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[
"Secure Computation for Machine Learning With SPDZ",
"Secure Computation for Machine Learning With SPDZ"
] |
[
"Valerie Chen [email protected] \nYale University\nYale University\nYale University\n\n",
"Valerio Pastro [email protected] \nYale University\nYale University\nYale University\n\n",
"Mariana Raykova [email protected] \nYale University\nYale University\nYale University\n\n"
] |
[
"Yale University\nYale University\nYale University\n",
"Yale University\nYale University\nYale University\n",
"Yale University\nYale University\nYale University\n"
] |
[] |
Secure Multi-Party Computation (MPC) is an area of cryptography that enables computation on sensitive data from multiple sources while maintaining privacy guarantees. However, theoretical MPC protocols often do not scale efficiently to real-world data. This project investigates the efficiency of the SPDZ framework [1], which provides an implementation of an MPC protocol with malicious security, in the context of popular machine learning (ML) algorithms. In particular, we chose applications such as linear regression and logistic regression, which have been implemented and evaluated using semi-honest MPC techniques[2,7]. We demonstrate that the SPDZ framework outperforms these previous implementations while providing stronger security.
| null |
[
"https://arxiv.org/pdf/1901.00329v1.pdf"
] | 57,373,856 |
1901.00329
|
7a56aba1a4d4020c4933319588b9ed2b34d51125
|
Secure Computation for Machine Learning With SPDZ
Valerie Chen [email protected]
Yale University
Yale University
Yale University
Valerio Pastro [email protected]
Yale University
Yale University
Yale University
Mariana Raykova [email protected]
Yale University
Yale University
Yale University
Secure Computation for Machine Learning With SPDZ
Secure Multi-Party Computation (MPC) is an area of cryptography that enables computation on sensitive data from multiple sources while maintaining privacy guarantees. However, theoretical MPC protocols often do not scale efficiently to real-world data. This project investigates the efficiency of the SPDZ framework [1], which provides an implementation of an MPC protocol with malicious security, in the context of popular machine learning (ML) algorithms. In particular, we chose applications such as linear regression and logistic regression, which have been implemented and evaluated using semi-honest MPC techniques[2,7]. We demonstrate that the SPDZ framework outperforms these previous implementations while providing stronger security.
Introduction
Many machine learning techniques, including regression analysis, aim to build a model that fits a set of predictors to a dependent variable. Such techniques are widely used to model and analyze big data. In many settings, however, the input data for such ML analysis tools is partitioned among different parties, which have strict privacy policies. For example, if the Center for Disease Control is interested in identifying disease outbreak, they might want to incorporate patient data from many individual hospitals. The problem is that openly sharing this data for prediction or model-building purposes is against modern day privacy laws as it would leak private individual data. This is one of many real-world applications that could benefit from MPC, which allows parties to evaluate the output of the analysis without revealing more about the private inputs.
Multi-Party Computation
Secure MPC addresses the above problem by providing a mechanism through which different parties can run a joint computation over their private inputs with guarantees that the only thing revealed about the inputs is the output of the computation and whatever can be inherently inferred from it. There are two main types of MPC protocols in terms of their security guarantees: semi-honest and malicious protocols [8,4]. In semi-honest security, it is assumed that the parties will follow the protocol as specified, but they can try to infer information about the input from the protocol messages. In malicious security, dishonest parties may attempt to deviate from the specified protocol, and the protocol must guarantee that these parties cannot learn about the inputs. Since malicious protocols have to satisfy stronger guarantees in general, such construction are less efficient than semi-honest protocols.
Two recent works [2,7] propose efficient implementation of several central machine learning building blocks including conjugate gradient decent (CGD) and stochastic gradient descent (SGD) as well as their applications in linear and logistic regression. The work of Gascon et al. [2] uses the framework for semi-honest computation Obliv-C [9] and proposes several different methods for solving systems of linear equations. Their main premise is to use an iterative method such as CGD and demonstrate trade-offs that saves computation and hence efficiency for the MPC in return for a small accuracy loss. They further propose a modification of CGD that has stable behavior using fixed-point arithmetic because emulating floating point with the underlying MPC representation introduces substantial efficiency overhead.
The work of Mohassel and Zhang [7] utilizes stochastic gradient descent as a method for linear regression and logistic regression with the incorporation of different activation functions. The authors consider arithmetic representation for the computation and propose new secure computation techniques for matrix computation, which generalizes the approach for generating multiplicative triples in a preprocessing step. Similarly to the work of Gascon et al. [2], this paper considers techniques for approximation that save computation. For example, they use a piece-wise approximation of the logistic function. The authors also propose new techniques for more efficient approximate computation of fixed-point encodings.
The techniques in both of the above works are restricted to the setting of two party computation. We selected the SPDZ [1,5,6] framework for our experiments since it is one of the main and most comprehensive implementations for multiparty computation protocols, which provide malicious security and support more than two parties.
ML Functionalities
For our implementation we consider the same algorithms and functionalities as in the above two papers. Next we provide a brief overview of these classic algorithms (details can be found in [3]).
Direct vs. Iterative Decomposition for Solving a Linear System
Solving a system of linear equations, which underlies linear regression learning, can be done using techniques for direct and indirect decomposition. LDLT and Cholesky are both variants of direct decomposition methods which decompose a Hermitian, positive-definite matrix into a lower triangular matrix and its conjugate transpose. The algorithms are cubic in complexity with asymptotic run time of O(d 3 ), where d is the dimension of the input matrix. The difference between LDLT and Cholesky is that Cholesky requires a square root. The representation of square root computation as an arithmetic circuit used in the MPC computation in SPDZ introduces considerable overhead. That is why we used the iterative Newton method as a way of approximating the square root computation. It computes x i , where x 2 i = S, with repetition of the following update function x n+1 = 1 2 (x n + S/x n ). In terms of an iterative approach to regression, we used the approach proposed by Gascon et al. [2], which uses a normalized version of CGD that preserves stability and convergence rate with fixed-point number representation. Similarly to other MPC implementations using floating point representation in SPDZ introduces substantial efficiency overhead.
Stochastic Gradient Descent
Stochastic gradient descent is an iterative approximation method that converges to the global minimum for convex problems, like linear and logistic regression. It is also a driving mechanism for non-convex problems like neural networks. An SGD iteration updates a weight vector w using a randomly selected sample from the training input as follows: w j := w j − α(∂C i (w)/∂w j ) with learning rate α. In this update C i is the cost function, which can be instantiated with different concrete functions to obtain computation for linear regression and logistic regression. A common technique for SGD computation is called mini-batch -instead of selecting one sample per iteration, a small batch of size B samples are selected and the update function is performed averaging the partial derivatives across all samples. We use the mini-batch SGD in our implementation to obtain accuracy benefits. While the work of Mohassel and Zhang [7] has optimizations for matrix computation, which can be used with mini-batch, for SPDZ this does not lead to additional savings.
Linear and Non-Linear Activation Functions
To obtain a solution for linear regression using SGD, we instantiate the update function of a learned weight as w j = w j − α(X i · w * −y i )X ij , where X is the input matrix and y is the input vector.
In this update function, the weights are adjusted element-wise by the error from the predicted and expected value at a rate determined by α.
Logistic regression is a classification algorithm for modeling a binary dependent variable. Logistic regression fits the logistic function f (u) = 1 1+e − u to the input. The corresponding update function for mini-batched SGD for logistic regression is
w = w − 1 |B| αX T B × f (X B × w − Y B ),
where f maps the predicted value into the binary output space. Mohassel and Zhang [8] proposed the following piecewise function as approximation for f :
f (u) = 0 if u < −0.5 u + 0.5 if −0.5 ≤ u ≤ 0.5 1 if u > 0.5
We compare the results of this MPC-friendly piecewise function to a more standard approximation approach of taking the Taylor Series expansion to varying degrees.
Experiments
Experimental Setup
For our evaluation, we implemented all algorithms both in the SPDZ framework as well as in python as a plaintext verification of the algorithm. The main metrics of evaluations were the latency of the MPC computation and the accuracy error, and we aimed to explore the trade-offs between accuracy and efficiency. We varied the precision after the decimal point depending on what was used in the works that we compared against (32 and 64 bits for the linear regression, less for SGD).
We evaluated our methods on both real-world datasets (MNIST, Arcene, and 9 other UCI open-source datasets) as well as synthetically generated data. These real-world datasets allow us to compare the accuracy results to existing works and to demonstrate that SPDZ can be used in practical settings. We used synthetic data in order to explore larger ranges of data characteristics such as dimension (d = 10 Most of our experiments were ran using machines on the same local area network where there is no network latency. We performed tests where both parties were deployed on separate Amazon EC2 m4.large instances (see Figure 4). We also ran experiment with up to four parties (see Figure 3).
Results
In this section we present empirical results for our SPDZ implementations evaluated with real and synthetically generated databases. We compare the five different algorithms in terms of accuracy and run time for various parameters.
For LDLT, Cholesky and various iterations of CGD, we evaluated on synthetically generated data of varying sizes and condition numbers. The larger the condition number is, the larger the error in approximations of the solution is. The direct decomposition methods grew exponentially in run time as input size increases, which is shown in the left column of Figure 1 -this unlikely to be suitable for large size real data. Alternatively, the iterative CGD runtime increases at a much slower rate.
In the middle column of Figure 2, we find that about 20 iterations are sufficient to reach maximum accuracy given the number of allocated bits even with varying condition numbers. Particularly for the 64-bit case, shown on the bottom right, the accuracy is identical for CGD after 15 iterations and Cholesky/LDLT. Figure 2 compares SGD on MNIST and Arcene results. It shows that the number of bits of precision needed to get good accuracy is highly dependent on the dataset. For MNIST, 13 bits was sufficient to match plaintext accuracy, but 28 bits were needed for Arcene. The MNIST data contains only 784 features while there are 10,000 in the Arcene data, 3,000 of which are considered "probes" with no predictive power, which could explain the lower overall accuracy of [7]. While the numbers in the MNIST data ranged from 0 to 9, Mohassel and Zhang [7] only used 0s and 1s labels from the dataset, reducing it to a binary problem. We replicated this approach and present the results below. We did run the computation to predict all 10 digits, but found that SGD only achieved a much lower accuracy of about 19%. We also compared the root mean squared error (RMSE) of SGD on 9 UCI open-sourced datasets of ranging sizes to results in [2]. Our results in the SPDZ secure setting typically increased RMSE by about 5 − 20% compared to plaintext computation, but still outperformed RMSE results from [2] in both CGD and SGD.
In terms of logistic regression, for SPDZ, we did not find that the new activation function was a better alternative to taking a Taylor Series approximation for the exponential function as shown in Table 1. We found that for SPDZ, which is based on arithmetic circuits, the extra time to take a few extra degrees in the approximation was negligible.
Conclusion
SPDZ was able to achieve comparable accuracy for LDLT, Cholesky, and CGD when compared to Obliv-C with runtime faster by an order of magnitude on larger matrix sizes even in a distributed machine setting. SPDZ also achieved lower RMSE than Obliv-C using SGD. SPDZ was able to match accuracy and latency results for SecureML on SGD and Logistic Regression. This result is promising for SPDZ to be extended to more complex algorithms including neural networks with hidden layers.
Supplementary Material
Three and four party runtimes
In addition to the 2 player case, we also ran experiments with 3 and 4 players on the local network, where the input was vertically partitioned between parties.
EC2 runtimes
We deployed 2 AWS EC2 Instances that ran the SPDZ protocol. The EC2 runtime results are comparable to the local network results for smaller matrix sizes but increases at a faster rate for larger matrix sizes.
Activation function runtime comparison
To determine the difference in efficiency in the two activation functions, the new one proposed by [7] and the other method which they claimed to be inefficient, we ran results for SPDZ on two different datasets for a few different variations of the activation functions. The compile and run times correspond to the offline and online phase for SPDZ respectively.
RMSE results
SGD was also evaluated on 9 different datasets selected from the UCI repository. Further details about the specific datasets, including the regularization parameter used, are detailed in [2]. The dimensions of the problems ranged from 7 to 384 with the number of examples ranging from over 200 to almost 3 million.
, 20 ,
2050, 100, 200, 500), condition number (cd = [1,10]), and number of examples (n = 1000, 100000).
Figure 1 :Figure 2 :
12(Left) Run time as a function of input dimension. (Middle) Condition number as a function of accuracy. (Right) Accuracy as a function of the input dimension. (Top) Fixed-point with 60 bits of precision. (Bottom) Fixed-point with 28 bits of precision Comparing accuracy of privacy preserving linear regression with various fixed point precisions and plaintext training on floating point for MNIST (left) and Arcene (right).
Figure 3 :
3Run time results for 3 and 4 players in 32 bits. (Left) 3 Players. (Right) 4 Players.
Figure 4 :
4Run time results for 2 players deployed on EC2 instances averaged over 15 runs. (Left) 32 bits. (Right) 64 bits.
Table 1 :
1Comparing the validation accuracy for different activation functions for logistic regression.Plaintext New Activation function Polynomial Approximation
degree 2 degree 5 degree 7 degree 10
MNIST 99.9%
95%
97%
85%
91%
99.5%
Arcene 72.0%
44.0%
44.0%
44.5%
65%
72%
Table 2 :
2Comparison of efficiency for the proposed activation function in[7] and Taylor Series approximation in SPDZ.MNIST
Arcene
Compile Run Compile
Run
New Activation Function
26.01
18.8
220.14
224.47
2 Polynomial
27.79
19.28
222.85
239.98
5 Polynomial
30.32
19.8
225.33
241.59
7 Polynomial
31.99
20.65
228.33
239.11
10 Polynomial
34.24
20.71
230.81
240.79
Table 3 :
3SGD Results for 9 open sourced UCI datasets.Dataset Name
SGD Plaintext
SPDZ (28 bit)
Error
Time
SPDZ (13 bit)
Error
Time
Student Performance
0.11
0.12 (+8.34%) 2.174
0.12(+8.34%) 2.616
Auto MPG
0.56
0.68 (+21.4%) 0.663
0.68(+21.4%) 0.704
Communities and Crime 0.06
0.17 (+183%) 10.352 0.19 (+216%) 12.363
Wine Quality
0.18
0.19 (+5.55%) 0.809
0.19 (+5.55%) 0.969
Bike Sharing
0.23
0.24 (+4.34%) 0.910
0.25 (+8.69%) 1.097
Blog Feedback
0.04
0.04 (+0.0%)
9.455
0.04 (+0.0%)
9.091
CT Slices
0.22
0.22 (+0.0%)
16.458 0.22 (+0.0%)
16.295
Year Prediction MSD
0.06
0.06 (+0.0%)
7.146
0.06 (+0.0%)
8.513
Gas Sensor Array
0.20
0.36 (+20.0%) 1.151
0.36 (+20.0%) 1.559
Multiparty computation from somewhat homomorphic encryption. Ivan Damgård, Nigel P Valerio Pastro, Sarah Smart, Zakarias, Advances in Cryptology -CRYPTO 2012 -32nd. Ivan Damgård, Valerio Pastro, Nigel P. Smart, and Sarah Zakarias. Multiparty computation from somewhat homomorphic encryption. In Advances in Cryptology -CRYPTO 2012 -32nd
Annual Cryptology Conference, Proceedings. nullSanta Barbara, CA, USAAnnual Cryptology Conference, Santa Barbara, CA, USA, August 19-23, 2012. Proceedings, pages 643-662, 2012.
Privacy-preserving distributed linear regression on high-dimensional data. Adrià Gascón, Phillipp Schoppmann, Borja Balle, Mariana Raykova, Jack Doerner, Samee Zahur, David Evans, PoPETs. 20174Adrià Gascón, Phillipp Schoppmann, Borja Balle, Mariana Raykova, Jack Doerner, Samee Zahur, and David Evans. Privacy-preserving distributed linear regression on high-dimensional data. PoPETs, 2017(4):345-364, 2017.
The elements of statistical learning: data mining, inference, and prediction, 2nd Edition. Springer series in statistics. Trevor Hastie, Robert Tibshirani, Jerome H Friedman, SpringerTrevor Hastie, Robert Tibshirani, and Jerome H. Friedman. The elements of statistical learning: data mining, inference, and prediction, 2nd Edition. Springer series in statistics. Springer, 2009.
A note on the relation between the definitions of security for semi-honest and malicious adversaries ?. Carmit Hazay, Yehuda Lindell, Carmit Hazay and Yehuda Lindell. A note on the relation between the definitions of security for semi-honest and malicious adversaries ?, 2010.
Overdrive: Making SPDZ great again. Marcel Keller, Valerio Pastro, Dragos Rotaru, Advances in Cryptology -EUROCRYPT 2018 -37th Annual International Conference on the Theory and Applications of Cryptographic Techniques. Tel Aviv, IsraelProceedings, Part IIIMarcel Keller, Valerio Pastro, and Dragos Rotaru. Overdrive: Making SPDZ great again. In Advances in Cryptology -EUROCRYPT 2018 -37th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Tel Aviv, Israel, April 29 -May 3, 2018 Proceedings, Part III, pages 158-189, 2018.
An architecture for practical actively secure mpc with dishonest majority. Marcel Keller, Peter Scholl, Nigel P Smart, Proceedings of the 2013 ACM SIGSAC Conference on Computer & Communications Security, CCS '13. the 2013 ACM SIGSAC Conference on Computer & Communications Security, CCS '13Marcel Keller, Peter Scholl, and Nigel P. Smart. An architecture for practical actively secure mpc with dishonest majority. In Proceedings of the 2013 ACM SIGSAC Conference on Computer & Communications Security, CCS '13, 2013.
Secureml: A system for scalable privacy-preserving machine learning. Payman Mohassel, Yupeng Zhang, 2017 IEEE Symposium on Security and Privacy. San Jose, CA, USAPayman Mohassel and Yupeng Zhang. Secureml: A system for scalable privacy-preserving machine learning. In 2017 IEEE Symposium on Security and Privacy, SP 2017, San Jose, CA, USA, May 22-26, 2017, pages 19-38, 2017.
Basic Applications. Goldreich Oded. Foundations of Cryptography. 2Cambridge University Press1st editionGoldreich Oded. Foundations of Cryptography: Volume 2, Basic Applications. Cambridge University Press, New York, NY, USA, 1st edition, 2009.
Obliv-c: A language for extensible data-oblivious computation. Samee Zahur, David Evans, Cryptology ePrint Archive. ReportSamee Zahur and David Evans. Obliv-c: A language for extensible data-oblivious computation. Cryptology ePrint Archive, Report 2015/1153, 2015. https://eprint.iacr.org/2015/ 1153.
|
[] |
[
"PROBLEMS WITH DUALITY IN N=2 SUPER-YANG-MILLS THEORY",
"PROBLEMS WITH DUALITY IN N=2 SUPER-YANG-MILLS THEORY"
] |
[
"Martin Cederwall \nInstitute for Theoretical Physics\nGöteborg University\nChalmers University of Technology\nS-412 96GöteborgSweden\n"
] |
[
"Institute for Theoretical Physics\nGöteborg University\nChalmers University of Technology\nS-412 96GöteborgSweden"
] |
[] |
Actual calculations of monopole and dyon spectra have previously been performed in N=4 SYM and in N=2 SYM with gauge group SU(2), and are in total agreement with duality conjectures for the finite theories. These calculations are extended to N=2 SYM with higher rank gauge groups, and it turns out that the SU(2) model with four fundamental hypermultiplet is an exception in that its soliton spectrum supports duality. This may be an indication that the other perturbatively finite N=2 theories have non-perturbative contributions to the β-function. This talk contains a short summary of recent results [1].
| null |
[
"https://arxiv.org/pdf/hep-th/9606096v1.pdf"
] | 2,341,972 |
hep-th/9606096
|
358407739f91064e04269c670ec6482cb7e7f300
|
PROBLEMS WITH DUALITY IN N=2 SUPER-YANG-MILLS THEORY
arXiv:hep-th/9606096v1 17 Jun 1996 June, 1996
Martin Cederwall
Institute for Theoretical Physics
Göteborg University
Chalmers University of Technology
S-412 96GöteborgSweden
PROBLEMS WITH DUALITY IN N=2 SUPER-YANG-MILLS THEORY
arXiv:hep-th/9606096v1 17 Jun 1996 June, 1996
Actual calculations of monopole and dyon spectra have previously been performed in N=4 SYM and in N=2 SYM with gauge group SU(2), and are in total agreement with duality conjectures for the finite theories. These calculations are extended to N=2 SYM with higher rank gauge groups, and it turns out that the SU(2) model with four fundamental hypermultiplet is an exception in that its soliton spectrum supports duality. This may be an indication that the other perturbatively finite N=2 theories have non-perturbative contributions to the β-function. This talk contains a short summary of recent results [1].
The last years have seen big progress in the understanding of non-perturbative aspects of quantum field theory and string theory. In supersymmetric enough theories, some non-perturbative properties may actually be calculated exactly [2,3,4,5,6]. Intimately connected with the new techniques developed is the somewhat older concept of duality [7,8]. Finite quantum field theories may, and seem to, exhibit a duality symmetry, which is the realization and extension of the old puzzling symmetry between electricity and magnetism in Maxwell theory. Z 2 duality transforms the coupling constant of the theory to its inverse, and exchanges (non-abelian) electric and magnetic charges. When the theta angle is included in the complex coupling constant, the group of duality transformations is Sl(2; Z), acting on the vector of electric and magnetic charge, and projectively on the coupling constant. Then also dyons, states carrying both electric and magnetic charge, are involved. The relevant magnetically charged states in a model containing Yang-Mills and a Higgs field are the magnetic (multi-)monopoles and dyons. If a theory is finite, and the coupling constant does not run, the transformations can make sense, and there is the possibility that duality is an exact symmetry of the theory. Then there is no fundamental difference between elementary and solitonic excitations, just that we need to chose one element in the duality group, i.e. a set of "elementary" excitations, in order to formulate the theory. A geometric understanding of this kind of duality, in the context of string theory called S-duality, is still lacking, and is probably needed for our understanding of what string theory "really is".
There is an infinity of perturbatively finite quantum field theories. Of special interest among these are supersymmetric Yang-Mills theories with at least two supersymmetries, i.e. N = 2, 4. All N = 4 super-Yang-Mills theories are perturbatively finite [9,10,11,12,13,14], and this property has been shown to hold also non-perturbatively [15]. The models with N = 2 are perturbatively finite if the matter content is appropriate, e.g. the SU (N c ) models with N f = 2N c hypermultiplets in the fundamental representation. It is of course important to decide whether these theories are non-perturbatively finite.
The questions of finiteness and duality are closely related. A non-running coupling constant is required for duality symmetry, and conversely, if there is a number of excitations that become massless as the coupling constant goes to some specific value in a finite theory, these should play the role of elementary excitations in a perturbation expansion of a finite theory. Examining the spectrum of solitons in a theory may therefore give information both about duality and finiteness.
The procedure for finding (part of) the spectra of monopoles and dyons in the theories at hand is well established [16,17,18,19,20,21,22]. Starting from a classical solution that is BPS-saturated, i.e. has a certain relation between mass and magnetic charge and breaks half of the supersymmetry, one makes a low-energy approximation. Here, the N = 2 supersymmetry is crucial, in that the electric and magnetic charges enter the central charges in the supersymmetry algebra. BPS-saturated states come in short multiplets, with half the size of a full multiplet (this is identical to the difference between massless and massive multiplets, only that "m = 0" is shifted in the presence of the central extensions). The number of states in a multiplet should not be quantum corrected and the BPS property is therefore valid also outside the low-energy approximation [23]. Due to the presence of zero-modes, bosonic and fermionic, and the presence of a mass-gap, the low-energy approximation gets only a finite number of degrees of freedom, namely the moduli parameters of the relevant topological sector (specified by magnetic charge) and fermionic zero-modes. The task of finding BPS states is, roughly speaking, reduced to that of finding ground states of a supersymmetric quantum mechanics model, which amounts to a cohomology problem. The riemannian geometry of the monopole moduli spaces and the geometry of the index bundles of fermionic zero-modes over these moduli spaces are essentially determined from the form of the kinetic terms in the field theory lagrangian, and the geometric considerations can be pushed quite far on a purely formal basis. What restricts the actual calculation of ground states, is that it, at least this far, requires concrete knowledge of the moduli spaces. Only at the lowest values of the magnetic charge is the metric structure known explicitely, which restricts the calculations to these cases (note, however, the method of [17], which seems to circumvent this statement).
These calculations have been performed for the N = 4 theories [16,17,21,22], where the results are in perfect agreement with the predictions from duality. Also for the N = 2 theory with gauge group SU (2) and four hypermultiplets in the fundamental representation, duality (as predicted in [3]), is in agreement with the obtained spectrum [20,19]. We wanted to continue this investigation to include also the other perturbatively finite N = 2 theories. The expectation was to find spectra that would lend themselves to a direct interpretation in terms of a dual finite theory. For a number of reasons, this did not happen.
It is quite easy to point at at least two problems with duality in N = 2 theories with gauge groups of rank higher that one. The electric charges of the elementary excitation lie on the weight lattice of the gauge group, and the magnetic charges on the dual to this lattice (more precisely, on the dual to the weight lattice of the universal cover of the group), the coroot lattice, which for simply laced groups coincide with the root lattice. These two lattices are in general different, also modulo a rescaling (for SU (4), they are the fcc and bcc lattices). Note that this only is an argument against Z 2 symmetry of electric-magnetic interchange in the cases where the lattices differ, but in a finite theory this would also mean that one is not expecting to find any BPSsaturated states with purely magnetic charge, which we actually do. Also, there is a serious problem already with the classical moduli spaces, namely that only some sectors of the coroot lattice are allowed as magnetic charges. These are the coroots that in a suitable basis are sums of the simple coroots with only positive or only negative coefficients. The other coroots would correspond to superpositions of monopoles and anti-monopoles, i.e. of selfdual and antiselfdual field configarations. Such a configuaration can not be BPS-saturated. In the N = 4 case, where only the trivial conjugacy class of the weight lattice is populated with elementary excitations, this presents no problems. All the magnetic charges align with the magnetic ones, and the forbidden sectors are avoided. For N = 2 theories with gauge groups of rank higher than one this is a problem.
Of course, indirect arguments like these, how convincing they might sound, can turn out to be wrong or irrelevant for reasons difficult to predict. Therefore, it is important to perform the actual calcualtion of the spectra in order to examine if there is some obvious sign of duality in some parts of them. We did not find any such signs. Some of the spectra for gauge group SU (3) are presented diagrammatically in [1].
What are the interpretations of these results? One must bear in mind that the non-perturbative finiteness of all these N = 2 models has not been proven, and the results we have obtained would become very natural if there were instanton contributions to the β-function (with the exception of the finite SU (2) theory). We do not think our evidence is strong enough for a claim of this type, but we feel that the possibility must be considered, even if it is against the common lore. Presumably, if the perturbatively finite models discussed here actually exhibit non-perturbative divergencies, they should manifest themselves as diverging sums over instanton number, since the contribution from a sector with given instanton number should be perturbatively finite, and the integrals over instanton moduli converge in the presence of a Higgs expectation value.
For concrete calculations, formulas and presentation of the spectra, I refer to [1], which also contains a more exhaustive list of references.
AcknowledgementThe work summarized here was done in collaboration with Magnus Holm. I would also like to thank the organizers of the Second International Sakharov Conference on Physics for their great hospitality and all the help they provided during this exciting week.
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[] |
[
"A DISCRETE HARDY'S UNCERTAINTY PRINCIPLE AND DISCRETE EVOLUTIONS",
"A DISCRETE HARDY'S UNCERTAINTY PRINCIPLE AND DISCRETE EVOLUTIONS"
] |
[
"Aingeru Fernández-Bertolin "
] |
[] |
[] |
In this paper we give a discrete version of Hardy's uncertainty principle, by using complex variable arguments, as in the classical proof of Hardy's principle. Moreover, we give an interpretation of this principle in terms of decaying solutions to the discrete Schrödinger and heat equations.
|
10.1007/s11854-019-0002-1
|
[
"https://arxiv.org/pdf/1506.00119v1.pdf"
] | 119,142,459 |
1506.00119
|
fd9f1981c6c65ed46233a9a0943754d947c9c5c2
|
A DISCRETE HARDY'S UNCERTAINTY PRINCIPLE AND DISCRETE EVOLUTIONS
30 May 2015
Aingeru Fernández-Bertolin
A DISCRETE HARDY'S UNCERTAINTY PRINCIPLE AND DISCRETE EVOLUTIONS
30 May 2015
In this paper we give a discrete version of Hardy's uncertainty principle, by using complex variable arguments, as in the classical proof of Hardy's principle. Moreover, we give an interpretation of this principle in terms of decaying solutions to the discrete Schrödinger and heat equations.
Introduction
In functional analysis, uncertainty principles are, in general, results that state that a function and its Fourier transform cannot decay too fast simultaneously. The most studied uncertainty principle, which comes back to Heisenberg, says that
2 d R d |xf (x)| 2 dx 1/2 R d |∇f (x)| 2 dx 1/2 ≥ R d |f (x)| 2 dx,
and, moreover, the equality is attained if and only if f (x) = Ce −α|x| 2 /2 , for α > 0. In the discrete setting there are some versions of this inequality. For instance, in a previous paper [9], we studied an inequality discretizing the position and momentum operators (see also [1,2,10] for more references to this uncertainty principle). We saw how we can recover the classical minimizer (i.e. the Gaussian) from the minimizer of the discrete uncertainty principle, which is given in terms of modified Bessel functions I k (x) = 1 π π 0 e z cos θ cos(kθ) dθ, k ∈ Z.
In this paper, we are interested in giving a discrete version of Hardy's uncertainty principle [4,13] in one dimension |f (x)| ≤ Ce −x 2 /2α , |f (ξ)| ≤ Ce −ξ 2 /2β , with αβ < 1 ⇒ f ≡ 0, and, in the case αβ = 1 then f (x) = Ce −x 2 /2α . Thus, this uncertainty principle is telling us that a function and its Fourier transform cannot have both Gaussian decay for some coefficients α, β The original proof of this principle is strongly based on complex variable arguments (Phragmen-Lindelöf's principle and Liouville's theorem). Moreover, if we consider a solution to the onedimensional free Schrödinger equation we can write the solution as
∂ t u = i∂ xx u, u(x, 0) = u 0 (x),u(x, t) = e ix 2 /4t √ it e i(·) 2 /4t u 0 ∧ x 2t .
Basically, this says that the solution is the Fourier transform of the initial datum, so we can write Hardy's uncertainty principle in terms of solutions to the Schrödinger equation,
(1) u(x, 0) = O(e −x 2 /2α ), u(x, 1) = e i∂xx u(x, 0) = O(e −x 2 /2β ), αβ < 4 ⇒ u ≡ 0,
having a precise expression for u 0 in the case αβ = 4. On the other hand, it can also be stated in terms of solutions to the heat equation, as follows:
(2) f ∈ L 2 (R), e x 2 /2δ e ∂xx f ∈ L 2 (R) for some δ ≤ 2 ⇒ f ≡ 0.
Under these terms, there is a series of papers [5][6][7][8] and [3] where the authors prove Hardy's uncertainty principle for perturbed Schrödinger and heat equations just by using real calculus arguments. In the case of the Schrödinger equation, they are able to get a result up to the endpoint case. However, for the heat equation their result does not cover the whole case.
Here first we are going to see, using complex variable arguments, a version of Hardy's uncertainty principle in a discrete setting, saying that if a complex function is controlled in some region of the complex plane by a function that will be closely related to the Gaussian, and its Fourier coefficients are controlled precisely by the minimizer we got in [9] (i.e. by the modified Bessel function I k (x)), then in some cases we have that the function is zero, or we can give a precise expression for the function. This is not exactly stated as Hardy's classical principle, since now, by using Fourier series we relate the sequence to a periodic function in an interval of the form [−π/h, π/h], while in the continuous case the relation one gets using the Fourier transform is between functions in the real line. Due to this periodicity, just knowing the behavior of the function in that interval does not give us any information.
Once we have this result, we want to give some uncertainty principles in terms of the free Schrödinger and the heat equations, concluding that solutions to these equations cannot decay too fast at two different times. We are going to use two different approaches to prove these results. First, we take advantage of what is known in the continuous case, so we need to assume that the initial datum of the discrete equation is similar to a function on the real line. In this case, it is not reasonable to expect that the solution to the discrete equation is identically zero, but it tends to the zero function in some sense as h, the mesh step, tends to zero. However, when this mesh step is fixed, we can use the discrete version of the uncertainty principle we mention above in order to see that under some cases this solution is identically zero. As it is pointed out in (1), the condition on the decay coefficients in the continuous case is αβ < 4, while, the discrete approach we are going to study here leads to a more restrictive condition for α and β, α + β < 2. Although this is a clear difference between the continuous and the discrete case, we are going to see here that this is the sharp condition for our results.
When preparing this manuscript, we learn about a recent and independent result in this direction [12]. There, the authors use complex variable arguments to prove a sharp analog of Hardy's uncertainty principle considering solutions to the discrete Schrödinger equation. On the other hand, they also use the real variable approach in [5][6][7][8] in order to add real-valued timedependent potentials. There is overlap between their results and those presented here in Section 4, although the proofs of the results are different.
The paper is organized as follows: In Section 2 we give a general result in the discrete case that is related to Hardy's uncertainty principle. In Section 3 we give a version of Hardy's uncertainty principle for the discrete Schrödinger and discrete heat equations, by relating these equations to the continuous equations, noticing that we should not expect the solutions to be identically zero using this approach. In Section 4 we use the results of Section 2 to give discrete versions of Hardy's uncertainty principle in the spirit of the results in Section 3 without using information about the continuous case and, moreover, we give some examples of discrete data that prove the sharpness of the decay conditions. Finally, in an appendix we give some examples of nonzero functions that satisfy the hypotheses of Theorem 2.1 when the statement of the theorem does not give any extra information about the function.
A discrete uncertainty principle using complex variable arguments
For h > 0, that represents the mesh size, we define the Fourier coefficients of a 2π
h -periodic function f h in the following way:
f h (k) = 1 √ 2π π/h −π/h f (ξ)e −iξkh dξ, f h (x) = h √ 2π k∈Zf h (k)e ihkx .
There is a result concerning the extension of this function f h to the complex plane. Roughly speaking, this result states that a function that decays faster than e −a|x| , ∀a > 0, can be extended to the complex plane as an entire function. When the Fourier coefficients are of the form I k (u/h 2 ) for u ∈ C we are in this case, and we have that the corresponding periodic function is
h √ 2π k∈Z I k (u/h 2 )e ihk(x+iy) = h √ 2π e u cos(x+iy)/h 2 .
So we are going to see that if the Fourier coefficients decay faster than the modified Bessel function, and on the other hand the periodic function f h is controlled by some function that is related to the one we have computed above, then in some cases the function will be zero, according to |u| and the argument of the Bessel function. Theorem 2.1. Assume that f h is a complex-valued function and that there are u = re iθ ∈ C, b ∈ [0, 2π), δ ∈ (0, π/2), and s > 0 such that, for all y ≤ 0,
f h b − θ + π 2 + δ h + iy ≤ Ce ℜu h 2 cos(θ− π 2 −δ) cosh(yh)− ℑu h 2 sin(θ− π 2 −δ) sinh(yh) , f h b − θ − π 2 − δ h + iy ≤ Ce ℜu h 2 cos(θ+ π 2 +δ) cosh(yh)− ℑu h 2 sin(θ+ π 2 +δ) sinh(yh) ,
and, for all y ≥ 0,
f h b + θ + π 2 + δ h + iy ≤ Ce ℜu h 2 cos(θ+ π 2 +δ) cosh(yh)+ ℑu h 2 sin(θ+ π 2 +δ) sinh(yh) , f h b + θ − π 2 − δ h + iy ≤ Ce ℜu h 2 cos(θ− π 2 −δ) cosh(yh)+ ℑu h 2 sin(θ− π 2 −δ) sinh(yh) .
Moreover, assume that the Fourier coefficients of f h satisfy
|f h (k)| ≤ CI k 1 sh 2 , ∀k ∈ Z.
Then:
(1) rs < 1 ⇒ There are nonzero functions that satisfy the hypotheses. (See the appendix below)
(2) rs = 1 ⇒ f h (z) = Ce u h 2 cos(zh−b) . (3) rs > 1 ⇒ f h (z) ≡ 0.
Remark 2.1. The hypothesis on the periodic function is telling us that we know how the function behaves near the critical lines where the function e u h 2 cos(zh−b) has the greatest decay.
Remark 2.2. In the applications of this theorem, we are going to consider δ close to π/2.
Proof. To begin with, we show that the case rs > 1 can be proved as a consequence of the case rs = 1, as in Hardy's original theorem. Indeed, if r > 1 s , the hypotheses of the theorem are satisfied forũ = e iθ s and s, and in this casers = 1. To see this, let us consider y ≤ 0, so that (C is a constant that may change from line to line)
f h b − θ + π 2 + δ h + iy ≤ Ce ℜu h 2 cos(θ− π 2 −δ) cosh(yh)− ℑu h 2 sin(θ− π 2 −δ) sinh(yh) ≤ Ce re −yh cos(π/2+δ)/2h 2 ≤ Ce e −yh cos(π/2+δ)/2sh 2 ≤ Ce ℜũ h 2 cos(θ− π 2 −δ) cosh(yh)− ℑũ h 2 sin(θ− π
since y is negative and cos(π/2 + δ) is also negative. For the other three lines that we have to control, we can use the same argument. Hence, by (2) in the theorem, we have that f h (z) = Ceũ h 2 cos (z−b)h . But then the hypotheses of the theorem are not satisfied, unless
C = 0 ⇒ f h ≡ 0.
Thus, we only need to prove that when rs = 1 we can determine completely the function by the properties of the hypotheses. The proof of this fact relies on the maximum principle and Liouville's theorem, so it is quite similar to the original proof of Hardy's uncertainty principle. As we have pointed out above, the condition for the Fourier coefficients implies that f h is a 2π/h-periodic function and the extension
f h (z) = h √ 2π k∈Zf h (k)e ihkz , z = x + iy,
is an entire function, so we have, for all z in the plane
|f h (z)| ≤ Ch k∈Z I k r h 2 e −hky = Che r h 2 cosh(yh) ,
We split up a strip of length 2π/h in six regions following the figures below (being the first one for the upper half plane and the second one for the lower half plane, and the boundary of each region is determined by the dashed red lines and the line {ℜz = 0}):
ℜz b+θ h b+θ+3π/2−δ h b+θ+π/2+δ h b+θ−π/2−δ h 1 2 3 ℜz b−θ h b−θ+3π/2−δ h b−θ+π/2+δ h b−θ−π/2−δ h 4 5 6
The procedure is very simple. In each region, we first multiply f h by a nice function (which will depend on a parameter ǫ) that decays when y tends to infinity in the unbounded part of the region. Then we see that the product is bounded at the boundary of the region, so we can apply the maximum principle and let ǫ tend to zero. We do not want the sign of the real part of this function to change in each region, so we look for the behavior of sin x cos x on [π/2, π]. We illustrate this in region 1, since the procedure is exactly the same. In this region we consider the function
g ǫ (x + iy) = f h (x + iy)e − u h 2 cos(zh−b)−iǫ cos 2 (T z) , where T z = a 1 z + b 1 is the linear transformation which maps the interval [ b+θ−π/2−δ h , b+θ h ] into [π/2, π].
Hence, we have that
|g ǫ (x + iy) = |f h (x + iy)|e −ℜu h 2 cos(xh−b) cosh(yh)− ℑu h 2 sin(xh−b) sinh(yh)+2ǫφ(x,y) , where φ(x, y) = cos(a 1 x + b 1 ) sin(a 1 x + b 1 ) cosh(a 1 y) sinh(a 1 y).
We can see that at the boundary of the region we are considering, this function is bounded. Moreover, a simple computation shows that 2a1 h = π π/2+δ > 1, and this fact tells us that when y tends to infinity, the leading term is given by the ǫ part, which is negative. Hence the function is bounded when y is large and we can apply the maximum principle to get that |g ǫ (z)| ≤ C. Letting ǫ tend to zero we conclude that, for all z in region 1,
|f h (z)e − u h 2 cos(zh−b) | ≤ C.
When we repeat this procedure in the other regions, we have that |f h (z)e − u h 2 cos(zh−b) | ≤ C in the two strips described in the figure above. One is a strip of length 2π/h of the upper half plane, and the other one is another strip of length 2π/h of the lower half plane. By periodicity, this estimate holds for all z in the plane, and by Liouville's theorem this implies that f (z) = Ce In Section 4 we will use this theorem in order to see that a solution to a discrete equation cannot decay faster than some modified Bessel functions at two different times. What we have to see is the evolution of this equation in the periodic setting, by considering that the solution is given as the Fourier coefficients of some periodic function. The behavior at one time and the expression of the solution in the periodic setting will give us bounds for the function in some vertical lines of the complex plane, so we will use these bounds and the decay of the function at the other time to conclude that the function is zero.
A discrete Hardy's theorem based on the classical Hardy's uncertainty principle
We are going to consider a solution to the Schrödinger equation
(3) ∂ t f h k (t) = i f h k+1 (t) − 2f h k (t) + f h k−1 (t) h 2 ,
where, in order to use the continuous uncertainty principle in the discrete setting, we are going to assume that the initial datum f h k (0) is not far from the evaluation of a continuous function u 0 (x), that we will use as initial datum for the continuous Schrödinger equation. More precisely, we require that there is a function u 0 such that
k∈Z |f h k (0) − u 0 (kh)| 2 < h µ ,
for some µ > 0. In order to be able to evaluate the function, we are going to require u 0 ∈ H s (R), for some s > 1/2. Moreover, this is telling us that at any tme t, the solution to the discrete Schrödinger equation with initial datum f h k (0) is not far from the solution to the discrete equation with initial datum z h k (0) = u 0 (kh), since the ℓ 2 norm is preserved in time.
Now what we want to see is how we can relate the solution to a discrete equation to a solution to its continuous version. Since we are going to give an ℓ ∞ version of the uncertainty principle, we want to estimate the ℓ ∞ norm of the difference between the evaluation of the solution to the continuous equation at time t in the mesh {kh : k > 0} with initial data u 0 (x) and the solution to the discrete equation with initial data u 0 (kh). Since u 0 ∈ H s with s > 1/2, we have that R (1 + |ξ|) (2s−1)/4 |û 0 (ξ)| dξ < +∞, and using similar arguments as in [14], where this type of equations is studied when the time is also discrete, we can get that
(4) |u(kh, 1) − z h k (1)| ≤ h (2s−1)/8 R (1 + |ξ|) (2s−1)/4 |û 0 (ξ)| dξ.
This is the easiest way to talk about convergence of the discrete equation to the continuous one. However, in [11] the authors give estimates for a vast variety of Banach spaces.
Under these circumstances, we have the following result:
Theorem 3.1. Let f h m (t)
be a solution to the discrete Schrödinger equation (3), and assume that there is u 0 ∈ H s with s > 1/2 such that for some µ > 0,
(5) k∈Z |f h k (0) − u 0 (kh)| 2 < h µ .
If for some α, β satisfying αβ < 4 we have, for all k ∈ Z and h > 0,
(6) |f h k (0)| ≤ C I k (α/h 2 ) I 0 (α/h 2 ) , |f h k (1)| ≤ C I k (β/h 2 ) I 0 (β/h 2 ) ,
then u 0 ≡ 0 and |f h k (t)| tends to zero uniformly in k when h tends to zero.
Proof. Since we want to use the continuous result, what we need to prove is that u 0 and u(x, 1) = e i∂xx u 0 are controlled by some Gaussians. We have to distinguish several cases according to x.
First, if x = 0 we have that |f h 0 (0) − u 0 (0)| ≤ h µ/2 , so |u 0 (0)| ≤ |f h 0 (0)| + h µ/2 ≤ C + h µ/2 ,
and letting h tend to zero, we conclude that |u 0 (0)| ≤ C.
Now, if 0 < x ≤ 1 we have that u 0 (x) = u 0 j x j
for all j ∈ N, and then |f
x/j j (0) − u 0 (x)| ≤ x µ/2 j µ/2 ≤ 1 j µ/2 . Hence |u 0 (x)| ≤ |f x/j j (0)| + 1 j µ/2 ≤ C I j (αj 2 /x 2 ) I 0 (αj 2 /x 2 ) + 1 j µ/2 ,
and it was proved in [9] that this quotient of Bessel functions tend uniformly in x to the Gaussian e −x 2 /2α , so letting j tend to infinity we have |u 0 (x)| ≤ Ce −x 2 /2α . Changing j by −j we can argue in the same fashion to get the same result in −1 ≤ x < 0.
Finally, if 1 ≤ x we use that u 0 (x) = u 0 ⌈x⌉ 4 j x ⌈x⌉ 4 j for all j ∈ N, and then, as in the previous case
|u 0 (x)| ≤ |f x/⌈x⌉ 4 j ⌈x⌉ 4 j | + x µ/2 ⌈x⌉ 2µ j µ/2 ≤ C I ⌈x⌉ 4 j (αj 2 ⌈x⌉ 8 /x 2 ) I 0 (αj 2 ⌈x⌉ 8 /x 2 ) + 1 j µ/2 ,
and we can use the same reasoning as in [9] to see that this quotient also converges to the Gaussian uniformly in x. Actually, this case is easier than the previous one, since in that case one has to show a uniform estimate over compact sets |x| ≤ M and then see that if M is large, for M < |x| both functions are very small. In this case, we do not need to do this and it can be proved the uniform convergence in one step. Now letting j tend to infinity we get that |u 0 (x)| ≤ Ce −x 2 /2α in this region. Changing j by −j we prove the same when x ≤ −1 and gathering all the results we conclude that |u 0 (x)| ≤ Ce −x 2 /2α for all x ∈ R. Now what we need to see is the evolution of u 0 (x) under the continuous Schrödinger equation. Since u 0 ∈ H s , with s > 1/2, we have that the solution to the continuous Schrödinger equation at the mesh is similar to the solution to the discrete Schrödinger equation when we discretize u 0 (x) by taking its values at the mesh, according to (4). On the other hand, since the initial data f h m (0) satisfies (5), that is, it is close to the evaluation at the mesh of u 0 , we have that at time t f h m (t) is also close to z h m (t), since we have
|f h m (t) − z h m (t)| < k∈Z |f h k (t) − z h k (t)| 2 1/2 = k∈Z |f h k (0) − u 0 (kh)| 2 1/2 < h µ/2 .
Hence, we can do the same as we have done with u 0 (x) to see that it is bounded by another Gaussian. For instance, if 0 < x ≤ 1,
|u(x, 1)| = |u j x j , 1 | ≤ |z x/j j (1)| + 1 j (2s−1)/8 ≤ |f x/j j (1)| + 1 j (2s−1)/8 + 1 j µ/2 ≤ C I j (βj 2 /x 2 ) I 0 (βj 2 /x 2 ) + 1 j (2s−1)/8 + 1 j µ/2 ,
and letting j tend to infinity we get |u(x, 1)| ≤ e −x 2 /2β . Doing the same for the other regions we finally get that |u(x, 1)| ≤ e −x 2 /2β for all x ∈ R, so we can use Hardy's uncertainty principle in its version for Schrödinger evolutions (1) to conclude that u ≡ 0. In particular, this tells us that u 0 ≡ 0 and then what we have is that |f h k (t)| < h µ/2 for all k ∈ Z, so the solution to the discrete equation tends to zero uniformly in k when h tends to zero.
As we see in the statement of the theorem, we do not prove that the discrete solution is f h k (t) ≡ 0, but it is close to zero as h is getting smaller. We can find examples where we see that we cannot expect to have a complete analog version to Hardy's uncertainty principle using this approach. For instance, if we take as initial datum
f h k (0) = I k (i/2h 2 ) I 0 (5/2h 2 ) ⇒ |f h k (0)| ≤ I k (1/2h 2 ) I 0 (5/2h 2 ) < I k (1/2h 2 ) I 0 (1/2h 2 ) ,
since |I k (z)| ≤ I k (|z|) and the modified Bessel function I 0 (x) is increasing when 0 < x < ∞. Thus α = 1 2 , and, if we solve the discrete Schrödinger equation we get at time t = 1,
f h k (1) = e −2i/h 2 ∞ m=−∞ I m (i/2h 2 ) I 0 (5/2h 2 ) I k−m (2i/h 2 ) = e −2i/h 2 I m (5i/2h 2 ) I 0 (5/2h 2 ) ⇒ |f h k (1)| ≤ I m (5/2h 2 ) I 0 (5/2h 2 ) ,
so in this case β = 5 2 and αβ = 5 4 < 4. Then by the theorem, if there exists u 0 such that (5) is satisfied, u 0 has to be identically zero. In this case it is quite easy to see that this happens. Indeed,
k∈Z |f h k (0)| 2 = 1 I 2 0 (5/2h 2 )
, and seeing the asymptotic behavior of I 0 (t) when t is large, we see that (5) is satisfied for any µ > 0. Hence we have an example of a nonzero sequence that satisfies the hypotheses of the theorem. However, in the next section we will see that if we relax the condition on the coefficients α and β, at least when h is fixed we can give a result that tells us when f h k is going to be identically zero.
On the other hand, if we consider as initial datum the sequence
g h k (0) = (−1) k I k (1/h 2 ) I 0 (1/h 2 ) ,
then in this case we are not going to have a function u 0 satisfying (5), since the subsequences of the odd members and even members are discrete version of the functions −e −x 2 /2 and e −x 2 /2 respectively, as we can see in Figure 1. In this case, the solution to the Schrödinger equation is and, at least numerically, the decay coefficient for g h k (1) seems to be (see Figure 2) β = 5. This is not very surprising, since the decay one obtains solving the continuous Schrödinger equation with initial datum e −x 2 /2 is precisely e −x 2 /10 , so the coefficients in this case are α = 1, β = 5 as well. On the other hand, this is telling us that maybe one can remove the hypothesis (5) on the existence of a proper function u 0 and having then that just the decay conditions (6) imply that the solution to the discrete Schrödinger equation is getting smaller uniformly when h tends to zero. We see that it seems to be controlled by I k (β/h 2 ) 5 1/4 I0(β/h 2 ) for β = 5 (green dots), but this is not the case if we take β = 4.9 (red dots). Now, considering the discrete heat equation, we have similar results. Although in this case since we have a parabolic equation we can get better estimates for the convergence of the discrete system to the continuous equation, we are going to use the same estimates (4) as for the Schrödinger equation. Recall that we are comparing the discrete datum with the samples of a continuous function, and this is well-defined when the function belongs to H s , s > 1/2, so although it could be possible to give some estimates depending on the norm of the function in a weaker space (L 2 for instance), the function has to have some degree of smoothness that makes useless the advantage of using these better estimates. On the other hand, in Theorem 3.1 we have used an ℓ 2 estimate for the difference between the discrete initial datum and the continuous function, while now we can replace it by an ℓ ∞ estimate. Indeed, if we assume that a sequence v h m satisfies |v h m | ≤ h µ for some µ > 0 and for all m ∈ Z, then, solving the discrete heat equation we get
g h k (1) = e −2i/h 2 ∞ m=−∞ (−1) m I m (1/h 2 ) I 0 (1/h 2 ) I k−m (2i/h 2 ) = e 2i/h 2 I m ((2i − 1)/h 2 ) I 0 (1/h 2 ) ,e t∆ d v h m = e −2t/h 2 ∞ k=−∞ v h k I m−k (2t/h 2 ) ⇒ |e t∆ d v h m | ≤ h µ e −2t/h 2 ∞ k=−∞ I m−k (2t/h 2 ) = h µ ,
so we have that the evolution of v h m also satisfies the same estimate. In the case of the Schrödinger equation we do not have this property, and this is the reason why we have to consider an ℓ 2 estimate, since we have the conservation of this norm. Then the result we have is:
Theorem 3.2. Let v h m (t)
be a solution to the discrete heat equation, and assume that there is u 0 ∈ H s with s > 1/2 such that for some µ > 0,
(7) sup k∈Z |v h k (0) − u 0 (kh)| < h µ .
If for some α satisfying α < 2 we have, for all k ∈ Z and h > 0,
(8) |v h k (1)| ≤ C I k (α/h 2 ) I 0 (β/h 2 ) ,
then u 0 ≡ 0 and |v h k (t)| tends to zero uniformly in k when h tends to zero.
Proof. Since we have that u 0 ∈ H s we only need to see if e x 2 /2δ u(x, 1) belongs to L 2 (R) for some δ ≤ 2. Since we have the uniform estimate (7) and the convergence of the solution to the discrete heat equation with initial datum z h k = u 0 (kh) to the solution to the continuous heat equation with initial datum u 0 (x) (4), we can repeat the procedure we use in Theorem 3.1 to conclude that |u(x, 1)| ≤ e −x 2 /2α . Now, since α < 2, que can take δ such that α < δ ≤ 2 and then
R e x 2 /δ |u(x, 1)| 2 dx ≤ R e x 2 ( 1 δ − 1 α ) < +∞,
so, by Hardy's uncertainty principle for heat evolutions (2) we conclude that u 0 ≡ 0.
A weaker discrete version of Hardy's uncertainty principle
As in the classical result of Hardy, we want to have decay conditions on solutions to the discrete Schrödinger equation that make the solution to be identically zero. For this purpose, we are going to use results coming from Section 2, where we use complex variable arguments to conclude that if a periodic function and its Fourier coefficients satisfy certain decay conditions, then the function is identically zero.
The first question is to choose the parameters b and θ in Theorem 2.1 properly. For that, we rewrite the discrete Schrödinger equation (3) in a periodic setting, thinking of f h k (t) as the Fourier coefficients of a function g h (t). It is easy to see then that solving (3) is the same as solving
∂ t g h (x, t) = 2i(cos(xh) − 1) h 2 g h (x, t) ⇒ g h (x, t) = e 2i(cos(xh)−1)/h 2 g h (x, 0),
and then we observe that solving the discrete Schrödinger equation is basically multiply by the exponential e i cos(xh) , so we are going to take b, θ so that the conclusion of Theorem 2.1 is that if rs = 1, then the function is a constant times e ir h 2 cos(zh) , that is, we take b = 0 and θ = π 2 . Then we can rewrite Theorem 2.1 in order to have: Corollary 4.1. Assume that g h is a complex-valued function and that there are r, s > 0, δ ∈ (0, π/2) such that,
g h δ h + iy , g h π − δ h + iy ≤ Ce r h 2 sin δ sinh(yh) , for all y ≤ 0 g h −δ h + iy , g h π + δ h + iy ≤ Ce − r h 2 sin δ sinh(yh) , for all y ≥ 0 |ĝ h (k)| ≤ CI k 1 sh 2 , ∀k ∈ Z.
Then:
(1) rs < 1 ⇒ There are nonzero functions that satisfy the hypotheses.
(2) rs = 1 ⇒ g h = Ce ir h 2 cos(zh) (andĝ h (k) = CI k (ir/h 2 )). (3) rs > 1 ⇒ g h ≡ 0.
This is the version we are going to use in order to have a discrete version of Hardy's theorem.
As we have pointed out in Section 3, it is not reasonable to think that the condition on the coefficients α and β that measure the decay of the solution at each time will be the same as in the classical result, so we can only talk about weaker versions in this setting. More precisely, we have the following result:
Theorem 4.1. Let f h m (t)
be a solution to the discrete Schrödinger equation (3), and assume that
|f h k (0)| ≤ I k (α/h 2 ) I0(α/h 2 ) , |f h k (1)| ≤ I k (β/h 2 ) I0(β/h 2 )
with α and β positive numbers satisfying α + β < 2, then
f h = (f h k ) ≡ 0.
Proof. We define the function g h (x, t) such thatĝ h (k, t) = f h k (t). As we have seen in Section 2, the decay of the initial datum implies that the periodic function g h can be extended to a entire function and that
|g h (z, 0)| ≤ h √ 2πI 0 (α/h 2 ) e α h 2 cosh(yh) ,
for z = x + iy. Therefore, solving the discrete Schrödinger equation in this periodic setting, since it is a multiplication by another periodic and entire function, we have that at time t = 1 the corresponding function g h (z, 1) is going to be periodic and entire as well. Moreover,
|g h (z, 1)| = |e 2i h 2 cos(zh) g h (z, 0)| ≤ h √ 2πI 0 (α/h 2 ) e 2 h 2 sin(xh) sinh(yh)+ α h 2 cosh(yh) .
The key point here is how to choose δ in the corollary in order to hold all the hypotheses. Since α + β < 2, there is ǫ such that α + β < 2 − ǫ, so α < 2 − ǫ. For this ǫ we take δ ǫ close but less than π/2 such that 1 sin δǫ = 1 + ǫ α . and r ǫ = 2 − α sin δǫ = 2 − α − ǫ > 0. Now, if y ≤ 0,
g h δ ǫ h + iy, 1 ≤ h √ 2πI 0 (α/h 2 ) e 2 h 2 sin δǫ sinh(yh)+ α h 2 cosh(yh) = h √ 2πI 0 (α/h 2 ) e α h 2 e yh + rǫ h 2 sin δǫ sinh(yh) ≤ h √ 2πI 0 (α/h 2 ) e α h 2 + rǫ h 2 sin δǫ sinh(yh) ,
so we have exactly the kind of bounds we use as hypothesis in Corollary 4.1. In order to get the result, we may let the constant C in the corollary depending on h and α. However, using the asymptotic expression of I 0 (t) we get rid of the dependence in h, since we have, at least for
h small enough h √ 2πI 0 (α/h 2 ) e α h 2 ≤ 3 √ α 2 .
Considering now the heat equation, we can use the same argument as for the Schrödinger equation in order to have a very similar result, again asking about decay conditions at two different times. Now solving the heat equation is equivalent to multiply in the periodic setting by e cos(xh) , so we use Theorem 2.1 with b = 0 and θ = 0 in order to have: Corollary 4.2. Assume that g h is a complex-valued function and that there are r, s > 0, δ ∈ (0, π/2) such that, for all y ∈ R,
g h π/2 + δ h + iy , g h − π/2 + δ h + iy ≤ Ce − r h 2 sin δ cosh(yh) , |ĝ h (k)| ≤ CI k 1 sh 2 , ∀k ∈ Z.
Then:
(1) rs < 1 ⇒ There are nonzero functions that satisfy the hypotheses.
(2) rs = 1 ⇒ g h = Ce r h 2 cos(zh) (andĝ h (k) = CI k (r/h 2 )). (3) rs > 1 ⇒ g h ≡ 0.
And by means of this Corollary, we have the result:
Theorem 4.2. Let v h m (t)
be a solution to the discrete heat equation (3), and assume that
|v h k (0)| ≤ I k (α/h 2 ) I0(α/h 2 ) , |v h k (1)| ≤ I k (β/h 2 ) I0(β/h 2 )
and α and β are positive numbers satisfying α + β < 2,
then v h = (v h k ) ≡ 0.
Proof. The proof of this theorem follows the same argument as Theorem 4.1, just rewriting it for solutions to the discrete heat equation instead of the discrete Schrödinger equation.
Moreover, we can see again that the condition α + β < 2 is optimal, but now we can construct examples that are not discrete versions of the identically zero function and are close to the optimal condition α + β = 2. For instance, if we take the function u 0 (x) = e −x 2 /2ǫ , then u(x, t) = e t∂xx u 0 (x) is the function given by
u(x, t) = e −x 2 /(4t+2ǫ)
2t/ǫ + 1 , so in this case, we can take δ on Hardy's uncertainty principle for the heat equation bigger than, but as close as we want to 2 + ǫ. Then, if we take a discrete version by using the modified Bessel functions, we have a result with α = ǫ and β = 2 + ǫ, so α + β is as close as we want to the optimal value 2.
Appendix: Examples
In this Appendix we are going to see some examples of nonzero functions that satisfy the hypotheses of Theorem 2.1 when we are in the first case of the theorem rs < 1.
First example
The first example, and the most basic example, is given by the function f h (z) = Ce v h 2 cos(zh−b) , where v = ae iθ , for r < a < 1/s.
Second example
A calculation shows that the Fourier coefficient of Ce u h 2 cos(zh−b) is I k (u/h 2 )e −ikb . Here we are going to take u such that |u| = 1 and a > 1, in order to define the function whose
More examples
In some particular cases, like the particular cases of Corollary 4.1 or Corollary 4.2, we can get more examples by studying the evolution of discrete equations. For instance, if we assume that we are in the case of the discrete heat equation, that is, we have b = θ = 0, and, for the sake of simplicity, we assume that h = 1, once we have an example of a function that satisfies the hypotheses, say f , we can generate more functions of this type by using the original function as initial datum of the discrete heat equation. Indeed, for the evolution of the Fourier coefficients we havef
(k, t) = e −2t m∈Zf (m, 0)I m−k (2t) ⇒ |f (k, t)| ≤ I k 1 s + 2t .
On the other hand |f (π/2 + δ + iy, t)| = e −2t−(2t+r) sin δ cosh(y) , so the function f (z, t) also satisfy the hypotheses of the corollary, with r t = r +2t and s t = s 1+2ts . Notice that since rs < 1, we have immediately that r t s t < 1 for all t.
Furthermore, if we have a function not necessarily on the class of functions that satisfy the hypotheses, but it satisfies the condition on the Fourier coefficients, let us say a function g such that for some a > 0, |ĝ(k)| ≤ CI k (a), and we solve again the discrete heat equation then we have that |ĝ(k, t)| ≤ CI k (t + a), |g(t, x + iy)| ≤ Ce (t cos(x)+a) cosh y .
If t ≤ a we do not have the existence of δ such that the assumptions hold. However, if t > 2a, there exists δ such that t cos(π/2 + δ) = a(cos(π/2 + δ) − 1) (Notice that cos(π/2 + δ) = cos(3π/2 + δ)).
Hence for those t we have that |f (t, π/2 + δ + iy)| ≤ Ce acos(π/2+δ) cosh y .
So f (t) is in the class of functions which satisfy the assumptions of the corollary. Notice that the product of the two constants is a t+a < 1.
On the other hand, we can fix δ and then, when t ≥ a cos(π/2+δ)−1 cos(π/2+δ) the hypothesis holds.
Figure 1 .
1Graphic representation of g h k (0) when h = 1 20 and k ∈ [−50, 50] The odd coefficients are plotted in red dots and the even coefficients in black dots. We see the convergence of those subsequences to their respective Guassians ±e −x 2 /2 represented in blue and green lines.
Figure 2 .
2Representation of |g h k (1)| (black dots) when h = 1 20 and k ∈ [200, 250]
Fourier coefficients arefh (k) = (ae −ibh ) k I k u ah 2 .This function satisfies the hypotheses for s = 1 andũ = u a 2 . Indeed, one can compute the function f h to getf h (z) = e u 2 (e i(zh−b) +e −i(zh−b) /r 2 ), and now it is easy to check that all the hypotheses are satisfied.
−δ) sinh(yh) ,
Roughly speaking, what this theorem is saying is that in order to have a nonzero solution, if one of the coefficients is small, then the best possible coefficient one can have at time 1 is close to 2, thinking about being "best" as being the smallest possible coefficient, since the smaller the coefficient α or β, the better the decay. On the other hand what the classical Hardy's principle states is that the smaller one coefficient is, the bigger the other coefficient has to be in order to have a nonzero function.
Acknowledgments.This paper is part of the Ph. D. thesis of the author, who is supported by the predoctoral grant BFI-2011-11 of the Basque Government and the projects MTM2011-24054 and IT641-13. The author would like to thank L. Vega, without whose help this paper would not have been possible.It is easy to check that for the other three lines in the plane we need to consider we can do the same, so g h (z, 1) satisfies the hypotheses of the corollary with r = r ǫ . On the other hand, the decay of the Fourier coefficients at time 1 implies that s = 1 β and again the constan C in the statement of the corollary is independent of h. Finally, we have that,Remark 4.1. The condition on α and β is, as it should be expected, weaker than the original condition αβ < 4. Actually α + β < 2 ⇒ αβ < 1. Going back to the example we considered in Section 3 to show that there are nonzero functions satisfying the hypothesis of Theorem 3.1, recall that then α = 1/2 and β = 5/2, so in that case α + β = 3, hence, that example does not make any contradiction with this result.Remark 4.2. We can easily see that the condition α + β < 2 is optimal, in the sense that we can give examples of solutions satisfying the hypothesis when α + β = 2. Indeed, considerHence, we have a non-zero function that satisfies the decay conditions with α = β = 1. On the other hand, looking at Theorem 3.1 we have that if the function u 0 of the hypothesis(5)exists, it has to be zero, and it is as easy to check as in the example in Section 3 that this is the case.
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Hardy uncertainty and unique continuation properties of covariant Schrödinger flows. M Cowling, L Escauriaza, C E Kenig, G Ponce, L Vega, J. Funct. Anal. 26410M. Cowling, L. Escauriaza, C.E. Kenig, G. Ponce, L. Vega: Hardy uncertainty and unique continuation properties of covariant Schrödinger flows, J. Funct. Anal. 264 (2013), no. 10, 2386-2425.
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Hardy's uncertainty principle, convexity and Schrödinger equations. L Escauriaza, C E Kenig, G Ponce, L Vega, Journal European Math. Soc. 10L. Escauriaza, C.E. Kenig, G. Ponce, L. Vega: Hardy's uncertainty principle, convexity and Schrödinger equations, Journal European Math. Soc. 10 (2008), 883-907.
The sharp Hardy uncertainty principle for Schrödinger evolutions. L Escauriaza, C E Kenig, G Ponce, L Vega, Duke Math. J. 1551L. Escauriaza, C.E. Kenig, G. Ponce, L. Vega: The sharp Hardy uncertainty principle for Schrödinger evolutions., Duke Math. J. 155 (2010), no. 1, 163-187.
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Philippe Jaming, Yurii Lyubarskii, Eugenia Malinnikova, Karl-Mikael Perfekt, arXiv:1505.05398Uniqueness for discrete Schrödinger evolutions. math.APPhilippe Jaming, Yurii Lyubarskii, Eugenia Malinnikova, Karl-Mikael Perfekt, Uniqueness for discrete Schrödinger evolutions, arXiv:1505.05398 [math.AP].
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. A Fernández-Bertolin, 644BilbaoDepartamento de Matemáticas, Universidad del País Vasco UPV/EHUSpain E-mail address: [email protected]. Fernández-Bertolin:Departamento de Matemáticas, Universidad del País Vasco UPV/EHU, apartado 644, 48080, Bilbao, Spain E-mail address: [email protected]
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[] |
[
"Evidence for neutron star triaxial free precession in Her X-1 from Fermi/GBM pulse period measurements",
"Evidence for neutron star triaxial free precession in Her X-1 from Fermi/GBM pulse period measurements"
] |
[
"Dmitry Kolesnikov \nMoscow State University\nSternberg Astronomical Institute\nUniversitetskij pr. 13119234MoscowRussia\n",
"Nikolai Shakura \nMoscow State University\nSternberg Astronomical Institute\nUniversitetskij pr. 13119234MoscowRussia\n",
"Konstantin Postnov \nMoscow State University\nSternberg Astronomical Institute\nUniversitetskij pr. 13119234MoscowRussia\n\nKazan Federal University\nKremlyovskaya 18420008KazanRussia\n"
] |
[
"Moscow State University\nSternberg Astronomical Institute\nUniversitetskij pr. 13119234MoscowRussia",
"Moscow State University\nSternberg Astronomical Institute\nUniversitetskij pr. 13119234MoscowRussia",
"Moscow State University\nSternberg Astronomical Institute\nUniversitetskij pr. 13119234MoscowRussia",
"Kazan Federal University\nKremlyovskaya 18420008KazanRussia"
] |
[
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Her X-1/HZ Her is one of the best studied accreting X-ray pulsars. In addition to the pulsating and orbital periods, the X-ray and optical light curves of the source exhibit an almost periodic 35-day variability caused by a precessing accretion disk. The nature of the observed long-term stability of the 35-day cycle has been debatable. The X-ray pulse frequency of Her X-1 measured by the Fermi/GBM demonstrates periodical variations with X-ray flux at the Main-on state of the source. We explain the observed periodic sub-microsecond pulse frequency changes by the free precession of a triaxial neutron star with parameters previously inferred from an independent analysis of the X-ray pulse evolution over the 35-day cycle. In the Fermi/GBM data, we identified several time intervals with a duration of half a year or longer where the neutron star precession period describing the pulse frequency variations does not change. We found that the NS precession period varies within one per cent in different intervals. Such variations in the free precession period on a year time scale can be explained by 1% changes in the fractional difference between the triaxial neutron star's moments of inertia due to the accreted mass readjustment or variable internal coupling of the neutron star crust with the core.
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10.1093/mnras/stac1107
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[
"https://arxiv.org/pdf/2204.06408v1.pdf"
] | 248,157,309 |
2204.06408
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29641039d6253ab1d4cff1e0e1e941deb170a3e0
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Evidence for neutron star triaxial free precession in Her X-1 from Fermi/GBM pulse period measurements
2021
Dmitry Kolesnikov
Moscow State University
Sternberg Astronomical Institute
Universitetskij pr. 13119234MoscowRussia
Nikolai Shakura
Moscow State University
Sternberg Astronomical Institute
Universitetskij pr. 13119234MoscowRussia
Konstantin Postnov
Moscow State University
Sternberg Astronomical Institute
Universitetskij pr. 13119234MoscowRussia
Kazan Federal University
Kremlyovskaya 18420008KazanRussia
Evidence for neutron star triaxial free precession in Her X-1 from Fermi/GBM pulse period measurements
MNRAS
0002021Accepted XXX. Received YYY; in original form ZZZPreprint 14 April 2022 Compiled using MNRAS L A T E X style file v3.0X-rays: binaries -X-rays: individual: Her X-1 -stars: neutron
Her X-1/HZ Her is one of the best studied accreting X-ray pulsars. In addition to the pulsating and orbital periods, the X-ray and optical light curves of the source exhibit an almost periodic 35-day variability caused by a precessing accretion disk. The nature of the observed long-term stability of the 35-day cycle has been debatable. The X-ray pulse frequency of Her X-1 measured by the Fermi/GBM demonstrates periodical variations with X-ray flux at the Main-on state of the source. We explain the observed periodic sub-microsecond pulse frequency changes by the free precession of a triaxial neutron star with parameters previously inferred from an independent analysis of the X-ray pulse evolution over the 35-day cycle. In the Fermi/GBM data, we identified several time intervals with a duration of half a year or longer where the neutron star precession period describing the pulse frequency variations does not change. We found that the NS precession period varies within one per cent in different intervals. Such variations in the free precession period on a year time scale can be explained by 1% changes in the fractional difference between the triaxial neutron star's moments of inertia due to the accreted mass readjustment or variable internal coupling of the neutron star crust with the core.
INTRODUCTION
Her X-1 is an accreting X-ray pulsar with a pulse period of * = 1.24 s around the optical star HZ Her with an orbital period of 1.7 days (Tananbaum et al. 1972;Cherepashchuk et al. 1972). The binary system is viewed almost edge-on. This causes different eclipsing features, including periodic orbital eclipses by the optical star and Xray dips due to gas streams shielding the line of sight ( e.g., Shakura et al. 1999). The source also demonstrates a long-term 35-day X-ray flux modulation (Giacconi et al. 1973). It consists of an X-ray bright Main-on state lasting about seven binary orbital periods, followed by a first low state with an almost zero flux (about four orbits), a Shorton state less prominent than the Main-on (about four orbits), and a second low-on state (about four orbits), see Shakura et al. (1998a); Leahy & Wang (2020) for more detail.
The nature of the 35-day modulation has been debatable. One of the first explanation involved a freely precessing neutron star (NS) Brecher (1972); Novikov (1973). For the observed 35-day period to be the NS free precession period pr , an axially symmetric NS should maintain a tiny ellipticity of the order of Δ / ∼ * / pr ∼ 10 −6 (here Δ is the difference in the NS's moments of inertia). In the case of a single NS, the unavoidable internal dissipation would tend to secularly align the spin and precession axes. This argument has been considered disfavoring the NS free precession as the reason for the long-term periodicity in pulsars (e.g., Shaham (1977)). The precession of an accretion disk around NS provides another explanation ★ E-mail: [email protected] to the 35-day cycle (e.g., Katz 1973;Roberts 1974;Petterson 1975, and subsequent papers). Presently, a rich phenomenology, both in the X-ray and optical, supports the presence of a tilted, retrograde, precessing accretion disk in Her X-1 (e.g., Boynton et al. 1973;Leahy 2003;Klochkov et al. 2006;Brumback et al. 2021). In the middle of the Main-on and Short-on states, the disk is maximum open to the observer's view, while during the low states, the outer parts of the tilted disk block the X-ray source.
Extensive X-ray observations of Her X-1 demonstrate that there can occur long (with a duration of up to 1.5 years) anomalous low states of the X-ray source during which the X-ray flux is completely extinguished but the X-ray irradiation of the optical star HZ Her persists (Parmar et al. 1985;Vrtilek et al. 1994;Coburn et al. 2000;Boyd et al. 2004;). These anomalous low states are likely due to vanishing the disk tilt to the orbital plane. As long as the disk tilt is close to zero, the X-ray source remains blocked from the observer's view by the disk's outer parts. An analysis of archive optical observations of HZ Her using photo plates showed that in the past there were periods when the X-ray irradiation effect was absent altogether (Jones et al. 1973;Hudec & Wenzel 1976). This means that sometimes in Her X-1/HZ Her binary system, the accretion onto the neutron star can cease completely (Bisnovatyi-Kogan et al. 1978). The cessation of accretion could occur, for example, because of a sudden jump in the NS magnetic field, which sometimes are observed in Her X-1 (Staubert et al. 2019), or a decrease in the mass inflow from the optical star, which can turn-off accretion due to the propeller effect.
The fact that the 35-day cycle re-appears in phase with the av-erage 35-day ephemeris after the end of anomalous low states and the stable periodic behavior of X-ray pulse profiles Staubert et al. (2013) requires a 'stable clock' mechanism operating in Her X-1/HZ Her (Staubert et al. 2009), which may be the NS free precession. Indeed, a model of two-axial NS free precession can reproduce the observed regular X-ray pulse profile changes with the 35-day phase . This model involves a complex non-dipole magnetic field structure near the surface of accreting NS in Her X-1 and pencil-beam local emitting diagram. The non-dipole surface fields includes an additional quadrupole component producing ringlike structures around the NS magnetic poles (Shakura et al. 1991). This additional field doesn't distort the NS's form which is assumed to be shaped by a much stronger internal magnetic field ∼ 10 14 G (Braithwaite 2009). The model also can explain the complicated optical variability of HZ Her over the 35-day cycle, which is primarily shaped by the irradiation effect of the optical star's atmosphere by the X-ray emission from NS (Kolesnikov et al. 2020). A triaxial NS precession in Her X-1 was proposed earlier by us (Shakura et al. 1998b) to explain an anomalously narrow 35-day cycle of Her X-1 observed by HEAO-1. Presently, there is a growing empirical evidence that NS free precession could be responsible for different long-term periodicities in single magnetized NS, such as magnetars and fast radio bursts (FRBs) (see, e.g. Levin et al. 2020;Zanazzi & Lai 2020;Cordes et al. 2021;Wasserman et al. 2021;Makishima et al. 2021).
A precessing, pulsating NS should exhibit regular pulse period (or frequency) variations with a fractional amplitude change of Δ * / * ≈ * / pr ∼ 10 −6 (e.g, Ruderman 1970;Truemper et al. 1986;Bisnovatyi-Kogan et al. 1989;Bisnovatyj-Kogan & Kahabka 1993;Shakura 1995). This tiny pulse frequency variations of Her X-1 can be searched for by the continuous monitoring of X-ray sources.
In this paper, we show that the periodic sub-microsecond pulse period variability observed in Her X-1 at the 35-day cycle maxima (the Main-on state) by Fermi/GBM (Gamma-ray Burst Monitor) (Meegan et al. 2009) can be explained by the motion of X-ray emitting region on the NS surface during the free precession of a triaxial NS. A preliminary analysis of the Fermi/GBM data for the two-axial NS free precession was reported in Shakura et al. (2021).
FERMI/GBM X-RAY PULSAR HER X-1 FREQUENCY MEASUREMENTS
Fermi/GBM X-ray pulsar Her X-1 frequency measurements are publicly available 1 and updated on daily basis. The measured frequency ( ) of Her X-1 can be represented as a sum of non-periodic long-term frequency variability 0 ( ) and periodic 35-day frequency variability ( ):
( ) = 0 ( ) + ( )(1)
or, equivalently, in terms of the angular frequency:
Ω( ) = Ω 0 ( ) + Ω( ) .(2)
In accreting pulsars like Her X-1, the long-term pulsar frequency trend Ω 0 ( ) can be due to changing accretion torques, see Fig. 2.
Assuming that the pulsating flux is emitted near the north magnetic pole of a rotating solid body, the 35-day periodic variations Ω( ) are defined by the rate of change of the angle Φ of the spherical 1 https://gammaray.nsstc.nasa.gov/gbm/science/pulsars/ lightcurves/herx1.html triangle 3 Ω , see Fig. 1:
Ω( ) = Φ( ) .(3)
Here we show that the periodic change of the angle Φ with parameters as in Her X-1 can be explained by a freely precessing NS. We start with considering a two-axial NS precession, which can be treated analytically, and continue with a more general case of triaxial NS free precession.
FREE PRECESSION OF THE NEUTRON STAR
Precession of an axially symmetric NS
It is straightforward to calculate analytically the pulse frequency variations from a freely precessing axially symmetric NS when the NS moments of inertia 1 = 2 ≠ 3 (see, e.g, Ruderman 1970;Truemper et al. 1986;Bisnovatyi-Kogan et al. 1989;Bisnovatyj-Kogan & Kahabka 1993;Shakura 1995). Below we will assume that the precession frequency is much lower than the spin frequency of the NS so that the total angular momentum vector to a high accuracy coincides with the NS spin vector. When the NS spin frequency vector is misaligned with the principal inertia axis 3 by angle , the free precession angular frequency reads
= Ω 1 − 3 1 cos .(4)
The observed pulse frequency is modulated by the time derivative of the angle Φ marking the NS precession phase (see Fig. 1). For the angle between the north magnetic pole and 3 axis, the phase Φ can be found from the sine and cosine theorem for spherical triangles:
cos Φ( ) = sin sin ( ) √︁ 1 − [cos cos + sin sin cos ( )] 2 ,(5)
where ( ) is the azimuthal angle of the vector in a rigid coordinate frame related to the NS's principal inertia axes (the light grey lines in Fig. 1). In the course of NS free precession, ( ) is a linear function of time:
( ) = 0 + Ω .(6)
The amplitude of the periodic sub-microsecond pulse frequency periodic variations observed by Fermi/GBM in Her X-1 can be easily adjusted by assuming a two-axial NS free precession with the appropriate choice of the NS ellipticity Δ / (Shakura et al. 2021). However, the shape of the measured pulse frequency variations as a function of the 35-day phase can be better reproduced by assuming a slight NS triaxiality, 1 ≠ 2 ≠ 3 .
Precession of a triaxial NS
Given the moments of inertia 1 < 2 < 3 and angular velocity , the NS rotational energy is
2 = 1 Ω 2 1 + 2 Ω 2 2 + 3 Ω 2 3 ,(7)
and the angular momentum is Following Landau & Lifshitz (1976), the motion of the angular momentum vector is described by the equations
2 = 2 1 Ω 2 1 + 2 2 Ω 2 2 + 2 3 Ω 2 3 .(8)Ω 1 = √︄ 2 3 − 2 1 ( 3 − 1 ) cn (9) Ω 2 = √︄ 2 3 − 2 2 ( 3 − 2 ) sn (10) Ω 3 = √︄ 2 − 2 1 3 ( 3 − 1 ) dn ,(11)
where cn , sn , dn are elliptic Jacobi functions, and the dimensionless time is
= √︄ ( 3 − 2 ) ( 2 − 2 1 ) 1 2 3 .(12)
The free precession period reads
= 4 √︄ 1 2 3 ( 3 − 2 ) ( 2 − 2 1 ) ∫ /2 0 √︁ 1 − 2 sin 2 ,(13)
where the parameter is defined as
2 = ( 2 − 1 ) (2 3 − 2 ) ( 3 − 2 ) ( 2 − 2 1 ) .(14)
For a given NS rotational period, the fractional moment inertia differences Δ 2 = ( 2 − 1 )/ 1 and Δ 3 = ( 3 − 1 )/ 1 fully determine the NS free precession period . However, a realistic NS is not a fully rigid body. In Her X-1, the NS free precession period can change due to the action of external torques, mass accretion, non-rigid coupling between the crust and the core, etc.
In the Fermi/GBM data, we identified 10 time intervals Δ , = I, . . . , X comprising ∼ 5 − 20 consecutive cycles that can be described by approximately constant (see below Fig. 3 and Table 2). (Due to scarce points in some cycles, the data chunks Δ with constant 35-day cycle duration are not always contingent and can be separated by time intervals, which we excluded from the analysis; their inclusion does not change the results but worsens the 2 of the fit). We assigned equal values of Δ 2 for all 35-day cycles to minimize residuals between the model and observations. The parameter (Δ 3 ) was calculated individually from Eq. (13) inside each data intervals Δ with constant period . Thus, inside each data interval, for given NS parameters and free precession period we can numerically calculate positions of the vector , the phase angle Φ (see Fig. 1) and derivative Φ/ defining the pulse frequency variations.
MODELLING OF HER X-1 PULSAR FREQUENCY VARIATIONS
In accreting X-ray pulsars, the long-term pulse frequency variations 0 ( ) are caused by various factors, e.g. by variable accretion torque which are difficult to predict. Here, in order to subtract the long-term pulse frequency variations, we model 0 ( ) as a cubic spline passing through nodes , 0 ( ) as follows.
We introduce the residuals between the observed pulsar frequency measurements at moments and the theoretical model ( ):
= ∑︁ ( ( ) − ) 2 .(15)
where the index runs through all frequency measurements, index corresponds to the 35-day cycles considered, see Table A1 in Appendix A. Our theoretical model ( ) is the sum of the periodic 35day pulsar frequency variations Φ/ due to the NS free precession and long-term trend 0 ( ):
( ) = 1 2 Φ( ) + 0 ( )(16)
The time coordinate of the spline nodes is defined as the mean time of the pulse frequency measurement within the -th Main-on:
= 1 ∑︁ ,(17)
Here, is the number of observations within the -th Main-on. The spline value 0 ( ) is the difference between the mean pulse frequency and the model NS free precession frequency at the moment :
0 ( ) = 1 ∑︁ − 1 2 Φ( ) ,(18)
Parameters of the long-term evolution 0 ( ) and 35-day variations of X-ray pulse frequency ( ) were evaluated by minimizing the residuals , equation 15. Parameters of the triaxial NS free precession are listed in Tables 1 and 2. The minimizing of the residuals were done using the LMFIT package (Newville et al. 2014).
Inside each -th data interval with constant 35-day cycle duration , the fractional NS moment of inertia difference Δ 3 was optimized to fit the observed pulse frequency variations measured by Fermi/GBM. The parameters Δ 2 and the NS principal axis of inertia 3 misalignment with the angular momentum 0 were fixed for all 35-day cycles.
The trajectory of the NS angular momentum on the surface (see Fig. 1) can be defined by Δ 2 , Δ 3 and the misalignment angle 0 at the NS precession zero phase (cf. Eq. (4) for two-axial case, where this angle is constant). With fixed Δ 2 and 0 , the NS free precession period (Eq. 13) is defined by Δ 3 only. As seen from Table 2, the 35-day period in Her X-1 changes within the range Table 2). Also, measurements with error larger than 0.1 Hz or outside Main-on state were removed. 34 .8 − 35 .2, i.e. |Δ / | 1% on a timescale of half a year or longer. Variations of the moments of inertia are possible for a not fully rigid NS body; variations of the misalignment between the NS principal inertia axis 3 and angular momentum can be due to the internal coupling between the NS crust and core. Both cases are physically plausible for a realistic NS. In our model with fixed 0 , the changes in NS moments of inertia can be due to the redistribution of mass accreted onto the NS. Indeed, on a year timescale, the accreted mass in Her X-1 is ∼ 10 17 [g/s] × 3 × 10 7 [s] ∼ 3 × 10 24 g, i.e. the fractional change in the NS moment of inertia is / ≈ / ∼ 10 −9 . Thus, the mass redistribution in the non-rigid NS body with a mean ellipticity of 10 −6 could be sufficient to produce 1% variations in the relative difference of the NS moments of inertia (see Table 2).
The NS free precession period as a function of Δ 3 in our model for Her X-1 with parameters from Table 1 is shown in Fig. 4. It is seen that a 1% variations in Δ 3 alter the free precession period correspondingly. Therefore, the NS free precession model for Her X-1 suggests a 1% change in the NS body parameters on a year timescale. Similar indications have been obtained earlier from the analysis of O-C behaviour of the mean 35-day cycle duration ).
The best-fit modeling of the periodic X-ray pulse variations of Her X-1 by the triaxial NS free precession with parameters from Table 1 and Table 2 is shown in Figs. 5 and 6. The solid black line presents the model ( ), with the 35-day cycle duration adjusted using the fractional moment inertia difference Δ 3 and the NS free precession zero phase at the beginning of each data interval Δ listed in Table 2.
6.650 6.675 6.700 6.725 6.750 6.775
Relative moment of inertia ∆I 3 × 10 −7 Free precession period P , days Figure 4. Triaxial NS free precession period as a function of relative moment of inertia difference Δ 3 with other parameters fixed, see Table 1.
DISCUSSION & CONCLUSION
The 0.3-0.5 microsecond variability of the X-ray pulse period of Her X-1 measured by Fermi/GBM 2 with ∼ 0.1 microsec accuracy suggests the emitting region radial velocity amplitude / ≤ Δ * / * = Δ * / * ∼ 3 × 10 −7 . In the present paper, we have shown that such variations are possible for a freely precessing, likely triaxial NS in Her X-1. In the Fermi/GBM data, we have identified several time intervals with a duration of half a year or longer (see Fig. 3 and Table 2) where the NS precession period does not change noticeably. The NS precession period varies within 1% in different intervals. Such variations can be explained by 1% changes in the NS moment inertia difference due to accreted mass readjustment or variable internal coupling of the NS crust with the core. In principle, besides the NS free precession, the pulsar frequency variations could be generated by the reflection from the warped accretion disk precessing with the angular velocity Ω . In that case, the maximum radial velocity of the reflector should be , = Ω ≈ 2 × 10 2 cm/s for the assumed inner disk radius ≈ 10 8 cm. This velocity would give rise to the Doppler frequency modulation with an amplitude of Δ / * ∼ 10 −8 , much smaller than the observed value. The Doppler broadening of the reprocessed pulsations on the accretion disk flow would smear the precession-induced frequency variations. There is another point of concern with the disk reflection model. In Her X-1, the beginning of the 35-day cycle is known to be due to the central X-ray source opening by the outer parts of the precessing accretion disk (Kuster et al. 2005). If the pulse period variations were produced by the reflection from the disk, one would expect correlation between the 35-day cycle beginning and the pulse frequency maximum, which is not found. Therefore, the possibility that the observed pulsar period change in Her X-1 is due to reprocession of the X-ray pulses on the disk seems unlikely.
In our model, the inner part of the disk should align with the NS's equator due to magnetic forces (Lipunov & Shakura 1980;Lipunov et al. 1981;Lai 1999) during the 35-day cycle Main-on. The pulsar period 1.24 s should be close to the equilibrium value (the magnetospheric radius is close to the corotation radius), suggesting the inner disk radius ∼ 100 ns . Therefore, the accreting plasma gets frozen into the magnetic field and is canalised onto the NS's surface in regions defined by the local magnetic field structure. In this case, the precession of the outer parts of the disk should not produce variations of the hot spot geometry.
During the Short-on stage, the X-ray flux from Her X-1 is several times as low as at the Main-on, and the pulse period determination from Fermi/GBM data is less certain. However, on several occasions (e.g., on MJD 54952, 55757, 56418, 57532) the pulse period is found to be at the approximately the same level as at the Main-on 2 . In our model, the Short-on pulse is shaped by emitting arcs located symmetrically to the inertia axis 3 but phase-separated by (see Fig. 2 and 3 in Postnov et al. 2013). Therefore, the expected pattern of the pulse profile variations during the Short-on should be similar to the Main-on. Future accurate measurements of the X-ray pulse timing in Her X-1 Short-on are valuable to test this prediction.
We conclude that a freely precessing NS in Her X-1 with parameters inferred from an independent analysis of X-ray pulse profile evolution with 35-day phase can explain regular sub-microsecond pulse period changes observed by Fermi/GBM. To explain a 1% variations in the NS free precession period on a year timescale, the model requires the corresponding change in the NS parameters (relative difference in the moments of inertia or the NS angular momentum misalignment with the principal moment of inertia). These changes might be related to the variable internal coupling of the NS crust with the core. The model has also proved successful in explaining the HZ Her optical light curves over the 35day cycle as well (Kolesnikov et al. 2020). Therefore, after about half century of studies, the NS free precession as the inner clock mechanism for the observed 35-day cycle in Her X-1/HZ Her is further supported by the X-ray pulse period frequency variations observed by Fermi/GBM. Table 2). The dots show the Fermi/GBM X-ray pulse frequency measurements.
DATA AVAILABILITY
The data underlying this article are available in the article, Fermi/GBM X-ray data are freely available at https://gammaray.nsstc.nasa.gov/gbm/science/ pulsars/lightcurves/herx1.html, Swift/BAT X-ray data are freely available at https://swift.gsfc.nasa.gov/ results/transients/HerX-1/.
APPENDIX A: HER X-1 LONG-TERM PULSE FREQUENCY EVOLUTION
Here we present the table of the spline values , 0 ( ) of the long-term evolution of Her X-1 pulse frequency. The method of calculation of , 0 ( ) is described in Section 4. The numbering of 35-day cycles follows the convention introduced by Staubert et al. (1983).
Figure 1 .
1A schematic view of free precession of a triaxial neutron star. The surface coordinates (light grey lines) are related to NS inertial axes 1 , 2 , 3 . The path of the NS angular momentum vector (the spin axis) on the NS surface during the triaxial free precession is shown by the solid line. The path of during a two-axial free precession is shown by the dashed line. The asterisk marks the north magnetic pole . Φ is the phase angle of the north magnetic pole .
Figure 2 .Figure 3 .
23Her X-1 pulse frequency as a function of time. The dots show Fermi/GBM measurements (in black during the 35-day cycle Main-on phases 0.0-0.35, in red otherwise). The grey solid line shows the long-term pulse frequency variations 0 ( ) approximated as described in Section 4. The same as in Fig. 2 with grey zones showing intervals Δ with constant NS free precession period (see
Figure 5 .
5The best-fit modeling (the solid line) of the periodic X-ray pulse frequency variations of Her X-1 by the triaxial NS free precession (intervals I-VII from
Figure 6 .
6The same as inFig. 5for intervals VIII-X fromTable 2.
Table 1 .
1Triaxial free precession model parameters fixed during the fitting inside -th data intervals with constant 35-day cycle durationParameter
Symbol Value
and 3 axis misalignment
0
50°a
t zero free precession phase
Coordinates of the
90°m
agnetic pole
30°F
ractional moment inertia difference ( 2 − 1 )/ 1
Δ 2
3 × 10 −7
Table 2 .
2Triaxial NS free precession model parameters inside -th data intervals with constant 35-day cycle duration marked inFig. 3. * initial phase 0 (see Eq. 6) is calculated for the time of first Fermi/GBM data point for Her X-1 0 = MJD 54722.15470Interval number
Δ , MJD
Cycle duration,
Δ 3 × 10 −7
*
0
Reduced 2
I
54722.15 -55568.84
35.14
6.68
−0.45
7.2
II
55597.73 -56022.78
35.02
6.70
−0.305
9.3
III
56085.66 -56543.01
34.85
6.73
0.024
4.9
IV
56641.62 -56893.27
35.25
6.67
−1.13
3.2
V
56956.11 -57136.34
35.05
6.70
−0.63
4.4
VI
57408.42 -57552.93
34.83
6.73
0.01
1.8
VII
57583.53 -57731.39
35.1
6.69
−0.88
4.8
VIII
58137.80 -58625.74
34.8
6.73
−0.01
4.4
IX
58690.33 -58975.96
35.01
6.70
0.0
3.6
X
59039.85 -59392.50
35.0
6.70
−0.015
10.0
Table A1 .
A1Pulse frequency long-term evolutionCycle number
MJD
0 ( ) × 10 −5 + 0.8079, Hz
383
54726.91
2.3476
385
54760.36
2.4355
386
54796.05
2.5070
387
54831.80
2.5380
388
54865.80
2.5659
389
54900.66
2.6082
390
54936.37
2.6978
391
54971.21
2.7340
392
55006.07
2.8031
393
55041.76
2.8834
394
55076.63
2.9694
395
55112.33
3.0662
396
55146.33
3.0910
397
55182.04
3.1806
398
55216.88
3.2679
399
55251.74
3.3362
400
55287.45
3.3517
401
55321.46
3.3598
402
55357.15
3.4129
403
55392.85
3.4379
404
55428.56
3.4796
405
55464.27
3.5462
406
55498.27
3.4341
407
55531.43
3.2819
408
55567.13
3.4153
409
55601.13
3.3524
410
55635.97
3.4482
Table A1 -
A1continued Pulse frequency long-term evolutionCycle number
MJD
0 ( ) × 10 −5 + 0.8079, Hz
411
55670.85
3.4878
412
55705.68
3.4367
413
55741.39
3.4997
414
55777.10
3.5698
415
55810.62
3.6085
416
55845.94
3.4974
417
55880.79
3.5782
418
55916.50
3.6168
419
55951.36
3.6936
420
55986.21
3.7729
421
56019.37
3.5627
422
56053.36
3.5510
422
56089.08
3.5909
424
56123.93
3.6445
425
56157.93
3.7143
426
56193.66
3.8105
427
56229.34
3.8463
428
56264.19
3.8520
429
56299.04
3.7043
430
56333.04
3.7663
431
56368.75
3.8619
432
56403.61
3.9323
433
56438.46
3.9829
434
56473.31
3.9383
435
56508.17
3.9673
436
56541.32
3.7576
437
56575.32
3.7831
438
56609.32
3.7088
439
56645.02
3.8269
440
56679.87
3.9120
441
56714.75
3.9438
442
56750.44
3.9712
443
56785.31
3.9958
444
56821.00
4.0038
445
56856.69
4.0148
446
56891.56
3.9804
447
56924.70
3.7997
448
56959.57
3.9324
449
56994.40
3.9863
450
57030.10
4.0276
451
57064.13
4.0673
452
57099.84
4.1381
453
57133.83
4.2445
454
57169.52
4.3165
455
57204.38
4.2516
456
57236.69
3.9947
457
57269.85
3.8602
458
57305.56
3.8793
459
57339.56
3.6466
460
57375.24
3.7617
461
57410.11
3.8909
462
57444.96
3.9009
463
57480.66
3.9054
464
57516.37
3.9028
465
57551.23
3.8788
466
57586.08
3.9012
467
57621.77
3.9046
468
57657.47
3.9111
469
57693.46
3.9164
470
57728.89
3.9432
471
57762.88
3.9992
472
57798.58
3.9025
473
57832.60
3.9695
Table A1 -
A1continued Pulse frequency long-term evolutionCycle number
MJD
0 ( ) × 10 −5 + 0.8079, Hz
474
57865.75
3.7353
475
57900.60
3.7525
476
57935.46
3.8704
477
57969.45
3.8382
478
58003.45
3.6421
479
58036.61
3.6337
480
58070.63
3.7774
481
58105.48
3.8695
482
58141.17
3.9264
483
58175.18
3.7653
484
58208.36
3.5862
485
58243.18
3.6102
486
58277.18
3.6743
487
58312.04
3.6719
488
58346.89
3.7699
489
58380.89
3.8702
490
58416.61
3.9037
491
58450.59
3.8997
492
58486.31
3.9490
493
58521.13
3.9164
494
58556.02
3.8763
495
58589.17
3.8031
496
58624.88
3.8219
497
58658.09
3.7112
498
58692.78
3.7505
499
58726.88
3.8553
500
58762.58
3.9048
501
58797.45
3.9344
502
58832.30
3.9583
503
58868.00
3.9290
504
58902.00
3.9213
505
58937.69
3.9152
506
58973.42
3.9562
507
59006.56
3.7438
508
59041.41
3.7264
509
59075.41
3.8179
510
59110.27
3.8715
511
59144.27
3.9584
512
59179.14
4.0506
513
59213.98
4.0771
514
59248.84
4.1086
515
59283.68
4.1487
516
59318.54
4.2293
517
59352.54
4.2662
518
59389.16
4.3000
D.Kolesnikov et al.
MNRAS 000, 1-8 (2021)
https://gammaray.nsstc.nasa.gov/gbm/science/pulsars/ lightcurves/herx1.html
ACKNOWLEDGEMENTSWe thank the anonymous referee for useful comments. The work of DK and NS was supported by the RSF grant 21-12-00141 (modelling of Her X-1 pulsar frequency variations; calculation of Swift/BAT 35-day cycle turn-on times). The authors acknowledge the Interdisciplinary Scientific Educational School of Moscow University 'Fundamental and applied space research'. KP acknowledges support by the Kazan Federal University Strategic Academic Leadership Program ("PRIORITY-2030").
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[
"BOUNDS FOR TWISTS OF GLp3q L-FUNCTIONS",
"BOUNDS FOR TWISTS OF GLp3q L-FUNCTIONS"
] |
[
"Yongxiao Lin "
] |
[] |
[] |
Let π be a fixed Hecke-Maass cusp form for SLp3, Zq and χ be a primitive Dirichlet character modulo M , which we assume to be a prime. Let Lps, π b χq be the L-function associated to π b χ. In this paper, introducing some variants to previous methods, we establish the bound Lp1{2`it, π b χq ! π,δ pM p|t|`1qq 3{4´δ for any δ ă 1{36.
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[
"https://arxiv.org/pdf/1802.05111v2.pdf"
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1802.05111
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11f4d7efe8a1c1536d88bab30481c5ca5357c40a
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BOUNDS FOR TWISTS OF GLp3q L-FUNCTIONS
14 Feb 2018
Yongxiao Lin
BOUNDS FOR TWISTS OF GLp3q L-FUNCTIONS
14 Feb 2018arXiv:1802.05111v1 [math.NT]
Let π be a fixed Hecke-Maass cusp form for SLp3, Zq and χ be a primitive Dirichlet character modulo M , which we assume to be a prime. Let Lps, π b χq be the L-function associated to π b χ. In this paper, introducing some variants to previous methods, we establish the bound Lp1{2`it, π b χq ! π,δ pM p|t|`1qq 3{4´δ for any δ ă 1{36.
Introduction and Statement of Results
The subconvexity problem, which asks for an estimate of an automorphic L-function on the critical line s " 1{2`it that is better by a power saving than the bound implied by the functional equation and the Phragmen-Lindelöf principle, is a central problem in analytic number theory. Many cases have been treated in the past; see [12] for results with full generality on GLp2q. It has only been recently that people have started making progress on GLp3q with the introduction of new techniques.
In this paper, we are interested in certain degree 3 L-functions. Let π be a fixed Hecke-Maass cusp form of type pν 1 , ν 2 q for SLp3, Zq with normalized Fourier coefficients λpm, nq. Let χ be a primitive Dirichlet character modulo M . Let Lps, πq " 8 ÿ n"1 λp1, nq n s and Lps, π b χq " 8 ÿ n"1 λp1, nqχpnq n s be the L-series associated with π and π b χ; these series can be continued to entire functions of s P C with functional equations. One aims to beat the convexity bound Lp1{2`it, π b χq ! π,ε pM p|t|`1qq 3{4`ε . For the L-function Lps, πq, the first breakthrough was made by Li [9] who resolved the subconvexity problem in the t-aspect. Using a first moment method, Li showed that Lp1{2`it, πq ! p|t|`1q 3{4´δ`ε with δ " 1{16, for the symmetric square lift π of an SLp2, Zq Maass cusp form. The method depends on the non-negativity of central values of certain L-functions, which necessitates in the self-duality assumption on the cusp form π. Li's exponent of saving δ " 1{16 was later improved to δ " 1{12 by Mckee, Sun and Ye [10], and to δ " 1{8 by Nunes [19]. Later, Munshi [17] generalized Li's result [9] to arbitrary fixed cusp forms with the same exponent of saving δ " 1{16, by taking an approach other than the moment method, namely, Kloosterman's variant of the circle method, enhanced by a "conductor lowering" trick. Munshi's approach does not have to use the assumption on the non-negativity of central values of L-functions, which enables him to deal with more general cusp forms.
For the case where M , the conductor of χ, is varying, in the special case that π is self-dual and χ is quadratic, a subconvex bound was obtained by Blomer [1]. He showed that Lp1{2, πbχq ! M 3{4´δ , for any δ ă 1{8, by using the first moment method as in Li's work. Introducing a variant of the circle method, the GL 2 Petersson delta method, Munshi [16,18] established a subconvexity result L p1{2, π b χq ! M 3{4´δ , for any δ ă 1{308. Again, the approach that Munshi took does not require the non-negativity of certain L-functions, which removes the self-duality assumption on the forms π and χ in Blomer's work. Recently Holowinsky and Nelson [5] discovered a new look at Munshi's delta method, which removes the use of Petersson trace formula in [16] altogether as well as improves the exponent of saving to any δ ă 1{36.
It is then natural to ask the question of establishing a subconvex bound with two simultaneously varying parameters, for example, the conductor and t-aspects. For the conductor aspect, we will be content by considering the special case of the twists π b χ of a fixed cusp form π by Dirichlet characters χ of varying conductor M . Our task is then to solve the subconvexity problem for L p1{2`it, π b χq, simultaneously in M and t. In the special case that π is self-dual and χ is quadratic, a bound Lp1{2`it, π b χq ! pM p|t|`1qq 3{4´δ , for some δ ą 0, was obtained by Huang [6], by combining the treatment of Li and Blomer, with input from [22]. It is now desirable to ask, "Can one prove a subconvex bound for the Dirichlet twist L-functions L ps, π b χq, simultaneously in the conductor and t-aspects, for a general SLp3, Zq Hecke cusp form and general primitive Dirichlet characters?" Our main result answers this affirmatively.
Theorem 1.1. Let π be a Hecke-Maass cusp form for SLp3, Zq and χ be a primitive Dirichlet character modulo M , which we assume to be prime. Given any ε ą 0, we have
Lˆ1 2`i t, π b χ˙! π,ε pM p|t|`1qq 3{4´1{36`ε .(1)
Remark 1.2. Below we will carry out the proof under the assumption |t| ą M ε for any ε ą 0. We make such assumption so as to control the error term of the stationary phase analysis in our approach. For the case |t| ă M ε , the bound (1) follows from the work [5], since there their bound L p1{2`it, π b χq ! t,π,ε M 3{4´1{36`ε is of polynomially dependence in t.
For subconvexity bounds on GLp3q in other aspects, see [2,3,15,21]. Our approach is a variant of the methods introduced in the works [16] and [5]. In Section 2, we will give a brief outline of our approach for the simpler case L p1{2`it, πq, to guide the readers through.
Notation. We use epxq to denote expp2πixq. We denote ε an arbitrary small positive constant, which might change from line to line. In this paper the notation A -B (sometimes even A « B) means that B{pM |t|q ε ! A ! BpM |t|q ε . We reserve the letters p and ℓ to denote primes. The notations p " P and ℓ " L denote primes in the dyadic segments rP, 2P s and rL, 2Ls respectively.
2. An outline of the proof For any N ě 1, let
(2) SpN q " ÿ ně1 λp1, nqχpnqn´i t w´n N¯,
where w is a smooth function with support in r1, 2s satisfying w pjq pxq ! j 1. By symmetry, we assume t ą 2 from now on. Using a standard approximate functional equation argument ([8, Theorem 5.3]) and the estimate ř nďX |λp1, nq| ! X 1`ε , one can derive the following. Lemma 2.1. For any δ ą 0 and ε ą 0, we have
Lˆ1 2`i t, π b χ˙! pM tq ε sup N |SpN q| ? N`p M tq 3{4´δ{2`ε ,
where the supremum is taken over N in the range pM tq 3{2´δ ă N ă pM tq 3{2`ε .
From the lemma, it suffices to beat the convexity bound SpN q ! N 1`ε , for N in the range pM tq 3{2´δ ă N ă pM tq 3{2`ε , which we henceforth assume, where 0 ă δ ă 1{2 is a constant to be optimized later.
Our approach is inspired by the work [16] and is a further variant to the recent work [5]. We now give a brief introduction to the approach in [16]. Let p be a prime number, and let k " 3 mod 4 be a positive integer. Let ψ be a character of Fp satisfying ψp´1q "´1 " p´1q k . One can consider ψ as a character modulo pM . Let H k ppM, ψq be an orthogonal Hecke basis of the space of cusp forms S k ppM, ψq of level pM , nebentypus ψ and weight k. For f P H k ppM, ψq, let λ f pnq be its Fourier coefficients. Denote P ‹ " ř P ăpă2P ř ψ mod p p1´ψp´1qq. Then we have the following averaged version of the Petersson formula:
δpr, nq " 1 P ‹ ÿ p"P ÿ ψ mod p p1´ψp´1qq ÿ f PH k ppM,ψq ω´1 f λ f prqλ f pnq 2πi P ‹ ÿ p"P ÿ ψ mod p p1´ψp´1qq 8 ÿ c"1 S ψ pr, n; cpM q cpM J k´1ˆ4 π ? rn cpM˙,(3)
where δpr, nq denotes the Kronecker symbol, ω´1 f " Γpk´1q p4πq k´1 }f } 2 is the spectral weight, and S ψ pr, n; cq " ÿ ‹ α mod c ψpαqe`r α`nᾱ c˘i s the generalized Kloosterman sum. Let L be the set of primes in the interval rL, 2Ls and let L ‹ " |L| denote the cardinality of L. By writing his main sum of interest ř ř 8 m,n"1 λpm, nqχpnqW´n m 2 N¯V`n N˘a s
1 L ‹ ÿ ℓPLχ pℓq 8 ÿ ÿ m,n"1 λpm, nqWˆn m 2 N˙8 ÿ r"1 χprqV´r N ℓ¯δ pr, nℓq,
and then substituting the formula (3) with δpr, nℓq in, Munshi expressed the sum as the summation of two terms, say F ‹ and O ‹ . Successfully bounding F ‹ and O ‹ simultaneously with suitable choices of P and L to balance the contribution enables him to get his main result L p1{2, π b χq ! π,ε M 3{4´1{308`ε . Now we turn to our case. We give a sketch of our argument for the simpler case L p1{2`it, πq, for which the argument will be more transparent. The general case L p1{2`it, π b χq follows along the same line of proof. From Lemma 2.1, it suffices to beat the convexity bound N for the smoothed sum
S ‹ pN q :" ÿ ně1 λp1, nqn´i t w´n N¯, for t 3{2´δ ă N ă t 3{2`ε .
For the purpose of this sketch, we focus on the case N « t 3{2 and assume the Ramanujan bound |λpm, nq| ! pmnq ε . Let P and L be two large parameters to be specified later. We will show that without using the Petersson formula (3) we can still write, up to some scalars,
S ‹ pN q " F`O`O`N t´1 00˘, where F " 1 P 2 ÿ p"P p it ÿ ℓ"L ℓ´i t ÿ r"t 1{2 P {L r´i t 8 ÿ n"1 λp1, nqe´´n p ℓr¯w´n N¯, O " t 1{2 P L 8 ÿ n"1 λp1, nqw´n N¯ÿ p"P ÿ ℓ"L ÿ r‰0 J it pr, np{ℓq,(4)
with
J it pr, np{ℓq " ż R x´i t eˆ´n t N x˙V pxq eˆ´r N p ℓt x˙dx.
Here V pxq is a smooth compactly supported function satisfying V pjq pxq ! 1 for all j ě 0. Now our task is to beat the bound N « t 3{2 for F and O simultaneously. We estimate the term O first. The integral J it pr, np{ℓq restricts the length of the r-sum to 0 ‰ |r| ď N ε t 2 L N P « t 1{2 L P . From the second derivative test we have J it pr, np{ℓq ! t´1 {2 .
Estimating trivially using the bound J it pr, np{ℓq ! t´1 {2 , we find that
O ! t 1{2 P L N P L t 1{2 L P t´1 {2 ! N t 1{2 L P ,
so we need to save a little more than t 1{2 L{P for O. We apply the Cauchy-Schwarz inequality to reduce the task to saving t 1{2 L{P from t 1{2 P L N 1{2¨ÿ
n"Nˇÿ p"P
where 1 ! R ! t 1{2 L P .
For the moment we pretend R " t 1{2 L P . Opening the square and applying Poisson summation to the n-sum, only the zero frequency contributes. For the diagonal term, we save P LR « t 1{2 L 2 , which is satisfactory as long as t 1{2 L 2 ą tL 2 {P 2 , i.e., P " t 1{4 . For the off-diagonal, after Poisson summation we encounter the integral
J " ż R J it pr 1 , N p 1 y{ℓ 1 qJ it pr 2 , N p 2 y{ℓ 2 q w pyq dy.
We save a t from bounding the integral, by using stationary phase and the first derivative test (which is the content of Lemma 5.1), so that the off-diagonal is satisfactory as long as P " L. Hence O is fine for our purpose if P ą maxtt 1{4 , Lu.
Next, we try to bound the F term in (4). Estimating trivially, we have
F ! 1 P 2 P Lt 1{2 P {LN ! N t 1{2
, so our job is to save a little more than t 1{2 .
We apply Voronoi summation to the n-sum, to get
F « t 1{2 P 4 ÿ p"P p it ÿ ℓ"L ℓ´i t ÿ r"t 1{2 P {L r´i t ÿ n"P 3
λpn, 1qS pp, n; ℓrq .
Using Weil's bound for the Kloosterman sum we get F ! t 5{4 P 3{2 , which gives us a saving of t 3{4 {P 3{2 over the original bound N t 1{2 , and we need to save P 3{2 {t 1{4 from the above sum. Pulling the r and n-sums outside, and applying the Cauchy-Schwarz inequality, our job is to save P 3 {t 1{2 from the sum
ÿ r"t 1{2 P {L ÿ n"P 3ˇÿ ℓ"L ℓ´i t ÿ p"P p it S pp, n; ℓrqˇˇˇˇ2 .
Our final step is to open the square and apply Poisson summation to the n-sum. The diagonal is satisfactory if P L ą P 3 {t 1{2 , that is, L ą P 2 {t 1{2 . For the off-diagonal, the zero frequency (which vanishes unless ℓ 1 " ℓ 2 ) makes a contribution which is dominated by the diagonal contribution. The non-zero frequencies contribute an amount of P 2 L 4 pt 1{2 P {Lq 5{2 " P 9{2 L 3{2 t 5{4 , from which we earn a saving of
P 7 Lt{pP 9{2 L 3{2 t 5{4 q " P 5{2 L 1{2 t 1{4 , which is satisfactory if P 5{2 L 1{2 t 1{4 ą P 3 {t 1{2 , i.e., LP ă t 1{2 . Hence F is fine for our purpose if t 1{2 {P ą L ą P 2 {t 1{2 .
Now it turns out that we have a choice for the parameters P and L to simultaneously beat the convexity bound for F and O, which in turn implies a subconvexity bound for L p1{2`it, πq.
Below we will carry out details for the general case L p1{2`it, π b χq, where χ is a primitive Dirichlet character modulo M .
Some lemmas
In this section, we collect some lemmas that we are going to use in our proof. Let pα 1 , α 2 , α 3 q be the spectral parameters associated to the Maass form π. Let
G δ psq :" # 2p2πq´sΓpsq cospπs{2q, if δ " 0, 2ip2πq´sΓpsq sinpπs{2q, if δ " 1, and let G pα,δq psq " 3 ź j"1 G δj ps`α j q,
where α " pα 1 , α 2 , α 3 q, and δ " pδ 1 , δ 2 , δ 3 q. Define
j pα,δq pxq " 1 2πi ż C G pα,δq psqx´sds, x ą 0,
where C is a curved contour such that all the singularities of G˘psq are to the left of C, defined as in Def.
3.2 of [20].
Let J π p˘xq :" J pα,δq p˘xq " 1 2`j pα,δq pxq˘j pα,δ`eq pxq˘, where e " p1, 1, 1q, and δ`e is taken modulo 2.
The Bessel function J π p˘xq satisfies the following property.
Lemma 3.1. (1). Let ρ ą maxt´ℜα 1 ,´ℜα 2 ,´ℜα 3 u. For x ! 1, we have x j J pjq π p˘xq ! α1,α2,α3,ρ,j x´ρ. (2). Let K ě 0 be a fixed nonnegative integer. For x ą 0, we may write J π p˘x 3 q " ep˘3xq x Wπ pxq`Eπ pxq,
such that Wπ pxq and Eπ pxq are real analytic functions on p0, 8q satisfying
Wπ pxq " K´1 ÿ m"0 Bmpπqx´m`O K,α1,α2,α3`x´K˘, and E˘, pjq π pxq ! α1,α2,α3,j expp´3 ? 3πxq x , for x " α1,α2,α3 1,
where Bmpπq are constants depending on α 1 , α 2 and α 3 .
Proof. See [20, Theorem 14.1]; note that our J π p˘xq is the J pλ,δq px 1{3 q in the notation of [20]. Q.E.D.
Now we recall the Voronoi formula for GLp3q, in which the Bessel function J π p˘xq appears naturally.
Lemma 3.2 ( [14]). For pa, cq " 1,āa " 1pmod cq, we have In particular, replacing wpnq by w`n N˘g ives If w pjq pyq ! 1, then from the oscillation of J π pxq when |x| ą N ε , W˘´N m 12 n mc 3¯i s negligibly small as long as m 12 n is such that N m 12 n mc 3 " N ε . If we write
U˘pxq " xW˘pxq, then (5) becomes
(6) 8 ÿ n"1 λpm, nqe´´n a c¯w pnq " c ÿ ÿ m 1 |mc 8 ÿ n"1 λpn, m 1 q m 1 n Spām,˘n; mc{m 1 q U˘ˆm 12 n mc 3˙,
which is the usual version of Voronoi formula given in the work [14] and others.
Remark 3.3. Here the normalization of (5) is different from the usual version (6). With this normalization, the weight function on the right is the Hankel transform of the original Schwarz class function, matching the rank one and rank two cases. We thank Zhi Qi for making us aware of this.
Lemma 3.4 (Miller's bound, [13]). Uniformly in α, we have
(7) ÿ nďX λp1, nqepαnq ! π,ε X 3 4`ε .
Lemma 3.5 ([5, Lemma 2]). Let s 1 , s 2 be natural numbers. Let t 1 , t 2 , n be integers. Set
C :" ÿ xprs1,s2sq Spt 1 x, 1; s 1 qSpt 2 x, 1; s 2 qeˆn x rs 1 , s 2 s˙. Write s i " w i ps 1 , s 2 q, i " 1, 2, and set ∆ " w 2 2 t 1´w 2 1 t 2 . Then |C| ď 2 Opωprs1,s2sqq ps 1 s 2 rs 1 , s 2 sq 1{2 p∆, n, s 1 , s 2 q pn, s 1 , s 2 q 1{2 ,
where ωprs 1 , s 2 sq denotes the number of distinct prime factors of rs 1 , s 2 s, and the implied constant in O-symbol is absolute.
Following [22] and [11], we say a smooth function f px 1 , ..., x n q on R n to be inert if
(8) x j1
1¨¨¨x jn n f pj1,...,jnq px 1 , ..., x n q ! j1,...,jn 1.
Lemma 3.6. Let V be a smooth function with compact support on R ą0 , satisfying V pjq pxq ! j 1 for all j ě 0. Assume pM, rq " 1 and n -N , one has
8 ÿ r"1 χprqr´i t eˆ´nM r˙Vˆr N {M t" N M 3{2 t 3{2 gχ ? Mˆ2 π M t˙´i t ep´t{2πqχpnqn´i t V Aˆ2 πn N˙`OˆN M 3{2 t 1`A1 MˆN M t˙1´i t ÿ r‰0 S χ pr, n; M q ż R x´i t eˆ´n t N x˙V pxq eˆ´r N M 2 t x˙dx,(9)
where S χ pr, n; M q is the generalized Kloosterman sum, V A pxq is an inert function supported on x -1, and A ě 1 is any positive constant.
Proof. Writing eˆ´nM r˙" e´nr M¯e´´n M r¯, which follows from reciprocity, and applying Poisson summation, the r-sum becomes
8 ÿ r"1 χprqe´nr M¯r´i t e´´n M r¯Vˆr N {M t" N M 2 t ÿ rPZ ÿ apMq χpaqe´nā M¯e´a r M¯ż RˆN M t x˙´i t eˆ´n t N x˙V pxq eˆ´r N M 2 t x˙dx.
In particular, the zero frequency r " 0 is
1 MˆN M t˙1´i t gχχ pnq ż R x´i t eˆ´n t N x˙V pxq dx.
Considering the integral, by [11,Main Theorem], there is an inert function V A supported on
x 0 -1 such that ż R x´i t eˆ´n t N x˙V pxq dx " ż R eˆ´t log x 2π´n t N x˙V pxq dx " epf px 0 qq ? t V A px 0 q`O A`t´A˘,
where f pxq "´t log x 2π´n t N x , and x 0 " 2πn N is the unique solution for f 1 pxq " 0, and A ě 1 is any large constant. Therefore, ż
R x´i t eˆ´n t N x˙V pxq dx "ˆ2 π N˙´i t ep´t{2πq ? t n´i t V Aˆ2 πn N˙`O pt´Aq. Hence 8 ÿ r"1 χprqe´nr M¯r´i t e´´n M r¯Vˆr N {M t" 1 MˆN M t˙1´i t gχ χ pnqˆ2 π N˙´i t ep´t{2πq ? t n´i t V Aˆ2 πn N˙`OˆN M 3{2 t 1`A1 MˆN M t˙1´i t ÿ r‰0 S χ pr, n; M q ż R x´i t eˆ´n t N x˙V pxq eˆ´r N M 2 t x˙dx,
and (9) follows. Q.E.D.
From the lemma, assuming pM, ℓrq " 1 and n -N , one has
8 ÿ r"1 χprqr´i t eˆ´n pM ℓr˙Vˆr N p{M ℓt" N p M 3{2 t 3{2 ℓ gχ ? Mˆ2 πp M ℓt˙´i t ep´t{2πqχ`pl˘χpnqn´i t V Aˆ2 πn N1 MˆN p M ℓt˙1´i t ÿ r‰0 S χ pr, npl; M qJ it pr, np{ℓ; M q`OˆN p M 3{2 ℓt 1`A˙,(10)
where (11) J it pr, np{ℓ; M q :" whereV denotes the Fourier transform of the Schwartz function V which is normalized such thatV p0q " 1, and R ą 0 is a parameter. Inserting the identity, with suitable amplification, one can express the smoothed sum ř ně1 λp1, nqχpnqw`n N˘a s F`O. Balancing the contribution of F and O properly, the authors of
[5] obtained L p1{2, π b χq ! M 3{4´1{36`ε .
Lemma 3.8. For any ε ą 0, one has ÿ
p1"P ÿ p2"P ÿ ℓ1"L ÿ ℓ2"L ÿ r1"R ÿ r2"R ℓ1r2p2‰ℓ2r1p1 1 |ℓ 1 r 2 p 2´ℓ2 r 1 p 1 | ! pLP Rq 1`ε`m intL 2`ε , P 2`ε , R 2`ε u.
Proof. The sum is bounded by ÿ
ℓ1"L ÿ ℓ2"L ÿ m1"P R ÿ m2"P R ℓ1m2‰ℓ2m1 τ pm 1 qτ pm 2 q |ℓ 1 m 2´ℓ2 m 1 | ď ÿ 1ďdď2L 1 d ÿ ℓ1"L{d ÿ ℓ2"L{d pℓ1,ℓ2q"1 ÿ m1"P R ÿ m2"P R ℓ1m2‰ℓ2m1 τ pm 1 qτ pm 2 q |ℓ 1 m 2´ℓ2 m 1 | ,
where τ denotes the divisor function. Given d, for a fixed pair pℓ 1 , ℓ 2 q, and a fixed i with 1 ď |i| ! LP R{d, suppose pm 1 2 , m 1 1 q is a solution of the equation ℓ 1 m 2´ℓ2 m 1 " i. Since pℓ 1 , ℓ 2 q " 1, all the other solutions must be of the form pm 1 2 , m 1 1 q` Zpℓ 2 , ℓ 1 q. Therefore, for each fixed i, the total number of pairs pm 2 , m 1 q satisfying ℓ 1 m 2´ℓ2 m 1 " i is at most O p1`P Rd{Lq. In total, ÿ ℓ1"L ÿ ℓ2"L ÿ m1"P R ÿ m2"P R ℓ1m2‰ℓ2m1
1 |ℓ 1 m 2´ℓ2 m 1 | ! ÿ 1ďdď2L 1 d ÿ ℓ1"L{d ÿ ℓ2"L{d ÿ 1ďiďLP R{d 1 iˆ1`P Rd L! LP R`L 2 .
Clearly, the roles of ℓ i , p i and r i are symmetric in the above argument. One can replace the bound OpL 2`ε q by OpP 2`ε q or OpR 2`ε q. The lemma follows.
Using the same argument, we also have ÿ
p1"P ÿ p2"P ÿ ℓ1"L ÿ ℓ2"L ÿ r1"R ÿ r2"R ℓ1r2p2‰ℓ2r1p1 ℓ1r2p2"ℓ2r1p1pMq 1 |r 1 p 1 ℓ 2´r2 p 2 ℓ 1 | ! pLP Rq 1`ε {M`mintL 2`ε , P 2`ε , R 2`ε u{M.
Q.E.D.
Other relevant lemmas will be stated during the course of the proof.
Reducing SpN q to F 1 and O
Our basic strategy is to introduce more 'points' of summation to mimic the smoothed sum SpN q (2), which is our main object of study. Through out the paper we assume that |t| ą M ε for any ε ą 0.
Let P and L be two large parameters. We begin by introducing the following sum
F 1 " M 3{2 t 3{2 N P 2 ÿ p"Pχ ppqp it ÿ ℓ"L χpℓqℓ´i t 8 ÿ r"1 χprqr´i t Vˆr N p{M ℓt˙8 ÿ n"1 λp1, nqeˆ´n pM ℓr˙w´n N¯,(12)
where p " P and ℓ " L denote primes in the dyadic segments rP, 2P s and rL, 2Ls, respectively; w and V are smooth functions with compact supports on R ą0 satisfying w pjq pxq, V pjq pxq ! j 1 for all j ě 0.
We shall see that if one applies Poisson summation to the r-sum (which is the content of Lemma 3.6), then contribution of the zero frequency r " 0 will give rise to the sum SpN q that we are initially interested in. In order to bound SpN q, it suffices to bound F 1 and the sum arising from the non-zero frequencies r ‰ 0 (if we apply Poisson summation to the r-sum), which we denote by O. This observation is initially due to Holowinsky and Nelson [5, B.4], in their work in the Dirichlet character twist case.
Plugging the identity (10) in, we get We have shown the following.
F 1`O`O´N t 1{2´A¯,(13)
where A ě 1 is any constant, V A pxq is an inert function (see (8)) depending on A, supported on x -1 and J it pr, np{ℓ; M q is given by (11).
For any given ε ą 0, we can make the error term O`N t 1{2´A˘t o be negligibly small by assuming t ą M ε and taking A to be sufficiently large. From the lemma, to bound
8 ÿ n"1 λp1, nqχpnqn´i t w´n N¯V Aˆ2 πn N˙,
which is essentially our original object of study SpN q, it suffices to bound the terms F 1 and O. We shall do this in the next two sections separately.
Treatment of O
This section is devoted to giving a nontrivial bound for the sum where R satisfies
O " t 1{2 M 1{2 P L1 ! R ! N ε M 2 t 2 L N P .
By the Cauchy-Schwarz inequality and the Rankin-Selberg estimate Taking into account the oscillations of J it pr 1 , N p 1 y{ℓ 1 ; M q and J it pr 2 , N p 2 y{ℓ 2 ; M q, the integral is arbitrarily small for n ‰ 0 (since N " pM tq 1`ε ). Hence there is only zero frequency after Poisson in n:
ř 8 n"1 |λp1, nq| 2 w`n N˘! N 1`ε , OpRq ! N 1{2`ε t 1{2 M 1{2 P L¨88 ÿ n"1 S χ pr 1 , np 1l1 ; M qS χ pr 2 , np 2l2 ; M qJ it pr 1 , np 1 {ℓ 1 ; M qJ it pr 2 , np 2 {ℓ 2 ; M q w´n N" N M C J`OpN´2 018 q,(16)
where C "
ÿ apMq S χ pr 1 , ap 1l1 ; M qS χ pr 2 , ap 2l2 ; M q "M χ`p 1p2l1 ℓ 2˘ÿ ‹ βpMq eˆp r 1´r2p1 p 2 ℓ 1l2 qβ M" M χ`p 1p2l1 ℓ 2˘" M δ ℓ2r1p1"ℓ1r2p2pMq´1 ‰ , and J " ż R J it pr 1 , N p 1 y{ℓ 1 ; M qJ it pr 2 , N p 2 y{ℓ 2 ; M q w pyq dy.(17)
One readily sees that
(18) C " # OpM 2 q, ℓ 1 r 2 p 2 " ℓ 2 r 1 p 1 pM q OpM q,
otherwise.
For the integral J, if we use the previously mentioned second derivative bound J it pr, np{ℓ; M q ! t´1 {2 , we get J ! t´1. However, there are more cancellations beyond Opt´1q, as long as the parameters pr i , p i , ℓ i q satisfy r 1 p 1 ℓ 2 ‰ r 2 p 2 ℓ 1 . Indeed, we have the following bound.
Lemma 5.1. For J defined as in (17), we have
J ! min " t´1, M 2 ℓ 1 ℓ 2 N |ℓ 1 r 2 p 2´ℓ2 r 1 p 1 | * .
Proof. From (11), we write
J it pr, N py{ℓ; M q " ż R e pf pxqq V pxq dx, where f pxq "´t log x 2π´y t x´r N p M 2 ℓt
x. Set f 1 px 0 q " 0 and solve for x 0 to find the stationary point. There are several cases, but for our demonstration we concentrate on the following case:
x 0 "´1`b 1`1 6π 2 rN py M 2 t 2 ℓ 4πrN p M 2 t 2 ℓ .
Expanding the integral J it pr, N py{ℓ; M q at the stationary point x 0 (by [11,Main Theorem]), we get J it pr, N py{ℓ; M q "˜´1`c1`1 6π 2 rN py
M 2 t 2 ℓ¸´i t e˜´t 2π c 1`1 6π 2 rN py M 2 t 2 ℓ¸p 4πrN p M 2 t 2 ℓ q it ? t r V px 0 q`O B`t´B˘,
where r V pxq " r V B pxq is an inert function (see (8)) supported on x -1, and B ě 1 is any constant. Therefore
J " 1 t p r 1 N p 1 M 2 t 2 ℓ 1 q it p r 2 N p 2 M 2 t 2 ℓ 2 q´i t ż R wpyq r V px 0,1 q r V px 0,2 q¨´1`b 1`1 6π 2 r1N p1 M 2 t 2 ℓ1 ý 1`b1`1 6π 2 r2N p2 M 2 t 2 ℓ2 y‚´i t e˜´t 2π d 1`1 6π 2 r 1 N p 1 M 2 t 2 ℓ 1 y`t 2π d 1`1 6π 2 r 2 N p 2 M 2 t 2 ℓ 2 y¸dy`O`t´B˘,(19)
where x 0,i " p´1`b1`1 6π 2 riN piy M 2 t 2 ℓi q{ 4πriN pi M 2 t 2 ℓi , i " 1, 2. Denote z 1 " 16π 2 r1N p1
M 2 t 2 ℓ1
and z 2 " 16π 2 r2N p2 M 2 t 2 ℓ2 . Considering the compactness of the support of the weight function r V , we necessarily have z 1 -1, z 2 -1. If z 1 " z 2 , then we have J ! t´1 by estimating trivially. From now on we assume that z 1 ‰ z 2 .
We can rewrite the integral in (19) as ż
R w 1 pyqeˆ´t 2π φpyq˙dy,
where w 1 pyq " wpyq r V px 0,1 q r V px 0,2 q, and φpyq " logˆ´1`? 1`z 1 ý 1`?1`z 2 y˙`a 1`z 1 y´a1`z 2 y.
It turns out that B By φpyq " 1 2y
a 1`z 1 y´1 2y a 1`z 2 y " z 1´z2 2p ? 1`z 1 y`?1`z 2 yq .
Since z 1 y -1 and z 2 y -1, we have B By φpyq " |z 1´z2 |. Now by the first derivative test ([7, Lemma 5.
1.2]), ż R w 1 pyqeˆ´t 2π φpyq˙dy ! 1 t|z 1´z2 | ! M 2 t ℓ 1 ℓ 2 N |ℓ 1 r 2 p 2´ℓ2 r 1 p 1 | , upon plugging in z 1 " 16π 2 r1N p1 M 2 t 2 ℓ1
and z 2 " 16π 2 r2N p2 M 2 t 2 ℓ2 . The lemma readily follows. Q.E.D.
Remark 5.2. For ℓ 1 r 2 p 2 ‰ ℓ 2 r 1 p 1 , typically
M 2 ℓ 1 ℓ 2 N |ℓ 1 r 2 p 2´ℓ2 r 1 p 1 | « t´2,
so that the second bound of the lemma shows that we save an extra t over the 'trivial bound' t´1. The estimation of this lemma is an analytic analogue of the bound (18).
Now we return to the estimate of OpRq, in (15). Plugging the n-sum (16) into OpRq, up to a negligible error, we have
OpRq ! N 1{2`ε t 1{2 M 1{2 P L˜ÿ p1"P ÿ p2"P ÿ ℓ1"L ÿ ℓ2"L ÿ r1"R ÿ r2"R N M |C| |J|¸1 {2 ! N 1`ε t 1{2 M P Lˆÿ p1"P ÿ p2"P ÿ ℓ1"L ÿ ℓ2"L ÿ r1"R ÿ r2"R ℓ1r2p2"ℓ2r1p1pMq M 2 |J| ÿ p1"P ÿ p2"P ÿ ℓ1"L ÿ ℓ2"L ÿ r1"R ÿ r2"R ℓ1r2p2ıℓ2r1p1pMq M M 2 ℓ 1 ℓ 2 N |ℓ 1 r 2 p 2´ℓ2 r 1 p 1 |˙1 {2 ,(20)
by using (18) and Lemma 5.1. We remind the reader that R satisfies 1 ď R ! N ε M 2 t 2 L N P .
Using Lemma 5.1 again, the first term inside the parentheses is bounded by
M 2 ÿ p1"P ÿ p2"P ÿ ℓ1"L ÿ ℓ2"L ÿ r1"R ÿ r2"R ℓ1r2p2"ℓ2r1p1 t´1`M 4 L 2 N ÿ p1"P ÿ p2"P ÿ ℓ1"L ÿ ℓ2"L ÿ r1"R ÿ r2"R ℓ1r2p2‰ℓ2r1p1 ℓ1r2p2"ℓ2r1p1pMq 1 |ℓ 1 r 2 p 2´ℓ2 r 1 p 1 | ,
which is further dominated by
! N ε M 2 t´1P LR`N ε M 4 L 2 NˆL P R M`L 2 M! N ε M 4 tL 2 N`N ε M 5 t 2 L 4 N 2 ,
by using Lemma 3.8 and by noting that R ! N ε M 2 t 2 L N P . Similarly, the second term inside the parentheses of (20) is bounded by
M 3 L 2 N ÿ p1"P ÿ p2"P ÿ ℓ1"L ÿ ℓ2"L ÿ r1"R ÿ r2"R ℓ1r2p2‰ℓ2r1p1 1 |ℓ 1 r 2 p 2´ℓ2 r 1 p 1 | ! N ε M 3 L 2 N pP LR`L 2 q ! N ε M 5 t 2 L 4 N 2 ,
upon using Lemma 3.8.
Returning to the estimate of OpRq, we have shown for any 1 ď R ! N ε M 2 t 2 L N P , that
OpRq ! N 1`ε t 1{2 M P LˆM 4 tL 2 N`M 5 t 2 L 4 N 2˙1 {2 ! N 1{2`ε M t P`N ε M 3{2 t 3{2 L P .
We summarize the main result of this section.
Proposition 5.3. For any ε ą 0, we have the bound
O ! N 1{2`ε M t P`N ε M 3{2 t 3{2 L P ,
for O defined as in (14).
Remark 5.4. If we only use the 'trivial' bound J ! t´1 for the estimate of the integral J, then one will see that for the second term we get O`N ε M 3{2 t 2 L{P˘instead. It is thus crucial to use Lemma 5.1 to get an extra t 1{2 saving in order to beat the convexity bound in the t-aspect.
Treatment of F 1
The purpose of this section is to give a nontrivial bound for
F 1 " M 3{2 t 3{2 N P 2 ÿ p"Pχ ppqp it ÿ ℓ"L χpℓqℓ´i t 8 ÿ r"1 χprqr´i t Vˆr N p{M ℓt˙8 ÿ n"1
λp1, nqeˆ´n pM ℓr˙w´n N¯, defined in (12), where w and V are smooth compactly supported functions with bounded derivatives. Bounding the sum directly with Miller's bound (7), we have F 1 ! N 3{4`ε pM tq 1{2 , which is not satisfactory yet for our purpose.
We shall apply a Voronoi summation to the n-sum. To this end, one may assume pp, rq " 1 in F 1 , as the contribution from the terms pp, rq ą 1 is negligible, compared to the generic terms pp, rq " 1. We briefly justify this. Denote the terms with p|r in F 1 by F 7 1 . Then, Hence we can estimate F 7 1 as follows.
F 7 1 " M 3{2 t 3{2 N P 2 ÿ p"PF 7 1 - M 3{2 t 3{2 N P log P ÿ ℓ"L χpℓqℓ´i t 8 ÿ r"1 χprqr´i t Vˆr N {M ℓtl r ÿ m|ℓr 8 ÿ n"1 λpm, nq mn SpM, n; ℓr{mq U`ˆm 2 n pℓrq 3 {N! N ε M 3{2 t 3{2 N P ÿ ℓ"L ÿ r"N {MLt ℓr ÿ ÿ m 2 năpℓrq 3 {N |λpm, nq| mnˆℓ r m˙1 {2 ! N ε N 3{2 P M t ,
upon using Weil's bound, which is satisfactory for our purpose. From now on we assume that pp, ℓrq " 1. Application of Voronoi summation (6) to the n-sum yields Here contribution from the terms with m 2 n " N ε pℓrq 3 {N is negligibly small. Thus, we can truncate the pm, nq-sum at m 2 n ! N 2`ε P 3 {M 3 t 3 , at the cost of a negligible error. For those m 2 n ! N 2`ε P 3 {M 3 t 3 , the result of Jacquet and Shalika gives us the bound
U˘ˆm 2 nN ℓ 3 r 3˙! c m 2 nN ℓ 3 r 3 ,
while in general we have y j U˘, pjq pyq ! ? y. Considering for example, the plus case, we have Pulling the ℓ-sum inside the pm, nq-sum and applying the Cauchy-Schwarz inequality to F 1 , we obtain
by noting that ÿ ÿ m,ně1
m 2 n!N 2 P 3 {M 3 t 3 |λpn, mq| 2 mn ! N ε .
Now it remains to estimate Σ. Opening the square and interchanging the order of summations, we find
Σ ď ÿ r"N P {MLt ÿ măN P 3{2 {M 3{2 t 3{2 1 m ÿ ℓ1"L m|ℓ1r ÿ ℓ2"L m|ℓ2r ÿ p1"P ÿ p2"P ÿ n!N 2 P 3 {m 2 M 3 t 3 1 n Spp 1 M, n; ℓ 1 r{mqSpp 2 M, n; ℓ 2 r{mq U`ˆm 2 nN ℓ 3 1 r 3˙U`ˆm 2 nN ℓ 3 2 r 3˙ˇ.
Our next step is to apply Poisson summation to the n-sum. To this end, one can insert an nonnegative smooth function F which is supported on, say r1{2, 3s, and constantly 1 on r1, 2s, into the n-sum.
We apply Poisson summation with modulus rℓ 1 r{m, ℓ 2 r{ms, to get ÿ ně1 1 n Spp 1 M, n; ℓ 1 r{mqSpp 2 M, n; ℓ 2 r{mq U`ˆm 2 nN
ℓ 3 1 r 3˙U`ˆm 2 nN ℓ 3 2 r 3˙Fˆn N m" 1 rℓ 1 , ℓ 2 sr{m ÿ nPZ C ℓ1,ℓ2 pnqT pn, ℓ 1 , ℓ 2 q, where (22) N m ! N 2 P 3 {m 2 M 3 t 3 ,
(23) C ℓ1,ℓ2 pnq " ÿ aprℓ1,ℓ2s r m q Spp 1 M, a; ℓ 1 r{mqSpp 2 M, a; ℓ 2 r{mqeˆa n rℓ 1 , ℓ 2 sr{m˙, and T pn, ℓ 1 , ℓ 2 q "
ż R F pxq U`ˆm 2 N m N x ℓ 3 1 r 3˙U`ˆm 2 N m N x ℓ 3 2 r 3˙eˆ´n N m rℓ 1 , ℓ 2 sr{m x˙d x x .
By integrating by parts repeatedly, the integral T pn, ℓ 1 , ℓ 2 q is negligibly small, unless |n| ! N ε rℓ1,ℓ2sr{m
Nm ! N ε N P L{Mt
Nm . Therefore, we can truncate the dual n-sum at N ε N P L{Mt
Nm
(with the convention that if this is ă 1, then only the zero frequency n " 0 survives), at the cost of a negligible error. While in the range |n| ! N ε N P L{Mt Nm , we use the bound y j U`, pjq pyq ! ? y to obtain
(24) T pn, ℓ 1 , ℓ 2 q ! m 2 N m N pN P {M tq 3 .
In particular, T pn, ℓ 1 , ℓ 2 q ! 1, by consideration of the bound (22). Let us also observe in particular that
N 1 ! N 2 P 3 {M 3 t 3 ,
for later convenience. We arrive at
F 1 ! N 1{2`ε P 1{2 L 1{2 Ω 1{2`N 3{2`ε P M t ,(25)
where Ω " ÿ r"N P {MLt ÿ măN P 3{2 {M 3{2 t 3{2 ÿ ℓ1"L m|ℓ1r ÿ ℓ2"L m|ℓ2r ÿ p1"P ÿ p2"P ÿ |n|ăN P L{NmMt 1 rℓ 1 , ℓ 2 sr |C ℓ1,ℓ2 pnqT pn, ℓ 1 , ℓ 2 q|.
We have essentially square-root cancellation for the character sum C ℓ1,ℓ2 pnq, defined in (23). The details of this calculation were carried out in [5]. We have collected their results relevant to our present setting in Lemma 3.5.
Bounding our sum (23) using Lemma 3.5, we get
(26) |C ℓ1,ℓ2 pnq| ď 2 Opωprqq´r m¯3 {2 ℓ 1 ℓ 2 pℓ 1 , ℓ 2 q 1{2
p∆, n, ℓ 1 r{m, ℓ 2 r{mq pn, ℓ 1 r{m, ℓ 2 r{mq 1{2 , where ∆ :"p 1 ℓ 2 2´p 2 ℓ 2 1 pℓ 1 , ℓ 2 q 2 M, andp 1 andp 2 denote the multiplicative inverses of p 1 and p 2 modulo ℓ 1 r{m and ℓ 2 r{m, respectively.
We write Ω " Ω 0`Ω1 , where Ω 0 denotes contribution from the terms n " ∆ " 0, and Ω 1 denotes the complement.
Remark 6.1. In fact, Ω 0 is the diagonal contribution pℓ 1 , p 1 q " pℓ 2 , p 2 q to the sum (21), and Ω 1 is the off-diagonal contribution.
If ∆ " 0, thenp 1 ℓ 2 2´p 2 ℓ 2 1 " 0. Necessarily, ℓ 1 " ℓ 2 :" ℓ and p 1 " p 2 :" p. Under this condition,
|C ℓ,ℓ pnq| ď 2 Opωprqqˆℓ r m˙3 {2ˆn , ℓr m˙1 {2 .
In particular, |C ℓ,ℓ p0q| ď 2 Opωprqq`ℓr m˘2 . Therefore, Ω 0 ! ÿ r"N P {MLt ÿ ℓ"L ÿ m|ℓr ÿ p"P 2 Opωprqq 1 ℓrˆℓ r m˙2 |T p0, ℓ, ℓq| ! N 2`ε P 3 M 2 t 2 .
Meanwhile for Ω 1 , we further write Ω 1 " Ω 1a`Ω1b , where Ω 1a denotes the contribution coming from the n ‰ 0 terms, and Ω 1b denotes the contribution of the zero frequency: n " 0, ∆ ‰ 0. Plugging the bounds (24) and (26) in, we first see that ÿ ℓ2"L m|ℓ2r ÿ p1"P ÿ p2"P 1 ∆‰0 rℓ 1 , ℓ 2 sr |C ℓ1,ℓ2 p0qT p0, ℓ 1 , ℓ 2 q|.
A direct evaluation of C ℓ1,ℓ2 p0q from the definition (23) shows that it vanishes unless ℓ 1 " ℓ 2 :" ℓ. In the later case we have C ℓ,ℓ p0q " ℓr m ÿ ‹ βpℓr{mq eˆp 1´p2 ℓr{m β˙.
Recall for ℓ 1 " ℓ 2 " ℓ, ∆ " pp 1´p2 qM , wherep 1 andp 2 are the multiplicative inverses of p 1 and p 2 modulo ℓr{m, respectively. As ∆ ‰ 0, we havep 1 ‰p 2 , and hence in particular,p 1 ıp 2 mod ℓr{m. Therefore |C ℓ,ℓ p0q| ď ℓr m τ pℓr{mq,
where τ denotes the divisor function. We thus have Ω 1b !N ε ÿ r"N P {MLt ÿ ℓ"L ÿ m|ℓr ÿ p1"P ÿ p2"P 1p 1‰p2 ℓr ℓr m |T p0, ℓ, ℓq| ! N 1`ε P 3 M t .
This is dominated by the diagonal contribution Ω 0 (27), since M t ă N .
Hence we obtain the bound Ω " Ω 0`Ω1a`Ω1b
! N 2`ε P 3 M 2 t 2`N ε pN M tq 1{2 pP Lq 3{2 .(28)
Combining (25) and (28), we retrieve the bound on F 1 in the following. Proposition 6.2. For any given ε ą 0,
F 1 ! N 3{2`ε P M tL 1{2`N 3{4`ε pM tP Lq 1{4 .
Remark 6.3. We will assume L ă P , so that the term O´N Plugging these bounds into (13), SpN q ! N 3{2`ε P M tL 1{2`N 3{4`ε pM tP Lq 1{4`N 1{2`ε M t P`N ε M 3{2 t 3{2 L P .
Substituting this into Lemma 2.1 and noting that pM tq 3{2´δ ă N ă pM tq 3{2`ε , one gets Lˆ1 2`i t, π b χ˙! pM tq 1{2`ε P L 1{2`p M tq 5{8`ε pP Lq 1{4`p M tq 1`ε P`p M tq 3{4`δ{2`ε L P`p M tq 3{4´δ{2`ε " pM tq 5{8`ε P 1{4 L 1{2ˆP 3{4 pM tq 1{8`L 3{4˙`p M tq εˆM t P`p M tq 3{4´δ{2˙, upon assuming L ă pM tq 1{4´δ{2 . Equate the first two terms by letting L " P pM tq´1 {6 to get Lˆ1 2`i t, π b χ˙!pM tq 7{12`ε P 1{2`p M tq 1`ε {P`pM tq 3{4´δ{2`ε .
Letting P " pM tq 5{18 ,
Lˆ1
2`i t, π b χ˙!pM tq 13{18`ε`p M tq 3{4´δ{2`ε .
Finally, by choosing δ " 1{18, (29) implies that
Lˆ1
2`i t, π b χ˙!pM tq 3{4´1{36`ε .
Note that with such choices, L " pM tq 1{9 satisfies the assumption L ă pM tq 1{4´δ{2 " pM tq 2{9 . Theorem 1.1 follows.
J
it pr, np{ℓqˇˇˇˇ2 ,
. 7 .S
7The identity (9) is a further variant to the following key identity in[5, (3.6)]. χ pr, n; M qVˆr M {R˙,
F 1 S
1χ pr, npl; M qJ it pr, np{ℓ; M q`OpN t pr, npl; M qJ it pr, np{ℓ; M q.
Lemma 4. 1 .
1Asymptotically
pr, npl; M qJ it pr, np{ℓ; M q,
S
χ pr, npl; M qJ it pr, np{ℓ; M q, introduced in(14), where J it pr, np{ℓ; M q is defined in(11).For r ‰ 0, integrating by parts implies that the integral J it pr, np{ℓ; M q is negligibly small, unless 0 ‰ |r| ď N ε M 2 t 2 L N P (by[4, Lemma 8.1]). Moreover, using the second derivative test ([7, Lemma 5.1.3]) we find that J it pr, np{ℓ; M q ! t´1 {2 . To estimate O, it suffices to bound the sum pr, npl; M qJ it pr, np{ℓ; M q,
S
pr, npl; M qJ it pr, np{ℓ; M qˇˇˇˇ2 w´n N¯‚ 1{2 " N 1{2`ε t 1{2 M 1{2 P Lˆÿ p1"χ pr 1 , np 1l1 ; M qS χ pr 2 , np 2l2 ; M qJ it pr 1 , np 1 {ℓ 1 ; M qJ it pr 2 , np 2 {ℓ 2 ; M q w´n N¯˙1 {2 . pr 1 , np 1l1 ; M qS χ pr 2 , np 2l2 ; M qJ it pr 1 , np 1 {ℓ 1 ; M qJ it pr 2 , np 2 {ℓ 2 ; pr 1 , ap 1l1 ; M qS χ pr 2 , ap 2l2 ; M q e´a n Mż R J it pr 1 , N p 1 y{ℓ 1 ; M qJ it pr 2 , N p 2 y{ℓ 2 ; M q w pyq eˆ´n N M y˙dy.
An application of Voronoi summation (6) takes the n-sum to the following dual sum SpM,˘n; ℓr{mq U˘ˆm 2 n pℓrq 3 {N˙, where the new length can be truncated at m 2 n ă N ε pℓrq 3 {N , at the cost of a negligible error.
ppqp it SppM, n; ℓr{mq U`ˆm 2 nN ℓ 3 r 3˙`OˆN 3{2`ε P M t! N 1{2`ε P 1{2 L 1{2 Σ 1{2`
P
Mt¯i n (25) is negligible. 7. The choices of the parameters P and L Recall from Proposition 6.2, one has
r 1{2 m
1{23{2 p∆, n, r{mq pn, r{mq 1{2m 2 N m N pN P {M tq 3 {2 N 1 N pN P {M tq 3 !N ε pN M tq 1{2 pP Lq 3{2 .Now we treat the case of Ω 1b , which by our definition isΩ 1b " ÿ r"N P {MLt ÿ măN P 3{2 {M 3{2 t 3{2!N ε N P
M Lt
L 2 P 2 N P L
N 1 M tˆN
P
M Lt˙1
ÿ
ℓ1"L
m|ℓ1r
Acknowledgements. The author is most grateful to his advisor, Professor Roman Holowinsky, for all his guidance and support through out the project. He wishes to thank Zhi Qi for sharing his notes on Bessel functions and answering a few questions that the author had, and Runlin Zhang for helpful discussions related to the proofs of Lemmas 3.8 and 5.1. The author also thanks Professor Paul Nelson for a few insightful comments and Alex Beckwith for suggestions which help improve the presentation.
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231 W 18th Avenue. Columbus, OhioDepartment of Mathematics, The Ohio State University43210-1174 E-mail address: [email protected] of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, Ohio 43210-1174 E-mail address: [email protected]
|
[] |
[
"Improving Quality of Clustering using Cellular Automata for Information retrieval",
"Improving Quality of Clustering using Cellular Automata for Information retrieval",
"Improving Quality of Clustering using Cellular Automata for Information retrieval",
"Improving Quality of Clustering using Cellular Automata for Information retrieval"
] |
[
"P Kiran Sree ",
"Associate Prof \nDepartment of CSE, S.R.K.Institte of Technolgy\nVijayawadaIndia\n",
"P Kiran Sree \nDepartment of CSE\nSwarnandhara Engineering College\nNarasapurIndia\n",
"G V S Raju \nDepartment of CSE\nAcharya Nagarjuna University\nGunturIndia\n",
"I Ramesh Babu \nDepartment of CSE, Gokaraju Rangaraju Institute of Engg and Technology\nIndia\n",
"S Viswanadha Raju ",
"\nDet of CSE\nS.R.K.Institte of Technolgy\nVijayawadaIndia\n",
"P Kiran Sree ",
"Associate Prof \nDepartment of CSE, S.R.K.Institte of Technolgy\nVijayawadaIndia\n",
"P Kiran Sree \nDepartment of CSE\nSwarnandhara Engineering College\nNarasapurIndia\n",
"G V S Raju \nDepartment of CSE\nAcharya Nagarjuna University\nGunturIndia\n",
"I Ramesh Babu \nDepartment of CSE, Gokaraju Rangaraju Institute of Engg and Technology\nIndia\n",
"S Viswanadha Raju ",
"\nDet of CSE\nS.R.K.Institte of Technolgy\nVijayawadaIndia\n"
] |
[
"Department of CSE, S.R.K.Institte of Technolgy\nVijayawadaIndia",
"Department of CSE\nSwarnandhara Engineering College\nNarasapurIndia",
"Department of CSE\nAcharya Nagarjuna University\nGunturIndia",
"Department of CSE, Gokaraju Rangaraju Institute of Engg and Technology\nIndia",
"Det of CSE\nS.R.K.Institte of Technolgy\nVijayawadaIndia",
"Department of CSE, S.R.K.Institte of Technolgy\nVijayawadaIndia",
"Department of CSE\nSwarnandhara Engineering College\nNarasapurIndia",
"Department of CSE\nAcharya Nagarjuna University\nGunturIndia",
"Department of CSE, Gokaraju Rangaraju Institute of Engg and Technology\nIndia",
"Det of CSE\nS.R.K.Institte of Technolgy\nVijayawadaIndia"
] |
[
"Journal of Computer Science",
"Journal of Computer Science"
] |
Clustering has been widely applied to Information Retrieval (IR) on the grounds of its potential improved effectiveness over inverted file search. Clustering is a mostly unsupervised procedure and the majority of the clustering algorithms depend on certain assumptions in order to define the subgroups present in a data set .A clustering quality measure is a function that, given a data set and its partition into clusters, returns a non-negative real number representing the quality of that clustering. Moreover, they may behave in a different way depending on the features of the data set and their input parameters values. Therefore, in most applications the resulting clustering scheme requires some sort of evaluation as regards its validity. The quality of clustering can be enhanced by using a Cellular Automata Classifier for information retrieval. In this study we take the view that if cellular automata with clustering is applied to search results (query-specific clustering), then it has the potential to increase the retrieval effectiveness compared both to that of static clustering and of conventional inverted file search. We conducted a number of experiments using ten document collections and eight hierarchic clustering methods. Our results show that the effectiveness of query-specific clustering with cellular automata is indeed higher and suggest that there is scope for its application to IR.
|
10.3844/jcssp.2008.167.171
|
[
"https://arxiv.org/pdf/1401.2684v1.pdf"
] | 17,582,493 |
1401.2684
|
5fe3c15dcf78519910b0d18608f934ae259c8502
|
Improving Quality of Clustering using Cellular Automata for Information retrieval
2008
P Kiran Sree
Associate Prof
Department of CSE, S.R.K.Institte of Technolgy
VijayawadaIndia
P Kiran Sree
Department of CSE
Swarnandhara Engineering College
NarasapurIndia
G V S Raju
Department of CSE
Acharya Nagarjuna University
GunturIndia
I Ramesh Babu
Department of CSE, Gokaraju Rangaraju Institute of Engg and Technology
India
S Viswanadha Raju
Det of CSE
S.R.K.Institte of Technolgy
VijayawadaIndia
Improving Quality of Clustering using Cellular Automata for Information retrieval
Journal of Computer Science
422008167Cellular automatainformation retrievalclustering
Clustering has been widely applied to Information Retrieval (IR) on the grounds of its potential improved effectiveness over inverted file search. Clustering is a mostly unsupervised procedure and the majority of the clustering algorithms depend on certain assumptions in order to define the subgroups present in a data set .A clustering quality measure is a function that, given a data set and its partition into clusters, returns a non-negative real number representing the quality of that clustering. Moreover, they may behave in a different way depending on the features of the data set and their input parameters values. Therefore, in most applications the resulting clustering scheme requires some sort of evaluation as regards its validity. The quality of clustering can be enhanced by using a Cellular Automata Classifier for information retrieval. In this study we take the view that if cellular automata with clustering is applied to search results (query-specific clustering), then it has the potential to increase the retrieval effectiveness compared both to that of static clustering and of conventional inverted file search. We conducted a number of experiments using ten document collections and eight hierarchic clustering methods. Our results show that the effectiveness of query-specific clustering with cellular automata is indeed higher and suggest that there is scope for its application to IR.
INTRODUCTION
Locating interesting information is one of the most important tasks in Information Retrieval (IR). An IR system accepts a query from a user and responds with a set of documents. The system returns both relevant and non-relevant material and a document organization approach is applied to assist the user in finding the relevant information in the retrieved set. Generally a search engine presents the retrieved document set as a ranked list of document titles. The documents in the list are ordered by the probability of being relevant to the user's request. The highest ranked document is considered to be the most likely relevant document; the next one is slightly less likely and so on. This organizational approach can be found in almost any existing search engine. A number of alternative document organization approaches have been developed over the recent years. These approaches are normally based on visualization and presentation of some relationships among the documents, terms, or the user's query. One of such approaches is document clustering. Document clustering has been studied in the information retrieval for several decades.
Clustering is a mostly unsupervised procedure and the majority of the clustering algorithms depend on certain assumptions in order to define the subgroups present in a data set. Moreover, they may behave in a different way depending on the features of the data set and their input parameters values. Therefore, in most applications the resulting clustering scheme requires some sort of evaluation as regards its validity. Our experimental results confirm the reliability of our index showing that it performs favorably in all cases selecting independently of clustering algorithm the scheme that best fits the data under consideration
CELLULAR AUTOMATA (CA) AND FUZZY CELLULAR AUTOMATA (FCA)
A CA [4][5][6] , consists of a number of cells organized in the form of a lattice. It evolves in discrete space and time. The next state of a cell depends on its own state and the states of its neighboring cells. In a 3neighborhood dependency, the next state qi (t + 1) of a cell is assumed to be dependent only on itself and on its two neighbors (left and right) and is denoted as:
qi(t + 1) = f (qi−1(t), qi(t), qi+1(t))(1)
where, qi (t) represents the state of the i th cell at t th instant of time, f is the next state function and referred to as the rule of the automata. The decimal equivalent of the next state function, as introduced by Wolfram, is the rule number of the CA cell [8] . In a 2-state 3neighborhood CA, there are total 256 distinct next state functions.
FCA fundamentals: FCA [2,6] is a linear array of cells which evolves in time. Each cell of the array assumes a state qi, a rational value in the interval [0,1] (fuzzy states) and changes its state according to a local evolution function on its own state and the states of its two neighbors. The degree to which a cell is in fuzzy states 1 and 0 can be calculated with the membership functions. This gives more accuracy in finding the coding regions. In a FCA, the conventional Boolean functions are AND, OR, NOT. [5,6,8] , FMACA we use another vector for representation of chromosome.
Transition from one state to other: Once we formulated the transition function, we can move form one state to other. For the example 1 if initial state is P (0) = (0.80, 0.20, 0.20, 0.00) then the next states will be: Search strategy: In this research, we select the steepest descent strategy. It can make an evaluation for every solution in a neighborhood of P, then choose one which Table 1: FA rules Non-complemented rules Complemented rules can make the objective criterion function have maximal gain as a new solution. That is to say, it searches a candidate solution that can improve results furthest. Suppose neighborhood of P is Neighbour (P). The steepest descent strategy is to search a P` P` = Argmax (E(P`)-E(P)|P`∈Neighbour(P)) in Neighbour(P).
P (1) = (1.------------------------------------- ---------------------------------------- Rule
For any p∈Neighbour(P), E(P`)≥E(P) and E(P`)-E(P)>0.
( ) i i i i 2 i i i i E(S {d}) E(S ) D d D D 2 D d c 1 D − − = − − = − ⋅ + − (2) j j j j 2 j j j j E(S {d}) E(S ) D d D D 2 D d c 1 D − = + − = + ⋅ + −(3)
The text clustering algorithm based on Cellular automata based local search (TCLS) algorithm is composed of the following steps (one step):
• For one clustering partition P = (S 1 ,S 2 ,…,S K ) • Suppose max ∆ = 0, movedDoc = null, target = null, (movedDoc is the text that need to be moved, target is the target class that movedDoc moves to); For every text d∈S i in S For all j, j = 1,2…,K ∧ j ≠ i calculate j E(P) E(P ) E(P)
′ ∆ ≡ −
In P, let text d moves from S i to S j , then get the P`
Let b = argmax {∆ j E (P) > 0| j ≠ i} max max(max ,b) ∆ = ∆ , movedDoc d = , j t arg et S = • If movedDoc null ≠
(a best optimal solution has already been found), then let d move from S i to target and recalculate D i and D target • Return the partition P`
The difference between Cellular automata based local search strategy and K-Means is:
Suppose P = (S 1 ,S 2 ,…,S K ) P` = (S`1,S`2,…,S`K) ∈ Neighbour (P) the target is. It can be noted that, in every iterative of K-Means, the i D and d-c i , d∈S, i = 1,2,…,K are all need to be calculated.
That is to say, the time, space and calculate complexity in every iterative of Cellular automata based local search or K-Means are almost the same.
i i P i i d S i i K d S E(P) d (c c ) d (c c ) ′ ′ ∈ ′ ∈ ′ ′ ∆ = − − ′ ′ + − − + ∆ (4) p k E(P) ′ ∆ ≥ ∆(5)
Information retrieval system evaluation: To measure ad hoc IR effectiveness in the standard way, we need a test collection consisting of three things:
• A document collection • A test suite of information needs, expressible as queries • A set of relevance judgments, standardly a binary assessment of either relevant or non relevant for each query-document pair
EXPERIMENTAL RESULTS
In order to evaluate the classification without taking into account the position of clusters, we use the list of relevant documents supplied by NIST for TREC-7 and, for each query, select the best clusters according to the number of relevant documents they contain. Hence, we can measure how much the classification groups relevant documents. Let Lq be the list of documents constructed from the succession of the clusters ranked according to the number of relevant items they contain. The evaluations presented below have been obtained by means of the trec_eval application over the 50 queries (351-400) of TREC-7. The corpus was the one used for TREC-7 (528.155 documents extracted from the Financial Times, Federal Register, Foreign Broadcast Information Service and the LA Times. We have used the IR system developed by LIA and Bertin and Cie to obtain the lists of documents to cluster. Whenever possible, the first 1000 documents retrieved for each query have been kept to cluster them.
Number of classes:
The number of classes is defined at the start of the process. It cannot grow but can be reduced when a class empties. In next figures, the indicated numbers of clusters correspond to the values initially chosen. The documents that are not assigned to a class at the end of the classification are allocated to a new one (at the last position in the ranked list of clusters). By choosing the same number of classes from 2-13 for all queries, the levels of the average precision over all relevant documents are lower than those without classification with lists.
The differences between results indicated in Fig. 2 and 3 measure how much the above defined distance 4 We choose ni > ni+1 to favor the first ranked classes ranks the clusters. The average precision decrease is about 5% when clusters are ranked according to the computed distances and not according to the number of relevant documents they contain.
Let L q be the list of documents constructed from the succession of the clusters ranked according to the number of relevant items they contain.
Let L c be the list of documents constructed from the succession of the clusters ranked according to their distances with the query using CA classifier: By choosing the same number of classes from 5-25 for all queries, the levels of the average precision over all relevant documents are lower than those without classification with lists L q and L C ( Fig. 2 and 3). The decrease rate varies from 2.2-3% . Figure 2 and 3 show that those lists do not allow to globally improve results of the retrieval. The average precision decreases since the relevant documents that are not in the first cluster are ranked after all items of that one. The differences between results indicated in Fig. 2 and 3 measure how much the above defined distance 4 We choose ni > ni+1 to favor the first ranked classes. ranks the clusters. The average precision decrease is about 5% when clusters are ranked according to the computed distances and not according to the number of relevant documents they contain.
L c = C1×C2 ×C3×K
However, the first ranked cluster according to the distances to the queries is very often better than the next ones as shown in Fig. 4 where we have compared lists C1C2 and C2C1 . With this second list, the relative decrease of the average precision over the 145 queries equals 28% (from 0.21 to 0.089). In Fig. 5, we can see that the first ranked cluster is the best 3 times for the 5 queries indicated among 6 clusters. Figure 5 and 6 show some results obtained with 203 clusters. One can see that precision at low level of recall with lists Ln are better than those of list LC (succession of each cluster's contents). However, only list Lq allows to obtain better results than without classification. At recall 0.21, the relative increase of precision of list L5 over list LC equals 19.5% (from 0.297 to 0.3892). Figure 5 and 6 shows the results obtained by using the title field as queries and then the whole topic (list LCn with 2 clusters). Not surprisingly, the best results are obtained with longer queries even if in some cases the narrative field contains words not wanted.
CONCLUSION
We see that clustering with CA with local search can greatly improve the effectiveness of the ranked list. In fact it can be as effective as the interactive relevance feedback based on query expansion. Surprisingly this high performance can be achieved by following a very simple strategy. Given a list of clusters created by the CA based local search algorithm starts at the top of the list and follows it down examining the documents in each cluster. The experimental results proves the improvement of clustering quality with addition o f Cellular Automata.
It increases the effectiveness of retrieval by providing to users at least one cluster with a precision higher than the one obtained without using CA. We have examined, with TREC-7 corpora and queries, the impact on the classification results of the cluster numbers and of the way to browse them. We have shown that a variation of the number of clusters according to the query size improves the results. By automatically constructing a new ranked list according to the distances between clusters and queries, the precision is lower than without CA. The evaluation of other distances is in progress.
Fig. 1: Matrix representation
Clustering algorithm based on cellular automata based local search: Cellular automata based local search Based Clustering (LSC):• Give an initial clustering partition P = (S 1 ,S 2 ,…,S K ) • Run TCLS • If satisfy stop condition, then exit, else run step 2When the algorithm is running, the required space and time in every iterative are the same, the bottleneck of calculation is to calculate P E(P) E(P )
Fig. 3 :
3L C with different numbers of classes
Fig. 4 :Fig. 6 :
46Precision Precision for different browsing aspects
Dependency matrix for FCA: Rule defined in Eq. 1 should be represented as a local transition function of FCA cell. That rules(Table 1)are converted into matrix form for easier representation of chromosomes. indicates on which neighboring cells the state should depend. So cell 254 depends on its state, left neighbor and right neighbor Fig. 1. Now we represented the transition function in the form of matrix. In the case of complementExample 1: A 4-cell null boundary hybrid FCA with
the following rule <238, 254, 238, 252> (that is,
<(qi+qi+1), (qi−1+qi+qi+1), (qi+qi+1), (qi−1+qi)>)
applied from left to right, may be characterized by the
following dependency matrix.
While moving from one state to other, the
dependency matrix
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An interface for navigating clustered document sets returned by queries. R B Allen, P Obry, M Littman, Proceedings of the ACM Conference on Organizational Computing Systems. the ACM Conference on Organizational Computing SystemsMilpitas, CAAllen, R.B., P. Obry and M. Littman, 1993. An interface for navigating clustered document sets returned by queries. In: Proceedings of the ACM Conference on Organizational Computing Systems, Milpitas, CA., pp: 166-171.
The retrieval effectiveness of five clustering algorithms as a function of indexing exhaustivity. R Burgin, J. Am. Soc. Inform. Sci. 46Burgin, R., 1995. The retrieval effectiveness of five clustering algorithms as a function of indexing exhaustivity. J. Am. Soc. Inform. Sci., 46: 562-57.
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[
"Phase transitions of antiferromagnetic Ising spins on the zigzag surface of an asymmetrical Husimi lattice",
"Phase transitions of antiferromagnetic Ising spins on the zigzag surface of an asymmetrical Husimi lattice"
] |
[
"Purushottam D Gujrati ",
"Ran Huang [email protected] \nSchool of Life Sciences and Biotechnology\nState Key Laboratory of Microbial Metabolism\nShanghai Jiao Tong University\n200240ShanghaiPeople's Republic of China\n\nDepartment of Materials Technology and Engineering\nResearch Institute of Zhejiang University-Taizhou\n318000TaizhouZhejiangPeople's Republic of China\n",
"Purushottam D Gujrati \nDepartment of Physics\n\n\nDepartment of Polymer Science\nThe University of Akron\n44325AkronOHUSA\n"
] |
[
"School of Life Sciences and Biotechnology\nState Key Laboratory of Microbial Metabolism\nShanghai Jiao Tong University\n200240ShanghaiPeople's Republic of China",
"Department of Materials Technology and Engineering\nResearch Institute of Zhejiang University-Taizhou\n318000TaizhouZhejiangPeople's Republic of China",
"Department of Physics\n",
"Department of Polymer Science\nThe University of Akron\n44325AkronOHUSA"
] |
[] |
An asymmetrical two-dimensional Ising model with a zigzag surface, created by diagonally cutting a regular square lattice, has been developed to investigate the thermodynamics and phase transitions on surface by the methodology of recursive lattice, which we have previously applied to study polymers near a surface. The model retains the advantages of simple formulation and exact calculation of the conventional Bethelike lattices. An antiferromagnetic Ising model is solved on the surface of this lattice to evaluate thermal properties such as free energy, energy density and entropy, from which we have successfully identified a first-order order -disorder transition other than the spontaneous magnetization, and a secondary transition on the supercooled state indicated by the Kauzmann paradox.The diagonal interaction J PThe diagonal interaction J P between the two top sites in the triangle unit is the only competition to the nearest-neighbour J. Transition temperatures with J P ¼ +0:2, +0.4 and 0.6 are summarized in table 2. As expected, same polarity of J P and J obstructs the ordering degree with reduced T C and T K , while positive J P complies with J and gives higher Ts and larger supercooled region. A similar phenomenon of sharply reduced Ts and enlarged T C /T K ratio is also observed with J P ¼ À0:4, which implies that a vigorous competition between J P and J has the same effect of weak J to make the surface preferring supercooled state.The triplet interactionJ 0To avoid analysing the complex three body interactions, the triplet interaction term J 0 is introduced to count a compacted polarity of triangle unit, which may act similarly to the magnetic field H.Table 3summarizes the Ts with J 0 ¼ À0:3, 20.1 and 0.1. It is found that negative J 0 increases the transition temperatures or vice versa, and the thermodynamics is very sensitive to J 0 , i.e. a slight variation will
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10.1098/rsos.181500
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1809.03491
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0d66e19e9288424a5b939c89ede638a9b7b979e9
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Phase transitions of antiferromagnetic Ising spins on the zigzag surface of an asymmetrical Husimi lattice
Purushottam D Gujrati
Ran Huang [email protected]
School of Life Sciences and Biotechnology
State Key Laboratory of Microbial Metabolism
Shanghai Jiao Tong University
200240ShanghaiPeople's Republic of China
Department of Materials Technology and Engineering
Research Institute of Zhejiang University-Taizhou
318000TaizhouZhejiangPeople's Republic of China
Purushottam D Gujrati
Department of Physics
Department of Polymer Science
The University of Akron
44325AkronOHUSA
Phase transitions of antiferromagnetic Ising spins on the zigzag surface of an asymmetrical Husimi lattice
10.1098/rsos.181500Received: 16 September 2018 Accepted: 29 January 2019Research Cite this article: Huang R, Gujrati PD. 2019 Phase transitions of antiferromagnetic Ising spins on the zigzag surface of an asymmetrical Husimi lattice. R. Soc. open sci. 6: 181500. Authors for correspondence: Ran HuangSubject Category: Physics Subject Areas: thermodynamics/statistical physics Keywords: Zigzag surfaceasymmetrical Husimi latticephase transitions on surface
An asymmetrical two-dimensional Ising model with a zigzag surface, created by diagonally cutting a regular square lattice, has been developed to investigate the thermodynamics and phase transitions on surface by the methodology of recursive lattice, which we have previously applied to study polymers near a surface. The model retains the advantages of simple formulation and exact calculation of the conventional Bethelike lattices. An antiferromagnetic Ising model is solved on the surface of this lattice to evaluate thermal properties such as free energy, energy density and entropy, from which we have successfully identified a first-order order -disorder transition other than the spontaneous magnetization, and a secondary transition on the supercooled state indicated by the Kauzmann paradox.The diagonal interaction J PThe diagonal interaction J P between the two top sites in the triangle unit is the only competition to the nearest-neighbour J. Transition temperatures with J P ¼ +0:2, +0.4 and 0.6 are summarized in table 2. As expected, same polarity of J P and J obstructs the ordering degree with reduced T C and T K , while positive J P complies with J and gives higher Ts and larger supercooled region. A similar phenomenon of sharply reduced Ts and enlarged T C /T K ratio is also observed with J P ¼ À0:4, which implies that a vigorous competition between J P and J has the same effect of weak J to make the surface preferring supercooled state.The triplet interactionJ 0To avoid analysing the complex three body interactions, the triplet interaction term J 0 is introduced to count a compacted polarity of triangle unit, which may act similarly to the magnetic field H.Table 3summarizes the Ts with J 0 ¼ À0:3, 20.1 and 0.1. It is found that negative J 0 increases the transition temperatures or vice versa, and the thermodynamics is very sensitive to J 0 , i.e. a slight variation will
Introduction
The recursive lattice has been a classical methodology in statistical physics for several decades since its invention by Bethe [1] and Husimi [2]. With its impressive advantages of exact calculation and simple formulation due to the feature of recursive structure, many statistical or physical problems can seek an alternative but reliable solution from this modelling method, e.g. the travelling-salesman problem [3], the Ksatisfiability [4], the glass transition [5] and so on. For most cases, especially when involving complex structures, it is impossible to solve arbitrary models on a square lattice so one is forced to find approximated solutions. Usually, one attempts to solve the model in a mean-field approximation, while Gujrati [6] established that recursive lattice solutions are more reliable than the mean-field solutions, especially for the antiferromagnetic models.
On the other hand, the recursive feature of such lattices implies that the system is naturally homogeneous and suitable to describe bulky systems. Our group has already used the Bethe lattice to study inhomogeneous structures near a surface [7][8][9]. Here we extend to a one-/two-dimensional hybrid recursive lattice to study the phase transitions of Ising model on surface/interface, which is nonetheless still based on a symmetrical structure [10].
In this study, we have constructed an asymmetrical two-dimensional recursive lattice with a one-dimensional surface. By fractalizing the analogue of a diagonally cut regular square lattice, we can obtain a simple model with partial Husimi trees hung on a zigzag surface. Since this zigzag structure is infinite and homogeneous, it can be handled by recursive calculation [11,12], with approximating partial Husimi trees to be constant statistical contributions, which was derived from previous classical works, the exact calculation of this model appears to be feasible, like other Bethe-like models. An antiferromagnetic Ising model has been solved on the lattice, and we can locate the first-order phase transition and the Kauzmann paradox in the +1 spin system.
Lattice and model
Conventionally, the surface of a two-dimensional square lattice is just the one-dimensional line confining the bulk with a lower coordination number of 3. Imagine we cleave the lattice along square diagonals, a zigzag surface can be generated for the one-half bulk, and the basic unit of this lattice is square in the bulk and triangle on the surface. The triangle unit is surrounded by two identical triangles and a bulk square, thus a straight connection of triangle units can be adopted to represent the surface. This triangle chain is infinite and has the recursive feature required for calculation. Considering that bulk parts are merely structures hung on the zigzag line, on each triangle we may attach an infinite partial Husimi tree to represent the bulk part. The scheme of fractalizing the halved structure to be a recursive lattice is shown in figure 1.
Since this zigzag surface recursive lattice (ZSRL) has coordination of four inside the bulk and alternating two or four (averagely three) on the surface, we hope that it is a good approximation to the regular square lattice with a diagonal boundary. The site labelling is shown in figure 2: the vertices on surface triangle are labelled S and S 0 , where S 0 marks the site closer to an imaginary origin point on the surface, following the direction of recursive calculation. Note the S 0 in one triangle is S in the unit of lower level while the origin is marked as level 0. The base site that links the surface triangle and bulk tree is labelled S B .
Corresponding to the notation of neighbour interaction J, diagonal interaction J P and three spins interaction (triplet) J 0 , on the surface unit we label a bar on each term to differ them from bulk parameters. Then there are two neighbour interactions J, one diagonal interaction J P and one triplet ( J 0 ) in the Hamiltonian of one triangle unit a, which is
e a ¼ À J(S B Á S þ S B Á S 0 ) À J P Á S Á S 0 À J 0 Á S B Á S Á S 0 À H(S þ S B ),(2:1)
where H is the magnetic field applied to each spin. Note the H of site S 0 is not included in the last term, because it will be counted in the unit of next level. In equation (2.1), a negative value of J will raise the lowest energy with different neighbouring spins states, the system is thereafter the antiferromagnetic case, or vice versa. The detailed effects of energy parameters will be discussed in §4.2.
Calculation
Solutions on the surface
The general calculations of Ising model on Husimi square lattice can be found in previous works [6,13]. Here we follow the similar method of partial partition function (PPF) to achieve the solution x, which presents the probability of one site to be occupied by + spin on the surface site. The triangle unit has 2 3 ¼ 8 possible configurations, four of which are with the base site S 0 ¼ þ1 and the others are with royalsocietypublishing.org/journal/rsos R. Soc. open sci. 6: 181500 S 0 ¼ 21. We can derive two PPFs of the triangle on a level m from PPFs of the higher level m þ 1, the contribution from bulk Z B (S B ), and the local weight w(G),
Z m (þ) ¼ X 4 G¼1 Z mþ1 (S mþ1 )Z B (S B )w(G) ( 3 :1)
and
Z m (À) ¼ X 4 G¼5 Z mþ1 (S mþ1 )Z B (S B )w(G),(3:2)
where G is the index of configuration. Note that the total partition function (PF) of the entire system at the origin site S 0 is given by
Z 0 ¼ Z 0 (þ) 2 e ÀbH þ Z 0 (À) 2 e bH : (3:3)
This relationship is fundamental when we derive thermal quantities, e.g. free energy from the PPF. We introduce ratios
x m ¼ Z m (þ) Z m (þ) þ Z m (À)
and
y m ¼ Z m (À) Z m (þ) þ Z m (À) (3:4)
as the solutions on surface at level m. By denoting royalsocietypublishing.org/journal/rsos R. Soc. open sci. 6: 181500
z m (S m ) ¼ x m if S m ¼ þ1 y m if S m ¼ À1, ( (3:5)we have Z m (þ) ¼ B m x m and Z m (À) ¼ B m y m with B m ¼ Z m (þ) þ Z m (2), then with equations (3.1) and (3.2) we have z m (S m ) ¼ B mþ1 B B B m X G z mþ1 (S mþ1 )z B (S B )w(G): (3:6)
By introducing a set of polynomials
Q mþ ( x mþ1 , x B ) ¼ X 4 G¼1 z mþ1 (S mþ1 )z B (S B )w(G), Q mÀ ( y mþ1 , y B ) ¼ X 4 G¼5 z mþ1 (S mþ1 )z B (S B )w(G)
and
Q m ( x mþ1 , x B ) ¼ Q mþ ( x mþ1 , x B ) þ Q mÀ ( y mþ1 , y B ) ¼ B m B mþ1 B B , 9 > > > > > > > > > > = > > > > > > > > > > ; (3:7)
we can derive the solution x m at the mth level as a function of the ratio x mþ1 on higher level and the bulk solution x B ,
x m ¼ Q mþ ( x mþ1 , x B ) Q m ( x mþ1 , x B ) : (3:8)
Taking x B as the solution on a joint site between surface triangle and bulk square, imagine we start the recursive calculation from a position deep in the bulk, then approach the thermodynamic contributions to the surface, thereby x B is simply the fix-point solution of Husimi lattice. In this way, we can count the contribution of the infinite bulk tree as a constant input and focus on the recursive approach of x along the surface. It should be addressed that this set-up of constant x B ignores the possible backward effect from surface to the bulk, which may bias the numerical value of x B . However, this approximation is acceptable to make the model simple and solvable.
With a constant x B , by equation (3.8) we can recursively calculate the solution on surface for a number of iterations until we reach a fix-point solution. The form of equation (3.8) implies that, regardless of the system being antiferro-or ferromagnetic, a uniform 1-cycle solution is expected on the surface, while antiferromagnetic Ising model presents an alternating 2-cycle solution as the ordered state [13]. A negative neighbour interaction J prefers to anti-align the S versus S B and S 0 versus S B pair. Unless we set the diagonal interaction J P also to be negative and large enough to outweigh J, the system will prefer the same spin states on S and S 0 .
Recall that bulk solutions can be either a 1-cycle solution to present the metastable state, or a set of 2cycle solutions as the ordered state [13]. Taking the 1-cycle x B , which is usually 0.5 with H ¼ 0, we will obtain a fixed x ¼ 0:5 also corresponding to a metastable surface; for the 2-cycle solutions, we can substitute either fixed solution as x B , and it will affect the calculation on x to bias the surface fix-point solution to 0 or 1. For example, due to antiferromagnetism, at T ¼ 0 with H ¼ 0, we will have 0 and 1 solutions in the bulk, then if we take x B ¼ 0, the surface solution will be captured as x ¼ 1, and vice versa. Our results confirmed that the thermodynamics calculated based on either selection are identical. royalsocietypublishing.org/journal/rsos R. Soc. open sci. 6: 181500 Figure 3 shows the ordered solution on the surface with comparison of 2-cycle bulk solutions, the energy terms are set as J ¼ 21, J ¼ À1 and others to be 0. The metastable 0.5 solution is not presented in figure 3. For convenience, we are still going to call the stable solution and corresponding thermodynamics '2-cycle' in the following discussion, although both stable and metastable x are actually in 1-cycle form for ZSRL.
It can be observed that at high T all the solutions are 0.5, that spins anywhere have an equal probability to be +1. Below the Curie point, the spins undergoes self-magnetization (even without an external field H), and the neighbouring spins prefer the þ/2 alternating arrangement, which is the lowest energy state. The value of 2-cycle x B of bulk Ising model is referred from previous reports [10,13]. Along with the bulk solutions, the spins on surface also present preference on unitary direction under the bulk Curie point; however, we will show that the actual phase transition occurs far below the T C (bulk).
One more thing should be addressed here. We know from exact solutions that for the square lattice the self-magnetization occurs at T C ¼ 2= log 1 þ ffiffi ffi 2 p 2:27. In the recursive lattice (RL), the bulk transition temperature is 2.8 as shown in figure 3, which is not all that close to the known T C , but closer than the results of mean-field theory. We believe this overestimate is acceptable. The aspect that the RL method provides results between the 'real' exact solution and the mean-field theory seems to be generic. Similar results had also been observed in other works, e.g. the three-dimensional cube lattice [13]. However, the nature of this overestimate has not been investigated yet.
Free energy calculation
We follow the Gujrati trick to calculate the free energy of a local area by recursive approach [6,13]. The scheme will be briefly described here, as shown in figure 4. Owing to the uniform structure and solution on the surface, we can randomly select a site as the origin point O. The local area is chosen to be two triangle units joint on the origin. Imagine we cut off two sub-trees contributing to the point A and A 0 then rejoin them together to make an identical but smaller ZSRL, and hook up two partial bulk trees hung on B and B 0 together to make a full Husimi square lattice; therefore we have F total ¼ F local þ royalsocietypublishing.org/journal/rsos R. Soc. open sci. 6: 181500
F bulk þ F smaller .
With the definition of Helmholtz free energy F ¼ 2k B T log Z, the F per site in the local area is given as
F site ¼ À 1 3 T log Z 0 Z 1 Á Z B , ( 3 :9)
since there are three full sites in the local area (on whole site S 0 and four half-shared sites S 1 , S B ) and k B is normalized to be 1. Recalling equations (3.1)-(3.3), it is easy to break down the local free energy into a function of PPFs. Then from the relations between Z(+), Q and x derived in the previous section, we can calculate the above function as
F site ¼ À 1 3 T log Q 0 2 x B 2 e bH þ (1 À x B ) 2 e ÀbH , ( 3 :10)
then the entropy and energy ( per site) can be easily achieved by S ¼ 2dF/dT and E ¼ F þ TS.
Results and discussion
The thermodynamics and transitions on the surface
The thermal behaviours of the reference set-up with J ¼ 21, J ¼ À1 and other parameters to be 0 is shown in figure 5. The free energy of two solutions differ at T ¼ 2.8. Usually this bifurcation indicates the spontaneous magnetization (Curie point) in normal spin models. However, here on the surface the magnetization does not bend the free energy of alternating spins arrangement below the disordered 0.5 solution but upward, which is unphysical with a sharply increased entropy. Therefore, the 2-cycle solution at this point is only a numerical existence and the system must follow the curve of 1-cycle solution, until a cross point is reached at T ¼ 1.33, where the system makes a transition from the amorphous to the crystalline ordering, i.e. the order-disorder transition at critical temperature T C . Unlike the conventional self-magnetization where entropy is continuous, the entropy here must undergo a discontinuous jump like a first-order transition.
In this way, we may conclude that the critical order-disorder transition on the surface is not the Curie point, but much lower than that of the bulk. This can be understood as that, due to the asymmetry and Below the real T C (bulk) and the apparent Curie point on the surface, the spins on surface can maintain their 'melted' state in a deep temperature region, and this does not fall into the concept of supercooled liquid, since the corresponding crystal state in particular temperature region is unfeasible. Intuitively, this phenomenon may easily refer to the analogue of ice premelting [14]; nonetheless, this work only focuses on the preliminary modelling and we are not going to further expand this point.
For the 1-cycle solution below T C , the system can undergo cooling without any phase transition and shares features of a supercooled state. With further cooling process, the entropy of 2-cycle solution approaches zero, while the entropy of 1-cycle becomes negative at T ¼ 0.69, which is the Kauzmann paradox at T K [13].
The result indicates that a free surface dramatically decreases the transition temperatures. Figure 6 shows the free energy comparison of Husimi bulk system and ZSRL. This observation agrees with others' work on phase transitions on the surface or thin film, for example, the glass transition of polymer system in confined geometry [7][8][9]15]. The fact that similar reduction can be observed in our monoatomic model implies that the lower transition temperature on surface/thin film basically originates from the dimension reduction and less interaction constraints.
The effects of secondary energy parameters
Besides mathematical curiosity, there is always a primary expectation to describe and study real systems for establishing a theoretical model. In this way, other than the interaction J and J between the nearest neighbours, further interactions are included to make the model more versatile to describe various systems, as shown in equation (2.1). With adjustable combinations of energy parameters set-up, we can manipulate the thermal behaviour of system and the transition temperatures to better match the reality in particular situations. In other words, more adjustable parameters may serve as useful tools for the theoretical modelling to be correlated with experimental parameters. By the control of J ¼ 21 and other parameters to be 0 inside the bulk, we explored the effects of J, J P and J 0 . The secondary parameters are expected to either comply or compete with J, and some interesting phase behaviours are found with particular set-ups.
The surface nearest-neighbour interaction J
Considering the feature of asymmetry on the boundary, we set the nearest-neighbour interaction on the surface, denoted as J, to be differentiated and adjustable from the J, to make the model capable to describe some particular situations, e.g. the surface tension. The transition temperatures with J ¼ 21, other parameters to be 0, and J ¼ À0:5, 20.7, 20.9, 21.1, 21.3 and 21.5 are shown in table 1. As negative J complies with antiferromagnetic set-up J ¼ 21, the larger absolute value of J makes the system more stable and increases both critical and ideal glass transition temperature; and the relative length of supercooled region, indicated by the ratio T C /T K , gradually increases. However, for the J ¼ À0:5 decreased from 20.7, both T C and T K fall while the latter changes more dramatically, royalsocietypublishing.org/journal/rsos R. Soc. open sci. 6: 181500 raising a much larger supercooled region. This critical phenomenon observes that a loosened surface with too weak interactions is easier to be supercooled.
&
2019 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.
Figure 1 .
1The diagonal cutting on a regular square lattice to obtain a zigzag surface, and the construction of recursive lattice on the zigzag surface. The thick line presents the surface edge.
Figure 2 .
2The site labelling and calculation scheme on ZSRL. Starting from a random point on the surface, the recursive approach is to proceed to an imaginary origin.
Figure 3 .
3Solution on the surface and its comparison to the 2-cycle bulk solutions.
Figure 4 .
4The cutting scheme around the surface origin O for free energy calculation.
Figure 5 .
5The thermodynamics behaviours on the surface with J ¼ 21, J ¼ À1 and other parameters to be 0. royalsocietypublishing.org/journal/rsos R. Soc. open sci. 6: 181500 smaller coordination numbers on the surface, spins on the outer layer are less dragged by the bulk portions.
Figure 6 .
6The free energy comparison of Husimi bulk system and ZSRL with J ¼ 21, J ¼ À1 and other parameters to be 0.
Figure 7 .Figure 8 .
78A special case of ZSRL with J ¼ À1 and J 0 ¼ 0.A special case of ZSRL with J ¼ À1 and J 0 ¼ 20.5: (a) the free energy of 1-and 2-cycle solutions; (b) the enlargement of free energy around T ¼ 2.8. royalsocietypublishing.org/journal/rsos R. Soc. open sci. 6: 181500
Table 1 .Table 2 .Table 3 .
123The transition temperature variations with different J. The transition temperature variations with different J P . The transition temperature variations with different J 0 . royalsocietypublishing.org/journal/rsos R. Soc. open sci. 6: 181500J
T C
T K
T C /T K
20.5
0.85
0.40
2.13
20.7
1.10
0.60
1.83
20.9
1.20
0.63
1.90
21
1.33
0.69
1.93
21.1
1.50
0.77
1.95
21.3
1.80
0.90
2.00
21.5
2.10
1.01
2.08
J P
T C
T K
T C /T K
20.4
0.85
0.40
2.13
20.2
1.10
0.60
1.83
0
1.33
0.69
1.93
0.2
1.54
0.75
2.05
0.4
1.73
0.85
2.04
0.6
1.90
0.92
2.07
J 0
T C
T K
T C /T K
20.3
1.90
0.76
2.50
20.1
1.55
0.70
2.21
0
1.33
0.69
1.93
0.1
1.05
0.69
1.52
dramatically change the overall thermal behaviours. The set-up with absolute value of J 0 larger than 0.3 can converge the system to some bizarre states, which will be detailed in the following section.Two special cases with various J 0Since the J 0 plays a dominant role in ZSRL, the value of J 0 is limited to be relatively small. Abnormal behaviours can be observed with J 0 ¼ 0:3 and J 0 ¼ À0:5. In the first case, as shown infigure 7, the 2cycle solution will never have a lower free energy than 1-cycle. Even it has a lower entropy at low temperature, the order -disorder transition cannot be located since there is no cross point. On the other hand, the 1-cycle solution still undergoes Kauzmann paradox. Therefore, the only reasonable understanding is that, under this condition, regardless of the thermal state in the bulk, crystal state is not achievable on the surface. With temperature decreasing, we will only have supercooled liquid and the subsequent glassy state.The system goes to another extremity for J 0 ¼ À0:5:figure 8shows that, below the free energy differentiating point, the 2-cycle solution consistently becomes more stable than 1-cycle, like the normal behaviour of regular antiferromagnetic Ising models, and thereby the order -disorder transition becomes spontaneous magnetization; below T C the system can either be in crystal ordering or in the metastable supercooled state.ConclusionA zigzag surface recursive lattice has been constructed to describe a regular square lattice with one-dimensional boundary. The zigzag structure is taken as a surface assembled by triangle units, and halved Husimi trees are hung on the triangle units to represent the bulk portions. With the coordination number of four inside the bulk and average three on the surface, this model is considered to be a good approximation to a regular square lattice with surface. The antiferromagnetic Ising model is solved on the lattice, with the constant x B retrieved from regular Husimi lattice to count the bulk contribution, and a uniform solution is obtained on the surface to represent the ordered state. Then the thermodynamics of local area around the origin on surface can be derived by conventional techniques from x and x B . The transition temperatures are found to be dramatically reduced on the surface compared with in the bulk, and this reduction is simply due to the dimension downgrade and less interaction constrains on the surface.The effects of various interaction energy parameters other than nearest-neighbour J are investigated. These interactions could either increase or decrease the stability of system and change the transition temperatures according to the Hamiltonian. In addition to the effect of parameters, we have found several interesting behaviours with particular energy set-up.Data accessibility. The code and data are available from the Dryad Digital Repository: https://doi.org/10.5061/dryad. 99t5k4s[16].Authors' contributions. R.H. designed the model, did the programming, calculation and analysis, and wrote the manuscript. P.D.G. directed the research, derived the theoretical formulation and calculation method, edited the manuscript and approved its submission.Competing interests. We declare no competing interests. Funding. This work is financially supported by the National Natural Science Foundation of China (11505110), the China Postdoctoral Science Foundation (2016M591666) and the Taizhou Municipal Science and Technology Program (1701gy15 and 1801gy16).
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|
[] |
[
"Word Embedding Based Correlation Model for Question/Answer Matching",
"Word Embedding Based Correlation Model for Question/Answer Matching"
] |
[
"Yikang Shen \nSino-French Engineer School\nBeihang University\nChina\n",
"Wenge Rong \nSchool of Computer Science and Engineering\nBeihang University\nChina\n",
"Nan Jiang \nSchool of Computer Science and Engineering\nBeihang University\nChina\n",
"Baolin Peng [email protected] \nDepartment of System Engineering and Engineering Management\nThe Chinese University of Hong Kong\nChina\n",
"Jie Tang [email protected] \nDepartment of Computer Science and Technology\nTsinghua University\nChina\n",
"Zhang Xiong [email protected] \nSchool of Computer Science and Engineering\nBeihang University\nChina\n"
] |
[
"Sino-French Engineer School\nBeihang University\nChina",
"School of Computer Science and Engineering\nBeihang University\nChina",
"School of Computer Science and Engineering\nBeihang University\nChina",
"Department of System Engineering and Engineering Management\nThe Chinese University of Hong Kong\nChina",
"Department of Computer Science and Technology\nTsinghua University\nChina",
"School of Computer Science and Engineering\nBeihang University\nChina"
] |
[] |
The large scale of Q&A archives accumulated in community based question answering (CQA) servivces are important information and knowledge resource on the web. Question and answer matching task has been attached much importance to for its ability to reuse knowledge stored in these systems: it can be useful in enhancing user experience with recurrent questions. In this paper, a Word Embedding based Correlation (WEC) model is proposed by integrating advantages of both the translation model and word embedding. Given a random pair of words, WEC can score their co-occurrence probability in Q&A pairs, while it can also leverage the continuity and smoothness of continuous space word representation to deal with new pairs of words that are rare in the training parallel text. An experimental study on Yahoo! Answers dataset and Baidu Zhidao dataset shows this new method's promising potential.
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10.1609/aaai.v31i1.11002
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[
"https://arxiv.org/pdf/1511.04646v2.pdf"
] | 11,455,429 |
1511.04646
|
72107f7d4ef9a9ab331c614f40e799bb973fa2ad
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Word Embedding Based Correlation Model for Question/Answer Matching
Yikang Shen
Sino-French Engineer School
Beihang University
China
Wenge Rong
School of Computer Science and Engineering
Beihang University
China
Nan Jiang
School of Computer Science and Engineering
Beihang University
China
Baolin Peng [email protected]
Department of System Engineering and Engineering Management
The Chinese University of Hong Kong
China
Jie Tang [email protected]
Department of Computer Science and Technology
Tsinghua University
China
Zhang Xiong [email protected]
School of Computer Science and Engineering
Beihang University
China
Word Embedding Based Correlation Model for Question/Answer Matching
The large scale of Q&A archives accumulated in community based question answering (CQA) servivces are important information and knowledge resource on the web. Question and answer matching task has been attached much importance to for its ability to reuse knowledge stored in these systems: it can be useful in enhancing user experience with recurrent questions. In this paper, a Word Embedding based Correlation (WEC) model is proposed by integrating advantages of both the translation model and word embedding. Given a random pair of words, WEC can score their co-occurrence probability in Q&A pairs, while it can also leverage the continuity and smoothness of continuous space word representation to deal with new pairs of words that are rare in the training parallel text. An experimental study on Yahoo! Answers dataset and Baidu Zhidao dataset shows this new method's promising potential.
Introduction
Community Question Answering (CQA) services are websites that enable users to share knowledge by asking and answering different kinds of questions. Over the last decade, websites, such as Yahoo! Answers, Baidu Zhidao, Quora, and Zhihu, have accumulated large scale question and answer (Q&A) archives, which are usually organised as a question with a list of candidate answers and associated with metadata including user tagged subject categories, answer popularity votes, and selected correct answer (Zhou et al. 2015). This user-generated content is an important information repository on the web and makes Q&A archives invaluable resources for various tasks such as question-answering (Jeon et al. 2005;Xue et al. 2008;2016) and knowledge mining (Adamic et al. 2008).
To make better use of information stored in CQA systems, a fundamental task is to properly matching potential candidate answers to the question, since many questions recur enough to allow for at least a few new questions to be answered by past materials (Shtok et al. 2012). There are several challenges for this task among which the lexical gap or lexical chasm between the question and candidate answers is a difficult one (Berger et al. 2000). Lexical gap describes the distance between dissimilar but potentiality related words Copyright c 2017, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. in questions and answers. For example, given the question "What is the fastest car in the world?", a good answer might be "The Jaguar XJ220 is the dearest, fastest and most sought after car on the planet." This Q&A pair share no more than 4 words in common, including "the" and "is", but they are strongly associated by synonyms, hyponyms, or other weaker semantic associations (Yih et al. 2014). Due to the heterogeneity of question and answer, the lexical gap is more significant in Q&A matching task than other paraphrases detection task or information retrieval task.
A possible approach for the lexical gap problem is to employ translation model, which will leverage the Q&A pairs to learn the semantically related words (Jeon et al. 2005;Xue et al. 2008;Zhou et al. 2012;Guzmán et al. 2016). The basic assumption is that Q&A pairs are "parallel text" and relationship between words (or phrases) can be established through word-to-word (or phrase-to-phrase) translation probabilities by representing words in a discrete space. In spite of its wide use in many natural language processing tasks, discrete space representation has two majors disadvantages: 1) the curse of dimensionality (Bengio et al. 2003), for a natural language with a vocabulary V of size N , we need to learn at most N N word-to-word translation probabilities; 2) the generalisation structure is not obvious: it is difficult to estimate the probability of exact word if they are rare in the training parallel text (Zou et al. 2013).
An alternative method is to use a semantic-based model. Some work proposed to learn the latent topics aligned across the question-answer pairs to bridge the lexical gap, with the assumption that a question and its answer should share similar topic distribution (Cai et al. 2011;Ji et al. 2012). Recently, inspired by the success of word embedding, some papers propose to leverage the advantage of the vector representation of words to overcome the lexical gap (Shen et al. 2015;Zhou et al. 2015) by using similarity of word vector to represent the word-to-word relation. In other words, this method calculates Q&A matching probability based on semantic similarities between words. Because local smoothness properties of continuous space word representations, generalisation can be obtain more easily (Bengio et al. 2003). However question and answers are heterogeneous in many aspects, semantic similarities can be weak between questions and answers (Zhou et al. 2015).
Inspired by the pros and cons of the translation model and semantic model, in this paper we propose a Word Embedding Correlation (WEC) model, which integrates the advantages of both the translation model (Jeon et al. 2005;Xue et al. 2008) and word embedding (Bengio et al. 2003;Mikolov et al. 2013a;). In this model, a word-level correlations function C(q i , a j ) is designed to capture the word-to-word relation. Similar to traditional translation probability, this function calculates words co-occurrence probability in parallel text (Q&A pairs). Instead of using word's discrete representation and maintaining a big and sparse translation probability matrix, we map input words q i and a j into vectors and use a low dimension dense translation matrix M to capture the co-occurrence relationship of words. If co-occurrences of exact words are rare in the training parallel text, C(q i , a j ) can also estimate their correlations strength because of the local smoothness properties of continuous space word representations (Bengio et al. 2003). Based on the word-level correlations function, we propose a sentence-level correlations functions C(q, a) to calculate the relevance between question and answer. This sentence-level correlation function also makes it possible to learn the translation matrix M directly from parallel corpus. Furthermore, we combine our model with convolution neural network (CNN) (LeCun et al. 1998;Shen et al. 2015) to integrate both lexical and syntactical information stored in Q&A to estimate the matching probability. Experimental study on Yahoo! Answers dataset and Baidu Zhidao dataset has shown WEC model's potential.
The proposed model will be illustrated in detail in Section 2. Section 3 will elaborate on the experimental study. Section 4 will present related work in solving the CQA matching problem and Section 5 concludes the paper and highlights possible future research directions.
Methodology
Problem Definition Given a question q = q 1 ...q n , where q i is the i-th word in the question, and a set of candidate answers A = {a 1 , a 2 , ..., a n }, where a j = a j 1 ...a j m and a j k is the k-th word in j-th candidate answer, the goal is to identify the most relevant answer a best .
In order to solve this problem, we calculate the matching probability between q and each answer a i , and then rank candidate answers by their matching probabilities, which are calculated through three steps: 1) words in questions and answers are represented by vectors in a continuous space; 2) word-to-word correlation score is calculated by using a word-level correlation function; 3) Q&A matching probability is obtained by employing a phrase-level correlation function. Furthermore, we also propose to incorporate the proposed WEC model with convolution neural network (CNN) to achieve a better matching precision.
Word Embedding
In order to properly represent words in a continuous space, the idea of a neural language model (Bengio et al. 2003) is employed to enable jointly learn embedding of words into an n-dimensional vector space and to use these vectors to predict how likely a word is given its context. Skip-gram model (Mikolov et al. 2013a) is a widely used approach to compute such an embedding. When skip-gram networks are optimised via gradient ascent, the derivatives modify the word embedding matrix L ∈ R (n×|V |) , where |V | is the size of the vocabulary. The word vectors inside the embedding matrix capture distributional syntactic and semantic information via the word co-occurrence statistics (Bengio et al. 2003;Mikolov et al. 2013a). Once this matrix is learned on an unlabelled corpus, it can be used for subsequent tasks by using each word's vector v w (a column in L) to represent that word.
Word Embedding based Correlation (WEC) Model
Word-level Correlation Function In this paper we try to discover a correlation scoring function that uses word embedding as input and can also model the co-occurrence of words at the same time. In order to achieve this goal, we use a translation matrix M to transform words in the answer into words in the question. Given a pair of words (q i , a j ), their WEC scoring function is defined as:
C(q i , a j ) = cos < v qi , Mv aj >= v T qi ||v qi || Mv aj ||Mv aj ||(1)
where v qi and v aj represent q i and a j 's d-dimensional word embedding vectors; || · || is Euclidean norm; correlations matrix M ∈ R d×d . M is called translation matrix, because it maps word in the answer into a possible correlated word in the question. Then the cosine function will be used to capture the semantic similarity between origin words in the question and the mapped words. Previous cosine similarity is a special case of WEC Scoring Function when M is set to identity matrix. Meanwhile, C(q i , a j ) does not necessarily equal to C(a j , q i ), because the probability of q i existing in question and a j existing in answer may not equal to the probability of a j existing in question and q i existing in answer.
Sentence-level Correlation Function Based on the wordlevel correlation function, we further propose the sentencelevel correlation function, which integrates word-to-word correlation scores into the Q&A pair correlation score. For a Q&A pair (q, a), their correlation score is defined as:
C(q, a) = 1 |a| j max i C(q i , a j )(2)
where |a| represents the length of answer a, C(q i , a j ) is the correlations score of the i-th word in question and the jth word in the answer. The max-operator choose one most related word in question for each word in answer. Sentence level correlation score is calculated by averaging selected word-level scores. According to our experiments, this max-average function perform better than simply average all word-level correlation score, and maximizing more efficiently.
WEC + Convolution Neural Networks (CNN)
WEC is based on the bag-of-word schema, which puts the syntactical information aside, e.g., the word sequence information. As such in worst cases, two phrases may have same bag-of-words representation, their real meaning could be completely opposite (Socher et al. 2011).
To overcome this limitation, several approaches have been proposed and one possible solution is to use the convolution neural network (CNN) model (He et al. 2015;Mou et al. 2016). (Kalchbrenner et al. 2014) proposed that the convolutional and dynamic pooling layer in CNN can relate phrases far apart in the input sentence. In (Shen et al. 2015), S+CNN model is proposed for Q&A matching to integrate both syntactical and lexical information to estimate the matching probability. In their model, the input Q&A pair is transformed into a similarity matrix S, generated through function:
S ij = cos(q i mod |q| , a j mod |a| )(3)
where |q| and |a| are the respective lengths of question and answer, S is a n f × m f fix size matrix, and n f and m f are the number of rows and columns respectively. Thus, the maximum length of questions and answers should be limited to be no longer than n f and m f . Then the similarity matrix is used as an input of a CNN instead of an image in (LeCun et al. 1998), the output of the CNN is the matching score of the Q&A pair. Fig. 1 shows the architecture of the employed CNN. Figure 1: Architecture of CNN. It has two convolution layers, C1 and C2, each convolution layer is followed by a maxpooling layer, P1 and P2, and fully connected layer F. The input matrix (S or C) is an n f × m f fixed size matrix.
Instead of word embedding cosine similarity, the wordlevel correlation scores in WEC can be used in the formation of the input matrix of CNN. Similar to the S-matrix, we propose a correlations matrix C, generated through function:
C ij = C(q i mod |q| , a j mod |a| )(4)
where C is an n f × m f fixed size matrix, and used as the input matrix of CNN. In this way, we obtain a new combination model, called the WEC+CNN model. The complete training process comprises two supervised pre-training steps and a supervised fine-tuning step. In the first supervised pre-training step, we maximize the margin of the output of WEC function to pre-train M . In the second supervised pre-training step, we fix M , then maximize the margin of the output of CNN to train the CNN part. In the fine-tuning step, we maximize the margin of the output of CNN to fine-tune all parameters in WEC and CNN.
Experimental Study Dataset
To evaluate the proposed WEC model, two datasets Yahoo! Answer and Baidu Zhidao are employed in this research. The dataset from Yahoo! Answers is available in Yahoo! Webscope 1 , including a large number of questions and their corresponding answers. In addition, the corpus contains a small amount of meta data, such as, which answer was selected as the best answer, and the category and sub-category assigned to this question (Surdeanu et al. 2008). To validate the proposed WEC model, we generate three different sub-datasets. As shown in Table 1, the first subset contains 53,261 questions which are categorized as being about travel. The second subset contains 57,576 questions that are categorized as being about relationships. The third subset contains 66,129 questions that are categorized as being about finance. Because the CNN model needs to limit the maximum length of questions and answers, all selected questions are no longer than 50 words, selected answers a no longer than 100 words. Thus, for the Yahoo! Answer dataset, n f = 50 and m f = 100. We also limit the minimum length of an answer to 5 words, to avoid answers like "Yes", "It doesn't work" or simply a URL. More than half of the questions and answers in Yahoo! Answers satisfy these limitations.
The
Experimental Settings
Evaluation Metrics We use the same evaluation method employed by (Lu and Li 2013;Shen et al. 2015) to evaluate the accuracy of matching questions and answers. A set of candidate answers is created with size 6 (one positive + five negative) for each question in the test data. We compare the performance of our approach in ranking quality of the six candidate answers against that of others baselines. Discounted cumulative gain (DCG) (Rvelin and Inen 2000) is employed to evaluate the ranking quality. The premise of DCG is that highly relevant documents appearing lower in a ranking list should be penalised as the graded relevance value is reduced logarithmically proportional to the position of the result. DCG accumulated at a particular rank position p is defined as:
DCG@p = rel 1 + p i=2 rel i log 2 (i)(5)
where the best answer rel = 1, for other answers rel = 0.
We choose DCG@1 to evaluate the precision of choosing the best answer and DCG@6 to evaluate the quality of ranking. Baseline We compare WEC model against Translation model (
S q,a = t∈q ((1 − λ)(β w∈a P (t|w)P ml (w|a) + (1 − β)P ml (t|a) + λP ml (t|Coll)))(7)
where P (t|w) denotes the probability that question word t is the translation of answer word w. IBM translation model 1 is used to learn P (t|w) with answer as the source and question as the target. In (Shen et al. 2015), word vector cosine similarity is used as word translation probabilities in TM and TRLM. Their results are also included in the experimental study.
Hyperparameter To train the skip-gram model, we use the hyper-parameters recommended in (Mikolov et al. 2013a): the dimension of word embedding is set to 500, and the window size is 10.
CNN model contains two convolution layers labelled as C1 and C2, each convolution layer is followed by a maxpooling layers P1 and P2, and fully connected layer F. Each unit in a convolution layer is connected to a 5 × 5 neighbourhood in the input. Each unit in max-pooling layer is connected to a 2 × 2 neighbourhood in the corresponding feature map in C1. Layer C1 and M1 contains 20 feature maps each. Layer C2 and M2 contains 50 feature maps each. The fully connected layer contains 500 units and is fully connected to M2. Table 2 shows the Q&A matching performance of WEC based methods, translation probability based methods and traditional retrieval methods on the Yahoo! Answer dataset.
Experiment Results
For top candidate answer precision, WEC slightly outperforms the translation probability based models. For candidate answer ranking qualities, WEC outperforms TRLM and TM. By adding CNN into the model, WEC+CNN outperforms all other models. It is possible to interpret that WEC model can perform better than TRLM and TM model, but merely using lexical level information limits its ability in selecting the best answer. Thus, WEC+CNN is able to improve the result by adding syntactical information into the model. Table 3 shows the Q&A matching performance of different approaches on Baidu Zhidao dataset. We find that WEC and WEC+CNN outperform all other models. Furthermore, IBM-1 based models outperform cos-similarity based models. This is possibly is due to the heterogeneity of question and answer, since both WEC and IBM translation model 1 can directly model the word-to-word co-occurrence probability instead of semantic similarity. On both datasets, traditional retrieval models obtain the worst result because they suffer from the lexical gap problem.
Examples
To better understand the behavior of WEC, we illustrated a number of example translations from answer words to a given question word in Table 4. Three different methods, e.g., WEC, cos-similarity, IBM Model 1, are employed to estimate the translation probabilities. Interestingly, these methods provide semantically related target words with different characters. To clarify this difference, consider the word "where". IBM model 1 provides "hamlets", "prefecture", "foxborough" and "berea". They are rarely appeared (comparing with "middle" and "southern") generic or specific name for settlement. Cos-similarity provides "what", "how", and "which". They are question words like "where". WEC model provides "middle", "southern", "southeastern", and "situated". These words are semantically related to the target word, and likely to appear in a suitable answer. The difference between three models reflect differences in their learning processes.
IBM translation model 1 leverage the co-occurrence of words in parallel corpus to learn translation probabilities. The sum of translation probabilities for a question word equal to 1. Therefore, answer words with low document frequency get relatively higher translation probabilities with certain question words, because these words co-occur with a smell set of question words, hence its translation probabilities concentrate on this set of words. Skip-gram model learn embedding of words into an n-dimensional vector and to use these vectors to predict how likely a word is given its context. Thus, the cos-similarity captures the probability that a pair of words appear with similar contexts. WEC tries to combines advantages of IBM model 1 and word embedding. The word vector capture distribu-tional syntactic and semantic information via the word cooccurrence statistics (Bengio et al. 2003;Mikolov et al. 2013a). Word-to-word correlation score are learned via maximizing the result of sentence-level correlation function Eq. (2). Meanwhile, WEC do not normalize the correlation score, which is more feasible for QA tasks.
The sentence-level correlation function Eq.
(2) is also capable of identifying important relevance between questions and answers. As shown in the Fig. 2, for each word in an answer, the max operator in Eq. (2) chooses the most relevant word in question, based on the correlation score calculated by Eq. (1). Interestingly, both words "try" and "looking" are linked with "find", while the relevance between "try" and "find" is more obscure than the obvious relevance between "looking" and "find". Although, the link between "a" and "the" is inappropriate in this context, but in many other contexts, this relationship may be correct. The relation between words given a certain context is left for future work.
Related Work Lexical Gap Problem in CQA
To fully use Q&A archives in CQA systems, there are two important tasks for a newly submitted question, including question retrieval, which focuses on matching new questions with archived questions in CQA systems (Jeon et al. 2005;Xue et al. 2008;Cao et al. 2010;Cai et al. 2011;Zhou et al. 2015) and answer locating, which focuses finding a potentially suitable answer within a collection of candidate answers (Berger et al. 2000;Surdeanu et al. 2008;Lu and Li 2013;Shen et al. 2015). One major challenge in both tasks is the lexical gap (chasm) problem (Chen et al. 2016). Potential solutions to overcome this difficulty include 1) query expansion, 2) statistical translation, 3) latent variable models (Berger et al. 2000).
Much importance has been attached to statistical translation in the literature. Classical methods include translation model (TM) (Jeon et al. 2005) and translation language model (TRLM) (Xue et al. 2008). Both use IBM translation model 1 to learn the translation probabilities between question and answer words. Apart from word-level translation, phrase-level translation for question and answer retrieval has also achieved promising results (Cai et al. 2011).
Latent variable models also attract much research in recent years. Proposals have been made to learn the latent topics aligned across the question-answer pairs to bridge the lexical gap, on the assumption that question and its answer should share a similar topic distribution (Cai et al. 2011;Ji et al. 2012). Furthermore, inspired by the recent success of word embedding, several approaches have been proposed to leverage the advantages of the vector representation to overcome the lexical gap (Shen et al. 2015;Zhou et al. 2015).
Different from previous models, our work aims at combining the idea of both statistical translation and latent variable model. We proposed a latent variable model, but parameters are learned to model the word-level translation probabilities. As a result, we can keep the generalisability of latent variable model, while achieving better precision than a brutal statistical translation model and provide more reasonable results in word-to-word correlation examples.
Translation Matrix
Distributed representations for words have proven its success in many domain applications. Its main advantage is that the representations of similar words are close in the vector space, which makes generalisation to novel patterns easier and model estimation more robust. Successful follow-up work includes application to statistical language modelling (Bengio et al. 2003;Mikolov et al. 2013a).
Inspired by vector representation of words, the translation matrix has been proposed to map vector representation x from one language space to another language space, using cosine similarity as a distance metric (Mikolov et al. 2013b). Our word-level WEC model uses the same translation functions to map vector x from answer semantic space to question semantic space. We further propose a sentence-level WEC model to calculate the Q&A matching probability, and a method to learn the translation matrix through maximising the matching accuracy in a parallel Q&A corpus.
Similarly neural tensor network (NTN) is also implemented to model relational information Qiu and Huang 2015). A tensor matrix is employed to seize the relationship between vectors. The NTN's main advantage is that it can relate two inputs multiplicatively instead of only implicitly through non-linearity as with standard neural networks where the entity vectors are simply concatenated ). Our model is conceptually similar to NTN and use a translation matrix to model the word-to-word relation in Q&A pairs. Similar to NTN, the translation matrix in our model makes it possible to explicitly relate the two inputs, and cos in Eq. (1) adds non-linearity.
Conclusion and Future Work
This paper presents a new approach for Q&A matching in CQA services. In order to solve the lexical gap between question and answer, a word embedding based correlation (WEC) model is proposed, where the co-occurrence relation between words in parallel text is represented as a matrix (or a set of matrices). Given a random pair of words, WEC model can score their co-occurrence probability in Q&A pairs like the previous translation model based approach. And it also leverages the continuity and smoothness of continuous space word representation to deal with new pairs of words that are rare in the training parallel text. Our experiments show that WEC and WEC+CNN outperform state-of-the-art models.
There are several interesting directions which deserve further exploration in the future. It is possible to apply this model in question-question matching tasks, or multilanguage question retrieval task. It is also interesting to explore the possibility of using this approach to solve other parallel detection problems (e.g., comment selection on a given tweet).
Baidu Zhidao dataset is provided in(Shen et al. 2015), and contains 99,909 questions and their best answers. Following their settings, 4 different datasets are generated, each contains questions from a single category: 'Computers & Internet', 'Education & Science', 'Games', and 'Entertainment & Recreation'. each dataset contains 90,000 training triples, the random category dataset contains 10,000 test questions, other datasets contain 1,000 test questions each.In this dataset n f = 30 and m f = 50.
TM) (Jeon et al. 2005), Translation based language model (TRLM) (Xue et al. 2008), Okapi model (Jeon et al. 2005) and Language model (LM) (Jeon et al. 2005). TM and TRLM use translation probabilities to overcome the lexical gap, while Okapi and LM only consider words that exist in both question and answer. Given a question q and answer a, TM (Jeon et al. 2005) can be define as: Xue et al. 2008) can be define as:
Table 1 :
1Compositions of Yahoo! Answer dataset. We randomly split questions from each category into training sets, validation sets and test sets by 4:1:1. For each question q in training (or validation) sets, 10 different (q, a + , a − ) triples are generated, where a + is q's best answer, y − is randomly selected answer from other question in the same category. Same q and a + appear in the 10 triples with different a − .Category
Question# Training set# Training triple# Validation set# Validation triple# Test set#
Travel
53,261
35,504
355,040
8,876
88,760
8,881
Relationships
57,576
38,384
383,840
9,596
95,960
9,596
Finance
66,129
44,084
440,840
11,021
110,210
11,024
Table 2 :
2Performance of different approaches on Yahoo! Answers dataset (WEC denote sentence-level WEC function)Approach
Travel
Relationships
Finance
DCG@1 DCG@6 DCG@1 DCG@6 DCG@1 DCG@6
WEC + CNN
0.761
0.946
0.709
0.938
0.780
0.952
WEC
0.734
0.946
0.698
0.936
0.761
0.949
TRLM
0.727
0.922
0.683
0.910
0.755
0.927
TM
0.698
0.914
0.676
0.912
0.742
0.926
Okapi
0.631
0.875
0.517
0.823
0.646
0.866
LM
0.592
0.848
0.525
0.825
0.595
0.838
Table 3 :
3Performance of different approaches on Baidu Zhidao dataset. IBM-1 denotes that translation probabilities are learned
using IBM translation model 1, cos denotes that translation probabilities are calculated through word vector's cosine-similarity
Approach
Computers &
Education &
Games
Entertainment &
Internet
Science
Recreation
DCG@1
DCG@6
DCG@1
DCG@6
DCG@1
DCG@6
DCG@1
DCG@6
WEC+CNN
0.826
0.970
0.870
0.980
0.703
0.941
0.780
0.963
WEC
0.821
0.968
0.838
0.975
0.692
0.937
0.778
0.962
TRLM(IBM-1) 0.780
0.937
0.843
0.948
0.654
0.894
0.709
0.918
TM(IBM-1)
0.732
0.925
0.766
0.931
0.598
0.876
0.626
0.892
S+CNN
0.658
0.912
0.734
0.939
0.619
0.894
0.543
0.866
TRLM(cos)
0.601
0.885
0.698
0.924
0.562
0.865
0.492
0.843
TM(cos)
0.596
0.885
0.691
0.922
0.560
0.863
0.486
0.841
Okapi
0.567
0.806
0.702
0.869
0.467
0.747
0.446
0.723
LM
0.624
0.830
0.746
0.881
0.544
0.765
0.488
0.740
Table 4 :
4Word-to-word translation examples learned from the Yahoo! Answer travel category dataset. Each column show the top 5 related answer words for a given question word. TTable denotes the type of word-to-word correlations model table used. The cos denote word vector's cosine similarity.Target
where
http://research.yahoo.com/Academic Relations
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|
[] |
[] |
[
"Anand Pillay \nUniversity of Notre Dame\nFudan University\n\n",
"Ningyuan Yao \nUniversity of Notre Dame\nFudan University\n\n"
] |
[
"University of Notre Dame\nFudan University\n",
"University of Notre Dame\nFudan University\n"
] |
[] |
It is known [2] that a group G definable in the field Q p of p-adic numbers is definably locally isomorphic to the group H(Q p ) of p-adic points of a (connected) algebraic group H over Q p . We observe here that if H is commutative then G is commutative-by-finite. This shows in particular that any one-dimensional group G definable in Q p is commutative-by-finite. This extends to groups definable in p-adically closed fields. We situate the results in a geometric structures environment.
|
10.1007/s00153-019-00673-y
|
[
"https://arxiv.org/pdf/1807.09079v1.pdf"
] | 119,639,224 |
1807.09079
|
b8a30c7e49f4d4a034439185944d9d85d6b1ce02
|
24 Jul 2018 July 25, 2018
Anand Pillay
University of Notre Dame
Fudan University
Ningyuan Yao
University of Notre Dame
Fudan University
24 Jul 2018 July 25, 2018arXiv:1807.09079v1 [math.LO] A note on groups definable in the p-adic field
It is known [2] that a group G definable in the field Q p of p-adic numbers is definably locally isomorphic to the group H(Q p ) of p-adic points of a (connected) algebraic group H over Q p . We observe here that if H is commutative then G is commutative-by-finite. This shows in particular that any one-dimensional group G definable in Q p is commutative-by-finite. This extends to groups definable in p-adically closed fields. We situate the results in a geometric structures environment.
Introduction and preliminaries
We here consider analogous questions vis-a-vis the p-adic field (or more generally p-adically closed fields) to questions long studied for the real field (more generally real closed fields, o-minimal structures). Namely the classification and description of definable groups. We emphasize definable rather than interpretable. These coincide in the real case because of elimination of imaginaries, but not in the p-adic case. Common features are that definable groups have naturally the structure of real and p-adic Lie groups [6], [7], as well as being locally isomorphic to real/p-adic algebraic groups [2]. But from this point on, things diverge: the p-adic dimension on definable sets is less well-behaved, in particular definable groups are in general far from being definably connected, and arguments from the real case do not go through.
We prove here that when G is locally commutative, equivalently the (connected) algebraic group H over Q p such that G is definably locally isomorphic to H(Q p ), is commutative, then G is commutative-by-finite. When G is onedimensional (in the sense of p-adic dimension) then H will be a connected algebraic group of algebraic-geometric dimension 1, so commutative. Thus we deduce from our results that one-dimensional groups definable in the padics are commutative-by-finite.
T h(Q p ) (i.e. the home sort) is dp-minimal (see [8]), hence any onedimensional group G definable in Q p is also dp-minimal, so by a result of Simon [8], G is commutative-by-bounded exponent. So we could deduce commutative-by-finiteness of one-dimensional G definable in Q p if we knew that there are no bounded exponent groups interpretable in the p-adics other than finite groups. This must be true but we do not know a proof at the moment.
The proof of our main result, which is in effect passing from being locally commutative to being commutative-by-finite, is relatively soft, and we situate it in the context of geometric structures.
A commutative-by-finite group G is amenable (as a discrete group), hence definably amenable with respect to any ambient structure in which G happens to be definable. As also T h(Q p ) is NIP (see Section 4.2, [1]), this puts us in a position to be able to apply some nice results from [4] which refine the "algebraic group configuration theorem" of [2], and we (and others) intend to carry this out in a future work. Even though the current paper is short we thought it makes sense as a "stand-alone" paper as it is self-contained, the methods are elementary, and it may be useful for future work by the authors and others.
We use fairly basic model theory. We refer to the excellent survey [1] as well as [2] and [5] for the model theory of the p-adic field (Q p , +, ×, 0, 1). In fact both [2] and [5] are also good references for the model theoretic background required for the current paper. A geometric structure (see Section 2 of [2]) is a one-sorted structure M such that in any model N of T h(M), algebraic closure satisfies exchange (so gives a so-called pregeometry on N) and there is a finite bound on the sizes of finite sets in uniformly definable families. The structure (Q p , +, ×, 0, 1) is an example of a geometric structure ([2], Proposition 2.11), as model-theoretic algebraic closure coincides with field-theoretic algebraic closure.
In a geometric structure M, if a is a finite tuple from M and B a subset of M then dim(a/B) denotes the size of a maximal algebraically independent over B subtuple of a. If M is saturated and X is a B-definable subset of M n (where B is finite) then dim(X) = max{dim(a/B) : a ∈ X}. It is important to know that when M is (Q p , +, ×, 0, 1), and X ⊆ M n is definable, then its dimensioin in the above sense coincides with its "topological dimension" , namely the greatest k ≤ n such that the image of X under some projection from M n to M k contains an open set.
In one of the general results below we make an assumption on the existence of G 0 . So we explain what this means, although it is discussed in the first section of [5]. Work in a saturated structureM and suppose G is definable inM . Let A be a small (of cardinality strictly less than the degree of saturation ofM ) subset ofM such that G is defined over A. Then by G 0 A we mean the intersection of all A-definable subgroups of G of finite index. We say that G 0 exists if G 0 A does not depend on A, namely cannot get smaller by increasing A. Note that the existence of G 0 is equivalent to the nonexistence of an infinite uniformly definable family of subgroups of G of some fixed finite index, which amounts to the DCC on intersections of uniformly definable subgroups of (a given) finite index. As mentioned in [5], the existence of G 0 follows from the ambient theory having NIP .
Finally let us state clearly the local isomorphism results alluded to earlier. [3] in the real closed field situation which works word for word in the p-adically closed field case.
Acknowledgements. Both authors would like to thank the Institut Henri Poincaré, Paris, for its hospitality and support during the trimester on model theory in early 2018 when this work was done. The second author would like to thank the IHES, Orsay, for its hospitality during the academic year 2017-18. Both authors would like to thank Immi Halupczok for discussions, in particular for telling us some valued-field arguments around finite index of centralizers which we use in the paper (although we situate it in a more general environment).
Results
We start with an easy lemma about geometric structures.
Lemma 2.1. Suppose M is a geometric structure, X ⊆ M n a definable set of dimension k say, and f is a definable function from X to M m . Suppose that f −1 (b) has dimension k for all b ∈ Im(f ) then Im(f ) is finite.
Proof. We may assume M to be saturated and work over the parameters over which X and f are defined. Suppose for a contradiction Im(f ) to be infinite. Then we can find b ∈ Im(f ) such that dim(b) ≥ 1. As dim(f −1 (b)) = k we can find a ∈ f −1 (b) such that dim(a/b) = k. Hence by subadditivity, dim(a, b) > k. As b ∈ dcl(a) it follows that dim(a) > k, contradicting that a ∈ X and dim(X) = k.
Remark 2.2. The conclusion of the lemma is weaker than stating that there is no definable equivalence relation on a definable k-dimensional set with infinitely many classes of dimension k. The latter is false in Q p Proposition 2.3. Let M be a geometric structure, and let G ⊆ M n be a group definable in M with dim(G) = k. Assume that (working in a saturated model) G 0 exists. Suppose that G contains a definable subset X of dimension k such that ab = ba for all a, b ∈ X. Then G is commutative-by-finite, namely G has a (definable) subgroup H of finite index such that H is commutative.
Proof. For a ∈ X, let f a be the (definable) function from G to G defined by f a (g) = gag −1 a −1 . Claim I. Fix a ∈ X. For any g 1 , g 2 ∈ G, f a (g 1 ) = f a (g 2 ) iff g 1 C G (a) = g 2 C G (a).
Proof of Claim I. This is obvious but we do it anyway. g 1 ag −1 1 a −1 = g 2 ag −1 2 a −1 iff g −1 2 g 1 ag −1 1 g 2 = a, namely g 1 g −1 2 ∈ C G (a). Claim II. For a ∈ X, Im(f a ) is finite. Proof of Claim II. Note that C G (a), the centralizer of a in G contains X (by assumption) so has dimension k. Hence for any g ∈ G, dim(gC G (a)) = k. By the right implies left implication in Claim I, f a is constant on gC G (a) for all g ∈ G. So we conclude by Lemma 2.1.
Claim III. For any a ∈ X, C G (a) has finite index in G. Proof of Claim III. By Claim I, Im(f ) is in bijection with G/C G (a), so by Claim II, C G (a) has finite index in G.
We may assume that we have been working in a saturated model M. So by compactness there is a finite bound on the index of C G (a) in G for a ∈ X. Our assumption that G 0 exists (as discussed in the previous section) implies that C G (X) = ∩ a∈X C G (a) is a finite subintersection, so is a definable subgroup H say of G, of finite index. Now for any a ∈ H, C G (a) contains X, so has dimension k. So repeating Claims I, II, and III, for a in H rather than X we conclude that C G (a) has finite index in G for all a ∈ H, and so again that C G (H) has finite index in G. So H ∩ C G (H) has finite index in G and is commutative.
Remark 2.4. One really needs only a finite-valued subadditive dimension on types of real tuples, for Lemma 2.1 and Proposition 2.3. And for Proposition 2.3, one only needs a definable set X ⊆ G of dimension k such that for all a ∈ X, C G (a) has dimension k.
In any case, from Proposition 2.3 we conclude: Theorem 2.5. (i) Let G be a group definable in Q p . Let H be a connected algebraic group over Q p as in Fact 1.1. Suppose that H is commutative, then G is commutative-by-finite.
(ii) Let G be a group of dimension 1 definable in Q p . Then G is commutativeby-finite.
Proof. (i) Let U, V be definable open neighbourhoods of the identity of G, H(Q p ) respectively and f : U → V as given by Fact 1.2. By choosing a smaller definable open neigbourhood U 1 of the identity contained in G such that ab ∈ U for a, b ∈ U 1 , we see from the assumptions that ab = ba for a, b ∈ U 1 . Now the p-adic topological) dimension of U 1 coincides with that of G, but both coincide with the dimension with respect to Q p as a geometric structure. So, bearing in mind that G 0 exists (working in a saturated model), as remarked in the introduction. We can apply Proposition 2.3 to conclude that G is commutative-by-profinite. (ii) If G has dimension 1 then by Fact 1.2 the connected algebraic group H has dimension 1 as an algebraic group, so is commutative. So part (i) implies.
Remark 2.6. By Remark 1.3 (ii), Theorem 2.4 goes through for groups definable in arbitrary p-adically closed fields (i.e. models of T h(Q p )).
Question 2.7. (i) Do we need the assumption that G 0 exists in Proposition 2.3? We presume yes, namely there is a counterexample without it. (ii) Let F be a geometric field in the sense of Definition 2.9 of [2]. Let G be a group definable in F and let H be a connected algebraic group over F given by Proposition 3.1' of [2]. Suppose H is commutative. Can one find a definable subset X of G of dimension equal to dim(G) such that for all a ∈ X, C G (a) has dimension equal to dim(G)?
Fact 1. 1 .
1([7]) Let G be a group definable in the field Q p . Then G has definably the structure of a p-adic Lie group. Moreover if G has dimension k as a definable group then it has dimension k as a p-adic Lie group.
Fact 1. 2 .
2([2]) Suppose G is a group definable in the field Q p . Consider G with its topology given by Fact 1.1. Then there is a connected algebraic group H over Q p , where the algebraic-geometric dimension of H equals the dimension of G, and a definable homeomorphism f between an open neigbourhood U of the identity in G and an open neighbourhood V of the identity of (the p-adic Lie group) H(Q p ) such that f (ab) = f (a)f (b) whenever a, b ∈ U and ab ∈ U.
Panorama of p-adic model theory. Luc Belair, Annales des Sciences Mathematiques du Quebec. 36Luc Belair, Panorama of p-adic model theory, Annales des Sciences Mathematiques du Quebec, 36 (2012), 43-75.
