 3D models can be found in [6] [12].
In this paper, we choose fully-convolutional network (FCN) [13], a 2D segmentation model based on a pre-trained deep neural network [20], as the baseline model. Deeper and/or more sophisticated networks can lead to higher segmentation accuracy [2] [15], but they are often more computationally expensive and risk over-fitting [14]. As we focus on segmenting small organs (e.g., the pancreas), the standard loss function computed per pixel/voxel may cause the model be heavily biased towards predicting a pixel/voxel to fall on background. Solutions include performing class-balancing [18], or directly designing a new loss function which is consistent to the evaluation metric [15]. The latter often works better.
The Proposed Approach
The Baseline Framework
We use the 2D fully-convolutional network (FCN) [13] as our baseline. FCN inherits the down-sampling process of popular networks for image classification, and then applies an up-sampling process named deconvolution to restore the output to the original size. Throughout this paper, we specify the down-sampling layers using a pre-trained 16-layer VGGNet [20]. We do not use deeper network structures like [9], which produce slightly higher accuracy at the expense of heavier computation and a higher risk of over-fitting.
Let a CT-scanned image be a 3D volume X of size W ×H×L, where W , H and L denote the width, height and length, respectively. Each volume is annotated with a ground-truth segmentation Y of the same dimensionality, where Y i = 1 indicates a foreground voxel. FCN is a 2D segmentation model M : O = f (I; Θ), where I denotes the input image, Θ the weights of the network, and O the output segmentation result which has the same spatial resolution as I. To fit the 3D volume X into a 2D network M, we cut it into a set of 2D slices. This process can be performed along three axes, i.e., in the coronal, sagittal and axial views. We denote these 2D slices as x C,w (w = 1, 2, . . . , W ), x S,h (h = 1, 2, . . . , H) and x A,l (l = 1, 2, . . . , L), where the subscripts C, S and A stand for "coronal", "sagittal" and "axial", respectively. We train three FCN models M C , M S and M A to perform segmentation through three views individually and integrate these cues at the testing stage.
Simply cutting a 3D volume into 2D slices ignores rich information, e.g., the correlation of structure between the neighboring slices. Motivated by this, we propose a 3-slice-segmentation model, which makes predictions on 3 successive slices simultaneously. In this case, each input image is composed of 3 successive slices, e.g., the w-th 2D image on the coronal view contains 3 slices (channels), i.e., x C,w−1 , x C,w and x C,w+1 . Thus, each 2D slice can appear in at most 3 input images. Applying this modification only slightly increases the number of network parameters, but provides us with the opportunity of incorporating visual cues on the neighboring slices during prediction. We can certainly process more slices at the same time, but in practice, this does not yield much accuracy gain. The original FCN model uses a loss function to sum up the cross-entropy loss at each voxel. For a training datum (x, y) and prediction p .
= f (x; Θ), the loss is computed as L voxel = i (−y i logp i − (1 − y i ) log(1 − p i )),
where i sums through all the voxels in y and p. However, as the pancreas often occupies a very small fraction of each slice, the voxel-wise loss is heavily biased to predicting a voxel as non-target (e.g., a simple model guesses that all voxels are background gets > 98% voxel-wise accuracy), which results in terrible performance as measured by the Dice-Sørensen Coefficient (DSC). To avoid this, we follow [15] to directly design a DSC-loss layer. The DSC between the ground-truth set G and the prediction set A is defined by DSC(A, G) = 2×|A∩G| |A|+|G| . Suppose the groundtruth annotation of each voxel is y i ∈ {0, 1} where 1 indicates the target, and the prediction value of each voxel is p i ∈ [0, 1], the DSC-loss function can be computed as
L DSC = 1 − DSC(p, y) = 1 − 2× i piyi i pi+ i yi . Note that this function is equivalent to the original DSC function if p i ∈ {0, 1} for all i. The gradient computation is straightforward: ∂LDSC pj = −2 × yj ( i pi+ i yi)− i piyi ( i pi+ i yi) 2 .
The Fine-Tuning Approach
We focus on segmenting small organs (e.g., the pancreas), which often occupy a very small part (e.g., < 0.5%) of the CT volume. It was observed [19][21] that deep segmentation networks such as FCN produce less satisfying results in these scenarios, arguably because the network is easily distracted by the varying contents in background regions, e.g., numerical instability caused by different vessel-contrast in the CECT scanning. In other words, a considerable number of neurons in the deep network are trained to detect the rough position of the pancreas, and thus the ability to precisely capture its shape and appearance becomes weak. Figure 1 shows the comparison of segmentation results when the input is either the entire image or a small area around the pancreas. We see that the latter strategy leads to more accurate segmentation. Motivated by this, we propose a fine-tuning approach to polish the initial segmentation. The idea is straightforward. In addition to the coarse-scaled deep network which sees the entire image, we train an additional fine-scaled deep network which only sees a small region covering the pancreas. In the training phase, this is done by taking the ground-truth annotation of each slice, and add a small frame around the minimal bounding box covering the segmentation mask. Note that, the input image size may differ from case to case. In the testing phase, we first feed the entire image to the coarse-scaled deep network and obtain the initial segmentation. Based on this, we estimate the bounding box, add a frame of a fixed width, crop the image region accordingly, and feed it to the fine-scaled models. Note that this fine-tuning process can be performed in an iterative manner at the testing stage without training new models. In practice, this process often terminates very quickly (e.g. in 2-3 iterations).
We illustrate the overall flowchart of our approach in Figure 2. We denote the coarse-scaled models trained on the coronal, sagittal and axial views as M C , M S and M A , respectively, and three fine-scaled models as M F C , M F S and M F A , where the superscript F stands for "fine-tuning". When a testing volume X is given, we first send it through three coarse-scaled models, and fuse their results into the initial segmentation P (0) = 1 3 P
C + P (0) S + P (0) A , where P (0) C , P(0)
S and P (0) A are the probabilistic outputs from three different views. We compute the initial segmentation mask S (0) by thresholding P (0) with 0.5. Then the fine-tuning process is performed iteratively. At the t-th iteration, the segmentation result of the previous iteration, i.e., S (t−1) is used to estimate the current 3D bounding box, and the volume within the 3D bounding box is cropped, framed and denoted as X (t) , and fed into the fine-scaled models. The results of three fine-scaled models are fused, i.e.,
P (t) = 1 3 P (t) C + P (t) S + P(t)
A , and the segmentation mask S (t) is updated. This process terminates when the maximal number of iterations T is reached, or the the similarity between successive segmentation results is sufficiently high, e.g., when DSC S (t−1) , S (t) > 0.95. Fig. 2. The flowchart of coarse-scaled segmentation (above) and fine-tuning (below). For simplicity, we only illustrate one iteration in the fine-tuning process. In practice, there are at most 10 iterations in total.
Implementation Details
In both coarse-scaled and fine-scaled models, we apply the FCN-8s configuration [13], which achieves the highest validation performance on the PascalVOC segmentation task. The down-sampling stage is directly borrowed from, and initialized by, a pre-trained 16-layer VGGNet. This is followed by several deconvolutional layers to up-sample the image to the original resolution. We train the network with a fixed learning rate of 10 −5 . We run 80,000 and 50,000 minibatches for the coarse-scaled and fine-scaled models, respectively. Each minibatch contains only one training sample (a 2D datum (x, y)).
In training the coarse-scaled models, we only select those 2D slices in which the pancreas occupies at least 1% pixels. This is to prevent the model from being heavily impacted by the noisy background contents. In training the finescaled models, we first use the ground-truth annotation to find the 3D bounding box of the pancreas. When each slice is generated within the box, we add a frame around it, and filled with the original image data. The top, bottom, left and right margins of the frame are random numbers uniformly sampled from {10, 11, . . . , 20}. This strategy, known as data augmentation, helps to regularize the network and prevent it from over-fitting. In the testing stage, the input image
Method View Fusion Coronal
Sagittal Axial 1-slice-segmentation 64.60 ± 11.42 69.71 ± 11.06 71.54 ± 8.87 75.51 ± 9.87 3-slice-segmentation 66.88 ± 11.08 71.41 ± 11.12 73.08 ± 9.60 75.74 ± 10.47 Table 1. Baseline segmentation accuracy (measured by DSC, %). We test both the 1-slice-segmentation and 3-slice-segmentation models. We report accuracies obtained by individual models and the fused model (see Section 3.1 for details).
is first fed into the coarse-scaled models to obtain an initial segmentation. After that, we find the minimal 3D bounding box containing the estimated pancreas. After each slice is cropped according to the box, we add a fixed frame of 15 pixels wide around it, and send it to the fine-scaled model.
Experiments
Dataset and Evaluation
We evaluate our algorithms on the NIH pancreas segmentation dataset [19], which contains 82 contrast-enhanced abdominal CT volumes. These data are acquired using Philips and Siemens MDCT scanners (120kVp tube voltage, scanning about 70 seconds after intravenous contrast injection in portal-venous). The subjects are normals, i.e., they are neither major abdominal pathologies nor pancreatic cancer lesions. Their ages range from 18 to 76 years with a mean age of 46.8±16.7. The resolution of each CT scan is 512×512×L, where L ∈ [181, 466] is the number of sampling slices along the long axis of the body. The slice thickness varies from 0.5mm-1.0mm. Following the standard cross-validation strategy, we split the dataset into 4 fixed folds, each of which contains approximately the same number of samples. We use the leave-one-out evaluation method, i.e., training the model on 3 out of 4 subsets and testing it on the remaining one. We measure the segmentation accuracy by computing the Dice-Sørensen Coefficient (DSC) for each sample. This is a similarity metric between the prediction voxel set A and the ground-truth set G. The mathematical form is DSC(A, G) = 2×|A∩G| |A|+|G| . We report the average DSC score together with the standard deviation over 82 testing cases.
Baseline Results
We first evaluate the baseline (coarse-scaled) approach, i.e., performing either 1-slice-segmentation or 3-slice-segmentation using the models M C , M S and M A trained from different views. We also fuse the these three models by averaging their results at the testing stage. The results are summarized in Table 1.
We observe that 3-slice-segmentation works better than 1-slice-segmentation in each single case, either when three views are considered individually or after their results are fused. This verifies that integrating information on the neighboring slices is useful for segmentation. In fact, this is an alternative way of introducing 3D cues into segmentation, which is less computationally expensive than 3D-based networks [17] [15]. When the amount of training data is limited (in the NIH pancreas segmentation dataset, only about 60 samples are used for training), this strategy also alleviates the risk of over-fitting, since the number of parameters only increases by a little compared to the 1-slice-segmentation model, a.k.a., the vanilla 2D FCN model. In the meantime, we find that fusing the results from three views largely boosts the segmentation accuracy. This suggests that complementary information is captured by combining different views. We also observe that in fusion, the 1-slice-segmentation approach enjoys significantly more accuracy gain than the 3-slice-segmentation approach. We believe this is a marginal effect, since both fusion and 3-slice-segmentation take the advantage of 3D visual cues.
In the fine-tuning section, to take the advantage of 3D, we train 3-slicesegmentation models, and perform fusion at the end of each iteration.
Fine-Tuning Results
Next, we evaluate the fine-tuning approach. As illustrated in Section 3.2, finetuning is performed in an iterative manner. Since deep neural networks are often sensitive to small noise, it is often very difficult for the iteration process to reach a complete stop, instead, after a sufficient number of iterations, the segmentation result does not change significantly. To measure convergence, we define the inter-iteration DSC to be d (t) .
= DSC S (t−1) , S (t) , where S (t) is the segmentation mask after the t-th iteration. Note that d (t) = 1 implies perfect convergence which is unlikely to happen. We compute the average d (t) value Table 2. Fine-tuned segmentation accuracy (measured by DSC, %). We start from the 3-slice-segmentation results, and explore different terminating conditions, including a fixed number of iterations and a fixed threshold of inter-iteration DSC. The last two lines show two upper-bounds of our approach, i.e., "Best of All Iterations" means that we choose the highest DSC value over 10 iterations, and "Oracle Bounding Box" corresponds to using the ground-truth segmentation in extracting the bounding box.
(t = 1, 2, . . . , T ) over 82 testing samples. The results are shown in Figure 3. Based on these, we conclude that the fine-tuning approach is generally stable: after 10 iterations, the average d (t) value over all samples is 0.9767, the median of the d (t) values is 0.9794, and the minimum is 0.9362. We also record the number of iterations required for each sample to reach a given threshold of d (t) . Not surprisingly, increasing the threshold leads to a larger number of iterations. Now, we consider two types of conditions for terminating the iteration, more precisely, after a fixed number of iterations, or after the inter-iteration DSC reaches a fixed threshold. When using the latter condition, we set the maximal number of iterations to be 10, i.e., after 10 iterations, the fine-tuning process is terminated even if the threshold is not reached. The results are summarized in Table 2. Our fine-tuning approach significantly boosts the baseline accuracy (by 6.63%). This is impressive given the relatively high baseline accuracy (76.15%).
Regarding different terminating conditions, we find that performing merely 1 iteration is enough to significantly boost the segmentation accuracy (+6.42%). This verifies our hypothesis, i.e., training a fine-scaled model helps to depict small organs more accurately. The best performance comes from setting a proper threshold (e.g., 0.95) of d (t) . Using a large threshold (e.g., 0.98 or 0.99) does not produce accuracy gain, but requires a larger number of iterations and, consequently, heavier computation at the testing stage. For all 82 testing samples, it takes on average less than 3 iterations to reach the threshold 0.95, which guarantees the efficiency of our approach. On a modern GPU, we need about 3 minutes on each testing sample, comparable to recent work [18], but we report much higher segmentation accuracy (see Table 3). Setting a threshold of d (t) is Table 3. Comparison of our algorithm with the state-of-the-art approaches. Our coarse-to-fine approach achieves the best number under each performance statistics. more efficient than using a fixed number of iterations, e.g., if we fix the number of iterations to be 3, the average DSC goes down by nearly 0.3%.
As an diagnostic experiment, we use the ground-truth bounding box of the testing cases to generate the input volume for fine-tuning. This results in a 83.18% accuracy (no iteration is needed). In comparison, we report a comparable 82.37% accuracy, indicating that we are good at estimating the bounding box.
We show two challenging cases in Figure 4, and observe how fine-tuning gradually improves the segmentation accuracy by iterations.
Comparison to the State-of-the-Art
In Table 3, we compare our segmentation result with the state-of-the-art approaches. Using DSC as the evaluation metric, our approach outperforms the recent published work [18] significantly. The average accuracy over 82 samples increases remarkably from 78.01% to 82.37%, and the standard deviation decreases from 8.20% to 5.68%, implying that fine-tuning generates more stable results. Especially, [18] reports 34.11% on the worse case, and this number is impressively boosted to 62.43% by our algorithm. We point out that these improvements are mainly owed to the fine-tuning approach. Without them, the average accuracy is 76.15%, and the accuracy on the worst case is merely 39.99%. Please refer to Figure 4 for an intuitive illustration of how fine-tuning improves the accuracy of challenging cases.
Conclusions
We present a coarse-to-fine approach for segmenting the pancreas from CTscanned images. Our motivation is straightforward: deep networks such as FCN are not good at segmenting very small objects compared to the input image size. Therefore, we train an additional fine-scaled model to deal with small inputs. In the testing stage, the coarse-scaled model is used to roughly locate the pancreas, and an iterative process is performed on the fine-scaled model to refine the initial segmentation. In practice, this process often comes to an end after 2-3 iterations.
We evaluate our algorithm on the NIH pancreas segmentation dataset with 82 samples, and outperforms the state-of-the-art by more than 4%, measured by the Dice-Sørensen Coefficient (DSC). Most of the benefit of the fine-tuning approach comes from the first iteration. The remaining iterations only improve the segmentation accuracy by a little (about 0.3%). We believe that our algorithm can achieve even higher accuracy when more powerful deep networks are equipped. We point out that the idea of fine-tuning can be applied to other small organs, e.g., the spleen. In the future, we will also explore the possibility of incorporating fine-tuning into an end-to-end training process.
Fig. 1 .
1The comparison of segmentations with different input regions (best viewed in color PDF), namely using the entire image or the bounding box (marked by the red frame). The right images show the segmentations within the frame, in which red, green and yellow indicate the prediction, ground-truth and overlapped regions, respectively.
Fig. 3 .
3Left: the average d (t) = DSC S (t−1) , S (t) over 82 samples with respect to t. Right: the histogram of samples that require a specific number of iterations in the fine-tuning process. 10 is the maximal number of iterations.
Input Image Initial Segmentation After 1 st Fine-Tuning After 2 nd Fine-Tuning Fig. 4. Examples of segmentation results before and after fine-tuning (best viewed in color PDF). We only show a small region covering the pancreas in the axial view. The terminating condition is d (t) > 0.95. Red, green and yellow indicate the prediction, ground-truth and overlapped regions, respectively. Method Mean DSC (%) Max DSC (%) Min DSC (%) Roth et.al, MICCAI'2015 [19] 71.42 ± 10.11 86.29 23.99 Roth et.al, MICCAI'2016[18] NIH Case #03
DSC = 57.66%
DSC = 81.39%
DSC = 81.45%
Input Image
Initial Segmentation After 1 st Fine-Tuning After 2 nd Fine-Tuning
NIH Case #09
DSC = 42.65%
DSC = 54.39%
DSC = 57.05%
Final (3 Iterations)
DSC = 82.19%
Final (10 Iterations)
DSC = 76.82%
78.01 ± 8.20
88.65
34.11
Ours, without Fine-Tuning
75.74 ± 10.47
88.12
39.99
Ours, with Fine-Tuning
82.37 ± 5.68
90.85
62.43
AcknowledgementsThis work is supported by the Lustgarten Foundation for Pancreatic Cancer research.
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[] |
[
"Oeljeklaus-Toma manifolds admitting no complex subvarieties",
"Oeljeklaus-Toma manifolds admitting no complex subvarieties"
] |
[
"L Ornea ",
"M Verbitsky ",
"Liviu Ornea ",
"Misha Verbitsky "
] |
[] |
[] |
The Oeljeklaus-Toma (OT-) manifolds are complex manifolds constructed by Oeljeklaus and Toma from certain number fields, and generalizing the Inoue surfaces S m . On each OT-manifold we construct a holomorphic line bundle with semipositive curvature form ω 0 and trivial Chern class. Using this form, we prove that the OT-manifolds admitting a locally conformally Kähler structure have no non-trivial complex subvarieties. The proof is based on the Strong Approximation theorem for number fields, which implies that any leaf of the null-foliation of ω 0 is Zariski dense.
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10.4310/mrl.2011.v18.n4.a12
|
[
"https://arxiv.org/pdf/1009.1101v5.pdf"
] | 54,944,438 |
1009.1101
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adfc8ad012f7acc5bbad971329f05d69a24fda23
|
Oeljeklaus-Toma manifolds admitting no complex subvarieties
2 Feb 2011
L Ornea
M Verbitsky
Liviu Ornea
Misha Verbitsky
Oeljeklaus-Toma manifolds admitting no complex subvarieties
2 Feb 2011Subvarieties in Oeljeklaus-Toma manifolds To Professor Vasile Brînzȃnescu at his sixty-fifth birthday
The Oeljeklaus-Toma (OT-) manifolds are complex manifolds constructed by Oeljeklaus and Toma from certain number fields, and generalizing the Inoue surfaces S m . On each OT-manifold we construct a holomorphic line bundle with semipositive curvature form ω 0 and trivial Chern class. Using this form, we prove that the OT-manifolds admitting a locally conformally Kähler structure have no non-trivial complex subvarieties. The proof is based on the Strong Approximation theorem for number fields, which implies that any leaf of the null-foliation of ω 0 is Zariski dense.
Introduction
OT-manifolds and their subvarieties
The Oeljeklaus-Toma (OT-) manifolds are an important class of compact complex manifolds not admitting a Kähler metric. They were discovered by Oeljeklaus and Toma in 2005 ([OT]). The construction of OT-manifolds uses the Dirichlet unit theorem from number theory (Subsection 1.2; see [PV] for additional details of this construction and many related questions). Starting from a degree 3 number field, one obtains a 2-dimensional OTmanifold known as Inoue surface S m (see [I]).
For some number fields, the OT-manifolds are locally conformally Kähler. A locally conformally Kähler (LCK) structure on a complex manifold is a Kähler metric on its universal coverM , such that the deck transform maps act onM by homotheties. The OT-manifolds serve an important function in the theory of LCK manifolds, providing a counterexample to a longstanding conjecture of I. Vaisman, [Va], who asked whether there exists a compact, non-Kähler LCK-manifold M with all odd Betti numbers even: b 2p+1 (M ) ≡ 0 mod 2. The Oeljeklaus-Toma manifolds in dimension 3 are the only known examples of compact LCK-manifolds with even odd Betti numbers, b 1 = b 5 = 2, b 3 = 0.
An OT-manifold is LCK if it is constructed from a number field K which has precisely 2 complex (non-real) embeddings, that is, two distinct homomorphisms K σ,σ −→ C. If the OT manifold has at least 4 complex embeddings and exactly one real, then it is not LCK. The remaining case is not yet decided.
Inoue surfaces S m have no curves. We give a generalization of this theorem, proving that an OT-manifold which is locally conformally Kähler has no non-trivial complex subvarieties. In particular, it has no non-constant meromorphic functions (as a meromorphic function without polar set would be holomorphic, and hence constant).
Question 1.1: Is there any OT (non-LCK) manifold that has non-constant meromorphic functions?
The idea of the proof of this result is quite simple. We construct a holomorphic Hermitian line bundle, called the weight bundle, on any OT-manifold M . This bundle is topologically trivial, and has semipositive curvature form ω 0 . The weight bundle also admits a flat connection, compatible with the holomorphic structure.
To learn about complex subvarieties of an OT-manifold, we study the zero-foliation Σ of ω 0 , proving that all its leaves are Zariski dense in M . For an OT-manifold M constructed from a number field K admitting exactly 2t distinct complex (non-real) embeddings to C, the leaves of Σ are t-dimensional. When t = 1, M is locally conformally Kähler, and Σ is one-dimensional. In this case, we prove that for any positive-dimensional complex subvariety Z ⊂ M , Z contains with each point z ∈ Z a leaf Σ z passing through z. Since all leaves of Σ are Zariski dense, the same is true for Z.
The weight bundle L is quite useful for many other purposes. As it was done in [Ve3], one can take the α-th tensor power of L, denoted by L α , for any real α; this power is well defined, because L is equipped with a natural C ∞ -trivialization. The Gauduchon degree deg g of L α , taken with respect to any Gauduchon metric, satisfies 1 α deg g L α = deg g L > 0, hence M admits a line bundle with any prescribed Gauduchon degree. This implies, in particular, that the connected component of the Picard group P ic(M ) is non-compact. Also, this implies that any vector bundle on M has degree zero after tensoring with an appropriate power of L; this is useful for the study of Hermitian-Einstein bundles on M , providing useful tools for the classification of stable bundles, and, eventually, coherent sheaves on M .
A similar argument was used in [Ve3] to study holomorphic vector bundles and subvarieties on homogeneous elliptic fibrations, such as Calabi-Eckmann manifolds and quasi-regular Vaisman manifolds. We pose two questions, very much unsolved, but quite natural in the context presented by [Ve3] and the present paper. Notice that from their construction it is clear that OT-manifolds are affine flat, that is, equipped with a flat, affine, torsion-free connection.
It is shown in [OT,Remark 1.7] that some OT manifolds admit a holomorphic foliation with compact leaves which are again OT manifolds. Hence, it is natural to pose the following: Question 1.2: Are the ones described in [OT,Remark 1.7] the only OT manifolds with compact complex subvarieties? Can we classify these subvarieties? Are they always totally geodesic with respect to the flat affine connection? Question 1.3: Does there exist a stable holomorphic vector bundle of rank > 1 on any OT-manifold of dimension > 2? Do all holomorphic vector bundles admit a flat connection, compatible with the holomorphic structure?
Remark 1.4: It is well known that generic complex tori have no non-trivial complex subvarieties. In [Ve2], it was shown that all stable bundles on a generic complex torus of dimension > 2 have rank 1, and all holomorphic vector bundles admit flat connections. As for compact complex surface of non-Kählerian type, it is proven in [Vu] that stable holomorphic 2-bundles with c 1 = 0 and c 2 = n exist for any n > 0.
Number theory and the construction of OT-manifolds
Let [K : Q] be a number field, that is, a finite extension of Q, of degree n, with σ 1 , ..., σ s the real embeddings of K into C, and σ s+1 , ..., σ n the complex embeddings. Since the complex embeddings of K into C occur in pairs of complex conjugate embeddings, the number n − s is even, n − s = 2t. Let σ = (σ 1 , . . . , σ n ) : K → C s+t be the corresponding group homomorphism.
Let O K be the ring of algebraic integers of K, O * K its multiplicative group of units and O * ,+ K the group of units which are positive in all the real embeddings of K.
Denote by H the upper complex half-plane. Using the Dirichlet's unit theorem, Oeljeklaus and Toma proved that
O K ⋊ O * ,+ K acts freely on H s × C t by T a (z i ) = (z i + σ i (a)), i = 1, . . . , s + 2t, a ∈ O K , R u (z i ) = (σ i (u)z i ), i = 1, . . . , s + 2t, u ∈ O * ,+ K .
(see [OT], [PV]). Moreover, an admissible subgroup U ⊂ O * ,+ K can always be found such that the action of Γ := O K ⋊ U is also properly discontinuous. For t = 1, every U of finite index in O * ,+ K has this property.
Definition 1.5: The manifold M K := (H s × C t )/Γ is called an Oeljeklaus-Toma manifold. It is a compact complex manifold of dimension s + 2t.
For s = t = 1, M K reduces to an Inoue surface S m (where m is a matrix in SL(3, Z)), see [I]. The corresponding number field K is Q[T ]/P (t), where P m (t) is the characteristic polynomial of the matrix m. It is shown in [OT] that the manifolds M K are never Kähler, but that for t = 1, M K is a locally conformally Kähler (LCK) manifold (see [DO] and the more recent survey [OV] for definitions and results in LCK geometry). We briefly explain the construction of this LCK metric.
Clearly, the function ψ(z) = s i=1 (im z i ) + |z s+1 | 2 is plurisubharmonic on H s × C.
It defines the Kähler form Ω := ∂∂ ψ on H s × C. The group Γ acts on (H s × C, Ω) by homotheties:
T * a Ω = Ω, R * u Ω = |σ s+1 (u)| 2 Ω.
Let now χ : Γ → R >0 be the character χ(γ) = γ * Ω Ω . We call automorphic any p-form η ∈ Λ p (H s ×C) which satisfies γ * η = χ(γ)η. For any automorphic function ϕ on H s × C, the quotient Ω ϕ is Γ-invariant and hence projects to an LCK metric ω on M K . This form satisfies the equation dω = θ ∧ ω, for the closed 1-form θ (called the Lee form) which is the projection on M K ofθ = −d log ϕ:
dω = − dϕ ϕ 2 ∧ω = −d(log ϕ) ∧ ω.
It is easily seen that the function ϕ = s i=1 (im z i ) −1 is automorphic, and hence it produces a LCK metric on M K as described above. This LCK metric generalizes the one constructed by Tricerri on S m , [Tr].
The main result of this paper shows that, just as Inoue surfaces S m have no complex curves, OT-manifolds have no complex subvarieties: Theorem 1.6: Let [K : Q] be a number field of degree n = s + 2, with s real embeddings and 2 complex embeddings, and M K the corresponding LCK OT-manifold. Then M K has no non-trivial complex subvarieties.
Proof: See Theorem 3.1.
Corollary 1.7: The LCK OT-manifold M K has no non-constant meromorphic functions.
The weight bundle of an OT-manifold
Definition 2.1: Let [K : Q] be a number field of degree n = s + 2t, with s real embeddings and 2t complex embeddings, and M K = H s × C t /Γ the associated OT-manifold. Denote by z 1 , ..., z s the standard complex coordinates on H s , and letθ := −d log s i=1 (im z i ). It is easy to see that the form θ is Γ-invariant. Therefore it is obtained as a lift of a form θ, called the Lee form of the OT-manifold. When t = 1, this is the Lee form constructed above.
Let M K be an OT-manifold, and θ its Lee form. Consider a trivial Hermitian line bundle L with connection ∇ := ∇ 0 + √ −1 θ c , where θ c := Iθ, and ∇ 0 is the trivial connection on L. Clearly, ∇ is Hermitian, and ∇ 0,1 = ∂ + θ 0,1 , where θ 0,1 is the (0,1)-part of θ.
Claim 2.2: In these assumptions, the curvature ω 0 of ∇ is − √ −1 dθ c . Moreover, this form is of type (1,1).
Proof: A simple computation shows that in the standard coordinates z 1 , ...z s , z s+1 , ...z s+t , ω 0 can be written as follows:
ω 0 = √ −1 ∂∂ log ϕ = √ −1 s i=1 dz i ∧ dz i | im z i | 2 ,
Definition 2.3: Let M K be an OT-manifold, and L the holomorphic Hermitian bundle defined above. Then L is called the weight bundle of M K .
We restate Claim 2.2 as Theorem 2.4: Let M K be an OT-manifold, and L its weight bundle with the holomorphic Hermitian structure and the Chern connection ∇ defined above. Consider the form ω 0 := √ −1 ∇ 2 . Then ω 0 is a semi-positive form, which can be written in the standard coordinates z 1 , ...z s , z s+1 , ..., z s+t as follows:
ω 0 = √ −1 ∂∂ log ϕ = √ −1 s i=1 dz i ∧ dz i | im z i | 2
Remark 2.5: The Vaisman manifolds are, by definition, LCK manifolds (M, I, g) satisfying the additional condition ∇ g θ = 0, where ∇ g is the Levi-Civita connection of an LCK metric g. For all Vaisman manifolds, the 2-form ω 0 = dθ c is semi-positive, being zero only on the direction of θ ♯ − Iθ ♯ . This is a general fact, proven in [Ve1], independent of the particular form of θ.
OT-manifolds are far from being Vaisman (they never admit any Vaisman metric), but the particular expression of their Lee form gives ω 0 the same property as for Vaisman manifold. This is what inspired our construction.
Remark 2.6: An object of interest in conformal geometry and, in particular, LCK geometry is the weight bundle. It is the real line bundle L −→ M associated to the representation GL(2n, R) ∋ A → | det A| 1 n (see [OV]). Then L can be complexified and endowed with the Chern connection ∇ 0 + √ −1 θ c (where ∇ 0 is the trivial connection). It can be verified that ω 0 = √ −1∇ 2 , and hence ω 0 can be seen as the curvature form of this Chern connection. When t = 1 and M K is the corresponding LCK OT-manifold, this construction gives the weight bundle introduced in Definition 2.3.
Remark 2.7: For any OT-manifold M , in addition to the Chern connection ∇ = ∇ 0 + √ −1 θ c , the weight bundle L also admits the connection ∇ 0 + θ, which is flat because dθ = 0. It is clear that the (0, 1)-part of ∇ coincides with the (0, 1)-part of this flat connection.
The following claim is obvious from the explicit form of ω 0 (Theorem 2.4).
Claim 2.8: In the assumptions of Theorem 2.4, letΣ be the holomorphic foliation on the coveringM K = H s × C t generated by the vector fields ∂ ∂z s+1 , ..., ∂ ∂z s+t . Then:
(i) The foliationΣ is Γ-invariant, hence it is obtained as the pullback of a holomorphic foliation Σ on M K =M K /Γ.
(ii) The foliation Σ is the null-space of the form ω 0 constructed above.
Claim 2.9: Let [K : Q] be a number field of degree n = s + 2, with s real embeddings and 2 complex embeddings, M K the corresponding LCK OTmanifold, and Σ ⊂ T M K the holomorphic foliation defined in Claim 2.8. Consider a complex closed subvariety Z ⊂ M K . Then Σ is tangent to Z at any point of Z:
∀z ∈ Z, Σ z ⊂ T z Z. (2.1)
Proof: The form ω 0 has (n−1) positive eigenvalues, where n = dim C M K , and its zero eigenspace at z is Σ z . Unless (2.1) holds at z ∈ Z, the restriction ω 0 Z has m = dim Z positive eigenvalues at z. Then Z ω m 0 > 0. This is impossible, because ω 0 is exact.
Corollary 2.10: In assumption of Claim 2.9, let Σ z be a leaf of Σ passing through z ∈ Z. Then Σ z ⊂ Z.
Complex subvarieties in LCK OT-manifold
Using Corollary 2.10, we can easily prove the main result of this paper.
Theorem 3.1: Let [K : Q] be a number field of degree n = s + 2, with s real embeddings and 2 complex embeddings, and let M K be the corresponding OT-manifold. Then M K has no non-trivial complex subvarieties.
Proof: Theorem 3.1 follows from Corollary 2.10 and the following more general proposition.
Proposition 3.2: Let [K : Q] be a number field of degree n = s + 2t, t > 0, with s real embeddings and 2t complex embeddings, and let M K = H s ×C t /Γ be the associated (non-Kähler) OT-manifold. Let Σ ⊂ T M K be the foliation defined in Claim 2.8. Consider a leaf of Σ, and let Z be its closure. Then (i) The preimage π −1 (Z) of Z toM K = H s × C t contains the set Z α 1 ,...,αs := {(z 1 , ..., z s , z s+1 , . . . , z s+t ) | im z i = α i } for some positive numbers (α 1 , . . . , α s ) ∈ R s .
(ii) Any complex subvariety of M K containing Z must coincide with M K .
Proof: The implication (i) ⇒ (ii) is clear, because any complex manifold containing Z α 1 ,...,αs must have the same dimension as M K . The proof of (i) is a bit more elaborate.
Let O be the ring of integers in K. By construction, the group Γ = π 1 (M K ) is a cross-product of the additive group O + of O with a subgroup of the multiplicative group O * . LetΣ be the pullback of the foliation Σ tõ M K = H s × C t . A leaf ofΣ is given as T t 1 ,...,ts := {(z 1 , ..., z s , z s+1 , ..., z s+t ) | z i = t i } for some (t 1 , ..., t s ) ∈ H s . LetZ := π −1 (Z) be the preimage of the corresponding closure of a leaf of Σ. Clearly,Z is the closure of Γ(T t 1 ,...,ts ). Therefore, to prove Proposition 3.2 (i) it is sufficient to show that the closure of Γ(T t 1 ,...,ts ) contains Z α 1 ,...,αs . In fact, even the smaller group O + ⊂ Γ will suffice, as seen from the following lemma, which proves Proposition 3.2. Proof: Equivalently, we may state that the closure of an orbit of the standard action of O + in H s is the set {(z 1 , . . . , z s , z s+1 , . . . , z s+t ) | im z i = α i }. This in turn is equivalent to the following Proof: 1 Let K be a number field, O K its ring of integers, P the set of all prime ideals of O K , V the product of all archimedean completions of K, and V 1 the product of some, but not all, archimedean completions. Denote by O ν the completion of O K at ν ∈ P, and let K ν be the corresponding local field. Consider the adele space A, obtained as a subset of the product V × ν∈P K ν , where all components, except finitely many, belong to O ν , and let A 1 be the image of projection of A to V 1 × ν∈P K ν . Denote by τ : K −→ A 1 the natural homomorphism, which is tautological componentwise.
From the Strong Approximation theorem (see [K] or [NT,Theorem 20.4.4] 2 ) it follows that the image τ (K) of K is dense in A 1 . Let
O A 1 := A 1 ∩ V 1 × ν∈P O ν 1
We are grateful to Marat Rovinsky, who kindly explained to us this proof 2 http://modular.fas.harvard.edu/papers/ant/html/node84.html be the set of points of A, corresponding to the integer adeles. Clearly, O A 1 is open in A 1 . Therefore, the intersection τ (K) ∩ O A 1 is dense in O A 1 . On the other hand, τ (K) ∩ O A 1 consists of those elements of the number field which are integer at all non-archimedean places. This gives τ (K) ∩ O A 1 = τ (O K ). Therefore, the image of O K to V 1 is dense.
Remark 3.5: The above argument actually proves that the image of O K in the product V 1 of all archimedean completions of K except one is dense in V 1 .
Lemma 3. 3 :
3Let [K : Q] be a number field of degree n = s + 2t, t > 0 with s real embeddings and 2t complex embeddings, andM K := H s × C t , equipped with the action of O + as in Subsection 1.2. Consider the subset T t 1 ,...,ts := {(z 1 , ..., z s , z s+1 , . . . , z s+t ) | z i = t i } inM K . Then the closure of O + (T t 1 ,...,ts ) coincides with Z α 1 ,...,αs := {(z 1 , ..., z s , z s+1 , . . . , z s+t ) | im z i = α i , } with α i := im t i .
Lemma 3.4: (cf. [OT, Claim following Lemma 2.4]) Let [K : Q] be a number field of degree n = s + 2t, t > 0 with s real embeddings σ 1 , . . . , σ s and 2t complex embeddings. Consider the additive group O + of the corresponding ring of integers. Let σ : O + −→ R s map ξ to σ 1 (ξ), . . . , σ s (ξ). Then the image of O + is dense in R s .
version 4.0, Jan 22, 2011
Acknowledgements:We are grateful to Katia Amerik for her support. Much gratitude to Marat Rovinsky for his invaluable help in proving the approximation lemma. Part of this work was done in Oberwolfach during the Research in Pairs programme; we are grateful to Oberwolfach Foundation for making it possible. Many thanks to Victor Vuletescu and Matei Toma for insightful email correspondence.
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Academiei Str, 70109 Bucharest, Romania. and Institute of Mathematics. Bucharest, Romania Liviu21Calea Grivitei Street 010702Academiei str., 70109 Bucharest, Romania. and Institute of Mathematics "Simion Stoilow" of the Romanian Academy, 21, Calea Grivitei Street 010702-Bucharest, Romania [email protected], [email protected]
. Vavilova Str, Moscow, Russia117312 [email protected], [email protected] Str. Moscow, Russia, 117312 [email protected], [email protected]
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"Title: An expansion of zeta(3) in continued fraction with parameter",
"Title: An expansion of zeta(3) in continued fraction with parameter",
"Title: An expansion of zeta(3) in continued fraction with parameter",
"Title: An expansion of zeta(3) in continued fraction with parameter"
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We present here continued fraction for Zeta(3) parametrized by some family of points (F,G) on projective line. This family of points can be obtained if from full projective line would be removed some no more than countable nowhere dense exeptional set of finite points. A countable nowhere dense set, which contains the above exeptional set of finite points, is specified also. L.A.Gutnik, An expansion of zeta(3) in continued fraction with parameter.
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"https://arxiv.org/pdf/1307.1125v2.pdf"
] | 119,125,112 |
1307.1125
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4c3da77a822944a51ce0712efb7f98f764e1b395
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Title: An expansion of zeta(3) in continued fraction with parameter
16 Jul 2013
Title: An expansion of zeta(3) in continued fraction with parameter
16 Jul 2013Authors:L.A.Gutnik Comments: 40 pages with 0 figures Subj-class: Number Theory MSC-class: 11-XX
We present here continued fraction for Zeta(3) parametrized by some family of points (F,G) on projective line. This family of points can be obtained if from full projective line would be removed some no more than countable nowhere dense exeptional set of finite points. A countable nowhere dense set, which contains the above exeptional set of finite points, is specified also. L.A.Gutnik, An expansion of zeta(3) in continued fraction with parameter.
To the thirty fifth anniversary of Apéry's discovery.
Table of contents §0. Foreword. §1. Introduction. Begin of the proof of Theorem B. §2. Transformation of the system considered in §1. §3. Calculation of the matrix A * * 1,0 (z, ν). §4. Further properties of the functions considered in §1. §5. Auxiliary difference equation. §6. Auxiliary continued fraction. §7. End of the Proof of theorem B. §0. Foreword . 1 We say that two infinite continuous fractions are equivalent if the set of their common convergents is infinite. We say that two infinite continuous fractions are essentially distinct if the set of their common convergents is finite or empty.
If
δ 0 = 1, δ ν ∈ C {0}, b (2) ν = b (1) ν δ ν , a (2) ν = a (1) ν δ ν δ ν−1
for ν ∈ N, then easy induction show that P
(2) ν = P
(1) ν ν κ=1 δ κ , Q
(2)
ν = Q (1) ν ν κ=1 δ κ ,
and continued fraction
(0.1) b (i) 0 + a (i) 1 | |b (i) 1 + a (i) 2 | |b (i) 2 + ... + a ∨ ν | |b (i) ν + ...,
with i=1 is equivalent to continued fraction (0.1) with i = 2. According to the famous result of R. Apéry [R. Apéry, 1981],
(0.2) ζ(3) = b ∨ 0 + a ∨ 1 | |b ∨ 1 + a ∨ 2 | |b ∨ 2 + ... + a ∨ ν | |b ∨ ν + ... with (0.3) b ∨ 0 = 0, b ∨ 1 = 5, a ∨ 1 = 6, b ∨ ν+1 =
34ν 3 + 51ν 2 + 27ν + 5, a ∨ ν+1 = −ν 6 , for ν ∈ N. We denote by r A,ν the ν-th convergent of continued fraction (0.2). Yu.V. Nesterenko in [Yu.V. Nesterenko, 1996] has offered the following expansion of 2ζ(3) in continuous fraction:
(0.4) 2ζ(3) = 2 + 1| |2 + 2| |4 + 1| |3 + 4| |2
..., with (0.5) b 0 = b 1 = a 2 = 2, a 1 = 1, b 2 = 4, (0.6) b 4k+1 = 2k + 2, a 4k+1 = k(k + 1), b 4k+2 = 2k + 4, a 4k+2 = (k + 1)(k + 2)
for k ∈ N, (0.7) b 4k+3 = 2k + 3, a 4k+3 = (k + 1) 2 , b 4k+4 = 2k + 2, a 4k+4 = (k + 2) 2 for k ∈ N 0 .
We denote by r N,ν the ν-th convergent of continued fraction (0.4). The continued fractions (0.4) and (0.2) are equivalent because 2r A,ν = r N,4ν−2 for all ν ∈ N.
Elementary proof of Yu.V. Nesterenko result can be found in [L.A. Gutnik,16, 2010]. Let me to formulate my result now. Let u and v are variables, τ = τ (ν) = ν + 1, σ = σ(ν) = τ (τ − 1) = ν(ν + 1),
where ν ∈ N. Let further (0.8) c u,v,2 (ν) = −τ (τ + 1) 2 (2τ − 1)× (−3(2τ − 1) 2 u 2 − (10τ 2 − 10τ + 3)uv + 2(3τ 4 − 6τ 3 + 4τ 2 − τ )v 2 = −3(4σ(ν) + 1)u 2 − (10σ(ν) + 3)uv + 2σ(3σ(nu) + 1)v 2 ∈ N [u, v].
(0.9) c u,v,1 (ν) = −12(68τ 6 − 45τ 4 + 12τ 2 − 1)u 2 − 8(157τ 6 − 106τ 4 + 30τ 2 − 3)uv+ 4(102τ 8 − 170τ 6 + 89τ 4 − 24τ 2 + 3)v 2 ∈ N[u, v], (0.10) c u,v,0 (ν) = τ (τ − 1) 2 (2τ + 1)×
(3(4τ 2 + 4τ + 1)u 2 + (10τ 2 + 10τ + 3)uv − 2(3τ 4 + 6τ 3 + 4τ 2 + τ )v 2 ) = τ (τ − 1) 2 (2τ + 1)×
(3(4σ(ν + 1) + 1)u 2 + (10σ(ν + 1) + 3)uv − 2σ(ν + 1)(3σ(ν + 1) + 1)v 2 ).
(0.11) b u,v (ν + 1) = −c u,v,1 (ν) ∈ Q [u, v] for ν ∈ N, (0.12) a u,v (ν + 1) = −c u,v,0 (ν)c u,v),2 (ν − 1) for ν ≥ 2, ν ∈ N, (0.13) a u,v (2) = −c u,v,0 (1), (0.14) P u,v (0) = b u,v (0) = 4(3u + 2v), Q u,v (0) = 1, (0.15) Q u,v (1) = b u,v (1) = (34u + 52v)/(u + v)
(0.16) P u,v (1) = (327u + 500v) (0.17) a u,v (1) = P u,v (1) = −b u,v (0)b u,v (1).
Calculations described in [L.A.Gutnik,18, 2013] lead to the following continued fraction over the field Q(u, v) :
(0.18) b u,v (0) + a u,v (1)| |b u,v (1) + a u,v (2)| |b u,v (2) + a u,v (3)| |b u,v (3) + ....
We denote by r u,v (ν) the ν-th convergent of continuous fraction (0.18). We denote by P u,v (ν) and Q u,v (ν) the respectively nominator and denominator of r u,v (ν). Let further ∆(x) = 18x 2 (2x + 1) + (7x + 3) 2 ,
ρ k (x) = 5x + 3 + (−1) k ∆(x) x(3x + 2) ,
where x = 2ν(ν + 1), ν ∈ N, k = 1, 2, and let A = {ρ k (x) : x = 2ν(ν + 1), ν ∈ N, k = 1, 2}.
B = − β * (2) 2 (1; ν)/β * (1) 2 (1; ν) : ν ∈ N 0 , where (0.19) β * (r) 2 (z; ν) = ν+1 k=0 ν + 1 k ν + k k 2 t r z k .
Then we have Theorem B. Let F + G = 0, (F + G)G ≥ 0, G/F ∈ B ∪ A, if F = 0. Then the specialization of (0.18) for u = F, v = G is well defined (i.e all convergents r u,v (ν) are well defined for u = F, v = G) and the following equality holds (0.20) 8(F + G)ζ(3) = b F,G (0) + a F,G (1)| |b F,G (1) + a F,G (2)| |b F,G (2) + a F,G (3)| |b F,G (3) + .... moreover P u,v (ν) and Q u,v (ν) are homogeneous polynomials in Z[u, v], and (0.21) max(2ν, 1) = deg u (P u,v (ν)) = deg v (P u,v (ν)) = deg(P u,v (ν)),
(0.22) max(2ν − 1, 0) = deg u (Q u,v (ν)) = deg v (Q u,v (ν)) = deg(Q u,v (ν)), where ν ∈ N 0 .
Remark. The values ρ k (x) with k = 1, 2 are zeros of the following trinomial a 0 (x) + 2a 1 (x)ρ + a 2 (x)ρ 2 , where a 0 (x) = −6(2x + 1), a 1 (x) = −2(5x + 3), a 2 (x) = x(3x + 2).
Since −a 0 (x)/a 2 (x) = 3/x + 3/(3x + 2), −a 1 (x)/a 2 (x) = 3/x + 1/(3x + 2), decrease together with increasing of x > 0 it follows that ∆(x)/(a 2 (x)) 2 = (−a 1 (x)/a 2 (x)) 2 − (−a 0 (x)/a 2 (x)),
and ρ 2 (x) decrease with increasing of x. Moreover, 0 < −ρ 1 (x) < ρ 2 (x) for any x > 0 and lim x→∞ r 2 (x) = 0. Consequently, for given F > 0 and G > 0 the condition of the Theorem B must be checked for finite family of ν; for example, if G/F > r(4), then condition of the Theorem B is fulfilled. We note that r(4) < 0, 36.
I prove Theorem B in sections 1 -7. Initial variants of this article can be found in [L.A.Gutnik,17, 2009], [L.A.Gutnik,18, 2013. §1. Introduction. Begin of the proof of Theorem B.
Let (1.1) |z| > 1, −3π/2 < arg(z) ≤ π/2, log(z) = ln(|z|) + i arg(z). Clearly, log(−z) = log(z) − iπ, when ℜ(z) > 0, and log(z) = log(−z) − iπ, when ℜ(z) < 0. Let (1.2) f * 1 (z, ν) := ν+1 k=0 (z) k ν + 1 k 2 ν + k ν 2 , (1.3) R(t, ν) = ν j=1 (t − j) ν+1 j=0 (t + j) , (1.4) f * 2 (z, ν) = +∞ t=1 z −t (ν + 1) 2 (R(α, t, ν)) 2 , (1.5) f * 4 (z, ν) = − +∞ t=1 z −t (ν + 1) 2 ∂ ∂t (R 2 ) (t, ν), (1.6) f * 3 (z, ν) = (log(z))f * 2 (z, ν) + f * 4 (z, ν), (1.7) f k (z, ν) = f * k (z, ν)/(ν + 1) 2 where k = 1, 2, 3, 4, ν ∈ N 0 . Let (1.8) τ = τ (ν) = ν + 1, µ = µ(ν) = τ 2 = (ν + 1) 2 , (1.9) a * 1,1 (z, ν) = 1 2 (−5µ + 3µ − 6µ 2 ) − zµ(1 + 18µ)+ (3µ + µ 2 + z(7µ + 16µ 2 ))τ,
(1.10) a * 1,2 (z; ν) = −8µ − 4µ 2 − z(12µ + 20µ 2 ) + (2 + 10µ + z(2 + 26µ))τ,
(1.11) a * 1,3 (z; ν) = −1 − 14µ + z(−1 + 2µ) + (7 + 8µ + z(3 − 8µ))τ, (1.12) a * 1,4 (z; ν) = (z − 1)(2 + 12µ − 10τ ), (1.13) a * 2,1 (z; ν) = −zµ − z20µ 2 − z12µ 3 + τ (7zµ + 26zµ 2 ), zµ(24 − 22α + 5α 2 + 28µ − 2αµ), (1.14) a * 2,2 (z; ν) = −µ − 3µ 2 + zµ(−13 − 38µ)+ (3µ + µ 2 + 2z + 33zµ + 16zµ 2 )τ, (1.15) a * 2,3 (z; ν) = −8µ − 4µ 2 − z − 6zµ + 4zµ 2 + (2 + 10µ + 5z − 2zµ)τ (1.16) a * 2,4 (z; ν) = (1 + 14µ − 7τ − 8µτ )(z − 1), (1.17) a * 3,1 (z; ν) = −zµ − 21zµ 2 − 26zµ 3 + (7zµ + 33zµ 2 + 8zµ 3 )τ, (1.18) a * 3,2 (z; ν) = −14zµ − 58zµ 2 − 12zµ 3 + (2z + 40zµ + 42zµ 2 )τ, (1.19) a * 3,3 (z; ν) = −µ − 3µ 2 − z − 17zµ − 6zµ 2 + (3µ + µ 2 + 7z + 17zµ)τ (1.20) a * 3,4 (z; ν) = (8µ + 4µ 2 − 2τ − 10µτ )(z − 1), (1.21) a * 4,1 (z; ν) = −zµ − 21zµ 2 − 38zµ 3 + (7zµ + 35zµ 2 + 18zµ 3 )τ, (1.22) a * 4,2 (z; ν) = −15zµ − 79zµ 2 − 38zµ 3 + (2z + 47zµ + 75zµ 2 + 8zµ 3 )τ,
(1.23) a * 4,3 (z; ν) = −z − 31zµ − 48zµ 2 − 4zµ 3 + (9z + 53zµ + 22zµ 2 )τ,
(1.24) a * 4,4 (z; ν) = −µ − 3µ 2 − z − 9zµ − 2zµ 2 + (3µ + µ 2 + 5z + 7zµ)τ
We denote by A * (z; ν) the 4 ×4-matrix with a * i,k (z; ν) in its i-th row and k-th column for i = 1, ..., 4, k = 1, ..., 4. Clearly,
(1.25) A * (z; ν) = A * (1; ν) + (z − 1)V * (ν),
where the matrix V * (ν) does not depend from z. Let
(1.26) X k (z; ν) = f k (z, ν) δf k (z, ν) δ 2 f k (z, ν) δ 3 f k (z, ν) , X * k (z; ν) = (ν + 1) 2 X k (z; ν) for k = 1, 2, 3, |z| > 1, ν ∈ N 0 . Let further (1.27) X k (z; −ν − 2) = X k (z; ν),
where ν ∈ N 0 . Let us consider the row
(1.28) R(ν) = (r 1 (ν), r 2 (ν), r 3 (ν), r 4 (ν)), where (1.29) r 1 (ν) = µ(ν) 2 , r 2 (ν) = 0, r 3 (ν) = −2µ(ν), r 4 (ν) = 0.
We have the following equalities:
A * (z; ν) = A * 1,0 (z; ν), X k (z; ν) = X 1,0,k (z; ν), R(ν) = R 1,0 (ν),
where A * α,0 (z; ν), X α,0,k (z; ν) and R α,0 (ν) are studied in [L.A. Gutnik,5, 2006] − [L.A. Gutnik,15, 2006]. We take α = 1 in (105), [L.A.Gutnik,13, 2007], in (1), [L.A.Gutnik,15, 2006], in §10.1, [L.A.Gutnik,14, 2007], §11.3,[L.A.Gutnik,15, 2006]. Then we have the following Theorem:
Theorem 1. The column X k (z; ν) satisfies to the equation
(1.30) ν 5 X k (z; ν − 1) = A * (z; ν)X k (z; ν),
for ν ∈ M * 1 = (−∞, −2] ∪ [1, +∞)) ∩ Z, k = 1, 2, 3, |z| > 1; moreover, the matrix A * (z; ν) has the following property:
(1.31) − ν 5 (ν + 1) 5 E 4 = A * (z; −ν − 1)A * (z; ν),
where E 4 is the 4 × 4 unit matrix, z ∈ C, ν ∈ C. The Lemma 11.3.1 in [L.A. Gutnik,15, 2006] have the following formulation for α = 1 : Theorem 2. The row R(ν) has the following property:
(1.32) R(ν − 1)A * (1; ν) = ν 5 R(ν), where ν ∈ C. §2. Transformation of the system considered in §1.
In view of (1.8), (1.29)
(2.1) r 1 (ν) = µ(ν) 2 = (ν + 1) 4 = τ 4 , r 2 (ν) = 0, r 3 (ν) = −2µ(ν) = −2(ν + 1) 2 = −2τ 2 , r 4 (ν) = 0.
Let E 4 denotes 4 × 4-unit matrix, and let C(ν) is result of replacement of first row of the matrix E 4 by the row in (1.28). Let further D(ν) denotes the adjoint matrix to the matrix C(ν). Then
(2.2) C(ν) = r 1 (ν) r 2 (ν) r 3 (ν) r 4 (ν) 0 1 0 0 0 0 1 0 0 0 0 1 , (2.3) D(ν) = 1 −r 2 (ν) −r 3 (ν) −r 4 (ν) 0 r 1 (ν) 0 0 0 0 r 1 (ν) 0 0 0 0 r 1 (ν) .
Clearly,
(2.4) C(ν)D(ν) = (µ(ν)) 2 E 4 , C(−ν − 2) = C(ν), D(−ν − 2) = D(ν).
Let
(2.5) A * * (z; ν) = C(ν − 1)A * 1,0 (z; ν)D(ν).
Then
(2.6)
A * * (z; −ν − 1) = C(−ν − 2)A * (z; −ν − 1)D(−ν − 1) = C(ν)A * (z; −ν − 1)D(ν − 1),
and, in view of (2.4), (1.31), (2.5),
(2.7) A * * (z; −ν − 1)A * * 1,0 (z; ν) = C(ν)A * (z; −ν − 1)D(ν − 1)C(ν − 1)A * (z; ν)D(ν) = −(µ(ν)µ(ν − 1)) 2 (ν(ν + 1)) 5 E 4 . Let (2.8) Y k (z; ν) = C(ν)X k (z; ν),
where k = 1, 2, 3, |z| > 1, ν ∈ M * * * 1 = ((−∞, −2] ∪ [0, +∞)) ∩ Z. Then, in view of (1.27), (2.4), (1.30),
(2.9) Y k (z; −ν − 2) = Y k (z; ν), (2.10) A * * (z; ν)Y k (z; ν) = C(ν − 1)A * (z; ν)D(ν)C(ν)X k (z; ν) = µ(ν) 2 C(ν − 1)A * (z; ν)X k (z; ν) = µ(ν) 2 ν 5 C(ν − 1)X k (z; ν − 1) = µ(ν) 2 ν 5 Y k (z; ν − 1), where k = 1, 2, 3, |z| > 1, ν ∈ M * 1 = ((−∞, −2] ∪ [1, +∞)) ∩ Z. Replacing in the equality (2.10) ν ∈ M * 1 = ((−∞, −2] ∪ [1, +∞)) ∩ Z by ν := −ν − 2 ∈ M * * 1 = ((−∞, −3] ∪ [0, +∞)) ∩ Z,
and taking in account (2.9) we obtain the equality
(2.11) − A * * (z; −ν − 2)Y k (z; ν) = µ(ν) 2 (ν + 2) 5 Y k (z; ν + 1), where k = 1, 2, 3, |z| > 1, ν ∈ M * * 1 = ((−∞, −3] ∪ [0, +∞)) ∩ Z. §3. Calculation of the matrix A * * 1,0 (z; ν).
We denote by a * * i,j (1; ν), where i, j = 1, 2, 3, 4, the expressions, which stand on intersection of i-th row and j-th column in the matrix A ast * (1; ν). Let
(3.1) V * * (ν) = C(ν − 1)V * (ν)D(ν).
Then, in view of (1.25),
(3.2) A * * (z; ν) = A * * (1; ν) + (z − 1)V * * (ν),
where the matrix V * * (ν) does not depend from z. Clearly, the first row of the matrix C(ν − 1)A * (1, ν) coincides with the row R(ν − 1)A * (z, ν) and, according to the Theorem 2 coincides with the row ν 5 R(ν), i.e. with the first row of the matrix ν 5 C(ν). Therefore, in view of (2.4), the first row of the matrix A * * (1, ν) is equal to (µ 1 (ν) 2 )ν 5ē 4,1 , whereē 4,l denotes the l-th row of the matrix E 4 for l = 1, 2, 3, 4. Hence
(3.3) a * * 1,1 (1; ν) = τ 4 (τ − 1) 5 , a * * 1,k (1; ν) = 0, where k = 2, 3, 4.
Clearly, the second, third and fourth row of the matrix C(ν − 1)A * (1, ν) coincides with respectively the second, third and fourth row of A * (1, ν).
In view of (1.13), (1.17) and (1.21),
(3.4) a * * 2,1 (1; ν) = a * 2,1 (1; ν) = −(12τ 6 − 26τ 5 + 20τ 4 − 7τ 3 + τ 2 ) = −τ 2 × (τ − 1)(12τ 3 − 14τ 2 + 6τ − 1) = −τ 2 (τ − 1)(2τ − 1)(6τ 2 − 4τ + 1).
(3.5) a * * 3,1 (1; ν) = a * 3,1 (1; ν) = 8τ 7 − 26τ 6 + 33τ 5 − 21τ 4 + 7τ
3 − τ 2 = τ 2 × (τ − 1)(8τ 4 − 18τ 3 + 15τ 2 − 6τ + 1) = τ 2 (τ − 1) 2 × (8τ 3 − 10τ 2 + 5τ − 1) = τ 2 (τ − 1) 2 (2τ − 1)(4τ 2 − 3τ + 1). (3.6) a * * 4,1 (1; ν) = −τ 2 (τ − 1) 3 (2τ − 1)(2τ 2 − 2τ + 1) = a * 4,1 (1; ν) = −τ 2 (τ − 1)(4τ 5 − 14τ 4 + 20τ 3 − 15τ 2 + 6τ − 1) = −τ 2 (τ − 1) 2 (4τ 4 − 10τ 3 + 10τ 2 − 5τ + 1) = −τ 2 (τ − 1) 3 (4τ 3 − 6τ 2 + 4τ − 1).
In view of (2.2), (2.3), (2.5)
(3.7) a * * k,j (1; ν) = −r j (ν)a * k,1 (1; ν) + r 1 (ν)a * k,j (1; ν)
where j, k = 2, 3, 4. In view of (1.14), (1.18), (1.22), (1.16), (1.20), (3.7) and (1.29),
(3.8) a * * 2,2 (1; ν) = τ 4 a * 2,2 (1; ν) = τ 5 (τ − 1)(17τ 3 − 24τ 2 + 12τ − 2) = (17τ 5 − 41τ 4 + 36τ 3 − 14τ 2 + 2τ )τ 4 . (3.9) a * * 3,2 (1; ν) = τ 4 a * 3,2 (1; ν) = −2τ 5 (τ − 1) 2 (6τ 3 − 9τ 2 + 5τ 2 − 1) = −(12τ 6 + 42τ 5 − 58τ 4 + 40τ 3 − 14τ 2 + 2τ )τ 4 = −2τ 5 (6τ 5 − 21τ 4 + 29τ 3 − 20τ 2 + 7τ − 1) = −2τ 5 (τ − 1)(6τ 4 − 15τ 3 + 14τ 2 − 6τ + 1), (3.10) a * * 4,2 (1; ν) = τ 4 a * 4,2 (1; ν) = τ 5 (τ − 1) 3 (8τ 3 − 14τ 2 + 9τ − 2) = (8τ 7 − 38τ 6 + 75τ 5 − 79τ 4 + 47τ 3 − 15τ 2 + 2τ )τ 4 = τ 5 (τ − 1)(8τ 5 − 30τ 4 + 45τ 3 − 34τ 2 + 13τ − 2) = τ 5 (τ − 1) 2 (8τ 4 − 22τ 3 + 23τ 2 − 11τ + 2), (3.11) a * * k,4 (1; ν) = τ 4 a * 1,0,k,4 (1; ν) = 0
for k = 2, 3. In view of (1.24), (3.7) and (1.29),
(3.12) a * * 4,4 (1; ν) = τ 4 a * 4,4 (1; ν) = τ 4 × (τ 5 − 5τ 4 + 10τ 3 − 10τ 2 + 5τ − 1) = τ 4 (τ − 1) 5 .
In view of (1.15),
(3.13) a * 2,3 (1; ν) = 8τ 3 − 14τ 2 + 7τ − 1 = (τ − 1)(8τ 2 − 6τ + 1), In view of (1.19), (3.14) a * α,0,3,3 (1; ν) = τ 5 − 9τ 4 + 20τ 3 − 18τ 2 + 7τ − 1 = (τ − 1)(τ 4 − 8τ 3 + 12τ 2 − 6τ + 1 = (τ − 1) 2 (τ 3 − 7τ 2 + 5τ − 1),
In view of (1.23),
(3.15) a * 4,3 (1; ν) = −4τ 6 + 22τ 5 − 48τ 4 + 53τ 3 − 31τ 2 + 9τ − 1 = (τ − 1)(−4τ 5 + 18τ 4 − 30τ 3 + 23τ 2 − 8τ + 1) = (τ − 1) 2 (−4τ 4 + 14τ 3 − 16τ 2 + 7τ − 1) = (τ − 1) 3 (−4τ 3 + 10τ 2 − 6τ + 1)
In view of (3.4), (3.13), (3.5), (3.14), (3.6), (3.15), (3.7) and (1.29),
(3.16) a * * 2,3 (1; ν) = 2τ 2 a * 2,1 (1; ν) + τ 4 a * 2,3 (1; ν) = (τ − 1)× (−2τ 4 (2τ − 1)(6τ 2 − 4τ + 1) + τ 4 (2τ − 1)(4τ − 1)) = τ 4 (τ − 1)(2τ − 1)(−12τ 2 + 12τ − 3) = −3τ 4 (2τ − 1) 3 , (3.17) a * * 3,3 (1; ν) = 2τ 2 a * 3,1 (1; ν) + τ 4 a * 3,3 (1; ν) = (τ − 1) 2 × (2τ 4 (8τ 3 − 10τ 2 + 5τ − 1) + τ 4 (τ 3 − 7τ 2 + 5τ − 1)) = (τ − 1) 2 (17τ 3 − 27τ 2 + 15τ − 3) = τ 4 (τ − 1) 2 ((τ − 1) 3 + 2(2τ − 1) 3 ), (3.18) a * * 4,3 (1; ν) = 2τ 2 a * 4,1 (1; ν) + τ 4 a * 4,3 (1; ν) = (τ − 1) 3 × (−2τ 4 (4τ 3 − 6τ 2 + 4τ − 1) + τ 4 (−4τ 3 + 10τ 2 − 6τ + 1)) = −τ 4 (τ − 1) 3 (12τ 3 − 22τ 2 + 14τ − 3) = −τ 4 (τ − 1) 3 (2τ − 1)(6τ 2 − 8τ + 3). §4. Properties of the functions considered in §1.
The function t r (R(t, ν)) 2 (see (1.3)) is regular at t = ∞ for r = 0, 1, 2, and has a pole of first order at t = ∞ for r = 3. So, in the case r = 0, 1, 2 we have the equalities
(4.1) Res(t r (R(t, ν)) 2 , t = ∞) = −[r/3] for r = 0, 1, 2, 3,
(4.2) lim t→∞ t r (R(t, ν)) 2 = 0 for r = 0, 1, 2, 3.
In view of (1.4),
(4.3) δ r f * 2 (z, ν) = +∞ t=1 z −t (ν + 1) 2 (−t) r (R(t, ν)) 2 ,
where we consider r = 0, 1, 2, 3. Expanding (ν + 1) 2 (−t) r (R(t, ν)) 2 into partial fractions relatively t, we obtain
(4.4) (ν + 1) 2 (−t) r (R(t; ν)) 2 = 2 i=1 ν+1 k=0 β (r) i,k,ν (t + k) −i ,
where ν ∈ N 0 , r = 0, 1, 2, 3,
(4.5) β (r) 2−j,k,ν = (ν + 1) 2 1 j! lim t→−k ∂ ∂t j ((−t) r (R(t, ν)(t + k)) 2 )
for j = 0, 1. In view of (4.1) and (4.4),
(4.6) ν+1 k=0 β (r) 1,k,ν = −[r/3](ν + 1) 2 for r = 0, 1, 2, 3.
In view of (4.4),
(4.7) − (ν + 1) 2 ∂ ∂t ((−t) r (R(t; ν)) 2 ) = 2 i=1 ν+1 k=0 β (r) i,k,ν i(t + k) −i−1 ,
where ν ∈ N 0 , r = 0, 1, 2, 3. Let
(4.8) S i,k (ν) = − ν+k κ=k+1 1/κ i − ν+1−k κ=1 1/κ i + k κ=1 1/κ i , where ν ∈ N 0 , i ∈ N, k ∈ [0, ν + 1] ∩ Z. In particular, (4.9) S 1,0 (0) = −1, S 1,1 (0) = 1, (4.10) S 1,0 (1) = −5/2, S 1,1 (1) = −1/2, S 1,2 (1) = 7/6, (4.11) S 1,0 (2) = −(1 + 1/2) − (1 + 1/2 + 1/3) = −10/3, (4.12) S 1,1 (2) = −(1/2 + 1/3) − (1 + 1/2) + 1 = −4/3
(4.13) S 1,2 (2) = −(1/3 + 1/4) − 1 + (1 + 1/2) = −1/12, (4.14) S 1,3 (2) = −(1/4 + 1/5) + (1 + 1/2 + 1/3) = 83/60.
In view of (4.5), (1.3) and (4.8)
(4.15) β (0) 2,k,ν = (ν + k)! k! × ν + 1 (ν + 1 − k)!k! 2 = ν + 1 k 2 ν + k k 2 , (4.16) β (0) 1,k,ν = 2β (0) 2,k,ν S 1,k (ν), where ν ∈ N 0 , i ∈ N, k ∈ [0, ν + 1] ∩ Z. In particular, (4.17) β (0) 2,0,0 = β (0) 2,1,0 = β (0) 2,0,1 = 1, β (0) 2,1,1 = 16, β(0)
2,2,1 = 9,
(4.18) β (0) 2,0,1 = 1, β (0) 2,1,1 = 16, β(0)
2,2,1 = 9,
(4.19) β (0) 2,0,2 = 1, β (0) 2,1,2 = 81, β(0)
2,2,2 = 324, β
2,3,2 = 100.
(4.20) β (0) 2,2,2 = 324, β(0)
2,3,2 = 100.
In view of (4.16), (4.17) -(4.19) and (4.9) -(4.14),
(4.21) β (0) 1,k,0 = 2β (0) 2,k,0 S 1,k (0) = −2 × 1 × (−1) k for k = 0, 1, (4.22) β (0) 1,0,1 = 2β (0) 2,0,1 S 1,0 (1) = 2 × 1 × (−5/2) = −5, (4.23) β (0) 1,1,1 = 2β (0) 2,1,1 S 1,1 (1) = 2 × 16 × (−1/2) = −16, (4.24) β (0) 1,2,1 = 2β(0)
2,2,1 S 1,2 (1) = 2 × 9 × (7/6) = 21,
(4.25) β (0) 1,0,2 = 2β(0)
2,0,2 S 1,0 (2) = 2(−10/3) = −20/3,
(4.26) β (0) 1,1,2 = 2β (0) 2,1,2 S 1,1 (2) = 2 × 81 × (−4/3) = −216, (4.27) β (0) 1,2,2 = 2β (0) 2,2,2 S 1,2 (2) = 2 × 324 × (−1/12) = −54, (4.28) β (0) 1,3,2 = 2β(0)
2,3,2 S 1,3 (2) = 2 × 100 × (83/60) = 830/3.
We put in (4.4) r = 0, and multiply both sides of obtained equality by (−t) r for r = 0, 1, 2, 3. Then we see that
(4.29) − t(ν + 1) 2 (R(t; ν)) 2 = 2 i=1 ν+1 k=0 β (0) i,k,ν (−t − k + k) (t + k) i = ν+1 k=0 kβ (0) 2,k,ν (t + k) 2 + ν+1 k=0 kβ (0) 1,k,ν − β (0) 2,k,ν t + k − ν+1 k=0 β (0) 1,k,ν , (4.30) (−t) 2 (ν + 1) 2 (R(α, t; ν)) 2 = 2 i=1 ν+1 k=0 β (0) i,k,ν (t + k − k) 2 (t + k) i = ν+1 k=0 k 2 β (0) 2,k,ν (t + k) 2 + ν+1 k=0 k 2 β (0) 1,k,ν − 2kβ (0) 2,k,ν t + k + ν+1 k=0 (β (0) 2,k,ν + (t − k)β (0) 1,k,ν ), (4.31) (−t) 3 (ν + 1) 2 (R(α, t; ν)) 2 = 2 i=1 ν+1 k=0 β (0) i,k,ν (−t − k + k) 3 (t + k) i = ν+1 k=0 k 3 β (0) 2,k,ν (t + k) 2 + ν+α k=0 k 3 β (0) 1,k,ν − 3k 2 β (0) 2,k,ν t + k − ν+1 k=0 (t − 2k)(β (0) 2,k,ν − ν+1 k=0 (t 2 − kt + k 2 )β (0) 1,k,ν .
The equality (4.6) with r = 0 again follows from (4.2) with r = 1 and (4.29); moreover, in view of (4.4) with r = 1, and (4.29),
(4.32) β (1) 2,k,ν = kβ (0) 2,k,ν , β (1) 1,k,ν = kβ (0) 1,k,ν − β (0) 2,k,ν
for k = 0, ..., ν + 1. The equality (4.6) with r = 1 again follows from (4.2) with r = 2, (4.6) with r = 0, (4.30) and (4.32); moreover, in view of (4.4) with r = 2, and (4.30),
(4.33) β (2) 2,k,ν = k 2 β (0) 2,k,ν , β (2) 1,k,ν = k 2 β (0) 1,k,ν − 2kβ (0) 2,k,ν
for k = 0, ..., ν + 1. The equality (4.6) with r = 2 again follows from (4.2), (4.6) with both r ∈ {0, 1}, (4.31), (4.32) and from (4.33); moreover, in view of (4.4) with r = 3, and (4.31),
(4.34) β (3) 2,k,ν = k 3 β (0) 2,k,ν , β(3)1,k,ν = k 3 β (0) 1,k,ν − 3k 2 β (0) 2,k,ν for k = 0, ..., ν + 1. In view of (1.4) -(1.6), (4.35) (δ r )f * 3 (z, ν) = (log(z))(δ) r )f * 2 (z, ν)+ r(δ) r−1 f * 2 (z, ν) + (δ) r f * 4 (z, ν) = (log(z))(δ) r )f * 2 (z, ν)+ +∞ t=1 z −t (ν + 1) 2 r(−t) r−1 − (−t) r ∂ ∂t R 2 (t, ν) = (log(z))(δ) r )f * 2 (z, ν) − +∞ t=1 z −t (ν + 1) 2 ∂ ∂t ((−t) r R 2 )(t, ν).
In view of (4.4), (4.7), (4.3) and (4.35),
(4.36) δ r f * 2+j (z; ν) − j(log(z))δ r f * 2 (z; ν) = 2 i=1 +∞ t=1 ν+1 k=0 (1 − j + ij)β (r) i,k,ν z k z −t−k (t + k) −i−j = 2 i=1 ν+1 k=0 (1 − j + ij)β (r) i,k,ν z k +∞ t=1 z −t−k (t + k) −i−j = 2 i=1 ν k=0 (1 − j + ij)β (r) i,k,ν z k L i+1 (1/z) − k τ =1 z −τ i(τ ) −i−j = 2 i=1 (1 − j + ij)β * (r) i (z; ν)L i+j (1/z) − β * (r) 3+j (z; ν), where j = 0, 1, r = 0, 1, 2, 3, |z| > 1, (4.37) L s (1/z) = ∞ n=1 1/(z n n s ), β * (r) i (z; ν) = ν+1 k=0 β (r) i,k,ν z k , for s ∈ Z, i ∈ {1, 2}, ν ∈ N 0 , (4.38) β * (r) 3+j (z; ν) = 2 i=1 ν+1 k=0 (1 − j + ij)β (r) i,k,ν k τ =1 z k−τ (τ ) −i−j = ν+α−1 σ=0 z σ ν+1−σ τ =1 2 i=1 (1 − j + ij)β (r) i,σ+τ,ν (τ ) −i−j .
In fiew of (1.2),
(4.39) f * 1 (z, ν) = β * (0) 2 (z; ν).
In view of (4.6) and (4.37), if r = 0, 1, 2, then
(4.40) β * (r) 1 (z; ν) = (z − 1)β * ∨(r) 1 (z; ν), where β * ∨(r) 1 (z; ν) ∈ Q[z], when ν ∈ N 0 .
In view of (4.6) and (4.37),
(4.41) β * (3) 1 (z; ν) = −(ν + 1) 2 + (z − 1)β * ∨(3) 1 (z; ν), where β * ∨(3) 1 (z; ν) ∈ Q[z], when ν ∈ N 0 .
In view of (4.32) -(4.34), (4.37),
(4.42) β * (1) 2 (z; ν) = δβ * (0) 2 (z; ν) = δf * 1 (z, ν), β * (1) 1 (z; ν) = δβ * (0) 1 (z; ν) − β * (0) 2 (z; ν), (4.43) β * (2) 2 (z; ν) = δ 2 β * (0) 2 (z; ν) = δ 2 f * 1 (z, ν), β * (2) 1 (z; ν) = δ 2 β * (0) 1 (z; ν) − 2δβ * (0) (4.44) β * (3) 2 (z; ν) = δ 3 β * (0) 2 (z; ν), β * (3) 1 (z; ν) = δ 3 β * (0) 1 (z; ν) − 3δ 2 β * (0) 2 (z; ν).
Clearly,
(4.45) (−δ) k L n (1/z) = L n−k (1/z), where k ∈ [0, +∞) ∩ Z, n ∈ Z, |z| > 1, (4.46) L 1 (1/z) = − log(1 − 1/z), −δL 1 (1/z) = 1/(z − 1) = L 0 (1/z), δ 2 L 1 (1/z) = 1/(z − 1) + 1/(z − 1) 2 = L −1 (1/z), −δ 3 L 1 (1/z) = L −2 (1/z) = 1/(z − 1) + 3/(z − 1) 2 + 2/(z − 1) 3 .
We apply the operator δ to the equality (4.36) for r = 0, 1, 2. Then, in view of (4.45), we obtain the equality
(4.47) δ r+1 f * 2+j (z; ν) − j(log(z))δ r+1 f * 2 (z, ν) = jδ r f * 2 (z; ν)+ 2 i=1 ((1 − j + ij)δβ * (r) i (z; ν))L i+j (1/z) − δβ * (r) 3+j (z; ν)− 2 i=1 (1 − j + ij)β * (r) i (z; ν)L i+j−1 (1/z) = j 2 i=1 β * (r) i (z; ν)L i (1/z) − β * (r) 3 (z; ν) + 2 i=1 ((1 − j + ij)δβ * (r) i (z; ν))L i+j (1/z) − δβ * (r) 3+j (z; ν)− 2 i=1 (1 − j + ij)β * (r) i (z; ν)L i+j−1 (1/z) .
It follws from (4.47) with j = 0 that
(4.48) δ r+1 f * 2 (z; ν) = −δβ (r) 3 (z; ν)+ 2 i=1 (δβ * (r) i (z; ν))L i (1/z) − β * (r) i (z; ν)L i−1 (1/z) = (δβ * (r) 2 (z; ν))L 2 (1/z) + (δβ * (r) 1 (z; ν) − β * (r) 2 (z; ν))L 1 (1/z)− δβ * (r) 3 (z; ν) − β * (r) 1 (z; ν)L 0 (1/z).
In view of (4.36) with j = 0, (4.48), (4.40),
(4.49) β * (r) 2 (z; ν) = δβ * (r−1) 2 (z; ν)) = δ r β * (0) 2 (z; ν), (4.50) β * (r) 1 (z; ν)) = δβ * (r−1) 1 (z; ν) − β * (r−1) 2 (z; ν) = δ r β * (0) 1 (z; ν) − rδ r−1 β * (0) 2 (z; ν), (4.51) β * (r) 3 (z; ν) = δβ * (r−1) 3 (z; ν) + β * (r−1) 1 (z; ν))L 0 (1/z) = δβ * (r−1) 3 (z; ν) + β * ∨(r−1) 1 (z; ν),
where r = 1, 2, 3. The equalities (4.42) -(4.44) follow from the equalities (4.49) and (4.50) again. In view of (4.15), (4.37), (1.2) and (4.49)
(4.52) β * (r) 2 (z; ν) = δ r f * 1 (z; ν) ∈ N[z],
where ν ∈ N 0 , r = 0, 1, 2, 3. It follws from (4.47) with j = 1 that
(4.53) δ r+1 f * 3 (z, ν) = (log(z))δ r+1 f * 2 (z, ν)+ 2 i=1 β * (r) i (z; ν)L i (1/z) − β * (r) 3 (z; ν) + 2 i=1 i(δβ * (r) i (z; ν))L i+1 (1/z) − δβ * (r) 4 (z; ν)− 2 i=1 iβ * (r) i (z; ν)L i (1/z) = (log(z))δ r+1 f * 2 (z; ν)+ 2 i=1 i(δβ * (r) i (z; ν))L i+1 (1/z) − δβ * (r) 4 (z; ν)− β * (r) 2 (z; ν)L 2 (1/z) − (β * (r) 3 (z; ν) + δβ(r)
4 (z; ν)) = (log(z))δ r+1 f * 2 (z; ν) + 2(δβ * (r) 2 (z; ν))L 3 (1/z)+ (δβ * (r)
1 (z; ν) − β * (r) 2 (z; ν))L 2 (1/z) − (δβ * (r) 4 (z; ν) + β * (r) 3 (z; ν)).
In view of (4.36) with j = 1, (4.53),
(4.54) β * (r+1) 2 (z; ν) = δβ * (r) 2 (z; ν) = δ r+1 β * (0) 2 (z; ν), (4.55) β * (r+1) 1 (z; ν) = δβ * (r) 2 (z; ν) − β * (r) 2 (z; ν) = δ r+1 β (0) 1 (z; ν) − δ r β(0)
2 (z; ν), where r = 0, 1, 2, and we obtain (4.49) -(4.49) again. Moreover, (4.56) β * (r+1) 4 (z; ν) = δβ * (r) 4 (z; ν) + β * (r)
3 (z; ν),
where r = 0, 1, 2. If we take now z ∈ (1, +∞) and will tend z to 1, then, in view of (4.36), (4.40), (4.41) and (4.46)
(4.57) δ r f * α,0,2+j (1, ν) = lim z→1+0 δ r f * 2 (z, ν) = (1 − j + ij)β * (r) 2 (1; ν)ζ(2 + j) − β * (r) 3+j (1; ν) = (1 + j)β * (r) 2 (1; ν)ζ(2 + j) − β * (r) 3+j (1; ν), where r = 0, 1, 2, i = 2, j = 0, 1, (4.58) lim z→1+0 (z − 1)δ 3 f * 2 (z, ν) = 0.
In view of (4.37), (4.38), (4.17 -(4.24),
(4.59) β * (0) 1 (z; 0) = β(0)
1,0,0 + β
1,1,0 z = −2 + 2z,
(4.60) β * (0) 2 (z; 0) = β (0) 2,0,0 + β (0) 2,1,0 z = 1 + z, (4.61) β * (0) 3 (z; 0) = β (0) 1,1,0 + β (0) 2,1,0 = 3, (4.62) β * (0) 4 (z; 0) = β(0)
1,1,0 + 2β
2,0,1 + β
2,1,1 z + β
2,2,1 z 2 = 1 + 16z + 9z 2 ,
(4.65) β * (0) 3 (z; 1) = β(0)
1,1,1 + β (4.67) β * (0) 1 (z; 2) = β
1,0,2 + β
1,1,2 z+
β (0) 1,2,2 z 2 + β (0) 1,3,2 z 3 = − 20 3 − 216z − 54z 2 + 830 3 z 3 = (z − 1)(830z 2 + 668z + 20)/3. (4.68) β * (0) 2 (z; 2) = β (0) 2,0,2 + β (0) 2,1,2 z + β (0) 2,2,2 z 2 + β(0)
2,3,2 z 3 = 1 + 81z + 324z 2 + 100z 3 .
(4.69) β * (0) 3 (z; 2) = β
1,1,2 + β (4.77) β * (1) 3 (z; 1) = δβ * (0) 3 (z; 1) + β * ∨(0) 1 (z; 1) = 30z + 21z + 5 = 51z + 5, (4.78) β * (1) 4 (z; 1) = δβ * (0) 4 (z; 1) + β * (0) 3 (z; 1) = 39z + 51 4 + 30z = 69z + 51 4 ,
(4.79) β * (1) 1 (z; 2) = δβ * (0) 1 (z; 2) − β * (0) 2 (z; 2) = −216z − 108z 2 + 830z 3 − (1 + 81z + 324z 2 + 100z 3 ) = −1 − 297z − 432z 2 + 730z 3 = (z − 1)(730z 2 + 298z + 1),
(4.80) β * (1) 2 (z; 2) = δβ * (0) 2 (z; 2) = 81z + 648z 2 + 300z 3 , (4.88) β * (2) 2 (z; 1) = δβ * (1) 2 (z; 1) = 16z + 36z 2 , (4.89) β * (2) 3 (z; 1) = δβ * (1) 3 (z; 1) + β * ∨(1) 1 (z; 1) = 51z + 33z + 1 = 84z + 1, (4.90) β * (2) 4 (z; 1) = δβ * (1) 4 (z; 1) + β * (1) 3 (z; 1) = 69z + 51z + 5 = 120z + 5, (4.91) β * (2) 1 (z; 2) = δβ * (1) 1 (z; 2) − β * (1) 2 (z; 2) = −297z − 864z 2 + 2190z 3 − (81z + 648z 2 + 300z 3 ) = z(−378 − 1512z + 1890z 2 ) = 378(z − 1)z(5z + 1), (4.92) β * (2) 2 (z; 2) = δβ * (1) 2 (z; 2) = 81z + 1296z 2 + 900z 3 , (4.93) β * (0) 3 (z; 2) = δβ * (1) 3 (z; 2) + β * ∨(1) 1 (z; 2) = 2060z 2 + 656z + 730z 2 + 298z + 1 = 2790z 2 + 954z + 1, (4.94) β * (2) 4 (z; 2) = δβ * (1) 4 (z; 2) + β * (1) 3 (z; 2) = 6729z/6 + 2660z 2 + 1030z 2 + 656z + 20/3 = 3690z 2 + 3555z/2 + 20/3, §5. Auxiliary difference equation.
Let y i,k (z; ν) denotes i−th element of the column Y k (z; ν) in (2.8). Then, in view of (1.26), (2.2), (2.8),
(5.1) y j+1−κ,k (z, ν) = δ j f k (z, ν), y 4,k (z, ν) = δ 3 f k (z, ν),
where j = 1, 2, k = 1, 2, 3, |z| > 1, ν ∈ N 0 . We denote v * * i,j (ν) the expression, which stands in the matrix V * * (ν) in intersection of i-th row and j-th column, where i = 1, 2, 3, 4, j = 1, 2, 3, 4. Let (5.2) D(z, ν, w) = z(w 2 − µ) 2 − w 4 , µ = (ν + 1) 2
In view of (1.29)
(5.3) (1/z)D(z, ν, w) = (1 − 1/z)w 4 + 3 k=0 r k+1 (ν)w k .
It follows from general properties of Mejer's functions that
(5.4) D(z, ν, δ)f k (z, ν) = 0,
where |z| > 1, −3π/2 < arg(z) ≤ π/2, log(z) = ln(|z|) + i arg(z), k = 1, 2, 3. Therefore, in view of (1.26), (2.2), (2.8),
(5.5) y k (z, ν) = −(1 − 1/z)δ 4 f k (z, ν)
where |z| > 1, −3π/2 < arg(z) ≤ π/2, log(z) = ln(|z|) + i arg(z), k = 1, 2, 3.
In view of (1.2) -(1.6), (4.35),
(5.6) lim z→1+0 (z − 1)δ 4 f 2 (z, ν) = lim z→1+0 (z − 1)(O(1) ln(1 − 1/z) + 1/(z − 1)) = 1, (5.7) lim z→1+0 (z − 1)δ 4 f k (z, ν) = 0, if k − 2 = ±1, (5.8) lim z→1+0 (log(z))δ i f k (z, ν) = lim z→1+0 (z − 1)δ i f k (z, ν) = 0,
if i = 0, 1, 2, 3, k = 1, 2, 3. Hence, if we tend z ∈ (1, +∞) to 1, then, in view of (1.26), (2.2), we obtain the equalities (5.9) y 1,1 (1, ν) = y 1,3 (1, ν) = 0, y 1,2 (1, ν) = −1.
In view of (2.10), (3.3) -(3.18), (5.1), (5.5),
(5.10) − a * * i,1 (1; ν)(1 − 1/z)δ 4 f k (z, ν) + 2 j=1 a * * i+1,j+1 (1; ν)δ j f k (z, ν) − (z − 1)v * * i,1 (ν)(1 − 1/z)δ 4 f k (z, ν)+ (z − 1) 2 j=1 v * * i+1,j+1 (ν)δ j f k (z, ν) = µ 1 (ν) 2 ν 5 δ i f k (z, ν − 1),
where i = 1, 2, k = 1, 2, 3, |z| > 1, −3π/2 < arg(z) ≤ π/2 and ν run over the set M * 1 = ((−∞, −2] ∪ [1, +∞)) ∩ Z. We tend z ∈ (1, +∞) to 1 now and obtain the equalities
(5.11) a * * i+1,1 (1; ν)(k − 1)(k − 3) + 2 j=1 a * * i+1,j+1 (1; ν)(δ j f k )(1, ν) = µ 1 (ν) 2 ν 5 δ i f k (1, ν − 1),
where i = 1, 2, k = 1, 2, 3 and ν ∈ M * 1 = ((−∞, −2] ∪ [1, +∞)) ∩ Z. Let are given
(5.12) F = {F (ν)} +∞ ν=−∞ and G = {G(ν)} +∞ ν=−∞ such that (5.13) F (−ν − 2) = F (ν), G(−ν − 2) = G(ν), F (ν) ∈ R, G(ν) ∈ R for ν ∈ Z. Let further (5.14) y * * F,G (z, ν) = F (ν)δf k (z, ν) + G(ν)δ 2 f k (z, ν)
for k = 1, 2, 3 and ν ∈ M * * * 1 = ((−∞, −2] ∪ [0, +∞)) ∩ Z. Since F and G have the property (5.13), it follows from (2.9)) that (5.15) y * * F,G (z, −ν − 2) = y * * F,G (z, ν)
for k = 1, 2, 3 and ν ∈ M * * *
1 = ((−∞, −2] ∪ [0, +∞)) ∩ Z. Let (5.16) a * * * F,G,j (z; ν) = F (ν − 1)a * * 2,j (z; ν) + G(ν − 1)a * * 3,j (z, ν)
for ν ∈ M * 1 = ((−∞, −2] ∪ [1, +∞)) ∩ Z, j = 1, 2, 3. In view of (5.11) (5.17), and taking in account (2.9) we obtain the equalities
(5.17) a * * * F,G,1 (1; ν)(k − 1)(k − 3)+ 2 j=1 a * * * F,G,j+1 (1; ν)(δ j f 1,0,k )(1, ν) = µ 1 (ν) 2 ν 5 y * * F,G (1, ν − 1), with ν ∈ M * 1 = ((−∞, −2] ∪ [1, +∞)) ∩ Z, k = 1, 2, 3. Replacing ν ∈ M * 1 by ν := −ν − 2 ∈ M * * 1 = ((−∞, −3] ∪ [0, +∞)) ∩ Z in(5.18) a * * * F,G,1 (1; −ν − 2)(k − 1)(k − 3)+ 2 j=1 a * * * F,G,j (1; −ν − 2)(δ j f k )(1, ν) = −µ 1 (ν) 2 (ν + 2) 5 y * * F,G (z, ν + 1),
where k = 1, 2, 3 and ν ∈ M * * 1 = ((−∞, −3] ∪ [0, +∞)) ∩ Z. Let
(5.19) w F,G,j (ν) = a * * * F,G,j+1 (1; −ν − 2) F (ν)(2 − j) + G(ν)(j − 1) a * * * F,G,j+1 (1; ν) , where j = 1, 2, ν ∈ M * * * * 1 = ((−∞, −3] ∪ [1, +∞)) ∩ Z, (5.20) W F,G (ν) = a * * * F,G,2 (1; −ν − 2) a * * * F,G,3 (1; −ν − 2) F (ν) G(ν) a * * * F,G,2 (1; ν) a * * * F,G,3 (1; ν) = w F,G,1 (ν) w F,G,2 (ν) , Y * * * k (ν) = (δf k )(1, ν) (δ 2 f k )(1, ν) , (5.21) Y * * * * F,G,k (ν) = µ 1 (−ν − 2) 2 (−ν − 2) 5 y * * F,G (z, −ν − 3) y * * F,G (z, ν) µ 1 (ν) 2 ν 5 y * * F,G (z, ν − 1) ,
where k = 1, 3, ν ∈ M * * * * 1 = ((−∞, −3] ∪ [1, +∞)) ∩ Z. Let further
(5.22) w F,G,3 (ν) = w F,G,3,1 (ν) w F,G,3,2 (ν) w F,G,3,3 (ν) = [ w F,G,1 (ν), w F,G,2 (ν)].
is vector product of w F,G,1 (ν) and w F,G,2 (ν). Letw F,G,3 (ν) = ( w F,G,3 (ν)) t is the row conjugate to the column w F,G,3 (ν). Then for scalar products ( w F,G,3 (ν), w F,G,j (ν)) we have the equalities
w F,G,3 (ν) w F,G,j (ν) = ( w F,G,3 (ν), w F,G,j (ν)) = 0, where ν ∈ M * * * * 1 = ((−∞, −3] ∪ [1, +∞)) ∩ Z, j = 1, 2. Therefore (5.23)w F,G,3 (ν)W F,G (ν) = 0 0 , where ν ∈ M * * * * 1 = ((−∞, −3] ∪ [1, +∞)) ∩ Z.
In view of (5.11) (5.18) and (5.23),
(5.24)w F,G,3 (ν)Y * * * * F,G,k (ν) =w F,G,3 (ν)W F,G,3 (ν)Y * * * k (ν) = 0,
where k = 1, 3 and ν ∈ M * * * *
= −ν − 1 = −τ ν = −τ, it follows that (5.25) a * * * F,G,1 (1; ν) = F (ν − 1)a * * 2,1 (1, ν) + G(ν − 1)a * * 3,1 (1, ν) = −F (ν − 1)τ 2 (τ − 1)(2τ − 1)(6τ 2 − 4τ + 1)+ G(ν − 1)τ 2 (τ − 1) 2 (2τ − 1)(4τ 2 − 3τ + 1),
(5.26) a * * * F,G,1 (1; −ν − 2) = F (ν + 1)a * * 2,1 (1, −ν − 2) + G(ν + 1)a * * 3,1 (1, −ν − 2) = −F (ν + 1)τ 2 (τ + 1)(2τ + 1)(6τ 2 + 4τ + 1)− −G(ν + 1)τ 2 (τ + 1) 2 (2τ + 1)(4τ 2 + 3τ + 1),
(5.27) a * * * F,G,2 (1; ν) = F (ν − 1)a * * 1,0,2,2 (1, ν) + G(ν − 1)a * * 3,2 (1, ν) = F (ν − 1)τ 5 (τ − 1)(τ 3 + 2(2τ − 1) 3 )− 2G(ν − 1)τ 5 (τ − 1) 2 (2τ − 1)(τ 3 − (τ − 1) 3 ),
(5.28) a * * * F,G,2 (z; −ν − 2) = F (ν + 1)a * * 2,2 (1, −ν − 2) + G(ν + 1)a * * 3,2 (1, −ν − 2) = −F (ν + 1)τ 5 (τ + 1)(τ 3 + 2(2τ + 1) 3 )− G(ν + 1)2τ 5 (τ + 1) 2 (2τ + 1)((τ + 1) 3 − τ 3 ),
(5.29) a * * * F,G,3 (1; ν) = F (ν − 1)a * * 2,3 (1, ν) + G(ν − 1)a * * 3,3 (1, ν) = −3F (ν − 1)τ 4 (τ − 1)(2τ − 1) 3 + G(ν − 1)τ 4 (τ − 1) 2 ((τ − 1) 3 + 2(2τ − 1) 3 ),
(5.30) a * * * F,G,3 (z; −ν − 2) = F (ν + 1)a * * 2,3 (1, −ν − 2) + G(ν + 1)a * * 3,3 (1, −ν − 2) = −3F (ν + 1)τ 4 (τ + 1)(2τ + 1) 3 − G(ν + 1)τ 4 (τ + 1) 2 (2τ + 1)((τ + 1) 3 + 2(2τ + 1) 3 ),
We consider the case now, when F and G are constant sequences, or, equivalently, real constants. In view of (5.19), (5.27) -(5.30)
(5.31) w 1,0,j (ν) = a * * * 1,0,j+1 (1; −ν − 2) 2 − j a * * * 1,0,j+1 (1; ν), , where j = 1, 2, ν ∈ M * * * * 1 = ((−∞, −3] ∪ [1, +∞)) ∩ Z, (5.32) w 1,0,1 (ν) = a * * * 1,0,2 (1; −ν − 2) 1 a * * * 1,0,2 (1; ν) = −τ 5 (τ + 1)(τ 3 + 2(2τ + 1) 3 ) 1 τ 5 (τ − 1)(τ 3 + 2(2τ − 1) 3 ) , (5.33) w 1,0,2 (ν) = a * * * 1,0,3 (1; −ν − 2) 0 a * * * 1,0,3 (1; ν) = −3τ 4 (τ + 1)(2τ + 1) 3 0 −3τ 4 (τ − 1)(2τ − 1) 3 , (5.34) w 0,1,j (ν) = a * * * 0,1,j+1 (1; −ν − 2) j − 1 a * * * 0,1,j+1 (1; ν), , where j = 1, 2, ν ∈ M * * * * 1 = ((−∞, −3] ∪ [1, +∞)) ∩ Z,(5.
35) w 0,1,1 (ν) = a * * * 0,1,2 (1; −ν − 2) 0 a * * * 0,1,2 (1; ν)
= −2τ 5 (τ + 1) 2 (2τ + 1)((τ + 1) 3 − τ 3 ) 0 −2τ 5 (τ − 1) 2 (2τ − 1)(τ 3 − (τ − 1) 3 ) , (5.36) w 0,1,2 (ν) = a * * * 0,1,3 (1; −ν − 2) 1 a * * * 0,1,3 (1; ν) = −τ 4 (τ + 1) 2 (2τ + 1)((τ + 1) 3 + 2(2τ + 1) 3 ) 1 τ 4 (τ − 1) 2 ((τ − 1) 3 + 2(2τ − 1) 3 ) ,
and,since F, G are constants now, it follows that (5.37) w F,G,j (ν) = F w 1,0,j (ν) + G w 0,1,j (ν)
where j = 1, 2, ν ∈ M * * * * 1 = ((−∞, −3] ∪ [1, +∞)) ∩ Z. In view of (5.22),
(5.38) w F,G,3 (ν) = F 2 [ w 1,0,1 (ν), w 1,0,2 (ν)]+ F G([ w 1,0,1 (ν), w 0,1,2 (ν)] + [ w 0,1,1 (ν), w 1,0,2 (ν)]) + G 2 [ w 0,1,1 (ν), w 0,1,2 (ν)].
For any
a = a 1 a 2 a 3
we put ( a) i = a i for i = 1, 2, 3. Let further (5.39) w i,j,4 (ν) = [ w i,1−i,1 (ν), w j,1−j,2 (ν)], with i = 0, 1, j = 0, 1. In view of (5.39),(5.27)-(5.30), w 1,1,4 (ν) = [ w 1,0,1 (ν), w 1,0,2 (ν)] = w 1,0,3 (ν)], (5.40) ( w 1,1,4 (ν)) 1 = ( w 1,0,3 (ν)) 1 = det 1 0 a * * * 1,0,2 (1; ν) a * * * 1,0,3 (1; ν) = a * * * 1,0,3 (1; ν) = a * * 2,3 (1, ν) = −3τ 4 (τ − 1)(2τ − 1) 3 , (5.41) ( w 1,1,4 (ν)) 2 = ( w 1,0,3 (ν)) 2 = − det a * * * 1,0,2 (1; −ν − 2) a * * * 1,0,3 (1; −ν − 2) a * * * 1,0,2 (1; ν) a * * * 1,0,3 (1; ν) = − det a * * 2,2 (1; −ν − 2) a * * 2,3 (1; −ν − 2) a * * 2,2 (1; ν) a * * 2,3 (1; ν) = a * * 2,2 (1; ν)a * * 2,3 (1; −ν − 2) − a * * 2,3 (1; ν)a * * 2,3 (1; −ν − 2) = τ 5 (τ − 1)(τ 3 + 2(2τ − 1) 3 )(−3τ 4 (τ + 1)(2τ
+ 1) 3 )− (−3τ 4 (τ − 1)(2τ − 1) 3 )(−τ 5 (τ + 1)(τ 3 + 2(2τ + 1) 3 )) = −3τ 9 (τ 2 − 1)(τ 3 ((2τ − 1) 3 + (2τ + 1) 3 ) + 4(4τ 2 − 1) 3 ) = −12τ 9 (τ 2 − 1)(68τ 6 − 45τ 4 + 12τ 2 − 1),
(5.42) ( w 1,1,4 (ν)) 3 = ( w 1,0,3 (ν)) 3 = det a * * * 1,0,2 (1; −ν − 2) a * * * 1,0,3 (1; −ν − 2) 1 0 = −a * * * 1,0,3 (1; −ν − 2) = −a * * 2,3 (1, −ν − 2) == 3τ 4 (τ + 1)(2τ + 1) 3 . In view of (5.39),(5.27)-(5.30), w 0,0,4 (ν) = [ w 0,1,1 (ν), w 0,1,2 (ν)] = w 0,1,3 (ν),
(5.43) ( w 0,0,4 (ν)) 1 = ( w 0,1,3 (ν)) 1 = det 0 1 a * * * 0,1,2 (1; ν) a * * * 0,1,3 (1; ν) = −a * * * 0,1,2 (1; ν) = −a * * 3,2 (1, ν) = 2τ 5 (τ − 1) 2 (2τ − 1)(τ 3 − (τ − 1) 3 ),
(5.44) ( w 0,0,4 (ν)) 2 = ( w 0,1,3 (ν)) 2 = − det a * * * 0,1,2 (1; −ν − 2) a * * * 0,1,3 (1; −ν − 2) a * * * 0,1,2 (1; ν) a * * 1 * 0,1,3 (1; ν) = − det a * * 3,2 (1; −ν − 2) a * * 3,3 (1; −ν − 2) a * * 3,2 (1; ν) a * * 3,3 (1; ν) = a * * 3,2 (1; ν)a * * 3,3 (1; −ν − 2) − a * * 3,3 (1; ν)a * * 3,2 (1; −ν − 2) = −2τ 5 (τ − 1) 2 (2τ − 1)(τ 3 − (τ − 1) 3 )× (−τ 4 (τ + 1) 2 ((τ + 1) 3 + 2(2τ + 1) 3 )− (5.45) ( w 0,0,4 (ν)) 3 = ( w 0,1,3 (ν)) 3 = det a * * * 0,1,2 (1; −ν − 2) a * * * 0,1,3 (1; −ν − 2) 0 1 = a ( 3, 2) * * (1; −ν − 2) = −2τ 5 (τ + 1) 2 (2τ + 1)((τ + 1) 3 − τ ) 3 ),
(−2τ 5 (τ + 1) 2 (2τ + 1)((τ + 1) 3 − τ 3 ))× (τ 4 (τ − 1) 2 ((τ − 1) 3 + 2(2τ − 1) 3 ) = 4τ 9 (τ 2 − 1) 2 (102τ 6 − 68τ 4 + 21τ 2 − 3),
In view of (5.39),(5.27)-(5.30),
w 0,1,4 (ν) = [ w 0,1,1 (ν), w 1,0,2 (ν)],
(5.46) ( w 0,1,4 (ν)) 1 = ([ w 0,1,1 (ν), w 1,0,2 (ν)]) 1 = det 0 0 a * * * 0,1,2 (1; ν) a * * * 1,0,3 (1; ν) = 0, (5.47) ( w 0,1,4 (ν)) 2 = ([ w 0,1,1 (ν), w 1,0,2 (ν)]) 2 = − det a * * * 0,1,2 (1; −ν − 2) a * * * 1,0,3 (1; −ν − 2) a * * * 0,1,2 (1; ν) a * * * 1,0,3 (1; ν) = −a * * 3,2 (1; −ν − 2)a * * 2,3 (1; ν) + a * * 3,2 (1; ν)a * * 2,3 (1; −ν − 2) = −12t 9 (τ 2 − 1)(4τ 2 − 1)(12τ 4 − 6τ 2 + 1),
(5.48) ( w 0,1,4 (ν)) 3 = ([ w 0,1,1 (ν), w 1,0,2 (ν)]) 3 , det a * * * 0,1,2 (1; −ν − 2) a * * * 1,0,3 (1; −ν − 2) 0 0 = 0,
In view of (5.39),(5.27)-(5.30),
w 1,0,4 (ν) = [ w 1,0,1 (ν), w 0,1,2 (ν)],
(5.49) ( w 1,0,4 (ν)) 1 = ([ w 1,0,1 (ν), w 0,1,2 (ν)]) 1 det 1 1 a * * * 1,0,2 (1; ν) a * * * 0,1,3 (1; ν) = a * * 3,3 (1; ν) − a * * 2,2 (1; ν) = −t 4 (t − 1)(2t − 1)(10t 2 − 10t + 3), (5.50) ( w 1,0,4 (ν)) 2 = ([w 1,0,1 (ν), w 0,1,2 (ν)]) 2 = − det a * * * 1,0,2 (1; −ν − 2) a * * * 0,1,3 (1; −ν − 2) a * * * 1,0,2 (1; ν) a * * * 0,1,3 (1; ν) = −a * * 2,2 (1; −ν − 2)a * * 3,3 (1; ν) + a * * 2,2 (1; ν)a * * 3,3 (1; −ν − 2) = −4t 9 (t 2 − 1)(170t 6 − 104t 4 + 30t 2 − 3), (5.51) ( w 1,0,4 (ν)) 3 = ([ w 1,0,1 (ν), w 0,1,2 (ν)]) 3 = det a * * * 1,0,2 (1; −ν − 2) a * * * 0,1,3 (1; −ν − 2) 1 1 = a * * 2,2 (1; −ν − 2) − a * * 3,3 (1; −ν − 2) = t 4 (t + 1)(2t + 1)(10t 2 + 10t + 3),
(5.52) ( w 0,1,4 (ν)) 2 + ( w 1,0,4 (ν)) 2 = −8t 9 (t 2 − 1)(157t 6 − 106t 4 + 30t 2 − 3).
Therefore,
(5.53) ( w F,G,3 (ν)) 1 = −3τ 4 (τ − 1)(2τ − 1) 3 F 2 − t 4 (τ − 1)(2τ − 1)(10τ 2 − 10τ + 3)F G+ 2τ 5 (τ − 1) 2 (2τ − 1)(τ 3 − (τ − 1) 3 )G 2 = τ 4 (τ − 1)(2τ − 1)× (−3(2τ − 1) 2 F 2 − (10τ 2 − 10τ + 3)F G + 2(τ − 1)(3τ 3 − 3τ 2 + τ )G 2 ) = τ 4 (τ − 1)(2τ − 1)× (−3(2τ − 1) 2 F 2 − (10τ 2 − 10τ + 3)F G + 2(3τ 4 − 6τ 3 + 4τ 2 − τ )G 2 , (5.54) ( w F,G,3 (ν)) 2 = −12τ 9 (τ 2 − 1)(68τ 6 − 45τ 4 + 12τ 2 − 1)F 2 − 8t 9 (t 2 − 1)(157t 6 − 106t 4 + 30t 2 − 3)F G+ 4τ 9 (τ 2 − 1) 2 (102τ 6 − 68τ 4 + 21τ 2 − 3)G 2 = −12τ 9 (τ 2 − 1)(68τ 6 − 45τ 4 + 12τ 2 − 1)F 2 − 8τ 9 (τ 2 − 1)(157τ 6 − 106τ 4 + 30τ 2 − 3)F G+ 4τ 9 (τ 2 − 1)(102τ 8 − 170τ 6 + 89τ 4 − 24τ 2 + 3)G 2 ,
(5.55) ( w F,G,3 (ν)) 3 = 3τ 4 (τ + 1)(2τ + 1) 3 F 2 + t 4 (t + 1)(2t + 1)(10t 2 + 10t + 3)F G − 2τ 5 (τ + 1) 2 (2τ + 1)((τ + 1) 3 − τ 3 )G 2 = τ 4 (τ + 1)(2τ + 1)× 3(2τ + 1) 2 F 2 + (10τ 2 + 10τ + 3)F G − 2(τ + 1)(3τ 3 + 3τ 2 + τ )G 2 = τ 4 (τ + 1)(2τ + 1)× 3(4τ 2 + 4τ + 1)F 2 + (10τ 2 + 10τ + 3)F G − 2(3τ 4 + 6τ 3 + 4τ 2 + τ )G 2 .
According to (5.14), (5.24), (5.21), (5.53), (5.54), (5.55), (5.56) − τ 4 (τ + 1) 5 w F,G,3,1 (ν)y * * F,G,k (ν + 1)+ w F,G,3,2 (ν)y * * F,G,k (ν) + τ 4 (τ − 1) 5 w F,G,3,3 (ν)y * * F,G,k (ν − 1) = 0. Since f 1,0,k (1, ν) = f * 1,0,k (1, ν)/(ν + 1) 2 , it follows from (5.56), (0.8) -(0.10) that (5.57) c F,G,2 (ν)x(ν + 1) + c F,G,1 (ν)x(ν) + c F,G,0 (ν)x(ν − 1) = 0
for x(ν) = x F,G,k (ν), where (5.58) x F,G,k (ν) = F δf * k (1, ν) + Gδ 2 f * k (1, ν)), k = 1, 3. Let (5.59) β * * F,G,i (z; ν) := F β * (1) 2i (z; ν) + Gβ * (2)
2i (z; ν)
for i = 1, 2. In view of (4.52),
β * * F,G,1 (1; ν) := F β * (1) 2 (1; ν) + Gβ * (2) 2 (1; ν) = F δf * 1 (1, ν) + Gδ 2 f * 1 (1, ν) = x F,G,1 (ν)
In view of (4.36) with j = 1 and (5.58),
(5.60) x F,G,3 (ν) = 2ζ(3)β * * F,G,1 (1; ν) − β * * F,G,2 (1; ν)
Before to complete the proof of Theorem B, we want to check equality (5.57) for ν = 1, k = 3. In view of (5.60), to check the equation (5.57) for ν = 1 it is sufficient to check the equalities 1,0,4 (1; 0) = 2, β * * F,G,1 (1, 2) = 1029F + 2277G, (5.61) β * * F,G,1 (1, 1) = 34F + 52G, β * * F,G,1 (1, 0) = F + G, β * * F,G,2 (1, 2) = (14843F + 32845G)/6, (5.62) β * * F,G,2 (1, 1) = (327F + 500G)/4, β * * F,G,2 (1, 0) = 3F + 2G.
c F,G,2 (1)β * * F,G,i (2) + c F,G,1 (1)β * * F,G,i (1) + c F,G,0 (1)β * * F,Let further ∆ i,ν (F, G) = c F,G,2 (ν)β F,G,i * * (ν + 1)+ c F,G,1 (ν)β F,G,i * * (ν) + c F,G,0 (ν)β F,G,i * * (ν − 1)
for i = 1, 2 ν ∈ N. We check now that ∆ i,1 (F, G) = 0 for i = 1, 2. We note that ∆ i,1 (F, G) is homogenous polynomial relatively variables F, G of degree equal to 3. To establish the equalities ∆ i,1 (F, G) = 0with i = 1, 2 it is sufficient to check them in four points (F, G) = (0, 1), (1, 0), (1, 1), (1, 2).
We have c 0,1,2 (1) = −54 × 28 = −12 × 126, c 0,1,1 (1) = 12 × 5521, c 0,1,0 (1) = −12 × 190, β * * 0,1,1 (2) = 2277, β * * 0,1,1 (1) = 52, β * * 0,1,1 (0) = 1, β * * 0,1,2 (2) = 32845/6, β * * 0,1,2 (1) = 125, β * * 0,1,2 (0) = 2, (2) = 2 × 1653, β * * 1,1,1 (1) = 2 × 43, β * * 1,1,1 (0) = 2, β * * 1,1,2 (2) = 7948, β * * 1,1,2 (1) = 827/4, β * * 1,1,2 (0) = 5,
c 1,1,0 (1) = −18 × 395, β * * 1,2,1 (2) = 3 × 1861, β * * 1,2,1 (1) = 3 × 46, β * * 1,2,1 (0) = 3,
where ν ∈ N. If F + G = 0, then the equation 6.10 is equivalent to the equation
(6.11) c a F,G,2 st(ν)x ν+1 + c * F,G,1 (ν)x ν + c * F,G,0 (ν)x ν−1 = 0,
It follows from (5.57) that
x ν = x F,G,k (ν) = F δf * 1,0,k (1, ν) + Gδ 2 f 1,0,k (1, ν)
satisfies to the equation (6.10) for ν ∈ N and fixed k ∈ {1, 3}. If G = 0, then, in view of (0.8) -(0.10),
c F,G,2 (ν) = −12τ 8 G 2 (1 + o(1))(τ → ∞), c F,G,1 (ν) = 408τ 8 G 2 (1 + o(1))(τ → ∞), c F,G,0 (ν) = −12τ 8 G 2 (1 + o(1))(τ → ∞).
If G = 0, then, in view of (0.8) -(0.10),
c F,G,2 (ν) = 24τ 6 F 2 (1 + o(1))(τ → ∞), c F,G,1 (ν) = −24 × 64τ 6 F 2 (1 + o(1))(τ → ∞), c F,G,0 (ν) = 24τ 6 F 2 (1 + o(1))(τ → ∞).
In any case the equation (6.10) is difference equation of Poincaré type with characteristic polynomial λ 2 −34λ+1. Hence, if {x ν } +∞ ν=1 is a non-zero solution of (6.10), ε ∈ (0, 1), then there are C 1 (ε) > 0 and C 2 (ε) > 0 such that only two possibilities exist:
(6.12) C 1 (ε) 1 + √ 2 −4ν(1+ε) ≤ |x ν | ≤ C 2 (ε) 1 + √ 2 −4ν(1−ε)
for all ν ∈ N or (6.13)
C 1 (ε) 1 + √ 2 4ν(1−ε) ≤ |x ν | ≤ C 2 (ε) 1 + √ 2 4ν(1+ε)
. for all ν ∈ N. In view of (4.52), if x ν = β * (r) (1; ν) = δ r f * 1 (1, ν) with r = 0, 1, 2, then (6.12) is impossible. Therefore (6.14)
C 1 (ε) 1 + √ 2 4ν(1−ε) ≤ β * (r) (1; ν) ≤ C 2 (ε) 1 + √ 2 4ν(1+ε)
.
for r = 0, 1, 2, and all ν ∈ N. Lemma 6.1. The following equalities hold:
(6.15) lim ν→∞ β * (1) 1,0,2 (1; ν)) = +∞, lim ν→∞ β * (2) 2
(1; ν)/β * (1) 2
(1; ν) = +∞,
Proof. The first of equalities (6.15) is obvious. We prove the second of equalities (6.15). According to Stirling's formula: log(n!) = (n + 1/2) log(n + 1) − n + O(1).
Let β > 0, n = βν + η 1,ν ∈ Z, (6.16) ν ∈ [(2(β + 1)/β, +∞) ∩ Z, |η ν | < 2.
Then (for fixed β) (6.17) log(n!) = (βν + η ν + 1/2) log(βν + η ν + 1) − βν + O(1) = (βν + η ν + 1/2) log(βν) − βν + O(1) = (βν) log(βν) − βν + O(1) log(ν).
Clearly, log((ν + 1)!) = ν log(ν) − ν + O(1) log(ν + 1).
Let further γ ∈ (0, 1), k = [γν], where ν ∈ [2(γ + 1)/γ, +∞) ∩ Z. Then, in view of (6.16) -(6.17) with β = γ and k in the role of n, (6.18) log(k!) = γν log(γν) − γν + O(1) log(ν).
Further we have ν + 1 − k = ν + 1 − γν + {γν} = (1 − γ)ν + 1 + {γν}. If ν ∈ [2(2 − γ)/(1 − γ), +∞) ∩ Z, then, according to (6.16) -(6.17) with β = 1 − γ, and n + 1 − k in the role of n, we have the equality (6.19) log((ν + 1 − k)!) = (1 − γ)ν log((1 − γ)ν) − (1 − γ)ν + O(1) log(ν).
Since ν + k = ν + γν − {γν} = (1 + γ)ν − {γν} it follows that, if ν ∈ [2(2 + γ)/(1 + γ), +∞) ∩ Z, then, in view of (6.16) -(6.17) with β = 1 + γ and n + k in the role of n, we have the equality (6.20) log((ν + k)!) = (1 + γ)ν log((1 + γ)ν) − (1 + γ)ν + O(1) log(ν).
Let ν ∈ [2(1/ min(γ, 1 − γ) + 1). Then (6.18) -(6.20) hold, and, moreover, log((ν + 1)!) = ν log ( Therefore, if ν ∈ [2/ min(γ, 1 − γ) + 2, +∞), k = [γν], then (6.21) log ν + 1 k ν + k k = ψ 1 (γ)ν + O(1) log(ν),
where ψ 1 (γ) = (1 + γ) log(1 + γ) − (1 − γ) log(1 − γ) − 2γ) log(γ). Clearly, d dγ ψ 1 (γ) = log((1 − γ 2 )/γ 2 )), and ψ 1 (γ) < ψ 1 1/ √ 2 = log 1 + 1/ √ 2 / 1 − 1/ √ 2 + 1/ √ 2 log 1 + 1/ √ 2 1 − 1/ √ 2 − 1/ √ 2 log(1/2) = 2 log 1 + √ 2 ,
if γ = 1/ √ 2. We can rewrite (6.14) in the form (6.22) C 1 (ε) exp 2ψ 1 1/ √ 2 ν(1 − ε) ≤ β * (r) (1; ν) ≤ C 2 (ε) exp 2ψ 1 1/ √ 2 ν(1 + ε)
for r = 0, 1, 2, and all ν ∈ N.
In view of (4.15), let ψ 2,ν (k) = β
2,k+1,ν /β
2,k,ν = (ν + k + 1)(ν − k + 1)/(k + 1) 2 . Then ψ 2,ν (k) = 1 if and only if 2k 2 + 2k − ν(ν + 2) = 0. Let r(ν) = −1/2 + 1/4 + ν(ν + 2)/2 = (1 + O(1)/ν)ν/ √ 2 Clearly, β
2,k,ν increases together with increasing of k ∈ [0, r(ν)] ∩ Z and β (0) 2,k,ν decreases together with increasing of k ∈ (r(ν), ν + 1] ∩ Z. We fix γ 1 ∈ 0, 1/ √ 2 and γ 2 ∈ 1/ √ 2, 1 . Clearly, there exists ν 0 ∈ N such that γ 1 < r(ν)/ν < γ 2 for all ν ∈ [ν 0 , +∞) ∩ N. Let ν > max(ν 0 , 1/γ 1 , 1/(1 − γ 2 )). Then, in view of (6.21) we have 2,k,ν = exp(2ψ 1 (γ 2 )ν + O(1) log(ν)).
Let γ 3 = min(ψ 1 (1/ √ 2) − ψ 1 (γ 1 ), ψ 1 (1/ √ 2) − ψ 1 (γ 2 )).
Then, in view of (6.22) β (r) 2 (1, ν) = γ 1 ν<k<γ 2 ν k r β (0) 2,k,ν
×
(1 + O(1) exp(−2γ 3 ν + O(log(ν)).
Hence, there exists ν 1 ∈ N such that for all ν ∈ [ν 1 , +∞)capZ we have So, β
2 (1, ν) ≥ [γ 1 ν]β
(2) 2 (1, ν)(1 + o(1)), when ν → +∞. . Let conditions Theorem B are fulfilled. Then, in view of (0.19), (6.23) β * * F,G,1 (1; ν) = (F + G)β * (1) 1,0,2 (1; ν) + G(β * (2) 1,0,2 (1; ν) − β * (1) 1,0,2 (1; ν)) = 0 for ν ∈ N 0 , and therefore, in view of (6.15), (6.24) β * F,G,1 (1; ν) = β * (1) 2
(1; ν)(F + Gβ * (2) 2
(1; ν)/β * (1) 2 (1; ν)) → ∞, when ν → ∞. Moreover, if F = 0 and G/F ∈ B then (6.23) and (6.24) hold, and, if in this case x ν = x F,G,1 (ν) = β * * F,G,1 (1; ν), then (6.12) is impossible. In view of (1.3) with α = 1, (4.3) and (4.36) with j = 1, δ r f * 1,0,3 (1, ν) = (ν + 1) 2 O(1);
hence, if x ν = x F,G,3 (ν) = F δf * 1,0,3 (1, ν) + Gδ 2 f * 1,0,3 (1, ν), then (6.13) is impossible. Therefore (6.25) C 1 (ε)/C 2 (ε)
1 + √ 2 8ν(1+ε) ≤ 2ζ(3) − β * * F,G,2 (1, ν) β * * F,G,1 (1, ν) ≤ C 2 (ε)/C 1 (ε) 1 + √ 2 −8ν(1−ε) .
; 1) = δβ * (0) 1 (z; 1) − β * (0) 2 (z; 1) = −16z+ 42z 2 − (1 + 16z + 9z 2 ) = −1 − 32z + 33z 2 = (z − 1)(33z + 1),
+ 66z 2 − (16z + 18z 2 ) = −48z + 48z 2 = 48z(z − 1).
1 =
1((−∞, −3] ∪ [1, +∞)) ∩ Z. In view of (5.16), (5.22) -(5.20), (3.3) -(3.18), and, since τ −ν−2
c
F,G,2 (1) = −54(−27F 2 − 23F G + 28G 2 ), c F,G,1 (1) = 12(−3679F 2 − 5646F G + 5459G 2 ), c F,G,0 (1) = 10(75F 2 + 63F G − 228G 2 ) = 30(25F 2 + 21F G − 76F 2 ),in view of (4.72) -(4.94),
12 × (−21 × 32845 + 5521 × 125 − 190) = 60(−21 × 6569 + 5521 × 25 − 76) = 60(−137949 + 138025 − 76)
33 × 1653 − 1268 × 43 − 25) = 72(54559 − 54524 − 25) = 0, ∆ 2,1 (1, 1) = 36(33 × 7948 − 317 × 827 − 125) = 36(262284 − 262159 − 125) = 0.
1) = −54 × (39) = −18 × 117 c 1,2,1 (1) = 18 × 4742,
ν) − ν + O(1) log(ν), log(ν!) = ν log(ν) − ν + O(1) log(ν). So, if ν ∈ [2/ min(γ, 1 − γ) + 2, +∞), k = [γν], then log ν + 1 k = ν log(ν) − ν − (γν log(γν) − γν)− ((1 − γ)ν log((1 − γ)ν) − (1 − γ)ν) + O(1) log(ν) =νlog(ν) − (γν log(γν) − (γν log(γ)− ((1 − γ)ν log(ν) − ((1 − γ)ν log(1 − γ) + O(1) log(ν) = (γ log(1/γ) + (1 − γ) log(1/(1 − γ)))ν + O(1) log(ν), and, analogously, log ν + k k = ((1 + γ) log(1 + γ) + γ log(1/γ))ν + O(1) log(ν)
k,ν < [γ 1 ν] r+1 β (0) 2,[γ 1 ν],ν = exp(2ψ 1 (γ 1 )ν + O(1) log(ν)
O(1) exp(−2γ 3 ν + O(log(ν)).
§6. Auxilliary continued fraction.for ν ∈ N,Let us consider the continued fraction,....Let r * u,v (ν) be the ν-th convergent of this continued fraction. Let P * u,v (ν) and Q * u,v (ν) be respectively nominator and denominator of convergent r * u,v (ν). Let us consider the equations (6.10) c F,G,2 (ν)x ν+1 + c F,G,1 (ν)x ν + c F,G,0 (ν)x ν−1 = 0,
−39 × 80533 + 271 × 1327 − 7 × 790) = (−3140787 + 3146317 − 5530 = 0∆ 2,1 (1, 2) = −351 × 80533 + 9times2371 × 1327 − 63 × 790) = 9. ∆ 2,1 (1, 2) = −351 × 80533 + 9times2371 × 1327 − 63 × 790) = 9(−39 × 80533 + 271 × 1327 − 7 × 790) = (−3140787 + 3146317 − 5530 = 0.
G) = 0 for any F and G. Therefore the equality (5.57) holds ,v,2 (j) for ν ∈ N 0. ∆ So, 1So ∆ i,1 (F, G) = 0 for any F and G. Therefore the equality (5.57) holds ,v,2 (j) for ν ∈ N 0 .
) = δ * F,G (0) = 1. Let c * * F,G,1 (ν) = c * F,G,1 (ν) for all ν ∈ N. * F So, G , letSo, δ * F,G (1) = δ * F,G (0) = 1. Let c * * F,G,1 (ν) = c * F,G,1 (ν) for all ν ∈ N, let
If conditions Theorem B are fulfilled, then δ F,G (ν) = 0 far all the ν ∈ N, and the equation (6.10) and the system y ν+1 + c * F,G,1 (ν)y(ν) + c * F. * * F , G 0 (ν) = C * F, G , 0 (ν)c * F,G,0 (ν − 1) for all ν ∈ [2, +∞) ∩ N, and let c * * F,G,0 (1) = c * F,G,0 (1). 2 st(ν − 1)y(ν − 1) = 0c * * F,G,0 (ν) = c * F,G,0 (ν)c * F,G,0 (ν − 1) for all ν ∈ [2, +∞) ∩ N, and let c * * F,G,0 (1) = c * F,G,0 (1). If conditions Theorem B are fulfilled, then δ F,G (ν) = 0 far all the ν ∈ N, and the equation (6.10) and the system y ν+1 + c * F,G,1 (ν)y(ν) + c * F,G,0 (ν)c a F,G,2 st(ν − 1)y(ν − 1) = 0
ν)/β * * F,G,1 (1, 0) satisfy to the first of equations (6.10) and the same initial conditions. Therefore (6.27) P * F,G (ν) = δ F,G (ν). F , G (ν) * X Ν, P * F Moreover, G , * F , G (ν)β * * F, G , 2β * * F,G,2 (1, ν)/β * * F,G,1 (1, 0y ν = δ F,G (ν) * x ν , ν ∈ N are equivalent. Moreover, P * F,G (ν) and δ * F,G (ν)β * * F,G,2 (1, ν)/β * * F,G,1 (1, 0) satisfy to the first of equations (6.10) and the same initial conditions. Therefore (6.27) P * F,G (ν) = δ F,G (ν)β * * F,G,2 (1, ν)/β * * F,G,1 (1, 0),
1 (1, ν)/β * * F,G,1 (1, 0), r * F,G (ν) = β * * F,G,2 (1, ν)/β * * F,G,1 (1, ν), for all ν ∈ N 0. ; Q * F Analogously, G (ν) = Δ, F , G (ν)β * * F, G , we have (6.28. In view of (6.25) lim ν→∞ r * F,G (ν) = 2ζ(3Analogously, we have (6.28) Q * F,G (ν) = δ F,G (ν)β * * F,G,1 (1, ν)/β * * F,G,1 (1, 0), r * F,G (ν) = β * * F,G,2 (1, ν)/β * * F,G,1 (1, ν), for all ν ∈ N 0 . In view of (6.25) lim ν→∞ r * F,G (ν) = 2ζ(3).
1) directly follows from (6.5) -(6.7) and (6.26). view of (0.16) If ν = 0, 1, then the equalities. 7In view of (6.1), (7.2) c * u,v,k (ν) = c * u/v,1,k (νProof. In view of (0.16) If ν = 0, 1, then the equalities (7.1) directly follows from (6.5) -(6.7) and (6.26) In view of (6.1), (7.2) c * u,v,k (ν) = c * u/v,1,k (ν)
Therefore the last equality in (7.1) holds for all ν ∈ N. for k = 0, 1, 2, ν ∈ N. In view (6.8)), (6.5), (6.6), (6.3), (6.2) and (7.2), a u,v (ν) = a u/v,1 (ν), b u,v (ν) = b u/v,1 (νfor k = 0, 1, 2, ν ∈ N. Therefore the last equality in (7.1) holds for all ν ∈ N. In view (6.8)), (6.5), (6.6), (6.3), (6.2) and (7.2), a u,v (ν) = a u/v,1 (ν), b u,v (ν) = b u/v,1 (ν)
1) hold for all ν − κ ,v (ν) = b * u,v (ν)P * u,v (ν − 1) + a * u. 2, +∞) ∩ Z. Let ν ∈ [2, +∞) ∩ Z, and letv (ν)P ( u,v ν − 2) = b * u/v,1 (ν)P * u/v,1 (ν − 1) + a * u/v,1 (ν)P * u/v,1 (ν − 2) = P * u/v,1 (νfor ν ∈ [2, +∞) ∩ Z. Let ν ∈ [2, +∞) ∩ Z, and let (7.1) hold for all ν − κ ,v (ν) = b * u,v (ν)P * u,v (ν − 1) + a * u,v (ν)P ( u,v ν − 2) = b * u/v,1 (ν)P * u/v,1 (ν − 1) + a * u/v,1 (ν)P * u/v,1 (ν − 2) = P * u/v,1 (ν)
. Q * U, v (ν) = b * u,v (ν)Q * u,v (ν − 1) + a * u,v (ν)Q * u,v (ν − 2) = b * u/v,1 (ν)Q * u/v,1 (ν − 1) + a * u/u,1 (ν)Q * u/v,1 (ν − 2) = Q u/v,1 (ν)Q * u,v (ν) = b * u,v (ν)Q * u,v (ν − 1) + a * u,v (ν)Q * u,v (ν − 2) = b * u/v,1 (ν)Q * u/v,1 (ν − 1) + a * u/u,1 (ν)Q * u/v,1 (ν − 2) = Q u/v,1 (ν).
In view of (0.10) and (0.8), c F,G,0 (ν) = −f rac(τ − 1) 2 (2τ + 1)(τ + 1) 2 (2τ − 1)× τ (τ + 1)c F,G,2 (ν + 1) = 0 view of. We have to prove the last equality in (7.4), because other assertions of the Lemma are obvious. 616.28) and (7.3), the following equalities hold: P u,v (ν) = 4P * u,v (ν)(u + v) 2ν = 16(u + v)δ u,v (ν)β * * u,v,2 (1, ν), Q u,v (ν) = Q * u,v (ν)(u + v) 2ν−1 = 4δ u,v (ν)β * * u,v,1 (1, νProof. We have to prove the last equality in (7.4), because other asser- tions of the Lemma are obvious. In view of (0.10) and (0.8), c F,G,0 (ν) = −f rac(τ − 1) 2 (2τ + 1)(τ + 1) 2 (2τ − 1)× τ (τ + 1)c F,G,2 (ν + 1) = 0 view of (5.61) -(5.62), (6.27) -(6.28) and (7.3), the following equalities hold: P u,v (ν) = 4P * u,v (ν)(u + v) 2ν = 16(u + v)δ u,v (ν)β * * u,v,2 (1, ν), Q u,v (ν) = Q * u,v (ν)(u + v) 2ν−1 = 4δ u,v (ν)β * * u,v,1 (1, ν)
v (ν)) = deg v (P u,v (ν)) = deg(P u,v (ν)), max. 2ν, 1) = deg u (P u. 2ν − 1, 0) = deg u (Q u,v (ν)) = deg v (Q u,v (ν)) = deg(Q u,v (ν)max(2ν, 1) = deg u (P u,v (ν)) = deg v (P u,v (ν)) = deg(P u,v (ν)), max(2ν − 1, 0) = deg u (Q u,v (ν)) = deg v (Q u,v (ν)) = deg(Q u,v (ν)),
Interpolation des fractions continues et irrationalite de certaines constantes. R Apéry, C.T.H. 3Bulletin de la section des sciences duR.Apéry, Interpolation des fractions continues et irrationalite de certaines constantes, Bulletin de la section des sciences du C.T.H., 1981, No 3, 37 - 53.
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. E-Mails, l − a − [email protected], [email protected]: l − a − [email protected], [email protected].
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[
"CONVERSE KAM THEORY REVISITED",
"CONVERSE KAM THEORY REVISITED"
] |
[
"Lin Wang "
] |
[] |
[] |
For an integrable Hamiltonian with d (d ≥ 2) degrees of freedom, we show the conditions on perturbations, for which invariant tori can be destructed.
| null |
[
"https://arxiv.org/pdf/1208.2840v1.pdf"
] | 119,144,096 |
1208.2840
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9546a4a930c29cc7809ea5993a2c93095531db8b
|
CONVERSE KAM THEORY REVISITED
14 Aug 2012
Lin Wang
CONVERSE KAM THEORY REVISITED
14 Aug 2012
For an integrable Hamiltonian with d (d ≥ 2) degrees of freedom, we show the conditions on perturbations, for which invariant tori can be destructed.
References 54
Introduction
By the Kolmogorov, Arnold and Moser (KAM) theory, we know that under certain non-degeneracy, most (full Lebesgue measure) invariant tori of an integrable Hamiltonian system are persisted under small perturbations. As the sizes of the perturbations increase, those persisted invariant tori are destructed progressively. The problems of determining the critical boundary between the persistence and destruction of invariant tori motivates so called converse KAM theory.
Roughly speaking, the converse KAM theory consists of two parts. The first part, with more physical flavor, is concerned about destruction of invariant tori under the perturbations with positive lower bound, which is started by the analysis of invariant circles of the generalized standard maps (see e.g. [Ci], [H2] and [Ma2]). Besides, numerical results go further than theoretical ones. More sharp boundaries are obtained by numerical methods (see e.g. [Gr] and [MP]). The second part is concerned about destruction of invariant tori under arbitrarily small perturbations in certain topology and it seems that numerical method is not as efficient as the first part. In this paper, we are devoted to develop the second part.
In 1962, Moser proved that the invariant circles with Diophantine rotation numbers of an integrable twist map is persisted under arbitrarily small perturbations in the C 333 topology ( [Mo1]). By the efforts of Moser,Rüssman,Herman and Pöschel ([H2,H3], [Mo2,Mo3], [P] and [R1, R2]), for Hamiltonian systems with d-degrees of freedom, it is obtained that certain invariant tori are persisted under arbitrarily small perturbations in the C 2d+δ topology, where δ is a small positive constant. Especially, Herman proved in [H3] that for twist maps on annulus, certain invariant circles can be persisted under arbitrarily small perturbations in the C 3 topology.
In contrast with the results on persistence of invariant tori, for exact areapreserving twist maps on annulus, it is proved by Herman in [H2] that invariant circles with given rotation numbers can be destructed by C 3−δ arbitrarily small C ∞ perturbations. For certain rotation numbers, it is obtained by Mather (resp. Forni) in [Ma5] (resp. [F]) that the invariant circles with those rotation numbers can be destroyed by small perturbations in finer topology respectively. More precisely, Mather considers Liouville rotation numbers and the topology of the perturbation induced by C ∞ metric. Forni is concerned about more special rotation numbers which can be approximated by rational ones exponentially and the topology of the perturbation induced by the supremum norm of C ω (real-analytic) function. Bessi extended the result to the systems with multi-degrees of freedom. He found that the invariant Lagrangian torus with certain rotation vector can be destructed by an arbitrarily small C ω perturbation for certain positive definite systems with multidegrees of freedom in [Be], where Lagrangian torus is a natural analogy to invariant circle in multi-degrees of freedom (see Definition 3.1 below).
On the other hand, it is also proved by Herman in [H5] that all Lagrangian tori of an integrable symplectic twist map with d ≥ 1 degrees of freedom can be destructed by C d+2−δ arbitrarily small C ∞ perturbations of the generating function. Equivalently ( [Go,Mo4]), it shows that all Lagrangian tori of an integrable Hamiltonian system with d ≥ 2 degrees of freedom can be destructed by C ∞ perturbations which are arbitrarily small in the C d+1−δ topology. Roughly speaking, there is a balance among the arithmetic property of the rotation vector, the regularity of the perturbation and its topology.
Comparing the results on both sides, it is natural to ask the following questions:
• for every given rotation vector ω, if the Lagrangian torus with ω can be destructed by an arbitrarily small C ∞ perturbation in the C r topology, then what is the maximum of r?
• for every given rotation vector ω, if the Lagrangian torus with ω can be destructed by an arbitrarily small analytic perturbation in the C r topology, then what is the maximum of r?
• if all of Lagrangian tori can be destructed by an arbitrarily small real-analytic perturbation in the C r topology, then what is the maximum of r?
Based on [CW] and [W1, W2, W3, W4], we have the following theorems. For positive definite Hamiltonian systems with d ≥ 2 degrees of freedom, together with Herman's result in [H5], we have the following Table 1. Unfortunately, except for the destruction of a Lagrangian torus by the C ∞ perturbations, we still don't know whether the other results are optimal. Some further developments of KAM theory are needed to verify the optimality. This paper is outlined as follows. In Section 2, we consider the destruction of a Lagrangian torus with given rotation vector (Theorem 1.1 and Theorem 1.2). Based on the difference of topology of phase spaces between d = 2 and d ≥ 3, we divide the arguments into two cases. In terms of the correspondence between exact area-preserving twist maps and Hamiltonian systems with 2 degrees of freedom, the problem on destruction of the Lagrangian torus for the Hamiltonian system is transformed into the one on destruction of the invariant circle for the twist map. Using variational method developed by Mather, a new proof of Herman's result ( [H2]) is provided. Moreover, from Jackson's approximation, the result on C ω perturbation is obtained. For the case with d ≥ 3 degrees of freedom, the minimality of the orbits on Lagrangian torus plays a crucial role in destruction of the torus. Combining with Melnikov method, the destruction of Lagrangian torus under C ω perturbations is achieved. In Section 3, we are concerned about the destruction of all Lagrangian tori (Theorem 1.3). Similar to Section 2, according to the correspondence between the Hamiltonian system and the symplectic twist map, we focus on destruction of Lagrangian tori for symplectic twist map. From a criterion of total destruction of Lagrangian tori found by Herman and an approximation lemma, total destruction of Lagrangian tori under C ω perturbations is completed.
r C ∞ C ω Single Destruction 2d − δ d + 1 − δ Total Destruction d + 1 − δ d − δ
2. Destruction of a Lagrangian torus 2.1. Case with 2 degrees of freedom Based on the correspondence between exact area-preserving twist maps and Hamiltonian systems with 2 degrees of freedom, it is sufficient to consider the destruction of invariant circle for exact area-preserving twist map. Let f : T × R → T × R (T = R/Z) be an exact area-preserving monotone twist map and h: R 2 → R 2 be a generating function for the lift F of f to R 2 , namely F is generated by the following equations
y = −∂ 1 h(x, x ′ ), y ′ = ∂ 2 h(x, x ′ ),
where F (x, y) = (x ′ , y ′ ). The lift F gives rise to a dynamical system whose orbits are given by the images of points of R 2 under the successive iterates of F . The orbit of the point (x 0 , y 0 ) is the bi-infinite sequence
{..., (x −k , y −k ), ..., (x −1 , y −1 ), (x 0 , y 0 ), (x 1 , y 1 ), ..., (x k , y k ), ...},
where (x k , y k ) = F (x k−1 , y k−1 ). The sequence (..., x −k , ..., x −1 , x 0 , x 1 , ..., x k , ...) denoted by (x i ) i∈Z is called stationary configuration if it stratifies the identity
∂ 1 h(x i , x i+1 ) + ∂ 2 h(x i−1 , x i ) = 0, for every i ∈ Z.
Given a sequence of points (z i , ..., z j ), we can associate its action
h(z i , ..., z j ) = i≤s<j h(z s , z s+1 ).
A configuration (x i ) i∈Z is called minimal if for any i < j ∈ Z, the segment of (x i , ..., x j ) minimizes h(z i , ..., z j ) among all segments (z i , ..., z j ) of the configuration satisfying z i = x i and z j = x j . It is easy to see that every minimal configuration is a stationary configuration. By [Ba], minimal configurations satisfy a group of remarkable properties as follows:
• Two distinct minimal configurations cross at most once, which is so called Aubry's crossing lemma.
• For every minimal configuration x = (x i ) i∈Z , the limit
ρ(x) = lim n→∞ x i+n − x i n
exists and doesn't depend on i ∈ Z. ρ(x) is called the rotation number of x.
• For every ω ∈ R, there exists a minimal configuration with rotation number ω. Following the notations of [B], the set of all minimal configurations with rotation number ω is denoted by M h ω , which can be endowed with the topology induced from the product topology on R Z . If x = (x i ) i∈Z is a minimal configuration, considering the projection pr :
M h ω → R defined by pr(x) = x 0 , we set A h ω = pr(M h ω ).
• If ω ∈ Q, say ω = p/q (in lowest terms), then it is convenient to define the rotation symbol to detect the structure of M h p/q . If x is a minimal configuration with rotation number p/q, then the rotation symbol σ(x) of x is defined as follows
σ(x) = p/q+, if x i+q > x i + p for all i, p/q, if x i+q = x i + p for all i, p/q−, if x i+q < x i + p for all i.
Moreover, we set
M h p/q + = {x a is minimal configuration with rotation symbol p/q or p/q+}, M h p/q − = {x a is minimal configuration with rotation symbol p/q or p/q−},
then both M h p/q + and M h p/q + are totally ordered. Namely, every two configurations in each of them do not cross. We denote pr(M h p/q + ) and pr(M h p/q − ) by A h p/q + and A h p/q − respectively.
• If ω ∈ R\Q and x is a minimal configuration with rotation number ω, then σ(x) = ω and M h ω is totally ordered. • A h ω is a closed subset of R for every rotation symbol ω.
Peierls's barrier
In [Ma4], Mather introduced the notion of Peierls's barrier and gave a criterion of existence of invariant circle. Namely, the exact area-preserving monotone twist map generated by h admits an invariant circle with rotation number ω if and only if the Peierls's barrier P h ω (ξ) vanishes identically for all ξ ∈ R. The Peierls's barrier is defined as follows:
• If ξ ∈ A h ω , we set P h ω (ξ)=0. • If ξ ∈ A h ω , since A h ω is a closed set in R, then ξ is contained in some complemen- tary interval (ξ − , ξ + ) of A h ω in R. By the definition of A h ω , there exist minimal configurations with rotation symbol ω, x − = (x − i ) i∈Z and x + = (x + i ) i∈Z sat- isfying x − 0 = ξ − and x + 0 = ξ + . For every configuration x = (x i ) i∈Z satisfying x − i ≤ x i ≤ x + i , we set G ω (x) = I (h(x i , x i+1 ) − h(x − i , x − i+1 )),
where I = Z, if ω is not a rational number, and
I = {0, ..., q − 1}, if ω = p/q. P h ω (ξ) is defined as the minimum of G ω (x) over the configurations x ∈ Π = i∈I [x − i , x + i ] satisfying x 0 = ξ. Namely P h ω (ξ) = min x {G ω (x)|x ∈ Π and x 0 = ξ}.
By [Ma4], P h ω (ξ) is a non-negative periodic function of the variable ξ ∈ R with the modulus of continuity with respect to ω.
h 0 (x, x ′ ) = 1 2 (x − x ′ ) 2 x, x ′ ∈ R.
We construct the perturbation consisting of two parts. The first one is
(2.1.1) u n (x) = 1 n a (1 − cos(2πx)) x ∈ R,
where n ∈ N and a is a positive constant independent of n. The second one is a non negative function v n (x) satisfying
(2.1.2) v n (x + 1) = v n (x), supp v n ∩ [0, 1] ⊂ [ 1 2 − 1 n a , 1 2 + 1 n a ], max v n = n −s , ||v n || C k ∼ n −s ′ ,
where f ∼ g means that 1
C g < f < Cg holds for a constant C > 1. For further deduction, we need s ′ > a. It is enough to take s = (k + 2)a for achieving that. The generating function of the nearly integrable system is constructed as follow:
(2.1.3) h n (x, x ′ ) = h 0 (x, x ′ ) + u n (x ′ ) + v n (x ′ ),
where n ∈ N. Moreover, we have the following theorem.
Theorem 2.1 For ω ∈ R\Q and n large enough, the exact area-preserving monotone twist map generated by h n does not admit any invariant circles with the rotation number satisfying
|ω| < n − a 2 −δ ,
where δ is a small positive constant independent of n.
We will prove Theorem 2.1 in the following sections. First of all, based on the theorem, we verify that our example has the property aforementioned in Section 1.
If ω ∈ Q, then the invariant circles with rotation number ω could be easily destructed by an analytic perturbation arbitrarily close to 0. Therefore it suffices to consider the irrational ω. The case with a given irrational rotation number can be easily reduced to the one with a small enough rotation number. More precisely, Lemma 2.2 Let h P be a generating function as follow
h P (x, x ′ ) = h 0 (x, x ′ ) + P (x ′ ),
where P is a periodic function of periodic 1. Let Q(x) = q −2 P (qx), q ∈ N, then the exact area-preserving monotone twist map generated by h Q (x,
x ′ ) = h 0 (x, x ′ ) + Q(x ′ )
admits an invariant circle with rotation number ω ∈ R\Q if and only if the exact area-preserving monotone twist map generated by h P admits an invariant circle with rotation number qω − p, p ∈ Z.
We omit the proof and for more details, see [H2]. For the sake of simplicity of notations, we denote Q qn by Q n and the same to u qn , v qn and h qn . Let
Q n (x) = q n −2 (u n (q n x) + v n (q n x)),
where (q n ) n∈N is a sequence satisfying Dirichlet approximation
(2.1.4) |q n ω − p n | < 1 q n ,
where p n ∈ Z and q n ∈ N. Since ω ∈ R\Q, we say q n → ∞ as n → ∞. Let
h n (x, x ′ ) = h 0 (x, x ′ ) + Q n (x ′ ), we have Corollary 2.
3 For a given rotation number ω ∈ R\Q and every ε, there exists N such that for n > N , the exact area-preserving monotone map generated byh n admits no invariant circle with rotation number ω and
||h n − h 0 || C 4−δ ′ < ε,
where δ ′ is a small positive constant independent of n.
||h n (x, x ′ ) − h 0 (x, x ′ )|| C r = ||Q n (x ′ )|| C r , ≤ q n −2 (||u n (q n x ′ )|| C r + ||v n (q n x ′ )|| C r ), ≤ q n −2 (q n −a (2π) r q n r + C 1 q n −s ′ q n r ), ≤ C 2 q n r−a−2 ,
where C 1 , C 2 are positive constants only depending on r.
To complete the proof, it is enough to make r − a − 2 < 0, which together with (2.1.55) implies r < a + 2 ≤ 4 − 2δ.
We set δ ′ = 2δ, then the proof of Corollary 2.3 is completed.
The following sections are devoted to prove Theorem 2.1. For simplicity, we don't distinguish the constant C in following different estimate formulas.
Estimate of lower bound of P hn
+
In this section, we will estimate the lower bound of P hn 0 + at a given point. To achieve that, we need to estimate the distances of pairwise adjacent elements of the minimal configuration.
A spacing lemma
Lemma 2.4 Let (x i ) i∈Z be a minimal configuration ofh n with rotation symbol 0 + , then
x i+1 − x i ≥ C(n − a 2 ), for x i ∈ 1 4 , 3 4 , whereh n (x i , x i+1 ) = h 0 (x i , x i+1 ) + u n (x i+1 ).
Proof Without loss of generality, we assume x i ∈ [0, 1] for all i ∈ Z. By Aubry's crossing lemma, we have
0 < ... < x i−1 < x i < x i+1 < ... < 1.
We consider the configuration (ξ i ) i∈Z defined by
ξ j = x j , j < i, x j+1 , j ≥ i. Since (x i ) i∈Z is minimal, we have i∈Zh n (ξ i , ξ i+1 ) − i∈Zh n (x i , x i+1 ) ≥ 0.
By the definitions ofh n and (ξ i ) i∈Z , we have
0 ≤ i∈Zh n (ξ i , ξ i+1 ) − i∈Zh n (x i , x i+1 ) =h n (x i−1 , x i+1 ) −h n (x i−1 , x i ) −h n (x i , x i+1 ) = (x i+1 − x i )(x i − x i−1 ) − u n (x i ). Moreover, u n (x i ) ≤ (x i+1 − x i )(x i − x i−1 ) ≤ 1 4 (x i+1 − x i−1 ) 2 .
Therefore,
x i+1 − x i−1 ≥ 2 u n (x i ). For x i ∈ [ 1 4 , 3 4 ], u n (x i ) ≥ n −a , hence, (2.1.6) x i+1 − x i−1 ≥ 2n − a 2 .
Since (x i ) i∈Z is a stationary configuration, we have
x i+1 − x i = −∂ 1hn (x i , x i+1 ), = ∂ 2hn (x i−1 , x i ), = x i − x i−1 + u ′ n (x i ). Since u ′ n (x) = 2π
n a sin(2πx), it follows from (2.1.6) that
x i+1 − x i ≥ C(n − a 2 ), x i ∈ 1 4 , 3 4 .
The proof of Lemma 2.4 is completed. By the definition of v n , supp v n ∩ [0, 1] ⊂ [ 1 2 − 1 n a , 1 2 + 1 n a ] and v n (x + 1) = v n (x). Let (x i ) i∈Z be the minimal configuration ofh n (
x i , x i+1 ) = h 0 (x i , x i+1 ) + u n (x i+1 ) with rotation symbol 0 + satisfying x 0 = 1 2 − 1 n a , then (x i ) i∈Z ∩ suppv n = ∅.
Moreover, for all i ∈ Z, v n (x i ) = 0.
Let (ξ i ) i∈Z be a minimal configuration of h n defined by (2.2.38) with rotation symbol 0
+ satisfying ξ 0 = η, where η satisfies v n (η) = max v n (x) = n −s , then i∈Z (h n (ξ i , ξ i+1 ) − h n (ξ − i , ξ − i+1 )) ≥ v n (η) + i∈Zh n (ξ i , ξ i+1 ) − i∈Z h n (ξ − i , ξ − i+1 ), ≥ v n (η) + i∈Zh n (x i , x i+1 ) − i∈Z h n (x i , x i+1 ), = v n (η) − i∈Z v n (x i+1 ), = v n (η).
Therefore,
P hn 0 + (η) = min x 0 =η i∈Z (h n (x i , x i+1 ) − h n (x − i , x − i+1 )) ≥ v n (η) = n −s .
We conclude that there exists a point ξ ∈ [ 1 2 − 1 n a , 1 2 + 1 n a ] such that (2.1.7) P hn 0 + (ξ) ≥ n −s .
2.1.2.3. The approximation from P hn 0 + to P hn ω In this section, we will prove the improvement of modulus of continuity of Peierls's barrier based on the hyperbolicity of h n . Namely Lemma 2.5 For every irrational rotation symbol ω satisfying 0 < ω < n − a 2 −δ , we have
|P hn ω (ξ) − P hn 0 + (ξ)| ≤ C exp −2n δ 2 .
where ξ ∈ 1 2 − 1 n a , 1 2 + 1 n a and δ is a small positive constant independent of n.
Some counting lemmas
To prove the lemma, we need to do some preliminary work. First of all, we count the number of the elements of a minimal configuration (x i ) i∈Z with arbitrary rotation symbol ω in a given interval. With the method of [F], we can conclude the following lemma.
Lemma 2.6 Let (x i ) i∈Z be a minimal configuration of h n with rotation symbol ω > 0, J n = exp −n δ 2 , 1 2 and Λ n = {i ∈ Z| x i ∈ J n }, then
♯Λ n ≤ Cn a 2 + δ 2 ,
where ♯Λ n denotes the number of elements in Λ n and δ is a small positive constant independent of n.
Proof Let
x − = exp −n δ 2 , x + = 1 2 and σ = x + x − 1 N , hence, ln σ = ln(x + ) − ln(x − ) N .
We choose N ∈ N such that 1 ≤ ln σ ≤ 2, then N = Ω n δ 2 . We consider the partition of the interval J n = [x − , x + ] into the subintervals
J k n = [σ k x − , σ k+1 x − ] where 0 ≤ k < N . Hence, J n = ∪ N −1 k=0 J k n . We set S k = {i ∈ Λ n |(x i−1 , x i+1 ) ⊂ J k
n } and m k = ♯S k . By the similar deduction as the one in Lemma 2.4, we have
x i+1 − x i−1 ≥ 2 u n (x i ) + v n (x i ) ≥ Cn − a 2 x i , for x i ∈ 0, 1 2 .
For simplicity of notation, we denote Cn − a 2 by α n .
If there exists k such that i ∈ S k for (x i ) i∈Z , then x i+1 − x i−1 ≥ α n σ k x − , moreover, m k α n σ k x − ≤ 2L(J k n ) = 2(σ − 1)σ k x − , where L(J k n )
denotes the length of the interval of J k n . Hence m k ≤ 2(σ − 1)α −1 n . On the other hand, if i ∈ Λ n \ ∪ N −1 k=0 S k , then there exists l satisfying 0 ≤ l < N such that
x i−1 < σ l x − < x i+1 .
Hence, ♯{i ∈ Λ n |i ∈ S k for any k} ≤ 2N.
Therefore,
♯(Λ n ) ≤ 2N (σ − 1)α −1 n + 2N.
Since 1 ≤ ln σ ≤ 2 and N = Ω n δ 2 , then we have
♯Λ n ≤ Cn a 2 + δ 2 .
The proof of Lemma 2.6 is completed.
Remark 2.7 Let (x i ) i∈Z be a minimal configuration of h n defined by (2.2.38) with rotation symbol ω > 0, An argument as similar as the one in Lemma 2.6 implies that
♯ i ∈ Z x i ∈ exp −n δ 2 , 1 − exp −n δ 2 ≤ Cn a 2 + δ 2 .
It is easy to count the number of the elements of a minimal configuration with irrational rotation symbol. More precisely, we have the following lemma.
Lemma 2.8 Let (x i ) i∈Z be a minimal configuration with rotation number ω ∈ R\Q. Then for every interval I k of length k, k ∈ N,
k ω − 1 ≤ ♯{i ∈ Z|x i ∈ I k } ≤ k ω + 1.
Proof For every minimal configuration (x i ) i∈Z with rotation number ω, there exists an orientation-preserving circle homeomorphism φ such that ρ(Φ) = ω, where Φ : R → R denotes a lift of φ. Since ω ∈ R\Q, thanks to [H1], φ has a unique invariant probability measureμ on T such that
Φ(x) x dμ = ω for every x ∈ R. We denote Φ(x) x dμ by µ(x, Φ(x)). In particular, µ(x i , x i+1 ) = ω, for every i ∈ Z. From µ(I k ) = k, it follow that ω(♯{i ∈ Z|x i ∈ I k } − 1) ≤ k, ω(♯{i ∈ Z|x i ∈ I k } + 1) ≥ k,
which completes the proof of Lemma 2.8.
Based on Lemma 2.6 and Lemma 2.8, if 0 < ω < n − a 2 −δ and ω is irrational, then
(2.1.8) ♯{i ∈ Z|x i ∈ I 1 } ≥ 1 ω − 1 ≥ C 1 n a 2 +δ > C 2 n a 2 + δ 2 ,
where I 1 denotes the closed interval of length 1. Moreover, we have the following conclusion.
Lemma 2.9 Let (x i ) i∈Z be a minimal configuration of h n defined by (2.2.38) with rotation symbol 0 < ω < n − a 2 −δ , then there exist j − , j + ∈ Z such that
0 < x j − −1 < x j − < x j − +1 ≤ exp(−n δ 2 ), 1 − exp(−n δ 2 ) ≤ x j + −1 < x j + < x j + +1 < 1.
Proof By contradiction, we assume that there exist at most two points of (
x i ) i∈Z in [0, exp(−n δ 2 )], say x m and x m+1 . It follows that x m−1 < 0 and x m+2 > exp(−n δ 2 ). Hence, among the intervals [x m−1 , x m ], [x m , x m+1 ] and [x m+1 , x m+2 ],
there exists at least one such that its length is not less than 1 3 exp(−n δ 2 ). Without loss of generality, say [x m+1 , x m+2 ].
Since (x i ) i∈Z is a stationary configuration, we have
x m+2 − x m+1 = x m+1 − x m + u ′ n (x m+1 ), where u ′ n (x m+1 ) = 2π n a sin(2πx m+1 ). From x m+1 ∈ [− exp(−n δ 2 ), exp(−n δ 2 )], it fol- lows that |u ′ n (x m+1 )| ≤ Cn −a exp(−n δ 2 ), which implies there exists N independent of n such that [− exp(−n δ 2 ), exp(−n δ 2 )] contains at most N points of (x i ) i∈Z .
On the other hand, by (2.1.8), we have that for n large enough, the number of points of (
x i ) i∈Z in [− exp(−n δ 2 ), exp(−n δ 2 )] is also large enough, which is a contra- diction. Therefore, there exists j − ∈ Z such that 0 ≤ x j − −1 < x j − < x j − +1 < exp(−n δ 2 ). Similarly, there exists j + ∈ Z such that 1 − exp(−n δ 2 ) ≤ x j + −1 < x j + < x j + +1 < 1.
The proof of Lemma 2.9 is completed.
Remark 2.10 From the proof of Lemma 2.9, it is easy to see that each of [0, exp(−n δ 2 )] and [1 − exp(−n δ 2 ), 1] contains a large number of points of the minimal configuration (x i ) i∈Z for n large enough.
By Lemma 2.8 and Lemma 2.9, without loss of generality, one can assume that
(2.1.9) j + − j − ≥ C n a 2 + 2δ 3 .
If ξ ∈ A hn ω , then P hn ω (ξ) = 0. Hence, it suffices to consider the case with ξ ∈ A hn ω for destruction of invariant circles. Let (ξ − , ξ + ) be the complementary interval of A hn ω in R and contains ξ. Let ξ ± ξ ± ξ ± = (ξ ± i ) i∈Z be the minimal configurations with rotation symbol ω satisfying ξ ± 0 = ξ ± and let (ξ i ) i∈Z be a minimal configuration of h n with rotation symbol ω satisfying ξ 0 = ξ and ξ − i ≤ ξ i ≤ ξ + i . By the definition of Peierls barrier, we have
P hn ω (ξ) = i∈Z (h n (ξ i , ξ i+1 ) − h n (ξ − i , ξ − i+1 )).
Since P hn ω (ξ) is 1-periodic with respect to ξ, without loss of generality, we assume that ξ ∈ [0, 1]. We set d(x) = min{|x|, |x − 1|} and denote exp(−n δ 2 ) by ǫ(n). By Lemma 2.9, there exist i − , i + such that
(2.1.10) d(ξ − i ) < ǫ(n) and ξ − i+1 − ξ − i−1 ≤ ǫ(n) for i = i − , i + .
Thanks to Aubry's crossing lemma, we have
ξ − i ≤ ξ i ≤ ξ + i ≤ ξ − i+1 . Hence, ξ i − ξ − i ≤ ǫ(n) for i = i − , i + .
Proof of lemma 2.5
In the following, we will prove Lemma 2.5 with the method similar to the one developed by Mather in [M3]. The proof can be proceeded in the following two steps.
Step 1 We consider the number of the elements in a segment of the configuration as the length of the segment. In the first step, we approximate P hn ω (ξ) for ξ ∈ 1 2 − 1 n a , 1 2 + 1 n a by the difference of the actions of the segments of length i + −i − +1. To achieve that, we define the following configurations
x i = ξ i , i = i − , i + , ξ − i , i = i − , i + , and y i = ξ i , i − < i < i + , ξ − i , i ≤ i − , i ≥ i + , where (ξ i ) i∈Z is a minimal configuration.
It is easy to see that ξ 0 = ξ is contained both of (x i ) i∈Z and (y i ) i∈Z up to the rearrangement of the index i since ξ ∈ 1 2 − 1 n a , 1 2 + 1 n a . Hence, by the minimality of (ξ i ) i∈Z satisfying ξ 0 = ξ, we have (2.1.11)
P hn ω (ξ) ≤ i∈Z (h n (y i , y i+1 ) − h n (ξ − i , ξ − i+1 )).
Since ω is irrational, then (x i ) i∈Z is asymptotic to (ξ − i ) i∈Z , which together with the minimality of (ξ
− i ) i∈Z yields (2.1.12) i∈Z (h n (y i , y i+1 ) − h n (ξ − i , ξ − i+1 ) ≤ i∈Z (h n (x i , x i+1 ) − h n (ξ − i , ξ − i+1 )). We set h(x i , ..., x j ) = i≤s<j h(x s , x s+1 ), then i∈Z (h n (x i , x i+1 ) − h n (ξ i , ξ i+1 )) = i=i − ,i + (h n (ξ i−1 , ξ − i , ξ i+1 ) − h n (ξ i−1 , ξ i , ξ i+1 )).
By the construction of v n and Lemma 2.
9, we have v n (ξ i − ), v n (ξ − i − ) = 0. It follows that h n (ξ i − −1 , ξ − i − , ξ i − +1 ) − h n (ξ i − −1 , ξ i − , ξ i − +1 ) = h n (ξ i − −1 , ξ − i − ) + h n (ξ − i − , ξ i − +1 ) − h n (ξ i − −1 , ξ i − ) − h n (ξ i − , ξ i − +1 ), = (ξ i − − ξ − i − )(ξ i − −1 + ξ − i − + ξ i − + ξ i − +1 ) + u n (ξ i − ) − u n (ξ − i − ), ≤ 4(ξ i − − ξ − i − )ǫ(n) + u ′ n (η)(ξ i − − ξ − i − ), ≤ 4ǫ(n) 2 + 2π n a sin(2πη)ǫ(n), ≤ Cǫ(n) 2 , where η ∈ (ξ i − , ξ − i − ). It is similar to obtain h n (ξ i + −1 , ξ − i + , ξ i + +1 ) − h n (ξ i + −1 , ξ i + , ξ i + +1 ) ≤ Cǫ(n) 2 . Hence, (2.1.13) i∈Z (h n (x i , x i+1 ) − h n (ξ i , ξ i+1 )) ≤ Cǫ(n) 2 . Moreover, i∈Z (h n (x i , x i+1 ) − h n (ξ − i , ξ − i+1 )) = i∈Z (h n (x i , x i+1 ) − h n (ξ i , ξ i+1 ) + h n (ξ i , ξ i+1 ) − h n (ξ − i , ξ − i+1 )), = i∈Z (h n (x i , x i+1 ) − h n (ξ i , ξ i+1 )) + P hn ω (ξ), ≤ P hn ω (ξ) + Cǫ(n) 2 .
Therefore, it follows from (2.1.11) and (2.1.12) that
(2.1.14)
P hn ω (ξ) ≤ i∈Z (h n (y i , y i+1 ) − h n (ξ − i , ξ − i+1 )) ≤ P hn ω (ξ) + Cǫ(n) 2 , where (2.1.15) i∈Z (h n (y i , y i+1 ) − h n (ξ − i , ξ − i+1 )) = h n (y i − , ..., y i + ) − h n (ξ − i − , ..., ξ − i + ).
Step 2 It follows from [M4] that the Peierls's barrier P hn 0 + (ξ) could be defined as follows
(2.1.16) P hn 0 + (ξ) = min η 0 =ξ i∈Z h n (η i , η i+1 ) − min i∈Z h n (z i , z i+1 ),
where (η i ) i∈Z and (z i ) i∈Z are monotone increasing configurations limiting on 0, 1. We set
K(ξ) = min η 0 =ξ i∈Z h n (η i , η i+1 ), K = min i∈Z h n (z i , z i+1 ).
First of all, it is easy to see that K(ξ) and K are bounded. Second, P hn 0 + (ξ) = 0 for ξ = 0 or 1. Hence, we only need to consider the case with ξ ∈ (0, 1). Following the ideas of [M6], let ξ − ξ − ξ − and ξ + ξ + ξ + be minimal configurations of rotation symbol 0 + and let (ξ − 0 , ξ + 0 ) be the complementary interval of A hn 0 + and contains ξ. Based on the definition P hn 0 + (ξ) = min
x 0 =ξ {G 0 + (x)|ξ − i ≤ ζ i ≤ ξ + i }, where G 0 + (ζ ζ ζ) = i∈Z (h n (ζ i , ζ i+1 ) − h n (ξ − i , ξ − i+1 )) = −K + i∈Z h n (ζ i , ζ i+1 ),
the proof of (2.1.16) will be completed when we verify that the configuration (ζ i ) i∈Z achieving the minimum in the definition of K(ξ) satisfies ξ − i ≤ ζ i ≤ ξ + i . It can be easily obtained by Aubry's crossing lemma. In fact, since (ξ − i ) i∈Z and (ζ i ) i∈Z are minimal and both are α-asymptotic to 0 as well as ω-asymptotic to 1, by Aubry's crossing lemma, (ξ − i ) i∈Z and (ζ i ) i∈Z do not cross. Similarly (ξ + i ) i∈Z and (ζ i ) i∈Z do not cross. It follows from
ζ 0 ∈ (ξ − 0 , ξ + 0 ) that (ζ i ) i∈Z achieving the minimum in the definition of K(ξ) satisfies ξ − i ≤ ζ i ≤ ξ + i . In the following, we will compare K, K(ξ) with h n (ξ − i − , ..., ξ − i + ), h n (y i − , ..., y i + ) respectively, here h n (ξ − i − , ..., ξ − i + ) and h n (y i − , .
.., y i + ) are as the same as the notations in (2.1.15).
First, we consider K and h n (ξ − i − , ..., ξ − i + ). Let (z i ) i∈Z be a monotone increasing configuration limiting on 0, 1 such that K = i∈Z h n (z i , z i+1 ). By Lemma 2.6,
♯{i ∈ Z|z i ∈ [ǫ(n), 1 − ǫ(n)]} ≤ Cn a 2 + δ 2 .
On the other hand, since (z i ) i∈Z has the rotation number 0 + , then from (2.1.9), it follows that up to the rearrangement of the index i, there exists a subset of length
i + − i − of (z i ) i∈Z , denoted by {z i − , z i − +1 , . . . , z i + −1 , z i + } such that z i − +1 ≤ ǫ(n), z i + −1 ≥ 1 − ǫ(n). By Lemma 2.9, ξ − i − +1 > 0 and ξ − i + −1 < 1 for the minimal configuration (ξ − i ) i∈Z . We consider the configuration (x i ) i∈Z defined by x i = ξ − i , i − < i < i + , x i = 0, i ≤ i − , x i = 1, i ≥ i + . By the definition of h n , h n (x i ,x i+1 ) = 0 for i < i − or i ≥ i + , then i∈Z h n (x i ,x i+1 ) = h n (x i − , ...,x i + ).
Moreover, by the minimality of (z i ) i∈Z , we have
(2.1.17) K ≤ i∈Z h n (x i ,x i+1 ) = h n (x i − , ...,x i + ). By the construction of h n , we have v n (x i − +1 ) = v n (ξ − i − +1 ) = 0. Hence, h n (x i − ,x i − +1 ) − h n (ξ − i − , ξ − i − +1 ) = 1 2 (x i − −x i − +1 ) 2 + u n (x i − +1 ) − 1 2 (ξ − i − − ξ − i − +1 ) 2 − u n (ξ − i − +1 ), = 1 2 (ξ − i − +1 ) 2 − 1 2 (ξ − i − +1 − ξ − i − ) 2 , = 1 2 ξ − i − (2ξ − i − +1 − ξ − i − ), ≤ Cǫ(n) 2 . (2.1.18) It is similar to obtain (2.1.19) h n (x i + −1 ,x i + ) − h n (ξ − i + −1 , ξ − i + ) ≤ Cǫ(n) 2 . Since h n (x i − , ...,x i + ) − h n (ξ − i − , ..., ξ − i + ) = h n (x i − ,x i − +1 ) − h n (ξ − i − , ξ − i − +1 ) + h n (x i + −1 ,x i + ) − h n (ξ − i + −1 , ξ − i + ), then (2.1.20) h n (x i − , ...,x i + ) − h n (ξ − i − , ..., ξ − i + ) ≤ Cǫ(n) 2 .
From (2.1.17) and (2.1.20) we have
(2.1.21) K ≤ h n (ξ − i − , ..., ξ − i + ) + Cǫ(n) 2 .
To obtain the reverse inequality of (2.2.45), we consider the configuration as follows
x i = z i , i − < i < i + , x i = 0, i ≤ i − , x i = 1, i ≥ i + . From the definition of h n , it follows that v n (z i − +1 ) = 0 and h n (z i , z i+1 ) ≥ 0 for all i ∈ Z. Moreover, we have h n (x i − , ...,x i + ) − K = h n (x i − ,x i − +1 ) + h n (x i + −1 ,x i + ) − i<i − ,i≥i + h n (z i , z i+1 ), ≤ 1 2 (z i − +1 ) 2 + u n (z i − +1 ) + 1 2 (z i + −1 − 1) 2 , ≤ u ′ n (η)z i − +1 + C 1 ǫ(n) 2 , ≤ 2πn −a sin(2πη)z i − +1 + C 1 ǫ(n) 2 , ≤ C 2 n −a (z i − +1 ) 2 + C 1 ǫ(n) 2 , ≤ Cǫ(n) 2 . where η ∈ (0, z i − +1 ). Namely (2.1.22) h n (x i − , ...,x i + ) ≤ K + Cǫ(n) 2 .
Furthermore, we consider the finite segment of the configuration defined by
η i =x i , i − < i < i + , η i = ξ − i , i = i − , η i = ξ − i , i = i + .
Then, the minimality of (ξ
− i ) i∈Z implies h n (ξ − i − , ..., ξ − i + ) ≤ h n (η i − , ..., η i + ). Hence, by (2.1.22), we have (2.1.23) h n (ξ − i − , ..., ξ − i + ) ≤ K + Cǫ(n) 2 + h n (η i − , ..., η i + ) − h n (x i − , ...,x i + ), where h n (η i − , ..., η i + ) − h n (x i − , ...,x i + ) = h n (η i − , η i − +1 ) − h n (x i − ,x i − +1 ) + h n (η i + −1 , η i + ) − h n (x i + −1 ,x i + ).
By the deduction as similar as (2.1.18), we have
h n (η i − , η i − +1 ) − h n (x i − ,x i − +1 ) ≤ Cǫ(n) 2 , h n (η i + −1 , η i + ) − h n (x i + −1 ,x i + ) ≤ Cǫ(n) 2 .
Moreover,
(2.1.24) h n (η i − , ..., η i + ) − h n (x i − , ...,x i + ) ≤ Cǫ(n) 2 .
Hence, from (2.1.23) and (2.13), it follows that
(2.1.25) h n (ξ − i − , ..., ξ − i + ) ≤ K + Cǫ(n) 2 ,
which together with (2.2.45) implies
(2.1.26) |h n (ξ − i − , ..., ξ − i + ) − K| ≤ Cǫ(n) 2 .
Next, we compare h n (y i − , ..., y i + ) with K(ξ).
Since (ξ i ) i∈Z is minimal among all configurations with rotation symbol ω satisfying ξ 0 = ξ. By (2.1.10) and Aubry's crossing lemma, we have
d(ξ i ) ≤ ǫ(n), for i = i − , i + , where d(ξ i ) = min{|ξ i |, |ξ i − 1|}.
By an argument as similar as the one in the comparison between K and h n (ξ
− i − , ..., ξ − i + ), we have (2.1.27) |h n (ξ i − , ..., ξ i + ) − K(ξ)| ≤ Cǫ(n) 2 .
By the construction of (y i ) i∈Z , namely
y i = ξ i , i − < i < i + , ξ − i , i ≤ i − , i ≥ i + , we have h n (y i − , ..., y i + ) − h n (ξ i − , ..., ξ i + ) = h n (ξ − i − , ξ i − +1 ) − h n (ξ i − , ξ i − +1 ) + h n (ξ i + −1 , ξ − i + ) − h n (ξ i + −1 , ξ i + ).
By the deduction as similar as (2.1.20), we have
(2.1.28) |h n (y i − , ..., y i + ) − h n (ξ i − , ..., ξ i + )| ≤ Cǫ(n) 2 .
Finally, from (2.1.14), (2.1.26), (2.1.27) and (2.1.28) we obtain
|P hn ω (ξ) − P hn 0 + (ξ)| ≤ |h n (y i − , ..., y i + ) − h n (ξ − i − , ..., ξ − i + ) + K − K(ξ)| + C 1 ǫ(n) 2 , ≤ |h n (ξ i − , ..., ξ i + ) − K(ξ)| + |h n (ξ − i − , ..., ξ − i + ) − K| + |h n (y i − , ..., y i + ) − h n (ξ i − , ..., ξ i + )| + C 1 ǫ(n) 2 , ≤ Cǫ(n) 2 , = C exp −2n δ 2 ,
which completes the proof of Lemma 2.5.
Proof of Theorem 2.1
Based on the preparation above, it is easy to prove Theorem 2.1. We assume that there exists an invariant circle with rotation number 0 < ω < n − a 2 −δ for h n , then P hn ω (ξ) ≡ 0 for every ξ ∈ R. By Lemma 2.5, we have
(2.1.29) |P hn 0 + (ξ)| ≤ C exp −2n δ 2 , for ξ ∈ 1 2 − 1 n a , 1 2 + 1 n a .
On the other hand, (2.1.7) implies that there exists a pointξ ∈ 1 2 − 1 n a , 1 2 + 1 n a such that
P hn 0 + (ξ) ≥ n −s . Hence, we have n −s ≤ C exp −2n δ 2 .
It is an obvious contradiction for n large enough. Therefore, there exists no invariant circle with rotation number 0 < ω < n − a 2 −δ . For −n − a 2 −δ < ω < 0, by comparing P hn ω (ξ) with P hn 0 − (ξ), the proof is similar. We omit the details. Therefore, the proof of Theorem 2.1 is completed.
C ω case
We will prove the following theorem:
Theorem 2.11 Given an integrable generating function h 0 , a rotation number ω and a small positive constant δ, there exists a sequence of real-analytic (h n ) n∈N such that h n → h 0 in the C 3−δ topology and the exact monotone area-preserving twist maps generated by (h n ) n∈N admit no invariant circles with the rotation number ω.
Construction of the generating functions
Consider a completely integrable system with the generating function
h 0 (x, x ′ ) = 1 2 (x − x ′ ) 2 , x, x ′ ∈ R.
We construct the perturbation consisting of two parts. The first one is
(2.1.30) u n (x) = 1 n a (1 − cos(2πx)), x ∈ R,
where n ∈ N and a is a positive constant independent of n.
We construct the second part of the perturbation in the following. Let p N (x) be a trigonometric polynomial of degree N . It is easy to see that for any r > 0,
(2.1.31) ||p N (x)|| r ≤ e rN ||p N (x)||,
where ||p N (x)|| r denotes the maximum of |p N (z)| in the strip S r = {z ∈ C| |Imz| ≤ r} of width 2r in the complex plane and ||p N (x)|| denotes the maximum of |p N (x)| on the real line. Without loss of generality, we take r = 1, a.e.
(2.1.32)
||p N (x)|| 1 ≤ e N max |p N (x)|.
Then, by Cauchy estimates, for any fixed s > 0, we have
(2.1.33) ||p N (x)|| C s ≤ C s e N max |p N (x)|,
where C s is a constant depending only on s. Based on Lemma 2.4, we need to construct a real analytic function with a "bump" in correspondence with the interval Λ n satisfying
(2.1.34) L(Λ n ) ∼ n − a 2 and Λ n ⊂ 1 4 , 3 4 ,
where L(Λ n ) denotes the Lebesgue measure of Λ n and f ∼ g means that 1 C g < f < Cg holds for some constant C > 1.
The "bump" will be accomplished by using Jackson's approximation theorem (see [Z, p115]). It states that let φ(x) be an k-times differentiable periodic function on R, then for every N ∈ N, there exists a trigonometric polynomial p N (x) of degree N such that
max |p N (x) − φ(x)| ≤ A k N −k ||φ(x)|| C k ,
where A k is a constant depending only on k ∈ N.
We take a C ∞ bump function φ supported on the interval Λ n , whose maximum is equal to 2. By (2.1.34), the length of Λ n is bounded by Cn − a 2 , one can choose φ(x) such that
(2.1.35) ||φ(x)|| C k ∼ n a 2 k = n ak 2 .
Then, choosing N large enough to achieve
(2.1.36) A k N −k ||φ(x)|| C k < σ ≪ 1,
where σ will be determined in the following, by Jackson's approximation theorem, we can construct a trigonometric polynomial p N (x) of degree N such that:
(2.1.37) max p N (x) ≥ 1, attained on Λ n , |p N (x)| ≤ σ, on [0, 1]\Λ n .
By (3.4.2), we have
(2.1.38) N ≥ Cσ − 1 k n a 2 .
Finally, we consider the normalized trigonometric polynomial
(2.1.39)p N (x) = e −2N p N (x) max p N (x) 2 .
From (2.1.33),p N (x) satisfies:
(2.1.40) p N (x) ≥ 0, ||p N (x)|| C s ≤ C, maxp N (x) = e −2N , attained on Λ n , |p N (x)| ≤ σ 2 e −2N , on [0, 1]\Λ n .
Based on preparations above, we can construct the second part of the perturbation as follow
(2.1.41) v n (x) = u n (x)p N (x) = 1 n a (1 − cos 2πx)p N (x).
It is easy to see v n satisfies the following properties:
(2.1.42)
v n (x) ≥ 0, ||v n (x)|| C s ≤ Cn −a , max v n (x) ≥ e −2N n −a , attained on Λ n , |v n (x)| ≤ Cσ 2 e −2N n −a , on [0, 1]\Λ n .
So far, we complete the construction of the generating function of the nearly integrable system, a.e.
(2.1.43)
h n (x, x ′ ) = h 0 (x, x ′ ) + u n (x ′ ) + v n (x ′ ),
where n ∈ N.
Proof of Theorem 2.11
If ω ∈ Q, then the invariant circles with rotation number ω could be easily destructed by an analytic perturbation arbitrarily close to 0. Therefore it suffices to consider the irrational ω. Firstly, we prove the non-existence of invariant circles with a small enough rotation number. More precisely, we have the following Lemma:
Lemma 2.12 For ω ∈ R\Q and n large enough, the exact area-preserving monotone twist map generated by h n admits no invariant circle with the rotation number satisfying
|ω| < n −a−δ ,
where δ is a small positive constant independent of n.
Proof First of all, we estimate the lower bound of P hn 0 + . Let (ξ i ) i∈Z be a minimal configuration of h n defined by (2.2.38) with rotation symbol 0 + satisfying ξ 0 = η, where η satisfies v n (η) = max v n (x) and let (x i ) i∈Z be the minimal configuration
ofh n (x i , x i+1 ) = h 0 (x i , x i+1 ) + u n (x i+1 ) with rotation symbol 0 + satisfying x 0 ∈ [0, 1]\Λ n , then i∈Z (h n (ξ i , ξ i+1 ) − h n (ξ − i , ξ − i+1 )) ≥ v n (η) + i∈Zh n (ξ i , ξ i+1 ) − i∈Z h n (ξ − i , ξ − i+1 ), ≥ v n (η) + i∈Zh n (x i , x i+1 ) − i∈Z h n (x i , x i+1 ), = v n (η) − i∈Z v n (x i+1 ).
Therefore, we have shown:
(2.1.44) P hn 0 + (η) = min
ξ 0 =η i∈Z (h n (ξ i , ξ i+1 ) − h n (ξ − i , ξ − i+1 )) ≥ v n (η) − i∈Z v n (x i+1 ).
By (2.1.42), we have
(2.1.45) v n (η) ≥ e −2N n −a .
It follows that
(2.1.46) i∈Z v n (x i+1 ) ≤ σ 2 e −2N i∈Z u n (x i+1 ) ≤ σ 2 e −2N i∈Z 1 4 (x i+1 − x i−1 ) 2 ≤ σ 2 e −2N .
Hence,
(2.1.47) P hn 0 + (η) ≥ e −2N (n −a − σ 2 ),
we choose then σ (consequently N ) in such a way that
1 4 n −a − σ 2 ≥ 0, which implies σ ≤ 1 2 n − a 2 .
By (3.4.4), if follows that
(2.1.48) N ≥ Cn a 2 + a 2k .
Therefore,
(2.1.49) max N P hn 0 + (η) ≥ n −a exp −Cn a 2 + a 2k .
Second, following a similar argument as [W1], we have
(2.1.50) |P hn ω (ξ) − P hn 0 + (ξ)| ≤ C exp −2n a 2 + δ 2 .
where ξ ∈ Λ n and δ is a small positive constant independent of n. Here Λ n is as the same as the notation in (2.1.34). Based on the preparations above, it is easy to prove Lemma 2.12. We assume that there exists an invariant circle with rotation number 0 < ω < n −a−δ for h n , then P hn ω (ξ) ≡ 0 for every ξ ∈ R. By (2.1.50), we have
(2.1.51) |P hn 0 + (ξ)| ≤ C exp −2n a 2 + δ 2 , for ξ ∈ Λ n .
On the other hand, (2.1.49) implies that there exists a point η ∈ Λ n such that
P hn 0 + (η) ≥ n −a exp −Cn a 2 + a 2k .
Hence, we have
(2.1.52) n −a exp −Cn a 2 + a 2k ≤ C exp −2n a 2 + δ 2 .
To achieve the contradiction, it suffices to take
k > 3a 2δ , which implies a 2k < δ 3 < δ 2 .
Hence, for n large enough
(2.1.53) n −a exp −Cn a 2 + a 2k ≥ C exp −2n a 2 + δ 2 ,
which contradicts (2.1.52). Therefore, there exists no invariant circle with rotation number 0 < ω < n −a−δ . For −n −a−δ < ω < 0, by comparing P hn ω (ξ) with P hn 0 − (ξ), the proof is similar. We omit the details. This completes the proof of Lemma 2.12.
By Lemma 2.2, the case with a given irrational rotation number can be easily reduced to the one with a small enough rotation number. For the sake of simplicity of notations, we denote Q qn by Q n and the same to u qn , v qn and h qn . Let Q n (x) = q n −2 (u n (q n x) + v n (q n x)),
where (q n ) n∈N is a sequence satisfying Dirichlet approximation
(2.1.54) |q n ω − p n | < 1 q n ,
where p n ∈ Z and q n ∈ N. Since ω ∈ R\Q, we say q n → ∞ as n → ∞. Let h n (x, x ′ ) = h 0 (x, x ′ ) + Q n (x ′ ), we prove Theorem 2.11 for (h n ) n∈N as follow:
Proof Based on Lemma 2.12 and Dirichlet approximation (2.1.54), it suffices to take 1 q n ≤ 1 q n a+δ , which implies
(2.1.55) a ≤ 1 − δ.
From the constructions of u n and v n , it follows that
||h n (x, x ′ ) − h 0 (x, x ′ )|| C r = ||Q n (x ′ )|| C r , ≤ q n −2 (||u n (q n x ′ )|| C r + ||v n (q n x ′ )|| C r ), ≤ q n −2 (q n −a (2π) r q n r + C 1 q n −a q n r ), ≤ C 2 q n r−a−2 ,
where C 1 , C 2 are positive constants only depending on r.
To complete the proof, it is enough to make r − a − 2 < 0, which together with (2.1.55) implies r < a + 2 ≤ 3 − δ.
This completes the proof of Theorem 2.11.
Case with d ≥ 3 degrees of freedom
Preliminaries
In T * T d , a submanifold T d is called Lagrangian torus if it is diffeomorphic to the torus T d and the symplectic form vanishes on it. For positive definite Hamiltonian systems, if a Lagrangian torus is invariant under the Hamiltonian flow, it is then the graph over T d (see [BP]). An example of Lagrangian torus is the KAM torus.
Definition 2.13T d is called d dimensional KAM torus if •T d is a Lipschitz graph over T d ;
•T d is invariant under the Hamiltonian flow Φ H t generated by the Hamiltonian function H;
• there exists a diffeomorphism φ :
T d →T d such that φ −1 • Φ t H • φ = R t ω for any t ∈ R, where R t ω :
x → x + ωt and ω is called the rotation vector ofT d . For positive definite Hamiltonian systems, each KAM torusT d supports a minimal measure µ. The rotation number ρ of µ is well defined and ρ(µ) = ω. The rotation vector of the Lagrangian torus T d is not well defined. If T d supports several invariant measures with different rotation vectors. In this paper, we are only concerned with Lagrangian tori with well defined rotation vectors.
Definition 2.14 T d is called d dimensional Lagrangian torus with the rotation vector ω if • T d is a Lagrangian submanifold;
• T d is invariant for the Hamiltonian flow Φ t H generated by H.
• each orbit on T d has the same rotation vector.
In [H4], it is proved that each orbit on T d is an action minimizing curve. An arithmetic approximation of the rotation vector is found in [Ch]. For any given vector ω ∈ R d with d ≥ 2, there is a sequence of integer vectors k n ∈ Z d with |k n | → ∞ such that
(2.2.1) | ω, k n | < C |k n | d−1 , where C is a constant independent of n, |k| = d j=1 k 2 i 1 2 , for k = (k 1 , k 2 , . . . , k d ).
A rotation vector ω ∈ R d is called resonant if there exists k ∈ Z d such that ω, k = 0. Otherwise, it is non-resonant. Obviously, a Lagrangian torus with the resonant rotation vector can be destructed by analytic perturbation arbitrarily close to zero. Hence, it is sufficient to consider the Lagrangian torus with the non resonant rotation vector. In that case, one can assume that the Lagrangian torus T d supports a uniquely ergodic minimizing measure. Moreover, by [Ma3], T d is a Lipschitz graph over the underlying manifold T d .
C ∞ case
We will prove the following theorem:
Theorem 2.15 Given an integrable Hamiltonian H 0 with d (d ≥ 3) degrees of freedom, a rotation vector ω and a small positive constant δ, there exists a sequence of C ∞ Hamiltonians {H n } n∈N such that H n → H 0 in C 2d−δ topology and the Hamiltonian flow generated by H n does not admit the Lagrangian torus with the rotation vector ω.
This theorem implies that the rigidity of the Lagrangian torus is as the same as the KAM torus. Roughly speaking, the maximum of r is closely related to the arithmetic property of the rotation vector ω. If ω is a Diophantine vector, then r is at most 2d − δ. If ω is a Liouville vector, then r can be arbitrarily large. If ω can be approximated exponentially by rational vectors, then the Lagrangian torus with the rotation vector ω can be destructed by an arbitrarily small perturbation in C ω (analytic) topology (see [Be]).
L. WANG
Destruction of Lagrangian torus with a special rotation vector
The Hamiltonian function we consider here is nearly integrable
(2.2.2) H n (q, p) = H 0 (p) − P n (q),
where (q, p) ∈ T d × R d . Without loss of generality, we assume
H 0 (p) = 1 2 |p| 2 ,
for which (2.2.2) is a typical mechanical system. Since H n is strictly convex with respect to p, by the Legendre transformation, the Lagrangian function corresponding to H n is
(2.2.3) L n (q,q) = 1 2 |q| 2 + P n (q), whereq = ∂H 0 ∂p . Let P n (q) = 1 n a (1 − cos q 1 ) + v n (q 1 , q 2 )
, where a is a positive constant independent of n. For the rotation vector ω = (ω 1 , . . . , ω d ), v n (q 1 , q 2 ) is constructed as follow
(2.2.4) v n is 2π-periodic, supp v n ∩ {[0, 2π] × [−π, π]} = B Rn (q * ), max (q 1 ,q 2 )∈[0,2π]×[−π,π] v n = v n (q 0 ) = |ω 1 | s , ||v n || C r ∼ |ω 1 | s ′ ,
where R n = |ω 1 | n 2 , q * = (π, 0) and we require s ′ > 3, it can be satisfied if s > r + 3. For (2.2.2), we have the following lemma.
Lemma 2.16 For n large enough, the Hamiltonian flow generated by H n (q, p) does not admit any Lagrangian torus with rotation vector ω = (ω 1 , . . . , ω d ) satisfying
|ω 1 | < n − a 2 −ǫ ,
where ǫ > 0 is independent of n.
Lemma 2.24 will be proved with variational method. First of all, we put it into the Lagrangian formalism. Let σ n = n −a . The Lagrangian function corresponding to (2.2.3) is L n q 1 , Q,q 1 ,Q = 1 2 |Q| 2 + 1 2 |q 1 | 2 + σ n (1 − cos(q 1 )) + v n (q 1 , q 2 ), (2.2.5) where Q = (q 2 , . . . , q d ). L n (q 1 , Q,q 1 ,Q) can be considered as a perturbation coupling of a rotator with d − 1 degrees of freedom and a perturbation with the Lagrangian function (2.2.6) A n (q 1 ,q 1 ) = 1 2 |q 1 | 2 + σ n (1 − cos(q 1 )), which corresponds to the Hamiltonian via Legendre transformation (2.2.7) h n (q 1 , p 1 ) = 1 2 |p 1 | 2 − σ n (1 − cos q 1 ).
The action of the simple pendulum
Each solution of the Lagrangian equation determined by A n , denoted by q 1 (t), determines an orbit (q 1 (t), p 1 (t)) of the Hamiltonian flow generated by h n . Each orbit stays in certain energy level set (q 1 , p 1 ) ∈ h −1 n (e). Under the boundary condition that t 0 = 0, t 1 = π (or t 1 = π, t 2 = 2π), there is a unique correspondence between t 1 − t 0 and the energy, denoted by e(t 1 − t 0 ), such that the determined orbit stays in the energy level set h −1 n (e(t 1 − t 0 )). More precisely, we have the following lemma.
Lemma 2.17 Letq 1 be the solution of A n on (t 0 ,t 1 ) satisfying the boundary conditions q 1 (t 0 ) = 0, q 1 (t 1 ) = π, e(t 1 − t 0 ) be the energy ofq 1 , i.e. (q 1 ,p 1 ) ∈ h −1 n (e(t 1 − t 0 )) and ω 1 be the average speed ofq 1 on (t 0 ,t 1 ), then
(2.2.8) e(t 1 − t 0 ) ∼ σ n exp − C √ σ n |ω 1 | ,
where f ∼ g means that 1 C g < f < Cg holds for some constant C > 0, σ n = n −a . Proof By the definition, we have
1 2 |q 1 | 2 − σ n (1 − cos(q 1 )) = e(t 1 − t 0 ),
hence |q 1 | = 2(e(t 1 − t 0 ) + σ n (1 − cos(q 1 ))).
Since the average speed ofq 1 is ω 1 , by a direct calculation, we have
π |ω 1 | = t 1 t 0 dt = π 0 dq 1 2(e(t 1 − t 0 ) + σ n (1 − cos(q 1 ))) ∼ 1 √ σ n ln σ n e(t 1 − t 0 ) , moreover, e(t 1 − t 0 ) ∼ σ n exp − C √ σ n |ω 1 | ,
which complete the proof of Lemma 2.17.
It is easy to see that Lemma 2.17 also holds for q 1 (t 1 ) = π, q 1 (t 2 ) = 2π.
The following lemma implies that the actions along orbits in the neighborhood of the separatix of the pendulum does not change too much with respect to a small change in speed (time).
Lemma 2.18 Lett 1 ,t 1 ∈ [t 0 , t 2 ]. Letq 1 (t) be a solution of A n on (t 0 ,t 1 ) and (t 1 , t 2 ) with boundary conditions respectively q 1 (t 0 ) = 0, q 1 (t 1 ) = π, q 1 (t 1 ) = π, q 1 (t 2 ) = 2π, and letq 1 (t) be a solution of A n on (t 0 ,t 1 ) and (t 1 , t 2 ) with boundary conditions respectively q 1 (t 0 ) = 0, q 1 (t 1 ) = π, q 1 (t 1 ) = π, q 1 (t 2 ) = 2π.
Letω ′ 1 andω ′′ 1 be the average speed ofq 1 on (t 0 ,t 1 ) and (t 1 , t 2 ) respectively. Letω ′ 1 andω ′′ 1 be the average speed ofq 1 on (t 0 ,t 1 ) and (t 1 , t 2 ) respectively. We set
|ω 1 | = max |ω ′ 1 |, |ω ′′ 1 |, |ω ′ 1 |, |ω ′′ 1 | , then (2.2.9) t 2 t 0 A n (q 1 ,q 1 )dt − t 2 t 0 A n (q 1 ,q 1 )dt ≤ C 1 |t 1 −t 1 |σ n exp − C 2 √ σ n |ω 1 | .
Proof The proof follows the similar idea of Lemma 4 in [Be]. Let q 1 (t) be a solution of A n on (t 0 , t 1 ) and (t 1 , t 2 ) with boundary conditions respectively q 1 (t 0 ) = 0, q 1 (t 1 ) = π, q 1 (t 1 ) = π, q 1 (t 2 ) = 2π.
We consider the function
L(t 1 ) = t 1 t 0 A n (q 1 ,q 1 )dt + t 2 t 1 A n (q 1 ,q 1 )dt, = π 0 2(e(t 1 − t 0 ) + V (q 1 ))dq 1 − e(t 1 − t 0 )(t 1 − t 0 ) + 2π π 2(e(t 2 − t 1 ) + V (q 1 ))dq 1 − e(t 2 − t 1 )(t 2 − t 1 ),
where V (q 1 ) = σ n (1 − cos(q 1 )), and e(∆t) denotes the energy of the orbit of the pendulum moving half a turn in time ∆t. The quantity e(∆t) is differentiable with respect to ∆t, then
dL(t 1 ) dt 1 = π 0ė (t 1 − t 0 ) 2(e(t 1 − t 0 ) + V (q 1 )) dq 1 −ė(t 1 − t 0 )(t 1 − t 0 ) − e(t 1 − t 0 ) − 2π πė (t 2 − t 1 ) 2(e(t 2 − t 1 ) + V (q 1 )) dq 1 +ė(t 2 − t 1 )(t 2 − t 1 ) + e(t 2 − t 1 ), = t 1 t 0ė (t 1 − t 0 )dt −ė(t 1 − t 0 )(t 1 − t 0 ) − e(t 1 − t 0 ) − t 2 t 1ė (t 2 − t 1 )dt +ė(t 2 − t 1 )(t 2 − t 1 ) + e(t 2 − t 1 ), =e(t 2 − t 1 ) − e(t 1 − t 0 ).
(2.2.10)
Thus, we have dL(t 1 ) dt 1 ≤ |e(t 2 − t 1 )| + |e(t 1 − t 0 )|.
Integrate fromt 1 tot 1 and from (2.2.8), it follows that
t 2 t 0 A n (q 1 ,q 1 )dt − t 2 t 0 A n (q 1 ,q 1 )dt ≤ C 1 |t 1 −t 1 |σ n exp − C 2 √ σ n |ω 1 | ,
which completes the proof of Lemma 2.18.
The velocity of the action minimizing orbit
Once the function q 1 (t) is fixed, the function Q(t) is the solution of the Euler-Lagrange equation with the non autonomous Lagrangian
(2.2.11) 1 2 |Q(t)| 2 + v n (q 1 (t), q 2 (t)) ,
where Q(t) = (q 2 (t), . . . , q d (t)).
Lemma 2.19
Let (q 1 (t), Q(t)) be the orbit of L n with rotation vector ω, then for any t ′ , t ′′ ∈ R and t ∈ [t ′ , t ′′ ] we have
(2.2.12) Q (t) − Q(t ′′ ) − Q(t ′ ) t ′′ − t ′ ≤ C|ω 1 | 2 .
Proof By the Euler-Lagrange equation, we havë q i (t) = 0, for i = 3, . . . , d, henceq i (t) = const., (2.2.12) is verified obviously for q i (t), i = 3, . . . , d. We just need to consider q 2 (t). Let q 1 (t 0 ) = 0 and q 1 (t 2 ) = 2π. It suffices to prove that for
t ∈ [t ′ , t ′′ ] ⊂ [t 0 , t 2 ] (2.2.13) q 2 (t) − q 2 (t ′′ ) − q 2 (t ′ ) t ′′ − t ′ ≤ C|ω 1 | 2 .
From the Euler-Lagrange equation, q 2 (t) = ∂v n ∂q 2 (q 1 (t), q 2 (t)), together with ||v n || C r ∼ |ω 1 | s ′ , we obtain
q 2 (t) ≤ C 1 |ω 1 | s ′ .
Integrate the two sides of the inequality above from t ′ to t ′′ , we have
|q 2 (t ′′ ) −q 2 (t ′ )| ≤ C 2 |ω 1 | s ′ |t ′′ − t ′ |.
It follows from (2.2.1) that |t ′′ − t ′ | ≤ C 3 |ω 1 | −1 . Hence
|q 2 (t ′′ ) −q 2 (t ′ )| ≤ C 4 |ω 1 | s ′ −1 . Since s ′ > 3, we have |q 2 (t ′′ ) −q 2 (t ′ )| ≤ C 4 |ω 1 | 2 .
This completes the proof.
Proof of Lemma 2.24
Based on the minimal property of the orbits on an invariant Lagrangian torus and its graph property, passing through each x ∈ T d , there is a unique minimal curve q(t) with rotation vector ω if the Hamiltonian flow generated by H n admits a Lagrangian torus with rotation vector ω. Hence, it is sufficient to prove the existence of some point in T d where no minimal curve passes through.
Indeed, any minimal curve does not pass through the subspace (π, 0) × T d−2 . It implies Lemma 2.24. Let us assume the contrary, namely, there existst 1 such that q 1 (t 1 ) = π, q 2 (t 1 ) = 0, where q(t) = (q 1 , q 2 , . . . , q d )(t) is a minimal curve in the universal covering space R d . Because of ω 1 = 0, there exist t 0 and t 2 such that q 1 (t 0 ) = 0, q 1 (t 2 ) = 2π.
Obviously, t 0 <t 1 < t 2 and
t 2 − t 0 ∼ 1 |ω 1 | .
Lett 1 be the last time beforet 1 or the first time aftert 1 such that
|q 2 (t 1 ) − q 2 (t 1 )| = π.
It is easy to see that,
|t 1 −t 1 | ∼ 1 |ω 2 | .
Since |ω 2 | ∼ 1, then |t 1 −t 1 | ≤ C 0 .
Without loss of generality, one can assume ω 1 > 0 and ω 2 > 0. Consider a solutioñ q 1 of A n on (t 0 ,t 1 ) and on (t 1 , t 2 ) with boundary conditions respectively q 1 (t 0 ) = q 1 (t 0 ) = 0, q 1 (t 1 ) = q 1 (t 1 ) = π, q 1 (t 1 ) = q 1 (t 1 ) = π, q 1 (t 2 ) = q 1 (t 2 ) = 2π.
Since q is assumed to be a minimal curve, we have (2.2.14)
t 2 t 0 L n (q 1 , Q,q 1 ,Q)dt − t 2 t 0 L n (q 1 , Q,q 1 ,Q)dt ≥ 0.
See Fig.1, where x 1 = (q 1 (t 1 ), q 2 (t 1 )) = (π, 0), x 0 = (q 1 (t 0 ), q 2 (t 0 )) = (0, q 2 (t 0 )), x 2 = (q 1 (t 2 ), q 2 (t 2 )) = (2π, q 2 (t 2 )),x ′ 1 = (π, −π) andx ′′ 1 = (π, π). (q 1 (t), q 2 (t)) passes through the pointx ′ 1 orx ′′ 1 . Thus, by the construction of L n , we obtain from (2.2.14) that
(2.2.15) t 2 t 0 A n (q 1 ,q 1 )dt − t 2 t 0 A n (q 1 ,q 1 )dt ≥ t 2 t 0 v n (q 1 , q 2 )dt − t 2 t 0 v n (q 1 , q 2 )dt.
By the definition of v n as (2.2.4), we find (q 1 (t), q 2 (t)) ∩ supp v n = ∅, for t ∈ (t 0 , t 2 ). · · · · · · · · · · · · 0 −π π q1 q2 2π 0 x0 x2 x1x ′′ 1 x ′ 1 Figure 1: The projections of the curves (q 1 (t), Q(t)) and (q 1 (t), Q(t)) on [0, 2π] × R In fact, if there would existt such that (q 1 (t), q 2 (t)) ∈ supp v n , without loss of generality, one can assumet >t 1 . By Lemma 2.19, for any t ∈ [t 1 ,t],
q 2 (t) ≤ C 1 , hence,t −t 1 ≥ C 2 ,
where C 1 , C 2 are constants independent of n. Consequently
|q 1 (t) −q 1 (t 1 )| ≥ C 3 |ω 1 | > R n ,
where R n is the radius of the support of v n . It is impossible. Hence, we have
t 2 t 0 v n (q 1 , q 2 )dt − t 2 t 0 v n (q 1 , q 2 )dt = t 2 t 0 v n (q 1 , q 2 )dt.
By the construction of v n and the minimality of (q 1 , Q), a simple calculation shows (2.2.16)
t 2 t 0 v n (q 1 , q 2 )dt ≥ |ω 1 | λ ,
where λ is a positive constant. Consequently, if follows from (2.2.15) that
(2.2.17) t 2 t 0 A n (q 1 ,q 1 )dt − t 2 t 0 A n (q 1 ,q 1 )dt ≥ |ω 1 | λ .
On the other hand, consider a solutionq 1 of A n on (t 0 ,t 1 ) and on (t 1 , t 2 ) with boundary conditions respectively q 1 (t 0 ) = q 1 (t 0 ) = 0, q 1 (t 1 ) = q 1 (t 1 ) = π, q 1 (t 1 ) = q 1 (t 1 ) = π, q 1 (t 2 ) = q 1 (t 2 ) = 2π.
Along both of which the action of A n achieves its minimum. Thus, we have
t 2 t 0 A n (q 1 ,q 1 )dt ≥ t 2 t 0 A n (q 1 ,q 1 )dt.
We compare the action t 2 t 0 A n (q 1 ,q 1 )dt with the action t 2 t 0 A n (q 1 ,q 1 )dt in the alternative cases, which is based on the choices oft 1 . See Fig.2, wheret = t 0 +t 2 2 .
L. WANḠ Case 1: |t 1 −t| ≤ C 0 . In this case, the average speed ofq 1 on (t 0 ,t 1 ) and (t 1 , t 2 ) have the same quantity order as |ω 1 |. By |t 1 −t 1 | ≤ C 0 , we have |t 1 −t| ≤ 2C 0 . Hence the average speed of q 1 on (t 0 ,t 1 ) and (t 1 , t 2 ) have also the same quantity order as |ω 1 |. Thus, Lemma 2.18 implies
t 2 t 0 A n (q 1 ,q 1 )dt − t 2 t 0 A n (q 1 ,q 1 )dt ≤ C 4 σ n exp − C 5 √ σ n |ω 1 | .
Case 2: |t 1 −t| > C 0 . In this case, we taket 1 such that |t 1 −t| ≤ |t 1 −t|, which can be achieved by the suitable choice of the position ofq 1 (t 1 ). More precisely,
• ift 1 >t + C 0 (Case 2a in Fig.2), we chooset 1 as the last time beforet 1 , i.e.
(q 1 (t 1 ), q 2 (t 1 )) =x ′ 1 in Fig.1;
• ift 1 <t − C 0 (Case 2b in Fig.2), we chooset 1 as the first time aftert 1 , i.e. (q 1 (t 1 ), q 2 (t 1 )) =x ′′ 1 in Fig.1.
For both cases 2a and 2b, it follows from (2.2.10) that
t 2 t 0 A n (q 1 ,q 1 )dt − t 2 t 0 A n (q 1 ,q 1 )dt ≤ 0.
Hence, for anyt 1 ∈ (t 0 , t 2 ), we can findt 1 such that
t 2 t 0 A n (q 1 ,q 1 )dt − t 2 t 0 A n (q 1 ,q 1 )dt ≤ t 2 t 0 A n (q 1 ,q 1 )dt − t 2 t 0 A n (q 1 ,q 1 )dt, ≤ C 4 σ n exp − C 5 √ σ n |ω 1 | . Since |ω 1 | ≤ n − a 2 −ǫ .
It is easy to see that for n large enough,
C 4 σ n exp − C 5 √ σ n |ω 1 | ≤ |ω 1 | λ ,
where σ n = n −a , which contradicts to (2.2.17) for large n. This completes the proof of Lemma 2.24.
Destruction of Lagrangian torus with an arbitrary rotation vector
By (2.2.1), for every non resonant rotation vector ω = (ω 1 , . . . , ω d ) (d ≥ 2), there exists a sequence of integer vector k n ∈ Z d satisfying |k n | → ∞ as n → ∞ such that | k n , ω | < C |k n | d−1 .
Transformation of coordinates
We choose a sequence of k n ∈ Z d satisfying (2.2.1) and an integer vector sequence k ′ n such that k ′ n , k n = 0. In addition, select d − 2 integer vectors l n3 , . . . , l nd such that k n , k ′ n , l n3 , . . . , l nd are pairwise orthogonal. Let
(2.2.18) K n = (k n , k ′ n , l n3 , . . . , l nd ) t .
We choose the transformation of the coordinates
q = K n x.
Let p denotes the dual coordinate of q in the sense of Legendre transformation, i.e. p = ∂L ∂q , it follows that y = K t n p, where K t n denotes the transpose of K n . We set
Φ n = K n K −t n , then q p = Φ n x y .
It is easy to verify that Φ t n J 0 Φ n = J 0 , where
J 0 = 0 1 −1 0 ,
where 1 denotes a d × d unit matrix. Hence, Φ n is a symplectic transformation in the phase space.
Lemma 2.20 If the Hamiltonian flow generated byH n (x, y) admits a Lagrangian torus with rotation vector ω, then the Hamiltonian flow generated by H n (q, p) also admits a Lagrangian torus with rotation vector K n ω, where (q, p) t = Φ n (x, y) t .
Proof LetT d be the Lagrangian torus admitted byH n (x, y), a symplectic form Ω vanishes on T xT d for every x ∈T d . Since K n consists of integer vectors, then T d := K nT d is still a torus. Φ n is a symplectic transformation, hence T d is a Lagrangian torus. From Definition 3.1, each orbit onT d has the same rotation vector ω. Letγ(t) be a lift of an orbit onT d , it follows that
ω = lim t→∞γ (t) −γ(−t)
2t .
L. WANG
For γ(t) = K nγ (t), we have
lim t→∞ γ(t) − γ(−t) 2t = lim t→∞ K nγ (t) − K nγ (−t) 2t ; = K n lim t→∞γ (t) −γ(−t) 2t ; = K n ω.
This completes the proof.
Remark 2.21 ForT d and T d in the proof of Lemma 2.20, we have
Vol (T d ) = | det K n |Vol (T d ),
where Vol(·) denotes the volume of (·).
Proof of Theorem 2.15
We constructH n (x, y) as follow:
(2.2.19)H n (x, y) = 1 2 |y| 2 −P n (x), whereP n (x) = 1 |k n | a+2 (1 − cos k n , x ) + 1 |k n | 2 v n k n , x , k ′ n , x ,
where k ′ n is the second row of K n and v n is defined by (2.2.4). Let q = K n x. In particular, we have (2.2.20) q 1 = k n , x , q 2 = k ′ n , x .
By the transformation of coordinates and the Legendre transformation, the Lagrangian function corresponding to (2.2.19) is
L n q 1 , Q,q 1 ,Q = 1 2 d i=3 |q i | 2 |l ni | 2 + |q 2 | 2 2|k ′ n | 2 + 1 |k n | 2 1 2 |q 1 | 2 + 1 |k n | a (1 − cos(q 1 )) + v n (q 1 , q 2 ) , (2.2.21)
where Q = (q 2 , . . . , q d ).
For the Hamiltonian flow generated by (2.2.19), by Lemma 2.20, for the destruction of the Lagrangian torusT d with rotation vector ω, it is sufficient to prove that the Euler-Lagrange flow generated by (2.2.21) admits no the Lagrangian torus T d := K nT d with rotation vector K n ω. Let K n ω = (ω 1 , ω 2 , . . . , ω d ). It is easy to see that there exists an integer vector k ′ n such that (2.2.22) k n , k ′ n = 0 and | k ′ n , ω | ∼ 1,
i.e. ω 2 ∼ 1. In fact, it suffices to consider k ′ n ∈ Z 3 . Let k ′ n = (k ′ n1 , k ′ n2 , k ′ n3 ), then for k ′ n ∈ Z d , one can take k ′ n = (k ′ n1 , k ′ n2 , k ′ n3 , 0, . . . , 0) to verify (2.2.22).
Since ω is non-resonant, then |k ′ n | → ∞, for n → ∞. Let θ be the angle between k ′ n and ω, then | k ′ n , ω | = |k ′ n ||ω|| cos θ|, where |k ′ n | is determined by (2.2.24) below. Let Π be the plane orthogonal with respect to k n . Let S Rα ⊂ Π be the sector with cental point (0, 0, 0), central angle α and radius R satisfying α = C 1 |k ′ n | and the angle between ω and one of the radii is equal to π 2 − C 2 |k ′ n | , where C 2 > C 1 . To achieve (2.2.22), it is sufficient to find an integer point m = (m 1 , m 2 , m 3 ) ∈ Z 3 satisfying (2.2.23) |m| ∼ |k ′ n | and m ∈ S Rα . In deed, we take k ′ n = (m 1 − 0, m 2 − 0, m 3 − 0). Since
cos θ ∼ cos( π 2 − 1 |k ′ n | ) = sin 1 |k ′ n | ∼ 1 |k ′ n | ,
we have | k ′ n , ω | ∼ 1. It is easy to see that there exists a suitable constant r(|k ′ n |) only depending on |k ′ n | such that the square of area (r(|k ′ n |)) 2 contains at least one integer point. We take (2.2.24) |k ′ n | ∼ (r(|k ′ n |)) κ , where κ ≫ 1, then it can be concluded that the integer satisfying (2.2.34) does exist.
Replacing n by |k n | in the proof of Lemma 2.24, we have that the Euler-Lagrange flow generated by (2.2.21) does not admit any Lagrangian torus with rotation vector satisfying |ω 1 | < |k n | − a 2 −ǫ . From the construction of K n , |ω 1 | = | k n , ω | which together with (2.2.1) implies
|ω 1 | < C |k n | d−1 .
Based on Lemma 2.20, it suffices to take C |k n | d−1 ≤ |k n | − a 2 −ǫ ,
||H n (x, y) − H 0 (y)|| C r = ||P n (x)|| C r , = |k n | −2 |k n | −a ||1 − cos k n , x || C r + ||v n ( k n , x , k ′ n , x )|| C r , ≤ |k n | −2 C 1 |k n | −a+r + C 2 |k n | −s ′ (d−1)+r , ≤ C 3 |k n | r−a−2 + |k n | r−3(d−1)−2 .
To complete the proof of Theorem 2.15, it is enough to make r − a − 2 < 0 and r − 3(d − 1) − 2 < 0, which together with (2.2.25) implies r < 2d − 2ǫ.
Taking δ = 3ǫ, this completes the proof of Theorem 2.15.
C ω case
We will prove the following theorem:
Theorem 2.22 Given an integrable Hamiltonian H 0 with d (d ≥ 3) degrees of freedom, a rotation vector ω and a small positive constant δ, there exists a sequence of C ω Hamiltonians {H n } n∈N such that H n → H 0 in C d+1−δ topology and the Hamiltonian flow generated by H n admits no Lagrangian torus with the rotation vector ω.
2.2.3.1. Construction of H n P n (x) is constructed as follow:
P n (x) = 1 |k n | d+1−ǫ (1 − cos k n , x ) + µ n 1 |k n | d+1−ǫ (1 − cos k n , x ) cos k ′ n , x ,
where k n , k ′ n are the first two rows of K n defined as (2.2.18), ǫ is a given small positive constant and µ n satisfies
(2.2.26) µ n ∼ exp −|k n | d 2 − 1 2 + ǫ 3 .
A simple calculation implies that for δ = 3ǫ
||H n (x, y) − H 0 (y)|| C d+1−δ = ||P n (x)|| C d+1−δ → 0 as n → ∞.
From the transformation matrix of coordinates (2.2.18), let δ n = 1 |kn| d−1−ǫ , the Lagrangian function corresponding to (2.2.3) is
L n q 1 , Q,q 1 ,Q = 1 2 d i=3 |q i | 2 |l ni | 2 + |q 2 | 2 2|k ′ n | 2 + 1 |k n | 2 1 2
|q 1 | 2 + δ n (1 − cos(q 1 )) + 1 |k n | 2 (µ n δ n (1 − cos(q 1 )) cos(q 2 )) , where Q = (q 2 , . . . , q d ).
L n (q 1 , Q,q 1 ,Q) can be considered as a perturbation coupling of a rotator with d − 1 degrees of freedom and a perturbation with the Lagrangian function (2.2.28)
A n (q 1 ,q 1 ) = 1 2 |q 1 | 2 + δ n (1 − cos(q 1 )), which corresponds to the Hamiltonian via Legendre transformation (2.2.29) h n (q 1 , p 1 ) = 1 2 |p 1 | 2 − δ n (1 − cos q 1 ).
We denote the coupling perturbation by (2.2.30)P n (q 1 , Q) = µ n δ n (1 − cos(q 1 )) cos(q 2 ).
Melnikov function
In the following, we give some approximation lemmas on the actions of (2.2.28) and (2.2.30) along the minimal orbits of L n by the calculation of Melnikov function.
For t ∈ R, the separatrixq 1 (t) of A n as (2.2.28) satisfyingq 1 > 0 andq 1 (0) = π is (2.2.31) q 1 (t) = 4 arctan exp( √ δ n t) ,
q 1 (t) = 2 √ δn cosh( √ δnt) .
Let the separatrix of A n takes value π at t 1 and Q(t 1 ) = Q 1 , then for a given rotation vector ω, we define the Melnikov function as
M n (ω, Q 1 , t 1 ) = δ n R (1 − cos(q 1 (t − t 1 ))) cos k ′ n , ω(t − t 1 ) + q 2 (t 1 ) dt.
Namely, M n is the integral of the coupling perturbation (2.2.30) along the separatrix of A n . It can be explicitly calculated as follow
(2.2.32) M n (ω, Q 1 , t 1 ) = 2π ω, k ′ n sinh( π ω,k ′ n 2 √ δn )
cos(q 2 (t 1 )).
It is easy to see that M n only depends on ω 2 = ω, k ′ n and q 2 (t 1 ) on which M n is 2π periodic. For the simplicity of notations, we denote M n (ω, Q 1 , t 1 ) by (2.2.33) M n (ω 2 , Q 1 , t 1 ) = 2π ω 2 sinh( πω 2 2 √ δn ) cos(q 2 (t 1 )).
Next, we work on the universal covering space of T d . By (2.2.33), a simple calculation implies the next lemma Lemma 2.23 Ifq 2 (t ′ ) mod 2π = 0 andq 2 (t ′′ ) mod 2π = π. Let Q ′ = (q 2 (t ′ ), q 3 (t ′ ), . . . , q d (t ′ )) and Q ′′ =q 2 (t ′′ ), q 3 (t ′′ ), . . . , q d (t ′′ )), then for n large enough
(2.2.34) M n (ω 2 , Q ′ , t ′ ) − M n (ω 2 , Q ′′ , t ′′ ) ≥ exp − λ √ δ n ,
where λ is a positive constant independent of n.
Proof By (2.2.22), we have
|ω 2 | ∼ 1. A simple calculation gives M n (ω 2 , Q ′ , t ′ ) − M n (ω 2 , Q ′′ , t ′′ ) ≥ exp − C 2 |ω 2 | √ δ n ≥ exp − λ √ δ n ,
where λ is a positive constant independent of n. This completes the proof of (2.2.34).
An approximation lemma
We use M n to approximate the action of the perturbation (2.2.30) along the minimal orbits of L n .
Lemma 2.24 Let (q 1 (t), Q(t)) be the minimal orbit of L n satisfying q 1 (t 1 ) = π with rotation vector ω, then (i) there exist τ > 0 and t 0 , t 2 satisfying
(2.2.35) t 0 ≤ t 1 − τ δ ′ n < t 1 + τ δ ′ n ≤ t 2 , such that q 1 (t 0 ) = 0, q 1 (t 2 ) = 2π. where δ ′ n = 1 |k n | d−1− ǫ 4 ; (ii) let (2.2.36)ω = ω 1 , Q(t 2 ) − Q(t 0 ) t 2 − t 0 , then δ n t 2 t 0
(1 − cos(q 1 (t))) cos(q 2 (t))dt − M n (ω 2 , Q(t 1 ), t 1 )
≤ C δ n exp − λ √ δ n ,
where C is a positive constant independent of n and λ is the same as the one in (2.2.34).
Proof The proof of Lemma 2.24 follows from the ideas of [Be]. We will prove (i) and (ii) respectively in the following. For the simplicity of notations, we will use u v (resp. u v) to denote u ≤ Cv (resp. u ≥ Cv) for some positive constant C.
Proof of (i)
We set τ = π C 0 , where C 0 is the constant in (2.2.1). From (2.2.1), it follows that
2τ δ ′ n < 2π |ω 1 | .
Hence, we have either
t 0 ≤ t 1 − τ δ ′ n or t 1 + τ δ ′ n ≤ t 2 .
Without loss of generality, we assume t 1 + τ δ ′ n ≤ t 2 and prove t 0 ≤ t 1 − τ δ ′ n in the following.
Let t −1 and t 3 be the last time to the left of t 1 such that q 1 (t −1 ) = −π and the first time to the right of t 1 such that q 1 (t 3 ) = 3π respectively. Consider the solution q of A n on (t −1 , t 1 ) satisfying the boundary condition q(t −1 ) = q 1 (t −1 ) = −π, q(t 1 ) = q 1 (t 1 ) = π.
We denote the energy ofq byē. Since t 1 − t −1 |k n | d−1 , by the deduction as similar as the one in Lemma 2.17, we have
(2.2.37) 0 <ē ≤ δ n exp − C|k n | ǫ √ δ n .
We set e(t) = 1 2 |q 1 (t)| 2 − δ n (1 − cos(q 1 (t))),
it is easy to see that there existst ∈ [t −1 , t 1 ] such that e(t) =ē. Indeed, without loss of generality, we assume by contradiction that q 1 (t) lie aboveq(t) in the phase plane for all t ∈ [t −1 , t 1 ], namelyq 1 (t) >q(t) for all t ∈ [t −1 , t 1 ], which is contradicted by the boundary conditionsq(t −1 ) = q 1 (t −1 ) andq(t 1 ) = q 1 (t 1 ). Hence, there exists t ∈ [t −1 , t 1 ] such that q 1 (t) =q(t) andq 1 (t) =q(t), moreover we have e(t) =ē. By Euler-Lagrange equation, we can estimateė(t). More precisely, e(t) µ n δ nq1 (t).
Hence, integrating fromt to t, we have
(2.2.38) sup t∈[t −1 ,t 1 ] |e(t) −ē| ≤ γµ n δ n ,
where γ is a positive constant independent of n.
We proceed the proof of (i) by the following three steps. a)q 1 (t 1 ) > 0.
we assume by contradiction thatq 1 (t 1 ) ≤ 0. q 1 (t 1 ) = π together with (2.2.38) implies that for n large enough,q 1 (t 1 ) < 0. Let us denote by (t 1 − ∆t, t 1 ) the maximal interval on the left of t 1 on which q 1 (t) ≥ π. Sinceq 1 (t 1 ) < 0, then ∆t > 0. From q 1 (t −1 ) = −π < π = q 1 (t 1 ), it follows that t 1 − ∆t > t −1 . Let
q(t) = q 1 (t) t ∈ [t −1 , t 1 − ∆t), 2π − q 1 (t) t ∈ [t 1 − ∆t, t 1 ].
It is easy to see that (q, Q) is still an action minimizing orbit on [t −1 , t 1 ]. By the definition of ∆t, we have thatq(t 1 − ∆t) = π anḋ q((t 1 − ∆t)−) ·q((t 1 − ∆t)+) ≤ 0, where (t 1 − ∆t)− denotes t tends to t 1 − ∆t from the left side and (t 1 − ∆t)+ denotes t tends to t 1 − ∆t from the right side. By Euler-Lagrange equation, we have thaṫ q(t) is continuous for t ∈ [t −1 , t 1 ]. Hence,q(t 1 − ∆t) = 0. On the other hand, from (2.2.37) and (2.2.38), it follows that for n large enough,q(t 1 − ∆t) = 0. Therefore, we haveq
1 (t 1 ) > 0. b)q 1 (t) > 0 for t ∈ t 1 − τ δ ′ n , t 1 .
Let (t, t 1 ] be the maximal interval to the left of t 1 on whichq 1 > 0, hence we can denote the inverse function by t(q 1 ). Let Γ = γµ n δ n . It follows from (2.2.38) that for q 1 < π, (s))) .
t 1 + q 1 π ds 2(ē − Γ + δ n (1 − cos(s))) ≤ t(q 1 ), ≤ t 1 + q 1 π ds 2(ē + Γ + δ n (1 − cos
On the other hand, denoting the inverse function ofq 1 (t − t 1 ) byt(q 1 ), we havê t(q 1 ) = t 1 + q 1 π ds 2(δ n (1 − cos(s))) .
Moreover, a simple calculation implies
|t(q 1 ) −t(q 1 )| ē + Γ (δ n (1 − cos(q 1 ))) 3 2
, which together with (2.2.26) implies that
(2.2.39) |t(q 1 ) −t(q 1 )| Γ (δ n (1 − cos(q 1 ))) 3 2 .
It is easy to see that (2.2.40) |q 1 | + |q 1 | δ n .
Let F (t) = min{1 − cos(q 1 (t)), 1 − cos(q 1 (t − t 1 ))}.
It follows from (2.2.39) and (2.2.40) that for t ∈ [t, t 2 ]
(2.2.41)
|q 1 (t) −q 1 (t − t 1 )| Γ √ δ n (δ n F (t)) 3 2 . Hence, we havet ≤ t 1 − τ δ ′ n .
In fact, we assume by contradiction thatt > t 1 − τ δ ′ n . It follows from (2.2.41) that for n large enough,
(2.2.42) q 1 (t) ≥ 1 2q 1 − τ δ ′ n .
If there exists t * ∈ [t, t 1 ] such thatq 1 (t * ) = 0, then
|e(t * )| ≥ δ n exp − C|k n | 3ǫ 4 √ δ n ,
which is contradicted by (2.2.38) and (2.2.26). Sinceq 1 (t) ≥ 0, we getq 1 (t) > 0, contradicted by the maximality of [t, t 1 ]. Therefore, we havet ≤ t 1 − τ δ ′ n . i.e.
q 1 (t) > 0 for t ∈ t 1 − τ δ ′ n , t 1 . c) q 1 (t 1 − τ δ ′ n ) > 0.
From (2.2.41) and (2.2.42), we have that for
t ∈ t 1 − τ δ ′ n , t 1 (2.2.43) |q 1 (t) −q 1 (t − t 1 )| Γδ −1 n exp C|k n | 3ǫ 4 √ δ n .
In terms of the definition ofq 1 (t), it follows from (2.2.26) that for n large enough,
(2.2.44) 1 2q 1 − τ δ ′ n ≤ q 1 t 1 − τ δ ′ n ≤ 3 2q 1 − τ δ ′ n . Hence, q 1 (t 1 − τ δ ′ n ) > 0, which together withq 1 (t) > 0 on t 1 − τ δ ′ n , t 1 implies t 0 ≤ t 1 − τ δ ′ n .
Similarly, we have t 1 + τ δ ′ n ≤ t 2 . The proof of (i) is completed.
Proof of (ii)
We let Ω = [t 1 − τ δ ′ n , t 1 + τ δ ′ n ]. Since δ n t 2 t 0
(1− cos(q 1 (t))) cos(q 2 (t))dt − M n (ω 2 , Q(t 1 ), t 1 ) ≤C 1 δ n Ω | cos(q 2 (t)) − cos (q 2 (t) + q 2 (t 1 ) −q 2 (t 1 )) |dt + Ω | cos(q 1 (t)) − cos(q 1 (t − t 1 ))|dt + [t 0 ,t 2 ]\Ω |1 − cos(q 1 (t))|dt + R\Ω |1 − cos(q 1 (t − t 1 ))|dt , whereq 2 (t) = q 2 (t 0 ) + ω 2 (t − t 0 ). Hence, it suffices to prove the following estimates (2.2.45) δ n Ω | cos(q 2 (t)) − cos (q 2 (t) + q 2 (t 1 ) −q 2 (t 1 )) |dt ǫ n ,
(2.2.46) δ n Ω | cos(q 1 (t)) − cos(q 1 (t − t 1 ))|dt ǫ n , (2.2.47) δ n [t 0 ,t 2 ]\Ω |1 − cos(q 1 (t))|dt ǫ n , (2.2.48) δ n R\Ω |1 − cos(q 1 (t − t 1 ))|dt ǫ n ,
where we set ǫ n = √ δ n exp(− λ √ δn ). We will prove (2.2.45)-(2.2.48) in the following.
First of all, we prove (2.2.45). We set
q(t) = q 1 (t − t 1 )φ t − (t 1 − τ δ ′ n ) , t 0 ≤ t ≤ t 1 , (q 1 (t − t 1 ) − 2π) φ −t + (t 1 + τ δ ′ n ) + 2π, t 1 ≤ t ≤ t 2 ,
where φ is a C ∞ function as follow
φ(t) = 0, t ≤ 0, 1, t ≥ 1.
We setQ(t) = Q(t 0 ) +ω(t − t 0 ), hence (q,Q)(t 0 ) = (q 1 , Q)(t 0 ) and (q,Q)(t 2 ) = (q 1 , Q)(t 2 ). From the minimality of (q 1 , Q), it follows that
t 2 t 0 L n q 1 , Q,q 1 ,Q dt ≤ t 2 t 0 L n q,Q,q,Q dt. Since t 2 − t 0 ≥ 2τ /δ ′ n , we letΩ = [t 1 − τ δ ′ n + 1, t 1 + τ δ ′ n − 1]
, then (2.2.31) and a direct calculation implies
t 2 t 0 (1 − cos(q(t)))dt = Ω 1 − cos(q 1 (t − t 1 ))dt + [t 0 ,t 2 ]\Ω (1 − cos(q(t)))dt, 1 √ δ n 2π 0 1 − cos(q 1 )dq 1 , 1 √ δ n ,
hence,
(2.2.49)
t 2 t 0
(1 − cos(q))dt 1 √ δ n .
Moreover, letω = (ω 2 , . . . ,ω d ), in terms of the definition of L n as (2.2.27), we have
t 2 t 0 L n q,Q,q,Q dt ≤ t 2 t 0 1 2 d i=3 |ω i | 2 |l ni | 2 + |ω 2 | 2 2|k ′ n | 2 + 1 |k n | 2 1 2 |q| 2 + δ n (1 − cos(q)) dt + 1 |k n | 2 µ n δ n t 2 t 0
(1 − cos(q))dt, hence, from (2.2.49), it follows that
t 2 t 0 L n q,Q,q,Q dt ≤ t 2 t 0 1 2 d i=3 |ω i | 2 |l ni | 2 + |ω 2 | 2 2|k ′ n | 2 + 1 |k n | 2 1 2
|q| 2 + δ n (1 − cos(q)) dt + C |k n | 2 µ n δ n .
Letq 1 be the solution of A n satisfying boundary conditions
q 1 t −1 + τ δ ′ n = q 1 t −1 + τ δ ′ n , q 1 t 1 − τ δ ′ n = q 1 t 1 − τ δ ′ n .
From the minimality of (q 1 , Q), it follows that Ω ′ L n q 1 , Q,q 1 ,Q dt ≤ Ω ′ L n q 1 , Q,q 1 ,Q dt.
By (2.2.27), we have Ω ′ 1 2 |q 1 | 2 + δ n (1 − cos(q 1 )) + µ n δ n (1 − cos(q 1 )) cos(q 2 )dt ≤ Ω ′ 1 2 |q 1 | 2 + δ n (1 − cos(q 1 )) + µ n δ n (1 − cos(q 1 )) cos(q 2 )dt.
From the construction ofq 1 , letẽ be the energy ofq 1 , a similar argument as the one in Lemma 2.17 implies
0 <ẽ ≤ δ n exp − C|k n | ǫ √ δ n .
Based on the change of variable dq =qdt, a direct calculation gives Ω ′ 1 2 |q 1 | 2 + δ n (1 − cos(q 1 )) + µ n δ n (1 − cos(q 1 )) cos(q 2 )dt exp − C|k n | ǫ √ δ n .
For n large enough, we have 1 2 Ω ′ |q 1 | 2 + δ n (1 − cos(q 1 ))dt ≤ Ω ′ 1 2 |q 1 | 2 + δ n (1 − cos(q 1 )) + µ n δ n (1 − cos(q 1 )) cos(q 2 )dt, which together with |q 1 | ≥ 0 implies
δ n Ω ′ |1 − cos(q 1 (t))|dt ≤ Ω ′ |q 1 | 2 + δ n (1 − cos(q 1 ))dt ǫ n ,henceδ n R\Ω |1 − cos(q 1 (t − t 1 ))|dt =δ n t 1 − τ δ ′ n −∞ 1 − cos(q 1 (t − t 1 ))dt + δ n ∞ t 1 + τ δ ′ n
Obviously, t 0 <t 1 < t 2 . From (2.2.1), we have
t 2 − t 0 ≥ 2π C 0 |k n | d−1 ,
where C 0 is the constant in (2.2.1). Lett 1 be the last time beforet 1 or the first time aftert 1 such that |q 2 (t 1 ) − q 2 (t 1 )| = π.
By (2.2.22), we have
|t 1 −t 1 | ∼ 1 |ω 2 | ≤ C 1 .
Without loss of generality, one can assume ω 1 > 0 and ω 2 > 0. Consider a solutioñ q 1 of A n on (t 0 ,t 1 ) and on (t 1 , t 2 ) with boundary conditions respectively q 1 (t 0 ) = q 1 (t 0 ) = 0, q 1 (t 1 ) = q 1 (t 1 ) = π, q 1 (t 1 ) = q 1 (t 1 ) = π, q 1 (t 2 ) = q 1 (t 2 ) = 2π.
Since q = (q 1 , q 2 , . . . , q d ) is assumed to be a minimal curve, setting Q = (q 2 , . . . , q d ), we have Fig.3, where x 1 = (q 1 (t 1 ), q 2 (t 1 )) = (π, 0), x 0 = (q 1 (t 0 ), q 2 (t 0 )) = (0, q 2 (t 0 )), x 2 = (q 1 (t 2 ), q 2 (t 2 )) = (2π, q 2 (t 2 )),x ′ 1 = (π, −π) andx ′′ 1 = (π, π).
(2.2.53) t 2 t 0 L n (q 1 , Q,q 1 ,Q)dt − t 2 t 0 L n (q 1 , Q,q 1 ,Q)dt ≥ 0. See
· · · · · · · · · · · · 0 −π π q1 q2 2π 0 x0 x2 x1x ′′ 1 x ′ 1 Figure 3: The projections of the curves (q 1 (t), Q(t)) and (q 1 (t), Q(t)) on [0, 2π] × R (q 1 (t), q 2 (t)) passes through the pointx ′ 1 orx ′′ 1 . Thus, by the construction of L n , we obtain from (2.2.53) that
t 2 t 0 A n (q 1 ,q 1 )dt − t 2 t 0 A n (q 1 ,q 1 )dt ≥ µ n t 2 t 0P n (q 1 , Q)dt − t 2 t 0P n (q 1 , Q)dt ,
(2.2.54) whereP n (q 1 , Q) = µ n δ n (1 − cos q 1 ) cos q 2 . By Lemma 2.24, we have
t 2 t 0P n (q 1 , Q)dt − t 2 t 0P n (q 1 , Q)dt ≤ M n (ω 2 , Q(t 1 ),t 1 ) − M n (ω 2 , Q(t 1 ),t 1 ) + C δ n exp − λ √ δ n ,
Destruction of all Lagrangian tori
In this section, we are concerned with Lagrangian tori as follow: [BP]).
In the following subsections, we consider the destruction of all Lagrangian tori of symplectic twist maps. Based on the correspondence between symplectic twist maps and Hamiltonian systems, it can be achieved to destruct all Lagrangian tori of Hamiltonian systems.
A toy model
To show the basic ideas, we are beginning with a toy model whose generating function is as follow:
(3.1.1) h n (x, x ′ ) = h 0 (x, x ′ ) − 5 4n 2 sin(nx ′ ) − 1 16n 2 cos(2nx ′ ), where h 0 (x, x ′ ) = 1 2 (x − x ′ ) 2 . Let f n (x, y) = (x ′ , y ′ )
be the exact area-preserving twist map generated by (3.1.1), then
y = −∂ 1 h n (x, x ′ ) = x ′ − x, y ′ = ∂ 2 h n (x, x ′ ) = x ′ − x − 5 4n cos(nx ′ ) + 1 8n sin(2nx ′ ).
We set φ n (x) = − 5 4n cos(nx) + 1 8n sin(2nx), then (3.1.2) f n (x, y) = (x + y, y + φ n (x + y)).
In [H2], Herman found a criterion of total destruction of invariant circles. By Birkhoff graph theorem (see [H4]), if f n admits an invariant circle, then the invariant circle is a Lipschitz graph. We denote the graph by ψ n , then it follows from [H5] that
ψ n • g n − ψ n = φ n • g n ,
where g n = Id + ψ n . This is equivalent to
(3.1.3) 1 2 (g n + g −1 n ) = Id + 1 2 φ n .
Let D n be the set of differentiable points of g n , then D n has full Lebesgue measure on R since g n is a Lipschitz function. For x ∈ D n , we differentiae (3.1.3), 1 2 (Dg n (x) + (Dg n ) −1 (g −1 n (x))) = 1 + 1 2 Dφ n (x).
Let G n = ||Dg n || L ∞ . It is easy to see that for ε > 0, there existsx ∈ D n such that Dg n (x) ≥ G n − ε. Let M n = max Dφ n , we have
1 2 G n + 1 G n − ε ≤ 1 + 1 2 M n .
Since ε > 0 is arbitrarily small, then
1 2 G n + 1 G n ≤ 1 + 1 2 M n .
Hence,
(3.1.4) G n ≤ 1 + 1 2 M n + M n + 1 4 M n 2 1 2 .
Obviously, for x ∈ D n , we have
1 G n ≤ 1 + 1 2 Dφ n (x).
Let m n = min Dφ n , then we have
1 G n ≤ 1 + 1 2 m n ,
which together with (3.1.4) implies that
1 1 + 1 2 m n ≤ 1 + 1 2 M n + M n + 1 4 M n 2 1 2 .
Therefore, it is sufficient for total destruction of invariant circles to construct φ n (x) such that (3.1.5) 1 1 + 1 2 min Dφ n > 1 + 1 2 max Dφ n + max Dφ n + 1 4 (max Dφ n ) 2 1 2 .
In our construction,
(3.1.6) Dφ n (x) = 5 4 sin(nx) + 1 4 cos(2nx).
A simple calculation implies min Dφ n (x) = − 3 2 , attained at x = 3 2n + 2πk n , max Dφ n (x) = 1, attained at x = π 2n + 2πk n ,
where k ∈ Z. Hence, (3.1.5) holds. Moreover, the exact area-preserving twist map generated by (3.1.1) admits no invariant circles. By interpolation inequality ( [H2]), for a small positive constant δ, we have
||φ n || C 1−δ ≤ 2||φ n || δ C 0 ||Dφ n || 1−δ C 0 .
From the construction of φ n , it follows that ||φ n || C 0 → 0, as n → ∞ and ||Dφ n || C 0 is bounded. Hence, ||φ n || C 1−δ → 0 as n → ∞, which implies that ||h n − h 0 || C 2−δ → 0 as n → ∞.
C ∞ case
In [H5], Herman extended the criterion (3.1.5) to multi-degrees of freedom. More precisely, for exact symplectic twist map of T * T d of the following form (3.2.1)
f (x, y) = (x + y, y + dΨ(x + y)),
where Ψ ∈ C r (T d , R), r ≥ 2 and dΨ = ∂Ψ ∂x 1 , · · · , ∂Ψ ∂x d .
Let E(x) be the derivative matrix of dΨ and T (x) = 1 d trE(x), where tr denotes the trace of E(x). From a similar argument as the deduction of (3.1.5), it follows that it is sufficient for total destruction of Lagrangian tori to construct T (x) such that
(3.2.2) 1 1 + 1 2 min T (x) > 1 + 1 2 max T (x) + max T (x) + 1 4 (max T (x)) 2 1 2 . Moreover, for T (x) → 0, (3.2.2) implies (3.2.3) − 1 2 min T (x) > max T (x) + O(max T (x)).
Herman constructed a sequence {Ψ n } n∈N that satisfies (3.2.3). It is easy to see T n (x) = 1 d ∆Ψ n where ∆ denotes the Laplacian. Since T n (x) is 2π-periodic, it is enough to construct it on [−π, π] d . More precisely,
T n (x) = T + n (x), x ∈ [0, π] d , T − n (x), x ∈ [−π, 0] d , 0, others.
where T n (x) is C ∞ function, T + n (x) and T − n (x) have the following forms respectively. T + n (x) satisfies: supp T + n (x) ⊂ [0, π] d , max T + n (x) = 1 n , T − n (x) satisfies:
supp T − n (x) = B Rn (x 0 ), max T − n (x) = 1 √ n , R n ∼ 1 √ n 1 d ,
x 0 = − π 2 , · · · , − π 2 , where f ∼ g means that 1
C g < f < Cg for a constant C > 1. Hence, we obtain a sequence of {T n (x)} n∈N with bounded C d norms and satisfying T d T n (x)dx = 0.
Form interpolation inequality, it follows that T n (x) → 0 as n → ∞ in the C d−δ topology for any δ > 0. Let Ψ n be the unique function in C ∞ (T d , R) such that T d Ψ n (x)dx = 0 and 1 d ∆Ψ n (x) = T n (x).
By Schauder estimates one knows that for any δ > 0, Ψ n (x) → 0 as n → ∞ in the C d+2−δ topology. From the construction of T n (x), it is easy to see that (3.2.3) is verified. Above all, we have the following theorem Theorem 3.2 All Lagrangian tori of an integrable symplectic twist map with d ≥ 1 degrees of freedom can be destructed by C ∞ perturbations of the generating function and the perturbations are arbitrarily small in the C d+2−δ topology for a small given constant δ > 0.
Based on the correspondence between symplectic twist maps and Hamiltonian systems, we have the following corollary.
Corollary 3.3 All Lagrangian tori of an integrable Hamiltonian system with d ≥ 2 degrees of freedom can be destructed by C ∞ perturbations which are arbitrarily small in the C d+1−δ topology for a small given constant δ > 0.
An approximation lemma
In this subsection, we will prove a lemma on C ∞ functions approximated by trigonometric polynomials. First of all, we need some notations. Define f (x + 2e j ) sin(mt) sin t 2 dt, where x ∈ R d , m ∈ N, j ∈ {1, . . . , d} and e j is the j-th vector of the canonical basis of R d . F
[j] m (f )(x) is a trigonometric polynomial in x j of degree at most m − 1. By [Z], 1 mπ π/2 −π/2 sin(mt) sin t 2 dt = 1, hence, from (3.3.1), we have
||F [j] m (f )|| C 0 ≤ ||f || C 0 . We denote P [j] m (f ) := 2F [j] 2m (f ) − F [j] m (f ).
It is easy to see that P
[j] m (f ) is a trigonometric polynomial in x j of degree at most 2m − 1. Moreover,
(3.3.2) ||P [j] m (f )|| C 0 ≤ 3||f || C 0 , (3.3.3) P [j] m (af + bg) = aP [j] m (f ) + bP [j] m (g),
where a, b ∈ R and f, g ∈ C ∞ 2π (R d , R). For k ∈ {1, . . . , d}, j 1 , . . . , j k ∈ {1, . . . , d} with j p = j q for p = q. Let m 1 , . . . , m k ∈ N and f ∈ C ∞ 2π (R d , R), we define (3.3.4) P [j 1 ,...,j k ] m 1 ,...,m k (f ) :
= P [j 1 ] m 1 P [j 2 ] m 2 · · · P [j k ] m k (f ) · · · .
It is easy to see that for all l ∈ {1, . . . , k}, P
[j 1 ,...,j k ] m 1 ,...,m k (f ) are trigonometric polynomials in x l of degree at most 2m l − 1. We have the following lemma.
Lemma 3.4 Let f ∈ C ∞ 2π (R d , R), r 1 , . . . , r d ∈ N, m 1 , . . . , m d ∈ N, then we have
(3.3.5) f − P [1,...,d] m 1 ,...,m d (f ) C 0 ≤ C d d j=1 1 m j r j ∂ r j f ∂x j r j C 0 ,
where C d is a constant only depending on d.
Proof From [A], we will prove Lemma 3.4 by induction. The case d = 1 is covered by Jackson's approximation theorem. More precisely, for f ∈ C ∞ 2π (R, R), m, r ∈ N, we have
(3.3.6) f − P [1] m (f ) C 0 ≤ C 1 1 m r ∂ r f ∂x r C 0 .
Let the assertion be true for some d ∈ N. We verify it for d + 1. Consider the functions f (x 1 , ·) with x 1 as a real parameter. Then by the assertion for d, we have Letx j ∈ R d denote the vector x ∈ R d+1 without its j-th entry. For the functions f (·,x 1 ), from (3.3.6), it follows that
f (·,x 1 ) − P [1] m 1 (f )(·,x 1 ) C 0 ≤ C 1 1 m 1 r 1 ∂ r 1 f ∂x 1 r 1 C 0 , hence, (3.3.8) f − P [1] m 1 (f ) C 0 ≤ C 1 1 m 1 r 1 ∂ r 1 f ∂x 1 r 1 C 0 .
By (
(f ) C 0 , ≤ C 1 1 m 1 r 1 ∂ r 1 f ∂x 1 r 1 C 0 + 3C d d+1 j=2 1 m j r j ∂ r j f ∂x j r j C 0 , ≤ C d+1 d+1 j=1 1 m j r j ∂ r j f ∂x j r j C 0 .
This finishes the proof.
Obviously, there exist mj, rj such that
1 mj rj ∂ rj f ∂xj rj C 0 = max 1≤j≤d 1 m j r j ∂ r j f ∂x j r j C 0 .
Hence, we have
f − P [1,...,d] m 1 ,...,m d (f ) C 0 ≤ dC d 1 mj rj ∂ rj f ∂xj rj C 0 , ≤ C ′ d 1 mj rj f C rj .
For the simplicity of notations, we denote
f (x) − p N (x) C 0 ≤ A dk N −k f (x) C k ,
where A dk is a constant depending on d and k.
C ω case
Similar to Herman's construction, we consider C ∞ functionT n (x) as follow:
T n (x) = T + n (x), x ∈ [0, π] d , T − n (x), x ∈ [−π, 0] d , 0,
others.
T + n (x) satisfies: suppT + n (x) ⊂ [0, π] d , maxT + n (x) ∼ 1,
T − n (x) satisfies:
suppT − n (x) = B Rn (x 0 ), maxT − n (x) = n, R n ∼ 1 n 1 d , x 0 = − π 2 , · · · , − π 2 .
By Lemma 3.4, there exists a trigonometric polynomial p N (x 1 , · · · , x d ) in x l (1 ≤ l ≤ d) of degree at most N such that T n (x 1 , · · · , x d ) − p N (x 1 , · · · , x d ) C 0 ≤ A dk N −k T n (x 1 , · · · , x d ) C k .
By the construction ofT n , we have (3.4.1) ||T n (x 1 , · · · , x d )|| C k ∼ n k d +1 .
Then, choosing N large enough such that
(3.4.2) A dk N −k ||T n (x 1 , · · · , x d )|| C k < σ ≪ 1,
where σ is a small enough positive constant. Hence, we have where C is a constant independent of n. We consider the normalized trigonometric polynomial (3.4.5)p N (x) = 1 n 1−ε p N (x) max |p N (x)| , where x = (x 1 , . . . , x d ). It is easy to see that maxp N (x) ∼ 1 n 2−ε , minp N (x) ∼ − 1 n 1−ε .
Hence, (3.2.3) is verified. Next, we estimate ||p N (x)|| C r . By a simple calculation, we have (3.4.6) ||p N (x)|| C r ≤ CnN r+1 .
Then,
||p N (x)|| C r ≤ 1 n 1−ε 1 max |p N (x)| ||p N (x)|| C r , ≤ 1 n 2−ε · CnN r+1 , = C 1 n 1−ε N r+1 .
To achieve n −(1−ε) N r+1 → 0 as n → ∞, it suffices to make 1 n 1−ε N r+1 ≤ 1 n ε .
Hence, we have r ≤ log N n 1−2ε − 1.
47 3.1 A toy model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2 C ∞ case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3 An approximation lemma . . . . . . . . . . . . . . . . . . . . . . . . 50 3.4 C ω case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Figure 2 :
2The choices oft 1
C
∞ 2π (R d , R) := f : R d → R|f ∈ C ∞ (R d , R) and 2π − periodic in x 1 , . . . , x d . Let f (x) ∈ C ∞ 2π (R d ,R). The m-th Fejér-polynomial of f with respect to x j
f (x 1 ,
1·) − P [2,...,d+1] m 2 ,...,m d+1 (f )(x 1 , ·) C 0 ≤ C f − P [2,...,d+1] m 2 ,...,m d+1 (f ) C 0 ≤ C
p
N (x) = P [1,...,d] m 1 ,...,m d (f )(x), where x = (x 1 , . . . , x d ) and N = 2mj − 1. Moreover, we denote k := rj, then (3.3.9)
p N (x) ∼ n, attained on B Rn (x 0 ), max p N (x) ∼ 1, on [−π, π] d \ B Rn (x 0 ).
Case with 2 degrees of freedom . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Preliminaries of exact area-preserving twist map . . . . . . . 4 2.1.1.1 Minimal configuration . . . . . . . . . . . . . . . . . 4 2.1.1.2 Peierls's barrier . . . . . . . . . . . . . . . . . . . . 5 2.1.2 C ∞ case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.2.1 Construction of the generating functions . . . . . . 6 2.1.2.2 Estimate of lower bound of P hn 0 + . . . . . . . . . . . 8 2.1.2.3 The approximation from P hn 0 + to P hn Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . 17 2.1.3 C ω case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.3.1 Construction of the generating functions . . . . . . 18 2.1.3.2 Proof of Theorem 2.11 . . . . . . . . . . . . . . . . . 20 2.2 Case with d ≥ 3 degrees of freedom . . . . . . . . . . . . . . . . . . . 22 2.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.2 C ∞ case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.2.1 Destruction of Lagrangian torus with a special rotation vector . . . . . . . . . . . . . . . . . . . . . . . 24 2.2.2.2 Destruction of Lagrangian torus with an arbitrary rotation vector . . . . . . . . . . . . . . . . . . . . . 31 2.2.3 C ω case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.2.3.1 Construction of H n . . . . . . . . . . . . . . . .1 Introduction
2
2 Destruction of a Lagrangian torus
4
2.1 ω
. . . . . . . .
9
2.1.2.4 . . 34
2.2.3.2 Melnikov function . . . . . . . . . . . . . . . . . . . 35
2.2.3.3 An approximation lemma . . . . . . . . . . . . . . . 36
2.2.3.4 Proof of Theorem 2.22 . . . . . . . . . . . . . . . . . 44
3 Destruction of all Lagrangian tori
Theorem 1.1 Given an integrable Hamiltonian H 0 with d (d ≥ 2) degrees of freedom and a rotation vector ω, there exists a sequence of C ∞ Hamiltonians {H n } n∈N such that H n → H 0 in C 2d−δ topology and the Hamiltonian flow generated by H n does not admit the Lagrangian torus with the rotation vector ω. Theorem 1.2 Given an integrable Hamiltonian H 0 with d (d ≥ 2) degrees of freedom and a rotation vector ω, there exists a sequence of C ω Hamiltonians {H n } n∈N such that H n → H 0 in C d+1−δ topology and the Hamiltonian flow generated by H n does not admit the Lagrangian torus with the rotation vector ω. Theorem 1.3 For an integrable Hamiltonian H 0 with d (d ≥ 2) degrees of freedom, all Lagrangian tori can be destructed by analytic perturbations which are arbitrarily small in the C d−δ topology.
Table 1 :
1Values of r for destruction of Lagrangian torus (tori) in the C r topology
2.1.1. Preliminaries of exact area-preserving twist map 2.1.1.1. Minimal configuration
2.1.2. C ∞ case 2.1.2.1. Construction of the generating functions Consider a completely integrable system with the generating function
, (2.2.47) is proved. Finally, the inequality (2.2.48) can be obtained by a direct calculation. More precisely,
Definition 3.1 T d is called d dimensional Lagrangian torus with the rotation vector ω if • T d is a Lagrangian submanifold; • T d is invariant for the Hamiltonian flow Φ t H generated by H. For positive definite Hamiltonian systems, if a Lagrangian torus is invariant under the Hamiltonian flow, it is then the graph over T d (see
(f ) − P [1,...,d+1] m 1 ,...,m d+1 (f ) C 0 = P [1] m 1 (f ) − P [1] m 1 P [2,...,d+1] m 2 ,...,m d+1 (f ) C 0 , = P [1] m 1 f − P [1] m 1 P [2,...,d+1] m 2 ,...,m d+1 (f ) C 0 ,which together with (3.3.8) implies thatf − P [1,...,d+1] m 1 ,...,m d+1 (f ) C 0 ≤ f − P [1] m 1 (f ) C 0 + P [1] m 1 (f ) − P [1]m 1 P [2,...,d+1] m 2 ,...,m d+13.3.2), (3.3.3)),(3.3.4) and (3.3.7), we have
P [1]
m 1 ≤ 3C d
d+1
j=2
1
m j r j
∂ r j f
∂x j r j
C 0
,
t t ′(1 − cos(q 1 ))dt, for any t, t ′ ∈ [t 0 , t 2 ].
− cos(q 1 (t − t 1 ))dt.
Since µ n is small enough for n large enough, we have(|q 1 | 2 + δ n (1 − cos(q 1 )))dt.It is easy to see that 1 2 |q i | 2 = 1 2 |ω i | 2 + ω i ,q i −ω i + 1 2 |q i −ω i | 2 , for i = 2, . . . , d, and the mean value ofq i −ω i vanishes on [t 0 , t 2 ] for i = 2, . . . , d. Hence,(|q 1 | 2 + δ n (1 − cos(q 1 )))dt.From the deduction above, we haveFrom the definition ofq and (2.2.31), a direct calculation implies(1 − cos(q(t)))dt,From Euler-Lagrange equation on (q 1 , Q), it follows that |Q| ≤ µ n δ n (1 − cos(q 1 )).Hence,Moreover, |Q(t) −ω| µ n δ n , for any t ∈ [t 0 , t 2 ], which yieldsif we integrateQ(t) from t 1 to t. In particular, we havethen the proof of (2.2.45) is completed if we take n large enough. Second, we prove (2.2.46). By (2.2.44), we havehence, from (2.2.31), it follows thatMoreover, the above inequality and (2.2.43) imply that for n large enough and] is similar, more precisely,Therefore, from (2.2.51), it follows thathence, we complete the proof of (2.2.46). Third, we prove (2.2.47). It suffices to prove for Ωwhere t −1 satisfies q 1 (t −1 ) = −π.By (2.2.31),q 1 (t) = 4 arctan exp( √ δ n (t − t 1 )) , we haveIt is similar to getHence, (2.2.48) is proved for n large enough,. So far, we complete the proof of (2.2.45)-(2.2.48) and also the proof of Lemma 2.24.Remark 2.25 In the proof of (2.2.45), as a bonus we obtain an important estimate onQ(t). Namely, let (q 1 (t), Q(t)) be the action minimizing orbit of L n , then for any t ′ , t ′′ ∈ R and t ∈ [t ′ , t ′′ ] we haveFrom Remark 2.25 and the definition of Melnikov function (2.2.32), it follows that for n large enough,where λ is the same as the one in (2.2.34).Proof of Theorem 2.22Based on the minimal property of the orbits on an invariant Lagrangian torus and its graph property, passing through each x ∈ T d , there is a unique minimal curve q(t) with rotation vector ω if the Hamiltonian flow generated by H n admits a Lagrangian torus with rotation vector ω. Hence, it is sufficient to prove the existence of some point in T d where no minimal curve passes through.Indeed, any minimal curve does not pass through the subspace (π, 0) × T d−2 . It implies Lemma 2.24. Let us assume contrary, namely, there existst 1 such that q 1 (t 1 ) = π, q 2 (t 1 ) = 0, where q(t) = (q 1 , q 2 , . . . , q d )(t) is a minimal orbit in the universal covering space R d . Because of ω 1 = 0, there exist t 0 and t 1 such that q 1 (t 0 ) = 0, q 1 (t 2 ) = 2π. whereω = (Q(t 2 )− Q(t 0 ))/(t 2 − t 0 ). Here, the approximation from M n (ω 2 , Q(t 1 ),t 1 ) to t 2 t 0P n (q 1 , Q)dt can not be obtained directly by Lemma 2.24, since (q 1 , Q(t)) may be not minimal. But based on the simplicity ofq 1 , a much simpler calculation than the one in Lemma 2.24 implies that the approximation in Lemma 2.24 is still verified for the orbit (q 1 , Q(t)). We omit it.By Lemma 2.23 and (2.2.52), we haveFrom (2.2.54), it follows thatOn the other hand, consider a solutionq 1 of A n on (t 0 ,t 1 ) and on (t 1 , t 2 ) with boundary conditions respectively q 1 (t 0 ) = q 1 (t 0 ) = 0, q 1 (t 1 ) = q 1 (t 1 ) = π, q 1 (t 1 ) = q 1 (t 1 ) = π, q 1 (t 2 ) = q 1 (t 2 ) = 2π.We haveSince |t 1 −t 1 | ≤ C 1 , An argument similar as the one toFig. 2impliesFrom (2.2.26),Hence, for anyt 1 ∈ (t 0 , t 2 ), we can findt 1 such that (1 − 2ε) − 1.Since k can be made large enough, we take k = d 2ε . LetIt follows thatp N (x) → 0 as n → ∞ in the C d−1−δ for any δ > 0. LetΨ n be the unique function in C ω (T d , R) such that T dΨ n (x)dx = 0 andBy Schauder estimates one knows that for any δ > 0,Ψ n (x) → 0 as n → ∞ in the C d+1−δ topology. From the construction ofp N (x), it is easy to see that (3.2.3) is verified. Hence, we have that the symplectic twist maps generated by the generating functionh n (x, x ′ ) = 1 2 (x− x ′ ) 2 +Ψ n (x ′ ) do not admit any Lagrangian tori for n large enough.So far, we prove the following theorem Theorem 3.5 All Lagrangian tori of an integrable symplectic twist map with d ≥ 2 degrees of freedom can be destructed by C ω perturbations of the generating function and the perturbations are arbitrarily small in the C d+1−δ topology for a small given constant δ > 0.Based on the correspondence between symplectic twist maps and Hamiltonian systems, together with the toy model corresponding to the case with d = 1, we have the following corollary.Corollary 3.6 All Lagrangian tori of an integrable Hamiltonian system with d ≥ 2 degrees of freedom can be destructed by C ω perturbations which are arbitrarily small in the C d−δ topology for a small given constant δ > 0.
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"A GRAPHICAL CATEGORIFICATION OF THE TWO-VARIABLE CHEBYSHEV POLYNOMIALS OF THE SECOND KIND",
"A GRAPHICAL CATEGORIFICATION OF THE TWO-VARIABLE CHEBYSHEV POLYNOMIALS OF THE SECOND KIND"
] |
[
"Wataru Yuasa "
] |
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We show that the A 2 clasps in the Karoubi envelope of A 2 spider satisfy the recursive formula of the two-variable Chebyshev polynomials of the second kind associated with a root system of type A 2 . The A 2 spider is a diagrammatic description of the representation category for Uq(sl 3 ) and the A 2 clasps are projectors. Our categorification also gives a natural definition of a q-deformation of the two-variable Chebyshev polynomials. This paper is constructed based only on the linear skein theory and graphical calculus.Question 1.1. Give diagrammatic categorifications of multi-variable generalizations of the Chebyshev polynomials.We treat with a two-variable generalization of the Chebyshev polynomials, called the A 2 Chebyshev polynomials, of the second kind appearing in[Koo74a,Koo74b,Koo74c,Koo74d]and give a solution of the above question. The A 2 Chebyshev polynomials of the second kind is a family of two-variable polynomials {S (k,l)
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"https://arxiv.org/pdf/1903.01099v1.pdf"
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1903.01099
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096ecd09ce1af3041362fa6a352792687992a051
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A GRAPHICAL CATEGORIFICATION OF THE TWO-VARIABLE CHEBYSHEV POLYNOMIALS OF THE SECOND KIND
Wataru Yuasa
A GRAPHICAL CATEGORIFICATION OF THE TWO-VARIABLE CHEBYSHEV POLYNOMIALS OF THE SECOND KIND
We show that the A 2 clasps in the Karoubi envelope of A 2 spider satisfy the recursive formula of the two-variable Chebyshev polynomials of the second kind associated with a root system of type A 2 . The A 2 spider is a diagrammatic description of the representation category for Uq(sl 3 ) and the A 2 clasps are projectors. Our categorification also gives a natural definition of a q-deformation of the two-variable Chebyshev polynomials. This paper is constructed based only on the linear skein theory and graphical calculus.Question 1.1. Give diagrammatic categorifications of multi-variable generalizations of the Chebyshev polynomials.We treat with a two-variable generalization of the Chebyshev polynomials, called the A 2 Chebyshev polynomials, of the second kind appearing in[Koo74a,Koo74b,Koo74c,Koo74d]and give a solution of the above question. The A 2 Chebyshev polynomials of the second kind is a family of two-variable polynomials {S (k,l)
INTRODUCTION
In this paper, we give a categorification of a two-variable Chebyshev polynomial of the second kind inspired by Queffelec and Wedrich [QW18a,QW18b]. They categorified the Chebyshev polynomials and power-sum symmetric polynomials through diagrammatic categories. In [QW18a], it is shown that the Jones-Wenzl projectors satisfy the recursive formula of the Chebyshev polynomials of the second kind (resp. first kind) in the split Grothendieck group of the Karoubi envelope of the Temperley-Lieb category (resp. an affine version of the Temperley-Lieb category).
From representation theoretical point of view, many mathematician and physicist have studied multi-variable generalizations of the Chebyshev polynomials associated with root systems. For examples, Koornwinder [Koo74a,Koo74b,Koo74c,Koo74d], Hoffman and Withers [HW88] for type A, and [NPST10,NPT11] in general. We consider the following problem:
products. An invariant vector is described as a linear combination of directed uni-trivalent planar graphs with source and sink vertices. We call it A 2 web. By using A 2 webs, we can consider the A 2 version of the Temperley-Lieb category, namely a linear category of intertwining operators between tensor powers of V + 's and V − 's, through Hom(A, B) ∼ = Inv(A * ⊗ B). We call it also the A 2 spider in this paper, and denote it by Sp q . Kuperberg also defined the (internal) A 2 clasp P + k − l + k − l in Hom(V ⊗k + ⊗ V ⊗l − , V ⊗k + ⊗ V ⊗l − ) for k, l ≥ 0 as a generalization of the Joens-Wenzl idempotent. Ohtsuki and Yamada gave a recursive definition of P + k − l + k − l in [OY97]. Then, P + k − l + k − l give the A 2 Chebyshev polynomial in the following sense. Let P (k,l) be an object in the Karoubi envelope Kar(Sp q ) of Sp q corresponding to the idempotent P + k − l + k − l . Theorem 1.2. The isomorphism classes of {P (k,l) } k,l≥0 satisfy the recursive formula of {S (k,l) } k,l≥0 in the split Grothendieck group K 0 (Kar(Sp)) where Sp is the A 2 spider at q = 1.
In the above, we consider the C-linear category Sp q at q = 1. However, we will show a reccursive formula of P (k,l) for the C(q 1 6 )-linear category Sp q and obtain the proof by specializing q = 1. This recursive formula of P (k,l) gives a natural q-deformation of the A 2 Chebyshev polynomial. Many kinds of q-deformations of the Chebyshev polynomials have been studied in various contexts, for example, [Dup10], [Cig12a,Cig12b].
This paper is organized as follows. In Section 2, we recall definitions and properties of the A 2 spider and the A 2 clasps. They are an A 2 version of the Temperley-Lieb category and the Jones-Wenzl projectors. We categorify the A 2 Chebyshev polynomials in Section 3. All proofs are given by diagrammatic calculations in the A 2 spider.
THE A 2 SPIDER AND THE A 2 CLASP
In this section, we introduce an A 2 version of the Temperley-Lieb category Sp q based on Kuperberg's A 2 spider [Kup96]. Our definition of Sp q is only for proof of Theorem 1.2. Evans and Pugh studied the details about the A 2 version of the Temperley-Lieb category and the A 2 planar algebra in [EP10,EP11]. We also define the A 2 clasps which play a role analogous to the Jones-Wenzl projectors and show some fundamental properties. This section is constructed by the linear skein theory and diagrammatic calculations.
2.1. The A 2 spider. An A 2 web is a linear combination of graphs in a disk D = [0, 1] × [0, 1] with marked points on [0, 1] × {0, 1} which represents an intertwining operator of U q (sl 3 ) in manner of Reshetikhin and Turaev [RT91]. See a Turaev's book [Tur94] and [BK01] for details. We only give a combinatorial definition of the A 2 spider which is a linear category constructed from A 2 webs. See [Kup94,Kup96] for details on relation to representation theory of U q (sl 3 ).
Let us recall the A 2 web defined by Kuperberg [Kup96]. For any n ∈ Z ≥0 , we denote sets of marked points on [0, 1] × {0, 1} by P n = { p 1 , p 2 , . . . , p n } and Q n = { q 1 , q 2 , . . . , q n } where p i = (i/(n + 1), 0) and q i = (i/(n + 1), 1) for 1 ≤ i ≤ n. If n = 0, then P 0 = Q 0 = ∅. A sign of P n is a map ε Pn : P n → {+, −}. The sign ε Pn is defined by the sequence ε Pn(1) ε Pn(2) . . . ε Pn(n) of + and −. A sign of Q n is defined in the same way. We consider D with marked points P k and Q l with signs ε P k and ε Q l where k, l ∈ Z ≥0 . A bipartite uni-trivalent graph G in D is a directed graph embedded into D such that every vertex is either trivalent or univalent and the vertices are divided into sinks or sources as follows:
• v or + v if v is a sink, • v or − v if v is a source.
An A 2 basis web is the boundary-fixing isotopy class of a bipartite trivalent graph G in D such that any internal face of D \ G has at least six sides. Let k and l be sequences of + and − with length k and l, respectively. We denote the set of A 2 basis webs in D with signed marked points ε P k = k and ε Q l = l by B( k , l ). The A 2 web space W ( k , l ) is the C(q 1 6 )-vector space spanned by B( k , l ). An A 2 web is an element in the A 2 web space.
For example, B(+ − +, − + −) has the following A 2 basis webs:
+ − + − + − , + − + − + − , + − + − + − , + − + − + − , + − + − + − , + − + − + − .
Definition 2.1 (The A 2 spider). The A 2 spider Sp q is a C(q 1 6 )-linear category defined as follows.
• An object of Sp q is a finite sequence of signs, that is, a map n : n → {+, −} where n = {1 < 2 < · · · < n} is a finite totally ordered set for n ∈ Z >0 where the map 0 is the map from 0 = ∅. A morphism Sp q ( k , l ) = Hom Sp q ( k , l ) is the A 2 web space W (¯ k , l ) where¯ k is the opposite sign of k . We remark that Sp q ( 0 , 0 ) is the 1-dimensional vector space spanned by the empty web D ∅ . • For A 2 basis webs F ∈ Sp q ( k , l ) and G ∈ Sp q ( l , m ), the composition G • F = GF ∈ Sp q ( k , m ) is defined by gluing top side of the disk of F and bottom side of the disk of G. If GF makes 4-, 2-, and 0-gons, we reduce them by
- = + , - = [2] , - = [3] , = [3] , where [n] = (q n 2 − q − n 2 )/(q 1 2 − q − 1 2 ). • The identity morphism 1 k in Sp q ( k , k )
is the k parallel edges with no trivalent vertex. • A tensor product of two objects is defined by their concatenation, that is,
k ⊗ k is a map k + k → {+, −} such that k ⊗ k (i) = k (i) if 0 ≤ i ≤ k and k ⊗ k (i) = k (i − k) if k + 1 ≤ i ≤ k + k .
The tensor product of two A 2 basis webs F ∈ Sp q ( k , l ) and G ∈ Sp q ( k , l ) is defined by gluing right side of the disk of F and left side of the disk of G.
We remark that an A 2 web f ∈ Sp q ( k , l ) represents an intertwining operator in Hom(V k , V l ) where V k = V k (1) ⊗V k (2) ⊗· · ·⊗V k (k) and V l = V l (1) ⊗V l (2) ⊗· · ·⊗V l (l) .
As an invariant vector f is an element in Inv(V * k ⊗ V l ) and V * k ∼ = V¯ k (k) ⊗ · · · ⊗ V¯ k (2) ⊗ V¯ k (1) because V * ± ∼ = V ∓ . See [Kup96] in detail. The signs of marked points of the A 2 web space are labeled according to the signs of the invariant space.
2.2.
A 2 clasps. We give a diagrammatic definition of an A 2 version of the Jones-Wenzl projectors, called the A 2 clasps, introduced in [Kup96,OY97]. The purpose of this section is to construct the general form of the A 2 clasp and to prove its properties by only using the linear skein theory.
In what follows, we omit "⊗" to describe the tensor product of objects in Sp q and + (resp. −) means the constant map from 1 to + (resp. −). Thus, + k (resp. − k ) is the constant map from k to + (resp. −) for any positive integer k.
Definition 2.2 (The A 2 clasp in Sp q (+ k , + k )).
(1)
1 = = 1 + ∈ Sp q (+, +) (2) k = k−1 − [k−1] [k] k−1 k−1 k−2 ∈ Sp q (+ k , + k )
The above recursive formula defines the A 2 clasp in Sp q (+ k , + k ) and we denote it by
P + k + k . The A 2 clasp in Sp q (− k , − k )
is also defined by the same way and we denote it by
P − k − k .
We introduce the following A 2 webs:
(2.1) t ++ − = , t −− + = , t + −− = , t − ++ = , b −+ = , b +− = , d +− = , d −+ =
An A 2 basis web B(¯ k , k ) provides a tiling of D. The following Lemma is easily shown by calculating the Euler number of the tiling.
Lemma 2.3 (Ohtsuki and Yamada [OY97, Lemma 3.3]). For any
A 2 basis web D in Sp q ( k , k ) other than 1 k has t ++ − , t −− + , b −+ , or b +− in the top side of D and t + −− , t − ++ , d −+ , or d +− in the bottom side of D.
Proposition 2.4 (Kuperberg [Kup96], Ohtsuki and Yamada [OY97]). Let k be a nonnegative integer, then
(1) P + k + k (1 + a ⊗ P + l + l ⊗ 1 + b ) = P + k + k = (1 + a ⊗ P + l + l ⊗ 1 + b )P + k + k for a + b + l = k, (2) P + k + k (1 + a ⊗ t ++ − ⊗ 1 + b ) = 0 = (1 + a ⊗ t − ++ ⊗ 1 + b )P + k + k for a + b + 2 = k.
The above is true for the opposite sign.
Proposition 2.5 (Uniqueness). If a non-trivial element T ∈ Sp q ( k , k ) satisfies T 2 = T and Proposition 2.4 (2), then T = P k k for ∈ {+, −}.
Proof. P + k + k can be expanded as a linear sum of A 2 basis webs P + k + k = c1 + k + x by Lemma 2.3 where c is a constant and x have t ++ − and t − ++ in the top side and in the bottom side. We can see that c = 1 from (P + k + k ) 2 = P + k + k and Proposition 2.4 (2). In the same way, we know T = 1 + k + x . Therefore,
P + k + k T = (1 + k + x)T = T P + k + k T = P + k + k (1 + k + x ) = P + k + k
We can prove it for = − in the same way.
For an A 2 basis web w ∈ B(¯ k , l ), we define w * ∈ B(¯ l , k ) as the reflection of w through the horizontal line [0, 1] × {1/2} with the opposite direcion. For the coefficient C(q 1 6 ), the star operator acts on C by complex conjugate and (q 1 6 ) * = q − 1 6 . In this way, we define a linear map * :
Sp q ( k , l ) → Sp q ( l , k ).
Remark 2.6. Proposition 2.5 implies (P + k
+ k ) * = P + k + k and (P − k − k ) * = P − k − k . We introduce the A 2 clasp P + k − l + k − l in Sp q (+ k − l , + k − l ) based on [OY97].
Definition 2.7 (The A 2 clasp in Sp q (+ k − l , + k − l )).
P + k − l + k − l = k k l l = min{k,l} i=0 (−1) i k i l i k+l+1 i k−i l−i i i k l k l . P − k + l − k + l is also defined by the same way.
One can prove a similar statement to Proposition 2.4.
Proposition 2.8 (Kuperberg [Kup96], Ohtsuki and Yamada [OY97]). Let k and l be nonnegative integers, then
(1) P + k − l + k − l (1 + a ⊗ P + s − t + s − t ⊗ 1 − b ) = P + k − l + k − l = (1 + a ⊗ P + s − t + s − t ⊗ 1 − b )P + k − l + k − l for a + s = k and b + t = l, (2) P + k − l + k − l (1 + a ⊗ b +− ⊗ 1 − b ) = 0 = (1 + a ⊗ d +− ⊗ 1 − b )P + k − l + k − l for a + 1 = k and b + 1 = l.
The same equalities hold for P − k + l − k + l .
One can prove the uniqueness of P + k − l + k − l in a similar way to Proposition 2.5.
Proposition 2.9 (Uniqueness). A non-trivial idempotent element in Sp q (+ k − l , + k − l ) satisfying Proposition 2.8 (2) is uniquely determined.
Proof. In the same way as the proof of Proposition 2.5.
Remark 2.10. Proposition 2.9 implies (P
+ k − l + k − l ) * = P + k − l + k − l and (P − k + l − k + l ) * = P − k + l − k + l .
We give an explicit definition of a general form of the A 2 clasp appear in Kuperberg [Kup96] and Kim [Kim07]. This A 2 clasp is an A 2 web, no longer idempotent, in Sp q ( , δ) such that k = # −1 (+) = #δ −1 (+) and l = # −1 (−) = #δ −1 (−). We introduce the following A 2 basis webs:
(2.2) H −+ +− = , H +− −+ = .
Let k and l be non-negative integers. We take an arbitrary object : k + l → {±} in Sp q satisfying k = # −1 (+) and l = # −1 (−). Then, we define an A 2 basis web σ + k − l as follows. We consider the disk D = [0, 1] × [0, 1] with marked points signed by + k − l and . Join the marked points labeled by + in the bottom side with ones of the upper side by straight arcs. In the same way, join the mark points labeled by − by straight arcs. Then, one can obtain the A 2 basis web σ + k − l by replacing all crossing points by H −+ +− .
Definition 2.11. Let k and l be non-negative integers. Then, an
A 2 clasp P + k − l in Sp q (+ k − l , ) is defined by P + k − l = σ + k − l P + k − l + k − l . Proposition 2.12. Compositions of P + k − l with A 2 basis webs t and d vanish.
Proof. We prove it by induction on the number h(σ) of
H −+ +− contained in σ = σ + k − l . If h(σ) = 0, it is clear since P + k − l = P + k − l + k − l and Proposition 2.8. If h(σ) = 1, then σ = 1 + k−1 ⊗ H −+ +− 1 − l−1 .
One can prove by easy calculations. When h(σ) = n + 1 (n ≥ 1), σ is described as a composition of σ with These terms has t − ++ and d −+ on the web σ . Therefore, ···
1 α ⊗H −+ +− ⊗1 β whrere = α−+β and σ = σ α+−β + k − l such that h(σ ) = n. Thus, σ is ··· ··· σ , ······ σ • P + k − l + k − l = 0,
by the induction hypothesis for σ • P + k − l + k − l . For other cases, we can prove in the same way.
Proposition 2.13. Let us decompose into three subsequences αβγ such that β is expressed as the form ++ · · · +−− · · · − or −− · · · −++ · · · +. Then,
(1) P + k − l = (1 α ⊗ P β β ⊗ 1 γ )P + k − l , (2) (P + k − l ) * P + k − l = P + k − l + k − l .
Especially, tP + k − l = 0 and dP + k − l = 0 where t (resp. d) is a tensor product of identity morphisms and t + −− or t − ++ (resp. d +− or d −+ ).
Proof. (1) is easily shown by expanding P β β . We can desctibe P + k − l as a product τ P + k − l where τ is a tensor product of identity morphisms and only one H −+ +− or H +− −+ . Then, (P + k − l ) * P + k − l = (P + k − l ) * τ * τ P + k − l and one can finish the proof of (2) by applying the defining relation of the A 2 web to τ * τ .
Let us define an A 2 clasp P in Sp q ( , ) by P + k − l (P + k − l ) * . Then, Proposition 2.14.
(1) (P ) 2 = P , (2) tP , dP , P b, and P t vanish.
In the above, t (resp. b) is a tensor product of identity morphisms and t ++ − or t −− + (resp. b −+ or b +− ).
Proof.
(P ) 2 = P + k − l (P + k − l ) * P + k − l (P + k − l ) * = P + k − l P + k − l + k − l (P + k − l ) * = P + k − l (P + k − l ) * = P
Proposition 2.15 (Uniqueness). If T in Sp q ( , ) satisfies the conditions (1) and (2) of Proposition 2.14, then T = P .
Proof. In the same way as the proof of Proposition 2.5.
Remark 2.16. Proposition 2.15 implies (P ) * = P .
For any non-negative integers k and l, we consider a subset Sp (k,l)
q = { ∈ Sp q | k = # −1 (+), l = # −1 (−) } of Sp q . Proposition 2.17. There exist a set of A 2 webs { P β α ∈ Sp q (α, β) | α, β ∈ Sp (k,l) q } satisfying (1) P γ β P β α = P γ α for any α, β, γ ∈ Sp (k,l) q ,
(2) tP β α , dP β α , P β α b, and P β α t vanish for any α, β ∈ Sp (k,l) q .
Proof. Let us define P β α = P β + k − l (P α + k − l ) * for any α, β, γ ∈ Sp (k,l) q . Then, it is obvious that P β α satisfies (2) because of Proposition 2.13 (1). We show the equation of (1):
P γ β P β α = P γ + k − l (P β + k − l ) * P β + k − l (P α + k − l ) * = P γ + k − l P + k − l + k − l (P α + k − l ) * by Proposition 2.13 (2) = P γ + k − l (1 + k − l + x)(P α + k − l ) * = P γ + k − l (P α + k − l ) * = P γ α .
2.3. Braidings in Sp q . We introduce A braiding {c δ, : δ ⊗ → ⊗ δ} which is a family of isomorphisms in Sp q . A definition of the braiding in the diagrammatic category Sp q is given by Kuperberg [Kup94,Kup96]. In detail about a general theory of braidings in monoidal categories, for example, see [Tur91]. Let us define a description of an A 2 web by using a crossing with over/under information introduced in Kuperberg [Kup96]:
c +,+ = = q 1 3 − q − 1 6 , c −,− = = q 1 3 − q − 1 6 , c +,− = = q − 1 3 − q 1 6 , c −,+ = = q − 1 3 − q 1 6 .
We also describe their inverses as By the above description, we can consider A 2 webs with over/under crossings. These A 2 webs satisfies the Reidemeister moves (R1)-(R4) for framed tangled trivalent graphs, that is, we can confirm the following local moves of A 2 webs:
(R1) , (R2) , (R3) , (R4) , .
For any objects δ and in Sp q , we define
c δ, = δ δ¯ ∈ Sp q (δ , δ), c −1 δ, = δ ¯ δ ∈ Sp q ( δ, δ ),
where an edge labeled by δ (resp. ) mean an embedding of 1 δ (resp. 1 ) along it with the same over/under information at every crossings. The invariance under the Reidemeister moves (R1)-(R4) provides the invariance under (R1)-(R3) for any labeled edges. By the same reason, we can slide A 2 clasps across over/under crossings, namely,
c δ, (1 δ ⊗ P ) = (P ⊗ 1 δ )c δ, , c δ, (P δ δ ⊗ 1 ) = (1 ⊗ P δ δ )c δ, , (2.3) c −1 δ, (1 ⊗ P δ δ ) = (P δ δ ⊗ 1 )c −1 δ, , c −1 δ, (P ⊗ 1 δ ) = (1 δ ⊗ P )c −1 δ, .
A CATEGORIFICATION OF S (k,l) (x, y)
Let us briefly recall the definition of the Karoubi envelope and the split Grothendieck group.
Definition 3.1 (The Karoubi envelope). The Karoubi envelope of a category A, denoted by Kar(A), is defined as follows:
• Objects in Kar(A) is pairs (X, f ) of objects X in A and idempotents f ∈ Hom A (X, X).
• Morphisms in Hom Kar(A) ((X, f ), (Y, g)) is morphisms φ ∈ Hom A (X, Y ) satisfying g • φ • f = φ for any objexts (X, f ) and (Y, g).
We remark that the identity in Hom Kar(A) ((X, f ), (X, f )) is given by f . If A is monoidal, then the Karoubi envelope Kar(A) inherits a tensor product with (X, f ) ⊗ (Y, g) = (X ⊗ Y, f ⊗ g).
Definition 3.2 (The split Grothendieck group). The split Grothendieck group of an additive category A with ⊕ is the abelian group K 0 (A) generated by isomorphism classes X of objects X in A modulo the relations X 1 ⊕ X 2 = X 1 + X 2 . If A is monoidal, that is A equipped with a tensor ⊗ and the identity object e, then K 0 (A) inherits a ring structure with the unit e and the multiplication X ⊗ Y = X · Y .
In this section, we consider the pairs {( , P )} in the Karoubi envelope Kar(Sp q ) of the A 2 spider and its split Grothendieck group. Firstly, it is easy to see that we can take a standard representatives for the isomorphism class of {( , P )}.
Lemma 3.3. For any ∈ Sp (k,l) q , ( , P ) ∼ = (+ k − l , P + k − l + k − l ) in Kar(Sp q ).
Proof. P + k − l and P + k − l give an isomorphism between ( , P ) and (+ k − l , P + k − l + k − l ). In fact, Proposition 2.17 shows P + k − l + k − l P + k − l P = P + k − l and P P + k − l P + k − l + k − l = P + k − l .
Thus, P + k − l and P + k − l are mophisms in Kar(Sp q ). By the same reason, P + k − l P + k − l = P and P + k − l P + k − l = P + k − l + k − l .
The multiplication in K 0 (Kar(Sp q )) is commutative because of a property (2.3) of a braiding.
Lemma 3.4. For any objects and δ in Sp q , P ⊗ P δ δ ∼ = P δ δ ⊗ P by c ,δ .
Let us denote objects (+ k − l , P + k − l + k − l ) in Kar(Sp q ) by P (k,l) for k, l ≥ 1. We denote the pair ( 0 , D ∅ ) of the empty sign and the empty web by P (0,0) . The split Grothendieck group K 0 (Kar(Sp q )) is a ring with with the unit 1 = P (0,0) . We denote (+, 1 + ) and (−, 1 − ) by X and Y in K 0 (Kar(Sp q )), respectively.
We will show the isomorphism classes P (k,l) satisfy the recursive formula of the A 2 Chebyshev polynomials in K 0 (Kar(Sp q )):
P (1,1) = XY − 1, (3.1) P (k+1,0) = X P (k,0) − P (k−1,1) for k ≥ 1, (3.2) P (k+1,l) = X P (k,l) − P (k−1,l+1) − P (k,l−1) for k, l ≥ 1, (3.3) P (0,l+1) = Y P (0,l) − P (1,l−1) for l ≥ 1, (3.4) P (k,l+1) = Y P (k,l) − P (k+1,l−1) − P (k−1,l) for k, l ≥ 1. (3.5)
We only have to prove (3.1)-(3.14) because of (P + k − l + k − l ) * ∼ = P − k + l − k + l . We use the following well-known fact about an additive category, see [ML98,KS06], for example.
Lemma 3.5. Let A be an additive category and X 1 , X 2 , Y objects in A. Y ∼ = X 1 ⊕ X 2 if and only if there exists morphisms p 1 : Y → X 1 , p 2 : Y → X 2 , ι 1 : X 1 → Y , and ι 2 : X 2 → Y satisfying the following conditions:
ι 1 • p 1 + ι 2 • p 2 = 1 Y , p 1 • ι 2 = p 2 • ι 1 = 0, p 1 • ι 1 = 1 X1 , p 2 • ι 2 = 1 X2 .
Proposition 3.6. P (1,1) = XY − 1 Proof. By Definition 2.7,
P +− +− + 1 [3] b +− d +− = 1 + ⊗ 1 − .
Let us prove (+−,
P +− +− + 1 [3] b +− d +− ) ∼ = P (1,1) ⊕ P (0,0)
. It is only necessary to construct projections p 1 : (+−,
P +− +− + 1 [3] b +− d +− ) → P (1,1) and p 2 : (+−, P +− +− + 1 [3] b +− d +− ) → P (0,P +− +− (P +− +− + 1 [3] b +− d +− ) = (P +− +− ) 2 + 1 [3] (P +− +− b +− )d +− = P +− +−
by Proposition 2.8, and
1 [3] d +− (P +− +− + 1 [3] b +− d +− ) = 1 [3] d +− P +− +− + 1 [3] 2 d +− b +− d +− = 1 [3] d +−(P +− +− + 1 [3] b +− d +− )P +− +− = (P +− +− ) 2 + 1 [3] (b +− )d +− P +− +− = P +− +− and (P +− +− + 1 [3] b +− d +− )b +− = P +− +− b +− + 1 [3] b +− d +− b +− = b +− .
In fact, it is easily to see that
ι 1 • p 2 = ι 2 • p 1 = 0, ι 1 • p 1 = P +− +− , ι 2 • p 2 = 1 [3] b +− d +− , p 1 • ι 2 = p 2 • ι 1 = 0, p 1 • ι 1 = P +− +− , p 2 • ι 2 = D ∅ .
Thus, we obtain P (1,1) ⊕ P (0,0) ∼ = (+−,
P +− +− + 1 [3] b +− d +− ). In K 0 (Kar(Sp q )), it is interpreted as P (1,1) + 1 = (+−, P +− +− + 1 [3] b +− d +− ) = XY .
Proposition 3.7. For any positive integer k, P (k+1,0) = X P (k,0) − P (k−1,1) .
Proof. By Definition 2.2,
P + k+1 + k+1 + [k] [k + 1] (P + k + k ⊗ 1 + )(1 + k−1 ⊗ t ++ − t − ++ )(P + k + k ⊗ 1 + ) = P + k + k ⊗ 1 + (3.6)
Let us denote the LHS of (3.6) by f . We show that the pair (+ k+1 , f ) is isomorphic to the direct sum P (k+1,0) ⊕ P (k−1,1) in Kar(Sp q ). We only have to construct projections p 1 : (+ k+1 , f ) → P (k+1,0) and p 2 : (+ k+1 , f ) → P (k−1,1) , and inclusions ι 1 : P (k+1,0) → (+ k+1 , f ) and ι 2 : P (k−1,1) → (+ k+1 , f ) satisfying (3.5). We confirm that
p 1 = P + k+1 + k+1 and p 2 = [k] [k + 1] P + k−1 − + k−1 − (1 + k−1 ⊗ t − ++ )(P + k + k ⊗ 1 + ),
provide projections and
ι 1 = P + k+1 + k+1
and ι 2 = (P + k
+ k ⊗ 1 + )(1 + k−1 ⊗ t ++ − )P + k−1 − + k−1 −
inclusions. It is easy to see that p 1 and ι 1 is a morphism between (+ k+1 , f ) and P (k+1,0) . We show that p 2 is a morphism in Kar(Sp q ), that is,
P + k−1 − + k−1 − P + k−1 − + k−1 − (1 + k−1 ⊗ t − ++ )(P + k + k ⊗ 1 + )f = P + k−1 − + k−1 − (1 + k−1 ⊗ t − ++ )(P + k + k ⊗ 1 + ).
By Proposition 2.8, the LHS of the above equation is
[k] [k + 1] P + k−1 − + k−1 − (1 + k−1 ⊗ t − ++ )(P + k + k ⊗ 1 + )(1 + k−1 ⊗ t ++ − t − ++ )(P + k + k ⊗ 1 + ).
We diagrammatically calculate it using Definition 2.2:
[k] [k + 1] k−1 1 k−1 k k−1 = [k] [k + 1] 1 k−1 k k−1 − [k − 1] [k] 1 k−1 k k−2 (3.7) = [k] [k + 1] [2] − [k − 1] [k] 1 k−1 k k−1 = 1 k−1 k k−1 = P + k−1 − + k−1 − (1 + k−1 ⊗ t − ++ )(P + k + k ⊗ 1 + ).
The third equation uses
[2][k] = [k + 1] + [k − 1]. To prove ι 2 is a morphism in Sp q ,
we have to compute f (P + k
+ k ⊗ 1 + )(1 + k−1 ⊗ t ++ − )P + k−1 − + k−1 − P + k−1 − + k−1 − . By Proposition 2.8, it reduce to [k] [k + 1] (P + k + k ⊗ 1 + )(1 + k−1 ⊗ t ++ − t − ++ )(P + k + k ⊗ 1 + )(1 + k−1 ⊗ t ++ − )P + k−1 − + k−1 − .
We can calculate it by turning the diagrams in (3.7) upside down. Moreover, it can be confirmed that
ι 2 • p 2 = [k] [k+1] (P + k + k ⊗ 1 + )(1 + k−1 ⊗ t ++ − t − ++ )(P + k + k ⊗ 1 + ) = P + k + k ⊗ 1 +by
Definition 2.7 and Poroposition 2.8 and p 2 • ι 2 = P + k−1 − + k−1 − by inserting P + k−1 − + k−1 − into the diagrams of (3.7). Therefore, these projections and inclusions satisfy the condition in Lemma 3.5 and give (+ k+1 , f ) ∼ = P (k+1,0) ⊕ P (k−1,1) in Kar(Sp q ). In terms of K 0 (Kar(Sp q )), P (k,0) X = (+ k+1 , f ) = P (k+1,0) + P (k−1,1) .
Let us prepare some lemmata to prove (3.14). We introduce an A 2 basis web represented by a rectangle with a diagonal line which consist of H −+ +− and H +− −+ . Definition 3.8. Let m and n be positive integers.
(1) for 1 < i < k + 1 and
X(k; 1) = (−1) k [k + 1] k 1 k = (−1) k [k + 1] (1 − ⊗ P + k − l −+ k − l−1 )(t −− + ⊗ P + k − l−1 + k − l−1 ).
Proof. A 2 webs appearing in the RHS of Lemma 3.9 contains t ++ − if j ≥ 1 in the top side. Thus,
k i k i k+1−i = k i k i k−i − [k] [k + 1] k i k−1 i k−i
by applying Lemma 3.9 to P + k + k . The A 2 web in the second term has X(k − 1; i). We obtain proof by induction on k. Proof. We decompose
P + k+1 − l + k+1 − l into (1 + ⊗ P + k − l + k − l )P + k+1 − l + k+1 − l (1 + ⊗ P + k − l + k − l )
and expand the middle A 2 clasp P + k+1 − l + k+1 − l by using Definition 2.7. We calculate the following A 2 web:
X(k, l; i) = We can easily calculate X(k, l; 0) from Definition 2.2:
X(k, l; 0) = (1 + ⊗ P + k − l + k − l )(P + k+1 + k+1 ⊗ 1 + )(1 + ⊗ P + k − l + k − l ) = 1 + ⊗ P + k − l + k − l − [k] [k + 1] (1 + ⊗ P + k − l + k − l )(t ++ − t − ++ ⊗ 1 + k−1 − l )(1 + ⊗ P + k − l + k − l )
We complete the proof by substituting the above solutions of X(k, l; i) into
P + k+1 − l + k+1 − l = min{k+1,l} i=0 (−1) i k+1 i l i k+l+2 i X(k, l; i) = X(k, l; 0) − [k + 1][l]
[k + l + 2] X(k, l; 1). Proof. Apply Proposition 3.11 to P + k − l + k − l as follows:
k−1 l k l−1 = k−1 l k l−1 = k−1 l k l−1 − [k − 1] [k] k−1 l k l−1 − [l] [k][k + l + 1] k−1 l k l−1 = [2] − [k − 1] [k] k−1 l k l−1 − [k − 1] [k] k−1 l k l−1 − [l] [k][k + l + 1] k−1 l k l−1 = [k + 1] [k] k−1 l k l−1 − [l] [k][k + l + 1] k−1 l k l−1 .
Once again, we apply Proposition 3.11 to P − l + k−1 − l + k−1 :
+ k − l
Proof. The A 2 web in the left-hand side should be expressed as a scalar multiplication of P + k − l + k − l . This constant can be calculated by taking the closure of the A 2 webs in the both sides. We remark that the value of the closure of P
+ k − l + k − l is [k+1][l+1][k+l+2]
[2]
. Proposition 3.14. For any positive integers k, l ≥ 1, P (k+1,l) = X P (k,l) − P (k−1,l+1) − P (k,l−1) .
Proof. By Proposition 3.11, It is easy to see that the left-hand side is an idempotent and X P (k,l) − P (k+1,l) in K 0 (Kar(Sp q )). We denote the right-hand side of the above A 2 web by g. We consider morphisms p 1 : (+ k+1 − l , g) → P (k−1,l+1) and p 2 : (+ k+1 − l , g) → P (k,l−1) defined by (d +− ⊗ P + k − l−1 + k − l−1 )(1 + ⊗ P + k − l + k − l ).
1 + ⊗ P + k − l + k − l − P + k+1 − l + k+1 − l = [k] [k + 1]p 1 = [k] [k + 1] P + k−1 − l+1 + k−1 − l+1 (t − ++ ⊗ P + k−1 − l + k−1 − l )(1 + ⊗ P + k − l + k − l ),
Let us confirm that p 1 and p 2 are morphisms in Kar(Sp q ). By a similar way to (3.8) and using Lemma 3.12, = P + k−1 − l+1 + k−1 − l+1 (t − ++ ⊗ P + k−1 − l + k−1 − l )(1 + ⊗ P + k − l + k − l ).
P + k−1 − l+1 + k−1 − l+1 P + k−1 − l+1 + k−1 − l+1 (t − ++ ⊗ P + k−1 − l + k−1 − l )(1 + ⊗ P + k − l + k − l )g = [k] [k + 1]
By Lemma 3.12 and Lemma 3.13, = (d +− ⊗ P + k − l−1 + k − l−1 )(1 + ⊗ P + k − l + k − l ).
P + k − l−1 + k − l−1 (d +− ⊗ P + k − l−1 + k − l−1 )(1 + ⊗ P + k − l + k − l )g = [k] [k + 1]
Let us define ι 1 : P (k−1,l+1) → (+ k+1 − l , g) and ι 2 : P (k,l−1) → (+ k+1 − l , g) by (1 + ⊗ P + k − l + k − l )(t ++ − ⊗ P + k−1 − l + k−1 − l )P + k−1 − l+1 + k−1 − l+1 and (1 + ⊗ P + k − l + k − l )(b +− ⊗ P + k − l−1 + k − l−1 ), respectively. Because these webs are obtained by turning the webs in p 1 and p 2 upside down, we can confirm these maps are morphisms in Kar(Sp q ) by the same calculation in the above. The rest of the proof is to confirm that p 1 , p 2 , ι 1 , and ι 2 satisfy the condition in Lemma 3.5. It is easy to see that p i • ι j = 0 if i = j. p 1 • ι 1 = P + k−1 − l+1 + k−1 − l+1 and p 2 • ι 2 = P + k − l−1 + k − l−1 are
.
In the first case, one can show by easy calculations and the induction hypothesis. In the second case, we only have to prove ··· ··· σ • P + k − l + k − l = 0. By construction of σ, the right leg of H −+ +− and the up-pointing arc on one's right should have a crossing (H −+ +− ). Then, there exists σ such that h(σ ) = n − 1 and
,− = = c * +,− , c −1 −,+ = = c * −,+ .
0) , and inclusions ι 1 :P (1,1) → (+−, P +− +− + 1 [3] b +− d +− ) and ι 2 : P (0,0) → (+−, b +− d +− )satisfying the conditions in Lemma 3.5. These projections are given by p 1 = P +− +− and p 2 = 1 [3] d +− because
∈∈..
Sp q (+ n −, −+ n ) Sp q (+ n − m , − m + n )Lemma 3.9 (Kim [Kim07, Proposition 3.1]). For any positive integer k, Let k be a positive integer and X(k; i) Then, X(k; i) = 0
Proposition 3 .
311 (Kim [Kim07, Theorem 3.3]). For any positive integers k and l,
Lemma 3.12.k−1 l
k l−1
=
[k + 1]
[k]
1 −
[l + 1]
[k + 1][k + l + 1]
k−1 l
k l−1
X(k, l; 1) = − [k] [k + 1] (−1) k−1 [k]
derived from (3.8) and Lemma 3.13, respectively. By Proposition 3.11,The last equation uses a formula [a][b] =where a and b are integers. Thus, the condition in Lemma 3.5 is satisfied and we obtained the isomorphism (+ k+1 − l , g) ∼ = P (k−1,l+1) ⊕ P (k,l−1) in Kar(Sp q ). In terms of K 0 (Kar(Sp q )), the equation (3.9) is interpreted as X P (k,l) − P (k+1,l) = (+ k+1 − l , g) = P (k−1,l+1) + P (k,l−1) .
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. Kyoto Department Of Mathematics, Kitashirakawa University, Oiwake-Cho, Sakyo-Ku, KYOTO 606-8502, JAPAN E-mail address: [email protected] OF MATHEMATICS, KYOTO UNIVERSITY, KITASHIRAKAWA OIWAKE-CHO, SAKYO-KU, KYOTO 606-8502, JAPAN E-mail address: [email protected]
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[] |
[
"Mining the UKIDSS GPS: star formation and embedded clusters",
"Mining the UKIDSS GPS: star formation and embedded clusters"
] |
[
"O Solin [email protected] \nDepartment of Computer Science\nUniversity of Helsinki\nUniversity of Helsinki\nP.O. Box 68FI-00014Finland\n\nDepartment of Physics, Division of Geophysics and Astronomy\nUniversity of Helsinki\nUniversity of Helsinki\nP.O. Box 64, FI-00014Finland\n",
"E Ukkonen \nDepartment of Computer Science\nUniversity of Helsinki\nUniversity of Helsinki\nP.O. Box 68FI-00014Finland\n",
"L Haikala \nDepartment of Physics, Division of Geophysics and Astronomy\nUniversity of Helsinki\nUniversity of Helsinki\nP.O. Box 64, FI-00014Finland\n\nFinnish Centre for Astronomy with\nESO University of Turku\nVäisäläntie 20FI-21500PIIKKIÖFinland\n"
] |
[
"Department of Computer Science\nUniversity of Helsinki\nUniversity of Helsinki\nP.O. Box 68FI-00014Finland",
"Department of Physics, Division of Geophysics and Astronomy\nUniversity of Helsinki\nUniversity of Helsinki\nP.O. Box 64, FI-00014Finland",
"Department of Computer Science\nUniversity of Helsinki\nUniversity of Helsinki\nP.O. Box 68FI-00014Finland",
"Department of Physics, Division of Geophysics and Astronomy\nUniversity of Helsinki\nUniversity of Helsinki\nP.O. Box 64, FI-00014Finland",
"Finnish Centre for Astronomy with\nESO University of Turku\nVäisäläntie 20FI-21500PIIKKIÖFinland"
] |
[] |
Context. Data mining techniques must be developed and applied to analyse the large public data bases containing hundreds to thousands of millions entries. Aims. To develop methods for locating previously unknown stellar clusters from the UKIDSS Galactic Plane Survey catalogue data. Methods. The cluster candidates are computationally searched from pre-filtered catalogue data using a method that fits a mixture model of Gaussian densities and background noise using the Expectation Maximization algorithm. The catalogue data contains a significant number of false sources clustered around bright stars. A large fraction of these artefacts were automatically filtered out before or during the cluster search. The UKIDSS data reduction pipeline tends to classify marginally resolved stellar pairs and objects seen against variable surface brightness as extended objects (or "galaxies" in the archive parlance). 10% or 66 × 10 6 of the sources in the UKIDSS GPS catalogue brighter than 17 m in the K band are classified as "galaxies". Young embedded clusters create variable NIR surface brightness because the gas/dust clouds in which they were formed scatters the light from the cluster members. Such clusters appear therefore as clusters of "galaxies" in the catalogue and can be found using only a subset of the catalogue data. The detected "galaxy clusters" were finally screened visually to eliminate the remaining false detections due to data artefacts. Besides the embedded clusters the search also located locations of non clustered embedded star formation. Results. The search covered an area of 1302 deg 2 and 137 previously unknown cluster candidates and 30 previously unknown sites of star formation were found.
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10.1051/0004-6361/201118531
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"https://arxiv.org/pdf/1203.5292v1.pdf"
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1203.5292
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7ec8c6ebf6bf58fcfa5ce7236e124fa2b587b2a3
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Mining the UKIDSS GPS: star formation and embedded clusters
May 5, 2014
O Solin [email protected]
Department of Computer Science
University of Helsinki
University of Helsinki
P.O. Box 68FI-00014Finland
Department of Physics, Division of Geophysics and Astronomy
University of Helsinki
University of Helsinki
P.O. Box 64, FI-00014Finland
E Ukkonen
Department of Computer Science
University of Helsinki
University of Helsinki
P.O. Box 68FI-00014Finland
L Haikala
Department of Physics, Division of Geophysics and Astronomy
University of Helsinki
University of Helsinki
P.O. Box 64, FI-00014Finland
Finnish Centre for Astronomy with
ESO University of Turku
Väisäläntie 20FI-21500PIIKKIÖFinland
Mining the UKIDSS GPS: star formation and embedded clusters
May 5, 2014Astronomy & Astrophysics manuscript no. arXiv-aa18531-11 c ESO 2014open clusters and associations: general -methods: statistical -catalogs -surveys -infrared: stars
Context. Data mining techniques must be developed and applied to analyse the large public data bases containing hundreds to thousands of millions entries. Aims. To develop methods for locating previously unknown stellar clusters from the UKIDSS Galactic Plane Survey catalogue data. Methods. The cluster candidates are computationally searched from pre-filtered catalogue data using a method that fits a mixture model of Gaussian densities and background noise using the Expectation Maximization algorithm. The catalogue data contains a significant number of false sources clustered around bright stars. A large fraction of these artefacts were automatically filtered out before or during the cluster search. The UKIDSS data reduction pipeline tends to classify marginally resolved stellar pairs and objects seen against variable surface brightness as extended objects (or "galaxies" in the archive parlance). 10% or 66 × 10 6 of the sources in the UKIDSS GPS catalogue brighter than 17 m in the K band are classified as "galaxies". Young embedded clusters create variable NIR surface brightness because the gas/dust clouds in which they were formed scatters the light from the cluster members. Such clusters appear therefore as clusters of "galaxies" in the catalogue and can be found using only a subset of the catalogue data. The detected "galaxy clusters" were finally screened visually to eliminate the remaining false detections due to data artefacts. Besides the embedded clusters the search also located locations of non clustered embedded star formation. Results. The search covered an area of 1302 deg 2 and 137 previously unknown cluster candidates and 30 previously unknown sites of star formation were found.
Introduction
Several large digital data archives have become publicly available during the last decade. The archive data of stars and extragalactic objects has been extracted from dedicated large imaging surveys in wavelengths from optical (e.g. SDSS) through nearinfrared (NIR) (e.g. the Two Micron All Sky Survey (2MASS; Skrutskie et al. (2006)), the UKIRT Infrared Deep Sky Survey (UKIDSS; Lawrence et al. (2007)) to mid-infrared (MIR) (e.g. GLIMPSE) and far-infrared (FIR) (e.g. MIPSGAL). The catalogues contain hundreds of millions (e.g. SDSS) to thousands of millions (e.g. UKIDSS when finished) objects. Extracting information from a survey containing terabytes of data can naturally be done in the traditional way, case by case, in small restricted areas. But to really optimise the use of all the data, data mining techniques have to be applied. Data mining will allow to identify 'hidden' patterns and relations, which are not obvious, within the data (Brunner et al. 2001). In astronomy data mining methods have been applied to various research areas such as object classification, forecasting sunspots, and selection of quasar candidates (McConnell 2007).
The major part of star formation, be it low-or high-mass stars, takes place in clusters. The clusters are not bound and will eventually disrupt e.g. because of the Galactic differential rota-Appendices A, B and C are only available in electronic form via http://www.edpsciences.org tion (Blaauw 1952). The stellar clusters trace therefore the recent Galactic star formation. The younger the clusters are the more compact they are and the more closely they are associated with the interstellar gas and dust clouds they formed in. Detailed study of young clusters still associated with their parent cloud will provide information on the star formation process and the stellar initial mass function (IMF).
At the moment some 2000 Galactic stellar clusters are known. This is only a small fraction of the estimated total population of which a major part is obscured by interstellar dust to us and can not be observed in optical wavelengths. However, the extinction decreases at longer wavelengths and already at 2.2 microns in the NIR the extinction in magnitudes is only 11 percent of that in the V band (e.g. Rieke & Lebofsky 1985). The ongoing UKIDSS Galactic plane survey (GPS) is three magnitudes deeper than 2MASS and offers the possibility of detecting stellar clusters which are either more distant and/or more extincted than those visible in 2MASS. The UKIDSS GPS will cover 1800 deg 2 of the northern Milky Way in JHK to the limiting magnitudes of J=20. m 0,H=19. m 1,K=18. m 1. The survey began in May 2005 and when finished will provide an estimated ∼ 1 − 2 × 10 9 detections (mainly stellar sources) in three passbands, i.e. ∼ 3 times that of the 2MASS whole sky survey. Searching automatically for a stellar cluster in the complete UKIDSS GPS is possible only using data mining techniques.
Clusters from infrared archive data have been searched for by Dutra et al. (2003) and Bica et al. (2003a,b) by visual inspection of 2MASS images. Mercer et al. (2005) searched the GLIMPSE mid-infrared survey for clusters using an automated algorithm and visual inspection of images. Froebrich et al. (2007) (FSR) used 2MASS star density maps to locate clusters. Samuel & Lucas (2008) and Lucas (2008Lucas ( , 2009Lucas ( , 2011 applied the ideas from Mercer et al. (2005) to look for clusters from the the UKIDSS GPS. In a recent paper Froebrich et al. (2010) applied the code by Samuel & Lucas (2008) developed for the cluster search in UKIDSS GPS data to investigate the old star clusters in the FSR list (Froebrich et al. 2007). Many of these newly detected clusters or cluster candidates have not yet been studied in detail.
This paper presents an application of Gaussian mixture modelling, optimised with the Expectation Maximization (EM) algorithm (Dempster et al. 1977) to automatically locate stellar clusters in the UKIDSS GPS. The search algorithm and filtering of the catalogue artefacts have been described in detail. The search has so far been applied to the UKIDSS GPS DR7 covering an area of 1302 deg 2 . The data is described in Sect. 2 and the search method and results in Sects. 3 and 4. In Sect. 5 the data mining approach to cluster search, the results, supplementary information on the cluster candidates and selected individual cluster candidates are discussed. Conclusions are drawn in Sect. 6.
The data
UKIDSS is conducted with the Wide Field Camera (WFCAM; Casali et al. (2007)) mounted on the United Kingdom Infrared Telescope (UKIRT) on Mauna Kea. WFCAM consists of four 2048x2048 Rockwell devices and a single exposure covers an area of 0.21 deg 2 . The photometric system used by UKIDSS is described in Hodgkin et al. (2009) and Hewett et al. (2006). The WFCAM Science Archive (WSA; Irwin et al. in preparation; Hambly et al. 2008) holds the UKIDSS image and catalogue data products. The catalogue data is used for the automated search, and the image data for visual inspection of the cluster candidate areas given by the detection algorithm. The WSA releases the data in stages. The current 7th release for GPS covers 1302 deg 2 for the UKIDSS GPS K filter. Of this 819 deg 2 are covered in the J and H filters. This study uses all the data covered in the K band. Stars brighter than K = 10 m from the 2MASS survey are used for locating potential false positive clusters (see Figs. A.1 and A.2).
Search method
The search method takes pre-filtered catalogue data, divided into overlapping bins of size 4 by 4 , and performs a maximum likelihood fitting of a mixture of a Gaussian density and a uniform background. On each bin the fitting is done using the standard Expectation Maximization (EM) algorithm that is widely applied in a variety of sciences, and generally for data clustering in machine learning. The EM-algorithm has been applied for clustering in astronomy by Mercer et al. (2005) to discover new star clusters in the GLIMPSE survey, Uribe et al. (2006) to solve the stellar membership in open clusters, and Samuel & Lucas (2008) and Lucas (2008Lucas ( , 2009Lucas ( , 2011 to discover new star clusters in the UKIDSS GPS survey. Other applications in astronomy are by Martínez-González et al. (2003) to estimate the power spectrum of the cosmic microwave background, and by Yuan (2005) for the calibration of a high resolution spectrometer.
The algorithm
The catalogue data is treated using a mixture model consisting of two-dimensional Gaussian densities to model the stellar clusters and of homogeneous Poisson background to model the stars not belonging to the clusters.
P(X|µ, Σ, τ) = N i=1 τ 0 A + K k=1 τ k p(x i |k) ,(1)
where N is the number of sources within region A, K the number of modeled clusters, X the catalogue sources {x 1 , ..., x N }, and for Gaussian clusters the multivariate normal Gaussian density is
p(x i |k) = Φ k (x i |µ k , Σ k ) = 1 2π |Σ k | exp − 1 2 (x i − µ k ) T Σ −1 k (x i − µ k ) .
(2)
The model has three parameters: the mixing coefficients τ for the Gaussian clusters and the noise, and the means µ and covariances Σ for the Gaussian clusters. Coefficient τ 0 gives the proportion of stars belonging to the background and τ i gives the proportion of stars belonging to the ith cluster: K k=0 τ k = 1. After initializing the model parameters (see Sect. 3.2 step 5), the EM-algorithm works by repeating two alternating steps, the E-step and M-step. The E-step evaluates the responsibilities i.e. the posterior probabilities of each point x i belonging to each group k using the current parameter values.
Υ(z ik ) = τ 0 A τ 0 A + K j=1 τ j Φ(x i |µ j , Σ j ) for k = 0 Υ(z ik ) = τ k Φ(x i |µ k , Σ k ) τ 0 A + K j=1 τ j Φ(x i |µ j , Σ j ) for k > 0(3)
The M-step re-estimates the parameters using the current responsibilities
µ new k = 1 N N i=1 Υ(z ik )x i for k > 0 Σ new k = 1 N N i=1 Υ(z ik )(x i − µ new k )(x i − µ new k ) T for k > 0 τ new k = N k N for k ≥ 0 (4) where N k = N i=1 Υ(z ik ).
After each iteration round the log likelihood, that indicates how well the current model parameters fit the data, is evaluated:
l(X|Θ) = ln(P(X|µ, Σ, τ)) = N i=1 ln τ 0 A + K k=1 τ k p(x i |k) . (5)
The E-and M-steps are repeated until convergence is reached for the log likelihood.
The above formulation in Eqs. 1−5 of the mixture model and its estimation with the EM-algorithm is as in Fraley & Raftery (2002), Bishop (2006), and Mercer et al. (2005). In our model we fix the number of Gaussian clusters K to 1: we search smaller bins of data for one cluster at a time. We choose the diagonal covariance over the spherical and full covariance. The shape of a stellar cluster is usually not a circle. On the other hand the full covariance tends to suggest strong clusters in case of sparsely populated data bins or it might trap diffraction patterns of bright stars (see point i in Sect. A.1) or other beam-like artefacts.
For the diagonal covariance Eq.
(2) changes to
p(x i ) = Φ(x i |µ, σ) = 1 2π √ σ xx σ yy exp − 1 2 (x ix − µ x ) 2 σ xx + (x iy − µ y ) 2 σ yy(6)
and for the covariance matrix Σ = σ xx σ xy σ yx σ yy in Eq. (4) we have σ xy = σ yx = 0 and
σ new xx = 1 N N i=1 Υ(z i )(x ix − µ new x ) 2 σ new yy = 1 N N i=1 Υ(z i )(x iy − µ new y ) 2 (7)
Automated search
Clustering points in a two-dimensional space with the Gaussian mixture model is straightforward. However, the challenge in the UKIDSS GPS case is that searching for spatial densities among all catalogue data points without any other considerations locates clusters very poorly. Even if the data artefacts were filtered out it is likely that the clustering algorithm can locate only spatially compact star rich clusters if no additional data filtering takes place. This is because of the observed high spatial stellar density, strongly modulated by interstellar extinction, in the galactic plane. Sparse extended clusters do not sufficiently raise the spatial stellar number density to be caught by the algorithm. Therefore, suitable criteria must be applied to the data before clusters are searched for, i.e. the data must be pre-filtered. Clustered star formation, be it low-or high-mass stars, takes place in dense molecular clouds. The dust in these clouds reflects the light from the newly born stars and this can be observed as localised surface brightness. If the stars are still embedded in the parental cloud or if they are obscured by foreground dust clouds the surface brightness is best observed in the K band in which the interstellar extinction is the smallest of the UKIDSS filters. The presence of surface brightness is however not a proof of embedded clustered star formation. K band surface brightness can also be due to e.g. formation of an embedded single star, planetary nebula or scattering of the interstellar radiation field from dust clouds (see e.g Lehtinen & Mattila (1996) and Juvela et al. (2006Juvela et al. ( , 2008). K band surface brightness can also be due to line emission from shocked molecular hydrogen resulting from interaction of molecular outflows from newly born low mass stars with the surrounding molecular cloud. Searching for surface brightness offers means to detect embedded star formation and stellar clusters.
The WSA catalogue data table gpsSource used in this study lists magnitudes of stars and galaxies detected in the survey but not explicit information on surface brightness. The catalogue parameter mergedClass is given for every object: -3 for a probable galaxy, -2 for a probable star, -1 for a star, 1 for a galaxy, and 0 for noise.
The star/galaxy classification is based on the object image profile used by the pipeline source detection algorithm (Irwin et al. in preparation). The objects with intensity profiles similar to the UKIDSS WFCAM point spread function are classified as stars, the rest as galaxies (or noise). Scrutiny of the data base and the survey images reveals that the detection algorithm tends to classify most of the objects within regions of variable surface brightness as galaxies. Thus the classification is more precisely star/non-stellar than star/galaxy.
The pipeline feature of classifying objects seen superposed on variable surface brightness as galaxies can be utilised in the search of stellar clusters either embedded in or near molecular/dust clouds. Besides the clusters also single embedded stars with associated nebulosities, either due to outflow activity or reflection, will produce "cluster" detections. Even though the galactic plane is in the centre of the zone of avoidance a large number of extragalactic sources are seen in the catalogue. These are also, quite rightly, classified as galaxies.
A fraction of the catalogue sources are due to data artefacts. These artefacts are discussed in detail in Irwin et al. (in preparation) and Appendix 2 of Lucas et al. (2008). The artefacts cause highly varying extended surface brightness which causes the pipeline to classify most of the sources within the artefact as mergedClass = +1 sources. In addition sharp features in the artefacts produce non-existent mergedClass = +1 sources. The data artefacts must be filtered out from the data before the EMalgorithm is applied as otherwise too many false clusters due to artefacts will be located.
The following artefacts have been addressed: Diffraction patterns of bright stars and diffraction spikes due to secondary mirror supports, bright stars at or near the border of the detector array, beams, 'bow-ties', cross-talk images and persistence images. The steps taken to allow for these artefacts are described in detail in online Appendix A.1.
The classification of sources fainter than 17 m in K as star/non-stellar objects is highly unreliable. These sources were filtered out from the data.
The parameter k 1ppErrBits contains the quality error information for each source detection of the K filter. We accept sources with k 1ppErrBits < 524288. The value 131072 ≤ k 1ppErrBits < 524288 for the K band quality error bit flag refers to a GPS photometric calibration problem, that has been fixed in DR8 (WSA 2012). This value of k 1ppErrBits falls under the severity category of "Important Warning", but true sources can be found within such regions (e.g. clusters 107, 153, 154 and 155 in the list by Lucas (2009)). We recognise that stars with k 1ppErrBits ≥ 65536 (the value 65536 corresponds to "close to saturated") are considered to have unreliable photometry. In order not to loose true positives we apply the higher limit of k 1ppErrBits < 524288.
The catalogue data table gpsSource contains 125 attributes for each detected object. We tested the usefulness of other parameters in our clustering effort, but ultimately our method makes use only of the star/non-stellar classifier mergedClass.
UKIDSS DR7 contains 631 117 002 sources measured in the K filter. Out of these 343 737 754 i.e. 54% satisfy the criteria K magnitude brighter than 17 m and k 1ppErrBits < 524288. These sources are divided according to the mergedClass Table 1 and the 30 star formation location candidates (open circles) in Table 2. The grey area marks the UKIDSS DR7 K filter coverage. Only the northern edge of the Taurus-Auriga-Perseus star formation complex below the Galactic plane at l ∼ 170 • is shown in the lower figure. No cluster candidates and only one star formation location candidate were found in this area.
parameter so that a negligible fraction are probable galaxies or noise, 5% probable stars, 74% stars, and 19% galaxies. We end up using for the detection algorithm sources with K magnitude brighter than 17 m , k 1ppErrBits < 524288 and mergedClass = +1. These amount to 66 149 194 sources (∼ 10% out of all sources in UKIDSS DR7). Besides for excluding objects with K magnitude fainter than 17 m and point 3 below the magnitudes listed in the UKIDSS catalogue are in no way used in the automated search.
The automated search proceeds in the following steps. Only the K band data is used in the search.
1. The pre-filtered catalogue data is divided into smaller overlapping spatial bins of size 4 by 4 . Apart from bins at the dataset edges each bin overlaps one half of its neighbouring bins. 4 by 4 was chosen as a suitable size for the bin based on experiments with the cluster candidates in the list by Lucas (2009). 2. Remove false mergedClass = +1 classifications around bright stars and in the direction of the 8 diffraction spikes as explained in online Appendix A.1. 3. In order to track clusters with bright members the detection algorithm is run five times: once with all (filtered) input data and then using 80, 60, 40 and 20% of these sources arranged in descending order of the K magnitude. 4. The spatial coordinates are rescaled to the interval [0,1] to make all bins equally important but still allowing them to have differing means and variances. This step is relevant only for bins at the dataset edges and which are smaller than 4 by 4 .
5. In order to initialise the model parameters the data bin is divided into 16 subgrids to find the area with the highest spatial density. The initial value of the cluster mean µ is the center point of the subgrid with the highest density. The covariance matrix of the data points assigned to the subgrid with the highest density give the initial values for the cluster covariance Σ. The weights τ have as initial values the same value: τ 0 = τ 1 = 0.5. 6. Each data bin is represented by a mixture model of a background component and one Gaussian cluster component according to Sect. 3.1. 7. The EM-algorithm returns for each data bin a candidate cluster, i.e. an ellipse with the center point at the mean µ and half-axes determined by the covariance Σ. 8. Remove false positives created by bright stars at or just outside an array edge as explained in online Appendix A.1. 9. Rearrange the candidates in descending order of the Bayesian information criterion (BIC, Schwarz (1978)). The BIC is used for rough comparison between competing models, and is defined as
BIC = 2l(X|Θ) − dln(n),(8)
where d denotes the number of degrees of freedom of the model, and n is the number of data points. For this model with background noise and one Gaussian cluster with a diagonal mode covariance matrix, d sums to 5: -Weights τ 0 and τ 1 with one constraint of normality: τ 0 + τ 1 = 1. -Means µ x and µ y .
-Elements of the covariance matrix σ xx and σ yy . 10. Merge cluster candidates closer than one arcmin to each other.
11. Remove from the list the cluster candidates catalogued in Bica et al. (2003a,b)
Source screening
Images of the cluster candidate areas with BIC > 20 were retrieved from the database for visual inspection. Choosing 20 as the threshold value for the BIC gives 27599 cluster candidates which is a feasible number to inspect visually. Despite the effort to filter out data artefacts only ∼ 2% of these 27599 candidates are true cluster candidates, locations of embedded star formation, true galaxies or reflection nebulae. With the decreasing value of BIC the proportion of true candidates decreases strongly.
The cluster candidates were visually inspected and the obvious false clusters produced by data artefacts were excluded. The remaining candidates were screened by inspecting more thoroughly visually the grey scale J, H, K and the false colour images (J coded in blue, H in green and K in red) obtained from the WSA. The WSA images are automatically produced and the grey/colour scales are not necessarily optimised for resolving the high stellar densities in many of the clusters and the extended surface brightness in the locations of star formation. In such cases the J, H and K fits files obtained from the WSA were used to produce grey scale and false colour images with better optimised intensity levels. Examples of such false colour images are shown in online Appendix C.
Visual inspection of many candidates revealed them to be galaxies or single stars with a reflection nebula. These we do not list. Such sources are expected since we search particularly for embedded clusters of non-stellar sources using the mergedClass = +1 criterion. Almost any object with surface brightness produces mergedClass = +1 classifications. Examples of false positive candidates are shown in online Appendix A.2.
The selection criteria for a source to be accepted as a cluster or location of star formation candidate were conservative and doubtful objects were excluded. The selection was based solely on the optical appearance of the candidates and no attention was paid on the brightness of the stars in the direction of the candidates and no minimum number of stars in a possible cluster was required.
Finally SIMBAD data base was used to search for astronomical objects within 2 of the remaining candidates and SIMBAD literature links to these sources were inspected to exclude sources previously classified as stellar clusters or star forming locations.
Results
The search located 137 cluster and 30 star formation location candidates which, to our knowledge, are previously unknown. The cluster candidates are listed in Table 1 and the candidate locations of star formation in Table 2. The columns list (1) a running number, (2) source identification, (3,4) Galactic coordinates, (4,5) J2000.0 equatorial coordinates, (6) description of selected SIMBAD sources within 2 of the direction of the candidate and (8) references to selected publications in Table 3.
The distribution of the candidates is shown superposed on the observed GPS area in Fig. 1. The cluster candidates are marked with filled circles and the star formation location candi-dates as open circles. The candidates are distributed quite symmetrically around the Galactic midplane. As expected most of the candidates lie within two degrees from the Galactic plane. The surface density of the candidates is higher in the direction of the inner Galaxy (15 • < l < 107 • ) than of the outer Galaxy (141 • < l < 230 • ). The area in the direction of the inner Galaxy includes 20 times more sources than of the outer Galaxy. Only the northern edge of the Taurus-Auriga-Perseus star formation complex below the plane scanned by the UKIDSS GPS is shown in Fig. 1. No cluster candidates and only one star formation location candidate were found in this area. 4 by 4 images in JHK bands of the new cluster candidate areas are available in electronic form 1 . Most images show clear signs of reflected light in particular in the K band thus indicating embedded clusters or sites of star formation.
Cluster candidate 80 is shown in Fig. 2. In the lower panel are the 4 by 4 UKIDSS JHK images of the cluster candidate area. Reflected light from surrounding dust is visible in the K image. Above the UKIDSS JH images is the same area from 2MASS. A faint nebulosity can be spotted but no cluster. The cluster becomes visible as a spatial density when 60% of the sources with mergedClass = +1 arranged in descending order of the K magnitude (the large circles in the catalogue plot above the UKIDSS K image) are fed to the algorithm. A millimetre radio source has been detected in this direction (Rosolowsky et al. 2010) but it has not been identified as a cluster.
Further K band images of typical candidates are shown in Fig. 3. Location of star formation candidate 15 (Fig. 3a) has so far been identified only as an IRAS and a millimetre source. No stellar cluster is visible. Cluster candidate 116 (Fig. 3b) is visible as a galactic nebula in SDSS, but in the UKIDSS image there is a compact cluster. A second possibly associated cluster is seen NW of cluster candidate 116 in Fig. 3b. Cluster candidate 3 (Fig. 3c1) has around its location an IRAS source, an MSX source, an HII region, a submillimetre source, and a millimetre source. Expanded view of the cluster area using grey levels different from the WSA image is shown in Fig. 3c2. The cluster structure is better visible than in the image provided by WSA.
Further example cluster candidates including their colourcolour diagrams are shown in Appendix B.
Discussion
Searching for spatial overdensities only in the number of stars in UKIDSS GPS is not fruitful. The number of stars in the Galactic plane is high and as a consequence sparse clusters do not increase the number of stars sufficiently to be detected. Also as strong extinction takes place in the Galactic plane the actually observed number of stars is highly modulated, and this modulation produces structures which trigger our model. One major culprit for the difficulty of an automated search for stellar clusters lies in the UKIDSS data base. From automated search point of view, the data base is plagued by strong clustered artefacts which overshadow real structures.
Straightforward clustering of all objects without filtering of the data fails, and therefore additional search criteria must be adopted. The requirement of associated surface brightness via the mergedClass = +1, i.e. non-stellar classification, chosen in this work directs the search to embedded stellar clusters. This criterion takes advantage of the UKIDSS catalogue feature of classifying stars superposed on variable background as non stellar objects thus producing clusters of such objects. Additionally, besides the clusters, the search targets also the locations of nonclustered star formation. Other criteria which were tested did not prove out successfully. The search was conducted in 16 square arcmin bins at a time which means that spatially extended clusters are not detected unless they are strongly centrally concentrated. We choose the diagonal covariance over the spherical and full covariance. Using the threshold value of 20 for the BIC gives 27599 candidates out of which ∼ 2% are regarded true positive candidates through visual inspection of the candidate images. Out of these 167 (∼ 30%) candidates are stellar clusters or sites of non-clustered star formation not previously verified as such. The EM method, as applied in this work, performs well with the Bica et al. (2003a,b) and Lucas (2009) catalogue objects that are compact enough to fit the 4 by 4 window, but very poorly with the catalogue of Froebrich et al. (2007) that was compiled by searching for statistical over-densities only. Among the cluster candidates in the list by Lucas (2009) found also by our system are both nebulae and clusters with some or no nebulosity. Cluster candidates in the list by Lucas (2009) not found by our system were not found either because they did not represent themselves as clusters of non-stellar sources, or the BIC value given by our system fell under our cut-off value of 20.
Surface brightness due to embedded stellar clusters or star formation is only one indication of presence of such objects. Young, embedded stars are usually associated with infrared objects (e.g. IRAS or Spitzer), masers (e.g. H2O, SiO, methanol) and extended or point like (sub)mm sources. Early type stars are associated with HII regions. Numerous surveys for these star formation indicators have been conducted in the direction of e.g. colour selected IRAS sources. None of these indicators was used in the EM search. It is therefore of interest to study if one or more of these indicators have been detected in the direction of the new clusters and embedded star formation locations.
SIMBAD was used to search for sources within 2 from the cluster or embedded star formation candidates with the following results (the number of sources are given in parenthesis): IRAS point source (100), MSX source (38), (sub)millimetre source (60), maser (24), outflow candidate (4) and HII region (39). 31 candidates are seen in the direction of a Spitzer infrared dark cloud (IRDC). Cirrus-like IRAS point souces (IRAS detection only at 100 microns) were excluded. If the 100 micron IRAS flux was of low quality or an upper limit only a good quality flux rising from 12 microns to 60 microns was required. All the IRAS point sources listed as associated sources in Table 1 have IRAS fluxes rising from 12 microns to 100 microns, i.e. typical for embedded sources in star forming clouds. Mostly more than one of these indicators were seen in the direction of the candidates. 32 cluster candidates and 7 embedded star formation candidates were not associated with any object in the SIMBAD data base. The number of indicators seen in the direction of the candidates gives confidence that most of the new clusters or embedded star formation locations are real entities and not produced by chance nor are due to catalogue artefacts.
Although the EM-algorithm returns the half-axes of the ellipse covering the cluster candidate area, this method of clustering non-stellar sources produced by surface brightness is not adequate for deriving estimates for the cluster radii and the number of members. The radii could be estimated through visual examination of the images, but accurate estimates are outside the scope of this study. The UKIDSS GPS covers the direction of the Galactic anticentre. This region of sky is well visible from the northern hemisphere and has been intensively studied optically, in IR and in radio domain. The optical extinction in this general direction is also by far not so severe as the general direction of the Galactic centre. One would therefore not expect to find many previously undetected large and star rich clusters. Contrary to expectations a number of such clusters were detected (candidates 106, 107, 109, 110, 103, 114, 116 and 126). Of these the clusters pairs 106-107, 115-116 and 122-123 are seen in the same 4 by 4 image. Clusters 107 and 109 have many different SIMBAD indicators of star formation seen in their direction, 114 an IRAS point source and an HII region, 116 and 126 an IRAS point source and reflection nebulosity. Cluster 115 has no associated indicator. The small apparent size indicates that these clusters lie far in the outer Galactic plane. IRAS point sources in the direction of cluster candidates 107-110, 114-120, 124-128, 130, 132, 134, 136 and 137 and location of star formation candidates 25, 26, 28 and 30 have been included in the CO survey of , who have extensively studied the star formation in the outer Galaxy during the past two decades. Further detailed study of these sources will shed light on the star formation history of the outer Galaxy.
Further insight to the cluster candidates can be obtained by investigating their (H−K, J −H) colour-colour diagrams. Images and colour-colour diagrams of selected cluster candidates are shown in online Appendix B, Figs. B.1 to B.6. The colour-colour diagram is a useful tool to investigate the cluster properties and membership of individual stars if the photometric data is accurate. The background surface brightness and crowded stellar fields in the direction of many of the new cluster candidates makes accurate photometry, especially for the faint stars, difficult. UKIDSS sources brighter than 17 magnitudes in K and classified as non-stellar were used to find the cluster candidates. In the following "cluster indicator" refers to UKIDSS sources (both stellar and non-stellar) in the cluster direction brighter than 17 m in K. Of the example clusters in online Appendix B for cluster candidates 114 and 116 a large fraction of the cluster indicators lie within the reddening band. For the other candidates either a large (cluster candidates 20 and 110) or a major fraction of the cluster indicators lie to the right (cluster candidates 9 and 43) of the reddening band. The position of objects right of the reddening band could be explained by infrared excess caused by circumstellar dust but such a high number of these stars in the clusters is not expected. According to the colour-colour diagrams the foreground extinctions towards the cluster indicators are up from 10 magnitudes (early spectral type assumed) up to 30 or more magnitudes. This is a reasonable value. If the extinctions were significantly lower these clusters would have already been detected in the optical. Dedicated NIR imaging of these clusters is needed to obtain accurate cluster indicator photometry.
Notes on individual candidates
Cluster candidate 5 is seen at the edge of a dense dust cloud. Cluster candidate 18: the nebulous object seen to NW of the cluster position of this candidate is associated with a methanol maser that has been listed in three studies: Caswell et al. (1993), Błaszkiewicz & Kus (2004) and Szymczak et al. (2000). The maser from the Caswell list is used as an example on pp.17−20 in a presentation by Lucas (2008 (2009) is 2.5 NE of this candidate. The HII region has a diameter of 10 making this candidate part of SH2-75. No stellar cluster is visible in the image, but instead an object that could be e.g. an outflow cone. Location of star formation candidates 12 and 15 and cluster candidate 128 are possible molecular hydrogen objects based on UKIDSS images. Location of star formation candidate 25: two other nebulosities are seen nearby this candidate. Location of star formation candidate 29: northeast of this candidate are a second nebulous source 1.7 away and cluster candidate [IBP2002] CC09 in Ivanov et al. (2002) 3.2 away. Location of star formation candidate 30: the SH 2-287 A HII region.
With the exception of cluster candidates 5,17,23,56,63,66,69,72,75,76,81,82,83,86,93,97,98,99,102,103,105,111,113,115,119,121,122,125,129,130,131,133,135 and 136 and location of star formation candidates 2, 6,11,12,17,19,25 and 29 the rest of the candidates are associated at least with an IRAS point source, most also with other indicators of star formation. Many of our candidates are included in various studies:
-Classified as star forming regions (SFRs) based on sub-mm continuum imaging of IRAS sources selected from radio ultracompact HII region surveys ). (Cluster candidates 3, 6, 14, 15, 24 and 29 and location of star formation candidates 1 and 5). -Suspected sites of massive star formation based on millimetre continuum emission survey toward regions previously identified as harbouring a methanol maser and/or a radio ultracompact HII region . (Cluster candidates 1, 3 and 18 and location of star formation candidates 1 and 5).
-Included in a millimetre continuum and CS spectral line study of massive star forming regions in very early stages of evolution, most of them prior to building up an ultracompact HII region . (Cluster candidates 4,12,19,20 and 27 and location of star formation candidate 8). -Included in a study of 850 µm and 450 µm continuum emission seen towards a sample of high-mass protostellar objects (HMPOs) ). (Cluster candidates 4, 12, 19, 20 and 26 and location of star formation candidate 8). -Included in the APEX submillimetre survey that searches for massive pre-and proto-stellar clumps in the Galaxy in order to shed light on the early stages of star formation ). (Cluster candidates 4 and 6). -Identified as Extended Green Objects in a mid-IR survey by their extended 4.5 µm emission that may be an indicator of outflows specifically from massive protostars . (Cluster candidates 19, 48 and 54 and location of star formation candidate 1). -Included in a submillimetre survey whose primary goal is to identify and characterise HMPOs ) (Location of star formation candidates 8 and 9). The region studied is near the open cluster NGC 6823 to which candidate 8 has a distance of 4.3 and candidate 9 15 . -Bubble candidates from GLIMPSE .
(Cluster candidates 8, 27, 28, 31, 33 and 53). -Identified as zone of avoidance galaxies ZOAG G166.81-03.20, ZOAG G167.06+03.46, ZOAG G167.42+03.45 (Saurer et al. 1997) and ZOAG G228.10+00.80 (Seeberger et al. 1996). (Cluster candidates 109, 110, 112 and 137).
In general radio surveys find circumstellar dust envelopes and disks, and cold cores of molecular clouds. In areas where a radio telescope sees only a point source or signs of e.g. an ultracompact HII region, the UKIDSS images show structures of surface brightness and single stars thus verifying the results of the millimetre/submillimetre radio surveys of suspected star forming regions.
Zone of avoidance galaxies (ZOAG) have been identified in the direction of four of the new cluster candidates (109, 110, 112 and 137). False colour images produced using the WSA fits files in online Figs. C.1, C.2, C.3 and C.4 (cluster candidate 110 is presented also in Fig. B.3 in Appendix B) show that instead of being extragalactic sources they are Galactic clusters. A cluster of individual stars are seen in all figures. This would not be the case if the objects were extragalactic.
The discovery of the UKIDSS stellar cluster in the direction of SH2-105 was serendipitious. It is not surprising to detect a stellar cluster in a HII region. On the contrary, it would be surprising not to find one. The cluster was discovered while inspecting the effect of bright stars to the GPS catalogue. The UKIDSS catalogue data is plotted in Fig. 4. The large filled circles are non-stellar sources brighter than 17 m in K and the crosses are sources that are listed in 2MASS but not in UKIDSS GPS. The UKIDSS K band image is shown in the right panel. Even though an obvious cluster is seen in the K band image practically no stars are listed in the catalogue. The presence of the bright star prevents automated star detection and thus produces a void into the catalogue. The small "cluster" NE of the bright star in the UKIDSS catalogue data is produced mainly by the diffraction rings around the star. The serendipitous detection of the SH2-105 cluster indicates that GPS images may hold many objects, clusters or other, which can not be found using the GPS catalogue because the data is either missing or corrupted.
Summary
While the mixture model used here is an effective method to automatically locate clusters in a large amount of data, the ratio of true positive to false positive candidates given by our system, even though the input data was heavily filtered, is still poor. The processing of the data before it is given to the algorithm is at this stage quite limited: we start by choosing the non-stellar sources and proceed with removing potential false positives before and after the algorithm gives a list of candidates. Also judging from all the false positive examples presented here the mergedClass stellar/non-stellar classifier is often unreliable. This is to be expected within nebulous regions and in the vicinity of very bright stars, where the surface brightness also has a steep gradient. Also such classifiers are unreliable for low signal to noise ratio detections and marginally resolved stellar pairs are often misclassified as a single non-stellar source.
Conclusions
We have used Gaussian mixture modelling, optimised with the Expectation Maximization algorithm to locate embedded stellar clusters and locations of star formation from the UKIDSS Galactic Plane Survey data release 7. Taking advantage of a feature of the UKIDSS stellar classification method which tends to classify stars superposed on variable surface brightness as non-stellar objects we have targeted clusters associated with enhanced sky surface brightness, i.e. mainly embedded clusters. Approximately 10% (66 million objects) of the UKIDSS GPS DR7 objects in K band brighter than 17 m are classified nonstellar. However, the UKIDSS catalogue artefacts due to e.g. bright stars mimic true high surface brightness areas and produce clusters of objects classified as non-stellar by the UKIDSS pipeline. Without a proper filtering image artefacts strongly overshadow true clusters in an automated search. Despite heavy filtering only a few percent of the cluster candidates produced by the automated search turn out not to be data artefacts or false positives. Besides clusters also a number of candidates for locations of star formation were found. The real clusters and locations of star formation had to be visually selected from the list suggested by the automated search.
After discarding the already known clusters and protostars 137 previously unknown stellar clusters and 30 locations of star formation were found. An IRAS point source is seen in the direction of most of the new clusters and locations of star formation. An IRAS source is considered to be associated with a candidate only if it is close enough to the candidate, and the flux density increases towards 100 microns. Besides the IRAS point sources other indications of a still ongoing star formation (e.g. (sub)mm, MSX and maser sources) are detected in the direction or near a large part of the detected clusters. As expected most of the detected clusters or star formation locations are tightly concentrated on the Galactic plane. Relatively few clusters were detected in the direction of the northern Galactic plane. This possibly indicates that most of the northern clusters have already been discovered as this part of the plane has been more thoroughly investigated than the southern plane. However, some of the new northern clusters in the direction of the Galactic anticentre are massive and deserve to be investigated in more detail.
We will continue our search with the future UKIDSS releases. The part of the galactic plane which is not visible at Mauna Kea is being surveyed in the NIR by the VISTA telescope at Paranal observatory. Search for southern clusters down to the same limiting magnitude as the UKIDSS data will thus be possible in near future. Clustering all sources without filtering the data fails. Clustering only sources with K magnitude brighter than 17 m , k 1ppErrBits < 524288 and mergedClass = +1 improves the results remarkably, but visual inspection of the images of the candidate areas revealed a large fraction of the cluster candidates to be blatant false positives:
i) Bright stars tend to create artefacts in the catalogue data appearing as mergedClass = +1 classifications. This happens specially in the direction of the 8 spikes of the diffraction pattern from the two spiders supporting the secondary and the guider auxiliary lens (see Sect. 7.6 in Dye et al. (2006)). We fetched 2MASS stars brighter than 10 m in K, and examined their surroundings in the UKIDSS GPS images and catalogues creating thus rules according to which non-stellar sources are discarded both very near the bright star, and also farther away in the direction of the 8 diffraction spikes. The brighter the star, the greater the distance to which it produces false classifications. Here we note that bits 2 and 20 for the quality error bit flag are not yet implemented. The issue for the former is 'Close to a bright source' and for the latter 'Possible diffraction spike artefact/contamination' (WSA 2012). An example is shown in Fig. A.1. ii) Bright stars at or just outside an array edge tend to create mergedClass = +1 classifications. To find such potential locations we compare the coordinates of 2MASS stars brighter than 8 m in K against parameters minRa, minDec, maxRa and maxDec in the UKIDSS table CurrentAstrometry, and check that parameter multiframeID equals in the tables CurrentAstrometry and gpsDetection. Each cluster candidate is compared to these locations in order to automatically remove false positives. This method might remove also true positives as is the case with e.g.
[BDB2003] G094.60-01.80. An example is shown in Fig. A.2. iii) Beams, 'bow-ties', cross-talk images and persistence images create clusters of non-stellar sources. The 'bow-tie' is a low-level feature in the PSF produced by haloes of bright stars (see Sect. 7.6 in Dye et al. (2006)). As for the cross-talk images the WSA states that the GPS cannot ever be cross-talk flagged with the current algorithm parameters as its fields are just simply too crowded (WSA 2012). The first three types of these false positives are not numerous. It would be useful to remove the persistence image clusters but at present we cannot separate them from true positive clusters using the catalogue data. Examples are shown in Fig 126 • ) at the center of the image is a millimetre radio-source and classified as a possible planetary nebula. This seems to be an outflow coming from a hole in a dark cloud. Excess surface brightness due to the bright central source makes the stars appear as non-stellar and in addition seems to produce non-existent sources.
Fig. 1 .
1Galactic distribution of the 137 cluster candidates (filled circles) in
Fig. 2 .
2Cluster candidate 80. In the lower panel 4 by 4 UKIDSS J, H and K (from left to right) images of the cluster candidate area. Above the UKIDSS J and H images the same area in J, H and K from 2MASS. Image orientation is North up and East left. Panel upper right: The UKIDSS K band catalogue data. The non-stellar sources brighter than 17 m in K are plotted using large filled circles. The confidence ellipse provided by the EM algorithm is shown.
Fig. 3 .
3Typical K band images of cluster and star formation candidates. a) location of star formation candidate 15; b) cluster candidate 116; c1) and c2) cluster candidate 3. Image size is 4 by 4 and image orientation North up and East left.
Fig. 4 .
4The SH2-105 HII region (l = 75.834 • , b = 0.402 • ) is located below a bright star. The catalogue data is shown in the panel on the left. The large filled circles are non-stellar sources brighter than 17 m in K and the crosses are sources that are listed in 2MASS but not in UKIDSS GPS. The 4 by 4 UKIDSS K band image is shown in the panel on the right. The SH2-105 HII region is surrounded with an ellipse to indicate its position in the catalogue data plot.
Fig. A. 1 .
1Example of a false positive cluster caused by the bright K = 4. m 4 star 2MASS J18360783-0644313. The red points and crosses are UKIDSS GPS sources brighter than 17 m and classified as non-stellar. The red crosses mark the filtered out points. Appendix A: False positive clusters A.1. False positive clusters caused by artefacts
A. 2 .
2False positive clusters caused by surface brightness In Figs. A.4 and A.5 two examples of a false positive candidate caused by surface brightness are shown. In Fig. A.4 the false positive cluster at (l = 12.841 • , b = 0.544 • ) is caused by the interplay of extinction and the reflection of the interstellar radiation field from the dust cloud. In Fig. A.5 the object at (l = 16.799 • , b = 0.
). Cluster candidate 29: the nebulous structure southwest is probably associated. Cluster candidate 71: [BDB2003] G077.46+01.76 is 2.8 away from this candidate. Cluster candidate 121: 3.1 southwest of this candidate is a concentration of stars. Cluster candidate 131: cluster candidate nro 15 in the list by Lucas (2009) is 5.1 away from this candidate. Location of star formation candidate 3: the HII region W 48C is 1.8 away from this candidate. Location of star formation candidate 6: the SH2-75 HII region and an IRAS source classified as a cluster by Faustini et al.
Table 1 .
1List of cluster candidates.Notes. (S ) Source classification from SIMBAD: IRDC stands for infrared dark cloud, of? for outflow candidate, bub for bubble, Mas for maser, (s)mm for (sub-)millimetre source, 2MASX for 2MASS extended source, RNe for reflection nebula and DNe for dark nebula.#
ID
l
b
α (J2000)
δ (J2000)
Associated sources S
References
[ • ]
[ • ]
[h m s]
[ • ' "]
1
G011.495−1.483
11.495
−1.483
18 16 21
−19 41 31
IRAS,MSX,HII,smm,mm,Mas
3,6,11,12,17
2
G013.076−0.309
13.076
−0.309
18 15 11
−17 44 35
IRAS,mm,IRDC
18
3
G018.303−0.392
18.303
−0.392
18 25 43
−13 10 23
IRAS,MSX,HII,smm,mm
2,3
4
G018.655−0.059
18.655
−0.059
18 25 11
−12 42 22
IRAS,MSX,smm,mm,IRDC
4,5,7,18
5
G018.850+2.023
18.850
+2.023
18 18 02
−11 33 18
. . .
. . .
6
G019.073−0.286
19.073
−0.286
18 26 48
−12 26 31
IRAS,MSX,HII,smm,mm,IRDC
2,7,18
7
G020.711−0.291
20.711
−0.291
18 29 56
−10 59 38
IRAS,mm
. . .
8
G021.343−0.137
21.343
−0.137
18 30 34
−10 21 47
mm,IRDC,bub
13,18
9
G022.257−0.880
22.257
−0.880
18 34 57
−09 53 42
IRAS
. . .
10
G022.952−0.316
22.952
−0.316
18 34 13
−09 01 05
IRAS,HII,mm,IRDC
18
11
G023.882−0.353
23.882
−0.353
18 36 05
−08 12 36
IRAS,mm,IRDC
18
12
G024.397−0.188
24.397
−0.188
18 36 27
−07 40 34
IRAS,smm,mm,IRDC
4,5,18
13
G025.462−0.159
25.462
−0.159
18 38 19
−06 43 01
mm
. . .
14
G026.544+0.414
26.544
+0.414
18 38 16
−05 29 35 IRAS,MSX,HII,smm,mm,2MASX,IRDC
2,18
15
G028.592−0.365
28.592
−0.365
18 44 49
−04 01 41
IRAS,HII,smm,mm
2
16
G028.693+0.177
28.693
+0.177
18 43 04
−03 41 28
MSX,HII,mm,DNe
8
17
G029.815+2.224
29.815
+2.224
18 37 50
−01 45 25
IRAS,MSX
. . .
18
G029.858−0.060
29.858
−0.060
18 46 02
−02 45 47
MSX,smm,mm,Mas,IRDC
3,18
19
G029.888−0.779
29.888
−0.779
18 48 40
−03 03 50
IRAS,smm,mm,of?,IRDC
4,5,9,18
20
G030.385−0.107
30.385
−0.107
18 47 10
−02 18 58
IRAS,MSX,HII,smm,mm,IRDC
4,5,18
21
G032.152+0.131
32.152
+0.131
18 49 33
−00 38 02
IRAS,MSX,HII,mm,Mas,IRDC
18
22
G034.132+0.472
34.132
+0.472
18 51 57
+01 16 59
IRAS,MSX,HII,mm
. . .
23
G034.583−0.238
34.583
−0.238
18 55 18
+01 21 40
. . .
. . .
24
G037.876−0.400
37.876
−0.400
19 01 54
+04 12 58
IRAS,MSX,HII,smm,mm,Mas,IRDC
2,18
25
G038.937−0.459
38.937
−0.459
19 04 04
+05 07 55
MSX,mm,DNe
8
26
G039.903−1.351
39.903
−1.351
19 09 02
+05 34 48
HII,Mas
. . .
27
G042.111−0.447
42.111
−0.447
19 09 54
+07 57 22
IRAS,HII,smm,mm,bub
4,5,13
28
G042.834−0.151
42.834
−0.151
19 10 11
+08 44 02
IRAS,smm,IRDC,bub
13,18
29
G043.186−0.525
43.186
−0.525
19 12 11
+08 52 23
MSX,HII,smm,mm,Mas,IRDC
1,2,6,18
30
G043.889−0.784
43.889
−0.784
19 14 26
+09 22 34
IRAS,MSX,HII,smm,Mas,IRDC
1,6,18
31
G045.397−0.709
45.397
−0.709
19 17 01
+10 44 42
HII,IRDC,bub
13,18
32
G045.417−0.105
45.417
−0.105
19 14 53
+11 02 38
IRAS,mm,IRDC
18
33
G048.845−0.544
48.845
−0.544
19 23 03
+13 52 05
IRDC,bub
13,18
34
G049.288−0.056
49.288
−0.056
19 22 08
+14 29 17
IRAS,mm,2MASX,Mas,IRDC
17,18
35
G049.430−0.011
49.430
−0.011
19 22 15
+14 38 06
IRAS,MSX,mm,IRDC
18
36
G049.721−0.016
49.721
−0.016
19 22 50
+14 53 20
IRAS,mm,IRDC
18
37
G050.317+0.675
50.317
+0.675
19 21 28
+15 44 24
IRAS,MSX,HII,Mas
1,6
38
G050.490+0.994
50.490
+0.994
19 20 38
+16 02 35
IRAS
. . .
39
G051.210−0.799
51.210
−0.799
19 28 37
+15 49 37
IRAS
. . .
40
G051.401−0.890
51.401
−0.890
19 29 20
+15 57 07
IRAS
. . .
41
G051.426−0.615
51.426
−0.615
19 28 23
+16 06 18
IRAS
. . .
42
G052.367−1.044
52.367
−1.044
19 31 50
+16 43 30
IRAS
. . .
43
G052.753+0.335
52.753
+0.335
19 27 32
+17 43 26
IRAS,MSX,HII,mm,IRDC
18
44
G052.847−0.664
52.847
−0.664
19 31 24
+17 19 41
IRAS
. . .
45
G053.594−0.249
53.594
−0.249
19 31 23
+18 10 59
mm
. . .
46
G053.819−0.059
53.819
−0.059
19 31 08
+18 28 19
IRAS,mm,IRDC
18
47
G054.192−0.691
54.192
−0.691
19 34 14
+18 29 35
IRAS
. . .
48
G054.236+0.257
54.236
+0.257
19 30 49
+18 59 20
IRAS
. . .
49
G054.426+0.991
54.426
+0.991
19 28 28
+19 30 29
of?,IRDC
9,18
50
G054.493+1.579
54.493
+1.579
19 26 25
+19 50 49
IRAS
. . .
51
G054.522+0.919
54.522
+0.919
19 28 56
+19 33 29
IRAS,IRDC
18
52
G056.239−0.342
56.239
−0.342
19 37 10
+20 27 04
IRAS
. . .
53
G057.546−0.273
57.546
−0.273
19 39 40
+21 37 26
IRAS,MSX,HII,mm,IRDC,bub
13,18
54
G057.573+0.221
57.573
+0.221
19 37 52
+21 53 24
IRAS,IRDC
18
55
G057.608+0.024
57.608
+0.024
19 38 41
+21 49 26
IRAS,mm,of?
9
56
G061.193−0.299
61.193
−0.299
19 47 40
+24 46 23
. . .
. . .
57
G061.720+0.863
61.720
+0.863
19 44 24
+25 48 40
IRAS,MSX,HII,2MASX
. . .
58
G064.152+1.283
64.152
+1.283
19 48 15
+28 07 30
IRAS,MSX,HII
. . .
59
G068.239+0.960
68.239
+0.960
19 59 13
+31 27 47
IRAS,MSX,HII
. . .
60
G071.151+0.399
71.151
+0.399
20 08 50
+33 37 34
IRAS,MSX,HII,smm,mm,2MASX
. . .
61
G071.312+0.827
71.312
+0.827
20 07 32
+33 59 35
IRAS,Mas
11,12,17
62
G071.523−0.386
71.523
−0.386
20 12 58
+33 30 29
IRAS,MSX,mm,Mas
6,11,17
63
G071.804+0.846
71.804
+0.846
20 08 44
+34 25 01
. . .
. . .
64
G073.878+1.026
73.878
+1.026
20 13 34
+36 15 04
IRAS,MSX,HII,2MASX
. . .
65
G074.159+1.645
74.159
+1.645
20 11 46
+36 49 34
IRAS,smm,Mas
17
66
G074.213+1.650
74.213
+1.650
20 11 54
+36 52 26
. . .
. . .
67
G074.753+0.913
74.753
+0.913
20 16 27
+36 54 58
IRAS,MSX,HII,2MASX
. . .
68
G075.295+1.324
75.295
+1.324
20 16 16
+37 35 42
IRAS,HII,Mas
17
69
G077.127−1.228
77.127
−1.228
20 32 07
+37 37 30
. . .
. . .
70
G077.405−1.213
77.405
−1.213
20 32 54
+37 51 29
IRAS,mm
. . .
71
G077.437+1.720
77.437
+1.720
20 20 45
+39 35 18
MSX,HII
. . .
72
G077.568+3.693
77.568
+3.693
20 12 33
+40 47 49
. . .
. . .
73
G077.821−1.310
77.821
−1.310
20 34 33
+38 08 02
IRAS,mm
. . .
74
G078.703+1.243
78.703
+1.243
20 26 35
+40 21 04
mm
. . .
Table 2 .
2List of location of star formation candidates that cannot be verified as clusters. Notes and references are as inTable 1.#
ID
l
b
α (J2000)
δ (J2000)
Associated sources S
References
[ • ]
[ • ]
[h m s]
[ • ' "]
1
G035.028+0.351
35.028
+0.351
18 54 01
+02 01 34 IRAS,MSX,HII,smm,mm,Mas,of?,IRDC
2,3,6,9,18
2
G035.265+1.365
35.265
+1.365
18 50 50
+02 41 53
. . .
. . .
3
G035.359−1.772
35.359
−1.772
19 02 11
+01 21 04
HII
. . .
4
G036.401−1.763
36.401
−1.763
19 04 03
+02 16 52
IRAS,HII
. . .
5
G037.545−0.111
37.545
−0.111
19 00 16
+04 03 14
IRAS,MSX,HII,smm,mm,Mas,IRDC
1,2,3,6,18
6
G040.080+1.510
40.080
+1.510
18 59 07
+07 02 56
. . .
. . .
7
G055.364+0.186
55.364
+0.186
19 33 23
+19 56 35
IRAS,MSX,mm
. . .
8
G059.360−0.206
59.360
−0.206
19 43 18
+23 13 59
IRAS,MSX,HII,smm,mm
4,5,10
9
G059.640−0.181
59.640
−0.181
19 43 48
+23 29 17
IRAS,MSX,HII,smm,mm,IRDC
10,18
10 G061.315−2.062
61.315
−2.062
19 54 36
+23 58 41
IRAS
. . .
11 G068.858−0.041
68.858
−0.041
20 04 44
+31 27 27
. . .
. . .
12 G076.855+0.761
76.855
+0.761
20 23 06
+38 33 47
. . .
. . .
13 G077.901+1.769
77.901
+1.769
20 21 55
+39 59 49
IRAS,MSX,2MASX
. . .
14 G078.121+3.632
78.121
+3.632
20 14 26
+41 13 28
IRAS,MSX,HII,smm,Mas
4,5,6,11,12,17
15 G078.235+0.901
78.235
+0.901
20 26 37
+39 46 16
IRAS,mm
. . .
16 G079.151+1.830
79.151
+1.830
20 25 25
+41 03 22
IRAS
. . .
17 G079.155+2.222
79.155
+2.222
20 23 44
+41 17 04
. . .
. . .
18 G079.482−0.718
79.482
−0.718
20 37 15
+39 49 01
mm
. . .
19 G079.852−1.507
79.852
−1.507
20 41 41
+39 37 48
. . .
. . .
20 G081.309−0.112
81.309
−0.112
20 40 35
+41 38 13
mm
. . .
21 G081.516+0.192
81.516
+0.192
20 39 58
+41 59 12
MXS,2MASX
. . .
22 G085.034+0.361
85.034
+0.361
20 51 19
+44 50 31
mm,DNe
. . .
23 G162.459−8.674 162.459
−8.674
04 21 38
+37 34 44
IRAS
. . .
24 G171.531+2.445 171.531
+2.445
05 34 00
+37 24 28
IRAS
. . .
25 G173.185+2.356 173.185
+2.356
05 38 04
+35 58 00
smm,Mas
16,17
26 G207.312−2.538 207.312
−2.538
06 32 07
+03 50 05
IRAS
16
27 G209.561+0.577 209.561
+0.577
06 47 20
+03 15 47
IRAS
. . .
28 G212.064−0.739 212.064
−0.739
06 47 13
+00 26 06
IRAS,MSX,Mas
16,17
29 G217.496−0.070 217.496
−0.070
06 59 32
−04 05 34
. . .
. . .
30 G218.053−0.117 218.053
−0.117
07 00 23
−04 36 36
IRAS,HII,Mas
16,17
arXiv:1203.5292v1 [astro-ph.GA] 23 Mar 2012 O. Solin, E. Ukkonen and L. Haikala: Mining the UKIDSS GPS: star formation and embedded clusters
http://www.helsinki.fi/˜osolin/clusters
Acknowledgements. This work was funded by the Finnish Ministry of Education under the project "Utilizing Finland's membership in the European Southern Observatory". This work was supported by the Academy of Finland under grants 118653 (ALGODAN) and 132291, and by the Finnish Funding Agency for Technology and Innovation (TEKES) under the project MIFSAS. This work uses data products from the Two Micron All Sky Survey, and the United Kingdom Infrared Telescope Infrared Deep Sky Survey. This research uses the SIMBAD astronomical database service operated at CCDS, Strasbourg. We thank the referee Philip Lucas for useful comments and suggestions.The automated search uses by default the AperMag3 magnitudes (2.0 aperture diameter). For the colour-colour plots we experimented also with the AperMag1 (1.0 aperture diameter) and AperMag4 (2.8 aperture diameter) extended source magnitudes. For cluster candidates 20, 114 and 116 the colour-colour plots use the AperMag1 magnitudes because they seem to give better precision. For the remaining cluster candidates (9, 43 and 110) the colour-colour plots use the AperMag3 magnitudes. For ppErrbits we apply the same < 524288 limit as in the automated search knowing that by using this limit we don't take advantage of all the photometric warning flags. However for these six cluster candidates only for a negligible portion of the data ppErrbits > 255.We use in the figures a reddening slope of 1.6. We recognise that reddening bands in colour-colour diagrams are delimited by curves rather than vectors (e.g.Golay (1974)andStead & Hoare (2009)). The value of 1.6 is the mean of all the reddening tracks inStead & Hoare (2009). Irrespective of the uncertainty of the reddening vector the colour-colour plots allow to estimate the reddening in the direction of the cluster candidates. The notable difference between cluster candidates is the larger number of field stars, especially giants, in the direction of the inner Galaxy (cluster candidates 9, 20 and 43) with respect to the number of stars in the outer Galaxy (cluster candidates 110, 114 and 116). In the inner Galaxy the field stars, i.e. the stars not classified as non-stellar, lie within the approximate reddening path outlined by the reddening lines. In the outer Galaxy the statistics is poor because of the small number of field stars and the high extinction. The spread of the sources in the direction of the cluster candidates and classified as non-stellar is much higher than for the field stars. The photometry of these sources suffers from the faintness of the stars and the high background surface brightness. However, these were the objects used to locate the cluster candidates. In general it can be noted that the extinction towards these sources is on the average higher than for the general field stars population. The extinctions range from 10 magnitudes up to 20 magnitudes and more. Dedicated observations are needed for detailed analysis of the cluster candidates.Appendix C: Zone of avoidance galaxiesA zone of avoidance galaxy has been reported in the direction of three of the new cluster candidates: 109, 110, 112 and 137. False colour images of these candidates
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|
[] |
[
"Brighter-fatter effect in near-infrared detectors -I. Theory of flat auto-correlations",
"Brighter-fatter effect in near-infrared detectors -I. Theory of flat auto-correlations"
] |
[
"Christopher M Hirata [email protected] \nCenter for Cosmology and AstroParticle Physics\nThe Ohio State University\n191 West Woodruff Avenue43210ColumbusOhioUSA\n",
"Ami Choi \nCenter for Cosmology and AstroParticle Physics\nThe Ohio State University\n191 West Woodruff Avenue43210ColumbusOhioUSA\n"
] |
[
"Center for Cosmology and AstroParticle Physics\nThe Ohio State University\n191 West Woodruff Avenue43210ColumbusOhioUSA",
"Center for Cosmology and AstroParticle Physics\nThe Ohio State University\n191 West Woodruff Avenue43210ColumbusOhioUSA"
] |
[] |
Weak gravitational lensing studies aim to measure small distortions in the shapes of distant galaxies, thus placing very tight demands on the understanding of detectorinduced systematic effects in astronomical images. The Wide-Field Infrared Survey Telescope (WFIRST) will carry out weak lensing measurements in the near infrared using the new Teledyne H4RG-10 detector arrays, which makes the range of possible detector systematics very different from traditional weak lensing measurements using optical CCDs. One of the non-linear detector effects observed in CCDs is the brighterfatter effect (BFE), in which charge already accumulated in a pixel alters the electric field geometry and causes new charge to be deflected away from brighter pixels. Here we describe the formalism for measuring the BFE using flat field correlation functions in infrared detector arrays. The auto-correlation of CCD flat fields is often used to measure the BFE, but because the infrared detector arrays are read out with the charge "in place," the flat field correlations are dominated by capacitive cross-talk between neighboring pixels (the inter-pixel capacitance, or IPC). Conversely, if the BFE is present and one does not account for it, it can bias correlation measurements of the IPC and photon transfer curve measurements of the gain. We show that one can compute numerous cross-correlation functions between different time slices of the same flat exposures, and that correlations due to IPC and BFE leave distinct imprints. We generate a suite of simulated flat fields and show that the underlying IPC and BFE parameters can be extracted, even when both are present in the simulation. There are some biases in the BFE coefficients up to 12%, which are likely caused by higher order terms that are dropped from this analysis. The method is applied to laboratory data in the companion Paper II.
|
10.1088/1538-3873/ab44f7
|
[
"https://arxiv.org/pdf/1906.01846v2.pdf"
] | 174,799,400 |
1906.01846
|
4f6937837a28bea86e521d2d30ab013f6e2bb1c9
|
Brighter-fatter effect in near-infrared detectors -I. Theory of flat auto-correlations
January 27, 2020 23 Jan 2020
Christopher M Hirata [email protected]
Center for Cosmology and AstroParticle Physics
The Ohio State University
191 West Woodruff Avenue43210ColumbusOhioUSA
Ami Choi
Center for Cosmology and AstroParticle Physics
The Ohio State University
191 West Woodruff Avenue43210ColumbusOhioUSA
Brighter-fatter effect in near-infrared detectors -I. Theory of flat auto-correlations
January 27, 2020 23 Jan 2020Subject headings: instrumentation: detectors
Weak gravitational lensing studies aim to measure small distortions in the shapes of distant galaxies, thus placing very tight demands on the understanding of detectorinduced systematic effects in astronomical images. The Wide-Field Infrared Survey Telescope (WFIRST) will carry out weak lensing measurements in the near infrared using the new Teledyne H4RG-10 detector arrays, which makes the range of possible detector systematics very different from traditional weak lensing measurements using optical CCDs. One of the non-linear detector effects observed in CCDs is the brighterfatter effect (BFE), in which charge already accumulated in a pixel alters the electric field geometry and causes new charge to be deflected away from brighter pixels. Here we describe the formalism for measuring the BFE using flat field correlation functions in infrared detector arrays. The auto-correlation of CCD flat fields is often used to measure the BFE, but because the infrared detector arrays are read out with the charge "in place," the flat field correlations are dominated by capacitive cross-talk between neighboring pixels (the inter-pixel capacitance, or IPC). Conversely, if the BFE is present and one does not account for it, it can bias correlation measurements of the IPC and photon transfer curve measurements of the gain. We show that one can compute numerous cross-correlation functions between different time slices of the same flat exposures, and that correlations due to IPC and BFE leave distinct imprints. We generate a suite of simulated flat fields and show that the underlying IPC and BFE parameters can be extracted, even when both are present in the simulation. There are some biases in the BFE coefficients up to 12%, which are likely caused by higher order terms that are dropped from this analysis. The method is applied to laboratory data in the companion Paper II.
Introduction
Weak gravitational lensing (WL) -the distortion of the shapes and sizes of distant galaxies by the curvature of the intervening space-time -is a powerful method for probing the matter distribution in the Universe (for recent results, see e.g. Heymans et al. 2013;Abbott et al. 2016;and Hildebrandt et al. 2017). However, the signal is small and must be measured to a fraction of a percent to meet the science goals of current and future WL surveys. Therefore, WL programs place a very strong emphasis on understanding every systematic error that can occur in measuring the shape of the galaxy -this includes the contribution of the atmosphere, optics, and image motion to the smearing of an image, as well as imprints of the detector system and data processing.
The brighter-fatter effect (BFE; e.g. Antilogus et al. 2014) is one of these subtle effects that has been observed in silicon CCD detectors. This is a non-linear effect in which a brighter point source produces a larger image (as measured by e.g. full width at half maximum) in the CCD than a fainter point source. It is caused by changes in the electric field geometry in the CCD as a well fills up with electrons: if at any instant during the exposure a pixel (i, j) contains more charge than its neighbors, then due to self-repulsion of the electrons, additional photo-electrons generated will be less likely to land in pixel (i, j) and more likely to land in its neighbors. This is described phenomenologically for CCDs by supposing that the pixel boundaries 1 move as a function of accumulated charge. In thick CCDs, the BFE has been observed to have a significant range, e.g. in the Dark Energy Camera (DECam) CCDs, the pixel boundary shifts have been measured from charge up to ∼ 10 pixels away (Gruen et al. 2015). The BFE also manifests itself in correlation properties (variance and correlation function) of flat-field images, where the shifting pixel boundaries break the usual assumption that each photo-electron behaves independently from previous electrons and hence causes non-Poisson correlations in the flat images (Guyonnet et al. 2015). Indeed, this provided one of the early hints to the existence of the BFE (Downing et al. 2006).
The BFE and techniques for modeling it have been well-established in current WL surveys such as the Dark Energy Survey (DES; Gruen et al. 2015) and the Hyper-Suprime Cam (HSC; Coulton et al. 2018). Higher precision will be demanded of the next generation of WL surveys: the Large Synoptic Survey Telescope (LSST), the Euclid space mission, and the Wide-Field Infrared Survey Telescope (WFIRST) space mission. The BFE has been observed and characterized in Euclid development CCDs (Niemi et al. 2015) and in candidate sensors for LSST (Baumer & Roodman 2015;Lage et al. 2017). In setting requirements on any detector effect, it is important to study all of the ways that effect can enter into the analysis; in the case of the brighter-fatter effect in weak lensing, the stars used for determination of the point spread function (PSF) are much brighter (and have steeper intensity gradients) than either the galaxies or the sky background, and so the stars are most affected by the BFE. This means that the principal effect of the BFE is to make the measured (star-based) PSF larger than the correct PSF for faint galaxies. Subsequent stages of analysis will then over-correct for the smearing out of galaxy ellipticities by the finite size of the PSF, and hence will over-estimate the shear signal. Note that we consider the BFE to be a calibration problem, in the sense that WFIRST will need to develop a model for it; we do not need to eliminate it.
WFIRST plans to measure weak lensing in the near infrared (NIR), and thus silicon CCDs are not an option. Instead, it will use infrared detector arrays: each of the 18 detector arrays will use Teledyne's H4RG-10 readout integrated circuit (4088 × 4088 active pixels, 10 µm pitch) with mercury cadmium telluride (HgCdTe; 2.5 µm cutoff) as the light-absorbing component. 2 Just as for CCDs, the boundaries between adjacent pixels are defined by the solution to the drift-diffusion equation rather than a physical barrier, and so a BFE in WFIRST detectors would be physically plausible; however one would expect the details to be very different. It is therefore important to understand whether the BFE, or other non-linear detector effects, are present in WFIRST prototype devices, and if so how the BFE fits into the overall WFIRST calibration plan. A discussion of the physics of the BFE as applied to NIR detectors, as well as some characterization efforts on earlier generations of Teledyne detectors (H1RG and H2RG, 18 µm pitch) can be found in Plazas et al. (2017). The BFE has been measured in point source illumination tests on an H2RG tested for the Euclid program (Plazas et al. 2018).
In order to develop calibration and data reduction procedures for WFIRST data, it is essential to characterize non-linear effects such as the BFE. Such characterization efforts also provide us with the opportunity to learn about what calibration procedures work and what pitfalls exist. Such studies must be undertaken early in the life cycle of the project, especially in the context of a space mission where late changes to the calibration requirements could be expensive or impossible. Detailed characterization of subtle effects such as the BFE is also outside the scope of the technology development milestones (Spergel et al. 2015). These milestones were recently completed for the WFIRST NIR detectors, but focus on basic performance (e.g. quantum efficiency, read noise, persistence), production yield, and environmental testing (e.g. thermal cycling, vibrational, radiation). However, the studies for these milestones do not explore the detector-related systematic effects that are relevant to control shear errors at the few ×10 −4 level.
The two major methods of measuring the BFE are (i) measurement of spots projected onto the detector (either with a laboratory spot projector or using real stars observed through a telescope) and (ii) flat field statistics. This paper considers flat fields, since spots were not yet available for the H4RG-10 HgCdTe detectors at the time we began this project. 3 The data were acquired as part of general detector characterization and made available to the science teams; the test procedure was not specifically optimized for BFE studies.
The use of flat field statistics for BFE measurements presents some special challenges for infrared arrays, most notably that the flat field auto-correlation function is dominated by the effect of inter-pixel capacitance (IPC), which gives a positive correlation between adjacent pixels (Moore et al. 2004). IPC can have a linear component and a non-linear component, the latter of which is known as non-linear inter-pixel capacitance (NL-IPC, a signal dependent coupling that occurs when one converts from charge to voltage; e.g., Cheng 2009;Donlon et al. 2016Donlon et al. , 2017Donlon et al. , 2018. 4 On the other hand, the H4RG-10 provides a non-destructive read capability, which is very useful for BFE studies as it enables intermediate stages of the image to be observed: the flat field is then a 3D data cube, and correlations across different time slices ("frames") of the image can be measured. We will find that this capability allows us to simultaneously measure the IPC and BFE using flat fields. We will further find that the gain measurement from the photon transfer curve (e.g., Mortara & Fowler 1981;Janesick et al. 1985) must be corrected for the BFE in addition to the now-standard IPC correction (Moore et al. 2006;Brown et al. 2006).
The cleanest method for BFE measurements from H4RG-10 flat fields is to cross-correlate two correlated double sample (CDS) images, obtained from non-overlapping parts of the ramp. This eliminates any possible correlations from Poisson noise -including those that couple between neighboring pixels through linear IPC -as well as any read noise correlations that occur within a single frame. It does leave correlations due to classical non-linearity, which must be removed based on the standard non-linearity curve analysis. The non-overlapping correlation function method cannot tell the difference between the brighter-fatter effect (which occurs during the process of collecting charge into a well) from NL-IPC. To distinguish the BFE from NL-IPC, we must resort to correlations of CDS images over the same (or at least an overlapping) time interval, and observe how pixel variances or covariances of adjacent pixels change as one varies the time interval of interest. While these tests mix together many different detector effects, the BFE and NL-IPC hypotheses make distinct predictions. NL-IPC appears in these methods with a factor of 2 different from BFE, because in the BFE each electron 5 collected only affects the behavior of subsequent electrons, but in NL-IPC (which acts on collected charge) every electron affects every other electron. 6 using observations of stars to the WFIRST Detector Working Group. While WFC3-IR is a useful guide to some of the issues WFIRST will encounter, it is an 18 µm pitch device and so it is important to measure BFE parameters on H4RG-10 HgCdTe devices. This paper is organized as follows. In §2, we build up our description of the brighter-fatter effect as well as other detector effects relevant to the flat field (IPC and non-linearity) and the formalism for correlation functions among the frames of a flat field. In §3, we work out the theoretical predictions for the 2-point correlation function of the flat fields in the presence of the various effects. In §4, we describe a simulation incorporating IPC, non-linearity, and the BFE that allows us to test the characterization methods in §5. We conclude in §6. The appendix contains some technical material on the covariances of clipped data (Appendix A). The application to laboratory H4RG-10 data -and associated evidence for the BFE -is presented in a companion paper (Paper II).
Formalism
Brighter-fatter effect
Autocorrelation measurements are sensitive to the BFE via changes in the effective pixel area. 7 We suppose that a pixel (i, j) has effective area that changes depending on the charge in neighboring pixels:
A i,j = A 0 i,j 1 + ∆i,∆j a ∆i,∆j Q(i + ∆i, j + ∆j) ,(1)
where i and j denote column and row indices (0...4095 for the H4RG), Q(i, j) is the charge (in number of elementary charges) in pixel (i, j), and a ∆i,∆j denotes a coupling matrix. In a general case, one might allow a to also depend on i and j, which would correspond to a BFE that varies from one pixel to another. However with flat autocorrelation data and a plausible number of flats, it is possible only to measure averages of the a coefficients in groups of pixels. Therefore we will assume discrete translation invariance; note, however, that the ability to perform autocorrelations with different sets of pixels allows some (limited) ability to test for translation invariance.
While a ∆i,∆j is formally dimensionless, we will normally quote a ∆i,∆j in units of 10 −6 e −1 , ppm/e, or %/10 4 e (all of which are equivalent). These units are convenient because 10 4 e is a typical integrated signal level in the central pixel of a PSF star for WFIRST, so a measured value of a in ppm/e maps into the expected order of magnitude of the effect on a star in percent.
In a phenomenological BFE model, one specifies how much of the area change comes from each 7 In the BFE literature, the operational definition of pixel "area" is that Ai,j = QE −1 ref pi,j(x, y) dx dy, where pi,j(x, y) is the probability that a photon incident at position (x, y) on the detector leads to an electron collected in pixel (i, j). This integral is divided by a reference value of the quantum efficiency, QE ref , so that at low flux levels the sum of pixel areas in some region corresponds to the geometrical area. If the probability of collecting a charge in any well i,j pi,j(x, y) remains fixed, then the BFE conserves total pixel area. of the boundaries by writing
a ∆i,∆j = a R ∆i,∆j + a T ∆i,∆j + a L ∆i,∆j + a B ∆i,∆j ,(2)
where the superscripts R, T , L, and B refer to the right, top, left, and bottom boundaries respectively. If the quantum efficiency depends on the charge in the well, then we would include an additional term a QE ∆i,∆j . An image simulation requires as input all of these components individually, and they can be probed with spot illumination or individual pixel resets; however flat autocorrelations are only sensitive to the total.
A true BFE that works by moving pixel boundaries conserves total area, so we should have ∆i,∆j a ∆i,∆j = 0.
(3)
(We expect that a 0,0 would be negative, and the a coefficients for the neighbors would be positive.) However, the sum in Eq. (3) is ill-behaved, since the noise diverges as we continue to add pixels. It can therefore be tested only in the context of fitting a model to a ∆i,∆j . Moreover, if adding charge to a pixel changes the QE or charge collection probability, then Eq. (3) may be violated. Therefore, it is important to measure all of the a ∆i,∆j , without assuming Eq. (3). In general we will define:
Σ a = ∆i,∆j a ∆i,∆j .(4)
We will find it useful later to define a ∆i,∆j ≡ a ∆i,∆j − δ ∆i,0 δ ∆j,0 Σ a .
By definition, the a coefficients sum to zero. Some of the BFE tests that we will conduct are not sensitive to Σ a , and hence can only measure the a ∆i,∆j .
Finally, we define symmetry-averaged functions over coefficients (∆i, ∆j) related by the rotation or reflection symmetries of the pixel grid:
a ∆i,∆j ≡ 1 8 a ∆i,∆j + a ∆j,∆i + a −∆j,∆i + a −∆i,∆j + a −∆i,−∆j + a −∆j,−∆i + a ∆j,−∆i + a ∆i,−∆j . (6)
There is no law of physics requiring the BFE to respect the rotation and reflection symmetries (indeed, in CCDs it does not), so some test results are provided for, e.g. a ∆i,∆j and a ∆j,∆i separately.
Gains, nonlinearities, and IPC
"Raw" data from the detector arrays are not in electrons but in data numbers (DN), which are voltages quantized as 16-bit integers. As each pixel is exposed to light, the voltage difference across the photodiode decreases; for detectors that collect holes, the voltage on the readout (p-type) side of the diode increases. The observed signal S (units: DN) may increase or decrease depending on the polarity of the analog-to-digital converter. 8 In this paper we work with the convention that S decreases during integration. Ideally the relation between the accumulated charge and signal drop would be linear, but in practice it is not. This effect can contain contributions both from the non-linearity of the p − n junction itself as well as any step in the readout chain, and is generically observed in NIR detectors (e.g. Bohlin et al. 2005;Deustua et al. 2010;Hill et al. 2010); see Plazas et al. (2016) for a study of its impact on the WFIRST weak lensing program.
A polynomial model is typically used to describe the non-linearity curve; the most important correction is typically the quadratic term. In flat illumination, where each pixel accumulates charge Q, the drop in signal level is given by
S initial − S final = 1 g Q − βQ 2 ,(7)
where g is the gain (units: e/DN) and β is the leading-order non-linearity coefficient. Note that β has the same units as a, and so it will be convenient to quote it in ppm/e. We define "initial" for the purposes of Eq. (7) to mean immediately following a reset, which we take to be t = 0. Note that non-linearity parameters will likely depend on the reset voltage.
In thick CCD detectors, it is common to use auto-correlations of the flat field to measure the BFE. However, in infrared detectors the auto-correlation of a flat is instead dominated by interpixel capacitance (IPC). IPC is an electrical coupling between neighboring pixels, in which the voltage on one pixel is sensitive to the charge in its neighbors (e.g. Moore et al. 2004Moore et al. , 2006. This coupling increases the apparent size of the image of a star on the detector; see for a study of its impact on WFIRST. This means a more complex procedure is needed to probe the BFE in infrared arrays. Furthermore, determination of the gain from variance vs. mean plots must be corrected for IPC to obtain meaningful results (e.g. Moore et al. 2006;Fox et al. 2008;Crouzet et al. 2012). Normally the IPC is described by replacing the linear term in Eq. (7) with a kernel describing the capacitive cross-talk among the pixels:
S initial (i, j) − S final (i, j) = 1 g ∆i,∆j K ∆i,∆j Q i−∆i,j−∆j + [nonlinear terms],(8)
where the kernel matrix K satisfies ∆i,∆j K ∆i,∆j = 1.
In the case where the IPC only talks to the nearest neighbors and does so equally, we have K 0,0 = 1 − 4α, K 0,±1 = K ±1,0 = α, and all others are zero. However, asymmetries between the horizontal and vertical directions (K 0,±1 = K ±1,0 ) are commonly observed in NIR detectors. Therefore we measure separately α H = K ±1,0 and α V = K 0,±1 ; if these are different then we define α to be their average (α H +α V )/2. We will also allow for diagonal IPC, α D = K ±1,±1 (when this notation is used, we will not distinguish between the "northeast-southwest" and "northwest-southeast" directions, although in principle their IPC may be different).
Inter-pixel capacitance in a semiconductor device may depend on signal since such devices do not obey the principle of superposition. This "non-linear inter-pixel capacitance" (NL-IPC) can be phenomenologically similar to the BFE: if the IPC grows with signal level, then this will also lead to brighter stars showing a larger observed FWHM on account of the greater amount of coupling. However, NL-IPC is a different mechanism -it arises in the conversion of charge to voltage, whereas the BFE arises in the collection of charge -and as such there are subtle differences in how it impacts both flat field statistics and science data. Disentangling the two effects proves to be one of the most difficult part of the BFE analysis.
Discussions of NL-IPC are complicated by the fact that NIR detectors both have non-linear charge-to-signal conversion (Eq. 7) and IPC. In the presence of both of these effects, we generically expect some kind of non-linear cross-talk between neighboring pixels of order αβ, and great care is needed to even define a quantitative measure of NL-IPC. In most astronomical data processing, the non-linearity correction is applied to individual pixels as one of the first steps -and certainly before any attempt at IPC correction (if the latter is done at all). This is equivalent to the assumption that all of the non-linearity acts on the signal after IPC. In this paper, we use "NL-IPC" to denote any non-linearity in the charge-to-signal conversion that deviates from this assumption.
For the purposes of flat fields, we parameterize NL-IPC by a mean signal-level-dependent kernel,
S initial (i, j) − S final (i, j) = 1 g ∆i,∆j [K ∆i,∆j + K ∆i,∆jQ ]Q i−∆i,j−∆j ,(10)
whereQ is the mean accumulated charge (It in a flat exposure). One can equivalently write this in terms of α , α H , α V , etc.:
S initial (i, j) − S final (i, j) = 1 g Q i,j + (α H + α HQ )(Q i+1,j − Q i,j ) + (α H + α HQ )(Q i−1,j − Q i,j ) +(α V + α VQ )(Q i,j+1 − Q i,j ) + (α V + α VQ )(Q i,j−1 − Q i,j ) .(11)
In the flat illumination case, there is no ambiguity of what mean accumulated chargeQ should be used. In other cases such as spot illumination or pixel reset tests, neighboring pixels can have wells filled to very different levels and K is no longer the appropriate concept. Some studies have indicated that NL-IPC is a function of both contrast and signal level -see, e.g., Donlon et al. (2016Donlon et al. ( , 2017Donlon et al. ( , 2018 -and in this case one should write an IPC coupling constant α(Q i,j , Q i+1,j ) that is a function of charge in both pixels. 9 The flat field test probes the case of Q i+1,j ≈ Q i,j ≈Q, whereas single pixel reset and hot pixel tests measure the case where Q i+1,j ≈ 0.
Correlation functions
In a CCD flat, there is only a single read of the detector in each flat exposure. However, in an infrared array flat, one typically obtains N samples up the ramp, and correlation functions can be defined not just between different pixels but between different frames. If one denotes the frames by indices abc... then let us define:
C abcd (∆i, ∆j) = Cov [S a (i, j) − S b (i, j), S c (i + ∆i, j + ∆j) − S d (i + ∆i, j + ∆j)] .(12)
If one obtained N = 2 samples and took the autocorrelation of the CDS image S 1 − S 2 , this would correspond to C 1212 (∆i, ∆j). This is the procedure that is most similar to a CCD flat autocorrelation. However, as noted above, it contains IPC as well as BFE and therefore cannot distinguish the two. Fortunately, with multiple up-the-ramp samples an infrared array flat is much richer in information than a CCD flat, and the temporal structure (abcd indices) is the key to disentangling the various effects.
In what follows, we will simplify some expressions by writing
S ab (i, j) ≡ S a (i, j) − S b (i, j). (Note the sign convention!)
The correlation functions satisfy the trivial properties:
• C abcd (∆i, ∆j) = 0 if a = b or c = d.
• C abcd (∆i, ∆j) = C cdab (−∆i, −∆j).
• C abcd (∆i, ∆j) = −C bacd (∆i, ∆j) = −C abdc (∆i, ∆j) = C badc (∆i, ∆j).
• C abcd (∆i, ∆j) = C aecd (∆i, ∆j)+C ebcd (∆i, ∆j) and C abcd (∆i, ∆j) = C abcf (∆i, ∆j)+C abf d (∆i, ∆j).
The last property means that all of the correlation functions can be composed of "elementary" correlation functions C a,a+1,c,c+1 (∆i, ∆j).
Equation (12), like all covariance matrices, requires more than one realization to make a measurement. The standard approach, followed here, is to compare a pair of two flats. The "measured covariance" of any two observables O and O is then
Cov meas O, O = 1 2 (O A − O B )(O A − O B ) ,(13)
where the average is taken over pixels (i, j) in the region of interest. The differencing removes small deviations such as imperfect illumination patterns, intrinsic variations in pixel area or QE, etc.
In a flat or dark field with many samples, we may also construct a correlation function averaged in the time direction. If we take the average of n time-translations of the time windows abcd, then we findC abcd[n] (∆i, ∆j) = n−1 ν=0 C a+ν,b+ν,c+ν,d+ν (∆i, ∆j).
Since a flat field is not time-stationary (gain, non-linearity, and possibly other quantities will change as the voltage across the p − n junction decreases), we must keep track of all the time indices abcdn when fitting a model to the time-translation-averaged correlation function.
While in this study we consider individual time samples, space-based infrared surveys are often data rate limited and thus not every sample can be downlinked. Therefore future work should also examine how the BFE and other effects appear in the cross-correlation functions of flat field data with compression along the time axis, e.g., using the first few Legendre coefficients (Rauscher et al. 2019). If compression by linear combinations is used (Legendre coefficients, group averaging, etc.), any such cross-correlation function can be written trivially as an appropriate weighted sum over the C abcd (∆i, ∆j).
Multiple exposures
When we discuss statistical algorithms, it will be essential to describe operations acting on multiple exposures. Here a specific exposure will be denoted with a | separator, followed by the exposure type and number. For example, we write S a (i, j|F k ) to denote the signal in time step a and pixel (i, j) in the k th flat field, and S a (i, j|D k ) for the k th dark exposure. This formalism could be extended in the future to include other types of tests (besides flats and darks).
Theory
We now embark on the main calculation in this paper: the determination of the correlation function C abcd (∆i, ∆j) including IPC, classical non-linearity, and the brighter-fatter effect to leading order. We will summarize the general result in Eq. (51). The measured correlation functions can then be used to simultaneously constrain the gain, IPC, and BFE parameters of a detector array or sub-region thereof -a step we will take on real data in Paper II. We will assume a < b and c < d, since these functions contain all the information because of symmetries, but we do not assume anything else about the ordering. In particular, the exposure intervals a...b and c...d may be the same, may overlap, or may be non-overlapping. The "same interval" case (a, b) = (c, d) will be the most familiar to readers who have worked with 2D image products before (e.g., CCD images, or CDS images from infrared arrays). However, the non-overlapping case turns out to be of particular use for measuring the sum of BFE and NL-IPC. It is the combination of these many different cases (itself possible due to the non-destructive read capability) that allows us to constrain so many parameters.
Rather than trying to solve everything at once, we begin this section by considering a "perfect" detector (no IPC, BFE, or any form of non-linearity; §3.1), and then adding layers of physical and mathematical complexity. In particular, we add linear inter-pixel capacitance, which simply introduces a convolution kernel ( §3.2). Next is the classical non-linearity ( §3.3), where covariances of higher-order moments of the charge appear; then we put the classical non-linearity together with the IPC ( §3.4). We introduce the brighter-fatter effect in §3.5; there the buildup of charge in each pixel Q(i, j; t) is a stochastic process with interactions between pixels, and we write and solve a differential equation for the moments of the charge. Non-linear IPC is briefly discussed and included in §3.6. The main result, Eq. (51), is presented in §3.7, and we discuss some special cases in §3.8. A reader not interested in the details of the derivation may skip directly to Eq. (51) and the simplifications in §3.8 (though one will still have to refer to the definitions for the quantities that summarize the time intervals: t abcd , T abcd , σ abcd , and τ abcd , defined in Eqs. 18, 25, 46, and 47, respectively).
In this calculation, we suppose that the flat illumination provides current I per pixel (units: e/s) and that the frame a is saved at time t a . We also assume that the flat field illumination uses a wavelength long enough for quantum yield effects to be insignificant (i.e., where one photon produces at most one electron-hole pair). At wavelengths blueward of the quantum yield threshold, it is possible for multiple carriers to be produced, and then (by diffusion) end up in separate wells, leading to an additional contribution to the flat field autocorrelation function (e.g., McCullough et al. 2008) as well as errors in gain determination.
Perfect detector
In a perfect detector, with α, β, and a all zero, each pixel operates independently. The mean charge accumulated in pixel (i, j) in frame a is
Q a (i, j) = It a(15)
and the covariance structure is
Cov [Q a (i, j), Q b (i + ∆i, j + ∆j)] = It min(a,b) δ ∆i,0 δ ∆j,0 .(16)Since the signal difference S ab (i, j) = g −1 [Q b (i, j) − Q a (i, j)]
, we then find a covariance structure:
C abcd (∆i, ∆j) = I g 2 t min(a,c) − t min(a,d) − t min(b,c) + t min(b,d) δ ∆i,0 δ ∆j,0 .(17)
It is convenient then to define
t abcd ≡ t min(a,c) − t min(a,d) − t min(b,c) + t min(b,d) and t ab ≡ t b − t a ;(18)
by inspection if a < b and c < d, then t abcd is the amount of time in the intersection of the intervals (t a , t b ) ∩ (t c , t d ). In this case, we also have t abab = t ab .
Equation (17) is behind the usual concept of obtaining a system gain from a variance vs. mean plot: we have C abab (0, 0) = I g 2 t ab and S ab (i, j) =
I g t ab .(19)
For a detector with no read noise, the ratio of variance to mean is then 1/g. In practice C abab (0, 0) contains a contribution from read noise, which can be removed by taking the slope of the variance vs. mean plot.
We now consider the various non-ideal detector effects. We begin by considering the effects one at a time, but we also need to consider interactions between the IPC and non-linearity, i.e. effects of order αβ and αa.
Inter-pixel capacitance
In the presence of IPC, the covariance structure of Eq. (17) is modified via smoothing by the IPC kernel. The IPC kernel acts locally in time, so we may write
C abcd (∆i, ∆j) = S ab (i, j)S cd (i + ∆i, j + ∆j) = i 1 ,j 1 ,i 2 ,j 2 K i 1 ,j 1 K i 2 ,j 2 S ab (i − i 1 , j − j 1 )S cd (i − i 2 + ∆i, j − j 2 + ∆j) = i 1 ,j 1 ,i 2 ,j 2 K i 1 ,j 1 K i 2 ,j 2 I g 2 t abcd δ (i−i 1 )−(i−i 2 +∆i),0 δ (j−j 1 )−(j−j 2 +∆j),0 = i 1 ,j 1 K i 1 ,j 1 K i 1 +∆i,j 1 +∆j I g 2 t abcd .(20)
If the IPC kernel is represented by nearest-neighbor parameters α H,V , and diagonal-neighbor parameters α D , then
C abcd (0, 0) = I g 2 t abcd [(1 − 4α − 4α D ) 2 + 2α 2 H + 2α 2 V + 4α 2 D ], C abcd (±1, 0) = I g 2 t abcd [2α H (1 − 4α − 4α D ) + 4α V α D ] , C abcd (0, ±1) = I g 2 t abcd [2α V (1 − 4α − 4α D ) + 4α H α D ] , and C abcd (±1, ±1) = C abcd (±1, ∓1) = I g 2 t abcd [2α H α V + 2α D (1 − 4α − 4α D )] .(21)
(There are other non-zero terms.) The nearest-neighbor correlations are thus useful for measuring α H and α V , and the diagonal-neighbor correlations for α D .
Note that regardless of K, the IPC-induced correction to C abcd (∆i, ∆j) remains proportional to t abcd . Therefore, if IPC is the only non-ideal effect in the detector, the correlation function will be zero if t abcd = 0. The "disjoint correlation functions" with t abcd = 0 are therefore diagnostics of other effects -including, as we shall see, the brighter-fatter effect.
Classical non-linearity
The classical non-linearity -that arising from the nonlinearity of the electrons to data numbers conversion, Eq. (7) -contributes a correction to the correlation function that involves the third moment of the Poisson distribution. In the presence of only classical non-linearity, but no IPC or BFE, the pixels still operate independently, so for simplicity we will consider only one pixel. The connected skewness 10 of charges at different times is
Q a (i, j)Q b (i, j)Q c (i, j) conn = It min(a,b,c) ,(22)
since the connected skewness of a Poisson distribution is its mean, and all counts received after t min(a,b,c) are independent of Q min (a,b,c) . This leads, after some algebra, to the ancillary result
Cov[Q a (i, j), Q b (i, j) 2 ] = 2I 2 t b t min(a,b) + It min(a,b) .(23)
Then we find (suppressing i and j indices to avoid clutter):
C abcd (0, 0) = 1 g 2 Cov Q b − βQ 2 b − Q a + βQ 2 a , Q d − βQ 2 d − Q c + βQ 2 c = 1 g 2 {Cov(Q a , Q c ) − Cov(Q a , Q d ) − Cov(Q b , Q c ) + Cov(Q b , Q d )} + β g 2 Cov(Q 2 a , Q d ) − Cov(Q 2 a , Q c ) − Cov(Q 2 b , Q d ) + Cov(Q 2 b , Q c ) + Cov(Q 2 c , Q b ) − Cov(Q 2 c , Q a ) − Cov(Q 2 d , Q b ) + Cov(Q 2 d , Q a ) = 1 g 2 It abcd − 2 β g 2 It abcd +2 βI 2 g 2 (t a + t d )t min(d,a) − (t b + t d )t min(d,b) − (t a + t c )t min(c,a) + (t b + t c )t min(c,b) .(24)
We will define
T abcd ≡ −(t a + t d )t min(d,a) + (t b + t d )t min(d,b) + (t a + t c )t min(c,a) − (t b + t c )t min(c,b)(25)
(units: s 2 ) so that
C abcd (0, 0) = 1 g 2 (1 − 2β)It abcd − 2 β g 2 I 2 T abcd .(26)
Here the "1 − 2β" correction term is of little interest, since the correction is tiny even compared to WFIRST requirements -indeed, it represents the nonlinearity generated by a single electron, and if β ∼ O(1) ppm/e, then this is a correction of order 10 −6 . The T abcd term can be much larger.
Note the following special cases of T abcd :
10 Connected skewnesses are defined by ABC conn = ∆A∆B∆C , where ∆A ≡ A − A , etc.
• If a ≤ b ≤ c ≤ d, then we have t abcd = 0 and T abcd = t ab t cd ≥ 0.
• If a ≤ c ≤ b ≤ d, then we have t abcd = t bc and T abcd = t ab t cd + (t b + t c )t bc ≥ 0.
• If a = c and b = d, then we have t abcd = t ab and T abcd = 2t b t ab .
Interdependence of IPC and non-linearity
Because the IPC corrections to flat results are often large (e.g. α = 1.25% leads to an ∼ 10% correction to the gain!) we need to consider the way in which IPC interacts with the non-linearity curve. This is particularly true given that IPC-non-linearity interactions affect both of the flat auto-correlation measurements of the BFE presented in this document. In particular, we want to capture the order αβ terms in the flat auto-correlation function.
IPC and non-linearity may interact in a complicated way because the two steps do not in general commute. The approach taken here is the mathematical point of view: one chooses an ordering, and any additional effects -including issues associated with order of operations -are packaged into "non-linear IPC" ( §3.6). The ordering we choose here is IPC first and then non-linearity (consistent with "standard" pipelines that treat non-linearity as the last step in the signal chain and thus the first correction implemented in data processing). From the physical point of view, the non-linear capacitance coming from the depletion region in the photodiode and the capacitive links between pixels should be thought of as a non-linear capacitor network that is solved simultaneously (the non-linearities coming from the rest of the signal chain and the analog-to-digital converter really do come later). Further consideration of this physical point of view is deferred to future work.
With these assumptions, the non-linearity interacts with the IPC according to
[S initial − S final ](i, j) = 1 g ∆i,∆j K ∆i,∆j Q(i + ∆i, j + ∆j) − β ∆i,∆j K ∆i,∆j Q(i + ∆i, j + ∆j) 2 ≈ 1 g (1 − 4α − 4α D )Q(i, j) + α H [Q(i + 1, j) + Q(i − 1, j)] +α V [Q(i, j + 1) + Q(i, j − 1)] +α D [Q(i + 1, j + 1) + Q(i + 1, j − 1) + Q(i − 1, j + 1) + Q(i − 1, j − 1)] −β(1 − 8α)Q 2 (i, j) − 2α H βQ(i, j)[Q(i + 1, j) + Q(i − 1, j)] −2α V βQ(i, j)[Q(i, j + 1) + Q(i, j − 1)] ,(27)
where the approximation includes terms of order α H,V β but not α D β or α 2 β.
Our principal interest is in the contributions of order αβ to the correlation function, which occur at either zero lag (∆i, ∆j) = (0, 0) or for nearest-neighbor pixels, (∆i, ∆j) ∈ N . In general, the contribution of order αβ to C abcd (∆i, ∆j) (denoted below as ∆C abcd (∆i, ∆j)| αβ ) has four parts: the covariance of the order αβ term in (i, j) with the order 1 term in (i + ∆i, j + ∆j) (which we will call the "αβ × 1" term); the α × β term; the β × α term; and the 1 × αβ term. These can be read off from Eq. (27), and covariances can be computed using the fact that (i) the charges in each pixel are independent, and (ii) the Poisson statistics needed are in Eqs. (15), (16), and (23); this is an algebraically lengthy but straightforward exercise. The result is
∆C abcd (0, 0)| αβ = αβ g 2 (16I 2 T abcd + 12It abcd )(28)
for zero lag,
∆C abcd (±1, 0)| αβ = − 4α H β g 2 (I 2 T abcd + It abcd )(29)
for the horizontal neighbors, and
∆C abcd (0, ±1)| αβ = − 4α V β g 2 (I 2 T abcd + It abcd ).(30)
for the vertical nearest neighbors. (The order α H β and α V β contributions beyond the 4 nearest neighbor pixels are zero.) Note that we normally have I 2 T abcd It abcd , so that term is dominant.
Brighter-fatter effect; moving pixel boundaries
The effect of the BFE on pixel correlation functions in infrared arrays can be treated by considering the charge Q(i, j; t) at time t in pixel (i, j) as a stochastic function. The new ingredient is that the charge deposited in each pixel between t and t + δt depends on the charge already present at time t. The BFE process is Markovian, in the sense that the state at t + δt depends on the state at time t, but has no further dependence on the state of the system at any earlier times. 11 This allows us to write the moments of the charge (here we consider the first two moments, the mean and covariance) at time t + δt in terms of those at time t; by working to order δt, we can construct a system of differential equations for the moments, which we solve starting from the initial condition at t = 0. This is analogous to what has been done for CCDs (see Coulton et al. 2018 for an implementation of a very similar method; see also Astier et al. 2019, who work directly with derivatives rather than very short but finite time steps). We extend this method to apply to unequal-time correlation functions by writing a similar set of differential equations for the covariance between a pixel at time t 1 and another pixel at time t ≥ t 1 , again with t as the independent variable. We can take the covariance at t 1 as an initial condition.
Let us define the area defect of a pixel at time t to be
W (i, j; t) ≡ 1 + ∆i,∆j a ∆i,∆j Q(i + ∆i, j + ∆j, t);(31)
this is close to 1, with deviations controlled by the BFE. Then -given the state of the system Q(i, j; t) at time t -we can find the mean charge in pixel (i, j) at time t + δt as
Q(i, j; t + δt) | t = Q(i, j; t) + IW (i, j; t)δt,(32)
where δt is taken to be small, and the subscript | t denotes that the state of the detector at time t is fixed. Here IW (i, j; t)δt is the probability that an electron is collected in pixel (i, j) between times t and t + δt; we assume Iδt 1 (one electron at a time), and will take the limit as δt → 0 so that this approximation becomes arbitrarily good. Since we are turning the result into a first-order differential equation for each moment, we can drop terms of order δt 2 and higher in what follows. The change in 2nd moment is
Q(i, j; t + δt)Q(i , j ; t + δt) | t = Q(i, j; t)Q(i , j ; t) + IW (i, j; t)Q(i , j ; t)δt +IW (i , j ; t)Q(i, j; t) δt + IW (i, j; t)δ ii δ jj δt,(33)
where we have expanded Q(i, j; t + ∆t) = Q(i, j; t) + ∆Q(i, j; t), and the four terms on the right hand side correspond to the expectation values of
Q(i, j; t)Q(i , j ; t), ∆Q(i, j; t)Q(i , j ; t), Q(i, j; t)∆Q(i , j ; t)
, and ∆Q(i, j; t)∆Q(i , j ; t) respectively. The last term is only non-zero if the two pixels are identical (δ ii δ jj = 1), since then a single electron can increment both Q(i, j) and Q(i , j ).
It is now possible to solve the above system of equations to first order in a. Let us first consider Eq. (32). Taking the average of the right-hand side over possible realizations at time t, we see that
Q(i, j; t + δt) = Q(i, j; t) + I δt + ∆i,∆j a ∆i,∆j Q(i + ∆i, j + ∆j, t) I δt.(34)
Recalling that Σ a = ∆i,∆j a ∆i,∆j , and using translation invariance to show that the Q(i, j; t) all have the same expectation value, we see that
Q(i, j; t + δt) = Q(i, j; t) + I δt + Σ a Q(i, j; t) I δt.(35)
This becomes a differential equation for Q(i, j; t) :
d dt Q(i, j; t) = I(1 + Σ a Q(i, j; t) ),(36)
with solution starting from Q(i, j; t) = 0 at t = 0:
Q(i, j; t) = e IΣat − 1 Σ a ≈ It + 1 2 Σ a I 2 t 2 .(37)
(The approximation holds to first order in the a coefficients.)
The next step is to solve for the covariance matrix. This proceeds in two steps. First, one tracks the full second moment from time 0 to some later time t 1 . Then one tracks a conditional second moment to a later time t 2 ≥ t 1 . The mean of Eq. (33) is
Q(i, j; t + ∆t)Q(i , j ; t + ∆t) = Q(i, j; t)Q(i , j ; t) + I Q(i , j ; t) ∆t + I Q(i, j; t) ∆t +Iδ ii δ jj ∆t + I ∆i,∆j a ∆i,∆j Q(i + ∆i, j + ∆j; t)Q(i , j ; t) + Q(i + ∆i, j + ∆j; t)Q(i, j; t) +δ ii δ jj Q(i + ∆i, j + ∆j; t) ∆t.(38)
This can be turned into a differential equation. The first moment solution from Eq. (37) can be substituted in, and all second order terms in a dropped:
d dt Q(i, j; t)Q(i , j ; t) = 2I 2 t + Σ a I 3 t 2 + Iδ ii δ jj + I 2 Σ a tδ ii δ jj +I ∆i,∆j a ∆i,∆j Q(i + ∆i, j + ∆j; t)Q(i , j ; t) + Q(i + ∆i, j + ∆j; t)Q(i, j; t) .(39)
The initial condition is that Q(i, j; t)Q(i , j ; t) = 0 at t = 0. To solve Eq. (39) to first order in a, we use the standard method of first solving the equation at a = 0 (the zeroth order solution), then substituting this into any term multiplying a (or Σ a ) and solving again. This gives Q(i, j; t)Q(i , j ; t) = I 2 t 2 +Σ a I 3 t 3 +Itδ ii δ jj + 1 2
I 2 Σ a t 2 δ ii δ jj + 1 2 (a i−i ,j−j +a i −i,j −j )I 2 t 2 .(40)
In our case, however, we need not just equal-time but also unequal-time correlation functions of the charge. This means we need to propagate the second moment at time t 1 to the covariance between times t 1 and t 2 (without loss of generality, t 2 > t 1 ). Since the system is Markovian, if t 1 ≤ t we can take the expectation value in Eq. (32) to be conditioned not only on the state of the system at t but also at t 1 . Then we can multiply Eq. (32) by Q(i , j ; t 1 ). Since Q(i , j ; t 1 ) is fully determined by the state of the system at t 1 , we can pull it inside the expectation value:
Q(i , j ; t 1 )Q(i, j; t + δt) | t 1 ,t = Q(i , j ; t 1 )Q(i, j; t) + IW (i, j; t)Q(i , j ; t 1 )δt.(41)
We next average this over states of the system at t 1 and t (i.e., we remove the condition, but then get an expectation value on the right-hand side). Turning the result into a differential equation, we have d dt Q(i, j; t)Q(i , j ; t 1 ) = I W (i, j; t)Q(i , j ; t 1 )
= I Q(i , j ; t 1 ) + I ∆i,∆j a ∆i,∆j Q(i + ∆i, j + ∆j; t)Q(i , j ; t 1 ) .
with initial condition from Eq. (40), Q(i, j; t 1 )Q(i , j ; t 1 ) | a=0 = I 2 t 2 1 + It 1 δ ii δ jj . The solution to first order in a is
Q(i, j; t)Q(i , j ; t 1 ) = I 2 t 1 t + 1 2 Σ a I 3 t 1 t(t + t 1 ) + It 1 + 1 2 I 2 Σ a t 2 1 δ ii δ jj + 1 2 (a i−i ,j−j + a i −i,j −j )I 2 t 2 1 + a i −i,j −j I 2 t 1 (t − t 1 ).(43)
Subtracting out Q(i, j; t) Q(i , j ; t 1 ) from Eq. (37) gives the covariance matrix:
Cov Q(i, j; t), Q(i , j ; t 1 ) = It 1 + 1 2 I 2 Σ a t 2 1 δ ii δ jj + 1 2 (a i−i ,j−j + a i −i,j −j )I 2 t 2 1 + a i −i,j −j I 2 t 1 (t − t 1 ). (44)
Recall that this is for t ≥ t 1 ; for t < t 1 , one can use the symmetry of the covariance matrix to obtain the result. If one considers only the linear response of the detector, this maps directly into the flat autocorrelation function:
C abcd (∆i, ∆j) = 1 g 2 Cov [Q(i, j; t a ), Q(i + ∆i, j + ∆j; t c )] − Cov [Q(i, j; t a ), Q(i + ∆i, j + ∆j; t d )] −Cov [Q(i, j; t b ), Q(i + ∆i, j + ∆j; t c )] + Cov [Q(i, j; t b ), Q(i + ∆i, j + ∆j; t d )] = 1 g 2 It abcd + 1 2 I 2 Σ a σ abcd δ ∆i,0 δ ∆j,0 + 1 2 (a ∆i,∆j + a −∆i,−∆j )I 2 t ab t cd − 1 2 (a ∆i,∆j − a −∆i,−∆j )I 2 τ abcd ,(45)
where we define the auxiliary quantities
σ abcd = t 2 min(a,c) − t 2 min(a,d) − t 2 min(b,c) + t 2 min(b,d)(46)
and τ abcd = t ac t min(a,c) − t ad t min(a,d) − t bc t min(b,c) + t bd t min(b,d) .
Here σ abcd and τ abcd have units of s 2 and satisfy the following rules:
• σ abcd = σ cdab and τ abcd = −τ cdab .
• If a ≤ b ≤ c ≤ d, then σ abcd = 0 and τ abcd = t ab t cd ≥ 0.
• If a = c ≤ b = d, then σ abcd = t ab (t a + t b ) and τ abcd = 0.
Note that in Eq. (45), τ abcd describes the response of a correlation function to the odd part of a while the response to the even part is described by t ab t cd . The response to the summed effect Σ a is encoded in σ abcd .
In the presence of IPC, the BFE contribution to the correlation function should be convolved twice with the IPC kernel:
C BFE with IPC abcd (∆i, ∆j) = i 1 ,j 1 ,i 2 ,j 2 K i 1 ,j 1 K i 2 ,j 2 C BFE without IPC abcd (∆i + i 1 + i 2 , ∆j + j 1 + j 2 ).(48)
Non-linear inter-pixel capacitance (NL-IPC)
The contribution of NL-IPC to the covariance of signals is, to order K ,
Cov[S a (i, j), S c (i + ∆i, j + ∆j)]| K = 1 g 2 i ,j K i−i ,j−j K i+∆i−i ,j+∆j−j Q a Cov[Q a (i , j ), Q c (i , j )] i ,j K i+∆i−i ,j+∆j−j K i−i ,j−j Q c Cov[Q a (i , j ), Q c (i , j )] = 1 g 2 [KK ] ∆i,∆j I 2 (t a + t c )t min(a,c) ,(49)
where in the first expression the first term comes from the order K contribution to S a (i, j) and the second term from the contribution to S c (i + ∆i, j + ∆j). The final expression used the symmetry of K ∆i,∆j under (∆i, ∆j) → (−∆i, −∆j), and has defined the convolution [KK ] ∆i,∆j = i 1 ,j 1 K i 1 ,j 1 K ∆i−i 1 ,∆j−j 1 . The contribution to the correlation function is
C abcd (∆i, ∆j)| K = 1 g 2 [KK ] ∆i,∆j I 2 T abcd .(50)
Combined correlation function
Putting together all of the combinations -IPC, non-linearity, BFE, NL-IPC, and the leading order interactions -we have the following expression, including corrections of order α, α 2 , β, αβ, a, αa, α , and αα :
C abcd (∆i, ∆j) = 1 g 2 It abcd + 1 2 I 2 Σ a σ abcd [K 2 ] ∆i,∆j + 1 2 ([K 2 a] ∆i,∆j + [K 2 a] −∆i,−∆j )I 2 t ab t cd − 1 2 ([K 2 a] ∆i,∆j − [K 2 a] −∆i,−∆j )I 2 τ abcd − 2β(I 2 T abcd + It abcd )δ ∆i,0 δ ∆j,0 +αβ(16I 2 T abcd + 12It abcd )δ ∆i,0 δ ∆j,0 − 4α H β(I 2 T abcd + It abcd )δ |∆i|,1 δ ∆j,0 −4α V β(I 2 T abcd + It abcd )δ ∆i,0 δ |∆j|,1 + [KK ] ∆i,∆j I 2 T abcd ,(51)
where we have defined [K 2 ] ∆i,∆j to be the auto-convolution of K, and [K 2 a] ∆i,∆j to be the convolution of K 2 and a.
Special cases used in detector characterization
We now turn our focus to the special cases that are used in detector characterization. We consider two special cases of the correlation function: the equal-interval correlation functions (a = c < b = d -most similar to the auto-correlation that one would obtain from a CCD) and the nonoverlapping correlation functions (a < b < c < d -which exhibits new features only accessible with a non-destructive read capability). We also consider the mean-variance plot, which is a common diagnostic of the gain of a detector system, with a particular emphasis on how IPC, nonlinearity, and BFE affect the gain measurement.
In what follows, terms of order α, α 2 , β, a, αβ, and αa are kept. Higher terms in the non-ideal detector effects are dropped.
Equal-interval correlation function
The case of a = c < b = d corresponds to the auto-correlation of a single difference image S a − S b . It is therefore most comparable to what one would obtain with a CCD. The contributions at zero lag sum to:
C abab (0, 0) = I g 2 t ab (1 − 4α − 4α D ) 2 + 2(α 2 H + α 2 V ) + 4α 2 D − 4(1 − 8α)βIt b − 2(1 − 6α)β +[K 2 a] 0,0 It ab + 1 2 (1 − 8α)Σ a I(t a + t b ) + 2[KK ] 0,0 It b ,(52)
while the horizontal nearest neighbors are
C abab (±1, 0) = I g 2 t ab 2α H (1 − 4α − 4α D ) + 4α V α D − 8α H β It b + 1 2 + α H Σ a I(t a + t b ) +[K 2 a] H It ab + 2[KK ] 1,0 It b ,(53)
where we define a H = (a 1,0 + a −1,0 )/2. A similar equation holds for the vertical nearest neighbors. For the diagonal neighbors, we have
C abab ( 1, 1 ) = I g 2 t ab 2α D (1 − 4α − 4α D ) + 4α H α V + [K 2 a] 1,1 It ab + 2[KK ] 1,1 It b .(54)
The equal-interval correlation function, especially but not exclusively at zero lag, contains a large contribution from read noise (from various sources), and this must be removed before interpreting it.
The time-translation-averaged versions of Eqs. (52-54) can be evaluated with straightforward algebra; they arē
C abab[n] (0, 0) = I g 2 t ab (1 − 4α − 4α D ) 2 + 2(α 2 H + α 2 V ) + 4α 2 D − 4(1 − 8α)βI t b + n − 1 2 ∆t −2(1 − 6α)β + [K 2 a] 0,0 It ab + 1 2 (1 − 8α)Σ a I[t a + t b + (n − 1)∆t] +2[KK ] 0,0 I t b + n − 1 2 ∆t(55)
for zero lag;
C abab[n] (±1, 0) = I g 2 t ab 2α H (1 − 4α − 4α D ) + 4α V α D − 8α H β It b + n − 1 2 I∆t + 1 2 +α H Σ a I[t a + t b + (n − 1)∆t] + [K 2 a] H It ab + 2[KK ] 1,0 I t b + n − 1 2 ∆t(56)
for the nearest neighbor; and
C abab[n] ( 1, 1 ) = I g 2 t ab 2α D (1 − 4α − 4α D ) + [K 2 a] 1,1 It ab + 2[KK ] 1,1 I t b + n − 1 2 ∆t(57)
for the diagonal neighbor.
Non-overlapping correlation function
The case of a < b < c < d is special, because then σ abcd = t abcd = 0 and the correlation function -including contributions of IPC, classical non-linearity, and BFE -simplifies to
C abcd (∆i, ∆j)| a<b<c<d = I 2 t ab t cd g 2 [K 2 a] −∆i,−∆j + [KK ] ∆i,∆j − 2(1 − 8α)βδ ∆i,0 δ ∆j,0 −4α H βδ |∆i|,1 δ ∆j,0 − 4α V βδ ∆i,0 δ |∆j|,1 .(58)
That is, the non-overlapping correlation function is directly sensitive to the coefficients a ∆i,∆j , has no sensitivity to linear IPC at order α, and only has sensitivity to the classical non-linearity β at zero lag. At order αβ, there is a contribution in the nearest neighbors. There is a trivial mapping from the pixel-space lag in the correlation function (∆i, ∆j) to the lag in the BFE kernel a ∆i,∆j . Thus this should be a "clean" measurement of the inter-pixel non-linear effects (BFE and NL-IPC), insensitive to small errors in the determination of I and g. Any source of noise that is uncorrelated across frames is also removed. The reset (kT C) noise is also removed, since the correlation function is constructed from correlated double samples. The main drawback is that the method is only sensitive to a combination of BFE and NL-IPC, and cannot distinguish between the two mechanisms.
The one large correction that is necessary is that a 0,0 must be corrected for the classical non-linearity (which is a larger effect than the BFE). Therefore we need to measure β from the non-linearity (t 2 term) of the signal vs. time plot of the flat. Interestingly, this is sensitive to the combination β − 1 2 Σ a . It follows that the flat non-linearity and the non-overlapping correlation functions contain an intrinsic degeneracy where β and a 0,0 are both changed but holding the combination β − 1 2 a 0,0 constant. Other correlation functions are needed to break this degeneracy. A secondary correction is that C abcd (∆i, ∆j) in Eq. (58) is the correlation function of the signal, but the measurement contains signal+noise. Therefore any noise that is correlated across frames must be characterized and removed from C abcd (∆i, ∆j).
Mean-variance slope
A common method to estimate the gain of a system is to determine the ratio of the mean signal in a pair of matched flats to the variance. In practice, since the measured variance contains read noise, one measures the slope of the variance as a function of the mean, e.g.:
g raw abcd ≡ M cd − M ab V cd − V ab ,(59)
where M ab = S a (i, j) − S b (i, j) and V ab = C abab (0, 0) is the variance of a difference frame. This construction only makes sense for (a, b) = (c, d) (a common case is a = c < b < d). The mean is
M ab = It ab g 1 − β − Σ a 2 I(t a + t b ) .(60)
We obtain the variance from Eq. (52). The mean-variance slope is related to the gain bŷ
g raw abcd = g (1 − 4α − 4α D ) 2 + 2(α 2 H + α 2 V ) + 4α 2 D 1 + 2βI t cd t d − t ab t b t cd − t ab + β + (1 + 8α)[K 2 a] 0,0 I(t cd + t ab ) + 2(1 + 2α)β +2(1 + 8α)[KK ] 0,0 I t cd t d − t ab t b t cd − t ab .(61)
In the special case of a = c < b < d (which will be used herein), we find
g raw abad = g (1 − 4α − 4α D ) 2 + 2(α 2 H + α 2 V ) + 4α 2 D 1 + 2β − 8(1 + 3α)α It a + 3β − (1 + 8α)[K 2 a] 0,0 + 8(1 + 3α)α I(t ad + t ab ) + 2(1 + 2α)β .(62)
Here, we have used that to order αα ,
(1 + 8α)[KK ] 0,0 ≈ −4(1 + 4α)α + 4αα = −4(1 + 3α)α .
In Eq. (62), the pre-factor is the traditional IPC correction to the gain. Following this is a non-linear correction term that depends on the "start time" t a of the measurement. Then comes a second non-linear correction term that depends on the "duration" t ad + t ab of the measurement. Both are proportional to β (or to the a ∆i,∆j ); they have the same dependence in the special case of t a = t c = 0. The last term is formally of order β, but is smaller than the previous two correction terms as it does not contain a factor of accumulated charge It.
Flat field simulations
We construct simulations for validation and interpretation. This simulated data set contains flats and darks that are designed to resemble the real data cubes from the Detector Characterization Laboratory (DCL) at NASA Goddard Space Flight Center, with an implementation of the key effects described in the earlier sections of this paper. The procedure consists of three main steps, visualized in a flowchart in Fig. 1: first, user inputs and specifications are read from a configuration file (top); second, charge is accumulated via draws from a Poisson distribution and modified by a BFE kernel, if the BFE is turned on (loop on lower left); third, all other effects including linear IPC, classical non-linearity, and noise are applied to each time step of the charge array, which is ultimately converted to a signal and stored in an output fits data cube (lower right). The remainder of this section delves into the specifics of how the simulated flat fields are constructed.
Details of simulation procedure
The first part of the script sets up the simulation that will be created by ingesting a configuration file and using defaults when selections are not specified by the user. The default settings create a datacube with dimensions of 4096 2 pixels 2 with 66 time samples with the bounding 4 rows and columns designated as reference pixels. Substeps set the total number of time slices at which the charge is computed between the stored time slices with the default set to substep=2 (for this default the computation is done for 2 × 66 = 112 time steps). This setting exists to ensure convergence when the BFE mode is turned on. The user specifies quantities like gain g, current per pixel per second I, length of time sample in seconds, quantum efficiency QE, and IPC α. Reset frames and reset levels can also be set in the config file.
After reading in user specifications and initializing arrays, charge is drawn and accumulated over the total time steps. For the initial time frame, a random realization of charge is drawn from a Poisson distribution with a mean of QE × I × δt, with δt being the time between each time step. If the BFE is turned on, a matrix of pixel area defects W (i, j; t) given by Eq. 31 is calculated by convolving a user-specified input kernel a ∆i∆j with the charge distribution over the pixel grid at the given time t. Subsequent time frames compound the previous time frames with charge drawn from a Poisson distribution with the mean modified by the pixel area defects, i.e. charge is drawn from a Poisson distribution with a mean of W × QE × I × δt. If the BFE is turned off, all time frames are accumulated with fully random realizations of charge.
After the charge has accumulated over all time frames, a linear IPC can be applied by convolving the full charge data cube with an IPC kernel. Note that from this stage onward, operations are only performed on the time samples that will be saved (i.e. in the default settings, IPC is applied to the 66 time samples and not the intermediate substeps). Non-linearity β can also be applied after the IPC, where Q(i, j; t) → Q(i, j; t) − β[Q(i, j; t)] 2 .
We create realizations of noise datacubes using nghxrg 12 , the HxRG Noise Generator written in Python by Bernard Rauscher (Rauscher 2015). This software produces white read noise, pedestal drifts, correlated and uncorrelated pink noise, alternating column noise, and picture frame noise and was based on a principle components analysis of the James Webb Space Telescope NIRSpec detector subsystem. Here, we use input parameters tuned to WFIRST configurations.
The final step is to convert the charge into DN by dividing by g and save in an array of unsigned 16-bit integers. The output datacube is saved in fits format with a header containing information about the configuration settings and parameter values used to run the simulation.
The main flat field generation is done as part of the solid-waffle pipeline, which is further described in Section 5; the noise file must be generated separately using the nghxrg package.
Test bed of simulated flats and darks
We created a set of 10 simulated flat fields and 10 simulated darks to test the characterization and BFE measurement framework presented in this paper. The input parameters were chosen to resemble the real detector data analyzed in Paper II and are summarized in the 'truth' column of Table 1. All simulated data cubes are ascending ramps where the signal level in DN increases as time increases and have NAXIS1=NAXIS2=4096 (spatial dimensions) and NAXIS3=66 (readout time frames). The time between each time frame is 2.75 s, g = 2.06, QE=0.95, and β = 0.58. The IPC kernel is as described in Eq. 9 so that K 0,0 = 1 − 4α, K 0,±1 = K ±1,0 = α, and all others are zero, with α = 0.0169. The BFE is turned on, and has a zero-lag component a 0,0 = −1.372 ppm/e. We conservatively set substep=20 to ensure convergence. Table 1 provides symmetrized mean values of [K 2 a] ∆i∆j , the convolution of K 2 (the auto-convolution of K) and a ∆i∆j . The simulated flats each have illumination I = 559 e/s/pixel, while the simulated darks have illumination I = 0.191 e/s/pixel, which was chosen so that the resulting slope of the signal vs time matched a typical real dark. Random seeds from 1001-1010 and 2001-2010 were set for the flats and darks, respectively.
Finally, we generated 10 noise data cubes using nghxrg with NAXIS dimensions as specified above, n out=32 (number of detector outputs), nroh=8 (row overhead in pixels), rd noise=4 (standard deviation of white read noise in e), pedestal=4 (pedestal drift in e), c pink=3 (standard deviation of correlated pink noise), u pink=1 (standard deviation of uncorrelated pink noise), c ACN=1 (standard deviation of alternating column noise). For simplicity, we did not add any picture frame noise, and we set a bias offset of 19222 e to match a typical real dark. See Rauscher (2015) for further details of the noise recipe. Each of the 10 noise data cubes are combined with a flat and dark each so that each flat/dark pair have a noise realization in common; this simplification should not affect the results of this analysis in any significant way. These simulations were run on Pitzer (Ohio Supercomputer Center 2018), a supercomputer at the Ohio Supercomputing Center (Ohio Supercomputer Center 1987).
Characterization based on flat fields
We now turn to the practical problem at hand: extracting the calibration parameters (g, α, β, a ∆i,∆j , etc.) from a suite of flat field and dark exposures. We first provide an overview of our characterization pipeline (known as solid-waffle), and then describe in detail the modules therein. The tools are written in Python 2, with data stored in numpy arrays. Due to the large file size associated with flat fields using multiple up-the-ramp samples (2.2 GB per file for a 66-frame H4RG flat), the full data set is not stored in RAM; instead the fitsio package was used to enable rapid access to small subsets of the data from disk without reading the entire file. 13
Our analysis takes as input N flat fields and N dark images, where N ≥ 2. The SCA is broken into a grid of N x × N y "super-pixels," each of size ∆ x × ∆ y physical pixels. Statistical properties such as medians, variances, and correlation functions are understood to be computed in each superpixel. Note that N x ∆ x = N y ∆ y = 4096 for an H4RG (and 2048 for an H2RG). Super-pixels may be made larger to improve S/N, but this implies more averaging over the SCA so localized features and patterns may be washed out (we will see examples of this in Paper II).
Each super-pixel is processed through "basic" characterization. Following this, it passes through inter-pixel non-linearity (IPNL) determination using the non-overlapping correlation function, and then (optionally) through advanced characterization and other tests. We now describe these steps.
Basic characterization
The basic characterization step for a super-pixel is a prerequisite to studying all of the more subtle effects in the NIR detectors. It uses four time frames t a , t b , t c , and t d , and it does not take into account the diagonal IPC, the brighter-fatter effect, non-linear IPC, or signal-dependent QE.
We first construct the CDS images S ab (i, j|F k ) and S ad (i, j|F k ) within the range of column i and row j in the super-pixel, for each flat F k . We build a median (over flats k) image f (i, j) = med N k=1 S ad (i, j|F k ), and then a pixel mask based on requiring f (i, j) to be within 10% of its median (this time taken over i, j). This rejects disconnected or low-response pixels. Our next step is to perform a reference pixel subtraction. The procedure used here was obtained after some experimentation with DCL data, and is the default in our code, but may require some readjustment for other setups. We first find the range of rows j min ...j max = j min +∆ y −1 corresponding to the super-pixel, and find the two 4 × ∆ y blocks of reference pixels on the left and right sides of the SCA. For each flat exposure F k , and for each of our two CDS difference images (S ab and S ad ), we find the median of these 8∆ y pixels, and subtract this from the entire super-pixel. A similar procedure is applied to the dark images D k . Note that this procedure only adds or subtracts a constant in the super-pixel, and does not correct each individual row. 14 We next want to compute the raw gain,ĝ raw abad . To do this, we need to compute the mean signal levels M ab and M ad . The current default is to take the reference-corrected image S ab (i, j|F k ), and compute a mean in k followed by a median in (i, j). The variance V ab is obtained by taking each of the N (N − 1)/2 pairs of flats (k, ), with 1 ≤ k < ≤ N . For each pair, we compute the difference S ab (i, j|F k ) − S ab (i, j|F ), and compute the inter-quartile range (IQR) of the ∆ x ∆ y pixels. 15 The variance is estimated as (IQR/1.349) 2 /2, as appropriate for a Gaussian (but note that the IQR estimator is robust against outliers, unlike the standard variance estimator), and with a factor of 2 to account for the fact that the flat difference has noise from both flats. The V ab used in the raw gain estimator is the average of the N (N − 1)/2 estimates obtained from the various flat pairs. These means and variances are then plugged into Eq. (59).
Inter-pixel capacitance is addressed through the flat field auto-correlation method, which we implement as follows. For each of the N (N −1)/2 flat pairs, we construct a difference T (i, j|F k , F ) = S ad (i, j|F k ) − S ad (i, j|F ). We clip the top 100 % and bottom 100 % of the T (i, j|F k , F ) map, leaving 100(1 − 2 )% of the pixels unmasked. Then we define a horizontal correlation (64) where the average is over pixels where both that pixel (i, j) and its horizontal neighbor (i + 1, j) are unmasked. We then compute an averaged horizontal correlation
C H (|F k , F ) = 1 # pix (i, j) (i,j) [T (i, j|F k , F ) −T (|F k , F )][T (i + 1, j|F k , F ) −T (|F k , F )] ,C H = 1 2 × 1 f corr × 1 N (N − 1)/2 1≤k< ≤N [C H (|F k , F ) − C H (|D k , D )] .(65)
Here we have subtracted the correlation from a pair of dark frames (to remove the contribution of correlated read noise), and averaged over the flat pairs. The factor of 1 2 takes into account the fact that by subtracting two flats, we have doubled the correlation function. Finally, the factor of f corr takes into account the suppression of correlations by the histogram clipping of T . It depends on ; for our default choice of = 0.01, we have f corr = 0.7629. See Appendix A for a derivation of f corr . A similar calculation is used to obtain the vertical correlation function C V and the diagonal correlation function C D .
14 Correcting each row would print noise from the reference pixels as additional horizontal correlations. There are row-dependent drifts in the electronics, however we found that these are better eliminated at the correlation function level by either subtracting the correlation function in the darks or by the "baseline subtraction" method described in §5.2. 15 We use difference images because they are robust against permanent structure in the flat fields, e.g. variations in pixel area or quantum efficiency.
Finally, we need a measure of ramp curvature. We construct the difference box
R(i, j|F k ) = S cd (i, j|F k ) − t cd t ab S ab (i, j|F k )(66)
and perform the usual reference pixel subtraction (based on all 8∆ y "left+right" reference pixels in the same range of rows as the super-pixel). We clip the pixels corresponding to the top and bottom 100 % of the histogram of S ab (i, j|F k ) and of R(i, j|F k ), and then compute
frac dslope = N k=1 R(i, j|F k )/t cd N k=1 S ab (i, j|F k )/t ab .(67)
Note that 1 + frac dslope is the ratio of the slope of the signal (in DN/frame) in the cd interval relative to the ab interval. For a perfectly linear detector, frac dslope should be zero. For a non-linear detector, the mean signal is S a (i, j) = [It a − β r (It a ) 2 ]/g, where
β r = β − 1 2 Σ a(68)
is the ramp curvature (here Σ a denotes the signal-dependent QE, and enters via Eq. 37). Via straightforward algebra, we can see that the slope difference ratio frac dslope is expected to be
−βI(t c + t d − t a − t b ).
We may now construct an IPC + non-linearity corrected (αβ-corrected) gain g, estimated current per pixel I, horizontal and vertical IPC α H and α V , and ramp curvature β r by iteratively solving the system of equations:
g raw abad = g 1 + β r I(3t b + 3t d − 4t a ) (1 − 2α H − 2α V ) 2 + 2α 2 H + 2α 2 V ; C H = 2It ad α H g 2 (1 − 2α H − 2α V − 4β r It d ); C V = 2It ad α V g 2 (1 − 2α H − 2α V − 4β r It d ); M ad = It ad g [1 − β r I(t a + t d )]; and frac dslope = −β r I(t c + t d − t a − t b ).(69)
This is 5 equations for 5 unknowns; note that the difference between β and β r (i.e., the signaldependent QE term Σ a ) has been neglected in the gain and IPC determination. Initializing the system with g =ĝ raw abad , α H = α V = β = 0, I = gM ad /t ad , and solving the above equations in turn for g, α H , α V , I, and β leads to rapid convergence.
The resulting parameters g, α H , α V , I, and β contain small residual biases due to the BFE, nonlinear IPC, and signal-dependent QE if these phenomena are present. These will be explored in more detail in Paper II.
IPNL determination via the non-overlapping correlation function
With the basic parameters in each super-pixel measured, we may now measure the nonoverlapping correlation function, C abcd (∆i, ∆j) for a < b < c < d. This is almost a direct test for the presence of inter-pixel non-linearities (BFE and NL-IPC), since it contains no contribution from linear IPC, and only small corrections for classical non-linearity (β) are required. In particular, at zero lag, Eq. (58) can be rearranged to give
[K 2 a] 0,0 + [KK ] 0,0 = g 2 I 2 t ab t cd C abcd (0, 0) + 2(1 − 8α)β.(70)
The ramp curvature does not yield an estimate directly for β, but rather the combination β ramp = β − 1 2 Σ a . We also recall that to order αa, we have [K 2 a ] 0,0 = [K 2 a] 0,0 − (1 − 8α)Σ a . We can thus write, to O(αa):
[K 2 a ] 0,0 + [KK ] 0,0 = g 2 I 2 t ab t cd C abcd (0, 0) + 2(1 − 8α)β r .(71)
Similarly, one may compute the adjacent pixel correlation functions:
[K 2 a ] ±1,0 + [KK ] ±1,0 = g 2 I 2 t ab t cd C abcd (∓1, 0) + 4α H β r ,(72)
and similarly for the vertical directions.
Equations (71) and (72) show that the non-overlapping correlation function method, as we have implemented it, is sensitive to the [K 2 a ] ∆i,∆j + [KK ] ∆i,∆j coefficients. Note that the BFE and NL-IPC appear together, both with t ab t cd time dependence, and the non-overlapping correlation function method provides no way to separate them. This method has only a small correction on the right-hand side due to the ramp curvature β r , so this method of IPNL determination is not subject to spurious detection due to small errors in the basic parameters (g, I, and α). In most practical situations, we will find that the 4α H β r correction is smaller than the IPNL, and the 2(1 − 8α)β r correction is similar to the IPNL (see Paper II for quantitative details on a WFIRST development detector).
Our pipeline provides results out to a separation of 2 pixels in either the horizontal or vertical directions, i.e., it reports a 5 × 5 kernel [K 2 a + KK ].
We now turn to the implementation details of C abcd (∆i, ∆j) in the pipeline itself. The correlation function can be determined by the same methods used to compute C H and C V . However, we found in initial studies on DCL data that the measurements showed statistically significant deviations depending on which flat was used, which are suspected to be low frequency noise in the data (see horizontal stripes in the dark image and discussion in Paper II). Therefore, the default setting in our pipeline is to filter out the low frequencies via a baseline correction: instead of using the raw correlation function,
C raw abcd (∆i, ∆j) = 1 N pair i,j [S a − S b ](i, j) − S a − S b [S c − S d ](i + ∆i, j + ∆j) − S c − S d(73)
(where the overbar denotes an average and N pair is the number of pixel pairs in the sum), we find a "baseline" contribution:
C baseline abcd (∆j) = 1 N pair i,j,∆i [S a − S b ](i, j) − S a − S b [S c − S d ](i + ∆i , j + ∆j) − S c − S d ,(74)
where the pair summation runs over 6 ≤ |∆i | ≤ 10, and again N pair is the number of pixel pairs in the sum. That is, the baseline is the correlation function obtained by replacing pixel (i+∆i, j +∆j) with the average of pixels in the same row but 6-10 pixels left or right (ahead or behind in the readout sequence). Both the leading and trailing regions are used with equal weight, except that (i) the standard 1% outlier rejection is used before taking the covariance, and (ii) the implementation in the code rejects one of these regions if the pixel pair (i, j) ↔ (i + ∆i , j + ∆j) would span an output channel boundary. The correction regions are shown schematically in Figure 2. We then define a corrected correlation function:
C corrected abcd (∆i, ∆j) = C raw abcd (∆i, ∆j) − C baseline abcd (∆i, ∆j).(75)
Advanced characterization
While the basic characterization stage is sufficient to provide a pixel mask and the properties (gain, IPC, non-linearity) needed to convert the non-overlapping correlation function to an IPNL measurement, there are several ways it could be improved. The statistical uncertainties in the gain and IPC are significant, especially with small super-pixels. Moreover, if the BFE exists in these detectors (and we will see in Paper II that it does), then it imprints a bias on g, α, etc., and an iterative process is required to de-bias the final result. The "advanced characterization" tool handles both of these issues.
To motivate our approach to the first issue (noise in the parameters), and understand the improvement in knowledge of gain and IPC that can be achieved, let us first recall the uncertainty in gain and IPC achievable by the "basic" approach. If t ab t ad , then in the computation ofĝ raw abad from Eq. (59), the uncertainty is dominated by V ad . The variance of a Gaussian distribution with n pix = ∆ x ∆ y samples has a fractional uncertainty of 2/n pix . Similarly, the correlation coefficient ρ ∼ 2α of two adjacent pixels has 2n pix samples (counting both vertical and horizontal pairs) and hence an uncertainty of 1/(2n pix ). With N − 1 flat pairs, we should thus in principle achieve
σ(g) g perfect ≈ 2 (N − 1)n pix and σ(α)| perfect ≈ 1 2(N − 1)n pix .(76)
In practice, our pipeline does not do this quite well -the uncertainty in V ab is not negligible, and the use of the IQR carries a factor of 1.64 penalty in error for a Gaussian relative to the "idealized"
case. 16 However, if the flat field has N frame samples, and we break it into "sub-flats" of length µ, one might expect that by combining the sub-flats we could achieve an uncertainty that is reduced by a factor of N frame /µ. Our pipeline does not quite achieve this, but it nevertheless can beat the estimate in Eq. (76). One expects that if µ is decreased, we should see a reduction in the error (down to the fundamental limit of µ = 1). However, the magnitudes of the correlation functions decrease as one decreases µ, and hence we become more sensitive to the subtraction of noise from C H and C V . Therefore there is a trade-off in the choice of µ (and µ defined below), and we allow the user to set these in the configuration file.
The implementation of these ideas in our pipeline is as follows. First, the user sets the range of frames used (earliest frame a and latest frame d), as well as two integers µ and µ (with µ < µ) in the configuration file describing the spacing of time slices used in the gain and IPC determination; typical values would be µ = 1 and µ = 3. We then compute an averaged correlation function
C H =C a,a+µ,a,a+µ,[d−a−µ] (±1, 0) = 1 d − a − µ + 1 d−a−µ j=0 C H,a+j,a+j+µ ,(77)
where C H,a+j,a+j+µ is obtained using the same methodology as in basic characterization using the difference image of frames a + j and a + j + µ. Something similar is performed to computeC V .
(V a+j,a+j+µ − V a+j,a+j+µ ) d − a − µ + 1 ,(78)
where V ef is the variance of the difference of frames e and f as obtained using the same methodology as in basic characterization.
In the advanced characterization stage, the mean information on the ramp is obtained by taking the sequence of differences M a,a+1 , M a+1,a+2 , ... M d−1,d , and performing a linear fit:
M j,j+1 = c 0 + c 1 j + residuals,(79)
where the sum of the square of residuals is minimized. One then wants to simultaneously solve the equations:
∆V = I∆t g 2 [(1 − 4α) 2 + 2α 2 H + 2α 2 V ](µ − µ ) − 4(1 − 8α)β r (I∆t) 2 g 2 µ(a + µ) − µ (a + µ ) + d − a − µ 2 (µ − µ ) + Err[∆V ], C H = 2 I∆t g 2 µ 1 − 4α − 4α D − 4β r d + a + µ 2 I∆t + 1 2 α H + 4 I∆t g 2 µα V α D + Err[C H ], C V = 2 I∆t g 2 µ 1 − 4α − 4α D − 4β r d + a + µ 2 I∆t + 1 2 α V + 4 I∆t g 2 µα H α D + Err[C V ], C D = 2 I∆t g 2 µ [(1 − 4α − 4α D )α D ] α V + 2 I∆t g 2 µα H α V + Err[C D ], c 1 = −2β r (I∆t) 2 g + Err[c 1 ], and c 0 = 1 g [I∆t − β r (I∆t) 2 ] + Err[c 0 ].(80)
Here "Err[...]" denotes the contribution to the specified quantity coming from BFE, NL-IPC, and signal-dependent QE (we will consider these shortly; in future versions of the pipeline we may add other effects). Once again, these are 6 equations for 6 unknowns (g, I, α H , α V , α D , and β r ). A straightforward and effective method is to alternately use the ∆V , c 1 , and c 0 equations to solve algebraically for I, g, and β r ; and then to use theC H ,C V , andC D equations to solve for α H , α V , and α D .
The advanced characterization pipeline can run in two modes for computing the error terms Err [...]; these are none, bfe, and nlipc. The none mode is the simplest: it sets the error terms to zero. When run on a detector that has, e.g., the BFE, the none mode is subject to similar biases as the "basic" characterization, but can give smaller statistical error.
Given that we will see in Paper II that the BFE is significant for the H4RGs, we included the bfe mode. This computes the error terms Err [...] under the assumption that there is a BFE (a ∆i,∆j = 0), but with no non-linear IPC (K ∆i,∆j = 0) or signal-dependent QE (Σ a = 0). Under these assumptions:
Err[∆V ] = [K 2 a + KK ] 0,0 (I∆t) 2 g 2 (µ 2 − µ 2 ), Err[C H ] = [K 2 a + KK ] 1,0 + [K 2 a + KK ] −1,0 2 (I∆t) 2 g 2 µ 2 , Err[C V ] = [K 2 a + KK ] 0,1 + [K 2 a + KK ] 0,−1 2 (I∆t) 2 g 2 µ 2 , and Err[c 0 ] = Err[c 1 ] = 0.(81)
One must iteratively perform the advanced characterization computation in this section and solve for the [K 2 a + KK ] kernel via the procedure in §5.2 until all parameters are converged.
A similar approach is used for the nlipc mode, where the IPNL kernel is attributed entirely to NL-IPC instead of the BFE. In this case:
Err[∆V ] = [K 2 a + KK ] 0,0 (I∆t) 2 g 2 µ(a + µ) − µ (a + µ ) + d − a − µ 2 (µ − µ ) , Err[C H ] = [K 2 a + KK ] 1,0 + [K 2 a + KK ] −1,0 2 (I∆t) 2 g 2 µ d + a + µ 2 , Err[C V ] = [K 2 a + KK ] 0,1 + [K 2 a + KK ] 0,−1 2 (I∆t) 2 g 2 µ d + a + µ 2 , and Err[c 0 ] = Err[c 1 ] = 0.(82)
Characterization of simulated detector data
In Fig. 3, we show the results of applying the aforementioned advanced characterization steps to pairs of simulated flats and darks using the specifications described in § 4. Mean quantities over N good good super-pixels and their statistical errors are provided in Table 1. The latter values are computed as standard deviations on the mean of the N good super-pixels. Table 1 contains the values of the recovered BFE coefficients obtained after iterative application of the advanced characterization described in this section and the method described in § 5.2 (labeled 'Method 1'). The time frames used for our fiducial scheme are 3, 11, 13, and 21. solid-waffle solves for [K 2 a + KK ], which reduces to [K 2 a ] since K = 0 in the simulations. [K 2 a ] values are provided as symmetrical averages for stacks of 3 and 10 simulated flats and compared against the simulation input, where the input a has been convolved with the input K 2 (auto-convolution of K) to get values comparable to what is actually measured in the correlation analysis. In the central value at zero-lag, [K 2 a ] 0,0 , we can see there is a bias of 0.1398 ppm/e for the 10 flat stack relative to the input into the simulation (12.1% bias compared to the input value). We compute the Method 1 BFE coefficients for two alternative time intervals; the first uses time intervals of half the duration of the fiducial scheme and results in a bias of 11.2% in the zero-lag coefficient, while the second uses time intervals of twice the fiducial duration and results in a bias of 20.5%. We note that the changes to β ramp in these alternative time setups are much less than a percent.
We have also run the simulation with only BFE and no IPC and no classical non-linearity. In the fiducial 3, 11, 13, 21 time frame analysis setup for 10 simulated flats and darks, we obtain [K 2 a ] 0,0 = −1.3225 ± 0.0077 (stat) ppm/e, which is biased compared to the input value of -1.3720 ppm/e by 3.6%. In this setup, the correct charge per time slice, gain, α and β are consistent with the input values (where the latter two are consistent with 0). We suggest the likely source of bias in the BFE coefficients extracted from the simulations is due to exclusion of higher order terms in the interactions among the BFE, IPC, and classical non-linearity, and we will revisit this investigation in future work. Note that such an investigation of higher-order effects has recently been completed for CCDs (Astier et al. 2019 Table 1: Averaged results for the simulations, based on stacks of flat ramps. These values were obtained with advanced characterization with ncycle=3.
Raw gain and equal-interval correlation tests
The above techniques enable us to correct the measured properties (gain, IPC, and nonlinearity) for the IPNL -if we know whether to interpret the non-overlapping correlation function as BFE, NL-IPC, or a mixture of the two. Fortunately, the flat field auto-correlations carry enough information to distinguish the sources of IPNL. We cannot do this based on the non-overlapping correlation function, since in that case both BFE and NL-IPC scale as ∝ t ab t cd , but we can use the scalings of the raw gainĝ raw abad and the adjacent-pixel correlations C adad ( 1, 0 ) as a function of which intervals in the flat field are taken.
Raw gain vs. interval duration
In this case, the key observable is the mean-variance slope, in the formĝ raw abad . From Eq. (62), one sees that there should be two time dependences: one that depends on the start time t a and contains only the classical non-linearity β, and one that depends on the duration pattern (t ab and t ad ) and depends on both β and a 0,0 . In this section, we consider the first dependence. We fix t a and fit a linear equation of the form:
lnĝ raw abad = C 0 + C 1 I(t ad + t ab ),(83)
where C 0 is the intercept and C 1 is the slope. 17 From Eq. (62), we interpret the slope as
C 1 = 3β − (1 + 8α)[K 2 a] 0,0 + 8(1 + 3α)α = 3β r none 3β r − (1 + 8α)[K 2 a ] 0,0 bfe 3β r − 2(1 + 8α)[KK ] 0,0 nlipc ,(84)
where the three possibilities on the right are for no IPNL (none), and for the cases where the IPNL is pure BFE (bfe) or pure NL-IPC (nlipc). If there is a measurement of [K 2 a + KK ] from the non-overlapping correlation function, then Eq. (84) can be used to test these hypotheses about its origin.
We compute the raw gain for frame triplets from [1,3,5], [1,3,6],..., [1,5,18] as a function of the signal level accumulated between the first time slice and the time slice d = 5...18 for the simulated detector data. The top panel of Figure 4 visualizes the results of this test. Each data point is a mean over all super-pixels, with an error bar based on the error on the mean. The dashed line is the bfe interpretation of quantities from Method 1, as given by Eq. 84, and the solid line is the 17 An alternative, which we tried first, is to do a linear fitĝ raw abd = B0 + B1I(t ad + t ab ), and use the slope-to-intercept ratio B1/B0. This procedure is not stable because the intercept B0 is obtained by extrapolating to t ad + t ab = 0. There is therefore a strong anti-correlation between the slope and intercept, which results in a noise bias: B1/B0 is biased upward by an amount −Cov(B0, B1)/B 2 0 . The amount of bias increases as subsets of the data are used. The formulation of Eq. (83) avoids this problem. nlipc interpretation. These lines are plotted such that the central values pass through the center of the measurements. The simulated data agree firmly with the bfe slope, as is expected.
In each panel of Figure 4, we also show a systematic error related to the modeling of the non-linearity ("sys nl"). This is based on fitting a 5th order polynomial to the median signal levels in the detector. For both this 5th order curve and the quadratic (β) model, we computed the expected raw logarithmic gain ln g raw a,b,d for Poisson statistics, compute the difference, and plot an error bar showing the peak−valley range. For the case of these simulated data this systematic is negligible, however we include it in anticipation of the analysis of the real data in Paper II where there may be deviations of the classical non-linearity from the β model.
We can also make an estimate of the zero-lag BFE coefficient by re-arranging the left part of Eq. 84 and substituting β r = β − 1 2 Σ a :
a 0,0,M2 ≡ a 0,0 + 8αa <1,0> − 3 2 Σ a − 8(1 + 3α)α = 3β r − C 1(85)
Since we did not include non-linear IPC in the simulations, Eq. 85 simplifies to a 0,0 + 8αa <1,0> . For the 10 simulated flats,â 0,0,M 2 = −1.3513 ± 0.0144 (stat) ppm/e. The input value is -1.3341 ppm/e, so these values agree to within 1.2σ.
Raw gain vs. interval center
A similar test can be carried out by measuring how the raw gainĝ raw abad varies with t a as t ab and t ad are held fixed. We fit: lnĝ raw abad = C 0 + C 1 It a .
In this case, we see that one should have
C 1 = 2β − 8(1 + 3α)α = 2β r none 2β r bfe 2β r − 2(1 + 8α)[KK ] 0,0 nlipc .(87)
Note that the slope C 1 has no sensitivity to the BFE -the none and bfe cases give identical predictions. It is however sensitive to NL-IPC.
We compute the raw gain for frame triplets from [1,3,5], [2,4,6],..., [14,16,18], as a function of the signal level accumulated between the first time slice and the time slice a = 1, ..., 14 for the simulated detector data. The middle panel of Figure 4 visualizes the results of this test, showing that the simulated data are again consistent with the bfe slope.
Re-writing Eq. 87 and using the fact that α = 0, we can also compute β = 1 2 C 1 and Σ a = C 1 − 2β r . β = 0.5677 ± 0.0079 ppm/e, which is very close to the input value of 0.58 ppm/e (within 1.6σ). Likewise, Σ a = −0.0211 ± 0.0158 is very close to the expected value of 0.
CDS autocorrelation vs. signal
This method uses the equal-interval correlation function in adjacent pixels, Eq. (53). Once the preliminary characterization of the detector has been performed, we may fix the starting time t a and fit the combination g 2 C abab (±1, 0)/(It ab ) as a function of t ab , i.e., we fit g 2 It ab C abab ( ±1, 0 ) = C 0 + C 1 It ab .
The slope is given by
C 1 = −8αβ + αΣ a + [K 2 a] 1,0 + 2[KK ] 1,0 = −8αβ r none −8αβ r + [K 2 a ] 1,0 bfe −8αβ r + 2[KK ] 1,0 nlipc .(89)
Adding 8α H β r to the left hand part of Eq. 89 gives
C 1 + 8αβ r = [K 2 a] 1,0 + 2[KK ] 1,0 − 3αΣ a = [K 2 a + 2KK ] 1,0 − αΣ a .(90)
We measure the IPC via basic characterization of frame triplets from [1,2,3], [1,2,4],..., [1,2,18], and CDS auto-correlations for [frame 3 -frame 1], [frame 4 -frame 1],..., [frame 18 -frame 1]. The bottom panel of Figure 4 visualizes the results of this test on the simulated detector data, which are consistent with the bfe interpretation.
We expect that [K 2 a + 2KK ] 1,0 − αΣ a simplifies to [K 2 a ] 1,0 for the simulated data. This value is 0.1855 ± 0.0032 ppm/e and can be compared with the value obtained from Method 1 of [K 2 a ] 1,0 = 0.1980 ± 0.0033 and the input value of 0.2034 ppm/e (∼9% difference between the input and the value obtained with the CDS autocorrelation method).
Discussion
In this paper, we present formalism to connect flat field correlations to various detector effects in infrared detector arrays, including non-linear effects such as the BFE and NL-IPC. This formalism is built up through first considering the Poisson statistics in a perfect detector and then including contributions from the IPC kernel, classical non-linearity, BFE, and NL-IPC. In the expression for the combined cross-correlation of two CDS images (sampled at time frames a, b, c, d), we consider the leading order interactions, namely α, α 2 , β, αβ, a, αa, α , and αα . We discuss two special cases of the combined correlation function: the non-overlapping correlation function (a < b < c < d), which has the most sensitivity to the inter-pixel non-linear effects, but cannot by itself distinguish between the BFE and NL-IPC; the equal-interval correlation function (a = c < b = d), which is the auto-correlation of a CDS image and is most similar to the flat field statistics available for CCDs. We also discuss features of the raw gain for the case of (a = c < b < d), which provides a means of distinguishing between the BFE and NL-IPC interpretations through the different behaviors of these mechanisms as a function of time.
We describe a procedure for characterizing detector arrays and extracting measurements of the IPNL. This involves constructing CDS images, performing a reference pixel subtraction, computing the raw gain, IPC, correlations (in the horizontal, vertical, and diagonal directions), and ramp curvature; we use these to solve for g, α H , α V , I, and β. We show how to use the non-overlapping correlation function to obtain the IPNL and also how to apply an iterative scheme to correct the g, α H , α V , I, and β for residual biases imprinted by the IPNL.
We validate our methodology on simulated flat fields, which are constructed to imitate characteristics (g, α H , α V , I, and β) of the real detector array tested in Paper II. For this first investigation, we input a BFE kernel (but no NL-IPC). We extract parameters that match the inputs with high accuracy, except for the BFE kernel, for which we obtain a zero-lag component which is biased by 12%. We also show that the raw gain and equal-interval correlation function interpretation tests are successful in distinguishing between the BFE and NL-IPC as the underlying mechanism for the IPNL in the simulations. Given the success in obtaining equivalent inputs and outputs of the other key parameters, namely β and α, we suggest the 12% bias in the extracted BFE kernel could likely be explained by unaccounted interactions at higher orders that were dropped in the approximations used in this work. The impact of these higher order terms is under investigation and will be addressed in future work. . We carry out measurements of the nonoverlapping correlation function C abcd (∆i, ∆j) with pixels at separation (∆i, ∆j). We are interested in measurements of the BFE in a 5 × 5 pixel region centered on zero lag (yellow shaded region). The "baseline" is measured in the blue shaded regions; each yellow measurement pixel is corrected using blue baseline pixels in the same row. The fast-read direction is horizontal.
Finally, we compute the difference of variances ∆V =C a,a+µ,a,a+µ,[d−a−µ] (0, 0) −C a,a+µ ,a,a+µ ,[d−a−µ]
Fig. 1 .
1-Flowchart showing the construction procedure for basic flat simulations containing the BFE, IPC, and classical-nonlinearity.
Fig. 2 .
2-The baseline correction scheme used in Eq.(75)
Fig. 3 .
3-Advanced characterization of 3 pairs of simulated flats and darks.
Fig. 4 .
4-Visual comparison of BFE predictions from Method 1 vs measurements from Methods 2 and 3 for simulated detector data (3 flats).
). Non-overlapping correlation function (Method 1) BFE Coefficients -frames 3,11,13,21, baseline-corrected [K 2 a ] 0,0Quantity
Units
Flat type, number
Uncert
Notes
sim,n3
sim,n10
truth
stat.(3) stat.(10) sys.(3)
Charge, It n,n+1
ke
1.4607
1.4615
1.4604
0.0006
0.0003
Gain g
e/DN
2.0606
2.0620
2.0600
0.0008
0.0004
IPC α
%
1.6764
1.6793
1.6900
0.0055
0.0025
IPC α H
%
1.6809
1.6806
1.6900
0.0039
0.0018
IPC α V
%
1.6720
1.6779
1.6900
0.0038
0.0018
IPC α D
%
-0.0002
-0.0021
0.0000
0.0027
0.0012
Non-linearity βramp
ppm/e
0.5835
0.5782
0.5800
0.0003
0.0001
0.0091
Alternative intervals
Non-linearity βramp
ppm/e
0.5862
0.5794
0.5800
0.0006
0.0003
0.0191
Frames 3,7,9
Non-linearity βramp
ppm/e
0.5806
0.5801
0.5800
0.0002
0.0001
0.0052
Frames 3,19,21
ppm/e -1.0373
-1.0192
-1.1590
0.0145
0.0064
0.0103
Central pixel
[K 2 a ] <1,0>
ppm/e
0.1838
0.1980
0.2034
0.0073
0.0033
Nearest neighbor
[K 2 a ] <1,1>
ppm/e
0.0362
0.0428
0.0505
0.0072
0.0032
Diagonal
[K 2 a ] <2,0>
ppm/e
0.0155
0.0133
0.0120
0.0074
0.0032
[K 2 a ] <2,1>
ppm/e
0.0049
0.0010
0.0027
0.0052
0.0023
[K 2 a ] <2,2>
ppm/e
0.0271
0.0179
0.0185
0.0075
0.0033
BFE Coefficients -frames 3,7,9,13 baseline-corrected
[K 2 a ] 0,0
ppm/e -1.0400
-1.0293
-1.1590
0.0288
0.0130
0.0216
Central pixel
[K 2 a ] <1,0>
ppm/e
0.2381
0.2195
0.2034
0.0152
0.0066
Nearest neighbor
[K 2 a ] <1,1>
ppm/e
0.0479
0.0392
0.0505
0.0151
0.0066
Diagonal
BFE Coefficients -frames 3,19,21,37 baseline-corrected
[K 2 a ] 0,0
ppm/e -0.9156
-0.9214
-1.1590
0.0068
0.0031
0.0059
Central pixel
[K 2 a ] <1,0>
ppm/e
0.1850
0.1818
0.2034
0.0034
0.0015
Nearest neighbor
[K 2 a ] <1,1>
ppm/e
0.0422
0.0450
0.0505
0.0035
0.0016
Diagonal
Mean-variance relation (Method 2)
a 0,0,M 2
ppm/e -1.3120
-1.3513
-1.3720
0.0383
0.0144
0.0273
β − 4(1 + 3α)α
ppm/e
0.5613
0.5677
0.5800
0.0218
0.0079
a −8(1 + 3α)α
ppm/e -0.0445
-0.0211
0.0000
0.0436
0.0158
0.0182
Adjacent pixel correlations (Method 3)
[K 2 a ] <1,0> − α a
ppm/e
0.1816
0.1855
0.2034
0.0072
0.0032
Is the BFE turned on? yes Is IPC turned on? no Calculate BFE area defects and determine BFE-modified mean charge yes Draw charge from Poisson w/mean charge no Convolve charge data cube with IPC kernel at each t frameStart
input config
Read config,
set defaults,
initialize
Reset first frame?
Reset frame to
specified reset level
yes
Start from t=0
no
Is current time
less than
final time?
yes
Is classical non-
linearity turned on?
no
Draw charge from
Poisson w/BFE-modified
mean charge
Accumulate charge
Apply classical non-
linearity to charge
data cube at each t
frame
yes
Apply noise?
no
Reset this frame?
Add noise at
each t frame
yes
Divide by gain,
store in 16-bit int
no
Reset frame to
specified reset level
yes
Is this frame
saved?
no
Save frame in
charge data cube
yes
Increment t
no
End, output
in fits datacube
with header
Signal level It 1,a [ke] lng raw a,a +2,a +4 sys nl CDS ACF vs. signal pure BFE pure NL-IPC beta only5
10
15
20
25
Signal level It 1,d [ke]
0.87
0.88
0.89
0.90
0.91
0.92
0.93
0.94
0.95
0.96
lng raw
1,3,d
sys nl
Raw gain vs. interval duration
pure BFE
pure NL-IPC
0
5
10
15
0.885
0.890
0.895
0.900
0.905
0.910
0.915
sys nl
Raw gain vs. interval center
pure BFE
pure NL-IPC
5
10
15
20
25
Signal level It 1,d [ke]
0.029
0.030
0.031
0.032
0.033
0.034
0.035
0.036
0.037
g 2
C
1d1d ( 1,0 )/[It
1d ]
A "pixel boundary" in an either astronomical CCD or a NIR detector is not a physical barrier between neighboring pixels, but rather is defined by the electric fields and diffusion coefficients that determine which well ultimately collects a photo-electron.
For more background on the HxRG series devices, we refer the reader to the overview(Beletic et al. 2008;Blank et al. 2011), and reports on the JWST/NIRSpec H2RGs(Rauscher et al. 2007(Rauscher et al. , 2014) and on the WFIRST H4RG development(Piquette et al. 2014).3 Jay Anderson (private communication) has presented some results on the BFE in the HST WFC3-IR channel
There are also reports of NL-IPC in H2RG detectors (Arielle Bertrou-Cantou, private communication).5 The Teledyne HxRG detectors actually collect holes, rather than electrons, although the statistical techniques in this paper are agnostic to the sign of the charge. We will use the standard nomenclature of charge in "electrons"this is common usage in the astronomical community and HST WFC3-IR documentation, even though the WFC3-IR detector also collects holes.6 In §3, the mathematics of this is worked out in great detail, but the underlying reason for the factor of 2 is the simple combinatorial effect.
We have worked with raw data of both polarities, and it is easy enough to switch between them by inserting an optional mapping S → 2 16 − 1 − S in the input script.
If viewed as a capacitor network, voltage in the pixel might be a more fundamental variable than the charge.
If we also tried to include charge trapping effects such as persistence, then they would not be Markovian. These are not treated in the present formalism.
https://github.com/BJRauscher/nghxrg
With standard FITS routines and "usual astronomer writing Python" level of attention to data handling, reading the files can completely swamp the computation time!
See, e.g., DasGupta (2011), §9.5 for a general discussion of this issue.
This preprint was prepared with the AAS L A T E X macros v5.2.
AcknowledgementsWe thank the Detector Characterization Laboratory personnel, Yiting Wen, Bob Hill, and Bernie Rauscher at NASA Goddard Space Flight Center for their efforts enabling the existence and access to the data analyzed in this series of papers, and we thank Chaz Shapiro, Andrés Plazas, and Eric Huff for helpful discussions. We thank Jay Anderson and Arielle Bertrou-Cantou for useful presentations to the Detector Working Group on their analyses of non-linearities in the HST/WFC3-IR and Euclid H2RG detectors. We thank the anonymous referee for helpful suggestions that improved the clarity of this paper. We are also grateful for the use of Ohio Supercomputer Center (1987) for computing the results in this work. AC and CMH acknowledge support from NASA grant 15-WFIRST15-0008. During the preparation of this work, CMH has also been supported by the Simons Foundation and the US Department of Energy. Software: Astropy (Astropy Collaboration et al. 2013, fitsio (Sheldon 2019), Matplotlib(Hunter 2007), NumPy(Oliphant 2006-), SciPy(Jones et al. 2001-)A. Clipping correction to the covarianceThis appendix considers the correction to the covariance matrix of two jointly Gaussian distributed variables, X and Y , when those distributions are clipped. We are interested in the parameter f corr defined byWe assume that a fraction of the data are clipped from both the top and the bottom of the distribution in X and Y ; if X and Y were independent, this would mean that a fraction (1 − 2 ) 2 of the data points survive the clipping, but the fraction that survives may be larger if X and Y are covariant.The determination of f corr is invariant to linear rescaling of X and Y , so without loss of generality, we assume that X and Y both have mean 0 and variance 1. Their "true" covariance is then the correlation coefficient ρ. The clipping is equivalent to the restriction of the data at |X|, |Y | < ξ, whereSince the clipped distribution still has X = Y = 0 by symmetry, we are interested in the mean value of XY :where the denominator is the survival probability of a data point (X, Y ), and the probability distribution isThe covariance can be expanded in a power series in ρ; the leading term isso thatThe clipped covariances used in this paper to measure IPC are corrected using the leading constant term in f corr . The correction factor should converge to 1 as → 0; this is easily verified.Note that the "correction" is not small: for = 0.01 (i.e. clipping the top 1% and bottom 1% of the distribution) we have f corr = 0.7629. If one clips more of the distribution, the correction becomes enormous: at = 0.025 we have f corr = 0.5758. Going the other way, even for = 10 −3 , the correction is f cor = 0.9587.
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|
[
"https://github.com/BJRauscher/nghxrg"
] |
[
"Quintessential Inflation and Cosmological Seesaw Mechanism: Reheating and Observational Constraints",
"Quintessential Inflation and Cosmological Seesaw Mechanism: Reheating and Observational Constraints"
] |
[
"L Aresté Saló \nSchool of Mathematical Sciences\nQueen Mary University of London\nMile End RoadE1 4NSLon-donUnited Kingdom\n",
"D Benisty \nPhysics Department\nBen-Gurion University of the Negev\n84105Beer-ShevaIsrael\n\nDAMTP\nCentre for Mathematical Sciences\nUniversity of Cambridge\nWilberforce RoadCB3 0WACambridgeUnited Kingdom\n\nFrankfurt Institute for Advanced Studies (FIAS)\nRuth-Moufang-Strasse 160438Frankfurt am MainGermany\n",
"E I Guendelman \nPhysics Department\nBen-Gurion University of the Negev\n84105Beer-ShevaIsrael\n\nFrankfurt Institute for Advanced Studies (FIAS)\nRuth-Moufang-Strasse 160438Frankfurt am MainGermany\n\nBahamas Advanced Study Institute and Conferences\n4A Ocean Heights\nHill View Circle\n\nStella Maris\nLong IslandThe Bahamas\n",
"J D Haro \nFrankfurt Institute for Advanced Studies (FIAS)\nRuth-Moufang-Strasse 160438Frankfurt am MainGermany\n\nDepartament de Matemàtiques\nUniversitat Politècnica de Catalunya\nDiagonal 64708028BarcelonaSpain\n"
] |
[
"School of Mathematical Sciences\nQueen Mary University of London\nMile End RoadE1 4NSLon-donUnited Kingdom",
"Physics Department\nBen-Gurion University of the Negev\n84105Beer-ShevaIsrael",
"DAMTP\nCentre for Mathematical Sciences\nUniversity of Cambridge\nWilberforce RoadCB3 0WACambridgeUnited Kingdom",
"Frankfurt Institute for Advanced Studies (FIAS)\nRuth-Moufang-Strasse 160438Frankfurt am MainGermany",
"Physics Department\nBen-Gurion University of the Negev\n84105Beer-ShevaIsrael",
"Frankfurt Institute for Advanced Studies (FIAS)\nRuth-Moufang-Strasse 160438Frankfurt am MainGermany",
"Bahamas Advanced Study Institute and Conferences\n4A Ocean Heights\nHill View Circle",
"Stella Maris\nLong IslandThe Bahamas",
"Frankfurt Institute for Advanced Studies (FIAS)\nRuth-Moufang-Strasse 160438Frankfurt am MainGermany",
"Departament de Matemàtiques\nUniversitat Politècnica de Catalunya\nDiagonal 64708028BarcelonaSpain"
] |
[] |
Recently a new kind of quintessential inflation coming from the Lorentzian distribution has been introduced in [1, 2]. The model leads to a very simple potential, which basically depends on two parameters, belonging to the class of α-attractors and depicting correctly the early and late time accelerations of our universe. The potential emphasizes a cosmological seesaw mechanism (CSSM) that produces a large inflationary vacuum energy in one side of the potential and a very small value of dark energy on the right hand side of the potential. Here we show that the model agrees with the recent observations and with the reheating constraints. Therefore the model gives a reasonable scenario beyond the standard ΛCDM that includes the inflationary epoch.
|
10.1088/1475-7516/2021/07/007
|
[
"https://arxiv.org/pdf/2102.09514v2.pdf"
] | 231,951,497 |
2102.09514
|
75fe4e35a685d2d3e5254d9c1e4b30a917328a63
|
Quintessential Inflation and Cosmological Seesaw Mechanism: Reheating and Observational Constraints
30 Jun 2021
L Aresté Saló
School of Mathematical Sciences
Queen Mary University of London
Mile End RoadE1 4NSLon-donUnited Kingdom
D Benisty
Physics Department
Ben-Gurion University of the Negev
84105Beer-ShevaIsrael
DAMTP
Centre for Mathematical Sciences
University of Cambridge
Wilberforce RoadCB3 0WACambridgeUnited Kingdom
Frankfurt Institute for Advanced Studies (FIAS)
Ruth-Moufang-Strasse 160438Frankfurt am MainGermany
E I Guendelman
Physics Department
Ben-Gurion University of the Negev
84105Beer-ShevaIsrael
Frankfurt Institute for Advanced Studies (FIAS)
Ruth-Moufang-Strasse 160438Frankfurt am MainGermany
Bahamas Advanced Study Institute and Conferences
4A Ocean Heights
Hill View Circle
Stella Maris
Long IslandThe Bahamas
J D Haro
Frankfurt Institute for Advanced Studies (FIAS)
Ruth-Moufang-Strasse 160438Frankfurt am MainGermany
Departament de Matemàtiques
Universitat Politècnica de Catalunya
Diagonal 64708028BarcelonaSpain
Quintessential Inflation and Cosmological Seesaw Mechanism: Reheating and Observational Constraints
30 Jun 2021
Recently a new kind of quintessential inflation coming from the Lorentzian distribution has been introduced in [1, 2]. The model leads to a very simple potential, which basically depends on two parameters, belonging to the class of α-attractors and depicting correctly the early and late time accelerations of our universe. The potential emphasizes a cosmological seesaw mechanism (CSSM) that produces a large inflationary vacuum energy in one side of the potential and a very small value of dark energy on the right hand side of the potential. Here we show that the model agrees with the recent observations and with the reheating constraints. Therefore the model gives a reasonable scenario beyond the standard ΛCDM that includes the inflationary epoch.
Introduction
After the discovery of the current cosmic acceleration [3][4][5][6], several theoretical mechanisms were developed in order to explain it. One of them is quintessence (see for instance [7][8][9][10][11][12][13][14][15]), where a scalar field is the responsible for the late time acceleration of our universe. The next step was to unify both acceleration phases: the early acceleration of the universe, named inflation [16][17][18], with the current acceleration. One of the simplest ways to do it is the so-called quintessential inflation, introduced for the first time by Peebles and Vilenkin in [19], where the inflaton field is the only responsible for both inflationary phases.
Several authors developed and improved the original Peebles-Vilenkin model obtaining models whose theoretical results match very well with the observational data provided by the Planck's team [53][54][55]. A simple model, constructed from the well-known Lorentzian distribution, was recently presented in [1,2]. We show that the model provides the same spectral index and ratio of tensor to scalar perturbation as the α-attractors models [17,[56][57][58][59][60][61], meaning that it yields a power spectrum of perturbations agreeing with the observation data and is able to depict correctly the current cosmic acceleration. Another property of the model is that it provides a seesaw mechanism. In the theory of grand unification of particle physics and, in particular, in theories of neutrino masses and neutrino oscillation, the seesaw mechanism is a generic model used to understand the relative sizes of observed neutrino masses of the order of eV , compared to those of quarks and charged leptons, which are millions of times heavier [62][63][64][65][66][67]. The approach adopted in [39,[68][69][70] explains the difference between the inflationary vacuum energy density value and the late dark energy density value. As in the case of masses difference in particle physics, here the model predicts that as long as the one epoch has a very low energy density the other one has to have very large energy density. In [1,2] a potential is constructed from the Lorentzian form of the parameter. Here we consider some scalar potential from the beginning which implements the same features of the original models, with much less parameter numbers. We study in detail this simple model, which only depends on two parameters, and we show its viability.
The paper is organized as follows: In Section 2 we introduce the Lorentzian quintessential model, studying its power spectrum during inflation and providing the theoretical value of the parameters involved in the model. Sections 3 and 4 are devoted to the study of all the evolution of the inflaton field, showing that the theoretical results provided by the model agree with the current observational data. Then, in Section 5 we discuss the viability of other similar models and in Section 6 we use a combination of cosmological probes from different data sets to constrain further our model and verify its viability. Finally, Section 7 summarises the results. The units used throughout the paper are = c = 1 and we denote the reduced Planck's mass by M pl ≡ 1 √ 8πG ∼ = 2.44 × 10 18 GeV.
The Lorentzian Quintessential Inflation model
Based on the Cauchy distribution (Lorentzian in the physics language) in [1,2] the ansatz to be considered is the following one,
(N ) = ξ π Γ/2 N 2 + Γ 2 /4 , (2.1)
where is the main slow-roll parameter and N denotes the number of e-folds. From this ansatz, we can find the exact corresponding potential of the scalar field, namely
V (ϕ) = λM 4 pl exp − 2ξ π arctan (sinh (γϕ/M pl )) · 1 − 2γ 2 ξ 2 3π 3 1 cosh (γϕ/M pl ) , (2.2)
where λ is a dimensionless parameter and the parameter γ is defined by
γ ≡ π Γξ .
This potential can be derived by using equations (37) in [71]. However, in this work we are going to use a more simplified potential, keeping the same properties as the original potential but not coming from the ansatz (2.1),
V (ϕ) = λM 4 pl exp − 2ξ π arctan (sinh (γϕ/M pl )) . (2.3)
We can see the shape of the potential on Fig. 1, where the inflationary epoch takes place on the left hand side of the graph, while the dark energy epoch occurs on the right hand side.
The main slow roll parameter is given by
≡ M 2 pl 2 V ϕ V 2 = 2ξ/(Γπ) cosh 2 γ ϕ M pl = 2γ 2 ξ 2 /π 2 cosh 2 γ ϕ M pl (2.4)
and, since inflation ends when EN D = 1, one has to assume that 2ξ Γπ > 1 to guarantee the end of this period.
In fact, we have and we can see that, for large values of γ, one has that ϕ EN D is close to zero. Thus, we will choose γ 1 =⇒ Γξ 1, which is completely compatible with the condition 2ξ Γπ > 1. On the other hand, the other important slow roll parameter is given by
ϕ EN D = M pl γ ln 2ξ Γπ − 2ξ Γπ − 1 = (2.5) M pl γ ln √ 2ξ π γ − γ 2 − π 22ξη ≡ M 2 pl V ϕϕ V = 2ξγ 2 π tanh γ ϕ M pl cosh γ ϕ M pl + 4γ 2 ξ 2 /π 2 cosh 2 γ ϕ M pl . (2.6)
Both slow roll parameters have to be evaluated when the pivot scale leaves the Hubble radius, which will happen for large values of cosh (γϕ/M pl ), obtaining * =
2γ 2 ξ 2 /π 2 cosh 2 γ ϕ * M pl , η * ∼ = 2ξγ 2 π tanh γ ϕ * M pl cosh γ ϕ * M pl (2.7)
with ϕ * < 0. Then, since the spectral index is given in the first approximation by n s ∼ = 1 − 6 * + 2η * , one gets after some algebra
n s ∼ = 1 + 2η * ∼ = 1 − γ r/2,(2.8)
where r = 16 * is the ratio of tensor to scalar perturbations. Now, we calculate the number of e-folds from the leaving of the pivot scale to the end of inflation, which is given by
N = 1 M pl ϕ EN D ϕ * 1 √ 2 dϕ = π 2γ 2 ξ [sinh (γϕ EN D /M pl ) − sinh (γϕ * /M pl )] ∼ = ξ √ 2 * ,
so we have that
n s ∼ = 1 − 2 N , r ∼ = 8 N 2 γ 2 , (2.9)
meaning that our model leads to the same spectral index and tensor/scalar ratio as the α-attractors models with α = 2 3γ 2 (see for instance [17]).
Remark 2.1 For the original potential coming from the ansatz (2.1), i.e., for the potential given in the formula (2.2), one can use the slow roll parameters 1 = and 2 = d ln 1 dN = 2(2 − η) to obtain, for large values of the number of e-folds,
1 = 1 2γ 2 N 2 and 2 = − 2 N , (2.10)
and thus, taking into account that n s = 1 − 2 1 − 2 , one easily gets the result given in the formula (2.9). Unfortunately, since our potential (2.3) is a simplified version of the original potential coming from the ansatz (2.1), in order to justify the expression of the spectral index and the ratio of tensor to scalar perturbations, we cannot do this simple calculation, which only holds for the original potential, and we must perform all the calculation presented in this section.
Finally, it is well-known that the power spectrum of scalar perturbations is given by where we have chosen as a value of n s its central value 0.9649. Summing up, we will choose our parameters satisfying the condition (2.12), with γ 1 and ξ 1, which will always fulfill the constraints Γξ 1 and 2ξ Γπ that we have imposed. Then, to find the values of the parameters one can perform the following heuristic argument:
P ζ = H 2 * 8π 2 * M 2 pl ∼ 2 × 10 −9 .
Taking for example γ = 10 2 , the constraint (2.12) becomes λe ξ ∼ 7 × 10 −15 . On the other hand, at the present time we will have γϕ 0 /M pl 1 where ϕ 0 denotes the current value of the field. Thus, we will have V (ϕ 0 ) ∼ λM 4 pl e −ξ , which is the dark energy at the present time, meaning that
0.7 ∼ = Ω ϕ,0 ∼ = V (ϕ 0 ) 3H 2 0 M 2 pl ∼ λe −ξ 3 M pl H 0 2 . (2.13)
Taking the value H 0 = 67.81 Km/sec/Mpc = 5.94 × 10 −61 M pl , we get the equations λe ξ ∼ 7 × 10 −15 and λe −ξ ∼ 10 −120 , (2.14) whose solution is given by ξ ∼ 122 and λ ∼ 10 −69 . If we choose γ ∼ 10 2 , we see that the values of ξ and γ could be set equal in order to obtain the desired results from both the early and late inflation. From now on we will set ξ = γ, since it may help to find a successful combination of parameters because it reduces the number of effective parameters. As we will see later, numerical calculations show that, in order to have Ω ϕ,0 ∼ = 0.7 (observational data show that, at the present time, the ratio of the energy density of the scalar field to the critical one is approximately 0.7), one has to choose ξ = γ ∼ = 121.8.
To end this section we aim to find the relation between the number of e-folds and the reheating temperature, namely T rh .
For this derivation, we are going to use the same procedure as in [33], which resembles the ones in [72] and [73]. We start with the formula
k * a 0 H 0 = e −N H * H 0 a EN D a kin a kin a rh a rh a matt a matt a 0 , (2.15)
where a is the scale factor and a EN D , a kin , a rh , a matt and a 0 denote respectively its value at the end of inflation, at the beginning of kination, radiation and matter domination, and finally at the present time.
Taking into account that a kin a rh 16) and noting that H 0 ∼ 2 × 10 −4 Mpc −1 ∼ 6 × 10 −61 M pl and k * = a 0 k phys , where we have chosen k phys = 0.02Mpc −1 , we have obtained
6 = ρ rh ρ kin and a rh a matt 4 = ρ matt ρ rh ,(2.N = −4.61 + ln H * H 0 + ln a EN D a kin + 1 4 ln g matt g rh + 1 6 ln ρ rh ρ kin + ln T 0 T rh ,(2.17)
where we have used that after the matter-radiation equality the evolution is adiabatic, that is, a 0 T 0 = a matt T matt as well as the relations ρ matt = π 2 30 g matt T 4 matt and ρ rh = π 2 30 g rh T 4 rh being g matt = 3.36 the degrees of freedom at the matter-radiation equality and we have chosen as degrees of freedom at the reheating time the ones of the Standard Model, i.e., g rh = 106.75. Now, from the formula of the power spectrum (2.11) we infer that
H * ∼ 4×10 −4 √ * M pl , obtaining N = 125.37 + 1 2 ln * + ln a EN D a kin + 1 6 ln ρ rh ρ kin + ln T 0 T rh ,(2.18)
and introducing the current value of the temperature of the universe T 0 ∼ 9.6 × 10 −32 M pl we get
N = 54.36 + 1 2 ln * + ln a EN D a kin − 1 3 ln H kin T rh M 2 pl . (2.19)
As we will see in next section we have numerically checked that H kin ∼ 4 × 10 −8 M pl , which leads to
N + ln N = 54.82 − 1 3 ln T rh M pl ,(2.20)
where we have used that * = 1 2γ 2 N 2 and we have also numerically computed that ln a EN D a kin ∼ = −0.068.
Since the scale of nucleosynthesis is 1 MeV and in order to avoid the late time decay of gravitational relic products such as moduli fields or gravitinos which could jeopardise the nucleosynthesis success, one needs temperatures lower than 10 9 GeV. So, we will assume that 1MeV ≤ T rh ≤ 10 9 GeV, which leads to constrain the number of e-folds to 58 N 67. And for this number of e-folds, 0.966 n s 0.970, which enters within its 2σ CL range.
Dynamical evolution of the scalar field
In this section, we want to calculate the value of the scalar field and its derivative. In this model, as always happens in quintessential inflation, the early inflation is followed up by a kination phase, which is essential to match the model with the Hot Big Bang. Effectively, immediately after the end of inflation the potential is so low that the kinetic energy density of the inflaton field dominates, that is, the universe enters in a kination phase, which is characterised by an effective Equation of State (EoS) parameter w ef f equal to 1 because the potential is negligible. Thus, the energy density of the scalar field decreases as a −6 , being a the scale factor. On the contrary, the particles produced during the phase transition between inflation and kination, whose energy density decreases as a −4 , will eventually dominate and the universe will become reheated.
Remark 3.1 Note that this kination phase is not needed in standard inflation where the inflaton field loses all its energy oscillating in the deep well of the potential and producing the particles that will reheat the universe.
Then, taking into account the importance of the kination phase in quintessential inflation, analytical calculations can be done disregarding the potential during kination because during this epoch the potential energy of the field is negligible. Then, since during kination one has a ∝ t 1/3 =⇒ H = 1 3t , using the Friedmann equation the dynamics in this regime will beφ
2 2 = M 2 pl 3t 2 =⇒φ = 2 3 M pl t =⇒ (3.1) ϕ(t) = ϕ kin + 2 3 M pl ln t t kin ,
where we use by definition [74,75] as the beginning of the kination the moment when the Equation of State parameter is close to 1, which coincides when the derivative of the field is maximum, corresponding to ϕ kin ≈ −0.03M pl and w ϕ ≈ 0.99 for γ = ξ = 122.
Recall that for our choice of the parameters ϕ EN D is very close to zero and, looking at the shape of the potential, this regime has to start very near from ϕ = 0. In order to check it numerically, we have integrated the dynamical system ϕ + 3Hφ + V ϕ = 0, with initial conditions when the pivot scale leaves the Hubble radius, that is, with
ϕ i = ϕ * andφ i = 0, where (1 − n s ) 2 8γ 2 = 2γ 2 ξ 2 π 2 1 cosh 2 (γϕ * /M pl ) , n s = 0.9649.
Thus, at the reheating time, i.e., at the beginning of the radiation phase, one has
ϕ rh = ϕ kin + 2 3 M pl ln H kin H rh ,(3.2)
where we assume, as usual, that there is not drop of energy from the end of inflation to the beginning of kination, i.e.,
H kin = H EN D = √ V (ϕ EN D ) √ 2M pl
, which is numerically satisfied, both being of the order of 4 × 10 −8 M pl .
And, using that at the reheating time (i.e., when the energy density of the scalar field and the one of the relativistic plasma coincide) the Hubble rate is given by H 2 rh = 2ρ rh 3M 2 pl , one gets
ϕ rh = ϕ kin + 2 3 M pl ln H kin π 2 g rh 45 T 2 rh M pl (3.3) andφ rh = π 2 g rh 15 T 2 rh ,(3.4)
where we have used that the energy density and the temperature are related via the formula ρ rh = π 2 30 g rh T 4 rh , where the number of degrees of freedom for the Standard Model is g rh = 106.75 [76]. Because of the smoothness of the potential, since the gravitational particle production [77][78][79][80][81] only works for potentials with an abrupt phase transition leading to a non-adiabatic process which allows the production of particles (see for instance the Peebles-Vilenkin potential [19]), we consider "Instant Preheating" [82][83][84][85] and, thus, we will choose as the reheating temperature T rh ∼ = 10 9 GeV, which is its usual value when the mechanism to reheat the universe is this one.
Effectively, considering a massless scalar X-field conformally coupled with gravity and interacting with the inflaton field as follows, L int = − 1 2 gϕ 2 X 2 [82,83], where g is the dimensionless coupling constant and where the Enhanced Symmetry Point (ESP) has been chosen at ϕ = 0 because, as we have already shown numerically, the beginning of the kination starts at ϕ kin ∼ −0.03M pl . Then, at the beginning of the kination, the adiabaticity is broken and X-particles are produced with a number density equal to [86] n X,kin = g 3/2φ 3/2 kin 8π 3 (3.5) and, since these particles acquire a very heavy effective mass equal to gM pl , in order to reheat the universe they have to decay into lighter ones forming a relativistic plasma, whose energy density will eventually dominate the one of the inflaton field (recall that during kination the energy density of the field decays as a −6 and the one of the relativistic plasma as a −4 ), obtaining a reheated universe with a reheating temperature given by [84] T rh = 30 g * π 2 1/4 ρ 1/4 X,dec ρ X,dec ρ ϕ,dec (3.6)
∼ 10 14 g 15/8 M pl Γ 1/4 GeV,
where g * = 106.75 are the degrees of freedom for the Standard Model, Γ is the decay rate and the sub-index "dec" denotes the moment when the X-field decays completely. Assuming now that the X-field decays into fermions via a Yukawa type of interaction hψψX with decay rate Γ =
Numerical simulation
First of all, we consider the central values obtained in [53] (see the second column in Table 4 of [53]) of the red-shift at the matter-radiation equality z eq = 3365, the present value of the ratio of the matter energy density to the critical one Ω m,0 = 0.308, and, once again, H 0 = 67.81 Km/sec/Mpc = 5.94 × 10 −61 M pl . Then, the present value of the matter energy density is ρ m,0 = 3H 2 0 M 2 pl Ω m,0 = 3.26 × 10 −121 M 4 pl , and at matter-radiation equality we will have ρ eq = 2ρ m,0 (1 + z eq ) 3 = 2.48 × 10 −110 M 4 pl = 8.8 × 10 −1 eV 4 . So, at the beginning of matter-radiation equality the energy density of the matter and radiation will be ρ m,eq = ρ r,eq = ρ eq /2 ∼ = 4.4 × 10 −1 eV 4 .
In this way, the dynamical equations after the beginning of the radiation can be easily obtained using as a time variable N ≡ − ln(1 + z) = ln a a 0 . Recasting the energy density of radiation and matter respectively as functions of N , we get ρ m (a) = ρ m,eq a eq a 3 → ρ m (N ) = ρ m,eq e 3(Neq−N ) (4.1) and ρ r (a) = ρ r,eq a eq a 4 → ρ r (N ) = ρ r,eq e 4(Neq−N ) , (4.2) where N eq ∼ = −8.121 denotes the value of the time N at the beginning of the matter-radiation equality. To obtain the dynamical system for this scalar field model, we will introduce the dimensionless variables
x = ϕ M pl , y =φ H 0 M pl . (4.3)
Taking into account the conservation equationφ + 3Hφ + V ϕ = 0, one arrives at the following dynamical system, x = y/H, y = −3y −V x /H, (4.4) where the prime is the derivative with respect to N ,
H = H H 0 andV = V H 2 0 M 2 pl
. It is not difficult to see that one can writē
H = 1 √ 3 y 2 2 +V (x) +ρ r (N ) +ρ m (N ) , (4.5)
where we have defined the dimensionless energy densities as
ρ r = ρ r H 2 0 M 2 pl ,ρ m = ρ m H 2 0 M 2 pl . (4.6)
Finally, we have to integrate the dynamical system (4.4), with initial conditions x(N rh ) = x rh = 20 and y(N rh ) = y rh = 2.4 × 10 42 imposing thatH(0) = 1, where N rh denotes the beginning of reheating, which is obtained imposing that ρ r,eq e 4(Neq−N rh ) = π 2 30 g rh T 4 rh , (4.7)
that is,
N rh = N eq − 1 4 ln g rh g eq − ln T rh T eq ∼ = −50.68, (4.8)
where we have used that ρ eq,r = π 2 30 g eq T 4 eq with g eq = 3.36 and, thus, T eq ∼ = 7.81 × 10 −10 GeV. We have numerically checked that, to obtain the conditionH(0) = 1, the parameters γ and ξ have to be equal to 121.8. Once these parameters have been properly selected, the obtained results are presented in Figure 3.
Other similar possible models
In this section, we are going to test different models that resemble the one that has been considered so far. An analogously built simplified model would be
V (ϕ) = λM 4 pl e −α arctan β ϕ M pl , (5.1)
for α and β being its positive parameters, where we have suppressed the sinh function. Its slow-roll parameters and η are
= 1 2 αβ 1 + β ϕ M pl 2 2 , (5.2) η = αβ 1 + β ϕ M pl 2 2 1 + 2 β α ϕ M pl .
Hence, the slow-roll parameter is also related to a Lorentzian distribution, in this case in function of ϕ instead of N and with an overall square involved, which makes this model an interesting case worth to study. Using that β ϕ * M pl max(1, |α), we get that
n s ∼ = 1 + 4 α β M pl ϕ * 3 , r ∼ = 8 α β 2 M pl ϕ * 4 . (5.3)
Using the same approximations, one can find that
N = 1 M pl ϕ EN D ϕ * 1 √ 2 dϕ ∼ = − β 3α ϕ * M pl 3 (5.4)
and, therefore,
n s ∼ = 1 − 4 3N and r ∼ = 8 α 9βN 2 2/3 . (5.5)
In order to study the viability of this model, let's start by fixing the value of the parameters in analogy to the main model considered in this work, namely β = γ and α = 2ξ π . In this case, the relation between the number of e-folds and the reheating temperature yields and
Ω ϕ = ρϕ 3H 2 M 2 pl
, from kination to future times. Lower: The effective Equation of State parameter w ef f , from kination to future times. As one can see in the picture, after kination the universe enters in a large period of time where radiation dominates. Then, after the matter-radiation equality, the universe becomes matter-dominated and, finally, near the present, it enters in a new accelerated phase where w ef f approaches −1.
which leads to 57.6 N 66.7, for which the spectral index clearly falls outside of the allowed range. So, this model does not work in the same way as we have shown for the previous one. However, this does not rule out that for other values of the parameters α and β viability could be proved as well. The same applies to the model named "arctan inflation" introduced in [87],
V (ϕ) = λM 4 pl 1 − α arctan β ϕ M pl , (5.7)
whose viability was proved in Section 4.19 of [88] for some given parameters α and β, which does not contradict our statement.
To finish this section a final comment is in order. One could also use the original model obtained in [1],
V (ϕ) = λM 4 pl exp − 2ξ π arctan (sinh (γϕ/M pl )) · 1 − 2γ 2 ξ 2 3π 3 1 cosh (γϕ/M pl )
, (5.8) which is negative around ϕ ∼ = 0. However, it leads to the exact same results, given that the only change is the behavior of the potential for ϕ ∼ = 0. Effectively, when the pivot scale leaves the Hubble radius one has
1 − 2γ 2 ξ 2 3π 3 1 cosh (γϕ * /M pl ) ∼ = 1 − r 64π ∼ = 1,(5.9)
because r ∼ = 8
N 2 γ 2 1.
Thus, the last term of the potential (5.8) does not affect to the power spectrum of perturbations. In the same way, one can easily check that at the end of inflation the potential is positive. Therefore, defining once again that kination starts when w ϕ ∼ = 1, which occurs when ϕ kin = 0.073M pl (corresponding now to the time when the potential becomes positive again), everything works as expected.
Cosmological Probes
In order to constrain our model, we use a few data sets: Cosmic Chronometers (CC) exploit the evolution of differential ages of passive galaxies at different redshifts to directly constrain the Hubble parameter [89]. We use uncorrelated 30 CC measurements of H(z) discussed in [90][91][92][93]. For Standard Candles (SC) we use measurements of the Pantheon Type Ia supernova dataset [94] that were collected in [95] and the measurements from Quasars [96] and Gamma Ray Bursts [97]. The parameters of the models are to be fitted with by comparing the observed µ obs i value to the theoretical µ th i value of the distance moduli, which is given by µ = m − M = 5 log 10 (D L ) + µ 0 , (6.1)
where m and M are the apparent and absolute magnitudes and µ 0 = 5 log H −1 0 /M pc + 25 is the nuisance parameter that has been marginalized. The distance moduli is given for different redshifts µ i = µ(z i ). The luminosity distance is defined by
D L (z) = c H 0 (1 + z) z 0 dz * E(z * ) , (6.2)
where E(z) = H(z) H 0 . Here, we are assuming that Ω k = 0 (flat space-time). We use uncorrelated data points from different Baryon Acoustic Oscillations (BAO) collected in [98] from [99][100][101][102][103][104][105][106][107][108][109][110]. Studies of the BAO feature in the transverse direction provide a measurement of D H (z)/r d = c/H(z)r d , where r d is the sound horizon at the drag epoch and it is taken as an independent parameter and with the comoving angular diameter distance [111,112] being
D M = z 0 c dz H(z )
.
(6.3)
In our database we also use the angular diameter distance D A = D M /(1 + z) and D V (z)/r d , which is a combination of the BAO peak coordinates above, namely
D V (z) ≡ [zD H (z)D 2 M (z)] 1/3 . (6.4)
Finally we take the CMB Distant Prior measurements [113]. The distance priors provide effective information of the CMB power spectrum in two aspects: the acoustic scale l A characterizes the CMB temperature power spectrum in the transverse direction, leading to the variation of the peak spacing, and the "shift parameter" R influences the CMB temperature spectrum along the line-of-sight direction, affecting the heights of the peaks, which are defined as follows:
l A = (1 + z) πD A (z) r s , R(z) = √ Ω m H 0 c (1 + z)D A (z). (6.5)
The observables that [113] reports are:
R z = 1.7502 ± 0.0046, l A = 301.471 ± 0.09, n s = 0.9649 ± 0.0043 (6.6)
with a corresponding covariance matrix (see table I in [113]). The points incorporate the expansion rate from the CMB epoch, and the observables from inflation. We also include other measurements from the late universe in addition to the CMB points. The combination yields a good test for the model with respect to the data. In our analysis we used r s as independent parameter. We take the complete analyses that combine the likelihoods from all of the datasets. We use a nested sampler as it is implemented within the open-source packaged P olychord [114] with the GetDist package [115] [6] has been incorporated into our analysis as an additional prior (R19). Figure 4 shows the posterior distribution of the data fit with the best fit values at table 1. One can see that the Gaussian prior of the Hubble parameter does not change the results by much. For both cases the χ 2 minimized value gives a good fit, since χ 2 /Dof = [255.7/273, 257.0/273] ∼ 1, where Dof are the degrees of freedom for the χ 2 distribution.
The statement from the fit shows that the QI models that we discuss here are viable models and can describe early times as well as late times.
Concluding remarks
In this paper we study the phenomenological implications of a Lorentzian Quintessential Model, depending only on two parameters, where the reheating of the universe, due to the smoothness of the corresponding potential, is produced via the well-known Instant Preheating mechanism. We have shown, analytically and numerically, that for reasonable value of these parameters, this simple model is able to depict correctly our universe unifying its early and late time acceleration. In fact, the model belongs to the class of the so-called α-attractors and, thus, matches very well with the observational data of the power spectrum of perturbation during inflation provided by the Planck's team. It leads to a current dark energy density around 70% of the total one.
In addition to the reheating constraints, we have tested the model with different measurements, some of them from the late universe such as Type Ia supernova, Gamma Ray Bursts and Quasars and the others from the Cosmic Microwave Background from the early universe. The model fits very well to the latest measurements and gives a reasonable scenario beyond the standard ΛCDM that includes the inflationary epoch. Further analysis of the α attractors with LQI background is studied in [116].
Figure 1 .
1The shape of the scalar potential (2.3) with ξ ∼ 122 and λ ∼ 10 −69 . The left side shows the inflationary energy density and the right side shows the late dark energy density.
since in our case V (ϕ * ) ∼ = λM 4 pl e ξ , meaning that H 2 * ∼ = λM 2 pl 3 e ξ , and taking into account that * ∼ = (1−ns) 2 8γ 2 , one gets the constraint λγ 2 e ξ ∼ 7 × 10 −11 , (2.12)
Figure 2 .
2The Marginalized joint confidence contours for (n s , r) at 1σ and 2σ CL, without the presence of running of the spectral indices. We have drawn the curve for the present model for γ 1 from N = 58 to N = 67 e-folds. (Figure courtesy of the Planck2018 Collaboration).
8π , where h is a dimensionless coupling constant, one gets T rh ∼ 10 14 g 13/8 h −1/2 GeV, (3.7) which leads for the narrow range of viable parameters g and h [84] to a reheating temperature around 10 9 GeV. So, finally, at the beginning of the radiation era we have ϕ rh ∼ = 20M plφrh ∼ = 1.4 × 10 −18 M 2 pl . (3.8)
Figure 3 .
3Upper: The density parameters Ω m = ρm3H 2 M 2 pl (orange curve), Ω r = ρr 3H 2 M 2 pl (blue curve)
Figure 4 .
4The posterior distribution for the LQI model with 1σ and 2σ. The data set include Baryon Acoustic Oscillations dataset, Cosmic Chronometers, the Hubble Diagram from Type Ia supernova, Quasars and Gamma Ray Bursts and the CMB. R19 denotes the Riess 2019 measurement of the Hubble constant as a Gaussian prior.
Figure 5 .
5The posterior distribution for the LQI model with 1σ and 2σ, for the Hubble parameter vs. the parameter ξ. The data set include Baryon Acoustic Oscillations dataset, Cosmic Chronometers, the Hubble Diagram from Type Ia supernova, Quasars and Gamma Ray Bursts and the CMB. R19 denotes the Riess 2019 measurement of the Hubble constant as a Gaussian prior.
to present the results. The prior we choose is with a uniform distribution, where Ω r ∈ [0; 1.], Ω m ∈ [0.; 1.], ϕ 0 ∈ [20; 25],φ 0 ∈ [0; 10 −70 ] Ω ϕ ∈ [0.; 1.], H 0 ∈ [50; 100]Km/sec/Mpc, ξ = γ =∈ [100; 130], r s ∈ [130; 160]Mpc. The measurement of the Hubble constant yielding H 0 = 74.03 ± 1.42 (km/s)/Mpc at 68% CL by
AcknowledgmentsWe would like to thank Fotios Anagnostopoulos for helpful comments and advice. D. Benisty and E. I. Guendelman thank Ben Gurion University of the Negev for a great support. D. Benisty also thanks to the Grants Committee of the Rothschild and the Blavatnik Cambridge Fellowships for generous supports. The investigation of J. de Haro has been supported by MINECO (Spain) grant MTM2017-84214-C2-1-P and in part by the Catalan Government 2017-SGR-247. This work has been supported by the European COST actions CA15117 and CA18108.
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|
[] |
[
"Efficient visible frequency comb generation via Cherenkov radiation from a Kerr microcomb",
"Efficient visible frequency comb generation via Cherenkov radiation from a Kerr microcomb"
] |
[
"Xiang Guo \nDepartment of Electrical Engineering\nYale University\n06511New HavenConnecticutUSA\n",
"Chang-Ling Zou \nDepartment of Electrical Engineering\nYale University\n06511New HavenConnecticutUSA\n\nDepartment of Applied Physics\nYale University\n06511New HavenConnecticutUSA\n",
"Hojoong Jung \nDepartment of Electrical Engineering\nYale University\n06511New HavenConnecticutUSA\n",
"Zheng Gong \nDepartment of Electrical Engineering\nYale University\n06511New HavenConnecticutUSA\n",
"Alexander Bruch \nDepartment of Electrical Engineering\nYale University\n06511New HavenConnecticutUSA\n",
"Liang Jiang \nDepartment of Applied Physics\nYale University\n06511New HavenConnecticutUSA\n",
"Hong X Tang \nDepartment of Electrical Engineering\nYale University\n06511New HavenConnecticutUSA\n"
] |
[
"Department of Electrical Engineering\nYale University\n06511New HavenConnecticutUSA",
"Department of Electrical Engineering\nYale University\n06511New HavenConnecticutUSA",
"Department of Applied Physics\nYale University\n06511New HavenConnecticutUSA",
"Department of Electrical Engineering\nYale University\n06511New HavenConnecticutUSA",
"Department of Electrical Engineering\nYale University\n06511New HavenConnecticutUSA",
"Department of Electrical Engineering\nYale University\n06511New HavenConnecticutUSA",
"Department of Applied Physics\nYale University\n06511New HavenConnecticutUSA",
"Department of Electrical Engineering\nYale University\n06511New HavenConnecticutUSA"
] |
[] |
Optical frequency combs enable state-of-the-art applications including frequency metrology, optical clocks, astronomical measurements and sensing. Recent demonstrations of microresonator-based Kerr frequency combs or microcombs pave the way to scalable and stable comb sources on a photonic chip. Generating microcombs in the visible wavelength range, however, has been limited by large material dispersion and optical loss. Here we demonstrate a scheme for efficiently generating visible microcomb in a high Q aluminum nitride microring resonator. Enhanced Pockels effect strongly couples infrared and visible modes into hybrid mode pairs, which participate in the Kerr microcomb generation process and lead to strong Cherenkov radiation in the visible band of an octave apart. A surprisingly high conversion efficiency of 22% is achieved from the pump laser to the visible comb. We further demonstrate a robust frequency tuning of the visible comb by more than one free spectral range and apply it to the absorption spectroscopy of a water-based dye molecule solution. Our work marks the first step towards high-efficiency visible microcomb generation and its utilization, and it also provides insights on the significance of Pockels effect and its strong coupling with Kerr nonlinearity in a single microcavity device. arXiv:1704.04264v1 [physics.optics]
|
10.1103/physrevapplied.10.014012
|
[
"https://arxiv.org/pdf/1704.04264v1.pdf"
] | 51,683,296 |
1704.04264
|
137f770c52c210d432ec85cfc39dad9390ff5870
|
Efficient visible frequency comb generation via Cherenkov radiation from a Kerr microcomb
Xiang Guo
Department of Electrical Engineering
Yale University
06511New HavenConnecticutUSA
Chang-Ling Zou
Department of Electrical Engineering
Yale University
06511New HavenConnecticutUSA
Department of Applied Physics
Yale University
06511New HavenConnecticutUSA
Hojoong Jung
Department of Electrical Engineering
Yale University
06511New HavenConnecticutUSA
Zheng Gong
Department of Electrical Engineering
Yale University
06511New HavenConnecticutUSA
Alexander Bruch
Department of Electrical Engineering
Yale University
06511New HavenConnecticutUSA
Liang Jiang
Department of Applied Physics
Yale University
06511New HavenConnecticutUSA
Hong X Tang
Department of Electrical Engineering
Yale University
06511New HavenConnecticutUSA
Efficient visible frequency comb generation via Cherenkov radiation from a Kerr microcomb
Optical frequency combs enable state-of-the-art applications including frequency metrology, optical clocks, astronomical measurements and sensing. Recent demonstrations of microresonator-based Kerr frequency combs or microcombs pave the way to scalable and stable comb sources on a photonic chip. Generating microcombs in the visible wavelength range, however, has been limited by large material dispersion and optical loss. Here we demonstrate a scheme for efficiently generating visible microcomb in a high Q aluminum nitride microring resonator. Enhanced Pockels effect strongly couples infrared and visible modes into hybrid mode pairs, which participate in the Kerr microcomb generation process and lead to strong Cherenkov radiation in the visible band of an octave apart. A surprisingly high conversion efficiency of 22% is achieved from the pump laser to the visible comb. We further demonstrate a robust frequency tuning of the visible comb by more than one free spectral range and apply it to the absorption spectroscopy of a water-based dye molecule solution. Our work marks the first step towards high-efficiency visible microcomb generation and its utilization, and it also provides insights on the significance of Pockels effect and its strong coupling with Kerr nonlinearity in a single microcavity device. arXiv:1704.04264v1 [physics.optics]
I. INTRODUCTION
The optical frequency combs are invaluable in diverse applications, including but not limited to precision metrology [1][2][3], optical communication [4], arbitrary waveform generation [5], microwave photonics [6,7], astronomical measurement [8,9] and spectroscopic sensing [10][11][12]. The large size and demanding cost of the mode-locked laser combs stimulate the need for a stable, low-cost and compact comb source, where the whispering gallery microresonator brings the breakthrough [13,14]. Microresonators provide an excellent device configuration for comb generation on a chip, benefiting from the enhanced nonlinear optic effect by the high quality factors and small mode volume, as well as the engineerable dispersion by the geometry control. Over the last decade, we have witnessed the exciting progresses of microcombs, including octave comb span [15,16], temporal dissipative Kerr solitons [17][18][19][20][21], dual-comb spectroscopy [12,22] and 2f − 3f self-referencing [23]. Beyond the promising applications, the microcombs also provide a new testing bed for intriguing nonlinear physics because of its roots on generalized nonlinear Schrodinger equations, and allow for the fundamental studies of solitons, breathers, chaos, and rogue waves [24][25][26][27][28].
Despite the demanding need of visible combs for applications such as bio-medical imaging [29], frequency locking [30], and astronomical calibration [8,9], demonstrating a microcomb in visible wavelength is rather challenging. The large material dispersion together with elevated optical loss in most materials appears to be the main obstacles for generating and broadening the visible microcomb. Great efforts have been devoted by the community to address these challenges. Only relatively narrow Kerr combs have been generated at the wavelength below 800 nm in polished calcium fluoride [31] and silica bubble [32] resonators, whose quality factors are challenging to achieve for typical integrated microresonators.
In this article, we demonstrate a scheme for highefficiency visible microcomb generation on a chip by combining two coherent nonlinear optical processes (Pockels and Kerr effects) in the microresonator. We realize a modified four-wave mixing process where the pump resides in the low loss infrared band but emits photons into visible band directly through the strongly coupled visible-infrared mode pairs. First, the strong second-order (Pockels effect) optical nonlinearity χ (2) in aluminum nitride (AlN) microring [33] coherently couples the visible and infrared optical modes, which form hybrid mode pairs [34]. Mediated by these hybrid mode pairs, the visible modes participate in the four-wave mixing processes, which is stimulated by a pump laser at infrared wavelength through Kerr nonlinearity χ (3) . This strong hybridization of χ (2) − χ (3) process enables efficient comb generation in the highly dispersive visible wavelength band. The nonlinear mode coupling between the visible and infrared optical modes leads to the observation of Cherenkov radiation in the visible comb spectrum, which is a new mechanism originated from the modified density of state by the coherent χ (2) nonlinear processes. This nonlinear-mode-coupling-induced Cherenkov radiation differentiates the current work from previous approaches of converting infrared comb to visible wavelengths by external frequency doubling [11,35] or weak intracavity χ (2) process [36][37][38][39], behaving as the backbone for the realized high pump-to-visible comb conversion efficiency. We further show that our visible microcomb can be robustly tuned by more than one free-spectral-range through thermal tuning, a vital property for f − 2f self-referencing [40] and frequency locking to atomic transmission [30]. Lastly, we perform a proof-of-principle experiment to showcase the visible comb spectroscopy of a water-based dye molecule solution, which is not accessible by the more commonly available near-infrared comb because of the strong water absorption.
+ = (a) (b) (c) χ (3) χ (2) ( ( Cherenkov Radiation ( m 0 ,ω 0 ) (2 m 0 ,Ω 0 ) (2 m 0 ,2 ω 0 )
Orbital mode number
χ (2) & χ (3)
Frequency
χ (2) χ (3) χ (2) & χ (3) χ (2) χ (3)
FIG. 1. AlN microring resonator for efficient comb generation and emission in visible wavelength. (a) Dual wavelength band frequency comb generation in a microring resonator. A single color pump is sent into a microring resonator with hybrid secondand third-order nonlinearity. After reaching the threshold of comb generation process, the infrared part of the comb is coupled out through the top bus waveguide while the visible part of the comb is coupled out through the bottom wrap-around waveguide. Background: false-color SEM image of the core devices. Eight (three shown in the SEM) microrings are cascaded using one set of bus waveguides. (b) The energy diagram of the χ (2) , χ (3) and the cascaded nonlinear interaction that involves the visible modes in the modified four wave mixing process. (c) The positions of the comb lines (red and blue solid circles) and their corresponding optical modes (red and blue open triangles) in the frequency-momentum space. The size of the circles represent the intensity of the comb lines. Cherenkov radiation appears in the position where the visible comb line has the same frequency as its corresponding optical mode. Inset: schematic of dual-band comb generation process in a hybrid-nonlinearity microring cavity. Figure 1(a) shows a false color scanning electron microscope (SEM) image of the fabricated microring systems. We design a series (typically eight) of microrings which share the same set of coupling waveguides but have a constant frequency offset. As a result, each microring resonator can be pumped independently, which dramatically enlarges the device parameter space that we can afford for optimal device engineering within each fabrication run. The middle inset of Fig. 1(a) shows the schematic illustration of the dual-band comb generation process in the microring. The AlN microring supports high quality-factor (Q) optical modes ranging from visible (blue lines) to infrared (red lines) wavelengths. These optical modes form a variety of energy levels interconnected by second-and third-order nonlinearity, giving rise to two kinds of coherent nonlinear processes ( Fig. 1(b)). First, driven by the χ (2) Pockels nonlinearity, optical modes in visible and infrared bands can be strongly coupled and form hybrid modes [34]. Here in our system the visible modes are higher-order transversemagnetic (TM) modes (TM 2 ) while the infrared modes are fundamental TM modes (TM 0 ). The amount of hybridization relies on the phase match condition of the χ (2) process, which can be engineered by tuning the width of the microring [41]. Second, due to the Kerr effect χ (3) , these hybrid modes participate in the microcomb generation process [13,17,20,42] and lase when the pump laser reaches a certain threshold. Therefore, the combination of strong χ (2) and χ (3) nonlinearity of AlN allows the efficient generation of both infrared and visible combs, as shown in the inset of Fig. 1(c).
II. THEORETICAL BACKGROUND AND DEVICE DESIGN
To describe the cascaded coherent nonlinear process in our system, we represent the infrared and visible mode families by bosonic operator a j and b j . The corresponding mode frequencies are ω j = ω 0 + d 1 j + d 2 j 2 /2 and Ω j = Ω 0 + D 1 j + D 2 j 2 /2, respectively, when neglecting the higher-order dispersion. Here, the central infrared (visible) modes a 0 (b 0 ) has a frequency of ω 0 (Ω 0 ) and an orbital mode number of m 0 (2m 0 ). j∈ Z is the relative mode number with respect to the central modes (a 0 , b 0 ). d 1 and D 1 are the free spectral ranges, while d 2 and D 2 describe the group velocity dispersion of the corresponding mode families. We can see from the above expressions that the optical modes of infrared and visible wavelength are not of equal spacing in frequency domain, which is illustrated by the open triangles in Fig. 1(c). On the other hand, the frequencies of infrared and visible comb lines are of equal spacing, which can be expressed by:
ω j,comb = ω 0 + d 1 j, Ω j,comb = 2ω 0 + d 1 j.
The position of the comb lines are represented by the dots in Fig. 1(c). We introduce the integrated dispersion D int , which describes the angular frequency difference between the optical modes and the corresponding comb lines. It is intuitive that when the integrated dispersion for infrared (D int,IR = ω j − ω j,comb ) or visible (D int,vis = Ω j − Ω j,comb ) mode approaches 0, the light generated in that mode will be enhanced by the resonance. As a result, in our system we should expect an enhanced comb generation in visible wavelength where D int,vis ≈ 0 (as noted in Fig. 1(c)), which is referred to the Cherenkov radiation and discussed later.
We first describe how the visible and infrared optical modes can be coupled through Pockels effect. The dynamics of modes in the resonator can be described by the Hamiltonian
H = N1 j=−N1 ∆ a j a † j a j + N2 j=−N2 ∆ b j b † j b j +H χ (2) + H χ (3) + 0 a 0 + a † 0 .(1)
where
H χ (2) = j,k,l g (2)
jkl a j a k b † l + a † j a † k b l is the threewave mixing interaction arising from Pockels effect of AlN with coupling strength of g (2) jkl , and H χ (3) includes the fourwave mixing interaction (Kerr effect) inside one mode family or between two mode families [43]. Note that g (2) jkl is nonzero only when j + k = l due to momentum conservation. With a pump field near a 0 (with a detuning δ), the frequency detunings between the comb lines and the opti-
cal modes are ∆ a j = d 2 j 2 − δ and ∆ b j = Ω 0 + (D 1 − d 1 ) j + D 2 j 2 − 2 (ω 0 + δ).
Under strong external pump, the cavity field of the pump mode (a 0 ) can be approximated by a classical coherent field a 0 ≈ N p with N p for the intracavity pump photon number. We can therefore linearize the three-wave mixing interaction and obtain the dominant coherent conversion between two mode families
H χ (2) ≈ j G (2) j a j b † j + a † j b j ,(2)
where G
(2) j = g (2) 0jj
N p . Despite a large difference in optical frequency, infrared (a j ) and visible (b j ) mode families are coupled through nonlinear interaction, which is essentially analogous to the linear coupling between two different spatial mode families of the same wavelength [44]. This nonlinear coupling leads to the formation of visible-infrared hybrid mode pairs, which can be described by the bosonic operators as superposition of visible and infrared modes
A j = 1 N A,j G (2) j a j + λ + j − ∆ a j b j ,(3)B j = 1 N B,j λ − j − ∆ b j a j + G (2) j b j ,(4)
where
λ ± j = ∆ a j +∆ b j 2 ± ∆ a j −∆ b j 2 2 + G (2) j 2
, N A,j and N B,j are the normalization factors.
Combing the χ (2) -induced mode coupling and the Kerr effect, an effective two-mode-family Kerr comb generation is obtained. The pump at infrared band generates emissions not only into the infrared wavelengths, but also into the visible wavelengths. For example, a possible photon emission at a frequency of ω in infrared wavelength can also be accumulated in a visible mode at a frequency of ω + ω 0 + δ. As discussed above, we expect an enhanced emission where D int,vis approaches 0, i.e. the visible comb line overlaps with its corresponding optical mode. It is convenient to quantify this on-resonance enhancement of comb generation in terms of the density of states (DOS) [43], which describes the field enhancement factor for a given optical mode and frequency detuning. By observing the DOS at the positions where comb lines reside, we can predict the relative intensity of the generated comb lines. Figure 2 (Fig. 2(a)), the DOS along the black dashed line is symmetric around the pump. D int,IR is of parabolic shape as represented by the red dashed line in Fig. 2(a). However, for the visible wavelength ( Fig. 2(b)), the DOS along the black dashed line is asymmetric, showing an enhanced DOS at 725 nm where the comb line's frequency matches the optical mode's frequency (D int,vis = 0). Here D int,vis is represented by the green dashed line in Fig. 2(b).
The enhanced DOS at the D int,vis = 0 greatly boosts the comb emission due to Cherenkov radiation, similar to those observations induced by higher order dispersion [20,45,46] (a) or linear mode coupling [44,47].
(c) (d) (e) (b) (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x)
III. EXPERIMENTAL MEASUREMENTS
A. Dual band frequency comb
In the experiment we pump our microring with a 100 kHz repetition rate, 10 ns-long laser system (See Appendix B and supplementary section IV for more details). Figure 2(c) and (d) are the typical measurement spectra of the dualband combs. The infrared comb spectrum is relatively symmetric around the pump wavelength, as predicted by the DOS in Fig. 2(a). For the visible combs, however, the spectrum is asymmetric and extends towards short wavelength side. The strong emission peaks near the second harmonic wavelength (777 nm) of the pump are attributed to the large intracavity photon number near pump wavelength, while the strong emissions centered around 725 nm (noted by the blue arrow in Fig. 2(d)) are attributed to the Cherenkov radiation, which is characterized by an enhanced DOS and D int,vis = 0 as shown in Fig. 2(b). We will show later that when the large intracavity pump photon number is combined together with Cherenkov enhancement, i.e. when the Cherenkov radiation wavelength is close to the second harmonic wavelength of the pump, very efficient visible comb generation can be obtained. As a further confirmation of this Cherenkov radiation mechanism, we carry out the numerical simulation of comb generation process. The numerical calculation is based on the Heisenberg equations of optical modes derived from the Hamiltonian shown in Eq. 1 [43]. Comparing the simulated results (Fig. 2(e) and (f)) with the experimental data, we find valid agreement which consolidates our analysis of the physical mechanism. The residual difference between the simulation (Fig. 2(f)) and the measured results (Fig. 2(d)) can be explained by a wavelength-dependent coupling efficiency between the microring and the visible light extraction waveguide, which increases with wavelength due to larger evanescent field but is not considered in our simulation model.
The optical mode number where the Cherenkov radiation appears (j CR ) should satisfy the linear phase match condition D int,vis (j CR ) = 0, which corresponds to
j CR = − (D 1 − d 1 ) D 2 ± 1 D 2 (D 1 − d 1 ) 2 − 2D 2 (Ω 0 − 2ω 0 ).
(5) According to Eq. 5, the wavelength of Cherenkov radiation is related to Ω 0 − 2ω 0 , which is the frequency detuning between the second harmonic of the pump and its corresponding visible optical mode. To verify this relation in experiment, we change the frequency detuning Ω 0 − 2ω 0 by controlling the width of the microring, which is varied from 1.12 µm to 1.21 µm. Figure 3(a) and (b) show the measured dual comb spectra generated from microrings with different widths. We find that the position of the Cherenkov radiation (as noted by the blue arrows in Fig. 3(b)) in the visible comb spectrum changes consistently from shorter to longer wavelength with the increase of the microring width. Figure 3(c) shows the measured central wavelength of Cherenkov radiation (dots) against the microring width, exhibiting a good agreement with the theoretical prediction according to Eq. 5 (solid line).
As easily observed from the comb spectra ( Fig. 3(a)), the power, span, and the envelope shape of the infrared combs of different devices are quite similar because the dispersion at infrared wavelength is not sensitive to the widths of the microring. In contrast, those of the visible combs change drastically ( Fig. 3(b)). In Fig. 3(d), the span of dual-band combs is summarized. We find that the appearance of Cherenkov radiation can help extend the span of the visible comb, which has been demonstrated in Kerr combs [20,46]. When the Cherenkov radiation appears far-away from the second harmonic wavelength of the pump (e.g. the first and last devices in Fig. 3(b)), the generated visible comb tends to have a broader comb span and more comb lines. On the other hand, when the wavelength of Cherenkov radiation is close to the second harmonic wavelength of the pump (e.g. the 5 th and 6 th devices in Fig. 3(b)), there are less visible comb lines but the total power of the generated visible comb is greatly enhanced. As clearly observed in Fig. 3(e), the visible comb power (blue dots) varies more than two orders of magnitudes from 5 × 10 −3 mW to 0.61 mW, while the power of infrared comb (red dots) keeps around 0.1 mW.
It is quite counter-intuitive that the power of the generated visible comb can be almost ten times larger than that of the infrared comb. Such results cannot be explained by the simple conversion from infrared comb to the visible comb, and reaffirms the important role of the visible-infrared strong coupling in the visible comb generation process. We further investigate the dependence of comb power on the pump power, as shown in Fig. 4(c) and (d). When the Cherenkov radiation matches the second harmonic wavelength of the pump, an increase of comb powers with pump is observed in both infrared and visible bands (Fig. 4(c)). The pump-to-comb power conversion efficiency saturates at 3% for infrared combs and 22% for visible combs (Fig. 4(d)). Such high conversion efficiency can be attributed to both the large cavity photon number near the pump wavelength and the Cherenkov radiation enhancement. As can be observed in Fig. 4(b), the DOS at D int,vis = 0 wavelength is greatly boosted, much larger than that can be observed when the Cherenkov radiation wavelength is far-away (e.g. Fig. 2(b)). Such large DOS finally enables the surprisingly high visible comb generation efficiency. The detailed comb spectra under different pump powers are shown in the supplementary section V.
B. Thermal tuning of optical comb
The ability of continuously tuning the frequency comb is vital for applications such as precision sensing, frequency locking to atomic transition, and f −2f self-referencing. By tuning the temperature of the device, we obtain a continuously tunable visible comb by more than one free spectral range through thermo-optic effect [48], which allows for a much larger frequency tuning range than the mechanical actuation [49] or electro-optic effects [50]. Figure 5(a) and (c) show the infrared and visible comb spectra under different temperature. The measured thermal shifting of the infrared comb lines are 2.62 GHz/K. Considering the free spectral range of 726.7 GHz, a temperature tuning range of 277.4 K is needed for shifting the infrared comb by one free spectral range. The visible comb lines, however, have a thermal shifting (5.24 GHz/K) twice as large as the infrared comb line. This doubled thermal shifting can be explained by the three-wave mixing process where two of the infrared photons combine together to generate one visible photon. The zoom in of the spectra in Fig. 5(b) and (d) clearly show that the visible comb has been tuned by one free spectral range with thermal tuning while the infrared comb is tuned by half free spectral range.
C. Visible comb spectroscopy
Spectroscopy is one of the important applications of optical frequency comb. For bio-medical sensing, which is predominantly in a water environment, visible optical combs are needed because of water's low absorption coefficient in this wavelength range. Here we show the proof-ofprinciple experiment of frequency comb spectroscopy using our broadband, high power visible comb. To validate this method, we first apply our visible comb to measure the transmission spectrum of a thin film bandpass filter near 780 nm. By tuning the angle of the bandpass filter, the transmission band can be tuned continuously. After generating the visible comb on-chip, we send the comb through a fiber-to-fiber u-bench (Thorlabs FBC-780-APC) where the thin film filter can be inserted. The experimental setup is shown in supplementary section IV. Here the visible comb spectrum through an empty u-bench is measured as a reference, as shown by the blue line in Fig. 6(a). We then insert the thin film filter inside the u-bench with either 0 • or 15 • tilting, and measure the transmitted visible comb spectra afterwards. As shown by the green and red lines in Fig. 6(a), the passband of the thin film filter is tuned to shorter wavelength with an increase of tilting angle. We can extract the transmission of the bandpass filter in the position of each comb line, as plotted in Fig. 6(b) with green and red circles. To independently calibrate the sample's absorption, we use a tunable Ti: sapphire laser (M2 Lasers SolsTiS) to measure the transmission spectrum of the bandpass filter, as shown by the dashed lines in Fig. 6(b). A good agreement between these two methods has been observed.
The visible microcomb is then used to measure the transmission spectrum of a water-solvable fluorescent dye molecule. The output of our visible comb is sent through a cuvette which contains either pure water or dye solution, and the transmitted comb spectra are measured as shown in Fig. 6(c). Comparing the comb's spectrum after passing through the dye solution (red line in Fig. 6(c)) with the reference spectrum (blue line in Fig. 6(c)), we can clearly see the wavelength-dependent absorption induced by the fluorescent dye molecule. We plot the comb spectroscopy measurement result of this dye solution in Fig. 6(d), together with an independent measurement result using Ti: Sapphire laser (dashed line in Fig. 6(d)). A good agreement is obtained between the comb spectroscopy and the tunable Ti: sapphire laser, showing the validity of the visible comb spectroscopy in a water-based environment.
IV. DISCUSSION AND CONCLUSION
Our experiment shows a novel scheme to generate high power microcomb in visible wavelength range, which is beneficial for realizing f − 2f self-reference on a single chip, for example by beating an octave spanning TM 0 mode Kerr combs and a TM 2 mode visible comb. The demonstrated thermal tuning can be an efficient way to control the carrier-envelope offset frequency. With an in situ, Cherenkov radiation enhanced frequency up-conversion process, the visible comb line power can be high enough, eliminating bulky equipment for external laser transfer and frequency conversion [23]. The ability to realize highefficiency χ (2) and χ (3) nonlinear process in a single microresonator opens the door for extending the Kerr frequency comb into both shorter and longer wavelength ranges and it is possible to realize multi-octave optical frequency comb generation from a single on-chip device. Future studies along this direction may include more coherent nonlinear effects in a single microresonator, such as third harmonic generation, Raman scattering, and electro-optical effects. Preliminary theoretical work [51] suggests the potential to realize triple-soliton states at three wavelength bands, uncovering the intriguing potential of the cascaded nonlinear process in a microcavity. Red dot: transmission spectrum extracted from the data shown in (c); pink dot: transmission spectrum extracted from the comb data measured by a high sensitivity but low resolution optical spectrum analyzer; dashed line: transmission spectrum measured by tunable Ti: sapphire laser. Inset: the chemical formula of the used dye molecule.
APPENDIX A: DEVICE DESIGN AND FABRICATION
For efficient frequency comb generation in visible wavelength, the device geometry should be engineered to realize the anomalous dispersion for the fundamental (TM 0 ) modes at the pump wavelength, as well as the phase match condition between the fundamental modes at infrared band and the high-order (TM 2 ) modes at visible band. We design the microring width varying from 1.12 µm to 1.21 µm, for which parameters the anomalous dispersion is always achieved while the Cherenkov radiation wavelength is continuously tuned. For the convenience of fabricating and characterizing the microring with different geometry parameters, there are eight microring resonators in each bus waveguide sets. To avoid the overlap of the resonances for different microring resonators in the same bus waveguide sets, the radii of the cascaded microrings are offset by 9 nm, which results in an offset of resonance wavelength by 0.4 nm. As a result, the resonances of the eight microrings are well separated in frequency domain and can be selectively pumped by tuning the pump laser wavelength. There are two waveguides coupled with the microring resonator. One wrap-around waveguide tapered from 0.175 µm to 0.125 µm or from 0.15 µm to 0.1 µm is used to efficiently extract the visible light from the resonator, with a coupling gap varying from 0.3 µm to 0.5µm. The width of the other bus waveguide is fixed to be 0.8 µm with a gap of 0.6 µm, realizing critical coupling for the pump light in infrared band. The radius of the microrings is fixed to be 30 µm.
Our device is fabricated using AlN on SiO 2 on silicon wafer. The nominal AlN film thickness is 1 µm, while the measured thickness is 1.055 µm. After defining the pattern with FOx 16 using electron beam lithography, the waveguide and microring resonators are dry etched using Cl 2 /BCl 3 /Ar chemistry, and then a 1 µm thick PECVD oxide is deposited on top of the AlN waveguide. The chip is annealed in N 2 atmosphere for 2 hours at 950 • C to improve the quality factors of optical modes. A critically-coupled quality factor of 1×10 6 has been achieved in infrared band, and the visible resonance has a typical intrinsic quality factor of 1.5 × 10 5 .
APPENDIX B: DETAILS OF MEASUREMENT PROCESS
The pump laser pulse is generated by amplifying 10 ns square pulse (duty cycle 1/1000) in two stages of EDFAs. Tunable bandpass filters are inserted after each amplification stage to remove the ASE noise. Due to the low average power of the pulses, the peak power of the optical pulse can be amplified to more than 10 W. The seeding pulse is obtained by modulating the output of a continuous-wave infrared laser (New Focus TLB-6728) with a electro-optic modulator. The 10 ns pulse duration time is much longer than the cavity lifetime (< 1 ns) of our microring cavity, leading to a quasi-continuous wave pump for the optical modes. The optical comb spectra are measured by optical spectra analyzer which has a measurement span of 600 nm to 1700 nm. To avoid crosstalk in the optical spectrum analyzer, we used a long-pass (short-pass) filter to block all the visible (infrared) light when we measure the infrared (visible) comb spectrum. Our chip sits on top of a close-loop temperature control unit (Covesion OC2) which has a thermal stability of 0.01 • C and a thermal tuning range from room temperature to 200 • C. The used thin film bandpass filter is 790/12 nm VersaChrome filter from Semrock and the fluorescent dye is sulfo-Cyanine7 from Lumiprobe.
FIG. 2 .
2Cherenkov radiation induced by nonlinear mode coupling. (a) The density of state for the infrared modes with a pump in a0 mode. Here the natural logarithm of the calculated density of state is plotted. An anomalous dispersion leads to a parabolic shape of frequency detuning between the frequency of each comb line and that of the optical modes. The dashed red line shows the frequency detuning D int(IR) between the infrared comb lines and the corresponding optical modes. (b) The density of state for the visible modes with a pump in a0 mode. Here the natural logarithm of the calculated density of state is plotted. The dashed green line shows the frequency detuning D int(vis) between the visible comb lines and the corresponding optical modes. The wavelength where the visible comb frequency detuning D int(vis) approaches zero corresponds to Cherenkov radiation, leading to an enhanced emission into this mode. (c)-(d) The measured spectrum of the infrared (c) and visible (d) frequency comb. The blue arrow indicates the position of Cherenkov radiation. (e)-(f) Numerical simulation of the infrared (e)and visible (f) frequency comb. The discrepancy between (d) and (f) can be explained by a wavelengthdependent coupling efficiency from microring to wrap-around waveguide, which is not considered in simulation.
(a) and (b) show the calculated DOS for infrared and visible modes, respectively. Here we are interested in the DOS along the D int = 0 line (black dashed lines) in the figures, which corresponds to the positions where the comb lines appear. For the infrared band
FIG. 3 .
3Dual-band frequency comb generated by microrings with different widths. (a) Infrared combs generated by devices with a width of 1.12 µm (bottom) to 1.21 µm (top). (b) The corresponding visible combs generated by devices with a width of 1.12 µm (bottom) to 1.21 µm (top). The blue arrows show the Cherenkov radiation wavelength. (c) Cherenkov radiation wavelength for devices with different microring widths. The circles correspond to the experimental data and the solid line represents the theoretical calculations. The pentagram marks the device which is used to measure the power dependence inFig. 4. (d)The number of infrared (red) and visible (blue) comb lines for devices with different widths. (e) The total power of the infrared (red) and visible (blue) comb lines for devices with different widths.
FIG. 4 .
4Dual band comb generation efficiency under different pump powers. (a) The density of state for the infrared modes (with a pump in the a0 infrared mode) when the Cherenkov radiation is close to the second harmonic wavelength of the pump. (b) The corresponding density of state for the visible mode. (c) infrared (red) and visible (blue) comb powers under different pump powers. (d) On-chip conversion efficiency of the generated infrared (red) and visible (blue) combs. Here both the pump and the generated comb powers refer to the average powers.
FIG. 5 .
5Wavelength tuning of both infrared and visible frequency combs. (a)-(b) The infrared frequency comb spectrum under different temperature of the device. (b) shows the zoomin of the dashed box region in (a). (c)-(d) The visible frequency comb spectrum under different temperature of the device. (d) shows the zoom-in of the dashed box region in (c).
ACKNOWLEDGMENTSH.X.T. acknowledges support from DARPA SCOUT program, an LPS/ARO grant (W911NF-14-1-0563), an AFOSR MURI grant (FA9550-15-1-0029), and a Packard Fellowship in Science and Engineering. Facilities used for device fabrication were supported by Yale SEAS cleanroom and Yale Institute for Nanoscience and Quantum Engineering. L.J. acknowledges support from the Alfred P. Sloan Foundation and Packard Foundation. X.G. thanks Chen Zhao for the discussion and help in visible comb spectroscopy measurement. The authors thank Michael Power and Dr. Michael Rooks for assistance in device fabrication.
FIG. 6 .
6Comb spectroscopy in visible range. (a) Visible comb spectra with and without band pass filter. Blue line: original visible comb spectrum; green line: visible comb spectrum after 0 degree tilted thin film bandpass filter; red line: visible comb spectrum after 15 degree tilted thin film bandpass filter. (b) The measured transmission spectrum of the thin film bandpass filter using comb spectroscopy (dots) and Ti: sapphire laser (dashed lines). Inset: thin film tunable filter. (c) Visible comb spectra passing through pure water (blue) or Cy-7 fluorescent dye solution (red). (d) The measured transmission spectrum of the Cy-7 fluorescent dye.
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[] |
[
"A Fully Stochastic Primal-Dual Algorithm",
"A Fully Stochastic Primal-Dual Algorithm"
] |
[
"Adil Salim ",
"Pascal Bianchi ",
"Walid Hachem "
] |
[] |
[] |
A new stochastic primal-dual algorithm for solving a composite optimization problem is proposed. It is assumed that all the functions/matrix used to define the optimization problem are given as statistical expectations. These expectations are unknown but revealed across time through i.i.d realizations. This covers the case of convex optimization under stochastic linear constraints. The proposed algorithm is proven to converge to a saddle point of the Lagrangian function. In the framework of the monotone operator theory, the convergence proof relies on recent results on the stochastic Forward Backward algorithm involving random monotone operators.
|
10.1007/s11590-020-01614-y
|
[
"https://arxiv.org/pdf/1901.08170v2.pdf"
] | 88,517,578 |
1901.08170
|
186c0ef6514322d6997077cb145ca1d4668eb5c1
|
A Fully Stochastic Primal-Dual Algorithm
3 Feb 2019
Adil Salim
Pascal Bianchi
Walid Hachem
A Fully Stochastic Primal-Dual Algorithm
3 Feb 2019Received: date / Accepted: dateNoname manuscript No. (will be inserted by the editor)
A new stochastic primal-dual algorithm for solving a composite optimization problem is proposed. It is assumed that all the functions/matrix used to define the optimization problem are given as statistical expectations. These expectations are unknown but revealed across time through i.i.d realizations. This covers the case of convex optimization under stochastic linear constraints. The proposed algorithm is proven to converge to a saddle point of the Lagrangian function. In the framework of the monotone operator theory, the convergence proof relies on recent results on the stochastic Forward Backward algorithm involving random monotone operators.
Consider c ∈ V and H = ι {c} , where ι C is the indicator function of the set C, i.e, the function equal to 0 on C and +∞ elsewhere. In this particular case, Problem (1) boils down to the linearly constrained problem min x∈X F(x) + G(x) s.t. Lx = c.
(2)
In order to solve Problem (1), primal-dual methods generate a sequence of primal estimates (x n ) n∈N and a sequence of dual estimates (λ n ) n∈N jointly converging to a saddle point of the Lagrangian function. As is well known, the qualification condition
c ∈ ri (L dom G − dom H)
where ri is the relative interior of a set, ensures the existence of such a point [4].
There is a rich literature on such algorithms which cannot be exhaustively listed [1,2,3]. In this paper, it is assumed that the quantities that enter the minimization problem are likely to be unavailable or difficult to compute numerically. More precisely, it is assumed that the functions F and G are defined as expectations of random functions. Given a probability space (Ξ, G , µ), consider two convex normal integrands (see below) f : Ξ × X → R and g : Ξ × V → (−∞, +∞]. Then, we consider that F(x) = E µ (f (·, x)) and G(x) = E µ (g(·, x)). In addition, let L be a measurable function from (Ξ, G , µ) to L(X , V) (i.e a random matrix), then it is assumed that L = E µ L(·). Finally, the Fenchel conjugate H ⋆ of H takes the form H ⋆ (λ) = E µ (p(·, λ)), where p is a normal convex integrand.
In the particular case of Problem (2), let us assume that c = E µ (c(·)) where c(·) : Ξ → V is a random vector. Then, since H ⋆ (λ) = λ, c , we simply put p(·, λ) = λ, c(·) .
In order to solve Problem (1), the observer is given the functions f , g, p, and L, along with a sequence of independent and identically distributed (i.i.d.) random variables (ξ n ) with the probability distribution µ. In this paper, a new stochastic primal dual algorithm based on this data is proposed to solve this problem.
The convergence proof for this algorithm relies on the monotone operator theory. The algorithm is built around an instantiation of the stochastic Forward-Backward algorithm involving random monotone operators that was introduced in [5]. It is proven that the weighted means of the iterates of the algorithm, where the weights are given by the step sizes of the algorithm, converges almost surely to a saddle point of the Lagrangian function. To the authors knowledge, the proposed algorithm is the first method that allows to solve Problem (1) in a fully stochastic setting. Existing methods typically allow to handle subproblems of Problem (1) in which some quantities used to define (1) are assumed to be available or set equal to zero [6,7,8,9,10]. In particular, the new algorithm generalizes the stochastic gradient algorithm (in the case where only F is non zero), the stochastic proximal point algorithm [11,10,12] (only G is non zero), and the stochastic proximal gradient algorithm [13,14] (only F + G is non zero).
To our knowledge, the proposed algorithm is also one of the first methods that allows to tackle stochastic linear constraints. The paper [8] studies stochastic inequality constraints for optimization over a compact set and provide regret bounds. Handling stochastic constraints online is suitable in various fields of machine learning like Neyman-Pearson classification or online portfolio optimization. For example, the Markowitz portfolio optimization problem is an instance of Problem (2) where ξ is a random variable with values in
X , F(x) = E ξ ( x, ξ 2 ), G(x) = ι ∆ (x)
where ∆ is the probability simplex, L = E ξ (ξ T ) and c is some real positive number. In this case, authors usually assume that L = E ξ (ξ T ) is fully known or estimated. The paper is organized as follows. The next section is devoted to rigorously state the main problem and the main algorithm. In section 3 the convergence proof of the algorithm is given. Some notations.The notation B(X ) will refer to the Borel σ-field of X . Both the operator norm and the Euclidean norm will be denoted as · . The distance of a point x to a set S is denoted as dist(x, S). As mentioned above, we denote as L(X , V) the set of linear operators, identified with matrices, from X to V. The set of proper, lower semicontinuous convex functions on X is Γ 0 (X ).
Problem description
Before entering our subject, we recall some definitions regarding set-valued functions and integrals. Let (Ξ, G , µ) be a probability space where the σ-field G is µ-complete. Given a Euclidean space X , let h : Ξ ⇒ X be a set valued function such that h(s) is a closed set for each s ∈ Ξ. The function h is said measurable if {s : h(s) ∩ S = ∅} ∈ G for any set S ∈ B(X ). An equivalent definition for the mesurability of h requires that the domain dom(h) := {s ∈ Ξ : h(s) = ∅} of h belongs to G , and that there exists a sequence of measurable functions ϕ n : dom(h) → X such that h(s) = cl {ϕ n (s)} n for all s ∈ dom(h), where cl is the closure of a set. Such functions are called measurable selections of h. Assume now that h is measurable and that µ(dom(h)) = 1. Given 1 ≤ p < ∞, let L p (µ) be the space of the G -measurable functions ϕ : Ξ → X such that ϕ p dµ < ∞, and let
S p h := {ϕ ∈ L p (µ) : ϕ(s) ∈ h(s) µ − almost everywhere (a.e.)} . If S 1 h = ∅, the function h is said integrable. The selection integral of h is the set hdµ := cl ß Ξ ϕdµ : ϕ ∈ S 1 h ™ .
(
In all the remainder, given a single-valued or a set-valued function h, the notation E µ h will refer to the integral of h with respect to µ. The meaning of this integral will be clear from the context.
We now state our problem. A function h :
Ξ × X → (−∞, ∞] is said a convex normal integrand if h(s, ·)
is convex, and if the set-valued mapping s → epi h(s, ·) is closed-valued and measurable, where epi is the epigraph of a function. Let f : Ξ ×X → (−∞, ∞] be a convex normal integrand, and assume that |f (s, x)| µ(ds) < ∞ for all x ∈ X . Consider the convex function F(x) defined on X as the Lebesgue integral F(x) = E µ f (·, x). Denoting as ∂f (s, x) the subdifferential of f (s, ·) with respect to x, it is known that the set-valued function ∂f (·, x) is measurable, S 1 ∂f (·,x) = ∅, and ∂F(x) = E µ ∂f (·, x) for each x ∈ X , where the integral is the selection integral defined above [15,16].
Let g : Ξ × X → (−∞, ∞] be another convex normal integrand, and let G(x) = E µ g(·, x), where the integral is defined as the sum
{s : g(s,x)∈[0,∞)} g(s, x) µ(ds) + {s : g(s,x)∈]−∞,0[} g(s, x) µ(ds) + I(x) ,
and
I(x) = ß +∞, if µ({s : g(s, x) = ∞}) > 0,
0, otherwise , and where the convention (+∞) + (−∞) = +∞ is used. The function G is a lower semi continuous convex function if G(x) > −∞ for all x, which we assume. We shall also assume that G is proper. Note that this implies that g(s, ·) ∈ Γ 0 (X ) for µ-almost all s. It is also known that ∂g(·, x) is measurable for each x [15]. We assume that ∂G(x) = E µ ∂g(·, x), where the right hand member is set to ∅ for the values of x for which S 1 ∂g(·,x) = ∅. Before proceeding in the problem statement, it is useful to provide sufficient conditions under which this interchange of the expectation and the subdifferentiation is possible. By [16], this will be the case if the following conditions hold: i) the set-valued mapping ) there exists x 0 ∈ X at which G is finite and continuous. Another case where this interchange is permitted is the following. Let m be a positive integer, and let C 1 , . . . C m be a collection of closed and convex subsets of X . Let C = ∩ m i=k C k = ∅, and assume that the normal cone N C (x) of C at x satisfies the identity N C (x) = m k=1 N C k (x) for each x ∈ X , where the summation is the usual set summation. As is well known, this identity holds true under a qualification condition of the type ∩ m k=1 ri C k = ∅ (see also [17] for other conditions). Now, assume that Ξ = {1, . . . , m} and that µ is an arbitrary probability measure putting a positive weight on each {k} ⊂ Ξ. Let g(s, x) be the indicator function
s → cl dom g(s, ·) is constant µ-a.e., where dom g(s, ·) is the domain of g(s, ·), ii) G(x) < ∞ whenever x ∈ dom g(s, ·) µ-a.e., iiig(s, x) = ι Cs (x) for (s, x) ∈ Ξ × X .(4)
Then it is obvious that g is a convex normal integrand, G = ι C , and ∂G(x) = E µ ∂g(·, x). We can also combine these two types of conditions: let (Σ, T , ν) be a probability space, where T is ν-complete, and let h : Σ ×X → (−∞, ∞] be a convex normal integrand satisfying the conditions i)-iii) above. Consider the closed and convex sets C 1 , . . . , C m introduced above, and let α be a probability measure on the set {0, . . . , m} such that α({k}) > 0 for each k ∈ {0, . . . , m}. Now, set Ξ = Σ × {0, . . . , m}, µ = ν ⊗ α, and define g :
Ξ × X → (−∞, ∞] as g(s, x) = ß α(0) −1 h(u, x) if k = 0, ι C k (x) otherwise,
where s = (u, k) ∈ Σ × {0, . . . , m}. Then it is clear that
G(x) = 1 α(0) Σ h(u, x)ν(du) + ι C (x) ,
and
∂G(x) = E µ ∂g(·, x) = 1 α(0) E ν ∂h(·, x) + m k=1 N C k (x) .
To proceed with our problem statement, we introduce another convex normal integrand p : Ξ × Z → (−∞, ∞] and assume that the function p has verbatim the same properties as g, after replacing the space X with V. We also denote H the Fenchel conjugate of P(λ) = E µ p(·, λ), so that H ⋆ (λ) = E µ p(·, λ).
Finally, let L : Ξ → L(X , V) be an operator-valued measurable function. Let us assume that L is µ-integrable, and let us introduce the Lebesgue integral L = E µ L.
Having introduced these functions, our purpose is to find a solution x ∈ X of Problem (1), where the set of such points is assumed non empty. To solve this problem, the observer is given the functions f, g, p, L, and a sequence of i.i.d random variables (ξ n ) n∈N from a probability space (Ω, F , P) to (Ξ, G ) with the probability distribution µ.
Denote as prox h (x) = arg min y∈X h(y)+ y −x 2 /2 the Moreau's proximity operator of a function h ∈ Γ 0 (X ). We also denote as ∂ 0 h(x) the least norm element of the set ∂h(x), which is known to exist and to be unique [4]. Similarly, ∂ 0 f (s, x) will refer to the least norm element of ∂f (s, x) which was introduced above. We shall also denote as ‹ ∇f (s, x) a measurable subgradient of f (s, ·) at x. More precisely, ‹ ∇f : (Ξ × X , G ⊗ B(X )) → (X , B(X )) is a measurable function such that for each x ∈ X , ‹ ∇f (·, x) ∈ S 1 ∂f (·,x) (recall that this set is non empty). A possible choice for ‹ ∇f (s, x) is ∂ 0 f (s, x) (see [5, §2.3 and §3.1] for the measurability issues). Turning back to Problem (1), our purpose will be to find a saddle point of the Lagrangian (x, λ) → F(x) + G(x) − H ⋆ (λ) + Lx, λ . Denoting as S ⊂ X × V the set of these saddle points, an element (x, λ) of S is characterized by the inclusions ß 0 ∈ ∂F(x) + ∂G(x) +L T λ,
0 = −Lx +∂H ⋆ (λ) .(5)
Consider a sequence of positive weights (γ n ) n∈N . The algorithm proposed here consists in the following iterations applied to the random vector (x n , λ n ) ∈ X × V.
We also give the instance of the main algorithm that allows to solve Problem (2) (which is a instance of Problem (1)).
The convergence of Algorithm 1 is stated by the following theorem. (1), and let the following assumptions hold true.
Theorem 1 Consider the Problem
Algorithm 1 The Main Algorithm : Solving Problem (1)
x n+1 = prox γ n+1 g(ξ n+1 ,·) xn − γ n+1 ( ∇f (ξ n+1 , xn) + L(ξ n+1 ) T λn) , λ n+1 = prox γ n+1 p(ξ n+1 ,·) (λn + γ n+1 L(ξ n+1 )xn) .
Algorithm 2 Stochastic Linear Constraints : Solving Problem (2)
x n+1 = prox γ n+1 g(ξ n+1 ,·) xn − γ n+1 ( ∇f (ξ n+1 , xn) + L(ξ n+1 ) T λn) , λ n+1 = λn + γ n+1 (L(ξ n+1 )xn − c(ξ n+1 )) .
There exists an integer m ≥ 2 that satisfies the following conditions:
-The function L is in L 2m (µ).
-There exists a point (x ⋆ , λ ⋆ ) ∈ S, and three functions ϕ f ∈ S 2m ∂f (·,x⋆) , ϕ g ∈ S 2m ∂g(·,x⋆) , and ϕ p ∈ S 2m ∂p(·,λ⋆) which
E µ ϕ f + E µ ϕ g + L T λ ⋆ = 0, and − Lx ⋆ + E µ ϕ p = 0.(6)
The last assumption is verified for m = 1 and for each point (x ⋆ , λ ⋆ ) ∈ S.
For any compact set
K of dom ∂G, there exist ε ∈ (0, 1] and x 0 ∈ dom ∂G such that sup x∈K E ∂ 0 g(·, x) 1+ε < +∞, and E ∂ 0 g(·, x 0 ) 1+1/ε < +∞. 4. Writing D ∂g (s) = dom ∂g(s, ·), there exists C > 0 such that for all x ∈ X , E µ dist(x, D ∂g (·)) 2 ≥ C dist(x, dom ∂G) 2 .
5. There exists C > 0 such that for any x ∈ X and any γ > 0,
prox γg(s,·) (x) − Π g (s, x) 4 µ(ds) ≤ Cγ 4 (1 + x 2m ),
where Π g (s, ·) is the projection operator onto cl(dom ∂g(s, ·)), and where m is the integer provided by Assumption 2.
Assumptions similar to 3-5 are made on the function p and P.
6. There exists a measurable Ξ → R + function β such that β 2m is µ-integrable, where m is the integer provided by Assumption 2, and such that for all
x ∈ X , ‹ ∇f (s, x) ≤ β(s)(1 + x ).
Moreover, there exists a constant
C > 0 such that E µ ‹ ∇f (·, x) 4 ≤ C(1 + x 2m ).
Consider the sequence of iterates (x n , λ n ) produced by the algorithm (1), and define the averaged estimates
x n = n k=1 γ k x k n k=1 γ k , andλ n = n k=1 γ k λ k n k=1 γ k .
Then, the sequence (x n , λ n ) is bounded in L 2m (Ω) and the sequence (x n ,λ n ) converges almost surely (a.s.) to a random variable (X, Λ) supported by S.
Let us now discuss our assumptions. Assumption 1 is standard in the decreasing step case. Assumption 2 is a moment assumption that is generally easy to check. Note that this assumption requires the set of saddle points S to be non empty. Notice the relation between Equations (6) and the two inclusions in (5). Focusing on the first inclusion, there exist a ∈ ∂F (x ⋆ ) = E µ ∂f (·, x ⋆ ) and b ∈ ∂G(x ⋆ ) = E µ ∂g(·, x ⋆ ) such that 0 = a + b + L T λ ⋆ . Then, Assumption 2 states that there are two measurable selections ϕ f and ϕ g of ∂f (·, x ⋆ ) and ∂g(·, x ⋆ ) respectively which are both in L 2m (µ) and which satisfy a = E µ ϕ f and b = E µ ϕ g . Not also that the larger is m, and the weaker is Assumption 5.
Assumption 3 is relatively weak and easy to check. This assumption on the functions g and p is much weaker than Assumption 6, which assumes that the growth of ‹ ∇f (s, ·) is not faster than linear. This is due to the fact that g and p enter the algorithm (1) through the proximity operator while the function f is used explicitly in this algorithm (through its (sub)gradient). This use of the functions f is reminiscent of the well-known Robbins-Monro algorithm, where a linear growth is needed to ensure the algorithm stability. Note that Assumption 6 is satisfied under the more restrictive assumption that ∇f (s, ·) is L-Lipschitz continuous without any bounded gradient assumption.
Assumption 4 is quite weak, and is studied e.g in [18]. This assumption is easy to illustrate in the case where g(s, x) = ι Cs (x) as in (4). Following [17], we say that the subsets (C 1 , . . . , C m ) are linearly regular if there exists C > 0 such that for every x,
max i=1...m dist(x, C i ) ≥ C dist(x, C).
Sufficient conditions for a collection of sets to satisfy the above condition can be found in [17] and the references therein. Note that this condition implies that N C (x) = m i=1 N Ci (x). Let us finally discuss Assumption 5. As γ → 0, it is known that prox γg(s,·) (x) converges to Π g (s, x) for every (s, x). Assumption 5 provides a control on the convergence rate. This assumption holds under the sufficient condition that for µ-almost every s and for every x ∈ dom ∂g(s, ·),
∂g 0 (s, x) ≤ β(s)(1 + x m/2 ) ,
where β is a positive random variable with a finite fourth moment [12].
Proof of Theorem 1
The proof of Theorem 1 employs the monotone operator theory. We begin by recalling some basic facts on monotone operators. All the results below can be found in [19,4] without further mention.
A set-valued mapping A : X ⇒ X on the Euclidean space X will be called herein an operator. An operator with singleton values is identified with a function. As above, the domain of A is dom(A) = {x ∈ X :
A(x) = ∅}. The graph of A is gr(A) = {(x, y) ∈ X × X : y ∈ A(x)}. The operator A is said monotone if ∀(x, y), (x ′ , y ′ ) ∈ gr(A), y − y ′ , x − x ′ ≥ 0.
A monotone operator with non empty domain is said maximal if gr(A) is a maximal element for the inclusion ordering in the family of the monotone operator graphs. Let I be the identity operator, and let A −1 be the inverse of A, which is defined by the fact that (x, y) ∈ gr(A −1 ) ⇔ (y, x) ∈ gr(A). An operator A belongs to the set M (X ) of the maximal monotone operators on X if and only if for each γ > 0, the so-called resolvent (I +γA) −1 is a contraction defined on the whole space X . In particular, it is single-valued. A typical element of M (X ) is the subdifferential ∂G of a function G ∈ Γ 0 (X ). In this case, the resolvent (I + γ∂G) −1 for γ > 0 coincides with the proximity operator prox γG . A skew-symmetric element of L(X , X ) can also be checked to be an element of M (X ).
The set of zeros of an operator A on X is the set Z(A) = {x ∈ X : 0 ∈ A(x)}. The sum of two operators A and B is the operator A+ B whose image at x is the set sum of A(x) and B(x). Given two operators A, B ∈ M (X ), where B is single-valued with domain X , the so-called Forward-Backward algorithm is an iterative algorithm for finding a point in Z(A + B). It reads
x n+1 = (I + γA) −1 (x n − γB(x n ))
where γ is a positive step.
In the sequel, we shall be interested by random elements of M (X ) as used in [12,5,14]. Consider a function A : Ξ → M (X ), where (Ξ, G , µ) is the probability space introduced at the beginning of Section 2. By the maximality of A(s), the graph gr(A(s)) is known to be a closed subset of X × X . By saying that A(·) is a M (X )-valued random variable, we mean that the function s → gr(A(s)) is measurable according to the definition of Section 2. When A(s) = ∂h(s, ·), where h : Ξ × X → (−∞, ∞] is a convex normal integrand such as h(s, ·) is proper µ-a.e., A is a random element of M (X ). Finally, when A(s) is a skew-symmetric element of L(X , X ) which is measurable in the usual sense (as a Ξ → L(X , X ) function), then it is also a random element of M (X ).
We now enter the proof of Theorem 1. Let us set Y = X × V, and endow this Euclidean space with the standard scalar product. By writing (x, λ) ∈ Y, it will be understood that x ∈ X and λ ∈ V.
For each s ∈ Ξ, define the set-valued operator A(s) on Y as
A(s, (x, λ)) = ï ∂g(s, x) ∂p(s, λ) ò ,
where A(s, (x, λ)) is the image of (x, λ) by A(s). Fixing s ∈ Ξ, the operator A(s, (x, λ)) coincides with the subdifferential of the convex normal integrand g(s, x)+p(s, λ) with respect to (x, λ). Thus, the map s → A(s) is a measurable Ξ → M (Y) function. Let us also define the operator B(s) as
B(s, (x, λ)) = ï ∂f (s, x) +L(s) T λ −L(s)x ò .
We can write B(s) = B 1 (s) + B 2 (s), where For the same reasons as for the operators A(s) and B(s), it holds that A, B, and A + B belong to M (Y). Moreover, recalling the system of inclusions (5), we also obtain that S = Z(A + B).
B 1 (s, (x, λ)) = ï ∂f (s, x) 0 ò , B 2 (s) = ï 0 L(s) T −L(s) 0 ò (B 2 (s) is
Defining the function b(s, (x, λ)) = ñ ‹ ∇f (s, x) +L(s) T λ −L(s)x ô (obviously, b(s, (x, λ)) ∈ B(s, (x, λ)) µ-a.e.), let us consider the following version of the Forward-Backward algorithm (x n+1 , λ n+1 ) = (I + γ n+1 A(ξ n+1 , ·)) −1 ((x n , λ n ) − γ n+1 b(ξ n+1 , (x n , λ n ))) .
On the one hand, one can easily check that this is exactly Algorithm (1). On the other hand, this algorithm is an instance of the random Forward-Backward algorithm studied in [5]. By checking the assumptions of Theorem 1 one by one, one sees that the assumptions of [5, Th. 3.1 and Cor. 3.1] are verified. Theorem 1 follows.
Remark 1
The convergence stated by Theorem 1 concerns the averaged sequence (x n ,λ n ). One can ask whether the sequence (x n , λ n ) itself converges to S. A counterexample is provided by the particular case X = V = R, f = g = p = 0, and L = 1 (proof omitted). A pointwise convergence would have been possible if A + B were so-called demipositive [5]. Note that in the previous counterexample, A + B = ï 0 1 −1 0 ò is not demipositive.
Remark 2 Constant step Forward-Backward algorithms usually require the operator B to be so-called cocoercive. This property is not needed if a decreasing step size is used [20,5].
a linear skew-symmetric operator written in a matrix form in Y). For each s ∈ Ξ, both these operators belong to M (Y), and dom B 2 (s) = Y. Thus, B(s) ∈ M (Y) by [4, Cor. 24.4]. Moreover, since both B 1 and B 2 are measurable, B is a M (Y)-valued random variable.Now, from the assumptions on the functions f, g, and p, we see that the operators A = E µ A and B = E µ B, where E µ is the selection integral (
. The step size sequence satisfies (γ n ) ∈ ℓ 2 \ ℓ 1 , and γ n+1 /γ n → 1 as n → ∞.
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|
[] |
[
"Time-correlated Photons from a In 0.5 Ga 0.5 P Photonic Crystal Cavity on a Silicon Chip",
"Time-correlated Photons from a In 0.5 Ga 0.5 P Photonic Crystal Cavity on a Silicon Chip"
] |
[
"Alexandre Chopin \nThales Research and Technology\nCampus Polytechnique, 1 avenue Augustin Fresnel91767PalaiseauFrance\n\nCentre de Nanosciences et de Nanotetchnologies\nCNRS\nUniversité Paris Saclay\nPalaiseauFrance\n",
"Inès Ghorbel \nThales Research and Technology\nCampus Polytechnique, 1 avenue Augustin Fresnel91767PalaiseauFrance\n",
"Sylvain Combrié \nThales Research and Technology\nCampus Polytechnique, 1 avenue Augustin Fresnel91767PalaiseauFrance\n",
"Gabriel Marty \nThales Research and Technology\nCampus Polytechnique, 1 avenue Augustin Fresnel91767PalaiseauFrance\n\nCentre de Nanosciences et de Nanotetchnologies\nCNRS\nUniversité Paris Saclay\nPalaiseauFrance\n",
"Fabrice Raineri \nCentre de Nanosciences et de Nanotetchnologies\nCNRS\nUniversité Paris Saclay\nPalaiseauFrance\n\nInstitut de Physique de Nice\nUMR 7010\nUniversité Côte d'Azur\nCNRS\nSophia AntipolisFrance\n",
"Alfredo De Rossi \nThales Research and Technology\nCampus Polytechnique, 1 avenue Augustin Fresnel91767PalaiseauFrance\n"
] |
[
"Thales Research and Technology\nCampus Polytechnique, 1 avenue Augustin Fresnel91767PalaiseauFrance",
"Centre de Nanosciences et de Nanotetchnologies\nCNRS\nUniversité Paris Saclay\nPalaiseauFrance",
"Thales Research and Technology\nCampus Polytechnique, 1 avenue Augustin Fresnel91767PalaiseauFrance",
"Thales Research and Technology\nCampus Polytechnique, 1 avenue Augustin Fresnel91767PalaiseauFrance",
"Thales Research and Technology\nCampus Polytechnique, 1 avenue Augustin Fresnel91767PalaiseauFrance",
"Centre de Nanosciences et de Nanotetchnologies\nCNRS\nUniversité Paris Saclay\nPalaiseauFrance",
"Centre de Nanosciences et de Nanotetchnologies\nCNRS\nUniversité Paris Saclay\nPalaiseauFrance",
"Institut de Physique de Nice\nUMR 7010\nUniversité Côte d'Azur\nCNRS\nSophia AntipolisFrance",
"Thales Research and Technology\nCampus Polytechnique, 1 avenue Augustin Fresnel91767PalaiseauFrance"
] |
[] |
Time-correlated photon pairs are generated by triply-resonant Four-Wave-Mixing in a In0.5Ga0.5P Photonic Crystal cavitiy. Maximal efficiency is reached by actively compensating the residual spectral misalignment of the cavity modes. The generation rate reaches 5 MHz in cavities with Q-factor ≈ 4 × 10 4 , more than one order of magnitude larger than what is measured using ring resonators with similar Q factors fabricated on the same chip. The Photonic Crystal source is integrated on a Si photonic circuit, an important asset for applications in quantum technologies.
| null |
[
"https://arxiv.org/pdf/2202.09622v1.pdf"
] | 247,011,146 |
2202.09622
|
0f511da72eca9243ce1c6b274248e427b4ea367c
|
Time-correlated Photons from a In 0.5 Ga 0.5 P Photonic Crystal Cavity on a Silicon Chip
Alexandre Chopin
Thales Research and Technology
Campus Polytechnique, 1 avenue Augustin Fresnel91767PalaiseauFrance
Centre de Nanosciences et de Nanotetchnologies
CNRS
Université Paris Saclay
PalaiseauFrance
Inès Ghorbel
Thales Research and Technology
Campus Polytechnique, 1 avenue Augustin Fresnel91767PalaiseauFrance
Sylvain Combrié
Thales Research and Technology
Campus Polytechnique, 1 avenue Augustin Fresnel91767PalaiseauFrance
Gabriel Marty
Thales Research and Technology
Campus Polytechnique, 1 avenue Augustin Fresnel91767PalaiseauFrance
Centre de Nanosciences et de Nanotetchnologies
CNRS
Université Paris Saclay
PalaiseauFrance
Fabrice Raineri
Centre de Nanosciences et de Nanotetchnologies
CNRS
Université Paris Saclay
PalaiseauFrance
Institut de Physique de Nice
UMR 7010
Université Côte d'Azur
CNRS
Sophia AntipolisFrance
Alfredo De Rossi
Thales Research and Technology
Campus Polytechnique, 1 avenue Augustin Fresnel91767PalaiseauFrance
Time-correlated Photons from a In 0.5 Ga 0.5 P Photonic Crystal Cavity on a Silicon Chip
Time-correlated photon pairs are generated by triply-resonant Four-Wave-Mixing in a In0.5Ga0.5P Photonic Crystal cavitiy. Maximal efficiency is reached by actively compensating the residual spectral misalignment of the cavity modes. The generation rate reaches 5 MHz in cavities with Q-factor ≈ 4 × 10 4 , more than one order of magnitude larger than what is measured using ring resonators with similar Q factors fabricated on the same chip. The Photonic Crystal source is integrated on a Si photonic circuit, an important asset for applications in quantum technologies.
Parametric down-conversion, either through a second order or third order nonlinear optical process, underlies the emission of correlated photon pairs, entanglement [1] and squeezing. The miniaturization and the integration of sources based on these processes plays an essential role in quantum technologies [2]. Maximizing the efficiency of the parametric interaction implies improved generation rate, entanglement, antibunching and the decrease of the pump power level [3,4]. Ultimate efficiency (i.e. single-photon nonlinearity) leads to deterministic quantum gates [5]. Considering resonantly enhanced pair generation through Spontaneous Four-Wave Mixing (SFWM), its rate R (or brilliance) depends [6] on the Kerr nonlinear index n 2 , on the Qfactor of the resonances, on the effective volume for the nonlinear interaction V χ [7] and on the pump power P delivered to the resonator, namely:
R ∝ n 2 2 Q 3 V 2 χ P 2(1)
Therefore, the rate can be enhanced by design (increasing Q, decreasing the size of the resonator) or choosing a material with a larger n 2 . Ring resonators made of AlGaAs, a group III-V semiconductor with sizable Kerr nonlinearity, have achieved a high (MHz) internal generation rate with 10 µW on-chip pump [3]. On the other hand, due to scattering at sidewalls, the radius of high-Q rings made of semiconductors is hardly smaller than 10 µm. In contrast, high-Q photonic crystal (PhC) resonators with similar Q-factor confine the field in a volume which is orders of magnitude smaller [8], thereby raising high expectations for a very strong nonlinear interaction. Yet, efficiency is reached only when the interacting "pump", "signal" and "idler" fields are simultaneously on resonance with the corresponding cavity modes. This condition, trivially satisfied in ring resonators, is extremely challenging in PhC. Triplyresonant SFWM was reported in a multi-mode resonator made of three coupled single-mode PhC cavities [9]. The normalized generation rate (300 Hz µW −2 ) is large, considering the moderate Q-factor (≈ 5×10 3 ), owing to the small volume of the resonator. Correlated photon pairs have been reported in PhC waveguides [10] and coupled cavities resonator waveguides [11], both with a pulsed pump and peak power level well above 100 mW. A novel tuning technique, exploiting the inhomogenous thermo-optic effect, has enabled triply-resonant Four-Wave-Mixing (FWM) in multi-mode high-Q (Q > 10 5 ) PhC, ultimately leading to the PhC Optical Parametric Oscillator [7]. We show that this approach, here modified to operate with a fixed wavelength pump, enables time-correlated pairs are emitted at maximal efficiency (i.e. following the Q 3 scaling) within a PhC resonator, integrated with a Silicon Photonic circuit.
The nanobeam PhC cavity [12] is made of In 0.5 Ga 0.5 P, a group III-V semiconductor alloy grown lattice-matched to GaAs. Its electronic bandgap (1.89 eV) is large enough to suppress Two-Photon-Absorption (TPA) of a pump in the Telecom spectral range [13]. Thus, large electric fields can be established, enabling the observation of temporal solitons at chip scale [14]. Moreover, the residual linear absorption rate is about 1.5 × 10 −8 s −1 , which is extremely small, compared to other direct gap semiconductors [15]. The period of the holes is tapered to establish an effective harmonic potential for the field, involving nominally spectrally equispaced resonances at frequencies (ω − , ω 0 , ω + ), Fig. 1(a). The PhC is coupled through evanescent field to an underlying Silicon photonic wire, Fig.1(b). The full details on the fabrication and FWM process are given in Ref. [16] and in the Supplemental Material. The cavities considered hereafter have quality factors ranging from 2 × 10 4 to 10 5 and resonances between 192 THz and 194 THz (1545 nm -1560 nm). The same chip contains ring resonators fabricated simultaneously with the PhC resonators, with a radius of 30 µm and comparable Q factors. Compared to SFWM in a ring resonator, a specific fea- ture here is the tuning process and the dependence of the emission rate on the injected pump power and the detuning. This stems from the markedly inhomogeneous and only partially overlapped distribution of the field of the interacting modes. Consequently [7], when the cavity heats up due to pump dissipation, the modes experi-
FIG. 2.
Achievement of equispaced resonances through thermal sweep and fixed pump wavelength. (a) Transmitted pump, after waveguide couplers (circles, line is a guide for the eyes) as the sample temperature is decreased; (b) corresponding resonance frequency ω0 (circles), solid and dashed lines represent the pump ωp and the thermo-optic drift of ω0 respectively; (c) corresponding measured FSR (markers), fit (solid lines) and value at uniform temperature (dashed lines); insets represent the aligned / not aligned configurations. ence a differential thermo-refractive effect, which compensates for the unavoidable fabrication tolerances. The pump laser is operated at a fixed frequency ω p , larger than the ω 0 resonance at sample temperature T=28°C, the overline symbol (¯) denotes hereafter the absence of internal dissipation, hence homogeneous temperature in the sample. The transmission decreases, as the temperature is decreased below T=26°C, until the bistable jump is reached at T=19°C, Fig. 2(a). Optical Coherent Tomography (OCT) (see Supplemental Material, Ref. [17]) is used to track the corresponding thermal drift of the resonances, shown in Fig. 2(b). As ω 0 approaches the pump ω p , the energy stored in the mode, hence the internal dissipation, increase and ω 0 is clamped to the pump (within 26°C -19°C); the drift ω 0 −ω 0 is linearly related to the sample temperature, and can be approximated by the pump offset ∆ 0 = ω p −ω 0 ≈ ω 0 −ω 0 . The differential thermo-optic effect is apparent in Fig. 2(c), as the intervals ∆ω ± = |ω ± − ω 0 | change until they equalize at a given temperature (25°C here). Let us consider the mismatch 2D = ∆ω + − ∆ω − . A welldefined temperature of the sample, at which the FSR are equalized and FWM is triply resonant, will exist only if the mismatch at uniform sample temperature is negative (2D < 0 ). This is related to the sign of the differential thermo-optic effect. If this condition is satisfied, the pump power needs only to be large enough to keep the resonance locked until triple resonance is reached. The fixed pump frequency is chosen such that signal and idler are both aligned with the channels of the DWDM (dense wavelength division multiplexing) filter at the output of the sample, Fig. 3(a). A bandpass (BP) filter is also used to remove the amplified spontaneous emission from the pump laser before entering the sample. Considering that the on-chip pump power P is roughly 250 µW and that most of it is reflected back on resonance (side-coupled cavity geometry), the 110 dB overall pump suppression at the detectors, ensured by cascading a BP filter on the signal and idler channels, is enough for our purpose. Two single-photon avalanche photodiodes (SPAD) (quantum efficiency η q = 20 %, deadtime τ D = 15 µs) are operated in gated mode (signal frequency f = 3 MHz, duty cycle σ = 0.5) and connected to a time correlator (TC), as shown in Fig. 3(a). The count of single photons on each channel (N 1 ,N 2 ) increases by orders of magnitude above the background simultaneously in both channels as the triply resonant FWM is established (see Supplemental Material). A raw coincidences histogram, Fig. 3(b), is collected as a function of the sample temperature with integration time T int . The raw count of coincidences C raw (the sum of all coincidences over the main peak of width 2T j = 300 ps, T j the timing jitter of the detectors) is corrected with the estimate of the accidental counts A (summing over the same width outside the peak) to generate the "true" coincidences C T = C raw −A. The Coincidence to Accidental Ratio is CAR = C T /A, following Ref. [18].
To cope with the saturation of the SPADs as the count rate increases, further correction is applied and real co- [19]. Finally, the coincidence rate is then
incidences are C = C T (1 − N 1 τ D ) −1 (1 − N 2 τ D ) −1 σ −1R det = C/T int (see Supplemental Material)
The on-chip pair generation rate R, that would be relevant for instance in a quantum chip, is deduced by taking into account the photon loss in the two channels α i and α s from the Silicon wire all the way to the detectors considering their quantum efficiency [6], hence R = R det /α i α s . The attenuation ranges between 25 dB and 30 dB, depending on the sample. The main contributions are the tunable BP filters, the chip to fiber grating couplers and quantum efficiency of the SPADs. Moreover, in contrast with ring resonators, side-coupled PhC emit signal and idler photons with equal probability in both directions of the Silicon waveguide, giving an additional 3 dB loss. Fig. 3(c) shows the coincidence rate R as the cavity is tuned towards its maximum efficiency by changing its baseline temperature, pump wavelength and power being fixed. A clear maximum, sharply emerging from noise is observed before a decrease due to a deviation from the triply resonant configuration. The sudden drop after the bistable jump clearly indicates that the system is then out of resonance. Yet, the maximum is reached after crossing the point of triple resonance, Fig. 3(c,d). This is explained by a model is introduced hereafter. Starting from the connection between spontaneous and stimulated FWM [20], the spontaneous emission rate is:
R =ˆη χ dν(2)
the probability of stimulated conversion η χ integrated over all the possible combinations of signal and idler frequencies satisfying the photon energy conservation. It can be expressed in the limit of undepleted pump and low parametric gain [7]. Combining this to eq. 2, under the approximation that the intra cavity energy scales linearly with the pump offset (true if the absorption is linear), gives:
R = η max χ 4 ∆ 2 0 ∆ 2 bist Γ − Γ + (Γ − + Γ + ) (Γ − + Γ + ) 2 + 16D 2 1 − ∆0 ∆opt 2 (3)
where η max χ the maximum conversion probability at a given pump power, Γ −,0,+ are the total cavity damping rate. ∆ bist (∆ opt ) is the value of ∆ 0 for which the bistable jump (triply resonant configuration) is reached (see Supplemental Material). The model (Eq. 3) is superimposed to the measurement in Fig. 3(c), with all these parameters being measured independently (see Supplemental Material) and none fitted. By inspecting the equation we note that R depends on ∆ 0 via two terms: one is maximized when ∆ 0 = ∆ opt , i.e. triply resonant FWM, the other increases with ∆ 2 0 . So, R continues to increase although the triply resonance is loosely satisfied (∆ 0 − ∆ opt < Γ). The agreement with the experimental points is remarkable. The model predicts that the optimal choice of the pump power is such that the bistable jump occurs just a little after the triply resonant FWM. This condition is shown in Fig. 3(d), using cavity B, where a 5 MHz maximum on chip generation rate is estimated with roughly the same on-chip power as cavity A. Crucially, it demonstrates that the device can be tuned to its maximum efficiency which ultimately depends on the Q factor. Let us compare the PhC with a ring resonator. The radius is 30µm, the same as in Ref. [3], and the Q factor is roughly the same as in the PhC (see Supplemental Material). As shown in Fig. 4(a), the generation rate R scales with the square of the pump, as expected. Yet, it is apparent that for PhC cavity made of the same material, R is much larger by at least one order of magnitude, when normalized to the pump power. Fig. 4(b) shows that in rings, the CAR decreases with the generation rate, indicating that accidental counts are mostly due to lost pairs, hence insertion loss [18]. The corresponding measurement for PhC reveals that the brillance is much larger at the same CAR. The generation rate is larger for PhC because of a much tighter confinement in PhC. This is apparent in Fig. 4(c), as the maximum generation rate, normalized to the pump power, scales as the square of the inverse of V χ , which is an order of magnitude larger in the rings. Let us note that V χ takes into account the spatial distribution and mutual overlap of the interacting modes, therefore it is distinct from the physical volume of the resonator; indeed, the difference in terms of footprint is even more pronounced. In summary, we have reported time-correlated pair generation from a PhC resonator systematically operated at its maximum efficiency, owing to a thermal tuning mechanism. This only requires a fixed wavelength laser pump, removing the need for a tunable source, which is convenient for integration. The agreement with theory is within experimental error, with no need of fitting parameters. Simple scaling with volume is demonstrated by comparison with a ring resonator fabricated on the same chip. As the generation rate R ∝ Q 3 , we extrapolate a substantial (two orders of magnitude) improvement when using resonators with Q = 2 × 10 5 as for the recently demonstrated PhC OPO [7]. This would essentially match the very recent achievement of ultrabright source made of AlGaAs [3]. The integration of the PhC with a photonic circuit leaves the possibility to include the essential functions of pump suppression and signal/idler separation, and in perspective, to operate several of such sources simultaneously.
FIG. 1 .
1Parametric PhC source. (a) Squared fields (calculated) of the first four modes (filled curves) of the PhC cavity and their envelopes (dashed lines) along the axis x, with x = 0 the center of the beam; pump (ωp), signal (ωs) and idler (ωi) photons are resonant with modes ω0, ω−, ω+ in the SFWM process. (b) SEM image of the In0.5Ga0.5P PhC resonator on the Silica layer (left) and representation of the hybrid III-V PhC on Silicon layer stack (right)
FIG. 3 .
3Photon pairs generation in PhC. (a) Simplified time-correlation measurement set-up: Dense wavelength division multiplexing (DWDM), band pass filters (BPF), Power meter (PM), single-photon avalanche photodiode (SPAD), Time Correlator (TC). (b) Raw coincidence (at max. rate 5 MHz) histogram with integration time Tint, time bin 90 ps. (c,d) On-chip pair generation rate R (bottom) and corresponding FSR (top) as a function of the pump offset, controlled by the temperature; symbols and error bars are experimental points, solid line is theory with no fitting parameters; the vertical dashed line marks the equal FSR; (c,d) correspond to cavities (A,B).
FIG. 4 .
4SFWM in PhC (circles) and ring (squares) resonators. (a) On-chip pair generation rate vs. power (markers) and P 2 fit (dashed line); (b) corresponding CAR vs. generation rate (same color and symbol code); (c) Generation rate on chip normalized with pump power vs. the nonlinear interaction volume Vχ and the footprint of the device. Circles (squares) denote PhC (rings).
This work was Funded by the French National Research Agency (ANR) under the contract COLOURS (ANR-21-CE24-0024). The authors thank Grégory Moille for the design of the rings used here. Thierry Debuisschert and Kamel Bencheikh for discussions and clarifications. Laurent Labonté and Sébastien Tanzilli for having provided assistance in preliminary experiments in their facilities. See Supplemental Material at [URL will be inserted by publisher] for the derivation of the theory and for details about the experiments.
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"\nDepartment of Electrical and Computer Engineering\nUniversity of Alberta\nCanada\n"
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"Department of Electrical and Computer Engineering\nUniversity of Alberta\nCanada"
] |
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In recent years, under deregulated environment, electric utility companies have been encouraged to ensure maximum system reliability through the employment of costeffective long-term asset management strategies. To help achieve this goal, this research proposes a novel statistical approach to forecast power system asset population reliability. It uniquely combines a few modified Weibull distribution models to build a robust joint forecast model. At first, the classic age based Weibull distribution model is reviewed. In comparison, this paper proposes a few modified Weibull distribution models to incorporate special considerations for power system applications. Furthermore, this paper proposes a novel method to effectively measure the forecast accuracy and evaluate different Weibull distribution models. As a result, for a specific asset population, the suitable model(s) can be selected. More importantly, if more than one suitable model exists, these models can be mathematically combined as a joint forecast model to forecast future asset reliability. Finally, the proposed methods were applied to a Canadian utility company for the reliability forecast of electromechanical relays and the results are discussed in detail to demonstrate the practicality and usefulness of this research.Index Terms-Weibull Distribution, Power System Reliability, Asset ManagementMing Dong (S'08, M'13, SM'18) received his doctoral degree from in 2013. Since graduation, he has been working in various roles in two major electric utility companies in West Canada as a Professional Engineer (P.Eng.) and Senior Engineer for more than 5 years. In 2017, he received the Certificate of Data Science and Big Data Analytics from Massachusetts Institute of Technology. His research interests include applications of artificial intelligence and big data technologies in power system planning and operation, power quality data analytics, power equipment testing and system grounding.Alexandre Nassif (S'05, M'09, SM'13) is a specialist engineer in ATCO Electric. He published more than 50 technical papers in international journals and conferences in the areas of power quality, DER, microgrids and power system protection and stability. Before joining ATCO, he simultaneously worked for Hydro One as a protection planning engineer and Ryerson University as a post-doctoral research fellow. He holds a doctoral degree from the University of Alberta and is a Professional Engineer in Alberta.
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10.1109/tpwrs.2018.2877743
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[
"https://arxiv.org/pdf/1805.10420v1.pdf"
] | 44,136,786 |
1805.10420
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42db1d5563a8541b57ad40e8e11bfb61db9f8e9f
|
Department of Electrical and Computer Engineering
University of Alberta
Canada
In recent years, under deregulated environment, electric utility companies have been encouraged to ensure maximum system reliability through the employment of costeffective long-term asset management strategies. To help achieve this goal, this research proposes a novel statistical approach to forecast power system asset population reliability. It uniquely combines a few modified Weibull distribution models to build a robust joint forecast model. At first, the classic age based Weibull distribution model is reviewed. In comparison, this paper proposes a few modified Weibull distribution models to incorporate special considerations for power system applications. Furthermore, this paper proposes a novel method to effectively measure the forecast accuracy and evaluate different Weibull distribution models. As a result, for a specific asset population, the suitable model(s) can be selected. More importantly, if more than one suitable model exists, these models can be mathematically combined as a joint forecast model to forecast future asset reliability. Finally, the proposed methods were applied to a Canadian utility company for the reliability forecast of electromechanical relays and the results are discussed in detail to demonstrate the practicality and usefulness of this research.Index Terms-Weibull Distribution, Power System Reliability, Asset ManagementMing Dong (S'08, M'13, SM'18) received his doctoral degree from in 2013. Since graduation, he has been working in various roles in two major electric utility companies in West Canada as a Professional Engineer (P.Eng.) and Senior Engineer for more than 5 years. In 2017, he received the Certificate of Data Science and Big Data Analytics from Massachusetts Institute of Technology. His research interests include applications of artificial intelligence and big data technologies in power system planning and operation, power quality data analytics, power equipment testing and system grounding.Alexandre Nassif (S'05, M'09, SM'13) is a specialist engineer in ATCO Electric. He published more than 50 technical papers in international journals and conferences in the areas of power quality, DER, microgrids and power system protection and stability. Before joining ATCO, he simultaneously worked for Hydro One as a protection planning engineer and Ryerson University as a post-doctoral research fellow. He holds a doctoral degree from the University of Alberta and is a Professional Engineer in Alberta.
I. INTRODUCTION
Owdays,under deregulated environment, electric utility companies are encouraged to reduce overall cost while maintaining system reliability risk at an acceptable level. To achieve this goal, understanding and forecasting the reliability trends of different asset populations is the key. Sophisticated and optimal asset management measures can only be established based on the accurate forecasting of asset reliability change in the future.
Previously, the standard age based Weibull distribution has been widely used in reliability engineering as a statistical tool to model equipment aging failures [1][2][3][4]. However, this classic model cannot effectively incorporate additional information such as asset health condition data, asset warranty, energization delay, asset infant mortality period and minimum spare requirements which many electric utility companies often have to consider. To resolve this problem, this paper proposes a novel statistical approach to forecast power system M. Dong is with ENMAX Power Corporation, Calgary, AB, Canada, T2G 4S7 (e-mail: [email protected])
A. Nassif is with ATCO Electric, Edmonton, AB, Canada, T5J 2V6 (emails: [email protected]) asset population reliability. The main contributions of this paper include: it proposes four modified Weibull distribution models in response to additional information and considerations a power system may have; it proposes a unique method to measure the forecast accuracy of different Weibull distribution models. Based on this method, suitable Weibull distribution model(s) can be identified for a specific asset population. it proposes a unique method to combine different Weibull distribution models as a joint forecast model. The flowchart of the proposed methods is shown in Fig.1. In the beginning, the operation status data of a specific asset population is analyzed, converted to the cumulative failure probability table. This table is then split into training records and testing records. Training records are used to model the asset failure progression using different Weibull distributions Combining Modified Weibull Distribution Models for Power System Reliability Forecast Ming Dong, Senior Member, IEEE and Alexandre B. Nassif, Senior Member, IEEE N models. Testing records are used to evaluate the forecast accuracy of the distribution models. The suitable Weibull distribution model(s) can be identified based on the test results. When there is more than one suitable Weibull distribution models, the models can be mathematically combined as a joint model. In the next stage, the combined joint model is applied to the broader asset population data and the future reliability change of this population can be forecasted.
This paper firstly describes the required asset operation status dataset as the foundation before applying the proposed methods. Then it reviewed the standard Weibull distribution for aging analysis. Based on this, a few modified Weibull distributions including health index based Weibull distribution, X-shifting Weibull distribution, Y-shifting Weibull distribution and XY-shifting Weibull distribution are explained with reference to practical scenarios in power systems. A few methods to determine Weibull distribution parameters such as Maximum Likelihood Estimation (MLE), Least Squares Estimation (LSE) are reviewed. After this, the proposed methods of evaluating and combining Weibull distribution models are presented. Then the process of forecasting asset reliability using the produced joint forecast model is explained. Finally, this paper provides a real example of applying proposed methods to forecast future electromechanical relay failures and spare requirements for a Canadian utility company.
II. ASSET OPERATION STATUS DATASET
Nowadays, in light of the significant value embedded in data, many electric utility companies have employed sophisticated Computerized Maintenance Management Systems (CMMS) to track and store various asset data [5][6]. For the purpose of asset reliability forecast, the asset operation status data is used. It is organized by asset type and asset population. Asset operation status data must include age and operating statuses (i.e. working or failed). The data is continuously recorded as new asset gets installed and old asset retires due to failures. Just like any data mining tasks, low quality input data will undoubtedly lead to inaccurate data observations and analysis [7].Understanding the data requirements of asset operation status dataset and preparing qualified asset operation status dataset is a key step towards the successful implementation of the methods proposed in this paper. It should be noted that: One type of asset often contains many sub types due to different technology adoptions and manufacturing standards. The sub types should be manually identified by utility asset engineers based on equipment domain knowledge and separated into different datasets. For example, underground cables normally have paper insulated cables, cross-linked Polyethylene (XLPE), Ethylene Propylene Rubber (EPR) [8] and these cables are often in service at the same time in the same utility company's power system because they were installed at different times in history adopting different technologies. These cables demonstrate different failure characteristics and should be viewed as different asset types and analyzed separately; another example is mineral oil immersed power transformers. Power transformers manufactured before 1970 are rated at 55 temperature rise. Power transformers manufactured in 1970s are rated 55/65 and became 65 after 1970s till today [9]. Since the failure characteristics of transformers are closely related to winding and top-oil temperature rises, power transformers should be separated into different datasets by temperature rating or approximately by manufacturing time. The same type of asset can be installed in different ways.
For example, underground cables can be direct buried in soil or installed in PVC conduit. In general, the direct buried cables are more prone to fail due to soil corrosion and moisture ingress [10] ; distribution and transmission power poles are constructed as tangent structures (carrying straight conductors), angle structures (having deflection angles), dead-end structures (at the end of the line) or transformer poles(carrying mounted transformers). These structures incur different mechanical forces which lead to different failure characteristics. Similarly to dealing with the sub types, asset with different installations should be separated into different populations and form different datasets. Utility companies may also perform inspection or testing for certain asset types and populations. In cases like this, the latest inspection/testing data can be linked with the age and status data. This is because the modified Weibull distribution models this paper proposes can incorporate asset condition data and derive health index based models. Asset operation status dataset used by the proposed methods does not have to be the entire population for a specific type of asset. Oftentimes, a randomly sampled population fraction with hundreds of records will be good enough to establish a confident Weibull distribution model. Data completeness is another consideration. Sometimes only limited members in a population have inspection or testing data and these members should be utilized for establishing the model. The established model can still be applied to the entire population for a later stage.
As an example, a 138 kV in-duct XLPE transmission underground cable operation status dataset is shown in Table I which follows the above discussed data requirements.
III. STANDARD AGE BASED WEIBULL DISTRIBUTION
The Weibull distribution is a continuous probability distribution widely used in the field of reliability engineering [1][2][3][4]. The standard probability density function of equipment age is
( ) ( ) ( ) ( )(1)
where >0 is the scale parameter; >0 is the shape parameter; A in this application is the equipment age; f(A) is the probability of failure. The corresponding cumulative Weibull distribution function is given as:
( ) ( )(2)
where F(A) is the cumulative probability of failure. shows an example of cumulative Weibull distribution function for a certain type of equipment. As age progresses, the cumulative probability of failure increases until it reaches 100% at a later age. This is a statistical description of an asset aging failure process and can be used for equipment survival analysis [11].
This model can be derived from the asset operation status dataset discussed in Section II. For a specific age A, the observed cumulative failure probability is
̂( ) ( )
where is the total number of failed asset in the dataset with an age less than or equal to A； is the total number of working asset in the dataset with an age greater than age A. Based on this calculation, a new table can be produced to depict the relationship of Age A and the observed corresponding cumulative failure probability ̂( ). Table II shows an example of cumulative failure probability vs. age. ̂( ) increases to 100% at the age of 35. It should be noted sometimes due to data availability, observed ̂( ) may not immediately change between consecutive years such as when A=3 and 4 in this case. This does not impact the modelling as the observed ̂( ) is based on frequency and does not always comply with ( ). Table II will be split into training and testing records. The training records will be fed into the Weibull distribution modeling module as shown in Fig.1, following the methods explained in Section V to estimate the parameters and . IV. MODIFIED WEIBULL DISTRIBUTIONS This paper proposes a few modified Weibull distribution models. The physical meanings of these distributions and their applications are discussed in this section.
A. Two-Parameter Health index based Weibull Distribution
Equation (2) only considers age as the sole factor in asset's failure progression modeling. This could be accurate for assets that are used in a near homogenous environment where operating conditions such as loading level and temperatures are about the same. However, for a large power system where operating conditions vary significantly, it is not uncommon to find a young asset with higher failure probability and an old asset with a lower failure probability. In cases like this, the classic age based Weibull distribution modeling will not be able to provide accurate forecast. However, as mentioned in Section II, if additional asset condition information is available, the independent variable A can be converted to health index H as below.
{ ( ) ( ) ∑ ∑ ( )
where A is normalized asset age; is the weighting factor for asset age;
is the weighting factor for condition attribute; is normalized condition attribute. The above equation uses different weighting factors to combine age and other condition data to generate an index between 0 and 100 [12]. Similar to using age alone, health index H progresses from 0(brand new, all healthy) to 100 (most unhealthy observation in the system). The higher H is, the higher the corresponding failure probability F(H) is. As can be seen, the standard age based Weibull distribution (2) is just a special case of (4) when other condition weighting factors = 0 and =1. Fig.3. shows an example of health index based Weibull distribution.
This model can be derived from the asset operation status dataset discussed in Section II. For a specific health index H, the observed cumulative failure probability
̂( ) ( )
where is the total number of failed asset in the dataset with a health index greater than or equal to H;
is the total number of working asset in the dataset with a health index greater than H. Based on this calculation, Table III can be produced to depict the relationship between health index H and cumulative failure probability F(H). Table III shows an example of cumulative failure probability vs. health index.
B. X-Shift Weibull Distribution
X-Shift Weibull distribution has one additional parameter (
). This parameter shifts the Weibull distribution curve along X axis. Fig.4. shows an example of X-Shift Weibull distribution. Mathematically, it is given as below:
{ ( ) ( ) ∑ ∑ ( )
The physical meanings of parameter in the power system context can mean the following: A failure-free period: some assets naturally have an extremely low failure probability during a certain initial period. This is more true for assets in protected, controlled and sometimes enclosed environment, for example a GIS switchgear almost never fails in the first few years. Asset warranty and insurance: some assets are warranted by manufacturers. This implies assuming free of failure during the warrantied period of time. Even if there is a failure, the manufacturer could cover the cost and loss of the failure and from the utility perspective, it is like "failure-free".
Energization delay: in power systems, after the installation of an asset, there could be an energization delay before this asset is energized and put into actual use. For example, an underground feeder system is often constructed by stages. The cable is often installed at an early stage and will stay in place for a few months or even years before the other parts of system get constructed. In the end, the entire system will be energized at the same time. This kind of construction is very common for the development of green field projects. In this case, cable will not fail before energization time. When >0, this distribution curve is shifted upward and the physical meaning is to capture the initial failure probability associated with an infant mortality period [13][14]. In reliability engineering, some assets could easily fail at the initial period and the failure probability will reduce to stable stage after this initial period, this period is called the infant mortality period. Reflected by the cumulative distribution function, this initial failure can be described as a great than zero cumulative failure probability when H is zero. Another occasion for applying a positive is when making spare forecast for critical asset. A minimum spare level is required even in the early time to ensure maximum system reliability.
D. XY-Shift Weibull Distribution
XY-Shift Weibull distribution has two additional parameter and to allow the Weibull distribution curve to shift along both X axis and Y axis Fig.6. shows an example of Y-Shift Weibull distribution. Mathematically, it is given as below. Table II and Table III contain the cumulative failure probability records that can be used to estimate Weibull distribution parameters and establish the models. However, not all the data should be used at the stage of model establishment. This is because this paper uses a unique method to evaluate and combine multiple Weibull distribution models. Similar to a typical supervised learning process [15], Table II and Table III should be split into two parts: Training records: the records that are used to estimate Weibull distribution parameters. A certain percentage, for example 80% of the data, can be randomly picked and grouped as training records. When picking the records, a uniform distribution can be applied to sample the entire range of H and F(H). This is to ensure the entire failure progression can be modeled with enough supporting data. Testing records: the remaining records that will be used later to measure the accuracy of a model. There are many existing ways of estimating two-parameter Weibull distribution parameters and . A traditional method is the Maximum Likelihood Estimation (MLE) method [16]. Considering the Weibull probability density function given in (1), the likelihood function is given as: (9) Taking the natural logarithm of both sides and partially differentiating with respect to and , we get:
{ ∑ ∑ ∑ ( )
Solving the above equations, we get:
∑ ∑ ∑ ( )
can be solved using Newton-Raphson method or any other numerical method. When ̂ is obtained, the value of ̂ can be obtained by using:
̂ ∑ ̂ ( )
Least squares estimation (LSE) is another common method used for Weibull distribution parameter estimation [16].The cumulative Weibull distribution function can be linearized by taking logarithmic transformation to the form of:
( )
Proper p and q should be chosen to minimize the sum of squared errors:
∑( ) ( )
The then unbiased estimators can be calculated as:
{ ̂ ∑ ∑ ∑ ∑ (∑ ) ̂ ̂ ( ̂) ( )
In addition to MLE and LSE, there are also some modern optimization algorithms for estimating Weibull distribution parameters such as using simulated annealing algorithm [17] and genetic algorithm [18] As discussed in Section IV., X-Shift, Y-Shift and XY-Shift Weibull distributions have additional shifting parameters. However, these parameters are often not random parameters and have physical meanings. Therefore based on human expert's domain knowledge, these parameters can be initialized. A practical way is to initialize these parameters to discrete values in a numerical range. For example, if we know that the typical energization delay for a transmission cable population is 1-2 years, can be set to 1.0,1.25, 1.5, 1.75 and 2.0. For any of these five values, the corresponding X-Shift Weibull distribution will become equivalent to a twoparameter Weibull Distribution model and can therefore estimate the remaining parameters and using the methods discussed above. Instead of having to choose optimal or at this stage, we can keep all established models with discrete or values. They can be all evaluated, selected and combined as one joint model, which will be discussed in Section VI.
VI. EVALUATING AND COMBINING WEIBULL DISTRIBUTION
MODELS
As discussed in Section V, the asset operation status dataset is split to two parts: training records and testing records. The testing records are now used to evaluate the accuracy of the established models.
A. Evaluating Weibull distribution models
Suppose S is the set of established Weibull distribution models with different forms and parameters. We have:
* + ( )
For each model , we can calculate its Mean Squared Error (MSE) for the testing records, which is given by:
∑[ ̂( ) ( )] ( )
Only testing records are used for evaluation because testing records were not used previously for model parameter estimation and they are unknown points to this established model. Traditionally, Weibull distribution models were not evaluated based on pre-selected testing data and therefore cannot effectively describe the model's forecasting capability when cope with unknowns. This paper proposes this new evaluation method for Weibull distribution and it has two advantages: A good MSE on testing records indicate the true forecast accuracy of this model and can be generalized to describe the forecasting capability for future unknown data using this model. This is especially important for health index based models because a calculated health index using (4) can be any real value in [0,100] and the original asset operating status dataset used for Weibull distribution modelling may not contain these values. Furthermore, as explained in Section V., the training records are intentionally selected following a uniform distribution so that the corresponding testing records will also be able to cover the entire range of available H and F(H) data. The MSE therefore can indicate the forecast capability of the entire failure progression. Similar ideas of using training and testing sets have been applied very successfully in many supervised learning applications [15].
Suitable models are defined as a subset where for an its MSE for testing records is less than a pre-defined maximum threshold. The other way to choose is to rank all MSEs for and only choose the top models with smallest MSEs, for example the top 3 models. Only is taken into the next step to establish the joint forecast model.
B. Combining Weibull Distribution Models as one joint model
On the basis of evaluating Weibull distribution models, this paper further proposes a powerful method to combine different Weibull distribution models as one joint forecasting model. This was never studied before when traditionally only single Weibull distribution model is applied for a forecasting task. One challenge utility asset engineers face is that they cannot pre-determine which Weibull distribution model will be able to yield the most accurate forecast. They also cannot predetermine the most optimal shifting parameters although they know a probable range these parameters should fall under. A joint model can keep all the suitable models and leverage the strength of each model towards the final forecast output. Similar ideas can be found in some ensemble learning methods which attempt to build an accurate joint classifier using multiple weaker classifiers [15]. The joint model can be created as:
( ) ∑ ( )( ∑ ) ( )
In ( ), given a new health index input H, a weak model's output ( ) with a higher MSE will be assigned with a smaller weighting factor while a strong model's output ( ) with a lower MSE will be assigned with a bigger weighting factor. All suitable models' outputs will be averaged by the weighting factors to produce a combined output ( ) This ( ) is the final forecasted cumulative failure probability for input H. In many cases, the joint model can have a smaller MSE than any single model. If , this method is equivalent to selecting the best model.
VII. ASSET RELIABILITY FORECAST USING THE MODELS
The joint forecasting model ( ) can be used to forecast asset reliability in the future [19].
For a specific future time span [ , a single asset's forecasted failure probability during this time span is:
{ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
where and are the time intervals in years from today till and ; H is today's asset health index; and are the future health indexes at and ; ( ) is a health index estimator and are dependent on the initial health index H and time interval . This estimator can be derived based on statistically checking historical asset health index values that are close to H and their average change after a time interval close to . It can also be approximated based on the current H, A and using the shown equations. This approximation is more accurate when is much higher than or when most condition attributes are strongly correlated with age, which in ma ny cases are true.
If only age is used in Weibull distribution modeling, the above equations can be simplified as:
{ ( ) ( ) ( ) ( )
For an asset population with N members, the forecasted total number of failures during time span [ is
∑ ( ) ( )
where ( ) is the forecasted failure probability using (20) for asset member i, assuming there are in total N members in this asset population.
can be further converted to consequence function C given the average loss factor per failure.
( )
If is unknown or the loss significantly differs between different population members, Monte Carlo simulation can be employed to forecast future system reliability consequences for a given time span [20]- [21].
VIII. CASE STUDY
The proposed method was recently applied to a utility company in West Canada for the reliability forecast of electromechanical relay population. The process and results are discussed as below.
A. Electromechanical relay operation status dataset
The electromechanical relay operation dataset contain age data, operating status data and three condition attributes: enclosure condition, mechanical condition and electrical condition. Utility substation crews annually inspect these electromechanical relays for these three conditions. Based on the inspection results, they assign health ratings for each condition attributes. For a limited population, they also perform a low-voltage simulation test and check if the relay can correctly react to the testing signals and control the trip circuits [23]. If the relay fails to make the correct actions, it will be labeled as a failed relay and will get replaced otherwise labeled as a working relay. Health index H is determined by the normalized relay age, enclosure condition, mechanical condition and electrical condition ratings. Weighting factors are pre-determined by asset engineers based on domain knowledge and experience. They are listed in Table IV. The dataset contain 560 relay records with operating statuses obtained from the historical low-voltage simulation tests. Their ages and the latest condition ratings are also included. For each relay, the health index is calculated using (4). Then the dataset can be converted to a cumulative failure probability vs. health index table in the format of Table III. This table is further split into 80 records for training and 20 records for testing.
B. Modified Weibull Distribution Models
In this utility company, the electromechanical relays typically have an approximate 2-year energization delay due to substation, transmission and distribution line construction. By checking the inspection records, it is found all relays less than 2-years old have a health index less than 5. Therefore <=5 and for modeling, it is discretized as {2.5,5}. In addition, the asset engineer is also certain that these electromechanical relays' infant mortality rate is less than 10%, which is discretized as {5%,10%}. Based on these priori knowledge and assumptions, 12 Weibull distribution models are established in the form of (3), (4), (6), (7) and (8). After evaluating each model using the 20 testing records, their MSEs are calculated and ranked in Table V. Table IV, top 3 models are selected as suitable models and are combined as ( ) using (18). Graphically, these three suitable models and ( ) are shown in Fig.7. The joint model is also evaluated using the testing records and its MSE is 0.0012. It is better than any single Weibull distribution model listed in Table V.
D. Reliability forecast for the next 5 years
This joint model is then applied to the entire 2334 electromechanical relay population. The health index composition is shown in Fig.8. From 2019 to 2023, the failure probability for each relay is calculated using (19) and the total number of failures are calculated using (21). The results show that there could be 277 relay failures from 2019 to 2023. Accordingly, the utility company could consider preparing at least 277 new relays and replace the failed relays proactively from 2019 to 2023.Some electromechanical relays may have to be replaced with digital relays. This finding is important for supply and construction resource planning for the utility company. This paper presents a novel approach for creating and combining a certain number of modified Weibull distribution models to forecast power system asset reliability. Compared to previous works, the proposed methods have the following advantages: It is not reliant on a single Weibull distribution model.
Instead, it uses a few modified Weibull distribution models to incorporate additional information and considerations; It uses a unique method to measure the forecast accuracy of different Weibull distribution models which can better describe the model's forecasting capability. it can effectively combine different Weibull distribution models as a joint forecast model which could have a better performance than individual model. The proposed method was applied to a utility company in West Canada to study electromechanical relay reliability and demonstrated success. The proposed methods can also be used for other power asset populations.
Fig. 1 .
1Flowchart of proposed approach for power system asset reliability forecast
Fig. 2 .
2Standard age based Weibull DistributionFig.2.
Fig. 3 .
3Health index based Weibull Distribution
Fig. 4 .
4X-Shift Weibull DistributionC. Y-Shift Weibull DistributionFig.5. Y-Shift Weibull Distribution Y-Shift Weibull distribution has one additional parameterThis parameter shifts the Weibull Distribution curve along Y axis.Fig.5. shows an example of Y-Shift Weibull distribution. Mathematically, it is given as below:When <0, this distribution curve is shifted downward and the physical meaning becomes very similar to the X-shift Weibull distribution where a starting period of time has no positive failure probability (negative failure probability does not have a statistical meaning and can be capped to zero).
Fig. 6 .
6XY-Shift Weibull DistributionXY-Shift Weibull distribution combines the characteristics of X-Shift Weibull distribution and Y-Shift Weibull distribution. For example, it can be used when an asset type has an energization delay as well as a failure probability in its infant mortality period.V. ESTIMATION OF MODEL PARAMETERS
Fig. 7 .
7Comparison of models
Fig. 8 .
8Health index percentage in the electromechanical relay population IX. CONCLUSIONS
TABLE I ASSET
IOPERATION STATUS DATASET FOR 138KV IN-DUCT XLPETRANSMISSION UNDERGROUND CABLE
Cable
ID
Age
Operating
Status
Partial
Discharge
Result
Neutral
Condition
Splice
Condition
1
10
Working
Good
Good
Good
2
11
Working
Good
Poor
Good
3
17
Working
Medium
Good
Medium
4
37
Failed
Medium
Good
Good
5
45
Working
Good
Poor
Medium
6
43
Working
Good
Good
Medium
7
52
Failed
Poor
Poor
Poor
8
25
Failed
Good
Good
Poor
9
35
Working
Good
Poor
Good
10
40
Working
Good
Good
Good
…
…
…
…
…
…
TABLE II EXAMPLE
IIOF CUMULATIVE FAILURE PROBABILITY VS. AGEA
̂( A)
0
0.0%
1
0.0%
2
0.8%
3
1.5%
4
1.5%
5
2.2%
6
3.5%
7
5.7%
…
…
32
33
98.2%
34
99.1%
35
100%
TABLE III EXAMPLE
IIIOF CUMULATIVE FAILURE OF PROBABILITY VS.HEALTH INDEX
H
F(H)
0
0.0%
1
0.3%
2
2.6%
3
2.9%
4
6.5%
5
9.3%
6
9.5%
….
…
97
98.7%
98
99.0%
99
99.3%
100
100%
TABLE IV WEIGHTING
IVFACTORS FOR HEALTH INDEX HItem
Weighting factor
Age
0.7
Enclosure Condition
0.1
Mechanical condition
0.1
Electrical Condition
0.1
TABLE V 12
VWEIBULL DISTRIBUTION MODELS AND TESTING MSEID
Model Description
MSE
Ranking
1
Two-parameter
0.0015
3
2
X-Shift( =2.5)
0.0015
4
3
X-Shift( =5)
0.0016
6
4
Y-shift( =0.05)
0.0015
5
4
Y-shift( =0.10)
0.0040
9
5
XY-shift( =2.5, =0.05)
0.0013
2
6
XY-shift( =2.5, =0.10)
0.0035
8
7
XY-shift( =5, =0.05)
0.0013
1
8
XY-Shift( =5, =0.10)
0.0034
7
C. Joint Weibull Distribution Model
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|
[] |
[
"Scatterbrain: Unifying Sparse and Low-rank Attention Approximation",
"Scatterbrain: Unifying Sparse and Low-rank Attention Approximation"
] |
[
"Beidi Chen ",
"Tri Dao ",
"Eric Winsor [email protected] ",
"Zhao Song [email protected] \nDepartment of Computer Science and Engineering\nAdobe Research\nUniversity at Buffalo\nSUNY\n",
"Atri Rudra ",
"Christopher Ré ",
"\nDepartment of Computer Science\nStanford University\n\n"
] |
[
"Department of Computer Science and Engineering\nAdobe Research\nUniversity at Buffalo\nSUNY",
"Department of Computer Science\nStanford University\n"
] |
[] |
Recent advances in efficient Transformers have exploited either the sparsity or low-rank properties of attention matrices to reduce the computational and memory bottlenecks of modeling long sequences. However, it is still challenging to balance the trade-off between model quality and efficiency to perform a one-size-fits-all approximation for different tasks. To better understand this trade-off, we observe that sparse and low-rank approximations excel in different regimes, determined by the softmax temperature in attention, and sparse + low-rank can outperform each individually. Inspired by the classical robust-PCA algorithm for sparse and low-rank decomposition, we propose Scatterbrain, a novel way to unify sparse (via locality sensitive hashing) and low-rank (via kernel feature map) attention for accurate and efficient approximation. The estimation is unbiased with provably low error. We empirically show that Scatterbrain can achieve 2.1× lower error than baselines when serving as a drop-in replacement in BigGAN image generation and pre-trained T2T-ViT. On a pre-trained T2T Vision transformer, even without fine-tuning, Scatterbrain can reduce 98% of attention memory at the cost of only 1% drop in accuracy. We demonstrate Scatterbrain for end-to-end training with up to 4 points better perplexity and 5 points better average accuracy than sparse or low-rank efficient transformers on language modeling and long-range-arena tasks. * Equal contribution. Order determined by coin flip. 1 arXiv:2110.15343v1 [cs.LG] 28 Oct 2021 85% 95% Kernel & Hash Construction , , CATEGORIZATION Input SPARSE LOWRANK SCATTERBRAINFigure 1: Left: regimes that sparse+low-rank approximation is more accurate, based on the entropy of the attention matrices. Right: Scatterbrain Workflow. For the attention layer in Transformers, after computing Query Q, Key K, and Value V matrices, we approximate softmax(QK )V with two components:We observe that sparse and low-rank approximations are complementary for many attention matrices in practice, and sparse+low-rank could outperform each individually(Figure 1 left). We empirically categorize the regimes in which sparse or low-rank approximation achieves better error based on the softmax temperature of attention (of which the entropy of softmax distribution can be used as a proxy). We expect that sparse methods perform well if the attention depends on a few entries (low entropy softmax). In contrast, low-rank methods do better if the attention depends on a mixture of many components (high entropy softmax). This explains the phenomenon that current sparse and low-rank-based approaches excel on different kinds of tasks. A natural question is whether one could understand and unify the strength of both approaches. While it is NP-hard to find the optimal combination of sparse and low-rank approximations, Robust PCA [9] is a polynomial-time solution with tight approximation error. We observe that Robust PCA achieves lower approximation error than sparse or low-rank alone on attention matrices. The difference is most pronounced for "mid-range" entropy, where we observe that up to 95% error reduction is possible.The connection between Robust PCA and attention matrix estimation provides an opportunity to realize a more robust approximation. Specifically, given an attention matrix, one could adaptively perform sparse+lowrank approximation to obtain a low error. However, it comes with three challenges: (i) How to decompose the attention matrices into sparse and low-rank components and estimate them efficiently and accurately; Robust PCA is accurate but slow and requires materializing the full attention, while straightforward addition of sparse and low-rank attention will be inaccurate due to double counting. (ii) It is not clear if there is a theoretical guarantee that sparse + low-rank approximation is strictly better than sparse or low-rank in some regimes, though we observe the separation empirically. (iii) How does the lower approximation error transfer to end-to-end performance in real tasks.In this paper, we propose Scatterbrain, an accurate and efficient robust estimation of attention matrices with theoretical guarantees to address the above challenges. Specifically: • In Section 3, we observe that sparse and low-rank approximation are complementary and demonstrate that sparse + low-rank structure arises naturally when elements in the input sequence form clusters. We theoretically characterize and analyze the regimes where sparse, low-rank, and sparse+low-rank excel, dictated by the softmax temperature of attention. • In Section 4, inspired by the classical Robust PCA algorithm, we propose Scatterbrain, which efficiently combines sparse and low-rank matrices to approximate attention. In particular, we use Locality Sensitive Hashing (LSH) to identify large entries of the attention matrix (after softmax) without materializing the full matrix and then leverage kernel approximation to parameterize the low-rank part. We prove that our method has a strictly lower approximation error than the low-rank baseline. • In Section 5, we empirically validate our theory and the proposed method, showing that Scatterbrain accurately approximates the attention matrix, is memory efficient for long sequences, and works well across different tasks. First, we show that its approximation accuracy is close to our oracle Robust PCA and 2 achieves 2.1× lower error compared to other efficient baselines on real benchmarks. This leads to a direct application of Scatterbrain as a drop-in replacement to pre-trained full attention, thus reducing up to 98% of the memory required for attention computations in pre-trained T2T-ViT and BigGAN while maintaining similar quality. Last we show that its superior accuracy and efficiency can improve the efficiency-accuracy trade-offs of Transformer end-to-end training. On the WikiText-103 language modeling task, Scatterbrain achieves up to 1 point better perplexity compared to Reformer and Performer. On 5 benchmark long-range tasks, Scatterbrain improves the average accuracy by up to 5 points. 1Problem Setting and Related WorkWe first define the approximation problem we aim to solve in this paper. Then we discuss the applications of sparse and low-rank techniques in efficient Transformers and introduce robust PCA algorithm. Problem Formulation: In the attention matrix approximation problem, we are given three matrices, query, key, and value, Q, K, V ∈ R n×d to compute softmax(QK )V . We seek to reduce the quadratic complexity of softmax(QK ) (applied row-wise) with low approximation error. More precisely, for an approximation procedure f , we minimize two objectives, the approximation error E f (Q, K) − softmax(QK ) 2 F , and the computation/memory cost C(f (·)).Sparse, Low-rank Approximation for Attention Matrices: Recent work exploits the sparsity patterns or finds a low-rank mapping of the original attention matrices to overcome the computational and memory bottlenecks in Transformers[17,22, 35, 36,53,65]. Generally, we can divide most of the techniques into two categories -sparse and low-rank approximations. Reformer [36] is a representative sparse variant that uses LSH [3] to retrieve or detect the locations of the attention matrices with large values and reduce the computation from O(n 2 ) to O(n log n). Performer[17]is an example of the low-rank variant, which uses kernelization to avoid explicit O(n 2 d) computation. One problem of either the sparse or low-rank approximation is that the structure of the attention matrices varies in practice, and it is challenging to perform robust approximation on a wide range of attention matrices. For example, Wang et al.[65]observes that attentions tend to have more low-rank structures in lower layers and Ramsauer et al.[51]shows that they are sparser in the later stage of the training. Ideally, we want to unify the strength of both techniques, but it is NP-hard to find the best combination of sparse and low-rank approximation.Sparse + Low-rank and Robust PCA: Fortunately, classical Robust PCA [9] presents a polynomial algorithm to find the approximately optimal or good combinations of sparse and low-rank approximation of the matrices. The sparse + low-rank matrix structure has been well studied in statistics and signal processing since the late 2000s [9]. This structure naturally generalizes low-rank [33,62], and sparse [60] matrices. Scatterbrain is built on a line of work, e.g., Bigbird [70], Longformer [5] with the theme of combining multiple types of attention. However, despite the multitude of papers, this sparse + low-rank matrix approximation has not been rigorously studied in the context of attention matrices. We undertake this study and show how we can relax the sparse + low-rank approximation from robust PCA, making it efficient while still retaining PCA's accuracy. In fact, our results shed further light on why Bigbird or Longformer work, as they are special cases of a single principled structure. An extended discussion of related work is in Appendix A.Characterization of Sparse + Low-rank Approx. to Attention MatricesWe motivate the use of sparse + low-rank approximation of the attention matrices with the key observation that for many attention matrices, sparse and low-rank approximation are complementary, and their ideal combination (via Robust PCA) can outperform both (Section 3.1). Furthermore, we argue that the sparse + low-rank structure can arise naturally when elements in the input sequence form clusters, as dictated by the softmax temperature (Section 3.2).
| null |
[
"https://arxiv.org/pdf/2110.15343v1.pdf"
] | 240,070,455 |
2110.15343
|
469828497d03004eab945b74e1c15cc2cee9e128
|
Scatterbrain: Unifying Sparse and Low-rank Attention Approximation
October 29, 2021
Beidi Chen
Tri Dao
Eric Winsor [email protected]
Zhao Song [email protected]
Department of Computer Science and Engineering
Adobe Research
University at Buffalo
SUNY
Atri Rudra
Christopher Ré
Department of Computer Science
Stanford University
Scatterbrain: Unifying Sparse and Low-rank Attention Approximation
October 29, 2021
Recent advances in efficient Transformers have exploited either the sparsity or low-rank properties of attention matrices to reduce the computational and memory bottlenecks of modeling long sequences. However, it is still challenging to balance the trade-off between model quality and efficiency to perform a one-size-fits-all approximation for different tasks. To better understand this trade-off, we observe that sparse and low-rank approximations excel in different regimes, determined by the softmax temperature in attention, and sparse + low-rank can outperform each individually. Inspired by the classical robust-PCA algorithm for sparse and low-rank decomposition, we propose Scatterbrain, a novel way to unify sparse (via locality sensitive hashing) and low-rank (via kernel feature map) attention for accurate and efficient approximation. The estimation is unbiased with provably low error. We empirically show that Scatterbrain can achieve 2.1× lower error than baselines when serving as a drop-in replacement in BigGAN image generation and pre-trained T2T-ViT. On a pre-trained T2T Vision transformer, even without fine-tuning, Scatterbrain can reduce 98% of attention memory at the cost of only 1% drop in accuracy. We demonstrate Scatterbrain for end-to-end training with up to 4 points better perplexity and 5 points better average accuracy than sparse or low-rank efficient transformers on language modeling and long-range-arena tasks. * Equal contribution. Order determined by coin flip. 1 arXiv:2110.15343v1 [cs.LG] 28 Oct 2021 85% 95% Kernel & Hash Construction , , CATEGORIZATION Input SPARSE LOWRANK SCATTERBRAINFigure 1: Left: regimes that sparse+low-rank approximation is more accurate, based on the entropy of the attention matrices. Right: Scatterbrain Workflow. For the attention layer in Transformers, after computing Query Q, Key K, and Value V matrices, we approximate softmax(QK )V with two components:We observe that sparse and low-rank approximations are complementary for many attention matrices in practice, and sparse+low-rank could outperform each individually(Figure 1 left). We empirically categorize the regimes in which sparse or low-rank approximation achieves better error based on the softmax temperature of attention (of which the entropy of softmax distribution can be used as a proxy). We expect that sparse methods perform well if the attention depends on a few entries (low entropy softmax). In contrast, low-rank methods do better if the attention depends on a mixture of many components (high entropy softmax). This explains the phenomenon that current sparse and low-rank-based approaches excel on different kinds of tasks. A natural question is whether one could understand and unify the strength of both approaches. While it is NP-hard to find the optimal combination of sparse and low-rank approximations, Robust PCA [9] is a polynomial-time solution with tight approximation error. We observe that Robust PCA achieves lower approximation error than sparse or low-rank alone on attention matrices. The difference is most pronounced for "mid-range" entropy, where we observe that up to 95% error reduction is possible.The connection between Robust PCA and attention matrix estimation provides an opportunity to realize a more robust approximation. Specifically, given an attention matrix, one could adaptively perform sparse+lowrank approximation to obtain a low error. However, it comes with three challenges: (i) How to decompose the attention matrices into sparse and low-rank components and estimate them efficiently and accurately; Robust PCA is accurate but slow and requires materializing the full attention, while straightforward addition of sparse and low-rank attention will be inaccurate due to double counting. (ii) It is not clear if there is a theoretical guarantee that sparse + low-rank approximation is strictly better than sparse or low-rank in some regimes, though we observe the separation empirically. (iii) How does the lower approximation error transfer to end-to-end performance in real tasks.In this paper, we propose Scatterbrain, an accurate and efficient robust estimation of attention matrices with theoretical guarantees to address the above challenges. Specifically: • In Section 3, we observe that sparse and low-rank approximation are complementary and demonstrate that sparse + low-rank structure arises naturally when elements in the input sequence form clusters. We theoretically characterize and analyze the regimes where sparse, low-rank, and sparse+low-rank excel, dictated by the softmax temperature of attention. • In Section 4, inspired by the classical Robust PCA algorithm, we propose Scatterbrain, which efficiently combines sparse and low-rank matrices to approximate attention. In particular, we use Locality Sensitive Hashing (LSH) to identify large entries of the attention matrix (after softmax) without materializing the full matrix and then leverage kernel approximation to parameterize the low-rank part. We prove that our method has a strictly lower approximation error than the low-rank baseline. • In Section 5, we empirically validate our theory and the proposed method, showing that Scatterbrain accurately approximates the attention matrix, is memory efficient for long sequences, and works well across different tasks. First, we show that its approximation accuracy is close to our oracle Robust PCA and 2 achieves 2.1× lower error compared to other efficient baselines on real benchmarks. This leads to a direct application of Scatterbrain as a drop-in replacement to pre-trained full attention, thus reducing up to 98% of the memory required for attention computations in pre-trained T2T-ViT and BigGAN while maintaining similar quality. Last we show that its superior accuracy and efficiency can improve the efficiency-accuracy trade-offs of Transformer end-to-end training. On the WikiText-103 language modeling task, Scatterbrain achieves up to 1 point better perplexity compared to Reformer and Performer. On 5 benchmark long-range tasks, Scatterbrain improves the average accuracy by up to 5 points. 1Problem Setting and Related WorkWe first define the approximation problem we aim to solve in this paper. Then we discuss the applications of sparse and low-rank techniques in efficient Transformers and introduce robust PCA algorithm. Problem Formulation: In the attention matrix approximation problem, we are given three matrices, query, key, and value, Q, K, V ∈ R n×d to compute softmax(QK )V . We seek to reduce the quadratic complexity of softmax(QK ) (applied row-wise) with low approximation error. More precisely, for an approximation procedure f , we minimize two objectives, the approximation error E f (Q, K) − softmax(QK ) 2 F , and the computation/memory cost C(f (·)).Sparse, Low-rank Approximation for Attention Matrices: Recent work exploits the sparsity patterns or finds a low-rank mapping of the original attention matrices to overcome the computational and memory bottlenecks in Transformers[17,22, 35, 36,53,65]. Generally, we can divide most of the techniques into two categories -sparse and low-rank approximations. Reformer [36] is a representative sparse variant that uses LSH [3] to retrieve or detect the locations of the attention matrices with large values and reduce the computation from O(n 2 ) to O(n log n). Performer[17]is an example of the low-rank variant, which uses kernelization to avoid explicit O(n 2 d) computation. One problem of either the sparse or low-rank approximation is that the structure of the attention matrices varies in practice, and it is challenging to perform robust approximation on a wide range of attention matrices. For example, Wang et al.[65]observes that attentions tend to have more low-rank structures in lower layers and Ramsauer et al.[51]shows that they are sparser in the later stage of the training. Ideally, we want to unify the strength of both techniques, but it is NP-hard to find the best combination of sparse and low-rank approximation.Sparse + Low-rank and Robust PCA: Fortunately, classical Robust PCA [9] presents a polynomial algorithm to find the approximately optimal or good combinations of sparse and low-rank approximation of the matrices. The sparse + low-rank matrix structure has been well studied in statistics and signal processing since the late 2000s [9]. This structure naturally generalizes low-rank [33,62], and sparse [60] matrices. Scatterbrain is built on a line of work, e.g., Bigbird [70], Longformer [5] with the theme of combining multiple types of attention. However, despite the multitude of papers, this sparse + low-rank matrix approximation has not been rigorously studied in the context of attention matrices. We undertake this study and show how we can relax the sparse + low-rank approximation from robust PCA, making it efficient while still retaining PCA's accuracy. In fact, our results shed further light on why Bigbird or Longformer work, as they are special cases of a single principled structure. An extended discussion of related work is in Appendix A.Characterization of Sparse + Low-rank Approx. to Attention MatricesWe motivate the use of sparse + low-rank approximation of the attention matrices with the key observation that for many attention matrices, sparse and low-rank approximation are complementary, and their ideal combination (via Robust PCA) can outperform both (Section 3.1). Furthermore, we argue that the sparse + low-rank structure can arise naturally when elements in the input sequence form clusters, as dictated by the softmax temperature (Section 3.2).
Introduction
Transformer models [63] have been adapted in a wide variety of applications, including natural language processing [7,26,50], image processing [10,47], and speech recognition [42]. Training large Transformers requires extensive computational and memory resources, especially when modeling long sequences, mainly due to the quadratic complexity (w.r.t. sequence length) in attention layers. Recent advances in efficient transformers [17,22,35,36,65] leverage attention approximation to overcome the bottleneck by approximating the attention matrices. However, it is challenging to find a robust approximation method that balances the efficiency-accuracy trade-off on a wide variety of tasks [57,58].
We categorize most of the existing approaches for efficient attention matrix computation into two major groups: exploiting either the sparsity, e.g., Reformer [36], SMYRF [22], or low-rank properties of the attention matrices, e.g., Linformer [65], Linear Transformer [35], and Performer [17]. However, these techniques usually have different strengths and focus on the performance of specific tasks, so their approximations still cause accuracy degradation on many other tasks. For instance, according to a recent benchmark paper [57] and our experiments, low-rank-based attention might be less effective on hierarchically structured data or language modeling tasks, while sparse-based variants do not perform well on classification tasks.
Motivating Observations: Low-rank and Sparse Structures of Attention Matrices
We empirically characterize regimes where sparse and low-rank approximation are well-suited, based on the softmax temperature (for which we use the softmax distribution entropy is a proxy). Specifically, in Fig. 1 (left), we present the approximation error of the original attention matrices and the approximation (sparse or low-rank) of matrices sampled from a 4-layer Transformer trained on IMDb reviews classification [57]. We make two observations: 1. Sparse and low-rank approximation are complementary: sparse excels when the softmax temperature scale is low (i.e., low entropy), and low-rank excels when the softmax temperature is high (i.e., high entropy). 2. An ideal combination of sparse and low-rank (orange line in Fig. 1 left), obtained with robust PCA, can achieve lower error than both. Similar observations on other benchmarks and details are presented in Appendix B. Figure 2: Visualization of the generative process, for three different values of the intra-cluster distance ∆ (small, medium, and large). The vectors from the input sequence (rows of Q) form clusters that lie approximately on the unit sphere. Different colors represent different clusters.
A Generative Model of How Sparse + Low-rank Structure Can Arise
Sparse + low-rank parameterization is more expressive than either sparse or low-rank alone. Indeed, in the Appendix, we construct a family of attention matrices to show the separation between the approximation capability of sparse + low-rank vs. sparse or low-rank alone: for an n × n attention matrix, sparse or low-rank alone requires a O(n 2 ) parameters to get approximation error in Frobenius norm, while sparse + low-rank only requires O(n) parameters.
Moreover, we argue here that sparse + low-rank is a natural candidate to approximate generic attention matrices. We describe a generative model of how the sparse + low-rank structure in attention matrices could arise when the elements of the input sequence form clusters. Under this process, we characterize how the softmax temperature dictates when we would need sparse, low-rank, or sparse + low-rank matrices to approximate the attention matrix. This result corroborates the observation in Section 3.1.
Generative process of clustered elements in input sequence We describe here a generative model of an input sequence to attention, parameterized by the inverse temperature β ∈ R and the intra-cluster distance ∆ ∈ R. Process 1. Let Q ∈ R n×d , where d ≥ Ω(log 3/2 (n)), with every row of Q generated randomly as follows: 1. For C = Ω(n), sample C number of cluster centers c 1 , . . . , c C ∈ R d independently from N (0, I d / √ d). 2. For each cluster around c i , sample n i = O(1) number of elements around c i , of the form z ij = c i + r ij for j = 1, . . . , n i where r ij ∼ N (0, I d ∆/ √ d). Assume that the total number of elements is n = n 1 + · · · + n C and ∆ ≤ O(1/ log 1/4 n). Let Q be the matrix whose rows are the vectors z ij where i = 1, . . . , C and j = 1, . . . , n i . Let A = QQ and let the attention matrix be M β = exp(β · A).
We visualize this generative process in Fig. 2.
Softmax temperature and approx. error We characterize when to use sparse, low-rank, or sparse + low-rank to approximate the attention matrices in Process 1, depending on the inverse temperature β. The intuition here is that the inverse temperature corresponds to the strength of interaction between the clusters. If β is large, intra-cluster interaction dominates the attention matrix, the softmax distribution is peaked, and so we only need a sparse matrix to approximate the attention. If β is small, then the inter-cluster attention is similar to intra-cluster attention, the softmax distribution is diffuse, and we can approximate it with a low-rank matrix. In the middle regime of β, we need the sparse part to cover the intra-cluster attention and the low-rank part to approximate the inter-cluster attention.
We formalize this intuition in Theorem 1 (in bounds below we think of as a constant). All the proofs are in Appendix D.
Theorem 1. Let M β , be the attention matrix in Process 1. Fix ∈ (0, 1). Let R ∈ R n×n be a matrix. Consider low-rank, sparse, and sparse + low-rank approximations to M β . 1. High temperature: Assume β = o(log n/log d).
(a) Low-rank: There exists R with n o(1) rank (and hence n 1+o(1) parameters) such that M β −R F ≤ n.
(b) Sparse: If R has sparsity o(n 2 ), then M β − R F ≥ Ω(n).
2. Mid temperature: Assume (1 − ∆ 2 ) log n ≤ β ≤ O(log n).
(a) Sparse + low-rank: There exists a sparse + low-rank R with n 1+o(1) parameters with M β −R F ≤ n.
(b) Low-rank: If R is such that n − rank(R) = Ω(n), then M β − R F ≥ Ω(n).
(c) Sparse: If R has sparsity o(n 2 ), then M β − R F ≥ Ω(n).
3. Low temperature: Assume β = Ω(log n).
(a) Low-rank: If R is such that n − rank(R) = Ω(n), then M β − R F ≥ Ω(e β(1−∆ 2 ) ). We present Scatterbrain, and show that it approximates attention accurately and efficiently. Section 4.1 describes the challenges of designing an accurate and efficient approximation, and how obvious baselines such as Robust PCA or a simple combination of sparse attention and low-rank attention fail to meet both criteria. Section 4.2 demonstrates how Scatterbrain address the challenges ( Fig. 1 contains a schematic of Scatterbrain). In Section 4.3, we show that Scatterbrain is unbiased with provably lower variance than low-rank baselines such as Performer. Fig. 3 shows a qualitative comparison between different methods of approximating the attention matrix: Robust PCA is accurate but slow, sparse (e.g., Reformer), and low-rank (e.g., Performer) attention are fast and memory-efficient but may not be very accurate, while Scatterbrain is more accurate than its sparse and low-rank counterparts while remaining just as efficient.
More details about the efficient implementation of Scatterbrain are in Appendix C.
Challenges of Designing an Accurate and Efficient Sparse + Low-rank Approximation
We seek a sparse + low-rank approximation of the attention matrix 2 A that is both accurate and efficient. The natural theoretical baseline of Robust PCA is too slow and requires too much memory, while the most straightforward way of combining sparse attention and low-rank attention fails due to double counting on the support of the sparse attention.
1. If the goal is accuracy, Robust PCA is the most studied algorithm to find a sparse + low-rank approximation to a given matrix. It relaxes the NP-hard problem of finding the best sparse + low-rank approximation into a convex optimization problem, with the nuclear norm and 1 constraints. Even though it can be solved in polynomial time, it is orders of magnitude too slow to be used in each iteration of a training loop. Moreover, it requires materializing the attention matrix, which defeats the main purpose of reducing compute and memory requirements. 2. On the other hand, one efficient way to get sparse + low-rank approximation of an attention matrix is to simply add the entries of a sparse approximation S (say, from Reformer) and a low-rank approximation Q K for Q, K ∈ R n×m (say, from Performer). The sparse matrix S typically has support determined randomly [16], by LSH [22,36], or by clustering [53]. On the support of S, which likely includes the locations of the large entries of the attention matrix A, the entries of S match those of A. One can multiply (S + Q K )V = SV + Q( K V ) efficiently because S is sparse, and grouping Q( K V ) reduces the matrix multiplication complexity when m n, from O(n 2 m) to O(nmd). The approximation S + Q K matches Q K outside supp(S), hence it could be accurate there if Q K is accurate. However, S + Q K will not be accurate on the support of S due to the contributions from both S and from Q K . Adjusting Q K to discount the contribution from S is difficult, especially if we want to avoid materializing Q K for efficiency.
Scatterbrain: Algorithm Intuition and Description
The simple insight behind our method is that on the support of the sparse matrix S, instead of trying to match the entries of the attention matrix A, we can set the entries of S to discount the contribution from the low-rank part Q K . This way, the approximation S + Q K will match A exactly on the support of S, and will match Q K outside supp(S), which means it will still be accurate there if Q K is accurate. We do not need to materialize the full matrix Q K as need a subset of its entries is required, hence our approximation will be compute and memory efficient.
Scatterbrain thus proceeds in three steps: we construct a low-rank approximation Q K ≈ A, and construct a sparse matrix S such that S + Q K matches A on the support of S, then finally multiply SV and Q( K V ) and combine the result. More specifically: 1. Low-rank Approximation. We define a procedure LowRank that returns two matrices Q, K ∈ R n×m such that Q K approximates A. In particular, we use a randomized kernel feature map φ :
R d → R m where φ(x) = 1 √ m exp(W x − x 2 /2)
with W ∈ R m×d randomly sampled, entry-wise, from the standard normal distribution N (0, 1). We apply φ to each row vector of Q, K matrices, and denote Q = φ(Q) and K = φ(K) (row-wise). Note that we do not materialize Q K . 2. Sparse Approximation. We define a procedure Sparse that returns a sparse matrix S that matches A − Q K on supp(S). In particular, using a family of locality sensitive hash functions, compute the hash codes of each query and key vectors in Q, K matrices (row-wise). Let S be the set of locations (i, j) where q i and k j have the same hash codes (i.e, fall into the same hash bucket). Let S be the sparse matrix whose support is S, and for each (i, j) ∈ S, define
S i,j = exp(q i k j ) − φ(q i ) φ(k j ) = exp(q i k j ) − q i k j ,(1)
where q i , k j , q i , k j are the i-th and j-th rows of Q, K, Q, K respectively. Note that we do not materialize Q K . 3. Scatterbrain Approximation. With Q, K returned from LowRank and S from Sparse, we compute the (unnormalized) attention output with
O = ( Q K + S)V = Q( K V ) + SV.(2)
The precise algorithm, including the normalization step, as well as the causal/unidirectional variant, is described in Appendix C. We also note Scatterbrain's flexibility: it can use different kinds of low-rank and sparse approximation as its sub-components. The combination of Reformer and Performer is simply one instance of Scatterbrain. Instead of using Reformer as a sparse component, we could use local attention [5] or random block-sparse attention [16]. Instead of using Performer [17] as a low-rank component, we could also use Linear attention [35] or global tokens as in BigBird [70].
MSE
Reformer (sparse) Performer (low-rank) Scatterbrain (sparse+low-rank) Figure 4: Per-entry MSE for different approximations, across a range of magnitude of q k.Scatterbrain has low MSE for both small and large entries, thus outperforming its sparse (Reformer) and low-rank (Performer) counterparts.
The Scatterbrain method would work exactly the same way. As long as the low-rank component is unbiased (e.g., Performer), its combination with any sparse component in Scatterbrain would yield an unbiased estimator of the attention matrix as shown below.
Scatterbrain: Analysis
Our method combines a low-rank approximation Q K (which has rank m n) with a sparse approximation S. We argue that it is accurate (lower approximation error than baselines) and efficient (scaling the same as sparse or low-rank alone). The main insight of the analysis is that our approximation is exact for entries on the support of S (picked by LSH), which are likely to be large. For entries not in the support of S (likely to be small), our approximation matches the low-rank part (Performer) Q K , which is unbiased and has low variance for these entries. As a result, Scatterbrain retains the unbiasedness of Performer [17] but with strictly lower variance.
We compare Scatterbrain to its low-rank baseline (Performer) and sparse baseline (Reformer). Performer is also based on the kernel approximation φ, and simply uses Q K to approximate the attention matrix A. Reformer uses LSH to identify large entries of A, then compute a sparse matrix S such that S ij = exp(q i k j ) for ij ∈ supp(S).
Accuracy: Because of the way S is defined in Eq. (1), Q K + S matches A = exp(QK ) exactly on locations (i, j) ∈ S, which are locations with likely large values. This addresses a weakness of low-rank methods (e.g., Performer) where the low-rank estimate is not accurate for locations with large values. We analyze the expectation and variance per entry of our estimator below (proof in Appendix D).
Theorem 2. Define σ(q, k) = exp(q k), σ pfe as Performer's estimator and σ sbe as Scatterbrain estimator. Denote S d−1 ⊂ R d as the unit sphere. Suppose q, k ∈ S d−1 are such that q − k < τ . Then:
E[ σ sbe (q, k)] = σ(q, k), Var[ σ sbe (q, k)] = (1 − p) · Var[ σ pfe (q, k)] < Var[ σ pfe (q, k)](3)
where p = exp(− τ 2 4−τ 2 ln d − O τ (ln ln d)). Hence Scatterbrain is unbiased, similar to Performer [17], but with strictly lower variance. The variance is small if exp(q k) is small (since Var( σ pfe (q, k)) will be small), or if exp(q k) is large (since the probability of not being selected by LSH, 1 − p, will be small). In Fig. 4, we plot the per-entry MSE of different methods from Theorem 2 when approximating the unnormalized softmax attention exp(QK ). Scatterbrain can approximate well both small entries (similar to the low-rank baseline, Performer), as well as large entries (similar to the sparse baseline, Reformer). Thus Scatterbrain has much lower MSE than Performer for large entries, and lower MSE than Reformer for small entries.