Groups definable in local and pseudofinite fields. E Hrushovski, A Pillay, Israel J. Math. 85E. Hrushovski and A. Pillay, Groups definable in local and pseudofinite fields, Israel J. Math., 85 (1994), 203-262.
Affiine Nash groups over real closed fields. E Hrushovski, A Pillay, Confluentes Math. 34E. Hrushovski and A. Pillay, Affiine Nash groups over real closed fields, Confluentes Math. vol 3, no. 4 (2011), 577-585.
S Montenegro, A Onshuus, P Simon, arXiv:1610.03150v2Stabilizers, groups with fgenerics in NT P 2 and P RC fields. preprint 2018S. Montenegro, A. Onshuus, and P. Simon, . Stabilizers, groups with f - generics in NT P 2 and P RC fields, preprint 2018, arXiv:1610.03150v2.
Definable groups and compact p-adic Lie groups. A Onshuus, A Pillay, J. London Math. Soc. 78A. Onshuus and A. Pillay, Definable groups and compact p-adic Lie groups, J. London Math. Soc. vol 78 (2008), 233-247.
Groups and fields definable in o-minimal structures. A Pillay, J. Pure Appl. Algebra. 53A. Pillay, Groups and fields definable in o-minimal structures, J. Pure Appl. Algebra, 53 (1988), 233-255.
On fields definable in Q p. A Pillay, Archive Math. Logic. 29A. Pillay, On fields definable in Q p , Archive Math. Logic, vol 29 (1989), 1-7.
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|
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[
"Probing spectral-temporal correlations with a versatile integrated source of parametric down-conversion states",
"Probing spectral-temporal correlations with a versatile integrated source of parametric down-conversion states"
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[
"Vahid Ansari [email protected] \nIntegrated Quantum Optics\nUniversity of Paderborn\nGermany\n",
"Benjamin Brecht \nIntegrated Quantum Optics\nUniversity of Paderborn\nGermany\n",
"Georg Harder \nIntegrated Quantum Optics\nUniversity of Paderborn\nGermany\n",
"Christine Silberhorn \nIntegrated Quantum Optics\nUniversity of Paderborn\nGermany\n"
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] |
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The spectral-temporal correlation and the correlation time of a biphoton wavepacket generated in the process of parametric down-conversion (PDC), is of great importance for a broad range of quantum experiments. We utilise an integrated PDC source to generate biphotons with different types of spectral-temporal correlations and probe their respective correlation times. The outcomes confirms that the correlation time is independent of the coherence time of the pump light, and it is only determined by the waveguide length and its dispersion properties. Furthermore, we investigate the properties of the PDC biphoton wavepacket exhibiting different types of spectraltemporal correlations and their suitability for quantum-enhanced applications.
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1404.7725
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Probing spectral-temporal correlations with a versatile integrated source of parametric down-conversion states
20 Jan 2015 21 January 2015
Vahid Ansari [email protected]
Integrated Quantum Optics
University of Paderborn
Germany
Benjamin Brecht
Integrated Quantum Optics
University of Paderborn
Germany
Georg Harder
Integrated Quantum Optics
University of Paderborn
Germany
Christine Silberhorn
Integrated Quantum Optics
University of Paderborn
Germany
Probing spectral-temporal correlations with a versatile integrated source of parametric down-conversion states
20 Jan 2015 21 January 2015arXiv:1404.7725v3 [quant-ph]
The spectral-temporal correlation and the correlation time of a biphoton wavepacket generated in the process of parametric down-conversion (PDC), is of great importance for a broad range of quantum experiments. We utilise an integrated PDC source to generate biphotons with different types of spectral-temporal correlations and probe their respective correlation times. The outcomes confirms that the correlation time is independent of the coherence time of the pump light, and it is only determined by the waveguide length and its dispersion properties. Furthermore, we investigate the properties of the PDC biphoton wavepacket exhibiting different types of spectraltemporal correlations and their suitability for quantum-enhanced applications.
Introduction
Parametric down-conversion (PDC) [1,2] is a well-established experimental tool for the generation of photon pair states, which are deployed in various experiments ranging from fundamental tests of quantum mechanics [3,4,5] to elaborate quantum information processing protocols [6,7,8,9]. In this context, interesting traits of PDC states are, on the one hand, the spectral-temporal correlation between the generated photons and, on the other hand, the biphotons correlation time T c . In the past few years, considerable effort has been focused to control and manipulate the PDC biphoton correlations to realise specifically tailored states exploiting different methods, e.g. in bulk PDC sources by manipulating attributes of the pump field such as beam waist [10], spatial chirp and profile [11] and coherence time [12] in bulk crystal PDC sources, or the use of different crystals [13,14] and tailored phasematching [15,16].
The correlation time of PDC biphotons is typically in the range of a few hundreds of femtoseconds and measures not only the degree of simultaneity between the pair photons, but also defines the shortest timing information extractable from the PDC state. Thus, the correlation time poses an inherent limitation to the achievable precision of quantumenhanced applications such as quantum clock synchronisation [17,18], quantum optical coherence tomography [19] or quantum interferometric optical lithography [20]. A thorough investigation of its underlying physics is indispensable for pushing those applications further towards their ultimate limits.
With state-of-the-art single photon detectors it is not possible to directly measure T c , due to their limited timing resolution of at least several tens of picoseconds [21]. However, Hong, Ou and Mandel (HOM) showed in their seminal paper that this obstacle can be overcome, when deploying quantum interference of photons at a beamsplitter [22]. With this method, they measured a correlation time T c of about 100 fs. More precisely, they sent the two generated PDC photons into the input ports of a balanced beamsplitter and measured the coincidence events between its output ports when delaying one of the photons with respect to the other. When two photons are identical in all degrees of freedom (arrival time, spectrum and polarisation), they bunch together and emerge from the same output port. Consequently the detected coincidence rate drops to zero. Making the photons distinguishable, for instance by introducing a relative time delay, degrades the quantum interference and the typical HOM dip can be observed. The amount of delay, which is required to render the photons completely distinguishable is proportional to the correlation time T c . Typically, due to experimental considerations, a narrow-band spectral filtering of PDC photons is applied prior to the detection [22,23]. However, the use of narrow-band filters will alter the created PDC state and thus the width of interference pattern only samples the bandwidth of the applied filters and not the true PDC correlation time.
In this paper we utilise a waveguided PDC source developed in our group which is capable of generating PDC biphotons with well-controlled, tunable spectral-temporal correlations [24,25]. We probe their correlation times T c and show that the properties of the PDC pump field, including its coherence time and chirp, have actually no impact on the correlation time T c of the generated biphotons. Then, we present a special case where T c is in fact larger than the coherence time of the pump. We give an intuitive explanation for this by analytical calculations which are valid for a wide range of PDC sources and can readily be extended to any PDC. Our results allow us to identify the potential gain or loss of timing information relative to the pump field for different PDC sources.
Theoretical background
Earlier theoretical studies have shown that the correlation time of the PDC biphotons does not depend on the spectral bandwidth of the pump [26,27]. Here, we extend these studies by considering chirped pump pulses, which is realistic with consideration of the experimental conditions. We derive the interference pattern between PDC photons and write T c only in terms of crystal properties. Following the standard quantum first order perturbative approach we write the PDC state as [28,29]
|ψ PDC ∝ dω s dω i f (ω s , ω i )â † s (ω s )â † i (ω i ) |0 ,(1)
where theâ † s (ω s ) andâ † i (ω i ) are the standard creation operators for a signal photon at frequency ω s and an idler photon at frequency ω i , respectively. The joint spectral amplitude (JSA) function f (ω s , ω i ) is a complex-valued function that describes the spectral-temporal structure of the created photon pair. The JSA function comprises the pump envelope function α(ω s + ω i ) and the phasematching function φ(ω s , ω i ), which reflect energy and momentum conservation of the PDC process, respectively. For our analysis, we assume a Gaussian pump spectrum and allow for a possible frequency chirp which is typically present in experimental settings due to the dispersion of optical components. Hence we write
α(ν s + ν i ) = e − νs+ν i σp 2 e ıβ(νs+ν i ) 2 ,(2)
where σ p is the spectral width of the pump pulse and β characterizes its chirp at the input of the nonlinear medium. Moreover we defined the frequency detunings ν µ = ω µ − ω 0 µ (with µ ∈ {s, i}) from the perfectly phasematched central PDC frequency ω 0 µ . The phasematching function is governed by the dispersion properties and the length of the nonlinear waveguide L, and is given by
φ(ω s , ω i ) = e −γ( L 2 ∆k(ωs,ω i )) 2 e ı L 2 ∆k(ωs,ω i ) ,(3)
where ∆k(ω s , ω i ) is the phase-mismatch between pump, signal and idler in the waveguide. Here the sinc-profile of the phasematching function is approximated with a Gaussian function. This can be realised by appropriate engineering of the nonlinearity of the waveguide [30,16]. Using a Taylor expansion up to the first order around the perfectly phasematched frequencies ω 0 µ , we rewrite the phase-mismatch as
L∆k ≈ τ s ν s + τ i ν i . Here τ µ = L(u −1 µ − u −1 p ),
where u denotes the group velocity of the corresponding fields [29]. In our calculations we dismiss linear phase terms in ν µ in the JSA function, since they only cause a temporal shift of the biphoton state. Following these calculation we can express the JSA as [31,32]
f (ν s , ν i ) = exp − T 2 ss ν 2 s + T 2 ii ν 2 i + 2T 2 si ν s ν i ,(4)
where we defined
T 2 λµ ≡ 1 σ 2 + γ 4 τ λ τ µ − ıβ λ, µ = s, i.(5)
To derive the HOM interference pattern, we take the PDC biphoton state in Eq. (1) and apply a time delay τ to one input arm of the beamsplitter. The rate of coincidence events between the output ports of the beamsplitter is then given by [28]
R c (τ ) ∝ 1 − dωdω e ı(ω−ω)τ f (ω,ω)f * (ω, ω).(6)
For τ = 0, i.e. perfect temporal overlap of the photons at the beamsplitter, the integrand of Eq. (6) measures the symmetry of the JSA function describing the PDC state under the exchange of signal and idler. Therefore the visibility of the interference probes the indistinguishability of interfering photons. By inserting the JSA function into Eq. (6) we find
R c (τ ) ∝ 1 − dωdω e ı(ω−ω)τ e −(Aν 2 s +Aν 2 i +Bνsν i ) ,(7)
where we define
A ≡ 2 σ 2 p + γ 4 (τ 2 s + τ 2 i ), B ≡ 2 σ 2 p + γ 2 τ s τ i .(8)
From Eqs. (7) and (8) we see that the chirp of the pump field does not affect the interference at all. Performing the frequency integrations in Eq. (7) finally yields
R c (τ ) ∝ 1 − h(σ p , τ s , τ i ) exp −τ 2 2 √ γ L 2 (u −1 s − u −1 i ) 2 ,(9)
in correspondence to earlier theoretical studies [33,27]. Here, h(σ p , τ s , τ i ) is a coefficient which depends on both pump and crystal properties and determines the visibility of the interference. However, the shape of the dip is contained in the exponential function, which is completely independent of the pump. This means that for a given nonlinear crystal, the correlation time T c is solely defined by crystal parameters. We interpret this result in an intuitive manner: in the process of PDC, a pump photon propagates through a waveguide of length L and coherently decays into a pair of photons. These, in turn, travel through the waveguide at different group velocities and thus acquire a temporal walk-off. Post-selecting on coincidence detection events we find that, on average, the PDC process takes place at the centre of the waveguide. The associated walk-off is the correlation time T c , which consequently is proportional to half the waveguide length L/2, as visualized in Fig. 1. This situation simply exemplifies the fact that the correlation time T c is independent of the pump coherence time.
We investigate this behaviour in Fig. 2, where we plot the JSA function f (ω s , ω i ) for a waveguide with fixed dispersion properties but at different waveguide lengths L and pump spectral widths σ p (left) and analyse the resulting HOM interference pattern (right). The grey shaded areas in the JSA plots correspond to signal and idler marginal spectra, whereas the shaded areas in the HOM dip plots highlight the FWHM of the respective interference dip, which by definition is the correlation time T c . Note that although we focus on a specific waveguide dispersion, our results are generally applicable to any PDC. Figure 2. Investigation of the relation between spectral width of the pump field, waveguide length and the correlation time of PDC biphoton wavepacket. All parameters are given in arbitrary units. The JSA reflects PDC processes exhibiting symmetric group velocity matching (SGVM) [29] for different values of the pump spectral width σ p , and the waveguide length L (left). From these we retrieved the HOM dip interference patterns for the respective parameters (right). A detail explanation is presented in the text.
(a) (b) (c) (d) (e)
In Fig. 2-(a,b,c) we assumed a fixed waveguide length L and varied the spectral pump width σ p and thus the pump coherence time. Despite the different coherence times of the pump field, it is evident that the correlation times of all three biphoton states, as given by the FWHM of the HOM dips, are identical. In contrast, graphs 2-(c,d,e) juxtapose three waveguides with different lengths L that are pumped with similar pump pulses of spectral width σ p . It is obvious that a longer waveguide results in a broader HOM dip, which can be easily understood with the picture of the temporal walk-off between signal and idler.
Furthermore, a comparison between Fig. 2-(a,b) shows that in general, the correlation time is not simply the convolution of signal and idler marginal distributions. It is important to recognize that the JSA is fundamentally a two-dimensional distribution function, whereas marginal spectra correspond to the traced-out system. Therefore, the marginals do not contain full information of any biphoton state. Only in the case of a spectrally decorrelated PDC we find that T c is proportional to the convolution of the signal and idler marginal distributions [21,22].
Let us now consider the temporal information of the PDC state. It has been shown that entanglement, including spectral entanglement, can be exploited to enhance the precision of quantum metrology protocols. Comparing the cases in Fig. 2 (a) and (e), we first note that they have the same amount of spectral entanglement and the same marginal distributions. However, by comparing the resulting HOM interference patterns we realise that spectrally anticorrelated states provide a considerably shorter correlation time. To understand this, it is helpful to consider the joint temporal amplitude (JTA) function corresponding to the JSA, which essentially has the shape of the JSA but is rotated about 90 • . The correlation time between signal and idler photons measures the simultaneity of the photons, which denotes |t s −t i |. This corresponds to the width of the JTA function along −45 • in the (t s , t i )-plane. Thus, the spectrally anticorrelated state from Fig. 2 (a) is ideal for measuring arrival time differences, whereas the spectrally correlated state from Fig. 2 (e) is adapted to measuring absolute times t s + t i [17].
Experimental results and discussion
To examine the theory, we use a periodically poled potassium titanyl phosphate (ppKTP) waveguided PDC source recently presented in [24,25]. Our source provides a frequency-degenerate type-II PDC at telecommunication wavelengths, where the phasematching function is oriented along +59 • in the (ω s , ω i )-plane. This allows us to realise PDC states with all possible types of spectral correlations, simply by changing the spectral width of the pump pulse, similar to Fig. 2-(a-c). A schematic of the experimental setup is shown in Fig. 3.
We deploy a Ti:Sapphire oscillator with a repetition rate of 80 MHz which provides ultrafast pulses with duration of 200 fs at the central wavelength of 767.5 nm. To change the spectral width σ p or equivalently the coherence time of the pump pulses we use a 4f-setup as a tunable spectral filter. The 4f-setup consists of two diffraction Figure 3. Experimental setup. A 4f-setup is employed as tunable spectral filter to control the coherence time of the pump field and thereby the spectral correlation of the biphoton state. An 8mm long ppKTP waveguide is used as degenerate type-II PDC source. We use a longpass filter (LPF) to block out the pump beam. A broad bandpass filter (BPF) with a transmission band of approximately 8 nm centred at the central PDC wavelength suppresses the background noise, but does not cut the biphoton spectrum. The PDC photons are separated by a polarising beamsplitter (PBS). A half waveplate (HWP) is used to make the polarisation of signal and idler photons identical. Then the PDC photons are interfered at a balanced fibre coupler. gratings, two lenses and one adjustable slit in the centre, as demonstrated in Fig. 3. This arrangement is a one dimensional Fourier processor. The diffracted light beam is focused on the Fourier plane, where we use a slit to manipulate the spectral width of the pump field. However, this arrangement cuts off the spectrum as a rectangular bandpass filter which distorts the spectrum from its initially Gaussian shape. Nevertheless, a simple modelling shows that it is possible to recover the Gaussian shape of the spectrum only by tilting the slit. We use this experimental technique to maintain the original shape of the pump pulses.
For the HOM measurement we interfere the signal and idler photons on a singlemode balanced fibre coupler after separating them with a polarizing beamsplitter (PBS) and rotating the signal polarisation with a half-wave plate (HWP). Using a translation Figure 4. The joint spectral intensity (JSI) and HOM interference measurements for spectrally (a) anticorrelated, (b) decorrelated and (c) correlated PDC states. Left: A fibre spectrometer with the resolution of 1.8 × 1.8 nm 2 is used to measure the JSI of the PDC biphoton state [34] for different spectral widths of the pump field ∆λ p listed in TABLE 1. Right: The corresponding HOM interference patterns, performed with pump energies as low as 0.6 pJ per pulse leading to a mean photon number of 0.002 pairs per pulse. The error bars are smaller than the points. In the correlated case we removed the bandpass filter (BPF), since the marginal spectra are broader than the filter transmission in this case.
stage we introduce a path difference between the interfering photons while recording the coincidence events between the output ports of the coupler with avalanche photo diodes (APDs) connected to a homemade field programmable gate array (FPGA).
In Fig. 4 we plot the measured joint spectral intensity functions |f (ω s , ω i )| 2 and the corresponding HOM dips for cases of a spectrally anticorrelated, a decorrelated and correlated PDC [34]. In agreement with our expectations, the widths of the dips and hence T c are nearly identical upon significantly different coherence times of the pump field and are in excellent agreement with the theoretical prediction of 1.16 ± 0.03 ps as computed from Eq. (9). The small differences of the correlation times can be explained by the inherent sinc-shape of the phasematching function. This can be verified by numerical modelling of the exact JSA function and the interference pattern, which is plotted in Fig. 4 as the grey fit functions. In TABLE 1 a summary of the crucial parameters and outcomes of our measurement is presented.
In spectrally decorrelated and correlated states, T c is actually larger than the coherence time of the pump ∆τ p . This corresponds to the schematic in Fig. 1, where the photons are on average detected outside the pump. Only for the case of the spectrally anticorrelated PDC, the photons are found inside the pump. A similar observation was reported previously in [12], where the correlation time of biphoton wavepacket was compared utilising pulse and continuous wave pumping. Moreover we verify that T c equals the convolution of the signal and idler marginal durations only for the case of a decorrelated PDC and differs otherwise.
With these results we can also quantify the gain or loss of timing information relative to the pump field, by observing the HOM interference between PDC photons. Comparing T c to the pump duration ∆τ p , we find that the spectrally anticorrelated PDC state is tailored to precisely retrieve the correlation time |t s − t i |. The most prominent example for this situation is of course the original work of Hong, Ou and Mandel [22]. On the other hand, the spectrally correlated PDC state is appropriate to measure the sum of arrival times t s + t i [17].
Conclusion
We theoretically and experimentally investigated the properties of PDC states with different types of spectral-temporal correlations. Using a tunable integrated source, we generated spectrally decorrelated, positively and negatively correlated PDC states. The results shows that the coherence time and chirp of the pump field indeed has no effect on the correlation time of the generated biphoton state. We presented an intuitive model for the origin of the correlation time of PDC photons, which is in complete agreement with our experimental outcomes. Furthermore, the appropriateness of different PDC states in respect to derive timing information were discussed. We shown, depending on the spectral correlation of the PDC state, by measuring the HOM interference one can gain or lose timing information compared with the classical pump field. This fundamental insight can be useful to the diverse number of applications relying on the interference of PDC photons.
Acknowledgements
We acknowledges helpful discussions with Malte Avenhaus.
Figure 1 .
1Temporal visualisation of the correlation time of the biphoton. A pump pulse with coherence time τ p decays inside the waveguide and creates daughter photons which travel with different velocities through the medium. On average, the downconversion process happens at the middle of the waveguide, thus the correlation time T c between the down-converted photons is proportional to L/2.
Table 1 .
1The correlation time for spectrally decorrelated and (anti-)correlated PDC states. The measured correlation time T c , the spectral FWHM of individual fields ∆λ, the temporal FWHM of each fields ∆τ and the temporal convolution of daughter pulses ∆τ conv.= ∆τ 2
s + ∆τ 2
i are listed.
Spectral correlation
T c (ps)
∆λ s/i (nm) ∆λ p (nm) ∆τ s/i (ps) ∆τ p (ps) ∆τ conv (ps)
correlated
1.21 ± 0.03 5.84/10.14
4.5
0.59/0.34
0.19
0.68
decorrelated
1.16 ± 0.01
4.3/5.46
2
0.8/0.63
0.43
1.02
anticorrelated
1.10 ± 0.02 3.06/3.12
0.7
1.13/1.11
1.24
1.58
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Engineered optical nonlinearity for quantum light sources. M Agata, Alessandro Brańczyk, Fedrizzi, M Thomas, Timothy C Stace, Andrew G Ralph, White, Optics Express. 191Agata M Brańczyk, Alessandro Fedrizzi, Thomas M Stace, Timothy C Ralph, and Andrew G White. Engineered optical nonlinearity for quantum light sources. Optics Express, 19(1):55-65, 2011.
On the relationship between pump chirp and single-photon chirp in spontaneous parametric downconversion. Xochitl Sanchez-Lozano, Alfred , B U'ren, Jose Luis Lucio, Journal of Optics. 14115202Xochitl Sanchez-Lozano, Alfred B U'Ren, and Jose Luis Lucio. On the relationship between pump chirp and single-photon chirp in spontaneous parametric downconversion. Journal of Optics, 14(1):015202, December 2011.
Control, measurement, and propagation of entanglement in photon pairs generated through type-II parametric down-conversion. Yasser Jeronimo, - Moreno, Alfred B U' Ren, Physical Review A. 79333839Yasser Jeronimo-Moreno and Alfred B U'Ren. Control, measurement, and propagation of entanglement in photon pairs generated through type-II parametric down-conversion. Physical Review A, 79(3):033839, March 2009.
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|
[] |
[
"Localization landscape theory of disorder in semiconductors II: Urbach tails of disordered quantum well layers",
"Localization landscape theory of disorder in semiconductors II: Urbach tails of disordered quantum well layers"
] |
[
"Marco Piccardo \nLaboratoire de Physique de la Matière Condensée\nCNRS\nEcole polytechnique\n\nUniversité Paris Saclay\n91128Palaiseau CedexFrance\n\nMaterials Department\nUniversity of California\n93106Santa BarbaraCaliforniaUSA\n",
"Chi-Kang Li \nGraduate Institute of Photonics and Optoelectronics\nDepartment of Electrical Engineering\nNational Taiwan University\n10617TaipeiTaiwan\n",
"Yuh-Renn Wu \nGraduate Institute of Photonics and Optoelectronics\nDepartment of Electrical Engineering\nNational Taiwan University\n10617TaipeiTaiwan\n",
"James S Speck \nMaterials Department\nUniversity of California\n93106Santa BarbaraCaliforniaUSA\n",
"Bastien Bonef \nMaterials Department\nUniversity of California\n93106Santa BarbaraCaliforniaUSA\n",
"Robert M Farrell \nMaterials Department\nUniversity of California\n93106Santa BarbaraCaliforniaUSA\n",
"Marcel Filoche \nLaboratoire de Physique de la Matière Condensée\nCNRS\nEcole polytechnique\n\nUniversité Paris Saclay\n91128Palaiseau CedexFrance\n",
"Lucio Martinelli \nLaboratoire de Physique de la Matière Condensée\nCNRS\nEcole polytechnique\n\nUniversité Paris Saclay\n91128Palaiseau CedexFrance\n",
"Jacques Peretti \nLaboratoire de Physique de la Matière Condensée\nCNRS\nEcole polytechnique\n\nUniversité Paris Saclay\n91128Palaiseau CedexFrance\n",
"Claude Weisbuch \nLaboratoire de Physique de la Matière Condensée\nCNRS\nEcole polytechnique\n\nUniversité Paris Saclay\n91128Palaiseau CedexFrance\n\nMaterials Department\nUniversity of California\n93106Santa BarbaraCaliforniaUSA\n"
] |
[
"Laboratoire de Physique de la Matière Condensée\nCNRS\nEcole polytechnique",
"Université Paris Saclay\n91128Palaiseau CedexFrance",
"Materials Department\nUniversity of California\n93106Santa BarbaraCaliforniaUSA",
"Graduate Institute of Photonics and Optoelectronics\nDepartment of Electrical Engineering\nNational Taiwan University\n10617TaipeiTaiwan",
"Graduate Institute of Photonics and Optoelectronics\nDepartment of Electrical Engineering\nNational Taiwan University\n10617TaipeiTaiwan",
"Materials Department\nUniversity of California\n93106Santa BarbaraCaliforniaUSA",
"Materials Department\nUniversity of California\n93106Santa BarbaraCaliforniaUSA",
"Materials Department\nUniversity of California\n93106Santa BarbaraCaliforniaUSA",
"Laboratoire de Physique de la Matière Condensée\nCNRS\nEcole polytechnique",
"Université Paris Saclay\n91128Palaiseau CedexFrance",
"Laboratoire de Physique de la Matière Condensée\nCNRS\nEcole polytechnique",
"Université Paris Saclay\n91128Palaiseau CedexFrance",
"Laboratoire de Physique de la Matière Condensée\nCNRS\nEcole polytechnique",
"Université Paris Saclay\n91128Palaiseau CedexFrance",
"Laboratoire de Physique de la Matière Condensée\nCNRS\nEcole polytechnique",
"Université Paris Saclay\n91128Palaiseau CedexFrance",
"Materials Department\nUniversity of California\n93106Santa BarbaraCaliforniaUSA"
] |
[] |
Urbach tails in semiconductors are often associated to effects of compositional disorder. The Urbach tail observed in InGaN alloy quantum wells of solar cells and LEDs by biased photocurrent spectroscopy is shown to be characteristic of the ternary alloy disorder. The broadening of the absorption edge observed for quantum wells emitting from violet to green (indium content ranging from 0 to 28%) corresponds to a typical Urbach energy of 20 meV. A 3D absorption model is developed based on a recent theory of disorder-induced localization which provides the effective potential seen by the localized carriers without having to resort to the solution of the Schrödinger equation in a disordered potential. This model incorporating compositional disorder accounts well for the experimental broadening of the Urbach tail of the absorption edge. For energies below the Urbach tail of the InGaN quantum wells, type-II well-to-barrier transitions are observed and modeled. This contribution to the below bandgap absorption is particularly efficient in near-UV emitting quantum wells. When reverse biasing the device, the well-to-barrier below bandgap absorption exhibits a red shift, while the Urbach tail corresponding to the absorption within the quantum wells is blue shifted, due to the partial compensation of the internal piezoelectric fields by the external bias. The good agreement between the measured Urbach tail and its modeling by the new localization theory demonstrates the applicability of the latter to compositional disorder effects in nitride semiconductors.
|
10.1103/physrevb.95.144205
|
[
"https://arxiv.org/pdf/1704.05524v1.pdf"
] | 119,203,590 |
1704.05524
|
bcbf66a38d01772a1b1b971367cfaa0085d99126
|
Localization landscape theory of disorder in semiconductors II: Urbach tails of disordered quantum well layers
Marco Piccardo
Laboratoire de Physique de la Matière Condensée
CNRS
Ecole polytechnique
Université Paris Saclay
91128Palaiseau CedexFrance
Materials Department
University of California
93106Santa BarbaraCaliforniaUSA
Chi-Kang Li
Graduate Institute of Photonics and Optoelectronics
Department of Electrical Engineering
National Taiwan University
10617TaipeiTaiwan
Yuh-Renn Wu
Graduate Institute of Photonics and Optoelectronics
Department of Electrical Engineering
National Taiwan University
10617TaipeiTaiwan
James S Speck
Materials Department
University of California
93106Santa BarbaraCaliforniaUSA
Bastien Bonef
Materials Department
University of California
93106Santa BarbaraCaliforniaUSA
Robert M Farrell
Materials Department
University of California
93106Santa BarbaraCaliforniaUSA
Marcel Filoche
Laboratoire de Physique de la Matière Condensée
CNRS
Ecole polytechnique
Université Paris Saclay
91128Palaiseau CedexFrance
Lucio Martinelli
Laboratoire de Physique de la Matière Condensée
CNRS
Ecole polytechnique
Université Paris Saclay
91128Palaiseau CedexFrance
Jacques Peretti
Laboratoire de Physique de la Matière Condensée
CNRS
Ecole polytechnique
Université Paris Saclay
91128Palaiseau CedexFrance
Claude Weisbuch
Laboratoire de Physique de la Matière Condensée
CNRS
Ecole polytechnique
Université Paris Saclay
91128Palaiseau CedexFrance
Materials Department
University of California
93106Santa BarbaraCaliforniaUSA
Localization landscape theory of disorder in semiconductors II: Urbach tails of disordered quantum well layers
(Dated: April 20, 2017)numbers: 7123An7215Rn0365Ge
Urbach tails in semiconductors are often associated to effects of compositional disorder. The Urbach tail observed in InGaN alloy quantum wells of solar cells and LEDs by biased photocurrent spectroscopy is shown to be characteristic of the ternary alloy disorder. The broadening of the absorption edge observed for quantum wells emitting from violet to green (indium content ranging from 0 to 28%) corresponds to a typical Urbach energy of 20 meV. A 3D absorption model is developed based on a recent theory of disorder-induced localization which provides the effective potential seen by the localized carriers without having to resort to the solution of the Schrödinger equation in a disordered potential. This model incorporating compositional disorder accounts well for the experimental broadening of the Urbach tail of the absorption edge. For energies below the Urbach tail of the InGaN quantum wells, type-II well-to-barrier transitions are observed and modeled. This contribution to the below bandgap absorption is particularly efficient in near-UV emitting quantum wells. When reverse biasing the device, the well-to-barrier below bandgap absorption exhibits a red shift, while the Urbach tail corresponding to the absorption within the quantum wells is blue shifted, due to the partial compensation of the internal piezoelectric fields by the external bias. The good agreement between the measured Urbach tail and its modeling by the new localization theory demonstrates the applicability of the latter to compositional disorder effects in nitride semiconductors.
Urbach tails in semiconductors are often associated to effects of compositional disorder. The Urbach tail observed in InGaN alloy quantum wells of solar cells and LEDs by biased photocurrent spectroscopy is shown to be characteristic of the ternary alloy disorder. The broadening of the absorption edge observed for quantum wells emitting from violet to green (indium content ranging from 0 to 28%) corresponds to a typical Urbach energy of 20 meV. A 3D absorption model is developed based on a recent theory of disorder-induced localization which provides the effective potential seen by the localized carriers without having to resort to the solution of the Schrödinger equation in a disordered potential. This model incorporating compositional disorder accounts well for the experimental broadening of the Urbach tail of the absorption edge. For energies below the Urbach tail of the InGaN quantum wells, type-II well-to-barrier transitions are observed and modeled. This contribution to the below bandgap absorption is particularly efficient in near-UV emitting quantum wells. When reverse biasing the device, the well-to-barrier below bandgap absorption exhibits a red shift, while the Urbach tail corresponding to the absorption within the quantum wells is blue shifted, due to the partial compensation of the internal piezoelectric fields by the external bias. The good agreement between the measured Urbach tail and its modeling by the new localization theory demonstrates the applicability of the latter to compositional disorder effects in nitride semiconductors.
I. INTRODUCTION
The absorption edge in semiconductors is usually characterized by a long energy tail with an absorption coefficient α exponentially varying with photon energy and temperature along the Urbach tail rule 1 α (hν) = α 0 · e (hν−E1)/E U (T ) ,
where α 0 is a constant, E 1 determines the onset of the tail, usually the bandgap energy E g in semiconductors, and E U is the Urbach energy. Note that Urbach tails were proposed in 1953 essentially as a phenomenological description of thermally activated belowgap absorption (see e.g. Ref.
2 and references therein) and in Urbach's original expression E U = kT /γ, γ being a constant. Several attempts have since then been made trying to derive a universal theory. 3,4 By extension, exponential absorption edges are also called Urbach tails although they might not be associated with thermally activated phenomena. A variety of phenomena, intrinsic or extrinsic, have been invoked, and sometimes demonstrated, as physical mechanisms leading to Urbach tails such as the Franz-Keldysh (FK) effect associated with the random electric fields of impurities, defectinduced smearing of band edges 5 and phonon-assisted absorption. 2 The impact of compositional alloy disorder on thermally activated Urbach tails has been clearly identified in InAs x Sb 1−x 6 while in the case of amorphous Si Cody et al. could characterize both the effects of structural disorder and thermal disorder. 7 In the InGaN alloy system, very large Urbach energies E U of few hundreds of meV have been observed, [8][9][10] attributed to phase separation of the InGaN alloy. This is to be contrasted with the luminescence linewidth in LED materials which is most often quite smaller, of the order of 100 meV or less. The Urbach energy observed in pure GaN is 12 meV 11 or 15 meV. 12 As will be shown in the present paper, the Urbach tail of high quality In x Ga 1−x N quantum wells (QWs) has Urbach energies in the range 15-30 meV and is determined by the disorder introduced by the indium compositional fluctuations.
Understanding disorder phenomena in nitride ternary alloys, such as InGaN and AlGaN, is of utmost importance as they constitute the core of several modern optoelectronic devices due to their wide tunability of energy bandgap. It is indeed well established by atom probe tomography (APT) that in these alloys atoms are randomly distributed and thus induce random potential fluctuations. 13 Although APT provides imaging of the compositional disorder with near atomic resolution, the relation between the measured atomic maps and the resulting energy fluctuations occurring near the band gap edge of the alloy, where all physical phenomena of devices occur, is an area of ongoing research -including the work reported here.
Theoretical and numerical studies indicate that com-positional disorder in nitride ternary alloys is large enough to induce carrier localization which, in turn, may strongly influence carrier transport, optical properties and efficiency of a device. In the case of nitride lightemitting diodes (LEDs), simulations of I-V characteristics including random alloy fluctuations in the InGaN QWs and AlGaN barriers predict a voltage threshold which is almost 1 V lower with respect to the case of a homogeneous active region, in better agreement with the experimental values for actual devices, due to percolation in the disordered layers. 14-16 Two companion papers, referred to as LL1 (Ref. 17) and LL3 (Ref. 16), describe the basis of the localization landscape (LL) theory and its application to transport and recombination in nitride heterostructure LEDs, respectively. Atomistic calculations by Schulz et al. 18 show that low-band electron and hole states produce a strong inhomogeneous broadening of luminescence spectra of In 0.25 Ga 0.75 N/GaN QWs. Firstprinciple calculations of non-radiative Auger rates in nitride semiconductors by Kioupakis et al. 19 indicate that indirect Auger recombination mediated by alloy disorder may play an important role in determining the efficiency loss of nitride optoelectronic devices. On the experimental side, besides the Urbach tail measurement mentioned above, a well-known effect often interpreted as an evidence of carrier localization induced by disorder in InGaN QW layers is the "S-shaped" emission shift of the peak energy of photoluminescence observed as a function of temperature. 20,21 This phenomenon indicates relaxation of the carrier ensemble at low temperatures towards deeply localized centers, signaled by a red-shift of the emission energy, but does not provide information on the disorder effects impacting device operation.
In contrast, optical absorption spectroscopy should provide a more direct access to the disorder-induced energy fluctuations near the band edge, as it is directly linked to the electronic densities of states. In the present study the role of compositional disorder in a series of In x Ga 1−x N QW structures of different indium content is characterized by means of biased photocurrent spectroscopy (BPCS). Assuming that the collection efficiency of photocurrent is independent of photon energy, BPCS measurements are representative of optical absorption spectra. 22 For instance, in GaAs/AlGaAs QWs a series of distinct excitonic peaks could be observed by Collins et al. 23 using BPCS, indicating that excitons, after photocreation, are ionized and transport through the structure contributing to photocurrent. To the extent that the optical interband matrix element has a weak dependence on photon energy in the narrow energy range of the Urbach tail, then below-gap absorption is determined by the joint density of states (JDOS) of the disordered system weighted by the overlap integral between the envelope functions of the localized states. 24 The contribution of internal piezoelectric fields on the absorption spectra can be evaluated by comparing data collected with different external applied biases. Previous photocurrent studies were carried out in InGaN/GaN QW structures under applied bias but did not aim at characterizing disorderrelated effects on below-gap absorption. 25,26 In the present work we are able to characterize the effect of compositional disorder in InGaN layers with different indium content by analyzing their Urbach tails as measured by BPCS. QWs emitting from the violet ([In]=11%) to the green ([In]=28%) range exhibit small values of the Urbach energy, typically around 20 meV, while a much larger broadening would be expected from the fluctuations of the calculated random potential corresponding to the alloy disorder. An independent absorption model is derived on the basis of a recent mathematical theory of localization that takes into account the "effective" confining potential seen by localized states in disordered systems without having to resort to the Schrödinger equation. 27,28 A development of the LL theory in the framework of semiconductors is reported in LL1 (Ref. 17) and its predictions account well for the experimental broadening of the below-gap absorption edge.
The description of the samples and details on the experimental setup are given in Sec. II. In Sec. III the main results of BPCS for In x Ga 1−x N QW structures with indium composition ranging between 0% and 28% will be presented. It will be shown that three different regions may be identified in the photocurrent spectra corresponding to different absorption mechanisms. The focus will be on the experimental characterization of Urbach tails as a function of indium composition, applied bias, sample temperature and type of device incorporating InGaN QWs. In Sec. IV theoretical models will be derived to study the effect of electric fields and compositional disorder on Urbach tails. First the influence of the polarization-induced QW electric field on the optical transition matrix element by means of the FK effect at below-gap photon energies will be evaluated. Then the absorption model based on the LL theory will be introduced and its predictions will be compared to the measured Urbach tails giving an insight into the effect of compositional disorder in InGaN layers. Finally calculations of type-II well-to-barrier transitions will be presented in the Appendix to model an additional absorption mechanism observed in the below-gap responsivity of near-UV emitting QWs.
II. EXPERIMENT
A. Samples
The samples are a series of In x Ga 1−x N/GaN multiple QW (MQW) solar cells grown by metal-organic chemical-vapor deposition (MOCVD) on (0001) sapphire substrates with mean indium content ranging between 6% and 28% measured by x-ray diffraction (XRD) with a PANalytic MRD PRO diffractometer. The structures are schematized in Fig. 1(a) and consist of a 1 µm unintentionally doped GaN template layer, a 2 µm Si-doped The nand p-layer adjacent to the active region are δ-doped to reduce the polarization fields in the QWs. APT experiments were performed to confirm the random distribution of indium atoms in the InGaN layers in the different samples. Specimens were prepared with a FEI Helios 600 dual beam FIB instrument following standard procedure. APT analyses were performed with a Cameca 3000X HR Local Electrode Atom Probe (LEAP) operated in laser-pulse mode with a green laser (532 nm). The laser pulse energy was 1 nJ and a detection rate of 0.005 atoms per pulse was set during each sample analysis. APT 3D reconstruction was carried out using commercial software IVASTM. Statistical distribution analysis were performed on sampling volumes of 30 × 30 × 2 nm 3 taken in the center of each InGaN layers and for each sample. The random distribution of indium in each InGaN layer was confirmed by the success of the χ 2 test between experimental distributions and the binomial distribution predicted for random alloys.
A UV LED with pure GaN QW layers is also studied for comparison. This sample is grown by molecular beam epitaxy (MBE) on a semipolar (2021) GaN substrate and its structure consists of an AlGaN buffer layer, a Si-doped n-Al 0.20 Ga 0.80 N layer, a 5 period undoped MQW active region with 5 nm GaN QWs and 6 nm Al 0.12 Ga 0.88 N barriers, a Mg-doped p-Al 0.20 Ga 0.80 N layer and a highly Mg-doped p + -GaN contact layer. The samples are processed by standard contact lithography and the final de-vices consist of 1 mm × 1 mm mesas, 30/300 nm Pd/Au p-contact grids on the top of each mesa with a center-tocenter grid spacing of 200 µm, and a 30/300 nm Al/Au n-contacts around the base of each mesa [ Fig. 1(b)].
For comparison, commercial-grade LEDs produced by Nichia Corporation are also investigated: a UV LED (NCSU275 U395), in which the glass window has been removed to avoid partial absorption in the UV range of the exciting optical beam in BPCS measurements and in which the Zener diode has been removed to apply a wider range of reverse biases to the device; a bluish-green LED (NCSE 119AT), in which the silicone dome and the Zener diode were not removed. 29
B. Experimental set-up
The BPCS set-up consists of an Energetiq EQ-99 light source coupled to a TRIAX 180 monochromator with an output wavelength bandwidth of 2 nm (corresponding to about 15 meV in the range of interest of the present study). We note that for such value the convolution of an exponential decay with the Gaussian spectral line shape of the exciting beam does not affect the slope of the Urbach tail (i.e., the measurement of E U ). The spot size of the optical beam on the sample was ∼ 1 mm 2 , being comparable with the area of the mesas of the devices. The samples were biased with a Keithley 2400 current-voltage source. The optical beam was modulated with a chopper wheel at a frequency of 82 Hz and the photocurrent signal was measured with a Princeton Applied Research 5209 lock-in amplifier. Low-temperature measurements were carried out in a JANIS SHI-4 closed cycle cryostat.
III. RESULTS
BPCS measurements were carried out on the series of In x Ga 1−x N QW structures with indium composition varying between 0% and 28%, covering the UV to green range of the spectrum, and for applied biases varying from +1 V to -4 V (where the negative sign indicates a reverse bias). The results are presented in Fig. 2. The responsivity is obtained normalizing the photocurrent measured at a given photon energy to the incident optical power on the sample. Three different regions, named 'A', 'B' and 'C' (enclosed in Fig. 2 by boundaries), can be identified in the series of spectra according to their bias dependence. The bias dependence in these three regions is shown more in detail in Fig. 3, where the responsivity at some fixed energy value is recorded for a continuous variation of the applied bias from -4 to +1 V. The specific energy values chosen for each sample are indicated in Fig. 2. The photoluminescence (PL) peak energy of the solar cells measured at a bias of 0 V with different above-gap excitation laser sources is marked as stars in Fig. 2. The A-region is characterized by a weak increase in responsivity at increasing reverse bias [ Fig. 3(a)]. It is observed for all indium compositions (Fig. 2) and corresponds to absorption above the band gap of the QWs, E g . The latter is defined, similarly to Ref. 22, as the transition energy between the exponential onset of the Urbach tail and the saturation region. From this definition it can be seen in Fig. 2(b)-(f) that E g lies close to the intersection of the BPCS curves at the boundary between the A-and B-region. 30 A detailed description of the carrier escape mechanism from the QWs for excitation in the A-region is given in Ref. 31 according to which both tunneling and thermionic emission are expected to contribute, considering a barrier thickness of 7 nm as in our structures. No, or very little, disorder-induced effects are expected for this process, so the A region will not be discussed in the following.
The B-region is identified by an exponential decrease in responsivity at increasing reverse bias [ Fig. 3(b)], corresponding in the photocurrent spectra to an Urbach tail undergoing a blue-shift with increased reverse voltage (Fig. 2). Let us recall that in c-plane wurtzite InGaN/GaN structures, the electric field E QW points towards the substrate, as shown in the schematic of Fig. 2(f). 32 Increasing the reverse bias then partly compensates E QW leading to a blue-shift in the absorption edge. The B-region appears in the spectra for [In]=6% and its energy range progressively extends at larger indium concentrations. We remark that the PL peak energy of all the In x Ga 1−x N/GaN solar cells lies in the B-region which then is associated with transitions occurring in the QWs. Below-gap absorption processes giving origin to the B-region may depend on compositional disorder and QW electric field, as it will be further analyzed in Sec. IV.
None of these effects are expected to significantly affect below-gap absorption in semipolar GaN QWs, due to the reduced internal electric fields and absence of alloy fluctuations, explaining why the B-region is not observed in this sample [absorption curves in Fig. 2(a)]. Let us point out that this result does not exclude the occurrence of compositional disorder effects in semipolar (or non-polar) ternary InGaN layers, which were not investigated in the present work. 33,34 Finally, the C-region, in contrast with the B-region, is characterized by an exponential increase in responsivity at increasing reverse bias [ Fig. 3(c)] and is most clearly observed in the UV diodes. We interpret it as being due to type II well-to-barrier transitions, as detailed in the Appendix.
We remark that the B-and C-region originate from competing mechanisms of below-gap absorption which dominate, respectively, at high and low indium concentration. The sample with [In]=11% corresponds to the intermediate case in which the two mechanisms have comparable efficiencies as may be observed in the deviation from the typical responsivity trend of the B-and Cregion at large and small reverse bias, respectively [purple curves in Fig. 3(b and (c)].
In the following the dependence of the Urbach tail observed in the B-region will be examined as a function of different factors, such as bias, indium composition, sample temperature and type of device, specifically solar cells or LEDs both incorporating InGaN QWs.
Urbach energies are fitted in a typical interval of photon energies ranging from E g − 150 meV to E g − 20 meV for samples with mean indium composition varying between 11% and 28%. If carrier collection by BPCS were substantially dependent on the energy of the confined states in the QW, then one would expect that for a larger reverse bias applied to the solar cells the extraction efficiency of low-localization energy carriers would increase, while that of high-localization energy carriers would remain roughly constant, thus changing the slope of the measured Urbach tails. In contrast, the experimental fact that the E U values appear to be almost independent on the applied bias [ Fig. 4(a)] indicates that carrier collection is only weakly sensitive to the exciting photon energy in the B-region, corroborating our hypothesis of the responsivity spectra as proportional to optical absorption. The average of E U taken over the different applied biases and denoted as E U in Fig. 4(b) varies weakly with indium concentration for QWs emitting from the violet to the green range with the typical value being around 20 meV. In Fig. 5(a) BPCS measurements of the sample with [In] = 11% at room temperature and at 5 K can be compared. The increase in band gap energy at low temperature causes the photocurrent spectra to shift towards higher photon energy. But the bias dependence of the three different regions remains essentially unaffected [ Fig. 5(b)-(d)], except for the C-region in the lowbias range, a feature that is discussed in the Appendix. Remarkably, the slope of the Urbach tail is substantially unchanged by the sample temperature: E U is 15.8 ± 0.7 meV at 300 K and 14.7 ± 0.9 meV at 5 K, in agreement with the interpretation of the Urbach tail as being related with compositional disorder, a structural property of the QWs that is temperature independent, 35 therefore not due to phonon-assisted broadening.
Next we compare BPCS of UCSB solar cells with commercial LEDs to verify whether commercial-grade wafers and a different optimization of the optoelectronic devices, i.e., luminescence in LEDs vs. carrier collection in solar cells, may give rise to different features in the photocurrent spectra. Two commercial LEDs with electroluminescence (EL) spectra similar to those of two UCSB solar cells with [In]=11% and 28% are investigated [ Fig. 6(a),(c)]. The corresponding sets of BPCS measurements exhibit strong similarities, apart from a small energy shift likely due to the difference in QW composition. In particular, in the case of UV devices [ Fig. 6(b)] the same A, B, C regions characterized by their bias dependence, as previously discussed, are observed. Green-emitting devices also exhibit common features related to the A and B regions [ Fig. 6(d)], though the application of a negative bias to the commercial green LED is severely limited by a Zener diode in parallel with the diode. The Urbach energies of the commercial devices are 21.4 ± 1.1 meV and 20.5 ± 0.5 meV for the UV and green LED, respectively, being closely in the range of E U measured in the UCSB solar cells. This indicates that E U is essentially unaffected by any possible difference in material quality. The overall close response in the measured spectra of the different devices suggests that BPCS allows carrying out a spectroscopy of intrinsic optoelectronic properties of InGaN QWs.
IV. THEORETICAL MODEL
In this section the possible absorption mechanisms giving origin to the Urbach tail of the B-region are analyzed by means of different models. Considering the strong polarization-induced electric fields present in c-plane In-GaN/GaN QWs, typically of the order of few MV/cm in the range of [In] considered here, the first hypothesis we formulate is that the measured Urbach tails orig- inate from FK, a well-known mechanism of below-gap absorption in bulk semiconductors. In presence of an electric field E QW the electron and hole wave functions, which are Airy functions with tails extending in the classically forbidden region, overlap even for transition energies hν < E g .
Since we are dealing with QW structures, we should consider the quantum-confined Franz-Keldysh effect (QCFK), 36 shown schematically in Fig. 7(a), which corresponds to the FK in the limit of thin slabs of material. The absorption curve of an ideal QW should exhibit a step-like increase with photon energy due to the discrete nature of the allowed interband transitions. As a consequence of the symmetry breaking caused by the electric field, the major effect of QCFK is to change the selection rules in the QW and permit "forbidden" transitions.
To evaluate the influence of E QW in below-gap absorption we simulate the absorption curves of the In 0.17 Ga 0.83 N/GaN MQW solar cell structure as a function of bias using a 1D Schrödinger-Poisson-driftdiffusion solver 37,38 that assumes homogeneous QWs, i.e., without alloy disorder, but including thickness fluctuations of ±1 monolayer among the different QWs. 39 The polarization fields are calculated from the spontaneous and piezoelectric polarizations of nitride QWs (parameters given in Ref. 16). The quantum mechanical expression of the absorption coefficient of Ref. 36 is used, taking into account both transitions from the heavy hole and light hole bands. The simulated absorption spectra are shown in Fig. 7(b). The blue-shift of the absorption edge at increasing reverse bias due to the partial compensation of E QW is quantitatively well reproduced in the simulations. Nevertheless the simulated curves exhibit a step-like increase due to the small thickness of the QW layers 36 and QCFK alone is not sufficient to explain the observed experimental broadening of the absorption edge [cf. Fig. 2(d)]. Based on low temperature measurements discussed in Sec. III phonon-assisted broadening of the absorption edge has already been excluded. The main remaining mechanism of broadening is compositional disorder. Therefore to properly describe the Urbach tails of InGaN layers we develop a 3D absorption model incorporating both the effects of the electric field and compositional disorder in the QW.
In principle, to compute the absorption coefficient of a QW, solving the Schrödinger equation is necessary to obtain the localized eigenstates and corresponding eigenenergies needed to calculate the electron-hole overlap and energies of the different interband optical transitions. 24 In addition to this, when simulating biased devices, such as the solar cells considered in this work, also the Poisson and drift-diffusion (DD) equations have to be solved to account for the charge distribution and carrier transport. Note that solving the coupled Schrödinger-Poisson-DD equations for a 3D device structure incorporating compositional disorder can be very demanding in terms of computational resources (see LL3, Ref. 16). Here we employ an alternative approach based on the LL theory 27 which is able to capture the confining properties of a disordered system without solving the associated Schrödinger equation
Hψ = E ψ ,(2)
where H = − 2 /2m ∆ + V is the Hamiltonian, V is the potential energy of the disordered system and ψ and E are the eigenstate and eigenenergy, respectively. Instead, a much simpler linear problem defined by
Hu = 1(3)
is solved. Arnold et al. recently demonstrated that the inverse of this landscape, i.e. 1/u, acts as an "effective" confining potential seen by the localized eigenstates. 28 In other words, the quantum properties of a disordered potential such as quantum wave interference, quantum confinement and tunneling are directly translated into 1/u in the form of a classical potential. An important property of the function 1/u (or u) is that its crest (respectively, valley) lines partition the domain into the localization subregions, i.e., the regions of lower energy solutions of the Schrödinger equation (LL1, Ref. 17). This allows a very efficient computation of quantum effects in disordered semiconductors, and will be used in the following to compute the absorption spectra of the measured nitride solar cells. A schematic of the typical simulated InGaN/GaN QW structure is shown in Fig. 8(a). The size of the simulated domain is 30 nm × 30 nm × 255 nm. For the ease of computation the MQW region of the measured InGaN/GaN solar cells is substituted by an equivalent intrinsic layer with a single QW. Since the 10 QWs of each solar cell are nominally identical with very similar electric field conditions, the single QW model represents a fair approximation of the MQW structure in terms of the absorption spectrum. For each solar cell, the simulations are repeated for 10 different random configurations of the indium map and finally averaged to produce an average absorption curve. The indium atoms are first randomly distributed in the QW on an atomic grid with spacing a=2.833Å corresponding to the average distance between cations in GaN. Then at each grid site the local indium composition is averaged via the Gaussian averaging method (see LL3, Ref. 16), allowing to smooth the rapidly oscillating distribution of atoms and to obtain a continuous fluctuating potential. In order to preserve the disorder effects on the electronic properties of the QW, the length scale of the averaging must be smaller or comparable to the typical scale of the effective potential fluctuations seen by the carriers. We fixed the Gaussian broadening parameter to a value of 2.0a≈0.6 nm, which satisfies the aforementioned condition (LL3, Ref. 16). By taking the plane-averaged values of the resulting indium composition we obtain an indium profile along the growth direction, whose maximum defines the indium concentration and whose FWHM defines the thickness of the QW. In the following simulations we model four solar cell structures with InGaN QWs having an indium composition of 12%, 18%, 22%, 28% and a corresponding FWHM of 1.6 nm, 1.8 nm, 1.8 nm and 2.1 nm. These values allow to align the threshold of the simulated absorption spectra with that of the experimental curves, while remaining within ∼1% of the indium content and 0.1 nm of the QW thickness values determined by XRD in the experimental structures. Moreover, to include in the simulations the fluctuations in the width of the QWs, we consider in the process of random indium atoms generation a circular region with one additional monolayer [ Fig. 8(b)], similarly to Ref. 18. The disk occupies roughly 1/3 of the entire QW area.
All the material parameters needed in the simulations are assigned according to the local composition by an interpolation method as bias applied to the simulated solar cell the 3D LLs are computed. As an example, 2D cuts of the "effective" confining potentials of electrons and holes, 1/u e and 1/u h , in the middle of the QW of an In 0.11 Ga 0.89 N/GaN structure at 0 V are shown in Fig. 8(c), where the network of localization subregions is highlighted (thick lines). We shall now see how to exploit the LLs to compute absorption in these structures.
E InxGa1−xN g = (1 − x) E GaN g + x E InN g − 1.4 x(1 − x), E AlxGa1−xN g = (1 − x) E GaN g + x E AlN g − 0.8 x(1 − x), ε InxGa1−xN r = (1 − x) ε GaN r + x ε InN r , ε AlxGa1−xN r = (1 − x) ε GaN r + x ε AlN r , m * ,InxGa1−xN = (1 − x)/m * ,GaN + x/m * ,InN −1 , m * ,AlxGa1−xN = (1 − x)/m * ,GaN + x/m * ,
In the companion paper LL1 (Ref. 17), the LL theory is applied to the framework of semiconductors to derive a model accounting for quantum effects in disordered materials. In particular it is shown that the landscape u allows estimating in each localization subregion Ω m the fundamental localized eigenfunction and its corresponding eigenenergy as:
ψ (m) 0 ≈ u u (5) E (m) 0 ≈ u|1 u 2(6)
where the expressions above must be evaluated in the considered subregion and the L 2 -normalization of u is used. Moreover the following expression for the absorption coefficient of a disordered QW is derived
α(hv) = 2 3 · C · JDOS (m,n) 3D (hv) I (m,n) 0,0(7)
where C = πe 2 |p cv | 2 /m 2 0 n r ε 0 hν is a prefactor depending on the real part of the surrounding refractive index n r and the interband momentum matrix element p cv , and where the summation is carried out over all electron and hole subregions (labeled m and n, respectively). Superimposing the maps of the two landscapes [ Fig. 8(c)], several subregions which consist in intersections of the localization subregions of electrons and holes can be defined. The joint density of states of each of these electron-hole "overlapping" subregions is given by states of the m-th electron and n-th hole subregions. The overlap factor is determined by the electron and hole fundamental eigenstates as
JDOS (m,n) 3D (hν) = √ 2m 3/2 r π 2 3 hν − E g − E (m) e,0 − E (n) h,0 ,(8)I (m,n) 0,0 = | ψ (m) e,0 |ψ (n) h,0 | 2 ψ (m) e,0 2 ψ (n) h,0 2(9)
Note that the eigenenergies and eigenfunctions appearing in Eq. (8) and (9) can be directly computed from the LLs of electrons and holes using Eq. (5) and (6), respectively, without having to resort to the Schrödinger equation. In support of this absorption model, we have shown in Sec. IV B of LL1 (Ref. 17) that the absorption spectrum calculated in a 1D disordered superlattices by the landscape theory agrees very well with the exact 1D Schrödinger calculation. Urbach tails computed from the 3D absorption model for an In 0.17 Ga 0.83 N/GaN structure at different applied biases are shown in Fig. 9(a) and can be compared with the experimental Urbach tails measured by BPCS of the In 0.17 Ga 0.83 N/GaN solar cell in Fig. 9(b). The 3D nature of the model allows reproducing both the broadening and the bias-dependence of the experimental Urbach tails which are mainly defined by the in-plane compositional disorder and the QW electric field along the growth direction, respectively. The discrepancy in the magnitude of the blue-shift, differing by a factor ∼2 between theory and experiments, may be due to a difference in the fraction of reverse voltage dropped across the MQW active region of the experimental structures and the single QW embedded in an equivalent intrinsic layer used in the simulations. However it should be noted that a quantitative description of the bias-dependence is secondary in terms of the present study, which aims at determining the disorder-induced broadening of the absorption edge, and the observed qualitative agreement is already a good indication that the LL model includes as well QCFK effects. In Fig. 10 the simulated absorption curves of the InGaN/GaN solar cells at 0 V for a mean indium composition ranging between 11% and 28% can be compared with the corresponding experimental measurements. Note that the absorption model only considers optical transitions occurring within the QW which give origin to the A-and B-region observed in the experiments, while type-II well-to-barrier transitions responsible for the C-region and unrelated with disorder (see Appendix) are not taken into account. The simulations reproduce well the absolute energy position and the broadening of the absorption edge as a function of the mean indium content of the QWs. Let us remark that the absorption model incorporating compositional alloy disorder, well width fluctuations and electric field effects gives a lower bound on the Urbach energies, explaining why the simulated tails slightly underestimate the experimental measurements. There may be other broadening processes (e.g. defect-induced smearing of band edges) which could decrease the slope of the below-gap absorption tail (thus increase E U ) but these do not seem to play an important role as the agreement with the experiments appears to be already good.
Finally let us detail why the observed E U values of ∼20 meV [ Fig. 4(b)] may be surprising and the agreement between the theory and the experiments is remarkable. In the considered range of indium concentrations, compositional disorder is expected to produce large potential fluctuations in the InGaN/GaN QWs. This is clearly observed in the simulations of the conduction band potential E c for [In]=11% and [In]=28% shown in Fig. 11(a)-(c), where the 2D maps correspond to the mid-plane of the QW and the oscillating curves correspond to 1D cuts of these maps. The amplitude of the E c fluctuations, characterized by the standard deviation σ V , ranges between 38 meV ([In]=11%) and 55 meV ([In]=28%). Intuitively, one could think that the typical size of the potential energy fluctuations of the QW defined by σ V should be comparable to the broadening parameter E U of the below-gap absorption tails, both being directly related to the DOS of the system. However the E U values measured in the InGaN solar cells are quite smaller than σ V . Our model based on the LL theory allows explaining this discrepancy. Indeed by computing the effective potentials 1/u e [ Fig. 11(d)-(f)] corresponding to the conduction band maps [ Fig. 11(a)-(c)] of the InGaN QWs it can be observed that the amplitude of the fluctuations of the "effective" potential, characterized by the standard deviation σ 1/u , is much reduced with respect to σ V . The corresponding simulations for the case of holes show a smaller change between the amplitude of the fluctuations of E v [ Fig. 11(g)-(i)] and 1/u h [ Fig. 11(j)-(l)], due to the fact that holes have less confinement energy than electrons, therefore are more localized and thus more sensitive to the modulating potential. This is an indication of stronger hole localization than electron one, a phenomenon predicted by different models in the literature. 18,42 For both electrons and holes the values of σ 1/u lie in the 25-35 meV range, being closer than σ V to the experimental E U values. The quantum effects of confinement and tunneling, which diminish the extreme values of the potential fluctuations, have been taken into account by the wave equation Eq. (2) and thus diminish the effective energy fluctuations. This result is extended in Fig. 12 showing the dependence of σ V and σ 1/u on different indium concentrations for both the electron and hole landscapes. These observations suggest that the model based on 1/u finely captures the effective confining potential seen by the carriers in the disordered InGaN layers, which ultimately allows correctly computing the broadening of the absorption edge of the InGaN/GaN QWs.
V. CONCLUSION
In this work we investigated, both with experiments and theory, the effect of compositional disorder in the Urbach tails of InGaN layers with different indium composition. Experimentally, Urbach tails can be measured by means of biased photocurrent spectroscopy of InGaN/GaN solar cells and LEDs. The different devices produce very similar spectra indicating that a spectroscopy of intrinsic optoelectronic properties of InGaN QWs can be carried out by BPCS. A detailed analysis of the measured responsivity spectra is required in order to identify different below-gap absorption mechanisms. Below-gap absorption due to type-II well-tobarrier transitions, particularly efficient in near-UV emitting QWs, was observed and modeled. A different mechanism of below-gap absorption corresponding to transitions within the QW was observed to be responsible for the broadening of the absorption edge of InGaN QWs with [In] > 6%. A 1D model only considering the effect of QW electric field on absorption due to QCFK allows to quantitatively reproduce the blue-shift seen in the experiments but does not account for the experimental broadening of the absorption edge. Thermal disorder is excluded since the experimental broadening of the Urbach tails is essentially unchanged in measurements at 5 K. A 3D absorption model based on the LL theory is used to efficiently compute absorption spectra of InGaN/GaN QW structures without having to resort to the Schrödinger equation. The model takes into account compositional disorder, well width fluctuations and electric field effects in the QW and this is sufficient to reproduce quite closely the broadening and the bias-dependence of the experimental Urbach tails. The reason for the agreement between experiments and theory lies in the ability of the LL theory to describe the effective potentials seen by the carriers in the disordered InGaN layers.
Appendix: Type-II well-to-barrier transitions
The mechanism at the origin of the the C-region observed in the BPCS measurements at below-gap photon energies is modeled in the following as a type-II well-tobarrier transition, similarly to the process suggested in Ref. 25. The final state of the transition corresponds to a hole occupying the ground state of the QW and an electron excited in the barrier [ Fig. 13(a)], which should correspond to the electronic process giving the dominant contribution to the photocurrent signal for type-II absorption because of the light mass of the unbound carrier. Due to the presence of the barrier and QW electric fields both wave functions are Airy functions.
Already at this point we can remark that the present model may explain why the C-region is well observed in UV diodes but progressively disappears going in the visible range. Indeed increasing the indium content of the QW increases the well-barrier conduction band offset ∆E c and decreases the threshold photon energy for below-gap transitions. This in turn spatially shifts the final state electron Airy function further from the bound hole wave function, decreasing the e-h overlap and thus the probability amplitude of type-II transitions, as schematized in Fig. 13(b). For this reason other mechanisms of below-gap absorption, namely compositional disorder and QCFK in the QW, are observed to dominate in the spectra of the measured solar cells with high indium composition and large well-barrier conduction band offset.
The final electron state ψ e is calculated in the bulk approximation as an Airy function with parameters defined
· − E e | E b | + z (A.1)
where E b is the electric field in the barrier of the structure, m e and E e are the electron effective mass and energy, respectively, and z is the growth direction. The hole ground state ψ h,0 is calculated analytically from the Schrödinger equation in the approximation of an infinitely deep potential well of width L QW in presence of an electric field E QW : where Ai and Bi are Airy functions; Z h,0 (E h,0 , z) is defined as 2m h e| E QW |/ 2 ) 1/3 · E h,0 | E QW | + z ; m h is the hole effective mass; the coefficient c h,0 and the hole energy E h,0 are obtained from the boundary conditions ψ h,0 (E h,0 , 0) = ψ h,0 (E h,0 , L QW ) = 0. Then the optical absorption, or equivalently the imaginary part of the optical susceptibility Im(χ), will be proportional to the overlap integral I(E e , E h,0 ) as: Im(χ) = I(E e , E h,0 ) δ(E e + E h,0 + E g,QW
+ ∆E c − hν) dE e (A.3)
where I (E e , E h,0 ) = | L QW 0 ψ * e (E e , z)ψ h,0 (E h,0 , z) dz| 2 with the integral boundaries fixed by the boundary conditions of ψ h,0 ; E g,QW is the energy gap of the QW; ∆E c is assumed to be 80% of the band offset of the heterojunction.
Inserting the electron and hole wave functions obtained from Eq. (A.1) and (A.2) into Eq. (A.3) type-II absorption is calculated as a function of applied bias for a single QW structure representative of the measured solar cell with [In]=11%. It is assumed E g,QW =3.23 eV as the experimental value measured at 5 K, and the biasdependence of E QW and E b is obtained from 1D Poisson-DD simulations of the device. 37 The results of the model are shown in Fig. 13(c). A qualitative agreement is observed between the simulated absorption curves and the experimental features of the C-region shown in Fig. 5(a): increasing the reverse bias from +1 V to -4 V the absorption rapidly increases while the slope of the tail decreases. The change in absorption as a function of bias at a fixed photon energy [marked 'C' in Fig. 13(c)] is plotted in Fig. 13(d). Qualitatively, the nearly exponential increase in absorption resembles the experimental increase in responsivity measured at 5 K shown in Fig. 5(d). The larger signal observed in the experiments at 300 K in the ∼[-1,+1] V range is attributed to additional mechanisms of below-gap absorption ex-citing carriers in the QW, namely disorder and FK. Indeed, while type-II transitions directly create carriers in the barrier ready to contribute to the photocurrent signal (if we neglect subsequent recapture by other QWs), other mechanisms involving absorption in the QW will require thermal energy in order to allow thermionic emission from the QW and contribute to photocurrent. These different processes are schematized in Fig. 5(e).
The proposed model gives considerable qualitative insights into the mechanism responsible for the observation of the C-region. Finally, we remark that while the model only considers type-II well-to-barrier transitions, in the experimental C-region it may be present, even at 5 K, a residual contribution to the responsivity signal due to other mechanisms of absorption. This may be the source of quantitative disagreement on the rate of the increase of the modeled absorption vs. the experimental responsivity as a function of bias, in addition to the approximations used (ψ e in the bulk limit, ψ h,0 in the infinitely deep well limit) and the sensitivity of the model on different parameters, such as ∆E c , E QW , E b , which have been assumed here.
PACS numbers: 71.23.An, 72.15.Rn, 03.65.Ge
FIG. 1 .
1(a) Schematic of the InGaN/GaN solar cell structures grown by MOCVD. (b) Contact geometry of the processed devices.
n-GaN layer ([Si]=6 × 10 18 cm −3 ), a 10 nm highly Sidoped n + -GaN layer ([Si]=2 × 10 19 cm −3 ), a 10 period undoped MQW active region with ∼ 2 nm In x Ga 1−x N QWs (the QW thickness measured by XRD slightly increases with [In] in the different solar cells: 1.6 nm for 6%, 1.7 nm for 11%, 1.8 nm for 17%, 1.9 nm for 22% and 2.1 nm for 28%) and 7 nm GaN barriers, a 40 nm highly Mg-doped p + -GaN layer ([Mg]=5 × 10 19 cm −3 ), a 30 nm Mg-doped p-GaN layer ([Mg]=2 × 10 19 cm −3 ) and a 15 nm highly Mg-doped p + -GaN contact layer.
FIG. 2 .
2Responsivity measurements by BPCS of (a) m-plane UV GaN/AlGaN LED, and (b)-(f) c-plane InxGa1−xN/GaN solar cells with indium composition ranging between 6% and 28 The curves correspond to different applied biases to the devices (the negative sign indicates a reverse bias). Three regions having a different bias dependence and named 'A', 'B', 'C' are enclosed by boundaries. The PL peak energy of the solar cells measured at 0 V is marked as stars. The schematic in (f) shows the conduction band diagram of a cplane InGaN/GaN QW structure in which the QW electric field is partly compensated under application of a reverse bias.
FIG. 3 .
3(a)-(c) Change in responsivity measured by BPCS as a function of applied bias at exciting photon energies belonging to the A, B, C regions of the spectra for InxGa1−xN QW structures with different indium composition. The photon energies corresponding to the different curves are marked as 'A', 'B', 'C' in Fig. 2.
FIG. 4 .
4(a) Dependence on applied bias of the Urbach energy EU determined in the B-region of BPCS spectra for InxGa1−xN QW layers with different mean indium content. (b) Mean values of EU averaged over the applied biases and plotted as a function of the mean indium content of the corresponding InGaN/GaN solar cell.
FIG. 5 .
5(a) Responsivity measurements by BPCS of an In0.11Ga0.89N/GaN solar cell at 300 K (continuous lines) and 5 K (dashed lines) as a function of applied bias. The low temperature curves are blue-shifted due to the increase in bandgap energy. The A, B, C regions are enclosed by boundaries. (b)-(d) Change in responsivity as a function of applied bias at 300 K (continuous lines) and 5 K (dashed lines) for exciting photon energies belonging to the different regions and marked in (a) as A, B, C and A, B, C, at 300 K and 5 K, respectively. (e) Different mechanisms of carrier extraction from the QW. At low temperature tunneling outside the QW (T) may occur via type-II transitions. At room temperature, in addition to tunneling, absorption in the QW followed by thermionic emission (TE) may also occur.
FIG. 6 .
6(a),(b) EL spectra measured at 5 mA injection and BPCS measurements of an In0.11Ga0.89N/GaN solar cell (continuous lines) and a commercial UV LED (dashed lines). (c),(d) EL spectra measured at 5 mA injection and BPCS measurements of an In0.28Ga0.72N/GaN solar cell (continuous lines) and a commercial bluish-green LED (dashed lines).
FIG. 7 .
7(a) Schematic of QCFK absorption in a QW. In presence of an electric field EQW below-gap absorption at energy hν < Eg is allowed by the overlap of the electron and hole wave functions. (b) Simulation of QCFK absorption curves as a function of applied bias for an In0.17Ga0.83N/GaN MQW solar cell structure including thickness fluctuations among the different QWs. Alloy disorder is not taken into account in these simulations.
FIG. 8 .
8(a) Schematic of the typical InGaN/GaN QW structure used in the 3D absorption model. (b) The simulated InGaN QW incorporates a disk that is one monolayer thick representing a well width fluctuation. (c) "Effective" electron and hole confining potentials obtained from the LL theory for an In0.11Ga0.89N/GaN structure at 0 V and computed self-consistently with the Poisson-DD equations. The maps shown here correspond to 2D cuts of the 3D landscapes in the middle plane of the QW.
where m r = (1/m e + 1/m h ) −1 is the reduced effective mass, and E (m) e,0 and E (n) h,0 are the fundamental localized FIG. 9. Urbach tails of In0.17Ga0.83N/GaN structures: (a) computed from the 3D absorption model including compositional disorder and QW electric field, and (b) experimentally measured by BPCS. Curves correspond to different applied biases to the structure. The dashed line is shown as a reference and corresponds to an Urbach energy of 20 meV.
FIG. 10 .
10Absorption curves for InGaN/GaN structures of different indium composition at 0 V computed using the 3D theoretical model incorporating compositional disorder and QW electric field as a function of different smoothing parameters (dashed line). The responsivity curves measured by BPCS of InGaN/GaN solar cells of different indium composition at 0 V are also shown (continuous lines).
FIG. 11 .
11(a)-(b) Simulated conduction band potential maps of InGaN QWs for two different indium compositions (11% and 28%). All 2D cuts in this figure are performed just above z = 141.5 nm (see Fig. 8), i.e., at the bottom of the QW. (c) 1D cuts along the dashed lines of the Ec maps showing the fluctuating potential. (d)-(e) Electron effective confining potentials calculated from the LL theory based on the Ec maps shown in (a)-(b). (f) 1D cuts along the dashed lines of 1/ue. The amplitude of the fluctuations is much reduced with respect to that of the corresponding Ec potentials. The simulated maps and 1D profiles corresponding to the case of holes are shown in (g)-(l). The circular shaped fluctuation observed in the center of (a),(b),(d),(e),(g),(h),(j),(k) is due to the single monolayer disk fluctuation (see Fig. 8).
FIG. 12 .
12Standard deviations characterizing the amplitude of the fluctuations in InGaN QWs of different indium composition for (a) the conduction and valence band potentials, and (b) the corresponding electron and hole effective confining potentials.
FIG. 13 .
13(a) Schematic of type-II well-to-barrier transitions in an InGaN/GaN QW. In presence of barrier and QW electric fields the electron and hole final states, ψe and ψ h,0 , are Airy functions with tails extending in the gap and allowing below-gap absorption at photon energies hν < Eg,QW . (b) Schematic explaining why type-II transitions have a higher probability in InGaN/GaN structures with low indium content. Band diagrams are calculated with a 1D Poisson-DD solver and show realistic QW electric fields. (c) Simulated type-II absorption curves for an In0.11Ga0.89N/GaN QW as a function of applied bias. (d) Change in the simulated type-II absorption as a function of bias at a photon energy marked as 'C' in (c). by the barrier material: 36 ψ e (E e , z) = Ai 2m e · e · | E
ψ h, 0 (
0E h,0 , z) = Ai [Z h,0 (E h,0 , z)] + c h,0 Bi [Z h,0 (E h,0 , z)] (A.2)
AlGaN and InGaN parameters are given inTable I. The new self-consistent Poisson-DD-landscape approach presented in LL1 (Ref. 17) is used to solve the Poisson and DD equations self-consistently with Eq. (3) accounting for quantum effects of disorder. Before solving the Poisson equation, the spontaneous and piezoelectric polarization fields are calculated over the entire structure, as detailed in LL3 (Ref. 16). For each givenAlN −1
(4)
where the GaN, Eg
εr
m e m ⊥
e
m hh m lh
units (eV)
(m0) (m0) (m0) (m0)
GaN 3.437 10.4 0.21 0.20 1.87 0.14
InN 0.61 15.3 0.07 0.07 1.61 0.11
AlN
6.0 10.31 0.32 0.30 2.68 0.26
Bandgap alloy
InGaN: 1.4
bowing parameter
AlGaN: 0.8
TABLE I. Band structure parameters for wurtzite nitride al-
loys: bandgap, relative permittivity, and effective masses. 40,41
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"POSSIBLE INDICES FOR THE GALOIS IMAGE OF ELLIPTIC CURVES OVER Q",
"POSSIBLE INDICES FOR THE GALOIS IMAGE OF ELLIPTIC CURVES OVER Q"
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"6, 8, 10, 12, 16, 20, 24, 30, 32, 36, 40, 48, 54, 60, 72, 84, 96, 108, 120, 144, 240, 288, 336112, 192, 220, 360, 384, 504, 576, 768, 864, 1152, 1200, 1296, 1536"
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For a non-CM elliptic curve E/Q, the Galois action on its torsion points can be expressed in terms of a Galois representation ρE : Gal Q := Gal(Q/Q) → GL2( Z). A well-known theorem of Serre says that the image of ρE is open and hence has finite index in GL2( Z). We will study what indices are possible assuming that we are willing to exclude a finite number of possible j-invariants from consideration. For example, we will show that there is a finite set J of rational numbers such that if E/Q is a non-CM elliptic curve with j-invariant not in J and with surjective mod ℓ representations for all ℓ > 37 (which conjecturally always holds), then the index [GL2( Z) : ρE(Gal Q )] lies in the set
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"https://arxiv.org/pdf/1508.07663v2.pdf"
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1508.07663
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91e2e2c3f7ec39f0fd7a8c9e37da0d8c3eb03e72
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POSSIBLE INDICES FOR THE GALOIS IMAGE OF ELLIPTIC CURVES OVER Q
18 Jan 2022
David Zywina
6, 8, 10, 12, 16, 20, 24, 30, 32, 36, 40, 48, 54, 60, 72, 84, 96, 108, 120, 144, 240, 288, 336112, 192, 220, 360, 384, 504, 576, 768, 864, 1152, 1200, 1296, 1536
POSSIBLE INDICES FOR THE GALOIS IMAGE OF ELLIPTIC CURVES OVER Q
18 Jan 2022arXiv:1508.07663v2 [math.NT]
For a non-CM elliptic curve E/Q, the Galois action on its torsion points can be expressed in terms of a Galois representation ρE : Gal Q := Gal(Q/Q) → GL2( Z). A well-known theorem of Serre says that the image of ρE is open and hence has finite index in GL2( Z). We will study what indices are possible assuming that we are willing to exclude a finite number of possible j-invariants from consideration. For example, we will show that there is a finite set J of rational numbers such that if E/Q is a non-CM elliptic curve with j-invariant not in J and with surjective mod ℓ representations for all ℓ > 37 (which conjecturally always holds), then the index [GL2( Z) : ρE(Gal Q )] lies in the set
it is uniquely determined up to conjugacy by an element of GL 2 (Z/N Z). By choosing bases compatibly for all N , we may combine the representations ρ E,N to obtain a single Galois representation ρ E : Gal Q → GL 2 ( Z) that describes the Galois action on all the torsion points of E, where Z is the profinite completion of Z. If E is non-CM, then the following theorem of Serre [Ser72] says that the image is, up to finite index, as large as possible.
Theorem 1.1 (Serre). If E/Q is a non-CM elliptic curve, then ρ E (Gal Q ) has finite index in GL 2 ( Z).
Serre's theorem is qualitative, and it natural to ask what the possible values for the index [GL 2 ( Z) : ρ E (Gal Q )] are. Our theorems address this question assuming that we are willing to exclude a finite number of exceptional j-invariants from consideration; we will see later that the index [GL 2 ( Z) : ρ E (Gal Q )] depends only on the j-invariant j E of E.
The most difficult part of Serre's proof of Theorem 1.1 is to show that there is an integer c E such that ρ E,ℓ (Gal Q ) = GL 2 (Z/ℓZ) for all ℓ > c E . In [Ser72, §4.3], Serre asks whether one can choose c E independent of the elliptic curve (moreover, he asked whether this holds with c E = 37 [Ser81,p. 399]). We formulate this as a conjecture.
(i) Assuming Conjecture 1.2, Theorem 1.4 and Serre's theorem implies that there is an absolute constant C such that [GL 2 ( Z) : ρ E (Gal Q )] ≤ C for all non-CM elliptic curves E over Q. (ii) The set J in Theorem 1.4 contains more than the thirteen j-invariants coming from those elliptic curves over Q with complex multiplication. For example, the set J contains −7 · 11 3 and −7 · 137 3 · 2083 3 which arise from the two non-cuspidal rational points of X 0 (37), see [Vél74]. If E/Q is an elliptic curve with j-invariant −7 · 11 3 or −7 · 137 3 · 2083 3 , then one can show that [GL 2 ( Z) : ρ E (Gal Q )] ≥ 2736. (iii) In our proofs of Theorems 1.3 and 1.4, the finite set J that arises is ineffective. The ineffectiveness arises from an application of Faltings' theorem to a finite number of modular curves of genus at least 2.
1.2. Overview. In §2, we show that the index of ρ E (Gal Q ) in GL 2 ( Z) depends only on its commutator subgroup. In §3, we give some background on modular curves; for a fixed group G of GL 2 (Z/N Z) containing −I, its rational points will describe the elliptic curves E/Q with j E / ∈ {0, 1728} for which ρ E,N (Gal Q ) is conjugate to a subgroup of G.
In §4, we prove a version of Theorem 1.3 with I replaced by another finite set I that is defined in terms of the congruence subgroups of SL 2 (Z) with genus 0 or 1. Here we use Faltings' theorem to deal with rational points of several modular curves with genus at least 2.
In §5, we describe how to compute the set I ; it agrees with our set I. Here, and throughout the paper, we avoid computing models for modular curves. For a genus 0 modular curve, we use the Hasse principle to determine whether it is isomorphic to P 1 Q . We compute the Jacobian of genus 1 modular curves, up to isogeny, by counting their F p -points via the moduli interpretation. We also make use of the classification of genus 0 and 1 congruence subgroups due to Cummin and Pauli.
Finally, in §6 we complete the proofs of Theorems 1.3, 1.4 and 1.5.
1.3. Notation. Fix a positive integer m. Let Z m be the ring that is the inverse limit of the rings Z/m i Z with respect to the reduction maps; equivalently, the inverse limit of Z/N Z, where N divides some power of m. We will make frequent use of the identifications Z m = ℓ|m Z ℓ and Z = ℓ Z ℓ , where ℓ denotes a prime. In particular, Z m depends only on the primes dividing m. For a subgroup G of GL 2 (Z/mZ), GL 2 (Z m ) or GL 2 ( Z) and an integer N dividing m, we denote by G(N ) the image of the group G in GL 2 (Z/N Z) under reduction modulo N .
All profinite groups will be considered with their profinite topologies. The commutator subgroup of a profinite group G is the closed subgroup G ′ generated by its commutators.
For each prime p, let v p : Q × ։ Z be the p-adic valuation. use of some of the Magma code from [Sut15]. The computations in §5 were performed using the Magma computer algebra system [BCP97]; code can be found at https://github.com/davidzywina/PossibleIndices
The commutator subgroup of the image of Galois
Let E be a non-CM elliptic curve defined over Q. Using the Weil pairing on the groups E[N ], one can show that the homomorphism det •ρ E : Gal Q → Z × is equal to the cyclotomic character χ. Recall that χ : Gal Q → Z × satisfies σ(ζ) = ζ χ(σ) mod n for any integer n ≥ 1, where ζ ∈ Q is an n-th root of unity and σ ∈ Gal Q .
We first show that index of ρ E (Gal Q ) in GL 2 ( Z) is determined by its commutator subgroup.
Proposition 2.1. We have [GL 2 ( Z) : ρ E (Gal Q )] = [SL 2 ( Z) : ρ E (Gal Q ) ′ ].
Proof. The character χ is surjective, so det(ρ E (Gal Q )) = Z × and hence ρ E (Gal Q ) ∩ SL 2 ( Z) = ρ E (Gal Q cyc ), where Q cyc is the cyclotomic extension of Q. We thus have
[GL 2 ( Z) : ρ E (Gal Q )] = [SL 2 ( Z) : ρ E (Gal Q ) ∩ SL 2 ( Z)] = [SL 2 ( Z) : ρ E (Gal Q cyc )].
It thus suffices to show that ρ E (Gal Q cyc ) equals ρ E (Gal Q ab ) = ρ E (Gal Q ) ′ , where Q ab ⊆ Q is the maximal abelian extension of Q. This follows from the Kronecker-Weber theorem which says that Q cyc = Q ab .
Remark 2.2.
(i) One can show that there are infinitely many different groups of the form ρ E (Gal Q ) as E varies over non-CM elliptic curves over Q; moreover, there are infinitely many such groups with index 2 in GL 2 ( Z). One consequence of Proposition 2.1 is that to compute the index
[GL 2 ( Z) : ρ E (Gal Q )] one does not need to know the full group ρ E (Gal Q ), only ρ E (Gal Q ) ′ .
Conjecturally, there are only a finite number of subgroups of SL 2 ( Z) of the form ρ E (Gal Q ) ′ with a non-CM E/Q. Indeed, suppose that Conjecture 1.2 holds. Remark 1.6(i) and Proposition 2.1 implies that the index of [SL 2 ( Z) : ρ E (Gal Q ) ′ ] is uniformly bounded for non-CM E/Q. The finite number of possible groups of the form ρ E (Gal Q ) ′ follows from their only being finitely many open subgroup of SL 2 ( Z) of a given index.
(ii) For a non-CM elliptic curve E over a number field K, a similar argument shows that
[GL 2 ( Z) : ρ E (Gal K )] ≤ [ Z × : χ(Gal K )] · [SL 2 ( Z) : ρ E (Gal K ) ′ ].
The inequality may be strict if K = Q (the cyclotomic extension of K does not agree with the maximal abelian extension of K).
The following corollary show that for an elliptic curve E/Q, the index of ρ E (Gal Q ) in GL 2 ( Z) depends only on the Q-isomorphism class of E. In particular, the j-invariant is the correct notion to use in Theorems 1.4 and 1.5.
Corollary 2.3. For an elliptic curve E over Q, the index [GL 2 ( Z) : ρ E (Gal Q )] depends only on the j-invariant of E.
Proof. Suppose that E 1 and E 2 are elliptic curves over Q with the same j-invariant (and hence isomorphic over Q). If E 1 (and hence E 2 ) has complex multiplication, then both indices are infinite. We may thus assume that E 1 and E 2 are non-CM. Since they have the same j-invariant, E 1 and E 2 are isomorphic over a quadratic extension L of Q. Fixing such an isomorphism, we can identify the representations ρ E 1 | Gal L and ρ E 2 | Gal L . We have L ⊆ Q ab , so the groups ρ E 1 (Gal Q ab ) = ρ E 1 (Gal Q ) ′ and ρ E 2 (Gal Q ab ) = ρ E 2 (Gal Q ) ′ are equal under this identification. The corollary then follows immediately from Proposition 2.1.
Modular curves
Fix a positive integer N and a subgroup G of GL 2 (Z/N Z) containing −I that satisfies det(G) = (Z/N Z) × . Denote by Y G and X G , the Z[1/N ]-schemes that are the coarse space of the algebraic stacks M
G) = (Z/N Z) × .
In later sections, we will mostly work with the generic fiber of X G , which we will also denote by X G , which is a smooth, projective and geometrically irreducible curve over Q (similarly, we will work with the generic fiber of Y G which will be a non-empty open subvariety of X G ).
Fix a field k whose characteristic does not divide N ; for simplicity, we will also assume that k is perfect. Choose an algebraic closure k of k and set Gal k := Gal(k/k).
In §3.1, we use the moduli property of M • G [1/N ] to give a description of the sets Y G (k) and Y G (k). In §3.2, we describe the natural morphism from Y G to the j-line. In §3.3, we give a way to compute the cardinality of the finite set X ∞ G (k) of cusps of X G that are defined over k. In §3.4, we determine when the set Y G (R) is non-empty. In §3.5, we will observe that Y G (C) as a Riemann surface is isomorphic to the quotient of the upper-half plane by the congruence subgroup Γ G consisting of A ∈ SL 2 (Z) for which A modulo N lies G. Finally in §3.6, we explain how to compute the cardinality of X G (F p ) for primes p ∤ 6N .
3.1. Points of Y G . For an elliptic curve E over k, let E[N ] be the N -torsion subgroup of E(k). A G-level structure for E is an equivalence class [α] G of group isomorphisms α : E[N ] ∼ − → (Z/N Z) 2 ,
where we say that α and α ′ are equivalent if α = g • α ′ for some g ∈ G. We say that two pairs (E, [α] G ) and (E ′ , [α ′ ] G ), both consisting of an elliptic curve over k and a G-level structure, are isomorphic if there is an isomorphism φ :
E → E ′ of elliptic curves such that [α] G = [α ′ • φ] G , where we also denote by φ the isomorphism E[N ] → E ′ [N ], P → φ(P ). From [DR73, IV Definition 3.2], M • G [1/N ](k)
is the category with objects (E, [α] G ), i.e., elliptic curves over k with a G-level structure, and morphisms being the isomorphisms between such pairs. Since Y G is the coarse space of M • G [1/N ], we find that Y G (k) is the set of isomorphisms classes in
M • G [1/N ](k).
The functoriality of M • G [1/N ], gives an action of the group Gal k on Y G (k). Take any σ ∈ Gal k . Let E σ be the base extension of E/k by the morphism Spec k → Spec k coming from σ. The natural morphism E σ → E of schemes induces a group isomorphism E σ [N ] → E[N ] which, by abuse of notation, we will denote by σ −1 . More explicitly, if E is given by a Weierstrass equation y 2 + a 1 xy + a 3 y = x 3 + a 4 x + a 6 with a i ∈ k, we may take E σ to be the curve defined by y 2 + σ(a 1 )xy + σ(a 3 )y = x 3 + σ(a 4 )x + σ(a 6 ); the isomorphism E σ [N ] → E[N ] is then given by (x, y) → (σ −1 (x), σ −1 (y)). For a point P ∈ Y G (k) represented by a pair (E, [α] G ), the point
σ(P ) ∈ Y G (k) is represented by (E σ , [α • σ −1 ] G ).
Since k is perfect, Y G (k) is the subset of Y G (k) stable under the action of Gal k . The following lemma describes Y G (k). For an elliptic curve E over k, let E[N ] be the N -torsion subgroup of
E(k). Each σ ∈ Gal k gives an isomorphism E[N ] ∼ − → E[N ]
, P → σ −1 (P ) that we will also denote by σ −1 .
Lemma 3.1. (i) Every point P ∈ Y G (k) is represented by a pair (E, [α] G ) with E defined over k. (ii) Let P ∈ Y G (k) be a point represented by a pair (E, [α] G ) with E defined over k. Then P is an element of Y G (k)
if and only if for all σ ∈ Gal k , we have an equality
α • σ −1 = g • α • φ of isomorphisms E[N ] ∼ − → (Z/N Z) 2 for some φ ∈ Aut(E k ) and g ∈ G.
Proof. First suppose that (E, [α] G ) represents a point P ∈ Y G (k). To prove (i) it suffices to show that E is isomorphic over k to an elliptic curve defined over k. So we need only show that j E is an element of k. For any σ ∈ Gal k , the point P = σ(P ) is also represented by
(E σ , [α • σ −1 ] G ).
This implies that E and E σ are isomorphic and hence σ(j E ) = j E . We thus have j E ∈ k since k is perfect. We now prove (ii). Let P ∈ Y G (k) be a point represented by a pair (E, [α] G ) with E defined over k. Take any σ ∈ Gal k . The point σ(P ) is represented by (E, [α • σ −1 ] G ); we can make the identification E = E σ since E is defined over k. We have σ(P ) = P if and only if there is an
automorphism φ ∈ Aut(E k ) such that [α • σ −1 ] G = [α • φ] G . Since k is perfect, we have P ∈ Y G (k) if and only if for all σ ∈ Gal k , we have [α • σ −1 ] G = [α • φ] G for some φ ∈ Aut(E k )
; this is a reformulation of part (ii).
3.2.
Morphism to the j-line. If G = GL 2 (Z/N Z), then there is only a single G-level structure for each elliptic curve. There is an isomorphism Y GL 2 (Z/N Z) = A 1 Z[1/N ] ; on k-points, it takes a point represented by a pair (E, Fix an elliptic curve E over k. By choosing a basis for E[N ] as a Z/N Z-module, the Galois action on E[N ] can be expressed in terms of a representation ρ E,N : Gal k → GL 2 (Z/N Z); this is the same as the earlier definition with k = Q. The representation ρ E,N is uniquely determined up to conjugation by an element of GL 2 (Z/N Z).
[α] G ) to the j-invariant j E ∈ k. If G ′ is a subgroup of GL 2 (Z/N Z) containing G, then there is a natural morphism Y G → Y G ′ . In particular, G ′ = GL 2 (Z/N Z) gives a morphism π G : Y G → A 1 Z[1/N ]
Proposition 3.2. Let E be an elliptic curve over k with j E / ∈ {0, 1728}. The group ρ E,N (Gal k ) is conjugate in GL 2 (Z/N Z) to a subgroup of G if and only if j E is an element of π G (Y G (k)).
Proof. First suppose that ρ E,N (Gal k ) is conjugate to a subgroup of G. There is thus an isomorphism α : E[N ] ∼ − → (Z/N Z) 2 such that α • σ • α −1 ∈ G for all σ ∈ Gal k . By Lemma 3.1(ii), with φ = 1, the pair (E, [α] G ) represents a point P ∈ Y G (k). Therefore, j E = π G (P ) is an element of π G (Y G (k)). Now suppose that j E = π G (P ) for some point P ∈ Y G (k). Lemma 3.1 implies that P is represented by a pair (E, [α] G ), where for all σ ∈ Gal k , we have α
• σ −1 • φ • α −1 ∈ G for some automorphism φ of E k . The assumption j E / ∈ {0, 1728} implies that Aut(E k ) = {±1}.
In particular, every automorphism of E k acts on E[N ] as ±I. Since G contains −I, we deduce that α • σ −1 • α −1 ∈ G for all σ ∈ Gal k . We may choose ρ E,N so that ρ E,N (σ) = α • σ • α −1 for all σ ∈ Gal k , and hence ρ E,N (Gal k ) is a subgroup of G.
Take any j ∈ k and fix an elliptic curve E over k with j E = j. Let M be the group of isomorphisms E[N ]
∼ − → (Z/N Z) 2 .
Composition gives an action of the groups G and Aut(E k ) on M ; they are left and right actions, respectively. The map α ∈ M → (E, [α] G ) induces a bijection
G\M/ Aut(E k ) ∼ − → {P ∈ Y G (k) : π G (P ) = j}. (3.1)
The group Gal k acts on M by the map Gal k ×M → M , (σ, α) → α • σ −1 . From the description of the Galois action in §3.1, we find that the bijection (3.1) respects the Gal k -actions. The following lemma is now immediate (again we are using that k is perfect).
Lemma 3.3. The set {P ∈ Y G (k) : π G (P ) = j} has the same cardinality as the subset of G\M/ Aut(E k ) fixed by the Gal k -action.
3.3. Cusps. In this section, we state an analogue of Lemma 3.3 for X ∞ G (k). Let M be the group of isomorphisms µ N × Z/N Z
∼ − → (Z/N Z) 2 , where µ N is the group of N -th roots of unity in k. The group Gal k acts on M by the map Gal k ×M → M , (σ, α) → α • σ −1 ,
where σ −1 acts on µ N as usual and trivially on Z/N Z. Let U be the subgroup of Aut(µ N × Z/N Z) given by the matrices ± ( 1 u 0 1 ) with u ∈ Hom(Z/N Z, µ N ). Composition gives an action of the groups G and U on M ; they are left and right actions, respectively. Construction 5.3 of [DR73,VI] shows that there is a bijection
X ∞ G (k) ∼ − → G\M/U
that respects the actions of Gal k . We thus have a bijection between X ∞ G (k) and the subset of G\M/U fixed by the action of Gal k .
Observe that the cardinality of X ∞ G (k) depends only on G and the image of the character χ N : Gal k → (Z/N Z) × describing the Galois action on µ N , i.e., σ(ζ) = ζ χ N (σ) for all σ ∈ Gal k and all ζ ∈ µ N . Let B be the subgroup of GL 2 (Z/N Z) consisting of matrices of the form b 0 0 1 with b ∈ χ N (Gal k ). Let U be the subgroup of GL 2 (Z/N Z) generated by −I and ( 1 1 0 1 ). The group B normalizes U and hence right multiplication gives a well-defined action of B on G\ GL 2 (Z/N Z)/U . The following lemma is now immediate.
Lemma 3.4. The set X ∞ G (k) has the same cardinality as the subset of G\ GL 2 (Z/N Z)/U fixed by right multiplication by B.
3.4. Real points. The following proposition tells us when Y G (R) is non-empty.
Proposition 3.5. The set Y G (R) is non-empty if and only if G contains an element that is conjugate in GL 2 (Z/N Z) to 1 0 0 −1 or 1 1 0 −1 . Proof. Let E be any elliptic curve over R. As a topological group, the identity component of E(R) is isomorphic to R/Z. So there is a point P 1 ∈ E(R) of order N . Choose a second point P 2 ∈ E(C) so that {P 1 , P 2 } is a basis of E[N ] as a Z/N Z-module. Define ρ E,N with respect to this basis.
Let σ ∈ Aut(C/R) be the complex conjugation automorphism. We have σ(P 1 ) = P 1 and σ(P 2 ) = bP 1 + dP 2 for some b, d ∈ Z/N Z, i.e., ρ E,N (σ) := 1 b 0 d ∈ GL 2 (Z/N Z). Using the Weil pairing, we find that det(ρ E,N (σ)) describes how σ acts on the N -th roots of unity. Since complex conjugation inverts roots of unity, we have det(ρ E,N (σ)) = −1 and hence d = −1. For a fixed m ∈ Z/N Z, define points P ′ 1 := P 1 and P ′ 2 := P 2 + mP 1 . The points {P ′ 1 , P ′ 2 } are a basis for E[N ], and we have σ(P ′ 1 ) = P ′ 1 and σ(P ′ 2 ) = (bP 1 − P 2 ) + mP 1 = −(P 2 + mP 1 ) + (b + 2m)P 1 = −P ′ 2 + (b + 2m)P ′ 1 . We can choose m so that b + 2m is congruent to 0 or 1 modulo N ; with such an m and the choice of basis {P ′ 1 , P ′ 2 }, the matrix ρ E,N (σ) will be 1 0 0 −1 or 1 1 0 −1 . We claim that both of the matrices 1 0 0 −1 and 1 1 0 −1 are conjugate to ρ E,N (σ) for some E/R with j E / ∈ {0, 1728}. This is clear if N is odd since the two matrices are then conjugate (we could have solved for m in either of the congruences above). If N is even, then it suffices to show that both possibilities occur when N = 2; this is easy (if E/Q is given by a Weierstrass equation y 2 = x 3 + ax + b, the two possibilities are distinguished by the number of real roots that x 3 + ax + b has).
Using
Proposition 3.2, we deduce that π G (Y G (R)) − {0, 1728} is non-empty if and only if G contains an element that is conjugate in GL 2 (Z/N Z) to 1 0 0 −1 or 1 1 0 −1 . To complete the proof of the proposition, we need to show that if π G (Y G (R)) ⊆ {0, 1728}, then π G (Y G (R)) is empty. So suppose that π G (Y G (R)) ⊆ {0, 1728} and hence Y G (R) is finite. However, since Y G over Q is a smooth, geometrically irreducible curve, the set Y G (R) is either empty or infinite.
3.5. Complex points. The complex points Y G (C) form a Riemann surface. In this section, we describe it as a familiar quotient of the upper half plane by a congruence subgroup.
Let H be the complex upper half plane.
For z ∈ H and γ = a b c d ∈ SL 2 (Z), set γ(z) := (az + b)/(cz + d). We let SL 2 (Z) act on the right of H by H × SL 2 (Z) → H, (z, γ) → γ t (z), where γ t is the transpose of γ. For a congruence subgroup Γ, the quotient H/Γ is a smooth Riemann surface.
We define the genus of a congruence subgroup Γ to be the genus of the Riemann surface H/Γ.
Remark 3.6. One could also consider the quotient Γ\H of H under the left action given by (γ, z) → γ(z); it is isomorphic to the Riemann surface H/Γ (use that
γ t = Bγ −1 B −1 for all γ ∈ Γ, where B = 0 −1 1 0
). In particular, the genus of Γ\H agrees with the genus of Γ.
Let Γ G be the congruence subgroup consisting of matrices γ ∈ SL 2 (Z) whose image modulo N lies in G. The image of Γ G modulo N is G∩SL 2 (Z/N Z) since the reduction map SL 2 (Z) → SL 2 (Z/N Z) is surjective. In particular, Γ G depends only on the group G ∩ SL 2 (Z/N Z) and we have
[SL 2 (Z) : Γ G ] = [SL 2 (Z/N Z) : G ∩ SL 2 (Z/N Z)].
Proposition 3.7. The Riemann surfaces Y G (C) and H/Γ G are isomorphic. In particular, the genus of Y G is equal to the genus of Γ G .
Proof. Set X ± := C − R; we let GL 2 (Z) act on the right in the same manner SL 2 (Z) acts on H. We also let GL 2 (Z) act on the right of
G\ GL 2 (Z/N Z) by right multiplication. From [DR73, IV §5.3], we have an isomorphism Y G (C) ∼ = X ± × (G\ GL 2 (Z/N Z)) / GL 2 (Z).
Using that det(G) = (Z/N Z) × and setting H := G ∩ SL 2 (Z/N Z), we find that the natural maps
(H × (G\ GL 2 (Z/N Z)))/ SL 2 (Z) → (X ± × (G\ GL 2 (Z/N Z)))/ GL 2 (Z) and (H × (H\ SL 2 (Z/N Z)))/ SL 2 (Z) → (H × (G\ GL 2 (Z/N Z)))/ SL 2 (Z)
are isomorphisms of Riemann surfaces. It thus suffices to show that H/Γ G and (H×(H\ SL 2 (Z/N Z)))/ SL 2 (Z) are isomorphic. Define the map
ϕ : H/Γ G → (H × (H\ SL 2 (Z/N Z)))/ SL 2 (Z) 7
that takes a class containing z to the class represented by (z, H · I). For γ ∈ SL 2 (Z), the pairs (z, H · I) and (γ t (z), H · γ −1 ) lies in the same class of (H × (H\ SL 2 (Z/N Z)))/ SL 2 (Z); from this one readily deduced that ϕ is well-defined and injective. It is straightforward to check that ϕ is an isomorphism of Riemann surfaces.
3.6. F p -points. Fix a prime p ∤ 6N and an algebraic closure F p of F p . The Galois group Gal(F p /F p ) is topologically generated by the automorphism Frob p : x → x p . In this section, we will describe how to compute |X G (F p )|.
For an imaginary quadratic order O of discriminant D, the j-invariant of the complex elliptic curve C/O is an algebraic integer; its minimal polynomial P D (x) ∈ Z[x] is the Hilbert class polynomial of O. For an integer D < 0 which is not the discriminant of a quadratic order, we set P D (x) = 1.
Fix an elliptic curve E over
F p with j E / ∈ {0, 1728}. Let a E be the integer p + 1 − |E(F p )|. Set ∆ E := a 2 E − 4p; we have ∆ E = 0 by the Hasse inequality. Let b E be the largest integer b ≥ 1 such that b 2 |∆ E and P ∆ E /b 2 (j E ) = 0; this is well-defined since we will always have P ∆ E (j E ) = 0. Define the matrix Φ E := (a E − ∆ E /b E )/2 ∆ E /b E · (1 − ∆ E /b 2 E )/4 b E (a E + ∆ E /b E )/2 ;
it has integer entries since ∆ E /b 2 E is an integer congruent to 0 or 1 modulo 4 (it is the discriminant of a quadratic order) and ∆ E ≡ a E (mod 2). One can check that Φ E has trace a E and determinant p. In practice, Φ E is straightforward to compute; there are many good algorithms to compute a E and P D (x).
The following proposition shows that Φ E describes ρ E,N (Frob p ), and hence also ρ E,N , up to conjugacy.
Proposition 3.8. With notation as above, the reduction of Φ E modulo N is conjugate in GL 2 (Z/N Z) to ρ E,N (Frob p ).
Proof. It suffices to prove the proposition when N is a prime power. For N a prime power, it is then a consequence of Theorem 2 in [Cen16].
We now explain how to compute |X G (F p )|. We can compute |X ∞ G (F p )| using Lemma 3.4 (with k = F p , the subgroup χ N (Gal Fp ) of (Z/N Z) × is generated by p modulo N ). So we need only describe how to compute |Y G (F p )|; it thus suffices to compute each term in the sum
|Y G (F p )| = j∈Fp |{P ∈ Y G (F p ) : π G (P ) = j}|.
Take any j ∈ F p and fix an elliptic curve E over F p with j E = j.
∼ − → (Z/N Z) 2 . Since −I ∈ G, we have G\M/ Aut(E F p ) = G\M . Lemma 3.3 implies that |{P ∈ Y G (F p )
: π G (P ) = j}| is equal to cardinality of the subset of G\M fixed by the action of Frob p . By Proposition 3.8 and choosing an appropriate basis of E[N ], we deduce that |{P ∈ Y G (F p ) : π G (P ) = j}| is equal to the cardinality of the subset of G\ GL 2 (Z/N Z) fixed by right multiplication by Φ E . In particular, note that we can compute |{P ∈ Y G (F p ) : π G (P ) = j}| without having to compute E[N ]. Now suppose that j ∈ {0, 1728} and recall that p ∤ 6. When j = 0, we take E/F p to be the curve defined by y 2 = x 3 − 1; the group Aut(E Fp ) is cyclic of order 6 and generated by (x, y) → (ζx, −y), where ζ ∈ F p is a cube root of unity. When j = 1728, we take E/F p to be the curve defined by y 2 = x 3 − x; the group Aut(E Fp ) is cyclic of order 6 and generated by (x, y) → (−x, ζy), where ζ ∈ F p is a fourth root of unity.
One can compute an explicit basis of E[N ]. With respect to this basis, the action of Aut(E F p ) on E[N ] corresponds to a subgroup A of GL 2 (Z/N Z) and the action of Frob p on E[N ] corresponds to a matrix Φ E,N ∈ GL 2 (Z/N Z). Lemma 3.3 implies that |{P ∈ Y G (F p ) : π G (P ) = j}| equals the number of elements in G\ GL 2 (Z/N Z)/A that are fixed by right multiplication by Φ E,N .
Preliminary work
Take any congruence subgroup Γ of SL 2 (Z) and denote its level by N 0 . Let ±Γ be the congruence subgroup generated by Γ and −I. Let N be the integer N 0 , 4N 0 or 2N 0 when v 2 (N 0 ) is 0, 1 or at least 2, respectively.
Definition 4.1. We define I (Γ) to be the set of integers
[SL 2 (Z N ) : G ′ ] · 2/gcd(2, N ),
where G varies over the open subgroups of GL 2 (Z N ) that are the inverse image by the reduction
map GL 2 (Z N ) → GL 2 (Z/N Z) of a subgroup G(N ) ⊆ GL 2 (Z/N Z) which satisfies the following conditions: (a) G(N ) ∩ SL 2 (Z/N Z) is equal to ±Γ modulo N , (b) G(N ) ⊇ (Z/N Z) × · I, (c) det(G(N )) = (Z/N Z) × , (d) G(N ) contains a matrix that is conjugate to 1 0 0 −1 or 1 1 0 −1 in GL 2 (Z/N Z), (e) the set X G(N ) (Q) is infinite.
The set I (Γ) is finite since there are only finitely many possible G(N ) for a fixed N . In the special case N = 1, we view GL 2 (Z N ) and SL 2 (Z N ) as trivial groups and hence we find that I (SL 2 (Z)) = {2}. Define the set of integers
I := Γ I (Γ),
where the union is over the congruence subgroups of SL 2 (Z) that have genus 0 or 1. The set I is finite since there are only finitely many congruence subgroups of genus 0 or 1, see [CP03].
The goal of this section is to prove the following theorem.
Theorem 4.2. Fix an integer c. There is a finite set J, depending only on c, such that if E/Q is an elliptic curve with j E / ∈ J and ρ E,ℓ surjective for all primes ℓ > c, then [GL 2 ( Z) : ρ E (Gal Q )] is an element of I .
In §5, we will compute I and show that it is equal to the set I from §1; this will prove Theorem 1.3. Let m be the product of the primes ℓ for which ℓ ≤ 5 or for which ρ E,ℓ is not surjective. The group G m ∩ SL 2 (Z m ) is open in SL 2 (Z m ). Let N 0 ≥ 1 be the smallest positive integer dividing some power of m for which
G m ∩ SL 2 (Z m ) ⊇ {A ∈ SL 2 (Z m ) : A ≡ I (mod N 0 )}. (4.2)
Let N be the integer N 0 , 4N 0 or 2N 0 when v 2 (N 0 ) is 0, 1 or at least 2, respectively.
Define Γ E := Γ G(N ) ; it is the congruence subgroup consisting of matrices in SL 2 (Z) whose image modulo N lies in G(N ). Note that the congruence subgroup Γ E has level N 0 and contains −I. Proof. Our congruence subgroup Γ E contains −I and was chosen so that Γ E modulo N equals
G(N ) ∩ SL 2 (Z/N Z). We have G ⊇ Z × · I, so G(N ) ⊇ (Z/N Z) × · I. We have det(ρ E (Gal Q )) = Z × , so det(G(N )) = (Z/N Z) × .
It remains to show that condition (d) holds. Since E/Q is non-CM and ρ E,N (Gal Q ) is a subgroup of G(N ), we have Y G(N ) (Q) = ∅ by Proposition 3.2. In particular, Y G(N ) (R) = ∅. Proposition 3.5 implies that G contains an element that is conjugate in GL 2 (Z/N Z) to 1 0 0 −1 or 1 1 0 −1 . The following lemma shows that G N is determined by G (N ).
Lemma 4.4. The group G N is the inverse image of G(N ) under the reduction modulo N map GL 2 (Z N ) → GL 2 (Z/N Z).
Proof. Take any A ∈ GL 2 (Z N ) satisfying A ≡ I (mod N ); we need only verify that A is an element of G N . Our integer N has the property that (1 + N 0 Z N ) 2 = 1 + N Z N . Since det(A) ≡ 1 (mod N ), we have det(A) = λ 2 for some λ ∈ 1 + N 0 Z N . Define B := λ −1 A; it is an element of SL 2 (Z N ) that is congruent to I modulo N 0 . Using (4.2), we deduce that B is an element of G N . From the definition of G, it is clear that G N contains the scalar matrix λI. Therefore, A = λI · B is an element of G N .
The following group theoretical lemma will be proved in §4.4.
Lemma 4.5. We have N ).
[SL 2 (Z N ) : G ′ ] = [SL 2 (Z m ) : G ′ m ] = [SL 2 (Z N ) : G ′ N ] · 2/ gcd(2,Moreover, G ′ = G ′ m × ℓ∤m SL 2 (Z ℓ ).
The following lemma motivates our definition of I . N ) is an element of I (Γ E ). To complete the proof of the lemma, we need to show that Γ E has genus 0 or 1 since then I (Γ E ) ⊆ I . The genus of Γ E is equal to the genus of X G(N ) by Proposition 3.7. Since X G(N ) has infinitely many rational point, it must have genus 0 or 1 by Faltings' theorem.
4.2.
Exceptional rational points on modular curves. Let S be the set of pairs (N, G) with N ≥ 1 an integer not divisible by any prime ℓ > 13 and with G a subgroup of GL 2 (Z/N Z) satisfying the following conditions:
• det(G) = (Z/N Z) × and −I ∈ G, • X G has genus at least 2 or X G (Q) is finite. Define the set J :=
(N,G)∈S π G (Y G (Q)).
We will prove that J is finite. We will need the following lemma. Proposition 4.8. The set J is finite.
Proof. Fix pairs (N, G), (N ′ , G ′ ) ∈ S such that N is a divisor of N ′ and such that reduction modulo N gives a well-defined map G ′ → G. This gives rise to a morphism ϕ :
Y G ′ → Y G of curves over Q such that π G • ϕ = π G ′ . In particular, π G ′ (Y G ′ (Q)) ⊆ π G (Y G (Q)). Therefore, J = (N,G)∈S ′ π G (Y G (Q)),
where S ′ is the set of pairs (N, G) ∈ S for which there is no pair (N ′ , G ′ ) ∈ S − {(N, G)} with N ′ a divisor of N so that the reduction modulo N ′ defines a map G → G ′ . For each pair (N, G) ∈ S ′ , the set Y G (Q), and hence also π G (Y G (Q)), is finite. The finiteness is immediate from the definition of S when Y G has genus 0 or 1. If Y G has genus at least 2, then Y G (Q) is finite by Faltings' theorem. So to prove that J is finite, it suffices to show that S ′ is finite.
Let m be the product of primes ℓ ≤ 13. For each pair (N, G) ∈ S ′ , letG be the open subgroup of GL 2 (Z m ) that is the inverse image of G under the reduction map GL 2 (Z m ) → GL 2 (Z/N Z). Note that we can recover the pair (N, G) fromG; N ≥ 1 is the smallest integer (not divisible by primes ℓ > 13) such thatG contains {A ∈ GL 2 (Z m ) : A ≡ I (mod N )} and G is the image ofG in GL 2 (Z/N Z). Define the set G := {G : (N, G) ∈ S ′ }. We have |G| = |S ′ |, so it suffices to show that the set G is finite.
Suppose that G is infinite. We now recursively define a sequence {M i } i≥0 of open subgroups of GL 2 (Z m ) such that (4.3). In particular, there are infinitely many congruence subgroup of genus 0 or 1. However, there are only finitely many congruence subgroups of SL 2 (Z) of genus 0 and 1; moreover, the level of such congruence subgroups is at most 52 by [CP03]. This contradiction implies that G, and hence S ′ , is finite.
M 0 M 1 M 2 M 3 . . .
For each prime ℓ, let J ℓ be the set of j-invariants of elliptic curves E/Q for which ρ E,ℓ is not surjective.
Proposition 4.9. The set J ℓ is finite for all primes ℓ > 13.
Proof. Fix a prime ℓ > 13. By Proposition 3.2, it suffices to show that X G (Q) is finite for each of the maximal subgroups G of GL 2 (Z/ℓZ) that satisfy det(G) = (Z/ℓZ) × . Fix such a group G and let Γ = Γ G be the congruence subgroup consisting of A ∈ SL 2 (Z) for which A modulo N lies in G. The curve X G has the same genus as Γ by Proposition 3.7. If Γ has genus at least 2, then X G (Q) is finite by Faltings' theorem.
We may thus suppose that Γ has genus 0 or 1. From the description of congruence subgroups of genus 0 and 1 in [CP03], we find that ℓ ∈ {17, 19} and that Γ modulo ℓ contains an element of order ℓ. Therefore, after replacing G by a conjugate in GL 2 (Z/ℓZ), we may assume that G is the subgroup of upper-triangular matrices. So we are left to consider the modular curve X 0 (ℓ) := X G with ℓ ∈ {17, 19}. The curve X 0 (ℓ), with ℓ ∈ {17, 19}, indeed has finitely many points (it has a rational cusp, so it is an elliptic curve of conductor ℓ ∈ {17, 19}; all such elliptic curves have rank 0). Take any elliptic curve E/Q with j E / ∈ J for which ρ E,ℓ is surjective for all ℓ > c. Since j E / ∈ J ℓ for 13 < ℓ ≤ c, the representation ρ E,ℓ is surjective for all ℓ > 13.
Let Γ E be the congruence subgroup from §4.1; denote its level by N 0 and define N as in the beginning of the section. Let G(N ) be the subgroup of GL 2 (Z/N Z) from §4.1 associated to E/Q. Proof. Take S as in §4.2. The integer N is not divisible by any prime ℓ > 13 since ρ E,ℓ is surjective for all ℓ > 13. If (N, G(N )) ∈ S, then j E ∈ π G(N ) (Y G(N ) (Q)) ⊆ J ⊆ J. Since j E / ∈ J by assumption, we have (N, G(N )) / ∈ S. We have det(G(N )) = (Z/N Z) × and −I ∈ G(N ), so (N, G(N )) / ∈ S implies that X G(N ) has genus 0 or 1, and that X G(N ) (Q) is infinite.
· 3 · 5. Since G m ∩ SL 2 (Z m ) contains {A ∈ SL 2 (Z m ) : A ≡ I (mod N 0 )}, we have G m ∩ SL 2 (Z m ) = W × SL 2 (Z d ).
for a subgroup W of SL 2 (Z N ) containing {A ∈ SL 2 (Z N ) :
A ≡ I (mod N 0 )}. Since G m ∩ SL 2 (Z m )
is a normal subgroup of G m , the group W is normal in G N . We have G d = GL 2 (Z d ), since G d ⊇ SL 2 (Z d ) and det(G d ) = Z × d (note that det(ρ E (Gal Q )) = Z × ).
Now consider the quotient map
ϕ : G N × G d → G N /W × G d / SL 2 (Z d ).
We can view G m as an open subgroup of G N × G d ; it projects surjectively on both of the factors.
The group G m contains W × SL 2 (Z d ), so there is an open subgroup Y of G N /W × G d / SL 2 (Z d ) for which G m = ϕ −1 (Y ).
Take any matrices B 1 , B 2 ∈ G d = GL 2 (Z d ) with det(B 1 ) = det(B 2 ); equivalently, with the same image in G d / SL 2 (Z d ). There is a matrix A ∈ G N such that (A, B 1 ) ∈ G m and hence also (A, B 2 ) ∈ G m since ϕ (A, B 1 ) = ϕ(A, B 2 ). Therefore, the commutator subgroup G ′ m contains the element
(A, B 1 ) · (A, B 2 ) · (A, B 1 ) −1 · (A, B 2 ) −1 = (I, B 1 B 2 B −1 1 B −1 2 )
. By Lemma 4.11(iv) below, the group GL 2 (Z d ) ′ is topologically generated by the set
B 1 B 2 B −1 1 B −1 2 : B 1 , B 2 ∈ GL 2 (Z d ), det(B 1 ) = det(B 2 ) , and hence G ′ m ⊇ {I} × GL 2 (Z d ) ′ . We have an inclusion G ′ m ⊆ G ′ N × G ′ d = G ′ N × GL 2 (Z d ) ′since G m ⊇ {I} × GL 2 (Z d ) ′ we find that G ′ m = G ′ N × GL 2 (Z d ) ′ . (4.4) Lemma 4.11. (i) For ℓ ≥ 5, we have SL 2 (Z ℓ ) ′ = SL 2 (Z ℓ ).
(ii) For ℓ = 2 or 3, let b = 4 or 3, respectively. Then reduction modulo b induces an isomorphism Finally consider (iv). Without loss of generality, we may assume that d is a prime, say ℓ. The topological group generated by the set C = {ABA −1 B −1 : A, B ∈ GL 2 (Z ℓ ), det(A) = det(B)} contains SL 2 (Z ℓ ) ′ , so it suffices to show that the image of C generates GL 2 (Z ℓ ) ′ / SL 2 (Z ℓ ) ′ . If ℓ ≥ 5, this is trivial since GL 2 (Z ℓ ) ′ and SL 2 (Z ℓ ) ′ both equal SL 2 (Z ℓ ) by (i). For ℓ = 2 or 3, it suffices by part (ii) to show that GL 2 (Z/bZ) ′ is generated by ABA −1 B −1 with matrices A, B ∈ GL 2 (Z/bZ) having the same determinant; this again is an easy calculation.
SL 2 (Z ℓ )/ SL 2 (Z ℓ ) ′ ∼ − → SL 2 (Z/bZ)/ SL 2 (Z/bZ) ′ of cyclic groups of order b. (iii) We have GL 2 (Z 3 ) ′ = SL 2 (Z 3 ) and [SL 2 (Z 2 ) : GL 2 (Z 2 ) ′ ] = 2.
Before computing G ′ , we first state Goursat's lemma; we will give a more general version than needed so that it can be cited in future work.
Lemma 4.12 (Goursat's Lemma). Let B 1 , . . . , B n be profinite groups. Assume that for distinct 1 ≤ i, j ≤ n, the groups B i and B j have no finite simple groups as common quotients. Suppose that H is a closed subgroup of n i=1 B i that satisfies p j (H) = B j for all j where p j : n i=1 B i → B j is the projection map. Then H = n i=1 B i . Proof. We proceed by induction on n. The case n = 1 is trivial, so assume that n = 2. The kernel of p 1 | H is a closed subgroup of H of the form {I} × N 2 , and similarly the kernel of p 2 | H is of the form N 1 × {I}. The group N = N 1 × N 2 is a closed normal subgroup of H. Since p 1 | H is surjective, we find that N 1 = p 1 (N ) is a closed normal subgroup of B 1 ; this gives an isomorphism H/N ∼ = B 1 /N 1 of profinite groups. Similarly, we have H/N ∼ = B 2 /N 2 and thus B 1 /N 1 and B 2 /N 2 are isomorphism.
Since we have assumed that B 1 and B 2 have no common finite simple quotients, we deduce that B 1 = N 1 and B 2 = N 2 . This proves the n = 2 case since H contains N 1 × N 2 = B 1 × B 2 . Now fix an n ≥ 3 and assume that the n − 1 case of the lemma has been proved. Then the imagẽ H of H in C := n−1 i=1 B i is a closed subgroup such that the projectionH → B i is surjective for all 1 ≤ i ≤ n − 1. By our inductive hypothesis, we haveH = C. So H is a closed subgroup of C × B n and the projections H → C and H → B n are surjective. By the n = 2 case, it suffices to show any finite simple quotient of C is not a quotient of B n . Take any open normal subgroup U of C such that C/U is a finite simple group. There is an integer 1 ≤ j ≤ n − 1 for which the projection U → B j is not surjective (if not, then we could use our inductive hypothesis to show that U = C). For simplicity, suppose j = 1; then U is of the form N 1 × B 2 × · · · × B n−1 where N 1 is an open normal subgroup of B 1 . Since C/U ∼ = B 1 /N 1 , we deduce from the hypothesis on the B i that C/U is not a quotient of B n .
We claim that G ′ ℓ = SL 2 (Z ℓ ) for every prime ℓ ∤ m. We have the easy inclusions G ′ ℓ ⊆ GL 2 (Z ℓ ) ′ ⊆ SL 2 (Z ℓ ). By [Ser89, IV Lemma 3] and ℓ > 5 (since ℓ ∤ m), we have G ′ ℓ = SL 2 (Z ℓ ) if and only if the image of G ′ ℓ in SL 2 (Z/ℓZ) is SL 2 (Z/ℓZ). It thus suffices to show that ρ E,ℓ (Gal Q ) ′ = SL 2 (Z/ℓZ). Since ℓ ∤ m, we have ρ E,ℓ (Gal Q ) = GL 2 (Z/ℓZ) and hence ρ E,ℓ (Gal Q ) ′ = SL 2 (Z/ℓZ) by Lemma 4.11(i); this proves our claim.
We can view G ′ as a subgroup of G ′ m × ℓ∤m SL 2 (Z ℓ ). The projection of G ′ to the the factors G ′ m and SL 2 (Z ℓ ) = G ′ ℓ with ℓ ∤ m are all surjective. Fix a prime ℓ ≥ 5. The simple group PSL 2 (F ℓ ) is a quotient of SL 2 (Z ℓ ). Since ℓ-groups are solvable and SL 2 (Z ℓ ) ′ = SL 2 (Z ℓ ) by Lemma 4.11(i), we find that PSL 2 (F ℓ ) is the only simple group that is a quotient of SL 2 (Z ℓ ). Note that the groups PSL 2 (F ℓ ) are non-isomorphic for different ℓ; in fact, they have different cardinalities.
Take any prime ℓ ∤ m, and hence ℓ > 5. We claim that the simple group PSL 2 (F ℓ ) is not isomorphic to a quotient of G ′ m . Indeed, any closed subgroup H of GL 2 (Z m ) has no quotients isomorphic to PSL 2 (F ℓ ) with ℓ > 5 and ℓ ∤ m (this follows from the calculation of the groups Occ(GL 2 (Z ℓ )) in [Ser98, IV-25]). We can now apply Goursat's lemma (Lemma 4.12) to deduce that
G ′ = G ′ m × ℓ∤m SL 2 (Z ℓ ). Therefore, [SL 2 ( Z) : G ′ ] = [SL 2 (Z m ) : G ′ m ]. By (4.4), we have [SL 2 (Z m ) : G ′ m ] = [SL 2 (Z N ) : G ′ N ] · [SL 2 (Z d ) : GL 2 (Z d ) ′ ]. By Lemma 4.11, [SL 2 (Z d ) : GL 2 (Z d ) ′ ] = ℓ|d [SL 2 (Z ℓ ) : GL 2 (Z ℓ ) ′ ] is equal to 1 if d is odd and 2 if d is even. Since N and d have opposite parities, we conclude that [SL 2 (Z m ) : G ′ m ] is equal to [SL 2 (Z N ) : G ′ N ] if N is even and [SL 2 (Z N ) : G ′ N ] · 2 if N is odd.
The lemma is now immediate.
Index computations
In §1.1, we defined the set In §4, we defined the set of integers
I := Γ I (Γ),
where Γ runs over the congruence subgroups of SL 2 (Z) of genus 0 or 1. The goal of this section is to outline the computations needed to verify the following.
Proposition 5.1. We have I = I.
The computations in this section were performed with Magma [BCP97]; code for the computations can be found at https://github.com/davidzywina/PossibleIndices Let S 0 and S 1 be sets of representatives of the congruence subgroups of SL 2 (Z) containing −I, up to conjugacy in GL 2 (Z), with genus 0 and 1, respectively. Set S := S 0 ∪ S 1 . Since the set I (Γ) does not change if we replace Γ by ±Γ or by a conjugate subgroup in GL 2 (Z), we have
I = Γ∈S I (Γ).
Cummin and Pauli [CP03] have classified the congruence subgroups of PSL 2 (Z) with genus 0 or 1, up to conjugacy in PGL 2 (Z). We thus have a classification of the congruence subgroups Γ of SL 2 (Z), up to conjugacy in GL 2 (Z), of genus 0 or 1 that contain −I. Moreover, they have made available an explicit list 1 of such congruence subgroups; each congruence subgroup is given by a level N and set of generators of its image in SL 2 (Z/N Z)/{±I}. In our computations, we will let S 0 and S 1 consist of congruence subgroups from the explicit list of Cummin and Pauli. 5.1. Computing indices. Fix a congruence subgroup Γ of SL 2 (Z) that contains −I and has level N 0 . Let N be the integer N 0 , 4N 0 or 2N 0 when v 2 (N 0 ) is 0, 1 or at least 2, respectively. For simplicity, we will assume that N > 1.
We first explain how we computed the subgroups G(N ) of GL 2 (Z/N Z) that satisfy conditions (a), (b) and (c) of Definition 4.1. Instead of directly looking for subgroups in GL 2 (Z/N Z), we will search for certain abelian subgroups in a smaller group.
Let We shall now describe how to compute the index [SL 2 (Z N ) : G ′ ]; this is needed in order to compute I (Γ). We remark that G ′ (M ) = G(M ) ′ .
Lemma 5.3. The group G ′ contains {A ∈ SL 2 (Z N ) : A ≡ I (mod N 2 )}. In particular, we have [SL 2 (Z N ) : G ′ ] = [SL 2 (Z/N 2 Z) : G(N 2 ) ′ ].
Proof. Since G ⊇ I + N M 2 (Z N ), it suffices to prove that (I + N M 2 (Z N )) ′ = SL 2 (Z N ) ∩ (I + N 2 M 2 (Z N )). So it suffices to prove that (I + qM 2 (Z q )) ′ = SL 2 (Z q ) ∩ (I + q 2 M 2 (Z q )) for any prime power q > 1; this is Lemma 1 of [LT76,p.163].
Lemma 5.3 allows us to compute [SL 2 (Z N ) : G ′ ] by computing the finite group G(N 2 ) ′ . In practice, we will use the following to reduce the computation to finding G(M ) ′ for some, possibly smaller, divisor M of N 2 . Let H be a closed subgroup of SL 2 (Z N ) whose image in SL 2 (Z/rM Z) contains {A ∈ SL 2 (Z/rM Z) : A ≡ I (mod M )}. We claim that H ⊇ S M ; the lemma will follow from the claim with H = G ′ . By replacing H with H ∩ S M , we may assume that H is a closed subgroup of S M . Since S M is a product of the pro-ℓ groups S ℓ v ℓ (M ) with ℓ|M , we may further assume that M is a power of a prime ℓ and hence r = ℓ.
So fix a prime power ℓ e > 1 and let H be a closed subgroup of S ℓ e for which H(ℓ e+1 ) = {A ∈ SL 2 (Z/ℓ e+1 Z) : A ≡ I (mod ℓ e )}; we need to prove that H = S ℓ e .
For each integer i ≥ 1, define H i := H ∩ (I + ℓ i M 2 (Z ℓ )) and h i :
= H i /H i+1 . For any A ∈ M 2 (Z ℓ ) with I + ℓ i A ∈ SL 2 (Z ℓ ), we have tr(A) ≡ 0 (mod ℓ). The map H i → M 2 (Z ℓ ), I + ℓ i A → A thus induces a homomorphism ϕ i : h i ֒→ sl 2 (F ℓ ),
where sl 2 (F ℓ ) is the subgroup of trace 0 matrices in M 2 (F ℓ ). Using that H is closed, we deduce that H = S ℓ e if and only if ϕ i is surjective for all i ≥ e.
We now show that ϕ i is surjective for all i ≥ e. We proceed by induction on i; the homomorphism ϕ e is surjective by our initial assumption on H. Now suppose that ϕ i is surjective for a fixed i ≥ e. Take any matrix B in the set B := {( 0 1 0 0 ) , ( 0 0 1 0 ) , 1 1 −1 −1 }. The matrix I + ℓ i B has determinant 1, so the surjectivity of ϕ i implies that there is a matrix A ∈ M 2 (Z ℓ ) with A ≡ B (mod ℓ) such that h := I + ℓ i A is an element of H.
Working modulo ℓ 2i+1 , we find that (ℓ i A) 2 = ℓ 2i A 2 ≡ ℓ 2i B 2 = 0, where the last equality uses that B 2 = 0. In particular, (ℓ i A) 2 ≡ 0 (mod ℓ i+2 ). Therefore,
h ℓ ≡ I + ℓ 1 ℓ i A ≡ I + ℓ i+1 A ≡ I + ℓ i+1 B (mod ℓ i+2 )
. Since h ℓ ∈ H, we find that B modulo ℓ lies in the image of ϕ i+1 . Since sl 2 (F ℓ ) is generated by the B ∈ B, we deduce that ϕ i+1 is surjective.
Remark 5.5. In practice, a useful way to compute G ′ is to first find open subgroups B ℓ of GL 2 (Z ℓ ) such that ℓ|N B ℓ ⊆ G. We can then compute B ′ ℓ ⊆ SL 2 (Z ℓ ) using Lemma 5.4. Let m ℓ the smallest power of ℓ for which B ℓ is determined by its image modulo m ℓ . We will then have [SL 2 Proof. The inclusion I (Γ) ⊆ I ′ (Γ) is obvious. So assume that G(N ) is any group satisfying conditions (a)-(d) of Definition 4.1 and that X ∞ G(N ) (Q p ) is empty for at most one prime p|N . To prove the inclusion I ′′ (Γ) ⊆ I (Γ), we need to verify that X := X G(N ) has infinitely many Qpoints. Note that the curve X Q is smooth and projective; it has genus 0 by our assumption on Γ and Proposition 3.7.
We claim that X(Q v ) is non-empty for all places v of Q; the places corresponds to the primes p or to ∞ where Q ∞ = R. Condition (d) and Proposition 3.5 imply that X(R) is non-empty. Now take any prime p ∤ N . As an Z[1/N ]-scheme X has good reduction at p and hence the fiber X over F p is a smooth and projective curve of genus 0. Therefore, X(F p ) is non-empty and any of the points can be lifted by Hensel's lemma to a point in X(Q p ). By our hypothesis on the sets X ∞ G(N ) (Q p ) with p|N , we deduce that there is at most one prime p 0 such that X(Q p 0 ) is empty. So suppose that there is precisely one prime p 0 for which X(Q p 0 ) is empty. The curve X Q has a model given by a conic of the form ax 2 + by 2 − z 2 = 0 with a, b ∈ Q × . The Hilbert symbol (a, b) v , for a place v, is equal to +1 if X(Q v ) = ∅ and −1 otherwise. Therefore, v (a, b) = (a, b) p 0 = −1. However, we have v (a, b) = 1 by reciprocity. This contradiction proves our claim that X(Q v ) is non-empty for all places v of Q.
The curve X Q has genus 0 so it satisfies the Hasse principle, and hence has a Q-rational point. The curve X Q is thus isomorphic to P 1 Q and has infinitely many Q-points. We shall use the explicit set S 0 due to Cummin and Pauli. For each Γ ∈ S 0 , it is straightforward to compute the set I ′ (Γ) using the method in §5.1.
Using Lemma 3.4 and the discussion in §5.1, we can also compute I ′′ (Γ). Fix a prime p dividing N . Take e so that p e N and set M = N/p e . The image of the character χ N : Gal Qp → (Z/N Z) × = (Z/p e Z) × × (Z/M Z) × arising from the Galois action on the N -th roots of unity is (Z/p e Z) × × p .
Our Magma computations show that Γ∈S 0 I ′′ (Γ) = I 0 and Γ∈S 0 I ′ (Γ) = I 0 , where Using the inclusions of Lemma 5.6, we deduce that I 0 = I 0 .
Remark 5.7. From our genus 0 computations, we find that S 0 has cardinality 121 which led to 331 total groups G(N ) that satisfied (a)-(d) with respect to some Γ ∈ S 0 . 5.3. Genus 1 computations. Now define the set of integers
I 1 := Γ∈S 1 (I (Γ) − I 0 ),
where I 0 is the set from §5.2. Instead of computing I (Γ), we will compute a related quantity. We define I ′′′ (Γ) to be the set of integers as in Definition 4.1 with condition (e) excluded and satisfying the additional condition that the Mordell-Weil group of the Jacobian J of the curve X G(N ) over Q has positive rank. For a congruence subgroup Γ of genus 1, we have an inclusion I (Γ) ⊆ I ′′′ (Γ) since a genus 1 curve over Q that has a Q-point is isomorphic to its Jacobian. Therefore,
I 1 ⊆ Γ∈S 1 (I ′′′ (Γ) − I 0 ).
We now explain how to compute I ′′′ (Γ) − I 0 for a fixed congruence subgroup Γ of genus 1. As described in §5.1, we can compute the subgroups G(N ) satisfying the conditions (a)-(d). For each group G(N ), it is described in §5.1 how to compute [SL 2 (Z N ) : G ′ ], where G is the inverse image of G(N ) under the reduction map GL 2 (Z N ) → GL 2 (Z/N Z). We may assume that [SL 2 (Z N ) : G ′ ] · 2/ gcd(2, N ) / ∈ I 0 since otherwise it does not contribute to I ′′′ (Γ) − I 0 . Let J be the Jacobian of the curve X G(N ) over Q; it is an elliptic curve since Γ has genus 1. Let us now explain how to compute the rank of J(Q) (and hence finish our method for computing I ′′′ (Γ) − I 0 ) without having to compute a model for X G . Moreover, we shall determine the elliptic curve J up to isogeny (defined over Q); note that the Mordell rank is an isogeny invariant.
The curve J has good reduction at all primes p ∤ N since the Z[1/N ]-scheme X G(N ) is smooth. If E/Q is an elliptic curve with good reduction at all primes p ∤ N , then its conductor divides N max := p|N p ep , where e 2 = 8, e 3 = 5 and e p = 2 otherwise. One can compute a finite list of elliptic curves E 1 , . . . , E n over Q that represent the isogeny classes of elliptic curves over Q with good reduction at p ∤ N . In our computations, we will have N max ≤ 2 8 · 3 5 = 62208 and hence the representative curves E i can all be found in Cremona's database [Cre] of elliptic curves which are included in Magma (it currently contains all elliptic curves over Q with conductor at most 500000). It remains to determine which curve E i is isogenous to J. Take any prime p ∤ N . Using the methods of §3.6, we can compute the cardinality of X G(N ) (F p ) and hence also the trace of Frobenius
a p (J) = p + 1 − |J(F p )| = p + 1 − |X G(N ) (F p )|.
If a p (E i ) = a p (J), then E i and J are not isogenous elliptic curves over Q. By computing a p (J) for enough primes p ∤ N , one can eventually eliminate all but one curve E i 0 which then must be isogenous to J. There are then known methods to determine the Mordell rank of E i 0 ; the rank is also part of Cremona's database. Therefore, we can compute the rank of J(Q).
Our Magma computations show that In particular, I 1 ⊆ {220, 240, 360, 504}.
We now describe how the values 220, 240, 360 and 504 arise in our computations. For an odd prime ℓ, let N − ℓ be the normalizer in GL 2 (Z/ℓZ) of a non-split Cartan subgroup and let N + ℓ be the normalizer in GL 2 (Z/ℓZ) of a split Cartan subgroup. Define G 1 := N − 11 . We can identify N − 3 × N − 5 and N − 3 × N + 5 with subgroups G 2 and G 3 , respectively, of GL 2 (Z/15Z). We can identify N − 3 × N − 7 with a subgroup G 4 of GL 2 (Z/21Z). Fix an n ∈ {220, 240, 360, 504}. Let Γ ∈ S 1 be any congruence subgroup such that n ∈ I (Γ). Let G(N ) be one of the groups such that the following hold:
• it satisfies conditions (a), (b), (c) and (d) of Definition 4.1, • the Jacobian J of the curve X G(N ) over Q has positive rank,
• we have [SL 2 (Z N ) : G ′ ] · 2/ gcd(2, N ) = n, where G is the inverse image of G(N ) under the reduction GL 2 (Z N ) → GL 2 (Z/N Z).
Our computations show that one of the following hold:
• We have n = 220, N = 11 and G(N ) is conjugate in GL 2 (Z/11Z) to G 1 .
• We have n = 240, N = 15 and G(N ) is conjugate in GL 2 (Z/15Z) to G 2 .
• We have n = 360, N = 15 and G(N ) is conjugate in GL 2 (Z/15Z) to G 3 .
• We have n = 504, N = 21 and G(N ) is conjugate in GL 2 (Z/21Z) to G 4 .
For later, we note that the index [GL 2 (Z/N Z) : G i ] is 55, 30, 45 or 63 for i = 1, 2, 3 or 4, respectively.
Lemma 5.8. We have I 1 = {220, 240, 360, 504}.
Proof. We already know the inclusion I 1 ⊆ {220, 240, 360, 504}. It thus suffices to show that the set X G i (Q) is infinite for all 1 ≤ i ≤ 4. So for a fixed i ∈ {1, 2, 3, 4}, it suffices to show that X G i (Q) is non-empty, since it then becomes isomorphic to its Jacobian which we know has infinitely many rational points. By Proposition 3.2, it suffices to find a single elliptic curve E/Q with j E / ∈ {0, 1728} for which ρ E,N (Gal Q ) is conjugate to a subgroup of G i .
Let E/Q be a CM elliptic curve. Define R := End(E Q ); it is an order in the imaginary quadratic field K := R ⊗ Z Q. Take any odd prime ℓ that does not divide the discriminant of R. One can show that ρ E,ℓ (Gal Q ) is contained in the normalizer of a Cartan subgroup C ⊆ GL 2 (Z/ℓZ) isomorphic to (R/ℓR) × , cf. [Ser97, Appendix A.5]. The Cartan group C is split if and only if ℓ splits in K.
Consider the CM curve E 1 /Q defined by y 2 = x 3 − 11x+ 14; R is an order in Q(i) of discriminant −16. The primes 3, 7 and 11 are inert in Q(i) and 5 is split in Q(i). Therefore, ρ E 1 ,11 (Gal Q ), ρ E 1 ,15 (Gal Q ) and ρ E 1 ,21 (Gal Q ) are conjugate to subgroups of G 1 , G 3 and G 4 , respectively.
Consider the CM curve E 2 /Q defined by y 2 + xy = x 3 − x 2 − 2x − 1; R is an order in Q( √ −7) of discriminant −7. The primes 3 and 5 are inert in Q( √ −7). Therefore, ρ E 2 ,15 (Gal Q ) is conjugate to a subgroup of G 2 .
Remark 5.9. From our genus 1 computations, we find that S 1 has cardinality 163 which led to 805 total groups G(N ) that satisfied (a)-(d) with respect to some Γ ∈ S 1 . We needed to determine the Jacobian of X G(N ) , up to isogeny, for 63 of these groups G(N ). 5.4. Proof of Proposition 5.1. In §5.2, we found that Γ∈S 0 I (Γ) = I 0 . By Lemma 5.8, we have Proof. Fix a non-CM elliptic curve E/Q with j E ∈ π G(N ) (Y G(N ) (Q)− W ) = π G(M ) (Y G(M ) (Q)− W ). There is a point P ∈ Y G (Q) − W for which π G(M ) (P ) = j E .
With notation as in §3, there is an isomorphism α :
E[M ]
ℓ is odd, we deduce that ρ E,ℓ (Gal Q ) contains an element of order ℓ. This contradicts that the order of ρ E,ℓ (Gal Q ) is not divisible by ℓ. Therefore, ρ E,ℓ is surjective as claimed.
Let W and S be the sets from Lemma 6.3 and Lemma 6.4, respectively. Take any elliptic curve E/Q with j E ∈ π G(N ) (S − W ). Lemma 6.4 implies that the representation ρ E,ℓ is surjective for all ℓ > 13. Lemma 6.3 then implies that [GL 2 ( Z) : ρ E (Gal Q )] = n. Therefore, J n ⊇ π G(N ) (S − W ). So to prove that J n is infinite, it suffices to show that the set S − W is infinite.
First suppose that X G(N ) has genus 0. The set W is a thin subset of X G(N ) (Q) ∼ = P 1 (Q) in the language of [Ser97, §9.1]; this uses that the union defining W is finite and that the morphisms ϕ B are dominant with degree at least 2. From [Ser97, §9.7], we find that W has density 0. Since S has positive density, we deduce that S − W is infinite.
Finally suppose that X G(N ) has genus 1. Since S is infinite, it suffices to show that W is finite. So take any proper subgroup B of G(M ) satisfying det(B) = (Z/M Z) × and −I ∈ B. It thus suffices to show that the set X B (Q) is finite. The morphism ϕ B : X B → X G(N ) is dominant, so X B has genus at least 1. If X B has genus greater than 1, then X B (Q) is finite by Faltings' theorem. We are left to consider the case where X B has genus 1. Let Γ B be the congruence subgroup associated to X B ; it has genus 1. We have Γ B ⊆ Γ and hence the level of Γ B is divisible by [CP03], we find that there are no genus 1 congruence subgroups of SL 2 (Z) containing −I whose level is divisible by N 0 and whose index in SL 2 (Z) has b as a proper divisor. So the case where X B has genus 1 does not occur and we are done.
Main results. Let E be an elliptic curve defined over Q. For each integer N > 1, let E[N ] be the N -torsion subgroup of E(Q). The group E[N ] is a free Z/N Z-module of rank 2 and has natural action of the absolute Galois group Gal Q := Gal(Q/Q). This Galois action on E[N ] may be expressed in terms of a Galois representation ρ E,N : Gal Q → Aut Z/N Z (E[N ]) ∼ = GL 2 (Z/N Z);
that maps a k-point represented by a pair (E, [α] G ) to the j-invariant of E.
First suppose that j / ∈ {0, 1728}. We have Aut(E Fp ) = {±I} and hence each automorphism acts on E[N ] by I or −I. Let M be the group of isomorphisms E[N ]
4. 1 .
1The congruence subgroup Γ E . Fix a non-CM elliptic curve E over Q. Define the subgroupG := Z × · ρ E (Gal Q ) of GL 2 ( Z).For each positive integer n, let G n be the image of G under the projection map GL 2 ( Z) → GL 2 (Z n ). By Serre's theorem, G is an open subgroup of GL 2 ( Z). We have an equality G ′ = ρ E (Gal Q ) ′ of commutator subgroups and hence (4.1) [GL 2 ( Z) : ρ E (Gal Q )] = [SL 2 ( Z) : G ′ ] by Proposition 2.1. There is no harm in working with the larger group G since we are only concerned about the index [GL 2 ( Z) : ρ E (Gal Q )].
Proposition 4. 3 .
3The subgroup G(N ) of GL 2 (Z/N Z) satisfies conditions (a), (b), (c) and (d) of Definition 4.1 with Γ = Γ E .
Lemma 4. 6 .
6If X G(N ) (Q) is infinite, then [GL 2 ( Z) : ρ E (Gal Q )] is an element of I . Proof. By Lemma 4.5 and (4.1), we have [GL 2 ( Z) : ρ E (Gal Q )] = [SL 2 (Z N ) : G ′ N ] · 2/ gcd(2, N ). The group G(N ) satisfies conditions (a), (b), (c) and (d) of Definition 4.1 with Γ = Γ E by Lemma 4.4. The group G(N ) satisfies (e) by assumption. Using Lemma 4.4, we deduce that [SL 2 (Z N ) : G ′ N ] · 2/ gcd(2,
Lemma 4. 7 .
7Fix an integer m ≥ 2. An open subgroup H of GL 2 (Z m ) has only a finite number of closed maximal subgroups and they are all open. Proof. The lemma follows from the proposition in [Ser97, §10.6] which gives a condition for the Frattini subgroup of H to be open; note that H contains a normal subgroup of the form I + m e M 2 (Z m ) for some e ≥ 1 and that I + m e M 2 (Z m ) is the product of pro-ℓ groups with ℓ|m.
that each M i has infinitely many subgroups in G. Set M 0 := GL 2 (Z m ). Take an i ≥ 0 for which M i has been defined and has infinitely many subgroups in G. Since M i has only finite many open maximal subgroups by Lemma 4.7, one of the them contains infinitely many subgroups in G; denote such a maximal subgroup by M i+1 .Take any i ≥ 0. Since there are elements of G that are proper subgroups of M i , we deduce that M i G for some pair (N, G) ∈ S ′ . The group G =G(N ) is thus a proper subgroup of M i (N ) ⊆ GL 2 (Z/N Z). We have det(M i (N )) = (Z/N Z) × and −I ∈ M i (N ) since G has these properties. We have (N, M i (N )) / ∈ S since otherwise (N, G) would not be an element of S ′ . Therefore, the modular curve X M i (N ) has genus 0 or 1. By Proposition 3.7, the congruence subgroupΓ i := Γ M i (N ) (which consists of A ∈ SL 2 (Z) with A modulo N in M i (N )) has genus 0 or 1. We have [SL 2 (Z) : Γ i ] = [SL 2 (Z/N Z) : M i (N ) ∩ SL 2 (Z/N Z)] = [GL 2 (Z/N Z) : M i (N )] = [GL 2 (Z m ) : M i ], so [SL 2 (Z) : Γ i ] → ∞ as i → ∞ by the proper inclusions
4. 3 . 2 .
32Proof of Theorem 4.2. Let J and J ℓ (with ℓ > 13) be the sets from §4.Define
Lemma 4 . 10 .
410The set X G(N ) (Q) is infinite.
4.10 together imply that [GL 2 ( Z) : ρ E (Gal Q )] is an element of I . 4.4. Proof of Lemma 4.5. Let d be the product of primes that divide m but not N ; it divides 2
and the projections of G ′ m onto the first and second factors are both surjective;
(iv) For each positive integer d, the group GL 2 (Z d ) ′ is topologically generated by the set{ABA −1 B −1 : A, B ∈ GL 2 (Z d ), det(A) = det(B)}.Proof. For part (i) and (ii), see [Zyw10, Lemma A.1]. To verify (iii), it suffices by (ii) to show that GL 2 (Z/3Z) ′ = SL 2 (Z/3Z) and [SL 2 (Z/4Z) : GL 2 (Z/4Z) ′ ] = 2; this is an easy computation.
H be the the image of ±Γ = Γ in SL 2 (Z/N Z). Define the subgroup H := (Z/N Z) × · H of GL 2 (Z/N Z). We may assume that H = H ∩ SL 2 (Z/N Z); otherwise, conditions (a) and (b) are incompatible. Let N be the normalizer of H (equivalently, of H) in GL 2 (Z/N Z) and set C := N / H. Since det( H) = ((Z/N Z) × ) 2 , the determinant induces a homomorphism det : C → (Z/N Z) × /((Z/N Z) × ) 2 =: Q N . Lemma 5.2. The subgroups G(N ) of GL 2 (Z/N Z) that satisfy conditions (a), (b) and (c) of Definition 4.1 are precisely the groups obtained by taking the inverse image under N → C of the subgroups W of C for which the determinant induces an isomorphism W ∼ − → Q N . Proof. Let B := G(N ) be a subgroup of GL 2 (Z/N Z) that satisfies conditions (a), (b) and (c). The group B contains H by (a) and (b). For any matrix A ∈ B with det(A) a square, there is a scalar λ ∈ (Z/N Z) × such that det(λA) = 1. Since B ∩ SL 2 (Z/N Z) = H by (a), we deduce that H consists precisely of the element of B with square determinant. The determinant thus gives rise to an H is a normal subgroup of B, and hence B ⊆ N , and the determinant map induces an isomorphism B/ H ∼ − → Q N . Let W be the image of the natural injection B/ H ֒→ N / H = C; it satisfies the conditions for W in the statement of the lemma. Now take any subgroup W of C for which the determinant gives an isomorphism W ∼ − → Q N . Let B be the inverse image of W under the map N → C. The short exact sequence (5.1) holds. Therefore, B ∩ SL 2 (Z/N Z) is equal to H ∩ SL 2 (Z/N Z) = H. We have B ⊇ (Z/N Z) × · I since B ⊇ H. So det(B) ⊇ ((Z/N Z) × ) 2 ; with det(B/ H) = Q N , this implies that det(B) = (Z/N Z) × . We have verified that G(N ) := B satisfies conditions (a), (b) and (c). We first compute the subgroups W of C for which the determinant map N /H → Q N gives an isomorphism W ∼ − → Q N . By Lemma 5.2, the subgroups G(N ) of GL 2 (Z/N Z) that satisfy the conditions (a), (b) and (c) of Definition 4.1 are precisely the inverse images of the groups W under the quotient map N → C. We can then check condition (d) for each of the groups G(N ). Now fix one of the finite number of groups G(N ) that satisfies conditions (a), (b), (c) and (d) of Definition 4.1.Let G be the inverse image of G(N ) under the reduction map GL 2 (Z N ) → GL 2 (Z/N Z). As usual, for an integer M dividing some power of N , we let G(M ) be the image of G in GL 2 (Z/M Z); note that G(N ) agrees with the previous notation.
Lemma 5 . 4 .
54Let r be the product of the primes dividing N . Let M > 1 be an integer having the same prime divisors as N . If G(rM ) ′ contains {A ∈ SL 2 (Z/rM Z) : A ≡ I (mod M )}, then [SL 2 (Z N ) : G ′ ] = [SL 2 (Z/M Z) : G(M ) ′ ]. Proof. For each positive integer m, define the group S m := {A ∈ SL 2 (Z m ) : A ≡ I (mod m)}.
(Z N ) : G ′ ] = [SL 2 (Z/M Z) : G(M ) ′ ] where M := ℓ|N m ℓ . 5.2. Genus 0 computations. In this section, we compute the set of integers computing I (Γ), we will compute two related quantities. Let I ′ (Γ) be the set of integers as in Definition 4.1 but with condition (e) excluded. Let I ′′ (Γ) be the set of integers as in Definition 4.1 with condition (e) excluded and satisfying the additional condition that X ∞ G(N ) (Q p ) is empty for at most one prime p|N . Lemma 5.6. For a congruence subgroup Γ of genus 0, we have I ′′ (Γ) ⊆ I (Γ) ⊆ I ′ (Γ).
Γ∈S 1 (
1I ′′′ (Γ) − I 0 ) = {220, 240, 360, 504}.
Γ∈S 1 I
1(Γ) − I 0 = Γ∈S 1 (I (Γ) − I 0 ) = {220, 240, 360, 504}. Therefore, I is equal to I 0 ∪ {220, 240, 360, 504} = I. 19 Take any proper subgroup B ⊆ G(M ) for which det(B) = (Z/M Z) × and −I ∈ B. We have a morphism ϕ B : Y B → Y G(M ) = Y G(N ) of curves over Q such that π B = π G(N ) • ϕ B . The morphism ϕ B has degree [G(M ) : varies over the proper subgroups of G(M ) for which det(B) = (Z/M Z) × and −I ∈ B. We have W ⊆ Y G(N ) (Q).Lemma 6.2. If E/Q is a non-CM elliptic curve with j E ∈ π G(N ) (Y G(N ) (Q)−W ), then ±ρ E,M (Gal Q ) is conjugate in GL 2 (Z/M Z) to G(M ).
N 0 .
0We have [SL 2 (Z) : Γ B ] = [GL 2 (Z/M Z) : B] and hence b := [GL 2 (Z/M Z) : G(M )] = [GL 2 (Z/N Z) : G(N )] is a proper divisor of [SL 2 (Z) : Γ B ]. From the computations in §5.3, we may assume that G(N ) is equal to one of the groups denoted G 1 , G 2 , G 3 or G 4 . In particular, we have (N 0 , b) ∈ {(11, 55), (15, 30), (15, 45), (21, 63)}. From the classification in
• G [1/N ] and M G [1/N ], respectively, from [DR73, IV §3]. We refer to [DR73, IV] for further details. The Z[1/N ]-scheme X G is smooth and proper and Y G is an open subscheme of X G . The complement of Y G in X G , which we denote by X ∞ G , is a finiteétale scheme over Z[1/N ], see [DR73, IV §5.2]. The fibers of X G are geometrically irreducible, see [DR73, IV Corollaire 5.6]; this uses our assumption that det(
See http://www.uncg.edu/mat/faculty/pauli/congruence/congruence.html
Acknowledgments. Thanks to Andrew Sutherland and David Zureick-Brown. We have made∼ − → (Z/M Z) 2 such that the pair (E, [α] G ) represents P . Since j E / ∈ {0, 1728}, the automorphisms of E Q act on E[N ] by I or −I. By Lemma 3.1(ii) and −I ∈ G(M ), we have α • σ −1 • α −1 ∈ G(M ) for all σ ∈ Gal Q . We may assume that ρ E,M was chosen so that ρ E,or ρ E,ℓ is not surjective for some prime ℓ > 13.The curve E is non-CM since ρ E,ℓ is surjective for ℓ > 13. Define the subgroupof GL 2 ( Z). By Lemma 6.2, we may assume that ±ρ E,M (Gal Q ) = G(M ). Since G(M ) contains the scalar matrices in GL 2 (Z/M Z), we have H(M ) = G(M ) and an inclusion H ⊆ G. In particular,Let m 0 be the product of the primes ℓ for which ℓ ≤ 5 or for which ρ E,ℓ is not surjective. Let H m and H m 0 be the image of H under the projection to GL 2 (Z m ) and GL 2 (Z m 0 ), respectively. The integer m 0 divides m since ρ E,ℓ is surjective for all ℓ > 13.Lemma 4.5 applied with G and m replaced by H and m 0 , respectively, implies that H ′ = H ′ m 0 × ℓ∤m 0 SL 2 (Z ℓ ). Therefore, we haveSince H ′ ⊆ G ′ ⊆ SL 2 ( Z), we deduce thatWe It remains to show that [SL 2 ( Z) : G ′ ] = n. We have G = G N × ℓ∤N GL 2 (Z ℓ ), so G ′ = G ′ N × ℓ∤N GL 2 (Z ℓ ) ′ . By Lemma 4.11, the index [SL 2 (Z ℓ ) : GL 2 (Z ℓ ) ′ ] is 1 or 2 when ℓ = 2 or ℓ = 2, respectively. Therefore,Recall that a subset S of P 1 (Q) has density δ ifwhere h is the height function. If X G(N ) has genus 0, then it is isomorphic to P 1 Q (from our assumptions on G(N ), the curve X G(N ) has infinitely many Q-points). Choosing such an isomorphism X G(N ) ∼ = P 1 Q allows us to define the density of a subset of X G(N ) (Q); the existence and value of the density does not depend on the choice of isomorphism.
Let E/Q be a non-CM elliptic curve and suppose ℓ > 37 is a prime for which ρ E,ℓ is not surjective. Lemma 6.1.. Then ℓ ≤ [GL 2 ( Z) : ρ E (Gal Q )Lemma 6.1. Let E/Q be a non-CM elliptic curve and suppose ℓ > 37 is a prime for which ρ E,ℓ is not surjective. Then ℓ ≤ [GL 2 ( Z) : ρ E (Gal Q )].
Gal Q ) is contained in the normalizer of a Cartan subgroup of GL 2 (Z/ℓZ). In particular, we have. Ser81, §8.4], we find that ρ E,ℓ. GL 2 (Z/ℓZ) : ρ E,ℓ (Gal Q )] ≥ ℓ(ℓ − 1)/2 ≥ ℓ. Therefore, ℓ ≤ [GL 2 (Z/ℓZ) : ρ E,ℓ (Gal Q )] ≤ [GL 2 ( Z) : ρ E (Gal Q )Proof. From [Ser81, §8.4], we find that ρ E,ℓ (Gal Q ) is contained in the normalizer of a Cartan subgroup of GL 2 (Z/ℓZ). In particular, we have [GL 2 (Z/ℓZ) : ρ E,ℓ (Gal Q )] ≥ ℓ(ℓ − 1)/2 ≥ ℓ. Therefore, ℓ ≤ [GL 2 (Z/ℓZ) : ρ E,ℓ (Gal Q )] ≤ [GL 2 ( Z) : ρ E (Gal Q )].
First suppose that there is a finite set J such that if E/Q is an elliptic curve with j E / ∈ J, then [GL 2 ( Z) : ρ E (Gal Q )] ∈ I. There is thus an integer c > 37 such that for any non-CM E/Q, we have. GL 2 ( Z) : ρ E (Gal Q )] ≤ c, this uses Serre's theorem (and Lemma 2First suppose that there is a finite set J such that if E/Q is an elliptic curve with j E / ∈ J, then [GL 2 ( Z) : ρ E (Gal Q )] ∈ I. There is thus an integer c > 37 such that for any non-CM E/Q, we have [GL 2 ( Z) : ρ E (Gal Q )] ≤ c, this uses Serre's theorem (and Lemma 2
Now suppose that Conjecture 1.2 holds for some constant c. Let J be the finite set from Theo. Now suppose that Conjecture 1.2 holds for some constant c. Let J be the finite set from Theo-
. ∈ J We Have ; ∈ I, GL 2 ( Z) : ρ E (Gal Q )∈ J, we have [GL 2 ( Z) : ρ E (Gal Q )] ∈ I.
= n. Lemma 6.1 implies that ρ E,ℓ is surjective for all primes ℓ > max{37, n}. Let J be the set from Theorem 1.3 with c := max{37, n}. Now take any elliptic curve E/Q with j E ∈ J n − J; note that J n − J is non-empty since J n is infinite and J is finite. The representation ρ E,ℓ is surjective for all ℓ > c and j E / ∈ J, so. E , equivalently, with [GL 2 ( Z) : ρ E (Gal Q ). GL 2 ( Z) : ρ E (Gal Q )j E ∈ J n , equivalently, with [GL 2 ( Z) : ρ E (Gal Q )] = n. Lemma 6.1 implies that ρ E,ℓ is surjective for all primes ℓ > max{37, n}. Let J be the set from Theorem 1.3 with c := max{37, n}. Now take any elliptic curve E/Q with j E ∈ J n − J; note that J n − J is non-empty since J n is infinite and J is finite. The representation ρ E,ℓ is surjective for all ℓ > c and j E / ∈ J, so [GL 2 ( Z) : ρ E (Gal Q )]
By Proposition 5.1, we have n ∈ I (Γ) for some congruence subgroup Γ of SL 2 (Z) of genus 0 or 1. From our computation of I 0 in §5.2, we may assume that Γ has genus 0 when n / ∈ {220, 240, 360, 504}. Denote the level of Γ by N 0. Now take any integer n ∈ I. To complete the proof of the theorem, we need to show that J n is infinite. Let N be the integer N 0 , 4N 0 or 2N 0 when v 2 (N 0 ) is 0, 1 or at least 2, respectively. The integer N is not divisible by any prime ℓ > 13 (if Γ has genus 0, this follows from the classification of genus 0 congruence subgroups in [CP03]; if Γ has genus 1, then we saw in §5.3 that N ∈ {11, 15, 21})Now take any integer n ∈ I. To complete the proof of the theorem, we need to show that J n is infinite. By Proposition 5.1, we have n ∈ I (Γ) for some congruence subgroup Γ of SL 2 (Z) of genus 0 or 1. From our computation of I 0 in §5.2, we may assume that Γ has genus 0 when n / ∈ {220, 240, 360, 504}. Denote the level of Γ by N 0 . Let N be the integer N 0 , 4N 0 or 2N 0 when v 2 (N 0 ) is 0, 1 or at least 2, respectively. The integer N is not divisible by any prime ℓ > 13 (if Γ has genus 0, this follows from the classification of genus 0 congruence subgroups in [CP03]; if Γ has genus 1, then we saw in §5.3 that N ∈ {11, 15, 21}).
(c), (d) and (e) of Definition 4.1 and also satisfies n =. Since n ∈ I (Γ), there is a subgroup G(N ) of GL 2 (Z/N Z) that satisfies conditions (a), (b). 2gcd(2, N ), where G N is the inverse image of G(N ) under the reduction map GL 2 (Z N ) → GL. Let G be the inverse image of G(N ) under GL 2 ( Z) → GL 2 (Z/N ZSince n ∈ I (Γ), there is a subgroup G(N ) of GL 2 (Z/N Z) that satisfies conditions (a), (b), (c), (d) and (e) of Definition 4.1 and also satisfies n = [SL 2 (Z N ) : G ′ N ] · 2/ gcd(2, N ), where G N is the inverse image of G(N ) under the reduction map GL 2 (Z N ) → GL 2 (Z/N Z). Let G be the inverse image of G(N ) under GL 2 ( Z) → GL 2 (Z/N Z).
N divides some power of m. Let G m be the image of G under the projection map GL 2 ( Z) → GL 2 (Z m ). Lemma 4.7 implies that there is a positive integer M , dividing some power of m, such that if H is an open subgroup of G m ⊆ GL 2 (Z m ), then H equals G m if and only if H(M ) equals G m (M ) = G(M ). Let m be the product of the primes ℓ ≤ 13; note thatLet m be the product of the primes ℓ ≤ 13; note that N divides some power of m. Let G m be the image of G under the projection map GL 2 ( Z) → GL 2 (Z m ). Lemma 4.7 implies that there is a positive integer M , dividing some power of m, such that if H is an open subgroup of G m ⊆ GL 2 (Z m ), then H equals G m if and only if H(M ) equals G m (M ) = G(M ).
There is an infinite subset S of Y G(N ) (Q), with positive density if X G(N ) has genus 0, such that if E/Q is an elliptic curve with j E ∈ π G(N ) (S). Lemma 6.4.. then ρ E,ℓ is surjective for all ℓ > 13Lemma 6.4. There is an infinite subset S of Y G(N ) (Q), with positive density if X G(N ) has genus 0, such that if E/Q is an elliptic curve with j E ∈ π G(N ) (S), then ρ E,ℓ is surjective for all ℓ > 13.
Now consider the case where X G(N ) has genus 1. If one point of X G(N ) (Q) was isolated in X G(N ) (Q v ), then using the group law of X G(N ) (Q) (by first fixing a rational point), we find that every point is isolated. So suppose that for each P ∈ X G(N ) (Q), there is an open subset U P ⊆ X G(N ) (Q v ) such that U P ∩ X G(N ) (Q) = {P }. The sets {U P } P ∈X G(N) (Q) along with the complement of the closure of X G(N ) (Q) in X G(N ) (Q v ) form an open cover of X G(N ) (Q v ) that has no finite subcover. This contradicts the compactness of X G(N ) (Q v ) and proves the claim. Since π G(N ) : Y G(N ) (R) → R is continuous, the above claim with v = ∞ implies that the set π G(N ) (Y G(N ) (Q)) is not a subset of Z. Choose a rational number j ∈ π G(N ) (Y G(N ) (Q)) that is not an integer. There is a prime p such that v p (j) is negative. the set X G(N ) (Q) has no isolated points in X G(N ) (Q v ), i.e., there is no open subset U of X G(N ) (Q v ), with respect to the v-adic topology. If X G(N ) has genus 0, then the claim follows since no point in P 1 (Q) is isolated in P 1 (Q v ). set e := −v p (j). Let U be the set of points P ∈ Y G(N ) (Q p ) for which π G(N ) (P ) = 0 and v p (π G(N ) (P )) = −e; it is an open subset of Y G(N ) (Q pProof. We claim that for any place v of Q, the set X G(N ) (Q) has no isolated points in X G(N ) (Q v ), i.e., there is no open subset U of X G(N ) (Q v ), with respect to the v-adic topology, for which U ∩ X G(N ) (Q) consists of a single point. If X G(N ) has genus 0, then the claim follows since no point in P 1 (Q) is isolated in P 1 (Q v ). Now consider the case where X G(N ) has genus 1. If one point of X G(N ) (Q) was isolated in X G(N ) (Q v ), then using the group law of X G(N ) (Q) (by first fixing a rational point), we find that every point is isolated. So suppose that for each P ∈ X G(N ) (Q), there is an open subset U P ⊆ X G(N ) (Q v ) such that U P ∩ X G(N ) (Q) = {P }. The sets {U P } P ∈X G(N) (Q) along with the complement of the closure of X G(N ) (Q) in X G(N ) (Q v ) form an open cover of X G(N ) (Q v ) that has no finite subcover. This contradicts the compactness of X G(N ) (Q v ) and proves the claim. Since π G(N ) : Y G(N ) (R) → R is continuous, the above claim with v = ∞ implies that the set π G(N ) (Y G(N ) (Q)) is not a subset of Z. Choose a rational number j ∈ π G(N ) (Y G(N ) (Q)) that is not an integer. There is a prime p such that v p (j) is negative; set e := −v p (j). Let U be the set of points P ∈ Y G(N ) (Q p ) for which π G(N ) (P ) = 0 and v p (π G(N ) (P )) = −e; it is an open subset of Y G(N ) (Q p ).
If X G(N ) has genus 0, then S clearly has positive density. Now take any elliptic curve E/Q with j E ∈ π G(N ) (S) and any prime ℓ > max{37, e}; it is non-CM ≤ max{37, e}. Suppose that ρ E,ℓ is not surjective. From Lemmas 16, 17 and 18 in [Ser81], we find that ρ E,ℓ (Gal Q ) is contained in the normalizer of a Cartan subgroup of GL 2 (Z/ℓZ). In particular, the order of ρ E,ℓ (Gal Q ) is not divisible by ℓ. We have v p (j E ) = −e < 0 since j E ∈ π G(N ) (S). Define S := U ∩ Y G(N ) (Q) = U ∩ X G(N ) (Q)Let E ′ /Q p be the Tate curve with j-invariant j E ; see. Ser98, IV Appendix A.1] for details. From the proposition in [Ser98, IV Appendix A.1.5] and our assumption ℓ > e, we find that ρ E ′ ,ℓ (Gal Qp ) contains an element of order ℓ. Since E ′ and E have the same j-invariant, they become isomorphic over some quadratic extension of Q p . Since ReferencesDefine S := U ∩ Y G(N ) (Q) = U ∩ X G(N ) (Q); it is non-empty by our choice of e (in particular, U is non-empty). The set S is infinite since otherwise there would be an isolated point of X G(N ) (Q) in X G(N ) (Q p ). If X G(N ) has genus 0, then S clearly has positive density. Now take any elliptic curve E/Q with j E ∈ π G(N ) (S) and any prime ℓ > max{37, e}; it is non-CM ≤ max{37, e}. Suppose that ρ E,ℓ is not surjective. From Lemmas 16, 17 and 18 in [Ser81], we find that ρ E,ℓ (Gal Q ) is contained in the normalizer of a Cartan subgroup of GL 2 (Z/ℓZ). In particular, the order of ρ E,ℓ (Gal Q ) is not divisible by ℓ. We have v p (j E ) = −e < 0 since j E ∈ π G(N ) (S). Let E ′ /Q p be the Tate curve with j-invariant j E ; see [Ser98, IV Appendix A.1] for details. From the proposition in [Ser98, IV Appendix A.1.5] and our assumption ℓ > e, we find that ρ E ′ ,ℓ (Gal Qp ) contains an element of order ℓ. Since E ′ and E have the same j-invariant, they become isomorphic over some quadratic extension of Q p . Since References
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Frobenius distributions in GL2-extensions. Serge Lang, Hale Trotter, Lecture Notes in Mathematics. 504Springer-VerlagDistribution of Frobenius automorphisms in GL2-extensions of the rational numbers. MR0568299 (58 #27900) ↑5.1Serge Lang and Hale Trotter, Frobenius distributions in GL2-extensions, Lecture Notes in Mathematics, Vol. 504, Springer-Verlag, Berlin, 1976. Distribution of Frobenius automorphisms in GL2-extensions of the rational numbers. MR0568299 (58 #27900) ↑5.1
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Jacques Vélu, Les points rationnels de X0(37), Journées Arithmétiques. Grenoble37Jacques Vélu, Les points rationnels de X0(37), Journées Arithmétiques (Grenoble, 1973), 1974, pp. 169-179. Bull. Soc. Math. France Mém., 37. MR0366930 (51 #3176) ↑ii
Elliptic curves with maximal Galois action on their torsion points. David Zywina, ↑4.4Bull. London Math. Soc. 425David Zywina, Elliptic curves with maximal Galois action on their torsion points, Bull. London Math. Soc. 42 (2010), no. 5, 811-826. ↑4.4
USA Email address: [email protected]. Ithaca, NY 14853Department of Mathematics, Cornell UniversityDepartment of Mathematics, Cornell University, Ithaca, NY 14853, USA Email address: [email protected] URL: http://www.math.cornell.edu/~zywina
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[
"https://github.com/davidzywina/PossibleIndices",
"https://github.com/davidzywina/PossibleIndices"
] |
[
"BERNSTEIN TYPE RESULTS FOR LAGRANGIAN GRAPHS WITH PARTIALLY HARMONIC GAUSS MAP",
"BERNSTEIN TYPE RESULTS FOR LAGRANGIAN GRAPHS WITH PARTIALLY HARMONIC GAUSS MAP"
] |
[
"Wei Zhang "
] |
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We establish Bernstein Theorems for Lagrangian graphs which are Hamiltonian minimal or have conformal Maslov form. Some known results of minimal (Lagrangian) submanifolds are generalized. arXiv:0707.0183v2 [math.DG]
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[
"https://arxiv.org/pdf/0707.0183v2.pdf"
] | 1,780,420 |
0707.0183
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82e52318f746e172de5d1e2c2f8a4628dbadd47b
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BERNSTEIN TYPE RESULTS FOR LAGRANGIAN GRAPHS WITH PARTIALLY HARMONIC GAUSS MAP
Wei Zhang
BERNSTEIN TYPE RESULTS FOR LAGRANGIAN GRAPHS WITH PARTIALLY HARMONIC GAUSS MAP
We establish Bernstein Theorems for Lagrangian graphs which are Hamiltonian minimal or have conformal Maslov form. Some known results of minimal (Lagrangian) submanifolds are generalized. arXiv:0707.0183v2 [math.DG]
Introduction
The classical Bernstein theorem asserts any global graphic minimal surface on R 2 must be a plane. This result has been generalized to R n for n ≤ 7, and higher dimensions or codimesions under various growth conditions, see [4], [7], [12] and their references.
Due to string theory, minimal Lagrangian submanifolds, especially the special Lagrangian submanifolds received much attention in recent years ( [9]). The Lagrangian graph in C n has the form of (x, ∇u(x)), where u(x) is a smooth function on R n . Some authors established Bernstein type results for minimal Lagrangian graph under various conditions ( [8], [15], [12], [3]). On the other hand, there are two nice generalizations of minimal Lagrangian submanifolds: Hamiltonian minimal Lagrangian submanifolds and Lagrangian submanifolds with conformal Maslov form. They are introduced respectively by [10] and [11]. It turns out that they have partially harmonic Gauss maps into the Lagrangian Grassmannian, and may be regarded as generalizations of submanifolds with parallel mean curvature vector. In this paper, we will establish Bernstein type results for these two classes of typical Lagrangian submanifolds.
Mathematics Classification Primary(2000): 53A10, 53A07, 53C38 Keywords: Bernstein type theorem, Hamiltonian minimal, conformal Maslov form, Lagrangian graphs.
The author thanks his advisor Professor Dong for suggesting him study this subject and some related methods. He is also grateful to the referee who had pointed out a mistake in the previous verison. Theorem 1. Let (x,∇u) represent a Hamiltonian minimal Lagrangian submanifold Σ in C n . If u is a smooth convex function on R n , i.e.
Hess(u) ≥ 0 then Σ is an affine plane.
Theorem 2. Σ = (x, ∇u) is a Lagrangian submanifold with conformal Maslov form. Suppose that there exists a number β 0 with
β 0 < cos −n ( π 2 √ κn ), κ = 1 n = 2 2 n ≥ 3
such that ∆ u := det(I + Hess 2 (u)) ≤ β 0 then Σ is a plane.
Theorem 1 was obtained in [15] under the condition that Σ is minimal Lagrangian and there is some kind of generalization of Yuan's Theorem in [3]; Theorem 2 was established in [6] for the submanifolds of parallel mean curvature in Euclidean space R N and improved in [7] later.
As we know, there is a fiberation of lagrangian grassmannian over S 1 used to introduce the Maslov form. The basic ideas of this paper is to project the Gauss map to S 1 or a convex domain in the standard fiber. The partial harmonicity implies that these projected map is harmonic. Therefore we may use the methods in [6] to prove our results.
Preliminary
Let L(n) be the Lagrangian Grassmannian consisting of all oriented Lagrangian subspaces in C n . It can be identified with U (n)/SO(n) in a natural way. More over it is totally geodesic in the usual oriented Grassmann manifold G(n, n) (cf. [8]). We will explain this in a more invariant way.
We define an automorphism F ω of G(n, n), which sends any n-subspace P to its oriented symplectic orthogonal with respect to the canonical symplectic form ω in C n , i.e. F ω (P ) = J(P ⊥ ), where J the complex structure. A subspace P ∈ G(n, n) is Lagrangian if and only if F ω (P ) = P (cf. Lemma I.2.1 of [1]), so L(n) is totally geodesic in G(n, n).
The determinant mapping det : U (n) → S 1 descends to the quotient by SO(n), so that U (n)/SO(n) is a bundle over S 1 with projection det : U (n)/SO(n) → S 1 and standard fiber SU (n)/SO(n). Following proposition describes some important gemetric properties of this fibration, which will be used to prove the main results.
Proposition 3. The projection det : U (n)/SO(n) → S 1 is a Riemannian submersion with totally geodesic fiber, and the horizontal lift of S 1 is a geodesic orthogonal to each fibers. Furthermore, for any open arc
c = {e iθ : α < θ < β, 0 < β − α < 2π}of S 1 , (det) −1 (c) is isometric to a Riemann product ( α √ n , β √ n ) × SU (n)/SO(n)
with metric dθ 2 ×h, where h denotes the standard metric on SU(n)/SO(n) as a symmetric space.
Proof. For convenience, from now on, always denote O the point in L(n) representing base plane R n . First show the fiber is totally geodesic.
Let F θ = det −1 (θ) be the fiber at e iθ ∈ S 1 , where θ ∈ [0, 2π). L(n) is symmetric, so it is only necessary to check the fiber F 0 . The tangent vectors to F 0 at O congruent to the form θ 1 . . . θ n , θ k = 0
The geodesics issue from these vectors are left cosets of diagonal matrices:
e itθ 1 . . . e itθn , θ k = 0
whose determinant is always 1, still contained in F 0 . Hence F 0 is totally geodesic, then all the fibers are totally geodesic in L(n). We will directly show the property of local Riemannian product, and Riemann submersion is an easy corolary.
There is a geodesic σ(t) in the form of scalar matrix
e it √ n . . . e it √ n
where t is the arc length parameter. This curve intersects with it'self over period 2π √ n, i.e. a close geodesic of length 2π √ n, topological n-cover of the S 1 . Its tangent vector at O is:
1 √ n . . . 1 √ n
The inner product in T L(n) is given by trAB * , where A, B are matrix represent the tangent vectors. Therefore this curve is orthonormal to F 0 and by symmetric, to every fiber.
Since the scalar matrices commute with any other matrices, the multiplication of scalar matrices descendent to L(n) and is an isometric transformation between fibers. Thus when β < 2π,
(det) −1 (0, β) is iso- morphic to F 0 · σ(0, β √ n ) ∼ = (0, β √ n ) × SU (n)/SO(n)
. The effect of act by σ(t) is rotating the planes by a given angle.
Consequently, the projection det is a Riemann submersion with constant dilation factor √ n. σ is one of the horizontal lift of S 1 .
Now we introduce the concepts of Hamiltonian minimal Lagrangian submanifolds and Lagrangian submanifolds with conformal Maslov form, then state several properties close related to harmonic map.
Let η be a normal vector field along the Lagrangian submanifold Σ. Denote by α η the 1-form on Σ defined by
α η (X) = ω(η, X), X ∈ T Σ
where ω is the symplectic structure of C n . Particularly, if H is the mean curvature vector of Σ, α H is called the Maslov form.
The normal vector field η is called Hamiltonian if the 1-form α η associated with η is a exact form, i.e. α η = df , where f ∈ C ∞ (Σ).
According to [10], a Lagrangian submanifold Σ is called Hamiltonian minimal (H-minimal for short) if it is a critical point of the volume functional with respect to all Hamiltonian variations. There is a holomorphic volume form dz = dz 1 ∧ dz 2 · · · ∧ dz n in C n , evaluating dz on Σ's tangent space has scalar value γ. ψ is called the Lagrangian angle of oriented Lagrangian submanifold Σ if γ = e iψ (cf. [13]). ψ is a well defined function takes value in 2πR/Z.
Wolfson had proved that dψ = α H (Theorem 1.2 in [13]). By Lemma 4, Σ is H-minimal if and only if ψ is a harmonic real value function on it. Because the exponential map is a totally geodesic map from R to S 1 , by the composition law of harmonic map, γ is a harmonic map to S 1 .
The Gauss map ν takes values in L(n) ∼ = U (n)/SO(n). Obviously, γ = det • ν, thus there is a commute diagram
L(n) det Σ ν = = { { { { { { { { γ / / S 1 By Proposition 3, γ : Σ → S 1 is harmonic if and only if τ (ν) H ≡ 0,
where τ (ν) H is the horizontal component of the tension field τ (ν) with respect to the projection det. Thus Σ is H-minimal if and only if its Gauss map ν is horizontally harmonic.
We had introduced the Maslov form α H . According to [11], a Lagrangian submanifold Σ in C n is said to have conformal Maslov form (CMF briefly), if JH is a conformal vector fields on Σ. The Gauss map of such manifolds is vertically harmonic, i.e. τ (ν) V ≡ 0, where τ (ν) V is the vertical component of the tension field τ (ν) with respect to the projection det.
Proof of the main theorems
Simple Riemannian manifold, is quasi-isomorphic to Euclidean space, where certain Liuville type theorem holds on. For instance, Theorem 5 ([6]). Let ν : Σ → M be a harmonic map, Σ a simple Riemannian manifold and M is Riemannian manifold whose sectional curvature is bounded from above by a constant κ ≥ 0. Denote B R (q) a geodesic ball of radius R < π 2 √ κ which does not meet the cut locus of q. If the range ν(Σ) is contained in B R (q), then ν is a constant map.
Apply this theorem to the Gauss map of a submanifold with parallel mean curvature, [6] got Bernstein type theorem.
Reader can find the precise definition of simple manifold in [6]. In the context of lagrangian graph, uniformly bounded Hess(u) assures the simpleness.
At the same time, so far as we know, the Lagrangian fiberation has many local nice properties. The Gauss image X of an oriented Lagrangian graph Σ lies in a special region of L(n), which makes it possible to extend this local properties to global. To do this ,we need a natural representation of the X in L(n), i.e. picking up an element in every coset of U (n)/SO(n) canonically, to represent the given tangent plane.
Lemma 6. If P is a tangent plane of Σ = (x, ∇u) in C n , denote λ k the eigenvalues of Hess(u), then in its corresponding coset, there exists V = Sdiag(e iθ 1 , · · · , e iθn )S −1 , where S ∈ SO(n), and θ k = arctan λ k are the critical angles between P and R n .
Proof. This kind of P in C n R 2n , contains no vectors orthogonal to R n , so we can define a linear map form R n to its compliment JR n : F P (a) = b, where a ∈ R n , b ∈ JR n , iff there is a vector v in P , s.t. π 1 v = a, π 2 v = b. F P is Hess(u) essentially. By eigenvalue decomposition, there is a orthonormal basis s k of R n , s.t. F P (s k ) = λ k J(s k ). According to [14], [7], θ k = arctan λ k are the critical angles. Then e k = e iθ k s k are orthonomal basis of P . Let S be the transformation sends s k to the standard basis of R n . {s k } and the standard basis of R n can be also viewed as complex basis of C n , while S a transform of complex space. Thus the matrix represent P in the form of:
S e iθ 1 . . . e iθn S −1
Since the graph is global oriented, we can arrange s k making corresponding e k give the due orientation, so S belongs to SO(n) rather than O(n).
Given any tangent plane P , corresponds to a unique set of {θ k } where − π 2 < θ k < π 2 . Write P θ 1 ,··· ,θn in stead of P .
Prove Theorem 1.
This representation is unique up to permutations of θ k . Defineγ = θ k , it is a well defined function taking value in R. The following diagram commutes:
R e it Σγ > > γ / / S 1 i.e.
γ is a representation of the Lagrangian angle ψ andγ is harmonic iff γ is.
From theorem 5, any bounded harmonic function on simple manifold must be constant. If Σ is simple,γ = const is easily concluded form θ k ∈ (− nπ 2 , nπ 2 ). Thus γ = const either. Recall Proposition 2.17 in [5], which asserts that a connected submanifold Σ ∈ R 2n = C n is both Lagrangian and minimal if and only if Σ is special Lagrangian with respect to one of the calibration Re{e iθ dz}. If γ is const, Re{e iγ dz} is the required calibration and Σ the SL submanifold. Therefore: Proposition 7. Σ = (x, ∇u) is Hamiltonian minimal. If Hess(u) is uniformly bounded, then Σ is minimal in the usual sense.
Chern ([2]) proved that any graphic hypersurface in R n+1 with parallel mean curvature must be minimal. proposition 7 generalizes Chern's theorem to Lagrangian case. Noticing the the fact after a certain kind of rotation(see [15]), Hess(u) ≥ 0 force Σ to be simple and Theorem 1.1 in [15] Theorem 8 ( [15]). Suppose Σ = (x, ∇u) is a minimal Lagrangian submanifold of C n and u is a smooth convex function on R n , then Σ is an affine plane. theorem 1 follows.
Prove Theorem 2.
For the representation is unique,
X ∩ F θ is divided into disjoint com- ponents X l,θ = {P θ 1 ,··· ,θn ∈ X| k θ k = 2lπ + θ, l ∈ Z, −[ n 4 ] − 1 ≤ l ≤ [ n 4 ]}.
We can define a global projection π : X → F 0 by π(P θ 1 ,··· ,θn ) = P θ 1 ,··· ,θn · σ(− P θ k √ n ). Intuitionally, pull back all the components in all the fibers to the neighborhood of X 0,0 .
Thus the Gauss image X looks like a tube around the close geodesic σ and has a global Riemann product structure. Proposition 9. Σ is a simple Lagrangian graph with conformal Maslov form. Denote X the image of it's Gauss map and T R the region { k (θ k −
P i θ i n ) 2 ≤ R 2 , R < π 2 √ 2 }. If X ⊂ T R , then Σ is a plane. Proof. Project X to F 0 via π, denote D = π(X), there is Σ ν − → X π − → D ⊂ F 0
If ν is vertical harmonic, π • ν is a harmonic map to F 0 by the composition formula.
The geodesic ball B R of radius R < π 2 √ 2 is a convex domain in G(n, n)(cf. [6]). For F 0 is the totally geodesic fiber of L(n) and L(n) is totally geodesic in G(n, n), so is F 0 in G(n, n).
Thus F 0 ∩ B R is a convex domain in F 0 and D ⊂ F 0 ∩ B R .
Following the theorem 5, the harmonic map π • ν is constant. Without losing generality, assuming the image is O. Then the image of ν is the cosets represented by e iθ Id, i.e.
Hess(u) = (tan θ)Id
Then ∂ 2 u ∂x i ∂x j = 0 when i = j. thus ∂u ∂x i is only function about x i , so does ∂ 2 u
∂x 2 i . But ∂ 2 u ∂x 2 i = ∂ 2 u ∂x 2 j
for any x i , x j . Both sides have to be constant, i.e. Hess(u) = cId.
Therefore u is an affine plane.
The region T R is not a strict tube with constant diameter, for when P i θ i n approaches ± π 2 , the diameter will tend to zero, but this makes no difference to our proof. Remark. Proposition 9 says nothing but if the Gauss map lies in a sufficient small tabular neighborhood of radius π 2 √ 2 around the closed geodesic, then it is plane. In [7], after delicate study of the structure about Grassmannian, Jost and Xin show there is a bigger convex domain B G . Hence above proposition still holds if the image of ν • π lies in F 0 ∩ B G .
We are in the position to complete the proof of Theorem 2. Denote u = (det(I + Hess 2 (u))) 1 2 , then u ≤ cos −n ( R √ n ) implies Σ is simple and θ k 2 ≤ R 2 , moreover k (θ k − P i θ i n ) 2 ≤ R 2 . By Proposition 9, u is affine plane. The special case n = 2 follows form the fact the fiber SU (2)/SO(2) is isomorphic to S 2 . The radius of F 0 's convex domain is π 2 rather than π 2 √ 2 .
Lemma 4 .
4[10] The Lagrangian submanifold Σ is H-minimal if and only if δα H = 0, i.e. the Maslov form is coclosed.
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A bernstein result for minimal graphs of controlled growth. K Ecker, G Huisken, J. Differential Geom. 312K. Ecker and G. Huisken. A bernstein result for minimal graphs of controlled growth. J. Differential Geom., 31(2):397-400, 1990.
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Bernstein type theorems for higher codimension. J Jost, Y L Xin, Calc. Var. Partial Differential Equations. 94J. Jost and Y. L. Xin. Bernstein type theorems for higher codimension. Calc. Var. Partial Differential Equations, 9(4):277-296, 1999.
A bernstein theorem for special lagrangian graphs. J Jost, Y L Xin, arXiv:math.DG/0101131J. Jost and Y. L. Xin. A bernstein theorem for special lagrangian graphs. arXiv:math.DG/0101131, January 2001.
D D Joyce, arXiv:math.DG/0108088Lectures on calabi-yau and special lagrangian geometry. D. D. Joyce. Lectures on calabi-yau and special lagrangian geometry. arXiv:math.DG/0108088, 2001.
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Minimal lagrangian diffeomorphism and the monge-ampere equation. J Wolfson, J. Differential Geometry. 46J. Wolfson. Minimal lagrangian diffeomorphism and the monge-ampere equa- tion. J. Differential Geometry, 46:335-373, 1997.
Differential geometry of grassmann manifolds. Y C Wong, Proc. Nat. Acad. Sci. USA. 57Y. C. Wong. Differential geometry of grassmann manifolds. Proc. Nat. Acad. Sci. USA, 57:589-594, 1967.
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[] |
[
"The Double Copy Structure of Soft Gravitons",
"The Double Copy Structure of Soft Gravitons"
] |
[
"Agustín Sabio Vera \nInstituto de Física Teórica UAM/CSIC\nUniversidad Autónoma de Madrid C/ Nicolás\nCabrera 15E-28049MadridSpain\n",
"Miguel A Vázquez-Mozo \nDepartamento de Física Fundamental & IUFFyM\nUniversidad de Salamanca Plaza de la Merced s/n\nE-37008SalamancaSpain\n"
] |
[
"Instituto de Física Teórica UAM/CSIC\nUniversidad Autónoma de Madrid C/ Nicolás\nCabrera 15E-28049MadridSpain",
"Departamento de Física Fundamental & IUFFyM\nUniversidad de Salamanca Plaza de la Merced s/n\nE-37008SalamancaSpain"
] |
[] |
The subleading corrections to factorization theorems for soft bremsstrahlung in nonabelian gauge theories and gravity are investigated in the case of a five point amplitude with four scalars. Building on recent results, we write the action of the angular momentum operators on scattering amplitudes as derivatives with respect to the Mandelstam invariants to uncover a double copy structure in the contribution of the soft graviton to the amplitude, both in the leading term and the first correction. Using our approach, we study Gribov's theorem as extended to nonabelian gauge theories and gravity by Lipatov, and find that subleading corrections can be obtained from those to Low's theorem by dropping the terms with derivatives with respect to the center-of-mass energy, which are suppressed at high energies. In this case, the emitted gravitons are not necessarily soft.
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10.1007/jhep03(2015)070
|
[
"https://arxiv.org/pdf/1412.3699v2.pdf"
] | 34,923,117 |
1412.3699
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ce09842723979e463dcd3db5c35e204e144c327f
|
The Double Copy Structure of Soft Gravitons
17 Mar 2015
Agustín Sabio Vera
Instituto de Física Teórica UAM/CSIC
Universidad Autónoma de Madrid C/ Nicolás
Cabrera 15E-28049MadridSpain
Miguel A Vázquez-Mozo
Departamento de Física Fundamental & IUFFyM
Universidad de Salamanca Plaza de la Merced s/n
E-37008SalamancaSpain
The Double Copy Structure of Soft Gravitons
17 Mar 2015
The subleading corrections to factorization theorems for soft bremsstrahlung in nonabelian gauge theories and gravity are investigated in the case of a five point amplitude with four scalars. Building on recent results, we write the action of the angular momentum operators on scattering amplitudes as derivatives with respect to the Mandelstam invariants to uncover a double copy structure in the contribution of the soft graviton to the amplitude, both in the leading term and the first correction. Using our approach, we study Gribov's theorem as extended to nonabelian gauge theories and gravity by Lipatov, and find that subleading corrections can be obtained from those to Low's theorem by dropping the terms with derivatives with respect to the center-of-mass energy, which are suppressed at high energies. In this case, the emitted gravitons are not necessarily soft.
Introduction
The scattering amplitudes of quantum field theories with massless intermediate gauge bosons
have an interesting infrared behavior, in particular in the soft limit where massless bosons are emitted with very small momenta. In this context, it was proved by Low [1] that in QED the leading behavior of an inelastic amplitude with an emitted soft photon is dominated by those contributions in which the photon is bremsgestrahlt by the external states (Low formulated the theorem for scalar charged particles, Burnett and Kroll generalized it to the case of charged fermions [2]). Subleading corrections to this result, sensitive to internal emissions, were also computed in [1] and found to have a particularly simple form. In the case of gravity, Weinberg showed [3] that a similar result holds for scattering amplitudes in which a soft graviton is emitted.
More recently, there has been a renewed interest in these results stemming from the realization that Weinberg's soft-graviton theorem can be regarded as the Ward identity associated with the symmetries of the gravitational theory at null infinity [4]. This has led to the formulation of a new soft-graviton theorem including next-to-leading and next-to-next-to-leading order corrections which have a universal expression in terms of the angular momentum of the hard particles [5]. Using obvious notation, M n+1 (k; p 1 , . . . , p n ) = κ n i=1 ε µν p iµ p iν
p i · k + n i=1 ε µν p iµ (k α J (i) να ) p i · k + n i=1 ε µν (k α J (i) µα )(k β J (i) νβ ) p i · k
M n (p 1 , . . . , p n ), (1.1) where
J (i) µν = p iµ ∂ ∂p ν i − p iν ∂ ∂p µ i (1.2)
is the angular momentum operator of the i-th particle. It has been argued that this result is not renormalized [6,7]. Generalizations of this new soft-graviton theorem to arbitrary dimensions were studied in [8].
In the case of Yang-Mills theories, the subleading corrections to Low's result can also be encoded in terms of the angular momentum operator acting on the n-point amplitude [9,10]:
A n+1 (k; p 1 , . . . , p n ) = g n i=1 ǫ · p i p i · k + n i=1 ǫ µ k ν J (i) µν p i · k A n (p 1 , . . . , p n ). (1.3)
In QED, the subleading corrections also admit an interpretation in terms of the asymptotic symmetries of the theory at null infinity [11].
Low's theorem was originally derived in the limit in which the momentum of the photon is taken to be very small, k → 0. It was later realized by Gribov [12] that the expression found by
Low has a broader range of validity if the scattering takes place at a large center-of-mass energy of the colliding particles, √ s. In this case the factorization can also hold for hard emissions as long as their transverse momentum with respect to the radiating particle is small compared to the momentum transfers typical of the scattering process. Thus, for two colliding hadrons of typical mass µ and momenta p and q, the amplitude is dominated by external bremsstrahlung in the kinematic region defined by
2 p · k, 2 q · k ≪ s k 2 ⊥ ≈ (2 p · k)(2 q · k) s ≪ µ 2 ,(1.4)
with k ⊥ the transverse momentum of the photon. The main difference with respect to the regime of validity of Low's theorem (which applies in the region 2 p · k, 2 q · k ≪ µ 2 ) is that now we assume a large center-of-mass energy √ s ≫ µ, without requiring the photon momentum to be soft. The theorem was generalized by Lipatov in [13] to the case of a Yang-Mills field or a graviton coupled to scalars.
In this note we study the corrections to soft gluon and graviton theorems for amplitudes containing scalar fields, and investigate the double copy structure of the latter one. In Section 2,
we generalize the analysis of [14] to the nonabelian case for the scattering amplitude of two different scalars with the emission of a gluon, showing that the first correction to Low's leading result is completely fixed by gauge invariance, as it happens in QED. Writing the action of the angular momentum operators on the four scalars amplitude using derivatives with respect to the Mandelstam s and t invariants, we find that the amplitude has a particularly simple form in terms of a set of gauge invariant coefficients. Due to the Jacobi identity satisfied by the color factors, these coefficients admit shift transformations that preserve the value of the amplitude and do play an important role when connecting the gauge theory amplitude to the corresponding gravitational one. In Section 3 we analyze the gravitational scattering amplitude of two scalars with the emission of a graviton in the soft limit. As in the gauge theory case, we express the first correction in terms of derivatives with respect to the Mandelstam invariants and find that the associated coefficients have a double copy structure. This is interpreted in the sense that the contribution of the soft graviton to the five-point gravitational amplitude can be factored as the square of the contribution of the soft gluon to its gauge theory counterpart, after removing the color factors. Section 4 is devoted to the study of the Gribov limit, which allows for not necessarily soft bremsstrahlung, both in gauge theories and gravity. We find that the first correction to the amplitude in this kinematic region computed in [13] can be obtained from the corresponding correction to the Low/Weinberg theorems by dropping derivatives with respect to the s Mandelstam invariant. Finally, in Section 5 we summarize our conclusions.
Soft gluons in scalar QCD
It has been shown in [14] that in scalar QED the first subleading correction to Low's theorem is completely fixed by gauge invariance. In this section we extend this result to the nonabelian case by considering scalar QCD (sQCD) with two flavors and the scattering amplitude of two distinct scalars in an arbitrary representation with radiation of a gluon. The five generic topologies contributing to this process are shown in Fig. 1: four of them correspond to the bremsstrahlung of a gluon by the external scalars, while in the fifth one the gluon is emitted from an internal propagator. Based on Lorentz and color covariance, the amplitude can be written as
A 5 = 2g c 1 p ′ · ǫ s 1 ′ A 4 (p, q, p ′ + k, q ′ ) − c 2 p · ǫ s 1 A 4 (p − k, q, p ′ , q ′ ) + c 4 q ′ · ǫ s 2 ′ A 4 (p, q, p ′ , q ′ + k) − c 5 q · ǫ s 2 A 4 (p, q − k, p ′ , q ′ ) (2.1) + g 2 c 3 ǫ µ B µ 1 (k; p, q, p ′ , q ′ ) + c 6 ǫ µ B µ 2 (k; p, q, p ′ , q ′ ) + c 7 ǫ µ B µ 3 (k; p, q, p ′ , q ′ ) .
Here A 4 (p, q, p ′ , q ′ ) is the color-stripped four-point scalar amplitude, while the three functions B µ 1 (k; p, q, p ′ , q ′ ), B µ 2 (k; p, q, p ′ , q ′ ), and B µ 3 (k; p, q, p ′ , q ′ ) parametrize the diagrams with internal
(q, n) (p, j) (q ′ , m) (p ′ , i) (a, k) + (q, n) (p, j) (q ′ , m) (p ′ , i) (a, k) (q, n) (p, j) (q ′ , m) (p ′ , i) (a, k) + (q, n) (p, j) (a, k) (q ′ , m) (p ′ , i) + (q, n) (p, j) (q ′ , m) (p ′ , i)
(a, k) Figure 1: Generic topologies contributing to the scattering of two distinct scalars with gluon emission. The momenta p and q are taken incoming, while k, p ′ , and q ′ are outgoing.
emissions. The color factors are given by [15]
c 1 = T a ik T b kj T b mn , c 5 = T b ij T b mℓ T a ℓn , c 2 = T b ik T a kj T b mn , c 6 = T b ij T a mℓ T b ℓn + T b ij T b mℓ T a ℓn , c 3 = T a ik T b kj T b mn + T b ik T a kj T b mn , c 7 = if abc T b ij T c mn , (2.2) c 4 = T b ij T a mk T b kn ,
where T ij and T mn are the gauge group generators associated with the two scalar flavors. We have also introduced the following kinematic invariants
s 1 = 2 k · p, s 2 = 2 k · q, s 1 ′ = 2 k · p ′ , s 2 ′ = 2 k · q ′ . (2.3)
The next step is to enforce gauge invariance. The gauge Ward identity reads,
c 1 A 4 (p, q, p ′ + k, q ′ ) − c 2 A 4 (p − k, q, p ′ , q ′ ) + c 4 A 4 (p, q, p ′ , q ′ + k) − c 5 A 4 (p, q − k, p ′ , q ′ ) + k µ 2 c 3 B µ 1 (k; p, q, p ′ , q ′ ) + c 6 B µ 2 (k; p, q, p ′ , q ′ ) + c 7 B µ 3 (k; p, q, p ′ , q ′ ) = 0. (2.4)
In the soft limit we expand this equation in powers of the gluon momentum. At leading order in this expansion, the Ward identity is automatically satisfied due to the Jacobi identity
c 1 − c 2 + c 4 − c 5 = 0.
In the linear approximation, on the other hand, we are led to the equation
k µ 2 c 1 ∂ ∂p ′µ + c 2 ∂ ∂p µ + c 4 ∂ ∂q ′µ + c 5 ∂ ∂q µ A 4 (p, q, p ′ , q ′ ) +c 3 B µ 1 (0; p, q, p ′ , q ′ ) + c 6 B µ 2 (0; p, q, p ′ , q ′ ) + c 7 B µ 3 (0; p, q, p ′ , q ′ ) = 0. (2.5)
To solve for the functions B µ i (0; p, q, p ′ , q ′ ), we have to keep in mind that the color factors are not independent. In fact, there are four independent Jacobi identities relating them,
c 1 + c 2 − c 3 = 0, c 4 + c 5 − c 6 = 0, (2.6) c 1 − c 2 + c 7 = 0, c 4 − c 5 − c 7 = 0,
which can be used to eliminate c 1 , c 2 , c 4 and c 5 in favor of c 3 , c 6 and c 7 . Using these relations in Eq. (2.5) we arrive at
k µ c 3 ∂A 4 ∂p ′µ + ∂A 4 ∂p µ + B 1µ (0; p, q, p ′ , q ′ ) + c 6 ∂A 4 ∂q ′µ + ∂A 4 ∂q µ + B 2µ (0; p, q, p ′ , q ′ ) + c 7 − ∂A 4 ∂p ′µ + ∂A 4 ∂p µ + ∂A 4 ∂q ′µ − ∂A 4 ∂q µ + B 3µ (0; p, q, p ′ , q ′ ) = 0. (2.7)
Since c 3 , c 6 and c 7 are independent, we arrive at the equations to obtain the three undetermined functions that are solved by
B µ 1 (0; p, q, p ′ , q ′ ) = − ∂A 4 ∂p ′µ − ∂A 4 ∂p µ , B µ 2 (0; p, q, p ′ , q ′ ) = − ∂A 4 ∂q ′µ − ∂A 4 ∂q µ , (2.8) B µ 3 (0; p, q, p ′ , q ′ ) = ∂A 4 ∂p ′µ − ∂A 4 ∂p µ − ∂A 4 ∂q ′µ + ∂A 4 ∂q µ .
To these solutions for B µ i (0; p, q, p ′ , q ′ ) we could add a function ∆ µ i satisfying k µ ∆ µ i = 0. However, its tensor structure implies that such a function must be at least linear in k and therefore can be ignored at this order.
Having calculated the leading behavior of the functions associated with the internal emission diagrams, we expand the five-point amplitude (2.1) to O(k 0 ) and substitute the expressions found in (2.8). Using the Jacobi identities to eliminate the color factors c 3 , c 6 , and c 7 we have
A 5 = 2g c 1 p ′ · ǫ s 1 ′ − c 2 p · ǫ s 1 + c 4 q ′ · ǫ s 2 ′ − c 5 q · ǫ s 2 + k µ c 1 p ′ · ǫ s 1 ′ ∂ ∂p ′µ + c 2 p · ǫ s 1 ∂ ∂p µ + c 4 q ′ · ǫ s 2 ′ ∂ ∂q ′µ + c 5 q · ǫ s 2 ∂ ∂q µ (2.9) − 1 2 ǫ µ c 1 ∂ ∂p ′µ + c 2 ∂ ∂p µ + c 4 ∂ ∂q ′µ + c 5 ∂ ∂q µ A 4 (p, q, p ′ , q ′ ) + O(k).
After a few manipulations, this can be recast in terms of the angular momentum operators as
A 5 = 2g c 1 p ′ · ǫ s 1 ′ − c 2 p · ǫ s 1 + c 4 q ′ · ǫ s 2 ′ − c 5 q · ǫ s 2 + c 1 ǫ µ k ν J (1 ′ ) µν s 1 ′ + c 2 ǫ µ k ν J (1) µν s 1 + c 4 ǫ µ k ν J (2 ′ ) µν s 2 ′ + c 5 ǫ µ k ν J (2) µν s 2 A 4 (p, q, p ′ , q ′ ). (2.10)
We have shown that the first correction to the amplitude in the soft limit is completely fixed by the requirement of gauge invariance. Compared to the scalar QED case analyzed in [14],
we have a larger number of unknown functions associated with the different color structures in the internal emission diagrams. However, this very fact implies that there is an equally larger number of independent constraints to determine these functions. At linear order in the gluon momentum, we have again three equations for the first derivatives of B i (k; p, q, p ′ , q ′ ) at k = 0, but as in the Abelian case these relations leave the curls
∂B µ i ∂k α − ∂B α i ∂k µ k=0 , i = 1, 2, 3,(2.11)
undetermined.
The corrections to Low's theorem in Eq. (2.10) can be rewritten using the Mandelstam invariants s and t, that we define in the following symmetric form
s = 1 2 (p + q) 2 + 1 2 (p ′ + q ′ ) 2 , t = 1 2 (p − p ′ ) 2 + 1 2 (q − q ′ ) 2 . (2.12)
The combinations containing the angular momentum operators can then be expressed in terms of derivatives with respect to s and t as
ǫ µ k ν J (1 ′ ) µν = A 1 ′ ∂ ∂s + B 1 ′ ∂ ∂t , ǫ µ k ν J (1) µν = A 1 ∂ ∂s + B 1 ∂ ∂t , ǫ µ k ν J (2 ′ ) µν = A 2 ′ ∂ ∂s + B 2 ′ ∂ ∂t , (2.13) ǫ µ k ν J (2) µν = A 2 ∂ ∂s + B 2 ∂ ∂t ,
where the gauge invariant coefficients A i and B i are defined by
A 1 ′ = (ǫ · p ′ )(q ′ · k) − (ǫ · q ′ )(p ′ · k), A 1 = (ǫ · p)(q · k) − (ǫ · q)(p · k),(2.
14)
A 2 ′ = (ǫ · q ′ )(p ′ · k) − (ǫ · p ′ )(q ′ · k), A 2 = (ǫ · q)(p · k) − (ǫ · p)(q · k),
and
B 1 ′ = (ǫ · p)(p ′ · k) − (ǫ · p ′ )(p · k), B 1 = (ǫ · p ′ )(p · k) − (ǫ · p)(p ′ · k),(2.15)B 2 ′ = (ǫ · q)(q ′ · k) − (ǫ · q ′ )(q · k), B 2 = (ǫ · q ′ )(q · k) − (ǫ · q)(q ′ · k).
Plugging these expressions in the amplitude, we arrive at
A 5 = 2g c 1 p ′ · ǫ s 1 ′ − c 2 p · ǫ s 1 + c 4 q ′ · ǫ s 2 ′ − c 5 q · ǫ s 2 + c 1 A 1 ′ s 1 ′ + c 2 A 1 s 1 + c 4 A 2 ′ s 2 ′ + c 5 A 2 s 2 ∂ ∂s (2.16) + c 1 B 1 ′ s 1 ′ + c 2 B 1 s 1 + c 4 B 2 ′ s 2 ′ + c 5 B 2 s 2 ∂ ∂t A 4 (s, t).
In this expression gauge invariance follows trivially from the invariance of the coefficients A i and B i . It is important to note that there exists a larger set of transformations of these coefficients that leaves the first correction to Low's theorem invariant. These are given by the shifts
A 1 ′ −→ A 1 ′ + s 1 ′ α(p, q, p ′ , q ′ ), A 1 −→ A 1 − s 1 α(p, q, p ′ , q ′ ), A 2 ′ −→ A 2 ′ + s 2 ′ α(p, q, p ′ , q ′ ),(2.
17)
A 2 −→ A 2 − s 2 α(p, q, p ′ , q ′ ), and B 1 ′ −→ B 1 ′ + s 1 ′ β(p, q, p ′ , q ′ ), B 1 −→ B 1 − s 1 β(p, q, p ′ , q ′ ), B 2 ′ −→ B 2 ′ + s 2 ′ β(p, q, p ′ , q ′ ), (2.18) B 2 −→ B 2 − s 2 β(p, q, p ′ , q ′ ),
where α(p, q, p ′ , q ′ ) and β(p, q, p ′ , q ′ ) are two arbitrary functions of the scalar momenta, not necessarily local. Note that these transformations resemble those of the original color-kinematics duality [17], although they are only affecting the factorizing soft factors and not the full amplitude. Moreover, we have not identified any relevant role for the Jacobi identities when investigating the double copy structure in the gravitational case. This is likely to be a feature of the soft limit alone.
The gravitational amplitude and its double copy structure
The amplitude for the scattering of two distinct scalars with emission of a graviton can be computed in the soft limit by considering the five generic topologies shown in Fig. 1 with the gluon replaced by a graviton. Following the general arguments given in [14], the result is
M 5 = κ − p · ε · p s 1 + p ′ · ε · p ′ s 1 ′ − q · ε · q s 2 + q ′ · ε · q ′ s 2 ′ (3.1) + p ′ µ ε µν k α J (1 ′ ) να s 1 ′ + p µ ε µν k α J (1) να s 1 + q ′ µ ε µν k α J (2 ′ ) να s 2 ′ + q µ ε µν k α J (2) να s 2 M 4 (p, q, p ′ , q ′ ),
where M 4 (p, q, p ′ , q ′ ) denotes the gravitational scattering amplitude of the four hard (scalar)
particles. The subleading term can be recast in terms of derivatives with respect to the kinematic invariants s and t using
p ′ µ ε µν k α J (1 ′ ) να = A 1 ′ ∂ ∂s + B 1 ′ ∂ ∂t , p µ ε µν k α J (1) να = A 1 ∂ ∂s + B 1 ∂ ∂t , q ′ µ ε µν k α J (2 ′ ) να = A 2 ′ ∂ ∂s + B 2 ′ ∂ ∂t , (3.2) q µ ε µν k α J (2) να = A 2 ∂ ∂s + B 2 ∂ ∂t ,
where the new coefficients A i and B i are given by
A 1 ′ = (p ′ · ε · p ′ )(q ′ · k) − (p ′ · ε · q ′ )(p ′ · k), A 1 = (p · ε · p)(q · k) − (p · ε · q)(p · k), A 2 ′ = (q ′ · ε · q ′ )(p ′ · k) − (q ′ · ε · p ′ )(q ′ · k),(3.
3)
A 2 = (q · ε · q)(p · k) − (q · ε · p)(q · k),
and
B 1 ′ = (p ′ · ε · p)(p ′ · k) − (p ′ · ε · p ′ )(p · k), B 1 = (p · ε · p ′ )(p · k) − (p · ε · p)(p ′ · k), B 2 ′ = (q ′ · ε · q)(q ′ · k) − (q ′ · ε · q ′ )(q · k),(3.
4)
B 2 = (q · ε · q ′ )(q · k) − (q · ε · q)(q ′ · k).
In terms of this set of gauge invariant coefficients, the amplitude (3.1) reads
M 5 = κ − p · ε · p s 1 + p ′ · ε · p ′ s 1 ′ − q · ε · q s 2 + q ′ · ε · q ′ s 2 ′ (3.5) + A 1 ′ s 1 ′ + A 1 s 1 + A 2 ′ s 2 ′ + A 2 s 2 ∂ ∂s + B 1 ′ s 1 ′ + B 1 s 1 + B 2 ′ s 2 ′ + B 2 s 2 ∂ ∂t M 4 (s, t).
Similarly to the gauge theory case, the gravitational amplitude also remains invariant under the following generalized transformations of the coefficients (with i = 1, 1 ′ , 2, 2 ′ ):
A i −→ A i + s i α i (p, q, p ′ , q ′ ), B i −→ B i + s i β i (p, q, p ′ , q ′ ),(3.6)
where the functions α i (p, q, p ′ , q ′ ) and β i (p, q, p ′ , q ′ ) satisfy the constraint
i α i (p, q, p ′ , q ′ ) = 0, i β i (p, q, p ′ , q ′ ) = 0. (3.7)
These transformations turn out to be useful in finding a relation between the gravitational and gauge theory amplitudes. Transforming A i and B i using the functions
α 1 ′ = − α 2 ′ = −(p ′ · ε · p ′ )(q ′ · k) + (p ′ · ε · q ′ )[(p ′ − q ′ ) · k] + (q ′ · ε · q ′ )(p ′ · k) 2k · (p ′ + q ′ ) , α 1 = − α 2 = −(p · ε · p)(q · k) + (p · ε · q)[(p − q) · k] + (q · ε · q)(p · k) 2k · (p + q) , β 1 ′ = − β 1 = −(p ′ · ε · p ′ )(p · k) + (p ′ · ε · p)[k · (p + p ′ )] − (p · ε · p)(p ′ · k) 2k · (p − p ′ ) , (3.8) β 2 ′ = − β 2 = −(q ′ · ε · q ′ )(q · k) + (q · ε · q ′ )[k · (q + q ′ )] − (q · ε · q)(q ′ · k) 2k · (q − q ′ ) ,
we find the new coefficients A ′ i and B ′ i given by
A 1 ′ −→ A ′ 1 ′ = 2 (p ′ · ε · p ′ )(q ′ · k) 2 − 2(p ′ · ε · q ′ )(p ′ · k)(q ′ · k) + (q ′ · ε · q ′ )(p ′ · k) 2 s 1 ′ + s 2 ′ , A 1 −→ A ′ 1 = 2 (p · ε · p)(p · k) 2 − 2(p · ε · q)(p · k)(q · k) + (q · ε · q)(q · k) 2 s 1 + s 2 , (3.9) A 2 ′ −→ A ′ 2 ′ = 2 (p ′ · ε · p ′ )(q ′ · k) 2 − 2(p ′ · ε · q ′ )(p ′ · k)(q ′ · k) + (q ′ · ε · q ′ )(p ′ · k) 2 s 1 ′ + s 2 ′ , A 2 −→ A ′ 2 = 2 (p · ε · p)(q · k) 2 − 2(p · ε · q)(p · k)(q · k) + (q · ε · q)(p · k) 2 s 1 + s 2 , and B 1 ′ −→ B ′ 1 ′ = − 2 (p ′ · ε · p ′ )(p ′ · k) 2 − 2(p ′ · ε · p)(p · k)(p ′ · k) + (p · ε · p)(p ′ · k) 2 t 1 − t 2 , B 1 −→ B ′ 1 = 2 (p · ε · p)(p · k) 2 − 2(p · ε · p ′ )(p · k)(p ′ · k) + (p ′ · ε · p ′ )(p · k) 2 t 1 − t 2 , (3.10) B 2 ′ −→ B ′ 2 ′ = 2 (q ′ · ε · q ′ )(q · k) 2 − 2(q · ε · q ′ )(q · k)(q ′ · k) + (q · ε · q)(q ′ · k) 2 t 1 − t 2 , B 2 −→ B ′ 2 = − 2 (q · ε · q)(q ′ · k) 2 − 2(q · ε · q ′ )(q · k)(q ′ · k) + (q ′ · ε · q ′ )(q · k) 2 t 1 − t 2 .
Here, to simplify the notation, we have introduced the invariants
t 1 = (p − p ′ ) 2 , t 2 = (q − q ′ ) 2 . (3.11)
This transformation is interesting because if we compare the new set of coefficients with the A i 's and B i 's of the gauge theory amplitude given in Eqs. (2.14) and (2.15) we find the relations
A ′ 1 ′ = 2ε µν A µ 1 ′ A ν 1 ′ s 1 ′ + s 2 ′ , A ′ 1 = 2ε µν A µ 1 A ν 1 s 1 + s 2 , (3.12) A ′ 2 ′ = 2ε µν A µ 2 ′ A ν 2 ′ s 1 ′ + s 2 ′ , A ′ 2 = 2ε µν A µ 2 A ν 2 s 1 + s 2 and B ′ 1 ′ = − 2ε µν B µ 1 ′ B ν 1 ′ t 1 − t 2 , B ′ 1 = 2ε µν B µ 1 B ν 1 t 1 − t 2 , (3.13) B ′ 2 ′ = 2ε µν B µ 2 ′ B ν 2 ′ t 1 − t 2 , B ′ 2 = − 2ε µν B µ 2 B ν 2 t 1 − t 2 .
Thus, up to a common kinematic denominator and a phase (which can be absorbed in a redefinition of the B i 's), the coefficients of the gravity amplitude can be written as a double copy of the ones of the gauge theory. This structure is manifest if we rewrite the scalar amplitude in the gauge theory
A 5 = 2 g ǫ µ c 1 p ′µ s 1 ′ − c 2 p µ s 1 + c 4 q ′µ s 2 ′ − c 5 q µ s 2 (3.14) + c 1 A µ 1 ′ s 1 ′ + s 2 ′ 1 s 1 ′ + c 2 A µ 1 s 1 + s 2 1 s 1 + c 4 A µ 2 ′ s 1 ′ + s 2 ′ 1 s 2 ′ + c 5 A µ 2 s 1 + s 2 1 s 2 (s 1 + s 2 ) ∂ ∂s + c 1 B µ 1 ′ t 1 − t 2 1 s 1 ′ + c 2 B µ 1 t 1 − t 2 1 s 1 + c 4 B µ 2 ′ t 1 − t 2 1 s 2 ′ + c 5 B µ 2 t 1 − t 2 1 s 2 (t 1 − t 2 ) ∂ ∂t A 4 (s, t)
and compare with the gravitational amplitude written as
M 5 = κ ε µν p ′µ p ′ν s 1 ′ − p µ p ν s 1 + q ′µ q ′ν s 2 ′ − q µ q ν s 2 (3.15) + 2 A µ 1 ′ A ν 1 ′ (s 1 ′ + s 2 ′ ) 2 1 s 1 ′ + A µ 1 A ν 1 (s 1 + s 2 ) 2 1 s 1 + A µ 2 ′ A ν 2 ′ (s 1 ′ + s 2 ′ ) 2 1 s 2 ′ + A µ 1 A ν 1 (s 1 + s 2 ) 2 1 s 2 (s 1 + s 2 ) ∂ ∂s + 2 − B µ 1 ′ B ν 1 ′ (t 1 − t 2 ) 2 1 s 1 ′ + B µ 1 B ν 1 (t 1 − t 2 ) 2 1 s 1 + B µ 2 ′ B ν 2 ′ (t 1 − t 2 ) 2 1 s 2 ′ − B µ 1 B ν 1 (t 1 − t 2 ) 2 1 s 2 (t 1 − t 2 ) ∂ ∂t M 4 (s, t).
We have used momentum conservation s 1 + s 2 = s 1 ′ + s 2 ′ .
Some remarks on the expressions (3.14) and (3.15) are in order. Factoring out s 1 + s 2 and t 1 − t 2 might seem a mere analytic trick to get the double copy to work better. However, this way of writing the amplitudes is quite natural once we take into account that these two terms are the expansion parameters in the soft limit around s and t, so the double copy representation affects the coefficients of the series expansion around the k = 0 term. Written in this way, it is clear how the contribution of the soft graviton to the five-point amplitude can be obtained by replacing the color factors in the gauge amplitude with a second copy of the corresponding kinetic coefficient. This prescription works not only for the correction but for the leading term as well, where the kinematic coefficient is just the momentum.
We should also stress the importance of using derivatives with respect to the kinematic invariants in uncovering the double copy structure of the subleading corrections in the soft limit.
In this case, the tensor structure of the amplitude is completely codified in the coefficients of these derivatives, in which the double copy is glaring. Expressing the amplitude in terms of derivatives with respect to the momenta obscures this feature.
Gribov's limit
We have considered so far amplitudes in the standard soft gluon and graviton regimes, as well as their first corrections. As explained in the Introduction, Gribov found that Low's result is valid in a kinematic region larger than the strict k µ → 0 limit. In our notation this is given by
s 1 , s 2 ≪ s, k 2 ⊥ ≪ µ 2 ≪ s,(4.1)
with µ the mass of the scalars. Moreover, in this limit it is also satisfied [12]
|t 1 − t 2 | ≪ µ √ −t 1 ≈ µ √ −t 2 . (4.2)
It is remarkable that in this region the associated radiation can be hard, i.e., we are not just limited to emission of soft particles. Gribov's result was extended in [13] to the scattering amplitude of two scalar flavors in nonabelian gauge theories in the high energy limit s ≫ t ∼ µ 2 .
This inequality has important consequences for the form of the amplitude. Since the four-scalar amplitude A 4 (s, t) is dimensionless, it has to be a homogeneous function of degree zero of its two arguments. This means that in this kinematic regime, derivatives with respect to s are much smaller than the derivatives with respect to t, due to a suppression factor t/s:
s ∂ ∂s + t ∂ ∂t A 4 (s, t) = 0 =⇒ ∂ ∂s A 4 (s, t) = − t s ∂ ∂t A 4 (s, t). (4.3)
As a consequence, writing Eq. (2.1) in terms of the kinematic invariants, the momentum shifts in the expression only affect the second argument of A(s, t), i.e.,
A 5 = 2g c 1 p ′ · ǫ s 1 ′ A 4 (s, t 2 ) − c 2 p · ǫ s 1 A 4 (s, t 2 ) + c 4 q ′ · ǫ s 2 ′ A 4 (s, t 1 ) − c 5 q · ǫ s 2 A 4 (s, t 1 ) (4.4) + g 2 ǫ µ c 3 B µ 1 (k; p, q, p ′ , q ′ ) + c 6 B µ 2 (k; p, q, p ′ , q ′ ) + c 7 B µ 3 (k; p, q, p ′ , q ′ ) .
Again, to determine the unknown functions associated with internal gluon emission we write the gauge Ward identity and take into account that in our kinematic regime
t 1 − t 2 2 ≪ t 1 + t 2 2 ≡ t. (4.5)
Then, we expand the expression to first order in t 1 − t 2 . Since this parameter is proportional to k, the leading contribution of the functions B µ i (k; p, q, p ′ , q ′ ) to the Ward identity comes from setting k = 0 in the argument. Proceeding as in Section 2, we find that gauge invariance fixes the unknown functions and the result of [13] is recovered in a slightly different notation:
A 5 = 2g c 1 p ′ · ǫ s 1 ′ − c 2 p · ǫ s 1 + c 4 q ′ · ǫ s 2 ′ − c 5 q · ǫ s 2 + c 1 B 1 ′ s 1 ′ + c 2 B 1 s 1 + c 4 B 2 ′ s 2 ′ + c 5 B 2 s 2 ∂ ∂t A 4 (s, t). (4.6)
A similar calculation can be carried out for the gravitational amplitude. The only caveat lies in whether we can neglect derivatives with respect to s, since now due to the dimensionful coupling κ it is no longer true that the amplitude is a function of s/t. Nevertheless, at a fixed order in perturbation theory the amplitude has the generic form
M 4 (s, t) = (κ 2 s) n 2 f s t . (4.7)
However, if at large energies f (s/t) ∼ (s/t) α , the s-derivative of M 4 (s, t) is suppressed with respect to its t-derivatives by a power of t/s. This is indeed the case of the tree-level amplitude (with α = 1), so we can take the amplitude as constant with respect to s and retrieve the expression found in [13]:
M 5 = κ p ′ · ε · p ′ s 1 ′ − p · ε · p s 1 + q ′ · ε · q ′ s 2 ′ − q · ε · q s 2 + B 1 ′ s 1 ′ + B 1 s 1 + B 2 ′ s 2 ′ + B 2 s 2 ∂ ∂t M 4 (s, t). (4.8)
We see how Gribov's limit of the gauge and gravitational scattering amplitudes (4.6) and (4.8) can be respectively obtained from our expressions for the corrections to the Low and Weinberg limits (2.16) and (3.5) by just ignoring derivatives with respect to s.
Concluding remarks
The idea of the existence of a double copy representation of gravity has received strong support, ranging from KLT identities [16] to color-kinematics duality [17] (see [18] for a recent review).
In the context of the soft limit, it was found in [19] that the infrared behavior of both gauge theories and gravity is consistent with an underlying double copy provided by color-kinematics duality to all orders in perturbation theory.
In this paper we have studied the double copy structure in the context of the soft gluon and graviton theorems. Our analysis shows clear evidence that there is a sense in which we can state that (soft graviton) = (soft gluon) 2 : the contribution of a soft graviton in a scalar scattering amplitude can be written as the double copy of the corresponding contribution of a soft gluon.
Let us try to be more precise. Our proposal strongly resembles color-kinematics duality of gauge theory amplitudes, in which the gravity amplitude is obtained by replacing color factors by a second copy of a kinematic factor. It has however the peculiarity that it does not affect the whole five-point amplitude, but just the coefficients of the operator acting on the amplitude of the four hard particles. The rationale behind this is that it is this prefactor which contains all the information about the emitted gluon/graviton. This is precisely the sense of the moral equation (soft graviton)=(soft gluon) 2 .
Interestingly, in the case of the scalar QCD five-point amplitude studied here, it was shown in [20] that a naive application of color-kinematics duality does not render the full gravitational amplitude for graviton emission. Even in multi-Regge kinematics, the full graviton amplitude does not factorize in terms of two QCD effective gluon vertices (Lipatov vertices), due to an extra term which is necessary for the cancellation of overlapping divergences [15]. Despite this, here we have seen how a certain double copy structure does indeed survive in the Low/Weinberg and Gribov limits, but that this only affects the soft graviton. Incidentally, the soft graviton limit does not contain overlapping divergences, so the offending term breaking factorization in the Regge limit does not contribute to it. Let us remark that by going to the Gribov limit it is possible to escape from the soft graviton condition and extend Low's factorization to the emission of harder radiation. This is likely to have non-trivial consequences for the interpretation of the Gribov limit in terms of symmetries in a gravitational theory at null infinity.
The scalar QCD theory used here admits modifications for which the Bern-Carrasco-Johansson prescription in a general kinematics works [21]. One of them consists in taking all scalars to transform in the adjoint representation and adding a quartic contact self-coupling between the two flavors. Nevertheless, this modification does not spoil the double copy structure of the subleading terms, since the new diagrams only add to the leading soft behavior. This is because the new couplings are contact terms, so they only contribute to the sector with zero angular momentum for which the correction vanishes.
There are a number of related questions that deserve attention. One of them concerns the generalization of our result to other amplitudes and theories at tree and loop level (e.g., is there a hidden double copy interpretation of the double logarithms studied in [22]?). In this sense, as already stated in Section 3, the use of derivatives with respect to invariants seems crucial to expose the double copy structure in the amplitude. This is an added complication when investigating higher-point amplitudes. Another interesting problem to address is the implications of our results for the possible relation between the asymptotic symmetries of gauge theories [11,23] and gravity [4]. These and other issues will be studied elsewhere.
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[] |
[
"Sesquilinear forms over rings with involution",
"Sesquilinear forms over rings with involution"
] |
[
"Eva Bayer-Fluckiger \nEcole Polytechnique Fédérale de Lausanne\nSwitzerland\n",
"Daniel Arnold Moldovan \nEcole Polytechnique Fédérale de Lausanne\nSwitzerland\n"
] |
[
"Ecole Polytechnique Fédérale de Lausanne\nSwitzerland",
"Ecole Polytechnique Fédérale de Lausanne\nSwitzerland"
] |
[
"Mathematics Subject Classification"
] |
Many classical results concerning quadratic forms have been extended to hermitian forms over algebras with involution. However, not much is known in the case of sesquilinear forms without any symmetry property. The present paper will establish a Witt cancellation result, an analogue of Springer's theorem, as well as some local-global and finiteness results in this context.
|
10.1016/j.jpaa.2013.06.012
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[
"https://arxiv.org/pdf/1301.0003v2.pdf"
] | 1,081,665 |
1301.0003
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b14c05ea62e7666ee022b5bdedcf896cdf9bd774
|
Sesquilinear forms over rings with involution
2000
Eva Bayer-Fluckiger
Ecole Polytechnique Fédérale de Lausanne
Switzerland
Daniel Arnold Moldovan
Ecole Polytechnique Fédérale de Lausanne
Switzerland
Sesquilinear forms over rings with involution
Mathematics Subject Classification
2000sesquilinear formshermitian formshermitian categories
Many classical results concerning quadratic forms have been extended to hermitian forms over algebras with involution. However, not much is known in the case of sesquilinear forms without any symmetry property. The present paper will establish a Witt cancellation result, an analogue of Springer's theorem, as well as some local-global and finiteness results in this context.
Introduction
Many classical results on quadratic forms have been extended to hermitian forms over some rings with involution (see for instance [9], [14]). However, not much is known about sesquilinear forms without any symmetry property. The aim of this paper is to show that the category of sesquilinear forms over rings with involution is equivalent to the category of unimodular hermitian forms over a suitable additive category (cf. §1 - §4), and then to apply the theory of Quebbemann, Scharlau, Scharlau and Schulte (cf. [11], [12]) as well as known results concerning hermitian forms.
This enables us to prove Witt's cancellation theorem (see theorem 6.1) and an analogue of Springer's theorem (see theorem 7.1) for sesquilinear forms over finitedimensional algebras with involution over fields of characteristic different from 2. We also obtain local-global results (see theorem 9.1 and its corollary) as well as finiteness theorems. §1. Sesquilinear forms over rings with involution Let A be a ring. An involution on A is by definition an additive map σ : A → A such that σ(ab) = σ(b)σ(a) for all a, b ∈ A and σ 2 is the identity. Let V be a right A-module of finite type. A sesquilinear form over (A, σ) is a biadditive map s : V × V → A satisfying the condition s(xa, yb) = σ(a)s(x, y)b for all x, y ∈ V and all a, b ∈ A. The orthogonal sum of two sesquilinear forms (V, s) and (V ′ , s ′ ) is by definition the form (V ⊕ V ′ , s ⊕ s ′ ) defined by
(s ⊕ s ′ )(x ⊕ x ′ , y ⊕ y ′ ) = s(x, y) + s ′ (x ′ , y ′ )
for all x, y ∈ V and x ′ , y ′ ∈ V ′ . Two sesquilinear forms (V, s) and (V ′ , s ′ ) are called isometric if there exists an isomorphism of A-modules f : V→V ′ such that s ′ (f (x), f (y)) = s(x, y) for all x, y ∈ V .
Let V * be the additive group Hom A (V, A) with the right A-module structure given by (f · a)(x) = σ(a)f (x) for all a ∈ A, f ∈ V * and x ∈ V . We say that V is reflexive if the homomorphism of right A-modules e V : V → V * * defined by e V (x)(f ) = σ(f (x)) for all x ∈ V and f ∈ V * is bijective.
A sesquilinear form (V, s) induces two homomorphisms of right A-modules V → V * , called its left, respectively right adjoint, namely s l : V → V * defined by s l (x)(y) = s(x, y) and s r : V → V * given by s r (x)(y) = σ(s(y, x)) for all x, y ∈ V . Note that these are related by s r = s * ℓ e V . Note also that the data s and s r (as well as s and s ℓ ) are equivalent. The sesquilinear form (V, s) is said to be unimodular if and only if s ℓ is bijective (equivalently, s r is bijective).