Efficiency: In Eq. (2), the computation SV is efficient because S is sparse, and Q( K V ) is efficient because of the way we associate matrix multiplication (scaling as O(nmd) instead of O(n 2 d), which is much bigger if m n). We validate these two properties of our approach in Section 5.
Experiments
We validate three claims that suggest Scatterbrain provides an accurate and efficient approximation to attention matrices, allowing it to outperform its sparse and low-rank baselines on benchmark datasets. • In Section 5.1, we evaluate the approximation error and testing accuracy of different approximation methods on pre-trained models such as BigGAN and Vision Transformer. We show that the approximation by Scatterbrain is close to the Robust PCA oracle and up to 2.1× lower approximation error than other efficient baselines. Figure 5: First: approximation comparison between Scatterbrain and its "lowerbound" Robust PCA. Second: comparison of error vs. entropy among SMYRF, Performer and Scatterbrain, three representatives of sparse, low-rank and sparse+low-rank approximations. Third and forth: Inception score (higher is better) and FID score (lower is better) of different attention variants for pretrained BigGAN.
• In Section 5.2, we validate that when trained end-to-end, Scatterbrain outperforms baselines (sparse or low-rank attention) on a wide variety of benchmark tasks, including language modeling, classification, and the Long-range Arena (LRA) benchmarks. Scatterbrain achieves up to 5 points higher average accuracy on the LRA benchmark compared to Performer and Reformer. • In Section 5.3, we demonstrate the scalability of Scatterbrain, showing that it has comparable memory and time usage with simpler baselines (sparse or low-rank alone) across a range of input sequence lengths (Section 5.3), while requiring up to 12× smaller memory than full attention. All details (hyperparameters, data splits, etc.), along with additional experiments, are in Appendix E. We evaluate Scatterbrain's approximation accuracy in three steps: (1) compare it with of Robust PCA (sparse+lowrank), our theoretical foundation and oracle (2) compare it with SMYRF 3 [22], Performer [17], which are popular variants of sparse and low-rank approximation to attention respectively and a naive baseline that directly adds SMYRF and Performer, (3) evaluate the inference accuracy when replacing full attention with Scatterbrain approximation. Scatterbrain achieves error within 20% of the oracle robust PCA, and up to 2.1× lower error than SMYRF and Performer. When serving as a drop-in replacement for full attention, even without training, Scatterbrain can reduce the attention memory of Vision Transformer by 98% at the cost of only 0.8% drop of accuracy. Setup: We use the attention matrices from pre-trained BigGAN and T2T-ViT. BigGAN is a state-of-theart model in Image Generation for ImageNet. BigGAN has a single attention layer at resolution 64 × 64 (4096 queries). T2T-ViT has 14 attention layers. Scatterbrain sets the ratio between SMYRF and Performer based on the entropy of an observed subset of attention matrices in different layers. We allocate more memory to the low-rank component compared to the sparse part if the entropy is high.
Scatterbrain's Approximation Accuracy
Scatterbrain and Robust PCA: We first show that Scatterbrain approximates pre-trained attention matrices 10 5 × faster while its approximation error is within 20% on average. We also provide an example visualization on 100 attention matrices from the BigGAN generation process in Figure 5 (left).
Scatterbrain vs. SMYRF and Performer: We show that Scatterbrain approximates pre-trained dense attention matrices with very low error compared to sparse (Reformer) or low-rank (Performer). Measuring Frobenius approx. error on the BigGAN image generation task, Scatterbrain achieves 2× lower error compared to Performer. Drop-in replacement for full attention: We show that accurate approximation directly leads to efficient Inference. We replace BigGAN's dense attention with a Scatterbrain layer without other modifications. In 5 (right two), we show Inception and FID scores for Scatterbrain and other baselines under different memory budgets. Similarly, we use T2T-ViT [69], which is a token-to-token vision Transformer pre-trained on ImageNet [25]. In Table 1, we show the average approximation error of Scatterbrain for each layer and the end-to-end testing accuracy after replacing full attention with Scatterbrain and other baselines. Notably, Scatterbrain achieves 80.7% Top-1 accuracy, which is only 1% drop from the original 81.7% by full attention reducing up to 98% of the memory usage.
End-to-end Training Performance
Scatterbrain's accurate approximation of attention matrices allows it to outperform other efficient Transformer methods on benchmark tasks. Across a range of diverse tasks, both commonly used autoregressive tasks (sequence modeling) and benchmark long-range classification tasks (Long-Range Arena), Scatterbrain outperforms Performer (low-rank baseline) and Reformer (sparse baseline) by up to 4 points.
Auto-regressive Tasks
On the standard language modeling task of Wikitext-103, Scatterbrain obtains 1 point better perplexity than Reformer (sparse baseline), coming within 1.5 points of full attention.
Settings: We compare the performance of Scatterbrain against Reformer and Performer on one popular synthetic task, Copy, and one large language modeling task: WikiText103 [45]. Reformer is a representative sparse-approximation-based variant and Performer is a low-rank-approximation-based variant. The base model is vanilla Transformer [63]. We observed that generally allocating more memory budget to sparse tends to perform better, so Scatterbrain sets the ratio to 3:1 (sparse: low-rank component) for simplicity. The statistics of each dataset and model hyper-parameters are in Appendix E. We report the best results of each method in perplexity.
Results: Table 2 shows the testing perplexity for Scatterbrain and other baselines under the same parameter budget (each approximation is only allowed to compute 1 8 of the full computation). Scatterbrain achieves comparable perplexity compared to the full attention Transformer model on Copy, and WikiText-103. Notably, Scatterbrain achieves 4 points lower perplexity on Copy and 1 point lower on WikiText-103 compared to Reformer, while Performer does not train stably on auto-regressive tasks (loss does not go down).
Analysis: We also analyze the results by visualizing the error of Reformer (sparse), Performer (low-rank), and Scatterbrain (sparse + low-rank) given the same number of parameters when approximating the full attention matrices for each attention layer during training (Appendix E). The conclusion is for language modeling tasks, sparse+low-rank has the smallest approximation error in most of the cases, and sparse has the largest error, which matches with the end-to-end results. It also confirms the observation in the popular benchmark paper [57] that kernel or low-rank based approximations are less effective for hierarchical structured data.
Classification Tasks
On a suite of long-range benchmark tasks (Long Range Area), Scatterbrain outperforms Reformer (sparse baseline) and Performer (low-rank baseline) by up to 5 points on average.
Settings: We compare the performance of Scatterbrain against Reformer and Performer on ListOps, two classifications: byte-level IMDb reviews text classification, image classification on sequences of pixels, a text retrieval, and pathfinder tasks. The datasets are obtained from the Long Range Arena (LRA) Benchmark [57], which is a recent popular benchmark designed for testing efficient Transformers. Similar to the auto-regressive tasks above, we use Reformer and Performer as baselines. The base model is also a vanilla Transformer. We follow the evaluation protocol from [57]. We report the best accuracy of each method.
Results: Table 2 shows the individual and average accuracy of each task for Scatterbrain and other baselines under the same parameters budget. Specially, each approximation is only allowed to use 12.5% of the full computation. We can see Scatterbrain is very close to full attention even with a large reduction in computation and memory. Further more, it outperforms all the other baselines consistently on every task and achieves more than 5 point average accuracy improvement than sparse-based approximation Reformer and more than 2 point average accuracy improvement than low-rank-based variant Performer. Analysis: Similarly, in order to analyze the performance of Reformer, Performer and Scatterbrain, we visualize their approximation error given the same number of parameters when approximating the full attention matrices for each attention layer during training (Appendix E). We again find that Scatterbrain has the smallest approximation error, while Performer is the worst on ListOps and Reformer has the largest error on classification tasks, which matches with the end-to-end results and confirms our observations earlier (sparse and low-rank approximation excel in different regimes).
Scatterbrain's Efficiency, Scaling with Input Sequence Length
We include ablation studies on the scalability of Scatterbrain in Fig. 6, showing that it is as computation and memory-efficient as simpler baselines such as SMYRF and Performer, while up to 3× faster and 12× more memory efficient than full attention for sequence length 4096. This demonstrates that our combination of sparse and low-rank inherits their efficiency.
We report run times and memory consumption of the sequence lengths ranging from 512 to 32768. We use a batch size of 16 for all runs and conduct experiments a V100 GPU. Since the efficiency would be largely conditioned on hardware and implementation details, we perform best-effort fair comparisons. We adapt the Pytorch implementation from pytorch-fast-transformers library for our baselines and implement Scatterbrain similarly without any customized cuda kernels.
Discussion
Limitations. As Scatterbrain has sparse attention as a component, it is not yet as hardware friendly (on GPUs and TPUs) as the low-rank component, which uses the very optimized dense matrix multiplication. This is the same limitation suffered by other sparse attention methods, but we are excited that more efficient sparse GPU kernels are being developed [29,31].
Potential negative societal impacts. Our work seeks to understand the role of matrix approximation (and potentially energy savings) in the attention layer, which may improve a wide range of applications, each with their own potential benefits and harms. For example, making it language modeling more compute and memory efficient might facilitate spreading misinformation, and better image and video processing may make automatic surveillance easier. To mitigate these risks, one needs to address application-specific issues such as privacy and fairness, going beyond the error metrics we considered. Specially, for language models (LMs), while our work partially addresses the issue of environmental cost of LMs raised in [6], it does not address other issues such as unfathomable training data [6].
Discussion and future work. In this work, we make an observation on the sparse + low-rank structure of the attentions in Transformer models and theoretically characterize the regimes where sparse, low-rank and sparse + low-rank excel, based on the softmax temperature of the attention matrices. Motivated by this observation, we present Scatterbrain, a novel way to unify the strengths of both sparse and low-rank methods for accurate and efficient attention approximation with provable guarantees. We empirically verify the effectiveness of Scatterbrain on pretrained BigGAN, vision transformers, as well as end-to-end training of vanilla transformer. We anticipate that the study of this core approximation problem can prove useful in other contexts, such as generalized attention layers with other non-linearity beside softmax, and wide output layer in language modeling or extreme-classification.
[21] Tri Dao, Nimit Sohoni, Albert Gu, Matthew Eichhorn, Amit Blonder, Megan Leszczynski, Atri Rudra, and Christopher Ré. Kaleidoscope: An efficient, learnable representation for all structured linear maps.
In The International Conference on Learning Representations (ICLR), 2020.
[22] Giannis Daras, Nikita Kitaev, Augustus Odena, and Alexandros G Dimakis. Smyrf: Efficient attention using asymmetric clustering. arXiv preprint arXiv:2010.05315, 2020.
[23] Christopher De Sa, Christopher Re, and Kunle Olukotun. Global convergence of stochastic gradient descent for some non-convex matrix problems. In International Conference on Machine Learning, pages 2332-2341. PMLR, 2015.
[24] Christopher De Sa, Albert Gu, Rohan Puttagunta, Christopher Ré, and Atri Rudra. A two-pronged progress in structured dense matrix vector multiplication. [31] Scott Gray, Alec Radford, and Diederik P Kingma. GPU kernels for block-sparse weights. arXiv preprint arXiv:1711.09224, 3, 2017.
[32] Albert Gu, Tri Dao, Stefano Ermon, Atri Rudra, and Christopher Ré. Hippo: Recurrent memory with optimal polynomial projections. In Advances in neural information processing systems (NeurIPS), 2020.
[33] Harold Hotelling. Analysis of a complex of statistical variables into principal components. Journal of educational psychology, 24 (6) It is a modification of PCA to accommodate corrupted observations (aka, noise). The sparse part covers the noise, while the low-rank part recovers the principle components. The most popular method to solve the problem is convex relaxation [8], where one minimizes the error M − S − L 2 F subject to 1 constraint on S 1 and nuclear norm constraint on L * , in order to promote the sparsity of S and the low-rankness of L. This convex problem can be solved with a variety of methods, such as interior point methods or the method of Augmented Lagrange Multipliers.
In our context, to find a sparse + low-rank decomposition of the attention matrix, one can also heuristically "peel off" the sparse part by finding the large entries of the attention matrix, then find a low-rank decomposition of the remainder. To avoid materializing the full attention matrix, one can use LSH to find potential locations of large entries, and use matrix completion [52] to find a low-rank decomposition. Gradient descent can find global optimum for this matrix completion problem [23]. However, it still requires too many iterations to be used in each training step.
A.2 Efficient Transformers
Sparse, Low-rank Approx.: Transformer-based model such as BERT [38] has achieved unprecedented performance in natural language processing. Recently, Vision Transformers [28,69] has also achieved comparable performance to the traditional convolutional neural network in computer vision tasks [66]. However, the quadratic computation of the attention layers constrains the scalability of Transformers. There are many existing directions to overcome this bottleneck, including attention matrix approximation such as Reformer [36], Performer [17], leveraging a side memory module that can access multiple tokens at once [38, 39,56] such as Longformer [5] and BigBird [70], segment-based recurrence such as Transformer-XL [19] and Compressive Transformer [49]. Please refer to a recent survey [58] for more details. In this paper, we mainly explore within the scope of approximating dense or full attention matrices.
Existing combination of Sparse and Low-rank Attention: Our focus on the classical and welldefined problem of matrix approximation, as opposed to simply designing an efficient model that performs well on downstream tasks (e.g., Longformer, Luna, Long-short transformer, etc.) affords us several advantages: (i) Easier understanding and theoretical analysis (Section 3, 4). We see that Scatterbrain yields an unbiased estimate of the attention matrix, and we can also understand how its variance changes. (ii) Clear-cut evaluation based on approximation error, as well as the ability to directly replace a full attention layer with Scatterbrain attention without re-training (Section 5). This setting is increasingly important as transformer models are getting larger and training them from scratch has become prohibitively costly. Other methods such as Luna and Long-short transformer are not backward compatible with pre-trained models.
Here we compare Scatterbrain with other work mentioned by the reviewer, showing how most of them are special cases of Scatterbrain. We will also add this discussion in the updated version of the manuscript. • Longformer [5]: a special case of Scatterbrain where the sparse component is local attention, and the low-rank component is the global tokens. Global tokens can be considered a restricted form of low-rank approximation. • BigBird [70]: a special case of Scatterbrain where the sparse component is local + random sparse attention, and the low-rank component is the global tokens. The use of global tokens makes the model unsuited for autoregressive modeling. On the other hand, Scatterbrain's generality allows it to use other kinds of low-rank attention (e.g., Performer), and thus Scatterbrain works on both the causal/autoregressive and the bidirectional/non-causal attention settings. BigBird's motivation is also quite different from ours: they aim to design efficient attention such that the whole Transformer model is still a universal approximator and is Turing complete. Our goal is more concrete and easier to evaluate: we approximate the attention matrices, to get a small Frobenius error between the Scatterbrain attention and the full attention matrices. • Luna [43] (concurrent work): they use a fixed-length extra sequence and two consecutive attention steps: the context sequence attends to the extra sequence, and then the query sequence attends to the extra sequence. This is similar in spirit to low-rank attention (Linformer) and global tokens, but it is not a low-rank approximation due to the non-linearity between the two attention steps. It is not clear to us that it combines different kinds of attention. • Long-short transformer [71] (concurrent work): a special case of Scatterbrain where the sparse component is local attention and the low-rank component is Linformer.
A.3 Locality Sensitive Hashing for Efficient Neural Network Training
Locality Sensitive Hashing (LSH) has been well-studied in approximate nearest-neighbor search [2,11,27,30,34,54]. Since the brute-force approach for similarity search is computationally expensive, researchers have come up with various indexing structures to expedite the search process. Usually this comes with trade-offs on the search quality. Based on these indexing structures, one can achieve sub-linear search time. LSH has been used in estimation problem as well [12,13].
Recently, there has been several work taking advantage of LSH data structures for efficient neural network training. During training process, the weight matrices are slowly modified via gradients derived from objective functions. If we consider the weights as the search data and input as queries, we can view neural network training as a similarity search problem. For example, [14,18,41] proposes an algorithm which performs sparse forward and backward computations via maximum inner product search during training. It is based on the observation that the model is usually over-parameterized so the activation for a given input could be sparse and LSH is used to find or impose the sparse structure. Similarly, LSH based algorithms have also been used in Transformers [14,15], where LSH is used to capture the sparse structure of the attention matrices. They can largely reduce the memory bottleneck of self-attention modules especially over long sequences in Transformer. Though [15] has done some exploration to improve LSH accuracy-efficiency trade-offs through learnable LSH, most of the above works have limited understanding on when and where LSH can perform well.
A.4 Structured Matrices for Efficient Machine Learning Models
Sparse + low-rank is an example of a class of structured matrices: those with asymptotically fast matrix-vector multiplication algorithm (o(n 2 ) time complexity) and few parameters (o(n 2 ) space complexity). Common examples include sparse, low-rank matrices, and matrices based on fast transforms (e.g., Fourier transform, circulant, Toeplitz, Legendre transform, Chebyshev transform, and more generally orthogonal polynomial transforms). These classes of matrices, and their generalization, have been used in machine learning to replace dense matrices in fully connected, convolutional, and recurrent layers [32, 55,61]. De Sa et al. [24] shows that any structured matrix can be written as product of sparse matrices, and products of sparse matrices even with fixed sparsity pattern have been shown to be effective at parameterizing compressed models [1,20,21].
In our setting, it remains difficult to approximate the attention matrix with these more general classes of structured matrices. This is because many of them are fixed (e.g., Fourier transform, orthogonal polynomial transforms), and there lacks efficient algorithms to find the closest structured matrix to a given attention matrix.
B Motivating Observations: Low-rank and Sparse Structures of Attention Matrices
We aim to build a deeper understanding of sparse and low-rank structures in real attention matrices: where each of them excel, and the potential for their combination. Specifically, we • show that sparse and low-rank approximation errors are negatively correlated (through statistical tests),
• characterize regimes where each of sparse and low-rank approximation are well-suited, as dictated by the entropy of the softmax attention distribution, and • demonstrate that sparse + low-rank has the potential to achieve better approximation than either.
B.1 Setup
Denote M as the attention matrix (after softmax) and H as entropy. We measure approximation error by the Frobenius norm or the original matrix and the approximation (sparse or low-rank). All the observed attention matrices in this section are from (1) a 4-layer vanilla Transformer trained from scratch on char-level IMDb reviews classification [57] (2) a 16-layer vanilla Transformer trained from scratch on WikiText103 [45] (3) a 1-layer (attention) pre-trained BigGAN on ImageNet [25]. To collect attention matrices for IMDb and WikiText103, we first save checkpoint of the models in every epoch; then evaluate 100 samples from validate data for each checkpoint and collect attention matrices from each layer each head. Note we take the median of the stats (error) for those 100 samples if it is difficult to visualize. To collect attention matrices for BigGAN, we generate 100 samples and collect the attention on the fly. We fixed the number of parameters, K, allowed for each attention matrix approximation and collect the errors from ideal sparse and low-rank approximations: top−K entries for each row of the matrix for sparse and top−K eigenvalues for low-rank. Then we run three standard statistical correlation tests [4,59], Spearman, Pearson and Kendall's Tau on sparse and low-rank approximation error for all the matrices. We can see from Table 3 that errors are significantly negatively correlated (p-value < 0.05). Further more, the left three plots on Figure 7 visualizes the correlation between the two errors on three datasets.
B.2 Observation 1: Sparse and low-rank approximation errors are negatively correlated
This negative correlation suggests that there is some property of the softmax attention distribution which determines when sparse or low-rank excels. We validate this claim in the next observation.
B.3 Observation 2: Sparse approximation error is lower when softmax entropy is low and low-rank approximation error is lower error when entropy is high
We visualize the sparse and low-rank approximation error against the entropy of attention matrices H(M ) (applied to each row, then averaged) on the right plot in Figure 7. The attention matrices are ∈ R 1024×1024 (padded) so the x-axis has range from [0, ln (1024)]. For high-entropy distributions (more diffused) low-rank Sparse + low-rank yields better approximation than sparse or low-rank alone, across the board.
matrices approximates the attention matrix well. For low-entropy distributions (more peaked), sparse matrices are better-suited. This implies that sparse and low-rank approximations could be complementary: if we can combine the strength of both, it is possible to come up with a better approximation across more general scenarios. Therefore, in the next observation, we try to combine sparse and low-rank approximations.
B.4 Observation 3: Sparse + Low-rank achieves better approximation error than sparse or low-rank alone
We find an approximation of the attention matrix of the form S + L, where S is sparse and L is low-rank. This problem has a rich history and is commonly solved with Robust PCA. As shown in 7, across the range of entropy, sparse + low-rank approximation can achieve lower error than either sparse or low-rank when choosing the correct mix ratio of sparse and low rank approximation ideally (with robust-PCA). Motivated by the fact that sparse and low-rank approximations of attention matrices have complementary strengths (Observations 1 and 2), one might want to combine them (Observation 3) in hope of yielding a more robust approximation that works well across different kinds of attention matrices. The above introduces three main challenges that we have addressed in the main paper: • how to find sparse + low-rank decomposition of an attention matrix that is compute efficient (the most studied algorithm, robust PCA, is orders of magnitude too slow to be done at each training iteration) and memory efficient (i.e., without materializing the full matrix) (Section 4), • if we can find such a sparse + low-rank decomposition, how accurate is the approximation (Section 4.3), • how expressive is the sparse + low-rank parameterization, i.e., are there natural classes of matrices where sparse + low-rank yields asymptotically better approximation than sparse or low-rank alone) (Section 3)?
C Scatterbrain Algorithm and Implementation Details
Let Q, K ∈ R n×d be the query and key matrices respectively, and V ∈ R n×d be the value matrix. Let the rows of Q be q 1 , . . . , q n , and the rows of K be k 1 , . . . , k n . The attention computes:
softmax(QK )V,
with softmax applied row-wise, where for each vector v ∈ R n , softmax(v) = 1 n j=1 e v j e v1 , . . . , e vn . Here we omit the usual scaling of QK √ d for simplicity since that could be folded into Q or K. Note that softmax(QK ) = D −1 exp(QK ), where the exponential function is applied element-wise and D is a diagonal matrix containing the softmax normalization constants (D i,i = n j=1 exp(q i k j )). Then attention has the form D −1 exp(QK )V .
We describe the Scatterbrain approximation algorithm in Algorithm 1. This includes the normalization step.
Algorithm 1 Scatterbrain Approximation of Attention 1: Input: Q, K, V ∈ R n×d , hyper-parameters m, k, l 2: procedure Init(m, k, l)
3:
Sample W ∈ R m×d where Wi ∼ N (0, 1) i.i.d.
4:
Kernels φ : R d → R m , φ(x) = exp(W x− x 2 /2) √ m 5: Hash ∀l ∈ [L], H l = {h l,k } k∈[K] , H = ∪ l∈[L] H l 6: end procedure 7: procedure LowRankApprox(Q, K, V, φ) 8: Q = φ(Q), K = φ(K)
applied to each row S ← sparse matrix whose support is S 14: for (i, j) ∈ S do 15: Sij = exp(q i kj) − φ(qi) φ(kj). 16: end for 17: return SV , S1n. 18: end procedure 19: procedure ScatterbrainApprox(Q, K, V ) 20: φ, h ← Init(m, k, l).
21:
O lr , D lr ← LowRankApprox(Q, K, V, φ).
22:
Os, Ds ← SparseApprox(Q, K, V, φ, h).
23:
return diag(D lr + Ds) −1 (O lr + Os). 24: end procedure Autoregressive / Causal / Unidirectional Attention To approximate autoregressive attention, we simply use the autoregressive variant of low-rank attention, and apply the autoregressive mask to the sparse attention. In particular, let M ∈ R n×n be the autoregressive mask, whose lower triangle is all ones and the rest of the entries are zero. The unnormalized attention matrix is exp((QK ) M ), and the unnormalized output is exp((QK ) M )V , where is elementwise multiplication.
The low-rank autoregressive variant computes (( Q K ) M )V , though with a custom GPU kernel / implementation so as not to materialize the n × n matrix. For the sparse component, we simply mask out locations S ij where i > j. That is, we can perform S M efficiently. As a result, we can compute the Scatterbrain output (( Q K ) M )V + (S M )V efficiently.
D Proofs
D.1 Expressiveness of Sparse + Low-rank Matrices
To motivate the use of sparse + low-rank matrices, we describe a family of attention matrices where sparse + low-rank matrices need asymptotically fewer parameters to approximate the attention matrix, compared to sparse or low-rank matrices alone. For there cases, either sparse or low-rank alone requires a quadratic number of parameters (O(n 2 ), where n × n is the dimension of the attention matrix) to get approximation error in Frobenius norm, while sparse + low-rank only requires O(n) parameters.
We construct a matrix family that shows the separation between the approximation capability of sparse + low-rank vs. sparse or low-rank alone. More specifically, we will use diagonal + low-rank (a special case of sparse + low-rank). Example 1. Let denote a parameter that satisfies ∈ (0, 1/2]. Consider the following randomized construction of a matrix Q ∈ R n×d with d ≥ 6 −2 log n and d = Θ( −2 log n), where each entry of Q is picked independently and uniformly at random from {±1/ √ d}. Let M = σ(QQ ) where σ is the elementwise exponential function (we first ignore the normalization term of softmax here).
It can be shown (e.g. by Hoeffding's inequality) that with high probability
(QQ ) i,j = 1, if i = j; ∈ [− , ], otherwise.
Since M = σ(QQ ) where σ is the elementwise exponential function,
M i,j = e, if i = j; ∈ [1 − O( ), 1 + O( )], otherwise.
Intuitively, as the attention matrix M has large diagonal entries, low-rank matrices will not be able to approximate it well. However, the off-diagonals are also of reasonable size, thus making sparse approximation difficult. With sparse + low-rank, we can use the sparse part to represent the diagonal, and the low-rank part to represent the remaining elements, allowing it to approximate this matrix well. We formalize this separation in the theorem below.
Theorem 3. Let M be the attention matrix from Example 1. For any γ ∈ [0, 1], with probability at least 1 − n −1 , there exists a sparse + low-rank estimator with O(γ −1 n 3/2 log n) parameters that achieve γ √ n Frobenius error. For any matrix R ∈ R n×n with rank such that n−rank = Ω(n) (e.g., R has o(n 2 ) parameters), with probability at least 1 − n −1 , we have M − R F ≥ Ω( √ n). Moreover, any matrix E S that has row sparsity k (each row has less than k non-zeros) such that n − k = ω(1) (e.g., E S has o(n 2 ) parameters) will have error M − E S F ≥ Ω( √ n) with probability at least 1 − n −1 .
We see that for any γ ∈ [0, 1], any low-rank or sparse estimator for M with (n 2 ) parameters has Ω(γ −1 ) times the error of the sparse + low-rank estimator with O(γ −1 n 1.5 log n) parameters.
Proof of Theorem 3. For each i ∈ [n], let q i denote the i-th row of Q ∈ R n×d . Define J ∈ R n×n to be the all 1s matrix. Define T = M − J − QQ . Therefore,
T i,j = e − 2 if i = j e q i qj − 1 − q i q j otherwise .
By Hoeffding's inequality, for a pair i = j, we have that
P q i q j − E[q i q j ] ≥ ≤ 2 exp − 2 2 1 √ d − −1 √ d 2 = 2 exp(−d 2 /2).
Note that E[q i q j ] = 0.
Sparse estimator: For our sparse estimator, it is easy to see that for any E S ∈ R n×n that has row sparsity k (each row has fewer than k non-zeros), − k)).
M − E S F ≥ Ω( n(n
This implies that in order to achieve error O( √ n), we would need n − k = O(1), which requires Ω(n 2 )
parameters.
Now we construct a matrix that shows better separation between the approximation capability of sparse + low-rank vs sparse or low-rank alone.
Example 2. Consider the following randomized construction of matrix Q ∈ R n×d with d ≥ 6 −2 r log n and d = Θ( −2 r log n) ( ∈ (0, 1] and close to 0 and r is Θ(log n)): each entry of Q is picked independently and uniformly at random from {± r/d}. Let M = σ(QQ ) where σ is the elementwise exponential function.
Similar to Example 1, with high probability, we have:
(QQ ) i,j = r, if i = j; ∈ [− , ], otherwise.
We also have:
M i,j = e r , if i = j; ∈ [1 − O( ), 1 + O( )], otherwise.
By setting r appropriately, we can formalize the separation between the approximation ability of sparse, low-rank, and sparse + low-rank matrices: Proof of Theorem 4. Similar to the proof of Theorem 3, by Hoeffding's inequality, for a pair i = j, we have that
P q i q j − E[q i q j ] ≥ ≤ 2 exp − 2 2 r √ d − −r √ d 2 = 2 exp − d 2 2r .
Note that E[q i q j ] = 0. By a union bound over all pairs i = j (there are n(n − 1)/2 such pairs), with probability at least 1 − n −1 (since d ≥ 6 −2 r log n), we have that
q i q j ∈ [− , ] for all i = j.
Since we assume that d ≥ 6 −2 log n, we have that For the rest of the proof, we only consider this case (where
q i q j ∈ [− , ] for all i = j). Let T = M − (e r − 1) · I + J,
where J is the all one matrix. We see that T is zero on the diagonal. Moreover, using the fact that e x ≤ 1 + 2 |x| for all x ∈ [−1, 1], the off-diagonal entries of T have of magnitude at most 2 .
We consider 3 different estimators.
Sparse + low-rank estimator: Our estimator is
E SL = (e r − 1) · I sparse + J low−rank ,
where (e − 1)I has row sparsity 1 and rank(J) = 1.
25
The Frobenious error of sparse + low-rank approximation is
M − E SL F ≤ O( 2 n(n − 1)) ≤ O( n).
We have that: (i) Sparse + low-rank parameter count is n · (1 + 1) ≤ O(n).
(ii) Its Frobenius error is ≤ O(n). Low-rank estimator: We want to argue that low-rank approximation would require more parameters. From a similar observation that any matrix R with rank that n − rank = Ω(1),
(e r − 1)I − R F ≥ Ω(e r ),
(by Eckart-Young-Mirsky theorem), we obtain a similar result to the proof of Theorem 3.
If R is a matrix with rank such that n − rank = Ω(1),
then M − R F ≥ Ω(n) − T F ≥ Ω(n) − O( n) ≥ Ω(n).
Hence any low-rank matrix with O(n 2 ) parameters would have error Ω(n).
Sparse estimator: Similar to the proof of Theorem 3, for our sparse estimator, it is easy to see that for any E S ∈ R n×n that has row sparsity k (each row has fewer than k non-zeros),
M − E S F ≥ Ω( n(n − k)).
This implies that to get O(n) error, we would need Ω(n 2 ) parameters.
D.2 Generative Model, Softmax Temperature, and Matrix Approximation
Here we show 3 cases where depending on the softmax temperature, either we'll need low-rank, low-rank + sparse, or sparse to approximate the attention matrix. We start with some notation first. To prove Theorem 1, we first define a more general matrix class, prove that the attention matrix in Process 1 is a subset of this class (with high probability), and then show that Theorem 1 holds for this more general class. We introduce an extra parameter l ∈ R, in addition to the inverse temperature β and the intro-cluster distance ∆. We now show that Process 1 is a subset of Matrix Class 1, with high probability.
Matrix
z ij 2 = c i + r ij 2 = c i 2 + 2c i r ij + r ij 2 .
Since
c i ∼ N (0, I d / √ d), c i 2 ∈ [1 − ∆ 2 , 1 + ∆ 2 ]
with probability at least 1 − 2e −d∆ 2 /8 (by the standard argument using the fact that χ 2 -random variables are sub-exponential). Similarly,
r ij 2 ∈ [∆ 2 − ∆ 4 , ∆ 2 + ∆ 4 ]
with probability at least 1 − 2e −d∆ 2 /8 . By concentration of measure, we can also bound 2c i r ij ∈ [2∆ − 2∆ 3 , 2∆ + 2∆ 3 ] as well. Therefore, we have that z ij
2 ∈ [1 − O(∆), 1 + O(∆)].
Now we show that the large entries of QQ form a block diagonal matrix. With high probability, the large entries come from intra-cluster dot product, and the small entries come from inter-cluster dot product.
We bound the intra-cluster dot product:
z ij z ik = (c i + r ij ) (c i + r ik ) = c i 2 + c i r ij + c i r ik + r ij r ik .
Similar to the argument above, by concentration of measure, c i 2 ∈ [1 + ∆, 1 − ∆] with high probability (we will pick = θ(∆)). The cross terms c i r ij and c i r ik can be bounded using Cauchy-Schwarz inequality to be in [− ∆, ∆] with high probability. And the fourth term r ij r ik is in [− ∆ 2 , ∆ 2 ] with high probability. Therefore, the inner product is in 1 ± O( ∆) with high probability. This satisfies the first condition in Matrix Class 1, for l = 1 1−∆ 2 , assuming ≤ ∆. We use a similar argument to bound the inter-cluster dot product. For i = i To prove Theorem 1 for Matrix Class 1, we start with some technical lemmas.
z ij z i k = (c i + r ij ) (c i + r i k ) = c i c i + c i r i k + c i r ij + r ij r i k .Lemma 6. Let F ∈ R N ×N ≥0
be a symmetric matrix. Let λ max be the largest eigenvalue of F . Assuming N ≥ 2, we have that
λ max ≥ min i =j F [i, j].
Proof. Since F is symmetric, λ max is real and
λ max = max u =0 u F u u T u .
Let u be the all 1's vector, then
λ max ≥ 1 N i=j F [i, j] ≥ 1 N i =j F [i, j] ≥ 1 N · N (N − 1) min i =j F [i, j] ≥ min i =j F [i, j],
where the second step follows from all the diagonal entries are non-negative, the last step follows from N ≥ 2
The above implies the following result: The claim then follows from the fact that any eigenvalue of B is also an eigenvalue of F . We'll need the following function for our low-rank argument:
Corollary 7. Let F ∈ R N ×Nf k (x) = k i=0 x i i! . Note that f ∞ (x) = e x .e x = f D (x) + ∞ i=D+1 x i i! ,
It is sufficient to show that |e y − f (y)| < if we have
x D+1 (D + 1)! ≤ 2 ,
We can show that
y D D! ≤ L D D! ≤ L D (D/4) D = ( 4L D ) D ≤ (1/2) D ≤ /10
where the first step follows from |y| ≤ L, the second step follows n! ≥ (n/4) n , the forth step follows from D ≥ 10L, the last step follows D ≥ 10 log(1/ ) and ∈ (0, 1/10).
We'll also use the following fact: Proof. We can upper bound rank(f D (A)) in the following sense:
rank(f D (A)) ≤ (rank(A)) D ≤ d D = 2 D·log d = 2 o(log n) = n o(1) .
where the second step follows from rank(A) ≤ d, the forth step follows from D = o( log n log d ). Finally we're ready to prove the theorem:
Proof. The basic idea is: (i) Use f k * (b · A) to get the low-rank approximation (ii) Use exp(b · H) to get the sparse part.
Small β range, i.e., β is o log n log d . Low rank approximation: R = f k * (b · A). Since each entry of A is in [−1, 1], each entry of β · A is in [−β, β]. But note that β in this case is o log n d = O(log n · ∆). By the definition of k * , each entry of exp(β · A) − f k * (β · A) has absolute value ≤ . Therefore the overall error is ≤ n.
For sparse only: By assumption, m = Ω( L 0 ) entries in A are ≥ 0, which are exactly the entries in exp(β · A) that are ≥ 1. Hence any (say) m 2 sparse approximation has error ≥ m 2 ≥ Ω( L 0 ). By our assumption, L 0 = Ω(n 2 ).
Mid-range β, i.e., β ≥ 1 l · log n and β is O(log n). Sparse only: the argument is the same as in the low β range. Sparse + low-rank: The low-rank part R = f st(β · A). By Lemma 9, this has rank n o(1) , so it has n (1+o (1) Low-rank only: Let R be rank r − n o(1) − 1 that approximates M β . Then using the same argument as our existing lower bound argument, we get that R − R ≈ E S (this means that the error
≤ E F + M β − R F ).
Now note that S = e β·H − (f k * (β · A)) suppH is a symmetric, block diagonal matrix with r = Ω(n) blocks. Corollary 7 implies that λ r (S) is at least the smallest non-diagonal value in S. Now the smallest non-diagonal value in e β·H is ≥ e 1 l log n = n. On the other hand, the largest value in (f k * (β · A)) suppH is Hence λ r (S) is Ω(n). The claimed result then follows since E F ≤ n and rank R − R ≤ r − 1 (Eckart-Young-Mirsky theorem).
≤ k * β k * k * ! ≤ β · eβ k * − 1 k * −1
Large β range, i.e., β ≥ ω(log n). Sparse only: S = e β·H . Note that each entry in E = M β − S is upper bounded by e ∆·β ≤ e o( β log d ) . Then
E F ≤ n · e o( β log d ) ≤ · e log n +o( β log d ) ≤ · e o(β)+o( β log d ) ≤ · e o(β)
≤ · e β/l .
Low-rank only: since E F is ≤ e β/l , it is enough to argue that any rank r-approximation to S has error ≥ e β/l . But the latter follows since λ r (S) ≥ e β/l . This is because e b·H is symmetric and each entry in H is ≥ 1 λ . Then we can use Corollary 7. Eckart-Young-Mirsky then completes the proof.
29
D.3 Scatterbrain: Analysis
Here we prove Theorem 2, which shows that Scatterbrain approximation is unbiased and analyses its variance. We restate the theorem here for the reader's convenience.
Theorem. Define σ(q, k) = exp(q k), σ pfe as Performer's estimator and σ sbe as Scatterbrain estimator. Denote S d−1 ⊂ R d as the unit sphere. Suppose q, k ∈ S d−1 are such that q − k < τ . Then:
E[ σ sbe (q, k)] = σ(q, k), Var[ σ sbe (q, k)] = (1 − p) · Var[ σ pfe (q, k)] < Var[ σ pfe (q, k)] where p = exp(− τ 2 4−τ 2 ln d − O τ (ln ln d)).
Proof. Let A ij = exp(q k k j ) be ij-entry of the unnormalized attention matrix, A lr ij = φ(q i ) φ(k j ) the entry of the low-rank approximation (Performer), and let A sb ij be the entry of the Scatterbrain (sparse + low-rank) approximation. By the construction of the Scatterbrain attention matrix (Eq. (1)), if ij ∈ S, where S is the set of indices selected by the LSH, then:
A sb ij = ( Q K + S) ij = φ(q i ) φ(k j ) + exp(q i k j ) − φ(q i ) φ(k j ) = exp(q i k j ). If ij / ∈ S, then A sb ij = ( Q K + S) ij = φ(q i ) φ(k j ) + 0 = φ(q i ) φ(k j )
. In other words, A sb matches A on the indices in S, and matches A lr on the indices not in S.
To show that A sb is an unbiased estimator of A, we simply use the fact that A lr is also an unbiased estimator of A [17, Lemma 1]:
E[A sb ij ] = P(ij ∈ S)E[A ij | ij ∈ S] + P(ij / ∈ S)E[A lr ij | ij / ∈ S] = P(ij ∈ S)A ij + P(ij / ∈ S)A ij = A ij .
In other words, E[ σ sbe (q, k)] = σ(q, k). Now we analyze the per-entry variance of A sb . Since A sb is an unbiased estimator of A, by the law of total variance,
V (A sb ij ) = P(ij ∈ S)V (A ij | ij ∈ S) + P(ij / ∈ S)V (A lr ij | ij / ∈ S) = P(ij ∈ S) · 0 + P(ij / ∈ S)V (A lr ij ) = P(ij / ∈ S)V (A lr ij ).
To compute the probability that the index ij is not in S (i.e., not selected by LSH), we use the standard bound on cross-polytope LSH [3, Theorem 1]:
p := P(ij ∈ S) = exp(− τ 2 4 − τ 2 ln d − O τ (ln ln d)). Therefore, V (A sb ij ) = (1 − p)V (A lr ij ) < V (A lr ij ). In other words, V [ σ sbe (q, k)] = (1 − p) · V [ σ pfe (q, k)] < V [ σ pfe (q, k)].
More explicitly, by plugging in the variance of A lr [17, Lemma 2], we have
V (A sb ij ) = (1 − p) 1 m exp q i + k j 2 exp(2q i k j ) 1 − exp − q i + k j 2 , where p = exp(− τ 2 4−τ 2 ln d − O τ (ln ln d))
30 Recall we observe that entropy of the softmax attention distribution (i.e., scale of logits) determines the regimes where sparse, low-rank, or sparse + low-rank perform well. Scatterbrain yields better approximation than reformer or performer in most of the cases; performer performs the worst on language modeling tasks while reformer performs the worst on classification tasks. These plots for approximation error analysis match with their performance on downstream tasks.
32
E.3 More Ablation Studies
E.3.1 Memory Budget
We present an ablation study on the parameter budget for the WikiText-103 language modeling task. We show that Scatterbrain outperforms its sparse and low-rank baselines across a range of parameter budgets. The results are presented in Table 5. Analysis: We have observed that Scatterbrain outperforms its sparse and low-rank baselines under different memory budgets. Similar to what we found in Section 5.2, Performer does not train stably even with 1 4 of the full attention memory. However, under the Scatterbrain framework, Performer can be combined with Reformer in an elegant way to achieve the same accuracy while using only half of the memory and faster than Reformer by exploiting the sparse+low-rank structure in attention matrices.
E.3.2 Different Sparse and Low-rank baselines
Scatterbrain is general enough to accommodate different kinds of sparse and low-rank approximations as its sub-components. In particular, we can combine Local attention or block sparse (from Sparse Transformer and BigBird) + Performer (instead of Reformer + Performer) in a similar fashion. The support of the sparse matrix S will thus be fixed and not adaptive to input, but all the other steps are exactly the same.
We have run additional experiments on the Local attention + Performer combination and BigBird. Recall that in Appendix E, we have shown Scatterbrain can reduce the attention memory of Vision Transformer by 98% at the cost of only 0.8% drop of accuracy when serving as a drop-in replacement for full attention without training on ImageNet. We show the results for local+performer variation with the same memory budget in Table 6.
We have also run additional experiments on Local attention on Copy and Wikitext-103 language modeling task ( Table 7). We see that Local attention is reasonably competitive on Wikitext-103 but does not perform well on Copy. The results are not surprising as noted in the Reformer paper that Copy requires non-local attention lookups.
E.3.3 Different Sparse and Low-rank baselines
E.4 Analysis
Recall in Section 5, we have reported the analysis after visualizing the error of reformer (sparse), performer (low-rank), and Scatterbrain (sparse + low-rank) given the same number of parameters when approximating the full attention matrices for each attention layer during training. In Figure 8, we present the visualization. The conclusion for language modeling tasks is that sparse+low-rank has the smallest approximation error in most of the cases, and sparse has the largest error, which matches with the end-to-end results. It also confirms the observation in the popular benchmark paper [57] that kernel or low-rank based approximations are less effective for hierarchical structured data. For classification tasks, we again find that Scatterbrain has the smallest approximation error, while performer is the worst on ListOps and reformer has the largest error on classification tasks, which matches with the end-to-end results and confirms our observations earlier (sparse and low-rank approximation excel in different regimes).
E.5 Additional Experiments of Fine-tuning Bert on GLUE
We provide additional experiments of fine-tuning Bert on GLUE in Table 8. We follow the similar setting as [22]. We replace all the attention layers in Bert base model with Scatterbrain and other baselines. Then we fine-tune Bert on 9 downstream tasks for 3 epochs with batch size 32 and learning rate 3e-5. The parameter budget is 1/2 of the full attention because sequence length 128 is not very long. We can see Scatterbrain outperforms all the other baselines in most of the downstream tasks.
F Further Discussions and Future Work
In this paper, we present Scatterbrain, unifying the strength of sparse and low-rank approximation. It is inspired by the observations on the attention matrix structures induced by the data and softmax function as well as the classical robust-PCA algorithm. In our implementation and analysis, we have reformer/Smyrf and performer as the back-bone for sparse and low-rank approximations because of their properties, e.g. Performer is unbiased. Scatterbrain is fundamentally a framework for combining the strength of sparse and low-rank variants, so it can be easily extended to other variants, such as Routing Transformer [53] or Nystromformer [67]. Further more, our observations on the connection between entropy and low-rank/sparse approximation error also provide an opportunity for efficiently detecting the approximation or compression method to choose for different architectures or benchmarks.
( b )Figure 3 :
b3Sparse: There exists R with sparsity O(n) such that M β − R F ≤ · e β(Qualitative comparison of approx. accuracy and efficiency, among Robust PCA, sparse (Reformer) and low-rank (Performer) attention, and Scatterbrain. Scatterbrain is more accurate while being efficient.
Figure 6 :
6Speed and memory required by different efficient attention methods. Scatterbrain is competitive with SMYRF (sparse baseline) and Performer (low-rank baseline), while up to 3× faster and 12× more memory efficient than full attention for sequence length 4096.
B 34 F
34Motivating Observations: Low-rank and Sparse Structures of Attention Matrices 19 B.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 B.2 Observation 1: Sparse and low-rank approximation errors are negatively correlated . . . . 19 B.3 Observation 2: Sparse approximation error is lower when softmax entropy is low and low-rank approximation error is lower error when entropy is high . . . . . . . . . . . . . . 19 B.4 Observation 3: Sparse + Low-rank achieves better approximation error than sparse or low-rank alone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . of Sparse + Low-rank Matrices . . . . . . . . . . . . . . . . . . . . . . . . 23 D.2 Generative Model, Softmax Temperature, and Matrix Approximation . . . . . . . . . . . 26 D.3 Scatterbrain: Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 E Additional Experiments and Details 31 E.1 Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 E.2 Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 E.3 More Ablation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 E.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 E.5 Additional Experiments of Fine-tuning Bert on GLUE . . . . . . . . . . . . . . . . . . . . Further Component Analysis (robust PCA) is the problem of finding a composition of a matrix M into a sum of sparse and low-rank components: M = S + L.
Figure 7 :
7Characterization of the relationship between the softmax distribution of each attention matrix row and approximation error of sparse, low-rank and sparse+low-rank. The top, middle and bottom plots are for IMDb, WikiText103 and BigGAN-ImageNet respectively. Left: The approximation error of sparse and low-rank are negatively correlated. Sparse performs well when low-rank does not, and vice versa. Right: Entropy of the softmax attention distribution (i.e., scale of logits) determines the regimes where sparse, low-rank, or sparse + low-rank perform well.
( K V ), Q( K )1n. 10: end procedure 11: procedure SparseApprox(Q, K, V, φ, H) 12: S = {(i, j)|H(Qi) = H(Kj)} 13:
Theorem 4 .
4Let M be the attention matrix from Example 2. Any sparse or low-rank estimator of M needs Ω(n 2 ) parameters for O(n) error with probability at least 1 − n −1 while a sparse + low-rank estimator needs O(n) parameters for O(n) error with probability at least 1 − n −1 .
Given a matrix B, let B[i, j] be the entry in the ith row and jth column. For a range [l, r], we define a matrix B [l,r] where B [l,r] [i, j] = B[i, j] if B[i, j] ∈ [l, r] and B [l,r] = 0 otherwise (that is, B [l,r] only keep entries for B that are in the range [l, r], with other entries zeroed out). We write supp(C) for the set of locations of non-zeros in C. We let λ i (D) be the i-th largest (in absolute value) eigenvalue of D.
Class 1 .
1Let Q ∈ R n×d with every row of Q having 2 -norm in [1 − O(∆), 1 + O(∆)], and let A = QQ . Further: 1. Let H = A [1/l,2−1/l] for some l ≥ Ω(1). Assume that H is block diagonal with Ω(n) blocks, and supp(H) is o(n 2 ). That is, the large entries of QQ form a block diagonal matrix. 2. Let L = A − H then L = A [−∆,∆] where ∆ = o(1/log d). Assume that there is a constant fraction of elements in supp(L) falling in [0, ∆]. Assume that supp(A [0,∆] ) is Ω(n 2 ). Let M β = exp(β · A).
Lemma 5 .
5The matrix M β in Process 1 is a subset of Matrix Class 1, where l = 1 1−∆ 2 . Proof. We first bound the norm of each row in Q in Process 1. For any i, j, we have
By concentration of measure, c i c i ∈ [− , ]. Similar to the argument in the intra-cluster case, we can bound the other three terms, so this dot product is in [−O( ), O( )]. This satisfies the second condition in Matrix Class 1.
≥0 be a block diagonal matrix. Let r be the number of m × m blocks in F for some m ≥ 2. The λ r (F ) is at least the smallest non-diagonal element in any m × m block (m ≥ 2) in F . Proof. By Lemma 6, each m × m block B (m ≥ 2) by itself has max eigenvalue at least min i =j∈[m] B[i, j].
Definition 1 .
1Let ∈ (0, 1/10) and L > 0. We say a function f :R → R is ( , L)-close to e y if |e y − f (y)| ≤ for any y ∈ [−L, L].Lemma 8. For any ∈ (0, 1/10) and L > 0. If D ≥ 10(L + log(1/ )) then function f D (y) is ( , L)-close to e y . Proof. Recall the definition of function f D ,
Lemma 9 .
9For any D = o(log n/ log d), we have rank(f D ) ≤ n o(1) .
) parameters. The sparse part is S = e β·H − R supp(H) . Clearly this needs |supp(H)| parameters. Let E = M β − (S + R). Then (i) in supp(H), E is all 0. (ii) output of supp(H), by definition, entries of β · A are in [−β∆, β∆], which in the current range of β is [−O(log n∆), O(log n∆)]. Therefore all the entries of E have absolute value ≤ . By the definition of k * , we have that E F ≤ n.
Figure 8 :
8Top two plots present Approximation Error vs. Entropy of attention matrices for reformer, performer and Scatterbrain on Copy (left) and WikiText103 (right). Bottom two plots present Approximation Error vs. Entropy of attention matrices for reformer, performer and Scatterbrain on Text-IMDb (left) and Image-Cifar10 (right).
Table 1 :
1Top-1 Accuracy of pre-trained T2T Vision Transformer on ImageNet with different attention replacements. Error represents the average normalized approximation error to full attention.Attention
Top-1 Acc
Error (avg)
Full Attention
81.7%
-
SMYRF
79.8%
11.4%
Performer
80.1%
7.5%
Baseline SMYRF + Performer
79.7%
12.6%
Scatterbrain
80.7%
5.3%
Table 2 :
2The performance of Scatterbrain, Reformer, Performer and Full-Attention on Long-Range-Arena benchmarks and 2 popular language modeling tasks. We fix the same number of parameters (1/8 of the full) used for approximating the attention matrix for each method.Attention
Copy (ppl)
WikiText-103 (ppl)
Full Attention
1
25.258
Reformer
6.8
27.68
Performer
49
66
Scatterbrain
2.58
26.72
Attention
ListOps Text Retrieval Image Pathfinder Avg
Full Attention
38.2
63.29
80.85
41.78
73.98
59.62
Reformer
36.85 58.12
78.36
28.3
67.95
53.9
Performer
35.75 62.36
78.83
39.71
68.6
57.05
Scatterbrain
38.6 64.55 80.22 43.65
69.91
59.38
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Table of Contents
ofRobust PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 A.2 Efficient Transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 A.3 Locality Sensitive Hashing for Efficient Neural Network Training . . . . . . . . . . . . . . 18 A.4 Structured Matrices for Efficient Machine Learning Models . . . . . . . . . . . . . . . . . 18A Extended Related Work
17
A.1
Table 3 :
3The Spearman's rank, Pearson and Kendall's Tau correlation coefficients between Sparse and Low-rank approx. error on IMDb, WikiText-103, and BigGAN-ImageNet. P-values of < 0.05 indicate statistical significance. The two errors are negatively correlated.IMDb
WikiText103
BigGAN-ImageNet
Coef
p-value
Coef
p-value
Coef
p-value
Spearman's rank
-0.89
< .00001
-0.63
< .00001
-0.21
< .00001
Pearson
-0.78
< .00001
-0.61
< .00001
-0.31
< .00001
Kendall's Tau
-0.74
< .00001
-0.51
< .00001
-0.18
< .00001
Table 4 :
4The performance of Scatterbrain, reformer, performer and Full-Attention on Long-Range-Arena benchmarks and 2 popular language modeling tasks. We fix the same number of parameters(1/8 of the full) used for approximating the attention matrix for each method. Full Attention 38.2±0.17 63.29±0.38 80.85±0.12 41.78±0.26 73.98±0.31 59.62 Reformer 36.85±0.37 58.12±0.42 78.36±0.29 28.3±0.39 67.95±0.28 53.9 Performer 35.75±0.29 62.36±0.49 78.83±0.33 39.71±0.48 68.6±0.36 57.05 Scatterbrain 38.6±0.22 64.55±0.34 80.22±0.31 43.65±0.46 69.91±0.25 59.38Attention
Copy (ppl)
WikiText-103 (ppl)
Full Attention
1
25.258±0.37
Reformer
6.8±0.64
27.68±0.53
Performer
49±2.7
66±5.8
Scatterbrain
2.58±0.21
26.72±0.44
Attention
ListOps
Text
Retrieval
Image
Pathfinder
Avg
Table 5 :
5We run WikiText-103 LM with a sweep of 1/4, 1/8, 1/16 memory budget. We show the validation perplexity and speed-up with respect to the full attention with different efficient Attention layers.1
4 Mem
1
8 Mem
1
16 Mem
Perplexity (Speed-up)
Perplexity
Perplexity
Smyrf
26.76 (1.6×)
27.68 (1.39×)
28.7(1.85×)
Performer
58(2.13×)
66 (2.01×)
85(1.77×)
Scatterbrain
26.26(1.58×)
26.72 (1.87×)
27.74(2.03×)
Table 6 :
6Top-1 Accuracy of pre-trained T2T Vision Transformer on ImageNet with different attention replacements. Error represents the average normalized approximation error to full attention.Attention
Top-1 Acc
Full Attention
81.7%
SMYRF
79.8%
Local
79.6%
Performer
80.1%
BigBird
80.3%
Scatterbrain (Local + Performer)
80.3%
Scatterbrain (SMYRF + Performer)
80.7%
Table 7 :
7Additional experiments for Local attention on the Copy and Wikitext-103 language modeling task.Attention
Copy (ppl)
WikiText-103 (ppl)
Full Attention
1
25.258
Reformer
6.8
27.68
Performer
49
66
Local
53
30.72
Scatterbrain
2.58
26.72
Table 8 :
8Results of GLUE when replacing dense attention matrices with smyrf, performer and Scatterbrain in BERT base model. We fix the same number ofparameters (1/2 of the full) used for approximating the attention matrix for each method.CoLA
SST-2
MRPC
STS-B
QQP
MNLI
QNLI
RTE
WNLI
mcc
acc
acc
corr
acc
acc
acc
acc
acc
Full
0.576
0.934
0.874
0.879
0.905
0.813
0.916
0.668
0.43
Smyrf
0.538
0.912
0.833
0.856
0.898
0.775
0.879
0.626
0.412
Performer
0.508
0.838
0.782
0.203
0.831
0.563
0.763
0.556
0.449
Scatterbrain
0.569
0.927
0.863
0.867
0.902
0.813
0.893
0.619
0.428
For simplicity of discussion, we consider the unnormalized attention matrix A = exp(QK ), omitting the usual scaling of √ d and the softmax normalization constant.
SMYRF is a variant of Reformer that does not require the key and query to be the same, which is necessary for experiments in this section.
AcknowledgmentsWe thank Xun Huang, Sarah Hooper, Albert Gu, Ananya Kumar, Sen Wu, Trenton Chang, Megan Leszczynski, and Karan Goel for their helpful discussions and feedback on early drafts of the paper.We gratefully acknowledge the support S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views, policies, or endorsements, either expressed or implied, of NIH, ONR, or the U.S. Government. Atri Rudra's research is supported by NSF grant CCF-1763481.AppendixBy a union bound over all pairs i = j (there are n(n − 1)/2 such pairs), with probability at least 1 − n 2 exp −d 2 /2 , we have that q i q j ∈ [− , ] for all i = j.Since we assume that d ≥ 6 −2 log n, we have thatHence q i q j ∈ [− , ] for all i = j with probability at least 1 − n −1 . For the rest of the proof, we only consider this case (where q i q j ∈ [− , ] for all i = j).Since 1 + x ≤ e x ≤ 1 + x + x 2 for |x| < 1, we can bound the off diagonal elements |T i,j | ≤ 2 . In particular, for all i = j,Sparse + low-rank estimator: We use the following sparse + low-rank estimator:where (e − 2)I has row sparsity 1 and rank(JNotice that the E SL estimate matches M exactly on the diagonal, and on the off-diagonal it differs from M by T ij . Thus, the Frobenious error of the sparse + low-rank estimator isLow-rank estimator: We want to argue that low-rank approximation would require more parameters. If we approximate the matrix (e − 2)I by a matrix R with rank r, then the difference matrix will have at least n − d singular values of magnitude e − 2 ≥ 1/2. As a result, by the Eckart-Young-Mirsky theorem,Define T = T − (e − 2) · I, then T is all 0 on the diagonal and has absolute value ≤ 2 on off-diagonal entries. Thus T F ≤ 2 n = γ √ n.We want to show that if R is a rank r matrix, then M − R F ≥ 1 2 √ n − r − d − 1 − T F . We argue by contradiction. Suppose that there exists some matrix R with rank r such thatWe see that:This contradicts the result above, which states that (e − 2) · I − R F ≥ 1 2 n − rank(R). Therefore any low-rank estimator with rank r such that n − r = Ω(n), which has Ω(n 2 ) parameters, will have error at least Ω( √ n − r − d − 1) − T F = Ω( √ n), which is Ω(γ −1 ) times the error of the sparse + low-rank estimator above.
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It has roughly 1.2 million training images and 50,000 validation images. WikiText103 [45] and Copy [36]: WikiText103 is a popular dataset for auto-regressive models. It is from a collection of over 100 million tokens extracted from the set of verified good and featured articles on Wikipedia. It has 28,475 training articles, 60 for validation and 60 for testing. Copy is a synthetic a synthetic sequence duplication task where inputs are of the form 0w0w and w. . . ∈ {0, ImageNet is one of the most widely-used image classification benchmarks. In our experiments in Section 5.1 of evaluating the approximation accuracy of Scatterbrain, both BigGAN and Vision Transformer are pre-trained on this dataset. This task is useful for demonstrating the effectiveness of long range attention: it requires non-local attention lookups. It cannot be solved by any model relying on sparse attention with a limited range such as, local attention. Long Range Arena (LRA) [57]: This is a recent benchmark for evaluating efficient transformers with long input sequence. We used ListOps [46], byte-level IMDb reviews text classification [44], byte-level document retrieval [48], image classification on sequences of pixels [37] and Pathfinder [40]. We follow the same evaluation mechanism from [57] but implement our own version in Pytorch (like data loaderImageNet [25]: ImageNet is one of the most widely-used image classification benchmarks. In our experiments in Section 5.1 of evaluating the approximation accuracy of Scatterbrain, both BigGAN and Vision Transformer are pre-trained on this dataset. It has roughly 1.2 million training images and 50,000 validation images. WikiText103 [45] and Copy [36]: WikiText103 is a popular dataset for auto-regressive models. It is from a collection of over 100 million tokens extracted from the set of verified good and featured articles on Wikipedia. It has 28,475 training articles, 60 for validation and 60 for testing. Copy is a synthetic a synthetic sequence duplication task where inputs are of the form 0w0w and w ∈ {0, ..., N } * . It is previously used in [15, 36]. This task is useful for demonstrating the effectiveness of long range attention: it requires non-local attention lookups. It cannot be solved by any model relying on sparse attention with a limited range such as, local attention. Long Range Arena (LRA) [57]: This is a recent benchmark for evaluating efficient transformers with long input sequence. We used ListOps [46], byte-level IMDb reviews text classification [44], byte-level document retrieval [48], image classification on sequences of pixels [37] and Pathfinder [40]. We follow the same evaluation mechanism from [57] but implement our own version in Pytorch (like data loader).
CoLA and SST-2; similarity and paraphrase tasks. Sts-B Mrpc, Qqp Mnli, Qnli , Wnli , GLUE is a standard multi-task benchmark in NLP. It has single-sentence tasks. For our additional experiments below. not enough space to be included in the main paper), we follow the tradition from [22, 26, 68] and truncate all the input sequences to 128 tokensGlUE [64]: GLUE is a standard multi-task benchmark in NLP. It has single-sentence tasks, CoLA and SST-2; similarity and paraphrase tasks, MRPC, STS-B, QQP; and inference tasks, MNLI, QNLI, RTE and WNLI. For our additional experiments below (not enough space to be included in the main paper), we follow the tradition from [22, 26, 68] and truncate all the input sequences to 128 tokens.
Similar to the prior work, we also replace its full attention layer with Scatterbrain at the same resolution. The model has a single attention layer at resolution 64 × 64 (4096). Figure 5 in the main paper shows the best-effort comparison with [1/32, 1/16, 1/8, 1/4, 1/2] of the parameter budget. For example, if given parameter budgetE.2 Settings BigGAN: We adapt the same pre-trained BigGAN model from [22] with no additional training. The model has a single attention layer at resolution 64 × 64 (4096). Similar to the prior work, we also replace its full attention layer with Scatterbrain at the same resolution. Figure 5 in the main paper shows the best-effort comparison with [1/32, 1/16, 1/8, 1/4, 1/2] of the parameter budget. For example, if given parameter budget
Auto-regressive Model: We follow the settings from the popular repo https://github.com/NVIDIA/ DeepLearningExamples for training vanilla Transformer from scratch on WikiText103, except for chunking WikiText103 into sequence length 1024 in order to simulate long input sequences. The model is 16 layer with 8 head and 512 model dimension. We train all the models for 30 epochs and report the best Testing Perplexity. The model we use for Copy task is simply a 2-layer-4-head transformer and sequence length is also 1024. We make 5 runs and report average. Table 4 presents the results with standard deviation. Classification Model: We follow the model setting from. Without any additional training, we just replace the attention layer with Scatterbrain and other baselines and evaluate the approximation error and classification accuracy on ImageNet testings. Again, we report the best-effort best performance of each approximation given the certain parameter budget. 57, 67]. We share the same finding with [67] that the acuracy for the Retrieval tasks is actually higher than reported in [57/2, we report the best performance of Smyrf from choice of 32/64/128 hash round 64/32/16 cluster size. T2-ViT: We use the pre-trained vision transformer model T2T-ViT-14 from [69] with 224 × 224 image size. Without any additional training, we just replace the attention layer with Scatterbrain and other baselines and evaluate the approximation error and classification accuracy on ImageNet testings. Again, we report the best-effort best performance of each approximation given the certain parameter budget. Auto-regressive Model: We follow the settings from the popular repo https://github.com/NVIDIA/ DeepLearningExamples for training vanilla Transformer from scratch on WikiText103, except for chunking WikiText103 into sequence length 1024 in order to simulate long input sequences. The model is 16 layer with 8 head and 512 model dimension. We train all the models for 30 epochs and report the best Testing Perplexity. The model we use for Copy task is simply a 2-layer-4-head transformer and sequence length is also 1024. We make 5 runs and report average. Table 4 presents the results with standard deviation. Classification Model: We follow the model setting from [57, 67]. We share the same finding with [67] that the acuracy for the Retrieval tasks is actually higher than reported in [57].
For inference, we set this ratio based on the entropy of an observed subset of attention matrices in different layers: we allocate more memory to the low-rank component compared to the sparse component if the entropy is high. For training, generally allocating more memory budget to sparse tends to perform better, so in the experiment, we set the ratio to 3:1 (sparse: low-rank component) for simplicity. Moreover, in future work, it could be useful to make this ratio adaptive during training. For example, in the early stage of the training and early layers, attention matrices are usually more uniform (higher entropy). Thus, the approximation error could be even lower if the ratio favors low-rank-based components. One approach could be to monitor the approximation error of sparse and low-rank components compared to full attention regularly and adjust the memory budget accordingly. Ratio between Sparse and Low-rank components: There are some rules that we used in our experiments to set this ratio. We will add the above discussion to the updated manuscriptRatio between Sparse and Low-rank components: There are some rules that we used in our experiments to set this ratio. For inference, we set this ratio based on the entropy of an observed subset of attention matrices in different layers: we allocate more memory to the low-rank component compared to the sparse component if the entropy is high. For training, generally allocating more memory budget to sparse tends to perform better, so in the experiment, we set the ratio to 3:1 (sparse: low-rank component) for simplicity. Moreover, in future work, it could be useful to make this ratio adaptive during training. For example, in the early stage of the training and early layers, attention matrices are usually more uniform (higher entropy). Thus, the approximation error could be even lower if the ratio favors low-rank-based components. One approach could be to monitor the approximation error of sparse and low-rank components compared to full attention regularly and adjust the memory budget accordingly. We will add the above discussion to the updated manuscript.
|
[
"https://github.com/NVIDIA/",
"https://github.com/NVIDIA/"
] |
[
"A novel time delay estimation algorithm of acoustic pyrometry for furnace",
"A novel time delay estimation algorithm of acoustic pyrometry for furnace"
] |
[
"Qi Liu \nSchool of Energy and Environment\nSoutheast University\n210096NanjingChina\n",
"Bin Zhou [email protected] \nSchool of Energy and Environment\nSoutheast University\n210096NanjingChina\n",
"Jianyong Zhang \nSchool of Computing, Engineering and Digital Technologies\nTeesside University\nTS1 3BAMiddlesbroughUK\n",
"Ruixue Cheng \nSchool of Computing, Engineering and Digital Technologies\nTeesside University\nTS1 3BAMiddlesbroughUK\n"
] |
[
"School of Energy and Environment\nSoutheast University\n210096NanjingChina",
"School of Energy and Environment\nSoutheast University\n210096NanjingChina",
"School of Computing, Engineering and Digital Technologies\nTeesside University\nTS1 3BAMiddlesbroughUK",
"School of Computing, Engineering and Digital Technologies\nTeesside University\nTS1 3BAMiddlesbroughUK"
] |
[] |
Acoustic pyrometry is a non-contact measurement technology for monitoring furnace combustion reaction, diagnosing energy loss due to incomplete combustion and ensuring safe production. The accuracy of time of flight (TOF) estimation of an acoustic pyrometry directly affects the authenticity of furnace temperature measurement. In this paper presented is a novel TOF (i.e. time delay) estimation algorithm based on digital lock-in filtering (DLF) algorithm. In this research, the time-frequency relationship between the first harmonic of the acoustic signal and the moment of characteristic frequency applied is established through the digital lock-in and low-pass filtering techniques. The accurate estimation of TOF is obtained by extracting and comparing the temporal relationship of the characteristic frequency occurrence between received and source acoustic signals. The computational error analysis indicates that the accuracy of the proposed algorithm is better than that of the classical generalized cross-correlation (GCC) algorithm, and the computational effort is significantly reduced to half of that the GCC can offer. It can be confirmed that with this method, the temperature measurement in furnaces can be improved in terms of computational effort and accuracy, which are vital parameters in furnace combustion control. It provides a new idea of time delay estimation with the utilization of acoustic pyrometry for furnace.
| null |
[
"https://arxiv.org/pdf/2111.12884v1.pdf"
] | 244,708,977 |
2111.12884
|
2af8ef65e073e79943bed530670dbeb63c6aee10
|
A novel time delay estimation algorithm of acoustic pyrometry for furnace
Qi Liu
School of Energy and Environment
Southeast University
210096NanjingChina
Bin Zhou [email protected]
School of Energy and Environment
Southeast University
210096NanjingChina
Jianyong Zhang
School of Computing, Engineering and Digital Technologies
Teesside University
TS1 3BAMiddlesbroughUK
Ruixue Cheng
School of Computing, Engineering and Digital Technologies
Teesside University
TS1 3BAMiddlesbroughUK
A novel time delay estimation algorithm of acoustic pyrometry for furnace
Received xxxxxx Accepted for publication xxxxxx Published xxxxxxdigital lock-intime delay estimationtime of flightacoustic pyrometryfurnace temperature
Acoustic pyrometry is a non-contact measurement technology for monitoring furnace combustion reaction, diagnosing energy loss due to incomplete combustion and ensuring safe production. The accuracy of time of flight (TOF) estimation of an acoustic pyrometry directly affects the authenticity of furnace temperature measurement. In this paper presented is a novel TOF (i.e. time delay) estimation algorithm based on digital lock-in filtering (DLF) algorithm. In this research, the time-frequency relationship between the first harmonic of the acoustic signal and the moment of characteristic frequency applied is established through the digital lock-in and low-pass filtering techniques. The accurate estimation of TOF is obtained by extracting and comparing the temporal relationship of the characteristic frequency occurrence between received and source acoustic signals. The computational error analysis indicates that the accuracy of the proposed algorithm is better than that of the classical generalized cross-correlation (GCC) algorithm, and the computational effort is significantly reduced to half of that the GCC can offer. It can be confirmed that with this method, the temperature measurement in furnaces can be improved in terms of computational effort and accuracy, which are vital parameters in furnace combustion control. It provides a new idea of time delay estimation with the utilization of acoustic pyrometry for furnace.
Introduction
In coal-fired power plants, the furnace temperature distribution is an essential factor, not only to safety, but also for monitoring combustion reaction [1,2], improving combustion efficiency [1], and reducing pollution [3]. With the development of technologies in recent years, the noncontact temperature measurement in industrial processes is increasingly utilized. Two leading technologies used in coalfired boilers for temperature distribution measurement are the acoustic [4,5] and laser pyrometers [6]. According to the acoustic temperature measurement theory [7], the error of time of flight (TOF) measurement can directly affect the reconstruction accuracy of the temperature field. Therefore, the delay estimation of TOF is crucial for application of the technology.
The acoustic TOF is typically estimated using the direct cross-correlation method (DCC) [8], which determines the time delay by identifying the peak position of the crosscorrelation function according to the similarity of two acoustic signals. Based on DCC, many extensions have also been developed, such as the generalized cross-correlation (GCC) algorithm [9], the Hilbert transform-based algorithm , the weighted cross-correlation [9,10] and the phase correction cross-correlation [11]. By using the crosscorrelation algorithm, one can achieve low measurement error even in a noisy signal background. However, such an algorithm requires a wide signal bandwidth. Theoretically, narrower the signal bandwidth is, lower the measurement error can be achieved.
In the zero-crossing method [12,13], the received signals are obtained for a given detection threshold in the time domain, the TOF is estimated by calculating the difference between the reception time of two received signals. This method has the advantages of high calculation speed because of reduced number of sampling points, so that less computational effort is required. However, a high signal-tonoise ratio (SNR) is a pre-request to this method. The accuracy of TOF estimation depends heavily on SNR. The MAGNITUDE-squared coherence (MSC) function [14] is widely used in signal detection and time delay estimation [15]. It is a normalized cross-spectral density function and measures the strength of association and relative linearity between two stationary stochastic processes on a scale from zero to one. However, a significant limitation encountered in application of the MSC estimate is that only in some particular conditions, for example when coherence is zero or one, it is possible to provide the closed-form expression for the confidence interval. The phase spectrum estimation algorithm [16,17] is another popular method, which uses correlation function and power spectral density (PSD) as the pair of mutual Fourier transforms. The TOF estimation is obtained by detecting the slope of the phase spectrum of the cross power frequency spectrum. For this method, the accuracy of TOF estimation is constrained by the limited sampling period and finite observation time. Long sampling period can significantly increase the computational complexity. The adaptive TOF estimation algorithm [18] has also been used, which does not require prior knowledge of the signal and noise and the algorithm can be realized through automatic adjustment of the adaptive filter parameters based on the statistical characteristics of the signal. Yet its convergence speed and steady-state misalignment are dependent upon the step size factor. Small step size can be used to improve the TOF estimation accuracy, though it can lead to a heavy computational load. The Least mean-square time delay estimate (LMSTDE) has also been mentioned by some authors [19,20], in which, an adaptive FIR filter is used to model the time difference, and the filter weights are interpolated to obtain the delay time. In order to solve the problem of noise input due to limited filter length, many adjustments and deformations for LMSTDE have been made [21][22][23], however it is still not possible to obtain accurate TOF with a small number of filter taps in low SNR conditions [18]. Another Fourier transform based algorithm is known as 'high-order statistical delay estimation algorithm' [24][25][26]. This high-order spectral method of extended multidimensional Fourier transform [27] can be used to suppress stationary and correlated Gaussian white noise and extract the signal amplitude and phase information. However, to achieve a given accuracy, there are strict requirements for sufficient sampling length and high resolution of A/D converter.
In general, a coal-fired boiler furnace has a large crosssection and there are severe noises, which poses serious challenges to the accurate estimation of acoustic TOF for measurement apparatus. With the in-depth study of the delay estimation algorithm [28], the GCC delay estimation algorithm is by far almost the best method in terms of accuracy at low SNR. But the problem of its large computational volume has been a cause of criticism as well. Therefore, an acoustic TOF estimation algorithm that can suppress noise interference while ensure a low computational effort and high accuracy is the key to true furnace temperature distribution mapping.
A TOF estimation algorithm based on a digital lock-in filter (DLF) has been developed by the authors of this paper. With this method, the relationship between the first harmonic and the sweeping moment of time for each frequency component is established via digital lock-in filtering of acoustic time-domain signal. The acoustic TOF is then obtained by finding the mean difference of the moment of time between the received signal and the acoustic source. The accuracy of the DLF algorithm was verified through experiments by comparing it with the classical GCC algorithm. The experimental results under multiple working conditions at different noise levels indicate that the algorithm has strong robustness, better accuracy and precision.
Principle of Acoustic Pyrometry and Time Delay Estimation
Principle of Acoustic Pyrometry
The acoustic pyrometry works based on the thermodynamic gas equation and the acoustic waveform equation. The temperature information of the gaseous medium is obtained according to the acoustic TOF and the sensor positions [7,29]. The measuring principle of a single acoustic path pyrometry in a furnace is shown in Figure 1, where the loudspeaker and the microphone are mounted in two separate acoustic waveguides with sealed rears. The waveguides are perpendicular to the mean flow direction, so that the direction of soundwave is at 90 o (orthogonal) with the direction of flue gas movement.
As is well known, the sound speed c in a gaseous flue medium at a temperature T is given by：
LR cT m (1) 3
where c is the sound speed in the given medium, L is the distance between the microphone and loudspeaker in Figure 1, is the TOF, γ is the isentropic exponent of the medium (flue gas), R is the universal gas constant of an ideal gas, m is the molar mass, and T is the gas temperature. (1), it is can be seen that if L is a known, for a given type of flue gas, T can be found by measuring. Obviously, an accurate measurement of acoustic TOF is the premise and guarantee for acoustic temperature measurement.
Time delay estimation
The TOF estimation algorithm established based on digital lock-in filter (DLF) links the two signals in the time and frequency domains according to their instantaneous frequencies. This algorithm can provide an accurate TOF estimation by using the mathematical expectation of characteristic frequency TOF. Different to other algorithms, the bandwidth of the acoustic source does not constraint the application of this algorithm. The feasibility of DLF algorithm will be validated under low SNR environment by comparing it with the GCC algorithm in the following sections.
Digital Lock-in Filter Algorithm
It is well known that the acoustic frequency is the number of times that sound wave completes periodic vibrations per unit time. A loudspeaker driven by different source signals [7,[30][31][32] through power amplifier can produce different acoustic emission signals. In this research, the sinusoidal linear frequency-sweeping signal [10,32] is used as the acoustic source, which is a sine signal with continuous frequency change, also known as the dominant variable bandwidth signal [33]. The angular frequency Linear of this sweeping waveform is expressed as:
Linear 0 ( ) 2 (0 ) s s B t t f t (2)
where f0 is the starting frequency, B is the bandwidth in hertz, and τs is the pulse width. The waveform of this acoustic source is shown in Figure 2, the following three diagrams in Figure 2 are enlargements of the dashed sections. As the pulse width τs elapses from 0 to 0.2 s, the frequency of the acoustic source signal linearly increases from f0 to B+f0, i.e. from 4 kHz to 8 kHz.
Figure 2．Acoustic source waveform and instantaneous frequency
The spectrum of the environment background is usually extracted prior to other signal processing in DLF to guarantee a superior SNR. The source signal frequency band is chosen according to that of background noise to avoid the frequency band overlap as much as possible. The acoustic source signal is amplified and emitted through the loudspeaker and propagated in the flue for a particular time before it is received by the microphone. As shown in Figure 6, although the acoustic waves are subject to attenuation and distortion by the soot particles in a gas medium [34][35][36][37], the frequency information of the acoustic signal is retained. Compared to the acoustic source, only the phase of the received signal has been shifted, and the amount of phaseshift depends on the distance between the loudspeaker and microphone. When the phase difference between the source and the received signal is found, the TOF of the acoustic on the propagation path can be determined.
From equation (2), it can be seen that within the pulse width τs, there is a linear relationship between the frequency and time. There is a single independent frequency at a given moment of time. In an acoustic measurement system, the received acoustic signals are nonstationary signals. Constrained by the characteristics of the Fourier transform, the traditional frequency-domain analysis method cannot establish a precise relationship between signal representations in the time and the frequency domains. In contrast, the instantaneous frequency can better represent the local characteristics of the nonstationary signal in different periods.
It is assumed that x1(n) is sampled before the loudspeaker, propagated in the flue and received by the microphone as signal x2(n). These two signals can be expressed by the following set of equations [32]:
11 22 ( ) ( ) ( ) ( ) ( ) ( ) x n s n n x n s n D n (3)
where s(n) is the acoustic source signal. μ1(n) and μ2(n) are the noise-contaminated random components in the received signals. is the attenuation coefficient of the acoustic signal. 4 D is the TOF between the two signals. s(n), μ1(n), and μ2(n) are assumed uncorrelated. The amplitude and phase of each sinusoidal frequency component in a signal can be obtained by phase-sensitive demodulation [38]. For a frequency-sweeping signal composed of multiple frequency components, the TOF on the propagation path can be found using the mean transit time of all frequency components. Since x1(n) is sampled in front of the loudspeaker, the SNR is very high, and for naming convenience, we call x1(n) the source signal. A brief flowchart in Figure 3 describes how the time-frequency relationship between the source signal x1(n) and received signal x2(n) is obtained. Both the source signal and the received signal are processed with a digital lock-in and a low-pass filter to extract the amplitude and phase of a signal operating at a known frequency [39], which is likely buried in a noisy background. The first harmonic R1f can be extracted from the source and received signals using a digital lock-in processing technique [40][41][42]. For R1f , its demodulated components X and Y, expressed as X1f (t) and Y1f (t) are obtained through a multiplier and a low-pass filter, where the source and received signals are multiplied with the two sets of orthogonal reference signals (cos(2fmt) and sin(2fmt)) [43], and the resultant signals are filtered. Figure 3． Flowchart of DLF algorithm Therefore, the modulus of the first harmonic R1f can be calculated using equation(4).
22 1 1 1 ( ) ( ) ( ) f f f R t X t Y t (4)
Through the peak detection of the first harmonic, the moment tfm corresponding to the frequency fm can be found, and the relationship between the time and frequency of the acoustic source and the received signal can then be established. The TOF (τ) is computed using the average of TOF at the characteristic frequencies over the pass-band B:
where treceived_fm and tsourse_fm are the moment of time of the received and source signals at the frequency of fm respectively.
Since it is constant for the signal source by the loudspeaker every time, and its time-frequency information is known. Therefore, during on-line monitoring, only digital lock-in processing is needed for each received signal xk(n) in Figure 3, and then its TOF (k) can be obtained by comparing with the known time-frequency relationship of x1(n). This makes the amount of calculation each time depend only on the length of the received signal. Moreover, by analysing and extracting the first harmonic R1f from the source and received signals through digital lock-in processing, it is possible to avoid both the complex mathematical operations of Fourier expansion and convolution of the cross-correlation algorithm discussed in section 2.2.2. Therefore, this method is very effective in reducing the computational burden.
Because the sinusoidal linear frequency-sweeping signal contains multiple frequency components, there exists intermodulation distortion in the output signal of the loudspeaker. Coupled with the harmonic distortion in the acoustic transmission, such intermodulation distortion can cause a nonlinear relationship between the instantaneous frequency and the time in the received signal. In this research, this problem is overcome by using the DLF technology, with which the time-frequency relationship can be fitted linearly. The least-squares regression method is the most commonly used linear fitting solution [44,45]. This method is featured with maximizing the fitting accuracy by minimizing the sum of squared errors so that an optimized relationship can be identified.
Assume that the linear equation of the time-frequency relationship is:
ii ft (6)
where α and β are the intercept and slope respectively, and are the residuals between the actual and the fitting frequency values. The residual sum of squared error f (α, β) can be determined as follows: 22 11 ( , ) ( )
nn ii ii f f t (7)
where fi is the actual value. +ti is the targeting value. By making the partial derivatives of the residual sum f(,) with respect to α and β to zeros, the optimized α and β can be identified to achieve the minimum f(,).
Generalized Cross-correlation Algorithm
The generalized cross-correlation algorithm is the most widely used TOF estimation method in acoustic measurement technology, which is usually further divided into DCC [8,46] and GCC [9,47]. In the GCC algorithm, signal is usually weighted in a way to reduce the sensitivity 5 to noise [9]. The TOF (τ) in equation (1) can be obtained using the GCC algorithm. A brief flowchart in Figure 4 describes its principle.
Figure 4. Flowchart of GCC algorithm
The cross-correlation between the source signal xl(n) and the received signal x2(n) is related to the power spectral density (PSD) function by the well-known Fourier transform relationship [9]:
12 12 ( )= ( ) ( ) xx G f F x n F x n (8) 1 2 1 2 - ( ) ( ) ( )exp( 2 ) x x x x R f G f j f df (9)
where F stands for Fourier transform. Rx1x2() is the GCC coefficient. Gx1x2( f ) is the cross PSD function. ( f ) denotes the general frequency weighting function. It can be seen from (9) that TOF(τ) can be obtained through peak detection of the GCC coefficient function Rx1x2().
The principle of the GCC algorithm in Figure 4 shows that, during on-line monitoring, each TOF (k) calculation requires the participation of the known source signal x1(n). The Fourier transform and Fourier inverse transform are performed by x1(n) with the k th received signal xk(n). Moreover, the data size of the cross-correlation coefficient is twice as large as that of the source signal, which will undoubtedly increase the system burden.
In comparison with GCC, the amount of data involved in the computation of the DLF algorithm is halved, and only digital lock-in processing is required for each received signal. Moreover, in the DLF, complex mathematical calculations such as Fourier transform and inverse transform are not required, which leads to a significant reduction in the computational effort for time delay estimation.
Experimental setup and result analysis
In this research, the feasibility of the DLF algorithm was investigated experimentally for acoustic waves at different distances and under different SNR conditions. The distance and noise intensity were controlled with reference to the actual operating conditions of utility boilers [29,48]. The maximum length is set as 10 m and the maximum TOF is controlled within 30 ms. The GCC algorithm was operated synchronously in parallel with DLF for comparison purpose. In order to avoid interference from other environmental sources in the experiments, the waveguides were used to ensure that the acoustic wave propagates inside the pipe only. Since the length of the waveguide and the temperature were known, the theoretical acoustic TOF can be obtained as a reference as shown in Figure 5, the experimental system mainly consists of an acoustic waveguide, two loudspeakers, a microphone receiver. A data acquisition card, a power amplifier and an industrial computer were also included, but not shown in this figure. The loudspeakers S0 and S1 were used to emit acoustic signals and the Gaussian white noise with adjustable energy levels respectively, to create different SNR conditions.
Figure 5. Experimental schematic diagram
The dashed box in Figure 5 indicates that the length of the acoustic waveguide was adjustable to create different theoretical delay time. The diameter of the acoustic waveguide used in the experiments was 45 mm. The microphone was 1/2-inch in diameter and non-directional with their sensitivity of 67.1 mv/pa, dynamic range of 20 dB to 136 dB, and frequency response range of 20 Hz to 20 kHz. The data acquisition card used was NI-USB6356, which contains eight simultaneous analog input channels, each channel having a sampling rate of 1.25 MS/s with a resolution of 16-bit.
Time Delay Estimation at Different Distances
The first part of experiments was conducted to verify the accuracy of the DLF algorithm under different distance conditions by adjusting the length of the acoustic waveguide. The temperature of the measurement environment was kept constant during the experiment. In these experiments, S1 did not emit noise, and the SNR of the experimental environment was constant and above 10 dB. The acoustic waveguide was set at 8 different lengths of 0. Figure 6 shows the time-domain waveforms and PSD functions of the received acoustic signals at waveguide length of 3 m. The red and black curves in Figure 6(a) are the source waveforms and filtered received signal with the normalized amplitudes. The enlarged portion of the dashed box in Figure 6(a) shows that both the sound source and the received signal are sinusoidal. Since the signal from the microphone was received synchronously when S0 started emitting the waveform, the received signal has a certain time delay to the acoustic source signal due to waveform propagation in the medium for the given distance. This delay is TOF, which depends on the length of the acoustic waveguide. From the PSD functions of the two signals in Figure 6(b), it is clear that both the source and the received signals are linear frequency-sweeping signals. Compared to the frequency components outside the signal band, the power of both the source and the received signal is strong. The signal is received after propagation, and its PSD is slightly attenuated. According to the time-frequency correspondence, it can be seen that the range of power reduction in the frequency domain is roughly the same as the attenuation range of the time-domain waveform plot. However, since the amount of PSD attenuation in the effective frequency band was much higher than that of the frequency components outside the signal band, i.e. the frequency component of interest was relatively little attenuated, the TOF estimation was not affected. The time-frequency relationship with different TOFs was obtained by using the DLF algorithm to process the received signals at previously mentioned eight different waveguide lengths. Figure 7 shows the time-frequency relationship of the received signals for the waveguide of 2 m and 10 m. From this figure, it can be seen that when the received signals were directly processed using the DLF algorithm, nonlinear relationship occurred around the starting and ending points of the time-frequency curves. This is due to the intermodulation distortion generated during the loudspeaker emission and the harmonic distortion produced during the acoustic transmission. The time-frequency curves were fitted with the least-squares linear regression method proposed in 2.2.1, and the fitting results are shown in Figure 7. Based on the fitting curves, the time-frequency relationship can be regarded as linear over the frequency range from 4 kHz to 8 kHz. The time-frequency relationship between the acoustic source and the received signals was tested and obtained at the given eight conditions. From Figure 8, it is evident that the time-frequency lines are parallel for these different waveguide lengths. With the waveguide length increases, the acoustic signal remains constant in frequency and only shifts with time. Larger the waveguide length is, greater the amount of TOF becomes. (11) where τ(i) ,τ and τ (i) denote measured values, average values and reference values; n is the number of measurements. Figure 9(a) shows the acoustic TOF for each of the eight conditions calculated by the two algorithms. Since the TOFs obtained using the two algorithms were very close, the ranges of the left and right axes in Figure 9(a) are adjusted slightly inconsistently to allow the respective calculation results and error bars to be clearly expressed. Moreover, in order to make the comparison of error bars obvious in the figure, the standard deviations of the both methods are enlarged by 20 times. Figure 9(a) shows that the TOFs calculated using the two algorithms were almost linearly related to the measurement distance, i.e. waveguide length, and the SD values increases with distance. The maximum errors were 0.030 ms and 0.039 ms for DLF and GCC respectively. The TOF standard deviations for DLF algorithm at all 8 distances were lower than that of GCC algorithm, which demonstrated that the precision of TOF measurement using DLF algorithm is superior over that of GCC algorithm, showing good robustness of DLF method. Figure 9. Time delay estimation results of GCC and DLF algorithms Figure 9(b) shows the TOF systematic error of the DLF and GCC algorithms. Similar to the variation, the values of the two algorithms also increase with distance. The maximum errors of 0.034 ms and 0.055 ms were recorded for DLF and GCC respectively. The DLF algorithm has achieved smaller bias value at each given distance, which shows DLF has better trueness over GCC.
Time Delay Estimation under Different SNR conditions
Since the experiments with variable distances in Section 3.1 were conducted without noise interference, it was impossible to verify their accuracy and applicability in noisy environments. For the purposes of applications, several key practical factors should be considered, such as environmental interference, main frequency bands, and the power level of background noise [32]. For the flue gas in power plants, the main component of noise is Gaussian White Noise [49], and according to the research [32], it has been known that the main frequency band of the internal noise in the power plant furnace is below 3 kHz.
In the experiments described in this section, the loudspeaker S0 was still used to emit acoustic source signal and Loudspeaker S1 was to emit Gaussian white noise signals with adjustable intensity to set the SNR of the measurement environment. The noise energy was controlled by varying the intensity of the noise source and the gain of the power amplifier. In the experiments, the following SNR: 0.086 dB, -3.020 dB, -6.322 dB, -9.380 dB, -11.698 dB, -14.937 dB, and -16.167 dB were used. The number of measurements n under each SNR condition is 2000. Figure 10. Time delay estimation results under different SNR conditions Figure 10 shows the statistical results of TOF estimation for the two algorithms in different energy level of background noise with a fixed waveguide length of 2.03 m. The temperature of the measurement environment was kept constant during the experiments. With the decrease of SNR, the systematic error of the both algorithms has an increasing trend as a whole, with slight fluctuation. It can be seen that although the difference of between the two algorithms under each SNR condition were very small, the DLF algorithm has better results at every given SNR conditions. This indicates that although both have certain anti-noise capability, the DLF algorithm is more robust to interference. The SD of the DLF is smaller than that of the GCC for all seven SNR conditions with the maximum SD deviation occurred at the SNR of -6.322 dB. Neither the nor the SD of both algorithms tended to increase significantly with the enhanced noise. Compared to the classical GCC
Conclusion
In this paper, the TOF estimation algorithm using the digital lock-in filter (DLF) is proposed. The TOF estimation by using this method is obtained via harmonic peak detection after the signals are processed by digital lock-in and low-pass filtering. Theoretical and experimental results demonstrate that the accuracy (both bias and precision) of the proposed algorithm is better than that of the GCC, and the computational effort is halved compared to the GCC algorithm, which greatly reduce the computational burden of the system. The computational error analysis also indicates that the TOF measurement with DLF is robust even in noisy environments. The average of the DLF algorithm is 0.012 ms lower than that of the GCC algorithm, which means that the accuracy is improved by 4.8% with the DLF algorithm. The maximum SD values of the two algorithms are 0.125 ms and 0.119 ms, respectively, in an environment with SNR ranging from 0.086 dB to -16.167 dB, showing good accuracy and robustness over GCC. With further development, this novel method can provide an alternative, more accurate and robust temperature distribution measurement for acoustic pyrometry used in power plant furnaces.
Figure 1 .
1Schematic of acoustic pyrometryFrom equation
10.0 m respectively. The laboratory environment temperature was controlled at 26.3 °C.
Figure 6 .
6Waveform and PSD of source and received signals with 3 m long waveguide
Figure 7 .
7Time-frequency relationship for 2 m and 10 m long acoustic waveguides
Figure 8 .
8Time-frequency relationship between the acoustic source and eight received signalsIn the experiments, the two algorithms analysed in sections 2.2.1 and 2.2.2 were used to solve the acoustic TOF for each of the eight conditions. The number of measurements n at each distance is 2000. The standard deviation (SD) and the systematic error are given by:
algorithm, the good measurement precision and robustness of DLF had been achieved.
AcknowledgementsORCID iDsQi Liu: https://orcid.org/ 0000-0002-5516-8543 Bin Zhou: https://orcid.org/ 0000-0003-1200-2663 Jianyong Zhang: https://orcid.org/ 0000-0002-9538-9332 Ruixue Cheng: https://orcid.org/ 0000-0002-0342-295x
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|
[] |
[
"Simulating spin-boson models with trapped ions",
"Simulating spin-boson models with trapped ions"
] |
[
"A Lemmer \nInstitut für Theoretische Physik and IQ ST\nUniversität Ulm\nAlbert-Einstein Alle 1189069UlmGermany\n",
"C Cormick \nIFEG, CONICET\nUniversidad Nacional de Córdoba\nX5000HUACórdobaArgentina\n",
"D Tamascelli \nInstitut für Theoretische Physik and IQ ST\nUniversität Ulm\nAlbert-Einstein Alle 1189069UlmGermany\n\nDipartimento di Fisica\nUniversità degli Studi di Milano\nVia Celoria 1620133MilanoItaly\n",
"T Schaetz \nSympathetic\nPhysikalisches Institut\nAlbert-Ludwigs-Universität Freiburg\nHermann-Herder-Str.379104Freiburg, TopologicalGermany\n",
"S F Huelga \nInstitut für Theoretische Physik and IQ ST\nUniversität Ulm\nAlbert-Einstein Alle 1189069UlmGermany\n",
"M B Plenio \nInstitut für Theoretische Physik and IQ ST\nUniversität Ulm\nAlbert-Einstein Alle 1189069UlmGermany\n"
] |
[
"Institut für Theoretische Physik and IQ ST\nUniversität Ulm\nAlbert-Einstein Alle 1189069UlmGermany",
"IFEG, CONICET\nUniversidad Nacional de Córdoba\nX5000HUACórdobaArgentina",
"Institut für Theoretische Physik and IQ ST\nUniversität Ulm\nAlbert-Einstein Alle 1189069UlmGermany",
"Dipartimento di Fisica\nUniversità degli Studi di Milano\nVia Celoria 1620133MilanoItaly",
"Sympathetic\nPhysikalisches Institut\nAlbert-Ludwigs-Universität Freiburg\nHermann-Herder-Str.379104Freiburg, TopologicalGermany",
"Institut für Theoretische Physik and IQ ST\nUniversität Ulm\nAlbert-Einstein Alle 1189069UlmGermany",
"Institut für Theoretische Physik and IQ ST\nUniversität Ulm\nAlbert-Einstein Alle 1189069UlmGermany"
] |
[] |
We propose a method to simulate the dynamics of spin-boson models with small crystals of trapped ions where the electronic degree of freedom of one ion is used to encode the spin while the collective vibrational degrees of freedom are employed to form an effective harmonic environment. The key idea of our approach is that a single damped mode can be used to provide a harmonic environment with Lorentzian spectral density. More complex spectral functions can be tailored by combining several individually damped modes. We propose to work with mixed-species crystals such that one species serves to encode the spin while the other species is used to cool the vibrational degrees of freedom to engineer the environment. The strength of the dissipation on the spin can be controlled by tuning the coupling between spin and vibrational degrees of freedom. In this way the dynamics of spin-boson models with macroscopic and non-Markovian environments can be simulated using only a few ions. We illustrate the approach by simulating an experiment with realistic parameters and show by computing quantitative measures that the dynamics is genuinely non-Markovian.
| null |
[
"https://arxiv.org/pdf/1704.00629v1.pdf"
] | 119,184,165 |
1704.00629
|
62f5a899dd66666cce203a54ca4b691fce20d575
|
Simulating spin-boson models with trapped ions
A Lemmer
Institut für Theoretische Physik and IQ ST
Universität Ulm
Albert-Einstein Alle 1189069UlmGermany
C Cormick
IFEG, CONICET
Universidad Nacional de Córdoba
X5000HUACórdobaArgentina
D Tamascelli
Institut für Theoretische Physik and IQ ST
Universität Ulm
Albert-Einstein Alle 1189069UlmGermany
Dipartimento di Fisica
Università degli Studi di Milano
Via Celoria 1620133MilanoItaly
T Schaetz
Sympathetic
Physikalisches Institut
Albert-Ludwigs-Universität Freiburg
Hermann-Herder-Str.379104Freiburg, TopologicalGermany
S F Huelga
Institut für Theoretische Physik and IQ ST
Universität Ulm
Albert-Einstein Alle 1189069UlmGermany
M B Plenio
Institut für Theoretische Physik and IQ ST
Universität Ulm
Albert-Einstein Alle 1189069UlmGermany
Simulating spin-boson models with trapped ions
(Dated: April 4, 2017)
We propose a method to simulate the dynamics of spin-boson models with small crystals of trapped ions where the electronic degree of freedom of one ion is used to encode the spin while the collective vibrational degrees of freedom are employed to form an effective harmonic environment. The key idea of our approach is that a single damped mode can be used to provide a harmonic environment with Lorentzian spectral density. More complex spectral functions can be tailored by combining several individually damped modes. We propose to work with mixed-species crystals such that one species serves to encode the spin while the other species is used to cool the vibrational degrees of freedom to engineer the environment. The strength of the dissipation on the spin can be controlled by tuning the coupling between spin and vibrational degrees of freedom. In this way the dynamics of spin-boson models with macroscopic and non-Markovian environments can be simulated using only a few ions. We illustrate the approach by simulating an experiment with realistic parameters and show by computing quantitative measures that the dynamics is genuinely non-Markovian.
The spin-boson model is an archetypical model of an open quantum system with applications ranging from chemical reactions [1] over biological aggregates [2] to solid state physics [3][4][5]. The model describes a single spin coupled to a dissipative environment comprised by an infinite set of harmonic oscillators. It is well known that the effect of thermal oscillator environments on a quantum system is fully described by a single scalar function, the spectral density (or spectral function) of the environment [4]. Although approximate analytic solutions have been found for some spectral densities [3,4] no closed analytic solution of the spin-boson model is known. Meanwhile, dynamics and thermodynamical properties of spin-boson models have been investigated by a number of numerical approaches including techniques based on the numerical renormalization group [5], time-dependent density matrix renormalization group [6,7], path integral Monte Carlo [8], or the quasi-adiabatic propagator path integral approach [9]. Numerical simulations are especially needed for environments with spectral densities where the reorganization energy is of the order of the spectral width or highly structured environments with long-lived vibrational modes that lead to highly non-trivial dynamics. These types of spectral densities are of particular relevance for the excitonic and electronic dynamics in biomolecular systems [10] and pose considerable challenges for numerical methods especially when the results of non-linear spectroscopy need to be predicted [11]. Therefore, an experimental simulator with a high degree of control is desirable.
Trapped atomic ions provide a clean and highly controllable system where many dynamical quantities are directly accessible. They have proven to be a versatile platform for the simulation of a wide range of physical models, such as defect formation in classical phase transitions [12][13][14] as well as open * Electronic address: [email protected] † Electronic address: [email protected] and closed quantum systems [15][16][17][18][19][20]. The simulation of spinboson models using trapped atomic ions has been proposed previously [21] requiring rather large crystals comprising 50-100 ions. Such crystals feature a large number of vibrational modes which can be used to act as a mesoscopic environment for the spin. However, for these large crystals the level of control needed to simulate spin-boson models is not available in the foreseeable future. In this work, we develop a proposal to simulate the dynamics of spin-boson models using small crystals of trapped ions. Our procedure also relies on the vibrational degrees of freedom to model the environment, but it makes use of the fact that a damped mode produces an effective Lorentzian spectral density [1]. While in [1] the damping is provided by an oscillator reservoir with Ohmic spectral density we show that the same spectral density can be obtained in certain regions of parameter space if the damping is modeled by a Lindblad equation extending the results of [22][23][24][25]. The resulting spectral densities are continuous functions of frequency and can thus be identified with an environment made up of a macroscopic number of modes as it occurs in the condensed phase. Controlling the couplings of the spin to the modes, the mode frequencies and the damping rates, the shape of the spectral density can be tailored, allowing one to mimic environments with continuous and highly-structured spectral densities using only a small number of oscillators to form the environment. This reduced overhead brings the simulation of spin-boson models to the realm of state-of-the-art trapped-ion setups.
Spin-boson model.-The spin-boson model describes a twolevel system (spin 1/2) in a dissipative environment which is modeled by an infinite set of non-interacting harmonic oscillators. Denoting by ε the energy splitting between the spin states | ↑ and | ↓ and byh∆ the coupling between them, the Hamiltonian of the global system reads [3]
H sb = ε 2 σ z −h ∆ 2 σ x − 1 2
σ z ∑ nh λ n (a n + a † n ) + ∑ nh ω n a † n a n (1) where σ z = | ↑ ↑ | − | ↓ ↓ | and σ x = | ↑ ↓ | + | ↓ ↑ |. arXiv:1704.00629v1 [quant-ph] 3 Apr 2017 a † n (a n ) denotes the raising (lowering) operator of environmental mode n and ω n the corresponding frequency while the real λ n describe the couplings of the spin to the environmental oscillators. The spectral density which determines the influence of the oscillator environment on the spin [4] reads J(ω) = π ∑ n λ 2 n δ (ω − ω n ) with δ the Dirac δ -function. For a macroscopic environment one assumes that the frequencies are so closely spaced that J(ω) becomes a continuous function of ω.
One is generally interested in finding the reduced dynamics of the spin for an environment with a certain spectral density. The path-integral formalism [26] provides us with an exact expression for the propagator of the spin state where the effects of the environment are already included. For factorizing initial conditions ρ 0 = ρ s ⊗ ρ β with some spin state ρ s and the environmental modes in a thermal state ρ β at inverse temperature β = (k B T ) −1 the propagator for the spin reads [27]
G(t, 0) = q f q 0 Dq q f q 0 Dq e ī h (S 0 [q]−S 0 [q ]) F[q, q ].(2)
Here the path integral q f q 0 Dq runs over all spin state trajectories connecting q(0) = q 0 and q(t) = q f , S 0 [q] is the action of the free spin evolution and F[q, q ] is the Feynman-Vernon influence functional [27]. The influence functional contains the effect of the environment on the spin dynamics. For an oscillator environment and the considered coupling it can be written as [3]
F[q, q ] = exp − t 0 dt t 0 ds[q(t ) − q (t )] [L(t − s)q(s) − L * (t − s)q (s)](3)
where L(t) = 1 h 2 X(t)X(0) β is the reservoir correlation function with X = ∑ nh λ n (a n + a † n ). Alternatively, L(t) can be expressed in terms of the spectral density J(ω):
L(t) = 1 π ∞ 0 dω J(ω) coth βhω 2 cos(ωt) − i sin(ωt) .
(4) Spectral density of damped harmonic oscillators.-The key idea of our approach is the fact that a damped oscillator provides a continuous effective spectral density, and the observation that different environments that produce the same influence functional have the same effect on the spin dynamics [27].
Let us first consider an environment consisting of a single harmonic oscillator which is damped by an oscillator reservoir with Ohmic spectral function. If we denote the free oscillation frequency of the damped oscillator by Ω and the bath causes damping at rate κ on the oscillator, the effective spectral density generated by the damped oscillator on the spin is Lorentzian [1,28]
J eff (ω) = λ 2 κ κ 2 + (ω − ω m ) 2 − κ κ 2 + (ω + ω m ) 2 . (5)
Here ω m = √ Ω 2 − κ 2 is the reduced frequency of the damped oscillator andhλ the spin-oscillator coupling as in Eq. (1).
Note that we restrict our considerations to the underdamped regime κ < Ω.
The combined influence functional of several independent damped harmonic oscillators is given by the product of the individual influence functionals [27]. Therefore, if the reservoirs have the same temperature, according to Eqs. (3) and (4) their spectral densities add up and one can construct effective spectral densities J(ω) = ∑ n J eff,n (ω). Here J eff,n (ω) is the spectral density due to oscillator n given by Eq. (5) with the corresponding λ n , κ n , ω n . If one wants to approximate a certain target spectral density J t (ω) the values for λ n , κ n , ω n are found by minimizing the functional E[{λ n , κ n , ω n }] = ∞ 0 dω|J t (ω) − J(ω)| 2 as has been shown in [40].
In trapped-ion experiments, the motion of the ions is usually expressed in terms of a set of normal modes, each of which is a harmonic oscillator. Cooling of the modes is commonly described by a Lindblad equation [41,42]. Therefore, it is not immediately clear if we can obtain an effective spectral density as for the oscillator damped by an Ohmic bath, Eq. (5). We will now show that this is possible and we obtain the same spectral function for appropriate parameters.
Let us start by considering the reservoir correlation function L(t) in Eq. (4). We note that L(t) = L (t) + iL (t) is a complex-valued function with real and imaginary parts L (t) and L (t). For the oscillator damped by an Ohmic bath the coordinate correlation function and thus L(t) can be calculated analytically [3,28,43] and we obtain L (t) = L 1 (t) + L 2 (t)
L 1 (t) = λ 2 sinh(βhω m ) cosh(βhω m ) − cos(hβ κ) cos(ω m t) + sin(hβ κ) cosh(βhω m ) − cos(hβ κ) sin(ω m |t|) e −κ|t| ,L 2 (t) = −λ 2 8κω m hβ ∞ ∑ n=1 ν n e −ν n |t| (Ω 2 + ν 2 n ) 2 − 4κ 2 ν 2 n(6)
with the Matsubara frequencies ν n = 2πn/(hβ ) and
L (t) = −λ 2 sin(ω m t)e −κ|t| .(7)
In Lindblad description, a damped harmonic oscillator coupled to a thermal reservoir at inverse temperature β evolves according toρ
= − ī h [H, ρ] + D κ,n ρ(8)
where here H =hω m a † a is the Hamiltonian of the oscillator and its frequency ω m already includes possible renormalizations due to the damping. The dissipator reads [44] D κ,n ρ = κ(n + 1)[aρa † − a † aρ] + κn[a † ρa − aa † ρ] + H.c. (9) Using the quantum regression theorem we can obtain the reservoir correlation function L L (t) = L L (t) + iL L (t) for the damped harmonic oscillator in Lindblad description. We find that the real part
L L (t) = λ 2 coth βhω m 2 cos(ω m t)e −κ|t|(10)
has a different functional form than L (t) in Eq. (6) while the imaginary part L L (t) coincides with L (t) in Eq. (7) which is determined by J eff (ω) of Eq. (5). Writing L L (t) as in Eq. (4) we obtain L L (t) = 1 π ∞ 0 dωJ eff (ω) coth(βhω/2) cos(ωt) wherẽ
J eff (ω) = λ 2 coth βhω m 2 coth βhω 2 κ κ 2 + (ω − ω m ) 2 + κ κ 2 + (ω + ω m ) 2 .
(11) Despite the differences it is possible to obtain a very good agreement between the real parts L (t) and L L (t) and their frequency space representations Eqs. (5) and (11). Ref. [45] estimates that the quantum regression theorem can only yield quantitatively correct predictions for the two-time correlation functions of the damped harmonic oscillator if κ ω m and hβ κ 1. Indeed, under these assumptions we find very good agreement between L (t) and L L (t). If we have good agreement between L (t) and L L (t), we also find good agreement in frequency space. Note that while κ ω m is a necessary condition to derive the Lindblad equation (8) with the dissipator in Eq. (9),hβ κ 1 puts a lower bound on the temperature where the identification of L(t) and L L (t) is possible. On the other hand, also too high temperatures lead to deviations such that there is an intermediate temperature range where the best agreement is achieved (see [28] for a more detailed discussion).
In order to confirm the above considerations we simulated the dynamics of σ z (t) for the full spin-boson Hamiltonian in Eq. (1) with spectral density J eff (ω) from Eq. (5) using the numerically exact TEDOPA algorithm [6] and compared them with those given by Eq. (8) with H = H sb from Eq. (1) for a single mode. We considered an initial product state | ↑ ↑ | ⊗ ρ β and ε = 0, ω m /2π = 100 kHz, κ/2π = 1.25 kHz as well as a spin-mode coupling λ /2π = 100 kHz. We chosē hβ = 5.91·10 −6 s which corresponds ton(ω m ) = 0.025 for the Lindblad-damped oscillator and computed the evolution for spin energies ∆/2π = 50 kHz and 100 kHz. For both values of ∆ we obtain very good agreement (see [28]) which shows that the analogy to the macroscopic environment also holds when we probe the spectral density away from the resonance. Note that one simulation for ∆/2π = 50 kHz takes 15 days using 16 cores on a computing cluster which once more indicates the value of a trapped-ion simulator especially for structured environments and complex observables.
Trapped ion implementation. -Let us now proceed to illustrate how the ideas discussed above can be implemented in an ion-trap experiment. We consider N singly charged atomic ions with masses m j confined in a linear Paul trap with effective harmonic trapping potential. We assume trapping conditions such that laser cooled ions form a linear Coulomb crystal along z with equilibrium positions r 0 j = (0, 0, z 0 j ) T . The motional degrees of freedom can then be described in terms of N uncoupled normal modes in each spatial direction [46,47] and the motional Hamiltonian reads H m = ∑ n,αh ω n,α a † n,α a n,α where ω n,α is the frequency of mode n in spatial direction α ∈ {x, y, z} with ladder operators a † n,α , a n,α . For simplicity, we will focus on the case of a spin coupled to a single damped mode which corresponds to a spin-boson model with Lorentzian spectral density as in Eq. (5). This system already exhibits an interesting phenomenology and has been studied with a variety of numerical and analytical approaches, see e.g. [48][49][50][51]. For this purpose we only need N = 2 ions: one ion is used to encode the spin while the other ion provides sympathetic cooling of the shared modes of motion. In order to avoid that the cooling lasers couple to the spin transition we choose to work with mixed species ion crystals. Alternatively, one could rely on single site addressing. The internal levels of the spin ion are described by the Hamiltonian H s =h ω 0 2 σ z while the internal levels of the coolant ion are adiabatically eliminated from the dynamics leading to the effective description in Eq. (9) of the cooling [41,42].
For concreteness we consider a crystal composed of 24 Mg + and 25 Mg + . 25 Mg + has a nuclear spin and we can use the states |F = 3, m F = 3 ≡ | ↓ and |F = 2, m F = 2 ≡ | ↑ of the 2 S 1/2 electronic hyperfine ground-state manifold to encode the spin. The spin can be driven by a microwave or in a two-photon stimulated Raman configuration while the desired coupling of the spin to the motional degrees of freedom in the σ z basis is provided by a "walking standing wave". In this configuration the spin states are off-resonantly coupled to the P manifold by two laser beams near 280 nm whose beat note is tuned close to one of the motional mode frequencies [52]. The interaction of the ion with the applied fields is described by [28]
H int =h Ω d 2 σ + e −iω d t +h Ω odf 2 e i(k L r+φ L ) e −iω L t σ z + H.c. (12)
where Ω d is the Rabi frequency of the applied microwave or stimulated Raman field and ω d ≈ ω 0 its frequency. Ω odf , k L , ω L , φ L are the effective laser Rabi frequency, wave vector, frequency and phase, respectively. Directing k L along z the laser only couples to the motion along this axis. A two-ion crystal features two axial modes, an in-and an out-of-phase mode of motion with frequencies ω 1,z ≡ ω 1 and ω 2,z ≡ ω 2 . The two modes are well separated in frequency such that choosing the laser frequency ω L ≈ ω 2 the spin only couples to the ouf-of-phase mode. In an interaction picture rotating with the microwave and motional frequencies and under the rotating wave approximation the system's Hamiltonian reads [28] (13) where δ = ω 0 − ω d is the detuning of the field driving the spin transition and δ m = ω 2 − ω L ω 2 the detuning of the laser from the motional mode. The spin-motion coupling is given by λ = −iη 2 Ω odf e i(|k L |z 0 2 +φ L ) with the Lamb-Dicke factor η 2 = h/(2m 2 ω 2 )M 22 |k L |. Note that the laser phase can be chosen such that λ is real.M 22 is the out-of-phase mode amplitude at the spin ion in mass weighted coordinates and m 2 its mass. Identifyinghδ = ε, Ω d = −∆ and δ m = ω m we obtain the spin-boson Hamiltonian of Eq. (1) for a single mode. Adding the cooling on the second ion the full system evolves according to Eq. (8) where H = H sb1 from Eq. (13).
H sb1 =h δ 2 σ z +h Ω d 2 σ x −h λ 2 (a 2 + a † 2 )σ z +hδ m a † 2 a 2
We simulate the dynamics of the system for experimentally realistic parameters. We consider an axial potential 0 20 where a single 24 Mg + ion has a center-of-mass frequency ω com /2π = 2.54 MHz which leads to an out-of-phase mode frequency ω 2 /2π = 4.36 MHz and η 2 ≈ 0.15 for the mixed crystal where we assumed that the lasers inducing the spindependent force are at right angles. Furthermore, we assume that EIT cooling [42] is applied to the 24 Mg + ion which has already been used to sympathetically cool mixed-species ion crystals [53]. We assume a cooling rate 2κ/2π = 2.5 kHz and a steady-state populationn = 0.025 of the mode which is realistic in light of the results in [53]. Note that one has to make sure that the correspondence to the macroscopic environment holds for the effective mode frequency ω m = δ m which is the detuning of the spin-motion coupling and thus much smaller than the physical mode frequency. We chose the field driving the spin to be resonant, i.e. ε = 0, and a detuning ω m /2π = 100 kHz of the spin-motion coupling such that we recover the parameters we used previously and the correspondence holds. In the simulations we truncate the motional Hilbert space at n max = 15 excitations which makes truncation errors negligible.
∆ · t -1 1 σ z (t) a) λ/2π (kHz) 10 100 200 0 20 ∆ · t -1 1 σ z (t) b) 0 50 150 200 λ/2π(kHz) 0 1 N RHP c) ohmic resonance 0 50 150 200 λ/2π(kHz) 0 4 N BLP d)
In Fig. 1 we show the dynamics of σ z (t) under Eq. (8) (where H = H sb1 with the parameters from the previous paragraph) for an initial state ρ 0 = | ↑ ↑ | ⊗ ρ β where the thermal state ρ β has a mean occupation numbern = 0.025. We vary the spin-motion coupling λ /2π = 10 − 200 kHz. In panel a) we show the dynamics for ∆/2π = 3 kHz. In this case the spin samples the low frequencies of the spectral density in Eq. (5). Expanding the spectral density for small ω we obtain Ohmic behavior J eff (ω) ∼ ω. We observe a transition from damped to overdamped oscillations with increasing spin-mode coupling λ . This behavior is expected for an Ohmic spectral density at finite temperatures [3,4]. Note, however, that our spectral density J eff (ω), even if Ohmic for small frequencies, does not yield the same correlation function as a strict Ohmic environment. Therefore we can only expect qualitatively similar dynamics [48,49]. In panel b) we show σ z (t) for the same initial conditions and ∆/2π = 100 kHz such that the spin is resonant with the mode. In this regime the spin dynamics shows a very complex behavior which one would intuitively call non-Markovian.
Quantification of the degree of non-Markovianity of the dynamics.-In order to assess the non-Markovian character of the dynamics we compute two measures of non-Markovianity. The first measure, N RHP , arises from defining Markovianity in terms of the divisibility of the dynamical map of the dynamics [54], while the second, N BLP , is based on the definition that a dynamics is Markovian when yielding a monotonic decrease of state distinguishability [55]. We evaluated N RHP numerically [28] for the parameters of parts a) and b) of In order to tailor more complex spectral densities than in this proof-of-principle experiment, one would need to couple the spin to two or more damped modes with the appropriate couplings and cooling rates that match the effective spectral density to the desired one. In case several modes are used it could be advantageous to use the transverse modes of motion. Due to the smaller bandwidth of the transverse phonon frequencies it is easier to couple to and cool several modes at the same time. It should be borne in mind that the cooling rates should be considerably smaller than the spacing between modes. Only then the damping of each mode can be described by a dissipator as in Eq. (9). In order to fill possibly unwanted gaps in the effective spectral density one could then use the modes of the second transverse direction of motion and place the effective frequencies of these modes between those of the first direction.
Let us finally note that the model can be extended not only to more complex spectral densities by including more ions and thus modes but also to include more spins. Then trapped ions could be used as a testbed for the dynamics of exciton transport in complex spectral densities and especially the determination of higher order spectral responses, e.g. 2D electronic spectroscopy, which are exceedingly hard to compute numerically even for only a few electronic sites and a structured spectral density [11].
In summary, we have shown that spin-boson models with continuous spectral densities can be simulated using damped oscillators in Lindblad description. This leads to a significant reduction of the technical requirements for the implementa-tion of this paradigmatic model for decoherence and dissipation employing trapped ions. The joint effect of different damped modes allows one to tailor a variety of spectral densities with rich non-Markovian features. We showed that it is possible to carry out simulations of non-trivial dynamics making use of just one motional mode, and illustrated the practicality of our approach by simulating an experiment with realistic parameters.
Acknowledgements.-A. L. and D. T. acknowledge very useful discussions with A. Smirne. This work was supported by an Alexander-von-Humboldt Professorship, the ERC synergy grant BioQ and EU projects EQUAM and QUCHIP. Computational resources were provided by the bwUniCluster and the bwForCluster JUSTUS. We start by briefly surveying the quantities that we need for the discussion of the effective spectral densities of damped harmonic oscillators. We consider the spin-boson model where a spin is coupled to a bath of harmonic oscillators. The spin constitutes the principal system and the bath consists of an infinite set of independent harmonic oscillators. This is an archetypical model for a two-state system coupled to a dissipative environment and is conveniently modeled by the Hamiltonian [1]
H sb = ε 2 σ z −h ∆ 2 σ x + 1 2 ∑ n p 2 n m n + m n ω 2 n x 2 n − c n q 0 σ z x n (A1) where σ z = | ↑ ↑ | − | ↓ ↓ | and σ x = | ↑ ↓ | + | ↓ ↑ | de-
note the usual Pauli matrices, ε the energy splitting of the spin states andh∆ their coupling. p n and x n denote the canonical momenta and coordinates of the environmental modes of frequency ω n , q 0 is some characteristic length scale and c n describes the coupling of mode n to the spin. Quantizing the environmental oscillators x n = h/(2m n ω n )(a n + a † n ) so that a n and a † n denote the ladder operators of oscillator n we can write the spin-mode coupling as hλ n = c n q 0 h/(2m n ω n ).
(A2)
The spin-boson Hamiltonian can then be written as
H sb = ε 2 σ z −h ∆ 2 σ x − 1 2 σ z ∑ nh λ n (a †
n + a n ) + ∑ nh ω n a † n a n (A3) which is Eq. (1) of the main text. Note that we have omitted the ground-state energies of the oscillators. For an initial product state of spin and environment where the environment is in a thermal state at inverse temperature β the influence of the oscillator environment on the spin is given by the influence functional F[q, q ] in Eq. (3) of the main text which is in turn determined by the reservoir correlation function [1]
L(t) = 1 h 2 X(t)X(0) β (A4)
with the collective coordinate X(t) = q 0 ∑ n c n x n = ∑ nh λ n (a n + a † n ). The reservoir correlation function can be equivalently given in terms of the spectral density J(ω)
L(t) = 1 π ∞ 0 dω J(ω) coth βhω 2 cos(ωt) − i sin(ωt) .
(A5) It is known that an oscillator damped by a bath with Ohmic spectral density produces an effective environment with Lorentzian spectral density [2]. Here, we inspect in more detail when the same can be done for the damped harmonic oscillator in Lindblad description. To this end, it is instructive to start from the time domain and consider L(t) for the two cases.
Time domain considerations
The reservoir correlation function L(t) in Eq. (A4) may be written explicitly in terms of the environmental coordinate correlation functions using X(t) = q 0 ∑ n c n x n
L(t) = q 2 0 ∑ n c 2 n h 2 x n (t)x n (0) β (A6)
where we have used that the oscillators are independent. In the following we consider only a single oscillator and therefore omit the index n from now on. The function x(t)x(0) β is in general a complex function and we can write it in terms of its real and imaginary parts
x(t)x(0) β = S(t) + iA(t) (A7) where S(t) = 1 2 {x(t), x(0)} β ,(A8)A(t) = 1 2i [x(t), x(0)] β .(A9)
The imaginary part A(t) is related to the damped oscillator's response function
χ(t) through χ(t) = − 2 h Θ(t)A(t) [1]
where Θ(t) is the Heaviside step function. Note that accordingly also L(t) is a complex function
L(t) = L (t) + iL (t).(A10)
Let us now consider a damped oscillator that evolves according to the Lindblad equation given in Eq. (8) of the main texṫ
ρ = −i[ω m a † a, ρ] + D κ,n ρ (A11) with dissipator D κ,n ρ =κ(n + 1)[aρa † − a † aρ] + κn[a † ρa − aa † ρ] + H.c. (A12)
given in Eq. (9) of the main text. The above dissipator takes the mode populations to a thermal state with mean occupation numbern at a rate 2κ. We can compute the coordinate correlation function x(t)x(0) β ,L = S L (t) + iA L (t) of the damped harmonic oscillator in Lindblad description using the quantum regression theorem:
S L (t) =h 2mω m coth βhω m 2 cos(ω m t)e −κ|t|(A13)
and
A L (t) = −h 2mω m sin(ω m t)e −κ|t| .(A14)
Here m is the mass of the oscillator. Note that the frequency ω m is taken to include possible renormalizations of the mode frequency due to the damping and κ ω m is necessary to derive the Lindblad equation above. Inserting the result into Eq. (A6) and using Eq. (A2) we obtain the real and imaginary parts L L (t) and L L (t) of L L (t) from Eqs. (10) and (7) of the main text
L L (t) = λ 2 coth βhω m 2 cos(ω m t)e −κ|t|(A15)
and L L (t) = −λ 2 sin(ω m t)e −κ|t| .
As we stated earlier we have χ(t) = − 2 h Θ(t)A(t). Inserting A L (t) into the previous equation yields the response function of the classical damped harmonic oscillator. Having in mind that an Ohmic spectral density leads to the classical equation of motion for a damped oscillator [1], and thus the same response function, it seems appropriate to compare the regression theorem results to that of the oscillator damped by an Ohmic bath.
Therefore, we move on to the harmonic oscillator damped by a thermal oscillator bath with Ohmic spectral density. For this case, it is also possible to calculate the coordinate correlation function x(t)x(0) β analytically [1,3]. We denote the free oscillation frequency of the oscillator by Ω while we denote the damping rate on the oscillator's coordinate by κ ohm . In the underdamped regime κ ohm < Ω the oscillator's frequency is reduced to ω r = Ω 2 − κ 2 ohm due to the damping. Since we want to compare the results to the Lindblad case where κ ω m we will always have κ ohm Ω such that we are in the underdamped regime. In this regime the real part of the coordinate correlation S(t) splits in two parts [1]
S(t) = S 1 (t) + S 2 (t)(A17)
with S 1 (t) =h 2mω r sinh(βhω r ) cosh(βhω r ) − cos(βhκ ohm ) cos(ω r t) + sin(βhκ ohm ) cosh(βhω r ) − cos(βhκ ohm ) sin(ω r |t|) e −κ ohm |t| (A18) and
S 2 (t) = − 4κ ohm mβ ∞ ∑ n=1 ν n e −ν n |t| (Ω 2 + ν 2 n ) 2 − 4κ 2 ohm ν 2 n (A19)
where the ν n = 2πn/(hβ ) are the Matsubara frequencies. The imaginary part reads
A(t) = −h 2mω r sin(ω r t)e −κ ohm |t| .(A20)
Comparing Eqs. (A20) and (A14) we see that the imaginary parts A(t) and A L (t) are exactly equal for ω r = ω m and κ ohm = κ which we will assume from now on. With this substitution and inserting Eqs.
(A18)-(A20) into Eq. (A6) we ob- tain L(t) = L (t) + iL (t) = L 1 (t) + L 2 (t) + iL (t) where L 1 (t) = λ 2 sinh(βhω m ) cosh(βhω m ) − cos(βhκ) cos(ω m t) + sin(βhκ) cosh(βhω m ) − cos(βhκ) sin(ω m |t|) e −κ|t| , L 2 (t) = −λ 2 8κω m hβ ∞ ∑ n=1 ν n e −ν n t (Ω 2 + ν 2 n ) 2 − 4κ 2 ν 2 n (A21)
and
L (t) = −λ 2 sin(ω m t)e −κ|t|(A22)
recovering Eqs. (6) and Eq. (7) of the main text. The symmetric parts S(t) and S L (t) do not coincide after the substitution ω r = ω m and κ ohm = κ. Hence, in the following we seek the regimes where the two functions coincide. In order to identify S L (t) with S(t) we need to be able to neglect S 2 (t) as well as the sine component in S 1 (t). We start by considering S 2 (t). The argument follows Refs. [3,4]. The Matsubara frequencies ν n determine the time scale on which S 2 (t) decays, the smallest decay rate being ν 1 . Accordingly, if the decay rate κ is much smaller than the smallest Matsubara frequency, S 2 (t) drops to zero much faster than S 1 (t). This is the regime where
κhβ 2π = κ ν 1 1.(A23)
In this regime one expects that S 2 (t) will only produce deviations on very short time scales and is negligible if we are interested in not too short time scales. This is the case in our considerations. If S 2 (0) S 1 (0) we can neglect S 2 (t) completely.
Assuming we can disregard S 2 (t) we need to find the regime where
S L (t) ≈ S 1 (t).(A24)
In the limit βhκ 1 we can expand the sine and cosine terms in S 1 (t) in this small parameter and to first order we obtain
S 1 (t) ≈h 2mω m sinh(βhω m ) cosh(βhω m ) − 1 cos(ω m t) +h β κ cosh(βhω m ) − 1 sin(ω m |t|) e −κ|t| ≈h 2mω m sinh(βhω m ) cosh(βhω m ) − 1 cos(ω m t)e −κ|t| (A25)
where we have usedhβ κ sinh(hβ ω m ) in the last step. Using the identity coth x 2 = sinh(x)/(cosh x − 1) finally yields S 1 (t) = S L (t) if the reservoirs are at the same inverse temperature β . Accordingly, we assume that the reservoir in the Lindblad description and the Ohmic oscillator bath have the same inverse temperature β from now on.
Thus, we have established a regime where the coordinate correlation function of the Lindblad damped harmonic oscillator approximately coincides with that of an oscillator damped by a reservoir with Ohmic spectral density. In this regime the Lindblad damped oscillator should act as a macroscopic reservoir with Lorentzian spectral density as in Eq. (5) of the main text.
Note that for a given cooling rate κ the condition in Eq. (A23) puts a lower bound on the temperature where we can neglect S 2 (t) and thus a lower bound on the temperature where the Lindblad damped oscillator produces the same coordinate correlation function as the oscillator damped by a reservoir with Ohmic spectral density. Thus, we require κ ω m and κhβ 2π 1 to make the identification. Indeed, Refs. [3,4] estimate that the quantum regression theorem can only yield quantitatively correct predictions for the two-time correlation functions of the damped harmonic oscillator if the two above conditions are met.
For ion-trap experiments one usually considers the mean occupation numbern of the bosonic modes rather than their temperature and therefore it is desirable to cast condition (A23) in a form where it depends onn. Assuming a thermal state for a bosonic mode we can associate the temperature T eff =hω/[k B log(1 + 1/n)] to the mode and the condition in Eq. (A23) becomes
log(1 + 1 n ) 2π κ ω m 1.(A26)
Note that in the ion-trap implementation the mode frequency is an effective frequency much smaller than the physical frequency of the mode (see App. D). Therefore, one has to make sure the above condition is met for the effective frequency such that the correspondence to the effective harmonic environment is not lost. In order to make the above considerations more quantitative and illustrate that the match of the reservoir correlation functions is indeed very good we make a numerical comparison of the functions L(t) and L L (t) in the regime κ ω m , ν 1 . Since the imaginary parts of the two functions are equal we focus on the real parts L (t) and L L (t). In Fig. 2 we plot L (t)/λ 2 including the first 10 4 Matsubara frequencies together with L L (t) for ω m /2π = 100 kHz, κ/2π = 1.25 kHz and a mean occupation numbern(ω m ) = 0.025 which corresponds tohβ = 5.91 · 10 −6 s. These parameters are realistic in an ion trap experiment. In part a) of the figure we compare L (t) and L L (t) on short and in part b) on intermediate time scales. One can appreciate excellent agreement between the two functions. In order to illustrate for which parameters the approximation works well we compute the distance
d = 1 λ 2 ∞ 0 dt[L(t) − L L (t)](A27)
between the functions L(t) and L L (t) which can be evaluated analytically to yield
d =c q κ κ 2 + ω 2 m + c cl ω m κ 2 + ω 2 m − 8κω m hβ ∞ ∑ n=1 1 (ω 2 m + κ 2 + ν 2 n ) 2 − 4κ 2 ν 2 n ,(A28)
where we used the abbreviations c q = sinh(βhω m ) cosh(βhω m ) − cos(hβ κ) − coth hβ ω m 2 , c cl = sin(βhκ) cosh(βhω m ) − cos(hβ κ) .
We evaluate the difference for different cooling rates and mean occupation numbers while keeping the mode frequency fixed at ω m /2π = 100 kHz. The results are depicted in Fig. 3. Note that higher bars in the figure correspond to smaller values of d. We observe that increasing κ increases the difference between the two functions. For a fixed cooling rate we observe that the distance is minimal for intermediate values ofn. This can be understood by considering Eqs. (A15) and (A21). In order to identify L(t) and L L (t) we need to be able to neglect L 2 (t) and the sine component in L 1 (t). The condition in Eq. (A23) provides the regime where L 2 (t) is negligible and favors higher temperatures. However, in order to suppress the sine component in L 1 (t) lower temperatures are more favorable. Thus, we obtain the best match for intermediate temperatures.n
Frequency space considerations
In [2] Garg et al. show that a harmonic oscillator which in turn is damped by an oscillator environment with Ohmic spectral density with infinite cutoff produces the effective spectral density
q 2 0 h J eff,ohm (ω) = q 2 0 h c 2 1 m 2κω (Ω 2 − ω 2 ) 2 + 4ω 2 κ 2 . (A29)
Here, κ is the damping induced by the bath on the coordinate of the oscillator, Ω and m its free oscillation frequency and mass, respectively, and q 0 c 1 its coupling to the spin [see Eq. (A1)]. Note that upon writing the influence functional as in Eq. (3) of the main text we have absorbed the prefactor q 2 0 /h into the spectral density. Using Eq. (A2) and setting ω 2 m = Ω 2 − κ 2 one obtains the spectral density J eff (ω) in Eq. (5) of the main text
J eff (ω) = λ 2 κ κ 2 + (ω − ω m ) 2 − κ κ 2 + (ω + ω m ) 2 . (A30)
In the previous section we have seen that in a certain parameter regime the Lindblad description of the damped harmonic oscillator reproduces the coordinate correlation function and thus L(t) of the oscillator damped by an Ohmic bath. L(t) can also be written in terms of the spectral density J(ω) according to Eq. (A5). In fact, in almost all cases environments are characterized by their spectral density rather than their correlation functions. Therefore, we analyze the effective spectral density of the Lindblad-damped oscillator and compare it to the Lorentzian spectral density J eff (ω) in Eq. (A30) above.
The Fourier representation of L L (t) in Eqs. (A15) and (A16) reads
L L (t) = 1 π ∞ 0 dω J eff (ω) coth hβ ω 2 cos(ωt) − i J eff (ω) sin(ωt)] (A31)
with J eff (ω) as in Eq. (A30) above and
J eff (ω) = λ 2 coth βhω m 2 coth βhω 2 κ κ 2 + (ω − ω m ) 2 + κ κ 2 + (ω + ω m ) 2 .
(A32) We note that in generalJ eff (ω) = J eff (ω) and hence we cannot write L L (t) as a function of a single spectral density as in Eq. (A5), in general. Yet, from our considerations in the previous section we expect that for appropriate parameters
J eff (ω) ≈ J eff (ω) (A33)
such that we obtain the form of L(t) in Eq. (A5) as for a macroscopic environment. In Fig. 4 we compare the left and right hand sides of Eq. (A33) for the parameters we use in the previous section and the main text, i.e. ω m /2π = 100 kHz, κ/2π = 1.25 kHz andn(ω m ) = 0.025 (hβ = 5.91 · 10 −6 s) where we found very good agreement between the correlation functions L(t) and L L (t) (see Fig. 2). Panel a) shows J eff (ω) (solid line) and J eff (ω) (circles) for low frequencies and part b) shows the behavior around the resonance ω m /2π = 100 kHz. Both parts of the figure show that we obtain very good agreement in frequency space, too. Part c) of the figure shows the relative error
ε J = |J eff (ω) − J eff (ω)| J eff (ω) (A34)
which is remarkably small over the whole range ω/2π = 0 − 150 kHz. Note that the increase in the relative error for higher frequencies is because the spectral density goes to zero more rapidly than the effective one. However, since both contributions are small the effect of this difference should be negligible. Thus, we confirm the result of the previous section: for appropriate choices of mode frequency, cooling rate and temperature, the damped oscillator evolving according to the Lindblad equation can be attributed the effective spectral density J eff (ω) of a macroscopic oscillator environment. Note that the treatment is not perturbative in the spin-motion coupling λ , so that this equivalence is valid for arbitrary values of λ .
H sys = ε 2 σ z −h ∆ 2 σ x ,(B2)H env =h ω max 0 dω ωa † ω a ω ,(B3)H int = −σ zh 2 ω max 0 dω h(ω)(a ω + a † ω ),(B4)
where we have introduced a hard cutoff ω max . The spectral density J(ω) is given by
J(ω) = πh 2 (ω). (B5)
To simulate the evolution of the spin-boson model, we resorted to the Time Evolving density matrix with orthogonal polynomials (TEDOPA) algorithm. In this section we briefly present the TEDOPA scheme and refer to [5,6] for a more detailed presentation of the algorithm. TEDOPA is a certifiable and numerically exact method to treat open quantum system dynamics [6,7]. In a two-stage process TEDOPA first employs a unitary transformation reshaping the spin-boson model into a onedimensional configuration. New oscillators with creation and annihilation operators b † n and b n are defined using the unitary transformations U n (ω)
U n (ω) = h(ω)p n (ω), (B6) b † n = ω max 0 dω U n (ω)a † ω ,(B7)
where p n (ω), n = 0, 1, . . . are orthogonal polynomials with respect to the measure dµ(ω) = h 2 (ω)dω [5]. While in certain cases it is possible to perform this transformation analytically [5], in general a numerically stable procedure is used [8]. This transformation maps the environment to a semi-infinite one-dimensional chain of oscillators with nearest-neighbor interactions. In this configuration the spin only interacts with the first site of the chain. The Hamiltonian (B1) becomes
H =H sys −h t 0 2 σ z (b 0 + b † 0 ) + ∞ ∑ n=0h ω n b † n b n + ∞ ∑ n=0h t n (b † n b n+1 + b n b † n+1 ).(B8)
The nearest-neighbor geometry as well as coefficients ω n and t n are directly related to the recurrence coefficients of the three-term recurrence relation defining the orthogonal polynomials p n (ω) [5]. This transformation from the spin-boson model to a one-dimensional geometry is depicted in Fig. 5.
In the second step this emerging configuration is treated by the Time Evolving Block Decimation (TEBD) method. TEBD generates a high fidelity approximation of the time evolution of a one-dimensional system subject to a nearestneighbor Hamiltonian with polynomially scaling computational resources. TEBD does so by dynamically restricting the exponentially large Hilbert space to its most relevant subspace thus rendering the computation feasible [9,10]. TEBD is essentially a combination of an MPS description for a one-dimensional quantum system and an algorithm that applies two-site gates that are necessary to implement a Suzuki-Trotter time evolution. Together with MPS operations such as the application of measurements this yields a powerful simulation framework. An extension to mixed states is possible by introducing a matrix product operator (MPO) to describe the density matrix, in complete analogy to an MPS describing a state [10]. Such an extension is needed in our simulations in order to build the thermal state of the oscillator chain.
A last step is necessary to adjust this configuration further to suit numerical needs. The number of levels for the environment oscillators is restricted to a value d max to reduce the required computational resources. A suitable value for d max is related to the sites average occupation which, in turn, depends on the environment structure and temperature. In our simulations we set d max = 5: this value provides converged results for all examples provided. The Hilbert space dynamical reduction performed by TEBD is determined to the bond dimension. The optimal choice of this parameter depends on the amount of long range correlations in the system. For all the simulations used in this work, a bond dimension χ = 200 provided converged results. At last, we observe that the mapping described above produces a semi-infinite chain that must be truncated in order to enable simulations. In order to avoid unphysical back-action on the system due to finite-size effects, i.e. reflections from the end of the chain, the chain has to be sufficiently long to completely give the appearance of a "large" reservoir. These truncations can be rigorously certified by analytical bounds [7]. For the examples provided in the paper, chains of n = 15 sites are enough to avoid boundary effects. In order to further optimize our simulations, we augmented our TEDOPA code with a Reduced-Rank Randomized Singular Value Decomposition (RRSVD) routine [11]. Singular value decomposition (SVD) is at the heart of the dimensionality reduction TEBD relies on. RRSVD is a randomized version of the SVD that provides an improved-scaling SVD, with the same accuracy as the standard state-of-the-art deterministic SVD routines.
In order to benchmark the quality of the effective model presented in the main text we compared the dynamics of the full spin-boson model in Eq. (B1) with spectral density as in Eq. (5) of the main text with those of a spin coupled to a damped harmonic oscillator in Lindblad description. In the latter case the system evolves according to Eq. (8) of the main text with H = H sb1 from Eq. (13) of the main text. As in the main text we chose the parameters ε = 0, κ/2π = 1.25 kHz and ω m /2π = 100 kHz while we considered a spin-mode coupling strength λ /2π = 100 kHz and set the hard cutoff in Eq. (B3) to ω max /2π = 200 kHz. We simulated the dynamics of σ z (t) for initial product states | ↑ ↑ | ⊗ ρ β where ρ β is a thermal state at inverse temperaturehβ = 5.91 · 10 −6 s for the macroscopic environment and a thermal state of a single mode of frequency ω m with mean occupationn(ω m ) = 0.025 for the Lindblad case.
The results for spin energies of ∆/2π = 50, 100 kHz are shown in Fig. 6. For both cases one can appreciate very good agreement between the two dynamics. This also shows that the correspondence to the macroscopic environment holds away from the environmental resonance ω m /2π = 100 kHz.
Note that the simulation of one curve for the case ∆/2π = 50 kHz takes 15 days with 16 cores on the bwForCluster JUS-TUS such that simulations for the case ∆/2π = 3 kHz presented in the main text are out of reach. 0 20 In order to implement the spin-boson Hamiltonian in Eq. (13) of the main text with trapped ions we make use of the so-called spin-dependent optical dipole forces. In this section we derive the Hamiltonian for the optical dipole forces.
∆ · t -1 1 σ z (t) ∆/2π = 50kHz 100kHz
For clarity, we consider a somewhat simplified level structure. We employ the formalism of Ref. [12] to obtain expressions for the effective operators of a ground-state manifold weakly coupled to a decaying excited state manifold.
We consider an ion where the internal levels form a Λ-type three-level system consisting of the ground states | ↑ and | ↓ which are separated in energy byhω 0 and have an electric dipole transition to a decaying excited state |e (see Fig. 7). The free Hamiltonian of the system reads
H at = ∑ i=↓,↑,e ε i |i i| (C1)
with ε i the energy of the corresponding state. We assume that the dipole transitions are driven by two laser fields with frequencies ω 1/2 which couple to both transitions. In a rotating wave approximation using |Ω l,s | ω l we obtain the interaction Hamiltonian
H L (t) =h ∑ l=1,2 ∑ s=↓,↑ Ω l,s 2 e −iω l t |e s| + H.c. (C2)
where Ω l,s is the Rabi frequency of laser l on transition |s → |e . Note that we have included the phase factors e i(k l r+φ l ) where r denotes the ion's position and k l (φ l ) the laser wave vector (phase) into the Rabi frequencies. Finally, we assume that spontaneous emission from the excited level to the ground states is properly described by a dissipator in Lindblad form
Dρ = ∑ s=↓,↑ L s ρL † s − 1 2 {L † s L s , ρ}(C3)
where L s = √ Γ s |s e| and Γ = Γ ↑ + Γ ↓ is the overall decay rate of the excited state. Putting the pieces together the system evolves according tȯ
ρ = − ī h [H at + H L (t), ρ] + Dρ.(C4)
Let us now introduce the detuning
δ l,s = (ε e − ε s )/h − ω l (C5)
of laser l for transition |s → |e . Here, we assume δ l,s ∆ R ω 0 , Ω l,s , Γ. In this case the lasers are far off resonant for all transitions such that the ground states are only weakly coupled to the excited state. We can then adiabatically eliminate the excited state from the dynamics and obtain an effective dynamics in the ground state manifold. Applying the formalism of [12] to our system we obtain the effective Lindblad equatioṅ
ρ = − ī h [H eff , ρ] + ∑ k L eff k ρ(L eff k ) † − 1 2 {(L eff k ) † L eff k , ρ} .
(C6) The effective Hamiltonian H eff has three contributions H eff = H g +H sr +H odf . The first part contains the shifted ground state levels where the ∆ε s are the ac-Stark shifts of the spin-levels due to the applied laser beams
H g = ∑ s (ε s + ∆ε s )|s s| (C7)∆ε s = − ∑ l,sh |Ω l,s | 2 δ l,s 4δ 2 l,s + Γ 2 .(C8)
The second part, H sr , describes two-photon stimulated Raman transitions between the spin states where a photon is absorbed from one laser beam followed by stimulated emission into the other beam
H sr =h ∑ l ,l Ω sr l ,l 2 σ + e −i(ω l −ω l )t + H.c.(C9)
Here, we have introduced σ + = | ↑ ↓ | = (σ − ) † and Ω sr l ,l = − Ω * l,↑ Ω l ,↓ (δ l ,↓ + δ l,↑ ) (2δ l ,↓ − iΓ)(2δ l,↑ + iΓ)
.
The third part of the effective Hamiltonian is a time-dependent ac-Stark shift that can be used to create the optical dipole force
H odf =h ∑ s Ω s 2 e i(ω 1 −ω 2 )t |s s| + H.c.(C11)
where Ω s = − Ω * 1,s Ω 2,s (δ 2,s + δ 1,s ) (2δ 2,s − iΓ)(2δ 1,s + iΓ)
.
The Hamiltonian H odf can be written in terms of σ z = | ↑ ↑ | − | ↓ ↓ | such that we obtain
H odf =h Ω rw 2 e −i(ω 1 −ω 2 )t 1 +h Ω odf 2 e −i(ω 1 −ω 2 )t σ z + H.c.(C13)
where have introduced the Rabi frequencies
Ω odf = 1 2 (Ω * ↑ − Ω * ↓ ), Ω rw = 1 2 (Ω * ↑ + Ω * ↓ ). (C14)
Thus, we obtain three effects on the spin states. The first is an ac-Stark shift of the spin levels due to the laser fields. The differential ac-Stark shift between spin levels can usually be canceled in experiments by adjusting polarization and intensity of the lasers [13]. Hence, we ignore this contribution. Alternatively, it could be absorbed into ω 0 . If one chooses the frequency difference between lasers close to the transition frequency between the spin states ω 1 − ω 2 ≈ ω 0 the second part of the Hamiltonian is resonant and one can drive coherent two-photon stimulated Raman transitions between the spin states. In this case, we usually have Ω odf , Ω rw ω 0 , the third contribution H odf is highly off-resonant and can be neglected in a rotating wave approximation.
Finally, there is the regime of the spin-dependent optical dipole forces where the beatnote between the two lasers matches one of the motional frequencies ω 1 − ω 2 ≈ ω k . Usually ω k ω 0 such that now the stimulated Raman processes in H sr are highly off-resonant and can be neglected in a rotating wave approximation. Hence, in this regime we arrive at the effective Hamiltonian
H eff =h ω 0 2 σ z + h Ω odf 2 e i(k L r+φ L ) e −iω L t σ z + H.c. (C15)
with the effective laser frequency ω L = ω 1 − ω 2 and phase φ L = φ 1 − φ 2 . Furthermore, we have written the phases e ik l r explicitly again and introduced the effective laser wave vector k L = k 1 − k 2 . Note that we have omitted the first part of H odf in Eq. (C13). For our choice of laser frequency this term would couple to the motion but it can be canceled choosing the appropriate laser intensities, polarizations and detunings [13]. Let us turn to the dissipative part. The effective Lindblad operators are found to read:
L eff ↓ = Γ ↓ Ω 1,↓ e −iω 1 t 2δ 1,↓ −iΓ + Ω 2,↓ e −iω 2 t 2δ 2,↓ −iΓ | ↓ ↓ | + Γ ↓ Ω 1,↑ e −iω 1 t 2δ 1,↑ −iΓ + Ω 2,↑ e −iω 2 t 2δ 2,↑ −iΓ | ↓ ↑ |, (C16) L eff ↑ = Γ ↑ Ω 1,↑ e −iω 1 t 2δ 1,↑ −iΓ + Ω 2,↑ e −iω 2 t 2δ 2,↑ −iΓ | ↑ ↑ | + Γ ↑ Ω 1,↓ e −iω 1 t 2δ 1,↓ −iΓ + Ω 2,↓ e −iω 2 t 2δ 2,↓ −iΓ | ↑ ↓ |. (C17)
By keeping only the dominant contributions, i.e. those parts of the action of the Lindblad operators that are time-independent, and using δ l,s ∆ R we obtain effective operators
L ↑↑ = 1 2 Γ ↑ ∑ l |Ω l,↑ | 2 4∆ 2 R σ z , L ↓↓ = 1 2 Γ ↓ ∑ l |Ω l,↓ | 2 4∆ 2 R σ z
(C18) and
L ↑↓ = Γ ↑ ∑ l |Ω l,↓ | 2 4∆ 2 R σ + , L ↓↑ = Γ ↓ ∑ l |Ω l,↑ | 2 4∆ 2 R σ − .(C19)
The first two terms describe Rayleigh scattering where the spin state is not altered upon a scattering event but can introduce dephasing. The other operators describe Raman scattering where the spin state is changed upon a scattering event. If we assume the modulus of the Rabi frequencies is approximately equal |Ω l,s | ≈ Ω 0 , we can estimate the effective scattering rate Γ eff ≈ ΓΩ L /∆ R where Ω L = Ω 2 0 /(2∆ R ) is the approximate effective laser Rabi frequency. Hence, decoherence can be largely suppressed if we choose ∆ R large enough.
where λ = −iη 2Ωodf can always be taken to be real and δ m = ω m . Thus, the mode frequency in our simulation is given by the detuning of the spin dependent force. Making the substitutionshδ = ε and Ω d = −∆ we obtain the spin-boson Hamiltonian for a single mode.
Note that experimentally a finite bias ε can easily be included by introducing a detuning to the field driving the spin transition. For the spin-motion coupling we consider one has to take care that the laser beams providing the spin-motion coupling are sufficiently detuned such that the simulation is not compromised by errors due to photon scattering (see previous section). In order to avoid this source of error one could also rotate the spin basis and provide spin-motion coupling e.g. by a Mølmer-Sørensen interaction [18].
Appendix E: Computation of non-Markovianity measures
There are several different ways to define non-Markovian dynamics. Here, we start by reviewing the definition presented in [19]. Let us consider a quantum system whose time evolution is described by a completely positive and trace preserving dynamical map E t,t 0 . Then for an initial state ρ(t 0 ) the system's state at a later time t ≥ t 0 is given by
ρ(t) = E t,t 0 ρ(t 0 ).(E1)
According to [19] the dynamical map describes a Markovian evolution if and only if the map E t 2 ,t 1 exists and is completely positive for all t 2 ≥ t 1 ≥ t 0 .The degree of non-Markovianity of a dynamics over an interval I, N RHP , is then obtained by quantifying the departure of E t 2 ,t 1 from complete positivity over that interval. In particular, we have
N RHP = I,ḡ>0ḡ (t)dt I,ḡ>0 χ[ḡ(t)]dt (E2)
where the integral extends over those subintervals of I wherē g(t) > 0. The function χ[x] = 1 for x > 0 and χ[x] = 0 else and by definition "0/0"= 0 . The functionḡ(t) is given bȳ
g(t) = tanh[g(t)] where g(t) = lim ε→0 + [E t+ε,t ⊗ 1]|ψ ψ| 1 − 1 ε (E3)
where . . . 1 denotes the trace norm and |ψ = 1 √ d ∑ d n=1 |n, n is a maximally entangled state of the open system of finite dimension d with an ancillary system of the same size. Note that we restrict our considerations to finite dimensional open systems. [E t+ε,t ⊗ 1]|ψ ψ| is the so-called Choi matrix and is positive if and only if E t+ε,t is completely positive. Note that g(t) vanishes if E t+ε,t is completely positive. Thus, for a Markovian dynamics g(t) = 0 for all times and N RHP evaluates to zero.
We evaluated N RHP numerically for the spin-boson system consisting of a spin coupled to a damped mode described by Eq. (8) of the main text with the Hamiltonian in Eq. (13). To this end we divide the time interval I = [0, T ] that we want to inspect for non-Markovian dynamics in N equally spaced discrete times t i (t 0 = 0,t N = T ) and compute the time evolution of the basis states |k j|, k, j =↑, ↓ for all t i . By writing the time-evolved states |k j|(t i ) = ρ k j (t i ) as a vector v k j (t i ) = [ρ k j,↑↑ (t i ), ρ k j,↑↓ (t i ), ρ k j,↓↑ (t i ), ρ k j,↓↓ (t i )] T we can write the dynamical map E (t,t 0 ) in matrix representation
E(t,t 0 ) = [v ↑↑ (t), v ↑↓ (t), v ↓↑ (t), v ↓↓ (t)].(E4)
The matrix for the time evolution from t 1 to t 2 where t 2 ≥ t 1 ≥ t 0 is then computed by
E(t 2 ,t 1 ) = E(t 2 ,t 0 )E −1 (t 1 ,t 0 ) (E5)
where E −1 (t 1 ,t 0 ) is the normal matrix inverse. The Choi matrix [E t 2 ,t 1 ⊗ 1]|ψ ψ| is proportional to the reshuffled matrix E R (t 2 ,t 1 ) of the matrix E(t 2 ,t 1 ) [20]. In particular, the Choi matrix is given by
[E t 2 ,t 1 ⊗ 1]|ψ ψ| = 1 d E R (t 2 ,t 1 ) (E6)
where d is the dimension of the finite dimensional open quantum system. For the case of a spin E R (t 2 ,t 1 ) reads
E R (t 2 ,t 1 ) = E 11 E 12 E 21 E 22 E 13 E 14 E 23 E 24 E 31 E 32 E 41 E 42 E 33 E 34 E 43 E 44 .(E7)
where E mn corresponds to entry m, n of the 4 × 4 matrix E(t 2 ,t 1 ). Now, in order to obtain N RHP we evaluated a discrete version of g(t) according to
g(t i ) = [E t i+1 ,t i ⊗ 1]|ψ ψ| 1 − 1 t i+1 − t i = 1 d E R (t i+1 ,t i ) 1 − 1 t i+1 − t i .
(E8) The difficulty in evaluating g(t i ) is to decide which values of the numerator count as zero and which are counted as finite. The numerical calculations were performed using Python's Numpy and Scipy libraries. The oscillator's Hilbert space was truncated at a maximal phonon number n max = 15. The states were evolved in time by vectorizing the Lindblad equation and applying the matrix exponential of the Liouvillian on the vectorized form of the density matrix using the scipy.sparse.linalg.expm multiply routine. For a number of parameters the resulting density matrices were compared to the density matrices obtained by performing the matrix exponential first with scipy.sparse.linalg.expm and then the matrix vector multiplication. For all of the spin basis states the resulting matrices typically showed trace distances of a few times 10 −16 . Summing the largest errors of all the basis states yielded a few times 10 −15 . Taking this value as a rough estimate of the numerical precision we set g(t) = 0 if the numerator was smaller than 10 −14 . Finally, N RHP in this numerical approximation is given by
N RHP = ∑ N i=1,g(t i )>0 tanh[g(t i )] N g(t i )>0 (E9)
where N g(t i )>0 is the number of events where g(t i ) > 0. For the "ohmic" case (∆/2π = 3 kHz) we chose T = 0.01/∆ and N = 10 4 and for the resonant case (∆/2π = 100 kHz) T = 0.1/∆ and N = 10 4 . Note that taking a too small time steps eventually leads to discontinuous behavior in N RHP . The computation of the measure of non-Markovianity N BLP [21] is somewhat easier. N BLP was originally proposed as a measure of non-Markovianity based on the monotonicity of the trace distance under completely positive and trace preserving evolutions and is given by
N BLP = max ρ 1/2 I,σ >0 σ (t)dt (E10)
where σ (t) = d dt D(E t,t 0 ρ 1 , E t,t 0 ρ 2 ) and D(·, ·) is the trace distance. The integral extends over those subintervals of I where σ (t) > 0. Thus, N BLP detects non-Markovianity of a dynamical map E t,t 0 if the trace distance between two initial states ρ 1 and ρ 2 increases in the course of the dynamics induced by E t,t 0 . A nonzero value of N BLP can be associated with a backflow of information from the environment to the system [21]. It is known that optimal state pairs ρ 1 , ρ 2 that saturate the maximum in Eq. (E10) are orthogonal and lie on the boundary of state space [22]. However, since we only want to witness non-Markovian dynamics we do not need to perform the maximization in Eq. (E10). Therefore, we can provide a useful lower bound on N BLP by computing the measure for the eigenstates | ↑ / ↓ , |± x and |± y of the Pauli matrices σ z , σ x and σ y , respectively.
For the numerical computation of N BLP we considered the whole interval [0, 20/∆]. As in the previous case we considered N = 10 4 equally spaced points t i in the interval and computed the time evolution for the spin starting in each of the eigenstates of the Pauli matrices. We then computed the discrete version of N BLP
N BLP = ∑ i,D t i+1 −D t i >0 (D t i+1 − D t i )(E11)
for each pair of eigenstates. Here the sum runs over those i where the term in brackets is larger than zero and D t i = D(E t i ,t 0 ρ 1 , E t i ,t 0 ρ 2 ). We note that due to the finite number of "measurements" there will be small deviation to the true value of N RHP [23]. The values shown in Fig. 2 of the main text are obtained for the initial spin states ρ s (0) = |± ±| x in the Ohmic case and ρ s (0) = | ↑ ↑ |, | ↓ ↓ | in the resonant case.
Figure 1 :
1Parts a) and b) of the figure show the dynamics of σ z (t) in natural time units ∆ · t under Eq. (8) with H = H sb1 from Eq. (13), which corresponds to a spin-boson model with a Lorentzian spectral density as in Eq. (5), for varying spin-motion coupling λ . In part a) the spin energy ∆/2π = 3 kHz is much smaller than the mode frequency ω m /2π = 100 kHz, so that the environment is approximately Ohmic. In part b) the mode is resonant with the spin (∆/2π = 100 kHz). The remaining parameters are given in the text. Part c) shows the measure of non-Markovianity N RHP in the intervals [0.01/∆] and [0.1/∆] for the ohmic and resonant cases, respectively. Part d) depicts the measure of non-Markovianity N BLP over the whole interval [0, 20/∆] for both cases.
Fig. 1over a time interval [0, 0.01/∆] and [0, 0.1/∆], respectively. The results are shown in part c) of the figure. In both cases the measure is non-zero for all couplings λ /2π > 0. An evaluation of N RHP requires process tomography and is therefore experimentally time-consuming already for a single spin. Hence, it might be easier to experimentally detect non-Markovian dynamics using N BLP which only requires state tomography. We numerically computed a lower bound on N BLP [28] for the parameters in parts a) and b) ofFig. 1for the whole interval [0, 20/∆]. The results are shown in part d) of the figure. N BLP witnesses non-Markovianity in all regions where N RHP does, too. The somewhat discontinuous behavior of the curve for the resonant case is due to the finite time interval we are sampling.
Figure 2 :
2Comparison of L (t) = L 1 (t) + L 2 (t) from Eq. (A21) including the first 10 4 Matsubara frequencies (blue solid lines) and L L (t) from Eq. (A15) (dashed-dot line and crosses) for ω m /2π = 100 kHz, κ/2π = 1.25 kHz andn(ω m ) = 0.025 (hβ = 5.91 · 10 −6 s). Panel a) shows the time evolution for short times, while panel b) illustrates the intermediate time behavior.
Figure 3 :
3The figure shows the distance d, Eq. (A27), between the correlation functions L(t) including the first 10 4 Matsubara frequencies and L L (t) for different values of the cooling rate κ and mean occupation numbern for fixed mode frequency ω m /2π = 100 kHz. Higher bars correspond to smaller values of d.
Figure 4 :
4The figure compares J eff (ω) (solid line) andJ eff (ω) (circles) from Eq. (A33) for ω m /2π = 100 kHz, κ/2π = 1.25 kHz and n = 0.025. Part a) shows the behavior for small frequencies while part b) depicts the two functions around the resonance ω m /2π = 100 kHz. In part c) we show the relative error ε J from Eq. (A34) over the relevant frequency range covered by the spectral density.
Figure 5 :
5Illustration of the spin-boson model's transformation into a one-dimensional configuration where the system is only coupled to the environment's first site.
Figure 6 :
6The figure shows the dynamics of σ z (t) for the spinboson Hamiltonian in Eq. (B1) with spectral density J eff (ω) as in Eq. (5) of the main text (solid lines) and for a spin coupled to a damped mode described by the Lindblad equation(8)of the main text with the Hamiltonian from Eq. (13) of the main text (triangles and circles). The spin energies are ∆/2π = 100 kHz and ∆/2π = 50 kHz. The remaining parameters are given in the text.Appendix C: Spin-dependent optical dipole forces
Figure 7 :
7The figure shows a three level Λ-system consisting of the ground states | ↓ and | ↑ which are separated in frequency by ω 0 and both feature a dipole-allowed transition to the decaying excited state |e . The transitions are driven by two lasers and Ω l,s denotes the Rabi frequency of laser l on transition |s → |e . ∆ R is roughly the detuning of the lasers from the excited state. Depending on the effective laser frequency ω L = ω 1 − ω 2 different operations on the ground states can be implemented (see text). Spontaneous emission from the excited to the ground states happens at rates Γ s and is indicated by the curly lines.
[ 1 ]
1U. Weiss, Quantum Dissipative Systems (World Scientific, Singapore, 2007), Third Edition.[2] A. Garg, J. N. Onuchic, and V. Ambegaokar, Effect of friction on electron transfer in biomolecules. J. Chem. Phys. 83, 4491
I. SUPPLEMENTAL MATERIAL TO "SIMULATING SPIN-BOSON MODELS WITH TRAPPED IONS"Appendix A: Effective spectral densities of damped harmonic oscillatorsContents
References
5
I. Supplemental Material to "Simulating spin-boson
models with trapped ions"
7
A. Effective spectral densities of damped harmonic
oscillators
7
1. Time domain considerations
7
2. Frequency space considerations
10
B. tDMRG simulations using the TEDOPA algorithm 11
C. Spin-dependent optical dipole forces
12
D. Spin-boson Hamiltonian with trapped ions
14
E. Computation of non-Markovianity measures
15
References
16
Appendix B: tDMRG simulations using the TEDOPA algorithm For macroscopic environments the Hamiltonian for the spin-boson model considered in this work becomes H = H sys + H env + H int , (B1)
Appendix D: Spin-boson Hamiltonian with trapped ionsIn this section we want to show how to obtain the spinboson Hamiltonian in Eq. (13) of the main text in an ion trap experiment. For definiteness we chose to consider a24Mg + − 25 Mg + crystal.25Mg + has electronic hyperfine ground states with total angular momentum F = 2, 3 for the valence electron in the 2 S 1/2 state whose degeneracy can be lifted by a magnetic field. A possible choice for a qubit are the states |F = 3, m F = 3 ≡ | ↓ and |F = 2, m F = 2 ≡ | ↑ . The hyperfine splitting between the F = 2 and F = 3 states is about ω 0 /2π 1.8 GHz. At a magnetic field B = 4 G the other hyperfine states are well-separated from the qubit states due to the Zeeman interaction and we can assume the Hamiltonianfor the internal levels of25The two ions interact through their Coulomb interaction and their motion is coupled. If the ions are sufficiently cold they form a so-called Coulomb crystal and perform only small oscillations about equilibrium. We assume trapping conditions such that the ions form a string along z and their equilibrium positions read r 0 j = (0, 0, z 0 j ) T . Their motion is then conveniently described in terms of normal modes[14,15]. For a crystal of N ions we obtain N modes in each direction such that, taking into account the coupled harmonic motion, the system's Hamiltonian becomesHere, ω α,n is the frequency of mode n in direction α and a † α,n (a α,n ) creates (annihilates) an excitation in the corresponding mode.24Mg + is used to sympathetically cool the ions' coupled motion. Since the internal levels are adiabatically eliminated in the description of laser cooling[16,17]we have omitted them here. The spin transition can be driven either directly by a microwave or in a two-photon stimulated-Raman configuration (see previous section). We adopt the convention that we will call the field driving the spin transition the "microwave" independent of the physical realization.Let us now assume the spin is driven by a microwave with frequency ω d and Rabi frequency Ω d and we apply a spindependent force as in Eq. (C15). The interaction Hamiltonian then reads(D3) where we have set the microwave phase to zero and performed a rotating wave approximation. Ω odf denotes the effective laser Rabi frequency, ω L , k L and φ L the effective laser frequency, wave vector and phase. We assume k L = ke z such that the laser only couples to the motion along z. We have r jz = z 0 j + z j where the z j can be written in terms of the quantized normal modes[15]:where m j is the mass of ion j,M jn the amplitude of motional mode n at ion j in mass-weighted coordinates and ω n = ω z,n (for the operators accordingly). The full Hamiltonian of the system then readsMoving to an interaction picture with respect toH 0 = h(ω d /2)σ z +h ∑ α,n ω α,n a † α,n a α,n we obtain the transformed interaction Hamiltoniañwhere δ = ω 0 − ω d ,Ω odf = Ω odf e i(kz 0 2 +φ L ) and we have introduced the Lamb-Dicke factors η n =M 2n k h/(2m 2 ω n ). Note that we have assumed that the 25 Mg + ion is located at site 2.Usually for an optical wave vector η n 1 such that we can expand the exponential to first order in the η n . In the axial direction the two-ion crystal features an in-and out-ofphase mode of motion that are well separated in frequency. More precisely, we consider a trapping potential such that a single24Mg + has a center-of-mass frequency ω m /2π = 2.54 MHz. The in-and out-of-phase mode frequencies of the24Mg + − 25 Mg + crystal are then given by ω 1 /2π = 2.51 MHz and ω 2 /2π = 4.36 MHz , respectively. If we choose the laser frequency close to the out-of-phase mode frequency ω L ≈ ω 2 and Ω odf 2ω L , η 1 Ω odf |ω 1 −ω L | we can neglect all terms except the coupling to the out-of-phase mode in a rotating wave approximation and arrive at the final Hamiltoniaña † 2 σ z e iδ m t + H.c. (D7) where δ m = ω 2 − ω L ω 2 is the detuning of the laser from the out-of-phase mode and we choose ω L such that δ m > 0. Finally, we can cast the above Hamiltonian in a timeindependent form and we recover Eq. (13) of the main text(a 2 + a † 2 )σ z + ω m a † 2 a 2 (D8)
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Improved scaling of time-evolving block-decimation algorithm through reduced-rank randomized singular value decomposition. D Tamascelli, R Rosenbach, M B Plenio, Phys. Rev. E. 9163306D. Tamascelli, R. Rosenbach, and M. B. Plenio, Improved scaling of time-evolving block-decimation algorithm through reduced-rank randomized singular value decomposition, Phys. Rev. E. 91, 063306 (2015).
Effective operator formalism for open quantum systems. F Reiter, A S Sørensen, Phys. Rev. A. 32111F. Reiter and A. S. Sørensen, Effective operator formalism for open quantum systems. Phys. Rev. A 032111 (2012).
Quantum information processing with trapped ions. D J Wineland, M Barrett, J Britton, J Chiaverini, B De-Marco, W M Itano, B Jelenković, C Langer, D Leibfried, V Meyer, T Rosenband, T Schaetz, Phil. Trans. R. Soc. Lond. A. 3611349D. J. Wineland, M. Barrett, J. Britton, J. Chiaverini, B. De- Marco, W. M. Itano, B. Jelenković, C. Langer, D. Leibfried, V. Meyer, T. Rosenband, and T. Schaetz, Quantum information processing with trapped ions, Phil. Trans. R. Soc. Lond. A 361, 1349 (2003)
Quantum dynamics of cold trapped ions with application to quantum computation. D F V James, Appl. Phys. B. 66181D. F. V. James, Quantum dynamics of cold trapped ions with ap- plication to quantum computation Appl. Phys. B 66, 181 (1998)
Two-species Coulomb chains for quantum information. G Morigi, H Walther, Eur. Phys. J. D. 13261G. Morigi and H. Walther, Two-species Coulomb chains for quantum information. Eur. Phys. J. D 13, 261 (2001).
Laser cooling of trapped ions in a standing wave. J I Cirac, R Blatt, P Zoller, W D Phillips, Phys. Rev. A. 462668J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, Laser cool- ing of trapped ions in a standing wave. Phys. Rev. A 46, 2668 (1992).
Cooling atomic motion with quantum interference. G Morigi, Phys. Rev. A. 6733402G. Morigi, Cooling atomic motion with quantum interference. Phys. Rev. A 67, 033402 (2003).
Quantum Computation with Ions in Thermal Motion. A Sørensen, K Mølmer, Phys. Rev. Lett. 821971A. Sørensen, and K. Mølmer, Quantum Computation with Ions in Thermal Motion. Phys. Rev. Lett. 82, 1971 (1999).
Quantum non-Markovianity: characterization, quantification and detection. A Rivas, S F Huelga, M B Plenio, Rep. Prog. Phys. 7794001A. Rivas, S. F. Huelga, M. B. Plenio, Quantum non- Markovianity: characterization, quantification and detection. Rep. Prog. Phys. 77, 094001 (2014).
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Colloquium: Non-Markovian dynamics in open quantum systems. H.-P Breuer, E.-M Laine, J Piilo, B Vacchini, Rev. Mod. Phys. 8821002H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Collo- quium: Non-Markovian dynamics in open quantum systems. Rev. Mod. Phys. 88, 021002 (2016).
Optimal state pairs for non-Markovian quantum dynamics. S Wißmann, A Karlsson, E.-M Laine, J Piilo, H.-P Breuer, Phys. Rev. A. 8662108S. Wißmann, A. Karlsson, E.-M. Laine, J. Piilo, and H.-P. Breuer, Optimal state pairs for non-Markovian quantum dy- namics. Phys. Rev. A 86, 062108 (2012)
M Wittemer, G Clos, H.-P Breuer, U Warring, T Schaetz, arXiv:1702.07518Probing Quantum Memory Effects with High Resolution. M. Wittemer, G. Clos, H.-P. Breuer, U. Warring, and T. Schaetz, Probing Quantum Memory Effects with High Resolu- tion, arXiv:1702.07518
|
[] |
[
"A 1.46-2.48 m Spectroscopic Atlas of a T6 Dwarf (1060 K) Atmosphere with IGRINS: First Detections of H 2 S and H 2 , and Verification of H 2 O, CH 4 , and NH 3 Line Lists",
"A 1.46-2.48 m Spectroscopic Atlas of a T6 Dwarf (1060 K) Atmosphere with IGRINS: First Detections of H 2 S and H 2 , and Verification of H 2 O, CH 4 , and NH 3 Line Lists"
] |
[
"1★Megan E Tannock \nDepartment of Physics and Astronomy\nThe University of Western Ontario\n1151 Richmond StN6A 3K7LondonOntarioCanada\n",
"Stanimir Metchev \nDepartment of Physics and Astronomy\nInstitute for Earth and Space Exploration\nThe University of Western Ontario\n1151 Richmond StN6A 3K7LondonOntarioCanada\n\nDepartment of Astrophysics\nAmerican Museum of Natural History\n200 Central Park West10024-5102New YorkNew YorkUSA\n",
"Callie E Hood \nDepartment of Astronomy & Astrophysics\nUniversity of California\n95064Santa CruzCAUSA\n",
"Gregory N Mace \nDepartment of Astronomy\nThe University of Texas\n78712AustinTXUSA\n",
"Jonathan J Fortney \nDepartment of Astronomy & Astrophysics\nUniversity of California\n95064Santa CruzCAUSA\n",
"Caroline V Morley \nDepartment of Astronomy\nThe University of Texas\n78712AustinTXUSA\n",
"Daniel T Jaffe \nDepartment of Astronomy\nThe University of Texas\n78712AustinTXUSA\n",
"Roxana Lupu \nEureka Scientific Inc\n94602OaklandCAUSA\n"
] |
[
"Department of Physics and Astronomy\nThe University of Western Ontario\n1151 Richmond StN6A 3K7LondonOntarioCanada",
"Department of Physics and Astronomy\nInstitute for Earth and Space Exploration\nThe University of Western Ontario\n1151 Richmond StN6A 3K7LondonOntarioCanada",
"Department of Astrophysics\nAmerican Museum of Natural History\n200 Central Park West10024-5102New YorkNew YorkUSA",
"Department of Astronomy & Astrophysics\nUniversity of California\n95064Santa CruzCAUSA",
"Department of Astronomy\nThe University of Texas\n78712AustinTXUSA",
"Department of Astronomy & Astrophysics\nUniversity of California\n95064Santa CruzCAUSA",
"Department of Astronomy\nThe University of Texas\n78712AustinTXUSA",
"Department of Astronomy\nThe University of Texas\n78712AustinTXUSA",
"Eureka Scientific Inc\n94602OaklandCAUSA"
] |
[
"MNRAS"
] |
We present Gemini South/IGRINS observations of the 1060 K T6 dwarf 2MASS J08173001−6155158 with unprecedented resolution ( ≡ /Δ = 45 000) and signal-to-noise ratio (SNR>200) for a late-type T dwarf. We use this benchmark observation to test the reliability of molecular line lists used up-to-date atmospheric models. We determine which spectroscopic regions should be used to estimate the parameters of cold brown dwarfs and, by extension, exoplanets. We present a detailed spectroscopic atlas with molecular identifications across the and bands of the near-infrared. We find that water (H 2 O) line lists are overall reliable. We find the most discrepancies amongst older methane (CH 4 ) line lists, and that the most up-to-date CH 4 line lists correct many of these issues. We identify individual ammonia (NH 3 ) lines, a hydrogen sulfide (H 2 S) feature at 1.5900 m, and a molecular hydrogen (H 2 ) feature at 2.1218 m. These are the first unambiguous detections of H 2 S and H 2 in an extra-solar atmosphere. With the H 2 detection, we place an upper limit on the atmospheric dust concentration of this T6 dwarf: at least 500 times less than the interstellar value, implying that the atmosphere is effectively dust-free. We additionally identify several features that do not appear in the model spectra. Our assessment of the line lists is valuable for atmospheric model applications to highdispersion, low-SNR, high-background spectra, such as an exoplanet around a star. We demonstrate a significant enhancement in the detection of the CH 4 absorption signal in this T6 dwarf with the most up-to-date line lists.
|
10.1093/mnras/stac1412
|
[
"https://arxiv.org/pdf/2206.03519v1.pdf"
] | 249,254,944 |
2206.03519
|
18536220c059e246b27bacf2af500397ce847a4b
|
A 1.46-2.48 m Spectroscopic Atlas of a T6 Dwarf (1060 K) Atmosphere with IGRINS: First Detections of H 2 S and H 2 , and Verification of H 2 O, CH 4 , and NH 3 Line Lists
2022
1★Megan E Tannock
Department of Physics and Astronomy
The University of Western Ontario
1151 Richmond StN6A 3K7LondonOntarioCanada
Stanimir Metchev
Department of Physics and Astronomy
Institute for Earth and Space Exploration
The University of Western Ontario
1151 Richmond StN6A 3K7LondonOntarioCanada
Department of Astrophysics
American Museum of Natural History
200 Central Park West10024-5102New YorkNew YorkUSA
Callie E Hood
Department of Astronomy & Astrophysics
University of California
95064Santa CruzCAUSA
Gregory N Mace
Department of Astronomy
The University of Texas
78712AustinTXUSA
Jonathan J Fortney
Department of Astronomy & Astrophysics
University of California
95064Santa CruzCAUSA
Caroline V Morley
Department of Astronomy
The University of Texas
78712AustinTXUSA
Daniel T Jaffe
Department of Astronomy
The University of Texas
78712AustinTXUSA
Roxana Lupu
Eureka Scientific Inc
94602OaklandCAUSA
A 1.46-2.48 m Spectroscopic Atlas of a T6 Dwarf (1060 K) Atmosphere with IGRINS: First Detections of H 2 S and H 2 , and Verification of H 2 O, CH 4 , and NH 3 Line Lists
MNRAS
0002022Accepted 2022 May 18. Received 2022 May 17; in original form 2022 January 15Preprint 9 June 2022 Compiled using MNRAS L A T E X style file v3.0brown dwarfs -stars: individual (2MASS J08173001−6155158) -stars: atmospheres -planets and satellites: atmospheres -techniques: spectroscopic -line: identification
We present Gemini South/IGRINS observations of the 1060 K T6 dwarf 2MASS J08173001−6155158 with unprecedented resolution ( ≡ /Δ = 45 000) and signal-to-noise ratio (SNR>200) for a late-type T dwarf. We use this benchmark observation to test the reliability of molecular line lists used up-to-date atmospheric models. We determine which spectroscopic regions should be used to estimate the parameters of cold brown dwarfs and, by extension, exoplanets. We present a detailed spectroscopic atlas with molecular identifications across the and bands of the near-infrared. We find that water (H 2 O) line lists are overall reliable. We find the most discrepancies amongst older methane (CH 4 ) line lists, and that the most up-to-date CH 4 line lists correct many of these issues. We identify individual ammonia (NH 3 ) lines, a hydrogen sulfide (H 2 S) feature at 1.5900 m, and a molecular hydrogen (H 2 ) feature at 2.1218 m. These are the first unambiguous detections of H 2 S and H 2 in an extra-solar atmosphere. With the H 2 detection, we place an upper limit on the atmospheric dust concentration of this T6 dwarf: at least 500 times less than the interstellar value, implying that the atmosphere is effectively dust-free. We additionally identify several features that do not appear in the model spectra. Our assessment of the line lists is valuable for atmospheric model applications to highdispersion, low-SNR, high-background spectra, such as an exoplanet around a star. We demonstrate a significant enhancement in the detection of the CH 4 absorption signal in this T6 dwarf with the most up-to-date line lists.
INTRODUCTION
Reliable determinations of the effective temperatures, radii, and masses of isolated, self-luminous brown dwarfs and giant exoplanets are dependent on accurate modelling of their spectra. However, it is known that the laboratory-based experimental line lists used to generate model spectra are inconsistent with each other and are even missing lines for some molecular species (e.g., Saumon et al. 2012;Canty et al. 2015). Even the most up-to date spectral models do not completely reproduce observed spectral features in cold brown dwarfs, limiting our ability to constrain their basic properties.
Methane and ammonia are of particular interest for T dwarfs. At the time of their discovery, the distinction between L and T dwarfs was based on whether methane lines were present in low-resolution ( ≡ A current hurdle in characterizing cold brown dwarfs and giant exoplanets are systematic uncertainties in the wavelengths and strengths of absorption lines in theoretical photospheres. Missing lines or inaccurate line lists make detections of molecules and determinations of radial velocities and projected rotation velocities difficult or impossible, especially in low signal-to-noise observations of exoplanet atmospheres. It is therefore necessary to confirm the accuracy of line lists by comparing to high signal-to-noise observations. Isolated brown dwarfs, free from the overwhelming light of a companion star, have atmospheres containing some of the key opacity sources in exoplanets, making them suitable laboratories for testing the accuracy of line lists. Improvements in the atmospheric opacity estimates for cold substellar atmospheres would also be invaluable for the characterization of potentially habitable exoplanets. Methane and ammonia have been suggested as biosignature gases in exoplanet atmospheres (e.g., Léger et al. 1996;Seager et al. 2013). Water, while not a biosignature gas, is also an important signature of habitability and is a major opacity source in brown dwarfs.
We present a high signal-to-noise (SNR > 200) spectrum of a T6 dwarf with unprecedented = 45 000 resolution and 1.45-2.48 m coverage, observed with the Immersion GRating INfrared Spectrometer (IGRINS;Yuk et al. 2010;Park et al. 2014;Mace et al. 2016Mace et al. , 2018 on Gemini South. We perform a detailed study of absorption features due to water, methane, ammonia, carbon monoxide, and hydrogen sulfide. Our target, 2MASS J08173001−6155158 (also known as DENIS J081730.0-615520; hereafter 2M0817) was discovered by Artigau et al. (2010) through a photometric cross match between the Two Micron All Sky Survey (2MASS) and the DEep Near-Infrared Survey of the Southern sky (DENIS) point-source catalogues, and spectroscopically identified as a T6 dwarf. It is at a heliocentric distance of only 5.2127 ± 0.0113 pc (Gaia Collaboration 2016, and is one of the brightest late-type T dwarfs ( -band magnitude 13.52; Skrutskie et al. 2006). Radigan et al. (2014) find a rotation period of 2.8 ± 0.2 h for 2M0817 from ground-based -band observations spanning four hours.
SPECTROSCOPY WITH IGRINS ON GEMINI SOUTH
We observed 2M0817 with IGRINS on Gemini South under Gemini program ID GS-2018A-Q-304 (PI: M. Tannock). IGRINS is a high-resolution ( = 45 000), cross-dispersed spectrograph that simultaneously covers the and bands from 1.45 to 2.48 m.
Observations took place over four nights in April and May 2018 while IGRINS was on Gemini South. The slit was oriented at the default position angle of 90 degrees (east-west) for IGRINS, and exposures were taken along an ABBA dither pattern. We observed an A0 V star before or after each observation of the target at a similar airmass, with the same telescope and instrument configuration. We summarize these observations in Table 1.
Data Reduction
The data were reduced with Version 2 of the IGRINS Pipeline Package (PLP; Lee et al. 2017), at each epoch individually. The PLP performs sky subtraction, flat-fielding, bad-pixel correction, aperture extraction, wavelength calibration, and telluric correction. The PLP outputs wavelength-calibrated, telluric-corrected fluxes and the signal-to-noise ratio (SNR) for each point in the spectrum. For each observing epoch, the wavelength solution was derived from a combination of OH emission lines and telluric absorption lines. OH emission lines in the observed spectrum were removed through A-B pair subtraction, and telluric absorption lines were removed by dividing the target spectrum by an A0 V standard spectrum. The target spectrum was also multiplied by a standard Vega model to remove any features from the A0 V standard itself.
We found that the strongest telluric features were not completely removed in our data reduction, and left residuals that affected the chi square statistic ( 2 ) when comparing to models (Section 3). To identify and mask strong telluric features, we generated transmission spectra for the Earth's atmosphere with the Planetary Spectrum Generator (PSG; Villanueva et al. 2018) 1 . We used the Earth's Transmittance template with the longitude, latitude, and altitude of the Gemini South Observatory. We found that masking atmospheric lines with > 65 per cent absorption strengths in the PSG Earth transmittance spectrum, along with strong OH emission lines (identified from the atlas of Rousselot et al. 2000) significantly improved the quality of our model photosphere fits. The PSG spectra and the 65 per cent absorption threshold beyond which we masked features are shown in the Figures of the Appendix.
We used a custom IDL code to combine the individual spectra. We first corrected for the barycentric velocity at each epoch. We then processed the and bands separately: we normalized the flux to peak at unity in each of the and bands, and then resampled the data to identical wavelength values. We computed the weights from the SNR values computed by the PLP ( = ( / ) 2 ), where is the flux at each epoch) and computed the weighted average (¯= =1 ( / )) where is the sum of the weights for epochs) and uncertainties ( = 1/2 ) across all epochs. We found that in some cases, the IGRINS PLP produced fluxes of ∼0, but with disproportionately high SNR values, resulting in large weights. This produced large downward spikes in the weighted average spectrum. We obtained the highest SNR combined spectrum free of such spikes when we combined the three highest SNR epochs: 2018 May 22 (both sequences) and 2018 May 23. In Fig. 1 we show the data from each epoch in an order at the centre of the band. Three of the nights stand out with their higher SNR. We performed the remainder of our analysis with the weighted average of these three epochs. Our final combined spectrum (Fig. 2) had a signal-to-noise of ∼300 at the peak of the band and ∼200 at the peak of the band.
There is some overlap between the orders in the spectrum (see Table 2 for a list of the orders and their wavelength coverage). For our analysis, we analysed each order individually. The instrument blaze profile results in the short-wavelength ends of the order having lower SNR than the long-wavelength ends (see the bottom panel of Fig. 1). We show the complete spectra with the orders stitched together in Fig. 2. For this stitched spectrum, in each region of overlap, we averaged the fluxes from the two orders.
Confirmation of Wavelength Calibration
We verified our wavelength calibration by comparing the telluric lines in the spectra of our A0 V standard stars to the Earth's transmittance spectrum from the PSG, generated over the wavelength coverage of IGRINS at 1.5 times the resolution of IGRINS. In each IGRINS order, between five and ten lines spread over the order were selected (very deep lines and blended lines were avoided), and we measured the wavelength at the minimum flux for each of these lines. We found an average offset of less than half an IGRINS pixel (0.110 Å at the centre of the band), confirming our wavelength calibration. The SNR of the neighbouring orders are also shown in grey, to show that IGRINS has good SNR coverage at all wavelengths. The IGRINS instrument transmission profile (blaze) is imprinted on the SNR spectrum, and is the reason for the fall-off in SNR at the edges of the order. The three highest-SNR spectra, obtained on 2018 May 22 and 23, were combined to create the final spectrum shown in Fig. 2.
MODEL FITTING AND PARAMETER DETERMINATION
We compared our observed spectra to the models of Allard et al. (2012, Morley et al. (2012, hereafter, Morley), Marley et al. (2021, hereafter, Sonora Bobcat), and an alternative version of the Sonora Bobcat models with updated molecular line lists from Hood et al. (in preparation, hereafter, Bobcat Alternative A). The BT-Settl models are based on the PHOENIX code (Allard & Hauschildt 1995;Hauschildt et al. 1999). The latter three model sets are all based on the same 1D radiative-convective equilibrium model atmosphere code (e.g., Marley et al. 1996;Fortney et al. 2008;Marley & Robinson 2015). The Morley models include the effect of clouds that may be relevant for T dwarf atmospheres by applying the Ackerman & Marley (2001) cloud model. In contrast, the Sonora Bobcat models assume a cloud-free atmosphere. The Sonora Bobcat models include post-2012 updates to the gas opacity database, described in Freedman et al. (2014), Lupu et al. (2014), and Marley et al. (2021). The Bobcat Alternative A models are thermal emission spectra generated from the Sonora Bobcat atmospheric structures with the code described in the Appendix of Morley et al. (2015). Figure 2. The full -and -band IGRINS spectra of 2MASS J08173001−6155158 with epochs combined and the orders stitched together. This figure does not include the quadratic correction described in Section 3.1. These data appear noisy, but in fact have SNR 300 at the peak of the -band spectrum and SNR 200 at the -band peak. The apparent noise spikes are all absorption features, and can be seen in detail in the full set of figures in the Appendix.
Fitting of Photospheric Models
The models are provided on fixed grids of effective temperature ( eff ) and surface gravity (log , with in units of cm s −2 ), and we do not interpolate between models to intermediate values. We allowed our model fitting to explore eff grids between 700 K and 1300 K (the expected range of 900-1100 K in eff for a T6 dwarf, ±200 K; Filippazzo et al. 2015), in steps of 50 K or 100 K, depending on the model family. For log we explored grids between log = 4.0 and 5.5, in steps of 0.5 dex for all model families except the Sonora Bobcat models, which are in steps of 0.25 dex. The Morley models also have a sedimentation efficiency ( sed ) parameter on a grid from 2 to 5 in integer steps. We explored a radial velocity (RV) grid by applying a Doppler shift to the wavelength of the models. We also expect that our observed spectrum will have significant rotational broadening from its known axial rotation. We explored a grid of projected rotation velocities ( sin ), by simulating rotational broadening in the model spectra. We convolved the model spectra with the standard rotation kernel from Gray (1992), as described in Tannock et al. (2021). For both RV and sin we first explored coarse grids with steps of 2 km s −1 over a broad range of values, then narrowed our grid and repeated the fitting with finer steps of 0.1 km s −1 . After shifting and broadening the model spectra, we resampled the model spectra to the wavelengths of the observed spectrum using IDL's interpol function. 2 We also observed an instrumental effect resulting in an upward curving in the residuals when compared to models at the ends of the 2 We have not broadened the model spectra by the ≈6.5 km s −1 line width of the instrument profile. The potential effect on the sin = 22.5 ± 0.9 km s −1 that we ultimately find (Table 3) would be to decrease it by ≈0.6 km s −1 .
orders. To minimize this effect and analyse the highest SNR regions of the data, we removed the ends of each order, leaving ∼1-2 nm overlap between orders. We additionally divided out a quadratic function that minimized the 2 statistic between the data and the model to further remove this instrumental effect. With these corrections we obtained the 2 values as:
2 = ∑︁ =1 /( 2 + + ) − √︃ 2 + 2 2 ,(1)
where is the observed flux, is the flux of the model, is the uncertainty of the data, is a systematic uncertainty assigned to the model, and is the wavelength of the corresponding data point. The coefficients of the quadratic correction are , , and . Following a similar process to the one described in Suárez et al. (2021), we set the partial derivatives of Equation 1 to zero and solved the resulting system of equations to find the values of , , and . We determined the coefficients of the quadratic for every model on the model grid individually.
We identified the best-fitting order of the entire spectrum (order = 85 of the band for the Bobcat Alternative A model) and determined the value of that produced a reduced 2 statistic of 1.0. The adopted value of was approximately half of the average uncertainty of the data. This systematic uncertainty was added in quadrature to the observational uncertainties in every order. As a final measure of the goodness of fit for each order, in our figures we report the Δ 2 reduced with respect to the minimum 2 reduced = 1 value for the best-fitting order. That is, for the best-fitting ( = 85) order the goodness of fit is Δ 2 reduced = 0, while for orders with poorer fits, the goodness of fit is Δ 2 reduced = 2 reduced − 1.
The total uncertainty, including the systematic uncertainty added in quadrature, is shown in grey in Fig. 3 and in all following figures, including in the Appendix. The total uncertainty is still very small in most orders, and appears indistinguishable from the wavelength axis in most figures, as our estimated S/N ratios can be well over 100. Nevertheless, we believe the overall uncertainties to be this small based on the above 2 analysis.
Determination of Physical Parameters
We show the results of the model fitting across all orders for all model families in Fig. 3 and 4. In the top panel, a Bobcat Alternative A model is used to separate the contribution of each molecular species, in order to identify the dominant molecule or molecules in each order. These 'single-molecule models' include a single molecule (e.g., water, methane), plus collision-induced absorption from molecular hydrogen and helium. To help identify particular features and molecules, a panel like this is included at the top of almost all of our figures.
We find that the Bobcat Alternative A models with the updated line lists provide the best fits to the data. We adopt the values given by the Bobcat Alternative A models, and present the weighted average of each parameter across all and band orders in Table 3. As described in Tannock et al. (2021), we compute the weighted average and the unbiased weighted sample standard deviation, where the weight is − 2 reduced , so that the better fits and more reliable orders are more heavily weighted. The parameter values given in Fig. 3 and 4 are computed in the same way, but for each order separately. The Bobcat Alternative A models are the most consistent across all orders, and give the smallest uncertainties on the measured parameters. Overall, all models do fairly well in regions dominated by water, while fits are poor in regions dominated by methane.
For the remainder of our analysis, we will focus on the results of the Bobcat Alternative A models, unless otherwise stated. We show the best fitting Bobcat Alternative A models for all orders of the and bands in Fig. A1 and A2, and in the following sections we highlight a few notable orders and regions.
MOLECULE-BY-MOLECULE ANALYSIS OF THE MODEL SPECTRA
In this section we assess the quality of the fits from each family of models. We examine the parameters determined for each region of the spectrum and what the dominant absorbers are in each region. Water (H 2 O) and methane (CH 4 ) are the most abundant absorbers in latetype T dwarf spectra (Burgasser et al. 2006). Carbon monoxide (CO) and ammonia (NH 3 ) also play a major role, and hydrogen sulfide (H 2 S) is the next most abundant absorber. The references for the line lists of the major molecules used in each family of models are listed in Table 4. As 2M0817 is a fairly rapid rotator ( sin = 22.5 ± 0.9 km s −1 ; Table 3), we see that most lines are in fact blends of the dominant absorbers, most often H 2 O and CH 4 . In Fig. 5 we show order = 85 of the band: the order where the models most accurately represent the data. The dominant absorbers in this order are H 2 O, and CH 4 . The Bobcat Alternative A model provides the best fit, and the residuals for this model are very flat. The other models also do a fair job in matching the major features. For comparison, in Fig. 6, we show order = 111 of the band: one of the orders where all models provide poor fits. The major absorber in this order is CH 4 . We see that the locations of the strongest CH 4 features are matched in the Bobcat Alternative A model, which has the most up-to-date CH 4 line list (Table 4). In the following sections, we discuss each molecular absorber separately.
Water
The water-dominated regions of the spectrum provide the most consistent results from fitting models to spectra across all model families ( Fig. 3 and 4). The short-wavelength end of the band (1.454-1.580 m) gives consistent results for each model family, and across the various families. The long-wavelength end of the band (1.750-1.812 m) and the short-wavelength end of the band (1.894-2.100 m) give consistent results within each family of models, but not necessarily across the various model families.
We note that the Sonora Bobcat and BT-Settl models give higher estimates of the RV, and there is a trend in RV where the RV increases with wavelength (the models are increasingly blue-shifted) in the short-wavelength end of the band (1.894-2.060 m; Fig. 4) for these two models. NH 3 is also an important absorber in this region but is likely not responsible for this trend in RV because Sonora Bobcat shares the same line lists for ammonia as the Morley models (Yurchenko et al. 2011, BYTe), and the Morley models do not show this trend.
The behaviour for the BT-Settl models indicates that the BT2 ( Barber et al. 2006) H 2 O line lists, when used alone, are unreliable for RV determinations in this wavelength region. The similar behaviour from Sonora Bobcat indicates that , supplemented with isotopologues from BT2, is also unreliable. The Bobcat Alternative A models use ExoMol/POKAZATEL (Polyansky et al. 2018) as the main H 2 O line list, and also use isotopologue data from BT2. However for this model, we obtain very consistent RV measurements in this wavelength region. The improved accuracy of ExoMol/POKAZATEL line lists appear to make up for any discrepancies in BT2. The HITRAN'08 (Rothman et al. 2009) and Partridge & Schwenke (1997) line lists used in the Morley models also give more self-consistent estimates of RV in this region.
Overall we consider water, specifically for the line list used in the Bobcat Alternative A models (ExoMol/POKAZATEL), to be the most reliable molecule for determining the physical parameters of cold brown dwarfs, producing values that we trust.
Methane
As seen in Fig. 3 and 4, there is much greater variation in the parameters estimated in the methane-dominated regions (1.60-1.73 m in the band and 2.11-2.40 m in the band) compared to the waterdominated regions, and the sin values are particularly discrepant. Each family of models uses a different set of line lists for CH 4 , though there is some overlap between the Sonora Bobcat, Morley, and BT-Settl models which use multiple sources for their CH 4 line lists (Table 4). Uncertainty has been reported for theoretical CH 4 band positions previously: Canty et al. (2015) report offsets between the absorption features in their observed data and the peaks of CH 4 opacity from the Exomol/10to10 line list (Yurchenko & Tennyson 2014) between 1.615 and 1.710 m.
In Fig. 7 we show a Sonora Bobcat model and a Bobcat Alternative A model with identical physical parameters for an order in the methane region of the band (order = 111 of the band, 1.608-1.624 m). The CH 4 lines used in the Sonora Bobcat models (the same as examined by Canty et al. 2015; Table 4) do not match the data well, and appear to have a stretch across this order. Both models poorly fit the weaker lines and the continuum in this region. Radial The models shown in this panel have eff = 1100 K, log = 5.0 (with in cm s −2 ), sin = 22.5 km s −1 , and RV = 6.1 km s −1 , and are also matched to the resolution of the IGRINS data. The IGRINS order numbers ( ) are given along the top horizontal axis. Second panel from the top: The full -band IGRINS spectrum, with the orders stitched together. Bottom four panels: The parameters of the best-fitting model for each order, from each family of models. From top to bottom the parameters are: RV, sin , eff , and log . The weighted average of each parameter is given on the right side of the figure. In some cases the best-fitting models are at the maximum and minimum values of the allowed grid, which indicates that these models produce inadequate fits in the particular order. These values are still included in the weighted mean, but have very little weight assigned to them due to their large 2 statistics. . The same layout as Fig. 3, but for the band. The log value is extremely consistent for the Bobcat Alternative A models, with log = 5.0 in every order of the band. The standard deviation on this weighted average is therefore zero (see Section 3.2 for details on this calculation). In Table 3 we compute the weighted average and standard deviation based on both the and bands, so the standard deviation is non-zero for the final adopted value. velocities estimated by the Sonora Bobcat models are discrepant in the methane-dominated regions, due to these inaccurate line positions. We find significant improvement from the line lists used in the Bobcat Alternative A models (HITEMP, Hargreaves et al. 2020) over older models in regions dominated by CH 4 , in particular in the band. However, the regions dominated by CH 4 , even in the Bobcat Alternative A models, still have the most variation in the estimates of the physical parameters. We summarize these regions in Table 5, noted as 'CH 4 regions.' Models using older CH 4 line lists should therefore be used with caution. Unaccounted for disequilibrium chemistry may impact these weaker features. We do not explore disequilibrium chemistry for CH 4 or H 2 O in this work, but see Section 4.3 for details on CO disequilibrium chemistry.
H 2 O CH 4 CO NH 3 H 2 S Data Uncert Sonora Bobcat BT-Settl Morley Bobcat Alt A Dc 2 reduced =468 Dc 2 reduced =388 Dc 2 reduced =332 Dc 2 reduced =21 d.o.f.=1439
Recent theoretical line lists are far more complete than the previously-used laboratory-measured line lists, which are designed to have very accurate line positions but capture fewer lines due to the limits on resolution in laboratory experiments. Therefore, theoretical line lists should improve accuracy in regions of the spectrum with weaker bands present, if those bands were unresolved in the laboratory lists. A recent improvement in the available line lists has been the combinations of theoretical line lists with laboratory measurements (e.g., HITEMP, Hargreaves et al. 2020). Such combination lists provide the best of both worlds, as we show here, where we find a dramatic improvement to high resolution spectroscopic fits.
Carbon Monoxide
For effective temperatures 1300 K (near the L/T transition), the dominant carbon-bearing molecule in the visible part of atmospheres of brown dwarfs switches from CO to CH 4 (Fegley & Lodders 1996;Burrows et al. 1997). There are still signatures of CO in the spectra of cold brown dwarfs, and carbon exists abundantly as CO deeper in the atmosphere, where temperatures are higher.
We found that at the CO bands our model fitting selected higher effective temperatures ( eff ∼ 1200 K) compared to other orders. Accordingly, we observed several notable features in the residuals of orders = 77 through = 73 of the band (2.294-2.445 m), as well as in order = 115 of the band (1.554-1.569 m), where a CO band head is present. The features in the residuals aligned with CO absorption features. We show an example of this in Fig. 8, along with a model with increased CO abundance, providing an improved fit. The model with increased CO abundance is also shown in Fig. 3, 4, and the Appendix figures, and is described in detail below.
This increased CO abundance implies disequilibrium chemistry, which can occur when vertical mixing occurs in the atmosphere (Lodders & Fegley 2002;Saumon et al. 2003). If CO is being brought from deeper, hotter layers to the upper atmosphere faster than the chemical reaction that converts CO to CH 4 , there will be more CO in the upper layers of the atmosphere than predicted from chemical equilibrium.
The Sonora Bobcat and Bobcat Alternative A models use the same cloudless, rainout chemical equilibrium structure models ). These structure models assume chemical equilibrium and give the pressure, temperature, and chemical abundances throughout the atmosphere. To improve our fitting, we generated a small grid of Bobcat Alternative A models with varied amounts of CO, deviat- ing from the chemical equilibrium assumptions used in the Sonora Bobcat structure models. We took a simple approach where we fixed the volume mixing ratio (VMR) for CO to values of 10 −6 , 3 × 10 −5 , 10 −5 , 3 × 10 −4 , 10 −4 , 3 × 10 −3 , and 10 −3 . This is a zeroth-order approximation, as 1) the CO VMR is not constant throughout the entire atmosphere, 2) other abundances like CH 4 and H 2 O will also be affected by disequilibrium chemistry, and 3) we are using the temperature-pressure profile from the chemical equilibrium Sonora Bobcat models, but a much higher CO abundance could affect the temperature-pressure profile.
We found a CO VMR of 3 × 10 −4 provided the best fits to our data. Fig. 8 shows a comparison of the original equilibrium chemistry model to the model with this fixed CO VMR value (Fig. 3, 4, and the Appendix figures also show models with this fixed CO VMR). In equilibrium models, the CO VMR ranges from 10 −7 to 2.5 × 10 −4 for pressures probed by the band. The CO VMR value of the best fitting model is slightly higher than the range in values expected for equilibrium, and explains why our initial fitting selected models with higher effective temperatures, as the CO abundance would be higher in the hotter models.
Following the method in Section 6.1 of Miles et al. (2020), we use the quench pressure to estimate the eddy diffusion coefficient (log , where has units of cm 2 s −1 ). We find log = 6.4 for 2M0817. This value is in line with the range of relatively low inferred values (∼100 × lower than expected from mixing length theory of convection) from Miles et al. (2020) for colder ( eff ≤ 750 K) brown dwarfs. Miles et al. attribute the low values to quenching in radiative regions of the atmosphere, where mixing is likely more sluggish than in convective regions. Interestingly, we find the same behaviour here at ∼ 300 K hotter eff . The Sonora model pressuretemperature profile is completely radiative down past the quench pressure of 20 bars, to a pressure of 30 bars, where the atmosphere transitions to the deep convective region.
Disequilibrium chemistry for CO has been observed spectroscopically and inferred photometrically in many other late-type T dwarfs and Y dwarfs (e.g., Noll et al. 1997;Oppenheimer et al. 1998;Golimowski et al. 2004;Geballe et al. 2009;Leggett et al. 2012;Sorahana & Yamamura 2012;Miles et al. 2020), and has been known in Jupiter for decades (Prinn & Barshay 1977;Noll et al. 1988). The growing number of T and Y dwarfs with evidence for CO disequilibrium chemistry indicates that vertical mixing is an important factor in accurately modelling brown dwarf spectra even at cold temperatures.
Ammonia
Water and methane are the dominant absorbers in the spectra of latetype T dwarfs, but ammonia is important too, especially at T<700 K (the coldest T dwarfs and Y dwarfs), where it becomes the dominant nitrogen-bearing molecule (Lodders & Fegley 2002). Ammonia is of special significance as it is the defining species in the spectra of Y dwarfs (Cushing et al. 2011).
The choice of an ammonia line list (among published lists) does not appear to significantly impact the physical parameters derived by comparing to models, but ammonia lines are clearly present in the observed spectrum and are important to include in the models. We are able to detect ammonia clearly in several regions of our spectrum.
This T6 dwarf joins the handful of T dwarfs with confirmed NH 3 detections in the near-infrared. Saumon et al. (2000) find evidence for NH 3 in the -and -band spectra of Gliese 229B (spectral type T6.5p, eff ∼ 950 K) and Canty et al. (2015) report the detection of several NH 3 absorption features in the and bands in a T8 and T9 dwarf. Bochanski et al. (2011) additionally report detections of NH 3 in a T9 dwarf, however, Saumon et al. (2012) question whether some of those detections are indeed attributable to NH 3 . Saumon et al. (2012) do confirm the stronger NH 3 features at ∼2 m in the spectrum of Bochanski et al. (2011). We re-confirm the strongest NH 3 identified in these works, but some of the weaker lines identified in these later spectral types do not appear in our warmer T6 dwarf. Cushing et al. (2021) indicate NH 3 features should be present in the infrared at 1.03, 1.21, 1.31, 1.51, 1.66, 1.98, and 2.26 m, but would be blended with stronger H 2 O and CH 4 lines making them difficult to detect. While the features at 1.03, 1.21, and 1.31 m are outside of our wavelength coverage, we do have clear detections of NH 3 at 1.51, 1.98, and 2.26 m using the Bobcat Alternative A models. The ammonia lines in our observed spectra are indeed blended with stronger H 2 O lines, but we are able to detect them nonetheless. We compared Bobcat Alternative A models, with and without NH 3 , and the presence of the NH 3 is clear in the comb-like residuals of Fig. 9.
We also see significant improvement in the reduced 2 statistic when NH 3 is included in the model. We find that the NH 3 at 1.66 m is far too weak to detect amongst the much stronger H 2 O and CH 4 features in this region for an object of this temperature. While NH 3 has been detected in early T dwarfs in the mid-infrared (Roellig et al. 2004;Cushing et al. 2006), 2M0817 is the warmest brown dwarf with individual NH 3 lines detected in the near-infrared.
More recently, Line et al. (2015Line et al. ( , 2017 and Zalesky et al. (2019) constrained the NH 3 abundance for multiple cold brown dwarfs (spectral types T7 and later, including several Y dwarfs) with lowresolution ( < 300 with IRTF/SpeX and HST/WFC3) retrievals. These studies are sensitive to how NH 3 opacities influence the spectroscopic appearance of cold brown dwarfs, but the low-resolution of the observations prevents identification of individual NH 3 lines in the spectra.
Hydrogen Sulfide
We present clear, unambiguous detections of H 2 S in 2M0817. Our most notable detection is a feature at 1.5900 m. This feature is blended with a weak H 2 O line at the same position, so we show our data compared to Bobcat Alternative A models with and without H 2 S in Fig. 10. We see the clear signature of this H 2 S line in the residuals, as well as the presence of other weaker H 2 S lines nearby at 1.5906 m and 1.5912 m.
There is only one other report of a possible H 2 S detection in a brown dwarf in the literature. Saumon et al. (2000) note an H 2 S absorption feature at 2.1084 m in the spectrum of Gliese 229B (spectral type T6.5p). However, we do not confirm this line in our data, nor do our updated models predict any H 2 S lines at this position.
H 2 S has been identified in the giant planets of our Solar System: Irwin et al. (2018) detect H 2 S in the atmosphere of Uranus and Irwin et al. (2019) present a tentative detection of H 2 S in Neptune, both in the 1.57-1.59 m region, the same region in which we have our clearest detection. Detections of H 2 S in Jupiter have also been debated (Noll et al. 1995;Niemann et al. 1998). Our spectrum of 2M0817 exhibits the only convincing detection of H 2 S in an extrasolar atmosphere to date.
We estimated the column density of H 2 S ( H 2 S ) in the atmosphere of 2M0817 to compare against the Irwin et al. (2018Irwin et al. ( , 2019 estimates for Uranus and Neptune. We first determined the pressure ( ) corresponding to the brightness temperature ( ) at the centre of the strongest H 2 S line in the Bobcat Alternative A model spectrum ( = 14 bars and = 1420 K). Then, for each layer in the model atmosphere, we calculated the local number density of all gas molecules using the ideal gas law. We obtained the local number density of H 2 S specifically by multiplying with the H 2 S VMR (H 2 S has an approximately constant equilibrium VMR of of 2.5 × 10 −5 throughout the Bobcat Alternative A model atmosphere). Integrating this H 2 S number density from the pressure of the absorbing layer to the top of the atmosphere gives us the column density of H 2 S, H 2 S ∼ 7.7 × 10 20 cm −2 . This value is 3 to 130 times higher than column amounts determined from retrievals for solar system planets: H 2 S varies between 6 × 10 18 to 4.9 × 10 19 molecules per cm 2 across the disc of Uranus (Irwin et al. 2018), and 9 × 10 18 to 2.8 × 10 20 molecules per cm 2 across the disc of Neptune (Irwin et al. 2019).
The detection of H 2 S also offers tentative evidence of iron rain-out in the atmosphere of 2M0817. Below temperatures of 2300 K, iron is predominantly in the form of metallic droplets that settle to deeper atmospheric layers ( Figure 9. The same layout as Fig. 8, but showing Bobcat Alternative A models with and without NH 3 . Arrows indicate where the model without NH 3 deviates from the data. The NH 3 lines are blended with stronger H 2 O lines, but we see significant improvement in the 2 reduced values when NH 3 is included in the model. Order = 89 of the band has many strong telluric lines, but is still well fit by the models. It is difficult to discern the data from the model containing NH 3 , and the quality of the fit is reflect in the flat residuals and low 2 reduced value.
form of FeS, leaving no sulphur to form H 2 S, meaning that H 2 S would be absent from spectra (Fegley & Lodders 1994;Burrows et al. 2001;Lodders & Fegley 2006). The surface chemical reactions for the conversion of FeS to solid iron and H 2 S are described in Helling et al. (2019). We detect H 2 S in the atmosphere of 2M0817, so iron must be in the process of raining out. Rain-out chemistry is indeed assumed in the Sonora Bobcat and Bobcat Alternative A models.
Molecular Hydrogen
In order = 84 of the band, we identified an absorption feature at 2.12187 m that does not appear in any model of any family. This line is indicated by a black arrow in Fig. 11. We first identified this line through our model deviation analysis (Sec. 4.7), and after correcting for 2M0817's radial velocity (6.1 ± 0.5 km/s, Table 3), we found the position of this line to be 2.12183 ± 0.00005 m (vacuum wavelength). This matches the 2.121834 m wavelength of the molecular hydrogen (H 2 ) 1-0 S(1) transition to six significant figures (Scoville et al. 1983;Roueff et al. 2019). The H 2 1-0 S(1) feature is among the strongest H 2 lines when present in emission in photon-dominated regions, shocks, planetary nebulae, young stellar objects, and starburst galaxies (e.g., Habart et al. 2005). It has never before been detected in absorption in an extra-solar atmosphere, although it has been seen in Jupiter, Saturn, and Neptune (Kim et al. 1995;Trafton et al. 1997). We added the HITRAN 2016 (Gordon et al. 2017) line list for H 2 to the Bobcat Alternative A model and found excellent agreement between order = 84 of our observed spectrum and the model. As we did for our NH 3 and H 2 S identification, we present models with and without H 2 in Fig. 11, and we see the clear signature of the H 2 line in the residuals. We investigated each of the other wavelength regions where the model showed strong H 2 absorption features, in particular the 1-0 S(3) transition (1.957559 m), the 1-0 S(2) transition (2.033758 m), and the strongest line of the -branch: the 1-0 Q(1) transition (2.406592 m; Gautier et al. 1976;Roueff et al. 2019). We were unable to detect additional H 2 features, due to either the much stronger absorption by other molecules (mostly CH 4 ), the lower SNR in other parts of the spectrum, or both.
The first detection of a molecular hydrogen absorption feature in a brown dwarf atmosphere gives a new semi-empirical upper limit on the atmospheric dust concentration in this T6 dwarf. We attain this limit by comparing to the concentration content of the interstellar medium (ISM), where an atomic hydrogen column density of H = 2.2 × 10 21 cm −2 corresponds to a visual extinction of = 1.0 mag, given a 100:1 gas-to-dust ratio (Gorenstein 1975). We expect the ISM gas-to-dust ratio to be orders of magnitude lower than in the upper atmosphere of this brown dwarf, which unlike the ISM is gravitationally differentiated. Silicates (e.g., SiO 3 , MgSiO 3 , Mg 2 SiO 4 ) have ∼38-70 times higher molecular weights than molecular hydrogen, and should have settled mostly below the H 2 -dominated layers of the atmosphere.
The -band extinction in the ISM is such that / = 0.114 (Table 3 of Cardelli et al. 1989), so in the ISM H / = 1.94 × 10 22 cm −2 mag −1 . Virtually all of the hydrogen in the brown dwarf atmosphere is expected to be bound in H 2 (e.g, Burrows et al. 2001), so the projected H 2 column density per magnitude of ISM-likeband extinction is H 2 / = 0.97 × 10 22 cm −2 mag −1 . Following the same prescription as for our calculation of the H 2 S column density (Section 4.5), we obtain that the H 2 column density in the visible atmosphere of 2M0817 is H 2 ∼ 5.1 × 10 24 cm −2 (the centre of the strongest line corresponds to = 4 bars and = 1070 K, and H 2 has an approximately constant equilibrium VMR of 0.836 throughout the Bobcat Alternative A model atmosphere). If the gas-to-dust ratio in the atmosphere of 2M0817 were ISM-like, this H 2 column density would correspond to ∼ 500 mag of extinction. Instead, the H 2 line is readily detectable, so must be less than 1 mag. This implies that the amount of dust in the atmosphere is >500 times less than the interstellar value, and that the atmosphere of 2M018 is almost completely dust-free, as expected for a late-T dwarf.
Finally, the presence of the H 2 line in absorption instead of in emission indicates that the H 2 layer is cooler than the layers underneath it. Hence, the upper atmosphere of 2M0817 does not have a strong thermal inversion, as might otherwise be expected in the presence of hot eddies (Showman et al. 2020).
Shortcomings of the Models and Unidentified Lines
A goal of this work is to identify regions where the photospheric models do not completely reproduce the features in the observed spectra. To identify regions and specific absorption lines in the data which are not well reproduced with the models, we performed two checks. First, we measured the standard deviation, , of the residuals in each order, and then selected regions with at least five consecutive 2.12187 mm Figure 11. The same layout as Fig. 9, but showing Bobcat Alternative A models with and without H 2 absorption (collision-induced absorption from molecular hydrogen and helium is included in both models). This order shows a clear H 2 absorption feature at 2.12187 m, indicated with a black arrow. Order = 84 of the band is very well fit by models, and we see significant improvement in the 2 reduced value for the model including H 2 absorption. pixels more than 2 away from zero. Second, we applied a matched filter to the residuals by convolving the residuals with a template of an inverted absorption feature. For each order, we selected a telluric absorption feature in the reduced A0 V standard's spectrum to use as our filter template. We selected features surrounded by a flat continuum and avoided lines with greater than 65 per cent absorption (the threshold for our continuum mask, Section 2.1). In orders which had very few, or very weak telluric features, we selected a line from a neighbouring order. We then identified regions in the spectra where both the pixel values were outside of two standard deviations, and the matched filter response was higher than the surrounding pixels. This helped to eliminate false detections due to noise. We performed these checks only for the Bobcat Alternative A models, as they are the most up-to-date and provide the best fits to the data. We show an example of this analysis in Fig. 12, and we summarize the regions of interest in Table 5, with a brief description of the potential issue affecting the model in each case. These discrepancies can be seen in Fig. A1 and A2, indicated with with vertical grey dashed lines for individual features, and black brackets for wider discrepant regions. The H 2 feature (Section 4.6) was initially identified through this type of analysis. Most notably, a line is clearly missing from the models at 2.20695 m in order = 81 of the band, shown in Fig. 13. None of the models includes a line at this wavelength, and we have not identified the element or molecule responsible for this feature. Additionally, we find no absorption or emission in the A0 V stars at the wavelengths given in Table 5 which could introduce these unidentified features to our T6 spectrum. The feature in band order =121 does line up with a weak telluric H 2 O feature, but given the difference in the line widths, we believe this discrepancy between the data and model is not caused by the telluric line.
The Bobcat Alternative A models we use to anlayze our data are comprised of the five most abundant molecules (H 2 O, CH 4 , CO, NH 3 , and H 2 S), plus collision-induced absorption from molecular hydrogen and helium. The Sonora Bobcat, Morley, and BT-Settl models consist of more complete sets of molecules. We have confirmed that the lines listed in Table 5 are indeed missing in all families of models. We cannot eliminate all molecules that are included in the more complete Sonora Bobcat, Morley, and BT-Settl model families as being responsible for these missing lines, as the line lists could be incomplete or inaccurate, or there could be disequilibrium chemistry taking place, as we observed with CO (Section 4.3). We also confirmed that H 2 is not responsible for any of the unidentified lines.
Disequilibrium chemistry could imply that other mixing-sensitive gases such as phosphine (PH 3 ; the next most abundant molecule in these cold atmospheres) could also be present at higher abundances than expected for chemical equilibrium (Fegley & Lodders 1996). We generated a Bobcat Alternative A model with a greatly overestimated abundance of PH 3 (VMR of 1 × 10 −4 , which is more than 300 times the amount expected for equilibrium chemistry, and would require far more phosphorus than would be available in a solar-composition atmosphere) to compare to our spectra, intending to match the locations of the PH 3 features to the unidentified lines. We use the SAlTY line list from Exomol (Sousa-Silva et al. 2015) for PH 3 . We found that the PH 3 features did not match with any of the unidentified lines, and PH 3 is likely not responsible for these features. A study by Miles et al. (2020) searched for PH 3 in atmospheres of cold brown dwarfs displaying disequilibrium CO absorption. This study was performed in the and bands (centred at 3.45 m and 4.75 m, respectively), where H 2 O, CH 4 , and NH 3 absorb less strongly, but PH 3 absorbs much more strongly, and so should give the best chance at detecting PH 3 . Unfortunately, they were also unable to detect PH 3 .
Among the list of unidentified lines in Table 5, we list nearly the full wavelength coverage of orders = 113 through = 107 of the band. These orders cover 1.596-1.681 m and the dominant absorber in these orders is CH 4 . As discussed in Section 4.2, while the strongest absorption features are very well modelled in the Bobcat Alternative A models, the weaker features and continua in the models deviate significantly from the observations. We observe some bumpiness in the residuals throughout the entire IGRINS wavelength coverage, especially in these CH 4 -dominant regions of the -band. In these CH 4 regions the model accurately represents the deepest features, but appears to be incorrect or incomplete in the weaker features and continuum. Given the accuracy of the H 2 O lines elsewhere in the spectrum, we suspect these discrepancies are due to weak CH 4 lines, and not H 2 O.
There are a host of weaker features that are not being taken into account in the models, and these features contribute a non-trivial amount to the atmospheric opacity.
LESSONS LEARNED
We find that atmospheric models that use state-of-the-art line lists represent observations well. We are now able to extract more precise information from our data than merely detect the most abundant molecules: we can detect absorption by trace species that have never been seen before (like H 2 S and H 2 ), see low abundance species, and more readily detect abundances of species (as we have done for CO here). In all cases we recommend using the most-up-to-date models available with the most recent molecular line lists. We have found that the line lists used in the Bobcat Alternative A models (Table 4) give the most reliable and consistent estimates of all physical parameters across all wavelength regions of this study. More generally, we have found that all models do an adequate job fitting the data in regions where H 2 O is the dominant absorber.
We summarize our main recommendations and warnings, organized by the information of interest in the following two subsections.
Fitted Spectroscopic Parameters
Effective Temperature ( eff ) measurements are most accurate and consistent in the band, in regions where H 2 O is the dominant opacity source. We recommend using the Bobcat Alternative A models for measuring eff anywhere in the and bands. The Morley models over-estimate eff in the band and the Sonora Bobcat models over-estimate eff in the band, while The BT-Settl models underestimate eff in the band. If disequilibrium chemistry effects are not taken into consideration, eff may also be over-estimated.
Surface Gravity (log ) measurements are the most accurate and consistent in regions where H 2 O is the dominant opacity source in the band. We recommend using the Bobcat Alternative A models for measuring log anywhere in the and bands. The Sonora Bobcat models over-estimate log in the band. log may also be over estimated in regions where the dominant molecule switches from H 2 O to CH 4 , near the peaks of and bands (1.6 m and 2.1 m, respectively). Projected Rotation Velocity ( sin ) measurements are the most accurate and consistent in regions where H 2 O is the dominant opacity source in both the and bands. We recommend using the Bobcat Alternative A models for measuring sin anywhere in the and bands. We recommend using the region from 1.45 to 1.57 m in the band, or 1.89 to 2.10 m in the band if measuring sin with any other model. sin may be over estimated in regions where the dominant molecule switches from H 2 O to CH 4 , near the peaks of and bands (1.6 m and 2.1 m, respectively).
Radial Velocity (RV) measurements are the most accurate and consistent in regions where H 2 O is the dominant opacity source in both the and bands. We recommend using the Bobcat Alternative A models for measuring RV anywhere in the and bands. We recommend using the region from 1.45 to 1.58 m in the band if measuring RV with any other model. RV measurements demonstrate a blueshift with wavelength when measured from the Sonora Bobcat and BT-Settl models between 1.894 and 2.060 m. Methane (CH 4 ) is the dominant opacity source between 1.60 and 1.73 m in the band, and between 2.10 and 2.48 m in the band. The CH 4 -dominant region of the band (2.10-2.48 m) gives consistent results for all parameters for the Bobcat Alternative A models. Weak CH 4 lines between 1.59 and 1.67 m are poorly matched to data in all model families and in all line lists. We recommend using the HITEMP (Hargreaves et al. 2020) Figure 12. An example of the analysis done on the residuals to identify discrepancies between the models and data. The top panel shows the IGRINS spectrum with the best-fitting Bobcat Alternative A model for this order. The middle panel shows the residuals on the same y-scale as the top panel. Horizontal blue lines delineate 2 threshold, and regions with more than five consecutive pixels beyond 2 are highlighted with green. The filter response of a matched filter using a clean telluric line surrounded by a flat continuum is shown in the bottom panel. There is a clear outlier region at 1.52094 m flagged by both the residuals analysis, and also giving a high filter response. Other regions with a high filter response (e.g., 1.51714 m and 1.52602 m) don't meet our residuals criteria, and are therefore more likely due to noise in the data. The dominant absorber in order = 118 of the band is H 2 O, which also appears to be the molecule responsible for the flagged feature. Figure 13. The same layout as Fig. 11, but showing the best-fitting Bobcat Alternative A model for this order. Order = 81 of the band is fit well by models, and shows an unknown absorption feature that doesn't appear in any model of any family at 2.20690 m. This line is indicated with a black arrow.
Specific Molecules
Water (H
The strongest ammonia (NH 3 ) features occur between 1.50 and 1.52 m in the band, and 1.95 to 2.09 m and 2.18 to 2.21 m in the band. The choice of NH 3 line list does not appear to significantly impact the measured parameters, and we recommend using the ExoMol/CoYuTe (Coles et al. 2019) line list when studying NH 3 .
The strongest hydrogen sulfide (H 2 S) features occur in the band between 1.58 and 1.60 m. The choice of H 2 S line list does not appear to impact the measured parameters, and we recommend using the combinations of line lists from ExoMol (Tennyson & Yurchenko 2012), Azzam et al. (2015), and HITRAN 2012 (Rothman et al. 2013), or the updated versions of HITRAN from 2016 and 2020 (Gordon et al. 2017(Gordon et al. , 2022, which we have not tested here.
APPLICATIONS TO EXOPLANETS
In high-dispersion spectroscopic observations of exoplanets, where the planet itself cannot be spatially resolved, cross-correlation is a powerful technique for detecting and characterizing the planet. In addition to the identification of specific molecules, the velocity relative to the host star, information about planetary spin ( sin ) and atmospheric wind speeds may be determined (e.g., Snellen et al. 2010Snellen et al. , 2014. However, when an observed spectrum combines the star and planet, individual lines from the planet can have SNR 1, and the ability to recover a planet is only as good as the model. If fitting an incorrect model to a low SNR spectrum, the planet may not be recovered, or even discovered.
We have confirmed that the older CH 4 line lists are inaccurate in the 1.60-1.73 m and 2.10-2.40 m regions (see Section 4.2), and the inaccurate line positions could result in a non-detection of an exoplanet. We cross-correlated our data against the various models to showcase the improvement that the newer CH 4 lines offer in a cross-correlation analysis. We used the IDL function c_correlate and included only the CH 4 -dominated orders of the and bands ( = 112-104, 1.594-1.730 m; = 85-77, 2.084-2.294 m). The model parameters were fixed to the values given in Table 3. The results of the cross-correlations are shown in Fig. 14.
To isolate the effects of the choice of CH 4 line list our crosscorrelation analysis includes two different Bobcat Alternative A models: one with the updated CH 4 line lists and one with the older CH 4 line lists used in the Sonora Bobcat models, while keeping the H 2 O, CO, NH 3 , and H 2 S line lists the same as in the newest models (see Table 4). As seen in the left panel of Fig. 14, the peak of the Bobcat Alternative A model with the newest CH 4 line lists is the highest. The peak for the Bobcat Alternative A model with the older CH 4 line lists is significantly lower. In fact, the latter is nearly identical to the cross-correlation peak for the Sonora Bobcat model. This is as expected, as these two models use the same underlying atmospheric model, and we showed previously that the choice of line lists for the other molecules has less impact on the quality of the model fits to the data. We see that the peak of the Bobcat Alternative A model with the older CH 4 line lists is slightly higher than the peak of the Sonora Bobcat model, likely due to the newer line lists used for the other molecules, particularly H 2 O. We also see that the CH 4 line lists used in the Sonora Bobcat models result in an offset in the peak towards negative velocities. This is consistent with the lower radial velocities we found for the Sonora Bobcat model in orders where CH 4 was the main absorber ( Fig. 3 and 4).
The importance of the choice of CH 4 line lists in the methanedominated IGRINS orders is demonstrated in the right panel of Fig. 14. That figure compares the cross-correlation functions of our data with versions of the models that include only methane absorption and ignore the contributions from H 2 O, CO, NH 3 , and H 2 S. The 'Updated CH 4 only' models uses the line list incorporated in the Bobcat Alternative A models, while the 'Older CH 4 only' uses the line lists incorporated in the Sonora Bobcat models. The 'updated' and 'older' cross-correlations are very similar in shape and peak strengths to the Bobcat Alternative A model and Sonora Bobcat cross-correlations, respectively. This is not surprising, as we have performed the cross-correlation specifically for the CH 4 -dominated orders, and so models which include additional molecules provide only a small improvement.
In their analysis of the -band spectra of HD 209458 b and Pictoris b, Snellen et al. (2010Snellen et al. ( , 2014 were successful in detecting CO through cross-correlation with atmospheric models, but failed to recover CH 4 . Inaccurate line lists could be responsible for these non-detections, as these studies used the older HITRAN'08 (Rothman et al. 2009) for their CH 4 line lists. The atmosphere of Pictoris b may also be too hot for the detection of CH 4 ( eff = 1724 K; Chilcote et al. 2017), but CH 4 may be detectable in HD 209458 b ( eq = 1449 K; Torres et al. 2008) with an improved line list. Indeed, more recently, Guilluy et al. (2019) and Giacobbe et al. (2021)
CONCLUSIONS
The data presented here are among the highest resolution spectra ever published for a cold brown dwarf. We found that model spectra with the most recent line lists showed significant improvement in fitting the observed spectrum of the T6 dwarf 2MASS J08173001−6155158. The updated line lists for water, methane, and ammonia allow for precise empirical determinations of physical parameters. We identified the most reliable regions for measuring physical parameters of cold brown dwarfs, and we summarized our findings in Section 5. In particular, we highlighted the excellent fits of the Sonora Bobcat Alternative A models (Hood et al. in preparation), and the accuracy of the ExoMol/POKAZATEL line list for H 2 O (Polyansky et al. 2018) and the HITEMP line list for CH 4 (Hargreaves et al. 2020) in matching the observed spectrum. We confirmed that like other late-type T and Y dwarfs, 2M0817 demonstrates CO disequilibrium chemistry. We identified individual NH 3 lines in the spectrum of 2M0817, and we presented the first unambiguous detections of H 2 S and H 2 absorption in an extra-solar atmosphere. Our molecular hydrogen detection allowed us to place a semi-empirical upper limit on the atmospheric dust concentration of this brown dwarf. We found that the atmosphere of 2M0817 has >500 times less dust than the ISM, implying that the atmosphere is almost completely dust-free. Additionally, we identified several spectroscopic features that are missing from, or are poorly fit by the models. Finally, our crosscorrelation analysis showed that the most up-to-date line lists are significantly more sensitive to CH 4 absorption in the atmosphere of this T6 dwarf. This will improve the detectability of CH 4 and other atmospheric absorbers in more challenging observations, such as the high-dispersion, low-SNR, high-background spectra of exoplanets around their host stars. Table 4). We performed this cross correlation across the CH 4 -dominated orders of both the and bands for models with parameters given in Table 3. As expected, the peak of the Bobcat Alternative A model that uses the newest CH 4 line lists is higher than for the other models. Right panel: The same as the left panel, but for Bobcat Alternative A models that incorporate only CH 4 . The 'Updated CH 4 -only' model uses the line list used in the new Bobcat Alternative A model, while the 'Older CH 4 -only' model uses the line list used in the Sonora Bobcat model. Support for this work was provided by an Ontario Graduate Scholarship, NSERC, and the Canadian Space Agency Flights and Fieldwork for the Advancement of Science and Technology (FAST) funding initiative. Part of the work for this paper was completed at the Other Worlds Laboratory Exoplanet Summer Program 2019. We thank the Heising-Simons Foundation and the University of California Santa Cruz for funding this program.
This paper contains data based on observations obtained at the international Gemini Observatory (Program ID GS-2018A-Q-304), a program of NSF's NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation on behalf of the Gemini Observatory partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigación y Desarrollo (Chile), Ministerio de Ciencia, Tecnología e Innovación (Argentina), Ministério da Ciência, Tecnologia, Inovações e Comunicações (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea). We also thank Gemini Observatory for the opportunity for the first author to visit the Gemini South facility in La Serena, Chile through their 'Bring One, Get One Student Observer Support Program.'
This work used the Immersion Grating Infrared Spectrometer (IGRINS) that was developed under a collaboration between the University of Texas at Austin and the Korea Astronomy and Space Science Institute (KASI) with the financial support of the US National Science Foundation under grants AST-1229522 and AST-1702267, of the University of Texas at Austin, and of the Korean GMT Project of KASI. We thank the IGRINS team for their continued support throughout this project, the opportunity to observe in person with IGRINS under their guidance, and for including 2MASS J08173001−6155158 as a commissioning target, allowing us to obtain valuable additional data.
DATA AVAILABILITY
The Table 5 are indicated in these figures with black arrows for discrepant lines, and black brackets for discrepant regions. The bottom panel shows the PSG Earth's transmittance to help assess the telluric lines in our spectra. The OH emission lines are also shown as boxes and indicate position only, not line strength. Wider boxes indicate blended OH emission lines. A dashed horizontal line indicates the 65 per cent absorption threshold used for our telluric mask. The Δ 2 reduced is the difference between the displayed order's 2 reduced , and the 2 reduced of the best fitting order ( = 85, Fig. A2). This figure continues for many pages, with two orders per page, to show all 26 orders of the band.
Figure 1 .
1A sample order near the centre of the band, showing spectra from each of the six observing epochs. The normalized flux is shown in the top panel, and the deep absorption features in this order are due to H 2 O. The SNR is shown in the bottom panel.
Figure 3 .
3Results of the model fitting for the band. Top panel: The Bobcat Alternative A model spectra of each major molecule with collision-induced absorption from molecular hydrogen and helium included. The Model Flux (the y-axis of the top panel) is what would be measured at the surface of the object.
Figure 4
4Figure 4. The same layout as Fig. 3, but for the band. The log value is extremely consistent for the Bobcat Alternative A models, with log = 5.0 in every order of the band. The standard deviation on this weighted average is therefore zero (see Section 3.2 for details on this calculation). In Table 3 we compute the weighted average and standard deviation based on both the and bands, so the standard deviation is non-zero for the final adopted value.
Figure 5 .
5In order = 85 of the band, all models are very well matched to the data. The dominant absorbers in this region are H 2 O and CH 4 . The top panel shows the Bobcat Alternative A model spectra including opacity from one major molecule at a time, in addition to H 2 /He collision-induced absorption. The middle panel shows the IGRINS data (black; uncertainty shown in grey) with the best fitting models from each model family. The Bobcat Alternative A model spectra in the top panel have the same eff and log values as the best fitting Bobcat Alternative A model, are broadened to the same sin , and have the same RV shift applied. The bottom panel shows the residuals (data -model) on the same vertical scale as the middle panel, with the same colour scheme. The data and residuals contain gaps in the plot where strong telluric lines have been masked out. The Δ 2 reduced statistic is the difference between the 2 reduced of the model for the current order and the 2 reduced = 1.0 of the best-fitting model (Bobcat Alternative A model) for order = 85. The degrees of freedom (d.o.f.) for the 2 reduced for each model are also shown.
Figure 6 .
6The same layout asFig. 5, but now showing order = 111 of the band, an order with a poor fit. The dominant absorber in this order is CH 4 . The most up-to-date line lists for CH 4(Hargreaves et al. 2020, used in the Bobcat Alternative A model) provide accurate wavelengths for the deepest lines, but the weaker features in the continuum (likely CH 4 blended with H 2 O) are poorly-fit.Table 3. Properties of 2MASS J08173001−6155158. Parameters estimated from the spectra presented in this paper are based on all and band orders.Artigau et al. (2010). This work. The units of are cm s −2 .
Figure 7 .
7The improvement made with the newer CH 4 line lists is most apparent in order = 110 of the band. The top panel is the same asFig. 5: the Bobcat Alternative A model spectra of each major molecule with H 2 /He collision-induced absorption are shown. Here the middle panel shows the IGRINS data (black), the Sonora Bobcat model (light blue), and the Bobcat Alternative A model (gold). The models have identical physical parameters, rotational broadening ( sin ), and RV shift. The bottom panel shows the residuals (data -model) on the same vertical scale, with the same colour scheme. The deepest features are CH 4 , and the weaker features in the continuum are mainly CH 4 or CH 4 blended with H 2 O. The Bobcat Alternative A model shows excellent agreement with the data in the major features, while the Sonora Bobcat model appears to have a stretch causing misalignment in the major features when compared to the IGRINS data.
Figure 8 .
8Order = 77 of the band is dominated by CH 4 , but CO and H 2 O are also important absorbers. Here the top panel is similar toFig. 5, with an extra line for a model with an increased CO abundance (CO volume mixing ratio of 3 × 10 −4 , labelled 'Extra CO'). The middle panel shows the observed spectrum with two versions of a Bobcat Alternative A model, one with the CO as estimated by chemical equilibrium (labelled 'Original') and one with an increased abundance of CO (labelled 'Extra CO'). The bottom panel shows the residuals for the two models. The CO strength in particular is important to improve the accuracy of the models in the long end of the band. It is clear that the depth of the features in the 'Original' model is too weak at the positions of the CO features, implying that vertical mixing must be taking place in this atmosphere.
Figure 10 .
10The same layout asFig. 9, but showing Bobcat Alternative A models with and without H 2 S. This order shows a clear H 2 S detection at 1.5900 m. Order = 113 of the band is well fit by models. The H 2 S line of interest is blended with an H 2 O line, but we see the impact of H 2 S in the residuals, and improvement in the 2 reduced value for the model including H 2 S.
2 O) is the dominant opacity source between 1.45 and 1.58 m in the band, and between 1.89 and 2.10 m in the band. The H 2 O-dominant region of the band (1.45-1.58 m) gives consistent results for all parameters across all model families. We recommend using the ExoMol/POKAZATEL (Polyansky et al. 2018) line list when studying water.
had success detecting CH 4 from HD 102195 b and HD 209458 b, respectively, with more up-to-date line lists. Guilluy et al. (2019) used HITRAN2012 (Rothman et al. 2013), and Giacobbe et al. (2021) used Hargreaves et al. (2020), the same CH 4 line list used in the Bobcat Alternative A models. A separate demonstration of the enhanced utility of the newer Hargreaves et al. (2020) CH 4 line lists is evident in Line et al. (2021), who determined the C/H, O/H, and C/O ratios of the hot Jupiter WASP-77Ab ( eff = 1740 K; Maxted et al. 2013) using cross-correlation methods with IGRINS data.
Figure 14 .
14Left panel: Cross-correlations of the data with each of the model families, including a version of the Bobcat Alternative A model modified to use the older CH 4 line list (the other molecules use the newer line lists given in
raw IGRINS data for 2MASS J08173001−6155158 are available on the Gemini Archive under Program ID GS-2018A-Q-304. The reduced spectrum and best-fitting Bobcat Alternative A spectra are available through the Harvard Dataverse (https://doi.org/ 10.7910/DVN/DV1ZLR) or Zenodo (https://doi.org/10.5281/ zenodo.6082001).
Figure A1 .
A1Every order of the band. The model shown is the best fitting Bobcat Alternative A model for each order. Each order is fit independently (Section 3), so the physical parameters may differ between orders. The top panel shows the molecule-by-molecule breakdown of the model. The second panel from the top shows the IGRINS data with the full model. The second panel from the bottom shows the residuals on the same -scale as the panel above it. The model discrepancies listed in
Figure A1 .
A1Continued
Figure A2 .
A2The same asFig. A1, but for the band. This figure continues for many pages, with two orders per page, to show all 23 orders of the band.
Figure A2 .
A2Continued
Figure A2 .
A2Continued
Figure A2 .
A2Continued
Figure A2 .
A2Continued
Figure A2 .
A2Continued.
Table 1 .
1Gemini South/IGRINS Spectroscopic Observations of 2MASS J08173001−6155158 under Gemini program ID GS-2018A-Q-304 (PI: M. Tannock).The given SNR values are for the final, combined spectra. The FWHM of the trace includes both atmospheric seeing and effects from telescope and instrument optics. Typical atmospheric seeing at Gemini South is 0.5 .Date
Exposure
Exposure
Target
Telluric
Telluric
-band
-band
FWHM of the
Observed
Time
Sequence
Airmass
A0 V
Standard
SNR (at
SNR (at
Trace in the
(s)
Standard
Airmass
1.589 m) 2.101 m)
Band ( )
2018 Apr 5
1200
AB
1.18-1.20 HIP 40621
1.14
85
44
0.9
2018 Apr 5
600
ABBA
1.20-1.25 HIP 35393
1.28
63
34
0.9
2018 May 7
600
ABBA
1.22-1.28 HIP 36489
1.31
56
33
0.8
2018 May 22
518
ABBAAB 1.27-1.37 HIP 40621
1.36
174
112
0.6
2018 May 22
518
ABBAAB 1.40-1.58 HIP 40621
1.58
181
113
0.6
2018 May 23
518
ABBAAB 1.26-1.37 HIP 40621
1.35
184
119
0.6
Data from every epoch for H band, Order m = 114
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Normalized Flux
1.565
1.570
1.575
1.580
Wavelength (mm)
Only a selection of opacities, that dominate at near infrared wavelengths, are included: H 2 O, CH 4 , CO, NH 3 , H 2 S, and collision-induced opacity of H 2 -H 2 , H 2 -He, and H 2 -CH 4 . The opacity data for these sources are the same as the Sonora Bobcat models, with the notable exceptions of updated H 2 O (Polyansky et al. 2018), CH 4 (Hargreaves et al. 2020), and NH 3 (Coles et al. 2019) line lists. Full H-and K-band IGRINS spectra of 2MASS J08173001-61551581.5
1.6
1.7
1.8
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Normalized Flux
1.9
2.0
2.1
2.2
2.3
2.4
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Normalized Flux
2MASS 0817 Data
2MASS 0817 Uncertainty
Table 2 .
2Thewavelengths of the IGRINS orders and the major molecular absorbers in each order. Diffraction order numbers, , were extrapolated from Stahl
et al. (2021).
Band Order
Wavelength
Major
Band Order
Wavelength
Major
( )
Coverage ( m) Absorbers
( )
Coverage ( m) Absorbers
124
1.454-1.460
H 2 O
94
1.894-1.910
H 2 O
123
1.459-1.470
H 2 O
93
1.909-1.930
H 2 O
122
1.469-1.483
H 2 O
92
1.929-1.950
H 2 O
121
1.482-1.494
H 2 O
91
1.949-1.972
H 2 O, NH 3
120
1.493-1.506
H 2 O
90
1.971-1.993
H 2 O, NH 3
119
1.504-1.519
H 2 O, NH 3
89
1.992-2.015
H 2 O, NH 3
118
1.517-1.531
H 2 O
88
2.014-2.038
H 2 O, NH 3
117
1.529-1.543
H 2 O
87
2.037-2.061
H 2 O, NH 3
116
1.541-1.556
H 2 O
86
2.060-2.085
H 2 O, CH 4 , NH 3
115
1.554-1.569
H 2 O, CO
85
2.084-2.109
H 2 O, CH 4
114
1.567-1.583
H 2 O
84
2.108-2.134
H 2 O, CH 4 , H 2
113
1.581-1.596
H 2 O, CH 4 , H 2 S
83
2.133-2.159
H 2 O, CH 4
112
1.594-1.610
H 2 O, CH 4
82
. 2.158-2.185
CH 4
111
1.608-1.624
H 2 O, CH 4
81
2.184-2.212
CH 4 , NH 3
110
1.622-1.639
H 2 O, CH 4
80
2.211-2.239
CH 4
109
1.637-1.653
H 2 O, CH 4
79
2.238-2.267
CH 4
108
1.651-1.668
H 2 O, CH 4
78
2.266-2.295
H 2 O, CH 4
107
1.666-1.683
H 2 O, CH 4
77
2.294-2.326
H 2 O, CH 4 , CO
106
1.681-1.699
H 2 O, CH 4
76
2.325-2.355
H 2 O, CH 4 , CO
105
1.697-1.715
H 2 O, CH 4
75
2.354-2.383
H 2 O, CH 4 , CO
104
1.713-1.730
H 2 O, CH 4
74
2.389-2.414
H 2 O, CH 4 , CO
103
1.728-1.747
H 2 O, CH 4
73
2.420-2.445
H 2 O, CH 4 , CO
102
1.745-1.764
H 2 O, CH 4
72
2.452-2.478
H 2 O, CH 4
101
1.762-1.781
H 2 O, CH 4
100
1.779-1.798
H 2 O, CH 4
99
1.797-1.812
H 2 O, CH 4
An order with a good fit for all model families: K band, Order m=85
0
2.0×10
11
4.0×10
11
6.0×10
11
8.0×10
11
1.0×10
12
1.2×10
12
Model Flux (W/m
2
/m)
0.0
0.2
0.4
0.6
0.8
1.0
Normalized Flux
2.085
2.090
2.095
2.100
2.105
Wavelength (mm)
-0.4
-0.2
0.0
0.2
0.4
Residuals
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Sonora
Bobcat
BT-Settl
Morley
Bobcat
Alt A
Dc
2
reduced =8.8
Dc
2
reduced =5.4
Dc
2
reduced =8.1
Dc
2
reduced =0.0
d.o.f.=1715
Table 4 .
4Literature references for the line lists for each of the model photosphere families. For information about specific isotopologues, line widths, and how these sources are combined for each family of models please see the original works listed in the column headers.Inaccuracies in the CH 4 line lists in the H band, Order m=110Molecule Bobcat Alternative A (Hood et
al. in preparation)
Sonora Bobcat (Marley et al.
2021)
Morley (Morley et al. 2012)
BT-Settl (Allard et al. 2012,
2014)
H 2 O
ExoMol/POKAZATEL
(Polyansky et al. 2018);
BT2 (Barber et al. 2006)
Tennyson & Yurchenko (2018);
BT2 (Barber et al. 2006)
Partridge & Schwenke (1997);
HITRAN'08 (Rothman et al.
2009)
BT2 (Barber et al. 2006)
CH 4
HITEMP (Hargreaves et al.
2020)
Yurchenko et al. (2013); Exo-
mol/10to10 (Yurchenko & Ten-
nyson 2014); Spherical Top
Data System (Wenger & Cham-
pion 1998)
Spherical Top Data System
(Wenger & Champion 1998);
HITRAN'08 (Rothman et al.
2009); Strong et al. (1993)
Spherical Top Data System
(Wenger & Champion 1998)
CO
HITEMP 2010 (Rothman et al.
2010); Li et al. (2015)
HITEMP 2010 (Rothman et al.
2010); Li et al. (2015)
Goorvitch (1994); R. Tipping
(1993, private communication);
HITRAN'08 (Rothman et al.
2009)
Goorvitch (1994)
NH 3
ExoMol/CoYuTe (Coles et al.
2019)
BYTe (Yurchenko et al. 2011)
BYTe (Yurchenko et al. 2011)
Sharp & Burrows (2007)
H 2 S
ExoMol
(Tennyson
&
Yurchenko 2012); Azzam
et al. (2015); HITRAN 2012
(Rothman et al. 2013)
ExoMol
(Tennyson
&
Yurchenko 2012); Azzam
et al. (2015); HITRAN 2012
(Rothman et al. 2013)
R. Wattson (1996, private
communication); HITRAN'08
(Rothman et al. 2009)
HITRAN 2004 (Rothman et al.
2005)
0
1×10
12
2×10
12
3×10
12
4×10
12
5×10
12
6×10
12
Model Flux (W/m
2
/m)
0.0
0.5
1.0
1.5
Normalized Flux
1.625
1.630
1.635
Wavelength (mm)
-0.5
0.0
0.5
Residuals
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Bobcat
Alt A
Sonora
Bobcat
Model
parameters:
T eff =1100 K
log(g)=5.5
v sin(i)=
22.0 km/s
RV=6.0 km/s
Dc
2
reduced =270
Dc
2
reduced =35
d.o.f.=1461
Fegley & Lodders 1994; Burrows & Sharp 1999). At temperatures 750 K, any iron in the atmosphere would take theCO disequilibrium chemistry in the K band, Order m=77
0
2×10
11
4×10
11
6×10
11
Model Flux (W/m
2
/m)
0.0
0.5
1.0
1.5
2.0
Normalized Flux
2.295
2.300
2.305
2.310
2.315
2.320
2.325
Wavelength (mm)
-1.0
-0.5
0.0
0.5
1.0
Residuals
H 2 O
CH 4
CO
Extra CO
NH 3
H 2 S
Data
Uncert
Original
Extra CO
Model
parameters:
T eff =1000 K
log(g)=5.0
v sin(i)=
21.8km/s
RV=6.2km/s
Dc
2
reduced =5.3
Dc
2
reduced =1.6
d.o.f.=2001
Table 5 .
5The wavelengths of discrepancies in the models and unidentified absorption features. These regions are identified inFig. A1and A2, with vertical grey dashed lines for individual features, and black brackets for regions.Band Order Wavelength
Notes
( )
( m)
121
1.48463
Potential issue with blended line or
something missing in the model
118
1.52090
A line appears in the model which
is missing in the data (see Fig. 12).
The model line appears to be a wa-
ter/ammonia blend.
116
1.55120
Potential issue with blended line
(H 2 O and NH 3 ?)
115
1.56396
Potential issue with blended line
(H 2 O and NH 3 or H 2 O and H 2 S?)
113
1.5875-1.5960 CH 4 region a
112
1.5980-1.6090 CH 4 region a
111
1.6080-1.6240 CH 4 region a
110
1.6244-1.6390 CH 4 region a
109
1.6375-1.6510 CH 4 region a
108
1.6515-1.6650 CH 4 region a
108
1.65355
Potential issue with blended line
(CH 4 and H 2 O?)
108
1.65446
Model over-estimates flux
108
1.66319
Model over-estimates flux
107
1.6675-1.6810 CH 4 region a
107
1.66960
Line too weak in model
107
1.67380
Model under-estimates flux
106
1.68443
Potential issue with blended line
(CH 4 and H 2 O?)
106
1.68672
Potential issue with blended line
(CH 4 and H 2 O?)
106
1.69600
Potential issue with blended line
(CH 4 and H 2 O?)
87
2.04020
Model over-estimates flux
87
2.05478
Model under-estimates flux
81
2.20690
Line missing from model (see
Fig. 13)
line list when studying CH 4 . To measure accurate and consistent parameters, especially eff , disequilibrium chemistry may need to be considered for CO. We recommend using the HITEMP 2010(Rothman et al. 2010) line list when studying CO.Carbon monoxide (CO) bands occur between 1.55 and 1.57 m
in the
band, and 2.29 to 2.45 m in the
band. Residuals Analysis for H band, Order m=118
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Normalized Flux
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Residuals
1.518
1.520
1.522
1.524
1.526
1.528
1.530
Wavelength (mm)
0.0
0.1
0.2
0.3
0.4
0.5
(Filter Response)
2
Data
Model
2 s
Flagged
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
22.2 km/s
RV=6.6 km/s
Dc
2
reduced =2.5
d.o.f.=1359
1.52090 mm
Unidentified absorption feature in the K band, Order m=810
2.0×10
11
4.0×10
11
6.0×10
11
8.0×10
11
1.0×10
12
1.2×10
12
1.4×10
12
Model Flux (W/m
2
/m)
0.0
0.5
1.0
1.5
2.0
Normalized Flux
2.185
2.190
2.195
2.200
2.205
2.210
Wavelength (mm)
-1.0
-0.5
0.0
0.5
1.0
Residuals
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
22.0km/s
RV=5.6km/s
Dc
2
reduced =3.6
d.o.f.=1866
2.20690 mm
reduced =17.1 d.o.f.=1368Figure A1. Continued.Figure A1. Continued.Figure A1. Continued.Figure A1. Continued.Figure A1. Continued. reduced =4.1 d.o.f.=844 Figure A1. Continued.1.506
1.508
1.510
1.512
1.514
1.516
1.518
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
21.8 km/s
RV=6.4 km/s
Dc
2
reduced =2.4
d.o.f.=1420
Figure A1. Continued.
H band, Order m=118
0
2.0×10
12
4.0×10
12
6.0×10
12
8.0×10
12
1.0×10
13
1.2×10
13
Model Flux (W/m
2
/m)
0.0
0.5
1.0
1.5
Normalized Flux
-0.5
0.0
0.5
Residuals
1.518
1.520
1.522
1.524
1.526
1.528
1.530
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
22.2 km/s
RV=6.6 km/s
Dc
2
reduced =2.5
d.o.f.=1359
H band, Order m=117
0
2.0×10
12
4.0×10
12
6.0×10
12
8.0×10
12
1.0×10
13
1.2×10
13
Model Flux (W/m
2
/m)
0.0
0.5
1.0
1.5
Normalized Flux
-0.5
0.0
0.5
Residuals
1.530
1.532
1.534
1.536
1.538
1.540
1.542
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
22.0 km/s
RV=6.4 km/s
Dc
2
reduced =1.3
d.o.f.=1347
Figure A1. Continued.
H band, Order m=116
0
2×10
12
4×10
12
6×10
12
8×10
12
1×10
13
Model Flux (W/m
2
/m)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Normalized Flux
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Residuals
1.542
1.544
1.546
1.548
1.550
1.552
1.554
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
22.2 km/s
RV=6.6 km/s
Dc
2
reduced =1.4
d.o.f.=1348
H band, Order m=115
0
2×10
12
4×10
12
6×10
12
Model Flux (W/m
2
/m)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Normalized Flux
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Residuals
1.556
1.558
1.560
1.562
1.564
1.566
1.568
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1000 K
log(g)=5.0
v sin(i)=
22.2 km/s
RV=6.2 km/s
Dc
2
reduced =2.1
d.o.f.=1349
Figure A1. Continued.
H band, Order m=114
0
2×10
12
4×10
12
6×10
12
Model Flux (W/m
2
/m)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Normalized Flux
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Residuals
1.570
1.575
1.580
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1000 K
log(g)=5.0
v sin(i)=
22.8 km/s
RV=6.6 km/s
Dc
2
reduced =2.2
d.o.f.=1474
H band, Order m=113
0
2×10
12
4×10
12
6×10
12
8×10
12
Model Flux (W/m
2
/m)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Normalized Flux
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Residuals
1.582
1.584
1.586
1.588
1.590
1.592
1.594
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1000 K
log(g)=4.5
v sin(i)=
27.2 km/s
RV=6.6 km/s
Dc
2
reduced =2.3
d.o.f.=1349
Figure A1. Continued.
H band, Order m=112
0
2×10
12
4×10
12
6×10
12
8×10
12
1×10
13
Model Flux (W/m
2
/m)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Normalized Flux
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Residuals
1.595
1.600
1.605
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
27.2 km/s
RV=6.6 km/s
Dc
2
reduced =8.4
d.o.f.=1462
H band, Order m=111
0
2×10
12
4×10
12
6×10
12
8×10
12
Model Flux (W/m
2
/m)
0.0
0.5
1.0
1.5
Normalized Flux
-0.5
0.0
0.5
Residuals
1.610
1.615
1.620
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
23.0 km/s
RV=4.2 km/s
Dc
2
reduced =21.2
d.o.f.=1439
Figure A1. Continued.
H band, Order m=110
0
1×10
12
2×10
12
3×10
12
4×10
12
5×10
12
6×10
12
Model Flux (W/m
2
/m)
0.0
0.5
1.0
1.5
Normalized Flux
-0.5
0.0
0.5
Residuals
1.625
1.630
1.635
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.5
v sin(i)=
22.0 km/s
RV=6.0 km/s
Dc
2
reduced =35.3
d.o.f.=1461
H band, Order m=109
0
2×10
12
4×10
12
6×10
12
Model Flux (W/m
2
/m)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Normalized Flux
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Residuals
1.640
1.645
1.650
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
22.8 km/s
RV=6.6 km/s
Dc
2
H band, Order m=108
0
1×10
12
2×10
12
3×10
12
4×10
12
5×10
12
Model Flux (W/m
2
/m)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Normalized Flux
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Residuals
1.655
1.660
1.665
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.5
v sin(i)=
19.6 km/s
RV=5.0 km/s
Dc
2
reduced =7.2
d.o.f.=1487
H band, Order m=107
0
1×10
12
2×10
12
3×10
12
4×10
12
5×10
12
6×10
12
Model Flux (W/m
2
/m)
0.0
0.5
1.0
1.5
2.0
Normalized Flux
-1.0
-0.5
0.0
0.5
1.0
Residuals
1.670
1.675
1.680
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
22.2 km/s
RV=5.2 km/s
Dc
2
reduced =5.9
d.o.f.=1450
H band, Order m=106
0
1×10
12
2×10
12
3×10
12
4×10
12
5×10
12
Model Flux (W/m
2
/m)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Normalized Flux
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Residuals
1.685
1.690
1.695
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
22.4 km/s
RV=6.2 km/s
Dc
2
reduced =2.7
d.o.f.=1560
H band, Order m=105
0
1×10
12
2×10
12
3×10
12
4×10
12
5×10
12
Model Flux (W/m
2
/m)
0.0
0.5
1.0
1.5
Normalized Flux
-0.5
0.0
0.5
Residuals
1.700
1.705
1.710
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
21.6 km/s
RV=6.4 km/s
Dc
2
reduced =3.4
d.o.f.=1542
H band, Order m=104
0
1×10
12
2×10
12
3×10
12
4×10
12
5×10
12
Model Flux (W/m
2
/m)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Normalized Flux
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Residuals
1.715
1.720
1.725
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
22.6 km/s
RV=6.2 km/s
Dc
2
reduced =3.0
d.o.f.=1418
H band, Order m=103
0
1×10
12
2×10
12
3×10
12
Model Flux (W/m
2
/m)
0.0
0.5
1.0
1.5
Normalized Flux
-0.5
0.0
0.5
Residuals
1.730
1.735
1.740
1.745
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1000 K
log(g)=5.0
v sin(i)=
22.8 km/s
RV=6.4 km/s
Dc
2
reduced =3.4
d.o.f.=1503
H band, Order m=102
0
1×10
12
2×10
12
3×10
12
4×10
12
Model Flux (W/m
2
/m)
0.0
0.5
1.0
1.5
Normalized Flux
-0.5
0.0
0.5
Residuals
1.750
1.755
1.760
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
22.2 km/s
RV=6.6 km/s
Dc
2
reduced =2.1
d.o.f.=1552
H band, Order m=101
0
1×10
12
2×10
12
3×10
12
4×10
12
Model Flux (W/m
2
/m)
0.0
0.5
1.0
1.5
2.0
Normalized Flux
-1.0
-0.5
0.0
0.5
1.0
Residuals
1.765
1.770
1.775
1.780
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
22.4 km/s
RV=7.0 km/s
Dc
2
reduced =2.2
d.o.f.=1502
H band, Order m=100
0
1×10
12
2×10
12
3×10
12
4×10
12
Model Flux (W/m
2
/m)
0.0
0.5
1.0
1.5
2.0
2.5
Normalized Flux
-1.0
-0.5
0.0
0.5
1.0
Residuals
1.780
1.785
1.790
1.795
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
23.0 km/s
RV=7.2 km/s
Dc
2
reduced =3.5
d.o.f.=1427
H band, Order m=99
0
1×10
12
2×10
12
3×10
12
Model Flux (W/m
2
/m)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Normalized Flux
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Residuals
1.798
1.800
1.802
1.804
1.806
1.808
1.810
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
22.0 km/s
RV=7.4 km/s
Dc
2
K band, Order m=94
0
5.0×10
11
1.0×10
12
1.5×10
12
2.0×10
12
2.5×10
12
Model Flux (W/m
2
/m)
0.0
0.5
1.0
1.5
2.0
Normalized Flux
-1.0
-0.5
0.0
0.5
1.0
Residuals
1.895
1.900
1.905
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
22.6 km/s
RV=5.2 km/s
Dc
2
reduced =2.1
d.o.f.=510
K band, Order m=93
0
5.0×10
11
1.0×10
12
1.5×10
12
2.0×10
12
2.5×10
12
Model Flux (W/m
2
/m)
0.0
0.5
1.0
1.5
2.0
2.5
Normalized Flux
-1.0
-0.5
0.0
0.5
1.0
Residuals
1.910
1.915
1.920
1.925
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
22.4 km/s
RV=5.8 km/s
Dc
2
reduced =1.2
d.o.f.=750
Figure A2. Continued.reduced =2.5 d.o.f.=1528 Figure A2. Continued. reduced =0.1 d.o.f.=1715Figure A2. Continued. reduced =0.6 d.o.f.=1766 Figure A2. Continued. reduced =3.6 d.o.f.=1866 Figure A2. Continued. reduced =1.7 d.o.f.=1880 Figure A2. Continued.1.975
1.980
1.985
1.990
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
22.2 km/s
RV=6.0 km/s
Dc
2
reduced =3.1
d.o.f.=1517
K band, Order m=89
0
5.0×10
11
1.0×10
12
1.5×10
12
2.0×10
12
Model Flux (W/m
2
/m)
0.0
0.5
1.0
1.5
Normalized Flux
-0.5
0.0
0.5
Residuals
1.995
2.000
2.005
2.010
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
23.2 km/s
RV=6.4 km/s
Dc
2
reduced =1.8
d.o.f.=1120
K band, Order m=88
0
5.0×10
11
1.0×10
12
1.5×10
12
2.0×10
12
Model Flux (W/m
2
/m)
0.0
0.5
1.0
1.5
Normalized Flux
-0.5
0.0
0.5
Residuals
2.015
2.020
2.025
2.030
2.035
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
23.6 km/s
RV=5.8 km/s
Dc
2
reduced =3.7
d.o.f.=1538
K band, Order m=87
0
5.0×10
11
1.0×10
12
1.5×10
12
2.0×10
12
Model Flux (W/m
2
/m)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Normalized Flux
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Residuals
2.040
2.045
2.050
2.055
2.060
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
23.4 km/s
RV=6.2 km/s
Dc
2
K band, Order m=86
0
2.0×10
11
4.0×10
11
6.0×10
11
8.0×10
11
1.0×10
12
1.2×10
12
Model Flux (W/m
2
/m)
0.0
0.2
0.4
0.6
0.8
1.0
Normalized Flux
-0.4
-0.2
0.0
0.2
0.4
Residuals
2.065
2.070
2.075
2.080
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1000 K
log(g)=5.0
v sin(i)=
23.2 km/s
RV=6.2 km/s
Dc
2
reduced =1.0
d.o.f.=1633
K band, Order m=85
0
2.0×10
11
4.0×10
11
6.0×10
11
8.0×10
11
1.0×10
12
1.2×10
12
Model Flux (W/m
2
/m)
0.0
0.2
0.4
0.6
0.8
1.0
Normalized Flux
-0.4
-0.2
0.0
0.2
0.4
Residuals
2.085
2.090
2.095
2.100
2.105
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1000 K
log(g)=5.0
v sin(i)=
23.4 km/s
RV=6.0 km/s
Dc
2
K band, Order m=84
0
5.0×10
11
1.0×10
12
1.5×10
12
Model Flux (W/m
2
/m)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Normalized Flux
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Residuals
2.110
2.115
2.120
2.125
2.130
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
H 2
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
23.2 km/s
RV=5.6 km/s
Dc
2
reduced =0.2
d.o.f.=1732
K band, Order m=83
0
5.0×10
11
1.0×10
12
1.5×10
12
Model Flux (W/m
2
/m)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Normalized Flux
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Residuals
2.135
2.140
2.145
2.150
2.155
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
22.4 km/s
RV=5.4 km/s
Dc
2
K band, Order m=82
0
5.0×10
11
1.0×10
12
1.5×10
12
Model Flux (W/m
2
/m)
0.0
0.5
1.0
1.5
Normalized Flux
-0.5
0.0
0.5
Residuals
2.160
2.165
2.170
2.175
2.180
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
23.2 km/s
RV=5.8 km/s
Dc
2
reduced =1.9
d.o.f.=1809
K band, Order m=81
0
2.0×10
11
4.0×10
11
6.0×10
11
8.0×10
11
1.0×10
12
1.2×10
12
1.4×10
12
Model Flux (W/m
2
/m)
0.0
0.5
1.0
1.5
2.0
Normalized Flux
-1.0
-0.5
0.0
0.5
1.0
Residuals
2.185
2.190
2.195
2.200
2.205
2.210
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
22.0 km/s
RV=5.6 km/s
Dc
2
K band, Order m=80
0
2.0×10
11
4.0×10
11
6.0×10
11
8.0×10
11
1.0×10
12
1.2×10
12
Model Flux (W/m
2
/m)
0.0
0.5
1.0
1.5
Normalized Flux
-0.5
0.0
0.5
Residuals
2.215
2.220
2.225
2.230
2.235
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
20.6 km/s
RV=6.4 km/s
Dc
2
reduced =2.3
d.o.f.=1823
K band, Order m=79
0
2.0×10
11
4.0×10
11
6.0×10
11
8.0×10
11
1.0×10
12
1.2×10
12
Model Flux (W/m
2
/m)
0.0
0.5
1.0
1.5
2.0
Normalized Flux
-1.0
-0.5
0.0
0.5
1.0
Residuals
2.240
2.245
2.250
2.255
2.260
2.265
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
21.4 km/s
RV=6.2 km/s
Dc
2
K band, Order m=78
0
2×10
11
4×10
11
6×10
11
8×10
11
1×10
12
Model Flux (W/m
2
/m)
0.0
0.5
1.0
1.5
2.0
2.5
Normalized Flux
-1.0
-0.5
0.0
0.5
1.0
Residuals
2.270
2.275
2.280
2.285
2.290
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1100 K
log(g)=5.0
v sin(i)=
20.6 km/s
RV=7.0 km/s
Dc
2
reduced =1.2
d.o.f.=1865
K band, Order m=77
0
2×10
11
4×10
11
6×10
11
Model Flux (W/m
2
/m)
0.0
0.5
1.0
1.5
2.0
2.5
Normalized Flux
-1.0
-0.5
0.0
0.5
1.0
Residuals
2.295
2.300
2.305
2.310
2.315
2.320
2.325
Wavelength (mm)
0.0
0.2
0.4
0.6
0.8
1.0
Earth Transmittance
H 2 O
CH 4
CO
NH 3
H 2 S
Data
Uncert
Model
H 2 O
CO 2
O 3
N 2 O
CO
CH 4
O 2
N 2
OH
Model
parameters:
T eff =1000 K
log(g)=5.0
v sin(i)=
21.8 km/s
RV=6.2 km/s
Dc
2
reduced =1.6
d.o.f.=2001
Tannock et al.
https://psg.gsfc.nasa.gov/
MNRAS 000, 1-20(2022)
ACKNOWLEDGEMENTSWe would like to thank the anonymous referee for their careful and constructive comments that helped us improve this paper. We thank Dr. Mark Marley and Dr. Michael Cushing for useful discussions about these data, and for their insights on which molecules to investigate. We thank Dr. Michael Line for providing the updated methane and ammonia line lists used in the Bobcat Alternative A models. We thank Dr. Genaro Suárez for performing a secondary check of our estimates of fundamental parameters with his custom fitting algorithm. We thank Chris Wyenberg for his help with the cross-correlation analysis.APPENDIX A: THE FULL SUITE OF MODEL FITS FOR EVERY IGRINS ORDERWe show the best fitting Bobcat Alternative A models for all orders in the IGRINS spectrum in figures A1 ( band) and A2 ( band). This paper has been typeset from a T E X/L A T E X file prepared by the author.
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[
"Expectile Asymptotics",
"Expectile Asymptotics"
] |
[
"Hajo Holzmann [email protected] \nFachbereich Mathematik und Informatik Philipps\nInstitut für Stochastik Karlsruher Institut für Technologie (KIT)\nUniversität Marburg\n\n",
"Bernhard Klar [email protected] \nFachbereich Mathematik und Informatik Philipps\nInstitut für Stochastik Karlsruher Institut für Technologie (KIT)\nUniversität Marburg\n\n"
] |
[
"Fachbereich Mathematik und Informatik Philipps\nInstitut für Stochastik Karlsruher Institut für Technologie (KIT)\nUniversität Marburg\n",
"Fachbereich Mathematik und Informatik Philipps\nInstitut für Stochastik Karlsruher Institut für Technologie (KIT)\nUniversität Marburg\n"
] |
[] |
We discuss in detail the asymptotic distribution of sample expectiles. First, we show uniform consistency under the assumption of a finite mean. In case of a finite second moment, we show that for expectiles other then the mean, only the additional assumption of continuity of the distribution function at the expectile implies asymptotic normality, otherwise, the limit is non-normal. For a continuous distribution function we show the uniform central limit theorem for the expectile process. If, in contrast, the distribution is heavy-tailed, and contained in the domain of attraction of a stable law with 1 < α < 2, then we show that the expectile is also asymptotically stable distributed. Our findings are illustrated in a simulation section.
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10.1214/16-ejs1173
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[
"https://arxiv.org/pdf/1509.06866v2.pdf"
] | 88,514,739 |
1509.06866
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d62e48d7e4be4cc8d69475f9210b437e468971d8
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Expectile Asymptotics
13 Jul 2016 July 14, 2016
Hajo Holzmann [email protected]
Fachbereich Mathematik und Informatik Philipps
Institut für Stochastik Karlsruher Institut für Technologie (KIT)
Universität Marburg
Bernhard Klar [email protected]
Fachbereich Mathematik und Informatik Philipps
Institut für Stochastik Karlsruher Institut für Technologie (KIT)
Universität Marburg
Expectile Asymptotics
13 Jul 2016 July 14, 2016arXiv:1509.06866v2 [stat.ME]M-estimatorexpectilesconvergence to stable distributionsasymptotic normal- ityuniform central limit theorem
We discuss in detail the asymptotic distribution of sample expectiles. First, we show uniform consistency under the assumption of a finite mean. In case of a finite second moment, we show that for expectiles other then the mean, only the additional assumption of continuity of the distribution function at the expectile implies asymptotic normality, otherwise, the limit is non-normal. For a continuous distribution function we show the uniform central limit theorem for the expectile process. If, in contrast, the distribution is heavy-tailed, and contained in the domain of attraction of a stable law with 1 < α < 2, then we show that the expectile is also asymptotically stable distributed. Our findings are illustrated in a simulation section.
Introduction
Expectile regression, that is, regression on a parameter that generalizes the mean and characterizes the tail behaviour of a distribution, has been introduced by Newey and Powell (1987) as an alternative to more standard quantile regression; Breckling and Chambers (1988) considered regression based on more general asymmetric M-estimators. For a recent comparison between quantile and expectile regression and references see Schulze-Waltrup et al. (2014).
Let Y be a random variable with distribution function F and finite mean E|Y | < ∞. For a fixed τ ∈ (0, 1), the τ -expectile µ τ = µ τ (F ) of Y has been introduced by Newey and Powell (1987) as the minimizer of an asymmetric quadratic loss µ τ (F ) = arg min x∈R E S τ (x, Y ),
S τ (x, y) = τ /2 ((y − x) + ) 2 − (y + ) 2 + (1 − τ )/2 ((y − x) − ) 2 − (y − ) 2 .
(1) Apparently, for τ = 1/2 one obtains the mean. Alternatively to S τ (x, y) in (1), one may use other scoring functions for the expectile; these were recently characterized by Gneiting (2011, Theorem 10).
Compared to quantiles, expectiles require the existence of a first moment and hence lack robustness. On the other hand, for any distribution with finite mean, the expectile is unique for each τ , and the expectile curve is always strictly increasing and continuous.
More importantly, as a risk measure it has been shown recently that expectiles have the attractive property of coherence (see Bellini et al. (2014)), while quantiles suffer from the lack of subadditivity. Indeed, expectiles were shown to be the only coherent, elicitable risk measures in Ziegel (2014); for a discussion and comparison between value at risk (quantiles), expectiles and expected shortfall see Emmer et al. (2015). Further discussion and application of expectiles as risk measures are given in Delbaen (2013) and Bellini and Di Bernardino (2015).
In this note we study in detail the statistical, that is, asymptotic properties of the sample expectiles. Somewhat surprisingly and in contrast to the mean, for τ = 1/2 we find that even under the assumption of a finite second moment, the sample expectile is only asymptotically normal if the distribution function F is continuous at µ τ (F ), otherwise, the limit distribution is non-normal.
First, in Section 2.1 we show uniform consistency under the assumption of a finite mean.
Next, in Section 2.2 we show that if the distribution function F is continuous at its τexpectile µ τ (F ), there is an asymptotic linearization of the sample expectile for this τ . In case of finite second moments, this implies asymptotic normality, but if F is in the domain of attraction of a stable law, the sample expectile is also asymptotically stable distributed.
If F has a jump at µ τ (F ), we show in Section 2.3 that also under the assumption of a finite second moment, the asymptotic distribution of the sample expectile is non-normal.
Finally, for a continuous distribution function with second moments, we show the uniform central limit theorem for the expectile process. We illustrate our findings in a simulation in Section 3, using the t-distribution with low degrees of freedom as a prototypical example for heavy-tailed distributions. Based on an explicit representation of the expectile for discrete distributions, we exemplify the nonstandard asymptotic behavior of the empirical expectile by a three-point distribution. Proofs are deferred to Section 4.
In a recent paper, Krätschmer and Zähle (2016) obtained results on the asymptotics of expec-tiles which are to some extend complementary to our results. Using a non-standard version of the functional delta-method allows them to treat both the case of dependent data as well as expectiles of parametric estimates of the distribution. However, they only consider the case of a finite second moment (they even assume slightly more) and a distribution which is continuous at the expectiles, and further do not investigate properties of the expectile process.
2 Asymptotic properties of sample expectiles Newey and Powell (1987) state a number of useful properties of expectiles, mainly for absolutely continuous distributions F . Below we state an extension, and in particular point out the assumptions on F which are actually required. Introducing the identification function
I τ (x, y) = τ (y − x)1 {y≥x} − (1 − τ ) (x − y)1 {y<x}(2)
of the expectile, it is well-known that µ τ (F ) can equivalently be defined as unique solution of the first-order condition
EI τ (x, Y ) = 0, x ∈ R.(3)
The following identity, obtained by a partial integration, is important for us:
I τ (x, F ) :=EI τ (x, Y ) = τ ∞ x 1 − F (y) dy − (1 − τ ) x −∞ F (y) dy.(4)
Proposition 1. Let F be a distribution function with finite mean.
(i) For each τ ∈ (0, 1) there is a unique solution µ τ (F ) to (1) or, equivalently, to (3).
(ii) The function µ · (F ) : (0, 1) → R, τ → µ τ (F ), is continuous, strictly increasing, and has range {y ∈ R : 0 < F (y) < 1}.
(iii) If F is continuous in a neighborhood of µ τ (F ) for a given τ ∈ (0, 1), then µ · (F ) is continuously differentiable in a neighborhood of τ with derivative
∂ τ µ τ (F ) = ∞ µτ 1 − F (y) dy + µτ −∞ F (y) dy τ 1 − F µ τ + (1 − τ ) F µ τ .
Sample expectiles and uniform consistency
In this section we show strong uniform consistency of sample expectiles. Let Y have distribution function F , with finite first moment E F |Y | = E|Y | < ∞, and let Y 1 , Y 2 , . . . be i.i.d.
copies of Y , and letF n be the empirical distribution function. The empirical τ -expectilê µ τ,n = µ τ F n can be defined as solution of the equation
I τ x,F n = 1 n n k=1 I τ (x, Y k ) = 0.(5)
This type of estimator is often termed Z-estimator, and a large amount of theory is available to obtain asymptotic properties for this type of estimators. Alternatively, asymptotic results can be derived using the representation as an M-estimator, that is,
µ τ,n = argmin x∈RŜn (x),Ŝ n (x) = 1 n n k=1 S τ (x, Y k ) = R S τ (x, y) dF n (y).(6)
Here, any other scoring function for the expectile (Gneiting, 2011) could be used instead, they all result in the same estimator, the expectile of the empirical distribution function.
The measurability ofμ τ,n follows from Theorem (1.9) in Pfanzagl (1969), who studied Mestimators under the heading of minimum contrast estimation. More directly, measurability follows from the explicit representation ofμ τ,n in Subsection 3.2.
Theorem 2. Let Y, Y 1 , Y 2 , . . . be i.i.d. with distribution function F , and assume E F |Y | < ∞.
For any τ l , τ u ∈ (0, 1), τ l < τ u , we have sup τ l ≤τ ≤τu μ τ,n − µ τ (F ) → 0 a.s.
Asymptotic linearization and convergence to stable distributions
Let us consider the representation (6) of the sample expectile as an M-estimator. Asymptotic normality or, more generally, asymptotic linearization, requires that the asymptotic contrast function has a second order Taylor expansion at the true parameter. Since |∂ x S τ (x, y)| = |I τ (x, y)| ≤ c(|x| + |y|) for a suitable constant c, we may differentiate the asymptotic contrast
function ψ τ (x) = ES τ (x, Y ) = S τ (x, y) dF (y)(7)
under the integral sign to obtain
ψ ′ τ (x) = − EI τ (x, Y ) =: −I τ (x, F ).
We see from (4) that ψ ′ τ (x) has
right derivative ψ ′′ + τ (x) = τ 1 − F (x) + (1 − τ ) F (x) left derivative ψ ′′ − τ (x) = τ 1 − F (x−) + (1 − τ ) F (x−) (8) at x, where F (x−) = P(Y < x)
is the left limit of F at x. For τ = 1/2 (i.e. the mean), these are always equal, but generally only coincide at µ τ (F ) if F has no point mass in its τ -expectile. From Theorems 1 and 10 in Arcones (2000) we deduce the following linearization.
Theorem 3 (Asymptotic linearization). Let Y, Y 1 , Y 2 , . . . be i.i.d. with distribution function F . Assume that E F |Y | < ∞ and that F is continuous at µ τ = µ τ (F ) for a given τ ∈ (0, 1).
Let {a n } be a sequence of positive numbers which converges to infinity with sup n≥1 n −1 a 2 n < ∞, such that a n n n k=1
I τ (µ τ , Y k ) = O P (1).(9)
Then a n (μ τ,n − µ τ ) = a n n
τ 1 − F (µ τ ) + (1 − τ ) F (µ τ ) −1 n k=1 I τ (µ τ , Y k ) + o P (1).(10)
Asymptotic normality
In case of finite second moments, (9) is satisfied with a n = √ n by the central limit theorem, and we obtain asymptotic normality for a finite number of expectiles. In the following, we
write Y n L → F as a short-hand notation for Y n L → Y ∼ F , where F denotes the distribution function of Y . Corollary 4. Suppose that EY 2 < ∞. Let τ i ∈ (0, 1), i = 1, . . . , m be such that F does not have a point mass at any of the µ τ i , i = 1, . . . , m. Then √ n μ τ 1 ,n − µ τ 1 , . . . ,μ τm,n − µ τm ′ L → N 0, Σ , where Σ i,j = E I τ i (µ τ i , Y ) I τ j µ τ j , Y τ i 1 − F (µ τ i ) + (1 − τ i ) F (µ τ i ) τ j 1 − F (µ τ j ) + (1 − τ j ) F (µ τ j )(11)
for i, j = 1, . . . , m.
Convergence to stable distributions
A random variable X has an α-stable distribution if its characteristic function is given by
E e iuX = exp −|u| α 1 − iβ tan πα 2 sign(u) , α = 1, exp −|u| 1 + iβ 2 π sign(u) log |u| , α = 1, where 0 < α ≤ 2, β ∈ [−1, 1].
Assume that Y belongs to the domain of attraction of an α-stable distribution (Y ∈ DA(α)) with 0 < α < 2 (see, e.g., Embrechts et al. (1997, Def. 2.2.7)). This is the case if and only if Y has tail probabilities that satisfy
P (Y > y) = c + + o(1) y α L(y) and P (Y < −y) = c − + o(1) y α L(y), y → ∞,(12)
where L is slowly varying and c + , c − ≥ 0 with c + +c − > 0 (Embrechts et al., 1997, Th. 2.2.8).
In the following, we assume 1 < α < 2 to ensure that E|Y | < ∞.
n 1−1/α L 1 (n) (μ τ,n − µ τ (F )) L − →Z τ 1 − F (µ τ ) + (1 − τ ) F (µ τ ) .
Here,Z follows an α-stable distribution, and L 1 is an appropriate slowly varying function.
Proof. Since F ∈ DA(α), from (12) we obtain that
P (I(µ τ , Y ) > y) = τ α c + + o(1) y α L(y) and P (I(µ τ , Y ) < −y) = (1 − τ ) α c − + o(1) y α L(y) as y → ∞.
Consequently, I(µ τ , Y ) ∈ DA(α), and the general CLT (Embrechts et al., 1997, Th. 2.2.15) yields
n 1/α L 1 (n) −1 n k=1 I(µ τ , Y k ) − nEI(µ τ , Y ) L − →Z as n → ∞,
whereZ follows an α-stable distribution and L 1 is an appropriate slowly varying function.
This implies that (9) is satisfied, and an application of Theorem 3 together with the general CLT yields the statement of the corollary.
Instead of using the assumptions Y ∈ DA(α), suppose more specifically that Y belongs to the domain of normal attraction of some α-stable distribution with 1 < α < 2, i.e. Y has tail probabilities that satisfy
y α P (Y > y) → c + and y α P (Y < −y) → c − , y → ∞,(13)
with c + + c − > 0 and 1 < α < 2.
Corollary 6. Let Y, Y 1 , Y 2 , . . . be i.i.d. r.v.
with distribution function F that belongs to the normal domain of attraction of an α-stable distribution, where 1 < α < 2, that is, satisfies (13). Assume further that F has no point mass in µ τ . Then
n 1−1/αc (μ τ,n − µ τ ) L − → S(α,β) τ 1 − F (µ τ ) + (1 − τ ) F (µ τ ) , wherec = 2Γ(α) sin(πα/2) π(τ α c + + (1 − τ ) α c − ) 1/α ,β = τ α c + − (1 − τ ) α c − τ α c + + (1 − τ ) α c − ) .
Proof. Since
y α P (I(µ τ , Y ) > y) → τ α c + and y α P (I(µ τ , Y ) < −y) → (1 − τ ) α c − as y → ∞,
this follows from the general CLT for distributions in the normal domain of attraction of a corresponding stable law (Nolan, 2015, p. 22).
Further asymptotics under finite second moments
Suppose that Y ∼ F with E Y 2 < ∞ and V ar Y > 0. In contrast to the mean, asymptotic normality of general expectiles as in Corollary 4 actually requires the additional assumption that Y has no point mass at µ τ (F ), otherwise, the limit distribution is non-normal, as the following result shows.
Theorem 7. Let Y, Y 1 , Y 2 , . . . be i.i.d. with distribution function F with E Y 2 < ∞. Let τ ∈ (0, 1) and denote µ τ = µ τ (F ). Then √ n (μ τ,n − µ τ ) L → σ 1 W 1 W >0 + σ 2 W 1 W <0 ,(14)
where
W ∼ N 0, E[I τ (µ τ , Y ) 2 ] , σ 1 = 1 τ 1 − F µ τ + (1 − τ ) F µ τ , σ 2 = 1 τ 1 − F µ τ − + (1 − τ ) F µ τ − ,(15)
and
F (x−) = P (Y < x) denotes the left limit of F at x.
We prove Theorem 7 by using empirical process methods and the argmax continuity theorem as presented in Van der Vaart (1998). Alternatively one could exploit the convexity of the contrast and modify the assumptions and the proof in Hjort and Pollard (1993, Theorem 2.1) to give an alternative argument.
In case of a continuous distribution function, we also have convergence of the expectile process.
Theorem 8. Let Y, Y 1 , Y 2 , . . . be i.i.d. with distribution function F with E Y 2 < ∞. Let 0 < τ l < τ u < 1 and suppose that F is continuous in a neighborhood of µ τ l , µ τu . Then the sequence of processes
τ → √ n (μ τ,n − µ τ ) n≥1 , τ ∈ τ l , τ u ,(16)
converges weakly in C τ l , τ u to a Gaussian process with continuous sample paths and covariance function given in (11). Tran et. al. (2014) also show convergence of the expectile process. They argue via convergence of an associated quantile process, and therefore require that F has a density, further, they do not specify the covariance function of the limit process.
Theorem 7 shows that process convergence, at least in C τ l , τ u or even in l ∞ τ l , τ u , cannot be expected if F has a discontinuity in [τ l , τ u ], since in this case the limit process would be discontinuous as well.
3 Some Simulations 3.1 Illustration of convergence to a stable distribution
As an example for a distribution with finite expectation but infinite variance we consider Student's t-distribution t α , 1 < α < 2, with symmetric density
f α (x) = Γ((α + 1)/2) Γ(α/2) √ απ 1 + x 2 α − α+1 2 , x ∈ R. For Y α ∼ t α , lim y→∞ y α P (Y α > y) = lim y→∞ y α P (Y α < −y) = Γ((α + 1)/2) Γ(α/2) α α/2−1 √ π .
Accordingly, t α belongs to the domain of normal attraction of some α-stable distribution. To compute the theoretical τ -expectile, which is the unique solution of
µ τ − EY = 2τ − 1 1 − τ E (Y − µ τ ) + ,(17)
one can use the identity
E (Y α − µ τ ) + = √ α Γ((α + 1)/2) √ π(α − 1)Γ(α/2) 1 + µ 2 τ α 1−α 2 − µ τ (1 − F α (µ τ )) ,
where F α (·) denotes the distribution function of t α . The limiting behavior of the empirical τ -expectile then follows directly from Corollary 5. Figure 1 shows the distribution function of n 1−1/αc (μ τ,n − µ τ ) (more precisely the empirical distribution function based on 10000
replications) for sample sizes of 20, 200 and 2000 for several values of τ and α. It can be observed that the quality of the approximation by the corresponding limiting stable law depends on both τ and α: the approximation improves for decreasing α (see Figure 1 (a)-(c)) and for τ approaching the value 0.5 (see Figure 1 (d)-(f)).
Illustration of nonstandard asymptotics under finite second moments
To illustrate the convergence to a non-normal distribution stated in Theorem 7, we first give an explicit formula for the empirical expectile which is interesting in itself. From (17), it follows directly that the τ -expectile satisfies the equivalent conditions Upper row: Data follow t α -distribution with different α, τ = 0.8 fixed.
τ = E [(Y − µ τ ) − ] E [|Y − µ τ |] ,(18)µ τ = (1 − τ )E Y 1 {Y ≤µτ } + τ E Y 1 {Y >µτ } (1 − τ )P (Y ≤ µ τ ) + τ P (Y > µ τ ) .(19)
Lower row: Data follow t α -distribution with α = 1.5, different values of τ .
The subsequent representation follows Bellini (2012), but formulated for the empirical distribution, and allowing for ties. Let Y (1) ≤ . . . ≤ Y (n) denote the order statistics of Y 1 , . . . , Y n .
From (19), the empirical expectile satisfieŝ
µ τ,n = (1 − τ ) k Y (k) 1 {Y (k) ≤μτ,n} + τ k Y (k) 1 {Y (k) >μτ,n} (1 − τ ) k 1 {Y (k) ≤μτ,n} + τ k 1 {Y (k) >μτ,n} . Hence, forμ τ,n ∈ [Y (i) , Y (i+1) ), where Y (i) < Y (i+1) , one haŝ µ τ,n = (1 − τ ) i k=1 Y (k) + τ n k=i+1 Y (k) (1 − τ )i + τ (n − i) .(20)
Defining
τ * i := iY (i) − i k=1 Y (k) n k=1 |Y (k) − Y (i) | , i = 1, . . . , n,(21)
we haveμ τ,n = Y (i) iff τ = τ * i for i = 1, . . . , n (and then, (21) is the empirical counterpart of (18)). Note that τ * 0 = 0, τ * n = 1, and sinceμ τ,n is nondecreasing in τ , we obtain that τ * i ≤ τ * i+1 , i = 1, . . . , n − 1. As a consequence,
µ τ,n ∈ [Y (i) , Y (i+1) ) ⇔ τ ∈ [τ * i , τ * i+1 ), i = 1, . . . , n − 1.
Remark. (i) Formulas (21) and (20) are especially well-suited for plotting purposes without the need of any numerical root-finding.
(ii) From (20),μ τ,n is piecewise differentiable in τ with
dμ τ,n dτ = i n k=i+1 Y (k) − (n − i) i k=1 Y (k) ((1 − τ )i + τ (n − i)) 2 for τ ∈ (τ * i , τ * i+1 ).
If Y has a discrete distribution on 0, 1, 2, . . . (say), an analogous reasoning leads to the following explicit formula for the theoretical expectiles µ τ . Define
τ * i := i−1 k=0 (i − k)P (Y = k) k≥0 |i − k|P (Y = k) , i = 0, 1, 2, . . . .(22)
For τ ∈ [τ * i , τ * i+1 ), and accordingly µ τ ∈ [i, i + 1), one has
µ τ = (1 − τ ) k≤i kP (Y = k) + τ k>i kP (Y = k) (1 − τ )P (Y ≤ i) + τ P (Y > i) .(23)
Now, assume that Y follows a three-point distribution with P (Y = i) = p i , i = 0, 1, 2, with p 0 , p 1 , p 2 > 0, p 0 +p 1 +p 2 = 1. Then, from (22) and (23), we get τ * 0 = 0, τ * 1 = p 0 /(p 0 +p 2 ), τ * 2 = 1 and
µ τ = τ (p 1 +2p 2 )
(1−τ )p 0 +τ (p 1 +p 2 ) , 0 < τ < τ * 1 ,
(1−τ )p 1 +2τ p 2
(1−τ )(p 0 +p 1 )+τ p 2 , τ * 1 ≤ τ < 1.
Proofs
Proof of Proposition 1. Parts (i) and (ii) are from Newey and Powell (1987) except for the general continuity of µ τ (F ) in τ . From (4) we see that I τ (x, F ) is a continuous function of (τ, x). To show continuity of the expectile, first let τ n ↓ τ , and letμ τ = lim n µ τn (F ) for which by monotonicity µ τ (F ) ≤μ τ . By continuity of I τ (x, F ) we have 0 = lim n I τn µ τn (F ), F = I τ μ τ , F , but since µ τ (F ) is the unique zero, it follows that µ τ (F ) =μ τ , that is, right-continuity. The argument for left-continuity is the same.
(iii) From (4) we see that if F is continuous in a neighborhood of x, then I τ (·, F ) is
continuously differentiable at x with derivative −τ 1 − F (x) − (1 − τ ) F (x)
. The conclusion follows from the implicit function theorem.
Proof of Theorem 2. We start with strong consistency of individual expectiles, that is,
µ τ,n − → µ τ (F ) a.s.(24)
We may use the representation (5) of the empirical expectile as a Z-estimator and strengthen Van der Vaart (1998, Lemma 5.10) to almost sure convergence. Since x → I τ (x, F ) is strictly decreasing, we have for every ε > 0 that
I τ (µ τ − ε, F ) > 0 > I τ (µ τ + ε, F ).
Since I(µ τ ± ε,F n ) → I τ (µ τ ± ε, F ) a.s. as n → ∞, we have a.s. that I(µ τ − ε,F n ) > 0 > I(µ τ + ε,F n ) for large n ∈ N. Since each map x → I(x,F n ), n ∈ N, is continuous and has exactly one zeroμ τ,n , this zero must a.s. lie between µ τ ± ε for large n ∈ N, that is,
lim sup n→∞ μ τ,n − µ τ (F ) ≤ ε a.s. ∀ ε > 0,
showing (24).
Using Proposition 1 (ii) and individual consistency, the classical Glivenco-Cantelli argument may be applied. Let d = µ τu (F ) − µ τ l (F ), m ∈ N, and choose by continuity τ l = τ 0 ≤ τ 1 ≤ . . . ≤ τ m = τ u such that µ τ k (F ) = µ τ l (F ) + kd/m, k = 1, . . . , m. By monotonicity, for
τ k ≤ τ ≤ τ k+1 ,μ τ,n − µ τ (F ) ≤μ τ k+1 ,n − µ τ k+1 (F ) + µ τ k+1 (F ) − µ τ k (F ). Therefore sup τ l ≤τ ≤τu μ τ,n − µ τ (F ) ≤ max 0≤k≤m μ τ k ,n − µ τ k (F ) + d/m. Similarly, sup τ l ≤τ ≤τu µ τ (F ) −μ τ,n ≤ max 0≤k≤m μ τ k ,n − µ τ k (F ) + d/m. Since sup τ l ≤τ ≤τu μ τ,n − µ τ (F ) = max sup τ l ≤τ ≤τu μ τ,n − µ τ (F ) , sup τ l ≤τ ≤τu µ τ (F ) −μ τ,n ,
we have for any m ∈ N that lim sup n sup τ l ≤τ ≤τu
μ τ,n − µ τ (F ) ≤ lim sup n max 0≤k≤m μ τ k ,n − µ τ k (F ) + d/m = d/m a.s.
We shall derive Theorem 3 from Theorems 1 and 10 in Arcones (2000). For convenience, we state a version of these results, tailored to our needs.
Theorem [Theorems 1 and 10 in Arcones (2000) ] Let Y, Y 1 , Y 2 , . . . be i.i.d. with distribution function F . Let g : R 2 → R be a function such that g(·, ϑ) : R → R is measurable for each ϑ ∈ R. Letθ n be a sequence of r.v.'s satisfying
n −1 n k=1 g(Y k ,θ n ) = inf ϑ∈R n −1 n k=1 g(Y k , ϑ).
Suppose that:
(A.1)θ n P − → ϑ 0 , ϑ 0 ∈ R. (A.2) There is a positive constant V such that E[g(Y, ϑ) − g(Y, ϑ 0 )] = V (ϑ − ϑ 0 ) 2 + o(|ϑ − ϑ 0 | 2 ), as ϑ → ϑ 0 . (A.
3) Let ϕ : R → R and let {a n } be a sequence of positive numbers which converges to infinity with sup n≥1 n −1 a 2 n < ∞ such that
a n n −1 n j=1 ϕ(Y j ) − E[ϕ(Y )] = O P (1). (A.4) There is a function ζ : R → R with E|ζ(Y )| < ∞ such that lim δ→0 E sup |ϑ|≤δ |r(Y, ϑ) − ϑ 2 ζ(Y )| ϑ 2 = 0,
where r(y, ϑ) = g(y, ϑ 0 + ϑ) − g(y, ϑ 0 ) − ϑϕ(y).
Then, a n (θ n − ϑ 0 ) + a n 2V
1 n n j=1 ϕ(Y j ) − E[ϕ(Y )] P − → 0.(25)
Proof of Theorem 3. We verify the conditions of the above theorem for ϑ 0 = µ τ (F ), g(y, ϑ) = S τ (ϑ, y), ϕ(y) = −I τ µ τ (F ), y and
ζ(y) = τ 2 1 {y>µτ } + 1 − τ 2 1 {y<µτ } .
(A1) follows from Theorem 2.
(A2) follows from (7), (8), the assumption of continuity of F at µ τ (F ), and Taylor's theorem, which holds under the minimal assumption of an existing second derivative.
(A3) is (9).
Finally, for (A4) we compute that for x > 0,
S τ (µ τ + x, y) − S τ (µ τ , y) + xI τ (µ τ , y) = − τ 2 (y − µ τ − x) 2 1 {µτ <y≤µτ +x} + τ 2 x 2 1 {y>µτ } + 1 − τ 2 (y − µ τ − x) 2 1 {µτ ≤y<µτ +x} + 1 − τ 2 x 2 1 {y<µτ }
and similarly for x < 0. Therefore for some c > 0 we may estimate
S τ (µ τ + x, y) − S(µ τ , y) + xI τ (µ τ , y) − x 2 ζ(y) ≤c (y − µ τ − x) 2 1 {µτ −|x|≤y≤µτ +|x|} ≤c x 2 1 {µτ −|x|≤y≤µτ +|x|} , and therefore E sup |x|≤δ |S τ (µ τ + x, Y ) − S τ (µ τ , Y ) + xI τ (µ τ , Y ) − x 2 ζ(Y )| x 2 ≤c P µ τ − δ ≤ Y ≤ µ τ + δ → 0, δ → 0,
since Y does not have a point mass at µ τ .
Proof of Theorem 7. We start by establishing Lipschitz continuity of S τ (x, y) as a function of x with square-integrable Lipschitz constant. Since ∂ x S τ (x, y) = −I τ (x, y), we have for
x 1 , x 2 ∈ B δ (µ τ ) |S τ (x 1 , y) − S τ (x 2 , y)| ≤ c m(y)|x 1 − x 2 |, m(y) := sup x∈B δ (µτ ) I τ (x, y) .(26)
Then, the inequality
m(y) ≤ sup x∈B δ (µτ ) |x − y| ≤ sup x∈B δ (µτ ) |x| + |y| yields E[m(Y ) 2 ] < ∞ if EY 2 < ∞,
that is, the Lipschitz constant has finite second moment.
Next, the asymptotic contrast in (7) is continuously differentiable with left and right derivatives in µ τ given in (8). From Taylors formula, we obtain
ψ τ (x) − ψ τ (µ τ ) = (x − µ τ ) 2 ψ ′′ + τ (µ τ )/2 + o(|x − µ τ | 2 ), x > µ τ , ψ τ (x) − ψ τ (µ τ ) = (x − µ τ ) 2 ψ ′′ − τ (µ τ )/2 + o(|x − µ τ | 2 ), x < µ τ ,(27)
where ψ ′′ ± τ (µ τ ) are right/left second derivatives.
Therefore, the assumptions of Theorem 5.52 in Van der Vaart (1998) are satisfied with α = 2 and β = 1 (see the argument in Corollary 5.53, that the Lipschitz property (26) implies the concentration inequality), and we obtain the √ n-rate of convergence:
√ n (μ τ,n − µ τ ) = O P (1).
To obtain the asymptotic distribution, we apply the argmax-continuity theorem, Corollary 5.58 in Van der Vaart (1998). To this end, for a measurable function f with Ef 2 (Y ) < ∞,
denote P n f = 1 n n k=1 f (Y k ), P f = EF (Y ), and G n (f ) = √ n P n − P f.G n √ n S τ (µ τ + h/ √ n, ·) − S τ (µ τ , ·) + h I τ (µ τ , ·) n→∞ → 0 (P).
Therefore, for any M > 0, the difference between the processes
h → √ n P n √ n S τ (µ τ + h/ √ n, ·) − S τ (µ τ , ·) , |h| ≤ M, and h → n ψ τ µ τ + h/ √ n − ψ τ (µ τ ) − h G n I τ (µ τ , ·), |h| ≤ M,
tends to 0 in probability in sup-norm. Using (27), the second process converges to the Gaussian process
h → 1 2σ 1 h 2 1 h>0 + 1 2σ 2 h 2 1 h<0 − h W,(28)
where W is normally distributed as in the theorem, hence so does the first. From the argmax -continuity theorem, we obtain weak convergence of the minimizers √ n (μ τ,n − µ τ ) to the minimizer of the limit process. Now, a parabola h → −hW + h 2 /(2σ) for some σ > 0 is minimized at h = σW , yielding the negative value −σW 2 /2. Therefore, the minimizer of (28) is at h = σ 1 W for W > 0 and at h = σ 2 W for W < 0, which gives the statement of the theorem.
Proof of Theorem 8. We shall apply Van der Vaart (1995, Theorem 1), which gives asymptotic normality of functional Z-estimators; see also Kosorok (2008, Theorem 13.4), which additionally implies validity of the bootstrap. First, Theorem 2 gives the uniform consistency. Given ν ∈ C τ l , τ u ⊂ l ∞ τ l , τ u , the functions τ → I τ ν(τ ), F and τ → I τ ν(τ ),F n are also in C τ l , τ u , and τ → I τ µ τ , F = 0, τ → I τ μ τ,n ,F n = 0.
Next, we check the conditions (2), (3) and (4) in Van der Vaart (1995). Suppose that ν ∈ C τ l , τ u is such that F is continuous on the image of ν (this is true by our assumption if τu] is small enough). Then we apply the mean value theorem for each τ ∈ [τ l , τ u ] to obtain
ν − µ [τ l ,I τ ν(τ ), F − I τ µ τ , F + τ 1 − F (µ τ ) + (1 − τ ) F (µ τ ) ν(τ ) − µ τ ≤ F (ξ τ ) − F (µ τ ) ν(τ ) − µ τ ,
where ξ τ is between ν(τ ) and µ τ . Since F is uniformly continuous in a compact neighborhood
of [τ l , τ u ], we obtain sup τ ∈[τ l ,τu] I τ ν(τ ), F −I τ µ τ , F + τ 1−F (µ τ ) +(1−τ ) F (µ τ ) ν(τ )−µ τ = o ν −µ [τ l ,τu] ,(29)
showing Fréchet differentiability, that is, (4) in Van der Vaart (1995). Note that the derivative, multiplication with the function τ → − τ 1 − F (µ τ ) + (1 − τ ) F (µ τ ) is continuously invertible.
Since by Proposition 1, (iii), µ τ (F ) is continuously differentiable in τ , we have for an appropriate constant c > 0 that I τ 1 µ τ 1 (F ), y − I τ 2 µ τ 2 (F ), y ≤ c|y| |τ 1 − τ 2 |, τ 1 , τ 2 ∈ [τ l , τ u ], y ∈ R,
so that
I τ µ τ (F ), · τ ∈[τ l ,τu](31)
is a Donsker class of functions, see Van der Vaart (1998, Example 19.7.), taking care of (2) in Van der Vaart (1995). √ n I τ µ τ + x,F n − I τ µ τ + x, F − I τ µ τ ,F n − I τ µ τ , F .
Now, for a constant C > 0, y ∈ R, τ, τ 1 , ∈ [τ l , τ u ], |x|, |x 1 | ≤ 1, I τ 1 µ τ 1 + x 1 , y − I τ 1 µ τ 1 , y − I τ µ τ + x, y − I τ µ τ , y ≤ C|y| |x 1 − x| + |τ 1 − τ | , I τ µ τ + x, y − I τ µ τ , y ≤ C |y| |x|.
Therefore, sup |x|≤δn sup τ ∈[τ l ,τu] E I τ µ τ + x, Y − I τ µ τ , Y 2 ≤ C EY 2 δ n , and each F n = (x, τ ) → I τ µ τ + x, y − I τ µ τ , y , |x| ≤ δ n , τ ∈ [τ l , τ u ] is a Lipschitz-class of functions. Therefore we may estimate (32) by the bracketing integral J [] δ n , F n , L 2 (F ) and an additional sequence converging to zero, by Van der Vaart (1998, Lemma 19.34 and Example 19.7), which together → 0 as n → ∞ and δ n ↓ 0.
Next, we show that weak convergence is actually in C τ l , τ u . The expectile processes (16) have continuous sample paths. As for the limiting Gaussian process, it suffices to show continuity of the sample paths of the limit Gaussian process of the empirical process corresponding to the function class (31), since the inverse of the Fréchet derivative in (29) is simply multiplication by a fixed continuous function. By Van der Vaart (1998, Lemma 18.15), the limit process can be constructed to have continuous sample paths w.r.t. its standard deviation semimetric. In order to check that continuity also holds w.r.t. the ordinary distance on [τ l , τ u ], we
show that E I τ 2 (µ τ 2 , Y ) − I τ 1 (µ τ 1 , Y ) 2 ≤ C 2 (τ 2 − τ 1 ) 2 , τ j ∈ µ τ l , µ τu , j = 1, 2
for some C > 0. But this follows immediately from (30) upon squaring and integrating. This concludes the proof of the theorem.
Figure 1 :
1Convergence of the cumulative distribution function (cdf) of the empirical expectile to the corresponding limiting stable cdf.
Figure 2 :
2Density function of the standardized empirical expectile for n = 500 and of the corresponding limiting distribution. Data follow a three point distribution in 0,1,2.(a) τ = 0.7, normal limiting distribution. (b) τ = 0.8, non-normal limiting distribution.Next, we make the choice p 0 = 4/10, p 1 = 5/10, p 2 = 1/10. Then, µ 0.8 = 1, i.e. the distribution of Y has a point mass in µ τ for τ = 0.8, but not for other values of τ .Figure 2(a) shows the density of √ n (μ τ,n − µ τ ) (estimated by a nonparametric density estimator based on 20000 replications) for sample size 500 and τ = 0.7 (hence, µ τ = 49/54) together with the limiting normal distribution given in Corollary 4.Figure 2(b) shows the corresponding plot for τ = 0.8 together with the limiting non-normal distribution given in Corollary 7.
Finally
, to show (3) inVan der Vaart (1995), we choose δ n ↓ 0, and estimate in the first stepsup ν−µ [τ l ,τu] ≤δn sup τ ∈[τ l ,τu] √ n I τ ν(τ ),F n − I τ ν(τ ), F − I τ µ τ ,F n − I τ µ τ , F ≤ sup |x|≤δn sup τ ∈[τ l ,τu]
By the Lipschitz property(26), from the proof of Lemma 19.31 inVan der Vaart (1998) weobtain for any M > 0 that
sup
|h|≤M
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[
"Universal map of gas-dependent kinetic selectivity in carbon nanotube growth",
"Universal map of gas-dependent kinetic selectivity in carbon nanotube growth"
] |
[
"K Otsuka \nDepartment of Mechanical Engineering\nThe University of Tokyo\n113-8656TokyoJapan\n\nNanoscale Quantum Photonics Laboratory\nRIKEN Cluster for Pioneering Research\n351-0198SaitamaJapan\n",
"R Ishimaru \nDepartment of Mechanical Engineering\nThe University of Tokyo\n113-8656TokyoJapan\n",
"A Kobayashi \nDepartment of Mechanical Engineering\nThe University of Tokyo\n113-8656TokyoJapan\n",
"T Inoue \nDepartment of Applied Physics\nOsaka University\n565-0871OsakaJapan\n",
"R Xiang \nDepartment of Mechanical Engineering\nThe University of Tokyo\n113-8656TokyoJapan\n",
"S Chiashi \nDepartment of Mechanical Engineering\nThe University of Tokyo\n113-8656TokyoJapan\n",
"Y K Kato \nNanoscale Quantum Photonics Laboratory\nRIKEN Cluster for Pioneering Research\n351-0198SaitamaJapan\n\nQuantum Optoelectronics Research Team\nRIKEN Center for Advanced Photonics\n351-0198SaitamaJapan\n",
"S Maruyama \nDepartment of Mechanical Engineering\nThe University of Tokyo\n113-8656TokyoJapan\n"
] |
[
"Department of Mechanical Engineering\nThe University of Tokyo\n113-8656TokyoJapan",
"Nanoscale Quantum Photonics Laboratory\nRIKEN Cluster for Pioneering Research\n351-0198SaitamaJapan",
"Department of Mechanical Engineering\nThe University of Tokyo\n113-8656TokyoJapan",
"Department of Mechanical Engineering\nThe University of Tokyo\n113-8656TokyoJapan",
"Department of Applied Physics\nOsaka University\n565-0871OsakaJapan",
"Department of Mechanical Engineering\nThe University of Tokyo\n113-8656TokyoJapan",
"Department of Mechanical Engineering\nThe University of Tokyo\n113-8656TokyoJapan",
"Nanoscale Quantum Photonics Laboratory\nRIKEN Cluster for Pioneering Research\n351-0198SaitamaJapan",
"Quantum Optoelectronics Research Team\nRIKEN Center for Advanced Photonics\n351-0198SaitamaJapan",
"Department of Mechanical Engineering\nThe University of Tokyo\n113-8656TokyoJapan"
] |
[] |
Single-walled carbon nanotubes have been a candidate for outperforming silicon in ultrascaled transistors, but the realization of nanotube-based integrated circuits requires dense arrays of purely semiconducting species. Control over kinetics and thermodynamics in tube-catalyst systems plays a key role for direct growth of such nanotube arrays, and further progress requires the comprehensive understanding of seemingly contradictory reports on the growth kinetics. Here, we propose a universal kinetic model and provide its quantitative verification by ethanol-based isotope labeling experiments. While the removal of carbon from catalysts dominates the growth kinetics under a low supply of precursors, our kinetic model and experiments demonstrate that chirality-dependent growth rates emerge when sufficient amounts of carbon and etching agents are co-supplied. As the model can be extended to create kinetic maps as a function of gas compositions, our findings resolve discrepancies in literature and offer rational strategies for chirality selective growth for practical applications.chirality dependence, and the growth rate significantly differs within the same chirality[22,24].
|
10.1021/acsnano.1c10569
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[
"https://arxiv.org/pdf/2111.08411v1.pdf"
] | 244,129,954 |
2111.08411
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610b597f001f34a4aa0df8f32de8f014910dd93c
|
Universal map of gas-dependent kinetic selectivity in carbon nanotube growth
K Otsuka
Department of Mechanical Engineering
The University of Tokyo
113-8656TokyoJapan
Nanoscale Quantum Photonics Laboratory
RIKEN Cluster for Pioneering Research
351-0198SaitamaJapan
R Ishimaru
Department of Mechanical Engineering
The University of Tokyo
113-8656TokyoJapan
A Kobayashi
Department of Mechanical Engineering
The University of Tokyo
113-8656TokyoJapan
T Inoue
Department of Applied Physics
Osaka University
565-0871OsakaJapan
R Xiang
Department of Mechanical Engineering
The University of Tokyo
113-8656TokyoJapan
S Chiashi
Department of Mechanical Engineering
The University of Tokyo
113-8656TokyoJapan
Y K Kato
Nanoscale Quantum Photonics Laboratory
RIKEN Cluster for Pioneering Research
351-0198SaitamaJapan
Quantum Optoelectronics Research Team
RIKEN Center for Advanced Photonics
351-0198SaitamaJapan
S Maruyama
Department of Mechanical Engineering
The University of Tokyo
113-8656TokyoJapan
Universal map of gas-dependent kinetic selectivity in carbon nanotube growth
Single-walled carbon nanotubes have been a candidate for outperforming silicon in ultrascaled transistors, but the realization of nanotube-based integrated circuits requires dense arrays of purely semiconducting species. Control over kinetics and thermodynamics in tube-catalyst systems plays a key role for direct growth of such nanotube arrays, and further progress requires the comprehensive understanding of seemingly contradictory reports on the growth kinetics. Here, we propose a universal kinetic model and provide its quantitative verification by ethanol-based isotope labeling experiments. While the removal of carbon from catalysts dominates the growth kinetics under a low supply of precursors, our kinetic model and experiments demonstrate that chirality-dependent growth rates emerge when sufficient amounts of carbon and etching agents are co-supplied. As the model can be extended to create kinetic maps as a function of gas compositions, our findings resolve discrepancies in literature and offer rational strategies for chirality selective growth for practical applications.chirality dependence, and the growth rate significantly differs within the same chirality[22,24].
I. INTRODUCTION
Control over length, density, and chirality of singlewalled carbon nanotubes (SWCNTs) at the synthesis stage is still of great interest for their applications. The fabrication of logic integrated circuits (ICs) [1,2], for example, requires high-density arrays of exclusively semiconducting species. To fulfill the requirements for digital ICs [3], significant progress has been made in the chirality sorting in nanotube dispersions and the subsequent assembly [4][5][6], as well as the selective removal of metallic species [7][8][9] from aligned nanotubes grown on crystalline substrates [10,11]. Chirality-sorted nanotubes usually suffer from the excessive interface states that originate from residual surfactants, degrading the switching capability of transistors [5,12]. Ultimately high performance should be achieved by using processing-free nanotube arrays from chemical vapor deposition (CVD) on wafers.
As proposed and experimentally demonstrated [13][14][15][16][17][18][19], the harmony of kinetic and thermodynamic control is crucial to acquire nanotube arrays of a desired electronic type. The abundance of nanotubes with a given chirality (n,m) is obtained from the product of lengths and population [14,20]; the understanding of the growth kinetics further involves decoupling of the lengths into growth rates, incubation time, and lifetime (Fig. 1a). Several studies have captured the one-by-one growth rates by optical or electron microscopy [21][22][23][24], but the results often disagree with each other. Even for the case of alcohols as carbon sources [25], the discrepancy is evident. Despite the claim that the growth rate difference is the key for selective growth methods [16,26], one-by-one measurements show that metallic (m-) and semiconducting (s-)SWCNTs exhibit a similar growth rate without a clear This confusion may arise from complicated decomposition of precursors [27,28]. Since the decomposed products include several carbon sources, as well as etching agents that remove carbon from the catalyst [26,29], the growth kinetics cannot be accurately captured without considering these opposing effects. A recent study has shed light on the role of etching by observing the (n,m)-independent growth rate in the absence of etching agents [23]. There remains vast room for the verification of such a kinetic model and the quantitative understanding of various experimental observations.
Here we propose a universal model that quantitatively describes supply and removal of carbon during a catalytic CVD process, inspired by the distinctive chirality distribution only appearing under a high ethanol vapor pressure. In order to decouple growth parameters usually inter-linked during CVD processes, we develop a duplex labeling technique using isotope ethanol and acetylene, and trace the growth modulation of individual SWCNTs from spatial and spectral points of view. The dominant role of etching agents in the growth kinetics is quantitatively elucidated, which is a key to selective growth according to the electronic structures. The kinetic model also predicts the growth conditions under which the chiral angle-dependence emerges and thereby allows us to experimentally observe the fast growth of near-armchair species. We finally classify the nanotube growth into five regimes depending on the pressures of carbon sources and etching agents, which not only explains various phenomena unresolved in previous studies, but also offers solid strategies for chirality control in catalytic CVD process. Schematic showing the three levels of analysis on nanotube growth achieved before: spectroscopy to evaluate relative abundance (top), its breakdown into length and population (middle), and the tracing of growth process (bottom). This study seeks to uncover beyond the growth rate. Colors of lines represent nanotube chirality. (b,c) Relative population of nanotubes with each chirality sorted by chiral angles in the low-pressure ethanol CVD (b) and the atmospheric-pressure ethanol CVD (c), corresponding to the number of SWCNTs whose length exceeds the dashed line in the middle panel of (a). Data in (c) are adopted from ref. [31]. Inset of (b): Schematic showing the micro-PL measurements of air-suspended SWCNTs over trenches. (d,e) Schematics of the kinetic modeling of SWCNT growth at the equilibrium state (d) of a metal nanoparticle and a nanotube wall and that at the non-equilibrium state (e). When no atom enters or leaves the system, carbon concentration N = Neq. When carbon atoms are supplied to or removed from the iron nanoparticle, carbon concentration N shifts from Neq, leading to the growth or shrinkage of the nanotube wall. Parameters in dark red is extrinsic and can be determined by experimental setting, while those in dark blue is inherent to the catalyst-nanotube (intrinsic parameters). Sketch of catalyst-nanotube geometries are obtained from our classical molecular dynamics simulation using Tersoff-Brenner potential [33].
From chirality distribution to growth kinetics of individual nanotubes. Before delving into the growth kinetics, we focus on the effect of gas pressure and composition on the chirality distribution of nanotubes grown from ethanol under two slightly different conditions. In both cases, we use evaporated iron as catalysts on thermal oxide of Si substrates and assign the tube chirality by photoluminescence (PL) spectroscopy [30,31]. The key difference lies in the pressure inside the reactor; in the case at a low pressure (LP), ethanol partial pressure is 130 Pa with the flow of Ar containing 3% of H 2 . In the other case at an atmospheric pressure (AP), Ar/H 2 is supplied through an ethanol bubbler, where ethanol accounts for ∼2.4 kPa. While the LP-CVD process yields SWCNTs with a rather uniform distribution (Fig. 1b), the clear preference towards near-armchair chirality species has been observed in the AP-CVD (Fig. 1c). Because we are observing PL from nanotubes suspended across tranches, chirality-dependent growth rate of nano-tubes might lead to the near-armchair preference through the different probability of reaching the other side of trenches. We need to take a closer look at the growth kinetics to understand the modulated chirality distribution.
To analyze the growth kinetics, we consider a steadystate model [23,32] for a nanotube-catalyst system. First, we define the concentration of carbon in a catalyst at an equilibrium state to be N eq . When no carbon atom is added or removed to/from the catalyst particle, N reaches N eq either by precipitation as a tube wall or by dissolution of the tube wall into the catalyst (Fig. 1d).
In the presence of carbon sources and etching agents, the adsorption and desorption of carbon occur and thereby shift N , resulting in the continuous growth or shrinkage of the nanotube (Fig. 1e).
In our model, the growth rate γ g is proportional to the kinetic constant for growth k g and the degree of super-saturation ∆N (= N − N eq ),
γ g = k g ∆N.(1)
With the focus shifted to the role of etching agents, such as oxygen and water [34][35][36], carbon atoms are eliminated from the catalyst at the rate Γ e , which is proportional to the surface area A, a kinetic constant k e , the pressure of etching agents P e , and N . The carbon supply rate Γ C is proportional to A, adsorption efficiency k ad , and the carbon source pressure P C . We note that a CVD process includes several carbon sources due to gas decomposition, whose k ad varies widely, but we do not distinguish those carbon sources and express them using single values for P C and k ad . At the steady state, the equation to describe "carbon bookkeeping" becomes,
dN dt ∝ Γ C − (γ g D + Γ e ) = 0,(2)
where D is the number of carbon atoms per unit length (∝ diameter), and therefore the supersaturation of carbon can be expressed as,
∆N = k ad P C − k e P e N eq k g D + k e P e ,(3)
with D being D/A. This kinetic model can describe the growth and shrinkage [24,29,37] of nanotubes that depend on the balance between P C and P e . Dominant role of etching agents in growth kinetics. By varying an extrinsic parameter during the growth and tracing the corresponding modulation of kinetics, we verify the model and determine the intrinsic parameters. Unlike the parameters unique to each catalyst-nanotube pair shown in dark blue in Fig. 1e, P C and P e can be controlled by experimental settings, but they are intricately coupled in the ethanol CVD process. To decouple these parameters, we add a small amount of acetylene gas in the middle of the synthesis, which results in the independent increase of P C (Fig. S3). The complementary experiment where we change only P e is summarized in Fig. S5.
In our duplex labeling technique to trace the growth modulation of individual SWCNTs, ethanol with natural abundance of carbon (hereafter called 12 C ethanol) and 13 C-enriched ethanol (hereafter called 13 C ethanol) are introduced alternately (Fig. 2a). We add acetylene of natural abundance (hereafter called 12 C acetylene) as a growth accelerator in the latter half of the CVD process. The lower panel shows a typical time dependence of a nanotube length, which is reconstructed from the Raman mapping (Fig. 2b). The growth rate before the acetylene addition (γ g,1 ) and that defined between the labels #4 and #5 (γ g,2 ) are 5.2 and 29.4 µm/min, respectively. The growth rate is even accelerated to γ g,3 = 61.2 µm/min after the label #5, reflecting the slow saturation of a small amount of acetylene in the reactor. The addition of <2% of acetylene to ethanol accelerates the growth by 12fold owing to its efficient adsorption on the catalyst [38].
To fully exploit the model, we should quantitatively determine the multiplication factor λ of the total supply rate of carbon, assuming the acetylene addition increases P C to λP C . The isotope ratio in the nanotube grown from the mixture of 13 C ethanol and 12 C acetylene provides accurate λ for each nanotube through Raman spectroscopy (the inset of Fig. 2c). The 12 C ratio α in the label #5 of the particular nanotube ( Fig. 2a) is calculated to be 59% from the G-band frequency. The P C multiplication factor is then λ = 2.42 (= 1/(1−α)). To understand how the increased P C affects the actual growth rate, we plot the growth acceleration factor γ g,i /γ g,1 (i = 2, 3) against λ for all nanotubes (Fig. 2c). It is somewhat surprising that the growth rates after the acetylene addition is noticeably larger than λγ g,1 (solid line). Carbon removal from the catalyst holds the key to this discrepancy because the simple rate equations are derived for the growth before and after the acetylene addition,
γ g,1 = γ C − γ e ,(4)γ g,2 = λγ C − sγ e ,(5)
where γ C and γ e are defined as Γ C /D and Γ e /D, respectively. The γ e multiplication factor due to increased P C is given by
s = N 2 /N 1 = λk ad PC+kgD Neq k ad PC+kgD Neq (1 < s < λ) with N 1 (N 2 ) being N before (after) the acetylene addition.
The left panel of Fig. 2d shows the accelerated growth rate γ g,2 plotted against γ g,1 with two different flow rates of additive acetylene. Interestingly, the linear fit of each set of experimental data has a large value of y-intercept. For better understating of this tendency, we derive analytical expressions of γ g,1 and γ g,2 by assigning P C and λP C to P C in Eq. 3, respectively, and then remove k e to yield the relationship,
γ g,2 = λk ad P C + k g D N eq k ad P C + k g D N eq γ g,1 + k ad P C k g N eq k ad P C + k g D N eq (λ−1),(6)
which assumes k e variance to account for the γ g difference. The experimental confirmation of large yintercepts, which correspond to the second term on the right-hand side of Eq. 6, is equivalent to a non-zero N eq . This supports the versatility of our kinetic model to include nanotube shrinkage as well as growth. Note that eliminating k e from the γ g,1 -γ g,2 relationship yields the best fit to the experimental results (see Fig. S9). With two different amounts of additive acetylene, the ratio of the y-intercepts (14.4/23.0) agrees to that independently obtained from the average λ − 1 values (1.39/2.28), also verifying our kinetic model.
Having obtained λ from Raman spectra, we reproduce the γ g,1 -γ g,2 relationship using our kinetic model. When we simulate γ g with the carbon source pressures of P C and λP C , the relationships in the right panel of Fig. 2d are obtained. Here, k e and k g are determined from the averages of γ g,1 and γ g,2 , and we assume both kinetic constants have relative standard deviations of 20%. By comparing the experiments and simulations, the variance in γ g,1 can be attributed to k e varying from the average by up to ±60%. Although the parameters are adjusted using the results with the flow rate of 0.077 sccm alone, we can reasonably predict another γ g,1 -γ g,2 distribution for the other amount of acetylene once λ is determined from the Raman spectra.
We take a closer look at the influence of Γ e in individual nanotubes. As the slope of the γ g,1 -γ g,2 relationship is equal to s (Eq. 6), we can breakdown the growth rate γ g,1 of each nanotube into the contribution of supply and removal at a catalyst using Eq. 4 and 5 (Fig. 2e). The growth rate is correlated with the rate of carbon removal from the catalyst, indicating the dominant influence of the etching agent derived from ethanol on growth rate determination under this growth condition.
Such carbon removal effects have been claimed to be the driving force behind the selective growth of s-SWCNTs [26,39]. Using the duplex labeling, we can successfully quantify the susceptivity to carbon removal in the form of γ e /γ C for s-and m-SWCNTs as shown in Fig. 2f. While s-SWCNTs have a wide distribution between 0 to 1, m-SWCNTs show a rather narrow distribution closer to 1. We expect that when P e is further increased, the nanotubes with a large γ e /γ C , i.e., most of m-SWCNTs and a part of s-SWCNTs, start to be shortened, leading to the preferential growth of semiconducting species.
Emergence of chirality-dependent growth kinetics. So far, we have discussed the carbon supply/removal for catalysts. One might wonder if the susceptivity to carbon removal is determined by chirality (n, m) [40], but that is excluded by the widely distributed γ e /γ C (0.15-0.68) within the (12,8) nanotubes (Fig. S11). We can confirm the chirality-independent Γ e from the growth rate γ g,2 after the acetylene addition, which is plotted against the chiral angle (Fig. 3a). The dispersion within similar chiral angles is large likely due to the non-uniformity of catalyst particles, and no overall trend can be found. Figure 3b and c show the diameter and chiral angle distribution. Judging from the more significant preference towards armchair chirality in terms of population than that in the growth kinetics, the weak bias in the chiralangle distribution in Fig. 1b should be attributed to the number of nucleation in this regime.
With the kinetic model, we can clearly understand the insignificance of (n,m) in the apparent growth rate. In Fig. 3d, we draw the deterministic growth rate
γ g = k g ∆N = k g (k ad P C − k e P e N eq ) k g D + k e P e .(7)
From the simulation in which k e follows a normal distribution with a deviation of 20%, the distribution of growth rate at a certain k g can be obtained and overlayed in Fig. 3d as a color contour. The growth rate easily saturates with respect to k g because carbon supply to the catalyst becomes the rate-determining step at a small P e . Furthermore, γ g varies widely depending on k e (at a fixed k g ), indicating that k g is not the dominant factor in determining the growth kinetics. The absence of an explicit trend in the experiment (Fig. 3a) can be reasonably explained by these two aspects. We can also simulate the k g -dependent growth rate for the condition corresponding to the ethanol CVD that yields γ g,1 (the upper panel of Fig. 3e). At the reduced P C /P e ratio, the relative dispersion at a fixed k g is even larger, in good agreement with our previous study [22]. According to the above comparison, the clear (n,m) dependence in growth rates should appear when both P e and P C /P e are increased, and the corresponding distribution of simulated growth rates is shown in the lower panel of Fig. 3e. Experimentally, such a growth condition is often seen in the ethanol-based AP-CVD, where a large amount of acetylene is generated through a longtime dwelling in the hot zone of furnace ( Fig. S2 and S3).
To test the above hypothesis, we conduct another isotope labeling experiment, where we emulate the conditions for the AP-CVD [30] by introducing ethanol without a carrier gas (Fig. 3f). As the total pressure is similar to the CVD process for Fig. 2 despite the absence of buffer gases, the partial pressure of ethanol and the gas heating time are 10× larger. In this experiment, after reducing the catalyst in Ar/H 2 atmosphere, we start the synthesis only with 12 C ethanol and gradually increase the fraction of 13 C ethanol. Figure 3g shows the peak frequency of G-band obtained at different positions along a nanotube, slope of which can be converted into a growth rate. As shown in Fig. 3h, the growth rate is more clearly correlated to the chiral angle, though armchair (n,n) nanotubes [14] and a few s-SWCNTs deviate from the overall trend (see Fig. S2). Compared with the result in Fig. 3a, the growth rate with a small chiral angle is relatively suppressed in this growth regime.
Kinetic regimes depending on gas compositions. By taking gas compositions as explanatory variables, we can generalize the above discussions to gain a full picture of growth kinetics. Figure 4a shows the growth rate at the fixed k g and k e as a function of P C and P e . In addition, the sensitivities S g (S e ) of γ g to the change in k g (k e ) are obtained from the equations S g = ∂γg ∂kg kg |γg| (S e = ∂γg ∂ke ke |γg| ). Similar two-dimensional maps for S g and S e are provided in Fig. S12. Using these sensitivities, we define the relative dominance of k g in determining growth rates by S g/e = S g + ζS e (a weight coefficient ζ of 0.2 is chosen) as shown in Fig. 4b. A large S g/e requires a large P C /P e ratio and a large P e to suppress S e while keeping S g large.
When hydrocarbon is supplied and P e is negligible, the growth rate is simply limited by the carbon supply rate. All nanotubes thus grow at a similar rate as seen in the previous studies ("homogeneous rate regime") [41][42][43]. In this regime, a smaller k g results in a short growth lifetime due to a large N , and the (n,m) dependence may appear rather in the lifetime difference [23]. In contrast, "randomly dispersed rate regime" should be observed in the low-pressure ethanol CVD, where P C and P e are small. Growth rates are largely influenced by k e , an uncontrolled property of catalyst-nanotube pairs, leading to a minor (n,m) dependence. Unlike the homogeneous rate regime, the catalyst with a large k e is expected to have a small N and hence a long lifetime, which can explain the negative correlation between γ g and lifetime [24] (Fig. S13). Historically, the above two regimes with a small P e appeared when one attempted to capture the suppressed growth rate under somewhat extreme conditions (e.g., in-situ electron microscopy) [23,44]. This may ironically led to the disappearance of the chiral angle-dependent growth rate particularly in the experiments that carefully studied individual SWCNTs [20,22,24].
When the partial pressure of ethanol increases, P C and P e accordingly increases with their ratio being constant as drawn by the solid line in Fig. 4b. We have previously observed that increasing ethanol pressure alone leads to a significant tube-to-tube dispersion in the growth rate [22,45] and saturation of its average value [46] (Fig. S14). Although P C and P e are inter-related when using only ethanol, independent control of P e has been achieved by adding water vapor or methanol [26,39]. With the addition of such etching agents, the reduced P C /P e ratio results in the "metallicity selective regime", where only the nanotubes with a small k e can grow longer. As shown in Fig. 2f, m-SWCNTs have a relatively high k e , and therefore the growth of metallic nanotubes is preferentially suppressed. When the P C /P e ratio becomes even smaller, all nanotubes will be shortened ("shrinking regime") as experimentally observed in the literature [29]. It is noteworthy that growth and shrinkage rates have different dependencies on k g and k e [37] (Fig. S13).
Finally, the chiral-angle dominant rate regime [21] emerges when both the P C /P e ratio and P e are large. The white circles in Fig. 4 corresponding to the growth condition for Fig. 3f-h falls under this regime. Note that when CO is used as a precursor, the reverse reaction of disproportionation easily happens, resulting in the preferred growth of near-armchair species from liquid catalysts without explicitly adding etching agents [23,47,48]. As the previous studies that reported the growth of (n,m)specific nanotubes employed the ethanol bubbler-based AP-CVD [15,16], we suspect that the growth rates highest in a (2n,n) chirality indeed played an important role in solid catalyst systems, in addition to a thermodynamic preference.
In conclusion, we have proposed the simple but universal kinetic model of nanotube growth and provided the experimental supports. Whereas the susceptivity to carbon removal governs the growth rates in the lowpressure ethanol CVD, the kinetic model predicts the emergence of the chirality-dependent growth rates with the increased ethanol pressure. Furthermore, by general- izing the catalytic growth from the viewpoints of carbon supply and removal, we are able to comprehensively explain various experimental results that previously seemed contradictory. Such kinetic maps could be extended for a wide range of temperature and catalyst systems. For the selective growth with the purity level required for logic IC applications, the dependence of k ad and k e on the electronic structure of catalyst-nanotube systems also needs to be studied to boost kinetic selectivity. To capture transient phenomena at an early growth stage, atomicscale experiments and theoretical study should be performed, which will provide rational strategies for the control over SWCNT arrays when combined with the present findings.
METHODS
Carbon nanotube growth. SWCNT arrays are grown on r-cut single-crystal quartz substrates (Hoffman Materials Inc.) [22], and air-suspended SWCNTs are grown across trenches that are formed on SiO 2 /Si substrates by dry etching [31]. Iron catalysts with a typical thickness of 0.1 nm are evaporated in lithographically patterned areas. The catalyst is reduced in an Ar atmosphere containing 3% H 2 at 800 • C for 10 min, followed by the supply of carbon sources. Total pressure during the growth is 1.3-1.5 kPa and 110 kPa for the LP-and AP-CVD process, respectively. The flow rates of Ar/H 2 are 50 and 500 sccm for isotope labeling experiments and normal growth experiments without labeling, respectively. In the AP-CVD process, ethanol is supplied from a bubbler kept at 5 • C using Ar/H 2 as a carrier gas. After a certain growth duration, the carbon source supply is stopped, and the furnace is cooled to room temperature. Ethanol and acetylene with the natural isotope ratio ( 12 C ∼99%), as well as 13 C-enriched ethanol (Cambridge Isotope Laboratories, Inc., 1,2-13 C 2 , 99%) are used as carbon sources. Details for each growth condition are summarized in Table S1.
Raman mapping and spectroscopy of aligned nanotubes. Raman spectroscopy is performed for nanotube samples transferred to SiO 2 /Si substrates. SWCNT arrays are transferred via poly(methyl methacrylate) to Si with a 100-nm-thick oxide layer. To locate SWCNTs, metallic markers (Ti and Pt) are patterned on SiO 2 /Si substrates before the transfer of the SWCNTs. We use Raman spectrometry (Renishaw, inVia) to determine the types and positions of isotope labels in SWCNTs transferred to SiO 2 /Si substrates and then converted them to time evolution of SWCNT lengths [22]. Raman spectra are obtained in 0.6 µm steps along directions both parallel and perpendicular to the SWCNT orientation. Excitation wavelengths of 488 and 532 nm are mainly used. This is because the strong power of the available laser enabled efficient Raman mapping measurement, and the photon energies are resonant with SWCNTs of various chirality grown under the current growth conditions. Typically, each Raman spectrum is acquired over 5 s with excitation by a ∼ 0.6 µm wide and ∼ 17 µm long laser spot with an intensity of ∼ 30 mW (power density ∼ 3 × 10 5 W/cm 2 ).
PL measurement of air-suspended nanotubes. A homebuilt confocal microscopy system is used to perform PL measurements at room temperature in air [30,31].
We use a wavelength-tunable Ti:sapphire laser for excitation. The laser beam is focused on the samples using an objective lens with a numerical aperture of 0.65, and a working distance of 4.5 mm. PL is collected through the same objective lens and detected using a liquid-nitrogen-cooled InGaAs diode array attached to a spectrometer. For the quick chirality assignment, excitation wavelengths of 780, 850, and 910 nm are used, which are nearly resonant to a wide range of nanotubes with diameters between 0.98-1.36 nm. Excitation powers of 10-20 µW are used (see Fig. S1).
FIG. 1 .
1g = k g ΔN Γ C = Ak ad P C Γ e = Ak e P e N N = N eq +ΔN From chirality distribution of ensembles to growth kinetics by "carbon bookkeeping" of individual nanotubes. (a)
. 2. "Carbon bookkeeping" to elucidate the dominance of carbon removal in low-pressure CVD. (a) Switching of feedstock gases during the nanotube growth (upper panel) and a corresponding time evolution of a nanotube length traced by a Raman mapping (lower panel). In addition to Ar/H2 buffer gas at 50 sccm, 12 C ethanol ( 12 C2H5OH), 13 C ethanol, and 12 C acetylene ( 12 C2H2) are introduced. Acetylene also serves as a second labeling agent that modulates the isotope ratio in nanotubes, allowing identification of the growth time.(b) Raman mapping image showing the peak position of G-band and its peak area. Catalysts are placed at the top of the image. Excitation wavelength is 532 nm. Scale bar is 20 µm. (c) The PC multiplication factor λ versus the growth rate acceleration γg,i/γg,1 (i = 2, 3). The λ values are obtained from G-band downshifts that originate from the mixture of 12 C acetylene and 13 C ethanol. Inset: Typical Raman spectra of the nanotube in (a) measured at four different positions. Black lines are the Lorentzian fits. (d) (Left panel) Experimental growth rates before (γg,1) and after (γg,2) the addition of acetylene with two different flow rates (0.077 and 0.092 sccm). (Right panel) Simulated growth rates for comparison at different levels of λ (=2.39 and 3.28). (e) Contribution of carbon adsorption rate γC and removal rate −γe on the catalyst nanoparticles to the growth rate γg,1. Data only for s-SWCNTs are plotted. (f) The ratio of removal rate γe to carbon adsorption rate γC in the growth of semiconducting (upper) and metallic (lower) nanotubes.
FIG. 3 .
3Absence and emergence of chiral angle-dependent growth kinetics. (a) Growth rate γg,2 right after the acetylene addition as a function of the chiral angle. (b,c) Diameter (b) and chiral angle (c) distribution of analyzed SWCNTs. (d) Deterministic growth rate (red solid line) and simulated growth rate distribution (color contour map) as a function of the kinetic constant kg of the catalyst-nanotube systems. The parameters that reproduce the growth rate distribution in Fig. 2d (λ = 2.39) are used with kg uniformly distributed between 0 and 1.7, where kg = 0.85 corresponds to the average in the experiments. (e) Simulated growth rate distribution with the conditions that emulate the purely ethanol-based CVD (upper panel) and with 15× carbon sources and 10× etching agents (lower panel). (f) Total pressure and ethanol flow rate during the isotope labeling CVD, where 13 C ratio monotonically increases with time. As the total flow rate is small, extended dwell time of ethanol in the furnace leads to an increased PC/Pe ratio due to acetylene generation (Fig. S3). (g) Typical G-band peak position ωG plotted against the position along the tube axis. (h) Growth rate of the nanotube grown with the isotope labeling as a function of the chiral angle. Orange circles and blue diamonds represent s-and m-SWCNTs, respectively. The gray line is a guide for the eye.
FIG. 4 .
4Growth kinetic regimes that depend on pressures of carbon sources and etching agents. (a) Model-based growth rate as a function of the carbon source pressure PC and etching agent pressure Pe. Equivalent pressures of acetylene and water are based on PC increase by added acetylene and the decomposition simulation of ethanol (Fig. S3), respectively. (b) Dominance of the kinetic constant for growth S g/e in determining growth rate as a function of PC and Pe. Diamond, triangle, and circle marks represent the growth conditions that yield γg,1 and γg,2 in Fig. 2a, and γg in Fig. 3h, respectively. Dashed line and solid line represent the balancing point with zero growth rates and the same PC/Pe ratio as in the ethanol-based LP-CVD (γg,1), respectively.
O. and S.M. conceived the project and designed the experiments. K.O., R.I., and A.K. synthesized the samples and carried out the Raman spectroscopy measurements. K.O. and Y.K.K. performed photoluminescence spectroscopy. K.O., R.I., and A.K. analyzed data. K.O. proposed the growth kinetic model and wrote the manuscript. T.I., R.X., S.C., Y.K.K., and S.M. commented on the manuscript.
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[] |
[
"A NOTE ON BRIDGELAND'S HALL ALGEBRA OF TWO-PERIODIC COMPLEXES",
"A NOTE ON BRIDGELAND'S HALL ALGEBRA OF TWO-PERIODIC COMPLEXES"
] |
[
"Shintarou Yanagida "
] |
[] |
[] |
We show that the Hall algebra of two-periodic complexes, which is recently introduced by T. Bridgeland, coincides with the Drinfeld double of the ordinary Hall bialgebra.
|
10.1007/s00209-015-1573-x
|
[
"https://arxiv.org/pdf/1207.0905v1.pdf"
] | 118,616,808 |
1207.0905
|
0be19904bbc88b978761b9744723e667622766b2
|
A NOTE ON BRIDGELAND'S HALL ALGEBRA OF TWO-PERIODIC COMPLEXES
4 Jul 2012
Shintarou Yanagida
A NOTE ON BRIDGELAND'S HALL ALGEBRA OF TWO-PERIODIC COMPLEXES
4 Jul 2012
We show that the Hall algebra of two-periodic complexes, which is recently introduced by T. Bridgeland, coincides with the Drinfeld double of the ordinary Hall bialgebra.
0. Introduction 0.1. The main object of this paper is the Hall algebra of Z 2 (:= Z/2Z)-graded complexes, which was introduced by Bridgeland [1].
Let A be an abelian category over a finite field k := F q with finite dimensional morphism spaces. Let P ⊂ A be the subcategory of projective objects. Let C(A) ≡ C Z2 (A) be the abelian category of Z 2 -graded complexes in A. An object of C(A) is of the form
M 1 f / / M 2 g o o , f • g = 0, g • f = 0.
Let C(P) be the subcategory of complexes consisting of projectives, and H(C(P)) be its Hall algebra. One can introduce the twisted Hall algebra H tw (C(P)) as the twisting of H(C(P)) by the Euler form of A. In [1], Bridgeland introduced an algebra DH(A), which is the localization of the twisted Hall algebra H tw (C(P)) by the set of acyclic complexes:
DH(A) := H tw (C(P)) [M • ] −1 | H * (M • ) = 0 .
The purpose of this note is to show the following theorem, which was stated in [ Then the algebra DH(A) is isomorphic to the Drinfeld double of the bialgebra H(A) as an associative algebra.
Here H(A) is the (ordinary) extended Hall bialgebra, which will be recalled in § §1.1.1 -1.1.4. For the review of the Drinfeld double, see §1. 1.5. The proof will be explained in §2.
The organization of this note is as follows. In § 1, we review Bridgeland's theory [1] and prepare notations and statements which are necessary for the proof of the main theorem. In the subsection §1.1, we recall the ordinary theory of Hall algebra introduced by Ringel [10]. The next subsection §1.2 is devoted to the recollection of Bridgeland's theory.
The section §2 is devoted to the proof of the main theorem. We close this note by mentioning some consequences of the theorem in §3.
Notations and conventions.
We indicate several global notations. k := F q is a fixed finite field unless otherwise stated, and all the categories will be k-linear. We choose and fix a square root t := √ q. For an abelian category A, we denote by Obj(A) the class of objects of A. For an object M of A, the class of M in the Grothendieck group K(A) is denoted by M . Let K ≥0 (A) ⊂ K(A) be the subset of K(A) consisting of the classes A ∈ K(A) of A ∈ A (rather than the formal differences of them).
For an abelian category A which is essentially small, the set of its isomorphism classes is denoted by Iso(A).
For a complex M • = (· · · → M i di − → M i+1 → · · · ) in an abelian category A, its homology is denoted by
H * (M • ).
For a set S, we denote by |S| its cardinality.
H(A) := A∈Iso(A) C[A]
linearly spanned by symbols [A] with A running through the set Iso(A) of isomorphism classes of objects in A.
Definition/Fact 1.2 (Ringel [10]). The following operation defines on H(A) the structure of a unital associative algebra over C:
[A] ⋄ [B] := C∈Iso(A) | Ext 1 A (A, B) C | | Hom A (A, B)| · [C]. (1.1) Here Ext 1 A (A, B) C ⊂ Ext 1 A (A, B)
is the set parametrizing extensions of B by A with the middle term isomorphic to C.
A, B := i∈Z (−1) i dim k Ext i A (A, B), (1.2)
where the sum is finite by our assumptions on A. As is well known, this form descends to the one on the Grothendieck group K(A) of A, which is denoted by the same symbol as (1.2):
·, · : K(A) × K(A) −→ Z.
We will also use the symmetrized Euler form:
(·, ·) : K(A) × K(A) −→ Z, (α, β) := α, β + β, α .
Definition/Fact 1.4.
(1) The twisted Hall algebra H tw (A) is the same vector space as H(A) with the twisted multiplication
[A] * [B] := t A, B · [A] ⋄ [B]
for A, B ∈ Iso(A). Here t := √ q is the fixed square root of q. Then the algebra H(A) is naturally graded by the Grothendieck group K(A) of A:
H(A) = α∈K(A) H(A)[α], H(A)[α] := A=α C[A]. For α, β ∈ K(A), set H(A)[α] ⊗ C H(A)[β] := A=α, B=β C[A] ⊗ C C[B], H(A) ⊗ C H(A) := α,β∈K(A) H(A)[α] ⊗ C H(A)[β]
.
Thus H(A) ⊗H(A) is the space of all formal linear combinations
A,B c A,B · [A] ⊗ [B].
This tensor product ⊗ is called a completed tensor product.
Definition/Fact 1.6.
(1) (Green [7]) The following maps
∆ : H(A) −→ H(A) ⊗ C H(A), ǫ : H(A) −→ C define a topological coassociative coalgebra structure on H(A): ∆([A]) := B,C t B,C | Ext A (B, C) A | | Aut A (A)| · [B] ⊗ [C], ǫ([A]) := δ A,0 . (1.3)
(2) (Xiao [14]) On the extended algebra, defining the maps
∆ : H(A) −→ H(A) ⊗ C H(A), ǫ : H(A) −→ C by (1.3) and ∆(K α ) := K α ⊗ K α , ǫ(K α ) := 1,
one has a topological coassociative coalgebra structure on H(A).
Here the word topological means that everything should be considered in the completed space. For example, the coassociativity in (1) means that the two maps (∆ ⊗ 1)
⊗ ∆ and (1 ⊗ ∆) ⊗ ∆ from H(A) to H(A) ⊗H(A) ⊗H(A) coincide.
1.1.4. Bialgebra structure and Hopf pairing. Now we have an algebra structure and a coalgebra structure on H(A) (and on H(A)). In order that these structures are compatible and give a bialgebra structure, we must impose one more condition on A. This bialgebra structure on H(A) has an additional feature, that is, it is self-dual. The self-duality is stated in terms of a natural nondegenerate bilinear form, called Hopf pairing.
Definition/Fact 1.8 (Green, [7]). Assume that the abelian cateogry A satisfies the conditions (a), (b), (c'), (d) and (e).
(1) The non-degenerate bilinear form
(·, ·) H : H(A) ⊗ C H(A) −→ C given by ([A], [B]) H := δ A,B | Aut A (A)|
is a Hopf pairing on the bialgebra H(A), that is, for any x, y, z ∈ Iso(A), one has
(1.4) (x ⋄ y, z) H = (x ⊗ y, z) H .
(2) The non-degenerate bilinear form
(·, ·) H : H(A) ⊗ C H(A) −→ C given by ([A]K α , [B]K β ) H := δ A,B | Aut A (A)| t (α,β)
is a Hopf pairing on the bialgebra H(A), that is, for any x, y, z ∈ Iso(A), one has
(1.5) (x * y, z) H = (x ⊗ y, z) H .
Remark 1.9. In the right hand sides of (1.4) and (1.5), we used the usual pairing on the product space:
(x ⊗ y, z ⊗ w) H := (x, z) H · (y, w) HH −→ H ⊗ C H, a −→ a ⊗ 1 and H −→ H ⊗ C H, a −→ 1 ⊗ a are injective homomorphisms of C-algebras.
(2) For all elements a, b ∈ H, one has
(a ⊗ 1) • (1 ⊗ b) = a ⊗ b.
(3) For all elements a, b ∈ H, one has
(1.6) (a (2) , b (2) ) H · a (1) ⊗ b (1) = (a (1) , b (1) ) H · (1 ⊗ b (2) ) • (a (2) ⊗ 1).
Here we used Sweedler's notation: ∆(a) = a (1) ⊗ a (2) .
Remark 1.11. If H is a topological bialgebra, then one should replace the tensor product ⊗ in the statement by the completed one ⊗.
1.2.
Hall algebras of complexes. We summarize necessary definitions and properties of Hall algebras of Z 2 -graded complexes. Most of the materials were introduced or shown in [1]. In this subsection §1.2, A denotes an abelian category satisfying the following three conditions.
M 1 d1 / / M 0 d0 o o , d i+1 • d i = 0.
Hereafter indices in the diagram of an object in C Z2 (A) are understood by modulo 2. A morphism s • : M • → N • consists of a diagram This involution shifts the grading and changes the sign of the differential as follows:
M 1 s1 d1 / / M 0 d0 o o s0 N 1 d ′ 1 / / N 0 d ′ 0 o o with s i+1 • d i = d ′ i • s i . Two morphisms s • , t • : M • → N • are said to be homotopic if there are morphisms h i : M i → N i+1 such that t i − s i = d ′ i+1 • h i + h i+1 • d i .M • = M 1 d1 / / M 0 d0 o o * ←→ M * • = M 0 −d0 / / M 1 −d1 o o
Now let us recall
Fact 1.12 ([1, Lemma 3.3]). For M • , N • ∈ C(P) we have Ext 1 C(A) (N • , M • ) ∼ = Hom Ho(A) (N • , M * • ). A complex M • ∈ C(A) is called acyclic if H * (M • ) = 0.
To each object P ∈ P, we can attach acyclic complexes
K P • = P Id / / P 0 o o , K P * • = P 0 / / P −Id o o .
Remark 1.13. The complexes K P • , K P * • are denoted by K P , K * P in [1]. Let us recall the following fact shown in [1].
H(C(P)) ⊂ H(C(A))
be the subspace spanned by complexes of projective objects.
Define H tw (C(P)) to be the same vector space as H(C(P)) with the twisted multiplication
(1.7) [M • ] * [N • ] := t M0,N0 + M1,N1 · [M • ] ⋄ [N • ].
Now let us recall the simple relations satisfied by the acyclic complexes K P • : . For any object P ∈ P and any complex M • ∈ C(P), we have the following relations in H tw (C(P)):
[K P • ] * [M • ] = t P , M• · [K P • ⊕ M • ], [M • ] * [K P • ] = t − M•, P · [K P • ⊕ M • ], [K P • ] * [M • ] = t ( P , M•) · [M • ] * [K P • ], [K P * • ] * [M • ] = t −( P , M•) · [M • ] * [K P * • ]
In particular, for P, Q ∈ P we have As explained in [1, §3.6], this is the same as localizing by the elements [K P • ] and [K * P • ] for all objects P ∈ P. For an element α ∈ K(A), we define
[K P • ] * [K Q • ] = [K P • ⊕ K Q • ], [K P • ] * [K Q * • ] = [K P • ⊕ K * Q • ], [[K P • ], [K Q • ]] = [[K P • ], [K Q * • ]] = [[K P * • ], [K Q * • ]] = 0.K α := [K P • ] * [K Q • ] −1 , K * α := [K P * • ] * [K Q * • ] −1 ,
where we expressed α = P − Q using the classes of some projectives P, Q ∈ P. This is well defined by Fact 1.15 Remark 1.16. We will denote two different elements K α ∈ H(A) and K α ∈ DH(A) by the same symbol, following [1].
By Fact 1.15, we immediately have Corollary 1.17. In the algebra DH(A), we have and decomposing P and Q into finite direct sums P = ⊕ i P i , Q = ⊕ j Q j , one may write f = (f ij ) in matrix form with f ij : P i → Q j . The resolution (1.9) is said to be minimal if none of the morphisms f ij is an isomorphism.
(1) K α * M • = t (α, M•) · M • * K α , K * α * M • = t −(α, M•) · M • * K α , for arbitrary α ∈ K(A) and M • ∈ C(P). (2) [K α , K β ] = [K α , K * β ] = [K * α , K * β ] = 0 for arbitrary α, β ∈ K(A).0 → R ⊕ P ′ 1⊕f ′ − −− → R ⊕ Q ′ (0,g ′ ) −−−→ A → 0
with some object R ∈ P and some minimal projective resolution
0 → P ′ f ′ − → Q ′ g ′ − → A → 0. Definition 1.20 ([1, §4.2])
. Given an object A ∈ A, take a minimal projective resolution
0 → P A fA − − → Q A g − → A → 0, (1.10)
We define a Z 2 -graded complex
C A• := P A fA / / Q A 0 o o ∈ C(P).
Remark 1.21. The complex C A• is denoted as C A in [1].
By Fact 1.19, arbitrary two minimal projective resolutions of A are isomorphic, so the complex C A• is well-defined up to isomorphism.
M • = C A• ⊕ C B * • ⊕ K P • ⊕ K Q * • .
Moreover, the objects A, B ∈ A and P, Q ∈ P are unique up to isomorphism.
.4]). Given an object A ∈ A, we define elements E A• , F A• ∈ DH(A) by
E A := t P , A · K − P * [C A• ], F A := E * A .
Here we used a projective decomposition (1.10) of A and the associated complex C A• in Definition 1.20. defined by
[A] −→ E A (A ∈ Iso(A)), K α −→ K α (α ∈ K(A)).
By composing I e + and the involution * , we also have an embedding Let us write the equation (1.6) in the present situation: (2) ) * I e + (a (2) ).
I e − : H(A) ֒−→ DH(A) defined by [A] −→ F A (A ∈ Iso(A)), K α −→ K * α (α ∈ K(A)).(2.1) (a (2) , b (2) ) H · I e + (a (1) ) * I e − (b (1) ) ? = (a (1) , b (1) ) H · I e − (b
What we must do is to check the equation for the cases
(2.2) (1) (a, b) = (K α , K β ), (2) (a, b) = ([A], K β ), (2 ′ ) (a, b) = (K α , [B]), (3) (a, b) = ([A], [B])
with arbitrary α, β ∈ K(A) and A, B ∈ Iso(A).
2.1.
Case (1). It is easy to check the equation (2.1) for the case (1) in (2.2). Since ∆(K α ) = K α ⊗ K α , the equation in this case becomes
(K α , K β ) H · K α * K * β ? = (K α , K β ) H · K * β * K α ,
which is valid by Corollary 1.17 (2). (2) and (2'). The cases (2) and (2') in (2.2) are trivial. In fact, for the case (2), we may write
Cases
∆([A]) = A1,A2 g A A1,A2 · [A 1 ] ⊗ [A 2 ]
with some g A A1,A2 ∈ C and ∆(K β ) = K β ⊗ K β . Then (2.1) reads
Here the complex M • sits in the commutative diagram
Q B1 0 / / P B1 −fB 1 o o M 1 / / M 0 o o P A1 fA 1 / / Q A1 0 o o
where both columns give short exact sequences in A. Then, since P A1 and Q A1 are projective, we have
M 1 ∼ = Q B1 ⊕ P A1 and M 0 ∼ = P B1 ⊕ Q A1 . Q B1 0 / / P B1 −fB 1 o o Q B1 ⊕ P A1 f1 / / P B1 ⊕ Q A1 f0 o o P A1 fA 1 / / Q A1 0 o o (2.7)
Now the commutativity of the diagram restricts the morphisms M 1 → M 0 and M 0 → M 1 of the following types: Note also that
f 1 = 0 s 1 0 f A1 , f 0 = −f B1Hom C(A) (C A1 • , C B1 * • ) ∼ = Hom A (P A1 , Q B1 )
, which can be easily check. Then the term | Hom C(A) (C A1 • , C B1 * • )| at the denominator of (2.6) is equal to
(2.8) | Hom C(A) (C A1 • , C B1
Here the complex N • is of the next form:
P A2 f A 2 / / Q A2 0 o o P A2 ⊕ Q B2 f ′ 1 / / Q A2 ⊕ P B2 f ′ 0 o o Q B2 0 / / P B2 −f B 2 o o (2.11) where f ′ 1 = f A2 s ′ 1 0 0 , f ′ 0 = 0 s ′ 0 0 −f B2 with s ′ 1 : Q B2 → Q A2 and s ′ 0 : P B2 → P A2 .Q A2 = Q B1 , Q B2 = P B1 , P B2 = P A1 , P A2 = Q A1 and (2.13) f B2 = −s 1 , f A2 = s 0 , s ′ 1 = −f B1 , s ′ 0 = f A1 .
Then from (2.7) and (2.11), we have the next combined diagram
P A1 fA 1 / / _ −f B 2 =s1 Q A1 / / / / f A 2 =s ′ 0 A 1 P B1 fB 1 / / Q B1 / / / / B 1 B 1 A 2
where all the columns and rows are short exact. We also have the short exact sequences
A 2 / / A / / A 1 A 2 / / A / / A 1 B 2 / / B / / B 1 B 2 / / B / / B 1
Considering the classes in the Grothendieck group K(A), we have the relations
A 1 = A − A 2 = A − ( Q B1 − Q A1 ) = B 1 = B − B 2 = B − ( P B1 − P A1 )
and
A 2 = A − A 1 = A − ( Q A1 − P A1 ) = B 2 = B − B 1 = B − ( Q B1 − P B1 ).
Then we have
A − B = ( Q B1 − Q A1 ) − ( P B1 − P A1 ) = ( Q A1 − P A1 ) − ( Q B1 − P B1 ),
so that in K(A) we have (2.14)
Q B1 + P A1 = Q A1 + P B1 , A 1 = B 1 , A 2 = B 2 .
Now we will finish the proof. By the correspondences (2.12) and (2.13), we have only to check that the coefficients
(2.15) A 1 + B 1 , A 2 + P A1 , A 1 + P B1 , B 1 − (P B1 , A 1 ) − P A1 , Q B1 + Q A1 , P B1
in (2.9) and
(2.16)
A 1 , A 2 + B 2 + P B2 , B 2 + P A2 , A 2 − (P A2 , B 2 ) + Q B2 , P A2 − P B2 , Q A2
in (2.10) coincide. But using (2.14) we have
(2.15) = A 1 + B 1 , A − A 1 + B 1 , A 1 + P A1 , A 1 + P B1 , B 1 − P B1 , A 1 − A 1 , P B1 − P A1 , Q B1 + Q A1 , P B1 = 2 A 1 , A 2 + P A1 , A 1 − A 1 , P B1 − P A1 , Q B1 + Q A1 , P B1 = 2 A 1 , A 2 + P A1 , A 1 − Q B1 + Q A1 − A 1 , P B1 = 2 A 1 , A 2 + P A1 , − P B1 + P A1 , P B1 = 2 A 1 , A 2 .
A similar calculation gives
(2.16) = 2 A 1 , A 2 .
Thus the proof is completed.
Concluding remarks
As mentioned in [1, §1.4], the work of Cramer [3] and Theorem 1.26 yield the following:
The unit is given by [0], where 0 is the zero object of A. This algebra (H(A), ⋄, [0]) is called the Hall algebra of A. Below we will denote it by H(A) for simplicity.Remark 1.3. We follow [1] to choose | Ext 1 A (A, B) C |/| Hom A (A, B)| for the structure constant of the multiplication. It is proportional to the usual structure constant |{B ′ ⊂ C | B ′ ∼ = B, C/B ′ ∼ = A}| appearing in [10] and [11]. See [1, §2.3] for the detail.
( 2 )
2The extended Hall algebra H(A) is defined as an extension of H tw (A) by adjoining symbols K α for classes α ∈ K(A), and imposing relationsK α * K β = K α+β , K α * [B] = t (α, B) · [B] * K αfor α, β ∈ K(A) and B ∈ Iso(A). Note that H(A) has a vector space basis consisting of the elements K α * [B] for α ∈ K(A) and B ∈ Iso(A). Remark 1.5. In[1], the extended Hall algebra is denoted by H e tw (A).1.1.3. Green's coproduct.To introduce a coalgebra structure, one should consider a completion of the algebra. Assume that the abelian category A satisfies the conditions (a), (b), (c') and(e) nonzero object defines nonzero class in the Grothendieck group.
Fact 1. 7 (
7Green [7], Xiao[14]). Assume that the abelian category A satisfies the conditions (a), (b), (c'), (e) and(d) hereditary, that is, of global dimension at most 1. Then the tuples (H(A), ⋄, [0], ∆, ǫ), ( H(A), * , [0], ∆, ǫ) are topological bialgebras defined over C. That is,the map ∆ : H(A) → H(A) ⊗H(A) and ∆ : H(A) → H(A) ⊗ H(A) are homomorphisms of C-algebras. Below, we simply denote by H(A) and H(A) the bialgebras (H(A), ⋄, [0], ∆, ǫ) and ( H(A), * , [0], ∆, ǫ) respectively.
(a) essentially small with finite morphism spaces, (b) linear over k, (c) of finite global dimension and having enough projectives. 1.2.1. Categories of two-periodic complexes. We shall recall the basic definitions in [1, §3.1]. Let C Z2 (A) be the abelian category of Z 2 -graded complexes in A. An object M • of this category consists of the following diagram in A:
For an object M • ∈ C Z2 (A), we define its class in the K-group byM • := M 0 − M 1 ∈ K(A).Denote by Ho Z2 (A) the category obtained from C Z2 (A) by identifying homotopic morphisms. Let us also denote by C Z2 (P) ⊂ C Z2 (A), the full subcategories whose objects are complexes of projectives in A. Hereafter we drop the subscript Z 2 and just write C(A) := C Z2 (A), C(P) := C Z2 (P), Ho(A) := Ho Z2 (A). The shift functor [1] of complexes induces an involution C(A) * ←→ C(A).
Fact 1 .
114 ([1, Lemma 3.2]). For each acyclic complex of projectives M • ∈ C(P), there are objects P, Q ∈ P, unique up to isomorphism, such that M • ∼ = K P • ⊕ K Q * • . 1.2.2. Definition of Hall algebras of complexes. Let H(C(A)) be the Hall algebra of the abelian category C(A) defined in §1.1. As noted in [1, §3.5], this definition makes sense since the spaces Ext 1 C(A) (N • , M • ) are all finite-dimensional by Fact 1.12. Let
last line we used the commutator [x, y] := x * y − y * x.
Bridgeland's Hall algebra. Now we can introduce the main object: Bridgeland's Hall algebra. We define the localized Hall algebra DH(A) to be the localization of H tw (C(P)) with respect to the elements [M • ] corresponding to acyclic complexes M • : DH(A) := H tw (C(P)) [M • ] −1 | H * (M • ) = 0 .
1. 3 .
3Hereditary case. In this subsection §1.3, we assume that A satisfies the conditions (a), (b), (c) and the following additional ones: (d) A is hereditary, that is of global dimension at most 1, (e) nonzero objects in A define nonzero classes in K(A). Then by [1, §4] we have a nice basis for DH(A). To explain that, let us recall the minimal resolution of objects of A. 1.3.1. Minimal resolution and the complex C A• . Definition 1.18 ([1, §4.1]). Assume the conditions (a),(c),(d) on A. Then every object A ∈ A
Fact 1 .
119 ([1, Lemma 4.1]). Any resolution (1.9) is isomorphic to a resolution of the form
Fact 1 .
122 ([1, Lemma 4.2]). Every object M • ∈ C(P) has a direct sum decomposition
( 2 )
2The multiplication map m : a ⊗ b −→ I e + (a) * I e − (b) defines an isomorphism of vector spaces m : H(A) ⊗ C H(A) ∼ − − → DH(A).As a corollary, we have Corollary 1.25. DH(A) has a basis consisting of elementsE A * K α * K * β * F B , A, B ∈ Iso(A), α, β ∈ K(A).1.4. Main theorem. Now we can state our main theorem. Theorem 1.26. Assume that the abelian category A satisfies the conditions (a)-(e). Then the algebra DH(A) is isomorphic to the Drinfeld double of the bialgebra H(A). 2. Proof of the main theorem Because of the description of the basis of DH(A) (Corollary 1.25) and the definition of Drinfeld double (Fact 1.10), the proof of Theorem 1.26 is reduced to check the equation (1.6) for the elements consisting of the basis of H(A).
1 : P A1 → P B1 and s 0 : Q A1 → Q B1 (see also the argument in the proof of [1, Lemma 3.3]).
Corollary 3. 1 .
1For the hereditary abelian cateogry, the algebra DH(A) is functorial with respect to derived invariance.Precisely speaking, let A and B be two abelian categories satisfying conditions (a)-(e). Assume that the bounded derived categories D b (A) and D b (B) of A and B are equivalent by the functor Φ:Φ : D b (A) ∼ −−→ D b (B).Then one can construct an algebra isomorphismΦ DH : DH(A) ∼ −−→ DH(B),and this construction is functorial:(Φ 1 • Φ 2 ) DH = Φ DH 1 • Φ DH 2 for all equivalences Φ 1 , Φ 2 .Now set T := D b (A) for some hereditary abelian category A satisfying (a)-(e). We also set DH(T ) := DH(A). This algebra depends only on the triangulated cateogry T by the above corollary. Denote by Auteq(T ) the group of autoequivalences of T . Then, setting Aut DH (T ) := {Φ DH | Φ ∈ Auteq(T )}, we have an embedding of groups Aut DH (T ) ⊂ Aut(DH(T )). Let us close this note by mentioning a non-trivial example. For an elliptic curve C defined over k, set A := Coh(C), the abelian category of coherent sheaves on C. This category satisfies the conditions (a)-(e). Set T := D b (A) as above. Then by the theory of Fourier-Mukai transforms, we have a short exact sequence 0 / / Z ⊕ (C × C) / / Auteq(T ) / / SL 2 (Z) / / 0 of groups. Here Z⊕(C × C) corresponds to the subgroup of Auteq(T ) generated by the shifts [n] of complexes, the pushforward t a * by translations on C with a ∈ C, and tensor products L ⊗ (−) with L ∈ C := Pic 0 (C). The cokernel part SL 2 (Z) consists of (non-trivial) Fourier-Mukai transforms Φ E := Rp 2 * (E L ⊗ p * 1 (−)) with E ∈ D b (C ⊗ C). The generators S := 0 −1 1 0 and T := 1 −1 0 1 correspond to the Fourier-Mukai transforms Φ E0 and L ⊗ (−), where E 0 is the Poincaré bundle on C ⊗ C, and L ∈ Pic(C) is a degree one line bundle. (See [8, §9.5] for the detailed explanation.) Now one can see that the operation DH : Φ −→ Φ DH
1, Theorem 1.2].Theorem. Assume that the abelian category A satisfies the conditions
• essentially small with finite morphism spaces,
• linear over k,
• of finite global dimension and having enough projectives,
• hereditary,
• nonzero object defines nonzero class in the Grothendieck group.
1.1.2. Euler form and extended Hall algebra. Let us recall the notations for Grothendieck group given in §0.2. For objects A, B ∈ A, the Euler form is defined by
1.1.5. Drinfeld double. Here we recall the Drinfeld double of the self-dual bialgebra. For the complete treatment of Drinfeld double construction, we refer [9, §3.2] and [11, §5.2].Fact 1.10 (Drinfeld). Let H be a C-bialgebra with a Hopf pairing (·, ·) H : H ⊗ C H → C. Then there is a unique algebra structure • on H ⊗ H satisfying the following conditions (1) The maps
Now we must compare the coefficients of the same term [M • ] = [N • ] in (2.9) and (2.10). A quick observation yields that we must have the correspondences(2.12)
But recalling the Hopf pairingin Definition/Fact 1.8, the equation readsThen the equation (2.1) readsBy Definition 1.23, it becomes = A1,A2,B1Combining (2.5), (2.6) and (2.8), the light hand side of (2.4) becomesA similar argument yields RHS of (2.4) =/ / 0 Here Z/2Z corresponds to the involution * of the algebra DH(T ). Thus SL 2 (Z) acts on DH(T ). This action is essentially the same as the SL 2 (Z)-automorphisms of the algebra appearing in[2]. The same algebra appeared in the works[4],[5],[6]and[12]. The SL 2 (Z)-automorphisms (precisely speaking, the counterpart in the degenerate algebra) play an important role in the argument of[13]in the context of the so-called AGT relation/conjecture.
T Bridgeland, arXiv:1111.0745Quantum groups via Hall algebras of complexes. Bridgeland, T., Quantum groups via Hall algebras of complexes, arXiv:1111.0745.
On the Hall algebra of an elliptic curve, I, Duke Math. I Burban, O Schiffmann, J. 1617Burban, I., Schiffmann, O., On the Hall algebra of an elliptic curve, I, Duke Math. J. 161 (2012), no. 7, 1171-1231.
Double Hall algebras and derived equivalences. T Cramer, Adv. Math. 2243Cramer, T., Double Hall algebras and derived equivalences, Adv. Math. 224 (2010), no. 3, 1097-1120.
Quantum continuous gl ∞ : tensor products of Fock modules and Wn-characters. B Feigin, E Feigin, M Jimbo, T Miwa, E Mukhin, Kyoto J. Math. 512Feigin, B., Feigin, E., Jimbo, M., Miwa, T., Mukhin, E., Quantum continuous gl ∞ : tensor products of Fock modules and Wn-characters, Kyoto J. Math. 51 (2011), no. 2, 365-392.
A commutative algebra on degenerate CP1 and Macdonald polynomials. B Feigin, K Hashizume, A Hoshino, J Shiraishi, S Yanagida, J. Math. Phys. 50942Feigin, B, Hashizume, K., Hoshino, A., Shiraishi, J., Yanagida, S., A commutative algebra on degenerate CP1 and Mac- donald polynomials, J. Math. Phys. 50 (2009), no. 9, 095215, 42 pp.
Equivariant K-theory of Hilbert schemes via shuffle algebra. B Feigin, A Tsymbaliuk, Kyoto J. Math. 514Feigin, B., Tsymbaliuk, A., Equivariant K-theory of Hilbert schemes via shuffle algebra, Kyoto J. Math. 51 (2011), no. 4, 831-854.
Hall algebras, hereditary algebras and quantum groups. J Green, Invent. Math. 1202Green, J., Hall algebras, hereditary algebras and quantum groups, Invent. Math. 120 (1995), no. 2, 361-377.
Fourier-Mukai transforms in algebraic geometry. D Huybrechts, Oxford Mathematical Monographs. Oxford University PressHuybrechts, D., Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, Oxford University Press, 2006.
Quantum groups and their primitive ideals. A Joseph, Ergebnisse der Mathematik und ihrer Grenzgebiete. 293Springer-VerlagJoseph, A., Quantum groups and their primitive ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 29, Springer-Verlag, Berlin, 1995.
Hall algebras and quantum groups. C Ringel, Invent. Math. 1013Ringel, C., Hall algebras and quantum groups, Invent. Math. 101 (1990), no. 3, 583-591.
O Schiffmann, arXiv:0611617v2Lectures on Hall algebras. Schiffmann, O., Lectures on Hall algebras, arXiv:0611617v2.
The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of A 2. O Schiffmann, E Vasserot, arXiv:0905.2555v2preprintSchiffmann, O., Vasserot, E., The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of A 2 , preprint, arXiv:0905.2555v2.
O Schiffmann, E Vasserot, arXiv:1202.2756Cherednik algebras, W algebras and the equivariant cohomology of the moduli space of instantons on A 2 , preprint. Schiffmann, O., Vasserot, E., Cherednik algebras, W algebras and the equivariant cohomology of the moduli space of instantons on A 2 , preprint, arXiv:1202.2756.
Drinfeld double and Ringel-Green theory of Hall algebras. J Xiao, J. Algebra. 1901Research Institute for Mathematical Sciences, Kyoto UniversityJapan E-mail address: [email protected], J., Drinfeld double and Ringel-Green theory of Hall algebras, J. Algebra 190 (1997), no. 1, 100-144. Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan E-mail address: [email protected]
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[] |
[
"Decentralized Data Fusion and Active Sensing with Mobile Sensors for Modeling and Predicting Spatiotemporal Traffic Phenomena",
"Decentralized Data Fusion and Active Sensing with Mobile Sensors for Modeling and Predicting Spatiotemporal Traffic Phenomena"
] |
[
"Jie Chen ",
"Kian Hsiang Low ",
"Colin Keng-Yan Tan ",
"Ali Oran ",
"Patrick Jaillet ",
"John Dolan \nDept. Computer Science\nUniversity of Southern California *\nUSA\n",
"Gaurav Sukhatme ",
"\nDept. Computer Science\nDept. Electrical Engineering and Computer Science\nMIT ‡ The Robotics Institute\nNational University of Singapore † Singapore-MIT Alliance for Research and Technology § Republic of Singapore\nCarnegie Mellon University\n\n"
] |
[
"Dept. Computer Science\nUniversity of Southern California *\nUSA",
"Dept. Computer Science\nDept. Electrical Engineering and Computer Science\nMIT ‡ The Robotics Institute\nNational University of Singapore † Singapore-MIT Alliance for Research and Technology § Republic of Singapore\nCarnegie Mellon University\n"
] |
[] |
The problem of modeling and predicting spatiotemporal traffic phenomena over an urban road network is important to many traffic applications such as detecting and forecasting congestion hotspots. This paper presents a decentralized data fusion and active sensing (D 2 FAS) algorithm for mobile sensors to actively explore the road network to gather and assimilate the most informative data for predicting the traffic phenomenon. We analyze the time and communication complexity of D 2 FAS and demonstrate that it can scale well with a large number of observations and sensors. We provide a theoretical guarantee on its predictive performance to be equivalent to that of a sophisticated centralized sparse approximation for the Gaussian process (GP) model: The computation of such a sparse approximate GP model can thus be parallelized and distributed among the mobile sensors (in a Google-like MapReduce paradigm), thereby achieving efficient and scalable prediction. We also theoretically guarantee its active sensing performance that improves under various practical environmental conditions. Empirical evaluation on real-world urban road network data shows that our D 2 FAS algorithm is significantly more time-efficient and scalable than state-ofthe-art centralized algorithms while achieving comparable predictive performance.• Analyzing the time and communication overheads of
| null |
[
"https://arxiv.org/pdf/1206.6230v2.pdf"
] | 2,551,301 |
1408.2046
|
042da39e531f55f810e0a2ba49b46ab4d2631144
|
Decentralized Data Fusion and Active Sensing with Mobile Sensors for Modeling and Predicting Spatiotemporal Traffic Phenomena
Jie Chen
Kian Hsiang Low
Colin Keng-Yan Tan
Ali Oran
Patrick Jaillet
John Dolan
Dept. Computer Science
University of Southern California *
USA
Gaurav Sukhatme
Dept. Computer Science
Dept. Electrical Engineering and Computer Science
MIT ‡ The Robotics Institute
National University of Singapore † Singapore-MIT Alliance for Research and Technology § Republic of Singapore
Carnegie Mellon University
Decentralized Data Fusion and Active Sensing with Mobile Sensors for Modeling and Predicting Spatiotemporal Traffic Phenomena
The problem of modeling and predicting spatiotemporal traffic phenomena over an urban road network is important to many traffic applications such as detecting and forecasting congestion hotspots. This paper presents a decentralized data fusion and active sensing (D 2 FAS) algorithm for mobile sensors to actively explore the road network to gather and assimilate the most informative data for predicting the traffic phenomenon. We analyze the time and communication complexity of D 2 FAS and demonstrate that it can scale well with a large number of observations and sensors. We provide a theoretical guarantee on its predictive performance to be equivalent to that of a sophisticated centralized sparse approximation for the Gaussian process (GP) model: The computation of such a sparse approximate GP model can thus be parallelized and distributed among the mobile sensors (in a Google-like MapReduce paradigm), thereby achieving efficient and scalable prediction. We also theoretically guarantee its active sensing performance that improves under various practical environmental conditions. Empirical evaluation on real-world urban road network data shows that our D 2 FAS algorithm is significantly more time-efficient and scalable than state-ofthe-art centralized algorithms while achieving comparable predictive performance.• Analyzing the time and communication overheads of
Introduction
Knowing and understanding the traffic conditions and phenomena over road networks has become increasingly important to the goal of achieving smooth-flowing, congestion-free traffic, especially in densely-populated urban cities. According to a 2011 urban mobility report (Schrank et al., 2011), traffic congestion in the USA has caused 1.9 billion gallons of extra fuel, 4.8 billion hours of travel delay, and $101 billion of delay and fuel cost. Such huge resource wastage can potentially be mitigated if the spatiotemporally varying traffic phenomena (e.g., speeds and travel times along road segments) are predicted accurately enough in real time to detect and forecast the congestion hotspots; network-level (e.g., ramp metering, road pricing) and user-level (e.g., route replanning) measures can then be taken to relieve this congestion, so as to improve the overall efficiency of road networks.
In practice, it is non-trivial to achieve real-time, accurate prediction of a spatiotemporally varying traffic phenomenon because the quantity of sensors that can be deployed to observe an entire road network is costconstrained. Traditionally, static sensors such as loop detectors (Krause et al., 2008a;Wang and Papageorgiou, 2005) are placed at designated locations in a road network to collect data for predicting the traffic phenomenon. However, they provide sparse coverage (i.e., many road segments are not observed, thus leading to data sparsity), incur high installation and maintenance costs, and cannot reposition by themselves in response to changes in the traffic phenomenon. Low-cost GPS technology allows the collection of traffic data using passive mobile probes (Work et al., 2010) (e.g., taxis/cabs). Unlike static sensors, they can directly measure the travel times along road segments. But, they provide fairly sparse coverage due to low GPS sampling frequency (i.e., often imposed by taxi/cab companies) and no control over their routes, incur high initial implementation cost, pose privacy issues, and produce highly-varying speeds and travel times while traversing the same road segment due to inconsistent driving behaviors. A critical mass of probes is needed on each road segment to ease the severity of the last drawback (Srinivasan and Jovanis, 1996) but is often hard to achieve on non-highway segments due to sparse coverage. In contrast, we propose the use of active mobile probes (Turner et al., 1998) to overcome the limitations of static and passive mobile probes. In particular, they can be directed to explore any segments of a road network to gather traffic data at a desired GPS sampling rate while enforcing consistent driving behavior.
How then do the mobile probes/sensors actively explore a road network to gather and assimilate the most informative observations for predicting the traffic phenomenon? There are three key issues surrounding this problem, which will be discussed together with the related works:
Models for predicting spatiotemporal traffic phenomena. The spatiotemporal correlation structure of a traffic phenomenon can be exploited to predict the traffic conditions of any unobserved road segment at any time using the observations taken along the sensors' paths. To achieve this, existing Bayesian filtering frameworks (Chen et al., 2011;Wang and Papageorgiou, 2005;Work et al., 2010) utilize various handcrafted parametric models predicting traffic flow along a highway stretch that only correlate adjacent segments of the highway. So, their predictive performance will be compromised when the current observations are sparse and/or the actual spatial correlation spans multiple segments. Their strong Markov assumption further exacerbates this problem. It is also not shown how these models can be generalized to work for arbitrary road network topologies and more complex correlation structure. Existing multivariate parametric traffic prediction models (Kamarianakis and Prastacos, 2003;Min and Wynter, 2011) do not quantify uncertainty estimates of the predictions and impose rigid spatial locality assumptions that do not adapt to the true underlying correlation structure.
In contrast, we assume the traffic phenomenon over an urban road network (i.e., comprising the full range of road types like highways, arterials, slip roads) to be realized from a rich class of Bayesian non-parametric models called the Gaussian process (GP) (Section 2) that can formally characterize its spatiotemporal correlation structure and be refined with growing number of observations. More importantly, GP can provide formal measures of predictive uncertainty (e.g., based on variance or entropy criterion) for directing the sensors to explore highly uncertain areas of the road network. Krause et al. (2008a) used GP to represent the traffic phenomenon over a network of only highways and defined the correlation of speeds between highway segments to depend only on the geodesic (i.e., shortest path) distance of these segments with respect to the network topology; their features are not considered. Neumann et al. (2009) maintained a mixture of two independent GPs for flow prediction such that the correlation structure of one GP utilized road segment features while that of the other GP depended on manually specified relations (instead of geodesic distance) between segments with respect to an undirected network topology. Different from the above works, we propose a relational GP whose correlation structure exploits the geodesic distance between segments based on the topology of a directed road network with vertices denoting road segments and edges indicating adjacent segments weighted by dissimilarity of their features, hence tightly integrating the features and relational information.
Data fusion. The observations are gathered distributedly by each sensor along its path in the road network and have to be assimilated in order to predict the traffic phenomenon. Since a large number of observations are expected to be collected, a centralized approach to GP prediction cannot be performed in real time due to its cubic time complexity.
To resolve this, we propose a decentralized data fusion approach to efficient and scalable approximate GP prediction (Section 3). Existing decentralized and distributed Bayesian filtering frameworks for addressing non-traffic related problems (Chung et al., 2004;Coates, 2004;Olfati-Saber, 2005;Rosencrantz et al., 2003;Sukkarieh et al., 2003) will face the same difficulties as their centralized counterparts described above if applied to predicting traffic phenomena, thus resulting in loss of predictive performance. Distributed regression algorithms Paskin and Guestrin, 2004) for static sensor networks gain efficiency from spatial locality assumptions, which cannot be exploited by mobile sensors whose paths are not constrained by locality. Cortes (2009) proposed a distributed data fusion approach to approximate GP prediction based on an iterative Jacobi overrelaxation algorithm, which incurs some critical limitations: (a) the past observations taken along the sensors' paths are assumed to be uncorrelated, which greatly undermines its predictive performance when they are in fact correlated and/or the current observations are sparse; (b) when the number of sensors grows large, it converges very slowly; (c) it assumes that the range of positive correlation has to be bounded by some factor of the communication range. Our proposed decentralized algorithm does not suffer from these limitations and can be computed exactly with efficient time bounds.
Active sensing. The sensors have to coordinate to actively gather the most informative observations for minimizing the uncertainty of modeling and predicting the traffic phenomenon. Existing centralized (Low et al., 2008(Low et al., , 2009a(Low et al., , 2011 and decentralized (Low et al., 2012;Stranders et al., 2009) active sensing algorithms scale poorly with a large number of observations and sensors. We propose a partially decentralized active sensing algorithm that overcomes these issues of scalability (Section 4). This paper presents a novel Decentralized Data Fusion and Active Sensing (D 2 FAS) algorithm (Sections 3 and 4) for sampling spatiotemporally varying environmental phenomena with mobile sensors. Note that the decentralized data fusion component of D 2 FAS can also be used for static and passive mobile sensors. The practical applicability of D 2 FAS is not restricted to traffic monitoring; it can be used in other environmental sensing applications such as mineral prospecting (Low et al., 2007), monitoring of ocean and freshwater phenomena (Dolan et al., 2009;Podnar et al., 2010;Low et al., 2009bLow et al., , 2011Low et al., , 2012) (e.g., plankton bloom, anoxic zones), forest ecosystems, pollution (e.g., oil spill), or contamination (e.g., radiation leak). The specific contributions of this paper include: D 2 FAS (Section 5): We prove that D 2 FAS can scale better than existing state-of-the-art centralized algorithms with a large number of observations and sensors; • Theoretically guaranteeing the predictive performance of the decentralized data fusion component of D 2 FAS to be equivalent to that of a sophisticated centralized sparse approximation for the GP model (Section 3): The computation of such a sparse approximate GP model can thus be parallelized and distributed among the mobile sensors (in a Google-like MapReduce paradigm), thereby achieving efficient and scalable prediction; • Theoretically guaranteeing the performance of the partially decentralized active sensing component of D 2 FAS, from which various practical environmental conditions can be established to improve its performance; • Developing a relational GP model whose correlation structure can exploit both the road segment features and road network topology information (Section 2.1); • Empirically evaluating the predictive performance, time efficiency, and scalability of the D 2 FAS algorithm on a real-world traffic phenomenon (i.e., speeds of road segments) dataset over an urban road network (Section 6): D 2 FAS is more time-efficient and scales significantly better with increasing number of observations and sensors while achieving predictive performance close to that of existing state-of-the-art centralized algorithms.
Relational Gaussian Process Regression
The Gaussian process (GP) can be used to model a spatiotemporal traffic phenomenon over a road network as follows: The traffic phenomenon is defined to vary as a realization of a GP. Let V be a set of road segments representing the domain of the road network such that each road segment s ∈ V is specified by a p-dimensional vector of features and is associated with a realized (random) measurement z s (Z s ) of the traffic condition such as speed if s is observed (unobserved). Let {Z s } s∈V denote a GP, that is, every finite subset of {Z s } s∈V follows a multivariate Gaussian distribution (Rasmussen and Williams, 2006). Then, the GP is fully specified by its prior mean µ s E[Z s ] and covariance σ ss cov[Z s , Z s ] for all s, s ∈ V . In particular, we will describe in Section 2.1 how the covariance σ ss for modeling the correlation of measurements between all pairs of segments s, s ∈ V can be designed to exploit the road segment features and the road network topology.
A chief capability of the GP model is that of performing probabilistic regression: Given a set D ⊂ V of observed road segments and a column vector z D of corresponding measurements, the joint distribution of the measurements at any set Y ⊆ V \ D of unobserved road segments remains Gaussian with the following posterior mean vector and covariance matrix
µ Y |D µ Y + Σ Y D Σ −1 DD (z D − µ D ) (1) Σ Y Y |D Σ Y Y − Σ Y D Σ −1 DD Σ DY (2)
where µ Y (µ D ) is a column vector with mean components µ s for all s ∈ Y (s ∈ D), Σ Y D (Σ DD ) is a covariance matrix with covariance components σ ss for all s ∈ Y, s ∈ D (s, s ∈ D), and Σ DY is the transpose of Σ Y D . The posterior mean vector µ Y |D (1) is used to predict the measurements at any set Y of unobserved road segments. The posterior covariance matrix Σ Y Y |D (2), which is independent of the measurements z D , can be processed in two ways to quantify the uncertainty of these predictions: (a) the trace of Σ Y Y |D yields the sum of posterior variances Σ ss|D over all s ∈ Y ; (b) the determinant of Σ Y Y |D is used in calculating the Gaussian posterior joint entropy
H[Z Y |Z D ] 1 2 log(2πe) |Y | Σ Y Y |D .(3)
In contrast to the first measure of uncertainty that assumes conditional independence between measurements in the set Y of unobserved road segments, the entropy-based measure (3) accounts for their correlation, thereby not overestimating their uncertainty. Hence, we will focus on using the entropy-based measure of uncertainty in this paper.
Graph-Based Kernel
If the observations are noisy (i.e., by assuming additive independent identically distributed Gaussian noise with variance σ 2 n ), then their prior covariance σ ss can be expressed as σ ss = k(s, s ) + σ 2 n δ ss where δ ss is a Kronecker delta that is 1 if s = s and 0 otherwise, and k is a kernel function measuring the pairwise "similarity" of road segments. For a traffic phenomenon (e.g., road speeds), the correlation of measurements between pairs of road segments depends not only on their features (e.g., length, number of lanes, speed limit, direction) but also the road network topology. So, the kernel function is defined to exploit both the features and topology information, which will be described next.
Definition 1 (Road Network) Let the road network be represented as a weighted directed graph G (V, E, m) that consists of • a set V of vertices denoting the domain of all possible road segments,
• a set E ⊆ V × V of edges where there is an edge (s, s )
from s ∈ V to s ∈ V iff the end of segment s connects to the start of segment s in the road network, and • a weight function m : E → R + measuring the standardized Manhattan distance (Borg and Groenen, 2005)
m((s, s )) p i=1 |[s] i − [s ] i |/r i of each edge (s, s ) where [s] i ([s ] i )
is the i-th component of the feature vector specifying road segment s (s ), and r i is the range of the i-th feature. The weight function m serves as a dissimilarity measure between adjacent road segments.
The next step is to compute the shortest path distance d(s, s ) between all pairs of road segments s, s ∈ V (i.e., using Floyd-Warshall or Johnson's algorithm) with respect to the topology of the weighted directed graph G. Such a distance function is again a measure of dissimilarity, rather than one of similarity, as required by a kernel function. Fur-thermore, a valid GP kernel needs to be positive semidefinite and symmetric (Schölkopf and Smola, 2002), which are clearly violated by d.
To construct a valid GP kernel from d, multi-dimensional scaling (Borg and Groenen, 2005) is applied to embed the domain of road segments into the p -dimensional Euclidean space R p . Specifically, a mapping g : V → R p is determined by minimizing the squared loss g * = arg min g s,s ∈V (d(s, s )− g(s)−g(s ) ) 2 . With a small squared loss, the Euclidean distance g * (s) − g * (s ) between g * (s) and g * (s ) is expected to closely approximate the shortest path distance d(s, s ) between any pair of road segments s and s . After embedding into Euclidean space, a conventional kernel function such as the squared exponential one (Rasmussen and Williams, 2006) can be used:
k(s, s ) = σ 2 s exp − 1 2 p i=1 [g * (s)] i − [g * (s )] i i 2 where [g * (s)] i ([g * (s )] i ) is the i-th component of the p - dimensional vector g * (s) (g * (s ))
, and the hyperparameters σ s , 1 , . . . , p are, respectively, signal variance and lengthscales that can be learned using maximum likelihood estimation (Rasmussen and Williams, 2006). The resulting kernel function k 1 is guaranteed to be valid.
Subset of Data Approximation
Although the GP is an effective predictive model, it faces a practical limitation of cubic time complexity in the number |D| of observations; this can be observed from computing the posterior distribution (i.e., (1) and (2)), which requires inverting covariance matrix Σ DD and incurs O(|D| 3 ) time.
If |D| is expected to be large, GP prediction cannot be performed in real time. For practical usage, we have to resort to computationally cheaper approximate GP prediction.
A simple method of approximation is to select only a subset U of the entire set D of observed road segments (i.e., U ⊂ D) to compute the posterior distribution of the measurements at any set Y ⊆ V \ D of unobserved road segments. Such a subset of data (SoD) approximation method produces the following predictive Gaussian distribution, which closely resembles that of the full GP model (i.e., by simply replacing D in (1) and (2) with U ):
µ Y |U = µ Y + Σ Y U Σ −1 U U (z U − µ U ) (4) Σ Y Y |U = Σ Y Y − Σ Y U Σ −1 U U Σ U Y . (5) Notice that the covariance matrix Σ U U to be inverted only incurs O(|U | 3 ) time, which is independent of |D|.
The predictive performance of SoD approximation is sensitive to the selection of subset U . In practice, random subset selection often yields poor performance. This issue can be resolved by actively selecting an informative subset U in an iterative greedy manner: Firstly, U is initialized to be an empty set. Then, all road segments in D \ U are scored based on a criterion that can be chosen from, for example, the works of (Krause et al., 2008b;Lawrence et al., 2003;Seeger and Williams, 2003). The highest-scored segment is selected for inclusion in U and removed from D. This greedy selection procedure is iterated until U reaches a predefined size. Among the various criteria introduced earlier, the differential entropy score (Lawrence et al., 2003) is reported to perform well (Oh et al., 2010); it is a monotonic function of the posterior variance Σ ss|U (5), thus resulting in the greedy selection of a segment s ∈ D \ U with the largest variance in each iteration.
Decentralized Data Fusion
In the previous section, two centralized data fusion approaches to exact (i.e., (1) and (2)) and approximate (i.e., (4) and (5)) GP prediction are introduced. In this section, we will discuss the decentralized data fusion component of our D 2 FAS algorithm, which distributes the computational load among the mobile sensors to achieve efficient and scalable approximate GP prediction.
The intuition of our decentralized data fusion algorithm is as follows: Each of the K mobile sensors constructs a local summary of the observations taken along its own path in the road network and communicates its local summary to every other sensor. Then, it assimilates the local summaries received from the other sensors into a globally consistent summary, which is exploited for predicting the traffic phenomenon as well as active sensing. This intuition will be formally realized and described in the paragraphs below.
While exploring the road network, each mobile sensor summarizes its local observations taken along its path based on a common support set U ⊂ V known to all the other sensors. Its local summary is defined as follows:
Definition 2 (Local Summary) Given a common support set U ⊂ V known to all K mobile sensors, a set D k ⊂ V of observed road segments and a column vector z D k of corresponding measurements local to mobile sensor k, its local summary is defined as a tuple (ż k
U ,Σ k U U ) wherė z k U Σ U D k Σ −1 D k D k |U (z D k − µ D k ) (6) Σ k U U Σ U D k Σ −1 D k D k |U Σ D k U(7)
such that Σ D k D k |U is defined in a similar manner to (5).
Remark. Unlike SoD (Section 2.2), the support set U of road segments does not have to be observed, since the local summary (i.e., (6) and (7)) is independent of the corresponding measurements z U . So, U does not need to be a subset of D = K k=1 D k . To select an informative support set U from the set V of all possible segments in the road network, an offline active selection procedure similar to that in the last paragraph of Section 2.2 can be performed just once prior to observing data to determine U . In contrast, SoD has to perform online active selection every time new road segments are being observed.
By communicating its local summary to every other sensor, each mobile sensor can then construct a globally consistent summary from the received local summaries:
Definition 3 (Global Summary) Given a common support set U ⊂ V known to all K mobile sensors and the local summary (ż k U ,Σ k U U ) of every mobile sensor k = 1, . . . , K, the global summary is defined as a tuple
(z U ,Σ U U ) wherez U K k=1ż k U (8) Σ U U Σ U U + K k=1Σ k U U .(9)
Remark. In this paper, we assume all-to-all communication between the K mobile sensors. Supposing this is not possible and each sensor can only communicate locally with its neighbors, the summation structure of the global summary (specifically, (8) and (9)) makes it amenable to be constructed using distributed consensus filters (Olfati-Saber, 2005). We omit these details since they are beyond the scope of this paper.
Finally, the global summary is exploited by each mobile sensor to compute a globally consistent predictive Gaussian distribution, as detailed in Theorem 1A below, as well as to perform decentralized active sensing (Section 4):
Theorem 1 Let a common support set U ⊂ V be known to all K mobile sensors.
A. Given the global summary (z U ,Σ U U ), each mobile sensor computes a globally consistent predictive Gaussian distribution N (µ Y , Σ Y Y ) of the measurements at any set Y of unobserved road segments where
µ Y µ Y + Σ Y UΣ −1 U Uz U (10) Σ Y Y Σ Y Y − Σ Y U (Σ −1 U U −Σ −1 U U )Σ U Y . (11) B. Let N (µ PITC Y |D , Σ PITC Y Y |D )
be the predictive Gaussian distribution computed by the centralized sparse partially independent training conditional (PITC) approximation of GP model (Quiñonero-Candela and Rasmussen, 2005)
where µ PITC Y |D µ Y + Γ Y D (Γ DD + Λ) −1 (z D − µ D ) (12) Σ PITC Y Y |D Σ Y Y − Γ Y D (Γ DD + Λ) −1 Γ DY (13) such that Γ BB Σ BU Σ −1 U U Σ U B(14)
and Λ is a block-diagonal matrix constructed from the K diagonal blocks of Σ DD|U , each of which is
a matrix Σ D k D k |U for k = 1, . . . , K where D = K k=1 D k . Then, µ Y = µ PITC Y |D and Σ Y Y = Σ PITC Y Y |D .
The proof of Theorem 1B is given in Appendix A. The equivalence result of Theorem 1B bears two implications:
Remark 1. The computation of PITC can be parallelized and distributed among the mobile sensors in a Google-like MapReduce paradigm (Chu et al., 2007), thereby improving the time efficiency of prediction: Each of the K mappers (sensors) is tasked to compute its local summary while the reducer (any sensor) sums these local summaries into a global summary, which is then used to compute the predictive Gaussian distribution. Supposing |Y | ≤ |U | for simplicity, the O |D|((|D|/K) 2 + |U | 2 ) time incurred by PITC can be reduced to O (|D|/K) 3 + |U | 3 + |U | 2 K time of running our decentralized algorithm on each of the K sensors, the latter of which scales better with increasing number |D| of observations.
Remark 2. We can draw insights from PITC to elucidate an underlying property of our decentralized algorithm: It is assumed that Z D1 , . . . , Z D K , Z Y are conditionally independent given the measurements at the support set U of road segments. To potentially reduce the degree of violation of this assumption, an informative support set U is actively selected, as described earlier in this section. Furthermore, the experimental results on real-world urban road network data 2 (Section 6) show that D 2 FAS can achieve predictive performance comparable to that of the full GP model while enjoying significantly lower computational cost, thus demonstrating the practicality of such an assumption for predicting traffic phenomena. The predictive performance of D 2 FAS can be improved by increasing the size of U at the expense of greater time and communication overhead.
Decentralized Active Sensing
The problem of active sensing with K mobile sensors is formulated as follows: Given the set D k ⊂ V of observed road segments and the currently traversed road segment s k ∈ V of every mobile sensor k = 1, . . . , K, the mobile sensors have to coordinate to select the most informative walks w * 1 , . . . , w * K of length (i.e., number of road segments) L each and with respective origins s 1 , . . . , s K in the road network G:
(w * 1 , . . . , w * K )= arg max (w1,...,w K ) H Z K k=1 Yw k Z K k=1 D k(15)
where Y w k denotes the set of unobserved road segments induced by the walk w k . To simplify notation, let a joint walk be denoted by w (w 1 , . . . , w K ) (similarly, for w * ) and its induced set of unobserved road segments be Y w K k=1 Y w k from now on. Interestingly, it can be shown using the chain rule for entropy that these maximumentropy walks w * minimize the posterior joint entropy (i.e.,
H[Z V \(D Y w * ) |Z D Y w * ])
of the measurements at the remaining unobserved segments (i.e., V \ (D Y w * )) in the road network. After executing the walk w * k , each mobile sensor k observes the set Y w * k of road segments and updates its local information:
D k ← D k Y w * k , z D k ← z D k Y w * k , s k ← terminus of w * k .(16)
Evaluating the Gaussian posterior entropy term in (15) involves computing a posterior covariance matrix (3) using one of the data fusion methods described earlier: If (2) of full GP model (Section 2) or (5) of SoD (Section 2.2) is to be used, then the observations that are gathered distributedly by the sensors have to be fully communicated to a central data fusion center. In contrast, our decentralized data fusion algorithm (Section 3) only requires communicating local summaries (Definition 2) to compute (11) for solving the active sensing problem (15):
w * = arg max w H[Z Yw ] ,(17)H[Z Yw ] 1 2 log(2πe) |Yw| Σ YwYw .(18)
Without imposing any structural assumption, solving the active sensing problem (17) will be prohibitively expensive due to the space of possible joint walks w that grows exponentially in the number K of mobile sensors. To overcome this scalability issue for D 2 FAS, our key idea is to construct a block-diagonal matrix whose log-determinant closely approximates that of Σ YwYw (11) and exploit the property that the log-determinant of such a block-diagonal matrix can be decomposed into a sum of log-determinants of its diagonal blocks, each of which depends only on the walks of a disjoint subset of the K mobile sensors. Consequently, the active sensing problem can be partially decentralized, leading to a reduced space of possible joint walks to be searched, as detailed in the rest of this section.
Firstly, we extend an earlier assumption in Section 3: Z D1 , . . . , Z D K , Z Yw 1 , . . . , Z Yw K are conditionally independent given the measurements at the support set U of road segments. Then, it can be shown via the equivalence to PITC (Theorem 1B) that Σ YwYw (11) comprises diagonal blocks of the form Σ Yw k Yw k for k = 1, . . . , K and off-diagonal blocks of the form Σ Yw k UΣ −1 U U Σ U Yw k for k, k = 1, . . . , K and k = k . In particular, each offdiagonal block of Σ YwYw represents the correlation of measurements between the unobserved road segments Y w k and Y w k along the respective walks w k of sensor k and w k of sensor k . If the correlation between some pair of their possible walks is high enough, then their walks have to be coordinated. This is formally realized by the following coordination graph over the K sensors:
Definition 4 (Coordination Graph) Define the coordination graph to be an undirected graph G (V, E) that comprises • a set V of vertices denoting the K mobile sensors, and • a set E of edges denoting coordination dependencies between sensors such that there exists an edge {k, k } incident with sensors k ∈ V and k ∈ V \ {k} iff
max s∈Y W k ,s ∈Y W k Σ sUΣ −1 U U Σ U s > ε(19)
for a predefined constant ε > 0 where W k denotes the set of possible walks of length L of mobile sensor k from origin s k in the road network G and
Y W k w k ∈W k Y w k .
Remark. The construction of G can be decentralized as follows: SinceΣ U U is symmetric and positive definite, it can be decomposed by Cholesky factorization intoΣ U U = ΨΨ where Ψ is a lower triangular matrix and Ψ is the transpose of Ψ. Then, Σ sUΣ −1 U U Σ U s = (Ψ\Σ U s ) Ψ\Σ U s where Ψ\B denotes the column vector φ solving Ψφ = B. That is, Σ sUΣ −1 U U Σ U s (19) can be expressed as a dot product of two vectors Ψ\Σ U s and Ψ\Σ U s ; this property is exploited to determine adjacency between sensors in a decentralized manner:
Definition 5 (Adjacency) Let Φ k {Ψ\Σ U s } s∈Y W k (20) for k = 1, . . . , K. A sensor k ∈ V is adjacent to sensor k ∈ V \ {k} in coordination graph G iff max φ∈Φ k ,φ ∈Φ k φ φ > ε .(21)
It follows from the above definition that if each sensor k constructs Φ k and exchanges it with every other sensor, then it can determine its adjacency to all the other sensors and store this information in a column vector a k of length K with its k -th component being defined as follows:
[a k ] k = 1 if sensor k is adjacent to sensor k , 0 otherwise.(22)
By exchanging its adjacency vector a k with every other sensor, each sensor can construct a globally consistent adjacency matrix A G (a 1 . . . a K ) to represent coordination graph G.
Next, by computing the connected components (say, K of them) of coordination graph G, their resulting vertex sets partition the set V of K sensors into K disjoint subsets V 1 , . . . , V K such that the sensors within each subset have to coordinate their walks. Each sensor can determine its residing connected component in a decentralized way by performing a depth-first search in G starting from it as root.
Finally, construct a block-diagonal matrix Σ YwYw to comprise diagonal blocks of the form Σ Yw Vn Yw Vn for n = 1, . . . , K where w Vn (w k ) k∈Vn and Y w Vn k∈Vn Y w k . The active sensing problem (17) is then approximated by
max w 1 2 log(2πe) |Yw| Σ YwYw ≡ max (w V 1 ,...,w V K ) K n=1 log(2πe) |Yw Vn | Σ Yw Vn Yw Vn = K n=1 max w Vn log(2πe) |Yw Vn | Σ Yw Vn Yw Vn ,(23)
which can be solved in a partially decentralized manner by each disjoint subset V n of mobile sensors:
w Vn = arg max w Vn log(2πe) |Yw Vn | Σ Yw Vn Yw Vn .(24)
Our active sensing algorithm becomes fully decentralized if ε is set to be sufficiently large: more sensors become isolated in G, consequently decreasing the size κ max n |V n | of its largest connected component to 1. As shown in Section 5.1, decreasing κ improves its time efficiency. On the other hand, it tends to a centralized behavior (17) by setting ε → 0 + : G becomes near-complete, thus resulting in
κ → K. Let ξ max n,w Vn ,i,i Σ Yw Vn Yw Vn −1 ii(25)
and 0.5 log 1 1− K 1.5 L 2.5 κξε 2 . In the result below, we prove that the joint walk w ( w V1 , . . . , w V K ) is guaranteed to achieve an entropy H[Z Y w ] (i.e., by plugging w into (18)) that is not more than from the maximum entropy H[Z Y w * ] achieved by joint walk w * (17):
Theorem 2 (Performance Guarantee) If K 1.5 L 2.5 κξε < 1, then H[Z Y w * ] − H[Z Y w ] ≤ .
The proof of Theorem 2 is given in Appendix B. The implication of Theorem 2 is that our partially decentralized active sensing algorithm can perform comparatively well (i.e., small ) under the following favorable environmental conditions: (a) the network of K sensors is not large, (b) length L of each sensor's walk to be optimized is not long, (c) the largest subset of κ sensors being formed to coordinate their walks (i.e., largest connected component in G) is reasonably small, and (d) the minimum required correlation ε between walks of adjacent sensors is kept low.
Algorithm 1 below outlines the key operations of our D 2 FAS algorithm to be run on each mobile sensor k, as detailed previously in Sections 3 and 4:
Algorithm 1: D 2 FAS(U, K, L, k, D k , z D k , s k ) while true do / * Data fusion (Section 3) * / Construct local summary by (6) & (7) Exchange local summary with every sensor i = k Construct global summary by (8) & (9) Predict measurements at unobserved road segments by (10) & (11) / * Active Sensing (Section 4) * / Construct Φ k by (20) Exchange Φ k with every sensor i = k Compute adjacency vector a k by (21) & (22) Exchange adjacency vector with every sensor i = k Construct adjacency matrix of coordination graph Find vertex set Vn of its residing connected component Compute maximum-entropy joint walk w Vn by (24) Execute walk w k and observe its road segments Y w k Update local information D k , z D k , and s k by (16)
Time and Communication Overheads
In this section, the time and communication overheads of our D 2 FAS algorithm are analyzed and compared to that of centralized active sensing (17) coupled with the data fusion methods: Full GP (FGP) and SoD (Section 2).
Time Complexity
The data fusion component of D 2 FAS involves computing the local and global summaries and the predictive Gaussian distribution. To construct the local summary using (6) and (7), each sensor has to evaluate Σ D k D k |U in O |U | 3 + |U |(|D|/K) 2 time and invert it in O (|D|/K) 3 time, after which the local summary is obtained in O |U | 2 |D|/K + |U |(|D|/K) 2 time. The global summary is computed in O |U | 2 K by (8) and (9). Finally, the predictive Gaussian distribution is derived in O |U | 3 + |U ||Y | 2 time using (10) and (11). Supposing |Y | ≤ |U | for simplicity, the time complexity of data fusion is then O (|D|/K) 3 + |U | 3 + |U | 2 K .
Let the maximum out-degree of G be denoted by δ. Then, each sensor has to consider ∆ δ L possible walks of length L. The active sensing component of D 2 FAS involves computing Φ k in O ∆L|U | 2 time, a k in O ∆ 2 L 2 |U |K time, its residing connected component in O κ 2 time, and the maximum-entropy joint walk by (11) and (24) with the following incurred time: The largest connected component of κ sensors in G has to consider ∆ κ possible joint walks. Note that Σ Yw Vn Yw Vn = diag (Σ Yw k Yw k |U ) k∈Vn + Σ Yw Vn UΣ −1 U U Σ U Yw Vn where diag(B) constructs a diagonal matrix by placing vector B on its diagonal. By exploiting Φ k , the diagonal and latter matrix terms for all possible joint walks can be computed in O κ∆(L|U | 2 + L 2 |U |) and O κ 2 ∆ 2 L 2 |U | time, respectively. For each joint walk w Vn , evaluating the determinant of Σ Yw Vn Yw Vn incurs O (κL) 3 time. Therefore, the time complexity of active sensing is
O κ∆L|U | 2 + ∆ 2 L 2 |U |(K + κ 2 ) + ∆ κ (κL) 3 .
Hence, the time complexity of our D 2 FAS algorithm is O((|D|/K) 3 + |U | 2 (|U | + K + κ∆L) + ∆ 2 L 2 |U |(K + κ 2 ) + ∆ κ (κL) 3 ). In contrast, the time incurred by centralized active sensing coupled with FGP and SoD are, respectively, O |D| 3 + ∆ K KL(|D| 2 + (KL) 2 ) and O |U | 3 |D| + ∆ K KL(|U | 2 + (KL) 2 ) . It can be observed that D 2 FAS can scale better with large |D| (i.e., number of observations) and K (i.e., number of sensors). The scalability of D 2 FAS vs. FGP and SoD will be further evaluated empirically in Section 6.
Communication Complexity
Let the communication overhead be defined as the size of each broadcast message. Recall from the data fusion component of D 2 FAS in Algorithm 1 that, in each iteration, each sensor broadcasts a O |U | 2 -sized summary encapsulating its local observations, which is robust against communication failure. In contrast, FGP and SoD require each sensor to broadcast, in each iteration, a O(|D|/K)-sized message comprising exactly its local observations to handle communication failure. If the number of local observations grows to be larger in size than a local summary of predefined size, then the data fusion component of D 2 FAS is more scalable than FGP and SoD in terms of communication overhead. For the partially decentralized active sensing component of D 2 FAS, each sensor broadcasts O(∆L|U |)sized Φ k and O(K)-sized a k messages. This section evaluates the predictive performance, time efficiency, and scalability of our D 2 FAS algorithm on a real-world traffic phenomenon (i.e., speeds (km/h) of road segments) over an urban road network (top figure) in Tampines area, Singapore during evening peak hours on April 20, 2011. It comprises 775 road segments including highways, arterials, slip roads, etc. The mean speed is 48.8 km/h and the standard deviation is 20.5 km/h. The performance of D 2 FAS is compared to that of centralized active sensing (17) coupled with the state-of-the-art data fusion methods: full GP (FGP) and SoD (Section 2). A network of K mobile sensors is tasked to explore the road network to gather a total of up to 960 observations. To reduce computational time, each sensor repeatedly computes and executes maximum-entropy walks of length L = 2 (instead of computing a very long walk), unless otherwise stated. For D 2 FAS and SoD, |U | is set to 64 . For the active sensing component of D 2 FAS, ε is set to 0.1, unless otherwise stated. The experiments are run on a Linux PC with Intel Core TM 2 Quad CPU Q9550 at 2.83 GHz.
Performance Metrics
The first metric evaluates the predictive performance of a tested algorithm: It measures the root mean squared error (RMSE) |V | −1 s∈V (z s − µ s ) 2 over the entire domain V of the road network that is incurred by the predictive mean µ s of the tested algorithm, specifically, using (1) of FGP, (4) of SoD, or (10) of D 2 FAS. The second metric evaluates the time efficiency and scalability of a tested algorithm by measuring its incurred time; for D 2 FAS, the maximum of the time incurred by all subsets V 1 , . . . , V K of sensors is recorded.
Results and Analysis
Predictive performance and time efficiency. Fig. 1 shows results of the performance of the tested algorithms averaged over 40 randomly generated starting sensor locations with varying number K = 4, 6, 8 of sensors. It can be observed that D 2 FAS is significantly more time-efficient and scales better with increasing number |D| of observations (Figs. 1d to 1f) while achieving predictive performance close to that of centralized active sensing coupled with FGP and SoD (Figs. 1a to 1c). Specifically, D 2 FAS is about 1, 2, 4 orders of magnitude faster than centralized active sensing coupled with FGP and SoD for K = 4, 6, 8 sensors, respectively.
Scalability of D 2 FAS. Using the same results as that in Fig. 1, Fig. 2 plots them differently to reveal the scalability of the tested algorithms with increasing number K of sen- sors. Additionally, we provide results of the performance of D 2 FAS for K = 10, 20, 30 sensors; such results are not available for centralized active sensing coupled with FGP and SoD due to extremely long incurred time. It can be observed from Figs. 2a to 2c that the predictive performance of all tested algorithms improve with a larger number of sensors because each sensor needs to execute fewer walks and its performance is therefore less adversely affected by its myopic selection (i.e., L = 2) of maximum-entropy walks. As a result, more informative unobserved road segments are explored.
As shown in Fig. 2d, when the randomly placed sensors gather their initial observations (i.e., |D| < 400), the time incurred by D 2 FAS is higher for greater K due to larger subsets of sensors being formed to coordinate their walks (i.e., larger κ). As more observations are gathered (i.e., |D| ≥ 400), its partially decentralized active sensing component directs the sensors to explore further apart from each other in order to maximize the entropy of their walks. This consequently decreases κ, leading to a reduction in incurred time. Furthermore, as K increases from 4 to 20, the incurred time decreases due to its decentralized data fusion component that can distribute the computational load among a greater number of sensors. When the road network becomes more crowded from K = 20 to K = 30 sensors, the incurred time increases slightly due to slightly larger κ. In contrast, Figs. 2e and 2f show that the time taken by FGP and SoD increases significantly primarily due to their centralized active sensing incurring exponential time in K. Hence, the scalability of our D 2 FAS algorithm in the number of sensors allows the deployment of a largerscale mobile sensor network (i.e., K ≥ 10) to achieve more accurate traffic modeling and prediction (Figs. 2a to 2c).
Scalability of data fusion. Fig. 3 shows results of the scalability of the tested data fusion methods with increasing number K of sensors. In order to produce meaningful results for fair comparison, the same active sensing component has to be coupled with the data fusion methods and its incurred time kept to a minimum. As such, we impose the use of a fully decentralized active sensing component to be performed by each mobile sensor k: w * k = arg max w k H[Z Yw k |Z D ]. For D 2 FAS, this corresponds exactly to (24) by setting a large enough ε (in our experiments, ε = 2) to yield κ = 1; consequently, computational and communicational operations pertaining to the coordination graph can be omitted.
It can be seen from Fig. 3a that the time incurred by the decentralized data fusion component of D 2 FAS decreases with increasing K, as explained previously. In contrast, the time incurred by FGP and SoD increases ( Fig. 3b and 3c): As discussed above, a larger number of sensors results in a greater quantity of more informative unique observations to be gathered (i.e., fewer repeated observations), which increases the time needed for data fusion. When K ≥ 10, D 2 FAS is at least 1 order of magnitude faster than FGP and SoD. It can also be observed that D 2 FAS scales better with increasing number of observations. So, the real-time performance and scalability of D 2 FAS's decentralized data fusion enable it to be used for persistent large-scale traffic modeling and prediction where a large number of observations and sensors (including static and passive ones) are expected to be available.
Varying length L of walk. Fig. 4 shows results of the performance of the tested algorithms with varying length L = 2, 4, 6, 8 of maximum-entropy joint walks; we choose to experiment with just 2 sensors since Figs. 2 and 3 reveal that a smaller number of sensors produce poorer predictive performance and higher incurred time with large number of Figure 4: Graphs of (a-c) predictive performance and (d-f) time efficiency vs. total no. |D| of observations gathered by 2 mobile sensors with varying length L of maximumentropy joint walks.
observations for D 2 FAS. It can be observed that the predictive performance of all tested algorithms improve with increasing walk length L because the selection of maximumentropy joint walks is less myopic. The time incurred by D 2 FAS increases due to larger κ but grows more slowly and is lower than that incurred by centralized active sensing coupled with FGP and SoD. Specifically, when L = 8, D 2 FAS is at least 1 order of magnitude faster (i.e., average of 60 s) than centralized active sensing coupled with SoD (i.e., average of > 732 s) and FGP (i.e., not available due to excessive incurred time). Also, notice from Figs. 2a and 2d that if a large number of sensors (i.e., K = 30) is available, D 2 FAS can select shorter walks of L = 2 to be significantly more time-efficient (i.e., average of > 3 orders of magnitude faster) while achieving predictive performance comparable to that of SoD with L = 8 and FGP with L = 6.
Conclusion
This paper describes a decentralized data fusion and active sensing algorithm for modeling and predicting spatiotemporal traffic phenomena with mobile sensors. Analytical and empirical results have shown that our D 2 FAS algorithm is significantly more time-efficient and scales better with increasing number of observations and sensors while achieving predictive performance close to that of state-ofthe-art centralized active sensing coupled with FGP and SoD. Hence, D 2 FAS is practical for deployment in a largescale mobile sensor network to achieve persistent and accurate traffic modeling and prediction. For our future work, we will assume that each sensor can only communicate locally with its neighbors (instead of assuming all-to-all communication) and develop a distributed data fusion approach to efficient and scalable approximate GP prediction based on D 2 FAS and consensus filters (Olfati-Saber, 2005).
A Proof of Theorem 1B
We have to first simplify the Γ Y D (Γ DD + Λ) −1 term in the expressions of µ PITC Y |D (12) and Σ PITC Y Y |D (13).
(Γ DD + Λ) −1 = Σ DU Σ −1 U U Σ U D + Λ −1 = Λ −1 − Λ −1 Σ DU Σ U U + Σ U D Λ −1 Σ DU −1 Σ U D Λ −1 = Λ −1 − Λ −1 Σ DUΣ −1 U U Σ U D Λ −1 .(26)
The second equality follows from matrix inversion lemma. The last equality is due to
Σ U U + Σ U D Λ −1 Σ DU = Σ U U + K k=1 Σ U D k Σ −1 D k D k |U Σ D k U = Σ U U + K k=1Σ k U U =Σ U U .(27)
Using (14) and (26),
Γ Y D (Γ DD + Λ) −1 = Σ Y U Σ −1 U U Σ U D Λ −1 − Λ −1 Σ DUΣ −1 U U Σ U D Λ −1 = Σ Y U Σ −1 U U Σ U U − Σ U D Λ −1 Σ DU Σ −1 U U Σ U D Λ −1 = Σ Y UΣ −1 U U Σ U D Λ −1
(28) The third equality is due to (27).
From (12),
µ PITC Y |D = µ Y + Γ Y D (Γ DD + Λ) −1 (z D − µ D ) = µ Y + Σ Y UΣ −1 U U Σ U D Λ −1 (z D − µ D ) = µ Y + Σ Y UΣ −1 U Uz U = µ Y .
The second equality is due to (28).
The third equality follows from Σ
U D Λ −1 (z D − µ D ) = K k=1 Σ U D k Σ −1 D k D k |U (z D k − µ D k ) = K k=1ż k U =z U . From (13), Σ PITC Y Y |D = Σ Y Y − Γ Y D (Γ DD + Λ) −1 Γ DY = Σ Y Y − Σ Y UΣ −1 U U Σ U D Λ −1 Σ DU Σ −1 U U Σ U Y = Σ Y Y − Σ Y UΣ −1 U U Σ U D Λ −1 Σ DU Σ −1 U U Σ U Y −Σ Y U Σ −1 U U Σ U Y − Σ Y U Σ −1 U U Σ U Y = Σ Y Y − Σ Y UΣ −1 U U Σ U D Λ −1 Σ DU −Σ U U Σ −1 U U Σ U Y −Σ Y U Σ −1 U U Σ U Y = Σ Y Y − Σ Y U Σ −1 U U Σ U Y − Σ Y UΣ −1 U U Σ U Y = Σ Y Y − Σ Y U Σ −1 U U −Σ −1 U U Σ U Y = Σ Y Y .
The second equality follows from (14) and (28). The fifth equality is due to (27).
B Proof of Theorem 2
Let Σ YwYw Σ YwYw − Σ YwYw and ρ w be the spectral radius of Σ YwYw −1 Σ YwYw . We have to first bound ρ w from above.
For any joint walk w, Σ YwYw
Figure 1 :Figure 2 :
12Graphs of (a-c) predictive performance and (d-f) time efficiency vs. total no. |D| of observations gathered by varying number K of mobile sensors. Graphs of (a-c) predictive performance and (d-f) time efficiency vs. total no. |D| of observations gathered by varying number K of mobile sensors.
Figure 3 :
3Graphs of time efficiency vs. total no. |D| of observations gathered by varying number K of sensors.
Total no. |D| of observations Incurred time (s) Total no. |D| of observations Incurred time (s) Total no. |D| of observations Incurred time (s)200
400
600
800
1000
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
K=4
K=6
K=8
K=10
K=20
K=30
0
200
400
600
800
1000
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
K=4
K=6
K=8
K=10
K=20
K=30
0
200
400
600
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0
0.1
0.2
0.3
0.4
0.5
K=4
K=6
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K=30
For spatiotemporal traffic modeling, the kernel function k can be extended to account for the temporal dimension.
Quiñonero-Candela and Rasmussen (2005) only illustrated the predictive performance of PITC on a simulated toy example.
Acknowledgments.This work was supported by Singapore-MIT Alliance Research and Technology (SMART) Subaward Agreement 14 R-252-000-466-592.
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|
[] |
[
"Modelling energy spot prices by volatility modulated Lévy-driven Volterra processes",
"Modelling energy spot prices by volatility modulated Lévy-driven Volterra processes"
] |
[
"Ole E Barndorff-Nielsen \nThiele Center\nDepartment of Mathematical Sciences & CREATES\nDepartment of Economics and Business\nAarhus University\nNy Munkegade 118DK-8000AarhusDenmark\n\nInstitute for Advanced Study\nTech-nische Universität München\n85748GarchingGermany\n",
"Fred Espen Benth [email protected] \nCentre of Mathematics for Applications\nUniversity of Oslo\nBlindern, NP.O. Box 10530316OsloNorway\n",
"Almut E D Veraart [email protected] \nDepartment of Mathematics\nImperial College London\n180 Queen's GateSW7 2AZLondonUnited Kingdom and Creates\n"
] |
[
"Thiele Center\nDepartment of Mathematical Sciences & CREATES\nDepartment of Economics and Business\nAarhus University\nNy Munkegade 118DK-8000AarhusDenmark",
"Institute for Advanced Study\nTech-nische Universität München\n85748GarchingGermany",
"Centre of Mathematics for Applications\nUniversity of Oslo\nBlindern, NP.O. Box 10530316OsloNorway",
"Department of Mathematics\nImperial College London\n180 Queen's GateSW7 2AZLondonUnited Kingdom and Creates"
] |
[
"Bernoulli"
] |
This paper introduces the class of volatility modulated Lévy-driven Volterra (VMLV) processes and their important subclass of Lévy semistationary (LSS) processes as a new framework for modelling energy spot prices. The main modelling idea consists of four principles: First, deseasonalised spot prices can be modelled directly in stationarity. Second, stochastic volatility is regarded as a key factor for modelling energy spot prices. Third, the model allows for the possibility of jumps and extreme spikes and, lastly, it features great flexibility in terms of modelling the autocorrelation structure and the Samuelson effect. We provide a detailed analysis of the probabilistic properties of VMLV processes and show how they can capture many stylised facts of energy markets. Further, we derive forward prices based on our new spot price models and discuss option pricing. An empirical example based on electricity spot prices from the European Energy Exchange confirms the practical relevance of our new modelling framework.
|
10.3150/12-bej476
|
[
"https://arxiv.org/pdf/1307.6332v1.pdf"
] | 13,836,049 |
1307.6332
|
f4c9c13bef581828027752c173d26a7331c0faff
|
Modelling energy spot prices by volatility modulated Lévy-driven Volterra processes
2013
Ole E Barndorff-Nielsen
Thiele Center
Department of Mathematical Sciences & CREATES
Department of Economics and Business
Aarhus University
Ny Munkegade 118DK-8000AarhusDenmark
Institute for Advanced Study
Tech-nische Universität München
85748GarchingGermany
Fred Espen Benth [email protected]
Centre of Mathematics for Applications
University of Oslo
Blindern, NP.O. Box 10530316OsloNorway
Almut E D Veraart [email protected]
Department of Mathematics
Imperial College London
180 Queen's GateSW7 2AZLondonUnited Kingdom and Creates
Modelling energy spot prices by volatility modulated Lévy-driven Volterra processes
Bernoulli
193201310.3150/12-BEJ476arXiv:1307.6332v1 [q-fin.PR]energy marketsforward pricegeneralised hyperbolic distributionLévy semistationary processSamuelson effectspot pricestochastic integrationstochastic volatilityvolatility modulated Lévy-driven Volterra process
This paper introduces the class of volatility modulated Lévy-driven Volterra (VMLV) processes and their important subclass of Lévy semistationary (LSS) processes as a new framework for modelling energy spot prices. The main modelling idea consists of four principles: First, deseasonalised spot prices can be modelled directly in stationarity. Second, stochastic volatility is regarded as a key factor for modelling energy spot prices. Third, the model allows for the possibility of jumps and extreme spikes and, lastly, it features great flexibility in terms of modelling the autocorrelation structure and the Samuelson effect. We provide a detailed analysis of the probabilistic properties of VMLV processes and show how they can capture many stylised facts of energy markets. Further, we derive forward prices based on our new spot price models and discuss option pricing. An empirical example based on electricity spot prices from the European Energy Exchange confirms the practical relevance of our new modelling framework.
Introduction
Energy markets have been liberalised worldwide in the last two decades. Since then we have witnessed the increasing importance of such commodity markets which organise the trade and supply of energy such as electricity, oil, gas and coal. Closely related markets include also temperature and carbon markets. There is no doubt that such markets will play a vital role in the future given that the global demand for energy is constantly increasing. The main products traded on energy markets are spot prices, futures and forward contracts and options written on them. Recently, there has been an increasing research interest in the question of how such energy prices can be modelled mathematically. In this paper, we will focus on modelling energy spot prices, which include day-ahead as well as real-time prices.
Traditional spot price models typically allow for mean-reversion to reflect the fact that spot prices are determined as equilibrium prices between supply and demand. In particular, they are commonly based on a Gaussian Ornstein-Uhlenbeck (OU) process, see Schwartz [62], or more generally, on weighted sums of OU processes with different levels of mean-reversion, see, for example, Benth, Kallsen and Meyer-Brandis [24] and Klüppelberg, Meyer-Brandis and Schmidt [53]. In such a modelling framework, the meanreversion is modelled directly or physically, by claiming that the price change is (negatively) proportional to the current price. In this paper, we interpret the mean-reversion often found in commodity markets in a weak sense meaning that prices typically concentrate around a mean-level for demand and supply reasons. In order to account for such a weak form mean-reversion, we suggest to use a modelling framework which allows to model spot prices (after seasonal adjustment) directly in stationarity. This paper proposes to use the class of volatility modulated Lévy-driven Volterra (VMLV) processes as the building block for energy spot price models. In particular, the subclass of so-called Lévy semistationary (LSS) processes turns out to be of high practical relevance. Our main innovation lies in the fact that we propose a modelling framework for energy spot prices which (1) allows to model deseasonalised energy spot prices directly in stationarity, (2) comprises stochastic volatility, (3) accounts for the possibility of jumps and spikes, (4) features great flexibility in terms of modelling the autocorrelation structure of spot prices and of describing the so-called Samuelson effect, which refers to the finding that the volatility of a forward contract typically increases towards maturity.
We show that the new class of VMLV processes is analytically tractable, and we will give a detailed account of the theoretical properties of such processes. Furthermore, we derive explicit expressions for the forward prices implied by our new spot price model. In addition, we will see that our new modelling framework encompasses many classical models such as those based on the Schwartz one-factor mean-reversion model, see Schwartz [62], and the wider class of continuous-time autoregressive moving-average (CARMA) processes. In that sense, it can also be regarded as a unifying modelling approach for the most commonly used models for energy spot prices. However, the class of VMLV processes is much wider and directly allows to model the key special features of energy spot prices and, in particular, the stochastic volatility component.
The remaining part of the paper is structured as follows. We start by introducing the class of VMLV processes in Section 2. Next, we formulate both a geometric and an arithmetic spot price model class in Section 3 and describe how our new models embed many of the traditional models used in the recent literature. In Section 4, we derive the forward price dynamics of the models and consider questions like affinity of the forward price with respect to the underlying spot. Section 5 contains an empirical example, where we study electricity spot prices from the European Energy Exchange (EEX). Finally, Section 6 concludes, and the Appendix contains the proofs of the main results.
Preliminaries
Throughout this paper, we suppose that we have given a probability space (Ω, F , P ) with a filtration F = {F t } t∈R satisfying the 'usual conditions,' see Karatzas and Shreve [52], Definition I.2.25.
The driving Lévy process
Let L = (L t ) t≥0 denote a càdlàg Lévy process with Lévy-Khinchine representation E(exp(iζL t )) = exp(tψ(ζ)) for t ≥ 0, ζ ∈ R and
ψ(ζ) = idζ − 1 2 ζ 2 b + R (e iζz − 1 − iζzI {|z|≤1} )ℓ L (dz)
for d ∈ R, b ≥ 0 and the Lévy measure ℓ L satisfying ℓ L ({0}) = 0 and R (z 2 ∧ 1)ℓ L (dz) < ∞. We denote the corresponding characteristic triplet by (d, b, ℓ L ). In a next step, we extend the definition of the Lévy process to a process defined on the entire real line, by taking an independent copy of (L t ) t≥0 , which we denote by (L * t ) t≥0 and we define L(t) := −L * (−(t−)) for t < 0. Throughout the paper L = (L t ) t∈R denotes such a twosided Lévy process.
Volatility modulated Lévy-driven Volterra processes
The class of volatility modulated Lévy-driven Volterra (VMLV) processes, introduced by Barndorff-Nielsen and Schmiegel [11], has the form
Y t = µ + t −∞ G(t, s)ω s− dL s + t −∞ Q(t, s)a s ds, t ∈ R,(1)
where µ is a constant, L is the two-sided Lévy process defined above, G, Q : R 2 → R are measurable deterministic functions with G(t, s) = Q(t, s) = 0 for t < s, and ω = (ω t ) t∈R and a = (a t ) t∈R are càdlàg stochastic processes which are (throughout the paper) assumed to be independent of L. In addition, we assume that ω is positive. Note that such a process generalises the class of convoluted subordinators defined in Bender and Marquardt [21] to allow for stochastic volatility. A very important subclass of VMLV processes is the new class of Lévy semistationary (LSS) processes: We choose two functions g, q : R → R + such that G(t, s) = g(t − s) and Q(t, s) = q(t − s) with g(t − s) = q(t − s) = 0 whenever s > t, then an LSS process Y = {Y t } t∈R is given by
Y t = µ + t −∞ g(t − s)ω s− dL s + t −∞ q(t − s)a s ds, t ∈ R.(2)
Note that the name Lévy semistationary processes has been derived from the fact that the process Y is stationary as soon as ω and a are stationary. In the case that L = B is a two-sided Brownian motion, we call such processes Brownian semistationary (BSS) processes, which have recently been introduced by Barndorff-Nielsen and Schmiegel [12] in the context of modelling turbulence in physics. The class of LSS processes can be considered as the natural analogue for (semi-) stationary processes of Lévy semimartingales (LSM), given by
µ + t 0 ω s− dL s + t 0 a s ds, t ≥ 0.
Remark. The class of VMLV processes can be embedded into the class of ambit fields, see Barndorf-Nilsen and Schmiegel [9,10], Barndorff-Nielsen, Benth and Veraart [5,6]. Also, it is possible to define VMLV and LSS processes for singular kernel functions G and g, respectively; a function G (or g) defined as above is said to be singular if G(t, t−) (or g(0+)) does not exist or is not finite.
Integrability conditions
In order to simplify the exposition, we will focus on the stochastic integral in the definition of an VMLV (and of an LSS ) process only. That is, throughout the rest of the paper, let
Y t = t −∞ G(t, s)ω s− dL s , Y t = t −∞ g(t − s)ω s− dL s , t ∈ R.(3)
In this paper, we use the stochastic integration concept described in Basse-O'Connor, Graversen and Pedersen [20] where a stochastic integration theory on R, rather than on compact intervals as in the classical framework, is presented. Throughout the paper, we assume that the filtration F is such that L is a Lévy process with respect to F, see Basse-O'Connor, Graversen and Pedersen [20], Section 4, for details. Let (d, b, ℓ L ) denote the Lévy triplet of L associated with a truncation function h(z) = 1 {|z|≤1} . According to Basse-O'Connor, Graversen and Pedersen [20], Corollary 4.1, for t ∈ R the process (φ t (s)) s≤t with φ t (s) := G(t, s)ω s− is integrable with respect to L if and only if (φ t (s)) s≤t is F-predictable and the following conditions hold almost surely:
b t −∞ φ t (s) 2 ds < ∞, t −∞ R (1 ∧ |φ t (s)z| 2 )ℓ L (dz) ds < ∞,(4)t −∞ dφ t (s) + R (h(zφ t (s)) − φ t (s)h(z))ℓ L (dz) ds < ∞.
When we plug in G(t, s) = g(t − s), we immediately obtain the corresponding integrability conditions for the LSS process.
Example 1. In the case of a Gaussian Ornstein-Uhlenbeck process, that is, when g(t − s) = exp(−α(t − s)) for α > 0 and ω ≡ 1, then the integrability conditions above are clearly satisfied, since we have
b t −∞ exp(−2α(t − s)) ds = 1 2α b < ∞.
Square integrability
For many financial applications, it is natural to restrict the attention to models where the variance is finite, and we focus therefore on Lévy processes L with finite second moment. Note that the integrability conditions above do not ensure square-integrability of Y t even if L has finite second moment. But substitute the first condition in (4) with the stronger condition
t −∞ E(φ t (s) 2 ) ds = t −∞ G 2 (t, s)E[ω 2 s ] ds < ∞,(5)then t −∞ G(t, s)ω s− d(L s − E(L s )) is square integrable. Clearly, E[ω 2 s ] is constant in case of stationarity. For the Lebesgue integral part, we need E t −∞ G(t, s)ω s ds 2 < ∞.(6)
According to the Cauchy-Schwarz inequality, we find
E t −∞ G(t, s)ω s ds 2 ≤ t −∞ |G(t, s)| 2a ds t −∞ |G(t, s)| 2(1−a) E[ω 2 s ] ds
for any constant a ∈ (0, 1). Thus, a sufficient condition for (6) to hold is that there exists an a ∈ (0, 1) such that
t −∞ |G(t, s)| 2a ds < ∞, t −∞ |G(t, s)| 2(1−a) E[ω 2 s ] ds < ∞, which simplifies to ∞ 0 g 2a (x) dx < ∞, t −∞ g 2(1−a) (t − s)E[ω 2 s ] ds < ∞,(7)
in the LSS case. Given a model for ω and g, these conditions are simple to verify. Let us consider an example.
Example 2. In Example 1, we showed that for the kernel function g(x) = exp(−αx) and in the case of constant volatility, the conditions (4) are satisfied. Next, suppose that there is stochastic volatility, which is defined by the Barndorff-Nielsen and Shephard [13] stochastic volatility model, that is ω 2 s = s −∞ e −λ(s−u) dU λs , for s ∈ R, λ > 0 and a subordinator U . Suppose now that U has cumulant function ∞ 0 (exp(iθz) − 1)ℓ U (dz) for a Lévy measure ℓ U supported on the positive real axis, and that U 1 has finite expectation. In this case, we have that E[ω 2 s ] = ∞ 0 zℓ U (dz) < ∞ for all s. Thus, both (5) and (6) are satisfied (the latter can be seen after using the sufficient conditions), and we find that Y t is a square-integrable stochastic process.
The new model class for energy spot prices
This section presents the new modelling framework for energy spot prices, which is based on VMLV processes. As before, for ease of exposition, we will disregard the drift part in the general VMLV process for most of our analysis and rather use
Y = (Y t ) t∈R with Y t = t −∞ G(t, s)ω s− dL s(8)
as the building block for energy spot price, see (1) for the precise definition of all components. Throughout the paper, we assume that the corresponding integrability conditions hold. We can use the VMLV process defined in (8) as the building block to define both a geometric and an arithmetic model for the energy spot price. Also, we need to account for trends and seasonal effects. Let Λ : [0, ∞) → [0, ∞) denote a bounded and measurable deterministic seasonality and trend function. In a geometric set up, we define the spot price S g = (S g t ) t≥0 by
S g t = Λ(t) exp(Y t ), t ≥ 0.(9)
In such a modelling framework, the deseasonalised, logarithmic spot price is given by a VMLV process. Alternatively, one can construct a spot price model which is of arithmetic type. In particular, we define the electricity spot price S a = (S a t ) t≥0 by
S a t = Λ(t) + Y t , t ≥ 0.(10)
(Note that the seasonal function Λ in the geometric and the arithmetic model is typically not the same.) For general asset price models, one usually formulates conditions which ensure that prices can only take positive values. We can easily ensure positivity of our arithmetic model by imposing that L is a Lévy subordinator and that the kernel function G takes only positive values.
Model properties
Possibility of modelling in stationarity
We have formulated the new spot price model in the general form based on a VMLV process to be able to account for non-stationary effects, see, for example, Burger et al. [38], Burger, Graeber and Schindlmayr [37]. If the empirical data analysis, however, sup-ports the assumption of working under stationarity, then we will restrict ourselves to the analysis of LSS processes with stationary stochastic volatility. As mentioned in the Introduction, traditional models for energy spot prices are typically based on meanreverting stochastic processes, see, for example, Schwartz [62], since such a modelling framework reflects the fact that commodity spot prices are equilibrium prices determined by supply and demand. Stationarity can be regarded as a weak form of mean-reversion and is often found in empirical studies on energy spot prices; one such example will be presented in this paper.
The initial value
In order to be able to have a stationary model, the lower integration bound in the definition of the VMLV process, and in particular for the LSS process, is chosen to be −∞ rather than 0. Clearly, in any real application, we observe data from a starting value onwards, which is traditionally chosen as the observation at time t = 0. Hence, while VMLV processes are defined on the entire real line, we only define the spot price for t ≥ 0. The observed initial value of the spot price at time t = 0 is assumed to be a realisation of the random variable S g 0 = Λ(0) exp(Y 0 ) and S a 0 = Λ(0) + Y 0 , respectively. Such a choice guarantees that the deseasonalised spot price is a stationary process, provided we are in the stationary LSS framework.
The driving Lévy process
Since VMLV and LSS processes are driven by a general Lévy process L, it is possible to account for price jumps and spikes, which are often observed in electricity markets. At the same time, one can also allow for Brownian motion-driven models, which are very common in, for example, temperature markets, see, for example, Benth, Härdle and Cabrera [23].
Stochastic volatility
A key ingredient of our new modelling framework which sets the model apart from many traditional models is the fact that it allows for stochastic volatility. Volatility clusters are often found in energy prices, see, for example, Hikspoors and Jaimungal [50], Trolle and Schwartz [64], Benth [22], Benth and Vos [26], Koopman, Ooms and Carnero [55], Veraart and Veraart [65]. Therefore, it is important to have a stochastic volatility component, given by ω, in the model. Note that a very general model for the volatility process would be to choose an VMLV process, that is, ω 2 t = Z t and
Z t = t −∞ i(t, s) dU s ,(11)
where i denotes a deterministic, positive function and U is a Lévy subordinator. In fact, if we want to ensure that the volatility Z is stationary, we can work with a function of the form i(t, s) = i * (t − s), for a deterministic, positive function i * .
Autocorrelation structure and Samuelson effect
The kernel function G (or g) plays a vital role in our model and introduces a flexibility which many traditional models lack: We will see in Section 3.2 that the kernel functiontogether with the autocorrelation function of the stochastic volatility process -determines the autocorrelation function of the process Y . Hence our VMLV -based models are able to produce various types of autocorrelation functions depending on the choice of the kernel function G. It is important to stress here that this can be achieved by using one VMLV process only, whereas some traditional models need to introduce a multi-factor structure to obtain a comparable modelling flexibility. Also due to the flexibility in the choice of the kernel function, we can achieve greater flexibility in modelling the shape of the Samuelson effect often observed in forward prices, including the hyperbolic one suggested by Bjerksund, Rasmussen and Stensland [31] as a reasonable volatility feature in power markets. Note that we obtain the modelling flexibility in terms of the general kernel function G here since we specify our model directly through a stochastic integral whereas most of the traditional models are specified through evolutionary equations, which limit the choices of kernel functions associated with solutions to such equations.
In that context, we note that a VMLV or an LSS process cannot in general be written in form of a stochastic differential equation (due to the non-semimartingale character of the process). In Section 3.3, we will discuss sufficient conditions which ensure that an LSS process is a semimartingale.
3.1.6. A unifying approach for traditional spot price models As already mentioned above, energy spot prices are typically modelled in stationarity, hence the class of LSS processes is particularly relevant for applications. In the following, we will show that many of the traditional spot price models can be embedded into our LSS process-based framework.
Our new framework nests the stationary version of the classical one-factor Schwartz [62] model studied for oil prices. By letting L be a Lévy process with the pure-jump part given as a compound Poisson process, Cartea and Figueroa [40] successfully fitted the Schwartz model to electricity spot prices in the UK market. Benth andŠaltytė Benth [27] used a normal inverse Gaussian Lévy process L to model UK spot gas and Brent crude oil spot prices. Another example which is nested by the class of LSS processes is a model studied in Benth [22] in the context of gas markets, where the deseasonalised logarithmic spot price dynamics is assumed to follow a one-factor Schwartz process with stochastic volatility. A more general class of models which is nested is the class of socalled CARMA-processes, which has been successfully used in temperature modelling and weather derivatives pricing, see Benth,Šaltytė Benth and Koekebakker [30], Benth, Härdle and López Cabrera [23] and Härdle and López Cabrera [49], and more recently for electricity prices by García, Klüppelberg and Müller [45], Benth et al. [25]. A CARMA process is the continuous-time analogue of an ARMA time series, see Brockwell [33], Brockwell [34] for definition and details. More precisely, suppose that for nonnegative integers p > q
Y t = b ′ V t ,
where b ∈ R p and V is a p-dimensional OU process of the form
dV t = AV t dt + e p dL t ,(12)
with
A = 0 I p−1 −α p −α p−1 · · · − α 1 .
Here we use the notation I p−1 for the (p − 1) × (p − 1)-identity matrix, e p the pth coordinate vector (where the first p − 1 entries are zero and the pth entry is 1) and
b ′ = [b 0 , b 1 , . . . , b p−1 ] is the transpose of b, with b q = 1 and b j = 0 for q < j < p. In
Brockwell [35], it is shown that if all the eigenvalues of A have negative real parts, then
(V t ) t∈R defined as V t = t −∞ e A(t−s) e p dL(s),
is the (strictly) stationary solution of (12). Moreover,
Y t = b ′ V t = t −∞ b ′ e A(t−s) e p dL(s),(13)
is a CARMA(p, q) process. Hence, specifying g(x) = b ′ exp(Ax)e p in (13), the logspot price dynamics will be an LSS process, but without stochastic volatility. García, Klüppelberg and Müller [45] argue for CARMA(2, 1) dynamics as an appropriate class of models for the deseasonalised log-spot price at the Singapore New Electricity Market. The innovation process L is chosen to be in the class of stable processes. From Benth,Šaltytė Benth and Koekebakker [30], Brownian motion-driven CARMA(3, 0) models seem appropriate for modelling daily average temperatures, and are applied for temperature derivatives pricing, including forward price dynamics of various contracts. More recently, the dynamics of wind speeds have been modelled by a Brownian motion-driven CARMA(4, 0) model, and applied to wind derivatives pricing, see Benth andŠaltytė Benth [28] for more details.
Finally note that the arithmetic model based on a superposition of LSS processes nests the non-Gaussian Ornstein-Uhlenbeck model which has recently been proposed for modelling electricity spot prices, see Benth, Kallsen and Meyer-Brandis [24].
We emphasis again that, beyond the fact that LSS processes can be regarded as a unifying modelling approach which nest many of the existing spot price models, they also open up for entirely new model specifications, including more general choices of the kernel function (resulting in non-linear models) and the presence of stochastic volatility.
Second order structure
Next, we study the second order structure of volatility modulated Volterra processes
Y = (Y t ) t∈R , where Y t = t
−∞ G(t, s)ω s− dL s , assuming the integrability conditions (4) hold and that in addition Y is square integrable. Let κ 1 = E(L 1 ) and κ 2 = Var(L 1 ). Recall that throughout the paper we assume that the stochastic volatility ω is independent of the driving Lévy process. Note that proofs of the following results are easy and hence omitted.
Proposition 1. The conditional second order structure of Y is given by
E(Y t |ω) = κ 1 t −∞ G(t, s)ω s ds, Var(Y t |ω) = κ 2 t −∞ G(t, s) 2 ω 2 s ds, Cov((Y t+h , Y t )|ω) = κ 2 t −∞ G(t + h, s)G(t, s)ω 2 s ds for t ∈ R, h ≥ 0.
Corollary 1. The conditional second order structure of Y is given by
E(Y t |ω) = κ 1 ∞ 0 g(x)ω t−x dx, Var(Y t |ω) = κ 2 ∞ 0 g(x) 2 ω 2 t−x dx, Cov((Y t+h , Y t )|ω) = κ 2 ∞ 0 g(x + h)g(x)ω 2 t−x dx for t ∈ R, h ≥ 0.
The unconditional second order structure of Y is then given as follows.
Proposition 2. The second order structure of Y for stationary ω is given by
E(Y t ) = κ 1 E(ω 0 ) t −∞ G(t, s) ds, Var(Y t ) = κ 2 E(ω 2 0 ) t −∞ G(t, s) 2 ds + κ 2 1 t −∞ t −∞ G(t, s)G(t, u)γ(|s − u|) ds du, Cov(Y t+h , Y t ) = κ 2 E(ω 2 0 ) t −∞ G(t + h, s)G(t, s) ds + κ 2 1 t+h −∞ t −∞ G(t + h, s)G(t, u)γ(|s − u|) ds du, where γ(h) = Cov(ω t+h , ω t ) denotes the autocovariance function of ω, for t ∈ R, h ≥ 0.
The unconditional second order structure of Y is then given as follows.
Corollary 2. The second order structure of Y for stationary ω is given by
E(Y t ) = κ 1 E(ω 0 ) ∞ 0 g(x) dx, Var(Y t ) = κ 2 E(ω 2 0 ) ∞ 0 g(x) 2 dx + κ 2 1 ∞ 0 ∞ 0 g(x)g(y)γ(|x − y|) dx dy, Cov(Y t+h , Y t ) = κ 2 E(ω 2 0 ) ∞ 0 g(x + h)g(x) dx + κ 2 1 ∞ 0 ∞ 0 g(x + h)g(y)γ(|x − y|) dx dy, where γ(x) = Cov(ω t+x , ω t ) denotes the autocovariance function of ω, for t ∈ R, h ≥ 0. Hence, we have Cor(Y t+h , Y t ) (14) = κ 2 E(ω 2 0 ) ∞ 0 g(x + h)g(x) dx + κ 2 1 ∞ 0 ∞ 0 g(x + h)g(y)γ(|x − y|) dx dy κ 2 E(ω 2 0 ) ∞ 0 g(x) 2 dx + κ 2 1 ∞ 0 ∞ 0 g(x)g(y)γ(|x − y|) dx dy .
Corollary 3. If κ 1 = 0 or if ω has zero autocorrelation, then
Cor(Y t+h , Y t ) = ∞ 0 g(x + h)g(x) dx ∞ 0 g(x) 2 dx .
The last corollary shows that we get the same autocorrelation function as in the BSS model. From the results above, we clearly see the influence of the general damping function g on the correlation structure. A particular choice of g, which is interesting in the energy context is studied in the next example.
Example 3. Consider the case g(x) = σ x+b , for σ, b > 0 and ω ≡ 1, which is motivated from the forward model of Bjerksund, Rasmussen and Stensland [31], which we shall return to in Section 4. We have that
∞ 0 g 2 (x) dx = σ 2 b .
This ensures integrability of g(t − s) over (−∞, t) with respect to any square integrable martingale Lévy process L. Furthermore,
∞ 0 g(x + h)g(x) dx = σ 2 h ln(1 + h b ). Thus, Cor(Y t+h , Y t ) = b h ln 1 + h b .
Observe that since g can be written as
g(x) = σ x + b = x 0 −σ ds (s + b) 2 + σ b , it follows that the process Y (t) = t −∞ g(t − s) dB
Semimartingale conditions and absence of arbitrage
We pointed out that the subclass of LSS processes are particularly relevant for modelling energy spot prices since they allow one to model directly in stationarity. Let us focus on this class in more detail. Clearly, an LSS process is in general not a semimartingale. However, we can formulate sufficient conditions on the kernel function and on the stochastic volatility component which ensure the semimartingale property. The sufficient conditions are in line with the conditions formulated for BSS processes in Barndorff-Nielsen and Schmiegel [12], see also Barndorff-Nielsen and Basse-O'Connor [4]. Note that the proofs of the following results are provided in the Appendix.
Proposition 3. Let Y be an LSS process as defined in (2). Suppose the following conditions hold:
(i) E|L 1 | < ∞. (ii)
The function values g(0+) and q(0+) exist and are finite.
(iii) The kernel function g is absolutely continuous with square integrable derivative g ′ . (iv) The process (g ′ (t − s)ω s− ) s∈R is square integrable for each t ∈ R. (v) The process (q ′ (t − s)a s ) s∈R is integrable for each t ∈ R. Then (Y t ) t≥0 is a semimartingale with representation Y t = Y 0 + g(0+) t 0 ω s− dL s + t 0 A s ds for t ≥ 0,(15)
where
L s = L s − E(L s ) for s ∈ R and A s = g(0+)ω s− E(L 1 ) + s −∞ g ′ (s − u)ω u− dL u + q(0+)a s + s −∞ q ′ (s − u)a u du.
Example 4. An example of a kernel function which satisfies the above conditions is given by
g(x) = J i=1 w i exp(−λ i x) for λ i > 0, w i ≥ 0, i = 1, . . . , J.
For J = 1, Y is given by a volatility modulated Ornstein-Uhlenbeck process.
In a next step, we are now able to find a representation for the quadratic variation of an LSS process provided the conditions of Proposition 3 are satisfied.
Proposition 4. Let Y be an LSS process and suppose that the sufficient conditions for Y to be a semimartingale (as formulated in Proposition 3) hold. Then, the quadratic variation of Y is given by
[Y ] t = g(0+) 2 t 0 ω 2 s− d[L] s for t ≥ 0.
Note that the quadratic variation is a prominent measure of accumulated stochastic volatility or intermittency over a certain period of time and, hence, is a key object of interest in many areas of application and, in particular, in finance.
The question of deriving semimartingale conditions for LSS processes is closely linked to the question whether a spot price model based on an LSS process is prone to arbitrage opportunities. In classical financial theory, we usually stick to the semimartingale framework to ensure the absence of arbitrage. Nevertheless one might ask the question whether one could still work with the wider class of LSS processes which are not semimartingales.
Here we note that the standard semimartingale assumption in mathematical finance is only valid for tradeable assets in the sense of assets which can be held in a portfolio. Hence, when dealing with, for example, electricity spot prices, this assumption is not valid since electricity is essentially non-storable. Hence, such a spot price cannot be part of any financial portfolio and, therefore, the requirement of being a martingale under some equivalent measure Q is not necessary.
Guasoni, Rásonyi and Schachermayer [47] have pointed out that, while in frictionless markets martingale measures play a key role, this is not the case any more in the presence of market imperfections. In fact, in markets with transaction costs, consistent price systems as introduced in Schachermayer [61] are essential. In such a set-up, even processes which are not semimartingales can ensure that we have no free lunch with vanishing risk in the sense of Delbaen and Schachermayer [42]. It turns out that if a continuous price process has conditional full support, then it admits consistent price systems for arbitrarily small transaction costs, see Guasoni, Rásonyi and Schachermayer [47]. It has recently been shown by Pakkanen [57], that under certain conditions, a BSS process has conditional full support. This means that such processes can be used in financial applications without necessarily giving rise to arbitrage opportunities.
Model extensions
Let us briefly point out some model extensions concerning a multi-factor structure, nonstationary effects, multivariate models and alternative methods for incorporating stochastic volatility.
A straightforward extension of our model is to study a superposition of LSS processes for the spot price dynamics. That is, we could replace the process Y by a superposition of J ∈ N factors:
J i=1 w i Y (i) t where w 1 , . . . , w J ≥ 0, J i=1 w i = 1,(16)
and where all Y (i) t are defined as in (8) for independent Lévy processes L (i) and independent stochastic volatility processes ω (i) , in both the geometric and the arithmetic model. Such models include the Benth, Kallsen and Meyer-Brandis [24] model as a special case. A superposition of factors Y (i) opens up for separate modelling of spikes and other effects. For instance, one could let the first factor account for the spikes, using a Lévy process with big jumps at low frequency, while the function g forces the jumps back at a high speed. The next factor(s) could model the "normal" variations of the market, where one observes a slower force of mean-reversion, and high frequent Brownian-like noise, see Veraart and Veraart [65] for extensions along these lines. Note that all the results we derive in this paper based on the one factor model can be easily generalised to accommodate for the multi-factor framework. It should be noted that this type of "superposition" is quite different from the concept behind supOU processes as studied in, for example, Barndorff-Nielsen and Stelzer [15].
In order to study various energy spot prices simultaneously, one can consider extensions to a multivariate framework along the lines of Barndorff-Nielsen and Stelzer [15,16], Veraart and Veraart [65].
In addition, another interesting aspect which we leave for future research is the question of alternative ways of introducing stochastic volatility in VMLV processes. So far, we have introduced stochastic volatility by considering a stochastic proportional of the driving Lévy process, that is, we work with a stochastic integral of ω with respect to L. An alternative model specification could be based on a stochastic time change
t −∞ G(t, s) dL ω 2+ s , where ω 2+ s = s 0 ω 2 u du.
Such models can be constructed in a fashion similar to that of volatility modulated non-Gaussian Ornstein-Uhlenbeck processes introduced in Barndorff-Nielsen and Veraart [17]. We know that outside the Brownian or stable Lévy framework, stochastic proportional and stochastic time change are not equivalent. Whereas in the first case the jump size is modulated by a volatility term, in the latter case the speed of the process is changed randomly. These two concepts are in fact fundamentally different (except for the special cases pointed out above) and, hence, it will be worth investigating whether a combination of stochastic proportional and stochastic time change might be useful in certain applications.
Pricing of forward contracts
In this subsection, we are concerned with the calculation of the forward price F t (T ) at time t ≥ 0 for contracts maturing at time T ≥ t. We denote by T * < ∞ a finite time horizon for the forward market, meaning that all contracts of interest mature before this date. Note that in energy markets, the corresponding commodity typically gets delivered over a delivery period rather than at a fixed point in time. Extensions to such a framework can be dealt with using standard methods, see, for example, Benth,Šaltytė Benth and Koekebakker [29] for more details.
Let S = (S) t≥0 denote the spot price, being either of geometric or arithmetic kind as defined in (9) and (10), respectively, with
Y t = t −∞ G(t, s)ω s− dL s , Z t = ω 2 t = t −∞ i(t, s) dU s ,
where the stochastic volatility ω is chosen as previously defined in (11). Clearly, the corresponding results for LSS processes can be obtained by choosing G(t, s) = g(t − s). We use the conventional definition of a forward price in incomplete markets, see Duffie [43], ensuring the martingale property of t → F t (T ),
F t (T ) = E Q [S T |F t ], 0 ≤ t ≤ T ≤ T * ,(17)
with Q being an equivalent probability measure to P . Here, we suppose that S T ∈ L 1 (Q), the space of integrable random variables. In a moment, we shall introduce sufficient conditions for this.
Change of measure by generalised Esscher transform
In finance, one usually uses equivalent martingale measures Q, meaning that the equivalent probability measure Q should turn the discounted price dynamics of the underlying asset into a (local) Q-martingale. However, as we have already discussed, this restriction is not relevant in, for example, electricity markets since the spot is not tradeable. Thus, we may choose any equivalent probability Q as pricing measure. In practice, however, one restricts to a parametric class of equivalent probability measures, and the standard choice seems to be given by the Esscher transform, see Benth,Šaltytė Benth and Koekebakker [29], Shiryaev [63]. The Esscher transform naturally extends the Girsanov transform to Lévy processes. To this end, consider Q θ L defined as the (generalised) Esscher transform of L for a parameter θ(t) being a Borel measurable function. Following Shiryaev [63] (or Benth, Saltytė Benth and Koekebakker [29], Barndorff-Nielsen and Shiryaev [14]), Q θ L is defined via the Radon-Nikodym density process
dQ θ L dP Ft = exp t 0 θ(s) dL s − t 0 φ L (θ(s)) ds(18)
for θ(·) being a real-valued function which is integrable with respect to the Lévy process on [0, T * ], and φ L (x) = log(E(exp(xL 1 ))) = ψ(−ix) = dx + 1 2
x 2 b + R (e xz − 1 − xzI {|z|≤1 })ℓ L (dz),
(for x ∈ R) being the log-moment generating function of L 1 , assuming that the moment generating function of L 1 exists. A special choice is the 'constant' measure change, that is, letting
θ(t) = θ1 [0,∞) (t).(19)
In this case, if under the measure P , L has characteristic triplet (d, b, ℓ L ), where d is the drift, b is the squared volatility of the continuous martingale part and ℓ L is the Lévy measure in the Lévy-Khinchine representation, see Shiryaev [63], a fairly straightforward calculation shows that, see Shiryaev [63] again, the Esscher transform preserves the Lévy property of L, and the characteristic triplet under the measure Q θ L on the interval [0,
T * ] becomes (d θ , b, exp(θ·)ℓ L ), where d θ = d + bθ + |z|≤1 z(e θz − 1)ℓ L (dz).
This comes from the simple fact that the logarithmic moment generating function
of L under Q θ L is φ θ L (x) φ L (x + θ) − φ L (x).(20)
Remark. It is important to note here that the choice of θ(t) (as, e.g., in (19)) forces us to choose a starting time since the function θ will not be integrable with respect to L on the unbounded interval (−∞, t). Recall that the only reason why we model from −∞ rather than from 0 is the fact that we want to be able to obtain a stationary process under the probability measure P . Throughout this section, we choose the starting time to be zero, which is a convenient choice since L 0 = 0, and it is also practically reasonable since this can be considered as the time from which we start to observe the process. With such a choice, we do not introduce any risk premium for t < 0.
In the general case, with a time-dependent parameter function θ(t), the characteristic triplet of L under Q θ L will become time-dependent, and hence the Lévy process property is lost. Instead, L will be an independent increment process (sometimes called an additive process). Note that if L = B, a Brownian motion, the Esscher transform is simply a Girsanov change of measure where dB t = θ(t) dt + dW t for 0 ≤ t ≤ T * and a Q θ L -Brownian motion W .
Similarly, we do a (generalised) Esscher transform of U , the subordinator driving the stochastic volatility model, see (11). We define Q η U to have the Radon-Nikodym density process
dQ η U dP Ft = exp t 0 η(s) dU s − t 0 φ U (η(s)) ds
for η(·) ∈ R being a real-valued function which is integrable with respect to U on [0, T * ], and φ U (x) = log(E(exp(xU 1 ))) being the log-moment generating function of U 1 . Since U is a subordinator, we obtain
φ U (x) = dx + ∞ 0 (e xz − 1)ℓ U (dz),
where d ≥ 0 and ℓ U denotes the Lévy measure associated with U .
Remark. Our discussion above on choosing a starting value applies to the measure transform for the volatility process as well, and hence throughout the paper we will work under the assumption that θ(s) = η(s) = 0, for s < 0. Note in particular, that this assumption implies that under the risk-neutral probability measure, the characteristic triplets of L and U only change on the time interval [0, T * ]. On the interval (−∞, 0), we have the same characteristic triplet for L and U as under P .
Choosing η(t) = η1 [0,∞) (t), with a constant η ∈ R, an Esscher transform will give a characteristic triplet ( d, 0, exp(η·)ℓ U ), which thus preserves the subordinator property of (U t ) 0≤t≤T * under Q η U . For the general case, the process U will be a time-inhomogeneous subordinator (independent increment process with positive jumps). The log-moment generating function of U 1 under the measure Q η U is denoted by φ η U (x). In order to ensure the existence of the (generalised) Esscher transforms, we need some conditions. We need that there exists a constant c > 0 such that sup 0≤s≤T * |θ(s)| ≤ c, and where |z|>1 exp(cz)ℓ L (dz) < ∞. (Similarly, we must have such a condition for the Lévy measure of the subordinator driving the stochastic volatility, that is, ℓ U ). Also, we must require that exponential moments of L 1 and U 1 exist. More precisely, we suppose that parameter functions θ(·) and η(·) of the (generalised) Esscher transform are such that
T * 0 |z|>1 e |θ(s)|z ℓ L (dz) ds < ∞, T * 0 |z|>1 e |η(s)|z ℓ U (dz) ds < ∞.(21)
The exponential integrability conditions of the Lévy measures of L and U imply the existence of exponential moments, and thus that the Esscher transforms Q θ L and Q η U are well defined.
We define the probability Q θ,η Q θ L × Q η U as the class of pricing measures for deriving forward prices. In this respect, θ(t) may be referred to as the market price of risk, whereas η(t) is the market price of volatility risk. We note that a choice θ > 0 will put more weight to the positive jumps in the price dynamics, and less on the negative, increasing the "risk" for big upward movements in the prices under Q θ,η .
Let us denote by E θ,η the expectation operator with respect to Q θ,η , and by E η the expectation with respect to Q η U .
Forward price in the geometric case
Suppose that the spot price is defined by the geometric model
S t := S g t = Λ(t) exp(Y t ),
where Y is defined as in (3). In order to have the forward price F t (T ) well defined, we need to ensure that the spot price is integrable with respect to the chosen pricing measure Q θ,η . We discuss this issue in more detail in the following. We know that ω is positive and in general not bounded since it is defined via a subordinator. Thus, G(t, s)ω s + θ(s) (for s ≤ t) is unbounded as well. Supposing that L has exponential moments of all orders, we can calculate as follows using iterated expectations conditioning on the filtration G t generated by the paths of ω s , for s ≤ t:
E θ,η [S T ] = Λ(T )E θ,η E θ,η exp T −∞ G(T, s)ω s− dL s G T = Λ(T )E η exp 0 −∞ φ L (G(T, s)ω s ) ds exp T 0 φ θ L (G(T, s)ω s ) ds .
To have that S T ∈ L 1 (Q θ,η ), the two integrals must be finite. This puts additional restrictions on the choice of η and the specifications of G(t, s) and i(t, s). We note that when applying the Esscher transform, we must require that L has exponential moments of all orders, a rather strong restriction on the possible class of driving Lévy processes. In our empirical study, however, we will later see that the empirically relevant cases are either that L is a Brownian motion or that L is a generalised hyperbolic Lévy process, which possess exponential moments of all orders.
We are now ready to price forwards under the Esscher transform.
Proposition 5. Suppose that S T ∈ L 1 (Q θ,η ). Then, the forward price for 0 ≤ t ≤ T ≤ T * is given by
F t (T ) = Λ(T ) exp t −∞ G(T, s)ω s− dL s E η exp T t φ θ L (G(T, s)ω s ) ds F t .
Change of measure by the Girsanov transform in the Brownian case
As a special case, consider L = B, where B is a two-sided standard Brownian motion under P . In this case we apply the Girsanov transform rather than the generalised Esscher transform, and it turns out that a rescaling of the transform parameter function θ(t) by the volatility ω t is convenient for pricing of forwards. To this end, consider the Girsanov transform
B t = W t + t 0 θ(s) ω s− ds for t ≥ 0, B t = W t for t < 0,(22)
that is, we set θ(t) = 0 for t < 0. Supposing that the Novikov condition
E exp 1 2 T * 0 θ 2 (s) ω 2 s ds < ∞,
holds, we know that W t is a Brownian motion for 0 ≤ t ≤ T * under a probability Q θ B having density process
dQ θ B dP Ft = exp − t 0 θ(s) ω s− dB s − 1 2 t 0 θ 2 (s) ω 2 s ds .
Suppose that there exists a measurable function j(t) such that
j(t) ≤ i(t, s) i(0, s)(23)
for all 0 ≤ s ≤ t ≤ T * , with
T * 0 θ 2 (s) j(s) ds < ∞.
Furthermore, suppose the moment generating function of ω −2 0 exists on the interval [0, C U ). Then, for all θ(t) such that 0.5 T * 0 θ 2 (s)/j(s) ds ≤ C U , the Novikov condition is satisfied, since by the subordinator property of U t (restricting our attention to t ≥ 0)
ω 2 t = t −∞ i(t, s) dU s ≥ 0 −∞ i(t, s) dU s ≥ j(t) 0 −∞ i(0, s) dU s = j(t)w 2 0 ,
and therefore
E exp 1 2 T * 0 θ 2 (s) ω 2 s ds ≤ E exp 1 2 T * 0 θ 2 (s) j(s) dsω −2 0 < ∞.
Specifying i(t, s) = exp(−λ(t − s)), we have that i(t, s)/i(0, s) = exp(−λt) = j(t), and condition (23) holds with equality.
Forward price in the geometric case
We discuss the integrability of S T = S g T with respect to Q θ,η Q θ B × Q η U . By double conditioning with respect to the filtration generated by the paths of ω t , we find
E θ,η [S T ] = Λ(T ) exp T 0 G(T, s)θ(s) ds E θ,η E θ,η exp T −∞ G(T, s)ω s− dW s G T = Λ(T ) exp T 0 G(T, s)θ(s) ds E η exp 1 2 T −∞ G 2 (T, s)ω 2 s ds .
From collecting the conditions on G, i, θ and η for verifying all the steps above, we find that if s → G(T, s)θ(s) is integrable on [0, T ) (recall that θ(s) = η(s) = 0 for s < 0) and s → G 2 (T, s)i(s, v) is integrable on [v, T ) for all −∞ < v < T , then S T ∈ L 1 (Q θ,η ) as long as
T −∞ |z|>1 exp z 1 2 T v G 2 (T, s)i(s, v) ds + |η(v)| ℓ U (dz) dv < ∞.(24)
We assume these conditions to hold. We state the forward price for the case L = B and the Girsanov change of measure discussed above. Proposition 6. Suppose that L = B and that Q θ B is defined by the Girsanov transform in (22). Then, for 0 ≤ t ≤ T ≤ T * ,
F t (T ) = Λ(T ) exp t −∞ G(T, s)ω s− dW s + 1 2 t −∞ T t G 2 (T, s)i(s, v) ds dU v + T 0 G(T, s)θ(s) ds + T t φ η U 1 2 T v G 2 (T, s)i(s, v) ds dv .
Let us consider an example.
Example 5. In the BNS stochastic volatility model, we have i(t, s) = exp(−λ(t − s)).
Hence, which yields,
t −∞ T t G 2 (T, v)i(v, s) dv dU s = Z t T t G 2 (T, v)e λ(t−v) dv.
This implies from Proposition 6 that the forward price is affine in Z, the (square of the) stochastic volatility. The stochastic volatility model studied in Benth [22] is recovered by choosing G(t, s) = exp(−α(t − s)).
On the case of constant volatility
Suppose for a moment that the stochastic volatility process ω t is identical to one (i.e., that we do not have any stochastic volatility in the model). In this case, the forward price becomes
F t (T ) = Λ(T ) exp t −∞ G(T, s) dW s + T 0 G(T, s)θ(s) ds = Λ(T ) exp t −∞ G(T, s) dB s + T t G(T, s)θ(s) ds ,
where W t = B t for t < 0. Hence, the logarithmic forward (log-forward) price is
ln F t (T ) = ln Λ(T ) + T t G(T, s)θ(s) ds + M t (T ), with M t (T ) = t −∞ G(T, s) dB s for t ≤ T . Note that t → M t (T ), for t ≥ 0, is a P -martingale with the property (for S t = S g t ) M t (t) = Y t = ln S t − ln Λ(t).
In the classical Ornstein-Uhlenbeck case, with G(t, s) = g(t − s), g(x) = exp(−αx) for α > 0, we easily compute that
M t (T ) = e −α(T −t) Y t = e −α(T −t) Y t ,
and the forward price is explicitly dependent on the current spot price.
In the general case, this does not hold true. We have that M T (T ) = Y T , not unexpectedly, since the forward price converges to the spot at maturity (at least theoretically). However, apart from the special time point t = T , the forward price will in general not be a function of the current spot, but a function of the process M t (T ). Thus, at time t, the forward price will depend on
M t (T ) = t −∞ G(T, s) dB s ,
whereas the spot price depends on
Y t = t −∞ G(t, s) dB s .
The two stochastic integrals can be pathwise interpreted (they are both Wiener integrals since the integrands are deterministic functions), and both Y t and M t (T ) are generated by integrating over the same paths of a Brownian motion. However, the paths are scaled by two different functions G(T, s) and G(t, s). This allows for an additional degree of flexibility when creating forward curves compared to affine structures.
In the classical Ornstein-Uhlenbeck case, the forward curve as a function of time to maturity T − t will simply be a discounting of today's spot price, discounted by the speed of mean reversion of the spot (in addition comes deterministic scaling by the seasonality and market price of risk). To highlight the additional flexibility in our modelling framework of semistationary processes, suppose for the sake of illustration that G(t, s) = g 1 (t)g 2 (s). Then
M t (T ) = g 1 (T ) g 1 (t) Y t .
If furthermore lim T →∞ g 1 (T ) := g 1 (∞) = 0, we are in a situation where the long end (i.e., T large) of the forward curve is not a constant. In fact, we find for t ≥ 0 that
lim T →∞ ln F t (T ) − g 1 (t) T t g 2 (s)θ(s) ds − ln Λ(T ) = (ln S t − ln Λ(t)) g 1 (∞) g 1 (t) .
Since ln S t is random, we will have a randomly fluctuating long end of the forward curve. This is very different from the situation with a classical mean-reverting spot dynamics, which implies a deterministic forward price in the long end (dependent on the seasonality and market price of risk only). Various shapes of the forward curve T → F t (T ) can also be modelled via different specifications of G. For instance, if g 1 (T ) is a decreasing function, we obtain the contango and backwardation situations depending on the spot price being above or below the mean. If T → g 1 (T ) has a hump, we will also observe a hump in the forward curve. For general specifications of G we can have a high degree of flexibility in matching desirable shapes of the forward curve. Observe that the time-dynamics of the forward price can be considered as correlated with the spot rather than directly depending on the spot. In the Ornstein-Uhlenbeck situation, the log-forward price can be considered as a linear regression on the current spot price, with time-dependent coefficients. This is not the case for general specifications. However, we have that M t (T ) and Y t are both normally distributed random variables (recall that we are still restricting our attention to L = B), and the correlation between the two is
Cor(M t (T ), Y t ) = t −∞ G(T, s)G(t, s) ds t −∞ G 2 (T, s) ds t −∞ G 2 (t, s) ds .
O.E. Barndorff-Nielsen, F.E. Benth and A.E.D. Veraart
Obviously, for G(t, s) = g(t − s) = exp(−α(t − s)), the correlation is 1. In conclusion, we can obtain a weaker stochastic dependency between the spot and forward price than in the classical mean-reversion case by a different specification of the function G.
Affine structure of the forward price
In the discussion above, we saw that the choice G(t, s) = g 1 (t)g 2 (s) yielded a forward price expressible in terms of Y t . In the next proposition, we prove that this is the only choice of G yielding an affine structure. The result is slightly generalising the analysis of Carverhill [41].
Proposition 7. The forward price in Proposition 6 is affine in Y t and Z t if there exist functions g 1 , g 2 , i 1 and i 2 such that G(t, s) = g 1 (t)g 2 (s) and i(t, s) = i 1 (t)i 2 (s). Conversely, if the forward price is affine in Y t and Z t , and G and i are strictly positive and continuously differentiable in the first argument, then there exists functions g 1 , g 2 , i 1 and i 2 such that G(t, s) = g 1 (t)g 2 (s) and i(t, s) = i 1 (t)i 2 (s).
Obviously, the choice of G and i coming from OU-models,
G(t, s) = g(t − s) = exp(−α(t − s)), i(t, s) = exp(−λ(t − s)),
satisfy the conditions in the proposition above. In fact, appealing to similar arguments as in the proof of Proposition 7 above, one can show that this is the only choice (modulo multiplication by a constant) which is stationary and gives an affine structure in the spot and volatility for the forward price dynamics. In particular, the specification g(x) = σ/(x + b) considered in Example 3 gives a stationary spot price dynamics, but not an affine structure in the spot for the forward price.
Risk-neutral dynamics of the forward price and the Samuelson effect
Next, we turn our attention to the risk-neutral dynamics of the forward price.
Proposition 8. Assume that the assumptions of Proposition 6 hold and that Q η U is given by the (simple) Esscher transform. Then the risk-neutral dynamics of the forward price F t (T ) is given by
dF t (T ) F t− (T ) = G(T, t)ω t− dW t + ∞ 0 exp 1 2 H T (t, t)z − 1 N U (dz, dt), 0 ≤ t ≤ T ≤ T * , where H T (t, t) = T t G 2 (T, s)i(s, t) ds. Moreover N U (dz, dt) = N U (dz, dt) − ℓ η U (dz) dt is a Q η U -martingale,
where N U denotes the Poisson random measure associated with U , and ℓ η U = exp(η·)ℓ U is the Lévy measure of U under Q η U .
We observe that the dynamics will jump according to the changes in volatility given by the process U t . As expected, the integrand in the jump expression tends to zero when T − t → 0, since the forward price must (at least theoretically) converge to the spot when time to maturity goes to zero.
The forward dynamics will have a stochastic volatility given by G(T, t)ω t− . Hence, whenever lim t↑T G(T, t) exists, and G(T, T ) = 1, we have a.s.,
lim t↑T G(T, t)ω t− = ω T − .
When passing to the limit, we have implicitly supposed that we work with the version of ω t− having left-continuous paths with right-limits. By the definition of our integral in Y t , where the integrand is supposed predictable, this can be done. Thus, we find that the forward volatility converges to the spot volatility as time to maturity tends to zero, which is known as the Samuelson effect. Contrary to the classical situation where this convergence goes exponentially, we may have many different shapes of the volatility term structure resulting from our general modelling framework.
In Bjerksund, Rasmussen and Stensland [31], a forward price dynamics for electricity contracts is proposed to follow
dF t (T ) F t (T ) = a + σ T − t + b dW t ,(25)
where a, b and σ are positive constants. They argue that in electricity markets, the Samuelson effect is stronger close to maturity than what is observed in other commodity markets, and they suggest to capture this by letting it increase by the rate 1/(T − t + b) close to maturity of the contracts. This is in contrast to the common choice of volatility being σ exp(−α(T − t)), resulting from using the Schwartz model for the spot price dynamics. There is no reference to any spot model in the Bjerksund, Rasmussen and Stensland [31] model. The constant a comes from a non-stationary behaviour, which can be incorporated in the VMLV framework. However, here we focus on the stationary case and choose a = 0. Then we see that we can model the spot price by the BSS process
Y t = t −∞ g(t − s) dB s with g(x) = σ x + b .
Thus, after doing a Girsanov transform, we recover the risk-neutral forward dynamics of Bjerksund, Rasmussen and Stensland [31]. It is interesting to note that with this spot price dynamics, the forward dynamics is not affine in the spot. Hence, the Bjerksund, Rasmussen and Stensland [31] model is an example of a non-affine forward dynamics. Whenever σ = b, we do not have that g(t, t) = 1, and thus the Bjerksund, Rasmussen and Stensland [31] model does not satisfy the Samuelson effect, either.
Option pricing
We end this section with a discussion of option pricing. Let us assume that we have given an option with exercise time τ on a forward with maturity at time T ≥ τ . The option pays f (F τ (T )), and we are interested in finding the price at time t ≤ τ , denoted C(t).
From arbitrage theory, it holds that
C(t) = e −r(τ −t) E Q [f (F τ (T ))|F t ],(26)
where Q is the risk-neutral probability. Choosing Q = Q θ,η as coming from the Esscher transform above, we can derive option prices explicitly in terms of the characteristic function of U by Fourier transformation.
Proposition 9. Let Q = Q θ,η be the probability measure obtained from the Esscher transform. Let p(x) = f (exp(x)), and suppose that p ∈ L 1 (R). By applying the definitions of Fourier transforms and their inverses in Folland [44], we have that p(x) = 1 2π R p(y)e ixy dy, with p(y) is the Fourier transform of p(x) defined by p(y) = R p(x)e −ixy dx. Suppose that p ∈ L 1 (R). Then the option price is given by
C t = e −r(τ −t) × 1 2π R p(y) exp iy ln Θ(τ, T ) + t −∞ G(T, s)ω s− dW s − t −∞ 1 2 G 2 (T, s)ω 2 s ds × exp t −∞ h(T, τ, v, y) dU v exp τ t φ η U (h(T, τ, v, y)) dv dy, where H T (v, v) = T v G 2 (T, u)i(u, v) du and Θ(τ, T ) = Λ(T ) exp T 0 G(T, s)θ(s) ds + T τ φ η U 1 2 H T (v, v) dv , h(T, τ, v, y) = − 1 2 y 2 τ v G 2 (T, s)i(s, v) ds + iy 1 2 T τ G 2 (T, s)i(s, v) ds.
One can calculate option prices by applying the fast Fourier transform as long φ η U is known. If p is not integrable (as is the case for a call option), one may introduce a damping function to regularize it, see Carr and Madan [39] for details.
The arithmetic case
Let us consider the arithmetic spot price model,
S t := S a t = Λ(t) + Y t .
We analyse the forward price for this case, and discuss the affinity. The results and discussions are reasonably parallel to the geometric case, and we refrain from going into details but focus on some main results. Under a natural integrability condition of the spot price with respect to the Esscher transform measure Q θ,η , we find the following forward price for the arithmetic model. Proposition 10. Suppose that S T ∈ L 1 (Q θ,η ). Then, the forward price is given as
F t (T ) = Λ(T ) + t −∞ G(T, s)ω s− dL s + E θ [L 1 ] T t G(T, s)E η [ω s |F t ] ds .
The price is reasonably explicit, except for the conditional expectation of the stochastic volatility ω s . By the same arguments as in Proposition 7, the forward price becomes affine in the spot (or in Y t ) if and only if G(t, s) = g 1 (t)g 2 (s) for sufficiently regular functions g 1 and g 2 .
In the case L = B, we can obtain an explicit forward price when using the Girsanov transform as in (22). We easily compute that the forward price becomes
F t (T ) = Λ(T ) + t −∞ G(T, s)ω s− dW s + T 0 G(T, s)θ(s) ds .(27)
We note that there is no explicit dependence of the spot volatility ω s except indirectly in the stochastic integral. This is in contrast to the Lévy case with Esscher transform. The dynamics of the forward price becomes
dF t (T ) = G(T, t)ω t− dW t .(28)
If we furthermore let G(t, s) = g 1 (t)g 2 (s) for some sufficiently regular functions g 1 and g 2 , we find that
F t (T ) = Λ(T ) + g 1 (T ) g 1 (t) (S t − Λ(t)) + T t G(T, s)θ(s) ds.(29)
Hence, the forward curve moves stochastically as the deseasonalised spot price, whereas the shape of the curve is deterministically given by g 1 (T )/g 1 (t). This shape is scaled stochastically by the deseasonalised spot price. In addition, there is a deterministic term which is derived from the market price of risk θ.
We finally remark that also in the arithmetic case one may derive expressions for the prices of options that are computable by fast Fourier techniques.
Empirical study
In this section, we will show the practical relevance of our new model class for modelling empirical energy spot prices. Here we will focus on electricity spot prices and we will illustrate that they can be modelled by LSS processes -an important subclass of VMLP processes. Note that the data analysis is exploratory in nature since the estimation theory for VMLP or LSS processes has not been fully established yet.
Data description
We study electricity spot prices from the European Energy Exchange (EEX). We work with the daily Phelix peak load data (i.e., the daily averages of the hourly spot prices for electricity delivered during the 12 hours between 8am and 8pm) with delivery days from 01.01.2002 to 21.10.2008. Note that peak load data do not include weekends, and in total we have 1775 observations. The daily data, their returns and the corresponding autocorrelation functions are depicted in Figure 1.
Deseasonalising the data
Before analysing the data, we have deseasonalised the spot prices. Here, we have worked with a geometric model, that is, S g t = Λ(t) exp(Y t ). Then log(S g t ) = log(Λ(t)) + Y t where, as suggested in, for example, Klüppelberg, Meyer-Brandis and Schmidt [53], log(Λ(t)) := β 0 + β 1 cos τ 1 + 2πt 261 + β 2 cos
τ 2 + 2πt 5 + β 3 t,
which takes weakly and yearly effects and a linear trend into account. In order to ensure that the spikes do not have a big impact on parameter estimation, we have worked with a robust estimation technique based on iterated reweighted least squares. We have then subtracted the estimated seasonal function from the logarithmic spot prices from the time series and have worked with the deseasonalised data for the remaining part of the Section. Figure 2 depicts the deseasonalised logarithmic prices and the corresponding returns.
Stationary distribution of the prices
The class of VMLV processes is very rich and hence in a first step we checked whether we can restrict it to a smaller class in our empirical work. We have carried out unit root tests, more precisely the augmented Dickey-Fuller test (where the null hypothesis is that a unit root is present in the time series versus the alternative of a stationary time series); we obtained a p-value which is smaller than 0.01 and, hence, clearly reject the unit root hypothesis at a high significance level. Also the Phillips-Perron test led to the same conclusion. Hence, in the following, we assume that Y t = Y t is an LSS process. Next, we study the question which distribution describes the stationary distribution of Y appropriately. We know that in the absence of stochastic volatility an LSS process is a moving average process driven by a Lévy process and hence the integral is itself infinite divisible. We are hence dealing with a stationary infinitely divisible stochastic process, see Rajput and Rosiński [59], Sato [60], Barndorff-Nielsen [3] for more details. The literature on spot price modelling suggest to use semi-heavy and, in some cases, even heavy-tailed distributions in order to account for the extreme spikes in electricity spot prices, see, for example, Klüppelberg, Meyer-Brandis and Schmidt [53] and Benth et al. [25] who suggested to use the stable distribution for modelling electricity returns.
Here we focus on a mixture of a normal distribution in the sense of mean-variance mixtures, see Barndorff-Nielsen, Kent and Sørensen [8]. In particular, we will focus on the generalised hyperbolic (GH) distribution, see Barndorff-Nielsen and Halgreen [7], Barndorff-Nielsen [1], Barndorff-Nielsen [2], which turns out to provide a good fit to the deseasonalised logarithmic spot prices as we will see in the following.
The generalised hyperbolic distribution
A detailed review of the generalised hyperbolic distribution can be found in, for example, McNeil, Frey and Embrechts [56] and details on the corresponding implementation in R based on the ghyp package is provided in Breymann and Lüthi [32].
Let d, k ∈ N and let X denote a k-dimensional random vector. X is said to have multivariate generalised hyperbolic (GH) distribution if
X law = µ + W γ + √ W AZ,
where Z ∼ N (0, I k ), A ∈ R d×k , µ, γ ∈ R d . Further, W ≥ 0 is a one-dimensional random variable, independent of Z and with Generalised Inverse Gaussian (GIG) distribution, that is, W ∼ GIG(λ, χ, ψ). The density of the GIG distribution with parameters (λ, χ, ψ)
is given by
f GIG (x) = ψ χ λ/2 x λ−1 2K λ ( √ χψ) exp − 1 2 χ x + ψx ,
where K λ denotes the modified Bessel function of the third kind, and the parameters have to satisfy one of the following three restrictions
χ > 0, ψ ≥ 0, λ < 0 or χ > 0, ψ > 0, λ = 0 or χ ≥ 0, ψ > 0, λ > 0.
Typically, we refer to µ as the location parameter, to Σ = AA ′ as the dispersion matrix and to γ as the symmetry parameter (sometimes also called skewness parameter). The parameters λ, χ, ψ of the GIG distribution determine the shape of the GH distribution. The parametrisation described above is the so-called (λ, χ, ψ, µ, Σ, γ)-parametrisation of the GH distribution. However, for estimation purposes this parametrisation causes an identifiability problem and hence we worked with the so-called (λ, α, µ, Σ, γ)-parametrisation in our empirical study. Note that the (λ, χ, ψ, µ, Σ, γ)-parametrisation can be obtained by from (λ, α, µ, Σ, γ)-parametrisation by setting
ψ = α K λ+1 (α) K λ (α) , χ = α 2 ψ = α K λ (α) K λ+1 (α) ,
and λ, Σ, γ remain the same, see Breymann and Lüthi [32] for more details.
Estimation results
In our empirical study, we work with the one-dimensional GH distribution. That is, d = k = 1 and µ, γ and Σ = σ are scalars rather than a matrix and vectors, respectively. We have fitted 11 distributions within the GH class to the deseasonalised log-spot prices using quasi-maximum likelihood estimation: The asymmetric and symmetric versions of the
• generalised hyperbolic distribution (GHYP): λ ∈ R, α > 0, (λ ∈ R, χ > 0, ψ > 0), • normal inverse Gaussian (NIG) distribution: λ = − 1 2 , α > 0, (λ = − 1 2 , χ > 0, ψ > 0), • Student-t distribution (with ν degrees of freedom): λ = −ν/2 < −1, α = 0, (λ < 0, χ > 0, ψ = 0), • hyperbolic distribution (HYP): λ = (d + 1)/2, α > 0, (λ = (d + 1)/2, χ > 0, ψ > 0),
• Variance gamma distribution (VG): λ > 0, α = 0, (λ > 0, χ = 0, ψ > 0), and the Gaussian distribution. We have compared these distributions using the Akaike information criterion, see Table 1, which suggests that the symmetric NIG distribution is the preferred choice for the stationary distribution of the deseasonalised logarithmic spot prices. The diagnostic plots of the empirical and fitted logarithmic densities and the quantile-quantile plots of the fitted symmetric NIG distribution are depicted in Figure 3. We see that the fit is reasonable.
Stationary BSS processes with generalised hyperbolic marginals
In our empirical study, we have seen that the symmetric normal inverse Gaussian distribution fits the marginal distribution of the deseasonalised logarithmic electricity prices well. Hence, it is natural to ask whether there is a stationary BSS or LSS process with marginal normal inverse Gaussian or, more generally, generalised hyperbolic distribution?
The answer is yes, as we will show in the following. Note that the following investigation extends the study of Barndorff-Nielsen and Shephard [13], where the background driving process of an Ornstein-Uhlenbeck process was specified, given a marginal infinitely divisible distribution. Let us focus on a particular BSS process given by
Y t = µ + c t −∞ g(t − s)ω s dB s + γ t −∞ q(t − s)ω 2 s ds(30)
for constants c, γ ∈ R and for stationary ω and a standard Brownian motion B independent of ω.
Remark. Note that we have introduced a drift term in the BSS process again in order to derive the general theoretical result. For our empirical example, however, it would be sufficient to set γ = 0 as suggested by our estimation results above.
The conditional law of Y t given ω is normal:
Y t |ω law = N µ + γ t −∞ q(t − s)ω 2 s ds, c 2 t −∞ g 2 (t − s)ω 2 s ds .
Now suppose that ω 2 follows an LSS process given by
ω 2 t = t −∞ i * (t − s) dU s ,
where U is a subordinator. Then, by a stochastic Fubini theorem we find
t −∞ q(t − s)ω 2 s ds = t −∞ t u q(t − s)i * (s − u) ds dU u = t −∞ k(t − u) dU u ,
where k = q * i * , the convolution of q and i * . Similarly,
t −∞ g 2 (t − s)ω 2 s ds = t −∞ m(t − u) dU u ,
with m = g 2 * i * . Let g(t; ν, λ) denote the gamma density with parameters ν > 0 and λ > 0, that is,
g(t; ν, λ) = λ ν Γ(ν) t ν−1 e −λt .
Now we define
g(t) = λ Γ(2ν − 1) Γ(ν) 2 −1/2 2 ν g t; ν, λ 2 = λ ν−1/2 Γ(2ν − 1) 1/2 t ν−1 exp − λ 2 t(31)
for ν > 1 2 , which ensures the existence of the integral (30); then we have
g 2 (t) = λ 2ν−1 Γ(2ν − 1) t (2ν−1)−1 exp(−λt) = g(t; 2ν − 1, λ).
Hence, if, for 1 2 < ν < 1,
i * (t) = 1 λ g(t; 2 − 2ν, λ),
and if, moreover,
q(t) = g(t; 2ν − 1, λ),
we obtain
k(t) = m(t) = e −λt .
In other words,
Y t |ω law = N (µ + γσ 2 t , c 2 σ 2 t ), where σ 2 t = t −∞ e −λ(t−u) dU u .
We define the subordinator U with Lévy measure ℓ U by U t = U t/λ . Then
σ 2 t = t −∞ e −λ(t−u) dU λu .
Then one can easily show that the marginal distribution of σ 2 does not depend on λ, and the parameter λ determines the autocorrelation structure of σ 2 . It follows that if the subordinator U is such that σ 2 t has the generalised inverse Gaussian law GIG(λ, χ, ψ) then the law of Y t is the generalised hyperbolic GH(λ, χ, ψ, µ, c 2 , γ).
Is there such a subordinator? The answer is yes. To see this, let θ ≥ 0 and note that σ 2 t is infinitely divisible with kumulant function
K{θ ‡ σ 2 t } = log(E(exp(−θσ 2 t ))) = log E exp −θ t −∞ e −λ(t−u) dU λu = ∞ 0K {θe −λu ‡ U 1 }λ du = ∞ 0K {θe −u ‡ U 1 } du.
On the other hand, the subordinator U (here assumed to have no drift) has kumulant functionK
{θ ‡ U 1 } = log(E(exp(−θU 1 ))) = − ∞ 0 (1 − e −θx )ℓ U (dx),
where ℓ U is the Lévy measure of U . Combining we find That is, the Lévy measure ℓ σ 2 of σ 2 t is
K{θ ‡ σ 2 t } = − ∞ 0 (1 − e −θy )ℓ σ 2 (dy) = ∞ 0 ℓ U (e u dy) du.(32)
Thus, the question is: Does there exist a Lévy measure ℓ U on R + such that ℓ σ 2 given by (32) is the Lévy measure of the GIG(λ, χ, ψ) law. That, in fact, is the case since the GIG laws are self-decomposable, cf. Halgreen [48] and Jurek and Vervaat [51].
Implied autocorrelation structure
Next, we focus on the autocorrelation structure implied by the choice of the kernel functions which lead to a marginal GH distribution of the BSS process.
Proposition 11. Let Y be the BSS process defined in the previous subsection with kernel function g as defined in (31). In the case when γ = 0 and ν > 1 2 , we have
Cor(Y t , Y t+h ) = 1 2 ν−3/2 Γ(ν − 1/2)K ν−1/2 λh 2 for h > 0,
whereK ν (x) = x ν K ν (x) and K ν denotes the modified Bessel function of the third kind.
We have estimated the parameters ν and λ using a linear least squares estimate based on the empirical and the theoretical autocorrelation function using the first ⌊ √ 1775⌋ = 42 lags. We obtain λ = 0.055 and ν = 0.672. Figure 4 shows the empirical and the corresponding fitted autocorrelation function.
Remark. Note that the estimate ν = 0.672 implies that the corresponding BSS process is not a semimartingale, see, for example, Barndorff-Nielsen and Schmiegel [12] for details.
In the context of electricity prices, this does not need to be a concern since the electricity spot price is not tradeable.
We observe that the autocorrelation function induced by the gamma-kernel mimics the behaviour of the empirical autocorrelation function adequately. However, it does not fit the first 10 lags as well as, for example, the CARMA-kernel which we have fitted in the following subsection, but performs noticeably better for higher lags. The fit could be further improved by choosing σ 2 t to be a GIG supOU process rather than a GIG OU process. Then one obtains an even more flexible autocorrelation structure.
Empirical performance of a CARMA model
The recent literature on modelling electricity spot prices has advocated the use of linear models, that is, CARMA models, as described in detail in Section 3.1.6. Since CARMA models are special cases of our general modelling framework, we briefly demonstrate their empirical performance as well. It is well known, see, for example, Brockwell, Davis and Yang [36], that a discretely sampled CARMA(p, q) process (for p > q) has a weak ARMA(p, p − 1) representation. An automatic model selection using the Akaike information criterion within the class of (discrete-time ARIMA) models suggests that an ARMA(2,1) model is the best choice for our data. We take that result as an indication that a CARMA(2, 1) process (which has a weak ARMA(2,1) representation) might be a good choice. However, it should be noted that the relation between model selection in discrete and continuous time still needs to be explored in detail. We have estimated the parameters of the kernel function g which corresponds to a CARMA(2, 1) process using quasi-maximum-likelihood estimation based on the weak ARMA(2,1) representation. Diagnostic plots for the estimated CARMA(2, 1) model are provided in Figure 5. First, we compare the empirical and the estimated autocorrelation function, see Figure 5(a). Recall that the autocorrelation of Y is given by (14) and it simplifies to
Cor(Y t+h , Y t ) = ∞ 0 g(x + h)g(x) dx ∞ 0 g(x) 2 dx ,
if either the driving Lévy process has zero mean or if the stochastic volatility process has zero autocorrelation. After deseasonalising (which also includes detrending) the data, we have obtained data which have approximately zero mean. The empirical and the estimated autocorrelation function implied by a CARMA(2, 1) kernel function g match very well for the first 12 lags. Higher lags were however slightly better fitted by the gamma kernel used in the previous subsection. Figure 5(b) depicts the corresponding residuals from the weak ARMA(2,1) representation and Figures 5(c) and 5(d) show the autocorrelation functions of the corresponding residuals and squared residuals. Overall, we see that the fit provided by the CARMA(2, 1) kernel function is acceptable.
Note that in addition to estimating the parameters of the g function coming from a CARMA process one can also recover the driving Lévy process of a CARMA process based on recent findings by Brockwell, Davis and Yang [36]. This will make it possible to also address the question of whether stochastic volatility is needed to model electricity spot prices or not. See Veraart and Veraart [65] for empirical work along those lines in the context of electricity spot prices, whose results suggest that stochastic volatility is indeed important for modelling electricity spot prices.
Conclusion
This paper has focused on volatility modulated Lévy-driven Volterra (VMLV) processes as the building block for modelling energy spot prices. In particular, we have introduced the class of Lévy semistationary (LSS ) processes as an important subclass of VMLV processes, which reflect the stylised facts of empirical energy spot prices well. This modelling framework is built on four principles. First, deseasonalised spot prices can be modelled directly in stationarity to reflect the empirical fact that spot prices are equilibrium prices determined by supply and demand and, hence, tend to mean-revert (in a weak sense) to a long-term mean. Second, stochastic volatility is regarded as a key factor for modelling (energy) spot prices. Third, our new modelling framework allows for the possibility of jumps and extreme spikes. Fourth, we have seen that VMLP and, in particular, LSS processes feature great flexibility in terms of modelling the autocorrelation function and the Samuelson effect.
We have demonstrated that VMLV processes are highly analytically tractable; we have derived explicit formulae for the energy forward prices based on our new spot price models, and we have shown how the kernel function determines the Samuelson effect in our model. In addition, we have discussed option pricing based on transform-based methods.
An exploratory data analysis on electricity spot prices shows the potential our new approach has and more detailed empirical work is left for future research. Also, we plan to address the question of model estimation and inference. It will be important to study effi-where we use the notation G T = ∂G/∂T and ξ T = ∂ξ/∂T for the corresponding partial derivatives with respect to the first argument. Hence, we must have that G(t, s) = G(s, s) exp t s ξ T (u, u) du , and the separation property holds.
Likewise, to have affinity in the volatility Z(t), we must have that which holds by the stochastic Fubini theorem. Using the independent increment property of U and double conditioning, we reach for | arg(h)| < π and Re(λ), Re(ν) > 0, where K ν is the modified Bessel function of the third kind. Hence,
∞ 0 g(t + h)g(t) dt = λ 2ν−1 Γ(2ν − 1) exp − λh 2 1 √ π h λ ν−1/2 exp λh 2 Γ(ν)K 1/2−ν λh 2 = (λh) ν−1/2 Γ(2ν − 1) 1 √ π Γ(ν)K 1/2−ν λh 2 .
Now we apply Gradshteyn and Ryzhik [46], Formula 8.335.1, to obtain
Γ(2ν − 1) = 2 2ν−2 √ π Γ ν − 1 2 Γ(ν). Then ∞ 0 g(t + h)g(t) dt = (λh) ν−1/2 (2 2ν−2 / √ π)Γ(ν − 1/2)Γ(ν) 1 √ π Γ(ν)K 1/2−ν λh 2 = (λh) ν−1/2 2 2ν−2 Γ(ν − 1/2) K 1/2−ν λh 2 =
(λh) ν−1/2 2 ν−1/2 2 ν−3/2 Γ(ν − 1/2) K 1/2−ν λh 2 .
Since K ν (x) = K −ν (x) according to Gradshteyn and Ryzhik [46], Formula 8.486.16, the result follows.
This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2013, Vol. 19, No. 3, 803-845. This reprint differs from the original in pagination and typographic detail.
Figure 1 .
1Daily electricity peak load spot prices in Euro/MWh from the EEX, recorded from 01.01.2002 to 21.10.2008.
Figure 2 .
2Daily deseasonalised logarithmic spot prices.
Figure 3 .
3Diagnostic plots for the estimated symmetric normal inverse Gaussian distribution.
∞ 0 ℓ
0U (e u dy) du.
Figure 4 .
4Empirical and estimated autocorrelation function using the gamma kernel function with λ = 0.055 and ν = 0.672.
Figure 5 .
5Diagnostic plots for the estimated CARMA(2, 1) model.
s is a semimartingale according to the Knight condition, see Knight[54] and also Basse[18], Basse and Pedersen[19], Basse-O'Connor, Graversen and Pedersen[20].
Table 1 .
1Model selection based on the Akaike information criterion within the class of generalised hyperbolic distributions. We compare both the asymmetric and the symmetric versions of the generalised hyperbolic (GHYP), normal inverse Gaussian (NIG), Student-t, hyperbolic (HYP), variance Gamma (VG) and the Gaussian distributionModel
Symmetric
λ
α
µ
σ
γ
AIC
Log-Likel.
NIG
TRUE
−0.5
0.431 −0.001
0.395
0
1313.14
−653.57
GHYP
TRUE
−0.183
0.438 −0.001
0.392
0
1314.13
−653.06
NIG
FALSE
−0.5
0.431 −0.003
0.395
0.002
1315.10
−653.55
GHYP
FALSE
−0.184
0.438 −0.002
0.392
0.002
1316.10
−653.05
Student-t
TRUE
−1.366
0
−0.001
0.458
0
1327.28
−660.64
Student-t
FALSE
−1.365
0
−0.002
0.458
0.002
1329.26
−660.63
HYP
TRUE
1
0.150
0.000
0.375
0
1331.38
−662.69
HYP
FALSE
1
0.147
0.003
0.375 −0.003
1333.33
−662.66
VG
TRUE
0.975
0
0.003
0.379
0
1333.85
−663.92
VG
FALSE
0.970
0
0.007
0.379 −0.007
1335.42
−663.71
Gaussian
TRUE
NA
Inf
−0.000
0.395
0
1742.94
−869.47
T t G 2 (T, v)e −λ(v−s) dv = e −λ(t−s) T t G 2 (T, v)e λ(t−v) dv
AcknowledgementsWe would like to thank Andreas Basse-O'Connor and Jan Pedersen for helpful discussions and constructive comments. Also, we are a grateful to the valuable comments by two anonymous referees and by the Editor.Appendix: ProofsProof of Proposition 3. In order to prove the semimartingale conditions suppose for the moment that Y is a semimartingale, so that the stochastic differential of Y exists. Then, calculating formally, we findwhich indicates that Y can be represented, for t ≥ 0, asClearly, under the conditions formulated in Proposition 3, the above integrals are well defined, and Y , defined by(15), is a semimartingale, and dY exists and satisfies equation(33). A direct rewrite now shows that(33)agrees with the defining equation(2)of Y , and we can then deduce that Y is a semimartingale.Proof of Proposition 4. The result follows directly from the representation(15)and from properties of the quadratic variation process, see, for example, Protter[58].Proof of Proposition 5. First, writeand observe that the first integral on the right-hand side is F t -measurable. The result follows by using double conditioning, first with respect to the σ-algebra G T generated by the paths of ω s , s ≤ T and F t , and next with respect to F t .Proof of Proposition 6. By the Girsanov change of measure, we have where we set B s = W s for s < 0. By following the argumentation in the proof of Proposition 5, we are led to calculate the expectationBut, by the stochastic Fubini theorem, see, for example, Barndorff-Nielsen and Basse-O'Connor[4],Using the adaptedness to F t of the first integral and the independence from F t of the second, we find the desired result.Proof of Proposition 7. If G(t, s) = g 1 (t)g 2 (s) it holds thatand affinity holds in both the volatility and the spot price. Opposite, to have affinity in Y t we must have thatfor some function ξ(T, t), which means that the ratio ξ(T, t) = G(T, s)/G(t, s) is independent of s. ξ(T, t) is differentiable in T as long as G is. Furthermore, ξ(T, T ) = 1 by definition. Thus, by first differentiating ξ with respect to T and next letting T = t, it holds thats) must be independent of s. Denote the ratio by ξ(T, t), and differentiate with respect to T to obtainDifferentiating with respect to T , and next letting T = t gives i T (t, s) = i(t, s)(J T (t, t) − I(t, t)).Whence,and the separation property holds for i. The proposition is proved.for a deterministic function Θ(t, T ) given byNote that the process M T (t) t −∞ G(T, s)ω s− dW s is a (local) Q θ,η -martingale for t ≤ T . Moreover, from the stochastic Fubini theorem it holds thatThe result is then a direct consequence of the Itô formula for semimartingales, see, for example, Protter[58].Proof of Proposition 9. From Proposition 6, we know that we can write the forward price asLet now p(x) = f (exp(x)), and suppose that p ∈ L 1 (R). Recall that p(x) = 1 2π R p(y)e ixy dy, with p(y) is the Fourier transform of p(x) defined by p(y) = R p(x)e −ixy dx. Suppose that p ∈ L 1 (R). Hence, we find f (F τ (T )) = f (exp(ln(F τ (T )))) = p(ln(F τ (T ))) = 1 2π R p(y)e iy ln(Fτ (T )) dyNext, by commuting integration and expectation using dominated convergence and F tadaptedness, we obtainwhere we define a := a(y) := − 1 2 y 2 − 1 2 iy. Using the stochastic Fubini theorem again, we getAltogether, we obtainThe above expression can be further simplified by noting that.Proof of Proposition 10. Observe that.We then proceed as in the proof of Proposition 5, and finally we perform the differentiation and let x = 0.Proof of Proposition 11. We havewhich is a probability density and hence ∞ 0 g 2 (t) dt = 1. Now we derive the explicit formula for the autocorrelation function. Note that according to Gradshteyn and Ryzhik[46], Formula 3.383.8
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|
[] |
[
"LOCATING THE MOST ENERGETIC ELECTRONS IN CASSIOPEIA A",
"LOCATING THE MOST ENERGETIC ELECTRONS IN CASSIOPEIA A"
] |
[
"Brian W Grefenstette ",
"Stephen P Reynolds ",
"Fiona A Harrison ",
"T Brian Humensky ",
"Steven E Boggs ",
"Chris L Fryer ",
"Tracey Delaney ",
"Kristin K Madsen ",
"Hiromasa Miyasaka ",
"Daniel R Wik ",
"Andreas Zoglauer ",
"Karl Forster ",
"Takao Kitaguchi ",
"Laura Lopez ",
"Melania Nynka ",
"E Finn ",
"Christensen ",
"William W Craig ",
"Charles J Hailey ",
"Daniel Stern ",
"William W Zhang "
] |
[] |
[] |
We present deep (>2.4 Ms) observations of the Cassiopeia A supernova remnant with NuSTAR, which operates in the 3-79 keV bandpass and is the first instrument capable of spatially resolving the remnant above 15 keV. We find that the emission is not entirely dominated by the forward shock nor by a smooth "bright ring" at the reverse shock. Instead we find that the >15 keV emission is dominated by knots near the center of the remnant and dimmer filaments near the remnant's outer rim. These regions are fit with unbroken power-laws in the 15-50 keV bandpass, though the central knots have a steeper (Γ ∼ −3.35) spectrum than the outer filaments (Γ ∼ −3.06). We argue this difference implies that the central knots are located in the 3-D interior of the remnant rather than at the outer rim of the remnant and seen in the center due to projection effects. The morphology of >15 keV emission does not follow that of the radio emission nor that of the low energy (<12 keV) X-rays, leaving the origin of the >15 keV emission as an open mystery. Even at the forward shock front we find less steepening of the spectrum than expected from an exponentially cut off electron distribution with a single cutoff energy. Finally, we find that the GeV emission is not associated with the bright features in the NuSTAR band while the TeV emission may be, suggesting that both hadronic and leptonic emission mechanisms may be at work.
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10.1088/0004-637x/802/1/15
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[
"https://arxiv.org/pdf/1502.03024v1.pdf"
] | 7,393,631 |
1502.03024
|
c77fb957bd3c081fbcffe0ee4260e5036231f735
|
LOCATING THE MOST ENERGETIC ELECTRONS IN CASSIOPEIA A
10 Feb 2015 Draft version February 11, 2015 Draft version February 11, 2015
Brian W Grefenstette
Stephen P Reynolds
Fiona A Harrison
T Brian Humensky
Steven E Boggs
Chris L Fryer
Tracey Delaney
Kristin K Madsen
Hiromasa Miyasaka
Daniel R Wik
Andreas Zoglauer
Karl Forster
Takao Kitaguchi
Laura Lopez
Melania Nynka
E Finn
Christensen
William W Craig
Charles J Hailey
Daniel Stern
William W Zhang
LOCATING THE MOST ENERGETIC ELECTRONS IN CASSIOPEIA A
10 Feb 2015 Draft version February 11, 2015 Draft version February 11, 2015arXiv:1502.03024v1 [astro-ph.HE] Preprint typeset using L A T E X style emulateapj v. 5/2/11Subject headings: acceleration of particles -ISM: individual objects (Cassiopeia A) -ISM: supernova remnants -X-rays: ISM -radiation mechanisms: non-thermal
We present deep (>2.4 Ms) observations of the Cassiopeia A supernova remnant with NuSTAR, which operates in the 3-79 keV bandpass and is the first instrument capable of spatially resolving the remnant above 15 keV. We find that the emission is not entirely dominated by the forward shock nor by a smooth "bright ring" at the reverse shock. Instead we find that the >15 keV emission is dominated by knots near the center of the remnant and dimmer filaments near the remnant's outer rim. These regions are fit with unbroken power-laws in the 15-50 keV bandpass, though the central knots have a steeper (Γ ∼ −3.35) spectrum than the outer filaments (Γ ∼ −3.06). We argue this difference implies that the central knots are located in the 3-D interior of the remnant rather than at the outer rim of the remnant and seen in the center due to projection effects. The morphology of >15 keV emission does not follow that of the radio emission nor that of the low energy (<12 keV) X-rays, leaving the origin of the >15 keV emission as an open mystery. Even at the forward shock front we find less steepening of the spectrum than expected from an exponentially cut off electron distribution with a single cutoff energy. Finally, we find that the GeV emission is not associated with the bright features in the NuSTAR band while the TeV emission may be, suggesting that both hadronic and leptonic emission mechanisms may be at work.
INTRODUCTION
Young supernova remnants such as Cassiopeia A (Cas A) with shock velocities above 1000 km s −1 provide excellent opportunities to study in detail the process of shock acceleration of electrons to high energies (see Reynolds 2008 for a review). As the youngest Galactic remnant of a core-collapse (CC) supernova with an estimated explosion date of 1670 (Thorstensen et al. 2001), Cas A particularly important for contrasting with the historical remnants of Type Ia events, such as Tycho, Kepler, SN 1006, and G1.9+0.3 (although the identification as a Type Ia is less secure for G1.9+0.3). Non-thermal X-ray emission from all these objects can be characterized both spectrally and spatially and can be used to infer properties of the acceleration process. Of special interest for shock acceleration physics is the maximum energy to which electrons are accelerated, E m , and its dependence on the local shock velocity and other parameters. For example, the "thin rims" of synchrotron X-rays found at the peripheries of some supernova remnants imply strong magnetic-field amplification (Vink & Laming 2003;Parizot et al. 2006;Ressler et al. 2014) which, along with spectral inferences, gives an assumed exponential cutoff in the electron spectrum with e-folding energies in the range 10 -100 TeV.
Cas A has now been shown by light echoes to be the result of a SN IIb explosion where the progenitor lost most of a massive H envelope prior to the explosion (Krause et al. 2008). This implies that the remnant is currently expanding into the progenitor's stellar wind and so the blast wave should be quasi-perpendicular over most of its surface. Since the properties of shock acceleration are strongly dependent on the magnetic obliquity angle θ Bn between the shock velocity and the upstream magnetic field, one expects substantial differences in both the morphology and spectrum of synchrotron emission between remnants of CC and Type Ia events. In particular, many models (e.g. Riquelme & Spitkovsky 2010) assert that acceleration at quasi-perpendicular shocks (θ Bn ∼ 90 • ) is considerably different from that at quasiparallel shocks, so that remnants expanding into stellar winds might have very different properties in their nonthermal emission.
The spectral properties and basic morphology of Cas A in X-rays between 0.5 and 12 keV are well known from many previous studies, especially with XMM-Newton (Bleeker et al. 2001;Willingale et al. 2002) and Chandra (Gotthelf et al. 2001;Hwang et al. 2004). Thermal emission from the shocked ejecta dominates the integrated spectrum at these energies, with electron temperatures typically between 1 and 3 keV and in no cases above 6 keV. Local variations have been mapped out on arcsecond length scales (Hwang & Laming 2012); in almost all regions, a power-law component must be added to one or more thermal components to obtain a satisfactory description of the spectrum. The morphology is dominated by a "bright ring" about 3.5 ′ in diameter which is commonly associated with the reverse shock or contact discontinuity between shocked ejecta and shocked circumstellar material (CSM). Fainter emission forms a faint rim outside the bright ring (Gotthelf et al. 2001). The bright ring remains prominent to the highest energies imaged with XMM-Newton (Bleeker et al. 2001), with an East/West asymmetry reported in observed using Beppo-Sax (Vink et al. 2000) and Suzaku (Maeda et al. 2009). These authors note that the western half of the remnant appears to be brighter in the highest energy band of both instruments. The faint rim is located at the edge of radio emission, though the outer radio emission appears to form the edge of a plateau rather than thin filaments. Many arguments suggest that the outer radio edge and the X-ray filaments are located at the outer blast wave.
The integrated spectrum of Cas A has long been known to continue above the thermal spectrum to energies of order 100 keV with a spectrum reasonably well described by a single power-law, based on data from non-imaging instruments (e.g. CGRO, The et al. 1996; RXTE, HEAO-2, and OSSE, Allen et al. 1997;Beppo-SAX, Vink et al. 2000;INTEGRAL, Renaud et al. 2006;Suzaku, Maeda et al. 2009). While the tail was originally thought to be thermal, at least out to energies of 30 keV or so (Pravdo & Smith 1979), better analysis of the spectrum below 10 keV has shown that the temperature of the thermal plasmas only extend to ∼3 keV (Hwang & Laming 2012). Subsequent explanations on the high-energy tail have included nonthermal bremsstrahlung from a power-law distribution of somewhat suprathermal electrons (e.g. Asvarov et al. 1990, Bleeker et al. 2001) and synchrotron emission from shock-accelerated electrons with much higher energies (e.g., Allen et al. 1997). While the hard tail can be associated with the power-law components required in the modeling of the emission below 10 keV, the temperature and flux of the thermal components vary with position in the remnant. It is not clear how much of the flux below 10 keV is associated with the non-thermal component. In relatively line-free regions of the spectrum such as 4.2 -6 keV, estimates range from a few percent to half or more (e.g. Rothschild & Lingenfelter 2003;Helder & Vink 2008).
The argument over the emission mechanism of the hard tail seems to have been settled in favor of synchrotron emission. Electrons in the 10 -100 keV energy range required for bremsstrahlung should be accelerated in the forward or reverse shocks, but Coulomb interactions ought to cool the electrons efficiently not far behind those shocks, producing a dip in the spec-trum below 100 keV which is not observed (Vink 2008(Vink , 2012. Electron energization by lower hybrid waves in weak shocks in the remnant's interior might ameliorate this problem (Laming 2001). The presumptive site for the production of the very high-energy electrons required to produce synchrotron emission to 10 keV is the forward blast wave marked by the faint rim filaments (e.g., Berezhko & Völk 2004), though a recent model proposes that almost all emission above 10 keV should come from the reverse shock (Zirakashvili et al. 2014). Therefore, current theories expect the >15 keV emission to be predominantly originate in either the forward shock or the bright ring.
High spatial resolution observations with XMM and Chandra show that in addition to the clear outer rim and bright ring, there are bright central knots and filaments of soft X-ray emission that are dominated by non-thermal continuum (DeLaney et al. 2004), with intensities that vary with time (Patnaude & Fesen 2009). Whether these features are actually interior to the remnant and evidence for particle acceleration at the reverse shock or knots on the forward shock face seen in projection has remained an open question. It is not known whether the bright ring of the reverse shock, the outer rim of the forward shock, or something else, dominates the emission above 15 keV, which has significant implications for the particle acceleration in the remnant.
In this paper we report on the first spatially resolved hard X-ray images and spectroscopy of Cas A. These collected observations represent over 2.4 Ms of integration time, with the exposure time primarily driven by the study of 44 Ti in the remnant (Grefenstette et al. 2014). This results in deep observations that provide un- Figure 1. Deconvolved NuSTAR images of Cas A: red (4-6 keV), green (8-10 keV) and blue (10-15 keV). For this and all the images below North is up and East is left. These images are 7.6 arcminutes on a side and all of the color bands have a sqrt stretch. As the energy bands increase, the emission is more consistently dominated by the central knots rather than any diffuse emission in the remnant. However, even in the highest energy bands accessible to NuSTAR there is still some residual diffuse emission in the center of the remnant not associated with any obvious point sources. See online version for color images. precedented sensitivity in the hard X-ray band. These observations allow us to make important new measurements: better spectral characterization of the hard continuum (results from previous missions are not entirely consistent with one another), spatial identification of the sites of electron acceleration to the highest energies, and detailed spectral modeling over most of the energy range of observed X-ray emission. Fully exploiting this new energy band requires extensive analysis combining spatially-resolved information from the radio, soft Xrays, and hard X-rays. In this paper we present the initial results from the NuSTAR observations. We compare and contrast our findings with the previously-known characteristics of Cas A and discuss the implications for particle acceleration models. In addition, we provide context for this work by performing a high-level comparison of our findings in the hard X-ray band with previous work in the radio and soft X-ray bands. While the broadband picture is by no means complete, we expect that this will motivate the future theoretical modeling of the Cas A continuum.
2. NuSTAR DATA AND METHODS Cas A was observed by NuSTAR, a NASA Small Explorer (SMEX) satellite launched on June 13, 2012 (Harrison et al. 2013). NuSTAR has two co-aligned Xray telescopes observing the sky in an energy range from 3-79 keV. The field of view is roughly 12 ′ x 12 ′ , with a point-spread function (PSF) with a full-width half maximum of 18 ′′ and a half-power diameter of 58 ′′ .
The observations were completed over the first 18 months of the mission (Table 1) with exposures spanning between 120 and 280 ks in a single pointing. The aim point for the telescope was generally selected to avoid the gaps in the instrument when possible, though later observations were chosen to target specific spatial regions for the study of 44 Ti. Thermal motions of the 10-m mast that separates the NuSTAR optics and detectors as well as small variations in the spacecraft pointing introduce natural "dither" in the pointing pattern, further smoothing the exposure map. There were no roll angle constraints on these observations, so the remnant was observed at different position angles over the course of the observations. We reduced the NuSTAR data with the NuSTAR Data Analysis Software (NuSTARDAS) version 1.3.1 and NuSTAR CALDB version 20131223. The NuSTAR-DAS pipeline software and associated CALDB files are fully HEASARC FTOOL compatible and are written and maintained jointly by the ASI Science Data Center (ASDC, Italy) and the NuSTAR Science Operations Center (SOC) at Caltech. The NuSTARDAS pipeline generates Good Time Intervals (GTIs) for each observation that exclude periods when the source is occulted by the Earth and when the satellite is transiting the South Atlantic Anomaly (SAA), a region of high particle background. The default detector "depth cut" is applied to reduce the internal background at high energies. The images, exposure maps, and response files produced by the NuSTARDAS consistently account for the natural thermal motions of the mast.
We produced images in various bandpasses using the XSELECT multi-mission FTOOL. Cas A is a bright, extended source so there are no regions in the field-ofview of the telescope that can be used to directly estimate the background in the source region. Instead we model the background and produce energy-dependent background images. We follow the procedure outlined in Grefenstette et al. (2014) and Wik et al. (2014) to estimate the background components and their spatial distributions using the nuskybgd IDL suite. In general, the flux from Cas A dominates over the diffuse backgrounds by several orders of magnitude until energies of ∼50 keV where the signal becomes comparable to the background.
We combined images from all epochs using XIMAGE, taking into account the unique time-dependent exposure and vignetting corrections for each pointing. We chose the energy bands of 4-6 keV, 8-10 keV, 10-15 keV, 15-20, 20-25 keV, 25-35, 35-45, and 45-55 keV for analysis. We omit the 6-8 keV energy band from this work, as it is dominated by Fe line emission from the shocked ejecta and has been explored previously in detail (e.g. Hwang & Laming 2012). Here we are primarily interested in the continuum emission. For energy bands up to the 25-35 keV we deconvolved the images with the on-axis NuSTAR PSF contained in the CALDB and the max likelihood AstroLib IDL script which is based on a Richardson-Lucy deconvolution for images with Poisson noise. We chose to halt the deconvolution after 50 iterations, since after this the resulting images became sufficiently self-similar. For the 35-45 and 45-55 keV bands there are insufficient statistics to support the image deconvolution, so we instead smoothed the images with top-hat smoothing kernels with radii of 10 pixels (∼25 ′′ ).
To perform spatially resolved spectroscopy we defined standard ds9 region files and used the nuproducts FTOOL to extract spectra and produce ancillary response files (ARFs) and redistribution matrix files (RMFs) for each epoch with background spectra simulated using nuskbygd. To reduce the complexity of joint-fitting many spectra (14 ObsIds × two telescopes = 28 spectra), we combined the source pulse-height amplitude (PHA) files, ARFs, and simulated background PHA files using the addascaspec FTOOL and combined the RMF using the addrmf FTOOL. This results in exposure-weighted RMF and ARF files for each region. Based on many observations of the Crab at different offaxis angles (Madsen et al., in prep), we estimated that the systematic effects caused by combining the spectra in this manner are energy-independent and introduce noise primarily in the normalization of the combined spectrum that is on the order of a few percent. As we are not interested here in in the absolute normalization of the spectra, we ignore this effect below. We also estimate that this introduces a systematic error on the power-law index of ± 0.01, which is significantly smaller than than the statistical errors quoted below. As we are mostly interested in the non-thermal emission processes, we only fit the data from 15-50 keV. The lower energy bound was chosen to so that the contribution from thermal bremsstrahlung from the shock-heated ejecta studied in detail by Chandra and XMM-Newton is negligible, while at high energies we chose 50 keV since above this energy the signal from our source regions becomes comparable to the background and mentioned above.
3. RESULTS 3.1. NuSTAR Imaging While the spatially integrated hard X-ray emission from Cas A has been measured by a number of instruments, NuSTAR provides the first opportunity to spatially resolve the emission above 15 keV (Figures 1 -3). We adopt the terminology of "exterior" emission features to be those seen near the outside of the remnant when seen in projection on the sky and "central" emission features to be those seen toward the center of the remnant when seen in projection on the sky. We also adopt a description of "knots" of emission as those regions that are unresolved by NuSTAR and "filaments" as those that appear to be linearly extended regions of emission.
We find that above 15 keV the morphology of the rem- nant begins to deviate from the emission below 12 keV observed by Chandra, XMM-Newton, and Suzaku. While the outer filaments (i.e. the "thin rim" of the forward shock) are clearly visible in the NuSTAR images, the emission is dominated by two central unresolved knots in the west. These western central knots dominate the hard X-ray emission above 15 keV, which is broadly consistent with the results from Helder & Vink (2008), who found a global east/west asymmetry in the hardness ratio of the remnant based on data from Chandra, implying that the west should be brighter at higher energies. However, we note that though the central western knots dominate the emission, the tricolor image above 15 keV ( Figure 2) demonstrates that the exterior filaments are harder (bluer) than the central knots, a fact we explore quantitatively below.
Spatially Resolved Spectroscopy
With NuSTAR we can separately analyze the nonthermal continuum originating from different spatial regions of the remnant. Figure 4 shows the 8-10 keV NuSTAR image along with the extraction regions that we used for this work.
We find that all of the regions are well fit by unbroken power-law models across the 15-50 keV band. The exterior filaments (the regions labeled "northeast" and "southeast" in Figure 4) have similar spectra, with power-law indices of Γ ∼ −3.06, while the interior bright knots (the "knot" regions in Figure 4) also have similar spectra, but in general have a softer power-law index of Γ ∼ −3.35. We fit all of the regions independently but, for clarity, show the combined spectra and best-fit models for the exterior and interior regions in Figure 5 while the best-fit model parameters and 2-σ error ranges are given in Table 2.
At soft X-ray energies (<10 keV), the presence of lines associated with ionized S, Fe, and Ni (as well as lighter species below the calibrated NuSTAR band) indicate the presence of a hot thermal plasma. We do not attempt to model these thermal plasmas since we know from analysis with Chandra that the plasma properties may vary on ∼arcsecond spatial scales (Stage et al. 2006;Hwang & Laming 2012) and the relatively large . Combined spectra of exterior filaments and central knots of Cas A. The combined spectrum from the northeast and southeast external filaments is shown in black (upper curve), while the combined spectrum of the central knots is shown in red (lower curve). The spectrum of the knots has been artificially offset downwards by a factor of 10 for clarity in the top frame. The shaded region to the right is the 15-50 keV band used to fit the powerlaw component, which has then been extrapolated down to 3 keV. Across the 15-50 keV band the spectrum is well fit by a single power-law. The normalization of the power-law is allowed to vary between the different regions, though the index is tied together for the two exterior filament regions and the three central knot regions. Best-fit parameters are given in Table 2. The central knots have a softer power-law spectrum than the exterior filaments. When extrapolated to low energies, the central knots overpredict the observed emission while the exterior filaments underpredict the observed emission as demonstrated in the lower panel, which shows the ratio of the data to the models. See online version for color images.
NuSTAR regions will sample many regions with different physical parameters. However, we note that when we extrapolate the power-law fits from the 15-50 keV band to soft X-ray energies (<10 keV) we see different behavior for the exterior filaments and the central knots; the power-law model for the exterior filaments underpredicts the observed spectrum at soft X-ray energies while the power-law model for the central knots overpredicts the observed spectrum. We discuss the implications of this below. Figure 6 shows a comparison between the radio (6 cm intensity maps obtained with the VLA), soft X-rays (4-6 keV continuum images taken with Chandra from Hwang et al. (2004)), and the 10-15 keV hard X-ray observed by NuSTAR.
Multiwavelength Comparisons
Hard X-rays and Radio
The hard X-ray and radio morphologies of Cas A are substantially different (Figure 7). The outer filaments in the northeast and southeast visible in NuSTAR are coincident with the edge of faint radio emission in the VLA images, but the radio morphology is not of thin tangential filaments but simply is simply described by a broad plateau. Any filamentary structure could be unobserved due to a lack of dynamic range in the radio images (i.e. the radio emission from this region is faint compared to the emission in the bright ring) or to a change in the shape of the continuum extending to the radio. We do not see any enhancements in the hard X-ray images near the center of the remnant where we see the bright ring in the radio emission. Instead, we see that in the eastern half of the remnant there is little hard X-ray emission associated with the bright ring, while in the western half of the remnant the bright central knots appear to be located near the bright ring.
3.3.2. Hard X-rays and Soft X-rays We find that the Chandra and NuSTAR images (Figure 8) generally agree on large spatial scales up to ∼10-15 keV, but show differences in the higher energy bands observed by NuSTAR. Up to 15 keV, the exterior filaments in the forward shock of Cas A in the north, northeast, south, and southeast seen with Chandra are all clearly visible in the <15 keV NuSTAR images, as are some of the interior emission which is likely to be residual flux from hot thermal plasmas leaking into the NuSTAR band. It is in the relative intensity of the central knots that we note the major differences, with the western knots dominating much of the hard X-ray flux in the 10-15 keV band in NuSTAR while they appear relatively unremarkable in the Chandra images, even when steps are taken to reduce the impact of the bright thermal emission in Cas A (e.g. using imaging tomography as per DeLaney et al. (2004)). This difference becomes more apparent in the images at higher energies (Figures 2 and Figure 3), with the central knots in the west dominating the emission above 15 keV and the bright ring completely disappearing in the these energy bands. At the highest energies ( Figure 3) the emission appears to be mostly attributable to several of the bright knots, with the outer rims fading away. Several of the central knots are also near regions noted by Patnaude & Fesen (2009) and Uchiyama & Aharonian (2008) to show variability in the 4-6 keV continuum emission on timescales of a year, which we discuss below.
3.3.3. Radio, Soft, and Hard X-rays Figure 9 shows the overlay of the radio, soft, and hard X-ray images. In the particular case of the western knots, we find that the two hard X-ray knots observed by NuSTAR differ from both the soft X-ray and radio images. Figure 10 shows a zoomed-in view around the two western knots, with the Chandra and the VLA observations smoothed to approximately the same spatial scales as the NuSTAR deconvolved images. We see that the radio is clearly strongest near the western of the two knots, Chandra shows some flux near the eastern knot but is still dominated by the western knot, while NuSTAR observes two knots of nearly equivalent brightness. These central knots are spatially consistent with two bright knots in high-resolution maps of the radio luminosity and radio spectral index (Anderson &DeLaney et al. 2014), with the brighter of the two knots associated with a region of steeper radio spectral index while the other is associated with a region of flatter radio spectral index. The fact that NuSTAR sees comparably bright X-ray emission in both of these regions is puzzling.
keV, GeV, and TeV emission
There has been improved work over the last few years localizing and studying the GeV to TeV emission from Cas A. This is important for understanding the acceleration of cosmic rays up to the "knee" of the cosmic ray spectrum. MAGIC (Albert et al. 2007) and VER-ITAS (Acciari et al. 2010) have observed TeV emission that peaks in the western half of the remnant, while Yuan et al. (2013) show that the Fermi GeV emission peaks more towards the center of the remnant. We compare and contrast the centroids of the TeV and GeV emission with features in the 15-20 keV band (Figure 11). At the 90% level the GeV and TeV centroids are consistent with each other and could be associated with any part of the remnant. However, if we speculatively use the stricter 1-σ error limits then we note that the GeV centroid is located closer to regions of the remnant known to be bright in the infrared while the TeV centroid is closer to a bright region in the NuSTAR band.
We reproduce the data represented in Figure 11, as they are often reported in different coordinate systems and are shown with different confidence contours. We adopt decimal degrees in (α, δ) as our coordinate system and show 1σ statistical errors added in quadrature with the systematic errors quoted in the literature. Albert et al. (2007) find the TeV emission centered at (350.79, 58.81), with asymmetric errors in the α direction of (0.003 stat + 0.001 sys ) hours, which, when added in quadrature and projected into the tangent plane at a declination of 58.81 degrees, corresponds to an error estimate of 0.0245 degrees. The δ error is (0.03 stat + 0.01 sys ) degrees, or 0.036 degrees, when added in quadrature. For VERITAS, the centroid shown in Acciari et al. (2010) is (350.825, 58.8025) with symmetric errors in the tangent plane of 0.01 stat + 0.02 sys degrees, or 0.022 degrees when added added in quadrature. The Fermi centroids reported in the literature vary slightly in the literature owing (we assume) to a difference in the data volume used by two authors (Yuan et al. 2013 andSaha et al. 2014). The two reductions appear to largely be consistent with one another: Yuan et al. (2013) gives the location in Galactic coordinates of (111.74, -2.12) corresponding to (α, δ) of (350.2876, 58.5513), while Saha et al. (2014) shifts the centroid slightly to the northwest at coordinates (350.87, 58.83). In both cases the errors are symmetric (0.01 stat + 0.005 sys ), or 0.0112 degrees.
DISCUSSION
Hard X-ray Morphology
Previous observations of the non-thermal continuum emission in the Chandra and XMM-Newton X-ray bands suggest a non-thermal thin rim associated with the "forward shock" (Gotthelf et al. 2001) with some additional filamentary structures through the center of the remnant (Hughes et al. 2000). Based on XMM-Newton data, Bleeker et al. (2001) conclude that the non-thermal emission traces the softer thermal component and that the emission does not originate from a few localized regions. We have shown for the first time that the >15 keV flux from Cas A is dominated by neither the forward shock nor the bright ring (presumably the reverse shock). Instead, we find that, contrary to all expectations, the >15 keV emission is dominated by the emission from several bright knots near the center of the remnant.
The nature of the central emission in Cas A has been a long-standing debate (e.g., see Aharonian 2008 andHelder &Vink 2008). If the central knots of emission were actually simply the exterior filaments from the outer rim of the remnant projected onto the center then we would expect the spectral shape of the central knots to be similar to the spectral shape of the exterior filaments. Our observations have conclusively shown that the spectra from the two regions are quantitatively different. This points to some systematic shift between the exterior and central spatial regions, be it a difference in the underlying electron population, a difference in how the electrons are being accelerated, a difference in how the electrons are producing X-rays, or all of these.
If the central (western) knots are located at the forward shock front then the difference in the power-law index between the different spatial regions must be attributable to some change in the physical environment (e.g. higher magnetic fields, different compression ratios in the shock front, etc). While we cannot explicitly rule out this possibility, we find it suggestive that the individual outer filaments share a similar spectrum while the individual central knots share a (different) similar spec- trum. If all of the emission were coming from the forward blast wave then there would need to be some large scale physical differences between the outer rim emission seen in the northeast and southeast and the central emission seen in the west. Morphologically the central knots also appear point like, which is not what we would expect if we were seeing a face-on version of the filamentary structures on the (eastern) outer edge of the remnant. Finally, it seems unlikely that a viewing-angle dependence for the emission mechanism would conspire to alter the observed spectral index for the outer rims as compared to the cen- Figure 9. Comparison of the Chandra, VLA, and NuSTAR continuum images of Cas A. Red: Chandra 4-6 keV data from the left panel of Figure 8; Green: VLA 6 cm data from the left panel of Figure 7; Blue: NuSTAR 10-15 keV data from Figure 6. See online version for color images. tral knots. We thus conclude that we are observing two relatively independent electron populations and therefore argue that the central knots are, in fact, located in the interior of the remnant.
This leaves the unsolved problem of the origin of the electrons responsible for this emission in the interior of the remnant. Helder & Vink (2008) argue, with some assumptions about the turbulence of the magnetic field, that electrons accelerated at the forward shock could not have advected to the interior of the remnant over timescales consistent with the age of the remnant and before the electrons would cool via synchrotron losses. A related observational fact is that both the emission near the exterior filaments and internal knots have been shown to vary on ∼yearly timescales (Patnaude & Fesen 2006;Uchiyama & Aharonian 2008).
It is not yet clear whether these variations are caused by changes in the environment surrounding a (relatively static) population of electrons or whether some unknown mechanism is accelerating electrons in the interior of the remnant. If the latter were true, then we may also need to revisit non-thermal bremsstrahlung as a possible emission mechanism at least for the interior emission. Zirakashvili et al. (2014) has put forward the idea that the secondary electrons from the radioactive decay of 44 Ti may be important for seeding the relativistic electron population responsible for the non-thermal X-ray emission, but as the morphology observed by NuSTAR is dramatically different than the morphology of the 44 Ti in Cas A (Grefenstette et al. 2014) we do not think that this scenario is likely. We also note that if the electrons are being accelerated in the interior of the remnant at the reverse shock, we do not understand why they are only being accelerated at particular locations near the reverse shock. Again, perhaps this is driven by variation in the nature of the ejecta or the gas in the interior of the remnant near the reverse shock. As of now, we do not see a definitive solution for this problem
The nature of the non-thermal emission
The morphology of the >15 keV emission is puzzling and does not appear to lend itself to a simple interpretation. Models which only produce accelerated electrons near a uniform forward shock (e.g. that only predict nonthermal emission in a thin rim surrounding the supernova remnant) are disfavored without any of the complications due to the presence of the thermal flux in the 4-6 keV band from the shocked ejecta. The non-thermal emission (both in the radio as well as in the soft and hard X-ray bands) implies that the electrons producing the >15 keV emission should be the high energy tail of the same electrons producing the radio emission. Therefore the hard X-ray morphology should follow some of the large-scale features of the radio map. We find that this not to be the case and do not have a plausible explanation for why the non-thermal X-ray and radio maps differ significantly. A detailed, spatially resolved comparison of the radio spectral index and flux with the hard X-rays is required, but is beyond the scope of our present work.
It is still possible that the emission process for the bright knots is not synchrotron emission from 10 -100 TeV electrons at all, but rather bremsstrahlung from electrons with only somewhat suprathermal energies of 10 -100 keV. Laming (2001) suggested that lower hybrid waves at low Mach-number reflected shocks in the interior could produce such electrons. While the primary objection to such models (e.g. Vink 2008), that the electrons above thermal energies would rapidly cool due to Coulomb losses, would still need to be met, the localization of the emission into two bright knots may indicate some unique conditions that hold in the knots but not in the rest of the remnant. The predicted spectral shape of such emission can be adjusted somewhat by varying the electron Alfvén velocity, but the particular calculations of Laming (2001) all give a cutoff above ∼100 keV. If the continuum reported by Beppo-SAX out to 300 keV were confirmed then this could disfavor this picture.
The particular case of the bright western knots may be a connection between the X-rays and the radio bands. DeLaney et al. (2005) previously noted that there are small, bright features that are found in both the arcsecond-resolution radio and soft X-ray data. One of these two knots is brighter and has a steeper radio spectrum while the other is fainter with a flatter radio spectrum. The fact that NuSTAR sees comparably bright X-rays from both of these knots may suggest that there may be some unknown localized physical process acting on small scales to produce the enhanced radio emission as well as the enhanced hard X-ray emission. However, the data that we are comparing are not contemporaneous and Cas A is known to vary in both the radio and in the soft X-rays (Patnaude & Fesen 2006;Uchiyama & Aharonian 2008). We note that there now exists Janksy VLA radio images (DeLaney et al. 2014) that are contemporaneous with these NuSTAR observations and that these knots have not significantly changed their brightness or morphology when compared to the older VLA image. It may be the case that the bright features in the Cas A hard X-ray images are also evolving on timescales of a few years, though this is unlikely to explain the large-scale discrepancies between the radio and the hard X-ray images.
While the >15 keV emission is clearly dominated by several central knots and the outer filaments, there is also diffuse emission that appears to permeate the remnant. While the central knots may be localized regions of enhanced magnetic field, density variations, or particle acceleration, the source of the diffuse central emission at energies >15 keV is unclear. It is possible that this is in fact tenuous diffuse emission from the forward shock of the supernova remnant seen in projection, or that there is a population of unresolved and relatively dim knots in the interior of the remnant. One additional possibility is that some electrons that happen to be in regions of lower mean magnetic field strength and thus are longer lived. Electrons radiating at ≈15 keV in a magnetic field of 20 µG or less could survive for more than the 330 year lifetime of the remnant and so could be responsible for this emission.
The hard X-ray morphology may also have implications for the interpretation of the TeV and GeV emission. The centroids of the TeV emission may be spatially consistent with the bright western knots of synchrotron emission observed by NuSTAR, while the Fermi GeV centroid has no obvious counterpart in the NuSTAR band. This might be expected if the TeV emission is leptonic in nature (i.e. inverse Compton scattering of CMB photons), which requires the presence of relativistic electrons which would also produce synchrotron emission in the NuSTAR bandpass. Likewise, if the GeV emission is instead produced via hadronic emission mechanisms (that is, π 0 decay) then we might not expect to observe a counterpart in the NuSTAR band, since the π 0 emission is produced by ions with much lower energies than the synchrotronemitting electrons observed by NuSTAR. This would be consistent with broadband spectroscopic analyses (e.g. Saha et al. 2014), which suggests the need for both hadronic and leptonic emission mechanisms. It's also suggestive that the GeV centroid falls near enhanced emission observed at 24 and 70 microns (Hines et al. 2004) as well as near bright optical features in this region, both of which imply the presence of dense material and further supports a hadronic emission mechanism in that region. However, as the difference in the centroid regions is not strictly statistically significant (recall that all of the centroids are consistent the 90% level), we cannot make firm claims about separating the emission mechanisms without further improvements in the angular resolution of future GeV and TeV instruments. When those data are eventually available, combining the GeV/TeV data with these NuSTAR observations should prove fruitful.
Curvature in the continuum spectrum
For the following discussion, we presume that the emission in the NuSTAR band is synchrotron radiation from electrons with 10 -100 TeV energies, produced by diffusive shock acceleration.
The shape of the spectrum carries information about the source electron spectrum. For synchrotron emission (which is the only viable mechanism for the emission at the forward shock) from a power-law electron spectrum with an exponential cutoff, the resulting spectrum drops roughly as exp (− √ ν/ν rolloff ) , where hν rolloff is 1.9 times the photon energy at which electrons with E m radiate the peak of their synchrotron spectrum (e.g. Pacholczyk 1969). However, more elaborate inhomogeneous models of shell supernova remnants can predict significantly different spectral shapes (e.g. Reynolds 1998). In the case of the XSPEC srcut model, the model produces a smooth continuum over many decades in energy, from the radio to the X-ray band, with a characteristic frequency at which the spectrum gently rolls over. These frequencies are in the range 10 16 -10 18 Hz. This is sufficiently slow that over a relatively narrow bandpass in the hard X-rays the resulting spectrum can be reasonably fit by a power-law (i.e. 15-50 keV), though we do expect some steepening of the spectrum from the energy band previous sampled by Chandra to the NuSTAR band.
At the exterior of the remnant we find that the 15-50 keV power-law model under predicts the emission at low energies and therefore does not explicitly require curvature in the spectrum. This is not surprising, as we expect there to be some "PSF bleed" of the central regions of of the remnant that provides an additional soft (<10 keV) component on top of the non-thermal continuum. However, in this region the non-thermal continuum spectrum as observed by Chandra is well-fit by a power-law spectrum with a substantially harder photon index (e.g., Γ ∼ −2.3 for the northeast filaments from Patnaude & Fesen 2009). Given that the Chandra data are uncontaminated by neighboring thermal emission, we can then assume that the power-law index steepens from Γ ∼ −2.3 to Γ ∼ −3.05 from the 4-6 keV band to the 15-50 keV band, with no further curvature required by our data across the 15-50 keV band.
In the central emission regions the NuSTAR data alone demonstrates that the softer power-law fit in the 15-50 keV band over-predicts the 3-15 keV observed emission. We actually expect more contamination by thermal plasma(s) in the central extraction regions. This implies that, if anything, there is more curvature in the central knots than in the exterior filaments.
We can naively apply the srcut model to analyze the data to compare the NuSTAR 15-50 keV results to the 4-6 keV results from Chandra. The srcut model predicts a gradually steepening spectrum above hν rolloff . We calculated a custom srcut model for the steep radio spectral index of Cas A, α = −0.77, and calculated the slopes over frequency ranges of a factor of 10 0.5 , roughly the photon energy range from 15-50 keV. Then the measured values of Γ uniquely predict hν rolloff for a simple single srcut component. We find that the central-knot value of Γ = −3.35 requires hν rolloff = 1.3 keV, while the outer-filament value of Γ = −3.06 requires hν rolloff = 2.3 keV. We can perform the same operation for the powerlaw fits in the range of 4.2-6 keV done with Chandra data (Patnaude et al. 2011) 14 From their 2011 data, they find Γ of −2.85 and −2.56 for a sample of exterior fila-ments and for a region similar to the NuSTAR central knots, respectively, implying hν rolloff values of 0.44 and 1.1 keV. Thus, while our 15-50 keV power-law indices are steeper than their 4.2-6 keV values, they do not steepen as much as one would expect from a single power-law electron spectrum with an exponential cutoff.
We conclude that while softening of the spectrum is evident, it requires either electron distributions with a range of cutoff energies, as might naturally be expected from integrating over multiple shock regions, or a modification of shock acceleration physics that produces a more gradual cutoff in the electron distribution than exponential. Since one careful calculation of this cutoff (Zirakashvili & Aharonian 2007) yields a distribution which at the shock is actually exponential in the square of electron energy, i.e., a much steeper cutoff, this latter possibility seems less likely. We infer that even over the spatially localized regions of the forward blast wave and of the central knots, conditions must vary sufficiently to provide a range of electron cutoff energies whose superposition gives us the very gradual steepening we observe. Future NuSTAR observations of the historical Type Ia SNRs may show if they share this property.
We do urge caution in using the hν rolloff frequency reported above as a proxy for the maximum energy of the emitting electrons. Doing so requires some knowledge of the radio-spectral index (assumed to be 0.77 above), which is known to vary across the remnant (Anderson & Rudnick 1996) as well as spatially resolved knowledge of the radio 1 GHz flux. The latter measurement often suffers from a lack of dynamic range in Cas A (i.e. the northeast rim is undetected at radio wavelengths but bright in synchrotron X-rays). This introduces enough degeneracies in the srcut model fits that we do not include fits here with all of the parameters allowed to vary. However, the above analysis is entirely self-consistent and our result is relatively independent of the particular parameters used in srcut.
Comparison with spatially-integrated measurements
No previous instrument could spatially resolve the different emission regions in Cas A above 10 keV. However by considering the power law spectral index integrated over the remnant it could be possible to confirm our finding that the softer inner region dominates the total flux. The integrated emission detected by previous hard X-ray non-imaging instruments (e.g., The et al. 1996;Allen et al. 1997;Renaud et al. 2006;Vink et al. 2000;Maeda et al. 2009) extends to high energy (>100 keV), so that, in principle, the spectral index could be wellconstrained. Table 3 compiles the results of these previous measurements of the integrated spectrum of Cas A. It is clear different measurements disagree at levels greater than the formal statistical errors. This is likely due to systematics associated with the background-dominated measurements or the difference in energy bands in which the different instruments are sensitive. What is clear is that previous spatially-integrated measurements are not in agreement with one another and do not have statistical or systematic precision sufficient to confirm or rule out NuSTAR's finding that the softer central emission dominates the integrated flux. (1996) ii: Vink et al. (2001) iii: Rothschild & Lingenfelter (2003) iv: Renaud et al. (2006) v: Maeda et al. (2009 5. CONCLUSIONS We have shown that the hard X-ray emission from Cas A up to 50 keV resolves into two main populations: fainter outer filaments and bright central knots which dominate the emission above 15 keV. These two populations show different unbroken power-law spectra over the 15-50 keV NuSTAR band, with the central bright knots having a significantly softer spectra than the dim outer rims. We view this as evidence for two distinct populations of electrons responsible for the exterior and central emission and therefore argue that the central knots are in fact located in the interior of the remnant rather than at the forward shock and seen in projection.
The origin of the population of energetic electrons in the interior of the remnant remains a mystery, especially as the morphology above 15 keV does not appear to follow that of any other waveband. Some of the bright central knots appear to be spatially coincident with regions known to show rapid (∼yearly) variability in both soft X-rays and the radio, which may evidence for active particle acceleration in the interior of Cas A or for significant changes in the physical environment in the center of the remnant. Future NuSTAR observations with a longer temporal baseline may be able to test for changes in the flux of particular regions above 15 keV.
The steepening of the non-thermal spectrum from the Chandra band to the NuSTAR band requires either an electron distribution that cuts off more gradually than an exponential. While this could be due to some modification in the shock acceleration physics, we instead conclude that conditions in the forward shock blast wave produce a range of cutoff energies in the electron spectrum even on small spatial scales.
We have shown that imaging in the NuSTAR band can be useful for interpreting results from the GeV/TeV energy bands. The association of the centroid of the TeV emission with a region bright in the NuSTAR band and the lack of any such association with the centroid of the GeV emission can be naturally explained by a leptonic emission mechanism for the former and a hadronic emission mechanism for the latter. The interpretation that there are both hardonic and leptonic emission mechanisms at work in Cas A is consistent with broad-band spectrum fitting and may bear out as the measurements in the GeV and TeV bands improve.
B.G. thanks Una Hwang for the Chandra 4-6 keV band image. This work was supported under NASA contract NNG08FD60C and made use of data from the NuSTAR mission, a project led by the California Institute of Technology, managed by the Jet Propulsion Laboratory, and funded by NASA. We thank the NuSTAR Operations, Software and Calibration teams for support with the execution and analysis of these observations. This research has made use of the NuSTAR Data Analysis Software (NuSTARDAS), jointly developed by the ASI Science Data Center (ASDC, Italy) and the California Institute of Technology (USA).
Facilities: NuSTAR, Chandra, VLA, Fermi, VERI-TAS, HESS,
Figure 2 .
2Deconvolved NuSTAR images of Cas A: red (15-20 keV), green (20-25 keV), and blue (25-35 keV). See online version for color images.
Figure 3 .
3NuSTAR images of Cas A in the 35-45 keV band (top) and 45-55 keV band (bottom). The images have been smoothed using a top-hat smooth kernels of 10 pixels (∼25 ′′ ) and both have a sqrt scaling. The color stretch has been modified for illustrative purposes.
Figure 4 .
4Spectral extraction regions. The 15-20 keV NuSTAR image of Cas A shows an 8×8 arcminute region around the remnant shown with an aggressive stretch to highlight the residual diffuse emission through the remnant. Also shown are the extraction regions used in the spectral analysis: the green circles are the representative central knot regions (all of which have similar spectra), while the red ellipses are the northeast and southeast exterior filaments. These latter regions roughly correspond to exterior filaments ofPatnaude & Fesen (2009) and filaments 1, 2, 3, and 9 of. The cyan dashed circles are the approximate locations of the forward and reverse shocks fromGotthelf et al. (2001). See online version for color images.
Figure 5
5Figure 5. Combined spectra of exterior filaments and central knots of Cas A. The combined spectrum from the northeast and southeast external filaments is shown in black (upper curve), while the combined spectrum of the central knots is shown in red (lower curve). The spectrum of the knots has been artificially offset downwards by a factor of 10 for clarity in the top frame. The shaded region to the right is the 15-50 keV band used to fit the powerlaw component, which has then been extrapolated down to 3 keV. Across the 15-50 keV band the spectrum is well fit by a single power-law. The normalization of the power-law is allowed to vary between the different regions, though the index is tied together for the two exterior filament regions and the three central knot regions. Best-fit parameters are given in Table 2. The central knots have a softer power-law spectrum than the exterior filaments. When extrapolated to low energies, the central knots overpredict the observed emission while the exterior filaments underpredict the observed emission as demonstrated in the lower panel, which shows the ratio of the data to the models. See online version for color images.
Figure 6 .
6NuSTAR continuum image of Cas A. The 10-15 keV NuSTAR shows the image after it has been deconvolved with the PSF. The image has been been aggressively stretched to highlight the dimmer, diffuse emission from the remnant. See online version for color images.
Figure 7 .
7Comparison of the VLA 6 cm radio and NuSTAR continuum images of Cas A. Left: VLA 6 cm image. Right: The overlay of the VLA image with NuSTAR image from Fig 6. See online version for color images.
Figure 8 .
8Comparison of the Chandra and NuSTAR continuum images of Cas A. Left: Chandra 4-6 keV image. Right: the overlay of the Chandra image with NuSTAR image from Fig 6. The images have been stretched to highlight the dimmer, diffuse emission from both satellites. See online version for color images.
Figure 10 .
10Comparison of the Chandra, VLA, and NuSTAR images of Cas A, zoomed in on the bright western knots. The color scheme is the same as forFig 9.Here the Chandra and VLA data have been smoothed to approximately the same resolution as the NuSTAR 10-15 keV data and all of the images have been stretched for ease of comparison. See online version for color images.
Figure 11 .
11Comparison of the NuSTAR 15-20 keV 8x8 arcminute image with the centroid locations from two Fermi reductions (green circle: Yuan et al. 2013; cyan circle: Saha et al. 2014), VERI-TAS (yellow circle: Acciari et al. 2010), and MAGIC (red ellipse: Albert et al. 2007). Note that this image is rotated with respect to the comparison of the VHE centroids to the radio image shown by Yuan et al. 2013 as those authors present the data in Galactic coordinates. All the GeV and TeV regions have a radius that corresponds to the 1σ statistical errors added in quadrature with the systematic errors. See text for details. See online version of color images. and leave the origin of this emission as an open mystery.
Table 1
1NuSTAR ObservationsOBSID
Exposure UT Start Date
40001019002
294 ks
2012 Aug 18
40021001002
190 ks
2012 Aug 27
40021001004
29 ks
2012 Oct 07
40021001005
228 ks
2012 Oct 07
40021002002
288 ks
2012 Nov 27
40021002006
160 ks
2012 Mar 02
40021002008
226 ks
2012 Mar 05
40021002010
16 ks
2012 Mar 09
40021003003
13 ks
2013 May 28
40021003003
216 ks
2013 May 28
40021011002
246 ks
2013 Oct 30
40021012002
239 ks
2013 Nov 27
40021015002
86 ks
2013 Dec 21
40021015003
160 ks
2013 Dec 23
Total
≈ 2.4 Ms
Table 2
2NuSTAR power-law fits over the 15-50 keV bandRegion
Photon Index (2-σ Error Range)
Central Knots
−3.35 (3.29-3.41)
Outer Filaments
−3.06 (2.98-3.13)
Table 3
3Comparison with previous Power-law Indices The et al.Observatory (Band)
Γ
Notes
CGRO (40-120 keV) i
−3.06 ± 0.41
1
Beppo-SAX (12-300 keV) ii
−3.3 ± 0.05
2
Beppo-SAX (30-100 keV) ii
−3.1 ± 0.4
2
RXTE (20-100 keV) iii
−3.125 ± 0.05
1
INTEGRAL (21-120 keV) iv
−3.3 ± 0.1
3
Suzaku (3.4-40 keV) v
−3.06 ± 0.05
2
NS Central Knots (15-50 keV)
−3.35 ± 0.06
2
NS Exterior Filaments (15-50 keV)
−3.06 ± 0.06
2
Notes on error estimates:
1: Unstated, assumed to be 1-σ; 2: 90% confidence intervals.
3: 1-σ.
References:
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|
[] |
[
"Upper critical field H c2 in Bechgaard salts (TMTSF) 2 PF 6",
"Upper critical field H c2 in Bechgaard salts (TMTSF) 2 PF 6"
] |
[
"Ana Domínguez Folgueras \nDepartamento de Física\nUniversidad de Oviedo\n33007OviedoSpain\n\nInstituto de Ciencia de Materiales de Madrid\nCSIC\n28049MadridCantoblancoSpain\n\nDepartment of Physics & Astronomy\nUniversity of Southern California\n90089-0484Los AngelesCAUSA\n",
"Kazumi Maki \nDepartment of Physics & Astronomy\nUniversity of Southern California\n90089-0484Los AngelesCAUSA\n"
] |
[
"Departamento de Física\nUniversidad de Oviedo\n33007OviedoSpain",
"Instituto de Ciencia de Materiales de Madrid\nCSIC\n28049MadridCantoblancoSpain",
"Department of Physics & Astronomy\nUniversity of Southern California\n90089-0484Los AngelesCAUSA",
"Department of Physics & Astronomy\nUniversity of Southern California\n90089-0484Los AngelesCAUSA"
] |
[] |
The symmetry of the superconducting order parameter in Bechgaard salts is still unknown, though the triplet pairing is well established by NMR data and large upper critical field Hc2(0) ∼ 5 Tesla for H a and H b ′ . Here we examine the upper critical field of a few candidate superconductors within the standard formalism. The present analysis suggests strongly chiral f-wave superconductor somewhat similar to the one in Sr2RuO4 is the most likely candidate.
| null |
[
"https://arxiv.org/pdf/cond-mat/0601065v1.pdf"
] | 117,858,753 |
cond-mat/0601065
|
c6b12add32c7a704200b8d0aa5e3ca50a271b178
|
Upper critical field H c2 in Bechgaard salts (TMTSF) 2 PF 6
4 Jan 2006
Ana Domínguez Folgueras
Departamento de Física
Universidad de Oviedo
33007OviedoSpain
Instituto de Ciencia de Materiales de Madrid
CSIC
28049MadridCantoblancoSpain
Department of Physics & Astronomy
University of Southern California
90089-0484Los AngelesCAUSA
Kazumi Maki
Department of Physics & Astronomy
University of Southern California
90089-0484Los AngelesCAUSA
Upper critical field H c2 in Bechgaard salts (TMTSF) 2 PF 6
4 Jan 2006Europhysics Letters PREPRINT7470Kn -Organic Superconductors 7420Rp -Pairing symmetries 7425Op -Mixed statescritical fieldsand surface sheaths
The symmetry of the superconducting order parameter in Bechgaard salts is still unknown, though the triplet pairing is well established by NMR data and large upper critical field Hc2(0) ∼ 5 Tesla for H a and H b ′ . Here we examine the upper critical field of a few candidate superconductors within the standard formalism. The present analysis suggests strongly chiral f-wave superconductor somewhat similar to the one in Sr2RuO4 is the most likely candidate.
Introduction. -The Bechgaard salts (TMTSF) 2 PF 6 is the first organic superconductor discovered in 1980 [1]. For a long time the superconductivity is believed to be conventional s-wave [2]. Recently the symmetry of the superconducting energy gap becomes the central issue [3,4]. The upper critical field at T = 0K, H c2 ∼ 5 Tesla for both H a and H b ′ for both (TMTSF) 2 PF 6 [5] and (TMTSF) 2 ClO 4 [6] are clearly beyond the Pauli limit [7,8] indicating the triplet pairing. More recently the NMR data from (TMTSF) 2 PF 6 [9] indicates clearly the triplet pairing. Therefore the candidates for the superconductivity in Bechgaard salts are more likely within p-wave and f-wave superconductors. In the following we shall examine the upper critical field of these superconductors following the standard method initiated by Gor'kov [10] and extended by Luk'yanchuk and Mineev [11] for unconventional superconductors. Also we take the quasiparticle energy in the normal state as in the standard model for Bechgaard salts [2]
ξ(k) = v(|k a | − k F ) − 2t b cos bk − 2t c cos ck (1) with v : v b : v c ∼ 1 : 1/10 : 1/300 and v b = √ 2t b b and v c = √ 2t c c.
There are earlier analysis of H c2 of Bechgaard salts starting from the one dimensional models [12,13]. However, those models predict diverging H c2 (T ) for T → 0K or the reentrance behaviour, which have not been observed in the experiments [5,6]. Also, the quasilinear T dependence of H c2 (T ) in both (TMTSF) 2 PF 6 and (TMTSF) 2 ClO 4 is very unusual. Among the models we have considered, c EDP Sciences the chiral f ′ -wave superconductor with ∆ k ∼ 1 √ 2 sgn (k a ) + ı sin χ 2 cos χ 2 , looks most promising, where χ 1 = b k and χ 2 = c k where b and c are crystal vectors.
Also if the superconductor belongs to one of the nodal superconductors [14] and if nodes lay parallel to k c within the two sheets of the Fermi surface, the angle dependent nuclear spin relaxation rate T −1 1 in a magnetic field rotated within the b ′ − c * plane will tell the nodal directions.
Before proceeding, we show |∆( k)| os two chiral f-wave superconductors in Fig. 1 a) and b).
where |∆(k)| ∼ (1 + cos 2χ 1 )(1 − 1 2 cos 2χ 2 ) [4]. In this case the spin component is characterised by a unit vectord. Alsod is most likely oriented parallel to c * . Let's assumed c * , though H c2 (T ) is independent ofd as long as the spin orbit interaction is negligible. Experimental data from both UPt 3 and Sr 2 RuO 4 indicate that the spin-orbit interactions in these systems are not negligible but extremely small [15]. We consider a variety of triplet superconductors (see Fig. 1):
A. Simple p-wave SC: ∆(k) ∼ sgn(k a ). Following [16] the upper critical field is determined by
− ln t = ∞ 0 du sinh u (1 − K 1 ) (2) −C ln t = ∞ 0 du sinh u (C − K 2 )(3)
where
K 1 = e −ρu 2 |s| 2 1 + 2 C ρ 2 u 4 s 4 (4)K 2 = e −ρu 2 |s| 2 1 6 ρ 2 u 4 s * 4 + C 1 − 8ρu 2 |s| 2 + 12 ρ 2 u 4 |s| 4 − 16 3 ρ 3 u 6 |s| 6 + 2 3 ρ 4 u 8 |s| 8(5)
and
t = T Tc , ρ = vav b eHc2(T ) 2(2πT ) 2 , s = 1 √ 2 sgn(k a ) + ı sin χ 2 , χ 2 = c k and . . . means average over χ 2 .
Here v a , v c are the Fermi velocities parallel to the a axis and the c axis respectively.
Here we assumed that ∆( r) is given by [16]
∆( r) ∼ 1 + C(a + ) 4 (6) where = C n e −eBx 2 −nk(x+ız)− (nk) 2 4eB is the Abrikosov state [17] and a + = 1 √ 2eB (−ı∂ z − ∂ x + 2ıeHz) is the raising operator.
Then in the vicinity of t → 1 we find ρ = 2 7ζ(3) (− ln t) = 0.237697(− ln t) and C = − 93ζ (5) 647ζ (3) ρ.
For t → 0 on the other hand we obtain
ρ 0 = lim t→0 ρ t 2 = v a v c eH c2 (0) 2 (2πT c ) 2 = 0.1583(7)
and C = −0.031. From these we obtain
h(0) = H c2 (0) ∂Hc2(t) ∂t | t=1 = 0.6659(8)
Both ρ 0 (t) and C(t) are evaluated numerically and shown in Fig. 2 a) and b) respectively.
Here ρ 0 (t) = t 2 ρ(t) = v v c eH c2 (t)/2(2πT c ) 2 . B. Chiral p-wave SC: ∆(k) = 1/ √ 2sgn(k a ) + i sin(χ 2 ). Here 1 √ 2 sgn(k a ) + i sin(χ 2 )
is the analogue of e ıφ in the 3D systems in the quasi 1D system. For a chiral state the Abrikosov function is written as [18] ∆( r, k) ∼ (s + Cs * (a † ) 2 ) (9)
K 1 = e −ρu 2 |s| 2 |s| 2 − 2 C |s| 4 (10) K 2 = e −ρu 2 |s| 2 −|s| 4 + C |s| 2 1 − 4ρu 2 |s| 2 + 2 ρ 2 u 4 |s| 4(11)
and the same expressions for t, ρ,. . . For t → 1 we find C = 1 − √ 1.5 = −0.2247 and ρ = 0.3838(− ln t). On the other hand, for t → 0 we obtain C = −0.3660 and ρ 0 = 0.27343. From these we obtain h(0) = 0.71324. We obtain ρ(t) and C(t) numerically. They are shown in Fig. 2 a) and b) respectively.
C. Chiral f-wave SC:∆(k) ∼ds cos χ 1 . H c2 (t) is determined from eq. 3 where now:
K 1 = (1 + cos 2χ 1 ) e −ρu 2 |s| 2 |s| 2 − 2ρu 2 |s| 4(12)K 2 = (1 + cos 2χ 1 ) e −ρu 2 |s| 2 −ρu 2 |s| 4 + C|s| 2 1 − 4ρu 2 |s| 2 + 2ρ 2 u 4 |s| 4(13)
Here now . . . means the average over both χ 1 and χ 2 . Then it is easy to see that the chiral f-wave SC has the same H c2 (t) and C(t) as the chiral p-wave SC, since the variable χ 1 is readily integrated out.
D. Chiral f ′ -wave SC:∆(k) ∼ds cos χ 2 . Now we have a set of equations similar to the chiral f-wave except 1 + cos 2χ 1 in both eqs. 13 has to be replaced by 4 3 (1 + cos 2χ 1 ). Then we obtain for t → 1 C = −0.2247 and ρ = 0.5181(− ln t). On the other hand, for t → 0 we find C = −0.3660 and ρ 0 = 0.3734.
We show ρ 0 and C(t) of the chiral f ′ -wave in Fig.2 a) and b) respectively. Note that C(t) is the same for three chiral states (chiral p-wave, chiral f-wave and chiral f ′ -wave) as well as chiral p-wave studied in [18] Therefore for the magnetic field H b ′ , the chiral f ′ -wave have the largest H c2 (t) if we assume T c and v, v c are the same. Also H c2 (t) of these states are closest to the observation.
Upper critical field for H a. -
A. Simple p-wave SC: ∆ k = sgn (k a ). The equation for H c2 (t) is given by [16] and can be written as in eq.3 with K 1 = e −ρu 2 |s| 2 1 + 2Cρ 2 u 4 |s| 4 (14)
K 2 = e −ρu 2 |s| 2 ρ 2 u 4 |s| 4 + C 1 − 8ρu 2 |s| 2 + 12ρ 2 u 4 |s| 4 − 16 3 ρ 3 u 6 |s| 6 + 2 3 ρ 4 u 8 |s| 8(15)
where t = T Tc , ρ = vav b eHc2(t)
2(2πT ) 2
and s = sin χ 1 + ı sin χ 2 with χ 1 = b k and χ 2 = c k.
Then for t → 1, we find C = − 93ζ (5) Both h(t) and C(t) are evaluated numerically and we show them in Fig. 3 a) and b) respectively. Now H c2 (t) is determined by a similar set of equations as in sec. 1.B. In particular we find for t → 1 C = −0.027735 and ρ = 0.212598(ln t) while for t → 0 C = −0.067684 and ρ 0 = 0.139672. We obtain h(0) = 0.6566. We show h(t) and C(t) in Fig. 3 a) and b) respectively.
C. Chiral f-wave SC:∆(k) ∼ds cos χ 1 . Again we use a similar set of equations as those discussed in sec. 1.C, we find for t → 1 C = −0.0356236 and ρ = 0.2744495(ln t) while for t → 0 C = 0.066 and ρ 0 = 0.1920 and h(0) = 0.6997. Both h(t) and C(t) are evaluated numerically and shown in Fig. 3 a) and b).
D. Chiral f ′ -wave SC:∆(k) ∼ds cos χ 2 . Now we find for t → 1 C = −0.05 and ρ = −0.2910(ln t), while for t → 0 C = −0.1019 and ρ 0 = 0.2090.
We have shown again h(t) and C(t) in Fig. 3 a) and b) respectively. Comparing these results with H c2 (T ) from (TMTSF) 2 PF 6 and (TMTSF) 2 ClO 4 [4,5], we can conclude both H b ′ and H a the chiral f ′ -wave SC is most consistent with experimental data. In particular these states have relatively large h(0) (see Table I).On the other hand almost the same H c2 (0) for H b ′ and H a has to be still accounted.
Nodal lines in ∆( k). -We have seen that from the temperature dependence of H c2 (T ), we deduce the chiral f-wave and chiral f ′ are the most favourable. They have nodal lines on the Fermi surface (i.e. the χ 1 − χ 2 plane), the chiral f-wave SC at χ 1 = ± π 2 , while chiral f ′ -wave SC at χ 2 = ± π 2 . These nodal lines may be detected if the nuclear spin relaxation rate is measured in a magnetic field rotated within the b ′ − c * plane.
Following the standard procedure given in [14] the quasiparticle density of states in the vortex state for T << T c and E = 0 is given by where χ 10 is the position of the nodal line on the χ 10 axis. So for the chiral f-wave SC we find χ 10 = π 2 and N 0, H exhibits the simple angular dependence. On the other hand when nodal lines are on the χ 2 axis, the θ dependence will be too small to see. Finally this gives
N 0, H = 2 π 2 v 2 √ eH 1 + cos θ 2 sin χ 10 2 1 2(16)T − 1 1 H /T −1 1N = 2 π 2 2 v 2 (eH) 1 + cos θ 2(17)
for the chiral f-wave SC. We show the θ dependence of T −1 1 in Fig. for a few candidates. The chiral f-wave SC has the strongest θ dependence (solid line) while the chiral h-wave SC (dashed line) and the chiral p-wave SC (dotted line) have a similar θ dependence.
Concluding remarks. -We have computes the upper critical field of Bechgaard salts for a variety of model superconductors with the standard microscopic theory. We find: a) Assuming all these superconductors have the same T c , the chiral f ′ -wave SC ( ∆(k) ∼ 1 √ 2 sgn(K a ) + ı sin χ 2 cos χ 2 ) appears to be the most favourable with largest H c2 's for both H b ′ and H a; b) however, non of these states exhibit the quasi T linear dependence of H c2 (T ) as observed in [4]; c) Also the present theory predicts H c2 (0) ∼ (v v c ) −1 and (v b v c ) −1 for H b ′ and H a respectively. This means H c2 (0) for H a is about 5 time larger than the one for H b ′ contrary to observation; d) from H c2 (0) ∼ 5T and T c = 1.5K we can extract V 2 = √ v v c ∼ 1.5 * 10 4 cms −1 , consistent with the known values of v, v c . * * *
We thank Stuart Brown and Paul Chaikin for useful discussion on possible detection of the nodal structure of ∆( k)s in Bechgaard salts through NMR. We have benefited from discussion with Stephan Haas and David Parker. ADF acknowledges gratefully the financial support of Ministerio de Educación y Ciencia (Spain) (AP2003-1383) and Jaime Ferrer and Paco Guinea for useful discussion.
Fig. 1 -
1|∆( k)| of chiral f-wave and chiral f ′ -wave SC are sketched in a) and b) respectively.
f 1 and chiral f 2 respectively. Upper critical field for H b ′ . -In the following we neglect the spin component of ∆( k). Most likely the equal spin pairing is realised in Bechgaard salts as in Sr 2 RuO 4
Fig. 2 -
2Normalised Hc2(t) and C(t) for H b ′ are shown in a) and b) respectively. Here solid, dashed and dotted lines are chiral f ′ -wave, chiral p-wave and simple p-wave respectively.
508ζ(3) ρ and ρ = 2 7ζ(3) (− ln t) = 0.2377(− ln t). exp [α 0 + 2Cβ 0 ] = 0.1751209, where α 0 = − ln |s| 2 = 0.220051 and β 0 = − s 4 |s| 4 = 4 π − 1 = 0.0170. From these we obtain h(0) = 0.73673.
Fig. 3 -
3Normalised Hc2(t) and C(t) for H a are shown in a) and b) respectively. Here solid, dashed, dashed-dotted and dotted lines are chiral f ′ -wave, chiral f-wave, chiral p-wave and simple p-wave respectively. B. Chiral p-wave SC: ∆(k) ∼ 1 √ 2 sgn(k a ) + ı sin χ 2 .
Fig. 4 -
4The angle dependent nuclear spin relaxation rate for a few nodal superconductors is shown. (Chiral f-wave, chiral h-wave and chiral p-wave are represented in solid, dashed and dotted lines.)
Table I -
ISummary of results. Here ρ0(0) =v2 eH c2 (0)
2(2πTc ) 2 and h(0) =
H c2 (0)
∂H c2 (t)
∂t
|t=1
symmetry
C(0)
C(1)
− ∂ρ
∂t |t = 1
ρ0(0)
h(0)
p-wave
-0.031
0
0.2377
0.1583
0.6659
H b ′ chiral p-wave -0.2247 -0.3660
0.3838
0.2734 0.71324
chiral f ′ -wave -0.2247 -0.3660
0.5181
0.3734 0.72073
p-wave
-0.017
0
0.2377
0.1751
0,7366
H a chiral p-wave
-0.066
-0.028
0.2126
0.1396
0,6566
chiral f-wave
-0.066
-0.035
0.2744
0.1920
0.6997
chiral f ′ -wave -0.1019
-0.05
0.2910
0.2090
0,7182
. D Jerome, A Mazard, M Ribault, K Bechgaard, J. Phys (France) Lett. 95Jerome D., Mazard A., Ribault M. and Bechgaard K., J. Phys (France) Lett, 41 (1980) L95
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. I J Lee, P M Chaikin, M J Naughton, Phys. Rev. B. 63180502Lee I.J., Chaikin P.M. and Naughton M.J., Phys. Rev. B, 63 (2002) R180502
. J I Oh, M J Naughton, Phys. Rev. Lett. 9267001Oh J.I. and Naughton M.J., Phys. Rev. Lett., 92 (2004) 067001
. A M Clogston, Phys. Rev. Lett. 9266Clogston A.M., Phys. Rev. Lett., 9 (1967) 266
. B S Chandrasekhar, Appl. Phys. Lett. 17Chandrasekhar B.S., Appl. Phys. Lett., 1 (1962) 7
. I J Lee, S E Brown, W G Clark, M J Strouse, M J Naughton, W Kang, P M Chaikin, Phys. Rev. Lett. 8817004Lee I.J., Brown S.E., Clark W.G., Strouse M.J., Naughton M.J., Kang W. and Chaikin P.M., Phys. Rev. Lett., 88 (2002) 017004
. L P Gor'kov, Soviet Phys. JETP. 1059Gor'kov L.P., Soviet Phys. JETP, 10 (1960) 59
. I Luk'yanchuk, V P Mineev, Soviet Phys. JETP. 661168Luk'yanchuk I. and Mineev V.P., Soviet Phys. JETP, 66 (1987) 1168
. A G Lebed, JETP Lett. 44114Lebed A.G., JETP Lett, 44 (1986) 114
. N Depuis, G Mantambaux, C A R Sa De Melo, Phys. Rev. Lett. 702613Depuis N., Mantambaux G. and Sa de Melo C.A.R., Phys. Rev. Lett., 70 (1993) 2613
H Won, S Haas, D Parker, S Telang, A Vanyolos, K Maki, Lectures on the Physics of Highly Correlated Electron Systems IX. Melville789Won H., Haas S. Parker D., Telang S., Vanyolos A. and Maki K., Lectures on the Physics of Highly Correlated Electron Systems IX, AIP Conference Proceedings 789 (Melville 2005)
. K Maki, S Haas, D Parker, H Won, cond-mat/0504635Maki K., Haas S., Parker D. and Won H., cond-mat/0504635, (2005)
. H Won, K Maki, Europhys. Lett. 305927Phys. Rev. BWon H. and Maki K., Europhys. Lett., 30 (1995) 421 Phys. Rev. B, 53 (1996) 5927
. A A Abrikosov, Soviet Phys. JETP. 51174Abrikosov A.A., Soviet Phys. JETP, 5 (1957) 1174
. G F Wang, K Maki, Europhys. Lett. 4571Wang G.F. and Maki K., Europhys. Lett., 45 (1999) 71
|
[] |
[
"Reassigning the CaH + 1 1 Σ → 2 1 Σ vibronic transition with CaD +",
"Reassigning the CaH + 1 1 Σ → 2 1 Σ vibronic transition with CaD +",
"Reassigning the CaH + 1 1 Σ → 2 1 Σ vibronic transition with CaD +",
"Reassigning the CaH + 1 1 Σ → 2 1 Σ vibronic transition with CaD +"
] |
[
"John Condoluci \nSchool of Chemistry and Biochemistry\nGeorgia Institute of Technology\n30332AtlantaGAUSA\n",
"Smitha Janardan \nSchool of Chemistry and Biochemistry\nGeorgia Institute of Technology\n30332AtlantaGAUSA\n",
"Aaron T Calvin \nSchool of Chemistry and Biochemistry\nGeorgia Institute of Technology\n30332AtlantaGAUSA\n",
"René Rugango \nSchool of Chemistry and Biochemistry\nGeorgia Institute of Technology\n30332AtlantaGAUSA\n",
"Gang Shu \nSchool of Chemistry and Biochemistry\nGeorgia Institute of Technology\n30332AtlantaGAUSA\n",
"Kenneth R Brown \nSchool of Chemistry and Biochemistry\nGeorgia Institute of Technology\n30332AtlantaGAUSA\n\nSchool of Computational Science and Engineering\nGeorgia Institute of Technology\n30332AtlantaGAUSA\n\nSchool of Physics\nGeorgia Institute of Technology\n30332AtlantaGAUSA\n",
"John Condoluci \nSchool of Chemistry and Biochemistry\nGeorgia Institute of Technology\n30332AtlantaGAUSA\n",
"Smitha Janardan \nSchool of Chemistry and Biochemistry\nGeorgia Institute of Technology\n30332AtlantaGAUSA\n",
"Aaron T Calvin \nSchool of Chemistry and Biochemistry\nGeorgia Institute of Technology\n30332AtlantaGAUSA\n",
"René Rugango \nSchool of Chemistry and Biochemistry\nGeorgia Institute of Technology\n30332AtlantaGAUSA\n",
"Gang Shu \nSchool of Chemistry and Biochemistry\nGeorgia Institute of Technology\n30332AtlantaGAUSA\n",
"Kenneth R Brown \nSchool of Chemistry and Biochemistry\nGeorgia Institute of Technology\n30332AtlantaGAUSA\n\nSchool of Computational Science and Engineering\nGeorgia Institute of Technology\n30332AtlantaGAUSA\n\nSchool of Physics\nGeorgia Institute of Technology\n30332AtlantaGAUSA\n"
] |
[
"School of Chemistry and Biochemistry\nGeorgia Institute of Technology\n30332AtlantaGAUSA",
"School of Chemistry and Biochemistry\nGeorgia Institute of Technology\n30332AtlantaGAUSA",
"School of Chemistry and Biochemistry\nGeorgia Institute of Technology\n30332AtlantaGAUSA",
"School of Chemistry and Biochemistry\nGeorgia Institute of Technology\n30332AtlantaGAUSA",
"School of Chemistry and Biochemistry\nGeorgia Institute of Technology\n30332AtlantaGAUSA",
"School of Chemistry and Biochemistry\nGeorgia Institute of Technology\n30332AtlantaGAUSA",
"School of Computational Science and Engineering\nGeorgia Institute of Technology\n30332AtlantaGAUSA",
"School of Physics\nGeorgia Institute of Technology\n30332AtlantaGAUSA",
"School of Chemistry and Biochemistry\nGeorgia Institute of Technology\n30332AtlantaGAUSA",
"School of Chemistry and Biochemistry\nGeorgia Institute of Technology\n30332AtlantaGAUSA",
"School of Chemistry and Biochemistry\nGeorgia Institute of Technology\n30332AtlantaGAUSA",
"School of Chemistry and Biochemistry\nGeorgia Institute of Technology\n30332AtlantaGAUSA",
"School of Chemistry and Biochemistry\nGeorgia Institute of Technology\n30332AtlantaGAUSA",
"School of Chemistry and Biochemistry\nGeorgia Institute of Technology\n30332AtlantaGAUSA",
"School of Computational Science and Engineering\nGeorgia Institute of Technology\n30332AtlantaGAUSA",
"School of Physics\nGeorgia Institute of Technology\n30332AtlantaGAUSA"
] |
[] |
We observe vibronic transitions in CaD + between the 1 1 Σ and 2 1 Σ electronic states by resonance enhanced multiphoton photodissociation spectroscopy in a Coulomb crystal. The vibronic transitions are compared with previous measurements on CaH + . The result is a revised assignment of the CaH + vibronic levels and a disagreement with CASPT2 theoretical calculations by approximately 700 cm −1 .
|
10.1063/1.5016556
|
[
"https://arxiv.org/pdf/1705.01326v1.pdf"
] | 118,971,408 |
1705.01326
|
ac12af0cc6d7c973da215f351b7c026a72546747
|
Reassigning the CaH + 1 1 Σ → 2 1 Σ vibronic transition with CaD +
John Condoluci
School of Chemistry and Biochemistry
Georgia Institute of Technology
30332AtlantaGAUSA
Smitha Janardan
School of Chemistry and Biochemistry
Georgia Institute of Technology
30332AtlantaGAUSA
Aaron T Calvin
School of Chemistry and Biochemistry
Georgia Institute of Technology
30332AtlantaGAUSA
René Rugango
School of Chemistry and Biochemistry
Georgia Institute of Technology
30332AtlantaGAUSA
Gang Shu
School of Chemistry and Biochemistry
Georgia Institute of Technology
30332AtlantaGAUSA
Kenneth R Brown
School of Chemistry and Biochemistry
Georgia Institute of Technology
30332AtlantaGAUSA
School of Computational Science and Engineering
Georgia Institute of Technology
30332AtlantaGAUSA
School of Physics
Georgia Institute of Technology
30332AtlantaGAUSA
Reassigning the CaH + 1 1 Σ → 2 1 Σ vibronic transition with CaD +
(Dated: 4 May 2017)
We observe vibronic transitions in CaD + between the 1 1 Σ and 2 1 Σ electronic states by resonance enhanced multiphoton photodissociation spectroscopy in a Coulomb crystal. The vibronic transitions are compared with previous measurements on CaH + . The result is a revised assignment of the CaH + vibronic levels and a disagreement with CASPT2 theoretical calculations by approximately 700 cm −1 .
I. INTRODUCTION
Co-trapping molecular ions with Doppler-cooled atomic ions sympathetically cools the molecular motion to millikelvin temperatures, enabling studies in spectroscopy and reaction dynamics. [1][2][3][4][5] Cold ionic ensembles in a variety of platforms have proved useful for astrochemical identification [6][7][8] and studies of internal state distributions, 3,9 while co-trapping with Doppler-cooled ions offers advantages for probing the possible time variation of fundamental constants, [10][11][12] and performing quantum logic spectroscopy. [13][14][15][16] Ionic metal-hydrides like CaH + and MgH + are promising candidates for these applications 17 due to their large rotational constants and Doppler-cooled dissociation products.
To date, spectroscopy on CaH + is limited to two vibrational overtones, 18 four vibrational levels within the 2 1 Σ state, 19 a photodissociative electronic transition, 20 and single-ion quantum logic probes of rotational state. 16 The vibronic transitions of Ref. 19 were previously assigned according to theory. The observed transition frequencies agreed to within 50 cm −1 of theory, but the v = 0 → v = 0 transition was not observed. A similar issue was encountered for the isoelectronic KH neutral, where the first few vibronic lines were experimentally absent and KD spectroscopy was required in order to correctly assign the KH transitions. [21][22][23][24] Isotopic substitution changes the reduced mass but maintains the adiabatic electronic potential energy. The resulting shift in vibrational energy levels can be compared to theory to confirm peak assignments.
Here we apply this method to the spectroscopy of CaH + and CaD + . Vibronic transitions were calculated using a CASPT2 internuclear potential 25 and then compared to measured transition frequencies obtained by resonance enhanced multiphoton photodissociation spectroscopy (REMPD). Instead of observing the predicted shifts for deuterium substitution based on our previous assignment, this study reveals a 687 cm −1 disagreement a) Electronic mail: [email protected] in the electronic energy from CASPT2 and leads to a revised labeling of the CaH + and CaD + vibronic transitions.
II. METHODS
A. Experimental Setup
The experiment employed a manually-tuned, frequency-doubled Ti:Sapphire laser to probe the vibronic transitions of CaD + co-trapped with lasercooled Ca + in a heterogeneous Coulomb crystal. By measuring the Ca + fluorescence increase upon laser exposure, a spectrum of molecular dissociation rate was generated. The experiments take place in an ultrahigh vacuum chamber with a background pressure of 10 −10 torr and the ions are held in a linear Paul trap. CaD + is generated by reaction of D 2 with excited Ca + at pressures of 10 −8 torr. The Ca + is observed and laser cooled by the laser-induced fluorescence at 397 nm. A repump laser at 866 nm is also required to close the transition. Details of the experimental setup may be found in our previous work 19 on the vibronic spectroscopy of CaH + . The CASPT2 potential energy surface of CaH + guided our spectroscopic search for CaD + vibronic transitions. 25 To achieve the desired 24390 to 27100 cm −1 range, the mode-locked Ti:Sapphire laser was frequency-doubled by a BBO crystal before being sent along the trap axis. Each fluorescence measurement was taken for 8 ms after ten sets of alternating 200 µs delay and 200 µs exposure to the AOM-shuttered 20 mW Ti:Sapph beam. A fit of the fluorescence intensity to the total exposure time t to the exponential equation
CaD + CaH + 2 1 Σ 1 1 Σ v' = 4 v' = 3 v' = 2 v' = 1 v' = 0 v = 0 v' = 3 v' = 2 v' = 1 v' = 0 Ca + +D v =A t = A ∞ − (A ∞ − A 0 )e −Γ(λ)t(1)
yields the dissociation rate Γ as a function of wavelength λ. A ∞ and A 0 are the steady-state and initial fluorescence counts, respectively. Scans exhibiting first-order CaD + dissociation are presented in Fig. 2. Dissociation rate plotted as a function of frequency yielded the spectrum in Fig. 3. 26
B. Theoretical Model for Parameter Estimation
To compare theory and experiment, CaD + dissociation spectra were modeled using theoretical and experimental parameters. Calculation of the Einstein A for each transition gave a dissociation rate, and the total rate at each wavelength was obtained by summing all transitions covered by the laser linewidth. 19 Although the experimental CaD + dissociation is a multi-photon process, we used a first-order model by assuming the dissociation rate is much greater than the first excitation rate.
The spectra labeled "Theory" in Fig. 3 rely heavily on CASPT2 ab initio calculations: the potential energy surfaces, transition dipole moments µ v→v , and vibronic transition frequencies G 0→1 (v ) come from Ref. 25. Harmonic, ω 1 , and anharmonic, ωχ 1 , constants were generated using the internuclear potential curves and R.J.
LeRoy's open-source project, Level 8.2. 27 Relative peak heights were determined by assuming a thermal distribution of the rotational states in 1 1 Σ. The first 15 J rotational levels contribute >99.99% of the initial state population and dominate the shapes of the vibronic peaks.
During spectral simulation, the spectral density of the laser was held at a constant linewidth of 80 cm −1 to match the laser linewidth based on spectrometer measurements. The model's invariant peak intensity of 4.77 × 10 8 W/m 2 was calculated from the laser power and beam waist diameter at the trap center. The ab initio calculated energy of the 1 1 Σ ground vibrational state was used as a reference for all transitions due to lack of experimental data on the ground state potential.
III. RESULTS AND DISCUSSION
A. Comparison of Spectra Fig. 3 compares the REMPD spectra of CaH + and CaD + . As expected, the CaD + transition frequencies were more tightly-packed, owing to the decrease in ω e accompanying the increase in reduced mass. Complication arose when matching the ab initio predictions to experimental values: the theory, shifted by -50 cm −1 to agree with CaH + , systematically underestimated the CaD + vibronic transition frequencies by >100 cm −1 . If the assignments proposed in Ref. 19 were correct, this study would imply a 150 cm −1 isotopic shift of the electronic energy potential and further evidence the experimentally unobserved transition to the v = 0 state. The isotopic shift of the electronic energy level is large compared to the 10 cm −1 shifts seen in KH. 22,24 These spectral
Σ(v = 0) → 2 1 Σ(v = v )
. Frequencies are determined by fitting a convolution of laser linewidth and parameter-dependent level structure to laser-induced dissociation rates of CaH + and CaD + . The resulting experimental fit is compared to an ab-initio spectrum in Figure 3. The frequency in parentheses was unobserved but predicted by the CASPT2 model.
Previous anomalies prompted a reassignment of the vibrational energy levels within the 2 1 Σ manifold.
Experimental Fit Theory +687 cm
The new assignment of vibrational quantum numbers, shown in Table I, reflects a 687 cm −1 departure from ab initio calculations. This shift manifests as a 687 cm −1 increase in T (1) − E 0 . Roughly a vibrational quantum in CaH + , the revision is greater than the 100 -150 cm −1 error window afforded by the mismatch of calculated and measured dissociation asymptotes for Ca + (3d 1 ) and H(1s 1 ). This revised assignment, however, features observable vibrational peaks through v = 4 and good agreement for both isotopologues.
B. Experimental Parameters
With the new CaH + and CaD + peak assignments, we determined the spectroscopic constants of the excited state by fitting both theory and experimental values to a second-order model of vibrational energy levels. Vibronic transitions to v of the 2 1 Σ state were modeled with the equation
v v = T (1) + ω 1 (v + 1 2 ) − ωχ 1 (v + 1 2 ) 2 − E 0(2)
where T (1) is the potential minimum of the 2 1 Σ state, and E 0 is the zero-point energy of the ground state. The parameters T (1), ω 1 , and ωχ 1 for both CaH + and CaD + are varied to fit experimental data points by quadratic regression. Regression curves of theory predictions and experimental fits were plotted (see Fig. 4) to obtain the constants listed in Table II. Since CaH + ground state information is limited, 18 the E 0 energy is confined to the ab initio prediction. The spectroscopic constants agree reasonably with theory, however we notice a large shift of the excited state potential. To maintain the assumption that the internuclear potentials of CaH + and CaD + are similar, the energy surfaces of the ab initio-calculated 1 1 Σ and 2 1 Σ states must be separated by an additional 687 cm −1 .
This sizable departure from ab initio calculations is currently unexplained. Deviations could be due to the frozen core electron approximation of CASPT2 or nonadiabatic effects. The latter effect is expected to be small based on estimations from other diatomic hydrides. LiH vibrational levels show a 10 cm −1 isotopic electronic shift 28,29 and the KH, KD system also features corrections around 10 cm −1 . 22,24 The disparity in experimental and calculated peak heights may be explained after further study of electronic levels relevant to the dissociation pathway. In addition, the different rates for CaH + and CaD + dissociation may be due to an enhanced forward reaction rate for Ca + + H 2 due to the kinetic isotope effect depressing the observed dissociation rate. In our computations, the 102 cm −1 uncertainty in ω 1 absorbs the CaD + ωχ 1 constant. The deviations in peak position and relative peak heights from our simple model may be due to an unknown dissociation pathway and is a topic of future research. In particular, the shape of the v'=0 peak was not well matched by our simple model leading to a large uncertainty in its calculated position.
IV. CONCLUSION
The vibronic spectrum of CaD + was obtained by scanning a frequency-doubled Ti:Sapph laser over the frequencies predicted to excite 2 1 Σ vibrational modes before coupling to unbound electronic states. Collection of Ca + fluorescence allowed us to quantify the rate of CaD + dissociation and plot it against frequency at constant laser linewidth and intensity. The harmonic constant ω 1 = 537 ± 102 cm −1 and anharmonicity ωχ 1 = 0 ± 20 cm −1 of the 2 1 Σ state were extracted by fitting our experiments to a second-order vibrational energy expression, Eqn. 2. Comparison of the simulated experiment with CASPT2 predictions revealed a 687 cm −1 average deviation from standing theory. CaH + vibrational levels within 2 1 Σ were consequently reassigned by reducing the vibrational quantum number by one relative to the previous assignment. 19 To understand this disagreement, we require further information on the ground and excited electronic states of CaH + . A theoretical approach is to recalculate the molecular potentials with unfrozen core electrons. Experiments using mid-infrared spectroscopy to measure lower vibrational transitions of the ground-state can be combined with previous vibrational overtone data 18 to construct a 1 1 Σ potential energy. Probing higher-lying electronic states could offer insight into photodissociation rates and explain why they differ from theoretical expectations.
Future experiments will also include examination of the 2 1 Σ state with rotational resolution, improving the precision of the constants presented here.
FIG. 3 .
3Comparisons of the experimental and ab initio-predicted spectra for each isotopologue. The optimized parameters include G1(v ), B 1,v , and µ v→v . Relative heights of theoretical vibronic peaks are governed by CASPT2 transition dipole moments scaled to fit the experimental v = 2 peak, while peak shapes come from the Boltzmann rotational state distribution 298 K.
0 FIG. 1. Energy level diagram of the CaH + and CaD + transitions probed by applying a doubled Ti:Sapphire laser and observing resonance enhanced photodissociation. The photodissociation of dark CaD + ions in the crystal into trapped Ca + and free H can be observed in the crystal image and by an increase in 397 nm Ca + fluorescence signal.Ca + +H
ν (cm -1
)
2.0×10 2
4.0×10 2
6.0×10 2
8.0×10 2
2.5×10 4
2.6×10 4
2.7×10 4
2.8×10 4
5.0×10 4
5.2×10 4
5.4×10 4
396 nm
396 nm fit
392 nm
392 nm fit
A/A∞
0.92
0.94
0.96
0.98
1.00
1.02
Time (ms)
0
100
200
300
400
FIG. 2. Observed Ca + fluorescence of composite Coulomb
crystals exposed to two laser frequencies. An increase in flu-
orescence indicates resonance-enhanced dissociation of CaD +
into Ca + and D. Dissociation rates extracted from Eqn. 1 are
plotted against frequency in Fig. 3.
TABLE I .
IObserved transitions from 1 1
TABLE II .
IIMolecular constants for the 1 1 Σ → 2 1 Σ vibronic
transitions of CaH + and CaD + based on the revised peak
assignments. All values are in cm −1 .
CaH +
CaH +
CaD +
CaD +
Experimental CASPT2 Experimental CASPT2
ω1
795 ± 12
803 ± 3
537 ± 102
574 ± 1
ωχ1 11.5 ± 3.0 8.0 ± 0.7
0 ± 20
3.5 ± 0.1
T (1) 24978 ± 10 24226 ± 2 24973 ± 108 24223 ± 1
E0
-
739
-
526
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[] |
[
"An initial-boundary value problem of the general three-component nonlinear Schrödinger equation with a 4 × 4 Lax pair on a finite interval",
"An initial-boundary value problem of the general three-component nonlinear Schrödinger equation with a 4 × 4 Lax pair on a finite interval",
"An initial-boundary value problem of the general three-component nonlinear Schrödinger equation with a 4 × 4 Lax pair on a finite interval",
"An initial-boundary value problem of the general three-component nonlinear Schrödinger equation with a 4 × 4 Lax pair on a finite interval"
] |
[
"Zhenya Yan \nInstitute of Systems Science\nSchool of Mathematical Sciences\nKey Laboratory of Mathematics Mechanization\nAMSS\nChinese Academy of Sciences\n100190BeijingChina\n\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Zhenya Yan \nInstitute of Systems Science\nSchool of Mathematical Sciences\nKey Laboratory of Mathematics Mechanization\nAMSS\nChinese Academy of Sciences\n100190BeijingChina\n\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n"
] |
[
"Institute of Systems Science\nSchool of Mathematical Sciences\nKey Laboratory of Mathematics Mechanization\nAMSS\nChinese Academy of Sciences\n100190BeijingChina",
"University of Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Systems Science\nSchool of Mathematical Sciences\nKey Laboratory of Mathematics Mechanization\nAMSS\nChinese Academy of Sciences\n100190BeijingChina",
"University of Chinese Academy of Sciences\n100049BeijingChina"
] |
[] |
We investigate the initial-boundary value problem for the general three-component nonlinear Schrödinger (gtc-NLS) equation with a 4 × 4 Lax pair on a finite interval by extending the Fokas unified approach. The solutions of the gtc-NLS equation can be expressed in terms of the solutions of a 4 × 4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. Moreover, the relevant jump matrices of the RH problem can be explicitly found via the three spectral functions arising from the initial data, the Dirichlet-Neumann boundary data. The global relation is also established to deduce two distinct but equivalent types of representations (i.e., one by using the large k of asymptotics of the eigenfunctions and another one in terms of the Gelfand-Levitan-Marchenko (GLM) method) for the Dirichlet and Neumann boundary value problems. Moreover, the relevant formulae for boundary value problems on the finite interval can reduce to ones on the half-line as the length of the interval approaches to infinity. Finally, we also give the linearizable boundary conditions for the GLM representation.
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10.1093/imamat/hxab007
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[
"https://arxiv.org/pdf/1704.08561v1.pdf"
] | 119,651,267 |
1704.08561
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ab76a826d4544751aa958165e6141a199bc5ffae
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An initial-boundary value problem of the general three-component nonlinear Schrödinger equation with a 4 × 4 Lax pair on a finite interval
27 Apr 2017
Zhenya Yan
Institute of Systems Science
School of Mathematical Sciences
Key Laboratory of Mathematics Mechanization
AMSS
Chinese Academy of Sciences
100190BeijingChina
University of Chinese Academy of Sciences
100049BeijingChina
An initial-boundary value problem of the general three-component nonlinear Schrödinger equation with a 4 × 4 Lax pair on a finite interval
27 Apr 2017Riemann-Hilbert problemGeneral three-component nonlinear Schrödinger equationInitial- boundary value problemGlobal relationMaps between Dirichlet and Neumann problemsGelfand-Levitan- Marchenko representation
We investigate the initial-boundary value problem for the general three-component nonlinear Schrödinger (gtc-NLS) equation with a 4 × 4 Lax pair on a finite interval by extending the Fokas unified approach. The solutions of the gtc-NLS equation can be expressed in terms of the solutions of a 4 × 4 matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. Moreover, the relevant jump matrices of the RH problem can be explicitly found via the three spectral functions arising from the initial data, the Dirichlet-Neumann boundary data. The global relation is also established to deduce two distinct but equivalent types of representations (i.e., one by using the large k of asymptotics of the eigenfunctions and another one in terms of the Gelfand-Levitan-Marchenko (GLM) method) for the Dirichlet and Neumann boundary value problems. Moreover, the relevant formulae for boundary value problems on the finite interval can reduce to ones on the half-line as the length of the interval approaches to infinity. Finally, we also give the linearizable boundary conditions for the GLM representation.
Introduction
In the theory of integrable systems, the powerful inverse scattering transform (IST) [1][2][3] (also called nonlinear Fourier transform) was presented to analytically study the initial value problems of the integrable nonlinear wave equations starting from the spectral analysis of their associated systems of linear eigenvalue equations (also known as the Lax pair [4]). After that, some significant extensions of the IST were gradually developed. For instance, Deift and Zhou [5] developed the IST to present the nonlinear steepest descent method to explicitly explore the long-time asymptotics of the Cauchy problems of (1+1)-dimensional integrable nonlinear evolution equations in terms of RH problems. Fokas [6] extended the idea of the IST to put forward a unified method studying boundary value problems of both linear and integrable nonlinear PDEs with Lax pairs [7][8][9][10]. Especially, the Fokas' method can be used to study integrable nonlinear PDEs in terms of the simultaneous spectral analysis Email address: [email protected] of both parts of the Lax pairs and the global relations among spectral functions. This approach obviously differs from the standard IST in which the spectral analysis of only one part of the Lax pairs was considered [9]. The Fokas' unified method has been used to explore boundary value problems of some physically significant integrable nonlinear evolution equations (NLEEs) with 2 × 2 Lax pairs on the half-line and the finite interval (e.g., the nonlinear Schrödinger equation [6,[11][12][13][14], the sine-Gordon equation [15,16], the KdV equation [17], the mKdV equation [18][19][20], the derivative nonlinear Schrödinger equation [21,22], Ernst equations [23,24], and etc. [25][26][27][28][29][30][31]) and ones with 3 × 3 Lax pairs on the half-line and the finite interval (e.g., [32], the Degasperis-Procesi equation [33], the Sasa-Satsuma equation [34], the coupled nonlinear Schrödinger equations [35][36][37][38], and the Ostrovsky-Vakhnenko equation [39]).
To the best of our knowledge, there was no report on the initial-boundary value (IBV) problems of integrable NLEEs with 4 × 4 Lax pairs on the half-line or the finite interval before. The aim of this paper is to develop a methodology for analyzing the IBV problems for integrable NLEEs with 4 × 4 Lax pairs on a finite interval. The extension will contain some novelties from 2 × 2 and 3 × 3 to 4 × 4 matrix Lax pairs, but the two key steps of this method [6][7][8][9] In this paper, we will exhibit how steps (i) and (ii) can be actualized for the integrable general three-component nonlinear Schrödinger (gtc-NLS) equation with a 4 × 4 Lax pair [41] iq 1t + q 1xx − 2 α 11 |q 1 | 2 + α 22 |q 2 | 2 + α 33 |q 3 | 2 + 2 Re (α 12q1 q 2 + α 13q1 q 3 + α 23q2 q 3 ) q 1 = 0, iq 2t + q 2xx − 2 α 11 |q 1 | 2 + α 22 |q 2 | 2 + α 33 |q 3 | 2 + 2 Re (α 12q1 q 2 + α 13q1 q 3 + α 23q2 q 3 ) q 2 = 0, iq 3t + q 3xx − 2 α 11 |q 1 | 2 + α 22 |q 2 | 2 + α 33 |q 3 | 2 + 2 Re (α 12q1 q 2 + α 13q1 q 3 + α 23q2 q 3 ) q 3 = 0, (1) where the complex-valued vector fields q j = q j (x, t), j = 1, 2, 3 are the sufficiently smooth functions defined in the finite region Ω = {(x, t)|x ∈ [0, L], t ∈ [0, T ]}, with L > 0 being the length of the interval and T > 0 being the fixed finite time, the overbar denotes the complex conjugate, Re (·) denotes the real part, and the six
coefficients α ij 's (1 ≤ i ≤ j ≤ 3)
The gtc-NLS equations contain the group velocity dispersion (GVD, i.e., q jxx ), self-phase modulation (SPM, e.g., |q j | 2 q j ), cross-phase modulation (XPM, e.g., |q j | 2 q s , j = s), pair-tunneling modulation (PTM, e.g., q 2 jq s , j = s), and three-tunneling modulation (TTM, e.g., q 1q2 q 3 ). System (1) admits the distinct cases for the six parameters α ij , (1 ≤ i ≤ j ≤ 3) such as the three-component focusing NLS equation for α jj = −1 and α ij = 0 with i < j, the three-component defocusing NLS equation for α jj = 1 and α ij = 0 with i < j, the three-component mixed NLS equation for (α 11 = −1, α 22 = α 33 = 1) or (α 11 = 1, α 22 = α 33 = −1) and α ij = 0 with i < j, and other general three-component NLS equation. Recently, the three-component defocusing NLS equation with nonzero boundary conditions was studied via the IST [40].
We would like to investigate the gtc-NLS equation (1) with the initial-boundary value problems Initial conditions : q j (x, t = 0) = q 0j (x), j = 1, 2, 3, Dirichlet boundary conditions : q j (x = 0, t) = u 0j (t), q j (x = L, t) = v 0j (t), j = 1, 2, 3, Neumann boundary conditions : q jx (x = 0, t) = u 1j (t), q jx (x = L, t) = v 1j (t), j = 1, 2, 3,
where the initial data q 0j (x), (j = 1, 2, 3), and Dirichlet and Neumann boundary data u 0j (t), v 0j (t) and
u 1j (t), v 1j (t), j = 1, 2, 3 are sufficiently smooth and compatible at points (x, t) = (0, 0), (L, 0), respectively. The rest of this paper is organized as follows. In Sec. 2, we investigate the spectral analysis of the associated representation of the eigenfunctions in terms of the global relation. Moreover, we also show that the GLM representation is equivalent to one in Sec. 4. Finally, we also give the linearizable boundary conditions for the GLM representation.
2 The spectral analysis of a 4 × 4 Lax pair
The exact one-form
The gtc-NLS system (1) can be regarded as the compatible condition of a 4 × 4 Lax pair [41] ψ x + ikσ 4 ψ = U (x, t)ψ,
ψ t + 2ik 2 σ 4 ψ = V (x, t, k)ψ,(4)
where ψ = ψ(x, t, k) is a complex 4×4 matrix-valued or 4 × 1 column vector-valued eigenfunction, k ∈ C is an iso-spectral parameter, σ 4 = diag(1, 1, 1, −1), and the 4 × 4 matrices U and V are defined by
U (x, t) = 0 0 0 q 1 (x, t) 0 0 0 q 2 (x, t) 0 0 0 q 3 (x, t) p 1 (x, t) p 2 (x, t) p 3 (x, t) 0 , V (x, t, k) = 2kU (x, t) + V 0 (x, t),(5)
with p 1 (x, t) = α 11q1 +ᾱ 12q2 +ᾱ 13q3 , p 2 (x, t) = α 12q1 + α 22q2 +ᾱ 23q3 , p 3 (x, t) = α 13q1 + α 23q2 + α 33q3 , and
V 0 (x, t) = −i(U x + U 2 )σ 4 = −i q 1 p 1 q 1 p 2 q 1 p 3 −q 1x
q 2 p 1 q 2 p 2 q 2 p 3 −q 2x q 3 p 1 q 3 p 2 q 3 p 3 −q 3x p 1x p 2x p 3x −(q 1 p 1 + q 2 p 2 + q 3 p 3 ) Define a new eigenfunction µ(x, t, k) by µ(x, t, k) = ψ(x, t, k)e i(kx+2k 2 t)σ4 ,
,(6)
such that the Lax pair (4) becomes the equivalent form for µ(x, t, k)
µ x + ik[σ 4 , µ] = U (x, t)µ, µ t + 2ik 2 [σ 4 , µ] = V (x, t, k)µ,(8)
where [σ 4 , µ] ≡ σ 4 µ − µσ 4 . Letσ 4 denote the commutator with respect to σ 4 and the operator acting on a 4 × 4 matrix X byσ 4 X = [σ 4 , X] such that eσ 4 X = e σ4 Xe −σ4 , then the Lax pair (8) can be written as a full derivative form
d e i(kx+2k 2 t)σ4 µ(x, t, k) = W (x, t, k),(9)
where the exact one-form W (x, t, k) is of the form W (x, t, k) = e i(kx+2k 2 t)σ4 [U (x, t)dx + V (x, t, k)dt]µ(x, t, k).
The definition and boundedness of eigenfunctions µ ′ j s
For any point (x, t) in the region Ω = {(x, t)|x ∈ [0, L], t ∈ [0, T ]}, let {γ j } 4 1 be four contours connecting fours vertexes (x 1 , t 1 ) = (0, T ), (x 2 , t 2 ) = (0, 0), (x 3 , t 3 ) = (L, 0), (x 4 , t 4 ) = (L, T ) to (x, t), respectively (see Fig. 1). Therefore we get the following inequalities on these contours:
γ 1 : (0, T ) → (x, t), x − x ′ ≥ 0, t − τ ≤ 0, γ 2 : (0, 0) → (x, t), x − x ′ ≥ 0, t − τ ≥ 0, γ 3 : (L, 0) → (x, t), x − x ′ ≤ 0, t − τ ≥ 0, γ 4 : (L, T ) → (x, t), x − x ′ ≤ 0, t − τ ≤ 0,(11)
By means of the Volterra integral equations, it follows from Eqs. (9) and (10)
e −i(kx+2k 2 t)σ4 W j (x ′ , τ, k),(12)
where I = diag(1, 1, 1, 1), the integral is over a piecewise smooth curve from (x j , t j ) to (x, t), and W j (x, t, k)
is given by Eq. (10) with µ(x, t, k) replaced by µ j (x, t, k). Since the one-form W j are closed, thus µ j are independent of the path of integration. If we take the paths of integration to be parallel to the x and t axes, then the integral Eq. (12) reduces to
µ j (x, t, k) = I + x xj e −ik(x−x ′ )σ4 (U µ j )(x ′ , t, k)dx ′ + e −ik(x−xj )σ4 t tj e −2ik 2 (t−τ )σ4 (V µ j )(x j , τ, k)dτ,(13)
It follows from Eq. (13) that the four columns of the matrix µ j (x, t, k) contain the following exponentials [µ j ] s : e 2ik(x−x ′ )+4ik 2 (t−τ ) , j = 1, 2, 3, 4; s = 1, 2, 3, (14a)
[µ j ] 4 : e −2ik(x−x ′ )−4ik 2 (t−τ ) , e −2ik(x−x ′ )−4ik 2 (t−τ ) , e −2ik(x−x ′ )−4ik 2 (t−τ ) , j = 1, 2, 3, 4(14b)
To analyze the bounded regions of the eigenfunctions {µ j } 4 1 , we need to use the curve {k ∈ C|(Re f (k))(Re g(k)) = 0, f (k) = ik, g(k) = ik 2 } to separate the complex k-plane into four regions (see Fig. 2):
D 1 = {k ∈ C | Re f (k) < 0 and Re g(k) < 0}, D 2 = {k ∈ C | Re f (k) < 0 and Re g(k) > 0}, D 3 = {k ∈ C | Re f (k) > 0 and Re g(k) < 0}, D 4 = {k ∈ C | Re f (k) > 0 and Re g(k) > 0},(15)
which implies that D 1 and D 3 (D 2 and D 4 ) are symmetric about the origin.
Thus it follows from Eqs. (11), (14) and (15) that the regions, where the different columns of eigenfunctions {µ j } 4 1 are bounded and analytic in the complex k-plane, are presented below:
µ 1 : (f − ∩ g + , f − ∩ g + , f − ∩ g + , f + ∩ g − ) =: (D 2 , D 2 , D 2 , D 3 ), µ 2 : (f − ∩ g − , f − ∩ g − , f − ∩ g − , f + ∩ g + ) =: (D 1 , D 1 , D 1 , D 4 ),µ 3 : (f + ∩ g − , f + ∩ g − , f + ∩ g − , f − ∩ g + ) =: (D 3 , D 3 , D 3 , D 2 ),µ 4 : (f + ∩ g + , f + ∩ g + , f + ∩ g + , f − ∩ g − ) =: (D 4 , D 4 , D 4 , D 1 ),(16)
where f + =: Re f (k) > 0, f − =: Re f (k) < 0, g + =: Re g(k) > 0, and g − =: Re g(k) < 0.
The definition of the new matrix-valued functions M n 's
To construct the jump matrix in a RH problem, we introduce the solutions M n (x, t, k) of Eq. (8)
(M n ) sj (x, t, k) = δ sj + (γ n )sj e −i(kx+2k 2 t)σ4 W n (x ′ , τ, k) sj , k ∈ D n , s, j = 1, 2, 3, 4,(17)
where W n (x, t, k) isdefined by Eq. (10) with µ(x, t, k) replaced with M n (x, t, k), and the contours (γ n ) sj 's are given by
(γ n ) sj = γ 1 , if Re f s (k) > Re f j (k) and Re g s (k) ≤ Re g j (k), γ 2 , if Re f s (k) > Re f j (k) and Re g s (k) > Re g j (k), γ 3 , if Re f s (k) ≤ Re f j (k) and Re g s (k) ≥ Re g j (k), γ 4 , if Re f s (k) ≤ Re f j (k) and Re g s (k) ≤ Re g j (k),(18)for k ∈ D n , where f 1,2,3 (k) = −f 4 (k) = −ik, g 1,2,3 (k) = −g 4 (k) = −2ik 2 .
Notice that to distinguish (γ n ) sj 's to be the contour γ 3 or γ 4 for the special cases, Re f s (k) = Re f j (k) and Re g s (k) = Re g j (k), we choose them in these cases as γ 3 (or γ 4 ) which must appear in the matrix γ n , otherwise, we choose them in all these cases as the same γ 3 (or γ 4 ).
The definition (18) of (γ n ) sj implies that γ n (n = 1, 2, 3, 4) are explicitly given by
γ 1 = γ 4 γ 4 γ 4 γ 2 γ 4 γ 4 γ 4 γ 2 γ 4 γ 4 γ 4 γ 2 γ 4 γ 4 γ 4 γ 4 , γ 2 = γ 3 γ 3 γ 3 γ 1 γ 3 γ 3 γ 3 γ 1 γ 3 γ 3 γ 3 γ 1 γ 3 γ 3 γ 3 γ 3 , γ 3 = γ 3 γ 3 γ 3 γ 3 γ 3 γ 3 γ 3 γ 3 γ 3 γ 3 γ 3 γ 3 γ 1 γ 1 γ 1 γ 3 , γ 4 = γ 4 γ 4 γ 4 γ 4 γ 4 γ 4 γ 4 γ 4 γ 4 γ 4 γ 4 γ 4 γ 2 γ 2 γ 2 γ 4 ,(19)
Proof. Similar to the proof for the 3 × 3 Lax pair in [32], we can also proof the bounedness and analyticity of M n . The substitution of
µ(x, t, k) = M n (x, t, k) = M (0) n (x, t, k) + ∞ j=1 M (j) n (x, t, k) k j , k → ∞,
into the x-part of the Lax pair (8) yields Eq. (20).
The above-defined matrix-valued functions M n 's can be used to formulate a 4 × 4 matrix Riemann-Hilbert problem.
The spectral functions and jump matrices
We introduce the spectral functions S n (k) (n = 1, 2, 3, 4) by S n (k) = M n (x = 0, t = 0, k), k ∈ D n , n = 1, 2, 3, 4.
Let M (x, t, k) denote the sectionally analytic function on the Riemann k-spere which is equivalent to M n (x, t, k)
for k ∈ D n . Then M (x, t, k) solves the jump equations
M n (x, t, k) = M m (x, t, k)J mn (x, t, k), k ∈D n ∩D m , n, m = 1, 2, 3, 4, n = m,(22)
with the jump matrices J mn (x, t, k) defined by
J mn (x, t, k) = e −i(kx+2k 2 t)σ4 (S −1 m (k)S n (k)).(23)
The minors or the transpose of the adjugates of eigenfunctions
To conveniently calculate the spectral functions S n (k) in the following sections, we need to use the cofactor matrix X A (or the transpose of the adjugate) of a 4 × 4 matrix X defined as
adj(X) T = X A = m 11 (X) −m 12 (X) m 13 (X) −m 14 (X) −m 21 (X) m 22 (X) −m 23 (X) m 24 (X) m 31 (X) −m 32 (X) m 33 (X) −m 34 (X) −m 41 (X) m 42 (X) −m 43 (X) m 44 (X) ,
where m ij (X) denotes the (ij)th minor of X and (X A ) T X = adj(X)X = det X.
It follows from the Lax pair (4) that the eigenfunction {µ A j } 4 1 of the matrices {µ j (x, t, k)} 4 1 satisfy the Lax equation
µ A x − ik[σ 4 , µ A ] = −U T (x, t)µ A , µ A t − 2ik 2 [σ 4 , µ A ] = −V T (x, t, k)µ A ,(24)
whose solutions can be written as the form
µ A j (x, t, k) = I − γj e i[k(x−x ′ )+2k 2 (t−τ )]σ4 U T (x ′ , τ )dx ′ + V T (x ′ , τ, k)dτ µ A j (x ′ , τ, k), j = 1, 2, 3, 4,(25)
in terms of the Volterra integral equations. It is easy to check that the regions of boundedness of µ A j :
µ A 1 (x, t, k) is bounded for k ∈ (D 3 , D 3 , D 3 , D 2 ), µ A 2 (x, t, k) is bounded for k ∈ (D 4 , D 4 , D 4 , D 1 ), µ A 3 (x, t, k) is bounded for k ∈ (D 2 , D 2 , D 2 , D 3 ), µ A 4 (x, t, k) is bounded for k ∈ (D 1 , D 1 , D 1 , D 4 ).
which are symmetric ones of µ j about the Re k-axis (cf. Eq. (16)).
Symmetries of eigenfunctions
LetǓ in the Lax pair (4). Then we have
(x, t, k) = −ikσ 4 + U (x, t),V (x, t, k) = −2ik 2 σ 4 + V (x, t, k).(26)PǓ (x, t,k)P = −Ǔ (x, t, k) T , PV (x, t,k)P = −V (x, t, k) T ,(27)
where the symmetric matrix P is taken as
P = α 11ᾱ12ᾱ13 0 α 12 α 22ᾱ23 0 α 13 α 23 α 33 0 0 0 0 −1 , P 2 = I, P = P † ,(28)
Notice that the symmetric matrix P used here differs from the diag ones used in 3 × 3 Lax pairs [35][36][37][38]. Similar to the proof in Ref. [14], based on Eq. (24) and (27) we have the following proposition:
Proposition 2.2.
The matrix-valued eigenfunctions ψ(x, t, k) of the Lax pair (4) and µ j (x, t, k) of the Lax pair (8) both possess the same symmetric relations
ψ −1 (x, t, k) = P ψ(x, t,k) T P, µ −1 j (x, t, k) = P µ j (x, t,k) T P, j = 1, 2, 3, 4,(29)
Moreover, In the domains where µ j is bounded, we have
µ j (x, t, k) = I + O 1 k , k → ∞, j = 1, 2, 3, 4(30)
and
det[µ j (x, t, k)] = 1, j = 1, 2, 3, 4(31)
since the traces of the matrices U (x, t, k) and V (x, t, k) are zero.
The relations between spectral functions and jump matrices J mn
Since these functions µ j are dependent, thus we can define three 4 × 4 matrix-valued functions S(k), s(k) and S(k) between µ 2 and µ j , j = 1, 3, 4 in the form (cf. Fig. 3)
µ 1 (x, t, k) = µ 2 (x, t, k)e −i(kx+2k 2 t)σ4 S(k), µ 3 (x, t, k) = µ 2 (x, t, k)e −i(kx+2k 2 t)σ4 s(k), µ 4 (x, t, k) = µ 2 (x, t, k)e −i(kx+2k 2 t)σ4 S(k),(32)
Evaluating system (32) at (x, t) = (0, 0) and the three equations in system (32) at (x, t) = (0, T ), (L, 0), (L, T ),
respectively, we have S(k) = µ 1 (0, 0, k) = e 2ik 2 Tσ4 µ −1 2 (0, T, k), s(k) = µ 3 (0, 0, k) = e ikLσ4 µ −1 2 (L, 0, k), S(k) = µ 4 (0, 0, k) = e i(kL+2k 2 T )σ4 µ −1 2 (L, T, k),(33)
Except for the defined three relations, it follows from Eqs. (32) and (33) that we can find other three relations:
(i) the relation between µ 3 (x, t, k) and µ 4 (x, t, k)
µ 4 (x, t, k) = µ 3 (x, t, k)e −i[k(x−L)+2k 2 (t−T )]σ4 µ −1 3 (L, T, k) = µ 3 (x, t, k)e −i[k(x−L)+2k 2 t]σ4 S L (k), with S L (k) = µ 4 (L, 0, k) = e 2ik 2 Tσ4 µ −1 3 (L, T, k),(34)
(ii) the relation between µ 1 (x, t, k) and µ 4 (x, t, k)
µ 3 (x, t, k) = µ 1 (x, t, k)e −i(kx+2k 2 t)σ4 S(k), with S(k) = S −1 (k)s(k),(35)
and (iii) the relation between µ 1 (x, t, k) and µ 4 (x, t, k)
µ 4 (x, t, k) = µ 1 (x, t, k)e −i(kx+2k 2 t)σ4 s T (k), with s T (k) = S −1 (k)S(k),(36)
It follows from Eqs. (33) and (34) that we have the relation
S(k) = s(k)e ikLσ4 S L (k),(37)
The map of these relations among µ j is exhibited in Fig. 3. According to the definition (13) of µ j , Eq. (33) and (34) imply that
s(k) = I − L 0 e ikx ′σ 4 (U µ 3 )(x ′ , 0, k)dx ′ = I + L 0 e ikx ′σ 4 (U µ 2 )(x ′ , 0, k)dx ′ −1 , S(k) = I − T 0 e 2ik 2 τσ4 (V µ 1 )(0, τ, k)dx ′ = I + T 0 e 2ik 2 τσ4 (V µ 2 )(0, τ, k)dτ −1 , S L (k) = I − T 0 e 2ik 2 τσ4 (V µ 4 )(L, τ, k)dτ = I + T 0 e 2ik 2 τσ4 (V µ 3 )(L, τ, k)dτ −1 , S(k) = I − L 0 e ikx ′σ 4 (U µ 4 )(x ′ , 0, k)dx ′ − e ikLσ4 T 0 e 2ik 2 τσ4 (V µ 4 )(L, τ, k)dτ = I + e 2ik 2 Tσ4 L 0 e ikx ′σ 4 (U µ 2 )(x ′ , T, k)dx ′ + T 0 e 2ik 2 τσ4 (V µ 2 )(0, τ, k)dτ −1 ,(38)
which leads to S(k) and s T (k) in terms of Eqs. (35) and (36), where µ j2 (0, t, k), j 2 = 1, 2, µ j3 (L, t, k), j 3 = 3, 4, µ j1 (x, 0, k), j 1 = 2, 3, 4, µ 2 (x, T, k), 0 < x < L, 0 < t < T are defined by the integral equations
µ 1 (0, t, k) = I + t T e −2ik 2 (t−τ )σ4 (V µ 1 )(0, τ, k)dτ, µ 2 (0, t, k) = I + t 0 e −2ik 2 (t−τ )σ4 (V µ 2 )(0, τ, k)dτ, µ 3 (L, t, k) = I + t 0 e −2ik 2 (t−τ )σ4 (V µ 3 )(L, τ, k)dτ, µ 4 (L, t, k) = I + t T e −2ik 2 (t−τ )σ4 (V µ 4 )(L, τ, k)dτ, µ 2 (x, 0, k) = I + x 0 e ikx ′σ 4 (U µ 2 )(x ′ , 0, k)dx ′ , µ 3 (x, 0, k) = I + x L e ikx ′σ 4 (U µ 3 )(x ′ , 0, k)dx ′ , µ 4 (x, 0, k) = I + x L e ikx ′σ 4 (U µ 4 )(x ′ , 0, k)dx ′ − e −ik(x−L)σ4 T 0 e 2ik 2 τσ4 (V µ 4 )(L, τ, k)dτ, µ 2 (x, T, k) = I + x 0 e −ik(x−x ′ )σ4 (U µ 2 )(x ′ , T, k)dx ′ + e −ikxσ4 T 0 e −2ik 2 (T −τ )σ4 (V µ 2 )(0, τ, k)dτ,
It follows from the properties of µ j and µ A j that the functions {S(k), s(k), S(k), S L (k)} and {S A (k), s A (k), S A (k), S A L (k)} have the following boundedness:
S(k) is bounded for k ∈ (D 2 ∪ D 4 , D 2 ∪ D 4 , D 2 ∪ D 4 , D 1 ∪ D 3 ), s(k) is bounded for k ∈ (D 3 ∪ D 4 , D 3 ∪ D 4 , D 3 ∪ D 4 , D 1 ∪ D 2 ), S(k) is bounded for k ∈ (D 4 , D 4 , D 4 , D 1 ), S L (k) is bounded for k ∈ (D 2 ∪ D 4 , D 2 ∪ D 4 , D 2 ∪ D 4 , D 1 ∪ D 3 ), S A (k) is bounded for k ∈ (D 1 ∪ D 3 , D 1 ∪ D 3 , D 1 ∪ D 3 , D 2 1 ∪ D 4 ), s A (k) is bounded for k ∈ (D 1 ∪ D 2 , D 1 ∪ D 2 , D 1 ∪ D 2 , D 3 ∪ D 4 ), S A (k) is bounded for k ∈ (D 2 , D 2 , D 2 , D 3 ), S A L (k) is bounded for k ∈ (D 2 ∪ D 4 , D 2 ∪ D 4 , D 2 ∪ D 4 , D 1 ∪ D 3 ), Proposition 2.3. The matrix-valued functions S n (x, t, k) (n = 1, 2, 3, 4) defined by M n (x, t, k) = µ 2 (x, t, k)e −i(kx+2k 2 t)σ4 S n (k), k ∈ D n ,(39)
with M n given by Eq. (17) can be determined by the entries of the data S(k) = (S ij ) 4×4 , s(k) = (s ij ) 4×4 , and S(k) = (S ij ) 4×4 given by Eq. (33) as follows:
S 1 (k) = S 11 S 12 S 13 0 S 21 S 22 S 23 0 S 31 S 32 S 33 0 S 41 S 42 S 43 1 m 44 (S) , S 2 (k) = s 11 s 12 s 13 S 14 (S T s A ) 44 s 21 s 22 s 23 S 24 (S T s A ) 44 s 31 s 32 s 33 S 34 (S T s A ) 44 s 41 s 42 s 43 S 44 (S T s A ) 44 , S 3 (k) = 0 S 44 ,(40)
where n i1j1,i2j2 (X) denotes the determinant of the sub-matrix generated by choosing the cross elements of i 1,2 th rows and j 1,2 th columns of X, and Proof. We introduce the matrix-valued functions R n (k), S n (k), T n (k), and P n (k), n = 1, 2, 3, 4) by M n (x, t, k) and µ j (x, t, k)
S( M n (x, t, k) = µ 1 (x, t, k)e −i(kx+2k 2 t)σ4 R n (k), M n (x, t, k) = µ 2 (x, t, k)e −i(kx+2k 2 t)σ4 S n (k), M n (x, t, k) = µ 3 (x, t, k)e −i(kx+2k 2 t)σ4 T n (k), M n (x, t, k) = µ 4 (x, t, k)e −i(kx+2k 2 t)σ4 P n (k),(41)
It follows from Eq. (41) that we have the relations
R n (k) = e 2ik 2 Tσ4 M n (0, T, k), S n (k) = M n (0, 0, k), T n (k) = e ikLσ4 M n (L, 0, k), P n (k) = e i(kL+2k 2 T )σ4 M n (L, T, k),(42)
and
S(k) = µ 1 (0, 0, k) = S n (k)R −1 n (k), s(k) = µ 3 (0, 0, k) = S n (k)T −1 n (k), S(k) = µ 4 (0, 0, k) = S n (k)P −1 n (k),(43)
which can in general obtain the functions {R n , S n , T n , P n } for the given functions {s(k), S(k), S(k)}. Moreover, we can also determine some entries of {R n , S n , T n , P n } in terms of Eqs. (17) and (41)
(R n (k)) ij = 0, if (γ n ) ij = γ 1 , (S n (k)) ij = 0, if (γ n ) ij = γ 2 , (T n (k)) ij = δ ij , if (γ n ) ij = γ 3 , (P n (k)) ij = δ ij , if (γ n ) ij = γ 4 ,(44)
Thus it follows from systems (43) and (44) that we can find Eq. (40). We introduce the above possible zeros by {k j } N 1 and suppose that they satisfy the following assumption.
Assumption 2.4. We assume that
• m 44 (S)(k) has n 1 possible simple zeros in D 1 denoted by {k j } n1 1 ;
• (S T s A ) 44 (k) has n 2 − n 1 possible simple zeros in D 2 denoted by {k j } n2 n1+1 ;
• (s T S A ) 44 (k) has n 3 − n 2 possible simple zeros in D 3 denoted by {k j } n3 n2+1 ;
• S 44 (k) has N − n 3 possible simple zeros in D 4 denoted by {k j } N n3+1 ; and that none of these zeros coincide. Moreover, none of these functions are assumed to have zeros on the boundaries od the D n 's (n = 1, 2, 3, 4).
We can deduce the residue conditions at these zeros in the following expressions:
Proposition 2.5.
Let {M n } 4 1 be the eigenfunctions given by Eq. (17) and suppose that the set {k j } N 1 of singularities is as the above-mentioned Assumption 2.4. Then we have the following residue conditions for M n :
Res k=kj [M 1 ] 4 = n 12,23 (S)(k j )[M 1 (k j )] 1 −n 11,23 (S)(k j )[M 1 (k j )] 2 +n 11,22 (S)(k j )[M 1 (k j )] 3 m 44 (S)(k j )m 34 (S)(k j ) e 2θ(kj ) , for 1 ≤ j ≤ n 1 , k ∈ D 1 ,(45)Res k=kj [M 2 ] 4 = [M 2 (k j )] 1 [S 14 (k j )n 22,43 (s)(k j )−S 24 (k j )n 12,43 (s)(k j )+S 44 (k j )n 12,23 (s)(k j )] (S T s A ) 44 (k j )m 34 (s)(k j )e −2θ(kj ) − [M 2 (k j )] 2 [S 14 (k j )n 21,43 (s)(k j )−S 24 (k j )n 11,43 (s)(k j )+S 44 (k j )n 11,23 (s)(k j )] (S T s A ) 44 (k j )m 34 (s)(k j )e −2θ(kj) + [M 2 (k j )] 3 [S 14 (k j )n 21,42 (s)(k j )−S 24 (k j )n 11,42 (s)(k j )+S 44 (k j )n 11,22 (s)(k j )] (S T s A ) 44 (k j )m 34 (s)(k j )e −2θ(kj) , for n 1 + 1 ≤ j ≤ n 2 , k ∈ D 2 ,(46)Res k=kj [M 3 ] l = m 14 (S)(k j )n 4l,14 (s)(k j )−m 24 (S)(k j )n 4l,24 (s)(k j )+m 34 (S)(k j )n 4l,34 (s)(k j ) (s T S A ) 44 (k j )s 44 (k j )e 2θ(kj ) ×[M 3 k j )] 4 , for n 2 + 1 ≤ j ≤ n 3 , k ∈ D 3 , l = 1, 2, 3,(47)Res k=kj [M 4 ] l = − S 4l (k j ) S 44 (k j ) [M 4 (k j )] 4 e −2θ(kj) , for n 3 + 1 ≤ j ≤ N, k ∈ D 4 , l = 1, 2, 3,(48)
where the overdot stands for the derivative with resect to the parameter k and θ = θ(k) = −i(kx + 2k 2 t).
Proof. It follows from Eqs. (39) and (40) that the four columns of M 1 are given by the matrices µ 2 and S 1 (k)
[M 1 ] 1 = [µ 2 ] 1 S 11 + [µ 2 ] 2 S 21 + [µ 2 ] 3 S 31 + [µ 2 ] 4 S 41 e −2θ ,(49a)[M 1 ] 2 = [µ 2 ] 1 S 12 + [µ 2 ] 2 S 22 + [µ 2 ] 3 S 32 + [µ 2 ] 4 S 42 e −2θ ,(49b)[M 1 ] 3 = [µ 2 ] 1 S 13 + [µ 2 ] 2 S 23 + [µ 2 ] 3 S 33 + [µ 2 ] 4 S 43 e −2θ ,(49c)[M 1 ] 4 = [µ 2 ] 4 m 44 (S) ,(49d)
the four columns of M 2 are given by the matrices µ 2 and S 2 (k)
[M 2 ] 1 = [µ 2 ] 1 s 11 + [µ 2 ] 2 s 21 + [µ 2 ] 3 s 31 + [µ 2 ] 4 s 41 e −2θ ,(50a)[M 2 ] 2 = [µ 2 ] 1 s 12 + [µ 2 ] 2 s 22 + [µ 2 ] 3 s 32 + [µ 2 ] 4 s 42 e −2θ ,(50b)[M 2 ] 3 = [µ 2 ] 1 s 13 + [µ 2 ] 2 s 23 + [µ 2 ] 3 s 33 + [µ 2 ] 4 s 43 e −2θ ,(50c)[M 2 ] 4 = [µ 2 ] 1 S 14 (S T s A ) 44 e 2θ + [µ 2 ] 2 S 24 (S T s A ) 44 e 2θ + [µ 2 ] 3 S 34 (S T s A ) 44 e 2θ + [µ 2 ] 4 S 44 (S T s A ) 44 ,(50d)
the four columns of M 3 are given by the matrices µ 2 and S 3 (k)
[M 3 ] 1 = [µ 2 ] 1 S (11) 3 + [µ 2 ] 2 S (21) 3 + [µ 2 ] 3 S (31) 3 + [µ 2 ] 4 S (41) 3 e −2θ ,(51a)[M 3 ] 2 = [µ 2 ] 1 S (12) 3 + [µ 2 ] 2 S (22) 3 + [µ 2 ] 3 S (32) 3 + [µ 2 ] 4 S (42) 3 e −2θ ,(51b)[M 3 ] 3 = [µ 2 ] 1 S (13) 3 + [µ 2 ] 2 S (23) 3 + [µ 2 ] 3 S (33) 3 + [µ 2 ] 4 S (43) 3 e −2θ ,(51c)
For the case that k j ∈ D 1 is a simple zero of m 44 (S)(k), it follows from Eqs. (49a)-(49c) that we have [µ 2 ] j , j = 1, 2, 4 and then substitute them into Eq. (49d) to yield
[M 1 ] 4 = n 12,23 (S)[M 1 ] 1 − n 11,23 (S)[M 1 ] 2 + n 11,22 (S)[M 1 ] 3 m 34 (S)m 44 (S) e 2θ − [µ 2 ] 3 m 34 (S) e 2θ ,
whose residue at k j yields Eq. (45) for k j ∈ D 1 , respectively.
Similarly, we solve Eqs (50a)-(50c) for [µ 2 ] j , j = 1, 2, 4 and then substitute them into Eq (50d) to yield
The global relation
The definitions of the above-mentioned spectral functions S(k), s(k), S L (k), and S(k) imply that they are dependent. It follows from Eqs. (32) and (34) that
µ 4 (x, t, k) = µ 2 (x, t, k)e −i(kx+2k 2 t)σ4 S(k) = µ 2 (x, t, k)e −i(kx+2k 2 t)σ4 [s(k)e ikLσ4 S L (k)] = µ 1 (x, t, k)e −i(kx+2k 2 t)σ4 [S −1 (k)s(k)e ikLσ4 S L (k)],(53)
which leads to the global relation
c(T, k) = µ 4 (0, T, k) = e −2ik 2 Tσ4 [S −1 (k)s(k)e ikLσ4 S L (k)],(54)
by evaluating Eq. (53) at the point (x, t) = (0, T ) and using µ 1 (0, T, k) = I.
The 4 × 4 matrix Riemann-Hilbert problem
By using the district contours γ j (j = 1, 2, 3, 4), the integral solutions of the revised Lax pair (8), and S n due to {S(k), s(k), S(k), S L (k)}, we have defined the sectionally analytic function M n (x, t, k) (n = 1, 2, 3, 4), which solves a 4 × 4 matrix Riemann-Hilbert (RH) problem. This RH problem can be formulated on basis of the initial and boundary data of the functions q 1 (x, t), q 2 (x, t) and q 3 (x, t). Thus the solution of Eq. (1) for all values of
x, t can be refound by solving the RH problem.
Theorem 3.1. Let (q 1 (x, t), q 2 (x, t), q 3 (x, t)) be a solution of Eq. (1) in the interval domain Ω = {(x, t)|x ∈ [0, L], t ∈ [0, T ]}.
Then it can be reconstructed from the initial data defined by
q j (x, t = 0) = q 0j (x), j = 1, 2, 3,
and Dirichlet and Neumann boundary values defined by
Dirichlet boundary data : q j (x = 0, t) = u 0j (t), q j (x = L, t) = v 0j (t), j = 1, 2, 3, Neumann boundary data : q jx (x = 0, t) = u 1j (t), q jx (x = L, t) = v 1j (t), j = 1, 2, 3,
We can use the initial and boundary data to define the jump matrices J mn (x, t, k), (n, m = 1, ...
(q 1 (x, t), q 2 (x, t), q 3 (x, t)) of Eq. (1) is given by M (x, t, k) in the form q j (x, t) = 2i lim k→∞ (kM (x, t, k)) j4 , j = 1, 2, 3,(55)
where M (x, t, k) satisfies the following 4 × 4 matrix Riemann-Hilbert problem:
• M (x, t, k) is sectionally meromorphic on the Riemann k-sphere with jumps across the contoursD n ∪D m , (n, m = 1, ..., 4) (see Fig. 2).
• Across the contoursD n ∪D m (n, m = 1, ..., 4), M (x, t, k) satisfies the jump condition (22).
• The residue conditions of M (x, t, k) are satisfied in Proposition 2.5.
• M (x, t, k) = I + O(1/k) as k → ∞.
Proof. System (55) can be deduced from the large k asymptotics of the eigenfunctions. We can follow the similar one in Refs. [8,14] to show the rest proof of the Theorem.
The nonlinearizable boundary conditions
The key difficulty of initial-boundary value problems is to find the boundary values for a well-posed problem. All boundary value conditions are required for the definition of S(k) and S L (k), and hence for the formulate the RH problem. Our main conclusion exhibits the unknown boundary condition on basis of the prescribed boundary condition and the initial condition in terms of the solution of a system of nonlinear integral equations.