If (V, s) and (V ′ , s ′ ) are two sesquilinear forms, then an isomorphism f : V → V ′ defines an isometry between (V, s) and (V ′ , s ′ ) if and only if we have s ℓ = f * s ′ ℓ f . Let R be the category of reflexive right A-modules. The morphisms of this category are homomorphisms of A-modules.
Let us denote by S R (A, σ) the category of sesquilinear forms over (A, σ) defined on objects of R. The morphisms of this category are isometries. §2. Hermitian categories
The aim of this section is to recall the notion of hermitian forms in additive categories as defined in [11], [12] (see also [9], [14]). Let C be an additive category. Let * : C → C be a duality functor, i.e. an additive contravariant functor with a natural isomorphism (E C ) C∈C : id → * * such that E * C E C * = id C * for all C ∈ C. A hermitian form in the category C is a pair (C, h), where C is an object of C and h : C → C * such that h = h * E C . The hermitian form is said to be unimodular if h is an isomorphism. Orthogonal sums are defined in the obvious way.
Let (C, h) and (C ′ , h ′ ) be two hermitian forms in C. We say that they are isometric if there exists an isomorphism f : C→C ′ in the category C such that h = f * h ′ f . We denote by H(C) the category of unimodular hermitian forms in the category C. The morphisms are isometries.
We observe that if we take C = R then the above notion coincides with the notion of hermitian form over (A, σ) defined on objects of R. §3. An additive category Let (A, σ) be a ring with involution and R be the category of reflexive right Amodules. Let us consider the category M R with objects of the form (V, W, f 1 , f 2 ), where V and W are objects of R of finite type and f 1 :
V → W , f 2 : V → W are homomorphisms of A-modules. A morphism (V, W, f 1 , f 2 ) → (V ′ , W ′ , f ′ 1 , f ′ 2 ) in M R is a pair (φ, ψ) of A-linear homomorphisms φ : V → V ′ and ψ : W → W ′ such that f ′ 1 φ = ψf 1 and f ′ 2 φ = ψf 2 .
The dual of (φ, ψ) is (ψ * , φ * ). By defining direct sums in the obvious way we see that M R is an additive category. It is called the category of double arrows between objects of R.
Let (W * , V * , f * 2 , f * 1 ) be the dual of (V, W, f 1 , f 2 ) and set E (V,W,f 1 ,f 2 ) = (e V , e W ). This defines a duality on the category M R . We denote by H(M R ) the category of unimodular hermitian forms in the additive category M R , as defined in the previous section. §4. An equivalence of categories Let (A, σ) be a ring with involution and R be the category of reflexive right Amodules. The aim of this section is to prove that the category of sesquilinear forms S R (A, σ) is equivalent to the hermitian category H(M R ) defined in §3. We will use the methods of [3].
We define a functor F from the category S R (A, σ) to the category H(M R ) as follows. Let (V, s) be an object of S R (A, σ), and let s ℓ : V → V * and s r : V → V * be the left, respectively the right adjoint of (V, s) (cf. §1). Then (V, V * , s ℓ , s r ) is an object of M R and the pair (e V , id V * ) defines a unimodular hermitian form on (V, V * , s ℓ , s r ); note that the unimodularity follows from the reflexivity of V . Set F(V, s) = ((V, V * , s ℓ , s r ), (e V , id V * )). For an isomorphism φ : (V, s) → (W, t) of sesquilinear forms, where V, W ∈ R, set F(φ) = (φ, φ * −1 ).
Theorem
The functor F is an equivalence of categories between S R (A, σ) and H(M R ).
Proof. We already know that F sends sesquilinear forms over (A, σ) defined on objects of R to objects of H(M R ). Let us check that F sends morphisms to morphisms. Let φ : (V, s) → (W, t) be an isomorphism between two objects of S R(A,σ) . Then (φ, φ * −1 ) : (V, V * , s ℓ , s r ) → (W, W * , t ℓ , t r ) is an isomorphism in M R . Moreover, as φ * * e V = e W φ, it is also an isomorphism in H(M R ).
Let us define a functor G from H(M R ) to S R (A, σ). Let (M, ζ) be an object of H(M R ) with M = (V, W, f 1 , f 2 ) and ζ = (φ, ψ). Let us define a sesquilinear form s : V × V → A by s(x, y) = (ψf 2 )(x)(y) for all x, y ∈ V and set G(V, W, f 1 , f 2 ) = (V, s) (in other words, s ℓ = ψf 2 ). Then (V, s) is an object of S R (A, σ) (by definition of the right A-module structure of V * ). For a morphism λ = (λ 1 , λ 2 ) between two objects of H(M R ) set G(λ) = λ 1 .
Let us check that G sends morphisms to morphisms. Let
λ = (λ 1 , λ 2 ) : ((V, W, f 1 , f 2 ), (φ, ψ)) → ((V ′ , W ′ , f ′ 1 , f ′ 2 ), (φ ′ , ψ ′ ))
be a morphism between two objects of H(M R ). Then
λ 1 : V → V ′ and λ 2 : W → W ′ are isomorphisms such that λ 2 f 1 = f ′ 1 λ 1 and λ 2 f 2 = f ′ 2 λ 1 . In addition we have (φ, ψ) = (λ 1 , λ 2 ) * (φ ′ , ψ ′ )(λ 1 , λ 2 ), from which we deduce that φ = λ * 2 φ ′ λ 1 and ψ = λ * 1 ψ ′ λ 2 . It follows that λ * 1 ψ ′ f ′ 2 λ 1 = ψλ −1 2 f ′ 2 λ 1 = ψf 2 , so λ 1 is a morphism from (V, ψf 2 ) to (V ′ , ψ ′ f ′ 2
). An easy computation (using the relationship between s r and s ℓ ) shows that FG is isomorphic to the identity in H(M R ). Moreover, it is clear that GF is isomorphic to the identity in S R (A, σ). Therefore
f = η −1 0 f * η 0 for all f ∈ E, where f * denotes the dual of f in M R . On the set E + = {f ∈ E × | f = f } we define the following equivalence relation: f ≡ f ′ if there exists a g ∈ E × such that gf g = f ′ .
Let H( , E × ) denote the set of equivalence classes. It is clear that H( , E × ) is in bijection with the set of isometry classes of unimodular hermitian forms of rank one over (E, ).
Theorem The set of isometry classes of sesquilinear forms
(M, s) ∈ S R (A, σ) such that q(M, s) ≃ Q 0 is in bijection with H( , E × ). Proof. For every sesquilinear form (M, s) ∈ S R (A, σ) such that q(M, s) ≃ Q 0 we consider a fixed isomorphism ϕ : q(M, h)→Q 0 . Through this isomorphism the unimodular hermitian form (e M , id M * ) on q(M, s) induces a hermitian form η M = ϕ * −1 (e M , id M * )ϕ −1 on Q 0 . Moreover, η M is a unimodular hermitian form : this is easy to check, noting that if φ : M → M 0 is an isomorphism, then φ * * e M = e M 0 φ.
We define the following map:
{[(M, s)] | (M, s) ∈ S R (A, σ), q(M, h) ≃ Q 0 } → H( , E × ) [(M, s)] → η −1 0 η M . The hermitianity property implies that f = η −1 0 η M is an element of E + .
It is easy to check that the class of f is independent of the choice of the isomorphism φ. The above map is well defined since
(M, s) ≃ (N, t) if and only if η −1 0 η M ≡ η −1 0 η N .
This equivalence also shows its injectivity. Its surjectivity is easy to check. §6. Witt's cancellation theorem for sesquilinear forms
The aim of this section is to prove a cancellation theorem for sesquilinear forms. Let K be a field of characteristic different from 2, let A be a finite-dimensional K-algebra and let σ be an involution on A. Let us denote by R the category of reflexive right A-modules which are finite dimensional K-vector spaces.
(M 1 , s 1 ) ⊕ (M, s) ≃ (M 2 , s 2 ) ⊕ (M, s). Then we have (M 1 , s 1 ) ≃ (M 2 , s 2 ).
By the equivalence between the categories S R (A, σ) and H(M R ) given by theorem 4.1, it is enough to prove that Witt's cancellation theorem holds in the category H(M R ). This is the purpose of proposition 6.3. For its proof we use a result of Quebbemann, Scharlau and Schulte (see [11]
If (M, h), (M 1 , h 1 ) and (M 2 , h 2 ) are unimodular hermitian forms in C such that (M 1 , h 1 ) ⊕ (M, h) ≃ (M 2 , h 2 ) ⊕ (M, h), then (M 1 , h 1 ) ≃ (M 2 , h 2 ).
This theorem has been originally formulated for quadratic forms in C. But since 2 is invertible in the endomorphism ring of every object of C, the categories of quadratic and unimodular hermitian forms in C are isomorphic. We deduce that Witt's cancellation holds for unimodular hermitian forms in C as well.
6.3. Proposition Let (Q, η), (Q 1 , η 1 ) and (Q 2 , η 2 ) be unimodular hermitian forms in the category M R such that
(Q 1 , η 1 ) ⊕ (Q, η) ≃ (Q 2 , η 2 ) ⊕ (Q, η).
It then follows that (Q 1 , η 1 ) ≃ (Q 2 , η 2 ).
Proof. As in [3], proposition 2, we check that the category M R satisfies the conditions (i), (ii) and (iii) above. It follows from theorem 6.2 that Witt's cancellation theorem is true in the category H(M R ). §7. Springer's theorem for sesquilinear forms
The classical theorem of Springer states that if two quadratic forms over a field of characteristic different from 2 become isometric over an extension of odd degree, then they are already isometric over the base field. In this section we prove an analogue of Springer's theorem for sesquilinear forms defined on finitedimensional algebras with involution over a field of characteristic = 2.
Let K be a field of characteristic different from 2, A be a finite-dimensional Kalgebra and σ be a K-linear involution. We also consider a finite extension L of K, the finite-dimensional L-algebra A L = A ⊗ K L and the L-linear involution In order to prove this result we will use techniques of hermitian categories. We denote by M R ( (M R ) L ) the category of double arrows between objects of R (respectively objects of R tensorised with L over K). There is an obvious notion of scalar extension from the category M R to the category (
σ L = σ ⊗ id L on A L . If (M,M R ) L : if Q = (V, W, f, g) is an object of M R , set Q L = (V L , W L , f L , g L ), where V L = V ⊗ K L, W L = W ⊗ K L
and the morphisms f L and g L are defined by extending f , respectively g to L.
Note that if Q and Q ′ are two objects of M R which become isomorphic over a field extension of K, then Q and Q ′ are isomorphic (indeed, we are dealing with a linear problem). In this section we prove results similar to those of sections 6 and 7 for sesquilinear forms defined on algebras of finite rank over complete discrete valuation rings.
Let R be a complete discrete valuation ring and A be an R-algebra of finite rank with an involution denoted by σ. Suppose that there exists an element a in the center of A satisfying the property a + σ(a) = 1. For example this condition is fulfilled if 2 is invertible in R.
Let R be the category of reflexive right A-modules of finite type. Proof Denote by M R the category of double arrows between objects of R, cf. §3. Due to the equivalence of categories between the categories S R (A, σ) and H(M R ) given by theorem 4.1, it is enough to prove that Witt's cancellation theorem holds in the category H(M R ). This follows from theorem 6.2, observing that the category M R satisfies conditions (i), (ii), (iii) (see e.g. [3], proposition 2).
Next we prove a result analogous to Springer's theorem in the following setting: Let K be a non-dyadic local field, O K be its ring of integers and A be an O Kalgebra of finite rank with an O K -linear involution σ. Let L be a finite extension of K, O L be its ring of integers and let The classical Hasse-Minkowski theorem states that if two quadratic forms defined over a global field k of characteristic different from 2 become isometric over all the completions of k, then they are already isometric over k. The aim of this section is to investigate this result in the case of sesquilinear forms defined on a finite-dimensional skew field with involution. The case of bilinear forms has been treated by Waterhouse (cf. [15]), and is based on a classification result of Riehm (cf. [13]).
A L = A ⊗ O K O L with the O L -linear involution σ L = σ ⊗ id O L . If (M,
Let K be a field of characteristic different from 2, D be a finite-dimensional skew field with center K and σ be an involution on D. Denote by k the fixed field of σ in K. Then either k = K (when σ is said to be of the first kind ) or K is a quadratic extension of k and the restriction of σ to K is the non-trivial automorphism of K over k (in which case σ is said to be of the second kind or a unitary involution).
Suppose that k is a global field. For every prime spot p of k, let k p be the completion of k at p, K p = K ⊗ k k p and D p = D ⊗ k k p . Then D p is an algebra with center K p and we consider on it the extended involution σ p = σ ⊗id kp . Then from any sesquilinear form (V, s) over (D, σ) we obtain by extension of scalars a sesquilinear form (V p , s p ) over (D p , σ p ), where V p = V ⊗ k k p is a free right D p -module of rank dim D (V ).
We say that the weak Hasse principle holds for sesquilinear forms over (D, σ) if any two sesquilinear forms (V, s) and (V ′ , s ′ ) over (D, σ) that become isometric over all the completions of k (i.e. (V p , s p ) ≃ (V ′ p , s ′ p ) over (D p , σ p ) for every prime spot p of k) are already isometric over (D, σ).
In the sequel we will be interested in determining when the weak Hasse principle holds and we will adopt the point of view of hermitian categories. Let R be the category of finite dimensional right D-vector spaces. We denote by M R ( (M R ) p ) the category of double arrows between objects of R (respectively objects of R extended to k p ). Let us fix a sesquilinear form s 0 ∈ S R (D, σ). Consider a sesquilinear form s ∈ S R (D, σ) that becomes isometric to s 0 over all the completions of k. Then for every prime spot p of k we have (s 0 ) p ≃ s p and by theorem 4.1 it follows that q((s 0 ) p ) ≃ q(s p ) for all p. Since q commutes with base change, we obtain q(s 0 ) p ≃ q(s) p . It follows that q(s 0 ) ≃ q(s).
We denote by E the endomorphism ring of q(s 0 ) in M R . The unimodular hermitian form η 0 = (e V 0 , id V * 0 ) defined on q(s 0 ) induces an involution on E. For every p let E p = E ⊗ k k p , on which we consider the involution p = ⊗ id kp . By theorem 5.1 it is clear that if the localisation map
Φ : H( , E × ) → p H( p , E × p )
is injective, then the weak Hasse principle holds for sesquilinear forms in S R (D, σ). Hence in the sequel we will study the injectivity of the map Φ.
The involution induces an involution on E/rad(E), still denoted by . Since H( , E × ) ≃ H( , (E/rad(E)) × ) (cf. [4], lemma 2.6), we can suppose that rad(E) = 0. The ring E being artinian, it is semi-simple. We can thus write E as a product of simple factors:
E ≃ M n 1 (D 1 ) × ... × M nr (D r ),
where for all 1 ≤ i ≤ r, D i is a finite-dimensional skew field and n i ≥ 1. Without loss of generality we can suppose that the first m factors are fixed by the involution and that the other factors are interchanged two by two. For every 1 ≤ i ≤ m we denote by σ i the restriction of to M n i (D i ). We then obtain the isomorphism
H( , E × ) ≃ m i=1 H(σ i , M n i (D i ) × ).
Hence the map Φ is injective if and only if for every 1 ≤ i ≤ s the localisation map
Φ i : H(σ i , M n i (D i ) × ) → p H((σ i ) p , M n i ((D i ) p ) × ), defined in the obvious way, is injective ((D i ) p = D i ⊗ k k p and (σ i ) p = σ ⊗ id kp ).
We thus obtain: As the weak Hasse principle holds for unimodular hermitian forms over a skew field with a unitary involution (cf. [14], theorem 10.6.1), the map Φ i is injective.
Consider now the case when D i is a quaternion algebra and σ i is a symplectic involution. Then we consider the canonical involution τ i on D i and we proceed as above. The weak Hasse principle holds for unimodular hermitian forms over quaternion algebras with the canonical involution. Indeed, let h and h ′ be two unimodular hermitian forms over (D i , τ i ). Then their trace forms q h and q h ′ are two quadratic forms defined on the center of D i . Since they become isometric everywhere locally, they are isometric by the usual weak Hasse principle. But two unimodular hermitian forms over (D i , τ i ) are isometric if and only if their trace forms are isometric. Hence it follows that h ≃ h ′ . We conclude that the weak Hasse principle holds for unimodular hermitian forms over quaternion algebras with the canonical involution and thus the map Φ i is injective.
Corollary
If σ is a unitary involution, then the weak Hasse principle holds for sesquilinear forms in S R (D, σ).
Proof.
The statement immediately follows from theorem 9.1, since if σ is unitary, then all the involutions σ i are unitary as well. §10. Finiteness results
In this section we generalize the finiteness results of [4] to the setting of sesquilinear forms. For a ring A we denote by T (A) the Z-torsion subgroup of A. If R is a ring then we say that A is R-finite if A R = A ⊗ Z R is a finitely generated R-module and T (A) is finite.
Let (A, σ) be a ring with involution, R be the category of reflexive right Amodules of finite type and M R be the category defined in §3. We recall from §4 that the functor
F : S R (A, σ) → H(M R ) (V, s) → ((V, V * , s l , s r ), (e V , id V * ))
is an equivalence of categories. Fix a sesquilinear form (V, s) ∈ S R (A, σ) and denote by q(V, s) the corresponding object (V, V * , s l , s r ) of M R and by E the endomorphism ring of q(V, s) in M R . Let (V ′ , s ′ ) be an orthogonal summand of (V, s). Then q(V ′ , s ′ ) is a direct summand of q(V, s). Since there is an obvious surjection End(q(V, s)) → End(q(V ′ , s ′ )), we obtain that End(q(V ′ , s ′ )) is Z[1/m]-finite too. By the next proposition there exist only finitely many isometry classes of sesquilinear forms (V ′′ , s ′′ ) ∈ S R (A, σ) such that q(V ′′ , s ′′ ) ≃ q(V ′ , s ′ ). Hence the assertion of the theorem follows from the next result:
Theorem If there exists a non-zero integer m such that End
A (V ) is Z[1/m]-
Proof.
By theorem 4.1, every sesquilinear form (V, s) ∈ S R (A, σ) defines a unimodular hermitian form on q(V, s). But by [4], theorem 1.2 the number of isometry classes of unimodular hermitian forms on N is finite. Hence the assertion follows.
We say that two sesquilinear forms over (A, σ) are in the same genus if they become isometric over A ⊗ Z Z p for all primes of Q. Proof. Since End A (V ) is Q-finite, E is Q-finite too. Then by [4], theorem 3.4
Theorem If End
Proof. This follows from theorem 10.1 and the above correspondence. Let (M, b) be a G-bilinear form over R. If End R[G] (M) is Q-finite, then the genus of (M, b) contains only finitely many isometry classes of G-bilinear forms.
Theorem
Proof. This follows from theorem 10.3 and the above correspondence. §12. Generalization to systems of sesquilinear forms Let A be a ring with an involution σ and I be a set. A system of sesquilinear forms over (A, σ) is (V, (s i ) i∈I ), where V is a reflexive right A-module of finite type and for all i ∈ I, (V, s i ) is a sesquilinear form over (A, σ). A morphism between two systems of sesquilinear forms (V, (s i ) i∈I ) and (V ′ , (s ′ i ) i∈I ) consists of an isomorphism of A-modules f : V→V ′ such that for every i ∈ I and every x, y ∈ V we have s ′ i (f (x), f (y)) = s i (x, y). The results of the preceding sections can be generalized to systems of sesquilinear forms (for related results on systems of quadratic forms, see [1]). It is necessary to modify the definition of the category M R : its objects will be of the form (V, W, (f i , g i ) i∈I ) and the morphisms between two such objects will be the obvious ones. To a system (V, (s i ) i∈I ) of sesquilinear forms over (A, σ), where V ∈ R, it will correspond the object (V, V * , (((s i ) l , (s i ) r )) i∈I ) of the category M R .
F : S R (A, σ) → H(M R ) is an equivalence of categories. §5. Sesquilinear forms corresponding to a given object of M R Let A be a ring with an involution σ and R be the category of reflexive right Amodules of finite type. If (M, s) is a sesquilinear form over (A, σ) defined on an object of R, then we denote by q(M, s) the corresponding object (M, M * , s l , s r ) of the category M R . In this section we describe the set of isometry classes of sesquilinear forms over (A, σ) corresponding by theorem 4.1 to a given object of the category M R . Let us fix a sesquilinear form (M 0 , s 0 ) over (A, σ), where M 0 ∈ R, and consider the unimodular hermitian form η 0 = (e M 0 , id M * 0 ) defined on Q 0 = q(M 0 , s 0 ). Let E be the endomorphism ring of the object Q 0 in M R , and let us denote by E × the invertible elements of E. The form η 0 induces an involution on E, defined by
6. 1 .
1Theorem Let (M, s), (M 1 , s 1 ) and (M 2 , s 2 ) be sesquilinear forms over (A, σ) defined over objects of R such that
2. Theorem Let C be a hermitian category satisfying the following conditions: (i) All idempotents of C split, i.e. for any object M of C and for any idempotent e ∈ End(M), there exist an object M ′ ∈ C and morphisms i : M ′ → M, j : M → M ′ such that ji = id M ′ and ij = e.
(
ii) Every object M of C has a decomposition of the form M ≃ N 1 ⊕...⊕N r with N i ∈ C indecomposable and End(N i ) a local ring in which 2 is invertible. (iii) For every object M of C the ring End(M) is J(M)-adically complete, where J(M) is the Jacobson radical of End(M).
s) is a sesquilinear form over (A, σ), then we denote by (M L , s L ) the sesquilinear form over (A L , σ L ) obtained by extension of scalars: M L = M ⊗ K L and s L (x ⊗ a, y ⊗ b) = s(x, y) ⊗ ab for all x, y ∈ M and a, b ∈ L. Let R be the category of reflexive right A-modules which are finite dimensional K-vector spaces. 7.1. Theorem Suppose that L/K is an extension of odd degree. Let (M, s) and (M ′ , s ′ ) be two sesquilinear forms over (A, σ) defined on objects of R. If (M L , s L ) and (M ′ L , s ′ L ) are isometric over (A L , σ L ), then (M, s) and (M ′ , s ′ ) are isometric over (A, σ).
Proof of theorem 7.1. The proof uses the equivalence of categories proven in §4. Consider the objects q(M, s) and q(M ′ , s ′ ) of the category M R and the objects q(M L , s L ) and q(M ′ L , s ′ L ) of the category (M R ) L . By hypothesis (M L , s L ) and (M ′ L , s ′ L ) are isometric, so by theorem 4.1, the objects q(M L , s L ) ≃ q(M, s) L and q(M ′ L , s ′ L ) ≃ q(M ′ , s ′ ) L of (M R ) L are isomorphic. From this it follows that the objects q(M, s) and q(M ′ , s ′ ) of the category M R are isomorphic. Denote by Q the object q(M, s) of M R and by E the endomorphism ring of Q in the category M R .According to theorem 5.1, the set of isometry classes of sesquilinear forms (M ′′ , s ′′ ) over (A, σ) such that q(M ′′ , s ′′ ) ≃ Q is in bijection with the set H( , E × ). This last set is in turn in bijection with the set of isometry classes of rank one unimodular hermitian forms over (E, ). Denote by µ and µ ′ the rank one unimodular hermitian forms over (E, ) that correspond by the above bijections to (M, s), respectively (M ′ , s ′ ). Since (M L , s L ) ≃ (M ′ L , s ′ L ), the forms µ and µ ′ extend to isometric hermitian forms over (E L , L ), where E L = E ⊗ K L and L = ⊗ id L . Applying Springer's theorem for unimodular hermitian forms over (E, ) (cf.[2], th. 4.1 or[5], th. 2.1), we deduce that µ ≃ µ ′ , hence (M, s) ≃ (M ′ , s ′ ). §8. Sesquilinear forms defined on algebras over complete discrete valuation rings
8. 1 .
1Theorem Let (M, s), (M 1 , s 1 ) and (M 2 , s 2 ) be sesquilinear forms over (A, σ) defined on objects of R such that (M 1 , s 1 ) ⊕ (M, s) ≃ (M 2 , s 2 ) ⊕ (M, s).It follows that (M 1 , s 1 ) ≃ (M 2 , s 2 ).
s) is a sesquilinear form over (A, σ), then we denote by (M L , s L ) the sesquilinear form over (A L , σ L ) obtained by extension of scalars (in particular M L = M ⊗ O K O L ).Let R be the category of reflexive right A-modules which are finite dimensional K-vector spaces.
8. 2 .
2Theorem Suppose that the extension L/K is of odd degree. Let (M, s) and (M ′ , s ′ ) be two sesquilinear forms over (A, σ) with M, M ′ ∈ R. If (M L , s L ) and (M ′ L , s ′ L ) are isometric over (A L , σ L ), then (M, s) and (M ′ , s ′ ) are isometric over (A, σ). Denote by M R ( (M R ) L ) the category of double arrows between objects of R (respectively objects of R extended to O L ). There is an obvious notion of scalar extension from the category M R to the category (M R ) L : for an object Q = (M, N, f, g) of M R set Q L = (M L , N L , f L , g L ), where M L = M ⊗ O K O L , N L = N ⊗ O K O L and f L , g L are defined by extending f , respectively g. The proof of theorem 8.2 is analogous to the one of theorem 7.1, applying [7], p. 4 instead of [4], th. 2.1. §9. Weak Hasse principle for sesquilinear forms
9. 1 .
1Theorem If for all 1 ≤ i ≤ m, either the involution σ i is unitary, or D i is a quaternion algebra and σ i is a symplectic involution, then the map Φ is injective and hence the weak Hasse principle holds for sesquilinear forms over (D, σ).Proof. It suffices to prove that for all1 ≤ i ≤ m, the map Φ i is injective. Let 1 ≤ i ≤ mand suppose that σ i is unitary. According to the Albert-Riehm-Scharlau theorem ([10], theorem 3.1, p. 31), there exists a unitary involution τ i on D i with the same restriction to the center of D i as σ i . Then by Morita theory ([9], Chap. 1, theorem 9.3.5), the unimodular hermitian forms of rank one over (M n i (D i ), σ i ) (which are parametrised by the set H(σ i , M n i (D i )) ) are in bijection with the unimodular hermitian forms of rank n i over (D i , τ i ). An analogous statement holds for the unimodular hermitian forms of rank one over (M n i ((D i ) p ), (σ i ) p ).
10. 2 .
2Proposition Let N be an object of M R and assume that there exists a non-zero integer m such that End(N) is Z[1/m]-finite. Then there exist only finitely many isometry classes of sesquilinear forms (V, s) ∈ S R (A, σ) such that q(V, s) ≃ N.
A (V ) is Q-finite then the genus of (V, s) contains only a finite number of isometry classes of sesquilinear forms.
finite, then (V, s) contains only finitely many isometry classes of orthogonal summands.Proof. From the fact that EndA (V ) is Z[1/m]-finite it follows that E is Z[1/m]-finite and hence Q-finite. Hence by[4], theorem 1.1, q(V, s) has only finitely many isometry classes of direct summands.
Acknowledgements The second named author would like to thank Emmanuel Lequeu for many interesting and useful discussions.
The functor F induces a bijection between the genus of (V, s) and the genus of F(V, s), hence the assertion follows. §11. Bilinear forms invariant by a group action. F Genus Of, V, s) contains only finitely many isometry classes of unimodular hermitian formsgenus of F(V, s) contains only finitely many isometry classes of unimodular hermitian forms. The functor F induces a bijection between the genus of (V, s) and the genus of F(V, s), hence the assertion follows. §11. Bilinear forms invariant by a group action
Denote by R[G] the group ring of G over R. A G-bilinear form over R is by definition a pair (M, b), where M is a right R[G]-module which is an R-module of finite type and b : M ×M → R is an R-bilinear form such that b(xg, yg) = b(x, y) for all g ∈ G and x, y ∈ M. We say that two G-bilinear forms (M, b) and (M ′ , b ′ ) are isometric if there exists an isomorphism of R[G]-modules f : M→M ′ such that b ′ (f (x), f (y)) = b(x, y) for all x. Let R be a commutative ring and G be a finite group. y ∈ M. Two G-bilinear forms over R are said to be in the same genus if they become isometric over R ⊗ Z Z p for all primes of QLet R be a commutative ring and G be a finite group. Denote by R[G] the group ring of G over R. A G-bilinear form over R is by definition a pair (M, b), where M is a right R[G]-module which is an R-module of finite type and b : M ×M → R is an R-bilinear form such that b(xg, yg) = b(x, y) for all g ∈ G and x, y ∈ M. We say that two G-bilinear forms (M, b) and (M ′ , b ′ ) are isometric if there exists an isomorphism of R[G]-modules f : M→M ′ such that b ′ (f (x), f (y)) = b(x, y) for all x, y ∈ M. Two G-bilinear forms over R are said to be in the same genus if they become isometric over R ⊗ Z Z p for all primes of Q.
The aim of this section is to apply the results of sections 6, 7 and 10 to G-bilinear forms. The aim of this section is to apply the results of sections 6, 7 and 10 to G-bilinear forms.
canonical involution σ, which is the R-linear involution such that σ(g) = g −1 for every g ∈ G. Analogously to [8], theorem 7.1 it is straightforward to prove that there is a correspondence between G-bilinear forms over R and sesquilinear forms over. We endow the group ring R[G] with the. R[G], σ). Indeed, if (M, b) is a G-bilinear form over R then the form S : M × M → R[G], (x, y) → g∈G b(xg, y)gWe endow the group ring R[G] with the canonical involution σ, which is the R-linear involution such that σ(g) = g −1 for every g ∈ G. Analogously to [8], theorem 7.1 it is straightforward to prove that there is a correspondence between G-bilinear forms over R and sesquilinear forms over (R[G], σ). Indeed, if (M, b) is a G-bilinear form over R then the form S : M × M → R[G], (x, y) → g∈G b(xg, y)g
∈ M is clearly sesquilinear with respect to σ. Conversely, if (M, S) is a sesquilinear form over (R[G], σ) then M has an obvious structure of R-module and the form P • S : M × M →. R is bilinear, where P is the projection of R[G] onto the e G componentfor all x, y ∈ M is clearly sesquilinear with respect to σ. Conversely, if (M, S) is a sesquilinear form over (R[G], σ) then M has an obvious structure of R-module and the form P • S : M × M → R is bilinear, where P is the projection of R[G] onto the e G component.
Theorem Suppose that R is a field of characteristic = 2. Let (M 1 , b 1 ), (M 2 , b 2 ) and (M, b) be G-bilinear forms over R such that (M 1 , b 1 ) ⊕ (M, b) ≃ (M 2 , b 2 ) ⊕ (M, b). Then (M 1 , b 1 ) ≃ (M. Theorem Suppose that R is a field of characteristic = 2. Let (M 1 , b 1 ), (M 2 , b 2 ) and (M, b) be G-bilinear forms over R such that (M 1 , b 1 ) ⊕ (M, b) ≃ (M 2 , b 2 ) ⊕ (M, b). Then (M 1 , b 1 ) ≃ (M 2 , b 2 ).
This follows from theorem 6.1 and the above correspondence. Proof. This follows from theorem 6.1 and the above correspondence.
Theorem Suppose that R is a field of characteristic = 2. If two Gbilinear forms become isometric over an extension of R of odd degree then they are already isometric over R. Theorem Suppose that R is a field of characteristic = 2. If two G- bilinear forms become isometric over an extension of R of odd degree then they are already isometric over R.
The result follows from theorem 7.1 and the above correspondence. 11.3. Theorem Let (M, b) be a G-bilinear form over the commutative ring R. If there exists a non-zero integer m. such that End R[G] (M) is Z[1/m]-finite, then (M, b) contains only finitely many isometry classes of orthogonal summandsProof. The result follows from theorem 7.1 and the above correspondence. 11.3. Theorem Let (M, b) be a G-bilinear form over the commutative ring R. If there exists a non-zero integer m such that End R[G] (M) is Z[1/m]-finite, then (M, b) contains only finitely many isometry classes of orthogonal summands.
Principe de Hasse faible pour les systèmes de formes quadratiques. E Bayer-Fluckiger, J. reine angew. Math. 378E. Bayer-Fluckiger, Principe de Hasse faible pour les systèmes de formes quadratiques, J. reine angew. Math. 378 (1987), 53-59.
Self-dual normal bases. E Bayer-Fluckiger, Indag. Math. 92E. Bayer-Fluckiger, Self-dual normal bases, Indag. Math. 92 (1989), 379-383.
Non unimodular hermitian forms. E Bayer-Fluckiger, L Fainsilber, Invent. Math. 123E. Bayer-Fluckiger, L. Fainsilber, Non unimodular hermitian forms, Invent. Math. 123 (1996), 233-240.
Hermitian forms in additive categories: finiteness results. E Bayer-Fluckiger, C Kearton, S M Wilson, J. Algebra. 1232E. Bayer-Fluckiger, C. Kearton, S.M. Wilson, Hermitian forms in additive categories: finiteness results, J. Algebra 123 (1989), no. 2, 336-350.
Jr, Forms in odd degree extensions and self-dual normal bases. E Bayer-Fluckiger, H W Lenstra, Amer. J. Math. 112E. Bayer-Fluckiger, H.W. Lenstra, Jr, Forms in odd degree extensions and self-dual normal bases, Amer. J. Math. 112 (1990), 359-373.
Methods of representation theory, with applications to finite groups and orders, I. C W Curtis, I Reiner, WileyNew YorkC.W. Curtis, I. Reiner, Methods of representation theory, with applications to finite groups and orders, I, Wiley, New York (1981).
Formes hermitiennes sur des algèbres sur des anneaux locaux. L Fainsilber, Publications Mathématiques de Besançon. L. Fainsilber, Formes hermitiennes sur des algèbres sur des anneaux locaux, Publications Mathématiques de Besançon, 1994.
Forms over rings with involution. A Fröhlich, A M Mcevett, J. Algebra. 12A. Fröhlich, A.M. McEvett, Forms over rings with involution, J. Algebra 12 (1969), 79-104.
Quadratic and hermitian forms over rings, Grundlehren der math. M Knus, Wiss. 294Springer-VerlagM. Knus, Quadratic and hermitian forms over rings, Grundlehren der math. Wiss. 294, Springer-Verlag (1991).
The book of involutions. M Knus, A Merkurjev, M Rost, J.-P Tignol, Coll. Pub. 44Amer. Math. SocM. Knus, A. Merkurjev, M. Rost, J.-P. Tignol, The book of involutions, Coll. Pub. 44, Amer. Math. Soc. (1998).
Quadratic and hermitian forms in additive and abelian categories. H.-G Quebbemann, W Scharlau, M Schulte, J. Algebra. 59H.-G. Quebbemann, W. Scharlau, M. Schulte, Quadratic and hermitian forms in additive and abelian categories, J. Algebra 59 (1979), 264-289.
Quadratische Formen in additiven Kategorien. H.-G Quebbemann, R Scharlau, W Scharlau, M Schulte, Bull. Soc. Math. France Mémoire. 48H.-G. Quebbemann, R. Scharlau, W. Scharlau, M. Schulte, Quadratische Formen in additiven Kategorien, Bull. Soc. Math. France Mémoire 48 (1976), 93-101.
The equivalence of bilinear forms. C Riehm, J. Algebra. 31C. Riehm, The equivalence of bilinear forms, J. Algebra 31 (1974), 45-66.
Quadratic and Hermitian Forms, Grundlehren der math. W Scharlau, Wiss. 270Springer-VerlagW. Scharlau, Quadratic and Hermitian Forms, Grundlehren der math. Wiss. 270, Springer-Verlag (1985).
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"Elliptic fibrations and the Hilbert Property",
"Elliptic fibrations and the Hilbert Property"
] |
[
"Julian Lawrence Demeio "
] |
[] |
[] |
For a number field K, an algebraic variety X/K is said to have the Hilbert Property if X(K) is not thin. We are going to describe some examples of algebraic varieties, for which the Hilbert Property is a new result.The first class of examples is that of smooth cubic hypersurfaces with a K-rational point in P n /K, for n ≥ 3. These fall in the class of unirational varieties, for which the Hilbert Property was conjectured by Colliot-Thélène and Sansuc.We then provide a sufficient condition for which a surface endowed with multiple elliptic fibrations has the Hilbert Property. As an application, we prove the Hilbert Property of a class of K3 surfaces, and some Kummer surfaces.
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10.1093/imrn/rnz108
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[
"https://arxiv.org/pdf/1808.09808v2.pdf"
] | 119,714,103 |
1808.09808
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a9d36253c2e87eb430bfed70fe4e2685409258d1
|
Elliptic fibrations and the Hilbert Property
29 Aug 2018 August 30, 2018
Julian Lawrence Demeio
Elliptic fibrations and the Hilbert Property
29 Aug 2018 August 30, 2018
For a number field K, an algebraic variety X/K is said to have the Hilbert Property if X(K) is not thin. We are going to describe some examples of algebraic varieties, for which the Hilbert Property is a new result.The first class of examples is that of smooth cubic hypersurfaces with a K-rational point in P n /K, for n ≥ 3. These fall in the class of unirational varieties, for which the Hilbert Property was conjectured by Colliot-Thélène and Sansuc.We then provide a sufficient condition for which a surface endowed with multiple elliptic fibrations has the Hilbert Property. As an application, we prove the Hilbert Property of a class of K3 surfaces, and some Kummer surfaces.
Introduction
This paper is concerned with providing some examples of varieties with the Hilbert Property, concerning the set of k-rational points X(k), for an algebraic variety X defined over a field k.
A geometrically irreducible variety X over a field k is said to have the Hilbert Property if, for any finite morphism π : E → X, such that X(k) \ π(E(k)) is not Zariski-dense in X, there exists a rational section of π (see [10,Ch. 3] for an introduction of the Hilbert Property).
Some motivation for the study of the Hilbert Property comes from the following conjecture, of which a proof would settle the Inverse Galois Problem, as noted in [2]: Conjecture 1.1 (Colliot-Thélène, Sansuc). Let X be a unirational variety over a number field, then X has the Hilbert Property.
The first result of this paper, in Section 3, is the Hilbert Property for smooth cubic hypersurfaces of dim ≥ 2, defined over a number field K and with at least one K-rational point. Since, under these hypothesis, cubic hypersurfaces are Kunirational [9], this result gives positive examples of Conjecture 1.1. In Section 4 we prove the following theorem, which generalizes [4,Theorem 1.4].
Theorem 1.2. Let E be a projective smooth geometrically connected algebraic surface, defined over a number field K. Suppose that there exist n ≥ 2 elliptic fibrations π 1 , . . . , π n : E → P 1 . Let F ⊂ E denote the union of the divisors of E that are contained, for each i = 1, . . . , n, in a fiber of π i . If E \ F is simply connected and E(K) is Zariski-dense in E, then E has the Hilbert Property.
We then use this theorem to give explicit examples of K3 surfaces with the Hilbert Property. The examples are produced starting from a construction presented in [5], by Garbagnati and Salgado. Finally, we prove the Hilbert Property for some Kummer surfaces, for which the Hilbert Property was suggested to be true by Corvaja and Zannier [3].
Acknowledgements The author would like to thank his advisor Umberto Zannier, for having given him the opportunity to work on these topics, and providing important insights. The author would also like to thank Pietro Corvaja for fruitful discussions.
Background
This section contains some preliminaries. In particular, in the last paragraph, we shall recall some known results, that will be used in later sections, concerning the Hilbert Property. Moreover, we shall take care here of most of the notation that will be used in the paper.
Notation Throughout this paper, except when stated otherwise, k denotes a perfect field and K a number field. A (k-)variety is an algebraic quasi-projective variety (defined over a field k), not necessarily irreducible or reduced. Unless specified otherwise, we will always work with the Zariski topology.
Given a morphism f : X → Y between k-varieties, and a point s : Spec(k(s)) → Y , we denote by f −1 (s) the scheme-theoretic fibered product Spec(k(s))× Y X, and call it the fiber of f in s. Hence, with our notation, this is not necessarily reduced.
A geometrically integral k-variety X is a k-variety such that Xk is integral.
A proper morphism f : Y → X between normal k-varieties is a cover if the fiber f −1 (η) is finite for every point η of codimension ≤ 1 in X.
When k ⊂ C, a smooth k-variety X is simply connected if X C is a simply connected topological space.
Given a morphism f : Y → X between integral k-varieties, with Y normal, we will make use of the notion of relative normalization of X in Y . We refer the reader to [11,Def. 28.50.3, Tag 0BAK] or [7,Def. 4.1.24] for its definition, and recall here the properties that are needed in this paper. Namely, we will need that the relative normalization of X in Y is a finite morphism n :X → X such thatX is normal, and such that there exists a factorization f = ϕ • n, where ϕ : Y →X has a geometrically integral generic fiber. The normalization of an integral variety X is the usual normalizationX of X [7, Section 4.1.2].
The domain Dom(f ) of a rational morphism f : Y X between integral k-varieties is the maximal open Zariski subset U ⊂ Y such that f | U extends to a morphism on U.
Given a rational morphism f : Y X between integral k-varieties, and a birational transformation b : Y ′ Y , we will denote with abuse of notation, when there is no risk of confusion, the morphism f • b by f . We say that f is
well-defined on Y ′ if Dom(f • b) = Y ′ .
Ramification We recall that a morphism f : Y → X between k-varieties is unramified (resp.étale) in y ∈ Y if its differential df y : T y Y → T f (y) X is injective (resp. an isomorphism). Otherwise we say that f is ramified at y. The set of points where f is ramified has a closed subscheme structure in Y , and we will refer to it as the ramification locus. The image of the ramification locus under f is the diramation locus. We recall that, by Zariski's Purity Theorem [11,Lem. 53.20.4, Tag 0BMB], when Y is normal and X is smooth, the diramation locus of a finite morphism f : Y → X is a divisor. Hence, in this case, we will also refer to the diramation locus as the diramation divisor. A simply connected variety X does not have any geometrically integral cover of degree > 1 which is unramified in codimension 1.
Cubic hypersurfaces Cubic hypersurfaces are hypersurfaces in P n defined by a cubic homogeneous polynomial. We recall the following: (Segre '43,Manin '72). Let X be a smooth cubic hypersurface, defined over a field k, with a k-rational point. Then X is unirational.
Theorem 2.1
Hilbert Property For a more detailed exposition of the basic theory of the Hilbert Property and thin sets we refer the interested reader to [10,Ch. 3]. We limit ourselves here to recalling the most common definition (which is not the one given in the introduction), and some recent results.
Definition 2.2. Let X be a geometrically integral variety, defined over a field k. A thin subset S ⊂ X(k) is any set contained in a union D(k) ∪ i=1,...,n π i (E i (k)),
where D X is a subvariety, and π i : E i → X are generically finite morphisms 1 of degree > 1, and the E i 's are irreducible. Remark 2.3. A geometrically integral k-variety X has the Hilbert Property if and only if X(k) is not thin.
Throughout this paper, we will use the abbreviation HP for Hilbert Property. Definition 2.5. Let E be a normal projective algebraic K-surface. We say that a morphism π : E → P 1 is an elliptic fibration if its generic fiber is a smooth, geometrically connected, genus 1 curve.
The following theorem is included just for completeness. In fact, Theorem 1.2, which we are going to prove in Section 4, is a stronger version of it.
Theorem 2.6. Let K be a number field, and E be a complete smooth simply connected algebraic K-surface, endowed with two elliptic fibrations π i : E → P 1 /K, i = 1, 2, such that π 1 × π 2 : E → P 1 × P 1 is a finite morphism. Suppose moreover that the following hold:
(a) The K-rational points E(K) are Zariski-dense in E;
(b) Let η 1 ∼ = Spec K(λ) be the generic point of the codomain of π 1 . All the diramation points (i.e. the images of the ramification points) of the morphism π 2| π −1 1 (η 1 ) are non-constant in λ, and the same holds upon inverting π 1 and π 2 .
Then the surface E/K has the Hilbert Property.
Proof. See [4, Theorem 1.4].
3 Hilbert Property for cubic hypersurfaces Theorem 3.1. Let X ⊂ P n /K, n ≥ 3 be a smooth cubic hypersurface, with a K-rational point. Then X has the Hilbert Property.
In this section our base field will always be a number field K. We need the following lemma, of which an explicitly computable version can be found in [8].
Lemma 3.2. Let π : E → P 1 be an elliptic fibration, defined over a number field K. Then, there exists an open Zariski subset U π ⊂ E such that, for any P ∈ U π (K), π −1 (π(P )) is smooth and #π −1 (π(P ))(K) = ∞.
Proof of Theorem 3.1. We note that X is K-unirational by Theorem 2.1, in particular it has Zariski-dense K-rational points.
We prove the result by induction on n. Case n = 3. We assume by contradiction that X does not have the HP. Then there exist irreducible covers ϕ i :
Y i → X, i = 1, . . . , m of degree deg ϕ i > 1 and a divisor D ⊂ X such that X(K) ⊂ ∪ i ϕ i (Y i (K)) ∪ D(K).
We may assume, without loss of generality, that the Y i 's are normal and geometrically integral 2 , and that the ϕ i 's are finite morphisms. Let us denote now by R i the diramation divisor of ϕ i . By Lefschetz' hyperplane Theorem, X is simply connected, hence we know that the R i 's are nonempty for each i = 1, . . . , m.
Let us denote by A * 4 the dual affine space of A 4 , minus the origin. To each 3. The morphism [h 1 :
element h ∈ A * 4 corresponds a hyperplane H(h) ⊂ P 3 in a canonical way 3 . Let (h 1 , h 2 ) ∈ A * 4 (K) × A * 4 (K) be such that 1. H(h 1 ) ∩ H(h 2 ) ∩ X is (h 2 ] : X \ H(h 1 ) ∩ H(h 2 ) → P 1 is non-constant on each of the irreducible components of the R i 's.
We note that, since all conditions are open and non-empty, such a couple (h 1 , h 2 ) always exists. Let {P 1 , P 2 , P 3 } be the intersection H(h 1 ) ∩ H(h 2 ) ∩ X, and let π : X \ {P 1 , P 2 , P 3 } → P 1 be the following morphism:
P −→ [h 1 (P ) : h 2 (P )].
2 In fact, possibly by enlarging D, one can substitute Y i with the normalization of X in Y i . A normal variety that is not geometrically integral over the base field K does not have any K-rational points. 3 Namely
, if λ ∈ A * 4 , the associated hyperplane is {x ∈ P 3 | λ(x) = 0}.
The map π extends naturally to a morphismπ :X → P 1 , whereX = Bl P 1 +P 2 +P 3 X denotes the blowup of X in the (smooth) subscheme P 1 +P 2 +P 3 ⊂ X. We note that, since X is a cubic surface, the morphismπ is an elliptic fibration.
We claim now that the morphisms π • ϕ i : Y i \ ϕ −1 i ({P 1 , P 2 , P 3 }) → P 1 have geometrically integral generic fiber for each i = 1, . . . , m. In fact, let us assume by contradiction that there existed an i ∈ {1, . . . , m} such that this is not true. Let
π • ϕ i : Y i \ ϕ −1 i ({P 1 , P 2 , P 3 }) π ′ − → C r − → P 1 (3.1) be the relative normalization factorization of π•ϕ i| Y i \ϕ −1 i ({P 1 ,P 2 ,P 3 }) . We would have that deg r > 1. The factorization 3.1 yields a morphism ϕ ′ i : Y i \ϕ −1 i ({P 1 , P 2 , P 3 }) → X \ {P 1 , P 2 , P 3 } × P 1 C
, and a factorization:
ϕ i : Y i \ ϕ −1 i ({P 1 , P 2 , P 3 }) ϕ ′ i − → X \ {P 1 , P 2 , P 3 } × P 1 C α − → X \ {P 1 , P 2 , P 3 }, where X \ {P 1 , P 2 , P 3 } × P 1 C denotes the normalization of X \ {P 1 , P 2 , P 3 } × P 1 C.
Hence the diramation of ϕ i would contains the diramation of α, which would be nonempty if deg r > 1 (since X \ {P 1 , P 2 , P 3 } is simply connected) and contained in a finite union of fibers of π. This contradicts our choice of (h 1 , h 2 ).
Let us denote now byŶ i the desingularization of
Y ′ i = Y i × XX , and by ψ i :Ŷ i → P 1 the composition of the desingularization morphismŶ i → Y ′ i , the projection Y ′ i →XV i ⊂ P 1 such that, for each t ∈ V i (K), ψ −1 i (t)
is smooth (and we may assume, by further restricting V i , geometrically connected as well, because ψ i has a geometrically integral generic fiber). Let us denote now by D ′ ⊂X the proper subvariety, which is the union of all the following:
1. the fibersπ −1 (x), for each x / ∈ V i , for each i = 1, . . . , m;
2. the proper transform of D ⊂ X, and the exceptional locus ofX → X;
3. the proper transform of X \ U, where U is defined as in Lemma 3.2 for (E, π) = (X,π).
Let us choose now a K-rational point P ∈ (X \ D ′ )(K), and let us denote by E P the fiberπ −1 (π(P )). We know, by Lemma 3.2, that E P has infinitely many K-rational points. We have assumed, however, that
X(K) ⊂ ∪ i ϕ i (Y i (K)) ∪ D(K), and hence E P (K) ⊂ ∪ i ψ −1 i (π(P ))(K) ∪ (E P ∩ D ′ )(K). (3.2)
We claim that the right hand side of 3.2 is finite. In fact, for each i = 1, . . . , m, the morphism ψ −1 i (π(P )) → E P is ramified by the invariance of the ramification locus under base change (see e.g. [11, Tag 0C3H]), and, since the curve ψ −1 i (π(P )) is a smooth geometrically connected complete curve, by Riemann-Hurwitz theorem, it is of genus > 1. As a consequence, by Falting's theorem, ψ −1 i (π(P ))(K) is finite for each i = 1, . . . , m. Moreover, (E P ∩ D ′ ) is obviously finite, hence we have proved that the right hand side of 3.2 is finite. As we noted before, however, E P (K) is infinite. We have obtained a contradiction, proving the theorem in the case n = 3.
Case n ≥ 4. By Bertini's theorem [6, Remark 10.9.2], we know that there exists a Zariskiopen subset U ⊂ X such that, for each P ∈ U(K), the generic hyperplane of P n passing through P cuts X in a smooth irreducible cubic of dimension n − 2.
We choose now a K-rational point P ∈ U(K) , and two K-rational (distinct)
hyperplanes H 0 := {h 0 = 0}, H ∞ := {h ∞ = 0} passing through P , such that H 0 ∩ X is smooth. Let L := H 0 ∩ H ∞ .
Let us consider now the following morphism:
ϕ : X \ L ∩ X −→ P 1 , P −→ [h 0 (P ) : h ∞ (P )],
which extends naturally to a morphismφ :
Bl L∩X X → P 1 . For t = [t 1 : t 2 ] ∈ P 1 , the scheme-theoretic fiberφ −1 (t) is isomorphic to the intersection H t ∩ X, where H t denotes the hyperplane t 1 H 0 + t 2 H ∞ = 0 in P n . Since H 0 ∩ X is smooth, the intersection H t ∩ X is smooth for t in a Zariski open subset V ⊂ P 1 (K), containing 0. For x ∈ V (K), the fiberφ −1 (x)
is a smooth cubic in an (n − 1)dimensional projective space, with a K-rational point in it (namely, P ). Hence, by the induction hypothesis, this fiber has the HP, and hence, since P 1 /K has the HP, Bl L∩X X (and, therefore, X) has the HP as well by Theorem 2.4.
Surfaces with the Hilbert Property
The proof of Theorem 1.2 uses the following lemma, which is Lemma 3.2 of [3]. Let E be a smooth geometrically connected k-surface, endowed with fibrations π 1 , . . . , π n : E → P 1 , n ≥ 2. We call the fixed locus of π 1 , . . . , π n the following reduced subvariety of E:
Fix(π 1 , . . . , π n ) = {D : D is a divisor in E and π i| D is constant ∀i = 1, . . . , n} Remark 4.3. The subvariety F ⊂ E described in Theorem 1.2 is exactly Fix(π 1 , . . . , π n ).
Proof of Theorem 1.2. Suppose by contradiction that there exist m ∈ N, irreducible covers ϕ i : Y i → E, i = 1, . . . , m and a proper subvariety D E, such that E(K) ⊂ D(K) ∪ ∪ i ϕ i (Y i (K)), and deg ϕ i ≥ 2. We may assume, without loss of generality, that the Y i 's are smooth and geometrically connected.
We say that a cover Y i , and the corresponding i, is {j 1 , . . . , j k }-unramified, where {j 1 , . . . , j k } ⊂ {1, . . . , m}, if, for each j ∈ {j 1 , . . . , j k }, the diramation divisor of ϕ i is contained in a finite union of fibers of π j . We say that it is {j 1 , . . . , j k }-ramified otherwise. By the simply connectedness hypothesis, no cover is {1, . . . , m}-unramified.
We say that a point P ∈ E(K) is j-good if the fiber π −1 j (π j (P )) is smooth and geometrically connected of genus 1, and #π −1 j (π j (P ))(K) = ∞. By Lemma 3.2, there exists an open subset U ⊂ E such that, for each P ∈ U(K), P is jgood for each j = 1, . . . , m. We assume moreover, without loss of generality, that
U ∩ D = ∅. For each 1 ≤ i ≤ m, 1 ≤ j ≤ n, let π j • ϕ i : Y i r ij − → C ij → P 1
be the relative normalization factorization of π j • ϕ i . We have that the geometric generic fiber of r ij is irreducible, C ij is a smooth complete geometrically connected curve and C ij → P 1 is a finite morphism. Let E × P 1 C ij → E × P 1 C ij be a desingularization of E× P 1 C ij . Then there exists a commutative diagram as follows:
Y i E × P 1 C ij E Y i ψ ij b i ψ i where b i :Ŷ i → Y i is the
composition of a finite sequence of blowups.
When i is j-ramified, since the diramation locus of ψ i contains at least one component transverse to the fibration π j , and the morphism E × P 1 C ij → E is j-unramified, ψ ij must have at least one irreducible component of the diramation locus which is transverse to the fibration E × P 1 C ij → C ij . Hence, by the invariance of the ramification locus under base change [11, Tag 0C3H, Lemma 30.10.1], when i is not j-unramified, the geometric generic fiber ofŶ i → C ij , which is isomorphic to the geometric generic fiber of r ij , is ramified over the geometric generic fiber of E × P 1 C ij → C ij , which has genus 1. Therefore, it is a curve of genus > 1. Hence, when i is j-ramified, the geometric generic fiber of π j • ϕ i is a finite union of curves of genus > 1. When i is j-unramified one shows analogously that the geometric generic fiber of π j • ϕ i is a finite union of curves of genus 1.
We have that, for each P ∈ U(K):
P ∈ π −1 n (π n (P ))(K) ∩ U ⊂ i n−unramified ϕ i (Y i (K)) ∩ π −1 n (π n (P )) ∪ i n−ramified ϕ i (Y i (K)) ∩ π −1 n (π n (P )) = i n−unramified ϕ i ((π n • ϕ i ) −1 (π n (P )(K))) ∪ i n−ramified ϕ i ((π n • ϕ i ) −1 (π n (P )(K))).
Hence, as noted before, when i is n-ramified, after restricting (without loss of generality) U to a smaller Zariski open subset, (π n • ϕ i ) −1 (π n (P )) is a curve of genus > 1. Therefore, by Falting's Theorem, we deduce that:
π −1 n (π n (P ))(K) ⊂ i n−unramified ϕ i ((π n • ϕ i ) −1 (π n (P )(K))) ∪ A 0 (P ),
where A 0 (P ) is a finite set. Moreover, when i is n-unramified, after restricting (without loss of generality) U to a smaller Zariski open subset, (π n • ϕ i ) −1 (π n (P )) is a curve of genus 1. Therefore, by the weak Mordell-Weil theorem, we have that, for each i = 1, . . . , m, (π n • ϕ i ) −1 (π n (P ))(K) ⊂ π −1 n (π n (P ))(K) is either empty or a finite index coset.
Hence, by Lemma 4.1, we deduce that:
P ∈ π −1 n (π n (P ))(K) ⊂ i n−unramified ϕ i (Y i (K)).
Therefore we have that
E(K) ⊂ i n−unramified ϕ i (Y i (K)) ∪ (E \ U).
We have now reduced to the case where all the covers Y i are n-unramified. Proceeding with an easy induction on n, we may reduce to the case where all the covers Y i are {1, . . . , n}-unramified. But there are no such covers by hypothesis, whence we deduce that E(K) is not Zariski-dense in E, leading to the desired contradiction.
A family of K3 surfaces with the Hilbert Property
As an application of Theorem 1.2 we describe now a family of K3 surfaces 4 with the Hilbert Property.
For λ ∈ K * , and c 1 , c 2 ∈ K[x, y, z] cubic homogeneous polynomials, let c 2 ) are, up to a birational transformation, endowed with multiple elliptic fibrations, usually defined overK, whose construction we recall in the next paragraphs. In some particular cases, when enough of these fibrations are defined over K, this allows us to use Theorem 1.2 to prove the Hilbert Property of these varieties. In fact, in this case, equation (4.1) describes, up to a birational transformation, the quotient of E 1 ×E 2 by the group {±1}, where E 1 and E 2 are the elliptic curves defined by the following Weierstrass equations:
X ′ λ (f 1 , f 2 ) := {([w 0 : w 1 ], [x : y : z]) ∈ P 1 × P 2 | w 2 0 f 1 (x, y, z) = λw 2 1 f 2 (x, y, z)}. (4.1) The surfaces X ′ λ (c 1 ,E 1 : w 2 = f 1 (x, z), E 2 : w 2 = f 2 (y, z). (4.2)
Construction of the Elliptic Fibrations
We give now an explicit construction of a smooth model of of X ′ λ (c 1 , c 2 ) and of the elliptic fibrations it is endowed with, under a genericity assumption on c 1 and c 2 . We avoid going into detail, as these constructions are described thoroughly by Garbagnati and Salgado in [5].
Let P 1 , . . . , P 9 be 9 (distinct) points in P 2 (K) such that:
1. P 1 , . . . , P 4 are the four points of intersection of two smooth conics Q 1 1 := {q 1 1 = 0}, Q 1 2 := {q 1 2 = 0} in P 2 , defined over K;
2. P 5 , . . . , P 8 are the four points of intersection of two smooth conics Q 2 1 := {q 2 1 = 0}, Q 2 2 := {q 2 2 = 0} in P 2 , defined over K;
3. The eight points P 1 , . . . , P 8 are in generic position 5 ;
4. P 1 , . . . , P 9 are the nine points of intersection of two smooth cubics C 1 := {c 1 = 0}, C 2 := {c 2 = 0} in P 2 , defined over K.
Definition 4.5. We say that a 9-tuple (P 1 , . . . , P 9 ) ∈ P 2 (K) 9 is good if it satisfies the four conditions above.
We note that, by choosing two sufficiently Zariski generic 4-tuples p 1 , . . . , p 4 and p 5 , . . . , p 8 such that both p 1 + · · · + p 4 and p 5 + · · · + p 8 are defined over K, and letting p 9 be the unique ninth intersection of the pencil of cubics through p 1 , . . . , p 8 , the 9-tuple p 1 , . . . , p 9 is good.
We assume hereafter that a choice of a good 9-tuple of points P 1 , . . . , P 9 , of the two cubics C 1 , C 2 and of the conics Q i j , i, j = 1, 2 has been made. Let R := Bl P 1 +···+P 9 P 2 be the blowup of P 2 in the nine points P 1 , . . . , P 9 . The two cubics C 1 , C 2 define an elliptic fibration on R, which we denote by C, defined as C(p) = [c 1 (p) : c 2 (p)].
The fibers of C are by construction the proper transforms of the elements of the pencil generated by C 1 , C 2 .
For λ ∈ K * , let f λ : P 1 → P 1 be the morphism defined by f λ ([w 0 : w 1 ]) = [w 2 0 : λw 2 1 ]. Let also X λ (c 1 , c 2 ) be the smooth surface defined as the fibered product R × C,f λ P 1 , α λ : X λ (c 1 , c 2 ) → R be the projection on the first factor, and ϕ λ : X λ → P 1 be the projection on the second factor. The surface X λ (c 1 , c 2 ) is a K3 surface.
Note 1. We observe that, by construction, X λ (c 1 , c 2 ) is birational to X ′ λ (c 1 , c 2 ). The surface X λ (c 1 , c 2 ) is endowed with at least three elliptic fibrations. The first one is ϕ λ . The second and third one, which we denote by Q 1 and Q 2 , are the proper transforms of the two pencils of conics generated, respectively, by Proposition 4.6. Let P 1 , . . . , P 9 ∈ P 2 (K) be a good 9-tuple of points, and C 1 := {c 1 = 0}, C 2 := {c 2 = 0} be two smooth cubics such that C 1 ∩ C 2 = {P 1 , . . . , P 9 }. If X λ (c 1 , c 2 ) has Zariski-dense K-rational points, then it has the Hilbert Property.
Proof. Since Fix(ϕ λ , Q 1 , Q 2 ) = ∅, this is an immediate consequence of Theorem 1.2 applied to X λ (c 1 , c 2 ) (which, being a K3 surface, is simply connected), with fibrations ϕ λ , Q 1 and Q 2 .
Remark 4.7. In general, looking at the explicit equations, it is easy to show that, given c 1 , c 2 cubic polynomials, there exist always infinitely many λ ∈ K * such that X λ (c 1 , c 2 ) has Zariski-dense K-rational points. Hence, as a corollary of Theorem 4.6, one obtains that, for any c 1 , c 2 , there exist always infinitely many X λ (c 1 , c 2 ) with the Hilbert Property.
Kummer surfaces
The following proposition is another application of Theorem Proof. A desingularization of S, which we denote byS, may be obtained as the quotient by {±1} of the blow up E 1 × E 2 of E 1 ×E 2 in the 16 2-torsion points. We denote the set of the images of these points in S with T , and the corresponding exceptional lines inS with L. Moreover, we denote by b :S → S the just described desingularization morphism, and by q :
E 1 × E 2 → S the quotient map.
The surfaceS has at least three elliptic fibrations, defined over K. Two of these, which we denote by π i , i = 1, 2 are the following compositions:
S → S = E 1 × E 2 /{±1} → E i /{±1} ∼ = P 1 , i = 1, 2.
If E 1 := {y 2 1 z 1 = f 1 (x 1 , z 1 )} and E 2 := {y 2 2 z 2 = f 2 (x 2 , z 2 )}, then, as noted in Remark 4.4, S is birational to the surface defined by the following equation for ([x : y : z], [w 1 : x 2 ]) ∈ P 2 × P 1 :
w 2 1 f 2 (x, z) = w 2 2 f 1 (y, z).
We have that:
[w 1 : w 2 ] • q = [y 1 z 2 : y 2 z 1 ]. (4.3)
We then define the third fibration, π 3 , to be the extension (as a rational map) We have that, since y i is a local parameter at points of order 2 in E i , and z i is a local parameter at O ∈ E i , the morphism [y 1 z 2 : y 2 z 1 ] : E 1 × E 2 → P 1 is non-constant on the exceptional lines lying over the points (T 1 , T 2 ), when both T 1 and T 2 have order 2 or both have order 0.
It follows that Fix(π 1 , π 2 , π 3 ) is the union of the 6 exceptional lines inS lying over the points (T 1 , T 2 ), when exactly one of T 1 , T 2 has order 2 and the other is O. We denote the union of these 6 points in S with T b , and the corresponding lines inS with L b := b −1 (T b ). Since, by hypothesis, S(K) ⊃ q(E 1 (K) × E 2 (K)) is Zariski-dense in S, the proposition follows from Theorem 1.2 and the following lemma. Proof. Let Λ 1 =< e 1 , e 2 > and Λ 2 =< e 3 , e 4 > be lattices in C such that E 1 ∼ = C/Λ 1 , and E 2 ∼ = C/Λ 2 as analytic spaces. We observe that the universal cover of S \ L = S \ T is the following composition
C 2 \ 1 2 (Λ 1 × Λ 2 ) → C/Λ 1 × C/Λ 2 \ q −1 (T ) ∼ = E 1 × E 2 \ q −1 (T ) q − → S \ T .
Therefore
π 1 (S \ T , p 0 ) ∼ = (Λ 1 × Λ 2 )⋊ < ι >,
where p 0 denotes a point infinitesimally near to q((O, O)) ∈ S, the action of ι on Λ 1 × Λ 2 is given by (a, b) → (−a, −b), and the element ι corresponds to a (single) loop around q((O, O)) ∈ S (and hence ι 2 = 1). For any 2-torsion point T in E 1 × E 2 , let ι T ∈ π 1 (S \ T , p 0 ) denote the element corresponding to a (single) loop around the point q(T ) ∈ S 6 . We have that, if T = 4 i=1 ǫ i 2 e i , where ǫ i ∈ {0, 1}, then
ι T = 4 i=1 ǫ i e i ι.
Let H ⊂ (Λ 1 ×Λ 2 )⋊Z/2Z denote the minimal normal subgroup containing the ι T 's, for T ∈ T g := T \ T b . We note that (e 1 + e 4 )/2, (e 1 + e 3 )/2, (e 1 + e 3 + e 4 )/2 ∈ T g , and hence e 1 = (e 1 + e 4 ) + (e 1 + e 3 ) − (e 1 + e 3 + e 4 ) ∈ H. Analogously one has that e i ∈ H for every 1 ≤ i ≤ 4. Therefore, H = (Λ 1 × Λ 2 )⋊ < ι >.
We now observe that, in the blown-up surfaceS \ L b , the loop ι T becomes trivial for any point T ∈ T g . In fact, a small topological neighborhood L ǫ T of the exceptional line L T := b −1 (T ) ⊂S is retractible on L T itself, which is simply connected.
Therefore, by Van Kampen's Theorem applied toS \ L b = (S \ L) ∪ T ∈T g L ǫ T , we have that
π 1 (S \ L b ) ∼ = (Λ 1 × Λ 2 )⋊ < ι > H ∼ = {1},
as we wanted to prove. 6 This element is well-defined only after a choice of a path between q(p 0 ) and a point infinitesimally near to q(T ) has been made. This choice can be done arbitrarily and it is in fact irrelevant for our purposes. We will assume anyway that the path chosen is the geodetic, using the distance induced by the universal cover
R 4 (e1|e2|e3|e4) − −−−−−−− → C × C → C/Λ 1 × C/Λ 2 ∼ = E 1 × E 2 q − → S,
where the R 4 on the left is endowed with the Euclidean metric.
Theorem 2. 4 (
4Bary-Soroker, Fehm, Petersen). Let f : X → S be a morphism of K-varieties. Suppose that there exists a non-thin subset A ⊂ S(K) such that, for each s ∈ A, f −1 (s) has the HP. Then X/K has the HP. Proof. See [1, Theorem 1.1].
a scheme consisting of) three distinct points (and hence, as a direct consequence, H(h 1 ) ∩ H(h 2 ) is a line), and it is disjoint from the union of the R i 's; 2. H(h 1 ) ∩ X and H(h 2 ) ∩ X are smooth curves;
Lemma 4. 1 .
1Let G be a finitely generated abelian group of positive rank. Let n ∈ N and {h u + H u } u=1,...,n be a collection of finite index cosets in G, i.e. h u ∈ G, H u < G and [G : H u ] < ∞ for each u = 1, . . . , n. If G \ u=1,...,n (h u + H u ) is finite, then u=1,...,n (h u + H u ) = G.
Remark 4 . 4 .
44When c 1 (x, y, z) = f 1 (x, z) does not depend on y, c 2 (x, y, z) = f 2 (y, z) does not depend on x, and both f 1 and f 2 do not have multiple roots, equation (4.1) describes a Kummer surface (i.e. a quotient of an abelian surface by the group of isomorphisms {±1}).
. I.e., Q i = Q i • α λ , where the maps Q i : R → P 1 are defined as Q i (p) = [q i 1 (p) : q i 2 (p)], i = 1, 2.
Lemma 4 . 9 .
49The surface (S \ L b )/C is (topologically) simply connected.
and the mapπ :X → P 1 . By the Theorem of generic smoothness [6, Corollary 10.7] we know that there exists an open subset
1.2. Proposition 4.8. Let E 1 and E 2 be two elliptic curves defined over a number field K, with positive Mordell-Weil rank. The Kummer surface S := E 1 × E 2 /{±1} has the Hilbert Property.
tõ S of the map ([x : y : z], [w 1 : w 2 ]) → [w 1 : w 2 ]. Let us check that Dom(π 3 ) =S. The map [y 1 z 2 : y 2 z 1 ] is well-defined on E 1 × E 2 . Moreover, since [y 1 z 2 : y 2 z 1 ] : E 1 × E 2 → P 1 is invariant by the action of {±1}, it induces indeed a well-defined morphism on the quotientS = E 1 × E 2 /{±1}.
When X is normal, one can substitute here "generically finite morphisms" with "covers" to get an equivalent definition.
K3 surfaces (and, in general, Calabi-Yau varieties) represent a "limiting case" for the study of rational points, at least conjecturally. In fact, the conjectures of Vojta suggest that on algebraic
varieties there should be "less" rational points as the canonical bundle gets "bigger". Hence, since for K3 surfaces the canonical bundle is trivial by definition, we expect the rational points here not to be "too much", yet their existence (and Zariski-density) is not precluded. In fact, proving the HP, we are providing some examples of abudance of rational points in such surfaces.5 By generic position, we mean that no three of these points lie on a line, and no six of these points lie on a conic.
On varieties of Hilbert type. L Bary-Soroker, A Fehm, S Petersen, Ann. Inst. Fourier (Grenoble). 645Bary-Soroker, L., Fehm, A., Petersen, S.: On varieties of Hilbert type. Ann. Inst. Fourier (Grenoble) 64(5), 1893-1901 (2014)
Points rationnels sur les fibrations. J L Colliot-Thélène, Higher dimensional varieties and rational points. Budapest; BerlinSpringer12Colliot-Thélène, J.L.: Points rationnels sur les fibrations. In: Higher dimen- sional varieties and rational points (Budapest, 2001), Bolyai Soc. Math. Stud., vol. 12, pp. 171-221. Springer, Berlin (2003)
On the hilbert property and the fundamental group of algebraic varieties. P Corvaja, U Zannier, Mathematische Zeitschrift. Corvaja, P., Zannier, U.: On the hilbert property and the fundamental group of algebraic varieties. Mathematische Zeitschrift pp. 1-24 (2016)
J L Demeio, Non-rational varieties with the Hilbert Property. ArXiv e-prints. Demeio, J.L.: Non-rational varieties with the Hilbert Property. ArXiv e-prints (2018)
Linear systems on rational elliptic surfaces and elliptic fibrations on K3 surfaces. A Garbagnati, C Salgado, Garbagnati, A., Salgado, C.: Linear systems on rational elliptic surfaces and elliptic fibrations on K3 surfaces. ArXiv e-prints (2017)
Algebraic geometry. R Hartshorne, Graduate Texts in Mathematics. New YorkSpringer-VerlagHartshorne, R.: Algebraic geometry. Springer-Verlag, New York (1977). Grad- uate Texts in Mathematics, No. 52
Algebraic Geometry and Arithmetic Curves. Q Liu, Oxford University PressOxford graduate texts in mathematicsLiu, Q.: Algebraic Geometry and Arithmetic Curves. Oxford graduate texts in mathematics. Oxford University Press (2002)
Density of rational points on elliptic surfaces. R Van Luijk, Acta Arithmetica. 1562van Luijk, R.: Density of rational points on elliptic surfaces. Acta Arithmetica 156(2), 189-199 (2012)
A note on arithmetical properties of cubic surfaces. B Segre, J. London Math. Soc. 18Segre, B.: A note on arithmetical properties of cubic surfaces. J. London Math. Soc 18, 24-31 (1943)
J P Serre, Topics in Galois theory. A K Peters, Ltd., MA1Serre, J.P.: Topics in Galois theory, Research Notes in Mathematics, vol. 1, second edn. A K Peters, Ltd., MA (2008)
Stacks Project Authors, T.: stacks project. Stacks Project Authors, T.: stacks project. http: // stacks. math. columbia. edu (2017)
|
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IntroductionWireless systems, such as PDAs, wireless phones and on-board computers in cars can now have permanent access to a large number of applications: mail, word processing, GPS navigation system, m-learning, photo applications, video applications, etc.. The deployments of these applications are becoming increasingly complex due to the diversity of access terminals and communication infrastructure. It is based on highly centralized tools and many manual operations for the administrator. These techniques are not really usable in the network without administrator (network ad-hoc).Platforms deployment starts to make deployment easier on desktop machines. The first standardization work (under the OMA[2]) have permitted the emergence of specifications for deployment. In this context, we would build a prototype platform deployment that would deploy a local (peer to peer) application on a wireless system (mobile phones).Software component and Wireless systemWireless systems have become more complex and numerous approaches take advantage of today's software components paradigm to develop applications embedded in these systems[6].J2ME [5] defines a model of components (MIDlets). The life cycle of these components is managed by software residing on the wireless system, AMS (Application Management System). This management is crucial for wireless systems because they have limited resources.OMA Device Management ProtocolOMA is a large organization of mobile phone manufacturers, which is responsible for managing standards for portable equipment. Based on the SyncML protocol [1] in 2004, the AMO has proposed two standard services: data synchronization (Data Synchronization -SD) and management of devices (Device Management -DM). The OMA DM [3] is a protocol that allows operating and configuring the equipment, access and control resources on the mobile. An advantage of the OMA DM is the ability to deploy applications. In this section, we present the protocol for management of OMA applications. It consists of two parts:• The model data available for remote handling.• The communication protocol between the management server and mobile client.
| null |
[
"https://arxiv.org/pdf/1205.4261v1.pdf"
] | 16,612,895 |
1205.4261
|
5a082eb61078852cf0fea25d9523e65f22f19fa9
|
IntroductionWireless systems, such as PDAs, wireless phones and on-board computers in cars can now have permanent access to a large number of applications: mail, word processing, GPS navigation system, m-learning, photo applications, video applications, etc.. The deployments of these applications are becoming increasingly complex due to the diversity of access terminals and communication infrastructure. It is based on highly centralized tools and many manual operations for the administrator. These techniques are not really usable in the network without administrator (network ad-hoc).Platforms deployment starts to make deployment easier on desktop machines. The first standardization work (under the OMA[2]) have permitted the emergence of specifications for deployment. In this context, we would build a prototype platform deployment that would deploy a local (peer to peer) application on a wireless system (mobile phones).Software component and Wireless systemWireless systems have become more complex and numerous approaches take advantage of today's software components paradigm to develop applications embedded in these systems[6].J2ME [5] defines a model of components (MIDlets). The life cycle of these components is managed by software residing on the wireless system, AMS (Application Management System). This management is crucial for wireless systems because they have limited resources.OMA Device Management ProtocolOMA is a large organization of mobile phone manufacturers, which is responsible for managing standards for portable equipment. Based on the SyncML protocol [1] in 2004, the AMO has proposed two standard services: data synchronization (Data Synchronization -SD) and management of devices (Device Management -DM). The OMA DM [3] is a protocol that allows operating and configuring the equipment, access and control resources on the mobile. An advantage of the OMA DM is the ability to deploy applications. In this section, we present the protocol for management of OMA applications. It consists of two parts:• The model data available for remote handling.• The communication protocol between the management server and mobile client.
Data Model
It is difficult for the server to manage mobiles because of their configuration and their specific characteristics that depend on each manufacturer [7]. OMA DM has proposed a standard framework to enable providers to identify the description of devices and communicate it to the server. The server is based on the description to send operations. This description is called "the tree of device management". Figure 1 depicts an example of a tree management device. This tree holds all the managed objects of the device as a tree structure where each node is addressed (uniquely) by a URI (Uniform Resource Identifier). Abstract. The wide variety of wireless devices brings to design mobile applications as a collection of interchangeable software components adapted to the deployment environment of the software. To ensure the proper functioning of the software assembly and make a real enforcement in case of failures, the introduction of concepts, models and tools necessary for the administration of these components is crucial. This article proposes a method for deploying components in wireless systems.
Tree management devices
Mots-clés : J2ME, SyncML, OMA DM.
Figure1
. Tree management device.
The node can take any form: a single parameter, a GIF or even a complete application. There are two types of objects:
• The permanent objects are incorporated into the device, they cannot be deleted. For example, the DevInfo object that specifies the basic information of the device. • Dynamic objects are objects that can be added or deleted. (For example, tones).
Features Node
Nodes are entities that can be manipulated by the OMA DM protocol. Each tree node has a set of properties defining characteristics:
• ACL (Access Control List) defines who can use the node. • Format: Determines how a node is interpreted.
• Name: the name of the node. • Size: The size of the information the node contains.
• Title: the name of the node visible.
• Timestamp: date and time of its last modification.
• Type: the MIME type of information the node contains. • Verno: the version number, automatically incremented with each modification.
Nodes Manipulation
The management nodes can be handled by SyncML DM messages containing the following commands:
• Get: allows the server management to explore the structure of the tree. If the node is accessed within a node, a list of all node names is returned and if the node is a leaf, its information is returned. • Add: Adds a node to a tree.
• Replace: Replaces a node in the tree.
• Delete: Deletes a node from the tree.
• Copy: Copies a node of the tree.
The server can modify the tree management during the execution using Add commands. The new node can be an internal or a leaf node but the parent must be an interior node. The device itself can also change the tree.
Standard Objects
OMA DM proposes a framework that allows manufacturers to set their own descriptions for their products. But it would be limited if the entities in the manageable devices have different formats and different behaviors. To avoid this, the OMA has defined a number of items required. These objects are mainly associated with OMA DM [3] and the configuration of SyncML: -DevInfo description of the device, sent from client to server. -DevDetail: General information about the device that benefits from standardization. -DMAcc: Institution for the DM client.
Protocol and mechanism
OMA provides a communication protocol between the client and the server for device management. The protocol consists of a sequence of messages between the client and the server to perform operations.
Protocol packets
The protocol consists of two main phases, the initiation phase and the management phase. The first phase is designed to authenticate and exchange information of the device. The second makes commands of the server on the client. These phases are considered as transactions of the SyncML packets.
The management phase consists of a number of iterations. The contents of the packet sent by the server determine whether the session should be continued or not. If the server sends the transactions in a package that needs a response (Status or Results), this phase continues with a new client package for the server containing the answers. The response packet of the client starts a new iteration. The server may send a new package management operation and then starts a new iteration if desired [13]. The processing of packets can make an unpredictable time. Therefore, the OMA DM protocol indicates no timeout between packets.
Deployment system
This system creates an arborescence on the tree management device. This structure is the basis on which management can make all applications. It makes the management of the application easier.
Application management Object description
The application management object provides functionalities for remote application management; node in the internal management object is called SCM (Software Component Management) [11] and can be found then from the root node ( Figure 3). • Inventory of a device: the discovery of device applications. In fact, all the nodes of the tree management are taken and classified. This process sends to the server a list of applications and their status. The application state depends on its position on the tree. The server is based on that state to decide what action can be executed with the appropriate application. To perform any action deployment, we must first identify the device.
• Update of an application: Exchange of information on an application and the runtime files. We can only update applications that have been delivered by the OMA DM server [3].
• Installing an application: when you deliver applications to the device, they cannot be activated.
To activate, you must first install. Indeed, each node contains a set of commands for application deployment. In this case, to install an application, we send the command Exec to operation node appropriate to the object that represents the application.
• Application download: send application information with its URI (file. Jad or. Jar). Objects that describe downloadable applications are in the interior node Dowload. To start the download, the server sends the command Exec to node operation appropriate to the subject.
• Enabling / disabling an application: the command Exec is sent to the operation node appropriate for the object that represents the application.
• Deleting an Application: Removes all nodes on the application on the management tree.
CONCLUSION
The wide variety of wireless devices brings to design mobile applications as an assembly of interchangeable software components adapted to the deployment environment of the software [8]. To ensure the proper functioning of the software assembly and make a real enforcement in case of failures, the establishment of concepts, models and tools necessary to manage these components is necessary [13]. J2ME defines a model of components (MIDlets). The life cycle of these components is managed by software residing on the wireless system, AMS (Application Management System). This management is crucial for wireless systems because they have limited resources. OMA is a large organization of mobile phone manufacturers, which is responsible for managing standards for portable equipment. Based on the SyncML protocol in 2004, the AMO has proposed two standard services: data synchronization (DS -Data Synchronization) and management of devices (Device Management-DM) [12]. The OMA DM is a protocol that allows exploiting and configuring devices, access and controlling resources on mobiles. One advantage of the OMA DM is the ability to deploy applications. In this paper, architecture for deploying software components in wireless systems was presented. The objective was to develop a system for local deployment for mobile devices; this system must take into account all deployment activities: [10] installation, upgrade, activation, deactivation and uninstalling. To focus on our problem of deploying components, we have built our approach on a simple component model based on MIDlet, it would be interesting to apply our solution to a component model like OSGi [9], which is becoming more and more popular in the Wireless World.
Figure 2 .
2Packages of the OMA DM protocol.
Figure 3 .
3Application management Object description [4]. • Delivery of Application: send an application from the server to the device. The consignment consists of information on the application (such as name, version, etc...) Description file. Jad and the execution file. Applications delivered by the OMA DM server are located in the interior node Inventory / Delivered. Be aware of the type and the size of the mobile application, especially the MIDlet.
•Figure4.Figure5Figure 6 .
6Architecture of the Deployment System (prototype) peer -peer according to the OMA DM (OMA DM over OBEX / Bluetooth). SyncML Adapter a. SyncML Genenrator(Serevr) : generates management commands as SyncML messages. b. SyncML Parser(Client) : SyncML analyzes massages to extract management commands. Management Tree: configuration file in XML format (Figure 2). Management Agent: XML parser (DOM) enables the creation; modification and removal of elements of the configuration file (Tree Management). Download Macanisme : FTP OBEX. OBEX/Bluetooth : Transport Layer. OMA DM Client: 1: Server Bluetooth Address 2: Name of service 3: The root element (Software Component Management) 4: Structure of the Tree Management Application 5: Messages syncml Server to Client 6: Data from one node (descriptor) Server interface.
H Uwe, SyncML Synchronizing and Managing Your Mobile Data. Prentice Hall PTR PubUwe, H.,: SyncML Synchronizing and Managing Your Mobile Data. Prentice Hall PTR Pub (2002)
Oma device management ddf for application management version 1.1. Forum Nokia. OMA DM (Device Management) Working GroupOMA DM (Device Management) Working Group, http://www.openmobilealliance.org/tech/wg_committees/dm.ht ml 4. Nokia. Oma device management ddf for application management version 1.1. Forum Nokia, dec 2006.
Sun java wireless toolkit for cldc. J Me, J. ME. Sun java wireless toolkit for cldc. http ://java.sun.com/products/sjwtoolkit/
Franck Barbier, Fabien Romeo, Handbook of research on Mobile Business, chapter Administration of Wireless Software Components. Idea Group PublishingFranck Barbier and Fabien Romeo, Handbook of research on Mobile Business, chapter Administration of Wireless Software Components, pages 200-215, Idea Group Publishing, 2006.
Nicoloas Belloir, composition conceptuelle basée sur la relation Tout-Partie. Université de Pau et des pays de l'AdourPhD thesisNicoloas Belloir, composition conceptuelle basée sur la relation Tout-Partie. PhD thesis, Université de Pau et des pays de l'Adour, 2004.
Intergiciel et Construction d'Applications Réparties, chapitre le modèle de composants Fractal. T Coupaye, V Quéma, L Seinturier, J-B Tefani, Creative CommonsGrenobleT. Coupaye, V. Quéma, L. Seinturier and J-B tefani. Intergiciel et Construction d'Applications Réparties, chapitre le modèle de composants Fractal, pages 49-76. Creative Commons, Grenoble Janvier 2007.
OSGi Service Platform Mobile specification. Osgi Alliance, Release 4, Version 4.0OSGi Alliance. OSGi Service Platform Mobile specification, Release 4, Version 4.0, July 2006.
Fabien Romeo, Franck Barbier, Jean-Michel Bruel, Stefanos Zachariadis, Cecilia Mascolo and Wolfgang Emmerich, the SATIN Component System, a metamodel for Engineering Adaptable Mobile Systems, IEEE Transactions on Software Engineering. Paphos, ChyprusObservability and Controllability of Wireless Software Components In proceedings of the 7th IFIP International Conference on Distributed Applications and Interoperable SystemsFabien Romeo, Franck Barbier and Jean-Michel Bruel, Observability and Controllability of Wireless Software Components In proceedings of the 7th IFIP International Conference on Distributed Applications and Interoperable Systems, pages 48-61, Paphos, Chyprus, June 2007 11. Stefanos Zachariadis, Cecilia Mascolo and Wolfgang Emmerich, the SATIN Component System, a metamodel for Engineering Adaptable Mobile Systems, IEEE Transactions on Software Engineering, october 2006.
Jurgen Schonwalder, traditional approachs to distributed management in 19 th meeting of the Network Management Research Group (NMRG) og the Internet Research Task Force (IRTF), Janvier 2006. 13. Distributed Management Task Force, Web-Based Entreprise Management (WBEM). Jurgen Schonwalder, traditional approachs to distributed management in 19 th meeting of the Network Management Research Group (NMRG) og the Internet Research Task Force (IRTF), Janvier 2006. 13. Distributed Management Task Force, Web-Based Entreprise Management (WBEM), 2007, http://www.dmtf.org/standards/wbem
. Luciano Baresi, I Elisabetta, Carlo Nitto, Ghezzi, Toward Open World Software: issue and challenges. 3910IEEE ComputerLuciano Baresi, Elisabetta I Nitto and Carlo Ghezzi, Toward Open World Software: issue and challenges, IEEE Computer, 39 (10): 36-43, 2006.
|
[] |
[
"Spin Dynamics in the Magnetoelectric Effect LiCoPO 4 Compound",
"Spin Dynamics in the Magnetoelectric Effect LiCoPO 4 Compound",
"Spin Dynamics in the Magnetoelectric Effect LiCoPO 4 Compound",
"Spin Dynamics in the Magnetoelectric Effect LiCoPO 4 Compound"
] |
[
"Wei Tian \nDepartment of Physics and Astronomy\nAmes Laboratory\nIowa State University\n50011AmesIowaUSA\n",
"Jiying Li \nNCNR\nNational Institute of Standards and Technology\n20899GaithersburgMDUSA\n\nDept. of Materials Science and Engineering\nUniversity of Maryland\n20742College ParkMDUSA\n",
"Jeffrey W Lynn \nNCNR\nNational Institute of Standards and Technology\n20899GaithersburgMDUSA\n",
"Jerel L Zarestky \nDepartment of Physics and Astronomy\nAmes Laboratory\nIowa State University\n50011AmesIowaUSA\n",
"David Vaknin \nDepartment of Physics and Astronomy\nAmes Laboratory\nIowa State University\n50011AmesIowaUSA\n",
"Wei Tian \nDepartment of Physics and Astronomy\nAmes Laboratory\nIowa State University\n50011AmesIowaUSA\n",
"Jiying Li \nNCNR\nNational Institute of Standards and Technology\n20899GaithersburgMDUSA\n\nDept. of Materials Science and Engineering\nUniversity of Maryland\n20742College ParkMDUSA\n",
"Jeffrey W Lynn \nNCNR\nNational Institute of Standards and Technology\n20899GaithersburgMDUSA\n",
"Jerel L Zarestky \nDepartment of Physics and Astronomy\nAmes Laboratory\nIowa State University\n50011AmesIowaUSA\n",
"David Vaknin \nDepartment of Physics and Astronomy\nAmes Laboratory\nIowa State University\n50011AmesIowaUSA\n"
] |
[
"Department of Physics and Astronomy\nAmes Laboratory\nIowa State University\n50011AmesIowaUSA",
"NCNR\nNational Institute of Standards and Technology\n20899GaithersburgMDUSA",
"Dept. of Materials Science and Engineering\nUniversity of Maryland\n20742College ParkMDUSA",
"NCNR\nNational Institute of Standards and Technology\n20899GaithersburgMDUSA",
"Department of Physics and Astronomy\nAmes Laboratory\nIowa State University\n50011AmesIowaUSA",
"Department of Physics and Astronomy\nAmes Laboratory\nIowa State University\n50011AmesIowaUSA",
"Department of Physics and Astronomy\nAmes Laboratory\nIowa State University\n50011AmesIowaUSA",
"NCNR\nNational Institute of Standards and Technology\n20899GaithersburgMDUSA",
"Dept. of Materials Science and Engineering\nUniversity of Maryland\n20742College ParkMDUSA",
"NCNR\nNational Institute of Standards and Technology\n20899GaithersburgMDUSA",
"Department of Physics and Astronomy\nAmes Laboratory\nIowa State University\n50011AmesIowaUSA",
"Department of Physics and Astronomy\nAmes Laboratory\nIowa State University\n50011AmesIowaUSA"
] |
[] |
Inelastic neutron scattering (INS) experiments were performed to investigate the spin dynamics in magnetoelectric effect (ME) LiCoPO4 single crystals. Weak dispersion was detected in the magnetic excitation spectra along the three principal crystallographic axes measured around the (0 1 0) magnetic reflection. Analysis of the data using linear spin-wave theory indicate that single-ion anisotropy in LiCoPO4 is as important as the strongest nearest-neighbor exchange coupling. Our results suggest that Co 2+ single-ion anisotropy plays an important role in the spin dynamics of LiCoPO4 and must be taken into account in understanding its physical properties. High resolution INS measurements reveal an anomalous low energy excitation that we hypothesize may be related to the magnetoelectric effect of LiCoPO4.
|
10.1103/physrevb.78.184429
|
[
"https://arxiv.org/pdf/0810.4471v1.pdf"
] | 119,205,826 |
0810.4471
|
449695e2675a6b2766b22adc6c5a439b9fa0911f
|
Spin Dynamics in the Magnetoelectric Effect LiCoPO 4 Compound
24 Oct 2008
Wei Tian
Department of Physics and Astronomy
Ames Laboratory
Iowa State University
50011AmesIowaUSA
Jiying Li
NCNR
National Institute of Standards and Technology
20899GaithersburgMDUSA
Dept. of Materials Science and Engineering
University of Maryland
20742College ParkMDUSA
Jeffrey W Lynn
NCNR
National Institute of Standards and Technology
20899GaithersburgMDUSA
Jerel L Zarestky
Department of Physics and Astronomy
Ames Laboratory
Iowa State University
50011AmesIowaUSA
David Vaknin
Department of Physics and Astronomy
Ames Laboratory
Iowa State University
50011AmesIowaUSA
Spin Dynamics in the Magnetoelectric Effect LiCoPO 4 Compound
24 Oct 2008numbers: 7510Jm7540Gb7530Et
Inelastic neutron scattering (INS) experiments were performed to investigate the spin dynamics in magnetoelectric effect (ME) LiCoPO4 single crystals. Weak dispersion was detected in the magnetic excitation spectra along the three principal crystallographic axes measured around the (0 1 0) magnetic reflection. Analysis of the data using linear spin-wave theory indicate that single-ion anisotropy in LiCoPO4 is as important as the strongest nearest-neighbor exchange coupling. Our results suggest that Co 2+ single-ion anisotropy plays an important role in the spin dynamics of LiCoPO4 and must be taken into account in understanding its physical properties. High resolution INS measurements reveal an anomalous low energy excitation that we hypothesize may be related to the magnetoelectric effect of LiCoPO4.
I. INTRODUCTION
LiCoPO 4 is an antiferromagnetic (AFM) insulator belonging to the olivine family of lithium orthophosphates that share the general chemical formula LiMPO 4 (M = Mn 2+ , Fe 2+ , Co 2+ , Ni 2+ ) with four formula units per unit cell 1,2 . These materials continue to attract much attention due to their exceptionally large magnetoelectric (ME) effect and the anomalies exhibited in the ME coefficients as a function of temperature and magnetic field 3,4,5 . To date, it remained an open question whether the anomalies observed in the ME effect of LiMPO 4 are intrinsic due to the particular local environment surrounding the transition metal ions in LiM PO 4 or due to domain formation structure. The local environment can be slightly distorted in a magnetic (electric) field by virtue of the spin-orbit coupling, giving rise to a collective ferroelectric response (magneto-ferroelastic effect) 6,7 . The recent domain structure observed by second harmonic generation (SHG) in LiCoPO 4 was attributed to coexisting AFM and ferrotoroidic domains which may play a role in giving rise to the ME effect 8 . In addition to the strong ME effect, this family of materials also exhibits intriguing magnetic properties. At low temperatures, LiMPO 4 systems undergo transitions to AFM long-range-order (LRO), adopting similar magnetic structures, differing only in spin orientation. For example, with increasing temperature, LiNiPO 4 first undergoes a first-order commensurate-incommensurate (C-IC) phase transition at T N ≈ 20.8 K changing from a co-linear AFM state to a long-range IC order state, followed by a second-order phase transition from long-range IC to short-range IC order at T IC ≈ 21.7 K 9 . On the other hand, LiMnPO 4 undergoes an AFM LRO transition at T N ≈ 34 K as well as a field induced spinflop transition 10 . Weak ferromagnetism 11 , an ME "butterfly loop" anomaly 12 , and strong magnetic anisotropy have been observed in LiCoPO 4 13 which also exhibits the largest ME coefficient 5 among its counterpart com- Only the Co 2+ magnetic ions are shown for clarity. The intraplane (solid) and inter-plane (dashed) magnetic exchange interactions considered in the spin Hamiltonian (Eq. 1) are labeled. It is assumed that the spin is oriented strictly along the crystallographic b-axis in the model calculation.
pounds, making it of particular interest to study. LiCoPO 4 crystallizes in an orthorhombic symmetry, space group Pnma (no. 62) at room temperature, with lattice parameters a = 10.093, b = 5.89, and c = 4.705Å. The structure consists of buckled CoO layers stacked along the crystallographic a-axis and the magnetic Co 2+ (S = 3/2) ions are surrounded by oxygen ions in a strongly distorted CoO 6 octahedral coordination. LiCoPO 4 develops AFM LRO at T N ≈ 21.8 K 12,14 . Earlier studies have indicated a simple two-sublattice AFM state below T N with spins aligned along the b-axis (whereas LiFePO 4 , LiMnPO 4 , and LiNiPO 4 have spins oriented along the b, a, and c-axis respectively) 2,15 . Fig. 1 illustrates the magnetic structure of LiCoPO 4 . For simplicity, only the Co 2+ magnetic ions are shown. As depicted in Fig. 1, different magnetic exchange pathways are at play in this compound. Within the buckled CoO layer, nearest-neighbor Co 2+ ions are strongly coupled (J 1 ) through Co-O-Co super-exchange interactions, while additional inplane magnetic interactions between nextnearest-neighbors (J 2 , J 3 ) are mediated through the P O 4 group. Between adjacent layers, the magnetic couplings of nearest-neighbor ions (J 4 , J 5 ) are also mediated via the P O 4 groups. The P O 4 coupling are found to be rather strong and cannot be neglected 9,16 . Due to its layered structure, LiCoPO 4 exhibits properties between two-and three-dimensional (2D and 3D) magnetic systems.
Although the physical properties of LiCoPO 4 have been studied extensively in the past, there remain a number of puzzles in this compound 17 .
Recent magnetoelectric 5,12 , magneto-optic 18 , and magnetic property 19 studies do not agree with the originally proposed co-linear AFM structure and suggest a more complex magnetic structure for LiCoPO 4 . Neutron diffraction studies suggest the moments in LiCoPO 4 in the AFM phase are not strictly aligned along the b-axis but are uniformly rotated from this axis by a small angle (∼ 4.6 • ) 14 . The observation of weak ferromagnetism and an ME "butterfly loop" anomaly further motivated studies of LiCoPO 4 both experimentally and theoretically 20,21,22 . In this paper, we report single crystal inelastic neutron scattering (INS) studies that yield the microscopic magnetic interactions in LiCoPO 4 . The data were analyzed within the linear spin wave approximation using a spin Hamiltonian explicitly including the intra-, inter-plane nearest neighbor, next-nearestneighbor exchange interactions and single-ion anisotropy, which are determined in this study.
II. EXPERIMENTAL TECHNIQUES
All measurements reported here were carried out on LiCoPO 4 single crystals. Large crystals were grown for INS experiments by a LiCl flux method similar to that reported in Ref. 23. High purity starting materials of CoCl 2 (99.999%, MV laboratory), Li 3 PO 4 (99.999% Aldrich) and LiCl (99.999%) were thoroughly ground together at a molar ratio of Li 3 PO 4 :CoCl 2 :LiCl = 1:1:1 and sealed in a Pt crucible under Ar atmosphere. Note that the reaction of Li 3 PO 4 + CoCl 2 + LiCl −→ LiCoPO 4 + 3LiCl yields a molar ratio of 1:3 between LiCoPO 4 and the flux material LiCl which we found crucial in growing large LiCoPO 4 single crystals. The mixed powder is pre-melted at 800 • C and slowly heated to 900 • C. The crucible was maintained at 900 • C for 10 hours and then slowly cooled to 640 • C at a cooling rate of 0.7 • C/hr and then furnace-cooled to room temperature. Large purple color LiCoPO 4 single crystals were obtained and extracted by dissolving LiCl in water. The crystals were characterized by X-ray diffraction measurements and found to be of pure single phase. The lattice parameters determined from our neutron diffraction mea-surements a = 10.159, b = 5.9, and c = 4.70Å at 8 K are in good agreement with prior results 14 .
Two LiCoPO 4 single crystals grown from the same batch were used for the INS experiments. Sample #1, m ≈ 0.8 g, and sample #2, m ≈ 0.4 g, were oriented in the (H K 0) and (0 K L) scattering plane, respectively. INS experiments were performed using the HB1A tripleaxis spectrometer (TAS) at the High Flux Isotope Reactor (HFIR) neutron scattering facility at Oak Ridge National Laboratory, the BT7 thermal TAS, and the SPINS cold TAS at the NIST Center for Neutron Research at the National Institute of Standards Technology (NCNR). The magnetic excitations along the (H 1 0) and (0 K 0) directions were measured using the HB1A TAS on sample #1 which was mounted on a thin aluminum disk, sealed in an aluminum sample can and kept under helium atmosphere and cooled using a closed-cycle He refrigerator. Collimations of 48 ′ -48 ′ -sample-40 ′ -68 ′ downstream from reactor to detector were used throughout the experiment. Constant wave-vector scans were performed at T = 8 K (fully ordered state) and T = 35 K (well above T N ). Spin excitations along the (0 1 L) direction were measured using the BT7 TAS at NIST on sample ing the HB1A and BT7 TAS are shown in Fig. 2. Note that error bars in this paper are statistical in origin and represent one standard deviation. Fig. 2 (a) shows the temperature dependence of the magnetic excitation measured at (0 1 0). At T = 8 K in the fully ordered phase, a single excitation with energy transfer ofhω ≈ 4.7 meV is detected. At a temperature well above T N (T = 35 K) the peak disappears demonstrating the excitation is magnetic in origin. Below T N , further detailed measurements at (0 1 0) as a function of temperature indicate no significant temperature dependence of this excitation. Fig. 2 (b) shows the T = 8 K constant wave-vector scans measured at (0 1 0) and (0 1 1.5), which typically correspond to the minimum and maximum spin wave excitations. The data clearly show that the excitation observed at hω ≈ 4.7 meV at (0 1 0) shifts to higher energy transfer hω ≈ 5.3 meV at (0 1 1.5), yielding a maximum energy shift of ∼ 0.6 meV. This indicates that the overall disper-sion along the (0 1 L) direction is modest compared to an exchange energy of kT N ∼ 2 meV. Similar behaviors were observed for the dispersions along the (0 K 0) and (H 1 0) directions. Note that data in Fig. 2 (a) shows strong scattering at 2.5 meV, the lowest data point plotted. We will show later that it is from scattering of a low energy excitation at ∼ 1.2 meV which has been observed in recent SPINS high resolution measurements.
To determine the magnetic excitation spectra along all three principal axes directions, a series of constant wavevector scans were carried out in the (H K 0) and (0 K L) scattering planes at T = 8 K around the (0 1 0) magnetic reflection. Fig. 3 depicts the ground state magnetic dispersion relations along the (H 1 0), (0 K 0), and (0 1 L) directions constructed from energy scans at constant wave-vector. We determined the peak positions assuming Gaussian peak-shapes that were fit to each of the constant wave-vector scans measured. For both the HB1A and BT7 triple-axis-spectrometers, the energy resolution at the elastic position was ∆E ≈ 1 meV. The experimental uncertainties had a significant effect in the theoretical modeling as described in the text below. The measured spectra indicate a spin wave excitation ofhω ≈ 4.7 ±0.24 meV at (0 1 0) which vanishes abruptly above T N , while modest dispersion was observed along all three principal symmetry directions, with the scale in Fig. 3 being chosen to best exhibit the dispersion that falls within a band of 0.8 meV. This relatively weak dispersion suggests an Ising-like model in LiCoPO 4 ; in a pure Ising model the magnetic excitations are completely dispersionless.
To analyze the measured spin wave dispersion curves of LiCoPO 4 using linear spin wave theory, we consider the different magnetic exchange interactions as illustrated in Fig. 1 and assume an AFM ground state with spins pointing strictly along the b-axis. Taking into account the intra-plane and inter-plane nearest-neighbor, next-nearest-neighbor interactions, and the single-ion anisotropy, the Spin Hamiltonian can be expressed in the following form:
H = i,j J ij S i · S j + i,α D α (S α i ) 2 ,(1)
where D α (α = x, y, z) represents the single-ion anisotropy along the x , y, and z-directions. In order to have the spins pointing along the z-axis in the model calculation, the Cartesian coordinate x, y, and z directions are defined to be along the crystallographic a, c, and b directions respectively. The zero point of the energy spectrum is chosen such that D z =0. Within a linear spin wave approximation, the derived spin wave dispersion from Eq. 1 is given by:
hω = A 2 − (B ± C) 2 .(2)
where A = 4S(J 1 + J 5 ) − 2S[J 2 (1 − cos(q · r 5 )) + J 3 (1 − cos(q · r 6 )) + J 4 (2 − cos(q · r 7 ) − cos(q · r 8 ))] + D x S + D y S, C = 2J 1 S(cos(q · r 1 ) + cos(q · r 2 )) + 2J 5 S(cos(q · r 3 ) + cos(q · r 4 )). and r i denotes the vectors directed between two Co 2+ The obtained microscopic interaction parameter J 1 is significantly larger than J 4 and J 5 consistent with previous observations that the magnetic behavior of LiCoPO 4 is intermediate between a 2D and 3D system. Moreover, the obtained positive value of J 1 indicates in-plane AFM nearest-neighbor coupling, whereas negative J 4 and J 5 values suggest inter-plane FM coupling along the a-axis consistent with the magnetic structure, where the magnetic unit cell is doubled along the b-and c-axes, but not along the a-axis. J 2 and J 3 have the same sign as J 1 indicating that they compete with J 1 and may cause frustration, however, they are relatively weak compared to J 1 (J 2 /J 1 ≈ 0.14, J 3 /J 1 ≈ 0.26). Both D x and D y are positive favoring a ground state with the magnetic moment along the b-axis consistent with the elastic magnetic neutron scattering results. An important result in our study is the large values of the single-ion anisotropy compared to the nearestneighbor coupling D x ∼ D y ∼ J 1 . Although strong single-ion anisotropy in LiCoPO 4 has been suggested by several models 6,27 , this study provides experimental evidence that the single-ion anisotropy is as important as the strongest magnetic exchange interaction in LiCoPO 4 . Such relatively strong anisotropy may split the S = 3/2 quartet of the Co 2+ ion into two doublets rendering the suggested Ising-like character to LiCoPO 4 14 .
B = D x S − D y S,
The "±" sign in Eq. (2) (B ± C) indicates that there are two non-degenerate spin wave branches which come directly from the different values of D x and D y . Using the obtained best-fit parameters, the calculated dispersion curves of the two branches are plotted as solid lines ("B − C" branch) and dash lines ("B + C" branch ) in Fig. 3. The two calculated spin wave branches predict a maximum separation of ∼ 0.3 meV at (0 1 0), (1 1 0), and (0 1 1). In the thermal neutron TAS measurements using HB1A and BT7 with a resolution of ∼ 1 meV, only one excitation was observed at these wave vectors.
We have two high resolution measurements using SPINS TAS, fixed E f = 5 meV with a resolution of ∼ 0.28 meV, at (1 1 0) and (0 1 0) with energy transfer up to 8 meV.
The constant wave vector scan at (0 1 0) at T = 9 K is shown in Fig. 4 (a). At (0 1 0), where the model predicts the maximum separation between these two branches, only one excitation around ∼ 4.7 meV is observed. The additional low energy excitation observed at ∼ 1.2 meV does not agree with the model and is discussed below. Our results could not resolve the two branches for two possible reasons. First, the second excitation may be very weak in intensity, and our model does not predict intensities. Second, the intrinsic linewidth of the observed excitations are broader than the resolution (∼ 1 meV) suggestive of contributions from both branches overlap and cause the broadening.
High resolution measurements on SPINS show an anomalous low energy excitation below T N that does not fit in the linear spin wave model. Fig. 4 (a) depicts the T = 9 K, 15 K, and 21 K data measured at (0 1 0). In addition to thehω ≈ 4.7 meV excitation, the T = 9 K SPINS data clearly show a low energy excitation athω ≈ 1.2 meV. The peak position of this excitation is practically temperature independent but the peak intensity decreases with increasing temperature and eventually the peak vanishes above T N . It also shows weak dispersion along all three reciprocal directions. Fig. 4 (b) shows constant wave-vector scans measured along the (0 K 0) direction at 9 K. Very weak dispersion was observed along this direction and the data show that this excita- tion weakens in intensity (significantly) with increasing K and could not be detected at large K. Similar results were obtained along the (H 1 0) and (0 1 L) directions. In order to clarify the origin of this excitation, polarized neutron scattering experiments were carried out using the BT7 TAS. As discussed in Ref. 24,25,26, coherent nuclear scattering is always non-spin-flip scattering (++) because it never causes a reversal, or spin flip, of the neutron spin direction upon scattering. On the other hand, magnetic scattering depends on the relative orientation of the neutron polarizationp and the scattering vector Q. Only those spin components which are perpendicular to the scattering vector are effective. Thus for a fully polarized neutron beam, with the horizontal field configuration,p Q, all magnetic scattering is spin-flip scattering (-+), and ideally no non-spin-flip scattering will be observed. Our polarized measurements were carried out by performing constant wave-vector scans at (0 1.2 0) at T = 7 K with (++) and (-+) configurations and a horizontal magnetic guide field at the sample position (p Q). Inset in Fig. 5 (a) was observed in (++) channel as expected. The observed weak (-+) intensity can be attributed to the finite instrumental flipping ratio which we estimate to be ∼ 1/9 by comparing the integrated intensity of the (-+) and (++) scans of the (0 2 0). The spin-flip (-+) and non-spin-flip (++) scans at (0 1.2 0) were plotted in Fig. 5 (a) betweenhω = -1.75 meV and 4 meV. Additional magnetic scattering was detected in the (-+) spin-flip channel. In order to show the peak clearly, the subtracted data, the non-spin-flip (++) data was subtracted from the spinflip (-+) data, is plotted together with the SPINS data in Fig. 5 (b). At (0 1.2 0), the SPINS data (resolution of ∼ 0.28 meV) shows an excitation centered at 1.37 ± 0.05 meV. The subtracted BT7 polarized data (resolution of ∼ 1 meV) shows a rather broad peak consistent with the SPINS data within experimental error. The polarized measurements indicate that this low energy excitation is magnetic in origin, which agrees with the temperature de-pendence measurements. However, as shown in Fig. 3, it does not fit in the current spin wave model. Attempting to analyze the combined dispersions with gaps at ∼ 1.2 meV and ∼ 4.7 meV simultaneously using Eq. 2 failed, in particular in accounting for the ∼ 1.2 meV excitation. At this time, the nature of the ∼ 1.2 meV excitation is not clear, and based on it being nearly dispersionless we can only hypothesize that it may be due to a local magnetic excitation. Further studies are necessary in order to unravel the nature of this excitation.
IV. SUMMARY
The spin dynamics of the ME compound LiCoPO 4 were determined by inelastic neutron scattering experiments.
Similar to LiNiPO 4 7 , LiMnPO 4 10 , and LiFePO 4 28 , the overall observed magnetic excitation spectra in LiCoPO 4 can be adequately described by a linear spin wave theory. Our results indicate that single-ion anisotropy is as important as the strong nearest-neighbor magnetic coupling and plays an essential role in understanding the spin structure and dynamics of LiCoPO 4 .
However, the observation of the second low energy dispersionless ∼ 1.2 meV magnetic excitation is unusual and is not contained in the spin wave Hamiltonian, suggesting that it may be closely related to the strong ME effect in LiCoPO 4 . The nature of this excitation is not understood yet and requires further detailed studies.
V. ACKNOWLEDGMENTS
FIG. 1 :
1(Color online) Magnetic unit cell of LiCoPO4 displaced (0.25 0.25 0) r.l.u compared to the atomic unit cell.
#2. A fixed final energy of E f = 14.7 meV and a open-50 ′ -sample-50 ′ -open collimation were used with pyrolytic graphite (PG(0 0 2)) analyzer crystals in flat mode. High resolution INS measurements were carried out on sample #1 using the SPINS TAS with a fixed final energy, E f = 5 meV. A collimation of open-80-sample-80 ′ -open was used with a cold Be filter in the scattering beam. Polarized neutron scattering experiments were also performed using the BT7 TAS with a fixed final energy of E f = 13.7 meV to clarify the nature of the ∼ 1.2 meV low energy excitation. The polarization analysis technique as applied to this study is discussed in Ref. 24,25,26. 3 He spin filters (polarizer) are mounted before and after the sample with a spin flipper in the incident beam. The sample is maintained in a horizontal or vertical magnetic guide field of ∼5 Oe such that the neutron polarization p is parallel to the momentum transfer Q,p Q when a horizontal field is applied at the sample position, orp⊥Q when a vertical magnetic field is applied. With the spin flipper off, we measure the (++) non-spin-flip scattering. On the other hand, with the spin flipper on, the (-+) spin-flip scattering is measured. Sample #2 oriented in the (0 K L) scattering plane and a open-50 ′ -sample-80 ′open collimation were used throughout the polarization measurements. Constant wave-vector scans were carried out with both (++) and (-+) configurations. All measurement results have been normalized to a beam monitor count.III. EXPERIMENTAL RESULTS AND MODELINGRepresentative constant wave-vector scans with energy transfer betweenhω = 2.5 meV and 8 meV measured us-online) Representative constant wave-vector scans plotted as scattering intensity versus energy transfer betweenhω = 2.5 meV and 8 meV. (a). Temperature dependence of the ∼ 4.7 meV excitation measured at (0 1 0) at T = 8 K and 35 K using the HB1A TAS. (b). T = 8 K constant wave-vector scans using the BT7 TAS measured at (0 1 0) and (0 1 1.5) showing a single magnetic excitation that exhibits modest but unambiguous dispersion along the (0 1 L) direction. Intensities were normalized to the incident neutron flux by counting against neutron monitor counts.
FIG. 3 :
3(Color online) Spin-wave dispersion curves along three reciprocal directions constructed from a series constant wavevector scans measured at T = 8 K. Data points are obtained based upon a Gaussian peak approximation. The solid and dash lines are calculations based upon a global fit to the linear spin wave approximation theory as described in the text.
ions r 1
1= (0, b/2, c/2), r 2 = (0, b/2, −c/2),r 3 = (a/2, b/2, 0), r 4 = (a/2, −b/2, 0), r 5 = (0, b, 0), r 6 = (0, 0, c), r 7 = (a/2, 0, c/2), r 8 = (a/2, 0, −c/2).Non-linear-least-squares fits of the spin-wave dispersion expressed by Eq. 2 to the observed magnetic spectra yields:J 1 = 0.743 ± 0.187 meV, J 2 = 0.105± 0.159 meV, J 3 = 0.194± 0.131 meV, J 4 = −0.163± 0.08 meV, J 5 = −0.181± 0.125 meV, D x = 0.718± 0.192 meV, D y = 0.802± 0.208 meV.
FIG. 4 :
4(Color online) SPINS high resolution measurements of LiCoPO4. (a) Constant wave-vector scans measured at (0 1 0) at T = 9 K, 15 K, and 21 K indicating a second low energy excitation athω ∼ 1.2 meV below TN . (b) Weak dispersion observed in the constant wave-vector scans measured along the (0 K 0) direction at T = 9 K.
first compares the (0 2 0) nuclear Bragg scattering measured in non-spin-flip (++) and spin-flip (-+) configurations. Strong intensity online) BT7 polarized neutron data measured at (0 1.2 0) in the horizontal field configuration,p Q. (a) Comparison of spin-flip (-+) and non-spin-flip (++) scattering measured at T = 7 K indicating the magnetic origin of the second low energy excitation. Inset: Spin-flip (-+) and non-spin-flip (++) scattering of the nuclear reflection (0 2 0). (b) Comparison of SPINS high resolution data and the BT7 subtracted polarized data, the non-spin-flip (++) data was subtracted from the spin-flip (-+) data.
We acknowledge discussions with T. Barnes
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[] |
[
"Learning Transformations for Clustering and Classification Learning Transformations for Clustering and Classification",
"Learning Transformations for Clustering and Classification Learning Transformations for Clustering and Classification"
] |
[
"Qiang Qiu [email protected] \nDepartment of Electrical and Computer Engineering\nDepartment of Electrical and Computer Engineering\nDepartment of Computer Science\nDepartment of Biomedical Engineering Duke University Durham\nDuke University Durham\n27708, 27708NC, NCUSA, USA\n",
"Guillermo Sapiro [email protected] \nDepartment of Electrical and Computer Engineering\nDepartment of Electrical and Computer Engineering\nDepartment of Computer Science\nDepartment of Biomedical Engineering Duke University Durham\nDuke University Durham\n27708, 27708NC, NCUSA, USA\n"
] |
[
"Department of Electrical and Computer Engineering\nDepartment of Electrical and Computer Engineering\nDepartment of Computer Science\nDepartment of Biomedical Engineering Duke University Durham\nDuke University Durham\n27708, 27708NC, NCUSA, USA",
"Department of Electrical and Computer Engineering\nDepartment of Electrical and Computer Engineering\nDepartment of Computer Science\nDepartment of Biomedical Engineering Duke University Durham\nDuke University Durham\n27708, 27708NC, NCUSA, USA"
] |
[] |
A low-rank transformation learning framework for subspace clustering and classification is here proposed. Many high-dimensional data, such as face images and motion sequences, approximately lie in a union of low-dimensional subspaces. The corresponding subspace clustering problem has been extensively studied in the literature to partition such highdimensional data into clusters corresponding to their underlying low-dimensional subspaces. However, low-dimensional intrinsic structures are often violated for real-world observations, as they can be corrupted by errors or deviate from ideal models. We propose to address this by learning a linear transformation on subspaces using nuclear norm as the modeling and optimization criteria. The learned linear transformation restores a low-rank structure for data from the same subspace, and, at the same time, forces a maximally separated structure for data from different subspaces. In this way, we reduce variations within the subspaces, and increase separation between the subspaces for a more robust subspace clustering. This proposed learned robust subspace clustering framework significantly enhances the performance of existing subspace clustering methods. Basic theoretical results here presented help to further support the underlying framework. To exploit the low-rank structures of the transformed subspaces, we further introduce a fast subspace clustering technique, which efficiently combines robust PCA with sparse modeling. When class labels are present at the training stage, we show this low-rank transformation framework also significantly enhances classification performance. Extensive experiments using public datasets are presented, showing that the proposed approach significantly outperforms state-of-the-art methods for subspace clustering and classification. The learned low cost transform is also applicable to other classification frameworks.
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10.5555/2789272.2789279
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[
"https://arxiv.org/pdf/1309.2074v2.pdf"
] | 287,318 |
1309.2074
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2742a61d32053761bcc14bd6c32365bfcdbefe35
|
Learning Transformations for Clustering and Classification Learning Transformations for Clustering and Classification
Qiang Qiu [email protected]
Department of Electrical and Computer Engineering
Department of Electrical and Computer Engineering
Department of Computer Science
Department of Biomedical Engineering Duke University Durham
Duke University Durham
27708, 27708NC, NCUSA, USA
Guillermo Sapiro [email protected]
Department of Electrical and Computer Engineering
Department of Electrical and Computer Engineering
Department of Computer Science
Department of Biomedical Engineering Duke University Durham
Duke University Durham
27708, 27708NC, NCUSA, USA
Learning Transformations for Clustering and Classification Learning Transformations for Clustering and Classification
Editor: *Subspace clusteringclassificationlow-rank transformationnuclear normfeature learning
A low-rank transformation learning framework for subspace clustering and classification is here proposed. Many high-dimensional data, such as face images and motion sequences, approximately lie in a union of low-dimensional subspaces. The corresponding subspace clustering problem has been extensively studied in the literature to partition such highdimensional data into clusters corresponding to their underlying low-dimensional subspaces. However, low-dimensional intrinsic structures are often violated for real-world observations, as they can be corrupted by errors or deviate from ideal models. We propose to address this by learning a linear transformation on subspaces using nuclear norm as the modeling and optimization criteria. The learned linear transformation restores a low-rank structure for data from the same subspace, and, at the same time, forces a maximally separated structure for data from different subspaces. In this way, we reduce variations within the subspaces, and increase separation between the subspaces for a more robust subspace clustering. This proposed learned robust subspace clustering framework significantly enhances the performance of existing subspace clustering methods. Basic theoretical results here presented help to further support the underlying framework. To exploit the low-rank structures of the transformed subspaces, we further introduce a fast subspace clustering technique, which efficiently combines robust PCA with sparse modeling. When class labels are present at the training stage, we show this low-rank transformation framework also significantly enhances classification performance. Extensive experiments using public datasets are presented, showing that the proposed approach significantly outperforms state-of-the-art methods for subspace clustering and classification. The learned low cost transform is also applicable to other classification frameworks.
Introduction
High-dimensional data often have a small intrinsic dimension. For example, in the area of computer vision, face images of a subject (Basri and Jacobs (February 2003), Wright et al. (2009)), handwritten images of a digit (Hastie and Simard (1998)), and trajectories of a moving object (Tomasi and Kanade (1992)) can all be well-approximated by a lowdimensional subspace of the high-dimensional ambient space. Thus, multiple class data often lie in a union of low-dimensional subspaces. The ubiquitous subspace clustering problem is to partition high-dimensional data into clusters corresponding to their underlying subspaces.
Standard clustering methods such as k-means in general are not applicable to subspace clustering. Various methods have been recently suggested for subspace clustering, such as Sparse Subspace Clustering (SSC) (Elhamifar and Vidal (2013)) (see also its extensions and analysis in Liu et al. (2010); Soltanolkotabi and Candes (2012); Soltanolkotabi et al. (2013); Wang and Xu (2013)), Local Subspace Affinity (LSA) (Yan and Pollefeys (2006)), Local Best-fit Flats (LBF) (Zhang et al. (2012)), Generalized Principal Component Analysis (Vidal et al. (2003)), Agglomerative Lossy Compression (Ma et al. (2007)), Locally Linear Manifold Clustering (Goh and Vidal (2007)), and Spectral Curvature Clustering (Chen and Lerman (2009)). A recent survey on subspace clustering can be found in Vidal (2011).
Low-dimensional intrinsic structures, which enable subspace clustering, are often violated for real-world data. For example, under the assumption of Lambertian reflectance, Basri and Jacobs (February 2003) show that face images of a subject obtained under a wide variety of lighting conditions can be accurately approximated with a 9-dimensional linear subspace. However, real-world face images are often captured under pose variations; in addition, faces are not perfectly Lambertian, and exhibit cast shadows and specularities (Candès et al. (2011)). Therefore, it is critical for subspace clustering to handle corrupted underlying structures of realistic data, and as such, deviations from ideal subspaces.
When data from the same low-dimensional subspace are arranged as columns of a single matrix, the matrix should be approximately low-rank. Thus, a promising way to handle corrupted data for subspace clustering is to restore such low-rank structure. Recent efforts have been invested in seeking transformations such that the transformed data can be decomposed as the sum of a low-rank matrix component and a sparse error one (Peng et al. (2010); Shen and Wu (2012); Zhang et al. (2011)). Peng et al. (2010) and Zhang et al. (2011) are proposed for image alignment (see Kuybeda et al. (2013) for the extension to multiple-classes with applications in cryo-tomograhy), and Shen and Wu (2012) is discussed in the context of salient object detection. All these methods build on recent theoretical and computational advances in rank minimization.
In this paper, we propose to improve subspace clustering and classification by learning a linear transformation on subspaces using matrix rank, via its nuclear norm convex surrogate, as the optimization criteria. The learned linear transformation recovers a low-rank structure for data from the same subspace, and, at the same time, forces a maximally separated structure for data from different subspaces (actually high nuclear norm, which as discussed later, improves the separation between the subspaces). In this way, we reduce variations within the subspaces, and increase separations between the subspaces for more accurate subspace clustering and classification.
For example, as shown in Fig. 1, after faces are detected and aligned, e.g., using Zhu and Ramanan (June 2012), our approach learns linear transformations for face images to restore for the same subject a low-dimensional structure. By comparing the last row to the first row in Fig. 1, we can easily notice that faces from the same subject across different poses are more visually similar in the new transformed space, enabling better face clustering and classification across pose. This paper makes the following main contributions:
• Subspace low-rank transformation (LRT) is introduced and analyzed in the context of subspace clustering and classification; • A Learned Robust Subspace Clustering framework (LRSC) is proposed to enhance existing subspace clustering methods; • A discriminative low-rank (nuclear norm) transformation approach is proposed to reduce the variation within the classes and increase separations between the classes for improved classification; • We propose a specific fast subspace clustering technique, called Robust Sparse Subspace Clustering (R-SSC), by exploiting low-rank structures of the learned transformed subspaces; • We discuss online learning of subspace low-rank transformation for big data;
• We demonstrate through extensive experiments that the proposed approach significantly outperforms state-of-the-art methods for subspace clustering and classification.
The proposed approach can be considered as a way of learning data features, with such features learned in order to reduce within-class rank (nuclear norm), increase between class separation, and encourage robust subspace clustering. As such, the framework and criteria here introduced can be incorporated into other data classification and clustering problems.
In Section 2, we formulate and analyze the low-rank transformation learning problem. In sections 3 and 4, we discuss the low-rank transformation for subspace clustering and classification respectively. Experimental evaluations are given in Section 5 on public datasets commonly used for subspace clustering evaluation. Finally, Section 6 concludes the paper.
Learning Low-rank Transformations (LRT)
Let {S c } C c=1 be C n-dimensional subspaces of R d (not all subspaces are necessarily of the same dimension, this is only here assumed to simplify notation). Given a data set Y = {y i } N i=1 ⊆ R d , with each data point y i in one of the C subspaces, and in general the data arranged as columns of Y . Y c denotes the set of points in the c-th subspace S c , points arranged as columns of the matrix Y c .
As data points in Y c lie in a low-dimensional subspace, the matrix Y c is expected to be low-rank, and such low-rank structure is critical for accurate subspace clustering. However, as discussed above, this low-rank structure is often violated for real data.
Our proposed approach is to learn a global linear transformation on subspaces. Such linear transformation restores a low-rank structure for data from the same subspace, and, at the same time, encourages a maximally separated structure for data from different subspaces. In this way, we reduce the variation within the subspaces and introduce separations between the subspaces for more robust subspace clustering or classification.
Preliminary Pedagogical Formulation using Rank
We first assume the data cluster labels are known beforehand for training purposes, assumption to be removed when discussing the full clustering approach in Section 3. We adopt In the second row, the input faces are first detected and aligned, e.g., using the method in Zhu and Ramanan (June 2012). Pose models defined in Zhu and Ramanan (June 2012) enable an optional crop-andflip step to retain the more informative side of a face in the third row. Our proposed approach learns linear transformations for face images to restore for the same subject a low-dimensional structure as shown in the last row. By comparing the last row to the first row, we can easily notice that faces from the same subject across different poses are more visually similar in the new transformed space, enabling better face clustering or recognition across pose (note that the goal is clustering/recognition and not reconstruction). matrix rank as the key learning criterion (presented here first for pedagogical reasons, to be later replaced by the nuclear norm), and compute one global linear transformation on all subspaces as
arg T min C c=1 rank(TY c ) − rank(TY), s.t.||T|| 2 = γ,(1)
where T ∈ R d×d is one global linear transformation on all data points (we will later discuss then T's dimension is less than d), ||·|| 2 denotes the matrix induced 2-norm, and γ is a positive constant. Intuitively, minimizing the first representation term C c=1 rank(TY c ) encourages a consistent representation for the transformed data from the same subspace; and minimizing the second discrimination term −rank(TY) encourages a diverse representation for transformed data from different subspaces (we will later formally discuss that the convex surrogate nuclear norm actually has this desired effect). The normalization condition ||T|| 2 = γ prevents the trivial solution T = 0.
We now explain that the pedagogical formulation in (1) using rank is however not optimal to simultaneously reduce the variation within the same class subspaces and introduce separations between the different class subspaces, motivating the use of the nuclear norm not only for optimization reasons but for modeling ones as well. Let A and B be matrices of the same dimensions (standing for two classes Y 1 and Y 2 respectively), and [A, B] (standing for Y) be the concatenation of A and B, we have (Marsaglia and Styan (1972))
rank([A, B]) ≤ rank(A) + rank(B),(2)
with equality if and only if A and B are disjoint, i.e., they intersect only at the origin (often the analysis of subspace clustering algorithms considers disjoint spaces, e.g., Elhamifar and Vidal (2013)). It is easy to show that (2) can be extended for the concatenation of multiple matrices,
rank([Y 1 , Y 2 , Y 3 , · · · , Y C ]) ≤ rank(Y 1 ) + rank([Y 2 , Y 3 , · · · , Y C ]) (3) ≤ rank(Y 1 ) + rank(Y 2 ) + rank([Y 3 , · · · , Y C ])
. . .
≤ C c=1 rank(Y c ),
with equality if matrices are independent. Thus, for (1), we have
C c=1 rank(TY c ) − rank(TY) ≥ 0,(4)
and the objective function (1) reaches the minimum 0 if matrices are independent after applying the learned transformation T. However, independence does not infer maximal separation, an important goal for robust clustering and classification. For example, two lines intersecting only at the origin are independent regardless of the angle in between, and they are maximally separated only when the angle becomes π 2 . With this intuition in mind, we now proceed to describe our proposed formulation based on the nuclear norm.
Problem Formulation using Nuclear Norm
Let ||A|| * denote the nuclear norm of the matrix A, i.e., the sum of the singular values of A. The nuclear norm ||A|| * is the convex envelop of rank(A) over the unit ball of matrices Fazel (2002). As the nuclear norm can be optimized efficiently, it is often adopted as the best convex approximation of the rank function in the literature on rank optimization (see, e.g., Candès et al. (2011) and Recht et al. (2010)).
One factor that fundamentally affects the performance of subspace clustering and classification algorithms is the distance between subspaces. An important notion to quantify the distance (separation) between two subspaces S i and S j is the smallest principal angle θ ij (Miao and Ben-Israel (1992), Elhamifar and Vidal (2013)), which is defined as
θ ij = min u∈S i ,v∈S j arccos u v ||u|| 2 ||v|| 2 ,(5)
Note that θ ij ∈ [0, π 2 ]. We replace the rank function in (1) with the nuclear norm, (6) with the nuclear norm as the key criterion. Three subspaces in R 3 are denoted as A(red), B(blue), C(green). We denote the angle between subspaces A and B as θ AB (and analogous for the other pairs of subspaces). Using (6), we transform A, B, C in (a),(c),(e) to (b),(d),(f) respectively (in the first row the subspace C is empty, being this basically a two dimensional example). Data points in (e) are associated with random noises ∼ N (0, 0.01). We denote the root mean square deviation of points in A from the true subspace as A (and analogous for the other subspaces). We observe that the learned transformation T maximizes the distance between every pair of subspaces towards π 2 , and reduces the deviation of points from the true subspace when noise is present, note how the individual subspaces nuclear norm is significantly reduced. Note that, in (c) and (d), we have the same rank values rank(A) = 1, rank(B) = 1, rank([A, B]) = 2, but different nuclear norm values, manifesting the improved between-subspaces separation.
arg T min C c=1 ||TY c || * − ||TY|| * , s.t.||T|| 2 = γ.(6)
The normalization condition ||T|| 2 = γ prevents the trivial solution T = 0. Without loss of generality, we set γ = 1 unless otherwise specified. However, understanding the effects of adopting a different normalization here is interesting and is the subject of future research. Throughout this paper we keep this particular form of the normalization which was already proven to lead to excellent results.
It is important to note that (6) is not simply a relaxation of (1). Not only the replacement of the rank by the nuclear norm is critical for optimization considerations in reducing the variation within same class subspaces, but as we show next, the learned transformation T using the objective function (6) also maximizes the separation between different class subspaces (a missing property in (1)), leading to improved clustering and classification performance.
We start by presenting some basic norm relationships for matrices and their corresponding concatenations. It is easy to see that theorems 1 and 2 can be extended for the concatenation of multiple matrices. Thus, for (6), we have,
C c=1 ||TY c || * − ||TY|| * ≥ 0.(7)
Based on (7) and Theorem 2, the proposed objective function (6) reaches the minimum 0 if the column spaces of every pair of matrices are orthogonal after applying the learned transformation T; or equivalently, (6) reaches the minimum 0 when the separation between every pair of subspaces is maximized after transformation, i.e., the smallest principal angle between subspaces equals π 2 . Note that such improved separation is not obtained if the rank is used in the second term in (6), thereby further justifying the use of the nuclear norm instead.
We have then, both intuitively and theoretically, justified the selection of the criteria (6) for learning the transform T. We now illustrate the properties of the learned transformation T using synthetic examples in Fig. 2 (real examples are presented in Section 5). Here we adopt a projected subgradient method described in Appendix C (though other modern nuclear norm optimization techniques could be considered, including recent real-time formulations Sprechmann et al. (2012)) to search for the transformation matrix T that minimizes (6). As shown in Fig. 2, the learned transformation T via (6) maximizes the separation between every pair of subspaces towards π 2 , and reduces the deviation of the data points to the true subspace when noise is present.
Note that, comparing Fig. 2c to Fig.2d, the learned transformation using (6) maximizes the angle between subspaces, and the nuclear norm changes from |
Comparisons with other Transformations
For independent subspaces, a transformation that renders them pairwise orthogonal can be obtained in a closed-form as follows: we take a basis U c for the column space of Y c for each subspace, form a matrix U = [U 1 , ..., U C ], and then obtain the orthogonalizing transformation as T = (U U) −1 U . To further elaborate the properties of our learned transformation, using synthetic examples, we compare with the closed-form orthogonalizing transformation in Fig. 3 and with linear discriminant analysis (LDA) in Fig. 4.
Two intersecting planes are shown in Fig. 3a. Though subspaces here are neither independent nor disjoint, the closed-form orthogonalizing transformation still significantly increases the angle between the two planes towards π 2 in Fig. 3b (note that the angle for the common line here is always 0). Note also that the closed-form orthogonalizing transformation is of size r × d, where r is the sum of the dimension of each subspace, and we plot just the first 3 dimensions for visualization. Comparing to the orthogonalizing transformation, our leaned transformation in Fig. 3c introduces similar subspace separation, but enables significantly reduced within subspace variations, indicated by the decreased nuclear norm values (close to 1). The same set of experiments with different samples per subspace are shown in the second row of Fig. 3. Our formulation in (6) not only maximizes the separations between the different classes subspaces, but also simultaneously reduces the variations within the same class subspaces.
Our learned transformation shares a similar methodology with LDA, i.e., minimizing intra-class variation and maximizing inter-class separation. Two classes Y + and Y − are shown in Fig. 4a, each class consisting of two lines. Our learned transformation in Fig. 4c shows smaller intra-class variation than LDA in Fig. 4b by merging two lines in each class, and simultaneously maximizes the angle between two classes towards π 2 (such two-class clustering and classification is critical for example for trees-based techniques Qiu and Sapiro (April, 2014)). Note that we usually use LDA to reduce the data dimension to the number of classes minus 1; however, to better emphasize the distinction, we learn a (d − 1) × d sized transformation matrix using both methods. The closed-form orthogonalizing transformation discussed above also gives higher intra-class variations as |Y + | * = 1.45 and |Y + | * = 1.68. Fig. 4d shows an example of two non-linearly separable classes, i.e., two intersecting planes, which cannot be improved by LDA, as shown in Fig. 4e. However, our learned transformation in Fig. 4f prepares the data to be separable using subspace clustering. As shown in Qiu and Sapiro (April, 2014), the property demonstrated above makes our learned transformation a better learner than LDA in a binary classification tree.
Lastly, we generated an interesting disjoint case: we consider three lines A, B and C on the same plane that intersect at the origin; the angles between them are θ AB = 0.08, The closed-form orthogonalizing transformation significantly increase the angle between the two planes towards π 2 in (b). Our leaned transformation in (c) introduces similar subspace separation, but simultaneously enables significantly reduced within subspace variation, indicated by the smaller nuclear norm values (close to 1). The same set of experiments with 75 points per subspace are shown in the second row.
θ BC = 0.08, and θ AC = 0.17. As the closed-form orthogonalizing approach is valid for independent subspaces, it fails by producing θ AB = 0.005, θ BC = 0.005, θ BC = 0.01. Our framework is not limited to that, even if additional theoretical foundations are yet to come. After our learned transformation, we have θ AB = 1.20, θ BC = 1.20, and θ AC = 0.75. We can make two immediate observations: First, all angles are significantly increased within the Original subspaces . Two classes Y + and Y − are shown in (a), each class consisting of two lines. We notice that our learned transformation (c) shows smaller intra-class variation than LDA in (b) by merging two lines in each class, and simultaneously maximizes the angle between two classes towards π 2 (such two-class clustering and classification is critical for example for trees-based techniques Qiu and Sapiro (April, 2014)). (d) shows an example of two non-linearly separable classes, i.e., two intersecting planes, which cannot be improved by LDA in (e). However, our learned transformation in (f) prepares data to be separable using subspace clustering.
(a) Two classes {Y+,Y−}, Y+ = {A(blue), B(cyan)}, Y− = {C(yellow), D(red)}, θAB = 1.1, θAC = 1.1, θAD = 1.1, θBC = 1.3, θBD = 1.4, θCD = 0.5 , |Y+| * = 1.58, |Y−| * = 1.27. −4 −3 −2 −1 0 1 2 3 4 −1 −0.
valid range of [0, π 2 ]. Second, θ AB +θ BC +θ AC = π (we made the same two observations while repeating the experiments with different subspace angles). Though at this point we have no clean interpretation about how those angles are balanced when pair-wise orthogonality is not possible, we strongly believe that some theories are behind the above persistent observations and we are currently exploring this.
Discussions about Other Matrix Norms
We now discuss the advantages of replacing the rank function in (1) with the nuclear norm over other (popular) matrix norms, e.g., the induced 2-norm and the Frobenius norm.
Proposition 3 Let A and B be matrices of the same row dimensions, and [A, B] be the concatenation of A and B, we have
||[A, B]|| 2 ≤ ||A|| 2 + ||B|| 2 ,
with equality if at least one of the two matrices is zero.
Proposition 4 Let A and B be matrices of the same row dimensions, and [A, B] be the concatenation of A and B, we have
||[A, B]|| F ≤ ||A|| F + ||B|| F ,
with equality if and only if at least one of the two matrices is zero.
We choose the nuclear norm in (6) for two major advantages that are not so favorable in other (popular) matrix norms:
• The nuclear norm is the best convex approximation of the rank function Fazel (2002), which helps to reduce the variation within the subspaces (first term in (6)); • The objective function (6) is optimized when the distance between every pair of subspaces is maximized after transformation, which helps to introduce separations between the subspaces.
Note that (1), which is based on the rank, reaches the minimum when subspaces are independent but not necessarily maximally distant. Propositions 3 and 4 show that the property of the nuclear norm in Theorem 1 holds for the induced 2-norm and the Frobenius norm. However, if we replace the rank function in (1) with the induced 2-norm norm or the Frobenius norm, the objective function is minimized at the trivial solution T = 0, which is prevented by the normalization condition ||T|| 2 = γ (γ > 0).
Online Learning Low-rank Transformations
When data Y is big, we use an online algorithm to learn the low-rank transformation T:
• We first randomly partition the data set Y into B mini-batches; • Using mini-batch subgradient descent, a variant of stochastic subgradient descent, the subgradient in (16) in Appendix C is approximated by a sum of subgradients obtained from each mini-batch of samples,
T (t+1) = T (t) − ν B b=1 ∆T b ,(8)
where ∆T b is obtained from (17) Appendix C using only data points in the b-th mini-batch; • Starting with the first mini-batch, we learn the subspace transformation T b using data only in the b-th mini-batch, with T b−1 as warm restart.
Subspace Transformation with Compression
Given data Y ⊆ R d , so far, we considered a square linear transformation T of size d × d. If we devise a "fat" linear transformation T of size r × d, where (r < d), we enable dimension reduction along with transformation. This connects the proposed framework with the literature on compressed sensing, though the goal here is to learn a "sensing" matrix T for subspace classification and not for reconstruction Carson et al. (2012). The nuclear-norm minimization provides a new metric for such compressed sensing design (or compressed feature learning) paradigm. Results with this reduced dimensionality will be presented in Section 5.
Subspace Clustering using Low-rank Transformations
We now move from classification, where we learned the transform from training labeled data, to clustering, where no training data is available. In particular, we address the subspace clustering problem, meaning to partition the data set Y into C clusters corresponding to their underlying subspaces. We first present a general procedure to enhance the performance of existing subspace clustering methods in the literature. Then we further propose a specific fast subspace clustering technique to fully exploit the low-rank structure of (learned) transformed subspaces.
A Learned Robust Subspace Clustering (LRSC) Framework
In clustering tasks, the data labeling is of course not known beforehand in practice. The proposed algorithm, Algorithm 1, iterates between two stages: In the first assignment stage, we obtain clusters using any subspace clustering methods, e.g., SSC (Elhamifar and Vidal (2013)), LSA (Yan and Pollefeys (2006)), LBF (Zhang et al. (2012)). In particular, in this paper we often use the new improved technique introduced in Section 3.2. In the second update stage, based on the current clustering result, we compute the optimal subspace transformation that minimizes (6). The algorithm is repeated until the clustering assignments stop changing.
The LRSC algorithm is a general procedure to enhance the performance of any subspace clustering methods (part of the beauty of the proposed model is that it can be applied to any such algorithm, and even beyond Qiu and Sapiro (April, 2014)). We don't enforce an overall objective function at the present form for such versatility purpose.
To study convergence, one way is to adopt the subspace clustering method for the LRSC assignment step by optimizing the same LRSC update criterion (6): given the cluster assignment and the transformation T at the current LRSC iteration, we take a point y i out of its current cluster (keep the rest assignments no change) and place it into a cluster Y c that minimize C c=1 ||TY c || * . We iteratively perform this for all points, and then update T using current T as warm restart. In this way, we decrease (or keep) the overall objective function (6) after each LRSC iteration. However, the above approach is computational expensive and only allow one specific subspace clustering method. Thus, in the present implementation, an overall objective function of the type that the LRSC algorithm optimizes can take a form such as,
arg T,{Sc} C c=1 min C c=1 y i ∈Sc ||Ty i − P TYc Ty i || 2 2 + λ[ C c=1 ||TY c || * − ||TY|| * ], s.t.||T|| 2 = γ,(9)
where Y c denotes the set of points y i in the c-th subspace S c , and P TYc denotes the projection onto TY c . The LRSC iterative algorithm optimize (9) through alternative minimization (with a similar form as the popular k-means, but with a different data model and with the learned transform). While formally studying its convergence is the subject of future research, the experimental validation presented already demonstrates excellent performance, with LRSC just one of the possible applications of the proposed learned transform.
In all our experiments, we observe significant clustering error reduction in the first few LRSC iterations, and the proposed LRSC iterations enable significantly cleaner subspaces for all subspace clustering benchmark data in the literature. The intuition behinds the observed empirical convergence is that the update step in each LRSC iteration decreases the second term in (9) to a small value close to 0 as discussed in Section 2; at the same time, the updated transformation tends to reduce the intra-subspace variation, which further reduces the first cluster deviation term in (9) even with assignments derived from various subspace clustering methods.
Input: A set of data points Y = {yi} N i=1 ⊆ R d in a union of C subspaces. Output: A partition of Y into C disjoint clusters {Yc} C c=1
based on underlying subspaces. begin 1. Initial a transformation matrix T as the identity matrix ;
repeat Assignment stage: 2. Assign points in TY to clusters with any subspace clustering methods, e.g., the proposed R-SSC;
Update stage: 3. Obtain transformation T by minimizing (6) based on the current clustering result ; until assignment convergence;
4. Return the current clustering result {Yc} C c=1 ; end Algorithm 1: Learning a robust subspace clustering (LRSC) framework.
Robust Sparse Subspace Clustering (R-SSC)
Though Algorithm 1 can adopt any subspace clustering methods, to fully exploit the lowrank structure of the learned transformed subspaces, we further propose the following specific technique for the clustering step in the LRSC framework, called Robust Sparse Subspace Clustering (R-SSC):
1. For the transformed subspaces, we first recover their low-rank representation L by performing a low-rank decomposition (10), e.g., using RPCA (Candès et al. (2011)), 1 arg L,S min ||L|| * + β||S|| 1 s.t. TY = L + S.
2. Each transformed point Ty i is then sparsely decomposed over L,
arg x i min Ty i − Lx i 2 2 s.t. x i 0 ≤ K,(11)
where K is a predefined sparsity value (K > d). As explained in Elhamifar and Vidal (2013), a data point in a linear or affine subspace of dimension d can be written as a linear or affine combination of d or d + 1 points in the same subspace. Thus, if we represent a point as a linear or affine combination of all other points, a sparse linear or affine combination can be obtained by choosing d or d + 1 nonzero coefficients. 3. As the optimization process for (11) is computationally demanding, we further simplify (11) using Local Linear Embedding (Roweis and Saul (2000), Wang et al. (2010)). Each transformed point Ty i is represented using its K Nearest Neighbors (NN) in L, which are denoted as L i ,
arg x i min Ty i − L i x i 2 2 s.t. x i 1 = 1.(12)
LetL i = L i − 1Ty T i . x i can then be efficiently obtained in closed form,
x i =L iL T i \ 1,
where x = A \ B solves the system of linear equations Ax = B. As suggested in Roweis and Saul (2000), if the correlation matrixL iL T i is nearly singular, it can be conditioned by adding a small multiple of the identity matrix. From experiments, we observe this simplification step dramatically reduces the running time, without sacrificing the accuracy. 4. Given the sparse representation x i of each transformed data point Ty i , we denote the sparse representation matrix as X = [x 1 . . . x N ]. It is noted that x i is written as an N -sized vector with no more than K << N non-zero values (N being the total number of data points). The pairwise affinity matrix is now defined as W = |X| + |X T |, and the subspace clustering is obtained using spectral clustering (Luxburg (2007)).
Based on experimental results presented in Section 5, the proposed R-SSC outperforms state-of-the-art subspace clustering techniques, in both accuracy and running time, e.g., about 500 times faster than the original SSC using the implementation provided in Elhamifar and Vidal (2013). Performance is further enhanced when R-SCC is used as an internal step of LRSC in Algorithm 1.
1. Note that while the learned transform T encourages low-rank in each sub-space, outliers might still exists. Moreover, during the iterations in Algorithm 1, the intermediate learned T is not yet the desired one. This justifies the incorporation of this further low-rank decomposition.
Classification using Single or Multiple Low-rank Transformations
In Section 2, learning one global transformation over all classes has been discussed, and then incorporated into a clustering framework in Section 3. The availability of data labels for training enables us to consider instead learning individual class-based linear transformation. The problem of class-based linear transformation learning can be formulated as (13).
arg {Tc} C c=1 min C c=1 [||T c Y c || * − λ||T c Y ¬c || * ],(13)
where T c ∈ R d×d denotes the transformation for the c-th class, Y ¬c = Y \ Y c denotes all data except the c-th class, and λ is a positive balance parameter.
When a global transformation matrix T is learned, we can perform classification in the transformed space by simply considering the transformed data TY as the new features. For example, when a Nearest Neighbor (NN) classifier is used, a testing sample y uses Ty as the feature and searches for nearest neighbors among TY.
To fully exploit the low-rank structure of the transformed data, we propose to perform classification through the following procedure:
• For the c-th class, we first recover its low-rank representation L c by performing lowrank decomposition (14), e.g., using RPCA (Candès et al. (2011)): 2 arg Lc,Sc
min ||L c || * + β||S c || 1 s.t. TY c = L c + S c .(14)
• Each testing image y will then be assigned to the low-rank subspace L c that gives the minimal reconstruction error through sparse decomposition (15), e.g., using OMP (Pati et al. (Nov. 1993)),
arg x min Ty − L i x 2 2 s.t. x 0 ≤ T,(15)
where T is a predefined sparsity value.
When class-based transformations {T c } C c=1 are learned, we perform recognition in a similar way. However, now we apply all the learned transforms T c to each testing data point and then pick the best one using the same criterion of minimal reconstruction error through sparse decomposition (15).
Experimental Evaluation
This section first presents experimental evaluations on subspace clustering using three public datasets (standard benchmarks): the MNIST handwritten digit dataset, the Extended YaleB face dataset (Georghiades et al. (2001)) and the Hopkins 155 database of motion segmentation. The MNIST dataset consists of 8-bit grayscale handwritten digit images of "0" through "9" and 7000 examples for each class. The Extended YaleB face dataset contains 38 subjects with near frontal pose under 64 lighting conditions. All the images are resized to 16 × 16. The classical Hopkins 155 database of motion segmentation, which is available at http://www.vision.jhu.edu/data/hopkins155, contains 155 video sequences along with extracted feature trajectories, where 120 of the videos have two motions and 35 of the videos have three motions.
Subspace clustering methods compared are SSC (Elhamifar and Vidal (2013)), LSA (Yan and Pollefeys (2006)), and LBF (Zhang et al. (2012)). Based on the studies in Elhamifar and Vidal (2013), Vidal (2011) and Zhang et al. (2012), these three methods exhibit state-of-the-art subspace clustering performance. We adopt the LSA and SSC implementations provided in Elhamifar and Vidal (2013) from http://www.vision.jhu.edu/code/, and the LBF implementation provided in Zhang et al. (2012) from http://www.ima.umn. edu/~zhang620/lbf/. We adopt similar setups as described in Zhang et al. (2012) for experiments on subspace clustering.
This section then presents experimental evaluations on classification using two public face datasets: the CMU PIE dataset (Sim et al. (2003)) and the Extended YaleB dataset. The PIE dataset consists of 68 subjects imaged simultaneously under 13 different poses and 21 lighting conditions. All the face images are resized to 20 × 20. We adopt a NN classifier unless otherwise specified.
Subspace Clustering with Illustrative Examples
For illustration purposes, we conduct the first set of experiments on a subset of the MNIST dataset. We adopt a similar setup as described in Zhang et al. (2012), using the same sets of 2 or 3 digits, and randomly choose 200 images for each digit. We set the sparsity value K = 6 for R-SSC, and perform 100 iterations for the subgradient updates while learning the transformation on subspaces. The subgradient update step was ν = 0.02 (see Appendix C for details on the projected subgradient optimization algorithm).
Unless otherwise stated, we do not perform dimension reduction, such as PCA or random projections, to preprocess the data, thereby further saving computations (please note that the learned transform can itself reduce dimensions if so desired, see Section 5.7). In the literature, e.g., Elhamifar and Vidal (2013), Vidal (2011) and Zhang et al. (2012), projection to a very low dimension is usually performed to enhance the clustering performance. However, it is often not obvious how to determine the correct projection dimension for real data, and many subspace clustering methods show sensitive to the choice of the projection dimension. This dimension reduction step is not needed in the framework here proposed. Fig. 5 shows the misclassification rate (e) and running time (t) on clustering subspaces of two digits. The misclassification rate is the ratio of misclassified points to the total number of points 3 . For visualization purposes, the data are plotted with the dimension reduced to 2 using Laplacian Eigenmaps Belkin and Niyogi (2003). Different clusters are represented by different colors and the ground truth is plotted using the true cluster labels. The proposed R-SSC outperforms state-of-the-art methods, both in terms of clustering accuracy and running time. The clustering error of R-SSC is further reduced using the proposed LRSC framework in Algorithm 1 through the learned low-rank subspace transformation. The clustering 3. Meaning the ratio of points that were assigned to the wrong cluster. (2013), LSA Yan and Pollefeys (2006), and LBF Zhang et al. (2012). For visualization, the data are plotted with the dimension reduced to 2 using Laplacian Eigenmaps Belkin and Niyogi (2003). Different clusters are represented by different colors and the ground truth is plotted with the true cluster labels. iter indicates the number of LRSC iterations in Algorithm 1. The proposed R-SSC outperforms stateof-the-art methods in terms of both clustering accuracy and running time, e.g., about 500 times faster than SSC. The clustering performance of R-SSC is further improved using the proposed LRSC framework. Note how the data is clearly clustered in clean subspaces in the transformed domain (best viewed zooming on screen (2006) and LBF Zhang et al. (2012). LBF is adopted in the proposed LRSC framework and denoted as R-LBF. After convergence, R-LBF significantly outperforms state-of-the-art methods. a low-rank structure for data from the same subspace, but also increases the separations between the subspaces for more accurate clustering. Fig. 6 shows misclassification rate (e) on clustering subspaces of three digits. Here we adopt LBF in our LRSC framework, denoted as Robust LBF (R-LBF), to illustrate that the performance of existing subspace clustering methods can be enhanced using the proposed LRSC algorithm. After convergence, R-LBF, which uses the proposed learned subspace transformation, significantly outperforms state-of-the-art methods. Table 1 shows the misclassification rate on clustering different number of digits, [0 : c] denotes the subset of c + 1 digits from digit 0 to c. We randomly pick 100 samples per digit to compare the performance when a fewer number of data points per class are present. For all cases, the proposed LRSC method significantly outperforms state-of-the-art methods.
Online vs. Batch Learning
In this set of experiments, we use digits {1, 2} from the MNIST dataset. We select 1000 images for each digit, and randomly partition them into 5 mini-batches. We first perform one iteration of LRSC in Algorithm 1 over all selected data with various γ values. As shown in Fig. 7a, we always observe empirical convergence for subspace transformation learning via (6). The projected subgradient method presented in Appendix C converges to Figure 7: Convergence of the objective function (6) using online and batch learning for subspace transformation. We always observe empirical convergence for both online and batch learning. In (a), we vary the value of γ in the norm constraint ("No normalization" denotes removing the norm constraint). More discussions on convergence can be found in Appendix C. In (b), to converge to the same objective function value, it takes 131.76 sec. for online learning and 700.27 sec. for batch learning. a local minimum (or a stationary point). More discussions on convergence can be found in Appendix C.
Starting with the first mini-batch, we then perform one iteration of LRSC over one minibatch a time, with the subspace transformation learned from the previous mini-batch as warm restart. We adopt here 100 iterations for the subgradient descent updates. As shown in Fig. 7b, we observe similar empirical convergence for online transformation learning. To converge to the same objective function value, it takes 131.76 sec. for online learning and 700.27 sec. for batch learning.
Application to Face Clustering
In the Extended YaleB dataset, each of the 38 subjects is imaged under 64 lighting conditions, shown in Fig. 8a. Under the assumption of Lambertian reflectance, face images of each subject under different lighting conditions can be accurately approximated with a 9-dimensional linear subspace (Basri and Jacobs (February 2003)). We conduct the face clustering experiments on the first 9 subjects shown in Fig. 8b. We set the sparsity value K = 10 for R-SSC, and perform 100 iterations for the subgradient descent updates while learning the transformation. Fig. 9 shows error rate (e) and running time (t) on clustering subspaces of 9 subjects using different subspace clustering methods. The proposed R-SSC techniques outperforms state-of-the-art methods both in accuracy and running time. As shown in Fig. 10, using the proposed LRSC algorithm (that is, learning the transform), the misclassification errors of R-SSC are further reduced significantly, for example, from 67.37% to 4.94% for the 9 subjects. Fig. 10n shows the convergence of the T updating step in the first few LRSC iterations. The dramatic performance improvement can be explained in Fig. 11. We observe, as expected from the theory presented before, that the learned subspace transformation increases the distance (the smallest principal angle) between subspaces and, at the same time, reduces the nuclear norms of subspaces. More results on clustering subspaces of 2 and 3 subjects are shown in Fig. 12. Table 2 shows misclassification rate (e) on clustering subspaces of different number of subjects, [1 : c] denotes the first c subjects in the extended YaleB dataset. For all cases, the proposed LRSC method significantly outperforms state-of-the-art methods. Note that without the low-rank decomposition step in (10), we obtain a misclassification rate 18.38% (n) Convergence of T updating Figure 10: Misclassification rate (e) on clustering 9 subjects using the proposed LRSC framework. We adopt the proposed R-SSC technique for the clustering step. With the proposed LRSC framework, the clustering error of R-SSC is further reduced significantly, e.g., from 67.37% to 4.94% for the 9-subject case. Note how the classes are clustered in clean subspaces in the transformed domain.
for clustering all 38 subjects in the Extended YaleB dataset, which is slightly lower than the 11.02% reported in Table 2 (e) Subspace nuclear norm. Figure 11: The smallest and mean principal angles between pairs of 9 subject subspaces and the nuclear norms of 9 subject subspaces before and after transformation. Note that each entry in (a) and (b) denotes the smallest principal angle, and each entry in (c) and (d) denotes the average cosine over all principal angles. We observe that the learned subspace transformation increases the angles between subspaces and also reduces the nuclear norms of subspaces. Overall, the average smallest principal angles between subspaces increased from 0.09 to 0.26, and the average subspace nuclear norm decreased from 21.43 to 8.53.
transformation plays a major role here; and the robustness in the low-rank decomposition enhances the performance even further.
In Fig. 3 and Fig. 4, using synthetic examples, we previously compared our learned transformation with the closed-form orthogonalizing transformation and LDA. In Table 3, we further compare three transformations using real data. We perform supervised transformation learning on all 38 subjects in the Extended YaleB dataset using three different transformation learning algorithms, and then perform subspace clustering on the transformed data. The proposed transformation learning significantly outperforms the other two methods. The proposed R-SSC outperforms state-of-the-art methods both in accuracy and running time. With the proposed LRSC framework, the clustering error of R-SSC is further reduced significantly. Note how the classes are clustered in clean subspaces in the transformed domain (best viewed zooming on screen).
Application to Motion Segmentation
The Hopkins 155 dataset consists of three types of videos: checker, traffic and articulated, and 120 of the videos have two motions and 35 of the videos have three motions. The main task is to segment a video sequence of multiple rigidly moving objects into multiple spatiotemporal regions that correspond to different motions in the scene. This motion dataset contains much cleaner subspace data than the digits and faces data evaluated above. To enable a fair comparison, we project the data into a lower dimensional subspace using PCA as explained in Vidal (2011);Zhang et al. (2012). Results on other comparing methods are taken from Vidal (2011). As shown in Vidal (2011); Zhang et al. (2012), the SSC method significantly outperforms all previous state-of-the-art methods on this dataset. From Table 4, we can see that our method shows comparable results to SSC for two motions and outperforms SSC for three motions. Note that our method is orders of magnitude faster than SSC as discussed earlier.
Application to Face Recognition across Illumination
For the Extended YaleB dataset, we adopt a similar setup as described in Jiang et al. (June 2011);Zhang and Li (June 2010). We split the dataset into two halves by randomly selecting 32 lighting conditions for training, and the other half for testing. We learn a global low-rank transformation matrix from the training data. We report recognition accuracies in Table 5. We make the following observations. First, the recognition accuracy is increased from 91.77% to 99.10% by simply applying the learned transformation matrix to the original face images. Second, the best accuracy is obtained by first recovering the low-rank subspace for each subject, e.g., the third row in Fig. 13a. Then, each transformed testing face, e.g., the second row in Fig. 13b, is sparsely decomposed over the low-rank subspace of each subject through OMP, and classified to the subject with the minimal reconstruction error. A sparsity value 10 is used here for OMP. As shown in Fig. 13c, the low-rank representation for each subject shows reduced variations caused by illumination. Third, the global transformation performs better here than class-based transformations, which can be due to the fact that illumination in this dataset varies in a globally coordinated way across subjects. Last but not least, our method outperforms state-of-the-art sparse representation based face recognition methods.
Application to Face Recognition across Pose
We adopt the similar setup as described in Castillo and Jacobs (2009) to enable the comparison. In this experiment, we classify 68 subjects in three poses, frontal (c27), side (c05), Table 4: Misclassification rate (e%) on two motions and three motions segmentation in the Hopkins 155 dataset. As shown in Vidal (2011);Zhang et al. (2012), the SSC method significantly outperforms all previous state-of-the-art methods on this dataset. The proposed LRSC shows comparable results to SSC for two motions and outperforms SSC for three motions. Note that our method is orders of magnitude faster than SSC. and profile (c22), under lighting condition 12. We use the remaining poses as the training data.
For this example, we learn a class-based low-rank transformation matrix per subject from the training data. It is noted that the goal is to learn a transformation matrix to help in the classification, which may not necessarily correspond to the real geometric transform. Table 6 shows the face recognition accuracies under pose variations for the CMU PIE dataset (we applied the crop-and-flip step discussed in Fig. 1.). We make the following observations. First, the recognition accuracy is dramatically increased after applying the learned transformations. Second, the best accuracy is obtained by recovering the low-rank subspace for each subject, e.g., the third row in Fig. 14a and Fig. 14b. Then, each transformed testing face, e.g., Fig. 14c and Fig. 14d, is sparsely decomposed over the low-rank subspace of each subject through OMP, and classified to the subject with the minimal reconstruction error, Section 4. Third, the class-based transformation performs better than the global transformation in this case. The choice between these two settings is data dependent. Last but not least, our method outperforms SMD, which the best of our knowledge, reported the best recognition performance in such experimental setup. However, SMD is an unsupervised method, and the proposed method requires training, still illustrating how a simple learned transform (note that applying it to the data at testing time if virtually free of cost), can significantly improve performance.
Subject 1 Class T (dadl-10) 20x20
Original face images
Application to Face Recognition across Illumination and Pose
To enable the comparison with Qiu et al. (Oct. 2012), we adopt their setup for face recognition under combined pose and illumination variations for the CMU PIE dataset. We use 68 subjects in 5 poses, c22, c37, c27, c11 and c34, under 21 illumination conditions for training; and classify 68 subjects in 4 poses, c02, c05, c29 and c14, under 21 illumination conditions.
Three face recognition methods are adopted for comparisons: Eigenfaces Turk and Pentland (June 1991), SRC Wright et al. (2009), andDADL Qiu et al. (Oct. 2012). SRC and DADL are both state-of-the-art sparse representation methods for face recognition, and DADL adapts sparse dictionaries to the actual visual domains. As shown in Fig. 15, the proposed methods, both the global LRT (G-LRT) and class-based LRT (C-LRT), signifi-Subject 2 Share T (eccv) 20x20
Low-rank transformation c02 c05 c29 c14
(a) Globally transformed testing samples for subject1
Subject 2 Share T (eccv) 20x20
Low-rank transformation c02 c05 c29 c14
(b) Globally transformed testing samples for subject2 Figure 16: Face recognition under combined pose and illumination variations using global low-rank transformation.
cantly outperform the comparing methods, especially for extreme poses c02 and c14. Some testing examples using a global transformation are shown in Fig. 16. We notice that the transformed faces for each subject exhibit reduced variations caused by pose and illumination.
Discussion on the Size of the Transformation Matrix T
In the experiments presented above, we learned a square linear transformation. For example, if images are resized to 16 × 16, the learned subspace transformation T is of size 256 × 256. If we learn a transformation of size r × 256 with r < 256, we enable dimension reduction while performing subspace transformation (feature learning). Through experiments, we notice that the peak clustering accuracy is usually obtained when r is smaller than the dimension of the ambient space. For example, in Fig. 12, through exhaustive search for the optimal r, we observe the misclassification rate reduced from 2.38% to 0% for subjects {2, 3} at r = 96, and from 4.23% to 0% for subjects {4, 5, 6} at r = 40. As discussed before, this provides a framework to sense for clustering and classification, connecting the work here presented with the extensive literature on compressed sensing, and in particular for sensing design, e.g., Carson et al. (2012). We plan to study in detail the optimal size of the learned transformation matrix for subspace clustering and classification, including its potential connection with the number of subspaces in the data, and further investigate such connections with compressive sensing.
Conclusion
We introduced a subspace low-rank transformation approach for subspace clustering and classification. Using matrix rank as the optimization criteria, via its nuclear norm convex surrogate, we learn a subspace transformation that reduces variations within the subspaces, and increases separations between the subspaces. We demonstrated that the proposed approach significantly outperforms state-of-the-art methods for subspace clustering and classification, and provided some theoretical support to these experimental results.
Numerous venues of research are opened by the framework here introduced. At the theoretical level, extending the analysis to the noisy case is needed. Furthermore, understanding the virtues of the global vs the class-dependent transform is both important and interesting, as it is the study of the framework in its compressed dimensionality form. Beyond this, considering the proposed approach as a feature extraction technique, its combination with other successful clustering and classification techniques is the subject of current research.
Appendix A. Proof of Theorem 1 Proof: We know that (Srebro et al. (2005))
||A|| * = min U,V A=UV 1 2 (||U|| 2 F + ||V|| 2 F ).
We denote U A and V A the matrices that achieve the minimum; same for B, U B and V B ; and same for the concatenation [A, B], U [A,B] and V [A,B] . We then have
||A|| * = 1 2 (||U A || 2 F + ||V A || 2 F ), ||B|| * = 1 2 (||U B || 2 F + ||V B || 2 F ).≤ 1 2 (||[U A , U B ]|| 2 F + ||[V A , V B ]|| 2 F ) = 1 2 (||U A || 2 F + ||U B || 2 F + ||V A || 2 F + ||V B || 2 F ) = 1 2 (||U A || 2 F + ||V A || 2 F ) + 1 2 (||U B || 2 F + ||V B || 2 F ) = ||A|| * + ||B|| * .
Appendix B. Proof of Theorem 2
Proof: We perform the singular value decomposition of A and B as
A = [U A1 U A2 ] Σ A 0 0 0 [V A1 V A2 ] , B = [U B1 U B2 ] Σ B 0 0 0 [V B1 V B2 ] ,
where the diagonal entries of Σ A and Σ B contain non-zero singular values. We have
AA = [U A1 U A2 ] Σ A 2 0 0 0 [U A1 U A2 ] , BB = [U B1 U B2 ] Σ B 2 0 0 0 [U B1 U B2 ] .
The column spaces of A and B are considered to be orthogonal, i.e., U A1 U B1 = 0. The above can be written as
AA = [U A1 U B1 ] Σ A 2 0 0 0 [U A1 U B1 ] , BB = [U A1 U B1 ] 0 0 0 Σ B 2 [U A1 U B1 ] .
Then, we have
[A, B][A, B] = AA + BB = [U A1 U B1 ] Σ A 2 0 0 Σ B 2 [U A1 U B1 ] .
The nuclear norm ||A|| * is the sum of the square root of the singular values of AA . Thus, ||[A, B]|| * = ||A|| * + ||B|| * .
Appendix C. Projected Subgradient Learning Algorithm
We use a simple projected subgradient method to search for the transformation matrix T that minimizes (6). Before describing it, we should note that the problem is nondifferentiable and non-convex, and it deserves a proper study for efficient optimization, keeping in mind that the development of more advanced optimization techniques will just further improve the performance of the proposed framework. We selected a simple subgradient approach since the goal of this paper is to present the framework, and already this simple optimization leads to very fast convergence and excellent performance as detailed in Section 5, significant improvements in performance when compared to state-of-the-art. To minimize (6), the proposed projected subgradient method uses the iteration
T (t+1) = T (t) − ν∆T,(16)
where T (t) is the t-th iterate, and ν > 0 defines the step size. The subgradient step ∆T is evaluated as
∆T = C c=1 ∂||TY c ||Y T c − ∂||TY||Y T ,(17)
where ∂||·|| is the subdifferential of the norm ||·|| (given a matrix A, the subdifferential ∂||A|| can be evaluated using the simple approach shown in Algorithm 2, Watson (1992)).
After each iteration, we project T via γ T ||T|| . The objective function (6) is a D.C. (difference of convex functions) program (Dinh and An (1997), Yuille and Rangarajan (2003), Sriperumbudur and Lanckriet (2012)). We provide here a simple convergence analysis to the projected subgradient approach proposed above.
We first provide an analysis to the minimization of (6) without the norm constraint ||T|| 2 = γ, using the following iterative D.C. procedure (Yuille and Rangarajan (2003)):
• Initialize T (0) with the identity matrix;
• At the t-th iteration, we update T (t+1) by solving a convex minimization sub-problem (18),
T (t+1) = arg T min C c=1 ||TY c || * − ∂||T (t) Y||Y T T T .(18)
where the first term in (18) is the convex term in (6), and the added second term is a linear term on T using a subgradient of the concave term in (6) evaluated at the current iteration. We solve this sub-objective function (18) using the subgradient method, i.e., using a constant step size ν, we iteratively take a step in the negative direction of subgradient, and the subgradient is evaluated as
C c=1 ∂||TY c ||Y T c − ∂||T (t) Y||Y T .(19)
Though each subgradient step does not guarantee a decrease of the cost function (Boyd et al. (2003); Recht et al. (2010)), using a constant step size, the subgradient method is guaranteed to converge to within some range of the optimal value for a convex problem (Boyd et al. (2003)) (it is easy to notice that (16) is a simplified version of this D.C. procedure by performing only one iteration of the subgradient method in solving the subobjective function (18), and we will have more discussion on this simplification later). Given T (t+1) as the minimizer found for the convex problem (18) using the subgradient method, we have for (18),
C c=1 ||T (t+1) Y c || * − ∂||T (t) Y||Y T T (t+1) T (20) ≤ C c=1 ||T (t) Y c || * − ∂||T (t) Y||Y T T (t) T ,
and from the concavity of the second term in (6), we have −||T (t+1) Y|| * ≤ −||T (t) Y|| * − ∂||T (t) Y||Y T (T (t+1) − T (t) ) T .
By summing 20 and 21, we obtain
C c=1 ||T (t+1) Y c || * − ||T (t+1) Y|| * ≤ C c=1 ||T (t) Y c || * − ||T (t) Y || * .(22)
The objective (6) is bounded from below by 0 (shown in Section 2), and decreases after each iteration of the above D.C. procedure (shown in (22)). Thus, the convergence to a local minimum (or a stationary point) is guaranteed. For efficiency considerations, while solving the convex sub-objective function (18), we perform only one iteration of the subgradient method to obtain a simplified method (16), and still observe empirical convergence in all experiments, see Fig.7 and Fig.10n. The norm constraint ||T|| 2 = γ is adopted in our formulation to prevent the trivial solution T = 0. As shown above, the minimization of (6) without the norm constraint always converges to a local minimum (or a stationary point), thus the initialization becomes critical while dropping the norm constraint. By initializing T (0) with the identity matrix, we observe no trivial solution convergence in all experiments, such as the no normalization case in Fig.7.
As shown in Douglas et al. (2000), the norm constraint ||T|| 2 = γ can be incorporated to a gradient-based algorithm using various alternatives, e.g., Lagrange multipliers, coefficient normalization, and gradients in the tangent space. We implement the coefficient normalization method, i.e., after obtaining T (t+1) from (18), we normalize T (t+1) via γ T (t+1) ||T (t+1) || . In other words, we normalize the length of T (t+1) without changing its direction. As discussed in Douglas et al. (2000), the problem of minimizing a cost function subject to a norm constraint forms the basis for many important tasks, and gradient-based algorithms are often used along with the norm constraint. Though it is expected that a norm constraint does not change the convergence behavior of a gradient algorithm (Douglas et al. (2000); Fuhrmann and Liu (1984)), Fig.7, to the best of our knowledge, it is an open problem to analyze how a norm constraint and the choice of γ affect the convergence behavior of a gradient/subgradient method.
Input: An m × n matrix A, a small threshold value δ Output: The subdifferential of the matrix norm ∂||A||. Algorithm 2: An approach to evaluate the subdifferential of a matrix norm.
Figure 1 :
1Learned low-rank transformation on faces across pose.
Figure 2 :
2The learned transformation T using
Theorem 1
1Let A and B be matrices of the same row dimensions, and [A, B] be the concatenation of A and B, we have ||[A, B]|| * ≤ ||A|| * + ||B|| * . Proof: See Appendix A. Theorem 2 Let A and B be matrices of the same row dimensions, and [A, B] be the concatenation of A and B, we have ||[A, B]|| * = ||A|| * + ||B|| * . when the column spaces of A and B are orthogonal. Proof: See Appendix B.
[A, B]| * = 1.41 to |[A, B]| * = 1.95 to make |A| * + |B| * − |[A, B]| * ≈ 0; However, in both cases, where subspaces are independent, rank([A, B]) = 2, and rank(A) + rank(B) − rank([A, B]) = 0.
|A| * = 1.08, |B| * = 1.07.
Figure 3 :
3Comparisons with the closed-form orthogonalizing transformation. Two intersecting planes are shown in (a), and each plane contains 200 points.
0.04, θAC = 1.54, θAD = 1.54, θBC = 1.55, θBD = 1.56, θCD = 0.01 , |Y+| * = 1.02, |Y−| * = 1.00. Two classes {A(blue),B(red)}, θAB = 0.31, |A| * = 1.91, |B| * = 1.88. |A| * = 1.08, |B| * = 1.03.
Figure 4 :
4Comparisons with the linear discriminant analysis (LDA)
LBF
Figure 5 :
5Misclassification rate (e) and running time (t) on clustering 2 digits. Methods compared are SSC Elhamifar and Vidal
Figure 6 :
6Misclassification rate (e) on clustering 3 digits. Methods compared are LSA Yan and Pollefeys
Figure 8 :
8The extended YaleB face dataset. ) R-SSC, e = 67.37%, t = 1.83 sec.
Figure 9 :
9Misclassification rate (e) and running time (t) on clustering 9 subjects using different subspace clustering methods. The proposed R-SSC outperforms state-of-the-art methods both in accuracy and running time. This is further improved using the learned transform, LRSC reduces the error to 4.94%, seeFig. 10.
Figure 12 :
12Misclassification rate (e) and running time (t) on clustering 2 and 3 subjects.
(c) Mean low-rank components for subjects in the training data
Figure 13 :
13Face recognition across illumination using global low-rank transformation.
Figure 14 :Figure 15 :
1415Face recognition across pose using class-based low-rank transformation. Note, for example in (c) and (d), how the learned transform reduces the pose-variability. Face recognition accuracy under combined pose and illumination variations on the CMU PIE dataset. The proposed methods are denoted as G-LRT in color red and C-LRT in color blue. The proposed methods significantly outperform the comparing methods, especially for extreme poses c02 and c14.
begin 1 .
1Perform singular value decomposition: A = UΣV ;2. s ← the number of singular values smaller than δ , 3. Partition U and V asU = [U (1) , U (2) ], V = [V (1) , V (2) ] ;where U (1) and V (1) have (n − s) columns.4. Generate a random matrix B of the size (m − n + s) × s, B ← B ||B|| ; 5. ∂||A|| ← U (1) V (1)T + U (2) BV (2)T ; 6. Return ∂||A|| ; end
(c) Original subspaces for digits {1, 7}.e=9.2417%
t=78.00 sec
Ground truth
LSA
e=9.0047%
t=11.37 sec
SSC
e=4.0284%
t=447.66 sec
R-SSC (iter=0)
e=3.3175%
t=0.93 sec
Unsupervised clustering digits [1 2] in MNIST (e: misclassification rate t: running time)
R-SSC: Robust Sparse Subspace clustering (our approach)
(a) Original subspaces for digits {1, 2}.
Clustering digits [1 2] using low-rank subspace transformation (iter: EM iterations)
iter=1
e=2.8436%
iter=2
e=1.8957%
Ground
truth
R-SSC
iter=3
e=1.4218%
iter=4
e=1.8957%
iter=5
e=1.8957%
(b) Transformed subspaces for digits {1, 2}.
LBF
e=16.4352%
t=74.94 sec
Ground truth
LSA
e=2.3148%
t=11.31 sec
SSC
e=2.0833%
t=458.15 sec
R-SSC (iter=0)
e=1.1574%
t=0.95 sec
Unsupervised clustering digits [1 7] in MNIST (e: misclassification rate t: running time)
R-SSC: Robust Sparse Subspace clustering (our approach)
Clustering digits [1 7] using low-rank subspace transformation (iter: EM iterations)
iter=1
e=0.69444%
iter=2
e=0.46296%
Ground
truth
R-SSC
iter=3
e=0.46296%
iter=4
e=0.46296%
iter=5
e=0.46296%
(d) Transformed subspaces for digits {1, 7}.
).converges after about 3 LRSC iterations. The learned transformation not only recovers
LBF
e=30.9904%
Ground truth
LSA
e=30.1917%
Unsupervised clustering digits [1 2 3] in MNIST (e: misclassification rate)
R-LBF
e=9.5847%
Ground truth
R-LBF: adopt LBF as the subspace clustering method for transformed subspaces
Transformed subspaces
Original subspaces
(a) Digits {1, 2, 3}.
LBF
e=35.0937%
Ground truth
LSA
e=21.2947%
Unsupervised clustering digits [2 4 8] in MNIST (e: misclassification rate)
R-LBF
e=6.9847%
Ground truth
R-LBF: adopt LBF as the subspace clustering method for transformed subspaces
Transformed subspaces
Original subspaces
(b) Digits {2, 4, 8}.
Table 1 :
1Misclassification rate (e%) on clustering different numbers of digits in the MNIST dataset, [0 : c] denotes the subset of c + 1 digits from digit 0 to c. We randomly pick 100 samples per digit. For all cases, the proposed LRSC method significantly outperforms state-of-the-art methods.Subsets [0:1] [0:2]
[0:3]
[0:4]
[0:5]
[0:6]
[0:7]
[0:8]
C
2
3
4
5
6
7
8
9
LSA
0.47 47.57 36.73 30.90 40.46
48.13 39.87 44.03
LBF
0.47 23.62 29.19 51.37 48.99
53.01 39.87 38.79
LRSC
0
3.88 3.89 5.31
14.04 13.79 14.50 16.05
. Thus, pushing the subspaces apart through our learned.00 .10 .05 .07 .11 .07 .07 .10 .09
.10 .00 .08 .10 .11 .07 .09 .09 .11
.05 .08 .00 .07 .13 .09 .08 .09 .09
.07 .10 .07 .00 .13 .09 .04 .10 .09
.11 .11 .13 .13 .00 .11 .11 .13 .13
.07 .07 .09 .09 .11 .00 .08 .08 .08
.07 .09 .08 .04 .11 .08 .00 .09 .10
.10 .09 .09 .10 .13 .08 .09 .00 .07
.09 .11 .09 .09 .13 .08 .10 .07 .00
s1
s2
s3
s4
s5
s6
s7
s8
s9
s 1 s 2 s 3 s 4 s 5 s 6 s 7 s 8 s 9
(a) Original smallest angles.
.00 .26 .20 .20 .29 .20 .24 .22 .21
.26 .00 .26 .26 .34 .26 .30 .27 .24
.20 .26 .00 .24 .34 .22 .25 .19 .24
.20 .26 .24 .00 .35 .18 .23 .25 .25
.29 .34 .34 .35 .00 .33 .32 .29 .31
.20 .26 .22 .18 .33 .00 .21 .21 .23
.24 .30 .25 .23 .32 .21 .00 .26 .23
.22 .27 .19 .25 .29 .21 .26 .00 .24
.21 .24 .24 .25 .31 .23 .23 .24 .00
s1
s2
s3
s4
s5
s6
s7
s8
s9
s 1 s 2 s 3 s 4 s 5 s 6 s 7 s 8 s 9
(b) Transformed smallest angles.
1.0 .64 .64 .69 .67 .63 .58 .72 .67
.64 1.0 .63 .62 .64 .65 .61 .62 .63
.64 .63 1.0 .66 .65 .64 .61 .67 .65
.69 .62 .66 1.0 .64 .64 .61 .68 .67
.67 .64 .65 .64 1.0 .64 .57 .70 .67
.63 .65 .64 .64 .64 1.0 .59 .67 .70
.58 .61 .61 .61 .57 .59 1.0 .61 .61
.72 .62 .67 .68 .70 .67 .61 1.0 .66
.67 .63 .65 .67 .67 .70 .61 .66 1.0
s1
s2
s3
s4
s5
s6
s7
s8
s9
s 1 s 2 s 3 s 4 s 5 s 6 s 7 s 8 s 9
(c) Original mean cosine angles.
1.0 .47 .47 .52 .49 .44 .42 .51 .47
.47 1.0 .46 .45 .47 .48 .45 .44 .44
.47 .46 1.0 .48 .47 .45 .43 .47 .45
.52 .45 .48 1.0 .48 .47 .46 .49 .47
.49 .47 .47 .48 1.0 .47 .42 .49 .46
.44 .48 .45 .47 .47 1.0 .42 .47 .50
.42 .45 .43 .46 .42 .42 1.0 .45 .43
.51 .44 .47 .49 .49 .47 .45 1.0 .46
.47 .44 .45 .47 .46 .50 .43 .46 1
Table 2 :
2Misclassification rate (e%) on clustering different number of subjects in the Extended YaleB face dataset, [1 : c] denotes the first c subjects in the dataset. For all cases, the proposed LRSC method significantly outperforms state-of-the-art methods.Subsets [1:10] [1:15] [1:20] [1:25] [1:30] [1:38]
C
10
15
20
25
30
38
LSA
78.25 82.11 84.92 82.98 82.32 84.79
LBF
78.88 74.92 77.14 78.09 78.73 79.53
LRSC
5.39 4.76 9.36 8.44 8.14 11.02
Table 3 :
3Misclassification rate (e%) on clustering 38 subjects in the Extended YaleB
dataset using supervised transformation learning. The proposed transformation learning
outperforms both the closed-form orthogonalizing transformation and LDA on clustering
the transformed data.
Methods
Misclassification (%)
orthogonalizing 61.36
LDA
9.77
Proposed
5.47
Table 6 :
6Recognition accuracies (%) under pose variations for the CMU PIE dataset.Method
Frontal Side
Profile
(c27)
(c05) (c22)
SMD Castillo and Jacobs (2009) 83
82
57
Original+NN
39.85
37.65 17.06
Original(crop+flip)+NN
44.12
45.88 22.94
Class LRT+NN
98.97
96.91 67.65
Class LRT+OMP
100
100
67.65
Global LRT+NN
97.06
95.58 50
Global LRT+OMP
100
98.53 57.35
The matrices [U A , U B ] and [V A , V B ] obtained by concatenating the matrices that achieve the minimum for A and B when computing their nuclear norm, are not necessarily the ones that achieve the corresponding minimum in the nuclear norm computation of the concatenation matrix[A, B]. Thus, together with ||[A, B]|| 2 F = ||A|| 2 F + ||B|| 2 F , we have||[A, B]|| * =
1
2
(||U [A,B] || 2
F + ||V [A,B] || 2
F )
. Note that this is done only once and can be considered part of the training stage. As before, this further low-rank decomposition helps to handle outliers not addressed by the learned transform.
AcknowledgmentsThis work was partially supported by ONR, NGA, NSF, ARO, DHS, and AFOSR. We thank Dr. Pablo Sprechmann, Dr. Ehsan Elhamifar, Ching-Hui Chen, and Dr. Mariano Tepper for important feedback on this work.
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|
[] |
[
"Electromagnetic Shower Reconstruction and Energy Validation with Michel Electrons and 0 Samples for the Deep-Learning-Based Analyses in MicroBooNE",
"Electromagnetic Shower Reconstruction and Energy Validation with Michel Electrons and 0 Samples for the Deep-Learning-Based Analyses in MicroBooNE"
] |
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"M Bishai ",
"A Blake ",
"T Bolton ",
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"L Camilleri ",
"D Caratelli ",
"I Caro Terrazas \nColorado State University\nFort Collins80523COUSA\n\nColumbia University\n10027New YorkNYUSA\n\nFermi National Accelerator Laboratory (FNAL)\nUniversity of Edinburgh\nEH9 3FD, 60510Edinburgh, BataviaILUnited Kingdom, USA\n\nUniversidad de Granada\nE-18071GranadaSpain\n\nIllinois Institute of Technology (IIT)\nHarvard University\n02138, 60616Cambridge, ChicagoMA, ILUSA, USA\n\nKansas State University (KSU)\n66506ManhattanKSUSA\n\nLA1 4YW, United Kingdom Los Alamos National Laboratory (LANL)\nLancaster University\n87545Lancaster, Los AlamosNMUSA\n\nMassachusetts Institute of Technology (MIT)\nThe University of Manchester\nM13 9PL, 02139Manchester, CambridgeMAUnited Kingdom, USA\n\nUniversity of Michigan\n48109Ann ArborMIUSA\n\nUniversity of Minnesota\n55455MinneapolisMnUSA\n\nNew Mexico State University (NMSU)\nLas Cruces, NM88003USA\n\nUniversity of Oxford\nOX1 3RHOxfordUnited Kingdom\n\nUniversity of Pittsburgh\n15260PittsburghPAUSA\n\nRutgers University\n08854PiscatawayNJ, PAUSA\n\nSouth Dakota School of Mines and Technology (SDSMT)\nSLAC National Accelerator Laboratory\n94025, 57701Menlo Park, Rapid CityCA, SDUSA, USA\n\nUniversity of Southern Maine\n04104PortlandMEUSA\n\nSyracuse University\n13244SyracuseNYUSA\n\nTel Aviv University\n69978Tel AvivIsrael\n\nUniversity of Tennessee\n37996KnoxvilleTNUSA\n\nUniversity of Texas\n76019ArlingtonTXUSA\n\nℎℎ Center for Neutrino Physics\nTufts University\n02155MedfordMAUSA\n\nVirginia Tech\n24061BlacksburgVAUSA\n\nDepartment of Physics\nWright Laboratory\nUniversity of Warwick\nCV4 7ALCoventryUnited Kingdom\n\nYale University\n06520New HavenCTUSA\n",
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"R Lazur \nColorado State University\nFort Collins80523COUSA\n\nColumbia University\n10027New YorkNYUSA\n\nFermi National Accelerator Laboratory (FNAL)\nUniversity of Edinburgh\nEH9 3FD, 60510Edinburgh, BataviaILUnited Kingdom, USA\n\nUniversidad de Granada\nE-18071GranadaSpain\n\nIllinois Institute of Technology (IIT)\nHarvard University\n02138, 60616Cambridge, ChicagoMA, ILUSA, USA\n\nKansas State University (KSU)\n66506ManhattanKSUSA\n\nLA1 4YW, United Kingdom Los Alamos National Laboratory (LANL)\nLancaster University\n87545Lancaster, Los AlamosNMUSA\n\nMassachusetts Institute of Technology (MIT)\nThe University of Manchester\nM13 9PL, 02139Manchester, CambridgeMAUnited Kingdom, USA\n\nUniversity of Michigan\n48109Ann ArborMIUSA\n\nUniversity of Minnesota\n55455MinneapolisMnUSA\n\nNew Mexico State University (NMSU)\nLas Cruces, NM88003USA\n\nUniversity of Oxford\nOX1 3RHOxfordUnited Kingdom\n\nUniversity of Pittsburgh\n15260PittsburghPAUSA\n\nRutgers University\n08854PiscatawayNJ, PAUSA\n\nSouth Dakota School of Mines and Technology (SDSMT)\nSLAC National Accelerator Laboratory\n94025, 57701Menlo Park, Rapid CityCA, SDUSA, USA\n\nUniversity of Southern Maine\n04104PortlandMEUSA\n\nSyracuse University\n13244SyracuseNYUSA\n\nTel Aviv University\n69978Tel AvivIsrael\n\nUniversity of Tennessee\n37996KnoxvilleTNUSA\n\nUniversity of Texas\n76019ArlingtonTXUSA\n\nℎℎ Center for Neutrino Physics\nTufts University\n02155MedfordMAUSA\n\nVirginia Tech\n24061BlacksburgVAUSA\n\nDepartment of Physics\nWright Laboratory\nUniversity of Warwick\nCV4 7ALCoventryUnited Kingdom\n\nYale University\n06520New HavenCTUSA\n",
"I Lepetic ",
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"Y Li ",
"K Lin ",
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"J Moon ",
"M Mooney \nColorado State University\nFort Collins80523COUSA\n\nColumbia University\n10027New YorkNYUSA\n\nFermi National Accelerator Laboratory (FNAL)\nUniversity of Edinburgh\nEH9 3FD, 60510Edinburgh, BataviaILUnited Kingdom, USA\n\nUniversidad de Granada\nE-18071GranadaSpain\n\nIllinois Institute of Technology (IIT)\nHarvard University\n02138, 60616Cambridge, ChicagoMA, ILUSA, USA\n\nKansas State University (KSU)\n66506ManhattanKSUSA\n\nLA1 4YW, United Kingdom Los Alamos National Laboratory (LANL)\nLancaster University\n87545Lancaster, Los AlamosNMUSA\n\nMassachusetts Institute of Technology (MIT)\nThe University of Manchester\nM13 9PL, 02139Manchester, CambridgeMAUnited Kingdom, USA\n\nUniversity of Michigan\n48109Ann ArborMIUSA\n\nUniversity of Minnesota\n55455MinneapolisMnUSA\n\nNew Mexico State University (NMSU)\nLas Cruces, NM88003USA\n\nUniversity of Oxford\nOX1 3RHOxfordUnited Kingdom\n\nUniversity of Pittsburgh\n15260PittsburghPAUSA\n\nRutgers University\n08854PiscatawayNJ, PAUSA\n\nSouth Dakota School of Mines and Technology (SDSMT)\nSLAC National Accelerator Laboratory\n94025, 57701Menlo Park, Rapid CityCA, SDUSA, USA\n\nUniversity of Southern Maine\n04104PortlandMEUSA\n\nSyracuse University\n13244SyracuseNYUSA\n\nTel Aviv University\n69978Tel AvivIsrael\n\nUniversity of Tennessee\n37996KnoxvilleTNUSA\n\nUniversity of Texas\n76019ArlingtonTXUSA\n\nℎℎ Center for Neutrino Physics\nTufts University\n02155MedfordMAUSA\n\nVirginia Tech\n24061BlacksburgVAUSA\n\nDepartment of Physics\nWright Laboratory\nUniversity of Warwick\nCV4 7ALCoventryUnited Kingdom\n\nYale University\n06520New HavenCTUSA\n",
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"R K Neely ",
"D A Newmark ",
"J Nowak ",
"M Nunes ",
"O Palamara ",
"V Paolone ",
"A Papadopoulou ",
"V Papavassiliou ",
"S F Pate ",
"N Patel ",
"A Paudel ",
"Z Pavlovic ",
"E Piasetzky ",
"I D Ponce-Pinto ",
"S Prince ",
"X Qian ",
"J L Raaf ",
"V Radeka ",
"A Rafique ",
"M Reggiani-Guzzo ",
"L Ren ",
"L C J Rice ",
"L Rochester ",
"J Rodriguez ",
"Rondon M Rosenberg ",
"M Ross-Lonergan ",
"G Scanavini ",
"D W Schmitz ",
"A Schukraft ",
"W Seligman ",
"M H Shaevitz ",
"R Sharankova ",
"J Shi ",
"J Sinclair ",
"A Smith ",
"E L Snider ",
"M Soderberg ",
"S Söldner-Rembold ",
"P Spentzouris ",
"J Spitz ",
"M Stancari ",
"J St ",
"T John ",
"K Strauss ",
"S Sutton ",
"A M Sword-Fehlberg ",
"N Szelc ",
"W Tagg ",
"K Tang ",
"C Terao ",
"D Thorpe ",
"M Totani ",
"Y.-T Toups ",
"M A Tsai ",
"T Uchida ",
"W Usher ",
"B Van De Pontseele ",
"M Viren ",
"H Weber ",
"Z Wei ",
"S Williams ",
"T Wolbers ",
"M Wongjirad ",
"K Wospakrik ",
"N Wresilo ",
"W Wright ",
"E Wu ",
"T Yandel ",
"G Yang ",
"L E Yarbrough ",
"H W Yates ",
"G P Yu ",
"J Zeller ",
"C Zennamo ",
"Zhang ",
"\nThe MicroBooNE Collaboration Universität Bern\nCH-3012BernSwitzerland\n",
"\nBrookhaven National Laboratory (BNL)\nUpton11973NYUSA\n",
"\nUniversity of California\n93106Santa BarbaraCAUSA\n",
"\nCentro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT)\nUniversity of Cambridge\nMadrid E-28040CB3 0HECambridgeUnited Kingdom, Spain\n",
"\nUniversity of Chicago\n60637ChicagoILUSA\n",
"\nUniversity of Cincinnati\n45221CincinnatiOHUSA\n"
] |
[
"Colorado State University\nFort Collins80523COUSA",
"Columbia University\n10027New YorkNYUSA",
"Fermi National Accelerator Laboratory (FNAL)\nUniversity of Edinburgh\nEH9 3FD, 60510Edinburgh, BataviaILUnited Kingdom, USA",
"Universidad de Granada\nE-18071GranadaSpain",
"Illinois Institute of Technology (IIT)\nHarvard University\n02138, 60616Cambridge, ChicagoMA, ILUSA, USA",
"Kansas State University (KSU)\n66506ManhattanKSUSA",
"LA1 4YW, United Kingdom Los Alamos National Laboratory (LANL)\nLancaster University\n87545Lancaster, Los AlamosNMUSA",
"Massachusetts Institute of Technology (MIT)\nThe University of Manchester\nM13 9PL, 02139Manchester, CambridgeMAUnited Kingdom, USA",
"University of Michigan\n48109Ann ArborMIUSA",
"University of Minnesota\n55455MinneapolisMnUSA",
"New Mexico State University (NMSU)\nLas Cruces, NM88003USA",
"University of Oxford\nOX1 3RHOxfordUnited Kingdom",
"University of Pittsburgh\n15260PittsburghPAUSA",
"Rutgers University\n08854PiscatawayNJ, PAUSA",
"South Dakota School of Mines and Technology (SDSMT)\nSLAC National Accelerator Laboratory\n94025, 57701Menlo Park, Rapid CityCA, SDUSA, USA",
"University of Southern Maine\n04104PortlandMEUSA",
"Syracuse University\n13244SyracuseNYUSA",
"Tel Aviv University\n69978Tel AvivIsrael",
"University of Tennessee\n37996KnoxvilleTNUSA",
"University of Texas\n76019ArlingtonTXUSA",
"ℎℎ Center for Neutrino Physics\nTufts University\n02155MedfordMAUSA",
"Virginia Tech\n24061BlacksburgVAUSA",
"Department of Physics\nWright Laboratory\nUniversity of Warwick\nCV4 7ALCoventryUnited Kingdom",
"Yale University\n06520New HavenCTUSA",
"Colorado State University\nFort Collins80523COUSA",
"Columbia University\n10027New YorkNYUSA",
"Fermi National Accelerator Laboratory (FNAL)\nUniversity of Edinburgh\nEH9 3FD, 60510Edinburgh, BataviaILUnited Kingdom, USA",
"Universidad de Granada\nE-18071GranadaSpain",
"Illinois Institute of Technology (IIT)\nHarvard University\n02138, 60616Cambridge, ChicagoMA, ILUSA, USA",
"Kansas State University (KSU)\n66506ManhattanKSUSA",
"LA1 4YW, United Kingdom Los Alamos National Laboratory (LANL)\nLancaster University\n87545Lancaster, Los AlamosNMUSA",
"Massachusetts Institute of Technology (MIT)\nThe University of Manchester\nM13 9PL, 02139Manchester, CambridgeMAUnited Kingdom, USA",
"University of Michigan\n48109Ann ArborMIUSA",
"University of Minnesota\n55455MinneapolisMnUSA",
"New Mexico State University (NMSU)\nLas Cruces, NM88003USA",
"University of Oxford\nOX1 3RHOxfordUnited Kingdom",
"University of Pittsburgh\n15260PittsburghPAUSA",
"Rutgers University\n08854PiscatawayNJ, PAUSA",
"South Dakota School of Mines and Technology (SDSMT)\nSLAC National Accelerator Laboratory\n94025, 57701Menlo Park, Rapid CityCA, SDUSA, USA",
"University of Southern Maine\n04104PortlandMEUSA",
"Syracuse University\n13244SyracuseNYUSA",
"Tel Aviv University\n69978Tel AvivIsrael",
"University of Tennessee\n37996KnoxvilleTNUSA",
"University of Texas\n76019ArlingtonTXUSA",
"ℎℎ Center for Neutrino Physics\nTufts University\n02155MedfordMAUSA",
"Virginia Tech\n24061BlacksburgVAUSA",
"Department of Physics\nWright Laboratory\nUniversity of Warwick\nCV4 7ALCoventryUnited Kingdom",
"Yale University\n06520New HavenCTUSA",
"Colorado State University\nFort Collins80523COUSA",
"Columbia University\n10027New YorkNYUSA",
"Fermi National Accelerator Laboratory (FNAL)\nUniversity of Edinburgh\nEH9 3FD, 60510Edinburgh, BataviaILUnited Kingdom, USA",
"Universidad de Granada\nE-18071GranadaSpain",
"Illinois Institute of Technology (IIT)\nHarvard University\n02138, 60616Cambridge, ChicagoMA, ILUSA, USA",
"Kansas State University (KSU)\n66506ManhattanKSUSA",
"LA1 4YW, United Kingdom Los Alamos National Laboratory (LANL)\nLancaster University\n87545Lancaster, Los AlamosNMUSA",
"Massachusetts Institute of Technology (MIT)\nThe University of Manchester\nM13 9PL, 02139Manchester, CambridgeMAUnited Kingdom, USA",
"University of Michigan\n48109Ann ArborMIUSA",
"University of Minnesota\n55455MinneapolisMnUSA",
"New Mexico State University (NMSU)\nLas Cruces, NM88003USA",
"University of Oxford\nOX1 3RHOxfordUnited Kingdom",
"University of Pittsburgh\n15260PittsburghPAUSA",
"Rutgers University\n08854PiscatawayNJ, PAUSA",
"South Dakota School of Mines and Technology (SDSMT)\nSLAC National Accelerator Laboratory\n94025, 57701Menlo Park, Rapid CityCA, SDUSA, USA",
"University of Southern Maine\n04104PortlandMEUSA",
"Syracuse University\n13244SyracuseNYUSA",
"Tel Aviv University\n69978Tel AvivIsrael",
"University of Tennessee\n37996KnoxvilleTNUSA",
"University of Texas\n76019ArlingtonTXUSA",
"ℎℎ Center for Neutrino Physics\nTufts University\n02155MedfordMAUSA",
"Virginia Tech\n24061BlacksburgVAUSA",
"Department of Physics\nWright Laboratory\nUniversity of Warwick\nCV4 7ALCoventryUnited Kingdom",
"Yale University\n06520New HavenCTUSA",
"The MicroBooNE Collaboration Universität Bern\nCH-3012BernSwitzerland",
"Brookhaven National Laboratory (BNL)\nUpton11973NYUSA",
"University of California\n93106Santa BarbaraCAUSA",
"Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT)\nUniversity of Cambridge\nMadrid E-28040CB3 0HECambridgeUnited Kingdom, Spain",
"University of Chicago\n60637ChicagoILUSA",
"University of Cincinnati\n45221CincinnatiOHUSA"
] |
[] |
A : This article presents the reconstruction of the electromagnetic activity from electrons and photons (showers) used in the MicroBooNE deep learning-based low energy electron search. The reconstruction algorithm uses a combination of traditional and deep learning-based techniques to estimate shower energies. We validate these predictions using two -sourced data samples: charged/neutral current interactions with final state neutral pions and charged current interactions in which the muon stops and decays within the detector producing a Michel electron. Both the neutral pion sample and Michel electron sample demonstrate agreement between data and simulation. Further, the absolute shower energy scale is shown to be consistent with the relevant physical constant of each sample: the neutral pion mass peak and the Michel energy cutoff.
|
10.1088/1748-0221/16/12/t12017
|
[
"https://arxiv.org/pdf/2110.11874v3.pdf"
] | 239,616,366 |
2110.11874
|
86590b743a297b85e7de3485ebc8e191a7553b13
|
Electromagnetic Shower Reconstruction and Energy Validation with Michel Electrons and 0 Samples for the Deep-Learning-Based Analyses in MicroBooNE
1 Mar 2022
P Abratenko
R An
J Anthony
L Arellano
J Asaadi
A Ashkenazi
S Balasubramanian
B Baller
C Barnes
G Barr
V Basque
L Bathe-Peters
O Benevides
Rodrigues S Berkman
A Bhanderi
A Bhat
M Bishai
A Blake
T Bolton
J Y Book
L Camilleri
D Caratelli
I Caro Terrazas
Colorado State University
Fort Collins80523COUSA
Columbia University
10027New YorkNYUSA
Fermi National Accelerator Laboratory (FNAL)
University of Edinburgh
EH9 3FD, 60510Edinburgh, BataviaILUnited Kingdom, USA
Universidad de Granada
E-18071GranadaSpain
Illinois Institute of Technology (IIT)
Harvard University
02138, 60616Cambridge, ChicagoMA, ILUSA, USA
Kansas State University (KSU)
66506ManhattanKSUSA
LA1 4YW, United Kingdom Los Alamos National Laboratory (LANL)
Lancaster University
87545Lancaster, Los AlamosNMUSA
Massachusetts Institute of Technology (MIT)
The University of Manchester
M13 9PL, 02139Manchester, CambridgeMAUnited Kingdom, USA
University of Michigan
48109Ann ArborMIUSA
University of Minnesota
55455MinneapolisMnUSA
New Mexico State University (NMSU)
Las Cruces, NM88003USA
University of Oxford
OX1 3RHOxfordUnited Kingdom
University of Pittsburgh
15260PittsburghPAUSA
Rutgers University
08854PiscatawayNJ, PAUSA
South Dakota School of Mines and Technology (SDSMT)
SLAC National Accelerator Laboratory
94025, 57701Menlo Park, Rapid CityCA, SDUSA, USA
University of Southern Maine
04104PortlandMEUSA
Syracuse University
13244SyracuseNYUSA
Tel Aviv University
69978Tel AvivIsrael
University of Tennessee
37996KnoxvilleTNUSA
University of Texas
76019ArlingtonTXUSA
ℎℎ Center for Neutrino Physics
Tufts University
02155MedfordMAUSA
Virginia Tech
24061BlacksburgVAUSA
Department of Physics
Wright Laboratory
University of Warwick
CV4 7ALCoventryUnited Kingdom
Yale University
06520New HavenCTUSA
R Castillo Fernandez
F Cavanna
G Cerati
Y Chen
D Cianci
J M Conrad
M Convery
L Cooper-Troendle
J I Crespo-Anadón
M Del
Tutto S R Dennis
P Detje
A Devitt
R Diurba
R Dorrill
K Duffy
S Dytman
B Eberly
A Ereditato
J J Evans
R Fine
G A Fiorentini
Aguirre R S Fitzpatrick
B T Fleming
N Foppiani
D Franco
A P Furmanski
D Garcia-Gamez
S Gardiner
G Ge
S Gollapinni
O Goodwin
E Gramellini
P Green
H Greenlee
W Gu
R Guenette
P Guzowski
L Hagaman
O Hen
C Hilgenberg
G A Horton-Smith
A Hourlier
R Itay
C James
X Ji
L Jiang
J H Jo
R A Johnson
Y.-J Jwa
D Kalra
N Kamp
N Kaneshige
G Karagiorgi
W Ketchum
M Kirby
T Kobilarcik
I Kreslo
R Lazur
Colorado State University
Fort Collins80523COUSA
Columbia University
10027New YorkNYUSA
Fermi National Accelerator Laboratory (FNAL)
University of Edinburgh
EH9 3FD, 60510Edinburgh, BataviaILUnited Kingdom, USA
Universidad de Granada
E-18071GranadaSpain
Illinois Institute of Technology (IIT)
Harvard University
02138, 60616Cambridge, ChicagoMA, ILUSA, USA
Kansas State University (KSU)
66506ManhattanKSUSA
LA1 4YW, United Kingdom Los Alamos National Laboratory (LANL)
Lancaster University
87545Lancaster, Los AlamosNMUSA
Massachusetts Institute of Technology (MIT)
The University of Manchester
M13 9PL, 02139Manchester, CambridgeMAUnited Kingdom, USA
University of Michigan
48109Ann ArborMIUSA
University of Minnesota
55455MinneapolisMnUSA
New Mexico State University (NMSU)
Las Cruces, NM88003USA
University of Oxford
OX1 3RHOxfordUnited Kingdom
University of Pittsburgh
15260PittsburghPAUSA
Rutgers University
08854PiscatawayNJ, PAUSA
South Dakota School of Mines and Technology (SDSMT)
SLAC National Accelerator Laboratory
94025, 57701Menlo Park, Rapid CityCA, SDUSA, USA
University of Southern Maine
04104PortlandMEUSA
Syracuse University
13244SyracuseNYUSA
Tel Aviv University
69978Tel AvivIsrael
University of Tennessee
37996KnoxvilleTNUSA
University of Texas
76019ArlingtonTXUSA
ℎℎ Center for Neutrino Physics
Tufts University
02155MedfordMAUSA
Virginia Tech
24061BlacksburgVAUSA
Department of Physics
Wright Laboratory
University of Warwick
CV4 7ALCoventryUnited Kingdom
Yale University
06520New HavenCTUSA
I Lepetic
K Li
Y Li
K Lin
B R Littlejohn
W C Louis
X Luo
K Manivannan
C Mariani
D Marsden
J Marshall
D A Martinez
Caicedo K Mason
A Mastbaum
N Mcconkey
V Meddage
T Mettler
K Miller
J Mills
K Mistry
A Mogan
T Mohayai
J Moon
M Mooney
Colorado State University
Fort Collins80523COUSA
Columbia University
10027New YorkNYUSA
Fermi National Accelerator Laboratory (FNAL)
University of Edinburgh
EH9 3FD, 60510Edinburgh, BataviaILUnited Kingdom, USA
Universidad de Granada
E-18071GranadaSpain
Illinois Institute of Technology (IIT)
Harvard University
02138, 60616Cambridge, ChicagoMA, ILUSA, USA
Kansas State University (KSU)
66506ManhattanKSUSA
LA1 4YW, United Kingdom Los Alamos National Laboratory (LANL)
Lancaster University
87545Lancaster, Los AlamosNMUSA
Massachusetts Institute of Technology (MIT)
The University of Manchester
M13 9PL, 02139Manchester, CambridgeMAUnited Kingdom, USA
University of Michigan
48109Ann ArborMIUSA
University of Minnesota
55455MinneapolisMnUSA
New Mexico State University (NMSU)
Las Cruces, NM88003USA
University of Oxford
OX1 3RHOxfordUnited Kingdom
University of Pittsburgh
15260PittsburghPAUSA
Rutgers University
08854PiscatawayNJ, PAUSA
South Dakota School of Mines and Technology (SDSMT)
SLAC National Accelerator Laboratory
94025, 57701Menlo Park, Rapid CityCA, SDUSA, USA
University of Southern Maine
04104PortlandMEUSA
Syracuse University
13244SyracuseNYUSA
Tel Aviv University
69978Tel AvivIsrael
University of Tennessee
37996KnoxvilleTNUSA
University of Texas
76019ArlingtonTXUSA
ℎℎ Center for Neutrino Physics
Tufts University
02155MedfordMAUSA
Virginia Tech
24061BlacksburgVAUSA
Department of Physics
Wright Laboratory
University of Warwick
CV4 7ALCoventryUnited Kingdom
Yale University
06520New HavenCTUSA
A F Moor
C D Moore
L Mora
Lepin J Mousseau
M Murphy
D Naples
A Navrer-Agasson
M Nebot-Guinot
R K Neely
D A Newmark
J Nowak
M Nunes
O Palamara
V Paolone
A Papadopoulou
V Papavassiliou
S F Pate
N Patel
A Paudel
Z Pavlovic
E Piasetzky
I D Ponce-Pinto
S Prince
X Qian
J L Raaf
V Radeka
A Rafique
M Reggiani-Guzzo
L Ren
L C J Rice
L Rochester
J Rodriguez
Rondon M Rosenberg
M Ross-Lonergan
G Scanavini
D W Schmitz
A Schukraft
W Seligman
M H Shaevitz
R Sharankova
J Shi
J Sinclair
A Smith
E L Snider
M Soderberg
S Söldner-Rembold
P Spentzouris
J Spitz
M Stancari
J St
T John
K Strauss
S Sutton
A M Sword-Fehlberg
N Szelc
W Tagg
K Tang
C Terao
D Thorpe
M Totani
Y.-T Toups
M A Tsai
T Uchida
W Usher
B Van De Pontseele
M Viren
H Weber
Z Wei
S Williams
T Wolbers
M Wongjirad
K Wospakrik
N Wresilo
W Wright
E Wu
T Yandel
G Yang
L E Yarbrough
H W Yates
G P Yu
J Zeller
C Zennamo
Zhang
The MicroBooNE Collaboration Universität Bern
CH-3012BernSwitzerland
Brookhaven National Laboratory (BNL)
Upton11973NYUSA
University of California
93106Santa BarbaraCAUSA
Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT)
University of Cambridge
Madrid E-28040CB3 0HECambridgeUnited Kingdom, Spain
University of Chicago
60637ChicagoILUSA
University of Cincinnati
45221CincinnatiOHUSA
Electromagnetic Shower Reconstruction and Energy Validation with Michel Electrons and 0 Samples for the Deep-Learning-Based Analyses in MicroBooNE
1 Mar 2022P JINSTK : Neutrino detectorsNoble liquid detectors (scintillation, ionization, double-phase)Time projection Chambers (TPC)Pattern recognition, cluster finding, calibration and fitting meth- ods A X P : 211011874
A : This article presents the reconstruction of the electromagnetic activity from electrons and photons (showers) used in the MicroBooNE deep learning-based low energy electron search. The reconstruction algorithm uses a combination of traditional and deep learning-based techniques to estimate shower energies. We validate these predictions using two -sourced data samples: charged/neutral current interactions with final state neutral pions and charged current interactions in which the muon stops and decays within the detector producing a Michel electron. Both the neutral pion sample and Michel electron sample demonstrate agreement between data and simulation. Further, the absolute shower energy scale is shown to be consistent with the relevant physical constant of each sample: the neutral pion mass peak and the Michel energy cutoff.
A : This article presents the reconstruction of the electromagnetic activity from electrons and photons (showers) used in the MicroBooNE deep learning-based low energy electron search. The reconstruction algorithm uses a combination of traditional and deep learning-based techniques to estimate shower energies. We validate these predictions using two -sourced data samples: charged/neutral current interactions with final state neutral pions and charged current interactions in which the muon stops and decays within the detector producing a Michel electron. Both the neutral pion sample and Michel electron sample demonstrate agreement between data and simulation. Further, the absolute shower energy scale is shown to be consistent with the relevant physical constant of each sample: the neutral pion mass peak and the Michel energy cutoff. K : Neutrino detectors; Noble liquid detectors (scintillation, ionization, double-phase); Time projection Chambers (TPC); Pattern recognition, cluster finding, calibration and fitting methods
Introduction
The primary goal of the MicroBooNE experiment is investigate the anomalous excess of electron-like events observed in the MiniBooNE detector [1]. The anomaly is an excess of single electromagnetic showers with a peak shower energy in the 200-500 MeV observed in the MiniBooNE Cherenkov detector. The anomaly is here referred to as the Low Energy Excess (LEE). The MicroBooNE collaboration has developed several analyses designed to isolate LEE events, where event here refers to one detector readout record. The reconstruction tools presented here make up the shower reconstruction phase of the deep learning (DL)-based analysis [2]. A preliminary version of the (DL)-based analysis utilized in this study can be seen in ref. [3]. This analysis isolates events with 1 electron and 1 proton (1 1 ) in the final state. The neutrino energy is reconstructed as:
= + + + − ( − ),(1.1)
where indicates kinetic energy, is mass, is nuclear binding energy, and and indices indicate the electron and proton, respectively. The proton kinetic energy is reconstructed from the length of the track and the known energy deposited per unit length in liquid argon [4]. This article describes the reconstruction of the electron kinetic energy, which, for our signal of interest, ranges from 35 MeV to 1200 MeV. Across this range, the topology of the deposited electron energy changes from track-like at low energy to shower-like at high energy where photon radiation dominates. In this article, all electron energy deposits will be called showers despite the variety of topologies. The discussion will include both electrons and photons. This article reports the method of electromagnetic shower reconstruction used in the DLbased analyses. The simulation-derived shower-charge-to-energy conversion value is presented in section 3. We then use two low-energy samples to validate the shower reconstruction. The first sample consists of photons produced by neutral pions ( 0 ) decays and is described in section 4. The second sample consists of Michel electrons produced when a stopped muon from a charged current (CC) interaction decays and is described in section 5. We provide data-simulation comparison plots for each sample that we use to validate the relative shower energy scale between data and simulation. The use of the simulation-derived energy calculation is then further verified on data by measuring the agreement of data and simulation with two well-measured physical quantities: the 0 invariant mass and the cutoff of the Michel energy spectrum.
Preliminary Reconstruction of Events in the Deep Learning Analysis
The MicroBooNE detector is a 2.6 m × 2.3 m × 10.4 m Liquid Argon (LAr) Time Projection Chamber (TPC) filled with 85 metric tons of LAr [5]. The ionization electrons produced by charged particles in the event drift with a velocity of 0.1098 cm/ s through an applied potential of -70 kV to three wire planes [6]. The orientation of wires in the induction planes, and , are +60 • and −60 • relative to vertical. The collection plane, , has vertical wires. The wire spacing in all planes is 0.3 cm. The detector also contains a light collection system made up of 32 photomultiplier tubes that are used for triggering and initial event selection. The light collection system is not leveraged in the shower reconstruction work presented here.
The studies in this article use MicroBooNE data taken from 2016 to 2018 which were recorded over three run periods with 1.75 × 10 20 protons-on-target (POT) in Run 1, 2.70 × 10 20 POT in Run 2 and 2.43 × 10 20 POT in Run 3. There are various Monte Carlo (MC) simulation samples that are used to build the reconstruction algorithm. The simulated neutrino events are overlaid with off-beam cosmic muon data. These "overlay samples" are used throughout the rest of this article. They contain all types of simulated neutrino events expected in the given POT. There are three samples of this type, one corresponding to cosmic information collected in each of the three MicroBooNE data runs used in the study which corresponds to 6.67 × 10 20 total POT. Additionally, for the 0 study we incorporate a specialized overlay sample containing a larger number of events with a 0 that decays to two photons in the final state (high POT 0 sample). The incorporation of this sample allows for a reduction of the statistical uncertainty on the simulation. For the Michel study, we incorporate an overlay sample of neutrino events interacting outside of the MicroBooNE detector, as muons from these external CC interactions which enter the detector, come to a stop, and then decay provide an additional source of Michel electrons.
Various final state topologies occur at the neutrino energies observed in MicroBooNE. Those relevant to the DL-based LEE analysis have two particles attached to a vertex: 1 1 and 1 1 events from and charged-current quasi-elastic or meson exchange current scattering, other types of induced events with only two particles reconstructed at the vertex (such as 0 events with disconnected photons), and cosmic ray muons. The first steps of the "low level reconstruction" is to isolate the two-particle-vertices of interest. The methods have been described elsewhere [3,4]; we briefly review them here.
Preparation for reconstruction begins with algorithms to tag and discard the charge associated with the cosmic rays [7,8]. Next the waveform data from the wires in each plane are converted to "images" which are two-dimensional distributions with wire number along the axis and drift time along the axis. The intensity of each image "pixel" is given by the integrated reconstructed charge waveform from six 0.5 s TPC time increments after applying noise-filtering [9] and signal processing [10,11]. The six 0.5 s TPC time increments is comparable to the 3 mm wire spacing after accounting for the electron drift velocity. The intensity of each pixel is referred to as "Q (charge)" in this article. An example event display of this data format is shown in figure 1 (a). The event display shown is of the collection plane ( plane) of a simulated CC 0 event which passes the shower reconstruction stage. The z-axis represents the charge ( ) at each (wire,time) pixel. To make the deposited charge more clear in the display, has been given a maximum value of 100. In the event shown, the image is cropped centering at the neutrino interaction vertex. Shower reconstruction is also shown which will be discussed in section 3.
Any charge tagged as associated with a cosmic ray is removed at this stage. The , , and images are then passed into the deep learning convolutional neural net, called "SparseSSnet,"which is a semantic segmentation algorithm that labels pixels as track-like or shower-like [12]. Figure 1 (b) shows the SparseSSNet shower scores of the same event as (a) with the track-like particles masked out. The next step is to identify a "two particle" vertex, followed by 3D reconstruction of each particle. This process is described in detail for the 1 1 sample in ref. [4], and the 1 1 reconstruction follows similar steps. In short, the vertex algorithm searches for a characteristic "vee" shape where two particles meet at a vertex identifying cases where the particles are longer than 3 cm and the opening angle in at least one plane is greater than 10 • . Based on the SparseSSnet pixel tagging, the vee may be formed of a "track-track" pair or a "shower-track" pair [12]. For any given vertex, more than one vee may be found if there are more than two particles emitted from the interaction point. Also, more than one vertex may be found at different points in the same event. An important example of this, used in this article, is the case of 1 1 final state in which the muon decays to a Michel electron. This results in one vertex at the interaction point and another at the decay point. Because vertices are also found on cosmic rays, particularly those that stop and decay, many vertices are identified in any given event. All are passed to 3D reconstruction. At this stage in the reconstruction, multiple vertices can be reconstructed in each event. In later selections, a single vertex from the event will be chosen.
3D reconstruction of track-like objects (primarily protons and muons) is described in detail in ref. [4] and summarized here. The algorithm begins at the reconstructed vertex and follows ionization trails outward in 3D, clustering the charge into "prongs." A prong is defined here as a collection of continuous charge, but it need not be a single line of charge. The prong may comprise connected branches of charge as will be the case for electromagnetic showers. Each prong is assumed to come from one particle. In the case where two prongs are identified, we identify the proton as the prong having the higher average pixel-based ionization density. A fiducial volume containment requirement is enforced on all prongs. This requirement uses the distance of a prong from the edge of the detector as the minimal distance from all the prong's 3D points to a detector edge. It is required that either the distance of both prongs is > 5 cm from the edge or else that the combined distance of both prongs is > 15 cm from the edge.
In order to associate a prong with the SparseSSnet identified pixels, the 3D prong is projected onto the 2D images described above. The pixels in this projection are then matched to the prong. The kinetic energies of track-like prongs are calculated using the track lengths. The direction of track-like particles is also reconstructed as described in ref. [4]. Energy deposits from electrons and photons will suffer gaps due to radiated photons which the 3D track reconstruction cannot handle. Shower-like prongs therefore need a seperate reconstruction as described in section 3.
Electromagnetic Shower Energy Reconstruction
The electromagnetic shower reconstruction algorithm is run to find any associated shower particles and reconstruct their kinetic energies once a candidate vertex is isolated. The method described here builds on a previous MicroBooNE shower reconstruction described in ref. [13]. The algorithm described retains many vital features of the previous version, especially the use of a semantic segmentation neural network for pixel labeling. Important updates have been added including many simplifications of the algorithm made possible by the improved SparseSSnet.
The first step of the reconstruction is to mask the image to only use the pixels identified as shower by SparseSSNet. SparseSSNet outputs a value for each pixel indicating how likely it is that the pixel is part of a shower. This value, called shower score, falls between 0.0 and 1.0 where 1.0 indicates a shower-like pixel. Pixels with a shower score of >0.5 and intensity >10 (charge) counts are kept, all other pixels are masked out for the rest of the shower reconstruction. A threshold of 10 is used to remove wire noise. This cut is standard among all tools used in this analysis. The value was chosen based on the distribution of from minimum ionizing particles (MIPs),which peaks at ≈ 40 . A template isosceles triangle is then placed with its apex at the reconstructed vertex position, pointing in the positive wire direction. The triangle is optimized to choose the shower direction, length, and opening angle for which the triangle contains the most pixels with non-zero charge. These parameters each start at the minimum value shown in Table 1. In order to allow for showers that are detached from the vertex, a gap parameter is introduced allowing the triangle to start further from the vertex.
Once a first shower candidate has been found by the reconstruction algorithm, the pixels found in the shower are masked out. If the total amount of charge remaining passing the cuts of a shower score of >0.5 and intensity > 10 (charge) is > 5000 , the second shower algorithm is run on the masked image. The total range of allowed values for each of the template triangle parameters is shown in Table 1. In this table "first shower" refers to the shower found in the first pass of the shower reconstruction which is aimed at finding showers near the reconstructed vertex. "Second shower" refers to the shower found on the second run of the algorithm and has expanded parameters to search for detached showers as described in the following paragraphs. The parameters in each case are optimized sequentially in the order of direction, gap size, opening angle, and length. Figure 1 shows an example display demonstrating the 2D shower reconstruction on a simulated CC 0 event. The SparseSSNet shower-like particles are shown in (b) with the track-like particles masked out. The final optimized showers are shown in red (first shower) and magenta (second shower). In this example, the algorithm found the proper gap of the first shower, but not the second shower. This is acceptable as gap size is not a value that is utilized in any other part of the DL analysis. The reconstructed energy of each shower is also reported. Energy reconstruction of electromagnetic showers is a crucial component of the LEE analysis this work supports, which is designed to measure electrons from CC neutrino interactions over a a broad energy range from 35 to 1200 MeV. This analysis aims to develop and validate a energy reconstruction procedure for showers in this broad energy range. To determine the energy of each shower, the charge of all shower pixels enclosed in the -view triangle is integrated. This total shower charge will hereby be denoted by ℎ . We use the -view, which is the collection plane, because it has the highest signal-to-noise ratio of the three planes [9].
To convert the reconstructed charge to energy, the ℎ in a sample of simulated events is compared to the generated energy of the electrons in the events. A ℎ -to-MeV conversion line is determined as described below and shown in figure 2. As the focus of the larger analysis is 1 1 events, a sample of simulated 1 1 events are used. The simulated electron energy is plotted versus the reconstructed -view total shower charge sum for events in a simulation sample selected by the MicroBooNE DL-based 1 1 analysis. In each vertical bin, the peak is found with the help of a Gaussian distribution (represented by the black points in figure 2). The edge bins which have smaller statistics are excluded. Two examples of the Gaussian fits are shown in figure 3.While the Gaussian fits capture the bulk of simulated events well, one can see a tail in each distribution out to higher simulated electron energies which is not captured by the Gaussian. This is expected, as these tails correspond to electron showers which are not fully reconstructed (i.e., showers which pass through unresponsive wires or exit the active volume). The points are then fit to a line. The slope of this line is used for the ℎ -to-MeV conversion and is referred to in the rest of this article as − . While a Gaussian is not a perfect fit to the data in, the use of − , derived from these points, will be validated in section 4 and section 5. The resulting equation is:
Electron: [MeV] = (1.26 ± 0.01 × 10 −2 ) × ℎ [ Counts]. (3.1)
where the error corresponds to the uncertainty on the linear fit, which represents the statistical error on the simulated electron sample used for the fit. Various detector effects could cause this fit value to change. Specifically the amount of detected energy will depend on the position, amount of energy deposited, and the orientation of the particle's trajectory with respect to the wires [10,11,14]. Additionally, the value of − derived here assumes good argon purity and could be affected by periods of low purity. Potential systematic uncertainty could be introduced by these effects. The results shown in Sec 6 give an estimate of the size of the detector effect.
Using this shower energy calculation, we look at the energy resolution for a sample of simulated CC events containing an electron and no final state 0 . This isolates electrons from photons which will be discussed further in section 4.1. The following selection criteria are used:
1. Reconstructed vertex is less than 5 cm from simulated neutrino interaction vertex; 2. One simulated electron contained in event; The energy resolution, defined here as:
= − (3.2)
for this sample of simulated events is seen in figure 4. The mean is at -0.07 and the RMS is 0.22. The shower energy reconstruction presented here will be utilized and validated in section 4 and section 5. As stated earlier, the shower energy is calculated using a MicroBooNE simulation sample comprising of simulated neutrino events overlay with off-beam cosmic ray data. It is therefore notable that the energy calculated with this sample works well when applied to data samples as shown in the next sections. It will be seen that the 0 and Michel − samples both have good data/simulation agreement using eq. (3.1). A further test is done for each sample that finds the charge-to-energy conversion factor that gives the best agreement to known physical values (the 0 rest mass and the Michel electron spectrum cutoff). This test shows great data/simulation agreement with each sample. The results are also comparable to − given in eq. (3.1), even at different energy scales, validating the use of this linear conversion factor.
Neutral Pion Sample
The first sample used to analyze the performance of the shower reconstruction is a sample of 0 events. The identification and reconstruction of 0 events is presented. This is followed by both a verification of the ℎ -to-MeV conversion value ( − ) agreement to data and simulation, and a verification of agreement between data and simulation which is accomplished by using a well-measured physical quantity: the 0 invariant mass (135 MeV).
Identification and Reconstruction
The reconstruction of 0 events relies on the shower reconstruction described in section 3 and introduces 3D shower reconstruction. The starting point is either a track-track or track-shower vertex. The shower reconstruction is then applied as discussed above, centering on clusters of electromagnetic charge, but not requiring that the charge be attached to the starting vertex. This leads to the bulk of 0 events selected for this analysis matching two specific topologies. The most prevalent 0 topology in this study is from charge current (CC) 0 where the scattered muon and the proton from the Δ decay form the vertex, and there are two disconnected electromagnetic showers from 0 decays. The second is from neutral current (NC) 0 where one photon converted within the 0.3 cm wire spacing and thus forms a vertex with the proton, while the second photon is displaced from the vertex. As a result, contributions to the 0 selection discussed here will come from NC 0 and CC 0 , as well as CC − and CC + events where the + / − undergoes charge exchange within the nucleus.
To fully calculate the kinematics of the 0 event we must determine both the energy and the 3D angle of the electromagnetic showers. This is essential for reconstructing the 0 invariant mass-a useful quantity to test the shower reconstruction described in section 3. In order to cluster a full 3D shower, the 2D projections on the different wire planes are compared for overlap in time. The overlap fraction is defined as the fraction of shower pixels in the collection plane shower that overlap in time with shower pixels from a 2D shower in another plane, in which the and planes are considered separately. If the overlap fraction is > 0.5 in either or both planes, the pixels that overlap between the collection plane shower and the shower in another plane with the highest overlap fraction are used to calculate a cluster of 3D shower points. The direction is found by using the calculated center of the 3D point cluster and the event vertex.
This 3D reconstruction leads to what is referred to as the " 0 pre-selection cuts". These are requirements that are necessary in order to reconstruct two 3D showers.
1. Vertex passes fiducial volume containment requirement (described in section 2); 2. Two collection plane showers, each with reconstructed energy greater than 35 MeV;
3. Both collection plane showers have an overlap fraction with a shower in another plane greater than 0.5;
4. If a collection plane shower matches with showers in both the and planes, the one with the highest overlap fraction is chosen; and 5. The two collection plane showers cannot match to the same shower in another plane.
After the " 0 pre-selection cuts" have been applied, a substantial number of selected events have showers that are mis-reconstructed. The sample also contains a large number of backgrounds such as cosmic muons. To improve the selection, the following requirements are introduced that reduce mis-reconstruction and remove backgrounds. These are referred to as "box cuts" as they are hard cuts designed to remove background events at the tails of the distributions of various variables.
One of the variables used in these "box cuts" is a Δ mass test variable. This value is calculated for all those events passing the " 0 pre-selection cuts". The 4-vector of the reconstructed showers are used along with the 4-vector of the proton-like prong. The reconstruction of the proton-like prong is discussed in ref. [4]. These three objects are assumed to have come from a Δ decay and are therefore used to reconstruct a Δ rest mass. The tails of this distribution are comprised of mis-reconstructed 0 events and cosmic muon backgrounds which allows for another box cut. The box cuts are then:
1. Reconstructed 0 mass is less than 400 MeV; 2. Reconstructed energy of the leading photon is greater than 80 MeV;
3. The charge sum of all pixels (both track and shower) within 2 cm of the vertex is greater than 250 Q counts;
4. Leading shower reconstructed angle w.r.t. beam direction is less than 1.5 radians;
5. The angle between the two photons is less than 2.5 radians; and 6. Δ mass test variable is between 1000 and 1400 MeV.
Here, "leading photon" refers to the simulated photon with the highest energy. The reconstructed leading shower is the shower with the highest ℎ . An additional requirement on the 1e1p Boosted Decision Tree (BDT) of < 0.7 is further added at this stage to maintain blindness to the LEE for this study as required by the MicroBooNE blindness procedure [3]. This BDT is used to select events with one electron and one proton, so this is a relatively small cut that is a flat ≈ 3% effect. Events above this threshold may be mis-reconstructed 1e1p events to which we are currently maintaining blindness.
Using the shower energy calculation from the electron sample on both electron and photon showers assumes that the energy of both shower types is reconstructed the same way. To demonstrate that this is valid, the same simulated energy vs ℎ plot and fit is performed on two samples of It is seen from the fit results that the leading photon fit very closely matches the electron fit as expected while the sub-leading photon fit does not match as closely. This is due to the worse reconstruction in the sub-leading photon. Therefore, the reconstruction uses the charge-to-energy ( ℎ -to-MeV) conversion value found for the simulation electron sample, − . For the purpose of studying this sample, Monte Carlo simulation has been broken into various categories. "NC 0 " are neutral current 0 events with a well reconstructed vertex, which is a vertex within 5 cm of the true generated vertex. "CC 0 " are defined similarly for charged current 0 . "Offvtx 0 " are 0 events with poorly reconstructed vertices. Here, off-vertex means that the reconstructed vertex is further than 5 cm from a true generated neutrino vertex. "Non 0 " events are broken into on and off vertex as well. events are all events that originated from a . Cosmic background events also remain after selection.
Simulated energy resolution of the photons is presented in figure 6, where resolution is defined in eq. 3.2. This plot is made using the specialized high POT simulation sample of containing only events with a 0 in the final state. For the purpose of scaling the different samples and background contributions, the events are POT scaled to match the total data exposure. This distributions in this plot have all of the 0 selection cuts applied. The energy resolution is shown separately for leading (highest energy) photon and sub-leading photon. Table 2 shows the mean and RMS of the distributions shown in figure 6. Both the leading and sub-leading photon mean is close to zero. The leading photon has a smaller RMS than the sub-leading photon, which is more broad and has a larger tail. The resolution of photons from 0 is worse than that seen in the electrons, but they have similar bias as indicated by the mean. There are two main causes for the difference between leading and sub-leading photons. The first is that the sub-leading photon is generally the lower energy of the two. Failing to reconstruct a small number of pixels will have a larger effect on a lower energy shower which has fewer true shower pixels associated with it. The other cause are events where the leading and sub-leading shower are close together. Part of the sub-leading shower is reconstructed as part of the other shower as the leading shower is prioritized. Of these two sub-leading photon reconstruction failure modes, the first is dominant. Some of the other mis-reconstructed photons in both the leading and sub-leading plot are caused by mistakes in SparseSSNet, overlapping cosmic rays that were not removed properly, and unresponsive wires in the collection plane. Figure 7 shows a data to MC simulation comparison of the reconstructed photon energies in the 0 sample. The uncertainty bars on the simulation distribution represent the statistical uncertainty.