The generalized global relation
By evaluating Eqs. (53) and (54) at the point (x, t) = (0, t), we have
c(t, k) = µ 2 (0, t, k)e −2ik 2 tσ4 [s(k)e ikLσ4 S L (k)],
which and Eq. (34) lead to
c(t, k) = µ 2 (0, t, k)e −2ik 2 tσ4 [s(k)e ikLσ4 e 2ik 2 tσ4 µ −1 3 (L, t, k)], = µ 2 (0, t, k)[e −2ik 2 tσ4 s(k)][e ikLσ4 µ −1 3 (L, t, k)],(56)
Asymptotic behaviors of eigenfunctions
It follows from the Lax pair (8) that the eigenfunctions {µ j } 4 1 possess the following asymptotics as k → ∞
µ j (x, t, k) = I + 2 i=1 1 k i µ (i) j,11 µ (i) j,12 µ (i) j,13 µ (i) j,14 µ (i) j,21 µ (i) j,22 µ (i) j,23 µ (i) j,24 µ (i) j,31 µ (i) j,32 µ (i) j,33 µ (i) j,34 µ (i) j,41 µ (i) j,42 µ (i) j,43 µ (i) j,44 + O( 1 k 3 ) = I + 1 k (x,t) (xj,tj ) ∆ (1) 11 (x,t) (xj,tj ) ∆ (1) 12 (x,t) (xj ,tj ) ∆ (1) 13 − i 2 q 1 (x,t) (xj,tj ) ∆ (1) 21 (x,t) (xj,tj ) ∆ (1) 22 (x,t) (xj ,tj ) ∆ (1) 23 − i 2 q 2 (x,t) (xj,tj ) ∆ (1) 31 (x,t) (xj,tj ) ∆ (1) 32 (x,t) (xj ,tj ) ∆ (1) 33 − i 2 q 3 i 2 p 1 i 2 p 2 i 2 p 3 (x,t) (xj ,tj ) ∆ (1) 44 + 1 k 2 (x,t) (xj ,tj ) ∆ (2) 11 (x,t) (xj ,tj ) ∆ (2) 12 (x,t) (xj ,tj) ∆ (2) 13 µ (2) j,14 (x,t) (xj ,tj ) ∆ (2) 21 (x,t) (xj ,tj ) ∆ (2) 22 (x,t) (xj ,tj) ∆ (2) 23 µ (2) j,24 (x,t) (xj ,tj ) ∆ (2) 31 (x,t) (xj ,tj ) ∆ (2) 32 (x,t) (xj ,tj) ∆ (2) 33 µ (2) j,34 µ (2) j,41 µ (2) j,42 µ (2) j,43 (x,t) (xj ,tj) ∆ (2) 44 + O( 1 k 3 ),(57)
where we have introduced the following functions
∆ (1) jl = i 2 q j p l dx + 1 2 (q j p lx − q jx p l )dt, j, l = 1, 2, 3, ∆(1)44 = − i 2 3 j=1 q j p j dx + 1 2 and µ (2) j,l4 = 1 4 q lx + 1 2i q l (x,t) (xj ,tj ) ∆ (1) 44 , l = 1, 2, 3, µ (2) j,4l = 1 4 p lx + i 2 3 s=1 p s (x,t) (xj,tj ) ∆ (1) sl , l = 1, 2, 3, ∆ (2) sl = 1 4 q s p lx + i 2 q s 3 n=1 p n (x,t) (xj ,tj ) ∆ (1) nl dx + 1 4 q s p lx + iq sx p lx − iq s p l 3 j=1 q j p j + 1 2 3 n=1 (q s p nx − q sx p n ) (x,t) (xj,tj ) ∆ (1) nl dt, s, l = 1, 2, 3, ∆ (2) 44 = 1 4 3 l=1 p l q lx − i 2 3 l=1 p l q l (x,t) (xj ,tj) ∆ (1) 44 dx + 1 4 3 l=1 (p l q lx − ip lx q lx ) + i 3 l=1 p l q l 2 + 1 2 3 l=1 (p l q lx − p lx q l ) (x,t) (xj ,tj ) ∆ (1) 44 dt,
The functions {µ
(i) jl = µ (i) jl (x, t)} 4 1 , i = 1, 2 are independent of k. We define the function {Ψ ij (t, k)} 4 i,j=1 as µ 2 (0, t, k) = (Ψ sj (t, k)) 4×4 = I + 2 l=1 1 k l Ψ (l) 11 (t) Ψ (l) 12 (t) Ψ (l) 13 (t) Ψ (l) 14 (t) Ψ (l) 21 (t) Ψ (l) 22 (t) Ψ (l) 23 (t) Ψ (l) 24 (t) Ψ (l) 31 (t) Ψ (l) 32 (t) Ψ (l) 33 (t) Ψ (l) 34 (t) Ψ (l) 41 (t) Ψ (l) 42 (t) Ψ (l) 43 (t) Ψ (l) 44 (t) + O( 1 k 3 ),(58)
Based on the asymptotic of Eq. (57) and the boundary data at x = 0, we find
Ψ (1) 14 (t) = − i 2 u 01 (t), Ψ(1)24 (t) = − i 2 u 02 (t), Ψ(1)34 (t) = − i 2 u 03 (t), Ψ(1)41 (t) = i 2 [α 11ū01 (t) +ᾱ 12ū02 (t) +ᾱ 13ū03 (t)] , Ψ(1)42 (t) = i 2 [α 12ū01 (t) + α 22ū02 (t) +ᾱ 23ū03 (t)] , Ψ(1)43 (t) = i 2 [α 13ū01 (t)+α 23ū02 (t)+α 33ū03 (t)] , Ψ(2)14 = 1 4 u 11 + 1 2i u 01 Ψ (1) 44 , Ψ(2)24 = 1 4 u 12 + 1 2i u 02 Ψ (1) 44 , Ψ(2)34 = 1 4 u 13 + 1 2i u 03 Ψ (1) 44 , Ψ (1) 44 = 1 2 t 0 u 11 [α 11ū01 (t)+ᾱ 12ū02 (t)+ᾱ 13ū03 (t)]+u 12 [α 12ū01 (t)+α 22ū02 (t)+ᾱ 23ū03 (t)] +u 13 [α 13ū01 (t) + α 23ū02 (t) + α 33ū03 (t)] − u 01 [α 11ū11 (t) +ᾱ 12ū12 (t) +ᾱ 13ū13 (t)] −u 02 [α 12ū11 (t) + α 22ū12 (t) +ᾱ 23ū13 (t)] − u 03 [α 13ū11 (t) + α 23ū12 (t) + α 33ū13 (t)] dt,(59)
Thus we have the the boundary data at x = 0:
u 01 (t) = 2iΨ (1) 14 (t), u 02 (t) = 2iΨ (1) 24 (t), u 03 (t) = 2iΨ (1) 34 (t), u 11 (t) = 4Ψ (2) 14 (t) + 2iu 01 (t)Ψ (1) 44 (t), u 12 (t) = 4Ψ (2) 24 (t) + 2iu 02 (t)Ψ (1) 44 (t), u 13 (t) = 4Ψ (2) 34 (t) + 2iu 03 (t)Ψ (1) 44 (t),(60)
Similarly, we assume that the asymptotic formula of
µ 3 (L, t, k) = {φ ij (t, k)} 4 i,j=1 is of the from µ 3 (L, t, k) = (φ sj (t, k)) 4×4 = I + 2 l=1 1 k l φ (l) 11 (t) φ (l) 12 (t) φ (l) 13 (t) φ (l) 14 (t) φ (l) 21 (t) φ (l) 22 (t) φ (l) 23 (t) φ (l) 24 (t) φ (l) 31 (t) φ (l) 32 (t) φ (l) 33 (t) φ (l) 34 (t) φ (l) 41 (t) φ (l) 42 (t) φ (l) 43 (t) φ (l) 44 (t) + O( 1 k 3 ),(61)
By using the asymptotic of Eq. (57) and the boundary data at x = L, we find
φ (1) 14 (t) = − i 2 v 01 (t), φ(1)24 (t) = − i 2 v 02 (t), φ(1)34 (t) = − i 2 v 03 (t), φ(1)41 (t) = i 2 [α 11v01 (t) +ᾱ 12v02 (t) +ᾱ 13v03 (t)] , φ(1)42 (t) = i 2 [α 12v01 (t) + α 22v02 (t) +ᾱ 23v03 (t)] , φ(1)43 (t) = i 2 [α 13v01 (t) + α 23v02 (t) + α 33v03 (t)] , φ(2)14 = 1 4 v 11 + 1 2i v 01 φ (1) 44 , φ(2)24 = 1 4 v 12 + 1 2i v 02 φ (1) 44 , φ(2)34 = 1 4 v 12 + 1 2i v 03 φ (1) 44 , φ (1) 44 = 1 2 t 0 v 11 [α 11v01 (t)+ᾱ 12v02 (t)+ᾱ 13v03 (t)]+v 12 [α 12v01 (t)+α 22v02 (t)+ᾱ 23v03 (t)] +v 13 [α 13v01 (t) + α 23v02 (t) + α 33v03 (t)] − v 01 [α 11v11 (t) +ᾱ 12v12 (t) +ᾱ 13v13 (t)] −v 02 [α 12ū11 (t) + α 22v12 (t) +ᾱ 23v13 (t)] − v 03 [α 13v11 (t) + α 23v12 (t) + α 33v13 (t)] dt,(62)
which generates the following expressions for the boundary values at
x = L v 01 (t) = 2iφ(1)14 (t), v 02 (t) = 2iφ(1)24 (t), v 03 (t) = 2iφ(1)34 (t), v 11 (t) = 4φ(2)14 (t) + 2iv 01 (t)φ(1)44 (t), v 12 (t) = 4φ (2) 24 (t) + 2iv 02 (t)φ (1) 44 (t), v 13 (t) = 4φ (2) 34 (t) + 2iv 03 (t)φ (1) 44 (t),(63)
For the vanishing initial values, it follows from Eq. (56) that we have the following asymptotic of the global relation c j4 (t, k) and c 4j (t, k), j = 1, 2, 3.
Proposition 4.1. Let the initial and Dirichlet boundary conditions be compatible at points x = 0, L (i.e., q 0j (0) = u 0j (0) at x = 0 and q 0j (L) = v 0j (0) at x = L, j = 1, 2, 3). Then, the global relation (56) with the vanishing initial data implies that the large k behaviors of c j4 (t, k) and c 4j (t, k), j = 1, 2, 3 are of the form
c 14 (t, k) = Ψ (1) 14 k + Ψ (2) 14 + Ψ (1) 14φ
(1) 44
k 2 + O 1 k 3 − α 11φ (1) 41 +ᾱ 12φ (1) 42 +ᾱ 13φ (1) 43 k + 1 k 2 α 11φ (2) 41 +ᾱ 12φ (2) 42 +ᾱ 13φ (2) 43 +Ψ (1) 11 α 11φ (1) 41 +ᾱ 12φ (1) 42 +ᾱ 13φ (1) 43 + Ψ (1) 12 α 12φ (1) 41 + α 22φ (1) 42 +ᾱ 23φ (1) 43 + Ψ (1) 13 α 13φ (1) 41 + α 23φ (1) 42 + α 33φ (1) 43 + O 1 k 3 e 2ikL , k → ∞,(64a)c 24 (t, k) = Ψ (1) 24 k + Ψ (2) 24 + Ψ (1) 24φ
(1) 44
k 2 + O 1 k 3 − α 12φ (1) 41 + α 22φ (1) 42 +ᾱ 23φ (1) 43 k + 1 k 2 α 12φ (2) 41 + α 22φ (2) 42 +ᾱ 23φ (2) 43 +Ψ (1) 21 α 11φ (1) 41 +ᾱ 12φ (1) 42 +ᾱ 13φ (1) 43 + Ψ (1) 22 α 12φ (1) 41 + α 22φ (1) 42 +ᾱ 23φ (1) 43 + Ψ (1) 23 α 13φ (1) 41 + α 23φ (1) 42 + α 33φ (1) 43 + O 1 k 3 e 2ikL , k → ∞, (64b) c 34 (t, k) = Ψ (1) 34 k + Ψ (2) 34 + Ψ (1) 34φ
(1) 44
k 2 + O 1 k 3 − α 13φ (1) 41 + α 23φ (1) 42 + α 33φ (1) 43 k + 1 k 2 α 13φ (2) 41 + α 23φ (2) 42 + α 33φ(2)+ O 1 k 3 e 2ikL , k → ∞,(64c)c 41 (t, k) = − α 11φ (1) 14 + α 12φ (1) 24 + α 13φ (1) 34 k + 1 k 2 α 11φ (2) 14 + α 12φ (2) 24 + α 13φ (2) 34 +Ψ (1) 44 α 11φ (1) 14 + α 12φ (1) 24 + α 13φ (1) 34 + O 1 k 3 e −2ikL + 1 k (α 2 11 + |α 12 | 2 + |α 13 | 2 )Ψ (1) 41 + (α 11 α 12 + α 12 α 22 + α 13ᾱ23 )Ψ (1) 42 +(α 11 α 13 + α 12 α 23 + α 13 α 33 )Ψ (1) 43 + 1 k 2 (α 2 11 + |α 12 | 2 + |α 13 | 2 )Ψ (2) 41 +(α 11 α 12 + α 12 α 22 + α 13ᾱ23 )Ψ (2) 42 + (α 11 α 13 + α 12 α 23 + α 13 α 33 )Ψ (2) 43 +Ψ (1) 41 α 11 α 11φ (1) 11 +ᾱ 12φ(1)
12 +ᾱ 13φ
(1) 13
+ α 12 α 11φ (1) 21 +ᾱ 12φ (1) 22 +ᾱ 13φ
(1) 23
+α 13 α 11φ (1) 31 +ᾱ 12φ (1) 32 +ᾱ 13φ (1) 33 + Ψ (1) 42 α 11 α 12φ (1) 11 + α 22φ (1) 12 +ᾱ 23φ
(1) 13
+α 12 α 12φ (1) 21 + α 22φ (1) 22 +ᾱ 23φ (1) 23 + α 13 α 12φ (1) 31 + α 22φ (1) 32 +ᾱ 23φ (1) 33 +Ψ (1) 43 α 11 α 13φ (1) 11 + α 23φ (1) 12 + α 33φ (1) 13 + α 12 α 13φ (1) 21 + α 23φ (1) 22 + α 33φ (1) 23 +α 13 α 13φ (1) 31 + α 23φ (1) 32 + α 33φ (1) 33 + O 1 k 3 , k → ∞,(65)c 42 (t, k) = − ᾱ 12φ (1) 14 + α 22φ (1) 24 + α 23φ (1) 34 k + 1 k 2 ᾱ 12φ (2) 14 + α 22φ (2) 24 + α 23φ (2) 34 +Ψ (1) 44 ᾱ 12φ (1) 14 + α 22φ (1) 24 + α 23φ (1) 34 + O 1 k 3 e −2ikL + 1 k (|α 12 | 2 + α 2 22 + |α 23 | 2 )Ψ (1) 42 + (α 11ᾱ12 +ᾱ 12ᾱ22 +ᾱ 13 α 23 )Ψ (1) 41 +(ᾱ 12 α 13 + α 22 α 23 + α 23 α 33 )Ψ (1) 43 + 1 k 2 (|α 12 | 2 + α 2 22 + |α 23 | 2 )Ψ (2) 42 +(α 11ᾱ12 +ᾱ 12ᾱ22 +ᾱ 13 α 23 )Ψ (2) 41 + (ᾱ 12 α 13 + α 22 α 23 + α 23 α 33 )Ψ (2) 43 +Ψ (1) 41 ᾱ 12 α 11φ (1) 11 +ᾱ 12φ(1)
12 +ᾱ 13φ
(1) 13
+ α 22 α 11φ (1) 21 +ᾱ 12φ (1) 22 +ᾱ 13φ (1) 23 +α 23 α 11φ (1) 31 +ᾱ 12φ (1) 32 +ᾱ 13φ (1) 33 + Ψ (1) 42 ᾱ 12 α 12φ (1) 11 + α 22φ(1)
12 +ᾱ 23φ
(1) 13
+α 22 α 12φ (1) 21 + α 22φ (1) 22 +ᾱ 23φ (1) 23 + α 23 α 12φ (1) 31 + α 22φ (1) 32 +ᾱ 23φ (1) 33 +Ψ (1) 43 ᾱ 12 α 13φ (1) 11 + α 23φ (1) 12 + α 33φ (1) 13 + α 22 α 13φ (1) 21 + α 23φ (1) 22 + α 33φ
(1) 23
+α 23 α 13φ (1) 31 + α 23φ (1) 32 + α 33φ (1) 33 + O 1 k 3 , k → ∞,(66)c 43 (t, k) = − ᾱ 13φ (1) 14 +ᾱ 23φ (1) 24 + α 33φ (1) 34 k + 1 k 2 ᾱ 13φ (2) 14 +ᾱ 23φ (2) 24 + α 33φ (2) 34 +Ψ (1) 44 ᾱ 13φ (1) 14 +ᾱ 23φ (1) 24 + α 33φ (1) 34 + O 1 k 3 e −2ikL + 1 k (|α 13 | 2 + |α 23 | 2 + α 2 33 )Ψ (1) 43 + (α 11ᾱ13 +ᾱ 12ᾱ23 +ᾱ 13 α 33 )Ψ (1) 41 +(ᾱ 13 α 12 + α 22ᾱ23 + α 13ᾱ23 )Ψ (1) 42 + 1 k 2 (|α 13 | 2 + |α 23 | 2 + α 2 33 )Ψ (2) 43 +(α 11ᾱ12 +ᾱ 12ᾱ22 +ᾱ 13 α 23 )Ψ (2) 41 + (ᾱ 12 α 13 + α 22 α 23 + α 23 α 33 )Ψ (2) 43 +Ψ (1) 41 ᾱ 13 α 11φ (1) 11 +ᾱ 12φ (1) 12 +ᾱ 13φ
(1) 13
+ᾱ 23 α 11φ (1) 21 +ᾱ 12φ (1) 22 +ᾱ 13φ
(1) 23
+α 33 α 11φ (1) 31 +ᾱ 12φ (1) 32 +ᾱ 13φ (1) 33 + Ψ (1) 42 ᾱ 13 α 12φ (1) 11 + α 22φ(1)
12 +ᾱ 23φ
(1) 13
+ᾱ 23 α 12φ (1) 21 + α 22φ (1) 22 +ᾱ 23φ (1) 23 + α 33 α 12φ (1) 31 + α 22φ (1) 32 +ᾱ 23φ (1) 33 +Ψ (1) 43 ᾱ 13 α 13φ (1) 11 + α 23φ (1) 12 + α 33φ (1) 13 +ᾱ 23 α 13φ (1) 21 + α 23φ (1) 22 + α 33φ (1) 23 +α 33 α 13φ (1) 31 + α 23φ (1) 32 + α 33φ (1) 33 + O 1 k 3 , k → ∞,(67)
Proof. The global relation (56) under the vanishing initial data can be simplified as
c 14 (t, k) = Ψ 14 (t, k)φ 44 (t,k) − e 2ikL Ψ 11 (t, k)(α 11φ41 (t,k) +ᾱ 12φ42 (t,k) +ᾱ 13φ43 (t,k)) +Ψ 12 (t, k)(α 12φ41 (t,k) + α 22φ42 (t,k) +ᾱ 23φ43 (t,k)) +Ψ 13 (t, k)(α 13φ41 (t,k) + α 23φ42 (t,k) + α 33φ43 (t,k)) ,(68a)c 24 (t, k) = Ψ 24 (t, k)φ 44 (t,k) − e 2ikL Ψ 21 (t, k)(α 11φ41 (t,k) +ᾱ 12φ42 (t,k) +ᾱ 13φ43 (t,k)) +Ψ 22 (t, k)(α 12φ41 (t,k) + α 22φ42 (t,k) +ᾱ 23φ43 (t,k)) +Ψ 23 (t, k)(α 13φ41 (t,k) + α 23φ42 (t,k) + α 33φ43 (t,k)) ,(68b)c 34 (t, k) = Ψ 34 (t, k)φ 44 (t,k) − e 2ikL Ψ 31 (t, k)(α 11φ41 (t,k) +ᾱ 12φ42 (t,k) +ᾱ 13φ43 (t,k)) +Ψ 32 (t, k)(α 12φ41 (t,k) + α 22φ42 (t,k) +ᾱ 23φ43 (t,k)) +Ψ 33 (t, k)(α 13φ41 (t,k) + α 23φ42 (t,k) + α 33φ43 (t,k)) ,(68c)whereφ ij (t,k) = φ ij (t,k).
Recalling the time-part of the Lax pair (8)
µ t + 2ik 2 [σ 4 , µ] = V (x, t, k)µ,(69)
It follows from the first column of Eq. (69) with µ = µ 2 that we have
Ψ 11,t (t, k) = (2ku 01 + iu 11 )Ψ 41 − iΨ 11 (α 11 |u 01 | 2 +ᾱ 12 u 01ū02 +ᾱ 13 u 01ū03 ) −iΨ 21 (α 12 |u 01 | 2 + α 22 u 01ū02 +ᾱ 23 u 01ū03 ) −iΨ 31 (α 13 |u 01 | 2 + α 23 u 01ū02 + α 33 u 01ū03 ), Ψ 21,t (t, k) = (2ku 02 + iu 12 )Ψ 41 − iΨ 11 (α 11 u 02ū01 +ᾱ 12 |u 02 | 2 +ᾱ 13 u 02ū03 ) −iΨ 21 (α 12 u 02ū01 + α 22 |u 02 | 2 +ᾱ 23 u 02ū03 ) −iΨ 31 (α 13 u 02ū01 + α 23 |u 02 | 2 + α 33 u 02ū03 ), Ψ 31,t (t, k) = (2ku 03 + iu 13 )Ψ 41 − iΨ 11 (α 11 u 03ū01 +ᾱ 12 u 03ū02 +ᾱ 13 |u 03 | 2 ) −iΨ 21 (α 12 u 03ū01 + α 22 u 03ū02 +ᾱ 23 |u 03 | 2 ) −iΨ 31 (α 13 u 03ū01 + α 23 u 03ū02 + α 33 |u 03 | 2 ), Ψ 41,t (t, k) = Ψ 11 [α 11 (2kū 01 − iū 11 ) +ᾱ 12 (2kū 02 − iū 12 ) +ᾱ 13 (2kū 03 − iū 13 )] +Ψ 21 [α 12 (2kū 01 − iū 11 ) + α 22 (2kū 02 − iū 12 ) +ᾱ 23 (2kū 03 − iū 13 )] +Ψ 31 [α 13 (2kū 01 − iū 11 ) + α 23 (2kū 02 − iū 12 ) + α 33 (2kū 03 − iū 13 )] +iΨ 41 [4k 2 + α 11 |u 01 | 2 +ᾱ 12 u 01ū02 +ᾱ 13 u 01ū03 + α 12 u 02ū01 +α 22 |u 02 | 2 +ᾱ 23 u 02ū03 + α 13 u 03ū01 + α 23 u 03ū02 + α 33 |u 03 | 2 ],(70)
The second column of Eq. (69) with µ = µ 2 yields
Ψ 12,t (t, k) = (2ku 01 + iu 11 )Ψ 42 − iΨ 12 (α 11 |u 01 | 2 +ᾱ 12 u 01ū02 +ᾱ 13 u 01ū03 ) −iΨ 22 (α 12 |u 01 | 2 + α 22 u 01ū02 +ᾱ 23 u 01ū03 ) −iΨ 32 (α 13 |u 01 | 2 + α 23 u 01ū02 + α 33 u 01ū03 ), Ψ 22,t (t, k) = (2ku 02 + iu 12 )Ψ 42 − iΨ 12 (α 11 u 02ū01 +ᾱ 12 |u 02 | 2 +ᾱ 13 u 02ū03 ) −iΨ 22 (α 12 u 02ū01 + α 22 |u 02 | 2 +ᾱ 23 u 02ū03 ) −iΨ 32 (α 13 u 02ū01 + α 23 |u 02 | 2 + α 33 u 02ū03 ), Ψ 32,t (t, k) = (2ku 03 + iu 13 )Ψ 42 − iΨ 12 (α 11 u 03ū01 +ᾱ 12 u 03ū02 +ᾱ 13 |u 03 | 2 ) −iΨ 22 (α 12 u 03ū01 + α 22 u 03ū02 +ᾱ 23 |u 03 | 2 ) −iΨ 32 (α 13 u 03ū01 + α 23 u 03ū02 + α 33 |u 03 | 2 ), Ψ 42,t (t, k) = Ψ 12 [α 11 (2kū 01 − iū 11 ) +ᾱ 12 (2kū 02 − iū 12 ) +ᾱ 13 (2kū 03 − iū 13 )] +Ψ 22 [α 12 (2kū 01 − iū 11 ) + α 22 (2kū 02 − iū 12 ) +ᾱ 23 (2kū 03 − iū 13 )] +Ψ 32 [α 13 (2kū 01 − iū 11 ) + α 23 (2kū 02 − iū 12 ) + α 33 (2kū 03 − iū 13 )]
+iΨ 42 [4k 2 + α 11 |u 01 | 2 +ᾱ 12 u 01ū02 +ᾱ 13 u 01ū03 + α 12 u 02ū01
+α 22 |u 02 | 2 +ᾱ 23 u 02ū03 + α 13 u 03ū01 + α 23 u 03ū02 + α 33 |u 03 | 2 ],(71)
The third column of Eq. (69) with µ = µ 2 yields
Ψ 13,t (t, k) = (2ku 01 + iu 11 )Ψ 43 − iΨ 13 (α 11 |u 01 | 2 +ᾱ 12 u 01ū02 +ᾱ 13 u 01ū03 ) −iΨ 23 (α 12 |u 01 | 2 + α 22 u 01ū02 +ᾱ 23 u 01ū03 ) −iΨ 33 (α 13 |u 01 | 2 + α 23 u 01ū02 + α 33 u 01ū03 ), Ψ 23,t (t, k) = (2ku 02 + iu 12 )Ψ 43 − iΨ 13 (α 11 u 02ū01 +ᾱ 12 |u 02 | 2 +ᾱ 13 u 02ū03 ) −iΨ 23 (α 12 u 02ū01 + α 22 |u 02 | 2 +ᾱ 23 u 02ū03 ) −iΨ 33 (α 13 u 02ū01 + α 23 |u 02 | 2 + α 33 u 02ū03 ), Ψ 33,t (t, k) = (2ku 03 + iu 13 )Ψ 43 − iΨ 13 (α 11 u 03ū01 +ᾱ 12 u 03ū02 +ᾱ 13 |u 03 | 2 ) −iΨ 23 (α 12 u 03ū01 + α 22 u 03ū02 +ᾱ 23 |u 03 | 2 ) −iΨ 33 (α 13 u 03ū01 + α 23 u 03ū02 + α 33 |u 03 | 2 ), Ψ 43,t (t, k) = Ψ 13 [α 11 (2kū 01 − iū 11 ) +ᾱ 12 (2kū 02 − iū 12 ) +ᾱ 13 (2kū 03 − iū 13 )] +Ψ 23 [α 12 (2kū 01 − iū 11 ) + α 22 (2kū 02 − iū 12 ) +ᾱ 23 (2kū 03 − iū 13 )] +Ψ 33 [α 13 (2kū 01 − iū 11 ) + α 23 (2kū 02 − iū 12 ) + α 33 (2kū 03 − iū 13 )]
+iΨ 43 [4k 2 + α 11 |u 01 | 2 +ᾱ 12 u 01ū02 +ᾱ 13 u 01ū03 + α 12 u 02ū01
+α 22 |u 02 | 2 +ᾱ 23 u 02ū03 + α 13 u 03ū01 + α 23 u 03ū02 + α 33 |u 03 | 2 ],(72)
The fourth column of Eq. (69) with µ = µ 2 yields
Ψ 14,t (t, k) = (2ku 01 + iu 11 )Ψ 44 − iΨ 14 (4k 2 + α 11 |u 01 | 2 +ᾱ 12 u 01ū02 +ᾱ 13 u 01ū03 ) −iΨ 24 (α 12 |u 01 | 2 + α 22 u 01ū02 +ᾱ 23 u 01ū03 )
−iΨ 34 (α 13 |u 01 | 2 + α 23 u 01ū02 + α 33 u 01ū03 ), Ψ 24,t (t, k) = (2ku 02 + iu 12 )Ψ 44 − iΨ 14 (α 11 u 02ū01 +ᾱ 12 |u 02 | 2 +ᾱ 13 u 02ū03 ) −iΨ 24 (4k 2 + α 12 u 02ū01 + α 22 |u 02 | 2 +ᾱ 23 u 02ū03 ) −iΨ 34 (α 13 u 02ū01 + α 23 |u 02 | 2 + α 33 u 02ū03 ), Ψ 34,t (t, k) = (2ku 03 + iu 13 )Ψ 44 − iΨ 14 (α 11 u 03ū01 +ᾱ 12 u 03ū02 +ᾱ 13 |u 03 | 2 ) −iΨ 24 (α 12 u 03ū01 + α 22 u 03ū02 +ᾱ 23 |u 03 | 2 ) −iΨ 34 (4k 2 + α 13 u 03ū01 + α 23 u 03ū02 + α 33 |u 03 | 2 ),
Ψ 44,t (t, k) = Ψ 14 [α 11 (2kū 01 − iū 11 ) +ᾱ 12 (2kū 02 − iū 12 ) +ᾱ 13 (2kū 03 − iū 13 )] +Ψ 24 [α 12 (2kū 01 − iū 11 ) + α 22 (2kū 02 − iū 12 ) +ᾱ 23 (2kū 03 − iū 13 )] +Ψ 34 [α 13 (2kū 01 − iū 11 ) + α 23 (2kū 02 − iū 12 ) + α 33 (2kū 03 − iū 13 )]
+iΨ 44 [α 11 |u 01 | 2 +ᾱ 12 u 01ū02 +ᾱ 13 u 01ū03 + α 12 u 02ū01 + α 22 |u 02 | 2
+ᾱ 23 u 02ū03 + α 13 u 03ū01 + α 23 u 03ū02 + α 33 |u 03 | 2 ],(73)
Suppose that Ψ j1 's, j = 1, 2, 3, 4 are of the form
Ψ 11 Ψ 21 Ψ 31 Ψ 41 = a 10 (t) + a 11 (t) k + a 12 (t) k 2 + · · · + b 10 (t) + b 11 (t) k + b 12 (t) k 2 + · · · e 4ik 2 t ,(74)
where the 4 × 1 column vector functions a 1j (t), b 1j (t) (j = 0, 1, ..., ) are independent of k.
By substituting Eq. (74) into Eq.(70) and using the initial conditions a 10 (0) + b 10 (0) = (1, 0, 0, 0) T , a 11 (0) + b 11 (0) = (0, 0, 0, 0) T ,
we have Ψ 11 Ψ 21 Ψ 31 Ψ 41 = 1 0 0 0 + 2 s=1 1 k s Ψ (s) 11 Ψ (s) 21 Ψ (s) 31 Ψ (s) 41 + O 1 k 3 + 1 2ik 0 0 0 α 11ū01 (0) +ᾱ 12ū02 (0) +ᾱ 13ū03 (0) + O 1 k 2 e 4ik 2 t ,(75)
Similarly, it follows from Eqs. (71)-(73) that we have the asymptotic formulae for Ψ ij , i = 1, 2, 3, 4; j = 2, 3, 4
in the form
Ψ 12 Ψ 22 Ψ 32 Ψ 42 = 1 0 0 0 + 2 s=1 1 k s Ψ (s) 12 Ψ (s) 22 Ψ (s) 32 Ψ (s) 42 + O 1 k 3 + 1 2ik 0 0 0 α 12ū01 (0) + α 22ū02 (0) +ᾱ 23ū03 (0) + O 1 k 2 e 4ik 2 t ,(76) Ψ 13 Ψ 23 Ψ 33 Ψ 43 = 1 0 0 0 + 2 s=1 1 k s Ψ (s) 13 Ψ (s) 23 Ψ (s) 33 Ψ (s) 43 + O 1 k 3 + 1 2ik 0 0 0 α 13ū01 (0) + α 23ū02 (0) + α 33ū03 (0) + O 1 k 2 e 4ik 2 t ,(77)
and
Ψ 14 Ψ 24 Ψ 34 Ψ 44 = 1 0 0 0 + 2 s=1 1 k s Ψ (s) 14 Ψ (s) 24 Ψ (s) 34 Ψ (s) 44 + O 1 k 3 + i 2k u 01 (0) u 02 (0) u 03 (0) 0 + O 1 k 2 e −4ik 2 t ,(78)
Similar to Eqs. (70)-(73) for µ 2 (0, t, k), we also know that the function µ(x, t, k) = µ 3 (L, t, k) at x = L satisfy the t-part of Lax pair (69). The first column of Eq. (69) with µ = µ 3 yields
φ 11,t (t, k) = (2kv 01 + iv 11 )φ 41 − iφ 11 (α 11 |v 01 | 2 +ᾱ 12 v 01v02 +ᾱ 13 v 01v03 ) −iφ 21 (α 12 |v 01 | 2 + α 22 v 01v02 +ᾱ 23 v 01v03 ) −iφ 31 (α 13 |v 01 | 2 + α 23 v 01v02 + α 33 v 01v03 ), φ 21,t (t, k) = (2kv 02 + iv 12 )φ 41 − iφ 11 (α 11 v 02v01 +ᾱ 12 |v 02 | 2 +ᾱ 13 v 02v03 ) −iφ 21 (α 12 v 02v01 + α 22 |v 02 | 2 +ᾱ 23 v 02v03 ) −iφ 31 (α 13 v 02v01 + α 23 |v 02 | 2 + α 33 v 02v03 ), φ 31,t (t, k) = (2kv 03 + iv 13 )φ 41 − iφ 11 (α 11 v 03v01 +ᾱ 12 v 03v02 +ᾱ 13 |v 03 | 2 ) −iφ 21 (α 12 v 03v01 + α 22 v 03v02 +ᾱ 23 |v 03 | 2 ) −iφ 31 (α 13 v 03v01 + α 23 v 03v02 + α 33 |v 03 | 2 ), φ 41,t (t, k) = φ 11 [α 11 (2kv 01 − iv 11 ) +ᾱ 12 (2kv 02 − iv 12 ) +ᾱ 13 (2kv 03 − iv 13 )] +φ 21 [α 12 (2kv 01 − iv 11 ) + α 22 (2kv 02 − iv 12 ) +ᾱ 23 (2kv 03 − iv 13 )] +φ 31 [α 13 (2kv 01 − iv 11 ) + α 23 (2kv 02 − iv 12 ) + α 33 (2kv 03 − iv 13 )] +iφ 41 [4k 2 + α 11 |v 01 | 2 +ᾱ 12 v 01v02 +ᾱ 13 v 01v03 + α 12 v 02v01 +α 22 |v 02 | 2 +ᾱ 23 v 02v03 + α 13 v 03v01 + α 23 v 03v02 + α 33 |v 03 | 2 ],(79)
The second column of Eq. (69) with µ = µ 3 yields
φ 12,t (t, k) = (2kv 01 + iv 11 )φ 42 − iφ 12 (α 11 |v 01 | 2 +ᾱ 12 v 01v02 +ᾱ 13 v 01v03 ) −iφ 22 (α 12 |v 01 | 2 + α 22 v 01v02 +ᾱ 23 v 01v03 ) −iφ 32 (α 13 |v 01 | 2 + α 23 v 01v02 + α 33 v 01v03 ), φ 22,t (t, k) = (2kv 02 + iv 12 )φ 42 − iφ 12 (α 11 v 02v01 +ᾱ 12 |v 02 | 2 +ᾱ 13 v 02v03 ) −iφ 22 (α 12 v 02v01 + α 22 |v 02 | 2 +ᾱ 23 v 02v03 ) −iφ 32 (α 13 v 02v01 + α 23 |v 02 | 2 + α 33 v 02v03 ), φ 32,t (t, k) = (2kv 03 + iv 13 )φ 42 − iφ 12 (α 11 v 03v01 +ᾱ 12 v 03v02 +ᾱ 13 |v 03 | 2 ) −iφ 22 (α 12 v 03v01 + α 22 v 03v02 +ᾱ 23 |v 03 | 2 ) −iφ 32 (α 13 v 03v01 + α 23 v 03v02 + α 33 |v 03 | 2 ), φ 42,t (t, k) = φ 12 [α 11 (2kv 01 − iv 11 ) +ᾱ 12 (2kv 02 − iv 12 ) +ᾱ 13 (2kv 03 − iv 13 )] +φ 22 [α 12 (2kv 01 − iv 11 ) + α 22 (2kv 02 − iv 12 ) +ᾱ 23 (2kv 03 − iv 13 )] +φ 32 [α 13 (2kv 01 − iv 11 ) + α 23 (2kv 02 − iv 12 ) + α 33 (2kv 03 − iv 13 )] +iφ 42 [4k 2 + α 11 |v 01 | 2 +ᾱ 12 v 01v02 +ᾱ 13 v 01v03 + α 12 v 02v01 +α 22 |v 02 | 2 +ᾱ 23 v 02v03 + α 13 v 03v01 + α 23 v 03v02 + α 33 |v 03 | 2 ],(80)
The third column of Eq. (69) with µ = µ 3 yields
φ 13,t (t, k) = (2kv 01 + iv 11 )φ 43 − iφ 13 (α 11 |v 01 | 2 +ᾱ 12 v 01v02 +ᾱ 13 v 01v03 ) −iφ 23 (α 12 |v 01 | 2 + α 22 v 01v02 +ᾱ 23 v 01v03 ) −iφ 33 (α 13 |v 01 | 2 + α 23 v 01v02 + α 33 v 01v03 ), φ 23,t (t, k) = (2kv 02 + iv 12 )φ 43 − iφ 13 (α 11 v 02v01 +ᾱ 12 |v 02 | 2 +ᾱ 13 v 02v03 ) −iφ 23 (α 12 v 02v01 + α 22 |v 02 | 2 +ᾱ 23 v 02v03 ) −iφ 33 (α 13 v 02v01 + α 23 |v 02 | 2 + α 33 v 02v03 ), φ 33,t (t, k) = (2kv 03 + iv 13 )φ 43 − iφ 13 (α 11 v 03v01 +ᾱ 12 v 03v02 +ᾱ 13 |v 03 | 2 ) −iφ 23 (α 12 v 03v01 + α 22 v 03v02 +ᾱ 23 |v 03 | 2 ) −iφ 33 (α 13 v 03v01 + α 23 v 03v02 + α 33 |v 03 | 2 ), φ 43,t (t, k) = φ 13 [α 11 (2kv 01 − iv 11 ) +ᾱ 12 (2kv 02 − iv 12 ) +ᾱ 13 (2kv 03 − iv 13 )] +φ 23 [α 12 (2kv 01 − iv 11 ) + α 22 (2kv 02 − iv 12 ) +ᾱ 23 (2kv 03 − iv 13 )] +φ 33 [α 13 (2kv 01 − iv 11 ) + α 23 (2kv 02 − iv 12 ) + α 33 (2kv 03 − iv 13 )] +iφ 43 [4k 2 + α 11 |v 01 | 2 +ᾱ 12 v 01v02 +ᾱ 13 v 01v03 + α 12 v 02v01 +α 22 |v 02 | 2 +ᾱ 23 v 02v03 + α 13 v 03v01 + α 23 v 03v02 + α 33 |v 03 | 2 ],(81)
The fourth column of Eq. (69) with µ = µ 3 yields
φ 14,t (t, k) = (2kv 01 + iv 11 )φ 44 − iφ 14 (4k 2 + α 11 |v 01 | 2 +ᾱ 12 v 01v02 +ᾱ 13 v 01v03 ) −iφ 24 (α 12 |v 01 | 2 + α 22 v 01v02 +ᾱ 23 v 01v03 ) −iφ 34 (α 13 |v 01 | 2 + α 23 v 01v02 + α 33 v 01v03 ), φ 24,t (t, k) = (2kv 02 + iv 12 )φ 44 − iφ 14 (α 11 v 02v01 +ᾱ 12 |v 02 | 2 +ᾱ 13 v 02v03 ) −iφ 24 (4k 2 + α 12 v 02v01 + α 22 |v 02 | 2 +ᾱ 23 v 02v03 ) −iφ 34 (α 13 v 02v01 + α 23 |v 02 | 2 + α 33 v 02v03 ), φ 34,t (t, k) = (2kv 03 + iv 13 )φ 44 − iφ 14 (α 11 v 03v01 +ᾱ 12 v 03v02 +ᾱ 13 |v 03 | 2 ) −iφ 24 (α 12 v 03v01 + α 22 v 03v02 +ᾱ 23 |v 03 | 2 ) −iφ 34 (4k 2 + α 13 v 03v01 + α 23 v 03v02 + α 33 |v 03 | 2 ), φ 44,t (t, k) = φ 14 [α 11 (2kv 01 − iv 11 ) +ᾱ 12 (2kv 02 − iv 12 ) +ᾱ 13 (2kv 03 − iv 13 )] +φ 24 [α 12 (2kv 01 − iv 11 ) + α 22 (2kv 02 − iv 12 ) +ᾱ 23 (2kv 03 − iv 13 )] +φ 34 [α 13 (2kv 01 − iv 11 ) + α 23 (2kv 02 − iv 12 ) + α 33 (2kv 03 − iv 13 )] +iφ 44 [α 11 |v 01 | 2 +ᾱ 12 v 01v02 +ᾱ 13 v 01v03 + α 12 v 02v01 +α 22 |v 02 | 2 +ᾱ 23 v 02v03 + α 13 v 03v01 + α 23 v 03v02 + α 33 |v 03 | 2 ],(82)
Similarly, we can also obtain the asymptotic formulae for φ ij , i, j = 1, 2, 3, 4. The substitution of these formulae into Eq. (68a) and using the assumption that the initial and boundary data are compatible at x = 0 and x = L, we find the asymptotic result (64a) of c 14 (t, k) for k → ∞. Similarly we can also show Eqs. (64b) and (64c) for c 24 (t, k) and c 34 (t, k) as k → ∞.
Similarly, we have the global relation (56) under the vanishing initial data as
c 41 (t, k) = −Ψ 44 (t, k)(α 11φ14 + α 12φ24 + α 13φ34 )e −2ikL
+Ψ 41 (t, k) α 11 (α 11φ11 +ᾱ 12φ12 +ᾱ 13φ13 ) + α 12 (α 11φ21 +ᾱ 12φ22 +ᾱ 13φ23 ) +α 13 (α 11φ31 +ᾱ 12φ32 +ᾱ 13φ33 ) + Ψ 42 (t, k) α 11 (α 12φ11 + α 22φ12 +ᾱ 23φ13 )
+α 12 (α 12φ21 + α 22φ22 +ᾱ 23φ23 ) + α 13 (α 12φ31 + α 22φ32 +ᾱ 23φ33 ) +Ψ 43 (t, k) α 11 (α 13φ11 + α 23φ12 + α 33φ13 ) + α 12 (α 13φ21 + α 23φ22 + α 33φ23 ) +α 13 (α 13φ31 + α 23φ32 + α 33φ33 ) ,(83)c 42 (t, k) = −Ψ 44 (t, k)(ᾱ 12φ14 + α 22φ24 + α 23φ34 )e −2ikL +Ψ 41 (t, k) ᾱ 12 (α 11φ11 +ᾱ 12φ12 +ᾱ 13φ13 ) + α 22 (α 11φ21 +ᾱ 12φ22 +ᾱ 13φ23 ) +α 23 (α 11φ31 +ᾱ 12φ32 +ᾱ 13φ33 ) + Ψ 42 (t, k) ᾱ 12 (α 12φ11 + α 22φ12 +ᾱ 23φ13 ) +α 22 (α 12φ21 + α 22φ22 +ᾱ 23φ23 ) + α 23 (α 12φ31 + α 22φ32 +ᾱ 23φ33 ) +Ψ 43 (t, k) ᾱ 12 (α 13φ11 + α 23φ12 + α 33φ13 ) + α 22 (α 13φ21 + α 23φ22 + α 33φ23 ) +α 23 (α 13φ31 + α 23φ32 + α 33φ33 ) ,(84)c 43 (t, k) = −Ψ 44 (t, k)(ᾱ 13φ14 +ᾱ 23φ24 + α 33φ34 )e −2ikL
+Ψ 41 (t, k) ᾱ 13 (α 11φ11 +ᾱ 12φ12 +ᾱ 13φ13 ) +ᾱ 23 (α 11φ21 +ᾱ 12φ22 +ᾱ 13φ23 ) +α 33 (α 11φ31 +ᾱ 12φ32 +ᾱ 13φ33 ) + Ψ 42 (t, k) ᾱ 13 (α 12φ11 + α 22φ12 +ᾱ 23φ13 ) +ᾱ 23 (α 12φ21 + α 22φ22 +ᾱ 23φ23 ) + α 13 (α 12φ31 + α 22φ32 +ᾱ 23φ33 ) +Ψ 43 (t, k) ᾱ 13 (α 13φ11 + α 23φ12 + α 33φ13 ) +ᾱ 23 (α 13φ21 + α 23φ22 + α 33φ23 )
+α 33 (α 13φ31 + α 23φ32 + α 33φ33 ) ,(85)
whereφ ij =φ ij (t,k) = φ ij (t,k), such that we can show Eqs. (65)-(67) for c 4j (t, k), j = 1, 2, 3 as k → ∞.
The relation between Dirichlet and Neumann boundary value problems
In what follows we show that the spectral functions S(k) and S L (k) can be expressed in terms of the prescribed Dirichlet and Neumann boundary data and the initial data using the solution of a system of integral equations.
Introduce the new notations as
F ± (t, k) = F (t, k) ± F (t, −k), Σ ± (k) = e 2ikL ± e −2ikL .(86)
The sign ∂D j , j = 1, 2, 3, 4 stands for the boundary of the jth quadrant D j , oriented so that D j lies to the left of ∂D j . ∂D 0 3 denotes the boundary contour which has not contain the zeros of Σ − (k) and ∂D 0 3 = −∂D 0 1 .
Theorem 4.2.
Let q 0j (x) = q j (x, t = 0) = 0, j = 1, 2, 3 be the initial data of Eq. (1) on the interval x ∈ [0, L] and T < ∞. (i) For the Dirichlet problem, the boundary data u 0j (t) and v 0j (t) (j = 1, 2, 3) on the interval t ∈ [0, T ) are sufficiently smooth and compatible with the initial data q 0j (x), (j = 1, 2, 3) at points (x 2 , t 2 ) = (0, 0) and (x 3 , t 3 ) = (L, 0), respectively, i.e., u 0j (0) = q 0j (0), v 0j (0) = q 0j (L), j = 1, 2, 3; (ii) For the Neumann problem, the boundary data u 1j (t) and v 0j (t) (j = 1, 2, 3) on the interval t ∈ [0, T ) are sufficiently smooth and compatible with the initial data q 0j (x), (j = 1, 2, 3) at the origin (x 2 , t 2 ) = (0, 0) and (x 3 , t 3 ) = (L, 0), respectively.
For simplicity, let n 33,44 (S)(k) have no zeros in the domain D 1 . Then the spectral functions S(k) and S L (k) are defined by
S(k)=e 2ik 2 Tσ4 P Ψ 11 (T,k) Ψ 21 (T,k) Ψ 31 (T,k) Ψ 41 (T,k) Ψ 12 (T,k) Ψ 22 (T,k) Ψ 32 (T,k) Ψ 42 (T,k) Ψ 13 (T,k) Ψ 23 (T,k) Ψ 33 (T,k) Ψ 43 (T,k) Ψ 14 (T,k) Ψ 24 (T,k) Ψ 34 (T,k) Ψ 44 (T,k) P ,(87)S L (k) = e 2ik 2 Tσ4 P φ 11 (T,k) φ 21 (T,k) φ 31 (T,k) φ 41 (T,k) φ 12 (T,k) φ 22 (T,k) φ 32 (T,k) φ 42 (T,k) φ 13 (T,k) φ 23 (T,k) φ 33 (T,k) φ 43 (T,k) φ 14 (T,k) φ 24 (T,k) φ 34 (T,k) φ 44 (T,k) P ,(88)
where the matrix P is given by Eq. (28), and the complex-valued functions {Ψ ij (t, k)} 4 i,j=1 have the following system of integral equations
Ψ 11,t (t, k) =1 + t 0
(2ku 01 + iu 11 )Ψ 41 − iΨ 11 (α 11 |u 01 | 2 +ᾱ 12 u 01ū02 +ᾱ 13 u 01ū03 ) −iΨ 21 (α 12 |u 01 | 2 +α 22 u 01ū02 +ᾱ 23 u 01ū03 )−iΨ 31 (α 13 |u 01 | 2 +α 23 u 01ū02 +α 33 u 01ū03 ) (t ′ , k)dt ′ , Ψ 21,t (t, k) = t 0 (2ku 02 + iu 12 )Ψ 41 − iΨ 11 (α 11 u 02ū01 +ᾱ 12 |u 02 | 2 +ᾱ 13 u 02ū03 ) −iΨ 21 (α 12 u 02ū01 +α 22 |u 02 | 2 +ᾱ 23 u 02ū03 )−iΨ 31 (α 13 u 02ū01 +α 23 |u 02 | 2 +α 33 u 02ū03 ) (t ′ , k)dt ′ , Ψ 31,t (t, k) = t 0 (2ku 03 + iu 13 )Ψ 41 − iΨ 11 (α 11 u 03ū01 +ᾱ 12 u 03ū02 +ᾱ 13 |u 03 | 2 ) −iΨ 21 (α 12 u 03ū01 +α 22 u 03ū02 +ᾱ 23 |u 03 | 2 )−iΨ 31 (α 13 u 03ū01 +α 23 u 03ū02 +α 33 |u 03 | 2 ) (t ′ , k)dt ′ ,
Ψ 41,t (t, k) = t 0 e 4ik 2 (t−t ′ ) Ψ 11 [α 11 (2kū 01 − iū 11 ) +ᾱ 12 (2kū 02 − iū 12 ) +ᾱ 13 (2kū 03 − iū 13 )] +Ψ 21 [α 12 (2kū 01 − iū 11 ) + α 22 (2kū 02 − iū 12 ) +ᾱ 23 (2kū 03 − iū 13 )] +Ψ 31 [α 13 (2kū 01 − iū 11 ) + α 23 (2kū 02 − iū 12 ) + α 33 (2kū 03 − iū 13 )]
+iΨ 41 [α 11 |u 01 | 2 +ᾱ 12 u 01ū02 +ᾱ 13 u 01ū03 + α 12 u 02ū01 +α 22 |u 02 | 2 +ᾱ 23 u 02ū03 + α 13 u 03ū01 + α 23 u 03ū02 + α 33 |u 03 | 2 ] (t ′ , k)dt ′ ,
Ψ 12,t (t, k) = t 0 (2ku 01 + iu 11 )Ψ 42 − iΨ 12 (α 11 |u 01 | 2 +ᾱ 12 u 01ū02 +ᾱ 13 u 01ū03 ) −iΨ 22 (α 12 |u 01 | 2 + α 22 u 01ū02 +ᾱ 23 u 01ū03 ) − iΨ 32 (α 13 |u 01 | 2 + α 23 u 01ū02 + α 33 u 01ū03 ) (t ′ , k)dt ′ , Ψ 22,t (t, k) =1 + t 0 (2ku 02 + iu 12 )Ψ 42 − iΨ 12 (α 11 u 02ū01 +ᾱ 12 |u 02 | 2 +ᾱ 13 u 02ū03 ) −iΨ 22 (α 12 u 02ū01 + α 22 |u 02 | 2 +ᾱ 23 u 02ū03 ) − iΨ 32 (α 13 u 02ū01 + α 23 |u 02 | 2 + α 33 u 02ū03 ) (t ′ , k)dt ′ , Ψ 32,t (t, k) = t 0 (2ku 03 + iu 13 )Ψ 42 − iΨ 12 (α 11 u 03ū01 +ᾱ 12 u 03ū02 +ᾱ 13 |u 03 | 2 ) −iΨ 22 (α 12 u 03ū01 + α 22 u 03ū02 +ᾱ 23 |u 03 | 2 ) − iΨ 32 (α 13 u 03ū01 + α 23 u 03ū02 + α 33 |u 03 | 2 ) (t ′ , k)dt ′ , Ψ 42,t (t, k) = t 0 e 4ik 2 (t−t ′ ) Ψ 12 [α 11 (2kū 01 − iū 11 ) +ᾱ 12 (2kū 02 − iū 12 ) +ᾱ 13 (2kū 03 − iū 13 )] +Ψ 22 [α 12 (2kū 01 − iū 11 ) + α 22 (2kū 02 − iū 12 ) +ᾱ 23 (2kū 03 − iū 13 )] +Ψ 32 [α 13 (2kū 01 − iū 11 ) + α 23 (2kū 02 − iū 12 ) + α 33 (2kū 03 − iū 13 )](89)
+iΨ 42 [α 11 |u 01 | 2 +ᾱ 12 u 01ū02 +ᾱ 13 u 01ū03 + α 12 u 02ū01 13 (α 11 |u 01 | 2 +ᾱ 12 u 01ū02 +ᾱ 13 u 01ū03 ) −iΨ 22 (α 12 |u 01 | 2 + α 22 u 01ū02 +ᾱ 23 u 01ū03 ) − iΨ 32 (α 13 |u 01 | 2 + α 23 u 01ū02 + α 33 u 01ū03 ) (t ′ , k)dt ′ , Ψ 23,t (t, k) = t 0 (2ku 02 + iu 12 )Ψ 42 − iΨ 13 (α 11 u 02ū01 +ᾱ 12 |u 02 | 2 +ᾱ 13 u 02ū03 ) −iΨ 23 (α 12 u 02ū01 + α 22 |u 02 | 2 +ᾱ 23 u 02ū03 ) − iΨ 33 (α 13 u 02ū01 + α 23 |u 02 | 2 + α 33 u 02ū03 ) (t ′ , k)dt ′ , Ψ 33,t (t, k) =1 + t 0 (2ku 03 + iu 13 )Ψ 43 − iΨ 13 (α 11 u 03ū01 +ᾱ 12 u 03ū02 +ᾱ 13 |u 03 | 2 ) −iΨ 23 (α 12 u 03ū01 + α 22 u 03ū02 +ᾱ 23 |u 03 | 2 ) − iΨ 33 (α 13 u 03ū01 + α 23 u 03ū02 + α 33 |u 03 | 2 ) (t ′ , k)dt ′ ,
+α 22 |u 02 | 2 +ᾱ 23 u 02ū03 + α 13 u 03ū01 + α 23 u 03ū02 + α 33 |u 03 | 2 ] (t ′ , k)dt ′ ,(90) Ψ 13,t (t, k) = t 0 (2ku 01 + iu 11 )Ψ 43 − iΨΨ 43,t (t, k) = t 0 e 4ik 2 (t−t ′ ) Ψ 13 [α 11 (2kū 01 − iū 11 ) +ᾱ 12 (2kū 02 − iū 12 ) +ᾱ 13 (2kū 03 − iū 13 )] +Ψ 23 [α 12 (2kū 01 − iū 11 ) + α 22 (2kū 02 − iū 12 ) +ᾱ 23 (2kū 03 − iū 13 )] +Ψ 33 [α 13 (2kū 01 − iū 11 ) + α 23 (2kū 02 − iū 12 ) + α 33 (2kū 03 − iū 13 )]
+iΨ 43 [α 11 |u 01 | 2 +ᾱ 12 u 01ū02 +ᾱ 13 u 01ū03 + α 12 u 02ū01 +α 22 |u 02 | 2 +ᾱ 23 u 02ū03 + α 13 u 03ū01 + α 23 u 03ū02 + α 33 |u 03 | 2 ] (t ′ , k)dt ′ ,
and
Ψ 14,t (t, k) = t 0 e −4ik 2 (t−t ′ ) (2ku 01 + iu 11 )Ψ 44 − iΨ 14 (α 11 |u 01 | 2 +ᾱ 12 u 01ū02 +ᾱ 13 u 01ū03 ) −iΨ 24 (α 12 |u 01 | 2 + α 22 u 01ū02 +ᾱ 23 u 01ū03 ) − iΨ 34 (α 13 |u 01 | 2 + α 23 u 01ū02 + α 33 u 01ū03 ) (t ′ , k)dt ′ , Ψ 24,t (t, k) = t 0 e −4ik 2 (t−t ′ ) (2ku 02 + iu 12 )Ψ 44 − iΨ 14 (α 11 u 02ū01 +ᾱ 12 |u 02 | 2 +ᾱ 13 u 02ū03 ) −iΨ 24 (α 12 u 02ū01 + α 22 |u 02 | 2 +ᾱ 23 u 02ū03 ) − iΨ 34 (α 13 u 02ū01 + α 23 |u 02 | 2 + α 33 u 02ū03 ) (t ′ , k)dt ′ , Ψ 34,t (t, k) = t 0 e −4ik 2 (t−t ′ ) (2ku 03 + iu 13 )Ψ 44 − iΨ 14 (α 11 u 03ū01 +ᾱ 12 u 03ū02 +ᾱ 13 |u 03 | 2 ) −iΨ 24 (α 12 u 03ū01 + α 22 u 03ū02 +ᾱ 23 |u 03 | 2 ) − iΨ 34 (α 13 u 03ū01 + α 23 u 03ū02 + α 33 |u 03 | 2 ) (t ′ , k)dt ′ , Ψ 44,t (t, k) =1 + t 0 Ψ 14 [α 11 (2kū 01 − iū 11 ) +ᾱ 12 (2kū 02 − iū 12 ) +ᾱ 13 (2kū 03 − iū 13 )] +Ψ 24 [α 12 (2kū 01 − iū 11 ) + α 22 (2kū 02 − iū 12 ) +ᾱ 23 (2kū 03 − iū 13 )] +Ψ 34 [α 13 (2kū 01 − iū 11 ) + α 23 (2kū 02 − iū 12 ) + α 33 (2kū 03 − iū 13 )]
+iΨ 44 [α 11 |u 01 | 2 +ᾱ 12 u 01ū02 +ᾱ 13 u 01ū03 + α 12 u 02ū01
+α 22 |u 02 | 2 +ᾱ 23 u 02ū03 + α 13 u 03ū01 + α 23 u 03ū02 + α 33 |u 03 | 2 ] (t ′ , k)dt ′ ,(92), u 1j } with {v 0j , v 1j }, (j = 1, 2, 3), that is, φ ij (t, k) = Φ ij (t, k) {u 0l (t)=v 0l (t), u 1l (t)=v 1l (t)} , (i, j = 1,+ ∂D 0 3 4k iπΣ − Ψ 14 (φ 44 − 1)e −2ikL − (Ψ 11 −1)(α 11φ41 +ᾱ 12φ42 +ᾱ 13φ43 ) −Ψ 12 (α 12φ41 + α 22φ42 +ᾱ 23φ43 )−Ψ 13 (α 13φ41 + α 23φ42 + α 33φ43 ) − dk,(93)u 12 (t) = ∂D 0 3 2Σ + iπΣ − (kΨ 24− + iu 02 ) + u 02 (2Ψ 44− −φ 44− ) dk + ∂D 0 3 4 πΣ − α 12 −ikφ 41− + α 11 v 01 + α 12 v 02 + α 13 v 03 +α 22 −ikφ 42− +ᾱ 12 v 01 + α 22 v 02 + α 23 v 03 +ᾱ 23 −ikφ 43− +ᾱ 13 v 01 +ᾱ 23 v 02 + α 33 v 03 dk + ∂D 0 3 4k iπΣ − Ψ 24 (φ 44 − 1)e −2ikL − Ψ 21 (α 11φ41 +ᾱ 12φ42 +ᾱ 13φ43 ) −(Ψ 22 −1)(α 12φ41 + α 22φ42 +ᾱ 23φ43 )−Ψ 23 (α 13φ41 + α 23φ42 + α 33φ43 ) − dk,(94)u 13 (t) = ∂D 0 3 2Σ + iπΣ − (kΨ 34− + iu 03 ) + u 03 (2Ψ 44− −φ 44− ) dk + ∂D 0 3 4 πΣ − α 13 −ikφ 41− + α 11 v 01 + α 12 v 02 + α 13 v 03 +α 23 −ikφ 42− +ᾱ 12 v 01 + α 22 v 02 + α 23 v 03 +α 33 −ikφ 43− +ᾱ 13 v 01 +ᾱ 23 v 02 + α 33 v 03 dk + ∂D 0 3 4k iπΣ − Ψ 34 (φ 44 − 1)e −2ikL − Ψ 31 (α 11φ41 +ᾱ 12φ42 +ᾱ 13φ43 ) −Ψ 32 (α 12φ41 + α 22φ42 +ᾱ 23φ43 )−(Ψ 33 −1)(α 13φ41 + α 23φ42 + α 33φ43 ) − dk,(95)
and v 1j (t) = 4φ
(2) j4 + 2 π ∂D 0 3 v 0j φ 44− dk, j = 1, 2, 3,(96)
where
φ (2) 14 φ (2) 24 φ (2) 34 = 1 4π ∂D 0 3 2iΣ + Σ − kφ 14− + iv 01 kφ 24− + iv 02 kφ 34− + iv 03 − Ψ (1) 44− v 01 v 02 v 03 dk+ M T 2iπ I 1 (t) I 2 (t) I 3 (t) ,
with I 1 (t) = ∂D 0 3 2 Σ − (α 2 11 + |α 12 | 2 + |α 13 | 2 )(kΨ 41− + i(ᾱ 11 u 01 + α 12 u 02 + α 13 u 03 )) +(α 11ᾱ12 +ᾱ 12 α 22 +ᾱ 13 α 23 )(kΨ 42− + i(ᾱ 12 u 01 + α 22 u 02 + α 23 u 03 )) +(α 11ᾱ13 +ᾱ 12ᾱ23 +ᾱ 13 α 33 )(kΨ 43− + i(ᾱ 13 u 01 +ᾱ 23 u 02 + α 33 u 03 )) dk 23 | 2 )(kΨ 43− + i(ᾱ 13 u 01 +ᾱ 23 u 02 + α 33 u 03 )) +(α 11 α 13 + α 12 α 23 +ᾱ 13 α 23 )(kΨ 41− + i(ᾱ 11 u 01 + α 12 u 02 + α 13 u 03 )) +(α 13ᾱ12 + α 22 α 23 +ᾱ 13 α 23 )(kΨ 42− + i(ᾱ 12 u 01 + α 22 u 02 + α 23 u 03 )) dk
+ ∂D 0 3 2k Σ − (1 −Ψ 44 )(α 11 φ 14 +ᾱ 12 φ 24 +ᾱ 13 φ 34 )e 2ikL +Ψ 41 α 11 (α 11 (φ 11 − 1) + α 12 φ 12 + α 13 φ 13 ) +ᾱ 12 (α 11 φ 21 + α 12 (φ 22 − 1) + α 13 φ 23 ) +ᾱ 13 (α 11 φ 31 + α 12 φ 32 + α 13 (φ 33 − 1)) +Ψ 42 α 11 (ᾱ 12 (φ 11 − 1) + α 22 φ 12 + α 23 φ 13 ) +ᾱ 12 (ᾱ 12 φ 21 + α 22 (φ 22 − 1) + α 23 φ 23 ) +ᾱ 13 (ᾱ 12 φ 31 + α 22 φ 32 + α 23 (φ 33 − 1)) +Ψ 43 α 11 (ᾱ 13 (φ 11 − 1) +ᾱ 23 φ 12 + α 33 φ 13 ) +ᾱ 12 (ᾱ 13 φ 21 +ᾱ 23 (φ 22 − 1) + α 33 φ 23 ) +ᾱ 13 (ᾱ 13 φ 31 +ᾱ 23 φ 32 + α 33 (φ 33 − 1)) − dk,(97)I 2 (t) = ∂D 0 3 2 Σ − (α 2 12 + α 2 22 + |α 23 | 2 )(kΨ 42− + iI 3 (t) = ∂D 0 3 2 Σ − (α 2 12 + α 2 22 + |α(98)+ ∂D 0 3 2k Σ − (1 −Ψ 44 )(α 13 φ 14 + α 23 φ 24 + α 33 φ 34 )e 2ikL +Ψ 41 α 13 (α 11 (φ 11 − 1) + α 12 φ 12 + α 13 φ 13 ) + α 23 (α 11 φ 21 + α 12 (φ 22 − 1) + α 13 φ 23 ) +α 33 (α 11 φ 31 + α 12 φ 32 + α 13 (φ 33 − 1)) +Ψ 42 α 13 (ᾱ 12 (φ 11 − 1) + α 22 φ 12 + α 23 φ 13 ) +α 23 (ᾱ 12 φ 21 + α 22 (φ 22 − 1) + α 23 φ 23 ) + α 33 (ᾱ 12 φ 31 + α 22 φ 32 + α 23 (φ 33 − 1)) +Ψ 43 α 13 (ᾱ 13 (φ 11 − 1) +ᾱ 23 φ 12 + α 33 φ 13 ) + α 23 (ᾱ 13 φ 21 +ᾱ 23 (φ 22 − 1) + α 33 φ 23 ) +α 33 (ᾱ 13 φ 31 +ᾱ 23 φ 32 + α 33 (φ 33 − 1)) − dk,(99)
(ii) For the known Neumann problem, the unknown Dirichlet boundary data {u 0j (t)} 3 j=1 and {v 0j (t)} 3 j=1 , 0 < t < T can be given by
u 01 (t) = ∂D 0 3 1 πΣ − Σ + Ψ 14+ − 2(α 11φ41 +ᾱ 12φ42 +ᾱ 13φ43 ) + dk + ∂D 0 3 2 πΣ − Ψ 14 (φ 44 − 1)e −2ikL − (Ψ 11 − 1)(α 11φ41 +ᾱ 12φ42 +ᾱ 13φ43 ) +Ψ 12 (α 12φ41 + α 22φ42 +ᾱ 23φ43 ) + Ψ 13 (α 13φ41 + α 23φ42 + α 33φ43 ) + dk, (100a) u 02 (t) = ∂D 0 3 1 πΣ − Σ + Ψ 24+ − 2(α 12φ41 + α 22φ42 +ᾱ 23φ43 ) + dk + ∂D 0 3 2 πΣ − Ψ 24 (φ 44 − 1)e −2ikL − Ψ 21 (α 11φ41 +ᾱ 12φ42 +ᾱ 13φ43 ) +(Ψ 22 − 1)(α 12φ41 + α 22φ42 +ᾱ 23φ43 ) + Ψ 23 (α 13φ41 + α 23φ42 + α 33φ43 ) + dk, (100b) u 03 (t) = ∂D 0 3 1 πΣ − Σ + Ψ 34+ − 2(α 13φ41 + α 23φ42 + α 33φ43 ) + dk + ∂D 0 3 2 πΣ − Ψ 34 (φ 44 − 1)e −2ikL − Ψ 31 (α 11φ41 +ᾱ 12φ42 +ᾱ 13φ43 ) +Ψ 32 (α 12φ41 + α 22φ42 +ᾱ 23φ43 ) + (Ψ 33 − 1)(α 13φ41 + α 23φ42 + α 33φ43 ) + dk,(100c)
and v 01 (t) = 2iφ
(1) 14 , v 02 (t) = 2iφ
(1) 24 , v 03 (t) = 2iφ
(1) 34 ,(101)
where
φ (1) 14 φ (1) 24 φ (1) 34 = − 1 2iπ ∂D 0 3 Σ + Σ − φ 14+ φ 24+ φ 34+ dk + M T 2iπ J 1 (t) J 2 (t) J 3 (t) , with J 1 (t) = ∂D 0 3 2 Σ − (α 2 11 + |α 12 | 2 + |α 13 | 2 )Ψ 41+ + (α 11ᾱ12 +ᾱ 12 α 22 +ᾱ 13 α 23 )Ψ 42+ +(α 11ᾱ13 +ᾱ 12ᾱ23 +ᾱ 13 α 33 )Ψ 43+ dk + ∂D 0 3 2 Σ − (1 −Ψ 44 )(α 11 φ 14 +ᾱ 12 φ 24 +ᾱ 13 φ 34 )e 2ikL +Ψ 41 α 11 (α 11 (φ 11 − 1) + α 12 φ 12 + α 13 φ 13 ) +ᾱ 12 (α 11 φ 21 + α 12 (φ 22 − 1) + α 13 φ 23 ) +ᾱ 13 (α 11 φ 31 + α 12 φ 32 + α 13 (φ 33 − 1)) +Ψ 42 α 11 (ᾱ 12 (φ 11 − 1) + α 22 φ 12 + α 23 φ 13 ) +ᾱ 12 (ᾱ 12 φ 21 + α 22 (φ 22 − 1) + α 23 φ 23 ) +ᾱ 13 (ᾱ 12 φ 31 + α 22 φ 32 + α 23 (φ 33 − 1)) +Ψ 43 α 11 (ᾱ 13 (φ 11 − 1) +ᾱ 23 φ 12 + α 33 φ 13 ) +ᾱ 12 (ᾱ 13 φ 21 +ᾱ 23 (φ 22 − 1) + α 33 φ 23 ) +ᾱ 13 (ᾱ 13 φ 31 +ᾱ 23 φ 32 + α 33 (φ 33 − 1)) + dk,(102)J 2 (t) = ∂D 0 3 2 Σ − (α 2 12 + α 2 22 + |α 23 | 2 )Ψ 42+ + (α 11 α 12 +ᾱ 23 α 22 +ᾱ 33 α 33 )Ψ 41+ +(α 12ᾱ13 + α 22ᾱ23 +ᾱ 23 α 33 )Ψ 43+ dk + ∂D 0 3 2 Σ − (1 −Ψ 44 )(α 12 φ 14 + α 22 φ 24 +ᾱ 23 φ 34 )e 2ikL
+Ψ 41 α 12 (α 11 (φ 11 − 1) + α 12 φ 12 + α 13 φ 13 ) + α 22 (α 11 φ 21 + α 12 (φ 22 − 1) + α 13 φ 23 ) +ᾱ 23 (α 11 φ 31 + α 12 φ 32 + α 13 (φ 33 − 1)) +Ψ 42 α 12 (ᾱ 12 (φ 11 − 1) + α 22 φ 12 + α 23 φ 13 )
+α 22 (ᾱ 12 φ 21 + α 22 (φ 22 − 1) + α 23 φ 23 ) +ᾱ 23 (ᾱ 12 φ 31 + α 22 φ 32 + α 23 (φ 33 − 1)) +Ψ 43 α 12 (ᾱ 13 (φ 11 − 1) +ᾱ 23 φ 12 + α 33 φ 13 ) + α 22 (ᾱ 13 φ 21 +ᾱ 23 (φ 22 − 1) + α 33 φ 23 ) +ᾱ 23 (ᾱ 13 φ 31 +ᾱ 23 φ 32 + α 33 (φ 33 − 1)) + dk,(103)J 3 (t) = ∂D 0 3 2 Σ − (α 2 12 + α 2 22 + |α 23 | 2 )Ψ 43+ + (α 11 α 13 + α 12 α 23 +ᾱ 13 α 23 )Ψ 41+ +(α 13ᾱ12 + α 22 α 23 +ᾱ 13 α 23 )Ψ 42+ dk + ∂D 0 3 2 Σ − (1 −Ψ 44 )(α 13 φ 14 + α 23 φ 24 + α 33 φ 34 )e 2ikL +Ψ 41 α 13 (α 11 (φ 11 − 1) + α 12 φ 12 + α 13 φ 13 ) + α 23 (α 11 φ 21 + α 12 (φ 22 − 1) + α 13 φ 23 ) +α 33 (α 11 φ 31 + α 12 φ 32 + α 13 (φ 33 − 1)) +Ψ 42 α 13 (ᾱ 12 (φ 11 − 1) + α 22 φ 12 + α 23 φ 13 ) +α 23 (ᾱ 12 φ 21 + α 22 (φ 22 − 1) + α 23 φ 23 ) + α 33 (ᾱ 12 φ 31 + α 22 φ 32 + α 23 (φ 33 − 1)) +Ψ 43 α 13 (ᾱ 13 (φ 11 − 1) +ᾱ 23 φ 12 + α 33 φ 13 ) + α 23 (ᾱ 13 φ 21 +ᾱ 23 (φ 22 − 1) + α 33 φ 23 ) +α 33 (ᾱ 13 φ 31 +ᾱ 23 φ 32 + α 33 (φ 33 − 1)) + dk,(104)
where Ψ 14 = Ψ 14 (t, k),φ 44 = φ 44 (t,k) =φ 44 (t,k) and other functions have the similar expressions.
Proof. We can show that Eqs. (87) and (88) hold by means of Eqs. (33) and (34) with replacing T by t, that is, S(k) = e −2ik 2 tσ4 µ −1 2 (0, t, k) and S L (k) = e −2ik 2 tσ4 µ −1 3 (L, t, k) and the symmetry relation (29). Moreover, Eqs. (89)-(92) for Ψ ij (t, k), i, j = 1, 2, 3, 4 can be obtained by using the Volteral integral equations of µ 2 (0, t, k). Similarly, the expressions of φ ij (t, k), (i, j = 1, 2, 3, 4) can be found by means of the Volteral integral equations of µ 3 (L, t, k).
In what follows we show Eqs. (93)-(101), that is the map between Dirichlet and Neumann boundary conditions. (i) The Cauchy's theorem is employed to study Eq. (58) to generate iπΨ (1)
44 (t) = − ∂D2 + ∂D4 [Ψ 44 (t, k) − 1]dk = ∂D1 + ∂D3 [Ψ 44 (t, k) − 1]dk = ∂D3 [Ψ 44 (t, k) − 1]dk − ∂D3 [Ψ 44 (t, −k) − 1]dk = ∂D3 Ψ 44− (t, k)dk,(105)
and iπΨ (2)
14 (t) = ∂D1 + ∂D3 kΨ 14 (t, k) + i 2 u 01 (t) dk, = ∂D3 kΨ 14 (t, k) + i 2 u 01 (t) − dk, = ∂D 0 3 kΨ 14 (t, k) + i 2 u 01 (t) + 2e −2ikL Σ − (k) kΨ 14 (t, k) + i 2 u 01 (t) − dk + C 1 (t), = ∂D 0 3 Σ + Σ − (kΨ 14− + iu 01 )dk + C 1 (t),(106)
where we have introduced the function C 1 (t) as
C 1 (t) = − ∂D 0 3 2e −2ikL Σ − kΨ 14 (t, k) + i 2 u 01 (t) − dk,
We use the global relation (68a) to further reduce C 1 (t) in the form
C 1 (t) = − ∂D 0 3 2e −2ikL Σ − kΨ 14 (t, k) + i 2 u 01 (t) − dk = ∂D 0 3 2e −2ikL Σ − −kc 14 + Ψ (1) 14 + Ψ (1) 14φ (1) 44 k − (α 11φ (1) 41 +ᾱ 12φ (1) 42 +ᾱ 13φ (1) 43 )e 2ikL − dk − ∂D 0 3 2e −2ikL Σ − Ψ (1) 14φ (1) 44 k + α 11 (kφ 41 −φ (1) 41 ) +ᾱ 12 (kφ 42 −φ (1) 42 ) +ᾱ 13 (kφ 43 −φ (1) 43 ) e 2ikL − dk + ∂D 0 3 2ke −2ikL Σ − Ψ 14 (φ 44 − 1) − (Ψ 11 − 1)(α 11φ41 +ᾱ 12φ42 +ᾱ 13φ43 ) +Ψ 12 (α 12φ41 + α 22φ42 +ᾱ 23φ43 ) + Ψ 13 (α 13φ41 + α 23φ42 + α 33φ43 ) e 2ikL − dk,(107)
By applying the Cauchy's theorem and asymptotic (64a) to Eq. (107), we find that C 1 (t) can reduce to
C 1 (t) = −iπΨ (2) 14 − ∂D 0 3 i 2 u 01φ44− + 2i Σ − α 11 −ikφ 41− + α 11 v 01 + α 12 v 02 + α 13 v 03 +ᾱ 12 −ikφ 42− +ᾱ 12 v 01 + α 22 v 02 + α 23 v 03 +ᾱ 13 −ikφ 43− +ᾱ 13 v 01 +ᾱ 23 v 02 + α 33 v 03 dk + ∂D 0 3 2k Σ − Ψ 14 (φ 44 − 1)e −2ikL − (Ψ 11 − 1)(α 11φ41 +ᾱ 12φ42 +ᾱ 13φ43 ) −Ψ 12 (α 12φ41 + α 22φ42 +ᾱ 23φ43 ) − Ψ 13 (α 13φ41 + α 23φ42 + α 33φ43 ) − dk,(108)
It follows from Eqs. (106) and (108) that we have 2iπΨ
(2)
13 (t) = ∂D 0 3 Σ + Σ − (kΨ 14− + iu 01 ) − i 2 u 01φ44− dk + ∂D 0 3 2i Σ − α 11 −ikφ 41− + α 11 v 01 +ᾱ 12 v 02 +ᾱ 13 v 03 +ᾱ 12 −ikφ 42− + α 12 v 01 + α 22 v 02 +ᾱ 23 v 03 +ᾱ 13 −ikφ 43− + α 13 v 01 + α 23 v 02 + α 33 v 03 dk + ∂D 0 3 2k Σ − Ψ 14 (φ 44 − 1)e −2ikL − (Ψ 11 − 1)(α 11φ41 +ᾱ 12φ42 +ᾱ 13φ43 ) −Ψ 12 (α 12φ41 + α 22φ42 +ᾱ 23φ43 ) − Ψ 13 (α 13φ41 + α 23φ42 + α 33φ43 ) − dk,(109)= ∂D3 α 11 (kφ 14 (t, k) − φ (1) 14 ) +ᾱ 12 (kφ 24 (t, k) − φ (1) 24 ) +ᾱ 13 (kφ 34 (t, k) − φ (1) 34 ) − dk, = ∂D 0 3 α 11 kφ 14 (t, k) − φ (1) 14 − 2e 2ikL Σ − (kφ 14 − φ (1) 14 ) +ᾱ 12 kφ 24 (t, k) − φ (1) 24 − 2e 2ikL Σ − (kφ 24 − φ (1) 24 ) +ᾱ 13 kφ 34 (t, k) − φ (1) 34 − 2e 2ikL Σ − (kφ 34 − φ (1) 34 ) − dk + C 2 (t), = ∂D 0 3 − Σ + Σ − α 11 (kφ 14− − 2φ (1) 14 ) +ᾱ 12 (kφ 24− − 2φ (1) 24 ) +ᾱ 13 (kφ 34− − 2φ(1)34 ) dk + C 2 (t),(110)
where we have introduced the function C 2 (t) as
C 2 (t) = ∂D 0 3 2e 2ikL Σ − α 11 (kφ 14 − φ (1) 14 ) +ᾱ 12 (kφ 24 − φ (1) 24 ) +ᾱ 13 (kφ 34 − φ (1) 34 ) − dk,
We use the global relation (83) to further reduce C 2 (t) in the form +Ψ 41 α 11 (α 11 (φ 11 − 1) + α 12 φ 12 + α 13 φ 13 ) +ᾱ 12 (α 11 φ 21 + α 12 (φ 22 − 1) + α 13 φ 23 ) +ᾱ 13 (α 11 φ 31 + α 12 φ 32 + α 13 (φ 33 − 1)) +Ψ 42 α 11 (ᾱ 12 (φ 11 − 1) + α 22 φ 12 + α 23 φ 13 )
C 2 (t) = ∂D 0 3 2 Σ − −kc 41 (t,k) − α 11 φ (1) 14 +ᾱ 12 φ (1) 24 +ᾱ 13 φ (1) 34 e 2ikL −Ψ(1)44 α 11 φ (1) 14 +ᾱ 12 φ (1) 24 +ᾱ 13 φ (1) 34 e 2ikL k + (α 2 11 + |α 12 | 2 + |α 13 | 2 )Ψ(1)+ᾱ 12 (ᾱ 12 φ 21 + α 22 (φ 22 − 1) + α 23 φ 23 ) +ᾱ 13 (ᾱ 12 φ 31 + α 22 φ 32 + α 23 (φ 33 − 1))
+Ψ 43 α 11 (ᾱ 13 (φ 11 − 1) +ᾱ 23 φ 12 +ᾱ 33 φ 13 ) +ᾱ 12 (ᾱ 13 φ 21 +ᾱ 23 (φ 22 − 1) + α 33 φ 23 )
+ᾱ 13 (ᾱ 13 φ 31 +ᾱ 23 φ 32 + α 33 (φ 33 − 1)) − dk,(111)
We need to further reduce C 2 (t) by using the asymptotic (65) and the Cauchy's theorem such that we have C 2 (t) in the form
C 2 (t) = −iπ[α 11 φ(2)
14 +ᾱ 12 φ
24 +ᾱ 13 φ
34 ] − ∂D 0 3 i 2Ψ(2)
44− (α 11 v 01 +ᾱ 12 v 02 +ᾱ 13 v 03 )dk + I 1 (t),
34 ] = − ∂D 0 3 Σ + (k) Σ − (k) [α 11 (kφ 14− + iv 01 ) +ᾱ 12 (kφ 24− + iv 02 +ᾱ 13 (kφ 34− + iv 03 )] dk − ∂D 0 3 i 2Ψ(1)
44− (α 11 v 01 +ᾱ 12 v 02 +ᾱ 13 v 03 )dk + I 1 (t),
where I(t) is given by Eq. (97). Similarly, we can also the expressions of iπ[α 12 φ
(2) 14 + α 22 φ(2)
24 +ᾱ 23 φ
14 (t) = ∂D1 + ∂D3 Ψ 14 (t, k)dk = ∂D3 Ψ 14− (t, k)dk = ∂D 0 3 Ψ 14− (t, k) + 2 Σ − (k) (e −2ikL Ψ 14 ) + dk + C 3 (t) = ∂D 0 3 Σ + (k) Σ − (k) Ψ 14+ dk + C 3 (t),(114)
where C 3 (t) is defined by
C 3 (t) = − ∂D 0 3 2 Σ − (k) (e −2ikL Ψ 14 ) + dk,(115)
By applying the global relation (68a), the Cauchy's theorem and asymptotics (64a) to Eq. (115), we find
C 3 (t) = − ∂D 0 3 2 Σ − (e −2ikL Ψ 14 ) + dk = ∂D 0 3 2 Σ − −c 14 e −2ikL − (α 11φ41 +ᾱ 12φ42 +ᾱ 13φ43 ) + dk + ∂D 0 3 2 Σ − Ψ 14 (φ 44 − 1)e −2ikL − (Ψ 11 − 1)(α 11φ41 +ᾱ 12φ42 +ᾱ 13φ43 ) +Ψ 12 (α 12φ41 + α 22φ42 +ᾱ 23φ43 ) + Ψ 13 (α 13φ41 + α 23φ42 + α 33φ43 ) + dk, = −iπΨ (1) 14 − ∂D 0 3 2 Σ − (α 11φ41 +ᾱ 12φ42 +ᾱ 13φ43 ) + dk + ∂D 0 3 2 Σ − Ψ 14 (φ 44 − 1)e −2ikL − (Ψ 11 − 1)(α 11φ41 +ᾱ 12φ42 +ᾱ 13φ43 ) +Ψ 12 (α 12φ41 + α 22φ42 +ᾱ 23φ43 ) + Ψ 13 (α 13φ41 + α 23φ42 + α 33φ43 ) + dk,(116)
Eqs. (114) and (116) imply that 2iπΨ (1)
14 (t) = ∂D 0 3 Σ + (k) Σ − (k) Ψ 14+ − 2 Σ − (α 11φ41 +ᾱ 12φ42 +ᾱ 13φ43 ) + dk + ∂D 0 3 2 Σ − Ψ 14 (φ 44 − 1)e −2ikL − (Ψ 11 − 1)(α 11φ41 +ᾱ 12φ42 +ᾱ 13φ43 ) +Ψ 12 (α 12φ41 + α 22φ42 +ᾱ 23φ43 ) + Ψ 13 (α 13φ41 + α 23φ42 + α 33φ43 ) + dk,(117)
The effective characterizations
Substituting the perturbated expressions for eigenfunctions and initial boundary conditions
Ψ ij (t, k) = Ψ [0]
ij (t, k) + ǫΨ [1] ij (t, k) + ǫ 2 Ψ [2] ij (t, k) + · · · , i, j = 1, 2, 3, 4,
φ ij (t, k) = φ [0]
ij (t, k) + ǫφ [1] ij (t, k) + ǫ 2 φ [2] ij (t, k) + · · · , i, j = 1, 2, 3, 4, u sj (t) = ǫu [1] sj (t) + ǫ 2 u [2] sj (t) + · · · , s = 0, 1; j = 1, 2, 3, v sj (t) = ǫv [1] sj (t) + ǫ 2 v
Ψ [1]
js = Ψ [1] 44 = 0, j, s, = 1, 2, 3,
Ψ [1] j4 = t 0 e −4ik 2 (t−t ′ ) 2ku [1]
0j + iu [1] 1j (t ′ )dt ′ , j = 1, 2, 3,
Ψ [1] 41 = t 0 e 4ik 2 (t−t ′ ) α 11 2kū [1] 01 −iū [1]
11 +ᾱ 12 2kū [1] 02 −iū [1] 12 +ᾱ 13 2kū
[1] 03 −iū [1] 13 (t ′ )dt ′ , Ψ [1] 42 = t 0 e 4ik 2 (t−t ′ ) α 12 2kū [1] 01 −iū [1]
11 +α 22 2kū [1] 02 −iū [1] 12 +ᾱ 23 2kū
[1] 03 −iū [1] 13 (t ′ )dt ′ , Ψ [1] 43 = t 0 e 4ik 2 (t−t ′ ) α 13 2kū [1]
01 −iū [1] 11 +α 23 2kū [1] 02 −iū [1] 12 +α 33 2kū
13 (t ′ )dt ′ ,(120)
Similarly, we can also obtain the analogous expressions for {φ [l] ij } 4 i,j=1 , l = 0, 1 by means of the boundary values at x = L, that is, {v
[l]
ij }, i = 0, 1; j = 1, 2, 3; l = 0, 1.
where 'lower order terms' stands for the result involving known terms of lower order.
The terms of O(ǫ n ) in Eqs. (89)-(92) and the similar equations for φ ij yield
Ψ [n] j4 (t, k) = t 0 e −4ik 2 (t−t ′ ) 2ku [n] 0j + iu [n]
1j (t ′ )dt ′ + lower order terms, j = 1, 2, 3,
φ [n] 41 (t,k) = t 0 e −4ik 2 (t−t ′ ) α 11 (
which leads to
Ψ [n] j4− (t, k) = 4k t 0 e −4ik 2 (t−t ′ ) u [n]
0j (t ′ )dt ′ + lower order terms, j = 1, 2, 3,
φ [n] 41− (t,k) = 4k t 0 e −4ik 2 (t−t ′ ) α 11 v [n] 01 + α 12 v [n] 02 + α 13 v [n]
03 (t ′ )dt ′ + lower order terms,
φ [n] 42− (t,k) = 4k t 0 e −4ik 2 (t−t ′ ) ᾱ 12 v [n] 01 + α 22 v [n] 02 + α 23 v [n]
03 (t ′ )dt ′ + lower order terms,
φ [n] 43− (t,k) = 4k t 0 e −4ik 2 (t−t ′ ) ᾱ 13 v [n] 01 +ᾱ 23 v [n] 02 + α 33 v [n]
03 (t ′ )dt ′ + lower order terms,
j4+ (t, k) = 2i t 0 e −4ik 2 (t−t ′ ) u [n]
1j (t ′ )dt ′ + lower order terms, j = 1, 2, 3,
φ [n] 41+ (t,k) = t 0 e −4ik 2 (t−t ′ ) α 11 v [n] 11 + α 12 v [n] 12 + α 13 v [n]
13 (t ′ )dt ′ + lower order terms,
φ [n] 42+ (t,k) = t 0 e −4ik 2 (t−t ′ ) ᾱ 12 v [n] 11 + α 22 v [n] 12 + α 23 v [n]
13 (t ′ )dt ′ + lower order terms,
φ [n]43+ (t,k) = t 0 e −4ik 2 (t−t ′ ) ᾱ 13 v [n] 11 ) +ᾱ 23 v [n] 12 + α 33 v [n]
13 (t ′ )dt ′ + lower order terms,
The large L limit from the interval to the half-line
The formulae for the initial and boundary value conditions u 0j (t) and u 1j (t), j = 1, 2, 2 of Theorem 4.2 in the limit L → ∞ can reduce to the corresponding ones on the half-line. Since when L → ∞,
v 0j → 0, v 1j → 0, j = 1, 2, 3, φ ij → δ ij , Σ + (k) Σ − (k) → 1 as k → ∞ in D 3 ,(126)u 1j (t) = 2 π ∂D 0 3 [u 0j (Ψ 44− + 1) − ikΨ j4− ] dk, j = 1, 2, 3,(127)
for the given Dirichlet boundary problem, and the unknown Dirichlet boundary data
u 0j (t) = 1 π ∂D 0 3 Ψ j4+ dk, j = 1, 2, 3,(128)
for the given Neumann boundary problem.
The GLM representation and equivalence
In this section we deduce the eigenfunctions Ψ(t, k) and φ(t, k) in terms of the Gel'fand-Levitan-Marchenko (GLM) approach [25][26][27][28]. Moreover, the global relation can be used to find the unknown Neumann (Dirichlet) boundary values from the given Dirichlet (Neumann) boundary values by means of the GLM representations.
Moreover, the GLM representations are shown to be equivalent to the ones obtained in Sec. 4. Finally, the linearizable boundary conditions are presented for the GLM representations.
The GLM representation
Proposition 5.1. The eigenfunctions Ψ(t, k) and φ(t, k) possess the GLM representation
Ψ(t, k) = I + t −t L(t, s) + k + i 2 U (0) σ 4 G(t, s) e −2ik 2 (s−t)σ4 ds, (129a) φ(t, k) = I + t −t L(t, s) + k + i 2 U (L) σ 4 G(t, s) e −2ik 2 (s−t)σ4 ds,(129b)
where the 4 × 4 matrix-valued functions L(t, s) = (L ij ) 4×4 and G(t, s)
= (G ij ) 4×4 , −t ≤ s ≤ t satisfy a Goursat system L t (t, s)+σ 4 L s (t, s)σ 4 = iσ 4 U (0) x L(t, s)− 1 2 (U (0) ) 3 +iU (0) σ 4 + [U (0) x , U (0) ] G(t, s), G t (t, s) + σ 4 G s (t, s)σ 4 = 2U (0) L(t, s) + iσ 4 U (0) x G(t, s),(130)
with the initial conditions
L lj (t, −t) = L 44 (t, −t) = 0, l, j = 1, 2, 3, G lj (t, −t) = G 44 (t, −t) = 0, l, j = 1, 2, 3, G 14 (t, t) = u 01 (t), G 24 (t, t) = u 02 (t), G 34 (t, t) = u 03 (t), G 41 (t, t) = α 11ū01 (t) +ᾱ 12ū02 (t) +ᾱ 13ū03 (t), G 42 (t, t) = α 12ū01 (t) + α 22ū02 (t) +ᾱ 23ū03 (t), G 43 (t, t) = α 13ū01 (t) + α 23ū02 (t) + α 33ū03 (t), L 14 (t, t) = i 2 u 11 (t), L 24 (t, t) = i 2 u 12 (t), L 34 (t, t) = i 2 u 13 (t), L 41 (t, t) = − i 2 (α 11ū11 (t) +ᾱ 12ū12 (t) +ᾱ 13ū13 (t)), L 42 (t, t) = − i 2 (α 12ū11 (t) + α 22ū12 (t) +ᾱ 23ū13 (t)), L 43 (t, t) = − i 2 (α 13ū11 (t) + α 23ū12 (t) + α 33ū13 (t)),(131)U (0) = 0 0 0 u 01 (t) 0 0 0 u 02 (t) 0 0 0 u 03 (t) p 01 (t) p 02 (t) p 03 (t) 0 , U (0) x = 0 0 0 u 11 (t) 0 0 0 u 12 (t) 0 0 0 u 13 (t) p 11 (t) p 12 (t) p 13 (t) 0 ,(132)
with p 01 = α 11ū01 +ᾱ 12ū02 +ᾱ 13ū03 , p 02 = α 12ū01 + α 22ū02 +ᾱ 23ū03 , p 03 = α 13ū01 + α 23ū02 + α 33ū03 , p 11 = α 11ū11 +ᾱ 12ū12 +ᾱ 13ū13 , p 12 = α 12ū11 + α 22ū12 +ᾱ 23ū13 , p 13 = α 13ū11 + α 23ū12 + α 33ū13 , Similarly, L(t, s), G(t, s) satisfy the similar Eqs. (130) and (131)
with u 0j → v 0j , u 1j → v 1j , U (0) → U (L) = U (0) u0j →v0j , U (0) x → U (L) x = U (0) x u1j →v1j .
Proof. We assume that the function
ψ(t, k) = e −2ik 2 tσ4 + t −t [L 0 (t, s) + kG(t, s)]e −2ik 2 sσ4 ds,(133)
satisfies the time-part of Lax pair (4) with the boundary data ψ(0, k) = I at x = 0, where L 0 (t, s) and G(t, s) are the unknown 4 × 4 matrix-valued functions. We substitute Eq. (133) into the time-part of Lax pair (4) with the boundary data (3) and use the identity
t −t F (t, s)e −2ik 2 sσ4 ds = i 2k 2 F (t, t)e −2ik 2 tσ4 − F (t, −t)e 2ik 2 tσ4 − t −t F s (t, s)e −2ik 2 sσ4 ds σ 4 ,(134)
where the function F (t, s) is a 4 × 4 matrix-valued function. As a consequence, we find
L 0 (t, −t) + σ 4 L 0 (t, −t)σ 4 = −iU (0) G(t, −t)σ 4 , G(t, −t) + σ 4 G(t, −t)σ 4 = 0, L 0 (t, t) − σ 4 L 0 (t, t)σ 4 = iU (0) G(t, t)σ 4 + V (0) 0 , G(t, t) − σ 4 G(t, t)σ 4 = 2U (0) , L 0t (t, s) + σ 4 L 0s (t, s)σ 4 = −iU (0) G s (t, s)σ 4 + V (0) 0 L 0 (t, s), G t (t, s) + σ 4 G s (t, s)σ 4 = 2U (0) L 0 (t, s) + V (0) 0 G(t, s),(135)
where U (0) is given by Eq. (132) and
V (0) 0 = −i(U (0) x + U (0)2 )σ 4 = −i L(t, s) = L 0 (t, s) − i 2 U (0) σ 4 G(t, s),(136)
such that the first four equations of system (135) become
L(t, −t) + σ 4 L(t, −t)σ 4 = 0, G(t, −t) + σ 4 G(t, −t)σ 4 = 0, L(t, t) − σ 4 L(t, t)σ 4 = V (0) 0 , G(t, t) − σ 4 G(t, t)σ 4 = 2U (0) ,
which leads to Eq. (131), and from the last two equations of system (135) we have Eq. (130). By means of transformation (7), that is, µ 2 (0, t, k) = Ψ(t, k) = ψ(t, k)e 2ik 2 tσ4 , we know that Ψ(t, k) is given by Eq. (129a). Similarly, we can also show that Eq. (129b) holds.
For convenience, we rewrite a 4 × 4 matrix C = (C ij ) 4×4 as
C = (C ij ) 4×4 = C 3×3Cj4 C 4j C 44 ,C 3×3 = (C ij ) 3×3 ,C j4 = (C 14 , C 24 , C 34 ) T ,C 4j = (C 41 , C 42 , C 43 ),
The Dirichlet and Neumann boundary values at x = 0, L are simply written as
u j (t) = (u j1 (t), u j2 (t), u j3 (t)), v j (t) = (v j1 (t), v j2 (t)
, v j3 (t)), j = 1, 2, 3, w j0 (t) = (p j1 (t), p j2 (t), p j3 (t)), w jL (t) = (p j1 (t), p j2 (t), p j3 (t))| usj →vsj , s = 0, 1; j = 1, 2, 3,
For a matrix-valued function F (t, s), we introduce theF (t, k) bŷ
F (t, k) = t −t F (t, s)e 2ik 2 (s−t) ds,
Thus, the GLM expressions (129a) and (129b) of {Ψ ij , φ ij } can be rewritten as
Ψ 3×3 (t, k) = I +L 3×3 − i 2 u T 0 (t)Ĝ 4j + kĜ 3×3 , Ψ j4 (t, k) =L j4 − i 2 u T 0 (t)Ĝ 44 + kĜ j4 , j = 1, 2, 3, Ψ 4j (t, k) =L 4j + i 2ū 0 (t)MĜ 3×3 + kĜ 4j , j = 1, 2, 3, Ψ 44 (t, k) = 1 +L 44 + i 2ū 0 (t)MĜ j4 + kĜ 44 ,(138a) φ 3×3 (t, k) = I +L 3×3 − i 2 v T 0 (t)Ĝ 4j + kĜ 3×3 , φ j4 (t, k) =L j4 − i 2 v T 0 (t)Ĝ 44 + kĜ j4 , j = 1, 2, 3, φ 4j (t, k) =L 4j + i 2v 0 (t)MĜ 3×3 + kĜ 4j , j = 1, 2, 3, φ 44 (t, k) = 1 +L 44 + i 2v 0 (t)MĜ j4 + kĜ 44 ,(138b)
For the given Eqs. (138a) and (138b) we have the following proposition.
Proposition 5.2. lim t ′ →t ∂D 0 1 ke 4ik 2 (t−t ′ ) Σ − F j4 e −2ikL − dk = ∂D 0 1 ik 2 u T 0 Ĝ 44 −Ḡ 44 + k Σ − F j4 e −2ikL − dk, (139a) lim t ′ →t ∂D 0 1 ke 4ik 2 (t−t ′ ) Σ −F 4j− dk = ∂D 0 1 ik 2 M T v T 0 Ĝ 44 −Ḡ 44 + k Σ −F 4j− dk,(139b)lim t ′ →t ∂D 0 1 e 4ik 2 (t−t ′ ) Σ − F j4 e −2ikL + dk = ∂D 0 1 1 Σ − F j4 e −2ikL + dk,(139c)lim t ′ →t ∂D 0 1 e 4ik 2 (t−t ′ ) Σ −F 4j+ dk = ∂D 0 1 1 Σ −F 4j+ dk,(139d)
where the vector-valued functionsF j4 (t, k) andF 4j (t, k) (j = 1, 2, 3) are defined bỹ
F j4 (t, k) = − i 2 u T 0Ĝ 44 + i 2 M TḠT 3×3 Mv T 0 e 2ikL + L j4 − i 2 u T 0Ĝ 44 + kĜ j4 L 44 − i 2Ḡ T j4 Mv T 0 + kḠ 44 − L 3×3 − i 2 u T 0Ĝ4j + kĜ 3×3 M T L T 4j − i 2Ḡ T 3×3 Mv T 0 + kḠ T 4j e 2ikL ,(140)F 4j (t, k) = − i 2Ḡ T 3×3 Mu T 0 + i 2 M T v T 0Ĝ44 e 2ikL +M T L 3×3 − i 2 v T 0Ĝ 4j + kĜ 3×3 M T L T 4j − i 2Ḡ T 3×3 Mu T 0 + kḠ T 4j −M T L j4 − i 2 v T 0Ĝ 44 + kĜ j4 L 44 − i 2Ḡ T j4 Mu T 0 + kḠ 44 e 2ikL ,(141)
Proof. Similar to the proof of Lemma 4.3 in Ref. [31], we here show Eq. (139a) in detail. We multiply Eq. (140) by k Σ− e 4ik 2 (t−t ′ ) with 0 < t ′ < t and integrate along along ∂D 0 1 with respect to dk to yield
∂D 0 1 k Σ − e 4ik 2 (t−t ′ ) (F j4 e −2ikL ) − dk = ∂D 0 1 ik 2 e 4ik 2 (t−t ′ ) u T 0Ĝ44 dk − ∂D 0 1 k 3 e 4ik 2 (t−t ′ )Ĝ j4Ḡ44 dk − ∂D 0 1 ke 4ik 2 (t−t ′ ) L j4 − i 2 u T 0Ĝ44 L 44 − i 2Ḡ T j4 Mv T 0 dk + ∂D 0 1 k 2 Σ + Σ − e 4ik 2 (t−t ′ ) L j4 − i 2 u T 0Ĝ 44 Ḡ 44 +Ĝ j4 L 44 − i 2Ḡ j4 Mv T 0 dk − ∂D 0 1 2k 2 Σ − e 4ik 2 (t−t ′ ) L 3×3 − i 2 u T 0Ĝ4j M TḠT 4j +Ĝ 3×3 M T L T 4j − i 2Ḡ T 3×3 Mv T 0 dk,(142)
To further analyse the above equation, the following identities are introduced
∂D1 ke 4ik 2 (t−t ′ )F (t, k)dk = π 2 F (t, 2t ′ − t), 0 < t ′ < t, π 4 F (t, t), 0 < t ′ = t,(143)
and
∂D 0 1 k 2 Σ − e 4ik 2 (t−t ′ )F (t, k)dk = 2 ∂D 0 1 k 2 Σ − t ′ 0 e 4ik 2 (t−t ′ )F (t, 2τ − t)dτ − F (t, 2t ′ − t) 4ik 2 dk,(144)
which also holds for the cases that k 2 Σ− is taken place by k 2 Σ+ Σ− or k 2 . It follows from the first integral on the RHS of Eq. (142) and Eq. (143) that we have
lim t ′ →t ∂D 0 1 ik 2 e 4ik 2 (t−t ′ ) u T 0Ĝ44 dk = lim t ′ →t iπ 2 u T 0G22 (t, 2t ′ − t) = iπ 4 u T 0G44 (t, t),(145a)lim t ′ →t ∂D 0 1 ik 2 e 4ik 2 (t−t ′ ) u T 0Ĝ 44 dk = ∂D 0 1 ik 2 u T 0Ĝ 44 dk = iπ 8 u T 0G 44 (t, t),(145b)
Therefore, we know that the first integral on the RHS of Eq. (142) yields the following two terms
lim t ′ →t ∂D 0 1 ik 2 e 4ik 2 (t−t ′ ) u T 0Ĝ 44 dk = ∂D 0 1 ik 2 u T 0Ĝ 44 dk (145a) + ∂D 0 1 ik 2 u T 0Ĝ 44 dk (145b) ,(146)
Nowadays we study the second integral on the RHS of Eq. (142). It follows from the second integral on the RHS of Eq. (142) and Eq. (144) that we have
− ∂D 0 1 k 3 e 4ik 2 (t−t ′ )Ĝ j4Ḡ44 dk = −2 ∂D 0 1 k 3 t 0 e 4ik 2 (τ −t ′ )G j4 (t, 2τ − t)Ḡ 44 dτ dk = −2 ∂D 0 1 k 3 t ′ 0 e 4ik 2 (τ −t ′ )G j4 (t, 2τ − t)dτ −G j4 (t, 2t ′ − t) 4ik 2 Ḡ 44 dk,(147)
Thus we take the limit t ′ → t of Eq. (147) to have
− lim t ′ →t ∂D 0 1 k 3 e 4ik 2 (t−t ′ )Ĝ j4Ḡ44 dk = − ∂D 0 1 k 3Ĝ j4Ḡ44 dk + ∂D 0 1 k 2i u T 0Ḡ44 dk
Finally, following the proof in Ref. [31] we can show the limits t ′ → t of the rest three integrals (i.e., the third, fourth and fifth integrals) of Eq. (142) can be deduced by simply making the limit t ′ → t inside the every integral, that is, no additional terms arise in these integrals. For example,
lim t ′ →t ∂D 0 1 ke 4ik 2 (t−t ′ ) L j4 − i 2 u T 0Ĝ 44 L 44 − i 2Ḡ T j4 Mv T 0 dk = ∂D 0 1 k L j4 − i 2 u T 0Ĝ 44 L 44 − i 2Ḡ T j4 Mv T 0 dk.
Thus we complete the proof of Eq. (139a). Similarly, we can show that Eqs. (139b), (139c) and (139d) also hold. 87) and (88) with Ψ(t, k) and φ(t, k) given by Eq. (129a) and (129b).
(i) For the given Dirichlet boundary values u 0 (t) and v 0 (t), the unknown Neumann boundary values u 1 (t) and v 1 (t) are given by
u T 1 (t) = 4 iπ ∂D 0 1 Σ + Σ − k 2Ĝ j4 (t, t) + i 2 u T 0 (t) − 2M T Σ − k 2ḠT 4j (t, t) + i 2 Mv T 0 (t) + ik 2 u T 0 Ĝ 44 −Ḡ 44 + k Σ − F j4 e −2ikL − dk, (148a) v T 1 (t) = 4 iπ ∂D 0 1 − Σ + Σ − k 2Ĝ j4 (t, t) + i 2 v T 0 (t) + 2M T Σ − k 2ḠT 4j (t, t) + i 2 Mu T 0 (t) + ik 2 v T 0 Ĝ 44 −Ḡ 44 + k Σ − M TF 4j− dk,(148b)
(ii) For the given Neumann boundary values u 1 (t) and v 1 (t), the unknown Dirichlet boundary values u 0 (t) and
v 0 (t) are given by
u T 0 (t) = 2 π ∂D 0 1 Σ + Σ −L j4 − 2M T Σ −L T 4j + 1 Σ − F j4 e −2ikL + dk, (149a) v T 0 (t) = 2 π ∂D 0 1 2M T Σ −L T j4 − 1 Σ −L 4j + M T Σ −F 4j+ dk,(149b)
whereF j4 (t, k) andF 4j (t, k) are defined by Eqs. (140) and (141).
Proof. By means of the global relation (56) and Proposition 5.1, we can show that the spectral functions S(k) and S L (k) are defined by Eqs. (87) and (88) with Ψ(t, k) and φ(t, k) given by Eq. (129a) and (129b).
(i) we firstly consider the Dirichlet problem. It follows from the global relation (56) with the vanishing initial data c(t, k) = µ 2 (0, t, k)e ikLσ4 µ −1 3 (L, t, k),
that we findc
j4 (t, k) = −Ψ 3×3 M TφT 4j (t,k)e 2ikL +Ψ j4φ44 (t,k), (151a) c 4j (t, k) =Ψ 4j M TφT 3×3 (t,k)M T −Ψ 44φ T j4 (t,k)M T e −2ikL ,(151b)
Substituting Eqs. (138a) and (138b) into Eq. (151a) yields
M TLT 4j e 2ikL −L j4 = kĜ j4 − kM TḠT 4j e 2ikL +F j4 (t, k) −c j4 (t, k),(152)
whereF j4 (t, k) is given by Eq. (140). Eq. (152) with k → −k further yields
M TLT 4j e −2ikL −L j4 = −kĜ j4 + kM TḠT 4j e −2ikL +F j4 (t, −k) −c j4 (t, −k),(153)
It follows from Eqs (152) and (153) that we get
L j4 = kΣ + Σ −Ĝ j4 − 2k Σ − M TḠT 4j + 1 Σ − [F j4 (t, k) −c j4 (t, k)]e −2ikL − .(154)
We multiply Eq. (154) by ke 4ik 2 (t−t ′ ) with 0 < t ′ < t and integrate them along ∂D 0 1 with respect to dk, respectively to yield
∂D 0 1 ke 4ik 2 (t−t ′ )L j4 dk = ∂D 0 1 e 4ik 2 (t−t ′ ) k 2 Σ + Σ −Ĝ j4 dk − ∂D 0 1 e 4ik 2 (t−t ′ ) 2k 2 Σ − M TḠT 4j dk + ∂D 0 1 ke 4ik 2 (t−t ′ ) Σ − [F j4 (t, k)e −2ikL ] − dk,(155)
where we have used
∂D 0 1 ke 4ik 2 (t−t ′ )c j4− (t, k)dk = ∂D 0 1 ke 4ik 2 (t−t ′ ) (c j4 (t, k)e −π 2L j4 (t, 2t ′ − t) = 2 ∂D 0 1 k 2 Σ + Σ − t ′ 0 e 4ik 2 (t−t ′ )G j4 (t, 2τ − t)dτ −G j4 (t, 2t ′ − t) 4ik 2 dk −4 ∂D 0 1 k 2 M T Σ − t ′ 0 e 4ik 2 (t−t ′ )ḠT 4j (t, 2τ − t)dτ −Ḡ T 4j (t, 2t ′ − t) 4ik 2 dk + ∂D 0 1 k Σ − e 4ik 2 (t−t ′ ) [F j4 (t, k)e −2ikL ] − dk,(156)
We choose the limit t ′ → t of Eq. (156) with the initial data (131) and Proposition 5.2 to find π 2L j4 (t, t) = 2 lim
t ′ →t ∂D 0 1 k 2 Σ + Σ − t ′ 0 e 4ik 2 (t−t ′ )G j4 (t, 2τ − t)dτ −G j4 (t, 2t ′ − t) 4ik 2 dk −4 lim t ′ →t ∂D 0 1 k 2 M T Σ − t ′ 0 e 4ik 2 (t−t ′ )ḠT 4j (t, 2τ − t)dτ −Ḡ T 4j (t, 2t ′ − t) 4ik 2 dk + lim t ′ →t ∂D 0 1 k Σ − e 4ik 2 (t−t ′ ) [F j4 (t, k)e −2ikL ] − dk = ∂D 0 1 Σ + Σ − k 2Ĝ j4 (t, t) + i 2G j4 (t, t) − 2M T Σ − k 2ḠT 4j (t, t) + i 2Ḡ T 4j (t, t) + ik 2 u T 0 Ĝ 44 −Ḡ 44 + k Σ − F j4 e −2ikL − dk,(157)
Since the initial data (131) are of the form
L j4 (t, t) = i 2 u T 1 (t) = i 2 (u 11 (t), u 12 (t), u 13 (t)) T ,(158)
then we know that Eq. (148a) holds ny means of Eqs. (157) and (158).
To show Eq. (148b) we rewrite Eq. (151b) in the form
c T 4j (t,k) = M Tφ 3×3 M TΨT 4j (t,k) − M Tφ j4Ψ T 44 (t,k)e 2ikL ,(159)
We substitute Eqs. (138a) and (138b) into Eq. (159) to have
−L T 4j + M TL j4 e 2ikL = kḠ T 4j − kM TĜ j4 e 2ikL +F 4j (t, k) −c T 4j (t,k),(160)
whereF 4j (t, k) is given by Eq. (141). Eq. (160) with k → −k yields
−L T 4j + M TL j4 e −2ikL = −kḠ T 4j + kM TĜ j4 e −2ikL +F 4j (t, −k) −c T 4j (t, −k),(161)
It follows from Eqs. (160) and (161) that we have
M TL j4 = 2k Σ −Ḡ T 4j − kΣ + Σ − M TĜ j4 + 1 Σ − [F 4j (t, k) −c T 4j (t,k)] −(162)
We multiply Eq. (162) by ke 4ik 2 (t−t ′ ) with 0 < t ′ < t, integrate them along ∂D 0 1 with respect to dk, and use these conditions given by Eqs. (143) and (144) to yield
π 2 M TL j4 (t, 2t ′ − t) = −2 ∂D 0 1 k 2 Σ + Σ − M T t ′ 0 e 4ik 2 (t−t ′ )G j4 (t, 2τ − t)dτ −G j4 (t, 2t ′ − t) 4ik 2 dk +4 ∂D 0 1 k 2 Σ − t ′ 0 e 4ik 2 (t−t ′ )ḠT 4j (t, 2τ − t)dτ −Ḡ T 4j (t, 2t ′ − t) 4ik 2 dk + ∂D 0 1 k Σ − e 4ik 2 (t−t ′ )F 4j− (t, k)dk,(163)
where we have used the relation
∂D 0 1 k Σ − e 4ik 2 (t−t ′ )cT 4j− (t,k)dk = 0
due to the analytical property of the integrand in D 0 1 . We consider the limit t ′ → t of Eq. (163) with the initial data (131) and Proposition 5.2 to have
π 2 M TL j4 (t, t) = ∂D 0 1 − Σ + Σ − M T k 2Ĝ j4 (t, t) + i 2G j4 (t, t) + 2 Σ − k 2ḠT 4j (t, t) + i 2Ḡ T 4j (t, t) + ik 2 M T v T 0 Ĝ 44 −Ḡ 44 + k Σ −F 4j− (t, k) dk,(164)
Since the initial conditions are of the form
L j4 (t, t) = i 2 v T 1 (t) = i 2 (v 11 (t), v 12 (t), v 13 (t)) T ,(165)
then we have Eq. (148b) by combining Eqs. (164) and (165).
(ii) We now turn to consider the Neumann problem. It follows from Eqs (152), (153), (160) and (161) that we haveĜ
j4 = 1 kΣ − Σ +Lj4 − 2M TLT 4j + (F j4 (t, k) −c j4 (t, k))e −2ikL + ,(166a)G j4 = 1 kΣ − 2M TLT j4 − Σ +Lj4 + M T F 4j (t, k) −c T 44 (t,k) + .(166b)
We multiply Eqs. (166a) and (166b) by ke 4ik 2 (t−t ′ ) with 0 < t ′ < t, integrate them along ∂D 0 1 with respect to dk, and use these conditions given by Eqs. (143) and (144) to yield
π 2G j4 (t, 2t ′ − t) = ∂D 0 1 2Σ + Σ − t ′ 0 e 4ik 2 (t−t ′ )L j4 (t, 2τ − t)dτ −L j4 (t, 2t ′ − t) 4ik 2 dk − ∂D 0 1 4M T Σ − t ′ 0 e 4ik 2 (t−t ′ )LT 4j (t, 2τ − t)dτ −L T 4j (t, 2t ′ − t) 4ik 2 dk + ∂D 0 1 e 4ik 2 (t−t ′ ) Σ − (F j4 e −2ikL ) + dk, (167a) π 2G j4 (t, 2t ′ − t) = ∂D 0 1 4M T Σ − t ′ 0 e 4ik 2 (t−t ′ )LT j4 (t, 2τ − t)dτ −L T j4 (t, 2t ′ − t) 4ik 2 dk − ∂D 0 1 2 Σ − t ′ 0 e 4ik 2 (t−t ′ )L 4j (t, 2τ − t)dτ −L 4j (t, 2t ′ − t) 4ik 2 dk + ∂D 0 1 M T Σ − e 4ik 2 (t−t ′ )F 4j+ dk,(167b)
where we have used the analytical property of the matrix-valued functions We consider the limits t ′ → t of Eqs. (167a) and (167b) with the initial data (131) and Proposition 5.2 to find π 2G j4 (t, t) =
∂D 0 1 Σ + Σ −L j4 − 2M T Σ −L T 4j + 1 Σ − (F j4 e −2ikL ) + dk, (168a) π 2G j4 (t, t) = ∂D 0 1 2M T Σ −L T j4 − 1 Σ −L 4j + M T Σ −F 4j+ dk,(168b)
Since the initial conditions are of the form G j4 (t, t) = u T 0 (t) = (u 01 (t), u 02 (t), u 03 (t)) T ,G j4 (t, t) = v T 0 (t) = (v 01 (t), v 02 (t), v 03 (t)) T ,
then we have Eqs. (149a) and (149b) by using Eqs. (168a) and (168b). This completes the proof of the Theorem.
Equivalence of the two distinct representations
We now show that the above-mentioned GLM representation for the Dirichlet and Neumann boundary data in Theorem 5.3 is equivalent to one in Theorem 4.2.
Case i. From the Dirichlet boundary conditions to the Neumann boundary ones
It follows from Eqs. (138a) and (138b) that we obtaiñ
G j4 = 1 2kΨ j4− ,Ĝ j4 = 1 2kφ j4− ,Ĝ 44 = 1 2kΨ 44− ,Ĝ 44 = 1 2kφ 44− ,(170)
Substituting Eqs. (140) and (170) into Eq. (148a) yields
u T 1 (t) = 4 iπ ∂D 0 1 Σ + Σ − k 2Ĝ j4 (t, t) + i 2 u T 0 (t) − 2M T Σ − k 2ḠT 4j (t, t) + i 2 Mv T 0 (t) +iku T 0Ĝ44 + k 2i u T 0Ḡ44 + k Σ − Ψ j4 (φ 44 − 1)e −2ikL − (Ψ 3×3 − I)M TφT 4j − dk = ∂D 0 1 2Σ + iπΣ − kΨ j4− + iu T 0 (t) + 4iM T πΣ − kφ T 4j− + iMv T 0 (t) + 1 π u T 0 (2Ψ 44− −φ 44− ) + 4k iπΣ − Ψ j4 (φ 44 − 1)e −2ikL − (Ψ 3×3 − I)M TφT 4j − dk,(171)
Since the integrand in Eq. (171) is an odd function about k, which makes sure that the contour ∂D 0 1 can be replaced by ∂D 0 3 , thus we can find the same Neumann boundary data u 1j (t) (j = 1, 2, 3) at x = 0 given by Eqs. (93)-(95) from Eq. (171). Similarly, we can also find the Neumann boundary data v 1j (t) (j = 1, 2, 3) at x = L given by Eq. (96) from Eq. (148b).
Case ii. From the Neumann boundary conditions to the Dirichlet boundary ones
Eqs. (138a) and (138b) imply that
L j4 = 1 2Ψ j4+ (t, k) + i 2 u T 0Ĝ 44 ,L T 4j = 1 2φ T 4j+ (t, k) + i 2Ḡ T 3×3 Mv T 0 ,(172)
The substitution of Eqs. (172) and (140) into Eq. (149a) yields
u T 0 (t) = 2 π ∂D 0 1 Σ + Σ −L j4 − 2M T Σ −L T 4j + 1 Σ − F j4 e −2ikL + dk = ∂D 0 1 Σ + πΣ −Ψ j4+ − 2M T πΣ −φ T 4j+ + 2 πΣ − Ψ j4 (φ 44 (t,k)−1)e −2ikL −(Ψ 3×3 −I)M TφT 4j + dk,(173)
Since the integrand in Eq. (173) is an odd function about k, which makes sure that the contour ∂D 0 1 can be replaced by ∂D 0 3 , thus Eq. (173) yields the Dirichlet boundary values u 0j (t), j = 1, 2, 3 again. Similarly, we can also deduce the Dirichlet boundary values v 0j (t), j = 1, 2, 3 from Eq. (149b).
Linearizable boundary conditions for the GLM representation
In what follows we further explore the linearizable boundary conditions for the GLM representation given in Theorem 5.3.
Proposition 5.4. Let q j (x, t = 0) = q 0j (x), j = 1, 2, 3 be the initial conditions of the gtc-NLS equation (1) on the interval x ∈ [0, L], and one of the following boundary conditions, either (i) the Dirichlet boundary conditions at x = 0, L, q j (x = 0, t) = u 0j (t) = 0 and q j (x = L, t) = v 0j (t) = 0, j = 1, 2, 3, or (ii) the Robin boundary conditions x = 0, L, q jx (x = 0, t) − χq j (x = 0, t) = u 1j (t) − χu 0j (t) = 0, j = 1, 2, 3 and q jx (x = L, t) − ϑq j (x = L, t) = v 1j (t) − ϑv 0j (t) = 0, j = 1, 2, where χ and ϑ are both real parameters.
Then the eigenfunctions Ψ(t, k) and φ(t, k) can be expressed as
where the 4 × 4 matrix-valued function L(t, s) = (L ij ) 4×4 satisfies a reduced Goursat system
L 3×3t +L 3×3s = iu T 1 (t)L 4j ,
L j4t −L j4s = iu T 1 (t)L 44 , j = 1, 2, 3, L 4jt −L 4js = −iū 1 (t)ML 3×3 , j = 1, 2, 3,
L 44t +L 44s = −iū 1 (t)ML j4 ,(175)
with the initial data (cf. Eq. (131)) L 3×3 (t, −t) = 0 3×3 ,L 44 (t, −t) = 0,L j4 (t, t) = i 2 u T 1 (t),L 4j (t, t) = − i 2ū
1 (t)M,(176)
Similarly, the 4 × 4 matrix-valued function L(t, s) = (L ij ) 4×4 satisfies the analogous system (175) with u 1 (t)
replaced by v 1 (t).
(ii)
Ψ(t, k) = I + L 3×3Lj4 L 4jL44 + − i 2 u T 0 (t)Ĝ 4j kĜ j4 kĜ 4j i 2ū 0 (t)MĜ j4 ,(177a)φ(t, k) = I + L 3×3Lj4 L 4jL44 + − i 2 v T 0 (t)Ĝ 4j kĜ j4 kĜ 4j i 2v 0 (t)MĜ j4 ,(177b)
where the 4 × 4 matrix-valued functions L(t, s) = (L ij ) 4×4 and G(t, s) = (G ij ) 4×4 satisfy the reduced nonlinear
Goursat system
L 3×3t +L 3×3s = iχu T 0 (t)L 4j + 1 2 iu T 0 (t) − u T 0 (t)ū 0 (t)Mu T 0 (t) G 4j , L 44t +L 44s = −iχū 0 (t)ML j4 − 1 2 iu 0 (t)M +ū 0 (t)Mu T 0 (t)ū 0 (t)M G j4 , L j4t −L j4s = iχu T 0 (t)L 44 , L 4jt −L 4js = −iχū 0 (t)ML 3×3 , G j4t −G j4s = 2u T 0 (t)L 44 , G 4jt −G 4js = 2ū 0 (t)ML 3×3 ,(178)
with the initial data (cf. Eq. (131))
L 3×3 (t, −t) = 0 3×3 , L 44 (t, −t) = 0, L j4 (t, t) = i 2 χu T 0 (t), L 4j (t, t) = − i 2 χū 0 (t)M, G j4 (t, t) = u T 0 (t), G 4j (t, t) =ū 0 (t)M,(179)
Similarly, the 4 × 4 matrix-valued functions L(t, s) = (L ij ) 4×4 and G(t, s) = (G ij ) 4×4 satisfy the similar system (178) with u 0 (t) and χ replaced by v 0 (t) and ϑ, respectively.
Proof. Let us show that the linearizable boundary data correspond to the special cases of Proposition 5.1. Case (i) The Dirichlet zero boundary data q j (x = 0, t) = u 0j (t) = 0. It follows from the second one of system (130) thatG ij (t, s) satisfy
G 3×3t +G 3×3s = iu T 1 (t)G 4j , G j4t −G j4s = iu T 1 (t)G 44 , G 4jt −G 4js = −iū 1 (t)MG 3×3 , G 44t +G 44s = −iū 1 (t)MG j4 ,(180)
with the initial data (cf. Eq. (131)) G 3×3 (t, −t) = 0 3×3 ,G 44 (t, −t) = 0,G j4 (t, t) = 0 j4 ,G 4j (t, t) = 0 4j ,
Thus the unique solution of Eq. (180) is trivial, that is,G 3×3 (t, s) = 0,G 4j (t, s) = 0,G j4 (t, s) = 0,G 44 (t, s) = 0 such that Eq. (129a) reduces to Eq. (174a) and the condition (130) with (131) becomes (175) with (176). Similarly, for the Dirichlet zero boundary data q j (x = L, t) = v 0j (t) = 0, j = 1, 2, 3, we can also show Eq. (174b).
(ii) Consider the Robin boundary data q jx (x = 0, t) − χq j (x = 0, t) = u 1j (t) − χu 0j (t) = 0, (j = 1, 2, 3), that is, the Dirichlet and Neumann boundary data have the linear relation u 1 (t) = χu 0 (t).
(182)
We introduce a 4
It follows from Eq. (130) and (183) with Eq. (182) thatQ ij (t, s),G ij (t, s), i, j = 1, 2 satisfy
Q 3×3t +Q 3×3s = iu T 0 (t) − u T 0 (t)ū 0 (t)Mu T 0 (t) + χ 2 u T 0 (t) G 4j , Q j4t −Q j4s = iu T 0 (t) − u T 0 (t)ū 0 (t)Mu T 0 (t) + χ 2 u T 0 (t) G 44 , Q 4jt −Q 4js = −ū 0 (t)Mu T 0 (t)ū 0 (t)M − iu 0 (t)M + χ 2ū 0 (t)M G 3×3 , Q 44t +Q 44s = −ū 0 (t)Mu T 0 (t)ū 0 (t)M − iu 0 (t)M + χ 2ū 0 (t)M G j4 , G 3×3t +G 3×3s = u T 0 (t)Q 4j , G j4t −G j4s = u T 0 (t)Q 44 , G 4jt −G 4js =ū 0 (t)MQ 3×3 , G 44t +G 44s =ū 0 (t)MQ j4 ,(184)
with the initial data (cf. Eq. (131)) G 3×3 (t, −t) = 0 3×3 ,G 44 (t, −t) = 0,G j4 (t, t) = u T 0 (t),G 4j (t, t) =ū 0 (t)M, Q 3×3 (t, −t) = 0 3×3 ,Q 44 (t, −t) = 0,Q j4 (t, t) = 0 j4 ,Q 4j (t, t) = 0 4j ,
Thus the unique solution of Eq. (184) is trivial, that is,Q j4 (t, s) =Q 4j (t, s) =G 3×3 (t, s) =G 44 (t, s) = 0 such that Eq. (129a) reduces to Eq. (177a) and the condition (130) Similarly, for the Robin boundary data q jx (x = L, t) − ϑq j (x = L, t) = v 1j (t) − ϑv 0j (t) = 0, j = 1, 2, 3, that is, v 1 (t) = ϑv 0 (t), we can also show Eq. (177b).
Based on the Theorem 5.3 and Proposition 5.4, we have the following Proposition.
Proposition 5.5 For the linearizable Dirichlet boundary data u 0 (t) = v 0 (t) = 0, we have the Neumann boundary data u 1 (t) and v 1 (t):
u T 1 (t) = 4i π ∂D 0 1 kΨ j4 (φ 44 − I)dk, v T 1 (t) = 4i π ∂D 0 1 kφ j4 (Ψ 44 − I)dk,(186)
where
Ψ j4t + 4ik 2Ψ j4 = iu T 1 (t)(Ψ 44 + I), Ψ 44t = −iū 1 (t)MΨ j4 , φ j4t + 4ik 2φ j4 = iv T 1 (t)(φ 44 + I),
φ 44t = −iv 1 (t)Mφ j4 .
Remark 5.6. The analogous analysis of the Fokas unified method will use also to explore the IBV problems for other integrable nonlinear evolution PDEs with 4 × 4 Lax pairs both on the the half-line and the finite interval, such as the three-component derivative-NLS equation and the three-component higher-order NLS equation, which will be considered in other papers.
keep invariant: (i) Finding an integral representation of the solution in terms of a matrix RH problem formulated in the complex k-plane (k is a spectral parameter of the associated Lax pair). The integral representation in general contains the unknown boundary data such that this expression of the solution is not effective yet; (ii) Applying a global relation to consider the unknown boundary values. The representation of the unknown boundary values in general involves the solution of a nonlinear problem. But, this problem for the linearizable boundary conditions can be ignored since the unknown boundary values can be avoided in terms of only algebraic operations.
α 12 α 13 α 12 α 22 α 23 α 13ᾱ23 α 33 , M = M † , M 2 = I.
4 × 4
4Lax pair of Eq. (1), such as the eigenfunctions, the jump matrices, and the global relation. Sec. 3 gives the corresponding 4 × 4 matrix RH problem by means of the jump matrices obtained in Sec. 2. The global relation is used to establish the map between the Dirichlet and Neumann boundary values in Sec. 4. Particularly, the relevant formulae for boundary value problems on the finite interval can reduce to ones on the half-line as the length of the interval approaches to infinity. In Sec. 5, we present the Gelfand-Levitan-Marchenko (GLM)
Figure 1 :
1Contours γ j (j = 1, 2, 3, 4) from points (x j , t j ) to (x, t) in the region Ω = {(x, t)|x ∈ [0, L], t ∈ [0, T ]}.
that we introduce the four eigenfunctions {µ j } 4 1 on the four contours {γ j
Figure 2 :
2The regions D n (n = 1, 2, 3, 4) separating the complex k-plane.
Proposition 2. 1 .
1For the matrix-valued functions M n (x, t, k) (n = 1, 2, 3, 4) defined by Eq. (17) for k ∈D n and (x, t) ∈ Ω, and any fixed point (x, t), M n (x, t, k)'s are the bounded and analytic functions of k ∈ D n away from a possible discrete set of singularity {k j } at which the Fredholm determinants vanish. Moreover, M n (x, t, k)'s admit the bounded and continuous extensions toD n and M n (x, t, k) = I + O 1 k , k ∈ D n , k → ∞, n = 1, 2, 3, 4.
Figure 3 :
3The relations among µ j (x, t, k), j = 1, 2, 3, 4.
24 (S)n 1l,24 (s) − m 34 (S)n 1l,34 (s) + m 44 (S)n 1l,44 (s) (s T S A ) 14 (S)n 2l,14 (s) − m 34 (S)n 2l,34 (s) + m 44 (S)n 2l,44 (s) (s T S A ) 14 (S)n 3l,14 (s) − m 24 (S)n 3l,24 (s) + m 44 (S)n 3l,44 (s) (s T S A ) 14 (S)n 4l,14 (s) − m 24 (S)n 4l,24 (s) + m 34 (S)n 4l,34 (s) (s T S A )
2. 8 .
8The residue conditions for M n Since µ 2 (x, t, k) is an entire function, it follows from Eq. (39) that M n (x, t, k) only have singularities at the points where the S n (k)'s have singularities. We find from the expressions of S n (k) given by Eq. (40) that the possible singularities of M n are as follows: • [M ] 4 could admit poles in D 1 at the zeros of m 44 (S)(k); • [M ] 4 could have poles in D 2 at the zeros of (S T s A ) 44 (k); • [M ] l , l = 1, 2, 3 could be of poles in D 3 at the zeros of (s T S A ) 44 (k); • [M ] l , l = 1, 2, 3 could have poles in D 4 at the zeros of S 44 (k).
[M 2
2] 4 = [M 2 ] 1 [S 14 n 22,43 (s) − S 24 n 12,43 (s) + S 44 n 12,23 (s)] (S T s A ) 44 m 34 (s) e 2θ − [M 2 ] 2 [S 14 n 21,43 (s) − S 24 n 11,43 (s) + S 44 n 11,23 (s)] (S T s A ) 44 m 34 (s) e 2θ + [M 2 ] 3 [S 14 n 21,42 (s) − S 24 n 11,42 (s) + S 44 n 11,22 (s)] (S T s A ) 44 m 34 (s) e 2θ − [µ 2 ] 3 m 34 (s) e 2θ , whose residues at k j yields Eq. (46) for k j ∈ D 2 , respectively. Similarly, we can show Eq. (47) for k j ∈ D 3 and Eq. (48) for k j ∈ D 4 by analyzing Eqs. (51a)-(52d).
, 4 )
4given by Eq. (23) as well as the spectral functions S(k), s(k) and S(k) defined by Eq. (33). Assume that the possible zeros {k j } N 1 of the functions m 44 (S)(k), (S T s A ) 44 (k), (s T S A ) 44 (k), and S 44 (k) are as in Assumption 2.4. Then the solution
Thus, the column vectors [c(t, k)] j , j = 1, 2, 3 are analytic and bounded in D 4 away from the possible zeros of S 44 (k) and of order O( 1+e −2ikL k ) as k → ∞, and the column vector [c(t, k)] 4 is analytic and bounded in D 1 away from the possible zeros of m 44 (S)(k) and of order O( 1+e 2ikL k ) as k → ∞,
The functions {φ ij (t, k)} 4 i,j=1 are of the same integral equations (89)-(92) by replacing the functions {u 0j
the given Dirichlet problem, the unknown Neumann boundary data {u 1j (t)} 3 j=1 and {v 1j (t)} 3 j=1 , iπΣ − (kΨ 14− + iu 01 ) + u 01 (2Ψ 44− −φ 44− ) − α 11 −ikφ 41− + α 11 v 01 + α 12 v 02 + α 13 v 03 +ᾱ 12 −ikφ 42− +ᾱ 12 v 01 + α 22 v 02 + α 23 v 03 +ᾱ 13 −ikφ 43− +ᾱ 13 v 01 +ᾱ 23 v 02 + α 33 v 03 dk
(ᾱ 12 u 01 + α 22 u 02 + α 23 u 03 )) +(α 11 α 12 +ᾱ 23 α 22 +ᾱ 33 α 33 )(kΨ 41− + i(ᾱ 11 u 01 + α 12 u 02 + α 13 u 03 )) +(α 12ᾱ13 + α 22ᾱ23 +ᾱ 23 α 33 )(kΨ 43− + i(ᾱ 13 u 01 +ᾱ 23 u 02 + α 33 u 03 )) −Ψ 44 )(α 12 φ 14 + α 22 φ 24 +ᾱ 23 φ 34 )e 2ikL +Ψ 41 α 12 (α 11 (φ 11 − 1) + α 12 φ 12 + α 13 φ 13 ) + α 22 (α 11 φ 21 + α 12 (φ 22 − 1) + α 13 φ 23 ) +ᾱ 23 (α 11 φ 31 + α 12 φ 32 + α 13 (φ 33 − 1)) +Ψ 42 α 12 (ᾱ 12 (φ 11 − 1) + α 22 φ 12 + α 23 φ 13 ) +α 22 (ᾱ 12 φ 21 + α 22 (φ 22 − 1) + α 23 φ 23 ) +ᾱ 23 (ᾱ 12 φ 31 + α 22 φ 32 + α 23 (φ 33 − 1)) +Ψ 43 α 12 (ᾱ 13 (φ 11 − 1) +ᾱ 23 φ 12 + α 33 φ 13 ) + α 22 (ᾱ 13 φ 21 +ᾱ 23 (φ 22 − 1) + α 33 φ 23 ) +ᾱ 23 (ᾱ 13 φ 31 +ᾱ 23 φ 32 + α 33 (φ 33 − 1)) − dk,
41 +
41(α 11ᾱ12 +ᾱ 12 α 22 +ᾱ 13 α 23 )Ψ(1) 42 +(α 11ᾱ13 +ᾱ 12ᾱ23 +ᾱ 13 α 33 )Ψ |α 12 | 2 + |α 13 | 2 )(kΨ 41 −Ψ (1) 41 ) + (α 11ᾱ12 +ᾱ 12 α 22 +ᾱ 13 α 23 )(kΨ 42 −Ψ 11ᾱ13 +ᾱ 12ᾱ23 +ᾱ 13 α 33 )(kΨ 43 −Ψ Ψ 44 )(α 11 φ 14 +ᾱ 12 φ 24 +ᾱ 13 φ 34 )e 2ikL
such that we can show that Eq. (96) holds. (ii) We now deduce the Dirichlet boundary value problems (100a)-(100c) at x = 0 from the Neumann boundary value problems. It follows from the first one of Eq. (60) that u 01 (t) can be expressed by means of Ψ
Thus, substituting Eq. (117) into the first one of Eq. (60) yields Eq. (100a). Similarly, by applying the expressions of Ψ
t) to the second one of Eq. (60), we can find Eqs. (100b) and (100c). Similarly we also can show that the Dirichlet boundary value problems (101) at x = L hold from the Neumann boundary value problems.
. (70)-(73), where ǫ > 0 is a small parameter, we have these terms of O(1), and O(ǫ)
Thus, according to Eq. (126), the L → ∞ limits of Eqs. (93), (94), (100a), and (100b) yield the unknown Neumann boundary data
u
01 p 01 u 01 p 02 u 01 p 03 −u 11 u 02 p 01 u 02 p 02 u 02 p 03 −u 21 u 03 p 01 u 03 p 02 u 03 p 03 −u 31 p 11 p 21 p 31 −(u 01 p 01 + u 02 p 02 + u 03 p 03 )To reduce system (135) we further introduce the new matrix L(t, s) by
Theorem 5. 3 .
3Let q 0j (x) = q j (x, t = 0) = 0, j = 1, 2, 3 be the initial data of Eq. (1) on the interval x ∈ [0, L] and T < ∞. For the Dirichlet problem, the boundary data u 0j (t) and v 0j (t) (j = 1, 2, 3) on the interval t ∈ [0, T ) are sufficiently smooth and compatible with the initial data q j0 (x) (j = 1, 2, 3) at the points (x 2 , t 2 ) = (0, 0) and (x 3 , t 3 ) = (L, 0), respectively. For the Neumann problem, the boundary data u 1j (t) and v 1j (t) (j = 1, 2, 3) on the interval t ∈ [0, T ) are sufficiently smooth and compatible with the initial data q 0j (x) (j = 1, 2, 3) at the points (x 2 , t 2 ) = (0, 0) and (x 3 , t 3 ) = (L, 0), respectively. For simplicity, let n 33,44 (S)(k) have no zeros in the domain D 1 . Then the spectral functions S(k) and S L (k) are defined by Eqs. (
2ikL ) − dk = 0 in terms of their analytical properties in D 0 1 . Based on these conditions given by Eqs. (143) and (144), Eq. (155) can become
e 4ik 2 (t−t ′ ) (c j4 (t, k)e −2ikL ) + dk = e 4ik 2 (t−t ′ )cT 4j+ (t,k)dk = 0.
3×3
t, s) = 2L(t, s) − iχσ 4 G(t, s) by the linear combinations of L and G such that we have (t, s) = 2L 3×3 (t, s) − iχG 3×3 (t, s), Q j4 (t, s) = 2L j4 (t, s) − iχG j4 (t, s), Q 4j (t, s) = 2L 4j (t, s) + iχG 4j (t, s), Q 44 (t, s) = 2L 44 (t, s) + iχG 44 (t, s),
with Eq. (131) becomes Eq. (178) with Eq. (179).
j=1 (p j q jx − p jx q j )dt,
AcknowledgmentsThis work was partially supported by the NSFC under Grant No.11571346 and the Youth Innovation Promotion Association, CAS.
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"ON p-GROUPS WITH AUTOMORPHISM GROUPS RELATED TO THE EXCEPTIONAL CHEVALLEY GROUPS",
"ON p-GROUPS WITH AUTOMORPHISM GROUPS RELATED TO THE EXCEPTIONAL CHEVALLEY GROUPS"
] |
[
"Saul D Freedman "
] |
[] |
[] |
LetĜ be the finite simply connected version of an exceptional Chevalley group defined over F q , with q a power of an odd prime p. Additionally, let V be a nontrivial irreducible F q [Ĝ]-module of minimal dimension. We determine the submodule structure of the exterior square of V , and we explore the overgroup structure ofĜ in GL(V ), with a focus on maximal subgroups of certain classical groups that containĜ. WhenĜ is of type E 6 or E 7 and p > 3, we also study the submodule structure of the third Lie power of V . Using the information about submodule and overgroup structures in the case q = p, we construct a p-group P of exponent p, exponent-p class r and nilpotency class r, where r := 3 ifĜ is of type E 6 or E 7 and r := 2 otherwise. Moreover, the group A(P ) induced by Aut(P ) on P/Φ(P ) is the normaliser ofĜ in GL(V ). For each groupĜ of type G 2 or E 8 , we also construct a p-group P of exponent p 2 , exponent-p class 2 and nilpotency class 2 such that A(P ) =Ĝ. In each case, no p-group Q with A(Q) = A(P ) is smaller than P , in terms of exponent-p class, exponent and nilpotency class, and also in terms of order when P has class 2.
|
10.1080/00927872.2020.1760873
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[
"https://arxiv.org/pdf/1810.08365v1.pdf"
] | 119,327,594 |
1810.08365
|
681adb6b54a771a515c8883cebd5eae61f50b02d
|
ON p-GROUPS WITH AUTOMORPHISM GROUPS RELATED TO THE EXCEPTIONAL CHEVALLEY GROUPS
19 Oct 2018
Saul D Freedman
ON p-GROUPS WITH AUTOMORPHISM GROUPS RELATED TO THE EXCEPTIONAL CHEVALLEY GROUPS
19 Oct 2018arXiv:1810.08365v1 [math.GR]
LetĜ be the finite simply connected version of an exceptional Chevalley group defined over F q , with q a power of an odd prime p. Additionally, let V be a nontrivial irreducible F q [Ĝ]-module of minimal dimension. We determine the submodule structure of the exterior square of V , and we explore the overgroup structure ofĜ in GL(V ), with a focus on maximal subgroups of certain classical groups that containĜ. WhenĜ is of type E 6 or E 7 and p > 3, we also study the submodule structure of the third Lie power of V . Using the information about submodule and overgroup structures in the case q = p, we construct a p-group P of exponent p, exponent-p class r and nilpotency class r, where r := 3 ifĜ is of type E 6 or E 7 and r := 2 otherwise. Moreover, the group A(P ) induced by Aut(P ) on P/Φ(P ) is the normaliser ofĜ in GL(V ). For each groupĜ of type G 2 or E 8 , we also construct a p-group P of exponent p 2 , exponent-p class 2 and nilpotency class 2 such that A(P ) =Ĝ. In each case, no p-group Q with A(Q) = A(P ) is smaller than P , in terms of exponent-p class, exponent and nilpotency class, and also in terms of order when P has class 2.
Introduction
Let P be a finite p-group, and let A(P ) denote the group induced by Aut(P ) on the Frattini quotient P/Φ(P ) of P . More explicitly, A(P ) is the image of the natural homomorphism from Aut(P ) to Aut(P/Φ(P )). By Burnside's Basis Theorem, we can identify this Frattini quotient with the vector space F d p , where d is the rank of P , i.e., the minimum size of a generating set for P . Thus we can identify A(P ) naturally with a subgroup of the general linear group GL(d, p).
Observe that if P is the elementary abelian p-group of rank d, then A(P ) = GL(d, p). Thus in order to induce a proper subgroup of GL(d, p) on a p-group, this p-group must either have nilpotency class at least 2 or exponent at least p 2 , i.e., exponent-p class at least 2. Recall that the nilpotency class of a p-group P is the length of the lower central series of P , defined by γ 1 (P ) := P and γ i+1 (P ) := [γ i (P ), P ] for each i 1. Similarly, the exponent-p class of P is the length of the lower exponent-p central series of P , defined by P 1 (P ) := P and P i+1 (P ) := [P i (P ), P ]P i (P )p for each i 1. Here, if G is a group, then we write Gp to denote the characteristic subgroup x p | x ∈ G of G.
We now summarise some known results about the group A(P ). First and foremost, Bryant and Kovács [9] showed that any given subgroup of GL(d, p), with d > 1, can be induced on the Frattini quotient of some p-group. However, the p-group in their proof of this result has an exponent-p class comparable to |GL(d, p)| (depending on both d and p), and hence a huge order. It is therefore natural to ask when these properties of the p-group can in fact be relatively small. With this question in mind, Bamberg, Glasby, Morgan and Niemeyer [4] investigated the maximal subgroups of GL(d, p), again with d > 1. They showed that if p > 3, then such a maximal subgroup can be induced on a p-group of nilpotency class (and exponent-p class) 2, 3 or 4, exponent p, and order at most p d 4 /2 , as long as the maximal subgroup does not contain SL(d, p) and lies in a certain subset of the Aschbacher classes of GL(d, p). The Aschbacher classes of classical groups describe all of their maximal subgroups, via Aschbacher's Theorem [1]; we will define these classes later in this paper.
Among the maximal subgroups considered by Bamberg et al. are the full groups of similarities in GL(d, p) of bilinear forms on F d p . Together with GL(d, p) itself, these groups are the normalisers in GL(d, p) of the universal covers (or particular quotients of these universal covers in finitely many cases) of the classical Chevalley groups of dimension d defined over F p . In [3], Bamberg, Freedman and Morgan began a programme to explore the exceptional groups of Lie type in a similar way. In particular, for each odd prime p, they constructed a p-group P of order p 14 , nilpotency class 2 and exponent p (and hence exponent-p class 2) such that A(P ) is the normaliser of the exceptional Chevalley group G 2 (p) in GL (7, p). Note that 7 is the minimal dimension of a nontrivial irreducible F p [G 2 (p)]-module.
In this paper, we extend the aforementioned programme of study by constructing a "small" p-group in the case of each exceptional Chevalley group defined over F p (with p odd, and with p > 3 in some cases). In order to make this more precise, we require the following definition. Definition 1.1. Let H be a subgroup of GL(d, p), and let P H be the set of p-groups P of rank d such that A(P ) = H. We say that a p-group P ∈ P H is optimal with respect to H if:
(i) no group in P H has a smaller exponent-p class than P ;
(ii) no group in P H with the same exponent-p class as P has a smaller exponent than P ; (iii) no group in P H with the same exponent-p class and exponent as P has a smaller nilpotency class than P ; and (iv) no group in P H with the same exponent-p class, nilpotency class and exponent as P has a smaller order than P .
We can now summarise our main theorem, which we prove in §7, and which incorporates the aforementioned main result of [3]. Theorem 1.2. LetĜ ∈ {G 2 (p), F 4 (p), E 8 (p)} and let r := 2, or letĜ be the universal cover of E 6 (p) or of E 7 (p) and let r := 3, where p > r in either case. Additionally, let d be the minimal dimension of a nontrivial irreducible F p [Ĝ]-module. Then each optimal pgroup with respect to N GL(d,p) (Ĝ) has exponent-p class r, nilpotency class r and exponent p. Furthermore, ifĜ ∈ {G 2 (p), E 8 (p)}, then each optimal p-group with respect to the subgroup G of GL(d, p) has exponent-p class 2, nilpotency class 2 and exponent p 2 .
We will see in §3 that the groupĜ is the (finite) simply connected version of the corresponding exceptional Chevalley group. In fact, when r = 2, we construct an optimal p-group of the normaliser ofĜ in GL(d, p) (and ofĜ itself whenĜ = F 4 (p)). WhenĜ is the universal cover of E 6 (p) or of E 7 (p), we construct a p-group that satisfies the first three conditions of Definition 1.1 and a stronger version of condition (iv). In each case, we construct the p-group as a particular quotient of a suitable universal p-group, i.e., the largest p-group that satisfies a certain set of properties. Although the optimal p-group with respect to N GL(7,p) (G 2 (p)) was previously constructed by Bamberg, Freedman and Morgan, they did not induce G 2 (p) itself on the Frattini quotient of a p-group. We will show later in the paper that the construction of each p-group related toĜ relies on knowledge of the stabilisers in GL(d, p) of the submodules of a certain F p [Ĝ]-module.
Let V be a nonzero vector space over a field F. For elements u and v of the tensor algebra T (V ) of V , we can define a bracket operation [·, ·] :
T (V ) × T (V ) → T (V ) by [u, v] := u ⊗ v − v ⊗ u.
We use left-normed notation to denote brackets of brackets, for example, [u, v, w] := [[u, v], w] for u, v, w ∈ T (V ). The free Lie algebra L(V ) on V is the smallest subspace of T (V ) that contains V and that is closed under [·, ·], while the i-th Lie power L i V of V is the intersection of L(V ) and the i-th tensor power of V (see [4, §2], [21, §1]). It is easy to show that L(V ) is indeed a Lie algebra with Lie bracket [·, ·]; that L k V is spanned by the set {[v 1 , v 2 , . . . , v k ] | v 1 , v 2 , . . . , v k ∈ V }; and that GL(V ) acts linearly on L k V , with [v 1 , . . . , v k ] α := [v α 1 , . . . , v α k ] for all v 1 , . . . , v k ∈ V and all α ∈ GL(V ). It follows immediately that when char(F) = 2, L 2 V is isomorphic as an F[GL(V )]-module to the exterior square A 2 V of V . Additionally, Bamberg et al. [4,Lemma 3.1] showed that if char(F) = 2 / ∈ {2, 3}, then L 3 V ∼ = (A 2 V ⊗ V )/A 3 V . The F p [Ĝ]-module of importance is the Lie power L r V , where V is a minimal F p [Ĝ]module, i.e., a nontrivial irreducible F p [Ĝ]-module of minimal dimension d. In fact, the structure of Lie powers of modules is of general interest -see [21, §5] for a discussion of cases where this structure is known, and examples of applications. We therefore consider the more general case whereĜ is defined over any field F q of odd characteristic p, and determine the stabilisers in GL(d, q) of eachĜ-submodule of L r V . In order to do so, we prove results about the groupsĜ and the modules L r V that are interesting in their own right. We now summarise some of our most significant findings. (i) The submodule structure of L 2 V ∼ = A 2 V is given in Table 2.
(ii) Suppose that p > 3, and thatĜ is of type E 6 (with q = p if p = 5) or E 7 (with q = p if p ∈ {7, 11, 19}). Then the submodule structure of L 3 V is given in Figures 1-6.
Theorem 1.4. Let β be a reflexive bilinear form on V preserved byĜ, if such a form exists, and otherwise let β be the zero form. Additionally, let S be the last group in the derived series of the full group of isometries of β in SL(d, q).
(i) The normaliser ofĜ in S is the unique maximal subgroup of S that containsĜ.
(ii) If β is nonzero, then the full group of similarities of β in SL(d, q) is the unique maximal subgroup of SL(d, q) that containsĜ.
The proofs of these theorems are given in §4 and §5, respectively. We also show that the submodule structure of A 2 V is equivalent (in terms of containments and dimensions) to that of the exterior square of the F p [Ĝ]-module constructed from V by extending the scalars, and that of the exterior square of a related minimal F p [G]-module, where G is the simply connected simple linear algebraic group associated withĜ. This equivalence also holds between the corresponding third Lie powers, as long as p is not an exceptional prime listed in Theorem 1.3. Note that the submodule structure of A 2 V in the case wherê G = G 2 (q) was previously explored, in less detail, in [3] and [32,Ch. 9.3.2].
We now specify the structure of our paper. First, we discuss in §2 how we can use the knowledge of the submodule structures of V , L 2 V and L 3 V (each defined over F p ) and the stabilisers in GL(d, p) of their submodules to construct a p-group P as a quotient of an appropriate universal p-group such that A(P ) is as required. In §3, we use highest weight theory to determine the composition factors of the second and third Lie powers of the minimal modules over F p for the simply connected simple linear algebraic groups associated with exceptional Chevalley groups. We then use this information in §4 in order to determine the submodule structures of the second and third Lie powers of these modules, and of the modules for the groupsĜ, as detailed above. Next, in §5, we determine part of the overgroup structure ofĜ in GL(d, q). This overgroup structure, and the submodule structures of the second and third Lie powers of V (defined over F q ) are then used in §6 to determine the stabiliser in GL(d, q) of each submodule of these Lie powers. We state and prove the full version of our main theorem in §7, and then conclude in §8 with remarks about some open questions.
Universal p-groups
Let d > 1 be an integer, let p be a prime, and let V := F d p . In this section, we discuss the relation between the submodule structures of V , L 2 V and L 3 V and the group induced by Aut(P ) on P/Φ(P ), where P is a certain type of quotient of an appropriate universal p-group.
First, let B be the free Burnside group B(d, p) of rank d and exponent p, i.e., the largest group of rank d and exponent p. For each positive integer r, we write Γ(d, p, r) := B/γ r+1 (B). This is a finite group of rank d, exponent p and nilpotency class at most r. In fact, if there exists a p-group of rank d, exponent p and nilpotency class r, then every such group is a quotient of Γ(d, p, r). In this case, we call Γ(d, p, r) the universal p-group of rank d, exponent p and nilpotency class r. The following theorem of Bamberg et al. [4, §2] describes how, when r ∈ {2, 3} and p > r, Γ(d, p, r) can be constructed using Lie powers of V . Here, [·, ·] is the Lie bracket associated with the free Lie algebra L(V ).
Theorem 2.1. (i) Let Γ 2 (V ) be the set V × L 2 V equipped with the multiplication defined by (a, b)(f, g) := (a + f, b + g + [a, f ]) (1) for (a, b), (f, g) ∈ Γ 2 (V ). When p > 2, Γ 2 (V )
is a group of nilpotency class 2, and
Γ 2 (V ) ∼ = Γ(d, p, 2). (ii) Let Γ 3 (V ) be the set V × L 2 V × L 3 V equipped with the multiplication defined by (a, b, c)(f, g, h) := (a + f, b + g + [a, f ], c + h + 3([b, f ] − [g, a]) + [a, f, f − a]) (2) for (a, b, c), (f, g, h) ∈ Γ 3 (V ). When p > 3, Γ 3 (V ) is a group of nilpotency class 3, and Γ 3 (V ) ∼ = Γ(d, p, 3).
Note that for r ∈ {2, 3} with p > r, the identity of Γ 3 (V ) is the element with each coordinate equal to 0, and the inverse of an element of Γ r (V ) is obtained by multiplying each coordinate by −1. Bamberg et al. also showed that when p > 3, Γ(d, p, 4) is isomorphic to a group of nilpotency class 4 with underlying set V × L 2 V × L 3 V × L 4 V . However, we will not consider that group in this paper.
It is clear that, when p > 2,
R := {(0, b) | b ∈ L 2 V } is a subgroup of Z(Γ 2 (V ))
isomorphic to the elementary abelian group L 2 V . Hence we can identify the subgroups of R with the subspaces of L 2 V . If U is a proper subgroup of R, then we can identify P U :
= Γ 2 (V )/U with the set V × (L 2 V )/U,(3)
equipped with the multiplication given by
(a, b + U)(f, g + U) := (a + f, b + g + [a, f ] + U)(4)
for a, f ∈ V and b, g ∈ L 2 V . Similarly, if p > 3 and if W is a proper subgroup of
S := {(0, 0, c) | c ∈ L 3 V } Z(Γ 3 (V )), then W is a proper subspace of L 3 V . Here, we can identify Q W := Γ 3 (V )/W with the set V × L 2 V × (L 3 V )/W,(5)
equipped with the multiplication given by
(a, b, c+W )(f, g, h+W ) := (a+f, b+g+[a, f ], c+h+3([b, f ]−[g, a])+[a, f, f −a]+W ) (6) for a, f ∈ V , b, g ∈ L 2 V and c, h ∈ L 3 W . Note that P {0} = Γ 2 (V ) and Q {0} = Γ 3 (V ).
The following propositions provide some important details about P U and Q W . In the cases where P U = Γ 2 (V ) or Q W = Γ 3 (V ), these results follow easily from [4, §2].
Proposition 2.2. Suppose that p > 2, and let U be a proper subspace of L 2 V . Then
P U ′ = Φ(P U ) = {(0, b + U) | b ∈ L 2 V }, which is isomorphic to (L 2 V )/U. Proof. It is clear that R := {(0, b + U) | b ∈ L 2 V } is a subgroup of P U isomorphic to (L 2 V )/U. Observe that the commutator of two elements (a, b + U), (f, g + U) ∈ P U is (0, 2[a, f ] + U). Hence P U ′ R. Let {e 1 , . .
. , e d } be a basis for V , let i, j ∈ {1, . . . , d}, and let m := (p + 1)/2. Then [(e i , U), (e j , U)] m = (0, [e i , e j ] + U). The bilinearity of [·, ·] implies that L 2 V is spanned by the Lie brackets [e i , e j ], and so R is generated by the commutators [(e i , U), (e j , U)]. Therefore, P U ′ = R. Finally, Γ 2 (V ) has exponent p, and hence its quotient P U = 1 also has exponent p. Thus Φ(P U ) = P U ′ .
Proposition 2.3. Suppose that p > 3, and let W be a proper subspace of L 3 V . Then:
(i) γ 3 (Q W ) = {(0, 0, c + W ) | c ∈ L 3 V }, which is isomorphic to (L 3 V )/W ; and (ii) Q W ′ = Φ(Q W ) = {(0, b, c + W ) | b ∈ L 2 V, c ∈ L 3 V }. Proof. Let a i ∈ V , b i ∈ L 2 V and c i ∈ L 3 V for each i ∈ {1, 2, 3}. Bamberg et al. [4, p. 2936] show that, in Γ 3 (V ), [(a 1 , b 1 , c 1 ), (a 2 , b 2 , c 2 ), (a 3 , b 3 , c 3 )] = (0, 0, 12[a 1 , a 2 , a 3 ])
. By comparing (2) and (6), we see that the elements of Q W satisfy a corresponding equality, i.e., with the third coordinate of each element of Γ 3 (V ) replaced by its coset of W in L 3 V . The group
γ 3 (Q W ) = [Q W , Q W , Q W ]= {(0, 0, c + W ) | c ∈ L 3 V },[(e i , 0, W ), (e j , 0, W ), (e k , 0, W )]. Thus S = γ 3 (Q W ). Next, the first two coordinates of [(a 1 , b 1 , c 1 + W ), (a 2 , b 2 , c 2 + W )] are equal to the two coordinates of [(a 1 , b 1 ), (a 2 , b 2 )] ∈ Γ 2 (V ). Thus by Proposition 2.2, Q W ′ lies in the subgroup T := {(0, b, c + W ) | b ∈ L 2 V, c ∈ L 3 V } of Q W .
Furthermore, it follows from this proposition that for each b ∈ L 2 V , we can multiply commutators of elements of Γ 3 (V ) to obtain an element (0, b, h + W ) of Q W , for some h ∈ L 3 V . Since Q W ′ contains γ 3 (Q W ), we have shown above that, for each c ∈ L 3 V , the element (0, 0, c − h + W ) is a product of commutators of elements of Γ 3 (V ). As (0, b, h + W )(0, 0, c − h + W ) = (0, b, c + W ), the subgroup S is generated by commutators of elements of Γ 3 (V ). Therefore, Q W ′ = T .
For the proof of the following result, note that if N is a normal subgroup of a group G and if i is a positive integer, then G/N is nilpotent of nilpotency class i if and only if N contains γ i+1 (G) but not γ i (G). In addition, G and G/N have the same rank if and only if N Φ(G). Lemma 2.4. Let P be a p-group.
(i) Suppose that p > 2. Then P has rank d, exponent p and nilpotency class 2 if and only if P ∼ = P U for some proper subspace U of L 2 V . (ii) Suppose that p > 3. Then P has rank d, exponent p and nilpotency class 3 if and only if P is a quotient of Γ 3 (V ) by a normal subgroup that lies in γ 2 (Γ 3 (V )), and that does not contain γ 3 (Γ 3 (V )). In particular, if P ∼ = Q W for some proper subspace W of L 3 V , then P has rank d, exponent p and nilpotency class 3.
Proof. Let r ∈ {2, 3}, suppose that p > r, and Let N be a normal subgroup of G := Γ r (V ). Since γ r+1 (G) = 1, G/N is nilpotent of nilpotency class r if and only if N does not contain γ r (G). Additionally, as G has rank d, G/N also has rank d if and only if N Φ(G). Moreover, G has exponent p, and hence so does each nontrivial quotient of G.
It follows from Proposition 2.2 that if r = 2, then G/N has rank d, exponent p and nilpotency class 2 if and only if N is a proper subgroup of {(0, b) | b ∈ L 2 V }, i.e., a proper subspace of L 2 V . The definition of P U implies that if P ∼ = P U for some proper subspace U of L 2 V , then P has rank d, exponent p and nilpotency class 2. Conversely, if P is a p-group of rank d, exponent p and nilpotency class 2, then P is a quotient of G. Hence P ∼ = P U for some proper subspace U of L 2 V .
If instead r = 3, then Proposition 2.3 implies that G/N has rank d, exponent p and nilpotency class 3 if and only if γ 3 (Γ 3 (V )) N γ 2 (Γ 3 (V )). In particular, this is the case if N is a proper subspace of L 3 V . The remainder of the result follows from the fact that each p-group rank d, exponent p and nilpotency class 3 is a quotient of Γ 3 (V ).
Note that if N is a normal subgroup of Γ 3 (V ) that lies in γ 2 (Γ 3 (V )) and neither contains nor lies in γ 3 (Γ 3 (V )), then Γ 3 (V )/N is a p-group of rank d, exponent p and nilpotency class 3 that cannot be expressed as Q W for any subspace W of L 3 V . Now, with r ∈ {2, 3} and p > r, the group GL(d,
p) ∼ = GL(V ) acts on Γ r (V ), with (a, b) α := (a α , b α ) and (a, b, c) α := (a α , b α , c α ) for each α ∈ GL(d, p), a ∈ V , b ∈ L 2 V and c ∈ L 3 V [4, Theorem 2.5]. Additionally, if B = B(d, p), and if r is any positive integer such that γ r (B) = 1, then γ r (B)/γ r+1 (B) is an F p [GL(d, p)]-module isomorphic to L r V .
It follows that we can identify the subspaces of L r V with the quotients M/γ r+1 (B), for the subgroups M of γ r (B) that contain γ r+1 (B). In each case, M is a normal subgroup of B, The following theorem describes A(P ) when P is a p-group of exponent p and nilpotency class 2, or when P is isomorphic to Q W for some proper subspace W of L 3 V . This result was used in [3] and [4] to construct p-groups of exponent p and nilpotency class 2 or 3.
Theorem 2.6. Suppose that P has exponent p.
(i) Suppose that p > 2, and that P has nilpotency class 2, i.e., that P ∼ = P U for some
proper subspace U of L 2 V . Then A(P ) = GL(d, p) U . (ii) Suppose that p > 3, and that P ∼ = Q W for some proper subspace W of L 3 V . Then A(P ) = GL(d, p) W .
Proof. Let r ∈ {2, 3}, suppose that p > r, and let X be a proper subspace of L r V . Additionally, let X be the proper subspace of L r V such that P is isomorphic to P X or Q X . Then X = M/γ r+1 (B), where M is a proper subgroup of γ r (B) that contains γ r+1 (B), and P ∼ = B/M. It follows from Theorem 2.5 that, up to conjugacy in GL(d, p),
A(P ) = A(B/M) = GL(d, p) X .
This theorem shows that in order to construct a p-group P as P U (respectively, as Q W ) such that A(P ) is a particular subgroup H of GL(d, p), then H must be the stabiliser in GL(d, p) of some proper subspace of L 2 V (respectively, of L 3 V ). We must therefore be able to distinguish between H and any proper overgroup of H in GL(d, p) by comparing the subspaces of L 2 V (respectively, of L 3 V ) stabilised by these linear groups. Observe that if r ∈ {2, 3}, if p > r, and if P = Γ r (V ), then A(P ) = GL(d, p) {0} = GL(d, p). Hence if X is a proper subspace of L r V , then the linear group induced on the Frattini quotient of Γ r (V )/X is a subgroup of the linear group induced on the Frattini quotient of Γ r (V ).
Suppose now that p > 3, that H is a subgroup of GL(d, p), and that there is no p-group P of exponent-p class 2 such that A(P ) = H. Additionally, suppose that W is a proper subgroup of L 3 V such that A(Q W ) = H, with W having the largest order of such a proper subgroup. Then Q W has minimal order among the groups that can be expressed as Q X , with X a proper subspace of L 3 V such that A(Q X ) = H, and Q W has exponent-p class 3. Moreover, if Q is an optimal p-group with respect to H as in Definition 1.1, then Q has exponent-p class 3, exponent p and nilpotency class 3. However, we are not able to determine the order of Q. This is because, if N is a normal subgroup of Γ 3 (V ) that satisfies the hypotheses of Lemma 2.4(ii) but does not lie in γ 3 (Γ 3 (V )) ∼ = L 3 V , then Theorem 2.6(ii) does not yield any information about A(Γ 3 (V )/N). We therefore introduce the following definition.
Definition 2.7. Let H be a subgroup of GL(d, p), and suppose that each p-group that is optimal with respect to H has exponent-p class 3, exponent p and nilpotency class 3. Additionally, let Q H be the set of p-groups Q W , for proper subspaces W of L 3 V , such that A(Q W ) = H. Then a p-group P ∈ Q H is quasi-optimal with respect to H if no group in Q H has a smaller order than P .
Observe that a p-group that is quasi-optimal with respect to H satisfies conditions (i)-(iii) of Definition 1.1 and a stronger version of condition (iv). Although we are aware of p-groups that are quasi-optimal with respect to certain linear groups (such as those described in §7 and [4, Table 6.1]), we do not know whether or not any of these p-groups are optimal with respect to the corresponding linear groups. Now, let Q be a p-group of rank n > 1 and exponent-p class c. We summarise some definitions from a paper by O'Brien [30, §2] related to the p-covering group Q * of Q. Here, Q * is the largest group among p-groups P of rank n that contain an elementary abelian subgroup M P , with M P Z(P ) ∩ Φ(P ) and P/M P ∼ = Q, and every such p-group P is a quotient of Q * . The subgroup M := M Q * of Q * is called the p-multiplicator of Q. A subgroup X of M is called allowable if R := Q * /X has rank n and exponent-p class c + 1, and if R/P c+1 (R) ∼ = Q, where P c+1 (R) is the (c + 1)-th group in the lower exponent-p central series of R. O'Brien [30,Theorem 2.4] shows that X is allowable if and only if X < M and XP c+1 (Q * ) = M.
Throughout the rest of this section, let E denote the elementary abelian p-group of rank d. In the following proposition, we consider the p-covering group E * of E. Proposition 2.8. Suppose that P is a p-group. Then P has rank d and exponent-p class 2 if and only if P is a quotient of E * by a proper subgroup of Φ(E * ). In particular, E * has exponent-p class 2. Moreover, M E * = Φ(E * ).
Proof. Let M := M E * . Since E * /M is elementary abelian, with M Φ(E * ), we have M = Φ(E * ).
Additionally, E has exponent-p class 1, and P 2 (E * ) = Φ(E * ). Therefore, every proper subgroup X of M is allowable, i.e., E * /X has rank d and exponent-p class 2. Note that the direct product of d copies of the cyclic group C p 2 has rank d and exponent p-class 2, which means that Φ(E * ) is nontrivial. It follows that E * itself has exponent-p class 2.
Conversely, if P is a p-group of rank d and exponent-p class 2, then Φ(P ) is an elementary abelian subgroup of Z(P ), and P/Φ(P ) ∼ = V by Burnside's Basis Theorem. Thus by the definition of E * , there exists a normal subgroup Y of E * such that P ∼ = E * /Y . As E * and P both have rank d, and as P is not elementary abelian, we have Y < Φ(P ).
We can therefore consider E * as the universal p-group of rank d and exponent-p class 2. Additionally, each automorphism α of E lifts to an automorphism α * of E * , via the natural epimorphism from E * to E with kernel Φ(E * ) [12, p. 2275]. In fact, Aut(E) acts on M E * = Φ(E * ), with x α := x α * for each x ∈ Φ(E * ) and each α ∈ Aut(E). In particular, if V is the vector space F d p , then the natural linear action of Aut(E) ∼ = GL(V ) on E ∼ = V induces an action of GL(V ) on Φ(E * ).
Observe that the exponent-p class of a p-group of exponent p is equal to its nilpotency class. Supposing that p > 2, Theorem 2.1 implies that the universal p-group Γ(d, p, 2) of rank d, exponent p and nilpotency class 2 is also the universal p-group of rank d, exponent p and exponent-p class 2. Thus Γ(d, p, 2) is the largest quotient of E * of exponent p, i.e., [13, §1-2]. Since A 2 V ∼ = L 2 V , and since E * /Φ(E * ) ∼ = V by Burnside's Basis Theorem, we deduce the following. Proposition 2.9. Suppose that p > 2. Then E * is an extension of V ⊕ L 2 V by V .
Γ(d, p, 2) ∼ = E * /(E * )p. Moreover, Φ(E * ) = (E * )p ⊕ (E * ) ′ , with (E * )p ∼ = V and (E * ) ′ ∼ = A 2 V as F p [GL(V )]-modules
We can also use the above decomposition of Φ(E * ) into GL(V )-submodules to describe A(P ) when P is a p-group of exponent-p class 2. A version of the following result, which is a generalisation of Theorem 2.6(i), previously appeared in [13, §2].
Theorem 2.10. Suppose that p > 2, and that P has exponent-p class 2, i.e., that P ∼ = E * /X for a proper subgroup X of Φ(E * ) ∼ = V ⊕ L 2 V . Then A(P ) = GL(d, p) X . Furthermore:
(i) P is abelian if and only if X contains the direct summand L 2 V . In this case, P has exponent p 2 ; and A(P ) = GL(d, p) X∩V . (ii) P has exponent p if and only if X contains the direct summand V . In this case, P has nilpotency class 2; A(P ) = GL(d, p) U , where U is the proper subspace X ∩ L 2 V of L 2 V ; and P ∼ = P U .
Proof. Let Q := E * /X. Since X is an allowable subgroup of E * , the image of the natural action θ : Aut(Q) → Aut(Q/P 2 (Q)) is the stabiliser of X in Aut(E) ∼ = GL(d, p) (see [12,Theorem 3.2] and [30, Theorem 2.10]). We have P 2 (Q) = Φ(Q), and hence the image of θ is A(Q). As P ∼ = Q, it follows that A(P ) = A(Q) = GL(d, p) X , up to conjugacy in GL(d, p). Now, if P is abelian, then it has exponent p 2 , and X contains (E * ) ′ ∼ = L 2 V . Here, A(P ) is the stabiliser in GL(d, p) of X = (X ∩ V ) ⊕ L 2 V . However, the full group GL(d, p) stabilises L 2 V , and hence A(P ) is the stabiliser of X ∩ V .
If instead P has exponent p, then it has nilpotency class 2, and X contains (E * )p ∼ = V . Here, A(P ) is the stabiliser in GL(d, p) of X = V ⊕ (X ∩ L 2 V ). However, GL(d, p) stabilises V , and hence A(P ) is the stabiliser of the proper subspace
U = X ∩ L 2 V of L 2 V . We have E * /X ∼ = (E * /(E * )p)/(X/(E * )p), with E * /(E * )p ∼ = Γ(d, p, 2) and X/(E * )p ∼ = (V ⊕ (X ∩ L 2 V ))/V ∼ = U, and hence P ∼ = P U , by the definition of P U .
Thus in order to construct a p-group P of exponent-p class 2 such that A(P ) is a particular subgroup H of GL(d, p), we must be able to distinguish between H and any proper overgroup of H in GL(d, p) by comparing the subspaces of V ⊕ L 2 V stabilised by these linear groups. Observe that if P is a p-group satisfying condition (i) of Theorem 2.10, with X ∩ V = {0}, then A(P ) acts reducibly on V . In fact, A(P ) is a maximal subgroup of GL(d, p) that lies in the Aschbacher class denoted
C 1 [4, Remark 6.3]. If instead P = E * , then A(P ) = GL(d, p) {0} = GL(d, p).
The final theorem in this section reveals more information about a given p-group of exponent-p class 2. A p-group with a unique (proper nontrivial) characteristic subgroup is called a UCS p-group [13]. Since each group in the lower exponent-p central series of a p-group is characteristic, each UCS p-group has exponent-p class at most 2.
Highest weight theory
We now use highest weight theory to determine information about Lie powers of the minimal modules of the linear algebraic groups associated with exceptional Chevalley groups. This will allow us, in the next section, to derive corresponding results about the Lie powers of the minimal modules of the Chevalley groups themselves (or their universal covers). Malle and Testerman [28] and Lübeck [25] give excellent introductions to linear algebraic groups and highest weight theory, and we assume that the reader is relatively familiar with these topics. In this paper, all linear algebraic groups are assumed to be simple of simply connected type.
Throughout this section and the next, we use the following notation:
• q is a power of a prime p;
•G = t Y ℓ (q)
is a finite simple group of Lie type defined over F q ;
• J is the universal cover ofG, with Z(J) the Schur multiplier ofG;
•Ĝ is the quotient of J by the unique Sylow p-subgroup of Z(J); • K := F p ;
• G = Y ℓ is the linear algebraic group over K such thatĜ is the group of fixed points of a particular Steinberg endomorphism of G; • T is a fixed maximal torus of G; • X(T ) ∼ = Z ℓ is the character group of T ; • W is the Weyl group of G; • ∆ = {α 1 , . . . , α ℓ } is a fixed base of the root system of G; and • {λ 1 , . . . , λ ℓ } is the set of fundamental dominant weights of T (with respect to ∆).
Note that we do not consider the Tits group 2 F 4 (2)
′ as a group of Lie type. We follow the . IfG = E 6 (q) with q ≡ 1 (mod 3), thenĜ = J = 3 · E 6 (q) [35,Ch. 4.10.6], and ifG = E 7 (q) with q odd, thenĜ = J = 2 · E 7 (q) [35,Ch. 4.12]. Here, and throughout the paper, we denote group extensions using Atlas [11] notation.
As the above notation suggests, if r is a power of p, and if s Y ℓ (r) is a simple group of Lie type, then G is the linear algebraic group associated with bothG and s Y ℓ (r). Moreover, G can be considered as a group of matrices over K. We can also identify X(T ) with a subset of the Euclidean space R ℓ , with ∆ ⊂ X(T ) a basis for R ℓ . The Weyl group W is the group generated by the reflections in each hyperplane orthogonal to a root in ∆, and the action of W on R ℓ induces an action on X(T ). Each character α ∈ X(T ) is a Z-linear combination of the fundamental dominant weights λ i , and if the coefficient of each λ i is nonnegative, then we say that α is dominant.
In this paper, whenever we refer to a K[G]-module, we mean a finite-dimensional rational K[G]-module. We recall the following standard definition.
Definition 3.1. Let V be a K[G]-module. For λ ∈ X(T ), we define V λ := {v ∈ V | v t = (t)λv for all t ∈ T }. If V λ = {0}, then λ is a weight of V with (respect to T ) with multiplicity dim(V λ ), and V λ is a weight space of V .
We will write Λ(V ) to denote the weight multiset for V , with each weight represented as many times as its multiplicity. Each weight space of V is clearly a T -submodule of V . In fact, V decomposes as the direct sum of its weight spaces, and thus |Λ(V )| = dim(V ). It is easy to show that if U is a K[G]-module isomorphic to V , then Λ(U) = Λ(V ). We will shortly summarise methods of calculating the weight multisets for K[G]-modules constructed from other K[G]-modules, but first we define one such construction that applies to a module for any group over any field.
If V is a module for a group H over a field F, and if α ∈ Aut(H), then we V α denotes the module obtained by twisting V by α. Specifically, if V affords the representation ρ, then V α affords the representation ρ α , where (h)ρ α := (h α )ρ for each h ∈ H. In particular, we will see that if V is a K[G]-module, and if φ is the field automorphism of G that maps each matrix entry in an element of G to its p-th power, then the modules V φ k for positive integers k are important. Note that φ is an abstract group automorphism, but not a linear algebraic group automorphism; in fact, it is a Frobenius endomorphism of G.
Hall [17,Lemma 10.37] states the first result of the following lemma; Samelson [31, p. 106-108] states the second and third; and the fourth follows from [28, p. 135].
Lemma 3.2. Let V be a K[G]-module. (i) Suppose that U is a submodule of V . Then Λ(V /U) = Λ(V ) \ Λ(U). (ii) Suppose that U is a K[G]-module. Then Λ(U ⊗ V ) is the multiset [λ + µ | λ ∈ Λ(U), µ ∈ Λ(V )],
with λ 1 + µ 1 = λ 2 + µ 2 if and only if λ 1 = λ 2 and µ 1 = µ 2 . (iii) Suppose that n is an integer at least 2 and at most n := dim(V ), and that
Λ(V ) = [λ 1 , . . . , λ m ]. Then Λ(A n V ) = [λ i 1 + · · · + λ in | 1 i 1 < · · · < i n m]. (iv) Suppose that n is a nonnegative integer. Then Λ(V φ n ) = [p n λ | λ ∈ Λ(V )].
Throughout this paper, when we refer to the composition factors of V , we mean the multiset of composition factors, with possible isomorphisms between elements of this multiset. Proof. Let {0} = V 0 ⊂ V 1 ⊂ · · · ⊂ V n = V be a composition series for V . Additionally, for each integer k between 1 and n − 1 inclusive, let L k be the disjoint union of Λ(V /V k ) and each Λ(V i /V i−1 ) with i ∈ {1, 2, . . . , k}. We prove the result by induction on k. First,
V 1 /V 0 = V 1 /{0} = V 1 , and so Λ(V /V 1 ) = Λ(V ) \ Λ(V 1 /V 0 ). Thus Λ(V ) is the disjoint union of V /V 1 and Λ(V 1 /V 0 ), i.e., Λ(V ) = L 1 .
For the inductive step, suppose that Λ(V ) = L k for some k between 1 and n − 2 inclusive. We have (
V /V k )/(V k+1 /V k ) ∼ = V /V k+1 , and so Λ(V /V k+1 ) = Λ((V /V k )/(V k+1 /V k )) = Λ(V /V k ) \ Λ(V k+1 /V k ). Therefore, Λ(V /V k ) is the disjoint union of Λ(V /V k+1 ) and Λ(V k+1 /V k )
. This means that Λ(V ) = L k+1 . Hence induction gives Λ(V ) = L n−1 , which is the required disjoint union of weight multisets.
Let
be the relation on X(T ) such that α β for α, β ∈ X(T ) if and only if β − α is a linear combination of roots in ∆, with the coefficient of each root nonnegative. Then is a partial order. The following well-known theorem (see [10,) shows that there is a 1-1 correspondence between isomorphism classes of irreducible K[G]-modules and dominant characters of T .
Theorem 3.4 (Chevalley).
(i) Let V be an irreducible K[G]-module. Then there is a unique weight λ ∈ Λ(V ), called the highest weight of V , such that µ λ for all µ ∈ Λ(V ). Moreover, λ is dominant and has multiplicity 1.
(ii) If λ ∈ X(T ) is dominant, then there exists an irreducible K[G]-module V with highest weight λ. (iii) Two K[G]
-modules are isomorphic if and only if they have the same highest weight.
For a dominant character λ ∈ X(T ), we write L(λ) to denote the unique (up to isomorphism) irreducible K[G]-module with highest weight λ. Lemma 3.2(iv) implies that if n is a positive integer, then L(λ) φ n is isomorphic to the irreducible module L(p n λ). The only weight of the trivial irreducible module is the trivial character 0, and hence this module is equal to
L(0). For a general K[G]-module V , we say that a weight λ in a subset D of Λ(V ) is a highest weight of D if µ λ for all µ ∈ D.
Lemma 3.5. Let C be a (possibly empty) disjoint union of weight multisets for composition factors of the K[G]-module V . Additionally, let λ ∈ D := Λ(V ) \ C be such that λ < µ for all µ ∈ D. Then L(λ) is a composition factor of V . In particular, this is the case if D has a highest weight λ, or if V itself has a highest weight λ.
Proof. By Proposition 3.3, Λ(V ) is the disjoint union of the weight multisets for the composition factors of V . The subset D of Λ(V ) is also a disjoint union of weight multisets for composition factors of V , and hence λ ∈ Λ(U) for some composition factor U of V . Theorem 3.4 implies that U = L(µ), for some µ ∈ D with λ µ. By the definition of λ, we therefore have µ = λ and U = L(λ).
We can apply this lemma iteratively to a K[G]-module to increase the multiset of known composition factors of V , corresponding to C, and decrease the multiset of unknown composition factors, corresponding to D.
Much of our discussion of modules of linear algebraic groups also applies when these linear algebraic groups are defined over C. The following result is based on the discussion in [25, §3]. Lemma 3.6. Let F be equal to C or the algebraic closure of a finite field, let H be a linear algebraic group defined over F, and let λ be a dominant character of a maximal torus of H. Then there exists an F[H]-module V (λ), called the Weyl module corresponding to λ, such that:
(i) dim(V (λ)) does not depend on F; (ii) the irreducible F[H]-module L(λ) is a quotient of V (λ); (iii) if F = C, then V (λ) ∼ = L(λ); and (iv) there exists a prime r such that, if p r and F = F p , then V (λ) ∼ = L(λ).
There may also exist primes p < r such that V (λ) ∼ = L(λ) when F = F p .
In the following lemma, w 0 is the longest element of W , i.e., the unique element of W such that
∆ w 0 = −∆. Note that the if λ ∈ X(T ) is dominant, then so is −λ w 0 [28, p. 125, p. 132]. In fact, −λ w 0 = λ unless G is equal to A ℓ (with ℓ > 1), D ℓ (with ℓ odd) or E 6 [28, p. 133].
Proposition 3.7 ([19, p. 24, p. 118]). Let λ, µ ∈ X(T ) be dominant, and let M be a K[G]module whose composition factors are exactly L(λ) and L(µ), with L(µ) isomorphic to a submodule U of M. If L(λ) is not isomorphic to a submodule W of M, with M = U ⊕ W , then either:
(i) µ < λ, and M is isomorphic to a quotient of the Weyl module V (λ); or (ii) λ < µ, and M is isomorphic to a submodule of the dual module (V (−µ w 0 )) * .
We recall that a module V (for any group, over any field) is multiplicity free if it is a semisimple module with pairwise non-isomorphic irreducible submodules. Moreover, V is multiplicity free if and only if its composition factors are pairwise non-isomorphic, with each isomorphic to a submodule of V .
Let S be a finite subset of X(T ), with each λ ∈ S dominant. It follows from Lemma 3.6 that there exists a prime r such that, for all λ ∈ S, the dimension and weight multiset for L(λ) remain constant for all p r. We can also construct a new K[G]-module M using irreducible modules L(λ) with λ ∈ S, via tensor products, exterior powers and quotients (of submodules constructed similarly). As long as p r, the dimension and weight multiset for M will not vary with p, since these depend only on the weight multisets for the module L(λ), by Lemma 3.2. If we redefine S to include all dominant weights of M, then we deduce that there exists a prime r such that, for all p r, M has the same number of composition factors, with each having a constant dimension and highest weight. In the following theorem, which gives a sufficient condition for M to be multiplicity free, r denotes this prime.
Theorem 3.8. Let M be a K[G]-module constructed from irreducible K[G]-modules via tensor products, exterior powers and quotients. Suppose that the composition factors of M are the same as those of the corresponding module defined over F r , in terms of dimensions and highest weights, and suppose that these composition factors are pairwise non-isomorphic. Then M is multiplicity free.
Proof. Let {0} = M 0 ⊂ M 1 ⊂ · · · ⊂ M n = M be a composition series for M. Then Theorem 3.4 implies that, for each i ∈ {1, 2, . . . , n}, M i /M i−1 ∼ = L i := L(µ i ) for some dominant µ i ∈ X(T ). Since the dimension and highest weight of each composition factor of M are equal to those of the corresponding module defined over F r , Lemma 3.6 implies that
L i ∼ = V (µ i ) for each i. Hence V (µ i ) is irreducible. Furthermore, L(−µ w 0 i ) ∼ = (L(µ i )) * [28, p. 132], and hence dim(L(−µ w 0 i )) = dim((L(µ i )) * ) = dim(L i ) is the same when defined over K, or over F q for any q r. Hence V (−µ w 0 i ) ∼ = L(−µ w 0 i ) by Lemma 3.6.
Since the duals of isomorphic modules are isomorphic to each other, we have (
V (−µ w 0 i )) * ∼ = (L(−µ w 0 i )) * , which is isomorphic to (L * i ) * . This is isomorphic to L i since dim(L i ) is finite, and thus (V (−µ w 0 i )) * is irreducible. Now, for a given j ∈ {2, 3, . . . , n}, the composition factors of M j /M j−2 are M j /M j−1 ∼ = L j and M j−1 /M j−2 ∼ = L j−1 , with the latter isomorphic to a maximal submodule of M j /M j−2 .
Since V (µ j ) and (V (−µ w 0 j−1 )) * are irreducible, Proposition 3.7 implies that L j is also isomorphic to a maximal submodule of M j /M j−2 . It follows that M has a composition series
{0} = M 0 ⊂ M 1 ⊂ · · · M j−2 ⊂ U ⊂ M j ⊂ · · · ⊂ M n = M, with U/M j−2 ∼ = L j .
This process of finding a new composition series can be performed a total of (j −1) times in order to obtain a composition series whose smallest nonzero module is isomorphic to L j . Hence each composition factor of V is isomorphic to a submodule of V . As these composition factors are pairwise non-isomorphic, M is multiplicity free.
McNinch [29, Corollary 1.1.1] proved a more uniform result about (not necessarily multiplicity free) semisimple modules: if G = Y ℓ , then each K[G]-module of dimension at most ℓp is semisimple. However, we will use Theorem 3.8 to prove the semisimplicity of K[G]-modules whose dimensions are higher than this upper bound.
For the remainder of this section, we assume that G is a linear algebraic group associated with an exceptional Chevalley group. We will shortly calculate the composition factors of the exterior square A 2 V of each minimal K[G]-module V , and in some cases, the composition factors of (
A 2 V ⊗ V )/A 3 V . Recall that A 2 V ∼ = L 2 V when p > 2, and that (A 2 V ⊗ V )/A 3 V ∼ = L 3 V when p > 3.
Although we are mainly interested in the cases where these isomorphisms hold, we also include the small prime cases where they do not hold. Our calculations use data from Lübeck's paper [25], which, for each G, lists the dimension and highest weight of each irreducible K[G]-module whose dimension is below a certain value. We also use Lübeck's supplementary data [27], which, for each of these modules, lists the multiplicity of each of the module's dominant weights. Since each orbit of the action of the Weyl group W on X(T ) contains exactly one dominant weight, and since weight multiplicity for a given module is constant on Weyl orbits, we can use this data to compute the multiplicity of every weight of the module. Note that Lübeck also considers the cases where G is a classical linear algebraic group of relatively small rank, and that in some cases, his supplementary data covers higher ranks and dimensions than his paper. Table 1 gives the highest weight and dimension of each minimal K-module for G, up to isomorphism and twisting by a field automorphism of G, based on the data from [25]. Observe that the highest weight in each case is a fundamental dominant weight of T . Our ordering of the fundamental dominant weights of T , which corresponds to the labelling of the Dynkin diagram of G, follows the convention of Malle and Testerman [28, Table 9.1] and the Magma [6] computer algebra system. In some cases, this is different from the ordering used by Lübeck [
25, Appendix A.1]. When p = 3, the K[G 2 ]-module L(λ 2 ) has dimension 14, and when p = 2, the K[F 4 ]-module L(λ 1 ) has dimension 52. Each irreducible K[G]-module is self-dual, unless G = E 6 [25, Appendix A.3]. If G = E 6 and λ = 6 i=1 c i λ i , then the highest weight of (L(λ)) * is 6 i=1 c (i)τ λ i , where τ is the permutation (16)(35) associated with the nontrivial automorphism of the Dynkin diagram of G [25, Appendix A.3].
It follows that (L(λ)) * is isomorphic to the module obtained by twisting L(λ) by the graph automorphism of E 6 . In particular, this automorphism interchanges L(λ 1 ) and L(λ 6 ) ∼ = (L(λ 1 )) * . Similarly, if G = G 2 and p = 3 (respectively,
G 2 λ 1 p = 2 6 p > 2 7 G 2 λ 2 p = 3 7 F 4 λ 1 p = 2 26 F 4 λ 4 p = 3 25 p = 3 26 E 6 λ 1 All p 27 E 6 λ 6 All p 27 E 7 λ 7 All p 56 E 8 λ 8 All p 248
if G = F 4 and p = 2), then the graph automorphism of G interchanges L(λ 1 ) and L(λ 2 ) (respectively, L(λ 1 ) and L(λ 4 )).
We use Magma to calculate the composition factors of (iv) Find a weight λ ∈ D such that λ < µ for all µ ∈ D.
W ∈ {A 2 V, (A 2 V ⊗ V )/A 3 V },
(v) Add L(λ) to the list of composition factors of W (per Lemma 3.5). (vi) Determine the weight multiset Λ(L(λ)) using Lübeck's data and Weyl orbit calculations. (vii) Exclude each element of Λ(L(λ)) from D. (viii) Repeat steps (iv)-(vii) until D is the empty set. The following theorem summarises our results for A 2 V in each case. For completeness, we consider the cases where (G, V ) ∈ {(G 2 , L(λ 2 )), (F 4 , L(λ 1 ))}, even when p is such that V is not a minimal module. Here, and in the subsequent theorem related to (A 2 V ⊗ V )/A 3 V , the statement that certain modules are multiplicity free is a consequence of Theorem 3.8.
Theorem 3.9. Suppose that (G, V ) ∈ {(G 2 , L(λ 2 )), (F 4 , L(λ 1 ))}, or that V is a minimal K[G]
-module whose highest weight is a fundamental dominant weight of T . Then the composition factors of A 2 V , for each prime p, are listed in the tables in Appendix A. Furthermore, for the values of p given in the final row of each table, V is multiplicity free. In particular, if G = E 6 and p > 3, then A 2 V is irreducible.
Note that the first isomorphism in Table 5 is a consequence of Steinberg's Tensor Product Theorem (see [25,Theorem 2.2]).
We will see later that the module V is closely associated with an irreducible F q [Ĝ]module, with the Lie powers of the former associated with the Lie powers of the latter. In particular, this correspondence preserves irreducibility. In the case where q = p, we would like to use Theorem 2.6 to construct a p-group P such thatĜ is a subgroup of relatively small index of the group A(P ) induced by Aut(P ) on P/Φ(P ). As L 2 V is irreducible when G = E 6 and p > 2, we must therefore consider the structure of L 3 V . Although L 2 V is not irreducible when G = E 7 , we will see later in this paper that in order to apply Theorem 2.6 in this case as described above, we must again consider the structure of L 3 V . We therefore determine the composition factors of (
A 2 V ⊗ V )/A 3 V when G ∈ {E 6 , E 7 }.
Theorem 3.10. Suppose that G ∈ {E 6 , E 7 }, and let V be a minimal K[G]-module whose highest weight is a fundamental dominant weight of T . Then the composition factors of (A 2 V ⊗ V )/A 3 V , for each prime p, are listed in the tables in Appendix B. Furthermore, for the values of p given in the final row of each table, V is multiplicity free.
Minimal modules for exceptional Chevalley groups
Here, we use the results of the previous section to study the minimal modules for the exceptional Chevalley groups (or their universal covers), and the Lie powers of these modules. We retain the notation outlined at the start of the previous section, and we initially letG = t Y ℓ (q) be any finite simple group of Lie type. Additionally, let u := 1 ifG is a Suzuki or Ree group, i.e., if (t, Y ) ∈ {(2, B), (2, G), (2, F )}, and otherwise let u := t.
We recall that a group H is quasisimple if H is perfect and H/Z(H) is non-abelian and simple. Our next proposition is from [34, §13] (see also [23,Proposition 5.4.4]).
Proposition 4.3. The field F q u is a splitting field forĜ. Now, ifG is not a Suzuki or Ree group, then let L be the set of irreducible
K[G]-modules L(λ) such that λ is q-restricted, i.e., λ = ℓ i=1 c i λ i with 0 c i < q for each i. Steinberg [33, §11, §13]
gives the following theorem, as well as the slightly more complicated definition of L that is required whenG is a Suzuki or Ree group (see also [23, p. 191]). Proof. Since F q u is a splitting field forĜ by Proposition 4.3, there is a 1-1 correspondence [20,Corollary 9.8]. Theorem 4.4 therefore implies that the restrictions toĜ of distinct elements of L are obtained from the distinct irreducible F q u [Ĝ]-modules by extending the scalars.
V i ←→ V K i between distinct irreducible F q u [Ĝ]-modules and distinct irreducible K[Ĝ]- modulesTheorem 4.6. Let V be an F q u [Ĝ]-module that is constructed from irreducible F q u [Ĝ]
modules via tensor products, exterior powers and quotients (of submodules constructed similarly). In addition, let W be the K[G]-module that is constructed via the same operations on the corresponding elements of L. Then W |Ĝ ∼ = V K , and the correspondence of Lemma 4.5 applies between the composition factors of V and the composition factors of W . Moreover, if W is multiplicity free, then so is V .
Proof. Suppose first that M and N are successive submodules in a composition series for W , with N ⊂ M. Then these submodules restrict to submodules of W |Ĝ, and M|Ĝ/N|Ĝ is isomorphic to the restriction toĜ of the composition factor M/N ∈ L. This restriction is irreducible by Theorem 4.4, and thus it is a composition factor of W |Ĝ. Hence each composition series for W restricts to a composition series for WĜ, and it follows that the composition factors of W |Ĝ are the restrictions toĜ of the composition factors of W .
Next, suppose that X and Y are successive submodules in a composition series for V , with Y ⊂ X. Then Y K and X K are submodules of V K , and X K /Y K ∼ = (X/Y ) K is irreducible since F q u is a splitting field forĜ by Proposition 4.3. Hence extending the scalars for a composition series for V results in a composition series for V K . It follows that the composition factors of (X/Y )|Ĝ ∼ = (X|Ĝ)/(Y |Ĝ) are the restrictions toĜ of the composition factors of X/Y . Therefore, the composition factors of V K are obtained from the composition factors of V by extending the scalars. Now, constructing a module via tensor product, exterior power and quotient operations on certain irreducible modules and then restricting the module to a subgroup is equivalent to performing the same operations on the irreducible modules restricted to the subgroup. Similarly, constructing a module via such operations on certain irreducible modules and then extending the scalars is equivalent to extending the scalars for each irreducible module and then performing the operations. Lemma 4.5 therefore implies that W |Ĝ ∼ = V K . Hence the correspondence of Lemma 4.5 applies between the composition factors of V and the composition factors of W .
Finally, if W is multiplicity free, then the composition factors of W appear as nonisomorphic submodules of W , which restrict to non-isomorphic irreducible submodules of WĜ by Theorem 4.4, and hence WĜ is also multiplicity free. This means that V K is multiplicity free, which implies that V is semisimple [2, Proposition 2.1.5]. Hence V is the direct sum of a set U of irreducible submodules of V . If A, B ∈ U, then A K and B K are irreducible, and A K ∼ = B K since V K is multiplicity free. It follows that A ∼ = B [20, Corollary 9.8], and therefore V is multiplicity free.
We say that a module for an arbitrary group H over a field F can be written over a subfield E of F if the module affords a representation that maps each element of H to a matrix with entries in E. The following propositions describes the smallest field over which we can write an absolutely irreducibleĜ-module, whenG is not a Suzuki or Ree group. See [23,Remark 5.4.7] for a similar result whenG is a Suzuki or Ree group. 23, p. 193-194]). Suppose that q = p e for some positive integer e, and thatG is not a Suzuki or Ree group. If t = 1, then let γ be the graph automorphism of G of order t. Additionally, let f be a positive integer, let U be an absolutely irreducible F p f [Ĝ]-module that cannot be written over a proper subfield of F p f , and let W ∈ L be the K[G]-module such that W |Ĝ = U K . Then either:
Proposition 4.7 ([(i) f | e, and there exists an irreducible K[Ĝ]-module V with dim(U) = dim(V ) e/f ; or (ii) t > 1; W ∼ = W γ ; f | te and f ∤ e; and there exists an irreducible K[Ĝ]-module V with dim(U) = dim(V ) te/f .
Using Lübeck's [25] lists of irreducible modules, we see that γ does not fix the minimal K[G]-modules whenG ∈ { 3 D 4 (q), 2 E 6 (q)}, and hence the corresponding absolutely irre-ducibleĜ-modules can not be written over F q , or over a proper subfield of F q . Furthermore, whenG is a Suzuki or Ree group, the absolutely irreducibleĜ-modules corresponding to minimal K[G]-modules cannot be written over a proper subfield of F q [23, Remark 5.4.7(b)]. Note that whenG is a Suzuki or Ree group, we have q > p. Thus there is no twisted exceptional group of Lie typeG such that an absolutely irreducibleĜ-module corresponding to a minimal K[G]-module can be written over F p . This means that we cannot apply Theorem 2.6 or Theorem 2.10 to the images of the representations afforded by these modules. Of course, we also cannot apply these theorems whenG ∈ { 2 B 2 (q), 2 F 4 (q)} because q is even in these cases.
The following definition applies to modules for an arbitrary group. Table 1. Then:
(i) d is the dimension of the minimal F q [Ĝ]-modules, and there is a unique quasiequivalence class Q of these modules;
(ii) each module in Q is absolutely irreducible and faithful, and cannot be written over any proper subfield of F q ; (iii) the images of the F q -representations afforded by modules in Q form a single conjugacy class of subgroups of GL(d, q); (iv) if V ∈ Q, and if α is a nontrivial field automorphism ofĜ, then V ∼ = V α ;
(v) ifG = E 6 (q) and (G, p) / ∈ {(G 2 (q), 3), (F 4 (q), 2)}, then Q is also the unique equivalence class of minimal F q [Ĝ]-modules up to isomorphism and twisting by a field automorphism; (vi) ifG = E 6 (q) or if (G, p) ∈ {(G 2 (q), 3), (F 4 (q), 2)}, then there are two equivalence classes of minimal F q [Ĝ]-modules with respect to isomorphism and twisting by a field automorphism, and these equivalence classes are interchanged by the graph automorphism ofĜ; and (vii) ifG = E 6 (q), then a given minimal F q [Ĝ]-module and its dual lie in different equivalence classes with respect to isomorphism and twisting by a field automorphism.
L(λ) φ i ∼ = L(p i λ),
where λ is a fundamental dominant weight and i ∈ {0, 1, 2, . . . , e − 1}, with q = p e . Moreover,
L(p i λ) ∼ = L(p j λ) if i = j by Theorem 3.4.
If α is an automorphism of G that induces an automorphism α (with slight abuse of notation) ofĜ, and if an F q [Ĝ]-module V corresponds to a K[G]-module W in the way specified in Lemma 4.5, then V α corresponds to W α . Since the field automorphisms of G induce those ofĜ, it follows from Lemma 4.5 that the distinct (up to isomorphism, or up to isomorphism and twisting by a field automorphism) minimal F q [Ĝ]-modules are those that correspond to the distinct (in the same way) minimal K[G]modules in L. In particular, ifG = E 6 (q) or if (G, p) ∈ {(G 2 (q), 3), (F 4 (q), 2)}, then there are two equivalence classes of theseĜ-modules up to isomorphism and twisting by a field automorphism. Otherwise, there is a unique such equivalence class, and hence a unique quasi-equivalence class of these modules. Now, whenG = E 6 (q), the highest weight of each minimal K[G]-module is not a scalar multiple of the highest weight of its dual, and hence these modules are not equivalent up to isomorphism and twisting by a field automorphism. We also saw previously that in this case, or when (G, p) ∈ {(G 2 (q), 3), (F 4 (q), 2)}, twisting the minimal K[G]-modules by the graph automorphism of G results in a module that is not equivalent up to isomorphism and twisting by a field automorphism. This graph automorphism induces the graph automorphism ofĜ, and so it is clear that these duality and twisting properties also hold for the corresponding minimal F q [Ĝ]-modules. Thus the aforementioned equivalence classes of minimal F q [Ĝ]-modules form a single quasi-equivalence class. In all cases, the quasi-equivalence of modules implies conjugacy in GL(d, q) of the images of the afforded representations [8, p. 39-40].
IfĜ is isomorphic to the simple groupG, then it is immediate that each minimal F q [Ĝ]module is faithful. In fact, even whenĜ ∼ =G, which can occur only ifG ∈ {E 6 (q), E 7 (q)}, the minimal F q [Ĝ]-modules are faithful [23, p. 202-203] Recall that if V is a nonzero module over a field of characteristic not equal to 2, then L 2 V ∼ = A 2 V , and if the characteristic of the field is also not equal to 3, then L 3 V ∼ = (A 2 V ⊗ V )/A 3 V . We will now describe the submodule structure of L 2 V , where V is a minimal F q [Ĝ]-module with p > 2, and the structure of L 3 V whenG ∈ {E 6 (q), E 7 (q)}, with p > 3. In most cases, we also show that the submodule structure of L 2 V or L 3 V is equivalent (in terms of containments and dimensions) to the submodule structure of the same Lie power of the corresponding minimal K[G]-module and the corresponding minimal K[Ĝ]-module. As the submodule structure of the exterior square of a module is important in many applications, we use the notation A 2 V in the following theorem, instead of L 2 V .
For the proof of the following theorem, recall that if M is the set of irreducible submodules of a multiplicity free module M, then isomorphism of modules gives a 1-1 correspondence between the composition factors of M and the submodules in M. Furthermore, the set of submodules of M is exactly { N ∈N N | N ⊆ M}. (i) The submodule structure of A 2 V is given in Table 2, and is equivalent to the submodule structure of each of A 2 W and (A 2 W )|Ĝ. If p is an "exceptional prime" for G, i.e., if (G, p) lies in the set
{(G 2 (q), 3), (F 4 (q), 3), (E 7 (q), 7), (E 8 (q), 3), (E 8 (q), 5)},
then A 2 V is uniserial. Otherwise, A 2 V is multiplicity free. In particular, ifG = E 6 (q), then A 2 V is irreducible. (ii) IfG ∈ {G 2 (q), E 8 (q)} and B ∈ {V, W, W |Ĝ}, then the quotient of A 2 B by its largest maximal submodule is isomorphic to B. (iii) Suppose thatG ∈ {E 6 (q), E 7 (q)} and that p > 3, with q = p ifG = E 6 (q) and p = 5, or ifG = E 7 (q) and p ∈ {7, 11, 19}. The submodule structure of L 3 V is given in Figures 1-6. If p is not a listed "exceptional prime" forG, then L 3 V is multiplicity free, and the submodule structure of L 3 V is equivalent to that of L 3 W and that of (L 3 W )|Ĝ. , (E 8 (q), 3), (E 8 (q), 5)}. By Theorem, the module A 2 W is multiplicity free, and it is also irreducible whenG = E 6 (q). Furthermore, whenG = E 6 (q) and p > 5, or whenG = E 7 (q) and p / ∈ {2, 3, 7, 11, 19}, L 3 W is multiplicity free by Theorem 3.10. In each case, Lemma 4.5 and Theorem 4.6 imply that the corresponding F q [Ĝ]-module A 2 V or L 3 V is irreducible or multiplicity free as specified, as is the restriction of , it follows that (A 2 U) K is the restriction of (A 2 V ) K tô X. Computations using Magma show that the submodule structure of A 2 U is as required, and in particular, that A 2 U is uniserial. As A 2 U is a module forX defined over F p , which is a splitting field forX, and since K is algebraic over F p , (A 2 U) K is also uniserial, with an equivalent submodule structure to that of A 2 U. Since the uniserial module (A 2 U) K is the restriction of each of A 2 W and (A 2 V ) K toX, and since these three modules have equivalent composition factors (in terms of dimensions), the submodule structures of these modules are also equivalent. Additionally, as F q is a splitting field forĜ, the F q [Ĝ]-module A 2 V has an equivalent submodule structure to that of (A 2 V ) K . Now, letG ∈ {G 2 (q), E 8 (q)}, and let S be the largest maximal submodule of A 2 W . Tables 4, 5 and 10 list the composition factors of W , with W defined up to isomorphism and twisting by a field automorphism. It follows from these tables and Lemma 3.2 that either (A 2 W )/S ∼ = W , or (G, p) = 3 and the two composition factors of S have the same highest weight. However, we have shown that S is uniserial, and hence Proposition 3.7 implies that its two composition factors have different highest weights. Hence (A 2 W )/S ∼ = W . It follows easily that W |Ĝ ∼ = (A 2 W )|Ĝ/S|Ĝ ∼ = ((A 2 V )/N) K , where SĜ is the largest maximal submodule of (A 2 W )|Ĝ, and N is the largest maximal submodule of A 2 V . Thus both V and (A 2 V )/N correspond to W via the 1-1 correspondence of Lemma 4.5, and so A 2 V /N ∼ = V .
Additional Magma computations were used to directly determine the submodule structure of L 3 V in the case whereG = E 6 (5). We also used Magma to show that wheñ G = E 7 (7), L 3 V contains a 56-dimensional submodule; a 51072-dimensional submodule; and a uniserial 7392-dimensional submodule with two nonzero proper submodules, of dimension 6480 and 912, respectively. We will write U k to denote these submodules, where dim(U k ) = k. It follows from the dimensions of the composition factors of L 3 V that U 56 and U 51072 are irreducible, and that L 3 V = U 56 ⊕ U 7392 ⊕ U 51072 . No two of these three direct summands have a common composition factor, and so the submodules of L 3 V are exactly the direct sums of the submodules of U 56 , the submodules of U 7392 and the submodules of U 51072 . Using this fact also allows us to determine the containments between the submodules of L 3 V . Similar computations were performed in the case ofG = E 7 (11), where L 3 V is the direct sum of an irreducible 56-dimensional submodule, an irreducible 912-dimensional submodule and a uniserial 57552-dimensional submodule; and in the case ofG = E 7 (19), where L 3 V is the direct sum of an irreducible 912-dimensional submodule, an irreducible 51072-dimensional submodule and a uniserial 6536-dimensional submodule. We provide additional details about these Magma computations in Appendix C.
As mentioned in §1, the submodule structure of A 2 V in the case whereG = G 2 (q) has been explored, in less detail, in [3] and [32, Ch. 9.3.2]. It is also known that there exists an irreducible 7-dimensional module over R for the group G 2 (R) whose exterior square is the direct sum of an irreducible 7-dimensional submodule and an irreducible 14-dimensional submodule [16, p. 9].
Observe that, except in the case whereG = F 4 (q) and d depends on whether or not p is an exceptional prime, the dimensions of the submodules of eachĜ-module described
L 3 V , where V is a minimal F q [Ĝ]- module, withG = E 7 (19).
case of smaller modules inspired the above proof of the submodule structure of L 3 V for the exceptional prime cases withG = E 7 (q). Our observation also leads to the following conjecture.
Conjecture 4.12. Theorem 4.11(iii) holds even if we do not assume that q = p, and the submodule structure of L 3 V is equivalent to that of L 3 W and that of (L 3 W )|Ĝ even if p is an exceptional prime forG.
Suppose now that p is an exceptional prime forG ∈ {E 6 (q), E 7 (q)}, withX the corresponding group of Lie type defined over F p ,X the simply connected version ofX, and U a minimal F p [X]-module. Then we can adapt the proof of the uniserial cases of Theorem 4.11(i) to show that if N is a uniserial submodule of L 3 U, then N K is uniserial. Since L 3 U is a direct sum of uniserial submodules that have no common composition factors (even whenX = E 6 (5)), it follows that L 3 U and (L 3 U) K have equivalent submodule structures. Hence the submodule lattice of each of L 3 W , (L 3 V ) K ∼ = (L 3 W )|Ĝ and L 3 V is equivalent to a sublattice of the submodule lattice of L 3 U. Moreover, the length of a composition series is the same for all of these modules by Theorem 4.6. In particular, if L 3 W stabilises a submodule of the same dimension of each of the uniserial direct summands of L 3 U, then so do (L 3 V ) K and L 3 V . In this case, Conjecture 4.12 holds.
Overgroups of exceptional Chevalley groups
We now determine some of the overgroups of the simply connected versions of the exceptional Chevalley groups in the general linear groups corresponding to their minimal modules. In the next section, we will use these results to deduce which overgroups are distinguishable from each exceptional Chevalley group in the context of Theorems 2.6 and 2.10. Throughout the remainder of this paper, we use the following notation:
• q is a power of an odd prime p; •G is an exceptional Chevalley group defined over F q ; •Ĝ is the simply connected version ofG; • V is minimal F q [Ĝ]-module; • d := dim(V ); and • Z GL := Z (GL(d, q)).
By Lemma 4.9, d is given in Table 1. Although this lemma applies with p = 2, some of the important results in this section do not hold when p = 2, and hence we will not consider this case. Lemma 4.9 also shows that all minimal F q [Ĝ]-modules are absolutely irreducible and faithful, and there is a unique GL(d, q)-conjugacy class of images of representations afforded by these modules. Hence no result in this section depends on the choice of V , and we can identifyĜ with the image in GL(d, q) of the representation afforded by V . In fact, Proposition 4.10 shows that we can choose V to be any faithful d-dimensional F q [Ĝ]module. Furthermore, we have from Lemma 4.9 that no conjugate ofĜ in GL(d, q) can be written over a proper subfield of F q . Recall also from Theorem 4.1 thatĜ is quasisimple, and thusĜ lies in (GL(d, q)) ′ = SL(d, q).
The following is another important fact aboutĜ.
Proposition 5.1. The centraliser ofĜ in GL(d, q) is equal to Z GL . In particular, Z(Ĝ) is a subgroup of Z GL .
Proof. The centre ofĜ is a subgroup of the centraliser C GL(d,q) (Ĝ). AsĜ is an absolutely irreducible subgroup of GL(d, q), it follows from Schur's Lemma (see [8, p. 38]) that this centraliser is equal to Z GL .
Throughout this section, we use the following lemma, often without reference. Here, if a group H is a group of isometries of a bilinear or unitary form β defined on a vector space, then we say that H preserves β (or preserves β absolutely). If H is a group of similarities of β, then we say that H preserves β up to scalars. In particular, ifĜ preserves a nonzero bilinear or unitary form on U up to scalars, then the above lemma holds with H =Ĝ and U = V . Proposition 5.3. IfG = E 7 (q), thenĜ preserves a non-degenerate alternating form on V . IfG = E 6 (q), thenĜ preserves no nonzero unitary or reflexive bilinear form on V . Otherwise,Ĝ preserves a non-degenerate orthogonal form on V .
Proof. The result follows by [23, p. 200] forG ∈ {G 2 (q), F 4 (q), E 8 (q)} and by [23, Proposition 5.4.18] forG = E 7 (q). IfG = E 6 (q), then Lemma 4.9 implies that V ∼ = V * and that V * ∼ = V α for any nontrivial field automorphism α ofĜ. Hence in this case, the result follows by [23, Lemma 2.10.15].
Let β denote the zero form whenG = E 6 (q), and otherwise, let β denote the nondegenerate bilinear form preserved byG. The above proposition shows thatĜ lies in the full group of isometries of β in GL(d, q). When β is the zero form, this group of isometries is GL(d, q) itself. When β is alternating, let Sp(d, q) denote its group of isometries, and let CSp(d, q) denote the full group of similarities of β in GL(d, q). Note that Sp(d, q) < SL(d, q). Finally when β is orthogonal of type ε ∈ {•, +, −}, let GO ε (d, q) denote the full group of isometries of β; let SO ε (d, q) denote the intersection of GO ε (d, q) and SL(d, q); and let Ω ε (d, q) denote the last subgroup in the derived series of GO ε (d, q). If ε = •, i.e., if d is odd, then we will omit the superscript •. SinceĜ is quasisimple, we in fact havê
G Ω ε (d, q) ifG ∈ {G 2 (q), F 4 (q), E 8 (q)}.
Note that ifG = F 4 (q) with p > 3, thenĜ ∼ =G is the automorphism group of the 27-dimensional Albert algebra, i.e., the algebra of 3 × 3 octonion Hermitian matrices, over F q [35,Ch. 4.8]. Here, we can choose V to be the 26-dimensional subspace of the algebra consisting of trace 0 matrices. In this case, if A and B are matrices in V , then (A, B)β is the trace of 1 2 (AB + BA). By considering the matrix of this form, it can be shown that β is of plus type if q ≡ 1, 7 (mod 12), and of minus type otherwise. However, we will not use this fact in this paper.
Recall that if H is a finite group, then H ∞ denotes the last group in the derived series of H. Throughout the rest of this section, we use the following notation:
• T is equal to either GL(d, q), or the full group of isometries in GL(d, q) of some non-degenerate reflexive bilinear form on V ; • S := T ∞ ; and • R is an arbitrary subgroup of T that contains S.
In particular, if T = GL(d, q), then S = SL(d, q); if T = Sp(d, q), then S = T ; and as above, if T = GO ε (d, q), then S = Ω ε (d, q). In each case, S SL(d, q).
Our next goal is to determine the maximal subgroups of the groups S that containĜ. In order to do so, we state the definitions of the Aschbacher classes of subgroups of T , and the related Aschbacher's Theorem [1]. Our formulation of these definitions and this theorem are from Bray, Holt and Roney-Dougal [8,Ch. 2]. Note that some adjustments must be made when considering classical groups of dimension less than 7 or of even characteristic, and that adjustments can also be made to include the case where T is the isometry group of a non-degenerate unitary form on V .
Definition 5.4. The geometric subgroups of R are the subgroups that belong to one of the following classes.
C 1 : Stabilisers of certain nonzero proper subspaces of V . C 2 : Stabilisers of direct sum decompositions of V into proper equidimensional subspaces. C 3 : Stabilisers of extension fields F q r of F q , for primes r dividing d. C 4 : Stabilisers of tensor product decompositions of V into two lower-dimensional vector spaces. C 5 : Stabilisers of subfields F q 1/r of F q , for primes r. C 6 : Normalisers of symplectic-type or extraspecial r-subgroups of R, for primes r = p such that d is a power of r. C 7 : Stabilisers of tensor product decompositions of V into equidimensional vector spaces smaller than V . C 8 : Full groups of similarities of non-degenerate unitary or reflexive bilinear forms (when T = GL(d, q)).
Note that the lower-dimensional vector spaces in the decomposition of V associated with C 4 -subgroups are not required to be equidimensional.
In the following definition, H ∞ denotes the last subgroup in the derived series of the finite group H. Recall also that a group H is almost simple if there exists a non-abelian simple group X satisfying X ⊳ H Aut(X). Note that Bray, Holt and Roney-Dougal write S to denote the Aschbacher class C 9 , but we use notation consistent with Bamberg et al. [4]. The subgroup classes C 1 , . . . , C 9 of R are the Aschbacher classes of R.
Theorem 5.6 (Aschbacher's Theorem). Suppose that a maximal subgroup H of R does not contain S and is not a geometric subgroup of R. Then H is a C 9 -subgroup of R.
In this paper, when we say "a maximal C i -subgroup of R", we mean a maximal subgroup of R that is also a C i -subgroup of R. All maximal subgroups of S have been classified by Bray, Holt and Roney-Dougal [8] for d 12. Note that Kleidman [22] previously presented a classification of the maximal geometric subgroups of S for d 12, but without proof. Additionally, Kleidman and Liebeck [23] classified the maximal geometric subgroups for all d > 12, while Schröder [32] classified the maximal C 9 -subgroups for d ∈ {13, 14, 15}. However, there is no known method of classifying the maximal C 9 -subgroups uniformly for all d [8, p. 2].
We will now determine exactly when a geometric subgroup or a maximal C 9 -subgroup of S can containĜ. The next result follows from the definitions of the Aschbacher classes, and from the fact that if X is a C 3 -subgroup of T , then X ∞ is not absolutely irreducible [8, p. 56].
Proposition 5.7. Let H be a subgroup of R such that H ∞ is absolutely irreducible and such that no conjugate of H ∞ can be written over a proper subfield of F q . Then H does not lie in any C 1 -, C 3 -or C 5 -subgroup of R. In particular, this holds if H =Ĝ, or if H is a C 9 -subgroup of R. . , m} such that the quotient (Ĝ)π j ofĜ is nontrivial. Moreover, (Ĝ)π j is a subgroup of GL(V j ) ∼ = GL(d/m, q). However, Proposition 4.10 shows that no nontrivial quotient ofĜ is a subgroup of GL(n, q) for any n < d. This is a contradiction, and thusĜ lies in no C 2 -subgroup of S. We now consider the maximal geometric subgroups of the remaining classes. 1 The inequality m < √ d holds because, in the cases under consideration, √ d is not an integer when d is even.
Lemma 5.8. No C 2 -subgroup of R containsĜ. Proof. Suppose that H is a C 2 -subgroup of R that containsĜ. Then H stabilises a de- composition of V as V = V 1 ⊕ · · · ⊕ V m ,
Lemma 5.10. IfG = E 6 (q), then the full group of similarities of β in SL(d, q) is the only geometric subgroup of SL(d, q) that containsĜ. Moreover, this group of similarities is a maximal C 8 -subgroup of SL(d, q). IfG = E 6 (q), or if S = SL(d, q), thenĜ does not lie in any geometric subgroup of S.
Proof. By Proposition 5.7 and Lemmas 5.8 and 5.9, we only need to consider subgroups of S that lie in C 6 ∪ C 7 ∪ C 8 . Definition 5.4 shows that C 6 -subgroups of S are only defined when d is a power of a prime, and that C 7 -subgroups of S are only defined when d is a power of a positive integer less than d. Thus, in the cases we are considering, We now see from Definition 5.4 that ifĜ lies in a geometric subgroup H of S, then S = SL(d, q) and H is a C 8 -subgroup of S, i.e., H is the full group of similarities in S of a non-degenerate unitary or reflexive bilinear form. If this is the case, then each non-degenerate unitary or bilinear form that H preserves up to scalars is preserved bŷ G absolutely. ThusG = E 6 (q) by Proposition 5.3. It also follows that H preserves no non-degenerate unitary form up to scalars, and that the scalar multiples of β are the only non-degenerate bilinear forms that H preserves up to scalars. Hence H is the full group of similarities of β in S. Moreover, H is maximal in S, as shown by Bray, Holt and Roney-Dougal [8, In the proof of the following theorem, if H is a subgroup of GL(d, q), then we write H to denote the subgroup Z GL H/Z GL of PGL(d, q).
Theorem 5.11. Suppose that T is the full group of isometries of β in GL(d, q). The C 9subgroup N S (Ĝ) of S is the unique maximal subgroup of S that containsĜ. Furthermore, ifG = E 6 (q), then the full group of similarities of β in SL(d, q) is the unique maximal subgroup of SL(d, q) that containsĜ.
Proof. By Theorem 5.6 and Lemma 5.10, it suffices to show that if a maximal C 9 -subgroup of S containsĜ, then that maximal subgroup is equal to N S (Ĝ), and that ifG = E 6 (q), then no maximal C 9 -subgroup of SL(d, q) containsĜ. Let U ∈ {S, SL(d, q)}, and suppose that H is a maximal C 9 -subgroup of U that containsĜ. The perfect groupĜ then lies in H ∞ , which is easily shown to be quasisimple, with H ∞ ∼ = H ∞ /(H ∞ ∩ Z GL ). Sincẽ G ∼ =Ĝ/Z(Ĝ), which is equal toĜ/(Ĝ ∩ Z GL ) ∼ =Ĝ by Proposition 5.1, we haveG H ∞ . Now, Definition 5.5 implies that H ∞ is absolutely irreducible and cannot be written over a proper subfield of F q . The tables in [18] can be used to determine the finite, quasisimple, absolutely irreducible subgroups A of GL(d, q) such that A/Z(A) is not a simple group of Lie type defined over a field of characteristic p. In each case, there is no such group A whose order is divisible by |Ĝ|. Hence H ∞ is a simple group of Lie type t Y ℓ (r), where r is some power of p. Since p > 2, (t, Y ) / ∈ {(2, B), (2, F )}. IfX denotes the simply connected version of the subgroup H ∞ of PGL(d, q), then we require that there exists an absolutely irreducible d-dimensional F q [X]-module that cannot be written over a proper subfield of F q (see [25, p. 135-136]). It follows from Lemma 4.5 that there exists an irreducible d-dimensional K[X]-module, where K := F p and X is the linear algebraic group associated with H ∞ . We also have from Proposition 4.7 that either H ∞ = 2 G 2 (r) or r ∈ {q, q 1/t , q 2 , q 2/t , q 3 , q 3/t }. Here, we have used the fact that if d = x y for integers x > 0 and y > 1, then the tuple (d, x, y) lies in the set {(25, 5, 2), (27,3,3)}. In particular, if H ∞ = 2 G 2 (r), then r ∈ {q 2 , q 2/t } is only possible ifG = F 4 (q) with p = 3, and r ∈ {q 3 , q 3/t } is only possible ifG = E 6 (q).
Suppose that H ∞ is a classical group of Lie type. If ℓ is sufficiently large, then either d is the dimension of the natural F q [H ∞ ]-module, or there is no absolutely irreducible d-dimensional F q [X]-module [23,Proposition 5.4.11]. In fact, in the former case, there is a unique minimal F q [H ∞ ]-module up to quasi-equivalence, and it has dimension d. For smaller values of ℓ, we can use Lübeck's [25] lists of irreducible modules for linear algebraic groups to show that there is only an absolutely irreducible d-dimensional K[X]-module when |G| does not divide |H ∞ |. Therefore, H ∞ is a classical group of dimension d. If [25,Appendix A.3]). By considering the permutation of the Dynkin diagram of X induced by the graph automorphism of X of order 2, we see from Proposition 4.7 that r = q 1/2 in the former case and r = q in the latter case. Since there is a unique minimal F q [X]-module up to quasi-equivalence, there is a unique conjugacy class of subgroups B of GL(d, q) such that
Z GL B and B ∼ = H ∞ [8, p. 39-40]. It follows that Z GL H ∞ is conjugate in GL(d, q) to Z GL M, where M ∈ {SL(d, q), SU(d, q 1/2 ), Sp(d, q), Ω(d, q), Ω ± (d, q)}.
As M is perfect [8, Proposition 1.10.3], the perfect group H ∞ is in fact conjugate in GL(d, q) to M. By Lemma 5.2, H ∞ preserves at most one nonzero unitary or bilinear form. It follows that ifG = E 6 (q) and U preserves β, then H ∞ = U. If instead U = SL(d, q), then Definition 5.5 implies that H ∞ preserves no nonzero unitary or reflexive bilinear form, and we again have H ∞ = U. In each case, this contradicts the maximality of H in U. Thus H ∞ is not a classical group, i.e., it is an exceptional group of Lie type.
Lübeck's [25] lists of irreducible modules for linear algebraic groups imply that ifG = In each other case, there is no irreducible module for the linear algebraic group X of dimension 2 or 3 [25], and hence r ∈ {q, q 1/t }. IfG = E 6 (q) and H ∞ = 2 E 6 (r), then Table 1 and Lemma 3.2 imply that the irreducible d-dimensional X-module corresponds to the irreducible K[E 6 ]-module L(aλ i ), where a is a power of p and i ∈ {1, 6} [25, Appendix A.51]. It follows from Proposition 4.7 that r = q 1/2 . By considering group orders, we require (t, Y, ℓ) = (1, Y ′ , ℓ ′ ), and hence r = q and H ∞ ∼ =G.
Y ′ ℓ ′ (q), then either Y ′ ℓ ′ = Y ℓ , or (Y ′ ℓ ′ , Y ℓ ) ∈ {(F 4 , G 2 ), (E 6 , G 2 ), (E 7 , D 4 ), (E 8 , G 2 )}. In particular, if H ∞ = 2 G 2 (r),
We have from above that
Z GLĜ /Z GL =Ĝ ∼ =G ∼ = H ∞ = Z GL H ∞ /Z GL . AsĜ H ∞ , we have Z GLĜ = Z GL H ∞ .
SinceĜ and H ∞ are perfect, it follows thatĜ = H ∞ . In particular, this means that H normalisesĜ. Furthermore, sinceĜ < U, and since U is simple, Z GL U does not normalise Z GLĜ . However, Z GL does normalise Z GLĜ , which means that U does not. Since U normalises Z GL , this means that U does not normaliseĜ. Thus N U (Ĝ) is a proper subgroup of U, and so H = N U (Ĝ). If U = S, then we are done. Otherwise, N U (Ĝ) preserves β up to scalars, and Definition 5.5 implies that this group is not in fact a C 9 -subgroup of U = SL(d, q). In this case, no maximal C 9 -subgroup of U containsĜ, as required.
Note that Bray, Holt and Roney-Dougal [8, Table 8.40] previously showed that G 2 (q) is a maximal C 9 -subgroup of Ω(7, q).
We now determine the normaliser ofĜ in GL(d, q). Proof. First, observe that Z GLĜ ⊳ N. Let θ : N → Aut(Ĝ) be the action of N onĜ induced by conjugation. Then ker θ = C N (Ĝ), which is equal to Z GL by Proposition 5.1.
Since the restriction of θ toĜ is an epimorphism fromĜ to Inn(Ĝ), it follows that Z GLĜ is the full preimage of Inn(Ĝ) under θ. Therefore, θ maps distinct cosets of Z GLĜ in N to distinct cosets of Inn(Ĝ) in (Ĝ)θ. Let C be the full group of similarities of β in GL(d, q). Then N = N C (Ĝ), where we have used Lemma 5.2 in the case where β is nonzero. Bray, Holt and Roney-Dougal [8,Lemma 4.4.3] show that, since V is a faithful, absolutely irreducible module for the quasisimple groupĜ, an outer automorphism α ofĜ is induced by an element x ∈ N C (Ĝ) = N if and only if V α ∼ = V . If α is a field or graph automorphism ofĜ, then V α ∼ = V by Lemma 4.9. However, if α is a diagonal automorphism ofĜ, then V α ∼ = V [8, Proposition 5.1.9(i)]. The groupĜ has two nontrivial diagonal automorphisms ifG = E 6 (q) and q ≡ 1 (mod 3); one ifG = E 7 (q); and none otherwise (see [35,Ch. 4]). Since any automorphism ofĜ is a product of inner, field, graph and diagonal automorphisms [8,Proposition 5.1.1], N is as required.
Suppose now thatG = E 6 (q). Since N/(Z GLĜ ) is soluble, we have N ∞ = (Z GLĜ ) ∞ , which is equal toĜ ∞ =Ĝ. Lemma 4.9 and Proposition 5.3 show that N ∞ satisfies all required properties from Definition 5.5 for N to be a C 9 -subgroup of GL(d, q). We also see, by considering group orders, that N does not contain SL(d, q). Moreover, it is easy to show that N/(N ∩ Z GL ) = N/Z GL is isomorphic to (Ĝ/Z(Ĝ)). (q − 1, 3), which is the almost simple groupG. (q − 1, 3). Thus N is a C 9 -subgroup of GL(d, q). Finally, ifG = E 6 (q), then by considering group orders, we see that N is a proper subgroup of C, which is a proper subgroup of GL(d, q).
Stabilisers of submodules of Lie powers
In this section, we determine the stabilisers in GL(d, q) of relevant subspaces of the Lie powers of F q [Ĝ]-modules given in Theorem 4.11. We retain the notation outlined at the start of the previous section.
Recall that ifG = E 7 (q), thenĜ lies in the symplectic group Sp(56, q). In §3, we claimed that applying Theorem 2.6 to theĜ-submodules of L 3 V (with q = p) is more useful than applying this theorem to theĜ-submodules of L 2 V . The following lemma explains the reason for this. (i) The group CSp(56, q) stabilises eachĜ-submodule of L 2 V . (ii) Suppose that p > 3. Then each of Sp(56, q) and CSp(56, q) stabilises exactly two nonzero proper subspaces of L 3 V , of dimension 56 and 58464, respectively. If p = 19, then the latter subspace contains the former, and otherwise, L 3 V splits as the direct sum of these subspaces.
Proof. LetX be the simple group of Lie type PSp(56, q) = C 28 (q). The linear algebraic group related toX is X = C 28 , and the simply connected versionX ofX is Sp(56, q) [28, p. 193]. Let K be the algebraic closure of the field F p . Up to isomorphism, twisting by a field automorphism, and duals, the module with highest weight 2 λ 1 , of dimension 56, is the unique minimal K[X]-module [24, §1], which we will denote by U. In fact, this module is self-dual [28, p. 132-133], and hence it is the unique minimal K[X]-module up to isomorphism and twisting by a field automorphism. It follows from Lemma 4.5 that there is a unique (in the same way) minimal F q [X]-module, of dimension 56. Hence the irreducible 56-dimensionalĜ-module V is the restriction toĜ of a minimal F q [X]-module W . Similarly, L r V = (L r W )|Ĝ for each r ∈ {2, 3}. Suppose now that p / ∈ {2, 7}. Then L 2 U ∼ = A 2 U has two composition factors, of dimension 1 and 1539, respectively [24, §1]. Hence L 2 U is multiplicity free by Theorem 3.8, and thus L 2 W is also multiplicity free by Theorem 4.6. In particular, L 2 W is the direct sum of a 1-dimensional submodule and a 1539-dimensional submodule. By Theorem 4.11,Ĝ stabilises the same subspaces of L 2 V as Sp(56, q). Moreover, since Sp(56, q) ⊳ CSp(56, q), and since the irreducibleĜ-submodules of the multiplicity free module L 2 V are not equidimensional, CSp(56, q) stabilises the same subspaces of L 2 V as Sp(56, q).
If instead p = 7, then theĜ-module L 2 V is uniserial, and it has three composition factors, of dimension 1, 1 and 1538, respectively, by Theorem 4.11. In fact, these are exactly the dimensions of the Sp(56, q)-composition factors of L 2 W [24, §1]. Hence Sp(56, q) is uniserial, and in particular, Sp(56, q) stabilises the same subspaces of L 2 V asĜ. Since CSp(56, q) normalises Sp(56, q), and since L 2 W has a unique composition series, it follows from Clifford's Theorem that CSp(56, q) stabilises the same subspaces of L 2 V as Sp(56, q).
We now apply the methods used to derive Theorem 3.10 in order to determine the composition factors of the K[X]-module L 3 U. Magma calculations show that the Weyl orbit of λ 1 has size 56. Since the weight multiset for U must contain exactly dim(U) = 56 weights, and since it must contain the Weyl orbit of λ 1 , it follows that the weight multiset of U is precisely this Weyl orbit. The highest weight of L 3 U is λ 1 + λ 2 . The computations described in [25, §3] can be used to show that if p / ∈ {3, 19}, then the irreducible module L(λ 1 + λ 2 ) has dimension 58464, and the weight multiset of this module consists of one, two and 54 copies of the Weyl orbits of the weights λ 1 + λ 2 , λ 3 and λ 1 , respectively [26]. When these weights are excluded from the weight multiset of L 3 U, 56 weights remain, the highest of which is λ 1 . It follows that when p / ∈ {3, 19}, L 3 U has two composition factors, of dimension 56 and 58464, respectively. In this case, L 3 U is multiplicity free by Theorem 3.8, as is L 3 W by Theorem 4.6. Specifically, L 3 W is the direct sum of an irreducible submodule of dimension 56 and an irreducible submodule of dimension 58464. As above, CSp(56, q) stabilises the same subspaces of L 3 V as Sp(56, q).
Finally, when p = 19, the irreducible module L(λ 1 + λ 2 ) has dimension 58408, and the weight multiset of this module consists of one, two and 53 copies of the Weyl orbits of the weights λ 1 + λ 2 , λ 3 and λ 1 , respectively [26]. When these weights are excluded from the weight multiset of L 3 U, 112 weights remain, the highest of which is λ 1 , with multiplicity 2. Hence L 3 W has two 56-dimensional composition factors and one 58408-dimensional composition factor. TheĜ-submodule structure of L 3 V given in Figure 6 for the case of q = 19 implies that L 3 W is uniserial, with the dimensions of its submodules as required. A very similar argument to the one used in the uniserial cases of Theorem 4.11(i) shows that the submodule structure of L 3 W is as required even when q > p = 19. Again, CSp(56, q) stabilises the same subspaces of L 3 V as Sp(56, q).
We are therefore not able to distinguish between the simply connected versionĜ of E 7 (q) and CSp(56, q) by considering how these groups act on L 2 V . Thus applying Theorem 2.6(i) to the proper nontrivialĜ-submodules of L 2 V yields the same p-groups as applying the theorem to the proper nontrivial CSp(56, q)-submodules of L 2 V . Proposition 6.2. Let r ∈ {2, 3}, and suppose that p > r. Then Z GLĜ stabilises eacĥ G-submodule of L r V . Furthermore, if there is a subspace of L r V that is stabilised byĜ but not by N GL(d,q) (Ĝ), then r = 3, and:
(i)G = E 6 (q), p = 5, and q ≡ 1 (mod 3); or (ii)G = E 7 (q), p ∈ {7, 11, 19}, and q > p.
Proof. Since Z GL acts on V by scalar multiplication, the definition of the action of GL(d, q) on L r V implies that Z GL also acts on L r V by scalar multiplication. Hence Z GL stabilises every subspace of L r V , and thus Z GLĜ stabilises eachĜ-submodule of L r V . Let N := N GL(d,q) (Ĝ). Lemma 5.12 implies that ifG ∈ {G 2 (q), F 4 (q), E 8 (q)}, or ifG = E 6 (q) with q ≡ 1 (mod 3), then N = Z GLĜ . Otherwise, if L r V is multiplicity free, then we see from Theorem 4.11 that no two irreducible submodules of L r V are equidimensional. We also have Z GLĜ ⊳ N, and thus N stabilises the same subspaces of L r V as Z GLĜ .
We now consider the remaining cases where L r V is not multiplicity free, and either (i) and (ii) do not apply, or r = 2. Theorem 4.11 implies thatG = E 7 (q), with p = 7 if r = 2, and with q = p ∈ {7, 11, 19} if r = 3. Recall from Lemma 5.12 that N < CSp(56, q). Hence Lemma 6.1 implies that N stabilises eachĜ-submodule of L 2 V , and that N stabilises thê G-submodules of L 3 V of dimension 56 and 58464. We can show that N stabilises allĜsubmodules of L 3 V using the submodule structure of L 3 V shown in the fact that this is exactly the Z GLĜ -submodule structure of L 3 V ; the fact that N stabilises L 3 V , U 56 and U 58464 ; and the following: (a) if N stabilisesĜ-submodules X and U of L 3 V , and if there exists aĜ-submodule W of L 3 V such that X = U ⊕ W , then N stabilises W ; (b) if N stabilisesĜ−submodules U and W of L 3 V , with W ⊆ U, then N stabilises eachĜ-submodule in at least oneĜ-composition series for U containing W ; and (c) if N stabilises a multiplicity freeĜ-submodule U of L 3 V , then N stabilises eacĥ G-submodule of U.
These properties hold because no two submodules of L 3 V are equidimensional, and because the normal subgroup Z GLĜ of N has index coprime to p by Lemma 5.12. We detail our argument in the case of p = 7; the other cases are similar. Here, we write U k to denote the unique submodule of L 3 V of dimension k, when such a submodule exists. Figure 4 implies that each composition series for L 3 V that contains U 56 also contains either U 52040 , U 7448 or both U 6536 and U 57608 . By (b), N stabilises at least one of these composition series. As U 52040 = U 56 ⊕ U 912 ⊕ U 51072 is multiplicity free, (c) implies that if N stabilises U 52040 , then it also stabilises U 51072 . In addition, L 3 V = U 7448 ⊕ U 51072 and U 57608 = U 6536 ⊕ U 51072 , and hence if N stabilises either U 7448 or both U 6536 and U 57608 , then it also stabilises U 51072 by (a). Therefore, N does indeed stabilise U 51072 . It follows that N stabilises U 51128 = U 56 ⊕ U 51072 . As L 3 V = U 51128 ⊕ U 7392 , N also stabilises U 7392 by (a). The submodule U 7392 is uniserial, and hence (b) implies that each of its submodules is stabilised by N. Finally, L 3 V = U 56 ⊕ U 7392 ⊕ U 51072 , and N stabilises each submodule of each direct summand. Therefore, N stabilises all direct sums of submodules of these summands, which accounts for all submodules of L 3 V .
In particular,Ĝ and N GL(d,p) (Ĝ) stabilise the same set of subspaces of L r V whenever q = p. In fact, if Conjecture 4.12 holds, and the submodule structure of theĜ-module L 3 V depends only on p and not q, then the arguments in the above proof withG = E 7 (q) and p ∈ {7, 11, 19} hold even when q = p. Furthermore, if this conjecture holds, then a similar argument using Figure 2 shows that N GL(d,p) (Ĝ) stabilises all submodules of L 3 V wheneverG = E 6 (q) and p = 5. Theorem 6.3. Suppose that p > r, where r = 3 ifG ∈ {E 6 (q), E 7 (q)}, and r = 2 otherwise. Suppose also that q = p ifG = E 6 (q) with p = 5, or ifG = E 7 (q) with p ∈ {7, 11, 19}. Additionally, let X be a nonzero proper submodule of L r V . Then either GL(d, q) X = N GL(d,q) (Ĝ), orG = E 7 (q) and CSp(56, q) stabilises X.
Proof. Let N := N GL(d,q) (Ĝ), let M := N SL(d,q) (Ĝ), and let Z SL := Z(SL(d, q)). Lemma 5.12 and Dedekind's Identity imply that the normal subgroup Z SLĜ of M has index at most 3. Thus M ∞ = (Z SLĜ ) ∞ =Ĝ ∞ , which is equal toĜ by Theorem 4.1. HenceĜ is a characteristic subgroup of M. Moreover, SL(d, q) is a normal subgroup of GL(d, q), and so M = N ∩ SL(d, q) is normal in N. It follows that N GL(d,q) (M) = N. We also have GL(d, q) X N GL(d,q) (SL(d, q) X ). Since N stabilises X by Proposition 6.2, it suffices to show that SL(d, q) X = M.
Suppose thatG ∈ {G 2 (q), F 4 (q), E 8 (q)}. Recall thatĜ lies in SL(d, q) and preserves a non-degenerate orthogonal form on V . Let ε ∈ {•, +, −} be the type of this form. Then by Proposition 5.13, M is the largest subgroup of SL(d, q) that containsĜ but does not contain Ω ε (d, q). The group Ω ε (d, q) acts irreducibly on L 2 V [24, §1], and hence SL(d, q) X = M.
Next, suppose thatG = E 7 (q). Then Proposition 5.13 implies that M is the largest subgroup of SL(d, q) that containsĜ but does not contain Sp(56, q). Furthermore, if Sp(56, q) stabilises X, then so does CSp(56, q) by Lemma 6.1. Thus if CSp(56, q) does not stabilise X, then SL(d, q) X = M.
Finally, suppose thatG = E 6 (q). Then M is a maximal subgroup of SL(d, q) by Theorem 5.11. Bamberg et al. [4,Lemma 3.1] show that GL(d, q) acts irreducibly on L 3 V . Clifford's Theorem therefore implies that if SL(d, q) acts reducibly on L 3 V , then L 3 V is the direct sum of a set of proper equidimensional subspaces that are stabilised by SL(d, q), and hence bŷ G. However, Theorem 4.11 shows that no twoĜ-submodules of L 3 V are equidimensional. Therefore, SL(d, q) acts irreducibly on L 3 V , and hence SL(d, q) X = M.
If Conjecture 4.12 holds, then we do not need to assume that q = p in any case in the above lemma.
Recall that ifG ∈ {(G 2 (q), E 8 (q)}, thenĜ ∼ =G. The following lemma will allow us to apply 2.10 to these groups (with q = p) to yield p-groups that are not yielded by Theorem 2.6(i).
Lemma 6.4. Suppose thatG ∈ {(G 2 (q), E 8 (q)}. ThenG ∼ =Ĝ is the stabiliser in GL(d, q)
of a subspace of V ⊕ L 2 V that is isomorphic to L 2 V , and that does not contain V or L 2 V . Moreover, no properĜ-submodule of V ⊕ L 2 V has dimension larger than dim(L 2 V ).
Proof. Theorem 4.11 shows there exists an F[G]-epimorphism φ : L 2 V → V whose kernel is a maximal submodule of L 2 V . It follows that V ⊕ L 2 V contains the submodule M := {((u)φ, u) | u ∈ L 2 V }. This submodule does not contain the direct summand V , and is isomorphic, but not equal, to L 2 V . Since the composition factors of V ⊕ L 2 V are V and the composition factors of L 2 V , it follows from Theorem 4.11 that dim(L 2 V ) is the largest possible dimension of a proper submodule of V ⊕ L 2 V . Now, let H := GL(d, q) M . Since the subgroupG of H acts irreducibly on V , H also acts irreducibly on V . As H stabilises M, V is F q [H]-isomorphic to an H-composition factor of L 2 V . In particular, since dim(L 2 V ) > dim(V ), H acts reducibly on L 2 V . Lemma 5.12 and Theorem 6.3 imply thatG H Z GLG . If the action of an element z ∈ Z GL on V is equivalent to multiplication by a scalar µ ∈ F q \ {0}, then for each u ∈ L 2 V , we have ((u)φ, u) z = (µ(u)φ, µ 2 u) = ((µu)φ, µ 2 u). This is only a vector in M if µ = 1, i.e., if z = 1. Hence H =G.
Inducing exceptional Chevalley groups on P/Φ(P )
We can now induce on the Frattini quotient of a p-group the simply connected version of a given untwisted exceptional group of Lie type (defined over an appropriate field of prime order), or the normaliser of this simply connected group in GL(d, p). We again retain the notation outlined at the start of §5. Recall thatĜ = 3 ·G whenG = E 6 (q) with q ≡ 1 (mod 3); thatĜ = 2 ·G whenG = E 7 (q) (as p is odd); and thatĜ ∼ =G otherwise.
As in §2, we write then P U (respectively, Q W ) to denote the quotient of the universal p-group Γ(d, p, 2) (respectively, Γ(d, p, 3)) by a proper subgroup U of L 2 V (respectively, a proper subgroup W of L 3 V ). Also recall that if P is a p-group of rank d, then A(P ) GL(d, p) is the group induced by Aut(P ) on P/Φ(P ), and that the exponent-p class of P must be at least 2 in order for A(P ) to be a proper subgroup of GL(d, p). We begin by highlighting some proper subgroups of GL(d, p) that cannot be induced on the Frattini quotient of a p-group of low exponent-p class (in some cases, with low exponent or low nilpotency class also assumed).
Theorem 7.1. Suppose thatG is defined over a field of odd prime order p, with p > 3 if G ∈ {E 6 (p), E 7 (p)}.
(i) Assume that (G, H) ∈ {(F 4 (p), Z GL F 4 (p)), (E 6 (p), GL(27, p)), (E 7 (p), CSp(56, p))}, and let K be a proper subgroup of H that containsĜ. Then there is no p-group P of exponent-p class 2 such that A(P ) = K.
(ii) Assume thatG ∈ {G 2 (p), E 8 (p)}, and let K be a proper subgroup of Z GLĜ that containsĜ. Then there is no p-group P of exponent-p class 2 and exponent p such that A(P ) = K. Additionally, there is no abelian p-group P of exponent-p class 2 such that A(P ) = K. (iii) Assume thatG ∈ {E 6 (p), E 7 (p)}, and let K be a proper subgroup of N GL(d,p) (Ĝ) that containsĜ. Then there is no p-group P such that A(P ) = K and P ∼ = Q W for some proper subspace W of L 3 V .
Proof.
(i) Observe from Theorem 4.11 that no composition factor of theĜ-module L 2 V is isomorphic to the irreducibleĜ-module V . Thus the submodules of V ⊕ L 2 V are the submodules of L 2 V and the direct sums of V and the submodules of L 2 V . It follows that any overgroup ofĜ in GL(d, p) that stabilises allĜ-submodules of L 2 V also stabilises allĜ-submodules of V ⊕ L 2 V . IfG = E 6 (p), then Theorem 4.11 shows that L 2 V is irreducible, and thus each of its submodules is stabilised by H = GL (27, p). In the other two cases, H stabilises allĜ-submodules of L 2 V by Lemma 6.1 and Proposition 6.2. We have shown that, in each case, there is no proper subspace X of V ⊕ L 2 V such that K = GL(d, p) X . Hence the result follows from Theorem 2.10. (ii) Proposition 6.2 implies that Z GLĜ stabilises eachĜ-submodule of L 2 V . This means that there is no proper subspace U of L 2 V such that K = GL(d, p) U . It follows from Theorem 2.10(ii) that there is no p-group P of exponent-p class 2 and exponent p such that A(P ) = K. Furthermore, V is an irreducibleĜ-module, and so its only proper subspace is {0}, which is stabilised by GL(d, p). Thus Theorem 2.10(i) implies that there is no abelian p-group P of exponent-p class 2 such that A(P ) = K. (iii) Proposition 6.2 implies that N GL(d,p) (Ĝ) stabilises eachĜ-submodule of L 3 V . Hence there is no proper subspace W of L 3 V such that K = GL(d, p) W . Theorem 2.6(ii) therefore gives the required result.
We note that the groups Q W mentioned in Theorem 7.1(iii) have exponent p and nilpotency class 3 (and hence exponent-p class 3) by Lemma 2.4. However, it is possible that there exists a p-group P of exponent-p class 3 and exponent p, with P not isomorphic to Q W for any proper subspace W of L 3 V , such that A(P ) is a group K defined in Theorem 7.1(iii).
Note also that whenG = E 7 (p), the subgroup K in Theorem 7.1(i) is a proper subgroup of CSp(56, p). Indeed, Bamberg et al. [4, Table 6.1] constructed a p-group P of exponent p and nilpotency class 2 (and hence exponent-p class 2) such that A(P ) = CSp(n, p), for each integer n > 2 and each odd prime p such that CSp(n, p) is a maximal subgroup of GL(n, p).
We are now able to state and prove the main theorem of this thesis. Here, we refer to optimal and quasi-optimal p-groups, as defined in Definitions 1.1 and 2.7, respectively. Theorem 7.2. Suppose thatG is defined over a field of odd prime order p.
(i) IfG ∈ {G 2 (p), F 4 (p), E 8 (p)}, then each p-group that is optimal with respect to N GL(d,p) (Ĝ) has exponent-p class 2, nilpotency class 2 and exponent p. (ii) IfG ∈ {G 2 (p), E 8 (p)}, then each p-group that is optimal with respect toĜ has exponent-p class 2, nilpotency class 2 and exponent p 2 . (iii) IfG ∈ {E 6 (p), E 7 (p)} and p > 3, then each p-group that is optimal (or quasioptimal) with respect to N GL(d,p) (Ĝ) has exponent-p class 3, nilpotency class 3 and exponent p. Table 3 specifies the properties of each optimal p-group in case (i) or (ii), and each quasioptimal p-group in case (iii). Finally, each p-group in case (i) or (ii) has a unique proper nontrivial characteristic subgroup. Table 3. The properties of each optimal or quasi-optimal p-group P from Theorem 7.2. Here, t := (3, p − 1), and c denotes both the exponent-p class and nilpotency class of P .
G d c Exponent of P |P | A(P ) G 2 (p) 7 2 p p 14 Z GL G 2 (p) G 2 (p) 7 2 p 2 p 14 G 2 (p) F 4 (3) 25 2 3 3 77 Z GL F 4 (3) F 4 (p), p > 3 26 2 p p 78 Z GL F 4 (p) E 6 (p), p > 3 27 3 p p 456 (Z GL (t · E 6 (p))).t E 7 (p), p > 3 56 3 p p 2508 (Z GL (2 · E 7 (p))).2 E 8 (p) 248 2 p p 496 Z GL E 8 (p) E 8 (p) 248 2 p 2 p 496 E 8 (p)
Proof. Theorem 4.11 gives the dimensions of the properĜ-submodules of L 2 V (respectively, L 3 V ) in case (i) (respectively, in case (iii)). By Lemma 6.1 and Theorem 6.3, N := N GL(d,p) (Ĝ) is the stabiliser in GL(d, p) of the largest such proper submodule, except when G = E 7 (p), in which case N is the stabiliser in GL(d, p) of the second largest proper submodule. Similarly, Lemma 6.4 implies that in case (ii),Ĝ is the stabiliser in GL(d, p) of a proper subspace of V ⊕ L 2 V that is isomorphic to L 2 V and does not contain V or L 2 V , and that there is no properĜ-submodule of V ⊕ L 2 V of larger dimension. Let X be the specified proper submodule whose stabiliser in GL(d, p) is N orĜ. It follows from Theorem 2.6 that N is equal to A(P X ) or A(Q X ) in case (i) or case (iii), respectively, while Theorem 2.10 shows thatĜ is equal to A(E * /X) in case (ii), where E * is the p-covering group of the elementary abelian p-group E of rank d. Using the fact that V is an irreducibleĜ-module and the fact that (L 2 V )/X ∼ = (V ⊕L 2 V )/(V ⊕X) in case (i), Theorems 2.10 and 2.11 imply that each p-group P X or E * /X has a unique proper nontrivial characteristic subgroup. Theorems 2.6 and 2.10 show that, in each case, the p-group P X , E * /X or Q X has the exponent-p class, nilpotency class and exponent of the p-group given in Table 3. Here, we also use the fact that a p-group of exponent-p class 2 has nilpotency class at most 2 and exponent at most p 2 . Now, the order of Γ(d, p, r) is given in Theorem 2.1, while we have from Proposition 2.9 that E * is an extension of
V × L 2 V by V . Since L 2 V ∼ = A 2 V , we have |P X | = p d(d+1)/2−dim(X) , |E * /X| = p (d 2 +d)/2−dim(X) and |Q X | = p d(d+1)(2d+1)/6−dim(X) , with dim(X) = (d 2 − d)/2 in case (ii).
In each case, the order of this p-group is given in Table 3. The structure of A(P ) = N in cases (i) and (iii) is given by Lemma 5.12. Note that this lemma also implies that N is a proper subgroup of GL(d, p) in each case, and also a proper subgroup of CSp(d, p) whenG = E 7 (p).
We now show that the specified p-groups are optimal or quasi-optimal as required. In each case, the optimal p-group has exponent-p class at least 2. Since a p-group of exponent p has equal exponent-p class and nilpotency class, it follows that the p-group P X in case (i) is optimal with respect to N < GL(d, p). Additionally, Theorem 7.1(ii) implies that, in case (ii), E * /X is optimal with respect toĜ. Finally, Theorem 7.1(i) shows that, in case (iii), Q X is quasi-optimal with respect to N. By definition, each p-group that is optimal with respect to N in this case has the same exponent-p class, exponent and nilpotency class as Q X .
When calculating the order of A(P ) in cases (i) and (iii) of the above theorem, it is useful to know the order of Z GL ∩Ĝ. Proposition 5.1 implies that this order is 3 wheñ G = E 6 (p) with p ≡ 1 (mod 3), and 2 whenG = E 7 (p). Otherwise,Ĝ ∼ =G is non-abelian and simple, and hence Z GL ∩Ĝ = 1.
The p-groups in cases (i), (ii) and (iii) of Theorem 7.2 can be constructed as specific quotients of the universal p-group Γ(d, p, 2), the p-covering group E * of the elementary abelian p-group of rank d, and the universal p-group Γ(d, p, 3), respectively, as detailed in the proof of the theorem. The p-groups in case (i) can also be constructed as corresponding quotients of E * , by Theorem 2.10. Note also that case (i), withG = G 2 (p), was previously proved by Bamberg, Freedman and Morgan [3]. However, Theorem 7.2 is completely disjoint from the work of Bamberg et al. [4] mentioned in §1, which covers maximal subgroups of GL(d, p) that do not lie in the Aschbacher classes C 6 or C 9 . Indeed, Lemma 5.12 shows that A(P ) is a C 9 -subgroup of GL(d, p) whenG = E 6 (p), and otherwise A(P ) is not maximal in GL(d, p). Now, if Q is a p-group isomorphic to P , then A(P ) and A(Q) are the images of representations afforded by isomorphic modules, with V the restriction of the former module tô G. However, our method of constructing an optimal or quasi-optimal p-group in each case of Theorem 7.2 does not depend on the choice of the minimal F p [Ĝ]-module V . Therefore, if there are two isomorphism classes of minimal F p [Ĝ]-modules, then there are at least two isomorphism classes of optimal or quasi-optimal p-groups. By Lemma 4.9, this is the case whenG is equal to G 2 (3) or to E 6 (p) for some p > 3. If M is a minimal F p [Ĝ]-module isomorphic to V , then M = V x for some x ∈ GL(V ), and it is easy to see that L r M = (L r V ) x for each r ∈ {2, 3}. In particular, since no two submodules of L r V are equidimensional by Theorem 4.11, x maps the submodule X of L r V from the proof of Theorem 7.2 to the unique submodule of L r V of dimension dim(X), which we will denote by Y . It follows from the definition of the action of GL(d, p) on Γ(d, p, r) and the definitions of P X and Q X that x induces an isomorphism from P X to P Y or from Q X to Q Y , as appropriate. Therefore, in cases (i) and (iii) of Theorem 7.2, there are exactly two isomorphism classes of optimal or quasi-optimal p-groups, respectively, whenG is equal to G 2 (3) or to E 6 (p), and exactly one isomorphism class otherwise.
Concluding remarks
Let q be a power of an odd prime p, letĜ be the simply connected version of an exceptional Chevalley groupG defined over F q , and let G be the associated linear algebraic group. We have determined the submodule structure of the exterior square of each minimal F q [G]-module, and of the exterior square of the corresponding irreducible modules over F p forG and for G. We have done the same for the third Lie power of each irreducible module whenG ∈ {E 6 (q), E 7 (q)} and p > 3, except in the case of a few small values of p, where we have only determined the structure of the third Lie power of each minimal F p [Ĝ]-module. Conjecture 4.12 posits that the submodule structure of a minimal F p [Ĝ]module V in a given "exceptional prime" case is equivalent to that of the third Lie power of each corresponding irreducible module defined over F q or over F p . It would be interesting to determine whether this conjecture is true, and if not, then to determine the actual structures of the other third Lie powers. This would involve determining in each case which composition series of L 3 V correspond to composition series of the corresponding F p [G]-module.
Next, let q = p and p > r, where r := 3 ifG ∈ {E 6 (p), E 7 (p)} and r := 2 otherwise. In addition, let d be the dimension of a minimal F p [Ĝ]-module. Using information about the aforementioned submodule structures, we constructed a p-group P of exponent-p class r, nilpotency class r and exponent p such that the group A(P ) induced by Aut(P ) on P/Φ(P ) is the normaliser ofĜ in GL(d, p). In the casesG ∈ {G 2 (p), E 8 (p)}, we also constructed a p-group P of exponent-p class 2, nilpotency class 2 and exponent p such that A(P ) =Ĝ. The constructed p-group is optimal with respect to A(P ) in each case with r = 2, and quasi-optimal when r = 3. Roughly, this means that P has the smallest exponent-p class, exponent and nilpotency class of all p-groups Q with A(Q) = A(P ), and the smallest order when r = 2. However, it is possible that when r = 3, there exists a group Q of exponent-p class 3, nilpotency class 3 and exponent p such that |Q| < |P | and A(Q) = A(P ). If this is the case, then we would like to construct the smallest such group Q. By Proposition 2.3, Lemma 2.4 and Definition 2.7, this would involve studying the quotients of the universal p-group Γ(d, p, 3) by normal subgroups that lie in γ 2 (Γ(d, p, 3)), and neither lie in nor contain γ 3 (Γ(d, p, 3)). Now, in the cases whereG ∈ {G 2 (p), E 8 (p)}, we were able to induce preciselyG ∼ =Ĝ on the Frattini quotient of a p-group because the exterior square of each minimal module has a composition factor isomorphic to the minimal module. Note that whenG = E 8 (q), the Lie algebra of G is a minimal F p [G]-module. Tables 5 and 6 show that ifG ∈ {G 2 (q), F 4 (q)}, and if U is the irreducible F p [G]-module whose highest weight is the highest weight of the Lie algebra of G, then U is isomorphic to a composition factor of A 2 U. The methods used to derive Theorem 3.9 show that this holds ifG is any exceptional Chevalley group, and in fact also ifG ∈ { 3 D 4 (q), 2 E 6 (q)}. Note that in almost all cases, U is equal to the Lie algebra of G. In the twisted cases (with q odd), the graph automorphism of G of order 3 or 2, respectively, fixes U and each composition factor of A 2 U, and the dimension of each of these irreducible modules is not an integral power of any integer other than itself. Hence Proposition 4.7 implies that the corresponding absolutely irreducibleĜ-modules can be written over F q , even when F q is not a splitting field forĜ. Furthermore, in each case whereG is an exceptional group of Lie type other than a Suzuki or Ree group, ifG ∼ =Ĝ, then Z(G) acts trivially on U (see [25,Appendix A.2]). In this case, since Z(Ĝ) Z(G) [28,Corollary 24.13], and sinceG ∼ =Ĝ/Z(Ĝ) is the only nontrivial proper quotient ofĜ by Proposition 4.2, the absolutely irreducible F q [Ĝ]-modules are actually faithful modules for G. Therefore, when q = p, it may be possible to apply our methods to these modules, and the images of the afforded representations in the relevant general linear groups, in order to construct a p-group Q of exponent-p class 2 such that A(Q) is preciselyG. However, if G = E 8 (p), then A(Q) would be a subgroup of a general linear group of dimension higher than d, and hence the rank of Q would be greater than d.
Of course, the theorem of Bryant and Kovács [9] mentioned in §1 implies that, whenG is an exceptional group of Lie type, there is some p-group Q of rank d such that A(Q) is precisely the subgroupĜ of GL(d, p). By Theorem 7.1, in order to construct such a group Q, we must consider p-groups of a higher exponent or exponent-p class, or perhaps the aforementioned unexplored quotients of Γ(d, p, 3). It would also be of interest to construct p-groups Q such that A(Q) is a group of Lie type defined over a finite field whose order is not an odd prime (perhaps even a Suzuki or Ree group). It may be possible to do this by applying our methods to a subgroup of GL(n, p), for some n, that is isomorphic to such a group of Lie type (possibly defined over a field of characteristic other than p). However, our current methods of determining whether or not there exists a p-group Q of exponent p and nilpotency class r ∈ {2, 3}, such that A(Q) is a given subgroup of GL(n, p), apply only when p > r. Developing corresponding methods for the cases with p r would allow us to study additional families of groups. We now list, in Tables 11 and 12, the composition factors of the module (A 2 V ⊗ V )/A 3 V for each minimal K[G]-module V from Theorem 3.10. The module (A 2 V ⊗ V )/A 3 V is isomorphic to the third Lie power L 3 V of V when p > 3. In each case, the highest weights of all composition factors of (A 2 V ⊗ V )/A 3 V are comparable with respect to the partial order defined in §3, and we list these composition factors by descending highest weight. Table 11. The composition factors of U := (A 2 (V 1 )⊗V 1 )/A 3 (V 1 ) and W := (A 2 (V 2 ) ⊗ V 2 )/A 3 (V 2 ), with G = E 6 , V 1 = L(λ 1 ), and V 2 = L(λ 6 ). L(λ 2 ) 78 1 Table 12. The composition factors of (A 2 V ⊗ V )/A 3 V , with G = E 7 and V = L(λ 7 ). caseG = E 8 (5) completed after 4 CPU hours, and the maximum RAM usage during this calculation was 7.9 GB. Suppose now that p > 3, and let d := dim(V ). Recall that L 3 V is isomorphic to (A 2 V ⊗ V )/A 3 V . If {e 1 , . . . , e d } is a basis for V , then {(e i ∧ e j ) ⊗ e k + (e j ∧ e k ) ⊗ e i − (e i ∧ e k ) ⊗ e j | 1 i < j < k d} is a basis for the submodule A 3 V of A 2 V ⊗ V [4, p. 2938]. We can therefore construct L 3 V in Magma as the quotient of A 2 V ⊗ V by the submodule with this basis. Note that Magma orders its exterior square and tensor product basis vectors in descending canonical order. For example, if U is a 3-dimensional module with basis {e 1 , e 2 , e 3 }, then the ordered basis of A 2 U ⊗ U in Magma is In the case ofG = E 6 (5), we can determine the submodule structure of L 3 V using the MaximalSubmodules command iteratively. Here, the subset operator can be used to check that if a submodule U of L 3 V contains a submodule X, and if another submodule W of L 3 V contains a submodule Y with the same dimension as X, then Y = X.
We now assume thatG = E 7 (p) with p ∈ {7, 11, 19}. Let R be a minimal F 5 [X]-module, whereX is the simply connected version of E 7 (5). Observe from Figure 3 that if n is an integer such that an n-dimensional submodule of L 3 V is mentioned in the proof of Theorem 4.11, then there exists a unique n-dimensional submodule R n of L 3 R. In order to construct a submodule of L 3 V of a given dimension n, we construct R n and its maximal submodules using the MaximalSubmodules command iteratively. We then search for a vector r ∈ R n such that:
(i) r is an element of the basis for R n stored in Magma;
(ii) R n is the smallest submodule of L 3 R that contains r; and (iii) if r is expressed as a linear combination of basis vectors for L 3 R, then the coefficient of each basis vector is either 0, 1 or −1. Here, (iii) allows us to use exactly the same linear combination of basis vectors (in terms of coefficients) of L 3 V to construct its n-dimensional submodule. Using a 2.6 GHz CPU, our computations here completed after 49 CPU hours, with a maximum RAM usage of 31.8 GB. Note that after constructing L 3 R, we deleted several variables that were no longer necessary in order to minimise this maximum RAM usage.
In the cases with p ∈ {7, 19}, we used this method to construct the direct summands of L 3 V mentioned in the proof of Theorem 4.11. We then calculated the submodules of the reducible (and uniserial) direct summand using the MaximalSubmodules command iteratively. However, in the case of p = 11, the dimension of the reducible direct summand of L 3 V is very large. In this case, for a faster computation, we constructed the reducible direct summand of L 3 V , computed the irreducible submodules of L 3 V , and then computed the irreducible submodules of the quotient of the reducible direct summand by the irreducible submodule of dimension 6480. Together with the dimensions of the composition factors of L 3 V , these calculations imply that the submodules of the reducible direct summand are as required.
Our computations (including the construction of L 3 V but excluding the construction of the vectors r ∈ L 3 R), which we again ran using a 2.6 GHz CPU, completed after 0.6 CPU hours in the case of E 7 (7); 223 CPU hours in the case of E 7 (11); and 4 CPU hours in the case of E 7 (19). The maximum RAM usage during our computations for each of the three cases was 14.5 GB, 67.0 GB and 38.7 GB, respectively. As above, we deleted variables when they were no longer necessary in order to minimise RAM usage.
and we have Γ(d, p, r)/(M/γ r+1 (B)) = (B/γ r+1 (B))/(M/γ r+1 (B)) ∼ = B/M.
.
Let r be a positive integer, and let M be a proper subgroup of γ r (B) that contains γ r+1 (B). Then the group A(B/M) induced by Aut(B/M) on the Frattini quotient of B/M is the stabiliser of M/γ r+1 (B) in GL(d, p).
Theorem 2.11 ([13, Theorem 4]). Suppose that p > 2, and that P and X are as in Theorem 2.10. Then P has a unique proper nontrivial characteristic subgroup if and only if GL(d, p) X acts irreducibly on each of V and (V ⊕ L 2 V )/X.
convention of [ 8 ,
8Ch. 5.1.1] and refer toĜ as the (finite) simply connected version ofG. In fact, for all but a finite number of simple groups of Lie typeG, the Sylow p-subgroup of Z(J) is trivial, and henceĜ = J. Even when this is not the case,G ∼ =Ĝ/Z(Ĝ). Furthermore, if Z(J) is a p-group, thenĜ = J/Z(J) ∼ =G. In particular, ifG is an exceptional Chevalley group, thenĜ = J if and only ifG / ∈ {G 2 (3), G 2 (4), F 4 (2)}, andĜ ∼ =G if and only if G = E 6 (q) with q ≡ 1 (mod 3) orG = E 7 (q) with q odd [23, Theorem 5.1.4]
Proposition 3 . 3 .
33Let V be a K[G]-module. Then Λ(V ) is the disjoint union of the weight multisets for the composition factors of V .
via the following process. (i) Determine the weight multiset Λ(V ), e.g., by calculating the appropriate Weyl orbits from Lübeck's data. (ii) Calculate the weight multiset Λ(W ), using Lemma 3.2. (iii) Set D := Λ(W ).
Theorem 4 . 1 (
41Tits[28, Theorem 24.17]). The groupĜ is quasisimple.
Proposition 4 . 2 .
42IfĜ ∼ =G, then Z(Ĝ) is the only nontrivial proper normal subgroup of G. Proof. SinceĜ is quasisimple by Theorem 4.1, each proper normal subgroup ofĜ lies in Z(Ĝ) [5, p. 350]. IfĜ is not isomorphic toG ∼ =Ĝ/Z(Ĝ), then |Z(Ĝ)| is prime [28, Corollary 24.13]. Hence Z(Ĝ) is the only nontrivial proper normal subgroup ofĜ.
Theorem 4 . 4 .
44The irreducible K[Ĝ]-modules are the restrictions toĜ of the irreducible K[G]-modules in L. Furthermore, if L(λ), L(µ) ∈ L, then the restrictions of L(λ) and L(µ) toĜ are isomorphic if and only if λ = µ.
Up to isomorphism, there is a 1-1 correspondence between irreducible K[G]modules in L and irreducible F q u [Ĝ]-modules. Specifically, if U ∈ L, and if V is the corresponding irreducible F q u [Ĝ]-module, then U|Ĝ is isomorphic to the K[Ĝ]-module V K constructed by extending the scalars.
Lemma 4. 9 .
9Let d be the dimension of a minimal K[G]-module, as in
Theorem 4 . 11 .
411Let V be a minimal F q [Ĝ]-module, with p odd, and let W ∈ L be the irreducible K[G]-module corresponding to V , as in Lemma 4.5.
(iv) For a fixed groupG and a fixed prime p > 3, the dimensions of the composition factors of L 3 V do not depend on q or on the choice of V .Proof. We have from Lemma 4.9 that for a given combination ofG, d and q as allowed by this lemma, there is a unique quasi-equivalence class Q of minimal F q [Ĝ]-modules. If T ∈ Q and if α is an automorphism ofĜ, then A 2 (T α ) = (A 2 T ) α , and hence all modules in the set {A 2 T | T ∈ Q} are quasi-equivalent. In particular, all modules in this set have equivalent submodule structures. Similarly, this holds for all modules in the set {L 3 T | T ∈ Q}. Hence this structure does not depend on the choice of V . Furthermore, if (G, q, W ) = (G 2 (q), 3, L(λ 2 )), then the tables in Appendices A and B show that each composition factor of A 2 W or L 3 W lies in L. If (G, q, W ) = (G 2 (q), 3, L(λ 2 )), then the only composition factor of A 2 W that does not lie in L is L(3λ 1 ) ∼ = L(λ 1 ) φ . The field automorphism φ in this case induces the trivial automorphism ofĜ, and it follows that the restriction of L(3λ 1 ) toĜ is isomorphic to the restriction of L(λ 1 ) toĜ. Thus Theorem 4.6 implies that, in each case, the composition factors of A 2 V or L 3 V correspond to those of A 2 W or L 3 W , respectively. In particular, for a fixed groupG and a fixed prime p, the dimensions of the composition factors of A 2 V or of L 3 V do not depend on q or on the choice of V , nor do the composition factors of A 2 W , L 3 W or their restrictions toĜ.Suppose now that (G, p) / ∈ {(G 2 (q), 3), (F 4 (q), 3), (E 7 (q), 7)
A 2 W 6 .
26or L 3 W toĜ. The submodule structure of each multiplicity free K[G]-module, K[Ĝ]-module or F q [Ĝ]-module follows from the composition factors of the K[G]-modules given in the tables in Appendices A and B. Next, suppose thatG = Y ℓ (q), where (Y ℓ , p) ∈ {(G 2 , 3), (F 4 , 3), (E 7 , 7), (E 8 , 3), (E 8 , 5)}. Additionally, letX := Y ℓ (p), letX be the simply connected version ofX, and let U be a minimal F p [X]-module. Then G is the linear algebraic group associated withX. We have (A 2 U) K ∼ = (A 2 W )|X and (A 2 V ) K ∼ = (A 2 W )|Ĝ from Theorem 4.SinceX
Figure 1 .Figure 2 .Figure 3 .Figure 4 .Figure 5 .
12345The submodule structure of L 3 V , where V is a minimal F q [Ĝ]module, withG = E 6 (q) and q a power of a prime p > The submodule structure ofL 3 V , where V is a minimal F q [E 6 (5)]The submodule structure of L 3 V , where V is a minimal F q [Ĝ]module, withG = E 7 (q) and q a power of a prime p / ∈ {2, The submodule structure of L 3 V , where V is a minimal F q [Ĝ]module, withG = E 7 The submodule structure of L 3 V , where V is a minimal F q [Ĝ]module, withG = E 7 (11).
Figure 6 .
64.11 that is associated with an exceptional prime are exactly the same as the corresponding module when the prime is not exceptional. Furthermore, containments between these submodules in the exceptional prime case are exactly those that are allowed by the dimensions of the module's composition factors. In fact, this observation The submodule structure of
Lemma 5.2 ([8,). Let U be a vector space over F q , and let H be a perfect, absolutely irreducible subgroup of GL(U). Suppose also that no conjugate of H in GL(U) can be written over a proper subfield of F, and that H preserves a nonzero bilinear or unitary form β on U up to scalars. Then:(i) β is non-degenerate; (ii) H preserves β absolutely; (iii) H preserves no nonzero bilinear or unitary forms on U other than scalar multiples of β; and (iv) N GL(U ) (H) preserves β up to scalars.
Definition 5.5. A subgroup H of R lies in class C 9 if all of the following hold: (i) H/(H ∩ Z GL ) is almost simple; (ii) H does not contain S; (iii) H ∞ is absolutely irreducible; (iv) no conjugate of H ∞ in GL(d, q) can be written over a proper subfield of F q ; (v) H ∞ preserves no nonzero unitary or reflexive bilinear form if T = GL(d, q); (vi) H ∞ preserves no nonzero unitary or orthogonal form if T preserves a non-degenerate alternating form; and (vii) H ∞ preserves no nonzero unitary or alternating form if T preserves a non-degenerate orthogonal form.
where m > 1 is an integer dividing d, and where each V i is a subspace of V of dimension d/m. In particular, H permutes the components of this decomposition, and hence there exists a homomorphism ρ from H to the symmetric group S m such that (H)ρ is the permutation group induced by H on {V 1 , . . . , V m }. Since |G| does not divide |S m | for any m d, |G| does not divide the order of the quotient (Ĝ)ρ ofĜ. AsG ∼ =Ĝ/Z(Ĝ), it follows from Proposition 4.2 that (Ĝ)ρ = 1. Now, we can identify ker ρ with a subgroup of B := GL(V 1 ) ×· · ·×GL(V m ) [8, Ch. 2.2.2]. Let π i be the projection map from B to GL(V i ) for each i ∈ {1, . . . , m}. AsĜ ker ρ, there exists j ∈ {1, . .
Lemma 5 . 9 .
59No C 4 -subgroup of R containsĜ. Proof. Suppose that H is a C 4 -subgroup of R that containsĜ. Then H lies in the central product X := GL(m, q) • GL(n, q), where 1 1 < m < √ d and mn = d [8, Ch. 2.2.4]. Bamberg et al. [4, Lemma 5.5] show that X stabilises a subspace of A 2 V of dimension m 2 n+1 2 , and so the subgroupĜ of X also stabilises this subspace. However, Theorem 4.11 shows that A 2 V does not contain aĜ-submodule of dimension m 2 n+1 2 , for any permitted values of m and n. This is a contradiction, and thereforeĜ lies in no C 4 -subgroup of R.
C 6subgroups are only defined when d ∈ {7, 25, 27}, and C 7 -subgroups are only defined when d ∈ {25, 27}. If H 6 is a C 6 -subgroup of S with d ∈ {7, 25}, then S = SL(d, q) and H 6 has shape (A • r 1+m + ).Sp(m, r), where |A| 27 and (d, r, m) extraspecial group of order r 1+m and exponent r [8,
H ∞ = PSU(d, r), then the minimal F q [X]-module corresponds to the irreducible F p [X]module L(aλ i ), where a is a power of p and i ∈ {1, d − 1}, and if H ∞ = PΩ − (d, r), then the minimal F q [X]-module corresponds to L(aλ 1 ) [24, §1] (see also
then p = 3, and hence d ∈ {7, 27} [25, Appendix A.49]. In this case, r = q [23, Remark 5.7(b)].
Lemma 5 . 12 .
512Let N := N GL(d,q) (Ĝ). IfG = E 6 (q), then N = (Z GLĜ ).(q − 1, 3), and N is a C 9 -subgroup of GL(d, q). IfG = E 7 (q), then N = (Z GLĜ ).2, and N is a proper subgroup of CSp(56, q). Otherwise, N = Z GLĜ , and N is not maximal in GL(d, q).
Dedekind's Identity now implies that if S is as inTheorem 5.11, or if S := SL(d, q), then N S (Ĝ) lies in the set {Z(S)Ĝ, (Z(S)Ĝ).3} whenG = E 6 (q) with q ≡ 1 (mod 3); N S (Ĝ) lies in the set {Z(S)Ĝ, (Z(S)Ĝ).2} whenG = E 7 (q); and N S (Ĝ) = Z(S)Ĝ otherwise.Proposition 5.13. Suppose thatG = E 6 (q), and that T is the full group of isometries of β in GL(d, q). Additionally, let H be a subgroup of SL(d, q) that containsĜ. Then S H if and only if H N SL(d,q) (Ĝ). Proof. Suppose that S H. Then H ∩ S is a proper subgroup of S containingĜ, and so Theorem 5.11 implies that H ∩ S lies in the maximal subgroup N S (Ĝ) of S. SinceĜ is perfect, we haveĜ (H ∩ S) ∞ N ∞ , where N := N GL(d,q) (Ĝ). Lemma 5.12 shows that N/(Z GLĜ ) is soluble, and hence N ∞ = (Z GLĜ ) ∞ , which is equal toĜ ∞ =Ĝ. Therefore,Ĝ is the characteristic subgroup (H ∩ S) ∞ of H ∩ S. Furthermore, Theorem 5.11 implies that the proper subgroup H of SL(d, q) lies in the full group of similarities of β in SL(d, q). This group of similarities normalises S [23, p. 14], and hence H ∩ S ⊳ H. Thus H normalises the characteristic subgroupĜ of H ∩ S, as required. Conversely, as N S (Ĝ) is a proper subgroup of S, no subgroup of N SL(d,q) (Ĝ) contains S.
Lemma 6 . 1 .
61LetG = E 7 (q).
Table 4 .Table 5 .Table 6 .Table 7 .Table 8 .Table 9 .Table 10 .
45678910The composition factors of A 2 V , with G = G 2 and V = LThe composition factors ofA 2 V , with G = G 2 and V = L(λ 2 ). 1 ) ∼ = L(λ 1 ) ⊗ L(λ 1 ) φ 36 1 L(2λ 1 ) ∼ = L(λ 1 )The composition factors of A 2 V , with G = F 4 and V = L(λ 1 ). Condition on p Composition factor Dimension Multiplicity The composition factors of A 2 V , with G = F 4 and V = L(λ The composition factors of A 2 (V 1 ) and A 2 (V 2 ), with G = E 6 , V 1 = L(λ 1 ), and V 2 = L(λ 6 ). The composition factors of A 2 V , with G = E 7 and V = L(λ 7 ). Condition on p Composition factor Dimension Multiplicity The composition factors of A 2 V , with G = E 8 and V = L(λ Composition factors of third Lie powers of minimal K[G]-modules
{(e 2
2∧ e 3 ) ⊗ e 3 , (e 2 ∧ e 3 ) ⊗ e 2 , (e 2 ∧ e 3 ) ⊗ e 1 , (e 1 ∧ e 3 ) ⊗ e 3 , (e 1 ∧ e 3 ) ⊗ e 2 , (e 1 ∧ e 3 ) ⊗ e 1 , (e 1 ∧ e 2 ) ⊗ e 3 , (e 1 ∧ e 2 ) ⊗ e 2 , (e 1 ∧ e 2 ) ⊗ e 1 }.
which is a subgroup of Q W isomorphic to the elementary abelian group (L 3 V )/W . Now, let {e 1 , . . . , e d } be a basis for V . As p > 3, there is a positive integer m < p such that 12m ≡ 1 (mod p), and hence (0, 0, 12[e i , e j , e k ] + W ) m = (0, 0, [e i , e j , e k ] + W ). The bilinearity of [·, ·] implies that L 3 V is spanned by the Lie brackets [e i , e j , e k ], and hence S is generated by the commutators
Table 1 .
1The highest weights and dimensions of minimal K[G]-modules, up to isomorphism and twisting by a field automorphism of G.G Highest weight Condition on p Dimension
Let V and W be modules for a group H over the same field. We say that V and W are quasi-equivalent if there exists an automorphism α of H such that W ∼ = V α .Throughout the remainder of this section, we assume that:G is an exceptional Chevalley group. Recall that ifĜ ∼ =G, thenĜ is the universal cover ofG. In the latter case, each automorphism ofG lifts to a unique automorphism ofĜ, andĜ has no other automorphisms[15, Corollary 5.1.4].Definition 4.8 ([8, Ch. 1.8.2]).
module that lies in the set L can be written asProof. First, note that Lemma 4.5 implies that the minimal K[G]-modules and the minimal
F q [Ĝ]-modules have the same dimension. Table 1 lists all minimal K[G]-modules, up to
isomorphism and twisting by a field automorphism, where the highest weight of each of
these modules is a fundamental dominant weight. It follows from Lemma 3.2 that each
minimal K[G]-
. Moreover, each minimal F q [Ĝ]module is absolutely irreducible by Proposition 4.3. Finally, as there are no nontrivial irreducible K[G]-modules of dimension less than d, Proposition 4.7 implies that no minimal F q [Ĝ]-module can be written over a proper subfield of F q . In fact, Theorem 4.4 implies that the dimension of a minimal K[G]-module is equal to the dimension of a minimal K[Ĝ]-module.Proposition 4.10. Let d be the dimension of a minimal K[G]-module, as inTable 1. Additionally, let H = 1 be a quotient ofĜ, and let V be a faithful F q [H]-module of dimension at most d. Then dim(V ) = d, H =Ĝ, and V is irreducible.Proof. By Proposition 4.2, ifĜ is not simple and H =Ĝ, then H =Ĝ/Z(Ĝ) ∼ =G. Hence H is non-abelian in each case, and so the faithful H-module V has dimension at least 2. Suppose that V is reducible. Since each irreducible F q [H]-module is also an irreducible F q [Ĝ]-module, Lemma 4.9 implies that each F q [Ĝ]-composition factor of V is the trivial irreducible module. Each F q [H]-composition series for V is a normal series for V as an abelian p-group, and H is a group of automorphisms of this p-group since V is a faithful H-module. Therefore, H stabilises this normal series. This implies that H is a p-group[14, Corollary 5.3.3]. However, |G| divides |H|, andG is not a p-group. This is a contradiction, and thus V is irreducible. It follows immediately from Lemma 4.9 that dim(V ) = d and that V is a faithful F q [Ĝ]-module, i.e., H =Ĝ.
Table 2 .
2The submodule structure of A 2 V , where V is a minimal F q [Ĝ]module, with q a power of a prime p > 2.G
d
Exceptional
primes
Standard structure
of A 2 V
Structure of A 2 V for
exceptional primes
Table 2 .
29]. Additionally, if H 7 is a C 7 -subgroup of S with d ∈ {25, 27}, then the shape of H 7 is either B.PSL(n, q) t .C.S t or Ω(n, q) t .D.S t , where |B|Table 2.10]. By considering group orders, we see that, in each case, |G| does not divide |H 6 | or |H 7 |. Since |G| divides |Ĝ|,Ĝ does not lie in H 6 or in H 7 .5, |C|
125, |D|
4 and
Table 8 .
835] for d = 7, and by Kleidman and Liebeck [23, Proposition 7.8.1, Lemma 8.1.6] for the other values of d under consideration.
The weight associated with C 28 that we denote by λ i is denoted by λ 29−i by Lübeck[25].
Acknowledgements. The research in this paper was conducted while the author was in receipt of a Hackett Postgraduate Research Scholarship and an Australian Government Research Training Program at the University of Western Australia. The author is also grateful to Martin Liebeck, Donna Testerman and Gunter Malle for helpful discussions about highest weight theory, and to John Bamberg and Luke Morgan for general helpful discussions and detailed feedback.Condition on pComposition factor Dimension Multiplicity56 1C. Magma computationsIn this section, we discuss important details about the computations in the Magma [6] computer algebra system involved in the proof of Theorem 4.11. Let q be a power of an odd prime p, letG = Y m (q) be an untwisted exceptional group of Lie type, and letĜ be the simply connected version ofG. We can construct a minimal F q [Ĝ]-module V in Magma using the command GModule(ChevalleyGroup("Y",m,q)), unless (G, p) = (F 4 (q), 3), in which case V is a proper submodule of the original module.When q = p is an "exceptional prime" forG, we can determine the submodule structure of L 2 V ∼ = A 2 V using the SubmoduleLattice command. However, whenG ∈ {E 7 (p), E 8 (p)}, it is significantly faster to determine this structure using the MaximalSubmodules command iteratively. Using the latter method with a 2.6 GHz CPU, the calculation in the
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Composition factors of exterior squares of minimal K[G]-modules Tables 4-10 list the composition factors of the exterior square A 2 V of each irreducible K[G]-module V from Theorem 3.9. Here, V is a minimal K[G]-module, unless (G, V ) = (G 2 , L(λ 2 )) with p = 3, or (G, V ) = (F 4 , L(λ 1 )) with p > 2. A , In each case, the highest weights of all composition factors of A 2 V are comparable with respect to the partial order defined in §3, and we list these composition factors by descending highest weightA. Composition factors of exterior squares of minimal K[G]-modules Tables 4-10 list the composition factors of the exterior square A 2 V of each irreducible K[G]-module V from Theorem 3.9. Here, V is a minimal K[G]-module, unless (G, V ) = (G 2 , L(λ 2 )) with p = 3, or (G, V ) = (F 4 , L(λ 1 )) with p > 2. In each case, the highest weights of all composition factors of A 2 V are comparable with respect to the partial order defined in §3, and we list these composition factors by descending highest weight.
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[] |
[
"Partial Domination in Graphs",
"Partial Domination in Graphs"
] |
[
"Angsuman Das [email protected] \nDepartment of Mathematics\nSt. Xavier's College\nKolkataIndia\n"
] |
[
"Department of Mathematics\nSt. Xavier's College\nKolkataIndia"
] |
[] |
Let G = (V, E) be a graph. For some α with 0 < α ≤ 1, a subset S of V is said to be a α-partial dominating set if |N[S]| ≥ α|V |. The size of a smallest such S is called the αpartial domination number and is denoted by pd α (G). In this paper, we introduce α-partial domination number in a graph G and study different bounds on the partial domination number of a graph G with respect to its order, maximum degree, domination number etc., Moreover, α-partial domination spectrum is introduced and Nordhaus-Gaddum bounds on the partial domination number are studied.
|
10.1007/s40995-018-0618-5
|
[
"https://arxiv.org/pdf/1707.04898v2.pdf"
] | 119,178,481 |
1707.04898
|
6c429d24e824f98bd8a40bd3ce73d9838f5eab91
|
Partial Domination in Graphs
24 Jul 2017
Angsuman Das [email protected]
Department of Mathematics
St. Xavier's College
KolkataIndia
Partial Domination in Graphs
24 Jul 2017arXiv:1707.04898v2 [math.CO]α-dominationconnected graphedge-critical graphs 2008 MSC: 05C69
Let G = (V, E) be a graph. For some α with 0 < α ≤ 1, a subset S of V is said to be a α-partial dominating set if |N[S]| ≥ α|V |. The size of a smallest such S is called the αpartial domination number and is denoted by pd α (G). In this paper, we introduce α-partial domination number in a graph G and study different bounds on the partial domination number of a graph G with respect to its order, maximum degree, domination number etc., Moreover, α-partial domination spectrum is introduced and Nordhaus-Gaddum bounds on the partial domination number are studied.
Introduction
The domination in graphs has been an active area of research from the time of its inception. Two domination books [6,7] provide a comprehensive report of vastness of the area of the domination and its relation to other graph parameters. Many variations e.g., [1,3,4,5] etc., of the domination problem can be found in literature most of which are motivated by many real-life scenarios.
Consider the following scenario. Imagine that you are the curator of an art museum and you wish to determine the minimum number of guards you need to guard the exhibits. A guard can guard an exhibit that he/she is standing near, and any exhibit in the museum that they can clearly see. In order to model the security situation, you would construct a graph G as following: Each vertex represents an exhibit location and two vertices u and v are adjacent if and only if the locations they represent are visible from each other, that is, a person standing at the exhibit modelled by vertex u can clearly see the location of the exhibit modelled by vertex v, and vice versa. Suppose that security requirements mandate that a staff of guards are positioned at locations such that every art exhibit is protected by a guard that can see it, and budget restrictions make it desirable to hire as few guards as possible. In this case, the most economical solutions, that is, the minimum guards for possible guard location configurations, correspond to the γ-sets. Suppose that due to budgetary concerns, as curator, you are strictly limited to hiring exactly γ guards.
While this is the optimum solution economically, in a practical sense it leaves much to be desired. There will be days when guards are ill, guards need a day off or some of them institute a labour action and go on strike. As curator you can now at most secure a fraction or part of the exhibits and keep the rooms containing unguarded exhibits locked for that day. It is with this problem in mind, that we introduce in this paper the concept of the partial domination in a graph. A closely related problem based on algorithmic viewpoint can be found in [8].
Let G = (V, E) be a graph. For some α with 0 < α ≤ 1, a subset S of V is said to be a α-partial dominating set if |N[S]| ≥ α|V |. The size of a smallest such S is called the α-partial domination number and is denoted by pd α (G). Clearly 1 ≤ pd α (G) ≤ γ(G) and pd 1 (G) = γ(G). Also, α 1 < α 2 implies pd α 1 (G) ≤ pd α 2 (G).
Some Basic Results
We start with some basic results. As they are straightforward, they are given either without proof or with a minimalistic proof.
Proposition 2.1. Let G be a graph on n vertices. Then pd α (G) = 1 for all α ∈ (0, ∆+1 n ]. Proposition 2.2. Let G be a graph on n vertices. Then pd α (G) = γ(G) for all α ∈ (1 − 1 n , 1]. Proposition 2.3. pd α (C n ) = pd α (P n ) = ⌈ nα 3 ⌉ Proof: Let S be a pd α -set of C n . Then |N[S]| ≥ ⌈nα⌉. To dominate ⌈nα⌉ vertices in C n , we need at least ⌈nα⌉ 3 vertices. Thus pd α (C n ) = |S| = ⌈nα⌉
3 = ⌈ nα 3 ⌉. Similarly, pd α (P n ) = ⌈ nα 3 ⌉. Proposition 2.4. pd α (K n ) = 1 and for m ≥ n, pd α (K m,n ) = 1 if 0 < α ≤ m+1 m+n , 2 if α < m+1 m+n ≤ 1].
Some Bounds on Partial Domination Number
In this section, we study some bounds on partial domination number of a graph with respect to other graph parameters.
(G) ≤ γ ⌊ 1 α ⌋ . Proof: Let D be a γ-set of G and set t = ⌊ 1 α ⌋. Let D 1 , D 2 , . . . , D t be a partition of D such that |D i | ≤ γ ⌊ 1 α ⌋ for all i. Thus we have N[D] = N[D 1 ] ∪ N[D 2 ] ∪ · · · ∪ N[D t ]. Let
n be the order of G. Then
n = |N[D]| ≤ t i=1 |N[D i ]| ≤ t|N[D j ]|, where |N[D j ]| = max i |N[D i ]| i.e., |N[D j ]| ≥ n t = n ⌊ 1 α ⌋ ≥ nα
Thus, N[D j ] is an α-partial dominating set of G and hence pd α (G) ≤ γ ⌊ 1 α ⌋ . Corollary 3.4. If G is a graph of order n without any isolated vertex, then pd α (G) ≤ n 2⌊ 1 α ⌋
.
Proof: Since G does not have any isolated vertex, γ ≤ n/2. Thus the corollary follows from the previous theorem. Theorem 3.1. Let G be a graph with domination number γ(G). Then for all α ∈ (0, 1),
Corollary 3.5. If G is a graph with domination number γ and α ∈ (0, 1/γ], then pd α (G) = 1. Proof: Since α ≤ 1/γ, we have γ ≤ 1/α, i.e., γ ≤ ⌊1/α⌋. Hence γ ⌊1/α⌋ ≤ 1, i.e.,pd α (G) + pd 1−α (G) ≤ γ + 1.
Proof: Let S be a γ(G)-set and α ∈ (0, 1). Let S 1 be a subset of S with |N[S 1 ]| ≥ nα such that S 1 is a minimal subset of S with this property. Clearly pd α (G) ≤ |S 1 |. Let S 2 = S \ S 1 and v ∈ S 1 . Since S 1 is minimal with respect to the above property, we have
|N[S 1 \ {v}]| < nα. Now, as S = (S 1 \ {v}) ∪ (S 2 ∪ {v}), we get n = |V | = |N[S]| ≤ |N[(S 1 \ {v})]| + |N[(S 2 ∪ {v})]| < nα + |N[(S 2 ∪ {v})]| i.e., |N[(S 2 ∪ {v})]| > n(1 − α) Thus S 2 ∪{v} is an (1−α)-partial dominating set of G and pd 1−α (G) ≤ |S 2 ∪{v}| = |S 2 |+1. Hence, pd α (G) + pd 1−α (G) ≤ |S 1 | + |S 2 | + 1 = |S| + 1 = γ + 1.
In fact, it is possible to find a generalization of Theorem 3.1 in a natural way.
Theorem 3.2. Let G be a graph with domination number γ. For any positive integer k ≥ 2, with α 1 +α 2 +· · ·+α k ≤ 1 and α i ∈ (0, 1) for all i, pd α 1 (G)+pd α 2 (G)+· · ·+pd α k ≤ k 2 (γ +1). Proof: We prove it by induction on k. For k = 2, α 1 + α 2 ≤ 1. Hence, by Theorem 3.1, pd α 1 (G) + pd α 2 (G) ≤ pd α 1 (G) + pd 1−α 1 (G) ≤ γ + 1.
Assume that k > 2 and the theorem holds for integers less than k. Then at least one value of α i must satisfy
α i ≤ 1 2 . Without loss of generality, let α k ≤ 1 2 . Therefore, by Theorem 3.3, pd α k (G) ≤ γ 2 .
Finally, using the induction hypothesis, we get
[pd α 1 (G) + pd α 2 (G) + · · · + pd α k−1 (G)] + pd α k (G) ≤ (k − 1) 2 (γ + 1) + γ 2 i.e., pd α 1 (G) + pd α 2 (G) + · · · + pd α k (G) ≤ (k − 1) 2 (γ + 1) + γ 2 + 1 2 = k 2 (γ + 1). Theorem 3.3. Let G be a graph with components G 1 , G 2 , . . . , G k . Then pd α (G) ≤ k i=1 pd α (G i ) . Proof: Let S i be a pd α (G i )-set of G i , for i = 1, 2, . . . , k. Then |N[S i ]| ≥ α|V (G i )|, for i = 1, 2, . . . , k. Let S = S 1 ∪ S 2 ∪ · · · ∪ S k . Thus |N[S]| = k i=1 |N[S i ]| ≥ α k i=1 |V (G i )| = α|V (G)|,
and S is a α-partial dominating set of G and hence,
pd α (G) ≤ |S| = k i=1 |S i | = k i=1 pd α (G i ).
Vertex and Edge Removal
In this section , we focus on effect of removal and addition of edges and vertices of a graph on its partial domination number.
Theorem 4.1. Let G = (V, E) be a graph and e ∈ E. Then pd α (G) ≤ pd α (G − e) ≤ pd α (G) + 1. Proof: Clearly pd α (G) ≤ pd α (G − e)
. Thus we prove the other part of the inequality. Let S be a pd α (G)-set and e = xy where
x, y ∈ V . If x, y ∈ S, then N G [S] = N G−e [S] and hence S is an α-partial dominating set of G − e, i.e., pd α (G − e) ≤ pd α (G). Similarly, if x ∈ S and y ∈ S, then N G [S] = N G−e [S] and hence pd α (G − e) ≤ pd α (G). Finally, if x ∈ S and y ∈ S, then S ∪ {x} is an α-partial dominating set of G − e, i.e., pd α (G − e) ≤ pd α (G) + 1.
Combining all the cases, we get the upper bound.
Theorem 4.2. Let G = (V, E) be a graph and v ∈ V . Then pd α (G) − 1 ≤ pd α (G − v) ≤ pd α (G) + deg G (v) − 1. Proof: Let S be a pd α (G)-set. If v ∈ N[S], then S is an α-partial dominating set of G − v, as |N G−v [S]| = |N G [S]| ≥ |V |α > (|V | − 1)α and hence pd α (G − v) ≤ pd α (G). If v ∈ N[S] \ S, then N G−v [S] = N G [S] \ {v}, i.e., |N G−v [S]| = |N G [S]| − 1 ≥ |V |α − 1. Let u ∈ V \ N G−v [S] such that u = v and set S ′ = S ∪ {u}. Then |N G−v [S ′ ]| ≥ |N G−v [S]| + 1 ≥ |V |α ≥ (|V | − 1)α
i.e., S ′ is an α-partial dominating set of G − v and hence pd α (G − v) ≤ |S ′ | = pd α (G) + 1.
If
v ∈ S, then N G−v [S \ {v}] ⊇ N G [S] \ N[v], i.e., |N G−v [S \ {v}]| ≥ |N G [S]| − (deg G (v) + 1) ≥ |V |α − deg G (v) − 1. Now let T be a collection of deg G (v) many vertices in (V \ {v}) \ N G−v [S \ {v}] and let S 1 = (S \ {v}) ∪ T . Then N G−v [S 1 ] = N G−v [S \ {v}] ∪ N G−v [T ] and hence |N G−v [S 1 ]| ≥ |N G−v [S \{v}]|+|T | ≥ (|V |α−deg G (v)−1)+deg G (v) = |V |α−1 ≥ (|V |−1)α.
Thus S 1 is an α-partial dominating set of G − v, i.e.,
pd α (G − v) ≤ |S 1 | = |S| − 1 + deg G (v) = pd α (G) + deg G (v) − 1.
Combining all the above three cases, we have the proposed upper bound. For the lower bound, let S be a pd α (G − v)-set.
Then |N G−v [S]| ≥ (|V | − 1)α. Let u ∈ V \ N G−v [S] and set S 1 = S ∪ {u}. Then |N G [S 1 ]| ≥ |N G−v [S]| + 1 ≥ (|V | − 1)α + 1 ≥ |V |α.
Thus, S 1 is an α-partial dominating set of G and hence, pd α (G) ≤ |S 1
| = |S| + 1 = pd α (G − v) + 1, i.e., pd α (G) − 1 ≤ pd α (G − v).
We call a graph G, α-partial domination vertex critical, or just pd α -vertex critical if for any v ∈ V , pd α (G − v) < pd α (G). In the light of the above theorem, if a graph G is pd α -vertex critical, then pd α (G − v) = pd α (G) − 1.
(G) ≤ |T ∪ {v}| = pd α (G − v) + 1. Moreover as G is pd α -vertex critical graph, we have pd α (G − v) = pd α (G) − 1 for all v ∈ V . Thus pd α (G) = |T ∪ {v}|, i.e., S = T ∪ {v} is a pd α (G)-set containing v. If T ∩ N(v) = ∅, then |N G [T ]| = |N G−v [T ]| + 1 ≥ (|V | − 1)α + 1 ≥ |V |α, i.e.
, T is an α-partial dominating set of G. This contradicts pd α (G − v) < pd α (G) and hence
T ∩ N(v) = ∅. Thus pn G [v, S] = {v}.
Nordhaus-Gaddum Bounds
In this section, we study some Nordhaus-Gaddum bounds on partial domination number of a graph G. We start with recalling some known Nordhaus-Gaddum bounds on domination number of a graph G.
(G) + pd α (G) ≤ n ⌊ 1 α ⌋ + 1. Proof: From Theorem 3.3, we have pd α (G) ≤ γ(G) ⌊ 1 α ⌋ .
Thus
pd α (G) + pd α (G) ≤ γ(G) ⌊ 1 α ⌋ + γ(G) ⌊ 1 α ⌋ ≤ γ(G) + γ(G) ⌊ 1 α ⌋ + 1.
From Proposition 5.1, we get γ(G) + γ(G) ≤ n except when G = K n or G = K n . Thus apart from these two cases, we have pd α (G) + pd α (G) ≤ n ⌊ 1 α ⌋ + 1.
Now, consider the case when G = K n (or K n ). Then pd α (G) = 1 and pd α (G) = ⌈nα⌉.
Thus, pd α (G) + pd α (G) = ⌈nα⌉ + 1 ≤ n ⌊ 1 α ⌋ + 1. Combining all these cases, we have the theorem.
pd α (G) = pd α (G) = 1, i.e., pd α (G) + pd α (G) = 2 ≤ 4 − 1 ⌊ 1 α ⌋ + 1.
If 3/4 < α ≤ 1, we have pd α (G) = pd α (G) = 2. Also, ⌊1/α⌋ ≤ ⌊4/3⌋ = 1. Thus,
pd α (G) + pd α (G) = 4 = 3 + 1 ≤ 4 − 1 ⌊ 1 α ⌋ + 1.
Combining all these cases, we have the theorem.
Theorem 5.3. For graphs G and G without isolated vertices, pd α (G)+pd α (G) ≤ ⌊n/2⌋ + 2
⌊ 1 α ⌋ + 1.
Proof: The theorem follows exactly as the proof of Theorem 5.1 by using Proposition 5.3.
α-Partial Domination Spectrum of a Graph and its Consequences
We define α-partial domination spectrum of a graph G, denoted by Sp p α (G), to be the set of distinct values of pd α (G) as α runs over (0, 1], i.e., Sp p α (G) = {pd α : α ∈ (0, 1]}. Now, two cases may arise: either Sp p α (G) is singleton or not. It is known that if for a graph G, γ = 1, then pd α (G) = 1 for all α ∈ (0, 1], i.e., |Sp p α (G)| = 1. On the other hand, if γ ≥ 2, then 1, γ ∈ Sp p α (G), i.e., |Sp p α (G)| ≥ 2. Now, we move towards proving our main result that the α-partial domination number changes its value only at rational points as α runs over (0, 1]. However before doing that, we prove a lemma which we will use later. Thus there exists α ′ ∈ A such that pd α ′ = p. This imply that for all α ∈ [α ′ , α * ), pd α = p. Let (α k ) be a strictly monotonically increasing sequence in [α ′ , α * ) such that (α k ) converges to α * . Now as α k ∈ [α ′ , α * ), we have pd α k = p, i.e., for each α k , there exists S k ⊆ V with |S k | = p such that |N[S k ]| ≥ n · α k . Moreover as |S k | = p < q, S k is not a pd α * -set, i.e., |N[S k ]| < n · α * . Thus we get a sequence of S k of subsets of V such that
α k ≤ |N[S k ]| n < α * , ∀k ∈ N(1)
As G is a finite graph, the number of choices for subsets S k of size p is finite. Claim 2: α * ∈ Q. If possible, let α * ∈ (0, 1] \ Q. We observe that pd α * (G) = p, because if pd α * (G) < p, then for all α ∈ A, pd α (G) < p which contradicts the fact that p ∈ Sp p α (G). Since, α * is an irrational number, nα * is not an integer. Now as (0, 1] ∩ (R \ Q) is dense in (0, 1], there exists an irrational number α ∈ (0, 1] ∩ (R \ Q) with α > α * such that ⌈nα * ⌉ = ⌈nα⌉. (We omit the details of the proof) Now let S be a pd α * -set of G. Then |N[S]| ≥ nα * . As α * ∈ R \ Q, we have |N[S]| ≥ ⌈nα * ⌉ = ⌈nα⌉ ≥ nα. Thus S is a α-partial dominating set in G and |S| = p.
On the other hand, as α > α * , α ∈ B. But S being a α-partial dominating set of G must have cardinality ≥ q (by definition of B). This is a contradiction. Hence α * ∈ Q. Now we are in a position to prove the following theorem.
Theorem 6.1. Let G be a graph such that Sp p α (G) = {a 1 , a 2 , . . . , a t } with 1 = a 1 < a 2 < . . . < a t = γ and t > 1. Then there exists t − 1 rational numbers α 1 < α 2 < . . . < α t−1 in (0, 1) ∩ Q such that or equal to a t−1 and B 1 = {α ∈ (0, 1] : pd α (G) = a t } = (α t−1 , 1], we have {α ∈ (0, 1] :
pd α (G) = a t−1 } = (α t−2 , α t−1 ].
Continuing in this way, at one stage we substitute q = a 2 in Lemma 6.1 to get a rational number α 1 such that A t−1 = {α ∈ (0, 1] : pd α (G) < a 2 } = (0, α 1 ] and B t−1 = {α ∈ (0, 1] : pd α (G) ≥ a 2 } = (α 1 , 1]. By similar argument as that of above, we get {α ∈ (0, 1] : pd α (G) = a 2 } = (α 1 , α 2 ]. Moreover, as a 1 = 1 is the only value in Sp p α (G) which is less than a 2 , we have A t−1 = {α ∈ (0, 1] : pd α (G) = a 1 } = (0, α 1 ]. Hence the theorem.
We call the α i 's obtained in Theorem 6.1 as critical values of α. Theorem 6.1 has an immediate corollary. Corollary 6.2. Let G be a graph and α be a irrational number in (0, 1). Then there exists ǫ > 0, such that for all α ∈ (α − ǫ, α + ǫ), pd α (G) is constant.
Proof: The corollary follows from Theorem 6.1 and denseness of rationals and irrationals in R.
Our next goal is to find an upper bound on the size of the α-partial domination spectrum of a graph. Before that we prove a lemma. Lemma 6.3. Let G be a graph such that Sp p α (G) = {a 1 , a 2 , . . . , a t } with a 1 < a 2 < . . . < a t and let α i 's be as in Theorem 6.1. Then for each α i , there exists a pd α i -set S i ⊆ V such that
α i = |N[S i ]| n .
Proof: Since S i is a pd a i -set of G, we have |S i | = a i and
|N[S i ]| n ≥ α i .(2)
If possible, the inequality in Equation 2 is strict. But in that case, by denseness of real numbers, we can find α ′ > α i such that |N [S i ]| n ≥ α ′ > α i . Thus S i is α ′ -partial dominating set of G and hence pd α ′ (G) ≤ |S i | = a i < a i+1 . However as α ′ > α i , we have pd α ′ (G) ≥ a i+1 . This is a contradiction. Thus, there exists S i such that Equation 2 holds with equality. Hence the theorem. Theorem 6.2. For any graph G without isolated points, the critical values belong to the set { ∆+1 n , ∆+2 n , · · · , n−1 n } and |Sp p α (G)| ≤ n − ∆.
Proof: By Lemma 6.3, for every critical value α i , there exists a pd α i -set S i ⊆ V such that
α i = |N[S i ]| n .
Thus the first part of the theorem follows from Proposition 2.1, 2.2 and the observation that |N[S i ]| ≤ n.
For the second part, observe that |Sp p α (G)| is one more than the number of critical values. Thus, |Sp p α (G)| ≤ 1 + (n − ∆ − 1) = n − ∆.
Remark 6.1. The upper bound given in Theorem 6.2 is tight: Consider an n vertex graph which consists of a clique and some isolated vertices.
Conclusion
In this paper, we introduced a new graph invariant called the partial domination number of a graph. From an applications standpoint, it can be interpreted as the measure of the maximum surveillance possible if a fraction of minimum number of guards needed is available. We studied different bounds on the partial domination number of a graph G with respect to its order, maximum degree, domination number etc.
Proposition 3 . 1 .
31Let G = (V, E) be a graph on n vertices. Then pd α (G) = 1 if and only if there exists v ∈ V such that deg(v) ≥ ⌈nα⌉ − 1. Proof: pd α (G) = 1 if and only if there exists a vertex v ∈ V such that |N[v]| ≥ nα, i.e., deg(v) ≥ ⌈nα⌉ − 1.
Proposition 3 . 2 .
32Let G = (V, E) be a graph of order n and maximum degree ∆ such that∆ < ⌈nα⌉ − 1. Then nα ∆+1 ≤ pd α (G) ≤ ⌈nα⌉ − ∆. Proof: Let S be a pd α -set in G. Then nα ≤ |N[S]| ≤ v∈S deg(v) + |S| ≤ (∆ + 1)|S| = (∆ + 1)pd αand hence the lower bound follows.For the upper bound, let v be a vertex of maximum degree in G. Then v dominates ∆ + 1 vertices. Then v along with other ⌈nα⌉ − (∆ + 1) vertices outside N[v] forms a α-partial dominating set of G. Thus pd α (G) ≤ 1 + ⌈nα⌉ − (∆ + 1) = ⌈nα⌉ − ∆.
Proposition 3 . 3 .
33Let G be a graph with domination number γ. Then pd α
thus by Theorem 3.3, we have pd α (G) = 1.
Theorem 4 . 3 .
43If G = (V, E) be a pd α -vertex critical graph, then for every vertex v ∈ V , there exists a pd α (G)-set S containing v such that pn G [v, S] = {v}. Proof: Let T be a pd α (G − v)-set. Then |N G−v [T ]| ≥ (|V | − 1)α. Thus |N G [T ∪ {v}]| ≥ (|V | − 1)α + 1 ≥ |V |α,and hence T ∪ {v} is an α-partial dominating set of G, i.e., pd α
Proposition 5 . 1 (
51Cockayne and Hedeitniemi). For any graph G, γ(G) + γ G ≤ n + 1 with equality if and only if G = K n or G = K n .
Proposition 5 . 2 (
52Laskar and Peters). For connected graphs G and G, γ(G) + γ G ≤ n with equality if and only if G = P 4 .
Proposition 5 . 3 (
53Joseph and Arumugam). For graphs G and G without isolated vertices, γ(G) + γ G ≤ ⌊n/2⌋ + 2.
Theorem 5 . 1 .
51For any graph G, pd α
Theorem 5. 2 .
2For connected graphs G and G, pd α (G) + pd α (Similar to the proof of Theorem 5.1, we have pd α (G)+pd α (G) ≤ γ(G) + γ(G) 5.2, we get γ(G) + γ G ≤ n − 1 except when G = P 4 . Thus, apart from the case when G = P 4 , we have pd α (G) + pd α (G) consider the case when G = G = P 4 . If 0 < α ≤ 3/4, then by Proposition 2.3,
Lemma 6. 1 .
1Let G be a graph such that |Sp p α (G)| > 1. Let q ∈ Sp p α (G) such that q = 1. Let A = {α ∈ (0, 1] : pd α (G) < q} and B = {α ∈ (0, 1] : pd α (G) ≥ q}. Then there exists a rational number α * ∈ (0, 1) such that A = (0, α * ] and B = (α * , 1].Proof: Since q = 1, q is not the least element of Sp p α (G). Observe that both A and B are non-empty, because (0,∆+1 n ] ⊆ A and (1 − 1 n , 1] ⊆ B. In fact, both A and B are intervals. It follows from the fact that α ′ < α ′′ implies pd α ′ (G) ≤ pd α ′′ (G). Moreover, from the definition, it follows that A ∪ B = (0, 1] and A ∩ B = ∅. Thus there exists α * ∈ (0, 1] such that either A = (0, α * ), B = [α * , 1] or A = (0, α * ], B = (α * , 1]. Claim 1: We claim that both A and B are left-open, right-closed intervals, i.e., A = (0, α * ], B = (α * , 1]. If possible, let A = (0, α * ), B = [α * , 1]. Let p ∈ Sp p α (G) be the largest element in Sp p α (G) less than q.
= α * for all k ≥ t. This is a contradiction to Equation 1. Thus our claim is justified and hence A = (0, α * ] and B = (α * , 1].Thus,
the sequence |N [S k ]|
n
assumes finitely many values. Now, since (α k ) converges to α * , by
Sandwich Theorem, the sequence |N [S k ]|
n
converges to α * . As any convergent sequence
taking finitely many values is eventually constant, we have |N [S k ]|
n
to be an eventually
constant sequence. Thus there exists t ∈ N such that |N [S k ]|
n
AcknowledgementThe author is grateful to Geertrui Van de Voorde from University of Ghent for pointing out a mistake in an earlier version of the paper. We note that, authors in [2] independently, has proposed the same notion of partial domination. However, their research is mainly focussed on the case α =
. , . 2}, 1α i , α i+1∀i ∈ {1, 2, . . . , t − 2}, ∀α ∈ (α i , α i+1 ], pd α (G) = a i+1 .
Sp α (G). we have B 1 = {α ∈ (0, 1] : pd α (G) = a t } = (α t−1 , 1However, as a t is the largest element in Sp α (G), we have B 1 = {α ∈ (0, 1] : pd α (G) = a t } = (α t−1 , 1].
. Again, substituting q = a t−1 in Lemma 6.1, we get a rational number α t−2 such that A 2 = {α ∈ (0, 1. a t−1 } = (0, α t−2 ] and B 2 = {α ∈ (0, 1] : pd α (G) ≥ a t−1 } = (α t−2 , 1Again, substituting q = a t−1 in Lemma 6.1, we get a rational number α t−2 such that A 2 = {α ∈ (0, 1] : pd α (G) < a t−1 } = (0, α t−2 ] and B 2 = {α ∈ (0, 1] : pd α (G) ≥ a t−1 } = (α t−2 , 1].
Sp p α (G) which are greater. However, as a t and a t−1 are the only two elements in Sp p α (G) which are greater
Fair domination in graphs. Y Caro, A Hansberg, M Henning, Discrete Mathematics. 312Y. Caro, A. Hansberg, and M. Henning: Fair domination in graphs. Discrete Mathe- matics 312, (2012): 2905-2914.
B M Case, S T Hedetniemi, R C Laskar, D J Lipman, Partial Domination in Graphs. B.M. Case, S.T. Hedetniemi, R.C. Laskar and D.J. Lipman: Partial Domination in Graphs, https://arxiv.org/pdf/1705.03096.pdf.
Coefficient of Domination in Graphs. A Das, Discrete Math. Algorithm. Appl. 09212 pagesA. Das: Coefficient of Domination in Graphs, Discrete Math. Algorithm. Appl., Vol. 09, Issue 2, (2017) [12 pages]
Connected Fair Domination in Graphs. A Das, W J Desormeaux, Mathematics and Computing. ICMC 2017. CCIS. Giri D., Mohapatra R., Begehr H., Obaidat M.SingaporeSpringer655A. Das and W.J. Desormeaux: Connected Fair Domination in Graphs, In: Giri D., Mohapatra R., Begehr H., Obaidat M. (eds) Mathematics and Computing. ICMC 2017. CCIS Vol. 655. pp. 96-102, Springer, Singapore.
. J E Dunbar, D G Hoffman, R C Laskar, L R Markus, Discrete Mathematics. 211α-dominationJ.E. Dunbar, D.G. Hoffman, R.C. Laskar, and L.R. Markus: α-domination, Discrete Mathematics, Vol. 211, 11-26, 2000.
Fundamentals of Domination in Graphs. T W Haynes, S T Hedetniemi, P J Slater, Marcel Dekker IncT.W. Haynes, S.T. Hedetniemi and P.J. Slater: Fundamentals of Domination in Graphs, Marcel Dekker Inc., 1998.
T W Haynes, S T Hedetniemi, P J Slater, Domination in Graphs: Advanced Topics. Marcel Dekker IncT.W. Haynes, S.T. Hedetniemi and P.J. Slater (eds): Domination in Graphs: Ad- vanced Topics, Marcel Dekker Inc., 1998.
J Kneis, D Molle, P Rossmanith, Partial vs. Complete Domination: t-Dominating Set. Jan van Leeuwen et al.4362SOFSEM 2007J. Kneis, D. Molle and P. Rossmanith: Partial vs. Complete Domination: t- Dominating Set, In Jan van Leeuwen et al. (Eds.): SOFSEM 2007, LNCS 4362, pp. 367-376, 2007.
Introduction to Graph Theory. D B West, Prentice HallD.B. West: Introduction to Graph Theory, Prentice Hall, 2001.
|
[] |
[
"Structure of infinitely divisible semimartingales",
"Structure of infinitely divisible semimartingales"
] |
[
"Andreas Basse-O'connor \nAarhus University\nUniversity of Tennessee\n\n",
"Jan Rosiński \nAarhus University\nUniversity of Tennessee\n\n"
] |
[
"Aarhus University\nUniversity of Tennessee\n",
"Aarhus University\nUniversity of Tennessee\n"
] |
[] |
This paper gives a complete characterization of infinitely divisible semimartingales, i.e., semimartingales whose finite dimensional distributions are infinitely divisible. An explicit and essentially unique decomposition of such semimartingales is obtained. A new approach, combining series decompositions of infinitely divisible processes with detailed analysis of their jumps, is presented. As an ilustration of the main result, the semimartingale property is explicitely determined for a large class of stationary increment processes and several examples of processes of interest are considered. These results extend Stricker's theorem characterizing Gaussian semimartingales and Knight's theorem describing Gaussian moving average semimartingales, in particular.
| null |
[
"https://arxiv.org/pdf/1209.1644v3.pdf"
] | 116,927,388 |
1209.1644
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5eb9494ac4c68dbd005832a9c71682e21223df07
|
Structure of infinitely divisible semimartingales
May 2014
Andreas Basse-O'connor
Aarhus University
University of Tennessee
Jan Rosiński
Aarhus University
University of Tennessee
Structure of infinitely divisible semimartingales
May 2014SemimartingalesInfinitely divisible processesStationary processesFractional processes AMS Subject Classification: 60G4860H0560G5160G17
This paper gives a complete characterization of infinitely divisible semimartingales, i.e., semimartingales whose finite dimensional distributions are infinitely divisible. An explicit and essentially unique decomposition of such semimartingales is obtained. A new approach, combining series decompositions of infinitely divisible processes with detailed analysis of their jumps, is presented. As an ilustration of the main result, the semimartingale property is explicitely determined for a large class of stationary increment processes and several examples of processes of interest are considered. These results extend Stricker's theorem characterizing Gaussian semimartingales and Knight's theorem describing Gaussian moving average semimartingales, in particular.
Introduction
Semimartingales play a crucial role in stochastic analysis as they form the class of good integrators for the Itô type stochastic integral, cf. the Bichteler-Dellacherie Theorem [10], see also Beiglböck et al. [8] for a direct proof. Semimartingales also play a fundamental role in mathematical finance. Roughly speaking, the (discounted) asset price process must be a semimartingale in order to preclude arbitrage opportunities, see Beiglböck et al. [8,Theorems 1.4,1.6] for details, see also [20]. The question whether a given process is a semimartingale is also of importance in stochastic modeling, where long memory processes with possible jumps and high volatility are considered as driving processes for stochastic differential equations. Examples of such processes include various fractional, or more generally, Volterra processes driven by Lévy processes.
The problem of identifying semimartingales within given classes of stochastic processes has a long history. In the context of Gaussian processes, it was intensively studied in 1980s. Gal'chuk [16] investigated Gaussian semimartingales addressing a question posed by Prof. A.N. Shiryayev. Key results on Gaussian semimartingales are due to Jain and Monrad [18,Theorem 1.8] and Stricker [28,Théorème 1], see also Liptser and Shiryayev [22,Ch. 4.9] and [15,2,4,16]. A particularly interesting result is due to Knight [21,Theorem 6.5]: Let X = (X t ) t≥0 be a Gaussian moving average of the form
X t = t −∞ φ(t − s) dW s ,
where (W t ) t∈Ê is a Brownian motion and φ : Ê → Ê is a Lebesgue square integrable deterministic function vanishing on the negative axis. Then X is a semimartingale with respect to the filtration (F W t ) t≥0 if and only if φ is absolutely continuous on Ê + with a square integrable derivative. Then, X can be decomposed uniquely into a Brownian motion and a predictable process of finite variation. Extensions of Knight's result were given by Jeulin and Yor [19], see also [13,12,1,5]. The class of infinitely divisible processes is a natural extension of the class of Gaussian processes. A process X = (X t ) t≥0 is said to be infinitely divisible if its all finite dimensional distributions are infinitely divisible. This work is aimed to determine the structure of infinitely divisible semimartingales.
Recall that a semimartingale X = (X t ) t≥0 , by definition, admits a decomposition
X t = X 0 + M t + A t , t ≥ 0,(1.1)
where M = (M t ) t≥0 is a local martingale and A = (A t ) t≥0 is a process of finite variation. The problem of identifying infinitely divisible semimartingales can be divided into two questions for an infinitely divisible process X = (X t ) t≥0 :
Q1. Assuming that X is a semimartingale, what is the structure of the processes M and A in its decomposition (1.1)? In particular, do they have to be infinitely divisible?
Q2. How to verify whether or not a given infinitely divisible process X is a semimartingale?
The family of infinitely divisible processes constitutes a huge class, so we will focus on a parametrization that will be convenient to state our results as well as for applications. We will assume that a primary description of an infinitely divisible processes X = (X t ) t≥0 is its 'spectral' representation of the form
X t = (−∞,t]×V φ(t, s, v) Λ(ds, dv),
where V is a set, Λ is an independently scattered infinitely divisible random measure on Ê×V , and φ : Ê 2 ×V → Ê is a measurable deterministic function (see Section 2 for details). If we further assume that φ(t, s, x) = 0 whenever s > t, then X is adapted to the filtration
F Λ = (F Λ t ) t≥0 , where F Λ t is the σ-algebra generated by Λ restricted to (−∞, t] × V .
We characterize the semimartingale property of X with respect to F Λ , which, in the spirit, is similar to the above mentioned results of Stricker and Knight. However, in the non Gaussian case, our methods are completely different. We combine series representations of càdlàg infinitely divisible processes with detailed analysis of their jumps, which is a novel approach as far as we know. This is possible because such series representations converge uniformly a.s. on compacts, as shown in Basse-O'Connor and Rosiński [7,Theorem 3.1].
Section 2 contains preliminary definitions and facts. Our main result, Theorem 3.1, is stated and proved in Section 3. It gives the necessary and sufficient conditions for (X, F Λ ) to be a semimartingale, together with an essentially unique explicit decomposition of X into infinitely divisible components. It completely answers the above question Q1 and gives a framework how to approach question Q2 in concrete situations. In Section 4 we obtain explicit necessary and sufficient conditions for a large class of stationary increment infinitely divisible processes to be semimartingales. These conditions generalize, in a natural way, findings in Basse and Pedersen [3]. We conclude this paper with examples showing how these conditions can be verified for several processes of interest.
Preliminaries
Throughout the paper (Ω, F , P) stands for a probability space and (V, V) denotes a countably generated measurable space. Consider a process X = (X t ) t≥0 of the form
X t = (−∞,t]×V φ(t, s, v) Λ(ds, dv) (2.1) where φ : Ê 2 × V → Ê is a measurable deterministic function such that for every (s, v) ∈ Ê × V , φ(·, s, v) is càdlàg and Λ is an independently scattered infinitely divisible random measure on a σ-ring S of subsets of Ê × V such that [a, b] × B : [a, b] ⊂ Ê, B ∈ V 0 ⊂ S ⊂ B(Ê) ⊗ V, for some countable ring V 0 generating V. It follows that σ(S ) = B(Ê) ⊗ V.
For example, Λ can be a Poisson random measure. In general, we assume that for every A ∈ S , Λ(A) has an infinitely divisible distribution with the cumulant
log Ee iθΛ(A) = A iθb(u) − 1 2 θ 2 σ 2 (u) + Ê e iθx − 1 − iθ[[x]] ρ u (dx) κ(du),(2.2)
where u = (s, v) ∈ Ê × V , b : Ê × V → Ê is a measurable function, κ is a σfinite measure on Ê×V , {ρ u } u∈Ê×V is a measurable family of Lévy measures on Ê, and
[[x]] = x |x| ∨ 1 = x if |x| < 1 , sgn(x) otherwise (2.3)
is a truncation function. For example, if Λ is a Poisson random measure, then b(u) = 1, σ 2 (u) = 0, and ρ u = δ 1 . The integral in (2.1) is defined as in Rajput and Rosiński [25]. Accordingly to [
(a) Ê×V |B(f (u), u)| κ(du) < ∞, (b) Ê×V K(f (u), u) κ(du) < ∞, where B(x, u) = xb(u) + Ê [[xy]] − x[[y]] ρ u (dy) and (2.4) K(x, u) = x 2 σ 2 (u) + Ê [[xy]] 2 ρ u (dy), x ∈ Ê, u ∈ Ê × V.
The process X in (2.1) is infinitely divisible, i.e., all its finite dimensional distributions are infinitely divisible.
We will further require that
Λ({s} × B) = 0 a.s. for every s ∈ Ê, B ∈ V 0 . (2.5)
This condition is equivalent to that κ({s} × V ) = 0 for every s ∈ Ê.
Let F Λ = (F Λ t ) t≥0 be the augmented filtration generated by Λ, i.e. F Λ is the least filtration satisfying the usual conditions of right-continuity and completeness such that
σ Λ(A) : A ∈ S , A ⊆ (−∞, t] × V ⊆ F Λ t , t ≥ 0.
Definition (semimartingale): Let F = (F t ) t≥0 be a filtration. An F-adapted process X = (X t ) t≥0 is a semimartingale with respect to F if it admits a decomposition
X t = X 0 + A t + M t , t ≥ 0,(2.6)
where M = (M t ) t≥0 is a càdlàg local martingale with respect to F, A = (A t ) t≥0 is an F-adapted process with càdlàg paths of finite variation and A 0 = M 0 = 0. (Càdlàg means right-continuous with left-hand limits.) Moreover, X is called a special semimartingale if, in addition, A in (2.6) can be chosen predictable. In the later case, the decomposition (2.6) is unique and is called the canonical decomposition of X and processes M and A are called the martingale and finite variation components, respectively. We refer to Jacod and Shiryaev [17] and Protter [24] for basic properties of semimartingales. For stochastic processes X = (X t ) t≥0 and Y = (Y t ) t≥0 we will write X = Y when X and Y are indistinguishable, i.e., {ω : X t (ω) = Y t (ω) for some t ≥ 0} is a P-null set. For each càdlàg function g : Ê + → Ê let ∆g(t) = lim s↑t,s<t (g(t) − g(s)) denote the jump of g at t > 0 and ∆g(0) = 0.
Infinitely divisible semimartingales
The following is our main result.
Theorem 3.1. Let X = (X t ) t≥0 be a process of the form
X t = (−∞,t]×V φ(t, s, v) Λ(ds, dv),
specified by (2.1)-(2.3), and let B be given by (2.4). Assume that for every t > 0
(0,t]×V B φ(s, s, v), (s, v) κ(ds, dv) < ∞. (3.1)
Then X is a semimartingale with respect to the filtration
F Λ = (F Λ t ) t≥0 if and only if X t = X 0 + M t + A t , t ≥ 0, (3.2)
where M = (M t ) t≥0 is a continuous in probability semimartingale with independent increments given by
M t = (0,t]×V φ(s, s, v) Λ(ds, dv), t ≥ 0 (3.3)
(i.e., the integral exists), and A = (A t ) t≥0 is a predictable càdlàg process of finite variation given by
A t = (−∞,t]×V [φ(t, s, v) − φ(s + , s, v)] Λ(ds, dv). (3.4)
Decomposition (3.2) is unique in the following sense:
If X = X 0 + M ′ + A ′ ,
where M ′ is a continuous in probability semimartingale with independent increments and A ′ is a predictable càdlàg process of finite variation, then M ′ = M + g and A ′ = A − g for some continuous deterministic function g of finite variation, with M and A given by (3.3) and (3.4). If X is a semimartingale, then it is a special semimartingale if and only if E|M t | < ∞ for all t > 0; and in this case (M t − EM t ) t≥0 is a martingale and
X t = X 0 + (M t − EM t ) + (A t + EM t ), t ≥ 0
is the canonical decomposition of X.
Remark 3.2. Condition (3.1) is always satisfied when Λ is symmetric. Indeed, in this case B = 0.
Remark 3.3. Stricker's theorem [28] says that Gaussian semimartingales are those who admit a decomposition into a Gaussian martingale with independent increments and a predictable process of finite variation. Our theorem is an exact extension of this result to the infinitely divisible case. Example 3.5. Consider the setting in Theorem 3.1 and suppose that Λ is an α-stable random measure and α ∈ (0, 1). Then X is a semimartingale with respect to F Λ if and only if it is of finite variation. This follows by Theorem 3.1 because the process M given by (4.6) is of finite variation. Indeed, the Lévy-Itô decomposition, see [27,Theorem 19.3], yields that M =M + a, where a is a continuous deterministic function and processM is of finite variation, see [27,Eq. (20.24)]. Moreover since M is a semimartingale, the function a is of finite variation, cf. [17, Ch. II, Corollary 5.11], which implies that M is of finite variation.
Proof of Theorem 3.1. The sufficiency is obvious. To show the necessary part we need to show that a semimartingale X has a decomposition (3.2) where the processes M and A have the stated properties. We will start by considering the case where Λ does not have a Gaussian component, i.e. σ 2 = 0. Case 1. Λ has no Gaussian component: We divide the proof into the following six steps.
Step 1 :
Let X 0 t = X t − β(t), with β(t) = U B φ(t, u), u κ(du).
We will give the series representation for X 0 that will be crucial for our considerations. To this end, define for s = 0 and
u ∈ U = Ê × V R(s, u) = inf{x > 0 : ρ u (x, ∞) ≤ s} if s > 0, sup{x < 0 : ρ u (−∞, x) ≤ −s} if s < 0.
Choose a probability measureκ on U equivalent to κ, and let h(u) = 1 2 (dκ/dκ)(u). By an extension of our probability space if necessary, Rosiński [26], Proposition 2 and Theorem 4.1, shows that there exists three independent sequences
(Γ i ) i∈AE , (ǫ i ) i∈AE , and (T i ) i∈AE , where Γ i are partial sums of i.i.d. standard exponential random variables, ǫ i are i.i.d. symmetric Bernoulli ran- dom variables, and T i = (T 1 i , T 2 i ) are i.i.d. random variables in U with the common distributionκ, such that for every A ∈ S , Λ(A) = ν 0 (A) + ∞ j=1 [R j 1 A (T j ) − ν j (A) a.s. (3.5) where R j = R(ǫ j Γ j h(T j ), T j ), ν 0 (A) = A b(u) κ(du), and for j ≥ 1 ν j (A) = Γ j Γ j−1 E[[R(ǫ 1 rh(T 1 ), T 1 )]]1 A (T 1 ) dr.
It follows by the same argument that
X 0 t = ∞ j=1 R j φ(t, T j ) − α j (t) a.s., where α j (t) = Γ j Γ j−1 E[[R(ǫ 1 rh(T 1 ), T 1 )φ(t, T 1 )]] dr.
Step 2 : We will show that for every i ∈ AE
∆X T 1 i = R i φ(T 1 i , T i ) a.s. (3.6)
Since X is càdlàg, the series
X 0 t = ∞ j=1 R i φ(t, T i ) − α i (t)
converges uniformly for t in compact intervals a.s., cf. Basse-O'Connor and Rosiński [7,Corollary 3.2]. Moreover, β is càdlàg, see [7,Lemma 3.5], and by Lebesgue's dominated convergence theorem it follows that α j , for j ∈ AE, are càdlàg as well. Therefore, with probability one,
∆X t = ∆β(t) + ∞ j=1 R j ∆φ(t, T j ) − ∆α j (t) for all t > 0.
Hence, for every i ∈ AE almost surely
∆X T 1 i = ∆β(T 1 i )+ ∞ j=1 R j ∆φ(T 1 i , T j )−∆α j (T 1 i ) = ∞ j=1 R j ∆φ(T 1 i , T j ) (3.7)
Indeed, by assumption (2.5) the distribution of T 1 i is continuous. Since β may have only a countable number of discontinuities, with probability one
T 1 i is a continuity point of β. Hence ∆β(T 1 i ) = 0 a.s. Since (Γ j ) j∈AE are independent of T 1 i , the argument used for β also yields ∆α j (T 1 i ) = 0 a.s. This proves (3.7).
Furthermore, for i = j we get
P(∆φ(T 1 i , T j ) = 0) = U P(∆φ(T 1 i , T j ) = 0 | T j = u)κ(du) = U P(∆φ(T 1 i , u) = 0)κ(du) = 0
again because φ(·, u) may have only countably many jumps and the distri-
bution of T 1 i is continuous. If j = i then ∆φ(T 1 i , T i ) = lim h↓0, h>0 φ(T 1 i , T 1 i , T 2 i ) − φ(T 1 i − h, T 1 i , T 2 i ) = φ(T 1 i , T i )
as φ(t, s, v) = 0 whenever t < s. This simplifies (3.7) to (3.6).
Step 3 : Next we will show that M, defined in (3.3), is a well-defined càdlàg process satisfying
∆M T 1 i = ∆X T 1 i a.s. for all i ∈ AE. (3.8)
Since any semimartingale has finite quadratic variation, we get with probability one
∞ > 0<s≤t ∆X s 2 ≥ 0<T 1 i ≤t ∆X T 1 i 2 = ∞ i=1 R i φ(T 1 i , T i ) 2 1 {0<T 1 i ≤t} ,
where the last equation employs (3.6). Put for t, r ≥ 0 and (ǫ,
s, v) ∈ {−1, 1} × Ê × V H(t; r, (ǫ, s, v)) = R(ǫrh(s, v), (s, v))φ(s, s, v)1 {0<s≤t} .
The above bound shows that for each
t ≥ 0 ∞ i=1 |H(t; Γ i , (ǫ i , T 1 i , T 2 i ))| 2 < ∞ a.s.
That implies, by Rosiński [26,Theorem 4.1], that the following limit is finite
lim n→∞ n 0 E[[H(t; r, (ǫ 1 , T 1 1 , T 2 1 )) 2 ]] dr = ∞ 0 E[[H(t; r, (ǫ 1 , T 1 1 , T 2 1 )) 2 ]] dr.
Evaluating this limit we get
∞ > ∞ 0 E[[R(ǫ 1 rh(T 1 ), T 1 )φ(T 1 i , T i )1 {0<T 1 i ≤t} ]] 2 dr = ∞ 0 Ê×V E[[R(ǫ 1 rh(s, v), (s, v))φ(s, s, v)1 {0<s≤t} ]] 2κ (ds, dv)dr = 2 ∞ 0 Ê×V E[[R(ǫ 1 u, (s, v))φ(s, s, v)1 {0<s≤t} ]] 2 κ(ds, dv)du = Ê×V Ê [[xφ(s, s, v)1 {0<s≤t} ]] 2 ρ (s,v) (dx)κ(ds, dv) = (0,t]×V Ê min{|xφ(s, s, v)| 2 , 1} ρ (s,v) (dx)κ(ds, dv).
Finiteness of this integral in conjunction with (3.1) yield the existence of the stochastic integral
M t = (0,t]×V φ(s, s, v) Λ(ds, dv).
The fact that M has independent increments is obvious since Λ is independently scattered, and its continuity in probability is a consequence of (2.5).
We
ζ t = (0,t]×V B φ(s, s, v), (s, v) κ(ds, dv), t ≥ 0, see [25, Theorem 2.7].
For t ≥ 0 we can write M t as a series using the series representation (3.5) of Λ. It follows that
M t = ζ t + ∞ i=1 R i φ(T 1 i , T i )1 {0<T 1 i ≤t} − γ j (t) where γ j (t) = Γ j Γ j−1 E[[R(ǫ 1 rh(T 1 ), T 1 )φ(T 1 1 , T 1 )1 {0<T 1 j ≤t} )]] dr.
The assumption (2.5) yields that ζ and γ j are continuous. Moreover, by arguments as above we have
∆M T 1 i = R i φ(T 1 i , T i ) a.
s. and hence by (3.6) we obtain (3.8).
Step 4 : In the following we will show the existence of a sequence (τ k ) k∈AE of totally inaccessible stopping times such that all local martingales Z = (Z t ) t≥0 with respect to F Λ are purely discontinuous and with probability one
{t ≥ 0 : ∆Z t = 0} ⊆ {τ k : k ∈ AE} ⊆ {T 1 k : k ∈ AE}. (3.9)
To show the above choose a sequence (B k ) k≥1 of disjoint sets which generates V 0 and for all k ∈ AE let U k = (U k t ) t≥0 be given by
U k t = Λ((0, t] × B k ).
Assumption (2.5) implies that U k is a continuous in probability process with independent increments and has therefore a càdlàg modification. Hence U = {(U k according to [17], Ch. III, Theorem 1.14(b) and the remark after Ch. III, 4.35, µ has the martingale representation property, that is for all real-valued local martingales Z = (Z t ) t≥0 with respect to F Λ there exists a predictable Arguing as in Step 2 with φ(t, s, v) = 1 (0,t] (s)1 B k (v) shows that with probability one
function φ from Ω × Ê + × E into Ê such that Z t = φ * (µ − ν) t , t ≥ 0 (3.∆U k t = ∆ζ(t) + ∞ j=1 R j 1 {t=T 1 j } 1 {T 2 j ∈B k } − ∆γ j (t) for all t > 0 where ξ(t) = Ê×V 1 {0≤s≤t} 1 {v∈B k } b(s, v) κ(ds, dv), γ j (t) = Γ j Γ j−1 E[[R(ǫ 1 rh(T 1 ), T 1 )1 {T 1 j ≤t} 1 {T 2 j ∈B k } )]] dr.
By assumption (2.5), ξ and γ j are continuous and hence with probability one
∆U k t = ∞ j=1 R j 1 {t=T 1 j } 1 {T 2 j ∈B k } for all t > 0.
which shows that
{τ k : k ∈ AE} ⊆ {T 1 k : k ∈ AE} a.s.
and completes the proof of Step 4.
Step 5 : Fix r ∈ AE and let X ′ = (X ′ t ) t≥0 be given by
X ′ t = X t − s∈(0,t] ∆X s 1 {|∆Xs|>r} .
We will show that X ′ is a special semimartingale with martingale component
M ′ = (M ′ t ) t≥0 given by M ′ t =M t − EM t whereM t = M t − s∈(0,t] ∆M s 1 {|Ms|>r} . (3.11)
Recall that M is given by (3.3). To show above we note that X ′ is a special semimartingale since its jumps are bounded by r in absolute value; denote by W and N the finite variation and martingale compnents, respectively, in the canonical decomposition X ′ = X 0 + W + N of X ′ . That is, we want to show that N = M ′ . Process M ′ , given by (3.11), is obviously a martingale and by (3.8) we have for all i ∈ AE
∆M ′ T 1 i = ∆M T 1 i 1 {|∆M T 1 i |≤r} = ∆X T 1 i 1 {|∆X T 1 i |≤r} = ∆X ′ T 1 i a.s. (3.12)
Since W is predictable and τ k is a totally inaccessible stopping time we have that ∆W τ k = 0 a.s. cf. [17, Ch. I, Proposition 2.24] and hence
∆N τ k = ∆X ′ τ k − ∆W τ k = ∆X ′ τ k = ∆M ′ τ k a.s. (3.13)
the last equality follows by (3.12) and the second inclusion in (3.9). Since N and M ′ are local martingales they are in fact purely discontinuous local martingale, cf.
Step 3, and with probability one
{t ≥ 0 : ∆N t = 0} ⊆ {τ k : k ∈ AE}, {t ≥ 0 : ∆M ′ t = 0} ⊆ {τ k : k ∈ AE}.
According to (3.13) this shows that (∆N t ) t≥0 = (∆M ′ t ) t≥0 , and we conclude that N = M ′ since N and M ′ are purely discontinuous local martingales, see [17, Ch. I, Corollary 4.19]. This completes Step 4.
Step 6 : We will show that A, given by (3.4), is a predictable càdlàg process of finite variation. By linearity, A is a well-defined càdlàg process. According to Step 5 the process W := X ′ − X 0 − M ′ is predictable and has càdlàg paths of finite variation. Thus with V = (V t ) t≥0 given by
V t = s∈(0,t] ∆X s 1 {|Xs|>r} − s∈(0,t] ∆M s 1 {|Ms|>r}
we have by the definitions of W and V that
A t = X t − X 0 − M t = W t + V t − EM t .
(3.14)
This shows that A has càdlàg sample paths of finite variation. Next we will show that A is predictable. Since the processes W, V andM depend on the truncation level r they will be denoted W r , V r andM r in the following. As r → ∞, V r t (ω) → 0 point wise in (ω, t), which by (3.14) shows that
W r t (ω) − EM r t → A t (ω) point wise in (ω, t) as r → ∞. For all r ∈ AE, (W r t − EM r t ) t≥0
is a predictable process, which implies that A is a point wise limit of predictable processes and hence predictable. This completes the proof of Step 6. , which completes the proof in the Gaussian case. Case 3. Λ is general : Let us observe that it is enough to show the theorem in the above two cases. We may decompose Λ as Λ = Λ G + Λ P , where Λ G , Λ P are independent, independently scattered random measures. Λ G is a symmetric Gaussian random measure characterized by (2.2) with b ≡ 0 and κ ≡ 0 while Λ P is given by (2.2) with σ 2 ≡ 0. Observe that
F Λ = F Λ G ∨ F Λ P ,
which can be deduced from the Lévy-Itô decomposition used processes Y = (Y t ) t≥0 of the form Y t = Λ((0, t] × B) where B ∈ V 0 . We have X = X G + X P , where X G and X P are defined by (2.1) with Λ G and Λ P in the place of Λ, respectively. Since (Λ, X) and (Λ P − Λ G , X P − X G ) have the same distributions, the process X P − X G is a semimartingale with respect to
F Λ P −Λ G = F Λ P ∨ F −Λ G = F Λ .
Consequently, processes X G and X P are semimartingales with respect to F Λ , and so, they are semimartingales relative to F Λ G and F Λ P , respectively, and the general result follows from the above two cases.
The uniqueness part: Assume that X has a representation of the form X = X 0 + M ′ + A ′ where M ′ is a continuous in probability semimartingale with independent increments and A ′ is a càdlàg predictable process of finite variation. With M and A given by (3.3) and (3.4) we need to show that process Y, defined by Since Y is of finite variation we deduce that U = 0, that is, Y = f . This completes the proof of the uniqueness. The special semimartingale part: To prove the part concerning the special semimartingale property of X we note that the process A in (3.4) is a special semimartingale since it is predictable and of finite variation. Thus X is a special semimartingale if and only if M is special semimartingale. Due to its independent increments, M is a special semimartingale if and only if E|M t | < ∞ for all t > 0, cf. [17, Ch. II, Proposition 2.29(a)], and in that case M t = (M t − EM t ) + EM t is the canonical decomposition of M. This completes the proof. Remark 3.6. We conclude this section by recalling that the proof of any of the results on Gaussian semimartingales X mentioned in the Introduction relies on the approximations of the finite variation component A by discrete time Doob-Meyer decompositions A n = (A n t ) t≥0 given by
Y = M − M ′ = A ′ − A,(3.A n t = [2 n t] i=1 E[X i2 −n − X (i−1)2 −n |F (i−1)2 −n ], t ≥ 0
and showing that the convergence lim n A n t = A t holds in an appropriate sense, see [23]. This technique does not seem effective in the non-Gaussian situation since it relies on strong integrability properties of functionals of X.
Some stationary increment semimartingales
In this section we consider infinitely divisible processes which are stationary increment mixed moving averages (SIMMA). Specifically, a process X = (X t ) t≥0 is called a SIMMA process if it can be written in the form
X t = Ê×V f (t − s, v) − f 0 (−s, v) Λ(ds, dv), t ≥ 0, (4.1)
where the functions f and f 0 are deterministic measurable such that f (s, v) = f 0 (s, v) = 0 whenever s < 0, and f (·, v) is càdlàg for all v. Λ is an independently scattered infinitely divisible random measure that is invariant under translations over Ê. If V is a one-point space (or simply, there is no vcomponent in (4.1)) and f 0 = 0, then (4.1) defines a moving average (a mixed moving average for a general V , cf. [29]). If V is a one-point space and f 0 (x) = f (x) = x α + for some α ∈ Ê, then X is a fractional Lévy process.
The finite variation property of SIMMA processes was investigated in Basse-O'Connor and Rosiński [6] and these results, together with Theorem 3.1, are crucial in our description of SIMMA semimartingales.
The random measure Λ in (4.1) is as in (2.2) but the functions b and σ 2 do not depend on s and the measure κ is a product measure: κ(ds, dv) = ds m(dv) for some σ-finite measure m on V . In this case, for A ∈ S and Notice that Λ satisfies (2.5) since κ(ds, dv) = ds m(dv). The SIMMA process (4.1) is a special case of (2.1) if we take φ(t, s, v) = f (t − s, v) − f 0 (−s, v). Therefore, from Theorem 3.1 we obtain:
θ ∈ Ê log Ee iθΛ(A) (4.2) = A iθb(v) − 1 2 θ 2 σ 2 (v) + Ê (e iθx − 1 − iu[[x]]) ρ v (dx) ds m(dv). The function B in (2.4) is independent of s, so that with B(x, v) = B(x, (s, v)) we have B(x, v) = xb(v) + Ê [[xy]] − x[[y]] ρ v (dy), x ∈ Ê, v ∈ V.Theorem 4.1. Let X = (X t ) t≥0 be of the form X t = Ê×V f (t − s, v) − f 0 (−s, v) Λ(ds, dv), t ≥ 0,
specified by (4.1)-(4.2), and let B be given by (4.3). Assume that
V B f (0, v), v m(dv) < ∞. (4.4)
Then X is a semimartingale with respect to the filtration
F Λ = (F Λ t ) t≥0 if and only if X t = X 0 + M t + A t , t ≥ 0, (4.5)
where M = (M t ) t≥0 is a Lévy process given by
M t = (0,t]×V f (0, v) Λ(ds, dv), t ≥ 0,(4.6)
(i.e., the integral exists), and A = (A t ) t≥0 is a predictable process of finite variation given by
A t = Ê×V [g(t − s, v) − g(−s, v)] Λ(ds, dv) (4.7) where g(s, v) = f (s, v) − f (0, v)1 {s≥0} .
Now we will give specific and closely related necessary and sufficient conditions on f and Λ that make X a semimartingale.
v ∈ V , f (·, v) is absolutely continuous on [0, ∞) with a derivativeḟ (s, v) = ∂ ∂s f (s, v) satisfying V ∞ 0 |ḟ (s, v)| 2 σ 2 (v) ds m(dv) < ∞, (4.8) V ∞ 0 Ê |xḟ (s, v)| ∧ |xḟ (s, v)| 2 ρ v (dx) ds m(dv) < ∞. (4.9)
Then X is a semimartingale with respect to F Λ .
Proof. We need to verify the conditions of Theorem 4.1. We see that for m-a.e. v ∈ V , g(·, v) is absolutely continuous on Ê with derivativeġ(s, v) = f (s, v) for s > 0 andġ(s, v) = 0 for s < 0. By Jensen's inequality, for each fixed t > 0, the function
(s, v) → g(t − s, v) − g(−s, v) = t 0ġ (u − s, v) du,
when substituted forḟ (s, v) in (4.8)-(4.9), satisfies these conditions. Indeed, it is straightforward to verify (4.8). To verify (4.9) we use the fact that ψ : u → 2 |u| 0 (v ∧ 1) dv is convex and satisfies ψ(u) ≤ |ux| ∧ |ux| 2 ≤ 2ψ(u). In particular, (s, v) → g(t − s, v) − g(−s, v) satisfies (b) of the Introduction, and so does the function
(s, v) → f (0, v)1 (0,t] (s) = g(t − s, v) − g(−s, v) − [f (t − s, v) − f (−s, v)].
This fact together with assumption (4.4) guarantee that M of Theorem 4.1 is well-defined. Then A is well-defined by (4.5). The process A is of finite variation by [6, Theorem 3.1] because g(·, v) is absolutely continuous on Ê andġ(·, v) =ḟ (·, v) satisfies (4.8)-(4.9).
Theorem 4.3 (Necessity).
Suppose that X is a semimartingale with respect to F Λ and for m-almost every v ∈ V we have either
1 −1 |x| ρ v (dx) = ∞ or σ 2 (v) > 0.
(4.10)
Then for m-a.e. v, f (·, v) is absolutely continuous on [0, ∞) with a derivativė f (·, v) satisfying (4.8) and
∞ 0 Ê |xḟ (s, v)| ∧ |xḟ (s, v)| 2 (1 ∧ x −2 ) ρ v (dx) ds < ∞. (4.11)
If, additionally, then for m-a.e. v,
lim sup u→∞ u |x|>u |x| ρ v (dx) |x|≤u x 2 ρ v (dx) < ∞ m-a.e.∞ 0 Ê (|xḟ (s, v)| 2 ∧ |xḟ (s, v)|) ρ v (dx) ds < ∞. (4.13) Finally, if sup v∈V sup u>0 u |x|>u |x| ρ v (dx) |x|≤u x 2 ρ v (dx)
< ∞ (4.14)
thenḟ satisfies (4.8)-(4.9).
Proof. Assume that X is a semimartingale with respect to F Λ . By a symmetrization argument we may assume that Λ is a symmetric random measure. Indeed, let Λ ′ be an independent copy of Λ and X ′ be defined by (4.1) with Λ replaced by Λ ′ . Then X ′ is a semimartingale with respect to F Λ ′ . By the independence, both X and X ′ are semimartingales with respect to F Λ ∨ F Λ ′ and since F Λ−Λ ′ ⊆ F Λ ∨ F Λ ′ , the process X − X ′ is a semimartingale with respect to F Λ−Λ ′ . This shows that we may assume that Λ is symmetric. Then under assumption (4.12), and under assumption (4.14),ġ satisfies (4.9). Since
f (s, v) = g(s, v) + f (0, v)1 {s≥0} , f (·, v) is absolutely continuous on [0, ∞) with a derivativeḟ (·, v) =ġ(·, v) for m-a.e
. v satisfying the conditions of the theorem.
Remark 4.4. Theorem 4.3 becomes an exact converse to Theorem 4.2 when (4.10) holds and either (4.12) holds and V is a finite set, or (4.14) holds.
Remark 4.5. Condition (4.10) is in general necessary to deduce that f has absolutely continuous sections. Indeed, let V be a one point space so that Λ is generated by increments of a Lévy process denoted again by Λ. If (4.10) is not satisfied, then taking f = 1 [0,1] we get that X t = Λ t − Λ t−1 is of finite variation and hence a semimartingale, but f is not continuous on [0, ∞).
Next we will consider several consequences of Theorems 4.2-4.3. When there is no v-component, (4.4) is always satisfied and Λ is generated by a two-sided Lévy process. In what follows, Z = (Z t ) t∈Ê will denote a nondeterministic two-sided Lévy process, with characteristic triplet (b, σ 2 , ρ) and Z 0 = 0. F Z will denote the least filtration satisfying the usual conditions such that σ(Z u : u ∈ (−∞, t]) ⊆ F Z t for all t ≥ 0. The following proposition characterizes fractional Lévy processes which are semimartingales, and completes results of [3,Corollary 5.4] and parts of [9, Theorem 1]. Proposition 4.6 (Fractional Lévy processes). Let γ > 0, x + := max{x, 0} for x ∈ Ê, Z be a Lévy process as above, and X be a fractional Lévy process defined by
X t = t −∞ (t − s) γ + − (−s) γ + dZ s (4.15)
where the stochastic integrals exist. Then X is a semimartingale with respect to F Z if and only if σ 2 = 0, γ ∈ (0, 1 2 ) and
Ê |x| 1 1−γ ρ(dx) < ∞. (4.16)
Proof. First we notice that, as a consequence of X being well-defined, γ < 1 2 and |x|>1 |x| 1 1−γ ρ(dx) < ∞. Indeed, since the stochastic integral (4.15) is well-defined, [25,Theorem 2.7] shows that
t −∞ Ê 1 ∧ |{(t − s) γ − (−s) γ + }x| 2 ρ(dx) ds < ∞, t ≥ 0. (4.18)
This implies that γ < 1 2 if ρ(Ê) > 0. A similar argument shows that γ < 1 2 if σ 2 > 0, and thus, by the non-deterministic assumption on Z, we have shown that γ < 1 2 . Putting t = 1 in (4.18) and using the estimate
|(1−s) γ −(−s) γ + | ≥ |γ(1 − s) γ−1 | for s ∈ (−∞, 0] we get ∞ > 0 −∞ Ê 1 ∧ |γ(1 − s) γ−1 x| 2 ρ(dx) ds = Ê ∞ 1 1 ∧ |γs γ−1 x| 2 ds ρ(dx) ≥ Ê 1≤s≤|γx| 1 1−γ ds ρ(dx) ≥ |γx|>1 |γx| 1 1−γ − 1 ρ(dx).
This shows (4.17).
Suppose that X is a semimartingale. If σ 2 > 0, then according to Theorem 4.3, f is absolutely continuous on [0, ∞) with a derivativeḟ satisfying
∞ 0 |ḟ (t)| 2 dt = ∞ 0 γ 2 t 2(γ−1) dt < ∞
which is a contradiction and shows that σ 2 = 0. By the non-deterministic assumption on Z we have ρ(Ê) > 0. To complete the proof of the necessity part, it remains to show that
|x|≤1 |x| 1 1−γ ρ(dx) < ∞. (4.19) Sinceḟ (t) = γt γ−1 for t > 0, we have ∞ 0 |xḟ (t)| ∧ |xḟ (t)| 2 dt = C|x| 1 1−γ (4.20) where C = γ 1 1−γ (γ −1 + (1 − 2γ) −1 ). In the case |x|≤1 |x| ρ(dx) < ∞ (4.19) holds since 1 < 1 1−γ .
Thus we may assume that |x|≤1 |x| ρ(dx) = ∞, that is, (4.10) of Theorem 4.3 is satisfied. By Theorem 4.3 (4.11) and (4.20) we have
|x|≤1 |x| 1 1−γ ρ(dx) ≤ Ê |x| 1 1−γ (1 ∧ x −2 ) ρ(dx) < ∞
which completes the proof of the necessity part.
On the other hand, suppose that σ 2 = 0, γ ∈ (0, 1 2 ) and (4.16) is satisfied. By (4.16) and (4.20), f is absolutely continuous on [0, ∞) with a derivativė f satisfying (4.9) and hence X is a semimartingale with respect to F Z , cf. Theorem 4.2.
Below we will recall the conditions from [6] under which (4.12) or (4.14)
hold. Recall that a measure µ on Ê is said to be regularly varying if x → µ([−x, x] c ) is a regularly varying function; see [11].
v ∈ V (i) |x|>1 x 2 ρ v (dx) < ∞ or (ii) ρ v is regularly varying at ∞ with index β ∈ [−2, −1).
Suppose that ρ v = ρ for all v, where ρ satisfies (4.12) and is regularly varying with index β ∈ (−2, −1) at 0. Then (4.14) holds. 4.8. Suppose that Z = (Z t ) t∈Ê is a two-sided Lévy process as above, with paths of infinite variation on compact intervals. Let X = (X t ) t≥0 be a process of the form
X t = t −∞ f (t − s) − f 0 (−s) dZ s .
Suppose that the random variable Z 1 is either square-integrable or has a regularly varying distribution at ∞ of index β ∈ [−2, −1). Then X is a semimartingale with respect to F Z if and only if f is absolutely continuous on Example 4.9. In following we will consider X and Z given as in Corollary 4.8 where Z is either a stable or a tempered stable Lévy process.
[0, ∞) with a derivativeḟ satisfying ∞ 0 |ḟ (t)| 2 dt < ∞ if σ 2 > 0, ∞ 0 Ê |xḟ (t)| ∧ |xḟ (t)| 2 ρ(dx) dt < ∞.(4.
(i) Stable: Assume that Z is a symmetric α-stable Lévy process with index α ∈ (1, 2), that is, ρ(dx) = c|x| −α−1 dx where c > 0, and σ 2 = b = 0. Then X is a semimartingale with respect to F Z if and only if f is absolutely continuous on [0, ∞) with a derivativeḟ satisfying ∞ 0 |ḟ (t)| α dt < ∞.
(4.22)
We use Corollary 4.8 to show the above. Note that |x|≤1 |x| ρ(dx) = ∞ and ρ is regularly varying at ∞ of index −α ∈ (−2, −1). Moreover, the identity Ê |xy| ∧ |xy| 2 ρ(dx) = C|y| α , y ∈ Ê, (4.23) with C = 2c((2 − α) −1 + (α − 1) −1 ), shows that (4.21) is equivalent to (4.22). Thus the result follows by Corollary 4.8.
(ii) Tempered stable: Suppose that Z is a symmetric tempered stable Lévy process with indexs α ∈ [1, 2) and λ > 0, i.e., ρ(dx) = c|x| −α−1 e −λ|x| dx where c > 0, and σ 2 = b = 0. Then X is a semimartingale with respect to F Z if and only if f is absolutely continuous on [0, ∞) with a derivativeḟ satisfying ∞ 0 |ḟ (t)| α ∧ |ḟ (t)| 2 ds < ∞.
(4.24)
Again we will use Corollary 4.8. The conditions imposed on Z in Corollary 4.8 are satisfied due to the fact that |x|≤1 |x| ρ(dx) = ∞ and |x|>1 |x| 2 ρ(dx) < ∞. Moreover, using the asymptotics of the incomplete gamma functions we have that
Ê |xu| ∧ |xu| 2 ρ(dx) ∼ C 1 u α as u → ∞ C 2 u 2 as u → 0 (4.25)
where C 1 , C 2 > 0 are finite constants depending only on α, c and λ, and we write f (u) ∼ g(u) as u → ∞ (resp. u → 0) when f (u)/g(u) → 1 as u → ∞ (resp. u → 0). Eq. (4.25) shows that (4.21) is equivalent to (4.24), and hence the result follows by Corollary 4.8.
Example 4.10 (Multi-stable). In this example we extend Example 4.9(i) to the so called multi-stable processes, that is, we will consider X given by (4.1) with ρ v (dx) = c(v)|x| −α(v)−1 dx where α : V → (0, 2) and c : V → (0, ∞) are measurable functions, and b = σ 2 = 0. For v ∈ V , ρ v is the Lévy measure of a symmetric stable distribution with index α(v). Assume that there exists an r > 1 such that α(v) ≥ r for all v ∈ V . Then X is a semimartingale with respect to F Λ if and only if for m-a.e. v, f (·, v) is absolutely continuous on [0, ∞) with a derivativeḟ (·, v) satisfying
V ∞ 0 c(v) 2 − α(v) |ḟ (s, v)| α(v) ds m(dv) < ∞. (4.26)
To show the above we will argue similarly as in Example 4.9. By the symmetry, (4.4) is satisfied. For all v ∈ V , |x|≤1 |x| ρ v (dx) = ∞, which shows that (4.10) of Theorem 4.3 is satisfied. By basic calculus we have for v ∈ V that u |x|>u |x| ρ v (dx) = K(v) |x|≤u x 2 ρ v (dx) (4.27) where K(v) = (2 − α(v))/(α(v) − 1). Since α(v) ≥ r we have that K(v) ≤ 2/(r − 1) < ∞ which together with (4.27) implies (4.14). From (4.23) we infer that (4.9) is equivalent to (4.26), and thus Theorems 4.2-4.3 conclude the proof. where γ : V → (0, ∞) is a measurable function. Processes of the form (4.28) may be viewed as superpositions of fractional Lévy processes with (possible) different indexes; hence the name supFLP. If m-almost everywhere γ ∈ (0, 1 2 ), σ 2 = 0 and
V Ê |x| 1 1−γ(v) ρ v (dx) 1 2 − γ(v) −1 m(dv) < ∞,(4.29)
then X is a semimartingale with respect to F Λ . Conversely, if X is a semimartingale with respect to F Λ and |x|≤1 |x| ρ v (dx) = ∞ for m-a.e. v, then m-a.e. γ ∈ (0, 1 2 ), σ 2 = 0 and for all x ∈ Ê, where c,c > 0 are finite constants not depending v and x.
By Theorem 4.2 and (4.32), the sufficient part follows. To show the necessary part assume that X is a semimartingale with respect to F Λ and that |x|≤1 |x| ρ v (dx) = ∞ for m-a.e. v. By Theorem 4.3, f (·, v) is absolutely continuous with a derivativeḟ (·, v) satisfying (4.8) and (4.11). From (4.8) we deduce that σ 2 = 0 m-a.e. and from (4.11) and (4.32) we infer that |x|≤1 |x| 1 1−γ(v) ρ v (dx) < ∞ m-a.e. v.
(4.33)
By (4.31)-(4.33), condition (4.30) follows. Moreover, if ρ satisfies (4.14), then Theorem 4.3 together with (4.32) show (4.29). This completes the proof.
Remark 3. 4 .
4There is a slight inconsistency in the notations used in (1.1) and(3.2). In (3.2) M is a semimartingale with independent increments, which is a martingale when centered, in which case (1.1) and (3.2) coincide. However, if M does not have zero expectation, then further decomposition of M into a martingale and a process of finite variation is needed to obtain (1.1), but this is standard, see e.g.[27, Theorem 19.2].
(3.10) the symbol * denotes integration with respect to µ − ν as in[17]). Thus by definition, see[17, Ch. II, Definition 1.27(b)], Z is a purely discontinuous local martingale and (∆Z t ) t≥0 = (φ(t, ∆U t )1 {∆Ut =0} ) t≥0 , which shows that with probability one{t ≥ 0 : ∆Z t = 0} ⊆ {t ≥ 0 : ∆U t = 0}.All real-valued continuous in probability càdlàg processes Y = (Y t ) t≥0 with independent increments is quasi-left continuous cf. [17, Ch. II, Corollary 5.12], and hence there exists a sequence of totally inaccessible stopping times that exhausts the jumps of Y, cf. [17, Ch. I, Proposition 2.2]. Thus by a diagonal argument we may exhausts the jumps of U by a sequence of totally inaccessible stopping times (τ k ) k∈AE , that is {τ k : k ∈ AE} = {t ≥ 0 : ∆U t = 0}.
Case 2 .
2Λ is Gaussian: Assume that Λ is a symmetric Gaussian random measure. By Basse [2, Theorem 4.6] used on C t = (−∞, t] ×V , X is a special semimartingale in F Λ with martingale component M = (M t ) t≥0 given by
Theorem 4 . 2 (
42Sufficiency). Let X = (X t ) t≥0 be specified by (4.1)-(4.2). Suppose that (4.4) is satisfied and that for m-a.e.
(4.4) holds since B = 0. By Theorem 4.1 process A in (4.7) is of finite variation. It follows from [6, Theorem 3.3] that for m-a.e. v, g(·, v) is absolutely continuous on Ê with a derivativeġ(·, v) satisfying (4.8) and(4.11). Furthermoreġ satisfies(4.13)
Proposition 4. 7 ([ 6 ,
76Proposition 3.5]). Condition (4.12) is satisfied when one of the following two conditions holds for m-almost every
Theorems 4.2-4.3 and Proposition 4.7 gives the following:
Corollary
Example 4 .
411 (supFLP). Consider X = (X t ) t≥0 of the form
v) ρ v (dx) < ∞,(4.30) and if in addition ρ satisfies (4.14), then (4.29) holds.To show the above letf (t, v) = t γ(v) + for t ∈ Ê, v ∈ V . Since f (0, v) = 0for all v, (4.4) is satisfied. As in Example 4.6, we observe that the conditions|x|≥1 |x| 1 1−γ(v) ρ v (dx) < ∞ and γ(the fact that X is a well-defined. For γ(v) ∈ (0, 1 2 ), f (·, v)is absolutely continuous on [0, ∞). By(4.20)
t, v)| ∧ |xḟ (t, v)| 2 } dt ≤c |x|
may choose a càdlàg modification of M, cf. [27, Theorem 11.5]. Due to the independent increments, M is a semimartingale if and only if its shift component (ζ t ) t≥0 is of finite variation, cf. [17, Ch. II, Corollary 5.11], which follows from (3.1) and the fact that
second equality shows that Y is predictable, which together implies that Y is continuous, use [17, Ch. I, Proposition 2.24+Definition 2.25]. By the Lévy-Itô decompositions of M and M ′ it follows that Y = U + f where U is a continuous martingale and f is a continuous deterministic function of finite variation; that f is of finite variation follows by [17, Ch. II, Corollary 5.11].15)
is a continuous deterministic function of finite variation. The first equality in
(3.15) shows that Y is quasi-left continuous, cf. [17, Ch. II, Corollary 4.18],
and the
21 )
21Proof Corollary 4.8. The conditions imposed on Z 1 are equivalent to that ρ statisfies (i) or (ii) of Proposition 4.7, respectively, cf. [14, Theorem 1] and [27, Theorem 25.3]. Moreover, (4.10) of Theorem 4.3 is equivalent to that Z has sample paths of infinite variation on compacs and hence the result follows by Theorems 4.2-4.3.
t ) k∈AE : t ∈ Ê + } is a continuous in probability càdlàg Ê AE -valued process with no Gaussian component. Let E = Ê AE \ {0}. Then E is a Blackwell space and µ defined byµ(A) = ♯ t ∈ Ê + : (t, ∆U t ) ∈ A , A ∈ B(Ê + × E)is a Poisson random measure on Ê + × E. Let ν be the intensity measure of µ. By assumption (2.5) we have that ν({t} × E) = 0 for all t ≥ 0, moreover, F Λ is the least filtration for which µ is an optional random measure. Thus
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[
"CONSISTENT INTERACTIONS IN TERMS OF THE GENERALIZED FIELDS METHOD",
"CONSISTENT INTERACTIONS IN TERMS OF THE GENERALIZED FIELDS METHOD"
] |
[
"Omer F Dayi 1e-mailaddress:[email protected] \nDepartment of Physics\nTUBITAK -Marmara Research Center\nResearch Institute for Basic Sciences\nP.O. Box 2141470GebzeTURKEY\n"
] |
[
"Department of Physics\nTUBITAK -Marmara Research Center\nResearch Institute for Basic Sciences\nP.O. Box 2141470GebzeTURKEY"
] |
[] |
The interactions which preserve the structure of the gauge interactions of the free theory are introduced in terms of the generalized fields method of solving the Batalin-Vilkovisky master equation. It is shown that by virtue of this method the solution of the descent equations resulting from the cohomological analysis is provided straightforwardly. The general scheme is illustrated by applying it to spin-1 gauge field in 3 and 4 dimensions, to free BF theory in 2-d and to the antisymmetric tensor field in any dimension. It is shown that it reproduces the results obtained by cohomological techniques.
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10.1142/s0217751x97002383
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[
"https://arxiv.org/pdf/hep-th/9604167v2.pdf"
] | 13,141,206 |
hep-th/9604167
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c6ac106e395b67d5dc0964d419c191f5fcc4f58c
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CONSISTENT INTERACTIONS IN TERMS OF THE GENERALIZED FIELDS METHOD
Jan 1997
Omer F Dayi 1e-mailaddress:[email protected]
Department of Physics
TUBITAK -Marmara Research Center
Research Institute for Basic Sciences
P.O. Box 2141470GebzeTURKEY
CONSISTENT INTERACTIONS IN TERMS OF THE GENERALIZED FIELDS METHOD
Jan 1997arXiv:hep-th/9604167v2 9
The interactions which preserve the structure of the gauge interactions of the free theory are introduced in terms of the generalized fields method of solving the Batalin-Vilkovisky master equation. It is shown that by virtue of this method the solution of the descent equations resulting from the cohomological analysis is provided straightforwardly. The general scheme is illustrated by applying it to spin-1 gauge field in 3 and 4 dimensions, to free BF theory in 2-d and to the antisymmetric tensor field in any dimension. It is shown that it reproduces the results obtained by cohomological techniques.
In a local gauge theory if the number of the physical degrees of freedom is modified when the interactions are switched off, there may be some negative norm states which yield inconsistencies in the scattering theory where one uses the asymptotic states [1]. To avoid it one deals with the local interactions which preserve the structure of the gauge generators of the free theory. These are the consistent interactions [2]. By using the cohomological aspects of the Batalin-Vilkovisky (BV) method of quantization of gauge theories [3] one obtains the descent equations whose solution yields the consistent interactions [4]- [6].
The generalized fields method is proven to be powerful in the BV quantization of a large class of gauge theories [7]- [8]. It offers the proper solution of the (BV) master equation in a straightforward way.
We show that when one deals with a free gauge theory whose action can be written first order in the exterior derivative d and bilinear in fields, the generalized fields method can be applied. As far as the properties of gauge generators are unaltered when interactions are introduced the generalized fields method should still be applicable. Hence, it can be utilized to obtain the consistent interactions. Moreover, solution of the related descent equations is a by-product in this scheme, i.e. this method provides the solution of the cohomology problem in a straightforward and compact manner. We do not deal with the consistent interactions neither non-local nor possessing derivatives and obtainable by field redefinitions.
The general scheme is illustrated by applying it to the spin-1 gauge field in d = 4 and d = 3, whose consistent interactions yield Yang-Mills and Chern-Simons field theories. The consistent interactions of the free BF theory in d = 2 are shown to lead to the gauge action of the quadratic Lie algebras [9]. In all of the examples the solutions of the descent equations are presented. When we deal with the antisymmetric tensor field in D-dimension the application of the method is shown to provide in a straightforward manner the solution of the cohomology problem obtained in [6].
Let us deal with a free gauge theory whose action can be written as
A 0 = (B a ∧ dA a + 1 2 α ab B a ∧ B b )(1)
where α can be a vanishing or a constant matrix which will be suppressed in the following. It is invariant under the infinitesimal gauge transformations
δ Λ A a = dΛ a , δ Λ B a = 0.(2)
To quantize this theory in terms of the BV method, one introduces the ghost, ghost of ghost fields and the antifields (minimal sector), by inspecting the reducibility properties of the gauge transformations. One can attribute to each of the fields a total form degree defined as the sum of the differential form degree and the ghost number. To apply the generalized fields method one collects the original fields and the ones introduced for applying the BV method in the generalized fieldsà a ,B a whose total form degrees are the same with the differential form degrees of the original fields A a , B a . The action written in terms of the generalized fields
S 0 ≡ L 0 = (B a dà a + 1 2B aBa ) (D,0)(3)
(the first of the numbers in the parentheses is the differential form degree, the second is the ghost number and if there is only one number it denotes the total form degree) is the proper solution of the master equation
(S 0 , S 0 ) ≡ δ r S 0 δà a δ l S 0 ∂B a − δ r S 0 δB a δ l S 0 ∂à a = 0.(4)
To prove this observe that due to the gauge transformations (2), inB there is not any field possessing positive ghost number so that (BB) (D,0) = B ∧ B and there is not B ⋆ term in (BdÃ) (D,0) . BRST transformation of a functional is defined as
Ω 0 F (Ã,B) = ∂ r F ∂à a Ω 0à a + ∂ r F ∂B a Ω 0B a(5)
where
Ω 0à a = − ∂ l S 0 ∂B a , Ω 0B a = ∂ l S 0 ∂à a .(6)
In terms of the BRST charge Ω 0 (4) can equivalently be formulated as
Ω 0L0 + dω (D−1,1) = 0,(7)
where ω (D−1,1) is a suitable (D−1)-form possessing ghost number one. Moreover, due to the fact that (Ω 0 + d) 2 = 0, there are the descent equations
Ω 0 ω (D−1,1) + dκ (D−2,2) = 0, Ω 0 κ (D−2,2) + dλ (D−3,3) = 0, · · · ,(8)
with suitable κ, λ, · · · , vanishing at a certain step.
Obviously Ω 0 mapsà →dA andB →dB except
Ω 0 B ⋆a = dA a + B a .
However, there is not any B ⋆ term inL 0 , so that, the BRST transformation of the free (7) theory can be generalized as
(Ω 0 + d)(B a dà a ) (D) = 0.(9)
We dropped the B a ∧B a term which does not transform under Ω 0 . (9) written in components produces (8)
with the identification ω (D−1,1) ≡ (B a dà a ) (D−1,1) , κ (D−2,2) ≡ (B a dà a ) (D−2,
2) and so on. A compact notation similar to (9) is also presented in [10] The consistent interactions will be introduced as cubic or higher order in the generalized fieldsΦ ≡ (Ã,B):
S int ≡ k=1,M =3 g k M S M k = M =3 (g MΦ M + g 2 MΦ M +1 + · · ·) (D,0) = (g 3Φ 3 + g 2 3Φ 4 + · · · + g 4Φ 4 + g 2 4Φ 5 + · · ·) (D,0) ,(10)
such that
(S 0 + S int , S 0 + S int ) = 0.(11)
g M are independent coupling constants, hence, (11) leads to
(S 0 , S 0 ) = 0,(12)(S 0 , S M 1 ) = 0,(13)2(S 0 , S M 2 ) + (S M 1 , S M 1 ) = 0, (14) . . .
(13) can equivalently be written in terms of the BRST charge Ω 0 as
Ω 0Φ M (D,0) + dk (D−1,1) = 0,(15)
where k is a suitable (D − 1)-form possessing ghost number one. As it has already mentioned Ω 0 mapsà →dA andB →dB except
B ⋆ . Let the B ⋆ dependent terms in S M 1 are (Φ M (D,0) ) B ⋆ ∼ (B ⋆ΦM −1 + B ⋆2ΦM −2 + · · · + B ⋆M −1Φ ) (D,0)
. Thus in (15) the terms which does not possess an exterior derivative will be
Ω 0Φ M (D,0) + dk D−1,1 ∼ BΦ M −1 + 2BB ⋆ΦM −2 + · · · + (M − 1)B ⋆M −2Φ . (16)
There are two possibilities i) (16) vanishes due to symmetry properties, so that (15) is satisfied with k (D−1,1) ≡Φ 3 (D−1,1) , ii) (16) does not vanish so that, (15) is not satisfied. In the latter case the interactions are not consistent. Therefore, we conclude that as far as we deal with the consistent interactions (13) is equivalent to
Ω 0Φ M (D,0) + dΦ M (D−1,1) = 0,(17)
and it can be generalized to D-total form:
(Ω 0 + d)Φ M (D) = 0,(18)
Obviously some ofΦ M (a,b) can be vanishing. We conclude that the generalized fields method provide the consistent interactions as well as the solution of the descent equations.
Gauge invariance of the interacting theory can be obtained by replacing S 0 with S 0 +S int in (6) on the surface where the ghost fields and the antifields are vanishing. Spin-1 Gauge Field in D=4: Free theory given by the first order action
L 0 = −1 2 d 4 x (B aµν (∂ µ A a ν − ∂ ν A a µ ) − 1 2 B a µν B aµν ),(23)
is invariant under the infinitesimal gauge transformations δ Λ A a µ = ∂ µ Λ a , δ Λ B a µν = 0. The theory is irreducible, so that we need to introduce (in the minimal sector) the ghost field η (0,1) , and the antifields A ⋆ (3,−1) , η ⋆ (4,−2) , and B ⋆ (2,−1) . Here the star indicates the antifields as well as the Hodge-map.
The generalized fields arẽ
A a = A a (1,0) + η a (0,1) + B ⋆a (2,−1) , B a = −A ⋆a (3,−1) − η ⋆a (4,−2) + B a (2,0) .
The proper solution of the master equation is
S 0 = −1 2 d 4 x (B a dà a − 1 2B aBa ) (4.0) ,(24)= − d 4 x ( 1 2 B aµν (∂ µ A a ν − ∂ ν A a µ ) − A ⋆a µ ∂ µ η a − 1 4 B a µν B aµν ),(25)
from which we read the BRST transformations
Ω 0 A a µ = ∂ µ η a ; Ω 0 B aµν = 0; Ω 0 η a = 0; Ω 0 A ⋆aµ = ∂ ν B aµν ; Ω 0 B ⋆a µν = ∂ µ A a ν − ∂ ν A a µ − B a µν ; Ω 0 η a⋆ = −∂ µ A ⋆aµ .(26)κ (4,0) = f abc ( 1 2 B aµν A b µ A c ν − A ⋆aµ A b µ η c + B aµν η b B ⋆c µν − 1 2 η ⋆a η b η c ), κ (3,1) = f abc (B µν A b ν η c + 1 2 A ⋆aµ η b η c ), κ (2,2) = − 1 2 f abc B aµν η b η c .
If f abc satisfy the Jacobi identities, one can show that S 31 = g 3 κ (4,0) satisfies (S 31 , S 31 ) = 0.
Hence, there is no need of adding a quartic interaction with g 2 3 coupling constant. However, one can in principle add a 4-total formà 4 with another coupling constant g 4 . For being a consistent interaction S 41 should be BRST invariant. But one can show that Ω 0à 4 (4,0) = dK (3,1) , for any K. Therefore, we cannot add this interaction term. Observe that after a gauge fixing B ⋆ = 0 and using the equations of motion related to B µν , Yang-Mills theory follows with the required quartic interaction.
Spin-1 Gauge Field in D=3: The free gauge theory is
A 0 = A a ∧ dA a .(29)
It is invariant under the infinitesimal gauge transformations δ Λ A a = dΛ a . It is an irreducible theory, so that the generalized field is
A a = A a µ(1,0) + η a (0,1) + A ⋆a µ(2,−1) + η ⋆a (3,−2) .(30)
The solution of the master equation
S 0 = à a dà a ,(31)
leads to the BRST transformations
Ω 0 A a µ = ∂ µ η a , Ω 0 η a = 0, Ω 0 A ⋆aµν = −ǫ µνρ ∂ ν A a ρ , Ω 0 η ⋆a = −∂ µ A ⋆aµ .
The unique candidate for a consistent interaction is the ghost zero component of the 3-total form ξ ≡à 3 . Indeed, one can show that if f abc are antisymmetric in all of the indices the descent equations
(Ω 0 + d)(f abcà aÃbÃc ) (3) = 0,(32)
are satisfied, where the components are
ξ (3,0) = −f abc (A a ∧ A b ∧ A c + 6A ⋆a ∧ A b ∧ η c + 3η ⋆a η b η c ), ξ (2,1) = 3f abc (A a ∧ A b ∧ η c + A ⋆a ∧ η b ∧ A c ), ξ (1,2) = 3f abc A a ∧ η b ∧ η c , ξ (0,3) = −f abc η a ∧ η b ∧ η c .
There is no need of any further interaction term because, the total action S = S 0 + g 3 ξ (3,0) satisfies the master equation if f abc satisfy the Jacobi identities. Free BF Theory in D=2: Deal with the free gauge theory given by the Lagrange density
L 0 = − 1 2 ǫ µν Φ a (∂ µ h a ν − ∂ ν h a µ ),(33)
which is invariant under the infinitesimal gauge transformations
δh a µ = ∂ µ λ a , δΦ a = 0.(34)
Although a careful treatment of the global modes showed that at some points of the target manifold the theory is reducible [11], we deal with the regions of the target manifold which does not include these points. Hence, the generalized fields areh
a = h a (1,0) + η a (0,1) + Φ ⋆a (2,−1) ,(35)Φ a = −h ⋆a (1,−1) − η ⋆a (2,−2) + Φ a (0,0) .(36)
The BV quantized free action
S 0 = 1 2 Φ a dh a ,(37)
leads to the BRST transformations
Ω 0 h a µ = ∂ µ η a , Ω 0 Φ a = 0, Ω 0 η a = 0, Ω 0 h ⋆aµ = ǫ µν ∂ ν Φ a , Ω 0 Φ ⋆a = ǫ µν ∂ µ h a ν , Ω 0 η ⋆a = −∂ µ h ⋆aµ .
There is a unique candidate for a consistent cubic interaction:
S 31 = g 3 σ (2,0) ≡ g 3 1 2 d 2 xf abc (Φ ahbhc ) (2,0) .(38)
In fact, one can show that σ (D) written in components
σ (2,0) = f abc ( 1 2 ǫ µν h a µ h b ν Φ c − h ⋆aµ h b µ η c + Φ ⋆a Φ b η c − 1 2 η ⋆a η b η c ), σ (1,1) = f abc (h a µ η b Φ c − 1 2 h ⋆a µ η b η c ), σ (0,2) = − 1 2 f abc η a η b Φ c ,
satisfy the descent equations if f abc is antisymmetric in all of the indices. Moreover, if they satisfy the Jacobi identities one can show that
(S 1 , S 1 ) = 0.
Although, there is no need of quartic interaction with a coupling constant g 2 3 , one can in principle add quartic or higher interaction terms
M =4 g M S M 1 ≡ M =4 g M (Φ M −2h2 ) (2,0) ,
if they satisfy the descent equations (19)-(22). Let us deal with the quartic term S 41 = g 4 d 2 xV ad bc (Φ aΦdh bhc ) (2,0) .
Indeed one can show that
(Ω 0 + d)(V ad bcΦ aΦdh bhc ) (2) = 0,(40)(S 1 , S 2 ) = 0,(41)(S 2 , S 2 ) = 0,(42)
if the constants satisfy
V cd ab = −V cd ba , V cd ab = V dc ab , (43) f [ab d V ef c]d + V df [ab f c]d e + V ed [ab f c]d f = 0, (44) V de [ab V f g c]d = 0,(45)
where [ ] denotes that the indices within them are antisymmetrized.
The higher interactions can also be treated similarly.
for any κ (5,1) . Thus this is excluded.
3)HB 2 is a (D+1)-total form. Therefore it cannot be a Lagrange density. 4)H 2B is (2D − 4)-total form thus it is only permitted in D = 4. Before dealing with the case 4 let us see if there can be some other consistent interactions which are quartic or higher. There can be (H mBn ) (D,0) ; m, n ≤ 1, m + n > 3, which is a (m(D − 3) + 2n)-total form. Thus the dimensions should be
D = 3m − 2n m − 1 ,
if the interactions are consistent. However, one can easily observe that there is not any n, m which lead to an acceptable dimension. Hence we can conclude that the unique candidate for a consistent interaction is the case 4. In fact in D = 4, ω (4) ≡H 2B leads to the descent equations Ω 0 ω (4,0) + dω (3,1) = 0, Ω 0 ω (3,1) + dω (2,2) = 0, Ω 0 ω (2,2) = 0, ω (1,3) = ω (0,4) = 0.
The components written explicitly are ω (4,0) = f abc (H a ∧ H b ∧ B c + 2H a ∧ B ⋆b ∧ C a − 2H a ∧ C ⋆b ∧ η c + B ⋆a ∧ B ⋆b ∧ η c ), ω (3,1)
= f abc (−H a ∧ H b ∧ C c + 2H a ∧ B ⋆b ∧ η c ) ω (2,2) = f abc H a ∧ H b ∧ η c ,
where f abc is totally antisymmetric in its indices. Moreover, one can show that if f abc satisfy the Jacobi identities The result which we obtained agree with [6], where the same problem is considered in terms of cohomological techniques.
show that, in fact, (27) provides the solution of the descent equations Ω 0 κ (4,0) + dκ (3,1) = 0, Ω 0 κ (3,1) + dκ (2,2) = 0, Ω 0 κ (2,2) = 0, κ (1,3) = κ (0,4) = 0, if f abc is totally antisymmetric. Written explicitly
E-mail address: [email protected]
Antisymmetric Tensor Field in D-dimension: The first order free action of the two form field B = B µν dx µ ∧ dx ν in any D dimensions iswhere H is a D −3 differential form andĤ is its Hodge dual. (46) is invariant under the gauge transformationswhere Λ a is 1-form. The gauge transformations (47) are reducible because, they vanish for Λ a = dǫ a , where ǫ a is a scalar. Thus, in the minimal sector there are ghost and ghost of ghost fields. The generalized fields arẽwhere, now, star indicates also the Hodge map. Solution of the master equation for the free theory isHence, the BRST transformations areThere are four combinations of the generalized fields which are cubic: 1)H 3 is a 3(D −3)-total form. For being a Lagrange density the form degree should be D, which means that the dimension should be D = 9/2. Hence, it is not permitted. 2)B 3 is 6-total form so it is permitted in D = 6. In 6-dimensioñHowever, one can easily observe that Ω 0B 3 (6,0) = dκ(5,1)
. T Kugo, I Ojima, Prog. Theor. Phys. 661Suppl.T. Kugo and I. Ojima, Prog. Theor. Phys. (Suppl.) 66 (1979) 1.
Fang and C. Fronsdal. R Arnowitt, S K H Deser ; A, Bengtsson, J. Math. Phys. 2264, F.A. Berends, G.H. Burgers and H. Van Dam492031Phys. Rev. DR. Arnowitt and S. Deser, Nucl. Phys. 49 (1963), J. Fang and C. Fronsdal, J. Math. Phys. 20 (1979) 2264, F.A. Berends, G.H. Burgers and H. Van Dam, Nucl. Phys. B 260 (1985) 295, A.K.H. Bengtsson, Phys. Rev. D 32 (1985) 2031.
. I A Batalin, G A Vilkovisky, Phys. Lett. B. 1022567Phys. Rev. DI.A.Batalin and G.A.Vilkovisky, Phys. Lett. B 102 (1981) 27; Phys. Rev. D 28 (1983) 2567.
. G Barnich, M Henneaux, Phys. Lett. B. 311123G. Barnich and M. Henneaux, Phys. Lett. B 311 (1993) 123.
G Barnich, M Henneaux, R Tatar, PMIF/93-04Consistent Interactions Between Gauge Fields and Local BRST Cohomology: The Example of Yang-Mills Models. G. Barnich, M. Henneaux and R. Tatar, Consistent Interactions Between Gauge Fields and Local BRST Cohomology: The Example of Yang-Mills Models, ULB-PMIF/93-04, 1993.
Uniqueness of the Freedman-Townsend Interaction Vertex for Two-Form Gauge Fields. M Henneaux, hep-th/9511145M. Henneaux, Uniqueness of the Freedman-Townsend Interaction Vertex for Two-Form Gauge Fields, hep-th/9511145, 1995.
. Ö F Dayi, Mod. Phys. Lett. A. 8811ibid. 2087Ö.F. Dayi, Mod. Phys. Lett. A 8 (1993) 811; ibid. 2087.
. Ö F Dayi, Inter. J. Mod. Phys. A. 111Ö.F. Dayi, Inter. J. Mod. Phys. A 11 (1996) 1.
. N Ikeda, K I Izawa, 1077; ibid. 90Prog. Theor. Phys. 89237N. Ikeda and K.I. Izawa, Prog. Theor. Phys. 89 (1993) 1077; ibid. 90 (1993) 237.
N Dragon, hep- th/9602163BRS Symmetry and Cohomology. N. Dragon, BRS Symmetry and Cohomology, ITP-UH-3/96, hep- th/9602163.
. P Schaller, T Strobl, Mod. Phys. Lett. A. 93129P. Schaller and T. Strobl, Mod. Phys. Lett. A 9 (1994) 3129.
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[] |
[
"Analytical description of spin-Rabi oscillation controlled electronic transitions rates between weakly coupled pairs of paramagnetic states with S=1/2",
"Analytical description of spin-Rabi oscillation controlled electronic transitions rates between weakly coupled pairs of paramagnetic states with S=1/2"
] |
[
"R Glenn \nDepartment of Physics and Astronomy\nUniversity of Utah\n84112Salt Lake CityUT\n",
"W J Baker \nDepartment of Physics and Astronomy\nUniversity of Utah\n84112Salt Lake CityUT\n",
"C Boehme \nDepartment of Physics and Astronomy\nUniversity of Utah\n84112Salt Lake CityUT\n",
"M E Raikh \nDepartment of Physics and Astronomy\nUniversity of Utah\n84112Salt Lake CityUT\n"
] |
[
"Department of Physics and Astronomy\nUniversity of Utah\n84112Salt Lake CityUT",
"Department of Physics and Astronomy\nUniversity of Utah\n84112Salt Lake CityUT",
"Department of Physics and Astronomy\nUniversity of Utah\n84112Salt Lake CityUT",
"Department of Physics and Astronomy\nUniversity of Utah\n84112Salt Lake CityUT"
] |
[] |
We report on an analytical description of spin-dependent electronic transition rates which are controlled by a radiation induced spin-Rabi oscillation of weakly spin-exchange and spin-dipolar coupled paramagnetic states (S = 1 2 ). The oscillation components (the Fourier content) of the net transition rates within spin-pair ensembles are derived for randomly distributed spin resonances with account of a possible correlation between the two distributions that correspond to the two individual pair partners. The results presented here show that when electrically or optically detected Rabi spectroscopy is conducted under an increasing driving field B1, the Rabi spectrum evolves from a single resonance peak at s = ΩR, where ΩR = γB1 is the Rabi frequency (γ is the gyromagnetic ratio), to three peaks at s = ΩR, s = 2ΩR, and at low s ΩR. The crossover between the two regimes takes place when ΩR exceeds the expectation value δ0 of the difference of the Zeeman energies within the pairs, which corresponds to the broadening of the magnetic resonance lines in the presence of disorder caused by hyperfine field or distributions of Landé g-factors. We capture this crossover by analytically calculating the shapes of all three peaks at arbitrary relation between ΩR and δ0. When the peaks are well-developed their widths are ∆s ∼ δ 2 0 /ΩR.
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10.1103/physrevb.87.155208
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[
"https://arxiv.org/pdf/1207.1754v1.pdf"
] | 118,835,227 |
1207.1754
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0ddadd718b577362386d35eea20df29d3ee477f2
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Analytical description of spin-Rabi oscillation controlled electronic transitions rates between weakly coupled pairs of paramagnetic states with S=1/2
R Glenn
Department of Physics and Astronomy
University of Utah
84112Salt Lake CityUT
W J Baker
Department of Physics and Astronomy
University of Utah
84112Salt Lake CityUT
C Boehme
Department of Physics and Astronomy
University of Utah
84112Salt Lake CityUT
M E Raikh
Department of Physics and Astronomy
University of Utah
84112Salt Lake CityUT
Analytical description of spin-Rabi oscillation controlled electronic transitions rates between weakly coupled pairs of paramagnetic states with S=1/2
numbers: 4250Md7630-v7135Gg
We report on an analytical description of spin-dependent electronic transition rates which are controlled by a radiation induced spin-Rabi oscillation of weakly spin-exchange and spin-dipolar coupled paramagnetic states (S = 1 2 ). The oscillation components (the Fourier content) of the net transition rates within spin-pair ensembles are derived for randomly distributed spin resonances with account of a possible correlation between the two distributions that correspond to the two individual pair partners. The results presented here show that when electrically or optically detected Rabi spectroscopy is conducted under an increasing driving field B1, the Rabi spectrum evolves from a single resonance peak at s = ΩR, where ΩR = γB1 is the Rabi frequency (γ is the gyromagnetic ratio), to three peaks at s = ΩR, s = 2ΩR, and at low s ΩR. The crossover between the two regimes takes place when ΩR exceeds the expectation value δ0 of the difference of the Zeeman energies within the pairs, which corresponds to the broadening of the magnetic resonance lines in the presence of disorder caused by hyperfine field or distributions of Landé g-factors. We capture this crossover by analytically calculating the shapes of all three peaks at arbitrary relation between ΩR and δ0. When the peaks are well-developed their widths are ∆s ∼ δ 2 0 /ΩR.
I. INTRODUCTION
Over the past decade, pulsed electrically detected magnetic resonance (pEDMR) spectroscopy has been used increasingly for the investigation of the physical nature of spin-dependent electronic transitions which influence conductivity such as excess charge carrier recombination or transport transitions through localized paramagnetic states as seen in amorphous inorganic 1,2 , crystalline [3][4][5][6] as well as organic [7][8][9][10][11] semiconductor materials. In most of these experimental studies, pEDMR experiments are conducted within a pulse-probe scheme 12,13 where the current through the host materials of the spin systems of interest is measured while short and intensive magnetic resonant pulses are imposed. The pulsed magnetic resonant radiation prepares coherent spin non-eigenstates from initial, well defined eigenstates before the pulse. As the changed spin-states will also cause changes of the spin-dependent conductivity, one can gain information about the prepared coherent spin-state by integration of the electric current transient after the pulse which eventually, on long time scales (compared to the length of the coherent excitation), will return to its pre-pulse steady state. The probed charge (obtained from the integrated current transient) depends on the coherent spin state after the pulse which in turn depends on the pulse parameters (length, frequency, intensity) 12 . Therefore, a measurement of the charge as a function of the applied pulse length can reveal the propagation of a spin state in presence of the resonant radiation pulse. Thus, cur- Most EDMR detectable spin-dependent electronic transitions reported in the literature are due to Pauliblockade effects which occur for transitions between two paramagnetic states with S = 1 2 . These systems, illustrated in Fig. 1, usually require weak spin-orbit coupling as found in materials with low atomic order numbers (this means silicon and carbon materials) as well as sufficiently weak spin-spin coupling (which means exchange and dipolar interaction) within the formed pairs. In order to allow one of the electrons within this pair of paramagnetic states to undergo a transition into the other paramagnetic state (thereby forming a singlet spin manifold due to the Pauli exclusion principle), the spin pair state arXiv:1207.1754v1 [cond-mat.mtrl-sci] 7 Jul 2012 |ψ before the transition requires non-negligible singlet content ( ψ|S = 0) for the transition to have a nonnegligible probability. The special nature of such intermediate pair controlled spin-dependent transition rates was first recognized by Kaplan, Solomon and Mott 14 who explained the magnitude of continuous wave EDMR experiments at the time. With the advent of pEDMR about a decade ago, this model also became most significant for the understanding of many of the EDMR detected coherent spin motion experiments.
When pEDMR is applied to the intermediate pair processes described by Kaplan, Solomon and Mott, the observable applied to the spin ensemble is permutation symmetry (the singlet operator), it is not the magnetic polarization of the spin ensemble as it is the case for conventional magnetic resonance spectroscopies which are based on the detection of radiation. Some implications of this observable change have been discussed theoretically in previous studies on intermediate pairs for cases of no intrapair interactions 12,15 , cases of weak but non-negligible exchange interaction 16 (weak here means that the exchange interaction is much smaller than the spin-Zeeman splitting of the pair partners but not necessarily weaker than the difference of the pair partners Larmor frequencies), and for cases where disorder within the spin ensemble 17 is significant. These numerical studies have shown that an electrically detectable spin-Rabi oscillation can contain various harmonic components which can essentially form a "fingerprint" for the spin-Hamiltonian of the observed pairs. Thus, conducting a Fourier analysis of an observed spin-Rabi signal (one could call this Rabi spectroscopy) can give microscopic information about the nature of charge carrier states or of paramagnetic defects.
Most of the previously published pEDMR studies have been conducted as Rabi spectroscopy experiments 2,4,7,8,10,11 . For most of these experimental data, a correct interpretation would be impossible without the information provided by the existing theoretical studies 12,[15][16][17] . Nevertheless, these studies can only provide limited support for an experimental analysis since numerical simulations can only provide answers about the behavior of a simulated system for a fixed set of parameters, but they do not reveal analytical expressions that can be fit or directly compared to experimental data, and most importantly, they oftentimes do not enhance the qualitative understanding of a simulated system. It has been the goal of this study to overcome this problem by finding a closed analytical form for the description of spin-Rabi oscillation controlled spin-dependent transition rates within spin pairs with S = 1 2 . The expressions derived in the following reveal the dependence of all harmonic components found with electrically, and similarly, with intermediate spin-pair controlled optically detected transition rates on the parameters of both, the observed physical system as well as the performed Rabi-oscillation experiment.
The parameters of a performed Rabi-oscillation experiment are the spin S = 1 2 Rabi oscillation frequency Ω R = γB 1 which depends on the driving field strength B 1 as well as the gyromagnetic ratio γ. The parameters characterizing the observed spin pair are given by the spin-orbit controlled respective g-factors for each pair partner as well as local, and hyperfine fields which in general are different for the two pair partners as well. For our purposes, all these parameters can be taken into account by the difference δ of the pair partners Larmor frequencies. Taking this into account, we anticipate from the previously reported numerical simulations 12,15 that in the weak-driving regime where Ω R δ, the Fourier transformation F(s) of a Rabi oscillation transient exhibits only one peak at s = Ω R , while in the strongdriving regime Ω R δ) there is one peak at s = 2Ω R and no peak at s = Ω R . The crossover between these two situations occurs around Ω R ≈ δ.
This crossover with increasing Ω R was demonstrated experimentally for spin-dependent polaron pair recombination processes in different organic semiconductor materials 10,11 . Currently, it is rather difficult (if not impossible) to extract quantitative information from such experimentally measured Rabi spectra F(s) due to the lack of theoretical predictions. On the other hand, the theoretical problem is well-posed. When the exchange and dipolar coupling strength within a pair are smaller than δ 0 (which we assume), the shape of the Fourier transform depends only on two parameters: δ 0 , which is the r.m.s. value of δ, and Ω R .
In the following, the Fourier components of the Rabi spectrum F(s) are calculated analytically. We find that the width of the crossover region is broad and extends from Ω R δ0 ≈ 0.3, where the s = 2Ω R peak appears, to
Ω R δ0 ≈ 2,
where it dominates over s = Ω R peak. Most importantly, the analytical treatment reveals that F(s) consists not only of two peaks (as discussed in most experimental studies) but rather of three peaks. The origin of the third peak, which occurs at frequencies s Ω R , is due a disorder induced distribution of δ, which implies that even at strong Ω R , the two spins in the pair do not precess entirely in phase. This third peak is harder to observe experimentally compared to the two with higher frequencies, yet a prediction of its shape and evolution with Ω R is provided. Also, a correlation study of disorder within pairs affects the shape of an ensemble average of F(s). This follows from previous numerical study of the ratio between the magnitudes of the s = Ω R and s = 2Ω R Rabi oscillation peaks 17 which was conducted for particular value sets of δ 0 and Ω R .
The study presented here is outlined as follows. In Sect. II, a qualitative derivation of conductivity changes in pEDMR experiments is given which reproduces the result of rigorous consideration in Ref. 12. In Sect. III, the analytical expression for a disorder averaged Fourier transform of ∆σ(τ ) is presented before this is used in Sect. IV for the consideration of correlation effects between the two random distribution of the two pair partner resonances. An analysis and discussion of these results is then presented in Sect. V.
II. THE DEPENDENCE OF PEDMR INDUCED CONDUCTIVITY CHANGES ON THE PULSE DURATION τ
The pair partners within a spin pair are denoted with a and b respectively. Before a pulse is applied to a spin pair, it will rest in one of its four spin-eigenstates, as both a and b can be either in a | ↓ or in a | ↑ state. The important qualitative observation made in Ref. 18 is that only initial configurations | ↓↓ and | ↑↑ of the pair exist as the other two eigenstates with singlet content are very short lived. The | ↓↓ and | ↑↑ states are therefore responsible for the change of conductivity after the end of the pulse. Without confinement of generality, one can assume that at t = 0 both a and b are in the | ↓ state. The respective populations of the | ↓ states will then evolve according to the Rabi formula
n a,b (t) = 1 − Ω 2 R Ω 2 R + δ 2 a,b sin 2 1 4 Ω 2 R + δ 2 a,b t,(1)
where δ a = ω a − ω and δ b = ω b − ω are, the detuning frequencies of a and b, respectively. A detuning frequency is the difference of a Larmor frequency ω a or ω b and the frequency ω of the driving field. After the pulse ends at t = τ the pair is in the | ↓↓ state with probability P ↓↓ = n a (τ )n b (τ ) and in | ↑↑ state with probability P ↑↑ = 1 − n a (τ ) 1 − n b (τ ) . Then the probability to find the pair in one of the states | ↓↓ or | ↑↑ is equal to
P (τ ) = P ↑↑ + P ↓↓ = 1 − n a (τ ) − n b (τ ) + 2n a (τ )n b (τ ).(2)
It is easy to see that Eq. (2) also applies when the pair is initially in | ↑↑ state. Eq. (2) coincides with the corresponding expression for the τ -dependent part of the diagonal elements of the density matrix of the pair derived in Ref. 12. The probability P (τ ) serves as the initial condition for the transient restoration of the steady state current after the pulse 12 . Thus, ∆σ can be identified with P (τ ) within a factor. An expression for ∆σ(τ ) averaged over the contributions from all pairs within a pair ensemble is then obtained from
∆σ(τ ) = 1 2πδ 2 0 dδ a dδ b exp − δ 2 a + δ 2 b 2δ 2 0 ∆σ(δ a , δ b , τ ).
(3) Note that this expression for ∆σ(τ ) can also be written as
∆σ(τ ) = 1 − 2T (τ ) + 2T 2 (τ ),(4)
where the function T (τ ) is defined as
T (τ ) = 1 √ 2πδ 0 dδ e −δ 2 /2δ 2 0 Ω 2 R Ω 2 R + δ 2 × sin 2 1 4 Ω 2 R + δ 2 τ.(5)
In the limit of long pulses T (τ ) approaches a constant in an oscillatory fashion; the amplitude of the oscillations falls off slowly, as τ −1/2 , with the length of the pulse 19 . For strong disorder (δ Ω R ) the derivative can be expressed through the zero-order Bessel function 8
T (τ ) = 2 −3/2 π 1/2 Ω 3 R δ −1 0 J 0 (Ω R τ )
. While the second term in Eq. (4) describes Rabi oscillations within either component a or b of the pair, the third term "knows" about the collective spin precession of a and b. However, the T 2 -term also contains contributions from the individual precessions of pair partners a and b. We will therefore subtract these contributions and group them with the T -term in Eq. (4) prior to performing the Fourier transform. By substituting Eq. (1) into Eq. (2), we get
∆σ(τ ) = 1 2 + δ 2 a δ 2 b 2(Ω 2 R + δ 2 a )(Ω 2 R + δ 2 b ) + G 1 (δ a , δ b , τ ) + G 1 (δ b , δ a , τ ) + G − (δ a , δ b , τ ) + G + (δ a , δ b , τ ),(6)
where the functions describing the three harmonic Rabi oscillation peaks are defined as
G 1 (δ a , δ b , τ ) = ΩR 2 δ 2 b 2 cos Ω 2 R + δ 2 a τ (Ω 2 R + δ 2 a )(Ω 2 R + δ 2 b ) ,(7)G − (δ a , δ b , τ ) = ΩR 4 4 cos Ω 2 R + δ 2 a − Ω 2 R + δ 2 b τ (Ω 2 R + δ 2 a )(Ω 2 R + δ 2 b ) ,(8)G + (δ a , δ b , τ ) = ΩR 4 4 cos Ω 2 R + δ 2 a + Ω 2 R + δ 2 b τ (Ω 2 R + δ 2 a )(Ω 2 R + δ 2 b ) .
(9) The above terms G 1 , G + , G − describe the peaks s = Ω R , s = 2Ω R , and s Ω R , contained in the Fourier transform F(s), respectively.
III. AVERAGING OVER DISORDER WITHIN A SPIN PAIR ENSEMBLE
Variations of the magnetic resonance frequency of each pair partner in each individual pair can occur due to: (i) variations of the spin-orbit coupling which changes the g-factor 20 . This is seen in disordered materials where the lengths and angles of chemical bonds can vary strongly 1,2 . (ii) Due to random hyperfine fields which can strongly fluctuate throughout a material because of the small polarization nuclear spins even at low temperature and high magnetic fields 10,21 . We define the Fourier spectrum of the conductivity change from the steady state as
F(s) = ∞ 0 dτ cos(sτ ) ∆σ(τ ) − ∆σ(0) .(10)
The expression can be decomposed into three contributions F(s) = F 1 (s) + F 0 (s) + F 2 (s), which derive from the terms G 1 , G − , and G + terms in Eq. (6). Obviously, the time integration of each term yields a δ-function.
Our task is to perform the averaging of each δ-function over disorder, as in Eq. (3). We start from F 1 (s), which describes a peak near s = Ω R . For this contribution, the averaging over δ a , δ b reduces to the product of averages
F 1 (s) = Ω 2 R 8δ 2 0 dδ a e −δ 2 a /2δ 2 0 δ(s − Ω 2 R + δ 2 a ) Ω 2 R + δ 2 a × dδ b e −δ 2 b /δ 2 0 δ 2 b Ω 2 R + δ 2 b .(11)
It is convenient to evaluate the integral over δ a with the help of the δ-function. The second integral can be reduced to the error-function leading to
F 1 (s) = Ω 3 R 4δ 2 0 s s 2 − Ω 2 R exp − s 2 − Ω 2 R 2δ 2 0 f Ω 2 R 2δ 2 0 ,(12)
where the function f is defined as
f (b) = ∞ −∞ dy e −y 2 b y 2 1 + y 2 = √ π 1 b 1/2 − √ πe b erfc( √ b) .(13)
For F 2 (s), it is convenient, after substituting (9) into (10), to perform an integration over δ a , δ b in polar coordinates. Upon introducing the new variables
δ a = √ 2v cos φ, δ b = √ 2v sin φ,(14)
the expression for F 2 (s) acquires the form
F 2 (s) = Ω 4 R 8δ 2 0 ∞ 0 dv v e −v 2 /δ 2 0 2π 0 dφ × δ s − Ω 2 R + 2v 2 cos 2 φ − Ω 2 R + 2v 2 sin 2 φ Ω 2 R + 2v 2 cos 2 φ Ω 2 R + 2v 2 sin 2 φ .(15)
Without the denominator, it is straightforward to perform an integration over φ, which yields
2π 0 dφ δ s − Ω 2 R + 2v 2 cos 2 φ − Ω 2 R + 2v 2 sin 2 φ = 2|2Ω 2 R + 2v 2 − s 2 | s 2 2 − v 2 2 − Ω 2 R s 2 (Ω 2 R + v 2 ) − s 2 4 .(16)
With the denominator, Eq. (16) is to be divided by the value 1 4 (s 2 − 2Ω 2 R − 2v 2 ) 2 of the denominator where the argument of the δ-function is zero, which yields
F 2 (s) = Ω 4 R δ 2 0 s 2 2 −sΩ R 0 dv v e −v 2 /δ 2 0 s 2 − 2Ω 2 R − 2v 2 × 1 s 2 2 − v 2 2 − Ω 2 R s 2 (Ω 2 R + v 2 ) − s 2 4 .(17)
Eq. (17) is defined only for s > 2Ω R . As s approaches 2Ω R , both brackets under the square root turn to zero. At the same time, the integration interval also shrinks to zero. However, the factor |s 2 − 2Ω 2 R − 2v 2 | in the denominator is nonsingular near s = 2Ω R . To illuminate the behavior of F 2 (s) near the threshold, it is convenient to make the substitution w = v 2 − s 2 4 + Ω 2 R in the integral of Eq. (17). It then assumes the form
F 2 (s) = Ω 4 R 2δ 2 0 e − s 2 4 −Ω 2 R δ 2 0 ( s 2 −Ω R ) 2 0 dw e −w/δ 2 0 w s 2 − Ω R 2 − w × 1 | s 2 2 − 2w| s 2 + Ω R 2 − w .(18)
Now we see that only the first two factors in the denominator are singular when s is close to 2Ω R , where the s = 2Ω R peak occurs. In this domain we can set w = 0 in the last two factors and take them out of the integrand. Then, the remaining integral readily reduces to the modified Bessel function, I 0 (y), and we get
F 2 (s) = 2Ω 4 R δ 2 0 s 2 e − s 2 /4−Ω 2 R δ 2 0 s + 2Ω R G (s − 2Ω R ) 2 4δ 2 0 ,(19)
where
G(b) = b 0 dx e −x x(b − x) = πe −b/2 I 0 b 2 .(20)
The analysis of the shape of the s = 2Ω R peak given in Sect. V. will reveal that the approximation in Eq. (19) describes not only the vicinity (s − 2Ω R ) Ω R but the entire peak when Ω R is bigger than 0.3δ 0 .
Finally we turn our attention to the peak at s Ω R . The initial expression for F 2 (s) differs from Eq. (15) for F 0 (s) only in one respect. Instead of the sum, Ω 2 R + δ 2 a + Ω 2 R + δ 2 a , in the argument of the δ-function it contains a difference, Ω 2 R + δ 2 a − Ω 2 R + δ 2 a . The angular integration in polar coordinates is therefore performed in a similar way as Eq. (16)
2π 0 dφ δ s − Ω 2 R + 2v 2 cos 2 φ + Ω 2 R + 2v 2 sin 2 φ = |2Ω 2 R + 2v 2 − s 2 | s 2 2 − v 2 2 − Ω 2 R s 2 (Ω 2 R + v 2 ) − s 2 4 .(21)
Note that, compared to Eq. (21), the integral Eq. (16) has an extra factor of 2. This is because the argument of δ-function in Eq. (21) turns to zero at two values of φ, while in Eq. (16), it turns to zero at four values of φ.
To get the final expression for F 0 one has again to divide Eq. (21) by the value of denominator at the zeros points of the δ-function, which is equal to 1 4 (s 2 − 2Ω 2 R − 2v 2 ) 2 , i.e., the same as in F 2 (s). This leads to
F 0 (s) = Ω 4 R 2δ 2 0 ∞ s 2 2 +sΩ R dv v e −v 2 /δ 2 0 s 2 2 − v 2 2 − Ω 2 R s 2 × 1 s 2 − 2Ω 2 R − 2v 2 (Ω 2 R + v 2 ) − s 2 4 .(22)
We see again that only the first factor in the denominator is singular at the lower limit v = s 2 2 + sΩ R 1/2 . Thus, for small s, the peak is described by substituting s = 0 into the second and third factors in denominator, which leads to
F 0 (s) = Ω R 4δ 2 0 ∞ s 2 2 +sΩ R dv v e −v 2 /δ 2 0 s 2 2 − v 2 2 − Ω 2 R s 2 .(23)
The remaining integral can be expressed via the Macdonald function, K 0 (y), which yields
F 0 (s) = Ω R 8δ 2 0 e −s 2 /2δ 2 0 K 0 sΩ R δ 2 0 .(24)
We will see in Sect. V. that the expression in Eq. (24) describes the entire peak when Ω R is big enough, Ω R δ 0 .
IV. CORRELATION BETWEEN THE PAIR PARTNER DISORDER
Due to the proximity of the pair partners in each pair, the disorder related randomness of δ a and δ b may be correlated. Examples for such a correlation could be the common exposure of polaronic state in organic semiconductors to an overlapping nuclear spin bath or the correlation of the spin-orbit interaction in a disordered semiconductor due to local strain fields 22 . In such cases, δ a and δ b are not statistically independent and the degree of overlap can be expressed by a correlation parameter, x (0 < x < 1). Then the joint distribution function of δ a , δ b assumes the form
Φ(δ a , δ b ) = 1 2πδ 2 0 √ 1 − x 2 exp − δ 2 a + δ 2 b − 2xδ 2 a δ 2 b 2δ 2 0 (1 − x 2 )
.
(25) We study the effect of correlation for the limit Ω R δ 0 when all three peaks are well-developed and do not overlap. In this limit, we can use the expansion Ω 2
R + δ 2 a,b ≈ Ω R + δ 2 a,b
2Ω R in the arguments of the δ-functions. We can also replace Ω 2 R + δ 2 a,b by Ω R in the denominators of Eqs. (7) to (9). With these simplifications the expression for F 1 (s) assumes the form
F 1 (s) = 1 8δ 2 0 Ω 2 R √ 1 − x 2 dδ b dδ a δ 2 b × exp − δ 2 a + δ 2 b − 2xδ b δ a 2δ 2 0 (1 − x 2 ) δ s − Ω R − δ 2 a 2Ω R .(26)
Integrating over δ a with the help of the δ-function yields
F 1 (s) = exp − Ω R (s−Ω R ) δ 2 0 (1−x 2 ) 4Ω R δ 2 0 √ 1 − x 2 2Ω R (s − Ω R ) dδ b × δ 2 b exp − δ 2 b 2δ 2 0 (1 − x 2 ) cosh δ b x 2Ω R (s − Ω R ) δ 2 0 (1 − x 2 ) .(27)
Subsequent integration over δ b is straightforward leading to
F 1 (s) = √ πδ 0 (1 − x 2 ) 4Ω R Ω R (s − Ω R ) 1 + 2x 2 Ω R (s − Ω R ) δ 2 0 (1 − x 2 ) × exp − Ω R (s − Ω R ) δ 2 0 .(28)
We see that the correlation parameter, x, enters this expression only in the prefactor. In the limit of x → 0, Eq. (28) matches Eq. (12) as can be seen when Ω R δ 0 is assumed and Eq. (12) is expanded around s = Ω R .
In the limit of large Ω R the definition of F 2 (s) becomes
F 2 (s) = 1 16δ 2 0 √ 1 − x 2 dδ a dδ b × exp − δ 2 a + δ 2 b − 2xδ b δ a 2δ 2 0 (1 − x 2 ) δ s − 2Ω R − δ 2 a + δ 2 b 2Ω R .(29)
Both integrals over δ a and δ b can be taken explicitly upon introduction of polar coordinates
r = 2(δ 2 a + δ 2 b ), φ = arctan δ a − δ b δ a + δ b ,(30)
so that Eq. (29) assumes the form
F 2 (s) = 1 32δ 2 0 √ 1 − x 2 dr r dφ × exp − r 2 (1 + x cos 2φ) 4δ 2 0 (1 − x 2 ) δ s − 2Ω R − r 2 4Ω R .(31)
After integrating over r by using the δ-function, the remaining integral over φ reduces to I 0 (y), and we arrive at
F 2 (s) = πΩ R 8δ 2 0 √ 1 − x 2 exp − Ω R (s − 2Ω R ) δ 2 0 (1 − x 2 ) × I 0 Ω R x(s − 2Ω R ) δ 2 0 (1 − x 2 ) .(32)
Similarly to the non-correlated case, F 2 (s) is expressed through I 0 (y). However, note that the argument of I 0 (y) in Eq. (32) is completely different from Eq. (19). In fact, Eq. (32) was derived for the case when I 0 b 2 in Eq. (20) should be replaced by 1.
In the presence of disorder correlation, the shape of F 0 (s) can be also expressed via the Macdonald function with x-dependent argument. To see this, we take the definition
F 0 (s) = 1 16δ 2 0 √ 1 − x 2 dδ a dδ b × exp − δ 2 a + δ 2 b − 2xδ b δ a 2δ 2 0 (1 − x 2 ) δ s − δ 2 a − δ 2 b 2Ω R ,(33)
and introduce polar coordinates
r = δ 2 a + δ 2 b − 2xδ b δ a 2δ 2 0 (1 − x 2 ) , φ = arctan 1 + x 1 − x δ a − δ b δ a + δ b .(34)
Eq. (33) then becomes
F 0 (s) = 1 8 dr r e −r 2 dφ δ s − δ 2 0 r 2 √ 1 − x 2 Ω R sin 2φ .
(35) The integration over φ can be done explicitly by using the δ-function, yielding
F 0 (s) = 1 4s ∞ sΩ R δ 2 0 √ 1−x 2 1/2
dr r e −r 2
δ 4 0 (1−x 2 ) Ω 2 R s 2 r 4 − 1 .(36)
The reduction to the Macdonald function can then be achieved by the substitution r 2 = sΩ R
δ 2 0 √ 1−x 2 r 1 which re- veals F 0 (s) = Ω R 8δ 2 0 √ 1 − x 2 K 0 sΩ R δ 2 0 √ 1 − x 2 .(37)
V. ANALYSIS AND DISCUSSION
The most important results of this study are the analytical expressions for the lineshapes of Rabi-oscillation peaks as they can be found in the Fourier analysis (the Rabi spectra) of the pulse length dependent conductivity changes ∆σ(τ ) that are measured with pEDMR experiments. For the case of uncorrelated disorder, Eqs. (12), (19), and (24), reveal these peak shapes for the oscillation peaks F 1 (s), F 0 (s), and F 2 (s), respectively. For the case of correlated disorder, the same peaks are described by Eqs. (28), (32), and (37). We consider now the limit Ω R δ 0 , when all three peaks are well developed. It is easy to see that, for uncorrelated disorder, all three peaks exhibit the same exponential tail
F 1 (s) ≈ π 1/2 4Ω R ∆s s − Ω R e − s−Ω R ∆s , F 0 (s) ≈ √ π 8 √ 2s∆s e − s ∆s , F 2 (s) ≈ π 8∆s e − s−2Ω R ∆s ,(38)
where the characteristic width of the tail is given by
∆s = δ 2 0 Ω R .(39)
Naturally, all three peaks shrink with increasing Ω R . It is less trivial to realize that, for the same deviation from the origin, the peaks F 0 , and F 2 have a larger magnitude than F 1 , whose terms contain Ω R ∆s in the denominator. This relation between the peaks is illustrated in Fig. 2c.
We consider now the opposite limit of strong but uncorrelated disorder where δ 0 Ω R and x = 0. Eq. (19) implies that F 2 (2Ω R ) = Ω R 16δ 2 0 is always finite at its peak. At the same time, the value of F 1 (s) at s = 2Ω R is π 3/2 4 √ 6δ0 , i.e., it is bigger than F 2 (2Ω R ). Therefore, the s = 2Ω R peak is indistinguishable on the background of the s = Ω R peak. This behavior reflects the physics of the weak driving regime 12 where only one component of the pair can be in resonance with the driving field at any time. Numerically, however, the s = 2Ω R peak is pronounced already at Ω R > 0.3δ 0 , as shown in Fig. 2a. For the given strength of Ω R , the approximation Eq. (19) using the modified Bessel function is already justified.
The low frequency Rabi-oscillation peak which is described by Eq. (24) diverges logarithmically in the limit of s → 0 when F 0 (s) ∝ ln(1/s). Nevertheless, the peak still loses to F 1 (s) in the weak-driving regime due to its small prefactor Ω R 8δ 2 0 . The lineshape of this peak is described by Eq. (24) when the ratio Ω R /δ0 exceeds 1. This could be the reason why this peak has not received much attention in previous experimental studies since this regime is hard (yet not impossible) to attain experimentally. Fig. 3 illustrates how the intrapair correlation of disorder affects the shapes of the peaks in the strong-driving regime. One can see that the prime effect of correlation is a dramatic narrowing of the low-frequency peak. This narrowing reflects the fact that the low-frequency peak is entirely due to inequivalences of the pair partners, at x = 1, giving rise to a maximum. For small but finite (1 − x) such pairs exist and cause "normal" divergent behavior F 1 (s)
∝ 1−x √ s−Ω R
at s → Ω R , resulting in a minimum. Finally, the s = 2Ω R peak becomes enhanced by correlation near the origin for the same reason why F 1 (s) gets depleted. Conservation of the total area, which applies for all three peaks, is achieved due to depletion in the body.
We note that disorder broadens the peaks in F(s) only to the right (the higher frequencies) from corresponding thresholds. This is due to the adopted definition [Eq. (10)] of the Fourier transform. If we used the standard definition, F 2 (s) +F 2 (s) 1/2 , whereF(s) is defined as
F(s) = ∞ 0 dτ sin(sτ ) ∆σ(τ ) − ∆σ(0) ,(40)
the disorder will broaden the peaks both to the left and to the right (to higher and lower frequencies) from the threshold. This is illustrated in Fig. 4 for the central peak at s = Ω R . A formal expressioñ spectroscopy studies) leads to a significant unnecessary increase of the peak width which complicates the analysis of experimental data. For the illustration of the artificial broadening effect, we refer to an experimental data set which is displayed in the inset of Fig. 5. The data shows electrically detected spin-Rabi oscillation in a π-conjugated polymer diode measured at room temperatures on identically prepared samples and conditions as for the experiments described in Ref. 23. The diode consisted of an ITO/PEDOT/MEH-PPV/Ca/Al device stack. The signal was measured by detection of changes to a forward steady state current of I = 100µA current, under application of a 4V bias at room temperature. The experiment was conducted under an applied magnetic field of 344mT, the applied magnetic resonant excitation had a frequency of 9.6606GHz and a driving field strength B 1 = 0.519(36)mT produced by a 27W coherent pulsed microwave source. The inset of Fig. 5 shows the measured raw data. It displays a rapidly dephasing oscillation which is due to inhomogeneities of the applied B 1 field, Ref. 23. The main plot of Fig. 5 shows two different Fourier transformations of the data displayed in the inset plotted on a normalized scale as a function of frequency in units of Ω R = γB 1 . The black plot displays the absolute Fourier transform of the experimental data while the red plot displays one the in-phase (cos) component of the Fourier transform. From these plots, it becomes evident that the absolute Fourier transform displays significant broadening of the Rabi spectrum without providing additional insight. It shall be noted that the peak widths of the plot in Fig. 5 the peaks of the red plot are more asymmetric compared to the black plot. With the given length of the original data set (500ns), an additional symmetric broadening effect is introduced to both Fourier transforms which overlaps the intrinsic peak shapes. However, for the data sets displayed in Fig. 5, one can conclude, that the analysis of experimental spin-Rabi spectra measured with pEDMR and pODMR should be conducted on Fourier data obtained from Eq. (10). In order to scrutinize the analytical description of the Rabi-spectrum of a pair process given above, we can now fit the experimentally obtained Fourier transform of the electrically detected electron spin-pair Rabi-oscillation displayed in Fig. 5 with the analytical expressions in Eqs. (12), (19), and (24) for the Fourier transforms of an uncorrelated spin pair ensemble F 1 (s), F 2 (s), and F 0 (s), respectively. It is known that weakly coupled polaron pairs in MEH-PPV do not exhibit a correlation of their local hyperfine fields 12 so an agreement between the experimental data set and the sum of Eqs. (12), (19), and (24) is expected. Since the expressions for F i (s) are absolute, the fit of F (s) = F 0 (s) + F 1 (s) + F 2 (s) has only three fit parameters, namely (i) a global scaling factor of the unitless ordinate, (ii) the disorder factor Ω R /δ 0 as well as a third fit variable which is the width of a homogenous (Lorentzian) broadening of F (s). This broadening is due to an experimental artifact, coming from a Fourier transformation of a finite time interval as well as the measured decay of the Rabi-oscillation which is known to be due to inhomogeneities of the experimentally generated B 1 field 23 . Thus, the length (500ns) of the experimentally recorded Rabi oscillation determine the frequency resolution of the resulting Fourier transform while the decay (≈ 150ns) of the Rabi-oscillation further contributed to homogeneous broadening of the experimental Fourier transformed data.
F 1 (s) = Ω 3 R 2πδ 2 0 s dδ e −δ 2 /2δ 2 0 s 2 − Ω 2 R − δ 2 f Ω 2 R 2δ 2 0(41)) ) ( ) ( ( s F s F + ) (s F ) ( ) ( ) ( 2 1 0 s F s F s F + +
Using the three fit variables for F (s) mentioned above, we obtain a good fit of the experimental data as displayed by the blue line in Fig. 5. In this fit, the disorder parameter δ 0 /Ω R = 1.54 with an estimated error of less than 0.2. Given the Rabi nutation frequency Ω R = γB 1 = 14.5(1.0)MHz, we obtain δ 0 = 22.3(3.3)MHz. This value is equivalent to a hyperfine broadened polaron line width of δ 0 /γ = 0.80 (14)mT, a value that is in good agreement with the widths of EDMR detected polaron spin resonances in MEH-PPV 11,12,23 . We conclude from this that the analytical expressions Eqs. (12), (19), and (24) capture adequately the Rabi oscillations in weakly coupled spin pairs and should therefore be used for the fit of pEDMR data governed by non-correlated spin pair ensembles.
FIG. 1 :
1Schematic illustration of the Rabi oscillations with frequency ΩR in spin-1 2 pair. Components of the pair, a and b have different environments causing random shifts, δa and δb, from the resonant frequency, ω0. Relevant for PEDMR are the initial and final spin configurations | ↑↑ and | ↓↓ only. rent detectable observations of spin Rabi oscillations are possible.
FIG. 2 :
2The shapes of the three peaks s ΩR (red), s = ΩR (blue), and s = 2ΩR (green) in the Fourier transform of ∆σ(τ ) are plotted for three values of the dimensionless Rabi frequency w = ΩR/δ0: (a) w = 0.3, (b) w = 1.1, (c) w = 2.which are suppressed by the correlation. Another effect of strong correlation is that F 1 (s) develops a maximum at (s − Ω R ) = ∆s/2 and a minimum at (s − Ω R ) = (1 − x)∆s. The origin of the maximum is that, for full correlation (x = 1) the portion of resonant pairs in which only one partner participates to the Rabi oscillations vanishes. More precisely, F 1 (s) ∝ (s − Ω R ) 1/2 exp − s−Ω R ∆s FIG. 3: Illustration of the effect of intrapair correlation of disorder on the shapes of the three peaks s ΩR (red), s = ΩR (blue), and s = 2ΩR (green), in the Fourier transform of ∆σ(τ ). The dashed lines correspond to uncorrelated disorder (x = 0), while the full lines correspond to strongly correlated disorder (x = 0.95). The dimensionless Rabi frequency is w = ΩR/δ0 = 1.2.
FIG. 5 :
5Fourier transformation of an electrically detected spin-Rabi oscillation measured in an organic polymer diode. The inset displays the measured data set, for details about the measurement see text as well as Ref. 23. The black data of the main panel displays is the absolute Fourier transform F (s) 2 +F (s) 2 of the data in the inset. The red data displays is the real part F (s) of the Fourier transform of the data in the inset. The blue line represents a Fit of the experimental data with the analytical function F (s) = F0(s)+F1(s)+F2(s).
for two values of the dimensionless Rabi frequency w = ΩR/δ0. The dashed lines show the corresponding Fourier transform plotted using F(s) only.emerges as an obvious modification of Eq. (12). This
integral in Eq. (41) can be expressed through the error-
function. It is nonzero for s < Ω R and for s > Ω R . Near
s = Ω R it diverges as |s−Ω R | −1/2 . As shown in Fig. 4, the
inclusion ofF(s) into the Fourier transform (as done for
most previously published pEDMR and pODMR Rabi
FIG. 4: Solid lines: the Fourier transform of the central
peak plotted using the definition F 2 (s) +F 2 (s)
1/2 from Eqs.
(10), (40)
are too close to the frequency resolution of the Fourier transform to unambiguously identify that0 100 200 300 400 500
0
1
2
3
4
5
6
7
8
0.0
0.2
0.4
0.6
0.8
1.0
pulse length (ns)
∆σ (arb.
u.)
s/Ω R
FT[arb. u.]
2
/
1
2
2
AcknowledgmentsWe acknowledge D. P. Waters and R. Baarda for the preparation of the MEH-PPV diode pEDMR templates. We also acknowledge the support of this work by the National Science Foundation through the Materials Research Science and Excellence Center (#DMR-1121252). CB further acknowledges the support through a National Science Foundation CAREER award (#0953225).
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|
[] |
[
"Billiard representation for multidimensional multi-scalar cosmological model with exponential potentials",
"Billiard representation for multidimensional multi-scalar cosmological model with exponential potentials"
] |
[
"H Dehnen ",
"V D Ivashchuk ",
"V N Melnikov ",
"\nCenter for Gravitation and Fundamental Metrology\nUniversität Konstanz\nFakultät für Physik\nFach M 568D-78457Konstanz\n",
"\nInstitute of Gravitation and Cosmology, PFUR\nVNIIMS\n3/1 M. Ulyanovoy Str., 6 Miklukho-Maklaya Str119313, 117198Moscow, MoscowRussia, Russia\n"
] |
[
"Center for Gravitation and Fundamental Metrology\nUniversität Konstanz\nFakultät für Physik\nFach M 568D-78457Konstanz",
"Institute of Gravitation and Cosmology, PFUR\nVNIIMS\n3/1 M. Ulyanovoy Str., 6 Miklukho-Maklaya Str119313, 117198Moscow, MoscowRussia, Russia"
] |
[] |
Multidimensional cosmological-type model with n Einstein factor spaces in the theory with l scalar fields and multiple exponential potential is considered. The dynamics of the model near the singularity is reduced to a billiard on the (N − 1)-dimensional Lobachevsky space H N −1 , N = n + l. It is shown that for n > 1 the oscillating behaviour near the singularity is absent and solutions have an asymptotical Kasner-like behavior. For the case of one scale factor (n = 1) billiards with finite volumes (e.g. coinciding with that of the Bianchi-IX model) are described and oscillating behaviour of scalar fields near the singularity is obtained.
|
10.1023/b:gerg.0000032149.58989.e0
|
[
"https://arxiv.org/pdf/hep-th/0312317v1.pdf"
] | 15,388,260 |
hep-th/0312317
|
a49e6af496e0238f6f337d57ad7e3d3733cfeb52
|
Billiard representation for multidimensional multi-scalar cosmological model with exponential potentials
arXiv:hep-th/0312317v1 30 Dec 2003
H Dehnen
V D Ivashchuk
V N Melnikov
Center for Gravitation and Fundamental Metrology
Universität Konstanz
Fakultät für Physik
Fach M 568D-78457Konstanz
Institute of Gravitation and Cosmology, PFUR
VNIIMS
3/1 M. Ulyanovoy Str., 6 Miklukho-Maklaya Str119313, 117198Moscow, MoscowRussia, Russia
Billiard representation for multidimensional multi-scalar cosmological model with exponential potentials
arXiv:hep-th/0312317v1 30 Dec 2003
Multidimensional cosmological-type model with n Einstein factor spaces in the theory with l scalar fields and multiple exponential potential is considered. The dynamics of the model near the singularity is reduced to a billiard on the (N − 1)-dimensional Lobachevsky space H N −1 , N = n + l. It is shown that for n > 1 the oscillating behaviour near the singularity is absent and solutions have an asymptotical Kasner-like behavior. For the case of one scale factor (n = 1) billiards with finite volumes (e.g. coinciding with that of the Bianchi-IX model) are described and oscillating behaviour of scalar fields near the singularity is obtained.
Introduction
The study of different aspects of multidimensional models in gravitation and cosmology in arbitrary dimensions and with sources as fluids and fields we started more than a decade ago (see [1,2,3]). Special attention was devoted to the treatment of dilatonic interactions with electromagnetic fields and fields of forms of arbitrary rank [4]. Here we continue our investigations of multidimensional models, in particular with multiple exponential potential (MEP) [5] (for D = 4 case see [6]).
The models of such sort are currently rather popular (see, for example, [6,7,8,9,10] and refs. therein). Such potentials arise naturally in certain supergravitational models [10], in sigma-models [11] related to configurations with p-branes and in reconstruction from observations schemes [12]. They also appear when certain f (R) generalizations of Einstein-Hilbert action are considered [13].
Like in [5], here we consider D-dimensional model governed by the action This paper is devoted to the investigation of the possible oscillating (and probably stochastic) behaviour near the singularity (see [15]- [36] and references therein) for cosmological type solutions corresponding to the action (1.1).
S act = M d D Z |g|{R[g] − h αβ g M N ∂ M ϕ α ∂ N ϕ β − 2V ϕ (ϕ)} + S
We remind that near the singularity one can have an oscillating behavior like in the well-known mixmaster (Bianchi-IX) model [15]- [17] (see also [26]- [28]). Multidimensional generalizations and analogues of this model were considered by many authors (see, for example, [18]- [25]). In [29,30,31] a billiard representation for multidimensional cosmological models near the singularity was considered and the criterion for a volume of the billiard to be finite was established in terms of illumination of the unit sphere by pointlike sources. For multicomponent perfect-fluid this was considered in detail in [31] and generalized to p-brane case in [34] (see also [36] and refs. therein). Some topics related to general (non-homogeneous) situation were considered in [32,33].
Here we apply the billiard approach suggested in [29,30,31] to a cosmological model with MEP. We show that (as for the exact solutions from [5]) for n > 1 the oscillating behaviour near the singularity is absent. For n = 1 we find here examples of oscillating behavior for scalar fields but not for a scale factor.
The paper is organized as follows. In Sec. 2 the cosmological model with MEP is considered: Lagrange representation to equations of motion and the diagonalization of the Lagrangian are presented. In Sec. 3 a billiard approach in the multidimensional cosmology with MEP is obtained, and the case n > 1 is studied. Sec. 4 is devoted to description of billiards with finite volumes in the case of one scale factor (n = 1).
The model
Let
M = IR × M 1 × . . . × M n (2.1)
be a manifold equipped with the metric
g = we 2γ(u) du ⊗ du + n i=1 e 2φ i (u) g i ,(2.2)
where w = ±1, u is a distinguished coordinate; g i is a metric on d i -dimensional manifold M i , obeying:
Ric[g i ] = ξ i g i ,(2.
Lagrangian representation
It may be verified that the equations of motion (see Appendix A) corresponding to (1.1) for the field configuration (2.2)-(2.4) are equivalent to equations of motion for 1-dimensional σ-model with the action
S σ = 1 2 duN G ijφ iφj + h αβφ αφβ − 2N −2 V , (2.5) whereẋ ≡ dx/du, V = −wV ϕ (ϕ)e 2γ 0 (φ) + w 2 n i=1 ξ i d i e −2φ i +2γ 0 (φ) (2.6)
is the potential (V ϕ is defined in (1.2)) with
γ 0 (φ) ≡ n i=1 d i φ i ,(2.7)
and
N = exp(γ 0 − γ) > 0 (2.8)
is the lapse function. Here
G ij = d i δ ij − d i d j , G ij = δ ij d i + 1 2 − D ,(2.
9)
i, j = 1, . . . , n, are components of a gravitational part of minisupermetric and its dual [40].
Minisuperspace notations
In what follows we consider minisuperspace IR n+l with points
x ≡ (x A ) = (φ i , ϕ α ) (2.10)
equipped by minisuperspace metricḠ =Ḡ AB dx A ⊗dx B defined by the matrix and inverse one as follows:
(Ḡ AB ) = G ij 0 0 h αβ , (Ḡ AB ) = G ij 0 0 h αβ . (2.11)
The potential (2.6) reads (2.14) or, in components,
V = −w s∈S Λ s e 2U s (x) + n j=1 w 2 ξ j d j e 2U j (x) , (2.12) where U s (x) = U s A x A and U j (x) = U j A x A are defined as U s (x) = λ sα ϕ α + γ 0 (φ), (2.13) U j (x) = −φ j + γ 0 (φ),(U s A ) = (d i , λ sα ) (2.15) (U j A ) = (−δ j i + d i , 0) (2.16)
s ∈ S; i, j = 1, . . . , n.
The integrability of the Lagrange system (2.5) depends upon the scalar products of co-vectors U s , U i corresponding toḠ:
(U, U ′ ) =Ḡ AB U A U ′ B ,(2.17)
These products have the following form
(U i , U j ) = δ ij d j − 1, (2.18) (U s , U s ′ ) = −b + λ s · λ s ′ , (2.19) (U s , U i ) = −1, (2.20) where λ s · λ s ′ ≡ λ sα λ s ′ β h αβ , b = D − 1 D − 2 , (2.21)
s, s ′ ∈ S.
Diagonalization of the Lagrangian
Let the matrix (h αβ ) have the Euclidean signature. Then, the minisuperspace metric (2.11) has a pseudo-Euclidean signature (−, +, . . . , +) since the matrix (G ij ) has the pseudo-Euclidean signature [40].
Hence there exists a linear transformation
z a = e a A x A , (2.22)
diagonalizing the minisuperspace metric (2.11) where e a = (e a A ). Inverting the map (2.22) we get
G = η ac dz a ⊗ dz c = −dz 0 ⊗ dz 0 + N −1 k=1 dz k ⊗ dz k , (2.23) where (η ac ) = (η ac ) ≡ diag(−1, +1, . . . , +1),(2.x A = e A a z a ,(2.27)
where for components of the inverse matrix (e A a ) = (e a A ) −1 we obtain from
(2.26) e A a =Ḡ AB e c B η ca . (2.28)
Like in [31] we put
e 0 = q −1 U Λ , q = [−(U Λ , U Λ )] 1/2 = b 1/2 . (2.29) where U Λ (x) = U Λ A x A = γ 0 (φ)
is co-vector corresponding to the cosmological term, or, in components
(U Λ A ) = (d i , 0),(2.30)
and hence
z 0 = e 0 A x A = n i=1 q −1 d i x i . (2.31)
In z-coordinates (2.22) with z 0 from (2.31) the Lagrangian corresponding to (2.5) reads
L = L(z,ż, N ) = 1 2 N −1 η acż ażc − N V (z), (2.32) where V (z) = r∈S * A r exp(2u r a z a ) (2.33)
is a potential,
S * = {1, . . . , n} ∪ S (2.34)
is an extended index set and
A j = w 2 ξ j d j , A s = −wΛ s , (2.35) j = 1, . . . , n; s ∈ S. Here we denote u r a = e A a U r A = (U r , e c )η ca ,(2.u r 0 = −(U r , e 0 ) = ( n i=1 U r i )/q(D − 2), (2.37) r ∈ S * .
For the potential-term and curvature-term components we obtain from (2.29) and (2.37)
u s 0 = q > 0, u j 0 = 1/q > 0, (2.38) j = 1, . . . , n. We remind that (see (2.18)) (U j , U j ) = 1 d j − 1 < 0, (2.39)
for d j > 1, j = 1, . . . , n. For d j = 1 we have ξ j = A j = 0.
Billiard representation
Here we put the following restriction on parameters of the model:
− wΛ s > 0, (3.1) if (U s , U s ) = −b + λ 2 s > 0, (3.2) s ∈ S.
In what follows we denote by S + a subset of all s ∈ S satisfying (3.2). As we shall see below these restrictions are necessary for a formation of billiard "walls" (with positive infinite potential) in approaching to singularity. Due to relations (2.35), (2.38), (2.39) and (3.1) the parameters u r a in the potential (2.33) obey the following restrictions:
1. A r > 0 for (u r ) 2 = −(u r 0 ) 2 + ( u r ) 2 > 0; (3.3) 2. u r 0 > 0 for all r ∈ S * . (3.4)
Due to relations (3.3) and (3.4) the Lagrange system (2.32) for N ≥ 3 in the ("near the singularity") limit
z 0 → −∞, z 0 < −| z|, (3.5)
may be reduced to a motion of a point-like particle in N − 1-dimensional billiard belonging to Lobachevsky space [29,30,31,34]. For non-exceptional asymptotics (non-Milne-type) the limit (3.5) describes the approaching to the singularity. (in this case the volume scale factor vanishes exp( n i=1 d i x i ) = exp(qz 0 ) → +0). Indeed, introducing generalized Misner-Chitre coordinates in the lower light cone z 0 < −| z| [29,30]
z 0 = − exp(−y 0 ) 1 + y 2 1 − y 2 , (3.6) z = −2 exp(−y 0 ) y 1 − y 2 ,(3.
7)
| y| < 1, and fixing the time gauge
N = exp(−2y 0 ) = −z 2 . (3.8)
we get in the limit y 0 → −∞ (after separating y 0 variable) a "billiard" Lagrangian
L B = 1 2h ij ( y)ẏ iẏj − V ( y, B). (3.9) Hereh ij ( y) = 4δ ij (1 − y 2 ) −2 ,(3.N −1 = D N −1 ≡ { y : | y| < 1}. The "wall" potential V ( y, B) in (3.9) V ( y, B) ≡ 0, y ∈ B, +∞, y ∈ D N −1 \ B,(3.11)
corresponds to the open domain (billiard)
B = s∈S + B s ⊂ D N −1 ,(3.12)
where
B s = { y ∈ D N −1 || y − v s | > r s },(3.13)
and
v s = − u s /u s 0 , r s = ( v s ) 2 − 1, (3.14) (| v s | > 1) s ∈ S + .
The boundary of the billiard is formed by certain parts of m + = |S + | (N − 2)-dimensional spheres with centers in points v s and radii r s , s ∈ S + .
When S + = ∅ the Lagrangian (3.9) describes a motion of a particle of unit mass, moving in the (N −1)-dimensional billiard B ⊂ D N −1 (see (3.12)). The geodesic motion in B corresponds to a "Kasner epoch" while the reflection from the boundary corresponds to the change of Kasner epochs.
The billiard B has an infinite volume: volB = +∞ if and only if there are open zones at the infinite sphere | y| = 1. After a finite number of reflections from the boundary a particle moves towards one of these open zones. In this case for a corresponding cosmological model we get the "Kasner-like" behavior in the limit t → −∞ [34].
When volB < +∞ we get a never ending oscillating behaviour near the singularity.
In [31] the following simple geometric criterion for the finiteness of volume of B was proposed.
Proposition 1 [31]. The billiard B (3.12) has a finite volume if and only if point-like sources of light located at the points v s s ∈ S + (see (3.14)) illuminate the unit sphere S N −2 .
There exists a topological bound on a number of point-like sources m + illuminating the sphere S N −2 [41]:
m + ≥ N. (3.15)
Due to this restriction the number of exponential terms in potential obeying (3.2) m + = |S + | should at least exceed the value N = n + l for the existence of oscillating behaviour near the singularity.
Description in terms of Kasner-like parameters.
For zero potential V ϕ = 0 we get a Kasner-like solutions
g = wdτ ⊗ dτ + n i=1 A i τ 2α i g i , (3.16) ϕ β = α β ln τ + ϕ β 0 , (3.17) n i=1 d i α i = n i=1 d i (α i ) 2 + α β α γ h βγ = 1,(3.18)
where A i > 0 and ϕ β 0 are constants, i = 1, . . . , n; β, γ = 1, . . . , l. Let α = (α A ) = (α i , α γ ) obey the relations
U s (α) = U s A α A = n i=1 d i α i + λ asγ α γ > 0,(3.19)
for all s ∈ S + , then the field configuration (3.16)-(3.18) is the asymptotical (attractor) solution for a family of (exact) solutions, when τ → +0. Relations (3.19) may be easily understood using the following relations
Λ s exp[2λ s (ϕ) + 2γ 0 (φ)] = Λ s exp[2U s (x)] = C s τ 2U s (α) → 0, (3.20)
for τ → +0, where C s = 0 are constants, s ∈ S + . Other terms in the potential (2.6) are also vanishing near the singularity [29,30,31,34]. Thus, the potential (2.6) asymptotically tends to zero as τ → +0 and we are led to asymptotical solutions (3.16)- (3.18). Another way to get the conditions (3.19) is based on the isomorphism between S N −2 and the Kasner set (3.18) α A = e A a n a /q, (n a ) = (1, n), n ∈ S N −2 .
(3.21)
Here we use the diagonalizing matrix (e A a ) and the parameter q defined in the previous section (see (2.29)) [31,34]. Thus, we come to the following proposition.
Proposition 2. Billiard B (3.12) has a finite volume if and only if there are no α satisfying the relations (3.18) and (3.19).
So, we obtained a billiard representation for the model under consideration when the restrictions (3.1) are imposed.
Here we present also useful relations describing the billiard in terms of scalar products
v s v s ′ = u s u s ′ u s 0 u s ′ 0 = b −1 λ sα λ s ′ β h αβ , (3.22)
s, s ′ ∈ S + . They follow from the formulas u s u s ′ − u s 0 u s ′ 0 = (U s , U s ′ ) and (2.19).
Proposition 3. For n > 1 billiard B (3.12) has an infinite volume.
Proof. Due to Proposition 2 it is sufficient to present at least one set of Kasner parameters α = (α i , α γ ) obeying the relations (3.18) and (3.19). As an example of such set one may choose any Kasner set α (obeying (3.18) ) with α γ = 0, for example, with the following components
α 1 = d 1 ± √ R d 1 (d 1 + d 2 ) , α 2 = d 2 ∓ √ R d 2 (d 1 + d 2 )
, α i = 0 (i > 2). (3.23) where R = d 1 d 2 (d 1 + d 2 − 1). In this case inequalities (3.19) are satisfied, since U s (α) = 1 for all s. The proposition is proved. Thus, according to Proposition 3, for n > 1 we obviously have a "Kasnerlike" behavior near the singularity (as τ → 0). The oscillating behaviour near the singularity is impossible in this case.
Remark 1 (general "collision law"). The set of Kasner parameters (α ′ A ) after the collision with the s-th wall (s ∈ S + ) is defined by the Kasner set before the collision (α A ) according to the following formula [39]
α ′ A = α A − 2U s (α)U sA (U s , U s ) −1 1 − 2U s (α)(U s , U Λ )(U s , U s ) −1 ,(3.24)
where U sA =Ḡ AB U s B , U s (α) = U s A α A and co-vector U Λ is defined in (2.30). In the special case of one scalar field and 1-dimensional factor-spaces (i.e. l = d i = 1) this formula was suggested earlier in [35].
One factor-space
In this section we consider examples of l-dimensional billiards with finite volumes that occur in the model with l-scalar fields (l ≥ 2) and one scale factor (n = 1). Here we put h αβ = δ αβ and λ s = (λ s1 , . . . , λ sl ).
Thus, here we deal with the Lagrangian
L = R[g] − ∂ M ϕ∂ N ϕ − 2 s∈S Λ s exp[2 λ ϕ]. (4.1)
where ϕ = (ϕ 1 , . . . , ϕ l ).
In this (one-factor case) the following proposition takes place. Proof. According to relations (3.22) the set of vectors b −1/2 λ s ∈ IR l , s ∈ S + , is coinciding with the set v s ∈ IR l , s ∈ S + , up to O(l)-transformation, i.e. there exists orthogonal matrix A: A T A = 1, such that b −1/2 λ s = A v s ∈ IR l , s ∈ S + . Then the Proposition 4 follows from Proposition 1, since the sphere S l−1 is illuminated by sources v s , s ∈ S + , if and only if, it is illuminated by sources b −1/2 λ s , s ∈ S + . According to relations (3.16)-(3.18) we get the following asymptotical behavior for τ → 0
g as = wdτ ⊗ dτ + A 1 τ 2/(D−1) g 1 , (4.2) ϕ as = α ϕ ln τ + ϕ 0 , (4.3) ( α ϕ ) 2 = b −1 = (D − 2)/(D − 1). (4.4)
Here ϕ 0 and α ϕ change their values after the reflections from the billiard walls. Thus, here we obtained the oscillating behaviour of scalar fields near the singularity.
Remark 2 ("collision law"). From (3.24) we get the "collision law" relation in this case
α ′ ϕ = α ϕ − 2(1 + λ s α ϕ )(λ 2 s − b) −1 λ s 1 + 2(1 + λ s α ϕ )(λ 2 s − b) −1 b . (4.5)
The Kasner parameter for the scale factor is not changed after the "collision". l = 2 case. In the special case of two-component scalar field (l = 2), and m + = |S + | = 3 (i.e. when three "walls" appear) we find the necessary and sufficient condition for the finiteness of the billiard volume in terms of scalar products of the coupling vectors λ s ∈ IR 2 , s ∈ S + .
Proposition 5 . For n = 1, l = 2, h αβ = δ αβ and m + = |S + | = 3 the billiard B (3.12) has a finite volume if and only if the vectors λ s ∈ IR 2 , s ∈ S + , obey the following relations:
b −1 λ s λ s ′ ≥ 1 − b −1 λ 2 s − 1 b −1 λ 2 s ′ − 1, s < s ′ , (4.6) s<s ′ arccos λ s λ s ′ | λ s || λ s ′ | = 2π. (4.7)
Proof. According to Proposition 3 we should find the necessary and sufficient conditions for three points located in v s = b −1/2 λ s , s = 1, 2, 3, to illuminate the unit circle S 1 . Here we put S + = {1, 2, 3} for simplicity. It may be obtained from a simple geometrical consideration that such conditions may be chosen as the following ones:
θ ss ′ ≤ arccos 1 | v s | + arccos 1 | v s ′ | , s < s ′ ,(4.8)
and θ 12 + θ 23 + θ 13 = 2π (4.9)
where
θ ss ′ = arccos v s v s ′ | v s || v s ′ | (4.10)
is the angle between vectors v s and v s ′ s, s ′ = 1, 2, 3. Relation (4.8) means that the angle between two vectors v s and v s ′ should not exceed one half of the sum of two arcs on S 1 illuminated by source of light located in points v s and v s ′ (see Fig. 1). Relations (4.6) may be obtained from (4.8) by acting on both sides of the inequality by function cos. Relation (4.7) is just equivalent to (4.8). Relation (4.9) exclude the situation when points v s , s = 1, 2, 3, belong to a half-plane with a border-line containing the center of the unit circle. The proposition is proved.
An example of (sub-)compact triangle billiard with a finite area in the Lobachevsky space H 2 is depicted on Fig. 1. In the symmetric case when all λ 2 s = 4b and λ s λ s ′ = −2b for s = s ′ we get an example of non-(sub)-compact billiard with finite area. Such billiard appears in the well-known Bianchi-IX model, see Fig. 2.
For "quasi-Cartan" matrix defined as A ss ′ = 2(U s , U s ′ )/(U s ′ , U s ′ ) we get
(A ss ′ ) = 2 −2 −2 −2 2 −2 −2 −2 2 . (4.11)
This matrix coincides with the Cartan matrix of the hyperbolic of Kac-Moody algebra that is number 7 in classification of [42]. This Kac-Moody algebra is a subalgebra of A ∧∧ 1 [43] (see also [37,38]).
Discussions
In this paper we have considered the behavior near the singularity of the multidimensional cosmological-type model with n Einstein factor-spaces in the theory with scalar fields and MEP (multiple exponential potential). Using the results from [29,30,31,34] we have obtained the billiard representation on Here we have shown that for n > 1 the oscillating behavior near the singularity is absent, i.e. solutions have an asymptotical Kasner-like behavior.
For one-factor case we have described (in terms of illumination problem) the billiards with finite volume and hence with the oscillating behavior of scalar fields near the singularity.
In the model with two scalar fields and three potential walls we have found the necessary and sufficient conditions (in terms of dilatonic coupling vectors) for triangle billiards to be of finite volume. In (A.1)
T M N = h αβ ∂ M ϕ α ∂ N ϕ β − 1 2 g M N ∂ P ϕ α ∂ P ϕ β − V ϕ g M N . (A.3)
24) and here and in what follows a, c = 0, . . . , N − 1; N = n + l. The matrix of linear transformation (e a A ) satisfies the relation η ac e a A e c B =Ḡ AB (2.25) or, equivalently, η ac = e a AḠ AB e c B = (e a , e c ), (2.26)
10) i, j = 1, . . . , N − 1, are components of the canonical metric on the (N − 1)dimensional Lobachevsky space H
Proposition 4 .
4For n = 1 and h αβ = δ αβ the billiard B (3.12) has a finite volume if and only if point-like sources of light located at the points b −1/2 λ s ∈ IR l , s ∈ S + , (b = (D − 1)/(D − 2)) illuminate the unit sphere S l−1 .
Figure 1 .
1Triangle sub-compact billiard with finite volume for n = 1, l = 2 and m + = 3 .
Figure 2 .
2Triangle billiard coinciding with that of Bianchi-IX model. multidimensional Lobachevsky space for this cosmological-type model near the singularity.
AcknowlegmentsThis work was supported in part by the Russian Ministry of Science and Technology, Russian Foundation for Basic Research (RFFI-01-02-17312a)and DFG Project (436 RUS 113/678/0-1(R)).Authors thank colleagues from the Department of Physics, University of Konstanz, for the hospitality during their visits to Konstanz in August-December, 2003.Appendix A Equations of motionHere we outline for the sake of completeness the equations of motions corresponding to the action (1.1)
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|
[] |
[
"Deep Autoencoder-based Fuzzy C-Means for Topic Detection",
"Deep Autoencoder-based Fuzzy C-Means for Topic Detection"
] |
[
"Hendri Murfi [email protected] \nDepartment of Mathematics\nDepartment of Mathematics\nUniversitas Indonesia\n16424DepokIndonesia\n",
"Natasha Rosaline [email protected] \nDepartment of Mathematics\nUniversitas Indonesia\n16424DepokIndonesia\n",
"Nora Hariadi [email protected] \nUniversitas Indonesia\n16424DepokIndonesia\n"
] |
[
"Department of Mathematics\nDepartment of Mathematics\nUniversitas Indonesia\n16424DepokIndonesia",
"Department of Mathematics\nUniversitas Indonesia\n16424DepokIndonesia",
"Universitas Indonesia\n16424DepokIndonesia"
] |
[] |
Topic detection is a process for determining topics from a collection of textual data. One of the topic detection methods is a clustering-based method, which assumes that the centroids are topics. The clustering method has the advantage that it can process data with negative representations. Therefore, the clustering method allows a combination with a broader representation learning method. In this paper, we adopt deep learning for topic detection by using a deep autoencoder and fuzzy c-means called deep autoencoder-based fuzzy c-means (DFCM). The encoder of the autoencoder performs a lower-dimensional representation learning. Fuzzy c-means groups the lower-dimensional representation to identify the centroids. The autoencoder's decoder transforms back the centroids into the original representation to be interpreted as the topics. Our simulation shows that DFCM improves the coherence score of eigenspace-based fuzzy c-means (EFCM) and is comparable to the leading standard methods, i.e., nonnegative matrix factorization (NMF) or latent Dirichlet allocation (LDA).
|
10.1016/j.array.2021.100124
|
[
"https://arxiv.org/pdf/2102.02636v1.pdf"
] | 231,802,051 |
2102.02636
|
2a617014f0f260e59c7ea1e4ba9f8a6f8f7a8da0
|
Deep Autoencoder-based Fuzzy C-Means for Topic Detection
Hendri Murfi [email protected]
Department of Mathematics
Department of Mathematics
Universitas Indonesia
16424DepokIndonesia
Natasha Rosaline [email protected]
Department of Mathematics
Universitas Indonesia
16424DepokIndonesia
Nora Hariadi [email protected]
Universitas Indonesia
16424DepokIndonesia
Deep Autoencoder-based Fuzzy C-Means for Topic Detection
Topic detectionclusteringdeep learningautoencoderfuzzy c-means
Topic detection is a process for determining topics from a collection of textual data. One of the topic detection methods is a clustering-based method, which assumes that the centroids are topics. The clustering method has the advantage that it can process data with negative representations. Therefore, the clustering method allows a combination with a broader representation learning method. In this paper, we adopt deep learning for topic detection by using a deep autoencoder and fuzzy c-means called deep autoencoder-based fuzzy c-means (DFCM). The encoder of the autoencoder performs a lower-dimensional representation learning. Fuzzy c-means groups the lower-dimensional representation to identify the centroids. The autoencoder's decoder transforms back the centroids into the original representation to be interpreted as the topics. Our simulation shows that DFCM improves the coherence score of eigenspace-based fuzzy c-means (EFCM) and is comparable to the leading standard methods, i.e., nonnegative matrix factorization (NMF) or latent Dirichlet allocation (LDA).
Introduction
Topic detection is a process used to analyze words in a collection of textual data to determine the topics in the collection, how they relate to one another, and how they change over time. The topics usually are represented by a set of words. The coherence of the words usually measures the topic's interpretability. The standard topic detection methods are nonnegative matrix factorization (NMF) (Lee & Seung, 1999), clustering (Allan, 2002), and latent Dirichlet allocation (LDA) (Blei et al., 2003).
In the clustering method, the cluster centers or centroids are interpreted as a topic. In other words, the clustering method will group the textual data based on their topic similarity. Unlike the other two methods, the clustering method can process data with negative representations. Therefore, the clustering method allows a combination with broader representation learning or dimension reduction methods. Nur' aini et al. combines k-means and latent semantic analysis (LSA) for topic detection (Nur'aini, Najahaty, Hidayati, Murfi, & Nurrohmah, 2015). Firstly, the textual data are transformed into a lower-dimensional Eigenspace using the singular value decomposition (SVD). Next, k-means is performed on the Eigenspace to extract the topics that are then transformed back to the nonnegative subspace of the original space.
The k-means method splits the textual data into k clusters in which each textual data belongs to the nearest centroid. It means that the k-means method assumes that each textual data contains only one topic. This assumption is relatively weak and also different from the standard NMF and LDA, considering that the textual data may have many topics. Therefore, soft clustering is examined to be an alternative clustering method for topic detection. Fuzzy cmeans (FCM) is one of the famous soft clustering methods (Bezdek, Ehrlich, & Full, 1984). Using FCM, the textual data may belong to more than one cluster and may have more than one topic. The combination of FCM and LSA called Eigenspace-based fuzzy c-means (EFCM) is proposed for topic detection (Muliawati & Murfi, 2017). In general, some simulations show that EFCM gives the coherence scores between the ones of LDA and NMF (Murfi, 2018(Murfi, , 2019Praditya Nugraha, Rifky Yusdiansyah, & Murfi, 2019).
Currently, deep learning is the primary machine learning method for unstructured data such as images and text (Goodfellow, Bengio, & Courville, 2016;Zhang, Lipton, Li, & Smola, 2020). Deep learning has been extensively studied to extract a good representation of data by neural networks (Bengio, Courville, & Vincent, 2013). In this paper, we adopt deep learning to improve the performance of EFCM for topic prediction problems by using deep autoencoder (DAE) for the representation learning process. We call this topic detection method as deep autoencoder-based fuzzy c-means (DFCM). First, the encoder of DAE performs a lowerdimensional representation learning. Next, FCM groups the lower-dimensional representation to identify the centroids. Finally, the decoder of DAE transforms back the centroids into the original representation to provide the topics. Our simulation shows that DFCM improves the coherence score of EFCM and is comparable to the leading standard methods, i.e., NMF or LDA.
This paper's outline is as follows: In Section 2 and Section 3, we describe the related works and the methods, i.e., FCM, DAE, and DFCM. Section 4 describes the results and the discussion of our simulations. Finally, a general conclusion about the results is presented in Section 5.
Related Works
Topic detection methods are algorithms for discovering the topics or the themes from an unstructured collection of documents. Some recent publications show the gwowing use of topic detection for researchers in Library and Information Science to find the theme that they are interested in and then examine the documents related to that theme (Battsengel, Geetha, & Jeon, 2020;Lamba & Madhusudhan, 2019;Parlina, Ramli, & Murfi, 2020).
The standard topic detection methods are nonnegative matrix factorization (Cichocki & Phan, 2009;Févotte & Idier, 2011;Lee & Seung, 1999), clustering (Allan, 2002;Petkos, Papadopoulos, & Kompatsiaris, 2014) and latent Dirichlet allocation (Blei, 2012;Blei et al., 2003;Hoffman, Blei, & Bach, 2010;Hoffman, Blei, Wang, & Paisley, 2013). Unlike the other two methods, clustering is a general method of grouping data. Furthermore, this method can also process positive and negative data representation. Thus, the clustering method is more flexible to be combined with representation learning or dimension reduction.
Fuzzy clustering is one of the most widely used clustering methods because it has soft and flexible in grouping data to the cluster (Ruspini, Bezdek, & Keller, 2019). Bezdek developed FCM by extending the fuzzifier value m to m > 1 (Bezdek et al., 1984). This extension makes FCM a generalization from k-means, which is hard clustering. FCM is more suitable for the topic detection method because it allows adaptation to a document's condition with one or more topics, namely by finding the optimal fuzzifier value m.
In the era of big data, the existence of high-dimensional data is a big challenge for FCM (Winkler, Klawonn, & Kruse, 2011). By finding a new representation of the original data, two approaches are already used to reduce the difficulty of FCM in high-dimensional data. The first approach uses kernel methods to implicitly get more expressive features by formulating the data into the feature space constructed by some kernel functions (Huang, Chuang, & Chen, 2012;Shang, Zhang, Li, Jiao, & Stolkin, 2019). The second approach is an explicit transformation of the original data. In addition to the specified nonlinear data transformations (Zhu, Pedrycz, & Li, 2017), random projection (Rathore, Bezdek, Erfani, Rajasegarar, & Palaniswami, 2018) is commonly used to obtain low-dimensional data. The second approach is more suitable because the excellent design of kernel space for clustering is complicated. The ability of kernel methods to handle large-scale data is also a concern. In several empirical studies, the combination of FCM with the data transformation approach (Murfi, 2018;P. Nugraha, Rifky Yusdiansyah, & Murfi, 2019) provides better performance than the random projection approach (Yusdiansyah, Murfi, & Wibowo, 2019) for topic detection problems.
Currently, deep learning is the primary machine learning method for unstructured data such as images and text (Goodfellow et al., 2016;Zhang et al., 2020). Deep learning has been extensively studied to extract a good representation of data by neural networks (Bengio et al., 2013). The combination of the deep neural network and an unsupervised clustering method also becomes an active research field (Song, Huang, Liu, Wang, & Wang, 2014). In general, there are some approaches to incorporate deep learning. In general, there are several approaches to combining deep learning and clustering. The first approach is to combine representation learning and clustering in two steps, namely using a deep autoencoder for representation learning and then a clustering method for the next stage (Song et al., 2014;Song, Liu, Huang, Wang, & Tan, 2013). The second approach is to combine a deep autoencoder and a clustering method simultaneously (Guo, Gao, Liu, & Yin, 2017;Xie, Girshick, & Farhadi, 2016). The next approach is to combine clustering with a pretrained encoder, such as Bidirectional Encoder Representations from Transformers (BERT) proposed by Google (Guan et al.,5555). However, most of the clustering methods used in these approaches are hard clustering. Few studies work on the improvement of feature quality by deep learning for the fuzzy clustering. This study is to find a good deep representation for the fuzzy clustering, i.e. fuzzy c-means. In this research, we use the first approach and the second approach in combining representation learning and clustering. In this approach, representation learning and clustering are carried out separately, not simultaneously as in the second approach. This approach still requires the decoder part to transform the data back into original representation. In addition, our approach does not use a pretrained model because it is still difficult to determine the most important words to represent the resulting topics.
Methods
Let A be a word by document matrix and c be the number of topics. Given A and c, the topic detection problem is how to recover c topics from A. In the clustering-based topic detection method, the clustering centers or centroids are interpreted as topics. In this section, we describe deep autoencoder-based fuzzy c-means (DFCM) for topic detection. First, we review the core methods, i.e., fuzzy c-means (FCM) and deep autoencoder (DAE).
Fuzzy C-Means
Given a dataset in the form of a word by document matrix = [ 1 2 … ] and the number of centroids c, the goal of fuzzy c-means (FCM) can be formulated as the following constrained optimization:
min , = ∑ ∑ =1 =1 ‖ − ‖ 2 (1) . . ∑ = 1, ∀ =1 0 < ∑ < , ∀ =1 ∈ [0,1], ∀ , where
are centroids, is the membership of data point ak in cluster i, > 1 is the fuzzification constant, and ‖ . ‖ is any norm. The first constraint ensures that every data point has total membership in all clusters where each membership is in [0,1]. The second constraint guarantees that all clusters are nonempty (Bezdek et al., 1984).
The problem of the constrained optimization in Equation 1 is to find and that minimize the objective function J. The standard method to solve the constrained optimization is alternating optimization. First, we choose some initial values for the . Then we minimize J concerning the , keeping the fixed giving:
= �∑ � ‖ − ‖ � − � � 2 −1 � =1 � −1 , ∀ ,(2)
Next, we minimize J on the , keeping the fixed giving:
= ∑ �( ) � =1 ∑ ( ) =1 , ∀(3)
This two-step optimization are iterated until a stopping criterion is fulfilled, e.g., the maximum number of iteration, insignificant changes in the objective function J, the membership , or the centroids (Bezdek & Hathaway, 2003). The FCM algorithm is described in more detail in Algorithm 1.
According to Equation 2, the memberships tend to 0 or 1 when the fuzzification constant f approaches to 1. The bigger the fuzzification constant makes, the fuzzier the memberships . Therefore, the setting of the fuzzification constant is quite intuitive. The small fuzzification constant means that each textual data may contain a small number of topics. On the other hand, the bigger the fuzzification constant implies that each textual data may have more topics.
Algorithm 1. FCM Input : , , f, max iteration ( ), threshold ( ) Output : 1. set t = 0 2. initialize 3. update t = t + 1 4. calculate = �∑ � ‖ − ‖ 2 � − � 2 � 2 −1 � =1 � −1 , ∀ , 5. calculate = ∑ �( ) � =1 ∑ ( ) =1
, ∀ 6. if a stopping, i.e., > or ‖ − −1 ‖ < , is fulfilled then stop, else go back to step 3
Deep Autoencoder
Deep autoencoder (DAE) is a deep neural network for unsupervised learning problems. This unsupervised problem is solved using a supervised learning approach, where target labels are constructed from input features. This deep autoencoder architecture has the same output layer as the input layer, and the standard supervised learning can be applied.
The architecture of DAE for representation learning can be explained into three parts, i.e., encoder, code, and decode ( Figure 1). The encoder part consists of fully connected layers used to transform data input to the code part, a new data representation. The decoder part is used to transform the new data representation back to the original representation. The decoder part consists of fully connected layers having a symmetric structure with the encoder part. The reason is if an encoder requires a certain complexity (number of layers (depth), the number of neurons in each layer (units)) to represent data to new representation, then a decoder with the same complexity is needed to transform the new data representation back to the original data representation. For dimension reduction problem, the number of neurons in the code part is set less than the number of neurons in the input layer (Hinton & Salakhutdinov, 2006). DAE can be built layer by layer using greedy layer-wise pretraining, where each layer is built by a denoising autoencoder (Vincent et al., 2010). The denoising autoencoder is an autoencoder that reconstructs the input from a corrupted version to force the hidden layer to discover a more stable and robust representation.
Given textual dataset = { 1 , 2 , … , } with ∈ ℛ , ∀ = 1,2, … , , the denoising autoencoder consists of two layers as follows:
�~ 1 ( ) (4) = 1 (� , )(5)�~ 2 ( ) (6) = 2 � � , 2 �(7)
where 1 () and 2 () are methods to ignore some number of neuron outputs during training randomly, 1 () and 2 () are activation function, 1 and 1 are weights. The fitting is performed to minimize the loss ℒ( , ), i.e., errors between xi and yi. Next, hi becomes a new representation for the input data of the next layer. After training on each denoising autoencoder, the denoising autoencoder's weights become the corresponding weights of the autoencoder. Furthermore, the autoencoder is retrained to minimize a reconstruction loss for all layers. This DAE algorithm is described in more detail in Algorithm 2.
Deep Autoencoder-based Fuzzy C-Means
Deep autoencoder-based Fuzzy C-Means (DFCM) is a proposed topic detection method that combines DAE for representation learning and FCM for fuzzy clustering. FCM works well for low dimensional textual data and generates only one topic for high dimensional textual data. We can set the fuzzification constant of FCM with a small value to push FCM to produce more than one topic. However, this small fuzzification constant assumes that each textual data contains few topics and only one topic when the fuzzification constant approaches to one. Therefore, we use DAE to transform the data to lower-dimensional representation and keep the fuzzification constant adaptable for textual data with multi-topics. Figure 2 provides a general process of DFCM. Given a textual dataset = { 1 , 2 , … , } with ∈ ℛ , ∀ = 1,2, … , , the dimension of new data representation p, and the number of topic c. First, the textual data are transformed into a lower-dimensional representation using an encoder. We denote this transformation as follows:
� = ( , )(8)
where � ∈ ℛ � , ∀ = 1,2, … , . Next, we perform FCM on the dataset � with the lowerdimensional representation. In this step, centroids � ∈ ℛ � , ∀ = 1,2, … , are extracted from all c-given clusters as follows:
� = ( � , , , , )(9)
These centroids � are interpreted as the topics in the lower-dimensional representation. However, the topics have no meaning and will be meaningful if they are transformed back to the original representation. Therefore, it is necessary to transform the extracted topics back to the original representation as follows:
= max (0, (� ))(10)
where ∈ ℛ , ∀ = 1,2, … , , and max() is a function that gives a maximum between 0 and each element of (� ). This DFCM algorithm is described in more detail in Algorithm 3.
Results and Discussion
To examine the performance of DFCM, we apply the method to extract topics on two datasets, i.e., English email and Indonesian news. To prepare the textual data for the topic detection methods, we executed two main processes, i.e., cleaning and vectorizing. Firstly, we converted all words into lowercase, erase words containing domains such as www.* or https://*, and words containing @username, and erased # in #words. To standardize words with non-standard spelling, we replaced two or more repeating letters with only two occurrences. Stopwords and words with low-frequency terms occurring in fewer than t of the total m documents were also excluded, where the t threshold was set to max(10; m/1000). Finally, we use the term frequencyinverse document frequency for weighting. Given a tweet collection, the topic detection methods produce topics represented by its top 10 most frequent words. The standard quantitative method to measure the interpretability of the topics is topic coherence. In our simulations, we use one of the topic coherence measures called TC-W2V (O'Callaghan, Greene, Carthy, & Cunningham, 2015). Suppose a topic t consists of n words that are {t1,t2,…,tn}, the TC-W2V of the topic t is
− 2 ( ) = 1 � 2 � ∑ ∑ similarity ( , ) −1 =1 =2(11)
where wvj and wvi are vectors of word tj and ti constructed by a word2vec model.
The simulations are conducted in a Windows-based Python environment. For EFCM and DFCM parameters, we set the fuzzification constant f = 1.1, the maximum number of iteration T = 1000, and the threshold ε = 0.005. As mentioned before, the setting of the fuzzification constant is quite intuitive. The small fuzzification constant means that each tweet may contain a small number of topics. On the other hand, the bigger the fuzzification constant implies that each textual data may contain more topics. We initialize the centroids of FCM for both EFCM and DFCM using the best of the 10-run k-means clustering. In DFCM, DAE architecture consists of three symmetrical layers with each layer consisting of 500, 500, 2000 neurons. We implement this representation learning using Python-based Keras 1 . On the other hand, we use truncatedSVD implementation of scikit-learn for dimension reduction in EFCM (Pedregosa et al., 2011). Finally, we need to tune the rest parameters, i.e., the lower dimension p and the number of topics c, for EFCM and DFCM.
This simulation also compares DFCM with two standard topic detection methods: latent Dirichlet allocation (LDA) and nonnegative matrix factorization (NMF). We use the LDA and NMF implementations provided by scikit-learn (Pedregosa et al., 2011). The LDA algorithm uses the batch variational Bayes method for training LDA. Two parameters are usually optimized for this training method, namely α and η. α control the mixture of topics for a specific document. A smaller α means the document will likely have less of a mixture of topics. η control the distribution of words per topic. The larger η means the topic will likely have more words.
To optimize both parameters, we use a hyperparameter grid and run an algorithm for each combination [0.01, 0.1, 0.25, 0.5, 0.75, 1].
For NMF, the data vectors are normalized to unit length. The implementation of NMF uses a coordinate descent algorithm. There is no parameter we optimize for this algorithm. To reduce the instability of random initialization, the NNDSVD initialization is performed.
Enron -An English Email Dataset
The first dataset is Enron consisting of approximately 500,000 emails generated by employees of the Enron Corporation 2 . The Federal Energy Regulatory Commission obtained it during its investigation of Enron's collapse. To calculate the TC-W2V of the extracted topics, we use a pre-trained word2vec model trained on the Google News dataset for the English email dataset. The model contains 300-dimensional vectors for 3 million words and phrases 3 .
First, we analyze the effect of the number of DAE training epochs on the coherence scores of DFCM. Using a batch size of 256, the coherence scores for several epoch sizes are given in Figure 3. First, the number of epochs is set to 100. Then, this number of epochs is increased to 400 and 1000. The average coherence score of DFCM fluctuates and increases when the number of epochs is increased from 100 to 400. However, the average coherence score tends to decrease when the number of epochs is increased to 700. If we choose 400 as the number of epochs, then DFCM gives the mean coherence scores of 0.1771, 22% better than EFCM. Figure 4 provides simulation results similar to Figure 3, but for 10-dimensional representation. Compared to the five-dimensional representation, the mean coherence scores of DFCM fluctuate only slightly as the number of epochs is increased from 100, 400, and 700. DFCM gives the mean coherence scores of 0.1896 when the number of epochs is 400. Like the 5-dimensional data representation, DFCM still provides a better mean coherence score than EFCM, which is about 5% better. Figure 4 show that the 10-dimensional representation of data is more suitable for both the DFCM and EFCM methods. DFCM with the ten-dimensional representation gives the mean coherence scores 7% better than DFCM with the five-dimensional representation. Meanwhile, EFCM with the ten-dimensional representation provides an average coherence score of 26% better than EFCM with the five-dimensional representation. Furthermore, we also provide a comparison between the DFCM method with two other standard methods, namely NMF and LDA. Figure 5 includes coherence scores of 4 topic detection methods for the number of topics 10, 20, ..., 100. First, Figure 5 confirms the previous simulation results that EFCM provides coherence scores between NMF and LDA. From Figure 5, we can also see that DFCM can reach a coherence score that is slightly better than NMF in almost all number of topics. Only on the topic number of 10, NMF gives a significantly better coherence score. For all number of topics, DFCM provides better mean coherence scores of 3%, 7%, 34% than NMF, EFCM, LDA, respectively.
Figures 3 and
Berita -An Indonesian News Dataset
The second dataset is Berita, consisting of 50304 digital Indonesia news articles shared online through Twitter by nine Indonesian news portals widely known in Indonesia. These are Antara (antaranews.com), Detik (detik.com), Inilah (inilah.com), Kompas (kompas.com), Okezone (okezone.com), Republika (republika.co.id), Rakyat Merdeka (rmol.co), Tempo (tempo.co) and Viva (viva.co.id). The news articles contain published dates, titles, and some first sentences of contents. We construct the word2vec model using a corpus consisting of 750000 Indonesian documents from wiki, news, and tweets to measure the TC-W2V of the extracted topics. Unlike the word2vec model for the first English dataset, we train the Berita dataset to this word2vec model. Therefore, all vocabularies of the Berita dataset exist in the word2vec model.
The simulations for the Berita dataset are given in Figure 6, Figure 7, and Figure 8. Figure 6 is a simulation to see the effect of the number of epochs in the DAE learning to coherence scores of DFCM for the five-dimensional representation. Meanwhile, Figure 7 is a simulation to see the effect of the number of epochs in the DAE learning to coherence scores of DFCM for the 10-dimensional representation. The initial number of epochs is 50 and is increased to 100 and then 400. In general, increasing the number of epochs makes the coherence score lower for most topics. For the epoch number of 400, DFCM even provides a coherence score below the EFCM in almost all number of topics. The same conditions are seen for the 10-dimensional representation in Figure 7. If we use 50 epochs for DAE learning, then DFCM gives an average coherence score of 0.3730 for the five-dimensional representation. Meanwhile, EFCM provides an average coherence score of 0.3589. This means that DFCM achieves a slightly higher average coherence score than EFCM, which is about 4% better. A similar result is shown for the 10-dimensional representation where the DFCM gives an average coherence score of about 3%, slightly higher than the EFCM. From Figure 6 and Figure 7, we can also conclude that the five-dimensional representation provides a slightly better coherence score for both DFCM and EFCM. Figure 8 provides a comparison of DFCM on the Berita dataset with two other standard topic detection methods, namely NMF and LDA. In this comparison, both DFCM and EFCM use the five-dimensional representation. Figure 8 shows that DFCM, EFCM, NMF, and LDA provide mean coherence scores of 0.3730, 0.3588, 0.3560, and 0.2815, respectively. These results indicate that DFCM can better achieve an average coherence score than EFCM, NMF, and LDA. However, NMF still gives a better coherence score for the smallest number of topics. 10. In this Berita dataset, NMF and EFCM provide almost the same means coherence score.
Discussion
In the previous sub-chapter, the simulations show that DFCM achieves better mean coherence scores than EFCM. For the Enron dataset, DFCM provides mean coherence scores that are 7% better than the EFCM and 4% more than the EFCM for the News dataset. The main difference between these two methods is the lower-dimensional representation learning process where DFCM uses DAE, while EFCM uses truncatedSVD. DAE and truncatedSVD produce different lower-dimensional representations. TruncatedSVD creates lower-dimensional representation with orthogonal dimensions or features. Meanwhile, DAE produces lowerdimensional representations with dimensions or features that are not orthogonal. Topics generally consist of words that are not necessarily orthogonal, especially in the meaning of the words. Also, DAE implements denoising processes implicitly to produce these lowerdimensional representations. Thus, each of these lower-dimensional characteristics will more or less affect the resulting mean coherence scores.
DFCM also provides a higher average coherence score compared to NMF. DFCM achieved a 3% better average coherence score for the Enron dataset and a 5% better average coherence score for the News dataset. In contrast to EFCM, DFCM and NMF provide lowerdimensional representation with non-orthogonal dimensions or features. NMF carried out a topic extraction process in the original space, which consisted of words. Thus, the resulting topics can be directly interpreted, and their coherence scores are calculated. Meanwhile, DFCM extracts the topics in the lower-dimensional space and must be transformed back to the original space so that the extracted topics can be interpreted, and coherence scores are calculated. However, DFCM makes it possible to process a better representation for textual data that generally has a lot of noise and variation. Thus, the success of DFCM in achieving better coherence scores is mainly because DFCM processes textual data with better representations. The same condition for LDA, where LDA performs the topic extraction process in the original space, consists of words.
Deep learning is currently a popular supervised learning approach, especially for unstructured data such as images and text. Deep learning integrates the feature extraction process with classification or regression processes. In the context of unsupervised learning, deep autoencoder is a popular deep learning method for representation learning. This method makes it possible to do the denoising process while reducing dimensions to produce better lowerdimensional representation. However, the integration of representation learning methods with an unsupervised learning problem such as topic detection is still an opportunity to continue to be developed.
Conclusions
DFCM is a topic detection method that combines deep autoencoder for representation learning and fuzzy c-means for topic extraction. Therefore, deep autoencoder-fuzzy c-means makes it possible to process a better representation for textual data that generally has a lot of noise and variation. Unlike EFCM, DFCM extracts topics from lower-dimensional representations with dimensions or features that are not orthogonal. This representation is more realistic to represent the topics. Our simulation shows that DFCM gives a higher accuracy in terms of the coherence score than EFCM and the two standard methods, i.e., NMF and LDA.
Figure 1 .
1Deep Autoencoders
Algorithm 2 .Figure 2 .
22DAE Input : , the size of code p Output: encoder(w), decoder(w) 1. Initialize autoencoder(p) 2. (0) = 3. Let m be the number of layers of the autoencoder 4. FOR i = 1 TO m 5. Fitting the denoising autoencoder for the i-th layer: deAutoencoder( ( ) ), ( ) = min ℒ� ( −1) , � , Initialize weight of autoencoder with the corresponding weight of the denoising autoencoder: autoencoder( ( ) ), ∀ 8. Fitting the autoencoder: autoencoder( ), = min ℒ( , ) , ∀ Deep Autoencoders-based Fuzzy C-Means
Algorithm 3 .
3DFCM Input : , the size of code p, the number of topics , max number of iterations T, threshold Output: 1. Build autoencoder: encoder, decoder = DAE(X, p) 2. Transform X : � = ( ) 3. Perform FCM : � = � � , , , �,
Figure 3 .Figure 4 .
34Coherence scores in terms of TC-W2V for the Enron dataset on the number of topics 10, 20, …, 100 when the lower-dimensional representation is set to five. DFCM(100), DFCM(400), DFCM(700) mean the number of epoch of deep autoencoders are set to 100, 400, 700, respectively Coherence scores in terms of TC-W2V for the Enron dataset on the number of topics 10, 20, …, 100 when the lower-dimensional representation is set to ten. DFCM(100), DFCM(400), DFCM(700) mean the number of epoch of deep autoencoders are set to 100, 400, 700, respectively
Figure 5 .
5The comparison of coherence scores in terms of TC-W2V for LDA, NMF, EFCM, and DFCM on the number of topics 10, 20, …, 100 for the Enron dataset.
Figure 6 .Figure 7 .
67Coherence scores in terms of TC-W2V for the Berita dataset on the number of topics 10, 20, …, 100 when the lower-dimensional representation is set to five. DFCM(50), DFCM(100), DFCM(400) mean the number of epoch of deep autoencoders are set to 50, 100, 400, respectively Coherence scores in terms of TC-W2V for the Berita dataset on the number of topics 10, 20, …, 100 when the lower-dimensional representation is set to ten. DFCM(50), DFCM(100), DFCM(400) mean the number of epoch of DAE are set to 50, 100, 400, respectively
Figure 8 .
8The comparison of coherence scores in terms of TC-W2V for LDA, NMF, EFCM, and DFCM on the number of topics 10, 20, …, 100 for the Berita dataset.
https://keras.io 2 https://www.cs.cmu.edu/~./enron/ 3 https://code.google.com/archive/p/word2vec/
AcknowledgementThis paper was supported by Universitas Indonesia under PDUPT 2020 grant. Any opinions, findings, conclusion, and recommendations are the authors' and do not necessarily reflect those of the sponsor.
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|
[] |
[
"First experimental demonstration of a scalable linear majority gate based on spin waves",
"First experimental demonstration of a scalable linear majority gate based on spin waves"
] |
[
"Florin Ciubotaru [email protected] ",
"Giacomo Talmelli \nKU Leuven\nLeuvenBelgium\n",
"Thibaut Devolder \nCentre de Nanosciences et de Nanotechnologies\nUniv. Paris-Sud\nOrsayFrance\n",
"Odysseas Zografos ",
"Marc Heyns \nKU Leuven\nLeuvenBelgium\n",
"Christoph Adelmann ",
"Iuliana P Radu ",
"\n1 imecLeuvenBelgium\n"
] |
[
"KU Leuven\nLeuvenBelgium",
"Centre de Nanosciences et de Nanotechnologies\nUniv. Paris-Sud\nOrsayFrance",
"KU Leuven\nLeuvenBelgium",
"1 imecLeuvenBelgium"
] |
[] |
We report on the first experimental demonstration of majority logic operation using spin waves in a scaled device with an in-line input and output layout. The device operation is based on the interference of spin waves generated and detected by inductive antennas in an all-electrical microwave circuit. We demonstrate the full truth table of a majority logic function with the ability to distinguish between strong and weak majority, as well as an inverted majority function by adjusting the operation frequency. Circuit performance projections predict low energy consumption of spin wave based compared to CMOS for large arithmetic circuits.
|
10.1109/iedm.2018.8614488
|
[
"https://arxiv.org/pdf/2106.11192v1.pdf"
] | 58,675,103 |
2106.11192
|
31311e34c9f84bcff3c70774f604e646f4d36f92
|
First experimental demonstration of a scalable linear majority gate based on spin waves
Florin Ciubotaru [email protected]
Giacomo Talmelli
KU Leuven
LeuvenBelgium
Thibaut Devolder
Centre de Nanosciences et de Nanotechnologies
Univ. Paris-Sud
OrsayFrance
Odysseas Zografos
Marc Heyns
KU Leuven
LeuvenBelgium
Christoph Adelmann
Iuliana P Radu
1 imecLeuvenBelgium
First experimental demonstration of a scalable linear majority gate based on spin waves
We report on the first experimental demonstration of majority logic operation using spin waves in a scaled device with an in-line input and output layout. The device operation is based on the interference of spin waves generated and detected by inductive antennas in an all-electrical microwave circuit. We demonstrate the full truth table of a majority logic function with the ability to distinguish between strong and weak majority, as well as an inverted majority function by adjusting the operation frequency. Circuit performance projections predict low energy consumption of spin wave based compared to CMOS for large arithmetic circuits.
I. INTRODUCTION
Spintronic devices based on spin waves are promising alternatives to CMOS technology with high potential for power and area reduction per computing throughput [1,2]. The information can be encoded in either the amplitude or the phase of spin waves, while logic operation is based on their interference. Different spin-wave-based logic systems have been proposed, e.g. Mach Zender interferometers [3], magnonic transistors [4], or spin wave majority gates (SWMGs) [1]. The SWMG is the most promising concept as it possesses a higher expressive power than e.g. NAND or NOR gates and thus may reduce circuit complexity. The basic functionality of a such device has been proven at a mm scale using YIG films [5]. Device scalability to µm and nm dimensions has been predicted by micromagnetic simulations [6,7]. So far, the SWMG devices have been based on a "trident" shape. However, such a shape is difficult to scale due to increasing spin wave reflection at bends with µm and nm dimensions [6]. Moreover, such a shape is challenging to print at the nm scale using conventional lithography. In this work, we demonstrate a majority gate using spin wave interference in µm-sized ferromagnetic waveguides using a sequential "in-line" layout of input and output inductive antennas.
II. FABRICATION AND RF PROPERTIES
The device consisted of a magnetic stripe that served as a waveguide for the spin waves. Three inductive antennas were used to excite spin waves and an addition antenna was used to detect the resulting wave after interference, i.e. after computation (see Fig. 1).
For the waveguide, a Ta(3nm)/CoFeB(30nm)/Ta(3nm) film stack was sputtered onto 300 nm of SiO2 on Si (100). The stack was then patterned into stripes with a width of 4 µm using ionbeam etching. The ferromagnetic CoFeB acted as the waveguide for the spin waves while the Ta layers served as seed and cap layers to prevent from oxidation of the magnetic film. The waveguide was finally covered by 40 nm of SiNx for electric isolation (Fig. 2(a)). Subsequently, Ti/Au inductive antennas with a width of 500 nm were fabricated by electronbeam lithography and lift-off ( Fig. 2(b)). The antennas were connected to microwave coplanar waveguides and contacted by RF picoprobes (Fig. 2 (c)). A vector network analyzer (VNA) was used for both the excitation and characterization of the spin waves.
Spin waves were generated in the waveguide by the Oersted field generated by the RF currents flowing in the U-shaped input antennas and detected by a single-wire output antenna. This design provided a weak electromagnetic parasitic coupling between adjacent input antennas (see Figs. 3(a) and (b)), as well as between the input and output antennas, as indicated both by experiments and electromagnetic simulations [8]. Electromagnetic simulations of the Oersted field distribution (Fig. 3(c)) show that the U-shaped antennas can efficiently excite spin waves with wavelengths down to 700 nm ( Fig. 3(d)).
III. SPIN WAVE PROPERTIES AND INTERFERENCE
Spin wave transmission experiments were performed with the CoFeB waveguide magnetized by a magnetic field transverse to its long axis. In a first step, the propagation characteristics of spin waves emitted from each of the three input antennas towards the output antenna were determined. A schematic of the experimental configuration is shown in Fig. 4(a); Fig. 4(b) shows a typical transmitted signal from input I1 towards output O, corresponding to a spin wave propagation distance of 4.8 µm. The minimum spin wave frequency is given by the ferromagnetic resonance, whereas the maximum frequency depends on maximum wavenumber that an antenna can excite. The full frequency-field dependence of the transmitted signal from input I1 to output O is shown in Fig. 4(c). The device allowed for the generation and propagation of spin waves in a wide frequency range between 3 GHz and 22 GHz, depending on the external applied field. The dispersion relation calculated using parameters extracted from experiments (see Fig.4 (d)) demonstrates that the minimum spin wave frequency matches well the ferromagnetic resonance frequency, while the upper limit was set by the maximum wavevector (kmax ~ 8.9 rad/µm) that can be excited by the antenna (corresponding to a wavelength of = 700 nm). The spin wave transmission from inputs I2 and I3 to the output O is shown in Fig. 5. The dephasing due to the different propagation distances could clearly be observed.
Subsequent experiments studied the interference of the spin waves generated simultaneously by multiple input antennas (see Fig. 6(a)). Microwave currents with the same frequency were applied to all three antenna inputs. The output signal was studied as a function of the input frequency, bias field, and relative phase difference between the input signals. For a given set of field-frequency parameters, an oscillatory signal was detected by varying the phase of the input signals corresponding to the constructive or destructive interference of the three generated spin waves. For example, Fig. 6(b) shows the detected signal for a phase rotation of up to 4π of the signal at input I1, while I2 and I3 were kept in phase. The position of the maxima/minima could be tuned by varying the applied frequency, which changes the spin wave wavelength and thus modifies the interference pattern.
IV. LOGIC FUNCTIONS
Building logic functions based on spin wave interference requires the control of both the amplitude and the phase of the spin waves generated by each input. Signal matching at the output was obtained by phase shifters and attenuators in the microwave circuits of each input antenna (see Fig. 7(a)). Figure 7(b) shows that the amplitude and phase of the input signals could be synchronized over a 1 GHz bandwidth for a bias field of 40 mT.
Input 0 and 1 logic states of a SWMG were defined as the phase of the spin wave signals, i.e. as phases of 0 and π, respectively. The variation of the phase-sensitive S-parameter (here the imaginary part) measured by the VNA was used to define the logic output signal. Positive and negative variations correspond to output wave phases of π and 0, respectively (logic 1 and 0, respectively). Using the phase of one input as a control signal, for example setting the phase of I3 to π (Fig. 8(a)), and changing the phase of I1 and I2 between 0 and π, a logic OR function could be demonstrated, as shown in Fig. 8(b). In addition, a logic AND gate could be demonstrated by setting the phase of the control input I3 to 0.
By individually controlling the phase of each input, the truth table of the logic majority operation was demonstrated over a frequency bandwidth of ~300MHz, as shown in Fig. 9. Weak and strong majority states could be distinguished. The clear separation between the states suggests that adding additional inputs will allow to create an n-state logic. By tuning the applied frequency on the inputs, an oscillatory output signal was observed ( Fig. 10(a)). This fact is explained by the variation of the global phase due to the dependence of spin wave wavelength on frequency, leading to a change of the interference pattern at the output for different frequencies.
Thus, an inverted majority gate can be obtained in the same device by tuning the operation frequency ( Fig. 10(b)).
Benchmarking
Spin wave logic concepts, and more specifically spin wave devices (SWDs) [1] have been benchmarked several times [9,10]. All results show that using efficient voltage-driven spin wave generation and detection, SWDs can outperform state-ofthe-art CMOS technology in terms of energy consumption. To showcase this, based on previous benchmarking work [11], we adapt the energy calculations of the 10 designs described in Fig. 11. These benchmarks are combinational and represent a common subset of arithmetic designs used in digital integrated circuits. The energy consumption per operation of SWDs is compared in Fig. 12 to the 10nm CMOS technology node [12]. The energy consumption per operation of the SWD circuits is on average 7.6x times lower than for 10 nm CMOS. This benchmarking highlights the potential for ultralow-power logic built based on SWDs.
Weak & strong majority use cases
As shown in Fig. 9, the device allows for the distinction of strong and weak majority signals. This can be efficiently exploited by non-boolean or multilevel computational techniques. More specifically, it has been shown that strong/weak majority distinction can be applied to signal processing, such as pattern recognition [13]. Moreover, the above capability can be useful in applications where it is important to implement threshold functions (such as for neurons), where the thresholding sensitivity is more expressive than a binary component.
V. CONCLUSION
We have demonstrated a novel in-line spin wave majority gate concept that is both scalable and compatible with conventional CMOS patterning techniques. By individually controlling the phase and amplitude of signal at the three inputs, a full majority truth table was demonstrated. Due to the wave based nature of the operation, the output signal dependent on the applied frequency in an oscillatory way. This could be exploited to demonstrate an inverted majority function in the same device by adjusting the operation frequency. Circuit level benchmarking indicated the high potential of spin wave based devices for low-power electronics. We observe that for sufficiently large circuits, wave logic operate at lower energy than state-of-the-art CMOS.
Fig. 1 .
1Schematic of a spin wave in-line majority gate Fig. 2. (a) Schematic of device cross-section under an antenna. (b) SEM image of the active area of the device. The inputs are 3 U-shaped antennas and the output is a single wire antenna. (c) SEM image of the full microwave majority gate device with a sketch of the picoprobe connection to the transducers.
Fig. 3 .
3Scattering parameters of the device simulated by HFSS: (a) reflection coefficient for U-shape antennas, and (b) direct parasitic coupling between every input and the output antenna. (c) Magnetic field components generated by a U-shaped antenna simulated by HFSS and (d) the resulting bandwidth of the device.
Fig. 4 .
4(a) Schematic of the experiment for a single input and (b) the detected signal due to spin wave propagation at the output. (c) Frequency-field dependence of the spin wave transmission. Light blue color corresponds to zero spin-wave transmission, while the dark blue and the white band stands for propagating spin waves. (d) Spin wave dispersion relation calculated for three values of the magnetic field.
Fig. 5 .
5Spin wave transmission from inputs I2 and I3 to the output O.
Fig. 6 .
6(a) Sketch of a spin wave in-line majority gate showing phase control at input I1. (b) Output signal generated by the interference of the three spin waves. The change of the amplitude is determined by constructive and destructive interference. The bias field was set to 50 mT.
Fig. 7 .
7(a) Schematic of a spin wave in-line majority gate with phase control at each transducer. (b) Output signal generated by each input showing the possibility to match phase and amplitude of the three input spin waves, as required to build logic gates.
Fig. 8 .
8(a) Schematic of an OR gate where one input (I3) is fixed at a phase of π as a control gate. (b) Experimental signal demonstrating the functionality as an OR gate,Fig. 9. (a) Majority Gate truth table, indicating cases of strong and weak majority (b) Spin wave transmission due to interference of the 3 waves at the output tranducer. The 8 cases can be observed as well as a clear separation between strong and weak majority gate in a 200 MHz frequency bandwidth.
Fig. 10 .
10(a) Spin wave transmission due to interference of 3 waves at output in a broad frequency span for the two cases of strong majority. Due to the phase rotation it can be observed how a MAJ state transforms in a MIN (INV + MAJ) and then again in MAJ. (b) Experimental MIN truth table where strong and weak minority can be separated in a 200 MHz frequency range.
Fig. 11 .
11Descriptions of benchmark designs used to compare spin wave logic and CMOS technologies.
Fig. 12 .
12Energy per operation of spin wave logic circuits and CMOS (10nm) as technology reference. Benchmarks are ordered in increasing circuit size.
ACKNOWLEDGMENTThis work was performed as part of the imec IIAP program on Core CMOS and Beyond CMOS. Support from the H2020 project CHIRON (contract No. 801055) is gratefully acknowledged.
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|
[] |
[
"Optical detection of spin transport in non-magnetic metals",
"Optical detection of spin transport in non-magnetic metals"
] |
[
"F Fohr ",
"S Kaltenborn ",
"J Hamrle ",
"H Schultheiß ",
"A A Serga ",
"H C Schneider ",
"B Hillebrands ",
"Y Fukuma ",
"L Wang ",
"RIKENY Otani Asi ",
"Wako Hirosawa ",
"351-0198 ",
"\nFachbereich Physik and Forschungszentrum OPTIMAS\nTechnische Universität Kaiserslautern\nD-67663KaiserslauternGermany\n",
"\nUniversity of Tokyo\n5-15-5 Kashiwanoha277-8581KashiwaJapan\n"
] |
[
"Fachbereich Physik and Forschungszentrum OPTIMAS\nTechnische Universität Kaiserslautern\nD-67663KaiserslauternGermany",
"University of Tokyo\n5-15-5 Kashiwanoha277-8581KashiwaJapan"
] |
[] |
We determine the dynamic magnetization induced in non-magnetic metal wedges composed of silver, copper and platinum by means of Brillouin light scattering (BLS) microscopy. The magnetization is transferred from a ferromagnetic Ni80Fe20 layer to the metal wedge via the spin pumping effect. The spin pumping efficiency can be controlled by adding an insulating interlayer between the magnetic and non-magnetic layer. By comparing the experimental results to a dynamical macroscopic spin-transport model we determine the transverse relaxation time of the pumped spin current which is much smaller than the longitudinal relaxation time.
|
10.1103/physrevlett.106.226601
|
[
"https://arxiv.org/pdf/1011.4656v2.pdf"
] | 18,623,791 |
1011.4656
|
0e62555a44f04263a73e32f2b4db0d084f85e385
|
Optical detection of spin transport in non-magnetic metals
27 Apr 2011
F Fohr
S Kaltenborn
J Hamrle
H Schultheiß
A A Serga
H C Schneider
B Hillebrands
Y Fukuma
L Wang
RIKENY Otani Asi
Wako Hirosawa
351-0198
Fachbereich Physik and Forschungszentrum OPTIMAS
Technische Universität Kaiserslautern
D-67663KaiserslauternGermany
University of Tokyo
5-15-5 Kashiwanoha277-8581KashiwaJapan
Optical detection of spin transport in non-magnetic metals
27 Apr 2011(Dated: April 28, 2011)arXiv:1011.4656v2 [cond-mat.other]
We determine the dynamic magnetization induced in non-magnetic metal wedges composed of silver, copper and platinum by means of Brillouin light scattering (BLS) microscopy. The magnetization is transferred from a ferromagnetic Ni80Fe20 layer to the metal wedge via the spin pumping effect. The spin pumping efficiency can be controlled by adding an insulating interlayer between the magnetic and non-magnetic layer. By comparing the experimental results to a dynamical macroscopic spin-transport model we determine the transverse relaxation time of the pumped spin current which is much smaller than the longitudinal relaxation time.
Spin current injection from a magnetic to a nonmagnetic material is an important and central issue of magneto-electronics [1,2]. There are several ways to realize such an injection. Spin currents can be generated by spin polarized charge currents [3], the spin Hall effect [4], or spin pumping [5,6]. The spin accumulation in the non-magnet can be detected indirectly by an increased damping in the injection layer [6,7] or it can be probed by the conversion of spin current into voltage in a lateral spin valve [8,9] or via the inverse spin Hall effect [10,11].
In this letter we demonstrate that the dynamic magnetization, which is induced via spin pumping into a nonmagnetic material, can be observed directly by means of Brillouin light scattering (BLS) microscopy. We detect the spin polarization in metal wedges of Cu, Ag and Pt grown on top of a ferromagnetic layer (see Fig. 1). Light from a laser is focussed on the surface of the wedge and the inelastically scattered light is collected as a function of the local wedge thickness. This light originates from the non-magnetic layer due to inelastic scattering from the spin polarization as well as from the magnetic layer below the wedge as long as the optical path length through the non-magnetic layer is smaller, or at least comparable to the optical absorption length. The magnetic layer is excited externally by the RF field of a coplanar waveguide (CPW) at the ferromagnetic resonance frequency, and generates the spin polarization in the wedge layer via the spin pumping effect. In the nonmagnetic metals Cu, Ag and Pt, the magnetic interaction between the spin-polarized free electrons is rather small, so that the macroscopic spin polarization, which is induced into the non-magnetic layer due to the spin pumping process, cannot be described in terms of collective eigen-excitations but reflects the magnetization dynamics in the ferromagnetic layer. The BLS signal arises from the forced oscillating macroscopic spin polarization so that frequency and wavevector are determined by the magnons in the ferromagnetic film. Here we show that in the absence of a macroscopic magnetic ordering, interface impurities and roughness lead to a fast dephasing of the initially phase-aligned spins in the non-magnetic layer.
The CPW is prepared by means of maskless laser photolithography on an oxidized silicon substrate. It consists of a 300 nm gold layer with a signal line (S) of 20 µm width separated from the ground planes (G) by a 10 µm wide gap. A microwave current is applied to the CPW and generates an oscillating magnetic field in y-direction using a coordinate system as defined in Fig. 1. To reach high microwave power in a wide frequency band and to prevent reflections, the microwave current is terminated by a load at the end of the CPW with impedance matching.
On the CPW signal line a multilayer structure is deposited by electron beam evaporation. The multilayer has a width of 2 µm, a length of 5 mm and consists of: (i) a 7 nm thick MgO layer that prevents the microwave current from flowing into the metal wedge, because this would create a complicated current distribution and an unpredictable magnetic field disturbing the CPW magnetic field; (ii) a 30 nm thick Ni 80 Fe 20 layer that is excited externally by the CPW dynamic magnetic field and serves as the pumping layer for the attached metal wedge; (iii) an optional second 7 nm thick MgO interlayer to block spin pumping from the Ni 80 Fe 20 layer into the metal wedge; (iv) a metal wedge composed of either silver, copper or platinum. The optional MgO interlayer (iii) is used in a reference sample to separate the different contributions to the BLS intensity originating from the magnetic and the non-magnetic layer, respectively. In the sample without the MgO interlayer, spin pumping into the metal is expected to occur, whereas in the reference sample the pumping mechanism is blocked by the MgO layer. The latter is insulating but optically transparent, and therefore does not affect the detection by the probing laser light. For absolute height calibration of the metal wedge, the scan position in x-direction is calculated into a total thickness. The topography was scanned in y-direction with a mechanical profilometer for different points along the wedge, and for each of these profiles the thickness of the multilayer was extracted with the CPW level as reference level. Figure 1(d) shows the BLS spectrum taken on a pure Ni 80 Fe 20 film at an applied field of 20 mT in x-direction. The first and most pronounced maximum is visible at a frequency of 5.5 GHz but several other maxima develop at higher frequencies corresponding to higher laterally standing spin wave modes across the stripe [12,13].
A spatially uniform precession cannot be excited in a 2 µm wide stripe due to pinning effects at the boundaries. Standing spin waves build up across the width of the stripe in y-direction and the spin pumping efficiency becomes dependent on this coordinate ( Fig. 1(e)). The dynamic magnetization in the non-magnetic layer experiences additional dephasing due to the mixing of components pumped with different initial phases from neighbouring antinodes of higher-order standing waves.
To minimize this contribution to the dephasing, only the first standing spin wave mode, excited at a microwave frequency of 5.5 GHz and an external field of 20 mT was used in our BLS measurements. In Fig. 2 the measured BLS intensities of the maximum of the first mode are shown for different scan positions in x-direction. With increasing wedge thickness the BLS signal decays exponentially over a range of almost four decades in intensity. In the silver (Fig. 2(a)) and the copper wedge (Fig. 2(b)) the slopes of the exponential decay are different for the main (black dots) and the reference sample (red dots) whereas in the platinum sample both slopes are the same within the error bars (Fig. 2(c)).
The origin of the difference in silver and copper is the additional contribution to the BLS intensity due to the spin polarization pumped from the underlying Ni 80 Fe 20 layer. While the BLS signal of the main sample is determined by the optical decay from the signal originating in Ni 80 Fe 20 as well as by the induced magnetization in copper, the signal from the sample with the MgO interlayer, which prevents spin pumping, contains only the signal originating from the Ni 80 Fe 20 layer. In platinum this effect is not observable because the injected spin angular momentum is immediately transferred from the spin system to the lattice due to the high spin orbit interaction.
The total BLS intensity depends on the thickness of the metal wedge and consists of two contributions: One is due to the precessing magnetization in the ferromagnet, the other originates from the pumped spin polarization in the metal. The total BLS intensity is proportional to:
|E F + E N | 2 = |E F | 2 + 2 Re (E F E * N ) + |E N | 2(1)
where E F and E N are the electric fields of the probe laser light scattered inelastically in the magnetic and the nonmagnetic layer, respectively. Inside the metal wedge of thickness d, i.e. for 0 < z < d, the profile of the electric field originating from the incident light can be expressed as a damped wave (see inset of Fig. 2).
E F = E F,0 exp (2iñkd)(2)
Hereñ = n + i κ is the complex refractive index and k = ω/c is the vacuum wavevector of light. The factor 2 in the exponent takes into account that the BLS setup is prepared in backscattering geometry, and thus the light is passing through the structure twice.
The amplitude of the backscattered light from the nonmagnetic layer is a sum of contributions originating from different depths of the wedge, weighted by the decaying probe light amplitude as well as by the decaying contribution of the spin polarization to the scattered light:
E N = d 0 E N,0 exp [2iñk (d − z)] exp − z l 2 dz (3)
Here l 2 is the characteristic decay length of the dynamic spin polarization that gives rise to the BLS signal. In Fig. 2 the fit curves (black and red lines) and the extracted BLS intensity for the pure spin part (blue line), obtained by using Eqs. With knowledge of l 2 , a characteristic relaxation time T 2 can be calculated using wave-diffusion equations for the macroscopic spin density ρ s (z, t) = ρ ↑ (z, t) − ρ ↓ (z, t) and the spin-current density J s (z, t) = J ↑ (z, t) − J ↓ (z, t):
∂ ρ s (z, t) ∂t = −γ ρ s (z, t)× B− ∂ J s (z, t) ∂z − ∂ ρ s (z, t) ∂t int ,(4)
and
J s (z, t) = −D ∂ ρ s (z, t) ∂z −τ e γ J s (z, t)× B−τ e ∂ J s (z, t) ∂t .(5)
Here, τ e denotes the momentum relaxation time, γ is the absolute value of the electron (g ≈ 2) gyromagnetic ratio, B is the magnetic field and D is the diffusion constant. As a generalization of Ref. [14], we include different longitudinal (or spin-lattice) relaxation times T 1 and transverse (or spin-spin) relaxation times T 2 in the interaction contribution in Eq. (4): (∂ρ s (x) /∂t)| int = ρ s (x) /T 1 and (∂ρ s (y, z) /∂t)| int = ρ s (y, z) /T 2 .
The dynamical components of ρ s (z, t) in Eq. (4) decay with T 2 and the static component decays with T 1 . The latter is not accessible to Brillouin light scattering and is therefore neglected in the following.
It can be shown [14] that the relaxation time T 2 depends on the corresponding decay length l 2 via:
l 2 = v F √ 3 τ e T 2(6)
where v F is the Fermi velocity.
To obtain the transverse relaxation time, we solve Eqs. to match the experimentally determined value for l 2 (see Tab. I). According to Ref. [16] we use τ e (Cu) = 25 fs and τ e (Ag) = 40 fs for the momentum relaxation time and v F (Cu) = 1.57 nm/fs and v F (Ag) = 1.39 nm/fs for the Fermi velocity at room temperature. The influence of the small external magnetic field of 20 mT and the injection frequency of 5.5 GHz on the decay length is negligible in the calculations. This result is also confirmed by our BLS measurements of the copper sample: The decay is unchanged within the error bars at an injection frequency of 9.3 GHz and an applied field of 70 mT. Note that the accepted value of the longitudinal relaxation time T 1 , which is of the order of a few picoseconds [17,18], exceeds the value of T 2 determined by our BLS measurements by three orders of magnitude. This is a remarkable result because T 2 is usually considered equivalent to T 1 [19]. The magnitude of the discrepancy between T 1 and T 2 suggests an extrinsic effect acting differently on the transverse and the longitudinal component of the induced magnetization.
In order to understand this discrepancy we propose a relaxation mechanism, which is based on absorption of the transverse spin current due to magnetic impurities at the interface and dephasing of the transverse spin current in inhomogeneous magnetostatic fields arising from the interface roughness. Indeed, magnetic impurities at the interface were detected by means of secondary ion mass spectroscopy (SIMS). The magnetization of the paramagnetic impurities is aligned along the direction of the external magnetic field, so that the transverse spin current is effectively absorbed by exerting a maximum torque on the magnetization of the impurities. In case of spin current transmission from a nonmagnet into a ferromagnet [20], this effect is even more pronounced due to the influence of three processes: (i) spin-dependent reflection and transmission, (ii) rotation of reflected and transmitted spins, and (iii) spatial precession of spins in the ferromagnet. In our case, the spin current is transmitted from the ferromagnetic to the non-magnetic layer, so that the interface effect (ii) as well as effect (iii), which is based on the different kinetic energies of spin-up and spin-down electrons due to exchange splitting, can be neglected in our considerations. Therefore, only effect (i) contributes to relaxation and can be understood by expressing the transverse spin current as a linear combination of spin-up and spin-down components, which have different reflection and transmission amplitudes when they scatter on the paramagnetic impurities. As a result, the transverse spin current has a non-zero propagation length in contrast to the immediately absorbed spin current in [20].
The analysis of the Ni 80 Fe 20 layer topography via atomic force microscopy (AFM) reveals that Ni 80 Fe 20 clusters with a grain size of several tens of nanometers and an average Ni 80 Fe 20 surface roughness of 1.9 nm. This surface roughness is transferred to the Ni 80 Fe 20 /nonmagnet interface and causes spatially inhomogeneous magnetic fields, so that the spins, which precess initially with the same frequency and phase, experience additional dephasing due to different precession frequencies in these local magnetic fields [21].
In conclusion we have determined for the first time the existence of magnetization in non-magnetic metals by optical means. The transverse spin current is directly accessible to Brillouin light scattering microscopy and decays faster than the longitudinal spin current due to absorption and dephasing at the interface.
We acknowledge the financial support by the Deutsche Forschungsgemeinschaft and the Japan Science and Technology Agency. We also thank W. Bock from the IFOS institute for the SIMS measurements.
FIG. 1 .
1(Color online) (a) Scheme of the sample layout and (b) SEM picture of the waveguide. A multilayer structure is prepared on top of a coplanar waveguide. A static magnetic field µ0Hstatic of 20 mT is applied parallel to the signal line and perpendicular to the dynamic magnetic field hRF, which is caused by an alternating microwave current flowing through the coplanar waveguide. The magnetization in the Ni80Fe20 layer is excited by hRF and spins are pumped into the metal wedge. (c) MOKE hysteresis loop to determine the saturation field in x-direction. (d) BLS spectrum taken on a pure Ni80Fe20 film at a static magnetic field of 20 mT. (e) BLS scans across the structure for the first two maxima of (d) at microwave frequencies of 5.5 GHz and 7.8 GHz. The profiles correspond to the first and the third laterally standing spin wave mode. The second mode is not excited.
(1)-(3), are shown in addition to the measurement data. The fitting parameters in the simulation are the ratio of the field strengths E N,0 and E F,0 at the Ni 80 Fe 20 /Cu interface, the complex refractive indexñ in the metal and the decay length l 2 . The results of this simulation as well as the parameters of the fits inFig. 2are summarized in Tab. I. The contribution of the induced magnetization |E N | 2 to the BLS signal is at maximum for wedge thicknesses below 10 nm, even if the BLS signal originates from the Ni 80 Fe 20 layer, i.e. |E F | 2 is dominant. Above wedge thicknesses of 20 nm in silver (45 nm in copper) |E N | 2 becomes dominating over |E F | 2 .
FIG. 2 .
2(4) and(5) numerically by using T 2 as a fit parameter (Color online) BLS scan data of the silver (a), the copper (b) and the platinum (c) wedged sample. Each graph shows the measurement of the main sample with active spin pumping (black dots) and the respective reference sample with blocked spin pumping (red dots). A difference between main and reference sample is only visible for silver and copper but not for platinum (see text). The error bars reflect the uncertainty in thickness determination by the mechanical profilometer. The black and red lines are fits of the scan data according to Eq. (1). The evolution of the pure spin part of the BLS signal (blue line) is derived from the fitting parameters. The inset in (a) shows schematically that the optical absorption length zopt as well as the decay length of the transverse component l2 contribute to the total BLS intensity probed by the laser.
TABLE I .
IFitting parameters of the BLS data in Fig. 2 and
the resulting transverse relaxation time T2 according to the
macroscopic spin wave-equation.
Normal
metalññ lit in [15] l2 (nm) EF,0/EN,0 T2 (fs)
Ag
0.13+i3.4 0.13+i3.2 9 ±1
16
3 ±1
Cu
1.07+i3.3 1.07+i2.6 10 ±1 26
5 ±1
Pt
2.08+i5.2 2.08+i3.6 0 ±2
> 20
0 ±1
† current address: Materials Science Division. 70833Ostrava, Czech Republic; Argonne, Illinois 60439, USA* current address: Department of Physics, Ostrava University of Technology ; Argonne National Laboratory* current address: Department of Physics, Ostrava Univer- sity of Technology, 708 33 Ostrava, Czech Republic. † current address: Materials Science Division, Argonne Na- tional Laboratory, Argonne, Illinois 60439, USA.
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|
[] |
[
"Diagnosis of Supersymmetry Breaking Mediation Schemes by Mass Reconstruction at the LHC",
"Diagnosis of Supersymmetry Breaking Mediation Schemes by Mass Reconstruction at the LHC"
] |
[
"Bhaskar Dutta \nDepartment of Physics & Astronomy\nMitchell Institute for Fundamental Physics\nTexas A&M University\n77843-4242College StationTXUSA\n",
"Teruki Kamon \nDepartment of Physics & Astronomy\nMitchell Institute for Fundamental Physics\nTexas A&M University\n77843-4242College StationTXUSA\n\nDepartment of Physics\nKyungpook National University\n702-701DaeguSouth Korea\n",
"Abram Krislock \nDepartment of Physics & Astronomy\nMitchell Institute for Fundamental Physics\nTexas A&M University\n77843-4242College StationTXUSA\n\nDepartment of Physics\nAlbaNova\nStockholm University\nSE-106 91StockholmSweden\n",
"Kuver Sinha \nDepartment of Physics & Astronomy\nMitchell Institute for Fundamental Physics\nTexas A&M University\n77843-4242College StationTXUSA\n",
"Kechen Wang \nDepartment of Physics & Astronomy\nMitchell Institute for Fundamental Physics\nTexas A&M University\n77843-4242College StationTXUSA\n"
] |
[
"Department of Physics & Astronomy\nMitchell Institute for Fundamental Physics\nTexas A&M University\n77843-4242College StationTXUSA",
"Department of Physics & Astronomy\nMitchell Institute for Fundamental Physics\nTexas A&M University\n77843-4242College StationTXUSA",
"Department of Physics\nKyungpook National University\n702-701DaeguSouth Korea",
"Department of Physics & Astronomy\nMitchell Institute for Fundamental Physics\nTexas A&M University\n77843-4242College StationTXUSA",
"Department of Physics\nAlbaNova\nStockholm University\nSE-106 91StockholmSweden",
"Department of Physics & Astronomy\nMitchell Institute for Fundamental Physics\nTexas A&M University\n77843-4242College StationTXUSA",
"Department of Physics & Astronomy\nMitchell Institute for Fundamental Physics\nTexas A&M University\n77843-4242College StationTXUSA"
] |
[] |
If supersymmetry is discovered at the LHC, the next question will be the determination of the underlying model. While this may be challenging or even intractable, a more optimistic question is whether we can understand the main contours of any particular paradigm of the mediation of supersymmetry breaking. The determination of superpartner masses through endpoint measurements of kinematic observables arising from cascade decays is a powerful diagnostic tool. In particular, the determination of the gaugino sector has the potential to discriminate between certain mediation schemes (not all schemes, and not between different UV realizations of a given scheme). We reconstruct gaugino masses, choosing a model where anomaly contributions to supersymmetry breaking are important (KKLT compactification), and find the gaugino unification scale. Moreover, reconstruction of other superpartner masses allows us to solve for the parameters defining the UV model. The analysis is performed in the stop and stau coannihilation regions where the lightest neutralinos are mainly gauginos, to additionally satisfy dark matter constraints. We thus develop observables to determine stau and stop masses to verify that the coannihilation mechanism is indeed operational, and solve for the relic density.
|
10.1103/physrevd.85.115007
|
[
"https://arxiv.org/pdf/1112.3966v2.pdf"
] | 118,656,038 |
1112.3966
|
ca4a33252019a1f074eff95b8e83544a44bbbb16
|
Diagnosis of Supersymmetry Breaking Mediation Schemes by Mass Reconstruction at the LHC
Bhaskar Dutta
Department of Physics & Astronomy
Mitchell Institute for Fundamental Physics
Texas A&M University
77843-4242College StationTXUSA
Teruki Kamon
Department of Physics & Astronomy
Mitchell Institute for Fundamental Physics
Texas A&M University
77843-4242College StationTXUSA
Department of Physics
Kyungpook National University
702-701DaeguSouth Korea
Abram Krislock
Department of Physics & Astronomy
Mitchell Institute for Fundamental Physics
Texas A&M University
77843-4242College StationTXUSA
Department of Physics
AlbaNova
Stockholm University
SE-106 91StockholmSweden
Kuver Sinha
Department of Physics & Astronomy
Mitchell Institute for Fundamental Physics
Texas A&M University
77843-4242College StationTXUSA
Kechen Wang
Department of Physics & Astronomy
Mitchell Institute for Fundamental Physics
Texas A&M University
77843-4242College StationTXUSA
Diagnosis of Supersymmetry Breaking Mediation Schemes by Mass Reconstruction at the LHC
If supersymmetry is discovered at the LHC, the next question will be the determination of the underlying model. While this may be challenging or even intractable, a more optimistic question is whether we can understand the main contours of any particular paradigm of the mediation of supersymmetry breaking. The determination of superpartner masses through endpoint measurements of kinematic observables arising from cascade decays is a powerful diagnostic tool. In particular, the determination of the gaugino sector has the potential to discriminate between certain mediation schemes (not all schemes, and not between different UV realizations of a given scheme). We reconstruct gaugino masses, choosing a model where anomaly contributions to supersymmetry breaking are important (KKLT compactification), and find the gaugino unification scale. Moreover, reconstruction of other superpartner masses allows us to solve for the parameters defining the UV model. The analysis is performed in the stop and stau coannihilation regions where the lightest neutralinos are mainly gauginos, to additionally satisfy dark matter constraints. We thus develop observables to determine stau and stop masses to verify that the coannihilation mechanism is indeed operational, and solve for the relic density.
I. INTRODUCTION
The Large Hadron Collider (LHC) will soon be testing many ideas for physics beyond the Standard Model (SM). Of these, low energy supersymmetry [1] is the best motivated candidate for new TeV scale physics.
If supersymmetric partners are indeed detected at the LHC, the next step would be understanding the underlying scheme by which supersymmetry breaking is mediated to the visible sector. Of course, the full determination of an exact model of supersymmetry breaking and mediation may be challenging, but one can hope that the broad contours of the scheme can be understood.
How soon (if at all) will we be able to distinguish the underlying mechanism by which supersymmetry breaking is mediated to the visible sector?
It is useful to first understand the above question from the perspective of distinguishability studies undertaken from a top-down point of view [2]. Broadly, the program is to fit a set of UV parameters defining a sufficiently tractable model of supersymmetry breaking and mediation with observables measured in the experiment, and ask questions about distinguishability in the space of UV model parameters. Some models one can consider in such distinguishability tests are (i) mSUGRA or CMSSM [3] (defined at the GUT scale by universal scalar and gaugino masses and trilinear couplings m 0 , m 1/2 , and A 0 , as well as tanβ and ratio of Higgs vevs), (ii) the minimal anomaly mediated supersymmetry model [4] (defined by m 0 , tanβ, and universal superpartner mass scale m aux ), (iii) minimal gauge mediation (defined by the messenger scale M mess , supersymmetry breaking scale Λ and tanβ), as well as some examples inspired by string phenomenol-ogy: (iv) Large Volume Scenarios [5] (defined by m 0 and tanβ), and (v) mirage mediation models [6] in the context of KKLT [7].
However, the important point is that a given set of measured observables will in general not only point back to different UV models, it may point to different mediation schemes. It may be more optimistic to ask whether we can at least distinguish between more gross properties, such as mediation schemes.
It is important, then, to first discern whether a particular scheme of supersymmetry breaking mediation (such as, say, anomaly contributions) shows up unmistakably in the low energy spectrum, before locking into a particular UV model and undertaking a distinguishability study in the UV model parameter space or attempting to constrain it.
Which masses should one attempt to solve to this end?
One can look for relatively model-independent soft mass patterns that hold out clues for mediation schemes. As emphasised in [8], the gaugino sector offers a particularly promising portal for understanding the underlying mechanism of mediation. This is mainly because within the context of the Minimal Supersymmetric extension of the Standard Model (MSSM), the quantity M a /g 2 a does not run at one loop (where a = 3, 2, 1 refers to the SM gauge groups SU (3) × SU (2) × U (1), while M a refer to gaugino masses and g a to gauge couplings). If one makes two further assumptions: (i) that gaugino masses are dominated by contributions determined by tree level gauge kinetic functions and (ii) these tree level contributions are universal, then one obtains the well known mSUGRA pattern, with gaugino unification at the GUT scale mSUGRA : M 1 : M 2 : M 3 ∼ 1 : 2 : 6 .
(1)
This pattern is obtained in a variety of schemes: (i) gravity mediation, with different UV scenarios (ii) gauge mediation (iii) gaugino mediation [9]. We will elaborate further in the next Section. In a previous study [10], we have described a series of measurements of superpartner masses at the LHC for mSUGRA models using end-point techniques [11].
On the other hand, gaugino masses may be dominated entirely by one loop contributions coming from the SUGRA compensator, determined by the conformal anomaly of the effective theory at TeV scale. At some level, the flavor problem embedded in dominant tree level contributions is an indication that tree level is perhaps not the end of the story. For pure anomaly mediation, M a (Q)/g 2 a (Q) is proportional to the respective beta-function coefficients at the scale Q. Anomaly mediation effects are always present and become dominant over gravity mediation when the supersymmetry breaking sector is sequestered from the visible sector; however, the scheme by itself suffers from tachyonic sleptons.
Mirage mediation is a hybrid of the mSUGRA pattern and anomaly pattern, in which the tree level contribution and the one loop contribution are both equally competitive. The quantity M a /g 2 a then depends both on the universal mSUGRA contribution, as well as the RG beta coefficients, with comparative strengths given by a parameter α:
M a (Q) g 2 a (Q) = 1 + ln(M p /m 3/2 ) 16π 2 g 2 GU T b a α M 0 g 2 GU T ,(2)
with M 0 ∼ 1 TeV a mass scale. The above facts translate into the following ratio of low energy gaugino masses Anomaly :
M 1 : M 2 : M 3 ∼ 3.3 : 1 : 9 , Mirage : M 1 : M 2 : M 3 ∼ (1 + 0.66α) : (2 + 0.2α) : (6 − 1.8α) . (3)
Clearly, a deviation of the measured gaugino spectrum from the mSUGRA pattern would signal the relative importance of one loop contributions (which, of course, could have a plethora of model origins).
A. Mass Measurements at the LHC
In this paper, we take the first bottom-up steps toward the diagnosis of mediation schemes at the LHC, by choosing a representative model and attempting to reconstruct low energy masses.
Mass reconstruction gives us the following information: (i) As we have outlined above and spell out in more detail later, mass measurements can distinguish between mediation schemes. Of course, the choice of the UV model must be judicious -for example, reconstructing just the gaugino sector would still leave one incapable of distinguishing between the various models and schemes which give rise to the mSUGRA pattern of masses (this has also been verified by distinguishability tests [12]). We thus choose a model (KKLT compactification) where the imprints of anomaly mediation are unmistakable.
One important point in this paper is that in certain regions of parameter space, the crucial information from the gaugino spectrum regarding the scheme may be gleaned at relatively lower luminosity by a judicious choice of observables. Meanwhile, the details of a particular model only become apparent as other masses are reconstructed.
(ii) The measurement of low energy masses enables us to make statements about the dark matter relic density. In models with R−parity invariance, the lightest superpartner (typically the lightest neutralinoχ 1 0 ) is a dark matter candidate, and one would like to compute the relic density Ωh 2 associated with it. The direct measurement of stop (t) and stau (τ ) masses enables us to verify whether we are in regions of parameter space where stop-neutralino (t −χ 1 0 ) and stau-neutralino (τ −χ 1 0 ) coannihilation effects are important. We note that the investigation of mediation schemes, the first goal described above, typically allows one to be in much larger areas of parameter space, and one need not necessarily work in coannihilation regions. However, to satisfy relic density constraints, they are crucial.
Previously, we have undertaken studies of mass measurements in theτ −χ 1 0 coannihilation region in the context of the mSUGRA model [10,13]. The measurement of third generation squark masses presents its own challenges. Reconstruction of stop and sbotttom (b) is very hard in a cascade decay chain since both stops and sbottoms decay into b quarks. Further, to make the situation worse, in the stop coannihilation region the stop decay produces a lower p T jet due to the proximity of the lighter stop and the lightest neutralino masses. We invoke two new observables to measure lighter stop and sbottom masses in the cascade decays.
Thus, in this paper we fulfil the following objectives: (i) Constructing observables using different combinations of the jets, τ 's and W 's in the final states to determine the masses of supersymmetric particles in a model where anomaly contributions are important. We construct observables in the stop-neutralino and stauneutralino coannihilation regions.
(ii) Using the observables, we solve for the masses of the gluino, the two lighter neutralinos, the squarks (of the first two generation) and the lightest stau. We also determine the lighter stop and sbottom masses in the stop coannihilation region.
(iii) Using the neutralino and gluino masses we determine the gaugino unification scale, thereby establishing the imprint of one-loop anomaly contributions competitive with tree level contributions (mirage pattern) from experimentally measurable observables.
(iv) Using the masses of the other particles in conjunction with the gauginos, we (a) determine the parameters defining a particular UV completion of the mirage pattern (KKLT compactification with visible sector on D7 branes) and (b) test whether we are in a coanihilation region and dark matter relic density is satisfied.
A caveat is in order. The observables we construct in the current work enable us to measure the masses of the gluino and the two lighter neutralinos, which should be mainly gauginos for us to make statements on mediation within the structure outlined above. In the coannihilation regions, this is guaranteed, and thus we are able to simultaneously solve the relic density as well as determine the mediation scheme. It would be very interesting to probe similar issues in regions of parameter space where the Higgsino component in the neutralinos is not insignificant. However, such a study is beyond the scope of this paper.
Another comment pertains to the fact that we will be solving the model parameters from the masses. As we have stressed, our aim in this paper is to show that the determination of a given set of masses can unmistakably establish the contours of a mediation scheme (point (iii) above), while it may point back to different models. In point (iv) above, we will thus not attempt a distinguishability test of our particular model (KKLT) -rather, we will fit our model parameters with the masses we solve from the observables.
The plan of the paper is as follows. In Section II, we describe the contributions to gaugino masses, describe our choice of the UV model in more detail, and fix our benchmark points. In Section III, we develop the kinematic observables to measure sparticle masses in the stop coannihilation region. Included in this section is the development of specific observables required for the reconstruction of stop and sbottom masses. In Section IV, we give results for the gaugino mass pattern and unification scale and solve for the other masses and the full parameter set defining the UV model, along with the relic density. Moreover, we solve for the stop and sbottom masses. In Section V, we perform the above analysis in the stau coannihilation region of parameter space. We end with our conclusions.
II. MODEL AND BENCHMARK POINTS
As emphasised in Section I, a given pattern of low energy gaugino masses accommodates diverse mediation schemes, not to mention multiple models. We mention below a catalogue of such models and schemes, outlined in [8], to better situate our particular UV completion.
A. Contributions to Gaugino Masses
The various contributions to M a /g 2 a in 4D effective supergravity are
M a g 2 a =M a (0) +M a (1) | anomaly +M a (1) | other ,(4)
whereM a
(0) = 1/2F I ∂ I f (0)
a , with F I signifying the nonzero F −component of a hidden sector field X I breaking supersymmetry, and f a denoting the visible sector gauge kinetic functions. The one loop anomaly term receives contributions from the conformal and Konishi anomalies, with the conformal anomaly contribution given bỹ M a
(1) | conformal = (1/16π 2 )b a (F C /C), with C denoting the chiral compensator. We will not deal with the Konishi anomaly contribution in any detail in this paper. The other one loop contributionsM a (1) | other encapsulate field theoretic gauge threshold and UV-sensitive contributions like KK thresholds to the gaugino masses. We will assume that UV-sensitive contributions are subdominant (the assumption is purely because they are more model-dependent). The mSUGRA pattern, Eq. 1, arises in situations where M a /g 2 a is dominated by universal tree level piecẽ M a (0) . This happens in a variety of schemes: (i) gaugino mediation in higher dimensional brane models (ii) gravity mediation, with different UV scenarios such as dilaton/moduli mediation in heterotic string theory or large volume compactification in type IIB tring theory. Gauge mediation also gives the mSUGRA pattern of gaugino masses, since the gaugino masses are dominated by universal one-loop gauge threshold contributions. On the other hand, the mirage pattern in which tree level and loop level contributions are competitive occurs in gravity mediation in various UV scenarios, for example KKLT flux compactification with visible sector on D7 branes and explicit supersymmetry breaking by anti-D3 branes or spontaneous breaking by a matter sector. Mirage patterns are also obtained in deflected anomaly mediation [14].
However, the important point is that deviations from the mSUGRA pattern may signal some anomaly contribution at work.
We will take as an example the UV completion of KKLT with visible sector on D7 branes and explicit supersymmetry breaking by an anti-D3 brane.
B. KKLT with D7 branes: The UV Model Parameters
In this subsection, we sketch the outlines of the UV model, only with a view to defining the model parameters we will be working with. For details, we refer to [7].
The basic elements in a KKLT-type model of string compactification are: (i) background fluxes on a type IIB Calabi-Yau three-fold giving a Gukov-Vafa-Witten superpotential contribution, and (ii) gaugino condensation on D7-branes or Euclidean D3 instantons giving a non-perturbative superpotential contribution. These stabilize complex structure moduli and the dilaton, as well as Kähler moduli, in an AdS vacuum. Supersymmetry breaking by an anti-D3-brane then lifts the solution to a de Sitter vacuum. The visible sector is constructed with D7 branes in the bulk.
The soft masses at the GUT scale are
M a = M 0 + m 3/2 16π 2 b a g 2 a , m 2 i =m 2 i − m 3/2 16π 2 M 0 θ i − m 3/2 16π 2 2γ i(5)
where M 0 andm i are pure tree level modulus contributions, given as functions of the Kähler modulus T , while other terms are one-loop anomaly contributions. m 3/2 is the gravitino mass. In the above,
b a = −3tr T 2 a (Adj) + i tr T 2 a (φ i ) , γ i = 2 a g 2 a C a 2 (φ i ) − 1 2 jk |y ijk | 2 , γ i = 8π 2 dγ i d ln Q , θ i = 4 a g 2 a C a 2 (φ i ) − jk |y ijk | 2Ã ijk M 0 ,(6)
where the quadratic Casimir C a 2 (φ i ) = (N 2 − 1)/2N for a fundamental representation φ i of the gauge group SU (N ), C a 2 (φ i ) = q 2 i for the U (1) charge q i of φ i , and kl y ikl y * jkl is assumed to be diagonal. To list the set of parameters that give the low energy mass spectrum, we define the ratios
α ≡ m 3/2 M 0 ln(M P l /m 3/2 ) , 1 − n i ≡m 2 i M 2 0 ,(7)
where α represents the anomaly to modulus mediation ratio, while n i are modular weights, that parameterize the pattern of the pure modulus mediated soft masses. For convenience, we will henceforth choose to rescale α as follows
α henceforth = 16π 2 ln(M p /m 3/2 ) 1 α .(8)
From Eq. 5, then, it is clear that the parameters defining the low energy soft masses of the model are
m 3/2 , α, n m , n H , tanβ,(9)
where we choose m 3/2 in place of M 0 , split the matter modular weights into a universal sfermion weight n m and a Higgs weight n H . We have also replaced the Higgs mass parameters µ and B by tanβ and M Z .
C. Benchmark Points
The parameter space of the KKLT model has been studied in [15], and we choose points that satisfy the stop-neutralino and stau-neutralino coannihilation constraints, which appear in ample parts of model parameter space. We use darkSUSY [16] to select exact benchmark points in the above regions.
The benchmark point for the stop coannihilation region is shown in Tables I, II. The benchmark point for the stau coannihilation region is shown in Table III. The spectrum at the stau coannihilation benchmark point is shown in Table IV.
Note that our methods are valid in general, and the above benchmark points will be explored as an illustration. In particular, we will also study benchmark points with higher gluino mass, preferred by current LHC data, in Sections IV C and V C, where we will show that we need larger luminosity to establish the same set of observables.
Since tanβ is on the large side for the benchmark points, the lighter stau mass is between the lightest and next to lightest neutralinos. The lightest neutralino is mostly Bino and the next to lightest is mostly Wino. The Higgsino components are negligible since the dark matter relic density is satisfied by the coannihilation mechanism.
III. KINEMATIC OBSERVABLES IN THE STOP COANNIHILATION REGION
In this section, we present the measurement of physical observables that will be used to solve for the gaugino masses, as well as the masses of theτ andq. Moreover, we construct observables that will be needed to solve for thet andb masses. In the next section, we will present the mass and model parameter solutions, and also present gaugino unification. We will also give results for a benchmark point with heavier gluino there.
The mass spectrum of the model is determined using ISASUGRA [17]. The spectrum is then fed to PYTHIA [18], which generates the Monte Carlo hard scattering events and hadron cascade. These events are passed to the detector simulator PGS4 [19].
A. 2 Jets + 2τ + E /T
At the LHC, the main production processes for this model aregg,gq andqq.
The relevant decay chains are:
g −→ q LqL andq L −→χ 0 2 q −→τ ± 1 τ ∓ q −→χ 0 1 τ ± τ ∓ q .(10)
There is a high p T jet, missing transverse energy, and a pair of oppositely charged τ leptons.
The following cuts were set to select events for this signal:
(i) Missing transverse energy E / T ≥ 180 GeV; (ii) p T,jet1 + p T,jet2 + E / T 600 GeV; (iii) Leading jet cuts: At least two jets should be present, each with p T ≥ 200 GeV in |η| ≤ 2.5. The jets are both required to be non b-tagged jets, and the event is discarded when either of them is tagged as a b jet;
(iv) Soft jet cuts: Any jet with p T ≥ 30 GeV in |η| ≤ 2.5 is accepted in the analysis;
(v) τ cuts [20]: At least two τ leptons, with visible p T ≥ 15 GeV in |η| ≤ 2.5;
Various kinematic observables can be constructed in this signal. Of them, we choose four that give the most precise endpoint measurements with 50 fb −1 : M end jτ τ , M end τ τ , and the p T distributions of the higher and lower energy τ s.
The fifth observable (required to solve the five model parameters) is the peak of the M eff distribution.
1. M end τ τ
The main challenge in measuring the invariant di-tau mass is the background created by uncorrelated τ pairs. From the decay chain in question, it is clear that similarly charged τ 's are definitely uncorrelated and hence model this background quite well. Thus, we perform the opposite-sign (OS) minus like-sign (LS) subtraction on τ 's to obtain the signal. Figure 1 shows the histogram graph we obtained for the M τ τ distribution. The luminosity is 50 fb −1 , and we can see that a clear end-point is obtained. The τ leptons from each event are sorted into OS and LS pairs. Then, each pair is combined with the leading jets from the same event, and the invariant mass distribution M same jτ τ is obtained. At this stage the Bi-Event Subtraction Technique (BEST) [21] is initiated, by first combining the τ pair with leading jets from a separate event (which we also call a bi-event), to form the distribution M bi−event jτ τ . Since jets from a separate event are kinematically uncorrelated with the τ pair from the current event, this bi-event distribution models the jet background very well.
We note that the greater the number of different-event jets included in the bi-event distribution, the better the modelled background becomes. We include sufficient number of different-event jets to ensure a smooth background distribution.
After this, two subtractions are performed to obtain the final signal: (i) The OS−LS subtraction, which gets rid of uncorrelated τ pairs and (ii) The BEST subtraction, which gets rid of uncorrelated jets in the background. In Figure 2, we show the M jτ τ distribution where we find a very clear end point after the BEST subtraction.
FIG. 2: Distribution of Mjττ at a stop coannihilation benchmark point. The green (grey) same-event histogram is constructed by combining each OS−LS τ pair with a leading jet from the same event. The filled dot-dashed green (grey) bievent histogram is obtained by combining the OS−LS τ pair with a leading jet from a different event and is normalised to the long tail of the same-event histogram. The same-event minus bi-event subtraction (BEST) produces the black subtracted histogram. The subtracted histogram is fitted with a straight line to obtain the endpoint. The result for the endpoint is 269.09 ± 3.18(Stat.) GeV. The luminosity is 50 fb −1 .
Slope of pT of τ
The mass of theτ lies between the two lightest neutralinos, and its exact location approximately determines the p T of the τ 's in the ditau pair. The slope of the visible p T distribution of the lower energy visible τ from the di-tau pair (which we will denote by slope(p T )) is generally a good observable, that carries information about the masses ofτ andχ 0 1 (in the case that the lower energy τ is fromτ −→χ 0 1 + τ ). If the stau mass is very close to the neutralino mass (in the case of stau-neutralino coannihilation) we have a low energy τ and the slope of the p T distribution is a good observable.
Conversely, if the visible p T of a τ is large, the slope of the p T distribution is not a good observable. This is because the slope in the case of a higher energy τ doesn't show enough variation. In such cases, we use the kinematic information of the high energy τ by combining it with a leading jet to construct M jτ . We will use the endpoint of the M jτ distribution as an observable in the stau-neutralino coannihilation case, where one of the τ 's has large p T compared to the other.
If the stau is somewhat in between the two neutralinos, the slopes of the p T distributions of both taus become good observables (provided, of course, that neither p T is too large) as happens in this case. In such a scenario, we get two observables depending on two neutralinos and the stau mass without involving any jet. It is convenient to define the new observables p T,AM and p T,diff , which are the arithmetic mean and difference of the slopes of the higher and lower p T τ s. These variables can typically be measured down to lower luminosity than M jτ , since they do not involve the subtractions required for observables involving jets, as we describe below.
To obtain the distributions, first the OS−LS subtraction is performed. Then, we take the logarithm of the p T,high and p T,low distributions for the higher and lower energy τ respectively. The histograms are fitted using a linear function to find the slopes. Finally, the slopes are combined into the arithmetic mean p T,AM and the difference p T,diff :
p T,AM = 1 2 (slope(p T,high ) + slope(p T,low )) p T,diff = 1 2 (slope(p T,high ) − slope(p T,low )) .(11)
Note that these observables, formed by combining the p T information of the two τ 's, are particularly necessary in the case when they are close in mass, since we do not know if a given τ is originating from theτ decay or thẽ χ 0 2 decay. In Figure 3 and Figure 4, we show the p T distributions of two τ 's present in the sample.
B. 4 jets + E /T
The M eff distribution is formed by combining the p T of the first four leading jets and the missing energy These jets effectively come from the gluino and squark decays.
M eff = p T,jet1 + p T,jet2 + p T,jet3 + p T,jet4 + E / T (12)
The effective mass M eff is constructed from the 4jets + E / T sample, with the following cuts (i) Number of jets: At least four jets in the event;
(ii) Leading jet cuts: The first two leading jets each have p T ≥ 200 GeV in |η| ≤ 2.5;
(iii) Soft jet cuts: Jets with p T ≥ 30 GeV in |η| ≤ 2.5 are accepted in the analysis;
(iv) The jets are not b tagged;
(v) E / T ≥ 180 GeV;
(vi) p T,jet1 + p T,jet2 + E / T 600 GeV; (vii) No e s and µ s with p T ≥ 15 GeV; (viii) Transverse sphericity S T ≤ 0.2 . In Figure 5, we show the M eff distribution at the benchmark point. The peak of this distribution shows a well defined peak for 50 fb −1 luminosity. In Table V, we show the end points of M τ τ and M jτ τ distributions, peak of the M eff distribution, and the slopes of the p T distributions for the taus for our benchmark point. The statistical uncertainties range between 0.4% − 3.8%. To probe the third generation squark masses we need to involve b quarks. The relevant decay chain associated with the dominant production process for the reconstruction of third generation squarks in the stop coannihilation region is
g −→b + b −→t + W + b −→χ 0 1 + c + b + W.(13)
These signals are characterized by high p T jets accompanied by a W and E / T . We will be constructing the distributions M bW and M jW .
The cuts for the analysis are (i) E / T ≥ 180 GeV;
(ii) Number of jets: N jet ≥ 4; (iii) Leading jet cuts: The first two leading jets each have p T ≥ 200 GeV in |η| ≤ 2.5. They could be gluon, light-flavor, or b jets;
(iv) Soft jet cuts: Any jets with visible p T ≥ 30 GeV in |η| ≤ 2.5 are accepted in the analysis. This includes b-tagged jets;
(v) p T,jet1 + p T,jet2 + E / T 600 GeV; (vi) For M bW , at least one tight b jet is required. The first step in the analysis is the reconstruction of the W boson. The W appears in the detector as two jets whose invariant mass falls in the W mass window (65 GeV ≤ M jj ≤ 90 GeV). We thus choose soft jet pairs (from the third leading jet and below) which are not btagged, with 0.4 ≤ ∆R ≤ 1.5. The jets are put into two categories: those which are manifestly in the W window, and those that fall within the sideband window (40 GeV ≤ M jj ≤ 55 GeV or 100 GeV ≤ M jj ≤ 115 GeV). BEST is then performed for the two categories, to get rid of uncorrelated jet background. After this, the sideband subtraction is performed to obtain the W mass.
Once the W is reconstructed, it is paired up with jets to form the M bW and M jW distributions. W is first reconstructed with two jets whose invariant mass falls in the W window. The reconstructed W is then combined with a non b-tagged soft jet of rank three or lower from the same event, to produce the same-event blue (grey) histogram. The W is combined with a soft jet from a different event to produce the bi-event filled dot-dashed blue (grey) histogram, which is normalised to the shape of the long tail of the same-event histogram. The same-event minus bi-event subtraction (BEST) produces the black subtracted histogram. The subtracted histogram is fitted with a straight line to obtain the endpoint. The result for the endpoint is 287.55 ± 0.74(Stat.) GeV. The luminosity is 50 fb −1 .
For the M jW distribution, we pair the W with a non b-tagged soft jet, whose rank is three or lower. This is because we are in the stop coannihilation region. In Figure FIG. 7: Distribution of M bW at a stop coannihilation benchmark point. W is first reconstructed with two jets whose invariant mass falls in the W window. The reconstructed W is then combined with a b jet of any rank from the current event, to produce the same-event pink (grey) histogram. Events with M bW ≤ 200 GeV are discarded to remove the top peak. The W is combined with a b jet from a different event to produce the bi-event filled dot-dashed pink histogram, which is normalised to the shape of the long tail of the same-event histogram. The same-event minus bi-event subtraction (BEST) produces the black subtracted histogram. The subtracted histogram is fitted with a straight line to obtain the endpoint. The result for the endpoint is 325.67 ± 4.50(Stat.) GeV. The luminosity is 50 fb −1 .
6, we show the M jW distribution at the benchmark point, finding a well defined end-point for 50 fb −1 luminosity.
For the M bW distribution, we pair the W with a b jet of any rank from the current event. Note that the relevant b jet required to construct this observable need not be a leading jet; in fact, the leading jet will typically be non b-tagged. After pairing the W with the b jet, we do a further BEST to get rid of uncorrelated b jets. This gives the final signal for M bW . However, the b+W signal shows the presence of the unwanted top peak, which comes from t −→ b + W . The top window is removed from the final signal, by discarding events with M bW ≤ 200 GeV. In Figure 7, we show the M bW distribution obtained at the benchmark point, finding the end-point for 50 fb −1 luminosity.
In Table VI, we show the endpoint values obtained from these distributions. The statistical uncertainties range between 0.2% − 1.4%. The statistical uncertainty is larger for M bW due to the b jet.
IV. DETERMINATION OF PARTICLE MASSES, GAUGINO UNIFICATION SCALE, AND UV MODEL PARAMETERS IN THE STOP COANNIHILATION REGION
The observables measured in Section III are used to determine the masses of the gluino,χ 0 1 ,χ 0 2 ,τ , andq, which lead to the determination of model parameters (α, m 3/2 , tanβ, n m , n H ). We will also determine theb andt masses later in this section.
Theoretically, the functional dependences of the observables in Section III are expected to be:
M end τ τ = M end τ τ (mχ0 2 , mτ , mχ0 1 ) M end jτ τ = M end jτ τ (mq L , mχ0 2 , mχ0 1 ) slope(p T,AM ) = p T,AM (mχ0 2 , mχ0 1 ) slope(p T,diff ) = p T,diff (mτ , mχ0 2 , mχ0 1 ) M peak eff = M peak eff (mq L , mg)(14)
The masses are varied independently around the benchmark point, and the full simulation of the collider experiment and determination of the observables is performed each time. Thus, the functional dependence of the observables on the masses is determined, along with their uncertainties.
This set of equations can be used to solve for the masses. In general, the dependence of the observables on the masses is expected to be non-linear, corresponding to multiple solutions of the masses. However, near the benchmark point, we found that for most masses, linear functions fitted the dependence quite well. We obtained two solutions only for the stau mass, and one was quite far from the benchmark point and thus rejected. We will have more to say on these issues in Section V A.
In Table VII, we show the solution to theg,q L ,τ ,χ 0 2 , andχ 0 1 masses for 50 fb −1 luminosity. We also show the statistical uncertainties, which range between 1%−6.6%. Having determined the masses ofg,χ 0 1 ,χ 0 2 ,τ , andq, we can use these to numerically solve for the parameters of the full model. As we have stressed in Section I, the point of this work is to show that the contours of the mediation scheme (here, mirage pattern) show up unmistakably from the determination of certain masses. We will demonstrate this in Section IV C, where we will show the gaugino unification scale. However, a set of masses may point back at different models, and in particular, we will not attempt to perform a distinguishability study of our particular model. When we solve the model parameters for our particular example (KKLT), our methods will reflect this -we will start from the model parameters, fit with the spectrum, and zero in on the masses we solved.
We use the Nelder Mead method, a commonly used nonlinear optimization technique, to do so. An initial simplex is chosen on the parameter space {m 3/2 , α, n m , n H , tanβ, } and ISAJET [17] is used to find the corresponding mass. The scan over parameter space proceeds until the masses calculated by ISAJET are close to the solved values. Table VIII shows the corresponding model parameters we determined from mg, mq L , mχ0 2 , mτ 1 and mχ0 1 for 50 fb −1 and 100 fb −1 luminosities. At 50 fb −1 , the statistical uncertainties range between 5.6% − 17%. Using the model parameters from Table VIII, we calculate the relic density and we find:
Ωh 2 = 0.096 ± 0.029 .(15)
The relic density is determined to an accuracy of 31%.
B. Determination ofb andt Masses
The observables M end bW and M end jW are used to determine the third generation squark masses, once the other masses have been obtained with the observables previously determined.
Theoretically, the functional dependences are M bW = M bW (mb, mt, mg) and M jW = M jW (mb, mt, mχ0 1 ). As before, the masses are varied independently around the benchmark point, and the collider experiment and determination of M end bW and M end jW is performed each time. We show the masses of the third generation squarks with uncertainties in Table IX. The statistical uncertainties range between 2.5% − 11.3%. For this benchmark point, a luminosity of 200 fb −1 is required to solve for all the masses, following the techniques we have shown in this paper. We show the masses we obtained for this benchmark point in Table XI. The statistical uncertainties range between 0.9% − 14.7%.
Particle Mass
Stat.
t 690 ± 6 b 1002 ± 126 τ 717 ± 10 q 1133 −132, +167
The model parameters are solved as before, using the masses above as well as the gaugino masses solved at 200 fb −1 . The solution to the model parameters at 200 fb −1 for the new benchmark point are α = 3.84 ± 0.11, m 3/2 = 34463±805 GeV, n m = 0.078±0.007, n H = 0.592±0.036, and tanβ = 27.2 ± 1.94. The relic density is found to be Ωh 2 = 0.23 ± 0.13.
As mentioned in Section I, gaugino masses may be obtained at lower luminosity, as we elaborate below.
The solution of gaugino masses typically requires the values of four observables (p T,diff , p T,AM , M peak eff , and M end τ τ ), as is clear from the solutions we displayed in Table VII. One could have used the endpoint of the invariant mass distribution M jτ of the leading jet and the more energetic τ as an observable. Our analysis of this observable indicates that one requires more luminosity to find its endpoint, relative to the luminosity required to determine the observables p T,AM and p T,diff , because of the associated jet subtractions.
Since for the benchmark point the τ 's are quite close in mass, we use p T,AM and p T,diff , obviating the need to use observables which involve jet information. Then, χ 0 1 ,χ 0 2 , andτ masses can be obtained from p T,AM , p T,diff , and M end τ τ observables. Since these observables do not involve jet subtractions, they give acceptable endpoints at lower luminosity. On the other hand, M peak eff roughly gives the gluino mass.
In Table XII, we show the values of the gaugino masses at 50 fb −1 for the new benchmark point. The statistical uncertainties range between 1.2% − 4.2%.
In Figure 8, we show the running of the gauginos and the gaugino unification for our benchmark point. Clearly, gaugino unification occurs below the GUT scale, establishing the footprints of anomaly contributions to supersymmetry breaking. The theoretically expected gaugino unification scale, for this benchmark point, is given by
M unif = M GU T m 3/2 M p 2.47/α ∼ 10 7 GeV .(16)
Note that the expression differs from the ones in [8] by the rescaling in Eq. 8.
D. Systematic Errors
We also study an impact due to a ±3% uncertainty on the energy scale for jets, taus, and missing transverse energy to estimate the systematic uncertainties independently of luminosity. Table XIII shows the percentage systematic errors in the determination of the observables and masses. We find that the systematic errors for the observables are between 1% − 6%, which is larger than the statistical error 0.2% − 3% at 50 fb −1 . The systematic errors of masses, which vary from 2%−19%, are also large compared to the statistical errors, which range between 1% − 11% at 50 fb −1 . Since our assumption of ±3% on the energy scale could be changed in actual experimental conditions, we just show the size of errors in Table XIII as a reference and don't use them in the analyses performed in the previous sub-sections.
M end τ τ ± 3.0 mg ± 3.9 α ± 5.9 M end jτ τ ± 4.5 mχ0 2 −2.4,+4.2 m 3/2 ± 5.9 slope(p T,high ) ± 5.2 mχ0 1 ± 4 nm ± 12.5 slope(p T,low ) ± 1.0 mτ ± 2.6 nH ± 17 M peak eff ± 1.3 mq −13,+15 tanβ ± 6.1 M end jW ± 5.9 mt −2.5,+3.7 M end bW ± 3.7 mb ± 19
V. ANALYSIS IN THE STAU COANNIHILATION REGION
In this Section, we determine masses and model parameters for a different point in parameter space, where only stau coannihilation is operational. We first show our analysis for the benchmark model point shown in Table III. In Section V C, we will also analyse a benchmark point with heavier gluino mass preferred by current LHC data.
The distributions used in the stau coannihilation region are M τ τ , M jτ τ , M jτ , p T,low , and M eff . Note that we only use the p T,low corresponding to the lower energy τ fromτ −→χ 0 1 + τ , since the higher energy τ doesn't show enough variation as the masses are varied.
We use M jτ , which is formed by combining the higher energy τ with a leading jet from the same event. OS−LS and BEST are performed as usual. Theoretically, one expects M end jτ = M end jτ (mq L , mχ0 2 , mτ ). In Figures 9, 10, 11, 12, and 13, we present the distributions obtained. In Table XIV, we present the endpoints obtained from the M τ τ , M jτ τ , and M jτ distributions, the slope of the visible p T distribution of the lower energy τ , and the peak of the M eff distribution, at 100 fb −1 for the stau coannihilation benchmark point. The statistical uncertainties range between 0.6% − 4.7%.
FIG. 11: Distribution of p T,low at a stau coannihilation benchmark point. The notation is the same as in Figure 4. The slope obtained is −0.0849 ± 0.0041(Stat.). The luminosity is 100 fb −1 .
FIG. 12: Distribution of M eff at a stau coannihilation benchmark point. The notation is the same as in Figure 5. The peak obtained is 1257.26 ± 10.33(Stat.) GeV. The luminosity is 100 fb −1 .
A. Masses and Gaugino Unification
After obtaining the values of the observables at the benchmark point as quoted above, we varied the masses around the benchmark point. For each mass selection, we simulated the experiment and obtained all the observables. This gave us the observables as a function of the masses. We then iteratively solved for the masses. Obviously, one should expect more than one set of mass solutions in this method, since the dependence of the observables on the masses near the benchmark point is non-linear. In this case, we found two sets of solutions with quite disparate values of masses. This is a general problem, and we mention some arguments to reject the "wrong" solutions. Firstly, one set of solutions had masses far away from the benchmark point, where the expansion was not performed to begin with. Secondly, this set had a much larger stauneutralino mass difference, which did not satisfy the relic density constraint.
Finally, one can reject masses by appealing to the UV model and determining more observables, although this goes somewhat against the bottom-up philosophy we have pursued in this paper. Thus, we can use this set of mass solutions and solve for the UV model parameters, which we can then use to solve the spectrum of the model, in particular obtaining the b quark mass. Given the spectrum, we can obtain the theoretical value of a particular observable, say M bW , and on the other hand, measure it using the techniques outlined in the paper. For the incorrect mass solution set, the values thus obtained will be very different.
In any case, after rejecting the first set of solutions, the second set itself had two sets of values ofχ 0 1 andτ masses, which were close to each other. One set of solutions is shown in Table XV. The other solutions to theχ 0 1 and τ masses are mχ0 1 = 294 GeV and mτ = 337 GeV. All combinations of masses (four in all) are used to solve the model parameters, as we outline in the next subsection.
We display the solution set in Table XV, obtained at luminosity 100 fb −1 . The statistical uncertainties are 2.7% − 6.4%. Having determined the masses ofg,χ 0 1 ,χ 0 2 ,τ , andq, we can use these to numerically solve for the parameters of the full model.
We use the Nelder-Mead method to do so. An initial simplex is chosen on the parameter space {m 3/2 , α, n m , n H , tanβ, } and ISAJET is used to find the corresponding mass. The scan over parameter space proceeds until the masses calculated by ISAJET are close to the solved values.
In the previous subsection, we have discussed two solutions each forχ 0 1 andτ . We solved the model parameters for each combination of masses (four in all), and took the average of the values of the model parameters thus obtained. These average values are displayed in Table XVI. The statistical uncertainties are 9% − 40%. Using the model parameters from Table XVI, we calculate the relic density and we find:
Ωh 2 = 0.17 +0.12 −0.13 .(17)
The relic density is thus determined to an accuracy of 59%.
C. Gaugino Unification
Although upto this point, we have displayed our results in the stau coannihilation region at the benchmark point given in Table IV, our methods are applicable at other benchmark points with higher gluino mass, preferred by current LHC data. We display such a benchmark point for stau coannihilation below, and proceed to show gaugino unification. At this benchmark point, we found acceptable endpoints at 250 fb −1 , and solved for the observables and masses. The masses are mτ = 346 ± 12 GeV and mq = 1138 ± 72 GeV. The masses of the gauginos can also be determined, although we show their values at a lower luminosity below. After determining the masses, the observables can be solved and they are α = 9.95 ± 0.99, m 3/2 = 9636 ± 910 GeV, n m = 0.56 ± 0.09, n H = 1.14 +0. 20 −0.08 , and tanβ = 40.0 ± 7.0. The relic density for the heavy gluino point is found to be Ωh 2 = 0.05 +0. 21 −0.04 . The gaugino masses can be obtained at lower luminosity, as spelled out in Section I.
In the stau coannihilation region, one can use the proximity of the stau and lightest neutralino masses to solve gaugino masses using fewer observables, in particular avoiding the observable M end jτ , which typically requires higher luminosity due to jet subtractions. This is reminiscent of what we did in the previous sections with stop coannihilation, although there we used the proximity of τ p T 's.
We take approximately
mτ ∼ mχ0 1 + 15 .(18)
Thus, assuming that M peak eff is approximately determined by mg, it is clear from Eq. 14 that one can solve for mg, mχ0 2 , mχ0 1 from M peak eff , p T,low and M τ τ . These observables have acceptable endpoints and peaks down to 15 fb −1 . The main difference from the full set of observables is the M jτ distribution, which requires a higher luminosity to obtain acceptable endpoints due to jet subtractions.
In Table XVIII, we show the gaugino masses obtained at 15 fb −1 for the new stau coannihilation benchmark point with heavier gluino. The statistical uncertainties are 7% − 11.4%. In Figure 14, we show gaugino unification for the new benchmark point at 15 fb −1 .
VI. CONCLUSIONS
If supersymmetry is discovered at the LHC, the next step would be to understand as much as possible about the underlying model of supersymmetry breaking and mediation. How should one proceed?
One option is to start from specific models defined by a tractable number of parameters at high energy, and fit to observables measured in the collider. However, we have stressed that a given set of measured observables will generally point back at different UV models, distinguishing among which may be challenging and even intractable (depending on the complexity of the model and the number of reliable observables measured).
We have stressed on a bottom-up approach to this question: first identify sectors in the low energy spectrum which carry the imprints of mediation schemes in a relatively model-independent manner, next construct observables which carry information about these masses, and finally using the observables solve for the masses and identify a mediation scheme.
The gaugino sector offers a particularly fruitful sector in this program, since the determination of this sector can unambiguously show the imprints of anomaly contributions to supersymmetry breaking. We therefore take an example of a model where anomaly contributions are significant (supersymmetry breaking in KKLT compactification) and proceed to obtain the gaugino masses, by constructing suitable observables.
We rely on the construction of kinematic observables arising from cascade decays using endpoint techniques, since these are a particularly sharp tool for such diagnosis.
While the gauginos are important in the paradigm outlined above, they form only a subset of the masses we determined in this paper. Given a neutralino dark matter, the determination of other masses becomes a necessity when one wants to establish mechanisms to satisfy the relic density. This is perfectly amenable to the kind of bottom-up approach we have taken: in our examples, we needed to establish that the stop-neutralino and stauneutralino coannihilation mechanisms were operational. To do so demanded the determination of stop and stau masses, in addition to the neutralino.
We summarize our work below: (i) We constructed observables using different combinations of the jets, τ 's and W 's in the final states. We constructed these observables in the stop-neutralino and stau-neutralino coannihilation regions.
In the stop coannihilation region, we constructed two new observables, M end jW and M end bW , to determine the masses of thet and theb. The determination of thet andb masses is especially challenging, since both decay to b quarks. To make the problem worse, stop decays to missing energy by emitting a low energy jet in this coannihilation region. We showed how to reconstruct the stop mass in order to establish the fact that we are in the stop-neutralino coannihilation region.
We also constructed two other new observables, p T,AM and p T,diff which are combinations of the p T of two opposite sign τ 's. These observables are particularly useful when the sample shows the presence of two opposite sign τ 's that are close in mass.
(ii) Using the observables, we solved for the masses of the gluino, the two lighter neutralinos, squarks (of the first two generations) and the lightest stau. We also determined the lighter stop and sbottom masses in the stop coannihilation region.
(iii) We used the neutralino and gluino masses to determine the gaugino unification scale and discern the effects of anomaly mediation from a bottom-up approach, as elaborated above.
(iv) Using all the masses, we determined (a) the model parameters and (b) tested whether we are in a coanihilation region and dark matter relic density is satisfied. We find that relic density can be determined with an accuracy of 31% in the stop and 59% in the stau coannihilation region.
As far as the diagnosis of the supersymmetry breaking mediation scheme is concerned, the most important future direction is obviously to expand our techniques to parts of the parameter space away from the coannihilation regions. Our observables enabled us to solve for the lightest neutralinos, and for the statements on mediation scheme that we made above, they were assumed to be gauginos (this is guaranteed in the coannhilation regions). It would be very interesting to test the paradigm outlined in this paper at regions where the Higgsino component is not insignificant. Also interesting would be to find bottom-up techniques to further distinguish between different schemes that correspond to the same gaugino mass pattern. We leave these questions for future work.
FIG. 1 :
1Distribution of Mττ at a stop coannihilation benchmark point. The pink (grey) histogram is composed of OS τ pairs. The filled dot-dashed pink (grey) histogram is composed of LS τ pairs and is normalised to the shape of the long tail of the the pink (grey) OS histogram. The OS−LS subtraction produces the black subtracted histogram. This subtracted histogram is then fitted with a straight line to find the endpoint of the distribution. The result for the endpoint is 49.71 ± 0.2(Stat.) GeV. The luminosity is 50 fb −1 .
FIG. 3 :
3Distribution of p T,high at a stop coannihilation benchmark point. The pink (grey) histogram is composed of OS τ pairs. The filled dot-dashed pink (grey) histogram is composed of LS τ pairs and is normalised to the shape of the long tail of the OS histogram. The OS−LS subtraction produces the black subtracted histogram. This subtracted histogram is then fitted with a straight line to find the slope of the distribution. The value of the slope(p T,high ) observable is −0.0522 ± 0.0021(Stat.). The luminosity is 50 fb −1 .FIG. 4: Distribution of p T,low at a stop coannihilation benchmark point. The pink (grey) histogram is composed of OS τ pairs. The filled dot-dashed pink (grey) histogram is composed of LS τ pairs and is normalised to the shape of the tail of the OS histogram. The OS−LS subtraction produces the black subtracted histogram. This subtracted histogram is then fitted with a straight line to find the slope of the distribution. The value of the slope(p T,low ) observable is −0.1178 ± 0.0040(Stat.). The luminosity is 50 fb −1 .
FIG. 5 :
5Distribution of M eff at a stop coannihilation benchmark point. The distribution is fitted with a Gaussian function to find the peak. The value of the M peak eff observable is 1073.01 ± 8.72(Stat.) GeV. The luminosity is 50 fb −1 .
FIG. 6 :
6Distribution of MjW at a stop coannihilation benchmark point.
FIG. 8 :
8Running of gaugino masses. The vertical axis is masses in GeV. The luminosity is 50 fb −1 . The shaded lines from the top down represent, at the left edge of the graph, the gluino, wino and bino masses respectively.
FIG. 9 :
9Distribution of Mττ at a stau coannihilation benchmark point. The notation is the same as inFigure 1. The endpoint obtained is 90.70 ± 0.54(Stat.) GeV. The luminosity is 100 fb −1 .FIG. 10: Distribution of Mjττ at a stau coannihilation benchmark point. The notation is the same as inFigure 2. The endpoint obtained is 479.53±3.45(Stat.) GeV. The luminosity is 100 fb −1 .
FIG. 13 :
13Distribution of Mjτ in jτ τ system. The blue (grey) histogram is obtained by combining the higher energy τ of the OS−LS τ pair with a leading jet of the same event. The filled dot-dashed histogram is the bi-event histogram, constructed by combining the aforesaid τ with a leading jet from a different event and is normalised to the long tail of the sameevent histogram. The same-event minus bi-event subtraction (BEST) produces the black subtracted histogram. The subtracted histogram is fitted with a straight line to obtain the endpoint. The result for the endpoint is 448.40 ± 16.20(Stat.) GeV. The luminosity is 100 fb −1 .
FIG. 14 :
14Running of gaugino masses. The vertical axis is masses in GeV. The luminosity is 15 fb −1 . The shaded lines from the top down represent, at the left edge of the graph, the gluino, wino and bino masses respectively.
TABLE I :
IModel parameters chosen at a stop coannihilation benchmark point. All masses are in GeV.Parameter Value
α
4.5
m 3/2
14000
nm
0.0
nH
0.5
tanβ
30
TABLE II :
IISpectrum at a stop coannihilation benchmark point. All masses are in GeV.Particle Mass Particle Mass Particle Mass
dL
653.13ẽL
436.75χ 0
1
286.21
dR
635.86ẽR
411.28χ 0
2
338.21
uL
647.91τ1
315.08χ 0
3
477.35
uR
634.96τ2
417.70χ 0
4
502.68
b1
520.46χ ±
1
337.32
b2
596.25χ ±
2
500.41
t1
338.55g
649.78
t2
616.22
TABLE III :
IIIModel parameters chosen at a stau coannihilation benchmark point. All masses are in GeV.Parameter Value
α
7.5
m 3/2
10000
nm
0.5
nH
1.0
tanβ
30
TABLE IV :
IVSpectrum at a stau coannihilation benchmark point. All masses are in GeV.Particle Mass Particle Mass Particle Mass
dL
845.49ẽL
426.91χ 0
1
284.17
dR
813.52ẽR
367.70χ 0
2
389.17
uL
841.39τ1
309.75χ 0
3
548.88
uR
815.27τ2
425.68χ 0
4
569.04
b1
735.87χ ±
1
389.32
b2
791.30χ ±
2
568.10
t1
600.23g
897.55
t2
810.20
TABLE V :
VKinematic observables at 50 fb −1 for a stop coannihilation benchmark point. All masses are in GeV.Observable
Value 50 fb −1 Stat. 100 fb −1 Stat.
M end
τ τ
49.71
±0.20
±0.14
M end
jτ τ
269.09
±3.18
±2.25
slope(p T,high ) −0.0522
±0.0021
±0.0014
slope(p T,low ) −0.1178
±0.0040
±0.0028
M peak
eff
1073.01
±8.72
±6.17
C. Observables for the Determination of Third
Generation Squark Masses: M end
bW and M end
jW
TABLE VI :
VIKinematical observables M end jW and M end bW at 50 fb −1 for a stop coannihilation benchmark point. All masses are in GeV. Observable Value 50 fb −1 Stat. 100 fb −1 Stat.M end
jW
287.55
0.74
0.52
M end
bW
325.67
4.50
3.18
TABLE VII :
VIISolution to masses at 50 fb −1 for a stop coannihilation benchmark point. All masses are in GeV. Particle Mass 50 fb −1 Stat. 100 fb −1 Stat.g
646
−14,+19
−11,+14
qL
638
−34,+42
−23,+39
τ
318
−3,+3
−3,+3
χ 0
2
333
−7,+11
−6,+8
χ 0
1
276
−8,+13
−7,+10
A. Determination of UV Model Parameters and
Relic Density
TABLE VIII :
VIIISolution to model parameters at a stop coannihilation benchmark point at 50 fb −1 . Masses are in GeV. Parameter Value 50 fb −1 Stat. 100 fb −1 Stat.α
4.58
±0.21
±0.14
m 3/2
13717
±688
±517
nm
0.106
±0.015
±0.015
nH
0.578
±0.095
±0.091
tanβ
28.76
±1.65
±1.36
TABLE IX :
IXSolution to stop and sbottom masses at 50 fb −1 for a stop coannihilation benchmark point. All masses are in GeV. Particle Mass 50 fb −1 Stat. 100 fb −1 Stat.Upto this point, we have displayed our techniques of reconstructing masses at the benchmark point given in Table II. Our techniques work perfectly well at benchmark points with higher gluino mass, as preferred by current LHC data. Higher luminosity is of course required to obtain endpoints. Below, we choose such a benchmark point with mg ∼ 1.2 TeV, reconstruct gaugino masses, and show the gaugino unification scale.b
531
−60, +60
−47, +47
t
326
−5, +8
−4, +7
C. Gaugino Unification
TABLE X :
XModel parameters and spectrum at a new stop coannihilation benchmark point with heavier gluino. All masses are in GeV.Parameter Value Particle Mass
α
3.8g
1187
m 3/2
34800χ 0
2
740
nm
0.0χ 0
1
666
nH
0.5τ
721
tanβ
28q
1189
TABLE XI :
XISolution to masses at 200 fb −1 for a stop coannihilation benchmark point with heavier gluino. All masses are in GeV.
TABLE XII :
XIISolution to gaugino masses at 50 fb −1 for the new stop coannihilation benchmark point with heavier gluino. All masses are in GeV.Particle Mass 50fb −1 Stat.
g
1181
± 50
χ 0
2
738
± 15
χ 0
1
649
± 20
TABLE XIII :
XIIISystematic percentage errors in observables and
masses for a benchmark point in the stop coannihilation re-
gion.
Observable Error (%) Mass Error (%) Parameter Error (%)
TABLE XIV :
XIVKinematical observables at 100 fb −1 for the Stau coannihilation benchmark point. All masses in GeV. Observable Value 100 fb −1 Stat.M end
τ τ
90.70
±0.54
M end
jτ τ
479.53
±3.45
slope(pT,τ ) −0.0849
±0.0041
M peak
eff
1257.26
±10.33
M end
jτ
448.40
±16.20
TABLE XV :
XVSolution to masses at 100 fb −1 for the stau coannihilation benchmark point. All masses are in GeV.Particle Mass 100 fb −1 Stat.
g
895
−35, +50
qL
845
−36, +24
χ 0
2
388
−9, +25
τ
298
−8, +8
χ 0
1
274
−10, +10
B. Determination of UV Model Parameters and
Relic Density
TABLE XVI :
XVISolution to model parameters at the stau coannihilation benchmark point. For each mass solution, the model parameters are solved, and the average of all the different sets of solutions is shown in the table. Masses are in GeV.Parameter Value
Stat.
α
7.42
± 0.58
m 3/2
10171
± 882
nm
0.52
± 0.09
nH
1.17 −0.07, +0.22
tanβ
33.1
± 7.8
TABLE XVII :
XVIIModel parameters and spectrum at a new stau coannihilation benchmark point with heavier gluino. All masses are in GeV.Parameter Value Particle Mass
α
10g
1183
m 3/2
9500χ 0
2
499
nm
0.5χ 0
1
337
nH
1.0τ
361
tanβ
37q
1119
TABLE XVIII :
XVIIISolution to gaugino masses at 15 fb −1 for the new stau coannihilation benchmark point with heavier gluino. All masses are in GeV.Particle Mass 15fb −1 Stat.g
1186
± 84
χ 0
2
527
−30, +60
χ 0
1
317
± 33
VII. ACKNOWLEDGEMENTSWe would like to thank John Conley and JamieTatter
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|
[] |
[
"Thermoelectric Transport Coefficients for Massless Dirac Electrons in Quantum Limit",
"Thermoelectric Transport Coefficients for Massless Dirac Electrons in Quantum Limit"
] |
[
"Igor Proskurin \nDepartment of Physics\nUniversity of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan\n\nDepartment of Theoretical Physics\nInstitute of Natural Sciences\nUral Federal University\nMira 19620002EkaterinburgRussia\n",
"Masao Ogata \nDepartment of Physics\nUniversity of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan\n"
] |
[
"Department of Physics\nUniversity of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan",
"Department of Theoretical Physics\nInstitute of Natural Sciences\nUral Federal University\nMira 19620002EkaterinburgRussia",
"Department of Physics\nUniversity of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan"
] |
[
"Journal of the Physical Society of Japan LETTERS"
] |
We perform an analytic calculation of thermoelectric transport coefficients for massless Dirac electrons using the approach based on the Kubo-Středa formula and generalizedMott's relation. The main focus of the letter is made on the properties of the Nernst coefficient in the vicinity of the Dirac point in quantum limit. We calculate magnetic field and temperature dependencies of the Nernst coefficient and compare our results with recent experiments in α-(BEDT-TTF) 2 I 3 organic conductor. We argue that the Zeeman splitting is important to understand the experimental data at high magnetic fields.Unusual thermoelectric properties of graphene have attracted considerable interest.In graphene conducting electrons can be described by a Weyl equation that in quantizing magnetic field leads to relativistic Landau levels with energies ± ω c √ n (n = 0, 1, . . .)with ω c and being the cyclotron frequency and Plank's constant respectively. The important difference of relativistic Landau levels from the non-relativistic case is the existence of n = 0 level with zero energy. In quantizing magnetic field, when the chemical potential is close to n = 0 Landau level, Seebeck and Nernst coefficients show an anomalous behaviour. 1, 2) Massless Dirac fermions were also experimentally found in α-(BEDT-TTF) 2 I 3 organic conductor under pressure. 3, 4) Tight binding model 5, 6) and band structure calculations 7, 8) revealed the existence of the Dirac point in this material with conducting electrons obeying the tilted Weyl equation. Recently, an anomalously large Nernst signal *
|
10.7566/jpsj.82.063712
|
[
"https://arxiv.org/pdf/1305.3343v2.pdf"
] | 119,112,946 |
1305.3343
|
283fdb5082a1ad7f99979407f17d35a471ca6be6
|
Thermoelectric Transport Coefficients for Massless Dirac Electrons in Quantum Limit
15 May 2013
Igor Proskurin
Department of Physics
University of Tokyo
7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan
Department of Theoretical Physics
Institute of Natural Sciences
Ural Federal University
Mira 19620002EkaterinburgRussia
Masao Ogata
Department of Physics
University of Tokyo
7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan
Thermoelectric Transport Coefficients for Massless Dirac Electrons in Quantum Limit
Journal of the Physical Society of Japan LETTERS
15 May 2013Dirac electronsDirac pointthermoelectric coefficientsgrapheneα-(BEDT-TTF) 2 I 3Nernst effectgeneralized Mott's formula
We perform an analytic calculation of thermoelectric transport coefficients for massless Dirac electrons using the approach based on the Kubo-Středa formula and generalizedMott's relation. The main focus of the letter is made on the properties of the Nernst coefficient in the vicinity of the Dirac point in quantum limit. We calculate magnetic field and temperature dependencies of the Nernst coefficient and compare our results with recent experiments in α-(BEDT-TTF) 2 I 3 organic conductor. We argue that the Zeeman splitting is important to understand the experimental data at high magnetic fields.Unusual thermoelectric properties of graphene have attracted considerable interest.In graphene conducting electrons can be described by a Weyl equation that in quantizing magnetic field leads to relativistic Landau levels with energies ± ω c √ n (n = 0, 1, . . .)with ω c and being the cyclotron frequency and Plank's constant respectively. The important difference of relativistic Landau levels from the non-relativistic case is the existence of n = 0 level with zero energy. In quantizing magnetic field, when the chemical potential is close to n = 0 Landau level, Seebeck and Nernst coefficients show an anomalous behaviour. 1, 2) Massless Dirac fermions were also experimentally found in α-(BEDT-TTF) 2 I 3 organic conductor under pressure. 3, 4) Tight binding model 5, 6) and band structure calculations 7, 8) revealed the existence of the Dirac point in this material with conducting electrons obeying the tilted Weyl equation. Recently, an anomalously large Nernst signal *
was reported in this system at high magnetic fields. 9) Recently, the Nernst effect was intensively studied for massless Dirac fermions in graphene [10][11][12] and on the surface of a topological insulator. 13) Various theoretical approaches showed that the Nernst coefficient is greatly enhanced and can considerably exceed the Seebeck coefficient when the chemical potential is close to the Dirac point. [10][11][12] This behaviour is sharply contrasted from the case with finite chemical potential in which the behaviour of the the transport coefficients is consistent with previous theoretical predictions for the non-relativistic two-dimensional electron gas. 14,15) From a theoretical viewpoint the calculation of the Nernst coefficient is a rather challenging problem since the standard linear response approach, based on the Kubo formula, gives unphysical divergence at zero temperature. Corrections arising from the thermal magnetization should be taken into account. 16,17) Now it is well established that for the case of non-interacting electrons the thermopower tensor satisfies the generalized Mott's relation 16,17)
S = σ −1 (T, µ) eT ∞ −∞ dǫf ′ (ǫ) (ǫ − µ) σ(0, ǫ).(1)
where T is temperature, e > 0 is an electron charge, µ is chemical potential, σ is conduc-
tivity tensor, f (ǫ) = {1 + exp [(ǫ − µ) /k B T ]} −1 is Fermi-Dirac distribution function, f ′
denotes the derivative with respect to ǫ, and k B is Boltzmann constant. The important property of the Mott's formula is that, to calculate the thermopower tensor, one only needs to know the conductivity at T = 0 as a function of µ, σ(0, µ).
The present letter is mainly devoted to the properties of the Nernst coefficient of massless Dirac fermions in the vicinity of the Dirac point in quantum limit where the distance between Landau levels is greater than temperature and impurity broadening.
For this purpose we perform an analytic calculation of thermoelectric coefficients using the Kubo-Středa formula for conductivity and generalized Mott's relation. We ignore the possible tilting of the Dirac cone and consider a simple case of energy-independent damping Γ due to the impurity scattering. In the limiting case when Γ is much less than k B T and ω c , we obtain an analytical expressions for Seebeck and Nernst coefficients.
We show that the magnetic field dependence of the Nernst coefficient in the presence of the Zeeman splitting is different in Γ ≪ k B T and Γ k B T regimes.
In order to obtain the conductivity, σ(T, µ), we consider a system of free massless Dirac electrons confined to a two-dimensional (x, y)-plane moving in a magnetic field B perpendicular to the plane. The model Hamiltonian is given by
H = −v F i=x,y σ i [−i ∂ i + eA i (r)](2)
where v F is Fermi velocity, σ i is Pauli matrix, ∂ i denotes a derivative with respect to i = x, y, and A(r) = (−By, 0, 0) is a magnetic vector potential in the Landau gauge.
The solution of the eigenvalue problem for the Hamiltonian (2) leads to relativistic
Landau levels E nα = α ω c √ n (n = 0, 1, . . .) where α = ±1 is the band index and the cyclotron frequency is given by ω c = √ 2v F /l B with l B = /eB being the magnetic length. The corresponding eigenfunctions are
ψ k0 (x, y) = e ikx √ l B L 0 φ 0 y l B − kl B (3)
and
ψ knα (x, y) = e ikx √ 2l B L φ n−1 y l B − kl B αφ n y l B − kl B (4)
for n = 1, 2, . . ., where φ n are the eigenfunctions of a harmonic oscillator. We imply periodic boundary conditions in the x-direction with L and k being the system size in the x-direction and the wave number respectively.
For non-interacting electrons the conductivity can be calculated using the Kubo-
Středa formula 18) σ ij = ie 2 2π ∞ −∞ dǫf (ǫ)Tr v i dG + (ǫ) dǫ A(ǫ)v j − v i A(ǫ)v j dG − (ǫ) dǫ(5)
where Green functions are defined by
G ± (ǫ) = (ǫ − H ± iδ) −1 , A = i(G + − G − ), and v = (i/ ) [H, r]
is the velocity operator. Calculating the trace with eigenfunction (3) and (4) we obtain the following expressions for the diagonal and off-diagonal parts of the conductivity
σ xx = − e 3 v 2 F B 16π 2 αα ′ ∞ n=0 ∞ −∞ dǫf ′ (ǫ)A n+1α (ǫ)A nα ′ (ǫ),(6)σ xy = e 3 v 2 F B 8π 2 αα ′ ∞ n=0 ∞ −∞ dǫf (ǫ) d Re G n+1α (ǫ) dǫ A nα ′ (ǫ) − A n+1α (ǫ) d Re G nα ′ (ǫ) dǫ(7)
where
Re G nα (ǫ) = ǫ − E nα (ǫ − E nα ) 2 + Γ 2 ,(8)A nα (ǫ) = 2Γ (ǫ − E nα ) 2 + Γ 2 .(9)
Here, in order to take into account the impurity scattering, we introduce a damping parameter Γ.
In the low field limit ω c ≪ k B T , we can replace the summation over Landau levels in Eqs. (6) and (7) by an integration over continuous variable E. After performing the integration over E and using the Mott's relation (1), in the leading order in magnetic field, we obtain longitudinal and transversal components of the thermopower in terms of universal functions of ω c /Γ, k B T /Γ, and µ/Γ
S xx = − k B eK 0 xx K 0 xx ,(10)S xy = k B e ω c 2Γ 2 K 0 xxK 0 xy − K 0 xyK 0 xx (K 0 xx ) 2(11)
where
K 0 ij = ∞ −∞ dx cosh 2 1 2 x Φ ij k B T Γ x + µ Γ ,(12)K 0 ij = ∞ −∞ dx x cosh 2 1 2 x Φ ij k B T Γ x + µ Γ .(13)
These formulae are in an agreement with the results obtained previously using slightly different approaches. 10,12) The universal functions Φ ij are the same as obtained previously for the case of longitudinal and Hall conductivities calculations [19][20][21]
Φ xx (x) = 1 + x + 1 x tan −1 x,(14)Φ xy (x) = 1 x 8x 2 3(1 + x 2 ) 2 + 1 + x 2 x tan −1 x − 1 − x 2 1 + x 2 .(15)
From the Eqs. (14) and (15), using the expansion Φ xx ≈ 2, Φ xy ≈ 16x/3 (x ≪ 1) and
Φ xx ≈ (π/2)|x|, Φ xy ≈ (π/2) sgn x (x ≫ 1)
, the following asymptotic behaviour for the Nernst coefficient at µ = 0 can be obtained: 10, 12)
S xy = 2π 2 k 2 B T ( ω c ) 2 9eΓ 3 , for k B T ≪ Γ.(16)
and
S xy = ( ω c ) 2 4eΓT , for k B T ≫ Γ.(17)
On the other hand, in the quantum limit where Landau levels are well separated ω c ≫ max{k B T, Γ}, one needs to evaluate Eqs. (6) and (7) numerically, except for the case when Γ ≪ k B T . In this case one can approximate the Lorentzian in Eq. (9) by a δ-function. Applying this approximation to Eqs. (6) and (7) and using the Mott's formula (1) we obtain the following analytic results for thermoelectric coefficients
S xx = − k B e K xyKxy − (Γ/k B T ) 2 K xxKxx K 2 xy + (Γ/k B T ) 2 K 2 xx ,(18)S xy = Γ eT K xxKxy + K xyKxx K 2 xy + (Γ/k B T ) 2 K 2 xx(19)
where functions of ε nα = E nα /(2k B T ) and x = µ/(2k B T ) are introduced
K xx = 1 4 sech 2 x + 1 2 ∞ n=1 α n sech 2 (x − ε nα ) ,(20)K xx = 1 2 x cosh 2 x + ∞ n=1 α n (x − ε nα ) cosh 2 (x − ε nα ) ,(21)K xy = 1 2 tanh x + 1 4 ∞ n=1 α sinh 2x cosh (x − ε nα ) cosh (x + ε nα ) ,(22)K xy = ϕ (x) + ∞ n=1 α ϕ (x − ε nα ) ,(23)
and ϕ(z) = log (2 cosh z) − z tanh z. The corrections to K xy due to the impurity scattering are of order Γ/ ( ω c ), 22) and can be omitted.
In is important that, Eqs. (18) and (19)
S xx = − k B eK xy K xy ,(24)S xy = Γ eT K xxKxy + K xyKxx K 2 xy .(25)
Here the thermopower has a sequence of peaks near the Landau levels. At low temperatures at each Landau level S xx has a universal value − sgn n(k B /e) log 2/n, while the Nernst coefficient in this region is small in comparison with S xx .
In the latter case (µ ≈ 0), the behaviour of the thermoelectric coefficients changes significantly since K xy vanishes in the vicinity of n = 0 Landau level. As a result, the Nernst coefficient has a large peak at µ = 0 with the value given by
S xy = k 2 B T eΓK xy K xx .(26)
At low T the peak saturates at the value 4k 2 B T log 2/(eΓ), while S xx vanishes. In this region, for non-zero but small µ, the Nernst coefficient can considerably exceed the Seebeck coefficient.
In order to describe the behaviour of the thermoelectric coefficients in high magnetic fields we also take into account Zeeman splitting of Landau levels E nα → E nα ± ∆ where ∆ = (1/2)gµ B B, µ B and g are the Bohr magneton and g-factor respectively.
In magnetic field up to 10 T the Zeeman splitting is small compared with ω c . For magnetic field ω c ≫ k B T , the Nernst coefficient in the case without Zeeman splitting (g = 0) saturates as qualitatively predicted by Eq. (26). This saturating behaviour changes to a decay when Zeeman splitting becomes the same order as temperature and impurity broadening. In low magnetic field the asymptotic behaviour of S xy is described by Eqs. (16) and (17).
The magnetic field dependence of the Nernst coefficient in the presence of Zeeman splitting is different in the (a) Γ ≪ k B T and (b) Γ k B T limits which reflects the different mechanisms of Landau level broadening. Figure 2 shows the magnetic field dependence of the Nernst coefficient for several values of Γ/(k B T ). In the case (b) with large Γ/(k B T ) (dot-dash line in Fig. 2), S xy decreases monotonically except for the very vicinity of B = 0. In contrast, in the case (a) with small Γ/(k B T ) (solid line in Fig. 2),
there is a region in which S xy increases. In this case the increase the Nernst coefficient is understood from Eqs. (20), (23), and (26). Actually, the asymptotic behaviour for large ∆/(k B T ) is given by However, the Nernst coefficient starts to decrease in the large B region. This is understood as follows. For sufficiently large ∆ the impurity broadening at µ = 0, given by Lorentzian, becomes dominant over the temperature broadening which has exponential decay, as illustrated in Fig. 3 (a). This effect causes the decay of the Nernst coefficient in the large B region where the contribution at µ = 0 comes from the overlap of the split n = 0 Landau level. In case (b), the impurity broadening is always dominant over temperature broadening, as illustrated in Fig. 3 (b), and behaviour of the Nernst coefficient is similar to that shown in Fig. 1 and by dash-dot line in Fig. 2.
S xy = k 2 B T eΓ 1 + ∆ k B T + 3 2 + ∆ k B T e −∆/(k B T ) + O(e −2∆/(k B T ) ).(27)
The temperature dependence of the Nernst coefficient at µ = 0 for T = 1.5 K and v F = 0.5 × 10 5 m/s, Γ/k B = 3.75 K, and g = 2, calculated from Eqs. (1), (6) and (7), is shown in Fig. 4 for several values of magnetic field. For large magnetic fields, S xy shows activation behaviour at low temperature due to the Zeeman splitting of n = 0
Landau level, while for small B the temperature dependence at low temperature is approximately linear as predicted by Eq. (26). The position of the peak corresponds to the temperature when different Landau levels start to overlap which separates the quantum limit ( ω c ≫ k B T ) from the low field limit ( ω c ≪ k B T ). In the latter case the asymptotic behaviour at high temperatures is given by Eq. (17).
In summary, we have calculated longitudinal and transverse components of the thermopower in quantum limit. For the Nernst coefficient we have calculated the magnetic field and temperature dependencies at µ = 0. These results with the Zeeman term are qualitatively consistent with the recent experiments in α-(BEDT-TTF) 2 I 3 organic conductor, 9) although there are some quantitative discrepancies. First, for the g-factor of g = 2 the decay rate of S xy in Fig. 1 as a function of the magnetic field is about a factor 2 smaller than that of experiment. To reproduce the experimental decay rate in our theory, we need to assume g ≈ 6, which is similar to the effective g-factor discussed in Ref. 24) Second, the positions of the peaks on the temperature dependencies shown in Fig. 4 are shifted to higher temperatures than in the experiment. The origin of this shift, as pointed out in Ref., 25) may arise from magnetic field dependence of Γ due to the presence of charged impurities. 19,22) In order to achieve better agreement between the theory and experiments in α-(BEDT-TTF) 2 I 3 , it will be necessary to take into account the tilting of the Dirac cone and to use more realistic model for impurity scattering.
interpolate two typical cases: (I) when µ is away from the Dirac point and (II) when µ is close to the Dirac point (µ ≈ 0). In the former case, one can safely neglect the terms of order (Γ/k B T ) 2 , and obtain the results similar to the case of non-relativistic two-dimensional electron gas14,15)
Fig. 1 .
1Magnetic field dependence of the Nernst coefficient at µ = 0 for T = 1.5 K, v F = 0.5 × 10 5 m/s, and Γ/k B = 3.75 K for different values of g-factor.
vFig. 2 .
2F = 0.5 × 10 5 m/s, g = 2, and B = 1 T the ratio ∆/ ω c ≈ 0.03. The resulting magnetic field dependencies of the Nernst coefficient at µ = 0 for T = 1.5 K and v F = 0.5×10 5 m/s, and Γ/k B = 3.75 K for different values of g are shown in Fig. 1. To obtain Fig. 1 we perform the numerical summation over Landau levels in Eqs. (6), (7) and use the Mott's formula (1). The value Γ/k B = 3.75 K is chosen similar to that evaluated previously from the magnetoresistance calculations for α-(BEDT-TTF) 2 I 3 . 23) Magnetic field dependence of the Nernst coefficient at µ = 0 for T = 1.5 K, v F = 0.5 × 10 5 m/s, and g = 2 for different values of Γ/(k B T ).
Fig. 3 .
3The difference between the Landau level broadening in (a) Γ < k B T and (b) Γ > k B T case. The impurity (solid) and temperature (dashed) broadenings are given by Lorentzian Γ 2 / (µ ± ∆) 2 + Γ 2 and sech 2 [(µ ± ∆)/2k B T ] respectively.
Fig. 4 .
4Temperature dependence of the Nernst coefficient at µ = 0 for T = 1.5 K, v F = 0.5×10 5 m/s, and Γ/k B = 3.75 K for different values of magnetic field B.
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[
"Spontaneous Edge Accumulation of Spin Currents in Finite-Size Two-Dimensional Diffusive Spin-Orbit Coupled SF S Heterostructures",
"Spontaneous Edge Accumulation of Spin Currents in Finite-Size Two-Dimensional Diffusive Spin-Orbit Coupled SF S Heterostructures"
] |
[
"Mohammad Alidoust \nDepartment of Physics\nUniversity of Basel\nKlingelbergstrasse 82CH-4056BaselSwitzerland\n\nDepartment of Physics\nFaculty of Sciences\nUniversity of Isfahan\nHezar Jerib Avenue81746-73441IsfahanIran\n",
"Klaus Halterman \nMichelson Lab, Physics Division\nNaval Air Warfare Center\n93555China LakeCaliforniaUSA\n"
] |
[
"Department of Physics\nUniversity of Basel\nKlingelbergstrasse 82CH-4056BaselSwitzerland",
"Department of Physics\nFaculty of Sciences\nUniversity of Isfahan\nHezar Jerib Avenue81746-73441IsfahanIran",
"Michelson Lab, Physics Division\nNaval Air Warfare Center\n93555China LakeCaliforniaUSA"
] |
[] |
We theoretically study spin and charge currents through finite-size two-dimensional s-wave superconductor/uniform ferromagnet/s-wave superconductor (S/F /S) junctions with intrinsic spin-orbit interactions (ISOIs) using a quasiclassical approach. Considering experimentally realistic parameters, we demonstrate that the combination of spontaneously broken time-reversal symmetry and lack of inversion symmetry can result in spontaneously accumulated spin currents at the edges of finite-size two-dimensional magnetic S/F hybrids. Due to the spontaneous edge spin accumulation, the corners of the F wire host the maximum spin current density. We further reveal that this type edge phenomena are robust and independent of either the actual type of ISOIs or exchange field orientation. Moreover, we study spin current-phase relations in these diffusive spin-orbit coupled S/F /S junctions. Our results unveil net spin currents, not accompanied by charge supercurrent, that spontaneously accumulate at the sample edges through a modulating superconducting phase difference. Finally, we discuss possible experimental implementations to observe these edge phenomena.
|
10.1088/1367-2630/17/3/033001
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[
"https://arxiv.org/pdf/1504.05950v1.pdf"
] | 54,828,178 |
1504.05950
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0429349494df93bad4660b2622477c87a4e9d8c1
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Spontaneous Edge Accumulation of Spin Currents in Finite-Size Two-Dimensional Diffusive Spin-Orbit Coupled SF S Heterostructures
Mohammad Alidoust
Department of Physics
University of Basel
Klingelbergstrasse 82CH-4056BaselSwitzerland
Department of Physics
Faculty of Sciences
University of Isfahan
Hezar Jerib Avenue81746-73441IsfahanIran
Klaus Halterman
Michelson Lab, Physics Division
Naval Air Warfare Center
93555China LakeCaliforniaUSA
Spontaneous Edge Accumulation of Spin Currents in Finite-Size Two-Dimensional Diffusive Spin-Orbit Coupled SF S Heterostructures
(Dated: April 24, 2015)numbers: 7450+r7445+c7425Ha7478Na
We theoretically study spin and charge currents through finite-size two-dimensional s-wave superconductor/uniform ferromagnet/s-wave superconductor (S/F /S) junctions with intrinsic spin-orbit interactions (ISOIs) using a quasiclassical approach. Considering experimentally realistic parameters, we demonstrate that the combination of spontaneously broken time-reversal symmetry and lack of inversion symmetry can result in spontaneously accumulated spin currents at the edges of finite-size two-dimensional magnetic S/F hybrids. Due to the spontaneous edge spin accumulation, the corners of the F wire host the maximum spin current density. We further reveal that this type edge phenomena are robust and independent of either the actual type of ISOIs or exchange field orientation. Moreover, we study spin current-phase relations in these diffusive spin-orbit coupled S/F /S junctions. Our results unveil net spin currents, not accompanied by charge supercurrent, that spontaneously accumulate at the sample edges through a modulating superconducting phase difference. Finally, we discuss possible experimental implementations to observe these edge phenomena.
I. INTRODUCTION
Spintronics devices operate by spin transport mechanisms 1-6 rather than by utilizing charged carriers, as is done typically in conventional electronics devices. The use of spin currents can result in higher speeds and reduced dissipation 2,6 while exhibiting weak sensitivity to nonmagnetic impurities and temperature. [4][5][6] For functional spin-based devices, it is necessary to manipulate and generate spin-currents in a practical and efficient manner. For this reason, many investigations have focused on harnessing the spin-orbit interactions 12,13 (SOIs) present in many materials, including semiconductors. [7][8][9][10][11][14][15][16][17] The SOI is a quantum relativistic phenomenon that can be divided into two categories: i) intrinsic (originating from the electronic band structure of the material) and ii) extrinsic (originating from spin-dependent scattering of impurities). [4][5][6] The intrinsic spin-orbit interactions (ISOIs) such as Rashba 12 and Dresselhaus 13 , are experimentally controllable via tuning a gate voltage 16,[18][19][20][21][22][23][24] . This particular attribute has proliferated efforts striving for high-performance spin-based devices, including transistors, and new routes in information storage and transport. 3,15,16,20,[25][26][27] Similarly, ferromagnet (F ) and superconductor (S) heterostructures have received renewed interest lately due to the possibility of generating spin polarized triplet supercurrents [28][29][30][31][32][33][34][35][36][37] that can be used for practical purposes 32 . By considering a ferromagnet with an ISOI, the spin orbit interaction can couple with the magnetic exchange field, resulting in modified superconducting proximity effects and additional venues for new spin phenomena. Indeed, the ISOI can induce long-range proximity effects in uniformly magnetized S/F structures due to the momentum-dependence of the effective exchange field 73 . It is therefore of fundamental importance not only to find a simple, experimentally accessible structure that can support spin currents in F/S systems, but it is also crucial to determine the spatial behavior of the spin currents near the boundaries of the superconducting hybrids.
Many past works are based on the application of external electric or magnetic fields. One of the earliest such cases involved the combination of SOIs and an external electric field, giving rise to an accumulation of spin currents at the edges 41,42 (the so called spin-Hall effect 39,40 ). The spin currents generally tend to peak near the sample boundaries and vanish at the electrode/sample interfaces. [44][45][46][47] These theoretical predictions were later experimentally observed in semiconductor samples 43 . The spin-Hall phenomenon was also extensively studied in superconducting heterostructures where various types of spin-orbit coupling (SOC) play key roles. 23,[50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67] For example, the out-of-plane component of the spin density was theoretically investigated 50 in a spin-orbit coupled S/N/S junction [with normal metal (N ) interlayer] subject to an inhomogeneous external magnetic field. It was found that the spin density varies along the transverse direction, leading to a longitudinal phase difference between the S electrodes. The influence of extrinsic SOIs on the critical supercurrent in diffusive magnetic hybrid structures was also studied. 71,72 In nonmagnetic S/N /S Josephson junctions with SOC subject to a magnetic field, 0-π transitions may be induced. 29 In an earlier work 74 , singlet-triplet pair conversion was numerically investigated using a lattice model in a ballistic half metal ferromagnetic Josephson junction with an interfacial Rashba SOC. Several optimal configurations have also been theoretically studied for generating and detecting the predicted longrange triplet correlations in experiments. 73 In this paper, we study the local spin currents in uniformly magnetized S/F /S Josephson junctions with spatially uniform intrinsic SOIs, avoiding any external electric or magnetic fields. We employ a two-dimensional quasiclassical Keldysh-Usadel approach that incorporates a generic spin-dependent vector potential to study the behavior of the spin current components. We consider two types of ISOCs: i) Rashba and ii) Dresselhaus SOC, and the magnetization of the F wire can take arbitrary orientations. We find that the coupling of the F wire's exchange field and ISOIs leads to edge spin currents with three nonzero components. The spin current density components peak near the edges of the F strip and sharply decline when moving away from the edges. Therefore, the maximal spin current accumulation takes place near the F wire's corners. This phenomenon can be also observed in ISO coupled S/N /S junctions with a single spin active N /S interface as demonstrated in Ref. 75. Our results show that the spin current can be switched on or off at the S/F contacts, depending on the magnetization direction. The spatially averaged spin current components reveal a 2π-periodicity and even-functionality in ϕ, the phase difference between the S terminals. This is in contrast to the charge supercurrent which is a 2π-periodic odd function of ϕ (and consistent with previous studies 68 ). Note that for such junctions, the argument in the current phase relation for some situations can become modified by a simple ϕ 0 shift. 66,69,70 The simple hybrid structure proposed here relies solely on the intrinsic properties of the system, in contrast to other structures that rely inextricably on external fields to observe the edge spin currents [39][40][41][42][43][44][45][46]49,50,67 . Our device consists of a finitesize intrinsic SO coupled F wire (with uniform magnetization) sandwiched between two S banks. The spin currents then spontaneously accumulate at the sample edges, without the application of an external electric or magnetic field to the system. We demonstrate that the device allows for the realization of spin currents in the absence of charge supercurrent by simply modulating ϕ. The edge spin accumulation is a signature of the spin Hall effect 67 , and hence can be experimentally measured by optical experiments for instance 43 (see the discussions in Sec. III). Also, we discuss the symmetries present among the spin current components when varying the magnetization orientation with Rashba or Dresselhaus SOC present. Moreover, we compare our results with the charge and spin currents found in a nonmagnetic diffusive S/N /S Josephson junction with Rashba and/or Dresselhaus SOC. We find that the spin currents vanish in the S/N /S devices, consistent with previous works 67,75 , and that the charge current displays a spatially uniform profile without any transverse component, indicating conservation of charge current.
The paper is organized as follows. We outline the theoretical framework used to study hybrid structures with ISOCs in Sec. II. In Sec. III, the results of diffusive S/F /S Josephson junctions are presented for the case of Rashba ISOC. We next utilize the symmetries in the spin currents to give a simple prescription for finding the corresponding results for the Dresselhaus spin-orbit interaction. We finally present concluding remarks in Sec. IV.
II. THEORETICAL FORMALISM
The intrinsic SOI is a consequence of the moving carriers' spin interaction with an atomic potential V (r). Therefore, the total Hamiltonian of a moving electron in such an atomic potential can be expressed as, 20,76
H = P 2 2m + e 2 4m 2 0 c 2 P · τ × ∂V (r) ,(1)
where m 0 = 0.51Mev, is the free electron mass and and c is the velocity of light in vacuum. We define P to represent the electron's momentum vector, and ∂ ≡ (∂ x , ∂ y , ∂ z ). The vector of Pauli matrices, τ , is given in Appendix A. It has been shown that the linearized SOC term can be simply accounted for as an effective background field that follows SU(2) gauge symmetries. 47,73,77,78 Hence, it is sufficient to replace partial derivatives, appearing in the quasiclassical formalism, by their covariants. 47,73 Another advantage of the SU(2) approach is the convenient definition of physical quantities such as spin currents. 79 We start with the Usadel equations that enable us to study the charge and spin transport through diffusive F/S systems with the ferromagnetic regions having arbitrary magnetization patterns h(r) = h x (r), h y (r), h z (r) : 28,73,80
∂ ,Ĝ(r)[∂,Ĝ(r)] = −i D ερ 3 + diag[H(r), H T (r)],Ĝ(r) ,(2)H(r) = h(r) · σ,
where ρ and σ denote vectors comprised of 4 × 4 and 2 × 2 Pauli matrices (see Appendix A), and D represents the diffusive constant of the ferromagnetic medium. We have denoted the quasiparticles' energy by ε which is measured from the Fermi surface ε F . Throughout this work, we focus on the low proximity limit of the diffusive regime 28 . In this limit, the normal and anomalous components of the Green's function can be approximated by, F no (r) 1 and F (r) 1, respectively. Thus, the advanced component of total Green's function,Ĝ(r), takes the following form:
G A (r, ε) ≈ −1 −F (r, −ε) F * (r, ε) 1 ,(3)
where each entry stands for a 2×2 matrix block. Considering the Taylor expansion, the advanced component can be given by:Ĝ
A (r, ε) = −1 0 −f ↑↑ (r, −ε) −f − (r, −ε) 0 −1 −f + (r, −ε) −f ↓↓ (r, −ε) f * ↑↑ (r, ε) f * − (r, ε) 1 0 f * + (r, ε) f * ↓↓ (r, ε) 0 1 . (4)
Here we restrict our calculations to the equilibrium situations where the Retarded and Keldysh blocks of total Green's function are obtained by:Ĝ A (r) = − ρ 3Ĝ R (r)ρ 3 † , and
G K (r) = tanh(εk B T /2) Ĝ R (r) −Ĝ A (r) .
Here, k B and T denote the Boltzmann constant and system temperature, respectively. The Usadel equation, Eq. (2), leads to sixteen coupled complex partial differential equations in the low proximity limit that become highly complicated with the presence of intrinsic SOI terms. Unfortunately, the resultant system of coupled differential equations can only be simplified and decoupled under very limiting conditions, 28,29 leading to analytical results. However, for the systems considered in this paper, numerical methods are the most appropriate, and often the only possible routes to investigate the relevant transport properties. 73 The differential equations must be supplemented by the appropriate boundary conditions to properly capture the transport characteristics of S/F /S hybrid structures. We thus employ the Kupriyanov-Lukichev boundary conditions at the S/F interfaces 81 and control the intensity of induced proximity correlations using the barrier resistance parameter, ζ:
ζ Ĝ (r)∂Ĝ(r) ·n = [Ĝ BCS (θ),Ĝ(r)],(5)
wheren is a unit vector, directed perpendicular to a given interface. The solutions to Eqs. (2) for a bulk, even-frequency s-wave superconductor reads,
G R BCS (θ) = 1 cosh ϑ(ε) iσ 2 e iθ sinh ϑ(ε) iσ 2 e −iθ sinh ϑ(ε) −1 cosh ϑ(ε)
, (6) in which,
ϑ(ε) = arctanh( | ∆ | ε ),
is defined in terms of the superconducting gap ∆. Here the macroscopic phase of the bulk superconductor is denoted by θ, so that the difference between the macroscopic phases of the left and right S electrodes are given by θ l − θ r = ϕ. For more compact expressions in our subsequent calculations, we define the following piecewise functions:
s(ε) ≡ e iθ sinh ϑ(ε) = −∆ sgn(ε) √ ε 2 − ∆ 2 Θ(ε 2 − ∆ 2 ) − i √ ∆ 2 − ε 2 Θ(∆ 2 − ε 2 ) , c(ε) ≡ cosh ϑ(ε) = | ε | √ ε 2 − ∆ 2 Θ(ε 2 − ∆ 2 ) − iε √ ∆ 2 − ε 2 Θ(∆ 2 − ε 2 ),
where Θ(x) stands for the usual step function. It is clear that the general boundary conditions given by Eq. (5) do not permit current flow through the hard wall boundaries of the finite-size two-dimensional S/F /S Josephson junction, shown in Fig. 1.
To study the influence of differing types of ISOI on the system transport characteristics, we adopt a spin-dependent tensor vector potential A(r) = A x (r), A y (r), A z (r) , as follows: 47,73,75,77,78
A x (r) = 1 2 A x x (r)τ x + A y x (r)τ y + A z x (r)τ z ,(7a)A y (r) = 1 2 A x y (r)τ x + A y y (r)τ y + A z y (r)τ z ,(7b)A z (r) = 1 2 A x z (r)τ x + A y z (r)τ y + A z z (r)τ z .(7c)
FIG. 1. (Color online) Schematic of a finite-size two-dimensional magnetic S/F /S Josephson junction. The superconducting electrodes and rectangular ferromagnetic nano-wire are labelled S and F , respectively. We assume that the quasiparticle current experiences an intrinsic spin-orbit interaction (ISOI) solely inside the F region.
The thickness and width of the ferromagnetic strip are labeled dF and WF , respectively. The junction is located in the xy plane and the S/F interfaces are along the y axis. The F region has a uniform exchange field denoted by h and can take arbitrary orientations h x , h y , h z .
Using the above vector potential, we define the covariant derivatives by;∂
≡ ∂1 − ie A(r).(8)
Accordingly, the brackets seen in the Usadel equation, Eq. (2), and the boundary conditions, Eq. (5), (as well as the charge and spin currents that shall be discussed below, Eqs. (10) and (11)) take the following form:
[∂,Ĝ(r)] = ∂Ĝ(r) − ie[ A(r),Ĝ(r)].(9)
The spin and charge currents are key quantities that lend insight into the fundamental system transport aspects that provide valuable and crucial information for nanoscale elements in superconducting spintronics devices, as described in the introduction. Under equilibrium conditions, the vector charge ( J c ) and spin ( J sγ ) current densities can be expressed by the Keldysh block as follows: 47,78
J c (r, ϕ) = J c 0 +∞ −∞ dεTr ρ 3 Ǧ (r)[∂,Ǧ(r)] K ,(10)J sγ (r, ϕ) = J s 0 +∞ −∞ dεTr ρ 3 ν γ Ǧ (r)[∂,Ǧ(r)] K ,(11)
where J c 0 = N 0 eD/4, J s 0 = J c 0 /2e, and N 0 is the number of states at the Fermi surface. The vector current densities determine the local direction and amplitude of the currents as a function of coordinates inside the F strip. In other words, J(r), provides a spatial map to the currents inside the system. We designate γ = x, y, z for the three components of spin current, J sγ . The matrices we use throughout our derivations are given in Appendix A. To obtain the total Josephson charge current flowing through the magnetic strip, an additional integration over the y direction should be performed on Eq. (10) (see Fig. 1). The spin-dependent fields yield lengthly and cumbersome expressions, the details of which are not presented here for clarity. Having now outlined the theoretical approach utilized in this paper, we can now present our findings in the next section.
III. RESULTS AND DISCUSSIONS
In our computations below, we consider a uniform and coordinate-independent vector potential, A(r), i.e. ∂ · A(r) = 0, so that the spin vector potential is constant in the entire F region. A specific choice for the constant spin vector potential that results in Rashba (α) 12 and Dresselhaus (β) 13 types of SOC is,
A x x = −A y y = 2β, A y x = −A x y = 2α, A z x = A z y = 0, A z z = A x z = A y z = 0.(12)
By substituting the above set of parameters into Eqs. (7), we arrive at,
A x = βτ x − ατ y , (13a) A y = ατ x − βτ y .(13b)
The Rashba SOI 20 can be described through spatial inversion asymmetries while the Dresselhaus SOI 13 is described by bulk inversion asymmetries in the crystal structure. 20,21 Crystallographic inversion asymmetries 87 or lack of structural inversion symmetries 16,17,84,87 in heterostructures may cause the ISOIs
y = 0.25ξ S y = 0.5ξ S y = 0.75ξ S y = 1.0ξ S x = 0.25ξ S x = 0.5ξ S x = 0.75ξ S x = 1.0ξ S FIG. 2.
(Color online) Spatial profile of the spin current in a uniformly magnetized S/F /S Josephson (see Fig. 1) junction without ISOI. The magnetic exchange field is oriented along z, h = (0, 0, h z ), and therefore, solely the z component of spin current J sz
x (x, y) is nonvanishing. The junction length and width are set to dF = 2.0ξS and WF = 2.0ξS, respectively. The top panel exhibits the spin current variations along the x-position (the junction length) at four differing locations along the junction width, y = 0.25ξS, 0.5ξS, 0.75ξS, 1.0ξS. The bottom panel shows J sz
x (x, y) as a function of y-position along the junction width, at x = 0.25ξS, 0.5ξS, 0.75ξS, 1.0ξS. considered here. For example, strain can induce such inversion asymmetries 43,87,92,93 and thus, ISOIs, or the adjoining of two differing materials may generate the requisite interfacial SOIs 16,17,73,84,87 . Nonetheless, there is no straightforward method to measure SOIs in a hybrid structure. One possible approach would be first principle calculations 85 in conjunction with spin transfer torque experiments 73,86,87 . The intrinsic SOIs are often given by the first-order quasiparticle momentum, which is locked to their spins. This linearized approach is a simplification to the more generic picture dealing with higher orders of momentum, 21,82,[92][93][94][95] which can be observed in e.g., engineered materials. 92, 93 We here assume that ISOCs can be described by linear terms in the carriers' momentum. 13,20 Candidate materials to support spontaneous broken time-reversal and broken inversion symmetries include electron liquids with ISOIs, which naturally tend to have a Stoner-type magnetism at low densities, and a magnetically doped topological insulator surface (or by directly coating a topological insulator surface with magnetic insulators). [88][89][90] Other promising candidates involve the ferromagnetic semiconductors (Ga,Mn)As, where both the electronic structure and inherent magnetism make these materials well suited for experimental studies. 7,8,91 Our quasiclassical approach allows us to study systems involving nontrivial magnetizations and spin vector potentials with arbitrary spatial patterns. We thus consider a finite-sized, uniformly magnetized F wire whose exchange field can take arbitrary orientations. In order to determine systematically the behaviors of the spin and charge currents, we consider three orthogonal magnetization directions, namely along the x, y, and z axes. In addition, we incorporate pure Rashba (α = 0, β = 0) and Dresselhaus (β = 0, α = 0) SOCs that allow isolation of their effects relative to the physical quantities under study. When finding solutions to the Usadel equation, Eq. (2), and the corresponding current densities [Eqs. (10) and (11)], we have added a small imaginary part, δ ≈ 0.01∆ 0 , to the quasiparticles' energy, ε → ε + iδ, to enhance stability of the numerical solutions. The imaginary part can be physically viewed as accounting for inelastic scatterings. 68 Due to the presence of the finite parameter δ, we take the modulus of the currents in Eqs. (10) and (11). We normalize the quasiparticles' energy, ε, and exchange field h by the gap, ∆ 0 , at T = 0. Also, all lengths are measured in units of the superconducting coherence length ξ S . In our computations, we adopt natural units, so that k B = = 1.
To begin, we consider for comparison purposes, an S/F /S Josephson junction in the absence of SOCs. 28,29 The schematic of the S/F /S structure is depicted in Fig. 1. The parameters ζ = 4, | h| = 10∆ 0 and d F = 2.0ξ S , ensure the validity of low proximity limit considered throughout the paper. To have absolute comparisons, we set h = (0, 0, h z ) and compute the charge and spin currents using Eqs. (10) and (11), respectively. Figure 2 exhibits the spatial map of the spin current for W F = 2.0ξ S (see Fig. 1). Since the magnetization orientation is fixed along the z direction, J sz
x (x, y = y 0 ) is the only nonvanishing component of spin current for a given fixed location y 0 . The top panel of Fig. 2 illustrates the spatial variations of J sz
x (x, y = y 0 ) along the junction length in the x direction x (x, y), J sy x (x, y), and J sz x (x, y) in a uniformly magnetized Rashba S/F /S junction. The exchange field of the ferromagnetic strip points along the z direction: h = (0, 0, h z ) (see Fig. 1). The panels in the top row show the spin current components, J sγ
x (x, y), as a function of x at four differing locations along the junction width: y=0.25ξS, 0.5ξS, 0.75ξS, and 1.0ξS. The bottom row exhibits J sγ
x (x, y) versus y at x=0.25ξS, 0.5ξS, 0.75ξS, and 1.0ξS.
at differing positions along the junction width: y 0 = 0.25ξ S , 0.5ξ S , 0.75ξ S , and 1.0ξ S . The macroscopic phase difference between the S electrodes is set at a representative value, i.e., ϕ = π/2. The bottom panel in Fig. 2 shows J sz x (x = x 0 , y) as a function of y, at x 0 = 0.25ξ S , 0.5ξ S , 0.75ξ S , 1.0ξ S . The results demonstrate that the spin current is y independent in such hybrid junctions, namely J sz
x (x = x 0 , y) = const. (we also have found J sz y (x, y) = 0). In other words, it is appropriate to view this type of system as an effectively onedimensional junction. The variation of J sz
x (x, y) along the x direction is a consequence of spin torque transfer, and hence the spin current is not a conserved quantity. 44,45,47,59 The spin current is maximal at the S/F interfaces and vanishes at the middle of junction, x = 1.0ξ S = d F /2. This is contrast to the charge supercurrent in the F region, which is conserved, and thus has a constant value within the entire F strip (not shown).
To identify some of the salient features in Fig. 2, we consider now a simplified one-dimensional S/F /S system, which permits analytical expressions for the spin current density. To this end, we linearize the Usadel equation, and incorporate the Kupriyanov boundary conditions, where the superconducting electrodes have strong scattering impurities. We also still assume that the magnetization is oriented along z: h = (0, 0, h z ). Correspondingly, we define the dimensionless quantity, λ ± = 2i(ε±h z )/ε T , in which ε T is the Thouless energy, and the dimensionless x coordinate,x = x/d F ∈ [0, 1]. After some straightforward calculations, we obtain the follow-ing expression for the charge current [Eq. (10)]:
J c x (x, ϕ) = J c 0 sin ϕ +∞ −∞ dε 2i tanh(εk B T /2) ζ 2 λ + λ − [s * (−ε)] 2 λ + csc λ − + λ − csc λ + + [s * (ε)] 2 λ + cschλ − + λ − cschλ + .(14)
The charge current in this case is seen to exhibit the usual sin ϕ odd-functionality in the superconducting phase difference. Likewise, by substituting the solutions into Eq. (11), we arrive at the following expressions for the spin current components:
J sx x (x, ϕ) ≡ 0,(15a)J sy x (x, ϕ) ≡ 0,(15b)J sz x (x, ϕ) = J s 0 +∞ −∞ dε 2 tanh(εk B T /2) ζ 2 λ + λ − [s * (ε)] 2 λ + cosh 2xλ − cschλ − cos ϕ + [s * (−ε)] 2 λ + csc 2 λ − (cos λ − + cos ϕ) sin[λ − (1 − 2x)] − λ − csc 2 λ + (cos λ + + cos ϕ) sin[λ + (1 − 2x)] + [s * (ε)] 2 λ + coth λ − cschλ − (sinh[λ − (1 − 2x)] − sinh 2xλ − cos ϕ) − λ − csch 2 λ + (cosh λ + + cos ϕ) sinh[λ + (1 − 2x)] .(15c)
Equations (15a)-(15c) clearly demonstrate that the only nonvanishing component of spin current is J sz x , which is consistent with the exchange field aligned along z. 75 From Eq. (15c), it is also evident that J sz x is an odd function of the coordinatex relative to the middle of the junction (and thus vanishes there), and an even function of the phase difference, ϕ. These features are entirely consistent with the numerical results seen in Fig. 2.
We now incorporate Rashba and Dresselhaus SOCs, while keeping the magnetization orientation intact along the z direction. The ISOIs are confined within the F region and are not present within the S electrodes. Through exhaustive numerical investigations, we have found several symmetries among the components of spin current (discussed below) at three particular directions of the exchange field. Due to the symmetries available among the spin current components, we focus here on Rashba SOC. We emphasize that similar conclusions can be drawn for Dresselhaus SOC through the symmetries described below. Figure 3 exhibits the spatial profiles for the spin current density components, J sx
x (x, y), J sy x (x, y), and J sz
x (x, y). A square ferromagnetic strip is considered, with d F = W F = 2.0ξ S , and the superconducting phase difference is equal to ϕ = π/2. The Rashba SOC coefficient is set to a representative value α = 2.0ξ S , without loss of generality. 73 The top set of panels show J sγ
x (x, y = y 0 ) [γ = x, y, z] as a function of the x coordinate at y 0 = 0.25ξ S , 0.5ξ S , 0.75ξ S , and 1.0ξ S . Whereas the bottom panels represent the same quantities, but now as a function of y at x 0 = 0.25ξ S , 0.5ξ S , 0.75ξ S , and 1.0ξ S . As seen in Fig. 1, the junction length and width are parallel to the x and y axes, respectively. The components J sx
x (x, y = y 0 ) and J sy x (x, y = y 0 ), shown in the top row of Fig. 3, demonstrate that these spin current densities vanish at the S/F contacts. This finding is consistent with previous works involving nonsuperconducting heterojunctions [43][44][45]47,67 . The z component, J sz
x (x, y = y 0 ), however exhibits opposite behavior, and is nonzero at the S/F contacts due to the exchange field, which is oriented along the z axis. Similarly, as seen in Fig. 2, J sz
x (x, y = y 0 ) is finite at the S/F interfaces near the S reservoirs. One of the most important features of the results is seen in the top panels of Fig. 3, where two peaks in J sγ
x (x, y) emerge near the S/F contacts. We restrict the spatial profiles to 0 < x < d F /2 and 0 < y < W F /2, since the results are symmetrical with respect to x = 1.0ξ S = d F /2 and y = 1.0ξ S = W F /2, so that the maxima of J sγ
x (x, y = y 0 ) occurs near the edges of the F wire [at x = 0, and x = d F ]. Turning to the bottom row of panels in Fig. 3, we see that J sγ
x (x = x 0 , y) are nonzero at the vacuum boundaries, y = 0, and y = W F . Here also the largest values in the spin current density components take place near the transverse edges of the F wire (y = 0, and y = W F ). The magnitude of the spin current densities at x = 0.25ξ S are generally larger than the other x positions, in agreement with the results of J sγ
x (x, y = y 0 ) shown in the top row of panels.
We now consider the effects of changing the magnetization alignment in the ferromagnet. Thus, Fig. 4 represents the same Rashba spin-orbit coupled S/F /S junction as in Fig. 3, except the magnetization of the F wire is now ori-ented along the y axis. This specific direction of h leads to J sx
x (x, y = y 0 ) = J sz x (x, y = y 0 ) = 0 at the S/F interfaces and the spin current densities peak near the edges of F wire. The spin current density J sy
x (x, y = y 0 ) however is nonzero at the S/F contacts similarly to J sz
x (x, y = y 0 ) in the configuration where the magnetization points along the z direction (Fig. 3). As mentioned earlier, this nonvanishing behavior is directly related to the exchange field direction which lies now parallel to the y axis. Examining J sγ
x (x = x 0 , y) in the bottom row panels in Fig. 4, the maximal values of J sγ
x (x = x 0 , y) take place near the vacuum boundaries, i.e. y = 0, and y = W F . Our investigations demonstrate similar qualitative trends for the components of J sγ
x (x = x 0 , y) when the magnetization resides along the x axis. Note that the transverse components of the spin currents are nonzero inside the ISO coupled ferromagnetic wire, i.e., J sγ y (x, y) = 0, and vanish at the vacuum boundaries (y = 0, W F ). We mainly focus here on the J sγ
x (x, y), since the longitudinal components contain the relevant information needed to describe and understand the accumulation of spin current densities at the edges of the structures.
Considering now the previous characterization of the spin current components in systems with ISOCs, we schematically summarize the spatial maps in Fig. 5 for J sγ (x, y). The largest amplitudes of J sγ (x, y) reside near the edges of the F strip, i.e. x = 0, d F and y = 0, W F . We have qualitatively marked these regions by light yellow "ribbons". Therefore, the overlap of maximal amplitudes take place near the corner regions of the F strip. We have marked these areas by dashed curves and with a deeper yellow color. The spatial profiles found here are qualitatively similar to the existence of edge spin currents found in nonsuperconducting heterojunctions with ISOIs, 39-46 except with one crucial difference: here spin accumulation at the edges arises in the absence of an external field. As mentioned in other works 67 , the spin accumulation is a signature of the spin Hall effect. Therefore, the predicted spin accumulation in this paper might be measurable through optical experiments 43 , such as through Kerr rotation microscopy 43 , where spatial profiles of the spin polarizations near the edges can be imaged. An alternate experimental proposal involves multiterminal devices 44,45 . When transverse leads are attached to the lateral edges of a two-dimensional S/F /S junction (borders at y = 0, y = W N in Fig. 1(a)), the spin accumulations at the F wire's edges inject spin currents into the leads. 44,45 The transversely injected spin currents into the lateral leads in turn may induce a voltage drop between the additional leads. 44,45 The Josephson effect is a significant example of a macroscopic quantum phenomenon and is of fundamental importance in determining the properties of dissipationless coherent transport. Thus, the behavior of the spin currents upon varying the macroscopic phase difference is crucial to experiments and applications utilizing spin-Hall effects and spin transport. In Fig. 6, we therefore study the spin current components as a function of the macroscopic phase difference, ϕ, between the S banks. We consider the parameter set used in Fig. 4, including h = (0, h y , 0). In the top set of panels, the spatial variations of J sγ
x (x, y) are plotted at ϕ = 0, 0.2π, 0.4π, and x (x, y), J sy x (x, y), and J sz x (x, y) in an S/F /S system. The Rashba ferromagnetic wire's width and length are equal to WF = dF = 2.0ξS. The exchange field of the ferromagnetic strip is fixed along the y direction, h = (0, h y , 0). Top row: spatial behavior of J sγ
x (x, y) along the junction length, x, at y = 0.25ξS, 0.5ξS, 0.75ξS, 1.0ξS. Bottom row: spatial variations of J sγ
x (x, y) along the junction width in the y direction at x = 0.25ξS, 0.5ξS, 0.75ξS, 1.0ξS.
1.0π. We have also chosen a representative position along the junction width, corresponding to y = 1.0ξ S = W F /2, which simplifies the analysis while maintaining the generality of the discussion. Although J sγ x (x, y) has a minimum at y = 1.0ξ S , it exhibits the same trends as a function of ϕ compared to the other positions inside the F wire. By increasing the superconducting phase difference from 0 to π, the amplitudes of the spin current components decrease overall. In the bottom set of panels of Fig. 4, we illustrate J sγ
x (x, y) x as a function of ϕ, at y = 0.25ξ S , 0.5ξ S , 0.75ξ S , and 1.0ξ S . Here, we denote the spatial average over the x coordinate from 0 to d F by ... x . In order to better visualize the averaged profiles, we use logarithmic scales in the vertical axes of the bot- The light yellow ribbons display edge regions with maximum spin current densities. The induced spin currents can be considered as a response of the intrinsic spin-orbit coupled system to the presence of an exchange field (the combination of spontaneously broken time-reversal symmetries and the lack of inversion symmetries). As shown in Fig. 1, the exchange field of the ferromagnetic wire is uniform and can take arbitrary directions. The regions that carry maximal accumulation of spin currents are qualitatively shown by the semicircular regions. tom row of panels. As seen, the three components of spin current J sγ
x (x, y) x are even-functions of ϕ, with a period of 2π, namely J sγ (2nπ + ϕ) = J sγ (−ϕ), n ∈ Z. This is contrary to the charge supercurrent which is an odd-function of ϕ, i.e., J c (2nπ + ϕ) = −J c (−ϕ) regardless of a finite phase-shift ϕ 0 66,69,70 . These findings are entirely consistent with previous studies of S/F /S Josephson junctions with inhomogeneous magnetization patterns 68 . We here remark that an additional phase-shift ϕ 0 may appear in such junctions due to the coupling of exchange field and ISOIs. 66,69,70 Nonetheless, the explicit current-phase relations simply undergo a shift in ϕ 0 . 66,69 According to the current-phase relations, the charge supercurrent vanishes at certain ϕ that is quite different than the behavior of the spin current components which clearly show nonzero values at the same ϕ. Therefore, these differences in charge and spin currents allows for an examination of edge spin currents without any net charge current in an ISO coupled F wire sandwiched between two S banks.
We are now in a position to discuss symmetries that may arise among the spin current density components for differing magnetization orientations in systems with either Rashba or Dresselhaus SOCs. Our investigations have found that the out-of-plane spin current, J sz (x, y), remains unchanged upon exchanging the Rashba and Dresselhaus SOCs, regardless of the magnetization orientation. This follows from the form of the spin vector potential discussed at the beginning of this section. However, this picture changes for the in-plane J s{x,y} (x, y) components. The x and y components of the spin current become interchanged when transforming from one type of spin-orbit interaction to another. Precisely speaking, by going from Rashba to Dresselhaus spin-orbit coupling, x (x, y) as a function of x along the junction length at ϕ = 0, 0.2π, 0.4π, and 1.0π. The location along the junction width is fixed at the middle of the junction, y = WF /2 = 1.0ξS. The bottom row of panels represents the spatially averaged spin current components over the junction length (denoted by J sγ
x (x, y) x) vs ϕ. The average is performed along four positions: y=0.25ξS, 0.5ξS, 0.75ξS, and 1.0ξS. The ferromagnetic wire is a square strip with dF = WF = 2.0ξS, and exchange field h = (0, h y , 0). one simply needs to exchange indices x and y in the components of both the exchange field and the spin current. Otherwise, everything stays the same. By making use of the simple transformation rules described, one can easily deduce the results of Dresselhaus spin-orbit coupled systems from the plots presented for Rashba spin-orbit coupled S/F /S systems shown in Figs. 3, 4, and 6.
To conclude this section, we briefly discuss the importance of having a magnetic element in the Josephson junction for the effect of spin current edge accumulation to take place spontaneously. We thus take the limiting case of h = 0 in our previous calculations above involving S/F /S junctions. Using otherwise the same geometrical and material parameters, this case was found to produce no spin current, J sγ (x, y) = 0, in the presence of Rashba (α = 0, β = 0) and/or Dresselhaus (β = 0, α = 0) SOIs. These findings are consistent with previous works, 67 where several simplifying approximations were employed for Rashba-based S/N /S systems. Examining also the charge supercurrent, J c (x, y), for both the Rashba and Dresselhaus interactions, we observed a uniform spatial map for the charge current density for all ϕ, with J c x = const., and J c y = 0. In other words, the spin-dependent fields cannot induce transverse charge supercurrents in a diffusive S/N /S junction. This is in stark contrast to its ballistic S/N /S counterpart, where a transverse charge supercurrent (that is, equivalent to a supercurrent flowing along the y direction in our configuration depicted in Fig. 1) was theoretically predicted due to the presence of intrinsic SOIs 83 .
IV. CONCLUSIONS
We have theoretically studied the behavior of spin and charge currents in a finite-size two-dimensional S/F /S Josephson junction with intrinsic spin-orbit couplings. We utilized a two-dimensional Keldysh-Usadel quasiclassical approach that incorporates a generic spin-dependent vector potential. Our results demonstrate that the combination of a uniform magnetization and ISOIs drives the spin currents which spontaneously accumulate at the F wire's edges. The corners of the F wire were shown to host the maximum density of spin currents. (As demonstrated in Ref. 75, similar edge phenomena can be found in finite-size two-dimensional intrinsically spin orbit coupled S/N /S junctions with a single spin active interface. Additionally, it was shown that maximum singlettriplet conversions take place at the corners of N wire nearest the spin active interfaces 75 .) Our investigations show that the spontaneous edge accumulation of the spin currents are robust and can exist at all magnetization orientations, independent of the actual type of ISOIs. Our investigations have also found several symmetries among the spin current components upon varying magnetization orientations coupled to a Rashba or Dresselhaus SOI. By varying the superconducting phase difference, ϕ, between the S banks, we determined the spin and charge currents as a function of phase difference. We have found that net spin currents therefore emerge and accumulate spontaneously at the edges, in the absence of charge flow, when properly modulating ϕ in finite-size two-dimensional intrinsically spin-orbit coupled S/F /S hybrid structures. This work can be viewed as complementary to previous studies involving edge spin currents in non-superconducting spinorbit coupled structures where externally imposed fields were required [42][43][44][45]47,50,67 . We have shown that remarkably, edge spin currents can be spontaneously driven by the coupling of intrinsic properties of a system, i.e. spontaneously broken time-reversal and the lack of inversion symmetries in the absence of any externally imposed field.
online) Spatial behavior of the three spin current components, J sx
online) Spatial profiles of the spin current components; J sx
FIG. 5 .
5(Color online) Qualitative illustration of the edge spin current densities in a Rashba or Dresselhaus spin-orbit coupled S/F /S junction.
FIG
. 6. (Color online) Spin current components for differing values of the superconducting phase difference ϕ in a Rashba spin-orbit coupled S/F /S Josephson junction. The top row of panels shows the spatial variations of J sγ
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[
"A Tissue Engineered Model of Aging: Interdependence and Cooperative Effects in Failing Tissues OPEN",
"A Tissue Engineered Model of Aging: Interdependence and Cooperative Effects in Failing Tissues OPEN"
] |
[
"A Acun \nBioengineering Graduate Program\nUniversity of Notre Dame\nNotre Dame\n46556INUSA\n",
"D C Vural \nDepartment of Physics\n\n",
"& P Zorlutuna \nBioengineering Graduate Program\nUniversity of Notre Dame\nNotre Dame\n46556INUSA\n"
] |
[
"Bioengineering Graduate Program\nUniversity of Notre Dame\nNotre Dame\n46556INUSA",
"Department of Physics\n",
"Bioengineering Graduate Program\nUniversity of Notre Dame\nNotre Dame\n46556INUSA"
] |
[] |
Aging remains a fundamental open problem in modern biology. Although there exist a number of theories on aging on the cellular scale, nearly nothing is known about how microscopic failures cascade to macroscopic failures of tissues, organs and ultimately the organism. The goal of this work is to bridge microscopic cell failure to macroscopic manifestations of aging. We use tissue engineered constructs to control the cellular-level damage and cell-cell distance in individual tissues to establish the role of complex interdependence and interactions between cells in aging tissues. We found that while microscopic mechanisms drive aging, the interdependency between cells plays a major role in tissue death, providing evidence on how cellular aging is connected to its higher systemic consequences.
|
10.1038/s41598-017-05098-2
| null | 28,678,233 |
1707.01974
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52f8b4ded1c3b6115f2bd1fb24460fa6f1317609
|
A Tissue Engineered Model of Aging: Interdependence and Cooperative Effects in Failing Tissues OPEN
Published: xx xx xxxx
A Acun
Bioengineering Graduate Program
University of Notre Dame
Notre Dame
46556INUSA
D C Vural
Department of Physics
& P Zorlutuna
Bioengineering Graduate Program
University of Notre Dame
Notre Dame
46556INUSA
A Tissue Engineered Model of Aging: Interdependence and Cooperative Effects in Failing Tissues OPEN
Published: xx xx xxxx10.1038/s41598-017-05098-2Received: 31 January 2017 Accepted: 24 May 20171 Scientific RepoRts | 7: 5051 | University of Notre Dame, Notre Dame, IN, 46556, USA. 3 Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, 46556, USA. Correspondence and requests for materials should be addressed to P.Z. (email: [email protected])
Aging remains a fundamental open problem in modern biology. Although there exist a number of theories on aging on the cellular scale, nearly nothing is known about how microscopic failures cascade to macroscopic failures of tissues, organs and ultimately the organism. The goal of this work is to bridge microscopic cell failure to macroscopic manifestations of aging. We use tissue engineered constructs to control the cellular-level damage and cell-cell distance in individual tissues to establish the role of complex interdependence and interactions between cells in aging tissues. We found that while microscopic mechanisms drive aging, the interdependency between cells plays a major role in tissue death, providing evidence on how cellular aging is connected to its higher systemic consequences.