The total number of simulation events has been scaled to match the total number of data events. This plot uses both the MicroBooNE overlay simulation samples combined with the high POT 0 simulation sample and data as described in section 3. The spectra seen in this figure are at higher reconstructed energies than the Michel − in section 5 and closely mirror the shower energy scale of the MiniBooNE LEE. The 2 which is reported in the caption of this plot is a combined Neumann-Pearson (CNP) 2 [15]. The 2 is defined as:
2 = ∑︁ ( − ) 2 3 1/ +2/ ≠ 0 ( − ) 2 2 = 0 (4.3)
where and are the number of predicted and observed events in a given bin. This figure uses the same simulation sample with the same selection requirements as figure 6. The distribution is characterized in Table 2. The mean is close to zero indicating little bias. The 0 rest mass can now be reconstructed using the following equation:
0 = √︂ 4 sin 2 ( 2 ) ( 1 ) ( 2 ) (4.5)
where 1 is the leading photon energy and 2 is the sub-leading photon energy. The result of this reconstruction is shown in figure 9. As in figure 7, the total number of simulation events has been scaled to match the total number of data events. The distributions of both data and simulation peak around 135 MeV, which is the accepted 0 rest mass.
Validation of Agreement between Simulation, Data, and True Rest Mass
The 0 sample is next used to verify the agreement in data and simulation of the shower energy scale using the known 0 invariant mass 0 . Test points are found representing the best-fit of the ℎ -to-MeV conversion factor ( ) to 0 = 135 [ / 2 ]. This is done for a sample of each simulation and data. The goal for each is to find the value of that yields a 0 mass distribution that peaks closest to the true value of 135 [ / 2 ]. This value of is then compared between data and simulation and to the electron − value found in Section 3.
To find the optimal , the following 2 formula is minimized:
2 = ∑︁ (135[ / 2 ] − 0 ) 2 . (4.6)
where is each 0 event in the given sample, is 29.8 [ / 2 ] based on the width of a Gaussian fit to the good simulation 0 distribution (the NC 0 and CC 0 categories in figure 9).
0 = √︂ 4 sin 2 ( 2 ) ( × ( ℎ ) 1 ) ( × ( ℎ ) 2 ) (4.7)
where ℎ is the reconstructed shower charge and is the reconstructed opening angle between the two showers. The same 2 formula is minimized for both MC simulation and data. The 1-range is calculated on these fit points by looking at the range of values for each data and simulation which give a 2 value satisfying the Wilks' theorem condition | 2 ( ) − min [ 2 ( )] | < 1 [16]. The 2 distributions can be found in figure 10. The resulting values can be seen in the first two rows of Table 3. Excellent agreement is seen between the data and the simulation best fit points indicating that the simulation derived shower conversion factor is valid to use on data. Another important consideration is how closely the best fit values derived here match − in eq. (3.1). The best fit value of derived from the 0 sample has the potential to be affected by many factors. The largest of these factors is the amount of background events selected. The 0 selection presented in section 4.1 contains many background events, which should not necessarily reconstruct as a 0 mass of 135 [ / 2 ]. To account for this background in simulation a second fit is performed only using good simulation events. In this instance good simulation is defined as simulated events that pass all 0 cuts, have a simulated final state 0 , and have a reconstructed vertex within 5 cm of the true simulated neutrino interaction vertex. These are the only types of events in the selection which should result in a reconstructed 0 mass of 135 MeV. A further cut of 0 mass < 200 MeV is added to prevent mis-reconstructed events from having a large effect on the 2 .
To account for the background events in data, the optimal found previously is shifted by the same amount that the MC is shifted when backgrounds are included (7.11 × 10 −4 ) as seen the last two rows in Table 3. Data (un-shifted) are the results from minimizing eq. 4.6 over all data points. Data (shifted) shows the data points applying the shift found by comparing the fit over all simulation to the fit over good simulation. This shift retains the excellent data and simulation agreement. The result on data only differs by 1.6% from the value seen in eq. (3.1). This indicates that, at the photon energy scale seen in the 0 sample, the charge-to-energy conversion factor is valid. Data and simulation agreement of these results and agreement to eq. (3.1) is discussed further in section 6 and examined in combination with the results from the Michel − sample. Table 3: The best value of (MeV/ ℎ ) for each data and MC simulation sample and the range found using Wilks's theorem. Results are shown before accounting for background (top two rows) and after (bottom two rows).
Sample
[
Michel Electron Sample
The second sample used to validate the shower energy reconstruction consists of Michel electrons from decays of -sourced stopped muons in the detector. This is the first time a Michel sample has been reconstructed in a LArTPC which is dominated by Michels from interactions. As in section 4, we begin by describing the event selection criteria for this sample. Next, we examine the data/simulation agreement in the Michel shower energy spectrum. Finally, we assess the agreement of the Michel sample with the physical Michel cutoff of /2 = 52.8 MeV through a fit procedure. The data and MC simulation results of this fit agree, validating the use of the simulation-derived sh -to-MeV on data. Both the data and simulation fits also show consistency between the simulation-derived sh -to-MeV conversion value validating the absolute shower energy scale of the DL-based analysis in the low energy region ( 50 MeV).
Identification and Reconstruction
The Michel electron sample has been chosen in order to validate the reconstructed energy scale of lower-energy electrons, slightly below the energy scale of electrons in the low-energy excess search. Previous work has been performed in MicroBooNE using a larger sample than presented here, as seen in ref. [17]. The study presented here uses a different selection and is designed to test the reconstruction of showers used in the DL low-energy excess search. However, as shown below, our results are consistent with those shown in [17] in both the reconstructed Michel energy spectrum and corresponding energy resolution.
Muon-Michel vertices are identified through the following requirements:
1. Two prongs at the vertex;
2. Long prong track-length > 100 cm (candidate muon);
3. Short prong track-length < 30 cm (candidate Michel);
4. Long track consists of < 20% SparseSSNet shower-like pixels (candidate muon); 5. Short track consists of > 80% SparseSSNet shower-like pixels (candidate Michel); and 6. < 0.5 radians.
where is the azimuthal angle of the muon with respect to the horizontal plane, where = 0.5 rad corresponds to downward-going muons. Lastly, in events with more than one selected vertex, we keep only the first vertex. This strategy was chosen as it does not bias the Michel energy spectrum. There are two types of Michel electrons that are isolated before the selection. The first are Michels in neutrino events, which can be compared between simulation and data. The second are Michel electrons from stopped cosmic muons. As our simulation samples contain simulated neutrino events overlaid with cosmic data, all of the Michels from stopped cosmic muons come from actual data. Therefore, the only Michels that are truly simulated are those on simulated neutrino events. It is important for these studies to be certain that the Michels in the simulation sample do not come from the stopped cosmic muons. This ensures we are comparing Michels in neutrino events from data to simulated Michels. This is achieved by the final requirement on the muon polar angle , which removes a majority of the predominately downward-going cosmic muons as shown in figure 11. As discussed in section 2, the DL vertices search for the intersection of two "prongs".
A muon decay into a Michel electron forms this pattern and is therefore often reconstructed at the initial vertex stage of the event reconstruction.
For Michel electrons in events passing these cuts, the electromagnetic shower reconstruction algorithm from section 3 is applied to find ℎ . This is then converted to a shower energy via eq. 3.1. Figure 12 shows the shower energy distribution for Michel electrons in both data and simulation. Note that data here come from an open beam data set of corresponding to ≈ 5.3 × 10 19 POT. The simulated events are broken into various categories to indicate which type of event caused the muon. The majority of events come from interactions within the active detector volume. Some events from cosmic muons and muons from interactions outside the active volume remain in the sample. One can see that the high-end tails for the shower energy distribution in both data and simulation fall off around 60 MeV as expected. While not as sharp as the cut-off in ref. [17], the results are consistent. The high energy tail above the true value of 52.8 MeV likely is due to over estimation of shower energy reconstruction seen in figure 4. The shower energies of this sample are much lower than those seen in the 0 sample. The good data/simulation agreement within statistical uncertainty indicates that the shower algorithm performs well down to low energies. This agreement will be quantified further in the next sections.
Validation of Agreement between Simulation, Data, and True Michel Cutoff
We now perform a validation of the absolute shower energy scale and analyze the data/simulation agreement with the Michel sample analogous to the 0 study described in section 4.2. In the case of the Michel cross-check, we fit the sh (reconstructed shower charge) spectrum shown in figure 13 to the following five-parameter function:
( = sh cutoff ; , ) = ∫ 1 0 (3 2 − 2 3 ) 1 √ 2 2 exp −( − ) 2 2 2 (5.1)
where:
= √︄ 2 1 + 2 2 cutoff + 3 cutoff 2 . (5.2)
Here, ( ) represents a parameterization of the true Michel spectrum convoluted with a Gaussian representing charge resolution [18]. is a floating normalization parameter, and { 1 , 2 , 3 } represent contributions to the charge resolution corresponding to a constant noise term, a statistical charge-counting term, and a Gaussian noise term, respectively. cutoff represents the cutoff of the Michel shower energy spectrum, which, after the sh -to-MeV conversion, should correspond to the Michel energy cutoff of /2 ≈ 52.8 MeV. The integration over the variable represents a scan over the simulated shower charge spectrum. The expression is invariant under
∫ 1 0 ( ) → −1 cutoff ∫ cutoff 0 ( * ℎ / cutoff ) * sh ,
where (. . . ) represents the integrand in eq. (5.1). We first fit ( ) to the Michel spectrum in data and simulation by varying all five parameters cutoff , , 1 , 2 , 3 . This is done by minimizing the 2 :
2 ( cutoff , , 1 , 2 , 3 ) = ∑︁ ( − ( = ( sh ) cutoff ; , 1 , 2 , 3 ) i,stat. 2 (5.3)
where is the number of observed Michel events in sh bin and i,stat. = √ is the Poisson error. There are 12 shower charge bins ranging from 0 -6000 counts in this fit. Figure 14 shows 2D confidence regions for cutoff versus the different resolution parameters. They are calculated by fixing the remaining three parameters at their best fit values and using Wilks's theorem for two free parameters. As shown in figure 14, the fit generally prefers a large contribution from the flat resolution term 1 . In data, one can see that the 1 1 contour prefers a fractional resolution of ≈ 0.3, while the 1 2 and 3 contours are both consistent with zero. In simulation, the 1 1 contour prefers a fractional resolution of ≈ 0.25. The 1 3 contour is consistent with zero, but the 1 2 contour prefers a value of ≈ 13 [Q counts] 1/2 . This turns out to be a similar contribution when compared to the flat 1 term. A near-flat energy resolution for Michel showers is consistent with previous MicroBooNE work [17]. Next, the parameters { , 1 , 2 , 3 } are fixed at the minimum 2 values (respectively for data/simulation) and a 2 scan over cutoff is performed, this time only including the tail of the sh spectrum ( sh >3000 Q counts). The purpose of this one-dimensional scan is to obtain the 2 minimum and corresponding 1 interval on cutoff using Wilks's theorem for one fit parameter [16]. Figure 13 shows the observed Michel shower charge spectra in data and simulation along with their respective best fits from the 1D scan. Figure 15 shows 2 as a function of cutoff . The Wilks's theorem 1 interval on cutoff corresponds to the points for which 2 ( cutoff )−min{ 2 ; cutoff } ≤ 1. In order to get the charge to energy conversion factor from cutoff , = 52.8 MeV cutoff . The best fit and 1 intervals for are given in Table 4 along with the 2 /NDF of the best fit. One can see excellent agreement between data and simulation, demonstrating the consistency of the shower reconstruction. Note that the 1 interval on cutoff is larger in data than in simulation-this is because the statistical error on the data Michel sample is larger than that on the simulated Michel sample. The data/simulation agreement and consistency with eq. (3.1) demonstrated by this study are discussed further in section 6 in combination with the results from the 0 sample.
Combined Validation of Reconstructed Shower Energy
Both the data/simulation agreement of the shower reconstruction and the absolute scale of the simulation-derived ℎ -to-MeV conversion are validated using our two samples by utilizing the 0 invariant mass of ≈135 MeV and the Michel electron spectrum cut-off at ≈52.8 MeV. As described in sections 4.2 and 5.2, we have obtained comparison points separately for data and simulation in each sample. These points are shown with statistical uncertainties in figure 16. The ℎ -to-MeV conversion factor or values found in section 3 for electrons, leading photons, and sub-leading photons are shown by shaded bands in figure 16. In principle, one expects agreement between the points and the electron and leading photon calibration line. The 1 ranges in the value from both the data/simulation Michel cutoff study and the data/simulation 0 mass study agree well with the best-fit values from simulated electrons and leading photons. Agreement with the sub-leading photon line is not necessarily expected because of the reconstruction failure cases discussed previously. Table 5 shows the agreement of data and MC simulation for each point, as well as the agreement of each data and simulation point to the electron best fit value from eq. (3.1). It is seen here that the best fit values agree between data and simulation for each sample. This validates the use of the same simulation-derived sh -to-MeV conversion value ( − ) for both data and simulation. The sh -to-MeV photon calibration lines (eq. 4.1 and eq. 4.2) are also included for reference. The shaded regions represent the statistical uncertainty of this given calibration line.
While the best fit values for each sample do not exactly match − within statistical uncertainty, there are factors that may affect this value. These include: detector response modeling, sub-leading photon reconstruction in the 0 sample, and backgrounds in the Michel − sample. Therefore, the 2-6% difference gives an estimate of the scale of the possible data to simulation bias on the shower energy reconstruction. This size effect is acceptable for use in the DL LEE investigation. The < 6.5% difference between each ( , ) and − gives the scale of the detector systematic uncertainty in this reconstruction process.
In addition, as shown in figure 7 and figure 12, the showers found in the two samples are at different energy ranges. The assumption has been made in this analysis that the sh -to-MeV factor does not change with shower energy. The samples cover the range of values of interest for the DL 1e1p analysis. We conclude that the ℎ -to-MeV value given in eq. 3.1 is valid for EM showers in both data and simulation at the energy ranges and precision relevant for the MicroBooNE LEE search.
Conclusions
This article has reported the updated method of electromagnetic shower reconstruction for the MicroBooNE DL-based LEE analysis. Two samples that allow us to validate our shower reconstruction have been presented: photons produced by the decay 0 s, and Michel electrons produced when a -sourced stopped muon decays. The reconstruction and selection for each sample was described. The samples show good data/simulation agreement. The shower energy calculation uses a MC simulation-derived ℎ -to-MeV conversion factor. The absolute scale of the conversion factor, as well as its application to EM showers in both data and simulations, is validated using the 0 invariant mass of ≈135 MeV and the Michel electron cut-off at ≈53 MeV. Excellent data/simulation agreement is seen in this study. The ℎ -to-MeV conversion value used in the DL-based analysis is shown to be consistent with both of these physical quantities. The results we present here form the foundation for the MicroBooNE LEE DL-based analysis of 1 1 events that will be released in the future.
Figure 1 :
1Event displays of a simulated CC 0 event. (a) shows the raw Q image and (b) shows the SparseSSNet shower score with track-like particles masked out. In both (a) and (b) the leading reconstructed photon is represented by the red triangle and the sub-leading reconstructed photon (shower 2) is represented by the magenta triangle.
Figure 2 :
2Simulated electron energy vs ℎ for a sample of generated 1 1 events. The linear fit is used in the shower energy calculation. Decision Tree (BDT) score is greater than 0.7[3].
Figure 3 :
3Example distributions of simulated electron energies (solid lines) and corresponding Gaussian distributions (dashed lines) within two different shower charge sum ranges. The peaks of the Gaussian fits are used to generate the black points in figure 2.
Figure 4 :
4The energy resolution for a sample of simulated electrons as described by eq. 3.2. The axis has the raw number of simulated events without scaling. The dashed vertical line is included at = 0.0 for reference.
photons: leading and sub-leading photons from a sample of simulated CC 0 events. To ensure this fit is performed only over well-reconstructed events, the 0 selection box cuts are applied. The results are shown in figure 5. The resulting fit equations are: Leading Photon: [MeV] = (1.25 ± 0.02 × 10 −2 ) × ℎ [ Counts]. (4.1) Sub-leading Photon: [MeV] = (1.20 ± 0.02 × 10 −2 ) × ℎ [ Counts]. (4.2)
Figure 5 :
5Simulated photon energy vs ℎ for a sample of generated CC 0 events with the 0 selection applied. The best fit is shown for this sample as well as the best fit from the electron fit. (a): leading photon, (b): sub-leading photon.
Figure 6 :
6The energy resolution for each of the decay photon in the selected 0 sample. The leading photon is shown in (a) and the sub-leading photon is shown in (b). The events have been scaled to match the total data POT of 6.67 × 10 20 . Resolution is defined in eq. 3.2. The dashed vertical line is included at = 0.0 for reference.
2 :
2Characterization of the resolution distributions shown in figure 4, figure 6, and figure 8 .
Figure 7 :
7The reconstructed photon energies for events passing all selection cuts. The leading photon is shown in (a) and the sub-leading photon is shown in (b). The MC simulation samples have been normalized to the total number of data events. The data events are shown by black points. The number of events in each category is shown in the legend in parenthesises. The 2 /19( ) = 1.267 with a p-value of 0.193 for the leading shower and the 2 /19( ) = 0.973 with a p-value of 0.491 for the sub-leading shower.
Figure 8
8shows the resolution of . is the opening angle between the two showers, either simulated or reconstructed, used in the 0 mass reconstruction. The opening angle resolution is defined as:= − . (4.4)
Figure 8 :
8The opening angle ( ) resolution of the decay photons in the selected 0 sample. The events have been scaled to match total data POT of 6.67 × 10 20 . Resolution is defined in eq. 4.4.The dashed vertical line is included at = 0.0 for reference.
Figure 9 :
9The calculated 0 mass for events passing all selection cuts. The MC simulation samples have been normalized to total number of data events. The data events are shown by black points. The number of events in each category is shown in the legend in parenthesises. The 2 /19( ) = 0.976 with a p-value of 0.486 for the MC prediction. 0 represents the 0 mass and is given in this case by:
Figure 10 :
10Total 2 vs. distributions for all MC simulation(a), data(b), and good MC simulation(c) that pass the 0 selection criteria.
Figure 11 :
11distribution for selected events in both data and MC simulation, corresponding to ≈ 5.3 × 10 19 POT. The selection cut requiring < 0.5 radians is indicated by the dotted line. The MC simulation samples have been normalized to total number of data events. The data events are shown by black points. The number of events in each category is shown in the legend in parenthesises. The uncertainty bars here are statistical only. The 2 /9( ) = 0.822 with a p-value of 0.596 for the MC prediction.
Figure 12 :
12Electron energy distribution for Michels in both data and MC simulation after all selection criteria have been applied, corresponding to ≈ 5.3 × 10 19 POT. The MC simulation samples have been normalized to total number of data events. The data events are shown by black points. The number of events in each category is shown in the legend in parenthesises. The uncertainty bars here are statistical only. The 2 /9( ) = 0.608 with a p-value of 0.857 for the MC prediction.
Figure 13 :
13Top: Michel shower charge sum spectrum in data and MC simulation along with the corresponding best fit to eq. (5.1) (allowing only cutoff to vary in the fit). This sample corresponds to ≈ 5.3 × 10 19 POT. Bottom Ratio of the data/simulation to the corresponding fit. The MC simulation and fit result here have been normalized to match the data.
Figure 14 :Figure 15 :
1415Two-dimensional confidence regions for each resolution parameter in eq. (5.1) v.s. cutoff . 1 , 2 , and 3 regions are shown by red, green, and blue curves, respectively. The confidence regions for (a), (c), and (e) come the fit to data while those in (b), (d), and (f) come from the fit to simulation (MC). The units of each parameter in the plots are as follows: cutoff [Q counts], 1 [dimensionless], 2 [Q counts] 1/2 , 3 [Q counts]. 2 from (5.3) as a function of cutoff , for data (a) and MC simulation (b). This sample corresponds to ≈ 5.3 × 10 19 POT. The 1 allowed regions from Wilks' theorem are shown in shaded regions below each curve.
Figure 16 :
16The data and MC simulation points from each the 0 sample and the Michel − sample are compared with the sh -to-MeV electron calibration line used in the DL analysis from eq. 3.1.
Table 1 :
1Range of parameters for the shower reconstruction algorithm. Parameters are changed in the second shower search to allow for the capture of showers detached from the vertex.Minimum value Maximum value
Direction
0 degrees
360 degrees
Opening angle
17 degrees
75 degrees
Length (first shower)
3 cm
35 cm
Length (second shower)
3 cm
60 cm
Gap Size (first shower)
0 cm
17 cm
Gap Size (second shower)
0 cm
90 cm
Table
Table 4 :
4The best fit values and 1 ranges (via Wilks' theorem) for along the 2 /NDF of that fit given by eq. (5.3) for both data and MC simulation. The fit here is the one-dimensional scan over cutoff transformed into as described in the text.Data 1.341 × 10 −2 [1.282 × 10 −2 , 1.401 × 10 −2 ] 2.17/6 = 0.4 MC 1.308 × 10 −2 [1.279 × 10 −2 , 1.334 × 10 −2 ] 2.73/6 = 0.5Sample
[MeV/Q]
range
2 /NDF
Table 5 :
5Data and MC simulation best fit values from each sample and comparison to the charge-to-energy conversion factor from eq. (3.1) ( sh − − = − = 1.26 ± 0.01 × 10 −2 ). Uncertainties in ratios are calculated from the 1 range of each value. The background adjusted values are used for the 0 sample.Sample
[MeV/Q]
[MeV/Q]
/
/ −
/ −
0
1.230 +0.004
−0.006 × 10 −2 1.236 +0.005
−0.006 × 10 −2
1.005 +0.006
−0.006
0.984 +0.009
−0.009
0.979 +0.008
−0.009
Michel −
1.31 +0.03
−0.02 × 10 −2
1.34 +0.06
−0.06 × 10 −2
1.025 +0.051
−0.049
1.038 +0.022
−0.024
1.064 +0.048
−0.048
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|
[] |
[
"Stability of Boolean networks: The joint effects of topology and update rules",
"Stability of Boolean networks: The joint effects of topology and update rules"
] |
[
"Shane Squires \nUniversity of Maryland\nCollege ParkMDUSA\n",
"Andrew Pomerance \nUniversity of Maryland\nCollege ParkMDUSA\n",
"Michelle Girvan \nUniversity of Maryland\nCollege ParkMDUSA\n",
"Edward Ott \nUniversity of Maryland\nCollege ParkMDUSA\n"
] |
[
"University of Maryland\nCollege ParkMDUSA",
"University of Maryland\nCollege ParkMDUSA",
"University of Maryland\nCollege ParkMDUSA",
"University of Maryland\nCollege ParkMDUSA"
] |
[] |
We study the stability of orbits in large Boolean networks with given complex topology. We impose no restrictions on the form of the update rules, which may be correlated with local topological properties of the network. While recent past work has addressed the separate effects of nontrivial network topology and certain special classes of update rules on stability, only crude results exist about how these effects interact. We present a widely applicable solution to this problem. Numerical experiments confirm our theory and show that local correlations between topology and update rules can have profound effects on the qualitative behavior of these systems. pacs: 89.75.-k (complex systems), 05.45.-a (nonlinear dynamical systems), 64.60.aq (phase transitions in networks).
|
10.1103/physreve.90.022814
|
[
"https://arxiv.org/pdf/1310.1338v1.pdf"
] | 10,281,016 |
1310.1338
|
6c8819bf4f49ee706b176f88b70735081204e002
|
Stability of Boolean networks: The joint effects of topology and update rules
October 7, 2013
Shane Squires
University of Maryland
College ParkMDUSA
Andrew Pomerance
University of Maryland
College ParkMDUSA
Michelle Girvan
University of Maryland
College ParkMDUSA
Edward Ott
University of Maryland
College ParkMDUSA
Stability of Boolean networks: The joint effects of topology and update rules
October 7, 2013
We study the stability of orbits in large Boolean networks with given complex topology. We impose no restrictions on the form of the update rules, which may be correlated with local topological properties of the network. While recent past work has addressed the separate effects of nontrivial network topology and certain special classes of update rules on stability, only crude results exist about how these effects interact. We present a widely applicable solution to this problem. Numerical experiments confirm our theory and show that local correlations between topology and update rules can have profound effects on the qualitative behavior of these systems. pacs: 89.75.-k (complex systems), 05.45.-a (nonlinear dynamical systems), 64.60.aq (phase transitions in networks).
Introduction
Systems formed by interconnecting collections of Boolean elements have been successfully used to model the macroscopic behavior of a wide variety of complex systems. Examples include genetic control [1], neural networks [2], ferromagnetism [3], infectious disease spread [4], opinion dynamics [5], and applications in economics and geoscience [6]. Each of these diverse models share the same basic structure: a set of nodes, each of which has a binary state (0 or 1) at a given integer time t, and a set of update rules that determines the state of each node at time t + 1 given the states of the nodes at time t. The relationships between nodes define a graph, where an edge is drawn from node j to node i if the update rule for node i depends on the state of node j.
Depending on the desired application, the model's graph can be random [1], fully connected [2], a lattice [3], or have other complex topology [7,8]. The states of nodes can be updated deterministically or stochastically, synchronously or asynchronously [1][2][3]. Finally, in cases where the update rules are considered to be randomly generated, they can be drawn from many different ensembles [7,[9][10][11].
One important question about a Boolean network is whether or not it is stable, i.e., whether or not small perturbations of a typical initial state tend to grow or shrink as the system evolves. This question may have important ramifications for systems biology and neuroscience: it has been hypothesized that both gene networks [12] and neural networks [13] exist near the critical border separating the stable and unstable regimes. Recently, Pomerance et al. [8] introduced the additional hypothesis that orbital stability of the gene regulatory system may be causally related to cancer. Specifically, motivated by microdissection experiments showing genetic heterogeneity in tumors [14], they suggested that mutations that promote instability may be a contributing factor for some types of cancers.
In this paper, we present and numerically verify a general method for studying the stability of large, directed Boolean networks with locally tree-like topology. Here, by a locally tree-like network we mean that, if two nodes j and i are connected by a short directed path from j to i, it is very unlikely that there will exist a second such path of the same length. This allows us to make the approximation that two inputs to a node are uncorrelated. Analyses based on this approximation have been found to yield accurate results, even in cases where the network contains significant clustering [8,15], while making an analytic treatment of the system tractable. Our results offer a means of assessing the stability of a wide variety of Boolean network systems for which, up to now, no generally effective method has been available. We demonstrate the general utility of our approach with two examples illustrating that the joint effects of network topology and update rules can have profound effects on Boolean network dynamics, which cannot be captured by previous theories.
Model
Boolean networks are discrete-state dynamical systems in which each of the N nodes of a network has a binary state x i (t) = 0 or 1 at each integer-valued time t, and is updated at the next time t + 1 to a new binary state x i (t + 1) that is determined from the time t states of its network inputs. For now we assume that updates are synchronous, but in the Supporting Information (SI), we demonstrate that the stability criterion that we obtain is the same whether nodes are updated synchronously or asynchronously. Consider a node i which has K i network inputs, j 1 . . . j K i . The new state of node i is determined by a binary-valued update rule F i , according to
x i (t+1) = F i x j 1 (t), . . . , x j K i (t) .
Each F i may be specified in the form of a "truth table" listing all the 2 K i possible inputs and the corresponding output. Denoting the vector of input states to node i at time t as X i (t) = (x j 1 (t), . . . , x j K i (t)), we have
x i (t + 1) = F i (X i (t)) .(1)
The network structure is represented using an adjacency matrix A,
where A ij = 1 if there is an edge j → i [that is, if x i (t + 1) depends on x j (t)]
, and A ij = 0 otherwise. The question we address is whether the dynamics resulting from Eq. (1) are stable to small perturbations. To define stability, we assume N 1 and consider two states, x(t) = (x 1 (t), . . . , x N (t)) T andx(t) = (x 1 (t), . . . ,x N (t)) T . We define the normalized Hamming distance between these two states as the fraction of the nodal values that differ for the two states,
H(x(t),x(t)) = 1 N N i=1 |x i (t) −x i (t)| .(2)
We considerx(0) to be a state that is slightly perturbed from x(0), meaning that H(x(0),x(0)) 1. Stability is then defined by whether H(x(t),x(t)) decreases to zero or grows to O(1) as x(t) andx(t) evolve under Eq. (1). Our main theoretical result is a criterion for stability that accounts for the joint effects of network topology (i.e., the A ij ) and node dynamics (i.e., the functions F i ).
The stability of Boolean networks was addressed in the original work of Kauffman [1,12], where Boolean networks were first proposed as a model for genetic dynamics. Kauffman assumed a so-called N -K network topology in which K i was the same at each node, K i = K, and the K inputs to each node were chosen randomly from amongst the (N − 1) other nodes. Further, for each of the 2 K input states, F i was chosen to be 0 or 1 with probability 1 /2. Derrida and Pomeau [9] later generalized this model by introducing a truth table bias 0 <p < 1 such that, for a given input, F i = 1 with probabilityp. They also proposed a method of stability analysis for the case of "annealed" systems, described as follows. First, note that the problem that they and Kauffman were interested in was one in which the network (A ij ) and the node dynamics (F i ) are initially randomly chosen, fixed forever after, and then used to create the dynamics ("quenched randomness"). The annealed problem is different in that the random choices of the network and node dynamics are made at every time step. In contrast with the stability of the quenched system, the stability of the annealed system can be analytically determined [9]. Derrida and Pomeau conjectured that, in the large N limit, the stability boundaries for the quenched and annealed situations are approximately the same. This conjecture has been very well supported by the results of numerical experiments. Later authors generalized the Derrida-Pomeau annealing approach to include a distribution of in-degrees [16][17][18][19][20], joint in-degree/out-degree distributions [21], and "canalizing" update rules [7,22,23].
A further significant generalization was presented in Ref. [8] in which the network is quenched (not annealed), but the update rules (F i ) are annealed using a truth table biasp i that may vary from node to node. Reference [8] called this procedure "semi-annealing" and used it to study the effects of network topological properties on stability, including such factors as network degree assortativity, correlation between node degree and the node biasp i , and community structure. As in the case of annealing, numerical results strongly support the hypothesis that the stability of the semi-annealed (analytically treatable) system and a typical quenched system are similar [8,10].
In what follows, we generalize the results of Ref. [10] by using a semi-annealing procedure that enables the treatment of previously inaccessible cases of substantial interest in applications. We then illustrate this new procedure using two numerical examples. The first example is primarily pedagogical. The second is more applicationoriented and uses threshold rules of the form
x i (t + 1) = U j w ij x j (t) − θ i ,(3)
where U denotes the unit step function, θ i is a threshold value, and w ij is a signed weight whose magnitude reflects the strength of the influence of node j on node i (w ij = 0 if A ij = 0) and whose sign indicates whether node j "activates" or "represses" node i (i.e., promotes x i to be 1 or 0). (This model has been considered previously in the case where the network is N -K, θ i = 0, and w ij = ±1 [11,[24][25][26].) Threshold networks are also commonly used to model gene regulation [27,28], neural networks [2], and other applications. We begin by specifying our semi-annealing procedure, which is similar to the probabilistic Boolean networks described in [29]. We assign each node i an ensemble of update rules, T i , from which a specific update rule is randomly drawn at each time step t. This choice is made independently at each network node i, and we denote the probability of drawing update rule f as Pr[F i = f ]. The resulting dynamics may be described by the probabilities q i (X i ) that the state of node i, given inputs X i , will be 1 on the next time step,
q i (X i ) = f ∈T i Pr[F i = f ]f (X i ),(4)
where we have used the fact that f (X i ) = 0 or 1. It is important to note that q i (X i ) is solely determined from T i , independent of the update rule assignments at other nodes. Thus, computation of q i (X i ) is straightforward. The advantage of this semi-annealing procedure is that the resulting dynamics are simpler to analyze than those of systems with quenched update rules. Typically, the semi-annealed dynamics described by Eq. (4) possess a single ergodic attractor, and the stability of this attractor is similar to that of typical attractors in quenched systems. We assume the existence of a single ergodic attractor in our analysis below and briefly comment on cases for which this assumption fails in the SI.
When using the semi-annealed model, the selection of deterministic update rules is replaced by the selection of an update rule ensemble for each node i. The choice of T i , like the choice of F i in deterministic models, depends on the particular case under study. This will be illustrated in our numerical experiments.
To measure the stability of the semi-annealed dynamics generated by Eq. (4), we begin with many initial conditions x(0) and generate orbits x(t). We imagine that the initial conditions x(0) are selected randomly according to the natural measure of the attractor; in practice, this can be achieved by time-evolving another initial condition sufficiently long that transient behavior has ceased, and then using its final state as an initial condition. For each orbit x(t), we also consider a perturbed initial conditioñ x(0), obtained by randomly choosing a small fraction ε of the components of x(0) and "flipping" their states. That is, if node i is chosen to be flipped, thenx i (0) = 1 − x i (0). The perturbed initial condition is then used to generate a perturbed orbitx(t), where, for the semi-annealed case, the random update rule time sequence for each node is the same forx(t) and x(t). The growth or decay of the Hamming distance between x(t) andx(t) defines the stability of the system.
Analysis
Given an orbit on the ergodic attractor of the semi-annealed system, we define p i to be the fraction of time that the state x i (t) of node i is 1. We call p i the "dynamical bias" of node i and regard it as the probability that x i (t) = 1 at an arbitrarily chosen time. 1 In what follows, we first address the determination of the dynamical biases p i , which can then be used to determine the stability of the system. We first note that p i is determined by the set of probabilities Pr[X i ] of i receiving each input vector X i , using
p i = X i Pr [x i = 1 |X i ] Pr [X i ], or p i = X i q i (X i ) Pr [X i ] ,(5)
where q i is as defined in Eq. (4). Assuming that the network topology is locally treelike, the states of the inputs to node i can be treated as statistically independent [8,15]. Therefore, the probabilities Pr[X i ] are determined by the biases of i's inputs. Letting J i denote the set of indices of all nodes that are inputs to i,
Pr [X i ] = j∈J i [x j p j + (1 − x j ) (1 − p j )] ,(6)
where we have used the fact that x i = 0 or 1. Inserting (6) into (5) yields a set of N equations for the N node biases p i . In what follows, we envision that this set of equations has been solved for the dynamical biases p i at each node, and we will use these biases to evaluate the stability of the network. We find that for most practical purposes, one method for solving Eqs. (5-6) for the biases p i is by iteration: an initial guess for each p i can be inserted in (6), and (5) can then be used to obtain an improved guess, and so forth, until the p i have converged. We now consider the stability of the annealed system. We say that node i is "damaged" at time t if x i (t) andx i (t) differ at time t. We define a vector y(t) such that y i (t) is the probability that i is damaged at time t, i.e.,
y i (t) = Pr [x i (t) =x i (t)] .(7)
Next, let d i (X i ,X i ) be the probability that i will be damaged if its inputs in the two orbits are X i andX i ,
d i X i ,X i =Pr x i (t + 1) =x i (t + 1) X i (t),X i (t) .(8)
We have suppressed the time dependence of X i andX i in d i (X i ,X i ) since this can be expressed in terms of the time-independent update rule ensemble as
d i X i ,X i = f ∈T i Pr [F i = f ] · f (X i ) − f (X i ) ,(9)
where we have used the fact that f (X i ) = 0 or 1. Note that, like q i , d i depends only on T i , and thus is straightforward to calculate. Marginalizing over X i andX i in (7) and inserting (8),
y i (t + 1) = X i X i Pr X i (t),X i (t) d i X i ,X i .(10)
Because we are considering the question of stability, we have assumed that x(t) and
x(t) are close to each other in the sense of Hamming distance for small times t, so y i (t) 1 for all i. In this case we can ignore the possibility that X i (t) andX i (t) differ at two or more input states and drop all terms of O(y 2 ). Moreover, ifX i (t) and X i (t) are the same, d i = 0 via Eq. (9), so nothing is contributed to the sum in Eq. (10). Therefore, the only values ofX i which contribute significantly to the sum are the ones in whichX i (t) and X i (t) differ for exactly one node j. Let X j i (t) be a vector which is the same as X i (t) except that the state of input node j is flipped [x j (t) = 1 − x j (t)]. Using this notation, we can rewrite Eq. (10) as
y i (t + 1) = j∈J i X i Pr X i (t), X j i (t) d i X i , X j i .(11)
Furthermore, because the network is locally tree-like and the inputs to node i are therefore uncorrelated,
Pr X i (t), X j i (t) = Pr [X i ] Pr [x i (t) =x j (t)] = Pr [X i ] y j (t).(12)
When substituted into Eq. (11), this leads to
y i (t + 1) = j∈J i y j (t) X i Pr [X i ] d i X i , X j i .(13)
Since the second sum is time-independent, we can write
y i (t + 1) = j R ij y j (t) + O(y 2 ), (14a) R ij ≡ X i Pr [X i ] d i X i , X j i ,(14b)
where R ij = 0 when there is no edge from j to i. 2 R ij may be interpreted as the probability that damage will spread from node j to node i; in analogy with the terminology of Ref. [22], we call it the effective activity of j on i.
The average of the normalized Hamming distance over all possible perturbations and realizations of the semi-annealed dynamics is H(x(t),x(t)) = 1 N i y i (t), so the stability of the system is determined by whether or not the elements of y grow with time. This can be determined by writing Eq. (14a) in matrix form,
y(t + 1) = Ry(t).(15)
Since the effective activities R ij are non-negative, and R is typically a primitive matrix, the Frobenius-Perron theorem implies that the eigenvalue of R with largest magnitude is real and positive. We denote this eigenvalue λ R . If the initial perturbation has a nonzero component along the eigenvector associated with λ R , as is generally the case, then for t not too large, the expected Hamming distance will grow as (λ R ) t by Eq. (15). Therefore, λ R > 1 implies instability λ R < 1 implies stability .
One major advantage of our analysis is that, from a computational perspective, evaluating λ R is typically faster than finding the average Hamming distance through simulations. We discuss this further in the SI, along with other computational aspects of the above solution. Another potential advantage of the criterion (16) is that, in some cases, it can facilitate qualitative understanding. For example, in previous work [8], it was shown that network assortativity promotes instability for a special case of the above situation.
Numerical results
We now use the general framework presented above to analyze two cases that illustrate the effects of correlations between local topological features and update rules. In each example, we construct a single network with N = 10 5 nodes using the configuration model [30]. The in-degrees are Poisson-distributed with a mean of 4 and the out-degrees are scale-free with exponent γ ≈ 2.2. In Fig. 1, we plot the average Hamming distance H and λ R against a tuning parameter for each model. To calculate each Hamming distance H, we first time-evolve a random initial condition (using a quenched set of update rules) for 100 time steps to ensure that it is on an attractor, then apply a perturbation by flipping the values of a random fraction ε = 0.01 of the nodes. Next we time-evolve both the original and perturbed orbits for another 400 time steps, measuring the Hamming distance H over the last 100 of these to ensure that it has reached a steady state. We take the average H over 10 initial conditions for each quenched set of update rules. In the figures, we show H for both a single quenched system as well as an average over 50 sets of quenched update rules.
Example 1: XOR, OR, and AND update rules In our first example, we illustrate the effect of correlations by assigning a node either a highly "sensitive" update rule (XOR) or a less sensitive update rule (OR or AND) based on the node's in-degree. That is, the update rule at each node i is randomly drawn from three classes: (a) XOR, whose output is one (zero) if i has an odd (even) number of inputs that are one; (b) OR, whose output is one if and only if at least one input is one; or (c) AND, whose output is one if and only if all K i inputs are one. Following [22], we refer to XOR as highly sensitive because any single input flip will cause its output to flip, so R ij = 1 whenever there is an edge from j to i. On the other hand, if node i has OR or AND as its update rule, flipping node j will cause node i to flip only if every node other than j is zero or one, respectively. Thus in cases (b) and (c), R ij depends on the node biases p j which are obtained by solving Eqs. (5)(6). For OR, R ij = k (1 − p k ), and for AND, R ij = k p k , where the products are taken over all inputs k which are not equal to j. Figure 1(a-c) shows results for a network with these three classes of update rules. We assign a fraction of nodes α to have XOR update rules, and the remaining nodes are evenly split between OR and AND rules. We consider three cases: (1) XOR is assigned to the αN nodes with the largest in-degree; (2) XOR is randomly assigned to αN nodes irrespective of their degrees; or (3) XOR is assigned to the αN nodes with smallest in-degree. In all three cases, the remainder are randomly assigned OR or AND. In numerical simulations, all update rule assignments are quenched. In order to find appropriate semi-annealing probabilities to use in our theoretical prediction, we note that the initial assignment of XOR is deterministic in cases (1) and (3), but OR and AND are assigned randomly. Therefore, in the theory, we treat the network and the identity of the XOR nodes as fixed, and anneal over the OR and AND nodes, assigning a probability of 1 /2 to choosing either OR or AND on each time step. (Other annealing choices are also possible, but we choose this because it most straightforwardly resembles the quenched assignment of update rules above.) As can be seen in Fig. 1(a), the values of α at which the three cases become unstable are quite different, thus demonstrating that stability is strongly affected by correlation between the local topological property of nodal in-degree and the sensitive XOR update rule. As shown in Fig. 1(c), however, when we re-plot H against λ R , we see that in each case the network becomes unstable at λ R ≈ 1, as predicted by the theory. This is also strikingly illustrated by the vertical arrows in Fig. 1(a) marking the values of α at which λ R = 1 [c.f., Fig. 1(b)].
Example 2: Threshold networks
We now consider networks with threshold rules as given in Eq. (3); such threshold rules may be re-cast as Boolean functions F i by enumerating all possible X i and calculating whether the weighted sum of inputs exceeds the threshold θ i . Conversely, threshold rules are appropriate for Boolean network applications in which each edge has a fixed "activating" or "repressing" character.
Our results for threshold networks are shown in Fig. 1(d-f ) and are generated in the following manner. To assign the weight w ij for each edge j → i, we first assign half of the edges to be activating and half to be repressing. Then, the weight is drawn from a normal distribution with mean 1 (activating) or −1 (repressing) and standard deviation 1 /4. We also consider two additional cases where the weights are either correlated or anticorrelated to a topological property of the network, the product of a node's indegree and out-degree. (Nodes with a high degree product play a crucial role in the stability of Boolean networks [8,31,32].) We generate the correlated and anticorrelated cases by exchanging weights between pairs of edges in the original ("uncorrelated") case. Specifically, we repeat the following procedure. First, we select two random edges j 1 → i 1 and j 2 → i 2 in the network. Next, we identify the edge for which i has a higher degree product. Finally, in the correlated (anticorrelated) case, we exchange the values of the two weights if doing so would increase (decrease) the weight going to the node with the higher degree product. We repeat this procedure E/2 times, where E is the number of edges in the network, so that each edge is expected to be considered for one exchange.
In this example, we model the case in which the thresholds of different nodes are similar, but not necessarily equal. In the theory, we treat this case by annealing the thresholds θ i over a gaussian distribution with a meanθ and standard deviation σ θ = 1 /10. By Eq. (3),
q i (X i ) = Φ 1 σ θ j w ij x j −θ ,(17)
where
Φ(x) = (2π) −1/2 x −∞ exp(−t 2 /2)dt. Similarly, we find that d i (X i , X j i ) = |q i (X i )− q i (X j i )|.
These expressions can be used to calculate p i , R ij , and λ R using Eqs. (5)(6)14). (Here, as in many cases, it is not necessarily to list the ensemble of update rules T i explicitly, because q i and d i can be calculated directly.) In our numerical simulations, we treat θ i as quenched by writing θ i =θ + δθ i , where δθ i is drawn from a normal distribution with mean 0 and standard deviation σ θ .
In Fig. 1(d-f ), we show results for both a single quenched set of δθ i (hollow markers) as well as an average over 50 quenched sets of δθ i (filled markers). In each case, single quenched realizations show similar behavior to the average, in agreement with the semi-annealing hypothesis. More striking is the qualitative difference between the anticorrelated case and the two other cases. At low thresholds, the anticorrelated network is stable, whereas both of the other cases are unstable. As the threshold is increased, the anticorrelated network becomes unstable before becoming stable again at large thresholds. This behavior is explained in Fig. 1(e), where we see that in all three cases, λ R initially increases with increasingθ, but it is only in the anticorrelated that λ R < 1 initially. Finally, in Fig. 1(f ), we re-plot the same data for H against λ R . We see that in all three cases the stability transition clearly occurs at λ R = 1, confirming our analysis.
Discussion
We have presented a general framework for predicting orbit stability in large, locally tree-like Boolean networks, given arbitrary network topology and update rules. There are three main steps in this process: (1) select update rule ensembles T i (rather than deterministic rules F i ), and compute q i and d i ; (2) calculate the dynamical biases p i of the each node i by iterating Eqs. (5-6); and (3) calculate the activity matrix R with elements given by Eq. (14). The largest eigenvalue of this matrix, λ R , then determines the stability of the system, Eq. (16). As illustrated above, the first step requires a judicious selection of which aspects of the update rules should remain quenched, but is typically straightforward thereafter.
As examples of the application of our general stability criterion, we analyzed both a pedagogical case and the case of threshold networks, where all update rules are assumed to be of the form of Eq. (3). These results show that the stability of a Boolean network is strongly affected not only by the network topology and nodal update rules, but by correlations between the two. Although previous research into the stability of Boolean networks has primarily focused on either topology or update rules alone, Figs. 1(a,d) show that correlations can have profound qualitative effects on the dynamical properties of a network. Presumably, these aspects of biological networks interact strongly during evolution, and so joint effects in topology and update rules should be analyzed carefully when studying genetic or neural systems.
Acknowledgements: This work was funded by ARO grant W911NF-12-1-0101.
Supplemental Information
Abstract
In this supplement, we explore several topics related to our stability condition for Boolean networks. We show that the stability condition is unchanged for asynchronously updated networks, discuss the conditions under which our derivation is valid, analyze the computational complexity of our solution, and calculate the critical slope of the stability transition.
Asynchronous updating
Asynchronous updates may arise in discrete state systems for several reasons. For example, links may have nonuniform delays, δ ij , that model delays arising from, for example, the chemical kinetics of gene regulation. In this case, the dynamics would be described by a modified version of Eq. (4) in which the state of node i at time t depends on the states of its inputs j at times (t − δ ij ). Another alternative is a model in which nodes are individually chosen to be updated in a stochastically determined order. Here, we show that the stability condition given in the main text applies not only to the case of synchronous nodal updates, but to asynchronous models as well, including both of these examples.
In particular, we consider update times, τ 1 < τ 2 < ... < τ t < ..., where the update intervals, (τ t+1 − τ t ), are arbitrary, incommensurate and do not influence the analysis. Since the update times are incommensurate, we approximate the deterministic choice of node to update at each time, indexed by integer t, with a stochastic process where node i (and only node i) is chosen to be updated by Eq. (4) with probability ρ i . This is, of course, also appropriate to systems that are inherently stochastic. To analyze this case, we define the vector y(t) as in the main text; however, we must make some adjustments to the approximate update equation, Eq. (14). Since node i is chosen independently of the values of the nodes, the joint probability at time step t that node i is chosen for update and that node i differs between the two initial conditions after the update is given approximately by ρ i R ij y j (t). If node i is not chosen for update at this time step, y i does not change. Putting this together, we get, for small t and small initial perturbations, y i (t + 1) ≈ ρ i R ij y j (t) + (1 − ρ i )y i (t), which we rewrite in matrix form as
y(t + 1) = ρ(R − I)y(t) + y(t),(S1)
where ρ is a diagonal matrix with ρ i in each row, I is the identity matrix, and R is the activity matrix. In order to see that Eq. (16) also applies in this case, we note that, at criticality, y(t + 1) = y(t), so that Eq. (S1) reduces to ρ(R − I)y(t) = 0. This has a solution for y = 0 only if λ R = 1. Note, however, that in this case, for λ R > 1, the growth rate of the Hamming distance will be at a rate of the order of 1/N smaller than the rate of the synchronously updated networks, because N time steps of asynchronous update correspond to one time step of synchronous update.
Comments on Equations (5-6)
In the main text, we make three simplifying assumptions about the derivation and solution of Eqs. (5-6):
1. The correlations between the states of different inputs to a single node are negligible, because the network is locally treelike.
2. Equations (5-6) have a single stable solution, describing the attractor of the semi-annealed dynamics.
3. This solution can be found by iterating the equations.
Here we add some comments about these assumptions and the conditions under which they are valid. First, the locally treelike approximation has been effective in describing the structural and dynamical properties of a variety of complex networks, as documented in [15]. In particular, it has been applied successfully to Boolean networks and related percolation problems in [8,10,33,34]. In this context, we may argue as follows that it allows us to assume the independence of two nodes j 1 and j 2 which are both inputs to a third node i. We would expect correlations between x j 1 (t) and x j 2 (t) to arise mainly from the two nodes being mutually influenced by a previous state of a third node, x k (t − h), where node k has paths of length h to both j 1 and j 2 . But if this is the case, k has two independent paths of length h + 1 to i. In locally treelike networks, the number of such paths is an insignificant fraction of paths of the same length. It is hypothetically possible that this assumption may nonetheless break down in cases where the dynamics exhibit a long correlation length, as might be expected when there is a phase transition in p i . Numerically, however, we find no cause for concern. For example, such a case occurs for the threshold networks studied in Example 2 of the main text, when the stability transition at largeθ coincides with a phase transition in p i . In this case, as in all of our numerical work, we observe that the stability transition still occurs at λ R = 1, as predicted by theory.
We next consider the second and third assumptions above. The Brouwer fixed point theorem guarantees that Eqs. (5)(6) have at least one solution, but not that it is unique or stable. Non-uniqueness does not present any difficulties for the theory; in this case, each solution represents a separate attractor, and the stability of each attractor may be determined separately. (For example, it is possible to construct threshold networks which have one solution with p i = 0 for all nodes i and another solution where p i > 0 for most i.) A second, more troublesome scenario is that there are no stable solutions. In this case, iteration of Eqs. (5-6) would not converge to a solution but instead fluctuate periodically or chaotically. An example of this behavior occurs in networks where the update rules are chosen to approximate the logistic map. In this case, as the tuning parameter is changed, the system undergoes a period-doubling cascade. In principle the method could be extended to this situation; however, it seems unlikely that systems undergoing significant dynamics in the biases would be stable with respect to small perturbations. We note, however, that this is a rather pathological case, and that typical biological applications have stable solutions.
One final possibility is that there is a family of marginally stable solutions to Eqs. (5)(6). For example, this occurs in a loop with two nodes and the the copy update function (i.e., the output is the input). In this case, iteration oscillates and does not converge, since any solution where p 1 = p 2 is valid. We have never observed this phenomenon when 0 < q i (X i ) < 1 for all i and X i , but this sometimes occurs when Eqs. (5)(6) are applied directly to quenched, deterministic dynamics (i.e., q i (X i ) = 0 or 1 for all i and X i ). However, when analyzing deterministic dynamics, one may either consider a related semi-annealed problem (that reflects, say, realistic noise models or measurement uncertainty), as we do here, or one may measure Pr[X i ] for a particular attractor directly from numerical simulations, then find the stability condition using Eqs. (14) and (16).
Computational Complexity
We note that the procedure presented in the main text is applicable even to very large networks. The Frobenius-Perron eigenvalue λ R may typically be found through power iteration, which requires only O(E) operations, where E is the number of edges; for sparse networks, this is O(N ). Another advantage of our method is that the use of dynamical biases and the locally treelike approximation offers a tremendous computational improvement over previous theoretical treatments of similar systems. For example, the analysis of probabilistic Boolean networks in Ref. [29] relies upon a state transition matrix of size 2 N × 2 N , which is intractable for networks with more than a few dozen nodes. In contrast, iterating Eqs. (5-6) requires fewer than O(N 2 K ) steps, where K is the maximum in-degree of all nodes. This is numerically feasible for large-N networks as long as K ≤ 20. In many cases, additional simplifying assumptions may offer even greater computational speed.
Critical Slope
The second-order terms in the expansion in Eq. (10) may be used to derive the critical slope of H near λ R = 1. We include a sketch of this derivation because it may be useful for near-critical approximations or for designing networks with extreme behavior near the critical point.
To find the second-order terms in Eq. (10), we need to consider input combinations which differ for two distinct inputs j and k, which we denoteX = X j,k i in analogy with our definition of X j i in the main text. The probabilities of these input combinations are given, up to O(y 2 ), by
Pr X i (t), X j i (t) = Pr[X i ] y j (t) 1 − k =j y k (t)
Pr X i (t), X j,k i (t) = Pr[X i ] y j (t) y k (t).
(S2)
Following similar steps as those that led to Eq. (15), we obtain y i (t + 1) = j R ij y j (t) + j,k R ijk y j (t)y k (t),
R ijk ≡ 1 2 X i Pr[X i ]d i X i , X j,k i − R ij ,(S3)
where R ij is defined as in the main text. Note that when j = k, R ijk = 0. Now we may derive the critical slope. We consider each y i to have reached a steady state and hence drop the time-dependence in y i (t). Next we write a perturbation expansion for each variable near the critical point, y i = εHy 1 i + ε 2 y 2 i and λ R = 1 + ελ 1 R , where superscripts for y i and λ R refer to the level of the perturbation expansion. From Eq. (15), y 1 must be the right Frobenius-Perron eigenvector of the first-order R-matrix.
Here it has been normalized so that i y 1 i = 1. Inserting the second-order expansion and simplifying, we obtain
y 2 i = Hλ 1 R y 1 i + j R ij y 2 j + H 2 j,k R ijk y 1 j y 1 k (S4)
This expression may be further simplified by using left Frobenius-Perron eigenvector of R, which we denote u. Multiplying through by u i and summing over i, the left-hand side and the second term on the right-hand side of Eq. (S4) cancel to leading order in ε. With the remaining terms, we find that the critical slope m c = H/λ 1 R is
m c = − i u i y 1 i i,j,k R ijk u i y 1 j y 1 k .(S5)
This result may be used numerically to find the critical slope in particular cases, because the eigenvector y 1 may be found along with λ R . It may also be used to approximate the critical slope analytically, when good approximations for y 1 are known, as in Refs. [8,32].
Figure 1 :
1Normalized average Hamming distance H and λ R for a network with XOR, OR, and AND update rules (panels a-c) and a threshold network (panels d-f ). Filled markers are averaged over 50 quenched realizations of the thresholds, while hollow markers show a single quenched realization. Squares, circles, and triangles represent different correlations between network topology and update rules; see text for details. (a) When α is used as a tuning parameter, the stability transitions for each of the three cases are far apart. Locations where λ R = 1 are marked by arrows. (b) Viewing λ R as a function of α, we see that this behavior agrees with the theoretical prediction for critical stability, λ R = 1. (c) Plotting H against λ R directly, we see that the transition for all three curves occurs at λ R = 1. (d-f ) Results for threshold networks are shown, analogous to those in panels (a-c), usingθ as a tuning parameter rather than α.
This is in contrast to the "truth table bias," denotedp above, an external parameter used to define the ensemble of update rules in Ref.[9] and later work.
The second-order terms in this expansion are discussed further in the SI, where we derive an expression for the critical slope of the stability phase transition.
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The abstract of "Contextual Values of Observables in Quantum Measurements" by J. Dressel, S. Agarwal, and A. N. Jordan [Phys. Rev. Lett. 104 240401 (2010)] (called DAJ below), states:"We introduce contextual values as a generalization of the eigenvalues of an observable that takes into account both the system observable and a general measurement procedure. This technique leads to a natural definition of a general conditioned average that converges uniquely to the quantum weak value in the minimal disturbance limit."A counterexample to the claim of the last sentence was presented in [4], a 32-page paper discussing various topics related to DAJ, and a simpler counterexample in Version 1 of the present work. Subsequently Dressel and Jordan placed in the arXiv the paper of the title (called DJ below) which attempts to prove the claim of DAJ quoted above under stronger hypotheses than given in DAJ, hypotheses which the counterexample does not satisfy. The present work (Version 5) presents a new counterexample to this claim of DJ.A brief introduction to "contextual values" is included. Also included is a critical analysis of DJ. * For contact information, go to
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15 Oct 2012 June 24, 2011
Stephen Parrott 15 Oct 2012 June 24, 2011Counterexample to "Sufficient conditions for uniqueness of the Weak Value" by J. Dressel and A. N. Jordan, arXiv:1106.1871v1.
The abstract of "Contextual Values of Observables in Quantum Measurements" by J. Dressel, S. Agarwal, and A. N. Jordan [Phys. Rev. Lett. 104 240401 (2010)] (called DAJ below), states:"We introduce contextual values as a generalization of the eigenvalues of an observable that takes into account both the system observable and a general measurement procedure. This technique leads to a natural definition of a general conditioned average that converges uniquely to the quantum weak value in the minimal disturbance limit."A counterexample to the claim of the last sentence was presented in [4], a 32-page paper discussing various topics related to DAJ, and a simpler counterexample in Version 1 of the present work. Subsequently Dressel and Jordan placed in the arXiv the paper of the title (called DJ below) which attempts to prove the claim of DAJ quoted above under stronger hypotheses than given in DAJ, hypotheses which the counterexample does not satisfy. The present work (Version 5) presents a new counterexample to this claim of DJ.A brief introduction to "contextual values" is included. Also included is a critical analysis of DJ. * For contact information, go to
Introduction
A counterexample to a major claim of J. Dressel, S Agarwal, and A. N. Jordan, "Contextual values of observables in quantum measurements", Phys. Rev. Lett. 104 240401 (2010) (henceforth called DAJ) was given in [4], a 32-page paper discussing DAJ in detail. The claim in question is stated as follows in DAJ's abstract:
"We introduce contextual values as a generalization of the eigenvalues of an observable that takes into account both the system observable and a general measurement procedure. This technique leads to a natural definition of a general conditioned average that converges uniquely to the quantum weak value in the minimal disturbance limit." This wording (particularly, "minimal disturbance limit") is potentially misleading, as will be explained briefly below, and is discussed more fully in [4].
Version 1 presented a simple counterexample to the claim of the above quote based on my interpretation of the vague presentation of DAJ. A later paper by Dressel and Jordan, "Sufficient conditions for uniqueness of the Weak Value" [2] (henceforth called DJ) adjoined new (and very strong) hypotheses to DAJ which the counterexample did not satisfy and claimed to prove that the above quote was correct under these new hypotheses.
The present work presents a new counterexample to that claim. It also includes the introduction to the main ideas of DAJ of Version 1 and a critical analysis of DJ.
Notation and brief reprise of DAJ
To establish notation, we briefly summmarize the main ideas of DAJ. The notation generally follows DAJ except that DAJ denotes operators by both boldface and circumflex, e.g.,M , but we omit the boldface and "hat" decorations. Also, we use P f to denote the operator of projection onto the subspace spanned by a vector f . (DAJ usesÊ (2) f .) When we quote directly an equation of DAJ, we use DAJ's equation number, which ranges from (1) to (10), and also DAJ's original notation. Other equations will bear numbers beginning with (100).
Suppose we are given a set {M j } of measurement operators, where j is an index ranging over a finite set. We assume that the reader is familiar with the theory of measurement operators, as given, for example, in the book [5] of Nielsen and Chuang. By definition, measurement operators satisfy
j M † j M j = I ,(100)
where I denotes the identity operator. With such measurement operators is associated the positive operator valued measure (POVM) {E j } with E j := M † j M j . When the system is in a (generally mixed) normalized state ρ (represented as a positive operator of trace 1), the probability of a measurement yielding result j is Tr [M † j M j ρ] = Tr [E j ρ]. Moreover, after the measurement, the system will be in (unnormalized) state M j ρM † j , which when normalized is:
normalized post-measurement state = M j ρM † j Tr [M j ρM † j ] .(101)
For notational simplicity, we normalize states only in calculations where the normalization factor is material. We also assume given an operator A, representing what DAJ calls "the system observable" in the above quote. We ask if it is possible to choose real numbers α j , which DAJ calls contextual values, such that
A = j α j E j .(102)
This will not always be possible, but we consider only cases for which it is. When it is possible, it follows that the expectation Tr [Aρ] of A in the state ρ equals the expectation calculated from the probabilities Tr [E j ρ] obtained from the POVM {E j }, with the numerical value α j associated with outcome j:
Tr [Aρ] = j α j Tr [E j ρ] .(103)
The book [6] of Wiseman and Milburn defines a measurement to be "minimally disturbing" if the measurement operators M j are all positive (which implies that they are Hermitian). 1 DAJ uses a slightly more general definition to define their "minimal disturbance limit" of the above quote. We shall use the definition of Wiseman and Milburn [6] because it is simpler and sufficient for our counterexample. A counterexample under the definition of Wiseman and Milburn will also be a counterexample under any more inclusive definition, such as that of DAJ.
A particularly simple kind of measurement is one in which there are only two measurement operators, P f and I − P f . Intuitively, this "measurement" asks whether the (unnormalized) post-measurement state is P f or not. Here we are using the notation of mixed states. Phrased in terms of pure states, and assuming that the pre-measurement state ρ is pure, the measurement determines if the post-measurement state is the pure state f or a pure state orthogonal to f .
Suppose that we make a measurement with the original measurement operators M j and then make a second measurement with measurement operators P f , I − P f . In this situation, the second measurement is called a "postselection", and when it yields state P f , one says that the postselection has been "successful".
Such a compound measurement may be equivalently considered as a single measurement with measurement operators {P f M j , (I − P f )M j }. "Successful" postselection leaves the system in normalized state
(P f M j )ρ(P f M j ) † Tr [(P f M j )ρ(P f M j ) † ] ,(104)
which is pure state f (P f in mixed state notation). This result will occur with probability
p(j, f ) = Tr [(P f M j ) † P f M j ρ] = Tr [M † j P f M j ρ].
The probability p(j|f ) of first measurement result j given that the postselection was successful is:
p(j|f ) = p(j, f ) i p(i, f ) = Tr [M † j P f M j ρ] i Tr [M † i P f M i ρ] .(105)
Hence, if we assign numerical value α j to result j as above, the conditional expectation of the measurement given successful postselection is:
f A := j α j Tr [M † j P f M j ρ] i Tr [M † i P f M i ρ] .(106)
This is DAJ's "general conditioned average". Written in DAJ's original notation, this reads
f A = j α (1) j P j|f = j α (1) j Tr [Ê (1,2) jfρ ] j Tr [Ê (1,2) jfρ ] .(6)
DAJ's theory of contextual values was motivated by a theory of "weak measurements" initiated by Aharonov, Albert, and Vaidman [10] in 1988. Intuitively, a "weak" measurement is one which negligibly disturbs the state of the system. This can be formalized by introducing a "weak measurement" parameter g on which the measurement operators M j = M j (g) depend, and requiring that
lim g→0 M j (g)ρM † j (g) Tr [M j (g)ρM † j (g)]
= ρ for all ρ and j ,
This says that for small g, the post-measurement state is almost the same as the pre-measurement state ρ (cf. equation (104)). We shall refer to this as "weak measurement" or a "weak limit". The "minimal disturbance limit" mentioned in the above quote from DAJ's abstract presumably refers to (107) combined with their generalization of Wiseman and Milburn's "minimally disturbing" condition that the measurement operators be positive, and this is the definition that we shall use. 2 DAJ claims that in their "minimal disturbance limit" (which is implied by a weak limit with positive measurement operators), their "general conditioned average" f A (6), our (106), is always given by:
f A = 1/2Tr [P f {A, ρ}]] Tr [P f ρ] .(108)
Our equation (108) is equation (7) of DAJ:
A w = Tr [Ê (2) f {Â,ρ}] 2Tr [Ê (2) fρ ] ,(7)
Here A w is their notation for "weak value" of A. 3 The statement of DAJ quoted in the Introduction, that their ". . . general conditioned average . . . converges uniquely to the quantum weak value in the minimal disturbance limit", implies that for a weak limit of positive measurement operators, their (6) always evaluates to (7), or in our notation, our (106) always evaluates to (108). We shall give an example for which (106) does not evaluate to (108).
3 The counterexample
General discussion
We are assuming the "minimal disturbance" condition that the measurement operators be positive, so in the definition (106) of DAJ's "general conditioned average", we replace M † j with M j . First we examine its denominator. Let
η j (g) := Tr [M j (g)ρM j (g)] ,(109)
which are inverse normalization factors for the unnormalized post-measurement states M i (g)ρM i (g) (cf. (101). We shall assume that all η j (g) are bounded for mitted, the authors of DAJ confirmed that Wiseman and Milburn's definition implies theirs. DAJ uses but does not define the phrase "weak limit", but in the same message to PRL, the authors state that (107) corresponds to "ideally weak measurement". Since "ideally weak measurement" must be (assuming normal usage of syntax) a special case of mere "weak measurement", our counterexample which assumes (107) will also be a counterexample to the statement of DAJ quoted in the introduction.
I have made several direct inquiries to the authors of DAJ requesting a precise definition of their "minimal disturbance limit", but all have been ignored. 3 In the traditional theory of "weak measurement" initiated by [10], the weak limit (i.e., lim g→0 ) of (106) (equivalently, (6)) would be called a "weak value" of A, though the traditional "weak measurement" literature calculates it via different procedures. When ρ is a pure state, most modern authors calculate this weak value as (108) (equivalently (7)), though the seminal paper [10] arrived (via questionable mathematics) at a complex weak value of which (108) is the real part. (Only recently was it recognized that "weak values" are not unique [7][8] [9].) small g, which is expected (because we expect M j (g) to approach a multiple of the identity for small g in order to make the measurement "weak") and will be the case for our counterexample. We have lim g→0
j
Tr [P f M j (g)ρM j (g)] = lim g→0 j Tr [P f M j (g)ρM j (g) η j (g) − ρ ] η j (g) + lim g→0 j Tr [P f ρ] η j (g) = lim g→0 j Tr [P f ρ] Tr [M j (g)ρM j (g)] = Tr [P f ρ] lim g→0 Tr [ j M j (g)M j (g)ρ] = Tr [P f ρ] ,(110)
because
M 2 j = M † j M j = I and Tr [ρ] = 1.
. This is the denominator of DAJ's claimed result (108) (half the denominator of their (7) because both numerator and denominator of our (108) differ from (7) by a factor of 1/2).
Next we examine the numerator of the "general conditioned average" (106). We shall write it as a sum of two terms, the first term leading to DAJ's (108), and the second a term which does not obviously vanish in the limit g → 0.. The counterexample will be obtained by finding a case for which the limit of the second term actually does not vanish. Combining these gives
M ρM = 1 2 {M 2 , ρ} + 1 2 [M, [ρ, M ]] .(111)
Using (111) and the contextual value equation (102), A = j α j E j = j α j M 2 j , we can rewrite the numerator of (106) as numerator of (106) =
j α j Tr [M j P f M j ρ] = j α j Tr [P f M j ρM j ](112)= 1 2 Tr [P f {A, ρ}] + j 1 2 α j Tr [P f [M j , [ρ, M j ] ] .
After division by the denominator of (106), the first term gives DAJ's claimed (7) in the limit g → 0, our (108), and the second term gives difference between weak limit of (6) and (7)
= lim g→0 j 1 2 α j (g)Tr [P f [M j (g), [ρ, M j (g)] ] ] Tr [P f ρ] .(113)
We shall call (113) the "anomalous term". Since there is no obvious control over the size of the α j (g), a counterexample is expected, but was surprisingly hard to find. The Version 1 counterexample for the quoted claim of DAJ and the newer counterexample for the new claim of DJ are identical up to this point. The difference is that the Version 1 counterexample used three 2 × 2 diagonal matrices as measurement operators, resulting in a contextual value equation (102) with multiple solutions, whereas the newer counterexample uses three 3 × 3 diagonal matrices for which there is a unique solution to (102). The newer counterexample could supercede the Version 1 example, but we retain the original counterexample because of its simple and intuitive nature (e.g., all steps can be mentally verified).
The Version 1 counterexample
The "system observable" A for the counterexample will correspond to a 2 × 2 matrix
A := a 0 0 b(114)
There will be three measurement operators:
M 1 (g) := 1/2 + g 0 0 1/2 − g , M 2 (g) := 1/2 − g 0 0 1/2 + g ,(115)M 3 (g) := 1/2 − 2g 2 0 0 1/2 − 2g 2 .
Note that M 3 (g) is uniquely defined by the measurement operator equation 3 j=1 M 2 j (g) = 1 and that all three measurement operators approach multiples of the identity as g → 0, which assures weakness of the measurement. Note also that M 3 (g) is actually a multiple of the identity for all g, so the commutators in the expression (113) for the anomalous term which involve M 3 vanish. That is, M 3 , and hence α 3 , make no contribution to the anomalous term.
Writing out the contextual value equation (102) in components gives two scalar equations in three unknowns:
(1/2 + g) 2 α 1 (g) + (1/2 − g) 2 α 2 (g) + (1/2 − 2g 2 )α 3 (g) = a (116) (1/2 − g) 2 α 1 (g) + (1/2 + g) 2 α 2 (g) + (1/2 − 2g 2 )α 3 (g) = b .
The solution can be messy because of the algebraic coefficients. However, for the case a = 1 = b, a solution can be obtained without calculation. This choice of a and b corresponds to the system observable being the identity operator, so the measurement is not physically interesting, but it gives a mathematically valid example with minimal calculation. Later we shall indicate how counterexamples can be obtained for other choices of a and b from appropriate solutions of (116).
Assuming a = 1 = b, the system (116) can be rewritten
(1/2 + g) 2 α 1 (g) + (1/2 − g) 2 α 2 (g) = 1 − (1/2 − 2g 2 )α 3 (g) (117) (1/2 − g) 2 α 1 (g) + (1/2 + g) 2 α 2 (g) = 1 − (1/2 − 2g 2 )α 3 (g) .
We will think of α 3 (g) as a free parameter to be arbitrarily chosen, and as noted previously, the choice will not affect the anomalous term (113). Viewed in this way, (117) becomes a system of two linear equations in two unknowns which become the same equation if α 2 = α 1 , with solution
α 2 (g) = α 1 (g) = 1 − (1/2 − 2g 2 )α 3 (g) (1/2 + g) 2 + (1/2 − g) 2 = 1 − (1/2 − 2g 2 )α 3 (g) 1/2 + 2g 2 ) .(118)
Since α 3 can be chosen arbitrarily, also α 2 = α 1 can be arbitrary; we shall choose α 3 (g) so that
α 2 (g) = α 1 (g) = 1 g 2 .(119)
To see that this solution will produce a counterexample, note that for ρ = ρ 11 ρ 12 ρ 21 ρ 22
and for any diagonal matrix
D = d 1 0 0 d 2 , [D, ρ] = 0 (d 1 − d 2 )ρ 12 (d 2 − d 1 )ρ 21 0 , and [ D, [D, ρ] ] = 0 (d 1 − d 2 ) 2 ρ 12 (d 2 − d 1 ) 2 ρ 21 0 .
In particular for j = 1, 2,
[ M j (g), [M j (g), ρ] ] = 0 4g 2 ρ 12 4g 2 ρ 21 0 ,
and since M 3 (g) is a multiple of the identity, [M 3 (g), ρ] = 0. Hence (113) becomes:
−(1/2)Tr [P f j α j [M j (g), [M j (g), ρ] ] ]
Tr
[P f ρ = −Tr [P f 0 4ρ 12 4ρ 21 0 ] Tr [P f ρ] .
(120) This is easily seen to be nonzero for ρ 12 = 0 and appropriate P f . For a norm 1 vector f :
= (f 1 , f 2 ) weak limit of (6) = Tr [P f {A, ρ}] 2Tr [P f ρ] + −8ℜ(f * 2 f 1 ρ 21 ) |f 1 | 2 ρ 11 + 2ℜ(f * 2 f 1 ρ 21 ) + |f 2 | 2 ρ 22 .(121)
The counterexample just given assumed that the system observable A := diag{a, b} was the identity to make the calculations easy, but counterexamples can be obtained for any system observable. For example, if A is the one-dimensional projector A := diag{1, 0} , and if system (117) is solved with α 1 (g) := 1/g 2 , then α 2 (g) = 1/g 2 − 1/(2g), and the weak limit of the anomalous term is the same as just calculated for A = I. [4] DJ [2] adds additional (very strong) hypotheses to those of DAJ which the counterexample just given does not satisfy. 4 Assuming these additional conditions, DJ attempts to prove that (6) evaluates to (7) in their "minimal disturbance limit". The next sections will present a more powerful counterexample which disproves this new claim of DJ.
Originally a new paper with the more powerful counterexample was submitted to the arXiv, but a moderator rejected it. He thought that instead, Version 1 should be modified. Rather than waste time on a lengthy and unpleasant appeal, I decided that it would be easier to do that.
The paper to this point is Version 1. The sections following comprise essentially the rejected arXiv paper, which presents the more powerful counterexample and critically analyzes DJ.
The new counterexample is fairly simple, utilizing three 3 × 3 matrices, but not as intuitive as one would like. It was found by analyzing the properties that measurement operators might have in order that (7) could be shown false, and then playing with parametrized 3 × 3 measurement operators, trying to adjust the parameters so that (7) would not hold. The Version 4 counterexample is simpler and more powerful than the Version 2 counterexample. No doubt even simpler counterexamples could be found. Besides the new counterexample, we attempt to clarify some statements in DJ which we think might be misleading. be inferred or even guessed from DAJ. The closest reference in DAJ to something similar is the following. Nothing is said about this being a general assumption for the rest of the paper. Indeed, such an assumption would seriously restrict the applicability of the following definition (6) of "general conditioned average" f  , which requires no such assumption. I studied DAJ for months without ever being led to even consider the possibility that this might be an assumption for the general claims of its abstract.
The contextual values
α = (α 1 , . . . , α N ) are obtained from the eigenvalues a = (a 1 , . . . , a m ) of A = diag(a 1 , . . . , a m ) as α = F (+) a
where F is an N × m matrix satisfying F α = a and F (+) its Moore-Penrose pseudo-inverse. The Version 1 counterexample does not satisfy this condition.
Relying only on what is written in DAJ, it would be very hard for a reader to guess that this is supposed to be a hypothesis for (6), or for a claim that (6) implies (7), or both. (I did consider these possibilities, but rejected them as too unlikely, as will be explained later in more detail.) The only passage of DAJ which seems possibly relevant is:
". . . we propose that the physically sensible choice of [contextual values] is the least redundant set uniquely related to the eigenvalues through the Moore-Penrose pseudoinverse."
DAJ gives no reason why this should be the "physically sensible choice". (DJ does attempt to address this issue, but unconvincingly and badly incorrectly, as will be discussed later.) Again, to assume this would seem to artificially limit the applicability of (6), since (6) is correct independently of this assumption.
We do not list the other hypotheses for DJ's attempted proof that (6) implies (7) because they are more technical and less surprising than the two just discussed. Our counterexample will satisfy all of the hypotheses listed in DJ. The counterexample for Version 2 has been replaced by a simpler example in Version 4.
A counterexample to the claim of DJ
Section V of DJ entitled "General Proof" attempts to show that (6) implies (7) under their listed hypotheses. The present section presents a counterexample which satisfies all of their listed hypotheses, yet the weak limit of their "general conditioned average" (6) is not the "quantum weak value" (7). We follow identically the analysis of Section 3 through equation (113). This time, we use a system observable
A = 1 0 0 0 0 0 0 0 0 .(201)
and three measurement operators which are 3 × 3 diagonal matrices:
M 1 (g) := 1/2 + g 0 0 0 1/2 0 0 0 1/2 + g , M 2 (g) := 1/3 + g 2 0 0 0 1/3 + g 0 0 0 1/3 , M 3 (g) := 1/6 − g − g 2 0 0 0 1/6 − g 0 0 0 1/6 − g .(202)
The contextual values α = (α 1 , α 2 , α 3 ) satisfy F α = a := (1, 0, 0) T with
F = 1/2 + g 1/3 + g 2 1/6 − g − g 2 1/2 1/3 + g 1/6 − g 1/2 + g 1/3 1/6 − g (203)
The matrix F is invertible with inverse (which is also equal to the Moore-Penrose pseudoinverse F (+) )
F (+) = F −1 = 1−6g 6g 2 1−2g 2g −1+9g 6g 2 1−6g 6g 2 1+2g 2g −1+3g 6g 2 −5−6g 6g 2 1+2g 2g 3g+5 6g 2 .(204)
The important thing to note is that the first column, which is also (α 1 , α 2 , α 3 ) T , is of leading order 1/g 2 as g → 0, which is all that the subsequent proof will use:
α 1 (g) = α 2 (g) = 1 − 6g 6g 2 , α 3 (g) = −5 − 6g 6g 2 .(205)
The full inverse (204) was obtained from a computer algebra program, and the first column (which is all that the counterexample will use) was also checked by hand using Cramer's rule. Equations (110) through (113) write the "general conditioned average" f A of (6) as a sum of two terms, one of which evaluates to (7) in the weak limit g → 0. The other term, called the "anomalous term", is given by (113) as: difference between weak limit of (6) and (7)
= lim g→0 k 1 2 α k (g)Tr [P f [M k (g), [ρ, M k (g)] ] ]
Tr [P f ρ] .
(206)
To disprove the claim of DJ, we need to show that there exists a state ρ and vector f such that the anomalous term does not vanish. It is well-known that the only matrix S such that for all projection matrices P f , Tr [P f S] = 0, is the zero matrix S = 0. 5 Hence it will be enough to show that lim
g→0 k − 1 2 α k (g)[M k (g), [M k (g), ρ] ] = 0.(207)
for some mixed state ρ such that for all nonzero vectors f , Tr [P f ρ] = 0. First note that for any diagonal matrix D = diag(d 1 , d 2 , d 3 ) and any matrix
ρ = (ρ) ij , [D, [D, ρ]] ij = (d i − d j ) 2 ρ ij .(208)
In the cases of interest to us, D will be one of the measurement operators
M k (g) , (d i (g) − d j (g)) 2 = O(g 2 )
for all i, j, and for some i, j, the leading order of (d i (g) − d j (g)) 2 is actually g 2 . The α k (g) all diverge like 1/g 2 as g → 0. Thus we can see without calculation that we will obtain a counterexample unless some unrecognized relation forces the terms of (207) to exactly cancel. 6 That cancellation does not occur in this case can be seen with minimal calculation as follows. In (208), take (i, j) := (1, 2), and note that (d 1 − d 2 ) 2 is always non-negative. When D = M 3 (g), from the power series
√ c + x = √ c + x 2 √ c + O(x 2 ) ,
one sees that (d 1 − d 2 ) 2 = O(g 4 ), and since α 3 (g) is only O(g −2 ), the k = 3 term in (207) vanishes in the limit g → 0.
We also have α 1 (g) = α 2 (g) = 1 6g 2 − 1 g , 5 A computational proof is routine, but to see this without calculation, recall that S, T := Tr S † T defines a complex Hilbert space structure (i.e., positive definite complex inner product) on the set of n × n matrices. If S, T vanishes for all projectors T = P f , then (by the spectral theorem), it vanishes for all Hermitian T , and hence for all T , in which case S is orthogonal to all elements of this Hilbert space and hence is the zero element. and for either D = M 1 (g) or D = M 2 (g),
(d 1 − d 2 ) 2 = (g/ √ 2) 2 + O(g 2 )) 2 = g 2 /2 + O(g 3 )
So, in the limit g → 0, (207) evaluates to − 1 2 1 6
1 2 ρ 12 = − ρ 12 24 .(209)
Note that all we care about is that (209) does not always vanish, and this can be seen solely from the fact that α 1 and α 2 have the same sign, so that the k = 1, 2 terms in (207) are negative multiples of ρ 12 which do not vanish identically in the limit g → 0.
To finish the proof, let ρ be a positive definite state (i.e., all eigenvalues strictly positive) such that ρ 12 = 0. Such a state can be constructed by starting with a positive definite diagonal state and adding a small perturbation to assure ρ 12 = 0 (which will result in a positive definite state if the perturbation is small enough). Since ρ is positive definite, Tr [ρP f ] = 0 for all nonzero vectors f , and we are done.
6 Discussion of DJ 6
.1 Possible error in DJ's proof
The counterexample given above unfortunately relies on some detailed calculation. A conceptual counterexample would certainly be preferable. A reader interested in discovering the truth of the matter will be faced with the unpleasant choice of wading through DJ's dense proof or checking the boring details of the counterexample. For such readers, it may be helpful if we point out what seems a potentially erroneous step in DJ's proof.
A step which caused me to question their proof occurs at the very end of their Section V: ". . . to have a pole of order higher than g n [n = 1 in the counterexample] then there must be at least one relevant singular value with an order greater than g n . [The counterexample has a singular value of order g 2 .] However, if that were the case then the expansion of F to order g n would have a relevant singular value of zero and therefore could not satisfy (25) . . ." I have not been able to guess a meaning for "the expansion of F to order g n would have a relevant singular value of zero" under which the last sentence would be true.
Significance of the Moore-Penrose pseudo-inverse
The original paper DAJ [1] introduced the Moore-Penrose pseudo-inverse as follows:
". . . we propose that the physically sensible choice of [contextual values α] is the least redundant set uniquely related to the eigenvalues [ a = (a 1 , . . . , a m ) with A = diag(a 1 , . . . , a m )] through the Moore-Penrose pseudoinverse." I puzzled for a long time over this statement. Besides the fact that the meaning of "least redundant set" was obscure to me, they give no reason why this choice (which presumably means α = F (+) a, with F (+) the Moore-Penrose pseudoinverse) should be considered the unique "physically sensible" choice, or even a physically sensible choice. The arXiv paper DJ which we are discussing attempts to fill this gap, but the attempt relies on erroneous mathematics and is unconvincing.
Before starting the discussion of this attempt, let me remark that although the attempt seems partly aimed at invalidating the counterexample of [3], it is basically irrelevant to that aim. That counterexample is a valid mathematical counterexample to a mathematical claim of DAJ as I imagine the vague exposition of DAJ would probably be interpreted by most readers. Though the counterexample uses a particular solution of the contextual value equation F α = a, it was never claimed that this solution has any physically desirable properties. Though DJ does not show that the counterexample is unphysical as DJ claims, even if it were shown unphysical, it would still disprove the claim that (6) necessarily evaluates to (7) in the "minimal disturbance limit". A reader of DAJ cannot reasonably be expected to guess that the definition of "minimal disturbance limit" is supposed to include the pseudo-inverse prescription. Therefore, the discussion will be directed toward analyzing the claim of DJ that the pseudo-inverse solution should be preferred because DJ thinks (incorrectly) that ". . . the pseudo-inverse solution will choose the solution that generally provides the most rapid statistical convergence for observable measurements on the system."
A careful analysis of DJ's argument for this claim will reveal flaws which invalidate it.
DJ writes:
"With the pseudo-inverse in hand, we then find a uniquely specified solution α 0 = F (+) a that is directly related to the eigenvalues of the operator. Other solutions α = α 0 + x of (3) will contain additional components in the null space of F , and will thus deviate from this least redundant solution. [True if sympathetically interpreted, but tautological.] Consequently, the solution α 0 has the least norm of all solutions . . ."
The Euclidean norm || α|| 2 := j α 2 j in the real Hilbert space R n has no physical significance in quantum theory. Why is it relevant that α 0 has least norm? The discussion immediately following may possibly be intended to answer this, but when analyzed it only tautologically repeats what has already been said.
However, an inattentive reader could easily get the impression that something had been proved.
DJ thinks that this immediately following discussion (at the bottom of the first column of p.4) gives "mathematical reasons for using the pseudoinverse", but in fact no convincing reason has been given. The next paragraph continues:
"In addition to the mathematical reasons for using the pseudoinverse in this context [referring to the discussion just analyzed, which doesn't give any convincing mathematical reason], there is an important physical one that we will now describe. As mentioned, a fully compatible detector can be used together with the contextual values to reconstruct any moment of a compatible observable. However, since the detector outcomes are imperfectly correlated with the observable, the contextual values typically lie outside of the eigenvalue range and many repetitions of the measurement must be practically performed to obtain adequate precision for the moments. Importantly, the uncertainty in the moments is controlled by the the variance, not of the observable operator, but of the contextual values themselves. [emphasis mine]"
Consider a probability space with outcomes {1, 2, . . . , n} with probability p j for outcome j. A random variable v is an assignment j → v j of a real number v j to each outcome j. The meanv of v is defined as usual bȳ v := j v j p j , and the variance τ 2 of v is defined by
τ 2 := j (v j −v) 2 p j = j v 2 j p j −v 2 .
Here we use the symbol τ 2 instead of the customary σ 2 to denote the variance to avoid confusion with the different σ 2 defined in DJ (as the second moment). One can speak of the "variance" of a random variable on a classical probability space, or of the "variance" of quantum observable measured in a given state. But what can it mean to speak of the "variance" of contextual values α j ? Contextual values are are predefined to satisfy
A = j α j M † j M j ,(102)
where A is the "system observable" and {M j } a collection of measurement operators. What is measured are the outcomes j. However, even though we know the contextual values beforehand from (102), one might speak of "measuring" them in the following sense. To every outcome j corresponds a contextual value α j . A given state of the system ρ makes the set of all outcomes j into a probability space by assigning a probability p j to each outcome j:
p j = Tr [M † j M j ρ].
The assignment j → α j is a random variable on this probability space, and it is meaningful to speak of its "variance". The subsequent analysis assumes that this is the meaning that DJ intended. This discussion may seem inappropriately elementary, but I was initially puzzled about this point, and it cannot hurt to make it explicit.
Note that the meanᾱ = Tr [Aρ] of this random variable is the same no matter how the contextual values α j are chosen so long as they satisfy the contextual value equation (102). That implies that choosing the contextual values so as to minimize the true variance τ 2 in a given state is equivalent to minimizing the second moment σ 2 . Note also that the mean and variance implicitly depend on the state ρ, and that there is no reason to think that one might be able to choose the contextual values so as to minimize the variance in all states.
DJ continues:
"Consequently, it is in the experimenters best interests to minimize the second moment of the contextual values,
σ 2 = j α 2 j p j ,(210)
where p j is the probability of outcome j."
DJ correctly identifies σ 2 as the second moment, but unless read very carefully, the subsequent discussion could encourage confusion of σ 2 with the true variance τ 2 . Next DJ notes that || α|| 2 is a (very crude) upper bound for σ 2 :
σ 2 := j α 2 j p j ≤ j α 2 j = || α|| 2 .( * )
"In the absence of prior knowledge about the system one is dealing with, this is the most general bound one can make. Therefore, the pseudo-inverse solution will choose the solution that generally provides the most rapid statistical convergence for observable measurements on the system."
This is highly questionable. Although it may not be clear at this point, subsequent paragraphs make clear that DJ is claiming that it is legitimate to use || α|| 2 as a sort of estimate for σ 2 , the strange and invalid justification for the claim being the sentences of the quote following equation (*). DJ's next paragraph computes || α(g)|| 2 for both the α(g) used in the counterexample of [3] and for the pseudo-inverse solution α 0 (g) = F (+) (g) a, using || α(g)|| 2 as a kind of crude estimate for σ 2 = σ 2 (g) = σ 2 (g, ρ).
"For the case of the counterexample, the Parrott solution (13) [(13) should be (11)] has to leading order the bound on the variance
|| α|| 2 = 3 g 4 − 3(a − b) 2g 3 + O( 1 g 2 ),(15)
while the pseudoinverse solution (11) [(11) should be (13)] has to leading order the bound
|| α|| 2 = (a − b) 2 8g 2 + 2 3 (a + b) 2 + O(g 2 ).(16)
For any observable a, the Parrott solution has detector variance of order O(1/g 4 )[emphasis mine], which would swamp any attempt to measure an observable near the weak limit. . . . However, the pseudoinverse solution has a detector variance of order O(1/g 2 ) in the worst case; . . . "
What invalidates the argument is the use of the crude upper bound (*) as an estimate for the second moment σ 2 and the subsequent claim that "the Parrott solution has detector variance of order O(1/g 4 ) . . . ". 7 Solely from upper bounds for two quantities, one cannot draw any reliable conclusions about the relative size of the quantities themselves. To see this clearly in a simpler context which uses essentially the same reasoning, consider the upper bounds
x < x 4 and x 2 < x 3 for real numbers x > 1. From the fact that the first upper bound x 4 (for x) is larger than the second upper bound x 3 (for x 2 ), we cannot conclude that x is larger than x 2 for x > 1. Yet DJ's argument relies on this type of incorrect reasoning.
In the interests of following closely the exposition of DJ, we passed rapidly over (*). Let us return to analyze it more closely:
σ 2 := j α 2 j p j ≤ j α 2 j = || α|| 2 .( * )
"In the absence of prior knowledge about the system one is dealing with, this is the most general bound one can make. Therefore, the pseudo-inverse solution will choose the solution that generally provides the most rapid statistical convergence for observable measurements on the system."
Note once again that σ 2 = σ 2 (ρ) depends implicitly on the state ρ because the probabilities p j = Tr [ρM † j M j ] of outcome j depend on ρ. Keeping this in mind, one sees how crude the upper bound (*) really is. For a nonzero system observable A, equality holds in (*) (i.e., σ 2 (ρ) = || α|| 2 ) only in the trivial case in which one particular p J = 1 and the others vanish, and in addition, α j (g) = 0 for j = J. That corresponds to the trivial case in which there is effectively only one measurement operator M J (g) satisfying α J (g)M † J (g)M J (g) = A. (The other measurement operators play the role of assuring that j M † j M j = I, but do not contribute to the estimation of the expectation of A in the state ρ, Tr [Aρ].) Since one of DJ's hypotheses (which we did not discuss above) is that lim g→0 M j (g) is a multiple of the identity for all j, also the system observable A is a multiple of the identity.
The statement following (*), that "this is the most general bound one can make", seems a very strange form of reasoning. Doubtless, (*) was the most general bound that the authors knew how to make, but it seems unscientific to base an important argument on an unsupported personal belief that no one else can do better.
In fact, a better bound is possible. By the Cauchy-Schwartz inequality,
σ 2 = j α 2 j p j ≤ [ j (α 2 j ) 2 ] 1/2 [ j p 2 j ] 1/2 ≤ j α 4 j 1/2 , ( * * )
since 0 ≤ p j ≤ 1, so p 2 j ≤ p j and j p 2 j ≤ j p j = 1. That (**) is a better bound than (*) when at least two α j are nonzero follows from
j α 4 j 1/2 2 = j (α 2 j ) 2 < j α 2 j 2 = || α|| 2 2 ,
because for any collection of at least two positive numbers {q j }, q 2 j < [ j q j ] 2 . If the authors were to reformulate their proposal for the appropriate choice of the contextual values in terms of this better bound, it seems unlikely that DAJ's proposed Moore-Penrose pseudo-inverse solution would minimize (**), or possible bounds even better than (**). And as pointed out earlier, the physical meaning or appropriateness of minimizing a particular upper bound for the detector's second moment remains obscure.
What is the "physically sensible" choice of contextual values?
In many experiments (indeed, in all experiments known to me), the system always starts in a known state ρ. For such an experiment, it seems to me that the "physically sensible choice" of contextual values would be the choice that minimizes the detector variance τ 2 = τ 2 (ρ) in that initial state ρ.
It is a simple exercise to work out a necessary condition for this minimization, and the pseudo-inverse prescription does not necessarily satisfy it. For the reader's convenience, we sketch the details.
The contextual values equation (102) for contextual values α = (α 1 , . . . , α N ) can always be written as a linear system given by a vector equation
α = F ( a) ,(211)
where a is a vector associated with the system observable A and F a matrix whose size will depend on the dimension of a. 8 Given an initial state ρ, measurement operators M j , and associated probabilities p j = Tr [ρM † j M j ], we want to minimize the detector variance
τ 2 (ρ) := i p i α 2 i − i p i α i 2 .(212)
As noted in the preceding subsection, for a particular state ρ and taking into account the contextual value equation (102), this is the same as minimizing the second moment
σ 2 (ρ) := i p i α 2 i .(213)
(To avoid confusion, we continue using DJ's nonstandard notation σ 2 for the second moment instead of the variance.) Let α P denote a particular solution of F ( α) = a. Then the general solution of F ( α) = a is α = α P + η with η in the nullspace Null(F ) of F , and
σ 2 := i p i α 2 i = i p i (α P i ) 2 + 2 i p i α P i η i + i p i η 2 i .(214)
For small η, a nonvanishing linear second term will dominate the quadratic third term, 9 and we see that if α P is to minimize σ 2 , then the vector (p 1 α P 1 , . . . , p N α P N ) must be orthogonal to the nullspace of F . This is the necessary condition mentioned earlier.
Thus it seems to me that a "physically sensible" choice of contextual values in this situation should satisfy this necessary condition. However, the pseudo-inverse solution is abstractly defined by the different condition that α P = (α P 1 , . . . , α P N ) be orthogonal to Null(F ). 10 Even if the state ρ is not known from the start, to estimate the expectation of A as j α j Tr [M † j M j ρ] = j α j p j , one needs to estimate the p j as frequencies of occurence of outcome j, so the p j can be regarded as experimentally determined to any desired accuracy. Given these p j , one can then choose the solution α to the contextual value equation F ( α) = a to minimize (214) and the detector variance. This procedure for minimizing the detector variance will rarely result in the pseudo-inverse solution.
6.4 Does DAJ assume that contextual values α come from the Moore-Penrose pseudo-inverse, α = F (+) a?
We have seen that none of the reasons that DJ gives for determining contextual values by the pseudo-inverse construction,
α = F (+) a ,(215)
hold up under scrutiny. DAJ doesn't give any valid reasons, either. Its "general conditioned average" (6) does not require this hypothesis, nor the hypothesis that the system observable A and measurement operators M j mutually commute. Why assume something that is not needed? DJ gives the false impression that DAJ unequivocally assumes (215) as a hypothesis. For example, "The problem with Parrott's counterexample is that he ignores this discussion [of defining the contextual values by the pseudo-inverse prescription α := F (+) a] . . .".
The totality of this "discussion" is the single sentence:
". . . we propose that the physically sensible choice of CV is the least redundant set uniquely related to the eigenvalues through the Moore-Penrose pseudoinverse." DAJ does devote a long paragraph to a complicated method of defining and calculating the Moore-Penrose pseudo-inverse, but that has nothing to do with the reasons for using the pseudo-inverse in the first place. A reference to a mathematical text would have sufficed and saved sufficient space to have clearly stated their hypotheses for (6) and for the claimed implication that (6) implies (7) in their "minimal disturbance limit". If the authors don't tell us, how can we poor readers possibly guess that the pseudo-inverse prescription (215) is assumed as a hypothesis for (6) (if in fact it is, which to this day I don't know), or if not, as a hypothesis for a section which follows (6), such as the "Weak values" section? When I wrote [3] giving the counterexample, I did consider the possibility that DAJ might possibly be assuming the pseudo-inverse solution, but rejected it as implausible. This was partly because they had previously sent me an attempted proof that their (6) implies (7) in their "minimal disturbance limit" which if correct (it wasn't) would have applied to any solution α, not just the pseudo-inverse solution. (It also would have applied even if the measurement operators and system observable did not mutually commute.) So, I knew to a certainty that when DAJ was submitted, there was no reason for the authors to have assumed the pseudo-inverse prescription.
Also, in the sweeping claim of DAJ's abstract that their "general conditioned average" (6) ". . . converges uniquely to the quantum weak value in the minimal disturbance limit", by no stretch of the imagination could the reader guess that the technical pseudo-inverse prescription would be part of the definition of "minimal disturbance limit". And if the prescription is not part of the definition of "minimal disturbance limit", then to justify the claim, the prescription would have to be taken as part of the definition of their "general conditioned average" (6). But the latter alternative would artificially limit the applicability of (6), since (6) is correct no matter how the contextual values are chosen (subject to the contextual value equation (102)).
Section VI of DJ:
The last four paragraphs of Section VI of DJ (entitled "Discussion") are misleading and in some ways incorrect. The reasons are given in Section 11.1 of [4] and will not be repeated here.
Acknowledgments
I was surprised to see in DJ the acknowledgment: "We acknowledge correspondence with S. Parrott". That made me wonder if protocol required that I provide a similar acknowledgment. And if so, what should it say? Would it be proper to acknowledge negative contributions as well as positive ones, and if so should I? If I didn't, how would I explain why I didn't simply ask the authors about some of the questionable points in DAJ?
The (nearly unique) positive contribution of the authors of DAJ to [4], [3], and the present work was to furnish their original argument that (6) implies (7) in their "minimal disturbance limit". That argument brought to my attention the decomposition of equation (111), which was part of their attempted proofs.
That argument was definitely incorrect because I found a counterexample to one of its steps. I sent the counterexample to the authors in mid-February, but they never acknowledged it. I made several subsequent inquiries about other points in DAJ, but all were ignored. I have not heard from them since February 19. (It is now June 23). (What little correspondence we did exchange was uniformly courteous.) That is why I was unable to clarify other vague points in DAJ such as for which results (if any) the pseudo-inverse solution was assumed as a hypothesis.
I intend to eventually post on my website, www.math.umb.edu/∼sp , a complete account of the strange aspects of this affair, which has been unique in my professional experience. It will raise questions about the editorial practices of influential journals of the American Physical Society, among other issues.
DJ acknowledges that their work was supported by two grants, at least one of which was taxpayer-supported via the National Science Fountation. The present work was not supported by any grants, unless donation of the author's time might be considered a kind of "grant".
If so, it is a "grant" to society in general. I have spent months trying to unravel DAJ, mostly without any help. I submit this to the arXiv to save others similar time. It is painful to realize that I have largely wasted my time for a contribution so small, but it is satisfying to hope that the time saved by others may result in larger contributions than I could have made.
Added in version 8: Version 2 of [2], arXiv:1106.1871v2 replies to the present work. It was was published in J. Phys. A: Math. Theor. 45 015304. The published version will be called DJpub below.
I thank the authors for noting a typo in the definition of the (3, 3) entry of the 3 × 3 matrix M 2 (g) in the Section 5 counterexample on p.11. The original entry 1/3 should have been 1/3, and this correction has been made in this Version 7. The original analysis assumed the correct value, so apart from this substitution, no changes were necessary.
DJpub reinterprets (unjustifiably, in my view) one of the hypotheses of [2] and notes that the counterexample given above does not satisfy the reinterpreted hypothesis. An analysis of DJpub has been posted in arXiv:1202.5604, and an abbreviated version has been under consideration by J. Phys. A for over 10 months (as of this writing, October 14, 2012).
Note the trivial identity for operators M, ρ: M ρM = M [ρ, M ] + M 2 ρ and the similar M ρM = −[ρ, M ]M + ρM 2 .
"
To illustrate [emphasis mine] the construction of the least redundant set of [contextual values], we consider the case when {M j } and all commute."
This is a technical definition which can be misleading if one does not realize that normal associations of the English phrase "minimally disturbing" are not implied. Further discussion can be found in[6] and[4].
DAJ only partially and unclearly defines its "minimally disturbing" condition, but in a message to Physical Review Letters (PRL) in response to a "Comment" paper that I sub-
The new additional hypotheses for the claim that (6) implies(7) in the "minimal disturbance limit" Section 5 of DJ lists several additional assumptions, the most important of which are:1. The M j commute with each other and A (so they can all be represented by diagonal matrices).This is a strong assumption. It is hard to imagine how it could reasonably4 However, the fact that these additional conditions cannot reasonably be inferred from DAJ is not made clear by DJ, and a casual reader might well obtain the opposite impression.
One useful observation that we can make from what we have done so far without detailed calculation is that the attempted proof of DJ is likely wrong or at least seriously incomplete, since that attempted proof concludes the vanishing of (207) on the basis of order of magnitude arguments only. Though framed in different language, it essentially says that (207) must vanish because they think that α k (g) = (F (+) (g)(1, 0, 0) T ) k = O(1/g) (contradicting (205)), while [M j (g), [M j (g), ρ]] = O(g 2 ).
DJ incorrectly identifies σ 2 as the "variance" instead of the second moment, but this is a mere slip. Ignoring this slip, technically one could argue that this statement is correct because to say that a quantity is O(1/g 4 ) only means that it increases no faster than 1/g 4 as g → 0.For example, g 8 = O(1/g 4 ). However, in the context and taking into account the typically sloppy use of the "big-oh" notation in the physics literature, most readers would probably interpret this passage as claiming that the "Parrott solution" has variance of leading order 1/g 4 , which would be an invalid conclusion from the argument.
If A or some of the measurement operators are not diagonal, then a will not be the vector of eigenvalues of A as in DJ. For example, if A is a general 2 × 2 Hermitian matrix, then a = (a 1 , a 2 , a 3 ) may be taken to be the three-dimensional vector (A 11 , A 12 , A 22 ), and in general, a may be formed from the components of A on or above its main diagonal.9 More precisely, if for some η the linear term does not vanish, then replacing η by xη, with x real, gives a quadratic function in x with nonvanishing linear term, which cannot have a minimum at x = 0.10 This is discussed but not proved in the Appendix to[4]. A formal statement and proof can be found in[11], p. 9, Theorem 1.1.1.
Contextual Values of Observables in Quantum Measurements. J Dressel, S Agarwal, A N Jordan, Phys. Rev. Lett. 104240401J. Dressel, S. Agarwal, and A. N. Jordan, "Contextual Values of Observ- ables in Quantum Measurements", Phys. Rev. Lett. 104 240401 (2010)
Sufficient conditions for uniqueness of the Weak Value. J Dressel, A N Jordan, arXiv:1106.1871v1J. Dressel and A. N. Jordan, "Sufficient conditions for uniqueness of the Weak Value", arXiv:1106.1871v1
Introduction to "contextual values" and a simpler counterexample to a claim of Dressel. S Parrott ; Agarwal, Jordan , arXiv:1105.4188v1Phys. Rev. Lett. 104240401S. Parrott, "Introduction to "contextual values" and a simpler counterex- ample to a claim of Dressel, Agarwal, and Jordan [Phys. Rev. Lett. 104 240401 (2010)], arXiv:1105.4188v1
Contextual weak values' of quantum measurements with positive measurement operators are not limited to the traditional weak value. S Parrott, arXiv:1102.4407v6Parrott, S. , " 'Contextual weak values' of quantum measurements with positive measurement operators are not limited to the traditional weak value", arXiv:1102.4407v6
M A Nielsen, I L Chuang, Quantum Computation and Quantum Information. CambridgeCambridge University PressNielsen, M. A. , Chuang, I. L. . Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)
H M Wiseman, G J Milburn, Quantum Measurement and Control. Cambridge University PressH. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press, 2009
Complex weak values in quantum measurement. R Jozsa, Phys Rev A. 7644103Jozsa, R., "Complex weak values in quantum measurement" Phys Rev A 76 044103 (2007)
Quantum weak values are not unique. What do they actually measure?. S Parrott, arXiv:0908.0035S. Parrott, "Quantum weak values are not unique. What do they actually measure?", arXiv:0908.0035
What do quantum 'weak' measurements actually measure?. S Parrott, arXiv:0909.0295Parrott, S. "What do quantum 'weak' measurements actually measure? arXiv:0909.0295
How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Y Aharonov, D Z Albert, L Vaidman, Phys. Rev. Lett. 60Aharonov, Y. , Albert, D. Z. , Vaidman, L. . How the result of a measure- ment of a component of the spin of a spin-1/2 particle can turn out to be 100., Phys. Rev. Lett 60, 1351-1354 (1988)
S L Campbell, C D Meyer, Generalized Inverses of Linear Transformations. S. L. Campbell and C. D. Meyer, Generalized Inverses of Linear Transfor- mations, Society for Industrial and Applied Mathematics, 2008
|
[] |
[
"A SUBMILLIMETER HCN LASER IN IRC+10216",
"A SUBMILLIMETER HCN LASER IN IRC+10216"
] |
[
"Peter Schilke ",
"David M Mehringer ",
"Karl M Menten "
] |
[] |
[
"Astrophysical Journal (Letters)"
] |
We report the detection of a strong submillimeter wavelength HCN laser line at a frequency near 805 GHz toward the carbon star IRC+10216. This line, the J=9-8 rotational transition within the (04 0 0) vibrationally excited state, is one of a series of HCN laser lines that were first detected in the laboratory in the early days of laser spectroscopy. Since its lower energy level is 4200 K above the ground state, the laser emission must arise from the inner part of IRC+10216's circumstellar envelope. To better characterize this environment, we observed other, thermally emitting, vibrationally excited HCN lines and find that they, like the laser line, arise in a region of temperature ≈ 1000 K that is located within the dust formation radius; this conclusion is supported by the linewidth of the laser. The (04 0 0), J=9-8 laser might be chemically pumped and may be the only known laser (or maser) that is excited both in the laboratory and in space by a similar mechanism.
|
10.1086/312416
|
[
"https://arxiv.org/pdf/astro-ph/9911377v1.pdf"
] | 17,990,217 |
astro-ph/9911377
|
3c1a67b855c29fabdcebacdd28e31021dbdfabf2
|
A SUBMILLIMETER HCN LASER IN IRC+10216
Peter Schilke
David M Mehringer
Karl M Menten
A SUBMILLIMETER HCN LASER IN IRC+10216
Astrophysical Journal (Letters)
arXiv:astro-ph/9911377v1 19 Nov 1999Subject headings: circumstellar matter -masers -stars: AGB and post-AGB -stars: mass loss
We report the detection of a strong submillimeter wavelength HCN laser line at a frequency near 805 GHz toward the carbon star IRC+10216. This line, the J=9-8 rotational transition within the (04 0 0) vibrationally excited state, is one of a series of HCN laser lines that were first detected in the laboratory in the early days of laser spectroscopy. Since its lower energy level is 4200 K above the ground state, the laser emission must arise from the inner part of IRC+10216's circumstellar envelope. To better characterize this environment, we observed other, thermally emitting, vibrationally excited HCN lines and find that they, like the laser line, arise in a region of temperature ≈ 1000 K that is located within the dust formation radius; this conclusion is supported by the linewidth of the laser. The (04 0 0), J=9-8 laser might be chemically pumped and may be the only known laser (or maser) that is excited both in the laboratory and in space by a similar mechanism.
Introduction
Rotational lines from vibrationally excited states of various molecules are useful probes of the hottest and densest regions of circumstellar envelopes around asymptotic giant branch (AGB) stars. For example, intense maser emission from vibrationally excited SiO (up to the v = 4 state; Cernicharo, Bujarrabal, & Santarén 1993) and H 2 O (in the ν 2 bending mode; Menten & Young 1995) is a ubiquitous phenomenon in oxygen-rich red giants and supergiants. Interferometric observations show that these masers arise from a region of thickness a few stellar radii that is located between the photosphere and the inner edge of the dust formation zone (Diamond et al. 1994;Greenhill et al. 1995;see Habing 1996 for a recent review).
In carbon-rich AGB stars, (sub)millimeter-wavelength vibrationally excited rotational lines have been observed from, among others, the HCN, CS, and SiS molecules in particular toward the extreme carbon star IRC+10216 (CW Leo) (Turner 1987a,b;Ziurys & Turner 1986). This object is due to its proximity (D ≈ 140 pc) and high mass-loss rate (Ṁ ≈ 2 × 10 −5 M ⊙ yr −1 , Crosas & Menten 1997;Groenewegen, van der Veen, & Matthews 1998) one of the most prolific and best-studied molecular line sources in the sky (see, e.g., Avery et al. 1992;Groesbeck, Phillips, & Blake 1994;Kawaguchi et al. 1995;Cernicharo, Guélin, & Kahane 1999). Recent millimeter-wavelength interferometric observations have shown that the vibrationally excited emission from all three of the molecules mentioned above arises from the innermost parts of IRC+10216's circumstellar envelope, i.e. from a region with a radius of ≈ 5-10 stellar radii, r ⋆ (Lucas & Guilloteau 1992;Lucas & Guélin 1999); we assume a value of 0. ′′ 023 for r ⋆ (Danchi et al. 1994). Therefore, vibrationally excited lines are interesting probes of the hot (T ∼ 1000 K), dense regions in which dust grains form (see, e.g., Winters, Dominik, & Sedlmayr 1994;Danchi et al. 1994;Groenewegen 1997).
In the past, only two molecular lines, both from HCN, have been found to show strong maser emission in carbon stars. One of these, the 177 GHz J=2-1 transition from the (01 1c 0) excited bending mode was detected toward IRC+10216 by Lucas & Cernicharo (1989). The high flux density (≈ 400 Jy), asymmetric line profile, and time variability (Cernicharo et al. 1999) clearly prove that this line is masing. In this Letter, we report the detection of submillimeter laser 4 emission in the J=9-8 rotational line within the highly excited (04 0 0) vibrational state of HCN at a frequency near 805 GHz.
Observations
The observations were made in 1998 February as part of a systematic study of IRC+10216's submillimeter spectrum with the 810 GHz receiver (Kooi et al. 1998) of the Caltech Submillimeter Observatory (CSO) 10.4 m telescope on Mauna Kea, Hawaii. Typical single sideband system temperatures were about 5000 K under excellent weather conditions, i.e. time periods when the precipitable water vapor content was below 1 mm. Pointing was checked by observing the CO(7-6) line toward IRC+10216 itself, resulting in a pointing accuracy of ≈ 2 ′′ . At 805 GHz the beam width is 9 ′′ (FWHM) and the main beam efficiency was found to be 0.3 from observations of Mars. The spectrometer and observing procedure are described in Menten & Young (1995). We estimate our absolute calibration uncertainties to be ≈ 30%. Additional observations of HCN J=3-2, 4-3, and 8-7 rotational lines within different vibrational states at frequencies near 266, 355, and 709 GHz were made in 1998 April with the same telescope.
Results and Discussion
A HCN submillimeter laser
At millimeter wavelengths IRC+10216 has a rich molecular spectrum with emission from more than 50 species arising from a compact central region and/or from hollow shells centered on the star. Radii and thicknesses of these shells depend on the excitation requirements of the molecule in question and on the chemistry. Generally speaking, since submillimeter transitions need much higher densities and temperatures for their excitation than lower frequency lines, the submillimeter spectrum of IRC+10216 is considerably sparser than its millimeter spectrum. In particular, lines from long chain molecules and other radicals, which reside in shells of modest temperature and density (see, e.g., Lucas & Guélin 1999), are largely absent and most of the detected lines can be assigned to simple diatomic or triatomic species emitted from the inner envelope. Moreover, almost all of the submillimeter lines observed toward IRC+10216 are readily assigned to known carriers (Groesbeck et al. 1994), whereas in the mm-range many (weak) lines remain unidentified (Cernicharo et al. 1999).
We were thus surprised to discover a strong spectral line at a frequency of ≈ 804.751 GHz ( Fig. 1), which we could not readily identify in available line catalogs (e.g., Pickett et. al. 1998). The line's frequency was verified by checking its sideband assignment by shifting the local oscillator frequency. With a total width < ∼ 10 km s −1 (FWZP) the emission covers a much smaller velocity interval than most lines observed toward IRC+10216, which have full widths of 29 km s −1 , i.e. twice the terminal outflow velocity, v ∞ . This indicates that the emission arises from a region in which the outflow has not yet reached its terminal velocity and/or that the line shows laser action. The latter is also suggested by the asymmetric line profile which is distinctly different from the symmetric parabolic, rectangular, or double "horn"-shaped profiles expected and commonly observed in the IRC+10216 envelope.
A literature search resulted in the identification of the line with the J=9-8 transition of HCN within the ν 2 = 4 vibrationally excited bending mode 5 (04 0 0), whose lower (J=8) level is 4163 K above the ground state and whose frequency was measured by Hocker & Javan (1967) as 804.7509 GHz with an estimated uncertainty of 1 MHz. Most interestingly, this line was among the first submillimeter laser transitions detected in the laboratory (Mathias, Crocker, & Wills 1965) and is part of the extensively studied (11 1 0)/(04 0 0) coriolis-perturbed system around the J= 9 to 11 rotational levels in these states (Gebbie, Stone, & Findlay 1964;Lide & Maki 1967). These lasers, which are produced in gas discharges, were intensely studied in the early years of molecular laser research (for reviews see, e.g., Chantry 1971;Kneubühl & Sturzenegger 1980;Pichamuthu 1983). This results in rotation-vibration interactions, i.e. Coriolis coupling, and non-vanishing transition probabilities for cross-ladder transitions (Lide & Maki 1967).
To understand the occurence of laser action in these cross-ladder and connected transitions, we note that the (100) state is, practically speaking, metastable (Herzberg 1945), with vibrational transitions from the (100) to the (000) ground state being ∼ 2 -3 orders of magnitudes slower than (010)-(000) and (001)-(000) transitions (Kim & King 1979). In the laboratory, HCN molecules produced in a gas discharge are distributed over the various vibrational states, most of which [including the (04 0 0) state] will depopulate quickly by spontaneous emission directly to the ground state or by cascading down the vibrational ladder. However, the metastable (100) state, and all combination states building upon it, such as the (11 1 0) state, will become overpopulated relative to other vibrational states, including the (04 0 0) state. A consequence of this is laser action in the J=11-10 and 10-9 cross-ladder transitions, which in turn causes the (04 0 0), J=10 and J=9 states to be overpopulated and the (11 1 0), J=11 and 10 to be underpopulated, resulting in laser action in the (04 0 0), J=10-9 and 9-8 as well as in the (11 1 0), J=12-11 and 11-10 transitions.
It is interesting to note that of the more than 120 cosmic maser and laser transitions known so far, the HCN (04 0 0), J=9-8 line is, apart from the (J, K) = (3, 3) inversion line of ammonia (NH 3 , Mangum & Wootten 1994), the only one that has been reported to show maser action in an astronomical object as well as in the laboratory. In the case of the NH 3 (3,3) maser, inversion in the laboratory (Gordon, Zeiger, & Townes 1954) is achieved by completely different means than in interstellar space (Flower, Offer, & Schilke 1990). In contrast, quite remarkably, the HCN laser might be produced in IRC+10216's circumstellar envelope by essentially the same mechanism as in a gas discharge call. This will be discussed in §3.3 after a we have described the environment in which the laser arises.
Vibrationally excited HCN in IRC+10216's innermost envelope
Since the (04 0 0), J=9-8 line is emitted from a very highly excited vibrational state of the molecule, we try to characterize the excitation conditions of hot HCN in IRC+10216's inner envelope by using our own multi-line observations as well as data from the literature.
The first observations of vibrationally excited HCN in IRC+10216 were made by Ziurys & Turner (1986), who detected the J=3-2 transition in the (01 1c 0), (01 1 d 0), and (02 0 0) states and find the levels to be thermally populated. Guilloteau, Omont, & Lucas (1987) report strong maser emission in the (02 0 0), J=1-0 transition toward the carbon star CIT 6. Weaker maser emission in this line is found toward other carbon-rich stars (Lucas, Guilloteau, & Omont 1988) including IRC+10216 (Guilloteau et al. 1987). Interferometric observations toward the latter star suggest that the maser action arises from within a region of radius 0. ′′ 25, or ≈ 10 r ⋆ (Lucas & Guilloteau 1992). Strong maser action in the (01 1c 0), J=2-1 transition was found toward IRC+10216 by Lucas & Cernicharo (1989), who also observed thermally excited lines from the (01 1 d 0), (02 0 0) and (02 2 0) states. (100) and (001) states of HCN were first detected by Groesbeck, Phillips, & Blake (1994), who observed the J=4-3 transition in these as well as in several of the (01 ℓ 0) and (02 ℓ 0) states. Using the Infrared Space Observatory (ISO) Cernicharo et al. (1996) found emission in these vibrational states from the J=18-17 up to the J=48-47 rotational lines. These observations indicate a vibrational temperature of 1000 K. Fig. 3.-Vibration-rotation Boltzmann plot for HCN, comprising our data and the (020) data from Groesbeck et al. 1994. The entries for the (01 1c 0) maser and (04 0 0) laser lines are marked as an "x" and a triangle, respectively, and are not used in the fit.
Rotational lines from the
We complemented the data from the literature by using the CSO to observe various other thermally excited lines from vibrationally excited states with energies comparable to or even exceeding that of the (04 0 0), J=9-8 line. Details of these observations will be given in a future publication. Using the complete set of data we construct the vibrational-rotational excitation diagram shown in Fig. 3, for which we assume a point-like source and "normalize" the intensities, which were measured with different resolutions, to a 11 ′′ FWHM beam. Clearly, the abscissa values of the data points representing the (01 1c 0), J=2-1 maser and our (04 0 0), J=9-8 laser are much higher than the thermal equilibrium values expected for these lines. Excluding those points, a least square fit yields a vibrational temperature of 1000 K, similar to the value found by Cernicharo et al. (1996). Assuming that the brightness temperature of the thermally excited lines in the (04 0 0) state is equal to or smaller than this number, we find an lower limit of 0. ′′ 08, or ≈ 3.5r ⋆ for the radius of the emitting region. This is comparable to the upper limit found interferometrically for the radius of the (02 0 0), J=1-0 emission region and is close to the dust condensation radius, which according to the model calculations of Groenewegen (1997) is at 4.5 r ⋆ at a temperature of 1075 K. The high excitation vibrationally excited thermal lines have, in contrast to the laser line, symmetric profiles covering a velocity range that is very similar to that of the laser line, i.e. much narrower than that of lines from the vibrational ground state. We therefore conclude that the HCN laser line is formed at the dust condensation radius at a temperature of 1000 K.
Both the laser discussed here and the (01 1c 0), J=2-1 and (02 0 0), J=1-0 masers exhibit enhanced emission at velocities that are blueshifted relative to the stellar velocity. If a systematic outflow has already started in the part of the envelope giving rise to the HCN masers, one would expect the blueshifted emission to arise from between the star and the observer, allowing for the possibility that its greater intensity is due to amplication of background emission from the stellar photosphere. A significant reduction of the redshifted emission due to geometrical blocking by the star can be excluded if the size of the HCN emission region has dimensions of the order discussed above. On the other hand, we know from Very Long Baseline Interferometry of SiO masers in O-rich Mira stars that red-and blue-shifted SiO maser emission does not necessarily arise from, respectively, the back and front parts of the circumstellar shell; instead ring-like emission distributions are observed, indicating tangential amplification (Diamond et al. 1994). Recent high resolution infrared observations in the K ′ band (Weigelt et al. 1998) indicate that the dust shell of IRC+10216 is extremely clumpy and asymmetric to a radius of ≈ 0. ′′ 2 or 9 r ⋆ , which is comparable to the size of the dust formation and HCN laser/maser region. Maser pumping by such a highly anisotropic infrared field would certainly result in an inhomogenous emission distribution giving rise to an asymmetric line profile. Very high resolution interferometric observations with future instruments such as the Atacama Large Millimeter Array (ALMA) are needed to image the maser emission and study its dynamics.
Pumping Schemes
A direct radiative pump from the ground state into the (11 1 0) state does not seem likely, since in that case the other vibrational states would be preferentially populated by virtue of their higher transition probabilities. We were unable to check for possible line overlaps between infrared transitions from HCN and other molecules abundant in IRC+10216, such as CO, and pumping by that mechanism remains viable in principle. Collisional pumping might also be possible, since the critical density (i.e., the ratio of the Einstein A-coefficient to the collisional excitation coefficient) of the (100) stack, which includes, among others, the (11 1 0) combination state, is lower than that of the other vibrational states, if we assume comparable collision rates. A more detailed analysis of this mechanism requires knowledge of collisional excitation coefficients.
Most interestingly, chemical pumping, which has been invoked to explain the laboratory HCN lasers (see, e.g., Chantry 1971;Pichamuthu 1983), might also be at work in IRC+10216. The basic idea is that the HCN molecules are formed in random vibrational states, and a population inversion is achieved in the (11 1 0)/(04 0 0) system because of the wide difference in the radiative decay rates of the different vibrational levels. For this mechanism to work, the number of HCN creation events per time unit should exceed the number of HCN laser photons emitted by a considerable amount. HCN is formed in thermochemical equilibrium (see Tsuji 1964), where the reaction of CN with H 2 is likely to be a major channel. In the following we assume that HCN formation proceeds at a typical rate of 10 −10 s −1 in a shell between radii of 2 × 10 14 and 3 × 10 14 cm, i.e. just interior to the dust formation radius. If we further assume a CN abundance of 10 −11 (the value at 1000 K; Lafont, Lucas & Omont 1982) and an H 2 density of 10 12 cm −3 , we calculate 8 × 10 45 HCN creation events per second. From our spectrum we determine an integrated line flux of 6100 Jy km s −1 , which corresponds to an isotropic photon luminosity of 7 × 10 43 s −1 . Given that any alternative production mechanism would only increase the number of creation events, it appears that chemical pumping seems possible, although the uncertainties in the numbers used are considerable.
Finally, we note that if the circumstellar 805 GHz laser results from overpopulation of the (11 1 0) state and (11 1 0)-(04 0 0) cross-ladder transitions, as is highly likely, one would also expect the other transitions of this Coriolis-coupled system, like in the laboratory, to show strong laser action toward IRC+10216. Of these (see Fig. 2), the (11 1 0)-(04 0 0), J=10-9 cross ladder transition at 890.76 GHz and the 894.31 GHz J=10-9 line within the (04 0 0) state are, in principle, observable from high, dry mountain sites, but outside the frequency range of the receiving system currently available to us. The (11 1 0)-(04 0 0), J=11-10 and (11 1 0), J=11-10 transitions at 964.31 and 967.96 GHz, respectively, should be observable with the Stratospheric Observatory for Infrared Astronomy (SOFIA).
We would like to thank Tom Phillips, the director of the CSO, for granting us observing time, Darrel Dowell for help with some of the observations, and Mark Reid for comments on the manuscript. Work at the Caltech Submillimeter Observatory is supported by NSF contract AST 96-15025.
Fig. 1 .
1-Spectrum of the (04 0 0), J=9-8 laser line is shown as the bold line. The very narrow linewidth and asymmetric shape are conspicuous when compared to a spectrum of the HCN J=4-3 transition from the vibrational ground state, which is overlaid as the thin line. The flux density scale is appropriate for the (04 0 0), J=9-8 line only.
Fig. 2 .
2-Excerpt from the level diagrams of the (11 1 0) and (04 0 0) vibrationally excited states of HCN near the Coriolis resonance involving the J=8 to 12 rotational levels. Prominent laser lines detected in the laboratory are indicated by arrows along with their frequencies.The bold arrow indicates the (04 0 0), J=9-8 transition discussed in the present paper.
Fig. 2 shows
2HCN energy levels around the J=10 and J=9 rotational states within the (11 1 0) and (04 0 0) vibrational ladders. Fortuitously, several of the rotational levels in the (11 1 0) state are very close in energy to levels with identical J in the (04 0 0) state.
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, Bonn, D-53121, Germany 2 California Institute of Technology, MS 320-47, Pasadena, CA 91125 3 Present address: University of Illinois, Department of Astronomy, 1002 W. Green St., Urbana, IL 61801
In the laboratory spectroscopy literature, the term laser is used for stimulated emission in the far-infrared and submillimeter regimes, i.e. between microwave and optical wavelengths, in particular for the HCN line discussed here. We have followed this convention (see alsoStrelnitski et al. 1996), instead of using the term maser, which is common in radio astronomy.
The linear triatomic HCN molecule has three vibrational states, one bending mode ν 2 , with ν 2 = 1 being ≈ 1000 K above ground, and two stretching vibrations: ν 1 , corresponding to the CN stretch, where ν 1 = 1 is 3000 K above the ground state, and ν 3 , the CH stretch, with ν 3 = 1 being 4700 K above ground. [We follow the notation ofHerzberg (1945) and note that various papers differ in their usage of the ν 1 , ν 2 , and ν 3 quantum numbers.] The bending mode ν 2 is doubly degenerate; for ν 2 = 0 this degeneracy is lifted and the levels are split by rotation-vibration interaction. A new quantum number, ℓ (where ℓ = ν, ν − 2, . . . − ν) is needed to describe the system. Overtones and combination bands also exist. From here on, the notation (ν 1 ν ℓ 2 ν 3 ) is used.
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[] |
[
"Cross frequency coupling in next generation inhibitory neural mass models",
"Cross frequency coupling in next generation inhibitory neural mass models"
] |
[
"Andrea Ceni \nDepartment of Computer Science\nCollege of Engineering, Mathematics and Physical Sciences\nUniversity of Exeter\nEX4 4QFExeterUK\n",
"Simona Olmi \nInria Sophia Antipolis Méditerranée Research Centre\n2004, 06902Route des Lucioles, ValbonneFrance\n\nCNR -Consiglio Nazionale delle Ricerche -Istituto dei Sistemi Complessi\nvia Madonna del Piano 1050019Sesto FiorentinoItaly\n",
"Alessandro Torcini \nCNR -Consiglio Nazionale delle Ricerche -Istituto dei Sistemi Complessi\nvia Madonna del Piano 1050019Sesto FiorentinoItaly\n\nLaboratoire de Physique Théorique et Modélisation\nUMR 8089\nUniversité de Cergy-Pontoise\nCNRS\n95302Cergy-Pontoise cedexFrance\n",
"David Angulo-Garcia \nModelado Computacional -Dinámica y Complejidad de Sistemas\nInstituto de Matemáticas Aplicadas\nCarrera 6 #36 -100\n\nUniversidad de Cartagena. Cartagena de Indias\nColombia\n"
] |
[
"Department of Computer Science\nCollege of Engineering, Mathematics and Physical Sciences\nUniversity of Exeter\nEX4 4QFExeterUK",
"Inria Sophia Antipolis Méditerranée Research Centre\n2004, 06902Route des Lucioles, ValbonneFrance",
"CNR -Consiglio Nazionale delle Ricerche -Istituto dei Sistemi Complessi\nvia Madonna del Piano 1050019Sesto FiorentinoItaly",
"CNR -Consiglio Nazionale delle Ricerche -Istituto dei Sistemi Complessi\nvia Madonna del Piano 1050019Sesto FiorentinoItaly",
"Laboratoire de Physique Théorique et Modélisation\nUMR 8089\nUniversité de Cergy-Pontoise\nCNRS\n95302Cergy-Pontoise cedexFrance",
"Modelado Computacional -Dinámica y Complejidad de Sistemas\nInstituto de Matemáticas Aplicadas\nCarrera 6 #36 -100",
"Universidad de Cartagena. Cartagena de Indias\nColombia"
] |
[] |
Coupling among neural rhythms is one of the most important mechanisms at the basis of cognitive processes in the brain. In this study we consider a neural mass model, rigorously obtained from the microscopic dynamics of an inhibitory spiking network with exponential synapses, able to autonomously generate collective oscillations (COs). These oscillations emerge via a super-critical Hopf bifurcation, and their frequencies are controlled by the synaptic time scale, the synaptic coupling and the excitability of the neural population. Furthermore, we show that two inhibitory populations in a master-slave configuration with different synaptic time scales can display various collective dynamical regimes: namely, damped oscillations towards a stable focus, periodic and quasi-periodic oscillations, and chaos. Finally, when bidirectionally coupled the two inhibitory populations can exhibit different types of θ-γ cross-frequency couplings (CFCs): namely, phase-phase and phase-amplitude CFC. The coupling between θ and γ COs is enhanced in presence of a external θ forcing, reminiscent of the type of modulation induced in Hippocampal and Cortex circuits via optogenetic drive.
|
10.1101/745828
|
[
"https://arxiv.org/pdf/1908.07954v1.pdf"
] | 201,126,163 |
1908.07954
|
6b2ad634047a68c33a733f587df8ec3d37f091af
|
Cross frequency coupling in next generation inhibitory neural mass models
Andrea Ceni
Department of Computer Science
College of Engineering, Mathematics and Physical Sciences
University of Exeter
EX4 4QFExeterUK
Simona Olmi
Inria Sophia Antipolis Méditerranée Research Centre
2004, 06902Route des Lucioles, ValbonneFrance
CNR -Consiglio Nazionale delle Ricerche -Istituto dei Sistemi Complessi
via Madonna del Piano 1050019Sesto FiorentinoItaly
Alessandro Torcini
CNR -Consiglio Nazionale delle Ricerche -Istituto dei Sistemi Complessi
via Madonna del Piano 1050019Sesto FiorentinoItaly
Laboratoire de Physique Théorique et Modélisation
UMR 8089
Université de Cergy-Pontoise
CNRS
95302Cergy-Pontoise cedexFrance
David Angulo-Garcia
Modelado Computacional -Dinámica y Complejidad de Sistemas
Instituto de Matemáticas Aplicadas
Carrera 6 #36 -100
Universidad de Cartagena. Cartagena de Indias
Colombia
Cross frequency coupling in next generation inhibitory neural mass models
Coupling among neural rhythms is one of the most important mechanisms at the basis of cognitive processes in the brain. In this study we consider a neural mass model, rigorously obtained from the microscopic dynamics of an inhibitory spiking network with exponential synapses, able to autonomously generate collective oscillations (COs). These oscillations emerge via a super-critical Hopf bifurcation, and their frequencies are controlled by the synaptic time scale, the synaptic coupling and the excitability of the neural population. Furthermore, we show that two inhibitory populations in a master-slave configuration with different synaptic time scales can display various collective dynamical regimes: namely, damped oscillations towards a stable focus, periodic and quasi-periodic oscillations, and chaos. Finally, when bidirectionally coupled the two inhibitory populations can exhibit different types of θ-γ cross-frequency couplings (CFCs): namely, phase-phase and phase-amplitude CFC. The coupling between θ and γ COs is enhanced in presence of a external θ forcing, reminiscent of the type of modulation induced in Hippocampal and Cortex circuits via optogenetic drive.
In healthy conditions, the brain's activity reveals a series of intermingled oscillations, generated by large ensembles of neurons, which provide a functional substrate for information processing. How single neuron properties influence neuronal population dynamics is an unsolved question, whose solution could help in the understanding of the emergent collective behaviors arising during cognitive processes. Here we consider a neural mass model, which reproduces exactly the macroscopic activity of a network of spiking neurons. This mean-field model is employed to shade some light on an important and ubiquitous neural mechanism underlying information processing in the brain: the θ-γ cross-frequency coupling. In particular, we will explore in detail the conditions under which two coupled inhibitory neural populations can generate these functionally relevant coupled rhythms.
I. INTRODUCTION
Neural rhythms are the backbone of information coding in the brain 13 . These rhythms are the proxy of large populations of neurons that orchestrate their activity in a regular fashion and selectively communicate with other populations producing complex functional interactions 15,52 .
One of the most prominent rhythmic interaction appearing in the brain is the so-called θ-γ coupling, exhibited between the slow oscillating θ band (4Hz -12Hz) and the faster γ rhythm (25 -100 Hz) 34 . This specific frequency interaction is an example of a more general mechanism termed Cross-Frequency-Coupling (CFC) 15,28 . CFC has been proposed to be at the basis of sequence representation, long distance communication, sensory parsing and de-multiplexing 25,28 . CFC can manifest in a variety of ways depending on the type of modulation that one rhythm imposes on the other (Amplitude-Amplitude, Amplitude-Frequency, Phase-Amplitude, Frequency-Frequency, Frequency-Phase or Phase-Phase) 28 .
In this article, we will focus on Phase-Phase (P-P) and Phase-Amplitude (P-A) couplings of θ and γ rhythms. In particular, P-P coupling refers to n:m phase locking between gamma and theta phase oscillations 46 and it has been demonstrated to play a role in visual tasks in humans 24 and it has been identified in the rodent hippocampus during maze exploration 5 . The P-A coupling (or θ-nested γ oscillations) corresponds to the fact that the phase of the theta-oscillation modifies the amplitude of the gamma waves and it has been shown to support the formation of new episodic memories in the human hippocampus 33 and to emerge in various part of the rodent brain during optogenetic theta stimulations in vitro 2,10,11,42 .
In an attempt to understand the complex interactions emerging during CFC, several mathematical models describing the activity of a large ensemble of neurons have been proposed 19 . Widely known examples include the phenomenologically based Wilson-Cowan 55 and Jansen-Rit 27 neural mass models. While the usefulness of these heuristic models is out of any doubt, they are not related to any underlying microscopic dynamics of a neuronal arXiv:1908.07954v1 [nlin.AO] 21 Aug 2019 population. A second approach consists on the use of a mean-field (MF) description of the activity of a large ensemble of interacting neurons 8,9,16,39,40,45,49 . Although in this latter MF framework some knowledge is gained on the underlying microscopic dynamics that manifests at the macroscopic level, it comes with the downside of several, not always verified, assumptions about the statistics of single neuron firings 50 .
Recently, it has been possible to obtain in an exact manner a macroscopic description of an infinite ensemble of pulse-coupled Quadratic Integrate-and-Fire (QIF) neurons 18,36,38,43 . This result has been achieved thanks to the Ott-Antonsen ansatz, which allows to exactly obtain the macroscopic evolution of phase oscillator networks fully coupled via purely sinusoidal field 41 . In particular, in 38 the authors have been able to derive for QIF neurons with instantaneous synapses an exact neural mass model in terms of the firing rate and the average membrane potential. This exactly reduced model allows for the first time to understand the effects of the microscopic neural dynamics at the macroscopic level without making use of deliberate assumptions on the firing statistics.
In this paper, we employ the exactly reduced model introduced in 20 for QIF neurons with exponential synapses to study the emergence of mixed-mode oscillations in inhibitory networks with fast and slow synaptic kinetics. In particular, in sub-section III A we will characterize the dynamical behavior of a single population and review the conditions required to observe collective oscillations (COs). Afterwards, in sub-section III B, we will analyze the response of the population to a modulating signal paying special attention to the emergence of phase synchronization. We will then consider the possible macroscopic dynamics displayed by two coupled populations characterized by different synaptic time scales in a master-slave configuration (sub-section III C) and in a bidirectional set-up (sub-section III D). In this latter case we will focus on the conditions for the emergence of θ-γ CFCs among the COs displayed by the two populations.
II. MODEL AND METHODS
A. Network Dynamics
Through this paper we will consider either one or two populations of QIF neurons interacting via inhibitory post-synaptic potentials (IPSPs) with an exponential profile. In this framework, the activity of the population l ∈ {A, B} is described by the dynamics of the membrane potentials V (l) i of its neurons and of the associated synaptic fields S (l) i . Here we will assume a fully coupled topology for all networks, hence each neuron within a certain population is subject to the same synaptic field S (l) , where the neuron index has been dropped. Therefore, the dynamics of the network can be written as
τV (l) i = (V (l) i ) 2 + η (l) i + J ll τ S (l) + J kl τ S (k) + I (l) (t) τ (l) dṠ (l) = −S (l) + 1 N (l) t (l) j δ(t − t (l) j ) (1) i = 1, ..., N (l) l, k ∈ {A, B} ;
where τ = 10 ms is the membrane time constant which is assumed equal for both populations, η (l) i is the excitability of the i th neuron of population l, J lk is the strength of the inhibitory synaptic coupling of population l acting on population k and I (l) (t) is a time dependent external current applied on population l. The synaptic field S (l) (t) is the linear super-position of the all exponential IPSPs s(t) = e −t/τ (l) d emitted within the l population in the past. Due to the quadratic term in the membrane potential evolution, which allows the variable to reach infinity in a finite time, the emission of the j th spike in the network occurs at time t
(l) j whenever V (l) i (t (l)− j ) → +∞, while the reset mechanism is modeled by setting V (l) i (t (l)+ j ) → −∞, immediately after the spike emission.
For reasons that will be clear in the next paragraph we will also assume that the neuron excitability values η (l) i are randomly distributed according to a Lorentzian probability density function (PDF)
g l (η) = 1 π ∆ (l) (η −η (l) ) 2 + (∆ (l) ) 2 ,(2)
whereη (l) is the median and ∆ (l) is the half-width halfmaximum (HWHM) of the PDF. For simplicity, we set η = 1 throughout the paper, unless otherwise stated. In order to characterize the macroscopic dynamics we will employ the following indicators:
r (l) (t) = 1 N (l) ∆t t (l) j δ(t−t (l) j ), v (l) (t) = 1 N (l) N (l) j V (l) j (t),
which represent the average population activity of each network and the average membrane potential, respectively. In particular the average population activity of the l−network r (l) (t) is given by the number of spikes emitted in a time unit ∆t, divided by the total number of neurons. Furthermore, the emergence of COs in the dynamical evolution, correspondng to periodic motions of r (l) (t) and v (l) (t), will be characterized in terms of their frequencies ν (l) .
B. Mean-Field Evolution:
In the limit of infinitely many neurons N (l) → ∞ we can derive, for each population l, the evolution of the PDF ρ (l) (V |η, t) describing the probability distribution of finding a neuron with potential V and excitability η at a time t via the continuity equation:
∂ρ (l) (V |η, t) ∂t = − ∂F (l) (V |η, t) ∂V (l)(3)F (l) (V |η, t) = ρ (l) (V (l) ) 2 + η (l) + I (l) ,
where F (l) (V |η, t) is the probability flux while I (l) = J ll τ S (l) + J kl τ S (k) + I (l) (t) (for l, k ∈ {A, B}) is the contribution of all synaptic currents plus the external input. The continuity equation (3) is completed with the boundary conditions describing the firing and reset mechanisms of the network model (1), which reads as
lim V →−∞ F (l) (V |η, t) = lim V →∞ F (l) (V |η, t).(4)
Following the theoretical framework developed in 38 , one can apply the Ott-Antosen ansatz 41 to obtain an exact macroscopic description of the infinite dimensional 2-population system (1) in terms of collective variables. In order to obtain exact analytic results it is crucial to assume that the distribution of the neuronal excitabilities is a Lorentzian PDF, however the overall picture is not particularly influenced by considering others PDFs, like Gaussian and Erdös-Renyi ones 38 . In the case of the QIF model the macroscopic variables for each l−population are the instantaneous firing rate r (l) , the average membrane potential v (l) and the mean synaptic activity s (l) , which evolve according to:
r (l) = ∆ (l) τ 2 π + 2r (l) v (l) τ (5) v (l) = (v (l) ) 2 +η (l) + I (l) (t) τ + J ll s (l) + J kl s (k) − τ (πr (l) ) 2 s (l) = 1 τ (l) d [−s (l) + r (l) ],
for l, k ∈ {A, B}.
C. Dynamical Indicators
We make use of two dynamical indicators to characterize the evolution of the MF model (5): a Poincaré section and the corresponding Lyapunov Spectrum (LS).
The considered Poincaré section is defined as the manifoldṙ (A) (t) = 0 withṙ (A) (t − ) > 0, which amounts to identify the local maxima r max of the time trace
δṙ (l) = 2(r (l) δv (l) + v (A) δr (A) ) τ (6) δv (l) = 2v (l) δv (l) τ + J ll δs (l) + J kl δs (k) − 2π 2 τ r (l) δr (l) δṡ (l) = δs (l) + δr (l) τ (l) d .
The LS is thus composed by 6 Lyapunov Exponents (LEs) {λ i } which quantify the average growth rates of infinitesimal perturbations along the different orthogonal manifolds estimated as
λ i = lim t→∞ 1 t log |δ(t)| |δ 0 | ,(7)
by employing the well known technique described in Benettin et al. 6 to maintain the tangent vectors orthonormal during the evolution. From the knowledge of the LS one can obtain an estimation of the fractal dimension of the corresponding invariant set in terms of the socalled Kaplan-Yorke dimension D KY 29 , defined implicitly as follows:
D KY i=1 λ i ≡ 0 ,(8)
D. Locking Characterization
To investigate the capability of two interacting populations to lock their dynamics in a biologically relevant manner, we measure the degree of phase synchronization in the exactly reduced system (5). Therefore, we extract the phase of the population activity, by performing the Hilbert transform H[·] of the firing rate for each population l ∈ {A, B}, thus obtaining the imaginary part of the analytic signal: namely, φ (l) (t) := r (l) (t) + jH[r (l) (t)]. The evolution of the phase in time is then obtained as Φ (l) (t) = arg[φ (l) (t)]. A generalized phase difference of the n : m phase-locked mode can be defined as:
∆Φ nm (t) = nΦ (A) − mΦ (B) ,(9)
and the degree of synchronization in the phase locked regime can be quantified in terms of the Kuramoto order parameter for the phase difference, namely:
ρ nm = | e j∆Φnm(t) | ,(10)
where · denotes a time average, and | · | is the norm of the complex number 5,30 .
III. RESULTS
A. Self sustained oscillations in one population
Firstly, by following 20 we analyse the case of a single population with self-inhibition in absence of any external drive. Without lack of generality we take in account just population A: this amounts to have a set of only three equations in (5) with J BA = I (A) (t) = 0. For simplicity in the notation we finally drop the indices denoting the populations.
In the fully coupled QIF network oscillations can be observed only for IPSPs of finite duration, namely exponential in the present case. In particular, COs appear when the equilibrium point of the macroscopic system (r 0 , v 0 , s 0 ) undergoes a Hopf bifurcation. Simulations of the QIF network model and the corresponding MF dynamics are compared in Fig. 1 revealing a very good agreement both in the asynchronous and in the oscillatory state. In particular, for the parameters considered in Fig Due to the simplicity of the reduced model it is also possible to parametrize the Hopf boundaries where the asynchronous state loses stability as a function of the marginally stable solution (r 0 , v 0 , s 0 ). In particular taking τ d and J as bifurcation parameters, one can verify that the boundaries of the Hopf bifurcation curves are defined by:
τ (H) d = 9τ v 2 0η τ − π 2 r 2 0 τ 3 ± τ (η − π 2 r 2 0 τ 2 ) 2 + 2v 2 0 (9η − 41π 2 r 2 0 τ 2 ) + 17v 4 0 16 (π 2 r 2 0 τ 2 v 0 + v 3 0 )(11)J (H) =η + v 2 0 r 0 τ − π 2 r 0 τ(12)
The equilibrium values are related by the equalities v 0 = −∆/(2τ πr 0 ) and s 0 = r 0 , with r 0 acting as a free parameter (see the Appendix for more details).
The phase diagrams showing the existence of self sus- tained oscillations in the {r 0 , τ d } plane are displayed in Fig. 2 (A), for three values of ∆: the region inside the closed curves corresponds to the oscillating regime. Upon decreasing the dispersion of the excitability (∆), the region of oscillatory behavior increases. This result highlights that some degree of homogeneity in the neural population is required in order to sustain a collective activity. In particular, for dispersions larger than a critical value ∆ c it is impossible for the system to sustain COs (for the parameter employed in the figure ∆ c ≈ 0.1453).
The inset in Fig. 2 (A) displays the same boundaries in the {τ d , J} plane. From this figure one can observe that, upon increasing (decreasing) ∆, the range of inhibitory strength and of synaptic times required to sustain oscillations decreases (increase). Thus indicating that more heterogeneous is the system the more the parameters J and τ d should be finely tuned in order to have COs. It's worth mentioning that for instantaneous synapses (corresponding to τ d → 0) no oscillations can emerge autonomously in fully coupled systems with homogeneous synaptic coupling as shown in 20 and evident from Fig. 2. However, COs can be observed in sparse balanced networks also for instantaneous synapses and in absence of any delay in the signal transmission 21 .
In order to understand the role played by the different parameters in modifying the frequency of the COs, we have estimated this frequency for ∆ = 0.5 in the (τ d , J)plane. The results are shown as a heat-map in Fig. 2 (B). It turns out that the frequency tends to decrease for increasing values of τ d while it is almost independent on the value of J.
It is also worth noticing the fundamental role played by the self-inhibition in sustaining the autonomously generated oscillations, as it becomes clear from Eq. (11), since the Hopf bifurcations exist only for negative values of J.
From these results we can conclude that a single pop-ulation of QIF neurons can self sustain oscillations with a wide range of frequencies ν 5 − 30 Hz thanks to a finite synaptic time and to the self-inhibitory action of the neurons within the population.
B. One population under external forcing
Another relevant scenario in the framework of CFC 25 , is the case where an oscillatory drive I(t) is applied to a neural population exhibiting COs. The forcing term can represent an input generated from an another neural population or an external stimulus. Therefore, we examine the behavior of the MF model driven by the following harmonic signal
I(t) = −I 0 (1 + sin(2πν 0 t))(13)
characterized by a driving frequency ν 0 and an amplitude I 0 . Notice that we have chosen a strictly negative harmonic signal, to asses the effect on the population dynamics of a driving signal originating from a distinct inhibitory population. The results of this analysis are illustrated in Fig. 3. First, we study the phase locking of the population dynamics, characterized by an oscillatory frequency ν, to the modulatory input, for different forcing frequencies ν 0 and amplitudes I 0 (see Fig. 3 (A)). For small amplitudes, the external modulation is only able to lock the dynamics into a given n : m mode (measured in terms of the indicator (10)) for a limited range of forcing frequencies ν 0 , while the ratio n : m decreases for increasing ν 0 . Furthermore, the range where phase locking is observable increases with the amplitude I 0 , thus giving rise to the Arnold tongues shown in Fig. 3 (A).
To better understand how the locking emerges, we consider the ratio ν/ν 0 between the CO frequency and the forcing one for a large interval of ν 0 -values and for a fixed amplitude value I 0 = 0.4, denoted as a red dashed line in Fig. 3 (A). The results are reported in Fig. 3 (B), the ratio ν/ν 0 reveals a structure similar to a devil's staircase, presenting plateaus (corresponding to the locked modes) intermingled with regions where the ratio has not always a monotonic behaviour. For the same parameters, we also report the maxima r max of the instantaneous firing frequency of the forced system and the corresponding Lyapunov Spectrum (LS) in Figs. 3 (C) and (D), respectively. From these two indicators one can infer that most of the phase-locked regions correspond to regular periodic motion, as revealed by the single value of r max and by a single zero LE observables in a large portion of the devil's staircase plateaus. On the other hand, for ν 0 > 15 Hz the regions where the r(t) presents peaks of different heights are in correspondence with the nonflat regions of the devil's staircase. In particular these regions are associated to quasi-periodic motions, as confirmed by the existence of two zero LEs in the LS. A zoom in the region ν 0 = [4,14] Hz, corresponding to 1 : 1 locking, is reported in the insets of Figs. 3 (C) and (D). These enlargements show the emergence of chaotic windows, where r max assumes values over continuous in-tervals and the maximal LE is positive. Furthermore, in these chaotic windows D KY is slightly larger than two, as shown in 3 (E), indicating that the chaotic attractor is low dimensional. This is confirmed by the stroboscopic attractor reported in Fig. 4 (B), obtained by reporting the macroscopic variables at regular time intervals equal to integer multiples of the forcing period ν −1 0 . Indeed the points of the attractor cover a set with a dimension slightly larger than one, since one degree of freedom is lost due to the stroboscopic observation. Interestingly, the chaotic motion appears despite the 1 : 1 locking, this means that the time trace of r(t) presents always a single oscillation within a cycle of the external forcing but characterized by different amplitudes 44 , as shown in Fig. 4 (A).
C. Two populations in a master-slave configuration
Despite the fact that at a macroscopic level the network dynamics of a single population with exponential synapses is exactly described in the limit N → ∞ by three degrees of freedom (5) our and previous analysis 20 have not reported evidences of chaotic motions for a sin- gle inhibitory population. The situation is different for an excitatory population, as briefly discussed in 7 , or in presence of an external forcing as shown in the previous sub-section.
In this sub-section we want to analyze the dynamical regimes emerging when a fast oscillating population (indicated as A) is driven by a a slowly oscillating population (denoted by B) in a master-slave configuration corresponding to J BA = 0 and J AB = 0. Particular attention will be devoted to chaotic regimes. The different possible scenarios can be captured by considering only two sets of parameters, denoted as C 1 and C 2 and essentially characterized by different ratios of the synaptic time scales of the fast and slow family, namely: The coupling between the two population J BA and the network heterogeneity ∆ = ∆ (A) = ∆ (B) will be employed as control parameters, while we will assume for simplicityη which organizes the plane in four different regions. In these regions, labelled I-IV, the prevalent dynamics corresponds to stable foci in I, stable limit cycles in II and III, and to stable Tori T 2 in IV (see Fig. 5 (A) ). For each region we report in Fig. 5 BA is the locus of Hopf bifurcations (black solid) dividing foci (I) from stable oscillations (III), while at larger coupling J BA it becomes a secondary Hopf (or Torus) bifurcation line (blue solid) separating periodic (II) from quasi-periodic motions (IV). Moreover, the region III of stable limit cycles is divided by the region IV where Tori T 2 emerge from an another Torus bifurcation line (blue solid). Finally regions I and II are separated by a super-critical Hopf line (black solid).
We will now focus on the case ∆ = 0.01 (corresponding to the red dashed line in Fig. 5 (A)), to analyze the different regimes observable by varying J BA . To this aim, similarly to what done in the previous sub-sections, we characterize the dynamics of the system in terms of the values of the Poincaré map r max , which corresponds to periodic COs. This is confirmed by the values of the LS reported in 6 (B): the LEs are all negative except for the first one that is zero. At J BA ≈ −6.05 a broad band appears for the distribution of r (A) max indicating that the time trace now displays maxima of different heights. This is due to the Torus bifurcation leading from a periodic to a quasi-periodic motion. The emergence of quasi-periodic motions is confirmed by the fact that in the corresponding intervals the first two LEs are zero (Fig. 6 (D)). For larger values of the cross-coupling (namely, J BA ≈ [−5.8 : −5]) a period three window is clearly observable. Beyond this interval one observes quasi-periodic motions for almost all the negative values of the cross-coupling, apart narrow parameter intervals were locking of the two frequencies of the COs occur, as expected beyond a Torus bifurcation 31 .
We then proceed to study the parameter set C 2 . First, we show that the bidimensional bifurcation diagram in the plane {∆, J BA } presents a similar structure to that reported in Fig. 5 Fig. 7 (A)). As in the previous case, we select the value ∆ = 0.01 for analyzing the distribution of maxima of the firing rate r (A) and the associated Lyapunov spectra , see Fig. 7. For highly negative values of the crossinhibition we observe a periodic behaviour of the firing rate r (A) . At the intersection with the torus bifurcation (occurring at J BA = −7.48) quasi-periodicity emerges similarly to what observed for the parameter set C 1 . However, at larger values of the cross-inhibition (namely J BA ∈ [−7.3445, −7.3217]) we observe a period-doubling cascade leading to chaos for J BA −7.32. The systems stays chaotic in the interval J BA ∈ [−7.32, −7.18] apart for the occurrence of periodic windows. This is confirmed by the fact that the maximal LE becomes positive in the corresponding interval, as shown in Fig. 7 (D). An example of chaotic attractor is reported in Fig. 7 (F) with a fractal dimension slightly larger than two, as confirmed also by the estimation of D KY whose values are displayed in Fig. 7 (E). As in the case of the single forced population the macroscopic chaotic attractor is low dimensional.
For J BA > −7.18 we have periodic and quasi-periodic activity, but no more chaos, in particular for J BA > −4.2 we essentially observe mostly quasi-periodic motions up to J BA = 0. As mentioned before, the main difference between the parameter sets C 1 and C 2 is the ratio of the time scales associated to the synaptic filtering. For the parameter set C 1 the time scale ratio τ to be crucially dependent on inhibitory networks 14 , the origin of the θ-modulation is still under debate. It has been suggested to be due either to an external excitatory drive 12 or to a cross-disinhibition originating from a distinct inhibitory population 23,54 .
In this sub-section we analyze the possibility that two bidirectionally interacting inhibitory populations could be at the basis of the θ − γ CFC. Inspired by previous analysis, we propose the difference in the synaptic time kinetics as a possible mechanism to achieve CFC 54 . Therefore we set the synaptic time scale of the fast (slow) population to τ A,d = 9 ms (τ B,d = 50 ms), which corresponds approximately to the time scales of IPSPs generated via GABA A,f ast (GABA A,slow ) receptors. Regarding the other parameters, internal to each population, these are chosen in a such a way that the self-generated oscillations correspond roughly to θ and γ rhythms, respectively.
Firstly, we consider the case in which no external modulation is present, i.e I (l) (t) = 0. Depending on the value of the cross-coupling parameters J AB and J BA different type of m : n P-P coupling can be achieved. In particular, as shown in Fig. 8 (A), we observe 1 : 1 and 2 : 1 phase synchronization in large regions of the (J AB ; J BA ) plane, while 3 : 1 and 5 : 2 locking emerge only along restricted stripes of the plane. In particular, we focus on the values of cross-inhibition J lk for which it is possible to achieve a 3 : 1 phase synchronization, corresponding to a θ − γ coupling. As evident from the green area in Fig. 8 (A), this specific P-P coupling occurs only for low values of J AB , namely J AB ∈ [0.5, 2.5].
Among the parameter values corresponding to 3 : 1 locking we choose for further analysis the ones for which the order parameter ρ 31 is maximal (denoted as a red circle in Fig. 8 (A)). In particular, we performed simulation of the network (1) as well as of the corresponding MF model (5). The raster plot in Fig. 8 (B) confirms that the two populations display COs locked in a 3 : 1 fashion. During the burst emitted from the slow population the fast one displays irregular asynchronous activity followed by three rapid bursts (each lasting around 10 ms), before the next CO of the slow population. The fact that the slow population emits bursts of longer duration is confirmed by the analysis of the instantaneous firing rates reported in Fig. 8 (D). Indeed, r (A) has oscillation of amplitude much larger than r (B) indicating that more neurons are recruited for a burst of population (A) with respect to population (B). This difference in the oscillations amplitude can also explain why the 3 : 1 locked mode is observable for J AB J BA , indeed for larger J AB the activity of the slow population would be silenced. It is worth to notice in Fig. 8 Fig. 8 (D)) reveals that the amount of power in the θ band is quite small (see the peak around 10 Hz in the inset) with respect to the power in the γ band. This indicates that a CFC among the two bands is indeed present, but the interaction is limited.
Furthermore, varying the amplitude of the heterogeneity, as measured by ∆ = ∆ (A) = ∆ (B) , we can verify the capability of the network to sustain the 3 : 1 locked mode even in presence of disorder in the neural excitabilities. It can be seen that the system loses the ability to sustain such a locked state already for ∆ > 0.1, indicating that CFC can occur only for a limited amount of disorder in the distribution of the neuronal excitabilities in agreement with the results reported in 54 (see Fig. 8 (E). For large disorder the only possible locked state is that corresponding to 1 : 1 phase synchronization.
Sofar we have analyzed the possibility that θ and γ rhythms were locally generated in inhibitory populations with different synaptic scales. However, the results of several optogenetic experiments performed for different area of the hippocampus and of the enthorinal cortex suggest that a θ frequency drive is sufficient to induce in vitro θ-γ CFCs 2,10,11,42 . However, the interpretation of these experiments disagrees on the origin of the locally generated γ oscillations. Two mechanism have been suggested: namely, either inhibitory 2,42 or excitatory-inhibitory feedback loops 10,11 . Therefore, to clarify if recurrently coupled inhibitory populations, with different synaptic time scales, under a θ-drive can display θ-γ CFC we drive the slow population via an external current I (B) = I (B) 0 sin(2πν θ t) with ν θ = 10 Hz, while the rest of the parameters remains unchanged.
As before, we look for the range of cross-inhibitions in which a 3 : 1 phase-locked mode emerges: results are plotted in Fig. 9 (A). We observe that the region where θ − γ CFC can be observed definitely enlarge in presence of an external θ-modulation. Furthermore, as observable from the power spectrum reported in Fig. 9 (B) the power in the θ band is noticeably increased as a consequence of the external modulation with respect to the non-modulated case. Moreover, the θ − γ CFC is now observable over a wider range of disorder on the excitabilities, indeed the 3 : 1 locked state survives up to ∆ ≈ 0.2 − 0.3, as shown in Fig. 9 (C). These evidences are similar to what reported in 54 , where adding a slow modulation to the θ generating population was sufficient to render more robust the observed CFC to the presence of disorder in the excitability distribution.
Finally, if we look at the network activity we observe two different scenarios corresponding to the 3 : 1 locked mode: (1) a P-P locking at low disorder and (2) a P-A locking (or θ-nested γ oscillations) at larger disorder. The first scenario is characterized by the fast population displaying clear COs slightly modulated in their amplitudes by the activity of the slow population, but tightly locked in phase with the slow ones (see Fig. 10 (A,B)). The second scenario presents a firing activity of the fast population strongly modulated in its amplitude by the slow population as observable in Fig. 10 (A,B). In this latter case the neurons in population (A) fire almost asynchronously with a really low firing rate, however the coupling with by the activity of the slow population (B) is reflected in a clear modulation of the firing rate r (A) , analogously to what reported for θ-nested γ oscillations induced by optogenetic stimulation 11,42 .
IV. CONCLUDING REMARKS
Self sustained oscillatory behavior has been widely studied in the neuroscientific community, as several coding mechanisms rely on the rhythms generated by interneuronal populations 53 . In particular, the emergence of COs in inhibitory networks has been usually related to the presence of an additional timescale, beyond the one associated with the membrane potential evolution, which can be either the transmission delay 8,9 or a finite synaptic time 18,20,36,51 . The only example of COs emerging for instantaneous synapses and in absence of delay has been reported in 21 for a sparse network of QIF neurons in a balanced setup. In this case, for values of the in-degree sufficiently large the COs emerge due to endogenous fluctuations, which persist in the thermodynamic limit.
In this paper we have considered an heterogeneous inhibitory population with exponentially decaying synapses, which can be described at a macroscopic level by an exact reduced model of three variables: namely, the firing rate, the average membrane potential and the mean synaptic activity. As shown in 18,20 , the presence of the synaptic dynamics is at the origin of the COs, emerging via a super-critical Hopf bifurcation. In particular, we have shown that the period of the COs is controlled by the synaptic time scale and that, for increasing heterogeneity, the observation of COs require finer and finer tuning of the model parameters. Moreover, we have characterized in detail the effect of an in-hibitory periodic current on a single self-oscillating population. The external forcing leads to the appearance of locking phenomena characterized by Arnold tongues and devil's staircase. We have also identified low dimensional chaotic windows, where the instantaneous firing rate of the forced population display oscillations of irregular amplitudes, but tightly locked to the external signal oscillations 44 .
Furthermore, we considered two inhibitory populations connected in a master-slave configuration, i.e. unidirectionally coupled, where the fast oscillating population is forced by the one with slow synaptic dynamics. In a single population we observed, for sufficiently large heterogeneity, only focus solutions at the macroscopic level, which can turn in COs by reducing the disorder in the network. In presence of a second population the complexity of the macroscopic solutions increases for crossinhibitory coupling not too negative. In particular, by increasing the cross-coupling, the focus becomes a limit cycle via a super-critical Hopf bifurcation and the COs a quasi-periodic motion via a Torus bifurcation. Depending on the parameter values a period doubling cascade leading to chaotic behaviour is observable above the Torus bifurcation. In particular, the macroscopic attractor has a fractal dimension slightly larger than two with a single associated positive Lyapunov exponent despite the fact that the two coupled neural mass model are described by six degrees of freedom. The macroscopic solutions we have found in the master-slave configuration are similar to those identified in 37 for θ-neurons populations coupled via pulses of finite width.
Even though the dichotomy between chaos and reliability in brain coding is a debated topic 3,26,32,35 , it is of great importance to establish the conditions in which such behavior may appear. In the present case, chaotic behavior is found only when the synaptic time scales of the two inhibitory populations are quite different. However, these values are consistent with synaptic times observable in interneuron populations with fast and slow GABA A receptors, as shown in 4,47,54 .
We finally explored the possibility that two interacting inhibitory networks could give rise to specific crossfrequency-coupling mechanisms 25 . In particular, for its relevance in neuroscience we limited the analysis to θ-γ CFC emerging as a consequence of the interaction between fast and slow GABA A kinetics, as reported in 54 for populations of Hodgkin-Huxley neurons. In the original set-up was possible to observe θ-γ CFC coupling in a narrow region of parameters and for a limited range of heterogeneity. The addition of a θ-forcing on the slow population renders the system more robust to the disorder in the excitability distribution and enlarges the observability region of the θ-γ CFC. Furthermore, we observed two kind of CFC: namely, P-P (P-A) coupling for low (high) heterogeneity. Both these scenarios have been reported experimentally for θ-γ oscillations: namely, P-A coupling has been reported in vitro for optogenetic θ-stimulations of the the hippocampal area CA1 11 and CA3 2 , as well as of the medial enthorinal cortex 42 ; P-P coupling have been observed in the hippocampus in behaving rats 5,17 .
Our analysis shows, for the first time to our knowledge, that the θ-γ CFC, reported for Hodgkin-Huxley networks in 54 , can be reproduced also at the level of exact neural mass models for two coupled inhibitory populations. Our results pave the way for further studies of other CFC mechanisms present in the brain, e.g by employing analytically estimated macroscopic Phase Response Curves 22 to characterize phase synchronization in multiscale networks of QIF neurons.
In order to study the linear stability of the equilibrium point, we consider the corresponding eigenvalue problem, namely det 2v 0 /τ − Λ 2r 0 /τ 0 −2π 2 τ r 0 2v 0 /τ − Λ J 1/τ d 0 −Λ − 1/τ d = 0 (17) where Λ are the complex eigenvalues which can be found by solving the following characteristic polynomial p(Λ) = τ d τ 2 Λ 3 + AΛ 2 + Λ (τ d B − 4τ v 0 ) + (B − 2r 0 Jτ ) (18) where A = (τ 2 − 4v 0 τ d τ ) and B = (4v 2 0 + 4π 2 r 2 0 τ 2 ). In order to obtain a parametrization of the Hopf bifurcation curve we impose Λ = iΩ with Ω ∈ \ {0} and solve p(iΩ) = 0, which can only be satisfied if Re[p(iΩ)] = 0 and Im[p(iΩ)] = 0 .
Solving for Ω Eqs. (19) we end up with:
Ω Re = ± (B − 2r 0 Jτ ) A(20)Ω Im = ± (τ d B − 4τ v 0 ) τ d τ 2 .(21)
By equating (20) and (21) we can find the values of J (H) where the Hopf bifurcation occurs, namely
J (H) = 2v 0 τ 2 4π 2 τ 2 d r 2 0 + 1 + 4τ 2 d v 2 0 − 4τ d τ v 0 τ d r 0 τ 2 .
(22) Finally, by introducing Eq. (22) in (15), and noticing from Eq. (16) that s 0 = r 0 , one can derive the values of the synaptic time scale τ (H) d that bounds the oscillating region and that is reported in Eq. (11). Notice that the dependence on ∆ is implicitly introduced in the expression of v 0 , therefore, it possible to obtain also the critical value ∆ (H) associated to the Hopf transition by substituting (14) in (15) and solving for ∆. It should be also stressed that this approach cannot distinguish between super-critical and sub-critical Hopf bifurcations. As a matter of fact for an inhibitory QIF population with exponential synapses no sub-critical bifurcations have been reported for homogeneous synaptic couplings 20 , while these emerge whenever a disorder is introduced either in the synaptic couplings or in the link distribution 7 . 1) showing the fast and slow population in blue and red colors respectively for the same system analyzed in Fig. 9 with the optimal pair {JAB, JBA} = {−1, −6.63} and ∆ = 0.05. B) Time traces of r(t) for the fast (blue) and slow (red) populations for the case depicted in the raster plot in A). C-D) same as in A-B) for the same parameters except for ∆ = 0.2. Parameters as in Fig. 9.
FIG. 1 .
1Oscillations in a self inhibited QIF population: Left (right) column refers to damped (self-sustained) COs. From top to bottom: raster plots (A)-(D); istantaneous firing rate r(t) (B)-(E); average membrane potential v(t)(C)-(F). The network simulations are reported as black dots, while the MF dynamics (5) is shown as a red line. For the network simulation N = 10000 QIF units are simulated. Left (right) panels were obtained with τ d = 3 ms (τ d = 8 ms). Other parametersη = 1, ∆ = 0.05, J = −20 and τ = 10 ms.
r (A) (t). On the other hand to compute the LS one should consider the time evolution of the tangent vector δ = {δr (A) , δv (A) , δs (A) , δr (B) , δv (B) , δs (B) } resulting from the linearization of the original system Eq. (5), namely
and panels Fig. 1 (A-C) refer to a stable focus for the MF at τ d < τ (H) d , while panels Fig. 1 (D-E) to a stable limit cycle for τ d > τ (H) d .
FIG. 2 .
2Oscillation Stability Region: A) Hopf bifurcation boundaries in the (r0, τ d ) plane obtained from (11) for three different values of the heterogeneity: namely ∆ = 0.05 (blue), ∆ = 0.1 (red), ∆ = 0.14 (yellow). Inset: Hopf bifurcation boundaries in the (τ d , J) space for the same ∆-values. B) Heat map of the frequencies of oscillation of the instantaneous firing rate in the (τ d , J) plane, for ∆ = 0.05. The black curve coincides with the Hopf boundary denoted in blue in the inset of panel A). To quantify the frequencies of COs, a transient time tt = 1 s is discarded and then the number of peaks in r(t) are counted and divided by the simulation time ts = 2 s. For this figure, τ = 10 ms andη = 1.
FIG. 3 .
3Single population driven by a harmonic signal: A) Phase locking diagram of a single inhibitory population driven by a purely inhibitory harmonic external drive (13) in the (I0, ν0)-plane. The Arnold's tongues corresponding to the different regions of phase locking are plotted in colors according to the color code reported in the panel. The dashed line indicates the value I0 = 0.4 analysed in the subsequent panels. B) Ratio of the system's oscillation frequency ν and of the driving frequency ν0 as a function of ν0 showing a devil's staircase structure. Maxima of r (C) and the corresponding Lyapunov exponents (D) as a function of ν0. In particular, the first, second and third LEs are reported as blue, red and green lines, respectively. The dashed rectangles indicate the zoomed regions in C) and D) shown in the corresponding insets. E) Kaplan-Yorke fractal dimension DKY versus ν0, the dashed red line denotes a value of two. The employed parameters are J = −10,η, ∆ = 0.01, τ = 10 ms, τ d = 80 ms. For the estimation of rmax and of the Lyapunov exponents a transient time of tt = 1 s was discarded and then the maximum values were stored over a time interval ts = 3 s, while for the LS the tangent space was followed for a period ts = 20 s.
FIG. 4 .
4Chaotic attractor for a single driven population: A) Time trace of the average firing rate r(t) (blue solid line) and profile of the forcing term I(t) (red dashed line) are shown for a chaotic regime.The chaotic dynamics is clearly locked to the frequency of the external drive. B) Stroboscopic attractor: the values of (r, v, s) are recorded at regular time intervals corresponding to integer multiples of the forcing period ν −1 0 . Parameters for this figure as in Fig. 3 with ν0 = 10.4 Hz.
ms, J BB = −20}.
(A) =η (B) = 1. Let us first consider the set of parameters C 1 , in this case the analysis of the possible bifurcations arising in the {∆, J BA } plane reveals the existence of a codimension two bifurcation point at (∆ (H) , J (H) BA ) ≈ (0.078, −4.75)
(B) a corresponding sample trajectory projected in the sub-space {r (A) , v (A) } taken in proximity of the codimension two point. The critical vertical line observable at ∆ (H) 0.077 in Fig. 5 (A) is a direct consequence of the super-critical Hopf bifurcation already present at the level of single population discussed in sub-section III A. This line for J < J (H)
and of the associated LS. In particular in Fig. 6 (A) are reported the values of r (A) max in the range of cross-inhibition J BA = [−10, 0].
(A). Indeed also in the present case a codimension two point located at {∆ (H) , J (H) BA } ≈ {0.06, 0} divides the phase space in 4 regions analogous to those observed for the parameter set C 1 and separated by the same kind of bifurcations (see the inset of
5, while for the set C 2 becomes 1:32. We have been able to find chaotic motions only for inhibitory populations with this large difference in their synaptic time scale. However, these values are biologically plausible, indeed they can correspond to populations of interneu-FIG. 5. Bifurcation diagram for the parameter set C1. A) Bifurcation diagram in the plane (∆, JBA) for the masterslave coupled system. The codimension 2 bifurcation point (red circle) divides the plane in 4 different regions (I-IV) corresponding to different dynamical regimes. The Hopf bifurcations (black lines) separate foci from limit cycles, while the Torus bifurcations (blue lines) denote the emergence of Tori T 2 from limit cycles. A sample trajectory for each one of the four regions are shown in panels (B-E). The red dashed line in (A) indicates the value of ∆ considered in Fig. 6. rons generating IPSPs mediated via GABA A,f ast and GABA A,slow receptors 1 , which have been identified in the hippocampus 4 and in the cortex 47 . D. Cross-frequency-coupling in bidirectionally coupled populations As already mentioned a fundamental example of CFC, is represented by the coupling of the θ and γ rhythms. Gamma oscillations are usually modulated by theta oscillations during locomotory actions and rapid eye movement (REM) sleep in the hippocampus 34 as well as in the neocortex 48 . While gamma oscillations have been shown FIG. 6. Characterization of the dynamics for the parameter set C1. A) -C) Local maxima r (A) max of the firing rate of the A-population and B) -D) LEs as a function of the coupling strength JAB. In panels (B,D) the blue curve represents the first LE, the orange the second LE and the green one the third LE. The dashed rectangle in panel A (B) indicates the zoomed region presented in panel C (D). For the evaluation of the maxima, a transient time of tt = 10 s was discarded and then maximum values were stored during ts = 15 s. For the LEs estimation, after discarding the transient time tt, the evolution of the tangent space was followed for a period t = 500 s.
(D) the good agreement between the firing rates obtained from the network simulations (dots) and from the evolution of the MF model (line). An analysis of the power spectrum of r (A) (shown in FIG. 7. Characterization of the dynamics for the parameter set C2. A-C) Values of r (A) max as a function of JBA. In the inset it is shown the bifurcation diagram in the (∆, JBA) plane. This reveals the same structure of the diagram in Fig. 5 (A), here region (II) is not shown since it corresponds to the region of excitatory cross-coupling. B-D) First three LEs as a function of cross-inhibitory coupling JBA. The red dashed rectangle in panel A (B) denotes the zoomed region presented in panel C (D). E) Kaplan-Yorke dimension DKY in the parameter interval where chaos is present. F) Chaotic attractor in the (r (A) , v (A) , s (A) ) space at JBA = −7.25. For the evaluation of r (A) max, a transient time of tt = 10 s was discarded and then maximal values were stored during ts = 15 s. For the LEs estimation, after the same transient time as before, the evolution of the tangent space was followed during t = 50 s.
FIG. 8 .
8CFC in bidirectionally coupled populations. A) Heat map of the order parameter locking modes for different values of the cross-coupling. Colored symbols denote three couple of parameters {JAB, JBA} corresponding to 3 : 1 locking examined in (E) for different disorder values. In particular the red circle denotes the couple {JAB, JBA} for which one obtains the maximal value of ρ31 and these parameter values are employed for the simulations reported in (B,C). B) Raster plot of the network model Eq. (1) showing the fast and slow population in blue and red colors respectively. C) Instantaneous firing rates of the two populations obtained from the evolution of the MF dynamics (5) (same color code as in panel (B)). D) Power spectrum of the time trace r (A) (t) shown as a blue line in panel (C), in the inset an enlargement of the spectrum corresponding to the θ-band is shown. E) Ratio of the fast and slow frequencies of the COs ν (A) /ν (B) showing the extent of the 3 : 1 locking interval for the three values of {JAB, JBA} denoted by the symbols of the same color in (A) versus the disorder ∆ := ∆A = ∆B. Parameters for fast population are τ (A) d = 9 ms,ηA = 2, JAA = −2 and for the slow one τ (B) d = 50 ms,ηB = 1.5, JBB = −18. Common parameters ∆ = 0.05 (for panels A to D) and τ = 10. The time traces used for the phase locking analysis were taken over a period of t = 10 s after discarding an initial transient time tt = 10 s. For the network simulations in B), NA = NB = 10000 neurons, in the figure only half of them are depicted. The spectrum in (D) was obtained by averaging 50 power spectra each calculated over a time trace of duration t = 16.384 s with 2 15 equispaced samples, after a transient time of tt = 10 s.
ACKNOWLEDGMENTS
Authors are in debt with Ernest Montbrió for various enlightening interactions in the first phase of development of this project, furthermore they acknowledge fruitful discussions with Federico Devalle, Boris Gutkin, and Alex Roxin. A.T. received financial support by the Excellence Initiative A*MIDEX (Grant No. ANR-11-IDEX-0001-02) (together with D. A.-G.), by the Excellence Initiative I-Site Paris Seine (Grant No ANR-16-IDEX-008), by the Labex MME-DII (Grant No ANR-11-LBX-0023-01) (together with S.O.) and by the ANR Project ERMUNDY (Grant No ANR-18-CE37-0014), all part of the French programme "Investissements d'Avenir". D.A-G was also supported by CNRS for a research period at LPTM, UMR 8089, Université de Cergy-Pontoise, France. A.C was supported by Eras-mus+ Traineeship 2016/2017 contract between University of Florence, Department of Mathematics and Computer Science "Ulisse Dini" (DIMAI), and Centre de Physique Théorique (CPT) and LabEx Archimède, Marseille, France. APPENDIX: HOPF BOUNDARIES In the case of a single population of inhibitory neurons with no external input, the fixed point solutions (v 0 , r 0 , s 0 ) of the MF model (5) are given by the follow-− (πτ r 0 ) 2 + τ Js 0 = 0 (15) s 0 = r 0 .
FIG. 9 .
9CFC in bidirectionally coupled populations with an external θ modulation. A) Locking modes for different values of the cross-coupling. Red circle denotes the pair {JAB, JBA} giving rise to the maximum value of ρ31 which is used for simulations in B-C). Black diamond and blue square correspond also to 3:1 modes with smaller order parameter value. B) Power spectrum of the time trace r (A) (t) for the the case denoted by the red circle in panel (A), the inset displays an enlargement corresponding to the θ-band. C) Ratio of the fast and slow population frequencies ν (A) /ν (B) showing the extend of the 3:1 mode for the three values of {JAB, JBA} depicted by the symbols in A) at varying values of disorder ∆ := ∆ (A) = ∆ (B) . For the θ-forcing current we set I ν θ = 10 Hz, all the other parameters as in Fig. 8. FIG. 10. P-P and P-A couplings in presence of a θ forcing. A) Raster plot of the network model Eq. (
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Self-consistent analysis of asynchronous neural activity. E Ullner, A Politi, A Torcini, in preparationUllner, E., Politi, A., and Torcini, A. (2019). Self-consistent analysis of asynchronous neural activity. in preparation.
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The brainweb: phase synchronization and large-scale integration. F Varela, J.-P Lachaux, E Rodriguez, J Martinerie, Nature reviews neuroscience. 24229Varela, F., Lachaux, J.-P., Rodriguez, E., and Martinerie, J. (2001). The brainweb: phase synchronization and large-scale integration. Nature reviews neuroscience, 2(4):229.
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Networks of interneurons with fast and slow γ-aminobutyric acid type a (gabaa) kinetics provide substrate for mixed gamma-theta rhythm. J A White, M I Banks, R A Pearce, N J Kopell, Proceedings of the National Academy of Sciences. 9714White, J. A., Banks, M. I., Pearce, R. A., and Kopell, N. J. (2000). Networks of interneurons with fast and slow γ-aminobutyric acid type a (gabaa) kinetics provide substrate for mixed gamma-theta rhythm. Proceedings of the National Academy of Sciences, 97(14):8128-8133.
Excitatory and inhibitory interactions in localized populations of model neurons. H R Wilson, J D Cowan, Biophysical journal. 121Wilson, H. R. and Cowan, J. D. (1972). Excitatory and inhibitory interactions in localized populations of model neurons. Biophys- ical journal, 12(1):1-24.
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"A NEW SAMPLE OF LOW-MASS BLACK HOLES IN ACTIVE GALAXIES",
"A NEW SAMPLE OF LOW-MASS BLACK HOLES IN ACTIVE GALAXIES"
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"The Astrophysical Journal"
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We present an expanded sample of low-mass black holes (BHs) found in galactic nuclei. Using standard virial mass techniques to estimate BH masses, we select from the Fourth Data Release of the Sloan Digital Sky Survey all broad-line active galaxies with masses < 2 × 10 6 M ⊙ . BHs in this mass regime provide unique tests of the relationship between BHs and galaxies, since their late-type galaxy hosts do not necessarily contain classical bulges. Furthermore, they provide observational analogs of primordial seed BHs and are expected, when merging, to provide strong gravitational signals for future detectors such as LISA. From our preliminary sample of 19, we have increased the total sample by an order of magnitude to 174, as well as an additional 55 (less secure) candidates. The sample has a median BH mass of M BH = 1.3 × 10 6 M ⊙ , and in general the objects are radiating at high fractions of their Eddington limits. We investigate the broad spectral properties of the sample; 55 are detected by ROSAT, with soft X-ray luminosities in the range 10 40 to 7 × 10 43 ergs s −1 . Much like the preliminary sample, these objects are predominantly radio-quiet (R ≡ f 6cm / f 4400 < 10), but 11 objects are detected at 20 cm, with radio powers (10 21 − 10 23 W Hz −1 ) that may arise from either star formation or nuclear activity; only 1% of the sample is radio-loud. We further confirm that, with M g = −19.3 and g − r = 0.7 mag, the host galaxies are low-mass, late-type systems. At least 40% show disk-like morphologies, and the combination of host galaxy colors and higher-order Balmer absorption lines indicate intermediate-age stellar populations in a subset of the sample.
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"https://arxiv.org/pdf/0707.2617v1.pdf"
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0707.2617
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100cd1851bf97fc6cd435f0b70de266ed4c129ad
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A NEW SAMPLE OF LOW-MASS BLACK HOLES IN ACTIVE GALAXIES
18 Jul 2007
To
In
A NEW SAMPLE OF LOW-MASS BLACK HOLES IN ACTIVE GALAXIES
The Astrophysical Journal
18 Jul 2007arXiv:0707.2617v1 [astro-ph] Preprint typeset using L A T E X style emulateapj v. 26/01/00Subject headings: galaxies: active -galaxies: nuclei -galaxies: Seyfert
We present an expanded sample of low-mass black holes (BHs) found in galactic nuclei. Using standard virial mass techniques to estimate BH masses, we select from the Fourth Data Release of the Sloan Digital Sky Survey all broad-line active galaxies with masses < 2 × 10 6 M ⊙ . BHs in this mass regime provide unique tests of the relationship between BHs and galaxies, since their late-type galaxy hosts do not necessarily contain classical bulges. Furthermore, they provide observational analogs of primordial seed BHs and are expected, when merging, to provide strong gravitational signals for future detectors such as LISA. From our preliminary sample of 19, we have increased the total sample by an order of magnitude to 174, as well as an additional 55 (less secure) candidates. The sample has a median BH mass of M BH = 1.3 × 10 6 M ⊙ , and in general the objects are radiating at high fractions of their Eddington limits. We investigate the broad spectral properties of the sample; 55 are detected by ROSAT, with soft X-ray luminosities in the range 10 40 to 7 × 10 43 ergs s −1 . Much like the preliminary sample, these objects are predominantly radio-quiet (R ≡ f 6cm / f 4400 < 10), but 11 objects are detected at 20 cm, with radio powers (10 21 − 10 23 W Hz −1 ) that may arise from either star formation or nuclear activity; only 1% of the sample is radio-loud. We further confirm that, with M g = −19.3 and g − r = 0.7 mag, the host galaxies are low-mass, late-type systems. At least 40% show disk-like morphologies, and the combination of host galaxy colors and higher-order Balmer absorption lines indicate intermediate-age stellar populations in a subset of the sample.
INTRODUCTION
There are many strong observational connections between galaxy bulges and central supermassive black holes (BHs). BH masses correlate surprisingly tightly with bulge properties, including luminosity (e.g., Marconi & Hunt 2003) and stellar velocity dispersion (M BH − σ * ; Ferrarese & Merritt 2000;Gebhardt et al. 2000a;Tremaine et al. 2002). Also active galactic nuclei (AGNs) in the local Universe are predominantly found in massive, bulge-dominated galaxies (Ho et al. 1997b;Kauffmann et al. 2003). Perhaps the formation of BHs and bulges are related, in which case we might not expect bulgeless galaxies to obey scaling relations between BHs and galaxies, or even necessarily host a central BH. Indeed, dynamical study of the bulgeless spiral galaxy M33 places strong limits on the presence of a dark central massive object (Gebhardt et al. 2001), while any BH in the nucleus of the dwarf spheroidal galaxy NGC 205 has been shown to lie below the low-mass extrapolation of the M BH − σ * relation (Valluri et al. 2005). On the other hand, the M31 globular cluster G1 shows dynamical and radiative evidence for a central BH (Gebhardt et al. 2002(Gebhardt et al. , 2005Pooley & Rappaport 2006;Ulvestad et al. 2007). However, the mixed stellar population and high degree of rotational support in this massive cluster suggest that G1 is actually the nucleus of a tidally stripped dwarf galaxy (e.g., Meylan et al. 2001). Thus, dynamical studies present an ambiguous verdict on the presence of nuclear BHs in dwarf stellar systems. Unfortunately, it is not currently feasible to spatially resolve the gravitational sphere of influence of a ∼ 10 5 M ⊙ BH outside the Local Group, in order to search for BHs in more dwarf stellar systems using dynamical methods.
Although they are difficult to find, the occupation fraction of nuclear BHs in dwarf systems and the space density of lowmass BHs are of considerable interest. Apart from furnishing additional insight into the possible origin of the M BH − σ * relation, low-mass BHs provide low-redshift counterparts to the primordial seed BHs; the low-mass cut-off in the BH mass function today provides a constraint on the mass function of seed BHs. The merging of BHs in this mass range is expected to provide a strong signal for the gravitational wave experiment LISA (e.g., Hughes 2002). Furthermore, gravitational radiation recoil is expected to impart velocities to BH merger remnants that exceed the escape velocities of dwarf galaxies (e.g., Favata et al. 2004;Merritt et al. 2004). This effect alone might decrease the occupation fraction of BHs in dwarf galaxies.
In the absence of detectable dynamical signatures, we are forced to rely on less direct evidence for the presence of nuclear BHs, namely AGN activity. In fact, there are two well-studied AGNs in dwarf galaxies: NGC 4395 is a bulgeless spiral galaxy (Filippenko & Sargent 1989), while POX 52 (Kunth et al. 1987) is a dwarf spheroidal galaxy (Barth et al. 2004) 2 . BH mass determinations are far less certain in the absence of dynamical constraints, but a variety of techniques yield BH masses for each object of ∼ 10 5 M ⊙ (Filippenko & Ho 2003;Barth et al. 2004;Peterson et al. 2005). Greene & Ho (2004;GH hereafter) performed a systematic search for AGNs with 1 Hubble Fellow and Princeton-Carnegie Fellow. 2 Such objects are commonly referred to as dwarf elliptical galaxies in the literature. However, because their structure is quite different from that of classical elliptical galaxies, we prefer to refer to them as dwarf spheroidal systems. Table 1 is available in its entirety via the link to the machine-readable table above. The following is included only as a guide to content and presentation.
BH masses < 10 6 M ⊙ , using the First Data Release of the Sloan Digital Sky Survey (SDSS; York et al. 2000) and recovered 19 objects in that preliminary search. Remarkably, follow-up spectroscopy using ESI on Keck suggests that these sources, as well as NGC 4395, POX 52, and G1, are consistent with the lowmass extrapolation of the M BH − σ * relation (Barth et al. 2005). At the same time, many of the host galaxies are late-type galaxies without classical bulges (J. E. Greene, et al. in preparation). Clearly, more objects are needed to both confirm this preliminary result, and also to constrain the structural parameters of the host galaxies. To this end, we have repeated our analysis using the Fourth Data Release of the SDSS (DR4; Adelman-McCarthy et al. 2006), and present the updated sample here.
Our goal is to study the properties of the lowest-mass BHs that are identifiable with the SDSS. The upper mass limit on this sample is somewhat arbitrary. Within the inactive sample of galaxies with dynamical BH masses, the BH in M32, M BH =(2.5 +3.0 −2.0 ) × 10 6 M ⊙ , has the lowest mass (Tremaine et al. 2002; excluding the Milky Way, whose modern mass is 3.5 × 10 6 M ⊙ ; e.g., Ghez et al. 2005;Eisenhauer et al. 2005). We have adopted 2 × 10 6 M ⊙ as the upper limit for our search. Because dynamical methods cannot extend into this regime, we previously had constraints on neither the existence of lowermass BHs nor on whether they obey similar scaling relations with galaxy bulge properties as their high-mass cousins. Furthermore, while there are good reasons to believe that not every dwarf stellar system contains a BH, we do not know at what mass the occupation fraction departs from unity.
Throughout we assume the following cosmological parameters to calculate distances: H 0 = 100 h = 71 km s −1 Mpc −1 , Ω m = 0.27, and Ω Λ = 0.75 (Spergel et al. 2003).
SAMPLE SELECTION AND ANALYSIS
Our low-mass BHs are selected from the sample of broadline AGNs with z < 0.352 described in detail by Greene & Ho (2007b) and briefly reviewed here for completeness. We begin with DR4 of the SDSS and search for all AGNs with broad Hα, where "broad" in this context means a significant extra component relative to the narrow-line profile (based on the [S II] λλ6716, 6731 doublet; e.g., Ho et al. 1997c). Continuum subtraction, which is crucial to uncover low-contrast broad lines, is performed with the principle component analysis method developed by Hao et al. (2005). Our subsequent selec-tion technique comprises a two-step procedure: we first select objects with high root-mean-square (rms) deviations above the continuum in the region potentially containing broad Hα, and then perform more detailed profile fitting to isolate those objects with broad Hα profiles. Unfortunately, this selection process results in many sources of such low Hα luminosity that (1) their nature is ambiguous and (2) our ability to recover a reliable BH mass is severely compromised. Based on simulations, we thus impose a combined rms-weighted flux and equivalent width (EW) cut designed to minimize spurious detections. The resulting sample comprises 8435 objects.
Since we cannot use stellar dynamical methods to measure BH masses, we instead use the photoionized broad-line region (BLR) gas as a dynamical tracer of the BH mass. While the BLR velocity dispersion is derived from the line width, the BLR radius is inferred from the AGN luminosity (in this case Hα; Greene & Ho 2005b). The so-called radius-luminosity relation is derived from reverberation mapping of ∼ 30 AGNs, for which radii are measured based on the delay between variations in the AGN photoionizing continuum and BLR line emission (e.g., Kaspi et al. 2005;Greene & Ho 2005b;Bentz et al. 2006). The "virial" BH mass is simply M BH = f Rυ 2 /G, where f is a scaling factor that accounts for the unknown geometry of the BLR, assumed here to be spherical ( f =0.75 ;Netzer 1990). Although BH masses derived from reverberation mapping are susceptible to large systematic errors (due to uncertainties in the BLR geometry and kinematics; e.g., Krolik 2001), remarkably they have been shown to agree with σ * , in those cases for which σ * can be measured. The measured scatter is ∼ 0.3 dex (Gebhardt et al. 2000b;Ferrarese et al. 2001;Onken et al. 2004;Nelson et al. 2004). Virial masses are less direct, since they rely on the radius-luminosity relation; in the largest such comparison to date, find a scatter of 0.4 dex of single-epoch virial masses about the M BH − σ * relation. Our BH mass estimator is discussed in detail in the Appendix.
Here we simply note that Greene & Ho (2007b) used an old version of the radius-luminosity relation slope, while here we have updated the value to that found by Bentz et al. (2006). In general, the derived masses increase by ∼ 0.3 dex.
Selecting those broad-line AGNs with masses < 2 × 10 6 M ⊙ results in a sample of 174 objects, which are the main subject of this paper (Fig. 1). In addition, some (55) of the objects below the detection threshold are nevertheless interesting candidates. These objects have been selected by manual inspection of the spectra, and thus comprise a biased and incomplete sample. We worry, as explained above, that our BH masses are not stable for these Hα luminosities, but we deem these objects worthy of follow-up spectroscopy. In what follows we will refer to these objects as the c ("candidate") sample, and we will present all trends including and excluding these objects. Basic properties of the sample are summarized in Table 1.
Fe II Fitting
Our modeling of the Hβ region warrants some additional discussion. While the continua of our spectra, in general, are dominated by stellar light, in some cases there is an additional "pseudo-continuum" component contributed by broad Fe II multiplets. The Fe II emission extends over the entire optical and UV spectrum, but is especially troublesome in the wavelength ranges 4400-4800 Å and 5150-5500 Å. In principle, both the Hβ and [O III] λλ4959, 5007 fits can be severely compromised when the Fe II component is ignored (see Fig. 2). We follow standard procedure and model the Fe II emission with an empirical template (e.g., Boroson & Green 1992;Greene & Ho 2005b). Since many of our sources also require substantial galaxy subtraction, the Fe II, Hβ, and [O III] lines are modeled simultaneously in the galaxy-continuum-subtracted spectrum, as described below.
In addition to an Fe II template, our model includes narrow [O III], and narrow and broad Hβ. In the standard way, all relative wavelengths for central narrow components are fixed to laboratory values, and the narrow Hβ flux is fixed to be no more than 1/3.1 of the narrow component of Hα (Case B ′ recombination; Halpern & Steiner 1983;Gaskell & Ferland 1984). The [O III] lines are fit simultaneously with a core and a wing component, following Greene & Ho (2005a), with a relative strength of 2.96, while the narrow Hβ line is modeled with a single Gaussian due to its typical low signal-to-noise ratio.
Usually, Fe II templates are derived from 1 Zw I, a strong Fe II source with relatively narrow broad lines. With a broadline width of FWHM = 1240 km s −1 (Boroson & Green 1992), 1 Zw I is significantly broader than many of the objects in our sample. For this reason, we are compelled to build our own Fe II template from the SDSS data. Using a high S/N spectrum of SDSS J155909.62+350147.4, whose FWHM Hα is 860 km s −1 , we model Hβ+[O III] using the 4750-5100 Å region with no Fe II template included (Fig. 2a), and subtract this model. We further mask the regions 4840-4870 Å and 5010-5018 Å (note these wavelengths are in vacuum), since these regions are dominated by noise in the subtraction. While the resulting template is similar to 1 Zw I (Fig. 2b), it is clear that we do not recover the shape of the 5000 Å feature perfectly, and that there are slight differences in continuum slope between the two templates. To explore the importance of the former problem, we have repeated our fitting procedure using templates made with various masking regions, and find no difference in the derived fit parameters. The continuum shape, on the other hand, remains a systematic uncertainty inherent to our method; there is some degeneracy between the shape of the continuum and the amplitude of Fe II contamination, leading to uncertainties in the overall amplitude of Fe II. Nevertheless, our measured Hβ+ [O III] fits are robust to this uncertainty. When we fix the Fe II template to values ranging from one-half to twice the bestfit value, we find < 1% changes in all other measured parameters. Presumably the [O III] is sufficiently resolved from the Fe II component at 5000 Å so that small errors in the latter do not strongly impact our model of the former. Greene & Ho (2004) Let us examine how the original GH sample compares with this new larger sample. Objects from the original paper will be referenced using their identification number in that paper (e.g., GH02). There are no significant differences between the two works, but there are some issues worth noting. In general, we have improved our selection procedure (see Greene & Ho 2007b), so that our sample from DR1 is close to two and a half times larger than the original sample. We are now using the Hα rather than the L 5100 luminosity to derive the BLR radius, and we are using the updated radius-luminosity relation slope of Bentz et al. (2006). Only one (GH19) of the GH objects is not included in the final DR4 broad-line AGN sample of Greene & Ho (2007b), although GH07, GH11, GH15, GH16, and GH18 now have masses above our mass boundary of 2 × 10 6 M ⊙ . Finally, the original GH sample excluded galaxies with significant galaxy contamination, due to the fear that these objects would comprise more massive BHs with low-contrast broad lines that were rendered undetectable by the large galaxy luminosity. In this work we have not explicitly included such a cut, but our EW threshold is operationally similar. While some of the objects in the c sample may consist of high-mass interlopers, Zw I Fe II template (blue dotted) with that used in this work (solid). Spectra have been normalized to unity. The two templates are very similar, although our template is clearly narrower, and, as discussed in the text, the overall slope is somewhat different. (c) A sample fit to the Hβ+[O III] region. Spectra in solid are the continuum-subtracted (top) and residual (bottom) spectra. The total model is overplotted (red thin), along with individual components (blue dotted). Units the same as (a) above.
Comparison with
we find in general very good agreement in properties between the two samples. Dong et al. (2007a, b) Dong et al. (2007a have presented a low-mass BH with a mass of 7 × 10 4 M ⊙ . This source was selected from the Fifth Data Release of the SDSS, and so cannot be cross-checked with our sample. However, the object has been serendipitously observed with HST and XMM-Newton, as well as being detected with the Röentgen Satellite (ROSAT). There is tentative evidence for variability in the X-ray source (L X = 7 × 10 40 ergs s −1 ) and the host galaxy is a disk galaxy with M R = −17.8 mag and a nuclear bar. The same group has identified a sample of 245 lowmass BHs from DR4, with a mass range of 5 × 10 4 − 10 6 M ⊙ , and Eddington ratios ranging from 0.02 to 8, comparable to our sample (Dong et al. 2007b). Samples selected from two independent methods will provide important tests on the limitations of our method. Note, however, that the published masses are based on the radius-luminosity relation slope of Kaspi et al. (2005), and would increase by ∼ 0.3 dex according to the formalism adopted in this paper.
Comparison with
SAMPLE PROPERTIES
The final sample has a median redshift of 0.086 (0.099 if the c sample is excluded). The original sample has a slightly lower median z = 0.08 due in part to subsequent deep observations in the Southern Strip (targeting quasars to a limiting magnitude of i = 19.9 mag rather than i = 19.1 mag). In Figure 1 we show the distribution of "virial" BH mass and Eddington ratio for the sample, where L Edd ≡ 1.26 × 10 38 (M BH /M ⊙ ) ergs s −1 . Assuming that L bol = 9.8 L 5100 (McLure & Dunlop 2004), in terms of L Hα our bolometric correction is L bol = 2.34 × 10 44 (L Hα /10 42 ) 0.86 ergs s −1 (Greene & Ho 2005b). The median BH mass is M BH = 1.3 × 10 6 M ⊙ . With a median L bol /L Edd = 0.4, we are clearly dominated by sources radiating at substantial fractions of their Eddington limits (Fig. 1). This is not surprising; as discussed by Greene & Ho (2007b), we are only sensitive to the most luminous BHs in this mass regime.
Optical Spectral Properties
The statistical power of our new enlarged sample provides new constraints on the ensemble physical properties of both the radiating BHs and their host galaxies. We begin by placing the sources on the two-dimensional diagnostic diagrams that in combination discriminate between a stellar ionizing spectrum or one considerably harder (e.g., Baldwin et al. 1981;Veilleux & Osterbrock 1987). These line ratios are typically used to divide emission-line galaxies into H II galaxies and narrow-line AGNs (e.g., Ho et al. 1997a;Kauffmann et al. 2003;Hao et al. 2005; Table 2). In contrast, broad-line-selected objects are unbiased with regard to position on the diagnostic diagrams, while at the same time we know that some fraction of the ionizing continuum is contributed by an AGN.
We focus first on the [N II] λ6583/Hα diagnostic diagram (Fig. 3a). We do not see many objects in the lower-right quadrant. Objects there are low-ionization nuclear emission-line region (LINER; Heckman 1980) sources and typically are highly sub-Eddington AGNs (Ho 2004). Compared to classical Seyfert galaxies, which occupy the upper-right region of the diagnostic diagram, we see that our sources span a broader range in both [N II]/Hα and [O III]/Hβ. The decrease in [N II]/Hα is most easily explained as a decrease in the gas-phase metallicity of these AGNs (e.g., Kraemer et al. 1999;Groves et al. 2006). Since nitrogen is a secondary element, [N II] is particularly sensitive to changes in metallicity for > 0.1 Z ⊙ . In contrast, the [O III]/Hβ ratio changes much more slowly, because as the metallicity decreases, the temperature of the NLR increases correspondingly, which tends to boost the [O III] strength (e.g., Groves et al. 2004aGroves et al. , 2006; see metallicity tracks on Fig. 4). Relatively low metallicities for this sample are not unexpected, since local AGN hosts in general are known to be massive, bulgedominated galaxies (e.g., Ho et al. 1997b; Kauffmann et al. Table 2. Emission-line Measurements Note. -Col.
ID [O II] Fe II (Hβ)n (Hβ) b [O III] [O I] Hαn Hα b [N II] [S II] [S II] FWHM Hα FWHM [OIII] λ3727 λ4570 λ5007 λ6300 λ6583 λ6716 λ6731 (km s −1 ) (km s −1 ) (1) (2) (3) (4) (5) (6) (7) (8) (9)(10)
(1): Identification number assigned in this paper. Col.
(2)-(12): All fluxes are relative to that of [O III] λ5007, which is listed in units of 10 −17 ergs s −1 cm −2 . Note that these are observed values; only Galactic extinction correction has been applied. The subscripts "n" and "b" in Col. (4)-(5) and (8)-(9) refer to the narrow and broad components of the line, respectively. Col. (13)- (14): Linewidths have been corrected for instrumental resolution using the values measured from arc spectra and tabulated by the SDSS. Table 2 is available in its entirety via the link to the machine-readable table above. The following is included only as a guide to content and presentation.
2003), while these low-mass BHs tend to live in low-mass and correspondingly low-metallicity systems ( §3.4; Tremonti et al. 2004 (Fig. 4). Changes in a variety of NLR conditions, including metallicity, electron density (n e ), ionization parameter (U), ionizing spectral shape, and reddening might be invoked to explain the observed line ratios, while particular properties of either the AGN (e.g., luminosity or Eddington ratio) or the host galaxy (e.g., star formation) may be responsible for the changing conditions. As we will argue, it is unlikely that a single parameter is responsible for all of these trends. Correlated errors might also be expected to move line ratios in particular ways across the diagnostic diagrams. If, for instance, we are systematically overestimating the strength of narrow Hα in systems with weak NLR emission, then we might expect correlated tracks as observed. However, [O III] is never blended with broad Hβ in these sources, and thus [O III] errors are not correlated with the other lines, but nevertheless the [O III] line strength spans a wide range relative to Hβ.
One might imagine that the intrinsic line ratios are all located in the Seyfert locus, but that contamination from star formation spreads out their positions in the diagnostic diagrams. We perform a simple thought experiment to demonstrate that there must be an intrinsic spread in line ratios to reproduce the observed trends. Since star-forming galaxies follow a welldefined sequence in the diagnostic diagram (bounded by the Kauffmann et al. 2003 relation shown Fig. 5), we can easily investigate the type of bias that star formation would contribute. In Figure 5 we have chosen three fiducial AGN positions in the diagnostic diagram. Position A represents a typical high-ionization system, while B has low [O III]/Hβ and [N II]/Hα and C represents a LINER. To each AGN we add varying amounts of star formation with line ratios drawn from the Kauffmann line, and the Hβ strength in star formation varying from half to twice the AGN Hβ luminosity. The tracks (shown as dash-dot, solid, and dashed lines for positions A, B, and C, respectively) demonstrate that while star formation may increase the spread in positions on the diagnostic diagram, it alone cannot move objects from the high-ionization position A to the low-[O III]/Hβ positions we observe. There must be a true spread in the intrinsic AGN line ratios. Furthermore, position C is disfavored, since only with a large fraction of the light contributed by low-metallicity star formation might we move the objects from position C toward position B. If the objects in this sample follow the mass-metallicity relation of Tremonti et al. (2004), with typical luminosities of M g ≈ −19.3 (Table 3, §3.4), they would not, on average, have sub-solar metallicity.
Given that the dispersion in line ratios must be intrinsic to the NLR, we investigate various means to explain the observed line-ratio distributions. Ho et al. (1993) find that narrow-line AGNs display a similar trend in the occupation of diagnostic diagrams, which they ascribe to a trend of harder ionizing spectral shape at lower luminosity. Exactly this trend has been seen in the X-ray to optical slopes of broad-line AGNs over a wide range in luminosity (e.g., Strateva et al. 2005), including for a subset of the GH sample (Greene & Ho 2007a). A correlation between low-luminosities and harder spectral shape would lead to a correlation between AGN luminosity and line ratios, but we see no trends between AGN luminosity (as traced by L Hα ) and either [ Groves et al. 2004a, b). A single parameter seems unable to reproduce all of the observed trends simultaneously. Presumably this is due, in part, to stratification in the density and ionization parameter of the NLR with radius, which leads to different typical conditions in the emission regions of different lines (e.g., Filippenko & Halpern 1984).
In summary, a single parameter cannot be invoked to explain the observed positions in the diagnostic diagrams. Table 3 is available in its entirety via the link to the machine-readable table above. The following is included only as a guide to content and presentation.
Although changing physical conditions (such as electron density) within the NLR may be important, the objects lying close to the star-forming locus may simply have an extremely weak NLR, such that the majority of the narrow emission is coming from H II regions. A similar phenomenon, the "vanishing" NLR, has been seen in a substantial fraction of higher redshift AGNs at high Eddington ratios (e.g., Netzer et al. 2004Netzer et al. , 2006. If observed correlations between NLR size and AGN luminosity (e.g., Bennert et al. 2002;Schmitt et al. 2003) are extrapolated to the highest luminosities, then one would predict unbound nebulosities; Netzer et al. (2004) suppose that the NLRs in these systems grew so large that they were no longer bound to their host galaxies. The luminosities in our systems are significantly lower, and the expected NLR sizes are < 100 pc if they obey the NLR size-luminosity relation, but we cannot rule out that the AGN actually expelled some large fraction of the surrounding ISM, leading to overall weakness in the line intensities.
Finally, there is an apparent connection between [O III]/Hβ and Eddington ratio (e.g., Boroson 2002). However, for these objects there is no trend between [O III]/Hβ(narrow) and L bol /L Edd (ρ = 0.043, P = 0.5) or [O III] equivalent width and L bol /L Edd (ρ = 0.093, P = 0.2). These trends do not change when the c sample is excluded. On the other hand, although no physically motivated explanation exists, low [O III]/Hβ traditionally was a defining characteristic of the class of objects known as narrow-line Seyfert 1 galaxies (NLS1s; Osterbrock & Pogge 1985), Table 4 is available in its entirety via the link to the machine-readable table above. The following is included only as a guide to content and presentation.
Table 4. ROSAT Detections ID C log N H log f X log L X (1) (2) (3) (4)(5)
which are broad-line AGNs with relatively narrow broad-line widths. We turn now to a general comparison of our objects with NLS1s.
Comparison with NLS1s
While NLS1s are an observationally defined class, a picture has emerged, based on their strong Fe II/Hβ and weak [O III]/Hβ line ratios, strong soft X-ray excess (e.g., Boller et al. 1996) and X-ray variability (Leighly 1999), that NLS1s are in general low-mass BHs (thus accounting for the relatively narrow broad lines) in a high accretion state (e.g., Pounds et al. 1995). Since the current sample is selected based on BH mass (and indirectly L bol /L Edd ), it represents a very uniform, optically selected NLS1 sample. In this section, we highlight comparisons between the optical and broad spectral properties of this and other NLS1 samples.
One of the supporting pieces of evidence that NLS1s are radiating close to their Eddington limit comes from their observed high Fe II/Hβ ratios (e.g., Véron-Cetty et al. 2001; throughout this section we refer to total [broad+narrow] Hβ luminosity). Using principle component analysis, Boroson & Green (1992) found that NLS1s lie at one extreme of their first component (fondly known as Eigenvector 1), and later work has shown that Eigenvector 1 properties, including high Fe II/Hβ and low [O III]/Hβ ratios, as well as soft X-ray excess and radio weakness, tend to be correlated with Eddington ratio (e.g., Boroson 2002). On the other hand, GH found that their objects span a larger range in both Fe II and [O III] strengths relative to Hβ than classic NLS1s.
The current sample has somewhat lower Fe II/Hβ strengths than GH found, perhaps because there are objects with more significant galaxy contamination in this sample. Using the Kaplan-Meier product-limit estimator, which accounts for upper limits (Feigelson & Nelson 1985), we find Fe II/Hβ = 0.57 ± 0.03 (0.61 ± 0.03) with (without) the c sample, as compared to Fe II/Hβ = 0.67 ± 0.04 for the 56 NLS1s presented by Véron-Cetty et al. (2001) More generally, trends in spectral properties with Eddington ratio may be manifest in composite spectra. We investigate four bins of increasing Eddington ratio (similar to increasing bins in luminosity over such a restricted range in BH mass; Fig. 6), and we have included the c objects, which comprise the lowest L bol /L Edd bin. To compute the composite, each spectrum is normalized to the median value in the spectral regions 4100-4800 Å and 5100-5800 Å, resampled onto the same wavelength grid and then median combined. Median stacking preserves relative line ratios, but not the continuum shape (e.g., Francis et al. 1991). Although we do not explicitly weight the spectra, they are each normalized to their respective continuum level, which gives extra weight to faint sources. Errors at each pixel are calculated as the 68% interquartile range of the included pixel values, following, e.g., Fine et al. (2006). Note that this procedure is considerably simpler than that of, e.g., Francis et al. (1991), since the redshifts measured by SDSS are highly reliable and the spectral coverage and depth are so uniform for the majority of the sample. The clearest trend as we move to higher Eddington ratio is the decreasing importance of the galaxy continuum. At the same time, the Fe II features are much more apparent in the higher Eddington ratio bins. This appears to be in keeping with our expectations from Eigenvector 1. On the other hand, using the original, unbinned data, we do not find significant evidence for a correlation between Fe II/Hβ and L bol /L Edd (ρ = −0.07, P = 0.3). There is a decrease in the [O III]/Hβ(total) ratio with Eddington ratio in the composite spectra, which ranges from 3.6 ± 1.6 in the lowest L bol /L Edd bin to 0.65 ± 0.03 (or 2.4 ± 0.3 when considering only the narrow Hβ). Errors are dominated by the variance in line strengths for individual objects comprising the composite.
Radio and X-ray Spectral Properties
There is additional information about the accretion properties of the sample in their broader spectral energy distributions. NLS1s, in general, are known for their strong soft X-ray excess, but Greene & Ho (2007a; see also Williams et al. 2004) have found that objects selected purely on the basis of mass are not as extreme in this regard as classic NLS1s. In this case, using 5 ks observations of the 10 nearest GH objects, they found that the objects had very typical 0.5-2 keV power-law slopes of Γ s ≈ 1.8 [where N(E) ∝ E −Γs ]. Within the current sample, 55 are detected by the ROSAT All-Sky Survey (Voges et al. 1999), of which only two are in the c sample, with fluxes ranging from 5.9 × 10 −14 to 3.4 × 10 −12 ergs s −1 cm −2 . Fluxes are derived from the ROSAT count rates using WebPimms (Mukai 1993) FIG. 6.-Composite spectra in bins of increasing Eddington ratio, as shown. Each spectrum is the median combination of all spectra in the indicated L bol /L Edd bin, with the total number of galaxies indicated. Each spectrum is normalized to emission-line-free bands prior to combination (see text for details). Note the increasing strength of the Fe II features and decreasing galaxy continuum strength with increasing Eddington ratio. and assuming a slope of Γ s = 2. In calculating uncertainties in the fluxes, Γ s is allowed to vary from 1 to 3, which is the full range observed by Greene & Ho (2007a). The corresponding range of 0.5-2 keV luminosities is 10 40 to 7 × 10 43 ergs s −1 (Table 4). Since inactive galaxies with optical luminosities similar to those of the galaxies in our sample (L B ≈ 10 9 − 10 10 L ⊙ ) are expected to have typical X-ray luminosities of ∼ 10 39 ergs s −1 (e.g., Fabbiano 1989), the AGN most likely dominates the Xray emission from the majority of these sources.
Customarily, the ratio of optical to X-ray luminosity is expressed in terms of the slope of a supposed power-law between the optical and X-ray bands, α ox ≡ −0.3838 log( f 2500 / f 2keV ). The flux density at 2500 Å, f 2500 , is calculated using the measured L Hα , the relation between L Hα and L 5100 from Greene & Ho (2005b), and an assumed spectral slope of β = −1.56 ( f λ ∝ λ −β ; Vanden Berk et al. 2001). For our sample, the values lie in the range −1.5 <α ox < −0.69, with a median of α ox = −1.04. We would expect an α ox ≈ −1.2 corresponding to the median l 2500 = 6×10 27 ergs s −1 Hz −1 (as was seen for the GH objects observed with Chandra; Greene & Ho 2007a). While the observed median is slightly higher than one would expect (e.g., Strateva et al. 2005;Steffen et al. 2006), it is within the observed scatter, and given that these are the ROSAT-detected members of our sample, it is not surprising to find that they are X-ray bright. We see no obvious trends between α ox and L bol /L Edd or Fe II/Hβ, but we caution that the dynamic range of this sample is very limited, and that we do not have meaningful upper limits for the X-ray-undetected sources.
Radio emission is a complementary probe of the accretion process, if difficult to interpret cleanly. In general, NLS1s are radio-quiet (see review in , where radio-quiet typically means R < 10 (R ≡ f 6cm / f 4400 ). Weak radio emission is part of Eigenvector 1 (e.g., Boroson & Green 1992), and mirrors the general tendency for BHs of all masses to be radio-quiet when in high accretion states (e.g., McClintock & Remillard 2006). Also, Best et al. (2005) find a strong preference for radio sources to live in massive galaxies. A low incidence of radio activity in this sample is thus unsurprising. Eleven of the objects are detected by the VLA 3 FIRST survey at 20 cm although all but 18 objects are within the FIRST footprint (Becker et al. 1995. The radio sources at 20 cm are unresolved in all cases, with radio powers that range from 2 × 10 21 to 9 × 10 22 W Hz −1 (Table 5). Inactive star-forming galaxies can easily produce this level of radio emission (e.g., Condon 1989), which need not be AGN-dominated. The exception is GH10, which we have imaged with the VLA in A-configuration at 6 cm . We find that the source remains completely unresolved, suggesting that the AGN provides the dominant source of emission from this galaxy.
Of the objects with radio emission, four are in the c sample and five others are radio-quiet, which translates into a radioloud fraction of 1% (accounting for the 18 objects in the sample that are outside of the FIRST footprint), to be compared with the range of 0% − 6% for NLS1s in the literature . In Figure 7 the detected sources are placed on the relation between radio power and [O III] luminosity observed for local AGNs. They are roughly consistent with the relation measured for Seyfert galaxies (Ho & Peng 2001), but they are systematically higher than the fiducial relation. In addition, we have retrieved FIRST cut-outs around each non-detection. We use these images to derive upper limits on the radio luminosity (Table 5), and we also stack the images in two bins divided at an [O III] luminosity of L [OIII] =3 × 10 40 ergs s −1 . Because FIRST is a uniform survey, no weighting is applied, but in 16 cases Table 6 is available in its entirety via the link to the machine-readable table above. The following is included only as a guide to content and presentation.
there is a ∼ 2 σ detection in the inner five square pixels (where σ is measured in a box off the center of the cut-out). We exclude these from our stacked images, although the answer changes negligibly when they are included. We do detect radio emission in each stacked image, shown as crossed squares in Figure 7, also consistent with the general relation.
Host Galaxies
Naively, our expectation is that lower-mass BHs will be found, in general, in low-mass galaxies. However, the tight observed relations between BHs and galaxies pertain specifically to galaxy bulges. We are now in a position to test whether BHbulge relations break down in low-mass galaxies. The spectra provide an estimate for the AGN luminosity (we prefer to use the Hα luminosity because of its robustness in the presence of significant galaxy contamination; Greene & Ho 2005b), which we convert to magnitudes using SDSS filter functions and assuming a spectral shape of β = −1.56 (Vanden Berk et al. 2001). If we assume a flatter slope of β = −1, as suggested by GH, the median magnitude difference is only 0.06 mag. In addition to the power law, we also include the observed strong emission lines (Hβ, [O III], [N II], and Hα) in the calculation of the AGN luminosity (on average, line emission accounts for g ≈ 0.08 mag). Once we have removed our best estimate for the AGN luminosity from the SDSS Petrosian magnitudes, we calculate Kcorrections using the routines described in Blanton & Roweis (2007). The distribution of host galaxy luminosities thus derived has a median host galaxy luminosity of M g = −19.3 mag (with or without the c sample) and is shown in Figure 8. With this method of AGN removal, we tend to underestimate the AGN luminosity due to fiber losses, and thereby (conservatively) overestimate the galaxy luminosity. As a lower bound on the galaxy luminosity, we use the SDSS point-spread function (PSF; unresolved) magnitude as an alternate estimate of the AGN luminosity, and then apply a K-correction to the corrected colors in the same manner. This method, on average, overestimates the AGN luminosity, and results in a median host galaxy luminosity of M g = −19.0 mag (Fig. 8, dotted histogram). The true host luminosities lie somewhere between these two bounds. For reference, Blanton et al. (2003) find a characteristic luminosity of M * g = −20.1 mag at z = 0.1 (for our assumed cosmology), and thus our targets are ∼ 0.8 − 1.1 mag fainter than M * g . Morphological information is difficult to extract from the SDSS images alone, due to their limited depth and spatial resolution, but it is clear that BHs selected to have low masses are found in low-mass galaxies. However, if we restrict our attention to the 139 sources with z < 0.1, we find ∼ 80 clear disk galaxies (see GH for example SDSS images for the original sample). We can also calculate galaxy colors: g − r = 0.7 mag for the spectroscopic estimate of the AGN luminosity (0.6 mag for the PSF method). According to Fukugita et al. (1995), the host galaxies have the colors of typical Sab galaxies (g−r = 0.66 mag). HST images of the GH sample indicate that there are actually two different populations of host galaxies. One is composed of late-type spiral galaxies, typically with strong bars. The other-the majority-group is comprised of compact spheroids, not unlike POX 52, the prototypical dwarf spheroidal galaxy hosting an AGN, which has red colors (B − V = 0.8 or g − r ≈ 0.67 mag). We do not yet know if the spheroidal galaxies in the larger sample are red in general.
Stellar Populations and Ongoing Star Formation
It is tempting to imagine that the constant mass ratio observed between BHs and bulges results from concurrent (or at least orchestrated) star formation and AGN activity. Indeed, various theories have postulated that AGN activity is responsible for truncating star formation (e.g., Springel et al. 2005). While characterizing the star formation rates of broadline AGNs is notoriously challenging, evidence for ongoing star formation has been seen in some cases (e.g., Canalizo & Stockton 2001). With narrow-line objects the galaxy is easier to study directly, and Kauffmann et al. (2003) find a strong tendency for the most active objects in their sample of SDSS narrow-line AGNs to contain a significant post-starburst population. Ho (2005) use the [O II] doublet, which is intrinsically weak in high-ionization Seyfert galaxies, to find that star formation is suppressed in luminous AGNs, even when molecular gas is present (see also Kim et al. 2006); Ho speculates that this may be a concrete manifestation of AGN feedback. Our AGNs, while of low-mass and correspondingly low-luminosity, nevertheless have the highest growth rates of any local AGNs because of their high Eddington ratios (Greene & Ho 2007b). Unfortunately, we do not have a large amount of information about the stellar populations, both because of the AGN dominance and the low typical signal-to-noise ratio of the spectra.
From the host galaxy colors, it is clear that recent star formation has occurred in some fraction of the galaxies. A more compelling argument for intermediate-age stellar populations comes from the strength of the higher-order Balmer lines apparent in the stacked spectra (Fig. 6). Balmer absorption is present in all Eddington ratio bins, but we cannot, from these data alone, quantify whether there is any correlation between AGN luminosity and strength of Balmer absorption. We are wary to measure stellar population indices for these objects without a clean measurement of the AGN luminosity, but the absorption-line spectra (particularly in panels c and d of Fig. 6) clearly contain a contribution from A stars.
Each of these clues only places constraints on star formation in the past ∼ 10 9 yr. As always, it is far more challenging to find evidence for ongoing star formation, since the emission from the AGN itself generates nebular emission. The HST morphologies do suggest that star formation is ongoing at least in the spiral arms of the disk galaxies in the GH sample, but a more quantitative estimate is desirable. This was the motivation of Ho (2005) , and hence indirectly with galaxy color, a similar correlation may result. With these data alone, we cannot place strong constraints on the ongoing star formation in these systems, nor are constraints on star formation alone sufficient to implicate a causal connection with bulge growth. We would like to determine whether there is an excess of star formation in the AGN hosts as compared to inactive dwarf galaxies. A combination of high spatial resolution optical imaging with HST and broader spectral modeling of the host colors should be a first step toward this goal.
FIG. 8.-Distributions of g-band absolute magnitudes. The magnitudes pertain to the AGN alone (top), the host galaxy alone (middle), and the entire system (bottom). Filled histograms represent objects below the detection threshold defined in Greene & Ho (2007b; the c sample). The AGN contribution was estimated from L Hα , using the L Hα -L 5100 relation of Greene & Ho 2005b, and assuming a spectral shape of β = −1.56 ( f λ ∝ λ −β ; Vanden Berk et al. 2001). Alternatively, the AGN magnitude may be estimated from the SDSS PSF magnitude, which leads to the distribution shown in dotted lines in the middle panel.
SUMMARY AND FUTURE WORK
We have increased the sample of low-mass BHs in galactic nuclei by an order of magnitude (174+55 additional candidates). In general the sample properties are completely consistent with that of the original GH sample. The objects are radiating at ∼ one-third of their Eddington limit, and many of them have ROSAT counterparts. On the other hand, as expected, very few have radio detections in the FIRST survey.
Much like our previous sample, the current objects are found in sub-L * galaxies, suggesting that indeed low-mass BHs are found in low-mass stellar systems. The colors and composite spectral properties indicate that there are intermediate-age stellar populations, but we have no direct constraints on ongoing star formation. Although the SDSS imaging is neither deep enough nor of high enough angular resolution to determine the host galaxy structure, we do have relevant external data for the original GH sample. The bulge stellar velocity dispersions presented in Barth et al. (2005) show that the objects are consistent with the low-mass extrapolation of the M BH − σ * relation. There is a suggestion, from , that the slope of the AGN M BH − σ * relation flattens somewhat at these low masses.
While the systems (at least to first order) obey the M BH − σ * relation, they do not necessarily contain classical bulges. NGC 4395 is an Sdm spiral galaxy (i.e., no bulge), while POX 52 is probably a dwarf spheroidal. As a result, the low-mass systems do not appear to obey the low-mass extrapolation of the M BH --L bulge relation (e.g., Marconi & Hunt 2003). The total galaxy luminosities are certainly larger than one would expect, based on their BH masses, and this appears to be true even in the absence of disk components (Barth et al. 2004; J. E. Greene et al. in preparation). Therefore even near their Eddington limits these BHs are not capable of providing as much energy per unit host galaxy mass as more massive systems. The mystery, then, is why they appear to obey the same M BH − σ * relation.
Along these lines, we should note that the revised radiusluminosity relation slope (Bentz et al. 2006) substantially increases the virial masses at low BH mass, which in turn increases the apparent flattening in the M BH − σ * relation slope at low mass previously reported by . More insight should be gained into the reality of this slope change once we have measured σ * for the present sample. Ultimately, we wish to elucidate the degree to which the slope and (crucially) intrinsic scatter of the M BH − σ * relation has any mass dependence (e.g., Robertson et al. 2006;.
Indeed, there are a number of interesting applications to pursue with the improved statistics afforded by this new, larger sample. We plan to examine the environments and clustering properties of these objects, and see how they compare to inactive galaxies of similar mass and color. At the same time, we would like to constrain the stellar populations of the host galaxies in a more quantitative fashion. For the GH sample, at least, this will be enabled by a combination of HST imaging and Spitzer IRS spectra. Most immediately, we will use the HST imaging, in combination with stellar velocity dispersion measurements, to place the GH sample on the fundamental plane. In combination, we will use these data to examine differences in the evolution of low-mass systems as compared to supermassive BHs in massive bulges.
APPENDIX VIRIAL BLACK HOLE MASS MEASUREMENTS
The virial technique is quite indirect, and its calibration is a matter of ongoing research. Therefore, it is worth describing our methodology in some detail. In particular, we discuss the potential systematic errors incurred by uncertainties in the slope of the radius-luminosity relation on the one hand and the derivation of robust gas velocity dispersions on the other.
At the present time, the radius-luminosity relation is calibrated using the ∼ 30 objects with BLR radii determined from reverberation mapping. Bentz et al. have used HST to determine the true AGN luminosity, as free as possible of host-galaxy contamination. Their quoted radius-luminosity relation slope is significantly shallower than that of Kaspi et al. (2005). It is worth keeping in mind the important caveat that the luminosities of the objects in that study barely overlap with those in this work, and most are significantly higher than those considered here. While programs to measure BLR radii for lower-luminosity sources are underway, at the moment we have no choice but to adopt the results for the more luminous sources.
As described in detail in Greene & Ho (2005b), the line luminosity provides a more robust measure of the AGN luminosity in the presence of host galaxy contamination (or contamination from non-thermal jet emission). Therefore we have used the L 5100 -L Hα relation from Greene & Ho (2005b)
M ⊙ ,(A1)
where we assume a spherical BLR ( f = 0.75). Here we are using the FWHM of Hα as a measure of the gas velocity dispersion.
In principle the width of the broad emission lines represents the gas kinematics, but in practice it is unclear how to derive the most robust velocity dispersion measure. It has long been clear that the broad line shape depends systematically on other AGN properties including line width (or BH mass) and luminosity (e.g., Sulentic et al. 2000); in general, narrower lines have more Lorentzian shapes (e.g., Véron-Cetty et al. 2001). If the geometry and kinematics of the BLR change with Eddington ratio (as suggested by these observations) then the assumption of a single geometrical factor f breaks down. In other words, the measured virial mass for a given BH may vary as changes in the Eddington ratio lead to changes in the BLR kinematics and thereby f . An extreme example of this effect is reported for AGNs with double-peaked Balmer lines (Bian et al. 2007). Without independent information it is impossible to determine f as a function of L bol /L Edd . Luckily, the M BH − σ * relation provides an independent handle on the BH mass. Especially for the relatively low-luminosity AGNs considered at low redshift, it is safe to assume that the systems intrinsically obey the M BH − σ * relation, and adopt the M BH − σ * -derived mass as the true BH mass. Both Onken et al. (2004) and find evidence for an average offset 4 of −0.26 dex from masses derived with f = 0.75. However, as argued above, we expect that f is not a constant, but rather depends on L bol /L Edd . While looked for evidence of a trend between f and L bol /L Edd , their dynamic range in both BH mass and luminosity was too limited to draw robust conclusions. Collin et al. (2006) perform an exhaustive study of the relation between line shape and virial mass. They use the Onken et al. sample of 14 AGNs with both reverberation-mapped masses and σ * measurements to investigate the merits of different measures of the line width. Specifically, they compare virial BH masses estimated from the FWHM and the luminosity-weighted second moment of the velocity profile (σ) with the σ * -derived BH mass. As described above, they find that narrower lines are peaky (small values of FWHM/σ) while broader lines are boxy (large values of FWHM/σ). As a result, the inferred f may depend not only on whether FWHM or σ is used to measure line width, but also on the line shape. Using the comparison between virial mass and σ * , they derive a different value of f for each subset of objects (9 peaky and 5 boxy) using both FWHM and σ. They claim that while f is constant across line samples when σ is used as the velocity dispersion indicator, the FWHM provides a biased measure of the virial mass because f is smaller for the boxy subset. However, investigation of their table reveals that the values of f derived from FWHM measurements are also consistent (within the quoted errors) with a constant value. Although the result may very well be real, it is certainly not statistically significant at this time.
Moreover, we question the premise of Collin et al. that the best measure of line width is the one that minimizes changes in f . Rather, since the BLR geometry really does change with L bol /L Edd , one should calibrate different f values for different average line shapes, when sufficient statistics exist to do so in a robust fashion. Collin et al. present an additional piece of information that we believe favors use of the FWHM over σ. Because they are using reverberation-mapped AGNs, many epochs of observations are available. One may derive a so-called rms spectrum that isolates the variable part of the line. As argued in, e.g., Peterson et al. (2004), it is preferable to measure the width of the variable component of the line, which hopefully reflects the kinematics of gas at the measured radii. As it turns out, the FWHM measured from the mean and rms spectra are more self-consistent than the respective σ measurements. There appears to be a very broad, non-variable component of the line which biases σ more strongly than FWHM (perhaps an optically thin component; e.g., Shields et al. 1995). While the very broad component cannot be removed in single-epoch observations, at least the FWHM is less biased by it. Finally, it is quite clear from Figure 1 of Collin et al. that the scatter in σ measurements is considerably larger than in the FWHM. The same result was reported by .
Based on all of these considerations, the following choices were made in calculating the BH masses presented herein:
• We use the radius-luminosity relation reported by Bentz et al. (2006). Note that this differs from Greene & Ho (2007b).
• We use the FWHM to determine the gas velocity dispersion.
• Because we believe f is not a constant, but rather depends on L bol /L Edd , and in order to facilitate comparison with the literature, we do not apply an f correction but use the fiducial assumption of a spherical BLR ( f = 0.75; e.g., Netzer 1990).
Obviously, virial masses remain highly uncertain, and potentially systematically biased. In addition to the many uncertainties highlighted above, it is not yet clear the degree to which the BLR is a flattened structure, which means that inclination effects cannot be properly accounted for, even in a statistical sense. We are hopeful that an increased sample of reverberation-mapped sources (e.g., Kollatschny 2003), combined with an increase in the number of measured σ * measurements in AGNs, will decrease the outstanding systematic errors in these masses in the near future.
Note. -Col. (1): Identification number assigned in this paper. Col. (2): Official SDSS name. Col. (3): Objects that are beneath our detection threshold are indicated with c. Previous identifier, if relevant, is indicated. Col. (4): Redshift measured by the SDSS pipeline. Col. (5): Petrosian g magnitude. Objects with questionable Petrosian magnitudes (flagged as many petro or no petro by the SDSS pipeline) are noted with * , and the model magnitudes are tabulated. Col. (6): Petrosian g − r color. Col. (7): Galactic extinction in the g band.
FIG. 1 .
1-Distributions of "virial" BH mass and Eddington ratio for the entire sample. M BH is calculated from L Hα and FWHM Hα , using the formalism ofGreene & Ho 2005b. The Eddington ratio is derived assuming an average bolometric correction of L bol = 2.34 × 10 44 (L Hα /10 42 ) 0.86 ergs s −1 (see text). Filled histograms represent objects below the detection threshold defined inGreene & Ho 2007b (the c sample).
FIG. 2 .
2-(a) The Hβ+[O III] fit to SDSS J155909.62+350147 used to create the Fe II template shown in (b). Shown are the continuum-subtracted (top), best-fit (red thin) and residual (bottom) spectra. Units are 10 −17 ergs s −1 cm −2 Å −1 . (b) A comparison of the standard Boroson & Green 1992 1
FIG. 3 .
3-Diagnostic diagrams plotting log [O III] λ5007/Hβ versus (a) log [N II] λ6583/Hα, (b) log [S II] λλ6716, 6731/Hα, and (c) log [O I] λ6300/Hα. The c sample is shown in open symbols, and there are red boxes around ROSAT detections and blue triangles around FIRST detections. The line ratios have not been corrected for reddening, but this should not matter because of the small wavelength separation of the lines. The dotted lines mark the boundaries of the three main classes of emission-line nuclei, according to the convention ofHo et al. 1997a: "H" = H II nuclei, "S" = Seyferts, and "L" = LINERs.
Note. -Col. (1): Identification number assigned in this paper. Col. (2): Total g-band absolute magnitude. Col. (3): AGN g-band absolute magnitude, estimated from LHα given in Col. (5) and a conversion from LHα to Mg assuming f λ ∝ λ −1.56 . Col. (4): Host galaxy g-band absolute magnitude, obtained by subtracting the AGN luminosity from the total luminosity. Cases where the AGN accounts for the total galaxy luminosity are indicated with an ellipsis. Col. (5): AGN luminosity in broad Hα (ergs s −1 ). Col. (6): Virial mass estimate of the BH (M⊙). Col. (7): Ratio of the bolometric luminosity (see text) to the Eddington luminosity. Col. (8): Previously reported virial mass estimate of the BH (M⊙) from Greene & Ho 2007b. There are no entries for the c sample.
Note. -Col. (1): Identification number assigned in this paper. Col. (2): ROSAT count rate (counts sec −1 ). Col. (3): Galactic column N H (cm −2 ), calculated following Dickey & Lockman 1990. Col. (4): X-ray flux in the 0.5-2 keV band (ergs s −1 cm −2 ) assuming a power-law spectrum with Γs = 2 and N H from col. (3). Col. (5): X-ray luminosity in the 0.5-2 keV band (ergs s −1 ).
FIG. 4 .
4-Tracks in [O II]/[O III] for different metallicities, following Groves et al. 2006. At a given metallicity, the tracks span a range in ionization parameterŨ 0 ≡ S * /(cñ H ), where S * is the number of ionizing photons, and ñ H = Pgas/(k10 4 ). The model values are log U 0 =(−0.40, −1.03, −1.43, −1.88, −2.24, −2.57, −2.98) and the tracks are labeled at the low-Ũ 0 end, as indicated with dashed lines. It is clear that metallicity alone cannot account for the spread in [O II]/[O III] ratios we observe. Open symbols indicate the c sample.
FIG. 5 .
5-A diagnostic diagram highlighting how different effects move points in the parameter space. For pure AGNs with line ratios originating at positions A, B, and C, we have mixed star-forming galaxies with line ratios along theKauffmann et al. 2003 line (blue dotted), assuming that the Hβ strength from photoionization by starlight ranges from half (upper track; A in long dash-dot, B in solid, C in long dash) to twice (lower track) that by the AGN. The thin arrows indicate the trajectory of increasing contributions from star formation. Also shown are photoionization models fromGroves et al. 2006. Each track covers a single metallicity and the same range of ionization parameters asFigure 4above. While mixing stellar and AGN-excited line ratios may broaden the locus of points in the diagnostic diagrams, it is clear that the intrinsic line ratios do not all originate at position A.
Note. -Col. (1): Identification number assigned in this paper. Detections are followed by upper limits. Col. (2): Flux density (or upper limit) at 20 cm from FIRST (mJy). Col. (3): Corresponding radio power (W Hz −1 ). Col. (4): R ≡ f6cm/f 4400Å , assuming a spectral index of αr = 0.46 (radio; Ulvestad & Ho 2001) and αo = 0.44 (optical; Vanden Berk et al. 2001), where fν ∝ ν −αo .
FIG. 7 .
7-Adopted from Greene et al. 2006, this figure shows the relation between L [OIII] and radio power for a large sample of AGNs. Objects from this paper are shown as open stars, while stacked luminosities for undetected sources are shown as crossed squares and upper limits from the original GH sample are shown as open triangles. For clarity, our data are plotted alone in the inset. The solid line represents the fit from Ho & Peng 2001 to radio-quiet Seyfert galaxies and Palomar-Green (PG; Schmidt & Green 1983) quasars: P 6cm = (0.46 ± 0.15) L [OIII] + (2.68 ± 6.21). The Ho & Peng sample is overplotted with filled (radio-loud sources) and open (radio-quiet sources) symbols. PG quasars are shown as squares, and Seyferts are shown as circles. NGC 4395, with L [OIII] = 5.8 × 10 37 ergs s −1 (adjusted to a distance of 4.2 Mpc; Thim et al. 2004) is highlighted as a semifilled circle.
in combination with the revised radius-luminosity relation from Bentz et al. (and uncertainties from M. Bentz, private communication) to derive a final BH mass estimator: M BH = (3.0 +0.6 −0.5 ) × 10 6 L Hα 10 42 erg s −1 0.45±0.03 FWHM Hα 10 3 km s −1 2.06±0.06
1 Table 1 .
11The SDSS SampleID
SDSS Name
Flag
z
g
g − r
Ag
(1)
(2)
(3)
(4)
(5)
(6)
(7)
1
SDSS J000111.15−100155.5
0.0489
18.6
0.7
0.21
2
SDSS J000308.47+154842.3
0.118
18.2
0.5
0.16
3
SDSS J000605.59−092007.0
c
0.0699
17.9
0.7
0.34
). More striking is the apparent corresponding decrease in [O III]/Hβ, which as we have seen, cannot be ascribed to a decrease in metallicity. In fact, [O III]/Hβ appears to decrease in concert with [N II]/Hα, [S II]/Hα, and [O I] λ6300/Hα (Fig. 3b, c). At the same time, there is a clear trend of increasing [O II] λ3727/[O III] line ratios as the [O III]/Hβ ratio decreases
O III]/Hβ or [O I]/Hα. We have already discussed the impact of varying metallicity above, and since [N II]/Hα is most sensitive to its variation, metallicity is unlikely to explain the correlated decrease in [S II]/Hα, [O I]/Hα and [N II]/Hα. Furthermore, metallicity variations alone cannot explain the large observed range in [O II]/[O III] ratios (Fig. 4). Changes in either ionization parameter or density might explain the [O II]/[O III] ratios; increases in either leads to a relative increase in [O III]. Indeed, electron densities inferred from the ratio of the [S II] lines show a significant correlation with [O III]/Hβ. The Spearman rank correlation coefficient is ρ = 0.296, with a probability P < 10 −4 that no correlation is present. However, there is only a mildly significant trend between electron density and [N II]/Hα (ρ = 0.197, P = 0.003) and no trend with [S II]/Hα (ρ = 0.124, P = 0.06; as one might expect given their lower critical densities). On the other hand, changes in ionization parameter move, for example, [N II]/Hα vs. [O III]/Hβ on tracks perpendicular to those observed (e.g.,
Table 3 .
3Luminosity and Mass MeasurementsID
Mg(total)
Mg(AGN) Mg(host)
log LHα
log MBH
log L bol /L Edd
log MOLD
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
1
−18.02
−15.91
−17.82
40.11
6.2
−1.6
5.8
2
−20.16
−18.24
−19.91
41.22
6.1
−0.5
5.9
3
−19.44
−16.31
−19.37
40.26
6.2
−1.4
· · ·
, and Fe II/Hβ = 1.32 ± 0.16 for the original GH sample. The [O III] distributions are consistent with our previous findings: [O III]/Hβ = 2.6 ± 0.3 (1.9 ± 0.2), while we found [O III]/Hβ = 2.24 ± 0.72 for the original sample. Although it has been customary to cite [O III]/Hβ(total), the ratio of [O III]/Hβ(narrow) is more straightforward to interpret, and we report it as well: [O III]/Hβ(narrow) = 6.1 ± 0.5 (6.5 ± 0.6). In all other discussion, we report trends for [O III]/Hβ(narrow).
Table 5 .
5FIRST DetectionsID
S20cm
log P20cm
log R
(1)
(2)
(3)
(4)
47
1.77±0.14
21.41±0.04
0.65
69
1.43±0.15
22.87±0.05
0.96
87
1.03±0.22
22.19±0.10
0.60
101
2.41±0.14
22.60±0.03
1.90
106
5.96±0.17
22.19±0.01
1.30
140
1.03±0.15
21.94±0.07
1.20
146
1.29±0.15
22.31±0.05
0.44
158
1.55±0.15
21.87±0.04
0.89
163
2.33±0.14
21.37±0.03
1.10
174
1.91±0.15
22.95±0.04
1.20
203
3.39±0.14
21.87±0.02
0.00
1
<0.33±0.16
<21.27±0.23
<0.71
2
<0.45±0.15
<22.20±0.15
<0.70
3
<0.20±0.11
<21.37±0.28
<0.69
to use the [O II]/[O III] ratio as an indicator of concurrent star formation. Unfortunately, as we have seen above, we do not have a strong handle on the physical properties in the NLR, and therefore cannot uniquely interpret the observed spread in [O II]/[O III] ratio. There is a mild correlation between the [O II]/[O III] ratio and the g − r color (ρ = −0.3, P < 10 −4 ; including or excluding the c sample), which one might expect if the [O II] arises from current star formation. On the other hand, if the NLR conditions correlate with Hubble type (not an unreasonable possibility)
The VLA is operated by the National Radio Astronomy Observatory, which is a facility of the National Science Foundation, operated under cooperative agreement by Associated Universities, Inc
Note that the Greene & Ho measurement, while including many more AGNs, is still dominated by the reverberation-mapped sources, which have considerably smaller systematic errors in their virial BH mass estimates.
We acknowledge the useful suggestions of the anonymous referee that improved this work significantly. We thank Michael Blanton for making his K-correction code publicly available, and Brent Groves for kindly providing his metallicity models prior to publication. We thank the entire SDSS team for providing the fantastic data products that made this work possible. Support for J. E. G. was provided by NASA through Hubble Fellowship grant HF-01196 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555. L. C. H. acknowledges support by the Carnegie Institution of Washington and by NASA grant SAO 06700600. Funding for the SDSS has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS web site is http://www.sdss.org/. This research has made use of data obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA's Goddard Space Flight Center.
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|
[] |
[
"Pseudo-bundles of exterior algebras as diffeological Clifford modules",
"Pseudo-bundles of exterior algebras as diffeological Clifford modules"
] |
[
"Ekaterina Pervova "
] |
[] |
[] |
We consider the diffeological pseudo-bundles of exterior algebras, and the Clifford action of the corresponding Clifford algebras, associated to a given finite-dimensional and locally trivial diffeological vector pseudo-bundle, as well as the behavior of the former three constructions (exterior algebra, Clifford action, Clifford algebra) under the diffeological gluing of pseudo-bundles. Despite these being our main object of interest, we dedicate significant attention to the issues of compatibility of pseudo-metrics, and the gluing-dual commutativity condition, that is, the condition ensuring that the dual of the result of gluing together two pseudo-bundles can equivalently be obtained by gluing together their duals, which is not automatic in the diffeological context. We show that, assuming that the dual of the gluing map, which itself does not have to be a diffeomorphism, on the total space is one, the commutativity condition is satisfied, via a natural map, which in addition turns out to be an isometry for the natural pseudo-metrics on the pseudo-bundles involved. MSC (2010): 53C15 (primary), 57R35, 57R45 (secondary).Main definitions and known factsWe now go, as briefly as possible, over the main definitions that appear in what follows (for terms whose use is not as frequent, we will provide definitions as we go along).8 This means the topology underlying the diffeological structure, the so-called D-topology, see[4]. In most significant examples, however, the diffeology is put on a space already carrying a topological structure, and in way such that the D-topology coincides with it. 9 Notice that in the case when the gluing map f is invertible as map from its domain to its range, and only in this case, X 1 ∪ f X 2 is a span of the spaces X 1 and X 2 . However, a priori the gluing construction is more general. 10 They are defined by composition of the obvious inclusions into X 1 ⊔ X 2 , with the quotient projection π.11 Which is a natural consequence of the construction and also its merit, as it allows to treat, for instance, conical singularities as the results of gluing to a one-point space. For this reason, although in most cases we deal with gluings along invertible maps, hence symmetric ones, we treat them as if they were not, to keep the discussion as general as possible.12 According to a personal preference, one could use the language of category theory and describe the gluing operation as the pushforward of the pair of maps Id : X 1 → X 1 and f : X 1 ⊃ Y → X 2 to the category of smooth maps between diffeologies, that enjoys the bifunctoriality property.13Relative to the product diffeology on C ∞ (X 1 , Z) × C ∞ (X 2 , Z) which in turn comes from the functional diffeologies on C ∞ (X 1 , Z) and C ∞ (X 2 , Z).
|
10.1007/s00006-017-0769-z
|
[
"https://arxiv.org/pdf/1604.04861v2.pdf"
] | 119,619,680 |
1604.04861
|
6618f558ede7f026172fcf950387ad4732f16f58
|
Pseudo-bundles of exterior algebras as diffeological Clifford modules
9 Feb 2017 January 5, 2018
Ekaterina Pervova
Pseudo-bundles of exterior algebras as diffeological Clifford modules
9 Feb 2017 January 5, 2018
We consider the diffeological pseudo-bundles of exterior algebras, and the Clifford action of the corresponding Clifford algebras, associated to a given finite-dimensional and locally trivial diffeological vector pseudo-bundle, as well as the behavior of the former three constructions (exterior algebra, Clifford action, Clifford algebra) under the diffeological gluing of pseudo-bundles. Despite these being our main object of interest, we dedicate significant attention to the issues of compatibility of pseudo-metrics, and the gluing-dual commutativity condition, that is, the condition ensuring that the dual of the result of gluing together two pseudo-bundles can equivalently be obtained by gluing together their duals, which is not automatic in the diffeological context. We show that, assuming that the dual of the gluing map, which itself does not have to be a diffeomorphism, on the total space is one, the commutativity condition is satisfied, via a natural map, which in addition turns out to be an isometry for the natural pseudo-metrics on the pseudo-bundles involved. MSC (2010): 53C15 (primary), 57R35, 57R45 (secondary).Main definitions and known factsWe now go, as briefly as possible, over the main definitions that appear in what follows (for terms whose use is not as frequent, we will provide definitions as we go along).8 This means the topology underlying the diffeological structure, the so-called D-topology, see[4]. In most significant examples, however, the diffeology is put on a space already carrying a topological structure, and in way such that the D-topology coincides with it. 9 Notice that in the case when the gluing map f is invertible as map from its domain to its range, and only in this case, X 1 ∪ f X 2 is a span of the spaces X 1 and X 2 . However, a priori the gluing construction is more general. 10 They are defined by composition of the obvious inclusions into X 1 ⊔ X 2 , with the quotient projection π.11 Which is a natural consequence of the construction and also its merit, as it allows to treat, for instance, conical singularities as the results of gluing to a one-point space. For this reason, although in most cases we deal with gluings along invertible maps, hence symmetric ones, we treat them as if they were not, to keep the discussion as general as possible.12 According to a personal preference, one could use the language of category theory and describe the gluing operation as the pushforward of the pair of maps Id : X 1 → X 1 and f : X 1 ⊃ Y → X 2 to the category of smooth maps between diffeologies, that enjoys the bifunctoriality property.13Relative to the product diffeology on C ∞ (X 1 , Z) × C ∞ (X 2 , Z) which in turn comes from the functional diffeologies on C ∞ (X 1 , Z) and C ∞ (X 2 , Z).
Introduction
This work is intended as a supplement to [8], dealing with some issues regarding pseudo-bundles of exterior algebras associated to finite-dimensional diffeological vector pseudo-bundles, viewed as pseudobundles of diffeological Clifford modules, so endowed with the action of the corresponding diffeological pseudo-bundles of Clifford algebras. Let us explain as briefly as possible what all these objects are; the precise definitions are to be found in a dedicated section, or else in the references given therein.
First of all, the diffeology. The notion is due to J.M. Souriau [14], [15] and is a categorical extension of the notion of a smooth structure; in essence, or maybe as an example, it is a way to consider a topological space which is in no way a manifold, as if it were one. In and of itself, a diffeology on a set is a collection of maps into this set which are declared to be smooth; this collection must satisfy certain conditions. The set is then a diffeological space, and all the basic constructions follow; there is a notion of smooth maps between two diffeological spaces, that of the underlying topology, the so-called D-topology (introduced in [4], see also [2] for a recent treatment), and so on. The notion builds on existing ones, such as those of differentiable spaces, V-manifolds, and so on (see, for instance, [13], [1], to name a few). A particularly important aspect of diffeology is that all the usual topological constructions, notably subsets and quotients, have an inherited diffeological structure (unlike the case of smooth manifolds, where subsets and quotients are quite rarely smooth manifolds themselves).
The basic object for us, though, is not just a diffeological space, but a diffeological vector pseudobundle ( [4], [17], [5], [10]). The difference with respect to the standard notion is not only in the fact that the smooth structure is replaced by a diffeological one (under some respects this would be a minor difference), but also in that it does not have to be locally trivial, although in many contexts we do add this assumption, see the discussion of pseudo-metrics, which replace the usual Riemannian metricsdiffeological pseudo-bundles frequently do not carry the latter.
On the other hand, as explained in the references listed above, the usual operations on vector bundles have their diffeological counterparts; in particular, direct sums, tensor products, and taking duals all apply. From this, obtaining pseudo-bundles of tensor algebras, those of exterior algebras, or defining, in the abstract, pseudo-bundles of Clifford modules is automatic.
The point of view that we take in this paper has to do with studying the behavior of these concepts under the operation of diffeological gluing. This procedure is one of the many possible extensions of the concept of an atlas on a smooth manifold, and the resulting spaces are among the more obvious extensions of smooth manifolds and include some well-known singular spaces; for instance, a manifold with a conical singularity can be seen as a result of gluing of a usual smooth manifold to a single-point space.
For the basic operations on diffeological vector pseudo-bundles, the behavior under diffeological gluing was considered in [10] (see also [11] for some details); for tensor algebras and pairs of given Clifford modules, in [8]. What is lacking is a study of gluing of pseudo-bundles of the exterior algebras, in particular, the covariant version. This paper aims to fill this void (another motivation for it is to provide some necessary building blocks for defining the notion of a diffeological Dirac operator and studying its behavior under gluing, see [12]).
The content Section 1 goes over the main definitions used and introduces notation. In Section 2 we consider the compatibility of pseudo-metrics in terms of the assumptions on the gluing map(s); in Section 3 we relate this to the gluing-dual commutativity condition, and in Section 4 we show that the compatibility of dual pseudo-metrics implies that the commutativity condition must be satisfied. In Section 5 we show that the gluing-dual commutativity diffeomorphism is an isometry. All these allow us to consider, in Sections 6-8, Clifford algebras (the covariant case), the exterior algebras, and the corresponding Clifford actions; in particular, in Section 8 we establish, where appropriate, several equivalences showing that, again under the gluing-dual commutativity assumption, everything reduces to two cases, the contravariant case and the covariant one. Section 9 contains a couple of simple (but necessarily lengthy) examples.
Diffeology and diffeological vector spaces
Let X be a set. A diffeology on X (see [14], [15]) is a set D = {p : U → X} of maps into X, each defined on a domain of some R n (with varying n), that satisfies the following conditions: 1) it includes all constant maps, i.e., maps of form U → {x 0 }, for all open (possibly disconnected) sets U ⊆ R n and for all points x 0 ∈ X; 2) for any D ∋ p : U → X and for any usual smooth map g : V → U (again defined on some domain V ⊆ R m ) we have p • g ∈ D; and 3) if a set map p : U → X is such that its domain of definition U ⊆ R n has an open cover U = ∪ i∈I U i for which p| Ui ∈ D then p ∈ D.
The maps composing D are called plots. A standard example of diffeology/ diffeological space is a usual smooth manifold M , with diffeology composed of all usual smooth maps into M . On the other hand, any set (usually at a least a topological space) admits plenty of non-standard diffeologies, obtained via the concept of a generated diffeology.
Generated diffeologies Given a fixed set X, various diffeologies on it can be compared with respect to the inclusion; 1 for two diffeologies D and D ′ such that D ⊂ D ′ one says that D is finer than D ′ , whereas D ′ is said to be coarser. Frequently, for a given property P which a diffeology might possess, there is the finest and/or the coarsest diffeology with the property P (see [5], Sect. 1.25); this fact is often used in describing concrete diffeologies, or defining a class of them. A specific example of the former is the generated diffeology: for a set X and a set A = {p : U → X} of maps into X, the diffeology generated by A is the smallest diffeology on X that contains A. Notice that A can be any set; it might include non-differentiable maps, discontinuous ones, and so on.
Smooth maps A map f : X → Y between two diffeological spaces X and Y is considered smooth if for every plot p of X the composition f • p is a plot of Y . Note that it might easily happen that all f • p are plots of Y , but that vice versa is not true: Y may have plots that do not have form f • p, whatever the plot p of X. On the other hand, if for every plot q : U → Y of Y and for every point u ∈ U there is a plot p u of X such that in a neighborhood of u we have q = f • p u then we say that the diffeology of Y is the pushforward of the diffeology of X via f , and conversely, the diffeology of X is the pullback of the diffeology of Y by f . Subset, quotient, product, and disjoint union diffeologies All typical topological constructions admit diffeological counterparts. If X ′ is any subset of a diffeological space X, it carries the subset diffeology that consists of all plots of X whose range is contained in X ′ ; and if X/ ∼ is any 2 quotient of X, with π : X → X/ ∼ being the natural projection, then the standard choice of diffeology on X/ ∼ is the quotient diffeology, defined as the pushforward of the diffeology of X by π. As we said above, this means that locally each plot of X/ ∼ has form π • p, where p is a plot of X.
Let us now have several (a finite number of, although the definition can be stated more broadly) diffeological spaces X 1 , . . . , X n . Their usual direct product also has its standard diffeology, the product diffeology, defined as the coarsest diffeology such that the projection on each term is smooth. Locally any plot of this diffeology is just an n-tuple (p 1 , . . . , p n ), where each p i is a plot of the corresponding X i . Finally, the disjoint union ⊔ n i=1 X i of these spaces has the disjoint union diffeology, this being the finest diffeology such that the inclusion of each term into the disjoint union is smooth. Locally, any plot of this diffeology is a plot of precisely one of terms X i .
Functional diffeology The space C ∞ (X, Y ) of all smooth (in the diffeological sense) maps between two diffeological spaces X and Y also has its standard diffeology, called the functional diffeology. It consists of all possible maps q : U → C ∞ (X, Y ) such that for every plot p : U ′ → X of X the natural evaluation map U × U ′ ∋ (u, u ′ ) → q(u)(p(u ′ )) ∈ Y is smooth (with respect to the diffeology of Y ; the product U × U ′ is still a domain, therefore asking for the evaluation map to be smooth is equivalent to asking it to be a plot of Y ).
Diffeological vector spaces This is one specific instance of a diffeological space endowed also with an algebraic structure whose operations are smooth for the diffeology involved. 3 A (real) diffeological vector space is a vector space V endowed with a diffeology such that the addition map V × V → V and the scalar multiplication map R × V → V are smooth (for the product diffeology on V × V and R × V respectively). For a fixed V there can be many such diffeologies; any one of them is called a vector space diffeology (on V ). If V is finite-dimensional, and so as just a vector space is isomorphic to some R n , then the finest of all vector space diffeologies is the one consisting of all usual smooth maps into it. 4 This diffeology is called the standard diffeology; endowed with it, V is called a standard space.
All usual operations on vector spaces (taking subspaces, quotients, direct sums, tensor products, and duals) admit their natural diffeological counterparts ( [17], [19]; see also [7]), via the more general diffeological constructions described above. Thus, any vector subspace of a diffeological vector space V is automatically endowed with the subset diffeology; every quotient space carries the quotient diffeology; the direct sum carries the product diffeology (relative to the diffeologies of its terms); and the tensor product has the quotient diffeology of the finest vector space diffeology on the free product of the factors that contains the product diffeology on their direct product. Finally, the diffeological dual V * of V is defined as the space of all diffeologically smooth linear maps L ∞ (V, R) (where R is standard) endowed with the functional diffeology. Notice that, unless a finite-dimensional V is a standard space, we have dim(V * ) < dim(V ) (and in general, the space of smooth linear maps between two diffeological vector spaces is strictly smaller than the space of all linear maps).
Pseudo-metrics and characteristic subspaces A finite-dimensional diffeological vector space V does not admit a smooth scalar product, unless it is standard (see [5]). The best possible substitute for it is any smooth symmetric semi-definite positive bilinear form of rank dim(V * ); (at least one) such a form exists on any finite-dimensional V and is called a pseudo-metric.
A pseudo-metric g on a finite-dimensional diffeological vector space V allows to identify in V the unique vector subspace V 0 which is maximal, with respect to inclusion, for the following two properties: the subset diffeology of V 0 is that of a standard space, and V 0 splits off smoothly in V , which means there is a usual vector space direct sum decomposition V = V 0 ⊕ V 1 such that the corresponding direct sum diffeology on V relative to the subset diffeologies on V 0 and V 1 coincides with the initial diffeology of V . 5 This subspace can be described as the subspace generated by all the eigenvectors of g that are relative to the positive eigenvalues; however, it does not actually depend on the specific choice of a pseudo-metric and is instead an invariant of V itself. It is called the characteristic subspace of V .
Diffeological vector pseudo-bundles and pseudo-metrics on them
A diffeological vector pseudo-bundle is a diffeological counterpart of a usual smooth vector bundle. 6 Apart from the diffeological smoothness replacing the usual concept of smooth maps, they lack an atlas of local trivializations, although in many contexts, and in most of what follows, we do add this assumption.
Diffeological vector pseudo-bundles Let V and X be diffeological spaces, and let π : V → X be a smooth surjective map. The map π, or the total space V , is called a diffeological vector pseudobundle if for each x ∈ X the pre-image π −1 (x) carries a vector space structure such that the following three maps are smooth: the addition map V × X V → V (where V × X V is endowed with the subset diffeology as a subset of V × V ), the scalar multiplication map R × V → V , and the zero section X → V . An example of a diffeological vector pseudo-bundle which is not locally trivial, can be found in [3] (see Example 4.3).
The fibrewise operations and fibrewise diffeologies Since each fibre of a diffeological vector pseudo-bundle is a diffeological vector space, all the usual operations on vector bundles (direct sums, tensor products, dual bundles) can be performed on/with pseudo-bundles (see [17], and also [10] for some details), although not in an entirely similar way (the lack of local trivializations prevents that). Instead, these operations are performed by first carrying out the operation in question to each fibre, defining the total space of the new pseudo-bundle as the union of the resulting diffeological vector spaces (with the obvious fibering over the base), and finally defining the diffeology of this total space as the finest that induces on each fibre its existing diffeology. 7 As an example, and also because this instance will be particularly important for us, let us consider dual pseudo-bundles. Let π : V → X be a diffeological vector pseudo-bundle with finite-dimensional fibres. The dual pseudo-bundle of V is
V * = ∪ x∈X (π −1 (x)) * , where (π −1 (x)) * = L ∞ (π −1 (x), R)
is the diffeological dual of the diffeological vector space π −1 (x), the pseudo-bundle projection π * is given by π * ((π −1 (x)) * ) = {x} for all x ∈ X, and the diffeology on V * is characterized as follows: a map q : R l ⊇ U ′ → V * is a plot of V * if and only if for every plot p : R m ⊇ U → V of V the evaluation map (u ′ , u) → q(u ′ )(p(u)) ∈ R is smooth for the subset diffeology on its domain of definition {(u ′ , u) | π * (q(u ′ )) = π(p(u))} ⊆ R l+m and the standard diffeology on R. The collection of all possible maps q satisfying this property does form a diffeology, equipped with which, V * becomes a difffeological vector pseudo-bundle, and furthermore, the corresponding subset diffeology on each fibre (π * ) −1 (x) coincides with the usual (functional) diffeology on (π −1 (x)) * .
Finally, another useful observation (and maybe a peculiarity of diffeology) is that any collection of vector subspaces, one per fibre, in a diffeological vector pseudo-bundle is again a diffeological vector pseudo-bundle (called a diffeological sub-bundle), for the subset diffeology. Likewise, any collection of quotients, one of each fibre, is a diffeological vector pseudo-bundle for the quotient diffeology. We call these facts a peculiarity since they go well beyond what happens for the usual smooth vector bundles.
Pseudo-metrics on diffeological vector pseudo-bundles Let π : V → X be a diffeological vector pseudo-bundle with finite-dimensional fibres. A pseudo-metric on it is a smooth section of the pseudobundle V * ⊗ V * such that for each x ∈ X the bilinear form g(x) is a pseudo-metric, in the sense of diffeological vector spaces, on the fibre π −1 (x). Not all pseudo-bundles admit a pseudo-metric (see [10]; it is not quite clear yet under which conditions a pseudo-bundle admits a pseudo-metric), although if a pseudo-bundle is locally trivial with a finite atlas of local trivializations, the reasoning similar to that in the case of usual smooth vector bundles would allow to conclude its existence on any pseudo-bundle with the above two properties.
Diffeological gluing
On the level of the underlying topological 8 spaces, diffeological gluing (introduced in [10]) is just the usual topological gluing. The result is endowed with a canonical diffeology, called the gluing diffeology. It is the finest diffeology for several properties, and is usually finer than other natural diffeologies on the same space.
Gluing of spaces, maps, and pseudo-bundles
The basic ingredient in the definition of the diffeological gluing procedure is the operation of gluing of two diffeological spaces, where we must essentially specify which diffeology is assigned to the space obtained by the usual topological gluing. This basic construction is then extended to gluing of smooth maps between diffeological spaces, a particularly important instance of which is the gluing of diffeological vector pseudo-bundles.
Diffeological spaces Let X 1 and X 2 be two diffeological spaces, and let f : X 1 ⊇ Y → X 2 be a map defined on a subset Y of X 1 and smooth for the subset diffeology on Y . Let
X 1 ∪ f X 2 := (X 1 ⊔ X 2 ) / ∼ ,
where the equivalence relation ∼ is given by,
X 1 ⊔ X 2 ∋ x 1 ∼ x 2 ∈ X 1 ⊔ X 2 if and only if either x 1 = x 2 or x 1 ∈ Y and x 2 = f (x 1 )
. Denote by π : X 1 ⊔ X 2 → X 1 ∪ f X 2 the quotient projection, and define the gluing diffeology on X 1 ∪ f X 2 to be the pushforward of the disjoint union diffeology on X 1 ⊔ X 2 by π. 9 This construction is particularly adapted to endowing piecewise-linear objects with a diffeology (among others). An easiest example is a wedge of two lines, which can be identified with the union of the two coordinate axes in R 2 . It is interesting to notice that the gluing diffeology on this union is finer than the subset diffeology relative to its inclusion into R 2 , as demonstrated by an example due to Watts, see Example 2.67 in [18]. (From this, it is easy to extrapolate the existence of similar examples in other dimensions). Indeed, relative to the gluing diffeology the natural inclusions 10
i 1 : X 1 \ Y ֒→ X 1 ∪ f X 2 and i 2 : X 2 ֒→ X 1 ∪ f X 2
are smooth, and their ranges form a disjoint cover of X 1 ∪ f X 2 . Notice in particular that X 1 does not in general inject into X 1 ∪ f X 2 , while X 2 always does; in other words, a priori the operation of gluing is not symmetric. 11 This asymmetry is demonstrated, for instance, by the following (useful in practice) description of plots of the gluing diffeology. Since the latter is a pushforward diffeology, any plot of it lifts to a plot of the covering space X 1 ⊔ X 2 . By the properties of the disjoint union diffeology, this means that if p : U → X 1 ∪ f X 2 is a plot and U is connected then it either lifts to a plot p 1 of X 1 or a plot p 2 of X 2 . By construction of X 1 ∪ f X 2 , we obtain that in the former case
p(u) = i 1 (p 1 (u)) if p 1 (u) ∈ X 1 \ Y, i 2 (f (p 1 (u))) if p 1 (u) ∈ Y,
while in the latter case we simply have p = i 2 • p 2 .
Gluing of maps The operation of gluing of diffeological spaces, when performed on the domains and possibly the ranges of some given smooth maps, defines a gluing of these maps, provided the maps themselves satisfy a natural compatibility condition. 12 More precisely, suppose first that X 1 , X 2 , Z are diffeological spaces and ϕ i : X i → Z for i = 1, 2 are smooth maps. Let f : X 1 ⊇ Y → X 2 be a smooth map, and suppose that ϕ 2 (f (y)) = ϕ 1 (y) for all y ∈ Y (the maps ϕ 1 and ϕ 2 are then said to be f -compatible). Then the map ϕ 1 ∪ f ϕ 2 :
X 1 ∪ f X 2 → Z given by (ϕ 1 ∪ f ϕ 2 )(x) = ϕ 1 (i −1 1 (x)) if x ∈ Range(i 1 ), ϕ 2 (i −1 2 (x)) if x ∈ Range(i 2 )
is well-defined and smooth. In fact, assigning the map
ϕ 1 ∪ f ϕ 2 ∈ C ∞ (X 1 ∪ f X 2 , Z) to each pair (ϕ 1 , ϕ 2 ), with ϕ i ∈ C ∞ (X i , Z) for i = 1, 2, of f -compatible maps yields a map C ∞ (X 1 , Z) × comp C ∞ (X 2 , Z) → C ∞ (X 1 ∪ f X 2 , Z)
that is smooth for the functional diffeology on C ∞ (X 1 ∪ f X 2 , Z) and the subset diffeology 13 on the set of f -compatible pairs C ∞ (X 1 , Z) × comp C ∞ (X 2 , Z) (see [11] for details). Finally, all of this extends to the case of two maps with distinct ranges, that is, ϕ 1 : X 1 → Z 1 and ϕ 2 : X 2 → Z 2 , with appropriate gluings of X 1 to X 2 and Z 1 to Z 2 . Specifically, let again f : X 1 ⊇ Y → X 2 be smooth, but consider also a smooth g : Z 1 ⊇ ϕ 1 (Y ) → Z 2 ; assume that ϕ 2 (f (y)) = g(ϕ 1 (y)) for all y ∈ Y (the maps ϕ 1 and ϕ 2 are said to be (f, g)-compatible). Notice that the counterparts of i 1 and i 2 for the space Z 1 ∪ g Z 2 are the natural inclusion maps j 1 :
Z 1 \ ϕ 1 (Y ) ֒→ Z 1 ∪ g Z 2 and j 2 : Z 2 → Z 1 ∪ g Z 2 .
We define the map ϕ 1 ∪ (f,g) ϕ 2 by setting
(ϕ 1 ∪ (f,g) ϕ 2 )(x) = j 1 (ϕ 1 (i −1 1 (x))) if x ∈ Range(i 1 ), j 2 (ϕ 2 (i −1 2 (x))) if x ∈ Range(i 2 ).
All the analogous statements, in particular, the smoothness of the thus-defined map
C ∞ (X 1 , Z 1 ) × comp C ∞ (X 2 , Z 2 ) → C ∞ (X 1 ∪ f X 2 , Z 1 ∪ g Z 2 )
, continue to hold (see [11]).
Pseudo-bundles Let us now turn to gluing of two pseudo-bundles; this is of course a specific instance of gluing of (f, g)-compatible maps. We specifically indicate it in order to to fix notation and standard terminology, and also to give conditions under which the result of gluing is again a diffeological vector pseudo-bundle.
Let π 1 : V 1 → X 1 and π 2 : V 2 → X 2 be two such pseudo-bundles, let f : X 1 ⊇ Y → X 2 be a smooth map, and letf : V 1 ⊇ π −1 1 (Y ) → V 2 be any smooth lift of f that is linear on each fibre (where it is defined). The gluing of these pseudo-bundles consists in the already-defined operations of gluing V 1 to V 2 alongf , gluing X 1 to X 2 along f , and π 1 to π 2 along (f , f ). It is easy to check (see [10]) that the resulting map π 1 ∪ (f ,f ) π 2 : V 1 ∪f V 2 → X 1 ∪ f X 2 is a diffeological vector pseudo-bundle for the gluing diffeologies on V 1 ∪f V 2 and X 1 ∪ f X 2 ; in particular, the vector space structure on its fibres is inherited from either V 1 or V 2 (more precisely, it is inherited from V 1 on fibres over the points in i 1 (X 1 \ Y ), and from V 2 on fibres over the points in i 2 (X 2 )).
Standard notation for gluing of pseudo-bundles We now fix some standard notation that applies to pseudo-bundles specifically. We have already described the standard inclusions
i 1 : X 1 \ Y ֒→ X 1 ∪ f X 2 , i 2 : X 2 ֒→ X 1 ∪ f X 2 , j 1 : V 1 \ π −1 1 (Y ) ֒→ V 1 ∪f V 2 , j 2 : V 2 ֒→ V 1 ∪f V 2 .
When dealing with more than one gluing at a time, we will needed a more complicated notation, which is as follows. Let χ 1 ∪ (h,h) χ 2 : W 1 ∪h W 2 → Z 1 ∪ h Z 2 be any pseudo-bundle obtained by gluing; denote by Y ′ ⊂ Z 1 the domain of definition of h. Then the counterparts of i 1 and i 2 will be denoted by 14
i Z1 1 : Z 1 \ Y ′ ֒→ Z 1 ∪ h Z 2 and i Z2 2 : Z 2 ֒→ Z 1 ∪ h Z 2 for Z 1 ∪ h Z 2 j W1 1 : W 1 \ χ −1 1 (Y ′ ) ֒→ W 1 ∪h W 2 and j W2 2 : W 2 ֒→ W 1 ∪h W 2 for W 1 ∪h W 2 .
Obviously, i W1 1 and j W1 1 would mean the same thing, and the same goes for i W2 2 and j W2 2 ; however, we use the different letters so that there always be a clear distinction between the base space and the total space. Finally, since the base space will be the same for all our pseudo-bundles, and so we will usually use the abbreviated notation i 1 , i 2 for it.
The switch map As we mentioned many times already, the operation of gluing for diffeological spaces is asymmetric. However, if we assume that gluing map f is a diffeomorphism with its image then obviously, we can use its inverse to perform the gluing in the reverse order, with the two results, X 1 ∪ f X 2 and X 2 ∪ f −1 X 1 , being canonically diffeomorphic via the so-called switch map
ϕ X1↔X2 : X 1 ∪ f X 2 → X 2 ∪ f −1 X 1 .
Using the notation just introduced, this map can be described by
ϕ X1↔X2 (i X1 1 (x)) = i X1 2 (x) for x ∈ X 1 \ Y, ϕ X1↔X2 (i X2 2 (f (x))) = i X1 2 (x) for x ∈ Y, ϕ X1↔X2 (i X2 2 (x)) = i X2 1 (x) for x ∈ X 2 \ f (Y ).
This is well-defined, not only because the maps i X1 1 and i X2 2 are injective with disjoint ranges covering X 1 ∪ f X 2 , but also because f is a diffeomorphism with its image.
Gluing and operations
Diffeological gluing of pseudo-bundles is relatively well-behaved with respect to the usual operations on vector bundles. More precisely, it commutes with the direct sum and the tensor product, while he situation is somewhat more complicated for the dual pseudo-bundles, see [10] (the facts needed are recalled below).
14 The rule-of-thumb is that i smth Direct sum Gluing of diffeological vector pseudo-bundles commutes with the direct sum in the following sense. Given a gluing along (f , f ) of a pseudo-bundle π 1 : V 1 → X 1 to a pseudo-bundle π 2 : V 2 → X 2 , as well as a gluing along (f ′ , f ) of a pseudo-bundle π ′ 1 : V ′ 1 → X 1 to a pseudo-bundle π ′ 2 : V ′ 2 → X 2 , there are two natural pseudo-bundles that can be formed from these by applying the operations of gluing and direct sum. These are the pseudo-bundles
(π 1 ∪ (f ,f ) π 2 ) ⊕ (π ′ 1 ∪ (f ′ ,f ) π ′ 2 ) : (V 1 ∪f V 2 ) ⊕ (V ′ 1 ∪f ′ V ′ 2 ) → X 1 ∪ f X 2 and (π 1 ⊕ π ′ 1 ) ∪ (f ⊕f ′ ,f ) (π 2 ⊕ π ′ 2 ) : (V 1 ⊕ V ′ 1 ) ∪f ⊕f ′ (V 2 ⊕ V ′ 2 ) → X 1 ∪ f X 2 ;
they are diffeomorphic as pseudo-bundles, that is, there exists a fibrewise linear diffeomorphism
Φ ∪,⊕ : (V 1 ∪f V 2 ) ⊕ (V ′ 1 ∪f ′ V ′ 2 ) → (V 1 ⊕ V ′ 1 ) ∪f ⊕f ′ (V 2 ⊕ V ′ 2 ) (see below) that covers the identity map on the base X 1 ∪ f X 2 .
Tensor product What has just been said about the direct sum, applies equally well to the tensor product. Specifically, the two possible pseudo-bundles are
(π 1 ∪ (f ,f ) π 2 ) ⊗ (π ′ 1 ∪ (f ′ ,f ) π ′ 2 ) : (V 1 ∪f V 2 ) ⊗ (V ′ 1 ∪f ′ V ′ 2 ) → X 1 ∪ f X 2 and (π 1 ⊗ π ′ 1 ) ∪ (f ⊗f ′ ,f ) (π 2 ⊗ π ′ 2 ) : (V 1 ⊗ V ′ 1 ) ∪f ⊗f ′ (V 2 ⊗ V ′ 2 ) → X 1 ∪ f X 2 , and again, they are pseudo-bundle-diffeomorphic via Φ ∪,⊗ : (V 1 ∪f V 2 ) ⊗ (V ′ 1 ∪f ′ V ′ 2 ) → (V 1 ⊗ V ′ 1 ) ∪f ⊗f ′ (V 2 ⊗ V ′ 2 ) covering the identity on X 1 ∪ f X 2 .
The dual pseudo-bundle The case of dual pseudo-bundles is substantially different. For one thing, to even make sense of the commutativity question, we must assume that f is invertible (which in general it does not have to be). Moreover, even with this assumption, in general the operation of gluing does not commute with that of taking duals. The reason of this is easy to explain. Let π 1 : V 1 → X 1 and π 2 : V 2 → X 2 be two diffeological vector pseudo-bundles, and let (f , f ) be a gluing between them; consider the pseudo-bundle π 1 ∪ (f ,f ) π 2 : V 1 ∪f V 2 → X 1 ∪ f X 2 and the corresponding dual pseudo-bundle
(π 1 ∪ (f ,f ) π 2 ) * : (V 1 ∪f V 2 ) * → X 1 ∪ f X 2 ;
compare it with the result of the induced gluing (performed along the pair (f * , f )) of π * 2 : V * 2 → X 2 to π * 1 : V * 1 → X 1 , that is, the pseudo-bundle
π * 2 ∪ (f * ,f ) π * 1 : V * 2 ∪f * V * 1 → X 2 ∪ f −1 X 1 .
It then follows from the construction itself that for any y ∈ Y (the domain of gluing) we have
((π 1 ∪ (f ,f ) π 2 ) * ) −1 (i X2 2 (f (y))) ∼ = (π −1 2 (f (y))) * and (π * 2 ∪ (f * ,f ) π * 1 ) −1 (i X1 2 (y)) ∼ = (π −1 1 (y)) * ;
since i X2 2 (f (y)) and i X1 2 (y) are related by the switch map, for the two pseudo-bundles to be diffeomorphic in a natural way 15 the two vector spaces (π −1 2 (f (y))) * and (π −1 1 (y)) * must be diffeomorphic, and a priori they are not. 16 Thus, we obtain the necessary condition (for the pseudo-bundles (V 1 ∪f V 2 ) * and V * 2 ∪f * V * 1 to be diffeomorphic), which is that (π −1 2 (f (y))) * ∼ = (π −1 1 (y)) * for all y ∈ Y . We do note right away that this condition may not be sufficient, in the sense that two pseudo-bundles over the same base may have all respective fibres diffeomorphic without being diffeomorphic themselves (this can be illustrated by the standard example of open annulus and open Möbius strip, both of which, equipped with the standard diffeology 17 can be seen as pseudo-bundles over the circle). We will leave it at that for now, returning to the issue of the gluing-dual commutativity later in the paper. 15 For us this means, via a diffeomorphism covering the switch map. 16 They may have different dimensions. 17 That is, one determined by their usual smooth structure.
The commutativity diffeomorphisms
We now say more about the commutativity diffeomorphisms mentioned in the previous section. 18 The diffeomorphism Φ ∪,⊕ We have already mentioned the existence of this diffeomorphism, which is a pseudo-bundle map
Φ ∪,⊕ : (V 1 ∪f V 2 ) ⊕ (V ′ 1 ∪f ′ V ′ 2 ) → (V 1 ⊕ V ′ 1 ) ∪f ⊕f ′ (V 2 ⊕ V ′ 2 )
that covers the identity map on X 1 ∪ f X 2 . We now add that this map can be described (in fact, fully defined) by the following identities:
Φ ∪,⊕ • (j V1 1 ⊕ j V ′ 1 1 ) = j V1⊕V ′ 1 1 and Φ ∪,⊕ • (j V2 2 ⊕ j V ′ 2 2 ) = j V2⊕V ′ 2 2 .
The diffeomorphism Φ ∪,⊗ Once again, the case of the tensor product is very similar to that of the direct sum. The already-mentioned diffeomorphism
Φ ∪,⊗ : (V 1 ∪f V 2 ) ⊗ (V ′ 1 ∪f ′ V ′ 2 ) → (V 1 ⊗ V ′ 1 ) ∪f ⊗f ′ (V 2 ⊗ V ′ 2 )
is uniquely determined by the identities
Φ ∪,⊗ • (j V1 1 ⊗ j V ′ 1 1 ) = j V1⊗V ′ 1 1 and Φ ∪,⊗ • (j V2 2 ⊗ j V ′ 2 2 ) = j V2⊗V ′ 2 2 ;
notice that these, themselves, suffice to ensure that it covers the identity map on X 1 ∪ f X 2 .
The gluing-dual commutativity conditions, and diffeomorphism Φ ∪, * We have already described the situation regarding the gluing-dual commutativity, first of all the fact, that it is far from being always present. At this moment we concentrate on what it actually means for the gluing to commute with taking duals (once again leaving aside the question when this does happen). Specifically, we say that the gluing-dual commutativity condition holds, if there exists a diffeomorphism
Φ ∪, * : (V 1 ∪f V 2 ) * → V * 2 ∪f * V * 1
that covers the switch map, that is,
(π * 2 ∪ (f * ,f −1 ) π * 1 ) • Φ ∪, * = ϕ X1↔X2 • (π 1 ∪ (f ,f ) π 2 ) * ,
and such that the following are true:
Φ ∪, * • ((j V1 1 ) * ) −1 = j V * 1 2 on (π * 2 ∪ (f * ,f −1 ) π * 1 ) −1 (i X1 2 (X 1 \ Y )), Φ ∪, * • ((j V2 2 ) * ) −1 = j V * 1 2 •f * on (π * 2 ∪ (f * ,f −1 ) π * 1 ) −1 (i X1 2 (Y )), Φ ∪, * • ((j V2 2 ) * ) −1 = j V * 2 1 on (π * 2 ∪ (f * ,f −1 ) π * 1 ) −1 (i X2 1 (X 2 \ f (Y ))).
Gluing and pseudo-metrics
The behavior of pseudo-metrics under gluing depends significantly on whether the gluing-dual commutativity condition is satisfied. More precisely, if we glue together two pseudo-bundles carrying a pseudometric each, then under a certain natural compatibility condition (see below) for these pseudo-metrics, the new pseudo-bundle carries a pseudo-metric as well; but how the latter is constructed depends on the existence of the commutativity diffeomorphism Φ ∪, * .
The compatibility notion for pseudo-metrics Let π 1 : V 1 → X 1 and π 2 : V 2 → X 2 be two finitedimensional diffeological vector pseudo-bundles, endowed with pseudo-metrics g 1 and g 2 respectively, and let (f , f ) be a pair of smooth maps that defines gluing of X 1 to X 2 ; let Y ⊆ X 1 be the domain of definition of f . The pseudo-metrics g 1 and g 2 are said to be compatible with (the gluing along) the pair (f , f ) if for all y ∈ Y and for all v, w ∈ π −1 1 (y) the following is true:
g 1 (y)(v, w) = g 2 (f (y))(f (v),f (w)
).
If f is invertible, this means that g 2 and g 1 are (f −1 ,f * ⊗f * )-compatible as smooth maps X 2 → V * 2 ⊗ V * 2 and X 1 → V * 1 ⊗ V * 1 respectively. As can be expected, 19 the existence of compatible pseudo-metrics on two pseudo-bundles imposes substantial restrictions on their fibres over the domain of gluing. Later in the paper we will make precise statements to this effect.
The induced pseudo-metric in the presence of Φ ∪, * If we assume that the gluing-dual commutativity condition is satisfied, this implies also that f is invertible (with smooth inverse). In this case we can use the map
g 2 ∪ (f −1 ,f * ⊗f * ) g 1 : X 2 ∪ f −1 X 1 → (V * 2 ⊗ V * 2 ) ∪f * ⊗f * (V * 1 ⊗ V * 1 )
to construct a pseudo-metric on V 1 ∪f V 2 by taking the following composition of it with the switch map and the commutativity diffeomorphisms:
g = Φ −1 ∪, * ⊗ Φ −1 ∪, * • Φ ⊗,∪ • (g 2 ∪ (f −1 ,f * ⊗f * ) g 1 ) • ϕ X1↔X2 ,
where ϕ X1↔X2 is the switch map, Φ ∪, * (of which we need the inverse) is the just-seen gluing-dual commutativity diffeomorphism, while Φ ⊗,∪ :
(V * 2 ⊗ V * 2 ) ∪f * ⊗f * (V * 1 ⊗ V * 1 ) → (V * 2 ∪f * V * 1 ) ⊗ (V * 2 ∪f * V * 1 )
is the appropriate version of the tensor product-gluing commutativity diffeomorphism.
Constructing a pseudo-metric on V 1 ∪f V 2 when Φ ∪, * does not exist Although we will mostly deal with the cases where the gluing-dual commutativity condition is present (and so the above definition of the pseudo-metricg on V 1 ∪f V 2 is sufficient), we briefly mention that even if such condition does not hold, the flexibility of diffeology allows for a direct construction of a pseudo-metric on V 1 ∪f V 2 . This construction uses the fact that each fibre of V 1 ∪f V 2 is naturally identified with one of either V 1 or V 2 ; accordingly,g can be defined to coincide with either g 1 or g 2 on each fibre individually. The surprising fact is thatg coming from this construction is still diffeologically smooth across the fibres; see [11] for details.
The pseudo-bundle of smooth linear maps
The last more-or-less standard construction that we need is that of pseudo-bundle of smooth linear maps. Let π 1 : V 1 → X and π 2 : V 2 → X be two finite-dimensional diffeological vector pseudobundles with the same space X. For every x ∈ X the space L ∞ (π −1 1 (x), π −1 2 (x)) of smooth linear maps
π −1 1 (x) → π −1 2 (x) is a (finite-dimensional) diffeological vector space for the functional diffeology. The union L(V 1 , V 2 ) = ∪ x∈X L ∞ (π −1 1 (x), π −1 2 (x)
) of all these spaces has the obvious projection (denoted π L ) to X, and the pre-image of each point under this projection has vector space structure. It becomes a diffeological vector pseudo-bundle when endowed with the pseudo-bundle functional diffeology, that is defined as the finest diffeology containing all maps p : U → L(V 1 , V 2 ), with U ⊆ R m an arbitrary domain, that possess the following property: for every plot q : R m ′ ⊇ U ′ → V 1 of V 1 the corresponding evaluation map (u, u ′ ) → p(u)(q(u ′ )) ∈ V 2 defined on Y ′ = {(u, u ′ ) | π L (p(u)) = π 1 (q(u ′ ))} ⊂ U × U ′ is smooth for the subset diffeology of Y ′ . This is the type of object where the Clifford actions live.
The pseudo-bundles of Clifford algebras and Clifford modules
For diffeological pseudo-bundles, these have already been described in the abstract setting (see [8]). We briefly summarize the main points that appear therein, noting that the main conclusions do not differ from the usual case, or are as expected anyhow.
1.5.1 The pseudo-bundle cl(V, g)
Let π : V → X be a finite-dimensional diffeological pseudo-bundle endowed with a pseudo-metric g. The construction of the corresponding pseudo-bundle cl(V, g) of Clifford algebras is the immediate one, since all the operations involved have already been described. Specifically, the pseudo-bundle of Clifford algebras π Cl : cl(V, g) → X is given by
cl(V, g) := ∪ x∈X cl(π −1 (x), g(x))
and is endowed with the quotient diffeology coming from pseudo-bundle of tensor algebras π T (V ) : T (V ) → X. The latter pseudo-bundle has total space given by
T (V ) := ∪ x∈X T (π −1 (x)), where T (π −1 (x)) := r (π −1 (x)
) ⊗r is the usual tensor algebra of the diffeological vector space π −1 (x) (in particular, it is endowed with the vector space direct sum diffeology relative to the tensor product diffeology on each factor).
Remark 1.1. We will not make much use of the algebra structure on this pseudo-bundle, and so will actually consider all the direct sums involved to be finite (limited by the maximum of the dimensions of fibres of V in question -this includes the assumption that such maximum exists), thus considering, instead of the whole T (V ) its finite-dimensional sub-bundle T n (V ), with fibre at x the space T n (π −1 (x)) consisting of all tensors in T (π −1 (x)) of degree at most n. These fibres are not algebras, but of course each of them is a vector subspace of the corresponding fibre of T (V ).
Recall that the subset diffeology on each fibre of T (V ) is that of the tensor algebra 20 of the individual fibre π −1 (x). In each such fibre we choose the subspace W x that is the kernel of the universal map T (π −1 (x)) → Cℓ(π −1 (x), g(x)). Then, as is generally the case, W = ∪ x∈X W x ⊂ T (V ) endowed with the subset diffeology relative to this inclusion is a sub-bundle of T (V ). The fibre of the corresponding quotient pseudo-bundle at any given point x ∈ X is cl(π −1 (x), g(x)), and the quotient diffeology on the fibre is that of the Clifford algebra over the vector space π −1 (x). This is exactly the diffeology that we endow Cℓ(V, g) with.
1.5.2 The pseudo-bundle Cℓ(V 1 ∪f V 2 ,g) as the result of a gluing
The main result, that we immediately state and that appears in [8], is the following one. Theorem 1.2. Let π 1 : V 1 → X 1 and π 2 : V 2 → X 2 be two diffeological vector pseudo-bundles, let (f , f ) be two maps defining a gluing between these two pseudo-bundles, both of which are diffeomorphisms, and let g 1 and g 2 be pseudo-metrics on V 1 and V 2 respectively, compatible with this gluing. Letg be the pseudo-metric on V 1 ∪f V 2 induced by g 1 and g 2 . Then there exists a mapF Cℓ defining a gluing of the pseudo-bundles Cℓ(V 1 , g 1 ) and Cℓ(V 2 , g 2 ), and a diffeomorphism Φ Cℓ between the pseudo-bundles
Cℓ(V 1 , g 1 ) ∪F Cℓ Cℓ(V 2 , g 2 ) and Cℓ(V 1 ∪f V 2 ,g) covering the identity on X 1 ∪ f X 2 .
Let us briefly describe the mapsF Cℓ and Φ Cℓ . The construction ofF Cℓ is the immediately obvious one. It is defined on each fibre over a point y ∈ Y as the map Cℓ(π −1 1 (y), g 1 | π −1 1 (y) ) → Cℓ(π −1 2 (f (y)), g 2 | π −1 2 (f (y)) ) induced byf via the universal property of Clifford algebras. In practice, this means that on each fibrẽ F Cℓ is linear and multiplicative (with respect to the tensor product), so if v 1 ⊗ . . . ⊗ v k is a representative of an equivalence class in Cℓ(π −1 1 (y), g 1 | π −1 1 (y) ) (viewed as the appropriate quotient of T (π −1 1 (y))) then by this definitionF
Cℓ (v 1 ⊗ . . . ⊗ v k ) =F Cl (v 1 ) ⊗ . . . ⊗F Cℓ (v k ) =f (v 1 ) ⊗ . . . ⊗f (v k ).
20 Or of its appropriate vector subspace, see the remark above.
That this is well-defined as a map Cℓ(π −1 1 (y), g 1 | π −1 1 (y) ) → Cℓ(π −1 2 (f (y)), g 2 | π −1 2 (f (y)) ) follows from the compatibility of pseudo-metrics g 1 and
g 2 . Indeed, if v, w ∈ π −1 1 (y) theñ F Cℓ (v ⊗ w + w ⊗ v + 2g 1 (y)(v, w)) =f (v) ⊗f (w) +f (w) ⊗f (v) + 2g 1 (y)(v, w)
by the above formula, and 2g 1 (y)(v, w) = 2g 2 (f (y))(f (v),f (w)) by the compatibility. Thus,F Cℓ preserves the defining relation for Clifford algebras, so indeed it is well-defined (on each fibre; hence on the whole pseudo-bundle). The diffeomorphism Φ Cℓ , which we specify to be Cℓ(
V 1 , g 1 ) ∪F Cℓ Cℓ(V 2 , g 2 ) → Cℓ(V 1 ∪f V 2 ,g)
, is then the natural identification; namely, by definitions of the gluing operation and that of the induced pseudo-metricg, over a point of form
x = i X1 1 (x 1 ) both the fibre of Cℓ(V 1 , g 1 ) ∪F Cℓ Cℓ(V 2 , g 2 ) and that of Cℓ(V 1 ∪f V 2 ,g) are naturally identified with Cℓ(π −1 1 (x), g 1 | π −1 1 (x) ), while over any point of form x = i X2 2 (x 2 ) they are identified with Cℓ(π −1 2 (x), g 2 | π −1 2 (x) ).
Gluing of pseudo-bundles of Clifford modules
A statement similar to that of the Theorem cited in the previous section can also be obtained for given Clifford modules over the algebras Cℓ(V 1 , g 1 ) and Cℓ(V 2 , g 2 ). This requires some additional assumptions on these modules, and the appropriate notion of the compatibility of the actions.
The two Clifford modules Let π 1 :
V 1 → X 1 , π 2 : V 2 → X 2 , (f , f ), g 1 , and g 2 be as in Theorem 1.2.
Recall that this yields the following pseudo-bundles, π Cℓ
1 : Cℓ(V 1 , g 1 ) → X 1 , π Cℓ 2 : Cℓ(V 2 , g 2 ) → X 2 , and π Cℓ 1 ∪ (F Cℓ ,f ) π Cℓ 2 : Cℓ(V 1 , g 1 ) ∪F Cℓ Cℓ(V 2 , g 2 ) = Cℓ(V 1 ∪f V 2 ,g) → X 1 ∪ f X 2
. Now suppose that we are given two pseudo-bundles of Clifford modules, χ 1 : E 1 → X 1 and χ 2 :
E 2 → X 2 (over Cℓ(V 1 , g 1 ) and Cℓ(V 2 , g 2 ) respectively), that is, there is a smooth pseudo-bundle map c i : cl(V i , g i ) → L(E i , E i )
that covers the identity on the bases. Suppose further that there is a smooth fibrewise linear mapf ′ :
χ −1 1 (Y ) → χ −1 2 (f (Y )
) that covers f . We describe the pseudo-bundle E 1 ∪f ′ E 2 as a Clifford module over Cℓ(V 1 ∪f V 2 ,g), with respect to an action induced by c 1 and c 2 .
Compatibility of c 1 and c 2 Similar to how it occurs for smooth maps, there is always an action of Cℓ(V 1 ∪f V 2 ,g) on E 1 ∪f ′ E 2 induced by c 1 and c 2 , it is smooth on each fibre but in general, it might not be smooth across the fibres. For it to be so, we need a notion of compatibility for Clifford actions, which is as follows. Definition 1.3. The actions c 1 and c 2 are compatible (with respect toF Cℓ andf ′ ) if for all y ∈ Y , for all v ∈ (π Cℓ 1 ) −1 (y), and for all e 1 ∈ χ −1 1 (y) we havẽ
f ′ (c 1 (v)(e 1 )) = c 2 (F Cℓ (v))(f ′ (e 1 )).
We note that the compatibility of c 1 and c 2 as it has been just defined, does not automatically translate into their (F Cℓ ,f ′ ) compatibility as smooth maps in Cℓ [8] for a discussion on this.
(V i , g i ) → L(E i , E i ); see
The induced action Assuming now that the two given actions c 1 and c 2 are compatible in the sense just stated, we can define an induced action on E 1 ∪f ′ E 2 , that is, a smooth homomorphism
c : Cℓ(V 1 ∪f V 2 ,g) → L(E 1 ∪f ′ E 2 , E 1 ∪f ′ E 2 ).
Using the already-mentioned identification, via the diffeomorphism Φ Cℓ , of Cℓ(V 1 ∪f V 2 ,g) with Cℓ(V 1 , g 1 )∪F Cℓ Cℓ(V 2 , g 2 ), the action c can be described by defining first
c ′ (v)(e) = j E1 1 c 1 ((j Cℓ(V1,g1) 1 ) −1 (v))((j E1 1 ) −1 (e)) if v ∈ Im(j Cℓ(V1,g1) 1 ) ⇒ e ∈ Im(j E1 1 ), j E2 2 c 2 ((j cl(V2,g2) 2 ) −1 (v))((j E2 2 ) −1 (e)) if v ∈ Im(j cl(V2,g2) 2 ) ⇒ e ∈ Im(j E2 2 ).
Since the images of the inductions j Cℓ(V1,g1) 1 and j Cℓ(V2,g2) 2 are disjoint and cover Cℓ(V 1 , g 1 )∪F Cℓ Cℓ(V 2 , g 2 ), and those of j E1 1 and j E2 2 cover E 1 ∪f ′ E 2 (and are disjoint as well), this is a well-defined fibrewise action of the former on the latter. From the formal point of view, we must also pre-compose it with the inverse of Φ Cℓ , to obtain an action
c = c ′ • (Φ Cℓ ) −1 : Cℓ(V 1 ∪f V 2 ,g) → L(E 1 ∪f ′ E 2 , E 1 ∪f ′ E 2 ).
Then, the following is true (see [8]).
Theorem 1.4. The action c is smooth as a map Cℓ(V 1 ∪f V 2 ,g) → L(E 1 ∪f ′ E 2 , E 1 ∪f ′ E 2 ).
The induced pseudo-metrics on dual pseudo-bundles
Let π : V → X be a locally trivial finite-dimensional diffeological vector pseudo-bundle endowed with a pseudo-metric g; then its dual pseudo-bundle π * : V * → X admits an induced pseudo-metric g * (see [11]; we also recall the definition below). In this section we consider gluings of two pseudo-bundles endowed with compatible pseudo-metrics, and the corresponding dual constructions; starting from indicating which restrictions are imposed in the initial pseudo-bundles by the existence of compatible pseudo-metrics on them, we proceed to discuss when the induced pseudo-metrics on the dual pseudo-bundles are compatible in their turn, and finally, how it relates to the gluing-dual commutativity condition.
The induced pseudo-metric on V *
Let π : V → X be a locally trivial finite-dimensional diffeological vector pseudo-bundle, and let g be a pseudo-metric on it. Under the assumption of local triviality of V , 21 the dual pseudo-bundle V * carries an induced pseudo-metric g * , which is obtained by what can be considered a diffeological counterpart of the usual natural pairing.
Specifically, consider the pseudo-bundle map Φ :
V → V * defined by Φ(v) = g(π(v))(v, ·) for all v ∈ V.
It is not hard to show (see [9] for the case of a single diffeological vector space, and then [11] for the case of pseudo-bundles) that Φ is surjective, smooth, and linear on each fibre. The induced pseudo-metric g * is given by the following equality:
g * (x)(Φ(v), Φ(w)) := g(x)(v, w) for all x ∈ X and for all v, w ∈ V such that π(v) = π(w) = x.
This is well-defined, because whenever Φ(v) = Φ(v ′ ) (which obviously can occur only for v, v ′ belonging to the same fibre), the vectors v and v ′ differ by an element of the isotropic subspace of the fibre to which they belong. Furthermore, since each fibre of the dual pseudo-bundle (in the finite-dimensional case) carries the standard diffeology (see [9]), g * (x) is always a scalar product. Notice that without the requirement of local triviality, we cannot guarantee that g * is indeed a pseudo-metric, and more precisely, that it is smooth (the map Φ always has a right inverse, which a priori may not be smooth).
Lemma 2.1. Let π : V → X be a finite-dimensional diffeological vector pseudo-bundle endowed with a pseudo-metric g, let π * : V * → X be the dual pseudo-bundle, and let g * be the induced pseudo-metric. Then for all x ∈ X the symmetric bilinear form g * (x) on (π * ) −1 (x) is non-degenerate.
Existence of compatible pseudo-metrics: the case of a diffeological vector space
Before treating various issues regarding the induced pseudo-metrics, it makes sense to consider in more detail what the compatibility of two pseudo-metrics means. We do so starting with the case of just diffeological vector spaces (we consider the duals of vector spaces in the section that immediately follows, and pseudo-bundles in the one after that). Let V and W be finite-dimensional diffeological vector spaces, let g V be a pseudo-metric on V , and let g W be a pseudo-metric on W . We assume that we are given a smooth linear map f : V → W , with respect to which g V and g W are compatible,
g V (v 1 , v 2 ) = g W (f (v 1 ), f (v 2 )
). We show that, quite similarly to usual vector spaces and scalar products, there are pairs of diffeological ones such that no pair of pseudo-metrics is compatible with respect to any smooth linear f . The similarity that we are referring to has to do with the fact that the compatibility of two pseudo-metrics with respect to f essentially amounts to f being a diffeological analogue of an isometry onto a subspace. As is well-known, between two usual vector spaces such isometry may not exist (it is necessary that the dimension of the domain space must be less or equal to that of the target space), and something similar happens for diffeological vector spaces; and then further conditions are added in terms of their diffeological structures.
The characteristic subspaces of V and W
Assuming that two given pseudo-metrics g V and g W on diffeological vector spaces V and W respectively are compatible with a given f : V → W has several implications for the diffeological structures of V and W ; describing these requires the following notion.
Given a pseudo-metric g on a finite-dimensional diffeological vector space V , the subspace V 0 V generated by all the eigenvectors of g relative to the non-zero eigenvalues has subset diffeology that is standard; and among all subspaces of V whose diffeology is standard, it has the maximal dimension, which is equal to dim(V * ). In general, V contains more than one subspace of dimension dim(V * ) whose diffeology is standard. But the subspace V 0 is the only one that also splits off as a smooth direct summand. 22 Thus, V 0 does not actually depend on the choice of a pseudo-metric and is an invariant of the space itself (see [9]). We call this subspace the characteristic subspace of V .
Let us now return to the two diffeological vector spaces V and W above. Let V 0 and W 0 be their characteristic subspaces, and let V 1 V and W 1 W be the isotropic subspaces relative to g V and g W respectively, such that V = V 0 ⊕ V 1 and W = W 0 ⊕ W 1 with each of these decompositions being smooth. We also recall [9] that V 0 not only has the same dimension as V * , but for any fixed pseudo-metric is diffeomorphic to it, via (the restriction to V 0 of) the map Φ
V : v → g V (v, ·); likewise, W 0 is diffeomorphic to W * via Φ W : w → g W (w, ·).
The necessary conditions
Let us now assume that the given g V , g W , and f satisfy the compatibility condition. The corollaries of this assumption can be described in terms of the characteristic subspaces of V and W , and therefore in terms of their diffeological duals.
The kernel of f The first corollary is quite trivial, and starts with a simple linear algebra argument.
Let v ∈ V belong to the kernel of f . Then by the compatibility assumption for g V and g W we have
g V (v, v ′ ) = g W (0, f (v ′ )) = 0 for any v ′ ∈ V.
Thus, the kernel of f is contained in the maximal isotropic subspace V 1 , therefore the restriction of f to V 0 is a bijection with its image. This restriction is of course a smooth map, and since V 0 splits off as a smooth direct summand, it is an induction (that is, a diffeomorphism with its image). Finally, f itself is a diffeomorphism of V 0 ⊕ (V 1 /Ker(f )) with its image in W . In particular, we have the following. Notice that in the standard case V and W would be vector spaces, g V and g W scalar products on them, and f an isometry of V with its image in W . In particular, f would be injective; Lemma 2.2 is the diffeological counterpart of that.
The dimensions of V and W , and those of V * and W * Continuing the analogy with the standard case, we observe that the standard inequality dim(V ) dim(W ) does not have to hold in the diffeological setting. What instead is true, is the corresponding inequality for the dimensions of their dual spaces, which follows from the lemma below. Lemma 2.3. Let V and W be finite-dimensional diffeological vector spaces, let f : V → W be a smooth linear map, and suppose that V and W carry pseudo-metrics g V and g W respectively, compatible with respect to f . Then the subset diffeology of f (V 0 ) is the standard one.
Proof. Let e 1 , . . . , e n be a g V -orthonormal basis of V 0 ; then by compatibility f (e 1 ), . . . , f (e n ) is a g Worthonormal basis of f (V 0 ), which can be completed to a basis of eigenvectors of g W . It suffices to show that the projection of W on the line generated by each f (e i ) is a usual smooth function. Since this projection is given by w → g W (f (e i ), w), the claim follows from the smoothness of g W . Now, the fact that f (V 0 ) carries the standard diffeology, does not automatically imply that it is contained in W 0 -there are standard subspaces that are not (we will however show later on that this inclusion does hold for f (V 0 )). However, f (V 0 ) is still a standard subspace of W , and since W 0 has maximal dimension among such subspaces, we have
dim(V * ) = dim(V 0 ) = dim(f (V 0 )) dim(W 0 ) = dim(W * ).
Therefore we have the following statement.
Proposition 2.4. Let V and W be finite-dimensional diffeological vector spaces. If there exist a smooth linear map f : V → W and pseudo-metrics g V and g W on V and W respectively, compatible with respect to f , then
dim(V * ) dim(W * ).
In other words, if dim(V * ) > dim(W * ), then no two pseudo-metrics on V and W are compatible, whatever the map f (which obviously mimics the standard situation: there is no isometry from the space of a bigger dimension to one of smaller dimension). 23
The subspace f (V 0 ) in W We now show that the a priori case when f (V 0 ) is not contained in W 0 is actually impossible, that is, if f : V → W is such that V and W admit compatible pseudo-metrics then f sends the characteristic subspace of V to the characteristic subspace of W . Lemma 2.5. Let V and W be finite-dimensional diffeological vector spaces, let f : V → W be a smooth linear map, and suppose that V and W admit compatible pseudo-metrics g V and g W respectively. Then f (V 0 ) splits off smoothly in W .
Proof. Let e 1 , . . . , e n be a g V -orthonormal basis of V 0 . Then by assumption f (e 1 ), . . . , f (e n ) is a g Worthonormal basis of f (V 0 ). This can be completed to an orthogonal basis of W composed of eigenvectors of g W ; denote by u 1 , . . . , u k the elements added, ordered in such a way that the eigenvectors corresponding to the zero eigenvalue are the last m vectors. Let us show that the usual direct sum decomposition
W = f (V 0 ) ⊕ Span(u 1 , . . . , u k ) is a smooth one.
Let p : U → W be a plot of W , and let p ′ be its composition with the projection (associated to the direct sum decomposition just mentioned) of W to f (V 0 ). It suffices to show that p ′ is a plot of f (V 0 ). Notice that by the choice of the basis f (e 1 ), . . . , f (e n ), u 1 , . . . , u k of W (more precisely, by the g W -orthogonality of said basis) we have
p ′ (u) = g W (f (e 1 ), p(u))f (e 1 ) + . . . + g W (f (e n ), p(u))f (e n ),
where each coefficient g W (f (e i ), p(u)) is an ordinary smooth function U → R by the smoothness of the pseudo-metric g W . This means precisely that p ′ is a plot of f (V 0 ), whence the claim.
From the lemma just proven, we can now easily draw the following conclusion. Proof. The subspace W 0 is the only subspace of dimension equal to that of W * that has standard diffeology and splits off smoothly. Since f (V 0 )⊕W ′ 0 has all the same properties, we obtain that f
(V 0 )⊕W ′ 0 = W 0 .
The summary of necessary conditions We collect the conclusions of this section in the following statement.
Theorem 2.7. Let V and W be two finite-dimensional diffeological vector spaces, and let f : V → W be a smooth linear map. If there exist pseudo-metrics g V and g W on V and W respectively that are compatible with respect to f then the following are true:
1. dim(V * ) dim(W * ); 2. Ker(f ) ∩ V 0 = {0}, where V 0 is the characteristic subspace of V ;
3. The subset diffeology on f (V 0 ) relative to its inclusion into W is standard;
4. f (V 0 ) splits off smoothly in W .
Sufficient conditions
Suppose now that V and W are such that the just-mentioned necessary condition is satisfied, and let f :
V → W be a smooth linear map such that Ker(f ) ∩ V 0 = {0} (where V 0 is the characteristic subspace of V ). By definition of a pseudo-metric, if V = V 0 ⊕ V 1 is a smooth decomposition of V 24 then g V is
defined by its restriction to V 0 (which is a scalar product) and is extended by zero elsewhere. The same is true of g W and the corresponding smooth decomposition W 0 ⊕ W 1 . In this way we obtain the following.
g V (v i , v j ) = δ i,j for i, j = 1, . . . , k and g V (v i , v k+j ) = 0 for i = 1, . . . , n, j = 1, . . . , n − k,
and extend by bilinearity and symmetricity. To define g W , then, consider f (v 1 ), . . . , f (v k ) ∈ W 0 ; notice that they are linearly independent by the assumption on Ker(f ). Add first u 1 , . . . , u l ∈ W 0 to obtain the basis f (v 1 ), . . . , f (v k ), u 1 , . . . , u l of W 0 . Finally, choose a basis w 1 , . . . , w m of W 1 to obtain the basis f (v 1 ), . . . , f (v k ), u 1 , . . . , u l , w 1 , . . . , w m of the whole W . It then suffices to define g W to be
g W (f (v i ), f (v j )) = δ i,j , g W (u i , u j ) = δ i,j , g W (f (v i ), u j ) = 0, g W (f (v i ), w p ) = 0, g W (u i , w p ) = 0
and extend by bilinearity and symmetry. The bilinear maps g V and g W thus obtained are smooth, because each of the characteristic subspaces V 0 and W 0 splits off as a smooth direct summand, and by construction g V and g W are zero maps outside of V 0 and W 0 respectively. Finally, that they are pseudo-metrics and are compatible with each other is immediate from their definitions, whence the conclusion.
We are ready to establish the final criterion of the existence of compatible pseudo-metrics on a pair of diffeological vector spaces, that we state in the following form. Proof. The fact that these two conditions are necessary follows from Lemma 2.2 and Lemma 2.5, so let us show that they are sufficient. Let V = V 0 ⊕V 1 be a smooth decomposition, let g V be any pseudo-metric on V , and let W = f (V 0 ) ⊕ W ′ be a smooth decomposition that exists by assumption. Notice that, since W ′ is just another instance of a finite-dimensional diffeological space, it has its own smooth decomposition of form W ′ 0 ⊕ W 1 , where W ′ 0 is standard and W 1 has trivial diffeological dual; so the whole of W smoothly decomposes as
W = (f (V 0 ) ⊕ W ′ 0 ) ⊕ W 1 . Notice that this implies that W * = (f (V 0 ) ⊕ W ′ 0 )
* (by the smoothness of the decomposition), in particular, they have the same dimension.
Let us define a pseudo-metric g W , by setting it to coincide with g V (in the obvious sense) on f (V 0 ), choosing any scalar product for its restriction on W ′ 0 , while requiring W ′ 0 to be orthogonal to f (V 0 ), and finally setting W 1 to be an isotropic subspace. That this is indeed a pseudo-metric follows from the considerations above, so it remains to show that g W is indeed compatible with g V . This essentially follows from the construction, more precisely, from the fact that f (V 0 ) is orthogonal to any its direct
complement. Indeed, if v ′ = v ′ 0 + v ′ 1 and v ′′ = v ′′ 0 + v ′′ 1 are any two elements of V , then g V (v ′ , v ′′ ) = g V (v ′ 0 , v ′′ 0 ) = g W (f (v ′ 0 ), f (v ′′ 0 )) = g W (f (v ′ 0 ) + f (v ′ 1 ), f (v ′′ 0 ) + f (v ′′ 1 )) = g W (f (v ′ ), f (v ′′ )),
where the third equality is by the orthogonality just mentioned. This means that g V and g W are compatible with f , and the proof is finished.
Compatibility of the dual pseudo-metrics: diffeological vector spaces
We now consider the induced pseudo-metrics on the duals of diffeological vector spaces; the main question that we aim to answer is, under what conditions the pair of pseudo-metrics dual to (induced by) two compatible ones is in turn compatible.
The induced pseudo-metric g * on V * : definition Recall ( [9]) that, given a finite-dimensional diffeological vector space V endowed with a pseudo-metric g, the diffeological dual of V carries the induced pseudo-metric g * (actually, a scalar product, since the diffeological dual of any finite-dimensional diffeological vector space is standard) defined by
g * (v * 1 , v * 2 ) := g(v 1 , v 2 ),
where v i ∈ V is any element such that v * i (·) = g(v i , ·) for i = 1, 2. That this is well-defined, i.e., the result does not depend on the choice of v i (as long as g(v i , ·) remains the same), and that v * i always admits such a form, was shown in [9].
The compatibility for the induced pseudo-metrics Let g V and g W be pseudo-metrics on V and W respectively, compatible with respect to f . Let w * 1 , w * 2 ∈ W * ; then there exist w 1 , w 2 ∈ W , defined up to the cosets of the isotropic subspace of g W , such that w * i (·) = g W (w i , ·) for i = 1, 2, by definition of the dual pseudo-metric
g * W (w * 1 , w * 2 ) = g W (w 1 , w 2 ), and finally, f * (w * i )(·) = w * i (f (·)) = g W (w i , f (·)
). The compatibility condition that we need to check is the following one:
g * W (w * 1 , w * 2 ) = g * V (f * (w * 1 ), f * (w * 2 )).
Now, in order to calculate the right-hand term in this expression, we must choose v 1 and v 2 , again defined up to their cosets with respect to the isotropic subspace of g V , such that
g V (v i , v ′ ) = f * (w * i )(v ′ ) = g W (w i , f (v ′ ))
, for all elements of v ′ ∈ V and for w * 1 , w * 2 ∈ W * . The term on the right then becomes
g * V (f * (w * 1 ), f * (w * 2 )) = g V (v 1 , v 2 ).
The dual pseudo-metrics and compatibility Let us now consider the pseudo-metrics on V * and W * dual to a pair of compatible pseudo-metrics on V and W . We observe right away that in general, the induced pseudo-metrics are not compatible. This follows from Lemma 2.2, as well as from the standard theory, all diffeological constructions being in fact extensions of the standard ones.
Example 2.10. Let V be the standard R n , with the canonical basis denoted by e 1 , . . . , e n , and let W be the standard R n+k , with the canonical basis denoted by u 1 , . . . , u n , u n+1 , . . . , u n+k . Let f : V → W be the embedding of V via the identification of V with the subspace generated by u 1 , . . . , u n , given by e i → u i for i = 1, . . . , n. Let g V be any scalar product on R n ; this trivially induces a scalar product on f (V ) = Span(u 1 , . . . , u n ) W 0 , and let g W be any extension of it to a scalar product on the whole W . Let us consider the dual map on the dual the standard complement of the subspace Span(u 1 , . . . , u n ), that is, on the dual of Span(u n+1 , . . . , u n+k ). This dual is the usual dual, so it is Span(u n+1 , . . . , u n+k ).
Let v be any element of V ; since f (v) ∈ Span(u 1 , . . . , u n ), we have
f * (u n+i )(v) = u n+i (f (v)) = 0,
so in the end we obtain that Ker(f * ) = Span(u n+1 , . . . , u n+k ).
Finally, let us consider the compatibility condition. We observe that g * W (u n+i , u n+i ) = g W (u n+i , u n+i ) > 0, since g W is a scalar product, while, of course,
g * V (f * (u n+i ), f * (u n+i )) = 0.
Quite evidently, the compatibility condition cannot be satisfied (unless k = 0).
Sufficient conditions for compatibility of the induced pseudo-metrics It can be inferred from the above example that the induced pseudo-metrics on the duals of standard spaces are compatible only if the spaces have the same dimension (which is not surprising, since in this case the notion of the induced pseudo-metric itself coincides with the standard one). This can be generalized to the following statement. Proof. The only if part of the statement, illustrated by the example above, follows from standard reasoning. Indeed, g * W and g * V are usual scalar products on standard spaces W * and V * respectively, and their compatibility means that f * is a usual isometry, whose existence implies that W * and V * have the same dimension, and being standard spaces, this means that they are diffeomorphic as diffeological vector spaces.
Let us prove the if part, namely, that g * W and g * V are compatible under the assumptions of the proposition. Let e 1 , . . . , e n be a g V -orthogonal basis of V 0 . Since g V and g W are compatible, f (e 1 ), . . . , f (e n ) is a g W -orthogonal basis of f (V 0 ). Now, (e 1 ) * , . . . , (e n ) * (recall here that v * for v ∈ V stands for the map v * (·) = g V (v, ·)) form a basis of V * (which is also orthogonal with respect to the induced pseudo-metric g * V )
, while (f (e 1 )) * , . . . , (f (e n )) * are linearly independent elements of W * . Since V * and W * have the same dimension, (f (e 1 )) * , . . . , (f (e n )) * actually forms a basis of W * ; and so, g * W is entirely determined by its values on pairs (f (e i )) * , (f (e j )) * , and moreover, we have
g * W ((f (e i )) * , (f (e j )) * ) = g W (f (e i ), f (e j )) = g V (e i , e j ) = g * V (e * i , e * j ). Finally, f * ((f (e i )) * )(v) = (f (e i )) * (f (v)) = g W (f (e i ), f (v)) = g V (e i , v) = e * i (v) for all v ∈ V . There- fore g * V (e * i , e * j ) = g * V (f * ((f (e i )) * ), f * ((f (e j )) * )) = g * W ((f (e i ))
* , (f (e j )) * ), at which point the compatibility, with respect to f * , of the pseudo-metrics g * W and g * V follows from (f (e 1 )) * , . . . , (f (e n )) * being a basis of W * .
Compatibility of the dual pseudo-metrics: diffeological vector pseudobundles
Let us now consider the following question: if two given pseudo-metrics g 1 and g 2 are compatible with respect to the gluing along a given pair of maps (f,f ), when is it true that g * 2 and g * 1 are compatible with the gluing defined by (f −1 ,f * )? (Obviously, we assume here that f is invertible).
The compatibility condition for g * 2 and g * 1 Let π 1 : V 1 → X 1 and π 2 : V 2 → X 2 be locally trivial finite-dimensional diffeological vector pseudo-bundles, let (f , f ) be a gluing of the former to the latter such that f is a diffeomorphism with its image, and let g 1 and g 2 be pseudo-metrics on V 1 and V 2 respectively, compatible with respect to the given gluing. The latter induces a well-defined gluing, along the mapsf * and f −1 , of the dual pseudo-bundle π * 2 : V * 2 → X 2 to the pseudo-bundle π * 1 :
V * 1 → X 1 , the result of which is the pseudo-bundle π * 2 ∪ (f * ,f −1 ) π * 1 : V * 2 ∪f * V * 1 → X 2 ∪ f −1 X 1 , while g 2 and g 1 induce pseudo-metrics g * 2 : X 2 → (V * 2 ) * ⊗ (V * 2 ) * and g * 1 : X 1 → (V * 1 ) * ⊗ (V * 1 )
* on the dual pseudo-bundles. They satisfy the usual compatibility condition if
g * 1 (f −1 (y ′ ))(f * (v * ),f * (w * )) = g * 2 (y ′ )(v * , w * ) for all y ′ ∈ Y ′ = f (Y ) and for all v * , w * ∈ (π −1 2 (y ′ )) * .
The necessary condition The compatibility between g * 2 and g * 1 implies in particular that for all y ∈ Y the pseudo-metrics g * 2 (f (y)) and g * 1 (y) are compatible with the smooth linear mapf * | π −1
1 (y)
between diffeological vector spaces (π −1 2 (f (y))) * and (π −1 1 (y)) * . Thus, Theorem 2.11 implies thatf * is a diffeomorphism on each fibre. Proposition 2.12. Let π 1 : V 1 → X 1 and π 2 : V 2 → X 2 be diffeological vector pseudo-bundles, locally trivial and with finite-dimensional fibres, and let (f , f ) be an invertible gluing between them. Suppose that g 1 and g 2 are two pseudo-metrics on these pseudo-bundles compatible with the gluing along (f , f ). If the induced pseudo-metrics g * 2 and g * 1 are compatible with the gluing along (f * , f −1 ) then the restriction of f * on each fibre in its domain of definition is a diffeomorphism.
Criterion of compatibility The statement that follows shows that, for the two induced pseudometrics to be compatible with the induced gluing, the map dual to the gluing mapf must satisfy a rather stringent condition (although an expected one).
Theorem 2.13. Let π 1 : V 1 → X 1 and π 2 : V 2 → X 2 be two diffeological vector pseudo-bundles, locally trivial and with finite-dimensional fibres, let (f , f ) be a gluing between them, and let g 1 and g 2 be compatible pseudo-metrics on V 1 and V 2 respectively. Then the induced pseudo-metrics g * 2 and g * 1 on the corresponding dual pseudo-bundles are compatible if and only iff * is a pseudo-bundle diffeomorphism of its domain with its image.
Notice that diffeological vector spaces may have diffeomorphic duals without being diffeomorphic themselves, and the same is true for diffeological vector pseudo-bundles.
Proof. By assumption, g 1 and g 2 are compatible with the gluing given by the pair (f , f ), that is
g 2 (f (y))(f (v),f (w)) = g 1 (y)(v, w) for all y ∈ Y and for all v, w ∈ π −1 1 (y).
Suppose first thatf * is a diffeomorphism with its image. Then, first of all, by the definition of g * 2 we have
g * 2 (y ′ )(v * , w * ) = g 2 (y ′ )(v, w) for all y ′ ∈ f (Y ), for all v * , w * ∈ (π −1
2 (y ′ )) * , and for v, w ∈ (π −1 2 (y ′ )) 0 . Notice that v and w are uniquely defined by the latter condition; and they are such that v * (·) = g 2 (y ′ )(v, ·) and w * (·) = g 2 (y ′ )(w, ·). Notice also (we will need this immediately below) that this means
f * (v * )(·) = v * (f (·)) = g 2 (y ′ )(v,f (·)) = g 2 (y ′ )(f (v 1 ),f (·)) = g 1 (f −1 (y ′ ))(v 1 , ·),
where v 1 ∈ (π −1 1 (f −1 (y ′ ))) 0 is such that v =f (v 1 ); such an element exists and is uniquely defined becausẽ f * being a diffeomorphism is equivalent tof being a diffeomorphism between each pair of subspaces (π −1 1 (f −1 (y ′ ))) 0 and (π −1 2 (y ′ )) 0 . Similarly, we havẽ
f * (w * )(·) = g 2 (y ′ )(f (w 1 ),f (·)) = g 1 (f −1 )(w 1 , ·)
for w 1 ∈ (π −1 1 (f −1 (y ′ ))) 0 such that w =f (w 1 ). It remains now to consider the left-hand part of the compatibility condition. We have:
g * 1 (f −1 (y ′ ))(f * (v * ),f * (w)) = g 1 (f −1 (y ′ ))(v 1 , w 1 ) = g 2 (y ′ )(f (v 1 ),f (w 1 )) = g 2 (y ′ )(v, w) = g * 2 (y ′ )(v * , w * ),
as wanted. Let us prove the vice versa of the statement, that is, let us assume that g * 2 and g * 1 are compatible, and let us show thatf * is a diffeomorphism. We notice first of all that it follows from the considerations made for individual vector spaces thatf * is bijective and, as is the case for any dual map, it is smooth. Finally, the smoothness of its inverse follows from the fact that V * 1 and V * 2 are locally trivial and have standard fibres.
3 Compatibility of pseudo-metrics and the gluing-dual commutativity conditions
In this section we consider which correlations there might be between the notion of compatible pseudometrics on two given pseudo-bundles (with a specified gluing), and the gluing-dual commutativity conditions. We start by taking our two usual pseudo-bundles, π 1 : V 1 → X 1 and π 2 : V 2 → X 2 , and a gluing along (f , f ) between them. Assuming that these pseudo-bundles admit pseudo-metrics g 1 and g 2 compatible with the gluing, we consider the following questions: 1) what are the implications of the existence of compatible pseudo-metrics for the pseudo-bundles themselves? 2) if the gluing-dual commutativity condition holds for V 1 , V 2 , and (f , f ), does it necessarily hold for V * 2 , V * 1 , and (f * , f −1 )? 3) if g * 2 and g * 1 exist, under what conditions are they compatible with (f * , f −1 ), in particular, is their compatibility equivalent to the gluing-dual commutativity? 4) does taking the dual pseudo-metric commute (in the notation to be introduced, this will be the equalityg * = g * ) with the gluing of pseudo-metrics (as defined in [11])? We consider these questions in order, after quickly introducing a preliminary notion.
The characteristic sub-bundle of a finite-dimensional vector pseudo-bundle
Let π : V → X be a finite-dimensional diffeological vector pseudo-bundle, and let (π −1 (x)) 0 be the characteristic subspace of the fibre π −1 (x). Denote by V 0 the sub-bundle of V defined as
V 0 := ∪ x∈X (π −1 (x)) 0 .
We say that V 0 is the characteristic sub-bundle of the pseudo-bundle V . It is evident from the construction that the characteristic sub-bundle of a locally trivial pseudo-bundle is itself locally trivial. Furthermore, every pseudo-metric on V is uniquely defined by its restriction to V 0 . Finally, the vice versa of the latter statement is also true, if we assume V to be locally trivial.
Lemma 3.1. Let π : V → X be a locally trivial diffeological vector pseudo-bundle, let π 0 : V 0 → X be its characteristic sub-bundle, and let g 0 be a pseudo-metric on V 0 . Then there exists one, and only one, pseudo-metric g on V whose restriction on V 0 coincides with g 0 .
Proof. It suffices to define g(x)(v ′ , v ′′ ) = g 0 (x)(v ′ , v ′′ ) if v ′ , v ′′ ∈ π −1 0 (x) 0
otherwise; the conclusion then follows from the definitions of a pseudo-metric and that of the characteristic sub-bundle.
Proposition 3.2. Let π : V → X be a locally trivial finite-dimensional diffeological vector pseudo-bundle that admits a pseudo-metric g. Then its characteristic sub-bundle π 0 : V 0 → X is diffeomorphic to its dual pseudo-bundle π * : V * → X via the natural pairing map associated to g.
Proof. Let ψ g : V 0 → V * be the natural pairing associated to g, that is,
ψ g (v)(·) = g(π(v))(v, ·).
That this is a bijection follows from its fibrewise nature and it being a bijection on each individual fibre (see [9]); furthermore (see the same source), as a map on the characteristic subspace it is a diffeomorphism with the dual fibre. By the assumption of local triviality this implies that ψ g , as well as its inverse, are smooth across the fibres as well, so they are smooth as a whole, whence the conclusion. 25 3.2 Implications of compatibility of g 1 and g 2 for V 1 and V 2
This extends the criterion for diffeological vector spaces (Theorem 2.9). The pseudo-bundle version is an immediate consequence and is as follows.
Proposition 3.3. Let π 1 : V 1 → X 1 and π 2 : V 2 → X 2 be two diffeological vector pseudo-bundles, locally trivial and with finite-dimensional fibres, let (f , f ) be a gluing between them, and let g 1 and g 2 be compatible pseudo-metrics. Thenf determines, over the domain of gluing, a smooth embedding of the characteristic sub-bundle of V 1 into the characteristic sub-bundle of V 2 .
Proof. This follows directly from the already-mentioned Theorem 2.9, applied to the restriction off on each fibre in its domain of definition; the theorem affirms that such restriction is an embedding of each characteristic subspace, so the fibre of the characteristic sub-bundle, of V 1 into that of V 2 . We should only add that the restriction off onto the intersection of its domain of definition with the characteristic sub-bundle of V 1 is smooth across the fibres, becausef is so.
The gluing-dual commutativity condition and gluing along a diffeomorphism
We now recall a statement (which essentially appears in [10], Lemma 5.17) that (together with some results from the previous sections) will allow us to deduce the gluing-dual commutativity in a number of cases.
The statement basically is that if the gluing of two pseudo-bundles is performed along a diffeomorphism, then the gluing-dual commutativity condition always holds. We also add the explicit construction of the commutativity diffeomorphism (which was not specified in the above source).
Theorem 3.4. Let χ 1 : W 1 → X 1 and χ 2 : W 2 → X 2 be two diffeological vector pseudo-bundles, let h : X 1 ⊇ Y → X 2 be a smooth invertible map with smooth inverse, and leth be its smooth fibrewise linear lift that is a diffeomorphism of its domain with its image. Then the map
Ψ ∪, * : (W 1 ∪h W 2 ) * → W * 2 ∪h * W * 1 defined by Ψ ∪, * = j W * 1 2 • (j W1 1 ) * on ((χ 1 ∪ (h,h) χ 2 ) * ) −1 (i X1 1 (X 1 \ Y )) j W * 1 2 •h * • (j W2 2 ) * on ((χ 1 ∪ (h,h) χ 2 ) * ) −1 (i X2 2 (h(Y )) j W * 2 1 • (j W2 2 ) * on ((χ 1 ∪ (h,h) χ 2 ) * ) −1 (i X2 2 (X 2 \ h(Y )
) is a pseudo-bundle diffeomorphism covering the switch map ϕ X1↔X2 .
Proof. It is easy to see that Ψ ∪, * is a bijection, so let us show that it is smooth (the proof that its inverse is smooth is then analogous). Let us first consider the general shape of an arbitrary plot q * of (W 1 ∪h W 2 ) * , and that of an arbitrary plot s * of W * 2 ∪h * W * 1 . Let q * : U → (W 1 ∪h W 2 ) * be any plot; we can however assume that U is connected, so (χ 1 ∪ (h,h) χ 2 ) • q, which is a plot of X 1 ∪ h X 2 , lifts to a plot of X 1 or to a plot of X 2 . This means that q * acts only on fibres of W 1 ∪h W 2 that pullback to W 1 or to W 2 , respectively. In the former case we have that there exists a plot q * 1 of W * 1 such that
q * = ((j W1 1 ) * ) −1 • q * 1 over i X1 1 (X 1 \ Y ) ((j W2 2 ) * ) −1 • (h * ) −1 • q * 1 over i X2 2 (h(Y ));
in the latter case there exists a plot q * 2 of W * 2 such that
q * = ((j W2 2 ) * ) −1 • q * 2 .
25 It is easy prove that ψg is smooth even if we do not assume V to be locally trivial.
For analogous reasons, if s * : U ′ → W * 2 ∪h * W * 1 defined on a connected domain U ′ , then either there exists a plot s * 2 of W * 2 such that
s * = j W * 2 1 • s * 2 over i X2 1 (X 2 \ h(Y )) j W * 1 2 •h * • s * 2 over i X1 2 (h(Y )), or else there exists a plot s * 1 of W * 1 such that s * = j W * 2 2
• s * 1 . Let us consider Ψ ∪, * • q * for q * of the first and the second type. Assume that q * is of the first type. Then by direct calculation we obtain
Ψ ∪, * • q * = j W * 1 2 • q * 1 over i X1 2 (X 1 \ Y ), j W * 1 2 •h * • (j W2 2 ) * • ((j W2 2 ) * ) −1 • (h * ) −1 • q * 1 over i X1 2 (Y ), that is, Ψ ∪, * • q * = j W * 1 2
• q * 1 over the whole of i X1 2 (X 1 ), which corresponds to a plot s * of the second type, for s * 1 := q * 1 . Similarly, if q * has its second possible form, we obtain
Ψ ∪, * • q * = j W * 1 2 •h * • q * 2 over i X1 2 (Y ) j W * 2 1 • q * 2 over i X2 1 (X 2 \ h(Y ))
, that is, the first possible form of a plot s * , with s * 2 := q * 2 . Finally, the smoothness of (Ψ ∪, * ) −1 , whose formula
(Ψ ∪, * ) −1 = ((j W1 1 ) * ) −1 • (j W * 1 2 ) −1 on (χ * 2 ∪ (h,h) χ * 1 ) −1 (i X1 2 (X 1 \ Y )) ((j W2 2 ) * ) −1 • (h * ) −1 • (j W * 1 2 ) −1 on (χ * 2 ∪ (h,h) χ * 1 ) −1 (i X1 2 (Y ) ((j W2 2 ) * ) −1 • (j W * 2 1 ) −1 on (χ * 2 ∪ (h,h) χ * 1 ) −1 (i X2 1 (X 2 \ h(Y )
) is given by the inverses of the three maps that Ψ ∪, * itself, is established in a completely analogous fashion.
Gluing-dual commutativity conditions for (f , f ) and (f * , f −1 )
We now consider the gluing-dual commutativity condition for V * 2 , V * 1 , and (f * , f −1 ), under the assumption that such condition holds for V 1 , V 2 , and (f , f ). For the duals, this condition takes form of the existence of a diffeomorphism Φ ( * )
∪, * : (V * 2 ∪f * V * 1 ) * → V * * 1 ∪f * * V * * 2
covering the inverse of the switch map ϕ X1↔X2 and satisfying the following:
Φ ( * ) ∪, * • ((j V * 2 2 ) * ) −1 = j V * * 1 1 on (π * * 1 ∪ (f * * ,f ) π * * 2 ) −1 (i X1 1 (X 1 \ Y )), Φ ( * ) ∪, * • ((j V * 1 1 ) * ) −1 = j V * * 2 2 •f * on (π * * 1 ∪ (f * * ,f ) π * * 2 ) −1 (i X2 2 (f (Y ))), Φ ( * ) ∪, * • ((j V * 1 2 ) * ) −1 = j V * * 2 2 on (π * * 1 ∪ (f * * ,f ) π * * 2 ) −1 (i X2 2 (X 2 \ f (Y ))).
Notice that this formula can serve as a definition of a certain map Φ ( * )
∪, * between the domain and the range indicated; what we really need to do is to show that it is a diffeomorphism.
Theorem 3.5. Let π 1 : V 1 → X 1 and π 2 : V 2 → X 2 be two locally trivial finite-dimensional diffeological vector pseudo-bundles, let (f , f ) be a pair of smooth maps that defines a gluing of the former pseudobundle to the latter, and let Φ ∪, * : (V 1 ∪f V 2 ) * → V * 2 ∪f * V * 1 be the canonical gluing-dual commutativity diffeomorphism. Let g 1 and g 2 be pseudo-metrics on V 1 and V 2 respectively, compatible with respect to the gluing. Then there exists a diffeomorphism Φ ( * )
∪, * : (V * 2 ∪f * V * 1 ) * → V * * 1 ∪f * * V * * 2 covering the map (ϕ X1↔X2 ) −1 : X 2 ∪ f −1 X 1 → X 1 ∪ f X 2 .
The claim of the theorem could be restated by saying that the dual pseudo-bundles V * 2 and V * 1 satisfy the gluing-dual commutativity condition for the gluing pair (f * , f −1 ).
Proof. Recall that the gluing-dual commutativity condition for the initial pseudo-bundles, that is, for V 1 and V 2 , implies thatf * is a diffeomorphism of its domain with its image. It is then a direct consequence of Theorem 3.4 that the following map
Φ ( * ) ∪, * = j V * * 2 2 • j V * 2 1 * on (π * 2 ∪ (f * ,f −1 ) π * 1 ) * −1 (i X2 1 (X 2 \ f (Y ))), j V * * 2 2 •f * • j V * 1 2 * on (π * 2 ∪ (f * ,f −1 ) π * 1 ) * −1 (i X1 2 (Y )), j V * * 1 1 • j V * 1 2 * on (π * 2 ∪ (f * ,f −1 ) π * 1 ) * −1 (i X1 2 (X 1 \ Y ))
is the desired gluing-dual commutativity diffeomorphism.
4 Compatibility of g * 2 and g * 1 implies the gluing-dual commutativity condition for V 1 and V 2 So far we have spoken of the gluing-dual commutativity condition as a prerequisite to obtaining a canonical construction of the induced pseudo-metric on the pseudo-bundle obtained by gluing. In principle, it is not a necessary condition (a pseudo-metric on V 1 ∪f V 2 can be constructed directly out of compatible pseudometrics on V 1 and V 2 , using the flexibility of diffeology in piecing together smooth maps); however, if we want at the same time to consider the dual pseudo-bundles V * 2 and V * 1 , and to ensure that the induced pseudo-metrics on them are again compatible, the reasoning involved starts to come rather close to the gluing-dual commutativity. Indeed, in this section we show that there is essentially an equivalence between the compatibility of g * 2 with g * 1 , and the existence of a gluing-dual commutativity diffeomorphism Φ ∪, * . 4.1 From Φ ∪, * to the compatibility of g * 2 , g * 1 , and (f * , f −1 )
It is not difficult to show that if we assume the gluing-dual commutativity, and more precisely, the existence of the specific diffeomorphism Φ ∪, * : (V 1 ∪f V 2 ) * → V * 2 ∪f * V * 1 given by
Φ ∪, * = j V * 1 2 • (j V1 1 ) * on ((π 1 ∪ (f ,f ) π 2 ) * ) −1 (i X1 1 (X 1 \ Y )) j V * 1 2 •f * • (j V2 2 ) * on ((π 1 ∪ (f ,f ) π 2 ) * ) −1 (i X2 2 (f (Y ))) j V * 2 1 • (j V2 2 ) * on ((π 1 ∪ (f ,f ) π 2 ) * ) −1 (i X2 2 (X 2 \ f (Y )))
then the dual pseudo-metrics, if they exist, are compatible.
Theorem 4.1. Let π 1 : V 1 → X 1 and π 2 : V 2 → X 2 be two finite-dimensional locally trivial diffeological vector pseudo-bundles, let (f , f ) be a pair of smooth maps defining a gluing of V 1 to V 2 , and let g 1 and g 2 be pseudo-metrics on V 1 and V 2 respectively, compatible with this gluing; assume that V 1 , V 2 , and (f , f ) satisfy the gluing-dual commutativity condition. Then g * 2 and g * 1 are compatible with the gluing of V * 2 to V * 1 along the pair (f * , f −1 ).
Proof. By Theorem 2.13 it suffices to show thatf * is a diffeomorphism of its domain with its image, and this is a trivial consequence of the form in which we stated the gluing-dual commutativity condition, namely, as the smoothness of the specific diffeomorphism Φ ∪, * . Indeed, denoting for brevity Z * 2 := ((π 1 ∪ (f ,f ) π 2 ) * ) −1 (i X2 2 (f (Y ))), we immediately obtaiñ
f * = (j V * 1 2 ) −1 • Φ ∪, * | Z * 2 • ((j V2 2 ) * ) −1 ;
thus,f * is a composition of three diffeomorphisms, so a diffeomorphism itself.
4.2
The vice versa: the smoothness of Φ ∪, * out of compatibility of g * 2 and g *
1
The reverse implication, that is, obtaining a smooth Φ ∪, * assuming the compatibility of g * 2 and g * 1 , is now easily obtained by applying Theorem 3.4, and the criteria for compatibility of the dual pseudo-metrics. Do note that the application of Theorem 3.4 is not straightforward; indeed, it speaks of gluing along a diffeomorphism, and the assumptions formulated in terms of various pseudo-metrics do not provide for f being one. Therefore we need some preliminary considerations.
Characteristic sub-bundles and the respective dual pseudo-bundles
In order to obtain our desired conclusion, namely, that the compatibility of pseudo-metrics dual to a given pair of compatible pseudo-metrics provides for the gluing-dual commutativity, we need several preliminary statements. We collect them in this section; they are more or less of independent interest.
The pseudo-metricg on the pseudo-bundle W 1 ∪hW 2 Assuming that W 1 and W 2 admit compatible pseudo-metrics g 1 and g 2 allows (without any further assumptions onh) for a direct construction of a pseudo-metricg on W 1 ∪h W 2 , which fibrewise coincides with either g 1 or g 2 , as appropriate. This pseudo-metric is defined by the following formula:
g(x) = g 1 ((i X1 1 ) −1 (x)) • ((j W1 1 ) −1 ⊗ (j W1 1 ) −1 ) for x ∈ i X1 1 (X 1 \ Y ) g 2 ((i X2 2 ) −1 (x)) • ((j W2 2 ) −1 ⊗ (j W2 2 ) −1 ) for x ∈ i X2 2 (X 2 ).
The switch map for characteristic sub-bundles As we have already commented, a pseudo-metric on a pseudo-bundle is essentially determined by its behavior on the characteristic sub-bundle; and furthermore, assuming the local triviality and the existence of a pseudo-metric, the characteristic sub-bundle is diffeomorphic to the dual pseudo-bundle. Thus, we can expect significant correlations between these three notions; and in this paragraph we specify some of them, as needed to reach the final aim of this section.
Lemma 4.2. Let χ 1 : W 1 → X 1 and χ 2 : W 2 → X 2 be two locally trivial diffeological vector pseudobundles, let W 0 1 and W 0 2 be their characteristic sub-bundles, let h : X 1 ⊇ Y → X 2 be a smooth invertible map with smooth inverse, and leth be its smooth fibrewise linear lift such that its restrictionh 0 on Domain(h) ∩ W 0 1 is a diffeomorphism. Then there is a canonical diffeomorphism
ϕ W 0 1 ↔W 0 2 : W 0 1 ∪h 0 W 0 2 → W 0 2 ∪h−1 0 W 0 1 covering the switch map ϕ X1↔X2 .
Proof. The diffeomorphism in question is in fact the same concept as the switch map (which we implied is a diffeomorphism, but did not prove that). Indeed, we denote the map obtained by analogy with ϕ X1↔X2 , by ϕ W 0 1 ↔W 0 2 and define it to be
ϕ W 0 1 ↔W 0 2 = j W 0 1 2 • (j W 0 1 1 ) −1 on (χ 0 1 ∪ (h0,h) χ 0 2 ) −1 (i X1 1 (X 1 \ Y )) j W 0 1 2 • (h 0 ) −1 • (j W 0 2 2 ) −1 on (χ 0 1 ∪ (h0,h) χ 0 2 ) −1 (i X2 2 (h(Y ))) j W 0 2 1 • (j W 0 2 2 ) −1 on (χ 0 1 ∪ (h0,h) χ 0 2 ) −1 (i X2 2 (X 2 \ f (Y ))),
where by χ 0 i we denote the restriction of χ i to W 0 i . Notice that it is its own inverse; let us show that it is smooth.
Let p : U → W 0 1 ∪h 0 W 0 2 ; it suffices to consider the case when U is connected. Under such assumption, p lifts to either a plot p 1 of W 0 1 or to a plot p 2 of W 0 2 , therefore p itself either has form
p = j W 0 1 1 • p 1 on p −1 1 (W 0 1 \ Domain(h 0 )) j W 0 2 2 •h 0 • p 1 on p −1 1 (Domain(h 0 ))
in the former case, or it has form p = j W 0 2 2 • p 2 in the latter case. Accordingly, by direct calculation we obtain that ϕ W 0
1 ↔W 0 2 • p = j W 0 2 2
• p 1 for p that lifts to p 1 and
ϕ W 0 1 ↔W 0 2 • p = j W 0 1 2 • (h 0 ) −1 • p 2 on p −1 2 (Range(h 0 )) j W 0 2 1 • p 2 on p −1 2 (W 0 2 \ Range(h 0 ))
for p that lifts for p 2 . Clearly, in both cases the result is a plot of W 0 2 ∪ (h0) −1 W 0 1 , whence the conclusion.
The lemma just proven is a preliminary statement which will be needed to establish a link between the following two statements; all three put together will allows us to relate the gluing-dual commutativity to compatibility of (dual) pseudo-metrics.
The triple W 0 1 ∪h 0 W 0 2 ∼ = (W 1 ∪h W 2 ) 0 ∼ = (W 1 ∪h W 2 ) *
The facts that we prove here ensure a kind of total commutativity between () 0 (characteristic) and () * (dual); this phrase is of course very informal, what we really mean shall be clear from the two statements that follow.
Proposition 4.3. Let χ 1 : W 1 → X 1 and χ 2 : W 2 → X 2 be two locally trivial diffeological vector pseudobundles, let W 0 1 and W 0 2 be their characteristic sub-bundles, let h : X 1 ⊇ Y → X 2 be a smooth invertible map with smooth inverse, and leth be its smooth fibrewise linear lift. Let g 1 and g 2 be pseudo-metrics on W 1 and W 2 respectively, compatible with the gluing along (h, h), and assume that the dual pseudo-metrics g * 2 and g * 1 are compatible with (h * , h −1 ). Leth 0 be the restriction ofh on Domain(h) ∩ W 0 1 . Then:
1.h 0 is a diffeomorphism with values in W 0 2 ; 2. There is a pseudo-bundle diffeomorphism Φ 0 : W 0 1 ∪h 0 W 0 2 → (W 1 ∪h W 2 ) 0 covering the identity map X 1 ∪ f X 2 → X 1 ∪ f X 2 ;
3. The natural pairing map Ψ 0 g : (W 1 ∪h W 2 ) 0 → (W 1 ∪h W 2 ) * associated to the pseudo-metricg on W 1 ∪h W 2 is a diffeomorphism of the characteristic sub-bundle (W 1 ∪h W 2 ) 0 with the dual pseudobundle (W 1 ∪h W 2 ) * .
Proof. 1. The compatibility of g 1 and g 2 means that their restrictions g 1 (y) and g 2 (h(y)) on each fibre in the domain of definition and, respectively, the range ofh are compatible pseudo-metrics on the diffeological vector spaces χ −1 1 (y) and χ −1 2 (h(y)). By Theorem 2.9 this means that the restrictionh 0 ofh to Domain(h) ∩ W 0 1 is a smooth injection, whose range is contained in Range(h) ∩ W 0 2 . Furthermore, by compatibility of g * 2 and g * 1 and Theorem 2.11, the dual spaces (χ −1 2 (h(y))) * and (χ −1 1 (y)) * have the same dimension, which is equal to the dimension of the corresponding characteristic subspaces; thereforẽ h 0 is also surjective. Finally, that its inverse is smooth, follows from local triviality and the fact that its restriction onto each fibre is a smooth linear map between finite-dimensional vector spaces.
2. This is a direct consequence of the definition of a characteristic sub-bundle. The diffeomorphism
Φ 0 : W 0 1 ∪h 0 W 0 2 → (W 1 ∪h W 2 ) 0 is defined by Φ 0 = j W1 1 • (j W 0 1 1 ) −1 over i X1 1 (X 1 \ Y ) j W2 2 • (j W 0 2 2 ) −1 over i X2 2 (X 2 );
it is essentially the natural inclusion map.
3. The diffeomorphism in question is the pairing map
Ψ 0 g : (W 1 ∪h W 2 ) 0 → (W 1 ∪h W 2 ) *
restricted to the characteristic sub-bundle and defined in the usual way, i.e., by
if w ∈ (W 1 ∪h W 2 ) 0 ⇒ Ψ 0 g (w)(·) =g((χ 1 ∪ (h,h) χ 2 )(w))(w, ·).
That it is smooth, follows from smoothness ofg; that it is bijective, is easily obtained by examining its restriction on each fibre, where, since the fibres of characteristic sub-bundle have standard diffeology, it becomes the usual isomorphism-by-duality on some standard R n . Finally, the smoothness of its inverse follows from the assumption of local triviality.
Under the same assumptions as those of Proposition 4.3, we also have the following:
Proposition 4.4. There is a canonical diffeomorphism Ψ 0 g2 ∪ (h −1 0 ,h * ) Ψ 0 g1 : W 0 2 ∪h−1 0 W 0 1 → W * 2 ∪h * W * 1 ,
whose restrictions onto the factors of gluing coincide with the natural pairing maps associated to a pair of compatible pseudo-metrics g 2 and g 1 .
Proof. It follows from the assumptions, specifically, the assumption of compatibility of g * 2 and g * 1 , that h * is a diffeomorphism; by Proposition 4.3 theh −1 0 is also a diffeomorphism. Thus, the desired diffeomorphism of the two pseudo-bundles in the statement is the result Ψ 0 g2 ∪ (h −1 0 ,h * ) Ψ 0 g1 of a gluing (see Section 1.3.1 for definitions) of the natural pairing maps Ψ 0 g2 : W 0 2 → W * 2 and Ψ 0 g1 : W 0 1 → W * 1 . Since the gluing of two diffeomorphisms along a pair of diffeomorphisms yields again a diffeomorphism, we only need to check that Ψ 0 g2 and Ψ 0 g1 are (h −1 0 ,h 0 )-compatible, that is, that for any
w 2 ∈ Domain(h −1 0 ) we havẽ h * (Ψ 0 g2 (w 2 )) = Ψ 0 g1 (h −1 0 (w 2 )).
Let us verify why this is true. The left-hand side of the expression is by definitioñ
h * (Ψ 0 g2 (w 2 ))(·) =h * (g 2 (χ 2 (w 2 ))(w 2 , ·)) = g 2 (χ 2 (w 2 ))(w 2 ,h(·)) = g 1 (h −1 (χ 2 (w 2 )))(h −1 0 (w 2 ), ·),
where the last equality follows from the compatibility of pseudo-metrics g 1 and g 2 . The right-hand side of the expression is Ψ 0
g1 (h −1 0 (w 2 ))(·) = g 1 (χ 1 (h −1 0 (w 2 )))(h −1 0 (w 2 ), ·),
and it remains to observe that h −1 (χ 2 (w 2 )) = χ 1 (h −1 0 (w 2 )), simply becauseh 0 is a lift of h, and the statement is proven.
The map Ψ 0 g2 ∪ (h −1 0 ,h * ) Ψ 0 g1
We conclude this section by giving the precise formula for the map Ψ 0 g2 ∪ (h −1 0 ,h * ) Ψ 0 g1 , which we will need in the next section. As follows from the general construction of gluing of two smooth maps, we have
Ψ 0 g2 ∪ (h −1 0 ,h * ) Ψ 0 g1 (w 0 ) = j W * 2 1 • Ψ 0 g2 • (j W 0 2 1 ) −1 on (χ 0 2 ∪ (h −1 0 ,h −1 ) χ 0 1 ) −1 (i X2 1 (X 2 \ h(Y ))) j W * 1 2 • Ψ 0 g1 • (j W 0 1 2 ) −1 on (χ 0 2 ∪ (h −1 0 ,h −1 ) χ 0 1 ) −1 (i X1 2 (X 1 )).
Proving the gluing-dual commutativity
We now give our final statement, which is a sufficient condition (and, together with Theorem 4.1, a criterion) for the gluing-dual commutativity condition to be satisfied.
Theorem 4.5. Let π 1 : V 1 → X 1 and π 2 : V 2 → X 2 be diffeological vector pseudo-bundles, locally trivial and with finite-dimensional fibres, and let (f , f ) be a gluing of V 1 to V 2 , with an invertible f . Suppose that V 1 and V 2 admit pseudo-metrics compatible with this gluing, and let g 1 and g 2 be a fixed choice of such pseudo-metrics. Assume, finally, that the induced pseudo-metrics g * 2 and g * 1 on the dual pseudo-bundles V * 2 and V * 1 are compatible with the gluing along (f * , f −1 ). Then V 1 , V 2 , and (f , f ) satisfy the gluing-dual commutativity condition.
Proof. Let us first show that (V 1 ∪f V 2 ) * and V * 2 ∪f * V * 1 are diffeomorphic, and then explain why there is a canonical diffeomorphism between them. Applying Proposition 4.3(3) and then (2), we obtain
(V 1 ∪f V 2 ) * ∼ = (V 1 ∪f V 2 ) 0 ∼ = V 0 1 ∪f 0 V 0 2 ;
next, by Lemma 4.2 and Proposition 4.4 we obtain
V 0 1 ∪f 0 V 0 2 ∼ = V 0 2 ∪f−1 0 V 0 1 ∼ = V * 2 ∪f * V * 1 ,
as wanted. It remains to see that the diffeomorphisms involved produce in the end the canonical commutativity diffeomorphism Φ ∪, * . Let us now specify the final diffeomorphism that we obtain from the above sequence. Letg be the pseudo-metric on V 1 ∪f V 2 induced by g 1 and g 2 . Then the first diffeomorphism of the sequence is (Ψ 0 g ) −1 ; the second is (Φ 0 ) −1 , the inverse of the diffeomorphism described in the proof of Proposition 4.3, the third is the switch-like map ϕ V 0
1 ↔V 0 2 , and the fourth is Ψ 0 g2 ∪ (f −1 0 ,f * ) Ψ 0 g1
. We need to see that the composition
Φ = Ψ 0 g2 ∪ (f −1 0 ,f * ) Ψ 0 g1 • ϕ V 0 1 ↔V 0 2 • (Φ 0 ) −1 • (Ψ 0 g ) −1
coincides with the appropriate canonically defined map Φ ∪, * . Indeed, after some obvious cancelations the pointwise description of the diffeomorphism Φ is as follows:
Φ(v) = j V * 1 2 • Ψ 0 g1 • (j V1 1 ) −1 • (Ψ 0 g ) −1 (v) if (π 1 ∪ (f ,f ) π 2 ) * (v) ∈ i X1 1 (X 1 \ Y ) j V * 1 2 • Ψ 0 g1 •f −1 0 • (j V2 2 ) −1 • (Ψ 0 g ) −1 (v) if (π 1 ∪ (f ,f ) π 2 ) * (v) ∈ i X2 2 (f (Y )) j V * 2 1 • Ψ 0 g2 • (j V2 2 ) −1 • (Ψ 0 g ) −1 (v) if (π 1 ∪ (f ,f ) π 2 ) * (v) ∈ i X2 2 (X 2 \ f (Y )).
Let us consider the three cases indicated. Let first v be such that (
π 1 ∪ (f ,f ) π 2 ) * (v) ∈ i X1 1 (X 1 \ Y ); then we have Φ(v) = j V * 1 2 • Ψ 0 g1 • (j V1 1 ) −1 (Ψ 0 g ) −1 (v) , where (Ψ 0 g ) −1 (v) = v 0 ∈ j V1 1 (π −1 1 (X 1 \ Y )) ∩ (V 1 ∪f V 2 ) 0 such that v(·) = g 1 (x)((j V1 1 ) −1 (v 0 ), (j V1 1 ) −1 (·)) for x = π 1 ((j V1 1 ) −1 (v 0 )), where (·) stands for the argument of v ∈ (V 1 ∪f V 2 ) * , that is being taken in j V1 1 (V 1 \ π −1 1 (Y )). Hence we obtain Φ(v)(·) = j V * 1 2 g 1 (x)((j V1 1 ) −1 (v 0 ), ·) .
This time (·) stands for an element of V 0 1 ; it is related to the argument of v(·) by the map j V 0 1 1 , so we have in the end
Φ(v)(·) = j V * 1 2 v(j V 0 1 1 (·)) ⇒ Φ(v) = j V * 1 2 • (j V1 1 ) * (v) ⇒ Φ = j V * 1 2 • (j V1 1 ) * ,
i.e., the canonical form of the gluing-dual commutativity diffeomorphism. The other two cases are rather similar.
Let v ∈ (V 1 ∪f V 2 ) * be such that (π 1 ∪ (f ,f ) π 2 ) * (v) ∈ i X2 2 (f (Y )); we then have Φ(v) = j V * 1 2 • Ψ 0 g1 •f −1 0 • (j V2 2 ) −1 (Ψ 0 g ) −1 (v) , where (Ψ 0 g ) −1 (v) = v 0 ∈ j V2 2 π −1 2 (f (Y )) ∩ (V 1 ∪f V 2 ) 0 such that v(·) = g 2 (x)((j V2 2 ) −1 (v 0 ), (j V2 2 ) −1 (·)) for x = π 2 ((j V2 2 ) −1 (v 0 )
). From this, we obtain
Φ(v) = j V * 1 2 • Ψ 0 g1 •f −1 0 • (j V2 2 ) −1 (v 0 )(·) = j V * 1 2 g 1 (f −1 (x))((f −1 0 • (j V2 2 ) −1 )(v 0 ), ·) .
Once again, we should relate this to the expression for v(·), this time keeping in mind the compatibility of the pseudo-metrics g 1 and g 2 . It suffices to recall that the argument (·) in this case is being taken in V 0 1 and is related to that of v(·) by the map j V2 2 •f 0 . Therefore we can rewrite the expression for Φ(v)(·) as follows:
Φ(v)(·) = j V * 1 2 g 1 (f −1 (x))((f −1 0 • (j V2 2 ) −1 )(v 0 ), ·) = (j V * 1 2 •f * 0 ) g 2 (x)((j V2 2 ) −1 (v 0 ),f 0 (·)) ,
which then allows us to conclude that
Φ(v)(·) = j V * 1 2 •f * 0 • (j V2 2 ) * (v)(·) ⇒ Φ = j V * 1 2 •f * 0 • (j V2 2 ) * .
It remains to consider the third part of the definition of Φ.
Let v ∈ (V 1 ∪f V 2 ) * be such that (π 1 ∪ (f ,f ) π 2 ) * (v) ∈ i X2 2 (X 2 \ f (Y )); we then have Φ(v) = j V * 2 1 • Ψ 0 g2 • (j V2 2 ) −1 (Ψ 0 g ) −1 (v) , where (Ψ 0 g ) −1 (v) = v 0 ∈ j V2 2 (π −1 2 (X 2 \ f (Y ))) ∩ (V 1 ∪f V 2 ) 0 such that v(·) = g 2 (x)((j V2 2 ) −1 (v 0 ), (j V2 2 ) −1 (·)) for x = π 2 ((j V2 2 ) −1 (v 0 )).
We therefore have
Φ(v)(·) = j V * 2 1 • Ψ 0 g2 • (j V2 2 ) −1 (v 0 )(·) = j V * 2 1 g 2 (x)((j V2 2 ) −1 (v 0 ), ·) .
By the same considerations regarding the argument (·), which in this case is related to that of v(·) by the map j V2 2 , we obtain
Φ(v)(·) = j V * 2 1 v(j V 0 2 2 (·)) ⇒ Φ = j V * 2 1 • (j V2 2 ) * ,
therefore Φ has the canonical form also in the third case, whence the final claim.
Final remarks on the gluing-commutativity condition
To conclude the discussion on the gluing-dual commutativity, we stress that the crucial point 26 throughout was the dual mapf * being a diffeomorphism (of its domain with its image). As follows from our proofs, it is this condition that is equivalent to both
• the specific map Φ ∪, * being a diffeomorphism; and
• the dual pseudo-metrics g * 2 and g * 1 being compatible.
It remains to observe that this also justifies our choice to state right away the gluing-dual commutativity condition in terms of Φ ∪, * being smooth, rather than just asking for the existence of some diffeomorphism between (V 1 ∪f V 2 ) * and V * 2 ∪f * V * 1 : the two are equivalent.
5 The pseudo-metricsg * and g * on pseudo-bundles
(V 1 ∪f V 2 ) * and V * 2 ∪f * V * 1
Assuming that V 1 , V 2 , and (f , f ) satisfy the gluing-dual commutativity condition implies, in particular, that there are two ways to construct a pseudo-metric on the pseudo-bundle (V 1 ∪f V 2 ) * ∼ = V * 2 ∪f * V * 1 , which correspond, respectively, to the left-hand side and the right-hand side of this expression. Specifically, the (specific expression for the) pseudo-bundle on the left carries the pseudo-metricg * that is induced by, or dual to, the pseudo-metricg. The pseudo-bundle on the right is obtained from a given gluing of two pseudo-bundles carrying compatible pseudo-metrics; it therefore carries a pseudo-metric g * that corresponds to this gluing (we mentioned this in Section 1; the details can be found in [11], and we recall what we need immediately below). We show that this is actually the same pseudo-metric.
The pseudo-metricg
* on (V 1 ∪f V 2 ) *
This is the pseudo-metric dual to 27 the pseudo-metricg on the pseudo-bundle V 1 ∪f V 2 ; the latter is defined as the compositioñ
g = Φ −1 ∪, * ⊗ Φ −1 ∪, * • Φ ⊗,∪ • (g 2 ∪ (f −1 ,f * ⊗f * ) g 1 ) • ϕ X1↔X2 ,
where ϕ X1↔X2 is the switch map, and
Φ ∪, * : (V 1 ∪f V 2 ) * → V * 2 ∪f * V * 1 and Φ ⊗,∪ : (V * 2 ⊗ V * 2 ) ∪f * ⊗f * (V * 1 ⊗ V * 1 ) → (V * 2 ∪f * V * 1 ) ⊗ (V * 2 ∪f * V * 1 )
are the appropriate versions of the commutativity diffeomorphisms (see Section 4.2.1 for the explicit formula). The pseudo-metricg * is then defined as the pseudo-metric dual tog, which by the usual definition means that, if Ψg : V 1 ∪f V 2 → (V 1 ∪f V 2 ) * is the (already-seen) pairing map relative tog, that is,
Ψg(v) =g((π 1 ∪ (f ,f ) π 2 )(v))(v, ·), then for any x ∈ X 1 ∪ f X 2 and any v * , w * ∈ ((π 1 ∪ (f ,f ) π 2 ) * ) −1 (x) we have by definitiong * (x)(v * , w * ) :=g(x)(v, w),
where v, w ∈ (π 1 ∪ (f ,f ) π 2 ) −1 (x) are any two elements such that Ψg(v) = v * and Ψg(w) = w * . We can avail ourselves of the already-mentioned restriction Ψ 0 g of Ψg to the characteristic sub-bundle (V 1 ∪f V 2 ) 0 and thus defineg
* (x)(v * , w * ) :=g(x) (Ψ 0 g ) −1 (v * ), (Ψ 0 g ) −1 (w * )
. Finally, an even more explicit formula forg * is
g * (x)(v * , w * ) = g 1 ((i X1 1 ) −1 (x))(v 1 , w 1 ), if x ∈ Range(i X1 1 ), g 2 ((i X2 2 ) −1 (x))(v 2 , w 2 ), if x ∈ Range(i X2 2 ),
where we have by definition
v 1 = (j V1 1 ) −1 • (Ψ 0 g ) −1 (v * ), w 1 = (j V1 1 ) −1 • (Ψ 0 g ) −1 (w * ) v 2 = (j V2 2 ) −1 • (Ψ 0 g ) −1 (v * ), w 2 = (j V2 2 ) −1 • (Ψ 0 g ) −1 (w * ).
The pseudo-metric
g * on V * 2 ∪f * V * 1
The pseudo-metric g * is defined on the pseudo-bundle V * 2 ∪f * V * 1 fibrewise, by imposing it to coincide with g * 2 or g * 1 on the appropriate subsets. Specifically, for i = 1, 2 let, as before, Ψ gi : V i → V * i be the pairing map associated to g 1 and g 2 respectively; then for any given x ∈ X 2 ∪ f −1 X 1 and any two
v * , w * ∈ (π * 2 ∪ (f * ,f −1 ) π * 1 ) −1 (x) we define g * (x)(v * , w * ) = g * 2 ((i X2 1 ) −1 (x))((j V * 2 1 ) −1 (v * ), (j V * 2 1 ) −1 (w * )) = g 2 ((i X2 1 ) −1 (x))(v 2 , w 2 ) if x ∈ Range(i X2 1 ), g * 1 ((i X1 2 ) −1 (x))((j V * 1 2 ) −1 (v * ), (j V * 1 2 ) −1 (w * )) = g 1 ((i X1 2 ) −1 (x))(v 1 , w 1 ) if x ∈ Range(i X1 2 ),
where again we make reference to the characteristic sub-bundles and the corresponding invertible restrictions Ψ 0 g2 of Ψ g2 and Ψ 0 g1 of Ψ g1 , since by construction
v 2 = (Ψ 0 g2 ) −1 • (j V * 2 1 ) −1 (v * ), w 2 = (Ψ 0 g2 ) −1 • (j V * 2 1 ) −1 (w * ), v 1 = (Ψ 0 g1 ) −1 • (j V * 1 2 ) −1 (v * ), w 1 = (Ψ 0 g1 ) −1 • (j V * 1 2 ) −1 (w * ).
27 It would be more precise to say, induced by duality.
Comparingg * and g *
In the case we are considering, we have assumed 28 that V 1 , V 2 , and (f , f ) satisfy the gluing-dual commutativity condition. By the very definition of the latter, this means that the pseudo-bundles (V 1 ∪f V 2 ) * and V * 2 ∪f * V * 1 are diffeomorphic, and in a canonical way. Since both of these pseudo-bundles also carry the canonical pseudo-metrics, described in the two sections above, it is natural to ask next whether their canonical identification, via the gluing-dual commutativity diffeomorphism Φ ∪, * , is an isometry relative to these pseudo-metrics. In this section we prove that it is.
More precisely, sinceg * and g * are maps
g * : X 1 ∪ f X 2 → (V 1 ∪f V 2 ) * * ⊗ (V 1 ∪f V 2 ) * * and g * : X 2 ∪ f −1 X 1 → (V * 2 ∪f * V * 1 ) * ⊗ (V * 2 ∪f * V * 1 ) * ,
we show that by adding the diffeomorphism
(Φ −1 ∪, * ) * ⊗ (Φ −1 ∪, * ) * : (V 1 ∪f V 2 ) * * ⊗ (V 1 ∪f V 2 ) * * → (V * 2 ∪f * V * 1 ) * ⊗ (V * 2 ∪f * V * 1 ) * and the switch map ϕ X1↔X2 : X 1 ∪ f X 2 → X 2 ∪ f −1 X 2 , we obtain (Φ −1 ∪, * ) * ⊗ (Φ −1 ∪, * ) * •g * = g * • (ϕ X1↔X2 ).
The full statement is as follows.
Theorem 5.1. Let π 1 : V 1 → X 1 and π 2 : V 2 → X 2 be two finite-dimensional locally trivial diffeological vector pseudo-bundle, and let (f , f ) be a gluing between them such that f andf * are diffeomorphisms. Assume that there exist pseudo-metrics g 1 and g 2 on V 1 and V 2 respectively, that are compatible with the gluing along (f , f ). Then the following is true:
(Φ −1 ∪, * ) * ⊗ (Φ −1 ∪, * ) * •g * = g * • (ϕ X1↔X2 ).
Notice that the assumptions of the theorem provide for the existence of all the maps that appear in the claim, that is, for the existence of the smooth switch map ϕ X1↔X2 , that of the gluing-dual commutativity diffeomorphism Φ ∪, * , and the existence and compatibility of the dual pseudo-metrics g * 2 and g * 1 , through which the pseudo-metricsg * and g * are defined.
Proof. The actual meaning of the formula that we wish to prove is as follows: taken an arbitrary x ∈ X 1 ∪ f X 2 and arbitrary v * , w * ∈ (π * 2 ∪ (f * ,f −1 ) π * 1 ) −1 (ϕ X1↔X2 (x)) ∈ V * 2 ∪f * V * 1 , we should have
g * (ϕ X1↔X2 (x))(v * , w * ) =g * (x)(Φ −1 ∪, * (v * ), Φ −1 ∪, * (w * )).
Since this formula involves the switch map and a gluing-dual commutativity diffeomorphism, both of which are defined separately in three cases, we should check the desired equality in the same three cases as well. These cases are:
x ∈ i X1 1 (X 1 \ Y ), x ∈ i X2 2 (f (Y )), and x ∈ i X2 2 (X 2 \ f (Y )). Let x ∈ i X1 1 (X 1 \ Y ); then v * , w * ∈ Range(j V * 1 2 ) and ϕ X1↔X2 (x) = i X1 2 ((i X1 1 ) −1 (x))
. Notice that the corresponding v and w that appear in the definition of the pseudo-metric g * are then
v = (Ψ 0 g1 ) −1 • (j V * 1 2 ) −1 (v * ), w = (Ψ 0 g1 ) −1 • (j V * 1 2 ) −1 (w * ), therefore we have g * (ϕ X1↔X2 (x))(v * , w * ) = g * (i X1 2 ((i X1 1 ) −1 (x)))(v * , w * ) = = g 1 ((i X1 1 ) −1 (x)) ((Ψ 0 g1 ) −1 • (j V * 1 2 ) −1 )(v * ), ((Ψ 0 g1 ) −1 • (j V *1
2 ) −1 )(w * ) .
28 As is, in fact, necessary for the two pseudo-metrics just described to be well-defined.
On the other hand, wheng * is applied to two elements v * 1 , w * 1 in the fibre of (V 1 ∪f V 2 ) * over a point
in i X1 1 (X 1 \ Y ), its value is g 1 ((i X1 1 ) −1 (x))(v 1 , w 1 ), where v 1 = (j V1 1 ) −1 • (Ψ 0 g ) −1 (v * 1 ), w 1 = (j V1 1 ) −1 • (Ψ 0 g ) −1 (w * 1 );
in our case we will have v * 1 := Φ −1 ∪, * (v * ) and w * 1 := Φ −1 ∪, * (w * ). Observe now that
Φ −1 ∪, * (v * ) = ((j V1 1 ) * ) −1 • (j V * 1 2 ) −1 (v * ) and Φ −1 ∪, * (w * ) = ((j V1 1 ) * ) −1 • (j V * 1 2 ) −1 (w * ),
and that in the case we are considering the relation between (Ψ 0 g ) −1 and (Ψ 0 g1 ) −1 is as follows:
(Ψ 0 g ) −1 = j V1 1 • (Ψ 0 g1 ) −1 • (j V1 1 ) * .
Therefore we have
v 1 = (j V1 1 ) −1 • (Ψ 0 g ) −1 Φ −1 ∪, * (v * ) = (Ψ 0 g1 ) −1 • (j V * 1 2 ) −1 (v * ) w 1 = (j V1 1 ) −1 • (Ψ 0 g ) −1 Φ −1 ∪, * (w * ) = (Ψ 0 g1 ) −1 • (j V * 1 2 ) −1 (w * ), hencẽ g * (x)(Φ −1 ∪, * (v * ), Φ −1 ∪, * (w * )) = = g 1 (i X1 1 ) −1 (x) ((Ψ 0 g1 ) −1 • (j V * 1 2 ) −1 )(v * ), ((Ψ 0 g1 ) −1 • (j V * 1 2 ) −1 )(w * ) = = g * (ϕ X1↔X2 (x))(v * , w * ),
as wanted.
Turning to the second case, let x ∈ i X2 2 (f (Y )), so that v * , w * ∈ Range(j
V * 1 2 ) and ϕ X1↔X2 (x) = (i X1 2 • f −1 • (i X2 2 ) −1 )(x). To calculate g * (ϕ X1↔X2 (x))(v * , w * ), write v = (Ψ 0 g1 ) −1 • (j V * 1 2 ) −1 (v * ), w = (Ψ 0 g1 ) −1 • (j V * 1 2 ) −1 (w * ); this implies that g * (ϕ X1↔X2 (x))(v * , w * ) = = g 1 (f −1 • (i X2 2 ) −1 )(x) ((Ψ 0 g1 ) −1 • (j V * 1 2 ) −1 )(v * ), ((Ψ 0 g1 ) −1 • (j V * 1 2 ) −1 )(w * ) = = g 2 ((i X2 2 ) −1 )(x)) (f • (Ψ 0 g1 ) −1 • (j V * 1 2 ) −1 )(v * ), (f • (Ψ 0 g1 ) −1 • (j V * 1 2 ) −1 )(w * ) ,
where we have used the compatibility of the pseudo-metrics g 1 and g 2 and the fact that v and w belong to the characteristic sub-bundle, on whichf is invertible by assumption.
To calculate now the second part of the identity to verify, recall that by definitioñ
g * (x)(v * 2 , w * 2 ) = g 2 ((i X2 2 ) −1 (x))(v 2 , w 2 ), where v 2 := (j V2 2 ) −1 • (Ψ 0 g ) −1 (v * 2 ) and w 2 := (j V2 2 ) −1 • (Ψ 0 g ) −1 (w * 2 ),
with v * 2 , w * 2 denoting for brevity v * 2 := Φ −1 ∪, * (v * ), w * 2 := Φ −1 ∪, * (w * ).
We now recall that in the case we are considering,
Φ −1 ∪, * (v * ) = ((j V1 1 ) * ) −1 •f * • (j V * 2 2 ) −1 (v * ) and Φ −1 ∪, * (w * ) = ((j V1 1 ) * ) −1 •f * • (j V * 2 2 ) −1 (w * ),
and there are the following relations between Ψ 0 g , Ψ 0 g1 , and Ψ 0 g2 (which we state immediately for their inverses):
(Ψ 0 g ) −1 = j V2 2 • (Ψ 0 g2 ) −1 • (f * ) −1 • (j V1 1 ) * and (Ψ 0 g2 ) −1 =f • (Ψ 0 g1 ) −1 . Thus, by direct calculation v 2 = (Ψ 0 g2 ) −1 • (j V * 2 2 ) −1 (v * ) and w 2 = (Ψ 0 g2 ) −1 • (j V * 2 2 ) −1 (w * ),
which means thatg * (x)(v * 2 , w * 2 ) = g * (ϕ X1↔X2 (x))(v * , w * ), again as wanted.
Finally, let us consider the third case, that of
x ∈ i X2 2 (X 2 \ f (Y )); then v * , w * ∈ Range(j V * 2 1 ) and ϕ X1↔X2 (x) = i X2 1 • (i X2 2 ) −1 (x). Furthermore, g * (ϕ X1↔X2 (x))(v * , w * ) = g 2 ((i X2 2 ) −1 (x))(v, w), where v = (Ψ 0 g2 ) −1 • (j V * 2 1 ) −1 (v * ) and w = (Ψ 0 g2 ) −1 • (j V * 2 1 ) −1 (w * ). On the other side, we haveg * (x)(v * 2 , w * 2 ) = g 2 ((i X2 2 ) −1 (x))(v 2 , w 2 ), where v 2 = (j V2 2 ) −1 • (Ψ 0 g ) −1 (v * 2 ) and w 2 = (j V2 2 ) −1 • (Ψ 0 g ) −1 (w * 2 ), and in turn v * 2 = (Φ ∪, * ) −1 (v * ) = ((j V2 2 ) * ) −1 • (j V * 2 1 ) −1 (v * ), w * 2 = (Φ ∪, * ) −1 (v * ) = ((j V2 2 ) * ) −1 • (j V * 2 1 ) −1 (w * ).
Finally, we observe that there is the following relation between Ψ 0 g and Ψ 0 g2 (stated again for their inverses):
(Ψ 0 g ) −1 = j V2 2 • (Ψ 0 g2 ) −1 • (j V2
2 ) * . Thus, putting together consecutively the expressions for v 2 , v * 2 , and (Ψ 0 g ) −1 (and likewise, for w 2 , w * 2 , and (Ψ 0 g ) −1 ), we obtain
v 2 = (Ψ 0 g2 ) −1 • (j V * 2 1 ) −1 (v * ) = v and w 2 = (Ψ 0 g2 ) −1 • (j V * 2 1 ) −1 (w * ) = w,
which implies thatg * (x)(v * 2 , w * 2 ) = g * (ϕ X1↔X2 (x))(v * , w * ), and therefore concludes our consideration of the third case. All cases having thus been exhausted, the proof is finished.
We can re-phrase our main conclusion by stating the following.
Corollary 5.2. The gluing-dual commutativity diffeomorphism Φ ∪, * is a pseudo-bundle isometry between
(V 1 ∪f V 2 ) * ,g * and V * 2 ∪f * V * 1 , g * .
The covariant Clifford algebras
We choose this umbrella term to refer to the pseudo-bundles of Clifford algebras that are associated to whatever pseudo-bundles we obtain by interposing the operations of gluing and taking the dual pseudobundle. These are, first of all, the pseudo-bundles of Clifford algebras associated to (V 1 ∪f V 2 ) * and V * 2 ∪f * V * 1 ; as we have seen above, these pseudo-bundles are a priori different, but they are naturally identified under appropriate assumptions. Once these assumptions are imposed, we still have another a priori difference, that of the two natural pseudo-metricsg * and g * on them being different, the possibility treated in the preceding section. Finally, there is a third option, that of the result of gluing of two pseudobundles of Clifford algebras, those associated to V * 2 and V * 1 . In this section we complete our consideration of these three cases, and of how they are interrelated.
The diffeomorphism
Cℓ((V 1 ∪f V 2 ) * ,g * ) ∼ = Cℓ(V * 2 ∪f * V * 1 , g * )
The diffeomorphism in question is easily obtained from the gluing-dual commutativity diffeomorphism Φ ∪, * , whose existence we assume and which in this case both guarantees that the pseudo-metricsg * and g * exist and are well-defined, and, by Corollary 4.1, is an isometry with respect to them. Extending Φ ∪, * to a diffeomorphism between Cℓ((V 1 ∪f V 2 ) * ,g * ) and Cℓ(V * 2 ∪f * V * 1 , g * ) is then a standard procedure, whose result we denote by
Φ Cℓ ∪, * : Cℓ((V 1 ∪f V 2 ) * ,g * ) → Cℓ(V * 2 ∪f * V * 1 , g * ).
To describe this diffeomorphism, it suffices to recall that for any equivalence class of form
[v 1 ⊗ . . . ⊗ v k ] ∈ (π 1 ∪ (f ,f ) π 2 ) Cℓ −1 (Y ) ⊂ Cℓ((V 1 ∪f V 2 ) * ,g * ), with a representative v 1 ⊗ . . . ⊗ v k , its image is the equivalence class in Cℓ(V * 2 ∪f * V * 1 , g * ) of Φ ∪, * (v ′ 1 ) ⊗ . . . ⊗ Φ ∪, * (v ′ k ).
In other words, Φ Cℓ ∪, * is the pushdown, by the quotient projections
π T ((V1∪f V2) * ) : T ((V 1 ∪f V 2 ) * ) → Cℓ((V 1 ∪f V 2 ) * ,g * ) and π T (V * 2 ∪f * V * 1 ) : T (V * 2 ∪f * V * 1 ) → Cℓ(V * 2 ∪f * V * 1 , g * ),
of the map n Φ ⊗n ∪, * , so that we have
Φ Cℓ ∪, * • π T ((V1∪f V2) * ) = π T (V * 2 ∪f * V * 1 ) • n Φ ⊗n ∪, * . Theorem 6.1. The map Φ Cℓ ∪, * : Cℓ((V 1 ∪f V 2 ) * ,g * ) → Cℓ(V * 2 ∪f * V * 1 , g * ) given by the identity Φ Cℓ ∪, * • π T ((V1∪f V2) * ) = π T (V * 2 ∪f * V * 1 ) • n Φ ⊗n ∪, * is a well-defined diffeomorphism.
Proof. This is a consequence of Corollary 5.2.
The pseudo-bundle
Cℓ(V * 2 , g * 2 ) ∪ (F * ) Cℓ Cℓ(V * 1 , g * 1 )
There is a third pseudo-bundle, with all fibres Clifford algebras, that is naturally associated to our data, the pseudo-bundle Cℓ(V * 2 , g * 2 ) ∪ (F * ) Cℓ Cℓ(V * 1 , g * 1 ). It is obtained by gluing together the pseudo-bundles of Clifford algebras relative to, respectively, (V * 2 , g * 2 ) and (V * 1 , g * 1 ), along the map (F * ) Cℓ , induced by the map f * ; this is the construction described in Section 1.5.2. To indicate how this construction works specifically for this case, it suffices to say that, for any given equivalence class in
((π * 2 ) Cℓ ) −1 (f (Y )) ⊂ Cℓ(V * 2 , g * 2 ), with an arbitrary representative v * 1 ⊗ . . . ⊗ v * k , its image under (F * ) Cℓ is the equivalence class in Cℓ(V * 1 , g * 1 ) of the elementf * (v * 1 ) ⊗ . . . ⊗f * (v * k )
. The main point, however, is that the resulting pseudo-bundle
Cℓ(V * 2 , g * 2 )∪ (F * ) Cℓ Cℓ(V * 1 , g * 1 ) is naturally diffeomorphic to the pseudo-bundle Cℓ(V * 2 ∪f * V * 1 , g * ) via the diffeomorphism Φ Cℓ( * ) : Cℓ(V * 2 , g * 2 ) ∪ (F * ) Cℓ Cℓ(V * 1 , g * 1 ) → Cℓ(V * 2 ∪f * V * 1 , g * )
that is determined by the following formula:
Φ Cℓ( * ) = (j V * 2 1 ) Cℓ • (j Cℓ(V * 2 ) 1 ) −1 on (π * 2 ) Cℓ ∪ ((F * ) Cℓ ,f −1 ) (π * 1 ) Cℓ −1 (i X2 1 (X 2 \ f (Y ))) (j V * 1 2 ) Cℓ • (j Cℓ(V * 1 ) 2 ) −1 on (π * 2 ) Cℓ ∪ ((F * ) Cℓ ,f −1 ) (π * 1 ) Cℓ −1 (i X1 2 (X 1 )), where (j V * 2 1 ) Cℓ : Cℓ(V * 2 , g * 2 )\((π * 2 ) Cl ) −1 (f (Y )) → Cℓ(V * 2 ∪f * V * 1 , g * ) and (j V * 1 2 ) Cℓ : Cℓ(V * 1 , g * 1 ) → Cℓ(V * 2 ∪f * V * 1 , g * )
are the fibrewise extensions to Cℓ(V * 2 , g * 2 ) \ ((π * 2 ) Cl ) −1 (f (Y )) and to Cℓ(V * 1 , g * 1 ), respectively, of the two natural inclusions
V * 2 \ (π * 2 ) −1 (f (Y )) ֒→ Cℓ(V * 2 ∪f * V * 1 , g * ) and V * 1 ֒→ Cℓ(V * 2 ∪f * V * 1 , g * ).
The latter inclusions are in turn given by the compositions of either j
V * 2 1 : V * 2 \ (π * 2 ) −1 (f (Y )) → V * 2 ∪f * V * 1 or j V * 1 2 : V * 1 → V * 2 ∪f * V * 1 , with the standard inclusion V * 2 ∪f * V * 1 → Cℓ(V * 2 ∪f * V * 1 , g * ). Theorem 6.2. The map Φ Cℓ( * ) : Cℓ(V * 2 , g * 2 ) ∪ (F * ) Cℓ Cℓ(V * 1 , g * 1 ) → Cℓ(V * 2 ∪f * V * 1 , g * ) thus defined is a smooth diffeomorphism.
Proof. This is a consequence of Theorem 5.5 in [8], applied to V * 2 , V * 1 , andf * ; notice that g * is exactly the counterpart of the pseudo-metricg that appears in the statement of the theorem just cited, in that it is obtained from g * 2 and g * 1 in precisely the same way asg is obtained from g 1 and g 2 .
6.3 The three diffeomorphisms Φ Cℓ ∪, * , Φ Cℓ( * ) , and Φ
Cℓ( * ) ∪
We now summarize the above by listing the three possible forms of the Clifford algebra pseudo-bundle, together with the assumptions that allow us to identify them to each other, and with the corresponding diffeomorphisms.
The assumptions As (almost) everywhere throughout the paper, we consider two finite-dimensional diffeological vector pseudo-bundles π 1 : V 1 → X 1 and π 2 : V 2 → X 2 , and a gluing of the former to the latter along the pair (f , f ). In order for the three diffeomorphisms to exist, we must also assume the following:
• the two pseudo-bundles are locally trivial; 29
• f andf * are diffeomorphisms;
• V 1 and V 2 admit compatible pseudo-metrics g 1 and g 2 respectively.
The three shapes of the pseudo-bundle of covariant Clifford algebras, and their equivalences Under the assumptions just listed, the following three pseudo-bundles are well-defined:
Cℓ((V 1 ∪f V 2 ) * ,g * ), Cℓ(V * 2 ∪f * V * 1 , g * ), Cℓ(V * 2 , g * 2 ) ∪ (F * ) Cℓ Cℓ(V * 1 , g * 1 )
. Although a priori these could be three different pseudo-bundles, the same assumptions that guarantee that all three are well-defined at the same time, also guarantee that they are in fact equivalent, via the following diffeomorphisms, already described above:
• Φ Cℓ ∪, * : Cℓ((V 1 ∪f V 2 ) * ,g * ) → Cℓ(V * 2 ∪f * V * 1 , g * ); • Φ Cℓ( * ) : Cℓ(V * 2 , g * 2 ) ∪ (F * ) Cℓ Cℓ(V * 1 , g * 1 ) → Cℓ(V * 2 ∪f * V * 1 , g * ); and • Φ Cℓ( * ) −1 • Φ Cℓ ∪, * =: Φ Cℓ( * ) ∪ : Cℓ((V 1 ∪f V 2 ) * ,g * ) → Cℓ(V * 2 , g * 2 ) ∪ (F * ) Cℓ Cℓ(V * 1 , g * 1 ). 7
The pseudo-bundles of exterior algebras, and gluing
We now turn to considering the pseudo-bundles of exterior algebras, first in the contravariant case, and then in the covariant case. We recall their construction, which is standard, and concentrate on the interactions with the operation of gluing.
The contravariant case
We first consider the contravariant version of the exterior algebra, by which we mean the following. Let first V be a diffeological vector space; for each tensor power of V consider the alternating operator 30
Alt : V ⊗ . . . ⊗ V n → V ⊗ . . . ⊗ V n ,
acting, as usual, by
Alt(v 1 ⊗ . . . ⊗ v n ) = 1 n! σ (−1) sgn(σ) v σ(1) ⊗ . . . ⊗ v σ(n)
and extended by linearity. In this section the n-th exterior power of V is the image
n (V ) = Alt(V ⊗ . . . ⊗ V n );
the whole exterior algebra * (V ) is the direct sum of all n (V ). We obtain the pseudo-bundle * (V ) of exterior algebras associated to a given pseudo-bundle π : V → X by employing the same operations in the pseudo-bundle version, and defining the alternating operator fibrewise.
The induced gluing mapf *
This map is provided by the universal factorization property for alternating maps. Specifically, let π 1 : V 1 → X 1 and π 2 : V 2 → X 2 be two finite-dimensional diffeological vector pseudo-bundles, and let (f , f ) be a gluing between them. Then the restrictionf | π −1 1 (y) off to each fibre in its domain of definition is a smooth linear map between diffeological vector spaces π −1 1 (y) and π −1 2 (f (y)), and the direct sum of all tensor degrees off | π −1 1 (y) is again a smooth linear map between the tensor algebras of these spaces:
n f | π −1 1 (y) ⊗n
: T (π −1 1 (y)) → T (π −1 2 (f (y))).
By construction, this map commutes with the two respective alternating operators, so its restriction, that
we denote by f | π −1 1 (y) * , to * (π −1 1 (y)) is a smooth linear map between the exterior algebras of the two fibres:
f | π −1 1 (y) * : * (π −1 1 (y)) → * (π −1 2 (f (y))).
Finally, the collectionf * :=
y∈Y f | π −1 1 (y) * ,
where Y is the domain of definition of f , yields a smooth and fibrewise linear mapf * between the appropriate subsets of * (V 1 ) and * (V 2 ). Thus, it yields an induced gluing between the corresponding pseudo-bundles of contravariant exterior algebras.
The pseudo-bundles
* (V 1 ∪f V 2 ) and * (V 1 ) ∪f * * (V 2 )
In a similar manner, for the pseudo-bundle V 1 ∪f V 2 there is its own alternating operator Alt, whose image is the pseudo-bundle * (V 1 ∪f V 2 ). Since each fibre of T (V 1 ∪f V 2 ) coincides with either a fibre of T (V 1 ) or one of T (V 2 ), and fibrewise each of the three alternating operators under consideration (those relative to V 1 , V 2 , and V 1 ∪f V 2 ) is the usual one of a diffeological vector space, it makes sense to expect the two pseudo-bundles of exterior algebras, * (V 1 ∪f V 2 ) and * (V 1 ) ∪f * * (V 2 ), to be diffeomorphic. Indeed, they are, and it is not difficult to describe the natural diffeomorphism between them; it is based on the gluing-tensor product commutativity diffeomorphism Φ ∪,⊗ , as the next construction shows. 30 In other terms, the antisymmetrization operator.
A preliminary remark At this moment we explicitly impose the assumption that for each of our pseudo-bundles V 1 and V 2 (and accordingly, for all the results of their gluings, their duals, and any mixture of such), the set of the dimensions of their fibres has a finite upper limit. We denote it by
dim V = sup x1∈X1,x2∈X2
{dim(π −1 1 (x 1 )), dim(π −1 2 (x 2 ))};
we do not go into any detail about how this correlates with any other assumptions of ours, just note that we will need for one of the diffeomorphisms that we define in the next paragraph (specifically, we use to ensure that Φ Λ * is indeed onto).
The diffeomorphism Φ (⊗n) ∪,⊗ : V 1 ∪f V 2 ⊗n → V ⊗n 1 ∪f ⊗n V ⊗n 2
We first describe the construction of this diffeomorphism, which is by induction on n. The base of the induction is n = 2, in which case Φ (⊗n) ∪,⊗ = Φ ∪,⊗ , the already-mentioned gluing-tensor product commutativity diffeomorphism. Suppose that Φ (⊗(n−1)) ∪,⊗ has already been defined. Then Φ (⊗n) ∪,⊗ is obtained as the composition
V 1 ∪f V 2 ⊗(n−1) ⊗ V 1 ∪f V 2 → V ⊗(n−1) 1 ∪f ⊗n V ⊗(n−1) 2 ⊗ V 1 ∪f V 2 → V ⊗n 1 ∪f ⊗n V ⊗n 2 ,
where the first arrow stands for Φ
∪,⊗ = Φf ⊗(n−1) ,f ∪,⊗ • Φ (⊗(n−1)) ∪,⊗ ⊗ Id V1∪f V2 .
Note also that we will denote the inverse of Φ the construction is exactly the same, just using the gluing-direct sum commutativity diffeomorphism Φ ∪,⊕ in place of Φ ∪,⊗ .
The diffeomorphism Φ (⊗n) ∪,⊗ , and the alternating operators We write Alt (n) for the restriction of the alternating operator Alt on V 1 ∪f V 2 onto the n-th tensor degree; likewise, Alt (n) i stands for the same restriction of the alternating operator on V i , for i = 1, 2. It is then quite trivial to observe that we have
Alt (n) = Φ (⊗n) ⊗,∪ • Alt (n) 1 ∪ (f ⊗n ,f ⊗ ) Alt (n) 2 • Φ (⊗n) ∪,⊗ ; equivalently, Φ (⊗n) ∪,⊗ • Alt (n) = Alt (n) 1 ∪ (f ⊗n ,f ⊗ ) Alt (n) 2 • Φ (⊗n) ∪,⊗ . The diffeomorphism Φ * : * (V 1 ∪f V 2 ) → * (V 1 ) ∪f * * (V 2 )
We can now define the desired diffeomorphism as
Φ * = Φ (dimV ) ∪,⊕ • n Φ (⊗n) ∪,⊗ | * (V1∪f V2) .
It remains to observe that Φ * is indeed onto * (V 1 ) ∪f * * (V 2 ), as follows from its commutativity (in the sense explained in the previous paragraph) with the alternating operators.
Theorem 7.1. The map Φ * = Φ (dimV ) ∪,⊕ • n Φ (⊗n) ∪,⊗ | * (V1∪f V2)
is a pseudo-bundle diffeomorphism * (V 1 ∪f V 2 ) → * (V 1 ) ∪f * * (V 2 ) covering the identity map on
X 1 ∪ f X 2 .
The pseudo-bundles of covariant exterior algebras
We now consider the covariant case. The basic definition is simple: the covariant exterior algebra (V ) of a pseudo-bundle V is * (V * ), the contravariant exterior algebra of its dual pseudo-bundle. So the reason why we consider it separately is to study its behavior with respect to the gluing, which, as we know, is not always well-behaved with respect to duality.
7.2.1
The induced map between (V 2 ) and (V 1 ) Indeed, let π 1 : V 1 → X 1 and π 2 : V 2 → X 2 be two finite-dimensional diffeological vector pseudo-bundles, and let (f , f ) be a gluing between them such that f is invertible. The gluing map between (V 2 ) and (V 1 ) is defined exactly asf * , but it is based on the dual mapf * . This gluing map is denoted byf and is in factf := (f * ) * .
The pseudo-bundles
(V 1 ∪f V 2 ) and (V 2 ) ∪f (V 1 )
As in the contravariant case, there are two natural pseudo-bundles of exterior algebras to consider, namely those mentioned in the title of this section. It is also natural to wonder whether they are diffeomorphic; we show that indeed they are, under the assumption that the gluing-dual commutativity condition is satisfied, by constructing a certain pseudo-bundle diffeomorphism
Φ : (V 1 ∪f V 2 ) → (V 2 ) ∪f (V 1 )
covering the switch map.
The n-th degree component of Φ Let Φ (⊗n) ∪,⊗ : V * 2 ∪f * V * 1 ⊗n → (V * 2 ) ⊗n ∪ (f * ) ⊗n (V * 1 )
⊗n be the diffeomorphism constructed in the previous section. The n-th tensor degree component of Φ is the composition
Φ (⊗n) ∪,⊗ • Φ ⊗n ∪, * : (V 1 ∪f V 2 ) * ⊗n → V * 2 ∪f * V * 1 ⊗n → (V * 2 ) ⊗n ∪ (f * ) ⊗n (V * 1 ) ⊗n .
Notice that if
Alt ∪, * : (V 1 ∪f V 2 ) * ⊗n → (V 1 ∪f V 2 ) * ⊗n , Alt 2 : (V * 2 ) ⊗n → (V * 2 ) ⊗n and Alt 1 : (V * 1 ) ⊗n → (V * 1 ) ⊗n
are the n-th degree alternating operators on (V 1 ∪f V 2 ) * , V * 2 , and V * 1 respectively, then we have
Φ (⊗n) ∪,⊗ • Φ ⊗n ∪, * • Alt ∪, * = Alt 2 ∪ ((f * ) ⊗n ,(f * ) ⊗n ) Alt 1 • Φ (⊗n) ∪,⊗ • Φ ⊗n ∪, * .
The diffeomorphism Φ We now employ also the gluing-direct sum commutativity diffeomorphism Φ
(dim V * ) ∪,⊕
, also from the previous section, to obtain Φ . Indeed, we define
Φ = Φ (dim V * ) ∪,⊕ • (dim V * ) n=0 Φ (⊗n) ∪,⊗ • Φ ⊗n ∪, * | (V1∪f V2) .
This is a well-defined injective, smooth and fibrewise linear map on (V 1 ∪f V 2 ); that its image is
(V 2 ) ∪f (V 1 )
, follows from the commutativity between each Φ (⊗n) ∪,⊗ • Φ ⊗n ∪, * and the appropriate alternating operators (see above).
Theorem 7.2. The map
Φ = Φ (dim V * ) ∪,⊕ • (dim V * ) n=0 Φ (⊗n) ∪,⊗ • Φ ⊗n ∪, * | (V1∪f V2)
is a pseudo-bundle diffeomorphism (V 1 ∪f V 2 ) → (V 2 ) ∪f (V 1 ) covering the switch map ϕ X1↔X2 .
The Clifford actions
In this section we consider all possible (shapes of) Clifford actions, first outlining what acts on what, and then establishing the various natural equivalences.
The outline
As we have seen in the preceding sections, there is a multitude of formally distinct, but (as we are about to see) equivalent with respect to the diffeomorphisms described in the previous two sections, Clifford actions relative to a given gluing of (V 1 , g 1 ) and (V 2 , g 2 ). In this section we give a list of these actions and their equivalences, with proofs and details appearing in the two sections immediately following. As before, we assume that f andf * are diffeomorphisms, and g 1 and g 2 are compatible.
The contravariant case
Recall that in this case we have two natural Clifford algebras, 31 specifically
Cℓ(V 1 ∪f V 2 ,g) ∼ = Cℓ(V 1 , g 1 ) ∪F Cℓ Cℓ(V 2 , g 2 )
(recall that the diffeomorphism that we have between them is Φ Cℓ : Cℓ(V 1 , g 1 ) ∪F Cℓ Cℓ(V 2 , g 2 ) → Cℓ(V 1 ∪f V 2 ,g)) and two natural exterior algebras, *
(V 1 ∪f V 2 ) ∼ = * (V 1 ) ∪f * * (V 2 ).
Summary of actions The natural actions are, the standard action c * of Cℓ(V 1 ∪f V 2 ,g) on * (V 1 ∪f V 2 ), and the composite actionc
* := c 1 ∪ (F Cℓ ,f * ) c 2 of Cℓ(V 1 , g 1 ) ∪F Cℓ Cℓ(V 2 , g 2 ) on * (V 1 ) ∪f * * (V 2 ),
where c i is the standard action of Cℓ(V i , g i ) on * (V i ). We will show that this is a partial case of the construction considered in [8].
The equivalence of the two actions This is expressed by the formula:
Φ * (c * (v)(e)) =c * (Φ Cℓ ) −1 (v) Φ * (e)
for all v ∈ Cℓ(V 1 ∪f V 2 ,g) and e ∈ * (V 1 ∪f V 2 ) such that π Cℓ (v) = π * (e). Below we will explain why this relation does hold.
The covariant case
There are three Clifford algebras to consider:
Cℓ((V 1 ∪f V 2 ) * ,g * ) ∼ = Cℓ(V * 2 ∪f * V * 1 , g * ) ∼ = Cℓ(V * 2 , g * 2 ) ∪ (F * ) Cℓ Cℓ(V * 1 , g * 1 ),
and essentially two exterior algebras:
(V 1 ∪f V 2 ) ∼ = (V 2 ) ∪f (V 1 ),
to which we will also add the contravariant exterior algebra * (V * 2 ∪f * V * 1 ).
Thus, c V acts as the exterior product, which is smooth by definition (recall that the diffeology on each exterior product degree can be described as the pushforward of the tensor product diffeology by the alternating operator, which makes it, and the exterior product as a consequence, automatically smooth). The smoothness of the map c j follows from the smoothness of the pseudo-metric g. To be slightly more explicit, we note that on a small enough neighborhood U , we can write a plot of the k-th exterior degree as (p 1 , . . . , p k ), where each p i is a plot of V , acting by u → p 1 (u) ∧ . . . ∧ p k (u). Therefore the evaluation map that determines the smoothness of c j is locally of form (u ′ , u) → p 1 (u) ∧ . . . ∧ g(π(p(u ′ )))(p(u ′ ), p j (u)) ∧ . . . ∧ p k (u) for some other plot p : U ′ → V of V . Since (u ′ , u) → g(π(p(u ′ )))(p(u ′ ), p j (u)) is a smooth function, and the diffeology of * (V ) is a (vector) pseudo-bundle diffeology, we obtain a plot of * (V ), whence the claim.
The compatibility of two standard Clifford actions
Likewise, we can show that under certain assumptions, two standard Clifford actions are compatible with a given gluing; this happens precisely when the gluing itself is commutative. Here is the precise statement.
Proposition 8.2. Let π 1 : V 1 → X 1 and π 2 : V 2 → X 2 be two locally trivial finite-dimensional diffeological vector pseudo-bundles, let (f , f ) be a gluing between them such that f andf are diffeomorphisms, and let g 1 and g 2 be compatible pseudo-metrics on V 1 and V 2 respectively. Let c i for i = 1, 2 be the standard Clifford actions of Cℓ(V i , g i ) on * (V i ). Then for all v, v 1 , . . . , v k ∈ V 1 such that π 1 (v) = π 1 (v 1 ) = . . . =
π 1 (v k ) ∈ Y we havef * (c 1 (v)(v 1 ∧ . . . ∧ v k )) = c 2 (F Cℓ (v))(f * (v 1 ∧ . . . ∧ v k )).
Proof. By the definition ofF Cℓ and that off * , we have that c 2 (F Cℓ (v))(f * (v 1 ∧ . . . ∧ v k )) = c 2 (f (v))(f (v 1 ) ∧ . . . ∧f (v k )).
The desired condition easily follows from this. Indeed,
f * (c 1 (v)(v 1 ∧ . . . ∧ v k )) = =f * (v ∧ v 1 ∧ . . . ∧ v k − k j=1 (−1) j+1 g 1 (π 1 (v))(v, v j )v 1 ∧ . . . ∧ v j−1 ∧ v j+1 ∧ . . . ∧ v k ) = =f (v) ∧f (v 1 ) ∧ . . . ∧f (v k )− − k j=1 (−1) j+1 g 1 (π 1 (v))(v, v j )f (v 1 ) ∧ . . . ∧f (v j−1 ) ∧f (v j+1 ) ∧ . . . ∧f (v k )). Now, since c 2 (f (v))(f (v 1 ) ∧ . . . ∧f (v k )) =f (v) ∧f (v 1 ) ∧ . . . ∧f (v k )− − k j=1 (−1) j+1 g 2 (π 2 (f (v)))(f (v),f (v j ))f (v 1 ) ∧ . . . ∧f (v j−1 ) ∧f (v j+1 ) ∧ . . . ∧f (v k )).
The pseudo-metrics g 1 and g 2 being compatible ensures that g 2 (π 2 (f (v)))(f (v),f (v j )) = g 1 (π 1 (v))(v, v j ), whence the claim.
8.4
The contravariant case: the actions on * (V 1 ∪f V 2 ) and * (V 1 ) ∪f * * (V 2 )
We now describe the actionc * , and prove its equivalence (already stated above) to the action c * . Notice that c * is an instance of the standard Clifford action, so it is smooth by Lemma 8.1.
The actionc * = c 1 ∪ (F Cl ,f * ) c 2 This is a partial case of a more general construction described in [8].
The construction bears some similarity to that of the gluing of smooth maps, although, as mentioned in the same source, it is not quite the same thing. To describe this action, let v ∈ cl(V 1 , g 1 ) ∪F Cℓ Cℓ(V 2 , g 2 ); thenc * (v) is an endomorphism of the fibre of * (V 1 ) ∪f * * (V 2 ) over the point (π Cℓ 1 ∪ (F Cℓ ,f ) π Cℓ 2 )(v) ∈ X 1 ∪ f X 2 . Then the actionc * is defined as follows:
c * (v)(e) = j * (V1) 1 c 1 ((j Cℓ(V1,g1) 1 ) −1 (v))((j * (V1) 1 ) −1 (e)) over i X1 1 (X 1 \ Y ) j * (V2) 2 c 2 ((j Cℓ(V2,g2) 2 ) −1 (v))((j * (V2) 2 ) −1 (e))
over i X2 2 (X 2 ).
In other words, we just pull back v and e to the respective factors of gluing, apply c 1 or c 2 , as appropriate, and re-insert the result into the pseudo-bundle * (V 1 ) ∪f * * (V 2 ). It now suffices to note that by Proposition 8.2 c 1 and c 2 are compatible as Clifford actions, so it follows from [8] that the actionc * ,∪ is smooth.
The equivalence of c * toc * ,∪ We now prove the already-mentioned statement of equivalence for these actions.
Theorem 8.3. Let v ∈ Cℓ(V 1 ∪f V 2 ,g) and e ∈ * (V 1 ∪f V 2 ) be such that π Cℓ (v) = π * (e). Then
Φ * (c * (v)(e)) =c * (Φ Cℓ ) −1 (v) Φ * (e) .
Proof. The proof is almost trivial if we adopt the following viewpoint: since both actions are fibrewise based on the standard Clifford action, it suffices to assume that v is an element of the copy of V 1 ∪f V 2 naturally contained in Cℓ(V 1 ∪f V 2 ,g), and that e belongs to the k-th exterior degree of V 1 ∪f V 2 , that is, e = v 1 ∧ . . . ∧ v k for v 1 , . . . , v k ∈ V 1 ∪f V 2 . Formally, there are two cases to consider: that of π Cℓ (v) ∈ i X1 1 (X 1 \ Y ) and that of π Cℓ (v) ∈ i X2 2 (X 2 ). Thus, suppose that π Cℓ (v) ∈ i X1 1 (X 1 \ Y ). Since c * is the standard action, we have
c * (v)(e) = v ∧ v 1 ∧ . . . ∧ v k − k j=1 (−1) j+1 v 1 ∧ . . . ∧g(π Cℓ (v))(v, v j ) ∧ . . . ∧ v k .
Therefore Φ * (c * (v)(e)) = j * (V1)
1 ((j V1 1 ) −1 (v) ∧ (j V1 1 ) −1 (v 1 ) ∧ . . . ∧ (j V1 1 ) −1 (v k )− − k j=1 (−1) j+1 (j V1 1 ) −1 (v 1 ) ∧ . . . ∧ g 1 (π 1 ((j V1 1 ) −1 (v)))((j V1 1 ) −1 (v), (j V1 1 ) −1 (v j )) ∧ . . . ∧ (j V1 1 ) −1 (v k )).
It therefore suffices to note that j Cℓ(V1,g1) 1
((j V1 1 ) −1 (v)) = (Φ Cℓ ) −1 (v) and j * (V1) 1 ((j V1 1 ) −1 (v 1 ) ∧ . . . ∧ (j V1 1 ) −1 (v k )) = Φ * (v 1 ∧ . . . ∧ v k )
, to draw the desired conclusion. Since the treatment of the case π Cℓ (v) ∈ i X2 2 (X 2 ) is exactly the same, the proof is finished.
The covariant case
In this case we have three potential actions, corresponding to the three shapes of the Clifford algebra and those of the three exterior algebras (one of which is actually a contravariant algebra, trivially identified to a covariant one). After a detailed description of the actions, we prove their equivalences, already announced in Section 8.1.2.
8.5.1
The action c * of Cℓ((V 1 ∪f V 2 ) * ,g * ) on (V 1 ∪f V 2 ) This is a case of a standard Clifford action, considered in Lemma 8.1; this lemma, in particular, ensures, that c * is a smooth action. Recall indeed that (V 1 ∪f V 2 ) = * ((V 1 ∪f V 2 ) * ) by its definition.
1 f 1 (x)
· v * (e 2 ) · w * (e 2 ).
The compatibility condition for g * 2 and g * 1 is thus 1 f2(0) = a 2 f1(0) , and so is equivalent to the one for g 1 and g 2 . The pseudo-metricg * ≡ g * is thereforẽ
g * (x ′ , y ′ , 0) = 1 f1(x ′ ) ∂ ∂z ⊗ ∂ ∂z if y ′ = 0, 1 f2(x ′ ) ∂ ∂z ⊗ ∂ ∂z if x ′ = 0.
in the latter case, the only thing that changes with respect to the formula just given, is that the term g(0, 0, 0)((0, 0, z 2 ), (0, 0, z) is replaced by the term g 2 (0, 0)((0, z 2 ), (0, z)), whose value however is exactly the same.
The covariant case is analogous, although we have three exterior algebras, (V 1 ∪f V 2 ), * (V * 2 ∪f * V * 1 ), and (V 2 ) ∪ (f * ) (V 1 ), with the actions c, c * ,∪ , and c ∪, * of, respectively, Cℓ((V 1 ∪f V 2 ) * ,g * ), Cℓ(V * 2 ∪f * V * 1 , g * ), and Cℓ(V * 2 , g * 2 ) ∪ (F * ) Cℓ Cℓ(V * 1 , g * 1 ). Once again, these actions have the same form everywhere except over the point of gluing (the origin), where we would formally write the formulae for c((0, 0, z 2 , w 2 ))(0, 0, z, w), c * ,∪ ((0, 0, z 2 , w 2 ))(0, 0, z, w), and c ∪, * ((0, 0, z 2 , w 2 ))(0, 0, z, w) with respect tõ g * , g * , or g * 1 , respectively:
c((0, 0, z 2 , w 2 ))(0, 0, z, w) = (0, 0, z 2 , w 2 ) ∧ (0, 0, z, w) − (0, 0, 0,g * (0, 0, 0)((0, 0, z 2 ), (0, 0, z))) = = (0, 0, z 2 w + w 2 z, w 2 w + 1 f2(0) z 2 z), c * ,∪ ((0, 0, z 2 , w 2 ))(0, 0, z, w) = (0, 0, z 2 , w 2 ) ∧ (0, 0, z, w) − 0, 0, 0, g * (0, 0, 0)((0, 0, z 2 ), (0, 0, z)) = = (0, 0, z 2 w + w 2 z, w 2 w − g * 1 (0, 0)((0, z 2 ), (0, z))) = (0, 0, z 2 w + w 2 z, w 2 w + 1 f1(0) z 2 z), c ∪, * ((0, 0, z 2 , w 2 ))(0, 0, z, w) = (0, 0, z 2 , w 2 ) ∧ (0, 0, z, w) − (0, 0, 0, g * 1 (0, 0)((0, z 2 ), (0, z))) = = (0, 0, z 2 w + w 2 z, w 2 w + 1 f1(0) z 2 z).
A non-diffeomorphismf and diffeomorphismf *
Let π 2 : V 2 → X 2 be the same as in the previous section, i.e., the standard projection R 2 → R; define π 1 : V 1 → X 1 to be the projection of V 1 = R 3 to X 1 = R, where X 1 carries the standard diffeology, and V 1 = R × R × R carries the product diffeology relative to the standard diffeologies on the first two factors and the vector space diffeology generated by the plot R ∋ x → |x| on the third factor. 32 The projection π 1 is just the projection onto the first factor. The gluing map f for the bases is the same, {0} → {0}, and the one for the total spaces is almost the same, specifically,f (0, y, z) = (0, ay) with a = 0 (again, notice that zeroing out the third coordinate is necessary forf to be smooth). The pseudo-bundle π 2 : V 2 → X 2 carries the same pseudo-metric g 2 as in the previous example, while the pseudo-metric g 1 on π 1 : V 1 → X 1 extends the previous one in a trivial manner: g 1 (x)((x, y 1 , z 1 ), (x, y 2 , z 2 )) = f 1 (x)y 1 y 2 .
The compatibility condition remains the same. The entire covariant case coincides with that of the example treated in the previous section. We only consider the pseudo-bundle V 1 ∪f V 2 and the corresponding contravariant constructions.
The pseudo-bundle V 1 ∪f V 2 We represent it as a subset in R 4 , specifically as the union of the plane given by the equations x = 0 and w = 0 (the part corresponding to V 2 ), and of the set {y = 0} \ {x = 0, y = 0, w = 0}; this is the part corresponding to V 1 , where excising the line {x = 0, y = 0, w = 0} reflects how V 1 ∪f V 2 contains V 1 \ π −1 1 (Y ), and not the entire V 1 . Thus, the entire set can be described as (x, 0, z, w) except the points (0, 0, z, 0) (0, y, z, 0) for all y, z.
The two Clifford algebras The Clifford algebra of V 2 is the already seen one; relative to the presentation of V 1 ∪f V 2 given above, we could describe it as a subset of R 5 , adding the 5th coordinate u 1 for the scalar part of Cℓ(V 2 , g 2 ) ∼ = R ⊕ V 2 . Thus, Cℓ(V 2 , g 2 ) = {(0, y, z, 0, u 1 )}, 32 In fact, any non-standard vector space diffeology would be sufficient for our purposes.
The Clifford actions It remains to describe the corresponding Clifford actions. As is standard, in the case of Cℓ(V 1 , g 1 ), it suffices to consider the action of elements of form (x, 0, z, 0, 0, 0) and (x, 0, 0, w, 0, 0) on elements of form (x, 0, z, 0, 0, 0), (x, 0, 0, w, 0, 0), (x, 0, 0, 0, u 1 , 0), and (x, 0, 0, 0, 0, u 2 ). For these elements the multiplication is determined as follows c 1 (x, 0, z, 0, 0, 0)(x, 0, z ′ , 0, 0, 0) = (x, 0, 0, 0, −f 1 (x)z 2 , 0) c 1 (x, 0, z, 0, 0, 0)(x, 0, 0, w, 0, 0) = (x, 0, 0, 0, 0, zw) c 1 (x, 0, z, 0, 0, 0)(x, 0, 0, 0, u 1 , 0) = (x, 0, u 1 z, 0, 0, 0) c 1 (x, 0, z, 0, 0, 0)(x, 0, 0, 0, 0, u 2 ) = (x, 0, 0, −u 2 f 1 (x)z, 0, 0) c 1 (x, 0, 0, w, 0, 0)(x, 0, z, 0, 0, 0) = (x, 0, 0, 0, 0, −zw) c 1 (x, 0, 0, w, 0, 0)(x, 0, 0, w ′ , 0, 0) = (x, 0, 0, 0, −f 1 (x)ww ′ , 0) c 1 (x, 0, 0, w, 0, 0)(x, 0, 0, 0, u 1 , 0) = (x, 0, 0, u 1 w, 0, 0) c 1 (x, 0, 0, w, 0, 0)(x, 0, 0, 0, 0, u 2 ) = (x, 0, 0, 0, 0, 0).
In the case of Cℓ(V 2 , g 2 ), it suffices to consider the action of (0, y, z, 0, 0, 0) on elements of form (0, y, z, 0, 0, 0) and (0, y, 0, 0, u 1 , 0), and we have c 2 (0, y, z, 0, 0, 0)(0, y, z ′ , 0, 0, 0) = (0, y, 0, 0, −f 2 (y)zz ′ , 0) c 2 (0, y, z, 0, 0, 0)(0, y, 0, 0, u 1 , 0) = (0, y, u 1 z, 0, 0, 0)
Finally, the Clifford action on both * (V 1 ∪f V 2 ) and * (V 1 ) ∪f * * (V 2 ) is obtained by concatenating the two lists; the difference between the two pseudo-bundles is not seen on the level of defining the action, but rather in how we determine the two sets of points (as already been indicated above), underlying the commutativity between the gluing and the exterior product.
used respectively for the left-hand and the right-hand factor in the base space of the pseudo-bundle under consideration, while j smth 1 and j smth 2 refer to the left-hand and the right-hand factor of the total space.
Lemma 2 . 2 .
22Let V and W be finite-dimensional diffeological vector spaces, and let f : V → W be a smooth linear map. If V and W admit pseudo-metrics compatible with f then the Ker(f ) ∩ V 0 = {0}.
Under the assumptions of Lemma 2.5, there is the inclusion f (V 0 ) W 0 .
Proposition 2. 8 .
8Let V and W be finite-dimensional diffeological vector spaces such that dim(V * ) dim(W * ), and let f : V → W be a smooth linear map such that Ker(f ) ∩ V 0 = {0} and f (V 0 ) W 0 . Then V and W admit pseudo-metrics compatible with respect to f . Proof. Let us fix smooth decompositions V = V 0 ⊕ V 1 and W = W 0 ⊕ W 1 . To construct a pseudo-metric g V , choose a basis v 1 , . . . , v k of V 0 and a basis v k+1 , . . . , v n of V 1 ; then set
Theorem 2 . 9 .
29Let V and W be two finite-dimensional diffeological vector spaces, and let f : V → W be a smooth linear map. Then V and W admit compatible pseudo-metrics if and only if Ker(f ) ∩ V 0 = {0} and f (V 0 ) W 0 .
Theorem 2 . 11 .
211Let V and W be two finite-dimensional diffeological vector spaces, and let f :V → W be a smooth linear map such that Ker(f ) ∩ V 0 = {0} and f (V 0 ) W 0 .Let g V and g W be compatible pseudo-metrics on V and W respectively. Then the induced pseudo-metrics g * W and g * V are compatible with f * if and only if f * : W * → V * is a diffeomorphism.
⊗
Id V1∪f V2 , and the second one, for the version Φf tensor product commutativity diffeomorphism applied to the case of two factors, V 1 ∪f V 2 . We can summarize the whole construction as Φ (⊗n)
∪ . Finally, for all k (limited in practice by dim V ) we define
In fact, each diffeology is just a set of maps, which may wholly contain another diffeology, or be contained in one; in fact, there is a complete lattice on them on any X.
We stress that this can indeed be any quotient, with no restrictions on the equivalence relation ∼.3 There are analogous notions of a diffeological group, diffeological algebra, and so on.4 In contrast with non-vector space diffeologies, the finest of which is always the discrete one, consisting of constant maps only.
A priori, this direct-sum-diffeology-from-subset-diffeologies is finer, so not all usual direct sum decompositions of diffeological vector spaces are smooth; see an example in[9].6 The choice of the term diffeological vector pseudo-bundle is ours; the same object is called just a diffeological fibre bundle in[4], a regular vector bundle in[17], and a diffeological vector space over X in[3]. The choice of the term pseudo-bundle is meant to distinguish these objects from the numerous other versions of bundles that have appeared so far.7 This obviously poses the question of the existence of such diffeology; this was considered in[17]; see also[3], Proposition 4.6 for a relevant methodology.
If one wishes, these could be described as universal factorization properties in the category of diffeological pseudobundles
The notion of a pseudo-metric is designed to be a generalization from a scalar product on a vector space, where the compatibility with a given f is equivalent to f being an isometry (of its domain with its range).
Which at the moment does not appear to be particularly limiting, in the sense that we do not know of any non-locallytrivial pseudo-bundles that admit pseudo-metrics in the first place.
Which means that the direct sum diffeology coincides with V 's or W 's own diffeology, or, alternatively, that the composition of each plot of V (respectively W ) with the projection on V 0 (respectively W 0 ) is a plot of the latter.
The choices of f however could be plenty; it suffices to take V the standard R n and W any other diffeological vector space of dimension strictly smaller than n. Any linear map from V to W is then going to be smooth (see Section 3.9 in[5]).
As we have said already, V 0 is uniquely defined by V ; this is not necessarily true of V 1 , which is defined uniquely only when the pseudo-metric has been fixed already. However, there is always at least one choice of V 1 ; what we mean at the moment is that such a choice is fixed arbitrarily.
Under the assumption that the pseudo-bundles involved are finite-dimensional and locally trivial.
This is sufficient but not necessary. What we really need is that the right inverse of the pairing map, that takes values in the characteristic sub-bundle be smooth, and so it suffices that this sub-bundle split off as a smooth direct summand.
A note on slight change in terminology: in the remainder of the paper we will just say Clifford algebra instead of a pseudo-bundle of Clifford algebras, and exterior algebra instead of pseudo-bundle of exterior algebras; in the present context this is unlikely to cause confusion.
Acknowledgments This work benefitted from some assistance, for which I would like to thank Prof. Riccardo Zucchi, even if he always says that he is being overvalued, some of his doctorate students (Martina and Leonardo), although they do not expect this at all, and also Prof. Mario Petrini (he most definitely will be surprised).Summary of Clifford actionsWe now outline which Clifford algebra (or the result of gluing of such) acts on which pseudo-bundle of exterior algebras:• Cℓ((V 1 ∪f V 2 ) * ,g * ) acts on (V 1 ∪f V 2 ) via the standard Clifford action c * ;• Cℓ(V * 2 , g * 2 ) ∪ (F * ) Cℓ Cℓ(V * 1 , g * 1 ) acts on (V 2 ) ∪f (V 1 ) via the action c * ,∪ (see Proposition 6.8) induced by the standard Clifford actions c * 2 and c * 1 of Cℓ(V * 2 , g * 2 ) and Cℓ(V * 1 , g * 1 ) on (V 2 ) and (V 1 ) respectively;• Cℓ(V * 2 ∪f * V * 1 , g * ) has, again, the standard Clifford action, which we have not mentioned yet and which we now denote byc * ,∪ , on * (V * 2 ∪f * V * 1 ).The equivalence of actions As in the contravariant case, the actions c * , c * ,∪ , andc * ,∪ turn out to be equivalent, with the equivalence established via the diffeomorphisms Φ Cℓ( * ) , Φ Cℓ ∪, * , and Φ Cℓ( * ) ∪ , as well as Φ and Φ ∪, * :. Specifically, we have:. Notice that these equivalences imply also the equivalence of c * toc * ,∪ .The standard Clifford action is smoothThe basis for several versions of the Clifford(-type) actions listed above is the usual action of the (contravariant) Clifford algebra on the corresponding (also contravariant) exterior algebra. This means the following.Let π : V → X be any locally trivial finite-dimensional diffeological vector pseudo-bundle that admits a pseudo-metric; let g be a fixed choice of a pseudo-metric on it. The standard Clifford action ofOn each fibre π −1 (x) of V , this is the usual Clifford action of the Clifford algebra relative to the bilinear symmetric form g(x) on the exterior algebra of π −1 (x). Proof. Notice first of all that the pseudo-bundle * (V ) smoothly splits as the direct sum k k V . It then follows from the above presentation of the action c and the definition of the diffeology of Cℓ(V, g), that it suffices to show that the following two maps c V , c j : V → L( * (V ), * (V )) are smooth:) be the standard Clifford actions. By Proposition 8.2, they are compatible with the gluings that yield respectively, with respect to the mapsf and (F * ) Cℓ . Thus, the procedure described in[8]yields a smooth action c * ,. Then we will havewith the Clifford multiplication being defined byFrom this, it is also quite evident that the result trivially coincides with Cℓ(V 1 ∪f V 2 ,g), so much in fact, that we can only distinguish between the two by choosing two slightly different forms of designating the same subset in R 4 . Specifically, in the case of Cℓ(V 1 ∪f V 2 ,g) we describe the set of its points asObviously, this is the same set as we described as the set of points of Cℓ(V 1 , g 1 )∪F Cℓ Cℓ(V 2 , g 2 ); the chosen presentation of the latter emphasizes its structure as the result of a gluing.The pseudo-bundles of covariant Clifford algebras Consider again the subset in R 4 given by the equation xy = 0. This is the subset that is identified with all three (shapes of) the Clifford algebra. For all three possibilities, we identify the copy of V * 1 contained in either of them with the hyperplane {(x, 0, z, 0)}, and the copy of V * 2 , with the hyperplane {(0, y, z, 0)}; the fourth coordinate w corresponds to the scalar part of the Clifford algebra.The distinction between the various shapes of the Clifford algebra is the following one. When this subset is viewed as Cℓ((V 1 ∪f V 2 ) * ,g * ), we describe the Clifford multiplication as (x, 0, z 1 , w 1 ) · Cℓ (x, 0, z 2 , w 2 ) = (x, 0, z 1 w 2 + z 2 w 1 , − 1 f1(x) z 1 z 2 + w 1 w 2 ) for x = 0, (0, y, z 1 , w 1 ) · Cℓ (0, y, z 2 , w 2 ) = (0, y, z 1 w 2 + z 2 w 1 , − 1 f2(2) z 1 z 2 + w 1 w 2 ) otherwise.On the other hand, when we view the same subset as either Cℓ(V * 2 , g * 2 ) ∪ (F * ) Cℓ Cℓ(V * 1 , g * 1 ) or Cℓ(V * 2 ∪f * V * 1 , g * ), we describe the corresponding product by (x, 0, z 1 , w 1 ) · Cℓ (x, 0, z 2 , w 2 ) = (x, 0, z 1 w 2 + z 2 w 1 , − 1 f1(x) z 1 z 2 + w 1 w 2 ) for all x, (0, y, z 1 , w 1 ) · Cℓ (0, y, z 2 , w 2 ) = (0, y, z 1 w 2 + z 2 w 1 , − 1 f2(2) z 1 z 2 + w 1 w 2 ) for y = 0.The pseudo-bundles of exterior algebras These can be presented in exactly the same way as those of Clifford algebras. In both the contravariant and the covariant case we have a unique presentation, again as a subset of R 4 given by the equation xy = 0, with the exterior productThe Clifford actions In the contravariant case, we have two exterior algebras, * (V 1 ∪f V 2 ) and * (V 1 )∪f * * (V 2 ), with the actions c andc of, respectively, Cℓ(V 1 ∪f V 2 ,g) and Cℓ(V 1 , g 1 )∪F Cℓ Cℓ(V 2 , g 2 ).In the former case, we have c((0, 0, z 2 , w 2 ))(0, 0, z, w) = (0, 0, z 2 , w 2 ) ∧ (0, 0, z, w) − (0, 0, 0,g(0, 0, 0)((0, 0, z 2 ), (0, 0, z))) = = (0, 0, z 2 w + w 2 z, w 2 w + f 2 (0)z 2 z);with the Clifford multiplication given byThe Clifford algebra Cℓ(V 1 , g 1 ) is bigger; since the fibres of V 1 have dimension 2, each fibre of cl(V 1 , g 1 ) has dimension 4. Thus, we represent it as a subset in R 6 , by adding the coordinates u 1 , u 2 , where u 1 corresponds to the scalar part and u 2 corresponds to the degree 2 vector part. Thus,with the Clifford multiplication given byFinally, Cℓ(V 1 ∪f V 2 ,g) can be described as the following subset in R 6 :{(x, y, z, w, u 1 , u 2 ) such that xy = 0,is presented as the subset in R 6 of the following form:{(x, 0, z, w, u 1 , u 2 ) such that x = 0} ∪ {(0, y, z, 0, u 1 , 0) for all y, z}.The fibrewise multiplication is described by uniting the two formulae just given.The contravariant exterior algebras Likewise, the exterior algebras * (V 1 ) and * (V 2 ) are given by the same sets. Both of these we immediately represent as subsets of R 6 , with the 5-th coordinate being the scalar part and the 6-th coordinate being the exterior product corresponding to the exterior product relative to the 3-rd and the 4-th coordinates; in the case of V 2 , this part is obviously trivial. Thus, we have *with the exterior product given by(0, y, z ′ , 0, u ′ 1 , 0) ∧ (0, y, z ′′ , 0, u ′′ 1 , 0) = (0, y, u ′′ 1 z ′ + u ′ 1 z ′′ , 0, u ′ 1 u ′′ 1 , 0).The exterior algebras * (V 1 ∪f V 2 ) and * (V 1 ) ∪f * * (V 2 ) are then represented respectively by the sets * (V 1 ∪f V 2 ) = {(x, y, z, w, u 1 , u 2 ), where xy = 0, x = 0 ⇒ w = u 2 = 0}, * (V 1 ) ∪f * * (V 2 ) = {(x, 0, z, w, u 1 , u 2 ) such that x = 0} ∪ {(0, y, z, 0, u 1 , 0)}.It is obvious that the two presentations determine the same set, with the second one possibly giving a better idea of the structure of the set, and the first one allowing for the uniform description of the exterior product, in the following way:(x, y, z ′ , w ′ , u ′ 1 , u ′ 2 ) ∧ (x, y, z ′′ , w ′′ , u ′′ 1 , u ′′ 2 ) = = (x, y, u ′′ 1 z ′ + u ′ 1 z ′′ , u ′′ 1 w ′ + u ′ 1 w ′′ , u ′ 1 u ′′ 1 , u ′′ 1 u ′ 2 + u ′ 1 u ′′ 2 + z ′ w ′′ − z ′′ w ′ ).
V * The pseudo-bundles of Clifford algebras All fibres in our case are 1-dimensional, so each of Cℓ. * , 2∪ Of Cℓ, 2Cℓ. is thus a trivial fibering of R 3 over R; the result of their gluing can be described as2 The action c * ,∪ of Cℓ(V * The pseudo-bundles of Clifford algebras All fibres in our case are 1-dimensional, so each of Cℓ(V 1 , g 1 ), Cℓ(V 2 , g 2 ) is thus a trivial fibering of R 3 over R; the result of their gluing can be described as
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|
[] |
[
"Meaningful Context, a Red Flag, or Both? Users' Preferences for Enhanced Misinformation Warnings on Twitter",
"Meaningful Context, a Red Flag, or Both? Users' Preferences for Enhanced Misinformation Warnings on Twitter"
] |
[
"Filipo Sharevski [email protected] \nDePaul University Chicago\nILUnited States\n",
"Amy Devine [email protected] \nDePaul University Chicago\nILUnited States\n",
"Emma Pieroni [email protected] \nDePaul University Chicago\nILUnited States\n",
"Peter Jachim [email protected] \nDePaul University Chicago\nILUnited States\n"
] |
[
"DePaul University Chicago\nILUnited States",
"DePaul University Chicago\nILUnited States",
"DePaul University Chicago\nILUnited States",
"DePaul University Chicago\nILUnited States"
] |
[] |
Warning users about misinformation on social media is not a simple usability task. Soft moderation has to balance between debunking falsehoods and avoiding moderation bias while preserving the social media consumption flow. Platforms thus employ minimally distinguishable warning tags with generic text under a suspected misinformation content. This approach resulted in an unfavorable outcome where the warnings "backfired" and users believed the misinformation more, not less. In response, we developed enhancements to the misinformation warnings where users are advised on the context of the information hazard and exposed to standard warning iconography. We ran an A/B evaluation with the Twitter's original warning tags in a 337 participant usability study. The majority of the participants preferred the enhancements as a nudge toward recognizing and avoiding misinformation. The enhanced warning tags were most favored by the politically left-leaning and to a lesser degree moderate participants, but they also appealed to roughly a third of the right-leaning participants. The education level was the only demographic factor shaping participants' preferences. We use our findings to propose user-tailored improvements in the soft moderation of misinformation on social media.CCS CONCEPTS• Security and privacy → Social aspects of security and privacy; Usability in security and privacy.
|
10.48550/arxiv.2205.01243
|
[
"https://arxiv.org/pdf/2205.01243v1.pdf"
] | 248,506,011 |
2205.01243
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8500b01cbdb3c92b83cba6c4a4f6bc5eb46419b4
|
Meaningful Context, a Red Flag, or Both? Users' Preferences for Enhanced Misinformation Warnings on Twitter
Filipo Sharevski [email protected]
DePaul University Chicago
ILUnited States
Amy Devine [email protected]
DePaul University Chicago
ILUnited States
Emma Pieroni [email protected]
DePaul University Chicago
ILUnited States
Peter Jachim [email protected]
DePaul University Chicago
ILUnited States
Meaningful Context, a Red Flag, or Both? Users' Preferences for Enhanced Misinformation Warnings on Twitter
misinformationsoft moderationwarningsTwitter
Warning users about misinformation on social media is not a simple usability task. Soft moderation has to balance between debunking falsehoods and avoiding moderation bias while preserving the social media consumption flow. Platforms thus employ minimally distinguishable warning tags with generic text under a suspected misinformation content. This approach resulted in an unfavorable outcome where the warnings "backfired" and users believed the misinformation more, not less. In response, we developed enhancements to the misinformation warnings where users are advised on the context of the information hazard and exposed to standard warning iconography. We ran an A/B evaluation with the Twitter's original warning tags in a 337 participant usability study. The majority of the participants preferred the enhancements as a nudge toward recognizing and avoiding misinformation. The enhanced warning tags were most favored by the politically left-leaning and to a lesser degree moderate participants, but they also appealed to roughly a third of the right-leaning participants. The education level was the only demographic factor shaping participants' preferences. We use our findings to propose user-tailored improvements in the soft moderation of misinformation on social media.CCS CONCEPTS• Security and privacy → Social aspects of security and privacy; Usability in security and privacy.
INTRODUCTION
Warnings and secure user behavior seems to have a perennially fraught relationship, despite the rich mediation of usability [2,16], interaction/visual design [19], and behavioral insights [59,81]. It is understandable that the complexity of this problem requires patience and eventual alignment between the security literacy of the average user and the pace with which new security hazards are introduced into users' daily life [17,28]. Usable security has, for one, made noticeable advancements of warnings that users do actually heed in conformance with the security recommendations: avoiding phishy websites and questionable attachments [57], skipping unencrypted communication [75], warming up to multi-factor authentication [40], and following up on system updates [43]. Advancements such as adaptive strategies for getting accustomed to warnings and security advice also help users transition to an acceptable secure behavior [22,29].
What actually is a bit difficult to understand is why, despite these advancements in usable security, warnings about misinformation on social media have made little progress in fostering desirable security behavior [69]. One could argue that the nature of the security hazard differs between the two settings -traditional programmatic security is far more complex to grasp than picking up on a causal post that links the COVID-19 vaccines with infertility -and that makes designing misinformation warnings an entirely different challenge. True, the one-size-fits-all here won't work because yesterday were the elections [32,84], today is COVID-19 and QAnon [6,49], and who knows what alternative narratives will emerge tomorrow. Embracing this predicament as a challenge in a usable security context has been sporadic so far, with the focus largely placed on mapping the "sources of misinformation" [13].
Misinformation sources won't go away. They existed long ago, learned to adapt and thrive in new information environments, and so long as the Internet evolves, they will too [58]. In the context of social media, these sources generate misinformation content that includes all false or inaccurate information such as: disinformation, fake news, rumors, conspiracy theories, hoaxes, trolling, urban legends, and spam [83]. It took some time for mainstream platforms to acknowledge that they have a serious problem on hands when misinformation started piling up [45]. They responded with warnings which conformed to the aesthetics of their interfaces and with language presumably appearing as unbiased and non-judgmental to users with diverse perspectives [72]. But this so-called "soft moderation" was applied halfheartedly, turning the warnings into hazards themselves-users started believing the misinformation more, not less, when a warning was explicitly appended to it [10,50].
The design of the warnings, thus, requires adaptation of the approach to retain their usability in various misinformation scenarios while avoiding a "backfire effect" [71]. To help this effort, we developed enhancements to the misinformation warning tags used by Twitter and evaluated them with a sample of 337 regular users. These enhancements address two elements that mainly contribute to the aforementioned predicament of soft moderation: meaningful context of the intentionally spread misinformation [83] and sufficiently potent interruption of the regular social media consumption flow [15,23]. Therefore, we formulated the warnings' text to fit the scenario surrounding a misinformation tweet and introduced red flag watermarks as a characteristic iconography of the visual frictions that users encounter in every aspect of their daily life [11].
The results of an A/B evaluation study with the warning tags currently employed on Twitter show that the majority of users do welcome the usable security enhancements. The added meaningful context was praised in helping participants avoid, ignore, and skip misinformation "right away. " The red flag watermarks were lauded for their "attention-grabbing" effect. Expectantly, there were also groups of users that leveraged this opportunity to express their protest against soft moderation as a way of forceful oppositionopinion-forming by Twitter as a self-appointed truth authority. Therefore, we analyzed the sentiment the warning tags incited and found that the enhancements did tilt the overall sentiment toward more positive from the status quo of the original Twitter warnings.
Sentiment often reflects users' political leanings and is shaped by the structure of their demographic identity [38,74]. Our results suggest that the left-leaning participants overwhelmingly welcomed the meaningful context, and the moderates and rightleaning joined them in lauding both the context and the red flag watermarks. Users' age, gender, and race/ethnicity did not factor in the sentiment in a significant measure; only the education level did. While users with either a high school education/GED or a college diploma were evenly split in preferring the original warnings and their enhanced counterparts, the users with some college education overwhelmingly preferred the latter. All but one of the users with a post-graduate level of education were entirely in preference for both the context and the red flag watermarks.
Scope and contribution of this work. With this work we aim to materialize the wealth of usable security cues, nudges, and advises in a social media environment where towards curbing misinformation. Our contributions, respectively, are:
The first A/B evaluation of enhanced social media warnings providing meaningful context and introducing visual design frictions in interacting with misinformation; Analysis of users' sentiment toward soft moderation in general and enhanced warning tags in particular from a political and demographic perspective; Basis and recommendations for user-tailored adaptations of soft moderation toward mindful and safe interaction on social media.
Following this introduction, we delve into the current state of misinformation warnings on social media in Section 2. We then elaborate on our usable security enhancement approach in Section 4. Section 5 provides the results of our A/B evaluation study and sentiment analysis. We discuss the results in Section 6 and provide our recommendations for the future of soft moderation before we conclude the paper in Section 7.
ACHTUNG! MISINFORMATION
Warnings on social media usually come in two main forms: (i) interstitial covers which obscure the misinformation and require users to click through to see the information; or (ii) trustworthiness tags which appear under the content and do not interrupt the user or compel action [36]. The former are more suitable for sensitive content where the exposure to the hazards should be avoided in the first place and the latter are usually applied to disputed or unverified content where the decision whether it is or not of misinformation nature is left to the user. But the COVID-19 "infodemic" demanded all hands on deck for soft moderation, and so mainstream social media platforms applied both warning variants to warn users of misleading and harmful COVID-19 information [45,62].
Evidence suggest that only the interstitial covers, but not the trustworthiness tags, make the users heed the warnings of misinformation [64,65,69,84]. It is tempting to simply discard the trustworthiness tags and only use interstitial covers, however. The interstitial covers do require additional clicks to get to the content in question, which could make the users avoid the content, but leave with a feeling that the social media platform is overtly imposing, "biased, " "punitive" or "restrictive of free speech" [33,64]. The trustworthiness tags might be more usable and mitigate the overt intrusion by blending with the visual aesthetic of the platforms interface (e.g. same colors, fonts, and obscure text), but they do run into other problems. Next to the "backfiring effect" [10,15,48,74], the tags could desensitize users to soft moderation when applied too frequently or create an "illusory truth effect" [51]. The absence of the tags in some scenarios might even create an 'implied truth effect" and lead users deem any misinformation content they encounter as credible and accurate [50].
Other factors also contribute to these negative effects, for example users' political affiliations and demographic identities. When trustworthiness tags directly challenged political falsehoods, they had the intended effect on Democrats but the opposite effect (e.g. they "backfired) on Republicans [74]. In the context of the COVID-19 pandemic, the tags resulted in a "belief echo, " manifested as skepticism of adequate COVID-19 immunization particularly among Republicans and Independents [69]. Usually older users with a level of education corresponding to higher analytical thinking succumb less to these negative effects [38]. Another factor is the asymmetrical nature of soft moderation-the mere exposure to misinformation often generates a strong and automatic affective response, but the warning itself may not generate a response of an equal and opposite magnitude [23]. This is because the trustworthiness tags often lack meaning, have ambiguous wording, or ask users to find context themselves which is cognitively demanding and time consuming [15]. Therefore, a natural step toward minimizing the said negative effects, is enhancing the trustworthiness tags to counter this asymmetry while keeping the appeal relevant for users of all ages, analytical prowess, and political leanings [38].
ENHANCED MISINFORMATION WARNINGS
The trustworthiness tags applied by Twitter make an interesting case of usable security interventions. Appended under suspected misinformation, this brand of tags warns after a user is exposed to the potentially harmful content [62]. Choosing to warn a user after-the-fact goes somewhat against the practice of using warning screens in browsers that come before a user gets a chance to visit a questionable website [19] (this effect is achieved with the interstitial covers, but they are verbose and disruptive of the natural social media consumption flow [7]). One could argue that the after-the-fact notification is chosen to counter "habituation", or the diminished response with repetitions of the same warning screens like these, or perhaps break the effect of "generalization" that might occur when habituation to these screens carries over to novel security interventions that look like the warning tags [79]. Seemingly designed to camouflage itself amongst the existing interface features, the warning tags are blue and not red in color, they do not obscure the suspected misinformation tweet, nor do they occur predictably like the warning screens every time an Internet browser cannot verify the visiting website's certificate (the tweets in question have to be fact checked, if not automatically flagged [32]). Twitter's warning tags might compare to the lock icons at the beginning of an URL bar in a browser indicating a "secure" connection [56]. Besides the habituation and generalization, the lock icons are confusing and don't convey the threat to the users in the first place so proposals have been made to pair the usable security iconography with words when possible [20]. Thus, it seems reasonable to pair an exclamation mark with a generic short text for warning users about misinformation tweets. But both the icon and the text are colored in the specific Twitter blue and fail to provide contrast to attract user's attention like the lock icons do with either red for "insecure" or green for "secure" browsing (alternatively a display of a locked/broken golden lock or strike-through the word "HTTPS"). Deliberately avoiding contrast makes it easier for users to overlook, ignore, or simply mistrust the warning tags as honest security aids [40].
Perhaps pairing the generic warning text with a link to a Twittercurated page or external trusted source containing additional information on the claims made within a suspected tweet could compensate for the lack of contrast. Often with a one-liner, users are offered for example to "get the facts on the COVID-19 vaccine, " "learn why health officials say vaccines are safe for most people, " or "learn how the voting by mail is safe and secure" [62]. The Fear of Missing Out (FOMO) aside [4], the warning text in fact advises users to contextualize the tweet themselves on the particular (mis)information topic. Users, unfortunately, rarely heed this advice and largely refuse to investigate any (mis)information further [24].
Security advice is not entirely anathema to users, particularly when it comes to their online security hygiene [55]. So it is not unreasonable to expect that users might heed the suggestion brought forth, brandished in a warning tag, if the advice itself provides a meaningful context for a particular topic of contention on Twitter without asking users to follow a link (which conflicts Twitter's own idea of curating "more accounts, and less links" in user's feeds [7]). Balancing for comprehensibility, we developed enhanced warning tags that provide meaningful context in regards (1) fabricated facts; and (2) improbable interpretations of facts. The enhancement choice follows the misinformation front put forth by Twitter and allowed us to conduct an A/B usability evaluation with the current warning tags applied to misinformation hazard. The enhanced warnings, in their tag-only variant, incorporate catchy acronyms as frictions indented to grab users attention in the absence of contrast [11].
We paired the text-only warning tags with the hereto ignored usable security intervention when it comes to misinformation: red flags as watermarks over suspected misinformation tweets.
The tag-and-watermark variant provided option for us to also test users' receptivity to warnings that incorporate contrast (red), gestalt iconography for general warnings (flag), and actionable advice for inspection (watermark). The choice of red flag was made after an extensive deliberation concerning warning design [12,82], warning cognition (automatic or System I; deliberate or System II) [44], and user experience design [19]. We decided against a smaller red flag as smaller labeling symbols were ignored on social media, e.g. Facebook used a small red box on the left with an exclamation mark and was either ignored or users believe the flagged post more, not less [61]. We decided to use red and not other color flags because a "red flag" is a common signal of oncoming danger and requires users to switch from System I to System II of cognition. Green usually signals "no danger" while orange or yellow signal "caution" but are often processed by System I cognition [60]. Red also has the highest "perceived hazardousness" on the color palette [82].
Fabricated Facts (SPAM)
The first text-only warning tag is shown in Figure 1a. We crafted a tweet, based on [37], and tagged for fabricated facts and presented it under a generic name, username, and without a profile image to avoid any threat to the validity of our A/B evaluation. Instead of advising the users to "get the facts about the COVID-19 vaccine" [62], we coined a catchy, yet familiar acronym: SPAM or Strange, Potentially Adverse Misinformation. With SPAM we wanted to see if we can contextualize the tweet's content, with an analogy to an already meaningful aspect of spam email, something most Twitter users have experience with [8].
We did break the one-liner rule for the warning text, but we opted for a minor engagement pain for a major gain in increased attention and warning adherence behavior. Our warning text following the SPAM acronym read: "If this was an email, this would have ended up in your spam folder." The overarching idea with the SPAM warning was to harness the "availability" and "recognition" heuristics characteristic captured in a Twitter flow [1,47]. Misinformation and fabricated facts are not always spam or vice versa, but anyhow align on the actionable outcome: ignore, delete, or take it with a grain of salt [52], which we argue is preferable compared to the "backfiring effect" of the generic warning tags [10,69].
The upgraded SPAM warning tag with a 50% transparency red flag watermark over the entire tweet is shown in Figure 1b. The "upgrade" bolsters the warning tag context along the same lines of "availability" and "recognition" heuristics by invoking the wellknown analogy between red flags and calls for attention. We opted for a watermark and not a replacement of the exclamation point inline the warning tag to avoid confusion with the red flag emoji frequently used on social media. The watermarking, centered in a ratio over the entire tweet area, follows the paradigm for misinformation flagging proposed in [71] with a midpoint transparency to create a non-negligible design friction for anyone attempting to read the tweet. By this choice, we wanted to stretch the overall text-and-flag warning throughout the suspected misinformation tweet and not only after it.
Improbable Interpretations of Facts (FFS)
The second set of warning tags is shown in Figure 2a and Figure 2b for the text-only and text-and-flag variants, respectively. Here we crafted a tweet, based on [46], containing an improbable interpretation of facts, keeping the engagement and posting structure in the similar order. In this case, we chose to provide a meaningful choice of context when tweets attempt to "spin" facts as a refined way of promulgating misinformation [18]. This practice, for example, earned Representative Marjorie Taylor Greene a permanent ban from Twitter [3]. Since we want to draw users' attention to such practices, we decided to ask whether they consider such tweets for For Facts' Sake or FFS, if not for anything else. We deliberately selected the acronym FFS to blend with the characteristic communication on Twitter that utilizes "compact language" due to the tweets' length restriction [86].
The FFS warning tag intended to provoke a pause in "recognition" heuristics since there are multiple meanings associated this acronym. We were aware that this might cause brief confusion, but nonetheless proceeded, since we wanted to explore if a brief confusion followed by contextual advice would suffice in refraining from taking the improbable interpretations of facts at face value. We utilized the growing evidence of "design frictions" purposefully created to disrupt mindless automatic interactions, prompting moments of reflection [11]. The brief confusion, promptly, is resolved by the following warning text advising users that "In this tweet, facts are missing, out of context, manipulated, or missing a source." To gauge the limits of the warnings-as-friction, the red flag watermark provides another stimulus to capitalize on by seeing what works as a resolution against the questionable content: incomplete factual presentation [80], lack of contextual consistency [34], overt factual manipulation [68], or obscure factual provenance [30].
.1 Research Questions
The evaluation of the enhanced warning tags was intended to gauge a preferential approach to soft moderation as well as understand the underpinning reasoning for it's acceptance (or lack thereof). A/B testing is a regular practice in usable security studies that informs the design of interface affordances, cues, and frictions [25,63,67]. Building on the exposure to contextual warning tags, a qualitative inquiry of how they fare in the misinformation front is important because the soft moderation employed by social media in general, and Twitter in particular, so far has yielded far from desirable results [39]. Users' often materialize their identity and political personas within social media and Twitter discourse [26,70], therefore we also investigated how this materialization shapes the preferences for our proposed soft moderation nudges. Based on this argumentation, the resulting research questions were:
• RQ1: What are the preferences of Twitter users for the SPAM and the FFS enhanced misinformation warning tags in both the text-only and text-and-flag variants? • RQ2: How effective are the SPAM and the FFS, enhancements in dispelling fabricated facts and improbable interpretations of facts? • RQ3: What is the relationship between the Twitter users' preferences for the enhanced misinformation warning tags and users' political leanings? • RQ4: What is the relationship between the Twitter users' preferences for the enhanced misinformation warning tags and users' demographic identities (race/ethnicity, level of education, gender identity, age)?
Recruitment
Our study was approved by our Institutional Review Board (IRB) before any research activities began. Subsequently we set to sample a population that was 18 years or above old, regular Twitter users from the United States through the Amazon Mechanical Turk. Both reputation and attention checks were included to prevent input from bots and poor responses. The survey took around 20 minutes and participants were compensated with the standard participation rate ($18 per hour). Participants were randomly assigned to either the A/B evaluation of the text-only or text-and-flag enhanced warning variants. The survey was anonymous and allowed users to skip any question they were uncomfortable answering. We refrained from exposing the participants to similar stimuli to prevent from generalization and obtain a direct comparison to the original warning tags on Twitter. We also randomized the order of each of the SPAM and FFS text-only and text-and-flag segments for each participant. We selected the content of the tweets to be of relevance to the participants so they could meaningfully engage with the tweet's content and see a clear relationship between the tweet and the warning tag (i.e. to prevent arbitrary and irrelevant responses). The two COVID-19 related tweets represent the main target of soft moderation front by Twitter during the execution of the study [November 2021 -January 2022] [76]. We selected one misleading tweet by Nate Silver [46], and wrote a second one based on a common piece of vaccine misinformation [37]. To account for accessibility, we provided alternative text describing each of the tweets and interventions we used to avoid visual misinterpretation. We assumed participants understood the Twitter interface, the tweets, and the warning tags.
Method and Instrumentation
Participants first indicated the reasons they usually come to Twitter for. Next, each participant was asked to indicate if they encountered warning tags and what were the content and the warnings about. We were aware that not every participant might have been exposed to warning tags so we included a small training segment where we created exposure to the concept of soft moderation with generic warning tags. The pre-exposure training, shown in the Appendix, was used to ensure a baseline understanding of content moderation among the participants, i.e. that Twitter uses content indicators for various types of contents (misinformation, sensitive content, graphic content, etc.). The training was general and referred to only "content indicators" without any references to "misinformation to avoid any potential impact on user responses.
Participants then were asked to evaluate each of the enhancements in comparison to the original tag ("Get the facts about COVID-19") [62]. Participants were next asked if seeing an enhanced warning tag would influence their dismissal of the tweet or tweets on the same contested topic as misinformation. Finally we collected participants' political leanings, race/ethnicity, level of education, gender identity, and age.
The qualitative responses were coded and categorized in respect the preference and the justification for it. These categories later helped perform a chi-square statistical analysis ( ) of the relationships between the preferences and participants' political leanings as well as their demographic identities. We performed a basic exploratory analysis of the preferences and justification to uncover the aspects in which the enhanced warning tags fair well (or vice versa) as a usable security nudges against misinformation. For each of the justifications in the open-ended questions we performed a sentiment analysis using the Valence Aware Dictionary for Sentiment Reasoning (VADER) [14,31,35]. VADER yields a compound score between -1 for a very negative piece of text, and 1 for a very positive one. We also used a Linguistic Inquiry and Word Count (LIWC) analysis to qualify the sentiment expressions in the responses respective to clout and tone [73]. Each one ranges between 0 and 100 with scores close to 0 indicates less confidence and weak argumentation (clout) or negative emotions (tone).
Finally we performed a Correspondence Analysis (CA) on a contingency table with rows of adjectives/verbs as keywords and the justification text as columns. The CA projects the variance in justification onto two dimensions using a weighted single value decomposition [27]. In CA, the further away the keywords are from the origin of the plot, the more discriminating they are, and smaller angles between a pro/against preference and a keyword (connected through the origin) indicates an association of the two. In our case, the two axes correspond with justifications' keywords (y-axis) respective to the participants' pro/against preferences (x-axis).
RESULTS
After the consolidation and consistency checks, a total of 337 participants have completed the study, with 176 in the text-only and 161 in the text-and-flag warning tag groups, respectively. Users indicated that communication was the most frequent factor for coming to Twitter (85.4%), followed by entertainment/cultural awareness (71.8%), news (63.5%), politics (46.5%) and health (26.7%). Around every third participant (32.9%) has encountered some form of a warning tag as part of Twitter's soft moderation effort in general.
The distribution of participants per their self-reported political leanings was: 147 (43.6%) left-learning, 96 (28.5%) moderate, 61(18.1%) right-leaning, and 33 (9.8%) apolitical. In respect to race and ethnicity, 247 (73.3%) identified as White, 29 (8.6%) as Black or African American, 42 (12.5%) as Asian, 12 (3.6%) as Latinx To ensure consistency in the analysis and validity of the results, each of the open-ended responses in the survey was coded independently by three researchers. The codebook was simple and included a coding on the preference expressed for the A/B evaluation as well as codes for the preference justification quotes from the participants. The Fleiss's kappa , as a measure of inter-coder agreement, was 0.960 on average with a 0.878 lower bound for the 95% confidence, which indicates an "excellent" inter-coder agreement overall.
Fabricated Facts (SPAM)
5.1.1 A/B Evaluation. The breakdown of preferences for both variants of the SPAM warning tag is given in Table 1. In the text-only variant, more than a half of the participants who preferred the original warning tags explicitly echoed a protest against Twitter's intrusion in contested matters such as COVID-19 vaccination. Verbosity and confusion was cited by roughly one out of five participants as a preference against the SPAM. The same number of participants didn't provided any justification. A small number of participants judged the SPAM tag as misaligned with Twitter's aesthetic and therefore, illegitimate. Neither of the text-only warning tags was the choice of 12.6% of the participants.
The SPAM text-only warning tag (Figure 1b) received the highest preference (46.3%). The meaningful context provided by the extended security advice was welcomed by 43.2% of them indicating that "The SPAM explanation is a valid one, and makes sense in the context of the tweet's content. " The on-point warning of questionable content was cited by 36.8% in preference of "a direct misinformation label right there without having to dig further into it. " One tenth of the pro-SPAM participants found the acronym and the text catchy, cheeky, and positively attention-grabbing. Reluctance to follow the links in the original tag variant was cited by 6% of the participants. Only 4% didn't provide any justification.
The pairing of the red flag with the SPAM warning tag was either too distracting or an indicator of Twitter's intrusion into the way content should be consumed. The preference against the text-and-flag SPAM tag was expressed in terms of "visual clutter that makes the tweet more difficult to read", "Doomsday level of importance", or "symbol of political hate". The pro text-and-flag SPAM tag participants welcomed the attention grabbing of the red flag suggesting that "the flag gets your attention; the text tells you it is misinformation -I tend skim when reading twitter posts and the other one is not as noticeable.". The enhanced context and the on-point warning for misinformation was preferred because " the flag reinforces the positive information that the tweet is spam".
Sentiment
Analysis. The sentiment analysis of the preferences for SPAM warning tags is shown in Figure 3. The violin plots show a multimodal distribution of sentiments where the original warning tag received an equal number of positive sentiments for being "simple and straightforward" as well as negative sentiments that "rather not see Twitter's judgement on whether something is misinformation or not". The justifications showed low confidence (clout = 26.15) but positive emotions (tone = 60.65). The text-only SPAM positive sentiment outweighs the negative one that captures justifications indicating that "'B' does a better job letting you know that the tweet's information is bad", with a bit more confidence (clout = 32.59) and on par with the positive emotions (tone = 62.74).
The introduction of the red flag in the SPAM warning tag apparently induced more negative sentiment when justifying the choice for the original warning tag. The justifications were a bit more convincing (clout = 30.91) but the emotions were highly negative (tone = 7.61). The red flag increased the positive sentiment for the text-and-flag SPAM warning tag with the most confidence of all justifications (clout = 34.11) and positive emotions (tone = 55.52). While the participants that were neither "A" or "B" were evenly distributed in the text-only variant, the negative sentiment was dominant in the text-and-flag variant. Both being very low on confidence and high on negative emotions, the introduction of the flag might have exacerbated the feelings against the soft moderation for some of these participants. Additionally, we performed a CA to review the adjectives used in explanations for user preferences for the SPAM warning tags. In the first component on the x-axis, which accounts for 56.57% of the inertia in the justifications, all but three keywords show values larger than zero. This suggests a bit more consistency in the way that the preferences for both the "A" and "B" text-only options were worded. Put it simply, the predicative/comparative "more, " "clear, " and "better" adjectives were associated with the text-only SPAM tag, while the "own" and "true" with the original warning. The tendency for the prior is a praise of the enhancements themselves while the latter hints of a general contempt for soft moderation on Twitter. The less discriminating "wrong, " and "false" echo a similar sentiment by the neither "A" or "B" participants in the text-only variant. The remaining keywords show values less than a zero, indicating that the adjectives used to justify those selections were generally less consistent, outside of the trend of expressing the preferences for text-and-flag SPAM with the keywords "obvious" and "red. "
Dispelling Fabricated Facts.
The A/B evaluation only obtained the preference for the SPAM warning tags without explicitly asking the participants to consider the security advice as applied to the tweets containing fabricated facts. To see if the SPAM warning tags actually work, we ask the participants to indicate if the tags helped them dispel fabricated facts in the example tweet. The results shown in Figure 5 indicate that the SPAM warning tags doesn't have to be users' best choice in order to work. Roughly half of the ones that preferred the original warning tag commented that the text-only "helped them understand the meaning of the tweet in a broader context." In the text-and-flag participants found the warning tags helpful too rationalizing that "Twitter should just remove the whole post in general if it comes to a big red flag watermark. " Even some of the neutral participants noted that the warning tag was reassuring on the inaccuracy of the content. Overall, 62% of the participants indicated that the SPAM warning tags worked for them with the desired effect of dispelling the fabricated effects of the COVID-19 vaccines.
Preferences and Political
Leanings. The COVID-19 pandemic didn't escape deep politicization and that naturally was reflected [49]. We were interested, therefore, to see if participants' preferences are affected by their political leanings. For both SPAM warning tags variants, as indicated in Table 5, the Pearson's Chi-Square test yielded a statistically significant relationship between their choices and where they stand on the political spectrum: (3) = 24.934, = .000 * and (3) = 24.611, = .000 * , respectively. The original tags are appealing to left-learning with a 1:1 ratio to the moderate and 2:1 ratio to the right-leaning participants. The text-only SPAM variant has these ratios increased to a 4.5:1. Here, the neither "A" or "B" participants are uniformly distributed. The introduction of the flag tipped the left-leaning with a 1:1.3 ratio to the moderate and with a 1.3:1 ratio to the right-leaning ones that preferred the original warning tag. Left-leaning preferences for the text-and-flag SPAM warning tag were 1.56:1 with the moderates, but 4.875:1 with the right-leaning participants. The moderate and right-leaning were the most present for the neither "A" or "B" in the text-and-flag variant. Overall, the context is useful for the left-leaning and moderate participants the most, with a considerable portion of the moderates and right-leaning preferring a minimum intervention and distraction from Twitter.
Preferences and Demographic
Identities. The demographic identities, as the earlier evidence suggests [69,84], factor in the way (mis)information is consumed from Twitter. Our analysis didn't find any significant relationship between the demographic identities and the preferences except between the education level and the textand-flag SPAM variant: (3) = 17.328, = .008 * . The enhanced tag, as Table 3 reveals, roughly evenly splits the high school/GED and college graduates' preferences but almost entirely earns the preferences of the ones with a post-graduate degree. It also does so with a 3:1 ratio for the participants with some college degree. Verbosity and confusion was the reason for almost two thirds of the participants to dislike the text-only FFS warning-tag. Roughly one third disliked it because of an anti-soft-moderation stance and one tenth provided no justification. The meaningful context provided by the FFS text-only warning tag (Figure 2b) was welcomed by almost 70% of the participants "because it doesn't just say that the tweet is disputed, it mentions the various ways that the tweet is incorrect. ". One out of ten participants liked that the FFS text-only warning tag because of the "assertive statement as opposed to just one word 'disputed' in 'A'. 'B' is more specific.". A small number deemed the acronym as "funny/witty" and 14.5% simply just liked the FFS security advice.
The preference against the text-and-flag FFS tag was again expressed in terms of destruction by more than a half of the participants preferring the original tag. A third of them cited the contempt for Twitter's decision to patronize users about how to interpret facts. A bit more than one tenth of the pro-original warning tags didn't provide justification. The participants pro the text-and-flag FFS liked the attention grabbing effect of the red flag noting that they "like that the red flag is big; You can see right away there is a problem with the tweet. " in roughly half of the cases. The context (34.2%) and the on-point warning that the tweet is a form of misinformation (11%) was preferred because "knowing that something is missing context is more informative than knowing it's disputed; Everything is disputed by someone. ". Only 6.9% didn't provide justification pro the text-and-flag FFS warning tag.
Sentiment
Analysis. The sentiment analysis of the preferences for FFS warning tags is shown in Figure 6. As the violin plots demonstrate, the original warning tag received roughly an equal number of positive sentiments for the "simple and straightforward and it doesn't try to make a judgment of the tweet" as well as negative sentiments that "the red watermarking is overkill regardless of placement and size.". The justifications showed again showed low confidence (clout = 21.15) but positive emotions (tone = 67.72). The text-only FFS positive sentiment further outweighs the negative one praising the tag's way of "explaining why the facts are probably being used in a misleading way.". The praises show twice as more confidence as the ones for the original warning tag (clout = 51.93) and more positive emotions (tone = 72.32). The red flag in the FFS warning tag again caused a shift toward more negative sentiment as was cast as "condescending" and "too distracting". The confidence plummeted in response to the flag-andtext variant (clout = 19.3) with the emotions remaining negative (tone = 31.61). The positive sentiment is prevalent with the pro FFS text-and-flag tag participants, which wielded a tad better justifications (clout = 30.95) and expressed more positive emotions (tone = 60.52). The red flag again tilted the balanced sentiment of the neutral participants in the text-only variant toward a more negative one in the text-and-flag variant.
The CA for the FFS A/B evaluation is plotted in Figure 7. For brevity, we used verbs as keywords here as the adjectives showed very similar dimensionality in the SPAM case (and vice versa). Here, the first component on the x-axis, accounting for 48.67% of the variance in justifications, shows the responses in order of preference from left (least popular), to right (most popular). Verbs used in explanations for some of the less popular choices include "disputed," and "seems," which both are terms that indicate more ambiguity in the truth ( Option "A" in the text-only comparison), and "know," which indicated more confidence ( Option "A" in the text-and-flag comparison). Justification for more popular responses include the verbs "prefer, " and "like, " which suggests approval for both FFS variants rather than a dislike for the original text-only warning. The most closely associated keyword with the text-only FFS warning tag is "tells, " which is an appreciation for the informal yet meaningful context conveyed. The y-axis, accounting for 27.54% of the variance, shows the Option "B" preferences center around the origin as an indicator of higher consensus between the pro FFS. Figure 8 shows that the detailed context provided through the FFS security advice is even more potent in dispelling improbable interpretation of facts. Overall, 68% of the participants indicated that the FFS warning tags worked for them, which is a 6% increase from the dispelling rate for the SPAM warning tags. Roughly half of the participants preferring the original tag conceded that the FFS warning tags in both variants are helpful in discrediting the manipulative tweet. A small but noticeable increase in the dispelling effect is also present for the neither "A" nor "B" participants compared to the SPAM warning tags. Similarly, the participants preferring both FFS tags were slightly
Dispelling Fabricated Facts.
Preferences and Political
Leanings. The preferences for both FFS warning tags variants, as indicated in Table 5, were related with a statistical significance to participants' political leanings:
(3) = 27.732, = .000 * and (3) = 36.483, = .000 * , respectively. The original tags are appealing to left-learning participants with a 1:1 ratio to the moderate ones and with a 2.5:1 ratio to the rightleaning participants. The text-only FFS variant has these ratios increased to a 3.6:1 between the left-leaning and the moderate participants and 4.8:1 between the left-learning and right-leaning participants. Unlike the SPAM variants, here, the neither "A" or "B" participants are dominantly right-leaning with a 2:1 ratio to both the left-leaning and moderate participants.
The introduction of the flag again kept the balance between the left-learning and moderate participants, but increased the ratio to almost 2:1 to the right-leaning ones that preferred the original warning tag. The left-leaning preferences for the text-and-flag FFS warning tag were in a 1.65:1 ratio with the moderates, but in an overwhelming 5.42:1 ratio with the right-leaning participants. The right-leaning again dominate in the neither "A" or "B" preferences for the FFS text-and-flag variant. Compared to the SPAM case, the extended FFS context is even more useful for the left-leaning participants. The moderates are roughly evenly split, but the right-leaning participants show a more salient anti-soft-moderation preference when exposed to the FFS warning tags.
Preferences and Demographic
Identities. Same as before, only the level of education mattered when it comes to the preferences. The Pearson's Chi-square tests revealed a significant relationship in this case with (3) = 17.773, = .007 * . As the Table 6 reveals, the high school/GED and the college graduates are slightly more in preference for the original tags, with a considerable dismissal for the soft-moderation altogether by the college graduates. The participants with some college-level education are 2:1 in ratio to the preference for the FFS text-and-flag variant with the ones preferring the original tag and 3:1 with the ones without a preference. The biggest difference is in for the participant with a post-graduate education level -they are almost entirely in favor of the FFS way of warning against improbable interpretations, manipulation, or selective choice of facts.
DISCUSSION
In this study we were motivated to bring soft moderation closer to users' everyday experiences while minimizing imposition, which as witnessed, often backfire [71]. We distinguished between a need for context when the hazard comes from the fabrication of facts and when the hazard comes from the interpretation of facts in a rather improbable way. In the first case, we were careful to avoid the perception trap of "correction of feelings, not falshoods" [41] and used an analogy with spam emails. We did so because users, by now, can recognize spam when they see it [9] and accept that spam filtering, performed by email providers, works well [53]. Understanding this, we wanted to regain the trust in the provider -Twitter in the case of the warning tags -and signal absence of bias or judgment in their action [42]. With this in mind, the SPAM warning tag shows a very promising step toward unified interpretation and increased trust in soft moderation (only related to COVID-19 misinformation, for now). If support from left-leaning participants was already hinted at from previous studies on soft moderation, it was nonetheless strongly reinforced in both the text-only ("...it tells participants, rather quickly, that the tweet is garbage" and text-and-flag variants ("the red flag will alert me before I even read any of it". Moderates were evenly split, expectedly, but reassured that the text-only variant "really tells you more of what is going on" while the text-and-flag variant "gives more specifics and is thus tougher to refute". In significant numbers, right-leaning participants made it clear that the text-only variant seems more appropriate because it's far more specific; the original tag feels more like an ad and nothing that I didn't already know." and praised the text-and-flag variant as "a large visual cue that's hard to ignore and will bring attention to the idea that something is going on with this information. ".
More promising evidence for the SPAM approach is the support across all levels of education without distinction of age, gender, or race/ethnicity. In the text-and-flag variant, only 10% of the participant with only high school education/GED disliked both the "A" and "B" options while the rest gave equal support of 40% for each option. Even though the participants with college diplomas tilted toward the option "A" (a relative difference of 8%), the group unequivocally acknowledged that the text-and-flag " is more clear and strong, and tells you exactly what is incorrect". After all, the SPAM tags helped more than 60% of all the participants to dispel the fabricated facts about COVID-19 vaccine side effects.
In the second case, we wanted to avoid authoritative imposition and thus worded the warning not to personify senior public health experts, usually responsible for interpretation of facts [77]. We also opted for a "bold" acronym choice to lure users' attention to the text of the warning tag, for a moment, instead of the warning tag as soft moderation. Once "hooked, " the cost to read the entire warning tag text was less then avoiding it as the derivation of new meaning to acronyms and words is a pragmatic way of conveying context on social media -take for example the hashtags on Twitter [66]. The text wasn't asking the user to "get facts" or "learn more," but instead, it gave several convincing options for users themselves to pick why the context is fitting to the possibly misinformation tweet [54].