Many simple organisms such as ferns, hydra or jellyfish, do not age 1 . They have a constant mortality rate due to intrinsic or extrinsic "accidents". In contrast, in complex species, at any given time, an older individual is more likely to die than a young one. Furthermore the functional form of mortality rate μ(t) exhibits a remarkable cross-species universality from worms and insects to birds and mammals, characterized by a sharp drop early in life, followed by a near-exponential increase (Gompertz Law) followed by a late life plateau [1][2][3] . The lack of aging in simple systems, and the similarity of mortality curves in complex organisms suggest that aging is an emergent, systemic property.
Aging is the outcome of a long evolutionary history [4][5][6] and is microscopically driven by well-understood biomolecular processes 7,8 . However organisms die not because they run out of cells, but rather due to complex systemic problems. Thus, a complete theory of aging must go beyond cellular damage. In our recent theoretical model we described aging, failure, and death in terms of a cascade of failures taking place on a complex network of interdependent nodes 9 . The basic assumption in this model is that nodes malfunction when other nodes they rely on also malfunction. As a result some failures cause others, which in turn propagates further, ultimately leading to a singular system-wide catastrophe. In simulations the catastrophe occurs suddenly as soon as the malfunctions accumulate to a critical point. Furthermore it was shown in ref. 9 that the collapse curves, the damage and repair rate dependence of network lifetimes, and event size distributions are very weakly dependent on the structure of the interdependence network; and theoretically obtained mortality curves were shown to be in good agreement with that observed in a large variety of complex organisms and machines. In this work we experimentally verify the elements of this theory with the use of synthetic tissues and demonstrate for the first time the systemic spread of failures from cellular level to tissue level, and propose a fresh view of systemic aging.
Consider a hypothetical organism whose organs/tissues/cells are spread sufficiently far apart to prohibit any interaction, but maintained well enough to avoid immediate death. Will this organism exhibit a monotonically increasing probability of death or display age associated problems? According to the interdependence network picture, one should see no systemic effects, nor any cascading failures due to lack of intercellular interactions. In this case, one should see a nearly constant mortality rate m = Δn/n only due to aging at the cellular level, thus, an exponentially decaying cell population n(t)~e −mt . However, when cells are nearby, thus interacting, we expect to see deviations from an exponential decay of population. More precisely, we expect to see a monotonically increasing mortality rate Δn/n, which implies an increase of downwards slope of n(t) (for full theoretical details cf. Materials and Methods and ref. 9).
One of the outcomes of this study is to demonstrate the above ideas experimentally, and quantify the interactions between cells that lead to tissue-level failure. While it is not possible to control the cellular level aging characteristics and cell-cell interaction strengths of in vivo tissues parametrically, tissue engineering techniques allow for direct tests on synthetic live tissues and organoids that can recapitulate native ones. These structures are sufficiently complex and controllable that we can explore the hypothetical idea discussed above, to shed light on the nature of aging. Synthetic tissues allow controlling damage and repair rates, cell viability, thus, allow direct observation on how cells influence one other's performance upon failure. Most recently, synthetic tissues were used to investigate various diseases, including age related ones, and successfully established several platforms, such as Alzheimer's disease in-a-dish 10 and Barth syndrome on-a-chip 11 . Therefore, using synthetic tissues, i.e. aging in a dish, to parametrically study a phenomenon as complex as aging, is a highly promising approach.
In this work we study the hierarchical spread of failure from cells to tissues by growing synthetic tissues in well-controlled hydrogel microenvironments in which we vary intercellular distance, environmental stress, and the age of cells. We first establish that systemic aging, in contrast to cellular aging, is real and significant, and that the effect vanishes when well maintained cells are spread far apart. In other words, dense tissues display larger "age-specific mortality", whereas the mortality rate of sparse tissues is near constant. We then determine the relative importance of systemic aging to cell-level aging by comparing the population curves of synthetic tissues made of young cells of varying density (and thus, interaction strength), to that of synthetic tissues made of "aged" cells. Finally, we exchange the culture media of the synthetic tissues made of young and aged cells with different cell population densities, much like the recent parabiosis experiments 12,13 , to identify the mechanism behind systemic aging.
Results and Discussion
Our findings support that aging cannot be solely explained by failures of individual cells but is an emergent phenomenon involving strong intercellular interactions. Specifically: (i) We find that systemic aging is a more important factor than cellular aging (regardless of how aging is induced). A healthy young cell is more likely to die if its neighbors malfunction, than an old or stressed cell with intact neighbors. (ii) We find that cellular aging is tightly coupled to systemic aging, since aging in the cellular level causes cells to lose their ability to interact with surrounding cells. Specifically, we determined that one of the causes underlying systemic aging is the loss of ability to receive or internally process functional cooperative factors from surrounding cells, but not a loss of ability to produce cooperative factors, or a loss of function of the produced cooperative factors. Our aged tissue model consists of neonatal rat cells treated to exhibit senescence markers, and a synthetic polymer, poly(ethylene glycol) (PEG) 4-arm acrylate modified with cell attachment peptide arginine-glycine-aspartic acid (RGD) (Fig. S1A), that provides a controlled, biomimetic 3-D microenvironment [14][15][16][17] . RGD-modified PEG (PEG-RGD) allows for cells to attach and spread similar to native tissue structure, while it prevents them from dividing or migrating since it is not enzymatically degradable by the cells. This allows us to control the localization, hence the distance between the individual cells constituting the tissue. In addition, the PEG-RGD hydrogels provide a stiffness of around 10 kPa for all cell encapsulation densities used (Fig. S1B). This provides a physiologically relevant system for the cell type used in this study (i.e.primary neonatal rat cardiac fibroblasts (CFs)), as the native heart muscle stiffness is 10 kPa at the beginning of the diastole 18 . In addition, the synthetic tissues with different encapsulation densities did not show any significant difference in their stiffness, ruling out any possibility of mechanical microenvironment contributing to the observed differences in cell survival. In order to mimic cellular level senescence, we artificially aged CFs through applying different kinds of cell-level stresses, which we refer to as "pre-aging" conditions. This way we aimed to test the effect of different types of cellular level damage on systemic tissue level failure. Two of the most important hallmarks of aging are cellular senescence and genomic instability 19,20 . Importantly, cellular senescence and DNA damage are not only observed in in vitro culture systems but also observed in aged tissues of various animal models 21 and humans 22,23 . Therefore, to construct physiologically relevant aged tissue models, we induced cellular senescence and DNA damage in CFs either by restricting the cells to a defined attachment area for a month (Chronological Senescence), applying a low dose of oxidative stress in addition to restricting to a defined attachment area for a month (Oxidative Senescence), or by having cells reproduce multiple times until they reach their replicative limit (Replicative Senescence). Several other studies have shown that treatments similar to the pre-aging conditions used in this study induce senescence and DNA damage in vitro: Replicative senescence is induced through multiple population doublings in various cell types [24][25][26] ; senescence associated with oxidative stress was exemplified in literature 27,28 ; and the chronological aging in a skin tissue equivalent has been shown in vitro 29 . As control, we prepared tissues using the cells isolated from young animals and simply passaged twice in regular cell culture conditions (Young). At the end of the pre-aging period, we confirmed that the cells showed the characteristic symptoms of aging for all three pre-aged groups. The pre-aging conditions resulted in cellular senescence in more than 50% of the population, whereas less than 1% of the young cells showed senescence as shown through senescence associated β-galactosidase staining ( Fig. 1A and B). We also tested the cyclin-dependent kinase inhibitor 1 (p21) expression of the cells pre-aged using the 3 abovementioned techniques (Fig. S2A). p21 expression is known to be elevated in aged cells and tissues 30 , thus is used as an aging marker. Therefore, the increased expression of p21 in the pre-aged cells further confirmed the aged phenotype of the cells. In addition, the characteristic enlarged and flattened morphology of replicative senescent cells 31 was observed in our "replicative senescence" pre-aged group (Fig. S2B). We also determined the DNA damage using comet assay in pre-aged and young cells, and observed that all pre-aged groups showed a significantly higher level of DNA damage (p < 0.05) (Fig. 1C) further confirming the success of our artificial aging treatment in presenting symptoms that are associated with aging on cellular level. Once the pre-aging treatment was completed the cells were encapsulated in PEG-RGD hydrogels at different densities ( Fig. 1D) (Fig. S3).
Scientific RepoRts | 7: 5051 | DOI:10.1038/s41598-017-05098-2
To tune the interactions between cells we varied the cell density. We controlled intercellular distance by using 3 different encapsulation densities: 1 × 10 5 (100 K), 1 × 10 4 (10 K), 1 × 10 3 (1 K) per construct. These 3 densities were chosen to achieve different degrees of intercellular communication through different degrees of secreted factors among the cells, representing dependent (100 K) and non-dependent (1 K) populations, respectively. In addition, by using a synthetic hydrogel without enzymatic sequences that can be degraded by the cells, we made sure that this density is maintained throughout the course of the experiment, as the cells can neither move nor proliferate through the material. The synthetic tissues were then cultured in normal cell culture conditions, and stained for dead cells using ethidium homodimer-1 and imaged daily. The number of dead cells was counted daily using imageJ software and staining/imaging/counting continued until at least 98% of the population was recorded to be dead ( Fig. 1E). At this point the tissues were assumed to be dead and the day that this ratio was reached was recorded as the lifetime of the samples. We estimated the cell-cell distance by considering the volume of the whole construct and the total number of cells, assuming the cells were uniformly distributed throughout the gels. We also experimentally determined the probability distribution for nearest neighbor distance of cells for the 3 densities used (Fig. 1F). We observed that the estimated cell-cell distance (170 µm, 80 µm, and 35 µm for 1 K, 10 K, and 100 K, respectively) was achieved in the synthetic tissues with cell-cell distances averaging at 170.1 ± 46.7 µm, 90.7 ± 20.9 µm, and 41.1 ± 11.5 µm for 1 K, 10 K, and 100 K, respectively.
We observed that the synthetic tissues with pre-aged cells died sooner than the synthetic tissues with young cells only when the population was dense enough to interact: for the dense, 100 K, tissues, ones with young cells lived 25 days, while the tissues with pre-aged cells lived 8-10 days. As the population density decreased, the difference between the tissues composed of pre-aged and young cells disappeared ( Fig. 2A). This result suggests that the complex cell-cell interactions provided in a dense population have an effect on aging through cellular interdependence. In addition, aging at the cellular level prohibits these interactions only when the cell population is dense enough to interact while the kind of cellular damage makes little difference.
We also calculated the cell mortality rate and determined how it changes as a function of cell density (and thus, strength of intercellular interactions). We found that cell mortality rate increases as the populations get sparser in 100 K and 10 K tissues (Fig. 2B). Interestingly, for populations lower than 1 K the cells are sufficiently spread apart that systemic aging completely vanishes, and the mortality rate is nearly constant. Denser populations died at an increasing rate as the live cell number decreased due to the cascading failures propagating across the interdependent cell network. In contrast, our sparse groups displayed very little signs of systemic aging, regardless of how the cells were pre-aged in advance.
In addition, as functional cells become sparser over time, both pre-aged and young cells were able to interact less and therefore died at an accelerating rate, which suggests that the disruption of cell-cell interactions in an interdependent population is the essential cause of aging. In other words, systemic aging can happen even without cellular level aging. Thus, individual cell damage must only be a proximal cause of aging.
In a realistic biological setting, an increase in cell failure rate over time logically implies an increase in tissue/ organ mortality rate over time, in the demographic sense. Thus, our observation that larger interactions cause larger increases in mortality rates is in qualitative agreement with how simple organisms (with decentralized interdependence structures) have a near-constant mortality rate while complex organisms (with tightly knit long range interdependence structures) tend to have rapidly increasing mortality rates.
To see the systemic effects clearly, we plot the cell death rate in tissues with different population densities against normalized time (Fig. 2C). While denser populations live longer, they also display stronger aging features, as measured by the increase in cell mortality rate over time. The sparsest group, 1 K, displays weaker signs of aging, indicating that cell deaths are solely due to damage in individual cells but not due to complex interdependence of the entire system. This is further supported by the longer lifetimes of young tissues only at the highest population density, or where the most complex cell-cell interactions are expected to be established (Fig. 2D), whereas the lifetime of all groups was approximately the same at both 10 K and 1 K population densities, regardless the cells were pre-aged or not, and regardless which way the cells were pre-aged.
Another important question about aging is whether aged cells/tissues respond to stress different than young cells/tissues. Accumulation of oxidative stress has been proposed to be a major factor in aging [32][33][34] . It is also present in tissues in high amounts under life threatening conditions such as myocardial ischemia. Thus, we investigated how different pre-aging conditions affect resilience to oxidative stress (Fig. 3). We exposed the synthetic tissues made from young or pre-aged cells to oxidative stress (0.2 mM H 2 O 2 ) throughout the lifetime measurement experiments and determined their death characteristics. We observed that having functional neighbors is more important than being stress-free, as the young cells survived the oxidative stress better only when the population was dense. The ability of the young tissues to cope with stress decreased as the population density decreased, further suggesting that the complex interdependence formed at higher densities is the key for how tissues fail regardless of environmental stress (Fig. S4).
Similarly, the probability of death of sparse 1 K tissues remained constant while that of 10 K and 100 K increased in time, following the same pattern as the tissues that were not exposed to any post-stress (Fig. S5). The cell mortality rates followed a similar pattern whether stress was applied or not, among the same pre-aging condition groups (Fig. 3A). The 10 K and 1 K populations of the pre-aged and young groups also followed the same pattern for cell mortality rate, regardless of the stress applied (Figs S6 and S7). In addition, when the population was sparse, tissues with young cells coped with stress equally worse as tissues with pre-aged cells due to their lack of interactions. When the population was dense, however, tissues with young cells coped with stress significantly better than tissues with pre-aged cells. This indicates that cellular level aging decreases stress resilience by disrupting cellular interactions (Fig. 3B).
We further investigated the interaction between cellular level aging and systemic aging. Specifically, we determined how cellular level aging disrupts the interactions to cause the tissues to collapse sooner and how they show a lower resilience to stress. We considered two possibilities: (i) Aged cells lose their ability to produce (functional) cooperative factors. (ii) Aged cells do produce cooperative factors, but they cannot make use of them due to degrading surface receptors or degrading downstream pathways. As discussed below, we find evidence to suggest that hypothesis (ii) is more plausible.
To test between these two hypotheses we designed an experiment where we transferred the conditioned media from high (100 K) population density tissues with young cells to tissues with pre-aged cells (Chronological Senescence) with high (100 K) population density ( Fig. 4A and B). We observed that the aged cells did not live longer when they received the young conditioned media. Their population curve followed almost the same profile as the control (tissues with aged cells that did not receive young media).
This suggests either that the tissues with aged cells are not able to receive or process cooperative factors or that the factors are received and processed but their quantity is insufficient for the dense aged tissue. We eliminated the latter option by performing another experiment in which we transfer the conditioned media of 100 K tissues with young/aged cells into 1 K tissues with young/aged cells. If the young conditioned media was helpful to the aged tissue but was quantitatively insufficient, it should cause the aged 1 K tissues to live longer compared to controls receiving non-conditioned media (Fig. 4C). If the aged cells are still producing the required factors, then the conditioned media from aged 100 K cells should aid young 1 K cells to live longer, as 100 K populations live longer than 1 K populations, regardless of age (Fig. 2D).
Our results showed that the aged 1 K tissues died sooner than all young 1 K tissues, and receiving young 100 K conditioned media did not improve their lifetime (Fig. 4D). This suggests that the aged cells cannot receive/process cooperative factors even when factors are functional and sufficiently available suggesting that the aged tissues show a functional loss through the pre-aging period.
We verified the availability of functional factors by the improved survival of young 1 K tissues when they received the conditioned media from synthetic tissues made from young or pre-aged cells at 100 K density. This also suggests that the aged cells do produce cooperative factors that help each other survive, however, their ability to receive/process them is compromised. Furthermore, when we added oxidative stress to the conditioned media and non-conditioned media, neither the lifetime nor the aging profile of the synthetic tissues made from young or pre-aged cells were affected (Fig. S8).
Presently available aging research primarily focuses on the role of various cellular and bio-molecular mechanisms on cell damage [35][36][37] . Our experimental evidence supports the thesis in ref. 9 that a viable theory of aging must take into account the emergent nature of aging, and describe it as a universal property of a strongly interdependent collection of failure prone components. While aging is clearly driven by microscopic mechanisms, it is "more than the sum" of these microscopic factors. Grounded by empirical evidence, we have directly and quantitatively observed how the interdependency between cells play a role in tissue death, and bridged cellular aging to its higher systemic manifestations. The molecular mechanisms behind the functional loss in aged tissues we observed in this study need to be explored further including the differences in cooperative factors produced by young versus aged tissues. We aim to investigate the differences in media components of aged and young tissues under normal and stress conditions in our future studies.
Generally speaking, the picture of failure-prone interdependent components is not specific to multicellular life: some of the ideas presented here may also potentially provide further insight into the aging of mechanical, social, political, and ecological systems.
Materials and Methods
All animal experiments were performed using protocols approved by Institutional Animal Care and Use Committee (IACUC) of University of Notre Dame, in accordance to the guidelines of National Institutes of Health, Office of Laboratory Animal Welfare.
Cell Culture. Neonatal rat cardiac-fibroblasts (CFs) were isolated from 2-day-old Sprague-Dawley rats (Charles River Laboratories). The isolation was carried out following a protocol approved by the guidelines of the University of Notre Dame, which has an approved Assurance of Compliance on file with the National Institutes of Health, Office of Laboratory Animal Welfare. Briefly, the rats were sacrificed by decapitation after CO 2 treatment, and the hearts were immediately excised. CFs were isolated using a previously established protocol 38 . Briefly, the rat hearts were minced and incubated in trypsin (Life Technologies) (4 °C, 16 h) with gentle agitation. Next, the heart pieces were further digested by adding collagenase type II (Worthington-Biochem) (37 °C) and agitating several times with subsequent trituration in order to digest the extracellular matrix. The undigested tissue pieces were filtered and the filtrate was plated (37 °C, 2 h). The filtrate at this point contained CFs and cardiac myocytes (CMs), which were separated making use of their differential attachment during a 2 h plating period. The CFs attach to the culture plate at the end of this 2 h where CMs remain in solution. Next, the CMs in the solution were removed from the plate and DMEM supplemented with 10% fetal bovine serum (FBS) and 1% penicillin/streptomycin (p/s) (standard culture media) was added to CFs for maintaining the culture. At this point the CFs were labeled as passage 1 (P1) and media was changed every 3 days. CF cultures were passaged at approximately 80% confluency using trypsin-EDTA (0.05%) (Life Technologies) and maintained in standard culture media until the start of pre-aging treatments.
Pre-aging of CFs. CFs were aged in culture prior to synthetic tissue fabrication in 3 different scenarios (i):
CFs (P3) were maintained in 100% confluency without passaging for 30 days in standard culture media, with media changes every 3 days (Chronological Senescence), (ii); CFs (P3) were maintained in 100% confluency without passaging for 30 days under low oxidative stress conditions (standard culture media supplemented with 20 μM H 2 O 2 ) with media changes every 3 days (Oxidative Senescence); and (iii) CFs (P3) that were allowed to go through approximately 20 population doublings via passaging several times until they reach replicative senescence (Replicative Senescence). CFs (P3) that were not pre-aged were used as controls (Young). The schematic of media transfer experiment of synthetic tissues made from young or pre-aged cells at 1 K density, receiving 100 K young or 100 K pre-aged synthetic tissue conditioned media. Synthetic tissues maintained in non-conditioned media are used as controls. (E) The live cell percentages of 1 K synthetic tissues receiving non-conditioned (control) or conditioned media. (F) The lifetime of synthetic tissues made from young or pre-aged cells at 1 K density receiving 100 K young or 100 K pre-aged synthetic tissue conditioned or non-conditioned media.
Scientific RepoRts | 7: 5051 | DOI:10.1038/s41598-017-05098-2 Senescence Marker Staining and Comet Assay. At the end of the pre-aging periods, the CFs were collected using trypsin-EDTA and reseeded to glass cover slips at a density of 1.5 × 10 6 cells/mL (n = 3). Cellular senescence was determined through senescence-associated β-galactosidase staining using the Cellular Senescence Assay Kit (Millipore) following manufacturer's instructions. The senescent cell percent was determined by counting the total cell number and the number of cells positively stained for senescence-associated β-galactosidase using imageJ software. At least 4 images from each sample from at least 3 different samples were counted and averaged to yield average percentage values.
At the end of the pre-aging period the DNA damage was determined by using comet assay (Cell Biolabs) following the manufacturer's instructions. The assay uses the extent of DNA smear (tail) upon single cell electrophoresis to examine the amount of DNA damage in the cells. Briefly, following the agarose gel electrophoresis, the cells in the gels were stained with a DNA dye and imaged using a fluorescence microscope (Zeiss, Hamamatsu ORCA flash 4.0). Then the images were used to measure the tail length using imageJ software. Since the DNA damage is correlated with the DNA tail extent, the average tail extent for different groups was calculated to compare the amount of DNA damage upon various pre-aging conditions performed in this study. Passage 3 CFs (Young group) were used as control in both assays.
PEG-RGD Hydrogel Preparation and Characterization.
The YRGDS (H-tyr-Arg-Gly-Asp-Ser-OH, Mw 596.60, Bachem) sequence was conjugated to acryloyl-PEG-N-hydroxysuccinimide (acryl-PEG-NHS, Mw 3400, Jen Kem Technology) following a previously described protocol 39 . The RGD conjugation to PEG was confirmed through Nuclear Magnetic Resonance (H-NMR) -Spectroscopy. RGD conjugated PEG-NHS was mixed with 4-arm PEG-acrylate (Jen Kem Technology) in a 1.5:8.5 (w/w) ratio to prepare PEG-RGD hydrogels. The stiffness of the hydrogels with 3 different cell encapsulation densities (1 × 10 5 , 1 × 10 4 , and 1 × 10 3 ) was determined using a nanoindenter (Piuma Chiaro, Optics11, Amsterdam, The Netherlands) with an indentation probe (spring constant of 0.261 N/m, tip diameter of 16 µm). Elastic moduli of hydrogels were identified using a custom made MATLAB code.
Synthetic Tissue Preparation.
Pre-aged CFs were collected at the end of the pre-aging periods, and young cells were collected when they reach approximately 80% confluency, using trypsin-EDTA and were encapsulated in PEG-RGD hydrogels (n = 6). 3 different cell densities were used for encapsulation to control cell-cell distance in the synthetic tissues: 1 × 10 5 (100 K), 1 × 10 4 (10 K), and 1 × 10 3 (1 K) CFs per construct (5 μL total volume). The hydrogel solution was prepared by dissolving PEG-RGD (20% w/v) in PBS and photoinitiator (PI) (Irgacure 2959, BASF) (0.1% w/v in PBS) was added to achieve crosslinking. Next, the resulting solution was mixed with the respective cell suspension at a ratio of 1:1 (PEG-RGD:cell solution) and a final concentration of 10% PEG-RGD and 0.05% PI was achieved. The homogenous cell distribution within the hydrogel was ensured by thorough mixing of the hydrogel solution and cell suspension by gentle pipetting. The mixture was then sandwiched between 100 µm thick spacers using a glass slide and exposed to 6.9 mW/cm 2 UV irradiation for 20 sec. This dose of UV and photoinitiator is in the range that have been shown to be safe for cell encapsulation studies 40 . The synthetic tissues were washed once with PBS (1-2 mins) and 3 times with fresh standard culture media (15 mins each) right after crosslinking in order to get rid of excess PI.
Conditioned Media Experiments.
For the media transfer experiments, synthetic tissues made from young or pre-aged cells with 100 K cell density in each tissue were prepared as previously described. The synthetic tissues were maintained for 2 days prior to transferring their media to other synthetic tissues. On day 3 the synthetic tissues to receive the conditioned media were fabricated and following the initial washes they received the respective conditioned media. Controls were prepared for each condition where the same groups with the same population density received non-conditioned media at the same time points. Both 100 K and 1 K samples receiving the conditioned media were stained for dead cells and imaged after 24 hours of receiving the conditioned media. The samples were stained/imaged every 24 hours and the number of dead cells was determined for each group daily, as described in the following section.
Determining Cell-Cell Distance Distribution. The distance from each cell to its closest neighbor was determined using imageJ software for 1 K, 10 K, and 100 K tissue constructs. When they reached their lifetime, the constructs were imaged using a confocal microscope by taking z-stacks from 5 different points of each construct. The cell-cell distance are measured from each individual z-slice in order to determine the distance between the cells on the same plane.
Determining Live Cell Percentages and Death Rate of the Synthetic Tissues. Ethidium homodimer 1 (EthD-1) (Life Technologies), a nuclear stain that only stains the dead cells, was used to determine the real time live cell percentages and the death rate of the synthetic tissues, following manufacturer's instructions. After fabrication and the initial washes, the 10 K and 100 K synthetic tissues were incubated in normoxic or 0.2 mM H 2 O 2 containing standard culture media for 24 h. At the end of 24 h the synthetic tissues were washed off their media, stained with EthD-1 and imaged using a fluorescence microscope (Zeiss, Hamamatsu ORCA flash 4.0). Following imaging the tissues were washed off the stain and the respective media was added. This procedure was repeated every 24 h until the live cell percentage was at most 2%. 1 K synthetic tissues were stained with EthD-1 following the same procedure, right after the fabrication and initial washes to determine the t = 0 h viability. Then the staining was repeated at 2 h, 4 h, 6 h, 20 h and 24 h, unless stated otherwise.
At the end of each staining the imaging was done by taking z-stacks of the gels at 3 different points, covering approximately 1/8 of the whole gel. Through structural illumination (Apotome, Zeiss) we were able to image our tissues with optical sectioning and eliminated any signal coming from the above or below planes. This allowed us to determine the number of dead cells in each plane separately. For 10 K and 100 K synthetic tissues the number of dead cells in the 3 middle slices of each stack were counted separately (in total 9 images per tissue) and averaged. This average was then multiplied by the total number of slices present in a stack (10-11 slices in each stack) and by the ratio of the field of view of one stack to the area of the whole gel (1:8) to yield the average dead cell number per gel. For 1 K synthetic tissues the same procedure was used for imaging, however, the maximum projections for each stack were used to calculate the average dead cell number per gel. We used the maximum projections of the stacks as the sparsity of the cells at that density allowed us to count the dead cells in all planes at once, thus eliminating the error coming from assuming a constant cell number in each slice of a stack. Then the counted number of dead cells was multiplied by the ratio of the field of view of one stack to the area of the whole gel to yield the average dead cell number per gel. The dead cell numbers were normalized using a calibration curve (Fig. S9). The live cell number on day 0 was assumed to be equal to the theoretical number for the respective population density: i.e. the number of live cells in a 100 K tissue on day 0 was taken as 1 × 10 5 cells per construct. The average dead cell number calculated for each synthetic tissue was then subtracted from the theoretical total cell number in each construct e.g. for a 100 K synthetic tissue, live cell number on day 1 was calculated as: 100,000 -(calculated dead cell number on day 1). Then the live cell number on day 2 was calculated by subtracting the calculated dead cell number from the live cell number on day 1. The live and dead cell numbers for each synthetic tissue were determined at the end of every time point. Then we calculated the percent live cell every day using the theoretical total cell number and the calculated live cell number: Live cell (%) = [(live cell number)/(total cell number)]*100.
In addition, we calculated the mortality rate of cells (m) in different synthetic tissues. For a tissue with non-interacting cells, m would be constant. Thus, the probability of death per unit time − = ∆ ∆ ( )( ) m n n t 1 would yield an exponentially decreasing population, n(t) = n 0 e −mt . Thus, by measuring the deviations of n(t) from this null-hypothesis, we quantified the strength of interactions between cells. Specifically, we determined how = − m n dn dt 1 depends on n and t separately. If m(n) is an increasing function, this indicates that the cells compete for resources, and would have a higher chance of survival when their population density is lower. If m(n) is a decreasing function, this would indicate cooperativity between cells, i.e. that cells have a higher chance of survival if they are densely packed.
Statistical Analysis.
All experiments were conducted with n = 3 and the individual experiments were performed twice. All results were represented as mean ± standard deviation. Student's t-test was used for comparing two individual groups. All p values reported were two-sided, and statistical significance was defined as p < 0.05. Data Availability. All data generated or analyzed during this study are included in this published article (and its Supplementary Information files).
Figure 1 .
1The preparation and characterization of pre-aged cells and fabrication of respective synthetic tissues. (A) Senescence marker staining, (B) the senescent cell percent, and (C) DNA damage in pre-aged and young cells. (D) The schematic of fabrication and staining, and (E) the day 1 live/dead (top row) and dead cell staining (bottom row) (dead, red = ethidium homodimer-1; live, green = calcein-AM) images of 1 K (day 1), 10 K (day 6), and 100 K (day 8) chronologically aged tissue constructs. (F) The cell-cell distance distributions, of the chronologically aged synthetic tissues with 1 K, 10 K, and 100 K population densities (red dot represents the average cell-cell distance). (Scale bars = 100 µm for live/dead images and inset images, and 200 µm for dead cell images).
Figure 2 .
2Failure characteristics of synthetic tissues made from young or pre-aged cells. The change of (A) live cell percentage over time, (B) cell mortality rate with respect to live cell number, and (C) probability of death over time. (D) The lifetime of synthetic tissues made from young or pre-aged cells with different population densities (*indicates significance, p < 0.05). Denser tissues where cells can interact, display age specific mortality, whereas the mortality rate of sparse tissues is closer to constant.
Figure 3 .
3Determining the resilience of synthetic tissues made from young or pre-aged cells against oxidative stress and the effect of different types of cellular level damage. (A) The change in cell mortality rates of 100 K tissues with respect to the change in population density. (B) The lifetime of synthetic tissues made from young or pre-aged cells with or without oxidative stress. (*Indicates significance, p < 0.05).
Figure 4 .
4Determining the effect of secreted factors on failure of synthetic tissues made from young or pre-aged cells. (A) The timeline for conditioned media experiments. (B) The schematic of media transfer and (C) the live cell percentage change over time of synthetic tissues with 100 K pre-aged cells receiving 100 K young synthetic tissue conditioned or non-conditioned media. (D)
© The Author(s) 2017
AcknowledgementsThis study is supported by National Science Foundation (NSF) Award PHY-1607643 and NSF CBET CAREER Award 1651385.Author ContributionsA.A. performed the experiments, preparedFigures 1-4, wrote the materials and methods and results sections, and helped writing discussion section. D.C.V. and P.Z. designed and supervised the experiments and the data analysis, and wrote the manuscript.Additional InformationSupplementary information accompanies this paper at doi:10.1038/s41598-017-05098-2Competing Interests: The authors declare that they have no competing interests.Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
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|
[] |
[
"SPLIT-FACETS FOR BALANCED MINIMAL EVOLUTION POLYTOPES AND THE PERMUTOASSOCIAHEDRON",
"SPLIT-FACETS FOR BALANCED MINIMAL EVOLUTION POLYTOPES AND THE PERMUTOASSOCIAHEDRON"
] |
[
"Stefan Forcey ",
"ANDLogan Keefe ",
"William Sands "
] |
[] |
[] |
Understanding the face structure of the balanced minimal evolution (BME) polytope, especially its top-dimensional facets, is a fundamental problem in phylogenetic theory. We show that BME polytope has a sub-lattice of its poset of faces which is isomorphic to a quotient of the well-studied permutoassociahedron. This sub-lattice corresponds to compatible sets of splits displayed by phylogenetic trees, and extends the lattice of faces of the BME polytope found by Hodge, Haws, and Yoshida. Each of the maximal elements in our new poset of faces corresponds to a single split of the leaves. Nearly all of these turn out to actually be facets of the BME polytope, a collection of facets which grows exponentially.2000 Mathematics Subject Classification. 90C05, 52B11, 92D15.
|
10.1007/s11538-017-0264-7
|
[
"https://arxiv.org/pdf/1608.01622v3.pdf"
] | 3,991,391 |
1608.01622
|
068fd2de4758df1fa7a6874d038ff1a651e93640
|
SPLIT-FACETS FOR BALANCED MINIMAL EVOLUTION POLYTOPES AND THE PERMUTOASSOCIAHEDRON
15 Mar 2017
Stefan Forcey
ANDLogan Keefe
William Sands
SPLIT-FACETS FOR BALANCED MINIMAL EVOLUTION POLYTOPES AND THE PERMUTOASSOCIAHEDRON
15 Mar 2017arXiv:1608.01622v3 [math.CO]
Understanding the face structure of the balanced minimal evolution (BME) polytope, especially its top-dimensional facets, is a fundamental problem in phylogenetic theory. We show that BME polytope has a sub-lattice of its poset of faces which is isomorphic to a quotient of the well-studied permutoassociahedron. This sub-lattice corresponds to compatible sets of splits displayed by phylogenetic trees, and extends the lattice of faces of the BME polytope found by Hodge, Haws, and Yoshida. Each of the maximal elements in our new poset of faces corresponds to a single split of the leaves. Nearly all of these turn out to actually be facets of the BME polytope, a collection of facets which grows exponentially.2000 Mathematics Subject Classification. 90C05, 52B11, 92D15.
Introduction
Phylogenetics is the study of the reconstruction of biological family trees from genetic data. Results from phylogenetics can inform every facet of modern biology, from natural history to medicine. A chief goal of biological research is to find relationships between genes and the functional structures of organisms. Knowing degrees of kinship can allow us to decide whether or not an adaptation in two species is probably an inherited feature of a common ancestor, and thus help to isolate the roles of genes common to both.
Mathematically, a phylogenetic tree is a cycle-free graph with no nodes (vertices) of degree 2, and with a set of distinct items assigned to the degree one nodes-that is, labeling the leaves. We study a method called balanced minimal evolution. This method begins with a given set of n items and a symmetric (or upper triangular) square n × n dissimilarity matrix whose entries are numerical dissimilarities, or distances, between pairs of items. From the dissimilarity matrix, often presented as a vector of distances, the balanced minimal evolution (BME) method constructs a binary (degree of vertices ≤ 3) phylogenetic tree with the n items labeling the n leaves.
It is well known that if a distance vector is created by starting with a given binary tree T with lengths assigned to each of its edges, and finding the pairwise distances between leaves just by adding the edge lengths along the path that connects them, then the tree T is uniquely recovered from that distance vector. The distance vector (or matrix) is called additive in this case. One recovery process is called the sequential algorithm, described first in [17]. It operates by leaf insertion and is performed in polynomial time: O(n 2 ). Another famous algorithm is neighbor joining, which reconstructs the tree in O(n 3 ) time [16]. It has the advantage of being a greedy algorithm for the BME problem, when extended to the non-additive case [9].
An alternate method of recovery via minimization was introduced by Pauplin in [14] and developed by Desper and Gascuel in [4]. This BME method uses a linear functional on binary phylogenetic trees t (without edge lengths) defined using the given distance vector. The output of the function is the length of the original tree T (assuming that the distance vector was created from T.) The function is minimized when the input tree t is identical to T , as trees without edge lengths. Thus by minimizing this functional, we recover the original tree topology. The latter terminology is used to describe two trees that are identical if we ignore edge lengths. The value of this approach is that the given distance vector is often corrupted by missing or incorrect data; but within error bounds we can still recover the tree topology by the minimization procedure. Furthermore, the BME method is statistically consistent in that as the distance vector approaches the accuracy of a true tree T, the BME method's output approaches that tree's topology [5,1,10].
More precisely: Let the set of n distinct species, or taxa, be called S. For convenience we will often let S = [n] = {1, 2, . . . , n}. Let vector d be given, having n 2 real valued components d ij , one for each pair {i, j} ⊂ S. There is a vector c(t) for each binary tree t on leaves S, also having n 2 components c ij (t), one for each pair {i, j} ⊂ S. These components are ordered in the same way for both vectors, and we will use the lexicographic ordering: d = d 12 , d 13 , . . . , d 1n , d 23 , d 24 , . . . , d n−1,n . We define, following Pauplin [14]:
c ij (t) = 1 2 l ij (t)
where l ij (t) is the number of internal nodes (degree 3 vertices) in the path from leaf i to leaf j.
If a phylogenetic tree T with non-negative edge lengths is given, then we can define the distance vector d(T ) by adding the edge lengths between each pair of leaves. Then the dot product c(T ) · d(T ) is equal to the sum of all the edge lengths of T, a sum which is known as the tree length. T is uniquely determined by d(T ) (unless there are length zero edges, in which case there is a finite set of trees determined). Using any other tree t as the input of c(t) will give a sub-optimal, larger value for c(t) · d(T ).
The BME tree for an arbitrary positive vector d is the binary tree t that minimizes d·c(t) for all binary trees on leaves S. Now this dot product is the least variance estimate of treelength, as shown in [5]. The value of setting up the question in this way is that it becomes a linear programming problem. The convex hull of all the vectors c(t) for all binary trees t on S is a polytope BME(S), hereafter also denoted BME(n) or P n as in [6] and [11]. The vertices of P n are precisely the (2n − 5)!! vectors c(t). Minimizing our dot product over this polytope is equivalent to minimizing over the vertices, and thus amenable to the simplex method.
In Fig. 1 we see the 2-dimensional polytope P 4 . In that figure we illustrate a simplifying choice that will be used throughout: rather than the original fractional coordinates c ij we will scale by a factor of 2 n−2 , giving a new vector x(t) with coordinates:
x ij (t) = 2 n−2 c ij (t) = 2 n−2−l ij (t) .
The convex hull of the vectors x(t) is a combinatorially equivalent scaled version of the BME polytope, so we refer to it by the same name. Since the furthest apart any two leaves may be is a distance of n − 2 internal nodes, this scaling will result in integral coordinates for our polytope. The tree t that minimizes d · c(t) will also minimize d · x(t).
A clade is a subgraph of a binary tree induced by an internal (degree three) node and all of the leaves descended from it in a particular direction. In other words: given an internal node v we choose two of its edges and all of the leaves that are connected to v via those two edges. Equivalently, given any internal edge, its deletion separates the tree into two clades. Two clades on the same tree must be either disjoint or nested, one contained in the other. A cherry is a clade with two leaves. We often refer to a clade by its set of (2 or more) leaves. A pair of intersecting cherries {a, b} and {b, c} have intersection in one leaf b, and thus cannot exist both on the same tree. A caterpillar is a tree with only two cherries. A split of the set of n leaves for our phylogenetic trees is a partition of the leaves into two parts, one part called S 1 with m leaves and another S 2 with the remaining n − m leaves. A tree displays a split if each part makes up the leaves of a clade. A facet of a polytope is a top-dimensional face of that polytope's boundary, or a co-dimension-1 face. Faces of a polytope can be of any dimension, from 0 to that of the (improper) face which is the polytope itself.
New Results
Our most important new discovery is a large family of facets of the BME polytope, which we call split − f acets in Theorem 4.7. This collection of facets is shown to exist for all n, and the number of facets in this family grows like 2 n . (1, 2, 1, 1, 2, 1) (1, 1, 2, 2, 1,
( ) In Theorem 4.3 we show that any (non-binary) phylogenetic tree corresponds to a face of P n . This allows us to define a map from the permutoassociahedron to the BME polytope, taking faces to faces. In Theorem 4.6 we show that this map preserves the partial order of faces.
In Theorem 4.5 we show that a special case of these tree-faces are the clade-faces discovered earlier in [11]. In Theorem 4.7 we show that another special case of tree-faces is our new class of facets of P n .
Previous results
Until recently, little was known about the structure of the BME polytopes, but several newly discovered features were described in [12] and [11]. The coordinates of the vertices satisfy a set of n independent equalities, which we will refer to as the Kraft equalities, after an equivalent description in [3]. For each leaf i we sum the coordinates that involve it:
j:j =i x ij = 2 n−2 .
These equalities govern the dimension of the BME polytope, dim(P n ) = n 2 − n. In [11] the authors prove the first description of faces of the n th balanced minimal evolution polytope P n . They find a family of faces that correspond to any set of disjoint clades. In [8] we show that these clade-faces are not facets, but instead show several new familys of facets. We add to that list here with a family of facets that grows exponentially. (Our results are listed in columns 5-7 of Table 1.)
We show in [8] that any pair of intersecting cherries corresponds to a facet of P n . For each pair of cherries with leaves {a, b} and {b, c}, there is a facet of P n whose vertices correspond to trees that have either one of those two cherries.
In addition, any caterpillar tree with fixed ends corresponds to a facet of P n . Thus for each pair of species there is a facet of P n whose vertices correspond to trees which are caterpillars with this pair as far apart as possible. Also shown in [8]: for n = 5, for each necklace of five leaves there is a corresponding facet which is combinatorially equivalent to a simplex.
Connection to the Permutoassociahedron
The n th permutoassociahedron KP n , also known as the type-A Coxeter associahedron, is defined in [13]. It is discussed in detail in [15], and related to the space of phylogenetic trees in [2]. A face of the permutoassociahedron corresponds to an ordered partition of a set S of n elements, whose parts label the leaves, left to right, of a rooted plane tree. We often use S = {1, . . . , n}. Alternatively we may use S = {1, . . . , n + 1} − {r} where r ∈ S is the label for the root. Bijectively, one of these labeled plane trees can also be described as a partial bracketing of an ordered partition, such as (({3}, {4, 5}), {2}, {1, 6, 7}).
The inclusion of faces corresponds to refinement of the ordered-partition trees: refinement of the tree structure by adding branches at nodes with degree larger than 3 (so that the collapse of the added branches returns the original tree) or refinement of the ordered partition, in which parts of it are further partitioned (subdivided, with ordering). To display the subdivision, the parts of the refined partition label the ordered leaves of a new subtree: a corolla, which is a tree with one root, one internal node, and 2 or more leaves.
Tree refinement can also be described as adding parentheses to the bracketing, or subdividing a set in the bracketing. A covering relation is either adding a single branch (pair of parentheses) or subdividing a single part of the partition. For examples of covering relations, Fig. 2. The 3-dimensional KP 3 is shown in Fig. 3 number dim. vertices facets facet inequalities number of number of of of P n of P n of P n (classification) facets vertices species in facet 3 12 5 (cyclic ordering) 6 9 105 90262 Table 1. Technical statistics for the BME polytopes P n . The first four columns are found in [12] and [11]. Our new and recent results are in the last 3 columns. The inequalities are given for any a, b, c, · · · ∈ [n]. Note that for n = 4 the three facets are described twice: our inequalities are redundant.
(({3}, {4, 5}), {2}, {1, 6, 7}) > ((({3}, {4, 5}), {2}), {1, 6, 7}) and (({3}, {4, 5}), {2}, {1, 6, 7}) > (({3}, {4, 5}), {2}, ({1}, {6}, {7})) or (({3}, {4, 5}), {2}, {1, 6, 7}) > (({3}, {4, 5}), {2}, ({1, 7}, {6})). The 2-dimensional KP 2 is shown in0 1 0 - - - 4 2 3 3 x ab ≥ 1 3 2 x ab + x bc − x ac ≤ 2 3 2 5 5 15 52 x ab ≥ 1 10 6 (caterpillar) x ab + x bc − x ac ≤ 4 30 6 (intersecting-cherry) x ab + x bc + x cd + x df + x f a ≤ 13x ab ≥ 1 15 24 (caterpillar) x ab + x bc − x ac ≤ 8 60 30 (intersecting-cherry) x ab + x bc + x ac ≤ 16 10 9 (3, 3)-split n n 2 − n (2n − 5)!! ? x ab ≥ 1 n 2 (n − 2)! (caterpillar) x ab + x bc − x ac ≤ 2 n−3 n 2 (n − 2) 2(2n − 7)!! (intersecting-cherry) x ab + x bc + x ac ≤ 2 n−2 n 3 3(2n − 9)!! (m, 3)-split, m > 3 i,j∈S 1 x ij ≤ (k − 1)2 n−3 2 n−1 − n 2 (2m − 3)!! (m, k)-split, −n − 1 ×(2k − 3)!! m > 2, k > 2
There is a straightforward lattice map ϕ from the faces of KP n to a sub-lattice of faces of the BME polytope. Since it preserves the poset structure, its preimages are a nice set of equivalence classes.
Definition 4.1. Let t be a plane rooted tree with leaves an ordered partition π of S. First let t ′ be the tree achieved by replacing each leaf labeled by part U ∈ π such that |U| > 1 with a corolla labeled by the elements of U. This corolla is attached at a new branch node where the leaf labeled by U was attached. Now let ϕ(t) be the tree f (t ′ ), where f is described as un-gluing t ′ from the plane in order to preserve only its underlying graph.
An example of the map ϕ is shown in Fig. 4. Note that forgetting the plane structure of t ′ ensures that the map ϕ is well-defined. The corolla that replaces each leaf labeled by U ∈ π is immediately seen as unordered since it is not fixed in the plane. Fig. 5 shows the full action of ϕ on the 2-dimensional KP 2 .
Since our map preserves the face order, it takes vertices to vertices. It is a set projection on vertices, and the number of elements in a preimage has a nice formula:
Proposition 4.2.
Let T be a binary phylogenetic tree with n leaves. The number of ordered plane rooted binary trees t such that ϕ(t) = T is 2 n−2 .
Proof. We note that the map is a surjection from vertices to vertices, since any leaf of a binary phylogenetic tree may be chosen as the root. By symmetry of the labeling of leaves, the size of each preimage must be the same. Here n is the total number of leaves, so in the permutoassociahedron vertices we are considering plane binary trees with n − 1 leaves and a root. We divide the total numbers of vertices of the two polytopes:
C n−2 (n − 1)! (2n − 5)!! = 2 n−2 .
Here we have used the formula for Catalan numbers:
C n−2 = 1 n−1 2(n−2) n−2 .
We also use the formula (2n − 5)!! = (2n−4)! 2 n−2 . Now we show how the targets of the map ϕ are actually faces of the BME polytope. Note that the image of the (improper) face which is the entire permutoassociahedron (as well as any of its corrolla facets) is the phylogenetic tree which is a corolla, or star: it has only one node = Figure 4. Action of the map ϕ: the second step shows that we no longer preserve plane structure or rooted-ness.
with degree ≥ 3. This corolla corresponds to the (improper) face which is the entire BME polytope. In what follows we will assume we are speaking of proper faces.
Theorem 4.3. For each non-binary phylogenetic tree t with n leaves there is a corresponding face F (t) of the BME polytope P n . The vertices of F (t) are the binary phylogenetic trees which are refinements of t.
Proof. We show that for each non-binary t there is a distance vector d(t) for which the product d(t)·x(t ′ ) is minimized simultaneously by precisely the set of binary phylogenetic trees t ′ which refine t.
The distance vector d(t) is defined as follows: the component d ij (t) is the number of edges in the path between leaf i and leaf j. Next we show that, for any tree t ′ , we have the inequality:
i<j d ij (t)x ij (t ′ ) ≥ 2 (n−2) |E(t)|
where E(t) is the set of edges of t. Moreover, we will show that the inequality is precisely an equality if and only if the tree t ′ is a refinement of t.
Our vector d(t) is constructed to be a vector of distances (of paths between leaves) for any binary tree that refines t. This is seen by assigning a length of 1 to each edge of the tree t, and calculating the distances between leaves by adding the edge lengths on the path between them for any two leaves. A binary tree t ′ that refines t is similarly given lengths of 1 for its edges, except for those edges whose collapse would return t ′ to the tree t. These latter edges are assigned a length of zero. Figure 5. Action of the map ϕ: the shaded faces all map to the shaded vertices. Now our result follows: given a distance vector whose components are the distances between leaves on a binary tree, the dot product of this vector with vertices of the BME polytope is minimized at the vertex corresponding to that tree. In our case all the binary trees t ′ which refine t, with their assigned edge lengths, share the distance vector d(t). Thus they are simultaneously the minimizers of our product, and the value of that product is 2 (n−2) times their common tree length.
Definition 4.4. For a non-binary phylogenetic tree t we call the corresponding face of the BME polytope the tree-face F (t).
An example of a tree-face, its vertices, and its inequality as given in the proof of Theorem 4.3, are shown in Fig. 6.
Some special cases of tree faces are important. First we mention the case in which the tree t has only one non-binary node, that is, exactly one node with degree larger than 3. Thus t can be seen as a collection of clades (and some single leaves) all attached to the non-binary node.
Proposition 4.5. For t an n-leaved phylogenetic tree with exactly one node ν of degree m > 3, the tree-face F (t) is precisely the clade-face F C 1 ,...,Cp , defined in [11], corresponding to the collection of clades C 1 , . . . , C p which result from deletion of ν. Thus F (t) is combinatorially equivalent to the smaller dimensional BME polytope P m .
Proof. Any tree t ′ which is a binary refinement of t can be constructed by attaching the clades C 1 , . . . , C p to p of the leaves of a binary treet. Note that since we don't consider single leaves to be clades, we need to say thatt has m leaves where m − p is the number of single leaves attached to ν. Figure 6. The three binary trees shown are the vertices of the tree-face corresponding to the tree in the center. The inequality which defines this face is: 2x 12 + 2x 13 + 3x 14 + 3x 15 + 2x 23 + 3x 24 + 3x 25 + 3x 34 + 3x 35 + 2x 45 ≥ 48
Recall from [11] that the face F C 1 ,...,Cp is the image of an affine transformation of the BME polytope P m . As stated by those authors, this combinatorial equivalence follows since every tree in F C 1 ,...,Cp can be constructed by starting with a binary tree on m leaves and attaching the clades F C 1 ,...,Cp to p of the m leaves. See Fig. 6 for an example of a clade-face, in fact a cherry clade-face, where the single clade in question is the cherry {4, 5}.
In [11] it is pointed out that the clade-faces form a sub-lattice of the lattice of faces of P n . Containment in that sublattice is simply refinement, where a sub-clade-face of a clade-face F (t) can be found by refining the tree t, as long as the result still has only a single non-binary node.
Now it is straightforward to see that refinement of trees in general gives a partial ordering of tree-faces, and indeed another sub-lattice of faces of the BME polytope which contains the clade-faces as a sub-lattice. We note that the map ϕ from the permutoassociahedron is a lattice map.
Proposition 4.6. If x ≤ y (thus containment as faces in the face lattice of KP n , ) then ϕ(x) ≤ ϕ(y) (so containment as faces in the face lattice of P n , the BME polytope).
Proof. The refinement of a labeled plane rooted tree t, or the refinement of the ordered partition labeling the leaves, both correspond to the refinement of ϕ(t). The former is direct, the latter is seen via the replacement of parts in the partition by the corresponding corollas, before and after subdivision.
Next we look at what are perhaps the most important tree-faces: those which correspond to facets of the BME polytope. It turns out that these facets correspond to trees t which have exactly two adjacent nodes with degree larger than 3.
Theorem 4.7. Let t be a phylogenetic tree with n > 5 leaves which has exactly one interior edge {ν, µ}, with ν and µ each having degree larger than 3. Then the trees which refine t are the vertices of a facet of the BME polytope P n .
The proof is in Section 6. Note that this implies that there are clade-faces which are not contained in any tree-face facet, as seen in Fig. 7.
It is clarifying to refer to the new family of facets in Theorem 4.7 as split-facets. The binary phylogenetic trees which display a given split correspond precisely to the trees which refine a tree as described in that theorem.
In fact we can see all the tree-faces in terms of displayed splits, since a split always corresponds to an internal edge. Thus we have that requiring 2 or more splits which the binary trees must all display simultaneously corresponds to specifying a tree-face, all of which are subfaces of split-facets. We also found a formula for the number of vertices in a split-face with parts of the split being S 1 of size k and S 2 of size m = n − k. The number of vertices is:
(2m − 3)!!(2k − 3)!! .
This formula is found via the multiplication principle, in which all possible clades are counted for each part of the split.
5.3.
Number of facets that a given tree belongs to, in the Splitohedron. The splitfaces, intersecting-cherry facets, and caterpillar facets together outline a relaxation of the BME polytope. We define a new polytope:
Definition 5.1. The splitohedron Sp(n) is defined as the intersection of the half-spaces of R ( n 2 ) given by the following inequalities listed by name: the intersecting-cherry facets, the split-facets, the caterpillar facets and the cherry clade-faces-and also obeying the n Kraft equalities. The splitohedron is a bounded polytope because the cherry clade-faces, where the inequality is x ij ≤ 2 n−3 , and the caterpillar facets, where the inequality is x ij ≥ 1, show that it lies inside the hypercube [1, 2 n−3 ] ( n 2 ) . It has the same dimension as the BME polytope, and often has many of the same vertices.
Theorem 5.2. For an n-leaved binary phylogenetic tree, if the number of cherries is at least n/4 then the tree represents a vertex in the BME polytope that is also a vertex of the splitohedron. For n ≤ 11 the tree represents a vertex regardless of the number of cherries.
Proof. For a given binary phylogenetic tree t it is straightforward to count how many distinct facets of the splitohedron it belongs to. If that number is as large as the dimension, we know that the tree lies at a vertex of the polytope Sp(n). First we note that an inequality which defines a facet of the BME polytope and which is also obeyed by the splitohedron therefore defines a facet of the splitohedron as well, by the nature of relaxation.
For each cherry {a, b} of t we have that t lies within 2(n − 2) facets, an intersecting-cherry facet for each choice of either a or b and a third leaf that is neither. For each interior edge that does not determine a cherry clade, we have that t lies within a split-facet. There are n − 3 − p such interior edges, where p is the number of cherries. Finally, if t is a caterpillar then it lies within 4 caterpillar facets, determined by a choice of one leaf from each cherry to fix.
All together t lies within p(2n−4) + n−3 −p = (2n−5)p + n−3 facets of the splitohedron, if it is not a caterpillar. For any n this number increases with p, as p ranges from 2 to n 2 . The dimension of the polytope is n 2 − n = 1 2 (n 2 − 3n). Comparing the two expressions shows that the tree t will represent a vertex of Sp(n) as long as p ≥ n 2 −5n+6 4n−10 . This is true, for instance, when p ≥ n/4.
In the worst case scenario for non-caterpillar trees, we have p = 3 and t is a vertex when n 2 ≤ 17n − 36, or for n ≤ 14. For caterpillar trees, where p = 2, we have the extra four facets so t is a vertex when n 2 ≤ 13n − 22, or for n ≤ 11.
Thus for n ≤ 11 we have all the binary phylogenetic trees represented as vertices of the splitohedron.
Proof of Theorem 4.7
First we prove that the split-facet is always a face of the BME polytope. This is implied by Theorem 4.3. However it is more useful to prove the following simpler linear inequality. Lemma 6.1. Consider the split π = {S 1 , S 2 } of the set of leaves. Let |S 1 | = k ≥ 3 and |S 2 | = m ≥ 3. Then the following inequality becomes an equality precisely for the trees which display the split, and a strict inequality for all others.
i < j, leaves i, j ∈ S 1 x ij ≤ (k − 1)2 n−3 .
Proof. (of the face inequality.) It follows directly from the fact that the sum of all coordinates for any tree with n leaves is n2 n−3 . Thus, if we double-sum over the leaves, we have i ( j x ij ) = n2 n−2 ; twice the total since we add each coordinate twice. Now consider a tree with k + 1 leaves (anticipating a clade with k leaves) and the double sum is (k + 1)2 k−1 . If we only sum over the first k leaves, thereby ignoring all the coordinates involving the k + 1st leaf, the smaller double sum totals to (k − 1)2 k−1 . (Note that the additional internal node connecting to the k + 1st leaf is causing the perceived difference in results for our clade of k leaves from an entire tree of n leaves. ) Next consider the actual situation of interest, where there is a clade of size k whose coordinates we double-sum over, but we have replaced the extra leaf with another clade of size n − k . Here each coordinate in the double sum is multiplied by the power of 2 achieved by adding n − k − 1 leaves, so our total becomes
2 n−k−1 (k − 1)2 k−1 = (k − 1)2 n−2 .
Recall that we have been double counting, so our result is 2 times too much: the actual sum of the coordinates in any clade with leaves from S 1 is
i < j, leaves i, j ∈ S 1 x ij = (k − 1)2 n−3 .
It is clear that for any tree which does not contain a clade consisting only of the leaves in S 1 , it instead must contain a collection of clades whose leaves together make up the set S 1 (some of which may be singletons.) Since some of these must be further apart (separated by more internal nodes from each other) than if they formed a single clade, then summing all the coordinates using indices only from S 1 will give a total strictly smaller than in the case where S 1 makes up the leaves of a single clade.
Notice that using the second part of the split, S 2 , as the basis for the sum works just as well. In practice the smaller part of the split is chosen in order to provide a shorter inequality. Now we prove the dimension of these faces. Proof. of Theorem 4.7: Base case. The proof is inductive. We start by proving the base case in which one of the parts of the split has exactly k = 3 leaves, and the other has size m ≥ 3.
To do this, we fill in the flag which goes from this facet down to the clade face for a fixed combination of the 3-leaved section of the split.
The first inequality is that of the facet itself, where we simply have a split. If we label the leaves in our k = 3-leaf section a,b,c; then our simplified inequality from above is x a,b + x a,c + x b,c ≤ 2 n−2 . Let the leaves in the m-leaf section be labeled as 1, 2, ..., m. We now rely on the fact that to show a chain of subfaces, our subsequent face inequalities only need to be strict on trees which obey the previous face inequality exactly, as an equality. This raises a caveat: the inequalities used for subfaces of the flag in our proof may not be actual face inequalities of the entire polytope.
Our next inequality is:
3x a,1 − x b,1 − x c,1 + 2x a,b + 2x a,c ≤ 3 · 2 n−3 .
This is intended to include all trees with a in a cherry, and to require the leaf 1 to be near the leaf a when a is not in the cherry. See the set pictured in part (ii) of Fig. 8.
In the case when a is in the cherry, x b,1 or x c,1 will be the size of x a,1 and the other will be twice its size. So the sum 3x a,1 − x b,1 − x c,1 will be 0. Then, x a,b or x a,c must be 2 n−3 and the other 2 n−4 . These add to 3 · 2 n−4 . So 2x a,b + 2x a,c = 3 · 2 n−3 .
When a is not in the cherry, for out inequality to be maximal we must have x a,1 = 2 n−4 and hence x b,1 and x c,1 as 2 n−5 . So 3x a,1 − x b,1 − x c,1 = 3 · 2 n−4 − 2 · 2 n−5 = 2 n−3 . Then, since a is near b and c but not in the cherry, we have 2x a,b + 2x a,c = 2 · 2 n−3 . So, the left hand side of our equation is 3 · 2 n−3 when 1 is close to a, as wanted. If 1 were to be further, it is easy to see the expression would be smaller.
Our next set of steps is dependent upon the size of m. The intent here is to build off of previous steps by forcing specific leaves to be far from the k-leaf cluster in each step. See the sets pictured in parts (iii) − (v) of Fig. 8. Our inequalities will be:
3x a,i − x b,i − x c,i + 2 i−1 2 n−4 (x a,b + x a,c ) ≥ 3 · 2 i−1 when i ≥ 3.
When i = 2, we use the inequality:
3x a,2 − x b,2 − x c,2 + 2 3−1 2 n−4 (x a,b + x a,c ) ≥ 3 · 2 3−1 .
This is because 2 is in a cherry with 3 so they must satisfy the same inequality, albeit with different coordinates.
This works since 3x a,i − x b,i − x c,i is 0 when a is in the cherry, and it is half the size of 2 i−1 2 n−4 (x a,b + x a,c ) when it is not in the cherry. Also, Figure 8. Flag for the base case in proof of Theorem 4.7. The sets include all the trees that can be formed by completing the pictures with additional leaf labels. Dashed-circled corollas denote all possible binary structures on the leaves (which are not always shown). Dots between labeled leaves denote an ordered caterpillar structure, while dots between unlabeled leaves denote an unordered caterpillar.
2 i−1 2 n−4 (x a,b + x a,U U U { { { { { { { { { } } } } } } } } } . . . . . .(i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi)
we have equality. If the leaf i moves at all when a is in the cherry, we still have equality. If it moves when a is not in the cherry, 3x a,i − x b,i − x c,i will become larger. After this chain, we have a simple inequality which forces a to be in the cherry, as in the set pictured in part (vi) of Fig. 8. It looks like:
2x a,b + 2x a,c ≤ 3 · 2 n−3 .
Next, for the set pictured in part (vii) of Fig. 8, we have 3x
b,1 − x a,1 − x c,1 + 2x a,b + 2x b,c ≤ 3 · 2 n−3 .
This works like the inequality for the face below the facet. This meaning that, it forces 1 to be close to b when b is not in the cherry, and has no effect on the tree when b is in the cherry. We then have the same i-indexed chain after it with the roles of a and b reversed, since we are trying to achieve the same result as with a but with b. See the sets pictured in part (viii) − (x) of Fig. 8. So, the inequalities are:
3x b,i − x a,i − x c,i + 2 i−1 2 n−4 (x b,a + x b,c ) ≥ 3 · 2 i−1 when i ≥ 3 and when i = 2, 3x b,2 − x a,2 − x c,2 + 2 3−1 2 n−4 (x b,a + x b,c ) ≥ 3 · 2 3−1 .
To finish, we use the fixed clade face of dimension m+1 2 − (m + 1) as described in [11] where c is not in the cherry. See the set pictured in part (xi) of . . Figure 9. Flag for the inductive step in proof of Theorem 4.7. Picture notation is as above.
(i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x)
Inductive step. Next we assume the theorem for splits of respective sizes k − 1 and m, both larger than 3, and inductively prove it for all k, m. We consider all the trees which display a given split π into leaves S 1 = {1, . . . , m} and leaves S 2 = {a, b, c, f } ∪ {y 1 , . . . , y k−4 }.
The inductive assumption allows us to use Theorem 4.5 in our proof. We can calculate the dimension of a face which has as its vertices all the binary phylogenetic trees that both display the split π and also have a cherry {a, b}. These trees are a subset of the set of all the trees with the cherry {a, b}, which describes a clade-face of the BME polytope. That clade-face is equivalent to P n−1 , and using the argument of the proof of Theorem 4.5 as found in [11], the cherry can be considered as a leaf of the trees of P n−1 . Thus the trees that both display our split and also have a cherry {a, b} display a split π ′ into m and k − 1 "leaves" which gives, by induction, a facet of P n−1 . The dimension of this face is thus n−1 2 − (n − 1) − 1, and so it is the top-dimensional face in a flag of length n−1 2 − (n − 1). Next we show the existence of a chain of faces of length n − 2, beginning with the face of all trees that display our split π and ending with the face that has all trees displaying π and possessing the cherry {a, b}. Concatenating this chain to the flag shown by induction gives a flag of length n 2 − n, which implies that our split-face is indeed a facet. After the split face, the second face in our flag is described by all the trees that both display the split π and possess either cherry {a, b} or cherry {b, c}. These trees, as a sub-face of the split face, have the face inequality:
x ab + x bc − x ac ≤ 2 (n−3) .
Note that this is a face by virtue of being the intersection of the split-facet and the intersectingcherry facet: in fact the proof from here is inspired by the proof of Theorem 4 (the intersectingcherry facet) in [8]. Indeed the next face in our flag is described by containing the trees which both display the split π and possess either cherry {a, b} or the two cherries {b, c} and {a, f }. Again this is an intersection of faces: the split-face and the second face of the flag shown in the proof of Theorem 4 in [8]. For completeness, the inequality obeyed by this third face is:
x bc + x bf − x ac − x af ≥ 0.
Next we have a chain of k − 4 faces which correspond to ordering the remaining k − 4 leaves of S 2 . For j ∈ 1 . . . k − 4 we take the set of trees that have the split π and the cherry {a, b}, or which have the cherries {b, c} and {a, f } as the two cherries of a caterpillar clade made from S 2 , and for which the leaves y 1 . . . y j are attached in that order starting as close as possible to the cherry {a, f }. See the pictures of sets (iv) -(vi) in Fig. 9, noting how the caterpillar clade is attached to S 1 at any point among its unordered nodes. The j th term in this list of faces obeys the inequality:
(2 n−3 − 2 m−1 )(x ay j − x by j ) ≤ (2 n−3 − x ab )(2 n−3−j − 2 m+j−1 ).
To see that this is an equality for the sets of trees in question, note first that when {a, b} is a cherry then x ab = 2 n−3 and x ay j = x by j . Also, when S 2 is fixed as a caterpillar clade, then x ab = 2 m−1 and x ay j − x by j = 2 n−3−j − 2 m+j−1 . Finally, when y j is found in a location on the caterpillar clade closer to leaf b, (which is the only way to be in the previous face while avoiding being in the current face), then x ay j − x by j is forced to be a lesser value.
After the chain of caterpillar clades using S 2 , we add a chain using caterpillar clades on S 1 . This chain begins with the set pictured in part (viii) of Fig. 9, where the leaf 1 is in the cherry at the far end of the flag. This face obeys the equality:
(2 n−3 − 2 m−1 )(x a1 − x b1 ) ≤ (2 n−3 − x ab )(2 − 2 k−3 ).
The comments just made about the previous faces also apply here, to show that the equality holds on the face and that when S 2 is fixed as a caterpillar clade, then x ab = 2 m−1 . Now though we see that if the leaf 1 is any closer to b, then both x a1 and x b1 increase. However, since they are both powers of two then increasing both by a factor of another power of two means their difference will be even larger-and we are subtracting in the order that ensures the inequality.
The remaining links in the chain are formed by fixing the leaves 2, . . . , m in order along the caterpillar clade in S 1 , as in pictured sets (viii) and (ix) of Fig. 9. When the leaf i is fixed, the face obeys the inequality:
(2 n−3 − 2 m−1 )(x ai − x bi ) ≤ (2 n−3 − x ab )(2 i−1 − 2 i+k−5 ).
This inequality is an equality on the face and strict on the trees of the previous face excluded from the current face, by the same arguments as above.
Finally we exclude all the trees displaying the split except for those with the cherry {a, b}, as shown by the pictured set (x) in Fig. 9. This completes the proof by induction, as explained above.
Future work
We have shown that (for n ≤ 11) the splitohedron contains among its vertices all the possible phylogenetic trees. Therefore if the BME linear program is optimized in the splitohedron at a valid tree vertex for n ≤ 11, it is also optimized in the BME polytope.
More importantly, however, the binary phylogenetic trees for any n all lie on the boundary of several facets of the splitohedron which are also facets of the BME polytope. Our continuing research program involves writing code that uses various linear programming methods in sequence, with a branch-and-bound scheme, to find the BME tree.
Then by finding further facets we will improve this theorem, hopefully to a version that holds for all n > 11.
Acknowledgements
We thank the editors and both referees for helpful comments. The first author would like to thank the organizers and participants in the Working group for geometric approaches to phylogenetic tree reconstructions, at the NSF/CBMS Conference on Mathematical Phylogeny held at Winthrop University in June-July 2014. Especially helpful were conversations with Ruriko Yoshida, Terrell Hodge and Matt Macauley. The first author would also like to thank the American Mathematical Society and the Mathematical Sciences Program of the National Security Agency for supporting this research through grant H98230-14-0121. 1 The first author's specific position on the NSA is published in [7]. Suffice it to say here that he appreciates NSA funding for open research and education, but encourages reformers of the NSA who are working to ensure that protections of civil liberties keep pace with intelligence capabilities.
Figure 1 .
1The polytope P 4 is a triangle. At the top we label the vertices with the three binary trees with leaves 1 . . . 4. Each edge shows a nearest-neighbor interchange; for instance the exchange of leaves 3 and 4 on the bottom edge. At bottom left are Pauplin's original coordinates and at bottom right are the coordinates, scaled by 2 n−2 = 4, which we will use.
Figure 2 .
2The 2-dimensional permutoassociahedron with labeled faces.
Figure 3 .
3The 3-dimensional permutoassociahedron with a labeled facet. This picture is redrawn from a version in[15].
..
Number of split-facets. For n = 6 there are 31 splits in all, but only 10 splits which obey the requirement that there are at least three leaves in each part. For n leaves the number of splits is 2 (n−1) − 1. (This is half the number of nontrivial, proper subsets.) Discarding the splits with only one leaf and discarding the cherry clade-faces, we are left Number of vertices in a split-facet. For n = 6 each facet of this type has 9 vertices since there are three choices of binary structure on each side of the split. Thus the facet itself must be an 8-dimensional simplex.
Figure 7 .
7Examples of chains in the lattice of tree-faces of the BME polytope P 7 .
c ) is 3 2 the size when a is in the cherry as when a is not in the cherry. So in both cases, when we have what we want,
Fig. 8 .
8The total length of our chain is n 2 − n − 1, proving that the (m,3)-split face is a facet.
This manuscript is submitted for publication with the understanding that the United States Government is authorized to reproduce and distribute reprints.
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. Forcey) Department of Mathematics. The University of AkronForcey) Department of Mathematics, The University of Akron, Akron, OH 44325-4002
E-mail address: sf34@uakron. Akron, OHDepartment of Mathematics, The University of AkronE-mail address: [email protected] URL: http://www.math.uakron.edu/~sf34/ (L. Keefe) Department of Mathematics, The University of Akron, Akron, OH 44325-4002
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[
"Polynomial Interpretation of Multipole Vectors",
"Polynomial Interpretation of Multipole Vectors"
] |
[
"Gabriel Katz \nBennington College\n05201BenningtonVTUSA\n"
] |
[
"Bennington College\n05201BenningtonVTUSA"
] |
[] |
Copi, Huterer, Starkman and Schwarz introduced multipole vectors in a tensor context and used them to demonstrate that the first-year WMAP quadrupole and octopole planes align at roughly the 99.9% confidence level. In the present article the language of polynomials provides a new and independent derivation of the multipole vector concept. Bézout's Theorem supports an elementary proof that the multipole vectors exist and are unique (up to rescaling). The constructive nature of the proof leads to a fast, practical algorithm for computing multipole vectors. We illustrate the algorithm by finding exact solutions for some simple toy examples, and numerical solutions for the first-year WMAP quadrupole and octopole. We then apply our algorithm to Monte Carlo skies to independently re-confirm the estimate that the WMAP quadrupole and octopole planes align at the 99.9% level.
|
10.1103/physrevd.70.063527
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[
"https://arxiv.org/pdf/astro-ph/0405631v2.pdf"
] | 119,472,833 |
astro-ph/0405631
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7e7d9f82adc0798692c7ae15ad0e6a5b6234aabd
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Polynomial Interpretation of Multipole Vectors
Jun 2004 (Dated: November 17, 2018)
Gabriel Katz
Bennington College
05201BenningtonVTUSA
Polynomial Interpretation of Multipole Vectors
Jun 2004 (Dated: November 17, 2018)PACS numbers: 9880-k
Copi, Huterer, Starkman and Schwarz introduced multipole vectors in a tensor context and used them to demonstrate that the first-year WMAP quadrupole and octopole planes align at roughly the 99.9% confidence level. In the present article the language of polynomials provides a new and independent derivation of the multipole vector concept. Bézout's Theorem supports an elementary proof that the multipole vectors exist and are unique (up to rescaling). The constructive nature of the proof leads to a fast, practical algorithm for computing multipole vectors. We illustrate the algorithm by finding exact solutions for some simple toy examples, and numerical solutions for the first-year WMAP quadrupole and octopole. We then apply our algorithm to Monte Carlo skies to independently re-confirm the estimate that the WMAP quadrupole and octopole planes align at the 99.9% level.
I. INTRODUCTION
The first-year WMAP data [1] reveal a somewhat planar octopole which approximately aligns with the quadrupole [2]. More recent studies confirm these conclusions at roughly the 99.9% level [9] while revealing mysterious alignments with the ecliptic plane [3], suggesting either a hitherto unknown solar system effect on the microwave background or an error in the collection and/or processing of the WMAP data. Other researchers find the ℓ = 4 multipole is generic, the ℓ = 5 multipole is unusually non-planar at the 99.8% level, and the ℓ = 6 multipole is unusually planar at the 98.6% level [4]. No explanation is yet known for these strange results.
Multipole vectors provide a convenient means to quantify the planarity of a given multipole, as well as to compare the alignment of two different multipoles [5]. The present authors, coming from a background in pure mathematics, were unable to decipher the formalism and terminology of Ref. [5] and chose instead to re-create the multipole vector concept from scratch. The real-valued spherical harmonics of order ℓ are precisely the homogeneous harmonic polynomials of degree ℓ in the variables x, y and z (for example Y 0 2 is the polynomial x 2 +y 2 −2z 2 , up to normalization), so the present authors sought to understand the multipole vectors of Copi, Huterer and Starkman (CHS) from a polynomial point of view.
Translated to the language of polynomials, CHS's motivating goal (see Eqn. (10) of [5]) was to factor every homogeneous harmonic polynomial P of degree ℓ into a * Electronic address: [email protected] † Electronic address: www.geometrygames.org/contact.html product of linear factors P (x, y, z) = λ · (a 1 x + b 1 y + c 1 z) · (a 2 x + b 2 y + c 2 z) · · · · (a ℓ x + b ℓ y + c ℓ z).
(1)
Such a factorization is of course impossible in general, as CHS implicitly acknowledge by their introduction of suitable error terms. In the language of polynomials the correct statement of the theorem is Theorem 1. Every homogeneous polynomial P of degree ℓ in x, y and z may be written as
P (x, y, z) = λ · (a 1 x + b 1 y + c 1 z) · (a 2 x + b 2 y + c 2 z) · · · · (a ℓ x + b ℓ y + c ℓ z) + (x 2 + y 2 + z 2 ) · R,(2)
where the remainder term R is a homogeneous polynomial of degree ℓ − 2. The decomposition is unique up to reordering and rescaling the linear factors.
Notes: (a) Theorem 1 lives entirely in the realm of real polynomials: the coefficients of P , R and all the linear factors a i x+b i y+c i z are assumed to be real. (b) Theorem 1 does not require the polynomial P to be harmonic.
In cosmological applications we are interested only in the value of the polynomial on the unit sphere S 2 ; we ignore its value on the rest of Euclidean 3-space. On the unit sphere the factor x 2 +y 2 +z 2 is identically 1, so in this case Theorem 1 says that any homogeneous polynomial P may be written as a product of linear factors λ(a 1 x + b 1 y + c 1 z) · · · (a ℓ x + b ℓ y + c ℓ z) plus a remainder term R of lower degree. Applying this reasoning recursively gives the easy Corollary 2. When restricted to the unit sphere, every polynomial P of degree ℓ in x, y and z may be written as P(x,y,z) = λ ℓ · (a ℓ,1 x + b ℓ,1 y + c ℓ,1 z) · (a ℓ,2 x + b ℓ,2 y + c ℓ,2 z) · · · (a ℓ,ℓ x + b ℓ,ℓ y + c ℓ,ℓ z)
+ . . . + λ 2 · (a 2,1 x + b 2,1 y + c 2,1 z) · (a 2,2 x + b 2,2 y + c 2,2 z) + λ 1 · (a 1,1 x + b 1,1 y + c 1,1 z) + λ 0 .(3)
The decomposition is unique up to reordering and rescaling the linear factors within each term.
Note: Corollary 2 does not require the polynomial P to be either homogeneous or harmonic.
Proof of Corollary 2. Write P as a sum of homogeneous terms P = P ℓ +P ℓ−1 +· · ·+P 1 +P 0 . First apply Theorem 1 to the highest order term P ℓ , yielding a factorization λ ℓ · (a ℓ,1 x + b ℓ,1 y + c ℓ,1 z) · · · (a ℓ,ℓ x + b ℓ,ℓ y + c ℓ,ℓ z) along with a remainder term R ℓ−2 of homogeneous degree ℓ − 2. (The factor x 2 + y 2 + z 2 may be ignored on the unit sphere.) Fold R ℓ−2 in with P ℓ−2 , and proceed recursively, applying Theorem 1 to P ℓ−1 , then P ℓ−2 , and so on.
To prove uniqueness, consider the even and odd parts of P separately. That is, write P = P even + P odd , where P even contains all the even-powered terms and P odd contains all the odd-powered terms. Say we have two potentially different decompositions for the even part
P even = Π ℓ + Π ℓ−2 + Π ℓ−4 + · · · + Π 0 = Π ′ ℓ + Π ′ ℓ−2 + Π ′ ℓ−4 + · · · + Π ′ 0 .(4)
where each Π i is the i th term in a decomposition (3), and where the leading index will be ℓ or ℓ − 1 according to whether ℓ is even or odd. To make these decompositions homogeneous, multiply through by appropriate powers of Q = x 2 + y 2 + z 2 ,
P even = Π ℓ + QΠ ℓ−2 + Q 2 Π ℓ−4 + · · · + Q ℓ/2 Π 0 = Π ′ ℓ + QΠ ′ ℓ−2 + Q 2 Π ′ ℓ−4 + · · · + Q ℓ/2 Π ′ 0 . (5)
This does not affect the value of P on the unit sphere, because Q = 1 there. The uniqueness part of Theorem 1 implies that the leading order terms Π ℓ and Π ′ ℓ must be equal. So subtract off those leading terms, divide through by Q, and apply Theorem 1 again to conclude Π ℓ−2 = Π ′ ℓ−2 . Continue recursively to finally reach Π 0 = Π ′ 0 . The same argument then proves that the odd part of P has a unique decomposition as well. Q.E.D.
II. PROOF OF THE MAIN THEOREM
Even though the statement of Theorem 1 lives wholly in the world of real polynomials, its proof will dive deeply into the world of complex polynomials. So let the variables x, y and z range over the complex numbers, while insisting that the coefficients of the polynomial P remain real. Because P has homogeneous degree ℓ, whenever one point (x 0 , y 0 , z 0 ) satisfies P (x, y, z) = 0, every nonzero constant multiple (αx 0 , αy 0 , αz 0 ) satisfies it as well. Thus the equation P = 0 is well defined on each equivalence class of points {α(x 0 , y 0 , z 0 ) | α ∈ C − {0}}. In other words, the complex curve P = 0 is well defined on the complex projective plane CP 2 , which is the quotient of
C 3 − {(0, 0, 0)} under the equivalence relation (x 0 , y 0 , z 0 ) ∼ α(x 0 , y 0 , z 0 )
. This leads us into the realm of algebraic geometry and puts its powerful tools at our disposal.
The most useful tool for our purposes is
Bézout's Theorem. If P and Q are homogeneous polynomials of degree m and n, respectively, then the curves P = 0 and Q = 0 intersect in CP 2
• in exactly m · n points, counted with multiplicity, if P and Q share no common factor, or
• in infinitely many points, if P and Q do share a common factor.
For an elementary exposition of Bézout's Theorem, see [6].
In the present case, the only way the polynomial P may share a factor with the irreducible polynomial Q(x, y, z) ≡ x 2 + y 2 + z 2 is for P to contain Q as a factor, in which case Theorem 1 is trivially satisfied (take λ = 0). So henceforth assume P does not contain Q as a factor. Bézout's Theorem now tells us that the degree ℓ complex curve P (x, y, z) = 0 intersects the quadratic curve Q(x, y, z) = 0 in exactly 2ℓ points, counted with multiplicities. None of the intersection points may be purely real, because real values cannot possibly satisfy x 2 + y 2 + z 2 = 0 -recall that the definition of CP 2 explicitly excludes (0, 0, 0). Furthermore, because P and Q both have real coefficients, whenever (x 0 , y 0 , z 0 ) lies in the intersection P = Q = 0, its complex conjugate (x 0 , y 0 , z 0 ) must lie there too. So the 2ℓ points of intersection consist of ℓ pairs of non-real complex conjugates
{p 1 , p 1 , . . . , p ℓ , p ℓ }. We claim that each pair {p i , p i } determines a unique line a i x + b i y + c i z = 0 with real coefficients. The proof is easy. The conjugate pair {p i , p i } lies on the line a i x + b i y + c i z = 0 if
and only if the real and imaginary parts satisfy the following two totally real equations
a i Re p i,x + b i Re p i,y + c i Re p i,z = 0 a i Im p i,x + b i Im p i,y + c i Im p i,z = 0.(6)
Geometrically those two equations represent planes in R 3 . If the coefficient vectors (Re p i,x , Re p i,y , Re p i,z ) and (Im p i,x , Im p i,y , Im p i,z ) are non-collinear, then the two planes are distinct and their intersection, which defines the solution set for (a i , b i , c i ), is a line through the origin in R 3 . In other words, the line a i x + b i y + c i z = 0 is unique. Normalize the coefficients to unit length, i.e. a 2 i + b 2 i + c 2 i = 1, and the only remaining ambiguity is an overall factor of ±1.
But what if the coefficient vectors Re p i = (Re p i,x , Re p i,y , Re p i,z ) and Im p i = (Im p i,x , Im p i,y , Im p i,z ) had been collinear? In this case the line a i x + b i y + c i z = 0 would be ill-defined. Fortunately this case does not arise. For if Im p i were proportional to Re p i , say Im p i = β Re p i , then the point p i , as an element of CP 2 , could be rewritten as a scalar multiple p i ∼ 1 1+iβ p i = Re p i , showing that p i is totally real. In other words, p i would lie in RP 2 ⊂ CP 2 . In particular, p i would be its own complex conjugate, and we can hardly expect a single point p i = p i to determine a unique line. Fortunately this case cannot occur, because Q(x, y, z) = x 2 + y 2 + z 2 = 0 admits no real solutions.
So let L i denote the unique line a i x + b i y + c i z = 0 containing the conjugate pair {p i , p i }. More precisely, let L i = a i x + b i y + c i z be the unique (modulo rescaling) real linear polynomial whose roots include both p i and p i . The desired decomposition (2) becomes
P = λ L 1 L 2 · · · L ℓ + Q · R.(7)
To prove that this equality holds, we again turn to Bézout's Theorem. First recall that the complex curve P = 0 intersects the complex curve Q = 0 in precisely the 2ℓ points {p 1 , p 1 , . . . , p ℓ , p ℓ }. By construction, the product curve L 1 L 2 · · · L ℓ = 0 also intersects Q = 0 in those same 2ℓ points, and by Bézout's Theorem there are no other points of intersection. Now pick any other point q ∈ {Q = 0} and define λ to be the ratio λ = P (q) L 1 (q)L 2 (q) · · · L ℓ (q)
.
Write a new polynomial
F ≡ P − λL 1 L 2 · · · L ℓ .(9)
This new polynomial F has degree ℓ, yet has zeros at the 2ℓ+1 distinct points { q, p 1 , p 1 , . . . , p ℓ , p ℓ } ⊂ Q. In other words, the complex curve F = 0 intersects the complex curve Q = 0 at (at least) 2ℓ + 1 distinct points. By Bézout's Theorem the polynomials F and Q must share a common factor; because Q is irreducible the common factor must perforce be Q itself. Thus we may factor F as
F = Q · R(10)
for some remainder term R. Combining (9) and (10) yields the desired decomposition (7).
Let us now prove that λ is real. In light of the factorization (10), the polynomial F is clearly zero on the whole complex curve Q = 0. In particular, for the point q chosen earlier,
F (q) = F (q) = 0.(11)
On the one hand F (q) = P (q) − λL 1 (q)L 2 (q) · · · L ℓ (q).
On the other hand, because P and the L i all have real coefficients,
F (q) = P (q) − λL 1 (q)L 2 (q) · · · L ℓ (q)(13)
Comparing (11), (12) and (13), and recalling that q was chosen to ensure L i (q) = 0, proves that λ =λ, in other words, λ is real. An elementary argument then shows that for all real values of x, y and z, R(x, y, z) = R(x, y, z) = R(x, y, z), implying that the coefficients of the polynomial R must all be real.
This completes the proof of the existence part of Theorem 1.
Let us now prove that the decomposition ( 2) is unique. Assume we have two decompositions
P (x, y, z) = λ L 1 L 2 · · · L ℓ + Q · R = λ ′ L ′ 1 L ′ 2 · · · L ′ ℓ + Q · R ′ .(14)
Our goal is to show that each linear factor L ′ i ′ in the second decomposition occurs as a factor L i in the first decomposition as well, modulo a possible rescaling. A given line L ′ i ′ = 0 intersects the quadratic Q = 0 in a pair of conjugate points p andp. Because p andp satisfy both L ′ i ′ = 0 and Q = 0, they satisfy P = 0 as well. Turning our attention to the first decomposition, because p andp satisfy both P = 0 and Q = 0, they satisfy L 1 L 2 · · · L ℓ = 0 as well. Hence p must satisfy one of the lines L i = 0, and because the line's coefficients are real, p must satisfy that same line. But we saw earlier that a pair of conjugate points p andp determines a unique line modulo normalization (recall the essentially unique solution to Equations (6)). Therefore L i is a constant multiple of L ′ i ′ , and if the coefficients of each have been normalized to length one, then L i = ±L ′ i ′ . This proves the uniqueness of the factorization.
If we evaluate the two decompositions (14) on the complex curve Q = 0 we get
λ L 1 L 2 · · · L ℓ = λ ′ L ′ 1 L ′ 2 · · · L ′ ℓ(15)
proving that if the coefficients of the L i and the L ′ i ′ are consistently normalized, then λ = λ ′ . It then follows easily that R = R ′ as well.
This completes the proof that the decomposition ( 2) is unique, thus completing the proof of Theorem 1.
III. COMPUTATIONAL CONSIDERATIONS
The proof presented in Section II is almost constructive, but not quite. It relies on Bézout's Theorem for the existence of the root pairs {p 1 , p 1 , . . . , p ℓ , p ℓ } but does not say how to find them. This section fills the gap.
The key observation is that the quadratic curve Q = x 2 + y 2 + z 2 = 0 is topologically a 2-sphere. More to the point, it is a copy of the complex projective line CP 1 , which happens to be homeomorphic to the 2-sphere. Let us parameterize the curve Q = 0 as
( i (u 2 − v 2 ), −2i uv, u 2 + v 2 )(16)
where u and v are homogeneous coordinates in CP 1 . Clearly the mapping (16) takes all points (u, v) ∈ CP 1 to the curve Q = 0, by construction. The question is, which of those points happen to satisfy the given polynomial P as well? Write
P (x, y, z) = P ( i (u 2 − v 2 ), −2i uv, u 2 + v 2 )(17)
to express P as a function on CP 1 .
If v = 0, then (u, v) and ( u v , 1) represent the same point in CP 1 . If we define α ≡ u v then expression (17) effectively reduces to a polynomial in a single variable,
P (x, y, z) = P ( i (α 2 − 1), −2i α, α 2 + 1) (18)
The roots of this polynomial are the desired root pairs {p 1 , p 1 , . . . , p ℓ , p ℓ }.
If, on the other hand, v = 0, then (u, v) = (u, 0) ∼ (1, 0). Thus (u, v) = (1, 0) may represent an additional root, which would not be found as a root of P (α) in (18).
Once we have found the parameters (u, v) for all 2ℓ roots of P , the easiest way to group them into conjugate pairs is to observe that the parameterization (16) maps "antipodal points" (u, v), (−v,ū) ∈ CP 1 to conjugate points (x, y, z), (x,ȳ,z) ∈ CP 2 . In other words, (α, 1) and (−1,ᾱ) ∼ (−1/ᾱ, 1) map to a pair of conjugate points in CP 2 .
IV. EXAMPLES
To illustrate how the algorithm works in practice, let us apply Theorem 1 to several concrete examples.
A. Toy Quadrupole
Consider the quadratic polynomial
P (x, y, z) = xy + yz + zx − x 2 − z 2 .(19)
First dismiss the special case (u, v) = (1, 0) by noting that the parameterization (16) maps (u, v) = (1, 0) to (x, y, z) = (i, 0, 1) for which (19) gives P (i, 0, 1) = i = 0. Now consider the general case, for which Equation (18) becomes
iα 4 + 2(1 − i)α 3 − 4α 2 − 2(1 + i)α − i = 0 (20)
with roots
α 1 = 1 + √ 2 α 2 = 1 − √ 2 α 3 = i(1 + √ 2) α 4 = i(1 − √ 2).(21)
corresponding, respectively, to the four points of CP 2 ,
p 1 = (1, −1, −i √ 2) p 1 = (1, −1, +i √ 2) p 2 = (−i √ 2, 1, −1) p 2 = (+i √ 2, 1, −1).(22)
Solving the line equations (6) converts the preceding two pairs of conjugate points to the two lines
L 1 = 1 2 x + 1 2 y = 0 L 2 = 1 2 y + 1 2 z = 0,(23)
which give us the two multipole vectors ( 1 2 , 1 2 , 0) and (0, 1 2 , 1 2 ). To find the correct λ, evaluate Equation (8) for, say,
q = (1, i, 0), giving λ = P (q) L 1 (q)L 2 (q) = −1 + i − 1 2 + i 2 = 2.(24)
Of course any other choice for q would have given the same answer λ = 2, just so we make sure q lies on the curve Q(q) = x 2 + y 2 + z 2 = 0 and exclude q ∈ {p 1 , p 1 , p 2 , p 2 }.
We may now write down the polynomial F from Equation 9, namely
F = P − λL 1 L 2 = (xy + yz + zx − x 2 − z 2 ) −2( 1 2 x + 1 2 y)( 1 2 y + 1 2 z) = −x 2 − y 2 − z 2 ,(25)
and divide by Q = x 2 + y 2 + z 2 to get the remainder term R = F/Q = −1. Thus the final decomposition promised by Theorem 1 becomes
xy + yz + zx − x 2 − z 2 = 2( 1 2 x + 1 2 y)( 1 2 y + 1 2 z) + (x 2 + y 2 + z 2 )(−1).(26)
B. Toy Octopole
The cubic polynomial P (x, y, z) = x 2 y + y 3
illustrates some non-generic behavior which may arise, namely the possibilities of (a) a "missing root" and (b) multiple roots. We will follow the same algorithm as in Section IV A, pointing out only the differences. The first difference is that the special case (u, v) = (1, 0), corresponding to (x, y, z) = (i, 0, 1), is indeed a root of P in (27). So we record that root and proceed onward in search of the other roots.
The next difference we encounter is that the polynomial 2iα 5 + 4iα 3 + 2iα = 0.
(28) has degree only 5, not degree 2ℓ = 2·3 = 6 as one expects in the generic case. Happily, this polynomial's five roots supplement the one exceptional root (i, 0, 1) we found in the previous paragraph, to give the required total of six roots. In other words, the unexpectedly low degree of the polynomial is intimately tied to the existence of the exceptional root (i, 0, 1).
The roots of (28) turn out to be {−i, i, −i, i, 0}. Unlike more generic polynomials, this one has multiple roots, implying a repeated factor in the product L 1 L 2 L 3 . Specifically, those five roots correspond to
p 1 = (+i, 1, 0) p 1 = (−i, 1, 0) p 2 = (+i, 1, 0) p 2 = (−i, 1, 0) p 3 = (−i, 0, 1),(29)
and then the one exceptional root (i, 0, 1) completes the pattern p 3 = (+i, 0, 1).
From here the algorithm is routine. The lines are
L 1 = z = 0 L 2 = z = 0 L 3 = y = 0,(31)
the scalar multiple is λ = −1, and the final factorization is
x 2 y + y 3 = −1(y)(z)(z) + (x 2 + y 2 + z 2 )(y).(32)
C. WMAP Quadrupole and Octopole
Our first task here is to convert a given set of coefficients a ℓm to a homogeneous harmonic polynomial. Converting the standard spherical harmonics Y m ℓ to polynomials in x, y and z is easy. For example, for the quadrupole,
Y −2 2 = 15 32π sin 2 θ e −2iϕ = 15 32π (x − iy) 2 Y −1 2 = 15 8π sin θ cos θ e −iϕ = 15 8π (x − iy)z Y 0 2 = 5 16π (3 cos 2 θ − 1) = 5 16π (3z 2 − 1) = 5 16π (3z 2 − (x 2 + y 2 + z 2 )) = 5 16π (−x 2 − y 2 + 2z 2 ) Y 1 2 = − 15 8π sin θ cos θ e iϕ = − 15 8π (x + iy)z Y 2 2 = 15 32π sin 2 θ e 2iϕ = 15 32π (x + iy) 2(33)
in full agreement with those that CHS found using their tensor algorithm (Equation 3 of [3]). An analogous computation for the octopole yields multipole vectorsv (3,1) ,v (3,2) andv (3,3) , again in full agreement with those reported in Equation 3 of [3].
V. HOW WELL DO THE WMAP QUADRUPOLE AND OCTOPOLE ALIGN?
Following [3], we define the quadrupole plane normal vector w 2,1,2 ≡v (2,1) ×v (2,2) (38) and the three octopole plane normal vectors w 3,1,2 ≡v (3,1) ×v (3,2) w 3,2,3 ≡v (3,2) ×v (3,3) w 3,3,1 ≡v (3,3) ×v (3,1) .
Still following [3], we judge the alignment of the quadrupole plane with the three octopole planes via the dot products
A 1 = |w 2,1,2 · w 3,2,3 | A 2 = |w 2,1,2 · w 3,3,1 | A 3 = |w 2,1,2 · w 3,1,2 |.(40)
Finally, in contrast to [3] (which in its preprint form contained some statistical flaws), we let the sum
S = A 1 + A 2 + A 3(41)
provide a numerical measure of how well the quadrupole plane aligns with the octopole planes. For the DQcorrected Tegmark map, the sum evaluates to S 0 = 2.395.
To judge how unusually large S 0 is, we evaluated S for 100000 random quadrupoles and octopoles, and found that only 118 trials produced S > S 0 . This 99.9% confidence level, while weaker than the incorrectly stated results of [3], is completely consistent with Huterer and Starkman's revised statistical analysis [7]. Like Schwarz, Starkman, Huterer and Copi, we find this result astonishing. In particular we find it difficult to believe that the quadrupole and octopole align so well merely by chance. Whether the alignment is imposed by the global topology of a small finite universe, or whether it is due to some previously unknown solar system effect, or whether it is merely the result of an error in the collection and/or processing of the WMAP data, remains to be seen.
In the meantime, we emphasize that our simulations use an entirely different algorithm from that of [3,5] as well as completely independent computer code. This effectively rules out the possibility of error in computing the 1-in-1000 estimate, forcing us to take that estimate quite seriously.
We should point out that our Monte Carlo simulations chose random quadrupoles and octopoles independently of each other, relative to spherically symmetric distributions on the spaces of all spherical harmonics of degree 2 and 3, respectively. In other words, we used independent Gaussian coefficients a ℓm .
VI. AN OPEN QUESTION
Corollary 2 motivates a broader question, first raised by Copi, Huterer and Starkman [8]. One would like to decompose an arbitrary real-valued function f : S 2 → R, for example the temperature function on the microwave sky, as a sum f = ∞ ℓ=0 λ ℓ ℓ i=1 L ℓ,i . In other words, this approach would bypass the spherical harmonics entirely, and instead write the function f directly as the sum of totally factored polynomials λ ℓ ℓ i=1 L ℓ,i , one for each degree ℓ.
Corollary 2 almost makes such a factorization possible. For example, if we approximate the microwave sky temperature by the sum of its first 837 multipoles, T = 837 ℓ=0 ℓ m=−ℓ a ℓ,m Y m ℓ , then Corollary 2 lets us re-express it as T = 837 ℓ=0 λ ℓ ℓ i=1 L ℓ,i . The question is, what happens when we add in the 838 th spherical harmonic? For sure we will add an 838 th term λ 838 838 i=1 L 838,i to our factored-polynomial decomposition. And almost surely the 836 th term will change significantly as it inherits the remainder R from the newly added 838 th term. But what about the lower order terms? Will the second, fourth and sixth terms remain stable? Or will they swing wildly every time we add a new high-order term to the series? In other words, is the decomposition stable?
Using the coefficients a 2,m for the DQ-corrected Tegmark map of the first-year WMAP quadrupole gives P (x, y, z) = −0.01262739 x 2 + 0.02302019 xy + 0.00677625 y 2 + 0.00950698 xz + 0.01064014 yz + 0.00585113 z 2 . (34) Following the same algorithm, as illustrated in Sections IV A and IV B, we get the polynomial (0.01847852 − i 0.00950698) −(0.04604038 + i 0.02128027) α −0.04065752 α 2 +(0.04604038 − i 0.02128027) α 3 +(0.01847852 + i 0.00950698) α 4 (35) leading to the lines L 1 = 0.939660 x + 0.187066 y + 0.286437 z = 0 L 2 = −0.437088 x + 0.792820 y + 0.424724 z = 0. (36) Converting the coefficients of these lines to spherical coordinates gives multipole vectorŝ v (2,1) = ( 11.26 • , 16.64 • ) v (2,2) = (118.87 • , 25.13 • )
First year Wilkinson Microwave Anisotropy Probe (WMAP 1) observations: preliminary maps and basic results. C Bennett, Astrophysical Journal Supplement Series. 148C. Bennett et al., First year Wilkinson Microwave Anisotropy Probe (WMAP 1) observations: preliminary maps and basic results, Astrophysical Journal Supplement Series 148 (2003), 1-27.
Hamilton, A high resolution foreground cleaned CMB map from WMAP. M Tegmark, A Oliveira-Costa, A J , astro-ph/0302496Phys. Rev. 68M. Tegmark, A. de Oliveira-Costa and A.J.S. Hamil- ton, A high resolution foreground cleaned CMB map from WMAP, Phys. Rev. D68 (2003) 123523, astro-ph/0302496.
Is the low-ℓ microwave background cosmic?, submitted to. D J Schwarz, G D Starkman, D Huterer, C J Copi, astro-ph/0403353Phys. Rev. Lett. D.J. Schwarz, G.D. Starkman, D. Huterer and C.J. Copi, Is the low-ℓ microwave background cosmic?, submitted to Phys. Rev. Lett., astro-ph/0403353.
On foreground removal from the Wilkinson Microwave Anisotropy Probe data by an internal linear combination method: limitations and implications. H K Eriksen, A J Banday, K M Górski, P B Lilje, astro-ph/0403098ApJ. submitted toH.K. Eriksen, A.J. Banday, K.M. Górski and P.B. Lilje, On foreground removal from the Wilkinson Microwave Anisotropy Probe data by an internal linear combination method: limitations and implications, submitted to ApJ, astro-ph/0403098.
Multipole vectors -a new representation of the CMB sky and evidence for statistical anisotropy or non-Gaussianity at 2 ≤ ℓ ≤ 8, submitted to. C J Copi, D Huterer, G D Starkman, astro-ph/0310511Phys. Rev. D. C.J. Copi, D. Huterer and G.D. Starkman, Multipole vec- tors -a new representation of the CMB sky and evidence for statistical anisotropy or non-Gaussianity at 2 ≤ ℓ ≤ 8, submitted to Phys. Rev. D, astro-ph/0310511.
Undergraduate Algebraic Geometry. M Reid, Cambridge University PressM. Reid, Undergraduate Algebraic Geometry, Cambridge University Press, 1988.
. D Huterer, G D Starkman, personal communicationD. Huterer and G.D. Starkman, personal communication.
. G D Starkman, personal communicationG.D. Starkman, personal communication.
|
[] |
[
"Invariant compact finite difference schemes",
"Invariant compact finite difference schemes"
] |
[
"E Ozbenli \nSchool of Aerospace and Mechanical Engineering\nUniversity of Oklahoma\n865 Asp AveOK73019NormanUSA\n",
"P Vedula \nSchool of Aerospace and Mechanical Engineering\nUniversity of Oklahoma\n865 Asp AveOK73019NormanUSA\n"
] |
[
"School of Aerospace and Mechanical Engineering\nUniversity of Oklahoma\n865 Asp AveOK73019NormanUSA",
"School of Aerospace and Mechanical Engineering\nUniversity of Oklahoma\n865 Asp AveOK73019NormanUSA"
] |
[] |
In this paper, we propose a method, that is based on equivariant moving frames, for development of high order accurate invariant compact finite difference schemes that preserve Lie symmetries of underlying partial differential equations. In this method, variable transformations that are obtained from the extended symmetry groups of PDEs are used to transform independent and dependent variables, and derivative terms of compact finite difference schemes (constructed for these PDEs) such that the resulting schemes are invariant under the chosen symmetry groups. The unknown symmetry parameters that arise from the application of these transformations are determined through selection of convenient moving frames. In some cases, owing to selection of convenient moving frames, numerical representation of invariant (or symmetry preserving) compact numerical schemes, that are developed through the proposed method, are found to be notably simpler than that of standard, noninvariant compact numerical schemes. Further, the order of accuracy of these invariant compact schemes can be arbitrarily set to a desired order by considering suitable compact finite difference algorithms. Application of the proposed method is demonstrated through construction of invariant compact finite difference schemes for some common linear and nonlinear PDEs (including the linear advection-diffusion equation in 1D/2D, the inviscid and viscous Burgers' equations in 1D). Results obtained from our numerical simulations indicate that invariant compact finite difference schemes not only inherit selected symmetry properties of underlying PDEs, but are also comparably more accurate than the standard, non-invariant base numerical schemes considered here.
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10.1103/physreve.101.023303
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[
"https://arxiv.org/pdf/1901.00599v1.pdf"
] | 119,623,689 |
1901.00599
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27196633ed8dfd6413a2a95f1fe284aaf884a2e4
|
Invariant compact finite difference schemes
3 Jan 2019
E Ozbenli
School of Aerospace and Mechanical Engineering
University of Oklahoma
865 Asp AveOK73019NormanUSA
P Vedula
School of Aerospace and Mechanical Engineering
University of Oklahoma
865 Asp AveOK73019NormanUSA
Invariant compact finite difference schemes
3 Jan 2019Lie symmetrycompact finite differencingfinite difference schemessymmetry preserving schemes
In this paper, we propose a method, that is based on equivariant moving frames, for development of high order accurate invariant compact finite difference schemes that preserve Lie symmetries of underlying partial differential equations. In this method, variable transformations that are obtained from the extended symmetry groups of PDEs are used to transform independent and dependent variables, and derivative terms of compact finite difference schemes (constructed for these PDEs) such that the resulting schemes are invariant under the chosen symmetry groups. The unknown symmetry parameters that arise from the application of these transformations are determined through selection of convenient moving frames. In some cases, owing to selection of convenient moving frames, numerical representation of invariant (or symmetry preserving) compact numerical schemes, that are developed through the proposed method, are found to be notably simpler than that of standard, noninvariant compact numerical schemes. Further, the order of accuracy of these invariant compact schemes can be arbitrarily set to a desired order by considering suitable compact finite difference algorithms. Application of the proposed method is demonstrated through construction of invariant compact finite difference schemes for some common linear and nonlinear PDEs (including the linear advection-diffusion equation in 1D/2D, the inviscid and viscous Burgers' equations in 1D). Results obtained from our numerical simulations indicate that invariant compact finite difference schemes not only inherit selected symmetry properties of underlying PDEs, but are also comparably more accurate than the standard, non-invariant base numerical schemes considered here.
converge with increasing resolution, especially in the case of short waves [1]. In this regard, Hirsh [1] presented a detailed application of compact finite differencing which included development and application of fourth order accurate compact schemes to three test problems, namely viscous Burgers' equation, Howarth's retarded boundary layer flow, and the incompressible driven cavity problem. The author also provided a brief discussion on how to treat boundary conditions when developing compact finite difference schemes, which could be problematic in some cases. In another work, Lele [2] extended the earlier works on compact finite differencing by presenting finite difference schemes that provide a better representation of shorter length scales for use on problems with a range of spatial scales. In addition, the author provided a detailed discussion on how to obtain compact finite difference schemes of various orders (up to tenth order) and treat the relevant boundary conditions. In a more recent work, Shukla et al. [7] presented a family of high order compact schemes that are built on non-uniform grids with spatial orders of accuracy ranging from 4th to 20th. These compact schemes are constructed such that they maintain high-order accuracy not only in the interior of a domain, but also at its boundaries. The authors demonstrated the application of these compact schemes to the linear wave equation and two-dimensional incompressible Navier-Stokes equations, and verified the achievement of high order accuracy for these problems. They further showed (via comparisons with benchmark solutions for the two-dimensional driven cavity flow, thermal convection in a square box, and flow past an impulsively started cylinder) that these high order compact schemes are stable and produce highly accurate results on stretched grids with more points clustered at boundaries.
Although compact finite differencing is an efficient method for construction of high order accurate numerical schemes, these schemes often ignore geometric properties of underlying differential equations as the focus is usually on the accuracy when developing these schemes. Schemes that preserve certain geometric properties (such as energy, momentum, symplecticity, Hamiltonian and Poisson structures of equations) are usually considered as geometric integrators. It is well-documented in literature that geometric integrators, which account for certain geometric properties of underlying differential equations, are likely to perform better than schemes standard schemes that ignore such properties [11][12][13][14][15][16][17][18].
Lie symmetry groups of differential equations are also geometric properties that could be preserved in numerical schemes. Numerous researchers have proposed methods for construction of numerical schemes that preserve symmetry groups of underlying differential equations . Most of these works can be categorized into two major groups. In the first group [20][21][22][23][24], invariants of difference equations are determined through Lie's infinitesimal approach, and then, a set of these invariants are used to construct invariant schemes that converge to the original differential equations in the continuous limit. In the other group [33][34][35][36][37][38][39][40][41][42][43], point transformations based on symmetry groups of differential equations are applied to some base (non-invariant) numerical schemes, and the unknown symmetry parameters of these transformations are determined through moving frames that are based on Cartan's method of normalization [44].
In this paper, we propose a mathematical approach for construction of high order accurate compact finite difference schemes that retain Lie symmetry groups of underlying differential equations. In the pro-posed method that is based on equivariant moving frames, extended symmetry groups of partial differential equations are used to obtain point transformations not only for independent and dependent variables of differential equations, but also for their derivative terms (which is a novel aspect of this paper that was not considered in earlier works [35,36,42]). Once point transformations for derivatives of differential equations are determined, then these transformations are applied to some (non-invariant) base compact finite difference schemes (of a desired order of accuracy) to obtain final invariant (or symmetry preserving) forms of these schemes. Here, we note that the unknown symmetry parameters that appear in these point transformations are determined by choosing convenient moving frames for which numerical representations of base schemes simplify notably, and their accuracy improves. The proposed method is applied to some commonly used linear and nonlinear problems, and for all the test problems, the resulting invariant schemes are found to perform significantly better than selected non-invariant base compact schemes in terms of numerical accuracy, verifying the potential advantages of symmetry preservation.
We demonstrate the implementation of the proposed method by considering fourth order accurate invariant compact finite difference schemes for one-and two-dimensional linear advection-diffusion equations and Burgers' equations (i.e., inviscid, viscous). For numerical simplicity, we use forward differencing to discretize temporal derivatives, and fourth order compact schemes based on central differencing to discretize spatial derivatives. Note that the proposed construction of invariant schemes can also be extended to arbitrarily high order temporal and spatial discretizations. Results obtained from the proposed invariant compact schemes developed for these test problems suggest that symmetry preservation can lead to significant improvements in numerical accuracy, besides storing important geometric information (regarding the underlying differential equations) in associated numerical schemes. This paper is organized as follows. In Section 2, the formulation for the fourth order accurate compact schemes along with a detailed discussion on Lie symmetry analysis and invariantization of compact schemes are provided. The step by step development of invariant compact schemes for some linear and nonlinear problems are noted in Section 3. Performance of the constructed invariant compact schemes, along with a detailed discussion of the results obtained from these schemes are presented in Section 4. And finally, the concluding remarks and a brief summary of the work are given in Section 5.
Mathematical formulation
In this section, the procedure (that is based on equivariant moving frames) for construction of invariant compact schemes is presented in detail. Brief discussions on Lie symmetry analysis and compact schemes are also included.
Construction of compact schemes
Compact finite difference methods are widely used for high order computations, and in some cases are favored over standard finite difference methods, due to their ability to achieve high order accuracy over smaller stencils. For instance, while a standard central difference approximation of the first derivative of a function on a three-point stencil is second order accurate, an approximation based on a compact scheme (that is also derived through central differencing) of the same derivative could be of higher orders. The implementation of compact schemes is rather simple. To illustrate construction of compact schemes through an example, let us develop fourth order accurate compact finite difference schemes for the first and second derivatives of a function U . Consider the following Taylor series expansion of the function U at grid points (i ± 1):
U i±1 =U i ± hU i x + h 2 2 U i xx ± h 3 6 U i xxx + h 4 24 U i (IV) ± O(h 5 )(1)
where h is the discrete spatial step and the symbol (·) x denotes derivative with respect to variable x. Similarly, the first and second derivative of U can be expanded in a Taylor series as
U i±1 x =U i x ± hU i xx + h 2 2 U i xxx ± h 3 6 U i (IV) + h 4 24 U i (V) ± O(h 5 ) (2) U i±1 xx =U i xx ± hU i xxx + h 2 2 U i (IV) ± h 3 6 U i (V) + h 4 24 U i (VI) ± O(h 5 ).(3)
In order to eliminate the higher order derivatives (i.e., U xx , U xxx , U (IV) , and U (V) ) and obtain an implicit relationship between the first derivative U x and the function U at nodes i ± 1, one can multiply Eq. (1) with constant a at point i + 1, and with constant b at point i − 1, and multiply Eq. (2) with quantity c × h at point i + 1, and with quantity d × h at point i − 1, and sum up these resulting quantities to obtain the following equation:
aU i+1 + bU i−1 + chU i+1 x + dhU i−1 x =(a + b)U i + (a − b + c + d)hU i x + (a + b + 2c − 2d) h 2 2 U i xx + (a − b + 3c + 3d) h 3 6 U i xxx + (a + b + 4c − 4d) h 4 24 U i (IV) + (c + d) h 5 24 U i (IV) + O(h 5 ).(4)
The arbitrary constants a, b, c, and d can be obtained via elimination of high order derivatives as {a, b, c, d} =
{ 3 4 , − 3 4 , − 1 4 , − 1 4 }.
Hence, the final form of Eq. (4) is
1 6 U i+1 x + 2 3 U i x + 1 6 U i−1 x = U i+1 − U i−1 2h + O(h 4 )(5)
which relates the function U to its first derivative and provides a fourth order accurate implicit approximation for the first derivative of U . Through similar algebraic manipulations, one can obtain the following fourth order accurate implicit approximation for the second derivative of the function U as well
1 12 U i+1 xx + 5 6 U i xx + 1 12 U i−1 xx = U i+1 − 2U i + U i−1 h 2 + O(h 4 ).(6)
Both Eqs. (5)-(6) yield tridiagonal matrices that can easily be solved to accurately approximate the first and second derivatives of U at all grid points. More information on compact schemes along with compact algorithms for derivatives with higher orders of accuracy, and a discussion on the treatment of boundary conditions in this approach can be found in the literature [1,2].
Lie symmetry analysis
A differential equation is said to possess a symmetry property if one can transform every variable in the equation according to some transformations, such that the resulting output reads exactly the same as the original differential equation in new (transformed) variables. Further, a Lie point symmetry group is an algebraic structure that consists of a set of objects, which correspond to continuous symmetries (or coordinate transformations) that map a system to itself with a binary operation that satisfies the following group axioms: 1) closure, 2) existence of identity element, 3) existence of inverse element, and 4) associativity.
The procedure for determination of Lie point symmetries of equations is straightforward and well-documented in the literature [45][46][47][48][49].
In this context, consider a surface L(x, u, p) = 0 to be a partial differential equation, and let the following be a one-parameter (kth-extended) Lie group G:
x j =x j (x, u, s) u i =ũ i (x, u, s) u i j1 =ũ i j1 (x, u, u 1 , s)(7)
. . .
u i j1j2...j k =ũ i j1j2...j k (x, u, p, s)
where the arbitrary constant s is the symmetry (or group) parameter, and p = (u 1 , u 2 , . . . , u k ). Here, the vectors x = (x 1 , x 2 , . . . , x m ) and u = (u 1 , u 2 , . . . , u n ) denote the independent and dependent variables, respectively, and u k represents the vector of all possible kth order derivatives of u with respect to the independent variables. Also, the operator (·) j1j2···j k represents the partial derivative ∂ k (·) ∂xj 1 ∂xj 2 ···∂xj k . The smooth transformation functions (x j ,ũ i , . . .ũ i j1j2...,j k ) given in group G can be further expanded in a Taylor series about the point s = 0 to determine the infinitesimal form of the one-parameter Lie group G as
x j = x j + s [ξ j (x, u)] + O(s 2 ) , ξ j ≡ ∂x j ∂s s=0 u i = u i + s [η i (x, u)] + O(s 2 ) , η i ≡ ∂ũ i ∂s s=0 u i j1 = u i j1 + s [η i [j1] (x, u, u 1 )] + O(s 2 ) , η i [j1] ≡ ∂ũ i j1 ∂s s=0(8)
. . .
u i j1j2...j k = u i j1j2...j k + s [η i [j1...j k ] (x, u, p)] + O(s 2 ) , η i [j1...j k ] ≡ ∂ũ i j1j2...j k ∂s s=0 .
where ξ j and η i are known as the coordinate functions (or the group infinitesimals), which define the transformation of the coordinate variables under the action of the group G. Similarly, η i [j1···j k ] is the kthextended group infinitesimal that defines how the kth derivative is transformed under the action of G, and is given by the following relation:
η i [j1...j k ] = D j k η i [j1...j k−1 ] − u i j1...j k−1 r D j k ξ r(9)
where D j k is the total derivative operator [46].
The surface L(x, u, p) = 0 is said to be invariant under the action of the group G if the equation reads the same in new variables as represented below:
L(x, u, p) = 0 ⇐⇒ L(x,ũ,p) = 0 .(10)
In order to determine the Lie point symmetry group G that will leave the surface L(x, u, p) = 0 invariant (or unchanged), the following invariance condition is applied
X [k] L(x, u, p) = 0 , (mod L = 0)(11)
where X [k] is the k-extended group operator that is of the form
X [k] = ξ j ∂ ∂x j + η i ∂ ∂u i + · · · + η i [j1...j k ] ∂ ∂u i j1j2...j k .(12)
Solution of the invariance condition given in Eq. (11) through determination of the coordinate functions yields the Lie point symmetry group G associated with the surface L(x, u, p) = 0. A more detailed discussion on Lie symmetry analysis, particularly regarding how to solve the invariance condition, can be found in the reference [46].
Invariantization of compact schemes
In this work, a compact finite difference scheme (corresponding to a surface L(z) = 0) is considered as an invariant compact scheme if its form remains unchanged under the action of a point symmetry group G associated with the surface L(z) = 0. In this context, letÑ c (z) = 0 be an invariant compact finite difference scheme, andφ c (z) = 0 be a stencil equation for the surface L(z) = 0 where z = (x, u, p) is the vector of the independent/dependent variables and derivatives, respectively. The compact schemeÑ c (z) = 0 and the stencil equationφ c (z) = 0 are said to be invariant under the action of the group element g (where g ∈ G) if the following condition is satisfied [34][35][36]:
N c (ρ(z)· z) = 0 ⇐⇒Ñ c (z) = 0 φ c (ρ(z)· z) = 0 ⇐⇒φ c (z) = 0(13)
where ρ(z) represents right moving frames defined on a manifold M such that it is a topological map (ρ : M → G) that satisfies the following condition:
ρ(g · z) = ρ(z) g −1 for ∀ g ∈ G.
For any given non-invariant compact finite difference scheme N c (z) = 0 (constructed for a surface L(z) = 0), an invariant form of this schemeÑ c (z) = 0 can be obtained by simply transforming every coordinate variable and derivative of the base (non-invariant) compact scheme according to the symmetry group G asÑ c (z) = N c (g · z) for all g ∈ G. The unknown group parameters (that appear when the action of a particular group element g on the coordinate variables and derivatives is evaluated) can be determined via Cartan's method of normalization. A more detailed discussion on Cartan's method of normalization and equivariant moving frames can be found in the literature [33][34][35]44].
Construction of invariant numerical schemes
In this section, the invariantization of compact finite difference schemes is illustrated through examples.
In particular, fourth order accurate invariant compact schemes are constructed for some linear and nonlinear problems.
Inviscid Burgers' equation
As our first test problem, we consider the inviscid Burgers' equation (IBE), which is a model that describes nonlinear wave propagation, and is of the form
u t + u u x = 0.(14)
A non-invariant compact scheme can be constructed for the IBE using the compact algorithms developed for the spatial first, Eq. (5), and second, Eq. (6), derivatives. As for the time derivative, for simplicity, a classical first order forward differencing technique can be considered. The order of accuracy can be improved from first to second order via truncation error analysis or defect correction. Hence the final form of the compact scheme develop for the inviscid Burgers' equation can be found as
N c (z) = u (i,n+1) − u (i,n) τ + uu x + d c = 0 .(15)
Here, d c represents the defect correction terms (obtained from truncation error analysis) that are added to the scheme to improve accuracy and is given by
d c = − τ 2 (u 2 u xx + 2 u u 2 x ) + O(τ 2 , h 4 )(16)
where τ and h denote the discrete time and space steps, respectively.
Further, the symmetry group G associated with the inviscid Burgers' equation can be found (via Lie symmetry analysis) as
X 1 = t 2 ∂ ∂ t + x t ∂ ∂ x + (x − t u) ∂ ∂ u X 2 = t x ∂ ∂ t + x 2 ∂ ∂ x + u (x − t u) ∂ ∂ u X 3 = 2 t ∂ ∂ t + x ∂ ∂ x − u ∂ ∂ u X 4 = x ∂ ∂ t − u 2 ∂ ∂ u(17)X 5 = t ∂ ∂ x + ∂ ∂ u X 6 = ∂ ∂ t X 7 = ∂ ∂ x
where X r=1,...,7 , is the group operator that corresponds to that particular subgroup. The point transformations,z = (t,x,ũ,p), associated with a particular subgroup can be found using Lie series expansion as follows
z i = e (sj Xj ) z i = z i + s j (X j z i ) + s 2 j 2! X j (X j z i ) + s 3 j 3! X j (X j (X j z i )) + · · · .(18)
Here, we note that in order to find the extended point transformationsp = (ũx,ũxx), one should extend the group operators given in Eq. (17) such that it accounts for all the derivative terms before these groups are used in the Lie series given in Eq. (18). Alternatively, one can use the chain rule to find the extended point transformations. For instance, the transformation expression for the spatial first derivative can be found bersome and not practical for preservation in associated compact finite difference schemes [42]. Hence, for this particular problem, we only choose the subgroups X 1 , X 3 , X 6 , and X 7 for preservation in the associated (non-invariant) compact scheme given in Eq. (15). The global transformations obtained from these particular subgroups are found via Eq. (18) as
t = e 2s3 (t + s 6 ) λ x = e s3 x + s 7 λ u = e −s3 (λu + s 1 (x + s 7 )) (19) ux = e −2s3 (λ 2 u x + s 1 λ) uxx = e −3s3 λ 3 u xx where λ = 1−s 1 (t+s 6 ).N c (z) = N c (g · z) =ũ (i,n+1) −ũ (i,n) τ +ũũx −τ 2 (ũ 2ũxx + 2ũũ 2 x ) + O(τ 2 ,h 4 ) = 0 .(20)
Based on the point transformations given in Eq. (19), it appears that the symmetry parameter s 3 does not appear in the transformed scheme given in Eq. (20). All the other symmetry parameters can be determined through Cartan's method of normalization. First, we consider convenient normalization conditions that lead to simple stencils. For instance, normalization conditionst (i,n) = 0 andx (i,n) = 0, among infinite possibilities, yield a simple stencil where the symmetry parameters s 6 and s 7 are −t (i,n) and −x (i,n) , respectively. Second, we choose normalization conditions that remove terms from the truncation error of compact schemes under consideration and hence lead to a considerable improvement in numerical accuracy, besides simplifying their numerical representations [42]. In this context, the normalization conditionũ
(i,n) x = 0 can be used to determine the symmetry parameter s 1 u (i,n) x = 0 ⇒ u (i,n) x + s 1 = 0 ⇒ s 1 = −u (i,n) x(21)
as this particular normalization condition removes all the terms that include the spatial first derivative from the compact scheme given in Eq. (20) in the transformed space as shown in the following:
N c (z) = N c (g · z) =ũ (i,n+1) −ũ (i,n) τ −τ 2ũ 2ũxx + O(τ 2 ,h 4 ) = 0 .(22)
The compact scheme given in Eq. (22) is invariant under the symmetry groups X 1 , X 3 , X 6 , and X 7 and can also be expressed in original variables as follows
u (i,n+1) = 1 λ n+1 u (i,n) + τ 2 2λ 2 n+1 (u (i,n) ) 2 u xx(23)
where λ n+1 = 1 − s 1 τ . Note that for most of the test problems considered in this work, we use a time-space orthogonal mesh, t (i+1,n) − t (i,n) = 0 and x (i,n+1) − x (i,n) = 0, and hence, for simplicity, we will replace Here we also note that for this particular problem, for simplicity, we considered first order forward differencing for the time derivative and used the method of modified equations to improve the accuracy of the approximation from first to second order. However, one could also use higher order approximations or other discretization techniques (i.e., Runge-Kutta methods) for the time derivative if desired. A particularly interesting case occurs when a TVD-RK2 discretization technique is used for the time derivative in Eq. (14).
t (i,
In this case, the final form of the invariant compact scheme constructed using the transformations and moving frames considered for the IBE would be identical to the invariant compact scheme given in Eq. (23).
Linear advection-diffusion equation in 1D
As our second test problem, we choose the one-dimensional linear advection-diffusion equation of the
form u t + α u x = ν u xx .(24)
which describes the evolution of a quantity u due to linear advection and diffusion processes. The symbols α and ν denote the constant characteristic speed and diffusion coefficient, respectively. A non-invariant compact numerical scheme can be developed for Eq. (24) as
u (i,n+1) − u (i,n) τ + α u x = ν u xx(25)
where forward differencing is considered for the time derivative, and the spatial first and second derivatives are approximated according to Eq. (5) and Eq. (6), respectively. The symmetry group G associated with the one-dimensional advection-diffusion equation is
X 1 = 2 t 2 ∂ ∂ t + 2 x t ∂ ∂ x − u (t + (x − α t) 2 ) 2 ν ∂ ∂ u X 2 = 4 t ∂ ∂ t + 2 (x + α t) ∂ ∂ x X 3 = t ∂ ∂ x − u (x − α t) 2 ν ∂ ∂ u X 4 = u ∂ ∂ u X 5 = ∂ ∂ t X 6 = ∂ ∂ x .(26)
Considering the subgroups X 1 , X 5 , and X 6 , the following point transformations can be obtained
t = t + s 5 λ x = x + s 6 λ u = λ 1 2 u exp − s 1 γ 2 2λν (27) ux = λ 1 2 ν −1 (s 1 γu + λνu x ) exp − s 1 γ 2 2λν ũxx = λ 1 2 ν −2 (s 2 1 γ 2 u − s 1 λνu − 2s 1 λγνu x + λ 2 ν 2 u xx ) exp − s 1 γ 2 2λν
where λ = 1 − 2 s 1 (t + s 5 ) and γ = x + s 6 − α (t + s 5 ). The other subgroups are neglected as their inclusion leads to point transformations of cumbersome structures that are difficult to implement. The normalization conditionst n = 0 andx i = 0 can be used to determine the symmetry parameters s 5 and s 6 , respectively.
The symmetry parameter s 1 (corresponding to the projection group X 1 ) can be found by considering the normalization conditioñ
u (i,n) xx = 0 ⇒ s 1 = ν u (i,n) u (i,n) xx .(28)
As all the unknown symmetry parameters are defined, the point transformations given in Eq. (27) can be implemented to the base compact numerical scheme, Eq. (25). This implementation appears to reduce the scheme to a form of linear advection equation in the transformed space as
u (i,n+1) −ũ (i,n) τ + αũ (i,n) x = 0(29)
where the spatial second derivative is removed from the scheme owing to the normalization condition given in Eq. (28). Hence, the transformed compact scheme given in Eq. (29) that is constructed for the onedimensional linear advection-diffusion equation and is invariant under the subgroups X 1 , X 5 , and X 6 can be expressed in the original discrete variables as follows
u (i,n+1) = λ − 3 2 n+1 (λ n+1 u (i,n) − τ α u (i,n) x ) exp s 1 α 2 τ 2 2νλ n+1(30)
where λ n+1 = 1 − 2s 1 τ .
Viscous Burgers' equation
As our third test problem, let us consider the viscous Burgers' equation that is of the form
u t + u u x = ν u xx(31)u (i,n+1) − u (i,n) τ + uu x = νu xx .(32)
The symmetry group G associated with the viscous Burgers' equation is
X 1 = t 2 ∂ ∂ t + x t ∂ ∂ x + (x − t u) ∂ ∂ u X 2 = t ∂ ∂ x + ∂ ∂ u X 3 = 2t ∂ ∂ t + x ∂ ∂ x − u ∂ ∂ u X 4 = ∂ ∂ t X 5 = ∂ ∂ x .(33)
The point transformations that account for the projection group X 1 , Galilean transformation group X 2 , scaling group X 3 , and translation groups X 4 and X 5 can be found as
t = e 2s3 (t + s 4 ) λ x = e s3 x + s 5 + s 2 (t + s 4 ) λ u = e −s3 (λu + s 1 (x + s 5 ) + s 2 ) (34) ux = e −2s3 (λ 2 u x + s 1 λ) uxx = e −3s3 λ 3 u xx
where λ = 1 − s 1 (t + s 4 ). As similar to the inviscid Burgers' equation, the scaling symmetry parameter s 3 does not occur when these transformations are implemented to the compact scheme given in Eq. (32).
The symmetry parameters associated with the translation groups X 4 and X 5 can be found by considering the same normalization conditions used for the previous problems. The Galilean parameter s 2 can be found by using the normalization conditionũ (i,n) = 0. And finally, the projection parameter s 1 can be found by choosing a moving frame for which the approximation for the first derivative goes to zero in the transformed
spaceũ (i,n) x = 0 ⇒ s 1 = −u (i,n) x .(35)
The above normalization condition indicates that all terms in the base (non-invariant) compact scheme, Eq. (32), that include the spatial first derivative will be removed from the compact scheme in the transformed space leading to the following reduced form
u (i,n+1) = ντũxx(36)
whereτ =t (i,n+1) . The transformed compact numerical scheme, Eq. (36), that is invariant under all the symmetry groups of the viscous Burgers' equation can also be expressed in original variables as
u (i,n+1) = 1 λ n+1 u (i,n) − s 1 (x (i,n+1) − x (i,n) ) + τ ν λ n+1 u (i,n) xx(37)
where λ n+1 = 1 − s 1 τ .
Advection-diffusion equation in 2D
As our last test problem, we choose the two-dimensional linear advection-diffusion equation that is of the form u t + α u x + β u y = ν (u xx + u yy ) (38) to demonstrate the applicability of the proposed method to a multidimensional problem. Here α and β denote constant characteristic wave speeds along x and y coordinates, respectively. For this particular PDE, two different compact numerical schemes that are invariant under the same symmetry groups, but are constructed using different moving frames, are developed. Similar to the previous problems, the base (non-invariant) compact numerical scheme considered for this PDE is also developed considering forward differencing for the temporal derivative and fourth order compact finite difference algorithms, given in Eq. (5)- (6), for the spatial derivatives as shown in the following:
u (i,j,n+1) − u (i,j,n) τ + α u x + β u y = ν (u xx + u yy ) .(39)
Considering the symmetry group associated with the two-dimensional linear advection-diffusion equation,
X 1 = 4 ν t 2 ∂ ∂ t + 4 ν x t ∂ ∂ x + 4 ν y t ∂ ∂ y − u [(x − α t) 2 + (y − β t) 2 + 4νt] ∂ ∂ u X 2 = 2 ν t ∂ ∂ x + 2 ν t ∂ ∂ y − u (x − α t + y − β t) ∂ ∂ u X 3 = 2 ν y ∂ ∂ x − 2 ν x ∂ ∂ y − u (β x − α y) ∂ ∂ u X 4 = 4 ν t ∂ ∂ t + 2ν x ∂ ∂ x + 2ν y ∂ ∂ y + u[α(x − α t) + β(y − βt)] ∂ ∂u (40) X 5 = u ∂ ∂ u , X 6 = ∂ ∂ t X 7 = ∂ ∂ x X 8 = ∂ ∂ y
the following point transformations that are based on the subgroups X 1 , X 6 , X 7 , and X 8 , are found
t = t + s 6 λ ,x = x + s 7 λ ,ỹ = y + s 8 λ u = λ u exp − s 1 (γ 2 + θ 2 ) λ ũx = (2 λ γ s 1 u + λ 2 u x ) exp − s 1 (γ 2 + θ 2 ) λ ũỹ = (2 λ θ s 1 u + λ 2 u y ) exp − s 1 (γ 2 + θ 2 ) λ (41) uxx = (4 λ γ 2 s 2 1 u − 2 λ 2 s 1 u + 4 λ 2 γ s 1 u x + λ 3 u xx ) exp − s 1 (γ 2 + θ 2 ) λ ũỹỹ = (4 λ θ 2 s 2 1 u − 2 λ 2 s 1 u + 4 λ 2 θ s 1 u y + λ 3 u yy ) exp − s 1 (γ 2 + θ 2 ) λ where λ = 1 − 4 ν s 1 (t + s 6 ) γ = x + s 7 − α (t + s 6 ) θ = y + s 8 − β (t + s 6 ) .
The base compact scheme given in Eq. (39) can be transformed according to the above transformations as
followsũ (i,j,n+1) −ũ (i,j,n) τ + αũx + βũỹ = ν (ũxx +ũỹỹ) .(42)
Here we note that, for simplicity, we ignore the Galilean (X 2 and X 3 ) and scaling (X 4 and X 5 ) groups and do not consider them for determination of the point transformations as their inclusion (besides the other symmetry groups) result in transformations that are laborious to implement. The symmetry parameters s 6 , s 7 , and s 8 can be determined by considering the normalization conditionst n = 0,x i = 0, andỹ j = 0, respectively. As for the determination of the symmetry parameter s 1 , we consider two different normalization conditions to evaluate the effect of these selections on the numerical accuracy of the resulting invariant schemes. We choose
∂xxũ (i,j,n) = 0 ⇒ s 1 = ∂ xx u (i,j,n) 2 u (i,j,n)(43)
as the first normalization condition and construct an invariant compact scheme (referred to as SYM-1) as
followsũ (i,j,n+1) −ũ (i,j,n) τ + αũx + βũỹ = νũỹỹ .(44)
In the second case, we consider the normalization condition
∂xxũ (i,j,n) + ∂ỹỹũ (i,j,n) = 0 ⇒ s 1 = ∂ xx u (i,j,n) + ∂ yy u (i,j,n) 4 u (i,j,n)(45)
and construct another invariant compact scheme (referred to as SYM-2) as
u (i,j,n+1) −ũ (i,j,n) τ + αũx + βũỹ = 0 .(46)
Here we note that both Eq. (44) and Eq. (46) can also be expressed in the original variables by implementing the transformations given in Eq. (41).
Results
In The variation of L ∞ errors (obtained from the standard FTCS scheme, standard compact finite difference scheme, and the invariant scheme) with respect to the number of spatial grid points is demonstrated in figure 2. The proposed invariant scheme (SYM) appears to be two orders more accurate than the standard second order FTCS scheme and is at the same order as the standard compact finite difference scheme which is known to be fourth order accurate. Here, we note that a sufficiently small-time step is considered for this simulation as the fourth order compact algorithms (given in Eq. (5) and Eq. (6)) are only considered for the spatial derivatives.
u(t, x) = 1 √ 2 π σ 2 exp − (x − u(t, x) t) 2 2 σ 2(47)
Further, we evaluated the performance of the proposed method by developing a fourth order accurate invariant compact finite difference scheme for the one-dimensional linear advection-diffusion equation given in Eq. (24). The following analytical solution numerical solutions is also shown in this figure (right figure). The invariant compact scheme appears to be capturing the wave propagation significantly better than the FTCS scheme, and slightly better than the compact scheme. Additionally, L ∞ error and root mean square error measures corresponding to the proposed invariant compact scheme, FTCS scheme and standard compact finite difference scheme are presented in table
u(t, x) = 1 4 π (L 2 + ν t) exp − (x − α t) 2 4 (L 2 + ν t)(48)
2. It appears that the invariant compact scheme is two orders of magnitude more accurate than the FTCS scheme and is one order of magnitude more accurate than the standard compact finite difference scheme.
Additionally, figure 4 shows the variation of L ∞ errors associated to the invariant compact scheme, FTCS scheme, and standard noninvariant compact scheme with respect to the number of spatial grid points. The invariant scheme appears to be two orders more accurate than the standard second order FTCS scheme.
Moreover, although both the invariant and standard non-invariant compact schemes are fourth order accurate, the invariant scheme appears to have slightly less numerical error.
In our next test case, we considered the viscous Burgers' equation and developed a fourth order accurate invariant compact scheme that preserve the whole symmetry group, given in Eq. (33), associated with this PDE. The following analytical solution
u(t, x) = − 2ν φ ∂φ ∂x + 4 ,(49)φ = exp − (x − 4t) 2 4ν(t + 1) + exp − (x − 4t − 2π) 2 4ν(t + 1)
is considered over the spatial domain Γ[0, 2π] where the initial and boundary conditions are determined from this solution.
Snapshots of the propagating shock, at t = 0.25, along with the spatial distribution of numerical errors, obtained from the fourth order accurate invariant compact scheme (SYM), standard second order FTCS scheme, and non-invariant fourth order compact finite difference scheme (COMP) are depicted in figure 5.
Although a coarse grid with 101 nodes is used for this particular run, it appears that the invariant scheme performs well and captures the shock propagation significantly better than the standard FTCS scheme, particularly near the shock-front. Further, L ∞ error and root mean square error analysis given in table 3 also confirms that the invariant compact scheme performs better than the standard FTCS scheme. For this The variation of L ∞ errors obtained from these numerical schemes with respect to number of spatial grid points is shown in figure 6. As expected, the results obtained from the invariant scheme are indeed fourth order accurate and are two orders more accurate than the standard FTCS scheme, which is known to be a second order accurate scheme. Also, both the invariant scheme and the standard fourth order compact scheme yield results of comparable order of accuracy with negligible differences.
As the proposed invariant compact scheme given in Eq. (37) preserves all the symmetry groups of the viscous Burgers' equation (including the Galilean symmetry group), under transformations based on these symmetry groups, the invariant scheme is expected to perform significantly better than the standard numerical schemes that do not preserve these symmetry groups. For instance, under a Galilean transformation of the form
x = x + c t,t = t,û = u + c(50)
the proposed invariant scheme (SYM) is likely to capture the evolution of the velocity profile significantly better than both the standard FTCS and compact schemes. This is expected as the invariant scheme preserves the Galilean transformation group X 2 given in Eq. (33) whereas the standard schemes do not.
To test this particular advantage of the invariant scheme, we applied the Galilean transformation given in Eq. (50) to these numerical schemes and presented the snapshots of the evolution of the numerical solutions from (two different) given initial profiles in figure 7. Additionally, root mean square errors and L ∞ errors associated with these numerical solutions are given in table 4 and table 5. These particular initial conditions along with the associated analytical solutions considered for the left and right plots in figure 7 can be found in reference [38]. The following analytical solution The variation of L ∞ errors (obtained from the proposed invariant schemes, standard FTCS scheme, and non-invariant compact scheme) with respect to the number of spatial grid points is presented in figure 9.
u(t, x, y) = 1 4 π (L 2 + ν t) exp − (x − α t) 2 + (y − β t) 2 4 (L 2 + ν t) .(51)
As expected, both of the proposed invariant compact schemes constructed for the two-dimensional linear advection-diffusion equation are indeed fourth order accurate, and perform significantly better than the second order standard forward in time central in space finite difference scheme (FTCS). Moreover, these invariant schemes also perform with slightly less error compared to the non-invariant compact scheme which is known to be a fourth order accurate scheme. Further, the invariant scheme SYM-2 appears to be slightly more accurate than the invariant scheme SYM-1 which indicates that the selection of moving frames could affect the accuracy of resulting invariant schemes. Although for this particular problem, the differences in the results obtained from the invariant schemes appear to be minor, in general the moving frames must be chosen carefully.
Conclusion
Compact finite difference schemes are preferred over standard finite difference schemes as these schemes enable high order accuracy on stencils with comparably small number of grid points, and have good, spectrallike resolution. In this paper, we presented a method, that is based on moving frames, for construction of invariant compact finite difference schemes that preserve Lie symmetry groups of underlying partial differential equations. In this method, we first determine the extended symmetry groups of PDEs, and then obtain point transformations based on these symmetry groups. These transformations are then applied to some (non-invariant) base compact finite difference schemes such that all the system variables (i.e., independent and dependent variables) and derivatives of these compact schemes are transformed. We then determine the unknown symmetry parameters that exist in these symmetry-based point transformations by considering convenient moving frames that are obtained through Cartan's method of normalization. In most cases, such convenient moving frames not only result in significant improvement in numerical accuracy but also notably simplify the numerical representations of the resulting invariant schemes, and eventually make them easier to program. Performance of the proposed method was evaluated via construction of high order accurate invariant compact finite difference schemes (built on simple three-point stencils) for some linear and nonlinear PDEs. Based on our evaluations, we concluded that symmetry preservation has the potential to significantly improve numerical accuracy of compact schemes, besides embedding important geometric properties of underlying PDEs.
As our first test case, we considered the inviscid Burgers' equation and constructed a high order accurate invariant compact finite difference scheme for this PDE. Although the order of accuracy of compact schemes can be arbitrarily set by considering suitable compact finite difference algorithms, for this particular problem, we chose fourth order accurate compact algorithms to approximate the spatial derivatives and constructed an invariant scheme based on these algorithms. In all the test problems, the temporal derivatives were handled through standard forward differencing. For this particular PDE, in order to improve the numerical accuracy from first to second order in time, the base scheme was modified using defect correction techniques.
The results obtained from this fourth order accurate invariant compact scheme were found to be slightly better than the results obtained from the standard compact scheme and were notably better than those of the standard FTCS scheme. For all the test cases, the computation times, required to run a simulation with a numerical error of comparable order, were found to be similar for both the proposed invariant scheme and standard compact scheme, and the differences were negligible.
As our next test problem, we considered the one-dimensional linear advection-diffusion equation and developed a fourth order accurate invariant compact scheme for this problem as well. For this particular problem, through the use of convenient moving frames (i.e.,ũxx = 0), the numerical representation of the base scheme were reduced to a form of the linear advection equation (ũt + αũx = 0) in the transformed space.
Similar to the previous problem, the quality of results obtained from this invariant compact scheme (in terms of numerical accuracy) was found to be better than that of the standard FTCS and compact schemes.
Next we constructed a fourth order accurate invariant compact finite difference scheme for the viscous Burgers' equation (which is of the form of a linear heat equation,ũt = νũxx, in the transformed space for the normalization conditionũx = 0) that preserves all the symmetries of the Burgers' equation, and compared our results with the standard schemes. As expected, the proposed invariant compact scheme developed for this problem yielded more accurate results than standard schemes in this case as well. In particular, the performance of the proposed invariant scheme was significantly better than that of the standard schemes when a Galilean transformation is applied to these schemes (see figure 7 and tables 4-5) to test how these schemes are affected by such transformations that are based on symmetries of the underlying differential equation. This is due to the fact that the invariant scheme preserves the Galilean symmetry group of the viscous Burgers' equation, whereas the standard schemes do not.
In order to demonstrate the implementation of the proposed method to a multidimensional problem, as our last test case, we considered the two-dimensional linear advection-diffusion equation and constructed a couple of fourth order accurate invariant compact schemes for this problem, where different moving frames are used in the construction of each invariant scheme to evaluate how this action effects the accuracy of the resulting schemes. For the first invariant scheme SYM-1, a normalization condition of the formũxx = 0 is used to determine the projection group parameter s 1 whereas for the other invariant scheme (SYM-2), this particular parameter was determined using the normalization conditionũxx +ũỹỹ = 0. Although both normalization conditions simplify the base compact scheme considered for this PDE notably, the latter condition reduces the base scheme to the form of a two-dimensional linear advection equation (ũt + αũx + βũỹ = 0) in the transformed space. As for the results obtained from these invariant schemes, SYM-2 appears to be slightly more accurate than SYM-1 where both of these schemes are notably more accurate than standard schemes. Although for this particular problem, selection of different moving frames in the construction of invariant schemes did not affect the accuracy of these schemes significantly, this may not be the case for other problems as there are usually infinitely many applicable moving frames, and not all of them will result in accurate invariant schemes.
While the proposed method could be effectively used for construction of invariant compact finite difference schemes with desired order of accuracy, there are few issues that need to be addressed in more detail. schemes. This obstacle could be avoided by selecting base schemes that are better suited to handle such discontinuities.
once the point transformations for the independent and dependent variables are found. Similarly, point transformations associated with a multiple number of subgroups can be obtained by substituting each subgroup into Eq. (18) in an arbitrary order. Although it is possible to consider the full Lie algebra and obtain global transformations for the coordinate variables and derivatives, it is sometimes practical to choose only certain subgroups as the form of the point transformations obtained from the full Lie algebra could be cum-
n) with t n , and x (i,n) with x i in the following examples. Invariance of the compact scheme constructed for the inviscid Burgers' equation, Eq. (23), can be verified by transforming every variable in this scheme according to the transformations given in Eq. (19), and the resulting transformed scheme should be identical to Eq. (23).
and develop an invariant compact numerical scheme for this particular PDE. Similar to the one-dimensional linear advection-diffusion equation, we consider forward differencing for the time derivative and use Eqs. (5)-(6) for the spatial derivatives to construct the non-invariant base compact scheme for the solution of this PDE as shown in the following
this section, performance of the proposed invariant compact finite difference schemes developed for the inviscid Burgers' equation, linear advection-diffusion equation (in 1D and 2D), and viscous Burgers' equation is evaluated. Results obtained from these invariant schemes are compared with the standard schemes for numerical accuracy. We first evaluate the performance of the invariant compact scheme constructed for the inviscid Burgers' equation, Eq. (23), by comparing the results with the analytical solution
over the spatial domain Γ(x) where x ∈ [−3, 3]. The initial and boundary conditions are noted from the analytical solution. Snapshots of the propagating wave that are obtained from the exact solution, the proposed invariant (compact) scheme (SYM), standard fourth order accurate compact scheme (COMP), and the classical second order accurate forward in time central in space (FTCS) scheme are shown in figure 1 (left plot). The associated numerical errors of these schemes, which are estimated as N exact − N numeric , are also given in this figure 1 (right plot). It appears that the results obtained from the proposed invariant compact scheme (SYM) are significantly more accurate than those obtained from the standard finite difference scheme (FTCS) and are slightly better than those obtained from the standard compact finite difference scheme (COMP). Further, the root mean square error (RMSE), estimated as (u a − u n ) 2 /N , and L ∞ error, estimated as max(|u a − u n |), of these numerical schemes, for this particular run, are given in table 1. According to the error analysis presented in this table, the L ∞ errors obtained from the FTCS scheme, compact finite difference scheme, and the invariant scheme are 4.0×10 −2 , 5.8×10 −3 , and 5.1×10 −3 , respectively. Similarly, the root mean square errors for these numerical schemes are found as 9.7×10 −3 (FTCS), 1.3×10 −3 (COMP),
Fig. 1 .
1Inviscid Burgers' equation. Comparison of wave formation profiles, at t = 0.5, obtained from the analytical solution (Exact), the standard forward in time central in space scheme (FTCS), the standard compact scheme (COMP), and the proposed invariant compact scheme (SYM) is shown in the left figure. Spatial distribution of errors for these numerical schemes is displayed in the right figure. Parameter settings: h = 0.2, τ = 0.001, and σ = 0.5 . and 1.1 × 10 −3 (SYM). As expected, the proposed invariant scheme (SYM) which preserves the symmetries of the underlying PDE has significantly less error compared to the standard FTCS scheme and has slightly less error than the standard compact finite difference scheme.
Fig. 2 .
2is considered over the spatial domain Γ[−2, 4], where the initial and boundary conditions are obtained from the exact solution. Here, L is the characteristic width of the kernel and assumed to be equal to 0.4 in all test cases. For this particular problem, evolution of the profile u(t, x) (from a given Gaussian initial profile) obtained from the proposed invariant scheme (SYM), standard FTCS scheme and compact finite difference (COMP) scheme is depicted infigure 3 (left figure). The spatial distribution of errors obtained from these Inviscid Burgers' equation. Comparison of L∞ errors of numerical schemes as a function of number of grid points.
Fig. 3 .
3Advection-diffusion equation in 1D. Snapshots of wave profiles, at t = 1.0, obtained from the analytical solution (Exact), the classical forward in time central in space scheme (FTCS), the standard compact scheme (COMP), and the proposed invariant compact scheme (SYM) are displayed in the left figure. Spatial distribution of errors is displayed in the right figure. Parameter settings: h = 0.2, τ = 0.001, ν = 1/60. .
Fig. 4 .
4Advection-diffusion equation in 1D. Comparison of L∞ errors of numerical schemes as a function of number of grid points.
Fig. 5 .
5Viscous Burgers' equation. Snapshots of shock formation profiles, at t = 0.25, obtained from the analytical solution (Exact), the standard forward in time central in space scheme (FTCS), the standard compact scheme (COMP), and the proposed invariant compact scheme (SYM) are shown in the left figure. Spatial distribution of errors for these numerical schemes is displayed in the right figure. Parameter settings: h = 0.063, τ = 0.0001, ν = 1/12. . particular run, root mean square errors corresponding to the invariant compact scheme, standard FTCS scheme, and non-invariant compact finite difference scheme are found to be 0.0140, 0.1251, and 0.0143, respectively. Similarly, L ∞ errors of these schemes are determined as 0.1060 (SYM), 0.8962 (FTCS), and 0.0994 (COMP).
Fig. 6 .
6Viscous Burgers' equation. Comparison of L∞ errors of numerical schemes as a function of number of grid points.
Based on figure 7 and relevant error tables, it appears that when the Galilean parameter c is equal to zero, all the numerical scheme captures the evolution of the solution well which is expected. However, for the cases when the Galilean parameter c is nonzero, both the standard FTCS scheme and compact finite difference scheme appear to overpredict the solution leading to a significant lag in the solution, particularly for large values of c. On the other hand, the invariant scheme, as it preserves the Galilean symmetry group, captures the evolution of the solution well even for nonzero values of the Galilean parameter c. In fact, in the case of a numerical precision considered in table 4 and table 5, the results obtained from the invariant scheme for nonzero values of c are found to be identical to the results of the case where c = 0. The latter indicates that the Galilean invariance property of the viscous Burgers' equation is indeed preserved in the relevant difference equation. This property of symmetry preservation in numerical schemes can be particularly useful when differential equations associated to more complex symmetries are solved through difference equations.As our last test case, we considered the two-dimensional linear advection-diffusion equation and constructed two different fourth order accurate invariant compact finite difference scheme (SYM-1 and SYM-2) for this PDE. The main difference between the constructed invariant schemes are that both are developed via selections of different moving frames, and the details of these selections are given in Section 3. The objective is to investigate the effect of these selections on the accuracy of the resulting invariant schemes.
Fig. 7 .
7Viscous Burgers' equation. Snapshots of numerical solutions, obtained from the analytical solution (Exact), standard forward in time central in space scheme (FTCS), standard compact scheme (COMP), and proposed invariant compact scheme (SYM), evolving from various initial profiles for different values of the Galilean parameter c. Left: h = 0.1, τ = 0.0001, ν = 0.05, Right: h = 0.02, τ = 0.0005, ν = 0.01.
is used to evaluate the quality of results obtained from the invariant schemes SYM-1 and SYM-2. Spatial distribution of numerical errors corresponding to the proposed invariant compact finite difference scheme (SYM-2) and standard non-invariant FTCS scheme is given in figure 8. Based on this figure, it appears that the invariant scheme has significantly less numerical error compared to the standard noninvariant FTCS scheme in this case as well. This improvement in numerical accuracy is also verified by the error analysis given in table 6, where both invariant schemes (SYM-1 and SYM-2) perform better than the standard schemes. L ∞ errors obtained from the invariant scheme SYM-1, invariant scheme SYM-2, FTCS scheme, and standard non-invariant compact scheme are noted as 3.4 × 10 −5 , 3.3 × 10 −5 , 2.4 × 10 −3 , and 3.8 × 10 −5 , respectively. It appears that the invariant schemes are at least two orders of magnitude more accurate than the standard FTCS scheme. Root mean square error measures of these numerical schemes also yield similar results, which are 3.3 × 10 −6 and 3.1 × 10 −6 for the invariant schemes SYM-1 and SYM-2, 2.7 × 10 −4 for the FTCS scheme, and 3.4 × 10 −6 for the non-invariant compact finite difference scheme.
Fig. 8 .
8Linear advection-diffusion equation in 2D. Spatial distribution of numerical errors, at t = 0.1, obtained from the classical base scheme (left) and the proposed invariant scheme (right). Parameter settings: hx = 0.16, hy = 0.16, τ = 0.0001, α = 1.0, β = 1.0, ν = 1/60.
Fig. 9 .
9Advection-diffusion equation in 2D. Comparison of L∞ errors of numerical schemes as a function of number of grid points.
Further research is required to understand how the performance of invariant compact schemes (constructed through the proposed method) is affected by the choice of subgroups (considered for preservation in the difference equation), choice of moving frames among infinite number of possibilities, and the nature of initial/boundary conditions and their compatibility with the selected subgroups. Based on our simulations, we observed that although it is possible to consider the whole symmetry group of a PDE for preservation in difference equations, this often leads to cumbersome numerical representations without notably enhancing numerical accuracy. For instance, in the case of the viscous Burgers' equation, the whole symmetry group of the PDE is preserved in the related difference equation. However, the advantages owing to the inclusion of the Galilean subgroup only become significant when the invariant scheme is actually transformed under a Galilean transformation as demonstrated in figure 7. Further, the choice of moving frames which are used to determine the unknown group parameters could affect the accuracy of resulting invariant schemes. To our knowledge, there is no systematic approach to select the best moving frame and one must consider all the pros and cons of a particular moving frame before making a selection. Based on our observations, we found that a moving frame that removes the leading order terms from truncation error of a difference equation is usually a good choice as such a moving frame also simplifies the base scheme (in the transformed space) and make it easier to program. Moreover, the performance of the constructed invariant schemes might be affected by the chosen initial/boundary conditions, especially if these conditions are not compatible with the chosen subgroups. This might be due to the fact that some of the limitations of base difference equations carry over to the constructed invariant schemes. For instance, for cases where discontinuities develop in solutions, the performance of the constructed invariant schemes will undoubtedly depend on the chosen base numerical
The compact scheme constructed for the inviscid Burgers' equation, Eq. (15), can be invariantized by transforming every coordinate variable and derivative according to the above transformations
Table 1
1Root mean square error (RMSE) and L∞ error associated with numerical solutions for the inviscid Burgers' equation.Error
FTCS
COMP
SYM
L∞
4.0 × 10 −2
5.8 × 10 −3
5.1 × 10 −3
RMSE
9.7 × 10 −3
1.3 × 10 −3
1.1 × 10 −3
Table 2
2Root mean square error (RMSE) and L∞ error associated with numerical solutions for one-dimensional linear advection-diffusion equation.Error
FTCS
COMP
SYM
L∞
2.9 × 10 −2
1.2 × 10 −3
4.6 × 10 −4
RMSE
1.2 × 10 −2
3.7 × 10 −4
2.1 × 10 −4
Table 3
3Root mean square error (RMSE) and L∞ error associated with numerical solutions for the viscous Burgers' equation.Error
FTCS
COMP
SYM
L∞
0.8962
0.0994
0.1060
RMSE
0.1251
0.0143
0.0140
Table 4
4Variation of RMSE and L∞ errors associated with numeri-
cal solutions presented in figure 7 (left) with respect to the
Galilean parameter c .
c
Error
FTCS
COMP
SYM
0
L∞
0.1157
0.0100
0.0120
RMSE
0.0213
0.0023
0.0022
0.5
L∞
0.5543
0.5131
0.0120
RMSE
0.2424
0.2417
0.0022
1.0
L∞
0.9033
0.9166
0.0120
RMSE
0.3232
0.3206
0.0022
Table 5
5Variation of RMSE and L∞ errors associated with numerical
solutions presented in figure 7 (right) with respect to the
Galilean parameter c .
c
Error
FTCS
COMP
SYM
0
L∞
0.2384
0.0269
0.0217
RMSE
0.0339
0.0041
0.0034
0.3
L∞
2.1117
2.0058
0.0217
RMSE
0.7521
0.7451
0.0034
0.75
L∞
2.2750
2.0118
0.0217
RMSE
1.2066
1.2027
0.0034
Table 6
6Root mean square error (RMSE) and L∞ error associated with numerical solutions for two-dimensional linear advection-diffusion equation.Error
FTCS
COMP
SYM-1
SYM-2
L∞
2.4 × 10 −3
3.8 × 10 −5
3.4 × 10 −5
3.3 × 10 −5
RMSE
2.7 × 10 −4
3.4 × 10 −6
3.3 × 10 −6
3.1 × 10 −6
AcknowledgmentsThe first author (PhD candidate) is grateful for financial support provided by the Ministry of National Education of Turkey.
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|
[] |
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"Interpreting the Spitzer/IRAC Colours of 7 z 9 Galaxies: Distinguishing Between Line Emission and Starlight Using ALMA",
"Interpreting the Spitzer/IRAC Colours of 7 z 9 Galaxies: Distinguishing Between Line Emission and Starlight Using ALMA"
] |
[
"G W Roberts-Borsani \nDepartment of Physics and Astronomy\nUniversity of California\n430 Portola Plaza90095Los Angeles, Los AngelesCAUSA\n\nDepartment of Physics and Astronomy\nUniversity College London\nGower StreetWC1E 6BTLondonUK\n",
"R S Ellis \nDepartment of Physics and Astronomy\nUniversity College London\nGower StreetWC1E 6BTLondonUK\n",
"N Laporte \nDepartment of Physics and Astronomy\nUniversity College London\nGower StreetWC1E 6BTLondonUK\n\nKavli Institute for Cosmology\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK\n\nCavendish Laboratory\nUniversity of Cambridge\n19 JJ Thomson AvenueCB3 0HECambridgeUK\n"
] |
[
"Department of Physics and Astronomy\nUniversity of California\n430 Portola Plaza90095Los Angeles, Los AngelesCAUSA",
"Department of Physics and Astronomy\nUniversity College London\nGower StreetWC1E 6BTLondonUK",
"Department of Physics and Astronomy\nUniversity College London\nGower StreetWC1E 6BTLondonUK",
"Department of Physics and Astronomy\nUniversity College London\nGower StreetWC1E 6BTLondonUK",
"Kavli Institute for Cosmology\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUK",
"Cavendish Laboratory\nUniversity of Cambridge\n19 JJ Thomson AvenueCB3 0HECambridgeUK"
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[] |
Prior to the launch of JWST, Spitzer/IRAC photometry offers the only means of studying the rest-frame optical properties of z >7 galaxies. Many such high redshift galaxies display a red [3.6] -[4.5] micron colour, often referred to as the "IRAC excess", which has conventionally been interpreted as arising from intense [O III]+H β emission within the [4.5] micron bandpass. An appealing aspect of this interpretation is similarly intense line emission seen in star-forming galaxies at lower redshift as well as the redshift-dependent behaviour of the IRAC colours beyond z ∼7 modelled as the various nebular lines move through the two bandpasses. In this paper we demonstrate that, given the photometric uncertainties, established stellar populations with Balmer (4000 Å rest-frame) breaks, such as those inferred at z >9 where line emission does not contaminate the IRAC bands, can equally well explain the redshiftdependent behaviour of the IRAC colours in 7 z 9 galaxies. We discuss possible ways of distinguishing between the two hypotheses using ALMA measures of [O III] 88 micron and dust continuum fluxes. Prior to further studies with JWST, we show that the distinction is important in determining the assembly history of galaxies in the first 500 Myr.
|
10.1093/mnras/staa2085
|
[
"https://arxiv.org/pdf/2002.02968v1.pdf"
] | 211,069,488 |
2002.02968
|
20676fd21bea5b74d26680bd917d3d3ef4a606ce
|
Interpreting the Spitzer/IRAC Colours of 7 z 9 Galaxies: Distinguishing Between Line Emission and Starlight Using ALMA
G W Roberts-Borsani
Department of Physics and Astronomy
University of California
430 Portola Plaza90095Los Angeles, Los AngelesCAUSA
Department of Physics and Astronomy
University College London
Gower StreetWC1E 6BTLondonUK
R S Ellis
Department of Physics and Astronomy
University College London
Gower StreetWC1E 6BTLondonUK
N Laporte
Department of Physics and Astronomy
University College London
Gower StreetWC1E 6BTLondonUK
Kavli Institute for Cosmology
University of Cambridge
Madingley RoadCB3 0HACambridgeUK
Cavendish Laboratory
University of Cambridge
19 JJ Thomson AvenueCB3 0HECambridgeUK
Interpreting the Spitzer/IRAC Colours of 7 z 9 Galaxies: Distinguishing Between Line Emission and Starlight Using ALMA
Accepted XXX. Received YYY; in original form ZZZMNRAS 000, 1-9 (2019) Preprint 11 February 2020 Compiled using MNRAS L A T E X style file v3.0galaxies: evolution -galaxies: high-redshift -cosmology: reionization -cos- mology: early Universe
Prior to the launch of JWST, Spitzer/IRAC photometry offers the only means of studying the rest-frame optical properties of z >7 galaxies. Many such high redshift galaxies display a red [3.6] -[4.5] micron colour, often referred to as the "IRAC excess", which has conventionally been interpreted as arising from intense [O III]+H β emission within the [4.5] micron bandpass. An appealing aspect of this interpretation is similarly intense line emission seen in star-forming galaxies at lower redshift as well as the redshift-dependent behaviour of the IRAC colours beyond z ∼7 modelled as the various nebular lines move through the two bandpasses. In this paper we demonstrate that, given the photometric uncertainties, established stellar populations with Balmer (4000 Å rest-frame) breaks, such as those inferred at z >9 where line emission does not contaminate the IRAC bands, can equally well explain the redshiftdependent behaviour of the IRAC colours in 7 z 9 galaxies. We discuss possible ways of distinguishing between the two hypotheses using ALMA measures of [O III] 88 micron and dust continuum fluxes. Prior to further studies with JWST, we show that the distinction is important in determining the assembly history of galaxies in the first 500 Myr.
INTRODUCTION
The last few years has seen impressive progress in studies of galaxies in the so-called "reionisation era" corresponding to the redshift interval 7 < z < 10. However, the number of spectroscopicallyconfirmed examples remains limited and much has been deduced from spectral energy distributions (SEDs) of photometric samples. In addition to demographic studies based on star formation rate densities (Oesch et al. 2014;McLeod et al. 2016) and luminosity functions (Atek et al. 2015;Bouwens et al. 2015), a key area of interest is studies of the gaseous and stellar properties of early systems. The latter topic is central to understand both the ionising capability of early galaxies as well as the age of their stellar populations (for a recent review see Stark 2016).
Although much of the progress has been made using photometric samples based on Hubble imaging, both in deep fields (Grogin et al. 2011;Koekemoer et al. 2011;Ellis et al. 2013) and through lensing clusters (Bradley et al. 2014;Lotz et al. 2017;Salmon et al. 2018;Coe et al. 2019), the Spitzer Space Telescope has made a key contribution since, at z 5, the two bandpasses at 3.6 and 4.5 µm sample the rest-frame optical. At redshifts of E-mail: [email protected] z 6.6, it is claimed that the redshift-dependent trend of the IRAC colours is consistent with intense nebular emission lines shifting through the bandpasses (Labbé et al. 2013;Smit et al. 2015;Roberts-Borsani et al. 2016), although the precision of this exercise is dependent mostly on samples with only photometric redshifts. With this in mind, the surprising spectroscopic confirmation with Lyman-α Zitrin et al. 2015;Stark et al. 2017) of the 4 brightest z > 7 galaxies in the CANDELS survey selected to display red Spitzer/IRAC 3.6µm-4.5µm colours (and hence an "IRAC excess") of > 0.5 mag (Roberts-Borsani et al. 2016), reinforced the hypothesis that the IRAC excess arises from intense [O III] 5007 Å plus Hβ emission within the 4.5µm band. To explain the IRAC colours, the rest-frame equivalent widths (EWs) of [O III] must be of order 500 Å.
Although it will not be possible to confirm this suggestion with direct spectroscopy until the launch of JWST, the so-called " [O III] hypothesis" has been widely accepted for several reasons. Firstly, at lower redshift z ∼6.6 where [O III] passes through the 3.6µm bandpass, the required blue 3.6µm-4.5µm colour is seen for a sample of galaxies, several of which are now spectroscopicallyconfirmed with ALMA (Smit et al. 2018; see also Sobral et al. 2015,Pentericci et al. 2016,Matthee et al. 2017. Finally, as a proof of concept, galaxies whose rest-frame [O III] emission exceed EW 1000 Å , while difficult to reproduce via modelling except in very young star-forming systems, have been studied at z 2 (Maseda et al. 2014).
A more recent development has been the location of IRACexcess galaxies whose photometric redshifts lie at z >9; at these redshifts [O III] is shifted beyond both IRAC filters. Thus far only one system, MACS1149-JD1, hereafter JD1, has been spectroscopically-confirmed at z=9.11 (Hashimoto et al. 2018). Analysis of its SED attributes the IRAC excess to the Balmer break at 4000 Å consistent with a mature ∼200-300 Myr old stellar population providing a first tantalising glimpse of "cosmic dawn" at z 15±3. This discovery raises the question of the extent to which the IRAC excess seen in galaxies at 7< z <9 might also, in part, be due to a similar Balmer break. The distinction is important since it would imply many luminous z 7-9 galaxies may have older stellar populations and larger stellar masses than previously thought, with interesting consequences for the presence of earlier star formation. An additional issue is whether JD1 is representative of the galaxy population at z 7 − 9 (see Katz et al. 2019).
The goal of the present paper is to explore the extent to which an IRAC excess and its redshift-dependent trend might be due, in part, to starlight rather than solely intense [O III] line emission. A plan of the paper follows. In §2 we examine predicted IRAC colours in the context of both hypotheses, using carefully-chosen template galaxies as well as contemporary stellar population models that incorporate nebular line emission. In §3, we turn to what data might be needed to distinguish between the two hypotheses. Prior to spectroscopy with JWST, we consider the flux of [O III] 88µm which is accessible with ALMA and examine the IRAC colour for those spectroscopically-confirmed galaxies for which [O III] 88 µm fluxes are available. In §4 we discuss our results and the implications on the early assembly of galaxies. Throughout this paper we refer to the HST F160W and Spitzer/IRAC 3.6 and 4.5 micron bands as H 160 , [3.6] and [4.5], respectively, for simplicity. We also assume H 0 =70 km/s/Mpc, Ω m = 0.3, and Ω ∧ =0.7. All magnitudes are in the AB system (Oke & Gunn 1983).
MODELLING THE IRAC COLOURS OF 7< Z <9 GALAXIES
In exploring the redshift-dependent behaviour of the IRAC 3.6 and 4.5µm colours, under the hypotheses of contributions from [O III] line emission or a Balmer break due to a more mature stellar population, we begin by selecting two template SEDs fitted to actual data for spectroscopically-confirmed z >7 galaxies. For the case of intense [O III] emission, we use a spectroscopic template fitted to the ground-based and HST/Spitzer photometry of EGSY8p7 at z=8.68 (Zitrin et al. 2015), one of the four IRAC-excess bright sources first identified in the CANDELS survey (Roberts-Borsani et al. 2016). The template and HST/Spitzer photometry used here are taken directly from the latter study and we refer the reader to that paper for details on the construction of the templates and derivation of the photometric data points. Similarly, for the case of a mature stellar population with a prominent Balmer break, we select the SED fit to MACS1149-JD1 at z=9.11 from Hashimoto et al. (2018). This fit represents a composite of a mature ∼200-300 Myrs population augmented with a younger component invoked to match the intensity of [O III] emission at 88µm discovered with ALMA; full details can be found in Hashimoto et al. (2018) The adopted spectral templates and associated photometry are shown in Figure 1, and for all subsequent analysis, we normalise both spectra by their flux at 0.325 µm (rest-frame), where the spectra are free from emission or absorption features, in order to ensure their 3.6 µm and 4.5 µm photometry can be directly compared.
We are now in a position to explore how these template spectra affect the IRAC colours over the redshift range 7< z <9. In Figure 2 we present the redshift evolution of the [3.6]-[4.5] and H 160 -[3.6] colours in steps of ∆ z = 0.05 for each of our fiducial spectra, following a similar approach by Labbé et al. (2013). At each redshift interval, the colours are measured directly from the redshifted spectrum using the relevant filter response curves. The simulation shows that the Balmer break in JD1 can mimic the effect of intense line emission to within ±0.1 mag in the [3.6]- [4.5] colour, particularly at redshifts z 7.5, and even produce redder colours at z 8.5. Since the JD1 template includes a contribution from nebular emission lines (e.g., [O II], Hβ and [O III]), including some from Hβ and [O III]λ4959 Å at z=9.11 in the 4.5µm band, we explored suppressing all optical line emission in this template but find consistently red IRAC colours with virtually identical colourcolour evolution (particularly at z 8) and little difference in normalisation (the masked spectrum consistently produces red IRAC [3.6]-[4.5] colours 0.2 mag lower than its unmasked counterpart) on the simulated colours (see discussion below and Figure 2).
The similarity between the redshift-dependent trends of a 6] colours as a function of redshift from 7 z 9 for the spectral templates of spectroscopically-confirmed galaxies EGSY8p7 (diamonds; assumed to be an extreme [O III]+Hβ emitter) and JD1 (circles; a Balmer break galaxy). Since the template of JD1 includes some optical nebular emission, we also show the results for JD1 with masked emission lines (crosses) to highlight the contribution from the stellar continuum only. The gray shaded region indicates the colour space over which the different spectra produce similar colours for the selected redshift interval. Overplotted are the z ∼8 and z ∼7 results from stacked photometrically-selected galaxies from the study of Labbé et al. (2013), and the connecting vector (black line) between the two. A clear difference exists between spectroscopic and photometric results, for which blue Spitzer/IRAC colours at z ∼7 may reflect contamination of the stack by strong nebular line emitters at z ∼6.6.
Balmer break and intense line emission over the chosen redshift interval 7< z <9 may seem surprising given earlier conclusions of a similar exercise undertaken by Labbé et al. (2013). Those authors explored the effect by analysing average SEDs (containing nebular emission lines) of photometrically-selected z phot ≈7 and z phot ≈8 galaxies. They were able to reproduce the [3.6] -[4.5] and H 160 -[3.6] colours only with the presence of strong (EW([O III]+Hβ)∼560-670 Å) nebular emission lines, rejecting an increased dust content or stellar age. Their average z ∼ 7 and z ∼ 8 colours, as well as the connecting vector, are shown in Figure 2 together with our results. However, as can be seen using our templates based on spectroscopically-confirmed galaxies, the same overall trend and [3.6] -[4.5] colours can also be reproduced by the JD1 Balmer break template. Throughout the redshift interval 7< z <9, at no point does the [3.6] -[4.5] colour drop below zero, in contrast to what Labbé et al. (2013) found at z ∼7. Conceivably their average SED at z=7 was contaminated by a few z ∼6.6 extreme line emitters now known to have extremely blue colours (Smit et al. 2015) due to contamination of the 3.6 µm band by [O III]+Hβ. At this point we caution the reader that this exercise is not motivated to claim that the IRAC excess seen in many sources at 7 < z < 9 cannot be due to intense [O III]+Hβ emission. Rather, we wish to point out that the existence of a Balmer break for JD1 at z > 9 may imply some contribution of starlight to the IRAC excess seen in sources at 7 < z < 9 and to explore whether such starlight is prominent in existing spectroscopically-confirmed galaxies at 7 < z < 9. Of course, these results will depend on the choice of spectral template. In the case of JD1 we selected the only case known to date of a galaxy with an IRAC excess which cannot be explained solely via intense line emission (Hashimoto et al. 2018). For the line emitter template, the results will differ slightly depending on which object is chosen from the samples available in the literature. Given this uncertainty plus recent discussions on the possible exceptional case of JD1 (Bingelli et al. 2019), it is helpful to examine the redshift-dependent trends produced by stellar population synthesis models as well as to expand the discussion to compare all these predictions with actual z > 7 data in the literature.
For the population synthesis models we use the Pégase3 suite (Fioc & Rocca-Volmerange 2019) which includes self-consistent modelling of nebular line emission and dust evolution. Clearly such models have an abundance of free parameters but, for the present exercise, our main goal is to demonstrate that relative contributions of synthetic galaxy spectra selected at various ages from a simple star formation history can also reproduce the trends we see using our observed spectral templates. For the current experiment, our simulated galaxy adopts a Chabrier (2003) IMF with a constant star formation history beginning at z = 15 and ending at z = 2. We select simulated spectra at various time intervals corresponding to galaxy ages from 1 Myr (z ∼15) to 600 Myr (z ∼6). Nebular emission and dust evolution are included in the modelling. As with the EGSY8p7 and JD1 spectra, the Pégase3 spectra are normalised to their flux at 0.325 µm (rest-frame) prior to analysis. Figure 3 shows the IRAC [3.6] -[4.5] colour versus redshift trend for our fiducial galaxy templates as well as the Pégase3 models. For each of our SEDs (i.e., EGSY8p7, JD1 and each of the Pégase spectra corresponding to various time intervals of the simulated galaxy's evolution), the spectrum is redshifted across our range of interest and the colour measured through the relevant response filters, in order to assess the relative colour contributions from each spectrum's features. The figure also shows the JD1 template adjusted to exclude the contribution from optical nebular lines. In order to compare with actual data, Table 1 represents a compilation of 12 z >7 spectroscopically-confirmed galaxies drawn from the literature, each with available HST and Spitzer/IRAC photometry, and we plot their photometric data alongside the spectroscopic results in Figure 3. For completeness we also add four spectroscopically-confirmed z 6.8 sources from the studies of Smit et al. (2018) and Laporte et al. (2017b) which demonstrate the influence of [O III]+Hβ emission in the 3.6µm band at lower redshift.
Focusing initially on the comparison between Pégase3 and our chosen galaxy templates, we can see very similar trends, albeit with some difference in normalisation. Within the 7 < z < 9 redshift range, Pégase models corresponding to younger ages closely track the evolution of the EGSY8p7 template, whose red colours are dominated by strong nebular emission lines, whilst evolved stellar ages are required to explain the evolution of the JD1 templates (both with and without emission lines), where the red colour is primarily due to a Balmer break.
Considering next how the templates and synthesis models match the IRAC colours of 12 spectroscopically-confirmed z > 7 galaxies, we can see that line emission in both Pégase3 and the EGSY8p7 (line emitting) template are required to explain the strong dip in [3.6] -[4.5] colour at z 6.6 as indicated by Smit et al. (2018); as expected the JD1 template with masked emission lines has no dip. However, at higher redshift, where the [O III]+Hβ lines enter the 4.5 µm band, the red colour is initially more easily reproduced by cases with strong line emission. For SEDs with a flat continuum, as assumed by Roberts-Borsani et al. (2016), such a colour remains relatively constant until the lines leave the band at z = 9. However, in the case of a moderate to strong Balmer break and reduced (but not absent) line emission, the IRAC excess increases from z 7.5 onwards as the Balmer break moves redward, thereby removing flux from the 3.6µm band whilst simultaneously providing a relatively constant amount of flux in the 4.5 µm band. In the case where the rest-frame optical continuum is not flat, this impacts the slope of the [3.6] -[4.5] colour evolution. The impact of the Balmer break is particularly evident when comparing the evolution of the masked JD1 spectrum to the older-aged Pégase3 synthesis models which, despite including (normal) emission lines display virtually identical [3.6] -[4.5] evolution.
Examining the actual data, one can reasonably securely conclude that the large IRAC excesses at 7 z 7.5 (e.g. Finkelstein et al. 2013;Hashimoto et al. 2019) are difficult to reproduce without an extreme [O III]+Hβ contribution, as is clearly the case for IRAC colours for the sources at z 6.8 studied by Smit et al. (2018) and Laporte et al. (2017b). However, for the sources at 7.5 z 9 (e.g., Watson et al. 2015;Tamura et al. 2019;Laporte et al. 2017a plus GN-z10-3 and EGSY8p7), the paucity of spectroscopic data makes it premature to conclude that the IRAC excess in galaxies beyond z 7 arises entirely from line emission.
DISTINGUISHING BETWEEN A BALMER BREAK AND INTENSE LINE EMISSION WITH ALMA
In Section 2 we have shown that red Spitzer/IRAC [3.6]-[4.5] colours for galaxies lying between 7 < z < 9 could arise from contributions of both intense nebular line emission and starlight. We now consider whether it is possible to break this degeneracy prior to the use of spectroscopy with JWST. ALMA observations with Band 7 targeting the [O III] 88 µm line and dust continuum may provide a potential way forward. Using detailed SED modelling with synthetic spectra from Pégase3 we now investigate whether ALMA observations can place constraints on the contribution from young stars via [O III] 88 µm emission and mature stellar populations from the presence of a dust continuum. Currently, four of the 12 z >7 spectroscopically-confirmed galaxies with Spitzer/IRAC excesses listed in Table 1 have the appropriate ALMA data: JD1, A2744_YD4, MACS0416_Y1 and B14-65666 (henceforth YD4, Y1 and B14 for convenience), see Table 2. To determine accurate SEDs, for each of the aforementioned galaxies we use the relevant references in All HST upper limits and error bars represent 1σ uncertainties, whilst those redward of these are 2σ.
To evaluate the relative contributions of nebular emission lines and starlight we create a repertoire of Pégase3 spectra with which to fit the above data for the four spectroscopically-confirmed z >7 galaxies. We generate mass-normalised galaxy spectra for a young component dominated by a recent burst of constant star formation with duration τ young =10 Myrs, and for a component with a less recent phase of constant star formation for a range of durations τ old =[10, 50, 100, 200, 300, 400, 500] Myrs, where a Balmer break is allowed to form. We then extract spectra at 1 Myr intervals for the young component, and 20 Myr intervals from ages of 1 Myr to the age of the Universe at the redshift of each galaxy for the older component. For simplicity, we assume emission lines arise from the young component only, since these come from star-forming regions and are not seen in mature stellar populations. These models are used, sometimes in combination (i.e. recent burst + earlier star formation), with a custom SED-fitting code in a Bayesian framework, to maximise the log-likelihood of the model given the data, including an analytical treatment of upper limits (Sawicki 2012). The free parameters of the code are the mass of the galaxy system, M sys 1 (one for each stellar component) and a multiplicative factor, L neb , to scale the luminosity of the nebular emission lines, whose FWHM are fixed to that of the [O III] 88 µm line presented in Table 2. The adopted priors are log M sys =[5,15] M and L neb =[0,50], allowing for both normal and extreme nebular emission contributions.
To illustrate how the ALMA observations may differentiate between intense nebular emission and starlight in explaining the IRAC colours, we consider the case of YD4 since, for this source, all photometric points redward of the Lyman break and the ALMA spectroscopic constraints are robustly measured. First we fit the data with a single-component young model (permitting a dust contribution), once with HST+VLT/HAWK-I+Spitzer/IRAC data only and then again incorporating the ALMA Band 7 constraints. The best-fit SEDs are shown in Figure 4. The continuum fits to the HST and HAWK-I photometry are satisfactory and comparable in each case. However, there is a major difference in the predicted Spitzer/IRAC photometry. Ignoring the ALMA constraints, the IRAC excess demands the presence of strong nebular emission lines in which case the [O III] 88 µm line is considerably overpredicted (by a factor 10, not shown) and the continuum dust emission is similarly poorly matched. Additionally, the presence of nebular emission lines -namely the [O II] doublet at a rest-frame of ∼3730 Å -adds non-negligible boosting to the [3.6] band. Including the ALMA constraints, the nebular emission in the IRAC bands is modest, indicating the need for an additional component to fit these data, for example a Balmer break originating from star formation at earlier times.
We thus now proceed to fit all of the available data for each of the four galaxies with a two-component model comprising a contribution from young stars with intense nebular emission and an older stellar component. The two-component models are derived from all unique combinations of young and older spectra requiring only that age young < age galaxy − τ old so that the recent burst of star formation from the young component occurs only when star formation in the older component has completed. In these two-component models, for simplicity, we assume the dust contributions arise only from the older component and specifically only if there is a Band 7 continuum detection. By comparing these two-component fits to those for a single-component (with dust included following the guidelines indicated above), we can determine, as suggested in the case of YD4, (i) whether the two-component fits are significantly better than the single young component ones and (ii) whether the first indicate the presence of an older, more mature stellar population.
The results of our best fit two-component models are presented in Figure 5, where we find generally good agreement with the observed photometric data sets and ALMA constraints. For two of the four galaxies (JD1 and YD4), the best-fit model correctly predicts the presence or upper limit of dust mass based on the ALMA Black points represent spectroscopically confirmed galaxies at z >7 with red IRAC colours (see Table 1) with additional data for z ∼6.6 sources with blue IRAC colours. The evolution for the synthetic spectra of EGSY8p7 (orange), JD1 (red, solid), JD1 with masked emission lines (red, dashed) can be compared with synthesis models including line emission from Pégase3 (blue shades) for a variety of galaxy ages, assuming a constant SFR from z =15 to z =2. The light gray shaded region highlights an approximate redshift interval over which ambiguity exists as to the primary mechanism for red Spitzer/IRAC colours. Figure 4. SED fits to the spectroscopically confirmed galaxy, YD4, with HST, VLT/HAWK-I K s and Spitzer/IRAC photometry (red points and error bars) and single models of dusty 1-10 Myr Pégase3 spectra. The suite of young Pégase3 spectra are fit once without the ALMA Band 7 spectroscopic observations (light gray line and blue points) and once with them (dark gray line and orange points), both with the strength of nebular emission lines as a free parameter in addition to their stellar masses. The IRAC excess is well fit by contributions from nebular emission lines without inclusion of the ALMA data. However when such constraints are included, the nebular emission is suppressed and cannot account for the excess flux in the Spitzer/IRAC 4.5 µm band. Thus, a secondary component arising from starlight is necessary to match the data.
well-reproduced for JD1 and YD4, the single-component fits fail to simultaneously reproduce both the ALMA constraints and the Spitzer/IRAC excess. In the cases of Y1 and B14, the ALMA constraints are better matched by the one-component fit, however the Spitzer/IRAC excess is only partially matched.
In our two-component fits, a sizeable contribution to the IRAC flux arises from a more mature stellar component. For JD1, YD4 and Y1, the contribution to the IRAC fluxes from the recent burst of star formation is only ∼10-30% and the older component (characterised by Balmer ratios of ∼2) dominates the flux at 70%. This is due primarily to the relative weakness of the [O III]+Hβ lines, for which we measure an equivalent width, EW([O III]+Hβ)≈25-106 Å (compared to EW([O III]+Hβ)≈450-7700 Å for a singlecomponent). The exception to this trend is B14, whose IRAC fluxes remain dominated by the younger component by ∼60-75% (EW([O III]+Hβ)≈200 Å for the two-component model and EW([O III]+Hβ)≈3460 Å for the one-component model). The total stellar mass (corrected for lensing for those lensed sources) for all of these fits range from log M * =8.88-10.19 M , with virtually all of the total stellar mass also coming from the earlier period of star formation and the most recent burst contributing primarily through the presence of weak-to-moderate nebular emission. Furthermore, we also note a difference in the ages of the galaxies, as determined by the onset of the earlier burst in the twocomponent model or the age of the single-component model. We find an increase in age for each of the four galaxies when multiple components are used, with ages of 80, 260, 140 and 20 Myrs characterising the two-component fits of JD1, YD4, Y1, and B14, and such ages decreasing down to 1, 2, 1 and 3 Myrs assuming on a single burst of recent star formation. In the case of Y1, the age estimate from the two-component fit is in fact a lower limit, since we only have an upper limit for the Spitzer/IRAC 3.6µm photometry. We note here that our preferred τ old =10 Myrs value and the galaxy age estimate for JD1 are somewhat lower than those estimated by Hashimoto et al. (2018), whose best fit comprises an older stellar population with an episode of star formation lasting τ old =100 Myrs and a galaxy age of 290 +190 −120 Myrs. However, as in their analysis, we find considerable statistical similarity with the best two-component fit assuming τ old =100 Myrs, in which case our age estimate increases to 140 Myrs, closer to the lower limit of their assumed value.
Finally, to enable a quantitative comparison of the one-and two-component fits we examine the log-likelihoods and find, with the exception of Y1, that the two-component fits of each of the galaxies is considered a better fit. However, given the obvious danger of concluding better fits with an additional component with further free parameters, we compare the goodness-of-fit via a comparison of their Bayesian Information Criteria (BIC), which uses the log-likelihoods whilst penalising for additional free parameters. With this consideration, the additional free parameters in the fit to the B14 data are sufficiently penalised to justify only the onecomponent fit. Whilst such comparisons no doubt gloss over the complexities of defining the birth of such galaxies, they serve as an important illustration of the potential consequences from overlooking the consideration of multiple episodes of star formation. We provide a summary of the above comparison and the main properties of our fits in Table 3.
DISCUSSION
We have shown that a Balmer break, arising from a mature stellar population, may be a significant contributor to the IRAC excess seen in spectroscopically-confirmed z > 7 star-forming galaxies. While our analysis does not rule out the possibility that much of this excess arises, as has been conventionally assumed, from intense [O III] emission, using ALMA [O III] 88µm emission and dust mass measures, we have examined whether we can constrain the relative contributions of starlight and line emission.
The distinction between intense line emission, attributed to recent episodes of star formation from a young ( 10 Myr) stellar population, and a prominent Balmer break consistent with more mature stars, is important in considerations of the early assembly history of galaxies. Both the stellar masses and earlier star formation histories will differ depending on the relative contributions and this, in turn, will affect the inferred star formation activity beyond . The best fit two-component SED models (blue lines and black points) to observed data (dark red circles and error bars) from HST/VISTA + VLT/HAWK-I K s + Spitzer/IRAC photometry (left) and ALMA spectroscopy (upper right inset) for galaxies at z > 7 with red Spitzer/IRAC colours (JD1, YD4, Y1 and B14). The total fit consists of contributions from both old (darker gray) and young (lighter gray) stars. the current HST redshift horizon of z 10. This was first demonstrated for the z = 9.11 galaxy JD1 by Hashimoto et al. (2018) where the IRAC excess must arise primarily from starlight, leading to a stellar mass of 4.2±1.0×10 9 M (lens-corrected for the preferred gravitational magnification) only 550 Myr after the Big Bang with an implied epoch of first star formation as early as z ∼15.
As an illustration, if we adopt the significant contribution to the IRAC excess from starlight for those z > 7 sources in Table 2 for which we fit two-components, their stellar masses increase by an average factor of 30 compared to contributions from young starlight alone. Although a single-component fit is probably an extreme comparison in this context, nonetheless our two-component fits with an earlier period of star formation must imply an assem-bly history beyond z 10, as discussed by Hashimoto et al. (2018), with interesting consequences for the interpretation of 21cm experiments (Bowman et al. 2018) and the timing of "cosmic dawn". In Figure 6, we plot the fractional stellar mass assembly history, averaged over our 4 galaxies, up to the epoch of observation in which it can be seen that ∼44% of the stellar mass was produced before a redshift z 10. Although clearly a modest sample restricted largely to the brightest studied sources at z > 7, we can compare this fractional mass assembly history with the prediction of two star formation histories for galaxies with SFRs >0.7 M yr −1 discussed by Oesch et al. (2014) similarly normalised at the mean redshift of our galaxies. Whilst the uncertainties are large due to small number statistics, if our galaxies are representative this would indicate Table 3. Summary of the main properties and parameters (uncorrected for any lensing of the objects) of the favoured one-and two-component SED fits determined by our SED fitting code with Pégase3 spectra. The tabulated flux percentages for the two-component models in the middle section of the table are the relative contributions from the (young,old) components to the total flux measured in that band. fraction of galaxy stellar mass 44% of stellar mass formed at z>10 34% for (1+z) 3.6 5% for (1+z) 10.9 m e a n S F h is t o r y ( 1 + z ) 3 . 6 ( 1 + z ) 1 0 .9 0.3 0.4 0.5 0.6 0.7 0.80.9 age of the universe [Gyr] Figure 6. The fractional stellar mass assembly history averaged over the four galaxies in Table 2 (dark red), adopting the best two-component fits in Figure 5. The solid and dashed blue lines represent the equivalent fractional histories for the two cosmic SFR density relations presented in Figure 9 of Oesch et al. (2014). a more gradual decline in the star formation history beyond z 8 than Oesch et al. (2014) prefer (see McLeod et al. 2016). Whilst clearly a simplistic comparison, it serves to emphasise the importance of determining the true origin of the IRAC excess in z > 7 galaxies. Ultimately NIRSpec on JWST will be well-placed to resolve the ambiguities explored in this paper via direct spectroscopy of a large sample of 7 < z < 9 galaxies securing not only the strength of rest-frame optical lines such as [O III] 5007 Å but also absorption line measures such as Hδ which is a further indicator of stellar ages.
Figure 2 .
2The [3.6] -[4.5] vs H 160 -[3.
Figure 3 .
3The Spitzer/IRAC [3.6] -[4.5] colour evolution as a function of redshift.
Figure 5
5Figure 5. The best fit two-component SED models (blue lines and black points) to observed data (dark red circles and error bars) from HST/VISTA + VLT/HAWK-I K s + Spitzer/IRAC photometry (left) and ALMA spectroscopy (upper right inset) for galaxies at z > 7 with red Spitzer/IRAC colours (JD1, YD4, Y1 and B14). The total fit consists of contributions from both old (darker gray) and young (lighter gray) stars.
. Since this spectral fit includes nebular line emission which, while not contributing significantly to the IRAC0
1
2
3
4
5
6
0.0
0.5
1.0
observed f [ Jy]
EGSY8p7
z spec =8.68
1.6
[3.6]
[4.5]
0
1
2
3
4
5
6
observed wavelength [ m]
0.0
0.5
1.0
observed f [ Jy]
JD1
z spec =9.11
1.6
[3.6]
[4.5]
Figure 1. The two synthetic spectra (blue lines) and associated photome-
try (dark red points with error bars; Roberts-Borsani et al. 2016; Zheng et
al. 2017) used in this study to demonstrate the similarity of the redshift-
dependent Spitzer/IRAC [3.6]-[4.5] evolution. The spectra are of EGSY8p7
(Roberts-Borsani et al. 2016; Zitrin et al. 2015), a supposed extreme
[O III]+Hβ line emitter at z =8.68 (top), and JD1 (Hashimoto et al. 2018),
a Balmer break galaxy at z =9.11 (bottom). The JD1 spectrum shown here
is uncorrected for magnification. Shown at the bottom of each panel are the
H ST /H 160 and S pitzer/IRAC 3.6 µm and 4.5 µm response filters, for
reference.
bands at z >9 will do so at lower redshift, we also explore the
effect of suppressing all optical emission lines from the fitted spec-
trum of JD1. The observed EWs of the combined [O III] and Hβ
lines are EW([O III]+Hβ)≈6690 Å and EW([O III]+Hβ)≈3408 Å for
EGSY8p7 and JD1, respectively.
Table 1 (
1and references therein) to compile HST ACS+WFC3/IR, (B 435 , V 606 , I 814 , Y 105 , J 125 , JH 140 and H 160 bands), VLT/HAWK-I K s , as well as Spitzer/IRAC [3.6] and [4.5] photometry. For B14 we use nearinfrared z, Y , J, and H photometry from VISTA, in addition to data in the VLT/HAWK-I K s and Spitzer/IRAC [3.6] and [4.5] bands.
Table 1. A list of spectroscopically confirmed galaxies at z >7 with red IRAC colours.ID
z spec
H 160
3.6µm -4.5µm
Reference
MACS1149_JD1
9.11
25.70±0.01
0.91±0.10
Hashimoto et al. (2018)
GN-z10-3
8.78
26.74±0.12
0.24±0.32
Laporte et al. (in prep.)
EGS8p7
8.68
25.26±0.09
0.76±0.14
Zitrin et al. (2015)
A2744_YD4
8.38
26.42±0.04
0.4±0.18
Laporte et al. (2017a)
MACS0416_Y1
8.31
26.04±0.05
>0.38
Tamura et al. (2019)
EGS-zs8-1
7.73
25.03±0.05
0.53±0.09
Oesch et al. (2015)
MACS1423-z7p64
7.64
25.03±0.10
>0.19
Hoag et al. (2017)
z8_GND_5296
7.51
25.55±0.07
0.98±0.07
Finkelstein et al. (2013)
A1689-zD1
7.50
24.70±0.10
0.30±0.30
Watson et al. (2015)
EGS-zs8-2
7.48
25.12±0.05
0.96±0.17
Roberts-Borsani et al. (2016); Stark et al. (2015)
COSY
7.15
25.06±0.06
1.03±0.06
Stark et al. (2015)
B14-65666
7.15
24.60 +0.3
−0.2
> 0.5
Hashimoto et al. (2019)
ID
z spec
µ
λ [O III]
[O III] integrated flux [O III] FWHM
λ cont
continuum flux
[µm]
[Jy km/s]
[km/s]
[µm]
Jy
MACS1149_JD1
9.11
10
893.86
0.229±0.050
154±39
917.63
<3.54
A2744_YD4
8.38
2
829.55
0.030±0.008
49.8±4.2
842.70
9.90±2.30
MACS0416_Y1
8.31
1.43 823.32
0.660±0.160
141±21
850.57
13.70±2.60
B14-65666
7.15
-
720.78
1.500±0.180
429±37
733.68
47.00±12.80
Table 2. A list of z > 7 galaxies from Table 1 with [O III] 88µm detections and dust continuum constraints (detections and non-detections) from ALMA Band
7 observations. The columns represent, from left to right: the redshift of the galaxy, the assumed magnification factor, the observed wavelength of the [O III]
88 µm detection, its integrated flux, and the measured FWHM. The last two columns are the central observed wavelength of the dust continuum observations
and the associated flux. All error bars and upper limits quoted here are 2σ.
continuum detections, whilst simultaneously reproducing the HST
photometry, Spitzer/IRAC excess, and [O III] 88 µm flux, within the
error bars. For Y1 and B14, on the other hand, the HST photom-
etry and Spitzer/IRAC excess are well reproduced, but the mod-
els are unable to simultaneously match both the [O III] 88 µm flux
and dust continuum (in the case of B14, the [O III] 88 µm flux is
matched but the dust continuum is underpredicted, and in the case
of Y1 the [O III] 88 µm flux is underpredicted and dust continuum
overpredicted). In comparing these fits to those assuming a single-
component only, although the HST photometry can be reasonably
0
1
2
3
4
5
wavelength [ m]
10 2
10 1
10 0
flux density [ Jy]
[O II]
[Ne III]
H
[OIII]
( )
( )
age=3 Myr, log M * =9.01 M
age=2 Myr, log M * =9.24 M
YD4, z spec =8.38
without ALMA
with ALMA
820 840 860
10 2
10 3
MNRAS 000, 1-9 (2019)
For a proper definition of how this translates in the Pégase3 formalism to stellar mass, see Fioc & Rocca-Volmerange 2019. MNRAS 000, 1-9 (2019)
This paper has been typeset from a T E X/L A T E X file prepared by the author.MNRAS 000, 1-9 (2019)
ACKNOWLEDGEMENTSWe thank Rychard Bouwens, Ivo Labbé and Dan Stark for useful discussions regarding the interpretation of Spitzer/IRAC colours. We would also like to thank Michel Fioc and Brigitte Rocca-Volmerange for their help in generating Pégase3 models, as well as Isabella Lamperti for illuminating discussions on developing the SED-fitting code and Roger Wesson for valuable conversations regarding the [O III] line ratio. The authors acknowledge funding from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation programme (grant agreement No 669253). NL acknowledges support from the Kavli foundation.
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|
[] |
[
"Master-Slave Scheme and Controlling Chaos in the Braiman and Goldhirsch Method",
"Master-Slave Scheme and Controlling Chaos in the Braiman and Goldhirsch Method"
] |
[
"Han-Tzong Su \nDepartment of Physics National\nNonlinear Science Group\nCheng Kung University Tainan\n70101Taiwan, Republic of China\n",
"Yaw-Hwang Chen \nDepartment of Physics National\nNonlinear Science Group\nCheng Kung University Tainan\n70101Taiwan, Republic of China\n",
"Ron Hsu \nDepartment of Physics National\nNonlinear Science Group\nCheng Kung University Tainan\n70101Taiwan, Republic of China\n"
] |
[
"Department of Physics National\nNonlinear Science Group\nCheng Kung University Tainan\n70101Taiwan, Republic of China",
"Department of Physics National\nNonlinear Science Group\nCheng Kung University Tainan\n70101Taiwan, Republic of China",
"Department of Physics National\nNonlinear Science Group\nCheng Kung University Tainan\n70101Taiwan, Republic of China"
] |
[] |
This brief report presents a master-slave scheme to demonstrate how control chaos works in the Braiman and Goldhirsch method for the onedimensional map system. The scheme also naturally explain why the anomalous response arises in a periodically perturbed, unimodal map system. PACS number(s): 05.45.+b.
|
10.1103/physreve.59.4687
|
[
"https://arxiv.org/pdf/chao-dyn/9802001v1.pdf"
] | 122,501,285 |
chao-dyn/9802001
|
b88ad635e8b0d2fbee8ae605a4afe7da599bdf59
|
Master-Slave Scheme and Controlling Chaos in the Braiman and Goldhirsch Method
arXiv:chao-dyn/9802001v1 31 Jan 1998
Han-Tzong Su
Department of Physics National
Nonlinear Science Group
Cheng Kung University Tainan
70101Taiwan, Republic of China
Yaw-Hwang Chen
Department of Physics National
Nonlinear Science Group
Cheng Kung University Tainan
70101Taiwan, Republic of China
Ron Hsu
Department of Physics National
Nonlinear Science Group
Cheng Kung University Tainan
70101Taiwan, Republic of China
Master-Slave Scheme and Controlling Chaos in the Braiman and Goldhirsch Method
arXiv:chao-dyn/9802001v1 31 Jan 1998Typeset using L A T E X
This brief report presents a master-slave scheme to demonstrate how control chaos works in the Braiman and Goldhirsch method for the onedimensional map system. The scheme also naturally explain why the anomalous response arises in a periodically perturbed, unimodal map system. PACS number(s): 05.45.+b.
ate stable periodic orbits from a chaos using a weak periodic perturbation.
Though there are some successful numerical and experimental demonstrations of the BG method [3]- [5], the periodicity and the stability condition of the targeted stable state was not identified analytically until recently [6]. If one considers a generic one-dimensional chaotic map under the influence of a periodic perturbation,
z n+1 = f (z n ) − αy n ,(1)
where α is a small number and y n is the added weak perturbation with periodicity p. The desired stable states in response to the period-p perturbation in a chaotic map system can only have the periodicity q = kp, where k is an integer number. Furthermore, using the linear stability analysis, one can deduce that the stability condition, for the output with the periodicity q = kp,
is |M| = k j=1 p ℓ=1 ∂f ∂z z jℓ < 1.(2)
Here,z jℓ is the jℓ times mapping ofz 1 , andz 1 are the roots of the periodicity
condition z = f (f (...(f (f (z) − αy 1 ) − αy 2 ) − ...) − αy kp−1 ) − αy kp .(3)
Even though, the analysis is presented in an elegant mathematical form in the reference [6]. Providing a more intuitive picture to illustrate how a chaotic system can be controlled by weak periodic perturbation is still a worthwhile effort. In this brief report, we will introduce a conceptual picture, called the master-salve scheme, to explain how control chaos works in the BG method.
This picture will also gives us a new handle to understand the anomalous responses in a dynamical system under the influence of periodic perturbation.
For example, when a period-2 perturbation with elements {y 1 = a, y 2 = 0.2} is added to a chaotic logistic map, then the system becomes z n+1 = 4z n (1 − z n ) − y n . The antimonotonicity -concurrent creation and destruction of periodic orbits [7,8], appears in the bifurcation diagram for the variation of a, see Fig.1. It seems to be in the contrary to the well known numerical fact [9]: the antimonotonicity could never appear in a unperturbed onedimensional unimodal map system. In the next paragraph, we will see that this anomalous response can be interpreted naturally in the master-slave scheme.
To begin with, let us consider a generic map under the influence of a period-p orbit {y 1 , y 2 , ..., y p }, see Eq.(1) . For convenience, we will label the initial data and initial perturbation as z 1 and y 1 , respectively. The key idea of the master-slave scheme is as following. We divide the original dynamical
variables z n into p new variables, called {x (1) m , x (2) m , ..., x (p) m }.
The relation between z n and the new variables x (i) m is defined as
x (i) m = z pm+i , 1 ≤ i ≤ p.(4)
Hence, the original dynamical equation can be separated into p maps:
x (2) m = f (x (1) m ) − αy 1 , x (3) m = f (x (2) m ) − αy 2 , .
..
x (p) m = f (x (p−1) m ) − αy p−1 , x (1) m+1 = f (x (p) m ) − αy p .(5)
Plugging the first (p − 1) maps,
x (2) m ,x (3) m ,...x (p) m , into x (1) m+1 , one find the map between x (1) m+1 and x (1)
m , which characterise the dynamical properties of original system, and let us call it the master equation:
x (1) m+1 = F (x (1) m ; αy 1 , αy 2 , ..., αy p ) = f (f (...(f (f (x (1) m ) − αy 1 ) − αy 2 ) − ...) − αy p−1 ) − αy p ,(6)
The remained p−1 maps, which just are mappings of x (1) m , and are designated as the slave equations:
x (2) m = f (x (1) m ) − αy 1 , x (3) m = f (x (2) m ) − αy 2 , .
..
x (p) m = f (x (p−1) m ) − αy p−1 .(7)
Since, the dynamics of the slave equations are completely controlled by the master equation, hence the name-the master-slave scheme. As long as the master equation, Eq. (6), is in a stable period-k orbit, then the slave equations indicate that (p − 1) images will appear simultaneously. It means that there exists a period-kp orbit in the original system.
From linear stability analysis, one can deduce that the stability condition for the period-k orbit in the master equation is |M| < 1. The stability quantity M now simply is
M = ∂ ∂x F k (x (1) m ; αy 1 , αy 2 , ..., αy p ) x (1) ,(8)
wherex (1) is one of the roots of the periodicity condition
x (1) = F k (x (1) m ; αy 1 , αy 2 , ..., αy p ).(9)
Obviously, in terms of the original dynamical variable z n and the map f , Eq.(8) and Eq.(9) will reduce to Eq.(2) and Eq.(3), respectively. Now, to be more specific, let us take α = 1 and the perturbation y n be of period-2 with elements {y 1 = a, y 2 = 0.2}. We will further assume that the system is a chaotic logistic map, f (z) = 4z(1 − z), before we turn on the perturbation. In this special case, the master equation becomes
x (1) m+1 = 4(4x (1) m (1 − x (1) m ) − a)(1 − 4x (1) m (1 − x (1) m ) + a) − 0.2,(10)
and the slave map is
x (2) m = 4x (1) m (1 − x (1) m ) − a.(11)
Here, x (1) m (x (2) m ) denotes the odd (even) part of z m , i.e. x (1) m = z 2m+1 (x (2) m = z 2m+2 ), and the initial value is labelled as x Fig.3. One can see that the same periodic orbits also appear exactly at the same regions of the perturbation a.
Obviously, the combination of Fig.2 x n+1 = F (x n , α) has at least two critical points that lie in a chaotic attractor for a parameter α = α * , then generically, F is antimonotone at α * . However, as it has been mentioned in the last paragraph, the dynamical of a periodi-side of Eq.(10) is a fourth order polynomial of x (1) m , and this implies that the system may have three critical points. Therefore, the antimonotonicity could arise naturally in a periodically perturbed logistic map.
Finally, a brief note is given as a concluding remark. The master-slave scheme gives us a conceptual picture that the periodic perturbation indeed makes controlling chaos feasible. The scheme also helps us to understand how the anomalous responses arise in a periodically perturbed one-dimensional map.
z 1 .
1The bifurcation diagram of the master equation, Eq.(10), for the perturbation a with values between 0.0 and 0.55, is shown in Fig.2. The bifurcation diagram indicates that the desired stable period-k orbit will appear if a suitable perturbation is applied on this chaotic logistic map. For example, period-1 orbit occurs when a is between (0.194, 0.240); and period-2 orbits can be generated when a is at (0.170, 0.194), (0.240, 0.290), or (0.425, 0.464); etc. From Eq.(11), the bifurcation diagram the slave map is plotted in
and Fig.3 leads to Fig.1 exactly. Also, from the numerical simulation presented in the Fig.1, one can clearly see that the periodicity of the stable response in the chaotic map under the influence of a period-2 perturbation is 2k -as one would have expected. For a careful reader, she/he may have noticed that there are some anomalous responses in the bifurcation diagrams of this perturbed logistic map, which occurs at the region a ∈ (0.25, 0.45) , see Fig. 1-3. These anomalous responses are called the antimonotonicity that does not exist in a unperturbed logistic map. Since the logistic map f (z) = rx(1 − x) only has one critical point at x = 0.5, it seems to be a dilemma for those who are familiar with the work of Dawson, Grebogi and Koçak [9] : if a one-dimensional map
Figure Captions : Figure 1 .
:1Bifurcation diagram for a periodically perturbed chaotic logistic map, z n+1 = 4z n (1 − z n ) − y n , where y n is of period-2 and with elements {y 1 = a, y 2 = 0.2}. The initial point for z 1 is 0.54, and 100 data points are plotted after 4000 transient iterations.
Figure 2 .
2Bifurcation diagram for the master equation Eq.(10), versus the perturbation a. The initial point for x
Figure 3 .
3The image x (2) m , which is determined by the slave equation Eq.(11), of the master equation Eq.(10) verus a.
Fig.1
. Y Braiman, I Goldhirsch, Phys. Rev. Lett. 662545Y. Braiman and I. Goldhirsch, Phys. Rev. Lett. 66, 2545 (1991).
. E Ott, C Grebogi, J A Yorke, Phys. Rev. Lett. 641196E. Ott, C. Grebogi, and J.A. Yorke, Phys. Rev. Lett. 64, 1196 (1990).
. T Shinbrot, C Grebogi, E Ott, J A Yorke ; V. In, S E Mahan, W L Ditto, M L Spano, Phys. Rev. 363411NatureT. Shinbrot, C. Grebogi, E. Ott and J.A. Yorke, Nature 363, 411 (1993). V. In, S.E. Mahan, W.L. Ditto and M.L. Spano, Phys. Rev.
Also see Coping with chaos. Lett, E. Ott., T. Sauer and J.A. YorkeWiley744420New Yorkfor more referencesLett. 74, 4420 (1995). Also see Coping with chaos, ed. E. Ott., T. Sauer and J.A. Yorke (Wiley, New York, 1995) for more references.
. S T Vohra, L Fabing, F Bucholtz, Phys. Rev. Lett. 7565S.T. Vohra, L. Fabing and F. Bucholtz, Phys. Rev. Lett. 75, 65 (1995).
. P Colet, Y Braiman, Phys. Rev. E. 53200P. Colet and Y. Braiman, Phys. Rev. E 53, 200 (1996).
. H.-J Li, J.-L Chern, Phys. Rev. E. 542118H.-J. Li and J.-L. Chern, Phys. Rev. E 54, 2118 (1996).
. R.-R Hsu, H.-T Su, J.-L Chern, C.-C Chen, Phys. Rev. Lett. 782936R.-R. Hsu, H.-T. Su, J.-L. Chern, and C.-C. Chen, Phys. Rev. Lett. 78, 2936 (1997).
. I Kan, H Koçak, J A Yorke, Ann. Math. 136219I. Kan, H. Koçak, and J. A. Yorke, Ann. Math. 136, 219 (1992).
. T C Newell, V Kovanis, A Gavrielides, Phys. Rev. Lett. 771747T. C. Newell, V. Kovanis, and A. Gavrielides, Phys. Rev. Lett. 77, 1747 (1996).
. S P Dawson, C Grebogi, H Koçak, Phys. Rev. E. 481676S. P. Dawson, C. Grebogi, and H. Koçak, Phys. Rev. E 48, 1676 (1993).
|
[] |
[
"Statistical inference of long-term causal effects in multiagent systems under the Neyman-Rubin model",
"Statistical inference of long-term causal effects in multiagent systems under the Neyman-Rubin model",
"Statistical inference of long-term causal effects in multiagent systems under the Neyman-Rubin model",
"Statistical inference of long-term causal effects in multiagent systems under the Neyman-Rubin model"
] |
[
"Panos Toulis \nHarvard University\n\n",
"David C Parkes \nHarvard University\n\n",
"Panos Toulis \nHarvard University\n\n",
"David C Parkes \nHarvard University\n\n"
] |
[
"Harvard University\n",
"Harvard University\n",
"Harvard University\n",
"Harvard University\n"
] |
[] |
Estimation of causal effects of interventions in dynamical systems of interacting agents is under-developed. In this paper, we explore the intricacies of this problem through standard approaches, and demonstrate the need for more appropriate methods. Working under the Neyman-Rubin causal model, we proceed to develop a causal inference method and we explicate the stability assumptions that are necessary for valid causal inference. Our method consists of a temporal component that models the evolution of behaviors that agents adopt over time, and a behavioral component that models the distribution of agent actions conditional on adopted behaviors. This allows the imputation of long-term estimates of quantities of interest, and thus the estimation of long-term causal effects of interventions. We demonstrate our method on a dataset from behavioral game theory, and discuss open problems to stimulate future research.
| null |
[
"https://arxiv.org/pdf/1501.02315v2.pdf"
] | 17,934,149 |
1501.02315
|
cc35962d60b3a3bd60c78c0f60b05b4b0da4b0b8
|
Statistical inference of long-term causal effects in multiagent systems under the Neyman-Rubin model
January 13, 2015
Panos Toulis
Harvard University
David C Parkes
Harvard University
Statistical inference of long-term causal effects in multiagent systems under the Neyman-Rubin model
January 13, 2015
Estimation of causal effects of interventions in dynamical systems of interacting agents is under-developed. In this paper, we explore the intricacies of this problem through standard approaches, and demonstrate the need for more appropriate methods. Working under the Neyman-Rubin causal model, we proceed to develop a causal inference method and we explicate the stability assumptions that are necessary for valid causal inference. Our method consists of a temporal component that models the evolution of behaviors that agents adopt over time, and a behavioral component that models the distribution of agent actions conditional on adopted behaviors. This allows the imputation of long-term estimates of quantities of interest, and thus the estimation of long-term causal effects of interventions. We demonstrate our method on a dataset from behavioral game theory, and discuss open problems to stimulate future research.
Introduction
Multiagent systems, such as online ad auctions, are ubiquitous and make up a significant portion of the total economic activity. Yet, rather paradoxically, there is a limited number of established statistical methods for experimentation and causal evaluation of interventions in such systems. In this work, we consider problems where the experimental units are agents, the intervention is a format of an economic mechanism (or game), and the outcomes are agent actions. 1 As argued by R.A. Fisher, evaluation of intervention effects hinges on an unambiguous interpretation of all possible experiment outcomes, and so "it is always needful to forecast all possible results of the experiment" (Fisher, 1935). However, in dynamical multiagent systems, such forecast is complicated by strategic interference and temporal dynamic behavior.
Strategic interference exists because agents change their actions depending on the economic environment and the agents they are facing. In statistical terms, the potential outcomes of experimental units under different treatment assignments depend on the treatment assignment of other units. Interference limits the inferential power of an experiment because the observed outcomes do not provide information about outcomes that could be observed under a different treatment assignment. Current methods assume away such a possibility through a stable unit-treatment value assumption (Rubin, 1974) or assume that interference arises from a static network of units and employ network randomization designs, such as cluster randomization (Ugander et al., 2013) or sequential randomization (Toulis and Kao, 2013). However, causal inference under any form of interference is still an open research problem that largely remains unsolved.
In addition, agents adopt a temporal dynamic behavior because, over time, strategic interference makes them change their actions in response to observed actions by other agents. This is important for causal inference because, in principle, we are interested in long-term causal effects, i.e., the effect of an intervention after agents have adopted some sort of equilibrium behavior. For instance, raising the reserve price in an auction might increase revenue in the short-run, but as agents adapt their bids or switch to another platform altogether, the long-term effect could be a net decrease. Interestingly, the notion of long-term causal effects is absent from, although not incompatible with, the Neyman-Rubin model.
Naturally, equilibrium effects have received attention in the econometric literature. For example, Athey et al. (2011) developed a method to compare two formats of U.S. timber auctions. Their approach is to use data from one auction in order to estimate aggregate bidder value distributions that act as primitives in a structural model, and thus use those estimates to predict the equilibrium outcomes on the other auction. Although not causal, their estimates do capture the notion of longterm intervention effects. However, as we explain in more detail in Section 2.1, they assume away considerations of temporal behavior.
Our goal in this work is to estimate long-term causal effects of interventions that are valid under the Neyman-Rubin model in the setting of economic mechanisms with interacting agents. In Section 2, we introduce a concrete problem instance using data from a real-world behavioral experiment by Rapoport and Boebel (1992). We further provide an overview of related approaches to the problem and demonstrate the need for a valid causal inference method.
In Section 3, we begin by introducing two novel components in the Neyman-Rubin model. The first component is a behavioral state-space where each state is a distribution of a finite number of behaviors over the agent population in the mechanism; this behavioral state is allowed to vary temporally, but it is considered to be latent. Although the treatment assignment mechanism is allowed to affect the initial behavioral state, the features of the aforementioned temporal evolution are intrinsic to the mechanism and thus the evolution is stable under the treatment assignment. The second component is the game-theoretic model which defines the distribution of agent actions, given the payoff structure of the mechanism and the underlying behavioral state, and thus is also stable under the treatment assignment. Taken together, the two aforementioned stability assumptions equip the Neyman-Rubin model for definition and estimation of valid long-term causal effects.
In Sections 4 and 5, we develop and demonstrate our method by analyzing the dataset by Rapoport and Boebel (1992). In particular, we adopt an instance of the quantal k-level (QLk) behavioral model introduced by Stahl and Wilson (1994). The latent behavioral state is thus defined as a distribution on k distinct behavior types, and the game-theoretic model is defined through the quantal responses of the associated behaviors. By modeling the latent behavioral state evolution as a stochastic process, we are able to impute long-term behavioral states which, through the gametheoretic model, allows imputation of long-term distribution of agent actions. Since causal effects are defined as functions of long-term agent actions, our method provides causal estimates of the quantity of interest that are valid in the Neyman-Rubin sense.
2 Causal inference in multiagent systems: Application and Standard methods Rapoport and Boebel (1992) conducted a behavioral experiment on a zero-sum, two-person game shown in Table 1 in order to test the minimax behavioral hypothesis that was initially suggested by Von Neumann and Morgenstern (1944). The values W, L indicate payments from player A to player B. Rapoport and Boebel (1992) considered two versions of the same game with (W, L) = ($10, −$6) and (W, L) = ($15, −$1). For example, if player A picks action A 1 and agent B picks action B 3 in the first version, then agent A has to pay $6 to agent B. 2 Twenty subjects were randomized to each game, and each player played both as A and as B in a match-up with two different players. Every match-up lasted 2 sessions (periods) of 60 rounds, where each round consisted of a selection of a strategy from each agent and a (possible) payoff. The aggregate data for this experiment are shown in Table 2, which reports the distribution of actions adopted by players within each game and period. Rapoport and Boebel (1992).
B 1 B 2 B 3 B 4 B 5 A 1 W L L L L A 2 L L W W W A 3 L W L L W A 4 L W L W L A 5 L W W L L
Although this experiment had a different purpose, we can adapt it for our own research goal. In particular, let us assume that the revenue for the gamemaster is a well-defined function of agentactions e.g., there is some fee paid for choosing an action. To be concrete, we will assume that the gamemaster receives $1 dollar when actions A 2 , A 5 , B 4 , B 5 are played. Let R j,t be the revenue of game format j, where j = 0 denotes the game (W, L) = ($10, −$6) and j = 1 denotes the game (W, L) = ($15, −$1), and t ∈ {1, 2, 3, 4} is the period. Also let π A j,t be the frequency of actions (5 × 1 column vector) of player A in game j at period t, and π B j,t likewise. Thus, by our assumption the revenue of the games are R j,t = r (π A j,t , π B j,t ) where r = (0, 1, 0, 0, 1, 0, 0, 0, 1, 1). We ask the following question: "What is the long-term causal effect on the revenue of the game if we switch from (W, L) = ($10, −$6) to (W, L) = ($15, −$1)?". To further simplify things we will consider period 4 as long-term, and hold out data on that period; i.e., we wish to estimate the revenue of each game in period 4 had all agents been assigned to that game. Note that this is different than estimating the revenue of the games in period 4 under the current assignment; i.e., the quantity δ = R 1,4 − R 0,4 = $0.054, which we also assume to be unknown. We return to clarify this important distinction in Section 3. Rapoport and Boebel (1992) broken down by game and session. Gray color indicates that we assume the data as holdout. The frequencies for actions A 5 , B 5 can be inferred.
Game Period
A 1 A 2 A 3 A 4 B 1 B 2 B 3 B
Standard Methods
In this section, we review current approaches to our causal question of the revenue effects of the choice of game format. Our goal is not to make an exhaustive presentation but to illustrate the fundamental assumptions underpinning each method, and thus pinpoint their insufficiency. The simplest approach under the Neyman-Rubin model is to leverage the randomization, observe the outcomes at some arbitrary point, say t = 3, and consider the estimateδ n = R 1,3 −R 0,3 = −$0.051. This estimate, also known as Neymannian estimate, is unbiased for δ because of complete randomization and it is statistically significant. Clearly, the dynamic strategic evolution of potential outcomes (i.e., action frequencies) is ignored, but in general such an approach is typical in the treatment effects literature. An additional crucial assumption is the stable unit-treatment value assumption (SUTVA) which, in this case, posits that the agent outcomes depend only on the assignment of that agent; such assumptions are implausible under strategic interference. A more sophisticated approach under the Neyman-Rubin model is to analyze the action frequencies as time-series with observations at t = 1, 2, 3 and then multiply impute the action frequency at t = 4 through a Bayesian structural model. Such an approach was developed by Brodersen et al. (2014), who wanted to estimate the effects of ad campaigns on website visits. However, such an approach is essentially macroeconometric, and thus does not model explicitly neither the strategic interactions among agents or their dynamic behavior.
A common econometric approach in evaluating policy changes is the so-called difference-indifferences (DID) estimator (Card and Krueger, 1994;Donald and Lang, 2007;Ostrovsky and Schwarz, 2011). In our case, this is inapplicable because there are no observations before the intervention, but we can still entertain the idea by considering period t = 1 as the pre-intervention period. The DID estimator would be in this case (R 1,3 −R 1,1 )−(R 0,3 −R 0,1 ) = −$0.164. The DID estimator captures a trend in the data by assuming a common linear trend for both treatment arms that is canceled out in subtraction. However, in order to be interpreted as a valid causal estimate, strong stability assumptions are required that essentially assume away strategic interference or complex dynamic play (Abadie, 2005;Angrist and Pischke, 2008). Athey et al. (2011) studied the effects of timber auction format (ascending versus sealed bid) on competition for timber tracts. Their method was to estimate bidder valuations from observed data in one auction and impute counterfactual bid distributions in the other auction, under the assumption of equilibrium play in both auctions. 3 This approach makes two critical implicit assumptions. First, the bidder valuation distribution is stable under the treatment assignment, and it thus a primitive that can be used to impute counterfactuals in other treatment assignments. Second, although imputation is performed for potential outcomes in equilibrium which captures the notion of long-term effects, inference is performed under the assumption of equilibrium play in the observed outcomes, and thus temporal dynamic behavior is assumed away. Finally, another popular approach to causality is through directed acyclical graphs (DAGs) between the variables of interest (Pearl, 2000). For example, Bottou et al. (2013) study the causal effects of the machine learning algorithm that scores online ads in the Bing search engine on the search engine revenue. Their approach is to create a full DAG of the system including variables such as queries, bids, prices and so on, and make a Causal Markov assumption for the DAG. This allows to predict counterfactuals for the revenue under manipulations of the scoring algorithm, using only observed data generated from the assumed DAG. However, a key assumption of the DAG approach is that the underlying structural equation model is stable under the treatment assignment, and only edges coming from parents of the manipulated variable need to be removed. As pointed out by Dash and Druzdzel (2001), this might be implausible in equilibrium systems. Consider, for example, a system where X → Y ← Z, and a manipulation that sets the distribution of Y independently of X, Z. Then after manipulation the two edges will need to be removed. However, if in an equilibrium it is required that Y ≈ XZ, then the two arrows should be reversed after the manipulation. Proper causal inference in equilibrium systems remains an open area with no well-established methodology (Dash, 2005).
Causal inference in multiagent systems: Estimand and stability assumptions
In this section we define formally our estimand of interest and explicate the stability assumptions that are necessary for valid causal inference under the Neyman-Rubin model. First, we introduce our notation for the rest of this paper. Let P = {1, 2, . . . , n} be a population of n agents and G 0 , G 1 be two forms of the same game as in (Rapoport and Boebel, 1992). Each game has the same discrete action space A. Consider an assignment mechanism that initially samples a vector Z ∈ {0, 1} n of n elements, where Z i = 1 indicates that agent i was assigned to game G 1 and Z i = 0 indicates that agent i was assigned to game G 0 ; therefore, Z is the initial assignment of agents to games that stays the same throughout the experiment. We assume that the games progress in discrete time-steps t = 1, 2, · · · and let t = ∞ denote the time step that is considered long-term. As the game progresses, an agent i picks action Y it (Z) ∈ A. Let ∆ p be the p-dimensional simplex, and α j,t (Z) ∈ ∆ |A| be the frequency of agent actions {Y it (Z) : Z i = j} in game G j during time period t over the action space A. Finally let 1, 0 be the vector of n ones and zeroes respectively, and let R jt (Z) = r α j,t (Z) be the revenue of game j at period t under initial assignment Z, defined as a linear function of action frequencies α j,t (Z) and fixed action fees r. We define the estimand of long-term causal effects as
τ = R 1∞ (1) − R 0∞ (0) = r [α 1,∞ (1) − α 0,∞ (0)] .(1)
The estimand (1) shows that for valid causal inference, one needs to extrapolate along two dimensions. First, given observed data up to some period t, we need to extrapolate to t = ∞. Second, for every game G j we need to extrapolate from the initial assignment Z to the assignment Z = j1 i.e., to outcomes that would be observed had all agents been assigned to play G j . The former is the problem of temporal dynamic agent behavior, whereas the latter is the problem of strategic interference.
Stability assumptions for valid causal inference
In typical causal inference methods, extrapolation from a treatment assignment Z to another assignment Z is achieved through various stability assumptions. For example, the SUTVA assumption (Rubin, 1974)
posits Y it (Z) = Y i (Z i ) i.e.
, the potential outcome for unit i depends only on the treatment assignment of unit itself. The great virtue of this assumption is that an observed outcome Y it (Z) immediately informs us about potential outcomes {Y it (Z ) : Z i = Z i }; i.e., for a multitude of treatment assignments that could be realized, thus allowing us to estimate more accurately the desired treatment effects.
In our problem, strategic interference and dynamic behavior cannot justify such assumption. Because of this, we require stability assumptions that are defined on populations of agents. For that reason, we augment our parameter space by introducing a behavioral space B with a finite number of agent behaviors. We assume that each agent i adopts a behavior B it (Z) ∈ B at time period t under an initial assignment Z, and that B it (Z) completely determines the distribution of its actions at that particular time period t. As before, let β j,t (Z) ∈ ∆ |B| be the frequency of agent behaviors {B it (Z) : Z i = j} in game G j during time period t over the behavioral space B. In Section 4 we will make this more concrete by formally defining the behavioral space.
Assumption 3.1 (Stable behaviors). Under initial assignment Z, the distribution of adopted behaviors in game G j at period t is independent of Z given the distribution of behaviors at the previous time period t − 1; i.e.,
Z |= β j,t (Z) | β j,t−1 (Z), G j .
Assumption 3.2 (Stable actions). Under initial assignment Z, the distribution of adopted actions in game G j at period t is independent of Z given the distribution of behaviors at time period t i.e.,
Z |= α j,t (Z) | β j,t (Z), G j .
In addition, there exists a |A| × |B| matrix H j of coefficients for game G j for which
E [α j,t (Z)| β j,t (Z)] = H j · β j,t (Z).(2)
Remarks. Assumption 3.1 implies that the dynamic evolution of agent behaviors is not affected by the initial treatment assignment Z, but it is an intrinsic property of the game itself. This is similar to the policy invariance assumption (Heckman and Vytlacil, 2005;Heckman et al., 1998), in which given the choice of treatment by the agent, the initial treatment assignment mechanism does not affect the outcomes. Assumption 3.2 implies that the latent behavioral state β j,t (Z) is sufficient to inform us about the distribution of actions taken by the agents; of course, this distribution also depends on the payoff structure of the game.
Assumptions 3.1 and 3.2 also reveal the role of randomizing Z in order to get unbiased estimation of the estimand τ defined in (1). In particular, let E β j,∞ (Z) β j,0 (Z)] = µ(β j,0 (Z)) for some function µ(·) ∈ ∆ |B| . Thus, by (2) we obtain E [α j,∞ (Z)| β j,0 (Z)] = H j · µ(β j,0 (Z)). The actual expected value of the long-term agent actions α j,∞ (Z) is obtained when all agents are assigned to game G j ; i.e., when Z = j1 in which case E [α j,∞ (Z)| β j,0 (j1)] = H j · µ(β j,0 (j1)). However, since β j,t (Z) is a proportion, by randomization it holds
E β j,0 (Z) = β j,0 (j1),(3)
where the expectation is taken over the randomization distribution of Z. Thus, the bias in estimating τ will depend on the difference µ(β j,0 (Z)) − µ(β j,0 (j1)). Thus, there are two approaches to unbiased estimation. First, one can set up a dynamic behavioral model such that µ(·) is constant to its argument; this is the case when the long-term distribution β j,∞ (Z) does not depend on the initial condition β j,0 (Z) e.g., as in a Markov chain with a stationary distribution. Alternatively, one can set up a model where µ(·) is linear so that E µ(β j,0 (Z)) = µ(E β j,0 (Z) ) = µ(β j,0 (j1)).
In general, the role of randomization is often obscured in multiagent settings. For example, if we assumed perfectly rational agents then randomization would not even be necessary, as observed outcomes would be accurately predicted by concepts such as Nash equilibrium. Our framework assigns a more plausible role to randomization by assuming that it is selecting the initial distribution of latent agent behaviors β j,0 (Z) in the game. After this initial assignment, our stability assumptions imply that the evolution of the game is determined unconditionally to the treatment assignment. Since randomization is unbiased for the selection of β j,0 (Z), inference for the counterfactual β j,0 (j1) i.e., when all agents are assigned to G j , will be (nearly) unbiased. This idea is depicted in Figure 1. Under our stability assumptions it is possible to estimate the long-term distribution of actions α j,∞ (Z) because the data generative process is stable under a realized treatment assignment Z. Randomizing Z helps us to obtain an unbiased estimate β j,0 (j1) through β j,0 (Z).
Concrete Methodology
Building upon the insights of Section 3, we now develop a concrete methodology to estimate longterm causal effects defined in Equation (1). In Section 5 we will apply our method in the dataset by Rapoport and Boebel (1992). For our behavioral model we adopt the quantal k-response (QL k ) that was initially proposed by Stahl and Wilson (1994), and was shown to predict well the observed human behavior in real-world behavioral experiments (Wright and Leyton-Brown, 2010). In QL k , agents possess increasing levels of sophistication. Following this earlier work, we will adopt k = 3, and thus consider a behavioral space with three different behavior types B = {b 0 , b 1 , b 2 }. Let π b ∈ ∆ |A| denote the β j,0 (Z)
β j,1 (Z) β j,2 (Z) β j,∞ (Z) α j,1 (Z) α j,2 (Z) α j,∞ (Z)
short-term long-term Figure 1: The latent distribution of behaviors in the population β j,t (Z) form, essentially, a Hidden Markov Model, where only the action frequencies α j,t (Z) adopted by the agents are observed.
Our goal is to infer the long-term action distribution α j,∞ (Z).
distribution of actions that an agent will play after adopting behavior b. In QL k the distributions depend on an assumption of quantal response, which is defined as follows. Let u ∈ R |A| denote a vector such that u i is the expected utility of agent taking action a i ∈ A, and let P j denote the payoff matrix in game j; therefore, if an agent is facing a strategy profile π then it holds u = P j π. The quantal best-response with parameter λ determines the distribution of actions that the agent will take, and is given by QBR(u; λ) = logistic(λu), where, for some vector x we define logistic(x) = (exp(x i )/ exp(x i )). The parameter λ ≥ 0 is usually called the precision of the quantal best-response. If λ is very large then the response is closer to the classical Nash best-response, whereas if λ = 0 the agent has no preferences among actions. In QL 3 the agents adopt actions as follows:
• Agents who adopt b 0 , termed level-0 agents, have precision λ 0 = 0, and thus will randomly pick one action from the action space A. Thus, π b 0 = QBR(u; 0) = (1/|A|)1.
• Agents who adopt b 0 , termed level-1 agents, have precision λ 1 and assume that all other agents are of level-0 type. Thus they are facing a vector of utilities u 1 = P j · π b 0 and so π b 1 = QBR(u 1 ; λ 1 ).
• Agents who adopt b 2 , termed level-2 agents, have precision λ 2 and assume that all other agents are of level-1 with precision λ (1)2 . Thus, they estimate that they are facing a strategy profile given by π b (1)2 = QBR(u 1 ; λ (1)2 ), where u 1 = P j π b 0 as above. Their expected utility vector is u 2 = P j π b (1)2 , and thus their quantal best-response is π b 2 = QBR(u 2 ; λ 2 ).
Let λ = (λ 1 , λ (1)2 , λ 2 ) and Π j (λ) = [π b 0 π b 1 π b 2 ] be the |A| × 3 matrix with the aforementioned best-response action distributions as columns. Recall that β j,t (Z) is the distribution of behaviors for game j, at time t under initial assignment Z. Then, the QL 3 implies that the expected agent action frequencies are given by,
E [α j,t (Z)| β j,t (Z)] = Π j (λ) · β j,t (Z),(4)
which satisfies Assumption 3.2. There are several options to model β j,t (Z) and satisfy Assumption 3.1. In general, it is reasonable to model β j,t (Z) as a stochastic process, such as a time series or Markov chain. Since β j,t (Z) is a discrete distribution, relevant statistical methods are available in analysis of continuous proportions, or compositional data (Aitchison, 1986). For example, Grunwald et al. (1993) consider a time series of continuous proportions with a Dirichlet likelihood, and develop a Bayesian method for fast updates using a conjugate prior. However, such methods cannot be directly applied in our problem because the likelihood induced by the aforementioned quantal best-response cannot be written efficiently in closed form. For that reason we adopt a simple VAR(1) model; details are given in the following section.
Likelihood and posterior inference
We can now write down the likelihood of observations as in Table 2. Consider an initial assignment Z, and observed action frequencies for time periods t = 1, 2, . . . T denoted as a T × |A| matrix α j,1:T (Z), where the i-th row is α j,i (Z). Our model parameters are the behavior distribution β j,1:T (Z), where the i-th row is β j,i (Z) and ψ are the model parameters for the temporal evolution of β j,t (Z); furthermore, we have the quantal best-response parameters are λ = (λ 1 , λ (1)2 , λ 2 ). Thus the likelihood of our model is given by
L(α j,1:T (Z); ψ, λ) = B T t=1 f (α j,t (Z)|β j,t (Z), λ) × h(B|ψ)dB,(5)
where B β j,1:T (Z) takes all possible values on the latent behavior space. Given the definition of quantal best-response for some distribution of behaviors β and parameters λ, we can obtain the expected action distribution α. Thus, the observed action counts N · α are distributed as a multinomial distribution i.e., N α ∼ Multinom( α; N ), which provides the likelihood term f (α|β, λ) in expression (5). To model the latent temporal behavioral state, we use a simple approach by adopting a VAR(1) model. In particular, we transform the proportion into a new variable w j,t logit(β j,t (Z)) and assume that
w j,t = ψ 0 w j,t−1 + µ + ψ 1 t ,(6)
where ψ 0 ∈ (0, 1), ψ 1 ∈ R + , and µ ∈ R |B| is a fixed parameter vector and t ∼ N (0, I) is i.i.d. standard normal. Decomposing the density h(B|ψ) = T t=2 p(β j,t (Z)|β j,t (Z), ψ) and computing the conditional densities through (6) yields the remaining likelihood term.
Application
We now return to the dataset of Rapoport and Boebel (1992) presented in Table 2. We assume diffuse priors for the parameters ψ, λ; in particular, we consider π(λ i ) ∝ Expo; i.e., an exponential random variable with rate around 1/10 for the parameters of the quantal best-response, a diffuse beta for ψ 0 and a flat variance prior for ψ 1 i.e., p(ψ 1 ) ∝ (1/ψ 2 1 ). As exact conditionals are hard to obtain under this model, we employ a Metropolis-Hastings scheme with a proposal distribution that simply disturbs slightly the current model parameters. This is efficient because we know beforehand that the values of our parameters are constrained; for example, we know that ψ 0 ∈ (0, 1) and that, roughly, λ i ∈ (0, Λ) for some Λ > 0, because the logistic functions of quantal best-response (see Section 4) become flat above, or below, a certain threshold. Figure 2: Bottom-right: Proportions of behaviors (b 0 , b 1 ) for time periods t = 1(red, "+"), t = 2 (green, "x") and t = 3 (blue, "♦"). An overall decrease of level-0 agents and an increase in level-1 agents is observed across time periods. Other: MCMC estimates for selected model parameters. We observe good mixing for parameter ψ 0 (bottom-left), which is hard to achieve for parameters λ 1 , λ (1)2 (top 2 figures), since they are non-linear in the likelihood model (5).
Our ultimate goal is to obtain imputations of the agent actions in t = 4 through the posterior predictive distribution of our model. We run our chain for 1e5 iterations and assume the first half of the samples as burn-in period; traceplots of MCMC estimates of the model parameters are shown in Figure 2. Additionally, a summary of the marginal posterior distribution of the model parameters is shown in Table 3. There are a few interesting observations. First, note that although it was not explicitly specified, the model obtains λ 2 > λ 1 in general i.e., that level-2 agents have better precision than agents at level-1. Interestingly, in this dataset, level-2 agents play as-if level-1 agents are very precise (see values for λ (1)2 in Table 3). Furthermore, estimates on ψ 0 i.e., the coefficient for the lag-1 regression in our VAR model of agent behaviors is significant around 0.3, indicating a temporal trend in the latent behavioral state. We can verify this trend by inspection of the bottom-right plot in Figure 2. Note that, overall, there is a steady decreasing trend for the proportion of level-0 agents and a parallel steady increasing trend for level-1 agents, across game time periods. This provides evidence of agent learning.
Estimation of long-term causal effects
We now turn our attention to obtaining posterior estimates for the long-term causal effect defined in (1). Using the MCMC strategy outlined above we are able to obtain imputations of the causal effect (1) through the posterior predictive distribution of the model parameters (λ, ψ). Figure 3 depicts 1,000 posterior samples. The actual estimate considered to be at t = 4 is equal to $0.054. Our estimates are able to to catch the latent dynamic trend in agent behaviors. Recall also that the naive estimates presented in Section 2.1 were estimating a strongly significant negative effect of introducing game design G 1 compared to G 0 , because they were misguided by strong short-term effects in the opposite direction. Our estimates are in fact biased; the posterior summary for the long-term causal effect is 0.088 ± 0.021, thus being about 63% off the actual value. However, this is perfectly reasonable in our application for the following reasons. First, the dataset is limited to 3 time periods which we use to impute action distributions in the fourth period; therefore, the role of the prior becomes important for identifiability and this necessarily induces bias. Second, the choice of the ground-truth value of the estimand at t = 4 is rather arbitrary, and mostly serves to illustrate the methodology of this work; for instance, it is possible that the bias would be significantly reduced had we observed more time periods for each game. Last, as pointed out in the discussion of our stability assumptions in Section 3, the form of the regression function E β j,∞ (Z) β j,0 (Z)] necessarily introduces bias; in our model, this regression function is linear so theoretically we should have obtained an unbiased result. However, the regression function of the actual dynamic process may very well be nonlinear, which would bias our results.
Discussion
In this paper, we explored the problem of estimating long-term causal effects of interventions in multiagent systems under the Neyman-Rubin causal model. Two features make this problem conceptually and technically challenging. First, strategic interference among competing agents limits the inferential abilities of randomized experiments. Second, dynamic temporal behavior by agents introduces short-term effects, whereas one is typically interested in long-term effects; i.e., when some sort of equilibrium play has been reached. Our first conceptual contribution is to explicate a set of sufficient stability assumptions that lead to valid causal inference within the Neyman-Rubin model. We make two stability assumptions, one for a latent behavioral state of the population and one for the distribution of actions adopted by the agents, conditional on the aggregate behavior. Under certain conditions, randomization yields an unbiased initial behavioral state after the intervention, which can lead to unbiased estimation of long-term actions under the aforementioned stability assumptions. The technical challenge in this problem is to correctly model the evolution of the latent dynamic behaviors, and the observed action profiles conditional on realized behaviors. Our approach is modular. Our first model consisted of a simple VAR model of the population behavioral state. Our second model consisted of a quantal k-response which predicts the action distribution adopted by the agents, conditional on a distribution of behaviors in the population. Working on a real-world dataset from a behavioral experiment (Rapoport and Boebel, 1992), we showed how our method can be applied for estimation of a long-term effect of intervention in the payoff structure of a game design in normal-form.
There are several issues that are left open. One important issue is the strategic interference between games (or mechanisms). For instance, in our application it was reasonable two assume that actions in one game did not affect actions in the other one. However, in most interesting situations e.g., in a real-interconnected market, it is hard to achieve a setting with no interference (e.g., in an auction players out-bid each other, or switch to other platforms and so on). A second issue, more technical in nature, is how to scale up the inference in games or markets with many participants and transactions; for example, there are millions to billions of second-price auctions happening online on a daily basis, and inference through our model in such settings is a challenge. A third, more theoretical concern, is whether it is possible to establish certain optimality properties for our approach. For instance, it is important to know what is the interconnections between the choice of the latent behavioral model, the randomization and statistical inference. Progress in answering such questions will lead to new and fruitful interactions of game theory with experimental design and causal inference.
Figure 3 :
3Posterior predictive distribution of imputations of the long-term causal effect.
Table 1 :
1Normal-form game in the experiment by
Table 2 :
2Frequency of actions for players A and B in the experiment by
Table 3 :
3Summary of the posterior distribution of model parameters. param. mean (sd) [Q 1 , Q 3 ]λ 1
0.5 (0.1) [0.4, 0.6]
λ (1)2
3.0 (0.7) [2.5, 3.5]
λ 2
1.9 (1.2) [0.9, 2.8]
ψ 0
0.3 (0.1) [0.3, 0.4]
ψ 1
1.4 (0.7) [0.9, 1.5]
For instance, in estimating the causal effect of reserve price on auction revenue, the units are bidders in the auction, the treatments could be two auction formats with different reserve prices, and the outcomes are agent bids that are aggregated in some way to define the auction revenue.
However, actual payments happened only in 3 out of 120 rounds between any pair of players.
Note thatAthey et al. (2011) consider a different situation than ours. We consider a normal-form game i.e., the payoff structure throughTable 1is known and fixed, whereas bidders in an auction carry private valuations when entering an auction.
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[] |
[
"Instanton and Spectral Flow in Topological Conformal Field Theories",
"Instanton and Spectral Flow in Topological Conformal Field Theories"
] |
[
"Toshio Nakatsu \nDepartment of Physics\nUniversity of Tokyo\nBunkyo-ku113TokyoJapan\n",
"Yuji Sugawara \nDepartment of Physics\nUniversity of Tokyo\nBunkyo-ku113TokyoJapan\n"
] |
[
"Department of Physics\nUniversity of Tokyo\nBunkyo-ku113TokyoJapan",
"Department of Physics\nUniversity of Tokyo\nBunkyo-ku113TokyoJapan"
] |
[] |
A class of two-dimensional topological conformal field theories (TCFTs) is studied within the framework of gauged WZW models in order to obtain some insights on the global geometrical nature of TCFTs. The BRST quantizations of topological G/H gauged WZW models ( the twisted versions of supersymmetric gauged WZW models ) are given under fixed back-ground gauge fields. The BRST-cohomology of the system is investigated, and then the correlation functions among these physical observables are considered under the instanton back-grounds.As a consequence, two-dimensional BF gauge theoretical aspects of TCFTs are revealed. Especially, it is shown that two correlation functions under the different instanton back-grounds can change to each other. This process of transmutation is described by the spectral flow.The flow is formulated as a "singular" gauge transformation which creates an appropriate back-ground charge on the physical vacuum of the system. The field identification problem of the system is also discussed from the above point of view.
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10.1016/0550-3213(93)90564-6
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[
"https://arxiv.org/pdf/hep-th/9304029v1.pdf"
] | 15,335,894 |
hep-th/9304029
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b1e70be6ea2b6219dd7e8325bb3617b474bd9797
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Instanton and Spectral Flow in Topological Conformal Field Theories
Apr 1993 April 1993
Toshio Nakatsu
Department of Physics
University of Tokyo
Bunkyo-ku113TokyoJapan
Yuji Sugawara
Department of Physics
University of Tokyo
Bunkyo-ku113TokyoJapan
Instanton and Spectral Flow in Topological Conformal Field Theories
Apr 1993 April 1993arXiv:hep-th/9304029v1 8 1
A class of two-dimensional topological conformal field theories (TCFTs) is studied within the framework of gauged WZW models in order to obtain some insights on the global geometrical nature of TCFTs. The BRST quantizations of topological G/H gauged WZW models ( the twisted versions of supersymmetric gauged WZW models ) are given under fixed back-ground gauge fields. The BRST-cohomology of the system is investigated, and then the correlation functions among these physical observables are considered under the instanton back-grounds.As a consequence, two-dimensional BF gauge theoretical aspects of TCFTs are revealed. Especially, it is shown that two correlation functions under the different instanton back-grounds can change to each other. This process of transmutation is described by the spectral flow.The flow is formulated as a "singular" gauge transformation which creates an appropriate back-ground charge on the physical vacuum of the system. The field identification problem of the system is also discussed from the above point of view.
Introduction
Two-dimensional topological conformal field theories (TCFT) play an important role in our recent understanding of string theory and two-dimensional gravity [1,2]. An algebraic method to construct a model of TCFTs was given in [3]; the "twist" of a model of N = 2 superconformal field theories (SCFTs). It is known that a large class of N = 2 SCFTs is obtainable by the constructions of Kazama and Suzuki [4], that is, the supercoset constructions associated with compact Kähler homogeneous spaces G/H. The corresponding TCFTs were studied in [5] from the algebraic point of view.
It is also known that supercoset CFTs can be realized by "supersymmetric gauged WZW models" [1,6,7]. Much have been learned about coset models by realizing them as gauged WZW models. Especially the global geometry of coset models, which is hidden in the free field realizations [8], appears [9]. Thus the twisted versions of the supersymmetric gauged WZW models, that is, the topological gauged WZW models give us a chance to study the TCFTs from the geometrical point of view. In spite of this utility the topological gauged WZW models have been little examined. The purpose of this paper is to study these topological models from the global geometrical point.
We begin in section 2 by formulating the topological gauged WZW model for a general compact Kähler homogeneous space G/H. The path-integral quantization of the model is given under a fixed back-ground gauge field, by utilizing the techniques developed in [11,12,13]. Subsequently, based on the gauge-fixed form, the local operator formulation is studied and the algebraic structure of the model is shown. In section 3 we investigate the spectrum of the physical observables, that is, the BRST-cohomology of the gaugefixed system. The semi-classical analysis of the system provides a starting point. Then, for example, it arises the question whether this semi-classical approximation is valid in our model. Is the correlation function among these semi-classical observables meaningful under any back-ground gauge field ? This question is clarified in the next section. For this purpose we turn our eyes to the BRST-cohomology on the full Hilbert space of the system. We introduce spectral flows U as the symmetry transformations of our model.
In section 4 we study the correlation functions among the physical observables {O a } a∈I under the instanton back-grounds. It is shown that two correlation functions under the different instanton back-grounds change to each other (4. 17):
U γ 1 (O a 1 ) · · · U γ N (O a N ) c 1 = O a 1 · · · O a N c 1 + N i=1 γ i .
The vectors γ i label the spectral flows and the c 1 is the set of the Chern numbers of the back-ground gauge field. This relation tells us that the instantons transumute the physical observables and vice versa. It is described by the flows U. The field identification in the algebraic CFT approach is also discussed from this rather topological view point. Finally section 5 is devoted to our conclusions and speculations. In appendix A the quantizations of the supersymmetric gauged WZW models are summarized. The correspondence between the "twist" of the gauged WZW models and that of N = 2 SCFTs are explained. The former is performed by replacing the physical Weyl spinors of the models with unphysical ghost fields, while the latter is done by twisting N = 2 SCAs by their U(1)-currents, not the fermion number currents. In appendix B the basic properties of spectral flows are described. They are arranged in order to fit our study of the topological gauged WZW models.
Topological Gauged WZW Models Associated with General Compact Kähler Homogeneous Spaces
We shall begin with preparing some notations of Lie algebras needed for our discussion.
Let G be a compact simple Lie group, H be a closed subgroup of G,and g, h be the corresponding Lie algebras. Assume that the homogeneous space G/H is a compact Kähler space, which implies this space is a so-called "flag manifold". Especially H includes the maximal torus T of G. The topological conformal models we investigate in this paper will be defined associated with the coset space of this type, or the pair (G, H). So it may be useful to give here a short digression to survey the structure of g C and h C .
First h C can be decomposed as follows;
h C = Z(h C ) ⊕ h C 0 = Z(h C ) ⊕ h (1)C 0 ⊕ · · · ⊕ h (r)C 0 (2. 1)
where Z(h C ) is the center of h C and h C 0 is a complex semi-simple Lie algebra. We denote its simple factors by h × U(1) × · · · × U(1).
(2. 2)
H 0 is the compact semi-simple Lie group corresponding to h C 0 , and l = rankg − rankh ≡ dimZ(h C ). We denote the root system of g C , h C 0 by
∆ ≡ ∆ + ∆ − , ∆ h ≡ ∆ + h 0 ∆ − h 0
respectively. We also set ∆ ± m = ∆ ± \ ∆ ± h . g C can be decomposed into the following pieces;
g C = h C ⊕ m + ⊕ m − = Z(h C ) ⊕ h C 0 ⊕ m + ⊕ m − (2. 3)
where m ± = α∈∆ ± m g α . g α is the root space for α ∈ ∆. This is an example of parabolic decompositions of g C (generalizations of the Cartan decomposition);
[h C , m ± ] ⊂ m ± [m ± , m ± ] ⊂ m ± . (2. 4)
If the homogeneous space G/H is hermitian symmetric, we further obtain the additional commutation relations;
[m ± , m ∓ ] ⊂ h C , (2. 5) which implies that the subalgebras m + , m − necessarily become abelian. This case is rather easy to handle in many respects. In the following part of this paper we also assume that the group G is simply-laced. Now we can present the model we shall work on, the "topological gauged WZW model for the homogeneous space G/H". We may call it the "topological G/H model". It is no other than the twisted version of the N = 2 supersymmetric gauged WZW model [1], [13] corresponding to the Kazama-Suzuki supercoset model [4]. Let Σ a Riemann surface. The partition function Z of the model is given by
Z = DgDAD(χ,χ, ψ,ψ) exp −kS G (g : A) − 1 2πi Σ {(∂ A ψ, χ) − (χ, ∂ Aψ )} .
(2. 6) In this expression the chiral field g is G-valued, the gauge field A is h-valued, and the ghost system is defined as follows;
ghosts ψ : m + -valued (0,0)-form,ψ ≡ ψ † : m − -valued (0,0)-form, anti-ghosts χ : m − -valued (1,0)-form,χ ≡ χ † : m + -valued (0,1)-form.
D A = ∂ A +∂ A is the canonical splitting of the covariant exterior derivative D A defined by the complex structure "compatible" with the background metric g on Σ (i.e. such that g becomes Kähler). The inner product "( , )" is the Cartan Killing form normalized by (θ, θ) = 2 (θ is the highest root of g C .), " †" is the "hermitian conjugation" so that g = {u ∈ g C ; u † = −u}. S G (g : A) stands for the action of the G-WZW model gauged by the subgroup H; 10 ) + (A 01 , ∂gg −1 ) − (A 01 , Ad(g)A 10 ) + (A 01 , A 10 )}.
S G (g : A) = i 4π Σ (g −1∂ g, g −1 ∂g) − i 24π B (g −1 dg, [g −1 dg,g −1 dg]) + i 2π Σ {−(g −1∂ g, A
(2. 7) Besides the corresponding gauge symmetry the model (2. 6) satisfies the following BRST symmetry, which is originated in the SUSY of the untwisted model;
δ G/H χ = kΠ m − (∂ A g g −1 + [χ, ψ]) ,δ G/Hχ = −kΠ m + (g −1∂ A g − [χ,ψ]), δ G/H g = ψg,δ G/H g = −gψ, δ G/H ψ = 1 2 [ψ, ψ],δ G/Hψ = 1 2 [ψ,ψ],
other combinations are defined to vanish, (2. 8) where Π m + , Π m − mean the orthogonal projections onto the spaces m + , m − with respect to the inner product ( , ). δ G/H ,δ G/H satisfy the following nilpotencies;
δ 2 G/H =δ 2 G/H = 0, {δ G/H ,δ G/H } = 0. (2. 9)
Because the ghost system ψ,ψ, χ,χ corresponds to that of the BRST symmetry (2. 8), the physical Hilbert space of the model should be restricted by taking the BRST cohomology for this symmetry.
Path Integration
Let us perform the path-integration in (2. 6) along the line in [13]. For that purpose we should extract the gauge volume of the underlying H-gauge symmetry from (2. 6). We shall parametrize the h-gauge field A as
A = h −1 a ( h −1 a 01 = h −1 a 01 h + h −1∂ h, h −1 a 10 = −( h −1 a 01 ) † ), (2. 10)
where a is a background h-gauge field and a 10 , a 01 are the holomorphic and anti-holomorphic components of a. h is a H C -chiral gauge transformation [11]. Then we insert the following identity into (2. 6); iω, e 1 2 (X+iY ) are the components corresponding to Z(h C ) in (2. 1). ω is a R l -valued 1-form. X, Y are R l -valued scalar fields, which correspond to the axial and vectorial components of the Z(H) C -chiral gauge transformation. a 0 , h 0 are the components corresponding to h C 0 in (2. 1).
1 = DaDh † Dh δ(A − h −1 a)∆ FP ( h −1 a),(2.h 0 = ρ 1/2 U (ρ ≡ h 0 h † 0 ∈ H C 0 /H 0 ,U ∈ H 0 ) is the "polar decomposition" of H C 0 .
Under the parametrization A = h −1 a the model will suffer the chiral anomaly. We must estimate it for the dynamical variable g and the ghost system χ,χ, ψ,ψ independently.
For g, by means of the Polyakov-Wiegmann identity (see [10], [11,12,13]) we can get
S G (g : h −1 a) = S G ( h g : a) − S G (hh † : a) = S G ( h g : a) − S H 0 (ρ : a 0 ) − i 4π Σ (∂X, ∂X) + 1 2π Σ (X, F (ω)), (2. 13) where h g = hgh † . F (ω) ≡ dω is the Z(h)-component of curvature of the back-ground gauge field a = (iω, a 0 ) ≡ (iω, a(1)0 , . . . , a (r) 0 ). S H 0 (ρ, a 0 ) = r i=1 S H (i) 0 (ρ (i) , a (i) 0 ) with ρ ≡ (ρ (1) , . . . , ρ (r) ) ∈ r i=1 H (i)C 0 /H (i) 0
. For the ghost system, the same discussion as in [13] gives;
Z χψ = D(χ,χ, ψ,ψ) exp − 1 2πi Σ {(∂ h −1 a ψ, χ) − (χ, ∂ h −1 aψ )} = D(χ,χ, ψ,ψ) exp − 1 2πi Σ {(∂ a ψ, χ) − (χ, ∂ aψ )} × r i=1 exp (g ∨ − h ∨ i )S H (i) 0 (ρ (i) : a (i) 0 ) × exp ig ∨ 4π Σ {(∂X, ∂X) − 2i(X, F (ω)) + 2 ig ∨ ρ g/h (X)R(g)} ,
(2. 14)
where g ∨ (h ∨ i ) are the dual coxeter numbers of g C (h (i)C 0 ), and we introduce the notation ρ g/h by
ρ g/h = ρ g − r i=1 ρ (i) h 0 . (2. 15) ρ g (ρ (i) h 0 ) are the Weyl vectors of g C (h (i)C 0
). R(g) is the Riemannian curvature tensor, χ(Σ) = 1 2π R(g). The contribution of ρ in (2. 14) will be obtained by calculating the Schwinger term of the underlying h 0 -current algebra. The contribution of X in (2. 14) seems to be a "background charge". It will be estimated by some direct anomaly calculation or applying the index theorem to the corresponding Dolbeault complex. We must further estimate the anomaly of the FP determinant ∆ FP ( h −1 a). It can be rewritten in the local functional form by introducing the additional FP ghosts;
ξ (h C -valued (0,0)-form),ξ (h C -valued (0,0)-form), ζ (h C -valued (1,0)-form),ζ (h C -valued (0,1)-form); ∆ FP ( h −1 a) = D(ζ,ζ, ξ,ξ) exp − 1 2πi Σ {(∂ h −1 a ξ, ζ) − (ζ, ∂h−1 aξ )} .
(2. 16) The chiral anomaly of this ghost system can be computed in the same way as that of the χψ-ghosts;
∆ FP ( h −1 a) = D(ζ,ζ, ξ,ξ) exp − 1 2πi Σ {(∂ a 0 ξ, ζ) − (ζ, ∂ a 0ξ )} × r i=1 exp 2h ∨ i S H (i) 0 (ρ (i) : a (i) 0 ) . (2. 17)
Notice that this determinant does not depend on ω since Z(h C ) acts trivially on the space h C ≡ Z(h C ) ⊕ h C 0 . By summing up the above estimations of the chiral anomaly and then dropping the gauge volumes DU, DY off we can obtain the following gauge fixed form of the model (2. 6);
Z g.f. = DaZ g.f. [a] ≡ D(a 0 , ω) Z g.f. [a 0 , ω], Z g.f. [a 0 , ω] = D(g, ρ, X, χ,χ, ψ,ψ, ζ,ζ, ξ,ξ) × exp −kS G (g : a) − S χψ (χ,χ, ψ,ψ : a) × exp r i=1 (k + g ∨ + h ∨ i )S H (i) 0 (ρ (i) : a (i) 0 ) − S X (X : ω) − S ζξ (ζ,ζ, ξ,ξ : a 0 ) . (2. 18)
In (2. 18) we introduce the following notations;
S χψ (χ,χ, ψ,ψ : a) = 1 2πi Σ {(∂ a ψ, χ) − (χ, ∂ aψ )} (2. 19) S ζξ (ζ,ζ, ξ,ξ : a 0 ) = 1 2πi Σ {(∂ a 0 ξ, ζ) − (ζ, ∂ a 0ξ )} (2. 20) S X (X : ω) = 1 4πi Σ (∂X, ∂X) + 2iα + (X, F (ω)) + 2iα − ρ g/h (X)R(g) , (2. 21) (α + = k + g ∨ , α − = − 1 √ k + g ∨ ),
where we have rescaled the scalar field X as α + X → X in (2. 21).
Except for the "Chern number dependent term" ∼ Σ (X, F (ω)) in (2.
18), The similar expression for the gauge fixed form (2. 18) is given in [19]. But this "Chern number dependent term" possesses a topological information of the system, and will play an important role in this paper. We will argue on this point in section 4.
Local Operator Formulation
Nextly we will give some operator formulation of the gauge fixed system (2. 18). We shall work on some fixed local holomorphic coordinate patch (U, z) ⊂ Σ, where we set the background fields g, a 0 , ω trivial. The quantization of the system is straightforward, since everything is expressed either in free fields, or in terms of ungauged WZW models on this patch.
First of all, from (2. 18), the total energy-momentum (EM) tensor T tot of the gauge fixed system is given by ;
T tot = T g + T ρ + T X + T χψ + T ζξ ,(2. 22)
where T g , T ρ are the Sugawara EM tensors of the G k ,
r i=1 H C 0 /H 0 −(k+g ∨ +h ∨ i )
-WZW models, and T X , T χψ , T ζξ are those obtained from the actions (2. 19)-(2. 21). Their explicit forms are given as follows;
T g = 1 2(k + g ∨ ) • • (J g , J g ) • • , (2. 23) T ρ = r i=1 T (i) ρ , (2. 24) T (i) ρ = 1 2{−(k + g ∨ + h ∨ i ) + h ∨ i } • • (J (i) ρ , J (i) ρ ) • • = − 1 2(k + g ∨ ) • • (J (i) ρ , J (i) ρ ) • • (2. 25) T X = − 1 2 : (∂ z X, ∂ z X) : +α − ρ g/h (∂ 2 z X) (2. 26) T χψ = − : (χ z , ∂ z ψ) : (2. 27) T ζξ = − : (ζ z , ∂ z ξ) :,(2. 28)
where : : denotes the standard normal ordering prescription defined by mode expansions, and
• • A(w)B(w) • • = 1 2πi w dz A(z)B(w) z − w . We denote the G k , H (i) 0,−(k+g ∨ +h ∨ i ) -currents of the corresponding WZW models by J g = −k∂g g −1 , J (i) ρ = (k + g ∨ + h ∨ i )∂ρ (i) ρ (i)−1 .
One can easily check that T tot indeed has vanishing central charge;
c tot = c g + c ρ + c X + c χψ + c ζξ = kdimg k + g ∨ + r i=1 −(k + g ∨ + h ∨ i )dimh (i) 0 −(k + g ∨ + h ∨ i ) + h ∨ i + l + 12α 2 − ρ 2 g/h + (−2) × 1 2 (dimg − dimh 0 − l) + (−2) × (dimh 0 + l) = 0. (2. 29)
This estimation suggests the total system is topologically invariant. Let us introduce the BRST-charges which will characterize the physical Hilbert space of the gauge fixed system. The BRST symmetry (supersymmetry) (2. 8) should correspond to the following BRST charge;
Q G/H = 1 2πi dz G + G/H , (2. 30)
where the BRST current G + G/H is defined as
G + G/H = −α − : (ψ, J g + 1 2 J χψ ) : . (2. 31)
We set J χψ = −[χ z , ψ]. Meanwhile, the BRST-charges for the H C 0 and Z(H C )-chiral gauge transformations are given by
Q H C 0 = 1 2πi dz G + H C 0 , Q Z(H C ) = 1 2πi dz G + Z(H C ) ,(2. 32)
where the BRST currents G +
H C 0 , G + Z(H C ) are defined by G + H C 0 = − α − √ 2 : (ξ,Ĵ h 0 + J ρ + 1 2 J ζξ ) :, (2. 33) G + Z(H C ) = − α − √ 2 (ξ,Ĵ Z(h) + J X ) − 2ρ g/h (∂ z ξ) . (2. 34)
In the above expressions we introduce the currentĴ aŝ
J = J g + J χψ (2. 35)
and the notationsĴ h (=Ĵ h 0 +Ĵ Z(h) ),Ĵ h 0 ,Ĵ Z(h) denote the projections ofĴ onto the corresponding spaces. This combined currentĴ is Q G/H -invariant. The current J X = α + ∂ z X is obtained from S X (2. 21) by taking the variation of the Z(h)-back-ground gauge field iω. Because the currentsĴ h 0 (i) , J ρ , J ζξ , Ĵ Z(h) and J X have levels k +g ∨ −h ∨ i , −(k +g ∨ +h ∨ i ), 2h ∨ i , k +g ∨ and −(k +g ∨ ) respectively, the "total h C 0 , Z(h C )-currents" of the gauge fixed system,
J h 0 tot ≡Ĵ h 0 + J ρ + J h 0 ζξ , J Z(h) tot ≡Ĵ Z(h) + J X (2. 36)
generate h C 0 , Z(h C )-current algebras with vanishing levels. This implies the nilpotency of the BRST-charges Q H C 0 , Q Z(H C ) (2. 32) (c.f. [12]). These three BRST-charges Q G/H , Q H C 0 , Q Z(H C ) anti-commute with one another. This fact is indeed natural, since they correspond to independent gauge degrees of freedom.
Set the total BRST charge of the gauge fixed system as
Q tot = Q G/H + Q H C 0 + Q Z(H C ) . (2. 37)
Then we can show that the total EM tensor (2. 22) itself is BRST-exact,
T tot = {Q tot , G − tot }, (2. 38)
where G − tot has the following factorized form:
G − tot = G − G/H + G − H C 0 + G − Z(H C )
. Each element is given by ;
G − G/H = −α − : (χ z , J g + 1 2 J χψ ) :, (2. 39) G − H C 0 = − α − √ 2 (ζ z ,Ĵ h 0 − J ρ ) (2. 40) G − Z(H C ) = − α − √ 2 (ζ z ,Ĵ Z(h) − J X ) − 2ρ g/h (∂ z ζ z ) . (2. 41)
The total h C 0 , Z(h C )-currents (2. 36) are also BRST-exact;
J h 0 tot = {Q tot , √ 2α + ζ h 0 z }, J Z(h) tot = {Q tot , √ 2α + ζ Z(h) z },(2. 42)
where ζ h 0 , ζ Z(h) are the h C 0 , Z(h C )-components of the antighost field ζ. The BRST-exactness of T tot , J h 0 ,Z(h) tot will assure the topological invariance of the system.
To end this section, it is convenient to give several remarks. Firstly we notice that the total EM tensor (2. 22) can be decomposed into the following three commuting pieces (c.f (2. 38)-(2. 41));
T tot = T G/H + T H C 0 + T Z(H C ) . (2. 43)
Each element in the RHS of (2. 43) is given by;
T G/H ≡ 1 2(k + g ∨ ) • • (J g , J g ) • • − • • (Ĵ h ,Ĵ h ) • • + 1 k + g ∨ ρ g/h (∂ zĴ Z(h) )− : (χ z , ∂ z ψ) : = {Q G/H , G − G/H }, (2. 44) T H C 0 ≡ 1 2(k + g ∨ ) • • (Ĵ h 0 ,Ĵ h 0 ) • • − • • (J ρ , J ρ ) • • − : (ζ h 0 z , ∂ z ξ h 0 ) : = {Q H C 0 , G H C 0 }, (2. 45) T Z(H C ) ≡ 1 2(k + g ∨ ) • • (Ĵ Z(h) ,Ĵ Z(h) ) • • − 1 k + g ∨ ρ g/h (∂ zĴ Z(h) ) − 1 2 : (∂ z X, ∂ z X) : +α − ρ g/h (∂ 2 z X)− : (ζ Z(h) z , ∂ z ξ Z(h) ) : = {Q Z(H C ) , G − Z(H C ) },(2. 46)
where ξ h 0 and ξ Z(h) denote the h C 0 , Z(h C )-components of the ghost field ξ. By introducing the U(1)-currents;
J G/H = : (ψ, χ z ) : + 2 k + g ∨ ρ g/h (Ĵ Z(h) ), (2. 47) J H C 0 = : (ξ h 0 , ζ h 0 z ) :, (2. 48) J Z(H C ) = : (ξ Z(h) , ζ Z(h) z ) : − 2 k + g ∨ ρ g/h (J X ),(2.Q KS = dimm + − 4 k + g ∨ ρ 2
g/h , and Q CG = l − 4 k + g ∨ ρ 2 g/h respectively 4 . The OPEs of the "residual sector" {G ± H C 0 , T H C 0 , J H C 0 } also have an almost similar form to the TCA with back-ground charge Q = dimh C 0 . But it is not the completely same, since the nilpotency of the G − -operator is broken;
G − H C 0 (z)G − H C 0 (w) ∼ 0.
It may reflect the fact that the manifold H C 0 is not necesarily Kähler, while G/H, Z(H C ) = C * × · · · × C * have natural Kähler structures.
These three algebras commute (anti-commute for ferminonic currents) with one another. They correspond to independent degrees of freedom related to the different gauge symmetries. We may call the TCA {G ± G/H , T G/H , J G/H } as that of "Kazama-Suzuki sector", because the field realizations (2. 31), (2. 39),(2. 44), (2. 47) are same as the twisted version of the N = 2 SCA of the Kazama-Suzuki model [4] for G/H. The TCA {G ± Z(H C ) , T Z(H C ) , J Z(H C ) } will be called that of Coulomb-gas(CG) sector [13]. This is because one can write it in the following convenient form. Introduce a iZ(h) ∼ = R l -valued real compact boson ϕ with the radius α + (normalized by (u, ∂ z ϕ(z))(v, ∂ w ϕ(w)) ∼ 4 A TCA {G ± , T, J} will be called a "TCA with background charge Q" if they satisfy
T (z)J(w) ∼ − Q (z − w) 3 + 1 (z − w) 2 J(w) + 1 z − w ∂ w J(w). − (u, v) (z − w) 2 , u, v ∈ Z(h C )) and rewriteĴ Z(h) as 5 J Z(h C ) = iα + ∂ z ϕ.
(2. 50)
Combining the compact and non-compact bosons ϕ, X to a Z(h C )-valued complex bosonφ = ϕ−iX, the TCA of the CG sector {G ± Z(H C ) , T Z(H C ) , J Z(H C ) } can be expressed in terms ofφ, ζ z , ξ;
T Z(H C ) = − 1 2 : (∂ zφ † ∂ zφ ) : +iα − ρ g/h (∂ 2 zφ )− : (ζ Z(h) z , ∂ z ξ Z(h) ) :, G + Z(H C ) = i √ 2 (ξ Z(h) , ∂ zφ ) + √ 2α − ρ g/h (∂ z ξ), G − Z(H C ) = i √ 2 (ζ Z(h) z , ∂ zφ † ) + √ 2α − ρ g/h (∂ z ζ Z(h) z ), J Z(H C ) = : (ξ Z(h) , ζ Z(h) z ) : +iα − ρ g/h (∂ zφ ) − iα − ρ g/h (∂ zφ † ). (2. 51)
These precisely coincide with those obtained by twisting the N = 2 Coulomb gas model [18].
BRST Analysis and the Chiral Primary Ring
The BRST Cohomology on the Semi-Classical Hilbert Space
Here let us consider the physical states of the gauge fixed system. They are characterized by the total BRST-charge Q tot = Q G/H + Q H C 0 + Q Z(H C ) (2. 37). The total Hilbert space H is spanned by the state vectors having the form |WZW (g) ⊗ |WZW (ρ) ⊗ |X ⊗ |χψ ⊗ |ζξ , and the physical Hilbert space H phys is defined as the Q tot -BRST cohomology group in the standard manner;
H phys = H * Q tot (H). (3. 1)
Instead of considering this total cohomology directly, we shall only estimate it "step by step". Namely we consider H p
Q Z(H C ) •H q Q H C 0 •H r Q G/H (H), in order 5
To complete the definition of ϕ we should further define the zero-mode ϕ 0 appropriately. We take the following convention; [a 0 ,
ϕ 0 ] = − i 2 with a 0 = 1 2πi i∂ z ϕdz, and [N χψ , ϕ 0 ] = 0 with N χψ = 1 2πi : (ψ(z), χ(z)) : dz.
to make the problem simple. In general this may give only a subspace of the precise physical Hibert space H * Q tot (H), but, if we can expect the corresponding spectral sequence degenerates at the 2nd order, it coincides with the physical Hilbert space itself.
First we we shall restrict our attention to the states realized by the direct products of the primary states of all the dynamical variables. In other words we replace the total Hilbert space H by its "semi-classical subspace";
H s.c ≡ {|A ∈ H ; |A is primary } (3. 2) in (3. 1).
The most non-trivial part of the cohomology calculation is the estimation
of H r Q G/H (H s.c ).
It is clearly the same algebra as the chiral primary ring in the G/H-Kazama-Suzuki model, which was fully investigated in the papers [14,15,16]. In order to present these results it is convenient to introduce some notations of Lie algebra. Let W be the Weyl group of g C . For any
w ∈ W we set Φ w ≡ w(∆ − ) ∩ ∆ + and define l(w) ≡ ♯Φ w (the "minimal length" of w). We also set W (g/h) ≡ {w ∈ W ; Φ w ⊂ ∆ + m }. With these notations H r Q G/H (H s.c )
is spanned by the following elements;
J a 1 0 · · ·Ĵ an 0 |Λ, w G/H ⊗ any state vector of ρ, X, ζ, ξ, |Λ, w G/H ≡ |Λ, w(Λ) g ⊗ α∈Φw ψ α 0 |0 χψ , (3. 3)
where w is any element of W (g/h) such that l(w) = r [14,15,16]. Notice that [Q G/H ,Ĵ] = 0 holds. Of course, precisely speaking, (3. 3) only expresses one representative of the corresponding cohomology class. One can always add to it any Q G/H -exact term. (3. 3) satisfies the condition G − G/H,0 |Ψ = 0, besides the BRST invariance Q G/H |Ψ = 0. These states are called as "chiral primary states" in SCFT [14,15,16], and correspond to "harmonic cocycles" in mathematical terminology.
Nextly we should take further the cohomologies with respect to Q H C 0 and Q Z(H C ) .
Consider the H C 0 -part. Under the action ofĴ h 0 0 (2. 35) we can extract irreducible h C 0 -modules (with respect toĴ 0 ) from H r Q G/H (H s.c ); H r Q G/H (H s.c ) = Λ,w H G/H (Λ, w) ⊗ H s.c ρ,X,ζξ ,(3. 4)
where H s.c ρ,X,ζξ denotes the space of the primary states of ρ, X, ζξ and 15,16]. Here we introduce the notation;
H G/H (Λ, w) ≡ {a i } CĴ a 1 0 · · ·Ĵ an 0 |Λ, w G/H is the irreducible h C 0 -module with the highest weight vector |Λ, w G/H having the highest weight w * Λ| h C 0 [14,w * Λ = w(Λ + ρ g ) − ρ g ,(3. 5)
and
| h C 0 denotes the projection to h C 0 ∩ t C , i.e. the Cartan subalgebra (CSA) of h C 0 . t C is the CSA of g C . w * Λ| h C 0
is dominant integral with respect to h C 0 as is shown from the definition of W (g/h). Fix one of the pair (Λ, w) and consider Q H C 0 -cohomology on the corresponding space H
G/H Λ,w ⊗ H s.c ρ,X,ζξ , which is also a h C 0 -module with respect to the total h 0 -current J h 0 tot =Ĵ h 0 + J ρ + J h 0 ζξ
(2. 36), and then the desired Q H C 0 -cohomology is nothing but the Lie algebra cohomology of h C 0 . In order to proceed further it is necessary to fix an appropriate h C 0 -module as the (semi-classical) Hilbert space of ρ. Here we shall take a h C 0 -module with the highest weight 6 ;
λ(Λ, w) def = w * Λ| h C 0 (≡ the conjugate to w * Λ| h C 0 ), (3. 6)
and denote it by H ρ (λ(Λ, w)). In this choice the Q H C 0 -cohomology is easily solved;
H q Q H C 0 (H G/H (Λ, w) ⊗ H s.c ρ,X,ζξ ) ∼ = H q (h C 0 ; C) ⊗ Inv h C 0 [H G/H (Λ, w) ⊗ H ρ (λ(Λ, w))] ⊗ H s.c X,ζ Z(h C ) ξ Z(h C ) . (3. 7) This is because h C 0 is semi-simple and H G/H (Λ, w) ⊗ H s.c ρ,X,ζξ is finite dimen- sional. In the R.H.S of (3. 7) the first factor H q (h C 0 ; C)
corresponds to the contribution of the h C 0 -component of ζξ-ghost system, and the second piece is the singlet tensors for the global H 0 -rotations. H s.c
X,ζ Z(h C ) ξ Z(h C )
is the semi-classical Hilbert space of X and the Z(h C )-component of ζξ-ghost system. Finally we should take the cohomology for the BRST-charge Q Z(H C ) . First let us consider the sector with no ζξ-ghost. Obviously all we have to do is to construct the singlet states for global Z(H C )-rotation. We find that any element of the Q G/H and Q H C 0 -cohomology space above constructed has a definte Z(h C )-charge; w * Λ| Z(h C ) + charge of X. Hence the desired result is simple; if and only if the Z(H C )-charge of X is equal to the value −w * Λ| Z(h C ) , we get the non-trivial BRST-cohomology. The sector with the ζξ-ghosts is also simple. We point out the following fact: Let ξ 1 , · · · , ξ l be the
Z(h C )-components of the ghost field ξ. Assume |I satisfies J Z(h)i tot,0 |I = I i |I , I i = 0 for ∀ i ∈ S ⊂ {1, . . . , l}, then, for ∀ S ′ ⊃ S, i∈S ′ ξ i 0 |I is Q Z(H C ) -invariant.
But it is BRST-trivial except only the case S = ∅. In fact we find that
i∈S ′ ξ i 0 |I = Q Z(H C ) 1 I i 0 i∈S ′ \{i 0 } ξ i 0 |I (3. 8)
for any i 0 ∈ S. This observation leads to the fact that the ζξ-ghost sector is completely factorized like as the H C 0 -part, namely, the physical state with the Z(H C )-ghost number p can be explicitely written as 7 an element of H 0
Q Z(H C ) •H q Q H C 0 •H r Q G/H (H s.c ) ⊗ i∈S ξ i 0 |0 ζξ , S ⊂ {1, . . . , l}, ♯S = p. (3. 9)
To sum up, the desired physical states can be written as follows: Let us denote the cohomology state corresponding to Λ, w and having no ζξ-ghosts (of both the H C 0 and the Z(H C )-part) by |Λ, w , which has l(w) as the χψ-ghost number and invariant under the chiral
H C ≡ H C 0 × Z(H C )-gauge transformations. The semi-classical physical Hilbert space can be expressed as H s.c phys = Λ,w C|Λ, w ⊗ S C i∈S ξ i 0 |0 ζξ ⊗ H * (h C 0 ; C). (3. 10)
Now let us turn our interests to the physical observables. We shall write the "chiral primary operator" corresponding to the physical state |Λ, w as
O Λ,w (x); O Λ,w (0)|0 = |Λ, w . (3. 11)
What ring structure do these operators have? Because it reflects only the local structure of the model, we may describe it by using some technique of CFT. Defining its structure constant by
O Λ 1 ,w 1 O Λ 2 ,w 2 = Λ 3 ,w 3 C (Λ 3 ,w 3 ) (Λ 1 ,w 1 ) (Λ 2 ,w 2 ) O Λ 3 ,w 3 (modulo BRST-exact terms),
(3. 12) we will get the following result;
C (Λ 3 ,w 3 ) (Λ 1 ,w 1 ) (Λ 2 ,w 2 ) ∝ F (G k ) Λ 3 Λ 1 ,Λ 2 r i=1 F (H (i) 0,k+g ∨ −h ∨ i ) w 3 * Λ 3 | i w 1 * Λ 1 | i ,w 2 * Λ 2 | i × δ(w 1 (Λ 1 )| Z + w 2 (Λ 2 )| Z − w 3 (Λ 3 )| Z ) δ(w 1 * 0| Z + w 2 * 0| Z − w 3 * 0| Z ),
(3. 13) 7 One might think that, because ξ i 0 = {Q Z(H C ) , α + X i 0 } holds, any state including the zero-modes of ξ becomes BRST-trivial, even if it has the form of (3. 9). But, actually it is not the case, since the operator X i 0 cannot act on the Hilbert space of the states possessing the definite Z(
H C )-charge. where F (G k ), F (H (i) 0,k+g ∨ −h ∨ i
) mean the fusion coefficients of the corresponding current algebras, and "delta function" is defined by
δ(x) = 1 x = 0 0 x = 0. (3. 14)
The notations Λ| i , Λ| Z mean the orthogonal projections to h
(i)C 0 , Z(h C ) respectively. The appearance of F (G k ) Λ 3 Λ 1 ,Λ 2 in (3. 13) is due to the cur- rent algebra J g , that is, G k -WZW model. The current algebrasĴ h 0 ≡ (Ĵ h (1) 0 , . . . ,Ĵ h (r) 0 ) will give the factor r i=1 F (H (i) 0,k+g ∨ −h ∨ i ) w 3 * Λ 3 | i w 1 * Λ 1 | i ,w 2 * Λ 2 | i in (3. 13), which will include the contribution of the ρ-sector ( r i H (i) −(k+g ∨ +h ∨ i ) - WZW model). 8 δ(w 1 (Λ 1 )| Z +w 2 (Λ 2 )| Z −w 3 (Λ 3 )| Z ) is due to the conservation of the Z(h C )-charge of g-sector. δ(w 1 * 0| Z + w 2 * 0| Z − w 3 * 0| Z ) reflects the conservation of the Z(h C )-charge of the χψ-sector, which we can derive by using the identity ; −w * 0| Z = {ρ g −w(ρ g )}| Z = α∈Φw α| Z . The Z(h C )-charge
conservation of the X-sector is automatically ensured by those of g and χψsector because of the BRST-invariance. We also note that the conservation of the N=2
U(1)-charge (the eigenvalue of J G/H,0 ) is included in the above Z(h)-charge consesrvations.
The structure constant (3. 13) has a complicated form, but if we only consider some suitable subring, we can get more simple results. For example, let us consider the no-ghost sector, i.e. the physical operators of the form O Λ,1 , ("1" means the identitiy in the Weyl group, which trivially belongs to W (g/h));
C (Λ 3 ,1) (Λ 1 ,1) (Λ 2 ,1) ∝ F (G k ) Λ 3 Λ 1 ,Λ 2 δ(Λ 1 | Z + Λ 2 | Z − Λ 3 | Z ), (3. 15)
which is the structure constant introduced by Gepner in the case of G/H = CP N [17]. In particular, in the case of G/H = G/T (T is the maximal torus of G), we get C
(Λ 3 ,1) (Λ 1 ,1) (Λ 2 ,1) ∝ δ(Λ 1 + Λ 2 − Λ 3 ),(3. 16)
since in this case Z(H C ) = T C holds and H C 0 is absent. In this subsection we only considered the semi-classical physical observables which will correspond to the solutions of the equations of motion ; ∂(∂gg −1 ) = 0,∂ψ = 0 etc. But, under some non-trivial background, i.e.
c 1 ≡ i 2π
F (a) = 0, new observables other than the above semi-classical ones will appear. They may be interpreted as "instanton-sectors" which will 8 Because the fusion rule of a current algebra is deeply connected with the structure of null vectors the negative level current algebra H (i) 0,−(k+g ∨ +h ∨ i ) does not give stronger condition than its positive level counter part H
(i) 0,k+g ∨ −h ∨ i .
correspond to the solutions of equations of motion;∂ a (∂ a gg −1 ) = 0,∂ a ψ = 0 etc. with c 1 = 0. To study these instanton contributions we should take the full Hilbert space into account.
The BRST-Cohomology on the Total Hilbert Space and Spectral Flow
Let us return to the problem of solving the BRST-cohomology on the total Hilbert space. We will adopt the same strategy as for the semi-classical case. Namely we will consider H r
Q Z(H C ) •H q Q H C 0 •H r Q G/H (H) instead of studying the cohomology H p+q+r Q tot (H) directly.
The results in this section will be described by using the terminology of affine Lie algebra. Besides the notations for affine Lie algebra some mathematical formulae which we need in this section are summarized in appendix B. For example we will denote the sets of positive (negative) roots of the
g C , h C 0 -current algebras by∆ + (∆ − ),∆ + h 0 (∆ − h 0 ) respectively.∆ + (h 0 ) consists of elements, α (α ∈ ∆ + (h 0 ) ) and α + nδ (α ∈ ∆ (h 0 ) , n ∈ Z >0 )
. δ is the generator of imaginary roots. The modes of the coset, H C 0 -ghost fields will be labelled by the elements of∆
+ m =∆ + \∆ + h 0 ,∆ h 0 =∆ + h 0 ∆ − h 0 . The (affine)
Weyl groups of g C , h C 0 -current algebras will be denoted byŴ ,Ŵ (h 0 ). Any elementŵ ∈Ŵ can be uniquely expressed as t α w (w ∈ W, α ∈ Q : the root lattice of g C ). t α (α ∈ Q) is the "translation" by α.
Firstly we pay attention to the estimation of H r Q G/H . This cohomology problem was fully studied in the papers [14,15]. It was shown that the cohomology elements are labelled by
(Λ,ŵ) ∈P k + ×Ŵ (g/h),(3. 17)
whereP k + is the set of dominant integral weights of g C -current algebra with level k.Ŵ (g/h) is the subset ofŴ which elements satisfy the condition;
Φŵ ⊂∆ + m , where we set Φŵ =ŵ(∆ − ) ∩∆ + . H r Q G/H (H) can be described as follows; H r Q G/H (H) = Λ ,ŵ H G/H (Λ,ŵ) ⊗ H ρ,X,ζξ . (3. 18) w = t α w ∈Ŵ (g/h) in the R.H.S of (3. 18) are those elements which satisfy r = l(w) − 2(ρ, α). H G/H (Λ,ŵ)
is spanned by the following vectors 9 ;
J a 1 −m 1 · · ·Ĵ an −mn |Λ,ŵ G/H (m 1 , · · · , m n ∈ Z ≥0 ), (3. 19) |Λ,ŵ G/H = |Λ,ŵ(Λ) g ⊗ α∈Φŵ ψ −α |0 χψ . (3. 20)
Under the action of the h C 0 -current algebraĴ h 0 (2. 35) H G/H (Λ,ŵ) is the irreducibleĥ C 0 -module with the highest weight vector |Λ,ŵ G/H which has the weight for the h
(i)C 0 -direction; w * Λ| i + (k + g ∨ − h ∨ i )Λ 0 (ŵ * Λ ≡ŵ(Λ +ρ g ) −ρ g ),(3. 21)
whereρ g = ρ g + g ∨ Λ 0 . "| i " means taking the classical part ofŵ * Λ and then projecting it to the h
(i)C 0 -component. H ρ,X,ζ,ξ in the R.H.S of (3. 18) is the Hilbert space of ρ, X, (ζ, ξ) fields.
Nextly we should take the cohomology with respect to Q H C 0 (2. 32). For this purpose we should take an appropriate representation for ρ field. As is the semi-classical case we may choose, as the Hilbert space of ρ, thê h C 0 -module which highest weight is given bŷ
λ (i) (Λ,ŵ) def =ŵ * Λ| i + (−k − g ∨ − h ∨ i )Λ 0 (3. 22) for theĥ (i)C 0 -component. We write it as H ρ (λ(Λ,ŵ)) = CJ a 1 ρ,−m 1 · · · J an ρ,−mn |λ(Λ,ŵ),λ(Λ,ŵ) ρ . (3. 23)
With this choice there exist the following elements of H 0
Q H C 0 •H 0 Q G/H (H); Inv h C 0 [H G/H (s.c) (Λ,ŵ) ⊗ H (s.c) ρ (λ(Λ,ŵ))] ⊗ |0 ζξ ,(3. 24)
where
H G/H (s.c) (Λ,ŵ) = CĴ a 1 0 · · ·Ĵ an 0 |Λ,ŵ G/H , H (s.c) ρ (λ(Λ,ŵ)) = CJ a 1 ρ,0 · · · J an ρ,0 |λ(Λ,ŵ),λ(Λ,ŵ) ρ (3. 25) are the semi-classical Hilbert space of H G/H (Λ,ŵ), H (s.c) ρ (λ(Λ,ŵ)) respec- tively. The global h C 0 -invariance in (3. 24) ensures the Q H C 0 -invariance of (α = α + nδ ∈∆ m ), ψα =α+nδ = ψ −α n for α ∈ ∆ − m χ α,n for α ∈ ∆ + m
the state since the (ζ, ξ)-vacuum state |0 ζξ is characterized by the conditions 10 :
ζ α+nδ |0 ζξ = (h, ζ t n )|0 ζξ = 0 ( for n ≥ 0, α ∈ ∆ h 0 , h ∈ t), ξ α+nδ |0 ζξ = (h, ξ t n )|0 ζξ = 0 (for n > 0, α ∈ ∆ h 0 , h ∈ t)
. t is the CSA of g. Lastly we should take the cohomology with respect to Q Z(H C ) (2. 32). It will be achieved by imposing the global Z(h C )-invariance on the state (3. 24), which gives rise to the following element that belongs to H 0
Q Z(H C ) •H 0 Q H C 0 •H r Q G/H (H); |Λ,ŵ def = Inv h C 0 [H G/H (s.c) (Λ,ŵ) ⊗ H (s.c) ρ (λ(Λ,ŵ))] ⊗ | −ŵ * Λ| Z X ⊗ |0 ζξ .
(3. 26) Here "| Z " stands for the similar meaning as | i , i.e. theŵ * Λ| Z is the Z(h C )projection of the classical part ofŵ * Λ.
To proceed further let us introduce a powerful tool -the "spectral flow" U [14,15], which is a family of infinite symmetry transformations of our topological model in the sense that they make the BRST charge Q tot (2. 37) invariant;
U Q tot U −1 = Q tot ,(3. 27)
and that they change the background gauge fields of the gauge fixed model appropriately. The second point will be discussed in the next section. These transformations are essentially induced from the "translations" in the (affine) Weyl group of g C -current algebra. Let us describe them explicitly. For this purpose we introduce the following subset of the weight lattice P of g C which labels the spectral flow U;
P(g/h) = { γ ∈ P ; ∃ σ ∈ W (h 0 ) s.t σ(C aff 0,h 0 + γ) = C aff 0,h 0 }, (3. 28)
where W (h 0 ) is the Weyl group of h 0 and C aff 0,h 0 is the subdomain of t * which contains the Weyl alcove of h 0 ;
C aff 0,h 0 = { u ∈ t * ; (u, ∀ α (i) l i ) ≥ 0, (u, ∀ θ (i) ) ≤ 1 }.
(3. 29) 10 We label the modes of ζ(z) = n ζ n z −n−1 , ξ(z) = n ξ n z −n by ζ α+nδ , ξ α+nδ (α ∈ ∆ h0 ), ζ t n and ξ t n ;
ζ α+nδ = ζ α,n for α ∈ ∆ + h0 ζ −α n for α ∈ ∆ − h0 , ξ α+nδ = ξ α,n for α ∈ ∆ + h0 ξ −α n for α ∈ ∆ − h0 α (i) l i (1 ≤ l i ≤ rankh (i)C 0
) are the simple roots of h (i)C 0
(1 ≤ i ≤ r), and θ (i) is the maximal root of h (i)C 0 (1 ≤ i ≤ r). With these definitions it is clear that σ ∈ W (h 0 ) in (3. 28) is uniquely determined for γ ∈ P if it exists, and we shall denote it by σ γ . We further introduce the notation;
w(γ) = σ γ t γ (3. 30)
for any element of γ ∈ P(g/h). The spectral flow U γ will be defined as the action ofŵ(γ).
We are now in a position to write the defintion of spectral flow. Under the action of U γ (γ ∈ P (g/h)) the g C -current algebra J g and the ghost fields
(χ, ψ) should be transformed into U γ J g U −1 γ , U γ χ U −1 γ , U γ ψ U −1 γ ; U γ J g,α (z) U −1 γ = J g,σγ (α) (z)z −(α,γ) U γ J α g (z) U −1 γ = J σγ (α) g (z)z (α,γ) (3. 31) U γ (h, J g )(z) U −1 γ = (σ γ (h), J g )(z) − k γ, h δ n,0 U γ χ α (z) U −1 γ = χ σγ (α) (z)z −(α,γ) U γ ψ α (z) U −1 γ = ψ σγ (α) (z)z (α,γ) (3. 32)
Moreover, by expressingŵ(γ) asŵ(γ) =τ γωγ ∈ D ×Ŵ (ω γ ∈Ŵ ,τ γ ∈ D (= the group of extended Dynkin diagram automorphisms of g C )), the transformations of the primary states are given by
U γ |Λ,Λ g = |τ γ (Λ),ŵ(γ)(Λ) g (3. 33) U γ |0 χψ = α∈Φŵ (γ) ψ −α |0 χψ .
From the transformation rule (3. 34) the state |Λ,ŵ G/H (3. 20) will be transformed into another physical state |Λ ′ ,ŵ ′ G/H by U γ ;
U γ |Λ,ŵ G/H = |Λ ′ ,ŵ ′ G/H Λ′ =τ γ (Λ) ∈P k + w ′ =ŵ(γ)ŵτ −1 γ ∈Ŵ (g/h). (3. 34)
Remark that thisŵ ′ is indeed an element ofŴ (g/h). (See the proposition B.7 in appendix B.) In the standpoint of the coset CFT it is claimed that any states |Λ,ŵ G/H which are transformed into each other by some spectral flow should be identified [14,15]. Because there exists the isomorphism between P(g/h) and P/Q(h 0 ); Nextly we will describe the action of U γ (γ ∈ P(g/h)) on the H C /H-WZW sector (the ρ-sector) and the H C -ghost sector. The h C 0 , Z(h C )-current algebras J ρ , J X and the ghost fields (ζ, ξ) are transformed into
P(g/h) ∋ γ −→ [γ] ∈ P/Q(h 0 ),(3.U γ J ρ U −1 γ , U γ J X U −1 γ and (U γ ζ U −1 γ , U γ ξ U −U γ |λ,λ ρ = |τ h 0 γ (λ),τ h 0 γ (λ) ρ (3. 36) U γ |β X = |β − (k + g ∨ )γ X (3. 37) U γ |0 ζξ = |τ h 0 γ ζξ ,(3. 38)
where the state |τ h 0 γ ζξ is defined by
α∈∆ + h 0 ζ −τ h 0 γ (α) ξ α |0 ζξ .
Especially the (ζ, ξ) Fock vacuum can be written as |τ h 0 γ = id ζξ . Since the BRST cohomology state (3. 26) is composed of the primary states of all the sectors or their descendents, its transformation rule under the actions of spectral flows can be derived from the above formulae. Moreover, because of the property (3. 27), this transformed state is also BRST-invariant (and not BRST-trivial). If taking care of the transformation rules of the ρsector (3. 37) and the H C -ghost sector (3. 38), we find that the transformed state does not necessarily have the form of (3. 26). Namely we will obtain new cohomology classes by making the spectral flows act on (3. 26). We rewrite the BRST-cohomology state (3. 26) as |Λ,ŵ,τ h 0 = id . Then U γ transforms this states into U γ |Λ,ŵ, id , which we denote by |Λ ′ ,ŵ ′ ,τ h 0 γ .Λ ′ ,ŵ ′ are those given in (3. 34). In this way we have obtained a family of the physical states labelled by
(Λ,ŵ,τ h 0 ) ∈P k + ×Ŵ (g/h) × D(h 0 ). (3. 39)
The actions of the spectral flows are closed among them. This is because the flows have the following property: For γ 1 , γ 2 ∈ P(g/h),
U γ 1 U γ 2 = U γ 2 U γ 1 (3. 40) = U γ 1 + • γ 2 ,
where γ 1 + • γ 2 ∈ P(g/h) is defined via the isomorphism (3. 35);
P/Q(h 0 ) ∋ [γ 1 + γ 2 ] −→ γ 1 + • γ 2 ∈ P(g/h). (3. 41)
For γ ∈ P(g/h), we also define − • γ ∈ P(g/h) by the image of [γ].
We denote the physical observable corresponding to the physical state |Λ,ŵ,τ h 0 by OΛ ,ŵ,τ h 0 as in the semi-classical case (3. 11). The ring structure of these operators will be given by the generalization of that of the semiclassical operators (3. 13);
C (Λ 3 ,ŵ 3 ,τ h 0 3 ) (Λ 1 ,ŵ 1 ,τ h 0 1 ) (Λ 2 ,ŵ 2 ,τ h 0 2 ) ∝ F (G k )Λ 3 Λ 1 ,Λ 2 r i=1 F (H (i) 0,k+g ∨ −h ∨ i )λ (i) 3 λ (i) 1 ,λ (i) 2 × δ(ŵ 1 (Λ 1 )| Z +ŵ 2 (Λ 2 )| Z −ŵ 3 (Λ 3 )| Z ) × δ(ŵ 1 * 0| Z +ŵ 2 * 0| Z −ŵ 3 * 0| Z ) × δ(τ h 0 1τ h 0 2τ h 0 −1 3 ), (3. 42) whereλ (i) j ≡λ (i) (Λ j ,ŵ j ) and δ(τ h 0 1τ h 0 2τ h 0 −1 3 ) stands for the "delta function on D(h 0 )" defined by δ(τ h 0 ) = 1τ h 0 = id 0τ h 0 = id. (3. 43)
It is an important question whether the field identification by the spectral flow is compatible with this ring structure (3. 42). We rewrite the physical operators as O a (a ∈ I). I is the index set (3. 39). Define the action of spectral flow on {O a } a∈I by the standard field-state correspondence, that is, the operator U γ (O a ) ≡ O γ·a is given by
U γ (O a )|0 = U γ |a ,(3. 44)
where |a is the physical state corresponding to O a . Then we can show the following identity with respect to the structure constant C c ab ;
C (γ 1 + • γ 2 )·c γ 1 ·a,γ 2 ·b = C c ab ,(3. 45)
or equivalently,
U γ 1 (O a )U γ 2 (O b ) = c∈I C c ab U γ 1 + • γ 2 (O c ). (3. 46)
This can be proved by observing the explicit form of C c ab (3. 42). The most non-trivial part of the proof is the following relation for the G-WZW sector; [15,21,26].) In the next section we will derive this formula (3. 46) from a more physical viewpoint, that is, as a direct consequence of the topological invariance of the system. The identity (3. 45) implies that the field identifications are consistent with the ring structure of the BRST-cohomology. Especially if we set
F (G k )τ γ + • γ ′ (Λ 3 ) τγ (Λ 1 ),τ γ ′ (Λ 2 ) = F (G k )Λ 3 Λ 1 ,Λ 2 ,(3.[O a ] = { U γ (O a ) ; ∀ γ ∈ P(g/h) },(3. 48)
we can consistently introduce a product on them by
[O a ][O b ] def = [O a O b ] (3. 49)
without depending on the choice of representatives. Hence, on the level of local properties of our model, these field identifications completely work and one may always extract at most finite physical degrees of freedom. However, once we turn our attention to the global structure of the system, we will face with somer subtle problem. Especially, if we calculate the correlation functions with some fixed back-ground topology (Euler number and Chern classes in our case), we will find that
U γ (O a ) O b . . . O c g,c 1 = O a O b . . . O c g,c 1 , (3. 50)
because O a and U γ (O a ) have different ghost numbers although they have the same N = 2 U(1)-charges. In the next section we will discuss this problem in detail.
Correlation Functions of Physical Observables and a Geometrical Interpretation of Spectral Flow
In this section we will study the correlators of the physical observables constructed in section 3, ( the correlators among the 0-form components). We shall perform this estimation with the topology fixed, that is, the genus g of Σ and the "vector of Chern numbers" c 1 ≡ i 2π F (a) fixed to be some definite values. For the genus g, the higher genus correlation functions (g ≥ 2) always become zero. This is a simple result obtained from counting of the anomaly for the N = 2 U(1)-charge (2. 47)(of the Kazama-Suzuki sector). Because all the physical observables obtained above have non-negative N = 2 U(1)-charges, while its back-ground charge is equal to Q KS (1 − g), which is negative when g ≥ 2. (Q KS = dimm + − 4 k + g ∨ ρ 2 g/h > 0.) We shall consider the genus 0 case. We write the correlator on sphere as · · · c 1 . We should manifestate the domain in which c 1 takes its value. Notice that c 1 must belong to the weight lattice P . 11 The Z(h C )-components of c 1 will play as topological invariants, while the h 0 -components can be changed by chiral gauge transformations. These gauge degrees of freedom correspond to the shifts by the root lattice Q(h 0 ) of h 0 . Hence we can assume c 1 takes its value in P(g/h) (3. 28), which is isomorphic to the quotient lattice P/Q(h 0 ).
Let us study the correlation function in the operator formalism. In this approach the correlation function may be regarded as the matrix element under the following configuration of the back-ground gauge field a = (a 0 , iω) (2. 12);
F (a) = −πc 1 δ (2) (z − ∞)dz ∧ dz.
(4. 1)
We normalize the delta function as δ (2) (z)dz ∧ dz = 2i. So we can define the correlator by
O a 1 (x 1 ) O a 2 (x 2 ) . . . O a 3 (x 3 ) c 1 = 0, c 1 |O ξ (x 0 ) O a 1 (x 1 ) O a 2 (x 2 ) . . . O an (x n )|0 . (4.
2) The in-vacuum |0 has no back-ground charge. The out-vacuum 0, c 1 | is the BRST-invariant state having the suitable back-ground charges corresponding to c 1 . This is an analoguous situation as that in the Coulomb gas realization of CFT [8]. O ξ (x) denotes the BRST-invariant operator which state is a∈h ξ a 0 |0 ζξ ⊗ |0 others . This operator should be inserted in order to cancel the ζξ-ghost number anomalies. In the following we may omit to write it explicitly. We also notice that, since our EM tensor is BRST-exact (2. 38), the correlation function (4. 2) does not depend on the operator insertion points. So we may also omit to write these insertion points.
The back-ground charges in (4. 2) can be determined from the estimations of the ghost number or the chiral anomalies. Let N
(α) χψ ≡ 1 2πi dz : ψ α χ α : (α ∈ ∆ + m ), N (α)
ζξ (α ∈ ∆ h 0 ), and N In order to give the explicit form of the out-vacuum state we first notice that the state 0, c 1 = 0| still has the non-zero back-ground charges which are due to the Fegin-Fucks term ∼ ρ g/h (X)R(g) in S X (2. 21). This out-vacuum 0, c 1 = 0| is realized by
0, c 1 |(J g,0 , h) = 0, c 1 |{−k c 1 , h }, (∀h ∈ t C ) (4. 6) 0, c 1 |(J (i) ρ,0 , h) = 0, c 1 |{(k + g ∨ + h ∨ i ) c 1 , h }, (∀h ∈ t(h (i) 0 ) C ) (4. 7) 0, c 1 |(J X,0 , h) = 0, c 1 |{2 ρ g/h , h + (k + g ∨ ) c 1 , h }, (∀h ∈ Z(h C )).0, c 1 = 0| = kΛ 0 , kΛ 0 | g ⊗ r i=1 −(k + g ∨ + h ∨ i )Λ 0 , −(k + g ∨ + h ∨ i )Λ 0 | ρ (i) ⊗ 2ρ g/h | X ⊗ 0| χψ α∈∆ + m χ α,0 ⊗ 0| ζξ a∈h ζ a 0 ≡ kΛ 0 , w 0 , 0|,
(4. 10)
in the notation of the previous section: (kΛ 0 , w 0 , id) ∈P k + ×Ŵ (g/h) × D(h 0 ). w 0 is the element having the maximal length in W (g/h). For the description of 0, c 1 | with general values of c 1 we utilize the relation;
0, c 1 |U γ = 0, c 1 + • γ|,(4. 11)
which follows from the facts that the spectral flow U γ preserves the BRSTinvariance and that the state 0, c 1 |U γ satisfies the same conditions (4. 3)-(4. 8) as those of 0, c 1 + • γ|. This identity (4. 11) is siginificant from the view of topological theory because it enables us to interpret the spectral flows as the transformations which connect the sectors with different Chern numbers, that is, the different instanton sectors. By applying the equality (4. 11) to the state 0, c 1 = 0| (4. 10) we can obtain the following general expression of 0, c 1 | :
0, c 1 = γ| = 0, c 1 = 0|U γ = τ −1 γ (kΛ 0 ),ŵ(γ) −1 w 0τγ ,τ h 0 −1 γ |,(4. 12)
whereτ γ ,ŵ(γ) andτ h 0 γ are, as in the previous section, the elements uniquely defined by γ.
To begin with, we shall consider the one point function. There exists a unique physical operator O c 1 max (x) defined by the condition
O c 1 max c 1 = 1. (4. 13)
This is the operator which state is dual to the out-vacuum 0, c 1 = γ| (4. 12), that is, the state |τ −1 γ (kΛ 0 ),ŵ(γ) −1 w 0τγ ,τ h 0 −1 γ . It has the maximum U(1)-charge of our model, which is equal to Q KS . For example O c 1 =0 max is the operator corresponding to |kΛ 0 , w 0 , id . It includes the term if α ∈ ∆ + m satisfies (α, c 1 ) ≥ 0,
• χ α,−1 , . . . , χ α,(α,c 1 )+1 if α ∈ ∆ + m satisfies (α, c 1 ) ≤ −2, • no modes of χ α , ψ α if α ∈ ∆ + m satisfies (α, c 1 ) = −1.
These modes have the following geometrical meaning. Let L α be the line bundle in which ψ α lives. Then χ α is a section of K ⊗ L −1 α . K ≡ T 1,0 * Σ is the canonical bundle of Σ. The space of the solutions of the equation of motion for ψ α is H 0 (Σ, L α ), and that for χ α is H 0 (Σ, K ⊗L −1 α ). These spaces can be realized by
H 0 (CP 1 , L) ∼ = 0, deg L < 0 P n (X 0 , X 1 ), deg L = n ≥ 0, (4. 14)
where X 0 , X 1 denote the homogeneous coordinates of CP 1 , and P n (X 0 , X 1 ) is the set of homogeneous polynomials of order n. Since we are now considering the situation of F (a) ∼ c 1 δ (2) (z − ∞), we may regard as L = O(∞) n , so the n + 1 independent elements of H 0 (CP 1 , L) behave as 1, z, . . . , z n around z = 0. Moreover, by recalling deg
L α = (α, c 1 ), deg K ⊗ L −1 α = −2 − (α, c 1 )
and the mode expansions ψ α (z) = n∈Z ψ α n z n , χ α (z) = n∈Z χ α,n z n+1 , we can find that O c 1 max just includes all the solutions of the equations of motion for ψ α , χ α . Therefore we can say O c 1 max corresponds to the "top cohomology class" on the instanton moduli space. Because of the relation; (4. 15) which is easily shown from (4. 12) U − • γ maps the top cohomology class on the c 1 = 0 instanton moduli space to that on the c 1 = γ moduli space.
O c 1 =γ max = U − • γ (O c 1 =0 max ),
Nextly we shall study the correlator of the form N i=1 O a i c 1 . We first notice that this correlator is non-zero if and only if the following relation holds;
N i=1 O a i ∼ const O c 1 max + b O b (mod BRST). (4. 16)
Making use of this observation and combining the relations (4. 15), (3. 46) successively, we can show the formula;
U γ 1 (O a 1 ) U γ 2 (O a 2 ) . . . U γn (O an ) c 1 = O a 1 O a 2 . . . O an c 1 + n i=1 γ i . (4.
17) This means that the field identification rule (3. 48) by the spectral flow is still consistent at the level of correlators if summing them up with respect to the Chern numbers c 1 . Namely the correlation function among the identified observables should be defined by; 18) or equivalently, by fixing c 1 to be some definite value, one may take the next;
[O a 1 ] [O a 2 ] . . . [O an ] def = c 1 ∈P(g/h) O a 1 O a 2 . . . O an c 1 ,(4.[O a 1 ] [O a 2 ] . . . [O an ] def = γ∈P(g/h) U γ (O a 1 ) O a 2 . . . O an c 1 . (4. 19)
It is interesting to derive the identity (4. 17) from somewhat different viewpoints. We restrict ourselves to a simple case G/H = G/T . We start with the definition of correlator by path-integration ;
N i=1 O a i (z i ) c 1 = D(g, X, χ, ψ, ζ, ξ) N j=1
O a j (z j ) e −kS G (g: ω)−S X (X: ω)−S χψ (χ,ψ: ω)−S ζξ (ζ,ξ: ω) . Let us consider the case of
N j=1 O a j (z j ) c 1 with c 1 = N j=1 γ j , γ j ∈ P
and a j ≡ (Λ j , w j ) ∈P k + × W (g/h) for ∀j. Namely, we set all the inserted observables semi-classical. First we point out that the topological invariance of our theory implies that the correlation function (4. 20) does not depend on any configuration of the back-ground gauge field ω as far as the Chern numbers c 1 are fixed. It is assured by the BRST-exactness (2. 42) of the total current J t tot ≡ J t g + J t X + J t χψ . Hence one may choose the following configuration; (4. 22) and then c 1 = N j=1 γ j . We are now considering the situation such that the curvature of the back-ground gauge field has some delta function singularities at the points z j where the observables O a i have been inserted, while in the situation considered before we took F (ω) ∼ j γ j δ (2) (z − ∞). These curvature singularities will give some non-trivial modifications to the observables inserted at the points of singularities. To recognize the above mentioned effect directly it may be helpful to consider first the X-sector. Recall that the action of X-sector (2. 21) has the term ∼ (X, F (ω)). Then the substitution of (4. 22) into the action (2. 21) is equivalent to the insertion of some vertex operators at the points z j . This effect will add the value −(k + g ∨ )γ j on to the X-momentum of O a j (z j ). It signals the possibility that the instanton contribution modifies the physical observables. This story is, however, too naive to justify completely. Because the vertex operator insertions from the curvature singularities are at the same points as that O a j are inserted, we should treat carefully the OPE singularities among them. We will later discuss these OPE singularities.
ω 01 | U 0 = −i N j=1 γ j∂ log(z − z j ), ω 01 | U∞ = 0, (4. 21) where U 0 , U ∞ are the coordinate patches around 0, ∞ such that Σ = U 0 ∪U ∞ , {z j } ⊂ U 0 , {z j } ⊂ U ∞ . This configuration gives F (ω) = iπ N j=1 γ j δ (2) (z − z j ) dz ∧ dz,
In order to proceed further we shall consider the following chiral gauge transformation 13 ;
Ω(t)(z) = e tu(z) , u(z) = h z − w , (∀h ∈ t), (4. 23)
and we replace −kS G (g : ω) by −kS G (g : Ω(t) −1 ω) in (4. 20). Then, by changing the integration variable g to Ω(t) −1 g, we can get the identity;
D(g, X, χ, ψ, ζ, ξ) N j=1 O a j (z j ) exp{−kS G (g : Ω(t) −1 ω) −S X (X : ω) − S χψ (χ, ψ : ω) − S ζξ (ζ, ξ : ω)} = D(g, X, χ, ψ, ζ, ξ) N j=1 e t w j (Λ j ),u(z j ) O a j (z j ) exp{−kS G (g : ω) + kS G (Ω(t) : ω) −S X (X : ω) − S χψ (χ, ψ : ω) − S ζξ (ζ, ξ : ω)}.
(4. 24)
Here we have used the fact; D( (Ω(t) : ω). Differentiating the both hand sides of (4. 24) with respect to t and then setting t = 0, we obtain the Ward identity;
Ω(t) −1 g) = Dg, O a j (z j )[ Ω(t) −1 g] = e t w j (Λ j ),u(z j ) O a j (z j )[g] and the Polyakov-Wiegmann identity; S G ( Ω(t) −1 g : Ω(t) −1 ω) = S G (g : ω) − S GD(g, X, · · ·) (h, J t g (w)) N j=1 O a j (z j ) e −S tot (g,X,···: ω) = N j=1 w j (Λ j ) + kγ j , h w − z j D(g, X, · · ·) N j=1
O a j (z j ) e −S tot (g,X,···: ω) .
(4. 25) Notice that the net effect of the curvature singularity ∼ γ j δ (2) (z − z j ) is the shift of the weight of g-sector; w j (Λ j ) −→ w j (Λ j ) + kγ j . The similar arguments also work for the X-sector and the χψ-sector. By replacing S X (X : ω) (or S χψ (χ, ψ : ω)) by S X (X : Ω(t) −1 ω) (or S χψ (χ, ψ : Ω(t) −1 ω)), and making use of the next formulas
D( Ω(t) −1 X) = D(X), S X ( Ω(t) −1 X : Ω(t) −1 ω) = S X (X : ω) − (−k − g ∨ )S G (Ω(t) : ω), D( Ω(t) −1 χ, Ω(t) −1 ψ) = D(χ, ψ) e g ∨ S G (Ω(t): ω) , S χψ ( Ω(t) −1 χ, Ω(t) −1 ψ : Ω(t) −1 ω) = S χψ (χ, ψ : ω),(4. 26)
one can show the Ward identities;
D(g, X, · · ·) (h, J t X (w)) N j=1 O a j (z j ) e −S tot (g,X,···: ω) = N j=1 − w j * Λ j + (k + g ∨ )γ j , h w − z j D(g, X, · · ·) N j=1
O a j (z j ) e −S tot (g,X,···: ω) ,
(4. 27) D(g, X, · · ·) (h, J t χψ (w)) N j=1 O a j (z j ) e −S tot (g,X,···: ω) = N j=1 w j (ρ g ) − ρ g + g ∨ γ j , h w − z j D(g, X, · · ·) N j=1
O a j (z j ) e −S tot (g,X,···: ω) .
(4. 28)
What do the identities (4. 25), (4. 27), (4. 28) mean? These identities suggest that, under the singular back-ground F (ω) ∼ γ j δ (2) (z − z j ), the chiral primary field O a j (z j ) behaves as U γ j (O a j )(z j ) in the flat back-ground. Because the BRST-invariance is not broken even in the non-flat back-ground, the BRST-invariant operator O a j (z j ) should be changed to another BRSTinvariant operator by the effect of curvature singularity. Comparing the spectrum of physical observables studied in the previous section with the data of the Ward identities, we can conclude that O a j (z j ) should be changed to the operator U γ j (O a j )(z j ). So, we have again obtained the relation (4. 17). Notice that this derivation of (4. 17) does not need the result (3. 46) which played an important role to obtain the relation (4. 17) in the previous section. We can rather show this formula (3. 46) very easily by means of the above result. In fact, if noting that
U γ 1 (O a 1 ) U γ 2 (O a 2 ) U − • (γ 1 + • γ 2 ) (O a 3 ) others O b c 1 = O a 1 O a 2 O a 3 others O b c 1 , (4. 29)
which is a trivial identity derived from (4. 17), the desired result (3. 46) follows. Thus the relation (3. 46) is a natural consequence of the topological invariance of our model. Related with the above discussion based on the path integration it may be helpful to study the local operator formulation around the point where the back-ground gauge field is singular. Consider again (4. 21) as the configuration of the back-ground gauge field and pay attention to a particular point, say, z 1 . We take a holomorphic coordinate {z, U} around z 1 such that z 1 = 0, z j ∈ U (j = 1) and simply write as γ 1 = γ, namely,
ω 01 | U = −iγ∂ log z.
(4. 30)
Let us consider the operator formalism on U. Notice that, by setting Ω (γ) = z −ν −1 (γ) , ω 01 can be trivialized on U;
iω 01 | U = −∂Ω (γ) Ω −1 (γ) | U .O (γ) [g, X, χ, ψ] def = O[ Ω −1 (γ) g, Ω −1 (γ) X, Ω −1 (γ) χ, Ω −1 (γ) ψ] ≡ O[Ω −1 (γ) g, X + α + log Ω −1 (γ) , Ω −1 (γ) χΩ (γ) , Ω −1 (γ) ψΩ (γ) ].
(4. 32)
In particular the current J g is transformed to J (γ) g ;
J (γ) g (z) = J g (z)[Ω −1 (γ) g] = Ω −1 (γ) J g (z) Ω (γ) + kΩ −1 (γ) ∂Ω (γ) (z).
(4. 33)
If we substitute Ω (γ) (z) = z −ν −1 (γ) into (4. 33) and compare it with the defintion of spectral flows (3. 31), we can find the following identity;
J (γ) g (z) = U γ J g (z) U −1 γ . (4. 34)
It is easy to observe that the same relations also hold for other fields;
X (γ) (z) = U γ X(z) U −1 γ , χ (γ) (z) = U γ χ(z) U −1 γ , ψ (γ) (z) = U γ ψ(z) U −1 γ .