The FFS tag did just that and succeeded. Left-learning participants liked that the text-only variant "gives real reasons why this tweet is suspect" and moderates seconded that the FFS's context "goes more in depth and makes you more alert to the tweet". Rightleaning participants confirmed our idea to avoid any relationship to an imposing authority: "The context in 'B' is better because Facebook came out saying that most if not all of their fact checkers don't check for facts, they just do it on opinion base. I'm sure Twitter does the same". The consensus across the participants of all political leanings that the "red flag watermark was really draws more attention", lead by the left-leaning ones, supports the potency of the FFS acronym as the "hook" entirely absent in the current soft moderation on Twitter.
The FFS text-and-flag variant appealed almost entirely to all participants with a post-graduate level of education. Interestingly, they were concerned not just for themselves but other users on Twitter and misinformation in general, noting that "it's important people really pick up on the fact this information might be misleading". So were the participants with a college degree even though they again tilted toward the original tag "A": It seems like option 'B' would help resolve the problems that false news or fake profiles create. The participants with some college experience, in favor of the FFS tag, pointed out that a "disputed facts" warning is less informative than a "missing facts" warning. The support from the participants with only high school education/GED underlined the essential usability of the warning itself: "It makes it known that something is up with this post and I shouldn't trust it 100% without doing more research. ". Overall, the FFS achieved a 68% effectiveness in dispelling an improbable interpretation of COVID-19 related facts among all participants.
Observed Backfire Effect
We did observe, albeit anecdotally, the backfire effect in 1.48% of the participants' response (all politically right-leaning). One participant, who was pro enhanced warnings, even provided a testimony of the backfire effect: "I have seen some of my crazy friends of mine where they think if Twitter disputes it, then it makes it ever more correct.". The warning tags in the original option were blamed that "lead you to the lying, paid, 'fact checkers"'; The enhanced warning tags were dismissed because they "force an opinion on you and suppress a side that has been more accurate than the CDC and Fauci so far since COVID". Few participants even declared that the warning tags "makes them leave Twitter entirely", perhaps rappelled by the Twitter's sweeping COVID-19 misleading policy from December 2021 [78].
The contempt for Twitter's soft moderation was made clear in the responses of a considerable group of participants, stating that "Twitter is not a medial expert. " Roughly a half preferring the original warning tags versus the SPAM variants cited Twitter intrusion into opinion formation as a choice for the "lesser of the two evils. " This fell down to a third of the Option "A" supporters in the case of the FFS, but considering that around 15% of the overall participants did not have a preference for either of the options is an indication that soft moderation has still a lot to do to appear unbiased and non-judgmental to users with diverse perspectives [72].
User-tailored Soft Moderation
Our results reveal several aspects worth considering for improving the soft moderation appeal among the Twitter users. There is no doubt that the meaningful context is useful but runs the risk of being avoided due to verbosity/confusion. In the SPAM case, a possible variation would be keeping just the acronym with a bit of text rewording, for example SPAM: Content like this usually ends in spam folders. This improvement avoids the words "strange, " "adverse" and "misinformation" while indirectly hints that it should be handled on user's discretion. Plus, it becomes a one-liner warning appearing more of a suggestion than an opinion voiced by Twitter, as several of our participants complained about.
Because the warning plays on the experience with spam emails, we also think it's worth testing the email iconography in line with the warning as shown in Figure 9. In this example, we borrowed the icon from Gmail's spam folder, but certainly could use any hexagram with an exclamation mark that provides contrast. This could also be an alternative to the red flag watermark to avoid participant recoiling from the sudden splash of red while still having an attention grabbing effect. Similarly, this could address the concerns for illegitimacy of the enhancement cited by some of the participants. The context conveyed by the FFS tags was well received, but some participants, expectedly, expressed concerns about the "catchy" nature of the acronym. It is therefore worth testing dropping the acronym altogether or replacing it with simply the word "facts" as shown in Figure 10. Here, the preceding iconography changes to a question mark, retaining the element of "hook" we envisioned in the first FFS variant. The following text is essentially the same, blending the acronym to look more "professional," as the participants expected. The red flag watermark, the results confirm, produces the desired effect of attention-grabbing. However, adaptations could be made here too. Participants commented on the size and the transparency, so variations could include testing options where these two variables are determined by the level of confidence of fact checkers or the engagement it attracts over time, as suggested in [71]. The watermark display could vary based on a particular user's content preferences, e.g. one group of users could see a red flag and another could see the words "red flag" as a watermark. Twitter already uses this approach to suggest adds and content in the users' feeds [7].
Ethical Considerations
Ethical concerns do arise when dealing with misinformation, or allegedly harmful information, within the pluralistic social media population. The tension between impartiality, profitability, and social responsibility of the platforms might not always ensure that misinformation is dealt with using consistent soft moderation criteria. With the honest, yet inevitable false-positives/negatives, the proposed enhancements -if applied -might be seen as unfair at best or simply harmful at worst. We therefore are open for democratic participation in the design that allows for remediation of concerns in such instances. Soft moderation, at least in our view, is a form of honest communicative action rather than an authoritative and absolute determination of truth, and as such, beneficial to all Twitter users without discrimination [5]. We are aware that facts change, become irrelevant, or are refuted over time so a retroactive application should also be considered to enable versatile soft moderation to the best of our (and Twitter's) abilities.
Limitations
We note several limitations of our study, which could be addressed in future work. The size of the sample could be enlarged to obtain an as varied as possible Twitter population. We used only two examples of misinformation on COVID-19, which is a limitation steaming both from restricted financial resources and limited attention span of participants [36]. An extended, or perhaps a longitudinal study that incorporates more COVID-19 misinformation instances over a time could not just help generalize our findings, but reveal important behavioral patterns in dealing with soft moderation. Also, it could help with an A/B evaluation for warning tags pertaining other contested topics such as elections [84]. Participants were exposed to generic formatting of the tweets emphasizing the content and the warning tags. In reality, misinformation could come from individual accounts, influencers, or accounts controlled by nefarious actors [85]. Misinformation is often amplified by social bots, and appears in users' feeds next to other posts, adds, with variable degree of visual interference [21]. All of these aspects could influence the preferences for or against soft moderation. Controlling for them will require a study executed in partnership with Twitter where the enhancements are tested with selected users on the live platform.
Such a test could not just capture the preferences of the regular Twitter users but help closely observe the "backfiring," "implied truth, " and "illusory truth" effects. We didn't explicitly test for these in our study, but it is important to track how misinformation itself materializes in the individual Twitter consumption. Our A/B evaluation is limited by the current formatting of the original warning tags on Twitter [62]. If Twitter chooses to reformat the tags, eliminate the links, or place them elsewhere, the enhancements also should change and the results might not hold for these new conditions.
CONCLUSION
This paper conveys the first extensive A/B evaluation of enhancements for misinformation warnings on Twitter. Providing users a meaningful context and attention-grabbing iconography, our results suggest, does help users recognize and contain COVID-19 misinformation. We weren't poised to solve the predicament of soft moderation in one shot; rather, the goal was to utilize the usable security body of knowledge to trace a path toward "inoculation" against information hazards on social media.
Figure 1 :
1Warning Tags Contextualizing Fabricated Facts
Figure 2 :
2a) Text-only FFS Warning Tag (b) Text-and-flag FFS Warning Tag Warning Tags Contextualizing Improbable Interpretation of Facts 4 EVALUATION STUDY 4
, 3 ( 0 .
09%) as Native Hawaiian or Pacific Islander, and 4 (1.1%) as Other. Education-wise, 71 (21.1%) of the participants had a high-school level, 57 (16.9%) some college, 175 (51.9%) 2-or 4-year college, and 57 (10.1%) had a gradate level of education. Gender-wise, 154 (45.7%) of the participants were female, 169 (53.4%) were male, and 3 (0.9%) identified as non-binary. Age-wise, 9 (2.7%) were in the 18 -24, 87 (25.8%) in the 25 -34, 136 (40.3%) in the 35 -44, 64 (19%) in the 45 -54, 33 (9.8%) in the 55 -64, and 8 (2.4%) in the 65 -74 bracket.
Figure 3 :
3Sentiments: SPAM Warning Tags. The violin plots show a multimodal distribution of sentiments (number of responses) over the VADER sentiment score ranging from -1 (negative) to +1 (positive) sentiment.
Figure 4 :
4Correspondence Analysis: SPAM Warning Tags
Figure 5 :
5Dispells: SPAM Warning Tags around the soft moderation effort of Twitter following the ban of President Donald Trump
Figure 6 :
6Sentiments: FFS Warning Tags. The violin plots show a multimodal distribution of sentiments (number of responses) over the VADER sentiment score ranging from -1 (negative) to +1 (positive) sentiment.
Figure 7 :
7Correspondence Analysis: FFS Warning Tags more assertive of the desired effect compared to their responses for the SPAM tag.
Figure 8 :
8Dispels: FFS Warning Tags
Figure 9 :
9Update: Text-only SPAM Warning Tag
Figure 10 :
10Update: Text-only FACTS Warning Tag
Table 1 :
1SPAM: Preferences Telling me something is spam is an opinion concerning this topic and feels intrusive to trying to control my opinions.19.4% Verbosity/confusion Because it's simple and straight to the point. The SPAM is confusing and too wordy.It tells me right away why Twitter marked it as misinformation so I don't have to wonder the reason on my own. I can also easily decide if I agree and move on or research further outside of Twitter. I love this and would be happy to see this on posts.I prefer the red flag, as it is impossible to miss. I often read Twitter on my phone, while I take the dog out and such, so I find myself thinking I should look something up after reading a tweet, but then I get busy doing other things and don't follow up.25.7% Meaningful ContextI like the red flag for sure -and the warning tag beneath gives a better commentary on why there was a red flag.22.9% On-point warningThe red flag makes it very obvious that the material is potentially false and can't be trustedOption
Pct.
Justifications
Representative Quotes
Text-only Warning Tags
Original (A)
(41.1%)
55.6% Twitter intrusion
19.4% No justification
Warning Tag A.
5.6%
Legitimate
Get the facts seems more legit to me.
SPAM (B)
(46.3%)
43.2% Meaningful Context Letting me know something is SPAM and dangerous is more useful than telling me where to find facts;
36.8% On-point warning
10%
Attention Grabbing
"B" is better at catching the attention of the reader. "A" could just be a service announcement -it just isn't
strong enough.
6%
Link reluctance
It's more detailed and explains why its there without having to click on anything
4%
No justification
Warning Tag B.
Neither (12.6%)
Neutral
I wouldn't utilize either.
Text-and-Flag Warning Tags
Original (A)
(37.9%)
44.2% Distracting Flag
Seeing the red flag almost makes the tweet look like it is harmful or not true at all. It stands out too much.
42.6% Twitter intrusion
I would prefer the original warning tag. "B" is too opinionated and biased.
11.6% No justification
Warning Tag A
SPAM (B)
(46.0%)
41.9% Attention Grabbing
9.5%
No justification
Warning Tag B
Neither (16.1%)
Neutral
I wouldn't prefer any of them.
Table 2 :
2SPAM vs Political LeaningsPolitical Leanings
Option
Left
Moderate
Right
Apolitical
Text-only Warning Tags
Original (A)
27
27
14
0
Spam (B)
54
12
13
0
Neither
6
5
5
1
Text-and-Flag Warning Tags
Original (A)
18
23
14
2
Spam (B)
39
25
8
0
Neither
2
4
7
2
Table 3 :
3SPAM vs Education Level The breakdown of preferences for both variants of the FFS warning tag variants are given inTable 4. In the text-only variant, only one third preferred the original tag and more than half of the participants choose the FFS text-only tag.Education Level
Option
High
School/GED
Some
College
College
Post-
Graduate
Text-and-Flag Warning Tags
Original (A)
9
5
42
1
Spam (B)
9
15
35
13
Neither
2
1
11
0
5.2 Improbable Interpretation of Facts (FFS)
5.2.1 A/B Evaluation.
Table 4 :
4FFS: PreferencesOption
Pct.
Justifications
Representative Quotes
Text-only Warning Tags
Original (A)
(33.5%)
60%
Verbosity/confusion Because the other is just too many words. It just needs to be simple to understand.
31.6% Twitter intrusion
Twitter is bad enough when it tries to manipulate and control their own agendas. I don't want to see
more additional information.
9.4%
No justification
Warning Tag A.
FFS (B)
(50.8%)
69.2% Meaningful Context
I would rather see context. It would bother me that some facts are missing and that I don't have the whole
story. Vaccinations are too important of a topic to be misconstrued.
10%
On-point warning
Because it explains right off the bat that this content is manipulated or missing a source.
6.6%
Attention Grabbing
"B" is engaging with the funny acronym.
14.2% No justification
Warning Tag B.
Neither (15.7%)
Neutral
I wouldn't utilize either.
Text-and-Flag Warning Tags
Original (A)
(38.4%)
54.1% Distracting Flag
It doesn't have the big red watermark that might make people feel like victims.
34.4% Twitter intrusion
Human beings are simple creatures, and they do not respond well to being patronized. The latter is
patronizing.
11.5% No justification
Warning Tag A
FFS (B)
(46.0%)
47.9% Attention Grabbing
Option B really draws your attention and is impossible to miss or misunderstand.
34.2% Meaningful Context It's important people really pick up on the fact this information might be misleading.
11%
On-point warning
The flag big and bold and it will tell me easily what to avoid and what is problematic.
6.9%
No justification
Warning Tag B.
Neither (15.6%)
Neutral
I wouldn't prefer any of them.
Table 5 :
5FFS vs Political LeaningsPolitical Leanings
Option
Left
Moderate
Right
Apolitical
Text-only Warning Tags
Original (A)
25
23
10
0
Spam (B)
58
16
12
0
Neither
4
5
10
1
Text-and-Flag Warning Tags
Original (A)
20
26
12
2
Spam (B)
38
23
7
0
Neither
1
3
10
2
Table 6 :
6FFS vs Education LevelEducation Level
Option
High
School
Some
College
College
Post-
Graduate
Text-and-Flag Warning Tags
Original (A)
11
6
42
1
Spam (B)
8
12
35
13
Neither
1
4
11
0
Conference'17, July 2017, Washington, DC, USA Authors
APPENDIX Pre-Exposure TrainingContent indicator is defined as a label that is assigned by Twitter under a Tweet in blue font preceded by an exclamation mark as shown in theFigure 11. Content indicators could be assigned for various types of contents, such as: misinformation, sensitive content, graphic content, etc.
Nudges for Privacy and Security: Understanding and Assisting Users' Choices Online. Alessandro Acquisti, Idris Adjerid, Rebecca Balebako, Laura Brandimarte, Lorrie Faith Cranor, Saranga Komanduri, Pedro Giovanni Leon, Norman Sadeh, Florian Schaub, Manya Sleeper, Yang Wang, Shomir Wilson, 10.1145/305492650Alessandro Acquisti, Idris Adjerid, Rebecca Balebako, Laura Brandimarte, Lor- rie Faith Cranor, Saranga Komanduri, Pedro Giovanni Leon, Norman Sadeh, Florian Schaub, Manya Sleeper, Yang Wang, and Shomir Wilson. 2017. Nudges for Privacy and Security: Understanding and Assisting Users' Choices Online. 50, 3, Article 44 (aug 2017), 41 pages. https://doi.org/10.1145/3054926
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Twitter Permanently Suspends Marjorie Taylor Greene's Account. Davie Alba , Davie Alba. 2022. Twitter Permanently Suspends Marjorie Taylor Greene's Account. https://www.nytimes.com/2022/01/02/technology/marjorie-taylor- greene-twitter.html
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The COVID vaccine totally makes your boobs bigger and grows your pp AT LEAST 3 inches. WHO CAN CONFIRM? We have to spread awareness and the truth about these vaccines!. Kara Corvus, Kara Corvus. 2021. The COVID vaccine totally makes your boobs bigger and grows your pp AT LEAST 3 inches. WHO CAN CONFIRM? We have to spread awareness and the truth about these vaccines! https://mobile.twitter.com/karac orvus/status/1421498443802021897
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|
[] |
[
"Zero-mode analysis of quantum statistical physics",
"Zero-mode analysis of quantum statistical physics"
] |
[
"A Bessa \nInstituto de Física\nUniversidade de São Paulo\nCaixa Postal 6631805315-970São PauloSPBrazil\n",
"C A A De Carvalho \nInstituto de Física\nUniversidade Federal do Rio de Janeiro\nCaixa Postal 6852821941-972Rio de JaneiroRJBrazil\n",
"E S Fraga "
] |
[
"Instituto de Física\nUniversidade de São Paulo\nCaixa Postal 6631805315-970São PauloSPBrazil",
"Instituto de Física\nUniversidade Federal do Rio de Janeiro\nCaixa Postal 6852821941-972Rio de JaneiroRJBrazil"
] |
[] |
We present a unified formulation for quantum statistical physics based on the representation of the density matrix as a functional integral. We identify the stochastic variable of the effective statistical theory that we derive as a boundary configuration and a zero mode relevant to the discussion of infrared physics. We illustrate our formulation by computing the partition function of an interacting one-dimensional quantum mechanical system at finite temperature from the pathintegral representation for the density matrix. The method of calculation provides an alternative to the usual sum over periodic trajectories: it sums over paths with coincident endpoints, and includes non-vanishing boundary terms. An appropriately modified expansion into Matsubara modes provides a natural separation of the zero-mode physics. This feature may be useful in the treatment of infrared divergences that plague the perturbative approach in thermal field theory.
|
10.1103/physreve.81.011103
|
[
"https://arxiv.org/pdf/0812.1941v1.pdf"
] | 118,712,142 |
0812.1941
|
3ba8bda5f7822ca2244fc7bd7e9343c799426d3e
|
Zero-mode analysis of quantum statistical physics
10 Dec 2008 (Dated: December 10, 2008)
A Bessa
Instituto de Física
Universidade de São Paulo
Caixa Postal 6631805315-970São PauloSPBrazil
C A A De Carvalho
Instituto de Física
Universidade Federal do Rio de Janeiro
Caixa Postal 6852821941-972Rio de JaneiroRJBrazil
E S Fraga
Zero-mode analysis of quantum statistical physics
10 Dec 2008 (Dated: December 10, 2008)
We present a unified formulation for quantum statistical physics based on the representation of the density matrix as a functional integral. We identify the stochastic variable of the effective statistical theory that we derive as a boundary configuration and a zero mode relevant to the discussion of infrared physics. We illustrate our formulation by computing the partition function of an interacting one-dimensional quantum mechanical system at finite temperature from the pathintegral representation for the density matrix. The method of calculation provides an alternative to the usual sum over periodic trajectories: it sums over paths with coincident endpoints, and includes non-vanishing boundary terms. An appropriately modified expansion into Matsubara modes provides a natural separation of the zero-mode physics. This feature may be useful in the treatment of infrared divergences that plague the perturbative approach in thermal field theory.
I. INTRODUCTION
Quantum statistical physics provides the conceptual and computational framework for the treatment of interacting many-body systems in thermal equilibrium with a heat bath. It describes a great variety of systems and phenomena over a wide range of scales, thus covering practically all areas of physics, from cosmology to particle physics, with an extensive number of examples in condensed matter.
Its broad spectrum of applications includes special relativistic systems. For those, the quantum statistical treatment is known as finite-temperature field theory [1], which is synonymous to quantum statistical field theory. In fact, just as quantum statistical mechanics adds stochastic probabilities to the quantum probabilities of quantum mechanics, finite temperature field theory does likewise with respect to quantum field theory, the natural combination of special relativity and quantum mechanics.
Quantum statistical physics can be formulated in the language of euclidean (imaginary-time) functional integrals, quite natural for finite-temperature field theories, but also applicable in nonrelativistic contexts, since quantum mechanics can be viewed as a field theory in zero spatial dimensions. Functional integrals thus furnish a powerful unifying formalism to compute correlations.
Indeed, the formalism expresses the partition function of a given system as a generating functional of euclidean Green functions (correlations), similarly to zerotemperature field theory, but with the time direction made compact, and specific boundary conditions that constrain the domain of field configurations in the func- * [email protected] † [email protected] ‡ [email protected] tional integral.
In order to best exploit the unifying aspect of the formalism, we propose to view the partition function as an integral of the diagonal density matrix element of the theory. The integral is performed over the stochastic variable which characterizes the representation of the matrix element (position representation for quantum mechanics, field representation for second quantized quantum mechanics or field theory). The density matrix is just the Boltzmann operator, whose Hamiltonian operator may either come from (second quantized) quantum mechanics or from field theory.
The density matrix element may be written as a functional integral in euclidean time τ which starts from a given configuration (a point in quantum mechanics, a field configuration in field theory) at τ = 0, and returns to that same configuration at τ = β . Clearly, there is a sum to be performed over configurations which coincide at the endpoints of the euclidean time interval. Those boundary configurations are identified as the stochastic variable of the remaining integral. Furthermore, it will be shown that they are zero modes of a modified Matsubara series. The theory of the zero modes is the effective stochastic theory obtained from the original Hamiltonian.
This density matrix method for deriving an effective stochastic problem was already used in [2], where it led to the construction of dimensionally reduced effective actions. Likewise, in Ref. [3] we used the density matrix method, and a semiclassical approximation, to investigate the thermodynamics of scalar fields. In the latter reference, the boundary configuration of the field (written as a constant plus gaussian fluctuations) played a very nontrivial role in the semiclassical formulation for the partition function of scalar fields.
Our prior uses of the method drew upon generalizations of a thermal semiclassical treatment previously developed and applied to quantum statistical mechanics, which produced excellent results for the ground state en-ergy and the specific heat of the anharmonic (quartic) oscillator [4][5][6]. In those prior uses, however, the connection with the stochastic variable was indirect, through classical solutions of equations of motion. Now, we have opted instead for a direct connection with the stochastic variable, to be identified with a zero mode, because we believe this is physically relevant to the discussion of infrared problems in several applications of quantum statistical physics.
In the context of finite-temperature field theory, for instance, it is known that a naive implementation of perturbation theory for the calculation of Feynman diagrams is ill-defined in the presence of massless bosons. This is due to the appearance of severe infrared divergences, brought about by the vanishing bosonic Matsubara mode in thermal propagators. The divergences plague the entire series, making it essentially meaningless [1,7,8].
As a result, one is forced to resort to resummation techniques that reorganize the perturbative series, and resum certain classes of diagrams, in order to extract sensible results. There are several ways of performing resummations, and rewriting the degrees of freedom more efficiently in terms of quasiparticles; we refer the reader to the reviews [7,8], and to Ref. [3], for a discussion and a list of specific references.
All such techniques are designed to partially tame the infrared divergences, creating a non-zero domain of validity for weak-coupling expansions 1 . Nevertheless, the zero-mode problem remains, and the region of validity of resummed perturbation treatments can not be indefinitely enlarged.
The infrared problems of finite temperature field theory resemble those encountered in the functional integral treatment of various condensed matter systems [10], where they are connected to collective excitations. This suggests that one should explore the unifying feature of the formalism, which is its statistical nature, to profit from parallel physical interpretations and insights. This is yet another reason for resorting to the density matrix formulation.
In the present article, we resume the study of quantum statistical mechanics, viewed as an exercise in zerodimensional finite-temperature field theory. Our objective is to show that boundary configurations are the zeromode stochastic variables of the effective statistical theory, and to compute that theory in various approximate schemes. Although we restrict our analysis to quantum mechanical examples, this should be regarded as a preliminary to the field theory case.
We compute the partition function of an interacting one-dimensional quantum mechanical system at finite temperature from a path-integral representation for the density matrix. The method of calculation that we pro-pose provides an alternative to the usual sum over periodic trajectories: it sums over paths with coincident endpoints, and includes the contribution of non-vanishing boundary terms. Indeed, an appropriately modified expansion into Matsubara modes provides a natural separation of the zero-mode physics, which is connected to the infrared problems in thermal field theory and elsewhere, and relates the zero mode to the boundary value of the quantum-mechanical coordinate.
The paper is organized as follows: in Section II, we present the proposed modified series expansion in Matsubara frequencies, and make a detailed comparison of our density matrix procedure with the standard one, for clarity; in Section III, we relate the boundary value of the quantum-mechanical coordinate (the stochastic variable) to the zero mode; In Section IV, we compute several thermodynamic quantities for the quadratic and the quartic potentials, and compare our results to the semiclassical findings of [4], and to some exact results; Section V contains our conclusions and outlook.
II. A MODIFIED SERIES EXPANSION
The partition function Z for a one-dimensional quantum-mechanical system in contact with a thermal reservoir at temperature T = 1/(k B β) may be written as
Z = dx 0 ρ (β; x 0 , x 0 ) ,(1)
where ρ(β; x 0 , x 0 ) is the diagonal element of the density matrix. In a path-integral formulation, the density matrix ρ is obtained from the imaginary-time evolution of the trajectory x : [0, β ] → R determined by an euclidean action S:
ρ(β; x 0 , x 0 ) = x(0)=x(β )=x0 [Dx] e −S[x]/ ,(2)S[x] = β 0 dτ 1 2 mẋ 2 (τ ) + V (x(τ )) ,(3)
where V is the potential 2 . In such a formalism, it is natural to regard x 0 as the effective stochastic degree of freedom of the theory, and to compute the distribution of the variable x 0 by integrating over the auxiliary variable x, with boundary conditions determined by x 0 . In this article, we want to show that the effective theory for the static variable x 0 not only underscores its statistical mechanical character, but can also be identified with the theory of the zero mode of an appropriately modified Matsubara expansion.
We can show that the formalism requires a modified Matsubara expansion for x by calculating the partition function of the harmonic oscillator in two different manners. The action S, in this simple case, is given by
S[x] = 1 2 β 0 dτ mẋ 2 (τ ) + mω 2 x 2 (τ ) .(4)
Without any loss of generality, one can take m = 1 in the previous equation by rescaling x by a factor 1/ √ m. One may compute the partition function using the following path integral:
Z = x(0)=x(β) [Dx] e −S[x] .(5)
The condition x(0) = x(β) is usually implemented by expanding the path x in Matsubara modes:
x(τ ) = ∞ n=−∞ c n e −iωnτ with ω n = 2πn β .(6)
The standard procedure is, then, to integrate Eq. (4) by parts:
S[x] = 1 2 β 0 dτ x(τ ) − d 2 dτ 2 + ω 2 x(τ ) + 1 2 x(τ )ẋ(τ ) β 0 .(7)
Using the periodicity of Eq. (6), one shows that the boundary term in (7) is zero. The remaining term can be cast in the form
S[x] = 1 2 β 0 dτ dτ ′ x(τ ) K(τ, τ ′ ) x(τ ′ ) ,(8)
where the Green function K(τ,
τ ′ ) = ∆ F (τ − τ ′ ) is such that − d 2 dτ 2 + ω 2 ∆ F (τ ) = δ(τ ) ,(9a)∆ F (τ − β) = ∆ F (τ ) .(9b)
This corresponds to the standard free propagator [1]
∆ F (τ ) = 1 2ω (1 + n(ω)) e −ωτ + n(ω) e ωτ ,(10)
where n(ω) is the Bose-Einstein distribution. Using ∆ F , it is straightforward to obtain the partition function. Up to an infinite constant, one obtains the well known result for the free energy
− 1 β log Z = ω 2 + 1 β log(1 − e −βω ) .(11)
There exists, however, an alternative path-integral method to calculate the partition function of the harmonic oscillator, which has the advantage of being free of spurious divergences. We start with (1), and decompose the configuration x as
x(τ ) = x c (τ ) + y(τ ) ,(12)
where x c is the solution of the following Euler-Lagrange equation:
d 2 x c dτ 2 − ω 2 x c = 0 , (13a) x c (0) = x c (β) = x 0 ,(13b)
which is given by
x c (τ ) = x 0 cosh[ω(τ − β/2)] cosh(ωβ/2) .(14)
It is easy to show that
S[x] = 1 2 x 0 [ẋ(β) −ẋ(0)] + S[y] = x 2 0 ω(cosh βω − 1) sinh βω + S[y] .(15)
Using that y vanishes at τ = 0, β, one rewrites the action of the trajectory y as
S[y] = 1 2 dτ dτ ′ y(τ )G −1 y (τ, τ ′ )y(τ ′ )(16)
where the Green function G y satisfies the following equation:
− d 2 dτ 2 + ω 2 G y (τ, τ ′ ) = δ(τ − τ ′ ) ,(17a)G y (0, τ ′ ) = G y (β, τ ′ ) = 0 . (17b)
One can show [4] that
G y (τ, τ ′ ) = sinh[ω(τ > − β)] sinh(βτ < ) ω sinh(ωβ) ,(18)
and
det G y = ω 2π sinh(ωβ) .(19)
In Eq. (18), τ > = max(τ, τ ′ ) and τ < = min(τ, τ ′ ). The path integral over y produces, then,
Z = (det G y ) 1/2 dx 0 e −ω[(cosh βω−1)/ sinh βω]x 2 0 (20) = (2 cosh βω − 2) −1/2 .(21)
Using (2 cosh ωβ − 2) = [1 − exp(−βω)] 2 exp(βω), and taking the logarithm we recover Eq. (11). Notice, however, that the two methods treat in a different way the quantitẏ Let us focus on the usual Matsubara expansion of the classical path (Eq. (14)). Being periodic, the corresponding series has a periodic derivative, so that the quantity (22) is put to zero. That was used to eliminate the boundary term in Eq. (7). However, we see from (15) thatẋ c (β) −ẋ c (0) = 0, otherwise one would obtain an incorrect result for Z. The second method of calculation sums over paths which have coincident endpoints, and thus includes paths that are not periodic. It is easy to realize that we introduce an error in the value of the action of any nonperiodic path x by evaluating the action of the associate Matsubara series as given by Eq. (6). The reason is that the usual Matsubara expansion is not suitable to handle boundary terms involving the derivative of x at 0 and β. We stress that, as long as the only condition over paths in the path integral is to have coincident values at 0 and β, non-periodicity is allowed.
x(β) −ẋ(0) .(22)
One can remedy this situation by defining an extension of x to [−β, β] as follows (see Fig. 1):
x(τ ) = x(τ ) χ [0,β] + (2x 0 − x(−τ )) χ [−β,0) ,(23)
where χ A denotes the characteristic function of A ⊂ R.
Notice thatx(−τ ) − x 0 = −(x(τ ) − x 0 )
. The odd character of the extension implies thatx − x 0 has a series of sines:
x(τ ) − x 0 . = ∞ n=1x n sinω n τ , withω n = nπ β .(24)
The notation with a dot is a reminder that the r.h.s. of Eq. (24) is a Fourier approximation of the l.h.s. The desired restriction to [0, β] is:
x(τ ) . = x 0 + ∞ n=1 x n sinω n τ ,(25)
where
x n = 2 β β 0 dτ x(τ ) sinω n τ .(26)
The series (25) does not impose conditions on the derivative of x. It is easy to convince oneself that evaluating the euclidean action S with the r.h.s. of Eq. (25) one obtains exactly S[x]. In particular, we have for the classical solution:
x c (τ )
. = x 0 − x 0 n odd 4ω 2 nπ (ω 2 n + ω 2 ) sinω n τ ,(27)
so that the quantityẋ c (β) −ẋ c (0) is given by
n odd −4x 0 ω 2ω n nπ (ω 2 n + ω 2 ) (cosω n β − 1) = 2x 0 ω(cosh βω − 1) sinh βω ,(28)
as implied by Eq. (15). Obviously, the series defined by Eq. (25) does not correspond to the most general trajectory in Z. In particular, that series imposes a serious restriction on the second derivative of x, by forcing it to vanish at 0 and β. However, at least for the calculation of usual action functionals, the boundary terms do not involve second derivatives and the aforementioned restriction is actually unimportant.
III. THE BOUNDARY VALUE AS THE ZERO MODE
The first remarkable property of the decomposition (25) is that the boundary value x 0 is the zero, static component of x which, in the context of thermal field theories, plays a special role in the infrared (low-momentum) limit of the theory. As mentioned previously, the infrared physics of bosonic theories is not accessible to plain perturbation theory, and resummation techniques are required in order to produce sensible results. In common, all such techniques render a special treatment to the zero mode 3 .
It is important to stress that the zero mode x 0 is not the zero mode of the usual Matsubara expansion. In fact, from Eq. (6), one can notice that x 0 involves the coefficients of all Matsubara modes:
x 0 = ∞ n=−∞ c n .(29)
The present identification of the zero mode with the boundary field in the context of quantum statistical mechanics aggregates to the zero-mode physics yet another important feature: the connection with the classical behavior at high temperatures. That regime is characterized by the condition
ω ≪ k B T(30)
(or βω ≪ 1, in natural units), where ω is the typical spacing between quantum levels. In order to see why the static mode governs the classical limit of the partition function, let us consider x c , the solution of the equation of motion:
d 2 x c dτ 2 − V ′ (x c (τ )) = 0 ,(31a)x c (0) = x c (β) = x 0 .(31b)
In the high-temperature regime (βω ≪ 1), thermal fluctuations dominate. In this limit, x c → x 0 and S(x c ) → βV (x 0 ), and we obtain for the partition function
lim T →∞ Z = N dx 0 e −βV (x0) ,(32)
where N is a normalization factor which incorporates quantum fluctuations. A proper normalization of Z leads to N 2 = m k B T /(2π 2 ), rendering the semiclassical expression
lim T →∞ Z = dp 2π dx 0 e −H/kB T ,(33)
with H = p 2 /2m + V (x 0 ), m being the particle mass. As discussed in [11], for βω ≪ 1 the correlation x 2 (τ ) follows the classical linear scaling with T . This behavior is entirely associated with the static component of x, and represents a problem for the high-temperature limit of the usual perturbative expansion. In contrast, the subtracted thermal propagator (without the contribution from the static mode) goes to zero as T increases, so that one should strongly improve the convergence of the perturbation expansion by calculating diagrams with that subtracted propagator.
We conclude that, either to control infrared divergences in thermal field theories or to describe the classical limit in quantum statistical mechanics, we need a separate treatment of the zero mode. The main advantage of our approach is to provide this separation in a natural fashion.
In order to explore the zero-mode physics, it is convenient to write
x(τ ) = x 0 + y(τ ) ,(34)
where y corresponds to the sinusoidal modes of Eq. (25) and has the following property:
y(0) = y(β) = 0 .(35)
In terms of y one can express Z conveniently as
Z = dx 0 y(0)=y(β)=0 [Dy] e −S[x0+y] ,(36)
and different strategies can be used to handle S[x 0 + y] in order to obtain an effective theory 4 for x 0 :
Z = dx 0 2π 2 /(m k B T ) e −β V eff (x0) .(37)
Notice that the intermediate step, including the paths y, is built using quantities which vanish at the boundary of imaginary time, and the contribution from problematic static quantities is factorized in the final integration over x 0 . Besides, for interacting systems, the interaction potential evaluated at x(τ ) = x 0 + y(τ ) can produce a y-dependent quadratic term for the zero mode x 0 , which could prevent possible divergences in the effective theory.
In the sequel, we use a very simple expansion around x 0 in two cases: the harmonic potential, as a consistency check, and the single-well quartic potential, in order to test the quality of our alternative approach and compare it to previous descriptions.
IV. APPLICATIONS AND COMPARISON TO PREVIOUS RESULTS
A. The quadratic case revisited
The action is again the one given by Eq. (4). Now, we use the decomposition of x given by Eq. (34) to obtain
S[x] = βω 2 2 x 2 0 + 1 2 dτ dτ ′ y G −1 y y + 2ω 2 x 0 y = βω 2 2 x 2 0 + 1 2 dτ y + ω 2 x 0 G y G −1 y × y + ω 2 x 0 G y − ω 4 x 2 0 2 dτ dτ ′ G y ,(38)
where G y was defined in (17). A simple calculation gives
dτ dτ ′ G y (τ, τ ′ ) = β ω 2 − 2(cosh βω − 1) ω 3 sinh βω .(39)
Shifting the variable of integration to
ζ(τ ) = y(τ ) + ω 2 x 0 dτ ′ G y (τ, τ ′ ) ,(40)
which obeys ζ(0) = ζ(β) = 0, and performing the gaussian integration, we obtain
Z = (det G y ) 1/2 dx 0 e −ω(cosh βω−1)/ sinh βω)x 2 0 . (41)
Integrating over x 0 and taking the logarithm, one reproduces Eq. (20). Notice that the propagator used in the perturbation expansion of the integral over the path y is well-behaved in the limit of small ω:
|G y (τ, τ ′ )| ≤ tanh βω/2 2ω βω≪1 −→ β 4 .(42)
B. The anharmonic oscillator
Let us now consider the following action:
S[x] = β 0 dτ 1 2 mẋ 2 (τ ) + 1 2 mω 2 x 2 (τ ) + λ 4 x 4 (τ ) .
(43) It is convenient to define the dimensionless variables: q = λ/(mω 2 ) x, θ = ωτ , Θ = ωβ and g = λ/(m 2 ω 3 ). The associated action for the variable q is:
S[q] = 1 g Θ 0 dθ 1 2q 2 (θ) + 1 2 q 2 (θ) + 1 4 q 4 (θ) . (44)
Using the decomposition q(θ) = q 0 + √ g η(θ), we write for the action:
S[q] = S 2 [q 0 , η] + S I [q 0 , η] ,(45)
where S 2 contains S[q 0 ], the kinetic term, and those which are linear or quadratic in η. The remaining contributions to S[q] are collected in S I . It is a simple matter to show that
S 2 [q 0 , η] = S[q 0 ] + 1 2 Θ 0 dθ η(θ) − d 2 dθ 2 + ω 2 η(θ) + 2α √ g η(θ) ,(46)
and
S I [q 0 , η] = Θ 0 dθ √ g q 0 η 3 (θ) + g 4 η 4 (θ) ,(47)
where ω 2 = 3 q 2 0 + 1 and α = q 3 0 + q 0 . One obtains a quadratic approximation for the partition function by considering
Z 2 = dq 0 η(0)=η(Θ)=0 [Dη] e −S2[q0,η] .(48)
Proceeding as in the previous section, we perform the gaussian integration over the variable
ζ(θ) = η(θ) + α √ g Θ 0 dθ ′ G η (θ, θ ′ ) ,(49)
where the Green function G η , defined by
− d 2 dθ 2 + ω 2 G η (θ, θ ′ ) = δ(θ − θ ′ ) ,(50a)G η (0, θ ′ ) = G η (Θ, θ ′ ) = 0 ,(50b)
is given by
G η (θ, θ ′ ) = sinh[ω(θ > − Θ)] sinh(ω θ < ) ω sinh(ω Θ) .(51)
The result for Z 2 is then
Z 2 = dq 0 (det G η ) 1/2 exp{−S[q 0 ] + Σ η } ,(52)
where
det G η = ω 2π sinh(ω Θ)(53)
and
Σ η = α 2 2g Θ 0 dθdθ ′ G η (θ, θ ′ ) .(54)
To go beyond the quadratic order in η, we use the following approximation:
e −SI [q0,η] ≈ 1 − Θ 0 dθ √ g q 0 η 3 (θ) + g 4 η 4 (θ) . (55) Therefore, Z ≈ Z 2 + δ (1) Z, where δ (1) Z = − dq 0 Θ 0 dθ √ g q 0 η 3 (θ) + g 4 η 4 (θ) ,(56)
with
η(θ 1 ) . . . η(θ k ) = η(0)=η(Θ)=0 [Dη] e −S2[q0,η] η(θ 1 ) . . . η(θ k ) .(57)
As usual, we can introduce a linear coupling of η with an external current j:
Z 2 [j] = dq 0 η(0)=η(Θ)=0 [Dη] exp −S 2 [q 0 , η] + dθj(θ)η(θ) ,(58)
and obtain the correlations (57) as functional derivatives of Z 2 [j] with respect to j. With minor modifications on the recipe for Z 2 , one obtains
Z 2 [j] = dq 0 (det G η ) 1/2 exp{−S[q 0 ] + Σ η [j]} , (59) where Σ η [j] = 1 2 Θ 0 dθdθ ′ σ(θ)G η (θ, θ ′ )σ(θ ′ ) ,(60)
and σ(θ) = α/ √ g − j(θ). Finally, we use
e −Ση δ 3 δj 3 (θ) j=0 e Ση [j] = I 3 0 (θ) + 3 I 0 (θ) G η (θ, θ)(61)
and e −Ση where I 0 (θ) plays the role of an expectation value of η(θ):
δ 4 δj 4 (θ) j=0 e Ση [j] = I 4 0 (θ)+6 I 2 0 (θ) G η (θ, θ) + 3G 2 η (θ, θ) ,(62)I 0 (θ) = − α √ g Θ 0 dθ ′ G η (θ, θ ′ ) = α √ g cosh ωθ − sinh ωθ tanh ωθ/2 − 1 ω 2 .(63)
The remaining integrations over θ can be done analytically, providing all the ingredients for the calculation of the first correction to Z 2 . Notice that the integrand of (52) is dominated by the vicinity of q 0 = 0, where the quadratic action reaches its minimum value. For small temperatures, that integrand becomes sharply peaked at zero, while for large T a broad range of q 0 substantially contributes. Therefore, one can estimate the importance of the correction term (56) using the following quantity:
S I (0, η) = 3g 4 Θ 0 dθ G 2 η (θ, θ) Θ≫1 −→ 3g Θ 16 .(64)
The temperature T min where S I (0, η) goes to 1 is displayed in Fig. 2 as a function of the coupling constant λ. T min provides a rough estimate for the validity of the quadratic approximation. From Fig. 2, we see that T min is of order 1 for λ as large as 50.
The results for the free energy and for the specific heat are displayed in Fig. 3 and Fig. 4, respectively. In the figures, classical corresponds to calculation using Eq. (33), quadratic uses Eq. (52), improved includes the correction corresponding to Eq. (56), semiclassical refers to the semiclassical result obtained in Ref. [4], 1-loop stands for the usual 1-loop perturbative result, and exact corresponds to results from Refs. [12,13] combined with WKB estimates.
The first remarkable thing in the plots is the failure of the perturbative result in the high-temperature region (T ≫ m). As seen in Ref. [4], that region is well described by the semiclassical curve, and asymptotically, by the classical one. This is evident from both plots. It is interesting that, for g = 0.4, the agreement extends to temperatures down to T ≈ m/2, a region where the character of the system is far from classical. However, it is surprising that our simple calculation using (52) works as well as the semiclassical one in that region. Indeed, the cited semiclassical calculation is based on a quadratic expansion around exact solutions of (31) for the quartic potential. In practice, one has to deal with rather involved Jacobi elliptic functions [4,14]. In contrast, using the expansion (34) no knowledge of classical solutions is demanded. This simplification represents an economic alternative which can be crucial in contexts where exact classical solutions are not available. From the plots, one also concludes that, in the region where the approximation is supposed to be good, the improved calculation exhibits a stronger convergence, indicating that the approximation is consistent.
A detailed look at the quadratic curve in the free energy plot (Fig. 3) reveals a non-monotonic behavior for low temperatures: the free energy passes by a maximum and decreases towards the free value 0.5. This is a general feature regardless of the value of m and λ, and it is a clear signal of the breakdown of the approximation below T min (see Fig. 2). The improved free energy diverges at T min , because at this value the sum Z 2 + δ (1) Z vanishes. The exact curve for the free energy reaches its maximum value at the ground state energy, where it rests down to T = 0. Using the maximum value assumed by the quadratic curve as an approximate value for the ground state energy, we obtain unexpectedly reasonable results, as shown in Table I. Fig. 4 compares different predictions for the specific heat for λ = 0.4, m = 1 and ω = 1. Notice that all curves, except for the perturbative one, have the correct high-temperature limit. The anomalous behavior of the quadratic approximation near T min is even more evident in this plot. The improvement obtained in the low-temperature regime when one corrects the quadratic calculation with (56) is also clearly shown. Finally, Fig. 5 displays the quadratic result for the specific heat when λ = 0.4 and m = 1 in the limit of zero ω. For large temperatures, the curve converges to the classical value 0.75. As discussed in Section II, with the separation of the zero mode, the finiteness of the theory for the η variable is not affected in the infrared limit, and we see in this specific case that the interaction makes the effective zero-mode theory well defined.
V. CONCLUSIONS AND OUTLOOK
The theoretical description of the thermodynamics of a given interacting many-body system kept in thermal equilibrium by the contact with a heat bath is an issue of capital importance in most realms of physics, from cosmology to condensed matter settings. A very convenient framework is provided by finite-temperature field theory, where one can use the imaginary-time formalism and make a direct connection to the powerful formulation of thermal averages in terms of functional integrals. Nevertheless, whenever collective (massless) bosonic modes are relevant, as is the case in practically any plasma, infrared divergences are unavoidable in a plain perturbative treatment, forcing a reorganization of the series that grants special attention to the zeroth Matsubara mode. As discussed in this paper, we also need a separate treatment of the static mode already in the context of quantum statistical mechanics in order to improve the convergence of the perturbative expansion in the hightemperature limit.
The choice of the traditional Matsubara mode expansion, combined with the trace requirement that the paths (configurations) coincide at the two boundaries of the compact euclidean time, implies a restriction to periodic paths in the process of functional integration to compute the partition function. However, rigorously speaking, the paths need not be periodic in the path-integral representation of the partition function. Indeed, the necessity of paths which are not periodic is particularly clear in the formulation of the partition function as an integral of the diagonal density matrix element of the theory. We have shown that, in order to allow for non-periodicity, one has to define a modified expansion into Matsubara modes.
In this paper, we have followed our alternative strategy in the case of quantum statistical mechanics, viewed not only as a toy model for the case of finite-temperature field theory, but also as a prototype for various relevant systems in statistical physics. More specifically, we have explored an alternative way of computing the partition function in the path-integral formalism that includes non-periodic trajectories, for which the usual Matsubara expansion leads to an incorrect result for the action of the trajectory. As was shown above, one can properly incorporate the contribution of non-periodic paths by using a modified Matsubara series expansion in which the zero mode is identified with the boundary value of the path. The latter turns out to be the stochastic variable that survives in the final effective theory.
The approach proposed in this paper has, thus, the advantage of providing a natural separation of the physics of the zero mode. In fact, we built a very simple effective theory for the zero mode in the nontrivial problem of the anharmonic oscillator, obtaining very precise results for practically the whole range of temperatures, and in a framework that is free of spurious divergences. In particular, we were able to describe the semiclassical (hightemperature) limit of the anharmonic oscillator without any information about the classical solutions. We expect that other potentials can profit from the simplicity of the present method in quantum mechanical systems.
In the case of finite-temperature field theory there are, of course, subtle issues of renormalization, which are ab-sent in the quantum mechanical setting, that will have to be addressed. Nevertheless, the results obtained in quantum statistical mechanics are encouraging, and we hope that this new perspective may shed light onto the problem of infrared divergences. Results in this direction will be presented elsewhere [15]
FIG. 1 :
1Schematic construction of the extension of a configuration x defined originally in [0, β] to the interval [−β, β]. Notice the odd character of the extension of x − x0.
FIG. 2 :
2Plot of Tmin as a function of the coupling constant λ (m = 1, ω = 1). The quadratic approximation is applicable for temperatures larger than Tmin.
FIG. 3 :
3Free energy of the anharmonic oscillator as a function of the temperature for λ = 0.4, m = 1 and ω = 1 (see text for details).
FIG. 4 :
4Specific heat of the anharmonic oscillator as a function of the temperature for λ = 0.4, m = 1 and ω = 1 (see text for details).
FIG. 5 :
5Specific heat of the anharmonic oscillator with ω = 0, m = 1 and different values of λ.
TABLE I :
IGround state energy of the quartic oscillator for different values of the coupling λ (m = ω = 1). Exact data were extracted from Ref.[13], whereas E0 (quadratic) was obtained using Eq. (52).
In the case of thermal QCD, for instance, the domain of validity of the naive perturbative expansion is the empty set[9].
In order to simplify the notation, we will adopt natural units where = 1 and k B = 1.
Rigorously speaking, x 0 is not a true Fourier mode since it is not orthogonal to sinωnτ in [0, β], but in [−β, β]. With this caveat in mind, we will continue to refer to x 0 as a mode of x.
With the normalization factor used in (37), one has the property: Veff(x 0 ) → V (x 0 ) when T → ∞, as follows from (33).
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. A Bessa, C A A De Carvalho, E S Fraga, work in progressA. Bessa, C. A. A. de Carvalho and E. S. Fraga, work in progress.
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[] |
[
"Self-assembly of Brownian motor by reduction of its effective temperature",
"Self-assembly of Brownian motor by reduction of its effective temperature"
] |
[
"Alexander Feigel \nSoreq NRC\n81800YavneIsrael\n",
"Asaf Rozen \nSoreq NRC\n81800YavneIsrael\n"
] |
[
"Soreq NRC\n81800YavneIsrael",
"Soreq NRC\n81800YavneIsrael"
] |
[] |
Emergence, optimization and stability of a motor-like motion in a fluctuating environment are analyzed. The emergence of motion is shown to be a general phenomenon. A motor converges to the state with the minimum of effective temperature and with the corresponding minimum in the rate of conformation changes similarly as some stochastic processes converge to the states with minimum diffusion activity. This mechanism is important to bacterial foraging (chemotaxis). This work, therefore, raises an analogy between chemotaxis and the emergence of living-like systems. The implications include the deviation of stable natural or artificial machines from the minimum entropy production principle, with a novel self-assembly mechanism for the emergence of the first molecular motors and for mass fabrication of the future nanodevices.
| null |
[
"https://arxiv.org/pdf/1312.5279v1.pdf"
] | 118,506,703 |
1312.5279
|
58b2f6885b46187635e224a5032986f8306264b6
|
Self-assembly of Brownian motor by reduction of its effective temperature
(Dated: December 19, 2013) 18 Dec 2013
Alexander Feigel
Soreq NRC
81800YavneIsrael
Asaf Rozen
Soreq NRC
81800YavneIsrael
Self-assembly of Brownian motor by reduction of its effective temperature
(Dated: December 19, 2013) 18 Dec 2013
Emergence, optimization and stability of a motor-like motion in a fluctuating environment are analyzed. The emergence of motion is shown to be a general phenomenon. A motor converges to the state with the minimum of effective temperature and with the corresponding minimum in the rate of conformation changes similarly as some stochastic processes converge to the states with minimum diffusion activity. This mechanism is important to bacterial foraging (chemotaxis). This work, therefore, raises an analogy between chemotaxis and the emergence of living-like systems. The implications include the deviation of stable natural or artificial machines from the minimum entropy production principle, with a novel self-assembly mechanism for the emergence of the first molecular motors and for mass fabrication of the future nanodevices.
Analysis of microscopic ratchets is the focus of modern statistical physics. Energy and conformation fluctuations drive small systems away from macroscopic physical laws. The ratchet-like systems might rectify these fluctuations to generate mechanical propulsion or complex robotic functions [1][2][3]. This type of microscopic machines amuse us with behavior that resembles living systems, such as steady state operation when out of thermal equilibrium and with a seeming reduction of entropy.
A macroscopic ratchet includes a gear with asymmetric teeth and a pawl (see Fig. 1(a)). Asymmetry of the teeth and of the pawl only permits omni-directional rotation of the gear. Motion of the teeth switches the pawl between open and closed states. The pawl in its open state allows a single tooth of the gear to pass in the allowed direction, then immediately switches to the closed state that prevents backward motion. A ratchet is a common part of mechanical jacks and precise clocks because its backward motion is forbidden.
A microscopic ratchet in a fluctuating environment might operate as a Brownian motor that converts thermal fluctuations into directed motion [1,2]. The implementation of this motor used to be a puzzle because a microscopic ratchet, contrary to a macroscopic one, might rotate in both directions. Fluctuations might keep the pawl in its open state and at the same time drive the gear in any direction. Bidirectional motion of a ratchet at a micro scale is intrinsic because omnidirectional rotation in a uniform temperature environment resulting from thermal fluctuations might reduce entropy and, therefore, breaks the second law of thermodynamics.
A series of works showed the essential conditions for a realistic Brownian motor to operate [2][3][4][5][6][7]. The generation of mechanical work when out of thermal fluctuations requires spatial asymmetry with a temperature gradient, a time-dependent modulation or another form of energy flux. This mechanism is considered a leading physical model for molecular motors [8][9][10][11][12] and inspired various accomplishments using the transport of particles * [email protected] in nanopores, in cold atoms lattices, in microfluidic circuits, and in superconducting and quantum nanodevices (for review see [3] and corresponding references).
The question of stability and the closely related problem of the emergence of the Brownian motor, to the best of our knowledge, was not addressed yet. The stability of the Brownian motor is important to take analogy with microscopic ratchets, molecular motors and living system beyond the mere operation principle toward emergence and optimization. For the sake of the stability problem, one should consider fluctuations in basic properties of the ratchet, such as asymmetry, rather only in the position of its mechanical parts. For instance, what are the required conditions to ensure development of asymmetry with the corresponding motor function? Continual machinelike behavior is a common property of organisms. This question, therefore, addresses the fundamental problems of emergence and stability of living systems.
Here we show a universal mechanism for self-assembly of a Brownian motor. This mechanism is based on the reduction of the motor's effective temperature because of the emergence of spatial asymmetry and associated motion. The steady state of this self-assembly differs from the normally accepted minima whether by free energy [13,14] or entropy production [15]. The proposed self-assembly permits the emergence and optimization of the motor function, rather than its mere emergence as a side effect of another optimization process. Self-assembly of a Brownian motor is an analogy of emergence of the first biological machines during pre-Darwinian chemical evolution [16], a method for mass production of artificial nanomachines [13] and a framework for analysis of fundamental questions about the stability of physical systems when out of thermal equilibrium.
The Smoluchowski-Feynman [1,2] ratchet is, probably, the most famous implementation of the Brownian motor (see Fig. 1(a)). The ratchet consists of two symmetric and nonsymmetric connected paddle wheels, kept at various temperatures T 1 and T 2 . These temperature gradients and asymmetry cause the ratchet to move continually until the temperatures equilibrate. atures still equilibrate through the ratchet's fluctuations. Temperature difference might be replaced by interaction with granular gas [17], cold atoms [18] or even by interaction with living species [19]. Triangulita is an elegant implementation of onedimensional Brownian ratchet [20] (see Fig. 1(b)). This motor includes only two symmetric (rectangle) and asymmetric (triangle) heads. Triangulita has a linear average velocity if its head as a triangle breaks the symmetry along the axis of propagation and if there is a temperature difference between the heads of the ratchet. The various temperatures of triangle and rectangle parts correspond to the interaction with ideal gases. The temperatures and densities of the gases are (T 1 , ρ 1 ) and (T 2 , ρ 2 ) correspondingly. This model allows analytical treatment at the limit m/M << 1, where m is the mass of the gas particles and M is the mass of Triangulita.
To study self-assembly of Triangulita, we make a single modification to the model: the triangle part can revolve around the joint point (see Fig. 2). This modification facilitates the transition from the symmetric state with no motion to a position with maximum asymmetry and maximum propulsion velocity. The mechanism of rotation is chosen to be coupled to the effective temperature of the motor T ef f instead of one of the reservoirs. This assumption is true if rotation mechanism is hidden inside the structure of Triangulita.
Triangulita with rotating triangle part might converge to an asymmetric state with maximum velocity by mutations that depend on effective temperature of the motor. Let us assume that the rotation of the triangle is slow compared with its linear motion and that triangle's ori-FIG. 2. Self-assembly of Triangulita Brownian motor. Motion emerges with transition of triangle part from symmetric (α = π/2 or α = π/6) to asymmetric position (α = 0 or α = π/3). Motion depends on the geometric form factors < sin n (θ) >, where θ is an angle between tangential and a horizontal lines. In this work, this transition occurs because of reduction in effective temperature of the motor. Effective temperature affects the mutation rate of triangle's orientation. The motor converges to an asymmetric state with minimum effective temperature and maximum propulsion because the corresponding diffusion process leads the system toward the state with minimum effective diffusion coefficient. Chemotaxis of bacteria, for instance, occurs because of similar phenomena. entation α changes by discrete steps ∆α in clockwise or counterclockwise directions (see Fig. 2).
Step in either direction is a transition through effective potential barrier ∆E. In this case, the transition rate is [21]:
R ∝ exp(−∆E/(k b T ef f (α)),(1)
where k b is Bolzman constant. Effective temperature T ef f is:
T ef f (α) ∝< v 2 > − < v > 2 ,(2)
where v is the ratchet's velocity. Effective temperature T ef f depends on triangle's orientation α because Triangulita's velocity v is a function of α. Dynamics of Triangulita in α space then described by continuity equation:
∂P (α) ∂t = ∂ ∂α J α (α),(3)
where J α (α) is the effective flux between the states with several orientations of the triangle. The steady distribution P st (α) corresponds to zero flux J α (α) = 0. The flux depends on the rate (1) and, therefore, is a function of effective temperature. Some mutation processes converge to the states with minimum mutation activity [22,23]. For instance, in the case of constant step ∆α with varying transition rate (1), the flux in the space of α is:
J α = ∂D(α)P (α) ∂α ,(4)
where P (α) is a probability of the rotation state α. The corresponding diffusion coefficient is [22]:
D(T ef f ) ∝ exp − ∆E k b T ef f .(5)
This flux results in a steady probability P st (α):
P st (α) ∝ 1 D(T ef f (α))
.
The ratio of the probabilities to be in two states with different effective temperatures T ef f 1 and T ef f 2 is:
P st (T ef f 1 ) P st (T ef f 2 ) ∝ exp ∆E 1 T ef f 1 − 1 T ef f 2 .(7)
The system, therefore, tends to concentrate at the state with the minimum effective temperature. The degree of concentration depends on the difference in the temperatures and value of the barrier ∆E.
To derive effective temperature of Triangulita as a function of its orientation α let us summarize the required analytical results [24]. Triangulita Brownian motor is described by Boltzmann-Master equation:
∂P (v, t) ∂t = dv [W (v|v )P (v , t) − W (v |v)P (v, t)] ,(8)
where P (v, t) is probability density of velocity v and W (v|v ) is transition rate from v to v. The motor changes its velocity because of interaction with the gas particles. In each thermal bath i, the interaction of gas with the heads of the motor is a function of the gas temperature T i and density ρ i with the size and shape of the motor's head. The latter is described by perimeter S with the following form factors:
< sin n (θ) > i = 2π 0 dθF i (θ)sin n (θ),(9)
where θ is a tangent of the surface measured counterclockwise from the axis of motion and F (θ)dθ is the fraction of the surface between θ and θ +dθ (see Fig. 2). The lowest order velocity of Triangulita is:
< v >= (10) = m M πk b T scale ef f 8M ρ i S i Ti T scale ef f − 1 < sin 3 (θ) > ρ i S i Ti T scale ef f < sin 2 (θ) > + +O m M 3 2 1 M ,
where scaling temperature is:
T scale ef f = γ i T i γ i ,(11)
with friction coefficients:
γ i = 4S i ρ i k b T i m 2π 1 2 < sin 2 (θ) > i .(12)
Finally, the motion of Triangulita is described by the moments of its velocity < v n > expended as a series of small parameter = m/M . Contribution of triangle part to the effective temperature (2) of Triangulita follows from eqs. (39) and (B2) of the work [24]:
T ef f ∝ (13) ∝ k b T scale ef f M − − < sin 4 (θ) > tr 2k b T scale ef f m M 2 S 1 ρ 1 S i ρ i Ti T scale ef f 1 2 < sin 2 (θ) > i T 1 T scale ef f − 1 T scale ef f T 1 − 1 4 + + < sin 3 (θ) > 2 tr 7πk b T scale ef f m 8M 2 S 2 1 ρ 2 1 S i ρ i Ti T scale ef f 1 2 < sin 2 (θ) > i 2 T 1 T scale ef f − 1 T 1 T scale ef f − 4 7 ,
where < sin n (θ) > tr are the form factors (9) of the triangle part of the motor. These form factors depends on the rotation state α and define the temperature difference between symmetric and asymmetric states.
Transition from symmetric to nonsymmetric motor state occurs universally near T 1 = T 2 . In this case T 1 = T scale ef f and effective temperature of the motor is independent of triangle's orientation, see for example, (13). Therefore, general form of effective temperature (13) near point T 1 = T 2 is:
T ef f = T 0 + A 1 ∆T + A 2 ∆T ∆α,(14)
where ∆T = T 1 − T 2 and A i are system dependent coefficients. Rotation ∆α reduces effective temperature either for positive or negative ∆T depending on the sign of the coefficient A 2 . Therefore, it is always possible to facilitate the motion by tuning temperature difference ∆T . As there are two types of transitions between symmet-ric and asymmetric states of Brownian motor, transition of the first type depends on the change of the form of Brownian ratchet. In this case effective temperature of the motor might reduce because of change in the form factors < sin 4 (θ) >, < sin 2 (θ) > or perimeter S, see the < sin 4 (θ) > term of (13). The motion comes as a side effect of optimization of friction coefficients (12) along the axis of the motion.
Transition of the second type is driven solely by asymmetry < sin 3 (θ) > and associated velocity (10). For example, < sin 4 (θ) > and < sin 2 (θ) > of an equilateral triangle are independent of orientation α (see Fig. 2). Therefore, the change in effective temperature (13) is:
∆T ef f ∝ (15) ∝ ∆ < sin 3 (θ) > 2 tr 7πk b T scale ef f m 8M 2 S 2 1 ρ 2 1 S i ρ i Ti T scale ef f 1 2 < sin 2 (θ) > i 2 T 1 T scale ef f − 1 T 1 T scale ef f − 4 7 ,
Favorable conditions for transition to the asymmetric state with maximum propulsion and minimum effective temperature are:
4 7 < T 1 T scale ef f < 1,(16)
where scale temperature T scale ef f is defined by (11). In space of temperatures (T 1 , T 2 ), these conditions are presented in Fig. 3.
Transition of the second type results in dynamics of a general Brownian motor near T 1 ≈ T 2 as an Onsager linear equation [25] [26] with coefficients that depend on stochastic process:
v = L vT (α) ∆T T 2 ,(17)J α (α, ∆T ef f (α)) = 0,
where L vT is Onsager coefficient, ∆T ef f (α) reduces with development of motion (15) and J α is the flux of the system in α space towards a state with maximum motion ability, for example, see (4). Orientation α becomes general parameter that describes asymmetry and the corresponding velocity v(α), i.g. (10). In the limit of strong coupling to the effective temperature, v = 0 or v = v max are the only possible states of the motor. Generalization of (17) to many degrees of freedom is:
J i = L ij X j ,(18)
J Lij (L ij , T ij ef f ) = 0, The motor-likes steady state, see eqs. (10) and (17), corresponds to constant external forces X j . The system (18) might possess several solutions resembling the variety of stable states in biological systems. The steady states of (18) do not nesssarily hold Prigogine's principle of minimum entropy production ∂S/∂t = J i X i .
The notion of effective temperature [27] and associated transitions might be integrated in any description of a system with conformation transitions out of thermal equilibrium, for instance [28]. The parameters of the system cease to be constant and converge to the states of FIG. 3. Phase diagram of Triangulita with rotating equilateral triangle. Self-assembly of Triangulita with equilateral triangle is an example of a system driven only by the development of motion. Self-assembly because of reduction of effective temperature in the region T1 ≈ T2 is universal property of Brownian motors. Therefore, self-assembly driven by effective temperature is a general mechanism. minimum effective temperature. This convergence might be associated with emergence of a new property such as motion or more complex activity. Effective temperature is a useful tool that already served to describe behavior and stability of an active medium [29,30].
Macro-molecule with preferential ATP hydrolysis site is a complementary system to two temperature Brownian motor, such as Triangulita or Smoluchowski-Feynman ratchet. ATP releases about 9k b T during reaction with the hydrolysis site. The molecule, therefore, is coupled to high temperature T high thermal bath in addition to its environment temperature T . According to this work, these two temperatures define the effective temperature of the molecule and, under particular circumstances, drives its conformational changes toward development of motion. This sequence, with a two-temperature framework, holds for the Brownian motor models based on external noise correlated in time. Weak coupling to ATP hydrolysis might be described either by two temperatures or by external modulation [31][32][33][34][35].
The proposed mechanism for the emergence of motor abilities matches the phenomenon of chemotaxis [36]. The mutation rate of a Brownian motor reduces with the development of motor abilities just as the effective diffusion coefficient of bacteria reduces toward the areas with high concentration of food. This analogy indicates generality of the mutation modulation mechanism in the early living systems.
Thus, a general mechanism favoring self-assembly of Brownian motors is predicted. This mechanism is a common property of several temperature systems near thermal equilibrium. The steady state of a Brownian motor is described by the nonlinear stochastic version of Onsager relations. It could be a novel description of molecular motors and the future artificial nanomachines with complex robotic functions. These results open a new way to consider emergence and stability of living-like systems and to accomplish the similar phenomena using alternative technologies.
No continual motion occurs if all parts are symmetric, though temper-arXiv:1312.5279v1 [cond-mat.stat-mech] 18 Dec 2013 A B FIG. 1. (A) Smoluchowski-Feynman Brownian motor. The motor consists of two heads connected by a common axis. The heads are asymmetric ratchet and symmetric paddle wheel. The motor rotate its heads are immersed in gases at different temperatures T1 and T2 correspondingly. Rotation continues until the temperatures equilibrate. This type of motor was the subject of several paradoxes and puzzles during the previous century. The ratchet is a physical model for artificial nanodevices and molecular motors such as kinesin. (B) Triangulita Brownian motor. Similar to the Smoluchowski-Feynman motor, connected asymmetric triangle and symmetric rectangle propulse linearly if they are coupled to the various temperatures. Triangulita enables analytic analysis of the major characteristics of Brownian motors.
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[
"Interacting double dark resonances in a hot atomic vapor of helium",
"Interacting double dark resonances in a hot atomic vapor of helium"
] |
[
"S Kumar \nSchool of Physical Sciences\nJawaharlal Nehru University\n110067New DelhiIndia\n",
"T Lauprêtre \nLaboratoire Aimé Cotton\nCNRS\nUniversité Paris Sud 11\n91405Orsay CedexFrance\n",
"R Ghosh \nSchool of Physical Sciences\nJawaharlal Nehru University\n110067New DelhiIndia\n",
"F Bretenaker \nLaboratoire Aimé Cotton\nCNRS\nUniversité Paris Sud 11\n91405Orsay CedexFrance\n",
"F Goldfarb \nLaboratoire Aimé Cotton\nCNRS\nUniversité Paris Sud 11\n91405Orsay CedexFrance\n"
] |
[
"School of Physical Sciences\nJawaharlal Nehru University\n110067New DelhiIndia",
"Laboratoire Aimé Cotton\nCNRS\nUniversité Paris Sud 11\n91405Orsay CedexFrance",
"School of Physical Sciences\nJawaharlal Nehru University\n110067New DelhiIndia",
"Laboratoire Aimé Cotton\nCNRS\nUniversité Paris Sud 11\n91405Orsay CedexFrance",
"Laboratoire Aimé Cotton\nCNRS\nUniversité Paris Sud 11\n91405Orsay CedexFrance"
] |
[] |
We experimentally and theoretically study two different tripod configurations using metastable helium ( 4 He*), with the probe field polarization perpendicular and parallel to the quantization axis, defined by an applied weak magnetic field. In the first case, the two dark resonances interact incoherently and merge together into a single EIT peak with increasing coupling power. In the second case, we observe destructive interference between the two dark resonances inducing an extra absorption peak at the line center.
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10.1103/physreva.84.023811
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[
"https://arxiv.org/pdf/1104.3427v2.pdf"
] | 67,841,836 |
1104.3427
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b00a7bf0b1daf4ac28712317948bfe42cf82b53f
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Interacting double dark resonances in a hot atomic vapor of helium
S Kumar
School of Physical Sciences
Jawaharlal Nehru University
110067New DelhiIndia
T Lauprêtre
Laboratoire Aimé Cotton
CNRS
Université Paris Sud 11
91405Orsay CedexFrance
R Ghosh
School of Physical Sciences
Jawaharlal Nehru University
110067New DelhiIndia
F Bretenaker
Laboratoire Aimé Cotton
CNRS
Université Paris Sud 11
91405Orsay CedexFrance
F Goldfarb
Laboratoire Aimé Cotton
CNRS
Université Paris Sud 11
91405Orsay CedexFrance
Interacting double dark resonances in a hot atomic vapor of helium
(Dated: 8 July 2011)numbers: 4250Gy4225Bs4250Nn4250Ct4265-k
We experimentally and theoretically study two different tripod configurations using metastable helium ( 4 He*), with the probe field polarization perpendicular and parallel to the quantization axis, defined by an applied weak magnetic field. In the first case, the two dark resonances interact incoherently and merge together into a single EIT peak with increasing coupling power. In the second case, we observe destructive interference between the two dark resonances inducing an extra absorption peak at the line center.
I. INTRODUCTION
Electromagnetically induced transparency (EIT) in three-level Λ-systems is a phenomenon in which an initially absorbing medium is rendered transparent to a resonant weak probe laser when a strong coupling laser is applied to a second transition [1]. In addition to its being a quantum interference phenomenon of fundamental interest, EIT has been studied extensively for its numerous applications, such as in slow and fast light, light storage, sensitive magnetometry and optical information processing. An extension of the usual three-level EIT to a four-level double-EIT scheme has been shown to expand the utility of EIT in a number of additional, potentially useful and easily controllable coherent nonlinear effects. These include engineering atomic response by perturbing a dark state [2][3][4], and control of group velocity via interacting dark resonances [5]. The interest in four-level tripod-like atomic configurations started with an early work [6] showing that in a tripod medium, there exists an internal state subspace spanned by two orthogonal dark states, which is immune to spontaneous decay. Subsequently, the creation and measurement of a superposition of quantum states using stimulated Raman adiabatic passage in a tripod have been proposed and demonstrated [7,8]. Simultaneous enhancement and suppression of a dark resonance have been observed by nondegenerate four-wave mixing in a solid in a tripod-like level configuration [9]. Large cross-phase modulation induced by interacting dark resonances in a tripod system of cold 87 Rb has also been reported [10]. Tripod configurations with two probes and a common coupling beam have been studied in a variety of contexts -theoretically for the magneto-optical Stern-Gerlach effect [11], for the experimental demonstration of light storage at dual frequencies [12], in a proposal for all-optical quantum computation with efficient cross-phase modulation, and sensitive optical magnetometry [13], and experimentally for matched slow pulses using double EIT in Rb [14], and also for the * Electronic address: [email protected] study of nonlinear Faraday effect in an inverted Y model in Rb vapor [15]. Interacting dark resonances in tripod configurations with two coupling beams and a common probe have been used to obtain sub-Doppler and subnatural narrowing of an absorption line, theoretically by Goren et al. [16], and experimentally by Gavra et al. in Rb [17]. A similar scheme has been suggested for applications in logic gates and sensitive optical switches [18].
There are several such applications of controllable double dark resonances in four-level tripod systems, and there are not many experimental results on tripod systems reported in the literature so far. In this context, we wish to probe a simple system of 4 He* at room temperature. This medium has been shown to be an ideal candidate for achieving ultra-narrow (less than 10 kHz) EIT in a three-level Λ-system involving only electronic spins in the presence of Doppler broadening [19]. We have confirmed the true nature of the two-photon process of EIT by our observation of asymmetric Doppleraveraged Fano-like transmission profiles in the presence of single-photon optical detunings. 4 He* has some peculiar favorable properties: (i) Velocity-changing collisions enable us to span the entire Doppler profile [20]. (ii) The absence of nuclear spin simplifies the level scheme and eliminates the need for repumping lasers compensating for losses into the other ground state hyperfine levels. (iii) Diffusive motion increases the transit time of the atoms through the laser beam and hence the Raman coherence life-time. (iv) Collisions with the ground state atoms do not depolarize the colliding 4 He*. Thus there are no background atoms to contribute to noise. (v) Penning ionization among identically polarized 4 He* atoms is almost forbidden [21]. In the present work, we show that 4 He* is a suitable candidate for realizing a clean four-level tripod system in a room-temperature gas. The excited state 2 3 P 0 (m e = 0, |e ) of 4 He* can be coupled selectively to the 2 3 S 1 sublevels, m g = −1 (|g − ), 0 (|g 0 ) and +1 (|g + ), by co-propagating laser beams at around 1083 nm, with σ + , π and σ − polarizations, respectively. The energy separation between the 2 3 P 0 and the next lower sublevel 2 3 P 1 is large (29.6 GHz) compared to the Doppler width (≈ 1 GHz), allowing one to ensure that each transition is isolated. This is not the case, for ex-ample, in Rb [17].
We focus on two different tripod configurations based on the interaction of the atoms with linearly polarized coupling and probe beams in the presence of a horizontal transverse magnetic field. In the first case, the probe beam has vertical linear (V) polarization and the coupling beam has horizontal linear (H) polarization, while in the second case, the probe beam has H-polarization and the coupling beam has V-polarization. We can easily switch from one configuration to the other by just a change in the orientation of a wave-plate used in the setup. The difference in the configurations comes from the number of probe transitions used, yielding a distinctive interplay of double dark resonances in each case. With 4 He* at room temperature, using a weak magnetic field and polarization selective transitions mentioned above, we experimentally realize a tripod configuration with two probed transitions in the first case, and observe that the double dark resonances add incoherently, as there is no coherence between the two populated probe ground levels. This configuration serves as a useful reference for the second configuration studied with two coupling transitions. In the second case, double dark resonances are related to coherent population trapping in the ground states. As the number of excited states is less than the number of ground states, transfer of coherence does not play any role [22]. Thus these double dark resonances are not stimulated Raman peaks [16,23] but detuned EIT peaks, interfering destructively with each other leading to an absorption dip in-between for non-zero magnetic fields. We model the system successfully and verify our experimental results.
The paper is organized as follows. In Sec. II, we describe the experimental set-up. In Sec. III, we present the experimental results and compare them with our numerical simulations for the two different tripod configurations. Our conclusions are presented in Sec. IV, with hints of potential applications.
II. EXPERIMENTAL SET-UP
The experimental set-up is shown in Fig. 1. The helium cell is 6-cm long and has a diameter of 2.5 cm, and is filled with 4 He at 1 Torr. The cell is placed in a three-layer µ-metal shield to isolate the system from the earth's magnetic field inhomogeneities. Helium atoms are excited to the metastable state by an RF discharge at 27 MHz. We use the 2 3 S 1 → 2 3 P 0 transition of 4 He* (D 0 line) with the coupling and probe beams derived from a single laser at nearly 1082.9 nm wavelength (linewidth 10 MHz), with a beam diameter of 1 cm after the telescope. The maximum available power for the coupling beam is about 27 mW, which is large enough due to the fact that the saturation intensity in 4 He* is very low (0.167 mW/cm 2 ). A probe power of 100 µW has been used throughout. The frequencies and intensities of the coupling and probe beams are adjusted by the amplitudes and the frequencies of the RF signals driving the acousto-optic modulators AOM-1 and AOM-2. In our experiment, a variable weak magnetic field (B), generated by a pair of rectangular coils surrounding the helium cell, removes the degeneracy of the lower sublevels. These coils are able to produce a constant horizontal magnetic field perpendicular to the direction of propagation of the laser beams. We theoretically estimate the magnetic field, which is constant within the cell area, and experimentally verify it by a teslameter. We consider two different tripod configurations, with the magnetic field parallel or perpendicular to the probe beam polarization. The direction of the static magnetic field is taken as the quantization axis. For the helium 2 3 S 1 state, the Landé g-factor is 2.002. The magnetic field shifts the metastable 2 3 S 1 (m J ) state by µ B Bm J g, where µ B = e /2m e = 9.274 × 10 −24 J/T is the Bohr magneton, and B is the applied magnetic field. This gives the Zeeman splitting, ∆ Z ≡ µ B Bm J g/h = 2.8 kHz for B = 1 mG. The Rabi frequency of the coupling beam Ω C is much larger than the Zeeman splitting ∆ Z .
In the rotating-wave approximation [24], the Hamiltonian of the system can be expressed as
H = H 0 + H I .(1)
H 0 is the unperturbed Hamiltonian,
H 0 = i ω i |i i|,(2)
where i = e, g − , g 0 , g + corresponds to the different levels, labeled in Figs. 2(b) and 4(b). H I is the interaction Hamiltonian involving the coupling and probe transitions.
The time evolution of the density matrix operator, in the presence of decay, is obtained from the Liouville equation as
d dt ρ = − i [H, ρ] + Rρ,(3)
where R is the relaxation matrix. The density matrix elements obey the conditions i ρ ii = 1 and ρ li = ρ * il . The sources of relaxation in our system are spontaneous emission from the excited state to the lower states with equal decay rates Γ 0 /3 (Γ 0 = 10 7 s −1 ), transit relaxation of the atoms through the beams from all allowed states with a rate Γ t (≈ 10 3 s −1 ), and Raman coherence decay with a rate Γ R (≈ 10 4 s −1 ). In our simple model, we do not explicitly take into account the Doppler effect, but assume that the optical coherence decay rate Γ/2π would effectively be given by the width (≈ 1 GHz) of the transition in the Doppler-broadened medium. This approximation has already been shown to be valid in the case of EIT in a standard three-level system in 4 He* [19,20].
A. First configuration
When we set the λ/2 plate in front of our helium cell at 45 o to the incident polarizations, the probe beam has V-polarization (σ), perpendicular to the magnetic field, while the coupling beam has H-polarization (π), parallel to the magnetic field (Fig. 2). Levels |e and |g − (|g + ) are coupled by the σ + (σ − )-polarized components of the weak probe beam of frequency ω P and detunings ∆ P = ω eg∓ −ω P ∓∆ Z . The strong coupling beam of frequency ω C and detuning ∆ C = ω eg0 − ω C couples the same excited level |e with the level |g 0 . The Raman detuning is δ = ∆ P − ∆ C .
We experimentally measure the evolution of the transmitted probe intensity (in arbitrary units) versus Raman detuning (δ), as shown in Figs. 3(a)-(c), for coupling powers of 1 mW, 10 mW and 22 mW, respectively, with magnetic fields of 0, 10 and 30 mG at each coupling power. We model the system by writing the optical Bloch equations (3) with the relevant interaction Hamiltonian, for a coupling beam of Rabi frequency Ω C with horizontal linear polarization (π) and a probe beam of Rabi frequency Ω P with two counter-circular polarization components (σ ± ) with respect to the quantization axis (z):
H I = − 2 Ω P √ 2 e −iωPt |e g − | + Ω C e −iωCt |e g 0 | + Ω P √ 2 e −iωPt |e g + | + H.c. .(4)
We take Ω P Ω C and consider Ω P to first order. We assume that the coupling beam is at resonance (∆ C = 0), the populations ρ g−g− and ρ g+g+ are approximately equal to 0.50, and ρ g0g0 ≈ 0 ≈ ρ ee . Then the steady state solutions of the six coupled optical Bloch equations for the coherences ρ eg− =ρ eg− e −iωPt , ρ eg+ =ρ eg+ e −iωPt , ρ g0g− =ρ g0g− e i(ωC−ωP)t , ρ g0g+ = ρ g0g+ e i(ωC−ωP)t , ρ g+g− =ρ g+g− and ρ g−g+ =ρ g−g+ givẽ
D Z w P D P 3 P 0 3 S 1 s + D Z w P 0 +1 s - w C π m g = -1 m e = 0 |g -〉 | g 0 〉 | g + 〉 |e〉 s ± p B z y l/2 at 45 W C W P He cell x (a) Polarization selection (b) Level diagramρ eg∓ Ω P = w ∓ / √ 2 2(a ∓ +∆ C − i 3 ) − |ΩC| 2 2(a∓−iΓR) ,(5)
where w ∓ = (ρ g∓g∓ − ρ ee ) = 0.5, a ∓ =δ ∓∆ Z , and all rates and frequencies, scaled by Γ/2π ≈ 10 9 Hz, are denoted by a bar over the corresponding symbols. The probe absorption and dispersion are proportional to the imaginary and real parts of the susceptibility. We obtain an expression for the probe susceptibility as
χ(ω P ) = A 1 2 √ 2 1 (a + − i 3 ) − |ΩC| 2 4(a+−iΓR) + 1 (a − − i 3 ) − |ΩC| 2 4(a−−iΓR) ,(6)
where A 1 = N |µ eg | 2 w ∓ / 0 , µ eg− ≈ µ eg+ = µ eg is the dipole matrix element for the probe transitions, and N is the atomic density. With the above reasonable approximations, the imaginary part of the susceptibility from Eq. (6) is found to be
Im [χ(ω P )] = 3A 1 √ 2 1 − 3|Ω C | 2 8 Λ a 2 + + Λ 2 + Λ a 2 − + Λ 2 ,(7)
where Λ =Γ R + 3|ΩC| 2
4
. The right-hand side of (7) is a sum of two Lorentzians with centers atδ= ±∆ Z and full widths at half maxima of 2Λ. The transmission profiles are generated from exp [−kLIm [χ(ω P )]], where k is the magnitude of the wave vector of the probe beam, and L is the length of the helium cell. The transmission profiles versus scaled Raman detuning (δ) are shown in Figs. 3(d)-(f), withΩ C = 1.8 × 10 −3 , 5.7 × 10 −3 and 8.6 × 10 −3 , respectively, corresponding to the experimental coupling powers, with Zeeman shifts also corresponding to the experimental values of the magnetic field.
In this configuration, as seen in Fig. 3, at zero magnetic field, we observe a single EIT peak at the line center [25]. When we apply a weak magnetic field, we observe double dark resonances for low coupling powers. The two corresponding peaks add incoherently. As we increase the coupling power, these two peaks broaden and eventually merge together into a single peak at the line center. To observe double dark resonances, it is required that all population is optically pumped into the |g − and |g + levels. The fact that this configuration leads to an incoherent sum of two EIT peaks can be easily understood. Indeed, the weak probe (treated to first order in the Rabi frequency Ω P ) cannot create any coherence between the two populated probe ground levels, |g − and |g + . We can thus expect the system to behave as two independent three-level systems connected by the single coupling beam, with each three-level system exhibiting its respective EIT peak for its particular Raman resonance.
B. Second configuration
We now set the λ/2 plate in front of the helium cell at a specific angle so that this plate behaves as neutral for the incident polarizations -the probe beam has Hpolarization (π), parallel to the magnetic field, while the coupling beam has V-polarization (σ), perpendicular to the magnetic field (see Fig. 4). Levels |e and |g − (|g + ) are coupled by the σ + (σ − )-polarized component of the strong coupling beam of frequency ω C and detunings ∆ C = ω eg∓ − ω C ∓ ∆ Z . A weak probe beam of frequency ω P and detuning ∆ P = ω eg0 − ω P couples the same excited level |e with the level |g 0 . We again model the system by writing the density matrix equations (3), with the corresponding interaction Hamiltonian for a probe beam of Rabi frequency Ω P with horizontal linear polarization (π) and a coupling beam of Rabi frequency Ω C with two counter-circular polarization components (σ ± ) with respect to the quantization axis:
D Z D Z m g = -1 D P 3 P 0 3 S 1 m e = 0 |e〉 s + w C w C 0 1 s - w P π |g -〉 |g 0 〉 |g + 〉(H I = − 2 Ω C √ 2 e −iωCt |e g − | + Ω P e −iωPt |e g 0 | + Ω C √ 2 e −iωCt |e g + | + H.c. .(8)
The main approximations used are to take Ω P Ω C , and to consider Ω P to first order while taking Ω C in all orders.
In our case, ∆ C = 0, as before. We assume that the probe ground level population ρ g0g0 is approximately equal to unity, and ρ g−g− ≈ 0 ≈ ρ g+g+ ≈ ρ ee . Then the steady state solutions of the three coupled optical Bloch equations for the coherences ρ eg0 =ρ eg0 e −iωPt , ρ g−g0 =ρ g−g0 e −i(ωP−ωC)t and ρ g+g0 =ρ g+g0 e −i(ωP−ωC)t giveρ
eg0 Ω P = w 0 2b + |ΩC| 2 4 1 (iΓR−a−) − 1 (a+−iΓR) ,(9)
where w 0 = (ρ g0g0 −ρ ee ) = 1, a ∓ =δ∓∆ Z , b =δ+∆ C − i 3 , and all rates and frequencies, scaled by Γ/2π ≈ 10 9 Hz, are denoted by a bar on top, as before.
We obtain an expression for the probe susceptibility as
χ(ω P ) = A 2 (a − − iΓ R )(a + − iΓ R ) 2b(a − − iΓ R )(a + − iΓ R ) − q |ΩC| 2 2 ,(10)
where q = (δ − iΓ R ), A 2 = N |µ eg0 | 2 w 0 / 0 , and µ eg0 is the dipole matrix element for the probe transition. The absorption and dispersion of the probe beam are proportional to the imaginary and real parts of the susceptibility. With the above reasonable approximations, we get the imaginary part of the susceptibility from Eq. (10) as This configuration, at zero magnetic field, as shown in Fig. 5, is equivalent to the degenerate two-level system with a σ ± coupling and a π probe. In this case, we observe a single EIT peak (black, open square) at the line center [19]. Note that Kim et al. [26] observed electromagnetically induced absorption (EIA) for the two-level degenerate system (F e = F g − 1 with F e ≥ 1 and F e = F g ) and this anomalous EIA has been interpreted by the analysis of dressed-atom multiphoton spectroscopy [27]. In our degenerate two-level system (F e = F g − 1 with F e = 0), in place of anomalous EIA, we observe an EIT peak at the line center. This EIT peak can be explained by the following change of basis, which is a simple example of Morris-Shore transformation [28]: Replace the two sublevels |g − and |g + by the usual dark (|N C = (|g − − |g + )/ √ 2) and bright (|C = (|g − + |g + )/ √ 2) states for the CPT in a threelevel system (|g − , |g + and |e ) for the coupling beams. Since the transition |N C → |e is not allowed, we have Corresponding numerically calculated transmission profiles, withΩC ≡ 2πΩC/Γ,δ ≡ 2πδ/Γ, and∆Z ≡ 2π∆Z/Γ. essentially obtained a three-level system (|C , |g 0 and |e ) with a probe and a coupling beam. Hence we observe a single EIT peak at the line center. The widths and heights of the EIT windows increase with the coupling power. When we apply a weak transverse magnetic field, the degeneracy of the lower levels is removed. In this case, two dark resonance peaks appear and they shift from the zero detuning position with increasing magnetic field. The separation between the two dark resonances varies linearly with the applied magnetic field. The double dark resonance cannot be explained in terms of transfer of coherence from the excited level to the ground level [16,23] but these dark resonances are two EIT peaks at δ = ±∆ Z . When the coupling power is increased in the presence of the magnetic field, an absorption line appears, much narrower and deeper than that found in the first configuration, which is the signature of an interference phenomenon between the two induced EIT windows [3,18]. This is the main result of this paper. It is visible in both the experimental data and in simulations that these EIT peaks are asymmetric at 10 and 22 mW coupling powers: the absorption dip walls are sharper than the external transparency window lines. Although it is well known that EIT profiles become asymmetric when the coupling beam is optically detuned, the optical detuning given by the Zeeman shift cannot explain such a shape: indeed, the detuning in our case is less than 100 kHz while hundreds of MHz are necessary to obtain any significant asymmetry in our system [29]. When comparing these profiles with the ones obtained with the first configuration (see Fig. 3), one notices here that the double peak transmissions are much higher. For the two values of the Zeeman shift used and at high enough coupling powers (10 and 22 mW), the transmissions are nearly as large as the transmission of the single peak recorded without any magnetic field: transmissions seem to be given by the total Rabi frequency Ω C while the widths are much narrower than the width expected with such a coupling intensity. The resulting very deep absorption line seems to narrow with increasing coupling power instead of disappearing because of saturation. This is very different from the first case, where the transmissions corresponding to 80 kHz of Zeeman shift remain roughly half the transmission of this single peak, and the saturation broadening makes the dip disappear for 50 kHz of Zeeman shift. We have checked that an incoherent addition of susceptibilities or transmissions would give a behavior similar to the first case and cannot explain the data recorded in the second configuration: the absorption dip appearing at the line center is narrower and deeper than is possible by adding two independent bestfit EIT profiles. A common picture for EIT in three-level systems is to consider that it is the result of interference between two absorption paths, a direct absorption from the probed level, and the other which is followed by induced emission and reabsorption by the coupling beam. In our tripod configuration, there can be emission and reabsorption with both σ + and σ − coupling beams. The constructive or destructive nature of this interference mechanism depends on the signs of the superpositions in the two dark states corresponding to the two three-level systems. In our case, the components of the coupling beam lead to opposite signs for the |g − ↔ |e and |g + ↔ |e transition amplitudes. As a result, there is a destructive interference between two EIT peaks at the center and we observe a sharp absorption dip, looking like (but different in nature from) EIA [22], flanked by two EIT (detuned) peaks [16]. It is clear from Fig. 5 that for fixed magnetic fields, the widths and heights of all the EIT windows increase while the width of the absorption dip decreases with increasing coupling power (the fullwidths at half-maxima of the absorption dips are about 47, 42 and 34 kHz, for coupling powers of 1, 10 and 22 mW, respectively, at B = 10 mG). In the absence of the magnetic field, the narrow absorption dip disappears and the system becomes transparent at the line center.
Im [χ(ω P )] = 3A 2 2a 2 − a 2 + +Γ R y(2Γ R + x) +Γ 3 R x 4a 2 − a 2 + + 4Γ R xy + x 2Γ2 R + 9δ 2 |ΩC| 4 4 ,(11)
IV. CONCLUSIONS
We have been able to carve out a clean four-level tripod system in a simple system of room-temperature 4 He* us-ing a weak magnetic field and polarization-selective transitions. Interesting interplay between the double dark resonances has been recorded.
In the first tripod configuration with two probed transitions, when the coupling is parallel to the weak magnetic field, we have observed that two EIT (detuned) add incoherently, and for large coupling power, these two peaks merge into a single EIT like peak at the line center. Such a double-EIT configuration has potential application in light storage for two frequencies [12,13] and coupling-induced switch in the presence of a small magnetic field.
In the second tripod configuration with two coupling transitions, when the probe is parallel to the weak magnetic field, we have observed a remarkable destructive interference between the two EIT peaks, leading to an extra, narrow absorption peak in-between the EIT peaks. The absorption feature is seen to become narrower with increasing coupling power and could be made subnatural even in the presence of Doppler broadening [16]. We stress here that our results, shown in Fig. 5, cannot be obtained from two independent EIT systems, even if one allows for asymmetric EIT windows resulting from detuned (by a few tens of kHz) coupling fields. The absorption dip appearing at the line center is narrower and deeper than is possible by adding two independent asymmetric EIT fits. The separation between the observed EIT peaks is determined by the applied magnetic field. Such a system may be used as a magnetometer, although the field values used by us are rather high. The shape of the resonances, however, depends critically on the direction of the magnetic field, making the sensor anisotropic. For known directions of the magnetic field, the symmetry of the system offers a specific advantage based on measurements of differences in frequencies [30]. In the absence of the magnetic field, the narrow absorption maximum disappears and the system becomes transparent at the line center. It thus has the potential to be used as a magneto-optic switch, with pulsed operation.
We have successfully modeled the system. For mixed polarizations of the coupling and the probe along the quantization axis, the structure of the resonances becomes complex and the features are under further investigation.
online) Experimental set-up. PBS: polarizing beam-splitter, AOM: acousto-optic modulator, λ/2: halfwave plate, λ/4: quarter-wave plate III. DARK RESONANCE PROFILES IN BASIC TRIPOD CONFIGURATIONS
FIG. 2 :FIG. 3 :
23(Color online) Tripod configuration with V-polarized (σ ± ) probes and H-polarized (π) coupling. (Color online) Left panel: Experimentally measured transmitted intensity (arb. units) versus Raman detuning δ, corresponding to the configuration shown in Fig. 2, with magnetic field B at 0 (black, open square), 10 (red, circle) and 30 (blue, triangle) mG, at coupling powers of (a) 1 mW, (b) 10 mW and (c) 22 mW. Right panel, (d), (e) and (f): Corresponding numerically calculated transmission profiles, withΩC ≡ 2πΩC/Γ,δ ≡ 2πδ/Γ, and∆Z ≡ 2π∆Z/Γ.
FIG. 4 :
4(Color online) Tripod configuration with H-polarized (π) probe and V-polarized (σ ± ) coupling beams.
where x = 2Γ R + 3 |ΩC| 2 2 and y =δ 2 +∆ 2 Z . We plot the experimentally measured transmitted intensity (arb. units) in Figs. 5(a)-(c), for coupling powers of 1 mW, 10 mW and 22 mW. The corresponding numerically calculated transmission profiles versus scaled Raman detuning are reproduced in Figs. 5(d)-(f), with the corresponding values ofΩ C = 1.8 × 10 −3 , 5.7 × 10 −3 and 8.6 × 10 −3 , respectively. In each case, we plot the profiles matching the magnetic fields of 0, 10 and 30 mG, as in the experiment. Our numerical simulations are in good agreement with the experimental results.
FIG. 5 :
5(Color online) Left panel: Experimentally measured transmitted intensity (arb. units) versus Raman detuning δ, corresponding to the configuration shown in Fig. 4, with magnetic field B at 0 (black, open square), 10 (red, circle) and 30 (blue, triangle) mG for coupling powers of (a) 1 mW, (b) 10 mW, and (c) 22 mW. Right panel, (d), (e) and (f):
AcknowledgmentsThis work is supported by the Indo-French Centre for Advanced Research (IFCPAR/CEFIPRA). The work of SK is supported by the Council of Scientific and Industrial Research, India.
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. A V Taichenachev, A M Tumaikin, V I Yudin, Phys. Rev. A. 6111802A.V. Taichenachev, A.M. Tumaikin, and V.I. Yudin, Phys. Rev. A 61, 011802(R) (1999).
. R Meshulam, T Zigdon, A D Wilson-Gordon, H Friedmann, Opt. Lett. 322318R. Meshulam, T. Zigdon, A.D. Wilson-Gordon, and H. Friedmann, Opt. Lett. 32, 2318 (2007).
M O Scully, M S Zubairy, Quantum Optics. CambridgeCambridge University PressM.O. Scully and M.S. Zubairy, Quantum Optics (Cam- bridge University Press, Cambridge, 1997).
. A Lezama, S Barreiro, A Lipsich, A M Akulshin, Phys. Rev. A. 6113801A. Lezama, S. Barreiro, A. Lipsich, and A.M. Akulshin, Phys. Rev. A 61, 013801 (1999).
. S K Kim, H S Moon, K Kim, J B Kim, Phys. Rev. A. 6863813S.K. Kim, H.S. Moon, K. Kim, and J.B. Kim, Phys. Rev. A 68, 063813 (2003).
. H S Chou, J Evers, Phys. Rev. Lett. 104213602H.S. Chou and J. Evers, Phys. Rev. Lett. 104, 213602 (2010).
. E S Kyoseva, N V Vitanov, Phys. Rev. A. 7323420E.S. Kyoseva and N.V. Vitanov, Phys. Rev. A 73, 023420 (2006).
. F Goldfarb, T Lauprêtre, J Ruggiero, F Bretenaker, J Ghosh, R Ghosh, C.R. Physique. 10919F. Goldfarb, T. Lauprêtre, J. Ruggiero, F. Bretenaker, J. Ghosh and R. Ghosh, C.R. Physique 10, 919 (2009).
. V I Yudin, A V Taichenachev, Y O Dudin, V L Velichansky, A S Zibrov, S A Zibrov, Phys. Rev. A. 8233807V.I. Yudin, A.V. Taichenachev, Y.O. Dudin, V.L. Velichansky, A.S. Zibrov, and S.A. Zibrov, Phys. Rev. A 82, 033807 (2010).
|
[] |
[
"Identifying Nominals with No Head Match Co-references Using Deep Learning",
"Identifying Nominals with No Head Match Co-references Using Deep Learning"
] |
[
"Matthew Stone [email protected] \nDepartment of Computer Science\nStanford University\n\n",
"Ramnik Arora [email protected] \nDepartment of Computer Science\nStanford University\n\n"
] |
[
"Department of Computer Science\nStanford University\n",
"Department of Computer Science\nStanford University\n"
] |
[] |
Identifying nominals with no head match is a long-standing challenge in coreference resolution with current systems performing significantly worse than humans. In this paper we present a new neural network architecture which outperforms the current state-of-the-art system on the English portion of the CoNLL 2012 Shared Task. This is done by using a logistic regression on features produced by two submodels, one of which is has the architecture proposed in [CM16a] while the other combines domain specific embeddings of the antecedent and the mention. We also propose some simple additional features which seem to improve performance for all models substantially, increasing F 1 by almost 4% on basic logistic regression and other complex models.
| null |
[
"https://arxiv.org/pdf/1710.00936v1.pdf"
] | 20,679,535 |
1710.00936
|
c06f4e7b63eb9695b65ff310f0936de666385cee
|
Identifying Nominals with No Head Match Co-references Using Deep Learning
Matthew Stone [email protected]
Department of Computer Science
Stanford University
Ramnik Arora [email protected]
Department of Computer Science
Stanford University
Identifying Nominals with No Head Match Co-references Using Deep Learning
Identifying nominals with no head match is a long-standing challenge in coreference resolution with current systems performing significantly worse than humans. In this paper we present a new neural network architecture which outperforms the current state-of-the-art system on the English portion of the CoNLL 2012 Shared Task. This is done by using a logistic regression on features produced by two submodels, one of which is has the architecture proposed in [CM16a] while the other combines domain specific embeddings of the antecedent and the mention. We also propose some simple additional features which seem to improve performance for all models substantially, increasing F 1 by almost 4% on basic logistic regression and other complex models.
Introduction
Coreference resolution is a domain of Natural Language Processing which is focused on finding all of the references to the same real world entity within a body of text. Applications of coreference resolution include full text understanding (for discourse understanding), machine translation (gender, numbers), text summarization, and question-answering systems.
Mentions, of which pairs can potentially form coreferences, come in three forms: named, pronominal or nominal. Named mentioned include a specific name, such as "President Obama." Pronominal mentions include the use of a pronoun but without naming the entity, such as "he" when referring to Obama. A nominal is the most vague and is neither named nor pronominal. A nominal mention could be "the President". Another nominal mention could be "the man guarded by the secret service." The first nominal example would have a head match with "President Obama" because of the overlap of the word "President". A nominal without a head match would be the second example.
This paper focuses on this match type using deep learning. One of the largest challenges with matching nominals with no head match is that a much deeper understanding of the mention pair is needed, including some modeling of the semantic meaning of each mention.
Traditional approaches to coreference began with Hobb's naive algorithm [Hob86]. This approach embodies the semantic or structural methodology behind anaphora resolution through a rules-based system. This algorithm is one of the early baselines for coreference.
There are three major modeling approaches to coreference:
• Mention pair models, which is the type this paper will attempt to implement, make independent pair-wise decisions. Typically these models also implement a reconciliation mechanism as a final stage before evaluation. These mechanisms map links that are implied by the model predictions [Hos16].
• Mention ranking models are attempting to solve a ranking problem by listing all possible antecedents and optimizing a ranking system to find the best or most likely antecedent mention [HS11].
• Entity mention models attempt to link all mentions of one entity in a cluster model by selectively choosing which a clustering algorithm that reduces errors on large scale mention clusters [CM16b].
Current state of the art models have a CoNLL F 1 [PMX + 12] score of almost 69% [CM16a] in aggregate while only 18.9% for corefences with no head match. Humans on the other hand score over 60% on similar tasks. Nominals with no head match need to make much more use of context. In particular, they leverage the word vector embeddings for related semantic meaning to identify these types of coreferences. With this in mind, the approaches found in this paper will emphasize aspects of word vector alignment. We'll be using a mention-pair model treating all mentions-pairs as independent, without any post reconciliation implemented as yet. One of the advancements in [CM16a] was post reconciliation via cluster resolution which we did not replicate. If A and B, as well as B and C are identified as corefernces, then the subsequent step would evaluate A and C, and either cluster A and C or break one of the two other links.
In our model, we increase the test F 1 on a 2% negative sample downsample from 0.49 for logistic regression baseline and 0.54 for state-of-the-art architecture in [CM16a] to 0.61 using a combination of new features and intelligent architecture. Note that these are not CoNLL F 1 scores, since this only applies to independent intra-document pairwise predictions rather than the downstream clustering task.
Data & Evaluation
We use the data provided by the CoNLL 2012 Shared Task on English [PMX + 12].
CoNLL 2012 Shared Task
The CoNLL 2012 shared task dataset provides 2802 documents in training, 343 documents in dev and 348 documents in test. The mentions, and the co-reference pairs and types are tagged in this dataset. We filter on nominal mentions and co-reference pairs which have no head-match. We structure our dataset as an evaluation of direct pair matches, and build deep learning models directly around this subset. In this paper, we treat all mentions as independent and try to learn simply based on input features.
The class label counts in the dataset are heavily skewed. If there are N mentions in a documents, there will be N 2 total mention pairs and only a few of them will be co-references. This means that the dataset is heavily skewed with significantly more negative examples than positive examples. In our training data we have 7,643,471 negative samples while 8,010 positive samples. We've downsampled the negative samples to 2% while maintaining 100% of our positive samples. Even with this downsampling, only 5% of our modified samples are positive mentions. See figure 1 for schematic representation of data-generating process.
Evaluation Metrics and Nuances
For co-reference classification, CoNLL F 1 , which is an average of MUC [VBA + 95], B 3 1 [BB98] and CEAF [Luo05] F 1 scores, is common evaluation metric. While we've used F 1 as an evaluation metric, it has shortcomings. Since F 1 scores are based on a strict {0, 1} classification, while the model predicts a probability estimate, they are very sensitive to thresholds. This is evidenced by figure 2 where for the same probability outputs, we can have drastically varying F 1 's depending on the threshold. By choosing F 1 as evaluation metric, the threshold becomes another highly sensitive hyper-parameter of the model. We also report on AUC (ROC) since it's more intuitive and independent of thresholds. An AUC of 0.5 represents no information or random predicitions which 1.0 is perfect information. We have implemented four model architectures and the baseline is the logistic regression with two sets of features. The base copy of features are similar to [CM16a] namely:
• Simplified version of sentence distance, including indicators for distances < 5 as well as actual distance (rather than the 10 buckets found in the paper). • Embeddings (from word2vec, 50 dimensions) of the first word, head words, last word, previous word, and next word. • Speaker features, as described in the paper.
• Pair features, including whether or not there is an exact match, a relaxed head match or if there is the same speaker tagged.
Model 1
The starting point of our paper is the architecture proposed in [CM16a] (See figure 3). The basic model is 3 fully connected layers with ReLU activations. The various hidden layer sizes are (500, 200, 100) and the final transformation is a sigmoid function. We use weighted cross entropy as our loss function for training. We also experimented with L2 loss, but found that weighted cross entropy better expressed the cost of missing rare events -a fundamental challenge in our problem.
Model 2
A secondary structure focuses on taking the insight that we are essentially modeling some embedding interactions and creating 2 separate learning models, one for each mention. The input features are trimmed to remove features related to the other word (other than mention distance). The final hidden layer of each of these mention-level models are then combined in a final layer dot-product. We then run this final layer through the sigmoid function to come up with a final prediction. The 2 final layers that are inputs to the dot-product could be thought of as domain specific embeddings of each mention. The interaction of these embeddings are then translated to our probability of coreference. Refer to figure 4 for details. Figure 4: Model 2 architecture learns the candidate antecedent and mention embeddings independently before combining them
Composite Models (Model 3 & Model 4)
Model 3 is an ensemble of model 2 and model 3, where we simply add the 2 final scores from model 1 and model 2 and then apply a sigmoid function to that output. This model is also trained end-to-end rather than using pre-trained versions of model 1 and 2, otherwise they would end up over-predicting the outcome.
Model 4 is a modification onto model 3 where instead of simply adding the two models, we take the sigmoid function of each of model 1 and model 2. We then create a new submodel with the same set of input features to train a probabilistic model that puts a weight of prb on model 1 and a weight of 1-prb on model 2. This probabilistic model is a logistic regression on input features that determines what weight should be given to model 1. This is also trained end-to-end. This allows the data to determine when model 1 may be preferable to model 2 and vice versa rather than do a simple ensemble average.
We explore regularization with dropout, especially at the first hidden layer. We found that applying dropout at higher levels was detrimental to performance (see figure 5b). We believe that at the size of model we were testing, dropout ended up removing too many nodes to offset the benefit from generalizing.
Experiments, Results & Challenges
Results
In our experiments we compare the baseline model of logistic regression with the four proposed models. The proposed models were tuned using the train and dev set on varying loss function, weights to the loss function, learning rate, neural network architecture, dropout and number of epochs. The best model performance on the dev set for each of the model was chosen and the results produced on the test set which are summarized in table 1. It must be noted that the test set, like the (a) Average precision and recall for our models as we increase the training epochs.
(b) The train and dev F1 scores with varying levels of dropout.
training and dev set has been downsampled to 2% negative samples and uses the additional features as described in section 4.2. Model 4, training on 15 epochs with a slow learning rate of 0.0001 performs almost 12% better than the baseline logistic regression while almost 8% better than the tuned state-of-the-art model.
Unsurprisingly, as can be seen in figure 5a, given the label imbalance, as we increase the number of epochs, we are able to increase the precision at the expense of recall for all models. The confusion matrix for the best performing model gives a good understanding of the labels and relative performance (
Impact of Additional Features
Looking at some of the errors like table 3, especially false negatives for co-reference like "the molara tooth" and "my taxi-the car", we added the cosine distance of the word vector embeddings as raw features to the model. In particular, we added the dot-product between the first, last and the head word of the two mentions, creating four extra features for every mention pair based on the semantic relationship between the two mentions. This drastically improved the model performance, as can be seen in table 4. Simply adding these features to the logistic regression increased the model performance by 4%. The intuition behind adding these features is simple -given concatenated word vectors [m 1 m 2 ], a fully connected neural network can never learn m 1 m 2 which represents how similar m 1 is to m 2 . Unsurprisingly, a lot of nominals with no head match are synonyms or specific example of a general entity which is now captured by these new features (this is also attempted by model 2). We also choose weighted cross entropy as our loss function, so that we can give further weight to positive examples through a tunable hyperparameter. When evaluating F 1 metrics, we are then forced to define a threshold at which we consider the model probability to be true through two mechanisms.
• When dealing with downsampled data, we use an F 1 tuner that searches over the training set for the threshold that yields the highest F 1 , and apply this to our dev set evaluation.
• When training on downsampled data but evaluating on the full dataset (for our test set) we run our calibration process. This takes the distribution of predictions from the training set and finds the percentile of data such that the model will emit the correct true rate as though the data was not downsampled. For example, if the rate in our downsampled data is 33% and the downsampling was 50%, that means the true rate is around 17%. In this example, calibration of the prediction percentile would be with the goal of yielding true 17% of the time on the training set. The idea behind setting a higher threshold is that if we need to reduce the number of true predictions, we should only keep the highest conviction guesses from the model. In practice, the level of model certainty required to emit a True varied between 97.5% to 99.8% depending on the model.
Memory and storage:
Because of the size of our dataset, we had challenges creating a full dataset on disk as well as fitting one in memory for training. Mitigating factors included only using halfprecision for float and working with downsampled training data with a calibration layer. We also partitioned training data and running over all of the partitions each epoch. This greatly increased the time per epoch and experiment. A possibility to reduce the size of the data is to do in-memory lookups of word vectors, but this would have great slowed down the training speed as each example would require a read-and-process operation rather than a simple read. We could also use a databse to store the test/train and dev features.
Evaluation and prediction reconciliation: A full evaluation would involve creating a full set of CoNLL evaluation metrics on the entire dataset. Instead, we used F 1 on a simple pair-matching model without any reconciliation step.
6
We found that extending the deep learning model by doing a inner product of the embeddings, a probabilistic model ensemble and adding the aforementioned set of features improves the model over the baseline. The first two improvements over a simple feed forward network gave our model more capacity and flexibility for identifying nominal co-references with no head match.
We believe there are several promising dimensions of future enhancements that we can explore along the following lines:
• Downstream tasks: Coreference links between mentions form an upstream part of a coreference system, and nominals with no head match are a fraction of the coreferences. In this paper, while we improve upon the existing state-of-the-art algorithm, we don't know the impact on the downstream tasks. • Document level memory: We are looking at intra-document coreferences in this paper.
Memory and attention play a role in human understanding of semantic information, which could be extended to this class of problems using LSTM or RNNs. It can also help weight multiword mentions to better capture the key part of what makes the object a real world reference. • A separate layer for singleton mentions: Identifying which mentions are never repeated could help the model by reducing the number of evaluated candidates for future evaluation. This could be done end-to-end or within a continuous framework with a penalty or probability interaction. This is motivated by [RdMP13] and could incorporate more semantic features.
Figure 1 :
1Schematic diagram of the data
Figure 2 :
2F 1 is very sensitive to the threshold chosen 3 System Architecture
Figure 3 :
3Model 1 architecture with three layer feed-forward neural network 3
Table : 2
:).Model Name
Training F1 Dev F1 Test F1
Logistic Regression
.76
.36
.49
Model 1
.76
.53
.53
Model 2
.65
.51
.53
Model 3
.75
.52
.56
Model 4
.82
.52
.61
Table 1: F1 score on 2% downsampled negative examples data
Predicted NO Predicted YES
Actual NO
20272
424
Actual YES
559
531
Table 2 :
2Confusion matrix for Model 4 over test dataset.
Table 3 :
3Example Wins: Examples of False Negatives without Addn Features turning to True Positive with Additional FeaturesModel Name
Training F 1 Test F 1 Test AUC (ROC)
Without Addn. Features
.71
.45
.90
With Addn. Features
.76
.49
.91
∆
+.05
+.04
+.01
Table 4 :
4examples from negative examples. We take a multi-step approach to resolving this issue and guiding the model to learn to distinguish positive and negative examples. We begin by downsampling our training data -keeping all positive examples and only 2% of the negative examples.The F 1 score improves drastically for logistic regression (actually, for all models) with the
addition of the new features.
4.3 Challenges
Class imbalance: As mentioned previously, the overall true rate of the examples in our dataset is
approximately 0.1%, meaning that models trained on the raw dataset have difficulty learning what
distinguishes positive
Algorithms for scoring coreference chains. Amit Bagga, Breck Baldwin, The First International Conference on Language Resources and Evaluation Workshop on Linguistics Coreference. Amit Bagga and Breck Baldwin. Algorithms for scoring coreference chains. In In The First International Conference on Language Resources and Evaluation Workshop on Linguistics Coreference, pages 563-566, 1998.
Deep reinforcement learning for mentionranking coreference models. Kevin Clark, Christopher D Manning, Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing. the 2016 Conference on Empirical Methods in Natural Language ProcessingAustin, Texas, USAKevin Clark and Christopher D. Manning. Deep reinforcement learning for mention- ranking coreference models. In Proceedings of the 2016 Conference on Empirical Meth- ods in Natural Language Processing, EMNLP 2016, Austin, Texas, USA, November 1-4, 2016, pages 2256-2262, 2016.
Improving coreference resolution by learning entity-level distributed representations. Kevin Clark, Christopher D Manning, abs/1606.01323CoRRKevin Clark and Christopher D. Manning. Improving coreference resolution by learning entity-level distributed representations. CoRR, abs/1606.01323, 2016.
Readings in natural language processing. chapter Resolving Pronoun References. J Hobbs, Morgan Kaufmann Publishers IncSan Francisco, CA, USAJ Hobbs. Readings in natural language processing. chapter Resolving Pronoun Ref- erences, pages 339-352. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 1986.
The Mention-Pair Model. Veronique Hoste, Springer Berlin HeidelbergBerlin, HeidelbergVeronique Hoste. The Mention-Pair Model, pages 269-282. Springer Berlin Heidel- berg, Berlin, Heidelberg, 2016.
A generative entity-mention model for linking entities with knowledge base. Xianpei Han, Le Sun, Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies. the 49th Annual Meeting of the Association for Computational Linguistics: Human Language TechnologiesStroudsburg, PA, USAAssociation for Computational Linguistics1Xianpei Han and Le Sun. A generative entity-mention model for linking entities with knowledge base. In Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies -Volume 1, HLT '11, pages 945-954, Stroudsburg, PA, USA, 2011. Association for Computational Linguistics.
On coreference resolution performance metrics. Xiaoqiang Luo, Proceedings of the Conference on Human Language Technology and Empirical Methods in Natural Language Processing, HLT '05. the Conference on Human Language Technology and Empirical Methods in Natural Language Processing, HLT '05Stroudsburg, PA, USAAssociation for Computational LinguisticsXiaoqiang Luo. On coreference resolution performance metrics. In Proceedings of the Conference on Human Language Technology and Empirical Methods in Natural Lan- guage Processing, HLT '05, pages 25-32, Stroudsburg, PA, USA, 2005. Association for Computational Linguistics.
CoNLL-2012 shared task: Modeling multilingual unrestricted coreference in OntoNotes. Alessandro + 12] Sameer Pradhan, Nianwen Moschitti, Olga Xue, Yuchen Uryupina, Zhang, Proceedings of the Sixteenth Conference on Computational Natural Language Learning. the Sixteenth Conference on Computational Natural Language LearningJeju, Korea+ 12] Sameer Pradhan, Alessandro Moschitti, Nianwen Xue, Olga Uryupina, and Yuchen Zhang. CoNLL-2012 shared task: Modeling multilingual unrestricted coreference in OntoNotes. In Proceedings of the Sixteenth Conference on Computational Natural Language Learning (CooNLL 2012), Jeju, Korea, 2012.
The life and death of discourse entities: Identifying singleton mentions. Marta Recasens, Marie-Catherine De Marneffe, Christopher Potts, HLT-NAACL. Lucy Vanderwende, Hal Daum III, and Katrin KirchhoffMarta Recasens, Marie-Catherine de Marneffe, and Christopher Potts. The life and death of discourse entities: Identifying singleton mentions. In Lucy Vanderwende, Hal Daum III, and Katrin Kirchhoff, editors, HLT-NAACL, pages 627-633. The Asso- ciation for Computational Linguistics, 2013.
A model-theoretic coreference scoring scheme. Marc Vilain, John Burger, John Aberdeen, Dennis Connolly, Lynette Hirschman, Proceedings of the 6th Conference on Message Understanding, MUC6 '95. the 6th Conference on Message Understanding, MUC6 '95Stroudsburg, PA, USAAssociation for Computational LinguisticsVBA + 95[VBA + 95] Marc Vilain, John Burger, John Aberdeen, Dennis Connolly, and Lynette Hirschman. A model-theoretic coreference scoring scheme. In Proceedings of the 6th Conference on Message Understanding, MUC6 '95, pages 45-52, Stroudsburg, PA, USA, 1995. Association for Computational Linguistics.
|
[] |
[
"Testing quark mixing in minimal left-right symmetric models with b-tags at the LHC",
"Testing quark mixing in minimal left-right symmetric models with b-tags at the LHC"
] |
[
"Andrew Fowlie \nNational Institute of Chemical Physics and Biophysics\nRavala 1010143TallinnEstonia\n",
"Luca Marzola \nInstitute of Physics\nLaboratory of Theoretical Physics\nUniversity of Tartu\nRavila 14c50411TartuEstonia\n"
] |
[
"National Institute of Chemical Physics and Biophysics\nRavala 1010143TallinnEstonia",
"Institute of Physics\nLaboratory of Theoretical Physics\nUniversity of Tartu\nRavila 14c50411TartuEstonia"
] |
[] |
Motivated by a hint in a CMS search for right-handed W -bosons in eejj final states, we propose an experimental test of quark-mixing matrices in a general left-right symmetric model, based on counting the numbers of b-tags from right-handed W -boson hadronic decays. We find that, with our test, differences between left-and right-handed quark-mixing matrices could be detected at the LHC with √ s = 14 TeV. With an integrated luminosity of about 20/fb, our test is sensitive to right-handed quark-mixing angles as small as about 30 • and with 3000/fb, our test's sensitivity improves to right-handed mixing angles as small as about 7.5 • . Our test's sensitivity might be further enhanced by tuning b-tagging efficiency against purity. * [email protected] † [email protected] 1 arXiv:1408.6699v1 [hep-ph]
|
10.1016/j.nuclphysb.2014.10.009
|
[
"https://arxiv.org/pdf/1408.6699v2.pdf"
] | 118,365,049 |
1408.6699
|
916ac3dfc8f8ea51c5111a6620eeccf73a7da60a
|
Testing quark mixing in minimal left-right symmetric models with b-tags at the LHC
(Dated: August 29, 2014) 28 Aug 2014
Andrew Fowlie
National Institute of Chemical Physics and Biophysics
Ravala 1010143TallinnEstonia
Luca Marzola
Institute of Physics
Laboratory of Theoretical Physics
University of Tartu
Ravila 14c50411TartuEstonia
Testing quark mixing in minimal left-right symmetric models with b-tags at the LHC
(Dated: August 29, 2014) 28 Aug 2014
Motivated by a hint in a CMS search for right-handed W -bosons in eejj final states, we propose an experimental test of quark-mixing matrices in a general left-right symmetric model, based on counting the numbers of b-tags from right-handed W -boson hadronic decays. We find that, with our test, differences between left-and right-handed quark-mixing matrices could be detected at the LHC with √ s = 14 TeV. With an integrated luminosity of about 20/fb, our test is sensitive to right-handed quark-mixing angles as small as about 30 • and with 3000/fb, our test's sensitivity improves to right-handed mixing angles as small as about 7.5 • . Our test's sensitivity might be further enhanced by tuning b-tagging efficiency against purity. * [email protected] † [email protected] 1 arXiv:1408.6699v1 [hep-ph]
I. INTRODUCTION
An unexplained feature of the Standard Model (SM) [1][2][3] is that left-right symmetry is broken; only left-handed fermions take part in weak interactions [4]. In the 1970s, Georgi and Glashow [5], amongst others [6][7][8][9][10], realized that puzzling aspects of the SM could be explained if, at high energy, nature is symmetric under a simple or semi-simple Lie group.
This popular proposal became known as a grand unified theory (GUT) and provided an ideal framework in which to restore left-right symmetry at high energy [6,7,[11][12][13][14][15].
A left-right symmetric GUT gauge group can be spontaneously broken to the SM gauge group via a left-right symmetric product gauge group. Minimal realizations of the latter contain the product gauge group SU(2) L × SU(2) R , as well as a discrete symmetry that ensures that the representations and couplings for the SU(2) L and SU(2) R gauge groups are indeed left-right symmetric. Generalized parity, P, and generalized charge conjugation, C, are common candidates for that discrete symmetry. At low energy, the minimal left-right symmetric gauge group is broken to the familiar SM gauge group and neither P nor C symmetry is preserved.
After the spontaneous symmetry breaking of both SU(2) L and SU(2) R , a left-right symmetric model includes massive W R,L -bosons, 1 massive quarks and two distinct quark-mixing matrices (see e.g., Ref. [16]). These result from the misalignment between the quark mass eigenstates and the SU(2) R or SU(2) L interaction eigenstates, with the SM CKM matrix [17,18] (henceforth LH CKM matrix) describing the resulting quark mixing in the latter case. It was recently shown in Ref. [19], following earlier work in Ref. [20,21], that in minimal left-right symmetric GUTs, the LH CKM matrix and the RH mixing matrix are approximately identical, modulo complex phases. In a general left-right symmetric GUT, this is possible, though not compulsory.
Left-right symmetric models are nowadays particularly interesting in light of an experimental hint from the Large Hadron Collider (LHC). In a recent CMS analysis [22], the number of events presenting two electrons (with no charge requirement imposed, i.e., e − e + , e + e + or e − e − ) and two jets in the final state exceeded the prediction of the SM. Whilst the excess might have a mundane explanation, such as a statistical fluctuation or a systematic error, we regard it as an intriguing hint. In fact, the detected anomaly could be explained by the production and subsequent decay of a right-handed W -boson in a left-right symmetric model ( Fig. 1),
qq → W R → eν R e → eeW R → eejj,(1)
provided the right-handed W -boson has a mass of about 2 TeV and only the right-handed electron-neutrino is lighter than about 2 TeV [23,24]. With the recent experimental hint in mind [22], we consider a scenario in which there exists a heavy right-handed W -boson and show that the equality of the LH CKM matrix and the RH mixing matrix could be tested at the LHC. As shown below, our method categorizes the hadronic decays of the new gauge boson by their number of b-tags 2 and quantifies the probability of obtaining the same result under the assumption that the RH mixing matrix matches the LH CKM one. The proposed procedure is therefore able to quantify the discrepancy between the quark mixings of the two chiral sectors in a model-independent way and constitutes a new collider test of minimal left-right models that complements the model-dependent results brought by meson-oscillation experiments [26,27] and low-energy observables (see e.g., Ref. [28]).
II. METHODOLOGY
The RH mixing matrix affects the rate at which right-handed W -bosons are produced from two protons and the right-handed W -boson's branching fractions to quarks in the 2 The mass of the bottom quark is such that it travels within the LHC detectors before decaying at a displaced vertex to highly energetic jets. From these features, b-jets can be identified or "tagged" by b-tagging algorithms (see e.g., Ref. [25]). Because top quarks decay into bottom quarks, top quarks result in a b-jet which can be b-tagged.
decay chain in Eq. (1). The production cross section depends on three unknown quantities:
the RH mixing matrix, the right-handed gauge coupling at low energy, g R (M W ), and the right-handed W -boson mass. 3 The branching fractions in the final hadronic decay, however, depend on only the RH mixing matrix. Thus, to investigate the RH mixing matrix, the right-handed W -boson's hadronic decay is the best place to start.
We parameterize the RH mixing matrix in the standard way [29,30], i.e., as the product of rotations on three planes in the basis of the quark fields (d, s, b) T ;
V R = 1 0 0 0 c 23 s 23 0 −s 23 c 23 c 13 0 s 13 0 1 0 −s 13 0 c 13 c 12 s 12 0 −s 12 c 12 0 0 0 1 ,(2)
where c ij ≡ cos θ ij and s ij ≡ sin θ ij with a superscript 'R' left understood. In principle, the RH mixing matrix contains five physical phases on top of three mixing angles. However, as will become soon apparent, our work is not sensitive to these quantities and therefore we chose to disregard them for the sake of simplicity.
Given the above mixing matrix, we calculate the right-handed W -boson's hadronic branching fractions through the Feynman rulē
q R W R g¥ h q R = ig R V R qq γ µ ,(3)
and assume that the right-handed W -boson is much heavier than the top quark, M W R ≈ 2 TeV m t , such that all quark masses are negligible. Motivated by the experimental hint [22], we also assume that the right-handed muon-neutrino is heavier than the right-
handed W -boson, m ν R µ > M W R , but that m ν R e < M W R . On top of that we neglect W L -W R mixing.
Although the right-handed W -boson's hadronic branching fractions can be straightforwardly computed, the b-tagging algorithms adopted in an experimental analysis are still imperfect:
• The efficiency, , is the probability that a genuine b-jet is b-tagged. With an appreciable probability, 1 − , a genuine b-jet might not be b-tagged. We assume that = 0.7.
• The purity, ρ, is the probability that a genuine light-jet is not b-tagged. With a small probability, 1 − ρ, a genuine light jet might be b-tagged. 4 We assume that ρ = 0.99.
Thus, any right-handed W -boson hadronic decay could actually result in 0, 1 or 2 b-tags, even if the right-handed W -boson decayed to only light quarks.
By combining our knowledge of the imperfections of b-tagging with the tree-level righthanded W -boson hadronic branching fractions, we then expect that right-handed W -bosons that decay hadronically result in 0, 1 or 2 b-tags with the following probabilities,
p 0 ≡ p(0 b-tags from W R hadronic decay) ∝ ρ 2 C 0 + ρ (1 − ) C 1 + (1 − ) 2 C 2 ,(4)p 1 ∝ 2ρ (1 − ρ) C 0 + ρ C 1 + (1 − ρ) (1 − ) C 1 + 2 (1 − ) C 2 ,(5)p 2 ∝ (1 − ρ) 2 C 0 + (1 − ρ) C 1 + 2 C 2 ,(6)
which include all possible mistaggings with the appropriate weights. In the above expressions we omitted a normalization constant and defined
C 0 := V R 11 2 + V R 12 2 + V R 21 2 + V R 22 2 =1 + cos 2 θ 13 cos 2 θ 23 ,(7)C 1 := V R 31 2 + V R 32 2 + V R 23 2 + V R 13 2 =2 1 − cos 2 θ 13 cos 2 θ 23 ,(8)C 2 := V R 33 2 = cos 2 θ 13 cos 2 θ 23 .(9)
As anticipated, the probabilities are independent of the five phases in the RH mixing matrix and, interestingly, depend on only the θ 13 and θ 23 mixing angles of the former. The dependence on the remaining mixing angle, θ 12 , is lost because this quantity regulates the mixing of firstand second-generation light quarks and therefore cannot affect the expected fraction of b-jets or light jets.
We assume that the production cross section for the right-handed W -boson is such that we expect that 10 of the 14 events in the ∼ 2 TeV bin in Ref. [22] result from the decay chain in Eq. (1). This can be achieved by tuning the right-handed W -boson coupling and mass.
We expect that the remaining 4 events result from SM backgrounds, as indicated in Ref. [22].
However, with so few events, it is impossible to infer interesting information about the RH mixing matrix. Thus, we refer to a √ s = 14 TeV scenario with an integrated luminosity identical to that in Ref. [22], ∼ 20/fb and scale the numbers of signal and background events in Ref. [22] by the ratio of the corresponding cross sections at √ s = 14 TeV and √ s = 8 TeV.
Consequently, at √ s = 14 TeV we expect s = 67.0 signal events [31] and b = 15.6 background events [32] and the increased number of signal events makes it possible to study the RH mixing matrix in this scenario. With our Eq. 4, 5 and 6, we will show that, if our alternative hypothesis is correct, future LHC experiments would have the power to reject the null hypothesis that V L = V R with at least 95% confidence by counting the numbers of b-tags.
For this purpose we consider two cases, which we regard as hypotheses in the statistical test performed below:
• The null hypothesis, H 0 : the RH mixing matrix is equal to the LH CKM matrix,
V R = V L(10)
with V L fixed by the usual SM quark mixing (see e.g., Ref. [33]).
• The alternative hypothesis, H 1 : the RH mixing matrix is independent of the LH CKM matrix.
From our Eq. 4, 5 and 6, we then calculate the expected numbers of events with i b-tags from a right-handed W -boson hadronic decay,
s i = p i × s,(11)
where s is the total number of expected signal events. Unfortunately, the signal region is contaminated with SM background events. The dominant SM background is tt. We make the approximation that all SM backgrounds result in the same b-tag distribution as that of tt production, such that
b i = p(i b-tags from tt) × b,(12)
where b is the total number of expected background events and the probability is calculated in a manner analogous to that in Eq. 4, 5 and 6. This approximation is conservative, because tt contaminates all b-tag categories with appreciable probabilities.
In a counting experiment, such as which we propose, the numbers of observed events in each b-tag category, o i , are Poisson distributed,
o i ∼ Po(s i + b i )(13)
and independent of each other.
Throughout the following discussion, our notation is such that if a quantity is calculated under the null hypothesis, it is superscripted with a zero, whereas if it is calculated under the alternative hypothesis, it is superscripted with a one. Our methodology is that for given mixing angles in the RH mixing matrix: 2. For each of the 1000 MC measurements, we calculate a log-likelihood ratio test-statistic (LLR) associated with the null hypothesis that V L = V R and the alternative hypothesis;
LLR = −2 ln L(o 1 i | H 0 ) max L(o 1 i | H 1 ) (14) = −2 i ln (s 0 i + b i ) o 1 i e −(s 0 i +b i ) o 1 i ! + 2 i ln max (s 1 i + b i ) o 1 i e −(s 1 i +b i ) o 1 i ! .(15)
In the second term, the likelihood is maximized by tuning the RH mixing matrix elements.
By Wilks' theorem, because the expected numbers of events, s 0 i + b i , are greater than about 5, the LLR is approximately χ 2 -distributed with 3 degrees of freedom, 5
LLR ∼ χ 2 3 .(16)
The p-value is the probability of obtaining such a large test-statistic by chance, were the null hypothesis true.
3. Finally, we find the median and 68% confidence interval for the p-value, by considering all of our MC experiments. Our ordering rule for the 68% confidence interval is that 16% of our MC experiments resulted in p-values above the interval and that 16% resulted in p-values below the interval.
III. RESULTS
In Fig. 2a, we plot the median exclusion for the null hypothesis that the LH CKM matrix equals the RH mixing matrix, were the RH mixing matrix in fact described by independent θ 13 and θ 23 mixing angles. Because the p-value is invariant under the exchange θ 13 ↔ θ 23 , Fig. 2a is approximately symmetric about the diagonal. If either of the θ 13 and θ 23 RH mixing angles were greater than about 40 • or if both were greater than about 30 • , we expect that in at least 50% of circumstances the null hypothesis would be rejected with at least 95% confidence.
To aid understanding, we also plot in Fig. 2b the expected fraction of events in each b-tag category resulting from a right-handed W -boson hadronic decay and the median p-value as a function of the universal mixing angle. Because Fig. 2a is approximately spherically symmetric, the behavior of the p-value in the direction θ 13 = θ 23 is approximately equal to that in any other direction.
By θ 30 • , the expected fractions of events in each b-tag category significantly differ from those in the null hypothesis. In particular, the expected fraction of events with one b-tag increases from about 15% to about 20%. The median p-value falls from 50% to less than 5% and the 68% band narrows. If θ 40 • , one should expect that in at least 84% of circumstances, the null hypothesis will be rejected with at least 95% confidence. If θ 30 • , one should expect that in at least 50% of circumstances, the null hypothesis will be rejected with at least 95% confidence.
Thus, it appears that with limited integrated luminosity of about 20/fb at √ s = 14 TeV, it might be possible to reject the theory that the LH CKM matrix is equal to the RH mixing matrix. Whether this is possible is, of course, dependent on the size of the mixing angles in the RH mixing matrix. If the θ 13 and θ 23 mixing angles in the RH mixing matrix differ only slightly from those in the LH CKM matrix, it will be difficult to test the equality of the LH CKM matrix and the RH mixing matrix. On the other hand, as explained in Sec. II, nothing
can be inferred about the mixing angle between light quarks, θ 12 , or complex phases.
In Fig. 3 as small as about 7.5 • . Below about 7.5 • , the numbers of events in each b-tag category in the V R = V L and V L = V R hypotheses are too similar for the hypotheses to be discriminated.
In our final scenario, with an improved b-tagging efficiency, we make slight inroads into θ 7.5 • ; in fact, the improved efficiency results in sensitivity to a universal mixing angle as small as about 6.5 • . With current algorithms, the b-tagging efficiency in this scenario unrealistic, but this scenario suggests that slight improvements in b-tagging efficiency, even to the detriment of purity, could improve sensitivity to the RH mixing matrix.
V L = V R .
The blue band is the 68% interval for the p-value, over MC experiments. The pink dashed line indicates a p-value of 5%. If the p-value drops below 5%, we can reject the null hypothesis with at least 95% confidence.
IV. CONCLUSIONS AND OUTLOOK
In light of an experimental hint from the LHC, left-right symmetric models are attracting renewed interest. In minimal left-right symmetric models, the LH CKM matrix is approximately equal to the RH mixing matrix. We proposed an experimental test of this equality at the LHC at √ s = 14 TeV, in a scenario in which a right-handed W -boson with a mass of about 2 TeV had been discovered, as suggested by the hint.
Our test involved counting the numbers of b-tags resulting from the right-handed Wboson's hadronic decays. We found that at √ s = 14 TeV with a limited integrated luminosity of about 20/fb, minimal left-right symmetric models could be rejected at 95% confidence, if the mixing angles in the RH mixing matrix were greater than about 30 • . Our test was, however, insensitive to complex phases and the mixing angle between the light quarks.
With an increased integrated luminosity of about 3000/fb, our test was sensitive to RH mixing angles as small as about 7.5 • and less if b-tagging efficiencies could be improved or optimized for our test.
Because in this paper we simply proposed a method, we made conservative approximations in our analysis. In particular, rather than performing a full Monte-Carlo simulation of the test, we scaled background estimates from the quoted CMS study and modelled the former on the most dangerous background, tt in our case. In a forthcoming publication [34], as well as refining the current analysis, we will propose a similar experimental test of the unitarity of the RH mixing matrix.
Figure 1 :
1Feynman diagram for the production and decay of a right-handed W -boson W R at the LHC.
1 .
1From the Poisson distributions in Eq. (13), we sample 1000 Monte-Carlo (MC) measurements of the numbers of observed events in each b-tag category, o 1 i , with the alternative hypothesis. The number of signal events in each b-tag category is a function of the RH mixing angles.
5
There are 3 approximately Gaussian contributions to the likelihood. In the first term in Eq. (15), no parameters are tuned, resulting in 3 degrees of freedom. In the second term, 18 parameters in the RH mixing matrix are tuned, resulting in 0 degrees of freedom. Thus, there are 3 − 0 = 3 degrees of freedom in the LLR.
L
, we consider two additional scenarios: a scenario with an increased integrated luminosity of L ∼ 3000/fb and a scenario with L ∼ 3000/fb and an improved in b-tagging efficiency = 0.8 to the detriment of the purity, ρ = 0.98. With increased integrated luminosity, the increased numbers of events result in sensitivity to a universal mixing angle Median p-value with 68% range (b) Universal mixing angle.
Figure 2 :
2A √ s = 14 TeV scenario with L ∼ 20/fb, with efficiency = 0.7 and purity ρ = 0.99. (a) Exclusion of the null hypothesis that V L = V R on the (θ 23 , θ 13 ) plane. The LH CKM matrix is marked with an arrow. (b) The RH mixing matrix universal mixing angle against (upper) the expected fractions of signal events in the b-tag categories and (lower) the median p-value for the null hypothesis that V L = V R . The blue band is the 68% interval for the p-value, over MC experiments.The pink dashed line indicates a p-value of 5%. If the p-value drops below 5%, we can reject the null hypothesis with at least 95% confidence.
L
Median p-value with 68% range (b) L ∼ 3000/fb, with efficiency = 0.8 and purity ρ = 0.98
Figure 3 :
3As in Fig. 2b, though for θ ≤ 15 • and with L ∼ 3000/fb (left) and with an improved efficiency (right). The RH mixing matrix universal mixing angle against (upper) the expected fractions of events in the b-tag categories and (lower) the median p-value for the null hypothesis that
The label on a gauge boson refers to the handedness of the fermions with which it interacts.
The right-handed gauge coupling at low energy, g R (M W ), might differ from g L (M W ) by renormalization group running, even if they are equal at a high energy.
We refer to first-and second-generation quarks as light quarks.
ACKNOWLEDGMENTSWe thank E. Gabrielli, M. Heikinheimo, M. Raidal and C. Spethmann for the useful comments and discussions. AF was supported in part by grants IUT23-6, CERN+, and by the European Union through the European Regional Development Fund and by ERDF project 3.2.0304.11-0313 Estonian Scientific Computing Infrastructure (ETAIS). LM acknowledges the European Social Fund for supporting his work with the grant MJD387.
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[] |
[
"Channel-state duality and the separability problem",
"Channel-state duality and the separability problem"
] |
[
"K V Antipin \nDepartment of Physics\nMoscow State University\n119991MoscowRussia\n"
] |
[
"Department of Physics\nMoscow State University\n119991MoscowRussia"
] |
[] |
Separability of quantum states is analyzed with the use of the Choi-Jamiolkowski isomorphism.I. IntroductionEntanglement as a resource [1, 2] is a central notion in quantum information theory. The important question is to tell whether a given quantum composite system state is entangled or separable. One of the first remarkable results in this direction was the positive partial transposition (PPT) criterion [3] as necessary condition for separability of bipartite mixed states. This simple but extremely useful observation by Asher Peres has generated further considerable research. It was proved that PPT condition is necessary and sufficient for separability of 2 ⊗ 2 and 2 ⊗ 3 states[4]. Over time several other necessary or/and sufficient criteria were developed[4,5,6,7,8,9], among which entanglement witnesses[5,7]and the CCNR criterion[8,9]proved to be important tools in detecting entanglement. The Bloch representation was first introduced into the separability problem by de Vicente [10], the idea, extended in a recent paper[11]where a necessary and sufficient separability criterion was obtained in terms of inequalities for singular values of the correlation matrix.In this paper we present an approach to the separability problem inspired by the well-known correspondence between completely positive (CP) maps and states, the Choi-Jamiolkowski isomorphism[12,13]. We make use of the fact that a state is separable if and only if the corresponding CP map can have operator-sum representation with unit rank operators. Different operator-sum representations of the same CP map are connected by linear transformations of the specific type. Thus, the state is separable if and only if the operators 1 of the corresponding CP map can all be transformed to those of unit rank. We analyze the conditions under which such transformations exist and show that this approach can be a powerful tool in investigating the separability problem. The method is illustrated by examples. In addition, spectrum based separability criterion is derived.II. Channel-state duality and connections with separabilityWe begin this section with recalling the properties of the Choi-Jamiolkowski isomorphism between states and CP maps. Let ρ AB be a density operator acting on Hilbert space H AB = H A ⊗ H B with dimensions dim(H A ) = m, dim(H B ) = n, m ≤ n. Let σ be a density operator on H A . A CP map Λ ρ : L(H A ) → L(H B ), corresponding to ρ AB , can be defined by the action on an arbitrary state σ as follows:where σ T -transposed operator σ, I A -identity operator acting on H A . * [email protected] 1 If a CP map is trace-preserving, they are called Kraus operators.
| null |
[
"https://arxiv.org/pdf/1909.13309v2.pdf"
] | 203,593,601 |
1909.13309
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c0325efa3732dfefc4d63b22e30051612344537f
|
Channel-state duality and the separability problem
29 Sep 2019
K V Antipin
Department of Physics
Moscow State University
119991MoscowRussia
Channel-state duality and the separability problem
29 Sep 2019
Separability of quantum states is analyzed with the use of the Choi-Jamiolkowski isomorphism.I. IntroductionEntanglement as a resource [1, 2] is a central notion in quantum information theory. The important question is to tell whether a given quantum composite system state is entangled or separable. One of the first remarkable results in this direction was the positive partial transposition (PPT) criterion [3] as necessary condition for separability of bipartite mixed states. This simple but extremely useful observation by Asher Peres has generated further considerable research. It was proved that PPT condition is necessary and sufficient for separability of 2 ⊗ 2 and 2 ⊗ 3 states[4]. Over time several other necessary or/and sufficient criteria were developed[4,5,6,7,8,9], among which entanglement witnesses[5,7]and the CCNR criterion[8,9]proved to be important tools in detecting entanglement. The Bloch representation was first introduced into the separability problem by de Vicente [10], the idea, extended in a recent paper[11]where a necessary and sufficient separability criterion was obtained in terms of inequalities for singular values of the correlation matrix.In this paper we present an approach to the separability problem inspired by the well-known correspondence between completely positive (CP) maps and states, the Choi-Jamiolkowski isomorphism[12,13]. We make use of the fact that a state is separable if and only if the corresponding CP map can have operator-sum representation with unit rank operators. Different operator-sum representations of the same CP map are connected by linear transformations of the specific type. Thus, the state is separable if and only if the operators 1 of the corresponding CP map can all be transformed to those of unit rank. We analyze the conditions under which such transformations exist and show that this approach can be a powerful tool in investigating the separability problem. The method is illustrated by examples. In addition, spectrum based separability criterion is derived.II. Channel-state duality and connections with separabilityWe begin this section with recalling the properties of the Choi-Jamiolkowski isomorphism between states and CP maps. Let ρ AB be a density operator acting on Hilbert space H AB = H A ⊗ H B with dimensions dim(H A ) = m, dim(H B ) = n, m ≤ n. Let σ be a density operator on H A . A CP map Λ ρ : L(H A ) → L(H B ), corresponding to ρ AB , can be defined by the action on an arbitrary state σ as follows:where σ T -transposed operator σ, I A -identity operator acting on H A . * [email protected] 1 If a CP map is trace-preserving, they are called Kraus operators.
Let H R be a reference Hilbert space isomorphic to H A , then the initial state ρ is recovered by the action of the Choi operator [14]:
(I R ⊗ Λ ρ ) |Γ Γ| RA = ρ RB ,(2)
where |Γ RA -unnormalized maximally entangled vector:
|Γ RA = m−1 i=0 |i R ⊗ |i A .(3)
Suppose that ρ is realized by a specific ensemble of bipartite pure states |Ψ a with probabilities p a :
ρ RB = a p a |Ψ a Ψ a | RB ,(4)
then, from eq. 1 and eq. 4 it is straightforward to see that
Λ ρ [ |ϕ ϕ| A ] = a p a R ϕ * |Ψ a Ψ a |ϕ * R ,(5)
where we extract the action on ϕ A using the dual vector R ϕ * | in accordance with the relative-state method [1]. Now, given a vector |Ψ a RB , an operator M a mapping H A to H B can be defined by
M a |ϕ A = √ mp a R ϕ * |Ψ a RB .(6)
Eq. 5 gives an operator-sum representation of the CP map Λ ρ acting on a pure state projector |ϕ ϕ| A and hence, by linearity, on any density operator σ:
Λ ρ [σ] = a M a σM † a .(7)
Note that the dimensional factor √ m is added in eq. 6 (and, correspondingly, the factor 1/m should be put before |Γ Γ| RA in eq. 2) for consistency with the fact that ρ AB and its reductions ρ A , ρ B all have unit traces. One important property of the operators M a , which we will use in the present paper, is the connection with the reduced density matrices ρ A = Tr B {ρ AB } and ρ B = Tr A {ρ AB }. Suppose that the pure states |Ψ a from the ensemble admit the following decomposition in the given orthonormal bases |i A , |j B of H A and H B :
|Ψ a AB = i,j c (a) ij |i A ⊗ |j B , 0 ≤ i ≤ m − 1, 0 ≤ j ≤ n − 1.(8)
Consider matrix element i|M † a M a |j : with the use of eqs. 6, 8 it can be evaluated as follows:
A i|M † a M a |j A = mp a
where ρ
(a) A = c (a) c (a) † = Tr B { |Ψ a Ψ a | AB } -reduction of |Ψ a Ψ a | AB on subsystem A such that ρ A = a p a ρ (a) A . We see that M † a M a = mp a (ρ (a) A ) T ,(10)
and the following property holds: a CP map is trace-preserving ( a M † a M a = I) iff the reduced density operator ρ A of the corresponding state is maximally mixed: ρ A = 1 m I A . In a similar way, it can be obtained that
M a M † a = mp a ρ (a) B .(11)
In addition, substituting 1 m I A for |ϕ ϕ| A in eq. 5 gives:
Λ ρ [ 1 m I A ] = Tr A {ρ AB } = ρ B ,(12)
hence a CP map is unital iff the reduced density operator ρ B of the corresponding state is maximally mixed: ρ B = 1 n I B . The second important property is the transformations of M a [1]. According to the Hughston-Jozsa-Wootters theorem [15], two different ensemble realizations of the same density operator
a p a |Ψ a Ψ a | AB = µ q µ |Φ µ Φ µ | AB (13) are related by √ q µ |Φ µ = a √ p a V µa |Ψ a ,(14)
where V µa -a matrix with orthonormal columns 1 . Therefore, from eqs. 6, 14 it follows that operators N µ and M a , corresponding to |Φ µ and |Ψ a , are related by
N µ = a V µa M a .(15)
Eq. 15 plays the main role in the development of our paper. Now, to analyze the separability of quantum states, we need one more statement that can be deduced from Theorem 4 of Ref. [16]. Proof. If ρ is separable, then, according to eq. 2, the corresponding map Λ ρ transforms the maximally entangled state to a separable state, and, by the proposition (C) ⇒ (D) of Theorem 4 from Ref. [16], Λ ρ can be written in operator sum using only operators of rank one, i. e., eq. 15 holds with {N µ } of rank one. The converse statement is also true due to the equivalence of the clauses (C) and (D) of the above mentioned Theorem.
Note that if ρ is separable, the CP map Λ ρ is not an entanglement-breaking map unless it is trace-preserving, i.e., ρ A = 1 m I A . As an indirect application of Lemma 1, we consider a separability criterion based on spectral properties.
III. Applications of the main result A. Spectral separability criterion
For now, let eq. 4 express an eigenvector decomposition of the density operator ρ. The matrices c (a) of eq. 8, which correspond to the eigenvectors |Ψ a , are orthonormal with respect to the Hilbert-Schmidt inner product:
Tr{c (a) † c (b) } = δ ab .(16)
From eqs. 6, 8, 16 it follows, then, that operators {M a } are mutually orthogonal:
Tr{M † a M b } = mp a δ ab .(17)
Now, consider eq. 15. We can take advantage of the eigenvector decomposition and express the coefficients V µa using eq. 17:
V µa = Tr{M † a N µ } mp a .(18)
On the one hand, V µa are the entries of a matrix with orthonormal columns, and so we can write:
µ |V µa | 2 = 1.(19)
On the other hand, we can give an upper bound on |Tr{M † a N µ }| using the Cauchy-Schwarz inequality or an even sharper bound using the following property [17,18]:
|Tr{A † B}| q i=1 σ i (A)σ i (B),(20)σ i (M a ) = mp a λ (a) i .(21)
Eqs. 20, 21 give an upper bound on |V µa |:
|V µa | m i=1 q µλ (µ) i λ (a) i p a ,(22)
where q µ -probabilities of the second ensemble realization of the density operator ρ, as in eq. 13; λ
(µ) i -eigenvalues ofρ (µ) A = Tr B { |Φ µ Φ µ | AB }.
If ρ is separable, then, according to Lemma 1, there exist coefficients V µa such that {N µ } are rank one operators. Correspondingly,ρ (µ) A are also of rank one, and so each of them has only one non-vanishing eigenvalue:λ
(µ) 1 = 1,λ (µ) 2 = . . . =λ (µ) m = 0.(23)
Combining this fact with eqs. 19, 22, we obtain:
1 = µ |V µa | 2 µ q µ λ (a) 1 p a = λ (a) 1 p a µ q µ = λ (a) 1 p a .(24)
As a result, we have the following Separability criterion 1. If a density operator ρ AB with an eigenvector decomposition ρ AB = a p a |Ψ a Ψ a | AB is separable, then the largest eigenvalue of each ρ
(a) A = Tr B { |Ψ a Ψ a | AB } is greater than or equal to the corresponding ensemble probability: λ (a) 1 p a .
We note that the criterion was derived independently of Ref. [19], where the same property was obtained by a different method within the framework of the theory of entanglement witnesses.
As an example of application of the criterion, consider isotropic states [20] in arbitrary dimension d (m = n = d):
ρ iso (α) = α |Φ + Φ + | + 1 − α d 2 I d ⊗ I d ,(25)
where 0 α 1, and |Φ + -maximally entangled state:
Φ + = 1 √ d d−1 i=0 |i A ⊗ |i B .(26)
In order to obtain an eigenvector decomposition of ρ iso , we introduce mutually orthogonal auxiliary states
Φ + k = 1 √ d d−1 j=0 e i2πjk/d |j A ⊗ |j B , k = 0, . . . , d − 1, Ψ ± ij = 1 √ 2 ( |i A ⊗ |j B ± |j A ⊗ |i B ), i < j, i, j = 0, . . . , d − 1,(27)
such that |Φ + ≡ Φ + 0 ; d states Φ + k and d(d−1)/2 states Ψ + ij belong to the symmetric subspace of H AB , and d(d − 1)/2 states Ψ − ij belong to the antisymmetric subspace. Therefore, the identity operator I d ⊗ I d can be decomposed in terms of d 2 mutually orthogonal states from eq. 27:
I d ⊗ I d = d−1 k=0 Φ + k Φ + k + d−1 i<j i,j=0 |Ψ + ij Ψ + ij | + |Ψ − ij Ψ − ij | ,(28)
and the isotropic state density operator is expressed as follows:
ρ iso (F ) = F |Φ + 0 Φ + 0 | + 1 − F d 2 − 1 d−1 k=1 |Φ + k Φ + k | + d−1 i<j i,j=0 |Ψ + ij Ψ + ij | + |Ψ − ij Ψ − ij | , F = α(d 2 − 1) + 1 d 2 .
(29)
Applying separability criterion 1 to the first term of the decomposition in eq. 29, we obtain that if ρ iso is separable, then
λ 1 ( |Φ + 0 Φ + 0 |) = 1 d F.(30)
Conversely, if F > 1/d, i.e., α > 1/(d + 1), then ρ iso is entangled. This fact was established by application of other separability criteria [20]. Although separability criterion 1 is able to detect all entangled isotropic states, it is not as much efficient in many other cases where it can be applicable. As an example, consider a 2 ⊗ 2 state
ρ ± = p |ψ ± ψ ± | + (1 − p) |00 00| ,(31)
where
|Ψ ± = 1 √ 2 (|01 ± |10
). In this case the criterion detects entanglement only when p > 1/2, whereas ρ ± is entangled at each p = 0.
As a consequence of separability criterion 1, the following inequality holds for a separable state: a λ (a) 1
(32)
This inequality can also be obtained from the majorization criterion [21], which states that the density matrix of a separable state is majorized by both of its reductions:
ρ AB ≺ ρ A , ρ AB ≺ ρ B ,(33)
where majorization A ≺ B for arbitrary Hermitian operators A and B is defined in terms of their eigenvalues
λ 1 (A) . . . λ d (A) and λ 1 (B) . . . λ d (B): k j=1 λ j (A) k j=1 λ j (B),(34)
for k = 1, . . . , d − 1, and with the inequality holding with equality when k = d. Applying eq. 33 and eq. 34 (with k = 1) to the eigenvector decomposition of ρ AB , in which λ j (ρ AB ) = p j , we obtain:
p 1 λ 1 (ρ A ) = λ 1 a p a ρ (a) A a p a λ (a) 1 .(35)
Here the last is due to the Ky Fan's maximum principle [17,18] in application to a sum of operators. Combination of eq. 35 with the fact that p a p 1 for all a yields inequality 32.
Separability criterion 1 is not strong enough to detect entanglement in many important cases. In the next section we proceed to the direct application of Lemma 1, a more powerful tool.
B. Application to concrete states
Lemma 1 states that a bipartite density operator ρ AB is separable if and only if the corresponding completely positive map can be represented by unit rank operators {N µ }. Therefore, given any ensemble decomposition of ρ AB (not necessarily an eigenvector one) and a corresponding set {M a } of initial operators representing the CP map, by virtue of eq. 15 one can search for linear combinations of M a with coefficients V µa that produce unit rank operators. To do this, one can set the determinants of all second order minors of N µ to null and analyze the resulting system of polynomial equations in variables V µa . ρ AB is separable if and only if there exist solutions V µa which form a matrix with orthonormal columns. Example 1. A mixture of two 2 ⊗ 2 maximally entangled density operators
ρ = p |ψ + ψ + | + (1 − p) |φ − φ − | ,(36)
where |ψ + = 1 √ 2 (|01 + |10 ) and |φ − = 1 √ 2 (|00 − |11 ). According to eq. 6, the CP map corresponding to ρ can be represented by two operators M ψ + and M φ − related to the states |ψ + and |φ − . Their matrix representations are obtained by substituting the elements of the computational basis for |ϕ and the states |ψ + and |φ − for |Ψ a in eq. 6:
M ψ + = √ p 0 1 1 0 , M φ − = 1 − p 1 0 0 −1 .(37)
Eq. 15 gives the expression for N µ in terms of V µa :
N µ = √ 1 − p V µ2 √ p V µ1 √ p V µ1 − √ 1 − p V µ2 .(38)
Operator N µ is of unit rank when the determinant of its matrix vanishes:
(1 − p)V 2 µ2 = −pV 2 µ1 .(39)
Taking absolute value of both parts of eq. 39, one can see that it is consistent with the normalization condition 19 when 1 − p = p. Therefore, the density operator in eq. 36 is entangled when p = 1/2. When p = 1/2, an example of a unitary matrix V satisfying V 2 µ2 = −V 2 µ1 can easily be found:
V = 1 √ 2 1 i 1 −i .(40)
Therefore, when p = 1/2, ρ is separable. The proposed method also gives a separable decomposition in this case. Substituting the entries from the first row (µ = 1) of the matrix of eq. 40 into eq. 38, we obtain N 1 = 1 2 i 1 1 −i , and, by eqs. 10, 11 (with N 1 instead of M a ), the reductions ρ (1)
A = ρ (1) B = 1 2 1 i −i 1(41)
along with the ensemble probability q 1 = 1 m Tr{N † 1 N 1 } = 1/2.
In a similar way, it can be obtained that
ρ (2) A = ρ (2) B = 1 2 1 −i i 1 , q 2 = 1/2,(42)
and so the separable decomposition (one of the many possible) of ρ is:
ρ = 1 2 ρ (1) A ⊗ ρ (1) B + 1 2 ρ (2) A ⊗ ρ (2) B .(43)
Example 2. A mixture of a maximally entangled density operator and a pure one
ρ = p |ψ + ψ + | + (1 − p) |00 00| ,(44)
where |ψ + = 1 √ 2 (|01 + |10 ). In this case the CP map is represented by two operators M ψ + and M 00 :
M ψ + = √ p 0 1 1 0 , M 00 = 2(1 − p) 1 0 0 0 .(45)
Operators N µ , their linear combinations, are given by
N µ = 2(1 − p) V µ2 √ p V µ1 √ p V µ1 0 .(46)
Again, N µ will be of unit rank when the determinant of its matrix vanishes:
det N µ = −pV 2 µ1 = 0.(47)
The coefficients V µ1 cannot be equal to null simultaneously due to eq. 19, so the only possibility for ρ to be separable is p = 0. When p = 0, ρ is entangled.
As Examples 1 and 2 show, when the number of terms in the ensemble decomposition of a given density operator is small, the application of Lemma 1 is quite easy. The analysis of the next example will take much more effort.
Example 3. A qubit-qubit isotropic state ρ iso .
Ensemble decomposition of ρ iso , a 2 ⊗ 2 density operator, is given by eq. 29 with d = 2 and 1/4 F 1. It consists of 4 terms determined by the states Φ + 0 , Φ + 1 , Ψ + 01 , Ψ − 01 . Let M 1 , M 2 , M 3 , M 4 denote the respective operators connected to these states by eq. 6. Their matrix representations are:
M 1 = √ F 1 0 0 1 , M 2 = 1−F 3 1 0 0 −1 , M 3 = 1−F 3 0 1 1 0 , M 4 = 1−F 3 0 −1 1 0 .(48)
Operators {M i } give the representation of a CP map corresponding to ρ iso . Their transformations, {N µ }, are given by
N µ = √ F V µ1 + 1−F 3 V µ2 1−F 3 (V µ3 − V µ4 ) 1−F 3 (V µ3 + V µ4 ) √ F V µ1 − 1−F 3 V µ2 .(49)
The rank of N µ is 1 when the determinant vanishes:
F V 2 µ1 = 1 − F 3 (V 2 µ2 + V 2 µ3 − V 2 µ4 ).(50)
Taking absolute value of both parts of this equation gives inequality
F |V µ1 | 2 1 − F 3 (|V µ2 | 2 + |V µ3 | 2 + |V µ4 | 2 ),(51)
which, after summing by µ and using eq. 19, takes form:
F 1 − F 3 3 = 1 − F.(52)
Consequently, when F > 1/2, a two-qubit isotropic state is entangled. For a complete analysis, it is necessary to show that if 1/4 F 1/2, then ρ iso is separable. Finding a matrix satisfying eq. 50 along with the orthonormal condition for columns is not an easy task if one tries to approach the problem by straightforward solving a system of polynomial equations. We will use a method for construction of unitary matrices described in [22]: Then the following matrix is a nm × nm unitary matrix:
Statement. LetB = a 11 A 1 a 12 A 2 · · · a 1n A n a 21 A 1 a 22 A 2 · · · a 2n A n . . . . . . . . . . . . a n1 A 1 a 2n A 2 · · · a nn A n
With the use of this statement we will show that for any F : 1/4 F 1/2, a unitary matrix with entries V µa satisfying eq. 50 can be constructed.
At first, we consider some particular case of eq. 50: for example, it could be
2 V 2 µ1 = V 2 µ2 + V 2 µ3 − V 2 µ4 .(53)
This choice corresponds to F = 2/5. Let U, W and a b c d -2 × 2 unitary matrices. We construct our solution, the 4 × 4 unitary matrix V , in accordance with the above statement:
V = a U b W c U d W .(54)
The key is to keep all the coefficients as simple as possible. We can choose a, b, c, d to have the same absolute values: a b c d = 1
√ 2 1 i 1 −i .
Substituting the ansatz for V into eq. 53, we obtain the following equations for the entries of U and W :
(2U 2 11 − U 2 12 ) − (W 2 12 − W 2 11 ) = 0, (2U 2 21 − U 2 22 ) − (W 2 22 − W 2 21 ) = 0.(55)
If we choose U in the simplest form: U = 1 √ 2 1 −1 1 1 , then the matrix W satisfying eq. 55 can also be easily guessed:
W = 1 2 i √ 3 −i 1 √ 3 .
The solution for F = 2/5 then will be:
V = 1 2 1 −1 − 3 2 1 √ 2 1 1 i √ 2 3 2 i 1 −1 3 2 − 1 √ 2 1 1 − i √ 2 − 3 2 i .(56)
We can see a specific pattern here in V:
V = 1 2 1 −1 −α β 1 1 iβ iα 1 −1 α −β 1 1 −iβ −iα ,(57)
where α, β -some positive numbers satisfying the orthonormal condition for columns (and rows) of V : α 2 +β 2 = 2. This pattern works for the general solution, for any F, 1/4 F 1/2: substituting V from eq. 57 into eq. 50, we can easily solve the resulting equations in variables α and β. The solution is as follows:
V = 1 2 1 −1 − 2F +1 2−2F 3−6F 2−2F 1 1 i 3−6F 2−2F i 2F +1 2−2F 1 −1 2F +1 2−2F − 3−6F 2−2F 1 1 −i 3−6F 2−2F −i 2F +1 2−2F .(58)
A unitary matrix satisfying eq. 50 is found, and we obtain that if 1/4 F 1/2, then ρ iso is separable.
With the help of eq. 58 a separable decomposition of ρ iso can be obtained. First, substituting V µa from eq. 58 into eq. 49, we obtain the expressions for 4 unit rank operators {N µ }. Next, direct calculations with the use of eqs. 10, 11, and the expression for the ensemble probabilities q µ = 1 m Tr{N † µ N µ } show that when 1/4 F 1/2, ρ iso can be decomposed as:
ρ iso = 1 4 (ρ (1) A ⊗ ρ (1) B + ρ (2) A ⊗ ρ (2) B + ρ (3) A ⊗ ρ (3) B + ρ (4) A ⊗ ρ (4) B ),(59)
where ρ
(i) A = |ψ (i) A ψ (i) A |, ρ (i) B = |ψ (i) B ψ (i) B | -projectors on pure states |ψ (1) A = a |0 A − b |1 A , |ψ (1) B = c |0 B − d |1 B , |ψ (2) A = b |0 A − ia |1 A , |ψ (2) B = d |0 B + ic |1 B , |ψ (3) A = a |0 A + b |1 A , |ψ (3) B = c |0 B + d |1 B , |ψ (4) A = b |0 A + ia |1 A , |ψ (4) B = d |0 B − ic |1 B ,(60)with a = 1 √ 6 3 − √ 3 − 12F 2 − 2 √ 3 (1 − F )F , b = 1 √ 6 √ 3 − 12F 2 + 2 √ 3 (1 − F )F + 3, c = 1 √ 6 √ 3 − 12F 2 − 2 √ 3 (1 − F )F + 3, d = 1 √ 6 3 − √ 3 − 12F 2 + 2 √ 3 (1 − F )F .(61)
Being 2 ⊗ 2 states, the density operators from examples 1, 2, 3 could be analyzed with the use of the Peres-Horodecki criterion (when the separable decomposition is not needed). In the next example we consider a 3 ⊗ 3 state having a positive partial transpose. Let ρ be a 3 ⊗ 3 density operator constructed with the use of the unextendible product bases (UPB) method [23]:
ρ = 1 4 I 3 ⊗ I 3 − 4 i=1 |ψ i ψ i | − |S S| ,(62)
where the states |ψ i and |S are given by
|ψ 1 = 1 √ 2 |0 |0 − 1 , |ψ 2 = 1 √ 2 |2 |1 − 2 , |ψ 3 = 1 √ 2 |0 − 1 |2 , |ψ 4 = 1 √ 2 |1 − 2 |0 , |S = 1 3 |0 + 1 + 2 |0 + 1 + 2 .(63)
The density operator ρ has a positive partial transpose, but in a 3 ⊗ 3 case the Peres-Horodecki criterion is not sufficient for the separability of the state. One possible ensemble decomposition of ρ can be given as
ρ = 1/4 4 i=1 |φ i φ i | [24], where |φ 1 = 1 2 (|ψ 5 + |ψ 6 − |ψ 7 − |ψ 8 ), |φ 2 = 1 2 (|ψ 5 − |ψ 6 + |ψ 7 − |ψ 8 ), |φ 3 = 1 2 (|ψ 5 − |ψ 6 − |ψ 7 + |ψ 8 ), |φ 4 = 1 6 (|ψ 5 + |ψ 6 + |ψ 7 + |ψ 8 ) − 2 √ 2 3 |ψ 9 ,(64)
and five product states
|ψ 5 = 1 √ 2 |0 |0 + 1 , |ψ 6 = 1 √ 2 |2 |1 + 2 , |ψ 7 = 1 √ 2 |0 + 1 |2 , |ψ 8 = 1 √ 2 |1 + 2 |0 , |ψ 9 = |1 |1(65)
together with |ψ 1 , |ψ 2 , |ψ 3 , |ψ 4 from eq. 63 form a complete orthogonal product basis. Given the ensemble decomposition, we obtain the operator representation of a CP map corresponding to the density operator ρ:
M φ 1 = √ 3 4 √ 2 1 −1 −1 1 0 1 −1 −1 1 , M φ 2 = √ 3 4 √ 2 1 −1 −1 1 0 −1 1 1 −1 , M φ 3 = √ 3 4 √ 2 1 1 1 1 0 −1 −1 −1 −1 , M φ 4 = 1 4 √ 6 1 1 1 1 −8 1 1 1 1 .(66)
By eq. 15, operators N µ are expressed as follows:
N µ = 1 4 √ 6 3(V µ1 + V µ2 + V µ3 ) + V µ4 V µ4 + 3(V µ3 − V µ2 − V µ1 ) V µ4 + 3(V µ3 − V µ1 − V µ2 ) V µ4 + 3(V µ1 + V µ2 + V µ3 ) −8V µ4 V µ4 + 3(V µ1 − V µ2 − V µ3 ) V µ4 + 3(V µ2 − V µ1 − V µ3 ) V µ4 + 3(V µ2 − V µ1 − V µ3 ) V µ4 + 3(V µ1 − V µ2 − V µ3 ) . (67)
The matrix in eq. 67 is of rank 1 when the determinants of all its second order minors vanish. We obtain a system of 9 equations:
− 3V µ4 2 − 10V µ3 V µ4 − 8V µ2 V µ4 − 8V µ1 V µ4 − 3V µ3 2 + 3V µ2 2 + 6V µ1 V µ2 + 3V µ1 =0, (68h) V µ1 V µ4 − 3V µ2 V µ3 =0.(68i)
As it turns out, the system can be easily analyzed. Adding eqs. 68g, 68h and also eqs. 68c, 68f, with the use of eq. 68i we have:
V 2 µ2 = V 2 µ1 , V 2 µ3 = V 2 µ1 .(69)
Addition of eqs. 68b, 68d yields:
3V 2 µ4 − 8V µ1 V µ4 − 3V 2 µ3 = 0,(70)
whereas addition of eqs. 68a, 68e gives:
− 3V 2 µ4 − 8V µ1 V µ4 + 3V 2 µ1 = 0.(71)
From the last three equations it follows that V µ1 V µ4 = 0, which in combination with eq. 68i and eq. 69 gives that V 2 µ1 = 0, i. e., the system has only trivial solution. This contradicts with the normalization condition, eq. 19. Consequently, by Lemma 1, ρ is entangled.
IV. Conclusions
As it was shown in section III., the method developed in the present paper can be applied to a great variety of states. It seems to be quite operational when the number of terms in the ensemble decomposition of a density operator is small, but still in this case it can't be better than the Peres-Horodecki criterion which is known to be necessary and sufficient for low rank density operators [7]. In the low rank case it can serve as a complementary method which allows to obtain separable decompositions of density matrices. In the high rank case, as examples with isotropic and PPT states showed, our approach is by no means operational, and it demands a detailed analysis of systems of polynomial equations. In many cases these polynomial systems can be reduced to simple ones; besides, numerical methods can be applied. The presented approach is also interesting from the theoretical point of view and can be used in derivation of separability criteria.
The open question is the connection of the presented approach with the Peres-Horodecki criterion in the qubit-qubit case. If each term in the ensemble decomposition of a given density operator is defined, as in eq. 8, by c (a) ij , a 2 × 2 matrix in this case, then the transformed operators {N µ } have the following matrix representation:
(N µ ) ij = √ m a V µa √ p a c (a) ji ,(72)
and the unit rank condition yields only one equation for each µ in the qubit-qubit case:
a, b √ p a p b V µa V µb (c
For a 2 ⊗ 2 state positivity of a partial transpose is sufficient for separability, but it is unclear how this fact implies existence of the solutions V µa of the above equation, such that V µa form a matrix with orthonormal columns.
AB Ψ a |(|i j|) A |Ψ a AB = = m,n,k,l mp a c (a) * mn m|i A n| B c = mp a (c (a) c (a) † ) ji = = mp a (ρ (a) A ) ji ,
where A, B -complex m × n matrices, q = min{m, n}, σ i (A), σ i (B) -singular values of A and B arranged in non-increasing order: σ 1 (A) σ 2 (A) . . . σ q (A, taken in non-increasing order. From eq. 10 we can see the connection between the singular values σ i (M a ) of the operators M a (thought of as matrices) and the eigenvalues λ
A 1 , A 2 , . . . , A n be m × m unitary matrices and let (a ij ) n i,j=1 , be a unitary matrix.
Example 4 .
4Detecting entanglement of a 3 ⊗ 3 PPT state.
Lemma 1. A bipartite mixed state ρ is separable if and only if the operators {M a } from the operator-sum representation of the corresponding completely positive map can all be transformed by means of eq. 15 to rank one operators {N µ }.
It will be a unitary matrix if the numbers of terms in both ensembles are equal.
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|
[] |
[
"Behaviour Policy Estimation in Off-Policy Policy Evaluation: Calibration Matters",
"Behaviour Policy Estimation in Off-Policy Policy Evaluation: Calibration Matters"
] |
[
"Aniruddh Raghu ",
"Omer Gottesman ",
"Yao Liu ",
"Matthieu Komorowski ",
"Aldo Faisal ",
"Finale Doshi-Velez ",
"Emma Brunskill "
] |
[] |
[] |
In this work, we consider the problem of estimating a behaviour policy for use in Off-Policy Policy Evaluation (OPE) when the true behaviour policy is unknown. Via a series of empirical studies, we demonstrate how accurate OPE is strongly dependent on the calibration of estimated behaviour policy models: how precisely the behaviour policy is estimated from data. We show how powerful parametric models such as neural networks can result in highly uncalibrated behaviour policy models on a real-world medical dataset, and illustrate how a simple, non-parametric, k-nearest neighbours model produces better calibrated behaviour policy estimates and can be used to obtain superior importance sampling-based OPE estimates.
| null |
[
"https://arxiv.org/pdf/1807.01066v2.pdf"
] | 49,562,036 |
1807.01066
|
bf38b796a0f3da7c61b13846144c0961a5a40405
|
Behaviour Policy Estimation in Off-Policy Policy Evaluation: Calibration Matters
Aniruddh Raghu
Omer Gottesman
Yao Liu
Matthieu Komorowski
Aldo Faisal
Finale Doshi-Velez
Emma Brunskill
Behaviour Policy Estimation in Off-Policy Policy Evaluation: Calibration Matters
In this work, we consider the problem of estimating a behaviour policy for use in Off-Policy Policy Evaluation (OPE) when the true behaviour policy is unknown. Via a series of empirical studies, we demonstrate how accurate OPE is strongly dependent on the calibration of estimated behaviour policy models: how precisely the behaviour policy is estimated from data. We show how powerful parametric models such as neural networks can result in highly uncalibrated behaviour policy models on a real-world medical dataset, and illustrate how a simple, non-parametric, k-nearest neighbours model produces better calibrated behaviour policy estimates and can be used to obtain superior importance sampling-based OPE estimates.
Introduction
In many decision-making contexts, one wishes to take advantage of already-collected data (for example, website interaction logs, patient trajectories, or robot trajectories) to estimate the value of a novel decision-making policy. This problem is known as Off-Policy Policy Evaluation (OPE), where we seek to determine the performance of an evaluation policy, given only data generated by a behaviour policy. Most OPE procedures (Precup, 2000;Jiang & Li, 2015;Thomas & Brunskill, 2016;Farajtabar et al., 2018) rely (at least partially) on the technique of Importance Sampling (IS) which, when used in RL, requires the behaviour policy to be known. However, for observational studies in domains such as healthcare, we do not have access to this information. One way to handle this is to estimate the behaviour policy from the data, and then use it to do importance sampling-based OPE. However, the quality of the * Equal contribution 1 Cambridge University 2 Harvard University 3 Stanford University 4 Imperial College London. Correspondence to: Aniruddh Raghu <[email protected]>. resulting OPE estimate is critically dependent on the calibration of the behaviour policy -how precisely it is estimated from the data, and whether the probabilities of actions under the approximate behaviour policy model represent the true probabilities.
In this work, we evaluate the sensitivity of off-policy evaluation to calibration errors in the learned behaviour policy. In particular, we perform a series of careful empirical studies demonstrating that:
1. Uncalibrated behaviour policy models can result in highly inaccurate OPE in a simple, controlled navigation domain. 2. In a real-world sepsis management domain, powerful parametric models such as deep neural networks produce highly uncalibrated probability estimates. 3. A simple, non-parametric, k-nearest neighbours model is better calibrated than all the other parametric models in our medical domain, and using this as a behaviour policy model results in superior OPE.
Background
In the reinforcement learning (RL) problem, an agent's interaction with an environment can be represented by a Markov Decision Process (MDP), defined by a tuple S, A, R, P, P 0 , γ , where S is the state space, A is the action space, R(s, a, s ) is the reward function, P (·|s, a) is the transition probability distribution, P 0 is the initial state distribution, and γ ∈ [0, 1) is the discount factor. A policy is defined as a mapping from states to actions, with π(a|s) representing the probability of taking action a in state s.
Let H := (s 0 , a 0 , r 0 , . . . , s T −1 , a T −1 , r T −1 , s T ) be a trajectory generated when following policy π, and R(H) = T −1 t=0 γ t r t be the return of trajectory H. We can evaluate a policy π by considering the expected return over trajectories when following it: V π = E H∼P π H R(H) . The expectation is taken over the probability distribution of trajectories under policy π. Let the value and action-value functions of a policy π at a state s or state-action pair (s, a) be V π (s) and Q π (s, a) respectively. These are defined as the expected return of a trajectory starting at state s or state-arXiv:1807.01066v2 [cs.
LG] 10 Jul 2018 action pair (s, a), and then following policy π. We can write V π = E s0∼P0 V π (s 0 ) .
In off-policy policy evaluation (OPE), we seek to estimate, with low mean squared error (MSE), the value V πe of an evaluation policy π e given a set of trajectories D = {H (i) } n i=1 generated independently by following a (distinct) behaviour policy π b .
Defining the importance weight (Precup, 2000),
ρ t = t−1 i=0 πe(a H i |s H i ) π b (a H i |s H i )
1 , we can form the stepwise Weighted Importance Sampling (WIS) estimator of V πe :
V πe step-WIS = n i=1 T −1 t=0 γ t ρ (i) t n i=1 ρ (i) t r (i) t .
In this work, we consider using the Per-Horizon WIS (PHWIS) estimator, which can handle differing trajectory lengths (Doroudi et al., 2017), to evaluate medical treatment strategies for sepsis. We also provide results using the Per-Horizon Weighted Doubly Robust (PHWDR) estimator, which incorporates an approximate model of Q πe (s, a) to lower the variance of value estimates (Jiang & Li, 2015;Thomas & Brunskill, 2016). Further information is in the supplementary material.
Impact of Mis-Calibration: Toy Domain
We firstly consider the effect of poorly calibrated behaviour policy models on OPE in a synthetic domain. The domain is a continuous 2D map (s ∈ R 2 ) with a discrete action space, A = {1, 2, 3, 4, 5}, with actions representing a movement of one unit in one of the four coordinate directions or staying in the current position. Gaussian noise of zero mean and specifiable variance is added onto the state of the agent after each action. An agent starts in the top left corner of the domain and receives a positive reward within a given radius of the top right corner, and a negative reward within a given radius of the bottom left corner. We set the horizon to be 15 in all experiments. A k-Nearest Neighbours (kNN) model is used to estimate the behaviour policy distribution, and its accuracy is varied by adjusting the number of neighbours and training data points used.
The quality of OPE is strongly dependent on the quality of behaviour policy estimation. Figure 1 illustrates this via relating the average absolute error in the behaviour policy estimation 1 n n i=1 |π(a (i) |s (i) ) −π(a (i) |s (i) )|, to the fractional error in OPE using the WIS estimator, for two different behaviour policies. The error is calculated with respect to using WIS with the true behaviour policy. Average absolute errors in behaviour policy models of as small as 0.06 can incur errors of up over 50% in the estimated value -having a well-calibrated model of the behaviour policy is therefore critical for good OPE. |π(a (i) |s (i) ) −π(a (i) |s (i) )|, for two different behaviour policies. The error is calculated with respect to using WIS with the true behaviour policy. The quality of OPE is strongly dependent on the quality of behaviour policy estimation.
Model calibration in the sepsis domain
As a case-study, we consider the challenge of obtaining wellcalibrated behaviour models on a real-world dataset, used in Komorowski et al. (2016) and Raghu et al. (2017), dealing with the medical treatment of sepsis patients in intensive care units (ICUs). We use the same framing as Raghu et al. (2017), where the medical treatment process for a sepsis patient is framed as a continuous state-space MDP. A patient's state is represented as a vector of demographic features, vital signs, and lab values. Our state representation concatenates the the previous three timesteps' raw state information to the current time's state vector to capture trends over time. The action space, A, is of size 25 and is discretised over doses of two drugs commonly given to sepsis patients. The reward r t is positive at intermediate timesteps when the patient's wellbeing improves, and negative when it deteriorates. At the terminal timestep of a patient's trajectory, a positive reward is assigned for survival, and a negative reward otherwise.
Obtaining well-calibrated behaviour policy models
We consider modelling the behaviour policy, µ(a|s) via supervised learning. Importantly, IS uses probabilities (rather than class labels) and hence we require a well-calibrated model, not just an accurate one. To evaluate calibration, we draw a series of test states s i from a held out test set, and calculate the total variation distance between the predictive distribution over actions from the estimated model,μ(·|s i ), and a ground-truth distribution obtained by considering the empirical distribution over actions from the k-nearest neighbours of the state s i on the held-out test set, using a custom distance kernel that assesses physiological similarity. Intuitively, states that are physiologically similar should have similar treatment (behaviour policy) distributions. For more information, see the supplementary material.
Approximate kNN produces better calibrated probabilities than parametric models. Table 1 shows the average total variation distance (over 500 test states) between the estimated and target behaviour distributions for different approximate behaviour policy models: logistic regression (LR), random forest (RF), neural network (NN), and an approximate kNN model using random projections (Indyk & Motwani, 1998) (used instead of full kNN for its computational efficiency). The parametric models are poorly calibrated, especially for sampled states with high severity, where there are fewer data points available for estimation. Neural networks can produce overconfident and incorrect probability estimates. Figure 2 shows example predictive distributions over actions for the neural network and approximate kNN as compared to the ground truth, demonstrating over-confident predictions (a result noted by Guo et al. (2017)) and incorrect predictions produced by the neural network. Approx kNN may therefore be more appropriate as a behaviour policy model for OPE.
OPE in the sepsis domain
We now use these behaviour policy models for OPE in the sepsis domain. To obtain ground truth for evaluation, we divide our dataset into two subsets D 1 and D 2 . We can use the behaviour policy from D 1 , π 1 , as the evaluation policy with D 2 . As we have trajectories with π 1 as the behaviour policy in D 1 , we can average returns on these trajectories to get an on-policy estimate of V π1 . Low mean squared error between the OPE estimate and the on-policy estimate provides an indication of correctness.
Two methods of splitting the trajectories are considered: random and intervention splitting. In random splitting, we randomly select half the trajectories to go in one set, and half to go in the other. In intervention splitting, the evaluation set contains half of the patients who were never treated with vasopressors (chosen randomly from all such patients), and the training set contains the remainder of patients. For both methods, results are averaged over different behaviour/evaluation policy pairs -50 for PHWIS and 10 for PHWDR.
In the limit of infinite data, random splitting results in identical behaviour and evaluation policies. In our setting, with limited data, the two policies are close (average total variation distance ≈ 0.09) but this splitting method still permits basic assessment of OPE quality. The average total variation distance with intervention splitting is approximately 0.29.
We estimate MSE(V π1 ,V π1 ) using a bootstrapped method:
1. Sample n = 200 trajectories from D 2 . 2. ObtainV π1 via an OPE method. 3. Repeat this process k = 500 times, representing samples from the distribution ofV π1 . 4. Compute the MSE between these samples and V π1 .
The approximate kNN behaviour policy model often results in the best OPE. Table 2 presents the MSE when using the PHWIS and PHWDR estimators for OPE. The es-timate for Q πe (s, a) in the PHWDR estimator was obtained using Fitted-Q Iteration (FQI) with random forests (Ernst et al., 2005). When using the PHWIS estimator, approximate kNN gives appreciably lower MSE than the neural network (NN), reinforcing the idea that it is better calibrated models can result in better OPE. The results with the PH-WDR estimator do not show as clear a dependence on the behaviour policy. This is because the Approximate Model (AM) terms in one case (random splitting) give low MSE estimates (MSE = 0.177), and in the other case (intervention splitting) give high MSE estimates (MSE = 3.87). There is therefore less of a dependence on the behaviour policy; OPE is dominated by the AM terms.
Conclusion
In this work, we considered the problem of behaviour policy estimation for Off-Policy Policy Evaluation (OPE), focusing an application in healthcare -evaluating medical treatment strategies for patients with sepsis. Via a series of empirical studies, we showed how well-calibrated behaviour policy models are highly important for good-quality OPE, and powerful parametric models such as neural networks can often give uncalibrated probability estimates. We demonstrated that a simple, non-parametric, k-nearest neighbours (kNN) behaviour policy model has better calibration than parametric models and that using this kNN model for OPE led to improved results in this real world domain. The proposed procedure can be used in other situations where the behaviour policy is unknown, and could improve the quality of OPE estimates, which is an important step towards the use of reinforcement learning in real-world domains.
Acknowledgements
This work was supported in part by the Harvard Data Science Initiative, Siemens, and a NSF CAREER grant.
A Off-Policy Policy Evaluation estimators
In off-policy policy evaluation (OPE), we consider the situation where we would like to estimate the value V πe of an evaluation policy π e given a set of trajectories D = {H (i) } n i=1 generated independently by following a (distinct) behaviour policy π b . We would like the estimatorV πe to have low mean squared error (MSE), defined as follows:
MSE(V πe ,V πe ) = E P π b H (V πe −V πe ) 2 .
Note that when we have trajectories from π e , we can form an estimate of V πe
usingV πe = 1 N N i=1 R(H i ), which is the Monte-Carlo estimator. Let us define the quantity ρ t = t i=0 πe(a H i |s H i ) π b (a H i |s H i )
. This is the importance weight (Precup, 2000), and is equal the ratio of the probability of the first t + 1 steps of trajectory H under π e to the probability under π b . 2 . Using this definition, we can form the importance sampling estimator of V πe :V πe
IS = 1 n n i=1 ρ (i) T −1 T −1 t=0 γ t r (i) t
Let us also defineV π M (s) andQ π M (s, a) to be estimates of the state and action value functions for policy π respectively under the approximate model (AM) of the MDP, M . We can use an approximate model M to directly find V πe . For example, we can write:
V πe AM = 1 n n i=1
a∈A π e (a|s
(i) 0 )Q πe M (s (i) 0 , a).
To estimate the quantity V πe , prior work has mainly used one or both of the techniques of Importance Sampling (IS) and Approximate Model (AM) estimation (Thomas & Brunskill, 2016). The IS approach to evaluation relies on using the importance weights ρ t to adjust for the difference between the probability of a trajectory H under the behaviour policy π b and the probability under the evaluation policy π e . Two commonly used estimators in the IS family, which improve on the simple IS estimator are the step-wise IS and step-wise WIS estimators, defined as follows (with i indexing the trajectories in
D):V πe step-IS = 1 n n i=1 T −1 t=0 γ t ρ (i) t r (i) tV πe step-WIS = n i=1 T −1 t=0 γ t ρ (i) t n i=1 ρ (i) t r (i) t
The step-IS estimator is an unbiased estimator of V πe but suffers from high variance (due to the product of importance weights). The step-WIS estimator is biased, but consistent, and has lower variance than step-IS. However, its variance can often still be unacceptably high (Thomas & Brunskill, 2016). These IS estimators can have significant bias when the behaviour policy is unknown.
In AM estimation, we use the approximate model M to directly find V πe , as defined earlier. It may be difficult to trust these estimators, however, given that we cannot always find their bias and variance.
Doubly Robust methods (Jiang & Li, 2015;Thomas & Brunskill, 2016) combine IS and AM techniques together in order to reduce the variance of the resulting estimator. The Weighted Doubly Robust (WDR) estimator, which has demonstrated effective empirical performance (Thomas & Brunskill, 2016), is defined as follows, with w
(i) t = ρ (i) t n i=1 ρ (i) t : V πe WDR = 1 n n i=1 T −1 t=0 γ t w (i) t r (i) t − γ t w (i) tQ πe M (s (i) t , a (i) t ) − w (i) t−1V πe M (s (i) t )
Note that these estimators are valid for trajectories with the same length; extensions to handle trajectories of different length can be found in Doroudi et al. (2017) -this is called the Per-Horizon extension (resulting in the PHIS and PHWIS estimators). Doroudi et al. (2017) defined the Per-Horizon Weighted Importance Sampling (PHWIS) estimator as follows:
V πe PHWIS = l∈L W l 1 {τi|Ti=l} ρ (i) Ti−1 {τi|Ti=l} ρ (i) Ti−1 Ti−1 t=0 γ t r (i) t
where L is the set of all trajectory lengths, and W l is the fraction of the total number of trajectories n with length equal to l:
W l = | {τi|Ti=l} | n
This estimator has high variance; we can define a lower variance equivalent by considering a step-wise version:
V πe step-PHWIS = l∈L W l {τi|Ti=l} Ti−1 t=0 ρ (i) t {τi|Ti=l} ρ (i) t γ t r (i) t
We can also introduce control variates into the estimator and form the Per-Horizon Weighted Doubly Robust (PHWDR) estimator, as follows. First, let us defineV πe WDR, l to be the WDR estimator given all trajectories of length l. We can write this as follows, with w
(i) t,l = ρ (i) t {τ i |T i =l} ρ (i) t : V πe WDR, l = {τi|Ti=l} T −1 t=0 γ t w (i) t,l r (i) t − γ t w (i) t,lQ πe M (s (i) t , a (i) t ) − w (i) t−1,lV πe M (s (i) t )
Then, it is straightforward to write, with W l as defined before:
V πe PHWDR = l∈L W lV πe WDR, l
B Assessing Behaviour Policy Calibration
To evaluate the calibration of models, we can calculate the distance between the estimated behaviour policy and target behaviour policy. In order to calculate this distance, we require the target behaviour policy, which is unknown. However, we can use domain knowledge to inform the choice of the target distribution. In this medical setting, we propose that what governs the clinician's choice of action is the physiological state of the patient, and that patients with similar physiological states will be treated in similar ways. This is a reasonable approximation, given that the state encodes the patient's physiology effectively (Raghu et al., 2017).
We define similarity of patient states using a 'physiological distance kernel', which is based on Euclidean distance and upweights certain informative features of the patient's state. Informative features were the patient's SOFA score, lactate levels, fluid output, mean and diastolic blood pressure, PaO 2 /FiO 2 ratio, chloride levels, weight, and age. These are clinically interpretable: the SOFA score and lactate levels provide indications of sepsis severity; careful monitoring of a patient's fluid levels is essential when managing sepsis (Marik et al., 2017); and blood pressure indicates whether a patient is in septic shock. These features are upweighted by a factor of 2 in our distance kernel (where D = 198, the dimensionality of our state representation):
k(s, s ) = D i=1 w i (s i − s i ) 2 w i = 2, for informative i w i = 1, otherwise
To find the target distribution for a given test state, we use a k-nearest neighbour (kNN) estimate with this distance kernel and form an empirical distribution of the actions taken from the test set neighbours. We consider 150 neighbours to provide reasonable coverage in the estimate. A Ball Tree data structure is used for efficiency. Querying this data structure is computationally expensive (∼1 second per query), so we sample 500 states for patients at different severities (range of SOFA score) and average results for these sets. We use the total variation distance, defined as δ(π b (·|s),π b (·|s)) = 1 2 a∈A |π b (a|s) −π b (a|s)| for the discrete action space, as the distance metric. Our approximate behaviour policies are trained on a separate training dataset and we compare the predictive distribution over actions for the test states to the result from the kNN estimate.
Proceedings of the 35 th International Conference on Machine Learning, Stockholm, Sweden, PMLR 80, 2018. Copyright 2018 by the author(s).
Figure 1 .
1Mean and standard deviation of the fractional error in OPE,V −V V , as a function of the average absolute error in behaviour policy estimation,
Figure 2 .
2Examples of how neural networks can suffer from poor calibration as behaviour policy models, via overconfident (Figure 2a) and incorrect predictions(Figure 2b). The approximate kNN model does not suffer from these issues.
Table 1. Average total variation distance between the estimated and target behaviour policy distributions for different models; logistic regression (LR), random forest (RF), neural network (NN), and approximate kNN; stratified by patient severity score (SOFA). The parametric models are surprisingly uncalibrated. Approximate k-nearest neighbours has the best results.Severity
LR
RF
NN
Approx kNN
0 -4
0.249 0.214 0.213
0.129
5 -9
0.269 0.254 0.246
0.152
10 -13 0.309 0.309 0.399
0.210
14 -23 0.356 0.337 0.426
0.199
Table 2. Average MSE when using PHWIS and PHWDR to evaluate different behaviour policies (from random and intervention splitting) in the medical domain with different behaviour policy models: approximate kNN and neural network (NN). The approximate kNN behaviour policy model results in better OPE with PH-WIS (pure importance sampling estimator). Approximate model (AM) terms in the PHWDR estimator make the dependence on behaviour policy less clear as the AM terms dominate.Approx kNN NN
Random split,V πe
PHWIS
2.48
4.04
Intervention split,V πe
PHWIS
2.04
4.65
Random split,V πe
PHWDR
2.04
2.02
Intervention split,V πe
PHWDR
3.90
3.90
abs/1511.03722, 2015. URL http://arxiv.org/ abs/1511.03722. Komorowski, M., Gordon, A., Celi, L. A., and Faisal, A. A Markov Decision Process to suggest optimal treatment of severe infections in intensive care.Guo, Chuan, Pleiss, Geoff, Sun, Yu, and Weinberger, Kil-
ian Q. On calibration of modern neural networks. In
International Conference on Machine Learning, pp. 1321-
1330, 2017.
Indyk, Piotr and Motwani, Rajeev. Approximate nearest
neighbors: towards removing the curse of dimensionality.
In Proceedings of the thirtieth annual ACM symposium
on Theory of computing, pp. 604-613. ACM, 1998.
Jiang, N. and Li, L.
Doubly Robust Off-policy
Evaluation for Reinforcement Learning.
CoRR,
In Neural Information
Processing Systems Workshop on Machine Learning for
Health, December 2016.
Marik, Paul E, Linde-Zwirble, Walter T, Bittner, Edward A,
Sahatjian, Jennifer, and Hansell, Douglas. Fluid admin-
istration in severe sepsis and septic shock, patterns and
outcomes: an analysis of a large national database. Inten-
sive care medicine, 43(5):625-632, 2017.
Precup, Doina. Eligibility traces for off-policy policy evalu-
ation. Citeseer, 2000.
Raghu, Aniruddh, Komorowski, Matthieu, Celi, Leo An-
thony, Szolovits, Peter, and Ghassemi, Marzyeh. Contin-
uous state-space models for optimal sepsis treatment-a
deep reinforcement learning approach. arXiv preprint
arXiv:1705.08422, 2017.
Thomas, Philip and Brunskill, Emma. Data-efficient off-
policy policy evaluation for reinforcement learning. In
International Conference on Machine Learning, pp. 2139-
2148, 2016.
We assume henceforth that for all state-action pairs (s, a) ∈ S × A, if π b (a|s) = 0 then πe(a|s) = 0.
We assume henceforth that for all state-action pairs (s, a) ∈ S × A, if π b (s|a) = 0 then πe(s|a) = 0.
Importance sampling for fair policy selection. Doroudi, Shayan, Thomas, S Philip, Emma Brunskill, Doroudi, Shayan, Thomas, Philip S, and Brunskill, Emma. Importance sampling for fair policy selection. 2017.
Treebased batch mode reinforcement learning. Damien Ernst, Pierre Geurts, Louis Wehenkel, Journal of Machine Learning Research. 6Ernst, Damien, Geurts, Pierre, and Wehenkel, Louis. Tree- based batch mode reinforcement learning. Journal of Machine Learning Research, 6(Apr):503-556, 2005.
More robust doubly robust off-policy evaluation. Mehrdad Farajtabar, Yinlam Chow, Mohammad Ghavamzadeh, abs/1802.03493CoRRFarajtabar, Mehrdad, Chow, Yinlam, and Ghavamzadeh, Mohammad. More robust doubly robust off-policy evalu- ation. CoRR, abs/1802.03493, 2018.
|
[] |
[
"Convergence Time Evaluation of Algorithms in MANETs",
"Convergence Time Evaluation of Algorithms in MANETs"
] |
[
"Annapurna P Patil [email protected] \nDepartment of Computer Science and Engineering\nM.S. Ramaiah Institute of Technology\nBangalore-54India\n",
"Narmada Sambaturu [email protected] \nDepartment of Computer Science and Engineering\nM.S. Ramaiah Institute of Technology\nBangalore-54India\n",
"Krittaya Chunhaviriyakul [email protected] \nDepartment of Computer Science and Engineering\nM.S. Ramaiah Institute of Technology\nBangalore-54India\n"
] |
[
"Department of Computer Science and Engineering\nM.S. Ramaiah Institute of Technology\nBangalore-54India",
"Department of Computer Science and Engineering\nM.S. Ramaiah Institute of Technology\nBangalore-54India",
"Department of Computer Science and Engineering\nM.S. Ramaiah Institute of Technology\nBangalore-54India"
] |
[
"IJCSIS) International Journal of Computer Science and Information Security"
] |
Since the advent of wireless communication, the need for mobile ad hoc networks has been growing exponentially. This has opened up a Pandora's Box of algorithms for dealing with mobile ad hoc networks, or MANETs, as they are generally referred to.Most attempts made at evaluating these algorithms so far have focused on parameters such as throughput, packet delivery ratio, overhead etc. An analysis of the convergence times of these algorithms is still an open issue. The work carried out fills this gap by evaluating the algorithms on the basis of convergence time.Algorithms for MANETs can be classified into three categories: reactive, proactive, and hybrid protocols. In this project, we compare the convergence times of representative algorithms in each category, namely Ad hoc On-Demand Distance Vector (AODV)-reactive, Destination Sequence Distance Vector protocol (DSDV)-proactive, and Temporally Ordered Routing Algorithm (TORA)-hybrid.The algorithm performances are compared by simulating them in ns2. Tcl is used to conduct the simulations, while perl is used to extract data from the simulation output and calculate convergence time. The design of the experiments carried on is documented using Unified modeling Language. Also, a user interface is created using perl, which enables the user to either run a desired simulation and measure convergence time, or measure the convergence time of a simulation that has been run earlier.After extensive testing, it was found that the two algorithms AODV and DSDV are suited to opposite ends of the spectrum of possible scenarios. AODV was observed to outperform DSDV when the node density was low and pause time was high (sparsely populated very dynamic networks), while DSDV performed better when the node density was high and pause time was low (densely populated, relatively static networks). The implementation of TORA in ns2 was found to contain bugs, and hence analysis of TORA was not possible.Future enhancements include rectifying the bugs in TORA and performing convergence time analysis of TORA as well. Also, a system could be developed which can switch between protocols in real time if it is found that another protocol is better suited to the current network environment.
| null |
[
"https://arxiv.org/pdf/0910.1475v1.pdf"
] | 5,890,213 |
0910.1475
|
cb595dfb761ac78a1d7590c6d3ea4cde238a011e
|
Convergence Time Evaluation of Algorithms in MANETs
2009
Annapurna P Patil [email protected]
Department of Computer Science and Engineering
M.S. Ramaiah Institute of Technology
Bangalore-54India
Narmada Sambaturu [email protected]
Department of Computer Science and Engineering
M.S. Ramaiah Institute of Technology
Bangalore-54India
Krittaya Chunhaviriyakul [email protected]
Department of Computer Science and Engineering
M.S. Ramaiah Institute of Technology
Bangalore-54India
Convergence Time Evaluation of Algorithms in MANETs
IJCSIS) International Journal of Computer Science and Information Security
512009
Since the advent of wireless communication, the need for mobile ad hoc networks has been growing exponentially. This has opened up a Pandora's Box of algorithms for dealing with mobile ad hoc networks, or MANETs, as they are generally referred to.Most attempts made at evaluating these algorithms so far have focused on parameters such as throughput, packet delivery ratio, overhead etc. An analysis of the convergence times of these algorithms is still an open issue. The work carried out fills this gap by evaluating the algorithms on the basis of convergence time.Algorithms for MANETs can be classified into three categories: reactive, proactive, and hybrid protocols. In this project, we compare the convergence times of representative algorithms in each category, namely Ad hoc On-Demand Distance Vector (AODV)-reactive, Destination Sequence Distance Vector protocol (DSDV)-proactive, and Temporally Ordered Routing Algorithm (TORA)-hybrid.The algorithm performances are compared by simulating them in ns2. Tcl is used to conduct the simulations, while perl is used to extract data from the simulation output and calculate convergence time. The design of the experiments carried on is documented using Unified modeling Language. Also, a user interface is created using perl, which enables the user to either run a desired simulation and measure convergence time, or measure the convergence time of a simulation that has been run earlier.After extensive testing, it was found that the two algorithms AODV and DSDV are suited to opposite ends of the spectrum of possible scenarios. AODV was observed to outperform DSDV when the node density was low and pause time was high (sparsely populated very dynamic networks), while DSDV performed better when the node density was high and pause time was low (densely populated, relatively static networks). The implementation of TORA in ns2 was found to contain bugs, and hence analysis of TORA was not possible.Future enhancements include rectifying the bugs in TORA and performing convergence time analysis of TORA as well. Also, a system could be developed which can switch between protocols in real time if it is found that another protocol is better suited to the current network environment.
I. INTRODUCTION
Recently there has been tremendous growth in the number of laptops and mobile phones. With the increase in their number, their participation in people's basic needs to share information has also grown. In areas in which there is little or no communication infrastructure or the existing infrastructure is expensive or inconvenient to use, wireless mobile users will still be able to communicate through the formation of a Mobile Ad Hoc Network (MANET).
In a MANET, each node participates in an ad hoc routing protocol that allows it to discover multi-hop paths through network from itself to destination.
There are three types of routing protocols used in MANETs namely: proactive, reactive and hybrid routing protocols.
Each type of protocols performs differently under different network scenarios. One protocol might perform better than others in specific situation.
To compare performance of each type of protocol, a representative routing protocol is chosen from each type, namely: DSDV (proactive), AODV (reactive), and TORA (hybrid). These protocols are compared in terms of convergence time to uncover in which situations these types of algorithms have their strengths and weaknesses.
II. RELATED WORK
Extensive work has been done on evaluating algorithms for MANETs. In [16], AODV and DSDV have been compared with average throughput, packet loss ratio, and routing overhead as the evaluation metrics, [17] has compared AODV and DSDV in terms of delay and drop rate, [18] compares AODV and DSDV in terms of throughput, packets received, delay and overload. Similarly, [19] compares AODV, DSDV and DSR in terms of throughput, delay, drop rate. This paper has also varied the mobility model used as Random Waypoint and Social Networking models.
III. OVERVIEW OF THE DOCUMENT Section 1 gives a general introduction to the project while section 2 explores related work. An overview of MANETs is given in section 4, along with a brief explanation of the chosen algorithms. Section 5 gives the system design and Implementation. Section 6 describes the testing of the system.
. Conclusion and future enhancements is provided in Section 7 followed by references.
IV. BACKGROUND
A. MANETs
In MANETs, each mobile device operates as a host and also as a router [10], forwarding packets for other mobile device when they are not within direct wireless range of each other.
Each node participates in an ad hoc routing protocol that allows it to discover multi-hop paths through network from itself to a destination. MANET is an infrastructure less network [8] since the mobile nodes in the network dynamically establish routing among themselves to form their own network on the fly. Ad hoc networks are very different from wired networks since they have many constraints which are not present in wired networks e.g. limited bandwidth, limited battery and dynamic topology. Since the nodes are mobile, the network topology may change rapidly and unpredictably over time. The network is decentralized, and all network activity including discovering the topology and delivering messages must be executed by the nodes themselves, i.e., routing functionality is incorporated into the mobile nodes.
Since the topology in a MANET environment is highly dynamic, the path information the routing algorithm may have acquired can become invalid. The algorithm has to detect that the information it possesses is invalid, and find the new correct path to the destination. The time between a fault detection, and restoration of a new, valid path, is referred to as convergence time [6]. This is a very important metric in MANET scenarios, and reflects the ability of the algorithm to adapt to the changing network dynamics. The algorithms must aim to minimize convergence time as much as possible.
There are three types of routing protocols used in MANETs: proactive, reactive and hybrid routing protocol. Proactive protocols are the ones which maintain routing information for all mobile nodes and keep this updating information periodically. An example of proactive protocols is DSDV (Destination Sequenced Distance Vector). Reactive protocols are the protocols which do not maintain routing information. Rather, they discover the path to destination on demand. This saves memory, battery power, and bandwidth. An example is AODV, or Ad hoc On demand Distance Vector. Hybrid protocols combine the advantages of both proactive and reactive protocols. An example is TORA, or Temporally Ordered Routing Algorithm.
B. DSDV
DSDV is a table-driven routing scheme for ad hoc mobile networks based on the Bellman-Ford algorithm. The main contribution of the algorithm was to solve the Routing Loop problem which is present in Bellman-Ford algorithm. To do so, DSDV makes use of sequence numbers. Each entry in the routing table contains a sequence number; the sequence numbers are generally even if a link is present; else, an odd number is used. The number is generated by the destination, and the emitter needs to send out the next update with this number. Routing information is distributed between nodes by sending full dumps infrequently and smaller incremental updates more frequently.
C. AODV
AODV is a reactive routing protocol, meaning that it establishes a route to a destination only on demand. When there is no need of a connection, the network remains silent. It is a distance-vector routing protocol. AODV avoids the countto-infinity problem by making use of sequence numbers.
When a connection is needed, the network node that needs the connection broadcasts a request for finding a route to the destination .Other nodes forward the message, and record the id of the node that they heard it from, creating temporary routes back to the needy node. When a node receives such a message and already has a route to the destination, it sends a message backwards through a temporary route to the requesting node. The needy node then begins using the route that has the least number of hops to the destination. Unused entries in the routing tables are recycled after a time. When a link fails, a routing error is passed back to a transmitting node, and the process repeats.
D. TORA
TORA is a hybrid protocol which combines the advantages of both proactive and reactive protocols. TORA does not use a shortest path solution. It builds and maintains a Directed Acyclic Graph rooted at a destination. No three nodes may have the same height. TORA makes use of height in order to prevent loops in routing. Information may flow from nodes with higher heights to nodes with lower heights, but not viceversa [22].
V. SYSTEM DESIGN AND IMPLEMENTATION
A. Main design criteria
Simulator chosen: The network simulator ns2 [11] (version 2.33) was chosen, as this has become a de facto standard for networking research. It was necessary to use available implementations of algorithms rather than implement them freshly ourselves, as it is important for the acceptance of an evaluation that the implementation used for evaluation has been scrutinized and accepted as correct by the community. Else the evaluation results will not be accepted as . doubt will exist about the correctness of the implementation of the algorithms.
Algorithms chosen: One representative algorithm from each type was chosen for comparison, as this would give a broad picture of which type of algorithm performs well in which environments. Further experiments can be built based on the results of this project, to compare convergence time performance of algorithms within the same category as well.
The specific algorithm chosen within each category was DSDV for proactive, AODV for reactive, and TORA for hybrid. These algorithms embody the principles of the type of algorithm they represent. Also, they have been implemented previously in ns2, and have been under community scrutiny for a long time.
However it was found that ns2's implementation of TORA contains bugs [20. 21, 22], and forms loops. Hence only DSDV and AODV have been actually simulated. The hybrid algorithm has been left as a future enhancement.
Mobility mode:. The Random Waypoint Model was chosen as the mobility model for the simulations. According to this model, a node waits in its current position for a duration of time specified by pause time. At the expiration of this pause time, it chooses a destination randomly, and moves to it with a speed chosen from the uniform distribution [0, max_speed]. This process is repeated until the end of the simulation.
Specific scenarios: The parameters that define the MANET scenario are node density, and node mobility. In this project, the node density can be varied by varying the number of nodes, while the mobility can be varied by varying the pause time. That is, if a node pauses for a longer time at each waypoint, its overall mobility is less, while pausing for a very short time, say 1s, means its mobility is very high.
B. Implementation
• Network scenario
The simulations are conducted using the network simulator ns2 [11]. Random Waypoint mobility model is used. The physical layer simulates the behavior of IEEE 802.11 (as included with ns2). Each node has a radio range of 250 meter, and uses TwoRayGround as the radio propagation model. All the scenarios are based on the following basic parameters: cbr (constant bit rate) traffic topology of size 500 m x 500 m maximum speed of each node 20 m/s simulation time 180s transmission rate (packet rate) 10 m/s The number of nodes is varied in the range [10,100] in steps of 10 (to represent 10 node densities). Pause time is varied in the range [0,180] in steps of 20 (to represent 10 pause times).
• Convergence time measurement
In [5], convergence time has been defined as the time between detection of an interface being down, and the time when the new routing information is available. [6] defines a route convergence period as the period that starts when a previously stable route to some destination becomes invalid and ends when the network has obtained a new stable route for. Similarly, we define convergence time as the time between a fault detection, and restoration of new, valid, path information.
[5] calculates convergence time in the IP backbone. The authors arrive at the value of convergence time by deploying entities called 'listeners', which listen to every link state PDU being sent by the is-is protocol. The time when the first 'adjacency down' packet is observed is taken as the time of detection of an interface being down. This failure event is said to end when the listener receives link state PDUs from both ends of the link.
We arrive at the convergence time by measuring the interval between the detection of route failure and successful arrival of a packet at the destination over the newly computed route. This includes not only the routing convergence time, but also the time taken for the packet to traverse the network from the source to the destination over the newly discovered path. Since this is a comparative analysis, and both the routing protocols use shortest distance with number of hops as the metric for distance calculation, both protocols will arrive at the same new route, and the time taken to reach the destination over this new route will be the same (since all physical characteristics are the same). Hence this extra time measured does not affect the comparative analysis.
In any case, the time taken for a packet to travel from the source to the destination is negligible when compared to the time taken for the algorithm to discover the new route, either through route request -route reply sequences as in reactive protocols, or by waiting for an update that contains new route information as in proactive protocols. Also, this automatically verifies that the new path calculated is correct.
The cycle of invalidation of old path and discovery of a new path might occur many times, and for many sourcedestination pairs over the course of the simulation. Hence the average value of these times is taken as the convergence time of that algorithm for that scenario.
This procedure has been carried out in perl.
VI. TESTING
In order to be able to cover most if not all the types of scenarios the algorithms might face, we varied both the node density (number of nodes) and the node mobility (pause time). The node density (number of nodes) was varied in the range [10,100] in steps of 10 (10 different node densities).
The upper limit of this range was chosen to be 180 because the simulation time is 180s in all the cases. Thus a pause time of 180 implies that the nodes pause in their initial positions for 180 seconds -the entire duration of the simulation. Hence this represents the case where nodes are completely static. Similarly, pause time 0 represents very high mobility where the nodes are in constant motion. Thus we tested each algorithm over 10 node densities x 10 pause times = 100 scenarios.
Also, each scenario was generated 3 times with the same parameters but with different seeds to the random number generators. This gave us a different set of connections and a different mobility signature each time. The convergence time was measured for each of these three scenarios, and the average of these was used in the final analysis. Thus every single point in the graphs shown below is the result of 3 simulations. At the conclusion of the project, a total of 600 simulations ((100 scenarios x 3 runs of each scenario) x 2 algorithms) had been run. All the graphs shown below plot the value of convergence time against pause time. In each graph, the node density is fixed. . DSDV was observed to converge faster than AODV at pause times of 140s, 160s, and 180s (when nodes are almost immobile). When the pause time was low (high mobility), AODV continued to outperform DSDV.
In figures 8, 9 and 10, the number of nodes in the scenario is 80, 90 and 100, respectively. DSDV was observed to converge faster than AODV in all cases except in extremely high mobility (at 0s, 20s, 40s pause times). AODV's convergence time degraded to its worst, standing at 8s for 100 nodes and pause time 60s.
VII. CONCLUSION AND FUTURE ENHACEMENTS
A.Conclusion
The aim of this project was to compare representative algorithms -DSDV (proactive), AODV (reactive) and TORA (hybrid) in terms of convergence time, and uncover in which situations these types of algorithms have their strengths and weaknesses.
Ns2 [11] was chosen to simulate these algorithms, and a perl script was written to measure convergence time by parsing the output of the simulations. The implementation of TORA in ns2 was found to contain bugs which caused it to form long living loops [20,21,22]. Hence an analysis of TORA was not possible. 600 simulations of the algorithms DSDV and AODV were conducted, over a wide range of network scenarios.
When the results of the simulations were analyzed, it was found that AODV was able to converge in less than 1s for scenarios where there was low node density (of the order of 30 nodes or lower) and high to very high mobility (of the order of 60 pause time or lower). In these scenarios, DSDV was found to perform very poorly, taking upto 18s to converge.
However, with increase in node density, the convergence time of AODV was found to increase steadily, while that of DSDV was found to decrease. For the case when node density was high (of the order of 80 nodes or higher) and mobility was low (of the order of 100s pause time or higher), DSDV was found to converge faster than AODV, with AODV taking as much as twice as long to converge.
For intermediate node densities (40 to 70 nodes) and intermediate mobilities (60s to 80s pause time), the convergence time performances of the two algorithms were found to be comparable (of the order of 4s).
Thus AODV and DSDV were together found to cover the two ends of the spectrum of possible network scenarios, with AODV providing fast convergence with low node densities and high mobilities, and DSDV performing well with high node densities and low mobilities.
B. Future Enhancements
The project developed was a primitive attempt and can be further improved to include the following features:
• To correct the errors present in TORA routing protocol and perform convergence time analysis. • Analysis of various other hybrid routing protocols could be experimented. • The switching from one routing protocol to another routing protocol can be made possible when the current network scenario can perform better with another protocol. Hence in real time application, this switching mechanism can provide a better performance routing when the real time information is necessary. • Different routing protocols from each type of MANET routing algorithms can be determined and compared to find the performance.
Figure 1 .
110 nodes, varying pause time Figure 2. 20 nodes, varying pause timeFigure 3. 30 nodes, varying pause time
Figure 4 Figure 10 .
410100 nodes, varying pause time Figures 1, 2 and 3 give the graphs of convergence times when the number of nodes were fixed to 10, 20 and 30, respectively (low node density). AODV was found to converge in less than 1 second most of the time, while DSDVs convergence characteristics varied with changes in the pause time. However, DSDV's performance matched that of AODV when pause time was 160s, and 180s. Figures 4, 5, 6 and 7 describe the cases when there were 40, 50, 60 and 70 nodes in the scenario (intermediate node densities)
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ACKNOWLEDGMENT We wish to acknowledge the efforts of our Professor and Head of the Department at CSE Dr. V. K. Aanthashayana for his guidance which helped us work hard towards producing an innovative work.
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Ad hoc On Demand Distance vector (AODV) Routing protocol. RFC 3561, at www.irtf.org"Ad hoc On Demand Distance vector (AODV) Routing protocol", RFC 3561, at www.irtf.org.
Highly Dynamic Destination-Sequenced Distance-Vector Routing (DSDV) for Mobile Computers. Charles E Perkins, Pravin Bhagwat, Proceedings of SIGCOM '94 Conference on Communications Architecture, protocols and Applications. SIGCOM '94 Conference on Communications Architecture, protocols and ApplicationsCharles E.Perkins and Pravin Bhagwat, "Highly Dynamic Destination- Sequenced Distance-Vector Routing (DSDV) for Mobile Computers" In Proceedings of SIGCOM '94 Conference on Communications Architecture, protocols and Applications August 1994.
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Analysis of link failures over an IP backbone. G Iannaccone, C.-N Chuah, R Mortier, S Bhattacharyya, C Diot, ACM SIGCOMM Internet Measurement Workshop (IMW). G. Iannaccone, C.-N. Chuah, R. Mortier, S. Bhattacharyya, and C. Diot, "Analysis of link failures over an IP backbone," in ACM SIGCOMM Internet Measurement Workshop (IMW), Nov. 2002.
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Goals and challenges of the DARPA GloMo program. M Barry, Robert J Leiner, Ambatipudi R Ruth, Sastry, IEEE Personal Communications. 36Barry M. Leiner, Robert J. Ruth, and Ambatipudi R. Sastry. Goals and challenges of the DARPA GloMo program. IEEE Personal Communications, 3(6):34-43, December 1996.
Research priorities in wireless and mobile communications and networking. Airlie House, VirginiaNational Science FoundationReportNational Science Foundation. Research priorities in wireless and mobile communications and networking: Report of a workshop held March 24- 26, 1997, Airlie House, Virginia. Available at http://www.cise.nsf.gov/anir/ww.html.
The Tactical Internet Graybeard Panel briefings. U.S. Army Digitization Office. Neil Siegel, Dave Hall, Clint Walker, Rene Rubio, Neil Siegel, Dave Hall, Clint Walker, and Rene Rubio. The Tactical Internet Graybeard Panel briefings. U.S. Army Digitization Office. Available at http://www.ado.army.mil/Briefings/Tact%20Internet/index.htm, October 1997.
The National Institute of Standards and Technology. Mobile ad hoc networks (MANETs. The National Institute of Standards and Technology. Mobile ad hoc networks (MANETs) at http://www.antd.nist.gov/wahn_mahn.shtml
Bionics: from biology to technology at. RWTH university of AachenRWTH university of Aachen, Bionics: from biology to technology at http://www-i4.informatik.rwth- aachen.de/content/research/projects/sub/bionics/ [13] Explanation of new trace format at ttp://www.isi.edu/nsnam/ns/doc/node186.html.
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Performance Comparison of the AODV and DSDV Routing Protocols in Mobile Ad Hoc Networks. R Khalaf, A El-Haj-Mahmoud, A Kayssi, From Proceeding (371) Communication Systems and Networks -2002. LebanonR. Khalaf, A. El-Haj-Mahmoud, and A. Kayssi (Lebanon) "Performance Comparison of the AODV and DSDV Routing Protocols in Mobile Ad Hoc Networks", From Proceeding (371) Communication Systems and Networks -2002
CBR Performance Evaluation over AODV and DSDV in RW Mobility Model. Ebrahim Mahdipour, Ehsan Aminian, Mohammad Torabi, Mehdi Zare, International Conference on Computer and Automation Engineering. International Conference onEbrahim Mahdipour, Ehsan Aminian, Mohammad Torabi, Mehdi Zare, "CBR Performance Evaluation over AODV and DSDV in RW Mobility Model," Computer and Automation Engineering, International Conference on, pp. 238-242, International Conference on Computer and Automation Engineering (iccae 2009), 2009.
Challenging Ad-Hoc Networks under Reliable & Unreliable Transport with Variable Node Density. Khalid Muazzam Ali Khan Khattak, Prof Iqbal, Dr, Journal of Theoretical and Applied Information Technology. Sikandar Hayat KhiyalMuazzam Ali Khan Khattak, Khalid Iqbal, Prof Dr. Sikandar Hayat Khiyal, "Challenging Ad-Hoc Networks under Reliable & Unreliable Transport with Variable Node Density", Journal of Theoretical and Applied Information Technology.
TCP performance evaluation over AODV and DSDV in RW and SN mobility models. Yeung D Y Ren Wei, 430074Wuhan; China; Hong Kong, ChinaSchool of Computer Science and Technology, Huazhong University of Science and Technology ; Department of Computer Science, Hong Kong University of Science and TechnologyREN Wei, YEUNG D.Y, JIN Hai1" TCP performance evaluation over AODV and DSDV in RW and SN mobility models" (School of Computer Science and Technology, Huazhong University of Science and Technology, Wuhan 430074, China) (Department of Computer Science, Hong Kong University of Science and Technology, Hong Kong, China) 2005.
Failed simulation with TORA protocol, problem with creating nodes in wireless + TORA at. Failed simulation with TORA protocol, problem with creating nodes in wireless + TORA at http://www.mail-archive.com/ns- [email protected]/msg09534.html[ns].
Problem in TORA simulation at. [ns] Problem in TORA simulation at http://www.mail-archive.com/ns- [email protected]/msg06608.html..
AUTHORS PROFILE The authors are Faculty and Students at M S Ramaiah Institute of Technology, Bangalore working in the area of Bio-inspired computing in the RD labs. 22] [ns] MANET : TORA RoutingDepartment of Computer Science[22] [ns] MANET : TORA Routing Protocol at http://www.mail- archive.com/[email protected]/msg05173.html. AUTHORS PROFILE The authors are Faculty and Students at M S Ramaiah Institute of Technology, Bangalore working in the area of Bio-inspired computing in the RD labs, Department of Computer Science.
|
[] |
[
"Maximal ideals in countable rings, constructively",
"Maximal ideals in countable rings, constructively"
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[
"Ingo Blechschmidt [email protected] \nUniversität Augsburg\nUniversitätsstr. 1486159AugsburgGermany\n",
"Peter Schuster [email protected] \nUniversità di Verona\nStrada le Grazie 1537134VeronaItaly\n"
] |
[
"Universität Augsburg\nUniversitätsstr. 1486159AugsburgGermany",
"Università di Verona\nStrada le Grazie 1537134VeronaItaly"
] |
[] |
The existence of a maximal ideal in a general nontrivial commutative ring is tied together with the axiom of choice. Following Berardi, Valentini and thus Krivine but using the relative interpretation of negation (that is, as "implies 0 = 1") we show, in constructive set theory with minimal logic, how for countable rings one can do without any kind of choice and without the usual decidability assumption that the ring is strongly discrete (membership in finitely generated ideals is decidable). By a functional recursive definition we obtain a maximal ideal in the sense that the quotient ring is a residue field (every noninvertible element is zero), and with strong discreteness even a geometric field (every element is either invertible or else zero). Krull's lemma for the related notion of prime ideal follows by passing to rings of fractions. All this equally applies to rings indexed by any well-founded set, and can be carried over to Heyting arithmetic with minimal logic. We further show how a metatheorem of Joyal and Tierney can be used to expand our treatment to arbitrary rings. Along the way we do a case study for proofs in algebra with minimal logic. An Agda formalization is available at an accompanying repository. 3Let A be a commutative ring with unit. The standard way of constructing a maximal ideal of A is to apply Zorn's lemma to the set of proper ideals of A; but this method is less an actual construction and more an appeal to the transfinite.If A is countable with enumeration x 0 , x 1 , . . ., we can hope to provide a more explicit construction by successively adding generators to the zero ideal, skipping those which would render it improper:. A maximal ideal is then obtained in the limit as the union of the intermediate stages m n . For instance, Krull in his 1929 Annals contribution [33, Hilfssatz] and books on constructive algebra [37, Lemma VI.3.2], [34, comment after Theorem VII.5.2] proceed in this fashion. A similar construction concocts Henkin 3 https:/$\protect\leavevmode@ifvmode\kern-.1667em\relax$/github.com/ iblech/constructive-maximal-ideals/ arXiv:2207.03873v1 [math.AC] 8 Jul 2022
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10.1007/978-3-031-08740-0_3
|
[
"https://arxiv.org/pdf/2207.03873v1.pdf"
] | 250,267,901 |
2207.03873
|
c0ac0919d8c5e881b6d71b1bbcbf0511f81d6ecd
|
Maximal ideals in countable rings, constructively
Ingo Blechschmidt [email protected]
Universität Augsburg
Universitätsstr. 1486159AugsburgGermany
Peter Schuster [email protected]
Università di Verona
Strada le Grazie 1537134VeronaItaly
Maximal ideals in countable rings, constructively
The existence of a maximal ideal in a general nontrivial commutative ring is tied together with the axiom of choice. Following Berardi, Valentini and thus Krivine but using the relative interpretation of negation (that is, as "implies 0 = 1") we show, in constructive set theory with minimal logic, how for countable rings one can do without any kind of choice and without the usual decidability assumption that the ring is strongly discrete (membership in finitely generated ideals is decidable). By a functional recursive definition we obtain a maximal ideal in the sense that the quotient ring is a residue field (every noninvertible element is zero), and with strong discreteness even a geometric field (every element is either invertible or else zero). Krull's lemma for the related notion of prime ideal follows by passing to rings of fractions. All this equally applies to rings indexed by any well-founded set, and can be carried over to Heyting arithmetic with minimal logic. We further show how a metatheorem of Joyal and Tierney can be used to expand our treatment to arbitrary rings. Along the way we do a case study for proofs in algebra with minimal logic. An Agda formalization is available at an accompanying repository. 3Let A be a commutative ring with unit. The standard way of constructing a maximal ideal of A is to apply Zorn's lemma to the set of proper ideals of A; but this method is less an actual construction and more an appeal to the transfinite.If A is countable with enumeration x 0 , x 1 , . . ., we can hope to provide a more explicit construction by successively adding generators to the zero ideal, skipping those which would render it improper:. A maximal ideal is then obtained in the limit as the union of the intermediate stages m n . For instance, Krull in his 1929 Annals contribution [33, Hilfssatz] and books on constructive algebra [37, Lemma VI.3.2], [34, comment after Theorem VII.5.2] proceed in this fashion. A similar construction concocts Henkin 3 https:/$\protect\leavevmode@ifvmode\kern-.1667em\relax$/github.com/ iblech/constructive-maximal-ideals/ arXiv:2207.03873v1 [math.AC] 8 Jul 2022
Abstract. The existence of a maximal ideal in a general nontrivial commutative ring is tied together with the axiom of choice. Following Berardi, Valentini and thus Krivine but using the relative interpretation of negation (that is, as "implies 0 = 1") we show, in constructive set theory with minimal logic, how for countable rings one can do without any kind of choice and without the usual decidability assumption that the ring is strongly discrete (membership in finitely generated ideals is decidable). By a functional recursive definition we obtain a maximal ideal in the sense that the quotient ring is a residue field (every noninvertible element is zero), and with strong discreteness even a geometric field (every element is either invertible or else zero). Krull's lemma for the related notion of prime ideal follows by passing to rings of fractions. All this equally applies to rings indexed by any well-founded set, and can be carried over to Heyting arithmetic with minimal logic. We further show how a metatheorem of Joyal and Tierney can be used to expand our treatment to arbitrary rings. Along the way we do a case study for proofs in algebra with minimal logic. An Agda formalization is available at an accompanying repository. 3 Let A be a commutative ring with unit. The standard way of constructing a maximal ideal of A is to apply Zorn's lemma to the set of proper ideals of A; but this method is less an actual construction and more an appeal to the transfinite.
If A is countable with enumeration x 0 , x 1 , . . ., we can hope to provide a more explicit construction by successively adding generators to the zero ideal, skipping those which would render it improper:
m 0 = {0} m n+1 = m n + (x n ), if 1 ∈ m n + (x n ), m n , else.
A maximal ideal is then obtained in the limit as the union of the intermediate stages m n . For instance, Krull in his 1929 Annals contribution [33,Hilfssatz] and books on constructive algebra [ This procedure avoids any form of choice by virtue of being a functional recursive definition, but still requires some form of omniscience in order to carry out the case distinction. In the present text we study a variant of this construction, due to Berardi and Valentini [9], which avoids any non-constructive principles and decidability assumptions, similar to a construction which has been studied by Krivine [32,p. 410] and later Herbelin and Ilik [24, p. 11] in the context of Gödel's completeness theorem. In this generality, the resulting maximal ideal has an elusive quality to it, but useful properties can still be extracted; and not only do we recover the original construction under certain decidability assumptions, we can also exploit a relativity phenomenon of mathematical logic in order to drop, with some caveats, the assumption that A is countable.
Conventions. Throughout this note, we fix a ring A, and work in a constructive metatheory. In the spirit of Lombardi and Quitté [34], we employ minimal logic [29], where by "not ϕ" we mean "ϕ ⇒ 1 = A 0", and do not assume any form of the axiom of choice. Consequently, by "x ∈ M " we mean x ∈ M ⇒ 1 = A 0, and a subset M ⊆ A is detachable if and only if for all x ∈ A, either x ∈ M or x ∈ M . For general background on constructive mathematics, we refer to [8,7,12].
For an arbitrary subset M ⊆ A, not necessarily detachable, the ideal (M ) generated by M is given by n i=1 a i v i n ≥ 0, a 1 , . . . , a n ∈ A, v 1 , . . . , v n ∈ M . Notice that, for every element v ∈ (M ), either v = 0 or M is inhabited, depending on whether n = 0 or n > 0 in n i=1 a i v i . This can also be seen from the alternative inductive generation of (M ) by the following rules:
v = 0 v ∈ (M ) v ∈ M v ∈ (M ) v ∈ (M ) w ∈ (M ) v + w ∈ (M ) a ∈ A v ∈ (M ) av ∈ (M )
Here we adhere to the paradigm of generalized inductive definitions [1,2,44].
A construction
We assume that the ring A is countable, with x 0 , x 1 , . . . an enumeration of the elements of A. We do not assume that A is discrete (that is, that x = y or x = y for all elements of A) or that it is strongly discrete (that is, that finitely generated ideals of A are detachable). Up to Corollary 1.2(a) below we follow [9]. We study the following recursive construction of ideals m 0 , m 1 , . . . of A:
m 0 := {0} m n+1 := m n + ({x n | 1 ∈ m n + (x n )}).
Finally, we set m := n m n . The construction of m n+1 from m n is uniquely specified, requiring no choices of any form. The set M n := {x n | 1 ∈ m n +(x n )} occurring in this construction contains the element x n if and only if 1 ∈ m n + (x n ); it is obtained from the singleton set {x n } by bounded separation. This set M n is inhabited precisely if 1 ∈ m n + (x n ), in which case m n+1 = m n + (x n ). However, in the generality we work in, we cannot assume that M n is empty or inhabited.
We can avoid the case distinction by the flexibility of nondetachable subsets, rendering it somewhat curious that-despite the conveyed flavor of a conjuring trick-the construction can still be used to obtain concrete positive results.
The ideal (M n ) is given by (M n ) = {ax n | (a = 0) ∨ (1 ∈ m n + (x n ))}.
Lemma 1.1. (a) The subset m is an ideal. (b)
The ideal m is proper in the sense that 1 ∈ m.
(c) For every number n ∈ N, the following are equivalent:
(1) x n ∈ m n+1 . (2) x n ∈ m. (3) 1 ∈ m + (x n ). (4) 1 ∈ m n + (x n ).
Proof. (a) Directed unions of ideals are ideals.
(b) Assume 1 ∈ m. Then 1 ∈ m n for some number n ≥ 0. We verify 1 = 0 by induction over n. If n = 0, then 1 ∈ m 0 = {0}. Hence 1 = 0. If n > 0, then 1 = y + ax n−1 for some elements a, y ∈ A such that y ∈ m n−1 and such that a = 0 or 1 ∈ m n−1 + (x n−1 ). In the first case, we have 1 = y ∈ m n−1 , hence 1 = 0 by the induction hypothesis. In the second case we have 1 = 0 by modus ponens applied to the implication 1 ∈ m n−1 + (x n−1 ) and the fact 1 ∈ m n−1 + (x n−1 ) (which follows directly from the equation
1 = y + ax n−1 ). (c) It is clear that (3) ⇒ (4) ⇒ (1) ⇒ (2). It remains to show that (2) ⇒ (3).
Assume x n ∈ m. In order to verify 1 ∈ m + (x n ), assume 1 ∈ m + (x n ).
Since m + (x n ) ⊆ m, we have 1 ∈ m. Hence 1 = 0 by properness of m.
Corollary 1.2. (a)
The ideal m is maximal in the sense that it is proper and that for all elements x ∈ A, if 1 ∈ m + (x), then x ∈ m. (b) The ideal m is prime in the sense that it is proper and that for all elements x, y ∈ A, if xy ∈ m and x ∈ m, then y ∈ m. (c) The ideal m is radical in the sense that for every k ≥ 0, if x k ∈ m, then x ∈ m.
Proof. (a) Immediate by Lemma 1.1(c). (b) This claim is true even for arbitrary maximal ideals: By maximality, it suffices to verify that 1 ∈ m + (y). If 1 ∈ m + (y), then x = x · 1 ∈ (x) · m + (xy) ⊆ m by xy ∈ m, hence x ∈ m, thus 1 = 0 by x ∈ m. (c) Let x k ∈ m. Then 1 ∈ m + (x), for if 1 ∈ m + (x), then also 1 = 1 k ∈ (m + (x)) k ⊆ m + (x k ) ⊆ m. Hence x ∈ m by maximality. This first-order maximality condition is equivalent [9] to the following higherorder version: For every ideal n such that 1 ∈ n, if m ⊆ n, then m = n.
The quotient ring A/m is a residue field in that 1 = 0 and that every element which is not invertible is zero-as with the real or complex numbers in constructive mathematics. 4 Each of the latter is in fact a Heyting field, a residue field which also is a local ring: if a finite sum is invertible then one of the summands is.
Example 1.4. If we enumerate Z by 0, 1, −1, 2, −2, . . ., the ideal m coincides with (2). If the enumeration starts with a prime p, the ideal m coincides with (p).
Example 1.5. If A is a local ring with group of units A × , then m = A \ A × .
Example 1.6. We can also use an arbitrary ideal a as m 0 instead of the zero ideal. All results in this section remain valid once "not ϕ" is redefined as "ϕ ⇒ 1 ∈ a"; the resulting ideal m is then a maximal ideal above a; it is proper in the sense that 1 ∈ m ⇒ 1 ∈ a. It can also be obtained by applying the original version of the construction in the quotient ring A/a (which is again countable) and taking the inverse image of the resulting ideal along the canonical projection A → A/a.
Example 1.7. Assume that A is a field. Let f ∈ A[X]
be a nonconstant monic polynomial. Since f is monic, it is not invertible; thus Example 1.6 shows that there is a maximal ideal m above (f ). Hence A[X]/m is a field in which f has a zero, namely the equivalence class of X. Iterating this Kronecker construction, we obtain a splitting field of f . No assumption regarding decidability of reducibility has to be made, but in return the resulting fields are only residue fields.
If we can decide whether a finitely generated ideal contains the unit or not, we can improve on Corollary 1.2(a). For instance this is the case for strongly discrete rings such as the ring Z, more generally for the ring of integers of every number field, and for polynomial rings over discrete fields [37, Theorem VIII.1.5].
Proposition 1.8. Assume that for every finitely generated ideal a ⊆ A we have 1 ∈ a or ¬(1 ∈ a). Then:
(a) Each ideal m n is finitely generated. (b) The ideal m is detachable.
If even 1 ∈ a or 1 ∈ a for every finitely generated ideal a ⊆ A, then:
(c) The ideal m is maximal in the strong sense that for every element x ∈ A, x ∈ m or 1 ∈ m + (x), which is to say that the quotient ring A/m is a geometric field (every element is zero or invertible). 5
Proof. We verify claim (a) by induction over n. The case n = 0 is clear. Let n > 0. By the induction hypothesis, the ideal m n−1 is finitely generated, hence so is m n−1 + (x n−1 ). By assumption, 1 ∈ m n−1 + (x n−1 ) or ¬(1 ∈ m n−1 + (x n−1 )).
In the first case m n = m n−1 + (x n−1 ). In the second case m n = m n−1 . In both cases the ideal m n is finitely generated.
To verify claim (b), let an element x n ∈ A be given. By assumption, 1 ∈ m n + (x n ) or ¬(1 ∈ m n + (x n )). Hence x n ∈ m or x n ∈ m by Lemma 1.1(c).
For claim (c), let an element x n ∈ A be given. If 1 ∈ m n + (x n ), then also 1 ∈ m + (x n ). If 1 ∈ m n + (x n ), then x n ∈ m by Lemma 1.1(c).
Remarkably, under the assumption of Proposition 1.8, the ideal m is detachable even though in general it fails to be finitely generated. Usually in constructive mathematics, ideals which are not finitely generated are seldom detachable. For
instance the ideal {x ∈ Z | x = 0 ∨ ϕ} ⊆ Z is detachable if and only if ϕ ∨ ¬ϕ. Remark 1.9.
There is an equivalent description of the maximal ideal m which uses sets G n of generators as proxies for the intermediate ideals m n :
G 0 := ∅ G n+1 := G n ∪ {x n | 1 ∈ (G n ∪ {x n })}
An induction establishes the relation (G n ) = m n ; setting G := n∈N G n , the analogue of Lemma 1.1(c) states that for every number n ∈ N, the following are equivalent:
(1) x n ∈ G n+1 . (2) x n ∈ G. (3) 1 ∈ (G) + (x n ). (4) 1 ∈ (G n ) + (x n ).
In particular, not only do we have that (G) = m, but G itself is already an ideal. This description of m is in a sense more "economical" as the intermediate stages G n are smaller (not yet being ideals), enabling arithmetization in Section 3. Remark 1.10. All results in this section carry over mutatis mutandis if A is only assumed to be subcountable, that is, if we are only given a partially defined surjection N A. In this case, we are given an enumeration x 0 , x 1 , . . . where some x i might not be defined; we then define m n+1 := m n + ({x n | x n is defined ∧ 1 ∈ m n + (x n )}). The generalization to the subcountable case is particularly useful in the Russian tradition of constructive mathematics as exhibited by the effective topos [27,39,41,6], where many rings of interest are subcountable, including uncountable ones such as the real numbers [27, Prop. 7.2].
On the intersection of all prime ideals
Classically, Krull's lemma states that the intersection of all prime ideals is the nilradical, the ideal (0) of all nilpotent elements. In our setup, we have the following substitute concerning complements:
(0) c = p ⊆ A p prime p ¬¬-stable p c = p ⊆ A p prime p radical p c . Lemma 2.1. Let x ∈ A.
Then there is an ideal p ⊆ A which is 1. "x-prime" in the sense that 1 ∈ p ⇒ x ∈ (0) and ab ∈ p ∧ b ∈ p ⇒ x ∈ (0) =⇒ a ∈ p, that is, prime if the negations occurring in the definition of "prime ideal" are understood as "ϕ ⇒ x ∈ (0)", 2. "x-stable" in the sense that (a ∈ p ⇒ x ∈ (0)) ⇒ x ∈ (0) ⇒ a ∈ p, 3. radical, 4. and such that x ∈ p if and only if x is nilpotent.
Proof. The localization A[x −1 ] is again countable, hence the construction of Section 1 can be carried out to obtain a maximal (and hence prime) ideal m ⊆ A[x −1 ]. Every negation occurring in the terms "maximal ideal" and "prime ideal" refers to 1 = 0 in A[x −1 ], which is equivalent to x being nilpotent.
The preimage of m under the localization homomorphism A → A[x −1 ] is the desired x-prime ideal.
Corollary 2.2 (Krull [33]). Let x ∈ A be an element which is not nilpotent. Then there is a (radical and ¬¬-stable) prime ideal p ⊆ A such that x ∈ p.
Proof. Because x is not nilpotent, the notion of an x-prime ideal and an ordinary prime ideal coincide. Hence the claim follows from Lemma 2.1.
An important part of constructive algebra is to devise tools to import proofs from classical commutative algebra into the constructive setting. 6 The following two statements are established test cases exploring the power of such tools [50,51,40,42,56,52,4,45,14,15]. Both statements admit direct computational proofs which do not refer to prime ideals; the challenge is not to find such proofs, but rather to imitate the two classical proofs above constructively, staying as close as possible to the original. It is remarkable that the construction of Section 1 meets this challenge at all, outlined as follows, despite its fundamental reliance on nondetachable subsets.
We continue assuming that A is countable: Section 4 indicates how this assumption can be dropped in quite general situations, while for the purposes of specific challenges such as Proposition 2.3 we could also simply pass to the countable subring generated by the polynomial coefficients or employ the method of indeterminate coefficients.
Proof (of Proposition 2.3). The first claim follows from a simple induction if
A is a reduced ring. In the general case, write f = a n X n + a n−1 X n−1 + · · · + a 0 . Let p be a radical a n -prime ideal as in Lemma 2.1. Since A/p is reduced, the nilpotent coefficient a n vanishes over A/p. Thus a n ∈ p, hence a n is nilpotent.
Since the polynomial f − a n X n is again nilpotent, we can continue by induction.
The second claim follows by a simple inductive argument if A is an integral domain with double negation stable equality. In the general case, write f = a n X n + · · · + a 0 and assume n ≥ 1. To reduce to the integral situation, let p be an a n -prime ideal as in Lemma 2.1. With negation "¬ϕ" understood as "ϕ ⇒ a n ∈ (0)", the quotient ring A/p is an integral domain with double negation stable equality. Hence a n = 0 in A/p, so a n ∈ p whereby a n is nilpotent. The polynomial f − a n X n is again invertible in A[X] (since the group of units is closed under adding nilpotent elements) so that we can continue by induction.
Just as Corollary 2.2 is a constructive substitute for the recognition of the intersection of all prime ideals as the nilradical, the following proposition is a substitute for the classical fact that the intersection of all maximal ideals is the Jacobson radical. As is customary in constructive algebra [34, Section IX.1], by Jacobson radical we mean the ideal {x ∈ A | ∀y ∈ A. 1 − xy ∈ A × }. Proposition 2.4. Let x ∈ A. If x is apart from the Jacobson radical (that is, 1−xy ∈ A × for some element y), then there is a maximal ideal m such that x ∈ m.
Proof. The standard proof as in [34, Lemma IX.1.1] applies: There is an element y such that 1 − xy is not invertible. By Example 1.6, there is an ideal m above a := (1 − xy) which is maximal not only as an ideal of A/a (where "¬ϕ" means "ϕ ⇒ 1 ∈ a") but also as an ideal of A (where "¬ϕ" means "ϕ ⇒ 1 = 0"). If x ∈ m,
then 1 = (1 − xy) + xy ∈ m; hence x ∈ m.
The two test cases presented in Proposition 2.3 only concern prime ideals. In contrast, the following example crucially rests on the maximality of the ideal m. Proposition 2.5. Let M ∈ A n×m be a matrix with more rows than columns. Assume that the induced linear map A m → A n is surjective. Then 1 = 0.
Proof. By passing to the quotient A/m, we may assume that A is a residue field. In this case the claim is standard linear algebra: If any of the matrix entries is invertible, the matrix could be transformed by elementary row and column operations to a matrix of the form 1 0 0 M , where the induced linear map of the submatrix M is again surjective. Thus 1 = 0 by induction.
Hence by the residue field property all matrix entries are zero. But the vector (1, 0, . . . , 0) ∈ A n still belongs to the range of M = 0, hence 1 = 0 by n > 0.
Remark 2.6. A more significant case study is Suslin's lemma, the fundamental and originally non-constructive ingredient in his second solution of Serre's problem [61]. The classical proof, concisely recalled in Yengui's constructive account [68], reduces modulo maximal ideals. The construction of Section 1 offers a constructive substitute. However, since gcd computations are required in the quotient rings, it is not enough that they are residue fields; they need to be geometric fields. Hence our approach has to be combined with the technique variously known as Friedman's trick, nontrivial exit continuation or baby version of Barr's theorem in order to yield a constructive proof [21,38,5,10].
In Heyting arithmetic
The construction presented in Section 1 crucially rests on the flexibility of nondetachable subsets: In absence of additional assumptions as in Proposition 1.8, we cannot give the ideals m n by decidable predicates A → {0, 1}-without additional hypotheses on A, membership of the ideals m n is not decidable. As such, the construction is naturally formalized in intuitionistic set theories such as czf or izf, which natively support such flexible subsets.
In this section, we explain how with some more care, the construction can also be carried out in much weaker foundations such as Heyting arithmetic ha. While formulation in classical Peano arithmetic pa is routine, the development in ha crucially rests on a specific feature of the construction, namely that the condition for membership is a negative condition.
To set the stage, we specify what we mean by a ring in the context of arithmetic. One option would be to decree that an arithmetized ring should be a single natural number coding a finite set of ring elements and the graphs of the corresponding ring operations; however, this perspective is too narrow, as we also want to work with infinite rings.
Instead, an arithmetized ring should be given by a "formulaic setoid with ring structure", that is: by a formula A(n) with free variable n, singling out which natural numbers constitute representatives of the ring elements; by a formula E(n, m) describing which representatives are deemed equivalent; by a formula Z(n) singling out representatives of the zero element; by a formula P (n, m, s) singling out representatives s of sums; and so on with the remaining data constituting a ring; such that axioms such as ∀n. Z(n) ⇒ A(n)
"every zero representative belongs to the ring"
∃n. Z(n) "there is a zero representative" ∀n, m. Z(n) ∧ Z(m) =⇒ E(n, m) "every two zero representatives are equivalent" ∀z, n. Z(z) ∧ A(n) =⇒ P (z, n, n) "zero is neutral with respect to addition"
hold. This conception of arithmetized rings deviates from the usual definition in reverse mathematics [60, Definition III.5.1] to support quotients even when ha cannot verify the existence of canonical representatives of equivalence classes. Although first-order arithmetic cannot quantify over ideals of arithmetized rings, specific ideals can be given by formulas I(n) such that axioms such as
∀n. I(n) ⇒ A(n) "I ⊆ A" ∃n. Z(n) ∧ I(n) "0 ∈ I"
hold. It is in this sense that we are striving to adapt the construction of Section 1 to describe a maximal ideal.
In this context, we can arithmetically imitate any set-theoretic description of a single ideal as a subset cut out by an explicit first-order formula. However, for recursively defined families of ideals, we require a suitable recursion theorem: If we are given (individual formulas M n (x) indexed by numerals representing) ideals m 0 , m 1 , m 2 , . . ., we cannot generally form n∈N m n , as the naive formula " n∈N M n (x)" representing their union would have infinite length. We can take the union only if the family is uniformly represented by a single formula M (n, x) (expressing that x represents an element of m n ).
This restriction is a blocking issue for arithmetizing the construction of the chain m 0 ⊆ m 1 ⊆ · · · of Section 1. Because m n occurs in the definition of m n+1 in negative position, naive arithmetization results in formulas of unbounded logical complexity, suggesting that a uniform definition might not be possible.
This issue has a counterpart in type-theoretic foundations of mathematics, where the family (m n ) n∈N cannot be given as an inductive family (failing the positivity check), and is also noted, though not resolved, in related work [24, p. 11]. The issue does not arise in the context of pa, where the law of excluded middle allows us to bound the logical complexity: We can blithely define the joint indicator function g(n, i) for the sets G n (such that G n = {x i | i ∈ N, g(n, i) = 1}) of Remark 1.9 by the recursion g(0, i) = 0 g(n + 1, i) = 1, if g(n, i) = 1 ∨ (i = n ∧ 1 ∈ (g(n, 0)x 0 , . . . , g(n, n − 1)x n−1 , x n )) 0, else.
This recursion can be carried out within pa since the recursive step only references the finitely many values g(n, 0), . . . , g(n, i). Heyting arithmetic, however, does not support this case distinction. The formalization of the construction in ha is only unlocked by the following direct characterization.
v = [v 0 , . . . , v n−1 ], if n−1 i=0 (v i = 1 ⇔ 1 ∈ a [v0,...,vi−1] )
, then a v = (G n ) + (x n ). In particular, in this case x n ∈ G if and only if 1 ∈ a v . 2. For every natural number n ∈ N,
x n ∈ G ⇐⇒ ¬∃v ∈ {0, 1} n . 1 ∈ a v ∧ n−1 i=0 (v i = 1 ⇔ 1 ∈ a [v0,...,vi−1] ).
Proof. The first part is by induction, employing the equivalences of Remark 1.9. The second rests on the tautology ¬α ⇐⇒ ¬(α ∧ (ϕ ∨ ¬ϕ)):
xn ∈ G ⇐⇒ ¬ 1 ∈ (Gn) + (xn) ⇐⇒ ¬ 1 ∈ (Gn) + (xn) ∧ n−1 i=0 (xi ∈ G ∨ xi ∈ G) ⇐⇒ ¬∃v ∈ {0, 1} n . 1 ∈ (Gn) + (xn) ∧ n−1 i=0 (vi = 1 ⇔ xi ∈ G) ⇐⇒ ¬∃v ∈ {0, 1} n . 1 ∈ av ∧ n−1 i=0 (vi = 1 ⇔ 1 ∈ a [v 0 ,...,v i−1 ] )
Condition (2) is manifestly formalizable in arithmetic, uniformly in n.
For general rings
The construction in Section 1 of a maximal ideal applies to countable rings. In absence of the axiom of choice, some restriction on the rings is required, as it is well-known that the statement that any nontrivial ring has a maximal ideal implies (over Zermelo-Fraenkel set theory zf) the axiom of choice [58,25,3,19,26].
However, this limitation only pertains to the abstract existence of maximal ideals, not to concrete consequences of their existence. Mathematical logic teaches us by way of diverse examples to not conflate these two concerns. For instance, although zf does not prove the axiom of choice, it does prove every theorem of zfc pertaining only to natural numbers (by interpreting a given zfc-proof in the constructible universe L and exploiting that the natural numbers are absolute between V and L [22,49]); similarly, although intuitionistic Zermelo-Fraenkel set theory izf does not prove the law of excluded middle, it does prove every Π 0 2theorem of zf (by the double negation translation combined with Friedman's continuation trick [20]). A similar phenomenon concerns countability, as follows.
A metatheorem by Joyal and Tierney Set theory teaches us that whether a given set is countable depends not only on the set itself, but is more aptly regarded as a property of the ambient universe [23]: Given any set M , there is a (non-Boolean) extension of the universe in which M becomes countable. Remarkably, the passage to such an extension preserves and reflects first-order logic. Hence we have the metatheorem that countability assumptions from intuitionistic proofs of first-order statements can always be mechanically eliminated. 7 Crucially, the first-order restriction is only on the form of the statements, not on the form of the proofs. These may freely employ higher-order constructs.
"First-order" statements are statements which only refer to elements, not to subsets; for instance, the statements of Proposition 2.3 are first-order and hence also hold without the countability assumption. In contrast, the statement "there is a maximal ideal" is a higher-order statement; hence we cannot eliminate countability assumptions from proofs of this statement.
The metatheorem expands the applicability of the construction of Section 1 and underscores the value of its intuitionistic analysis-the metatheorem cannot be applied to eliminate countability assumptions from classical proofs. Taken together, they strengthen the view of maximal ideals as convenient fictions [54,Section 1]. Maximal ideals can carry out their work by any of the following possibilities: (1) For countable (or well-founded) rings, no help is required. Section 1 presents an explicit construction of a maximal ideal. (2) For arbitrary rings, the existence of a maximal ideal follows from the axiom of choice. (3) Intuitionistic first-order consequences of the existence of a maximal ideal are true even if no actual maximal ideal can be constructed.
Comparison with dynamical algebra The dynamical approach [34, Section XV.6], [15], [69], [18] is another technique for constructively reinterpreting, without countability assumptions, classical proofs involving maximal ideals. We sketch here how the dynamical approach is intimately connected with the technique of this section, even though it is cast in entirely different language.
Suppose that a given classical proof appeals to the maximality condition "x ∈ m or 1 ∈ m + (x)" ("x is zero modulo m or invertible modulo m") only for a finite number x 0 , . . . , x n−1 of ring elements fixed beforehand. In this case we can, even if no enumeration of all elements of A exists or is available, apply the construction in Section 1 to this finite enumeration and use the resulting ideal m n as a partial substitute for an intangible maximal ideal.
The tools from pointfree topology driving Joyal and Tierney's metatheorem widen the applicability of this partial substitute to cases where the inspected ring elements are not fixed beforehand, by dynamically growing the partial enumeration as the proof runs its course. If required, a continuation-passing style transform as in Remark 2.6 can upgrade the maximal ideal from only satisfying "1 ∈ m + (x) implies x ∈ m" to satisfying the stronger condition "x ∈ m or 1 ∈ m + (x)". Unfolding the construction of m and the proof of Joyal and Tierney's metatheorem, we arrive at the dynamical method.
When we apply the construction of Section 1 internally in this topos, the result will be a certain sheaf of ideals; it is in that sense that every ring constructively possesses a maximal ideal. This sheaf will not be constant, hence not originate from an actual ideal of the given ring; but first-order consequences of the existence of this sheaf of ideals pass down to the ring. Details are provided by Joyal and Tierney [31, pp. 36f.], and introductions to pointfree topology and topos theory can be found in [10,30,64,63]. A predicative account on the basis of [36,65,16] is also possible. The phenomenon that size is relative also emerges in the Löwenheim-Skolem theorem.
Remark 1. 3 .
3The ideal m is double negation stable: for every ring element x, if ¬¬(x ∈ m), then x ∈ m. This is because by Lemma 1.1(c) membership of m is a negative condition and ¬¬¬ϕ ⇒ ¬ϕ is a tautology of minimal logic.
Proposition 2. 3 .
3Let f ∈ A[X] be a polynomial.1. If f is nilpotent in A[X], then all coefficients of f are nilpotent in A.2. If f is invertible in A[X], then all nonconstant coefficients of f are nilpotent.These facts have abstract classical proofs employing Krull's lemma as follows.Proof of 1. Simple induction if A is reduced; the general case reduces to this one: For every prime ideal p, the coefficients of f vanish over the reduced ring A/p. Hence they are contained in all prime ideals and are thereby nilpotent.Proof of 2. Simple induction ifA is an integral domain; the general case reduces to this one: For every prime ideal p, the nonconstant coefficients of f vanish over the integral domain A/p. Hence they are contained in all prime ideals and are thereby nilpotent.
Lemma 3. 1 .
1(In the situation of Remark 1.9.) For every finite binary sequence v = [v 0 , . . . , v n−1 ], set a v := (v 0 x 0 , . . . , v n−1 x n−1 , x n ). Then: 1. For every such sequence
https:/$\protect\leavevmode@ifvmode\kern-.1667em\relax$/github.com/ iblech/constructive-maximal-ideals/
Residue fields have many of the basic properties of the fields from classical mathematics. For instance, minimal generating families of vector spaces over residue fields are linearly independent, finitely generated vector spaces do (up to ¬¬) have a finite basis, monic polynomials possess splitting fields and Noether normalization is available (the proofs in[37] can be suitably adapted). The constructively rarer geometric fields-those kinds of fields for which every element is either invertible or zero-are required to ensure, for instance, that kernels of matrices are finite dimensional and that bilinear forms are diagonalizable.5 This notion of a maximal ideal, together with the corresponding one of a complete theory in propositional logic, has been generalized to the concept of a complete coalition[53,55] for an abstract inconsistency predicate.
Forms of Zorn's Lemma similar to Krull's Lemma feature prominently in algebra; to wit, in ordered algebra there are the Artin-Schreier theorem for fields, Levi's theorem for Abelian groups and Ribenboim's extension to modules. Dynamical algebra aside, to which we will come back later, these statements have recently gained attention from the angle of proof theory at large; see, for example,[46,47,48,66,57,67,11,43].
For every set M , there is a certain locale X (the classifying locale of enumerations of M ) which is overt, positive and such that its constant sheaf M is countable in the sense of the internal language of the topos of sheaves over X. A given intuitionistic proof can then be interpreted in this topos[13,35,59]; since the constant sheaf functor preserves first-order logic (by overtness), the sheaf M inherits any first-order assumptions about M required by the proof; and since it also reflects first-order logic (by overtness and positivity), the proof's conclusion descends to M .
The opinions expressed in this paper are solely those of the authors.
Acknowledgments The present study was carried out within the project "Reducing complexity in algebra, logic, combinatorics -REDCOM" belonging to the program "Ricerca Scientifica di Eccellenza 2018" of the Fondazione Cariverona and GNSAGA of the INdAM. 8 Important steps towards this paper were made during the Dagstuhl Seminar 21472 "Geometric Logic, Constructivisation, and Automated Theorem Proving" in November 2021. This paper would not have come to existence without the authors' numerous discussions with Daniel Wessel, and greatly benefited from astute comments of Karim Becher, Nicolas Daans, Kathrin Gimmi, Matthias Hutzler, Lukas Stoll and the three anonymous reviewers.
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[
"The Panchromatic Hubble Andromeda Treasury. Progression of Large-Scale Star Formation across Space and Time in M 31",
"The Panchromatic Hubble Andromeda Treasury. Progression of Large-Scale Star Formation across Space and Time in M 31"
] |
[
"Dimitrios A Gouliermis ",
"Lori C Beerman ",
"Luciana Bianchi ",
"Julianne J Dalcanton ",
"Andrew E Dolphin ",
"Morgan Fouesneau ",
"Karl D Gordon ",
"Puragra Guhathakurta ",
"Jason Kalirai ",
"Dustin Lang ",
"Anil Seth ",
"Evan Skillman ",
"Daniel R Weisz ",
"Benjamin F Williams "
] |
[] |
[] |
We investigate the clustering of early-type stars younger than 300 Myr on galactic scales in M 31. Based on the stellar photometric catalogs of the Panchromatic Hubble Andromeda Treasury program that also provides stellar parameters derived from the individual energy distributions, our analysis focused on the young stars in three star-forming regions, located at galactocentric distances of about 5, 10, and 15 kpc, corresponding to the inner spiral arms, the ring structure, and the outer arm, respectively. We apply the two-point correlation function to our selected sample to investigate the clustering behavior of these stars across different timeand length-scales. We find that young stellar structure survives across the whole extent of M 31 longer than 300 Myr. Stellar distribution in all regions appears to be self-similar, with younger stars being systematically more strongly clustered than the older, which are more dispersed. The observed clustering is interpreted as being induced by turbulence, the driving source for which is probably gravitational instabilities driven by the spiral arms, which are stronger closer to the galactic centre.
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10.1007/978-3-319-10614-4_24
|
[
"https://arxiv.org/pdf/1407.0829v1.pdf"
] | 119,294,162 |
1407.0829
|
ea6c9b572135958ea273f1cb8331e0d57c4f8b81
|
The Panchromatic Hubble Andromeda Treasury. Progression of Large-Scale Star Formation across Space and Time in M 31
Dimitrios A Gouliermis
Lori C Beerman
Luciana Bianchi
Julianne J Dalcanton
Andrew E Dolphin
Morgan Fouesneau
Karl D Gordon
Puragra Guhathakurta
Jason Kalirai
Dustin Lang
Anil Seth
Evan Skillman
Daniel R Weisz
Benjamin F Williams
The Panchromatic Hubble Andromeda Treasury. Progression of Large-Scale Star Formation across Space and Time in M 31
We investigate the clustering of early-type stars younger than 300 Myr on galactic scales in M 31. Based on the stellar photometric catalogs of the Panchromatic Hubble Andromeda Treasury program that also provides stellar parameters derived from the individual energy distributions, our analysis focused on the young stars in three star-forming regions, located at galactocentric distances of about 5, 10, and 15 kpc, corresponding to the inner spiral arms, the ring structure, and the outer arm, respectively. We apply the two-point correlation function to our selected sample to investigate the clustering behavior of these stars across different timeand length-scales. We find that young stellar structure survives across the whole extent of M 31 longer than 300 Myr. Stellar distribution in all regions appears to be self-similar, with younger stars being systematically more strongly clustered than the older, which are more dispersed. The observed clustering is interpreted as being induced by turbulence, the driving source for which is probably gravitational instabilities driven by the spiral arms, which are stronger closer to the galactic centre.
Fig. 1
Hierarchy in the ISM and stellar clustering in the general area of W3 complex. (a) On 100 pc-scale, the nebular structures W4 and W5 are shown in the mid-IR image from NASA/WISE, which covers the general region of the OB association Cas OB6. (b) On 10 pc-scale, the starforming region W3, considered as part of W4, is shown from NASA/Spitzer images, with the active star-forming region W3 Main showing bright PAH emission at 8µm. (c) Finally, on the 1 pc-scale high-resolution near-IR images from LUCI camera on the LBT reveal the embedded star-forming cluster of W3 Main (Bik et al., 2012(Bik et al., , 2014. WISE image credit: NASA/JPL-Caltech/WISE Team. are structured into a hierarchical fashion (Elmegreen 2011), similar to that of the interstellar matter (ISM). Giant molecular clouds are indeed hierarchical structures (Elmegreen & Falgarone, 1996;Stutzki et al., 1998), indicating that scale-free processes determine their global morphology, turbulence being considered the dominant (Elmegreen & Scalo, 2004;Mac Low & Klessen, 2004). It stands to reason that mechanisms regulating star formation (McKee & Ostriker, 2007) consequently shape stellar structures, which then build up in a hierarchical fashion, due to the selfsimilar nature of turbulent cascade (e.g., Klessen & Burkert, 2000). This structuring behavior is exemplified in Fig. 1 for the Galactic star-forming complex W3/4/5.
The formation of stars proceeds hierarchically also in time. The duration of star formation tends to increase with the size of the region as the crossing time for turbulent motions (Efremov & Elmegreen, 1998). Small regions form stars quickly and large regions, which contain the small ones, form stars over a longer period. This relation between length-and time-scale underscores the perception that both, cloud and stellar structures, come from interstellar gas turbulence and suggests that star formation in a molecular cloud is finished within only few turbulent crossing times (e.g., Ballesteros-Paredes et al., 1999;Elmegreen, 2000;Hartmann et al., 2001). Arp (1964). Area 9 (red) covers the northeastern turn-over of the inner spiral arm at ∼ 5 kpc from the centre, and area 15 (green) that of the second arm on the 10 kpc star-forming ring of Andromeda. Area 21 (blue) located at distance ∼ 15 kpc from the centre, includes various star-forming regions at the outskirts of M 31.
While this picture of clustered star formation explains the formation of young stellar assembles across different scales, the relation between ISM structure and stellar clustering is not well understood. In addition, apart from star formation, environmental conditions (local feedback, galactic dynamics, etc) influence the morphology of stellar clustering, and the observed variety of stellar systems in size, shape, and compactness. Dissentangling the relative importance of the heredity of star formation to the stellar clustering ("nature") in comparison to the environmental influence on its morphology and survival ("nurture") plays, thus, an important role to our understanding of clustered star formation.
We present our investigation of stellar structures formation over galactic scales from the census of bright blue stars, distributed over the typical length-scale of spiral arms. Our dataset is observed with the most complete stellar survey ever performed of M 31, obtained by the Panchromatic Hubble Space Telescope Andromeda Galaxy Treasury program 1 . We describe our analysis and present first results on the clustering behaviour of young stars up to ages of ∼ 300 Myr in this galaxy.
Observational material
The Panchromatic Hubble Space Telescope Andromeda Treasury (PHAT) program provides deep coverage of 1/3 of M 31 galaxy in six filters with HST. The survey spans the north east quadrant of the galaxy, continuously imaging from the nucleus to the last obvious regions of star formation visible with GALEX (e.g., Thilker et al., 2005). This part of M 31 is selected because of its lowest internal extinction, the highest intensity regions of unobscured star formation, and the least contamination from M 32. Imaging is performed blue-ward of 4000Å, in filters F275W and F336W with the WFC3 camera, in the optical F475W and F814W filters with ACS/WFC, and in the near-IR, in the WFC3 filters F110W and F160W (Williams et al., 2014). This wide panchromatic coverage baseline allows us to confidently estimate stellar effective temperatures, masses, ages and reddenings through a self-consistent Bayesian Spectral Energy Distribution (SED) fitting technique (Gordon et al., in prep), and identify specific features on the color-magnitude diagrams (CMDs) for hot and cool stars for a wide range of extinctions. The homogeneity of the stellar photometric catalogs produced by PHAT over a wide spatial coverage provides the unique opportunity to address the clustering behavior of star formation at length-scales of few kpc, corresponding to spiral arms, down to the few pc scale, where individual star clusters reside (Johnson et al., 2012;Fouesneau et al., 2014;Simones et al., 2014).
Observations of the PHAT survey were performed in 3 × 6 mosaics of 18 parallel ACS and WFC3 pointings. Each mosaic builds a so-called 'brick', roughly corresponding to regions of 1.5 × 3 kpc at the distance of M 31. In total, 23 such bricks tile the complete survey area. The regions of interest in our study are covered by few such bricks, all including portions of the star-forming spiral arms and 10-kpc ring of Andromeda (Fig. 2). We name each area after the identification number of its first brick, e.g., areas 9, 15 and 21. Area 9 covers only one brick, while areas 15 and 21 cover somewhat more than two adjacent bricks. These (apart from a brick that covers part of the bulge) are the only areas for which preliminary stellar parameters are available through our SED fitting technique. We distinguish the bright hot stars with the best-fit values in each area, based on three selection criteria:
1. Stars with T eff ≥ 10, 000 K, i.e., spectral type earlier than ∼ A0V. 2. Stars detected in at least three of the bluest filters, i.e., in F275W (NUV), F336W (U), and F475W (B). 3. Stars with equivalent U-band magnitude m 336 ≤ 25.25.
With these criteria, we ensure to select the brightest young stars with L ∼ > 80 L , and also the most accurate determinations of physical parameters based on the preliminary assumed stellar models.
Evolution of young stellar structure in M 31
The two-point correlation function
The spatial distribution of the early-type stars in the selected areas is characterized in terms of the two-point correlation function (TPCF). Originally introduced for measuring cosmological structure (e.g., Peebles, 1980), the TPCF is a robust Area 9 Area 15 Area 21 ( Fig. 3 The TPCF of the three areas of interest constructed for stars in two different age ranges, i.e., the complete samples with stellar ages up to ∼ 300 Myr and a subsample of stars with ages up to 25 Myr. All TPCFs behave as power-laws up to specific separations, beyond which the TPCF index evidently changes. Vertical lines, coloured accordingly, correspond to the limiting separations beyond which the effects of the limited field-of-view become dominant, and thus TPCF measurements are not trustful (see, e.g., Gouliermis et al., 2014). measurement of the degree of clustering in a sample of sources. Based on the original estimator by Peebles & Hauser (1974) the TPCF is estimated by counting the pairs of sources with different separations r. This function is defined as (see, e.g., Scheepmaker et al., 2009):
r y M 5 2 < e g a ) b ( r y M 0 0 3 < e g a ) a1 + ξ (r) = 1 nN N ∑ i=1 n i (r),(1)
which measures the surface density enhancement n i (r) within radius r from star i with respect to the global average surface densityn of the total sample of N stars. For a fractal distribution the TPCF yields a power-law dependency with radius of the form 1 + ξ (r) ∝ r η . By definition, the total number of stars N r within an aperture of radius r will be then increasing as N r ∝ r η · r 2 = r η+2 . The power-law index η is thus related to the (two-dimensional) fractal dimension, D 2 , of the (projected) distribution as D 2 = η + 2 (Mandelbrot, 1983). This TPCF formalism allows for the direct interpretation of the absolute value of 1 + ξ (r), without any comparison with a reference random distribution (as, e.g., proposed by Gomez et al., 1993), because for a random (Poisson) stellar distribution the value of the TPCF is always 1 + ξ (r) = 1, independent of r. For truly clustered distributions the value of the TPCF is a measurement of the clustering degree of the stars, and is always > 1; higher value at a given separation, stands for higher degree of clustering in the sample (for a complete description see Gouliermis et al., 2014, and references therein). The behaviour of the TPCF is illustrated by the examples given in Fig. 3, where the TPCFs of the three areas for the complete sample of stars with ages up to ∼ 300 Myr and stars with ages up to ∼ 25 Myr, are shown.
The log-log plots of Fig. 3 show that all TPCFs have broken power-law shape. We determine the index η of each part of the TPCF by applying a Levenberg-Marquard nonlinear least square minimization fit (Levenberg, 1944;Marquardt, 1963). The fitting function has the form:
log (1 + ξ (r)) = α + β · log (r) for log (r) < δ α + (β − γ) · δ + γ · log (r) for log (r) > δ(2)
where β and γ are the power-law slopes and δ is the logarithm of the position of the separation break along the abscissa, S break , where the TPCF index changes. Both slopes and the separation break are free parameters in our fit.
From the comparison of the TPCFs of Fig. 3 we derive two interesting results. First, the younger stars (age ∼ < 25 Myr) show a more fractal, i.e., clumpy, clustering than stars in the whole sample (age ∼ < 300 Myr). This is demonstrated by both the higher values of 1 + ξ at small separations and the steeper slopes of their TPCFs (the latter only for areas 9 and 15). Second, while the TPCF index of areas 9 and 15 changes significantly, becoming steeper for the younger sample, that of area 21 remains unchanged between the two samples. This indicates that older populations (up to ∼ 300 Myr) in areas 9 and 15 have higher filling factors (flatter TPCF slopes) than those in area 21. Considering that the latter is the most remote area, away from the centre of M 31, we can only assume that the more dispersed distribution of older stars in areas 9 and 15 is due to the dynamics of the galaxy. We investigate in more detail the evolution of the TPCF with stellar age in the following sections.
Evolution of the TPCF with time
We construct the TPCF of the young stellar samples in the three areas of interest for stars of different ages. We divide each sample to several subsamples of stars, according to the best-fit ages assigned to each star with the SED-fitting. These ages are constrained by the stellar evolution models (Girardi et al., 2010) and atmosphere templates (Castelli & Kurucz, 2004) preliminary used in our technique. Therefore, we select our subsamples of stars according to their effective temperatures, which are somewhat better-constrained by the SED fitting, and are clear indicators of the stellar evolutionary stage. In order for our analysis to be performed in samples of equivalent statistical significance (i.e., not being affected by different number statistics) we divided each stellar catalog into T eff -determined subsamples, containing the same number of stars, corresponding to ∼ 10% of the total. We divided thus each stellar catalog into nine (in the case of area 9 into ten) subsamples that contain almost identical number of stars at different evolutionary stages. This number is ∼ 5,000 in areas 9 and 21 and ∼10,000 in area 15. An example of the subsamples and the derived TPCF parameters for area 15 is shown in Table 1. We determine the TPCF index η for both small and large separations, as well as the separation S break where η changes, for each subsample by applying again a Levenberg-Marquard nonlinear least square minimization fit. The results are given in Fig. 4, where we show the relation of index η (and the derived D 2 ) to the average age of the corresponding stellar subsample for small (r ≤ S break ; top panel) and large (r ≥ S break ; middle panel) separations.
For small separations the plots of η (or D 2 ) versus stellar age show that there is a clear time evolution of the TPCF, with D 2 being smaller at younger ages, and becoming larger for older ages. In all areas D 2 starts with values ∼ < 1 at the smaller age bin of age ∼ 9 Myr, and increases almost monotonically to a value of ∼ 1.6 at age ∼ 75 Myr for area 9 and 35 Myr for area 15 (see also Table 1), and to ∼ 1.4 at age ∼ 45 Myr for area 21. It is worth noting that while for older stars in areas 9 and 15 the TPCF slope 'stabilizes' at almost constant value through the whole remaining age extent, in area 21 it drops again to the value of D 2 ∼ 1. For separations r ≥ S break (Fig. 4, middle panel) the TPCF shows no important evolution with stellar age, having a value of D 2 ∼ 1.85 ± 0.02 (area 9), ∼ 1.85 ± 0.11 (area 15), and ∼ 1.79 ± 0.07 (area 21). While this value for area 21 is somewhat smaller than those in areas 9 and 15, all values are high, close to the geometrical (projected) dimension, and thus consistent with almost uniform (non-clustered) stellar distributions. Interestingly, we observe from the bottom panels of Fig. 4, that the limit of the fractal regime, S break , seems to also evolve with stellar age.
Concluding Remarks and Summary
The value D 2 ∼ < 1 found for the youngest stars at separations r ≤ S break indicates that star formation is clumpy, forming well-clustered (highly fractal) stellar distributions. This clustering behavior becomes somewhat scattered (larger D 2 ) for older stars, sta- Relation of the TPCF index η (or the equivalent fractal dimension D 2 , given on the right ordinate) to average stellar age for small separations (smaller than the length S break where the TPCF slope changes). Middle panel: The same relation for large separations, i.e., r ≥ S break . Bottom panel: Relation of S break to average stellar age for each subsample. The grey dash-dotted lines are linear fits to the data. The TPCF of stars in the youngest age range of area 21 is a single power-law, and therefore there is no measurement for D 2 at large separations and S break for this age.
bilizing at D 2 ∼ 1.4 -1.6, up to ∼ 300 Myr. This fractal dimension, however, is still characteristic of self-similarity, indicating that while stellar clustering does evolve with time, the original structure in M 31 persists over more than ∼ 300 Myr. This time-scale is in agreement with the lower limits placed for structure survival in dwarf galaxies (Bastian et al., 2011). Values of D 2 ∼ 1.5, measured in the distributions of size and luminosity of star-forming regions in the spiral galaxy NGC 628, and across the whole extend of star-forming galaxies of various types, are interpreted as indication of hierarchy induced by turbulence (Elmegreen et al., 2006;Elmegreen et al., 2014). Similarly, we conclude that the observed fractal dimensions in all three areas may be driven by turbulent processes on galactic scales.
The change of the TPCF index at r S break and the stability of D 2 to an almost constant value for separations r > S break suggests that possibly there is a maximum length-scale up to which stars sustain their clustering pattern, i.e., up to which their structures survive. Moreover, the apparent dependence of S break to stellar age implies that this scale differs between stars at different evolutionary stages. Although the physical meaning of this break is not yet understood, its presence might be related to the scale of the galactic disk. For example, a break in the power spectra of ISM emission in the Large Magellanic Cloud is interpreted as due to the line-ofsight thickness of the galactic disk (Block et al., 2010). Indeed, in spiral galaxies stars tend to group up to scales comparable to the disk scale height, and larger structures become spiral-like (flocculent) because of their longer dynamical time-scales in comparison to the shear time (see, e.g., Elmegreen, 2011).
Under these circumstances, with the use of the TPCF, we not only determine the time-scale for structure survival, but we may also specify the upper lengthscale limit across the galactic disk where structure survives, before it 'dissolves' in it.
The observed difference in the behavior of the TPCF from one area to other, suggests a possible dependence of the clustering behavior of stars to the position of the areas across the disk of M 31, and thus the dynamical influence of the galaxy. Area 9 coincides with the first arm at distance ∼ 5 kpc, area 15 is located at the 10 kpc starforming ring, and the more remote area 21 at a distance ∼ 15 kpc away from the galactic centre (see Fig. 2, right panel). Area 9, being closer to the centre, experiences stronger gravitational instabilities and possible disruption by the galactic potential. This is portrayed by the TPCF (at small separations) and S break variabilities, which are seen in Fig. 4 only for area 9, and not for areas 15 and 21.
Considering the rotation of the galaxy (e.g., Sofue & Rubin, 2001), based on their positions, all three areas have rotational velocities quite similar to each other varying between 236 km s −1 at ∼ 5 kpc, 260 km s −1 at ∼ 10 kpc, and 251 km s −1 at ∼ 15 kpc (see, e.g., Table 1 in Carignan et al., 2006). While all areas have similar velocities, being at different distances from the galactic centre, perform a complete orbit at different time-scales, the closest being fastest. As a consequence, the oldest stars in our samples, of ∼ 300 Myr, have performed thus far 10% of an orbit in area 9, 6% in area 15 and 4% in area 21. This suggests that these stars have experienced different degrees of streaming motions driven by the spiral arms, which scatter stars as they shear by (e.g., Elmegreen & Struck, 2013), with those in area 21 being less disrupted.
Based on the discussion above, the findings of this study can be summarized to the following points:
1. Stellar clustering occurs at length-scales that depend on both stellar age and position on the disk. 2. Star formation produces clumpy structures of young stars, on which galactic dynamics influence over time subsequent clustering processes or substructures. 3. Young stellar structures survive across the whole extend of M 31 for at least ∼ 300 Myr. 4. The distribution of young stars evolves as a function of stellar age, but remains fractal with D 2 ∼ 1.4 -1.6 up to this age limit. 5. The observed self-similarity in the stellar distribution is probably induced by large-scale turbulence. 6. Stars in the outer parts away from the galactic centre experience less disruption driven by the gravity of spiral arms.
Fig. 2
2Footprints of the selected regions on the Spitzer 8µm image of M 31 (left panel), and coverage of the selected regions in respect to the Andromeda disk corrected for the inclination of the galaxy, shown face-on (right panel), in respect to the spiral arms in accordance to
Fig. 4
4Evolution of the TPCF as a function of stellar age for the three areas of interest. Top panel:
Table 1
1Selected stellar subsamples and derived TPCF parameters for area 15. Two-dimensional fractal dimension derived from the TPCF index η as D = η + 2. f Separation limit, where the TPCF shows a significant change in its slope η.log T eff
a age b
age limits cnd (10 −4 D e (small
D (large
Separation
(Myr)
(Myr)
pc −2 )
separations)
separations)
break f (pc)
4.38
9 ± 7 0.2 − 29.0 2.7
0.955 ± 0.004 1.583 ± 0.012 83.37 ± 3.83
4.26
21 ± 15 1.0 − 52.7 3.3
1.485 ± 0.069 1.835 ± 0.016 51.40 ± 4.71
4.21
34 ± 24 0.9 − 75.4 3.5
1.626 ± 0.185 1.883 ± 0.020 43.70 ± 7.27
4.17
51 ± 30 1.0 − 84.1 3.3
1.646 ± 0.249 1.904 ± 0.028 46.37 ± 9.14
4.14
70 ± 37 3.8 − 109.0 2.7
1.624 ± 0.218 1.904 ± 0.026 47.61 ± 7.85
4.11
104 ± 40 7.7 − 151.0 2.2
1.729 ± 0.361 1.925 ± 0.024 39.79 ± 16.49
4.08
145 ± 44 10.2 − 188.0 2.6
1.668 ± 0.290 1.904 ± 0.021 36.28 ± 10.70
4.04
193 ± 51 11.0 − 242.0 2.5
1.628 ± 0.209 1.888 ± 0.020 37.58 ± 7.69
4.01
252 ± 63 13.1 − 318.0 2.6
1.601 ± 0.137 1.882 ± 0.022 50.90 ± 6.06
a Average logarithmic effective temperature of stars in the subsample.
b Average age of stars in the subsample.
c Age limits of stars in the subsample.
d Mean stellar surface density.
e
http://www.astro.washington.edu/groups/phat/Home.html
Acknowledgements Based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Science Institute (STScI). STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555. D.A.G. acknowledges support by the German Research Foundation through grant GO 1659/3-1. I also acknowledge very useful discussions with Bruce Elmegreen, Ralf Klessen, Sacha Hony and Lukas Konstandin, which helped improve this manuscript. Finally, I would like to thank the organizers for a very interesting meeting in a wonderful location, and wish to our honourees and their beloved all the best for good health and happiness!
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|
[] |
[
"IDENTITY FAMILIES OF MULTIPLE HARMONIC SUMS AND MULTIPLE ZETA STAR VALUES",
"IDENTITY FAMILIES OF MULTIPLE HARMONIC SUMS AND MULTIPLE ZETA STAR VALUES"
] |
[
"Jianqiang Zhao [email protected] \nDepartment of Mathematics\nEckerd College\n33711St. PetersburgFL\n"
] |
[
"Department of Mathematics\nEckerd College\n33711St. PetersburgFL"
] |
[] |
In this paper we present many new families of identities for multiple harmonic sums using binomial coefficients. Some of these generalize a few recent results of Hessami Pilehrood, Hessami Pilehrood and Tauraso[7]. As applications we prove several conjectures involving multiple zeta star values (MZSV): the Two-one formula conjectured by Ohno and Zudilin, and a few conjectures of Imatomi et al. involving 2-3-2-1 type MZSV, where the boldfaced 2 means some finite string of 2's.
|
10.2969/jmsj/06841669
|
[
"https://arxiv.org/pdf/1303.2227v4.pdf"
] | 118,433,800 |
1303.2227
|
6feb7d1c903152a0accf2d7a451f35f47175cb14
|
IDENTITY FAMILIES OF MULTIPLE HARMONIC SUMS AND MULTIPLE ZETA STAR VALUES
12 Dec 2013
Jianqiang Zhao [email protected]
Department of Mathematics
Eckerd College
33711St. PetersburgFL
IDENTITY FAMILIES OF MULTIPLE HARMONIC SUMS AND MULTIPLE ZETA STAR VALUES
12 Dec 2013
In this paper we present many new families of identities for multiple harmonic sums using binomial coefficients. Some of these generalize a few recent results of Hessami Pilehrood, Hessami Pilehrood and Tauraso[7]. As applications we prove several conjectures involving multiple zeta star values (MZSV): the Two-one formula conjectured by Ohno and Zudilin, and a few conjectures of Imatomi et al. involving 2-3-2-1 type MZSV, where the boldfaced 2 means some finite string of 2's.
Introduction
For over two hundred years, Euler's pioneering work on double zeta values [5] was largely neglected, until in the early 1990s when Zagier showed the importance of the more general multiple zeta values in his famous paper [25]. Since then these values have come up in many areas of current research in mathematics and physics, such as knot theory, motivic theory, mirror symmetry and Feynman integrals, to name just a few. One of the central problems is to determine various Q-linear relations among these values, many of which have been discovered numerically first and then proved rigorously later. One such family that still defies a proof until now is the celebrated Two-one formula discovered by Ohno and Zudilin [19].
The main goal of this paper is to give a comprehensive study of multiple zeta star values of a few special types using the corresponding identities established first for multiple harmonic sums. As one of the applications, we give a concise proof of the Two-one formula.
We now recall some definitions. Let N be the set of natural numbers, Z the set of integers, Z * the set of nonzero integers, and N 0 = N ∪ {0}. For any ℓ ∈ N and s = (s 1 , s 2 , . . . , s ℓ ) ∈ (Z * ) ℓ we define the (alternating) multiple harmonic sum (MHS for short) H n (s 1 , s 2 , . . . , s ℓ ) =
n≥k 1 >k 2 >...>k ℓ ≥1 ℓ i=1 sgn(s i ) k i k |s i | i ,(1)
H ⋆ n (s 1 , s 2 , . . . , s ℓ ) =
n≥k 1 ≥k 2 ≥...≥k ℓ ≥1 ℓ i=1 sgn(s i ) k i k |s i | i .(2)
This star-version has been denoted by S n in the literature but it seems to be more appropriate to use H ⋆ in this paper due to its close connection with multiple zeta star values to be defined momentarily. Conventionally, we call l(s) := ℓ the depth and |s| := ℓ i=1 |s i | the weight. For convenience we set H n (s) = 0 if n < l(s), H n (∅) = H ⋆ n (∅) = 1 for all n ≥ 0, and {s 1 , s 2 , . . . , s ℓ } r the set formed by repeating the composition (s 1 , s 2 , . . . , s ℓ ) exactly r times.
When s = (s 1 , s 2 , . . . , s ℓ ) ∈ (Z * ) ℓ with (s 1 , sgn(s 1 )) = (1, 1) we set, respectively, the (alternating) Euler
When s ∈ N ℓ they are called the multiple zeta value (MZV) and the multiple zeta star value (MZSV), respectively. Notice that we have abused the notation in the definitions (1), (2) and (3) since all these sums can be evaluated at negative integers, because, for example, a multiple zeta function can be analytically continued to the whole complex space and it is possible to define its values at negative integers [1,27]. Hence throughout the paper we will writen whenever −n appears as an argument. For instance, we really should write ζ(2) instead of ζ(−2) which usually means the Riemann zeta function at −2.
We now state the Two-one formula conjectured by Ohno and Zudilin [19]. where p runs through all indices of the form (2a 1 + 1) • · · · • (2a r + 1) with "•" being either the symbol "," or the symbol "+".
Until recently, not many nontrivial families of identities relating MZVs or MZSVs with truly alternating Euler sums have been proved. One such result is proved by Zlobin [32] ζ ⋆ ({2} n ) = −2ζ(2n) for all n ≥ 1.
Another is proved in [30]: ζ({3} n ) = 8 n ζ({2, 1} n ). Recently, two more appear as (27) and (28) of [7] one of which yields a new proof of an identity of [26]. Notice that [7, (22)] implies (4) easily (see Lemma 4.1). To provide more such families in this paper we need some book-keeping first. A boldface of a single digit number means the number is repeated a few times. We underline a string pattern to mean the whole pattern is repeated. Thus the Two-one formula should be written as 2-1 formula and in each repetition the 2's may have different lengths.
Besides the 2-1 formula in Theorem 1.1 we show many analogous formulas in this paper. For ease of reference we list them as follows:
(1) 2-1: §2 for MHS, §4 for MZSV;
(2) 2-1-2 (nontrivial substring 2 at the end): §2 for MHS, §4 for MZSV;
(3) 2-c-2 (c ≥ 3 and 2 at the end may be trivial): to appear in a joint work with my student Erin Linebarger [16]; For example, the following is the 2-1-2 formula.
Theorem 1.2. Let r ∈ N and s = ({2} a 1 , 1, . . . , {2} ar , 1, {2} a r+1 ) where a 1 , a r+1 ∈ N and a j ∈ N 0 for all 2 ≤ j ≤ r. Then we have ζ ⋆ (s) = − p 2 ℓ(p) ζ(p),
where p runs through all indices of the form (2a 1 + 1) • · · · • (2a r + 1) • 2a r+1 with "•" being either the symbol "," or the symbol O-plus "⊕" defined by
a ⊕ b = sgn(a)b + sgn(b)a for all a, b ∈ Z * .(5)
When r = 2, we have checked numerically the following identities for all 0 ≤ a, b, c ≤ 2 and ac = 0 with the help of EZ-face [3]:
ζ ⋆ ({2} a , 1, {2} b , 1, {2} c ) = − 2ζ(2(a + b + c) + 2) − 4ζ(2a + 2 + 2b, 2c) − 4ζ(2a + 1, 2b + 1 + 2c) − 8ζ(2a + 1, 2b + 1, 2c).
One of the main results contained in [7, Theorem 2.3] is the following theorem.
H ⋆ n ({2} a , 1, {2} b ) = −2 n k=1 (−1) k n k k 2a+1+2b n+k k − 4 n k=1 H k−1 (2b) n k k 2a+1 n+k k . (B)
We want to caution the reader that the convention of index ordering in [7] is opposite to ours in the definitions (1) and (2) ) and (B) respectively by allowing the arguments to contain an arbitrary number of 2-strings, which lead to the 2-1 and 2-1-2 formulas for MHS. The proofs are straight-forward, however, the difficult part is the discovery of the theorems (using a lot of Maple experiments). By taking limits in these two theorems so that MHS become MZSV we can prove the 2-1 formula and the 2-1-2 formula for MZSV in Theorems 1.1 and 1.2, respectively.
In §8 and §9 we provide new and concise proofs of a few conjectures first formulated by Imatomi et al. in [13] concerning MZSV of types 2-3-2-1 and 2-3-2-1-2.
In the last section we propose a few possible future research directions, one of which will be carried out in a sequel to this work in which we will study congruence properties of MHS as further applications of the results we have obtained in this paper.
Acknowledgement. We would like to thank Roberto Tauraso for sending us their preprint [7]. This work is partially supported by NSF grant DMS1162116 which enables me to work with my students more productively. In particular, this paper is inspired by a recent collaboration with one of my students, Erin Linebarger [16]. We are able to generalize Theorems 2.1 of [7] to arbitrary number of strings of 2's using similar ideas contained in this paper. We further define Π(s) to be the set of all indices of the form (s 1 • · · · • s m ) where "•" being either the symbol "," or the symbol O-plus"⊕" defined by (5
H ⋆ n ({2} a , 1, {2} b , 1) = 2 n k=1 n k k 2(a+b)+2 n+k k + 4 n k=1 H k−1 (2b + 1) n k k 2a+1 n+k k .
When r = 3 we have: for all n ∈ N and a, b, c ∈ N 0
H ⋆ n ({2} a , 1, {2} b , 1, {2} c , 1) = 2 n k=1 n k k 2(a+b+c)+3 n+k k + 4 n k=1 H k−1 (2c + 1) n k k 2a+2b+2 n+k k + 4 n k=1 H k−1 (2b + 2c + 2) n k k 2a+1 n+k k + 8 n k=1 H k−1 (2b + 1, 2c + 1) n k k 2a+1 n+k k .
Using Maple we have verified both formulas numerically for a, b, c ≤ 5 and n ≤ 100.
We now generalize Theorem 1.3(B).
A combinatorial lemma
In this short section we prove the following combinatorial identities which are similar in spirit to [7, Lemma 2.2]. We will need these results several times throughout this paper.
H k−1 (v)A (m) n,k k a = n k=1 H k−1 (v)A (m) n,k k a+c + j+|x|=a+c j≥0,xr>a m l(x) n k=1 H k−1 (x, v)A (m) n,k k j ,(8)where x r denotes the last component of x ∈ N r . (ii) We have n n k=1 H k−1 (v)B (2) n,k k a = n k=1 H k−1 (v)B (2) n,k k a−1 + 2 n k=1 H k−1 (a, v)kB (2) n,k . (9) (iii) We have 1 n c n k=1 H k−1 (v)B (m) n,k k a = n k=1 H k−1 (v)B (m) n,k k a+c + j+|x|=a+c j≥0,xr<−a m l(x) n k=1 H k−1 (x, v)A (m) n,k k j ,(10)where x r denotes the last component of x ∈ N r−1 × Z * . (iv) We have n n k=1 H k−1 (v)A (2) n,k k a = n k=1 H k−1 (v)A (2) n,k k a−1 + 2 n k=1 H k−1 (a, v)kB (2) n,k .(11)
Proof. We need to mention again that the ordering is reversed in this paper so s 1 in [7, Lemma 2.2] should be the last component of x in our setup. Now, equation (8) follows from [7, Lemma 2.2] directly. We may also use this proof for (10)
we see that
2 n k=1 H k−1 (a, v)kB (2) n,k = n l=1 H l−1 (v) l a n k=l+1 2kB (2) n,k = n l=1 H l−1 (v) l a (n − l)B (2) n,l = n n l=1 H l−1 (v)B (2) n,l l a − n l=1 H l−1 (v)B (2) n,l l a−1
which is (9). Similar argument yields (11). We leave the details to the interested reader.
Remark 3.2. In this paper we will always choose c
(1) n = 1 so that A (1) n,k = n k and c (2) n = (n!) 2 /(2n)! so that A (2) n,k = n k / n+k k .
Proof of Theorems 2.1 and 2.3, 2-1 and 2-1-2 formulas
In the case n = 1 both statements in Theorems 2.1 and 2.3 become 1 = 1 trivially. We now assume n ≥ 2. By definition, for s = ({2} a 1 , 1, . . . , {2} ar , 1) we have
H ⋆ n (s) = a 1 l=0 1 n 2a 1 −2l H ⋆ n−1 ({2} l , 1, {2} a 2 , 1, . . . , {2} ar , 1) + 1 n 2a 1 +1 H ⋆ n ({2} a 2 , 1, . . . , {2} ar , 1).
By induction on r + n we see that
H ⋆ n (s) = a 1 l=0 1 n 2a 1 −2l q∈Π(2l+1,2a 2 +1,...,2ar+1) 2 ℓ(q) H n−1 (q) + 1 n 2a 1 +1 p∈Π(2a 2 +1,...,2ar+1) 2 ℓ(p) H n (p) = a 1 l=0 1 n 2a 1 −2l (q 1 ,...,qm)∈Π(1,2a 2 +1,...,2ar+1) 2 m n−1 k=1 1 k 2l+q 1 n−1 k n−1+k k H k−1 (q 2 , . . . , q m ) + 1 n 2a 1 +1 p∈Π(2a 2 +1,...,2ar+1) 2 ℓ(p) H n (p)
By changing the order of summations and using the identity
a 1 l=0 n k 2l = 1 k 2a 1 · n 2a 1 +2 − k 2a 1 +2 (n − k)(n + k)(13)
we see easily that
H ⋆ n (s) = (q 1 ,...,qm)∈Π(1,2a 2 +1,...,2ar+1) n k=1 1 − k 2a 1 +2 n 2a 1 +2 2 m k 2a 1 +q 1 n k n+k k H k−1 (q 2 , . . . , q m ) + 1 n 2a 1 +1 p∈Π(2a 2 +1,...,2ar+1) 2 ℓ(p) H n (p) = p∈Π(2a 1 +1,2a 2 +1,...,2ar+1) 2 ℓ(p) H n (p) + (a,v)∈Π(2a 2 +1,...,2ar+1) 2 ℓ(v)+1 · · n n k=1 H k−1 (v) n k k a n+k k − n k=1 H k−1 (v) n k k a−1 n+k k − 2 n k=1 kH k−1 (a, v) n k n+k k .(14)
Here the middle (resp. the last) term of (14) is obtained by taking (q 1 , . . . , q m ) = (1 + a, v) (resp. (q 1 , . . . , q m ) = (1, a, v). Notice the expression inside the pair of parentheses above vanishes by (9) of Lemma 3.1. This completes the proof of Theorem 2.1. Exactly the same argument works almost word for word for Theorem 2.3 so we leave the details to the interested reader.
We can now prove Theorems 1.1 and 1.2 by using Theorems 2.1 and 2.3 and the following key lemma proved in [16].
for all s ∈ (Z * ) d (s = ∅ if d = 0) we have lim n→∞ n k=1 |H k−1 (s)| k e 1 − n k n+k k = 0.(15)
Proof of Theorem 1.
MHS and MZSV identities
: 2-c-2-1 formula or 2-1-2-c-2-1 formula
In this and the next three sections we utilize the ideas in the previous sections to derive more MHS identities involving arguments of ({2} a , 1)-type alternating with those of ({2} b , c)-type (c ≥ 3). In this section, we start by considering strings ending with ({2} a , 1)-type. As before, we study the MHS first and then derive the corresponding MZSV identities by invoking Lemma 4.1.
Theorem 5.1. Let a j , b j , c j − 3 ∈ N 0 for all j ≥ 0. Then (2-c-2-1): For s = ({2} b 1 , c 1 , {2} a 1 , 1, . . . , {2} br , c r , {2} ar , 1), r ≥ 1, we have H ⋆ n (s) = p∈Π(2b 1 +2, {1} c 1 −3 , 2a 1 +2, ..., 2br+2, {1} cr −3 , 2ar+2) 2 ℓ(p) H n (p). (2-1-2-c-2-1): For s = ({2} a 0 , 1, {2} b 1 , c 1 , {2} a 1 , 1, . . . , {2} br , c r , {2} ar , 1), r ≥ 0, we have H ⋆ n (s) = p∈Π(2a 0 +1, 2b 1 +2,{1} c 1 −3 , 2a 1 +2, ..., 2br+2, {1} cr −3 , 2ar+2) 2 ℓ(p) H n (p).
Proof. We proceed by induction on n. When n = 1 the theorem is clear. Assume now the theorem is true for all n + r ≤ N where N ≥ 2. Suppose we have n ≥ 2 and
n + r = N + 1. Let s = ({2} b 1 , c 1 , {2} a 1 , 1, . . . , {2} br , c r , {2} ar , 1). We have by definition H ⋆ n (s) = b 1 l=0 1 n 2b 1 −2l H ⋆ n−1 ({2} l , c 1 , {2} a 1 , 1, {2} b 2 , . . . , {2} br , c r , {2} ar , 1) + 1 n 2b 1 +c 1 H ⋆ n ({2} a 1 , 1, {2} b 2 , c 2 , {2} a 2 , 1, . . . , {2} br , c r , {2} ar , 1).
For ease of reading we define the following index sets: for any composition v of integers
I(v) =J(v, {1} c 1 −3 , 2a 1 + 2), J(v) =Π(v, 2b 2 + 2, {1} c 2 −3 , 2a 2 + 2, . . . , 2b r + 2, {1} cr−3 , 2a r + 2).
Then by induction assumption
H ⋆ n (s) = b 1 l=0 1 n 2b 1 −2l q∈I(2l+2) 2 ℓ(q) H n−1 (q) + 1 n 2b 1 +c 1 p∈J(2a 1 +1) H n (p) = (q 1 ,...,qm)∈I(2) b 1 l=0 2 m n 2b 1 −2l n−1 k=1 sgn(q 1 ) k k 2l+|q 1 | n−1 k n−1+k k H k−1 (q 2 , . . . , q m ) + 1 n 2b 1 +c 1 p∈J(2a 1 +1) 2 ℓ(p) H n (p),
By changing the order of summations and using the identity (13) we see easily that
H ⋆ n (s) = (q 1 ,...,qm)∈I(2b 1 +2) 2 m n k=1 sgn(q 1 ) k k |q 1 | 1 − k 2b 1 +2 n 2b 1 +2 n k n+k k H k−1 (q 2 , . . . , q m ) + 1 n 2b 1 +c 1 p∈J(2a 1 +1) 2 ℓ(p) H n (p).
Notice that the index set
I(2b 1 + 2) = (p 1 ,...,pm)∈J(2a 1 +1) 2b 1 + 2 • 1 • · · · • 1 c 1 −3 times •(1 ⊕ p 1 ), . . . , p m .(16)
For each (p 1 , . . . , p m ) we can partition the set (16) into the following subsets:
(2b 1 + c 1 ) ⊕ p 1 , v ∪ j + 2 + 2b 1 , x,ī ⊕ p 1 , v , v = (p 2 , . . . , p m ),
for i ≥ 1, j ≥ 0 and positive compositions y with i + j + |y| = c 1 − 2. We have two cases: (i) p 1 > 0 and (ii) p 1 < 0. Set a = |p 1 |. It suffices to prove that in case (i)
Equation (17) follows from (10) This proves case (2-c-2-1) when r + n = N + 1. Now we turn to case (2-1-2-c-2-1). Let s = ({2} a 0 , 1, {2} b 1 , c 1 , {2} a 1 , 1, . . . , {2} br , c r , {2} ar , 1). By definition {2} a 1 , 1, . . . , {2} br , c r , {2} ar , 1). For ease of reading, for any composition v of integers we set
H ⋆ n (s) = a 0 l=0 1 n 2a 0 −2l H ⋆ n−1 ({2} l , 1, {2} b 1 , c 1 , {2} a 1 , 1, . . . , {2} br , c r , {2} ar , 1) + 1 n 2a 0 +1 H ⋆ n ({2} b 1 , c 1 ,K(v) = Π(v, 2b 1 + 2, {1} c 1 −3 , 2a 1 + 2, . . . , 2b r + 2, {1} cr−3 , 2a r + 2).
By the induction assumption and the fact that we have just proved the case (2-c-2-1) with r + n = N + 1 we see that
H ⋆ n (s) = a 0 l=0 1 n 2a 0 −2l q∈K(2l+1) 2 ℓ(q) H n−1 (q) + 1 n 2a 0 +1 p∈K(∅) 2 ℓ(p) H n (p) = (q 1 ,...,qm)∈K(1) a 0 l=0 2 m n 2a 0 −2l n−1 k=1 sgn(q 1 ) k k 2l+|q 1 | n−1 k n−1+k k H k−1 (q 2 , . . . , q m ) + 1 n 2a 0 +1 p∈K(∅) 2 ℓ(p) H n (p).
By changing the order of summations and using the identity (13) we get
H ⋆ n (s) = (q 1 ,...,qm)∈K(2a 0 +1) 2 m n k=1 1 − k 2a 0 +2 n 2a 0 +2 sgn(q 1 ) k k |q 1 | n k n+k k H k−1 (q 2 , . . . , q m ) + 1 n 2a 0 +1 p∈K(∅) 2 ℓ(p) H n (p).
Notice that the partition of the index set
K(2a 0 + 1) = (p 1 ,...,pm)∈K(∅) 2a 0 + 1, p 1 , v ∪ (2a 0 + 1) ⊕ p 1 , v ,
where v = (p 2 , . . . , p m ). For each (p 1 , . . . , p m ) we have two cases: (i) p 1 > 0 and (ii) p 1 < 0. Set a = |p 1 |. It suffices to prove that in case (i)
H k−1 (v)(−1) k n k k a n+k k − 2 n k=1 kH k−1 (a, v) n k n+k k − n k=1 H k−1 (v)(−1) k n k k a−1 n+k k = 0
by (11) of Lemma 3.1. This proves case (2-1-2-c-2-1) when r + n = N + 1.
We have completed the proof of Theorem 5.1.
When r = 1 we get the following: for all n ∈ N and a, b ∈ N 0
H ⋆ n ({2} b , 3, {2} a , 1) = 2 n k=1 n k k 2(b+a)+4 n+k k + 4 n k=1 (−1) k H k−1 (2a + 2) n k k 2b+2 n+k k ,(19)
in case (2-c-2-1), and in case (2-1-2-c-2-1)
H ⋆ n ({2} a 1 , 1,{2} b , 3, {2} a 2 , 1)(20)
=2 n k=1 n k
k 2(a 1 +b+a 2 +5) n+k k + 4 n k=1 H k−1 (2b + 2a 2 + 4) n k k 2a 1 +1 n+k k +4 n k=1 (−1) k H k−1 (2a 2 + 2) n k k 2a 1 +2b+3 n+k k + 8 n k=1 H k−1 (2b + 2, 2a 2 + 2) n k k 2a 1 +1 n+k k . Theorem 5.2. Let r ∈ N and a j , b j , c j − 3 ∈ N 0 for all j ≥ 1. Then (2-c-2-1): For s = ({2} b 1 , c 1 , {2} a 1 , 1, . . . , {2} br , c r , {2} ar , 1), r ≥ 1, we have ζ ⋆ (s) = p∈Π(2b 1 +2, {1} c 1 −3 , 2a 1 +2, ..., 2br +2, {1} cr −3 , 2ar +2) 2 ℓ(p) ζ(p).(21)
Proof. This follows from Theorem 5.1 and Lemma 4.1 easily.
For example, by (19) and (20) we see that
ζ ⋆ ({2} b , 3, {2} a , 1) = 2ζ(2a + 2b + 4) + 4ζ(2b + 2, 2a + 2),(23)
and
ζ ⋆ ({2} a 1 , 1, {2} b , 3, {2} a 2 , 1) = 2ζ(2(a 1 + b + a 2 ) + 5) + 4ζ(2a 1 + 1, 2b + 2a 2 + 4)
+ 4ζ(2a 1 + 2b + 3, 2a 2 + 2) + 8ζ(2a 1 + 1, 2b + 2, 2a 2 + 2).
MHS and MZSV identities: 2-c-2-1-2 and 2-1-2-c-2-1-2 formula
We continue to study MHS identities involving arguments of ({2} a , 1)-type alternating with those of ({2} b , c)-type (c ≥ 3). Now we consider strings ending with ({2} a , 1)-type trailed by a non-empty substring of 2's at the very end.
H ⋆ n (s) = − p∈Π(2b 1 +2, {1} c 1 −3 , 2a 1 +2, ..., 2br+2, {1} cr −3 , 2ar+2,2t) (2-1-2-c-2-1-2). For s = ({2} a 0 , 1, {2} b 1 , c 1 , {2} a 1 , 1, . . . , {2} br , c r , {2} ar , 1, {2} t ), r ≥ 0, we have H ⋆ n (s) = − p∈Π(2a 0 +1, 2b 1 +2,{1} c 1 −3 , 2a 1 +2, ..., 2br+2, {1} cr −3 , 2ar +2,2t) 2 ℓ(p) H n (p).
Proof. The proof is very similar to the proof of Theorem 5.1. Thus we leave it to the interested reader.
By taking n → ∞ and using Lemma 4.1 we get immediately the following results.
Theorem 6.2. Let t, r ∈ N and a j , b j , c j − 3 ∈ N 0 for all j ≥ 1. Then
(2-c-2-1-2). For s = ({2} b 1 , c 1 , {2} a 1 , 1, . . . , {2} br , c r , {2} ar , 1, {2} t ), r ≥ 1, we have ζ ⋆ (s) = − p∈Π(2b 1 +2, {1} c 1 −3 , 2a 1 +2, ..., 2br+2, {1} cr −3 , 2ar +2,2t) 2 ℓ(p) ζ(p).(24)(2-1-2-c-2-1-2): For s = ({2} a 0 , 1, {2} b 1 , c 1 , {2} a 1 , 1, . . . , {2} br , c r , {2} ar , 1, {2} t ), r ≥ 0 and a 0 ≥ 1, we have ζ ⋆ (s) = − p∈Π(2a 0 +1, 2b 1 +2,{1} c 1 −3 , 2a 1 +2, ..., 2br+2, {1} cr −3 , 2ar+2,2t) 2 ℓ(p) ζ(p).(25)
For example, taking r = 1 and c 1 = 3 we get in case (2-c-2-1-2)
ζ ⋆ ({2} b , 3,ζ ⋆ ({2} a 1 , 1, {2} b , 3, {2} a 2 , 1, {2} t ) = − 2ζ(2a 1 + 2b + 2a 2 + 2t + 5) − 4ζ(2a 1 + 2b + 2a 2 + 5, 2t) − 4ζ(2a 1 + 2b + 3, 2a 2 + 2t + 2) − 8ζ(2a 1 + 2b + 3, 2a 2 + 2, 2t) − 4ζ(2a 1 + 1, 2b + 2a 2 + 2t + 4) − 8ζ(2a 1 + 1, 2b + 2a 2 + 4, 2t) − 8ζ(2a 1 + 1, 2b + 2, 2a 2 + 2t + 2) − 16ζ(2a 1 + 1, 2b + 2, 2a 2 + 2, 2t).
We have verified these formulas numerically for a 1 , a 2 , a, b, t ≤ 2 using EZ-face [3].
MHS identities: 2-1-2-c-2 formula and 2-c-2-1-2-c-2 formula
In this section we continue the theme in the preceding sections by deriving more MHS identities involving compositions of ({2} a , 1)-type alternating with those of ({2} b , c)type (c ≥ 3). This time we consider strings ending with ({2} a , c)-type trailed by {2} t (this tail may be empty). We can then derive the corresponding MZSV identities by applying Lemma 4.1. We omit the proofs since they are essentially the same as those of Theorem 5.1 and Theorem 5.2.
Theorem 7.1. Let r ∈ N and t, a j , b j , c j − 3 ∈ N 0 for all j ≥ 1. Then
(2-1-2-c-2). For s = ({2} a 1 , 1, {2} b 1 , c 1 , . . . , {2} ar , 1, {2} br , c r , {2} t ), we have H ⋆ n (s) = − p∈Π(2a 1 +1, 2b 1 +2, {1} c 1 −3 , 2a 2 +2, ..., 2ar +2, 2br +2, {1} cr −3 , 2t+1) 2 ℓ(p) H n (p). (2-c-2-1-2-c-2). For s = ({2} b 1 , c 1 , {2} a 1 , 1, . . . , {2} br , c r , {2} ar , 1, {2} b r+1 , c r+1 , {2} t ) we have H ⋆ n (s) = − p∈Π(2b 1 +2,{1} c 1 −3 , 2a 1 +2, ..., 2ar +2, 2b r+1 +2, {1} c r+1 −3 , 2t+1) 2 ℓ(p) H n (p).
Setting r = 0 in Theorem 7.1 we recover [7, Theorem 2.1]. When r = 1 and t = 0 we get the following: for all n ∈ N and a, b ∈ N 0
H ⋆ n ({2} a , 1, {2} b , 3) = − 2 n k=1 (−1) k n k k 2(a+b)+4 n+k k − 4 n k=1 H k−1 (2b + 3) n k k 2a+1 n+k k −4 n k=1 H k−1 (1)(−1) k n k k 2a+2b+3 n+k k − 8 n k=1 H k−1 (2b + 2, 1) n k k 2a+1 n+k k in case (2-1-2-c-2), and in case (2-c-2-1-2-c-2): H ⋆ n ({2} b 1 , 3, {2} a , 1, {2} b 2 , 3) = −2 n k=1 (−1) k n k k 2(b 1 +a+b 2 )+7 n+k k − 4 n k=1 H k−1 (2b 2 + 3) n k k 2b 1 +2a+4 n+k k −4 n k=1 H k−1 (2a + 2b 2 + 5)(−1) k n k k 2b 1 +2 n+k k − 8 n k=1 H k−1 (2a + 2, 2b 2 + 3)(−1) k n k k 2b 1 +2 n+k k −4 n k=1 H k−1 (1)(−1) k n k k 2b 1 +2a+2b 2 +6 n+k k − 8 n k=1 H k−1 (2a + 2b 2 + 4, 1)(−1) k n k k 2b 1 +2 n+k k −8 n k=1 H k−1 (2b 2 + 2, 1) n k k 2b 1 +2a+4 n+k k − 16 n k=1 H k−1 (2a + 2, 2b 2 + 2, 1)(−1) k n k k 2b 1 +2 n+k k .
By taking n → ∞ in Theorem 7.1 and using Lemma 4.1 we obtain
Theorem 7.2. Let r ∈ N and a j , b j , c j − 3 ∈ N 0 for all j ≥ 1. Then (2-1-2-c-2). For s = ({2} a 1 , 1, {2} b 1 , c 1 , . . . , {2} ar , 1, {2} br , c r , {2} t ) with a 1 ≥ 1, we have ζ ⋆ (s) = − p∈Π(2a 1 +1, 2b 1 +2, {1} c 1 −3 , 2a 2 +2, ..., 2ar+2, 2br+2, {1} cr −3 , 2t+1) 2 ℓ(p) ζ(p).(26)(2-c-2-1-2-c-2). For s = ({2} b 1 , c 1 , {2} a 1 , 1, . . . , {2} br , c r , {2} ar , 1, {2} b r+1 , c r+1 , {2} t ) we have ζ ⋆ (s) = − p∈Π(2b 1 +2,{1} c 1 −3 , 2a 1 +2, ..., 2ar +2, 2b r+1 +2, {1} c r+1 −3 , 2t+1) 2 ℓ(p) ζ(p).(27)
For example, taking r = 1 and t = 0 in case (2-1-2-c-2) we get
ζ ⋆ ({2} a , 1, {2} b , 3) = − 2ζ(2a + 2b + 4) − 4ζ(1 + 2a, 2b + 3) −4ζ(2a + 2b + 3, 1) − 8ζ(2a + 1, 2b + 2, 1).
and in case (2-c-2-1-2-c-2) we get
ζ ⋆ ({2} b 1 , 3, {2} a , 1, {2} b 2 , 3) = − 2ζ(2(b 1 + a + b 2 ) + 7) − 4ζ(2b 1 + 2a + 4, 2b 2 + 3) − 4ζ(2b 1 + 2, 2a + 2b 2 + 5) − 4ζ(2b 1 + 2a + 2b 2 + 6, 1) − 8ζ(2b 1 + 2, 2a + 2, 2b 2 + 3) − 8ζ(2b 1 + 2, 2a + 2b 2 + 4, 1)
− 8ζ(2b 1 + 2a + 4, 2b 2 + 2, 1) − 16ζ(2b 1 + 2, 2a + 2, 2b 2 + 2, 1).
We have numerically verified these formulas with a, b 1 , b 2 ≤ 2 using EZ-face [3]. [21]. Yamamoto proves a more precise version in [23]. We now give a different and concise proof using the identities we have found in the last two sections. We begin with a lemma first. Lemma 8.1. Let n 1 , . . . , n ℓ ∈ Z * be ℓ even integers and m = |n 1 | + · · · + |n ℓ |. Then g∈S ℓ ζ n g(1) , . . . , n g(ℓ) = e 1 +···+ep=ℓ
(−1) ℓ−p p s=1 (e s − 1)! ζ k∈π 1 n k . . . ζ k∈πp n k ∈ Qπ m
where the sum in the right is taken over all the possible unordered partitions of the set {1, . . . , ℓ} into p subsets π 1 , . . . , π p with e 1 , . . . , e p elements respectively.
Proof. When all the arguments n 1 , . . . , n ℓ are positive the lemma becomes [9, Theorem 2.2]. Its proof there can be used here almost word for word. Notice that [9, Theorem 2.2] is re-proved as [15,Proposition 9.4] whose proof is different from that of [9] but also works here. Thus we leave the details to the interested reader.
Theorem 8.2. Let r be a positive integer, and e 1 , . . . , e 2r+1 nonnegative integers.
(i) Put m = e 1 + · · · + e 2r . Then we have
τ ∈S 2r ζ ⋆ ({2} e τ (1) , 3, {2} e τ (2) , 1, {2} e τ (3) , . . . , 3, {2} e τ (2r) , 1) ∈ Q · π 2m+4r .
(ii) Put m = e 1 + · · · + e 2r+1 . Then we have
τ ∈S 2r+1 ζ ⋆ ({2} e τ (1) , 3, {2} e τ (2) , 1, . . . , 3, {2} e τ (2n) , 1, {2} e τ (2r+1) +1 ) ∈ Q · π 2m+4r+2 .
Proof. We start with (i) first. When r = 1 this follows quickly from (23) by stuffle relation. For general r let a j = e 2j and b j = e 2j−1 for all j ≤ r and let A j = 2e j + 2 for all j ≤ 2r . Then we can apply (21) of Theorem 5.2 to the string s = ({2} b 1 , 3, {2} a 1 , 1, . . . , {2} br , 3, {2} ar , 1) and get
ζ ⋆ (s) = τ ∈S 2r τ p∈Π(A 1 ,...,A 2r ) 2 ℓ(p) ζ(p) .
Let A = (A 1 , . . . , A 2r ) and P ℓ (2r) be the set of all partitions of [2r] := {1, 2, . . . , 2r} into ℓ consecutive subsets. If λ = (λ 1 , . . . , λ ℓ ) ∈ P ℓ (2r) then we set λ j (A) = (A i ) i∈λ j so that the concatenation ℓ j=1 λ j (A) = A. Because of the permutation we see that
ζ ⋆ (s) = τ ∈S 2r τ 2r ℓ=1 2 ℓ λ∈P ℓ (2r) ζ ⊕ λ 1 (A), . . . , ⊕λ ℓ (A) = τ ∈S 2r τ 2r ℓ=1 2 ℓ ℓ! λ∈P ℓ (2r) g∈S ℓ ζ ⊕ λ g(1) (A), . . . , ⊕λ g(ℓ) (A) ,
where ⊕t is the ⊕-sum of all the components of t for any composition t. Hence Theorem 8.2(i) follows readily from the Lemma 8.1 since all A j 's are even numbers. Theorem 8.2 (ii) follows from Theorem 6.2 in a similar fashion so we leave the details to the interested reader.
ζ ⋆ ({2} e 0 , 3, {2} e 1 , 1, {2} e 2 , 3, . . . , 3, {2} e 2r−1 , 1, {2} e 2r ) = 2i+k+u=2r j+l+v=m (−1) j+k k + l k u + v u 2i + j j β k+l β u+v π 4r+2m (2i + 1)(4i + 2j + 1)! ,(28)
where β n = (−1) n (2 − 2 2n )B 2n /(2n)!. It is possible to modify our proof of Theorem 8.2 to give this more quantified version.
9. More Conjetures of Imatomi et al.
The following results were first conjectured by Imatomi et al. [13,Conjecture 4.5].
Theorem 9.1. Let m and n be two nonnegative integers.
(i) We have ζ ⋆ ({2} n , 3, {2} m , 1) + ζ ⋆ ({2} m , 3, {2} n , 1) = ζ ⋆ ({2} n+1 )ζ ⋆ ({2} m+1 ).
(ii) We have
(2n + 1)ζ ⋆ ({3, 1} n , 2) = j+k=n ζ ⋆ ({3, 1} j )ζ ⋆ ({2} 2k+1 ).
(iii) If n ≥ 1 then we have
e 1 +e 2 +···+e 2n =1 e 1 ,e 2 ,...,e 2n ≥0 ζ ⋆ ({2} e 1 , 3, {2} e 2 , 1, . . . , {2} e 2n−1 , 3, {2} e 2n , 1) = j+k=n−1 ζ ⋆ ({3, 1} j , 2)ζ ⋆ ({2} 2k+2 ).
Proof. (i). This follows immediately from (4) and (23).
(ii). We notice that by taking a i = b i = 0 and c i = 3 for all i ≤ r = j in Theo-
rem 5.1(2-c-2-1) we get ζ ⋆ ({3, 1} j ) = p 2j ∈Π({2} 2j ) 2 ℓ(p 2j ) ζ(p 2j ).
All of the components a j of 2 • · · · • 2 must satisfy the following sign rule: a j > 0 if and only if 4|a j .
On the other hand, by Theorem 6.1(2-c-2-1-2) we have
ζ ⋆ ({3, 1} n , 2) = p 2n+1 ∈Π({2} 2n+1 ) 2 ℓ(p 2n+1 ) ζ(p 2n+1 ).
Hence by (4) we need to show that
p 2n+1 ∈Π({2} 2n+1 ) 2 ℓ(p 2n+1 ) ζ(p 2n+1 ) = n j=0 p 2j ∈Π({2} 2j ) 2 ℓ(p 2j ) ζ(p 2j ) · 2ζ(4(n − j) + 2). (30)
Suppose an index in p 2j has length t + 1 (0 ≤ t ≤ 2n) given as (a 1 , . . . , a t+1 ), a i ∈ Z * ∀i = 1, . . . , t + 1.
We now show that there are exactly 2 t (2n + 1) copies of such term produced by stuffle product on the right hand side of (30). Indeed, for each i = 1, . . . , t + 1 the entry a i has two possibilities: (1). a i = 4b i > 0. Then for each k = 1, . . . , b i we may produce such a term on the right hand side of (30) by stuffing 2 t ζ(a 1 , . . . , a i−1 , 4k − 2, a i+1 , . . . , a t+1 ) from p 2j having length t + 1 with the term 2ζ(4(b i − k) + 2) at the right end of (30). Notice no shuffle is possible since 4(n − j) + 2 is not a multiple of 4. Hence these contribute to 2 t+1 b i = 2 t−1 a i copies of ζ(a 1 , . . . , a t+1 ).
(2). a i = 4b i + 2. Then for each k = 1, . . . , b i we may produce such a term on the right hand side of (30) by stuffing 2 t ζ(a 1 , . . . , a i−1 , 4k, a i+1 , . . . , a t+1 ) from p 2j having length t + 1 with the term 2ζ(4(b i − k) + 2) at the right end of (30). Further, there is exactly one possible shuffle given by
2 t−1 ζ(a 1 , . . . , a i−1 , a i+1 , . . . , a t+1 ) x 2ζ(4b i + 2) ,
since the index (a 1 , . . . , a i−1 , a i+1 , . . . , a t+1 ) has only length t. Altogether these produce 2 t+1 b i + 2 t = 2 t−1 |a i | copies of ζ(a 1 , . . . , a t+1 ).
By combining (1) and (2) we see that the right hand side of (30) produces exactly
t+1 i=1 2 t−1 |a i | = 2 t−1 · |(a 1 , . . . , a t+1 )| = 2 t (2n + 1)
copies of ζ(a 1 , . . . , a t+1 ) since the weight is 4n + 2. This proves (ii).
(iii). We use the same analysis as above and see that we need to prove the following identity:
q 2n 2 ℓ(q 2n ) ζ(q 2n ) = n−1 j=0 p 2j+1 ∈Π({2} 2j+1 ) 2 ℓ(p 2j+1 ) ζ(p 2j+1 ) · 2ζ(4(n − j)),(31)
where q 2n runs through all indices of the form A 1 • · · · • A 2n with one of the A j 's (say A j 0 ) equal to 4 and all the other A j 's equal to 2. For each choice of 2 t ζ(a 1 , . . . , a t+1 ) with length t + 1 from the left hand of (31), all but one of the argument components a 1 , . . . , a t+1 must satisfy the sign rule (29). The only exceptional component, say a i , must involve a merge with the special entry A j 0 = 4. Now there are two possibilities: (1). a i = 4b i + 2 > 0. Then for each k = 0, . . . , b i − 1 we may produce such a term on the right hand side of (31) by stuffing 2 t ζ(a 1 , . . . , a i−1 , 4k + 2, a i+1 , . . . , a t+1 ) from p 2j+1 having length t + 1 with the term 2ζ(4(b i − k)) at the right end of (31). Notice no shuffle is possible since 4(n − j) is a multiple of 4. Hence these contribute to 2 t+1 b i copies of ζ(a 1 , . . . , a t+1 ). On the left hand side, such a term must be produced by setting all 2b i − 1 consecutive •'s around A j 0 = 4 to ⊕:
. . . , A i ⊕ A i+1 ⊕ · · · ⊕ A j 0 ⊕ · · · ⊕ A ℓ 2b i entries , . . . .
But A j 0 can be at any one of the 2b i possible positions, thus producing 2 t+1 b i copies of ζ(a 1 , . . . , a t+1 ) which match exactly the right hand side of (31).
(2). a i = 4b i . Then for each k = 1, . . . , b i − 1 we may produce such a term on the right hand side of (31) by stuffing 2 t ζ(a 1 , . . . , a i−1 , 4k, a i+1 , . . . , a t+1 ) from p 2j having length t + 1 with the term 2ζ(4(b i − k)) at the right end of (31). Further, there is exactly one possible shuffle given by 2 t −1 ζ(a 1 , . . . , a i−1 , a i+1 , . . . , a t+1 ) x 2ζ(4b i ) .
Hence these contribute to 2 t+1 (b i − 1) + 2 t = 2 t (2b i − 1) copies of ζ(a 1 , . . . , a t+1 ). Similar to (1), on the left hand side, such a term must be produced by setting all 2b i − 2 consecutive •'s around A j 0 to ⊕. And A j 0 can be at any one of the 2b i − 1 possible positions, thus producing 2 t (2b i − 1) copies of ζ(a 1 , . . . , a t+1 ) which match exactly the right hand side of (31).
This concludes the proof of theorem.
Note that Theorem 9.1(i) is the more precise version of the n = 1 case of Theorem 8.2(i). And Theorem 9.1(iii) can be written more compactly as
ζ ⋆ {2} x{3, 1} n = n k=0 ζ ⋆ ({3, 1} n−k , 2)ζ ⋆ ({2} 2k ),
which is the more precise version of the m = 1 case of the following result of Kondo et al. [14]: For all nonnegative integers m and n we have
ζ ⋆ ({2} m
x{3, 1} n ) ∈ Qπ 2m+4n . The case m = 0 case has the following precise formulation by Muneta [17]:
ζ ⋆ {3, 1} n ) = n i=0 2 (4i + 2)! n 0 +n 1 =2(n−i) n 0 ,n 1 ≥0 (−1) n 1 (2 2n 0 − 2)B 2n 0 (2n 0 )! (2 2n 1 − 2)B 2n 1 (2n 1 )! π 4n .
Muneta also found precise form in case m = 1. Of course, these are all special cases of Yamamoto's general formula (28).
MHS: 1-c-1 formula
In this section we turn to MHS of the type 1-c-1 where the trailing 1 may be vacuous and the c's may be any positive integers (which is different from the requirement c ≥ 3 in the previous sections). The corresponding MZSVs diverge when the leading 1 is non-empty, however, in a sequel to this paper we will study the congruence properties of MHS where the results of this section will be utilized.
The
Proof. When n = 1 it is clear that both sides in (32) are equal to 1. We proceed by induction on n + r. By definition
H ⋆ n (s) = a 1 l=0 1 n a 1 −l H ⋆ n−1 ({1} l , c 1 , . . . , {1} ar , c r , {1} t ) + 1 n a 1 +c 1 H ⋆ n ({1} a 2 , c 2 , . . . , {1} ar , c r , {1} t ).
For ease of reading we define the following index sets: for any composition v of integers
I(v) =Π(v, {1} c 1 −2 , a 2 + 2, {1} c 2 −2 , . . . , a r + 2, {1} cr−2 , t + 1), J =Π(a 2 + 1, {1} c 2 −2 , a 3 + 2, .
. . , a r + 2, {1} cr−2 , t + 1).
By induction assumption
For each (p 1 , p 2 . . . , p m ) we can partition the set (33) into the following subsets:
c 1 + p 1 , v ∪ j + 1, y, i + p 1 , v , v = (p 2 ,
n,k as in Remark 3.2, c = c 1 − 1, a = p 1 , x = (y, i + p 1 ) and v = ∅. This completes the proof of our theorem.
For example, when r = 1 we recover [7, Theorem 2.2] and when r = 2 we get for all a 1 , a 2 , t ∈ N 0 and c 1 ,
c 2 ∈ N H ⋆ n ({1} a 1 , c 1 , {1} a 2 , c 2 , {1} t ) = − n k=1
(−1) k n k k a 1 +c 1 +a 2 +c 2 +t − i 2 +j 2 +|x 2 |=c 2 , i 2 ,j 2 ≥1 n k=1 H k−1 (x 2 , i 2 + t) n k (−1) k k a 1 +c 1 +a 2 +j 2 − i 1 +j 1 +|x 1 |=c 1 , i 1 ,j 1 ≥1 n k=1 H k−1 (x 1 , i 1 + a 2 + c 2 + t) n k (−1) k k a 1 +j 1 − iα+jα+|xα|=cα, iα,jα≥1, α=1,2 n k=1 H k−1 (x 1 , i 1 + a 2 + j 2 , x 2 , i 2 + t) n k (−1) k k a 1 +j 1 .
Concluding Remarks
There are many recent studies on MZVs, MZSVs and even their q-analogs. Most of the MZSV relations in [11] and [12] involving special types of arguments like ours in this paper can be proved in a more straight-forward manner using our results. However, it seems that the techniques contained here are hard to generalize to deal with MZVs even though these two types of values are extremely closely related from the point of view of their algebraic structures (see [10,12,18,20]). Such a generalization should help us resolve more conjectures such as those listed in [2, §7.2].
There are three more directions of research that should be of great interest. One is a theory generalizing the MHS identities obtained in this paper to truly alternating ones. We are aware of only one such instance. Setting Another direction is to establish a corresponding theory for multiple q-zeta values [4,28]. Initial computations show it is quite a promising project, see [6,8].
As for the third direction we notice that the many MHS identities proved in this paper can be used not only to derive MZSV identities but also to prove many congruences of MHS. This idea has already been carried out in [7] to prove one of our conjectures from [29]. In general, these congruences should shed more light on the unsolved [31, Conjecture 2.6] and the conjectures at the end of [29]. This will be done in a sequel to this paper.
sum and the (alternating) star Euler sum by
Theorem 1. 1 .
1Let r ∈ N and s = ({2} a 1 , 1, . . . , {2} ar , 1) where a 1 ∈ N and a j ∈ N 0 for all j ≥ 2. Then we have ζ ⋆ (s) = p 2 ℓ(p) ζ(p),
§5 for MHS and MZSV; (5) 2-c-2-1-2 (nontrivial 2 at the end): §6 for MHS and MZSV; (6) 2-1-2-c-2-1: §5 for MHS and MZSV; (7) 2-1-2-c-2-1-2 (nontrivial 2 at the end): §6 for MHS and MZSV; (8) 2-1-2-c-2 (2 at the end may be trivial): §7 for MHS and MZSV; (9) 2-c-2-1-2-c-2 (2 at the end may be trivial): §7 for MHS and MZSV; (10) 1-c-1 (c ≥ 1 and 1 at the end may be trivial): §10 for MHS.
Theorem 1. 3 .
3([7, Theorem 2.3]) Let a ∈ N 0 and b ∈ N. Then for any n ∈ N H ⋆ n ({2} a ,
2 .
2MHS identities: 2-1 formula and 2-1-2 formula To state our main theorems we need some additional notations first. For s = (s 1 , . . . , s m ) ∈ (Z * ) m we define the mollified companion of H n (s−1 (s 2 , . . . , s m ). (6)
Theorem 2. 3 .
3Suppose r ∈ N 0 and s = ({2} a 1 , 1, . . . , {2} ar , 1, {2} a r+1 ) where a j ∈ N 0 for all j ≤ r and a r+1 ∈ N. Then we have H ⋆ n (s) = − p∈Π(2a 1 +1,...,2ar+1,2a r+1 ) 2 ℓ(p) H n (p). Remark 2.4. When r = 0 Theorem 2.3 implies [7, (19)]. When r = 1 Theorem 2.3 becomes Theorem 1.3(B). When r = 2 we get the following: for all n, c ∈ N and a, b ∈ N 0 H ⋆ n ({2} a , 1, {2} b , 1, {2} c )
Lemma 3. 1 ..
1Let k, n ∈ N, a ∈ N 0 , A Suppose a ∈ N 0 and c ∈ N. Then (i)
7 ,
7by taking the sign of x r into consideration. Now by the identity proved in [Lemma
Lemma 4.1. [16, Lemma 4.2] Let d ∈ N 0 and let e be a real number with e > 1. Then
of Lemma 3.1 when c = c 1 −2,m = 2 and x = (y, i + a). Equation (18) follows from (8) with the same choice of parameters except x = (y, i+ a).
9) of Lemma 3.1, and in case (ii)
:
For s = ({2} a 0 , 1, {2} b 1 , c 1 , {2} a 1 , 1, . . . , {2} br , c r , {2} ar , 1), r ≥ 0 and a 0 ≥ 1, we have ζ ⋆ (s) = p∈Π(2a 0 +1, 2b 1 +2,{1} c 1 −3 , 2a 1 +2, ..., 2br+2, {1} cr −3 , 2ar+2) 2 ℓ(p) ζ(p).
Theorem 6 . 1 .
61Let t, r ∈ N and a j , b j , c j − 3 ∈ N 0 for all j ≥ 1. Then (2-c-2-1-2). For s =({2} b 1 , c 1 , {2} a 1 , 1, .. . , {2} br , c r , {2} ar , 1, {2} t ) we have
{2} a , 1 , 4
14{2} t ) = −2ζ(2b + 2a + 2t + 4) − 4ζ(2b + 2a +
8 .
8Conjectures of Imatomi et al. on MZSV of type 2-3-2-1 and 2-3-2-1-2-1 Throughout this section the notations of §5 and §6 are still in force. The following Theorem 8.2 was first conjectured by Imatomi et al. [13, Conjectures 4.1 and 4.3]. Special cases have been proved in [13, Theorem 1.1] and by Tasaka and Yamamoto in
Remark 8 . 3 .
83We notice that in [23, Theorem 1.1] Yamamoto obtains a more precise formula by using partial sums and generating functions: e 0 ,e 1 ,...,e 2r ≥0 e 0 +e 1 +···+e 2r =m
following theorem generalizes [7, Theorem 2.2]. For s = (s 1 , . . . , s m ) ∈ (Z * ) m we define s 2 , . . . , s m ).
Theorem 10. 1 .
1Let r ∈ N and s = ({1} a 1 , c 1 , . . . , {1} ar , c r , {1} t ) where t, a j ∈ N 0 and c j ∈ N for all j ≥ 1. Then H ⋆ n (s) = − p∈Π(a 1 +1, {1} c 1 −2 , a 2 +2, ..., ar+2, {1} cr −2 , t+1) n (p).
1 ,...,pm)∈J 1 • 1 • · · · • 1 c 1 −2 times •(p 1 + 1), p 2 , . . . , p m .
− 1 .
1. . . , p m ), for i ≥ 1, j ≥ 0 and positive compositions y with i + j + |y| = c 1 Thus
n ({1} a , 1) = n k=1 (2 k − 1)(−1) k k a+1 n k ∀a ∈ N 0 .
of MHS. This is the reason why a and b in Theorem 1.3(B) is switched from the original statement in [7, Theorem 2.3].Theorems 2.1 and 2.3 generalize Theorem 1.3(A
1 and Theorem 1.2. We observe that in Theorem 2.1 the first component ≥ 2a 1 + 1 ≥ 3, and in Theorem 2.3 the absolute value of first component ≥ 2a 1 + 1 ≥ 3. Therefore both theorems follow from Lemma 4.1 immediately. Remark 4.2. In[24] Yamamoto considers some algebraic structures depending on a variable t which reflect the properties of MZV and MZSV when t = 0 and t = 1, respectively. As he pointed out[23, Conjecture 4.4] the validity of the 2-1 formula Theorem 1.1 implies that the the algebra structure of MZSVs of the form ζ ⋆({2} a 1 , 1, .. . , {2} ar , 1) is reflected by setting t = 1/2.
l(y)+1i+j+|y|=c 1 −2, i≥1,j≥0 n k=1 H k−1 (y, i + a, v)(−1) k n k k j n+k k = 1 n c 1 −2 n k=1 H k−1 (v) n k k a n+k k − n k=1 H k−1 (v) n k k c 1 +a−2 n+k k ,(17)and in case (ii) 2 l(y)+1i+j+|y|=c 1 −2, i≥1,j≥0 n k=1 H k−1 (y, i + a, v)(−1) k n k k j n+k k = 1 n c 1 −2 n k=1 H k−1 (v)(−1) k n k k a n+k k − n k=1 H k−1 (v)(−1) k n k k c 1 +a−2 n+k k . (18)
ℓ(p) H n (p).
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[] |
[
"The charm quark mass from non-relativistic sum rules",
"The charm quark mass from non-relativistic sum rules"
] |
[
"Adrian Signer \nInstitute for Particle Physics Phenomenology Durham\nDH1 3LEEngland\n"
] |
[
"Institute for Particle Physics Phenomenology Durham\nDH1 3LEEngland"
] |
[] |
We present an analysis to determine the charm quark mass from nonrelativistic sum rules, using a combined approach taking into account fixed-order and effective-theory calculations. Non-perturbative corrections as well as higher-order perturbative corrections are under control. For the PS mass we find m PS (0.7 GeV) = 1.50 ± 0.04 GeV which translates into a MS mass of m = 1.25 ± 0.04 GeV.
|
10.1016/j.physletb.2009.01.028
|
[
"https://arxiv.org/pdf/0810.1152v1.pdf"
] | 15,764,641 |
0810.1152
|
fe34ee0a04d64c03704fba278ad02ad388468b97
|
The charm quark mass from non-relativistic sum rules
7 Oct 2008
Adrian Signer
Institute for Particle Physics Phenomenology Durham
DH1 3LEEngland
The charm quark mass from non-relativistic sum rules
7 Oct 2008IPPP/08/74 DCPT/08/148
We present an analysis to determine the charm quark mass from nonrelativistic sum rules, using a combined approach taking into account fixed-order and effective-theory calculations. Non-perturbative corrections as well as higher-order perturbative corrections are under control. For the PS mass we find m PS (0.7 GeV) = 1.50 ± 0.04 GeV which translates into a MS mass of m = 1.25 ± 0.04 GeV.
Introduction
In the sum rule approach [1] to determine the mass m of heavy quarks, Q, the sensitivity of the cross section σ(e + e − → QQ) near threshold √ s ≃ 2m is exploited by comparison of the experimental value of the n-th moment M n to the theoretical prediction. The moments are defined as
M n ≡ ∞ 0 ds s n+1 R QQ (s) = 12π 2 e 2 Q n! d dq 2 n Π(q 2 )| q 2 =0(1)
where Π(q 2 ) is the vacuum polarization, e Q the electric charge of the heavy quark and R QQ (s) ≡ σ(e + e − → QQ)/σ(e + e − → µ + µ − ) the normalized cross section. Traditionally, there have been two complementary theoretical approaches to determine M n . If n is chosen to be small, n 4 the moments are evaluated using a fixed-order approach. Sometimes this approach is referred to as relativistic or low-n sum rules. Alternatively, if n is large, fixed-order perturbation theory breaks down due to the presence of terms (α s √ n) l . In order to get a reliable theoretical prediction these terms have to be resummed, counting v ∼ 1/ √ n ∼ α s , where v is the (small) velocity of the heavy quarks. This is usually done in an effective-theory approach (for a review see Ref. [2]), treating the heavy quarks in the non-relativistic approximation. Therefore, this approach is referred to as non-relativistic sum rules.
Relativistic sum rules have been used to determine the bottom and charm quark masses [3,4,5,6]. The first two moments are known at four loop [5,7,8,9], i.e. O(α 3 s ), higher moments are currently known at O(α 2 s ) [10,11]. The extracted mass and its error depend crucially on how the experimentally poorly known continuum cross section is treated and how the theoretical error is estimated. Recently the charm quark mass has been determined using this approach but replacing experimental data for σ(e + e − → cc) by input from lattice QCD [12], which via tuning uses different experimental input such as the η c mass.
Applications of the non-relativistic sum rules have been restricted to the determination of the bottom quark mass [13,14] so far. The moments are known to next-to-next-to leading order (NNLO), in the counting of the effective theory, and in the case of Ref. [14] also include the resummation of logarithms [15] of the form (α s ln √ n) l at next-to-leading (NLL) and partially at next-to-next-toleading logarithmic accuracy (NNLL). A complete NNNLO calculation is still missing but partial results are available [16,17]. This method of extracting m is virtually insensitive to the continuum cross section but suffers from large higherorder corrections.
The main reason why non-relativistic sum rules have not been used in the case of charm quarks is that the application of perturbation theory in this context is thought to be questionable. First, the typical non-relativistic momentum and energy scales, 2m/ √ n and m/n, are very small for large n. To some extent this is related to the large higher-order corrections mentioned above and in fact is already a problem for the case of bottom quarks. Second, the non-perturbative contributions from vacuum condensates increase with increasing n and decreasing m and potentially make a precise determination of M n impossible.
For what values of n can sum rules be used in the charm case and when is perturbation theory not applicable any longer? In order to answer this question we will consider M n for all n ≤ 16 in the charm case. We will use an approach that combines techniques for low-n and large-n sum rules. In the case of the bottom quark it has been shown [18] that even though these techniques are completely different, the results are remarkably consistent. Encouraged by this we perform an all n analysis in the case of the charm quark. We will see that using a combined approach helps to keep the size of higher-order corrections under control. Also, the non-perturbative corrections due to the gluon condensate turn out to be much smaller than expected, even for large n, if a threshold mass definition [19,20,21] is used. This will lead us to conclude that, contrary to common belief, the charm quark mass can be extracted from non-relativistic sum rules in a reliable way.
In Section 2 we will describe how to obtain the experimental moments, the theoretical moments and the non-perturbative contributions in turn. We then combine these results in Section 3 and determine the charm quark mass.
Determination of the moments 2.1 The experimental moment
We start with the determination of the experimental moments. This is conveniently split into three regions: the resonance region including the bound states J/Ψ and Ψ(2S) below threshold, the threshold region 2M D 0 = 3.73 GeV ≤ √ s ≤ 4.8 GeV and the continuum region √ s > 4.8 GeV.
The mass and leptonic width of J/Ψ and Ψ(2S) are known to a high accuracy which leads to a very precise determination of the resonance contribution. We use M J/Ψ = 3096.916(11) MeV, M Ψ(2S) = 3686.09(4) MeV, Γ J/Ψ = 5.55 (14) keV and Γ Ψ(2S) = 2.38(4) keV [22]. The resonance contribution is then given by
M (res) n = 9π α 2 i Γ i M 2n+1 i(2)
where i ∈ {J/Ψ, Ψ(2S)} and α = α(M i ) = 1/134.
The contribution from the threshold and continuum region are much more difficult to determine. However, for increasing n these contributions become less and less important. In fact, for n > 5 the combined threshold and continuum contribution to the moments is smaller than the error from the resonance contribution. Since our analysis will be driven by large n a rather crude determination of these contributions with a large error will not affect the final result. This is one of the big advantages of this approach compared to a fixed-order low-n analysis. In particular, there is no need to replace experimental data above threshold by theoretical input to (artificially) decrease the experimental error.
For the continuum contribution we use data points above and below threshold [23] to obtain a very crude parameterization R cc (s) = 1.4 ± 0.5 for √ s > 4.8 GeV. In the threshold region, we include Ψ(3770), Ψ(4040), Ψ(4160) and Ψ(4415) [22] according to Eq. (2) in addition to an underlying contribution parameterized by R cc (s) = −4.88 + 1.31 √ s. This corresponds to a linear in √ s extrapolation between R cc ( √ s = 3.73 GeV) = 0 and R cc ( √ s = 4.8 GeV) = 1.4. This is of course a very crude estimate. We take this into account by taking as the error the full size of the underlying contribution. With this procedure we obtain the results as given in Table 1 for the experimental moments. The total error has been obtained by adding the separate errors in quadrature. Entries smaller than 5 × 10 −5 are given as 0.
We stress that it is of course possible to get more precise results for small n, but in our approach this is not required. In fact, in what follows we will consider only n > 2 and for our final result only moments with n 8 are relevant.
The theoretical moment
The evaluation of the theoretical moments follows the discussion given in Ref. [18]. We consider three ways to evaluate the moments: fixed-order (FO) moments, moments evaluated using an effective-theory approach (ET) writing
M n = ∞ −∞ 2 dE (2m) 2n+1 e − nE m 1 − E 2m + nE 2 (2m) 2 + . . . R cc (E)(3)
with E = √ s − 2m and, finally, moments using a combined approach. The latter are obtained by adding the FO and ET moments and subtracting the doubly counted terms [18]. These moments should provide us with a reliable theoretical prescription for all n as long a non-perturbative corrections are under control.
Since we are dealing (at least partially) with large n where the moments are dominated by the lowest lying resonances, we have to use a mass definition adapted to the description of such resonances, i.e. a threshold mass [19,20,21]. We use the PS mass [20] with the associated factorization scale µ F = 0.7 GeV. For the strong coupling we set α s (M Z ) = 0.118 and use three-loop evolution.
The main issue for a reliable extraction of the charm mass will be a realistic estimate of the theoretical error due to missing higher-order corrections. This is a notorious problem and there is no generally applicable procedure. We will therefore use a combination of different methods and criteria, as described below.
Let us use the 10th moment as an example to illustrate our determination of the theoretical error. In Figure 1 we display M 10 as a function of the scale µ for m PS (µ F ) = 1.50 GeV. We consider a range of different theoretical predictions.
FO: this is a fixed-order calculation including all terms up to O(α 3 s ), keeping in mind that the constant term of O(α 3 s ) is actually not yet known for n > 2. We have fixed this constant to be the one of the first moment. This has only a very weak influence on the result.
ET-NNLL: this is a renormalization-group improved effective-theory calculation complete at NNLO. The resummation of logarithms is complete at NLL and partially done at NNLL. For details we refer to Ref. [14].
ET-N3LO: this is an effective-theory calculation where the logarithms have been re-expanded and kept up to NNNLO. This result contains all terms at NNLO and the logarithmically enhanced NNNLO terms. It is used to gauge the impact of corrections beyond NNLO and the importance of resumming the logarithms. CB-NNLL: this is a combined FO-ET calculation [18]. The ET input is as for ET-NNLL. In addition all terms of O(α 3 s ) have been included. This is the "best" available theoretical prediction.
CB-N3LO: this is also a combined FO-ET calculation, but the ET input has been taken as in ET-N3LO.
CB-NLL: this is also a combined FO-ET calculation, but the ET input is modified to be only at NLO/NLL. The subtraction to avoid double counting when combining with the FO result has to be adapted accordingly. This result is used to consider the convergence of the perturbative series.
The following points related to Figure 1 will be of importance and in fact are valid for all moments that are relevant to us, i.e. with n 5:
• The ET and CB results are very close, indicating that the relativistic corrections to the ET result are small. This is not surprising for large n. What is surprising to some extent is that the relativistic corrections turn out to be small also for small n.
• The ET-NNLL and CB-NNLL results have a peak slightly above µ ≃ 1 GeV. For scales below the peak, the results become very soon unreliable indicating a breakdown of perturbation theory. The peak is close to µ = 2m/ √ n, the typical momentum scale in the non-relativistic region.
• The ET-N3LO and CB-N3LO results are very similar to the ET-NNLL and CB-NNLL results except for small µ. It is in this region only, where the resummation of logarithms actually becomes important and, therefore, the ET-N3LO and CB-N3LO results cannot be used any longer.
• The FO prediction does remarkably well even for large moments, where it is supposed to be inapplicable. This hinges on the fact that a threshold mass has been used in the FO approach.
Taking into account these observations we proceed as follows to determine the mass and its theoretical error due to missing higher-order corrections. We start by taking our best prediction, CB-NNLL, and determine a band of m values by varying 1 GeV ≤ µ ≤ 2 GeV. The standard prescription would be to vary the scale by a factor two around the typical value µ = 2m/ √ n. This would result in scales below 1 GeV though, which according to Figure 1 are not acceptable. However, if the upper limit of the standard variation is larger than 2 GeV (i.e. for n ≤ 9) we use the larger value instead. Note that in any case the peak of the CB-NNLL result is included in the band of scale variation. We now extract the mass as the central value of the band with symmetric errors. The results are shown in the first column of Table 2. As anticipated small moments have a large error in our approach and will not play a significant role.
Given the remarkable consistency and the small errors of these results it would be tempting to simply take them at face value. However, the scale dependence alone is a dubious way to determine the theoretical error. In order to get a more reliable estimate we extend the analysis. We repeat the same exercise for the CB-N3LO case. The results are given in the second column of Table 2. As can be seen from Figure 1 the CB-N3LO calculation leads to larger moments (for small µ) and therefore somewhat larger values for m. Also, the error is dominated by small scales µ ≃ 1 where the CB-N3LO starts to become unreliable due to the importance of resumming logarithms. Therefore, taking the upper end of the m band of the CB-N3LO results in an overestimate of the theoretical error. We thus combine the CB-NNLL and CB-N3LO results by subtracting the CB-NNLL error from the (smaller) CB-NNLL result and adding it to the (larger) CB-N3LO result. From this range we determine m (as the central value) and symmetric errors. In this way we take into account the CB-N3LO tendency to give larger values of m while discarding the unreliable small scale region of the CB-N3LO results. The results are given in the third column of Table 2.
Finally we perform two further cross checks on our error. We determine the mass using the CB-NLL calculation and using the same scale variation as above. The central value of the results are shown in column 4 of all lie within the error band which gives further confidence in our results. The same is even true for the mass values determined by a FO approach, listed in the last column of Table 2. This could be taken as an indication that the error has been overestimated. However, the corrections in the ET approach are very large and the NLL results lie within the NNLL error band only because they have been improved using the FO results in a combined analysis. Thus we prefer to keep the larger error, anticipating relatively large NNNLO corrections in the effective theory.
To visualize the consistency of our approach, in Figure 2 we plot the extracted mass with its error for all n. The left (dark blue) bands show m PS as extracted using a FO approach with the central value indicated by a dot. The right (light green) bands show the corresponding values using a ET-NNLL and a ET-N3LO approach. The latter leads to slightly larger values and errors for m PS and is depicted by the dashed band. The two dots in this band indicate the two central values for ET-NNLL (lower) and ET-N3LO (higher) respectively. Finally, the middle (red) bands show our combined result, as given in in the third column of Table 2, with the central value again indicated by a dot.
As expected, the central value of the combined result is close to the ET result for large n. For small n, the combined result is also consistent with the FO result, at the price of having a huge error. The combined result makes use of all available information and gives the most reliable prediction for large n.
The non-perturbative contribution
One of the main reasons why non-relativistic sum rules so far have not been used to extract the charm quark mass is the common belief that non-perturbative corrections are not under control. As we will see this is actually not the case.
A first hint that the situation is in much better control than anticipated is the fact that the (central value of the) mass extracted, as indicated by the points in Figure 2 is remarkably consistent for all values of n ≥ 5. If there were large nonperturbative effects they would with all likelihood affect results with increasing n more dramatically.
In order to get a more quantitative picture, we will consider the effects of the gluon condensate [24] which gives us a handle for the leading non-perturbative correction. The corresponding contribution to the sum rule has been computed to two loops [25] and reads
δM (np,X) n = 12π 2 e 2 Q (4m 2 ) n+2 α s π G 2 a n 1 + α s π [b n − (2n + 4) δb X ](4)
where the one-loop coefficients are given by a n = − (2n + 2) 15 Γ(4 + n) Γ(4) Γ(7/2) Γ(7/2 + n) (5) and b n are the two-loop coefficients in the pole scheme, as listed in Ref. [25]. The shifts δb X take into account the change in the mass scheme. For the PS mass we
have b PS n ≡ b n − (2n + 4) δb PS = b n − (2n + 4) C F µ F m(6)
where C F = 4/3 is a colour factor.
There are two issues we have to consider: how large are the contributions due to the gluon condensate and how reliable is the prediction in Eq. (4)? The answer to both questions crucially depends on the mass scheme used. Regarding the former, the key features of Eq. (4) are that the coefficients grow like a n ∼ n 3/2 and that δM (np) n increases rapidly for decreasing mass. To assess the situation, we calculated for the case of the PS scheme the ratio of the non-perturbative contributions, as given in Eq. (4), to the experimental moment, using m PS = µ = 1.5 GeV and (α s /π)G 2 = 0.005 GeV 4 [26]. The results are shown in Table 3. As can be seen, the non-perturbative contributions are well below 10% for all n ≤ 16 indicating that they are in fact not (yet) very important. Another satisfactory feature of the PS scheme is that the series in Eq. (4) is well behaved. To show this we also list the relative importance of the two-loop corrections. Clearly, they should be small compared to the leading term, for Eq. (4) to be applicable. For the first moment, the higher order correction is 75% of the leading term which is uncomfortably large. However, for larger values of n, where the contribution starts to become somewhat more relevant, the corrections are smaller. Table 3: Importance of gluon condensate contribution to moments (first row) and relative importance of two-loop corrections to gluon condensate contribution (second row) in the PS scheme.
From the results in Table 3 we conclude that the gluon condensate contributions are under control for all values of n considered here. This seems to be in contradiction with what is commonly stated. However, we stress that the picture is completely different if either the pole mass or the MS mass is used. It is well known that there is a close interplay between vacuum condensates and mass definitions [27,26]. In the case of the pole mass, the corrections are also relatively small (mainly because m OS > m PS ) but the series in α s in Eq. (4) is completely unreliable because the two-loop corrections exceed the one-loop corrections for all n. In the case of the MS mass, the contributions are huge for large n (mainly because m MS < m PS ) and the corrections are also very large, unless extremely small scales µ 1 GeV are used. For other threshold masses, such as e.g. the RS mass [21] we checked that the main conclusions are the same as for the PS mass.
Of course one might wonder about the contribution of further suppressed condensates such as the dimension 6 operator G 3 . However, as we will see, the contribution and induced error due to the G 2 operator is so small that we can simply take this into account by increasing the error. Thus we conclude that if a mass definition adapted for quark pairs near threshold is used, the nonperturbative corrections are under control even in the charm case. We remark that this is also in agreement with a recent analysis [12] where contributions from the gluon condensate were found to be much smaller than expected.
Results
In this section we extract the PS charm quark mass and determine the various errors. The dominant error will be the error δm th due to missing higher-order corrections discussed in Section 2.2. We will consider all 3 ≤ n ≤ 16 even though from Figure 2 it is clear that values n 5 are "useless" in the sense that their error is too large. The results are summarized in Table 4.
The first column shows the central value for the mass. These entries differ slightly from the corresponding entries of Table 2 because the effect of the gluon condensate, as discussed in Section 2.3, has been included. Apart from the theoretical error, taken directly from Table 2, we include three further sources of errors.
First we consider the experimental error, δm exp . We simply vary the experimental moments in the range given in Table 1 and consider the effect on the extracted mass. As expected, the error decreases rapidly for increasing n and becomes very soon negligible.
A more relevant source of error is the uncertainty in the strong coupling. We vary 0.116 ≤ α s (M Z ) ≤ 0.120 [22]. The resulting error, δm α , is listed again in Table 4.
Finally we consider the contribution and induced error due to the gluon condensate, δm GG . As discussed in Section 2.3, this contribution is surprisingly small. To determine the error we vary (α s /π)G 2 = 0.005 ± 0.004 GeV 4 [26] and determine the corresponding change in m. We then double this error to take into account higher-order corrections to Eq. (4), higher dimensional vacuum condensates and the fact that previous determinations of (α s /π)G 2 resulted in somewhat larger values. Even so, the error does virtually not affect the final result.
The total error, listed in the last column of Table 4 is obtained by adding the various errors in quadrature. We also checked that the higher-order QED contributions have a negligible effect. In fact, they change the mass by a few MeV at most. n m δm th δm exp δm α δm GG δm 3 1508 229 11 41 2 233 4 1507 147 6 34 3 151 5 1508 103 4 29 3 107 6 1506 81 3 27 3 85 7 1505 64 3 24 4 69 8 1506 52 2 22 4 57 9 1504 44 2 20 4 49 10 1503 40 2 19 5 45 11 1503 37 2 18 5 41 12 1501 35 2 17 5 39 13 1500 33 1 16 5 36 14 1500 31 1 15 6 35 15 1500 29 1 14 6 33 16 1500 27 1 14 6 31 Table 4: Extracted charm quark mass with separate and total errors. All entries are in MeV and for the PS mass with µ F = 0.7 GeV.
The extracted mass is virtually independent of n. Thus, the only issue regarding how to combine the results of Table 4 is the determination of the final error. Given the remarkable consistency between the various results we argue it is safe to take a single moment result with a rather large n. Therefore we take as our final result m PS (0.7 GeV) = 1.50 ± 0.04 GeV
Since the determination of the dominant error, δm th is somewhat arbitrary, we think it is misleading to give more significant figures in the error.
Converting this to the MS mass we obtain m ≡ m MS (m MS ) = 1.25 GeV. The error of 40 MeV in Eq. (7) results in an error of 35 MeV for the MS mass. However, there is also an error in the conversion itself. As an indication of this error we take the size of the fourth order term in the conversion and obtain an additional error of 15 MeV. Finally, there is an error in the conversion induced by the uncertainty in α s . Varying 0.116 ≤ α s (M Z ) ≤ 0.120 in the conversion and taking into account the correlation of this with the corresponding variation in the determination of the PS mass, this results in an error of 15 MeV as well. These errors are relatively large because the coupling is large due to the small scale. Thus a reduction in the error in Eq. (7) would only partially impact on the error in the MS mass. Combining in quadrature the three errors in the conversion we obtain for the MS mass m = 1.25 ± 0.04 GeV (8) This value is in good agreement with the world average m = 1.27 +0.07 −0.11 [22] but has a larger error than recent determinations using low-n moments [5,6,9]. However, we would argue that our estimate of the theoretical error is more conservative and that this determination of the charm quark mass is in many respects complementary to the low-n sum rules.
Conclusions
The main result of this work is that the non-relativistic sum rule can be used to obtain a precise and reliable determination of the charm quark mass. The non-perturbative corrections are under control even for n ≃ 8 − 16 as long as a suitable threshold mass definition is used. The situation with respect to the large corrections in the effective theory is much improved if a combined analysis is performed, including available fixed order results.
We are aware that these statements are to a certain extent in contradiction with what would naively be expected. However, looking at the situation more carefully, they are actually not that surprising. It has been shown previously [18] that in the case of the bottom quark, the FO as well as the ET approach work much better than expected. In the charm case large moments were also found to give consistent results [12]. The value of the quark mass does not seem to be the driving force for the large corrections in the effective theory. In fact, the corrections are also large in the top case. Thus, the reduction in mass from bottom to charm does not completely alter the question regarding the applicability of perturbation theory. Given that completely different theoretical approaches give comparable results and that the size of the corrections are reasonable in a combined approach, we argue that the situation regarding non-relativistic sum rules in the charm case is similar to the bottom case. In spite of large partial NNNLO corrections to the sum rules in the bottom case [17] we expect that the total NNNLO correction is within our error estimate, implying similar cancellation between the various NNNLO contributions as for the top case. With a careful, conservative error estimate the quark mass can be determined reliably.
The by far largest contribution to the error in the present determination of the charm quark mass comes from unknown higher-order corrections. An estimate of this error is notoriously difficult and to a large extent arbitrary. It is for this reason that we deliberately refrained from pushing the error estimate to an extreme. In particular, to make our error estimate as reliable as possible, we do not take the considerably smaller errors of the CB-NNLL result, nor do we take the smallest error in Table 4.
It is clear that neither a FO nor a ET analysis alone can cover the whole range of n and only a combined analysis can make use of all available information. In this sense the present analysis can be considered as to a large extent complementary to low-n sum rules, since it is clearly dominated by a large-n approach. This approach uses a different theoretical input and the consistency of this result with other determinations provides useful information.
Figure 1 :
1Scale dependence of various theoretical predictions for M 10 evaluated with m PS (0.7 GeV) = 1.50 GeV. M exp 10 is shown as thin grey band and the range of scale variation used for δm th is indicated by the light-grey region.
Figure 2 :
2Extracted values with theoretical errors for m PS (0.7 GeV) for all moments n ≤ 16, using a FO (left, dark blue bands), an ET (right, light green bands) and a combined approach (middle, red bands).
Table 1 :
1Values of experimental moments (in [GeV] −2n ) and their errors.
Table 2 .
2These values n CB-NNLL CB-N3LO m(δm th ) CB-NLL FO O(α 3s )
Table 2 :
2Extracted mass and theoretical error in MeV, using various approaches. The central column shows the combined result with error.
AcknowledgementIt is a pleasure to thank Roman Zwicky for many useful discussions. This work is supported in part by the European Community's Marie-Curie Research Training Network under contract MRTN-CT-2006-035505 'Tools and Precision Calculations for Physics Discoveries at Colliders'.
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. I I Y Bigi, M A Shifman, N Uraltsev, arXiv:hep-ph/9703290Ann. Rev. Nucl. Part. Sci. 47I. I. Y. Bigi, M. A. Shifman and N. Uraltsev, Ann. Rev. Nucl. Part. Sci. 47, 591 (1997) [arXiv:hep-ph/9703290];
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[] |
[
"Boundary Condition of Polyelectrolyte Adsorption",
"Boundary Condition of Polyelectrolyte Adsorption"
] |
[
"Chi-Ho Cheng \nInstitute of Physics\nAcademia Sinica\nTaipeiTaiwan\n"
] |
[
"Institute of Physics\nAcademia Sinica\nTaipeiTaiwan"
] |
[] |
The modification of the boundary condition for polyelectrolyte adsorption on charged surface with short-ranged interaction is investigated under two regimes. For weakly charged Gaussian polymer in which the short-ranged attraction dominates, the boundary condition is the same as that of the neutral polymer adsorption. For highly charged polymer (compressed state) in which the electrostatic interaction dominates, the linear relationship (electrostatic boundary condition) between the surface monomer density and the surface charge density needs to be modified.
|
10.1103/physreve.73.012801
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[
"https://arxiv.org/pdf/cond-mat/0512571v1.pdf"
] | 8,329,854 |
cond-mat/0512571
|
7dbc6ec3a9f9ea66518a50ba1a84261c2c57996c
|
Boundary Condition of Polyelectrolyte Adsorption
22 Dec 2005 (Dated: February 8, 2020)
Chi-Ho Cheng
Institute of Physics
Academia Sinica
TaipeiTaiwan
Boundary Condition of Polyelectrolyte Adsorption
22 Dec 2005 (Dated: February 8, 2020)numbers: 8235Gh6125Hq8235Rs6141+e
The modification of the boundary condition for polyelectrolyte adsorption on charged surface with short-ranged interaction is investigated under two regimes. For weakly charged Gaussian polymer in which the short-ranged attraction dominates, the boundary condition is the same as that of the neutral polymer adsorption. For highly charged polymer (compressed state) in which the electrostatic interaction dominates, the linear relationship (electrostatic boundary condition) between the surface monomer density and the surface charge density needs to be modified.
I. INTRODUCTION
Polyelectrolyte adsorption on neutral (due to shortranged interaction) and charged (due to electrostatic interaction) surface is still active and important in recent years [1,2,3]. Theoretical approach on solving the continuum theory (Edwards equation and its derivatives) [4] on the adsorption problem requires a proper boundary condition.
The boundary condition for a pure short-ranged attraction was first given by de Gennes [5]. Later, the same boundary condition was adopted for problems with both short-ranged and electrostatic interaction between the polymer and the surface [6,7,8,9,10]. The treatment implicitly assumes that the short-ranged interaction between the polymer and the surface dominates over the electrostatic ones. However, it is still a question for the validity of this assumption.
On the other hand, it was recently identified that the boundary condition for a highly charged polymer adsorbed on the charged surface is governed by the electrostatic boundary condition, and it can simply be expressed in a linear form between the surface monomer density and the surface charged density in the adsorption regime (compressed state) [11,12]. With an extra perturbed short-ranged interaction, although the interaction is also dominated by the electrostatic one, it is still a puzzle whether the form of the boundary condition remains unchanged or its modification is needed.
In this paper, we are going to fill the above two gaps in the literature. We show that for a weakly charged polymer adsorption due to short-ranged attraction, an perturbed electrostatic interaction in general does not modify the boundary condition. For a highly charged polymer adsorption (compressed state) due to electrostatic interaction, an perturbed short-ranged interaction would induce a non-linear correction to the original boundary condition expressed in a linear form between the surface mononer density and the surface charge density.
II. SHORT-RANGED ATTRACTION REVISED
Before our main investigation, we first revise a Gaussian polymer adsorbed on the surface with short-ranged attraction. Suppose the short-ranged attraction between the monomers and the hard-wall surface is modelled by the δ-potential −γδ(z − b) located just above the hardwall at z = 0.
The continuum equation describing the density profile ρ(z) = ψ 2 0 (z) is determined by the Edwards equation
− a 2 6 d 2 dz 2 − βγδ(z − b) ψ 0 (z) = ε 0 ψ 0 (z)(1)
where a is the bond length, β = 1/(k B T ), and ε 0 is the ground state eigenvalue. The boundary condition imposed is ψ 0 (0) = 0 and ψ 0 (+∞) = 0. Similar to the usual eigenproblem appearing in Quantum Mechanics [13],
ψ 0 (z) = sinh(z/d 0 ), 0 ≤ z ≤ b A exp(−z/d 0 ), z ≥ b(2)
up to a normalization constant. d 0 describes the length scale of the diffusion layer of the adsorbed polymer. By fitting the boundary condition at z = b, we have
b d 0 1 + coth( b d 0 ) = 6βγb a 2(3)
The binding energy (in unit of k B T ), or the eigenvalue ε 0 = −a 2 /6d 2 0 . The idea suggested by de Gennes [5] to absorb the δpotential into the surface (by taking sufficiently small b) is to modify the boundary condition at the surface and to match with the asymptotic behavior away from the surface by identifying the same binding energy (eigenvalue). That is, we are looking at the profile
ψ 1 (z) = A exp(−z/d 1 ), 0 < z < +∞(4)
in which it is the solution of the eigenproblem − a 2 6
d 2 dz 2 ψ 1 (z) = ε 0 ψ 1 (z)(5)
with the boundary condition
1 ψ 1 dψ 1 dz z=0 + = − 1 d 1 (6) ψ 1 (+∞) = 0 (7)
which is adopted on neutral polymer adsorption. The binding energy (in unit of k B T ) ε 0 = −a 2 /6d 2 1 . Hence d 1 = d 0 . Notice that, the microscopic parameters γ and b are now replaced by the macroscopic quantity d 0 .
III. SHORT-RANGED ATTRACTION WITH PERTURBED ELECTROSTATIC INTERACTION
Suppose the weakly charged polymer can still keep its Gaussian features when an perturbed local electrostatic interaction V (z) from the charged surface is considered. In general the local potential V (z) = V 0 at z = 0, becomes linear at z > ∼ 0, and saturate to zero at large z > ∼ r s (r s is the Debye screening length). The Edwards equation is
− a 2 6 d 2 dz 2 − βγδ(z − b) + βV (z) ψ 0 (z) = ε 0 ψ 0 (z) (8)
with the boundary condition ψ 0 (0) = ψ 0 (+∞) = 0. Following the same spirit in previous section, we absorb the δ-potential into the surface such that the eigenproblem becomes − a 2 6
d 2 dz 2 + βV (z) ψ 1 (z) = ε 0 ψ 1 (z)(9)
with the boundary condition same as Eqs.(6)- (7). The binding energy ε 0 in both Eqs. (8) and (9) can be estimated by the first-order perturbation theory [13] to the solution in Eqs. (2) and (4), respectively. In Eq.(8), its corresponding eigenvalue
ε 0 = − a 2 6d 2 0 + b 0 + ∞ b dz ψ 2 0 (z)βV (z) ≃ − a 2 6d 2 0 + ∞ b dz ψ 2 0 (z)βV (z)(10)
at sufficiently small b. The eigenvalue in Eq.(9) shares the same form
ε 0 = − a 2 6d 2 1 + ∞ 0 dz ψ 2 1 (z)βV (z)(11)
except d 0 is replaced by d 1 . Hence, by identifying the same eigenvalue in both Eqs.(10)-(11), we get d 1 = d 0 .
Both the neutral and weakly charged Gaussian polymer share the same boundary condition due to short-ranged attractive surface. Notice that the discussion of the boundary condition was also made by Joanny in which the coupling of the monomer density to a further electrostatic equation of Poisson-Boltzmann type is considered. The effective d 1 would then be different from d 0 [7]. In order to investigate the validity of the Gaussian feature, we choose the local potential of the Debye-Hückel form V (z) = V 0 exp(−z/r s ), where V 0 = 4πl B τ σr s , with the Bjerrum length l B , line charge density of polymer τ , and surface charge density of the surface σ. Substituting this V (z) into Eq. (11),
ε 0 = − a 2 6d 2 0 + 2βV 0 r s 2r s + d 0(12)
The first term is the binding energy due to short-ranged attraction whereas the second term the electrostatic interaction. The condition for perturbed electrostatic interaction requires (13) is a necessary condition to identify whether the electrostatic interaction is still perturbatively small. If the surface charge density becomes strong such that |V 0 | no longer satisfy Eq.(13), the Gaussian polymer undergoes conformational changes. The corresponding boundary condition would deviate Eq.(6) very much.
a 2 6d 2 0 ≫ 2β|V 0 |r s 2r s + d 0 (13) where it becomes |V 0 | ≪ k B T a 2 /6d 2 0 for low ionic strength r s ≫ d 1 . For high ionic strength r s ≪ d 0 , it requires |V 0 | ≪ k B T a 2 /12r s d 0 . Eq.
IV. ELECTROSTATIC BOUNDARY CONDITION WITH PERTURBED SHORT-RANGED INTERACTION
In another regime that the polymer is highly charged such that the adsorbed polymer is in a compressed state on the substrate, the boundary condition is determined by the electrostatic boundary condition across the dielectric [11,12]. The continuum theory is described also by the Edwards equation
− a 2 2 d 2 dz 2 + βV (z) ψ 0 (z) = ε 0 ψ 0 (z)(14)
where the coefficient of the entropic term is −a 2 /2 instead of −a 2 /6 [12]. The boundary condition imposed is ψ 0 (0) = C 0 and ψ 0 (+∞) = 0. C 0 = 0 because the electrostatic boundary condition for a compressed adsorbed polyelectrolyte needs to be satisfied [11],
C 2 0 = − 2K ǫ ′ /ǫ − 1 σ + ǫ ′ /ǫ + 1 2 σ p(15)
where ǫ and ǫ ′ are the dielectric constant of the medium and the substrate, respectively. σ is the surface charge density just above the substrate. σ p is the polarization surface charge density induced by the polymer only. It depends on ǫ ′ /ǫ but not on σ. K is the proportional constant depending only on ǫ ′ /ǫ. Both K and σ p are model dependent; in other words, they depend on the microscopic details of the system. Similar to the diffusive layer thickness d appearing in the previous sections, the microscopic details are absorbed into these two marcroscopic quantities K and σ p . In the following, with the perturbed short-ranged interaction (attractive or repulsive) modelled by a δ-potential located just above the substrate, we are going to investigate how this perturbed term is adsorbed into the boundary condition. That is, we consider the Edwards equation
− a 2 2 d 2 dz 2 + βV (z) ψ 1 (z) = ε 1 ψ 1 (z)(16)
with the boundary condition ψ 1 (0) = C 1 and ψ 1 (+∞) = 0. Notice that ε 0 in Eq.(14) (without δ-potential) is not equal to ε 1 in Eq.(16) (with δ-potential). In fact, the binding energy ε 1 can be related to ε 0 by perturbation theory [13] up to first order, in which
ε 1 = ε 0 + ∞ 0 dzψ 2 0 (z)(−βγδ(z − b)) = ε 0 − βγψ 2 0 (b) → ε 0 − βγC 2 0 (17)
for sufficiently small b. The change of the surface mononer density due to the perturbed interaction can be further estimated by applying the WKB approximation [13]. Near the surface, we have
ψ 0 (z) = A (ε 0 − V (z)) 1/4 sin √ 2 a z 0 dz ε 0 − V (z) + α ≃ A (ε 0 − V 0 ) 1/4 sin( 2(ε 0 − V 0 ) a z + α)(18)
where α = 0 related to
C 0 = A (ε 0 − V 0 ) 1/4 sin α(19)
Notice that, in the usual case of Quantum Mechanics [13], because of the hard-wall boundary condition C 0 = 0, α is set to be zero. Similarly, we can also write
ψ 1 (z) ≃ A (ε 1 − V 0 ) 1/4 sin( 2(ε 1 − V 0 ) a z + α) (20)
where the coefficients A and α are assumed unchanged. Hence
C 1 = A (ε 1 − V 0 ) 1/4 sin α(21)
From Eqs. (19) and (21), we got the relation (ε 0 −V 0 )C 4 0 = (ε 1 − V 0 )C 4 1 , and hence
C 1 ≃ C 0 − C 0 4(ε 0 − V 0 ) (ε 1 − ε 0 ) = C 0 + βγ 4(ε 0 − V 0 ) C 3 0(22)
by applying Eq.(17). Remind that ε 0 −V 0 > 0. Eq.(22) is consistent with our picture that short-ranged attraction (repulsion), γ > 0 (< 0), increases (decreases) the surface mononer density. The next higher order correction for C 1 is O(C 3 0 ) [14]. The linear relation between the surface mononer density and the surface charge density is no longer valid after including the short-ranged interaction effect. However, the violation of the linear relation implies that part of the surface mononer density is not due to the electrostatic interaction in which the electrostatic boundary condition does not apply [15].
ACKNOWLEDGEMENTSThe author would like to thank P.Y. Lai and X. Wu for helpful comments. The work was supported by the Postdoctoral Fellowship of Academia Sinica.
. J L Barrat, J F Joanny, Adv. Chem. Phys. XCIV. 1and references thereinJ.L. Barrat and J.F. Joanny, Adv. Chem. Phys. XCIV, 1 (1996) and references therein.
. A Y Grosberg, T T Nguyen, B I Shklovskii, Rev. Mod. Phys. 74and references thereinA.Y. Grosberg, T.T. Nguyen, and B.I. Shklovskii, Rev. Mod. Phys. 74, 329 (2002) and references therein.
. R R Netz, D Andelman, Phys. Rep. 380and references thererinR.R. Netz and D. Andelman, Phys. Rep. 380, 1 (2003) and references thererin.
M Doi, S F Edwards, The Theory of Polymer Dynamics. New YorkOxford UniversityM. Doi and S.F. Edwards, The Theory of Polymer Dy- namics (Oxford University, New York, 1986).
. P G De Gennes, Rep. Prog. Phys. 32187P.G. de Gennes, Rep. Prog. Phys. 32, 187 (1969).
. X Châtellier, J F Joanny, J. Phys. II. 61669X. Châtellier and J.F. Joanny, J. Phys. II 6, 1669 (1996).
. J F Joanny, Eur. Phys. J. B. 9117J.F. Joanny, Eur. Phys. J. B 9, 117 (1999).
. A Shafir, D Andelman, Phys. Rev. E. 7061804A. Shafir and D. Andelman, Phys. Rev. E 70, 061804 (2004).
. P M Biesheuvel, Eur. Phys. J. E. 16353P.M. Biesheuvel, Eur. Phys. J. E 16, 353 (2005).
. Q Wang, Macromolecules. 388911Q. Wang, Macromolecules 38, 8911 (2005).
. C H Cheng, P Y Lai, Phys. Rev. E. 7061805C.H. Cheng and P.Y. Lai, Phys. Rev. E 70, 061805 (2004).
. C H Cheng, P Y Lai, Phys. Rev. E. 7160802C.H. Cheng and P.Y. Lai, Phys. Rev. E 71, 060802(R) (2005).
L D Landau, E M Lifshitz, Quantum Mechanics: Non-relativistic Theory. Pergamon PressL.D. Landau and E.M. Lifshitz, Quantum Mechanics: Non-relativistic Theory (Pergamon Press, 1977).
The author have tried several different "variational wavefunctions" ψ(z) to determine the next higher order correction, and it was found that the orders are different among those different variational wavefunctions. It seems that the order depends on the details of the potential V (z). However, the main point is that the correction should be at higher order, and hence the simple linear form of the boundary condition no longer holds after including the short-ranged interactionThe author have tried several different "variational wave- functions" ψ(z) to determine the next higher order cor- rection, and it was found that the orders are different among those different variational wavefunctions. It seems that the order depends on the details of the potential V (z). However, the main point is that the correction should be at higher order, and hence the simple linear form of the boundary condition no longer holds after in- cluding the short-ranged interaction.
Another way to understand it is as follows. The part of the adsorbed monomers due to short-ranged interaction −γδ(z −b) is adsorbed at z = b, and it is counted into the surface monomer density after taking sufficiently small b. Another way to understand it is as follows. The part of the adsorbed monomers due to short-ranged interaction −γδ(z −b) is adsorbed at z = b, and it is counted into the surface monomer density after taking sufficiently small b.
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[
"arXiv:q-bio/0507044v1 [q-bio.MN] 28 Jul 2005 , A core genetic module : the Mixed Feedback Loop",
"arXiv:q-bio/0507044v1 [q-bio.MN] 28 Jul 2005 , A core genetic module : the Mixed Feedback Loop"
] |
[
"Paul François \nLaboratoire de Physique Statistique *\nUMR 8550\nCNRS\nEcole Normale Supérieure\n24, rue Lhomond75231ParisFrance\n",
"Vincent Hakim \nLaboratoire de Physique Statistique *\nUMR 8550\nCNRS\nEcole Normale Supérieure\n24, rue Lhomond75231ParisFrance\n"
] |
[
"Laboratoire de Physique Statistique *\nUMR 8550\nCNRS\nEcole Normale Supérieure\n24, rue Lhomond75231ParisFrance",
"Laboratoire de Physique Statistique *\nUMR 8550\nCNRS\nEcole Normale Supérieure\n24, rue Lhomond75231ParisFrance"
] |
[] |
The so-called Mixed Feedback Loop (MFL) is a small two-gene network where protein A regulates the transcription of protein B and the two proteins form a heterodimer. It has been found to be statistically over-represented in statistical analyses of gene and protein interaction databases and to lie at the core of several computer-generated genetic networks. Here, we propose and mathematically study a model of the MFL and show that, by itself, it can serve both as a bistable switch and as a clock (an oscillator) depending on kinetic parameters. The MFL phase diagram as well as a detailed description of the nonlinear oscillation regime are presented and some biological examples are discussed. The results emphasize the role of protein interactions in the function of genetic modules and the usefulness of modelling RNA dynamics explicitly.
|
10.1103/physreve.72.031908
|
[
"https://arxiv.org/pdf/q-bio/0507044v1.pdf"
] | 17,611,493 |
q-bio/0507044
|
a3f8903887dc852862e889401d100d91562d4a90
|
arXiv:q-bio/0507044v1 [q-bio.MN] 28 Jul 2005 , A core genetic module : the Mixed Feedback Loop
(Dated: February 9, 2008)
Paul François
Laboratoire de Physique Statistique *
UMR 8550
CNRS
Ecole Normale Supérieure
24, rue Lhomond75231ParisFrance
Vincent Hakim
Laboratoire de Physique Statistique *
UMR 8550
CNRS
Ecole Normale Supérieure
24, rue Lhomond75231ParisFrance
arXiv:q-bio/0507044v1 [q-bio.MN] 28 Jul 2005 , A core genetic module : the Mixed Feedback Loop
(Dated: February 9, 2008)numbers: 8717Aa8716Yc8240Bj
The so-called Mixed Feedback Loop (MFL) is a small two-gene network where protein A regulates the transcription of protein B and the two proteins form a heterodimer. It has been found to be statistically over-represented in statistical analyses of gene and protein interaction databases and to lie at the core of several computer-generated genetic networks. Here, we propose and mathematically study a model of the MFL and show that, by itself, it can serve both as a bistable switch and as a clock (an oscillator) depending on kinetic parameters. The MFL phase diagram as well as a detailed description of the nonlinear oscillation regime are presented and some biological examples are discussed. The results emphasize the role of protein interactions in the function of genetic modules and the usefulness of modelling RNA dynamics explicitly.
I. INTRODUCTION
Biological cells rely on complex networks of biochemical interactions. Large scale statistical analyses have revealed that the transcriptional regulation networks and the networks of protein-protein interaction for different organisms are far from random and contain significantly recurring patterns [1] or motifs. Mathematical modelling is useful to determine if these motifs can by themselves fulfill useful functions and it has, for instance, helped to show that the common 'Feed Forward Loop' motif in transcriptional regulation [1] can act as a persistence detector [2]. More recently, a combined analysis of protein-protein interactions and transcriptional networks in yeast (Saccharomyces cerevisiae) has pointed out several motifs of mixed interactions [3,4]. The simplest such motif composed of both transcriptional and proteinprotein interactions is the two-protein Mixed Feedback Loop (MFL) depicted in Fig. 1. It is composed of a transcription factor A, produced from gene g a , and of another protein B , produced from gene g b . A regulates the transcription of gene g b and also directly interacts with protein B. The MFL has independently been obtained as the core motif of several networks produced in a computer evolutionary search aiming at designing small functional genetic modules performing specific functions [5]. Here, to better understand the possible functions of this basic module, we propose a model of the MFL based on the simplest biochemical interactions. This mathematical model is described in section II. This is used to show that there exists wide ranges of kinetic parameters where the MFL behaves either as a bistable switch or as an oscillator. For the convenience of the reader, an overview of the different dynamical regimes is provided * LPS is laboratoire associé aux universités Paris VI and VII.
in section III. They are delimited in parameter space and their main characteristics are summarized. We then give detailed descriptions of the bistable regime in section IV and of the nonlinear oscillations in section V. A comparison is also made with a simple auto-inhibitory gene model with delay. In the concluding section, the important role played by protein dimerization and the usefulness of explicitly modelling mRNA dynamics are underlined and biological examples of the two proposed functions of the MFL are discussed.
II. A MATHEMATICAL MODEL OF THE MFL.
A. Model formulation As previously described, the MFL consists of two proteins A and B and their genes g a and g b , such that A regulates the transcription of gene g b and also directly interacts with B. Our aim is to analyze the dynamics of this small genetic module and see what can be achieved in the simplest setting. Therefore, different cellular compartments and separate concentrations for the nucleus and cytoplasm are not considered and biochemical reactions are modelled by simple rate equations. The proposed MFL model is represented schematically in Fig. 2 and consists of four equations that are described and explained below. The concentration of a chemical species X is denoted by square brackets and the cell volume is taken as volume unit. So, [X] represents the effective number of X molecules present in the cell.
The first two equations model the transcriptional regulation of gene g b by protein A:
d[g b ] dt = θ[g b : A] − α[g b ][A](1)d[r b ] dt = ρ f [g b ] + ρ b [g b : A] − δ r [r b ](2)
where it is assumed that gene g b exists under two forms, with A bound to its promoter with probability [g b : A] and without A with probability [g b ]. Since [g b ] + [g b : A] = 1, the single Eq. (1) is sufficient to describe the transition between the two forms. Specifically, A proteins bind to the g b promoter at a rate α and when bound they are released at a rate θ. The regulation of transcription of gene g b by protein A is described by Eq. (2) where r b stands for g b transcripts. When A is bound to g b promoter, transcription is initiated at a rate ρ b and otherwise it is initiated at a rate ρ f . Thus, ρ b > ρ f corresponds to transcriptional activation by A and ρ b < ρ f to transcriptional repression. A first order degradation for g b mRNA at a constant rate δ r has also been assumed. As given, the description is strictly valid for a single copy gene. However, it also applies for a gene with g 0 copy (e.g g 0 = 2) provided that ρ f and ρ b are understood to be g 0 -fold greater than the corresponding elementary rates. The production of A and B proteins and their complexation are described by the following two equations that complete our description of the MFL module,
d[A] dt = ρ A − γ[B] [A] − δ A [A] +θ[g b : A] − α[g b ][A] (3) d[B] dt = β[r b ] − γ[B] [A] − δ B [B](4)
where [A] and [B] respectively denote the concentration of proteins A and B. Since, regulation of gene g a is not considered, separate descriptions of its transcription and translation are not needed and it is simply assumed in Eq. (3) that protein A is produced at a given basal rate ρ A . The second crucial interaction of the MFL, the direct interaction between protein A and B is taken into account by assuming that A and B associate at a rate γ. It is assumed that B does not interact with a protein A that is bound to g b promoter. In addition Eq. (3) assumes a first-order degradation of protein A at a constant rate δ A , and its last two terms come from the (small) contribution to the concentration of A in solution of the binding (unbinding) of A to (from) g b promoter. Eq. (4) assumes that B proteins are produced from the transcripts of gene g b at a rate β and are degraded at a rate δ B in addition to their association with A proteins. As described by Eq. (3,4), the complexation between A and B proteins proceeds at a rate γ. For simplicity, we suppose that the AB complex does not interact with genes g a and g b , we neglect its dissociation [6] and simply assume that it is fully degraded at a rate δ AB after its formation,
d[AB] dt =γ[B] [A] − δ AB [AB](5)
Since the complex AB does not feed back on the dynamics of the other species, its concentration does not need to be monitored and Eq. (5) is not explicitly considered in the following.
B. Values of kinetic parameters and a small dimensionless parameter
Even in this simple model, ten kinetic constants should be specified. It is useful to consider the possible range of their values both to assess the biological relevance of the different dynamical regimes and to orient the model analysis.
Half-lives of mRNA range from a few minutes to several hours and are peaked around 20 minutes in yeast [7]. δ r = 0.03min −1 can therefore be taken as a typical value.
For the transcription factor-gene promoter interaction, typical values appear to be a critical concentration θ/α = [A] 0 in the nanomolar range, a bound state lifetime of several minutes and activated transcription rates of a few mRNAs per minute. We therefore assume α = θ/40, that θ is of the same order as δ r and ρ f ∼ min −1 , ρ b ∼ min −1 .
Proteins half-lives vary from a few minutes to several days [8]. The hour appears as a typical value. We choose δ A = δ B = 0.01min −1 and more generally consider that the A and B protein half-lives are of the same order or longer than that of g b mRNA. For protein production, β = 3 protein molecules per mRNA molecule per minute appears a plausible value for an eucaryotic cell [9]. This gives ρ A ≃ 100min −1 when combined with the previous values for mRNA production.
We assume that protein interaction is essentially limited by diffusion. A diffusion constant D ≃ 2.5µm 2 s −1 .
[10] gives a time s 2 /D of about a minute for diffusion across a cell of a size s = 10 µm. We therefore choose γ ≃ 1 min −1 .
It is convenient to adimension Eqs. (1,3) to decrease as far as possible the number of independent parameters.
We first normalize the g b mRNA concentration by the concentration that gives a production of B protein equal to that of A. We thus define the dimensionless concentration r = β[r b ]/ρ A . We normalize the protein concentrations by the equilibrium concentration resulting from production at a rate ρ A and dimerization at a rate γ and
define A = γ/ρ A [A], B = γ/ρ A [B]
. We also take as time unit the inverse of g b mRNA degradation rate 1/δ r . With theses substitutions, Eqs. (1-4) read,
dg dt =θ (1 − g) − g A A 0 (6) dr dt = ρ 0 g + ρ 1 (1 − g) − r (7) dB dt = 1 δ (r − A B) − d b B (8) dA dt = 1 δ (1 − A B) + µθ (1 − g) − g A A 0 − d a A (9)
where we have defined the following rescaled parame-
ters δ = δ r / √ ρ A γ , ρ 1 = βρ b /(ρ A δ r ), ρ 0 = βρ f /(ρ A δ r ) θ = θ/δ r , µ = γ/ρ A , d a = δ A /δ r and d b = δ B /δ r .
We have also introduced the dimensionless critical binding concentration A 0 = γ/ρ A θ/α. The model still depends on seven parameters. In order to simplify its analysis, it is useful to note that δ is a small parameter (approximately equal to 3 × 10 −3 with the previous estimations). The influence of three key parameters of order one in Eq. (6-9) is particularly examined in the following. These are ρ 0 and ρ 1 which measure the strengths of the two possible states of B production (with or without A bound to gene g b ) as compared to that of A, andθ which compares the rates of A unbinding from DNA to that of mRNA degradation (α is supposed to vary with θ to maintain a fixed critical binding concentration A 0 ).
III. OVERVIEW OF THE DYNAMICS IN
DIFFERENT PARAMETER REGIMES.
We provide here an overview of the different dynamical regimes of the MFL which are depicted in Fig. 3 and summarize their characteristics. The behavior of the MFL depends on whether protein A is a transcriptional activator or a transcriptional repressor and on the strengths of the two transcription rates of gene g b (i.e. with A bound or not to its promoter). More precisely, the key parameters are the strengths of B protein production, βρ f /δ r and βρ b /δ r , as compared to the production rate ρ A of protein A. It is therefore convenient to consider the previously introduced ratios ρ 0 = βρ f /(δ r ρ A ) and
ρ 1 = βρ b /(δ r ρ A ).
We have observed that depending on the values of ρ 0 and ρ 1 the MFL can be monostable, exhibit bistability or display oscillations. We qualitatively describe these three different cases in the following. Their biological relevance is further discussed in section VI.
A. Monostable steady states
The simplest case occurs when the production rate of A is higher or lower than the production rate of B, irrespective of the state of g b promoter. That is when both βρ f /δ r and βρ b /δ r are either higher or lower than ρ A . In this case, the MFL has a single stable state to which it relaxes starting from any initial conditions.
When both B production rates are higher than the production rate of A (i.e. ρ 0 > 1 and ρ 1 > 1), A proteins are quickly complexed by B and are unable to interact with g b promoter. The concentration [A] of uncomplexed A proteins is therefore low and results from a simple balance between production and complexation. The high concentration of uncomplexed B proteins is the effective result from transcription at the free g b promoter rate, complexation and degradation,
[B] ≃ 1 δ B β ρ f δ r − ρ A , [A] ≃ ρ A γ[B](10)
An equally simple but opposite result holds when both B production rates are smaller than the production rate of A (ρ 0 < 1 and ρ 1 < 1). Then, the concentration of uncomplexed A is high , g b promoter is occupied by A and a low B concentration results from a balance between complexation and degradation
[A] ≃ 1 δ A ρ A − β ρ b δ r , [B] ≃ βρ b δ r [A]γ(11)
The dynamics of the MFL is richer when the production rate of A is intermediate between the two possible production rates of B, βρ f /δ r and βρ b /δ r i.e. when either ρ 0 > 1 > ρ 1 or ρ 1 > 1 > ρ 0 . We consider these two cases in turn.
B. Transcriptional repression and bistability.
We first discuss the case when A is a transcriptional repressor, ρ 0 > 1 > ρ 1 . Then, two stable steady states can coexist. Let us suppose first that no A is bound to g b promoter. Then the production rate of B is larger than the production rate of A, and all produced A proteins are quickly complexed. This stably prevents the binding of A proteins to g b promoter and maintain a steady state with low A and high B concentrations approximately equal to
[B] 1 ≃ β ρ f δ r − ρ A 1 δ B , [A] 1 ≃ ρ A γ[B] 1 ,(12)
The second opposite possibility is that A is sufficiently abundant to repress the transcription of gene g b . Then, since the production rate of A has been supposed to be higher than the production rate of B in the repressed state, B proteins are quickly complexed but uncomplexed A proteins are present to maintain the repression of gene g b transcription. This gives rise to a second stable state with high A and low B concentrations approximately equal to
[A] 2 ≃ ρ A − β ρ b δ r 1 δ A , [B] 2 ≃ βρ b γ[A] 2 δ r(13)
The bistability domain is only exactly given by the simple inequalities ρ 0 > 1 and ρ 1 < 1 when the ratio δ of g b mRNA degradation rate over the effective protein dynamics rate is vanishingly small. As shown in section IV and on Fig. 3, for small value of δ, it is more accurately given by
ρ 0 > 1 + 2 (1 − ρ 1 ) δ B /A 0 , ρ 1 ≤ 1, ρ 1 < 1 − 2 (ρ 0 − 1) A 0 δ A , ρ 0 ≥ 1.(14)
In this intermediate range of production of A proteins, the network reaches one or the other fixed points, depending on its initial condition. The existence of this bistability domain can serve to convert a graded increase in A production in a much more abrupt switch-like response in A (and B) concentration as shown in Fig. 4. The usefulness of this general feature of multistability has been recently discussed in different contexts [11,12].
C. Transcriptional activation and oscillations.
When A is a transcriptional activator, the complexation of B with A acts as a negative feedback and can serve to diminish the variation in B protein concentration when A varies. This leads also to oscillations when A production lies in the intermediate range ρ 1 > 1 > ρ 0 . This oscillatory behavior mainly depends on the ratios of protein production ρ 0 , ρ 1 and exists for a large range of DNA-protein interaction kinetics . However, a faster kinetics leads to a smaller oscillatory domain as shown in Fig. 3. Oscillations cannot be sustained whenθ becomes large and comparable to 1/δ (with A 0 fixed), or equivalently when θ ∼ δ r /δ. It can also be noted that for a sufficiently large activation of transcription by A, there exists a domain of coexistence between oscillations and steady behavior as shown in Fig. 5, i.e. depending on initial conditions concentrations will oscillate in time or reach steady levels.
For small δ, oscillations are nonlinear for most parameters. As can be seen in Fig. 6, an oscillation cycle comprises two phases in succession, a first phase of duration T 1 when the concentration of protein A is high followed by a second phase of duration T 2 where the concentration of B is high. A full oscillation cycle of period T = T 1 + T 2 proceeds as follows. Let us start with low concentrations of A and B proteins at the beginning of the first phase. When no A is bound to g b promoter, B production rate is lower than A production rate and complexation cannot prevent the increase of A concentration. When the concentration of A has reached a critical level, A start to bind to g b promoter and activate transcription. This results in a higher production of B than A and the diminution of free A by complexation. Since A concentration is high, the produced B's are quickly complexed and the concentration of uncomplexed B's remain low. Eventually, A concentration drops below the binding level and no longer activates g b transcription. This marks the transition between the two parts of the oscillation cycle. B continues for a while to be produced from the transcripts and since few A's are present this now leads to a rise of the concentration of free B's. Finally, the concentrations of B transcripts and B proteins drop and phase II of the oscillation cycle terminates. A new cycle begins with low concentrations of A and B proteins.
One remarkable feature of the nonlinear oscillations coming from the smallness of the parameter δ, is that the concentrations of A and B proteins in their respective high phase depend weakly on the complex association rate γ, as long as it is not too small for the oscillation to exist, and that the period of the oscillations is strikingly insensitive to the exact value of γ. For the parameters of Fig. 6, the oscillation period only changes by about 1% when γ is reduced from 1min −1 to 2 × 10 −2 min −1 . This remains true even if the formation of the complex AB is not irreversible as supposed in the present model. We have checked that the case when the complex AB forms with an association rate γ a , dissociates with a rate γ d and is degraded with a rate δ AB is well described by the present model when one takes the effective rate γ = γ a δ AB /(γ d + δ AB ) for the irreversible formation of the complex (as obtained from a quasi-equilibrium assumption).
In the two following sections, we provide a more detailed analysis of the MFL different dynamical regimes and derivations of the results here summarized.
IV. THE SWITCH REGIME.
We first determine the possible steady states for the MFL. The free gene, mRNA and B protein concentrations are given in a steady state as a function of A concentration by,
g = A 0 A + A 0 (15) r = ρ 1 A + ρ 0 A 0 A + A 0 (16) B = ρ 1 A + ρ 0 A 0 (A + A 0 )(A + δd b )(17)
The concentration of A itself satisfies the following equation
1 = δd a A + (ρ 1 A + ρ 0 A 0 )A (A + A 0 )(A + δd b )(18)
For δ = 0, the right-hand side (r. h. s. ) goes monotonically from ρ 0 at small A to ρ 1 at large A and a solution,
A 2 = A 0 (ρ 0 − 1)/(1 − ρ 1 ) exists only when ρ 0 < 1 < ρ 1 or ρ 1 < 1 < ρ 0 .
For small δ, two other steady states are possible. A steady state with a small concentration of A,
A 1 ≃ δd b /(ρ 0 −1)
, exists when ρ 0 > 1. Inversely a steady state with a large concentration of A, A 3 ≃ (1−ρ 1 )/(δd a ) exists when ρ 1 < 1. Therefore, Eq. (18) has multiple (i.e. three) fixed points only when A is a transcriptional repressor in the region ρ 1 < 1 < ρ 0 . This is the parameter regime which we examine in the rest of this section.
A. The high A state
We show that the state with a high concentration of A proteins is stable. The form of Eqs. (8) and (9) suggests to simplify the analysis by using the fact that protein quickly reach a quasi-equilibrium state. However, both proteins cannot be in quasi-equilibrium at the same time. For instance, when proteins A ≫ B, only B is in quasiequilibrium and A scale as 1/δ. When A concentration is high, in the limit of small δ, we set a = δA. Substitution in Eq. (1,4) shows that g and B reach on a fast time-scale their quasi-equilibrium concentration
g ≃ δ A 0 /a (19) B ≃ δ r/a(20)
Therefore the dynamics of the MFL reduces to the following two equations in the limit δ → 0 :
dr dt = ρ 1 − r (21) da dt = 1 − r − d a a(22)
Eqs. (21,22) clearly show that the high A fixed point is stable and as found above satisfies r ≃ ρ 1 , B ≃ g ≃ 0, a = (1 − ρ 1 )/d a at the lowest order. This steady state is possible only if ρ 1 < 1, that is when B production rate is not high enough to titrate A and to prevent the repression by A.
B. The high B state.
The high B state can be analyzed in a very similar way in the limit of small δ. When B concentration is high, we define b = δB. In that case, A quickly reaches its quasi-equilibrium concentration:
A ≃ δ/b(23)
and the MFL dynamics reduces to the following three equations:
dg dt =θ(1 − g)(24)dr dt = ρ 0 g + ρ 1 (1 − g) − r (25) db dt = r − 1 − d b b(26)
The concentrations tend toward those of the high B fixed point
g ≃ 1, r ≃ ρ 0 , b ≃ (ρ 0 − 1)/d b A ≃ 0.
This steady state exists only if ρ 0 > 1, when the production of B proteins is high enough to titrate A proteins and to prevent transcriptional repression by A.
C. Bifurcation between the monostable and bistable regimes.
The previous analysis has shown that bistability exists in the domain ρ 0 > 1 > ρ 1 when δ ≪ 1. We analyze more precisely the nature and position of the bifurcations between the monostable and bistable regimes.
We consider first the transition between the bistable region and the monostable region with a high concentration of A (bottom left corner in Fig. 3). It follows from Eq. (18) that the low (A 1 ) and middle (A 2 ) A concentration fixed points coalesce and disappear when ρ 0 − 1 = O( √ δ) and both A 1 and A 2 are of order √ δ.
For A ∼ √ δ Eq. (18) becomes at dominant order after expansion,
ρ 0 − 1 = (1 − ρ 1 ) A A 0 + δ d b A(27)
For a given repression strength ρ 1 < 1, the r. h. s. has a minimum for
A = δd b A 0 /(1 − ρ 1 ) (that is of order √ δ as assumed) such that ρ 0 = 1 + 2 δd b (1 − ρ 1 ) A 0 + O(δ)(28)
Above this value of ρ 0 , Eq. (27) has two solutions and below it has none. Eq. (28) therefore marks the saddlenode bifurcation line that separates the bistable domain from the monostable domain with a high A concentration fixed point. For δ → 0, the zeroth order boundary ρ 0 = 1 is of course retrieved. The approximate expression (28) agrees well with an exact numerical determination of the bifurcation line as shown in Figure 3.
The transition between the bistable regime and the low A concentration monostable regime (top-right corner of Fig. 3) can be similarly analyzed. The high (A 3 ) and middle (A 2 ) A concentration fixed points can coalesce when for both the concentration A is of order
1/ √ δ. Ex- pansion of Eq. (18) for A ∼ 1/ √ δ gives 1 − ρ 1 = (ρ 0 − 1) A 0 A + δd A A(29)
For a given ρ 0 > 1, the r. h. s. of Eq. (29) is minimum for A = A 0 (ρ 0 − 1)/(δd a ) and at this minimum ρ 1 is equal to
ρ 1 = 1 − 2 d a A 0 δ(ρ 0 − 1)(30)
For ρ 1 smaller than this value, Eq. (29) has two roots corresponding to A 2 and A 3 . The two roots merge and disappear on the saddle-node bifurcation line (30) that marks the boundary between the bistable domain and the low A concentration monostable regime. Comparison between the approximate expression (30) and an exact numerical determination of the bifurcation line is shown in Figure 3.
V. THE OSCILLATOR REGIME
We consider now the case when A is a transcriptional activator, that is when ρ 0 < ρ 1 . In this case, the MFL has a single steady state since the right-hand side (r.h.s.) of Eq. (18) is a monotonically increasing function. However, as we show below, this steady state is unstable in a large domain of parameters and the MFL behaves as an oscillator.
A. The linear oscillatory instability
We begin by analyzing the linear stability of the fixed point (g * , r * , B * , A * ) where A * is the single solution of Eq. (18) and g * , r * and B * are given as functions of A * by Eqs. (15)(16)(17). After linearization of the MFL dynamics [Eqs. (6)(7)(8)(9) around this fixed point, the complex growth rates σ of perturbation growing (or decreasing) exponentially in time like exp(σt) are found to be the roots of the following characteristic polynomial,
σ +θ(1 + A * A 0 ) [σ + 1] σ + d a + B * δ σ + d b + A * δ − A * B * δ 2 = g * θ A 0 A * δ 2 (ρ 0 − ρ 1 ) − µσ (σ + 1) σ + d b + A * δ (31)
Again, the fact that δ is small simplifies the analysis. We consider first the case when the rescaled concentrations g * , r * , B * , A * are of order one. This corresponds to the fixed point A 2 of the previous part in the parameter
regime ρ 0 < 1 < ρ 1 , with A * ≃ A 0 (1 − ρ 0 )/(ρ 1 − 1), B * ≃ 1/A * and g * ≃ (ρ 1 − 1)/(ρ 1 − ρ 0 )
. Let us assume that the roots σ diverge as δ tends to zero. Then, Eq. (31) reduces at dominant order to
σ 3 (δσ + A * + B * ) = A * A 0θ g * δ (ρ 0 − ρ 1 )(32)
Therefore, three roots σ k , k = 0, 1, 2, are of order δ −1/3 and proportional to the three cubic roots j k of unity
σ k = −j k A * A * + B * g * A 0θ δ (ρ 1 − ρ 0 ) 1/3
, k = 0, 1, 2.
(33) The fourth root σ 3 is of order 1/δ and corresponds to a stable mode of real part −(A * + B * )/δ. The two roots σ 1 and σ 2 are complex conjugate and have a positive real part. Thus, in this parameter domain, the MFL fixed point is oscillatory instable. The dynamics tends toward an attractive limit cycle that we will analyze in the next subsection. When A * or B * grows (i.e when ρ 1 → 1 or ρ 0 → 1) the fixed point instability disappears via a Hopf bifurcation. We analyze these two boundaries in turn.
1.
A * high When ρ 1 → 1, A * becomes large and the real parts of the roots become of order one. Thus, Eq. (31) needs to be approximated in a different way. We neglect δ(σ + d b ) and B * as compared to A * in Eq. (31) and obtain (σ +θ
ρ 1 − ρ 0 ρ 1 − 1 )(σ + 1)(σ + d a ) = −θ (ρ 1 − 1) δA 0(34)
where we have replaced A * and g * by their expressions at the A 2 fixed point. When the r.h.s is the dominant constant in Eq. (34), its three roots are proportional as above to the three roots of (minus) unity and two of them have positive real parts. On the contrary, when ρ 1 becomes sufficiently close to one, the r.h.s of Eq. (34) becomes negligible and Eq. (34) has obviously three real negative roots. Therefore, when ρ 1 → 1, one traverses the boundary of the linearly unstable region. Two roots of Eq. (34) traverses the imaginary axis on this boundary in a Hopf bifurcation. Assuming (and checking afterwards) that, on this boundary, these roots are small compared with 1/(ρ 1 − 1), their expressions is found perturbatively by expanding the first factor in Eq. (34),
σ ± = − 1 + d a 2 + (ρ 1 − 1) 3 2(ρ 1 − ρ 0 ) 2 A 0θ δ ± i 2 (ρ 1 − 1) 2 A 0 (ρ 1 − ρ 0 )δ(35)
This provides the location of the stability boundary,
ρ 1 = 1 + δθA 0 (d a + 1)(ρ 1 − ρ 0 ) 2 1/3(36)
In the limit δ → 0 the zeroth order boundary ρ 1 = 1 is of course retrieved. It can be checked that the obtained result ρ 1 − 1 ∼ δ 1/3 justifies a posteriori the use of the A 2 δindependent expression for the fixed point. This fixed point linear stability boundary (36) can also be directly obtained from the Routh-Hurwitz condition [14] on the third degree polynomial (34). Comparison of the small δ Eq. (36) with the exact linear stability boundary is provided in Fig. 3.
A * small
We now consider the upper boundary (ρ 0 ∼ 1) of the linearly unstable region. When ρ 0 tends toward one, B * grows 1/(ρ 0 −1) and becomes large compared to δ(σ+d a ) and A * . Eq. (31) then simplifies and reduces at leading order to:
(σ +θ)(σ + 1)(σ + d b ) = −E(37)
with
E = A * θ A 0 B * δ (ρ 1 − 1)(38)
Again when E is large the roots are proportional to the three roots of -1 and two have positive real parts. On the contrary, the three roots are real and negative when E vanishes. There is a critical value E c such that for E < E c the fixed point is linearly stable. This occurs when the real part of the two complex conjugate roots becomes negative at the value E c given by the Routh-Hurwitz criterion [14]
E c = (d b + 1 +θ)(θ + d b + d bθ ) − d bθ(39)
The allied value of A at the fixed point concentration is obtained from Eq. (38),
A * = E c A 0 δ θ(ρ 1 − 1)(40)
Using the fixed point Eq. (18), this in turn corresponds to the line in parameter space
ρ 0 = 1 − E c δ(ρ 1 − 1) A 0θ 1 −θ d b E c(41)
Eq. (41), of course, reduces to the zeroth order boundary ρ 0 = 1 when δ → 0. Comparison of the small-δ asymptotic expression (41) with the exact linear stability line is provided in Fig. 3.
B. A description of the nonlinear oscillations
The MFL oscillations quickly become strongly nonlinear away from threshold (or, of course, when the bifurcation is subcritical). As there is no auto regulatory direct or indirect positive feedback loops in the MFL, twovariable reductions have a negative divergence. It follows from the well-known Bendixson's criterion [13], that they cannot be used to describe the oscillatory regime. It is for instance not possible to properly describe the oscillations by only focusing on the protein dynamics. A specific analysis is therefore required and developed in this section.
A full period of the nonlinear oscillations can be split into two parts: a first phase with A >> 1, and a second phase with B >> 1 (see Fig. 7).
In the following, these two main phases of the limit cycle are described. We use simple continuity arguments to match the two phases and obtain a description of the whole limit cycle. The oscillation period is computed (at the lowest-order in δ). A full justification of this simple matching procedure and a detailed study of the transition regimes between the two main phases are provided in Appendix A using matched asymptotics techniques.
Phase I: High A, Low B phase.
This phase is defined as the fraction of the limit-cycle where the concentration of protein A is larger than [A] 0 i. e. when A >> 1 >> B. Eq. (9) leads us to assume that A is of the order 1/δ. So we define a = δA. In the limit of small δ, as A scales as 1/δ, both the binding to g promoter and the dynamics of B resulting from its complexation with A are fast compared to that of mRNA and A. Consequently, g and B are in quasi-equilibrium and obey at lowest order in δ,
g = δA 0 /a (42) B = δr/a(43)
The dynamics of the MFL therefore reduces to the following two equations at lowest order in δ,
dr dt = ρ 1 − r (44) da dt = 1 − r − d a a(45)
The beginning of phase I coincides with the end of phase II where, as explained below, A concentration is small. Continuity therefore requires that a vanishes at the start of phase I (a detailed study of the transition region between the two phases is provided in Appendix A). Denoting by r 1 , the value of r at the start of phase I, an easy integration of the linear Eq. (44,45) gives,
r I (t) = ρ 1 + (r 1 − ρ 1 )e −t (46) a I (t) = 1 − ρ 1 d a [1 − e −dat ]+ r 1 − ρ 1 d a − 1 [e −dat − e −t ](47)
Subscripts I have been added to r and a in Eq. (46) and (47) to emphasize that the corresponding expressions are valid during phase I only. Phase I ends with the fall of A concentration, after a time t 1 such that a I (t 1 ) = 0. The mRNA concentration is then equal to r I (t 1 ) = r 2 . One can note that the rise and fall of the concentration of protein A imposes restrictions on the parameters. The rise at the beginning of phase I is possible only if A production dominates over its complexation with B, that is if r 1 < 1 (Eq. (45). The following fall requires the reverse which can only happen if B production becomes sufficiently important, namely r > 1. This requires r 2 > 1 and a fortiori ρ 1 > 1 (Eq. (44)).
Phase II : High B phase.
This second phase is defined as the part of the limitcycle where the concentration of protein A falls below [A] 0 i.e. B >> 1 >> A. The form of Eq. (8) leads us to assume that B scales as 1/δ and we define b = δB.
In the limit of small δ, the dynamics of A is fast compared to that of the other species and A is in quasiequilibrium. At lowest order its concentration reads,
A ≃ δ/b(48)
Thus, the dynamics of the MFL reduces, at lowest order in δ, to the following three equations,
dg dt =θ(1 − g) (49) dr dt = ρ 0 g + ρ 1 (1 − g) − r (50) db dt = r − 1 − d b b(51)
Continuity with the previous phase I leads us to require that at the beginning of phase II b = 0, r = r 2 and g = 0 (see Appendix A for a detailed justification). With these boundary conditions, the linear Eqs. 49 -51 can readily be integrated to obtain:
g II (t) = 1 − e −θt r II (t) = ρ 0 + r 2 − ρ 0 + ρ 1 − ρ 0 θ − 1 e −t − ρ 1 − ρ 0 θ − 1 e −θt b II (t) = ρ 0 − 1 d b [1 − e −d b t ] − ρ 1 − ρ 0 θ − 1 e −d b t − e −θt θ − d b + r 2 − ρ 0 + ρ 1 − ρ 0 θ − 1 e −d b t − e −t 1 − d b(52)
Since b vanishes at the beginning of phase II, b should start by rising. This imposes that B production dominates over complexation and requires that the concentration r at the beginning of phase II is greater than 1, i. e. r 2 > 1 (see Eq. (51)). Thus, r always remains larger than ρ 0 and would decrease toward ρ 0 for a long enough phase II (Eq. (50) and (52)). At the end of phase II, b should decrease to 0 to continuously match the beginning of phase I. This requires that complexation then dominates over production of B that is r < 1 (Eq. ( 51)), and it is only possible when ρ 0 < 1. With these conditions met, phase II lasts a time t 2 and ends when b II (t 2 ) = 0.
Matching of the two phases and period determination
In order to complete the description of the limit cycle, it remains to determine the four unknowns r 1 , r 2 , t 1 and t 2 from the four conditions coming from the continuity of proteins and mRNA concentrations,
r I (t 1 ) = r 2 , a I (t 1 ) = 0, (53) r II (t 2 ) = r 1 , b II (t 2 ) = 0.(54)
One possibility to solve these equations is to use Eq. (53) to express r 1 and r 2 as a function of t 1 . It is then not difficult to see that the third equation (54) implicitly determines t 2 as a function of t 1 . Once r 1 , r 2 and t 2 are obtained as functions of t 1 , the last equation b(t 2 ) = 0 can be solved for t 1 by a one-dimensional root finding algorithm. In this way, we have determined r 1 , r 2 , t 1 , t 2 for different sets of kinetic constants and obtained the rescaled period of oscillation
T r = t 1 + t 2 ,(55)
the dimensionfull period being T = T r /δ r . Results are shown in Fig. 8. The analytical period compares well to results obtained by direct numerical integration of Eq. (6)(7)(8)(9) for different values of δ with, of course, a closer agreement for smaller δ.
Eqs. (53,54) are difficult to solve analytically but analytic expressions can be obtained for various limiting cases. For instance, if the degradation of protein B is
negligible (d b = 0), Eq. (52) simplifies to b II (t) = (ρ 0 − 1)t − ρ 1 − ρ 0 (θ − 1)θ (1 − e −θt ) + r 2 − ρ 0 + ρ 1 − ρ 0 θ − 1 (1 − e −t )(56)
If we further consider the limit of large ρ 1 , the condition b II (t 2 ) = 0, gives the following estimate for the duration of phase II, up to exponentially small terms,
t 2 ≃ r 2 − ρ 0 1 − ρ 0 + ρ 1 − ρ 0 θ(1 − ρ 0 ) ,(57)
as well as the transcript concentration at the end of phase II
r 1 ≃ ρ 0 .(58)
Similarly, the condition a I (t 1 ) = 0 together with Eq. (47) show that t 1 is small for large ρ 1 ,
t 1 ≃ 2(1 − r 1 )/ρ 1 ≃ 2(1 − ρ 0 )/ρ 1(59)
where expression (58) has been used in the second equality. Given the duration (59) of phase I, the concentration r 2 of transcript at its end is directly obtained from Eq. (46) and (58)
r 2 ≃ 2 − ρ 0 .(60)
This finally provides the estimate of the period for large
ρ 1 (with d b ≃ 0), T r ≃ t 2 ≃ 2 + ρ 1 − ρ 0 θ(1 − ρ 0 ) .(61)
A comparison between Eq. (61) and numerically determined oscillation periods is provided in Fig. 8. Popular models for genetic oscillators consist in a protein repressing its own production with a phenomenological delay [15,16,17]. Using a Hill-function to model this repression, the simplest model of this kind reads,
dB(t) dt = ρ 1 + [B(t − τ )/B 0 ] n − cB(62)
where ρ is the protein production rate, c the protein degradation rate and τ the phenomenological delay for repression. The phase diagram of this simple oscillator can be computed [18]. In the limit of long delays (cτ ≫ 1), the oscillations are nonlinear. In an expansion in the small parameter ǫ = 1/cτ , their period T is
T /τ = 2 + κ cτ + · · · (63)
where κ is a constant which is approximately equal to 2 for n = 2 and ρ/c of order one [19]. It is interesting to note some analogies between this simple model with delay and the previously studied MFL. When τ >> 1/c, the period of the delayed model scales as τ while the period of the MFL scales as δ −1 r . The mRNA half-life in the MFL thus plays the role of the phenomenological delay in Eq. (62). This is in line with the mRNA being the major pacemaker of the MFL oscillator . The rescaled parameter ǫ in the simple model with delay plays a role similar to the rescaled parameter δ in the MFL in the sense that, first, for high values of ǫ, oscillations disappear [18], and for small values of this parameter, the period of the oscillations is independent of 1/cτ at dominant order. In the delayed model, ǫ represents the ratio between the typical life-time of protein over the delay, while in the MFL, δ represents the ratio between the typical time scale of protein production and sequestration over the typical life-time of RNA. These analogies show that for the MFL √ ρ A γ plays the role of the protein degradation constant c in the delayed model. Thus protein sequestration by complexation is the MFL analog of protein degradation in the simple model with delay. The fastness of both these processes relative to a slow mechanism (delay τ , RNA dynamics) ensures the flipping of both oscillators between two states (two concentrations of B for the delayed model, A high and B high for the MFL).
VI. DISCUSSION
We have proposed a simple model of the Mixed Feedback Loop, an over-represented motif in different genomes. We have shown that by itself this motif can serve as a bistable switch or generate oscillations.
A. Importance of dimerization and of RNA dynamics
The MFL dynamics crucially depends on the posttranscriptional interaction between A and B proteins. The dynamics of dimerization is fast and allows a high concentration of only one of the two proteins, thus effectively creating a dynamical switch between the two species. If, for instance, A proteins are present at a higher concentration than B proteins, all B's are quickly titrated and there only remains free A and AB dimers.
A bistable system is obtained by coupling this posttranscriptional mechanism to a transcriptional repression. The created bistable switch is quite different from the classical 'toggle switch' [20] which is based on reciprocal transcriptional repressions between two genes. In the MFL model, the single transcriptional repression of gene g b is sufficient for a working switch. Moreover, dimerization bypasses the need for cooperativity in the toggle switch [21], and may render the present module simpler to implement in a biological system.
When the dimerization between A and B is coupled to a transcriptional activation, the system behaves as an oscillator if the production of A proteins is intermediate between the two possible for B. Then, the level of g b transcripts controls A protein dimerization and, of course, B production rates. At the beginning of the derepression phase (phase I), the low level of transcripts leads to a small production of B proteins and to a rise of A concentration. This first phase ends when the subsequent high production and accumulation of g b transcripts give rise to a production of B proteins sufficient for the titration of all free A's. During the ensuing repression phase (phase II), g b transcription rate is low and the transcripts degradation produces a continuous decrease of their concentration. The parallel decrease of B production rate ultimately leads to the end of this second phase when B production rate is no longer sufficient for the titration of all produced free A's.
In summary, the core mechanism of the clock is built by coupling the rapid switch at the protein level to the slow mRNA dynamics. The protein switch in turn controls mRNA dynamics via transcriptional activation.
It is interesting to note that the MFL model in the simple form here analyzed provides a genetic network that oscillates without the need for several specific features previously considered in the literature for related networks. As already mentioned, oscillations require neither an additional positive feedback loop [22], nor selfactivation of gene A [23] nor a highly cooperative binding [24] of transcription factors to gene promoters. Similarly, an explicit delay is not needed although delays of various origins have been considered in models of oscillatory genetic networks in different contexts, as further discussed below.
Besides the motif topology, one feature of the present MFL model that renders this possible is the explicit description of transcription and translation. The condensed representation of protein production by a single effective process that is often used, would necessitate supplementary interactions to make oscillations possible. The description of mRNA was for instance omitted in the first version of a previously proposed evolutionary algorithm [5]. Oscillatory networks with a core MFL motifs were nonetheless produced but much less easily than in a refined second version that incorporates mRNA dynamics. This role of mRNA dynamics may well be worth bearing in mind when considering the elaborate regulation of translation that is presently being uncovered [25].
B. Biological MFL oscillators and switches.
Given the proposed functions for the MFL motif, it is of primary interest to know whether cases of its use as an oscillator or a switch are already documented. The oscillator function seems the clearest. Circadian clocks are genetic oscillators which generate endogenous rhythms with a period close to 24 hours and which are locked to an exact 24 hours period by the alternation of day and night. A MFL motif is found at the core of all known eucaryotic circadian oscillators. Taking for illustrative purposes the fungus Neurospora crassa, one of the best studied model organisms, it has been established that a dimer of White-Collar proteins (WCC) plays the role of A and activates the transcription of the frequency gene, the gene g b in this example. In turn, the FRQ protein interacts with WCC and prevents this activation [26]. Analogous motifs are found in the circadian genetic networks of flies and mammals, as shown in Table I. The p53/Mdm2 module [27] provides a different example. The p53 protein is a key tumor suppressor protein and an important role is played in its regulation by the Mdm2 protein. On the one hand, p53, similarly to the A protein, binds to Mdm2 gene an activates its transcription. On the other hand, the Mdm2 protein, as the B protein, binds to p53 and both blocks transcriptional activation by p53 and promotes its rapid proteolytic degradation. Cells exposed to stress have indeed been observed to present oscillations in both p53 and Mdm2 levels [27]. In both these examples, many other genetic interactions exist besides the above described MFL motifs and have been thought to provide delays necessary for oscillations. These have been postulated to come from chains of phosphorylations in circadian networks [28] or from a yet unknown intermediate species [27] in the p53-Mdm2 system. The realization that the MFL can lead to oscillations without further interactions may be useful in suggesting alternative models of these particular genetic networks or in reassessing the role of known interactions. For instance, this has led one of us to formulate a model of the Neurospora crassa circadian clock [29] in which kinases and phosphorylation influence the cycle period by modifying proteins degradation rates but are not needed to create key delays. The prevalence of the MFL motif probably arises from the usefulness of negative feedback but it may also imply that oscillations in genetic network are more common than usually thought.
The evidence for uses of the MFL motif as a switch appears less clear-cut at present. The classic case of the E. Coli lactose operon [30] can be thought as an effective version of a bistable MFL. The lac repressor represses the lac gene and plays the role of A in the MFL. The lac gene directs the production of a membrane permease which itself drives the absorption of external lactose. Since allolactose binding to the lac repressor blocks its transcriptional activity, allolactose effectively plays the same role as B in the MFL. The search for a more direct switch example has led us to consider the different MFL motifs [3]. An annotation taken from the SGD database [32] has been added.
in yeast as reported in [3] and reproduced in Table I. It was noted in [3] that most of these MFL motifs were central modules in biochemical pathways. Most of them take part in the cell cycle or in differentiation pathways, which certainly are processes where the cell switches from one function to another. However, we have found it difficult to disentangle the MFL's from numerous other known interactions and to confidently conclude that any of these motifs implements the proposed switch function. Similar difficulties have been recently emphasized for general motifs determined on purely statistical grounds and have led to question their functional significance [31]. In the present case, the difficulty of identifying biological cases of MFL switches, of course arises from our own very partial knowledge of the networks listed in Table I, but it may also be due to the fact that the possible role of the MFL module as a switch was not fully realized in previous investigations. The present study will hopefully help and trigger direct experimental investigations of these questions. In the following, we analyze the transitions between phase I and II of an oscillation period and use matched asymptotics to precisely justify the assumptions made in section V. This also provides the leading order ( √ δ) correction to the zeroth order results of section V. We consider the transition between the end of the high A/low B phase I and the start of the high B/low A phase II. This protein switch occurs on a time scale of order δ/δ r . On this fast time scale the mRNA concentration does not have time to change. Thus, in Eqs. (6)(7)(8)(9), we assume r = r 2 and introduce τ = t/δ. Eqs. (8,9) simply become at dominant order
dB dτ = r 2 − A B (A1) dA dτ = 1 − A B (A2)
The imposed boundary conditions are
A → (1 − r 2 )(τ − τ 1 ) + o(1), B → 0 at τ = −∞ (A3) B → (r 2 − 1)(τ − τ 2 ) + o(1), A → 0 at τ → +∞ (A4)
with the dimensionless transcript concentration r 2 > 1 (see section V B 1) and τ 1 and τ 2 two constants to be determined. Eqs. (A1,A2) are autonomous in time and therefore invariant by time translation. For general non integrable equations, numerical integration would be required to obtain the difference τ 2 − τ 1 . Here, however, the difference of protein concentrations, A − B, is easily integrated across the transition region. Fixing the time origin at the instant when A = B, one readily obtains the exact formula
A − B = (1 − r 2 )τ.(A5)
Comparison with Eqs. (A3,A4) shows that with this choice of time origin τ 1 = τ 2 = 0 (and more generally τ 1 = τ 2 ). The asymptotic conditions (A3,A4) coincide with the limiting behavior of a I (t)/δ at the end of phase I near t = t 1 , and with the limiting behavior of b II (t) at the beginning of phase II near t = 0, provided that a I (t 1 ) = b II (0) = 0 as was required in the main text. Thus, Eqs. (A1, A2) provide a uniform approximation of A(t) and B(t) throughout the transition from phase I to phase II. Given the exact result (A5), the integration of Eqs. (A1, A2) can in fact be replaced to the integration of the single following Riccati equation:
dA dτ = 1 − [A + (r 2 − 1)τ ]A(A6)
A comparison between Eq. (A6) and full numerical evolution is shown in Fig. 10. There is one subtlety, however. The quasi-equilibrium approximation for g [Eq. (42)] diverges at the end of phase I when A → 0 and g becomes larger than 1 before the transition region (A1,A2) of order δ. This clearly signals the breakdown of the approximation (42) in a larger intermediate region at the end of phase I. In a region of size t 1 − t ∼ √ δ, the evolution of g reduces to dg dt =θ 1 − g a δA 0 (A7) Indeed, taking a = (1 − r 2 )(t 1 − t) + o( √ δ) shows that all three terms kept in Eq. (A7) are of the same magnitude when g ∼ t 1 − t ∼ √ δ and that the additional term θg in Eq. (6) is negligible. Explicit integration in this intermediate region gives
g(t 1 ) − g(t) exp −κ 2 (t 1 − t) 2 =θ 0 t−t1 du exp κ 2 u 2 (A8)
where κ 2 =θ(1 − r 2 )/(2A 0 δ). In particular, the limit t → −∞, determines the value of g at the end of phase I and beginning of phase II
g(t 1 ) = g 2 = πA 0θ δ 2(r 2 − 1) (A9)
Eq. (A9) is the source of a correction of order √ δ to the asymptotic result (55) as will be shown below.
2. From phase II to phase I.
The transition from phase II to phase I can be analyzed quite similarly to the transition from phase I to phase II. In a time of order δ/δ r at the end of phase II, r has no time to change and the transition is described by Eqs. (A1,A2) with r 2 replaced by r 1 < 1 (see section V B 1) and the boundary conditions
B = (r 1 − 1)(τ − τ ′ 1 ) + o(1), A → 0 at τ = −∞ (A10) A = (1 − r 1 )(τ − τ ′ 2 ) + o(1), B → 0 at τ = +∞ (A11)
Again, τ ′ 1 = τ ′ 2 = 0 if the time origin is taken when A = B. This transition regime is followed by a longer transition on a time scale t − t 2 ∼ √ δ where g decreases from a value of order one to values of order δ characteristic of phase I. This reflects the fact that the binding of protein A to the promoter of gene g b is not instantaneous and requires some accumulation of protein A at the beginning of phase I. Taking A = (1 − r 1 )t/δ, g evolution is described in this intermediate regime by
dg dt =θ 1 − t(1 − r 1 ) g A 0 δ (A12)
The corresponding evolution of g is
g(t) = exp −κ 1 t 2 g 1 +θ t 0 du exp κ 1 u 2 (A13)
where κ 1 =θ(1 − r 1 )/(2A 0 δ). It indeed describes the transition from g 1 = 1 − exp(θt 2 ), the value of g at the end of phase II, to the quasi-equilibrium regime since the asymptotic behavior of Eq. (A13) for t ≫ √ δ is
g(t) ∼ A 0 δ (1 − r 1 )t(
FIG. 1 :FIG. 2 :FIG. 3 :FIG. 4 :FIG. 5 :FIG. 6 :FIG. 7 :
1234567Schematic representation of the over-represented MFL motif [4]. The bold arrow represents a transcriptional regulation interaction and the dashed double arrow a proteinprotein interaction. The proposed model of the MFL motif. The Greek letters denote the model different kinetic constants. The large crosses symbolize the degradation of the corresponding species. Different dynamical regimes of the MFL as ρ0 and ρ1 are varied. The borders between different regimes are shown as computed from numerical solutions of Eq. (18) and (31) forθ = 1.33 (full lines) orθ = 26.6 (dashed) as well as given by the asymptotics expansions [Eqs. (28,30,36, 41)] forθ = 1.33 (+ symbols). Here, and in all following figures, except when explicitly specified, the others parameters are [A]0 = 40 mol, β = 3 min −1 , δr = 0.03 min −1 , δA = 0.01 min −1 , δB = 0.01 min −1 , ρA = 100 mol.min −1 , γ = 1 mol −1 .min −1 where mol stands for molecules and min for minutes. The corresponding dimensionless parameters are δ = 0.003, da = d b = 0.33, A0 = 4, In the bistable regime a graded increase of ρA, the production of A, results in a jump in A concentration. The parameters are ρ f = 0.2 mol.min −1 ρ b = 2 mol.min −1 , and θ = 0.04 min −1 . The other parameters are as in Fig. 3. The oscillation domain for the same parameters as in Fig. 3 but for a larger domain of ρ1, the ratio of activated production of B to that of A. As in Fig. 3, the short-dashed line marks the boundary of the domain where the steady stable is unstable to oscillations forθ = 26.6. Numerical simulations (diamonds) show that the oscillating regime is stable up to the long-dashed line. Therefore, both the limit-cycle and the fixed point are stable attractors forθ = 26.6 in the region between the short-dashed and long-dashed lines. MFL in the oscillatory regime. The concentrations of the different species are shown as a function of time and display sustained oscillations. Constants are the same as in Fig. 4 except that ρA = 100 mol.min −1 , ρ f = 0.1 mol.min −1 ρ b = 5 mol.min −1 . Distinction between the two phases for the adimensioned equations [Eqs. (6-9)]. Top panel : oscillation of r. Bottom panel : oscillations for A (full line) and B (dotted).In order to clearly depict phase I and II of an oscillation cycle, the parameters are here chosen so that the two phases have similar durations : δ = 0.001, ρ1 = 1.5,ρ0 = 0,θ = 2, A0 = 1, µ = 1, da = d b = 0.
FIG. 8 :
8Comparison between the lowest order theoretical rescaled period Tr (bold line), the approximate expression for large ρ1 [Eq. (61)] (dashed line) and numerically computed periods for different values of δ, as a function of parameter ρ1. Other parameters areθ = 10, A0 = 10, µ = 1, da = d b = 0.
FIG. 9 :FIG. 10 :FIG. 11 :
91011Comparison between the full dynamics (+) symbols and the asymptotic description at order √ δ (dashed and dotted lines) during one oscillation cycle. (a) Concentration of mRNA behavior . Phase I asymptotics (dotted lines) and phase II asymptotics (dashed lines) are shown separately. The product A.B/β is also plotted (full line). The quasi-static assumption is seen to be well-satisfied away from the transition regions between the two phases.(b) Concentration of protein B. (c) Concentration of protein A. Parameters are as in Fig. The evolution of A given by the Riccati Eq. (A6) (+) is compared to that given by the complete MFL dynamics (dashed) during the transition from phase I to phase II. Parameters are as in Fig. Comparison of [g b ] obtained from a full numerical integration (+) with the uniform approximation obtained by matching the different transition regimes (bold line). The exponential relaxation in phase II [Eq. (A15)] is also shown (dashed). An enlargment of phase I and the transition regimes is shown in the inset. Parameters are as in Fig. 6.
TABLE I :
ISome biological examples of MFL motifs. The yeast motifs are reproduced from ref.
1 .
1From phase I to phase II.
3. Dominant correction to the oscillation period.The form of Eqs.(6)(7)(8)(9)could lead to think that the oscillation period has an expansion in powers of the small parameter δ. However, here as often, boundary layers lead to more complicated expansions. The two transition regimes of duration √ δ give rise to a dominant correction of order √ δ to the oscillation period. As pointed out above, due to the transition regime at the end of phase I, the starting value of g in phase II is g 2 [Eq. (A9)] of order √ δ. Therefore, the first Eq. (52) is replaced at order √ δ by the corrected evolution,This leads in turn to corrected evolutions r II,c (t) and b II,c (t) for the mRNA and B protein concentrations that are obtained by simply replacing (The other transition regime between phase II and phase I does not lead to modifications of the form of Eq. (46,47) describing the evolutions of mRNA and A protein during the bulk of phase I. Instead, it results in a difference of order √ δ between r II,c (t 2 ) and r 1 the effective concentrations of transcripts at the start of phase I. The transcript concentration depends on the promoter states at previous times. So, integrating Eq. (7) from the end of phase II, one obtainsReplacing g(t) by its expression (A12) during the transition regime at the start of phase I leads for t ≫ √ δ to(A17) Namely, the previous Eq. (46) but withNote that only the g 1 -term between the square brackets in Eq. (A12) contribute to Eq. (A17), the integral term giving a subdominant contribution. Finally, the durations T 1 and T 2 of both phases and the total period T r = T 1 + T 2 are obtained to √ δ accuracy by solving as beforetogether with the corrected version of Eq.(54), namely Eq. (A18) and b II,c (t 2 ) = 0 . The asymptotic description of g b evolution agrees well with numerical simulations of the full MFL model as shown inFig. 11. The √ δ correction terms lengthen the zeroth-order asymptotic estimation. For moderate value of δ, a numerically comparable contribution is however coming from higher-order terms that tend to lengthen the period [coming, for instance, from the breakdown of the quasi-equilibrium approximation for g b Eq. (42)] as can be seen inFig. 8.
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[
"COMPLEX TROPICAL LOCALIZATION, COAMOEBAS, AND MIRROR TROPICAL HYPERSURFACES",
"COMPLEX TROPICAL LOCALIZATION, COAMOEBAS, AND MIRROR TROPICAL HYPERSURFACES"
] |
[
"Mounir Nisse "
] |
[] |
[] |
We introduce in this paper the concept of tropical mirror hypersurfaces and we prove a complex tropical localization Theorem which is a version of Kapranov's Theorem [K-00] in tropical geometry. We give a geometric and a topological equivalence between coamoebas of complex algebraic hypersurfaces dened by a maximally sparse polynomial and coamoebas of maximally sparse complex tropical hypersurfaces.
| null |
[
"https://arxiv.org/pdf/0806.1959v1.pdf"
] | 17,611,665 |
0806.1959
|
48f4743765ea290db20a0aab483d927903079a9f
|
COMPLEX TROPICAL LOCALIZATION, COAMOEBAS, AND MIRROR TROPICAL HYPERSURFACES
Mounir Nisse
COMPLEX TROPICAL LOCALIZATION, COAMOEBAS, AND MIRROR TROPICAL HYPERSURFACES
We introduce in this paper the concept of tropical mirror hypersurfaces and we prove a complex tropical localization Theorem which is a version of Kapranov's Theorem [K-00] in tropical geometry. We give a geometric and a topological equivalence between coamoebas of complex algebraic hypersurfaces dened by a maximally sparse polynomial and coamoebas of maximally sparse complex tropical hypersurfaces.
Introduction
Amoebas have proved to be a very useful tool in several areas of mathematics, and they have many applications in real algebraic geometry , complex analysis, mirror symmetry, algebraic statistics and in several other areas (see [M1-02], [M2-04], [M3-02], , , and [R-01]). They degenerate to a piecewise-linear object called tropical varieties, [M1-02], [M2-04], and . However, we can use amoebas as an intermediate link between the classical and the tropical geometry. Coamoebas have a close relationship and similarities with amoebas and can be also used as an intermediate link between the tropical and the complex geometry.
A tropical hypersurface is the set of points in R n where some piecewise ane linear function (called tropical polynomial) is not dierentiable. Such a tropical polynomial may contain a tropical monomials which are not essential for the construction of the tropical hypersurface, but in the classical polynomial those monomials have a contribution and they often play a vital role in the geometry and the topology of the 1 arXiv:0806.1959v1 [math.AG] 11 Jun 2008 complex tropical hypersurface coamoeba dened by that polynomial. In this paper, we give a process to constructing the coamoeba of a complex tropical hypersurface by using a construction of a symmetric tropical hypersurface, which we call a mirror tropical hypersurface, that allows us to see and to understand the contribution on the coamoeba of the non-essential monomials in the tropical polynomial. The construction consists to look at the deformation of the extending Newton polytope in R n ×R instead of the deformation of the tropical hypersurface itself. What is the same by duality, but in plus we have a geometric point of view of that deformation. A symmetry appears naturally in this deformation, whose center is the time when the dual subdivision τ of the Newton polytope ∆ is reduced to one element (i.e., τ = {∆} itself).
If V f is an algebraic hypersurface in (K * ) n with K the eld of the Puiseux series, then we obtain the following results: Theorem 1.1 (Complex tropical localization). Let H αγ be a hyperplane in R n codual to an edge E αγ of the subdivision τ , and C be a connected component of H αγ ∩Arg(V ∞, f ). Then we have one of the two following cases: (i) The dimension of C is n − 1 and its interior is contained in the interior of a regular part of Arg(V ∞, f ); (ii) the dimension of C is zero (i.e., discrete) and C is contained in the intersection of H αγ and a line codual to some proper face of ∆ v . Theorem 1.2. Let V f ⊂ (K * ) n be a hypersurface dened by a polynomial f with Newton polytope ∆ such that the subdivision τ f = {∆ 1 , . . . , ∆ l } dual to the tropical hypersurface Val(V f ) is a triangulation. Then the geometry and the topology of the complex tropical hypersurfaces W (V f ) coamoebas are completely determined and constructed by gluing those of the truncated complex tropical hypersurfaces W (V f ∆ i ) using the complex tropical localization.
If V f is a complex algebraic hypersurface, then we have the following result: Theorem 1.3. Let V f be a complex algebraic hypersurface dened by a maximally sparse polynomial f . Then there exist a deformation of V f given by a family of polynomials f t such that the coamoeba of the complex tropical hypersurface V ∞, f (which is the limit of the H t (V ft ) with respect of the Hausdor metric on compact sets of (C * ) n ) has the same topology as the coamoeba coA f of V f (i.e., they are homeomorphic).
We recall the denitions and some Theorems of tropical geometry in section 2 alongside with all necessary notation. In section 3, we give the denition of complex tropical hypersurface and we describe those dened by maximally sparse polynomial with Newton polytope a simplex, and we give some examples of complex algebraic plane curves. In section 4, we introduce the notion of mirror tropical hypersurface, we give some examples, and we prove Theorem 1.2. In section 5, we prove the complex tropical localization Theorem. In section 6, we give a geometric and a topological description from the complex tropical hypersurface coamoeba to that of the complex algebraic hypersurface, and we will prove Theorem 1.3. Finally in section 7, we give the geometric and topological description of the coamoebas of some complex algebraic plane curves.
Preliminaries
Let K be the eld of the Puiseux series with real power, which is the eld of the series a(t) = r∈Aa ξ r t r with ξ r ∈ C * = C \ {0} and A a ⊂ R is well-ordered set (which means that any subset has a smallest element); the smallest element of A a is called the order of a, and denoted by ord(a) := min A a . It is well known that the eld K is algebraically closed and has a characteristic equal to zero, and it has a non-Archimedean valuation val(a) = − min A a satisfying to the following properties.
val(ab) = val(a) + val(b) val(a + b) ≤ max{val(a), val(b)},
and we put val(0) = −∞. If we denote by K * = K \ {0} and we apply the valuation map coordinate-wise we obtain a map Val : (K * ) n → R ∪ {−∞} which we will also call the valuation map. If a ∈ K * is the Puiseux series a = j∈Aa ξ j t j with ξ ∈ C * and A a ⊂ R is a well-ordered set. We complexify the valuation map as follows :
w : K * −→ C * a −→ w(a) = e val(a)+i arg(ξ − val(a) )
Let Arg be the argument map K * → S 1 dened by: for any a ∈ K a Puiseux series so that a = j∈Aa ξ j t j , then Arg(a) = e i arg(ξ − val(a) ) (this map extends the map Arg : C * → S 1 dened by ρe iθ → e iθ ). Applying this map coordinate-wise we obtain a map :
W : (K * ) n −→ (C * ) n Denition 2.1. The set V ∞ ⊂ (C * ) n is a complex tropical hypersurface if and only if there exists an algebraic hypersurface V K ⊂ (K * ) n over K such that W (V K ) = V ∞ , where W (V K ) is the closure of W (V K ) in (C * ) n ≈ R n ×(S 1 )
n as a Riemannian manifold with the metric of the product of the Euclidean metric on R n and the at metric on (S 1 ) n .
Let V f ⊂ (K * ) n be the algebraic hypersurface dened by the non-Archimedean polynomial:
f (z) = α∈A a α z α , z α = z α 1 1 z α 2 2 .
. . z αn n with a α ∈ K * and A a nite subset of Z n . We denote by ∆ f the Newton polytope of f , which is the convex hull in R n of A. Let ν f be the map dened on A as follows:
ν f : A −→ R α −→ ord(a α ).
The Legendre transform L(ν f ) of the map ν f is the piecewise ane linear convex function dened by:
L(ν f ) : R n −→ R x −→ max{< x, α > −ν f (α)};
where <, > denotes the scalar product in the Euclidean space.
Denition 2.2. The Legendre transform L(ν f ) of the map ν f is called the tropical polynomial associated to f , and denoted by f trop . Theorem 2.3 (Kapranov, (2000)). The image of the algebraic hypersurface V f under the valuation map Val is the set Γ f of points in R n where the piecewise ane linear function f trop is not dierentiable.
We denote by∆ f the extended Newton polytope of f which is the convex hull of the subset {(α, ν f (α)) ∈ A × R} of R n × R. Let ρ be the following map:
ρ : ∆ f −→ R x −→ min{t | (x, t) ∈∆ f }.
It's clear that the linearity domains of ρ dene a convex subdivision τ f = {∆ 1 , . . . , ∆ l } of ∆ f (by taking the linear subsets of the lower boundary of∆ f , see [PR1-04], , and [IMS-07] for more details). Let y =< x, v i > +r i be the equation of the hyperplane H i ⊂ R n × R containing the points with coordinates (α, ν f (α)) with
α ∈ Vert(∆ i ).
There is a duality between the subdivision τ f and the subdivision of R n induced by Γ f (see , , and [IMS-07]), where each connected component of R n \ Γ f is dual to some vertex of τ f and each k-cell of Γ f is dual to some (n − k)-cell of τ f . In particular,
each (n − 1)-cell of Γ f is dual to some edge of τ f . If x ∈ E * αβ ⊂ Γ f , then < α, x > −ν f (α) =< β, x > −ν f (β), so < α − β, x − v i >= 0. This means that v i is a vertex of Γ f dual to some ∆ i having E αβ as edge.
Let V be an algebraic hypersurface in (C * ) n dened by the complex polynomial:
f (z) = α∈supp(f ) a α z α , z α = z α 1 1 z α 2 2 . . . z αn n ,
where a α are non-zero complex numbers and supp(f ) is the support of f , and we denote by ∆ the Newton polytope of f (i.e., the convex hull in R n of supp(f )).
The following denition is given by M. Gelfand Denition 2.4. The amoeba A of an algebraic hypersurface V ⊂ (C * ) n is the image of V under the map :
Log :
(C * ) n −→ R n (z 1 , . . . , z n ) −→ (log z 1 , . . . , log z n ).
It was shown by M. Forsberg, M. that there is an injective map between the set of components {E ν } of R n \ A and Z n ∩ ∆: (2000)). Each component of R n \ A is a convex domain and there exists a locally constant function:
ord : {E ν } → Z n ∩ ∆ Theorem 2.5 (Foresberg-Passare-Tsikh,ord : R n \ A −→ Z n ∩ ∆ which maps dierent components of the complement of A to dierent lattice points of ∆. The coordinates z j of z ∈ (C * ) n are parameterized by z j = ρ j e i arg(z j ) with ρ j = z j ∈]0, ∞[ and arg(z j ) ∈ [0, 2π[ for j = 1, .
. . , n. Passare and Tsikh introduced the following set associated to a complex algebraic varieties. Denition 2.6 (Passare-Tsikh). The Coamoeba coA ⊂ (S 1 ) n of f is the image of V under the argument map Arg dened by the following:
Arg : (C * ) n ≈ R n × (S 1 ) n −→ (S 1 ) n (z 1 , . . . , z n ) −→ (e i arg(z 1 ) , .
. . , e i arg(zn) ).
Complex tropical hypersurfaces with a simplex Newton polytope
Let a = (a 1 , . . . , a n ) ∈ (K * ) n and H a ⊂ (K * ) n be the hyperplane dened by the polynomial f a (z 1 , . . . , z n ) = 1 + n j=1 a j z j , then it's clear that H a = τ a −1 (H 1 ). Let L be an invertible matrix with integer coecients and positive determinant
L = α 11 . . . α 1n . . . . . . . . . α n1 . . . α nn ,
and let Φ L, a be the homomorphism of the algebraic torus dened as follow.
Φ L, a : (K * ) n −→ (K * ) n (z 1 , . . . , z n ) −→ (a 1 n j=1 z α j1 j , . . . , a n n j=1 z α jn j ). Let V f ⊂ (K * ) n be the hypersurface dened by the polynomial f (z 1 , . . . , z n ) = 1 + n k=1 a k n j=1 z α jk j ,
such that its Newton polytope is the simplex ∆ f that is the image by L of the standard simplex. The matrix L is invertible, so Φ L, a (V f ) = H a , and then t L −1 (Val(H a )) = Val(V f ). It was the same thing for the complex tropical hypersurface i.e., t L −1 (W (V f )) = W (H a ), (because for any k = 1, . . . , n we have arg(a k n j=1 z α jk j ) = arg(a k ) + n j=1 < α jk , arg(z j ) >), abuse of notations; to be more precise we have t L −1 (Log(Arg(W (V f )))) = Log(Arg(W (H a ))). Hence we have the following (for more details, see [N1-07]):
coA (V f ) = τt L −1 (a −1 ) • t L −1 (coA (H 1 )).
So, the coamoeba of any hypersurface dened by a maximally sparse polynomial (that the number of its coecients is equal to the number of its Newton polytope vertices) with a simplex as Newton polytope, can be easily drawn. We remark that the eld of Puiseux series K can be replaced by the eld of complex numbers and we have the same results with the same formulas.
Example 3.1. We draw on gure 1 the coamoeba of the complex curve dened
by the polynomial f 1 (z, w) = w 3 z 2 + wz 3 + 1 where the matrix t L −1 1 is equal to 1 7 3 −1 −2 3
and on gure 2 the coamoeba of the complex curve dened by the
polynomial f 2 (z, w) = w 2 z 2 + z + w where the matrix t L −1 2 is equal to 1 3 1 1 −2 1 .f 1 (z, w) = wz 3 + z 2 w 3 + 1
Tropical mirror hypersurfaces
Let V f ⊂ (K * ) n be an algebraic hypersurface dened by a polynomial f and we assume that ∆ f is a simplex, A = Vert(∆ f )∪{β}, and the coecient a β is monomial. In
f (z, w) = z + w + z 2 w 2 addition, we suppose that β ∈ Vert(τ f ). Let {f u } u∈]−1;1]
be the family of polynomials dened as follow:
f u (z) = a β, u z β + α∈Vert(∆ f ) a α z α ,
with a β, u such that:
a β, u = ξ β t uν f (β)+(1−u)(<β,v>+r) if u ∈ [0; 1] ξ β t (1−u)(<β,v>+r)− u u+1 if u ∈] − 1; 0]
where ξ β is the complex coecient of a β , and v is the vertex of the tropical hypersurface Val(V f ). We can assume that < β, v > +r ≥ 0 (multiplying f by a power of t if necessary); so the map u
−→ (1 − u)(< β, v > +r) − u u+1 is a decreasing function on ] − 1; 0].
Remark 4.1.
(a) The deformation given above is such that f 1 (z) = f (z); (b) for any u ∈] − 1; 0], the subdivision τ fu = {∆ f } (i.e., trivial); (c) with the same assumption as above, when the order of the monomial a β, u reaches the hyperplane of R n × R containing the points with coordinates (α, ν f (α)) and α ∈ Vert(∆ f ) (i.e., for u ≤ 0). Then we consider the family of polynomialsf u dened by: u ) . In this case, we have the convergence when t tends to the innity, because the induced transformation of K is given by t −→ t −1 . Let I : (K * ) n → (K * ) n be the transformation dened as (z 1 , . . . , z n ) → (z −1 1 , . . . , z −1 n ), then by making the change of the variable t = 1 τ , we can see thatf u (z) = f u • I (z), and then
f u (z) =ã β, u z −β + α∈Vert(∆ f )ã α z −α , such that if α ∈ Vert(∆ f ) and a α (t) = r≥ord(aα) ξ α,r t r , then we setã α (t) = r≥ord(aα) ξ α,r t −r and if a β, u (t) = ξ β t ord(a β, u ) , we setã β, u (t) = ξ β t − ord(a β,Vf u = V fu•I = I (V fu ).
The tropical polynomial associated tof u is given bỹ
f u, trop = max α∈Sym(A) {< x, γ > − val(a γ,u )}
with Sym(A) the subset of Z n symmetric to A relatively to the origin. There exist a positive number s ∈] − 1; 0] such that the non-Archimedean amoebas dened by the tropical polynomialsf u, trop with u ∈ [−s; 0] are symmetric to those dened by f u, trop with u ∈ [0; 1] (By an automorphism of (K * ) n if necessary, we can assume that val(a α ) = 0 for any α ∈ Vert(∆ f , and in this case s = −ν f (a β ).). So, we can apply now Kapranov's theorem to the tropical hypersurfaces Γf u , and from the equality Vf
u = I (V fu ), we deduced that the coamoeba of V ∞, fu = W (V fu ) is the symmetric of the coamoeba of V ∞,fu = W (Vf u ).
(d) One way to look at the deformation of a tropical hypersurface is to think of it as a deformation of the extending Newton polytope of its dening polynomial. More precisely, the deformationf u, trop can be seen as a continuation of the deformation of the normal vectors to the hyperplanes in R n × R containing the lifting of the ∆ i 's element of the subdivision τ fu dual to f u, trop with 0 ≤ u ≤ 1. Indeed, when u = 0, all the normal vectors are equal and then for u ≤ 0 the coecient of index β becomes inessential in the tropical polynomial f u, trop , but in the non-Archimedean polynomial f u , it has a contribution and plays a crucial role for the determination of the complex tropical hypersurface coamoeba. We can see that if Γ is a tropical hypersurface with only one vertex, and V 1 , V 2 are two hypersurfaces in (K * ) n such that Val(V i ) = Γ for i = 1, 2, then the two mirror tropical hypersurfaces Mirr trop (V 1 ) and Mirr trop (V 2 ) are not necessary the same. A similar algebraic construction is given by Z. Izhakian and L. Rowen in [IR-08].
Theorem 4.3. Let V f ⊂ (K * ) n be a hypersurface dened by a polynomial f with Newton polytope ∆, and assume that the subdivision τ f = {∆ 1 , . . . , ∆ l } dual to the tropical hypersurface Val(V f ) is a triangulation. Then the geometry and the topology of the complex tropical hypersurfaces W (V f ) coamoebas are completely determined and constructed by gluing those of the truncated complex tropical hypersurfaces W (V f ∆ i ) using the complex tropical localization.
Proof Suppose that V f ⊂ (K * ) n is dened by the polynomial f (z) = α∈A a α z α with Newton polytope ∆ equal to the convex hull of A, and let A i = A ∩ ∆ i . If f ∆ i denotes the truncation of f to ∆ i , then the assumption of Theorem 4.3, means that the spine of the hypersurface amoeba of V f ∆ i has only one vertex. Let τ i = ∪ j ∆ ij be the convex subdivision of ∆ i given by taking the upper bound of the convex hull of the set {(α, r) ∈ A i × R | r ≤ ord(a α )}, which we can suppose to be a triangulation (by a small perturbation of the coecients order if necessary). Let inv : C((t)) → C((ρ)) be the morphism sending t of valuation +1 to ρ of valuation −1, and letf be the polynomial dened byf
(z) = α∈A iã α z −α withã α = inv(a α ) (this means that if a α (t) = r≥ord(aα) ξ α,r t r thenã α (t) = r≥ord(aα)
ξ α,r t −r ). We use now induction on the volume of ∆, and we assume that the coamoeba of any V f ∆ ij is constructed for each index ij using the complex tropical localization which we develop in the next section.
By construction we have
Vf i = V f ∆ i •I = I (V f ∆ i ).
The coamoeba of Vf i can be constructed, because in this case, one can apply Kapranov's Theorem, and we can also build the coamoeba of V f ∆ i . Knowing now all the coamoebas of the V f ∆ i 's, the coamoeba of the hypersurface V f itself can be built by reusing Kapranov's Theorem.
Examples 4.4.
(a) Example of the parabola (see gures 3, 4, and 5), where the deformation is seen as a deformation of the normal vectors to the hyperplanes in R n × R containing the lifting of the ∆ i 's, and the points with coordinates (α, ord(α)) and α ∈ Vert(∆ f ) are xed. (b) We give here an example where β ∈ Int(∆ f ) (see gures 6, 7, and 8), and as in the previous example, the deformation is supposed to x the order of the coecients of index in the vertices of the Newton polygon.
Camoebas of complex tropical hypersurfaces
In this section we consider an algebraic hypersurface V over the eld of Puiseux series K dened by a polynomial f with Newton polytope ∆. We denote by Γ the non-Archimedean amoeba of V and by V ∞, f the complex tropical hypersurface image of V under the map W . Let us denote by τ the subdivision of ∆ dual to Γ which we . suppose to be a triangulation, and assume that f is dened as follows: where a α are non-zero complex Puiseux series and supp(f ) is the support of f .
f (z) = α∈supp(f ) a α z α , z α = z α 1 1 z α 2 2 . . . z αn n ,
Complex tropical localization.
Denition 5.2. Let R n be the universal covering of the real torus (S 1 ) n . Let α and β be in the support of f . A hypersurface H αβ ⊂ R n is called codual (or corresponding)
to an edge E αβ in τ if it is given by the following equation:
arg(a α ) − arg(a β )+ < α − β, x >= π.
In addition if E αβ is an external edge of τ (i.e., E αβ is a proper edge of the Newton polytope ∆), then H αβ is called an external hyperplane. Denition 5.
3. An open subset C of the coamoeba of a complex tropical hypersurface
V ∞, f is called regular if for any point x in C there exist an open subset V (x) in (S 1 ) n containing x with V (x) ⊂ C and an open subset U in R n such that V (x) ⊂ Arg(Log −1 (U ) ∩ V ∞, f ) i where (Log −1 (U ) ∩ V ∞, f ) i is one connected component of Log −1 (U ) ∩ V ∞, f .
We denote by Critv(Arg) the set of critical values points in the coamoeba coA of a complex tropical hypersurface V .
Denition 5.4. An extra-piece is a connected component C of coA \ Critv(Arg) such that the boundary of its closure ∂ C is not contained in the union of hyperplanes codual to the edges of the subdivision.
This means that its boundary contains at least one component (smooth) in the set of critical values of the argument map. In the following Lemma we assume that the subdivision τ of the Newton polytope ∆ dual to the non-Archimedean amoeba Γ is a triangulation and contains inner edges. We begin by proving the following Lemma which is a local version of the Theorem 6.1 in the complex tropical case.
Lemma 5.5. Let H αγ be a hyperplane in R n codual to an inner edge E αγ of the subdivision τ . Then any connected component C of H αγ ∩ Arg(V ∞, f ) has a dimension n − 1 and its interior is contained in the interior of a regular part of Arg(V ∞, f ). Proof Let ∆ 1 and ∆ 2 be two elements of τ with a common edge E αγ , and v 1 and v 2 be their dual vertices in the non-Archimedean amoeba Γ. Let {x m } be a sequence in
Arg(Log −1 (v 1 ) ∩ V ∞, f ∆ 1 ) \ Arg(Log −1 (v 2 ) ∩ V ∞, f ∆ 2 ) which converge to some point x in H αγ \Arg(Log −1 (v 2 )∩V ∞, f ∆ 2 ). Let C be a connected component of Arg −1 ({x m })∩V ∞, f
and {z m } ⊂ C be a sequence such that Arg(z m ) = x m for each m. We claim that the sequence {z m } (by taking a subsequence if necessary) converges to some point z in V ∞, f . Indeed, the sequence {Log(z m )} converge to v 2 because the argument of z m is x m which converges to x ∈ H αγ , and x is an innite point for Arg(V ∞, f ∆ 1 ). This means that {Log(z m )} converges asymptotically in the direction of E αγ to the innity of Log(V ∞, f ∆ 1 ). So z m converge to the point z of (C * ) n with argument x and the valuation v 2 . V ∞, f is closed, hence z ∈ V ∞, f . Then all the components
of H αγ \ Arg(Log −1 (v 1 ) ∩ V ∞, f ∆ 1 ) ∩ Arg(Log −1 (v 2 ) ∩ V ∞, f ∆ 2 ) are in the interior of Arg(V ∞, f ).
Let now x be a point in the interior of the following set:
H αγ ∩ Arg(Log −1 (v 1 ) ∩ V ∞, f ∆ 1 ) ∩ Arg(Log −1 (v 2 ) ∩ Arg(V ∞, f ∆ 2 ) , and {x m } be a sequence in Arg(Log −1 (v 1 ) ∩ V ∞, f ∆ 1 ) ∩ Arg(Log −1 (v 2 ) ∩ Arg(V ∞, f ∆ 2 )
such that x m converges to x. We claim that there is no sequence {z m } in V ∞, f such that Arg(z m ) = x m for any m and z m converges in V ∞, f to some point z such that Arg(z) = x. Indeed, assume on the contrary that there exists a sequence {z m } in V ∞, f satisfying the assumption and converging to z in V ∞, f . On one hand we know that Log(z m ) converges to v 2 , because the argument of z m converges to x ∈ H αγ which is an innite point for Arg(Log −1 (v 1 ) ∩ V ∞, f ∆ 1 ) and then the valuation of the z m 's tends to the innity asymptotically in the direction of E αγ to v 2 (because v 2 represents the innity for Log(V ∞, f ∆ 1 ) in the direction of E αγ ). On the other hand, for the same reasons, the sequence Log(z m ) converge to v 1 . Contradiction, because by assumption v 1 = v 2 . In this case we have the so-called extra-piece.
Proposition 5.6. Let H αγ be a hyperplane in R n codual to an external edge E αγ of the subdivision τ , and let C be a connected component of H αγ ∩ Arg(V ∞, f ). Then we have one of the two following cases: (i) The dimension of C is n − 1 and its interior is contained in the interior of a regular part of Arg(V ∞, f ); (ii) the dimension of C is zero (i.e., discrete) and C is contained in the intersection of H αγ and a line codual to some proper face of ∆ v .
If the edge E αγ is a common edge to more than one element of the subdivision τ (which can occur only if n > 2), then by Lemma 5.5 we have the rst case. Assume that E αγ is an edge of only one element ∆ v of τ , and we denote by v the vertex of the tropical hypersurface dual to ∆ v . Let z ∈ Log −1 (v)∩V ∞, f such that Arg(z) = x which we assume in H αγ . We denote by C the connected component of H αγ ∩ Arg(V ∞, f ) containing x. We have to consider the following cases:
(a) supp(f ) = Vert(∆ v ), in this case there is nothing to prove, and we have case (ii) of the Proposition.
(b) supp(f ) ∩ ∆ v = Vert(∆ v ) or supp(f ) = Vert(∆ v ) ∪ {β 1 , . . . , β l } with β j ∈
∆ v ∩ Z n for any j. All other cases will be easily deduced thereof. Assume that
supp(f ) = Vert(∆ v ) ∪ {β} with β ∈ ∆ v .
Lemma 5.7. With the above notations, let A be the interior of C, then for any
x ∈ A there exists an open neighborhood V (x) of x in (S 1 ) n such that V (x) ⊂ Arg(V ∞, f ).
Proof Indeed, assume on the contrary that there exists a small open neighborhood V (x) of x in R n such that V (x) ∩ coA f ∆α j is empty, where ∆ α j is the simplex with vertices {α 1 , . . . , α j , . . . , α n+1 , β} and E αγ = E α 1 α 2 with j = 1, 2 (here we use the same letter for x and its lifting to the universal covering of the torus; abuse of notation). This means that V (x) ∩ Arg(V ∞, f ) lies in one side of the hyperplane H α 1 α 2 . So the dominating monomials in W −1 (Arg −1 (V (x)) ∩ V ∞, f ) are a α 1 , a α 2 , . . . , a α n+1 ,β , because if the monomial a α 2 is a dominating one, then Arg(V ∞, f ) ∩ V (x) lies on both sides of H α 1 α 2 . From Remarks 4.1 (a), (b), (c) and , we obtain that the dominating monomials in W −1 (Log −1 (v) ∩ V ∞, f ) are a α 1 , a α 2 , . . . , a α n+1 . Hence z lies in the domain where the monomials a α 1 , a α 2 , . . . , a α n+1 are dominating (a proper face of the simplex ∆ v ), and then Arg(z) = x is contained in H α 1 α 2 ∩ Arg(V ∞, f ∆α 2 ). Contradiction, because H α 1 α 2 ∩ Arg(V ∞, f ∆α 2 ) is discrete and then the intersection of any open neighborhood of x in R n with Arg(V ∞, f ) lies on both sides of the hyperplane H α 1 α 2 . In this case we have some extra-piece. Theorem 1.1 is an immediate consequence of Lemma 5.5 and Proposition 5.6.
Coamoebas of complex algebraic hypersurfaces
We now turn our attention to complex algebraic hypersurfaces, so in this section we assume that the polynomial f is complex. We will give a caracterization of the argument map critical values set contained in the hyperplanes codual to the edges of the subdivision τ dual to the spine of the amoeba A of V f , and we have the following. Theorem 6.1. Let H αγ be a hyperplane in R n codual to an edge E αγ . Then the intersection H αγ ∩ Critv(Arg) is discrete and it is contained in the union of lines L α β codual to some faces of τ .
Proof Assume that there is an open subset
A of H αγ such that A ⊂ coA V f , then
we claim that A ⊂ coA V ∞, f . Indeed, assume that E αγ is a common edge for two simplices ∆ 1 and ∆ 2 . Let y =< x, a 1 > +b 1 be the equation of the hyperplane in R n × R containing the points with coordinates (α, ν(α)) and α ∈ Vert(∆ 1 ) and ν the Passare-Rullgård function.
Let f t (z) = a α (et) <α,a 1 >+b 1 z α = (et) b 1 a α ((et) a 1 z) α
with a 1 = (a 11 , . . . , a 1n ) and
((et) a 1 z) α = (et) a 11 α 1 z α 1 1 (et) a 12 α 2 z α 2 2 . . . (et) a 1n αn z αn n .
Hence V ft ⊂ (C * ) n is the image of V f under the self dieomorphism φ t of (C * ) n given by:
(z 1 , . . . , z n ) → ((et) a 11 z 1 , (et) a 12 z 2 , . . . , (et) a 1n z n )
which conserves the arguments. Assume now that A ⊂ coA V f ∩ Critv(Arg), so when t is so close to zero then the set Log(Arg −1 (A) ∩ V f ) take place on the two sides of the hyperplane E * αγ in Γ dual to E αγ , because it is the case for the truncation V f ∆ 1 which approximate our hypersurface when t tends to zero. So, if one chooses a coecients d α and d γ such that the holomorphic annulus Y of equation d α z α + d γ z γ = 0 has the hyperplane containing E * αγ as its amoeba, and the hyperplane H αγ as its coamoeba, then V f ∩Y is nonempty. Let z 0 be a point in V f ∩Y , hence φ −1 and then Arg(z 0 ) ∈ coA V ∞, f ∆ 1 . It contradicts Lemma 5.5 if the hyperplane H αγ is inner, and Proposition 5.6 if H αγ is external, and then A is contained in the interior of a regular part of the coamoeba or it is discrete.
Let t be a strictly positive real number in ∈]0; 1 e ], and H t be the following self dieomorphism of (C * ) n :
H t : (C * ) n −→ (C * ) n (z 1 , . . . , z n ) −→ ( z 1 − 1 log t z 1 z 1 , . . . , z n − 1 log t zn zn ).
which denes a new complex structure on (C * ) n denoted by J t = (dH t ) • J • (dH t ) −1 where J is the standard complex structure. A J t -holomorphic hypersurface V t is a hypersurface holomorphic with respect to the J t complex structure on (C * ) n . It is equivalent to say that V t = H t (V ) where V ⊂ (C * ) n is an holomorphic hypersurface with respect to the standard complex structure J on (C * ) n .
Recall that the Hausdor distance between two closed subsets A, B of a metric space (E, d) is dened by: Here we take E = R n ×(S 1 ) n , with the distance dened as the product of the Euclidean metric on R n and the at metric on (S 1 ) n . A complex tropical hypersurface can be dened as follows (see [M1-02] and [M2-04]). Denition 6.2. A complex tropical hypersurface V ∞ ⊂ (C * ) n is the limit when t tends to zero of a sequence of a J t -holomorphic hypersurfaces V t ⊂ (C * ) n (with respect to the Hausdor metric on compact sets in (C * ) n ).
Coamoebas of maximally sparse hypersurfaces.
Let V f ⊂ (C * ) n be a hypersurface dened by a maximally sparse polynomial f (z) = α∈Vert(∆ f ) a α z α (recall that a polynomial f is maximally sparse means that supp(f ) = Vert(∆ f )). Let f t be the family of polynomials dened by
f t (z) = α∈Vert(∆ f ) a α (et) − Log(aα) z α and V t their zero locus. We denote by V ∞, f = lim t→0 H t (V t )
with respect to the Hausdor metric on compact sets of (C * ) n . Theorem 6.4. With the above notations and assumptions, the deformation of V f given by the family of polynomials f t satises the following: the coamoeba of the complex tropical hypersurface V ∞, f has the same topology of the coamoeba coA f of V f (i.e., they are homeomorphic).
Proof We will prove that the deformation given by {f t } denes a bijection between the complement components of the V f 's coamoeba and the complement components of the V ∞, f 's coamoeba. More precisely, we prove that such deformation conserve the complement components of the coamoeba and thus its topology. Assume that a complement component of the coamoeba is created (resp. disappear) for some t. Then there is a created (resp. disappear) component of the argument map critical values boundary, it means that some edge of the subdivision τ f dual to the spine of the amoeba A f disappears (resp. created), but it cannot occur because the polynomials are maximally sparse, and thus, the spines of the amoebas A V f t are of the same combinatorial type. It remains to show that two dierent complement components of the V f 's coamoeba cannot be deformed to the same complement component of the V ∞, f 's coamoeba. Assume on the contrary that there is two complement components C 1 and C 2 of the V f 's coamoeba which are deformed to one complement component C of the V ∞, f 's coamoeba. It means that one of these two components disappears or the component C is not convex, and then we have a contradiction in both cases.
Examples of complex algebraic plane curves coamoebas
(1) Let V f λ be the curve in (C * ) 2 dened by the following polynomial: f λ (z, w) = w 2 − λw + 2zw − z 2 w + 1.
Let f λ, 1 (z, w) = w 2 − λw + 2zw − z 2 w, so V f λ, 1 is just the parabola of example 1. Let f λ, 2 (z, w) = −λw + 2zw − z 2 w + 1, hence V f λ, 2 is the set of points (z, w) ∈ (C * ) 2 such that :
w = 1 z 2 − 2z + λ .