(4. 35)
These mean that the singular gauge transformation Ω (γ) which connects the flat back-ground (on U) to the singular back-ground (4. 30) is realized by the spectral flow U γ . These transformed operators will characterize the state vector which is inserted at the origin of the coordinate patch U. Especially the vacuum vector under the singular back-ground (4. 30) will be given by
|0 (γ) = U γ |0 , (4. 36)
where |0 is the usual vacuum with the locally flat back-ground on U, that is,ω such thatω = Ω −1 (γ) ω. The gauge transformed physical observables can be written as
O (γ) a (z) = U γ O a (z) U −1 γ . (4. 37)
Hence the next readily follows;
O (γ) a (0)|0 (γ) = U γ (O a )(0)|0 . (4. 38)
This relation can be read as follows: In the L.H.S of (4. 38) the "topological charge" γ is attached to the vacuum vector |0 (γ) , which correspond to the existence of curvature singularity. While, in the RHS of (4. 38) the vacuum vector has no charge and the charge γ is absorbed into the operator O a , which means O a is changed to U γ (O a ). For a general operator O(z) its gauge transformed partner O (γ) (z) will have a singularity at z = 0. This is because, in general, O (γ) has the same singularity as that of Ω (γ) . Can we define the operator O (γ) (0) in the L.H.S of (4. 38)? This issue has its origin in the singular configuration of the back-ground gauge field (4. 30) and the same difficulity has appeared in the earlier discussion of the X-sector, where it is necessary to insert some vertex operators at the points other operators have been inserted. We cannot avoid this difficulity if we study each sector independently. However, fortunately we can find, from the defintions of spectral flow (3. 31), ..., that the singularity of each sector precisely cancells in all. This is not surprising, since the BRSTinvariance (in particular, the Q T C -invariance) of the operator O a leads to the chiral gauge invariance. Actually O (γ) a (z) should define the same cohomology class as that of O a (z), namely,
O (γ) a (z) = O a (z) + {Q tot , * } (4. 39)
should hold. So we may rewrite (4. 38) as
O a (0)|0 (γ) = U γ (O a )(0)|0 + Q tot | * . (4. 40)
To sum up, one may say: When (and only when) treating the g, X, χψ-sectors as a combined system and considering only the BRST-invariant oprators, one can construct a consistent operator formalism even under the singular background (4. 30). This reflects the fact that the total system is topologically invariant but each sector is not. It may be also interesting to consider the following configuration of the back-ground gauge field instead of (4. 30);
ω| U = −γ ω(g)| U ,(4. 41)
where ω(g) is the Levi-Civita connection. Since the Euler number of U ∼ = semisphere, is equal to 1, − 1 2π U F (ω) = γ holds. Under this configuration of ω (4. 41) the conformal fields in our model will gain some extra spins. For example, the ghost ψ α (z) will be transformed to
ψ (γ)α (z)(≡ U γ ψ α (z) U −1 γ ) = n∈Z ψ α n z n−(α,γ) , = n∈Z ψ (γ)α n z n ,(4. 42)
where
ψ (γ)α n (≡ U γ ψ α n U −1 γ ) = ψ α n+(α,γ) . (4. 43)
The first line of (4. 42) and the vacuum condition; ψ α n |0 (γ) = 0 iff n > (α, γ), mean that the field ψ (γ)α (z) behaves as a spin −(α, γ) primary field on the vacuum |0 (γ) with respect to the modes ψ α n . This aspect corresponds to the configuration (4. 41). one may say that the spectral flow is the transformation changing the spins of the elementary fields as is expected from the back-ground (4. 41). On the other hand, the 2nd line of (4. 42) and the vacuum condition; ψ (γ)α n |0 (γ) = 0 iff n > 0, say that it is a spin 0 primary field with respect to the gauge transformed modes ψ (γ)α n , which corresponds to the configuration (4. 30) considered before. In this way, we obtain the two physical interpretations of the spectral flow corresponding to the choice of the back-ground (4. 30) or (4. 41); one of them is a singular gauge transformation and the other is a spin changing transformation. They are of course consistent with each other.
Conclusions
We have investigated the topological gauged WZW models for the cases of general Kähler homogeneous spaces. The gauge fixing and the operator formalism based on it were presented in such a way that they are natural extensions of those in [13]. We investigated the BRST-cohomology of our system, defined by the total BRST-charge Q tot = Q G/H +Q Z(H C ) +Q H C 0 . The cornerstone of this study is the concept "spectral flows" [14,15], which are symmetry transformations preserving the BRST-charge Q tot . Because of the relation (3. 46);
U γ 1 (O a )U γ 2 (O b ) = c∈I C c a b U γ 1 + • γ 2 (O c )
, the identifications of the physical obervables by the spectral flows are compatible with the chiral ring structure. We further argued the problem of these field identifications from the global geometrical standpoint. The correlation function among the identified observables, which is valid under any back-ground gauge field, is described in (4. 18)(or (4. 19)).
Under these studies the geometrical and physical pictures of the spectral flows appeared. They give the following two insights on the roles of the gauge field.
Firstly, the spectral flow is capable to connect two arbitary instanton sectors in the system which are labelled by different Chern numbers. This is recoginized as a phenomenon that the physical observables are transmuted by the effect of the back-ground curvature singularity, and it gives the relation (4. 17). Moreover, the consistency condition (3. 45) of the ring structure under the field identifications (3. 48) was shown to be a natural consequence of the relation (4. 17). In this respect it may be important to remark the analogy with the two-dimensional BF gauge theory [22]: S BF = i 2π (φ, F (ω)).
In this BF theory the correlator among the vertex operators e i(β,φ) satisfies the same relation as in (4. 17). The above BF theoretical aspect of topological SU(2)/U(1) gauged WZW model played an important part in the study of two-dimensional topological gravity [1]. Secondly, related with the path-integral approach, it was shown that two physical interpretations were possible for the spectral flow. It can be interpreted as a transformation changing the spins of the elementary fields in the system, while it can also work as a singular gauge transformation which creates an appropriate back-ground charge on the physical vacuum. In this respect it is useful to note the analogy with the following Coulomb gas system: S CG= 1 4πi {(∂φ, ∂φ) + 2α + (φ, F (ω)) + 2α − ρ g (φ)R(g)}. If we set ω = −ω(g)ρ g , the EM tensor of the system T CG will suffer the twist by the U(1)-current J CG = 2iα + ∂φ; T CG →T CG = T CG + 1 2 ∂J CG . This causes the changes of the spins of the vertex operators e i(β,φ) . The important difference between this Coulomb gas system and our topological system is that, in the topological model, the procedure of changing the spins is performed BRST-invariantly, because both the EM tensor and the U(1)-current are BRST-exact. The above "twist" does not change the physical contents of the topological system, while it does in the Coulomb gas system.
In order that spectral flows work completely in a given model, the underlying current algebras must be integrable. (See appendix B.) Therefore, if they are non-integrable, the symmetry of spectral flow may be lost. In these cases the state identification (in the sence of this paper) will no longer work, and so the infinite BRST states will be left for us. It seems interesting to understand these infinite physical states in a unifying standpoint, for example, by some appropriate generalization of spectral flow. In [23] we plan to study the case of SU(2)/U(1) with fractional levels. Especially the infinte physical states in the system will be studied from the above point of view. The correspondence between this infinite dimensional cohomology and the Lian-Zuckerman cohomology [24] of Liouville theory in two-dimensional gravity will be discussed. charge term" ∼ ρ g/h (X)R. In this way we can finally arrive at the following expression of the gauge-fixed system ;
Z g.f. [g] = DaZ g.f. [a, g] ≡ D(a 0 , ω)Z g.f. [a 0 , ω, g], Z g.f. [a 0 , ω, g] = D(g, ρ, X, ψ,ψ, ζ,ζ, ξ,ξ) × exp{−kS G (g, a) − S ψ (ψ,ψ, a)} × exp r i=1 (k + g ∨ + h ∨ i )S H (i) 0 (ρ (i) , a (i) 0 ) − S ′ X (X, ω, ω(g)) − S ζξ (ζ,ζ, ξ,ξ, a 0 ) , (A. 3)
where we set
S ψ (ψ,ψ, a) = 1 π Σ dv(g) {(ψ,∂ az ψ) + (ψ, ∂ azψ )} (A. 4) S ζξ (ζ,ζ, ξ,ξ, a 0 ) = 1 2πi {(∂ a 0 ξ, ζ) − (ζ, ∂ a 0ξ )} (A. 5) S ′ X (X, ω, ω(g)) = 1 4πi Σ (∂X, ∂X) + 2iα + (X, F (ω)) (A. 6) (α + = k + g ∨ , α − = − 1 √ k + g ∨ ).
The fields X, ρ, ξ,ξ, ζ,ζ are the counterparts of those in (2. 18).
We can read the total EM tensor of the gauge fixed system from (A. 3);
T ′ tot = T g + T ρ + T ′ X + T ψ + T ζξ , (A. 7)
where T g , T ρ , T ζξ are those given in (2. 23), (2. 25), (2. 28) and
T ′ X = − 1 2 : (∂ z X, ∂ z X) :, (A. 8) T ψ = − 1 2 : (ψ, ∂ z ψ) : . (A. 9)
The total central charge is easily calculated;
c ′ tot = c g + c ρ + c ′ X + c ψ + c ζξ = kdimg k + g ∨ + r i=1 −(k + g ∨ + h ∨ i )dimh (i) 0 −(k + g ∨ + h ∨ i ) + h ∨ i + l + 1 2 dim (m + ⊕ m − ) + (−2) × (dimh 0 + l) = 3dimm + − 12 k + g ∨ ρ 2 g/h . (A. 10)
It is equal to the central charge of the Kazama-Suzuki model for G/H. In fact the total system is equivalent to the Kazama-Suzuki model modulo a BRST exact term. T ′ tot can be factorized to
T ′ tot = T KS + T ′ Z(H C ) + T H C 0 , (A. 11)
where T KS is the EM tensor of the Kazama-Suzuki model [4] T KS = 1 2(k + g ∨ )
• • (J g , J g ) • • − • • (Ĵ h ,Ĵ h ) • • − 1 2 : (ψ, ∂ z ψ) :, (A. 12)
and T ′ Z(H C ) is defined by
T ′ Z(H C ) = 1 2(k + g ∨ ) • • (Ĵ Z(h) ,Ĵ Z(h) ) • • − 1 2 : (∂ z X, ∂ z X) : . (A. 13)
T H C 0 is that given in (2. 45). The physical degrees of freedom in this SUSY model are characterized by the BRST-charge
Q ′ tot = Q Z(H C ) + Q H C 0 (Q Z(H C ) ,Q H C 0 (2. 32))
. It is easy to see
T ′ tot = T KS + {Q ′ tot , G − ′ Z(H C ) + G − H C 0 }. (A. 14) Here G − H C 0 is that given in (2. 40), and G − ′ Z(H C ) is G − ′ Z(H C ) = − α − √ 2 (ζ z ,Ĵ Z(h) − J XT tot − T ′ tot = T G/H + T Z(H C ) − T KS − T ′ Z(H C ) = 1 2 ∂ z J G/H + {Q Z(H C ) , * }. (A. 16)
Thus the "twist" of the EM tensor of this SUSY gauged WZW model is given by adding the term,
1 2 ∂ z J G/H ≡ 1 4 ∂ z : (ψ, ψ) : + 1 k + g ∨ ∂ z ρ g/h (Ĵ Z(h) ).
The first component reflects the difference of spin between ψ and χψ and the second one is due to the distinction of the abelian anomaly commented above.
B Notes on Algebra Automorphisms of Affine Lie Algebras and the Spectral Flow
The purpose of this appendix is to give the precise definitions of the spectral flow [14,15] which is introduced in section 4. We will present several results needed for our main subjects without proof. Refer the ref. [15,25,26] for the proofs and more complete discussions.
Let us start with preparing some notations and conventions of affine Lie algebras.
B.1 Notations for Affine Lie Algebras
Let g be a (complex) simple Lie algebra (rank l), andĝ ≡ Lg ⊕ CK ⊕ Cd be the corresponding affine Lie algebra. Lg ≡ g⊗C[z, z −1 ] is the loop algebra of g. K is the canonical central element and d(≡ z d dz ) is the scaling element.
We assume that g is simply-laced for simplicity. Let t,t ≡ t ⊕ CK ⊕ Cd be the Cartan subalgebras of g,ĝ respectively, ∆ = ∆ + ∆ − ,∆ =∆ + ∆ − be the root systems of g,ĝ and Π = {α 1 , . . . , α l },Π = {α 0 ≡ δ − θ, α 1 , . . . , α l } (θ is the highest root of g and δ is defined by δ(d) = 1, δ(t) = δ(K) = 0 ) be the simple roots of g,ĝ.
We denote by ( , ) the Cartan-Killing metric normalized so that the square length of each root is 2. It is well-known that this metric is naturally extended to an invariant metric onĝ [25], which we shall denote by ( | );
(u(z)|v(z)) = 1 2πi
1 z (u(z), v(z)) dz ( ∀ u(z), v(z) ∈ Lg),
(K|d) = 1, (K|u(z)) = (d|u(z)) = (K|K) = (d|d) = 0 ( ∀ u(z) ∈ Lg).
(B. 1)
This metric induces the dual metric onĝ * , especially ont
* ≡ t * ⊕ CΛ 0 ⊕ Cδ,
which we express by the same notation ( | );
(α|β) = (α, β) ( ∀ α, β ∈ t * ), (Λ 0 |δ) = 1, (α|Λ 0 ) = (α|δ) = (Λ 0 |Λ 0 ) = (δ|δ) = 0, ( ∀ α ∈ t * ). (B. 2)
The sets of real and imaginary roots ofĝ are denoted by∆ real =∆ + real ∆ − real ,
∆ im =∆ + im ∆ − im respectively; ∆ + real = ∆ + ∪ { α + nδ ; α ∈ ∆, n ∈ Z >0 }, ∆ − real = −∆ + real , ∆ + im = { nδ ; n ∈ Z >0 },∆ − im = −∆ + im , (B. 3)
where ∆(∆ + ) are the sets of (positive) roots of g. We introduce the root lattice Q and the weight lattice P of g;
Q = l i=1 Zα i , P = l i=1 ZΛ i , (B. 4)
where Λ 1 , · · · , Λ l are the fundamental weights of g which satisfy,
(Λ i |α j ) = δ ij , (Λ i |Λ 0 ) = (Λ i |δ) = 0, for 1 ≤ ∀ i, j ≤ l.
The set of dominant integral weights with level k ofĝ is given byP
k + = {Λ = Λ + kΛ 0 : Λ ∈ P + , (Λ|θ) ≤ k }, (B. 5)
where P + is the set of dominant integral weights of g. Let W,Ŵ be the Weyl groups of g,ĝ.Ŵ has the structureŴ = W ×Q, where the root lattice Q acts ont * by "translations";
t γ (μ) =μ + (μ|δ)γ − (μ|γ) + 1 2 |γ| 2 (μ|δ) δ (B. 6) ( ∀ γ ∈ Q, ∀μ ∈t * )
Under the linear isomorphism ν :t 14 the root lattice Q also acts ont by t γ (ĥ) =ĥ + (ĥ|K)ν −1 (γ) − (ĥ|ν −1 (γ)) + 1 2 |γ| 2 (ĥ|K) K (B. 7)
∼ = − −− →t * defined by ν(h), h ′ = (h|h ′ ),
( ∀ γ ∈ Q, ∀ĥ ∈t).
The Weyl group W of g is generated by the "reflections in simple roots" r α i (1 ≤ i ≤ l) which act onλ ∈t * by
r α i (λ) =λ − (λ|α i )α i (1 ≤ i ≤ l).
The Weyl group W may be realized as N(T )/T , where T is the Cartan torus of G and N(T ) is the normalizer of T . r α i ∈ W (1 ≤ i ≤ l) will correspond to e πi 2 (eα i +e α i ) ∈ N(T ), where e α ∈ g α and e α ∈ g −α (α ∈ ∆ + ) are the Cartan-Weyl base of g. Hence r α i ∈ W (1 ≤ i ≤ l) will act onx ∈ĝ by
r α i (x) = e πi 2 (eα i +e α i )x e − πi 2 (eα i +e α i ) .
14 , is the dual pairing, i.e ν(h), h ′ = ν(h)(h ′ ). Especially ν(K) = δ, ν(d) = Λ 0 .
B.2 Some Automorphisms of Affine Lie Algebras
According to [15,26], we shall extend the affine Weyl groupŴ to the following group;W = W ×P (⊃Ŵ ), (B. 8)
where the classical weight lattice P acts ont * (t) in the completely same way as the root lattice Q (see (B. 6), (B. 7).) Moreover we introduce the following subgroup ofW ;
D = {ŵ ∈W ;ŵ(∆ + ) =∆ + }. (B. 9)
With this preparation the next lemma holds;
Lemma B.1 1.W = D ×Ŵ , namely, any elementŵ ofW is uniquely expresseible asŵ =τŵ 0 ,τ ∈ D,ŵ 0 ∈Ŵ , andŴ is a normal subgroup ofW .
2. For ∀ γ ∈ P , define the elementτ γ of D by the above unique decomposition t γ =τ γŵγ ,τ γ ∈ D,ŵ γ ∈Ŵ , then it holds that
τ γ+α =τ γ for ∀ α ∈ Q, τ γ 1 +γ 2 =τ γ 1τ γ 2 ,
the map γ ∈ P −→τ γ ∈ D is onto.
This lemma implies that the group D is isomorphic to P/Q( ∼ = Z(G)) as abelian group. It is further known [26] that Aut(Π) ∼ = Aut(Π) ×D holds, so one may call D as the "group of proper extended Dynkin dyagram automorphisms". One can thinkW as a subgroup of Aut(ĝ). In fact, the action ofW on the CSAt (and its dualt * ) is already defined. Introduce a Cartan-Weyl base ofĝ: e α+nδ = e α z n ∈ĝ α+nδ , e α+nδ = e α z −n ∈ĝ −α−nδ for α + nδ ∈∆ real + and hz n ∈ g nδ for n ∈ Z \ {0}. Here e α ∈ g α , e α ∈ g −α (α ∈ ∆ + ) are the Cartan-Weyl base of g and h ∈ t. Thenŵ = wt γ ∈Ŵ (w ∈ W, t γ ∈ P ) will act on them asŵ (e α z n ) = e w(α) z n−(α,γ) w(e α z −n ) = e w(α) z −n+(α,γ) (B. 10) w(hz n ) = w(h)z n .
It is easy to check that these definitions besides (B. 7), (B. 8) give an automorphism ofĝ.
Let us turn our interests to the g-current algebra with level k ∈ Z >0 (the integrable representations ofĝ); The current J(z) = J t (z) + α∈∆ + {J α (z)e α + J α (z)e α } is acting on the Hibert space H k = Λ ∈P k + L(Λ), where L(Λ) is the integrableĝ-module with the highest weightΛ. We prepare the next notation;
for ∀ x = u(z)+aK, (u(z) ∈ Lg) J[x] def = 1 2πi (J(z), u(z)) dz+ak. (B. 11)
The following theorem is important for our purpose;
Theorem B.2 For an arbitrary elementŵ = wt γ ofW (w ∈ W , γ ∈ P ), there exists a unitary transformation Uŵ on H k such that
Uŵ J[x] U −1 w = J[ŵ(x)], (B. 12)
namely,
Uŵ J α (z) U −1 w = z −(α,γ) J w(α) (z), ( ∀ α ∈ ∆ + ) Uŵ J α (z) U −1 w = z (α,γ) J w(α) (z), ( ∀ α ∈ ∆ + ) Uŵ (h, J t (z)) U −1 w = (w(h), J t (z)) − k γ, h 1 z , ( ∀ h ∈ t)
Uŵ|Λ,Λ = |τ γ (Λ),ŵ(Λ) (B. 13)
whereτ γ ∈ D is determined by the unique decomposition t γ =τ γŵγ ∈ D ×Ŵ in lemma B.1.
For the Sugawara EM tensor T (z)(≡ n∈Z L n z n+2 )
def = 1 2(k + g ∨ ) • • (J(z), J(z)) •
• , one can obtain the following transformation formula by direct computations.
Corollary B.3 Letŵ be as above,
Uŵ L n U −1 w = L n − w(γ), J t n + k 2 |γ| 2 δ n,0 . (B. 14)
Generally Uŵ maps L(Λ) to another integrable module L(τ γ (Λ)), and Uŵ makes L(Λ) invariant ifŵ ∈Ŵ , or equivalently, γ ∈ Q. It is worth pointing out that the integrability (or locally nilpotency) of the representation is crucial for the existence of Uŵ (see [25]). So, in non-integrable cases we cannot construct the unitary transformations Uŵ.
B.3 Spectral Flow
Now we would like to introduce the concept of spectral flow which gives rise to automorphisms of TCA being investigated in section 3, 4 and has its origin in the Lie algebra automorphisms given in theorem B.2.
First of all, recall the parabolic decomposition (2. 3); Naively we would like to define the spectral flow associated with the automorphism groupW . However, there is a technical problem; because the symmetry of the ρ-sector in (2. 18) is described by a h 0 -current algebra with negative level, we cannot construct the automorphism as that given in theorem B.2. In order to avoid this difficulty we must define the transformations of spectral flow as the actions of a suitable subgroup ofW , which we will denote byD, rather thanW itself. This subgroup should be defined as;
g = h ⊕ m + ⊕ m − = Z(h) ⊕ h 0 ⊕ m + ⊕ m − = Z(h) ⊕ (h (1) 0 ⊕ · · · ⊕ hD = {ŵ = σt γ ∈W ;ŵ(∆ + h 0 ) ⊂∆ + h 0 , σ ∈ W (h 0 ), γ ∈ P }. (B. 18)
Namelyŵ ∈D means thatŵ acts along theĥ 0 -direction as a diagram automorphism, and also notice thatŵ(∆ m ) ⊂∆ m holds. So, by considering the action ofD, we will be able to get a well-defined spectral flow without the difficulty of negative level. For our porpose it is more convenient to consider the following subset P(g/h) of P instead ofD itself;
P(g/h) = { γ ∈ P ; ∃ σ ∈ W (h 0 ), s.t σt γ ∈D }. (B. 19)
It is easy to see that, for ∀ γ ∈ P(g/h), σ in the R.H.S of (B. 19) is unique, and we will denote it by σ γ . In other words, the set P(g/h) has a one-to-one correspondence withD. We write this correspondence by
P(g/h) ∋ γ −→ŵ(γ) def = σ γ t γ ∈D. (B. 20)
An equivalent defintion of P(g/h) which is more geometrical is given by
P(g/h) = { γ ∈ P ; ∃ σ ∈ W (h 0 ) s.t σ(C aff 0,h 0 + γ) = C aff 0,h 0 }, (B. 21)
where C aff 0,h 0 is the subdomain of t * which contains the Weyl alcove of h 0 ;
C aff 0,h 0 = { u ∈ t * ; (u, α) ≥ 0 ( ∀ α ∈ ∆ + h 0 ), (u, ∀ θ (i) ) ≤ 1 }.
Here we denote the maximal root of h (i)C 0 by θ (i) (1 ≤ i ≤ r). Notice that σ ∈ W (h 0 ) in the R.H.S of (B. 21) is the same element as that of (B. 19), i.e. σ = σ γ . For instance, in the case of G = SU(N) (and H is arbitrary), we can find P(g/h) = { γ ∈ P ; (γ, ∀ α (1 ≤ l i ≤ rankh (i) 0 ). To proceed further let us consider the quotient weight lattice P/Q(h 0 ). Take any element [γ] ∈ P/Q(h 0 ), it is easy to show from (B. 19) (or (B. 21)) that the set [γ] ∩ P(g/h) necessarily includes a unique non-zero element. We will denote it by γ 0 . In this way one can obtain a one-to-one correspondence [γ] ←→ γ 0 between P/Q(h 0 ) and P(g/h). Making use of this correspondence, we will define an addition (we express it by " + • ") on P(g/h) induced from that of P/Q(h 0 ). Notice that, on P (Z(h)) ∩ P (⊂ P(g/h)), + • coincides with the usual addition +, but generally does not. Under these preparations we can prove;ŵ (γ 1 + • γ 2 ) =ŵ(γ 1 )ŵ(γ 2 ).
We summarize the above results as a proposition.
For the Cartan directions we set the actions of U γ as
U γ (h, ζ t n ) U −1 γ = (σ γ (h), ζ t n ), U γ (h, ξ t n ) U −1 γ = (σ γ (h), ξ t n ), ( ∀ h ∈ t).
(B. 30)
The Fock vacuum |0 ζξ is the state which satisfies ζ α+nδ |0 ζξ = (h, ζ t n )|0 ζξ = 0 ( ∀ n ≥ 0, ∀ α ∈ ∆ h 0 , ∀ h ∈ t), ξ α+nδ |0 ζξ = (h, ξ t n )|0 ζξ = 0 ( ∀ n > 0, ∀ α ∈ ∆ h 0 , ∀ h ∈ t), and its transformation law which is consistent with (B. 29), (B. 30) is as follows;
U γ |0 ζξ = α∈∆ + h 0 ζ −τ h 0 γ (α) ξ α |0 ζξ . (B. 31)
For the coset part (g, χψ) these definitions coincide with those given in [14,15]. But one must also introduce the concept of spectral flow for the H C 0 and Z(H C )-parts in order to formulate the model as a Lagrangian field theory.
Our main results in this appendix are included in the following theorem;
Theorem B. This theorem implies that the BRST-cohomology states of our model are invariant under the spectral flow. This fact plays a crucial role in our discussions in section 4. Any physical state |Ψ and its transformed state U γ |Ψ possess the same KS U(1)-charges. We will summarize a few useful statements as propositions.
Proposition B.6 The map γ ∈ P(g/h) −→ U γ gives a homomorphism of abelian group. Namely, U γ 1 + • γ 2 = U γ 1 U γ 2 , ( ∀ γ 1 , γ 2 ∈ P(g/h)). holds. Especially, for any α ∈ P(g/h) ∩ Q,ŵ(α)ŵ ∈Ŵ (g/h). where we set W (g/h) = {w ∈ W ; Φ w ⊂ ∆ + m }.
(Refer [15] for the proof) This means that any elementŵ ofŴ (g/h) can be uniquely decomposed asŵ = ωw, where ω ∈D ∩Ŵ and w ∈ W (g/h), or equivalently one can uniquely express it asŵ =ŵ(α)w by an element α ∈ P(g/h) ∩ Q.
49) one can find out that the pairs {G ± G/H , T G/H , J G/H }, {G ± Z(H C ) , T Z(H C ) , J Z(H C ) } generate the topological conformal algebras (TCAs) with background charges
35) the equivalent class of H Q G/H (H) (3. 18) under the identification (3. 34) can be labelled byP k + ×Ŵ (g/h) P/Q(h 0 ) ≡P k + × W (g/h) D [14, 15]. Note thatŴ (g/h) ∼ = W (g/h) × Q/Q(h 0 ) (see the propositon B.8 and the propositon B.4 in appendix B) and D ∼ = P/Q.
explicit forms are given in (B. 26), (B. 27), (B. 29), (B. 30) in appendix B. From the definition (3. 30)ŵ(γ) acts on h C 0 as an extended Dynkin diagram automorphism of h C 0 . We denote its action asτ h 0 γ ∈ D(h 0 )(= the group of the extended Dynkin diagram automorphisms of h C 0 ). Then U γ transforms the primary states into
47) and the similar relations of the H(i) 0 -parts. This relation (3. 47) is shown by making use of the Verlinde's formula on the fusion coefficients [20] and the modular transformation properties of the affine characters. (Refer for example
11
(j = 1, . . . , rankg) be the ghost number operators for (χ, ψ), (ζ, ξ)-ghost sectors. The out-vacuum 0, c 1 | should be characterized by the conditionsPrecisely speaking, c 1 ∈ ν −1 (P ), where ν : t C → t C * is the isomorphism defined by the inner product. (See appendix B.) But we shall here omit it to avoid the complexity of notations.
worthwhile to notice that the relation, 0, c 1 |J G/H,0 = 0, c 1 |Q KS , (4. 9) holds without depending on c 1 , which suggests some geometrical meaning of the Kazama-Suzuki U(1)-current J G/H (2. 47). The charge of it generates a global U(1)-rotation on both the bosonic part g (and necessarily X) and the fermionic part χψ along the direction which does not depend on c 1 .
for the χψ-sector. This makes us interpret O c 1 =0 max as the top cohomology class of H * DR (G/H) defined by the volume element of G/H. How about a case of c 1 = 0? The physical state corresponding to O c 1 max includes the elements, • ψ α 0 , . . . , ψ α −(α,c 1 )
(4. 20) is the real valued back-ground gauge field; a = iω and c 1 =− 1 2π Σ F (ω).Since we study the holomorphic part only, we may set ω 10 = 0 by treating ω 10 and ω 01 as independent variables 12 .
this relation we can consistently give the operators appropriate to the back-ground (4. 30). Let O = O[g, X, χ, ψ] be a general operator under the flat back-ground, then the corresponding operator O (γ) which is appropriate to the singular back-ground (4. 30) (or (4. 31)) should be defined by the gauge transform O (γ) = Ω (γ) O, that is,
P
(Z(h)) = {ω ∈ Z(h) * : (ω|β) ∈ Z for ∀ β ∈ Q(Z(h))},
5 1 .
1All the BRST charges (Q G/H , Q Z(H C ) , Q H C 0 , Q tot ) which are defined in section 2 are invariant under the spectral flows, that is, for an arbitrary element γ of P(g/h),U γ Q * U −1 γ = Q * , (B. 32)where " * " means "G/H" or ... "tot".2. The total EM tensor (2. 22) is invariant under the spectral flow moduloBRST-exact terms, explicitely written as;U γ L tot,n U −1 γ = L tot,n − σ γ (γ), J t tot,n = L tot,n − {Q tot , √ 2α + σ γ (γ), ζ n }. (B. 33)3. {G ± G/H , T G/H , J G/H }, the TCA of Kazama-Suzuki sector given in section 2, is 'strictly" invariant under the spectral flow;U γ T G/H (z) U −1 γ = T G/H (z), U γ G ± G/H (z) U −1 γ = G ± G/H (z), U γ J G/H (z) U −1 γ = J G/H (z).(B. 34)
introduce the following subset ofŴ ;W (g/h) = {ŵ ∈Ŵ ; Φŵ ⊂∆ + m }, (B. 36)and then the next propositions hold;Proposition B.7 Ifŵ ∈Ŵ (g/h), γ ∈ P(g/h), thenŵ(γ)ŵτ −1 γ ∈Ŵ (g/h)
KS , J G/H (2. 47) } are non-trivial physical observables (i.e. Q ′ tot -invariant and not exact). They generates a N = 2 SCA. This means that the model is equivalent to the Kazama-Suzuki model.By comparing T ′ tot with T tot (2. 43) we can find).
(A. 15)
In this way we can see that {G ±
G/H (2. 31) (2. 39), T
In the ref.[19] a different choice is taken for this representation of h C 0 . Because the total BRST cohomology depends on this choice the results given in[19] do not fully coincide with these we present in this paper.
We label the modes of χ α (z) = n χ α,n z −n−1 , ψ α (z) = n ψ α n z −n (α ∈ ∆ + m ) by ψα
Strictly speaking, we should treat the holomorphic and the anti-holomorphic parts simultaneously by setting ω 10 † = ω 01 .
We set Ω † = 1 formally, since we are here dealing with the holomorphic sector only.
AcknowledgementsWe thank Mr.Y.Sugiyama for the discussion at the early stage of this work.Appendix A Quantization of the N = 2 SUSY GaugedWZW ModelIn this appendix we present the quantizations of N = 2 supersymmetric (SUSY) gauged WZW models[1,6]in order to comment the relation between the "twist" of the SUSY gauged WZW models and that of N = 2 SCFTs[3]. We note that these quantizations are essentially a special case of the work in[7], in which general (not necesarily of N = 2) SUSY gauged WZW models were considered and it was shown that they describe N = 1 supercoset CFTs. N = 2 SUSY gauged WZW model associated with G/H, where G/H is assumed to be Kähler, is given by(A. 1) In this expression the Weyl fermions ψ,ψ are m + ⊕ m − -valued, where we set m ± be those introduced in section 2 in the text. The gauge field A is h-valued. dv(g) denotes the canonical volume element defined by a fixed Kähler metric g.As in the topological case we must perform the gauge fixing for the H Cchiral gauge transformations, and for this aim we need to estimate the chiral anomalies of the ingredients. The solution of this problem is almost the same as the topological case in the text. The only exception is the abelian part (i.e. Z(H C )-part) of the anomaly of the coset fermions ψ,ψ. It is due to the fact that ψ,ψ are spinor fields and that we have to use the index theorem for the spin complex rather than the Dolbeault complex. Therefore we must replace the estimation of the anomaly of χψ-system (2. 14) by the following one;Notice that a distinction from (2. 14) is the absence of the "back-groundThe three abelian groupsD, P(g/h) and P/Q(h 0 ) are isomorphic with one another. These isomorphisms are given byIn addition, the subgroupsD ∩Ŵ , P(g/h) and Q/Q(h 0 ) are also isomorphic.We have arrived at the stage to present the explicit definition of spectral flow. As was already mentioned, we shall give it associated with each element of P(g/h) or equivalentlyD. Recall that the ingredients of the model in the text are g, ρ, X, χψ, ζξ. g, ρ and X are the variables of G,We then defineor more explicitely,χψ-sector; Define the following sets of affine roots;and introduce the following notation for the mode expansions of χψ;We then definewhere Φŵ (γ) is the subset of∆ + given by Φŵ =ŵ(∆ − ) ∩∆ + , for ∀ŵ ∈W .The Fock vacuum |0 χψ is the state which satisfies ψα|0 χψ = 0, ( ∀α ∈∆ + m ).ζξ-sector; Introduce the following notations;and then we define
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|
[] |
[
"Quantum-Secured Single-Pixel Imaging against Jamming Attacks",
"Quantum-Secured Single-Pixel Imaging against Jamming Attacks"
] |
[
"Jaesung Heo \nAgency for Defense Development\nDaejeonSouth Korea\n",
"Junghyun Kim \nAgency for Defense Development\nDaejeonSouth Korea\n",
"Taek Jeong \nAgency for Defense Development\nDaejeonSouth Korea\n",
"Yong Sup Ihn \nAgency for Defense Development\nDaejeonSouth Korea\n",
"Duk Y Kim \nAgency for Defense Development\nDaejeonSouth Korea\n",
"Zaeill Kim \nAgency for Defense Development\nDaejeonSouth Korea\n",
"Yonggi Jo \nAgency for Defense Development\nDaejeonSouth Korea\n"
] |
[
"Agency for Defense Development\nDaejeonSouth Korea",
"Agency for Defense Development\nDaejeonSouth Korea",
"Agency for Defense Development\nDaejeonSouth Korea",
"Agency for Defense Development\nDaejeonSouth Korea",
"Agency for Defense Development\nDaejeonSouth Korea",
"Agency for Defense Development\nDaejeonSouth Korea",
"Agency for Defense Development\nDaejeonSouth Korea"
] |
[] |
We propose a quantum-secured single-pixel imaging method exploiting non-classical correlations of a photonpair for preventing a possible jamming attack. Our method is based on a single-pixel imaging which exploits spatial correlations between target-illuminating photons and the number of measured photons after a target interaction. To reduce effects of external noises, time-correlation of non-classically correlated photon-pairs is exploited. Simultaneously, security of the imaging method is investigated by analyzing polarization-correlation of the photon-pairs. By photon heralding, our method can obtain an image under an imaging disrupting attack by using strong chaotic light. Also, it is able to detect a deceiving attack based on the polarization-correlation. As a result, images obtained from a proof-of-principle demonstration are provided, and we show that the statistical errors in polarization measurement can reveal a deceiving attack. The proposed method can be developed by adopting matured techniques used in quantum secure communication.arXiv:2209.06365v1 [quant-ph]
| null |
[
"https://export.arxiv.org/pdf/2209.06365v1.pdf"
] | 252,222,568 |
2209.06365
|
35c5f62e88d103c2499734ae2fed7fa4d7a73819
|
Quantum-Secured Single-Pixel Imaging against Jamming Attacks
Jaesung Heo
Agency for Defense Development
DaejeonSouth Korea
Junghyun Kim
Agency for Defense Development
DaejeonSouth Korea
Taek Jeong
Agency for Defense Development
DaejeonSouth Korea
Yong Sup Ihn
Agency for Defense Development
DaejeonSouth Korea
Duk Y Kim
Agency for Defense Development
DaejeonSouth Korea
Zaeill Kim
Agency for Defense Development
DaejeonSouth Korea
Yonggi Jo
Agency for Defense Development
DaejeonSouth Korea
Quantum-Secured Single-Pixel Imaging against Jamming Attacks
(Dated: September 15, 2022)
We propose a quantum-secured single-pixel imaging method exploiting non-classical correlations of a photonpair for preventing a possible jamming attack. Our method is based on a single-pixel imaging which exploits spatial correlations between target-illuminating photons and the number of measured photons after a target interaction. To reduce effects of external noises, time-correlation of non-classically correlated photon-pairs is exploited. Simultaneously, security of the imaging method is investigated by analyzing polarization-correlation of the photon-pairs. By photon heralding, our method can obtain an image under an imaging disrupting attack by using strong chaotic light. Also, it is able to detect a deceiving attack based on the polarization-correlation. As a result, images obtained from a proof-of-principle demonstration are provided, and we show that the statistical errors in polarization measurement can reveal a deceiving attack. The proposed method can be developed by adopting matured techniques used in quantum secure communication.arXiv:2209.06365v1 [quant-ph]
I. INTRODUCTION
Non-classical correlations are the essential source of a quantum advantage in various quantum information protocols. For example, in entanglement-based quantum key distribution (QKD) [1,2] and quantum-secured imaging, [3,4] the correlation provides security against a possible eavesdropping attack in the quantum channel, and quantum ghost imaging (QGI) exploits the correlation for enhancing signal-to-noise ratio (SNR) of an image beyond the classical limit. [5] In the original correlation-based quantum-secured imaging, [3] an attack for deceiving the imaging system can be detected by analyzing non-classical correlation, and SNR of an image is enhanced like QGI. After its first proposal, there was experimental demonstration of the quantum-secured ghost imaging in time-frequency domain. [6] The first proposal of quantum-secured imaging was based on prepare-and-measure manner, [4] and in the protocol, a photon exploited for the security check simultaneously contributes to the imaging protocol. However, in the correlationbased quantum-secured imaging, [3,6] the security check and the imaging protocol are sequentially performed, i.e., some photon-pairs are used for the security check, and the others are exploited for the imaging process. Also, for quantum ghost imaging, an electron-multiplying charge-coupled device (EM-CCD) is used to measure position of a single photon. Due to low temporal resolution of an EMCCD (µs), the measurement should be operated in a very low-photon regime to prevent saturation. This means that an imaging system including an EMCCD has limited range of acceptable noise. [7,8] In this article, we propose a quantum-secured single-pixel imaging (QS-SPI) that exploits non-classical correlations of a photon-pair for checking security and imaging simultaneously. Our imaging method is based on single-pixel imaging (SPI), also known as computational ghost imaging (CGI). [9] By adopting CGI rather than QGI, we can exclude an EM-CCD in our setup, and thus, a detection part becomes simpler to only single photon counting module (SPCM) with sub-ns time-resolution. [10][11][12] * [email protected] Our method is not only cost-efficient, but also immune to potential attacks. We show that our QS-SPI is robust against two major attack methods. The first attack is an imaging disrupting attack that illuminates a strong chaotic light on the imaging system to disturb the imaging protocol. In the original SPI, an image of a target cannot be constructed under very strong noise, [12][13][14] however, it was shown that 1000 times stronger chaotic light compared to signal light intensity can be rejected by using time-correlation of photon-pairs. [15][16][17] This heralding photon scheme can be exploited in our QS-SPI as well. The second attack is a deceiving attack that illuminates a beam carrying fake image signal to SPI setup to construct a fraud image. Analyzing correlations of photon-pairs can reveal this deceiving attempt. In our proof-of-principle demonstration, we used polarization and time correlations for checking security. We expect that adopting matured techniques used in quantum secure communication can enhance the security checking performance of QS-SPI.
This article is organized as follows. In Sec. II, we propose the QS-SPI setup and describe its security. Experimental realization will be shown in Sec. III, and finally, it is concluded in Sec. IV.
II. QUANTUM-SECURED SINGLE-PIXEL IMAGING
In SPI, an image is constructed by using a correlation between spatial information of a beam illuminating a target and intensity of the beam measured after interaction with the target. A spatial light modulator (SLM) is used to modulate the spatial profile of the incident beam, and single-pixel detector such as a photodiode is exploited to measure the intensity. A target image can be constructed based on the correlation obtained after several repetitions with various spatial patterns modulated by the SLM. QS-SPI exploits correlations of photon-pairs, so it needs to control or measure signals in a single-photon level. A digital micromirror device (DMD) has higher reflectivity compared to liquid crystal based SLM, thus beneficial for the usage in single-photon regime. Moreover, DMD is independent of the polarization of an incident photon, different from a liquid crystal based SLM. Thus, DMD is exploited for our setup, and the measurement setup consists FIG. 1. A schematic diagram of QS-SPI. Alice who has the imaging system generates polarization entangled state. One photon in the signal mode of the entangled state is sent to an SLM, and only photon that has allowed position is reflected at the SLM. The photon illuminates a target and is measured by SPCMs after its interaction with the target. The other photon in the idler mode is measured by the other SPCMs. Time-correlation and polarization-correlation of the two modes are analyzed from the detections of the SPCMs. of single-photon counting module (SPCM) as a single-pixel detector.
Let us denote k-th spatial pattern as P (k) and its corresponding measured intensity as I (k) . Then, spatial correlation function G for constructing an image is calculated from the following equation:
G(i, j) = P (k) (i, j)I (k) − P (k) (i, j) I (k) ,(1)
where i and j represent the pixel position of 2D image and · denotes averaging for the whole N patterns. [9,18] In QS-SPI, the intensity I (k) is obtained from the coincidence counts of the signal and idler modes. Like heralded SPI, [16] an external noise contribution in I (k) is significantly suppressed due to time-correlation of a photon-pair and narrow time window of SPCMs. Thus, QS-SPI is naturally immune to the imaging disrupting attack that strong chaotic light illuminates an imaging system to saturate the sensor. Fig. 1 shows a schematic diagram of QS-SPI. In our QS-SPI, the polarization entangled state generated by spontaneous parametric down-conversion (SPDC) is exploited for the security check, and the Bell state is written in the following equation:
|Φ + = 1 √ 2 (|H, H SI + |V,V SI ) ,(2)
where |· means polarization state of a single photon, H (V ) denotes horizontal (vertical) polarization, and the subscripts S and I denote signal and idler modes, respectively. The signal photon generated from SPDC is sent to the DMD. On the DMD, the photon is selected by its position, and only the photon which has allowed position is reflected by the DMD and illuminates a target. The idler photon is ideally FIG. 2. A schematic diagram of QS-SPI under a possible attack. An enemy, called Eve, tries to deceive the imaging system. Eve modulates the number of photons induced in an image sensor to make the imaging system constructing a fraud image. Simultaneously, when a photon is sent to the sensor, Eve performs an intercept-and-resend attack for the least disturbance of the polarization of the received photon.
retained and measured to analyze correlations with the signal photon. In the QS-SPI setup, four SPCMs are exploited to measure time-correlation and polarization-correlation.
A. Method of Image Deceiving Attack
Security analysis method of QKD has been wellestablished for protecting photon carrying information against eavesdropping, which is directly related to the generation of secret keys. For example, in polarization-based QKD, polarization-encoded information of a photon is critical to secret key. There are many advanced attacks for extracting polarization-encoded information of successively transmitted photons such as collective attack, which exploits demanding technologies including quantum cloning machines [19,20], quantum memories [21], and collective measurements. [22,23] However, in SPI, the main purpose of an attack is deceiving an imaging system to construct a fake image rather than eavesdropping secret keys. For this purpose, the meaningful attack is modulating intensity (photon number) of the light induced in the image sensor for fake image formulation. Under this circumstance, intercept-and-resend attack is the probable attack strategy for image deceiving attack. [4] Fig. 2 shows a schematic diagram of QS-SPI under a possible attack of an enemy called Eve. It is assumed that Eve can exploit all implementations allowed by the laws of physics and all processes of QS-SPI are known to Eve. For the deceiving attack, Eve possesses time-resolved single-photon detectors with polarization discrimination and an on-demand singlephoton source with polarization control. Eve intercepts Alice's signal photon and discriminate its polarization. Since it is not possible to measure a quantum state in conjugate bases simultaneously, disturbance in original photon state is inevitable. After the polarization measurement, without a delay, the on-demand single-photon source generates a photon with the measured polarization, and the photon is sent to Alice. SPI constructs an image by spatial pattern information and the number of received photons, so Eve should control n g /n m according to the DMD pattern to make QS-SPI construct a fraud image, where n g (n m ) is the number of generated (measured) photons of Eve.
As the signal and idler are polarization entangled state, expected polarization of signal is determined when polarization of idler is measured, and only such combination can be detected ideally. However, intercept-and-resend attack leads to detection of signal photons in unexpected polarization, which is what QS-SPI uses to check its security. Details of the security check will be described in the following section.
B. Security Check in QS-SPI
Presence of Eve is tested by Alice via measuring photons in mutually unbiased bases (MUBs). One basis, named rectilinear basis, consists of horizontal and vertical polarization, and the other basis, diagonal basis, does diagonal (D) and antidiagonal (A) polarization. For the two bases, the following relations are satisfied:
|D = 1 √ 2 (|H + |V ) |A = 1 √ 2 (|H − |V ) ,
and thus, the two bases are MUBs. Alice, who has a QS-SPI system, randomly chooses the measurement basis for the security check. Different from QKD, it is not necessary for the basis choice of signal and idler modes to be independently random, since the measurement setups of the both modes belong to Alice. Let us define r 1 := H, r 2 := V , d 1 := D, and d 2 := A, then the following relations are satisfied:
P(X i , X j ) = C(X i , X j ) ∑ 2 k,l=1 C(X k , X l ) ,(3)
where C(x, y) is the coincidence counts of xand y-polarized photons in the signal and idler modes, respectively, P(x, y) is the probability of the coincidence count to happen, X ∈ {r, d}, and i, j ∈ {1, 2}. From Eq. (2), P(X i , X i ) = 1/2 and P(X i , X j ) = 0 for i = j, indicating that the latter coincidence count is unexpected. Error rate can be defined by the ratio of unexpected coincidence count to all coincidence counts. Since idler photon is unhindered by Eve's attack, an error rate is defined with respect to polarization of idler. Thus, a polarization error rate when an idler photon is detected on the SPCM corresponding to the polarization X i is written as following:
e X i = C(X j , X i ) ∑ 2 k=1 C(X k , X i ) ,(4)
where i = j.
In ideal, the error rates are always zero. However, if there is an enemy who tries to disturb the imaging system, the probabilities are affected by enemy's attack, so Alice can notice the presence of Eve by analyzing the error rates.
Eve possesses its own MUBs for polarization measurement. Let us denote its constitutive polarizations in primed notation, i.e., H , V , D , and A . For the same choice of measurement basis of Alice and Eve, let the angle difference between the two as θ , measured in counterclockwise from one polarization of the Alice to that of the Eve, i.e., angle measured from H-polarization to H -polarization in counterclockwise. Then, the angle difference between different bases is θ ± π 4 . If idler photon is measured as H-polarization in the rectilinear basis by Alice, then Alice's rectilinear polarization measurement on the signal photon always gives H-polarization. If Eve chooses its own rectilinear basis, Eve's setup measures H -polarization and V -polarization with probabilities cos 2 θ and sin 2 θ , respectively. Regardless of Eve's result, Alice's error rate, i.e., detection of V -polarized signal, is cos 2 θ sin 2 θ . Thus, the error rate observed by Alice is following:
2 cos 2 θ sin 2 θ = 1 − cos 2 2θ 2 .(5)
If Eve chooses the other measurement basis, the error rate is calculated by replacing θ to θ ± π 4 :
1 − sin 2 2θ 2 .
Since Eve's basis choice is random, the H-polarization error rate is calculated as the following equation:
e H = 1 2 1 − cos 2 2θ 2 + 1 − sin 2 2θ 2 = 1 4 .(7)
The result indicates that the criterion error rate for determining the presence of an attack is 25% regardless of the angle difference θ . If the error rate is less (greater) than 25%, the protocol is reliable (compromised).
III. PROOF-OF-PRINCIPLE DEMONSTRATION
A. QS-SPI setup Fig. 3 shows setups for proof-of-principle demonstration of QS-SPI. A polarization-entangled state is generated from the Sagnac interferometer with periodically poled potassium titanyl phosphate (ppKTP) crystal. [24] The crystal is pumped by 405 nm continuous wave (CW) laser, generating 810 nm polarization-entangled photon-pairs via type-II SPDC process. The initial state generated from the Sagnac interferometer is |Ψ + = 1 √ 2 (|H,V SI + |V, H SI ), so to make the state be |Φ + , additional phase shifts on the idler mode is given. Fig. 4 shows the results of quantum state tomography [25] of the generated state. The figure shows that |Φ + is well-prepared with its fidelity 98.6%.
FIG. 3. Experimental setups of our QS-SPI.
A polarization-entangled photon-pair is generated by the Sagnac interferometer with ppKTP crystal, and polarization of its idler photon is directly detected by SPCMs. The signal photon is reflected by the DMD with post-selection of its position and sent to the target, an alphabet letter "A". After interaction with the target, the photon is counted by SPCMs in selected polarization. Eve's attack is demonstrated by blocking Alice's signal photon and sending polarization-controlled laser beam with intensity modulation. Accidental coincidence counts in Alice's TCSPC is occurred by Eve's light, so the fraud image, an alphabet letter "D", is constructed in Alice's system. PBS: polarizing beam splitter; QWP (HWP): quarter (half) wave plate; ND filter: neutral density filter. After the generation, the idler mode is detected by SPCMs (Excelitas Technologies, SPCM-780-13-FC) in selected polarization. SPCMs are connected to a time-correlated single photon counting (TCSPC) module to record the photon counts with detected time and polarization. The signal photon is sent to the DMD (Vialux GmbH, DLP650LNIR), and the photon is post-selected by a displayed pattern on the DMD. The DMD displays the Hadamard patterns for enhancing image quality with restricted number of shots. [26][27][28][29]
A 2 n+1 × 2 n+1
Hadamard matrix is calculated by the following equation:
H 2 n+1 = H 2 n ⊗ H 2 ,(8)
where
H 2 = 1 1 1 −1 ,(9)
and ⊗ denotes tensor product. Hadamard patterns are generated by reshaping each row of the Hadamard matrix H 2 2n into a 2 n × 2 n square matrix. Since the negative pixel value cannot be displayed, two shots are necessary to represent a Hadamard pattern. [18] The first intensity pattern has bright and dark pixels for the matrix elements 1 and -1, respectively, and the other one is its inverse. The resolution of our Hadamard patterns is 32 × 32, so the total number of shots becomes 2048. The spatially post-selected signal photons interact with the target, an alphabet letter "A", and the number of transmitted photons is counted by SPCMs. The four SPCMs are connected to a TCSPC module, and coincidence counts of one SPCM in the signal mode and the other SPCM in the idler mode are analyzed by the TCSPC to construct an image and analyze security. The other coincidence counts such as that of SPCMs in the same mode or of multiple SPCMs are discarded.
In the setup, power of pump laser for generation of the entangled photon-pairs was 5 mW. Single count rates of the signal and idler were 6 × 10 3 cps and 8 × 10 4 cps, respectively, when there is no target. Under the same condition, we set the coincidence window as 650 ps, and coincidence count rate of the signal and the idler in the same polarization was 300 cps. The accumulation time of one Hadamard pattern was 3.5 seconds.
B. Demonstration of Eve's attack As previously described, Eve's intercept-and-resend attack exploits on-demand single-photon generator to make a generated photon enter within the coincidence window. However, since the implementation is not feasible with current technologies, we demonstrate a realistic attack with implementable devices.
Since QS-SPI constructs an image based on coincidence counts information, Eve needs to control the coincidence count for deceiving the system. In our deceiving attack, Eve's 810 nm CW laser illuminates Alice's receiver to make accidental coincidence counts occur. For the accidental coincidence counts to be dominant, Eve blocks Alice's signal mode. Instead of polarization control based on the measured information, we fix the polarization of the illumination laser as H , which is set to have approximately 0.10 radian angle difference with H to show the effect of Eve's misaligned bases. This demonstration provides a simulation of the interceptand-resend attack, since the statistics of Alice's polarization measurement in the diagonal basis is similar to those of the intercept-and-resend attack when Alice's and Eve's bases are different.
To deceive Alice's setup, accidental coincidence count rate needs to be similar to the coincidence count rate of the entangled photon-pairs. To achieve this condition, the power of Eve's laser is determined as follows. The detection probabilities per window on the signal and idler mode SPCMs are n E τ and n I τ, respectively, where n E is the mean photon number of Eve's laser and τ is a coincidence window. In this case, the coincidence probability in the coincidence window is given by the product of the single probabilities, n E n I τ 2 . Then, the accidental coincidence count rate n acc can be calculated from the following equation:
n acc = n I n E τ.(10)
To make n acc ∼ 300 cps, the coincidence count rate of the entangled photon-pairs, with n I = 8 × 10 4 cps and τ = 650 ps, we obtain n E ∼ 5.8 × 10 6 cps, which is 1000 times greater than the original signal photon count rate without a target. This photon number corresponds approximately to the 1.41 pW for an 810 nm CW laser. Intensity modulation of Eve's laser, which is necessary to deceive SPI, is performed by using another DMD. The Eve's DMD displays the overlapped patterns between Alice's Hadamard patterns and a fraud image, directly. Then at the end of the protocol, the fraud image, an alphabet letter "D", is constructed by the QS-SPI setup from the accidental coincidence counts induced by Eve.
C. Results
Fig . 5 shows the images obtained under the imaging disrupting attack by the original SPI (left) and by our QS-SPI (right). Eve's jamming laser, of which the power is 1000 times greater than Alice's signal, illuminates the receiver to disturb Alice's imaging process. In the original SPI, only single counts in the signal mode is exploited to construct an image, and thus, the obtained image is ruined by the attack. However, due to time-correlation between the signal and idler photons, QS-SPI can overcome the attack, i.e., the target image is successfully obtained. [16] Fig . 6 shows the obtained images with our QS-SPI setup without an attack (left) and under Eve's attack (right). In both cases, the images are well-constructed, so Alice cannot notice the presence of an attack from the obtained images. Therefore, to check the security, we analyze average of the four polarization errors given in Eq. (4). When there is no attack, the average error rate is suppressed below 5%, however, under Eve's attack, the error rate becomes nearly 50%. Fig. 7 shows the images and corresponding polarization error rates with their standard deviations. When there is no Eve, all the error rates are suppressed below 5%, so the security of the obtained images is guaranteed. Under Eve's attack, since Eve's laser has H -polarization, H-polarization error rate is below 5% but V -polarization error rate exceeds 95%. In the diagonal basis, D-polarization error rate is nearly 40%, and A-polarization error rate exceeds 60%. The both error rates in the diagonal basis are larger than 25% in Eq. (7), so Alice can notice that the obtained image is fake.
In theory, D-polarization and A-polarization error rates with the H -polarized light are given by sin 2 (π/4 − θ ) ≈ 0.40 and cos 2 (π/4 − θ ) ≈ 0.60, respectively, and these values are wellmatched with our experimental results.
Note that our attack scenario does not directly simulate the intercept-and-resend attack. There is no basis choice in our setups in Eve's side, however, in the intercept-and-resend attack, there is a random basis choice step. Since Eve's correct basis choice does not induce an error in the polarization statistics, the probability of mismatched basis choice, 1/2, should be considered in the intercept-and-resend attack. Thus, to simulate the intercept-and-resend attack, our error rates in the diagonal basis would be considered to halve. Even we consider these differences, A-polarization error is still larger than 25%, so we can expect that Eve's intercept-and-resend attack also can be noticed with our QS-SPI.
Moreover, in the intercept-and-resend attack, Eve sends both polarizations in the mismatched basis, in this case, Hand V -polarization, however, in our demonstration, only the H -polarized laser illuminates Alice's receiver. If we modify the setup including a random choice of H -and V -polarized laser, then both error rates in the diagonal basis would become similar.
IV. SUMMARY AND DISCUSSION
In this article, we provide a methodology of quantumsecured single-pixel imaging (QS-SPI) against jamming attacks.
By exploiting time-correlation and polarizationcorrelation, all detected photon-pairs are used for imaging and security checking, simultaneously. QS-SPI is naturally immune to an imaging disrupting attack, an enemy illuminating sensor with strong chaotic light, since it is based on the heralded SPI scheme. [16] A deceiving attack, such as an intercept-and-resend attack can be detected from polarization statistics, since an attack induces errors in the statistics. We demonstrated QS-SPI setup and deceiving attack as well. The attack is able to completely deceiving SPI, however, we showed that QS-SPI can detect the attack by analyzing the statistics. We presented that an intercept-and-resend attack can be simulated from the results of our demonstrated deceiving attack and that QS-SPI can detect the intercept-and-resend attack as well.
To use QS-SPI as an application, implementation of an active random basis choice is very hard. Thus, our active basis choice setups and two SPCMs can be modified to pas-sive ones consisting of a 50:50 beam splitter, phase shifters, and four SPCMs corresponding to four polarization detection. In the modified scheme, the photon-pair detected at SPCMs with mismatched basis should be discarded under our security check. However, if we introduce a security check based on the Bell inequality, [30] especially the Clauser-Horne-Shimony-Holt (CHSH) inequality, [31] then all bases combinations are exploited to check a security [32] and construct an image without discarding photon-pairs. In the security check, the absence of an attack is guaranteed if the polarization statistics violates the Bell inequality, and this security provides a deviceindependent security. [23,33,34] Since our setup exploited time-correlation of signal and idler photons, time-of-flight information of a signal photon should be measured. Therefore, QS-SPI naturally includes a quantum-secured optical ranging protocol, [4] i.e., QS-SPI provides a method to securely acquire a target distance against jamming attacks.
We expect that QS-SPI can be developed with matured techniques exploited in quantum secure communication. For example, six polarization states in three possible MUBs can be used to enhance security [35] or for referenceframe-independent security analysis [36] and various degreesof-freedoms in a single photon can be used to exploit high-dimensional quantum states [37][38][39][40] or hyper-entangled states. [41,42] It is expected that our method can be developed to quantum-secured LiDAR with quantum-correlation-based free-space experiment techniques as well. [43] FIG. 7. Images obtained by QS-SPI with individual idler polarization. The error rates are given under the images with its standard deviations. (a) When there is no attack, all error rates are suppressed below 5%. (b) Under the attack, since Eve's laser has H -polarization, H-polarization error rate is below 5% but V -polarization error rate exceeds 95%. In the diagonal basis, D-polarization error rate is nearly 40%, and Apolarization error rate exceeds 60%.
Writing -review & editing (supporting). Duk Y. Kim: Funding acquisition (equal); Methodology (supporting); Project administration (supporting); Writing -review & editing (supporting). Zaeill Kim: Funding acquisition (equal); Methodology (supporting); Project administration (equal); Supervision (supporting); Writing -review & editing (supporting). Yonggi Jo: Conceptualization (lead); Data curation (supporting); Formal analysis (equal); Funding acquisition (supporting); Investigation (supporting); Methodology (lead); Project administration (equal); Supervision (lead); Validation (equal); Visualization (equal); Writing -original draft (supporting); Writing -review & editing (equal).
DATA AVAILABILITY
The datasets generated and/or analyzed during the current study are not publicly available due to the security policy of the Ministry of National Defense of South Korea but are available from the corresponding author upon reasonable request.
FIG. 4 .
4The result of quantum state tomography of our source: (a) the real values of the density operator; (b) the imaginary values. These graphs indicate that |Φ + is well-prepared, and its fidelity is 98.6%.
FIG. 5 .
5Images constructed under the imaging disrupting attack by the original SPI (left) and our QS-SPI (right). In the attack, Eve's laser, of which the power is 1000 times larger than the original signal, illuminates Alice's receiver to disturb the imaging process. The image obtained by SPI is ruined by the laser, however, due to the time-correlation of a photon-pair, the target image is successfully constructed by QS-SPI.FIG. 6. Images constructed by our QS-SPI setup without an attack (left) and under Eve's attack (right). Averages of the four polarization error rates are given under the images. Alice cannot notice the presence of the attack without the error rates analysis.
ACKNOWLEDGMENTSThis work was supported by the Agency for Defense Development Grant funded by the Korean Government.AUTHOR DECLARATIONSConflict of InterestThe authors have no conflicts to disclose.
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|
[] |
[
"Uncertainty aware multimodal activity recognition with Bayesian inference",
"Uncertainty aware multimodal activity recognition with Bayesian inference"
] |
[
"Mahesh Subedar \nIntel Labs\n\n",
"Ranganath Krishnan \nIntel Labs\n\n",
"Paulo Lopez \nIntel Labs\n\n",
"Meyer Omesh Tickoo \nIntel Labs\n\n",
"Jonathan Huang \nIntel Labs\n\n"
] |
[
"Intel Labs\n",
"Intel Labs\n",
"Intel Labs\n",
"Intel Labs\n",
"Intel Labs\n"
] |
[] |
Deep neural networks (DNNs) provide state-of-the-art results for a multitude of applications, but the use of DNNs for multimodal audiovisual applications is still an unsolved problem. The current approaches that combine audiovisual information do not consider inherent uncertainty or leverage true classification confidence associated with each modality in the final decision. Our contribution in this work is to apply Bayesian variational inference to DNNs for audiovisual activity recognition and quantify model uncertainty along with principled confidence. We propose a novel approach that combines deterministic and variational layers to estimate model uncertainty and principled confidence. Our experiments with in-and out-of-distribution samples selected from a subset of the Moments-in-Time (MiT) dataset show more reliable confidence measure as compared to the non-Bayesian baseline. We also demonstrate the uncertainty estimates obtained from this framework can identify out-of-distribution data on the UCF101 and MiT datasets. In the multimodal setting, the proposed framework improved precision-recall AUC by 14.4% on the subset of MiT dataset as compared to non-Bayesian baseline.
| null |
[
"https://arxiv.org/pdf/1811.10811v1.pdf"
] | 53,744,781 |
1811.10811
|
1cca57d2532a4085fb357c10237bdded69541310
|
Uncertainty aware multimodal activity recognition with Bayesian inference
Mahesh Subedar
Intel Labs
Ranganath Krishnan
Intel Labs
Paulo Lopez
Intel Labs
Meyer Omesh Tickoo
Intel Labs
Jonathan Huang
Intel Labs
Uncertainty aware multimodal activity recognition with Bayesian inference
Deep neural networks (DNNs) provide state-of-the-art results for a multitude of applications, but the use of DNNs for multimodal audiovisual applications is still an unsolved problem. The current approaches that combine audiovisual information do not consider inherent uncertainty or leverage true classification confidence associated with each modality in the final decision. Our contribution in this work is to apply Bayesian variational inference to DNNs for audiovisual activity recognition and quantify model uncertainty along with principled confidence. We propose a novel approach that combines deterministic and variational layers to estimate model uncertainty and principled confidence. Our experiments with in-and out-of-distribution samples selected from a subset of the Moments-in-Time (MiT) dataset show more reliable confidence measure as compared to the non-Bayesian baseline. We also demonstrate the uncertainty estimates obtained from this framework can identify out-of-distribution data on the UCF101 and MiT datasets. In the multimodal setting, the proposed framework improved precision-recall AUC by 14.4% on the subset of MiT dataset as compared to non-Bayesian baseline.
Introduction
Audio and vision are complementary inputs and fusing the audiovisual modalities can greatly benefit an activity recognition application. Multimodal audiovisual activity recognition is still an unsolved problem, and deep neural network (DNN) architectures are not successful in modeling the inherent ambiguity in the correlation between the two modalities. Our evaluation of audiovisual inputs indicate that both audio and visual modalities may not be correlated for all the activity categories. One of the modalities (e.g., sneezing in audio, writing in vision) can be more certain about the activity class than the other modality. It is important to model reliable uncertainty estimates for the individual modalities to benefit from multimodal fusion.
Deep learning utilizes complex models that are trained with large datasets [1,2,3] and have proven to be scalable and successful in solving many perception tasks providing state-of-the-art results. However, DNNs are trained to obtain the maximum likelihood estimates and disregard uncertainty around the model parameters that eventually can lead to predictive uncertainty. Deep learning models may fail in the case of noisy or out-of-distribution data, leading to overconfident decisions that could be erroneous as SoftMax probability does not capture overall model confidence. Instead, it represents relative probability that an input is from a particular class compared to the other classes.
Probabilistic Bayesian models provide principled ways to gain insight about data and capture reliable uncertainty estimates in predictions. Bayesian deep learning [4,5] has allowed bridging DNNs and probabilistic Bayesian theory to leverage the strengths of both methodologies. Bayesian deep learning framework with Monte Carlo (MC) dropout approximate inference [6] has been used in visual scene understanding applications including camera relocalization [7], semantic segmentation [8] and depth regression [9]. In this work, we propose a Bayesian deep learning framework applied to a multimodal activity recognition task using the variational inference [10,11] technique. Activity recognition is an active area of research with multiple approaches depending on the application domain and the types of sensors [12]. Human activity recognition using wearable sensors such as accelerometer/gyroscopes and heart-rate monitors is used to recognize everyday human activities that include walking, running, and swimming. Human pose-based activity recognition [13,14] methods aggregate motion and appearance information along tracks of human body parts to recognize human activity. Multimodal methods which combine optical flow or depth information along with RGB data [15,16] are shown to provide state-of-the-art results for generic (not just hu- Figure 1: Bayesian neural network man) activity recognition tasks. Methods which combine semantic level information [17] such as pose, object/scene context and other attributes including linguistic descriptors are proposed to detect group activities and shown to be more robust to sensor noise.
Multimodal models are proposed for audiovisual analysis tasks such as emotion recognition [18], audiovisual speech recognition [19], speech localization [20,21], cross-modal retrieval [22]. A deep Boltzmann machine (DBM) [23] based architecture has been used to learn a generative model for the joint image and text space that is useful in the information retrieval task for both unimodal and multimodal queries. The audiovisual speech recognition (AVSR) task is shown to benefit from multimodal training of the joint models. In [19], a deep autoencoder model for cross-modality feature learning is proposed, where better features for one modality can be learned if multiple modalities are present at training time. A deep audio-visual speech recognition model [24] using self-attention encoder architecture is proposed to recognize speech from talking faces using vision and audio inputs. Recent work on sound localization and separation [20,21] has shown the benefits of a joint audiovisual representation for cross-modal selfsupervised learning using only audio-visual correspondence as the objective function. These audiovisual methods apply joint modeling of the audio and vision inputs during the training phase for better generalizability of the models, but then use single modality during the inference phase. Also, these methods do not provide a quantifiable means to determine the relative importance of each modality.
In this work, we focus on multimodal audiovisual activity recognition and use Bayesian DNNs to reliably estimate uncertainty associated with the individual modalities. We propose an architecture which combines state-of-theart DNN architectures with Bayesian variational layers to obtain reliable uncertainty estimates along with principled confidence. The proposed architecture can be extended to train an end-to-end model which can use the uncertainty es-timates from individual modalities to gate the contribution from an uncertain input modality towards the classification results. To the best of our knowledge, this is the first research effort that applies a Bayesian deep learning framework with variational inference for a multimodal activity recognition task to capture reliable uncertainty measures.
Our main contributions include:
1. A Bayesian variational inference model applied to the multimodal framework. Specifically, we focus on the audiovisual activity recognition task to capture principled confidence and uncertainty estimates.
2.
A Bayesian DNN architecture which combines deterministic and variational layers modeled with meanfield distributions.
3. Uncertainty estimates obtained from the proposed method to identify out-of-distribution data for activity recognition.
4. Audiovisual activity recognition on subset of MiT dataset using our proposed architecture which outperforms the non-Bayesian baseline.
The rest of the document is divided into the following sections. The background on Bayesian DNNs and audiovisual activity recognition are presented in Section 2. In Section 3, the architecture of the proposed Bayesian variational inference framework is presented. The results are presented in Section 4, followed by conclusions in Section 5.
Background
Bayesian deep neural networks
Bayesian DNNs provide a probabilistic interpretation of deep learning models by placing distributions over the model parameters (shown in Figure 1). Bayesian Inference can be applied to estimate the predictive distribution by propagating over the model likelihood while marginalizing over the learned posterior parameter distribution. Bayesian DNNs also help in regularization by introducing distribution over network weights, capturing the posterior uncertainty around the neural network parameters. This allows transferring inherent DNN uncertainty from the parameter space to the predictive uncertainty.
Given training dataset D = {x, y} with inputs x = x 1 , ..., x N and their corresponding outputs y = y 1 , ..., y N , in parametric Bayesian settings we would like to infer a distribution over weights w as a function y = f w (x) that represents the DNN model. With the posterior for model parameters inferred during Bayesian neural network training, we can predict the output for a new data point by propagating over the model likelihood p(y|x, w) while drawing samples from the learned parameter posterior p(w|D). Equation 1
shows the posterior distribution of model parameters obtained from model likelihood.
p(w, D) = p(y | x, w) p(w) p(y | x)(1)
Computing the posterior distribution p(w|D) is often intractable, some of the previously proposed techniques to achieve an analytically tractable inference include: (i) Markov chain Monte Carlo sampling based probabilistic inference [4,25] (ii) variational inference techniques to infer the tractable approximate posterior distribution around model parameters [26,27,28] and (iii) Monte Carlo dropout approximate inference [6]. In our work, we use the variational inference approach to infer the approximate posterior distribution around the model parameters.
Variational inference [11] is an active area of research in Bayesian deep learning, which uses gradient based optimization. This technique approximates a complex probability distribution p(w | D) with a simpler distribution q θ (w), parameterized by variational parameters θ while minimizing the Kullback-Leibler (KL) divergence [29]. Minimizing the KL divergence is equivalent to maximizing the log evidence lower bound [6,29].
L := q θ (w) log p(y | x, w) dw − KL[q θ (w) || p(w)](2)
Predictive distribution is obtained through multiple stochastic forward passes through the network during the prediction phase while sampling from the posterior distribution of network parameters through Monte Carlo estimators. Equation 3 shows the predictive distribution of the output y * given new input x * :
p(y * | x * , D) = p(y * | x * , w) q θ (w)dw p(y * | x * , D) ≈ 1 T T i=1 p(y * | x * , w i ) , w i ∼ q θ (w)(3)
where, T is number of Monte Carlo samples. We evaluate the model uncertainty using Bayesian active learning by disagreement (BALD) [30] for the activity recognition task. BALD quantifies mutual information between parameter posterior distribution and predictive distribution, which captures model uncertainty, as shown in Equation 4.
BALD := H(y * | x * , D) − E p(w | D) [H(y * | x * , w)] (4)
where, H(y * | x * , D) is the predictive entropy given by:
H(y * | x * , D) = − K−1 i=0 p iµ * log p iµ(5)
and p iµ is predictive mean probability of i th class from T Monte Carlo samples.
Audiovisual Activity Recognition
Vision and audio are the ubiquitous sensor inputs which are complementary in nature and have different representations. Audiovisual methods apply joint modeling of the audio and vision inputs [19,24] to achieve higher accuracies for complex tasks such as action recognition.
Vision-based activity recognition techniques apply a combination of spatiotemporal models to capture pixellevel information and temporal dynamics of the scene. In recent years, visual activity recognition models often use ConvNets-based models for spatial feature extraction. The image-based models [31,32] are pre-trained on ImageNet dataset to represent the spatial features. The temporal dynamics for activity recognition is typically modeled either by using a separate temporal sequence modeling such as variants of RNNs [33,34] or by applying 3D Con-vNets [35], which extend 2D ConvNets to the temporal dimension.
Following the successes of ConvNets on vision tasks, they are shown to provide state-of-the-art results for audio classification as well. Many of the top performing methods from recent audio classification challenges [36,37] use DNN architectures [38,39,40] with convolutional layers. In [41], a model similar to the VGG architecture (VGGish model) from the vision domain was trained using log-Mel spectrogram features on the Audio Set [42] dataset. Audio Set contains over one million Youtube video samples labeled with a vocabulary of acoustic events.
In this work, we focus on audiovisual activity recognition using DNNs on the trimmed video samples. The 3D-ConvNet (C3D) architecture [43] is shown to provide generic spatiotemporal representation for multiple vision tasks. We use a variant of 3D-ConvNet ResNet-101 C3D [44] architecture for the visual representation. We use VGGish architecture [41] for audio representation, which is shown to provide generic features for audio classification tasks.
Bayesian Multimodal DNN Architecture
We present a Bayesian deep learning framework for audiovisual activity recognition to obtain principled confidence and capture predictive uncertainty. Bayesian DNN models provide an uncertainty measure which is valuable in multimodal setup to fuse different modalities. The block diagram of the proposed audiovisual activity recognition using Bayesian variational inference is shown in Figure 2. Recent approaches for audiovisual analysis tasks use DNN architectures to represent the vision and audio features. Likewise, we use the ResNet-101 C3D and VGGish architectures for visual and audio modalities, respectively.
For the Bayesian DNN multimodal framework, we replace the final fully connected layer for both visual and audio DNN models with three fully connected variational layers followed by the categorical distribution (shown in Figure 2). The weights and bias parameters in the fully connected variational layers are modeled through meanfield normal distribution, and the network is trained using Bayesian variational inference based on KL divergence [27,28]. We use Flipout [45], which is an efficient method that correlates the gradients within a mini-batch by implicitly sampling pseudo-independent weight perturbations for each input.
Bayesian DNN maintains a probability distribution for every parameter, which can be complex to scale for deeper models as they are compute and memory intensive. In [46], it is shown that applying approximate Bayesian inference with Monte Carlo dropout to final few layers can be effective in estimating the model uncertainty. In the proposed framework, during prediction we perform multiple forward passes for the final variational layers and the remaining deterministic layers require only one forward pass.
For the comparison with the non-Bayesian baseline, we maintain the same model depth as the Bayesian DNN model and use three deterministic fully connected final layers for the non-Bayesian DNN model. The dropout layer is used after every fully connected layer to avoid over-fitting of the model. In the rest of the document, we refer the non-Bayesian DNN model as simply the DNN model. In the following section, we present the results from our experiments showing the effectiveness of Bayesian DNN over conventional DNN models.
Results
We analyze the model performance on the Moments-in-Time (MiT) [3] dataset. The MiT dataset consists of 339 classes, and each video clip is 3 secs (˜90 frames) in length. In this work, we considered a subset of 54 classes as indistribution and another 54 classes as out of distribution samples which include audio samples. In order to check whether DNNs can provide a reliable confidence measure, the subset of 54 classes for each category are selected after subjective evaluation to confirm the activities fall into two distinct distribution of classes.This will allow the comparison of confidence measures between DNN and Bayesian DNN models for in-and out-of-distribution classes, and the uncertainty estimates for the Bayesian DNN models (as the DNN model does not provide uncertainty estimates).
The ResNet-101 C3D DNN model is initialized with pretrained weights for the Kinetics dataset [1]. This model is then optimized with transfer learning by training the final fourteen layers. The VGGish model is initialized with pretrained weights for the Audio set [42] dataset. This model is then optimized with transfer learning by training the final five layers. We used stochastic gradient descent (SGD) optimizer with an initial learning rate of 0.0001 and momentum factor of 0.9 along with rate decay when the loss is plateaued.
We trained the ResNet101-C3D vision and VGGish audio architectures using the in-distribution class samples, which include˜150K training and˜5.3K validation data. We select individual vision and audio paths from the model Figure 3 shows the comparison of precision-recall (top) and ROC (bottom) plots using the confidence measures for DNN and Bayesian DNN models. It is observed from the plots that Bayesian DNN model consistently outperforms the DNN model for the individual modalities and also for the combined audiovisual modalities. The Precision-Recall
Confidence measure
In this section, we compare the confidence measure obtained from the DNN and Bayesian DNN models. The confidence measure for the conventional DNN is the SoftMax probabilities used for the predictions. The mean of the categorical predictive distribution obtained from Monte Carlo sampling provides the confidence measure for Bayesian DNNs. The density histograms for the confidence measure are plotted in Figure 4. The density histogram is a histogram with area normalized to one. The height (y-axis) of density histograms indicate the distribution of confidence measure. A distribution skewed towards the right (near 1.0 on x-axis) indicates the model has higher confidence in the predictions and the distributions skewed towards left indicate lower confidence. In Figure 4 (a) & (b), the density histograms for the DNN and Bayesian DNN vision models are presented, respectively. For true (correct) predictions both DNN and Bayesian DNN models show confidence measure density histograms peaked near 1.0, indicating higher confidence in the predictions. In the case of false (incorrect) predictions, the DNN model still shows confidence measure density histograms peaked near 1.0. On the contrary, the Bayesian DNN model shows confidence measure density histogram skewed towards lower values, indicating the reliability in the predictions.
In density histograms for the audio classification results (shown in Figure 4), the DNN model for false predictions shows a peak near higher confidence value, whereas the Bayesian DNN model shows overall lower confidence. The results from Bayesian DNN model does not show a strong peak for the true predictions, which may be attributed to lower accuracies which imply lower confidence in the model predictions.
In the case of audiovisual inputs (shown in Figure 4), the density histogram plots for true predictions indicate DNN and Bayesian DNN models peak near higher confidence values. But in the case of false predictions, DNN model confidence histograms incorrectly represents higher values while the Bayesian DNN model indicates overall lower confidence values.
We compare the confidence measure for audiovisual inputs obtained using in-and out-of-distribution classes for the subset of MiT dataset. The confidence measure density histogram plots shown in Figure 5 (a) indicate the DNN model incorrectly estimates higher confidence for out-ofdistribution classes and a peak is observed near higher values. The Bayesian DNN model (shown in Figure 5 (b)) indicates a lower confidence for out-of-distribution classes and is skewed towards lower values. This signifies Bayesian DNNs are being transparent in their predictions.
These results confirm that the proposed Bayesian DNN model provides more reliable confidence measure for false predictions and out-of-distribution samples, and is applicable to multimodal audiovisual settings.
Uncertainty measure
Bayesian DNN models capture uncertainty quantification which is beneficial to identify out-of-distribution samples. Out-of-distribution samples are data points which fall far off from the training data distribution. In this section, we compare BALD uncertainty measure (details are in Section 2) using in-and out-of-distribution samples from MiT dataset. Additionally, we compare the uncertainty estimates using UCF101 [47] visual action recognition dataset and use MiT dataset as out-of-distribution samples.
In Figure 6, the density histogram of BALD uncertainty measure for the Bayesian DNN model is presented. The in- and out-of-distribution classes are selected from subset of MiT dataset. For in-distribution samples, the uncertainty estimates are skewed towards lower values. In the case of outof-distribution samples, the density histograms are skewed towards higher uncertainty values. This indicates the BALD measure is able to capture inherent model uncertainty for the out-of-distribution classes which were not seen during the training step. We use the UCF101 visual activity recognition dataset, which has 101 activity classes to compare with MiT dataset (vision input) as the out-of-distribution samples. The training of the UCF101 dataset for vision input is done similar to the details provided in Section 3. The DNN and Bayesian DNN accuracy for the UCF101 dataset is given in Table 2, which is comparable to other results obtained for UCF101 using ResNet-101 C3D model [44].
The comparison of uncertainty measures for UCF101 dataset as in-distribution samples and the MiT dataset as out-of-distribution samples is shown in Figure 7. BALD uncertainty measure from the plots indicate a clear separation of in-and out-of-distribution samples. This validates the benefit of Bayesian DNN model which has the potential to identify out-of-distribution samples.
Conclusions
Effective multimodal activity recognition requires the underlying system to intelligently decide the relative importance of each modality. Bayesian inference provides a systematic way to quantify uncertainty in the deep learning model predictions. Uncertainty estimates obtained from Bayesian DNNs can identify inherent ambiguity in individual modalities, which in turn can benefit multimodal fusion. In this work, we propose a Bayesian DNN architecture that combines deterministic and variational layers applied to multimodal settings. We evaluate the proposed approach on audiovisual activity recognition using Moments-in-Time dataset. The results indicate Bayesian DNN architecture can provide more reliable confidence measure compared to the conventional DNNs. The uncertainty estimates obtained from the proposed method have the potential to identify outof-distribution data.
The proposed Bayesian deep learning architecture can be extended to other multimodal applications. We envision extending this architecture through a principled Bayesian multimodal fusion framework that optimizes the loss function weighted by the uncertainty estimates from each modality.
Figure 2 :
2Bayesian audiovisual activity recognition: ResNet-101 C3D and VGGish DNN architectures are used to represent vision and audio information, respectively. The final layer of the DNN is replaced with three fully connected variational layers followed by categorical distribution. The Bayesian inference is applied to the variational layers through Monte Carlo sampling on the posterior of model parameters, which provides the predictive distribution.
Figure 3 :
3Precision-Recall (top) and ROC (bottom) AUC plots micro-averaged over all the MiT in-distribution classes. The audiovisual Bayesian DNN model shows an improvement of 14.4% Precision-Recall AUC and 2.7% ROC AUC over the audiovisual DNN model. AUC plot for the audiovisual Bayesian-DNN model shows an improvement of 14.4% over the audiovisual DNN model and an improvement of 9.5% over the vision only Bayesian DNN model. The ROC plot for audiovisual Bayesian-DNN model shows an improvement of 2.7% over the audiovisual DNN model.
Figure 4 :Figure 5 :
45Density histogram of confidence measures for DNN and Bayesian DNN models: A distribution skewed towards right (near 1.0 on x-axis) indicates the model has higher confidence in predictions than the distribution skewed towards left. [The density histogram is a histogram with area normalized to one. Plots are overlaid with kernel density curves for better readability.] Density histogram of confidence measures obtained from in-and out-of-distribution classes for subset of MiT dataset. DNN model indicates high confidence for both the categories (peaked to the right or higher values) vs. Bayesian DNN model indicates lower confidence for out-of-distribution classes (skewed to the left or lower values) while higher values for in-distribution classes.
Figure 6 :
6Density histogram of BALD uncertainty measure obtained from Bayesian DNN model. In-and out-ofdistribution samples are from the subset of MiT dataset.
Figure 7 :
7Density histogram for BALD uncertainty measure obtained from Bayesian DNN model. In-distribution samples are from the UCF101 activity recognition dataset and out-of-distribution are from the MiT dataset.
The classification accuracy for MiT in-distribution samples is presented inTable 1. Bayesian DNN model consistently provides higher accuracies for individual and combined audio-vision modalities. The Bayesian DNN audiovisual model provides top-1 accuracy improvement of 10.1% over the Bayesian DNN vision model and 3.6% over the DNN audiovisual model results. Similarly, the audiovisual Bayesian DNN model provides top-5 accuracy improvement of 3.4% over the Bayesian DNN vision model and 5.8% over the DNN audiovisual model results. The slight decrease in the accuracy of DNN audiovisual model over the vision only model can be attributed to the overconfident SoftMax probabilities between vision and audio modalities which after mean-pooling results in lower accuracy values.Model
Top1 (%) Top5 (%)
DNN: Vision
52.65
79.79
Bayesian DNN: Vision
53.3
81.20
DNN: Audio
34.13
61.68
Bayesian DNN: Audio
35.80
63.40
DNN: Audiovisual
56.61
79.39
Bayesian DNN: Audiovisual
58.68
84.00
Table 1: Accuracies for MiT activity recognition dataset
(In-distribution classes): Comparison of accuracies for
Bayesian DNN and DNN models for audio, vision and
audiovisual activity recognition is presented. Audiovisual
Bayesian DNN model shows an improvement of 3.6% top1
and 5.8% top5 accuracy improvement over the audiovisual
DNN model.
shown in Figure 2 to obtain single modality results. In the
case of Bayesian DNN model, we perform multiple stochas-
tic forward passes on the final three fully connected vari-
ational layers with Monte Carlo sampling on the weight
and bias posterior distributions. In our experiments, 40 for-
ward passes provide reliable estimates above which the fi-
nal results are not affected. For the audiovisual results, we
consider mean-pooling of the audio-vision confidence mea-
sure estimates obtained from predictive distribution through
Monte Carlo sampling for Bayesian DNN and the mean-
pooling of SoftMax outputs for DNN model. In this work,
our baseline model is a conventional DNN model which is
compared with the proposed Bayesian DNN model. The
proposed framework can be applied to other DNN architec-
tures.
Table 2 :
2Accuracies for the UCF101 activity recognition dataset.
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[
"Colorful SU(2) center vortices in the continuum and on the lattice",
"Colorful SU(2) center vortices in the continuum and on the lattice"
] |
[
"Thomas Schweigler \nInstitute of Atomic and Subatomic Physics\nAmerican Physical Society\nVienna University of Technology\nWiedner Hauptstr. 8-10, One Research Road1040, 11961Vienna, RidgeNew YorkAustria, USA\n",
"Roman Höllwieser \nInstitute of Atomic and Subatomic Physics\nAmerican Physical Society\nVienna University of Technology\nWiedner Hauptstr. 8-10, One Research Road1040, 11961Vienna, RidgeNew YorkAustria, USA\n",
"Manfried Faber \nInstitute of Atomic and Subatomic Physics\nAmerican Physical Society\nVienna University of Technology\nWiedner Hauptstr. 8-10, One Research Road1040, 11961Vienna, RidgeNew YorkAustria, USA\n",
"Urs M Heller \nInstitute of Atomic and Subatomic Physics\nAmerican Physical Society\nVienna University of Technology\nWiedner Hauptstr. 8-10, One Research Road1040, 11961Vienna, RidgeNew YorkAustria, USA\n"
] |
[
"Institute of Atomic and Subatomic Physics\nAmerican Physical Society\nVienna University of Technology\nWiedner Hauptstr. 8-10, One Research Road1040, 11961Vienna, RidgeNew YorkAustria, USA",
"Institute of Atomic and Subatomic Physics\nAmerican Physical Society\nVienna University of Technology\nWiedner Hauptstr. 8-10, One Research Road1040, 11961Vienna, RidgeNew YorkAustria, USA",
"Institute of Atomic and Subatomic Physics\nAmerican Physical Society\nVienna University of Technology\nWiedner Hauptstr. 8-10, One Research Road1040, 11961Vienna, RidgeNew YorkAustria, USA",
"Institute of Atomic and Subatomic Physics\nAmerican Physical Society\nVienna University of Technology\nWiedner Hauptstr. 8-10, One Research Road1040, 11961Vienna, RidgeNew YorkAustria, USA"
] |
[] |
The spherical vortex as introduced in [Phys. Rev. D77, 014515 (2008)] is generalized. A continuum form of the spherical vortex is derived and investigated in detail. The discrepancy between the gluonic lattice topological charge and the index of the lattice Dirac operator described in previous papers is identified as a discretization effect. The importance of the investigations for Monte Carlo configurations is discussed.
|
10.1103/physrevd.87.054504
|
[
"https://arxiv.org/pdf/1212.3737v2.pdf"
] | 119,154,431 |
1212.3737
|
cae14cc3e87a4618c007626199329d11ce109f58
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Colorful SU(2) center vortices in the continuum and on the lattice
9 Feb 2013
Thomas Schweigler
Institute of Atomic and Subatomic Physics
American Physical Society
Vienna University of Technology
Wiedner Hauptstr. 8-10, One Research Road1040, 11961Vienna, RidgeNew YorkAustria, USA
Roman Höllwieser
Institute of Atomic and Subatomic Physics
American Physical Society
Vienna University of Technology
Wiedner Hauptstr. 8-10, One Research Road1040, 11961Vienna, RidgeNew YorkAustria, USA
Manfried Faber
Institute of Atomic and Subatomic Physics
American Physical Society
Vienna University of Technology
Wiedner Hauptstr. 8-10, One Research Road1040, 11961Vienna, RidgeNew YorkAustria, USA
Urs M Heller
Institute of Atomic and Subatomic Physics
American Physical Society
Vienna University of Technology
Wiedner Hauptstr. 8-10, One Research Road1040, 11961Vienna, RidgeNew YorkAustria, USA
Colorful SU(2) center vortices in the continuum and on the lattice
9 Feb 2013(Dated: May 5, 2014)2PACS numbers: 1115Ha a
The spherical vortex as introduced in [Phys. Rev. D77, 014515 (2008)] is generalized. A continuum form of the spherical vortex is derived and investigated in detail. The discrepancy between the gluonic lattice topological charge and the index of the lattice Dirac operator described in previous papers is identified as a discretization effect. The importance of the investigations for Monte Carlo configurations is discussed.
I. INTRODUCTION
The distribution of topological charge density is closely linked to the phenomena of the axial anomaly and spontaneous chiral symmetry breaking. There are strong hints that the QCD vacuum is dominated by center vortices [1][2][3][4][5][6]. Center vortices can explain color confinement [7][8][9][10] and seem to be also of paramount importance for spontaneous chiral symmetry breaking [11][12][13][14][15]. Center vortices can contribute to the topological charge density through intersection and writhing points [11], but also through their color structure. The prototype of the later contribution in SU (2) gauge theory is the spherical vortex introduced in previous articles of our group [16][17][18]. In this paper, a generalization of the original spherical vortex is constructed. Subsequently, the generalized spherical vortex is investigated in detail on the lattice and in the continuum.
In section II, we start with a description of the original spherical vortex on the lattice as introduced in [16]. From this lattice gauge configuration, we then derive the corresponding continuum object. With the continuum form, a more general spherical vortex can be described. The action and topological charge density of such generalized spherical vortices in the continuum are discussed in section III and IV. We show that the large action of the original spherical vortex encountered in previous investigations can be significantly reduced by spreading the vortex over several time slices. In section V, a generalized spherical vortex on the lattice is derived from the continuum form. Subsequently its gluonic and fermionic properties are investigated. In particular, we show that the discrepancy between the gluonic lattice topological charge and the index of the lattice Dirac operator seen for the original spherical vortex is simply a discretization effect. We conclude with remarks on the contributions of colorful center vortices to topological charge and chiral symmetry breaking in Monte Carlo configurations.
II. THE ORIGINAL SPHERICAL VORTEX
The original spherical vortex on the lattice as introduced in [16] is given by the lattice links [19]
U i (x) = ½ U 4 (x) = cos [α (| r − r 0 |)] ½ − i e r · σ sin [α (| r − r 0 |)] for t = 1 ½ else .(1)
Here x = {x 1 , x 2 , x 3 , t} stands for the lattice site and r denotes the spatial components only. The unit vector pointing from the vortex midpoint r 0 to the spatial lattice site r is denoted by
e r = r − r 0 | r − r 0 | . U µ (x)
is the link connecting the lattice site x with the site x +μ, whereμ is the unit vector pointing in the µ direction. The gauge links on the lattice represent the adjoint gauge transporters and, therefore, transform like U µ (x) → Ω(x +μ) U µ (x) Ω(x) † under a gauge transformation Ω(x). Indices denoted by Latin letters run from 1 to 3 (spatial indices), while indices denoted by Greek letters run from 1 to 4 (spatial plus temporal indices). The function α(r) rises linearly from −π to 0, i.e.,
α(r) = −π · 1 for r < R − d 2 R+ d 2 −r d for R − d 2 ≤ r ≤ R + d 2 0 for r > R + d 2 .(2)
Note that the spherical vortex configuration posses only non-trivial temporal links U 4 . All spatial links U i are trivial and therefore the gluonic lattice topological charge Q = 0. On the other hand the index of the Dirac operator ind [D] = n − − n + , defined as the difference between the number of left-and right-handed chiral zeromodes, is always 1. This gives a discrepancy between the index and the gluonic topological charge. Note that no violation of the index theorem ind [D] = Q occurs as the theorem is only stated for the continuum limit. When cooling, the gluonic topological charge approaches 1 and the discrepancy ind [D] = Q gets resolved [16][17][18].
Naively, one would perform the continuum limit of the spherical vortex by simply splitting the temporal links and leaving the spatial links trivial. This would give gauge configurations of the form
U i (x) = ½, U 4 (x) = cos [w(t) α (| r − r 0 |)] ½ − i e r · σ sin [w(t) α (| r − r 0 |)] with t w(t) = 1 ,(3)
where w(t) determines how the temporal links are split. As one can easily see, a singularity occurs at the vortex midpoint when 0 = w(t) = 1. Moreover, the gluonic topological charge Q clearly remains 0 independent of w(t) which doesn't agree with the cooling history of the original vertex, Eq. (1). Therefore, this simple procedure seems not to be suitable for performing the continuum limit of the lattice spherical vortex. In order to find a better continuum limit and to gain understanding of the spherical vortex, one has to apply a gauge transformation to the link configuration given in Eq. (1). With the lattice gauge transformation
Ω(x) = g( r) for 1 < t ≤ t g ½ else ,
where
g( r) = cos [α (| r − r 0 |)] ½ + i e r · σ sin [α (| r − r 0 |)] ,(4)
the spherical vortex becomes
U i (x) = g r +î g ( r) † for 1 < t ≤ t g ½ else , U 4 (x) = g ( r) † for t = t g ½ else .(5)
From this it becomes clear that the spherical vortex represents a transition (in the temporal direction) between two pure gauge fields. The transition occurs between t = 1 and t = 2. No transition occurs between t = t g and t = t g + 1 since all plaquettes there are trivial. For t ≤ 1 the pure gauge field is trivial, i.e., the winding number n w = 0. For t > 1 the pure gauge field is generated by the hedgehog gauge transformation, Eq. (4), which has n w = 1 [20]. This means that in the continuum limit, the spherical vortex (representing a transition between a vacuum with winding number 0 and 1) has topological charge Q = 1 [20]. It can be regarded as a squeezed instanton with center vortex structure.
Assuming an infinitely big temporal extent of the lattice and taking t g → ∞, the continuum field corresponding to Eq. (5) can be written as
A µ = f (t) i (∂ µ g) g † ,(6)
where g is given in Eq. (4) and f(t), determining the transition in temporal direction t, changes from 0 to 1 within one lattice unit. Clearly, one could also use a function f(t) that changes more slowly. In this case, the corresponding lattice object cannot be represented by configurations of the form of Eq. (1) anymore. We refer to the construction with a smoother function f (t) as a generalized spherical vortex. One construction of such a generalized spherical vortex on the lattice will be investigated in section V. Before doing so, we discuss the action and topological charge density of the continuum gauge field, given in Eq. (6) in the next two sections.
III. ACTION OF THE SPHERICAL VORTEX IN THE CONTINUUM
In this section, we calculate the action for the generalized spherical vortex in the continuum given in Eq. (6). For simplicity, we set the vortex midpoint r 0 to 0 in this and the next section. Moreover, we use the notation | r| = r.
To explicitly evaluate the action S for the A µ given in Eq. (6), we have to evaluate i (∂ µ g) g † where g is given in Eq. (4). To do so, we first have a look at i (∂ µ g) g † for a general gauge transformation
g = q 0 σ 0 + i q · σ ,
where q 0 and q are constraint by q 2 0 + | q| 2 = 1. Using the identity σ a σ b = δ ab ½ + i ǫ abc σ c and
q 0 ∂ µ q 0 + q k ∂ µ q k = 1 2 ∂ µ (q 0 q 0 + q k q k ) = 1 2 ∂ µ (1) = 0 ,
one gets, after a few lines of calculation,
i (∂ µ g) g † = σ · [(∂ µ q 0 ) q − q 0 (∂ µ q) + q × (∂ µ q)] .(7)
For the g given in Eq. (4) we have q 0 = cos(α(r)) and q = e r sin (α(r)). Inserting this into Eq. (7) and multiplying with f (t) gives
(A µ = σ a 2 A a µ ) A a i = f (t) · 2 · x i x a r 3 sin (α(r)) cos (α(r)) + ǫ iak x k 1 r 2 sin 2 (α(r)) − δ ia 1 r sin (α(r)) cos (α(r)) − x i x a r 2 α ′ (r) , A a 4 = 0 .(8)
In the following, we use the notation
A a µ = f (t) A a µ+ .(9)
The field strength tensor is given by
F a µν = ∂ µ A a ν − ∂ ν A a µ − ǫ abc A b µ A c ν .
The different elements of the field strength tensor can be simplified considerably for our particular gauge field configuration. Let us first have a look at the elements of the form F a 4i . Using the fact that the gauge field has no temporal components, i.e., A a 4 = 0, we can simplify these elements to
F a 4i = ∂ 4 A a i .(10)
The F a ij can be simplified too. With the notation (9) we get
F a ij = f (t) ∂ i A a j+ − f (t) ∂ j A a i+ − f (t) 2 ǫ abc A b i+ A c j+ = f (t) ∂ i A a j+ − ∂ j A a i+ − ǫ abc A b i+ A c j+ + f (t) − f (t) 2 ǫ abc A b i+ A c j+ = f (t) (1 − f (t)) ǫ abc A b i+ A c j+ .(11)
Using these results, we can now evaluate the action. Let us first evaluate tr C [F 4i F 4i ]. With (10) and (9) we get
tr C [F 4i F 4i ] = 1 2 F a 4i F a 4i = 1 2 d dt f(t) 2 A a i+ A a i+ .
Inserting the explicit form of A a i+ one can evaluate
A a i+ A a i+ = 4 2 r 2 sin (α(r)) 2 + α ′ (r) 2 .
Similarly we get
tr C [F ij F ij ] = 1 2 F a ij F a ij = f (t) 2 (1 − f (t)) 2 ǫ abc A b i+ A c j+ ǫ ade A d i+ A e j+ ,
where Eq. (11) has been used in the last step. With the help of a computer algebra program, one can evaluate
ǫ abc A b i+ A c j+ ǫ ade A d i+ A e j+ = 32 r 4 sin (α(r)) 2 sin (α(r)) 2 + 2r 2 α ′ (r) 2 .
Combining these identities and switching to polar coordinates gives
S = 1 2g 2 d 4 x tr C [F µν F µν ] = 1 2g 2 d 4 x (tr C [F ij F ij ] + 2 tr C [F 4i F 4i ]) = 1 2g 2 dt f (t) 2 (1 − f (t)) 2 dr 64π r 2 sin (α(r)) 2 sin (α(r)) 2 + 2r 2 α ′ (r) 2 + dt d dt f(t) 2
dr 16π 2 sin (α(r)) 2 + r 2 α ′ (r) 2 .
So far, the calculation has been done for a general f (t). Now we will choose f (t) as the piecewise linear function
f ∆t (t) = 0 for t < 1 t−1 ∆t for 1 ≤ t ≤ 1 + ∆t 1 for t > 1 + ∆t ,(13)
where ∆t stands for the duration of the transition. Inserting f ∆t (t) into Eq. (12) gives
S = 1 2g 2 ∆t · dr 32π
15r 2 sin (α(r)) 2 sin (α(r)) 2 + 2r 2 α ′ (r) 2
+ 1 ∆t · dr 16π 2 sin (α(r)) 2 + r 2 α ′ (r) 2 .(14)
One can see from Eq. (14), that as long as none of the spatial integrations gives zero, the action diverges for both ∆t → ∞ and ∆t → 0. Note that the first term in Eq. (14) represents the magnetic and the second term the electric contributions to the action. This means that for ∆t → 0, the action is purely electric, and for ∆t → ∞, it is purely magnetic. In between, there is a minimum where the electric and magnetic contributions are equal.
Performing the spatial integration is a difficult task, which was done with a computer algebra program. The result for a general ratio of d/R is lengthy and therefore not explicitly given here. However, let us state some properties of the result. If we minimize the action for a given R and d with respect to ∆t, we get an expression that depends only on the ratio d/R. This expression falls monotonically with d/R in the allowed range 0 ≤ d/R ≤ 2. It diverges for d/R → 0 and approaches its minimum for d/R → 2. The absolute minimum of the action (minimized with respect to ∆t, R and d) is given by S min = 1.667 S Inst (R = d/2 ≈ 0.305∆t). Here we denoted the action of one instanton by S Inst = 1/(2g 2 ) 16π 2 . This action serves as a lower bound for objects with topological charge |Q| = 1. Fixing the ratio d/R gives the action as a function of R and ∆t. For d/R = 1 we get
S(∆t) S Inst = 0.4358 R · ∆t + 3.722 R ∆t .(15)
IV. TOPOLOGICAL CHARGE DENSITY OF THE SPHERICAL VORTEX IN THE CONTINUUM
In this section, we calculate the topological charge density q(x) of field configurations of the form of Eq. (6). For now, the calculation is done for a general gauge transformation g and a general f (t) with the restriction, that they have to be chosen in such a way, that the resulting fields are free of singularities. Moreover, we will assume that g is independent of the temporal coordinate t. In the following, we again use the shorthand notation A i+ = i (∂ i g) g † .
As is well known [20], q(x) can be calculated as the full derivative
q(x) = ∂ µ K µ (x) , with K µ = 1 16π 2 ǫ µαβγ A a α ∂ β A a γ − 1 3 ǫ abc A a α A b β A c γ .
Let us now split ∂ µ K µ into a spatial (∂ i K i ) and a temporal part (∂ 4 K 4 ). First, we treat the spatial part. With the assumption that g is independent of the temporal coordinate, one can easily see that the gauge field (6) has no temporal component, i.e. A 4 = 0. Therefore, K i evaluates to zero:
K i = 1 16π 2 ǫ iαβγ A a α ∂ β A a γ − 1 3 ǫ abc A a α A b β A c γ = 1 16π 2 ǫ iα4γ A a α ∂ 4 A a γ = − 1 16π 2 ǫ ijk A a j ∂ 4 A a k = − 1 16π 2 ǫ ijk A a j+ A a k+ f(t) f ′ (t) = 0 .
This means, that we can identify the topological charge density with the temporal derivative of K 4 , i.e.,
q(x) = ∂ 4 K 4 .
From the definition of K µ given above, we see that we can write K 4 as
K 4 = − 1 16π 2 f(t) 2 ǫ ijk A a i+ ∂ j A a k+ − f(t) 3 1 3 ǫ ijk ǫ abc A a i+ A b j+ A c k+ ,
where we have used ǫ 4ijk = − ǫ ijk . Using the fact that the field strength vanishes for pure gauge fields, i.e., F a µν = ∂ µ A a ν+ − ∂ ν A a µ+ − ǫ abc A b µ+ A c ν+ = 0, one can rewrite the term ǫ ijk A a i+ ∂ j A a k+ as 1 2 ǫ ijk ǫ abc A a i+ A b j+ A c k+ . Using this result, we get
K 4 = − 1 16π 2 1 2 f(t) 2 − 1 3 f(t) 3 ǫ ijk ǫ abc A a i+ A b j+ A c k+ .(16)
The topological charge density is simply the temporal derivative of this expression. Note that ǫ ijk ǫ abc A a i+ A b j+ A c k+ is proportional to the winding number density of the gauge transformation. This means that the spatial dependence of the topological charge density is given by the winding number density and the temporal dependence is given by
(1 − f(t)) f(t) f ′ (t).
With the g defined in Eq. (4), the α(r) defined in Eq. (2) and the f(t) defined in Eq. (13) the topological charge density evaluates to
q(r, t) = 3 dπr 2 cos 2 π(r − R) d 1 4∆t − t 2 ∆t 3 · 1 for 1 ≤ t ≤ 1 + ∆t and R − d 2 ≤ r ≤ R + d 2 0 else .(17)
As can easily be checked, integrating this expression (r 2 dr and dt) yields Q = 1.
V. THE GENERALIZED SPHERICAL VORTEX ON THE LATTICE
We now put the generalized continuum spherical vortex of Eq. (6) onto the lattice with periodic boundary conditions. Clearly, the field as given in Eq. (6) does not fulfill periodic boundary conditions in the temporal direction. A µ vanishes for t → −∞ (and also for r → ∞) but not for t → ∞. If we want to get a vanishing gauge field A µ for t → ∞ by gauge transforming the field of Eq. (6), we have to find a gauge transformation that equals ½ for t → −∞ and g † for t → ∞. As can be shown easily by continuity arguments, there is no continuous gauge transformation that fulfills that criteria for the g given in Eq. (4). However, there is still the possibility to put the field onto a lattice of infinite size. Then, one can transform the links for t → ∞ to unity and subsequently close the lattice by periodic boundary conditions. Performing this procedure for the continuum field of Eq. (6) with f(t) given in Eq. (13) yields
U i (x) = g r +î g ( r) † (t−1)/∆t for 1 < t < 1 + ∆t g r +î g ( r) † for 1 + ∆t ≤ t ≤ t g ½ else , U 4 (x) = g( r) † for t = t g ½ else .(18)
The functions g( r) and α(r) are again given by Eqs. (4) and (2). Let us now have a look at the gauge action and topological charge of this generalized spherical vortex on the lattice. The gauge action as function of the temporal extent ∆t is plotted in Fig. 1a). Note, that the action of the lattice object matches the action of the underlying continuum object pretty well. However, it systematically underestimates the continuum value. In Fig. 1b) we show the topological charge density as function of ∆t for three values of R = d. As one can see, the discrepancy between the topological charge Q on the lattice and in the continuum (which is always 1) is dramatic for small temporal extent ∆t of the spherical vortex. For ∆t = 1 the topological charge of the continuum object is not recognized. However, one still gets a zeromode for the corresponding lattice Dirac operator, i.e., the fermions still see the topological charge of the underlying continuum object. This is not unexpected as the vacuum to vacuum transition is still present in the lattice object. To be more precise, the lattice samples the field before and after the transition. The transition itself falls between two time-slices. As discussed in section IV, the transition carries the topological charge. Therefore, by missing the transition, one also misses the topological charge. From Fig. 1 we see, that also for very big ∆t (slow vacuum to vacuum transitions), the lattice topological charge doesn't quite approach one. However, increasing R, keeping R = d in the example shown, increases Q towards one. Since R is the only spacial scale, 1/R acts as the spacial lattice spacing, showing that the deviations from Q = 1 at large ∆t (∆t > ∼ 10) are discretization effects in the spatial directions.
To conclude this section we briefly discuss how the fermionic results depend on the temporal extent of the vortex. zeromode is more or less located at the vortex. For the calculations, we used the overlap Dirac operator. With the described setting, we see that, as expected, the scalar density of the zeromode is smeared out in the temporal direction for the vortices with bigger temporal extent. The spatial localization, however, is more or less independent of the temporal extent of the vortex. Thus, all what really matters is that the vacuum to vacuum transition occurs, not how fast it occurs. Let us discuss this in a little bit more detail. First, we note that the scalar density of the zeromode is distributed, almost perfectly, spherically symmetric around the midpoint r 0 of the vortex. Keeping this in mind, the discussion of the localization in 3 dimensional space reduces to a discussion of the localization in | r − r 0 |. Thus we study
ρ(r, t) = 1 N (r) r δ(r, | r − r 0 |) ψ † 0 ( r, t)ψ 0 ( r, t) with N (r) = r δ(r, | r − r 0 |) .(19)
In other words, the quantity ρ(r, t) stands for the mean value of the scalar density of the zeromode, calculated for points belonging to the same | r − r 0 | and t. In Fig. 2 the results for ρ(r, t) for generalized spherical vortices with ∆t = 1 and ∆t = 5 are compared. One can see the similarity between the result for the two different ∆t.
VI. DISCUSSION AND OUTLOOK
In this paper, we identified the continuum object corresponding to the previously considered spherical vortex as a vacuum to vacuum transition in temporal direction. The discrepancy between the gluonic lattice topological charge and the index of the lattice Dirac operator, described in previous papers, turned out to be a discretization effect in the temporal direction. Starting from the continuum spherical vortex, we constructed a generalized spherical vortex on the lattice. We demonstrated the similarity to the original spherical vortex by using fermions as probes. We also showed that, with an appropriate choice of parameters, the action of the generalized spherical vortex can be quite small, as small as about 5/3 of the one-instanton action. For more details see [21].
It is known that topological charge contributions from center vortices, due to intersection (and writhing) points, emerge from vortex structures lying in a single U(1) subgroup [22]. Color rotations in such structures are suppressed by the action. For example, intersections of vortices with orthogonal color structures give maximally negative plaquettes and are thus suppressed in the continuum limit. However, this does not mean that the whole vortex structure can be gauge transformed to a single U(1) subgroup. We conjecture, for entropic reasons, that in the confined phase the color structure of center vortices contributes to the topological charge and chiral symmetry breaking.
Further investigations should therefore aim at quantifying these kinds of contributions in Monte Carlo generated configurations. First measurements show that the total vortex surface definitely covers the full S 3 , i.e. vortex plaquettes are distributed uniformly among the entire SU (2) color palette. However, color vectors are not gauge invariant and therefore the number of full coverings is not so easy to measure, even though it is a gauge invariant, topological quantity. Further, writhing points dominate the topological susceptibility [23,24], even in SU (3) [25], and therefore color contributions might only play a sub-dominant role like intersection points. But the colorful spherical vortex may act as an ansatz for a model of chiral symmetry breaking, as it shows properties similar to instantons [26].
FIG
For the fermion fields we use antiperiodic boundary conditions in the temporal direction and periodic boundary conditions in the spatial directions. With these boundary conditions, the Dirac operator possesses always exactly one zeromode for gauge configurations of the form of Eq. (18), independent of the values of the parameters. . 1. a) Comparison of the gauge action (in units of the instanton action SInst) on the lattice, (blue) diamonds, with the gauge action of the continuum object, (red) line, given by Eq. (15) as function of the temporal extent ∆t of the spherical vortex. The radius as well as the thickness of the vortex are given by R = d = 6. The midpoint of the vortex is half integer. b) Comparison of the gluonic topological charge Q(∆t) for three values of R = d. The lattice results (calculated with the plaquette definition) are shown as (green) squares, (blue) diamonds and (black) circles. In the continuum, Q(∆t) = 1 for all ∆t as shown by the (red) horizontal line. The lattice sizes N 3 s × Nt are chosen to fit the spacial and temporal extents of the spherical vortex.
FIG. 2 . 1
21ρ(r, t) as defined in Eq.(19) with normalization ρmax = 1. The configuration investigated is given by Eq. (18) with ∆t = 1, (red) diamonds, and ∆t = 5, (blue) circles. The parameters of α(r) are given by R = d = 6 in both cases. The calculations have been performed on a 24 3 × 12 lattice. The different curves belong to different values of t. For ∆t = . The highest curve corresponds to the first pair of t-values, the lowest curve to the last pair of t-values. In between the curves fall monotonically. The solid (magenta) line represents the average over t for ∆t = 1, the dash-dotted (green) line represents the same for ∆t = 5.
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Note that throughout this paper Einstein's summation convention is used. As already mentioned, we will work with SU(2) gauge theory. The Pauli matrices will be denoted by σ. Throughout this paper, the lattice constant is set to a = 1Note that throughout this paper Einstein's summation convention is used. As already mentioned, we will work with SU(2) gauge theory. The Pauli matrices will be denoted by σ. Throughout this paper, the lattice constant is set to a = 1.
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[] |
[
"MAHONIAN PARTITION IDENTITIES VIA POLYHEDRAL GEOMETRY",
"MAHONIAN PARTITION IDENTITIES VIA POLYHEDRAL GEOMETRY"
] |
[
"Matthias Beck \nLeon Ehrenpreis\n",
"Benjamin Braun \nLeon Ehrenpreis\n",
"Nguyen Le \nLeon Ehrenpreis\n"
] |
[
"Leon Ehrenpreis",
"Leon Ehrenpreis",
"Leon Ehrenpreis"
] |
[] |
In a series of papers, George Andrews and various coauthors successfully revitalized seemingly forgotten, powerful machinery based on MacMahon's Ω operator to systematically compute generating functions λ∈P z λ 1 1 · · · z λn n for some set P of integer partitions λ = (λ1, . . . , λn). Our goal is to geometrically prove and extend many of the Andrews et al theorems, by realizing a given family of partitions as the set of integer lattice points in a certain polyhedron.
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10.1007/978-1-4614-4075-8_3
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"https://arxiv.org/pdf/1103.1070v2.pdf"
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1103.1070
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a38bcc8c9c595543cefb477431208db1965d33da
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MAHONIAN PARTITION IDENTITIES VIA POLYHEDRAL GEOMETRY
4 Apr 2011
Matthias Beck
Leon Ehrenpreis
Benjamin Braun
Leon Ehrenpreis
Nguyen Le
Leon Ehrenpreis
MAHONIAN PARTITION IDENTITIES VIA POLYHEDRAL GEOMETRY
4 Apr 2011
In a series of papers, George Andrews and various coauthors successfully revitalized seemingly forgotten, powerful machinery based on MacMahon's Ω operator to systematically compute generating functions λ∈P z λ 1 1 · · · z λn n for some set P of integer partitions λ = (λ1, . . . , λn). Our goal is to geometrically prove and extend many of the Andrews et al theorems, by realizing a given family of partitions as the set of integer lattice points in a certain polyhedron.
Introduction
In a series of papers starting with [1], George Andrews and various coauthors successfully revitalized seemingly forgotten, powerful machinery based on MacMahon's Ω operator [15] to systematically compute generating functions related to various families of integer partitions. Andrews et al's papers concern generating functions of the form f P (z 1 , . . . , z n ) := λ∈P z λ 1 1 · · · z λn n and f P (q) := f P (q, . . . , q) = λ∈P q λ 1 +···+λn , for some set P of partitions λ = (λ 1 , . . . , λ n ); i.e., we think of the integers λ n ≥ · · · ≥ λ 1 ≥ 0 as the parts when some integer k is written as k = λ 1 + · · · + λ n . If we do not force an order onto the λ j 's, we call λ a composition of k. Below is a sample of some of these striking results.
Theorem 1 (Andrews [2]). Let In words: the number of partitions of an integer k satisfying the "higher-order difference conditions" in P r equals the number of partitions of k into parts that are r'th-order binomial coefficients.
Theorem 2 (Andrews-Paule-Riese [3]). Let n ≥ 3 and τ := {(λ 1 , . . . , λ n ) ∈ Z n : λ n ≥ · · · ≥ λ 1 ≥ 1 and λ 1 + · · · + λ n−1 > λ n } , the set of all "n-gon partitions." Then
f τ (q) = q n (1 − q)(1 − q 2 ) · · · (1 − q n ) − q 2n−2 (1 − q)(1 − q 2 )(1 − q 4 )(1 − q 6 ) · · · (1 − q 2n−2 )
.
More generally,
f τ (z 1 , . . . , z n ) = Z 1 (1 − Z 1 )(1 − Z 2 ) · · · (1 − Z n ) − Z 1 Z n−2 n (1 − Z n )(1 − Z n−1 )(1 − Z n−2 Z n )(1 − Z n−3 Z 2 n ) · · · (1 − Z 1 Z n−2 n ) ,
where Z j := z j z j+1 · · · z n for 1 ≤ j ≤ n.
The composition analogue of Theorem 2 was inspired by a problem of Hermite [18,Ex. 31], which is essentially the case n = 3 of the following.
Theorem 3 (Andrews-Paule-Riese [4]). Let
H := (λ 1 , . . . , λ n ) ∈ Z n >0 : λ 1 + · · · + λ j + · · · + λ n ≥ λ j for all 1 ≤ j ≤ n .
Then
f H (q) = q n (1 − q) n − n q 2n−1 (1 − q) n (1 + q) n−1 .
A natural question is whether there exist "full generating function" versions of Theorems 1 and 3, in analogy with Theorem 2; we will show that such versions (Theorems 6 and 7 below) follow effortlessly from our approach. (Xin [21, Example 6.1] previously computed a full-generatingfunction related to Theorem 3.) Our main goal is to prove these theorems geometrically, and more, by realizing a given family of partitions as the set of integer lattice points in a certain polyhedron. This approach is not new: Pak illustrated in [16,17] how one can obtain bijective proofs by realizing when both sides of a partition identity are generating functions of lattice points in unimodular cones (which we will define below); this included most of the identities appearing in [2], including Theorem 1. Corteel, Savage, and Wilf [13] implicitly used the extreme-ray description of a cone (see Lemma 4 below) to derive product formulas for partition generating functions, including those appearing in [2]. Beck, Gessel, Lee, and Savage [7] used triangulations of cones to extend results of Andrews, Paule, and Riese [5] on "symmetrically constrained compositions." However, we feel that each of these papers only scratched the surface of a polyhedral approach to partition identities, and we see the current paper as a further step towards a systematic study of this approach.
While the Ω-operator approach to partition identities is elegant and powerful (not to mention useful in the search for such identities), we see several reasons for pursuing a geometric interpretation of these results. As discussed in [11], partition analysis and the Ω operator are useful tools for studying partitions and compositions defined by linear constraints, which is equivalent to studying integer points in polyhedra. An explicit geometric approach to these problems often reveals interesting connections to geometric combinatorics, such as the connections and conjectures discussed in Sections 6 and 7 below. Also, one of the great appeals of partition analysis is that it is automatic; Andrews discusses this in the context of applying the Ω operator to the 4-dimensional case of lecture-hall partitions in [1]:
The point to stress here is that we have carried off the case j = 4 with no effective combinatorial argument or knowledge. In other words, the entire problem is reduced by Partition Analysis to the factorization of an explicit polynomial.
As we hope to show, the geometric perspective can often provide a clear view of sometimes mysterious formulas that arise from the symbolic manipulation of the Ω operator.
Polyhedral cones and their lattice points
We use the standard abbreviation z m := z m 1 1 · · · z mn n for two vectors z and m. Given a subset K of R n , the (integer-point) generating function of K is
σ K (z 1 , . . . , z n ) := m∈K∩Z n z m .
We will often encounter subsets that are cones, where a (polyhedral) cone C is the intersection of finitely many (open or closed) halfspaces whose bounding hyperplanes contain the origin. (Thus the cones appearing in this paper will not all be closed but in general partially open.) A closed cone has the alternative description (and this equivalence is nontrivial [22]) as the nonnegative span of a finite set of vectors in R n , the generators of C.
An n-dimensional cone in R n is simplicial if we only need n halfspaces to describe it. All of our cones will be pointed, i.e., they do not contain lines. The following exercise in linear algebra shows how to switch between the generator and halfspace descriptions of a simplicial cone.
Lemma 4. Let A be the inverse matrix of B ∈ R n×n . Then {x ∈ R n : A x ≥ 0} = {B y : y ≥ 0} ,
where each inequality is understood componentwise.
The (integer-point) generating function of a simplicial cone C ⊂ R n can be computed from first principles when C is rational, i.e., its generators can be chosen in Z n . A closed cone C is unimodular if its generators form a basis of Z n ; for unimodular cones, which is all we will need in what follows, we have the following simple lemma (for much more general results, see, e.g., [8,Chapter 3]).
Lemma 5. Suppose C = k j=1 R ≥0 v j + n i=k+1 R >0 v i is a unimodular cone in R n generated by v 1 , . . . , v n ∈ Z n . Then σ C (z 1 , . . . , z n ) = n i=k+1 z v i n j=1 (1 − z v j ) .
Unimodular Cones
Recall from Theorem 1 that
P r = λ : t j=0 (−1) j t j λ k+j ≥ 0 for k ≥ 1, 1 ≤ t ≤ r (where we set undefined λ j 's zero). Let P n r := (λ 1 , . . . , λ n ) ∈ Z n : t j=0 (−1) j t j λ k+j ≥ 0 for 1 ≤ k ≤ n, 1 ≤ t ≤ r
consist of all partitions in P r with at most n parts. As a warm-up example we will compute the (full) generating function of P n r :
Theorem 6.
f P n r (z 1 , . . . , z n ) = 1 (1 − z 1 ) (1 − z r 1 z 2 ) 1 − z ( r+1 r−1 ) 1 z r 2 z 3 1 − z ( r+2 r−1 ) 1 z ( r+1 r−1 ) 2 z r 3 z 4 · · · 1 − z ( r+n−2 r−1 ) 1 z ( r+n−3 r−1 ) 2 · · · z r n−1 z n .
Note that Theorem 1 follows upon setting z 1 = · · · = z n = q, using the identity
r + j − 2 r − 1 + r + j − 3 r − 1 + · · · + r + 1 = r + j − 1 r ,
and taking n → ∞.
Proof. It is easy to see that the inequalities
t j=0 (−1) j t j λ k+j ≥ 0 for 1 ≤ k ≤ n, 1 ≤ t ≤ r ,
which define P n r , are implied by the inequalities for t = r. Thus the cone containing P n r as its integer lattice points is
K := (x 1 , . . . , x n ) ∈ R n : r j=0 (−1) j r j x k+j ≥ 0 for 1 ≤ k ≤ n = 1 r r+1 r−1 r+2 r−1 · · · r+n−2 r−1 0 1 r r+1 r−1 · · · r+n−3 r−1 0 0 1 r · · · r+n−4 r−1 . . . . . . . . . . . . . . . 0 0 1 r 0 · · · 0 1 y : y 1 , . . . , y n ≥ 0
(whose generators we can compute, e.g., with the help of Lemma 4). Thus K is unimodular and, by Lemma 5,
σ K (z 1 , . . . , z n ) = 1 (1 − z 1 ) (1 − z r 1 z 2 ) 1 − z ( r+1 r−1 ) 1 z r 2 z 3 1 − z ( r+2 r−1 ) 1 z ( r+1 r−1 ) 2 z r 3 z 4 · · · 1 − z ( r+n−2 r−1 ) 1 z ( r+n−3 r−1 ) 2 · · · z r n−1 z n .
The idea behind this approach towards Theorem 1 can be found, in disguised form, in [13] and [16]. See also [10,12] for bijective approaches to Theorem 1 and its asymptotic consequences. We included this proof here in the interest of a self-contained exposition and also because none of [2, 13, 16] contains a full generating function version of (analogues of) Theorem 1.
Differences of two cones
They key idea behind the proof of Theorem 2 is to observe that the non-simplicial cone K := {(x 1 , . . . , x n ) ∈ R n : x n ≥ · · · ≥ x 1 > 0 and x 1 + · · · + x n−1 > x n } , whose integer lattice points form Andrews-Paule-Riese's set τ of n-gon partitions, can be written as a difference K = K 1 \ K 2 of two simplicial cones. Specifically, set
K 1 := {(x 1 , . . . , x n ) ∈ R n : x n ≥ · · · ≥ x 1 > 0} = 1 0 0 · · · 0 0 1 1 0 · · · 0 0 1 1 1 · · · 0 0 . . . . . . . . . . . . . . . . . . 1 1 1 · · · 1 0 1 1 1 · · · 1 1 y : y 1 > 0 , y 2 , . . . , y n ≥ 0 and K 2 := {(x 1 , . . . , x n ) ∈ R n : x n ≥ · · · ≥ x 1 > 0 and x 1 + · · · + x n−1 ≤ x n } = 1 0 0 · · · 0 0 1 1 0 · · · 0 0 1 1 1 · · · 0 0 . . . . . . . . . . . . . . . . . . 1 1 1 · · · 1 0 n − 1 n − 2 n − 3 · · · 1 1 y : y 1 > 0 , y 2 , . . . , y n ≥ 0
(whose generators we can compute, e.g., with the help of Lemma 4). One can see immediately from the generator matrices that both K 1 and K 2 are unimodular. (In a geometric sense, this is suggested by the form of the identity in Theorem 2. A similar simplification-through-takingdifferences phenomenon is described in the fifth "guideline" of Corteel, Lee, and Savage [11], which inspired our proof.) By Lemma 5 σ K 1 (z 1 , . . . , z n ) = z 1 · · · z n (1 − z n )(1 − z n−1 z n ) · · · (1 − z 1 · · · z n ) and σ K 2 (z 1 , . . . , z n ) = z 1 · · · z n−1 z n−1 n 1 − z 1 · · · z n−1 z n−1 n 1 − z 2 · · · z n−1 z n−2
n 1 − z 3 · · · z n−1 z n−3 n · · · (1 − z n−1 z n ) (1 − z n ) = Z 1 Z n−2 n (1 − Z n )(1 − Z n−1 )(1 − Z n−2 Z n )(1 − Z n−3 Z 2 n ) · · · (1 − Z 1 Z n−2 n ) ,
and the identity σ K (z 1 , . . . , z n ) = σ K 1 (z 1 , . . . , z n ) − σ K 2 (z 1 , . . . , z n ) completes the proof.
Differences of multiple cones
The "cone behind" Theorem 3 is . . . , x n ) ∈ R n >0 : x j ≤ x 1 + · · · + x j + · · · + x n for all 1 ≤ j ≤ n} ; Theorem 3 follows from the following result upon setting z 1 = · · · = z n = q.
K := {(x 1 ,Theorem 7. σ K (z 1 , . . . , z n ) = z 1 · · · z n (1 − z 1 ) · · · (1 − z n ) − n k=1 z 1 · · · z k−1 z n k z k+1 · · · z n (1 − z k ) n j=1 j =k (1 − z k z j ) .
Proof. Let e j denote the jth unit vector in R n . Observe that the non-simplicial cone K is expressible as a difference K = O \ n k=1 C k , where O := n j=1 R >0 e j and C k is the cone
C k := {(x 1 , . . . , x n ) ∈ R n >0 : x k > x 1 + · · · + x j + · · · + x n } = R >0 e k + n j=1 j =k R >0 (e j + e k )
Note that if i = j, then C i ∩ C j = ∅. Thus, the closure of K is "almost" the positive orthant O, except that we have to exclude points in O that can only be written as a linear combination that requires a single e k (as opposed to a linear combination of the vectors e j + e k ). (A similar simplification-through-taking-differences phenomenon appeared in the original proof of Theorem 3.) In generating-function terms, this set difference gives, by Lemma 5,
σ K (z 1 , . . . , z n ) = σ O (z 1 , . . . , z n ) − n k=1 σ C k (z 1 , . . . , z n ) = z 1 · · · z n (1 − z 1 ) · · · (1 − z n ) − n k=1 z 1 · · · z k−1 z n k z k+1 · · · z n (1 − z k ) n j=1 j =k (1 − z k z j )
.
Three remarks on this theorem are in order. First, as already mentioned, Xin [21, Example 6.1] previously computed a different full-generating-function related to Theorem 3; Xin's generating function handles non-negative, rather than positive, k-gon partitions. Second, the cone K is related to the second hypersimplex, a well-known object in geometric combinatorics (see Section 7 for more details).
Third, K is a suitable candidate for the "symmetrically constrained" approach in [7]; however, one should expect that this approach would give a different form for the generating function σ K (z 1 , . . . , z n ) from the one given in Theorem 7. The symmetrically constrained approach produces a triangulation of the cone K that is invariant under permutation of the standard basis vectors in R n , and then uses this triangulation to express σ K (z 1 , . . . , z n ) as a positive sum of rational generating functions for these cones (after some geometric shifting). The terms in this sum will all have 1 1−z 1 z 2 ···zn as a factor, as each of the simplicial cones in the triangulation of K will have the all-ones vector as a ray generator; this will clearly produce a different form from that in Theorem 7.
Cayley Compositions
A Cayley composition is a composition λ = (λ 1 , . . . , λ j−1 ) that satisfies 1 ≤ λ 1 ≤ 2 and 1 ≤ λ i+1 ≤ 2λ i for 1 ≤ i ≤ j − 2. Thus, the Cayley compositions with j − 1 parts are precisely the integer points in
C j := (λ 1 , . . . , λ j−1 ) ∈ Z j−1 >0 : λ 1 ≤ 2 and λ i ≤ 2λ i−1 for all 2 ≤ i ≤ j − 1 .
Our apparent shift in indexing maintains continuity between our statements and [6], where Cayley compositions always begin with a λ 0 = 1 part. Let f C j (z 1 , . . . , z j−1 ) be the generating function for C j . The following theorem is quite surprising.
Theorem 8 (Andrews-Paule-Riese-Strehl [6]). Let C j := (λ 1 , . . . , λ j−1 ) ∈ Z j−1 >0 : λ 1 ≤ 2 and λ i ≤ 2λ i−1 for all 2 ≤ i ≤ j − 1 .
Then for j ≥ 2,
f C j (1, 1, . . . , 1, q) = j−2 h=1 b j−h−1 (−1) h−1 q 2 h −1 (1 − q)(1 − q 2 )(1 − q 4 ) · · · (1 − q 2 h−1 ) + (−1) j q 2 j−1 −1 (1 − q 2 j−1 ) (1 − q)(1 − q 2 )(1 − q 4 ) · · · (1 − q 2 j−2 )
where b k is the coefficient of q 2 k −1 in the power series expansion of
1 1 − q ∞ m=0 1 1 − q 2 m .
Theorem 8 is derived as a consequence of the following recurrence relation obtained via MacMahon's Ω calculus.
Theorem 9 (Andrews-Paule-Riese-Strehl [6]).
f C j (z 1 , . . . , z j−1 ) = z j−1 1 − z j−1 f C j−1 (z 1 , . . . , z j−2 ) − f C j−1 z 1 , . . . , z j−3 , z j−2 z 2 j−1 .
Once this formula is obtained, the proof of Theorem 8 in [6] proceeds by repeatedly iterating the recurrence, specialized to f C j (1, . . . , 1, q). The final step is to argue that the sum of rational functions in Theorem 8, as analytic functions, must exhibit cancellation. We remark that Corteel, Lee, and Savage [11, Section 3] gave an alternative proof of Theorem 9.
Via geometry, we can shed light on the initial recurrence relation from three perspectives. First, we recognize that the recurrence reflects expressing C j as a difference of two subspaces of R j−1 defined by linear constraints.
First proof of Theorem 9. As a subspace of R j−1 , C j = K 1,j \ K 2,j where
K 1,j := (x 1 , . . . , x j−1 ) ∈ R j−1 : 1 ≤ x 1 ≤ 2, 1 ≤ x i+1 ≤ 2x i for 1 ≤ i ≤ j − 3, and 1 ≤ x j−1 and K 2,j := (x 1 , . . . , x j−1 ) ∈ R j−1 : 1 ≤ x 1 ≤ 2, x i+1 ≤ 2x i for 1 ≤ i ≤ j − 3, x j−1 > 2x j−2 .
If we distribute the leading multiplier in the right-hand side of the recurrence for f C j , the first term is the generating function of K 1,j , as there are no restrictions on the size of x j−1 . On the other hand, the integer points m ∈ K 2,j are precisely those in K 1,j satisfying x j−1 > 2x j−2 , which is equivalent to the condition that z m be divisible by z j−2 z 2 j−1 . The second term of the recurrence records precisely these integer points.
Our second proof amounts to a simple observation regarding the integer-point transform of C j .
Second proof of Theorem 9. Since for any λ ∈ C j ∩ Z j−1 we have 1 ≤ λ j−1 ≤ 2λ j−2 ,
f C j (z 1 , . . . , z j−1 ) = λ∈C j ∩Z j−1 z λ = λ∈C j−1 ∩Z j−2 z λ (z j−1 + z 2 j−1 + · · · + z 2λ j−2 j−1 ) = z j−1 λ∈C j−1 ∩Z j−2 z λ 1 − z 2λ j−2 j−1 1 − z j−1 = z j−1 1 − z j−1 λ∈C j−1 ∩Z j−2 z λ − z λ z 2λ j−2 j−1 = z j−1 1 − z j−1 f C j−1 (z 1 , . . . , z j−1 ) − f C j−1 (z 1 , . . . , z j−3 , z j−2 z 2 j−1 ) .
Following their statement of Theorem 8, the authors of [6] make the following comment: It hardly needs to be pointed out that [this formula] is a surprising representation of a polynomial. Indeed, the right-hand side does not look like a polynomial at all. Such a statement suggests that Brion's formula [9] for rational polytopes is lurking in the background; our third proof of Theorem 9 is based on this formula. Given a rational convex polytope P , we first define the tangent cone at a vertex v of P to be
T P (v) := {v + α(p − v) : α ∈ R ≥0 , p ∈ P } .
Theorem 10 (Brion). Suppose P is a rational convex polytope. Then we have the following identity of rational generating functions:
σ P (z) = v a vertex of P σ T P (v) (z) .
Note that the sum on the right-hand side is a sum of rational functions, while the left-hand side yields a polynomial.
Third proof of Theorem 9. To interpret the recurrence as a consequence of Brion's formula, we first assume that the f C j−1 's are expressed in the form of the right-hand side of Brion's formula, i.e., as a sum of integer-point transforms of the tangent cones at the vertices of C j−1 . We next rewrite the recurrence as
f C j (z 1 , . . . , z j−1 ) = z j−1 1 − z j−1 f C j−1 (z 1 , . . . , z j−2 ) + 1 1 − z −1 j−1 f C j−1 z 1 , . . . , z j−3 , z j−2 z 2 j−1 .
The polytope C j is a combinatorial cube; this can be easily seen by induction on j after observing that in C j−1 × R the hyperplanes x j−1 = 1 and x j−1 = 2x j−2 do not intersect. Thus, the tangent cones for vertices of C j can be expressed in terms of the tangent cones for vertices of C j−1 . Given a vertex v = {v 1 , . . . , v j−2 } of C j−1 , the two vertices of C j obtained from v are (v, 1) and (v, 2v j−2 ). For the vertex (v, 1) in C j , it is immediate that
σ T C j−1 ((v,1)) (z) = 1 1 − z j−1 σ T C j−2 (v) (z) .
Our proof will be complete after we show that for the vertex (v, 2v j−2 ) in C j ,
σ T C j−1 ((v,2v j−2 )) (z) = 1 1 − z −1 j−1 σ T C j−2 (v) (z 1 , . . . , z j−3 , z j−2 z 2 j−1 ) .
This follows from the fact that the edges in C j emanating from (v, 2v j−2 ) terminate in the vertex (v, 1) and in the vertices (w, 2w j−2 ) for vertices w of C j−1 that are connected to v by an edge in C j−1 . Thus, Theorem 9 follows from Brion's formula and induction.
There is an interesting remark about Theorem 8 and Brion's formula; while one might hope that the expression in Theorem 8 is obtained by directly specializing Brion's formula to z 1 = · · · = z j−2 = 1 and z j−1 = q, this is not the case. This specialization is not actually possible, as some of the rational functions for tangent cones in C j have denominators that lack a z j−1 variable, and hence this specialization would require evaluating rational functions at poles. The authors of [6] use the recurrence in Theorem 9 in a more subtle way, in that they first specialize the recurrence to
f C j (1, . . . , 1, q) = q 1 − q f C j−1 (1, . . . , 1) − f C j−1 1, . . . , 1, q 2
and then iterate the recurrence. In doing this, they simultaneously use the interpretation of f C j (z) as a polynomial (for the all-ones specialization) and also the interpretation of f C j (z) as a rational function (for the specialization involving q 2 ). Thus, while Theorem 8 looks similar to a Brion-type result, it is obtained differently. We remark that by specializing z 1 = · · · = z j−1 = q in Brion's formula for C j , one would obtain a representation of the polynomial f C j (q, . . . , q) as a sum of rational functions of q. x i = 2 , in the following manner. The linear inequality x j ≥ x 1 + · · · + x j + · · · + x n is equivalent to n i=1 x i 2 ≤ x j . When n i=1 x i = 1, we are considering the "slice" of K that is constrained by 0 < x j ≤ 1 2 and n i=1 x i = 1, which is 1 2 of ∆(2, n) with the condition that 0 < x j for all j. From this perspective, we can view the n-gon compositions of t as H(t) := (λ 1 , . . . , λ n ) ∈ Z n ≥0 : λ 1 + · · · + λ n = t , λ j ≤ λ 1 + · · · + λ j + · · · + λ n for all 1 ≤ j ≤ n = (λ 1 , . . . , λ n ) ∈ Z n : λ 1 + · · · + λ n = t , 0 ≤ λ j ≤ t 2 for all 1 ≤ j ≤ n .
The second hypersimplex is a well-studied object; for example, in matroid theory ∆(2, n) is the matroid basis polytope for the 2-uniform matroid on n vertices, while in combinatorial commutative algebra ∆(2, n) is the subject of [20,Chapter 9]. It would be interesting to consider analogues of Theorem 3 for the general case of the k th hypersimplex ∆(k, n) := {(x 1 , . . . , x n ) ∈ [0, 1] n : n i=1 x i = k}. The associated composition counting function has a natural interpretation: in (λ 1 , . . . , λ n ) ∈ Z n : λ 1 + · · · + λ n = t , 0 ≤ λ j ≤ t k for all 1 ≤ j ≤ n are all compositions of t whose parts are at most t k (i.e., the parts are not allowed to be too large, where "too large" depends on k).
Cayley polytopes.
We refer to the polytopes C j from Section 6 as Cayley polytopes. By taking a geometric view of Cayley compositions as integer points in C j , we may shift our focus from combinatorial properties of the integer points to properties of C j itself. Recall that the normalized volume of C j is Vol(C j ) := (j − 1)! vol(C j ) , where vol(C j ) is the Euclidean volume of C j . Based on experimental data obtained using the software LattE [14] and the Online Encyclopedia of Integer Sequences [19], we make the following conjecture:
Conjecture 11. For j ≥ 2, Vol(C j ) is equal to the number of labeled connected graphs on j − 1 vertices. 1
7 .
7Directions for further investigation 7.1. Cones over hypersimplices. We can view the cone K of the previous section as a cone over a "half-open" version of the second hypersimplex ∆(2, n) := (x 1 , . . . , x n ) ∈ [0, 1] n : n i=1
After a preprint of the current article was made public, Matjaz Konvalinka and Igor Pak communicated to us that they resolved Conjecture 11 by a direct combinatorial argument.
We thank Carla Savage for pointing out several results in the literature that were relevant to our project. This research was partially supported by the NSF through grants DMS-0810105 (Beck), DMS-0758321 (Braun), and DGE-0841164 (Le).
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"Hugo Do Nascimento \nUniversidade Federal de Goiás -Instituto de Informática Alameda Palmeiras\nCampus Samambaia CEP74690-900-Goiânia -GoiásQuadra D\n",
"Wellington Martins \nUniversidade Federal de Goiás -Instituto de Informática Alameda Palmeiras\nCampus Samambaia CEP74690-900-Goiânia -GoiásQuadra D\n"
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"Universidade Federal de Goiás -Instituto de Informática Alameda Palmeiras\nCampus Samambaia CEP74690-900-Goiânia -GoiásQuadra D",
"Universidade Federal de Goiás -Instituto de Informática Alameda Palmeiras\nCampus Samambaia CEP74690-900-Goiânia -GoiásQuadra D"
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Reduction operations are extensively employed in many computational problems. A reduction consists of, given a finite set of numeric elements, combining into a single value all elements in that set, using for this a combiner function. A parallel reduction, in turn, is the reduction operation concurrently performed when multiple execution units are available. The current work reports an investigation on this subject and depicts a GPU-based parallel approach for it. Employing techniques like Loop Unrolling, Persistent Threads and Algebraic Expressions to avoid thread divergence, the presented approach was able to achieve a 2.8x speedup when compared to ???, using a generic, simple and easily portable code. Experiments conducted to evaluate the approach show that the strategy is able to perform efficiently in AMD and NVidia's hardware, as well as in OpenCL and CUDA.
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10.1109/wscad.2018.00013
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[
"https://arxiv.org/pdf/1710.07358v1.pdf"
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1710.07358
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fb6986f0b3b8fe471bc524b59487a0aedd2da879
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A Fast and Generic GPU-Based Parallel Reduction Implementation
October 23, 2017
Walid Jradi
Universidade Federal de Goiás -Instituto de Informática Alameda Palmeiras
Campus Samambaia CEP74690-900-Goiânia -GoiásQuadra D
Hugo Do Nascimento
Universidade Federal de Goiás -Instituto de Informática Alameda Palmeiras
Campus Samambaia CEP74690-900-Goiânia -GoiásQuadra D
Wellington Martins
Universidade Federal de Goiás -Instituto de Informática Alameda Palmeiras
Campus Samambaia CEP74690-900-Goiânia -GoiásQuadra D
A Fast and Generic GPU-Based Parallel Reduction Implementation
October 23, 2017
Reduction operations are extensively employed in many computational problems. A reduction consists of, given a finite set of numeric elements, combining into a single value all elements in that set, using for this a combiner function. A parallel reduction, in turn, is the reduction operation concurrently performed when multiple execution units are available. The current work reports an investigation on this subject and depicts a GPU-based parallel approach for it. Employing techniques like Loop Unrolling, Persistent Threads and Algebraic Expressions to avoid thread divergence, the presented approach was able to achieve a 2.8x speedup when compared to ???, using a generic, simple and easily portable code. Experiments conducted to evaluate the approach show that the strategy is able to perform efficiently in AMD and NVidia's hardware, as well as in OpenCL and CUDA.
Introduction
Widely used as a basic subroutine for a number of algorithms such as Counting Sort [6], Stream Compaction [2], Golden Section and Fibonacci Methods [18] and Radix Sort [6,Chapter 8.3].
The remainder of this paper is structured as follows. Section 1.1 presents some basic concepts. Section 2 briefly describes the techniques currently in use. Section 3 explains our approach. Section 4 details the experimental environment and the results. Finally, Section 5 gives general remarks about the presented strategy.
Problem Definition
Formally, a reduction can be defined as follows [24]: Given a set X with n values, X = {x 0 , x 1 , ..., x n−1 }, compute x 0 ⊗ x 1 ⊗ ... ⊗ x n−1 . The associative operator ⊗ (also known as combiner function) can be (but is not limited to) any one of the set {+, ×, ∧, ∨, ⊕, ∩, ∪, max, min}.
Consider the pseudo code shown in Algorithm 1. At first glance, it seems that the algorithm is inherently sequential, since the variable accumulator depends on the value computed in the previous step, preventing any attempt of parallelization. However, it is possible to avoid this problem by making use of two basic properties of addition and multiplication operations: Associativity and Commutativity 2 .
Algorithm 1: Summation(A)
Input: A set A = {a 1 , a 2 , . . . , a n } of numeric elements Output: The sum of all elements 1 accumulator ← 0 2 for i ← 1 to n do 3 accumulator ← accumulator + a i 4 return accumulator • Associativity means that, given three or more numbers, they can be linked in any order without changing the final result. Taking the sum as an example, it's possible to do a 1 + a 2 and, then, add a 3 , and the result will be the same as doing a 3 + a 2 and then adding a 1 . Formally, we have (a 1 + a 2 ) + a 3 ≡ a 1 + (a 2 + a 3 );
• Commutativity ensures that no matter the order in which an operation on two numbers a 1 and a 2 is performed, the result will always be the same. Formally, for multiplication, we have a 1 · a 2 ≡ a 2 · a 1 .
Considering that the order in which the elements are combined does not affect the final result 3,4 , these two properties can be used, dividing the problem into smaller subproblems and these, in turn, solved in parallel. After solving each subproblem, the partial results are combined to produce the final result. Figure 1 illustrates the process using the associative operator "+" in an array with 16 elements. 2 Other two properties, Neutral Element and Closeness, guarantee, respectively, that any number added to zero results in the number itself, and when we add/multiply two or more numbers within the same set (natural, for example), the result will always be a number within the same set. 3 Although, mathematically, this is true for numbers in any set, in computational terms things are a little more complicated. For instance, these properties hold for the set of integers, but the same does not happen for the floating point numbers due to the inherent imprecision that arises when combining (adding, multiplying, etc.) numbers with different exponents, which leads to the absorption of the lower bits during the combine operation. As an example, mathematically the result of (1.5 + 4 50 − 4 50 ) is always the same, no matter the order the terms are added, whereas the floating point computed value can result in 0 or 1.5, depending on the sequence in which operations are performed [7,10,15,21]. 4 Note that, although this is a complicating factor when a large numerical precision is necessary, it did not actually preclude its application in a problem when the accumulated error using single-precision floats did not exceed a certain pre-defined threshold. On the other hand, if such precision becomes necessary, the problem could be greatly minimized by adopting the use of double-precision floating points (which potentially can decrease the application performance for certain GPU models) or using some strategies to reduce truncation errors, like the one proposed by Kahan [17], among others.
Parallel Reduction in GPUs
Since the arrival of programmable GPUs, some strategies to accelerate the reduction operation on such devices have been proposed. The two most well known are those described by Mark Harris [14] and Bryan Catanzaro [3]. Most recently, Justin Luitjens [19] presented some improvements to the strategies described in [14]. Unfortunately, the strategies adopted by [14] and [19], although very efficient, are limited to hardware and software provided by NVidia, restricting their use.
On the other hand, the proposal of Catanzaro [3] is based on the open standard OpenCL [11], adopted by a myriad of manufacturers, what makes it portable. Nevertheless, the code presented in [3] also has a weakeness, as it does not adopt some strategies that could significantly improve its performance.
This section details how the associative and commutative properties can be used to implement efficient parallel reductions on GPUs. As highlighted at the end of Section 1.1, the basic idea is to "split" the problem into smaller pieces and solve them in parallel. However, the execution environment (GPU hardware) imposes some restrictions that must be considered to maximize the speedup. Therefore, the details of how GPUs are organized [5,27] will dictate the choices from now on.
The approaches of Harris [14] and Catanzaro [3] to deal with reductions in GPUs operate in a pretty similar way, using a tree-based structure.
One of the aspects to be considered is the number of elements in the collection (vector) in which the reduction will be applied. If this amount is sufficiently small and can be stored in the local memory of each SM, then the reduction becomes quite simple. In [3], Catanzaro presents some strategies for this case and conducts performance comparisons between them. Then, after describing how reductions can be efficiently performed in small sets, Catanzaro shifts his focus to the cases in which a large volume of data must be handled. Three strategies are presented and a winner, called "Two-Stage Parallel Reduction", is elected. Harris [14] deals only with parallel reduction in large datasets.
Our approach is mainly based on a proposal from Catanzaro [3]. Therefore, a more detailed description of it is presented. First, however, we also give an explanation of the strategies by Harris [14] and Luitjens [19], since some ideas for speeding up the computation came from them. Hence, unlike the rest of the thesis, here their original code is presented, and not just the pseudo code.
Mark Harris' Work
The work presented by Harris [14] focuses on techniques for performing reductions of large data volumes. The author shows, through successive versions of the same algorithm, how bad decisions or an incorrect way of mapping the problem to the target platform can negatively impact the application performance.
Problems like shared memory bank conflict, lack of communication between thread blocks (making it impossible for a kernel to reduce a large array at once) and highly divergent warps are addressed. Starting with a naive version, step by step improvements are described, reaching an implementation 30x faster than the first one. Next, we show how the author achieved such speedups.
Harris performed experiments using a G80 GPU. This video card has a 384bit memory interface, with a 900 MHz DDR memory, which leads to a theoretic 384 * 1800 8 = 86.4GB/s of memory bandwidth 5 . All tests were conducted using a vector with 2 22 (4M) integer values.
As a result of all the applied optimizations, the final version of the code runs in 0.268ms and the memory bandwidth usage reaches 62.671GB/s. All these improvements are summarized in Table 1.
Time (ms) Memory Bandwidth (GB/s)
Step speedup
Justin Luitjens' Work
In [19] Luitjens shows how a new feature of the NVidia's Kepler (and newer) GPU architecture can be used to make reductions even faster when compared to the strategies presented in [14]: the shuffle (SHFL) instruction. Usually, work-items inside the same SM use the local (shared) memory when they need to communicate (exchange information). This involves a three-step process: writing the data to local memory, perform a synchronization barrier and then read the data back from local memory. The Kepler and newer architectures implement the shuffle instruction, which enables a work-item to directly read private data from another work-item in the same wave-front. According to the author, there are four main advantages in using this instruction:
• It ultimately allows work-items inside a wave-front to collectively exchange or broadcast data;
• It replaces the three-step process by a single instruction, effectively increasing the bandwidth and decreasing the latency;
• It does not use the local memory at all;
• A sync barrier is implicit in the instruction and, hence, a synchronization step inside a workgroup is not necessary. Figure 2 shows how this instruction can be used to build a reduction tree. As pointed out by Luitjens, this figure only includes the arrows for the workitems actually doing useful work. All work-items are indeed shifting values even though these values are not necessary in the reduction process. Figure 2: Parallel reduction using the shuffle instruction (extracted from [19]).
Using this instruction, several versions of the reduction were proposed, implemented and compared. However, although Luitjens states that the adopted strategies lead to faster reductions than those described by Harris [14], no comparative studies between the two approaches were conducted.
Bryan Catanzaro's Work
Now, we describe Catanzaro's two-stage parallel reduction approach for large datasets, as presented in [3].
The technique is based on dividing the data set in p pieces (or "chunks"), where p is large enough to keep all GPU cores busy. It is also necessary to limit the number of work-items to the maximum amount that the GPU can handle in total without having to switch between them (from now on, that maximum will be called GS -or global size). Each chunk is then processed by a work-group.
Since the sum operation has the properties of associativity and commutativity, each work-item can perform its own reduction sequentially and intercalary with the others. A work-item takes, as the starting point, its global identifier and accumulates, in a private variable, its partial sum, skipping GS positions at every step in the vector stored in the GPU's global memory.
After having completed a pass through the data set, the work-items in each workgroup write the result of their own reduction in a scrap vector located in local/shared memory which, in turn, will also be reduced in parallel. At the end of the process, each working group will have its own scrap containing, in its position 0, the result of the reduction so far. This partial result is then copied to another vector, this time stored in the GPU global memory, which size must be equal to -SM-. The first stage is then complete. Its source code, extracted from [3], is presented in Listing 1. The second stage is simpler. Since now there is a vector with |SM | elements in the global memory -with the result of a partial sum in each position -just the first |SM | work-items of the first SM copy their corresponding value to an array allocated in local memory. Then the work-items perform a new parallel sum of the elements in the vector. After copying the value in position 0 back to global memory, the reduction is finally complete.
The next sections detail some advanced techniques to further explore parallelism and that were extensively used in the present work.
Loop Unrolling
Loop Unrolling (also known as Loop Unwinding and Loop Unfolding) is an optimization technique -performed by the compiler or manually by the programmer -applicable to certain kinds of loops in order to reduce (or even prevent) the occurrence of execution branches and minimize the cost of instructions for controlling the loop [1,8,16,25]. Its goal is to optimize the program's execution speed at the expense of increasing the size of the generated code (space-time tradeoff ). It is easily applicable to loops where the number of executions is previously known, like routines of vector manipulation where the number of elements is fixed.
Basically the technique consists in the reuse of the sequence of instructions being executed within the loop, so as to include more of an iteration of the code every time the loop is repeated, reducing the amount of these repetitions. This reuse is done by manually replicating the code inside the loop a certain amount of times or through the "#pragma unroll n" 6 positioned immediately before the beginning of the loop. The number of times the loop is unrolled is called Unrolling Factor and, with the pragma directive, it is given by the parameter "n".
It is worth noting that with the pragma directive we leave the decisions of how the loop should be unrolled to the compiler, which may lead to a not so optimized resulting code. In the experiments performed as part of this work, the best results were always achieved using manual loop unrolling.
As an example, consider the C code shown in Listing 2, which simply multiplies the elements of an array by its index (a i ← a i · i). In this example, we call L the loop size and F its unrolling factor. L here is equal to 100. The two extra lines of code and the "i += 3 " in Listing 3 performs the desired three-fold (F = 3) manual loop unrolling.
As it can be seen, the L F ratio does not necessarily need to be an integer. If it admits a remainder, the compiler can (since the number of iterations is previously known at compile time) add extra code to the end of the unrolled generated code in order to ensure its correctness.
Unrolling, when applicable, offers several advantages over non-unrolled code. Besides the decrease in the number of iterations, an increase occurs in the amount of work done each time through the loop. This also open ways for the exploration of parallelism by the compiler in machines with multiple execution units, since each instruction within the loop can be handled by an independent thread.
However, these are only the most easily perceivable benefits. Agner Fog [8] listed several others, as well as some observations about when this technique should be avoided. Such factors (advantages and disadvantages) must be considered by the programmer when deciding to use loop unrolling or not.
Persistent Threads
Since the launch of the first programmable GPUs and with all its basic architecture inspired by the SIMD model, the "Single Instruction Multiple Thread" (SIMT) and "Single Program Multiple Data" (SPMD) paradigms have become standards de facto. Both seek to hide the details of the underlying hardware where the code runs, attempting to facilitate the painful task of development [12].
Gupta et al. [12] argue that the usage of these "traditional" paradigms greatly limits the actions of the programmer, because all control of the execution flow is in the power of the scheduler's video card. This programming style, which delegates all the decisions to the scheduler, is called by the authors as "non-PT", or "non-Persistent".
It requires that the software developer abstracts units of work to virtual work-items. Since the number of wave-fronts to create is based on the number of virtual work-items, during a kernel launch usually there are several hundreds of even thousands more wave-fronts to be executed than the amount of physical processing elements to assign them to.
Such scheduling of wave-fronts is performed by the scheduler and the programmer has no means to interfere in the process, e.g., how, where, when and in which order the work-groups will be assigned.
Gupta et al. claim that, while these abstractions reduce the effort for new developers in the GPGPU field, they also create obstacles for experienced programmers, who normally face problems for which workload is inherently irregular, therefore making it much more difficult to efficiently parallelize when compared to problems whose parallel solution is more regular.
According to Gupta et al., this uncovers a serious drawback of the current SPMD programming style, which is not able to ensure order, location and timing. It also does not allow the software developer to regulate these three parameters without completely avoiding the GPU scheduler.
Thus, to overcome these limitations, developers have been using a programming style called Persistent Threads ("PT"), whose low level of abstraction allows performance gains by directly controlling the scheduling of work-groups. And although this style has been in use for some time, only in 2012 it was formally introduced, described and analyzed by Gupta et al. [12]. They also list several problems when adopting the traditional style.
Basically, what the PT style change is the lifetime of a work-item [23], by letting it keep running longer and giving it much more work than in the traditional "non-PT" style [26]. This is done circumscribing the logic kernel (or part of it) in a loop, so this loop remains running while there are items to be processed.
Briefly, from the point of view of the developer, all work-items are active while the kernel is running. As a direct consequence of PT, a kernel should be triggered using only the amount of work-items that can be executed concurrently by each Streaming Multiprocessor. All these actions will prevent constant return of control to the host and the cost of new kernel invocations [23].
Gupta et al. acknowledge, however, that the technique of Persistent Threads is not a panacea, and its use should be carefully evaluated [12]. In particular, the technique fits well when the amount of memory accesses is limited (i.e., few reading/writing to global memory and a large volume of computation) and the problem being solved has not many initial input elements or the growth in the number of elements in the input set is fairly limited. Beyond these conditions, the traditional non-PT style tends to outperform the PT style.
Thread Divergence
Current GPUs are able to deliver massive computational power at a reasonably low cost. However, due to the way they are constructed, some obstacles must be overcome for the effective use of such power. One of the main and hardest obstacles to avoid is the presence of conditional statements [28] potentially leading to branches in the execution flow of the various work-items [13].
By default, GPUs try to run all the work-items inside the wave-fronts in the SIMD model. However, if the code being executed has conditional statements that lead to divergences in program flow, the divergent work-items will be stalled and its execution will only happen after the non-stalled work-items have completed their runs, which ultimately compromises the desired speedup. This phenomenon is called Thread Divergence [4,13,22,28].
Trying to circumvent this problem, some strategies have been proposed in order to minimize or even eliminate the effects of such phenomena. Among them, we cite [4,9,13,20,22,28].
-::: Wherefore it became necessary to develop a method to prevent flow divergence, which could ultimately compromise the performance of such a step of computation. The method is detailed at the end of Section 3.
The New Approach
The improvements proposed in our work focus on Steps 1 and 3 of the first stage of the reduction presented in Section 2.3. The improvements employ the same strategies proposed by Harris [14] to increase the performance of the approach originally presented by Catanzaro [3] but with appropriately chosen interventions.
In step 1 of the original implementation, the vector in global memory containing the data to be reduced is entirely traversed by the work-items, each one performing its own reduction. This step already uses the "Persistent-Thread" strategy, but its performance can be improved by adopting loop unrolling (Section 2.4). As it can be seen, instead of doing the unroll when the data is in local memory, as proposed by Harris [14] (Listings ?? and ?? of Section 2.1), our improvement performs the unroll in the global memory.
The code presented in Listing 4 shows the modified loop, assuming an unrolling factor (F) equals to 4, iGlobalID as the work-item global identifier and iLength as the number of elements to be reduced. A special attention must be given to how the data is brought from the global memory (aVector ) to the private memory (accumulator ), through the use of algebraic expressions that prevent reading from invalid memory locations, thus avoiding the usage of "ifs" and potential divergences in the execution flow. The expression i n < iLength expands to integers 1 or 0 whether it is, respectively, true or false. In the first case (i n < iLength) * (aV ector[i n ]) is interpreted as (1) * (aV ector[i n ]), adding the value stored in location i n to the partial sum (accumulator ). In the second case, the expression is interpreted as (0) * (aV ector[0]), ensuring that -regardless of the data stored in the first position of the vector -value 0 is added to accumulator, keeping the partial sum correctness.
At the begining of Step 3, the resulting values of the previous sums are already stored in the local memory of the SMs. Then, each SM performs its own local reduction with its work-items.
In the solutions presented by Harris [14] and Catanzaro [3], in this step all work-items are kept synchronized through the use of barriers. However, with minor conceptual changes, it is possible to completely eliminate the overhead caused by the barriers, not only in the last 6 iterations of the loop, as proposed by Harris [14].
Our strategy is to use algebraic expressions to keep all the work-items in the same execution step, maintaining its desired behaviour and algorithm correctness.
Consider the highly divergent code presented in Listing ?? (Section 2.6). Using a simple algebraic expression, it can be rewriten in order to completely eliminate the conditional statement and still return the right result of the comparison, as can be seen in Listing 5.
Listing 5: Algebraic "if-then-else" int s m a l l e s t V a l u e ( int a , int b ) { return ( a < b ) * a + ( a >= b ) * b ; } Note that the two boolean operations ((a < b) and (a >= b)) are mutually exclusive, being interpreted internally by the compiler as 0 (false) or 1 (true). So, assuming that a is smaller than b, the result of the algebraic operation is (1) * a + (0) * b which, ultimately, will return only the value of a.
The same strategy can be applied to lines 18 to 24 of Listing 1, that represent the third step of the first stage. The new code is shown in Listing 6, where iLocalSize stores the number of active local work-items and iLI represents the work-item's local identifier. Here, in each iteration of the loop, iP os is divided by 2 (iPos ¿ ¿ = 1 ) and bF lag is expanded to either 1 or 0, thus reducing by half the number of work-items doing a useful job. If, for the current work-item, the expression iLI < iP os becomes true, then the expression in the last line will be interpreted as scratch[iLI]+ = (1) * (scratch[iLI+(1) * iP os]), ensuring that the value stored in position iLI + iP os will be added to the value in position iLI. On the other hand, if the expression becomes false, it will be interpreted as scratch[iLI]+ = (0) * (scratch[iLI + (0) * iP os]), ensuring that the value in position iLI will not be considered. Since all work-items are always in the same step of computation -doing exactly the same job (useful or not), independently of being in the same wavefront -sync barriers are unnecessary. Table 2 and Figures 3 and 4 represent the performance gains achieved against the algorithm described in [3], where F = 1 is the runtime of the original code. The machine used in the tests was the same one presented in Section ??.
Computational Experiments
All tests were run on two vectors, one of integers and one of single precision floating points, containing 5533214 elements. There were no measurable differences between the two vector types.
The times listed in Table 2 were obtained with the OpenCL profiler CodeXL, version 2.0.12400.0, and are the averages of five consecutive executions for each F .
As can be seen, these results show that the version of the algorithm with F = 8 reached a speedup pretty close to 2.8x, when compared with the proposal of [3]. It may also be noted that such speedup stabilizes around this value (F = 16 provided just over 1.5% gain when compared to F = 8).
The same code was implemented in CUDA and tests were performed against the Kernel 7 of Harris presented in Section 2.1. The GPU used in the experiments was a Tesla C2075 with 6GB of memory. The architecture of such a video card provides 448 CUDA cores, a GPU clock of 575MHz and a shader clock of 1150Mhz. Its memory is clocked at 750MHz (3.0GHz effective).
The experiments employed the same two vectors containing 5533214 elements (integers and single precision floating points). Several values of the un- rolling factor (F ) were used in order to find the optimal value for such a video board. It was determined that up to F = 6 the performance gains were substantial and, with F ≥ 8, the gains were very discrete. According to this, all experiments were conducted using F = 8. Table 3 presents the running time (in milliseconds) of both approaches and the percentage of performance (given by the formula 100 * Tnew T k7
).
Time -Kernel 7 Time -New Approach % of Performance 0.17766 ms 0.17867 ms 99.4 Table 3: Parallel reduction execution times -new approach (with unrolling factor equals to 8) compared against Harris' code.
General Remarks
Reduction operations are widely employed in many computational problems. This chapter showed how such operations can be performed in a parallel fashion using graphics processing units and detailed the main approaches for them nowadays. All parallel reduction techniques currently in use suffer from some basic issues. Several only reach their peak performance by employing proprietary strategies and/or technologies, what ends up limiting their use to the platform for which they were designed. Others, though generic, do not adopt certain procedures that could increase their performance without loss of generality.
The strategy presented here combines the best of both worlds: It is generic enough to be used with both CUDA and OpenCL and can run on hardware of the two major GPU manufacturers with minimal changes, just being adapted to the particularities of each platform. The implemented code, besides simpler, offered a performance equivalent to the best strategy described by Harris [14].
A good performance of this routine is essential for the efficient execution of the macroscopic urban traffic assignment algorithm described in Chapter ??, since it is used on two occasions: in the computation of shortest paths and in the golden ratio method.
Figure 1 :
1Parallel reduction -associative reduction tree.
{ r e s u l t [ g e t g r o u p i d ( 0 ) ] = s c r a t c h [ 0
00e x ] = ( mine < o t h e r ) ? mine : o t h e r ; } b a r r i e r (CLK LOCAL MEM FENCE ) ; } i f ( l o c a l i n d e x == 0 )
Listing 2 :
2Multiplying elements in a vector f o r ( int i = 0 ; i < 1 0 0 ; i ++) { a [ i ] = a [ i ] * i ; } It's possible to significantly improve the execution speed of this algorithm by unrolling it, as shown in Listing 3
<i L e n g t h ) * ( aVector [ i 1 ])+ ( i 2 <i L e n g t h ) * ( aVector [ i 2 ])+ ( i 3 <i L e n g t h ) * ( aVector [ i 3 ] ) ) ; }
I ] += ( bFlag ) * ( s c r a t c h [ i L I + ( bFlag ) * i P o s ] ) ; }
Figure 3 :
3Chart of the parallel reduction execution times.
Figure 4 :
4Chart of the parallel reduction speedup.
F Time (ms) SpeedupMemory Bandwidth (GB/s) Bandwidth Usage (%)1 0.249780
1
88.6094002722
26.63
2 0.173930
1.4360949807
127.2515149773
38.24
3 0.139260
1.7936234382
158.9318971708
47.76
4 0.127700
1.955990603
173.3191542678
52.08
5 0.113930
2.1923988414
194.2671464935
58.37
6 0.100810
2.4777303839
219.5502033528
65.97
7 0.093740
2.6646042245
236.1089822914
70.95
8 0.089490
2.7911498491
247.3221142027
74.32
16 0.088160
2.8332577132
251.0532667877
75.44
Table 2 :
2Parallel reduction execution times. New approach compared against Catanzaro's original code.
Memory bandwidth basically determines how fast is the memory. Usually, it is measured in gigabytes per second (GB/s). The more bandwidth of the memory and the more it is explored by the running program, the faster the computation.
A directive pragma is a language construct that provides additional information to the compiler, specifying how to process its input. This additional information usually is beyond what is conveyed in the language itself.
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|
[] |
[
"Limit Cycle Oscillations, response time and the time-dependent solution to the Lotka-Volterra Predator-Prey model",
"Limit Cycle Oscillations, response time and the time-dependent solution to the Lotka-Volterra Predator-Prey model"
] |
[
"M Leconte \nKorea Institute of Fusion Energy (KFE)\n34133DaejeonSouth Korea\n",
"P Masson \nIndependent researcher\n\n",
"Lei Qi \nKorea Institute of Fusion Energy (KFE)\n34133DaejeonSouth Korea\n"
] |
[
"Korea Institute of Fusion Energy (KFE)\n34133DaejeonSouth Korea",
"Independent researcher\n",
"Korea Institute of Fusion Energy (KFE)\n34133DaejeonSouth Korea"
] |
[] |
In this work, the time-dependent solution for the Lotka-Volterra Predator-Prey model is derived with the help of the Lambert W function. This allows an exact analytical expression for the period of the associated limit-cycle oscillations (LCO), and also for the response time between predator and prey population. These results are applied to the predator-prey interaction of zonal density corrugations and turbulent particle flux in gyrokinetic simulations of collisionless trapped-electron model (CTEM) turbulence. In the turbulence simulations, the response time is shown to increase when approaching the linear threshold, and the same trend is observed in the Lotka-Volterra model.
|
10.1063/5.0076085
|
[
"https://arxiv.org/pdf/2110.11557v3.pdf"
] | 239,616,189 |
2110.11557
|
7bf4483d8b7f36ad79ee517b06405f78d4151fa0
|
Limit Cycle Oscillations, response time and the time-dependent solution to the Lotka-Volterra Predator-Prey model
January 13, 2022
M Leconte
Korea Institute of Fusion Energy (KFE)
34133DaejeonSouth Korea
P Masson
Independent researcher
Lei Qi
Korea Institute of Fusion Energy (KFE)
34133DaejeonSouth Korea
Limit Cycle Oscillations, response time and the time-dependent solution to the Lotka-Volterra Predator-Prey model
January 13, 2022a Author to whom correspondence should be addressed: [email protected]
In this work, the time-dependent solution for the Lotka-Volterra Predator-Prey model is derived with the help of the Lambert W function. This allows an exact analytical expression for the period of the associated limit-cycle oscillations (LCO), and also for the response time between predator and prey population. These results are applied to the predator-prey interaction of zonal density corrugations and turbulent particle flux in gyrokinetic simulations of collisionless trapped-electron model (CTEM) turbulence. In the turbulence simulations, the response time is shown to increase when approaching the linear threshold, and the same trend is observed in the Lotka-Volterra model.
Introduction
Limit-cycle oscillations occur in many areas of Biology and Physics [1,2]. One popular model to describe these phenomena is the Lotka-Volterra model [1,3] and its extensions. Lotka [4] and Volterra [5] derived it independently to describe the nonlinear interaction of predator and prey populations. Therefore, it is now widely known as the 'Predator-Prey model'. In Physics, many nonlinear interactions can be described with this model. In Plasma Physics, on which we focus here, it describes the interaction between axisymmetric self-generated flows (zonal flows), which acts as the predator, and the microturbulence -the prey -that drives them [6]. In the plasma turbulence context, this model is a building-block for more extended models of the Low to High confinement (L-H) transition in fusion plasmas such as the Kim-Diamond model [7]. See e.g. Ref. [8] for a review. Refs. [9] and [10] analyzed such extended predator-prey models -which are not integrableusing bifurcation theory. The limit-cycles associated to the Lotka-Volterra model have one particular feature that is interesting: They exhibit multiple time-scales. Hence, such models may be useful to understand certain types of relaxation-oscillations, and intermittent transport in turbulent plasmas. The well-known Van-der-Pol oscillator [11] and the closely-related Rayleigh oscillator [12] are also examples of a system showing relaxation-oscillations. During the L-H transition, a dithering phase (also called Intermittent phase or I-phase) is often observed experimentally when the heating power is slowly ramped up [13,14]. It can be understood as limit-cycle oscillations (LCO) between turbulence energy and zonal flow energy, as observed in gyrokinetic simulations [15,16]. It is this feature that we focus on in this work. More precisely, we focus on the response-time between these two quantities. Refs. [17,18,19] showed -based on a different model -that this response-time is a key quantity to understand nonlinear interactions. In the simplified framework of Drift-wave turbulence, there are several candidates to explain the interaction between zonal density corrugations and the turbulence. One of the authors (M. Leconte) proposed a model based on the nonlinear modulation of the transport crossphase between density and potential perturbations [20,21]. Another model, based on stochastic noise due to turbulence was proposed in Ref. [22]. The main results of this article are: (i) a new analytical solution opens the possibility of directly fitting experimental data of LCO to extract its key predator-prey features, with only two fitting parameters, c.f. Eqs. (14,15,16,17). (ii) the response time between turbulence energy and zonal energy increases as marginality is approached, in collision-less trapped electron mode turbulence simulations. A similar trend is observed in the Lotka-Volterra model. The rest of this article is organized as follows: In Section 2, we describe the Lotka-Volterra Predator-Prey model, and we derive its time-dependent solution analytically. In Section 3, we apply the newly-found solutions to understand the predator-prey dynamics between zonal density perturbations and the turbulent particle flux observed in global gyrokinetic simulations of collision-less trapped-electron mode turbulence (CTEM) [23,24,25,26]. Finally in Section 4, we discuss the results and give a conclusion.
Model
We consider the following Lotka-Volterra Predator-Prey model:
x = γx − α 1 xy (1) y = α 2 xy − µy(2)
Here, x denotes the prey population, and y denotes the predator population, andu = du/dt, with u = x, y, denotes the time derivative. The parameter γ denotes the prey growth-rate (birth-rate) in the absence of predator, and µ is the predator damping rate (death-rate) in the absence of prey. The coefficients α 1 , α 2 are positive constants. In applications to plasma turbulence, the predator is usually taken as zonal flow energy, and the prey as turbulence energy [6,7,8]. Here, we take the predator as zonal density corrugations driven by nonlinear modulation of the transport crossphase [20] and the prey as turbulence energy. In section 3, this model will be applied to gyrokinetic simulations of CTEM turbulence. There are several candidate mechanisms for the nonlinear generation of zonal density by the turbulence [20,22] based on fluid models, although a specific application to CTEM has not been proposed yet. For this reason, we treat the Lotka-Volterra as a phenomenological model here. However, deriving a predator-prey like reduced-model directly from the bounce-averaged gyrokinetic equation would be an important task. One could use a similar method as in Ref. [19], where a derivation of the traffic-jam model from the nonlinear gyrokinetic equation is sketched.
Normalization of the model
It is convenient to re-define the variables, so as to decrease the number of independent parameters [27]. We make the following change of variables:
X = α 2 µ x,(3)Y = α 1 µ y(4)
Using this change of variables, one obtains after some algebra:
X = 1 δ X − XY,(5)Y = XY − Y,(6)
with δ = µ/γ, and where the time has been re-scaled to µt → t. Note that for typical values of parameters, δ 1, but our analysis is valid for any value of δ.
Energy conservation
It is well-known that the system (5,6) has an invariant associated to its limitcycle. Here, we briefly review the derivation of this invariant (a Lyapunov function). Dividing Eq. (6) by Eq. (5), one obtains:
dY dX = − Y X · 1 − X 1 δ − Y(7)
Since this is a separable ordinary differential equation, one obtains -after some algebra -the following energy integral:
X − ln X + Y − 1 δ ln Y = E,(8)
where E = Cst is the total energy determined by initial conditions X(t = 0) and Y (t = 0). It can be shown that this quantity is actually a generalized Hamiltonian [28]. Contours of the Hamiltonian are shown for two values of the parameter δ = 0.5 and δ = 0.2 [ Fig. 1]. For small values of δ [ Fig. 1b], one observes that the limit cycles become more elongated in the Y direction, i.e. the predator population has a very large amplitude compared to the prey population. Up to now, the nonlinear solutions to system (5,6) are thus obtained in an implicit form, through their representation as a projection of the 4D dynamical phase-space (X,Ẋ, Y,Ẏ ) onto the 2D space (X, Y ). One can make the analogy with the Jacobi elliptic functions (c.f. Appendix).
In the following, we will go one step further, to obtain the nonlinear timedependent solutions in explicit form. It was shown in Ref. [27] that the energy integral can be used to express either of the variable X or Y in terms of the other, using the Lambert W function [29]. Note that this result was also obtained independently in the latter Reference (page 336 of [29]). This function is solution to the transcendental equation: W e W = u. Applied to the energy integral Eq. (8), one obtains after some algebra:
X(Y ) = −W j (−Y − 1 δ e Y −E ),(9)Y (X) = − 1 δ W j (−δX −δ e δ(X−E) ),(10)
where the subscript j = 0, −1 denotes the relevant branch of the Lambert W function. The W 0 function is known as the principal branch, while the W −1 function is called negative branch.
Time-dependent solutions
Let us first consider the solution for the predator population Y (t). Starting from the predator evolution Eq. (6). we write it in the form:
dt = g(X, Y )dY(11)
Here, the function is given by g(X, Y ) = − 1 (1−X)Y . Now, we use Eq. (8) to express the prey X in terms of the predator Y and total energy E:
X = −W j (−Y − 1 δ e Y −E ),(12)
where W j denotes the Lambert W function [29], and j = 0, −1 is the associated branch.
Integrating both sides of Eq. (11) yields:
t = G j LV (Y, E)(13)
where we call G j LV the 'second Lotka-Volterra' integral, defined as:
G j LV (Y, E) = T Y min − Y Y min dY Y [1 + W j (−Y − 1 δ e Y −E )] ,(14)
where W j = W 0 or W j = W −1 , depending on the branch of the Lambert W function considered. The integrand of the Lotka-Volterra integral (14) for the branches j=0,-1 is shown v.s. Y for different values of the parameter δ, for an 14) and (17).
energy of E = 3−ln 2 [ Fig. 3]. This value of energy corresponds to the initial conditions (X 0 , Y 0 ) = (2, 1). Note the vertical asymptotes corresponding to the minima and maxima of the predator population Y . The integration constant T Y min is given by expression (B2) in Appendix. Next, we invert Eq. (13) to obtain the time-dependent solution Y (t, E) = Λ preda j (t, E) for the predator population, where the function Λ preda
j (t, E) is given by: Λ preda j (t, E) = G −1 LV (t, E),(15)
where the superscript −1 denotes the function inverse (not to be confused with the index j = 0, −1 of the branch), and we use the shortcut notation G LV = G j LV . Care must be taken when inverting Eq. (13), because the real-valued Lambert W function has two branches: W 0 (x) and W −1 (x). As a second step, one applies a similar procedure for the prey population X,
dt = f (X, Y )dX, with f (X, Y ) = 1 ( 1 δ −Y )X , to obtain the time-dependent solution X(t, E) = Λ prey j (t, E) for the prey population: Λ prey j (t, E) = F −1 LV (t, E),(16)
where the superscript −1 denotes the function inverse, and we define the first Lotka-Volterra integral F LV as: Figure 4: a) Inverse functions G j LV (Y, E) (red) and F j LV (X, E) (blue) given by Eqs. (14) and (17)
F j LV (X, E) = T X min − X X min dX 1 δ X [1 + W j (δX −δ e δ(X −E) )] ,(17)
Response-time comparison with gyrokinetic simulations of CTEM turbulence
The collisionless trapped-electron mode (CTEM) is an instability due to the electron toroidal precession-drift resonance -a process similar to inverse Landau damping -in the low-collisionality regime [30]. Its sources of energy are the electron temperature gradient and the density gradient. CTEM instability is in the ion-scale range, with a typical poloidal wave-number k θ ρ i ∼ 1, where ρ i = √ m i T i /eB is the ion-gyroradius, T i is the ion temperature and other notations are standard. From collisionless trapped-electron mode (CTEM) gyrokinetic simulations, zonal density perturbations δn ez and electron particle flux Γ e are obtained. Figure 5 shows the zonal density corrugations v.s. radius and time. One clearly observes the radially oscillating zonal pattern known as 'staircase' [17,18,19,23,24,31,32,33,34,35,36,37,38]. The time-trace of electron particle flux Γ e and zonal density energy n 2 ez are shown [Fig 6a]. These quantities are spectral averages, e.g. Γ e (t) = krρ i ∈[0.4,1] |Γ e (t, k r )|, around the radial wavenumber k r ρ i = 0.78, which is a characteristic scale of the zonal staircase pattern. The simulations are performed with the gyrokinetic code gKPSP [39] which solves the nonlinear gyrokinetic equations for ions [40] and bounce-averaged kinetics for trapped electrons [41]. In the simulations, the equilibrium gradients are R/L n = 2. At mid-radius, the inverse aspect ratio is r/R = 0.18, safety factor q = 1.4, magnetic shearŝ = q r/q = 0.78, T i = T e . Hydrogen is the main ion m i /m e = 1836 and plasma elongation κ = 2. The turbulent transport of this plasma is dominated by CTEM [24,25]. A limit-cycle type of dynamics between Γ e and n 2 ez is clearly observed in dynamical phase-space [Fig 6b], although its amplitude decreases with time, probably due to additional turbulent dissipation. This is probably the reason for the spiraling in Fig 6b.The Lotka-Volterra model does not take into account this additional dissipation, but the overall dynamics is similar to the model.
The response time is usually defined as the time-lag between maxima or minima of two signals. Here, for clarity, we define it as the time-lag between the first maximum of the two signals, as indicated by black arrows in 7a]. Here, the critical gradient is R/L c T = 3. In the Lotka-Volterra model, the response time is given analytically by:
τ (δ, E) = G −1 LV (Y max ) − G −1 LV ( 1 δ ),(18)
where G −1 LV (Y, E) is the second Lotka-Volterra integral (branch j = −1) given by expression (14).
For the Lotka-Volterra PP model, the response time is shown v.s. the parameter 1/δ (i.e. γ/µ), at fixed energy E = 3−ln 2 [ Fig.7b]. The analytical result is shown (full symbols) and is compared to the numerical result (open symbols) obtained by solving numerically the Lotka-Volterra Eqs. (5,6). The agreement is reasonable. One observes that both response times Fig. 7a and 7b increase with increasing drive. Conversely, both response time decrease with decreasing drive. This is consistent with a critical exponent behavior: the response time increases as marginal stability is approached, i.e. as R/L T → R/L c T (L T = L T e ) in the gyrokinetic simulation and 1/δ → 0 in the Lotka-Volterra model. Writing τ = |R/L T − R/L c T | −α and τ LV = ( 1 δ ) −β , the following scalings or obtained: α = 0.67 and β = 0.9. This trend is also predicted in the traffic-jam model for avalanches of Ref. [19] for a 1D model of zonal ion temperature corrugations. Based on this model, the scaling of a characteristic response time was given as: 1 2τ c 2 0 τ χ 2 γ max , where γ max ∼ V E×B , with V E×B the zonal flow shearing rate, c 0 the initial avalanche speed and χ 2 the heat diffusivity. Assuming that χ 2 scales like χ 2 ∼
Discussion and conclusions
Let us first discuss the analytical solution of the Lotka-Volterra model. To our knowledge, this is the first time that such a closed-form exact solution of the model was obtained. This analytical solution (14,15,16,17) may possibly be used to fit experimental data of limit cycle oscillations in fusion devices. It provides a simpler alternative compared to the method used in Ref. [16], which extracted directly predator-prey coefficients from gyrokinetic simulation data by solving the Lotka-Volterra system numerically. More precisely, our method extracts two fitting-parameters: the ratio of linear predator damping-rate to prey growth-rate δ = µ/γ, and the 'energy' E associated to the limit-cycle. This can be used by the experimental community to know if a signal is 'predator-prey', and to extract its key parameters. We sketch briefly how one could proceed: From the experimental prey & predator signals X(t) and Y (t), one could minimize the squared-difference between X(t) and Λ prey j (t), and between Y (t) and Λ preda j (t) -i.e. a least-square fit -with respect to the fitting-parameters δ and E. There are some limitations to our model. First, this model differs from the drift-wave zonal-flow model of Ref. [6] in that the self-damping term is ne-glected in the turbulence evolution equation. Although this term is important to obtain steady saturated states in the long time limit, and associated bifurcation between states, this parameter has no effect on the transient limit-cycle regime -apart from displacing its center -on which we focus here. Second, we acknowledge that we did not formally invert Eqs. (14) and (17) by solving e.g. Eq. (13) for Y . Instead, we inverted them graphically by plotting X v.s. F LV and Y v.s. G LV . The inversion could be easily accomplished by writing the Lotka-Volterra integrals (14,17) as a Taylor series, and then inverting this series, using the Taylor series inverse-function formula. This is left for future work. Third, we verified our analytical solution only for one value of the energy E and associated initial conditions (X 0 , Y 0 ) = (2, 1). It would be useful to verify it for different initial conditions (different energies), but this is beyond the scope of this article. Let us now discuss the comparison of the response-time between the Lotka-Volterra model and gyrokinetic CTEM simulations. In Ref. [44], a stochastic version of the Kim-Diamond model was studied. The temporal cross-correlation between zonal flow energy and turbulence energy was computed and the associated response-time was obtained (Fig 1c in this reference). Ref. [19] predicted -in the framework of the 'traffic-jam' model-that the response-time scales like τ ∼ |R/L T − R/L c T | −η , i.e. the response time increases as marginal stability is approached. We showed that a similar trend, i.e. τ increasing when the prey drive 1/δ decreases is obtained when applying the Lotka-Volterra predator-prey model to the interaction between zonal density corrugations and turbulence. This is also consistent with Ref. [34], where the probability of finding large scale (and thus slowly evolving) zonal structures is maximal near-marginality (cf. Fig 4 in the latter Reference). More connection between the zonal staircase generation and predator-prey modeling would be interesting for future work. In conclusion, we derived analytically the solutions to the well-known Lotka-Volterra Predator-Prey model. We applied the newly-found solutions to calculate analytically the Predator-Prey response time, and we compared its scaling with that of gyrokinetic simulations of CTEM turbulence. As the Lotka-Volterra model is one of the simplest model to describe self-organized systems, we believe that having an analytical insight into the dynamics of this model is the first step to allow a better understanding of more complicated models.
Here κ = a/b = E/b 2 is the elongation, where E = ab is the energy, proportional to the surface area of the ellipse, i.e. the action invariant, I = pdq = πab, where q = X and p=Y are the usual position and momentum conjugate variables. Hence elongation can also be viewed as 'normalized energy'. One clearly sees the analogy between Eq. (A3) -which represents limit cycles of normalized energy E/b 2 = κ -and Eq. (8) for the Lotka-Volterra system. Both represent energy conservation. The only difference is the topology of the limit-cycle, and hence the form of the energy integral. Note also that in both cases, the shape of the limit-cycle depends on the value of the energy.
Dividing by κ and defining the elliptic modulus k = 1 − 1 κ 2 , this can be written:
(1 − k 2 )X 2 + Y 2 = 1 (A4)
It is well-known that Eq. (A4) describes phase-space contours associated to the solutions X/κ = cn(t, k) and Y = sn(t, k), where cn(t, k) and sn(t, k) are Jacobi elliptic functions, and κ = 1/
√ 1 − k 2 . where T Xupper = − Xmax X min dX X/δ 1+W −1 (−δX −δ )e δ(X−E)
is the upper semi-period for the prey population. Note that similar formulas are given in Ref. [27] for the quantities T Yupper and T X lower . The exact period T LCO of the orbit (or limit cycle) is then given by the sum of the two semi-periods:
T LCO = T Xupper + T X lower = T Y lower + T Yupper (B3)
Figure 1 :
1Contours of the energy integral (8) for the Lotka-Volterra system, for a) δ = 0.5, and b) δ = 0.2. The red circle indicates the center of the associated limit-cycles.
Figure 2 :
2They correspond to the two roots of the transcendental equation W e W = u, for u real-valued. For clarity, the two branches of the Lambert W function are plotted [Fig. 2] Plot of the Lambert W function: W j (x), j = 0, −1.
Figure 3 :
3Integrand of the a) second and b) first Lotka-Volterra integrals Eqs. (
, and b) Analytical solutions Y (t) (red) and X(t) (blue) of the Lotka-Volterra system, for an energy of E = 3 − ln 2 = 2.3069, compared with numerical solutions (open symbols). The parameter is δ = 0.5.We call the analytical solutions Λ prey j (t, E), and Λ preda j (t, E) given by expressions(15) and(16)the 'Lotka-Volterra functions'. The result is shown for a value δ = 0.5 of the parameter and an energy of E 2 [Fig. 4].
Figure 5 :
5a) Colormap of zonal electron density δn ez v.s. radius and time showing the zonal staircase pattern, and b) close-up in log-scale around the time of staircase formation t = 20 − 80 R/V T i .
2, R/L T i = 2.2, and R/L T e = 4.0 − 12.0 where n, T e and T i denote the electron density, electron temperature and ion temperature, and e.g. L n = −[ 1 n dn dr ] −1 .
Fig 6a . The response-time τ is shown v.s. the distance to threshold R/L T − R/L c T [26], related to the linear growth-rate γ TEM via γ TEM ∼
Figure 6 :
6a) Time-trace of electron particle flux Γ e (t) and zonal density energy n 2 ez (t), and b) associated limit-cycle in dynamical phase-space (Γ e ,n 2 ez ).
Figure 7 :
7Comparison of response times: a) response-time between particle flux Γ e and zonal density energy n 2 ez v.s. distance to threshold, and b) response-time between predator and prey in the Lotka-Volterra model.|R/L T − R/L c T | η , as for stiff transport with η > 0 a scaling exponent, this predicts that the response-time of Ref.[19] scales like τ ∼ |R/L T − R/L c T | −η .
AcknowledgementsThe authors would like to thank P.H. Diamond, Y. Kosuga, Raghvendra Singh, X. Garbet, T. Kobayashi, Min-Jun Choi, Jae-Min Kwon and Sumin Yi for helpful discussions. We also thank the anonymous Referees for valuable comments. This work was supported by R & D Program through Korean Institute of Fusion Energy (KFE) funded by the Ministry of Science and ICT of the Republic of Korea (KFE-EN2141-7). The data that support the findings of this study are available from the corresponding author upon reasonable request.Appendix A: Analogy with Jacobi elliptic functionsFirst note that harmonic functions sin(θ) and cos(θ) can be defined as the inverse of integrals, via:respectively. Jacobi elliptic functions arise from a generalization of these formulas, when the trigonometric circle is generalized to an ellipse (although historically, Jacobi discovered these functions using a different approach). One can make an analogy between the solutions Λ preda j (t, E), Λ prey j (t, E), with j = 0, −1 and the Jacobi elliptic functions sn(t,k) and cn(t,k), respectively. Note that Jacobi elliptic functions appear naturally in describing the non-circular bounce/transit periodic motion of guiding centers in tokamak geometry (e.g. the 'banana' orbits, etc . . . ) and its applications to residual zonal flow problem[43]. One way to define the Jacobi elliptic functions is to view them as extended trigonometric functions[42]. Consider an ellipse for which:x 2 a 2 +Defining normalized variables X = x/b and Y = y/b, one obtains, after some algebra:Appendix B: Expression for the integration constants T Y min and T X minThe quantity T Y min takes the form[27]:whereis the lower semi-period for the predator population. Similarly, the quantity T X min reads:
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|
[] |
[
"Spatially separating the conformers of the dipeptide Ac-Phe-Cys-NH 2",
"Spatially separating the conformers of the dipeptide Ac-Phe-Cys-NH 2"
] |
[
"Nicole Teschmit \nCenter for Free-Electron Laser Science\nDeutsches Elektronen-Synchrotron DESY\nNotkestrasse 8522607HamburgGermany\n\nThe Hamburg Center for Ultrafast Imaging\nUniversität Hamburg\nLuruper Chaussee 14922761HamburgGermany\n\nDepartment of Chemistry\nUniversität Hamburg\nMartin-Luther-King-Platz 620146HamburgGermany\n",
"Daniel A Horke \nCenter for Free-Electron Laser Science\nDeutsches Elektronen-Synchrotron DESY\nNotkestrasse 8522607HamburgGermany\n\nThe Hamburg Center for Ultrafast Imaging\nUniversität Hamburg\nLuruper Chaussee 14922761HamburgGermany\n",
"Jochen Küpper \nCenter for Free-Electron Laser Science\nDeutsches Elektronen-Synchrotron DESY\nNotkestrasse 8522607HamburgGermany\n\nThe Hamburg Center for Ultrafast Imaging\nUniversität Hamburg\nLuruper Chaussee 14922761HamburgGermany\n\nDepartment of Chemistry\nUniversität Hamburg\nMartin-Luther-King-Platz 620146HamburgGermany\n\nDepartment of Physics\nUniversität Hamburg\nLuruper Chaussee 14922761HamburgGermany\n"
] |
[
"Center for Free-Electron Laser Science\nDeutsches Elektronen-Synchrotron DESY\nNotkestrasse 8522607HamburgGermany",
"The Hamburg Center for Ultrafast Imaging\nUniversität Hamburg\nLuruper Chaussee 14922761HamburgGermany",
"Department of Chemistry\nUniversität Hamburg\nMartin-Luther-King-Platz 620146HamburgGermany",
"Center for Free-Electron Laser Science\nDeutsches Elektronen-Synchrotron DESY\nNotkestrasse 8522607HamburgGermany",
"The Hamburg Center for Ultrafast Imaging\nUniversität Hamburg\nLuruper Chaussee 14922761HamburgGermany",
"Center for Free-Electron Laser Science\nDeutsches Elektronen-Synchrotron DESY\nNotkestrasse 8522607HamburgGermany",
"The Hamburg Center for Ultrafast Imaging\nUniversität Hamburg\nLuruper Chaussee 14922761HamburgGermany",
"Department of Chemistry\nUniversität Hamburg\nMartin-Luther-King-Platz 620146HamburgGermany",
"Department of Physics\nUniversität Hamburg\nLuruper Chaussee 14922761HamburgGermany"
] |
[] |
Atomic-resolution-imaging approaches for single molecules, such as coherent x-ray diffraction at free-electron lasers, require the delivery of high-density beams of identical molecules. However, even very cold beams of biomolecules typically have multiple conformational states populated. We demonstrate the production of very cold (T rot ∼ 2.3 K) molecular beams of intact dipeptide molecules, which we then spatially separate into the individual populated conformational states. This is achieved using the combination of supersonic expansion laser-desorption vaporisation with electrostatic deflection in strong inhomogeneous fields. This represents the first demonstration of a conformer-separated and rotationally-cold molecular beam of a peptide, and will enable future single biomolecule x-ray diffraction measurements.
| null |
[
"https://arxiv.org/pdf/1805.12396v1.pdf"
] | 103,842,434 |
1805.12396
|
163fc99e7f8921f957a765ccc8185b0c414c16c8
|
Spatially separating the conformers of the dipeptide Ac-Phe-Cys-NH 2
Nicole Teschmit
Center for Free-Electron Laser Science
Deutsches Elektronen-Synchrotron DESY
Notkestrasse 8522607HamburgGermany
The Hamburg Center for Ultrafast Imaging
Universität Hamburg
Luruper Chaussee 14922761HamburgGermany
Department of Chemistry
Universität Hamburg
Martin-Luther-King-Platz 620146HamburgGermany
Daniel A Horke
Center for Free-Electron Laser Science
Deutsches Elektronen-Synchrotron DESY
Notkestrasse 8522607HamburgGermany
The Hamburg Center for Ultrafast Imaging
Universität Hamburg
Luruper Chaussee 14922761HamburgGermany
Jochen Küpper
Center for Free-Electron Laser Science
Deutsches Elektronen-Synchrotron DESY
Notkestrasse 8522607HamburgGermany
The Hamburg Center for Ultrafast Imaging
Universität Hamburg
Luruper Chaussee 14922761HamburgGermany
Department of Chemistry
Universität Hamburg
Martin-Luther-King-Platz 620146HamburgGermany
Department of Physics
Universität Hamburg
Luruper Chaussee 14922761HamburgGermany
Spatially separating the conformers of the dipeptide Ac-Phe-Cys-NH 2
(Dated: 1 June 2018)
Atomic-resolution-imaging approaches for single molecules, such as coherent x-ray diffraction at free-electron lasers, require the delivery of high-density beams of identical molecules. However, even very cold beams of biomolecules typically have multiple conformational states populated. We demonstrate the production of very cold (T rot ∼ 2.3 K) molecular beams of intact dipeptide molecules, which we then spatially separate into the individual populated conformational states. This is achieved using the combination of supersonic expansion laser-desorption vaporisation with electrostatic deflection in strong inhomogeneous fields. This represents the first demonstration of a conformer-separated and rotationally-cold molecular beam of a peptide, and will enable future single biomolecule x-ray diffraction measurements.
Proteins are the workhorses of biological functionality in living cells and are at the heart of, for instance, the transport of oxygen, the catalysis of biochemical reactions and interactions, or the reproduction of cells and replication of DNA. This wide-ranging functionality is enabled by the unique and specific 3-dimensional (3D) structures of these systems. While every protein is composed of a sequence of the 20 amino acids encoded in RNA, the exact sequence and resulting intra-molecular interactions lead to a specific and unique 3D structure, determining a proteins functionality. Changes in this 3D structure, such as misfolding, can dramatically alter protein function with potentially wide-ranging consequences, such as neurodegenerative diseases. 1-3 Especially the strong hydrogen bonding between amino acids within the sequence has a profound effect on the resulting protein structure. [4][5][6][7] In order to study the underlying intramolecular and hydrogen-bonding interaction in detail, one often turns to studying isolated small peptide fragments in the gas-phase. [8][9][10][11][12] However, even single amino acids and dipeptides often populate several conformational states, 7,13-16 e. g., rotational structural isomers, complicating detailed analysis and the extrapolation from small model data to entire protein complexes. Mapping the structure-function relationship of these biomolecular machines thus requires reproducible samples in the gas-phase in well-defined initial states. [17][18][19][20] More generally, speciesand conformer-pure samples of peptides in the gas-phase would open the door for novel non-species-specific experimental techniques, such as atomic-resolution diffractive imaging with x-rays [21][22][23][24][25] or electrons, 26,27 attosecondelectron-dynamics experiments, 28 or kinetic studies of the chemical reactivity of a single conformer. 18 Such experiments inherently do not distinguish which conformer was probed, making it very difficult or even impossible to * ) Electronic mail: [email protected]; https://www.controlledmolecule-imaging.org interpret data collected with more than one conformer present in the interaction volume.
To investigate biomolecules in the gas-phase requires their vaporisation without fragmentation or ionisation. Laser desorption (LD) has been demonstrated as a technique to vaporise such thermally labile molecules, 29,30 and the combination with supersonic expansion allows for rapid cooling of the desorbed molecules. [30][31][32][33] However, even in such cold molecular beams different conformers, which differ by rotations about single bonds, can coexist. In order to produce a pure beam containing only a single conformer, we combine LD with electrostatic deflection. 34 This allows the spatial separation of molecular species based on their distinct interaction with the applied electric field. This so-called Stark effect is dependent on the quantum-state-specific effective dipole moment and this technique has been demonstrated to spatially separate conformers of small aromatic molecules, 35,36 and for very small molecules it can even produce single-quantum-state samples. [37][38][39] Furthermore, due to the rotational-statedependence of the Stark effect, 34,40,41 deflection allows the creation of very cold (T rot < 100 mK) molecular ensembles. This can significantly improve the degrees of laser alignment and mixed-field orientation of molecules in space 42 and thus enable ensemble-averaged single-molecule imaging. 24,43 Here, we present the first combination of laser desorbed biomolecules with electrostatic deflection and demonstrate the spatial separation of the two main conformers of the dipeptide Ac-Phe-Cys-NH 2 , shown in Figure 1 a. These two conformers differ in their hydrogen-bonding interactions and, hence, 3D structure. One conformer, indicated by red colour throughout the paper, forms a hydrogen bond from the SH group to the oxygen on the carboxamide group, while the other conformer, blue colour, forms a hydrogen bond from the SH to the delocalised π-system. These two "beautiful molecules" 44 have been previously identified using vibrational and electronic spectroscopy. 1645 In a cold molecular beam these two conformers cannot interconvert, however, their significantly Figure 1 b. This allows for their spatial separation with the electrostatic deflector if a sufficiently cold molecular ensemble can be created. 34 This would, furthermore, also separate the sample of interest from unwanted fragments or contaminants present in the beam, such as carbon clusters from the LD process. 33 Compared to the separation of molecular ions in ion mobility measurements, 46,47 our method enables the separation of neutral species, avoiding space-charge density limitations that severely affect diffractive-imaging experiments. 25,26 Furthermore, the low temperatures of the generated molecular ensembles allow for strongly fixing the molecules in space 42 -two prerequisites for the recording of atomically resolved molecular movies. 23,24
RESULTS AND DISCUSSION
Our implementation of the combination of LD with electrostatic deflection is shown schematically in Figure 2; details are given in Methods. Briefly, the laser-desorbed molecular beam enters a 15 cm long deflector sustaining electric field strengths on the order of 150 kV/cm −1 . The different conformers experience a different vertical deflection within this field, which originates from the Stark-effect interaction between the molecules' space-fixed dipole moment µ eff , Figure 1 b, and the applied electric field . This leads to a force F = −µ eff ( ) · ∇ acting on the molecules. 34,41 Thus, the observed deflection depends on the effective-dipole-moment-to-mass ratio and the two conformers experience different forces, i. e., transverse accelerations, in the electric field, leading to their spatial separation. The molecular beam and the separation of conformers was characterised by recording spatial profiles of the beam. This was achieved by vertically translating the ionisation laser beam through the horizontal molecular beam, and recording the relative density as a function of laser height. The ionisation laser was tuned to specific resonances to selectively detect a single conformer. Such spatial molecular beam profiles for the individual conformers in the absence of an electric field, i. e., with the deflector at 0 kV, are shown in Figure 3 a, to which all beam-profile intensities have been normalised. These show that both conformers are centered around y = 0 mm and exhibit the same spatial distribution. The measured width of the molecular beam is predominately defined by the apertures of skimmers and the electrostatic deflector placed in the molecular beam, see Methods. The relative population of the two conformers in the beam was assessed by placing the ionisation laser focus at the center of the profile, as indicated by the black arrow in Figure 3 a, and scanning the ionisation wavelength across the electronic-origin transitions of the two conformers around 37325 cm −1 and 37450 cm −1 , respectively. The resulting resonance-enhanced multiphoton-ionisation (REMPI) spectrum is shown in Figure 3 d and yielded an intensity ratio of ∼2 : 1 for the SH-O and SH-π bound conformers, respectively. Assuming identical ionisation probabilities for the REMPI process, this ratio can be taken as a measure of the relative conformer populations in the molecular beam.
Charging of the electrostatic deflector lead to deflection of the molecular beam in the positive, upward direction, as shown in Figure 3 b,c. Application of 4 kV to the deflector, Figure 3 b, lead to a clear shift of both spatial profiles, with the more polar SH-π-bound conformer shifting significantly more. This created an area, between ∼1.7-2.5 mm, were a highly enriched sample of this conformer was obtained, as confirmed by the REMPI spectrum collected at position y = 1.75 mm and shown in Figure 3 e. To separate and create a pure sample of the SH-O-bound conformer, a voltage of 10 kV was applied, leading to depletion of the SH-π-bound system from the interaction region, as shown in Figure 3 c. This is due to the large deflection experienced by this more polar conformer, such that these molecules collided with the deflector or following apertures and no clear beam was observable anymore. Instead, a position-independent small background signal was present. A REMPI spectrum recorded in the deflected beam is shown in Figure 3 f, confirming the highly-enriched sample of the SH-O-bound conformer created under these conditions. Using a calibrated ion detector, we estimated the number of ions produced per laser shot to be ∼1 for REMPI ionisation. By using more efficient strong-field ionisation (SFI) we extracted a lower limit for the absolute number density of 10 7 cm −3 , see appendix A for details. Derivation of this density assumes an ionisation efficiency of 1 for SFI and only takes into account the major assigned fragmentation channels for Ac-Phe-Cys-NH 2 33 and thus strictly represent a lower limit of the density.
Further to the deflection of the molecular beam, we observed a significant broadening of the spatial profiles. This is due to the dispersion of the different rotational states in the electric field, arising from the rotationalstate-dependence of the Stark effect. 34,41 This is shown in Figure 1 b for J = 0 . . . 3 states, indicating the larger effective dipole moment of lower-lying rotational states, leading to these states being deflected more, and hence the creation of a rotationally colder sample in the deflected beam. 23,40 To extract approximate rotational temperatures and quantum-state distributions in the deflected beam, we have simulated particle trajectories through our setup for the different populated rotational states, 34 details are given in appendix B. Resulting simulated deflection profiles are shown as solid lines in Figure 3 a-c, which were obtained by applying a thermal-distribution weighting to the individual-state simulations, corresponding to the rotational temperature distribution from our LD-molecular-beam source. We extract an approximate rotational temperature of 2.3±0.5 K for the laser-desorbed molecular beam.
Furthermore, we extract the quantum-state distribution within the deflected beam in Figure 3 e,f. These are shown in Figure 4 and indicate that the deflector creates a significantly colder ensemble. While this has a non-thermal rotational state distribution, the highest rotational states populated are approximately corresponding to a 1.5 K distribution. Even colder ensembles can be probed by moving the interaction region further into the deflected beam, this is indicated by the magenta and cyan distributions in Figure 4, evaluated at position 2.2 mm in the deflected beam, which are comparable to a 1.0 K average.
These results highlight the quantum-state-sensitivity of the electrostatic-deflection technique, allowing us to control conformer populations and rotational state distri- Figure 3) and 2.2 mm. Shown in grey are thermal distributions at various temperatures, as indicated. The red/magenta and blue/cyan lines indicate distributions for the SH-O and SH-π conformer, respectively.
butions within the interaction region and creating samples well-suited for further control techniques such as alignment and orientation. 23,42,48 Moreover, ultrafast imaging experiments benefit from the significantly, typically several orders of magnitude, reduced density of carrier gas in the interaction region, which does not experience deflection in the electric field. The presented approach is generally applicable to any polar molecule that can be vaporised by laser desorption and entrained in a molecular beam. The achievable degree of species separation depends crucially on the difference in dipole-moment-to-mass ratio, 34,49 and for small peptide systems we estimate that a difference of ∼20 % is sufficient for creating pure samples of the more polar species, whereas differences above ∼50 % should allow creation of a pure sample of either species, with improved setups enabling the separation for even smaller differences. 50,51 The main limitation here is the creation of initially rotationally-cold samples in the desorption and entrainment process, such that an appreciable fraction of population is in the lower lying rotational states that exhibit the largest Stark shift. If this can be further improved, for example through the use of specially designed and higher pressure supersonic expansion valves, 52 higher state purities or the separation of species with smaller dipole-moment differences will be achievable. CONCLUSION We demonstrated the combination of laser desorption for the vaporisation of labile biological molecules with the electric deflector for the spatial separation of conformational states and the creation of pure and rotationally-cold samples of individual conformers. Using the prototypical (di)peptide Ac-Phe-Cys-NH 2 as a model system, we showed that its two conformers, in the gas-phase, can be spatially separated and samples of either conformer can be obtained. The measured deflection was quantitatively understood using trajectory calculations, which furthermore allowed us to assign a rotational temperature of 2.3 ± 0.5 K for the beam from our laser desorption source. The generally good agreement between experiment and simulation also confirms the calculated dipole moments and that Stark effect calculations based on the rigid-rotor approximation are sufficient even for these large systems. 53 The created molecular samples will enable novel x-ray diffractive imaging experiments: they are conformer-pure beams that are well-separated from carrier gas and rotationally cold enough for strong laser alignment and orientation. The achieved densities of around 10 7 cm −3 are sufficient for high-resolution diffraction experiments at free-electron laser sources such as the European XFEL, which will deliver up to 26,000 pulses per second, allowing fast collection of data. This enables the collection of a diffraction image within 1 h, 24 and simulated alignedmolecule diffraction patterns for the two conformers, showing marked differences, are shown in appendix C. Our laser desorption source, with its low overall repetition rate, but reasonably long gas pulses of 100s of µs, 33 is well-suited to the pulse-train structure of superconducting-LINAC-based XFELs. 54 The produced rotationally cold samples are well suited to strong-field alignment, which can be achieved using the available in-house laser systems available at FELs. 55 Our developed technique will more generally enable experiments on conformer-selected biological molecules with inherently non-species-specific experimental techniques, such as (sub-)femtosecond dynamics, 28 reactive collision studies, 18 or diffractive imaging. 25 This will open new pathways to study the intrinsic structure-function relationship of these basic molecular building blocks of the complex biochemical machinery.
METHODS
A laser desorption source, described in detail elsewhere, 33 is used to vaporise the dipeptide Ac-Phe-Cys-NH 2 (APCN, 95% purity, antibodiesonline GmbH), which is used without further purification. The resulting cold supersonic molecular beam is skimmed twice before entering the strong inhomogeneous field of the electrostatic deflector: once by a 2 mm skimmer (Beam Dynamics Inc. Model 50.8) 75 mm downstream of the expansion, and again by a 1 mm skimmer (Beam Dynamics Inc. Model 2) 409 mm downstream of the expansion. Within the strong inhomogeneous electric field of the deflector, molecules are dispersed according to their effective dipole moment-to-mass-ratio. 34 The molecular beam is skimmed once more with a 1.5 mm skimmer (Beam Dynamics Inc. Model 2) prior to entering the interaction region. This skimmer can be translated in height to ensure no part of the molecular beam is cut off. During measurements, data is collected for two skimmer positions and subsequently combined by keeping the highest intensity measured. The relative density of the conformers is probed via resonance-enhanced multi-photon ionisation (REMPI). 16 The ultraviolet probe light is produced by frequency doubling the output of a dye laser (Radiant Dyes NarrowScan, using Coumarin 153 dye in methanol), pumped by the third harmonic of a Nd:YAG laser (Innolas, SpitLight 600). Typical laser-pulse energies were around 19 µJ loosely focused to a 100 µm spot in the interaction region.
The structures and dipole moments of Ac-Phe-Cys-NH 2 were calculated using the GAMESS software suite 56 Appendix A: Determination of a lower limit of the number density For the density determination a time-of-flight mass spectrum using strong-field ionisation by femtosecond laser pulses (800 nm central wavelength, 40 fs duration, typical pulse energies of 100 µJ) is recorded. 33 In the timeof-flight mass spectrum all peaks originating from the Ac-Phe-Cys-NH 2 molecule are integrated and the total ion current on the detector determined. This is compared to the known calibrated current for a single ion hit, which leads to approximately 18 ions/shot in the ω 0 = 50 µm focus of the laser. Assuming an ionisation efficiency of 1 for strong-field ionisation and a molecular-beam width of 1 mm, this yields a density of 9 x 10 6 cm 3 . to J = 70 in the calculation, using the freely available CMIstark software package. 41 Rotational constants and dipole moment vectors were taken from the DFT calculations and are summarized in Table A1. Subsequently, for molecules in each quantum state we carried out classical trajectory simulations through the experimental setup, taking into account apertures and applying the appropriate forces when molecules are within the electrostatic deflector. 34 Finally, histograms of the final particle-position densities were determined at the interaction point and the contributions from each quantum state weighted by a Maxwell-Boltzmann distribution for a given initial temperature. 40 Simulated intensities for given conditionsspecies and deflector voltage -were scaled with a single amplitude-scaling factor to compare with experimental data to account for additional losses and detection efficiency in the setup. The temperature that best described the experimental observations was determined by comparing the combined residuals, that is the absolute deviation between simulation and data, from all deflected data sets, excluding the SH-π conformer at 10 kV where only a constant low background was observed experimentally, for different rotational temperatures. These are shown in Figure A5 and from the combined residuals (black trace) a rotational temperature of 2.3 K for our molecular beam was extracted. Since the minima for individual deflection profiles deviate by ∼0.5 K, conservative error bounds for the rotational temperature are ±0.5 K.
Appendix C: Simulated X-Ray diffraction patterns Simulated x-ray diffraction patterns at 9.5 keV photon energy, 25,57 achievable at current XFEL sources such as LCLS and the European XFEL, for the two conformers of Ac-Phe-Cys-NH 2 are shown in Figure A6. The simulation assumes a detector distance of 80 mm and maximum scattering angle on the detector of 50.2 • , corresponding to a resolution of d ≈ 154 pm at the edge. These calculations assume perfectly aligned molecules and no contribution from background gas in the interaction region, with the most-polarizable axis of the molecules aligned vertically and the second-most-polarizable axis aligned horizontally within the image plane, as it would be obtained in a typical aligned-molecule-diffraction experiment. 25,55,57 A clear, detectable difference between the two patterns is visible. Utilizing the 10 Hz bunch-structure of the upcoming European XFEL will enable the recording of such patterns as well as the reaction path of conformer interconversion. 24
FIG
. 1. a: The two main conformers of the dipeptide Ac-Phe-Cys-NH 2 with their distinct hydrogen-bonding interactions of the cystine sidechain indicated. b: The Stark energy curves (left) and effective dipole moments (right) for the lowest rotational states of the two conformers. different dipole moments of 3.2 D and 8.1 D result in different Stark interactions, see
FIG. 2 .
2Schematic of the experimental setup combining the laser desorption (LD) with electrostatic deflection. The inset shows a cross-section of the deflector and the electric-field strength inside.
FIG. 3 .
3Spatial molecular beam profiles (a-c) and corresponding REMPI spectra (d-f) for the two main conformers of Ac-Phe-Cys-NH 2 . These are collected at deflector voltages of 0 kV (a,d), 4 kV (b,e) and 10 kV (c,e). Solid lines in the deflection profile plots (a-c) are taken from quantum-state resolved trajectory simulations with a 2.3 K thermal state weighting. REMPI spectra are taken at the spatial position indicated by the black arrow in the spatial profiles.
molecular-beam profiles were simulated by first calculating the Stark energies for each conformer for all rotational states up to J = 50, including all states up ×2 ×2 ×2 FIG. A5. Residuals from fitting deflection profiles for different voltages and conformers as a function of temperature. Data for the SH-π conformer at 10 kV are not shown as only a constant background was observed. The sum of these residuals (black line) yielded a rotational temperature of 2.3 K.
. A6. Simulated x-ray-diffraction patterns of the two conformers of Ac-Phe-Cys-NH 2 .
The Hamburg Center for Ultrafast Imaging -Structure, Dynamics and Control of Matter at the Atomic Scale" of the Deutsche Forschungsgemeinschaft (CUI, DFG-EXC1074), and by the Helmholtz Gemeinschaft through the "Impuls-und Vernetzungsfond". We gratefully acknowledge a Kekulé Mobility Fellowship by the Fonds der Chemischen Industrie (FCI) for Nicole Teschmit.us-
ing the B3LYP functional with a 6-311(p) basis set and
confirmed against published structures. 16
Acknowledgments
We thank Christof Weitenberg and the group of Klaus
Sengstock for support with the wavemeter and Thomas
Kierspel for the simulation of x-ray diffraction patterns.
This work has been supported by the European Re-
search Council under the European Union's Seventh
Framework Programme (FP7/2007-2013) through the
Consolidator Grant COMOTION (ERC-614507-Küpper),
by the excellence cluster "
TABLE A1. Rotational constants and dipole moment vectors used for calculating the Stark effect.SH-O isomer SH-π isomer
A (Mhz)
340.181593
345.067516
B (MHz)
203.443113
215.965933
C (MHz)
159.877010
175.850323
µA (D)
0.768
6.789
µB (D)
2.406
-2.701
µC (D)
1.975
3.406
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"Photocatalytic activity, optical and ferroelectric properties of Bi 0.8 Nd 0.2 FeO 3 nanoparticles synthesized by sol-gel and hydrothermal methods",
"Photocatalytic activity, optical and ferroelectric properties of Bi 0.8 Nd 0.2 FeO 3 nanoparticles synthesized by sol-gel and hydrothermal methods"
] |
[
"Hamed Maleki [email protected] \nFaculty of Physics\nShahid Bahonar University of Kerman\nKermanIran\n"
] |
[
"Faculty of Physics\nShahid Bahonar University of Kerman\nKermanIran"
] |
[] |
In this study, the effects of synthesis method and dopant Neodymium ion on the ferroelectric properties and photocatalytic activity of bismuth ferrite were studied. BiFeO3 (BFO) and Bi0.8Nd0.2FeO3 (BNFO) nanoparticles were prepared through a facile sol-gel combustion (SG) and hydrothermal (HT) methods. The as-prepared products were characterized by X-ray powder diffraction (XRD), Furrier transform infrared spectroscopy (FTIR) and transmission electron microscope (TEM) images. Both nanophotocatalysts have similar crystal structures, but the SG products have semi-spherical morphology. On the other hand, HT samples have rod-like morphology. TEM results indicated that the morphology of products was not affected by the doping process. The thermal, optical and magnetic properties of nanoparticles were investigated by thermogravitometry and differential thermal analysis (TG/DTA), UV-vis spectroscopy, and vibrating sample magnetometer (VSM). The ferroelectric properties of BNFO nanoparticles were improved compared to the undoped bismuth ferrite. The photocatalytic activity of as-synthesized nanoparticles was also evaluated by the degradation of methyl orange (MO) under visible light irradiation. The photocatalytic activity of nanoparticles prepared via sol-gel method exhibited a higher photocatalytic activity compared to powders obtained by hydrothermal method. Also substitution of Nd into the BFO structure increased the photocatalytic activity of products.
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10.1016/j.jmmm.2018.03.043
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"https://arxiv.org/pdf/1808.08427v1.pdf"
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1808.08427
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db0533ba37ac7dbacc024d973d6dfc593b6695f0
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Photocatalytic activity, optical and ferroelectric properties of Bi 0.8 Nd 0.2 FeO 3 nanoparticles synthesized by sol-gel and hydrothermal methods
Hamed Maleki [email protected]
Faculty of Physics
Shahid Bahonar University of Kerman
KermanIran
Photocatalytic activity, optical and ferroelectric properties of Bi 0.8 Nd 0.2 FeO 3 nanoparticles synthesized by sol-gel and hydrothermal methods
1Multiferroicsphotocatalytic activitybismuth ferritesol-gelNeodymium dopinghydrothermal
In this study, the effects of synthesis method and dopant Neodymium ion on the ferroelectric properties and photocatalytic activity of bismuth ferrite were studied. BiFeO3 (BFO) and Bi0.8Nd0.2FeO3 (BNFO) nanoparticles were prepared through a facile sol-gel combustion (SG) and hydrothermal (HT) methods. The as-prepared products were characterized by X-ray powder diffraction (XRD), Furrier transform infrared spectroscopy (FTIR) and transmission electron microscope (TEM) images. Both nanophotocatalysts have similar crystal structures, but the SG products have semi-spherical morphology. On the other hand, HT samples have rod-like morphology. TEM results indicated that the morphology of products was not affected by the doping process. The thermal, optical and magnetic properties of nanoparticles were investigated by thermogravitometry and differential thermal analysis (TG/DTA), UV-vis spectroscopy, and vibrating sample magnetometer (VSM). The ferroelectric properties of BNFO nanoparticles were improved compared to the undoped bismuth ferrite. The photocatalytic activity of as-synthesized nanoparticles was also evaluated by the degradation of methyl orange (MO) under visible light irradiation. The photocatalytic activity of nanoparticles prepared via sol-gel method exhibited a higher photocatalytic activity compared to powders obtained by hydrothermal method. Also substitution of Nd into the BFO structure increased the photocatalytic activity of products.
Introduction
Multiferroic materials have simultaneously the properties of ferroelectricity, ferromagnetism and also in some cases ferroelasticity [1,2]. These materials have attracted attention because of their potential applications in data storage, spintronic devices and sensors and photovoltaics and so on [3][4][5][6][7][8]. Bismuth ferrite is the only known multiferroics that has rhombohedrally-distorted perovskite structure with the space group R3C [4] and shows both ferroelectricity and ferromagnetism at room temperature (RT) [9,10]. BiFeO3 (BFO) has the ferroelectric order below the Curie temperate TC~1103K and antiferromagnetic behavior below the Neel temperature TN~643K [11,12].
In recent years, BFO has received new attention due to its narrow optical band-gap (2.19 eV) and excellent chemical stability, which allows the photocatalytic activity under visible light [3,[13][14][15][16][17][18]. BFO shows different catalytic activities such as oxidation of organic compounds and degradation of pollutants [19][20][21]. In addition, BFO nanoparticles are magnetic semiconductor materials which could be separable in aqueous organic media [14].
However, there are some difficulties for different applications of BFO. Weak ferroelectricity, remanent polarization, high leakage current density, poor ferroelectric reliability and inhomogeneous weak magnetization are some of these challenges [22][23][24].
Many theoretical and experimental studies of bismuth ferrite have been investigated to expand the applications and solve these problems hindering practical usage of BFO. In order to overcome such problems and improve photocatalytic activity of BFO, many modification have been investigated which includes doping rare earth elements instead of bismuth into the BFO structure [15,[25][26][27][28][29][30][31][32][33][34]. In addition, it was investigated that the substitution of rare earth (RE) elements into the bismuth ferrite, can alter the photocatalytic activity [35].
In recent years, there has been many reports on preparation methods for BFO nanoparticles with a focus on photocatalytic applications, which includes sol-gel route [36,37], coprecipitation [38] and hydrothermal reaction [39][40][41]. However, as the morphology of nanoparticles affects the physical properties of bismuth ferrite, we are interested to study the effect of synthesis process on the photocatalytic activity of pure and Nd-doped bismuth ferrite, as well as other physical properties of BFO and BNFO nanoparticles. Although several researches on the structural, optical and multiferroic properties of bismuth ferrite has been done, few studies have investigated on the photocatalytic activity of Nd-doped BFO and the influence of synthesis process [42,43]. The goal of this study is to quantify the effect of Nd 3 dopant on photocatalytic degradation rates of MO. Furthermore, BFO and BNFO were characterized for the structure, morphology, energy band-gap. Moreover thermal, ferroelectric, leakage current density and magnetic properties of samples are investigated and compared between SG and HT as-synthesized products.
Experimental method
Sol-gel preparation of pure and Nd-doped BiFeO3
In the sol-gel method, stoichiometric amount of Bi(NO3)3.5H2O, Fe(NO3)3.9H2O and Nd(NO3)3.6H2O were dissolved in deionized water separately. Meantime, ethylene glycol (EG) and 2-methoxyethanol were mixed under stirring. Then, acetic acid was added to the solution dropwise (The pH of the mixed solution was adjusted to 1.5). This solution was mixed together under vigorous stirring for 30 min. Then metal nitrate solutions were mixed with fuel solution under constant stirring. The mixture was stirred constantly for 30 min and a dark red mixture appears. After stirring at RT for an hour, the temperature was increased to 70 o C. After 3 hours heating and stirring, a clear brownish gel was obtained and following a few minutes with a temperature higher than 90 o C, a yellow suspension is formed. The suspension was kept at RT for 10 hours and then it was put at 115 o C for the water evaporation and fuel combustion.
Finally the obtained powder was calcined at 650 o C for 5 hours before investigating the characteristics.
Hydrothermal synthesis of BFO and BNFO
In the next part, BiFeO3 and Bi0.8Nd0.2FeO3 nanoceramics were synthesized by a hydrothermal process. Typically, bismuth (III) nitrate pentahydrate, ferric (III) nitrate nonahydrate, and neodymium nitrate hexahydrate (for BNFO) were dissolved in the minimum amount of deionized water as a specified stoichiometric ratio. The mixture wad dropped into potassium hydroxide (4M, 30 ml) under magnetic stirring. After stirring for 30 min, the mixture was placed in a teflon-lined steel autoclave of 50 ml for the hydrothermal reaction with a filling capacity of 80 % and performed at 200 o C for 12 hours in an oven and then cooled to RT naturally. The products are collected and washed several times with distilled water and ethanol and dried at 110 o C for 2 h before further characterization.
Characterization
The structural properties of pure and Nd-doped BFO nanoparticles which are synthesized via both sol-gel and hydrothermal methods, were characterized by using XRD analysis (Philips powder diffractometer) with Cu-Kα radiation (λ=1.5406 Å) and Furrier transform inferared 4 (FTIR) TENSOR27 spectrometer. Crystal sizes also were determined by the Scherrer method.
Transmission electron microscope (TEM Leo-912-AB) was performed to study the morphology and size of products. The thermal behavior of the as-prepared samples are monitored by thermogravitometric and differential thermal analysis (TG/DTA NETZSCH-PC Luxx 409) with the heating rate of 10 o C/min up to 1000 o C. For the magnetic properties, the hysteresis loops are recorded up to 20 KG with the vibrating sample magnetometer (VSM-Lake Shore model 7410, SAIF) at RT. The optical properties of products were studied by UVvis absorption spectra by using Lambda900 spectrophotometer. The polarization electric field P-E hysteresis loops of the prepared pellets were measured by the Sawyer-Tower circuit.
Photocatalytic activity measurements
The photocatalytic activity of the BFO and BNFO nanoparticles for decomposition of methyl orange (MO) was studied under irradiation of visible light source at the natural pH value. The reaction temperature was also kept at RT. The initial concentration of MO was 15 mgl -1 with dispersing 0.1g BFO or BNFO in 200ml aqueous solution. Before irradiation, in order to reach an adsorption equilibrium of MO on products surface, the aqueous suspension was magnetically stirred for 75 min in the dark. Then the lamp was turned on and changes of MO concentration were measured by measuring the absorbance of the solution at 554 nm using a UV-vis spectrophotometer. C/C0, where C was the concentration of MO at time t, and C0 was the initial concentration of MO, was the photocatalytic degradation ratio of MO which has been investigated in this study.
Results and discussion
X-ray diffraction investigation
Figs. 1 (a) and (b) show the powder XRD patterns of BiFeO3 (x=0) and Bi0.8Nd0.2FeO3 (x=0.2) nanoparticles prepared by SG method. Analysis of patterns indicated that all products have a single perovskite phase with distorted rhombohedral structure with the space group R3C.
All samples reveal peaks that can be assigned to the standard card of BFO perovskite structure (JCPDS card No. 86-1518). By substituting 20% neodymium, a slight shifting of peaks towards lower angles occurred and a phase transformation from rhombohedral to tetragonal structure was observed (two major peaks at 30<2θ<33 merged into a one peak) and intensity of peaks were decreased. The width of the BFO peaks also increased with merging the nearby peaks.
The size of products was calculated by using the Scherrer formula: D= , where K is the shape factor that normally measures to be about 0.89, λ stands for the wavelength of X-ray source, β is the width of the observed diffraction peak at its half intensity maximum, and θ is the Bragg angle of each peak. The obtained average nanocrystal sizes were 51 and 46 nm for BFO and BNFO respectively. However for the case of BNFO, there is a shift in main peaks at 30<2θ<33 and furthermore, it has the tendency to merge in order to form one single widened peak ( Fig. 2 (b)). One can relate this behavior to a smaller ionic radius in neodymium compared to bismuth. Moreover, comparison of Figs 1(b) and 2(b) indicates that width of major peacks in HT products are a little sharper compared to the as-synthesized SG samples.
Finally, the analysis of particle size, pointed out that in the presence of Nd, the average size of nanoparticles decreases (~40 nm for BFO and ~36 nm for BNFO). Neither the characteristic peaks of Bi2Fe4O9 nor of Bi25FeO40 were found in any of BFO and BNFO patterns.
Furrier transform infrared spectroscopy
In order to further confirm the crystallinity of as-prepared products, Fig. 3 shows the FT-
Transmission electron microscope
Transmission electron microscope (TEM) is used for the observation of morphology and particle distribution and size of BFO and BNFO nanoparticles. As shown in Fig. 4 show BFO and BNFO nanopowders prepared via HT route.
Thermal behavior
The results of differential thermal analysis (DTA) for pure and 20%Nd-doped bismuth ferrite nanoparticles synthesized by the sol-gel method are shown in Figure 5 (a). The inset also indicates the thermogravimetric (TG) analysis of samples. In the case of BFO, in DTA curve for a temperature in the range of 180 °C, an exothermic peak can be seen with an instance of 0.2% reduction in weight. This peak can be attributed to hydrate thermal and the nitrate present in the course of evaporation as well as evaporating water on nanoparticle's surface. In the range of 400-600°C, one can observe a small exothermic peak as well as a weight reduction of 0.15%. This peak is an indication of the oxidation reaction between Fe 3+ and Bi 3+ and it is considered as an evidence for the crystal phase of BFO [47]. In the range of 820-840 °C, an endothermic peak appears which is due to electrical transmission temperature (TC ~ 830 o C).
For Bi0.8Nd0.2FeO3 in DTA curve, the hydrate and nitrate decomposition peaks were eliminated and 0.15% of the weight was declined. Also the phase transition from ferroelectric to Paraelectric states occurred with exothermic peak in lower temperature (TC ~828 o C). Fig. 5 (a) Bi0.8Nd0.2FeO3 can be seen in the measurements, which is related to the Curie temperature.
Inset in
M-H hysteresis loops analysis of BFO and BNFO
The RT magnetic hysteresis loops for all samples are shown in Fig. 6. In contrast to the antiferromagnetism for bulk BFO, nanoparticle samples showed weak ferromagnetic behavior which is in agreement with other reports of BFO [10,48,49]. Table 1 shows the information of magnetic characterization tests for the as-prepared BFO and BNFO nanoparticles synthesized by SG and HT methods. According to literature, in bismuth ferrite the iron ions (Fe 3+ ) have a strong relationship with magnetization and these ions are responsible for magnetic properties of BiFeO3. Surrounding each (Fe 3+ ) ion with a certain spin there are six other ions with nonparallel spins. These spins are not completely nonparallel however they are organized in a spiral manner with a period of 62nm, which leads to a magnetization value zero. Breaking the spiral organization of the spin is due to nanoparticles size reduction to less than 62 nm and the rise of uncompensated spins on the surface of nanocrystals (because of a rise of area relative to volume). This can be a justification for the increase in magnetic properties and the decline in ferromagnetic properties of bismuth ferrite [5,10]. When neodymium enters in the structure of bismuth ferrite, saturated magnetization (Ms) decreases. However, coercive force (Hc)
increases. The remanent magnetization in HT-prepared samples is also increased, but for the SG-prepared samples, Mr decreased. By neodymium doping, the size of nanoparticles has declined. On the other hand, the morphology of nanoparticles are completely different according to synthesis methods. In general by decreasing the size of particles and domains, the energy for changing the magnetic moments increases, which is due to changes in mechanism of process. For the HT-prepared samples, loops are not saturated up to 20 KG. The saturated magnetization in SG-prepared samples is much higher compare to the other ones. On the other hand, the coercive force for samples obtained from HT method, is slightly higher than the SG prepared samples.
Ferroelectric properties of BiFeO3 and Bi0.8Nd0.2FeO3 nanoceramics
In order to study the ferroelectric properties of products, here, the as-synthesized BFO and BNFO nanoparticles were pressed and were coated with a thin layer (~30 nm) of silver as electrodes by using DC sputtering. Fig. 7 presents the RT polarization-electric field (P-E) hysteresis loops of BFO and BNFO nanoparticles synthesis by SG and HT methods under applied electric field up to 25 kV/cm. It can be seen that for all products loops are unsaturated.
By adding Nd, into the structure of BFO, the saturated polarization is increased. However in both synthesis method results, remanent polarization and coercive field are decreased and reduced the leakage current [50,51]. The reduction of leakage current by doping Nd, is due to a reduction of oxygen vacancies [52,53]. When Nd is doped to BFO, the P-E loop is improved showing an elongated loop compared to the BFO samples. In order to study the influence of Nd on the ferroelectric behavior of all products, leakage current density-electric field (J-E) curves of BiFeO3 and Bi0.8Nd0.2FeO3 nanoparticles synthesized by SG and HT at RT is plotted in Fig. 8. BFO (BNFO) nanoparticles synthesized via SG show high leakage current compared to the BFO nanorods which were synthesized by HT. Although the J is decreased with the dopant Nd in the nanoparticles and remained much lower than that of BFO, the synthesis method also affects the leakage current and in the case of BNFO, the products synthesized by HT route shows the bigger leakage current density.
Uv-vis spectroscopy analysis
The optical band-gap of BiFeO3 and Bi0.8Nd0.2FeO3 nanoparticles have been calculated with the help of absorbance spectra. Fig. 9 shows the UV-vis absorption spectra of all products.
By using Tauc's equation, the energy band-gap (Eg), absorption coefficient (α), is related by (for materials with direct band-gap): (αhν) 2 =K(hν-Eg), where K is a constant and hν is the photon energy. An inset in Fig.8 indicates the plot of (αhν) 2 vs hν for BiFeO3 and Bi0.8Nd0.2FeO3 nanoparticles synthesis by SG and HT methods. The extrapolated straight line fitted to the linear part of curves gives the value of Eg. The extracted values of Eg for BFO (BNFO) nanoparticles synthesis by SG method is ~ 2.13 eV (2.08 eV) and 2.11 eV (2.04 eV) 13 for products that obtained from HT reaction route. Slight decrease of Eg value by Nd doping was observed which indicates the narrowing of the optical band-gap and enhanced photocatalytic activity and photovoltaic effects.
Photocatalytic activity of BFO and BNFO
The photocatalytic activity of products were evaluated by degrading methyl orange (MO) BNFO is 61% (73%). This result indicates that the morphology of nanoparticles is an important factor that has effects on the photocatalytic activity of BFO or BNFO particles.
Conclusions
In this paper the influence of synthesis method on the physical properties and photocatalytic activity of BiFeO3 and Bi0.8Nd0.2FeO3 nanoparticles was investigated by different characterization methods. In the BFO samples the main peaks change their shape and location after neodymium is involved and they have a tendency to merge and form a unit peak. The results from TG/DTA showed that by doping nanoparticles of bismuth ferrite the peak related to phase transition from ferroelectric to para electric took place in a lower temperature. The photocatalytic degradation of methylene orange (MO) under visible light irradiation was also implemented. Moreover the effect of neodymium doping into the bismuth ferrite, is studied. Results showed that the Nd-substitution, enhanced the ferroelectric characteristics and reduced the leakage current. In addition ferromagnetic properties were depended to the synthesis method which is related to the morphology of nanoparticles and by substitution Nd, the saturation magnetization was decreased.
Fig. 1 .
1(a) XRD patterns of BiFeO3 and Bi0.8Nd0.2FeO3 nanoparticles prepared by SG method. (b) Same graph in the range of 30<2θ<33 XRD patterns of as-prepared samples by HT are depicted in Figs. 2 (a) and (b). Analysis of diffraction patterns for the samples confirmed the perovskite single phase. All the major peaks of the patterns were possible to index in the rhombohedral phase (R3C) for BiFeO3.
6 Fig. 2 .
62(a) XRD patterns of pure and Nd-doped BiFeO3 with x= 0.2 synthesized by HT route and (b) XRD patterns of pure BFO and BNFO in the range of 30<2θ<33.
7 Fig. 3 .
73IR spectra of BFO and BNFO nanoparticles in the range of 400-4000 cm -1 . In both SG and HT cases, analysis of curves and band range 400-600 cm -1 is related to metal-oxygen bond and confirms the existence of perovskite structure for all samples. The vibration of Fe-O at ~450 cm -1 and stretching vibration of O-Fe-O bonds at ~550 cm -1 present in the octahedral FeO6 group and in the framework can be observed below 600 cm -1[44]. The broad band at 3200-3600 cm -1 is due to antisymmetric and symmetric stretching of H2O and O-H bond groups[45]. A strong band at around 1380 cm -1 was due to the presence of trapped nitrates[46]. (a) FTIR spectra of BFO and BNFO nanoparticles synthesized by SG and (b) products synthesized by HT.
, all samples consisted of nano-scale particles, however the morphology of powders depending on the synthesis method is very different. Although nanoparticles which were synthesized by SG are semi-spherical, products obtained via HT are nanorods. Figs. 4 (a) and (b) are BFO and BNFO nanoparticles that were prepared by SG method. Particles in both images are semi-spheroid and rectangular particles with some irregular shape particles which look denser. In contrast, products which were obtained from HT method have rod-like shape and look looser (Figs. 4 (c) and (d)). After incorporating the BiFeO3 with Nd, it can be seen that there is no significant change in the morphology of both SG and HT products. The size of nanoparticles obtained from TEM is a little larger than that obtained from the Scherrer equation. According to analysis of images, the average size of SG as-prepared products were obtained 55 and 51 nm for BFO and BNFO nanoparticles and a thickness length ranging from 45-50 nm and 39-42 nm for HT as-prepared BFO and BNFO powders. The length of nanorods was also in a range of 80-300 nm. The results demonstrate a clear influence of the preparation method on the structural features of the prepared nanocomposites.
8 Fig. 4 .
84(a) and (b) TEM images of BiFeO3 and Bi0.8Nd0.2FeO3 synthesized by SG method. (c) and (d)
shows TG analysis of the BFO and BNFO nanoparticles. In the curves, up to 200 o C the loss of physisorbed water is observed. The second part (200-450 o C) 9corresponds to the loss of surface hydroxyl groups. Finally the weight loss is observed above 500 o C, which is due to the decomposition of the nitrate spices[45].
Fig. 5 .
5(a) DTA and TG (as an inset) curves of BFO and BNFO nanoparticles synthesized by solgel method. (b) Same curves for as-prepared BFO and BNFO nanoparticles synthesized by hydrothermal method. Same as Fig. 5 (a), in Fig.5 (b) differential thermal analysis curves of BiFeO3 and Bi0.8Nd0.2FeO3 nanocrystals are shown. These substances are synthesized by hydrothermal method. In DTA curves for a temperature at the range up to 200°C, an exothermic peak can be observed. This peak is due to available nitrate and evaporating water on the surface of nanoparticles. In the range of 400-600°C a small exothermic peak is seen. This peak indicates the oxidation reaction between Fe 3+ and Bi 3+ . This can confirm the existence of crystal phase for BFO and 20%Nd-doped BFO. A phase transition at 833 o C for BiFeO3 and 831 o C for
Fig. 6 .
6(a) Magnetic hysteresis loops of BNFO nanoparticles with x=0,0.05,0.1,0.15,0.2 at two calcination temperatures of 550°C. (b) 650°C
Fig. 7 .
7P-E hysteresis loops of BFO and BNFO nanoparticles synthesized by (a) sol-gel and (b) hydrothermal.
Fig. 8 .
8RT current density-applied electric field (J-E) BFO and BNFO nanoparticles synthesized by SG and HT.
Fig. 9 .
9UV-vis spectrum of BiFeO3 and Bi0.8Nd0.2FeO3 nanoceramics prepared by SG and HT methods. Inset shows (αhν) 2 versus hν plots of all samples.
in an aqueous solution under visible light irradiation. The reaction rate constant of samples are plotted in Fig. 10. All BFO and BNFO nanoparticles have visible light induced photocatalytic activity. The Nd-substitution has a significant effect on the photocatalytic activity of bismuth ferrite. The first-order rate constant is calculated by the equation ln(C0/C) = κt, where C0 and C are concentrations of MO in solution at the beginning of the tests and at time t. The reaction rate constant (κ) is the slope in the apparent of ln(C0/C) vs. time. The HT-synthesized (SGsynthesized) BFO nanoparticles showed ~ 16% (23%) degradation of MO in 350 min. On the other hand, the addition of neodymium inside BFO nanoparticles, enhanced the photocatalytic 14 activity of BFO nanopowders. The total removal of MO by HT-synthesized (SG-synthesized)
Fig. 10 .
10(a) Photocatalytic degradation rates of BFO and BNFO nanoparticles prepared by SG and HT methods under visible light irradiation. (b) Represents the pseudo-first-order kinetics curves of the MO degradation.
Table 1 .
1Information from magnetic characterization for BFO and BNFO synthesized by sol gel andhydrothermal
Hydrothermal
Sol gel
Sample
H (G)
M (emu/g)
M (emu/g)
H (G)
M (emu/g)
M (emu/g)
208.97
0.0074
0.1093
63.28
0.0027
0.63
BiFeO 3
332.69
0.0037
0.0096
167.90
0.0068
0.54
Bi 0.8 Nd 0.2 FeO 3
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|
[] |
[
"Relaxed Leverage Sampling for Low-rank Matrix Completion",
"Relaxed Leverage Sampling for Low-rank Matrix Completion"
] |
[
"Abhisek Kundu [email protected] \nRensselaer Polytechnic Institute Troy\n12180New York\n"
] |
[
"Rensselaer Polytechnic Institute Troy\n12180New York"
] |
[] |
We consider the problem of exact recovery of any m × n matrix of rank ̺ from a small number of observed entries via the standard nuclear norm minimization framework in (2)(Candes and Recht [2009]). Such low-rank matrices have degrees of freedom (m + n)̺ − ̺ 2 . We show that such arbitrary low-rank matrices can be recovered exactly from as small as Θ ((m + n)̺ − ̺ 2 )log 2 (m + n) randomly sampled entries, thus matching the lower bound on the required number of entries (in degrees of freedom), with an additional factor of O(log 2 (m+ n)). The above bound on sample size is achieved if each entry is observed according to probabilities proportional to the sum of corresponding row and column leverage scores, minus their product (see(4)). We show that this relaxation in sampling probabilities (as opposed to sum of leverage scores in Chen et al. [2014]) gives us an additive improvement on the (best known) sample size obtained byChen et al. [2014]for the optimization problem in (2). Experiments on real data corroborate the theoretical improvement on sample size.Further, exact recovery of (a) incoherent matrices (with restricted leverage scores), and (b) matrices with only one of the row or column spaces to be incoherent, can be performed using our relaxed leverage score sampling, via (2), without knowing the leverage scores a priori. In such settings also we achieve additive improvement on sample size.
|
10.1016/j.ipl.2017.04.002
|
[
"https://arxiv.org/pdf/1503.06379v2.pdf"
] | 7,093,682 |
1503.06379
|
a54b53a000ed551f73f6465a2ec65ae8f7cfae69
|
Relaxed Leverage Sampling for Low-rank Matrix Completion
31 Mar 2015
Abhisek Kundu [email protected]
Rensselaer Polytechnic Institute Troy
12180New York
Relaxed Leverage Sampling for Low-rank Matrix Completion
31 Mar 2015arXiv:1503.06379v2 [cs.IT]matrix completionnuclear normleverage scorerandomized algorithms
We consider the problem of exact recovery of any m × n matrix of rank ̺ from a small number of observed entries via the standard nuclear norm minimization framework in (2)(Candes and Recht [2009]). Such low-rank matrices have degrees of freedom (m + n)̺ − ̺ 2 . We show that such arbitrary low-rank matrices can be recovered exactly from as small as Θ ((m + n)̺ − ̺ 2 )log 2 (m + n) randomly sampled entries, thus matching the lower bound on the required number of entries (in degrees of freedom), with an additional factor of O(log 2 (m+ n)). The above bound on sample size is achieved if each entry is observed according to probabilities proportional to the sum of corresponding row and column leverage scores, minus their product (see(4)). We show that this relaxation in sampling probabilities (as opposed to sum of leverage scores in Chen et al. [2014]) gives us an additive improvement on the (best known) sample size obtained byChen et al. [2014]for the optimization problem in (2). Experiments on real data corroborate the theoretical improvement on sample size.Further, exact recovery of (a) incoherent matrices (with restricted leverage scores), and (b) matrices with only one of the row or column spaces to be incoherent, can be performed using our relaxed leverage score sampling, via (2), without knowing the leverage scores a priori. In such settings also we achieve additive improvement on sample size.
Introduction
Suppose we have a data matrix M ∈ R m×n with incomplete/missing entries, say, we have information about only a small number elements of M. The matrix completion problem (Candes and Recht [2009]) is to predict those missing entries as accurately as possible based on the observed entries. Such partially-observed data may appear in many application domains. For example, in a userrecommendation system (a.k.a collaborative filtering) we have incomplete user ratings for various products, and the goal is to make predictions about a user's preferences for all the products (e.g., the Netflix problem). Also, the incomplete data could represent some partial distance matrix in a sensor network, or missing pixels in digital images because of occlusion or tracking failures in a video surveillance system (Candes and Tao [2010]).
More mathematically, we have information about the entries M ij , (i, j) ∈ Ω, where Ω ⊂ [m] × [n] is a sampled subset of all entries, and [n] denotes the list {1, ..., n}. The problem is to recover the unknown matrix M in a computationally tractable way from as few observed entries as possible. However, without further assumption on M it is impossible to predict the unobserved elements from a limited number of known entries. One popular assumption is that M has lowrank, say rank ̺. Such matrices have degrees of freedom (m + n)̺ − ̺ 2 , i.e., the elements of such low-rank matrices are controlled by this many parameters. To see this, we consider the singular value decomposition (SVD) of M,
M = i∈[̺] σ i u i v T i ,(1)
where σ i is the i-th largest singular value, and the corresponding left and right singular vectors are u i ∈ R m and v i ∈ R n , respectively. Left and right singular vectors form two sets of orthonormal vectors. The first left singular vector has m−1 degrees of freedom because of unit norm constraint. The second singular vector has m − 2 degrees of freedom as it has two constraints: the unit norm, and the orthogonality to the first vector. In this way, all the left singular vectors have total (m − 1) + (m − 2) + ... + (m − ̺) = m̺ − ̺(̺ + 1)/2 degrees of freedom. Similarly, the number is n̺ − ̺(̺ + 1)/2 for the right singular vectors. Considering ̺ additional degrees of freedom for the singular values, total number of degrees of freedom for M is (m + n)̺ − ̺ 2 (Candes and Tao [2010]). This implies, if the number of observed entries s = |Ω| < (m + n)̺ − ̺ 2 , there can be infinitely many matrices of rank at most ̺ with exactly the same entries in Ω; therefore, exact recovery of unobserved entries is impossible. So, we need at least (m + n)̺ − ̺ 2 many observed entries for exact matrix completion.
The matrix M, with the observed entries, can be interpreted as an element in mn-dimensional linear space, with available information about O((m + n)̺ − ̺ 2 ) coordinates. Remaining mn − O((m + n)̺ − ̺ 2 ) many coordinates are unknown. The set of matrices compatible with the observed entries forms a large affine space. Then, exact matrix completion problem is to specify an efficient algorithm which uniquely picks M from this high-dimensional affine space (Gross [2011]). Since, our target matrix M is low-rank, a natural optimization problem for finding M would be, min X∈R m×n rank(X) subject to X ij = M ij , (i, j) ∈ Ω.
However, minimizing rank over an affine space is known to be NP-hard. Candes and Recht [2009] proposed to solve the heuristic optimization in (2) to recover the low-rank matrix M.
min X∈R m×n X * subject to X ij = M ij (i, j) ∈ Ω,(2)
where the nuclear norm X * of a matrix X is defined as the sum of its singular values,
X * = i σ i (X).
(2) is a convex optimization problem that is efficiently solvable via semi-definite programming. Exact matrix completion thus becomes proving that the nuclear norm restricted to the affine space has a strict and global minima at M. That is, if M + Z = M is a matrix in the affine space in (2), we need to show M + Z * > M * .
Candes and Recht [2009], Gross [2011], Recht [2011], Candes et al. [2011] developed the sufficient conditions and main probabilistic tools in order to recover M as a unique solution to (2). One natural question is: which elements of M should we observe in (2), i.e., how should we construct the sample set Ω? We want to define some probabilities on the entries of M. Most of the existing work focused on the case when Ω in (2) is constructed by observing the entries of M uniformly randomly (Candes and Recht [2009], Gross [2011], Recht [2011], Candes et al. [2011]). However, this data-oblivious sampling scheme has a cost. To see this, let M be rank-1 (̺ = 1), and m = n, u 1 = v 1 = e 1 , σ 1 is arbitrary in (1), where e i is the standard basis vector (i-th component 1, others are zeros). This M has only one non-zero entry, i.e., M 1,1 = σ 1 , and all other entries are 0. The probability that a sample set Ω of size 2n − 1 (degrees of freedom), via uniform sampling, containing only zero entries of M is 1 − O( 1 n 3 ) ≈ 1. That is, this matrix cannot be recovered from sampling its entries uniformly unless we see almost all the entries. This is because by observing only zeros it is impossible to predict non-zeros of a matrix. This suggests that M cannot be in the null space of the sampling operator (to be defined later) extracting the values of a subset of the entries. Matrices similar to the above example can be characterized by the structure of their singular vectors. The singular vectors are (closely) 'aligned' with the standard basis (i.e., having very high inner product with the basis vectors). Therefore, the components of singular vectors should be sufficiently spread to reduce the number of observations needed to recover a low-rank matrix. Such restrictions on the row and column spaces of a low-rank matrix are called the incoherence assumptions (to be defined later). Gross [2011], Recht [2011] showed that such restricted class of n × n matrices of rank ̺ can be recovered exactly, with high probability, by observing Θ(n̺ log 2 n) entries sampled uniformly.
Very recently, Chen et al. [2014] proposed non-uniform probabilities proportional to the sum of row and column leverage scores of M to observe its entries (leveraged sampling). They eliminated the need for those 'incoherence' assumptions, and showed that any arbitrary n × n matrix of rank ̺ can be recovered exactly, with high probability, from as few as Θ(n̺ log 2 n) observed elements via leveraged sampling.
Similar to Chen et al. [2014], we also incorporate the row and column leverage scores of the reconstructing matrix M into our proposed probability of observing an entry. However, we use a relaxed notion of leverage score sampling. Specifically, we propose to observe an entry with probability proportional to the sum of the corresponding row and column leverage scores, minus their product. Theorem 1 shows that observing entries according to this relaxed leverage score sampling in (4), we can recover any arbitrary m×n matrix of rank-̺ exactly, with high probability, from as few as Θ(((m + n)̺ − ̺ 2 )log 2 (m + n)) observed entries, via the standard nuclear norm minimization framework in (2). This bound on the sample size is optimal (up to log 2 (m + n) factor) in the number of degrees of freedom of a rank-̺ matrix. Also, this gives us an additive improvement on the sample size obtained by Chen et al. [2014].
For an n×n matrix M of rank-̺ whose column space is incoherent and row space is arbitrarily coherent, Chen et al. [2014] give a provable sampling scheme (using leveraged sampling) which requires no prior knowledge of the leverage scores of M. They show that this M can be recovered exactly, with high probability, using sample size as small as Θ(n̺ log 2 n). We can incorporate our relaxed leverage scores in such setting, with no prior knowledge of leverage scores, to achieve additive improvement on the sample size obtained by Chen et al. [2014], while recovering M exactly with high probability.
Finally, our notion of relaxation in sampling probabilities also achieves an additive improvement on the sample size even in case of uniform sampling for incoherent matrices.
Notations and preliminaries
We briefly describe the main notations used in this work. Natural number {1, ..., n} are denoted by [n]. Natural logarithm of x is denoted by log(x). Matrices are bold uppercase, vectors are bold lowercase, and scalars are not bold. We denote the (i, j)-th entry of a matrix X by X ij . e i denotes the i-th standard basis vector in R d , with i-th component 1 and other entries zero. The dimension of e i will be clear from the context. X T and x T denote the transpose of matrix X and vector x, respectively. Tr(X) denotes the trace of a square matrix X.
Spectral norm of X is denoted by X 2 . The inner product between two matrices is X, Y = Tr(X T Y). Frobenius norm X is denoted by X F , and X F = X, X . The nuclear norm of X is denoted by X * . The maximum entry of X is denoted by X ∞ = max i,j |X ij |. For vectors Euclidean ℓ 2 norm is denoted by x 2 .
Linear operators acting on matrices are denoted by calligraphic letters. The spectral norm (largest singular value) of such operator A will be denoted by A op = sup X A(X) F / X F . Also, we denote f (n) = Θ(g(n)) when α 1 · g(n) ≤ f (n) ≤ α 2 · g(n), for some positive universal constants α 1 , α 2 .
Main Results
Our focus is to define a probability distribution on the entries of M (i.e., to construct the sample set Ω in (2)) to reduce the sample size, such that M becomes the unique optimal solution to (2). In this work our sampling follows the Bernoulli model (Candes et al. [2011], Chen et al. [2014]), where each entry (i, j) is observed independently with some probability p ij . Before we state our main result and the distribution, we first need to define the normalized leverage scores (Candes and Recht [2009], Recht [2011], Chen et al. [2014]).
Definition 1 Let M ∈ R m×n be of rank ̺ with SVD M = UΣV T , where U and V are the left and right singular matrices, respectively, and Σ is the diagonal matrix of singular values. Normalized leverage scores for i-th row (denoted by µ i ) and j-th column (denoted by ν j ) are defined as follows:
µ i = m ̺ U T e i 2 2 , i = 1, ..., m, ν j = n ̺ V T e j 2 2 , j = 1, ..., n(3)
Normalized leverage scores are non-negative, and they depend on the structure of row and column spaces of the matrix. Also, we have i µ i ̺ m = j ν j ̺ n = ̺, because U and V have orthonormal columns. We state our main result.
Theorem 1 Let M ∈ R m×n of rank ̺. Suppose, we have a subset of observed entries Ω ⊂ [m] × [n], where each entry (i, j) is observed independently with probability p ij , such that,
p ij = max min c 1 µ i ̺ m + ν j ̺ n − µ i ̺ m · ν j ̺ n log 2 (m + n), 1 , 1 (mn) 5 ,(4)
for some universal constant c 1 > 0. Then, M is the unique optimal solution to (2) with probability at least 1 − 3 log 3 (m + n)(m + n) 3−c , for sufficiently large c > 3. Moreover, if the number of observed entries, according to (4), is
|Ω| = Θ ((m + n)̺ − ̺ 2 )log 2 (m + n) ,
then, M is the unique optimal solution to (2) with probability at least 1− 6 log 3 (m + n)(m + n) 3−c , for sufficiently large c > 3.
Probabilities in (4) are biased towards the leverage score structure of the reconstructing matrix. This suggests that the elements in important rows and columns, indicated by high leverage scores {µ i } and {ν j }, of a matrix should be observed more frequently in order to reduce the number of observations needed for exact matrix completion. Chen et al. [2014] also noticed this, and they proposed to sum up µ i ̺ m and ν j ̺ n in the sampling probabilities. However, our distribution in (4) reduces this bias by subtracting the term µ i ̺ m · ν j ̺ n while maintaining the leverage score pattern in p ij . This relaxation in sampling probabilities helps us to reduce number of observations comparing to Chen et al. [2014], in additive sense, to recover the low-rank matrix exactly, via (2).
As discussed earlier, we need a minimum of Θ((m + n)̺ − ̺ 2 ) elements to recover a matrix exactly, regardless of the choice of probabilities. Theorem 1 proves that if we observe elements according to our relaxed leverage scores, we match this lower bound, up to a factor of O(log 2 (m + n)).
Also, using the relaxed leverage score sampling we observe improvement on sample size even in case of uniform sampling for matrices with incoherence restrictions. Let M ∈ R n×n be the rank-̺ reconstructing matrix with SVD UΣV T . Candes and Recht [2009], Candes and Tao [2010], Recht [2011], Gross [2011] use two incoherence parameters, µ 0 and µ 1 , for exact matrix completion using uniform sampling, where, (a) max i,
j {µ i , ν j } ≤ µ 0 , and (b) UV T ∞ = µ 1 ̺/n 2 . A meaningful range of µ 0 is 1 ≤ µ 0 ≤ min{m, n}/̺.
Then, the best known result was that if the sampling probability is uniform, such that,
p ij ≡ p ≥ c u max{µ 0 , µ 2 1 }̺ log 2 n n , ∀i, j,
where c u is a constant, then M is the unique optimal solution of (2) with high probability. Actually, the lower bound achieved on the sample size in a sample-with-replacement model was O(max{µ 0 , µ 2 1 }n̺ log 2 n) (Recht [2011]). Above, µ 1 ≤ µ 0 √ ̺, and it could create a suboptimal dependence of sample size on ̺, in the worst case. Chen et al. [2014] showed that observing entries with uniform probability satisfying, p ≥ c 1 2µ 0 ̺ n log 2 n, ∀i, j for some constant c 1 , would recover the matrix exactly, with high probability. In this case, the bound on sample size is O(2µ 0 n̺ log 2 n). They eliminated the need for the parameter µ 1 , and consequently the suboptimal dependence on ̺. It follows from Theorem (1) that we can recover the matrix exactly, with high probability, if each entry is sampled uniformly with probability satisfying,
p = max min c 1 2µ 0 ̺ n − µ 2 0 ̺ 2 n 2 log 2 n, 1 , n −10 , ∀i, j.
This requires a sample size of O((2µ 0 n̺ − µ 2 0 ̺ 2 )log 2 n), achieving an additive improvement on all the existing bounds above.
Column-Space-Incoherent Matrix Completion
Here we discuss exact completion of a low-rank matrix whose column space is incoherent, and we have control over the sampling of matrix entries. This setting is interesting in application domains like recommendation systems and gene expression data analysis (Krishnamurthy and Singh [2013]).
Algorithm 1, adapted from Chen et al. [2014], performs exact completion of a matrix M with incoherent column space, without a priori knowledge of leverage scores of M.
Step 3 of Algorithm 1 computes the column leverage scores of M exactly, with high probability, from only a small number of (uniformly) observed rows. We construct an additional sample set Ω of observed entries using our relaxed leverage scores in Step 4.
Step 5 solves the nuclear norm minimization problem in (2) with Ω to recover M exactly, with high probability. Theorem 2 proves the correctness of Algorithm 1.
Algorithm 1 Column-Space-Incoherent Matrix Completion
1: Input: M ∈ R m×n , with max i µ i ≤ µ 0 , ∀i ∈ [m], s.t. 1 ≤ µ 0 ≤ m/̺. 2:
Observe all the entries of a row of M picked with probability
p = min c 2 µ 0 ̺ log m m , 1 ,
where c 2 is a constant. 3: Compute the leverage scores, {ν j } ∀j ∈ [n], of the space spanned by these rows, and use them as estimates for true {ν j }, ∀j ∈ [n] of M. 4: Construct a sample set Ω of entries (i, j) of M observed with probabilities
p ij = min c 1 µ 0 ̺ m +ν j ̺ n − µ 0 ̺ m ·ν j ̺ n log 2 (m + n), 1 , ∀i, j.(5)
5: Solve (2) using sample set Ω, and let X * be the unique optimal solution. 6: Output: X * .
Theorem 2 Algorithm 1 computes the column leverage scores of M exactly (step 3), i.e.,ν j = ν j , ∀j ∈ [n]. Using the sample set Ω, Algorithm 1 recovers M as the unique optimal solution of (2). The total number of samples required by Algorithm 1 is
Θ(µ 0 ((m + 2n)̺ − ̺ 2 )log 2 (m + n)).
The above results hold with probability at least, 1 − 6 log 3 (m + n)(m + n) 3−c , for sufficiently large c > 3.
We compare the bound on sample size in Theorem 2 with a couple of existing results. Let us assume m = n for simplicity. Theorem 2 achieves an additive improvement O(̺ 2 log 2 n) on the sample size of Chen et al. [2014] while recovering M exactly, with high probability, via (2). Krishnamurthy and Singh [2013] proposed an adaptive sampling algorithm that recovers M exactly, with probability at least 1 − O(̺δ), and a sample size Θ(µ 0 n̺ 3/2 log(̺/δ)). Assuming comparable failure probabilities, sample size in Theorem 2 is better when ̺ is not too small.
Coherent Matrix Completion using Two-Phase-Sampling
We have so far seen that any arbitrary m × n matrix M of rank ̺ can be recovered exactly using Θ((m + n)̺ − ̺ 2 )log 2 (m + n)) observed entries sampled according to relaxed leverage scores of M. However, in reality, we do not have knowledge about the leverage scores of M, i.e., {µ i } and {ν j }, even when we have control over how to choose entries. Chen et al. [2014] proposed a heuristic two-phase sampling procedure (Algorithm 1 of Chen et al. [2014]) for exact matrix completion with no a priori knowledge about the leverage scores. Here is an informal description of it.
Let, the total budget of samples be s, and β ∈ [0, 1] be a parameter. First, construct an initial set Ω 1 by sampling entries uniformly (without replacement), such that, |Ω 1 | = βs. LetM be the
matrix withM ij = M ij if (i, j) ∈ Ω 1 , andM ij = 0 if (i, j) /
∈ Ω 1 . Let the rank-̺ SVD ofM beŨΣṼ T . Compute the leverage scores ofM and use them as estimates for the leverage scores
of M, i.e., useμ i = m ̺ Ũ T e i 2 2 as µ i for i ∈ [m], andν j = n ̺ Ṽ T e j 2 2 as ν j for j ∈ [n].
In the second phase, use these estimates to sample (without replacement) remaining (1 − β)s entries of M with probabilities proportional to (μ i ̺/m +ν j ̺/n) log 2 (m + n), to form the sample set Ω 2 . Then perform matrix completion using sample set Ω = Ω 1 ∪ Ω 2 in (2).
This heuristic is shown to work well on synthetic data that are less coherent (Chen et al. [2014]). For highly coherent data, e.g., only few entries are non-zeros and others are zeros, it works poorly, as expected.
We can incorporate our notion of relaxed leverage scores into the second phase of the above procedure by observing (without replacement) the remaining (1 − β)s entries of M with probabilities
p ij ∝ μ i ̺ m +ν j ̺ n −μ i ̺ m ·ν j ̺ n log 2 (m + n)
to form sample setΩ 2 , and perform nuclear norm minimization in (2) using Ω = Ω 1 ∪Ω 2 . We expect our relaxed leverage score sampling to follow similar trend as above, although we do not evaluate this heuristic numerically.
Rest of the content is organized as follows. Section 3 shows experimental results on real datasets to support the theoretical gain on the sample size using relaxed leverage score sampling. We give proofs of Theorem 1 and Theorem 2 in Section 4 and Section 5, respectively. Section 6 contains a proof of the sufficient conditions for the optimization problem in (2). Finally, Section 7 contains proofs of intermediate lemmas.
Experiments
We show experimental performance of the exact recovery of real data matrices via nuclear norm minimization in (2) using our relaxed leverage score sampling. We use the software 'TFOCS' v1.2, written by Stephen Becker, Emmanuel Candes, and Michael Grant, to solve (2).
Experimental Design
Let M be the rank-̺ data matrix. We construct the sample set Ω relax by observing (i, j)-th entry of M according to the relaxed leverage score probabilities in (6):
p [relax] ij = min c r · µ i ̺ m + ν j ̺ n − µ i ̺ m · ν j ̺ n , 1 , ∀i, j(6)
where c r is a universal constant. Similarly, we construct the sample set Ω lev by observing (i, j)-th entry of M according to the leverage score probabilities in (7):
p [lev] ij = min c l · µ i ̺ m + ν j ̺ n , 1 , ∀i, j(7)
where c l is a universal constant. We use Ω relax and Ω lev in the optimization problem (2), separately, to recover M. Let X * be the optimal solution to (2) using a sample set Ω. We say X * recovers M exactly if M − X * F / M F < ε, where ε is a tiny fraction. We set ε = 0.001. We perform 10 independent trails (sampling and recovery) and declare success if M is recovered exactly at least 9 times (i.e., at least 90% success rate). Let s r and s l be the average sample size for successful recovery of M using Ω relax and Ω lev , respectively. We expect c r ≈ c l , and the gain in sample size (s l − s r ) for Ω relax to be strictly positive, as suggested by the theory. Further, we investigate how (s l − s r ) behaves with respect to the rank ̺. For this, we define
Normalized Gain (∆ s ) = s l − s r c r .(8)
We expect ∆ s to be close to ̺ as the theory suggests (s l −s r ) ∝ O(̺ 2 ). For fairness of comparison, we use the same random seed for both the sampling methods in (6) and (7).
Datasets
MovieLens: This collaborative filtering dataset was collected through the MovieLens web site (movielens.umn.edu). It contains 100,000 ratings between 1 and 5 by 943 users on 1682 movies. Each user has rated at least 20 movies. We note that this dataset is numerically not low-rank. We perform rank truncation to create an explicit low-rank matrix to apply the theory in (2). We observe the singular value spectrum of this data to heuristically choose two values for rank: ̺ = 10 and ̺ = 20.
TechTC: We use a dataset from the Technion Repository of Text Categorization Database (TechTC) (Gabrilovich and Markovitch [2004]). Here each row is a document describing a topic, and words (columns) are the features for the topics. The (i, j)-th entry of this matrix is the frequency of j-th word appearing in i-th document. We choose a dataset containing the topics with IDs 11346 ans 22294. We preprocessed the data by removing all words of length four or less. Then, each row is normalized to have unit norm. Also, we observe the singular value spectrum of this preprocessed 125 × 14392 data to heuristically choose two values for rank: ̺ = 10 and ̺ = 20, to make the data explicitly low-rank.
Results
Figures 1 and 2 plot the singular values and the normalized leverage scores for rank-10 approximation for MovieLens and TechTC data, respectively. Normalized leverage scores are close to 1 when they are incoherent in nature. MovieLens dataset is reasonably coherent, and TechTC dataset has extremely high coherence. Table 1 shows the constants c l and c r , and the normalized gain ∆ s for exact recovery of MovieLens data. We see c l = c r and ∆ s ≈ ̺, as expected. We observe similar results for TechTC data in Table 2.
Overall, these results support the accuracy of the theoretical analysis on the gain in sample size using the relaxed leverage score sampling for exact recovery of a low-rank matrix via (2). c l /c r ∆ s ̺ = 10 11/11 9.7 ̺ = 20 7/7 18.4 c l /c r ∆ s ̺ = 10 4/4 6.6 ̺ = 20 3/3 15.2
Proof of Theorem 1
The main proof strategy was outlined by Candes and Recht [2009], Recht [2011], Gross [2011]: it is sufficient to construct a dual certificate Y obeying specific sub-gradient inequalities in order to show that M is the unique optimal solution to (2) (see Section 6 for more detail). We give a proof of Theorem 1 closely following the proof strategy of Recht [2011], Chen et al. [2014]. Before stating the optimality conditions we need additional notations.
Recall, U and V are the left and right singular matrices of M, respectively. Let u k (respectively v k ) denote the k-th column of U (respectively V). Let T be a linear space spanned by elements of the form u k y T and xv T k , 1 ≤ k ≤ ̺, for arbitrary x, y, and T ⊥ be its orthogonal complement, i.e., T ⊥ is spanned by the family (xy T ), where x (respectively y) is any vector orthogonal to the space spanned by the left singular vectors (respectively right singular vectors). Then, orthogonal projection onto T is given by the linear operator P T : R m×n → R m×n , defined as P T (X) = UU T X + XVV T − UU T XVV T .
Similarly, orthogonal projection onto T ⊥ is
P T ⊥ (X) = X − P T (X) = U ⊥ U T ⊥ XV ⊥ V T ⊥ .
Note that any m × n matrix X can be expressed as a sum of rank-one matrices as follows:
X = m,n i,j=1 e i e T j , X e i e T j .(9)
We define the sampling operator R Ω : R m×n → R m×n as,
R Ω (X) = m,n i,j=1 1 p ij δ ij e i e T j , X e i e T j(10)
where, δ ij = I((i, j) ∈ Ω), I(·) being the indicator function. That is, R Ω extracts the terms, corresponding to the indices (i, j) ∈ Ω, from (9) to form a partial sum in (10). Let P Ω (X) be the matrix with (P Ω (X)) ij = X ij if (i, j) ∈ Ω, and zero otherwise.
Optimality Conditions
Following the proof road map of Recht [2011], Chen et al. [2014], we restate the sufficient conditions for M to be the unique optimal solution to (2) (Section 6 contains a proof of sufficiency).
Proposition 1 The rank-̺ matrix M ∈ R m×n with SVD M = UΣV T is the unique optimal solution to (2) if the following conditions hold:
1. P T R Ω P T − P T op ≤ 1/2.
2.
There exists a dual certificate Y which satisfies P Ω (Y) = Y, and
(a) P T (Y) − UV T F ≤ ̺(m + n) −15 , (b) P T ⊥ (Y) 2 ≤ 1/2.
Condition 1 of Proposition 1 suggests R Ω should be nearly the identity operator on the subspace T . Next we discuss the construction of a dual certificate Y.
Constructing the Dual Certificate
We follow the so-called golfing scheme Gross [2011], Candes et al. [2011], Chen et al. [2014] to construct a matrix Y (the dual certificate) that satisfies Condition 2 in Proposition 1. Recall, we assume that the set of observed elements Ω follows the Bernoulli model with parameter p ij , i.e., each index (i, j) is observed independently with P [(i, j) ∈ Ω] = p ij (p ij in eqn (4)). We denote this by Ω ∼ Bernoulli(p ij ). Further, we assume that Ω is generated from Ω = ∪ k 0 k=1 Ω k , where for each k, Ω k ∼ Bernoulli(q ij ), and we set q ij = 1 − (1 − p ij ) 1/k 0 . Clearly, this implies P [(i, j) ∈ Ω] = p ij which is the original Bernoulli model for Ω. Note that, q ij ≥ p ij /k 0 because of overlapping of Ω k 's. We set k 0 = 11 · log(m + n). Then,
q ij ≥ min c 0 · log(m + n) · µ i ̺ m + ν j ̺ n − µ i ̺ m · ν j ̺ n , 1 ,(11)
where c 0 = c 1 /11. Starting with W 0 = 0 and for each k = 1, ..., k 0 , we recursively define
W k = W k−1 + R Ω k P T (UV T − P T (W k−1 ))(12)
where the sampling operator R Ω k : R m×n → R m×n is defined as
R Ω k (X) = i,j 1 q ij I((i, j) ∈ Ω k ) e i e T j , X e i e T j .
We set Y = W k 0 . This Y is supported on Ω, i.e., P Ω (Y) = Y. Let the sample setΩ be such that
Ω ∈ {Ω k : Ω = ∪ k 0 k=1 Ω k , Ω k ∼ Bernoulli(q ij )}.(13)
Since Ω k ∼ Bernoulli(q ij ) implies Ω ∼ Bernoulli(p ij ), for each k = 1, ..., k 0 , we prove (in Lemma 1) Condition 1 of Proposition 1 using sample setΩ in (13).
Lemma 1 LetΩ be a sample set in (13). Then, for any universal constant c > 1, we have
P T RΩP T − P T op ≤ 1 2 (14)
holding with probability at least 1 − (m + n) 1−c .
Before we validate Condition 2 in Proposition 1 using the Y constructed above, we claim the following results to hold with high probability. First, we borrow the following definitions of weighted infinity norms for a matrix Z ∈ R m×n from Chen et al. [2014].
Z µ(∞,2) := max max i m µ i ̺ Z i, * 2 , max j n ν j ̺ Z * ,j 2 Z µ(∞) := max i,j |Z ij | m µ i ̺ n ν j ̺
where Z i, * and Z * ,j denote the i-th row and j-th column of Z, respectively.
Lemma 2 bounds the spectral norm of the matrix (RΩ − I)(Z) using the sample setΩ.
Lemma 2 Let Z ∈ R m×n be a fixed matrix. LetΩ be a sample set in (13). Then, for any universal constant c > 1, we have
RΩ − I Z 2 ≤ 2 c c 0 Z µ(∞,2) + c c 0 Z µ(∞)
holding with probability at least 1 − (m + n) 1−c .
Next two results control the µ(∞, 2) and µ(∞) norms of the projection of a matrix after random sampling.
Lemma 3 Let Z ∈ R m×n be a fixed matrix. LetΩ be a sample set in (13). Then, for any universal constant c > 2, we have
(P T RΩ − P T )Z µ(∞,2) ≤ 1 2 Z µ(∞,2) + Z µ(∞)
holding with probability at least 1 − (m + n) 2−c .
Lemma 4 Let Z ∈ R m×n be a fixed matrix. LetΩ be a sample set in (13). Then, for any universal constant c > 3, we have
(P T RΩ − P T )Z µ(∞) ≤ 1 2 Z µ(∞)
holding with probability at least 1 − (m + n) 3−c .
We now validate Condition 2 in Proposition 1 using the Y constructed above.
Bounding UV T − P T (Y) F
We set ∆ k = UV T − P T (W k ), for k = 1, ..., k 0 . Then, from definition of W k we have
∆ k = (P T − P T R Ω k P T )∆ k−1 .
We used P T (UV T ) = UV T and P T P T (X) = P T (X). Using the independence of ∆ k−1 and Ω k ,
∆ k F = (P T − P T R Ω k P T )∆ k−1 F ≤ P T − P T R Ω k P T op ∆ k−1 F .
We can bound this by recursively applying Lemma 1 with Ω k , for all k. Thus,
P T (Y) − UV T F = ∆ k 0 F = 1 2 k 0 UV T F ≤ ̺ (m + n) 15
The above result fails with probability at most (m + n) 1−c for each k; thus, total probability of failure is at most 11(m + n) 1−c log(m + n).
Bounding P T ⊥ (Y) 2
By definition, Y can be written as
Y = k 0 k=1 R Ω k P T UV T − P T (W k−1 ) = k 0 k=1 R Ω k P T (∆ k−1 )
It follows that,
P T ⊥ (Y) 2 = P T ⊥ k 0 k=1 (R Ω k P T − P T ) (∆ k−1 ) 2 ≤ k 0 k=1 (R Ω k − I)(∆ k−1 ) 2 We use P T (∆ k ) = P T (UV T − P T (W k )) = UV T − P T (W k ) = ∆ k , for all k.
We apply Lemma 2 to each summand in the above inequality, with corresponding Ω k , to obtain
P T ⊥ (Y) 2 ≤ 2 c c 0 k 0 k=1 ∆ k−1 µ(∞,2) + c c 0 k 0 k=1 ∆ k−1 µ(∞)(15)
holding with probability at least 1 − (m + n) 1−c , for each k. We can derive the following, applying Lemma 4 k times, with Ω k ,
∆ k µ(∞) = (P T − P T R Ω k )∆ k−1 µ(∞) ≤ 1 2 i ∆ k−i µ(∞) ≤ 1 2 k UV T µ(∞)(16)
For each k, the above fails with probability at most k · (m + n) 3−c . Similarly, applying Lemma 3 and Lemma 4 recursively, with Ω k , we can derive,
∆ k µ(∞,2) = (P T − P T R Ω k P T )∆ k−1 µ(∞,2) ≤ 1 2 ∆ k−1 µ(∞) + 1 2 ∆ k−1 µ(∞,2) (step j) ≤ j i=1 1 2 i ∆ k−i µ(∞) + 1 2 j ∆ k−j µ(∞,2) ≤ j i=1 1 2 i 1 2 k−i UV T µ(∞) + 1 2 j ∆ k−j µ(∞,2) ≤ j 1 2 k UV T µ(∞) + 1 2 j ∆ k−j µ(∞,2) (step k) ≤ k 1 2 k UV T µ(∞) + 1 2 k UV T µ(∞,2)(17)
Above, we apply Lemma 3 k times, and Lemma 4 up to k i=1 i = k(k + 1)/2 times. Thus, the above relation holds with failure probability at most k · (m + n) 2−c + k(k + 1) · (m + n) 3−c /2, for each k. Using (16) and (17), it follows from (15),
P T ⊥ (Y) 2 ≤ 2 c c 0 k 0 k=1 (k − 1) 1 2 k−1 UV T µ(∞) + 2 c c 0 k 0 k=1 1 2 k−1 UV T µ(∞,2) + c c 0 k 0 k=1 1 2 k−1 UV T µ(∞,2)
We note that, for all (i, j),
(UV T ) ij = e T i UV T e j ≤ µ i ̺ m ν j ̺ n ≤ 1, (UV T ) i, * 2 = e T i UV T 2 = µ i ̺ m , (UV T ) * ,j 2 = UV T e j 2 = ν j ̺ n Thus, UV T µ(∞,2) = max max i m µ i ̺ (UV T ) i, * 2 , max j n ν j ̺ (UV T ) * ,j 2 = 1
Therefore,
P T ⊥ (Y) 2 ≤ 2 c c 0 k 0 k=1 (k − 1) 1 2 k−1 + 1 2 k−1 + c c 0 k 0 k=1 1 2 k−1 < 2 c c 0 ∞ k=1 k 1 2 k−1 + c c 0 ∞ k=1 1 2 k−1 = 8 c c 0 + 2c c 0 ≤ 1 2 ,
by setting c 0 ≥ 264c. Let δ = (m + n) 3−c . The above holds with failure probability at most δ + (k − 1)δ + δ · (k − 1)(k + 2)/2, for each k. Summing over all k = 1, ..., k 0 , total failure probability does not exceed δ · k 3 0 . Here we upper bound the failure probability by simply counting the number of times the intermediate lemmas are called. A better bookkeeping could give us tighter bound. Also, we note that the constant c 1 in Theorem 1 depends on c 0 and k 0 , where c 0 depends on c (related to failure probability). It is easy to verify that the final bound on c 0 remains the same for both the leverage score sampling (Chen et al. [2014]) and the relaxed leverage score sampling in (4). Consequently, the final global constant c 1 is the same for both the sampling methods, for a fixed k 0 (we can set the same k 0 for both the sampling methods). Also, the failure probabilities are the same for both the sampling methods. Therefore, the sample size needed to solve (2) is strictly smaller for the relaxed leverage score sampling (4), comparing to the leveraged sampling in Chen et al. [2014]. Experimental results on real datasets in Section 3 is in compliance with this theoretical analysis.
We sample each (i, j)-th entry independently with probability p ij to form the set of observations Ω. That is, total number of sampled entries, denoted by s, is a random variable. Expected number of observed entries required to solve (2) is
E(s) = i,j p ij = O(((m + n)̺ − ̺ 2 )log 2 (m + n)).
Condition 2a and 2b in Proposition 1 fail with probabilities at most k 0 · (m + n) 1−c and k 3 0 · (m + n) 3−c , respectively. Also, Lemma 1, used in Lemma 6, fails with probability at most (m + n) 1−c . Summing all of them, failure probability never exceeds 3 log 3 (m + n)(m + n) 3−c (a tighter bound could be calculated), for sufficiently large c > 3. Finally, we can apply Hoeffding's inequality to show that s is sharply concentrated around its expectation, i.e., s = Θ(((m + n)̺ − ̺ 2 )log 2 (m + n)) with probability at least 1 − 6 log 3 (m + n)(m + n) 3−c , for sufficiently large c > 3.
This completes the proof of Theorem 1.
Proof of Theorem 2
We closely follow the proof given by Chen et al. [2014]. We pick each row of M with some probability p and observe all the entries of this sampled row. Let Γ ⊆ [m] be the set of indices of the row picked, and S Γ (X) be a matrix obtained from X by zeroing out the rows outside Γ. Recall, SVD of M is UΣV T . We use the following lemma (Lemma 14 of Chen et al. [2014]).
Lemma 5 Let µ i ≤ max i m ρ U T e i 2 2 ≤ µ 0 , ∀i ∈ [m]
, and p ≥ c 2 µ 0 ̺ m logm for some universal constant c 2 . Then, for any universal constant c > 1, and c 2 ≥ 20c,
U T S Γ (U) − I ̺ 2 ≤ 1/2,
holds with probability at least 1 − (m + n) 1−c , where I ̺ is the identity matrix in R ̺×̺ . Now, U T S Γ (U) − I ̺ 2 ≤ 1/2 implies that U T S Γ (U) is invertible and S Γ (U) ∈ R m×̺ has rank-̺. Using SVD of M, we can write S Γ (M) = S Γ (U)ΣV T , and this has full rank-̺. Therefore, S Γ (M) and M have the same row space, and we conclude thatν j = ν j , ∀j ∈ [n]. Thus, using the sample set Ω in Algorithm 1 we can recover M exactly via nuclear norm minimization in (2), with high probability. Expected number of entries observed in Algorithm 1 is
pmn + m,n i,j p ij = O(µ 0 ((m + 2n)̺ − ̺ 2 )log 2 (m + n)),
where, p ij as in (5). We apply standard Hoeffding inequality to bound the actual sample size, and Theorem 2 follows as a corollary of Theorem 1.
Proof of Optimality Conditions in Proposition 1
Let M be the low-rank target matrix with rank-̺ SVD M = UΣV T . Let Z be any matrix such that R Ω (Z) = 0, e.g., Z is in the null space of R Ω operator. We can choose U ⊥ and V ⊥ such that [U, U ⊥ ] and [V, V ⊥ ] are unitary matrices for which U ⊥ V T ⊥ , P T ⊥ (Z) = P T ⊥ (Z) * . Then it follows from standard inequality of trace norm, for some Y in the range of R Ω ,
M + Z * ≥ UV T + U ⊥ V T ⊥ , M + Z = M * + UV T + U ⊥ V T ⊥ , Z = M * + UV T − P T (Y), P T (Z) + U ⊥ V T ⊥ − P T ⊥ (Y), P T ⊥ (Z) (a) ≥ M * − UV T − P T (Y) F · P T (Z) F + P T ⊥ (Z) * − P T ⊥ (Y), P T ⊥ (Z) ≥ M * − UV T − P T (Y) F · P T (Z) F + (1 − P T ⊥ (Y) 2 ) P T ⊥ (Z) * (b) ≥ M * + 1 − P T ⊥ (Y) 2 − UV T − P T (Y) F max i,j 1 √ p ij 1 − P T R Ω P T − P T op 1 2 P T ⊥ (Z) * > M *
Above, (a) follows from Von-Neumann trace inequality, and (b) follows from Lemma 6. Using max i,j 1 √ p ij ≤ (mn) 5/2 ≤ (m + n) 5 , and the conditions in Proposition 1, we derive the final inequality. Thus, if there exists a dual certificate Y satisfying Condition 2a and 2b in Proposition 1, we have for any X obeying R Ω (M − X) = 0, i.e., X ab = M ab , for all (a, b) ∈ Ω, X * > M * . Hence, M is the unique minimizer of (2).
The following lemma is similar to Lemma 13 of Chen et al. [2014].
Lemma 6 For any Z ∈ R m×n , s.t., P Ω (Z) = 0,
P T (Z) F ≤ 1 − P T R Ω P T − P T op − 1 2 max i,j 1 √ p ij P T ⊥ ZΩ P T (Z) 2 F = R Ω P T (Z), P T (Z) = P T R Ω P T (Z), P T (Z) = P T R Ω P T (Z) − P T (Z), P T (Z) + P T (Z), P T (Z) ≥ (1 − P T R Ω P T − P T op ) · P T (Z) 2 F(18)
Also, we have R
0 = R 1/2 Ω (Z) F ≥ R 1/2 Ω P T (Z) F − R 1/2 Ω P T ⊥ (Z) F R 1/2 Ω P T (Z) F ≤ R 1/2 Ω P T ⊥ (Z) F ≤ max i,j 1 √ p ij P T ⊥ (Z) F ,(19)
where we use
R 1/2 Ω P T ⊥ (Z) F ≤ max i,j 1 √ p ij i,j δ ij e i e T j , P T ⊥ (Z) e i e T j F ≤ max i,j 1 √ p ij P T ⊥ (Z) F
Combining (18) and (19), and using X F ≤ X * ,
(1 − P T R Ω P T − P T op ) · P T (Z) F ≤ max i,j 1 √ p ij P T ⊥ (Z) F ≤ max i,j 1 √ p ij P T ⊥ (Z) *
The result follows. ⋄
Proof of Technical Lemmas
Here we prove Lemmas 1 through 5 using the matrix Bernstein inequality of Lemma 8 as the main tool. Also, we frequently use the fact in (21) and the result in Lemma 7. Note that P T is self-adjoint linear operator. Thus we can write the following for any X ∈ R m×n :
P T (X) = i,j P T (X), e i e T j e i e T j = i,j P T (X), P T (e i e T j ) e i e T j = i,j X, P T (e i e T j ) e i e T j(20)
We can derive, for all i and j,
P T e i e T j 2 F = P T e i e T j , e i e T j = µ i ̺ m + ν j ̺ n − µ i ̺ m · ν j ̺ n(21)
Also, we know for all i, j,
0 ≤ µ i ̺ m ≤ µ i ̺ m ≤ 1, 0 ≤ ν j ̺ n ≤ ν j ̺ n ≤ 1.(22)
Lemma 7 Using our notations, for all i, j,
µ i ̺ m + ν j ̺ n − µ i ̺ m · ν j ̺ n ≥ µ i ̺ m · ν j ̺ n
Proof: Let, x = µ i ̺ m and y = ν j ̺ n . Then, (x + y − xy) 2 = xy + (x 2 − x 2 y) + (y 2 − xy 2 ) + x 2 y 2 + xy − x 2 y − xy 2 = xy + x 2 (1 − y) + y 2 (1 − x) + xy(1 − x)(1 − y) ≥ xy using (22) Also, x + y − xy ≥ 0. Thus, x + y − xy ≥ √ xy.
⋄
Lemma 8 (Tropp [2012], [Theorem 16] of Chen et al. [2014]) Let X 1 , ..., X N ∈ R m×n be independent, zero-mean random matrices. Suppose
max N t=1 E X t X T t 2 , N t=1 E X T t X t 2 ≤ σ 2
and X t 2 ≤ γ almost surely for all t. Then for any c > 0, we have N t=1 X t 2 ≤ 2 cσ 2 log(m + n) + cγlog(m + n)
with probability at least 1 − (m + n) −(c−1) .
We consider sampling probabilities {q ij } of the form (11) to prove Lemmas 1 through 4.
Notation Overloading: For simplicity, we reuse some of the notations in Section 7.1 through 7.4. Specifically, we replaceΩ by Ω to denote a sample set in (13), and, δ ij = I((i, j) ∈Ω).
Proof of Lemma 1
For any matrix Z ∈ R m×n , we can write
(P T R Ω P T − P T ) (Z) = i,j 1 q ij δ ij − 1 P T e i e T j , Z P T e i e T j = i,j S ij (Z).
Using E [δ ij ] = q ij , we have E[S ij (Z)] = 0 for any Z. Thus, we conclude that E[S ij ] = 0. Also, S ij 's are independent of each other. Using probabilities in (11) (S ij 's vanish when q ij = 1, for all Z and (i, j)), and (21), we derive
S ij (Z) F ≤ 1 q ij P T e i e T j 2 F Z F ≤ Z F c 0 · log(m + n)
.
From definition of operator norm, S ij op ≤ 1 c 0 ·log(m+n) . Also, we derive
E S 2 ij (Z) = E 1 q ij δ ij − 1 2 e i e T j , P T (Z) e i e T j , P T (e i e T j ) P T e i e T j = 1 − q ij q ij e i e T j , P T (Z) e i e T j , P T (e i e T j ) P T e i e T j i,j E S 2 ij (Z) F ≤ max i,j 1 − q ij q ij P T (e i e T j ) 2 F i,j e i e T j , P T (Z) P T e i e T j F = max i,j 1 − q ij q ij P T (e i e T j ) 2 F P T i,j e i e T j , P T (Z) e i e T j F = max i,j 1 − q ij q ij P T (e i e T j ) 2 F P T (Z) F i,j E S 2 ij op ≤ max i,j 1 − q ij q ij P T (e i e T j ) 2 F ≤ 1 c 0 · log(m + n)
We apply Matrix Bernstein inequality in Lemma 8 using
σ 2 = 1 c 0 · log(m + n) , γ = 1 c 0 · log(m + n) ,
to obtain, for any c > 1, c 0 ≥ 20c,
P T R Ω P T − P T op ≤ 1/2
holding with probability at least 1 − (m + n) (1−c) .
Proof of Lemma 2
We can write the matrix (R Ω − I) Z as sum of independent matrices:
(R Ω − I) Z = i,j 1 q ij δ ij − 1 Z ij e i e T j = i,j S ij .
We note that, E[S ij ] = 0, and S ij 's are zero matrix when q ij = 1, for all (i, j). We have
S ij 2 ≤ |Z ij | q ij . Moreover, i,j E S ij S T ij = i,j Z 2 ij e i e T i E 1 q ij δ ij − 1 2 = i j Z 2 ij 1 − q ij q ij e i e T i Thus, i,j E S ij S T ij 2 ≤ max i n j=1 1 − q ij q ij Z 2 ij Similarly, i,j E S T ij S ij 2 ≤ max j m i=1 1 − q ij q ij Z 2 ij
Clearly, when q ij = 1 the above quantities are zero. Using q ij in (11), and Lemma 7, we have
S ij 2 ≤ 1 c 0 · log(m + n) |Z ij | m µ i ̺ n ν j ̺ ≤ Z µ(∞)
c 0 · log(m + n) .
Using q ij in (11), and noting that µ i ̺ m +
ν j ̺ n − µ i ̺ m · ν j ̺ n ≥ µ i ̺ m , we have n j=1 1 − q ij q ij Z 2 ij ≤ 1 c 0 · log(m + n) · m µ i ̺ n j=1 Z 2 ij ≤ Z 2 µ(∞,2) c 0 · log(m + n) .
Similarly,
m i=1 1 − q ij q ij Z 2 ij ≤ 1 c 0 · log(m + n) · n ν j ̺ m i=1 Z 2 ij ≤ Z 2 µ(∞,2) c 0 · log(m + n) .
The lemma follows from Matrix Bernstein inequality in Lemma 8, with
γ log(m + n) ≤ 1 c 0 Z µ(∞) , σ 2 log(m + n) ≤ 1 c 0 Z 2 µ(∞,2) .
Proof of Lemma 3
Let,
X = (P T R Ω − P T )Z = i,j δ ij q ij − 1 Z ij P T (e i e T j )
Weighted b-th column of X can be written as sum of independent, zero-mean column vectors.
n ν b ̺ X * ,b = i,j δ ij q ij − 1 Z ij P T (e i e T j )e b n ν b ̺ = i,j s ij
Clearly, E[s ij ] = 0. We need bounds on s ij 2 and i,j E s T ij s ij 2 to apply Matrix Bernstein inequality. First, we need to bound P T (e i e T j )e b 2 .
I − UU T e i e T j VV T e b 2 ≤ e T j VV T e b j = b,(23)
Above we use triangle inequality and definition of µ i and ν b . Note that, s ij is a zero vector when q ij = 1, for all (i, j). Otherwise, for q ij = 1, we consider two cases. Using bounds in (23), we have for j = b,
s ij 2 ≤ 1 q ib |Z ib | n ν b ̺ µ i ̺ m + ν b ̺ n
Using q ij in (11), q ib ≥ c 0 log(m + n) µ i ̺ m ν b ̺ n and q ib ≥ c 0 log(m + n) · µ i ̺ m . Combining these two inequalities, we have
s ij 2 log(m + n) ≤ 2 c 0 |Z ib | m µ i ̺ · n ν b ̺ µ i ̺ m + ν b ̺ n µ i ̺ m + ν b ̺ n ≤ 2 c 0 Z µ(∞)
For j = b, using q ib ≥ c 0 log(m + n) µ i ̺ m ν b ̺ n (Lemma 7) and e T j VV T e b ≤ ν j ̺ n · ν b ̺ n ,
s ij 2 ≤ 1 q ij |Z ij | n ν b ̺ · ν j ̺ n · ν b ̺ n ≤ 2 c 0 log(m + n) Z µ(∞)
Therefore, for all (i, j), we have s ij 2 ≤ 2 c 0 log(m+n) Z µ(∞) . On the other hand,
i,j E s T ij s ij = j=b,i + j =b,i 1 − q ij q ij Z 2 ij P T (e i e T j )e b 2 2 · n ν b ̺
The above quantity is zero for q ij = 1. Otherwise, for q ij = 1, we consider two cases.
For j = b, using (23) we have, P T (e i e T j )e b 2 2 ≤ µ i ̺ m + ν b ̺ n 2 ≤ 2 µ i ̺ m + ν b ̺ n . Using q ij in (11), we have,
j=b,i ≤ 2 i 1 − q ib q ib Z 2 ib µ i ̺ m + ν b ̺ n · n ν b ̺ ≤ 4 c 0 log(m + n) Z 2 µ(∞,2) ,
where we use the following bound in the second inequality. For all (i, j), q ij = 0,
µ i ̺ m + ν j ̺ n µ i ̺ m + ν j ̺ n − µ i ̺ m · ν j ̺ n = 1 + µ i ̺ m · ν j ̺ n µ i ̺ m + ν j ̺ n − µ i ̺ m · ν j ̺ n ≤ 1 + µ i ̺ m · ν j ̺ n max{ µ i ̺ m , µ j ̺ n } ≤ 2.
For j = b, using q ij ≥ c 0 log(m + n) ·
Proof of Lemma 4
Let, X = (P T R Ω − P T )Z = i,j δ ij q ij − 1 Z ij P T (e i e T j ) . We write rescaled (a, b)-th element of X as
[X] ab m µ a ̺ n ν b ̺ = i,j δ ij q ij − 1 Z ij P T (e i e T j ) ab m µ a ̺ n ν b ̺ = i,j s ij
This is a sum of independent, zero-mean random variables. we seek to bound |s ij | and i,j E s 2 ij . First, we need to bound e a e T b , P T (e i e T j ) .
e a e T b , P T (e i e T j ) = e T a UU T (e i e T j )e b + e T a (e i e T j )VV T e b − e T a UU T (e i e T j )
VV T e b = P T (e a e T b ) 2 F = µa̺ m + ν b ̺ n − µa̺ m · ν b ̺ n i = a, j = b, e T a (I − UU T )e a e T j VV T e b ≤ e T j VV T e b i = a, j = b, e T a UU T e i e T b (I − VV T )e b ≤ e T a UU T e i i = a, j = b, e T a UU T e i e T j VV T e b ≤ e T a UU T e i e T j VV T e b i = a, j = b(24)
where we use I − UU T 2 ≤ 1 and I − VV T 2 ≤ 1. Note that, s ij = 0 when q ij = 1. Otherwise, for q ij = 1,
|s ij | ≤ 1 q ij |Z ij | e a e T b , P T (e i e T j ) m µ a ̺ n ν b ̺
We consider four cases. For i = a, j = b, using q ab ≥ c 0 log(m + n) µa̺ m + ν b ̺ n − µa̺ m · ν b ̺ n |s ij | ≤ 1 q ab |Z ab | P T (e a e T b )
Above we use µ i ̺ m ≤ 1, ν j ̺ n ≤ 1, for all i, j. We conclude, for all (i, j),
|s ij | ≤ 1 c 0 log(m + n) Z µ(∞) .
On the other hand,
i,j E s 2 ij = i,j E δ ij q ij − 1 2 Z 2 ij e a e T b , P T (e i e T j ) 2 m µ a ̺ · n ν b ̺ = i,j 1 − q ij q ij Z 2 ij e a e T b , P T (e i e T j ) 2 m µ a ̺ · n ν b ̺ = i=a,j=b + i=a,j =b + i =a,j=b + i =a,j =b
The above quantity is zero for q ij = 1. We bound the above considering four cases for q ij = 1. For i = a, j = b, using q ab ≥ c 0 log(m + n) µa̺ m + ν b ̺ n − µa̺ m · ν b ̺ n ,
Figure 1 :
1[MovieLens] Plot of singular values, and the normalized leverage scores for ̺ = 10.
Figure 2 :
2[TechTC] Plot of singular values, and the normalized leverage scores for ̺ = 10.
any Z s.t. P Ω (Z) = 0. It follows,
P
T (e i e T j )e b 2 = UU T (e i e T j )e b + (e i e T j )VV T e b − UU T (e i e T j )VV T e T e i + I − UU T e i V T
Table 1 :
1[MovieLens] Gain in sample size for exact recovery using relaxed leverage score sampling.
Table 2 :
2[TechTC] Gain in sample size for exact recovery using relaxed leverage score sampling.
µ j ̺ n and (23),in a similar way. We apply Matrix Bernstein inequality in Lemma 8, withWe set c 0 ≥ 80c to deriveSimilarly, we can bound m µa̺ X a, * 2 by the same quantity. We take a union bound over all rows a and all columns b (i.e., total (m + n) events) to obtain, for any c > 2,holding with probability at least 1 − (m + n) 2−c .ACKNOWLEDGMENT:I thank Prof Petros Drineas and Prof Malik Magdon-Ismail for helpful discussions on this topic. Prof Drineas encouraged me to work on this problem.
Experimental results on real data sets corroborate the theoretical analysis. It would be an interesting problem to reduce the bound on the sample size by a logarithmic factor to Θ(((m + n)̺ − ̺ 2 )log(m + n)). This is a theoretical lower bound established by Candes and Tao. is possible to recover any arbitrary low-rank data matrix exactly via the optimization problem in (2) using the relaxed leverage score sampling proposed in this work. and further reduction is not possibleis possible to recover any arbitrary low-rank data matrix exactly via the optimization problem in (2) using the relaxed leverage score sampling proposed in this work. This notion of relaxation in leverage scores requires a strictly smaller sample size comparing to the best-known result of Chen et al. [2014]. Experimental results on real data sets corroborate the theoretical analysis. It would be an interesting problem to reduce the bound on the sample size by a logarith- mic factor to Θ(((m + n)̺ − ̺ 2 )log(m + n)). This is a theoretical lower bound established by Candes and Tao [2010], and further reduction is not possible.
Exact matrix completion via convex optimization. E J Candes, B Recht, Foundations of Computational Mathematics. 9E. J. Candes and B. Recht. Exact matrix completion via convex optimization. In Foundations of Computational Mathematics, pages 717-772, 9(6), 2009.
The power of convex relaxation: Near-optimal matrix completion. E J Candes, T Tao, IEEE Transactions on Information Theory. 56E. J. Candes and T. Tao. The power of convex relaxation: Near-optimal matrix completion. In IEEE Transactions on Information Theory, pages 2053-2080, 56(5), 2010.
Robust principal component analysis?. E J Candes, X Li, Y Ma, J Wright, Journal of the ACM. 11E. J. Candes, X. Li, Y. Ma, and J. Wright. Robust principal component analysis? In Journal of the ACM, page 58(3):11, 2011.
Coherent Matrix Completion. Y Chen, Bhojanapalli, R Sanghavi, Ward, Proceedings of International Conference on Machine Learning. International Conference on Machine LearningY Chen, S Bhojanapalli, S Sanghavi, and R Ward. Coherent Matrix Completion. Proceedings of International Conference on Machine Learning, pages 674-682, 2014.
Text categorization with many redundant features: using aggressive feature selection to make SVMs competitive with C4.5. E Gabrilovich, S Markovitch, Proceedings of International Conference on Machine Learning. International Conference on Machine LearningE. Gabrilovich and S. Markovitch. Text categorization with many redundant features: using ag- gressive feature selection to make SVMs competitive with C4.5. In Proceedings of International Conference on Machine Learning, 2004.
Recovering low-rank matrices from few coefficients in any basis. D Gross, IEEE Transactions on Information Theory. 57D. Gross. Recovering low-rank matrices from few coefficients in any basis. In IEEE Transactions on Information Theory, pages 1548-1566, 57(3), 2011.
Low-rank matrix and tensor completion via adaptive sampling. Akshay Krishnamurthy, Aarti Singh, Advances in Neural Information Processing Systems. 26Akshay Krishnamurthy and Aarti Singh. Low-rank matrix and tensor completion via adaptive sampling. In Advances in Neural Information Processing Systems 26, pages 836-844. 2013.
A simpler approach to matrix completion. B Recht, The Journal of Machine Learning Research. 12B. Recht. A simpler approach to matrix completion. In The Journal of Machine Learning Research, pages 3413-3430, 12, 2011.
User-friendly tail bounds for sums of random matrices. J Tropp, Foundations of Computational Mathematics. 12J. Tropp. User-friendly tail bounds for sums of random matrices. In Foundations of Computational Mathematics, pages 12(4):389-434, 2012.
|
[] |
[
"Break up of returning plasma after the 7 June 2011 filament eruption by Rayleigh-Taylor instabilities",
"Break up of returning plasma after the 7 June 2011 filament eruption by Rayleigh-Taylor instabilities"
] |
[
"D E Innes \nMax-Planck Institut für Sonnensystemforschung\n37191Katlenburg-LindauGermany\n",
"R H Cameron \nMax-Planck Institut für Sonnensystemforschung\n37191Katlenburg-LindauGermany\n",
"L Fletcher \nMax-Planck Institut für Sonnensystemforschung\n37191Katlenburg-LindauGermany\n\nSchool of Physics and Astronomy\nSUPA\nUniversity of Glasgow\nG12 8QQGlasgowUK\n",
"S K Solanki \nMax-Planck Institut für Sonnensystemforschung\n37191Katlenburg-LindauGermany\n\nSchool of Space Research\nKyung Hee University\n446-701Yongin, GyeonggiKorea\n"
] |
[
"Max-Planck Institut für Sonnensystemforschung\n37191Katlenburg-LindauGermany",
"Max-Planck Institut für Sonnensystemforschung\n37191Katlenburg-LindauGermany",
"Max-Planck Institut für Sonnensystemforschung\n37191Katlenburg-LindauGermany",
"School of Physics and Astronomy\nSUPA\nUniversity of Glasgow\nG12 8QQGlasgowUK",
"Max-Planck Institut für Sonnensystemforschung\n37191Katlenburg-LindauGermany",
"School of Space Research\nKyung Hee University\n446-701Yongin, GyeonggiKorea"
] |
[] |
Context. A prominence eruption on 7 June 2011 produced spectacular curtains of plasma falling through the lower corona. At the solar surface they created an incredible display of extreme ultraviolet brightenings.Aims. To identify and analyze some of the local instabilities which produce structure in the falling plasma. Methods. The structures were investigated using SDO/AIA 171Å and 193Å images in which the falling plasma appeared dark against the bright coronal emission. Results. Several instances of the Rayleigh-Taylor instability were investigated. In two cases the Alfvén velocity associated with the dense plasma could be estimated from the separation of the Rayleigh-Taylor fingers. A second type of feature, which has the appearance of self-similar branching horns was discussed.
|
10.1051/0004-6361/201118530
|
[
"https://arxiv.org/pdf/1202.4981v1.pdf"
] | 54,614,069 |
1202.4981
|
d64b19d46b4151acabf1bdfc2536261254a7e089
|
Break up of returning plasma after the 7 June 2011 filament eruption by Rayleigh-Taylor instabilities
February 23, 2012
D E Innes
Max-Planck Institut für Sonnensystemforschung
37191Katlenburg-LindauGermany
R H Cameron
Max-Planck Institut für Sonnensystemforschung
37191Katlenburg-LindauGermany
L Fletcher
Max-Planck Institut für Sonnensystemforschung
37191Katlenburg-LindauGermany
School of Physics and Astronomy
SUPA
University of Glasgow
G12 8QQGlasgowUK
S K Solanki
Max-Planck Institut für Sonnensystemforschung
37191Katlenburg-LindauGermany
School of Space Research
Kyung Hee University
446-701Yongin, GyeonggiKorea
Break up of returning plasma after the 7 June 2011 filament eruption by Rayleigh-Taylor instabilities
February 23, 2012Astronomy & Astrophysics manuscript no. RT˙archive2 c ESO 2012 Received ...; accepted ...Sun: activity Sun:coronal mass ejection -Instabilities
Context. A prominence eruption on 7 June 2011 produced spectacular curtains of plasma falling through the lower corona. At the solar surface they created an incredible display of extreme ultraviolet brightenings.Aims. To identify and analyze some of the local instabilities which produce structure in the falling plasma. Methods. The structures were investigated using SDO/AIA 171Å and 193Å images in which the falling plasma appeared dark against the bright coronal emission. Results. Several instances of the Rayleigh-Taylor instability were investigated. In two cases the Alfvén velocity associated with the dense plasma could be estimated from the separation of the Rayleigh-Taylor fingers. A second type of feature, which has the appearance of self-similar branching horns was discussed.
Introduction
One of the most spectacular solar events seen so far with the Atmospheric Imaging Assembly (AIA) on the Solar Dynamics Observatory (SD0) occurred when a filament erupted on 7 June 2011 1 . The eruption that was associated with a fast coronal mass ejection (CME), an M2 flare and dome-shaped extreme ultraviolet (EUV) front (Cheng et al. 2012), slung material across almost a quarter of the solar surface. The 'S' shaped filament erupted from the southern active region AR11226 as it was nearing the western limb. The eruption started at 6:00 UT, and reached its peak GOES soft X-ray brightness at 6:35 UT. Non-escaping material was seen in SOHO/LASCO C2 and STEREO/COR1 images falling back from heights up to 4 solar radii. The first returning material was seen at the solar surface at around 7:00 UT in SDO/AIA and STEREO/EUVI-A images.
Here we concentrate on the structure of the falling plasma. An hour and a half after the eruption it looked like a huge upsidedown crown extending over at least 600 (Fig. 1). The leading edge had broken-up into semi-regular arcs and spikes, similar to the rim of a splash (e.g. Edgerton 1987). Such long fingers and arcs are also seen in the Crab supernova remnant (Hester 2008). In the Crab, the main fingers are thought to be the result of the magnetic Rayleigh-Taylor (RT) instability (Hester et al. 1996). RT has also been invoked to explain filamentary structure (Isobe et al. 2005) and prominence bubbles (Berger et al. 2011) on the Sun.
In this letter we highlight features that lead us to conclude that the magnetic RT is also responsible for the structures seen in this event. Further we investigate properties of the magnetic RT instability to obtain diagnostics of the local plasma conditions. Send offprint requests to: D.E. Innes e-mail: [email protected] 1 http://www.thesuntoday.org/current-observations/a-spectacularevent-a-filamentprominence-eruption-to-blow-your-socks-off/
Observations
SDO/AIA takes images of the full solar disk with a resolution of about 0.6 pixel −1 through 10 filters, selected to single out specific strong lines in the corona and continuum emission from the lower chromosphere. We investigate the structure of cold filament plasma which was seen as dark structures in the 171,131,193,211,304, and 335Å images due to absorption of background EUV emission by neutral hydrogen, helium and singly ionized helium. Since we will be discussing the structure of the plasma as it appears against the coronal background, we only show 171Å and 193Å data because these have the best contrast. The data are presented as either intensity or ratio images. The ratio images are the log of the ratio of the image at the time given and the image taken 12 s earlier except where times are given specifically. In the ratio images the absorbing plasma has moved from the bright to the dark regions. Flow velocities have been computed with the optical flow code of Gissot & Hochedez (2007) and are represented by white arrows in the ratio images.
Onset
The overall development of the filament eruption is shown in Fig. 2. Fig. 2a gives the impression of a hot core with a cool cone-like shell expanding upwards above it. There is also a cloud of hot ejecta coming from the center (red arrow). Focusing on the northern edge of the dark cone, we see that initially this edge was essentially straight. Then, as the erupting material expanded, parts broke away (red box in Fig. 2b) and the upper edge started to corrugate. These corrugations later developed into the finger and arc structure in Fig. 1. Unfortunately they were too far off limb to clearly observe their evolution in the SDO/AIA images due to low contrast with the background.
Spikes
The part that broke away shows small regularly spaced spikes pointing outward along its edge (Fig. 2d, red arrow). They look similar to the spikes observed by SDO/AIA in the 131Å filter on the edge of a coronal eruption and interpreted by Foullon et al. (2011) as Kelvin-Helmholtz roll-ups. The ones here appear to be different because, as implied by the ratio image of this filament ( Fig. 3) and shown in the online appendix ( Fig. A.1), they grew upwards and did not turn over. It is interesting to note that the growth which was along the flow direction was perpendicular to the solar radial direction, hence not governed by gravity.
The spikes in Fig. 3 had a separation of 5 , length of 6 , and growth rate 12 km s −1 . About 1 min later they faded and were overtaken by plasma from behind. Similar spikes were seen along the edge of filamentary structures throughout the evolution and they were all pointing in the direction of motion and disappeared without turning over.
Fingers and Arcs
The falling filament plasma closest to the active region presented a sequence of well-exposed images showing arcs and some small-scale fingers. A series of processes occurred. Structure in the ejected plasma was initially stretched out by its large-scale expansion, creating small cavities that then expanded by pressure gradients probably associated with outflow from the eruption site. As the cavity expanded, new smaller-scale fingers and arcs formed on the inside. The movie, arcs movie shows the complete evolution of the cavities and small-scale fingers and arcs on the inside. A snapshot of the movie is shown in Fig. 4a. The red bar bridges one of these small-scale arcs. The outline of an expanding cavity is clearly visible in the ratio image (Fig. 4b).
To study the acceleration of the arcs, we show the space-time running ratio image in Fig. 5. It was taken along the red arrow drawn through the apex of the arc in Fig. 4b. The expanding edge of the arc is over-plotted with the red line d ∝ t 2 which is the relationship expected for constant acceleration of the interface. Space-time images along different directions through the arc give similar results but the edge is not as sharp. This is the background on which the small-scale RT fingers we are interested in develop.
Horns
Another set of features we would like to point out are the horns. The formation of three of these is illustrated in Fig. 6 where they are labelled H1, H2, and H3, and in the accompanying movie, horns movie. These horns sometimes formed out of sheets that distorted (H1 and H2) and sometimes when a thread tore away from the main stem (H3). At the time the central frame was taken, H1 was compressing as it moved northward as though being pushed from the south. Most the other structures are also moving north although the tips of the the denser/darker fingers (e.g. H2, H3) have a significant sunward component. These ones later developed arc-like horns. In Fig. 6f, a small spike has developed inside H2 (red arrow), repeating in a self-similar way the finger and arc structure.
The movie shows there were several background effects that could have been influencing the evolution and growth of the horns because (i) nearly all the structures are moving northward, (ii) there are hot outflows from the eruption site, and (iii) at the end of the movie there is a bright ridge off-limb along which dark plasma seems to be falling. The ridge and sideways motion can be attributed to the coronal magnetic field configuration which at the height of the falling plasma consisted of closed loops connecting the active regions north and south of the equator (Fig. 1). There are no obvious signs of a direct influence of the outflows on the evolution of the falling plasma, but the flows would probably have been directed along field lines connecting the active regions and could therefore have added momentum to the filament motion. The horns might thus be associated with the global field configuration rather than being a purely local phenomenon.
Discussion
Since both the time and spatial scales are large, the evolution of the instabilities studied here can be treated as MHD phenomena. The examples discussed in Sect. 1, 2.1, 2.3 are all cases of cool dense material sitting on top of lighter hotter plasma. In the features discussed in 2.2 and 2.3, a large-scale pressure gradient appears to be accelerating the structure (see Fig. 5 where constant acceleration of the interface can be readily inferred). The RT instability occurs whenever a denser fluid is accelerated against a less dense fluid by, for example, gravity or pressure gradients. So if we ignore the effects of the magnetic field, each of these cases is RT unstable. Indeed the large-scale structure of the falling filament plasma looks like the rim of a large splash, in which long fingers are connected by shadowy arcs (Fig. 1). Although the micro-physics is different it is probable that here, like in the splash (Allen 1975), RT instabilities were responsible for the break-up.
To proceed further we need to consider the structure of the magnetic field in each particular case. Although the dense, cool filament plasma is partially neutral there will be strong collisional coupling between the neutrals and the ionized component. It is also reasonable to assume that the hot and cold plasma lie on different field lines because thermal conduction is efficient along field lines. In this case the magnetic field must be parallel to the interface between the two plasmas. Magnetic tension then tends to inhibit transverse motions which vary along any field line, with shorter wavelength fluctuations being more strongly inhibited. For the case where the magnetic field in both fluids is oriented in the same direction, the instability only occurs for waves which have k < k c where k is the component of the wave vec-tor aligned with the magnetic field (Chandrasekhar 1961). The wavelength associated with k c is
λ c = B 2 g(ρ h − ρ l )(1)
where ρ h /ρ l > 1 is the ratio of the densities on the two sides of the interface, B is the strength of the uniform magnetic field, and g is the net acceleration due to gravity and additional acceleration, a, caused by pressure gradients across the interface. Modes with k = 0 are unaffected by the field, with the consequence that the instability acts to form sheets aligned with the field (Isobe et al. 2005). Here, however, the magnetic field orientation in the cold plasma and in the warm plasma will not, in general, be aligned. In this case the magnetic tension acts against the instability for all k.
Simulations of the magnetic RT instability with different magnetic field orientations on the two sides of the density jump have been performed by Stone & Gardiner (2007). If the fields on either side of the interface are oriented parallel to the interface but at an angle to each other (the case we are concerned with), bubbles separated by long fingers on the scale of the critical wavelength grow preferentially. Their simulation produced long fingers separated by smooth arcs, morphologically similar to those in Fig. 1. The spacing of the RT bubbles corresponds to λ c as described above. Hence by measuring the separation of the fingers we can use Eq. 1 to place constraints on the plasma properties.
For all our examples the heavy plasma is much denser than the light plasma, so we take ρ h ρ l , and using B 2 = 4πρ h V 2 A , where V A is the Alfvén speed of the falling filament material where the instability starts, we obtain V A = λ c g/(4π).
(2)
Thus the observed 100 Mm separation of the fingers seen in Fig. 1 gives an Alfvén speed of approximately 47 km s −1 for the falling material. We note that since we can measure the spacing between the fingers only in the plane perpendicular to the line-of-sight, this is a lower limit to the Alfvén velocity. A slightly more complicated example is that corresponding to Figs. 4 and 5. Here acceleration due to a large-scale pressure gradient is important. The acceleration of the interface can be measured by fitting
d = a 2 t 2(3)
to the location of the interface shown in Fig. 5, where d is the distance moved by the interface, t is time, and a is the constant acceleration, which we measure to be 140 m s −2 . It is directed away from the Sun, so increases the effective g which appears in Eq. 2. Based on the observed spacing between the main fingers, 10 Mm, in the plane perpendicular to the line-of-sight, the lower limit for the Alfvén velocity in the dense plasma is 18 km s −1 . Throughout many smaller fingers appear but these are quickly damped, presumably because the spacing is less than the critical wavelength. The spikes of Sect 2.2 seem to be also produced by the RT instability based on their apparent evolution. We suspect that they are a consequence of the blast, seen 10 min earlier as a dome-like EUV wave (Cheng et al. 2012), that overtook the filament plasma. The observed short wavelengths are consistent with rapid acceleration. After the acceleration the RT instability will cease and magnetic tension will act to remove the fingers. Lastly we note that there are certainly other instabilities besides the RT which could be analyzed. The above examples were chosen because they were relatively clean, simple examples where the entire evolution could be followed. The horns discussed in section 2.4 are presumably the result of some instability which we have not been able to properly identify. We see no examples of Kelvin-Helmholtz even along the edges of the fingers where they could be expected in the non-linear phase of the RT instability.
Conclusion
In this letter we have analyzed spatially and temporally localized instabilities associated with the event on June 7 2011.
Concentrating on examples of the RT instability, we have shown that reasonable values for the Alfvén velocity in the falling plasma can be derived.
Fig. 1 .
1Falling (dark) plasma after the 2011 June 7 filament eruption looks like the fingers on the rim of a splash: a) 193Å intensity; b) 193Å ratio of images at the times given (see Sect. 2 for details). The white arrows in (b) represent plane-of-sky velocities and the red bar the typical spacing between fingers.
Fig. 2 .
2Erupting filament plasma seen in 171Å intensity images. The red arrow in (a) points to hot ejecta. Details of the red box in (b) are shown in (d). The red box in (c) outlines the regions displayed in Figs. 4 and 6. In (d) the arrow points to the spikes shown in detail inFig. 3. The FOV is 500 x 550 except in (d) where it is 96 x 110 .
Fig. 3 .
3Growth of spikes along filament indicated by the red arrow inFig. 2d: 171Åintensity ratio image. The red bar indicates the spikes' separation. The FOV is 96 x 66 .
Fig. 4 .
4Expanding arc with small-scale fingers and arcs inside: (a) 193Å intensity (b) 193Å ratio image. The red bar in (a) spans an arc between two small-scale fingers. The white arrows represent plane-of-sky velocity and the red line shows the position of the time slice inFig. 5. The FOV is 180 x 185 . The evolution can be seen in the movie arcs movie
Fig. 5 .
5Upwards expansion of arc: space-time slice of 193Å running ratio images along the red line inFig. 4b. The best fit d ∝ t 2 (Eq. 3) relationship is shown as a red dashed line. A white line is drawn at the time of the image inFig. 4
Fig. 6 .
6Formation of horns: (left) 171Å intensity; (right) 171Å ratio. White arrows represent plane-of-sky velocity. FOV is 130 x120 . The structures are labelled H1, H2, H3. The red arrow in (f) points to a secondary spike. The evolution of these structures is shown in the movie, horns movie.
Fig. A. 1 .
1Series of 171Å snapshots showing the evolution of the spikes described in section 2.2 and Fig 3. Small cavities develop between the spikes (see images at 06:24:48 and 06:26:00) while they continue to grow without rolling over. After only a few minutes the spikes fade and are overtaken denser plasma at their base.
arXiv:1202.4981v1 [astro-ph.SR] 22 Feb 2012 Innes, Cameron, Fletcher, Inhester, Solanki: Break up of returning filament plasma
Acknowledgements. The authors are indebted to the SDO/AIA teams and the German Data Center at MPS for providing the data. This work has been supported by WCU grant No. R31-10016 funded by the Korean Ministry of Education, Science and Technology and by grant ST/1001808/1 from UKs Science and Technology Facilities Council, and Leverhulme Foundation Grant F00-179A.Appendix A: Spikes' evolution
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