This means that arg(w 2 ) = − arg(w 1 ) mod 2π. Hence the coamoeba of the curve dened by f λ is as in the gure 8 on the left. (2) Let V f λ be the curve in (C * ) 2 dened by the following polynomial: f λ (z, w) = zw 2 + z 2 w + z + w + λzw.
Let f λ, 1 (z, w) = zw 2 + z + w + λzw. Hence V f λ, 1 is just a reparametrization of the parabola of example 1. We can see that z = − w 1+w 2 +λw . (1): on the left the coamoeba when λ = 0 and the curve is non-Harnack, and the coamoeba when λ = 0 and the triangulation is trivial on the right.
Let f λ, 2 (z, w) = zw 2 + z 2 w + z + λzw = z(1 + zw + w 2 + λw), hence V f λ, 2 is the set of points (z, w) ∈ (C * ) 2 such that :
z = − 1 + w 2 + λw w .
It means that arg(z 2 ) = − arg(z 1 ) mod 2π, where z 1 (resp. z 2 ) denotes the rst coordinate of a point in V f λ, 1 (resp. in V f λ, 2 ) . As in example 1, we have the gures 9 on the top right. (3) We give an example of a Newton polygon ∆ that denes not any real curve with maximal number of coamoeba complement components, but this maximal number is realized by a complex curve. Let ∆ be the polygon with vertices (1; 0), (0; 1), (1; 2), and (3; 1) (see gure 10 for the polygon and its subdivision dual to the spine of the amoeba). In this case we prove that no real polynomial can realize the maximal number of coamoeba complement components (the maximal number in the real case is ve, and the coamoeba is given in gure 11 on the left for some real coecients ), but the complex curve dened by Figure 11. Coamoeba of example (2) in three cases, the rst coamoeba is the one of the Harnack case, the second coamoeba of a non-Harnack case and the coecient λ = 0, and the last case is the case when λ = 0 and the subdivision is trivial.
the complex polynomial f (z, w) = e iα w + z + zw 2 + z 3 w with 0 < α < π, has a coamoeba with maximal number of complement components (i.e. six components, see gure 11 on the right). (3): on the left the coamoeba of a real curve with the same maximal number of complement components (i.e., 5 components; the picture here is in one fundamental domain) and on the right the coamoeba of a complex curve with a maximal number of complement components (6 components; the picture here is in four fundamental domains).
Figure 1 .
1The coamoeba of the curve dened by the polynomial
Figure 2 .
2The coamoeba of the curve dened by the polynomial
Denition 4. 2 .
2The tropical hypersurfaces Val(Vf u ) dened by the tropical polynomialsf u, trop for u ∈]−1; 0], are called the tropical mirror for the hypersurfaces V fu dened by the polynomial f u . We denote this hypersurface by Mirr trop (V fu ) := Val(Vf u ).
Figure 3 .
3The tropical curves Val(V fu ) for u ∈ [0; 1].
Figure 4 .
4The tropical curves Val(V f 0 ) and Val(Vf 0 ).
Figure 5 .
5The tropical curves Val(Vf u ) for u ∈] − 1; 0]
Figure 6 .
6The tropical curves Val(V fu ) for u ∈ [0; 1].
Figure 7 .
7The tropical curves Val(V f 0 ) and Val(Vf 0 ).
Figure 8 .
8The tropical curves Val(Vf u ) for u ∈] − 1; 0].
Figure 9 .
9The Newton Polygon of example (1) and its subdivision.
Figure 10 .
10Example
Figure 12 .Figure 13 .
1213The subdivision of the Newton polygon and its dual Example
t (z 0 ) ∈ φ −1 t (V f )∩φ −1 t (Y )
Laurent determinants and arrangements of hyperplane amoebas. M Forsberg, M ; Passare, A Tsikh, Advances in Math. 151M. Forsberg, M; Passare and A. Tsikh, Laurent determinants and arrangements of hyperplane amoebas, Advances in Math. 151, (2000), 45-70.
I M Gelfand, M M Kapranov, A V Zelevinski, Discriminants, resultants and multidimensional determinants. BostonBirkhäuserI. M. Gelfand, M. M. Kapranov and A. V. Zelevinski, Discriminants, resultants and multidimensional determinants, Birkhäuser Boston 1994.
Z Izhakian, L Rowen, Completions, reversals, and duality for tropical varieties. Z. Izhakian and L. Rowen, Completions, reversals, and duality for tropical varieties, http://fr.arxiv.org/pdf/0806.1175
Tropical Algebraic Geometry. I Itenberg, G Mikhalkin, E Shustin, Oberwolfach Seminars. 35IMS-07[IMS-07] I. Itenberg, G. Mikhalkin, and E. Shustin, Tropical Algebraic Geometry, Oberwolfach Seminars, Volume 35, Birkhäuser Basel-Boston-Berlin 2007.
Amoebas over non-Archimedean elds. M M Kapranov, PreprintM. M. Kapranov, Amoebas over non-Archimedean elds, Preprint 2000.
Decomposition into pairs-of-pants for complex algebraic hypersurfaces. G Mikhalkin, Topology. 43G. Mikhalkin, Decomposition into pairs-of-pants for complex algebraic hypersur- faces,Topology 43, (2004), 1035-1065.
Enumerative Tropical Algebraic Geometry In R 2. G Mikhalkin, J. Amer. Math. Soc. 18G. Mikhalkin, Enumerative Tropical Algebraic Geometry In R 2 , J. Amer. Math. Soc. 18, (2005), 313-377.
Real algebraic curves, moment map and amoebas. G Mikhalkin, Ann.of Math. 151G. Mikhalkin , Real algebraic curves, moment map and amoebas, Ann.of Math. 151 (2000), 309-326.
Maximally sparse polynomials have solid amoebas. M Nisse, PreprintM. Nisse, Maximally sparse polynomials have solid amoebas, Preprint 2006, http://fr.arxiv.org/pdf/0704.2216
M Nisse, Coamoebas of complex algebraic hypersurfaces. PreprintM. Nisse, Coamoebas of complex algebraic hypersurfaces, Preprint, (2007).
M Nisse, Amoebas and Coamoebas Relations and Similarities. PreprintM. Nisse, Amoebas and Coamoebas Relations and Similarities, Preprint, (2007).
Amoebas, Monge-Ampère measures, and triangulations of the Newton polytope. Rullgårdm Passare, H Passare, Rullgård, Duke Math. J. 121PR1-04[PR1-04] Passare and RullgårdM. Passare and H. Rullgård, Amoebas, Monge-Ampère measures, and triangulations of the Newton polytope, Duke Math. J. 121, (2004), 481-507.
Multiple Laurent series and polynomial amoebas. M Passare, H Rullgård, Actes des rencontres d'analyse complexe. s des rencontres d'analyse complexeAtlantique; Poitou-CharentesÉditions de l'actualité scientiqueM. Passare and H. Rullgård, Multiple Laurent series and polynomial amoebas, pp.123- 130 in: Actes des rencontres d'analyse complexe, Atlantique, Éditions de l'actualité scientique, Poitou-Charentes 2001.
. L Pachter, B Sturmfels, Algebraic Statistics for Computational Biology. Cambridge University PressL. Pachter and B. Sturmfels, Algebraic Statistics for Computational Biology, Cam- bridge University Press, 2004.
First steps in tropical geometry, Idempotent mathematics and mathematical physics. Richter-Gebert, Theobald J Sturmfels, B Richter-Gebert, T Sturmfels, Theobald, Contemp. Math. 377Amer. Math. SocRichter-Gebert, Sturmfels, Theobald J. Richter-Gebert, B. Sturmfels et T. Theobald , First steps in tropical geometry, Idempotent mathematics and mathematical physics, Contemp. Math., 377, (2005), 289-317 , Amer. Math. Soc., Providence, RI, 2005.
Polynomial amoebas and convexity. H Rullgård, Research Reports In Mathematics Number. 8Department Of Mathematics Stockholm UniversityH. Rullgård, Polynomial amoebas and convexity, Research Reports In Mathematics Number 8,2001, Department Of Mathematics Stockholm University.
O Viro, Arxiv: AG/0611382Patchworking real algebraic varieties. O. Viro, Patchworking real algebraic varieties, preprint: http://www.math.uu.se/ oleg; Arxiv: AG/0611382
. Analyse Algébrique. 175Université Pierre et Marie CurieInstitut de Mathématiques de Jussieu (UMR 7586. rue du Chevaleret. 75013 Paris E-mail address: [email protected] de Mathématiques de Jussieu (UMR 7586), Université Pierre et Marie Curie, Analyse Algébrique, 175, rue du Chevaleret,, 75013 Paris E-mail address: [email protected]
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[] |
[
"Isometries of a Generalized Numerical Radius on Compact Operators",
"Isometries of a Generalized Numerical Radius on Compact Operators"
] |
[
"Maria Inez Cardoso Gonçalves \nDepartamento de Matemática\nUniversidade Federal de Santa Catarina -Trindade -Florianópolis -SC -88\n040-900Brazil\n",
"Vladimir G Pestov \nDepartamento de Matemática\nUniversidade Federal de Santa Catarina -Trindade -Florianópolis -SC -88\n040-900Brazil\n\nDepartment of Mathematics and Statistics\nUniversity of Ottawa\nK1N 6N5OttawaONCanada\n"
] |
[
"Departamento de Matemática\nUniversidade Federal de Santa Catarina -Trindade -Florianópolis -SC -88\n040-900Brazil",
"Departamento de Matemática\nUniversidade Federal de Santa Catarina -Trindade -Florianópolis -SC -88\n040-900Brazil",
"Department of Mathematics and Statistics\nUniversity of Ottawa\nK1N 6N5OttawaONCanada"
] |
[] |
We describe all isometries of the q-numerical radius on the space K(H) of compact operators on a (possibly infinite-dimensional) Hilbert space H.
|
10.1016/j.laa.2014.02.040
|
[
"https://arxiv.org/pdf/1310.5155v1.pdf"
] | 119,716,276 |
1310.5155
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a50ef705fd4e9c973b768b6ef99ca2d5a25e53f1
|
Isometries of a Generalized Numerical Radius on Compact Operators
18 Oct 2013
Maria Inez Cardoso Gonçalves
Departamento de Matemática
Universidade Federal de Santa Catarina -Trindade -Florianópolis -SC -88
040-900Brazil
Vladimir G Pestov
Departamento de Matemática
Universidade Federal de Santa Catarina -Trindade -Florianópolis -SC -88
040-900Brazil
Department of Mathematics and Statistics
University of Ottawa
K1N 6N5OttawaONCanada
Isometries of a Generalized Numerical Radius on Compact Operators
18 Oct 2013arXiv:1310.5155v1 [math.FA]q-numerical radiusisometriescompact operators 2010 MSC: 47A1215A6015A0447A3046B2046B28
We describe all isometries of the q-numerical radius on the space K(H) of compact operators on a (possibly infinite-dimensional) Hilbert space H.
Introduction
Let 0 < q ≤ 1. The q-numerical radius of a bounded linear operator A on a Hilbert space H is given by r q (A) = sup{ Ax, y : x = y = 1, x, y = q}.
The q-numerical radius is a norm on B(H), equivalent to the uniform (spectral) norm. For q = 1, this reduces to the classical numerical radius:
r(A) = sup{ Ax, x : x = 1}.
In this article, we describe the isometries of the space K(H) of compact operators on an infinite-dimensional complex Hilbert space with regard to the q-numerical radius. In the finite-dimensional case, the description was obtained in [1], while the case of the classical numerical radius (also in finite dimensions) was previously treated by Lešnjak [6], Li andŠemrl [7] and Li et al. [8].
Here is the main result. where A † denotes either A ot A t or A * orĀ.
Here A * is the usual adjoint operator, given by
x, Ay = A * x, y ,
A t is the traspose of A, x, A t y = ȳ, Ax ,
andĀ is the complex conjugate of A,
x,Āy = x, Aȳ .
While the strategy of the proof is broadly similar to that in [1], the infinite dimensional case poses numerous difficulties and requires new techniques. For instance, one has to simultaneously deal with a number of different topologies on various spaces of operators, which all coincide between themselves in the finite-dimensional situation. The main challenge was the usage of extreme points, which was the key component of the proofs in [8,1]. If one defines the operator C q = qE 11 + 1 − q 2 E 12 on H = ℓ 2 , then the unit ball of the dual norm to the generalized numerical radius is the convex hull of the saturated unitary orbit SU (C q ) = {λU * C q U : λ ∈ C, |λ| = 1, U ∈ U(H)}.
It is easy to see that all points of SU (C q ) are extreme points, indeed exposed points of the dual unit ball. However, we were unable to prove that the set of extreme (or exposed) points of the dual ball is exactly the set SU (C q ), like it is in a finite-dimensional situation. Thus, a key component of the proofs in [8,1] was missing, and we had to find an ingenious way around it, by using points of weak continuity with regard to the Hilbert-Schmidt norm topology combined with David Milman's converse to the Krein-Milman theorem.
We believe that this technique can be of interest in connection with the next natural open problem: that of characterizing isometries of the generalized numerical radius on a large space B(H) of all bounded operators.
Notice also that in order to simplify notation, in the sequel we will often only give proofs for H = ℓ 2 , but they remain valid for an arbitrary H = ℓ 2 (Γ).
Generalized numerical radius
Duality
Let H be an infinite-dimensional separable Hilbert space. As usual, we will identify the space C 1 (H) (the first Schatten class) of all trace class operators on H, equipped with the trace-class norm T 1 = tr (|T |), with a predual of the space B(H), equipped with the uniform operator norm.
Let us remind how to define a pairing ·, · between C 1 (H) and B(H). If C is a trace-class operator and T is a bounded linear operator on H, the value of C, T is given by C, T = tr (CT ).
(1)
One can now prove that elements of B(H) are exactly the bounded linear functionals on C 1 (H), and that C 1 (H) * ∼ = B(H). The space K(H) of all compact operators on H, equipped with the uniform operator norm, is, under the above duality, the predual of C 1 (H). In particular, one has K(H) * * = B(H).
For more on the duality and on the above classes of operators, see e.g. [11], 1.15 and 1.19.
In the finite dimensional situation (dim H < ∞), this duality allows to conveniently identify all the spaces K(H), C 1 (H), B(H), as well as its dual space B(H) * , between themselves. In the infinite dimensions one has to treat them separately.
C-Numerical radius
Fix an operator C ∈ C 1 (H). For any A ∈ B(H) the generalized Cnumerical range of A is defined by:
W C (A) = {tr (CU * AU) : U ∈ U(H)},
and the generalized C-numerical radius r C of A is defined by Proof. From the definition of r C (A) we have that r C (A) ≥ 0 for all A ∈ B(H), so we need to show that if r C (A) = 0 then A = 0. Suppose C is non-scalar and tr (C) = 0, then if r C (A) = 0 we have that tr (CU * AU) = 0 for all unitary U, this implies that W C (A) is a singleton. By lemma 4.2 in [5], we have that A is a scalar operator, say A = λI, where λ is a scalar and I denotes the identity operator.
r C (A) = sup{|tr (CU * AU)| : U ∈ U(H)}.
Therefore we have that: 0 = tr (CU * AU) = λtr (C), since tr (C) = 0, this implies that λ = 0 and therefore A is the zero operator.
One can easily verify that r C (αA) = |α|r C (A), for all A ∈ B(H) and r C (A + B) ≤ r C (A) + r C (B), for all A, B ∈ B(H).
Let us put the generalized numerical radius in the context of duality. Clearly,
r C (A) = sup T Re C, A ,
where the supremum is taken over the saturated unitary orbit of A in C 1 (H):
SU (A) = {λU * AU : λ ∈ C, |λ| = 1, U ∈ U(H)}.
That is to say,
r C (A) = sup |λ| = 1 U ∈ U(H)
Re(tr (λUC * U * A)).
Equivalently, r C (A) is the supremum of the linear functional −, A over the convex circled hull of the unitary orbit of C. In the case where C is nonscalar and tr (C) = 0, Theorem 2.1 implies that this hull is absorbing and bounded in the vector space C 1 (H), so is the unit ball of a certain norm, r * C . The dual norm to r * C is the norm r C on B(H). We do not know if in this generality the norm r * C is the dual norm to the restriction of r C to K(H), nor whether the norm r C is equivalent to the uniform operator (spectral) norm on B(H). In particular, we cannot assert that the norms r C and r * C are complete. However, the answers to all of these questions are positive in the special case of a q-numerical radius.
q-Numerical radius
In the important particular case where H = ℓ 2 and C = E 11 one recovers the classical numerical radius:
r(A) = sup x =1 | Ax, x |.
It is well known that the spectral radius is a norm on B(H) equivalent to the uniform norm, moreover
r(A) ≤ A ≤ 2r(A).
(See e.g. Th. 2.14 in [4].)
The case of interest for us is a more general case of the q-numerical radius, where 0 < q ≤ 1:
r q (A) = sup x = y =1, x,y =q | Ax, y |.
In this case, the matrix C assumes the form
C q = qE 11 + 1 − q 2 E 12 .
Geometric considerations in a two-dimensional Euclidean space imply the following.
Lemma 2.2. For all A ∈ B(H), r q (A) ≥ qr(A).
Proof. Let ξ = 1 and Aξ, ξ = ±α, where α > 0. Identify ξ with the second coordinate vector e 2 of the plane R 2 spanned by ξ, Aξ, so that Aξ belongs to either the first or the fourth quadrant ( Aξ, e 1 ≥ 0). Thus,
Aξ = ( √ 1 − α 2 , ±α)
. Choose ζ with ζ = 1, ξ, ζ = q, and ζ belonging to the first quadrant if so does Aξ, or the second quadrant if Aξ is in the fourth. One can write ζ = (± 1 − q 2 , q). Now one has
| Aξ, ζ | = |± √ 1 − α 2 1 − q 2 ± αq| = √ 1 − α 2 1 − q 2 + αq,
whence the result follows.
We conclude: r q (A) ≥ (q/2) A . At the same time it is evident that r q (A) ≤ A . Thus, the norm r q on B(H) is equivalent to the uniform (spectral) norm.
As a consequence, K(H) is weakly dense in B(H) relative to the qnumerical radius (because the same is true of the uniform operator norm). It follows that the norm r * q on C 1 (H) as defined earlier is the dual norm to the restriction of r q on K(H). In particular, r * q is equivalent to the trace class norm on C 1 (H). We also conclude that the norm r q on B(H) and on K(H) and the norm r * q on C 1 (H) are complete. On the space of compact operators, sharper equivalence constants follow from the results of [8].
r q (A) ≤ A ≤ βr q (A), A ∈ B(H),(2)
where
β = max{1/p, 1/q} if 1 2 ≤ p, √ 5 − 4p/|q|, if 1 4 ≤ p < 1 2 , 2/q, otherwise.
Proof. The above result was established in the case dim H < ∞ in [8], Theorem 3.5. If we fix an orthonormal basis for H, the inequalities in (2) will hold for all operators A represented by matrices with finitely many entries. The same is true for the operators belonging to the norm completion of the linear space of such operators, that is, K(H). Proof. If A ∈ SU (C q ), then there exist a scalar θ, |θ| = 1 and an unitary operator U ∈ U(H) such that A = θU * C q U. Since U is a unitary operator and C q is a rank one operator of Hilbert-Schmidt norm 1 and tr(C q ) = q, we have that A is a rank one operator, |tr(A)| = |q| and ||A|| = 1. Suppose now that A is a rank one operator, tr(A) = q and A 2 = 1. Let's show that A ∈ SU (C q ).
Saturated unitary orbit
Since A is a rank one operator, there exists x, y ∈ H such that A = x⊗y * . We can suppose without loss of generality that ||x|| = 1. Since tr(A) = q, from the definition of trace of a rank one operator we have that x, y = q. Since 1 = ||A|| 2 = ||x||||y||, we also have that ||y|| = 1.
If
|q| = 1, from x, y = q we have that there exists θ ∈ C, |θ| = 1 such that y = θx. Let U ∈ U(H) such that U * x = e 1 , then U * AU = U * (x⊗y * )U = θe 1 ⊗ e * 1 =θE 11 =θC 1 and therefore A ∈ SU (C q )
. Suppose now |q| < 1. In this case x and y are linearly independent. Let z = y − y, x x = y − qx. Then z and x are orthogonal and moreover ||z|| 2 = 1 − |q| 2 . Once again we can find U ∈ U(H) such that U * x = ||x||e 1 = e 1 and U * z = ||z||e 2 = 1 − |q| 2 e 2 . Therefore U * AU = U * (x ⊗ y * )U = e 1 ⊗ ( 1 − |q| 2 e 2 + qe 1 ) * = C q . Therefore A ∈ SU (C q ).
Lemma 3.2. Let k ∈ N.
The set R k of all operators from ℓ 2 to itself having rank ≤ k is closed in the strong operator topology.
Proof. For a T / ∈ R k , choose k + 1 orthonormal vectors in the range of T :
T (x i ) = y i , i = 1, 2, . . . , k + 1, y i ⊥ y j , i = j.
Define an open neighbourhood V of T in the strong operator topology by the condition
S(x i ) − y i < 1 2n .
For any S ∈ V the vectors z i = S(x i ) are linearly independent. Indeed, let us consider a nontrivial linear combination n i=1 λ i z i , assuming also
λ 2 i = 1. Now, n i=1 λ i z i = n i=1 λ i y i + n i=1 λ i (S(x i ) − y i ).
The norm of the first vector on the r.h.s. is 1, while the norm of the second is bounded by Proof. According to Theorem 3.1, the set SU (C q ) is the intersection of the set R 1 of all rank one operators (closed in the strong operator topology by Lemma 3.2, therefore in the trace class norm topology), the unit sphere with regard to the Hilbert-Schmidt norm (which is closed in the Hilbert-Schmidt norm topology, therefore in the finer trace class norm topology), and the level surface of the function T → |tr (T )| which is continuous with regard to the trace class norm.
n i=1 |λ i | S(x i ) − y i ≤ 1 2 .T − S 2 ≤ T − S 1 ≤ √ 2k T − S 2 .
Proof. Recall that, given a compact operator T , one can write Proof. By Corollary 3.3, SU (C q ) is complete with regard to the trace class metric, and since the two metrics are Lipschitz equivalent, the same holds for the Hilbert-Schmidt metric.
T (x) = ∞ i=1 λ i x, v i u i ,(3)
We will be dealing with two weak * topologies on the space C 1 (H): the one coming from the duality with K(H), which serves as the weak * -topology for the trace class norm (equivalently, for the norm r * q ) and which we denote w * 1 , and the weak * -topology for the completion of the space with regard to the Hilbert-Schmidt norm (that is, the weak topology), which we will denote w * 2 . The w * 1 -topology is finer than the w * 2 -topology, because there are more compact operators than Hilbert-Schmidt operators.
Lemma 3.7. At every point of the unit ball of the normed space (C 1 (H), r * q ) belonging to the saturated unitary orbit SU (C q ) the w * 2 -topology is finer than the Hilbert-Schmidt topology (and thus, the two topologies coincide).
Proof. It is well known and easily seen that every point of the unit sphere of a Hilbert space is a point of weak continuity, that is, the neighbourhood filters in the weak topology and in the norm topology coincide at this point. Moreover, the same is true if the neighbourhoods of the point are considered not just on the sphere, but in the entire Hilbert unit ball. Denote B 2 the unit ball of C 1 (H) with regard to the Hilbert-Schmidt norm, and B q the unit ball with regard to r * q . If x is a point on the unit sphere of B 2 , then for each ǫ > 0 there is δ so that whenever y ∈ B 2 and x, y > 1 − δ, one has x − y 2 < ǫ. Now let x ∈ SU (C q ). Then x is on the unit sphere of B 2 , and the functional φ(y) = x, y is w * 2 -continuous. If y ∈ B q and φ(y) > δ, where δ = δ(ǫ) as above, then y ∈ B 2 and so x − y 2 < ǫ.
Recall David Milman's (partial) converse to the Krein-Milman theorem ( [9], Theorem 1): if T is a subset of a convex compact subset of a topological vector space and the closed convex hull of T is all of K, then every extreme point of K belongs to the closure of T . As an immediate corollary, we obtain: Lemma 3.8. All the extreme points of the unit ball of the norm r * q are contained in the w * 1 -closure of the saturated unitary orbit of C q .
Proof. The unit ball of r * q is the closed convex hull of the saturated unitary orbit SU (C q ). Equipped with the weak * topology relative to the duality between K(H) and C 1 (H), the unit ball is compact. Now apply the converse to the Krein-Milman theorem with T = SU (C q ). Remark 3.9. Even if the topologies (in our case, weak * topology and the trace class topology) coincide on a set (the saturated unitary orbit), one cannot conclude that the closures of this set in the ambient space with regard to the two topologies are the same. We do not know whether the extreme points of the convex closure of the saturated unitary orbit are all contained in the saturated unitary orbit.
Isometry of the generalized numerical radius
Linearity
Let φ is a q-numerical radius isometry of (K(H), r q ). Then φ is the sum of a translation by some S 0 ∈ K(H) and a q-numerical radius isometry preserving zero. From now, we will therefore presume that φ(0) = 0. The Mazur-Ulam theorem (cf. e.g. Theorem 1.3.5 in [2]) says: if φ is an isometry from a normed linear space X onto a normed linear space Y , and if φ(0) = 0, then φ is real linear.
Denote ψ the dual linear map. We conclude: ψ is a real linear isometry of (C 1 (H), w * q ).
Representing ψ as P (·)Q
The following result is an infinite-dimensional analogue of Lemma 2.2 in [1].
Proof. Let R = u ⊗ v t . Lets show that M(R) = L 1 + L 2 , where L 1 = {u ⊗ y * : y ∈ ℓ 2 }, L 2 = {x ⊗ v * : x ∈ ℓ 2 }.
First assume that R = A + B for operators A, B in SU (C). If A = a ⊗ c t and B = b ⊗ d t for a, b, c, d ∈ ℓ 2 , then either a and b are linearly dependent (in which case, we may take a = b = u), or c and d are linearly dependent and again we may take c = d = v. This means that either both A and B are in L 1 or both are in L 2 . Therefore M(R) ⊆ L 1 + L 2 .
For the converse, we may replace R by any operator from its unitary orbit, hence we may assume that R = ξE 11 + ηE 12 and so u = e 1 . By assumption, we have |ξ| < 2q and |η| < 2p. For every q ′ and p ′ with |ξ|/2 < q ′ < q and |η|/2 < p ′ < p, there exist complex numbers z j , 1 ≤ j ≤ 4, such that
|z 1 | = |z 2 | = q ′ , |z 3 | = |z 4 | = p ′ . Let r = (1 − (p ′ ) 2 − (q ′ ) 2 ) 1/2 , t ∈ R, k ≥ 3 and A t = z 1 E 11 + z 2 E 12 + re it E 1k , B t = z 3 E 11 + z 4 E 12 − re it E 1k .
Then A t , B t ∈ SU (C) and A t + B t = R. With all the various choices of p ′ , q ′ and t, it is easy to see that M(R) contains E 1j and iE 1j for j ≥ 1, hence it contains e 1 ⊗ y * for every y ∈ ℓ 2 . This proves that L 1 ⊂ M(R). By symmetry we also get L 2 ⊂ M(R). This proves that M(R) = L 1 + L 2 . Each of L 1 and L 2 is a space of infinite dimension over R, therefore M(R) has infinite dimension over R.
The following is an infinite-dimensional version of Lemma 2.3 in [1]. If A + B has rank 2, then both inclusions must be equalities. Let now R be a rank two bounded linear operator. Denote R = range (R) and N = null (R). If R is a sum of two rank one bounded operators A = a ⊗ b * and B = c ⊗ d * , then from above we must have R = span {a, c} and N ⊥ = span {b, d}. So the operators A and B vanish on N and map the two-dimensional space N ⊥ into the two-dimensional space R.
This reduces the problem to the case n = 2, which is treated in an identical way to that in the proof of Lemma 2.3 in [1]. Now we can conclude that ψ preserves rank one operators, just like in [1], Lemma 2.4, which is the finite-dimensional analogue of our next result. Lemma 4.3. Assume that ψ is a bijective bounded real-linear operator on K(ℓ 2 ) and that ψ(SU (C q )) = SU (C q ). Then ψ preserves rank one bounded linear operators.
Proof. Let R ∈ K(ℓ 2 ) be of rank one and let R ′ = λR, where 0 < λ < (min{2p, 2q}) R −1 . The operator R ′ satisfies the norm condition of Lemma 4.2. Thus the space M (R ′ ) is infinite-dimensional. By real linearity, we have that ψ maps M (R ′ ) onto M (ψ(R ′ )). Therefore M (ψ(R ′ )) is infinitedimensional as well. By Lemma 4.1, ψ(R ′ ) is not of rank 2. Since ψ(R), just like R, can be represented as a sum of two operators from SU (C q ), its rank does not exceed 2. It is also nonzero since ψ is a bijection. Therefore ψ(R ′ ) and hence also ψ(R) have rank one. Now let us establish an analogue of Corollary 2.5 in [1]. Lemma 4.4. There exist operators P, Q ∈ GL(H) such that for all A ∈ C 1 (H),
ψ(A) = P A † Q, where A † denotes either A or A t or A * orĀ.
Proof. According to Theorem 3.1 in [10], there exist bijective linear operators P, Q from H to itself so that for all A ∈ K(H),
ψ(A) = PÃQ or ψ(A) = PÃ t Q,
whereà is obtained from A by applying a field automorphism c →c entrywise. Since the only real-linear automorphisms of C are the identity and complex conjugation, the result will follow once we establish that P, Q are bounded, with bounded inverse. We will only give an argument in the case A † = A, the rest is similar. In this case, for every x, y ∈ H, one has
ψ(x ⊗ y * ) = P (x ⊗ y * )Q * .
Let z ∈ H be a non-zero vector. For every x ∈ H the operator x ⊗ z * is of rank one. The mapping
i z : H ∋ x → x ⊗ z * ∈ C 1 (H)
is a linear isometric embedding with regard to the trace class norm if z = 1. This can be seen by choosing two orthonormal bases (v n ), (u n ) so that x = x · v 1 and u 1 = z:
x ⊗ z * 1 = x −, v 1 u 1 + 0 · ... 1 = x .
Therefore, if z = 0, the mapping i z is an isomorphic embedding of H, admitting an isomorphic inverse. We conclude:
P = i −1
Qz • ψ • i z , and this composition is bounded by the assumption on ψ. A similar argument establishes that Q is bounded, and applying the above to ψ −1 , one concludes that the inverses of P, Q are bounded as well.
Corollary 4.5. The map ψ is an isomorphism of C 1 (H) with regard to the Hilbert-Schmidt norm, and therefore uniquely extends to an isomorphism of C 2 (H).
Proof. It is enough to apply the property
AB 2 ≤ A B 2 .
Proving that P, Q are unitary
The following should be obvious. Proof. Let A ∈ SU (C) be any point. Since ψ(A) is an extreme point of B C , by the partial converse to the Krein-Milman theorem (cf. Lemma 3.8), ψ(A) belongs to the w * 1 -closure of SU (C). The application ψ, being an invertible trace class bounded dual linear map, is a homeomorphism of the unit ball of r * C with regard to the w * 1 -topology, and at the same time a homeomorphism the unit ball of r * C with regard to the Hilbert-Schmidt topology. By Lemmas 3.7 and 4.6, the two topologies on this ball must coincide at the point ψ(A). Since every w * 1 -neighbourhood of ψ(A) contains a point of SU (C q ), the same is true of every Hilbert-Schmidt neighbourhood of the same point, and so ψ(A) must belong to the Hilbert-Schmidt closure of SU (C) in the unit ball of r * C . But SU (C) is complete in the Hilbert-Schmidt topology by Corollary 3.6, hence closed in the Hilbert-Schmidt topology on every ambient space, including the ball in question. Now we can conclude that P, Q are unitary just like in [1], Proposition 2.8, whose proof remains true for the infinite dimensional case with a suitable adjustements, as follows.
Proposition 4.8. Let H be an infinite dimensional Hilbert space and let ψ(A) = P A † Q for every A ∈ C 1 (H), where A † is either A or A t or A * or A. If ψ(SU (C)) = SU (C), then each of P and Q is a scalar multiple of a unitary operator on H and P Q = λI for a complex number λ of modulus one. Consequently, there exists a unitary operator U ∈ U(H) such that ψ(A) = λU * AU for every A ∈ C 1 (H).
Proof. Since each of the maps A → A t , A →Ā and A → A * preserves the saturated unitary orbit, it suffices to prove the Proposition for the case ψ(A) = P AQ.
Let x, y ∈ H be such that x ⊥ y and x = y = 1. Let s ∈ R and z = qx + e is 1 − q 2 y. Then x ⊗ z * is a rank one operator of the Hilbert-Schmidt norm 1 and trace q, hence x ⊗ z * ∈ SU (C q ) by Theorem 3.1. Thus P (x ⊗ z * )Q = ψ(x ⊗ z * ) ∈ SU (C q ), which implies that (a) P (x ⊗ z * )Q = 1, amd (b) |tr (P (x ⊗ z * )Q)| = q.
From condition (a), we have 1 = P (x ⊗ z * )Q = P x · Q * z , therefore qQ * x + e is 1 − q 2 Q * y = K for all s ∈ R, where K is a positive constant. Lemma 2.6 in [1] says that if u, v are two vectors in a complex Hilbert space with the property u + e is v = 1 for all s ∈ R, then u ⊥ v. We conclude: Q * x ⊥ Q * y.
We conclude that Q * preserves orthogonality. By Lemma 2.7 in [1], a bounded linear operator on a Hilbert space is a scalar multiple of isometry if and only if it preserves orthogonality of vectors. We conclude that Q * is a scalar multiple of an isometry of H, and therefore a scalar multiple of a unitary operator on H. Similarly, P is a scalar multiple of a unitary operator.
The trace condition (b) implies that q QP x, x + 1 − q 2 e is QP x, y = q, for every s ∈ R.
This implies that QP x, x = 0 or | QP x, x | = 1. Since this is true for every unit vector x, we get that W (QP ), the classical numerical range for QP , is included in the union of the unit circle and {0}. The well-known convexity of the classical numerical range (the Toeplitz-Hausdorff theorem [3,12]) implies that W (QP ) is a singleton of modulus 1 or 0. But QP is evidently nonzero, hence QP = λI, for a complex number of modulus one. Now we are ready to establish the main result.
Proof of Theorem 1.1. For the "if" part, it is easy to see that r q (A) = r q (Ā) = r 1 (A t ) = r q (A * ).
Conversely, if φ is an isometry of the normed space (C 1 (H), r q ) satisfying φ(0) = 0, then φ is linear, and there exist a unitary operator U on H and a complex number λ of modulus 1 such that the dual operator ψ on C 1 (H) satisfies ψ(A) = λU * A † U for all A ∈ C 1 (H).
This implies, in view of the pairing between C 1 (H) and K(H) (Eq. (1)), φ(A) =λUA † A * for every A ∈ K(H).
Theorem 1. 1 .
1Let H be an infinite-dimensional complex Hilbert space. Let 0 < q ≤ 1 and let φ : K(H) → K(H). Then φ is an isometry of the qnumerical radius if and only if there exist a compact operator S 0 ∈ K(H), a unitary operator U ∈ U(H) and a complex number µ with |µ| = 1 such that for all A ∈ K(H) φ(A) = S 0 + µU * A † U,
Theorem 2.1. r C (·) is a norm in B(H) ifand only if C is non-scalar and tr (C) = 0.
Theorem 2 . 3 .
23The norm r q on K(H) is equivalent to the uniform operator norm, with the constants
Proposition 3. 3 .
3The saturated unitary orbit SU (C q ) is closed in the trace class norm on C 1 (H).
Lemma 3. 4 .
4The trace class metric and Hilbert-Schmidt metric are Lipschitz equivalent on the set R k of all operators of rank ≤ k, with constants which only depend on k: for all T, S ∈ R k ,
where (v i ) and (u i ) are suitably chosen orthonormal bases of H. Now the trace of T is given by tr (above quantities are independent of the choice of representation (3).If T, S ∈ R k , then the operator T − S has rank ≤ 2k, and therefore the number of non-zero coefficients λ i in the sum(3)representing T − S is at most 2k. The values T − S 1 and T − S 2 become the values of the ℓ 1 norm and ℓ 2 norm respectively of the same vector in the standard vector space of dimension 2k. But these are Lipschitz equivalent, with constants which only depend on the dimension.
Corollary 3. 5 .
5The trace class norm topology and the Hilbert-Schmidt norm topology coincide on the saturated unitary orbit SU (C q ). Moreover, the corresponding metrics on the orbit are Lipschitz equivalent.
Corollary 3. 6 .
6The saturated unitary orbit SU (C q ) is complete in the Hilbert-Schmidt metric.
Lemma 4 . 1 .
41Let C and R be two bounded rank one operators on ℓ 2 , where R < min{2q, 2p}, and p = (1 − q 2 ) 1/2 , and let S(R) := {A ∈ SU (C) : R = A + B for some B ∈ SU (C)}, and let M(R) = span S(R). Then M(R) is a vector space of infinite dimension over R.
Lemma 4. 2 .
2Let R be a bounded rank two operator on the Hilbert space ℓ 2 , and let R = R 1 denote the set of all bounded rank one operators on ℓ 2 . Then the set A(R) = {A ∈ R : R = A + B for some B ∈ R} spans a real vector space of dimension 7 and so dim(M(R)) ≤ 7, where M(R) is defined as in Lemma 4.1. Proof. Without loss in generality, we assume the space ℓ 2 complex and all the operators complex-linear. If A = a ⊗ b * is a rank one bounded linear operator, given by x → a b, x , then range (A) = span (a) and null (A) = {b} ⊥ . Therefore if A = a ⊗ b * and B = c ⊗ d * are rank one bounded operators, then range (A + B) ⊆ span {a, c} and (null (A + B)) ⊥ ⊆ span {b, d}.
Lemma 4. 6 .
6Let X be a set equipped with two topologies, τ and σ, and let a map f : X → X be a homeomorphism with regard to the both topologies. Denote T the set of all points x ∈ X at which the neighbourhood filter τ contains the neighbourhood filter of σ. Then T is invariant under f :f (T ) = T.
Lemma 4. 7 .
7The saturated unitary orbit SU (C) is invariant under ψ.
Theorem 3.1. An operator A ∈ B(H) belongs to SU (C q ) if and only if A is a rank one operator, |tr(A)| = |q| and A 2 = 1 (in the Hilbert-Schmidt, or Frobenius, norm).
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[] |
[
"On a Linearized Problem Arising in the Navier-Stokes Flow of a Free Liquid Jet",
"On a Linearized Problem Arising in the Navier-Stokes Flow of a Free Liquid Jet"
] |
[
"Shaun Ceci \nDepartment of Mathematics\nLe Moyne College Syracuse\n13214-1399NYUSA\n",
"Thomas Hagen \nDepartment of Mathematical Sciences\nThe University of Memphis Memphis\n38152-3240TNUSA\n"
] |
[
"Department of Mathematics\nLe Moyne College Syracuse\n13214-1399NYUSA",
"Department of Mathematical Sciences\nThe University of Memphis Memphis\n38152-3240TNUSA"
] |
[] |
In this work, we analyze a Stokes problem arising in the study of the Navier-Stokes flow of a liquid jet. The analysis is accomplished by showing that the relevant Stokes operator accounting for a free surface gives rise to a sectorial operator which generates an analytic semigroup of contractions. Estimates on solutions are established using Fourier methods. The result presented is the key ingredient in a local existence and uniqueness proof for solutions of the full nonlinear problem.
|
10.1016/j.jmaa.2012.05.023
|
[
"https://arxiv.org/pdf/1112.4029v1.pdf"
] | 119,693,378 |
1112.4029
|
4a5b8ae6e360fb8c2f9dcaea31e0f89f4818af76
|
On a Linearized Problem Arising in the Navier-Stokes Flow of a Free Liquid Jet
17 Dec 2011
Shaun Ceci
Department of Mathematics
Le Moyne College Syracuse
13214-1399NYUSA
Thomas Hagen
Department of Mathematical Sciences
The University of Memphis Memphis
38152-3240TNUSA
On a Linearized Problem Arising in the Navier-Stokes Flow of a Free Liquid Jet
17 Dec 2011AMS (MOS) Subject Classification 47D0635P0576D07
In this work, we analyze a Stokes problem arising in the study of the Navier-Stokes flow of a liquid jet. The analysis is accomplished by showing that the relevant Stokes operator accounting for a free surface gives rise to a sectorial operator which generates an analytic semigroup of contractions. Estimates on solutions are established using Fourier methods. The result presented is the key ingredient in a local existence and uniqueness proof for solutions of the full nonlinear problem.
Introduction
In this paper, we are concerned with solutions of the modified Stokes problem
D m 3 q| Γ ℓ = D m 3 q| Γ 0 , D k 3 v| Γ ℓ = D k 3 v| Γ 0 for 0 ≤ m ≤ s − 2, 0 ≤ k ≤ s − 1 (1.5)
for suitable initial data and sufficiently general body forces f . Here Ω denotes the set
Ω = D × (0, ℓ),(1.6)
where ℓ > 0 and D = (a 1 , a 2 ) ∈ R 2 : a 2 1 + a 2 2 < κ 2 for some radius κ > 0. We are primarily interested in thin fluid filaments (i.e., where κ is small relative to the axial period ℓ) and hence we can assume κ < 1. Throughout, Cartesian coordinates in R n will be written in the form (a 1 , . . . , a n ). S F denotes the portion of ∂Ω corresponding to the cylinder surface, given by S F = (a 1 , a 2 , a 3 ) ∈ ∂Ω : a 2 1 + a 2 2 = κ 2 ,
Γ 0 and Γ ℓ denote the opposing faces of ∂Ω (1.8) and s ≥ 2. The quantity v is the (Lagrangian) fluid velocity, q is the (Lagrangian) fluid pressure, n is the outward unit normal to Ω, and µ > 0 is the (constant) fluid viscosity. The moving free-boundary condition is abbreviated by S(v, q) = 0 on S F , where
Γ 0 = D × {0}, Γ ℓ = D × {ℓ},S(v, q) = qn i − µ 3 j=1 (D j v i + D i v j )n j 3 i=1
.
(1.9)
Note that the conditions (1.5) require that solutions be periodic of period ℓ in the a 3 -direction. Our objective in this work is to show that this Stokes problem allows unique solutions for given initial data and arbitrary T > 0. We now briefly motivate how the linear problem (1.1)-(1.5) arises in the study of free fluid jets. We consider the three-dimensional motion of a jet bounded by an evolving free surface under isothermal conditions and without surface tension. The fluid is assumed to be viscous, homogeneous, incompressible, and Newtonian. To model the fluid jet, the three-dimensional incompressible Navier-Stokes equations are coupled with periodic boundary conditions in the axial direction as in [19] and moving free-surface boundary conditions in the radial direction:
D t u + (u · ∇)u − µ∆u + ∇p = g e 3
on Ω(t) (1.10)
∇ · u = 0 on Ω(t) (1.11) (pI − µ(∇u + ∇u T )) · N = P 0 N on S F (t) (1.12) D m 3 p| Γ ℓ (t) = D m 3 p| Γ 0 (t) , D k 3 u| Γ ℓ (t) = D k 3 u| Γ 0 (t) for 0 ≤ m ≤ s − 2, 0 ≤ k ≤ s − 1 (1.13)
D t y(t, ·) = u(t, y(t, ·)) on Ω (1.14)
u(0, ·) = u 0 (·) on Ω (1. 15) y(0, ·) = I(·) on Ω.
(1. 16) In this Eulerian description u is the fluid velocity, p is the fluid pressure, y is the fluid parcel trajectory map, P 0 is the (constant) ambient pressure, and g is the acceleration due to gravity. At time t, the fluid domain, free surface, and periodic faces are given by Ω(t) = y(t, Ω), S F (t) = y(t, S F ), Γ 0 (t) = y(t, Γ 0 ), and Γ ℓ (t) = y(t, Γ ℓ ), respectively. N is the outward unit normal to Ω(t) and e 3 = (0, 0, 1) T . The periodic boundary condition is chosen because it leads to a simpler functional setting and avoids all axial boundary layer difficulties while retaining the primary mathematical challenges of the problem. In addition, the assumption of periodicity in the axial direction has been successfully used to study physical flow phenomena in the numerical simulation of drop dynamics for viscoelastic fluid jets [11].
To obtain a fixed fluid domain, it is useful to shift to a Lagrangian specification of the flow field.
The problem (1.10)-(1.16) then becomes
D t v i − µ 3 j,k,m=1 λ j,k D k (λ j,m D m v i ) + 3 k=1 λ i,k D k q = gδ 3,i for i ∈ {1, 2, 3} on Ω (1.17) 3 j,k=1 λ j,k D k v j = 0 on Ω (1.18) qn i − µ 3 j,k=1 (λ j,k D k v i + λ i,k D k v j )n j = 0 for i ∈ {1, 2, 3} on S F (1.19) D m 3 q| Γ ℓ = D m 3 q| Γ 0 , D k 3 v| Γ ℓ = D k 3 v| Γ 0 , D k 3 x| Γ ℓ = D k 3 x| Γ 0 for 0 ≤ m ≤ s − 2, 0 ≤ k ≤ s − 1 (1.20) D t x = v on Ω (1.21) v(0, ·) = u 0 (·) on Ω (1.22) x(0, ·) = 0 on Ω. (1.23)
Here v is the Lagrangian fluid velocity, q is the difference between the Lagrangian fluid pressure and the ambient pressure, x = y−I is the fluid parcel displacement map, and δ i,j denotes the Kronecker delta. One consequence of converting the governing equations to the Lagrangian specification is the introduction of a priori unknown quantities involving derivatives of the trajectory map y, which we denote by
λ i,j (t, a) : Ω → R where Λ = λ i,j = (∇y) −1 = D 1 y 1 D 1 y 2 D 1 y 3 D 2 y 1 D 2 y 2 D 2 y 3 D 3 y 1 D 3 y 2 D 3 y 3 −1 .
(1.24)
It readily follows from a continuity argument that x ≈ 0 for t ≪ 1 so that Λ is approximately equal to the 3 × 3 identity matrix for small times t. Taking λ i,j = δ i,j in (1.17)-(1.23) we obtain, with the exception of the initial data, the linearized Stokes problem (1.1)-(1.5). Details of the Lagrangian coordinate change for closely related problems are given by Beale [5] and Teramoto [19].
To obtain a local-in-time solution for the full nonlinear problem (1.17)-(1.23), a fixed point approach can be used, which requires unique solvability of a slightly more general version of the linearized problem discussed here. That general case, however, can be shown to reduce to the one treated in this paper. This overarching strategy of studying a modified Stokes problem in Lagrangian coordinates and its use in a fixed point argument was championed by Beale in [5] for a semi-infinite "ocean" of fluid having a free upper surface and fixed bottom. Teramoto subsequently adapted Beale's techniques to gain similar results for a free surface problem involving axisymmetric flow down the exterior of a solid vertical column of sufficiently large radius [19].
It is important, however, to note that while the fluid jet appears similar to the problem considered in [19], there are key differences which require that we build upon the work done by Beale and Teramoto. For example, unlike the fluid domains under consideration in [3,4,5,6,19], there is no stationary surface opposite the free surface to which a Dirichlet boundary condition can be assigned. Foremost among the consequences of not having such a condition are the loss of general applicability of the Poincaré inequality and the loss of invertibility of the modified Stokes operator, which is central to the analysis. Moreover, where Teramoto is able to exploit axisymmetry and cylindrical coordinates to reduce his problem to two dimensions, the same approach introduces significant challenges in the fluid jet case since the Navier-Stokes equations in cylindrical coordinates have singular coefficients when the axis at r = 0 is contained in the fluid domain.
Also in contrast to [3,4,5,6,19], we utilize an elegant semigroup approach to the linearized problem. This has the benefit of immediately providing a solution to the abstract Cauchy problem associated with the problem (1.17)-(1.23). We improve upon the spectral analysis of the corresponding operator −A found in [5] and show that it is, in this setting, a sectorial operator which generates an analytic semigroup of contractions. Additionally, we are able to establish explicit characterizations for both spaces in the modified Helmholtz decomposition of (L 2 (Ω)) 3 .
This paper is organized as follows: in Section 2, we introduce the setting of the problem and present some preliminary lemmas which adapt and extend standard results from [5] to fit this setting; in Section 3, we derive the relevant abstract Cauchy problem and restrict the spectrum of the underlying differential linear operator A to a sector in the right half of the plane as well as provide estimates on the resolvent operator of A; in Section 4, we establish that −A is the infinitesimal generator of an analytic semigroup of contractions and use this solve the linearized problem (1.1)-(1.5).
State Spaces and Estimates
To analyze equations (1.1)-(1.5), we take as our initial fluid domain (the space occupied by the fluid at t = 0) the infinite cylinder along the a 3 -axis,
Ω ∞ = (a 1 , a 2 , a 3 ) ∈ R 3 : a 2 1 + a 2 2 < κ 2 ,(2.1)
with free surface ∂Ω ∞ = (a 1 , a 2 , a 3 ) ∈ R 3 : a 2 1 + a 2 2 = κ 2 . We restrict our attention to flow which is periodic in the a 3 direction, hence we are interested primarily in functions of the form
f = nf n (a 1 , a 2 )e 2πina 3 /ℓ ∈ H k loc (Ω ∞ ) = W k,2 loc (Ω ∞ ),(2.2)
withf n ∈ H k (D). In practice however, we will find it more convenient to work with functions over a single period. It is natural then to interpret a 3 -periodic functions on Ω ∞ as being defined on a solid torus T ⊂ R 3 . It should be clear that H k (T ) is smoothly isomorphic to the space of functions of interest. While T is a natural choice for the domain given the periodic setting, we prefer to work in the physical space occupied by Ω ∞ . To this end, we notice that there is a C ∞ diffeomorphism from one period of Ω ∞ onto T and consider the bounded set Ω = D × (0, ℓ) with boundary ∂Ω = S F ∪ Γ 0 ∪ Γ ℓ . While the use of Ω in place of Ω ∞ does give rise to minor technical issues (as opposed to T ) concerning the regularity of functions as one approaches the "artificial" corners in the boundaries, most of these problems can be dealt with by temporarily exchanging Ω for a larger subset of Ω ∞ . As such we will occasionally find a use for the set
Ω 1 = D × (−ℓ, ℓ).
Given a spatial domain U ⊂ R 3 and a time interval I ⊂ R, the following notational conventions are adopted for arbitrary function spaces X(U ) and Y (I × U ):
X(U ) = (X(U )) 3 , Y(I × U ) = (Y (I × U )) 3 , (2.3) X σ (U ) = {u ∈ X(U ) : ∇ · u = 0} , Y σ (I × U ) = {u ∈ Y(I × U ) : ∇ · u = 0} , (2.4) X = X(Ω), Y = Y ((0, T ) × Ω), (2.5) 0 X = {u ∈ X : u = 0 on S F } , 0 Y = {u ∈ Y : u = 0 on S F } . (2.6)
Here the vector and tensor fields are equipped with the Euclidean and Frobenius norms, respectively. To keep the notation simple, we use the following rule: If a function space already has a subscript, its divergence-free subspace will be denoted by appending a σ to the existing subscript. Our notation is thus closely aligned to the one chosen by Beale in [5]. Spaces not following these conventions will be explicitly defined in each instance. We now introduce the spaces fundamental to this text; though each assumes Ω as its spatial domain, the extension to Ω 1 is obvious. For the set of functions on Ω whose a 3 -periodic extensions are continuously differentiable and bounded on Ω ∞ we simply
take C k p = u| Ω : u = ∞ n=−∞û n (a 1 , a 2 )e 2πina 3 /ℓ ∈ C k Ω ∞ . Note thatû n ∈ C k (D) necessarily. Similarly, we define C ∞ p (or C k,α p )
to be the set of all such functions which are bounded and smooth (or Hölder continuous with exponent α) on Ω ∞ . It is clear that the following space is isomorphic to H k (T ):
H k p = u = ∞ n=−∞û n (a 1 , a 2 )e 2πina 3 /ℓ ∈ H k (Ω) :û n ∈ H k (D) and (u, u) H k p < ∞ (2.7) for k ∈ N 0 , where (u, v) H k p = ∞ n=−∞ k m=0 (2πn) 2m ℓ 2m−1 (û n ,v n ) H k−m (D) and u H k p = (u, u) H k p . (2.8)
The norms · H k p and · H k are equivalent norms on H k p which are actually equal for k ∈ {0, 1}. Here · H k denotes the standard norm on H k (Ω). It then readily follows that H 0 p = L 2 and
H k p = {f ∈ H k : D j 3 f | Γ ℓ = D j 3 f | Γ 0 for all 0 ≤ j ≤ k − 1} (2.9)
for k ≥ 1. We define H s p , s ∈ R + , using complex interpolation. Note that, throughout the text, we typically use r and s to denote non-integer regularity and k and m when we restrict ourselves to integer regularity.
Instead of the standard Helmholtz decomposition of L 2 , we take our lead from [5] and pursue something slightly different. In particular, to incorporate a 3 -periodicity along with the divergencefree condition into the auxiliary space, we choose our decomposition so that L 2 can be projected onto H 0 pσ . For convenience, we set P s = H s pσ and introduce the space
V s = {v ∈ P s : S tan (v) = 0 on S F },(2.10)
where S tan = S − (S · n)n is the tangential part of S. Finally, to incorporate regularity with respect to time we define the space
K s p (I × Ω) = H s/2 (I; H 0 p ) ∩ H 0 (I; H s p ). (2.11)
In contrast to [5], we now provide an explicit characterization of the orthogonal complement (P 0 ) ⊥ arising in our Helmholtz decomposition of L 2 . This result will prove important later on.
Proposition 2.1. The orthogonal complement of P 0 in L 2 has the characterization (P 0 ) ⊥ = {∇q : q ∈ 0 H 1 p }. Proof. Let Y = {∇q : q ∈ 0 H 1 p }. It is sufficient to show two things: (i) Y is closed in L 2 so that Y = (Y ⊥ ) ⊥ , and (ii) P 0 = Y ⊥ .
In order to prove (i), we will first need to show that the
orthogonal complement of X = 0 C ∞ pσ · L 2 in L 2 has the characterization X ⊥ = {∇q : q ∈ H 1 p }. Let q ∈ H 1 p , u ∈ X. There exist u k ∈ 0 C ∞ pσ such that u k → u in L 2 .
Integration by parts yields
(∇q, u) L 2 = lim k→∞ (∇q, u k ) L 2 = lim k→∞ Γ ℓ qu k · e 3 + Γ 0 qu k · (−e 3 ) = 0. (2.12)
Thus ∇q ∈ X ⊥ . Conversely, let w ∈ X ⊥ . Then, in particular, (w, u) L 2 = 0 for all u ∈ C ∞ cσ . Thus there exists p ∈ H 1 such that w = ∇p, see [18, pp. 10-11]. Now consider
u = 0 0 u(a 1 , a 2 ) ∈ 0 C ∞ pσ , (2.13)
where u ∈ C ∞ c (D) is arbitrary. Then, applying integration by parts, we obtain
0 = (w, u) L 2 = (∇p, u) L 2 (2.14) = Γ ℓ pu · e 3 + Γ 0 pu · (−e 3 ) (2.15) = Γ ℓ pu − Γ 0 pu (2.16) = (p| Γ ℓ − p| Γ 0 , u) L 2 (D) . (2.17)
Since u is an arbitrary element of a dense subset of L 2 (D) (see [12, p. 13]), this implies that
p| Γ ℓ = p| Γ 0 on L 2 (D). Hence p ∈ H 1 p by (2.9). With this characterization in hand, we can now prove (i). Let q k ∈ 0 H 1 p such that ∇q k → f ∈ L 2 . Since ∇q k ∈ X ⊥ , we have (f , u) L 2 = lim k→∞ (∇q k , u) L 2 = 0 (2.18)
for all u ∈ X. Hence f ∈ X ⊥ and so there exists p ∈ H 1 p such that f = ∇p. Notice that for n = 0
(q k ) n −p n 2 H 1 (D) ≤ ℓ 2 n 2πn ℓ 2 (q k ) n −p n 2 H 1 (D) + 2 j=1 D j ((q k ) n −p n ) 2 L 2 (D) (2.19) ≤ ℓ 2 ∇(q k − p) 2 L 2 . (2.20)
Thus (q k ) n →p n in H 1 (D). Since (q k ) n ∈ H 1 0 (D), a closed subspace of H 1 (D), we obtainp n ∈ H 1 0 (D) for n = 0. For n = 0, applying the standard Poincaré inequality yields a constant C > 0 such that
(q k ) 0 − (q m ) 0 2 H 1 (D) ≤ C ∇((q k ) 0 − (q m ) 0 ) 2 (L 2 (D)) 2 ≤ Cℓ 2 ∇(q k − q m ) 2 L 2 (2.21)
which implies that (q k ) 0 converges in H 1 0 (D). Moreover, the limit is necessarilyp 0 + λ, for some λ ∈ R, since it is readily seen that (q k ) 0 converges to this limit in the weaker
L 2 -norm. Thus f = ∇q where q = p + λ ∈ 0 H 1 p . Hence Y is closed in L 2 . Finally, we show (ii). Let u ∈ Y ⊥ and ϕ ∈ C ∞ c . Then 0 = (∇ϕ, u) L 2 = − Ω ϕ(∇ · u). (2.22)
Hence ∇ · u acts as a bounded linear functional on C ∞ c and can be extended to all of L 2 by density. This unique operator must be the zero functional and thus u ∈ P 0 . Conversely, let v ∈ P 0 . Since
L 2 = Y ⊕ Y ⊥ , there are q ∈ 0 H 1 p andṽ ∈ Y ⊥ such that v =ṽ + ∇q.
Taking the divergence of both sides of this equation yields ∆q = 0 and, by Lax-Milgram, q must be the unique solution of this equation in 0 H 1 p . Thus q = 0 and v =ṽ ∈ Y ⊥ . Thus P 0 = Y ⊥ and the claim follows.
The following are analogous to results in [5] and are readily shown using similar techniques. They are given here explicitly for the reader's covenience.
Proposition 2.2. Let P be the orthogonal projection of L 2 onto P 0 .
(1) For s ≥ 0, we have P H s p = P s and P | H s p : H s p → P s is bounded.
(2) P | K s p : K s p → K s p is bounded with norm bounded independent of T .
(3) Suppose s ≥ 1. If f ∈ H s p , then there is a uniquef ∈ H s p such that
P (∇f ) = ∇f , f | S F =f | S F , and ∆f = 0. (2.23)
We will see that, just as in [5], many crucial quantities can be cast as solutions of a particular problem involving Laplace's equation. Adapting the boundary conditions to reflect periodicity and the absence of a fixed bottom surface, the relevant problem in our setting takes the form
∆u = f in Ω, u = 0 on S F , D k 3 u| Γ ℓ = D k 3 u| Γ 0 for k ∈ {0, 1},(2.24)
where f ∈ H s−2 p is given. The following result demonstrates that (2.24) has a unique solution and provides an estimate for it in terms of the inhomogeneity f . We note that the provided proof does not draw from the corresponding proof in [5].
Proposition 2.3. For f ∈ H s−2 p , s ≥ 2, there is a unique solution u ∈ 0 H s p of ∆u = f on Ω. Additionally, there exists C > 0, independent of f , such that u H s p ≤ C f H s−2 p .
Proof. Let f = nf n e 2πina 3 /ℓ . We first consider the boundary-value problem, L n u = −f n on D with u = 0 on ∂D, where L n and its associated sesquilinear form (B n :
H 1 0 (D) × H 1 0 (D) → C) are given by L n u = −∆u + 2πn ℓ 2 u (2.25) B n [u, v] = (∇u, ∇v) L 2 (D) + 2πn ℓ 2 (u, v) L 2 (D) . (2.26)
Clearly, B n is continuous and coercive on H 1 0 (D), thus we can apply Lax-Milgram to obtain a unique weak solution,û n ∈ H 1 0 (D). The construction u = nû n e 2πina 3 /ℓ is then our candidate for the solution of the boundary-value problem in Ω. We now restrict our discussion to the case when s = k ∈ Z. Given the regularity of ∂D we can immediately conclude that eachû n ∈ H k (D) is a strong solution. Our goal is to show that u ∈ H k p . First we obtain some preliminary estimates for u n where n = 0:
B n [û n ,û n ] = (−f n ,û n ) L 2 (D) (2.27) ∇û n 2 L 2 (D) + 2πn ℓ 2 û n 2 L 2 (D) ≤ f n L 2 (D) û n L 2 (D) (2.28) û n L 2 (D) ≤ ℓ 2πn 2 f n L 2 (D) (2.29)
Notice that from this estimate we can conclude
n 2πn ℓ 2k û n 2 L 2 (D) ≤ n 2πn ℓ 2(k−2) f n 2 L 2 (D) ≤ f 2 H k−2 p < ∞. (2.30)
This gives us an estimate on the lowest order terms in the H k p -norm. For the highest order terms, standard elliptic regularity theory (e.g., see [9], p. 323) provides an estimate of the form
û n H k (D) ≤ C 1 f n H k−2 (D) , (2.31)
though the constant C 1 here generally depends on the coefficients (and hence n) of L n . However, upon closer inspection of the proof of this result (e.g., in [9]), we observe that our above estimates on û n L 2 (D) can be used in place of the usual L ∞ -estimate on the coefficient (2πn/ℓ) 2 of L n . In consequence this allows C 1 to be chosen independently of n. Therefore
n û n 2 H k (D) ≤ C 2 1 n f n 2 H k−2 (D) ≤ C 2 1 f 2 H k−2 p (Ω) < ∞. (2.32)
Finally, we must show that the intermediate order terms in the H k p -norm are also summable. Exploiting complex interpolation between H 0 (D) and H k (D) and Young's inequality, we obtain for
each 0 < m < k n 2πn ℓ 2m û n 2 H k−m (D) ≤ n 2πn ℓ 2m û n m/k L 2 (D) û n 1−m/k H k (D) 2 (2.33) ≤ C 2 n 2πn ℓ 2m(k−2)/k f n 2m/k L 2 (D) f n 2(k−m)/k H k−2 (D) (2.34) ≤ C 2 n m k 2πn ℓ 2m(k−2)/k f n 2m/k L 2 (D) k/m (2.35) + k − m k f n 2(k−m)/k H k−2 (D) k/(k−m) (2.36) = C 2 n m k 2πn ℓ 2(k−2) f n 2 L 2 (D) + k − m k f n 2 H k−2 (D) (2.37) ≤ C 3 f 2 H k−2 . (2.38) Thus u ∈ 0 H k p with u 2 H k p ≤ C 4 (k + 1) f 2 H k−2 p
, which completes the proof for integer values of s.
Interpolation then provides the remaining cases.
A Spectral Result
Our first goal is to use the modified Helmholtz projection P to rewrite the problem (1.1)-(1.5) in a variational form which has the velocity as its only unknown. First we notice that for any solution (v, q) of the problem, (1.2) implies v(t) ∈ P 0 for each t. Since it is readily seen that P commutes with D t , applying P to (1.1) yields
D t v − µP ∆v + ∇q 1 = P f ,(3.1)
where ∇q 1 = P ∇q (with ∆q 1 = 0 on Ω and q 1 = q on S F ) by Proposition 2.2(3). This application of P removes the indeterminacy of the pressure term in the sense that the value of q 1 is uniquely determined by v. mapping v → q 1 where q 1 is the function provided by Proposition 2.2(3) with ∇q 1 = P ∇q.
Proof. To see this, we use the fact that q and q 1 agree on the free surface and observe that (1.4) implies that the normal component of S(v, q) must vanish on S F . Hence S(v, q 1 ) · n = 0 implies
q 1 = 2µκ −2 2 i,j=1 a i a j D j v i on S F . Given v ∈ H s p , we note that f = 2µκ −2 2 i,j=1 a i a j D j v i ∈ H s−1
p . For s = 2, we can apply Lax-Milgram to obtain the existence of a unique weak solution q 1 ∈ H 1 p of the problem
∆q 1 = 0 on Ω (3.2) q 1 = f on S F (3.3) D k 3 q 1 | Γ ℓ = D k 3 q 1 | Γ 0 for k ∈ {0, 1} (3.4) with q 1 H 1 p ≤ C 1 v H 2 p where C 1 > 0 is independent of v. For s ≥ 3, we consider the problem ∆φ = −∆f on Ω (3.5) φ = 0 on S F (3.6) D k 3 φ| Γ ℓ = D k 3 φ| Γ 0 for k ∈ {0, 1} (3.7)
which has a unique solution φ ∈ 0 H s−1 p , by Proposition 2.3, satisfying φ H s−1
p ≤ C 2 ∆f H s−3 p for some C 2 > 0 which is independent of v. Finally, we set q 1 = φ + f ∈ H s−1 p and observe that q 1 H s−1 p ≤ C 2 ∆f H s−3 p + f H s−1 p ≤ C 3 v H s p .
Interpolation now yields the claim for the remaining values of s. It readily follows that the constructed operator is linear in v.
We now take the general approach used in semigroup theory by treating (1.1) as an abstract ordinary differential equation with respect to time whose solution is, for each value of t, an element of the appropriate function space (V s ) on Ω. If we define an operator A : V s → P s−2 by Av = −µP ∆v + ∇Qv, Notice that (1.4) is satisfied since our construction of Q ensures that the normal component of S(v, q) will vanish on S F . The operator A is a modification of the standard Stokes operator. We tackle the matter of determining the spectrum of −A first. Unfortunately, in contrast to the problems treated in [3,4,5,6,19], A is not injective with our boundary conditions (implying that 0 lies in the spectrum of A) since A(v + c) = Av for any constant vector c. This, combined with the inability to apply the Poincaré inequality in general, makes the problem of determining the spectrum more challenging here than in the aforementioned cases. Restricting the spectrum of A to a sector in the right half of the plane and providing estimates on the resolvent operator, the following theorem is a key result of this work.
We will show that ρ(A) contains all λ ∈ C such that |Im(λ)| > Re(λ). Given g ∈ P s−2 and λ with |Im(λ)| > Re(λ), we find a unique solution (v, q) ∈ V s × H s−1 p of the problem − µ∆v − λv + ∇q = g (3.11) along with (1.2), (1.4), and (1.5). To see that this is equivalent to the statement about ρ(A), suppose that there exists v ∈ V s such that (A − λI)v = g. Using our decomposition of L 2 , we find q 0 ∈ 0 H 1 p such that ∇q 0 = µ(I − P )∆v. Setting q = Qv + q 0 , we obtain (3.11). It is now straightforward to verify that (v, q) also satisfies (1.2), (1.4), and (1.5). Conversely, given a solution (v, q) of the stationary problem we can apply P to (3.11) Theorem 3.2. For g ∈ P s−2 , the problem (1.2), (1.4), (1.5), (3.11) has a unique weak solution (v, q) ∈ P 1 × L 2 for each λ ∈ C such that |Im(λ)| > Re(λ).
Proof. Notice that for any v ∈ P 1 , v satisfies (1.2) and the portion of (1.5) referring to the velocity. The free surface condition (1.4) is not necessarily satisfied though and will need to be incorporated into a variational formulation directly. To that end, let us consider the sesquilinear form ·, · :
P 1 × P 1 → C defined by v, u = −λ(v, u) L 2 + µ 2 3 i,j=1 Ω (D j v i + D i v j )(D jūi + D iūj ). (3.12) Now suppose u ∈ H 1 p , v ∈ P 2 , q ∈ H 1 p and observe that Ω (−µ∆v − λv + ∇q) · u = −µ Ω i ∆v i u i − λ(v, u) L 2 + ∂Ω q(u · n) − Ω q∇ · u (3.13) = µ i,j Ω D j u i D j v i + i ∂Ω u i (qn i − µ j D j v i n j ) − λ(v, u) L 2 − Ω q∇ · u (3.14) = v, u + ∂Ω S(v, q) · u + µ i,j ∂Ω u i D i v j n j − Ω D j u i D i v j − Ω q∇ · u (3.15) = v, u + ∂Ω S(v, q) · u − Ω q∇ · u + µ i Ω u i D i (∇ · v) (3.16) = v, u + ∂Ω S(v, q) · u − Ω q∇ · u. (3.17)
Notice that the pair (v, q) currently satisfies (1.2) and (1.5). If we suppose that (v, q) additionally satisfies (3.11) and (1.4), then we obtain
(g, u) L 2 = v, u + Γ ℓ S(v, q) · u + Γ 0 S(v, q) · u = v, u (3.18)
for all u ∈ P 1 . Thus v, u = (g, u) L 2 can be seen as a weak formulation of the full problem which does not involve q. In an effort to apply Lax-Milgram, we verify that the sesquilinear form is both continuous and coercive. Applying Hölder, | v, u | ≤ C v H 1 p u H 1 p , where C > 0 depends on µ and λ. Hence the sesquilinear form is continuous. That it is also coercive follows from Korn's inequality:
| v, v | 2 = µ 2 3 i,j=1 Ω |D j v i + D i v j | 2 − Re(λ) v 2 L 2 2 + Im(λ) v 2 L 2 2 (3.19) ≥ 1 2 µ 2 3 i,j=1 Ω |D j v i + D i v j | 2 − Re(λ) v 2 L 2 + |Im(λ)| · v 2 L 2 2 (3.20) ≥ C v 4 H 1 p . (3.21) For Re(λ) ≥ 0, line (3.19) implies | v, v | 2 ≥ Im(λ) 2 v 4 L 2 ≥ 1 2 |λ| 2 v 4 L 2 . (3.22)
Moreover, the same estimate can be obtained for Re(λ) < 0 since the right-hand side of line (3.19) then expands to something of the form φ + |λ| 2 v 4 L 2 where φ ≥ 0. Since the sesquilinear form satisfies the conditions of Lax-Milgram, we obtain a unique weak solution v ∈ P 1 of (1.2), (1.4), (1.5), (3.11) along with the estimate v H 1 p ≤ C g L 2 .
(3.23)
We now seek an associated pressure, q, of v. Recall that an associated pressure need only satisfy (3.11) in the sense of distributions (i.e., when tested against arbitrary u ∈ C ∞ c ). As with the velocity, we begin by finding a weak formulation for the pressure. Notice that, for q ∈ H 1 p with (v, q) satisfying (1.4), we obtain from (3.17) that
Ω q∇ · u = v, u − (g, u) L 2 (3.24)
for all u ∈ H 1 p . Using continuity of the sesquilinear form we obtain immediately that the right-hand side is a bounded linear functional in u, F : C ∞ c → C, which vanishes when ∇ · u = 0. From [15], we know there is a uniqueq ∈ L 2 such that F = ∇q and
Ωq = 0. (3.25)
It is now straightforward to verify that q = −q satisfies (3.11) in the distributional sense and hence is an associated pressure of v. It is uniquely determined under the additional condition Ω q = 0, but otherwise is unique only up to a constant.
The proof of the preceding theorem allows us to draw the following conclusion about the case λ = 0. Proof. Reasoning as in (3.19), we have
0 = | v, v | = µ 2 3 i,j=1 Ω |D j v i + D i v j | 2 ≥ 2µ 3 i=1 Ω |D j v i | 2 .(3.26)
Hence v and a fortiori q are constant.
R(λ; A)g H s p ≤ C( g H s−2 p + (1 + ε −1 )(|λ| + 1) (s−2)/2 g L 2 ) (3.27)
for all g ∈ P s−2 . Here C > 0 is a constant which is independent of λ, ε, and g.
Proof.
We will now demonstrate that the weak solution provided by Theorem 3.2 can, in fact, be made into a strong solution of the problem (1.2), (1.4), (1.5), (3.11). In order to avoid the lack of regularity due to the "artificial" corners in our domain, we turn to the equivalent problem of finding a weak solution (v 1 , q 1 ) of the problem (1.2), (1.4), (1.5), (3.11) on the larger domain Ω 1 . By choosing q 1 such that Ω 1 q 1 = 0 we can ensure that (v 1 , q 1 ) is simply the periodic extension of (v, q) to Ω 1 . Then it follows from standard results [10] that (v 1 , q 1 ) has the additional regularity we seek on compactly contained subsets of Ω 1 . To obtain regularity all the way up to the boundary, we follow the approach in [16] which is applicable to the boundary provided that it is smooth locally. Thus the pair (v 1 , q 1 ) has the desired regularity near S F up to and including the intersections with Γ 0 , Γ ℓ since these regions occur on a smooth portion of the free surface on Ω 1 . It follows that v p ∈ H 2 loc (Ω ∞ ) and q p ∈ H 1 loc (Ω ∞ ), hence v ∈ P 2 and q ∈ H 1 p . To see that (v, q) provides us with a strong solution of our problem, we only need to verify that (1.4) and (3.11) are satisfied. Using (3.17), for all u ∈ P 1 we have
(−µ∆v − λv + ∇q − g, u) L 2 = ∂Ω S(v, q) · u = S F S(v, q) · u.
(3.28)
Taking u ∈ 0 C ∞ pσ implies that −µ∆v − λv + ∇q − g lies in the orthogonal complement of 0 C ∞ pσ · L 2 .
It was demonstrated in the proof of Proposition 2.1 that this orthogonal complement consists of the gradients of functions in H 1 p , so that −µ∆v − λv + ∇q − g = ∇p for some p ∈ H 1 p . However, (3.17) now yields
(g, u) L 2 = (−µ∆v − λv + ∇(q − p), u) L 2 = v, u + ∂Ω S(v, q − p) · u − Ω (q − p)∇ · u (3.29)
for all u ∈ H 1 p . Restricting u to C ∞ c and exploiting (3.25) reduces this to Ω p∇ · u = 0. Integrating by parts, we see that Ω ∇p · u vanishes for arbitrary u ∈ C ∞ c . Since this is a dense subset of L 2 , ∇p = 0 and q satisfies (3.11). All that remains is to show that (1.4), the free surface boundary condition, is also satisfied. From (3.28) we now immediately obtain
S F S(v, q) · u = 0 (3.30)
for all u ∈ P 1 . Following the lead of [17], we localize to a neighborhood Σ ⊂ S F and construct u ∈ P 1 such that u| S F = (S(v, q) − (S(v, q) · n)n)φ where φ is a smooth nonnegative function vanishing outside Σ. Then
S F S(v, q) · u = Σ |S(v, q) − (S(v, q) · n)n| 2 φ + (S(v, q) · n)n · (S(v, q) − (S(v, q) · n)n)φ (3.31) = Σ |S tan (v)| 2 φ (3.32) = 0 (3.33)
implies that S tan (v) = 0 on Σ. Since Σ was chosen arbitrarily, we obtain S(v, q) = (S(v, q) · n)n on S F . Let θ(v, q) = q − 2µκ −2 a i a j D j v i ∈ H 1 p . Since θ(v, q)| S F = S(v, q) · n, (3.28) yields
We can now show that −A is the infinitesimal generator of an analytic semigroup of contractions. This is the main step involved in constructing solutions to the linear problem (1.1)-(1.5). We refer the reader to [8] for standard results in semigroup theory.
Theorem 4.1. The operator −A, with domain V 2 , generates an analytic semigroup of contractions, J(t), on P 0 with J(t) = 1.
Proof. As we seek to apply Lumer-Phillips, we begin by showing that −A is dissipative. To do this, we must improve (slightly) upon the estimate provided by (3.40). For λ < 0, we obtain
|(g, v) L 2 | = | v, v | = −λ v 2 L 2 + µ 2 3 i,j=1 Ω |D j v i + D i v j | 2 ≥ −λ v 2 L 2 . (4.1)
Dissipativity now follows using the Hölder inequality. Since A + I is surjective by Theorem 3.4 and P 0 is reflexive (as a Hilbert space), we can apply Lumer-Phillips to obtain that V 2 is dense in P 0 and −A generates a C 0 semigroup of contractions, J(t), on P 0 . As the generator of a C 0 semigroup of contractions, −A is closed (see Theorem II.1.4 in [8], for example) and using (3.40) together with Theorem 12.31 from [14] we see that J(t) is actually an analytic semigroup on P 0 . Now, since J(t) is a semigroup of contractions, we have J(t) ≤ 1. However, 0 is contained in the point spectrum of −A (see the discussion preceding Theorem 3.4). Hence J(t) = 1 as required.
With this semigroup result in hand, we are finally ready to solve the inhomogeneous linear problem (1.1)-(1.5). Theorem 4.1 immediately provides a solution to the Cauchy problem (3.9)-(3.10) and makes the Paley-Wiener theory utilized in [5] unnecessary. Here we abbreviate the sets G = (0, T ) × Ω and ∂G F = (0, T ) × S F .
v K s p + ∇q K s−2 p + q| S F K s−3/2 p (∂G F ) ≤ C f K s−2 p , (4.2)
where C is a positive constant which is independent of T and f .
Proof. First we notice that P f ∈ C 0,(s−3)/2 ([0, T ]; P 0 ) by the Sobolev Embedding Theorem. Combining Corollary 4.3.3 and Theorem 4.3.5(iii) from [13], the abstract Cauchy probleṁ
v + Av = P f (4.3) v(0, ·) = 0 (4.4)
has a unique strong solution v ∈ C 1,(s−3)/2 ([0, T ]; P 0 ), with v(t) ∈ V 2 for each t ∈ [0, T ]. Here we are exploiting the fact that −A is the generator of an analytic semigroup on P 0 . Note that v is a strong solution in the sense of semigroups, i.e. , v is differentiable almost everywhere on [0, T ], witḣ v ∈ L 1 ((0, T ); P 0 ), such that v(0, ·) = 0 andv(t) = −Av(t) + P f (t) almost everywhere on [0, T ]. In fact, v is a classical solution in the semigroup sense since it is continuously differentiable with respect to time.
To show that v ∈ K s p , we reconsider the abstract Cauchy problem (now with a new unknown variableṽ) from another perspective. We begin by applying the periodic analog of Lemma 2.2 from [5] in order to extend P f to K s−2 p (R × Ω) in such a way that the extension is bounded independent of T and vanishes for t < 0. Multiplying through the abstract Cauchy problem by the weight w(t) = e −t and taking Fourier transforms in t, we obtain F w (ṽ)(ξ) = (A + (1 + iξ)I) −1 F w (P f )(ξ).
(4.5)
Since it is clear that F w (P f )(ξ) ∈ P s−2 , this uniquely defines F w (ṽ)(ξ) ∈ V s by Theorem 3.4. Making use of the Fourier transform characterization of H s -spaces for s ∈ R + (e.g., see [1]) and the fact that Fourier transforms are unitary transformations, we have where c 1 , c 2 , and c 3 are positive constants which are independent of ξ and f . Similarly, we can apply estimate (3.40) to the second term of the integral to get (1 + ξ 2 ) s/2 F w (ṽ)(ξ + 1) 2 L 2 ≤ 2(1 + ξ 2 ) s/2 |1 + i(ξ + 1)| −2 F w (P f )(ξ + 1) 2 L 2 (4.13) = 2 1 + ξ 2 1 + (1 + ξ) 2 (1 + ξ 2 ) (s−2)/2 F w (P f )(ξ + 1) 2 L 2 (4.14)
≤ 6(1 + ξ 2 ) (s−2)/2 F w (P f )(ξ + 1) 2 L 2 . where c 4 > 0 is a constant which is independent of ξ and f . By uniqueness, we must have v = v| G ∈ K s p . We now seek a suitable q so that (v, q) is the unique solution of (1.1)-(1.5). For fixed t, this amounts to finding a unique q ∈ H s−1 p such that
∇q = µ∆v + Av + f − P f
on Ω (4.20) q = Qv on S F . (4.21)
Since s > 3, this is easily accomplished by taking the divergence of the first equation and applying
D
t v − µ∆v + ∇q = f on (0, T )
to obtain (A − λI)v = g. Hence (A − λI)v = g has a unique solution v if and only if the problem (1.2), (1.4), (1.5), (3.11) has a unique solution (v, q).
Corollary 3 . 3 .
33For g = 0, (v, q) ∈ P 1 × L 2 is a weak solution of the problem (1.2), (1.4), (1.5), (3.11) with λ = 0 if and only if v and q are constant.
Theorem 3. 4 .
4Let s ≥ 2. Then σ(A) ⊂ {λ ∈ C : |Im(λ)| ≤ Re(λ)}. Moreover, for λ with |Im(λ)| > Re(λ) and |λ| ≥ ε > 0 the resolvent operator R(λ; A) = (A − λI) −1 satisfies
Theorem 4 . 2 .(
42Let 3 < s ≤ 4, T > 0, and f ∈ K s−2 psuch that P f (0, ·) = 0. Then the problem (1.1)-(1.5) has a unique solution (v, q) such that v ∈ K s p , ∇q ∈ K s−2 p , and q| S F ∈ K ∂G F ). Moreover, this solution satisfies
θ(v, q)n · u = Ω ∇θ(v, q) · u = 0(3.34)for all u ∈ P 1 . By density, ∇θ(v, q) ∈ (P 0 ) ⊥ and θ(v, q) = p + ω for some p ∈ 0 H 1 p and ω ∈ R. Since this implies S(v, q) · n = q − 2µκ −2 a i a j D j v i = ω on S F , we take q * = q − ω and obtain a unique strong solution (v, q * ) ∈ V 2 × H 1 p of the problem (1.2),(1.4),(1.5),(3.11). To further increase regularity, we turn to the standard a priori estimates of Agmon, Douglis, and Nirenberg (ADN)[2]. Here it is useful to work on T rather than Ω. The associated problem is readily seen to be uniformly elliptic in the sense of ADN with boundary conditions satisfying the complementing condition. The a priori estimates in T then lead to the following estimate in Ω:where c 1 and c 2 are positive constants which do not depend on λ. Here we have used complex interpolation between L 2 and H s p . Finally, we apply Hölder to (3.22) which yieldsNow let us restrict ourselves to |λ| > ε for arbitrary ε > 0. If s = 2, then (3.39) and (3.40) yield (3.27) directly. Otherwise, we can apply Young's inequality to(3.39where c 3 , c 4 , and c 5 are positive constants which do not depend on λ and ǫ. Since v = R(λ; A)g, this completes the proof.Since H s p is compactly embedded in H s−2 p for s ≥ 2, Riesz-Schauder theory and Corollary 3.3 imply the following result. It follows that the kernel of A contains constants only.Corollary 3.5. σ(A) consists of isolated eigenvalues of finite multiplicity. Moreover, the eigenvalue 0 of A has multiplicity 1. Proposition 2.3. All that remains is to show that (v, q) satisfies (4.2). To estimate q we first notice that ∇q = µ(I − P )∆v + ∇Qv + (I − P )f .(4.22)The only term which we do not yet know how to estimate is ∇Qv. However, since ∆Qv = 0 on Ω and Qv = φ on S F where φ = 2µκ −2 2 i,j=1 a i a j D j v i ∈ H s−1 p , it follows from Proposition 2.2(3) that ∇Qv = P (∇φ). Then by Proposition 2.2(2),where c 5 and c 6 are positive constants. Similarly, since Q was constructed so that q = Qv on S F ,where c 7 and c 8 are positive constants. Thus, combining estimates, we obtainwhere c 9 , c 10 , and c 11 are positive constants which do not depend on f (or T ).Concluding RemarksWith Theorem 4.2 in hand, it is now straightforward to follow through the fixed point approach outlined in[5], with minor revisions, to obtain the following local existence result. For details about these modifications we refer the reader to[7].Theorem 5.1. Suppose 3 < s < 7 2 . For any u 0 ∈ V s−1 there exists T > 0, depending on u 0 H s−1 p , so that the problem (1.17)-(1.23) has a solution (v, q) with v ∈ K s p , q ∈ K s−3/2 p (∂G F ), and ∇q ∈ K s−2 p .Since the same arguments can be successfully applied to yield a similar local existence result when the initial displacement is taken to be nonzero in (1.23), uniqueness of solutions can then be shown to follow in the standard way. It is also a simple matter to prove that the unique solution given by Theorem 5.1 is axisymmetric provided that u 0 is taken to be axisymmetric.[7]contains additional details.
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Small-time existence for the Navier-Stokes equations with a free surface and surface tension. G Allain, Free Boundary Problems: Application and Theory. BostonPitmanIVG. Allain, Small-time existence for the Navier-Stokes equations with a free surface and surface tension, In Free Boundary Problems: Application and Theory, Vol. IV, Pitman, Boston, 1985, 355-364.
Small-time existence for the Navier-Stokes equations with a free surface. G Allain, Applied Mathematics and Optimization. 161G. Allain, Small-time existence for the Navier-Stokes equations with a free surface, Applied Mathematics and Optimization, 16(1) (1987) 37-50.
The initial value problem for the Navier-Stokes equations with a free surface. J T Beale, Communications on Pure and Applied Mathematics. 343J. T. Beale, The initial value problem for the Navier-Stokes equations with a free surface, Communications on Pure and Applied Mathematics, 34(3) (1981) 359-392.
Large-time regularity of viscous surface waves, Archive for Rational Mechanics and Analysis. J T Beale, 84J. T. Beale, Large-time regularity of viscous surface waves, Archive for Rational Mechanics and Analysis, 84(4) (1984) 307-352.
Navier-Stokes flow for a fluid jet with a free surface, doctoral dissertation, The University of Memphis. S Ceci, S. Ceci, Navier-Stokes flow for a fluid jet with a free surface, doctoral dissertation, The Uni- versity of Memphis, 2011.
One-parameter Semigroups for Linear Evolution Equations. K.-J Engel, R Nagel, Springer-VerlagNew YorkK.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.
L C Evans, Partial Differential Equations. ProvidenceAmerican Mathematical SocietyL. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998.
O A Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Second En. New YorkGordon and Breach Science Publishersglish ed.O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Second En- glish ed., Gordon and Breach Science Publishers, New York, 1969.
J Li, M A Fontelos, Drop dynamics on the beads-on-string structure for viscoelastic jets: A numerical study. 15J. Li and M. A. Fontelos, Drop dynamics on the beads-on-string structure for viscoelastic jets: A numerical study, Physics of Fluids, 15(4) (2003) 922-937.
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A Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. New YorkSpringer-VerlagA. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
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The Navier-Stokes Equations. H Sohr, Birkhäuser VerlagBaselH. Sohr, The Navier-Stokes Equations, Birkhäuser Verlag, Basel, 2001.
The solvability of the second initial-boundary value problem for a linear nonstationary system of Navier-Stokes equations, Zapiski Naučnyh Seminarov Leningradskogo Otdelenija Matematičeskogo Instituta im. V A Solonnikov, Steklova Akademii Nauk SSSR (LOMI). 69V. A. Solonnikov, The solvability of the second initial-boundary value problem for a linear nonstationary system of Navier-Stokes equations, Zapiski Naučnyh Seminarov Leningradskogo Otdelenija Matematičeskogo Instituta im. V. A. Steklova Akademii Nauk SSSR (LOMI), 69 (1977) 200-218, 277.
A certain boundary value problem for the stationary system of Navier-Stokes equations. V A Solonnikov, V E Ščadilov, Akademiya Nauk SSSR. Trudy Matematicheskogo Instituta imeni V. A. Steklova. 235V. A. Solonnikov and V. E.Ščadilov, A certain boundary value problem for the stationary system of Navier-Stokes equations, Akademiya Nauk SSSR. Trudy Matematicheskogo Instituta imeni V. A. Steklova, 125 (1973) 196-210, 235.
R Temam, Navier-Stokes Equations. ProvidenceAMS Chelsea PublishingR. Temam, Navier-Stokes Equations, AMS Chelsea Publishing, Providence, 2001.
Navier-Stokes flow down a vertical column. Y Teramoto, Navier-Stokes Equations: Theory and Numerical Methods. Longman, HarlowY. Teramoto, Navier-Stokes flow down a vertical column, In Navier-Stokes Equations: Theory and Numerical Methods, Longman, Harlow, 1998, 139-151.
|
[] |
[
"Cross-lingual Distillation for Text Classification",
"Cross-lingual Distillation for Text Classification"
] |
[
"Ruochen Xu [email protected] \nCarnegie Mellon Universit\nCarnegie Mellon Universit\n\n",
"Yiming Yang [email protected] \nCarnegie Mellon Universit\nCarnegie Mellon Universit\n\n"
] |
[
"Carnegie Mellon Universit\nCarnegie Mellon Universit\n",
"Carnegie Mellon Universit\nCarnegie Mellon Universit\n"
] |
[
"Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics"
] |
Cross-lingual text classification(CLTC) is the task of classifying documents written in different languages into the same taxonomy of categories. This paper presents a novel approach to CLTC that builds on model distillation, which adapts and extends a framework originally proposed for model compression. Using soft probabilistic predictions for the documents in a label-rich language as the (induced) supervisory labels in a parallel corpus of documents, we train classifiers successfully for new languages in which labeled training data are not available. An adversarial feature adaptation technique is also applied during the model training to reduce distribution mismatch. We conducted experiments on two benchmark CLTC datasets, treating English as the source language and German, French, Japan and Chinese as the unlabeled target languages. The proposed approach had the advantageous or comparable performance of the other state-of-art methods.1415
|
10.18653/v1/p17-1130
|
[
"https://www.aclweb.org/anthology/P17-1130.pdf"
] | 3,013,625 |
1705.02073
|
06b60e8e8b1e952d98955f980bf78ede1dd43127
|
Cross-lingual Distillation for Text Classification
July 30 -August 4, 2017. July 30 -August 4, 2017
Ruochen Xu [email protected]
Carnegie Mellon Universit
Carnegie Mellon Universit
Yiming Yang [email protected]
Carnegie Mellon Universit
Carnegie Mellon Universit
Cross-lingual Distillation for Text Classification
Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics
the 55th Annual Meeting of the Association for Computational LinguisticsVancouver, Canada; Vancouver, CanadaJuly 30 -August 4, 2017. July 30 -August 4, 201710.18653/v1/P17-1130
Cross-lingual text classification(CLTC) is the task of classifying documents written in different languages into the same taxonomy of categories. This paper presents a novel approach to CLTC that builds on model distillation, which adapts and extends a framework originally proposed for model compression. Using soft probabilistic predictions for the documents in a label-rich language as the (induced) supervisory labels in a parallel corpus of documents, we train classifiers successfully for new languages in which labeled training data are not available. An adversarial feature adaptation technique is also applied during the model training to reduce distribution mismatch. We conducted experiments on two benchmark CLTC datasets, treating English as the source language and German, French, Japan and Chinese as the unlabeled target languages. The proposed approach had the advantageous or comparable performance of the other state-of-art methods.1415
Introduction
The availability of massive multilingual data on the Internet makes cross-lingual text classification (CLTC) increasingly important. The task is defined as to classify documents in different languages using the same taxonomy of predefined categories.
CLTC systems build on supervised machine learning require a sufficiently amount of labeled training data for every domain of interest in each language. But in reality, labeled data are not evenly distributed among languages and across domains. English, for example, is a label-rich lan-guage in the domains of news stories, Wikipedia pages and reviews of hotels, products, etc. But many other languages do not necessarily have such rich amounts of labeled data. This leads to an open challenge in CLTC, i.e., how can we effectively leverage the trained classifiers in a label-rich source language to help the classification of documents in other label-poor target languages?
Existing methods in CLTC use either a bilingual dictionary or a parallel corpus to bridge language barriers and to translate classification models (Xu et al., 2016) or text data (Zhou et al., 2016a). There are limitations and challenges in using either type of resources. Dictionary-based methods often ignore the dependency of word meaning and its context, and cannot leverage domainspecific disambiguation when the dictionary on hand is a general-purpose one. Parallel-corpus based methods, although more effective in deploying context (when combined with word embedding in particular), often have an issue of domain mismatch or distribution mismatch if the available source-language training data, the parallel corpus (human-aligned or machine-translation induced one) and the target documents of interest are not in exactly the same domain and genre (Duh et al., 2011). How to solve such domain/distribution mismatch problems is an open question for research. This paper proposes a new parallel-corpus based approach, focusing on the reduction of domain/distribution matches in CLTC. We call this approach Cross-lingual Distillation with Feature Adaptation or CLDFA in short. It is inspired by the recent work in model compression (Hinton et al., 2015) where a large ensemble model is transformed to a compact (small) model. The assumption of knowledge distillation for model compression is that the knowledge learned by the large model can be viewed as a mapping from in-put space to output (label) space. Then, by training with the soft labels predicted by the large model, the small model can capture most of the knowledge from the large model. Extending this key idea to CLTC, if we see parallel documents as different instantiations of the same semantic concepts in different languages, a target-language classifier should gain the knowledge from a welltrained source classifier by training with the targetlanguage part of the parallel corpus and the soft labels made by the source classifier on the source language side. More specifically, we propose to distillate knowledge from the source language to the target language in the following 2-step process:
• Firstly, we train a source-language classifier with both labeled training documents and adapt it to the unlabeled documents from the source-language side of the parallel corpus. The adaptation enforces our classifier to extract features that are: 1) discriminative for the classification task and 2) invariant with regard to the distribution shift between training and parallel data.
• Secondly, we use the trained source-language classifier to obtain the soft labels for a parallel corpus, and the target-language part of the parallel corpus to train a target classifier, which yields a similar category distribution over target-language documents as that over source-language documents. We also use unlabeled testing documents in the target language to adapt the feature extractor in this training step.
Intuitively, the first step addresses the potential domain/distribution mismatch between the labeled data and the unlabeled data in the source language. The second step addresses the potential mismatch between the target-domain training data (in the parallel corpus) and the test data (not in the parallel corpus). The soft-label based training of target classifiers makes our approach unique among parallel-corpus based CLTC methods (Section 2.1. The feature adaptation step makes our framework particularly robust in addressing the distributional difference between in-domain documents and parallel corpus, which is important for the success of CLTC with low-resource languages.
The main contributions in this paper are the following:
• We propose a novel framework (CLDFA) for knowledge distillation in CLTC through a parallel corpus. It has the flexibility to be built on a large family of existing monolingual text classification methods and enables the use of a large amount of unlabeled data from both source and target language.
• CLDFA has the same computational complexity as the plug-in text classification method and hence is very efficient and scalable with the proper choice of plug-in text classifier.
• Our evaluation on benchmark datasets shows that our method had a better or at least comparable performance than that of other stateof-art CLTC methods.
Related Work
Related work can be outlined with respect to the representative work in CLTC and the recent progress in deep learning for knowledge distillation.
CLTC Methods
One branch of CLTC methods is to use lexical level mappings to transfer the knowledge from the source language to the target language. The work by Bel et al. (Bel et al., 2003) was the first effort to solve CLTC problem. They translated the target-language documents to source language using a bilingual dictionary. The classifier trained in the source language was then applied on those translated documents. Similarly, Mihalcea et al. (Mihalcea et al., 2007) built cross-lingual classifier by translating subjectivity words and phrases in the source language into the target language. Shi et al. (Shi et al., 2010) also utilized a bilingual dictionary. Instead of translating the documents, they tried to translate the classification model from source language to target language. Prettenhofer and Stein. (Prettenhofer and Stein, 2010) also used the bilingual dictionary as a word translation oracle and built their CLTC system on structural correspondence learning, a theory for domain adaptation. A more recent work by (Xu et al., 2016) extended seminal bilingual dictionaries with unlabeled corpora in low-resource languages. Chen et al. (Chen et al., 2016) used bilingual word embedding to map documents in source and target language into the same semantic space, and adversarial training was applied to enforce the trained classifier to be language-invariant. Some recent efforts in CLTC focus on the use of automatic machine translation (MT) technology. For example, Wan (Wan, 2009) used machine translation systems to give each document a source-language and a target-language version, where one version is machine-translated from the another one. A co-training (Blum and Mitchell, 1998) algorithm was applied on two versions of both source and target documents to iterative train classifiers in both languages. MTbased CLTC also include the work on multi-view learning with different algorithms, such as majority voting (Amini et al., 2009), matrix completion (Xiao and Guo, 2013) and multi-view coregularization (Guo and Xiao, 2012a).
Another branch of CLTC methods focuses on representation learning or the mapping of the induced representations in cross-language settings (Guo and Xiao, 2012b;Zhou et al., 2016aZhou et al., , 2015Zhou et al., , 2016bXiao and Guo, 2013;Jagarlamudi et al., 2011;De Smet et al., 2011;Vinokourov et al., 2002;Platt et al., 2010;Littman et al., 1998). For example, Meng et al. (Meng et al., 2012) and Lu et al. (Lu et al., 2011) used a parallel corpus to learn word alignment probabilities in a pre-processing step. Some other work attempts to find a languageinvariant (or interlingua) representation for words or documents in different languages using various techniques, such as latent semantic indexing (Littman et al., 1998), kernel canonical correlation analysis (Vinokourov et al., 2002), matrix completion (Xiao and Guo, 2013), principal component analysis (Platt et al., 2010) and Bayesian graphical models (De Smet et al., 2011).
Knowledge Distillation
The idea of distilling knowledge in a neural network was proposed by Hinton et al (Hinton et al., 2015), in which they introduced a student-teacher paradigm. Once the cumbersome teacher network was trained, the student network was trained according to soften predictions of the teacher network. In the field of computer vision, it has been empirically verified that student network trained by distillation performs better than the one trained with hard labels. (Hinton et al., 2015;Romero et al., 2014;Ba and Caruana, 2014). Gupta et al.(Gupta et al., 2015) transfers supervision be-tween images from different modalities(e.g. from RGB image to depth image). There are also some recent works applied distillation in the field of natural language. For example, Lili et al. (Mou et al., 2015) distilled task specific knowledge from a set of high-dimensional embeddings to a lowdimensional space. Zhiting et al. used an iterative distillation method to transfer the structured information of logic rules into the weights of a neural network. Kim et al. (Kim and Rush, 2016) applied knowledge distillation approaches in the field of machine translation to reduce the size of neural machine translation model. Our framework shares the same purpose of existing works that transfer knowledge between models of different properties, such as model complexity, modality, and structured logic. However, our transfer happens between models working on different languages. To the best of knowledge, this is the first work using knowledge distillation to bridge the language gap for NLP tasks.
Preliminary
Task and Notation
CLTC aims to use the training data in the source language to build a model applicable in the target language. In our setting, we have labeled data in
source language L src = {x i , y i } L i=1
, where x i is the labeled document in source language and y i is the label vector. We then have our test data in the target language, given by T tgt = {x i } T i=1 . Our framework can also use unlabeled documents from both languages in transductive learning settings. We use
U src = {x i } M i=1 to denote source- language unlabeled documents,U tgt = {x i } N i=1 to denote target-language unlabeled documents, and U parl = {(x i , x i )} P i=1
to denote a unlabeled bilingual parallel corpus where x i and x i are paired document translations of each other. We assume that the unlabeled parallel corpus does not overlap with the source-language training documents and the target-language test documents.
Convolutional Neural Network (CNN) as a Plug-in Classifier
We use a state-of-the-art CNN-based neural network classifier (Kim, 2014) as the plug-in classifier in our framework. Instead of using a bag-ofwords representation for each document, the CNN model concatenates the word embeddings (vertical vectors) of each input document into a n × k matrix, where n is the length (number of word occurrences) of the document, and k is the dimension of word embedding. Denoting by
x 1:n = x 1 ⊕ x 2 ⊕ ... ⊕ x n
as the resulted matrix, with ⊕ the concatenation operator. One-dimensional convolutional filter w ∈ R hk with window size h operates on every consecutive h words, with non-linear function f and bias b. For window of size h started at index i, the feature after convolutional filter is given by:
c i = f (w · x i:i+h−1 + b)
A max-over-time pooling (Collobert et al., 2011) is applied on c over all possible positions such that each filter extracts one feature. The model uses multiple filters with different window sizes. The concatenated outputs from filters consist the feature of each document. We can see the convolutional filters and pooling layers as feature extractor
f = G f (x, θ f )
, where θ f contains parameters for embedding layer and convolutional layer. Theses features are then passed to a fully connected softmax layer to produce probability distributions over labels. We see the final fully connected softmax layer as a label classifier G y (f , θ y ) that takes the output f from the feature extractor. The final output of model is given by G y (G f (x, θ f ), θ y ), which is jointly parameterized by {θ f , θ y } We want to emphasize that our choice of the plug-in classifier here is mainly for its simplicity and scalability to demonstrate our framework. There is a large family of neural classifiers for monolingual text classification that could be used in our framework as well, including other convolutional neural networks by (Johnson and Zhang, 2014), the recurrent neural networks by (Lai et al., 2015;Johnson and Zhang, 2016;Sutskever et al., 2014;Dai and Le, 2015), the attention mechanism by , the deep dense network by (Iyyer et al., 2015), and more.
Proposed Framework
Let us introduce two versions of our model for cross-language knowledge distillation, i.e., the vanilla version and the full version with feature adaptation. Both are supported by the proposed framework. We denote the former by CLD-KCNN and the latter by CLDFA-KCNN.
Vanilla Distillation
Without loss of generality, assume we are learning a multi-class classifier for the target language. We have y ∈ 1, 2, ..., |v | where v is the set of all possible classes. We assume the base classification network produces real number logits q j for each class. For example, for the case of CNN text classifier, the logits can be produced by a linear transformation which takes features extracted max-pooling layer and outputs a vector of size |v |. The logits are converted into probabilities of classes through the softmax layer, by normalizing each q j with all other logits.
p j = exp(q j /T ) |v | k=1 exp(q k /T )(1)
where T is a temperature and is normally set to 1. Using a higher value of T produces a softer probability distribution over classes. The first step of our framework is to train the source-language classifier on labeled source documents L src . We use standard temperature T = 1 and cross-entropy loss as the objective to minimize. For each example and its label (x i , y i ) from the source training set, we have:
L(θ src ) = − (x i ,y i )∈Lsrc |v | k=1 1{y i = k} log p(y = k|x i ; θ src )(2)
where p(y = k|x; θ src ) is source model controlled by parameter θ src and 1{·} is the indicator function.
In the second step, the knowledge captured in θ src is transferred to the distilled model in the target language by training it on the parallel corpus. The intuition is that paired documents in parallel corpus should have the same distribution of class predicted by the source model and target model. In the simplest version of our framework, for each source-language document in the parallel corpus, we predict a soft class distribution by source model with high temperature. Then we minimize the cross-entropy between soft distribution produced by source model and the soft distribution produced by target model on the paired documents in the target language. More formally, we optimize θ tgt according to the following loss function for each document pair (x i , x i ) in parallel corpus.
L(θ tgt ) = − (x i ,x i )∈U parl |v | k=1 p(y = k|x i ; θ src ) log p(y = k|x i ; θ tgt )(3)
During distillation, the same high temperature is used for training target model. After it has been trained, we set the temperature to 1 for testing.
We can show that under some assumptions, the two-step cross-lingual distillation is equivalent to distilling a target-language classifier in the targetlanguage input space.
Lemma 1. Assume the parallel corpus {x i , x i } ∈ U parl is generated by x i ∼ p(X ; η) and x i = t(x i ), where η controls the marginal distribution of x i and t is a differentiable translation function with integrable derivative. Let f θsrc (t(x )) be the function that outputs soft labels of p(y = k|t(x ); θ src ). The distillation given by equation 3 can be interpreted as distillation of a target language classifier f θsrc (t(x )) on target language documents sampled from p(X ; η). f θsrc (t(x )) is the classifier that takes input of target documents, translates them into source documents through t and makes prediction using the source classifier. If we further assume the testing documents have the same marginal distribution P (X ; η), then the distilled classifier should have similar generalization power as f θsrc (t(x )).
Theorem 2. Let source training data x i ∈ L src has marginal distribution p(X; λ). Under the assumptions of lemma 1, further assume p(t(x ); λ) = p(x ; η), p(y|t(x )) = p(y|x ) and t (x ) ≈ C, where C is a constant.
Then f θsrc (t(x )) actually minimizes the expected loss in target language data E x ∼p(X;η),y∼p(Y |x ) [L y, f (t(x )) ].
Proof. By definition of equation 2, f θsrc (x) minimizes the expected loss E x∼p(X;λ),y∼p(Y |x) [L y, f (x) ], where L is cross-entropy loss in our case. Then we can write Although vanilla distillation is intuitive and simple, it cannot handle distribution mismatch issues. For example, the marginal feature distributions of source-language documents in L src and U parl could be different, so are the distributions of target-language documents in U parl and T tgt . According to theorem 2, the vanilla distillation works for the best performance under unrealistic assumption: p(t(x )|λ) = p(x |η). To further illustrate our point, we trained a CNN classifier according to equation 2 and used the features extracted by G f to present the source-language documents in both L src and U parl . Then we projected the highdimensional features onto a 2-dimensional space via t-Distributed Stochastic Neighbor Embedding (t-SNE)(Maaten and Hinton, 2008). This resulted the visualization of the project data in Figures 1 and 2.
E x∼p(X;λ),y∼p(Y |x) [L y, f (x) ] = p(x; λ) y p(y|x)L y, f (x) dx = p(t(x ); λ) y p(y|t(x ))L y, f (t(x )) t (x )dx ≈C p(x ; η) y p(y|x )L y, f (t(x )) dx =CE x ∼p(X;η),y∼p(Y |x ) [L y, f (t(x )) ]
Distillation with Adversarial Feature Adaptation
It is quite obvious in Figure 1 that the generalpurpose parallel corpus has a very different feature distribution from that of the labeled source training set. Even for machine-translated parallel data from the same domain, as shown in figure 2, there is still a non-negligible distribution shift from the source language to the target language for the extracted features. Our interpretation of this observation is that when the MT system (e.g. Google Translate) is a general-purpose one, it non-avoidably add translation ambiguities which would lead the distribution shift from the original domain. To address the distribution divergence brought by either a general-purpose parallel corpus or an imperfect MT system, we seek to adapt the features extraction part of our neural classifier such that the feature distributions on both sides should be close as possible in the newly induced feature space. We adapt the adversarial training method by (Ganin and Lempitsky, 2014) to the cross-lingual settings in our problems.
Given a set of training set of L = {x i , y i } i=1,...,N and an unlabeled set U = {x i } i=1,...,M , our goal is to find a neural classifier G y (G f (x, θ f ), θ y ), which has good discriminative performance on L and also extracts features which have similar distributions on L and U . One way to maximize the similarity of two distributions is to maximize the loss of a discriminative classifier whose job is to discriminate the two feature distributions. We denote this classifier by G d (·, θ d ), which is parameterized by θ d .
At training time, we seek θ f to minimize the loss of G y and maximize the loss of G d . Meanwhile, θ y and θ d are also optimized to minimize their corresponding loss. The overall optimization could be summarized as follows:
E(θ f , θ y , θ d ) = x i ,y i ∈L L y (y i , G y (G f (x i , θ f ), θ y )) − α x i ∈L L d (0, G d (G f (x i , θ f ), θ d )) − α x j ∈U L d (1, G d (G f (x j , θ f ), θ d ))
where L y is the loss function for true labels y, L d is loss function for binary labels indicating the source of data and α is the hyperparameter that controls the relative importance of two losses. We optimize θ f , θ y for minimizing E and optimize θ d for maximizing E. We jointly optimize θ f , θ y , θ d through the gradient reversal layer (Ganin and Lempitsky, 2014).
We use this feature adaptation technique to firstly adapt the source-language classifier to the source-language documents of the parallel corpus. When training the target-language classifier by matching soft labels on the parallel corpus, we also adapt the classifier to the target testing documents. We use cross-entropy loss functions as L y and L d for both feature adaptation.
Experiments and Discussions
Dataset
Our experiments used two benchmark datasets, as described below.
(1) Amazon Reviews We used the multilingual multi-domain Amazon review dataset created by Prettenhofer and Stein (Prettenhofer and Stein, 2010). The dataset contains Amazon reviews in three domains: book, DVD and music. Each domain has the reviews in four different languages: English, German, French and Japanese. We treated English as the source language and the rest three as the target languages, respectively. This gives us 9 tasks (the product of the 3 domains and the 3 target languages) in total. For each task, there are 1000 positive and 1000 negative reviews in English and the target language, respectively. (Prettenhofer and Stein, 2010) also provides 2000 parallel reviews per task, that were generated using Google Translate 1 , and used by us for cross-language distillation. There are also several thousands of unlabeled reviews in each language. The statistics of unlabeled data is summarized in Table 1. All the reviews are tokenized using standard regular expressions except for Japanese, for which we used a publicly available segmenter 2 .
(2) English-Chinese Yelp Hotel Reviews This dataset was firstly used for CLTC by (Chen et al., 2016). The task is to make sentence-level sentiment classification with 5 labels(rating scale from 1 to 5), using English as the source language and Chinese as the target language. The labeled English data consists of balanced labels of 650k Yelp reviews from Zhang et al. . The Chinese data includes 20k labeled Chinese hotel reviews and 1037k unlabeled ones from (Lin et al., 2015). Following the approach by (Chen et al., 2016), we use 10k of labeled Chinese data as validation set and another 10k hotel reviews as held-out test data. We a random sample of 500k parallel sentences from UM-courpus (Tian et al., 2014), which is a general-purpose corpus designed for machine translation.
Baselines
We compare the proposed method with other stateof-the-art methods as outlined below. (Vinokourov et al., 2002) learn latent document representations in a shared lowdimensional space by performing the Latent Semantic Indexing (LSI), the Oriented Principal Component Analysis (OPCA) and a kernel (namely KCAA) for the parallel text. PL-MC (Xiao and Guo, 2013) recovers missing features via matrix Completion, and also uses LSI to induce a latent space for parallel text. All these methods train a classifier in the shared feature space with labeled training data from both the source and target languages.
(2) MT-based CLTC Methods
The methods in this category all use an MT system to translate each test document in the target language to the source language in the testing phase. The prediction on each translated document is made by a source-language classifier, which can be a Logistic Regression model (MT+LR) (Chen et al., 2016) or a deep averaging network (MT+DAN) (Chen et al., 2016).
(3) Adversarial Deep Averaging Network
Similar to our approach, the adversarial Deep Averaging Network (ADAN) also exploits adversarial training for CLTC (Chen et al., 2016). However, it does not have the parallel-corpus based knowledge distillation part (which we do). Instead, it uses averaged bilingual embeddings of words as its input and adapts the feature extractor to produce similar features in both languages.
We also include the results of mSDA for the Yelp Hotel Reviews dataset. mSDA (Chen et al., 2012) is a domain adaptation method based on Table 2: Accuracy scores of methods on the Amazon Reviews dataset: the best score in each row (a task) is highlighted in bold face. If the score of CLDFA-KCNN is statistically significantly better (in one-sample proportion tests) than the best among the baseline methods, it is marked using a star. Table 3: Accuracy scores of methods on the English-Chinese Yelp Hotel Reviews dataset stacked denoising autoencoders, which has been proved to be effective in cross-domain sentiment classification evaluations. We show the results reported by (Chen et al., 2012), where they used bilingual word embedding as input for mSDA.
Implementation Detail
We pre-trained both the source and target classifier with unlabeled data in each language. We ran word2vec (Mikolov et al., 2013) 3 on the tokenized unlabeled corpus. The learned word embeddings are used to initialize the word embedding look-up matrix, which maps input words to word embeddings and concatenates them into input matrix.
We fine-tuned the source-language classifier on the English training data with 5-fold crossvalidation. For English-Chinese Yelp-hotel review dataset, the temperature T (Section 4.1) in distillation is tuned on validation set in the target language. For Amazon review dataset, since there is no default validation set, we set temperature from low to high in {1, 3, 5, 10} and take the average among all predictions. 3 https://code.google.com/archive/p/word2vec/
Main Results
In tables 2 and 3 we compare the results of our methods (the vanilla version CLD-KCNN and the full version CLDFA-KCNN) with those of other methods based on the published results in the literature. The baseline methods are different in these two tables as they were previously evaluated (by their authors) on different benchmark datasets. Clearly, CLDFA-KCNN outperformed the other methods on all except one task in these two datasets, showing that knowledge distillation is successfully carried out in our approach. Noticing that CLDFA-KCNN outperformed CLD-KCNN, showing the effectiveness of adversarial feature extraction in reducing the distribution mismatch between the parallel corpus and the train/test data in the target domain. We should also point out that in Table 2, the four baseline methods (PL-LSI, PL-KCCA, PL-OPCA and PL-MC) were evaluated under the condition of using additional 100 labeled target documents for training, according to the author's report (Xiao and Guo, 2013). On the other hand, our methods (CLD-KCNN and CLDFA-KCNN) were evaluated under a tougher condition, i.e., not using any labeled data in the target domains.
We also test our framework when a few training documents in the target language are available. A simple way to utilize the target-language supervision is to fit the target-language model with labeled target data after optimizing with our crosslingual distillation framework. The performance of CLD-KCNN and CLDFA-KCNN trained with different sizes of labeled target-language data is shown in figure 3. We also compare the performance of training the same classifier using only the target-language labels(Target Only in figure 3). As we can see, our framework can efficiently utilize the extra supervision and improve the performance over the training using only the targetlanguage labels. The margin is most significant when the size of the target-language label is relatively small. : Accuracy scores of methods using varying sizes of target-language labeled data on the Amazon review dataset. The target language is German and the domain is music. The parallel corpus has a fixed size of 1000 and the size of the labeled target-language documents is shown on the x-axis
Conclusion
This work introduces a novel framework for distillation of discriminative knowledge across languages, providing effective and efficient algorithmic solutions for addressing domain/distribution mismatch issues in CLTC. The excellent performance of our approach is evident in our evaluation on two CLTC benchmark datasets, compared to that of other state-of-the-art methods.
Figure 1 :
1Extracted features for source-language documents in the English-Chinese Yelp Hotel Review dataset. Red dots represent features of the documents in L src and green dots represent the features of documents in U parl , which is a generalpurpose parallel corpus.
Figure 2 :
2Extracted features for the source-language documents in the Amazon Reviews dataset. Red dots represent the features of the labeled training documents in L src , and green dots represent the features of the documents in U parl , which are the machine-translated documents from a target language. Below each figure is the target language and the domain of review (Section 5.1).
( 1 )
1Parallel-Corpus based CLTC Methods Methods in this category all use an unlabeled parallel corpus. Methods named PL-LSI (Littman 1 translate.google.com 2 https://pypi.python.org/pypi/tinysegmenter et al., 1998), PL-OPCA (Platt et al., 2010) and PL-KCAA
Figure 3
3Figure 3: Accuracy scores of methods using varying sizes of target-language labeled data on the Amazon review dataset. The target language is German and the domain is music. The parallel corpus has a fixed size of 1000 and the size of the labeled target-language documents is shown on the x-axis
Table 1 :
1Dataset Statistics for the Amazon reviews dataset
Target Language Domain PL-LSI PL-KCCA PL-OPCA PL-MC CLD-KCNN CLDFA-KCNNGerman
book
77.59
79.14
74.72
79.22
82.54
83.95*
DVD
79.22
76.73
74.59
81.34
82.24
83.14*
music
73.81
79.18
74.45
79.39
74.65
79.02
French
book
79.56
77.56
76.55
81.92
81.6
83.37
DVD
77.82
78.19
70.54
81.97
82.41
82.56
music
75.39
78.24
73.69
79.3
83.01
83.31*
Janpanese
book
72.68
69.46
71.41
72.57
74.12
77.36*
DVD
72.55
74.79
71.84
76.6
79.67
80.52*
music
73.44
73.54
74.96
76.21
73.69
76.46
Averaged Accuracy
75.78
76.31
73.64
78.72
79.33
81.08*
AcknowledgementWe thank the reviewers for their helpful comments. This work is supported in part by Defense Advanced Research Projects Agency Information Innovation Oce (I2O), the Low Resource Languages for Emergent Incidents (LORELEI) Program, Issued by DARPA/I2O under Contract No. HR0011-15-C-0114, by the National Science Foundation (NSF) under grant IIS-1546329.
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[
"Asymmetric Unimodal Maps: Some Results from q-generalized Bit Cumulants",
"Asymmetric Unimodal Maps: Some Results from q-generalized Bit Cumulants"
] |
[
"Uǧur Tırnaklı *e-mail:[email protected] \nCentro Brasileiro de Pesquisas Físicas\nDepartment of Physics\nFaculty of Science\nEge University\nRua Xavier Sigaud 15022290-180, 35100Rio de Janeiro -RJ, IzmirBrazil, Turkey\n"
] |
[
"Centro Brasileiro de Pesquisas Físicas\nDepartment of Physics\nFaculty of Science\nEge University\nRua Xavier Sigaud 15022290-180, 35100Rio de Janeiro -RJ, IzmirBrazil, Turkey"
] |
[] |
In this study, using q-generalized bit cumulants (q is the nonextensivity parameter of the recently introduced Tsallis statistics), we investigate the asymmetric unimodal maps xt+1 = 1 − a|xt| z i (i = 1, 2 correspond to xt > 0 and xt < 0 respectively, zi > 1, 0 < a ≤ 2, t = 0, 1, 2, ...). The study of the q-generalized second cumulant of these maps allows us to determine, for the first time in the literature, the dependence of the inflexion parameter pairs (z1, z2) to the nonextensivity parameter q. This behaviour is found to be very similar to that of the logistic-like maps (z1 = z2 = z) reported recently by Costa et al. [Phys. Rev. E 56 (1997) 245].PACS Number(s): 05.70.Ce
|
10.1103/physreve.62.7857
|
[
"https://arxiv.org/pdf/cond-mat/9911420v1.pdf"
] | 36,377,435 |
cond-mat/9911420
|
301def4edc429619655438fbc5650cededb08938
|
Asymmetric Unimodal Maps: Some Results from q-generalized Bit Cumulants
25 Nov 1999
Uǧur Tırnaklı *e-mail:[email protected]
Centro Brasileiro de Pesquisas Físicas
Department of Physics
Faculty of Science
Ege University
Rua Xavier Sigaud 15022290-180, 35100Rio de Janeiro -RJ, IzmirBrazil, Turkey
Asymmetric Unimodal Maps: Some Results from q-generalized Bit Cumulants
25 Nov 1999
In this study, using q-generalized bit cumulants (q is the nonextensivity parameter of the recently introduced Tsallis statistics), we investigate the asymmetric unimodal maps xt+1 = 1 − a|xt| z i (i = 1, 2 correspond to xt > 0 and xt < 0 respectively, zi > 1, 0 < a ≤ 2, t = 0, 1, 2, ...). The study of the q-generalized second cumulant of these maps allows us to determine, for the first time in the literature, the dependence of the inflexion parameter pairs (z1, z2) to the nonextensivity parameter q. This behaviour is found to be very similar to that of the logistic-like maps (z1 = z2 = z) reported recently by Costa et al. [Phys. Rev. E 56 (1997) 245].PACS Number(s): 05.70.Ce
In recent years, sensitivity to initial conditions of nonlinear dynamical systems has started to be studied with an increasing interest. Among them, dissipative systems of low-dimensional maps [1][2][3][4], self-organized criticality [5], symbolic sequences [6] and conservative systems of long-ranged many-body Hamiltonians [7,8], low-dimensional maps [9] can be enumerated. We focus here on the one-dimensional dissipative maps. As it has already been well studied in the literature, to investigate the sensitivity to initial conditions of any kind of one-dimensional maps on the onset to chaos, it is possible to introduce
ξ(t) = lim ∆x(t)→0 ∆x(t) ∆x(0) ,(1)
(where ∆x(0) and ∆x(t) are discrepancies of the initial conditions at times 0 and t) which, if the Lyapunov exponent λ 1 = 0, satisfies the differential equation dξ/dt = λ 1 ξ, thus
ξ(t) = e λ1t ,(2)
whereas, if λ 1 = 0, it satisfies dξ/dt = λ q ξ q , thus
ξ(t) = [1 + (1 − q)λ q t] 1 1−q (q ∈ R)(3)
which recovers the standard case (namely Eq.(2)) for q = 1. Here, q is the nonextensivity parameter of recently introduced Tsallis statistics [10] and for q = 1 case, it is evident that Eq.(3) yields a powerlaw sensitivity to initial conditions. At the onset of chaos, ξ(t) presents strong fluctuations reflecting the fractal-like structure of the critical attractor and Eq.(3) delimits the power-law growth of the upper bounds of ξ(t). These upper bounds (ξ ∝ t 1 1−q ) allow us to calculate the values of q for the map under consideration. This method of finding numerical values of q has been successfully used for the logistic map [1], a family of logistic-like maps [2], the circle map [3] and a family of circular-like maps [4]. Beside this method, another one has been developed by Lyra and Tsallis [3] by looking at the geometrical aspects of dynamical attractors at the treshold to chaos. Using the multifractal singularity spectrum f (α) [11], they proposed the scaling relation
1 1 − q = 1 α min − 1 α max(4)
where α max (α min ) is the most rarefied (concentrated) region of the multifractal singularity spectrum of the attractor. This relation presents the second method of calculating the q values once the scaling properties of the dynamical attractor are known. It has already been shown that for all above-mentioned one-dimensional dissipative maps, the values of q index calculated within these two different methods are the same (within a good precision) for any given value of the inflexion parameter z of the used map. In addition to this, another important information which comes from all these works is the determination of the behaviour of the q index as a function of the inflexion parameter z. As it can be seen from Fig.2 of [2] and from the Table of [4], it seems that when z → ∞ , q approaches 1. The purpose of this paper is to show numerically whether such kind of behaviour is verified for asymmetric unimodal maps (AUMs)
x t+1 = 1 − a|x t | z1 if x t > 0 1 − a|x t | z2 if x t < 0(5)
where z 1,2 > 1, 0 < a ≤ 2 and t = 0, 1, 2, .... Unfortunately, up to now, for the AUMs, neither the first nor the second method can yield satisfactory results for the prediction of q values of any inflexion parameter pair (z 1 , z 2 ) [12]. The problem of the first method might be originated from the prediction of the critical a c values at the chaos treshold with enough precision, whereas the problem of the second method might be related to the numerical procedure used to estimate the f (α) curve, whose most rarefied region (α max ) is usually poorly sampled. Another fact which yields this failure might be the nonuniversal behaviour of AUMs reported in [13,14] (for example, it is shown that AUMs fail to exhibit the metric universality of Feigenbaum). Since the values of q are not available for AUMs, clearly it is not possible to see the behaviour of q as a function of (z 1 , z 2 ) pairs by using the above-mentioned two methods. In this paper, in order to see this behaviour, without finding the precise values of q for (z 1 , z 2 ) pairs, we use another technique based on the very recent generalization of bit cumulants for chaotic systems within Tsallis statistics [15,16]. To explain this technique, let us recall the main results of [15,16]. The generalized second cumulant (or heat capacity) is given by
C (q) 2 = q (q − 1) 2 ρ 2q−1 − ρ q 2(6)
where ρ is the natural invariant density [17]. As q → 1, we have
C (1) 2 = (ln ρ) 2 − ln ρ 2(7)
which is equivalent to the standard definition of the second cumulant [17,18]. It is shown [15,16] that there is a kind of scaling between C and also provides us a tool to fulfill our aim in this paper. In Fig.1, we illustrate this slope plotting C
(1) 2 vs C (q) 2
curve for a representative (z 1 , z 2 ) pair. It is evident that most of the data points fall onto a straight line which yields a well-defined slope (in fact, we encountered that a few points deviate from this straight line especially when the values of the inflexion parameters start to be very different from the standard case (namely, z 1 = z 2 ), but since such data points are very few, we prefer to calculate the slope using a linear regression -with an error of ±0.01 for each estimation of the slope-to data points which fall onto this straight line). As it is evident from Fig.2, when q → 1, naturally C (q)
2 → C (1) 2
and thus the slope also tends to unity. Therefore, now we have another parameter (which behaves exactly the same as q) from where we can check the behaviour of the q index as a function of (z 1 , z 2 ) pairs without knowing the exact values of q! Calculating the above-mentioned slope for various values of (z 1 , z 2 ) pairs, we are able to estimate the behaviour of q index as a function of inflexion parameters without finding any q value. Fig.3 represents this behaviour for (2, z 2 ),(z 1 , 2),(3, z 2 ) and (z 1 , 3) cases. It seems from the figure that as z 2 − z 1 → ±∞, the above-mentioned slope (and thus, q index) approaches to unity! This tendency is exactly the same as that observed for a family of logistic-like maps [2] and a family of circular-like maps [4], namely, as z → ∞, q approaches 1.
Summing up, for the first time (to the best of our knowledge), we manage to study the asymmetric unimodal maps in such a way that it became possible to relate it with the nonextensivity parameter q of Tsallis statistics. We were able to show that the dependence of the inflexion parameter pairs (z 1 , z 2 ) of these maps to the q index is very similar to that observed for other one-dimensional maps reported so far. Although the determination of the exact values of q index for (z 1 , z 2 ) pairs is still lacking, we hope that present work would be considered as a first attempt on this line and will accelarate other studies which are, no doubt, highly welcome.
ACKNOWLEDGMENTS
I acknowledge the partial support of BAYG-C program of TUBITAK (Turkish agency) and CNPq (Brazilian agency) as well as the support from Ege University Research Fund under the project number 98FEN025. for a representative value of (z 1 , z 2 ) pairs. Figure 2 : The behaviour of the slope as a function of q index for a representative value of (z 1 , z 2 ) pairs.
Figure Captions
where it is easily seen that most of the points can be fitted to a straight line. The slope of this line gives the scaling factor between C
Figure 1 :
1The scaling between the standard second cumulant C
Figure 3 :
3The behaviour of the slope as a function of z 2 − z 1 values for a family of four different (z 1 , z 2 ) pairs.
Fig 1
. C Tsallis, A R Plastino, W.-M Zheng, Chaos, Solitons and Fractals. 8885C. Tsallis, A.R. Plastino and W.-M. Zheng, Chaos, Solitons and Fractals 8 (1997) 885.
. U M S Costa, M L Lyra, A R Plastino, C Tsallis, Phys. Rev. E. 56245U.M.S. Costa, M.L. Lyra, A.R. Plastino and C. Tsallis, Phys. Rev. E 56 (1997) 245.
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. U Tirnakli, C Tsallis, M L Lyra, Eur. Phys. J. B. 11309U. Tirnakli, C. Tsallis and M.L. Lyra, Eur. Phys. J. B 11 (1999) 309.
. F A Tamarit, S A Cannas, C Tsallis, Eur. Phys. J. B. 1545F.A. Tamarit, S.A. Cannas and C. Tsallis, Eur. Phys. J. B 1 (1998) 545;
. A R R Papa, C Tsallis, Phys. Rev. E. 573923A.R.R. Papa and C. Tsallis, Phys. Rev. E 57 (1998) 3923.
. M Buiatti, P Grigolini, L Palatella, Physica A. 268214M. Buiatti, P. Grigolini and L. Palatella, Physica A 268 (1999) 214.
. C Anteneodo, C Tsallis, Phys. Rev. Lett. 805313C. Anteneodo andd C. Tsallis, Phys. Rev. Lett. 80 (1998) 5313.
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. C Tsallis, J. Stat. Phys. 52479C. Tsallis, J. Stat. Phys. 52 (1988) 479;
. E M F Curado, C Tsallis, J. Phys. A. 241019corrigenda: 24E.M.F. Curado and C. Tsallis, J. Phys. A 24 (1991) L69 [corrigenda: 24 (1991) 3187 and 25 (1992) 1019].
. T C Halsey, Phys. Rev. A. 331141T.C. Halsey et al., Phys. Rev. A 33 (1986) 1141.
. U Tirnakli, C Tsallis, M L Lyra, unpublished resultsU. Tirnakli, C.Tsallis and M.L. Lyra, unpublished results.
. R V Jensen, L K H Ma, Phys. Rev. A. 313993R.V. Jensen and L.K.H. Ma, Phys. Rev. A 31 (1985) 3993.
. M C De Sousa, E Vieira, C Lazo, Tsallis, Phys. Rev. A. 35945M.C. de Sousa Vieira, E. Lazo and C. Tsallis, Phys. Rev. A 35 (1987) 945.
Nonextensive thermodynamic formalism for chaotic dynamical systems. R S Johal, R Rai, cond- mat/9909218R.S. Johal and R. Rai, 'Nonextensive thermodynamic formalism for chaotic dynamical systems', cond- mat/9909218.
Generalized bit cumulants for chaotic systems: Numerical results. R Rai, R S , cond-mat/9909288R. Rai and R.S. Johal, Generalized bit cumulants for chaotic systems: Numerical results', cond-mat/9909288.
. F Schlogl, E Scholl, Z. Phys. B. 72231F. Schlogl and E. Scholl, Z. Phys. B 72 (1988) 231.
C Beck, F Schlogl, Thermodynamics of chaotic systems: an introduction. CambridgeCambridge University PressC. Beck and F. Schlogl, Thermodynamics of chaotic systems: an introduction (Cambridge University Press, Cambridge, 1993).
|
[] |
[
"Soft Session Types",
"Soft Session Types"
] |
[
"Ugo Dal [email protected] \nLaboratoire d'Informatique de Paris Nord\nUniversità di Bologna INRIA Sophia Antipolis\n\n",
"Lago \nLaboratoire d'Informatique de Paris Nord\nUniversità di Bologna INRIA Sophia Antipolis\n\n",
"Paolo Di Giamberardino \nLaboratoire d'Informatique de Paris Nord\nUniversità di Bologna INRIA Sophia Antipolis\n\n"
] |
[
"Laboratoire d'Informatique de Paris Nord\nUniversità di Bologna INRIA Sophia Antipolis\n",
"Laboratoire d'Informatique de Paris Nord\nUniversità di Bologna INRIA Sophia Antipolis\n",
"Laboratoire d'Informatique de Paris Nord\nUniversità di Bologna INRIA Sophia Antipolis\n"
] |
[
"18th International Workshop on Expressiveness in Concurrency (EXPRESS 2011) EPTCS 64"
] |
We show how systems of session types can enforce interactions to be bounded for all typable processes. The type system we propose is based on Lafont's soft linear logic and is strongly inspired by recent works about session types as intuitionistic linear logic formulas. Our main result is the existence, for every typable process, of a polynomial bound on the length of any reduction sequence starting from it and on the size of any of its reducts.
|
10.4204/eptcs.64.5
|
[
"https://arxiv.org/pdf/1108.4467v1.pdf"
] | 14,844,943 |
1108.4467
|
b0bc355dc34f876d18eb2eb43911edb52071ccfd
|
Soft Session Types
2011
Ugo Dal [email protected]
Laboratoire d'Informatique de Paris Nord
Università di Bologna INRIA Sophia Antipolis
Lago
Laboratoire d'Informatique de Paris Nord
Università di Bologna INRIA Sophia Antipolis
Paolo Di Giamberardino
Laboratoire d'Informatique de Paris Nord
Università di Bologna INRIA Sophia Antipolis
Soft Session Types
18th International Workshop on Expressiveness in Concurrency (EXPRESS 2011) EPTCS 64
201110.4204/EPTCS.64.5
We show how systems of session types can enforce interactions to be bounded for all typable processes. The type system we propose is based on Lafont's soft linear logic and is strongly inspired by recent works about session types as intuitionistic linear logic formulas. Our main result is the existence, for every typable process, of a polynomial bound on the length of any reduction sequence starting from it and on the size of any of its reducts.
Introduction
Session types are one of the most successful paradigms around which communication can be disciplined in a concurrent or object-based environment. They can come in many different flavors, depending on the underlying programming language and on the degree of flexibility they allow when defining the structure of sessions. As an example, systems of session types for multi-party interaction have been recently introduced [9], while a form of higher-order session has been shown to be definable [12]. Recursive types, on the other hand, have been part of the standard toolset of session type theories since their inception [8].
The key property induced by systems of session types is the following: if two (or more) processes can be typed with "dual" session types, then they can interact with each other without "going wrong", i.e. avoiding situations where one party needs some data with a certain type and the other(s) offer something of a different, incompatible type. Sometimes, one would like to go beyond that and design a type system which guarantees stronger properties, including quantitative ones. An example of a property that we find particularly interesting is the following: suppose that two processes P and Q interact by creating a session having type A through which they communicate. Is this interaction guaranteed to be finite? How long would it last? Moreover, P and Q could be forced to interact with other processes in order to be able to offer A. The question could then become: can the global amount of interaction be kept under control? In other words, one could be interested in proving the interaction induced by sessions to be bounded. This problem has been almost neglected by the research community in the area of session types, although it is the manifesto of the so-called implicit computational complexity (ICC), where one aims at giving machine-free characterizations of complexity classes based on programming languages and logical systems.
Linear logic (LL in the following) has been introduced twenty-five years ago by Jean-Yves Girard [7]. One of its greatest merits has been to allow a finer analysis of the computational content of both intuitionistic and classical logic. In turn, this is possible by distinguishing multiplicative as well as additive connectives, by an involutive notion of negation, and by giving a new status to structural rules allowing them to be applicable only to modal formulas. One of the many consequences of this new, refined way of looking at proof theory has been the introduction of natural characterizations of complexity classes by fragments of linear logic. This is possible because linear logic somehow "isolates" complexity in the modal fragment of the logic (which is solely responsible for the hyperexponential complexity of cut elimination in, say intuitionistic logic), which can then be restricted so as to get exactly the expressive power needed to capture small complexity classes. One of the simplest and most elegant of those systems is Lafont's soft linear logic (SLL in the following), which has been shown to correspond to polynomial time in the realm of classical [10], quantum [6] and higher-order concurrent computation [5].
Recently, Caires and Pfenning [1] have shown how a system of session types can be built around intuitionistic linear logic, by introducing πDILL, a type system for the π-calculus where types and rules are derived from the ones of intuitionistic linear logic. In their system, multiplicative connectives like ⊗ and ⊸ allow to model sequentiality in sessions, while the additive connectives & and ⊕ model external and internal choice, respectively. The modal connective !, on the other hand, allows to model a server of type !A which can offer the functionality expressed by A multiple times.
In this paper, we study a restriction of πDILL, called πDSLL, which can be thought of as being derived from πDILL in the same way as SLL is obtained from LL. In other words, the operator ! behaves in πDSLL in the same way as in SLL. The main result we prove about πDSLL is precisely about bounded interaction: whenever P can be typed in πDSLL and P → n Q, then both n and |Q| (the size of the process Q, to be defined later) are polynomially related to |P|. This ensures an abstract but quite strong form of bounded interaction. Another, perhaps more "interactive" formulation of the same result is the following: if P and Q interact via a channel of type A, then the "complexity" of this interaction is bounded by a polynomial on |P| + |Q|, whose degree only depends on A.
We see this paper as the first successful attempt to bring techniques from implicit computational complexity into the realm of session types. Although proving bounded interaction has been technically nontrivial, due to the peculiarities of the π-calculus, we think the main contribution of this work lies in showing that bounded termination can be enforced by a natural adaptation of known systems of session types. An extended version with more details is available [4].
πDILL, an Informal Account
In this section, we will outline the main properties of πDILL, a session type system recently introduced by Caires and Pfenning [1,2]. For more information, please consult the two cited papers.
In πDILL, session types are nothing more than formulas of (propositional) intuitionistic linear logic without atoms but with (multiplicative) constants:
A ::= 1 | A ⊗ A | A ⊸ A | A ⊕ A | A&A | !A.
These types are assigned to channels (names) by a formal system deriving judgments in the form
Γ; ∆ ⊢ P :: x : A,
where Γ and ∆ are contexts assigning types to channels, and P is a process of the name-passing π calculus. The judgment above can be read as follows: the process P acts on the channel x according to the session type A whenever composed with processes behaving according to Γ and ∆ (each on a specific channel). Informally, the various constructions on session types can be explained as follows: • 1 is the type of an empty session channel. A process offering to communicate via a session channel typed this way simply synchronizes with another process through it without exchanging anything. This is meant to be an abstraction for all ground session types, e.g. natural numbers, lists, etc. In linear logic, this is the unit for ⊗.
• A ⊗ B is the type of a session channel x through which a message carrying another channel with type A is sent. After performing this action, the underlying process behaves according to B on the same channel x. • A ⊸ B is the adjoint to A ⊗ B: on a channel with this type, a process communicate by first performing an input and receiving a channel with type A, then acting according to B, again on x. • A ⊕ B is the type of a channel on which a process either sends a special message inl and performs according to A or sends a special message inr and performs according to B. • The type A&B can be assigned to a channel x on which the underlying process offers the possibility of choosing between proceeding according to A or to B, both on x. So, in a sense, & models external choice. • Finally, the type !A is attributed to a channel x only if a process can be replicated by receiving a channel y through x, then behaving on y according to A. The assignments in Γ and ∆ are of two different natures:
• An assignment of a type A to a channel x in ∆ signals the need by P of a process offering a session of type A on the channel x; for this reason, ∆ is called the linear context; • An assignment of a type A to a channel x in Γ, on the other hand, represents the need by P of a process offering a session of type !A on the channel x; thus, Γ is the exponential context. Typing rules πDILL are very similar to the ones of DILL, itself one of the many possible formulations of linear logic as a sequent calculus. In particular, there are two cut rules, each corresponding to a different portion of the context: Γ; ∆ 1 ⊢ P ::
x : A Γ; ∆ 2 , x : A ⊢ Q :: T Γ; ∆ 1 , ∆ 2 ⊢ (νx)(P | Q) :: T Γ; / 0 ⊢ P :: y : A Γ, x : A; ∆ ⊢ Q :: T Γ; ∆ ⊢ (νx)(!x(y).P | Q) :: T
Please observe how cutting a process P against an assumption in the exponential context requires to "wrap" P inside a replicated input: this allows to turn P into a server.
In order to illustrate the intuitions above, we now give an example. Suppose that a process P models a service which acts on x as follows: it receives two natural numbers, to be interpreted as the number and secret code of a credit card and, if they correspond to a valid account, returns an MP3 file and a receipt code to the client. Otherwise, the session terminates. To do so, P needs to interact with another service (e.g. a banking service) Q through a channel y. The banking service, among others, provides a way to verify whether a given number and code correspond to a valid credit card. In πDILL, the process P would receive the type / 0; y : (N ⊸ 1 ⊕ 1)&A ⊢ P ::
x : N ⊸ N ⊸ (S ⊗ N) ⊕ 1,
where N and S are pseudo-types for natural numbers and MP3s, respectively. A is the type of all the other functionalities Q provides. As an example, P could be the following process:
x(nm 1 ).x(cd 1 ).y.inl; (νnm 2 )y nm 2 .(νcd 2 )y cd 2 .
y.case(x.inl; (νmp)x mp .(νrp)x rp , x.inr; 0)
Observe how the credit card number and secret code forwarded to Q are not the ones sent by the client: the flow of information happening inside a process is abstracted away in πDILL. Similarly, one can write a process Q and assign it a type as follows: / 0; / 0 ⊢ Q :: y : (N ⊸ 1 ⊕ 1)&A. Putting the two derivations together, we obtain / 0; / 0 ⊢ (νx)(P | Q) :: x : N ⊸ N ⊸ (S ⊗ N) ⊕ 1. Let us now make an observation which will probably be appreciated by the reader familiar with linear logic. The processes P and Q can be typed in πDILL without the use of any exponential rule, nor of cut. What allows to type the parallel composition (νx)(P | Q), on the other hand, is precisely the cut rule. The interaction between P and Q corresponds to the elimination of that cut. Since there isn't any exponential around, this process must be finite, since the size of the underlying process shrinks at every single reduction step. From a process-algebraic point of view, on the other hand, the finiteness of the interaction is an immediate consequence of the absence of any replication in P and Q.
The banking service Q can only serve one single session and would vanish at the end of it. To make it into a persistent server offering the same kind of session to possibly many different clients, Q must be put into a replication, obtaining R =!z(y).Q. In R, the channel z can be given type !((N ⊸ 1 ⊕ 1)&A) in the empty context. The process P should be somehow adapted to be able to interact with R: before performing the two outputs on y, it's necessary to "spawn" R by performing an output on z and passing y to it. This way we obtain a process S such that
/ 0; z :!((N ⊸ 1 ⊕ 1)&A) ⊢ S :: x : N ⊸ N ⊸ (S ⊗ N) ⊕ 1,
and the composition (νz)(S | R) can be given the same type as (νx)(P | Q). Of course, S could have used the channel z more than once, initiating different sessions. This is meant to model a situation in which the same client interacts with the same server by creating more than one session with the same type, itself done by performing more than one output on the same channel. Of course, servers can themselves depend on other servers. And these dependencies are naturally modeled by the exponential modality of linear logic.
On Bounded Interaction
In πDILL, the possibility of modeling persistent servers which in turn depend on other servers makes it possible to type processes which exhibit a very complex and combinatorially heavy interactive behavior.
Consider the following processes, the first one parameterized on any i ∈ N:
dupser i . = !x i (y).(νz)x i+1 z .(νw)x i+1 w .; dupclient . = (νy)x 0 y .
In πDILL, these processes can be typed as follows:
/ 0; x i+1 :!1 ⊢dupser i :: x i :!1; / 0; x 0 :!1 ⊢dupclient :: z : 1.
Then, for every n ∈ N one can type the parallel composition mulser n+1 . = (νx 1 . . . x n )(dupser n || . . . ||dupser 0 ) as follows / 0; x n :!1 ⊢ mulser n :: x 0 :!1.
Informally, mulser n is a persistent server which offers a session type 1 on a channel x 0 , provided a server with the same functionality is available on x n . The process mulser n is the parallel composition of n servers in the form dupser i , each spawning two different sessions provided by dupser i+1 on the same channel x i+1 .
The process mulser n cannot be further reduced. But notice that, once mulser n and dupclient are composed, the following exponential blowup is bound to happen:
(νx 0 )(mulser n | dupclient) ≡ (νx 0 . . . x n )(dupser n || . . . ||dupser 0 | dupclient) → (νx 0 . . . x n )(dupser n || . . . ||dupser 1 | P 1 ) → 2 (νx 1 . . . x n )(dupser n || . . . ||dupser 2 | P 2 | P 2 ) → 4 (νx 2 . . . x n )(dupser n || . . . ||dupser 3 | P 3 || . . . ||P 3 4 times ) → * (νx n )(dupser n | P n || . . . ||P n 2 n times ) → 2 n 0.
Here, for every i ∈ N the process P i is simply (νy)x i y .(νz)x i z . Notice that both the number or reduction steps and the size of intermediate processes are exponential in n, while the size of the initial process is linear in n. This is a perfectly legal process in πDILL. Moreover the type !1 of the channel x 0 through which dupclient and mulser n communicate does not contain any information about the "complexity" of the interaction: it is the same for every n.
The deep reasons why this phenomenon can happen lie in the very general (and "generous") rules governing the behavior of the exponential modality ! in linear logic. It is this generality that allows the embedding of propositional intuitionistic logic into linear logic. Since the complexity of normalization for the former [13,11] is nonelementary, the exponential blowup described above is not a surprise.
It would be desirable, on the other hand, to be sure that the interaction caused by any process P is bounded: whenever P → n Q, then there's a reasonably low upper bound to both n and |Q|. This is precisely what we achieve by restricting πDILL into πDSLL.
πDSLL: Syntax and Main Properties
In this section, the syntax of πDSLL will be introduced. Moreover, some basic operational properties will be given.
The Process Algebra
πDSLL is a type system for a fairly standard π-calculus, exactly the one on top of which πDILL is defined:
Definition 1 (Processes)
Given an infinite set of names or channels x, y, z, . . ., the set of processes is defined as follows:
P ::= 0 | P | Q | (νx)P | x(y).P | x y .P | !x(y).P | x.inl; P | x.inr; P | x.case(P, Q)
The only non-standard constructs are the last three, which allow to define a choice mechanism: the process x.case(P, Q) can evolve as P or as Q after having received a signal in the form inl o inr through x. Processes sending such a signal through the channel x, then continuing like P are, respectively, x.inl; P and x.inr; P. The set of names occurring free in the process P (hereby denoted fn(P)) is defined as usual. The same holds for the capture avoiding substitution of a name x for y in a process P (denoted P{x/y}), and for α-equivalence between processes (denoted ≡ α ).
Structural congruence is an equivalence relation identifying those processes which are syntactically different but can be considered equal for very simple structural reasons:
Definition 2 (Structural Congruence)
The relation ≡, called structural congruence, is the least congruence on processes satisfying the following seven axioms:
P ≡ Q whenever P ≡ α Q; (νx)0 ≡ 0; P | 0 ≡ P; (νx)(νy)P ≡ (νy)(νx)P; P | Q ≡ Q | P; ((νx)P) | Q ≡ (νx)(P | Q) whenever x / ∈ fn(Q); P | (Q | R) ≡ (P | Q) | R.
Formal systems for reduction and labelled semantics can be defined in a standard way. We refer the reader to [1] for more details.
A quantitative attribute of processes which is delicate to model in process algebras is their size: how can we measure the size of a process? In particular, it is not straightforward to define a measure which both reflects the "number of symbols" in the process and is invariant under structural congruence (this way facilitating all proofs). A good compromise is the following:
Definition 3 (Process Size)
The size |P| of a process P is defined by induction on the structure of P as follows:
|0| = 0; |x(y).P| = |P| + 1; |x.inl; P| = |P| + 1; |P | Q| = |P| + |Q|; |x y .P| = |P| + 1; |x.inr; P| = |P| + 1; |(νx)P| = |P|; |!x(y).P| = |P| + 1; |x.case(P, Q)| = |P| + |Q| + 1.
According to the definition above, the empty process 0 has null size, while restriction does not increase the size of the underlying process. This allows for a definition of size which remains invariant under structural congruence. The price to pay is the following: the "number of symbols" of a process P can be arbitrarily bigger than |P| (e.g. for every n ∈ N, |(νx) n P| = |P|). However, we have the following:
Lemma 1 For every P, Q, |P| = |Q| whenever P ≡ Q. Moreover, there is a polynomial p such that for every P, there is Q with P ≡ Q and the number of symbols in Q is at most p(|Q|).
The Type System
The language of types of πDSLL is exactly the same as the one of πDILL, and the interpretation of type constructs does not change (see Section 2 for some informal details). Typing judgments and typing rules, however, are significantly different, in particular, in the treatment of the exponential connective !. Typing judgments become syntactical expressions in the form Γ; ∆; Θ ⊢ P ::
x : A.
First of all, observe how the context is divided into three chunks now: Γ and ∆ have to be interpreted as exponential contexts, while Θ is the usual linear context from πDILL. The necessity of having two exponential contexts is a consequence of the finer, less canonical exponential discipline of SLL compared to the one of LL. We use the following terminology: Γ is said to be the auxiliary context, while ∆ is the multiplexor context. Typing rules are in Figure 1. context is treated multiplicatively, while the multiplexor context is treated additively, as in πDILL 1 . Now, consider the rules governing the exponential connective !, which are ♭ ! , ♭ # , !L ! , !L # and !R:
• The rules ♭ ! and ♭ # both allow to spawn a server. This corresponds to turning an assumption x : A in the linear context into one y : A in one of the exponential contexts; in ♭ # , x : A could be already present in the multiplexor context, while in ♭ ! this cannot happen; • The rules !L ! and !L # lift an assumption in the exponential contexts to the linear context; this requires changing its type from A to !A; • The rule !R allows to turn an ordinary process into a server, by packaging it into a replicated input and modifying its type. 1 The reader familiar with linear logic and proof nets will recognize in the different treatment of the auxiliary and multiplexor contexts, one of the basic principles of SLL: contraction is forbidden on the auxiliary doors of exponential boxes. The channel names contained in the auxiliary context correspond to the auxiliary doors of exponential boxes, so we treat them multiplicatively. The contraction effect induced by the additive treatment of the channel names in the multiplexor context corresponds to the multiplexing rule of SLL.
Finally there are three cut rules in the system, namely cut, cut ! and cut # : • cut is the usual linear cut rule, i.e. the natural generalization of the one from πDILL. • cut ! and cut # allow to eliminate an assumption in one of the the two exponential contexts. In both cases, the process which allows to do that must be typable with empty linear and multiplexor contexts.
Back to Our Example
Let us now reconsider the example processes introduced in Section 3. The basic building block over which everything is built was the process dupser i =!x i (y).(νz)x i+1 z .(νw)x i+1 w .. We claim that for every i, the process dupser i is not typable in πDSLL. To understand why, observe that the only way to type a replicated input like dupser i is by the typing rule !R, and that its premise requires the body of the replicated input to be typable with empty linear and multiplexor contexts. A quick inspection on the typing rules reveals that every name in the auxiliary context occurs (free) exactly once in the underlying process (provided we count two occurrences in the branches of a case as just a single occurrence). However, the name x i+1 appears twice in the body of dupser i . A slight variation on the example above, on the other hand, can be typed in πDSLL, but this requires changing its type. See [4] for more details.
Subject Reduction
A basic property most type systems for functional languages satisfy is subject reduction: typing is preserved along reduction. For processes, this is often true for internal reduction: if P → Q and ⊢ P : A, then ⊢ Q : A. In this section, a subject reduction result for πDSLL will be given and some ideas on the underlying proof will be described. Some concepts outlined here will become necessary ingredients in the proof of bounded interaction, to be done in Section 5 below. Subject reduction is proved by closely following the path traced by Caires and Pfenning; as a consequence, we proceed quite quickly, concentrating our attention on the differences with their proof.
When proving subject reduction, one constantly work with type derivations. This is particularly true here, where (internal) reduction corresponds to the cut-elimination process. A linear notation for proofs in the form of proof terms can be easily defined, allowing for more compact descriptions. As an example, a proof in the form π : Γ 1 ; ∆; Θ 1 ⊢ P :: x : A ρ : Γ 2 ; ∆; Θ 2 , x : A ⊢ Q :: T Γ 1 , Γ 2 ; ∆; Θ 1 , Θ 2 ⊢ (νx)(P | Q) :: T cut corresponds to the proof term cut(D, x.E), where D is the proof term for π and E is the proof term for ρ. If D is a proof term corresponding to a type derivation for the process P, we write D = P. From now on, proof terms will often take the place of processes: Γ; ∆; Θ ⊢ D :: T stands for the existence of a type derivation D with conclusion Γ; ∆; Θ ⊢ D :: T . A proof term D is said to be normal if it does not contain any instances of cut rules. Subject reduction will be proved by showing that if P is typable by a type derivation D and P → Q, then a type derivation E for Q exists. Actually, E can be obtained by manipulating D using techniques derived from cut-elimination. Noticeably, not every cut-elimination rule is necessary to prove subject reduction. In other words, we are in presence of a weak correspondence between proof terms and processes, and remain far from a genuine Curry-Howard correspondence.
Those manipulations of proof-terms which are necessary to prove subject reduction can be classified as follows:
• First of all, a binary relation =⇒ on proof terms called computational reduction can be defined. At the logical level, this corresponds to proper cut-elimination steps, i.e. those cut-elimination steps in which two rules introducing the same connective interact. At the process level, computational reduction correspond to internal reduction. =⇒ is not symmetric. • A binary relation −→ on proof terms called shift reduction, distinct from =⇒ must be introduced. At the process level, it corresponds to structural congruence. As =⇒, −→ is not a symmetric relation. • Finally, an equivalence relation ≡ on proof terms called proof equivalence is necessary. At the logical level, this corresponds to the so-called commuting conversions, while at the process level, the induced processes are either structurally congruent or strongly bisimilar. . , x n ), x.!L ! (x.G)) be the proof obtained by composing a proof F (whose last rule is !R) with a proof G (whose last rule is !L ! ) through a cut rule. A shift reduction rule tells us that y.G)) . . .)), which corresponds to the opening of a box in SLL. The shift reduction does not have a corresponding reduction step at process level, since D ≡ E; nevertheless, it is defined as an asymmetric relation, for technical reasons connected to the proof of bounded interaction. • Let D = cut # (F, x.cut(G, y.H)). A defining rule for proof equivalence ≡, states that in D the cut # rule can be permuted over the cut rule, by duplicating F; namely D ≡ E = cut(cut # (F, x.G), y.cut # (F, x.H)). This is possible because the channel x belongs to the multiplexor contexts of both G, H, such contexts being treated additively. At the process level, D = (νx)((!x(y). F) | (νy)( G | H)) , while E = (νy)(((νx)(!x(y). F) | G)) | ((νx)(!x(y). F) | H))), D and E being strongly bisimilar. Before proceeding to Subject Reduction, we give the following two lemmas, concerning structural properties of the type system: Let us give a sketch of the proof of Theorem 1. We reason by induction on the structure of D. Since D = P → Q the only possible last rules of D can be: 1L, !L ! , !L # ,, a linear cut (cut) or an exponential cut (cut ! or cut # ). In all the other cases, the underlying process can only perform a visible action, as can be easily verified by inspecting the rules from Figure 1. With this observation in mind, let us inspect the operational semantics derivation proving that P → Q. At some point we will find two subprocesses of P, call them R and S, which communicate, causing an internal reduction. We here claim that this can only happen in presence of a cut, and only the communication between R and S must occur along the channel involved in the cut. Now, it's only a matter of showing that the just described situation can be "resolved" preserving types. And this can be done by way of several lemmas, like the following: Lemma 4 Assume that:
D −→ E = !L ! (x 1 .!L ! (x 2 . . . . !L ! (x n .cut ! (F,1. Γ 1 ; ∆; Θ 1 ⊢ D :: x : A ⊗ B with D = P (νy)x y −−−−→ Q; 2. Γ 2 ; ∆; Θ 2 , x : A ⊗ B ⊢ E :: z : C with E = R x(y) − − → S. Then: 1. cut(D, x.E) ֒→=⇒֒→ F for some F; 2. Γ 1 , Γ 2 ; ∆; Θ 1 , Θ 2 ⊢ F :: z : C, where F ≡ (νx)(Q | S).
The other lemmas can be found in [4]. By the way, this proof technique is very similar to the one introduced by Caires and Pfenning [1].
Proving Polynomial Bounds
In this section, we prove the main result of this paper, namely some polynomial bounds on the length of internal reduction sequences and on the size of intermediate results for processes typable in πDILL.
In other words, interaction will be shown to be bounded. The simplest formulation of this result is the following: Theorem 2 For every type A, there is a polynomial p A such that whenever / 0; / 0; x : A ⊢ D :: y : 1 and / 0; / 0; / 0 ⊢ E :: x : A where D and E are normal and (νx)( D | E) → n P, it holds that n, |P| ≤ p A (| D| + | E|) Intuitively, what Theorem 2 says is that the complexity of the interaction between two processes typable without cuts and communicating through a channel with session type A is polynomial in their sizes, where the specific polynomial involved only depends on A itself. In other words, the complexity of the interaction is not only bounded, but can be somehow "read off" from the types of the communicating parties.
How does the proof of Theorem 2 look like? Conceptually, it can be thought of as being structured into four steps:
1. First of all, a natural number W(D) is attributed to any proof term D. W(D) is said to be the weight of D. 2. Secondly, the weight of any proof term is shown to strictly decrease along computational reduction, not to increase along shifting reduction and to stay the same for equivalent proof terms. 3. Thirdly, W(D) is shown to be bounded by a polynomial on | D|, where the exponent only depends on the nesting depth of boxes of D, denoted B(D). 4. Finally, the box depth B(D) of any proof term D is shown to be "readable" from its type interface. This is exactly what we are going to do in the rest of this section. Please observe how points 1-3 above allow to prove the following stronger result, from which Theorem 2 easily follows, given point 4: Proposition 1 For every n ∈ N, there is a polynomial p n such that for every process P with Γ; ∆; Θ ⊢ P :: T , if P → m Q, then m, |Q| ≤ p B(P) (|P|).
Preliminary Definitions
Some concepts have to be given before we can embark in the proof of Proposition 1. First of all, we need to define what the box-depth of a process is. Simply, given a process P, its box-depth B(P) is the nesting-level of replications 2 in P. As an example, the box-depth of !x(y).!z(w).0 is 2, while the one of (νx)y(z) is 0. Analogously, the box-depth of a proof term D is simply B( D). Now, suppose that Γ; ∆; Θ ⊢ D :: T and that x : A belongs to either Γ or ∆, i.e. that x is an "exponential" channel in D. A key parameter is the virtual number of occurrences of x in D, which is denoted as FO(x, D). This parameter, as its name suggests, is not simply the number of literal occurrences of x in D, but takes into account possible duplications derived from cuts. So, for example, FO(w, cut ! (D, x.E)) = FO(x, E) · FO(w, D) + FO(w, E), while FO(w, ⊗R(D, E)) is merely FO(w, D) + FO(w, E). Obviously, FO(w, ♭ ! (x, w.D)) = 1 and FO(w, ♭ # (x, w.D)) = 1.
A channel in either the auxiliary or the exponential context can "float" to the linear context as an effect of rules !L ! or !L # . From that moment on, it can only be treated as a linear channel. As a consequence, it makes sense to define the duplicability factor of a proof term It's now possible to give the definition of W(D), namely the weight of the proof term D. Before doing that, however, it is necessary to give a parameterized notion of weight, denoted W n (D). Intuitively, W n (D) is defined similarly to | D|. However, every input and output action in D can possibly count for more than one:
• Everything inside D in !R(x 1 , . . . , x n , D) counts for n;
• Everything inside D in either cut ! (D, x.E) or cut # (D, x.E) counts for FO(x, E). For example, W n (cut # (D, x.E)) = FO(x, E) · W n (D) + W n (E), while W n (&L 2 (x, y.D)) = 1 + W n (D). Now, W(D) is simply W D(D) (D)
. The concepts we have just introduced are more precisely defined in [4].
Monotonicity Results
The crucial ingredient for proving polynomial bounds are a series of results about how the weight D evolves when D is put in relation with another proof term E by way of either =⇒, −→ or ≡. Whenever a proof term D computationally reduces to E, the underlying weight is guaranteed to strictly decrease: • Suppose that D = cut ! (F, x.♭ ! (x, y.G)) =⇒ cut(F ⇓ , y.cut # (F, x.G ⇓ )) = E. Then,
D(D) = max{D(F ⇓ ), D(G ⇓ )} = max{D(F), D(F), D(G)} = D(E); W(D) = W D(D) (D) = FO(x, ♭ ! (x, y.G)) · W D(D) (F ⇓ ) + W D(D) (♭ ! (x, y.G)) = W D(D) (F) + W D(D) (♭ ! (x, y.G)) = W D(D) (F) + 1 + W D(D) (G) ≥ W D(E) (F) + 1 + W D(E) (G) > W D(E) (F) + W D(E) (G) = W D(E) (F) + 0 · W D(E) (F) + W D(E) (G) = W D(E) (F) + FO(x, G) · W D(E) (F) + W D(E) (G) = W D(E) (E) = W(E). • Suppose that D = cut # (F, x.♭ # (x, y.G)) =⇒ cut(F ⇓ , y.cut # (F, x.G)) = E.
Then we can proceed exactly as in the previous case. This concludes the proof. Proof. By induction on the proof that D −→ E. Some interesting cases: • Suppose that
• Suppose that D = cut(!R(x 1 , . . . , x n , F), x.!L ! (x.G)) −→ !L ! (x 1 .!L ! (x 2 . . . . !L ! (x n .cut ! (F, y.G)))) = E.D = cut(!R(x 1 , . . . , x n , F), x.!L # (x.G)) −→ !L # (x 1 .!L # (x 2 . . . . !L # (x n .cut # (F, y.G)))) = E.
Then we can proceed as in the previous case. This concludes the proof. • Suppose that D = cut(F, x.cut(G, y.H xy )) ≡ cut(G, x.cut(F, y.H xy )) = E.
Then we can proceed as in the previous case. • Suppose that
D = cut # (F, x.cut(G x , y.H xy )) ≡ cut(cut # (F, x.G x ), y.cut # (F, x.H xy )) = E.
Then, This concludes the proof. 2
D(D) = max{D(F), D(G x ), D(H xy )} = D(E) W(D) = FO(x, cut(G x , y.H xy )) · W D(D) (F) + W D(D) (G x ) + W D(D) (H xy ) = (FO(x, G x ) + FO(x, H xy )) · W D(D) (F) + W D(D) (G x ) + W D
Now, consider again the subject reduction theorem (Theorem 1): what it guarantees is that whenever P → Q and D = P, there is E with E = Q and D ֒→=⇒֒→ E. In view of the three propositions we have just stated and proved, it's clear that W(D) > E. Altogether, this implies that W(D) is an upper bound on the number or internal reduction steps D can perform. But is W(D) itself bounded?
Bounding the Weight
What kind of bounds can we expect to prove for W(D)? More specifically, how related are W(D) and | D|? Proof. An easy induction on the structure of a type derivation π for Γ; ∆; Θ ⊢ D :: T . Proof. By induction on the structure of D. Some interesting cases:
• If D = cut ! (D, x.E), then:
W n (cut ! (D, x.E)) = FO(x, E) · (W n (D) + 1) + W n (E)
≤ FO(x, E) · (|D| · n B(D)+1 + 1) + |E| · n B(E)+1
≤ n · |D| · n B(D)+1 + n + |E| · n B(E)+1
≤ |D| · n B(D)+2 + n B(E)+1 + |E| · n B(E)+1
≤ (|D| + |E| + 1) · n max{B(D)+2,B(E)+1}
= |cut ! (D, x.E)| · n B(cut ! (D,x.E)) .
• If D = !R(x 1 , . . . , x n , E), then:
W n (!R(x 1 , . . . , x n , E)) = n · (W n (E) + 1)
≤ n · |E| · n B(E)+1 + n ≤ |E| · n B(E)+2 + n B(E)+2
= (1 + |E|) · n B(!R(x 1 ,...,x n ,E))+1
= |!R(x 1 , . . . , x n , E)| · n B(!R(x 1 ,...,x n ,E))+1 .
This concludes the proof. 2
Putting Everything Together
We now have almost all the necessary ingredients to obtain a proof of Proposition 1: the only missing tales are the bounds on the size of any reducts, since the polynomial bounds on the length of internal reductions are exactly the ones from Lemma 6. Observe, however, that the latter induces the former:
Lemma 7 Suppose that P → n Q. Then |Q| ≤ n · |P|.
Proof. By induction on n, enriching the statement as follows: whenever P → n Q, both |Q| ≤ n · |P| and |R| ≤ |P| for every subprocess R of Q in the form !x(y).S. 2
Let us now consider Theorem 2: how can we deduce it from Proposition 1? Everything boils down to show that for normal processes, the box-depth can be read off from their type. In the following lemma, B(A) and B(Γ) are the nesting depths of ! inside the type A and inside the types appearing in Γ (for every type A and context Γ). Proof. An easy induction on D.
Conclusions
In this paper, we introduced a variation on Caires and Pfenning's πDILL, called πDSLL, being inspired by Lafont's soft linear logic. The key feature of πDSLL is the fact that the amount of interaction induced by allowing two processes to interact with each other is bounded by a polynomial whose degree can be "read off" from the type of the session channel through which they communicate. What we consider the main achievement of this paper is definitely not the proof of these polynomial bounds, which can be obtained by adapting the ones in [6] or in [5], although this anyway presents some technical difficulties due to the low-level nature of the π-calculus compared to the lambda calculus or to higher-order π-calculus. Instead, what we found very interesting is that the operational properties induced by typability in πDSLL, bounded interaction in primis, are not only very interesting and useful in practice, but different from the ones obtained in soft lambda calculi: in the latter, it's the normalization time which is bounded, while here it's the interaction time. Another aspect that we find interesting is the following: it seems that the constraints on processes induced by the adoption of the more stringent typing discipline πDSLL, as opposed to πDILL, are quite natural and do not rule out too many interesting examples. In particular, the way sessions can be defined remains essentially untouched: what changes is the way sessions can be offered, i.e. the discipline governing the offering of multiple sessions by servers.
All the examples in [1] and the one from Section 2 are indeed typable in πDSLL.
Topics for future work include the accommodation of recursive types into πDSLL. This could be easier than expected, due to the robustness of light logics to the presence of recursive types [3].
Figure 1 :
1The rules governing the typing constant 1, the multiplicatives (⊗ and ⊸) and the additives (⊕ and &) are exact analogues of the ones from πDILL. The only differences come from the presence of two exponential contexts: in binary multiplicative rules (⊗R and ⊸ L) the auxiliary Γ; ∆; Θ ⊢ P :: T Γ; ∆; Θ, x : 1 ⊢ P :: T 1L Γ; ∆; / 0 ⊢ 0 :: x : 1 1R Γ; ∆; Θ, y : A, x : B ⊢ P :: T Γ; ∆; Θ, x : A ⊗ B ⊢ x(y).P :: T ⊗L Γ 1 ; ∆; Θ 1 ⊢ P :: y : A Γ 2 ; ∆; Θ 2 ⊢ Q ::x : B Γ 1 , Γ 2 ; ∆; Θ 1 , Θ 2 ⊢ (νy)x y .(P | Q) :: x : A ⊗ B ⊗R Γ 1 ; ∆; Θ 1 , y : A ⊢ P :: T Γ 2 ; ∆; Θ 2 , x : B ⊢ Q :: T Γ 1 , Γ 2 ; ∆; Θ 1 , Θ 2 , x : A ⊸ B ⊢ (νy)x y .(P | Q) :: T ⊸ L Γ; ∆; Θ, y : A ⊢ P :: x : B Γ; ∆; Θ ⊢ x(y).P :: x : A ⊸ B ⊸ R Γ; ∆; Θ, x : A⊢ P :: T Γ; ∆; Θ, x : B ⊢ P :: T Γ; ∆; x : A ⊕ B, Θ ⊢ y.case(P, Q) :: T ⊕L Γ; ∆; Θ ⊢ P :: x : A Γ; ∆; Θ ⊢ x.inl; P :: x : A ⊕ B ⊕R 1 Γ; ∆; Θ ⊢ P :: x : B Γ; ∆; Θ ⊢ x.inr; P :: x : A ⊕ B ⊕R 2 Γ; ∆; Θ, x : A ⊢ P :: T Γ; ∆; Θ, x : A&B ⊢ x.inl; P :: T &L 1 Γ; ∆; Θ, x : B ⊢ P :: T Γ; ∆; Θ, x : A&B ⊢ x.inr; P :: T &L 2 Γ; ∆; Θ ⊢ P :: x : A Γ; ∆; Θ ⊢ P :: x : B Γ; ∆; Θ ⊢ y.case(P, Q) :: x : A&B &R Γ; ∆, x : A; Θ, y : A ⊢ P :: T Γ; ∆, x : A; Θ ⊢ (νy)x y .P :: T ♭ # Γ; ∆; Θ, y : A ⊢ P :: T Γ, x : A; ∆; Θ ⊢ (νy)x y .P :: T ♭ ! Γ; ∆, x : A; Θ ⊢ P :: T Γ; ∆; Θ, x :!A ⊢ P :: T !L # Γ, x : A; ∆; Θ ⊢ P :: T Γ; ∆; Θ, x :!A ⊢ P :: T !L ! Γ; / 0; / 0 ⊢ Q :: y : A / 0; ∆; !Γ ⊢!x(y).Q :: x :!A !R Γ 1 ; ∆; Θ 1 ⊢ P :: x : A Γ 2 ; ∆; Θ 2 , x : A ⊢ Q :: T Γ 1 , Γ 2 ; ∆; Θ 1 , Θ 2 ⊢ (νx)(P | Q) :: T cut ∆; / 0; / 0 ⊢ P :: y : A Γ; ∆, x : A; Θ ⊢ Q :: T Γ; ∆; Θ ⊢ (νx)(!x(y).P | Q) :: T cut # Γ 1 ; / 0; / 0 ⊢ P :: y : A Γ 2 , x : A; ∆; Θ ⊢ Q :: T Γ 1 , Γ 2 ; ∆; Θ ⊢ (νx)(!x(y).P | Q) :: T cut ! Typing rules for πDSLL.
The reflexive and transitive closure of −→ ∪ ≡ is denoted with ֒→, i.e. ֒→= ( −→ ∪ ≡) * . There is not enough space here to give the rules defining =⇒, −→ and ≡. Let us give only some relevant examples: • Let us consider the proof term D = cut((⊗R(F, G)), x. ⊗ L(x, y.x.H)) which corresponds to the ⊗case of cut elimination. By a computational reduction rule, D =⇒ E = cut(F, y.cut(G, x.H)). From the process side, D = (νx)(((νy)x y .( F | G)) | x(y). H) and E = (νx)(νy)(( F | G) | H), where E is the process obtained from D by internal passing the channel y through the channel x. • Let D = cut(!R(F, x 1 , . .
Lemma 2 (
2Weakening lemma) If Γ; ∆; Θ ⊢ D :: T and whenever ∆ ⊆ Φ, it holds that Γ; Φ; Θ ⊢ D :: T . Proof. By a simple induction on the structure of D. 2 Lemma 3 (Lifting lemma) If Γ; ∆; Θ ⊢ D :: T then there exists an E such that / 0; Γ, ∆; Θ ⊢ E :: T where E = D. We denote E by D ⇓ . Proof. Again, a simple induction on the structure of the proof term D. 2 Finally: Theorem 1 (Subject Reduction) Let Γ; ∆; Θ ⊢ D :: T . Suppose that D = P → Q. Then there is E such that E = Q, D ֒→=⇒֒→ E and Φ; Ψ; Θ ⊢ E :: T , where Γ, ∆ = Φ, Ψ.
D, written D(D), simply as the maximum of FO(x, D) over all instances of the rules !L ! or !L # in D, where x is the involved channel. For example, D(!L ! (x.D)) = max{D(D), FO(y, D)} and D(⊸ L(x, D, y.E)) = max{D(D), D(E)}.
Proposition 2
2If Γ; ∆; Θ ⊢ D :: T and D =⇒ E, then Φ; Ψ; Θ ⊢ E :: T (where Γ, ∆ = Φ, Ψ), D(E) ≤ D(D) and W(E) < W(D).Proof. By induction on the proof that D =⇒ E. Some interesting cases:• Suppose that D = cut(⊸ R(y.F), x. ⊸ L(x, G, x.H)) =⇒ cut(cut(G, y.F), x.H) = E. Then, D(D) = max{D(F), D(G), D(H)} = D(E); W(D) = W D(D) (D) = 3 + W D(D) (F) + W D(D) (G) + W D(D) (H) > 2 + W D(E) (F) + W D(E) (G) + W D(E) (H) = W D(E) (E) = W(E).• Suppose that D = cut(&R(F, G), x.&L 1 (x, y.H)) =⇒ cut(F, x.H) = E. Then, D(D) = max{D(F), D(G), D(H)} = D(E); W(D) = W D(D) (D) = 3 + W D(D) (F) + W D(D) (G) + W D(D) (H) > 2 + W D(E) (F) + W D(E) (G) + W D(E) (H) = W D(E) (E) = W(E).
2
2Shift reduction, on the other hand, is not guaranteed to induce a strict decrease on the underlying weight which, however, cannot increase:Proposition 3 If Γ; ∆; Θ ⊢ D :: T and D −→ E, then Γ; ∆; Θ ⊢ E :: T , D(E) ≤ D(D) and W(E) ≤ W(D).
D) = max{D(F), D(G)} = D(E) W(D) = W D(D) (D) = D(D) · W D(D) (F) + W D(D) (G) ≥ FO(y, G) · W D(D) (F) + W D(D) (G) = FO(y, G) · W D(E) (F) + W D(E) (G) = W D(E) (E) = W(E).
2 Finally, equivalence leaves the weight unchanged: Proposition 4
24If Γ; ∆; Θ ⊢ D :: T and D ≡ E, then Γ; ∆; Θ ⊢ E :: T , D(E) = D(D) and W(E) = W(D).Proof. By induction on the proof that D ≡ E. Some interesting cases:• Suppose that D = cut(F, x.cut(G x , y.H y )) ≡ cut(cut(F, x.G x ), y.H y ) = E.Then: D(D) = max{D(F), D(G x ), D(H y )} = D(E) W(D) = W D(D) (D) = W D(D) (F) + W D(D) (G x ) + W D(D) (H y ) = W D(E) (F) + W D(E) (G x ) + W D(E) (H y ) = W D(E) (E) = W(E).
•
Suppose that D = cut(F, x.cut ! (G, y.H xy )) ≡ cut ! (G, y.cut(F, x.H xy )) = E.Then, since FO(y, F) = 0, D(D) = max{D(F), D(G), D(H xy )} = D(E) W(D) = W D(D) (D) = W D(D) (F) + FO(y, H xy ) · W D(D) (G) + W D(D) (H xy ) = W D(D) (F) + FO(y, cut(F, x.H xy )) · W D(D) (G) + W D(D) (H xy ) = W D(E) (F) + FO(y, cut(F, x.H xy )) · W D(E) (G) + W D(E) (H xy ) = W D(E) (E) = W(E).
(D) (H xy ) = (FO(x, G x ) · W D(D) (F) + FO(x, H xy )) · W D(D) (F) + W D(D) (G x ) + W D(D) (H xy ) = W D(D) (cut # (F, x.G x )) + W D(D) (cut # (F, x.H xy )) = W D(D) (E) = W D(E) (E) = W(E).
Lemma 5
5Suppose Γ; ∆; Θ ⊢ D :: T . Then D(D) ≤ |D|.
2 Lemma 6
26If Γ; ∆; Θ ⊢ D :: T , then for every n ≥ D(D), W n (D) ≤ | D| · n B( D)+1 .
Lemma 8
8Suppose that Γ; ∆; Θ ⊢ D :: x : A and that D is normal. Then B( D) = max{B(Γ), B(∆), B(Θ), B(A)}.
This terminology is derived from linear logic, where proofs obtained by the promotion rule are usually called boxes
Luís Caires, & Frank Pfenning, 10.1007/978-3-642-15375-4_16Session Types as Intuitionistic Linear Propositions. Springer6269CONCUR 2010Luís Caires & Frank Pfenning (2010): Session Types as Intuitionistic Linear Propositions. In: CONCUR 2010, LNCS 6269, Springer, pp. 222-236, doi:10.1007/978-3-642-15375-4_16.
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Soft Session Types (Long Version. Ugo Dal Lago & Paolo Di Giamberardino, Ugo Dal Lago & Paolo Di Giamberardino: Soft Session Types (Long Version). Available at http://arxiv. org/abs/1107.4478.
Ugo Dal Lago, Simone Martini & Davide, Sangiorgi, 10.4204/EPTCS.41.4Light Logics and Higher-Order Processes. In: EXPRESS'10. 41Ugo Dal Lago, Simone Martini & Davide Sangiorgi (2010): Light Logics and Higher-Order Processes. In: EXPRESS'10, EPTCS 41, pp. 46-60, doi:10.4204/EPTCS.41.4.
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[] |
[
"STABILITY ESTIMATES FOR THE INVERSE FRACTIONAL CONDUCTIVITY PROBLEM",
"STABILITY ESTIMATES FOR THE INVERSE FRACTIONAL CONDUCTIVITY PROBLEM"
] |
[
"Giovanni Covi ",
"Jesse Railo ",
"Teemu Tyni ",
"Philipp Zimmermann "
] |
[] |
[] |
We study the stability of an inverse problem for the fractional conductivity equation on bounded smooth domains. We obtain a logarithmic stability estimate for the inverse problem under suitable a priori bounds on the globally defined conductivities. The argument has three main ingredients: 1. the logarithmic stability of the related inverse problem for the fractional Schrödinger equation by Rüland and Salo; 2. the Lipschitz stability of the exterior determination problem; 3. utilizing and identifying nonlocal analogies of Alessandrini's work on the stability of the classical Calderón problem. The main contribution of the article is the resolution of the technical difficulties related to the last mentioned step. Furthermore, we show the optimality of the logarithmic stability estimates, following the earlier works by Mandache on the instability of the inverse conductivity problem, and by Rüland and Salo on the analogous problem for the fractional Schrödinger equation.
| null |
[
"https://export.arxiv.org/pdf/2210.01875v1.pdf"
] | 252,716,019 |
2210.01875
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75f00c38fcbbd02531e452ce6ebea883923a6e35
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STABILITY ESTIMATES FOR THE INVERSE FRACTIONAL CONDUCTIVITY PROBLEM
Giovanni Covi
Jesse Railo
Teemu Tyni
Philipp Zimmermann
STABILITY ESTIMATES FOR THE INVERSE FRACTIONAL CONDUCTIVITY PROBLEM
We study the stability of an inverse problem for the fractional conductivity equation on bounded smooth domains. We obtain a logarithmic stability estimate for the inverse problem under suitable a priori bounds on the globally defined conductivities. The argument has three main ingredients: 1. the logarithmic stability of the related inverse problem for the fractional Schrödinger equation by Rüland and Salo; 2. the Lipschitz stability of the exterior determination problem; 3. utilizing and identifying nonlocal analogies of Alessandrini's work on the stability of the classical Calderón problem. The main contribution of the article is the resolution of the technical difficulties related to the last mentioned step. Furthermore, we show the optimality of the logarithmic stability estimates, following the earlier works by Mandache on the instability of the inverse conductivity problem, and by Rüland and Salo on the analogous problem for the fractional Schrödinger equation.
Introduction
Stability estimates for inverse problems give important information on theoretical limitations of different imaging techniques appearing in various medical, engineering, and scientific applications. They are also useful for development of numerical methods. A common feature of many inverse problems is that they are ill-posed, which means that small measurement errors may lead to large errors in the reconstructed images. One of the most popular model problems is the inverse conductivity problem, known as the Calderón problem [Cal80], where one aims to recover the conductivity γ from the voltage/current measurements on the boundary ∂Ω of an object Ω. In mathematical terms, one defines the data as a Dirichlet-to-Neumann (DN) map Λ γ : f → γ∂ ν u f | ∂Ω , where ∂ ν is the outer boundary normal derivative, the electric potential u f is the unique solution of the boundary value problem div(γ∇u) = 0 in Ω, u = f on ∂Ω,
and the voltage f is the given Dirichlet boundary condition. The Calderón problem asks to recover γ from the knowledge of Λ γ , which corresponds to knowing the outer normal fluxes (i.e. boundary currents) generated by imposing different boundary voltages f .
The Calderón problem serves both as a mathematical model for electrical impedance tomography [Uhl14], and more generally as a prototypical model for inverse problems. In fact, methods and techniques originally developed for the classical Calderón problem have applications in a wide range of other inverse problems, among which the anisotropic Calderón problem [APL05,DSFKSU09], hyperbolic problems [RS88,Sun90] and inverse problems related to the theory of elasticity [NU94]. The work of Sylvester and Uhlmann proved a fundamental uniqueness theorem for the classical Calderón problem in dimension n ≥ 3, using a reduction to an analogous problem for the Schrödinger equation and constructing the so called complex geometrical optics (or CGO) solutions [SU87]. Nachman established a reconstruction method [Nac88], and Astala-Päivärinta showed a fundamental uniqueness result when n = 2 using methods from complex analysis and a reduction to the Beltrami equation [AP06]. We recall the stability theorem of Alessandrini [Ale88], which is an important motivation for our present work:
Theorem 1.1 (Alessandrini [Ale88, Theorem 1]). Let Ω be a bounded domain in R n , n ≥ 3, with C ∞ boundary ∂Ω. Given s and E, s > n/2, E > 0, let γ 1 , γ 2 be any two functions in H s+2 (Ω) satisfying the following conditions E −1 ≤ γ (x), for every x in Ω, = 1, 2.
γ H s+2 (Ω) ≤ E, = 1, 2.
The following estimate holds 1 γ 1 − γ 2 L ∞ (Ω) ≤ C E ω( Λ γ 1 − Λ γ 2 H 1/2 (∂Ω)→H −1/2 (∂Ω) ),
where the function ω is such that ω(t) ≤ | log t| −δ , for every t, 0 < t < 1/e, and δ, 0 < δ < 1, depends only on n and s.
Mandache showed that the logarithmic stability estimates are optimal up to the constants C, δ [Man01]. The works of Alessandrini and Mandache therefore show that the classical Calderón problem is ill-posed and furthermore accurately characterize this phenomenon. Mandache's work was recently systematically studied and extended by Koch, Rüland and Salo [KRS21] to many different settings. For the other recent works on the stability of the classical Calderón problem, we point to the following works [CDR16,CS14], where stability under partial data is obtained, and stability for recovery of anisotropic conductivies is considered. Under certain a priori assumptions, such as piecewise constant conductivities, the stronger result of Lipschitz stability holds [AV05]. Lipschitz stability is also possible with a finite number of measurements [AS22]. In a different direction, we mention [AN19] for an application of stability to the statistical Calderón problem.
In the present work, we study the stability properties of an inverse problem for a nonlocal analogue of the classical Calderón problem. There has been growing interest towards establishing the theory of inverse problems for elliptic nonlocal variable coefficient operators. Other recent studies include inverse problems for the fractional powers of elliptic second order operators [GU21] and inverse problems for source-to-solutions maps related to fractional geometric operators on manifolds [FGKU21,QU22]. We note that the exterior value inverse problems considered in [GU21] for the operators (div(γ∇)) s , 0 < s < 1, generated by the heat semigroups, give another possibility to define a nonlocal conductivity equation which is presumably different from the equation we study here.
Let s ∈ (0, 1) and consider the exterior value problem for the fractional conductivity equation div s (Θ γ ∇ s u) = 0 in Ω,
u = f in Ω e ,(1)
where Ω e := R n \ Ω is the exterior of the domain Ω and Θ γ : R 2n → R n×n is the matrix defined as Θ γ (x, y) := γ 1/2 (x)γ 1/2 (y)1 n×n . We say u ∈ H s (R n ) is a (weak) solution of (1) if u − f ∈ H s (Ω) and B γ (u, φ) := C n,s 2ˆR2n γ 1/2 (x)γ 1/2 (y) |x − y| n+2s (u(x) − u(y))(φ(x) − φ(y)) dxdy = 0 holds for all φ ∈ C ∞ c (Ω). For all f ∈ X := H s (R n )/ H s (Ω) in the abstract trace space there is a unique weak solutions u f ∈ H s (R n ) of the fractional conductivity equation (1). The fractional conductivity operator converges in the sense of distributions to the classical conductivity operator when applied to sufficiently regular functions when s ↑ 1 [Cov20, Lemma 4.2].
The exterior DN map Λ γ : X → X * is defined by Λ γ f, g := B γ (u f , g).
The inverse problem for the fractional conductivity equation asks to recover the conductivity γ from Λ γ , which maps as Λ γ : H s (Ω e ) → H −s Ωe (R n ) in the case of Lipschitz domains. We define m γ := γ 1/2 − 1 and call it the background deviation of γ. Let Ω ⊂ R n be bounded in one direction and n ≥ 1. (We suppose additionally that 0 < s < 1/2 when n = 1.) The uniqueness properties of this inverse problem are studied extensively in the recent literature and the following list summarizes these advances:
• Global uniqueness. If W ⊂ Ω e is an open nonempty set such that γ i | W are continuous a.e., and
m i ∈ H 2s, n 2s (R n ) ∩ H s (R n ), j = 1, 2, then γ 1 = γ 2 if and only if Λ γ 1 f | W = Λ γ 2 f | W for all f ∈ C ∞ c (W ) [CRZ22]
. This result generalizes and expands the scope of the earlier works [Cov20,RZ22b], which solved the inverse problem in certain special cases by means of the fractional Liouville transformation. This is a technique used to reduce the fractional conductivity equation to the fractional Schrödinger equation introduced in [GSU20], which is in turn better understood.
• Low regularity uniqueness. If W ⊂ Ω e is an open nonempty set such that γ i | W are continuous a.e., and m i ∈ H s,n/s (R n ), then [RZ22c]. See the original work on the construction of counterexamples with H 2s, n 2s (R n ) regularity assumptions in [RZ22a], with some limitations in the cases of unbounded domains when n = 2, 3. In this article, we obtain a quantitative stability estimate for the global inverse fractional conductivity problem on bounded smooth domains with full data. This is based on one of the possible global uniqueness proofs presented in [CRZ22,RZ22c]. There remain some nontrivial challenges in order to obtain a quantitative version of the partial data uniqueness results in [CRZ22,RZ22c], as well as to remove the regularity/boundedness assumptions of the domain even for the full data case.
γ 1 = γ 2 if and only if Λ γ 1 f | W = Λ γ 2 f | W for all f ∈ C ∞ c (W ) [, γ 2 ∈ L ∞ (R n ) ∩ C ∞ (R n ) such that γ 1 (x), γ 2 (x) ≥ γ 0 > 0, m 1 , m 2 ∈ H s,n/s (R n ) ∩ H s (R n ), and Λ γ 1 f | W 2 = Λ γ 2 f | W 2 for all f ∈ C ∞ c (W 1 )
We will next recall two earlier stability results related to the fractional Calderón problems. The first one considers the stable recovery of γ in the exterior, based on [CRZ22, Proposition 1.4]. The second one considers the stability of the analogous inverse problem for the fractional Schrödinger equation (−∆) s + q due to Rüland and Salo [RS20, Theorem 1.2]. The uniqueness properties of the Calderón problem for large classes of fractional Schrödinger type equations have been extensively studied starting from the seminal work of [GSU20]. These include perturbations to the fractional powers of elliptic operators [GLX17], first order perturbations [CLR20], nonlinear perturbations [LL22], higher order equations with local perturbations [CMR21,CMRU22], quasilocal perturbations [Cov21], and general theory for nonlocal elliptic equations [RS20,RZ22b].
In particular, the following results are needed in our proofs:
Theorem 1.2 ([RZ22c, Remark 3.3]).
Let Ω ⊂ R n be a domain bounded in one direction and 0 < s < 1. Assume that γ 1 , γ 2 ∈ L ∞ (R n ) satisfy γ 1 (x), γ 2 (x) ≥ γ 0 > 0, and are continuous a.e. in Ω e . There exists a constant C > 0 depending only on s such that 2
γ 1 − γ 2 L ∞ (Ωe) ≤ C Λ γ 1 − Λ γ 2 * .
Given a Sobolev multiplier q ∈ M (H s → H −s ) (cf. [RS20,CMRU22]), we define the following bilinear form
B q (u, v) :=ˆR n (−∆) s/2 u (−∆) s/2 v dx + qu, v , u, v ∈ H s (R n ),
related to the fractional Schrödinger operator (−∆) s + q.
Theorem 1.3 ([RS20, Theorem 1.2]).
Let Ω ⊂ R n be a bounded smooth domain, 0 < s < 1, and W 1 , W 2 ⊂ Ω be nonemtpy open sets. Assume that for some δ, M > 0 the potentials q 1 , q 2 ∈ H δ, n 2s (R n ) have the bounds q j H δ, n 2s (Ω) ≤ M, j = 1, 2. Suppose also that zero is not a Dirichlet eigenvalue for the exterior value problem
(2) (−∆) s u + q j u = 0, in Ω with u| Ωe = 0, for j = 1, 2. Then one has 3
q 1 − q 2 L n 2s (Ω) ≤ ω( Λ q 1 − Λ q 2 H s (W 1 )→( H s (W 2 )) * ), where Λ q j : X → X * with Λ q j f, g := B q j (u f , g)
is the DN map related to the exterior value problem for equation (2), and ω is a modulus of continuity satisfying
ω(x) ≤ C| log x| −σ , 0 < x ≤ 1
for some C and σ depending only on Ω, n, s, W 1 , W 2 , δ and M .
Lemma 1.4 (Liouville reduction, [RZ22c, Lemma 3.9]). Let 0 < s < min(1, n/2). Assume that γ ∈ L ∞ (R n ) with conductivity matrix Θ γ and background deviation m satisfies γ(x) ≥ γ 0 > 0 and m ∈ H s,n/s (R n ). Let q γ := (−∆) s m γ 1/2 . Then there holds
Θ γ ∇ s u, ∇ s φ L 2 (R 2n ) = (−∆) s/2 (γ 1/2 u), (−∆) s/2 (γ 1/2 φ)) L 2 (R n ) + q γ (γ 1/2 u), (γ 1/2 φ) for all u, φ ∈ H s (R n ).
In the sequel, we will call q γ above a potential.
1.1. Main results. We next state our main result, whose proof is based on a reduction to Theorems 1.2 and 1.3.
Theorem 1.5. Let 0 < s < min(1, n/2), > 0 and assume that Ω ⊂ R n is a smooth bounded domain. Suppose that the the conductivities γ 1 , γ 2 ∈ L ∞ (R n ) with background deviations m 1 , m 2 fulfill the following conditions:
(i) γ 0 ≤ γ 1 (x), γ 2 (x) ≤ γ −1 0 for some 0 < γ 0 < 1, (ii) m 1 , m 2 ∈ H 4s+2 , n
2s (R n ) and there exists C 1 > 0 such that
(3) m i H 4s+2 , n 2s (R n ) ≤ C 1 for i = 1, 2, (iii) m 1 − m 2 ∈ H s (R n ) and there exists C 2 > 0 (4) (−∆) s m i L 1 (Ωe) ≤ C 2
for i = 1, 2. If θ 0 ∈ (max(1/2, 2s/n), 1) and there holds Λ γ 1 − Λ γ 2 * ≤ 3 −1/δ for some 0 < δ < 1−θ 0 2 , then we have
γ 1/2 1 − γ 1/2 2 L q (Ω) ≤ ω( Λ γ 1 − Λ γ 2 * ) for all 1 ≤ q ≤ 2n n−2s , where ω(x) is a logarithmic modulus of continuity satisfying ω(x) ≤ C| log x| −σ , for 0 < x ≤ 1,
for some constants σ, C > 0 depending only on s, , n, Ω, C 1 , C 2 , θ 0 and γ 0 .
Remark 1.6. We make several comments about Theorem 1.5 and its assumptions to clarify some interesting points:
(i) Theorems 1.2 and 1.5 together imply that for any compact set K ⊂ R n there holds γ
1/2 1 − γ 1/2 2 L q (K) ≤ ω( Λ γ 1 − Λ γ 2 * )
where ω is a logarithmic modulus of continuity with a constant C additionally depending on K. In general, one has L ∞ control in Ω e and L q control in Ω.
(ii) The L 1 assumption (4) is required due to the noncompact, global, setting of the problem, and L ∞ stability in the exterior. The stability estimate in the exterior forces us to impose the additional condition that the related potentials q i := (−∆) s m i γ 1/2 i ∈ (L ∞ (Ω e )) * with a priori bounds in their norms, and hence (4) is a natural assumption. (iii) We assume that the domain has smooth boundary due to Theorem 1.3, as one has to impose the smoothness assumption in order to use the Vishik-Eskin estimates. In light of [RS20, Remark 7.1], the rest of their proof could be formulated under weaker regularity assumptions, as well as the one of Theorem 1.5 (see Section 5). This leaves the interesting open question of whether it is possible to obtain the stability results, i.e. Theorems 1.3 and 1.5, for less regular domains. (iv) We impose the assumption (3) so that the related potentials satisfy q i ∈ H δ, n 2s (Ω) for some δ > 0, and Theorem 1.3 applies. The assumption (3) also implies that γ 1 , γ 2 are continuous, so that Theorem 1.2 is known to apply. This assumption is much stronger than the ones required for the global uniqueness theorems [CRZ22,RZ22c].
(v) By formally taking s = 1 in Theorem 1.5 and comparing with Theorem 1.1, one sees that the latter has slightly sharper differentiability assumptions when n = 3, and the reverse is true for n ≥ 5. In dimension n = 4, the assumptions of the two theorems are comparable.
We complement our stability estimate with the following statement on exponential instability under partial data.
Theorem 1.7. Let B 1 ⊂ R n be the unit ball. For any ∈ R + \ N such that − 2s ∈ R + \ N as well there exists a constant β > 0 such that for all sufficiently small > 0 there are conductivities γ 1 , γ 2 ∈ C (R n ) such that
Λ γ 1 − Λ γ 2 H s (B 3 \B 2 )→(H s (B 3 \B 2 )) * ≤ exp − − n (2n+3) , γ 1 − γ 2 L ∞ (B 1 ) ≥ , γ i C (B 1 ) ≤ β, 1 ≤ γ i ≤ 2, i = 1, 2.
Remark 1.8. In the above theorem, we let B r , r > 0, be the ball of radius r centered at the origin. For the sake of simplicity, we restrict our analysis of instability to a very symmetric geometrical setting. This is convenient for the proof, as it is possible to explicitly construct a basis {f h,k,l } h,k∈N,0≤l≤l h of L 2 (B 3 \ B 2 ) with the special properties given in Lemma 2.1 of [RS18], where l h is the number of spherical harmonics of order h on ∂B 1 . However, it would suffice to consider the exterior DN maps in any annulus B R \ B r with 1 < r < R, as the rest of the construction can be easily adapted to this case. Whether instability holds in the case of full data remains to be proved.
1.2.
Organization of the article. The article is organized as follows. We begin Section 2 by defining the many needed function spaces and recalling the notation for the fractional conductivity equation. Section 3 concerns extension and multiplication lemmas for Sobolev functions. In Section 4, we prove our main stability estimate, Theorem 1.5. In Section 5, we discuss quantitative reduction to the Schrödinger problem with partial data. Finally, to completement the stability theorem, in Section 6 we prove the exponential instability result of Theorem 1.7. For clarity, some proofs of auxiliary results are postponed to Appendix A.
Acknowledgements. G.C. was supported by an Alexander-von-Humboldt postdoctoral fellowship. J.R. was supported by the Vilho, Yrjö and Kalle Väisälä Foundation of the Finnish Academy of Science and Letters.
Preliminaries
2.1. Function spaces. Throughout this article Ω ⊂ R n is always an open set. The classical Sobolev spaces of order k ∈ N and integrability exponent p ∈ [1, ∞] are denoted by W k,p (Ω) and for k = 0 we use the convention W 0,p (Ω) = L p (Ω). Moreover, we let W s,p (Ω) stand for the fractional Sobolev spaces, when s ∈ R + \ N and 1 ≤ p < ∞. These spaces are also called Slobodeckij spaces or Gagliardo spaces. If 1 ≤ p < ∞ and s = k + σ with k ∈ N 0 , 0 < σ < 1, then they are defined by
W s,p (Ω) := { u ∈ W k,p (Ω) ; [∂ α u] W σ,p (Ω) < ∞ ∀|α| = k },u W s,p (Ω) := u p W k,p (Ω) + |α|=k [∂ α u] p W σ,p (Ω) 1/p .
Next we recall the definition of the Bessel potential spaces H s,p (R n ) and introduce several local variants of them. For the Fourier transform, we use the following convention
Fu(ξ) :=û(ξ) :=ˆR n u(x)e −ix·ξ dx,
whenever it is defined. Moreover, the Fourier transform acts as an isomorphism on the space of Schwartz functions S (R n ) and by duality on the space of tempered distributions S (R n ). The inverse of the Fourier transform is denoted in each case by F −1 . The Bessel potential of order s ∈ R is the Fourier multiplier D s :
S (R n ) → S (R n ), that is D s u := F −1 ( ξ s u),
where ξ := (1 + |ξ| 2 ) 1/2 is the Japanese-bracket. For any s ∈ R and 1 ≤ p < ∞, the Bessel potential space H s,p (R n ) is defined by
H s,p (R n ) := {u ∈ S (R n ) ; D s u ∈ L p (R n )}, which we endow with the norm u H s,p (R n ) := D s u L p (R n ) . If Ω ⊂ R n , F ⊂ R n
are given open and closed sets, then we define the following local Bessel potential spaces:
H s,p (Ω) := closure of C ∞ c (Ω) in H s,p (R n ), H s,p F (R n ) := { u ∈ H s,p (R n ) ; supp(u) ⊂ F }, H s,p (Ω) := { u| Ω ; u ∈ H s,p (R n ) }.
The space H s,p (Ω) is equipped with the quotient norm
u H s,p (Ω) := inf{ w H s,p (R n ) ; w ∈ H s,p (R n ), w| Ω = v }.
We see that H s,p (Ω), H s,p F (R n ) are closed subspaces of H s,p (R n ). As customary, we set H s (Ω) := H s,2 (Ω) for any open set Ω ⊂ R n . If the boundary of the domain Ω ⊂ R n is regular enough then there is a close relation between the fractional Sobolev and Bessel potential spaces but also between two of the above introduced local Bessel potential spaces. For this purpose we next introduce the Hölder spaces and the notion of domains of class C k,α .
For all k ∈ N 0 and 0 < α ≤ 1, the space C k,α (Ω) consists of all functions u ∈ C k (Ω) such that the norm
u C k,α (Ω) := u C k (Ω) + |β|=k [∂ β u] C 0,α (Ω) is finite, where u C k (Ω) := |β|≤k ∂ β u L ∞ (Ω) and [u] C 0,α (Ω) := sup x =y∈Ω |u(x) − u(y)| |x − y| α .
We remark that the same notation will be used for R m -valued functions. We say that an open subset Ω ⊂ R n is of class C k,α for k ∈ N 0 , 0 < α ≤ 1 if there exists C > 0 such that for any x ∈ ∂Ω there exists a ball B = B r (x), r > 0, and a map T :
Q → B satisfying (i) T ∈ C k,α (Q), T −1 ∈ C k,α (B), (ii) T C k,α (Q) , T −1 C k,α (B) ≤ C, (iii) T (Q + ) = Ω ∩ B, T (Q 0 ) = ∂Ω ∩ B.
In the special case k = 0, α = 1, we say that Ω is a Lipschitz domain. Moreover, we say that a domain is of class C k if the above conditions hold for α = 0 and it is smooth if it is of class C k for any k ∈ N. Above we used the following notation:
Q := {x = (x , x n ) ∈ R n−1 × R ; |x | < 1, |x n | < 1 } Q + := {x = (x , x n ) ∈ Q ; x n > 0 } Q 0 := {x = (x , x n ) ∈ Q ; x n = 0 }
One can prove the following equivalence of local Bessel potential spaces:
Lemma 2.1 ([McL00, Theorem 3.29]).
Let Ω ⊂ R n be a Lipschitz domain with bounded boundary and s ∈ R then H s (Ω) = H s Ω (R n ). Next we note that the following embeddings hold between Bessel potential spaces H s,p and the fractional Sobolev spaces W s,p :
Theorem 2.2. Let s ∈ R + \ N, 1 < p < ∞ and assume Ω ⊂ R n is an open set. (i) If 1 < p ≤ 2, s = k + σ with k ∈ N 0 , 0 < σ < 1 and Ω = R n or Ω is of class C k,1 with bounded boundary, then W s,p (Ω) → H s,p (Ω). (ii) If 2 ≤ p < ∞, then H s,p (Ω) → W s,p (Ω).
Remark 2.3. In the range 0 < s < 1, this theorem is a standard result. For Ω = R n a proof can be found in [Ste70, Chapter V, Theorem 5] and by using the extension theorem in Slobodeckij spaces (cf. [DNPV12, Theorem 5.4]) it follows for Lipschitz domains with bounded boundary. In the higher order case s > 1 it seems to be less well-known and therefore we provide a proof in the Appendix A.
Remark 2.4. In particular, the above theorem asserts that for all s = k + σ with k ∈ N 0 , 0 < σ < 1 there holds H s (Ω) = W s,2 (Ω), when Ω ⊂ R n is an open set of class C k,1 with bounded boundary.
2.2.
Fractional Laplacians, fractional gradient and fractional divergence. For all s ≥ 0 and u ∈ S (R n ), we define the fractional Laplacian of order s by (−∆) s u := F −1 (|ξ| 2s u), whenever the right hand side is well-defined. One can easily show by using the Mikhlin multiplier theorem that the fractional Laplacian is a bounded
linear operator (−∆) s : H t,p (R n ) → H t−2s,p (R n ) for all t ∈ R and 1 < p < ∞.
In the special case u ∈ S (R n ) and s ∈ (0, 1), the fractional Laplacian can be calculated as the following singular integrals (see e.g. [DNPV12,
Section 3]) (−∆) s u(x) = C n,s p.v.ˆR n u(x) − u(y) |x − y| n+2s dy = − C n,s 2ˆRn u(x + y) + u(x − y) − 2u(x) |y| n+2s dy,
where C n,s > 0 is a normalization constant. One immediately sees that the above integral is in the range s ∈ (0, 1/2) for (local) Lipschitz functions not really singular. The fractional Laplacian has a distinguished property which simplifies the analysis of the inverse fractional conductivity problem compared to the classical Calderón problem, namely the unique continuation property (UCP), which asserts that if
r ∈ R, 1 ≤ p < ∞, s ∈ R + \ N and u ∈ H r,p (R n ) satisfies u| V = (−∆) s u| V = 0 in some nonempty open set V ⊂ R n , then there holds u ≡ 0 in R n (cf. [KRZ22, Theorem 2.2]).
Moreover, let us point out that a large part of the theory of the inverse fractional conductivity problem can be extended to a certain class of unbounded domains, which are called domains bounded in one direction (cf. [RZ22b, Definition 2.1]), since the fractional Laplacian satisfies on these domains a Poincaré inequality. But in this work, we restrict our attention to bounded domains and therefore the stablity of the inverse fractional conductivity problem on these domains is still open.
For the rest of this section, we fix s ∈ (0, 1). The fractional gradient of order s is the bounded linear operator ∇ s :
H s (R n ) → L 2 (R 2n ; R n ) given by (see [Cov20, DGLZ12, RZ22b]) ∇ s u(x, y) := C n,s 2 u(x) − u(y) |x − y| n/2+s+1 (x − y), and satisfies ∇ s u L 2 (R 2n ) = (−∆) s/2 u L 2 (R n ) ≤ u H s (R n )
for all u ∈ H s (R n ). The adjoint of ∇ s is called fractional divergence of order s and denoted by div s . More concretely, the fractional divergence of order s is the bounded linear operator div s :
L 2 (R 2n ; R n ) → H −s (R n ) satisfying the identity div s u, v H −s (R n )×H s (R n ) = u, ∇ s v L 2 (R 2n ) for all u ∈ L 2 (R 2n ; R n ), v ∈ H s (R n ). A simple estimate shows that there holds (see [RZ22b, Section 8]) div s (u) H −s (R n ) ≤ u L 2 (R 2n )
for all u ∈ L 2 (R 2n ; R n ), and also a comparison with the quadratic form definition for the fractional Laplacian implies (−∆) s u = div s (∇ s u) for all u ∈ H s (R n ) (see e.g. [Kwa17, Theorem 1.1] and [Cov20, Lemma 2.1]).
Extension and multiplication lemmas for fractional Sobolev spaces
In this section, we establish a higher extension theorem for the spaces W s,p (Ω), where Ω ⊂ R n is a sufficiently regular domain with bounded boundary. This is then used to extend a Gagliardo-Nirenberg inequality to these domains (cf. [BM01, Corollary 2, (iii)]). These results are needed to have access to suitable Hölder embeddings for the conductivities and have access to L ∞ estimates in Ω e for the conductivities via concrete extension operators and the Gagliardo-Nirenberg inequality. This need in turn is related to having only L ∞ control of conductivities in the exterior via Theorem 1.2. The proofs of Lemmas 3.1 and 3.2 are found in Appendix A.
Lemma 3.1 (Multiplication by Hölder functions).
Let Ω ⊂ R n be a Lipschitz domain with bounded boundary, 1 ≤ p < ∞ and s ∈ R + \ N.
(i) Let 0 < s < 1 and s < µ ≤ 1. If u ∈ W s,p (Ω) and φ ∈ C 0,µ (Ω), then φu ∈ W s,p (Ω) satisfies φu W s,p (Ω) ≤ C 1 1 + µ s(µ − s) 1/p φ C 0,µ (Ω) u W s,p (Ω)
for some C 1 > 0 only depending on n and p.
(ii) Let s = k + σ with k ∈ N, 0 < σ < 1 and σ < µ ≤ 1. If u ∈ W s,p (Ω) and φ ∈ C k (Ω) with ∂ α φ ∈ C 0,µ (Ω) for all |α| ≤ k, then φu ∈ W s,p (Ω) satisfying φu W s,p (Ω) ≤ C 1 C 2 k =0 ∇ φ C 0,µ (Ω) u W s,p (Ω)
for some C 2 > 0 only depending on n, k, p, Ω.
The following statement is a generalization of [DNPV12, Lemma 5.1], where the support of u is not necessarily compact and s is allowed to be larger than one:
Lemma 3.2 (Zero extension). Let Ω ⊂ R n be an open set, s > 0 and 1 ≤ p < ∞. Assume that u ∈ W s,p (Ω) satisfies d := dist(supp(u), ∂Ω) > 0 and letū : R n → R be its zero extension, that is
u(x) := u(x), x ∈ Ω 0,
otherwise.
Thenū ∈ W s,p (R n ) and there holds
ū W s,p (R n ) ≤ C u W s,p (Ω) .
Lemma 3.3 (Higher order extension theorem). Let 1 ≤ p < ∞, s = k + σ with k ∈ N 0 , 0 < σ < 1 and assume that Ω ⊂ R n is a domain of class C k,1 with bounded boundary. Then there exists an extension op-
erator E : W s,p (Ω) → W s,p (R n ) such that Eu| Ω = u and Eu W s,p (R n ) ≤ C u W s,p (Ω) .
Proof of Lemma 3.3. By assumption there is a finite collection of balls B j , j = 1, . . . , m, and maps T j :
Q → B j such that (i) T j ∈ C k,1 (Q), T −1 j ∈ C k,1 (B j ), (ii) T j C k,1 (Q) , T −1 j C k,1 (B j ) ≤ C for some C > 0, (iii) T j (Q + ) = Ω ∩ B j , T j (Q 0 ) = ∂Ω ∩ B j for all j = 1, . . . , m. By [Bre11, Lemma 9.3] there exist (φ j ) j=0,...,m ⊂ C ∞ (R n ) such that (I) 0 ≤ φ j ≤ 1 for all j = 0, . . . , m, (II) supp(φ 0 ) ⊂ R n \ ∂Ω, (III) φ j ∈ C ∞ c (B j ) for all j = 1, .
. . , m (IV) and m j=0 φ j = 1 on R n . Using the compactness of ∂Ω and the assertion (II) we see that
d := dist(supp(φ 0 | Ω ), ∂Ω) > 0.
On the other hand the properties (III), (IV) imply ∂ α φ 0 ∈ C 0,1 (Ω) for all α ∈ N n 0 . Hence, by Lemma 3.1 we know φ 0 u ∈ W s,p (Ω) and therefore we deduce from Lemma 3.2 that u 0 := φ 0 u ∈ W s,p (R n ). Next we want to extend the functions φ j u to elements of W s,p (R n ). In the proof of [Dob10, Satz 6.10, Satz 6.38], which establishes the result for bounded domains, it has been shown that there exists u j ∈ W s,p (R n ) such that u j | Ω = φ j u for all j = 1, . . . , m and u j W s,p (R n ) ≤ C u W s,p (Ω) . Therefore, the operator E : W s,p (Ω) → W s,p (R n ) given by Eu := m j=0 u j satisfies the asserted properties and we can conclude the proof.
Lemma 3.4 (Gagliardo-Nirenberg inequality). Let 1 < p < ∞, s = k + σ with k ∈ N 0 , 0 < σ < 1 and assume that Ω = R n or Ω ⊂ R n is a domain of class C k,1 with bounded boundary. Then for any 0 < θ < 1 there holds
u W θs,p/θ (Ω) ≤ C u θ W s,p (Ω) u 1−θ L ∞ (Ω) for all u ∈ W s,p (Ω) ∩ L ∞ (Ω).
Proof. In the case Ω = R n the result holds by [BM01, Corollary 2.c)]. If Ω ⊂ R n is a domain of class C k,1 with bounded boundary then by Lemma 3.3 for all u ∈ W s,p (Ω)∩L ∞ (Ω) there is an extension Eu ∈ W s,p (R n ). Moreover, the proof in [Dob10, Satz 6.10, Satz 6.38] shows that one has Eu L ∞ (R n ) ≤ C u L ∞ (Ω) as the extensions u j are obtained by a higher order reflection technique. Therefore, we deduce
u W θs,p/θ (Ω) ≤ Eu W θs,p/θ (R n ) ≤ Eu θ W s,p (R n ) Eu 1−θ L ∞ (R n ) ≤ C u θ W s,p (Ω) u 1−θ L ∞ (Ω)
. Hence, we can conclude the proof.
Stability estimates
We prove Theorem 1.5 in this section. In the proof, we make use of the exterior determination result stated in Theorem 1.2. Then we establish Hölder estimates for the function γ
−1/2 1 − γ −1/2 2 in terms of γ 1/2 1 − γ 1/2 2
and a quantitative version of [RZ22c, Corollary 3.6]. Afterwards, we prove a reduction theorem, which demonstrates that the difference of the DN maps corresponding to two potentials q 1 , q 2 can essentially be controlled by powers of the difference of the DN maps related to the conductivities γ 1 , γ 2 . Finally, using the stability result stated in Theorem 1.3 for the fractional Schrödinger opeorators, we can prove Theorem 1.5.
Reduction Lemma.
Lemma 4.1. Let Ω ⊂ R n be an open set and 0 < α ≤ 1. For all γ 1 , γ 2 ∈ L ∞ (Ω) satisfying γ 1 (x), γ 2 (x) ≥ γ 0 > 0, we have
(5) γ −1/2 1 − γ −1/2 2 L ∞ (Ω) ≤ C γ 1/2 1 − γ 1/2 2 L ∞ (Ω) ≤ C γ 1 − γ 2 1/2 L ∞ (Ω) .
Moreover, under the additional assumption γ 1/2
1 , γ 1/2 2 ∈ C 0,α (Ω), there holds γ −1/2 i ∈ C 0,α (Ω) with (6) γ −1/2 i L ∞ (Ω) ≤ 1/γ 1/2 0 , [γ −1/2 i ] C 0,α (Ω) ≤ [γ 1/2 i ] C 0,α (Ω) γ 0 for i = 1, 2 and γ −1/2 1 − γ −1/2 2 ∈ C 0,α (Ω) satisfying [γ −1/2 1 − γ −1/2 2 ] C 0,α (Ω) ≤ [γ 1/2 1 − γ 1/2 2 ] C 0,α (Ω) γ 0 + γ 1/2 1 − γ 1/2 2 L ∞ (Ω) γ 3/2 0 ([γ 1/2 2 ] C 0,α (Ω) + [γ 1/2 1 ] C 0,α (Ω) ).(7)
Proof. We have
|γ −1/2 1 (x) − γ −1/2 2 (x)| = γ 1/2 2 (x) − γ 1/2 1 (x) γ 1/2 1 (x)γ 1/2 2 (x) ≤ γ −1 0 γ 1/2 1 − γ 1/2 2 L ∞ (Ω)
for all x ∈ Ω and therefore the first estimate in (5) follows. The second part in (5) follows from the estimate |a 1/2 − b 1/2 | ≤ |a − b| 1/2 for all a, b ∈ R + .
From now on assume that the functions γ 1 , γ 2 satisfy additionally γ 1/2 1 , γ 1/2 2 ∈ C 0,α (Ω). Using the uniform ellipticity of γ 1 , γ 2 , we have γ
−1/2 i L ∞ (Ω) ≤ γ −1/2 0 and |γ −1/2 i (x) − γ −1/2 i (y)| = |γ 1/2 i (y) − γ 1/2 (x)| |γ 1/2 i (x)γ 1/2 i (y)| ≤ [γ 1/2 i ] C 0,α (Ω) γ 0 |x − y| α .
This establishes the estimate (6) and hence γ i ∈ C 0,α (Ω) for i = 1, 2. Next we prove the bound (7). We have
[γ −1/2 1 − γ −1/2 2 ] C 0,α (Ω) = γ 1/2 2 − γ 1/2 1 γ 1/2 1 γ 1/2 2 C 0,α (Ω) = γ 1/2 1 − γ 1/2 2 γ 1/2 1 γ 1/2 2 C 0,α (Ω)
.
We have
(γ 1/2 1 − γ 1/2 2 )(x) γ 1/2 1 (x)γ 1/2 2 (x) − (γ 1/2 1 − γ 1/2 2 )(y) γ 1/2 1 (y)γ 1/2 2 (y) = |(γ 1/2 1 − γ 1/2 2 )(x)γ 1/2 1 (y)γ 1/2 2 (y) − (γ 1/2 1 − γ 1/2 2 )(y)γ 1/2 1 (x)γ 1/2 2 (x)| γ 1/2 1 (x)γ 1/2 2 (x)γ 1/2 1 (y)γ 1/2 2 (y) = ((γ 1/2 1 − γ 1/2 2 )(x) − (γ 1/2 1 − γ 1/2
2 )(y))γ By assumption we get
(γ 1/2 1 − γ 1/2 2 )(x) γ 1/2 1 (x)γ 1/2 2 (x) − (γ 1/2 1 − γ 1/2 2 )(y) γ 1/2 1 (y)γ 1/2 2 (y) ≤ |(γ 1/2 1 − γ 1/2 2 )(x) − (γ 1/2 1 − γ 1/2 2 )(y)| γ 1/2 1 (x)γ 1/2 2 (x) + |(γ 1/2 1 − γ 1/2 2 )(y)| |γ 1/2 2 (x) − γ 1/2 2 (y)| γ 1/2 1 (x)γ 1/2 2 (x)γ 1/2 2 (y) + |γ 1/2 1 (x) − γ 1/2 1 (y)| γ 1/2 1 (x)γ 1/2 1 (y)γ 1/2 2 (y) ≤ [γ 1/2 1 − γ 1/2 2 ] C 0,α (Ω) γ 0 + γ 1/2 1 − γ 1/2 2 L ∞ (Ω) γ 3/2 0 ([γ 1/2 2 ] C 0,α (Ω) + [γ 1/2 1 ] C 0,α (Ω) ) · |x − y| α
for all x, y ∈ Ω and hence there holds
[γ −1/2 1 − γ −1/2 2 ] C 0,α (Ω) ≤ [γ 1/2 1 − γ 1/2 2 ] C 0,α (Ω) γ 0 + γ 1/2 1 − γ 1/2 2 L ∞ (Ω) γ 3/2 0 ([γ 1/2 2 ] C 0,α (Ω) + [γ 1/2 1 ] C 0,α (Ω) ).
Lemma 4.2 (Multiplication by Sobolev functions).
Let Ω ⊂ R n be an open set and 0 < s < 1. If u ∈ H s (Ω) and γ ∈ L ∞ (R n ) with background deviation m ∈ H s,n/s (R n ) satisfies γ(x) ≥ γ 0 > 0 then there holds
(8) γ 1/2 u H s (Ω) ≤ C(1 + m L ∞ (R n ) + m H s,n/s (R n ) ) u H s (Ω) and (9) γ −1/2 u H s (Ω) ≤ C(1 + m L ∞ (R n ) + m H s,n/s (R n ) ) u H s (Ω) .
Proof. Let Eu ∈ H s (R n ) be an extension of u such that Eu H s (R n ) ≤ 2 u H s (Ω) . This extension exists by the quotient space definition of H s (Ω). Thus, applying [RZ22c, Lemma 3.4] to Eu ∈ H s (R n ), we deduce
γ 1/2 u H s (Ω) ≤ γ 1/2 Eu H s (R n ) ≤ mEu H s (R n ) + Eu H s (R n ) ≤ C(1 + m L ∞ (R n ) + m H s,n/s (R n ) ) Eu H s (R n ) ≤ C(1 + m L ∞ (R n ) + m H s,n/s (R n ) ) u H s (Ω) .
This establishes (8). Arguing as in the proof of [RZ22c, Lemma 3.7] we can write γ −1/2 = 1 − m m+1 with m m+1 ∈ H s,n/s (R n ) and m m+1 H s,n/s (R n ) ≤ m H s,n/s (R n ) . Thus, we can repeat the above estimates to obtain (9). Theorem 4.3. Let 0 < s < min(1, n/2), θ 0 ∈ (s/n, 1), 0 < 1 and k ∈ N 0 satisfy k < 2s + θ 0 < k + 1 and s + / ∈ N ∀ = 1, 2.
Assume that Ω ⊂ R n is a domain of class C k,1 with bounded boundary and the conductivities γ 1 , γ 2 ∈ L ∞ (R n ) with background deviations m 1 , m 2 and potentials q 1 , q 2 fulfill the following conditions:
(i) γ 0 ≤ γ 1 (x), γ 2 (x) ≤ γ −1 0 for some 0 < γ 0 < 1, (ii) m 1 , m 2 ∈ H s,n/s (R n )∩W 2s+ ,n/s (Ω e ) with m 1 −m 2 ∈ W 2s+ θ 0 ,θ 0 n/s (Ω e ), (iii) there exist C 1 , C 2 , C θ 0 > 0 such that (10) m i H s,n/s (R n ) ≤ C 1 , m i W 2s+ ,n/s (Ωe) ≤ C 2 for i = 1, 2 and (11) m 1 − m 2 θ 0 W 2s+ θ 0 ,θ 0 n/s (Ωe) ≤ C θ 0 .
Then there holds
Λ q 1 − Λ q 2 * ≤ CC θ 0 ( Λ γ 1 − Λ γ 2 * + Λ γ 1 − Λ γ 2 1 2 * + Λ γ 1 − Λ γ 2 1−θ 0 2 * )
.
Proof. Let f, g ∈ H s (Ω e ) and for i = 1, 2 denote by v i f ∈ H s (R n ) the unique solution to the fractional Schrödinger equation (−∆) s + q i (see [RZ22c,Lemma 3.11]). Using Lemma 1.4, we deduce for i = 1, 2 and any extension e g ∈ H s (R n ) of g ∈ H s (Ω e ) the identity
Λ q i f, g = B q i (v i f , e g ) = B γ i (γ −1/2 i v i f , γ −1/2 i e g ) = Λ γ i (γ −1/2 i f ), γ −1/2 i g .
Therefore, we obtain
(Λ q 1 − Λ q 2 )f, g = Λ γ 1 (γ −1/2 1 f ), γ −1/2 1 g − Λ γ 2 (γ −1/2 2 f ), γ −1/2 2 g = Λ γ 1 (γ −1/2 1 f ), (γ −1/2 1 − γ −1/2 2 )g + Λ γ 1 (γ −1/2 1 f ), γ −1/2 2 g − Λ γ 2 (γ −1/2 2 − γ −1/2 1 )f, γ −1/2 2 g − Λ γ 2 (γ −1/2 1 f ), γ −1/2 2 g = Λ γ 1 (γ −1/2 1 f ), (γ −1/2 1 − γ −1/2 2 )g + (Λ γ 1 − Λ γ 2 )(γ −1/2 1 f ), γ −1/2 2 g + Λ γ 2 (γ −1/2 1 − γ −1/2 2 )f, γ −1/2 2 g = : I 1 + I 2 + I 3
for all f, g ∈ H s (Ω e ). Next note that the assumption (i) and the fact that solutions to the homogeneous fractional conductivity equation depend continuously on the data imply (12) Λ γ i * ≤ C for i = 1, 2 and some C > 0. On the other hand, using Lemma 4.2, the uniform ellipticity (i) and the uniform bound (10) , we deduce
(13) γ 1/2 f H s (Ωe) ≤ C f H s (Ωe) and γ −1/2 f H s (Ωe) ≤ C f H s (Ωe)
for all f ∈ H s (Ω e ) and some C > 0. Using (12), (13) and Lemma 3.1, we can estimate I 1 as follows:
|I 1 | ≤ Λ γ 1 * γ −1/2 1 f H s (Ωe) (γ −1/2 1 − γ −1/2 2 )g H s (Ωe) ≤ Λ γ 1 * γ −1/2 1 f H s (Ωe) γ −1/2 1 − γ −1/2 2 C 0,s+ (Ωe) g H s (Ωe) ≤ C γ −1/2 1 − γ −1/2 2 C 0,s+ (Ωe) f H s (Ωe) g H s (Ωe) .
By Lemma 4.1 we can upper bound the Hölder norm by
γ −1/2 1 − γ −1/2 2 C 0,s+ (Ωe) ≤ C γ 1 − γ 2 1/2 L ∞ (Ωe) + [γ 1/2 1 − γ 1/2 2 ] C 0,s+ (Ωe) γ 0 + γ 1 − γ 2 1/2 L ∞ (Ωe) γ 3/2 0 ([γ 1/2 2 ] C 0,s+ (Ωe) + [γ 1/2 1 ] C 0,s+ (Ωe) ) ≤ C(1 + [γ 1/2 2 ] C 0,s+ (Ωe) + [γ 1/2 1 ] C 0,s+ (Ωe) ) γ 1 − γ 2 1/2 L ∞ (Ωe) + C[γ 1/2 1 − γ 1/2 2 ] C 0,s+ (Ωe) .(14)− γ 1/2 2 ] C 0,s+ (Ωe) ≤ C m 1 − m 2 W 2s+ ,n/s (Ωe) ≤ C m 1 − m 2 θ 0 W 2s+ θ 0 ,θ 0 n/s (Ωe) m 1 − m 2 1−θ 0 L ∞ (Ωe) ≤ C m 1 − m 2 θ 0 W 2s+ θ 0 ,θ 0 n/s (Ωe) γ 1 − γ 2 1−θ 0 2 L ∞ (Ωe)
for all s/n < θ 0 < 1. Note that by assumption we have s/n < 1/2. Therefore, using the assertion (11) and (15) we deduce from the estimate (14) the following bound:
γ −1/2 1 − γ −1/2 2 C 0,s+ (Ωe) ≤ CC θ 0 ( γ 1 − γ 2 1 2 L ∞ (Ωe) + γ 1 − γ 2 1−θ 0 2 L ∞ (Ωe)
). Hence, we have shown
|I 1 | ≤ CC θ 0 ( γ 1 − γ 2 1 2 L ∞ (Ωe) + γ 1 − γ 2 1−θ 0 2 L ∞ (Ωe) ) f H s (Ωe) g H s (Ωe)
. Clearly the same estimate holds for I 2 . Finally, for the expression I 3 we use (13) to obtain
|I 3 | ≤ C Λ γ 1 − Λ γ 2 * f H s (Ωe) g H s (Ωe) .
Therefore, using exterior stability (cf. Theorem 1.2) we have
Λ q 1 − Λ q 2 * ≤ CC θ 0 ( Λ γ 1 − Λ γ 2 * + Λ γ 1 − Λ γ 2 1 2 * + Λ γ 1 − Λ γ 2 1−θ 0 2 * ).
4.2.
Proof of Theorem 1.5. Using the reduction lemma from Section 4.1, we give here a proof of Theorem 1.5. Throughout this section, we will assume without loss of generality that > 0 is such that 0 < 1 and s + ∈ N for = 1, 2. We split the proof into three smaller technical lemmas. The first lemma states that under assumptions of Theorem 1.5 the function m := m/γ 1/2 1 satisfies a fractional conductivity equation connected to the conductivities γ i and the difference of the potentials q i . This lemma is our main tool for connecting the fractional conductivity equation to the fractional Schrödinger equation, which will allow us to use Theorem 1.3, once the potentials q i are shown to be regular enough.
Lemma 4.4. Let 0 < s < min(1, n/2), > 0 and assume that Ω ⊂ R n is a smooth bounded domain. Suppose that the the conductivities γ 1 , γ 2 ∈ L ∞ (R n ) with background deviations m 1 , m 2 fulfill the following conditions:
(i) γ 0 ≤ γ 1 (x), γ 2 (x) ≤ γ −1 0 for some 0 < γ 0 < 1, (ii) m 1 , m 2 ∈ H 4s+2 , n
2s (R n ) and there exists C 1 > 0 such that
m i H 4s+2 , n 2s (R n ) ≤ C 1 for i = 1, 2, (iii) m := m 1 − m 2 ∈ H s (R n ) .
Then there holds
div s (Θ γ 1 ∇ s m) = γ 1/2 1 γ 1/2 2 (q 2 − q 1 ) in R n ,(16)
where m := m/γ 1/2 1 .
Proof. First note that by assumption we have m i ∈ H 2s, n 2s (R n ) for i = 1, 2 and thus we can calculate as in [RZ22b, Proof of Lemma 8.13]:
γ 1/2 1 γ 1/2 2 (q 2 − q 1 ) = −γ 1/2 1 γ 1/2 2 (−∆) s m 2 γ 1/2 2 − (−∆) s m 1 γ 1/2 1 = γ 1/2 2 (−∆) s m 1 − γ 1/2 1 (−∆) s m 2 = (1 + m 2 )(−∆) s m 1 − (1 + m 1 )(−∆) s m 2 = (1 + m 2 )(−∆) s m 1 + (1 + m 1 )(−∆) s m − (1 + m 1 )(−∆) s m 1 = γ 1/2 1 (−∆) s m − m(−∆) s m 1 .
Setting m := m/γ 1/2 1 , we obtain
(17) γ 1/2 1 (−∆) s (γ 1/2 1 m) + γ 1/2 1 (γ 1/2 1 m)q 1 = γ 1/2 1 γ 1/2 2 (q 2 − q 1 ).
Using m 1 ∈ H 2s, n 2s (R n ) then we deduce from [RZ22b, Corollary A.8] that there holds γ 1/2 1 ψ, γ −1/2 1 ψ ∈ H s (R n ) for all ψ ∈ H s (R n ) and in particular m ∈ H s (R n ). We next observe that by the Gagliardo-Nirenberg inequality in Bessel potential spaces (cf. [RZ22b, Corollary A.3,(iii)]) and the monotonicity of Bessel potential spaces, we have
m i H 2s+ ,n/s (R n ) ≤ m i 1/2 H 4s+2 , n 2s (R n ) m i 1/2 L ∞ (R n )(18)
for i = 1, 2. By the uniform ellipticity of γ i , i = 1, 2, this immediately implies γ 1/2 1 γ 1/2 2 (q 2 − q 1 ) ∈ L n/s (R n ). By the assumptions n/s > 2, (11) and the uniform ellipticity as well as interpolation in L p spaces, we see that γ 1/2 1 γ 1/2 2 (q 2 − q 1 ) ∈ L 2 (Ω e ). On the other hand, the boundedness of Ω and n/s > 2 gives γ 1/2 1 γ 1/2 2 (q 2 − q 1 ) ∈ L 2 (Ω). Therefore, we have γ 1/2 1 γ 1/2 2 (q 2 − q 1 ) ∈ L 2 (R n ). Hence, multiplying (17) by φ ∈ S (R n ) and integrating over
R n showsˆR n (−∆) s (γ 1/2 1 m)(γ 1/2 1 φ) dx +ˆR n (γ 1/2 1 m)q 1 (γ 1/2 1 φ) dx =ˆR n γ 1/2 1 γ 1/2 2 (q 2 − q 1 )φ dx.
Now the first integral is finite since m ∈ H 2s, n 2s (R n ), φ ∈ S (R n ) and γ i ∈ L ∞ (R n ) for i = 1, 2, the second integral by [RZ22b, Lemma A.10] and Hölder's inequality and the integral on the right hand side by the fact that γ 1/2 1 γ 1/2 2 (q 2 − q 1 ) ∈ L 2 (R n ). Next let (ρ ) >0 be the standard mollifiers and let m := ρ * m. It is well-known that m → m in H 2s, n 2s (R n ) and H s (R n ) as m satisfies m ∈ H 2s, n 2s (R n ) ∩ H s (R n ). On the other hand since the Bessel potential commutes with mollification, we deduce m ∈ H t (R n ) for all t ∈ R as m ∈ L 2 (R n ). Therefore, we can calculatê
R n (−∆) s (γ 1/2 1 m)(γ 1/2 1 φ) dx = lim →0ˆRn (−∆) s m (γ 1/2 1 φ) dx = lim →0ˆRn (−∆) s/2 m (−∆) s/2 (γ 1/2 1 φ) dx =ˆR n (−∆) s/2 m(−∆) s/2 (γ 1/2 1 φ) dx =ˆR n (−∆) s/2 (γ 1/2 1 m)(−∆) s/2 (γ 1/2 1 φ) dx.
In the first equality we used the convergence m → m in H 2s, n 2s (R n ) as → 0, the continuity of the fractional Laplacian and that γ 1/2 1 φ ∈ L n n−2s (R n ), in the second equality that m ∈ H 2s (R n ), γ 1/2 1 φ ∈ H s (R n ) and Plancherel's theorem, in the third equality that m → m in H s (R n ) as → 0 and finally the definition of m. Therefore, we obtain
(−∆) s/2 (γ 1/2 1 m), (−∆) s/2 (γ 1/2 1 φ) L 2 (R n ) + q 1 (γ 1/2 1 m), (γ 1/2 1 φ) L 2 (R n ) = γ 1/2 1 γ 1/2 2 (q 2 − q 1 ), φ L 2 (R n ) (19) for all φ ∈ S (R n ). Now, if φ ∈ H s (R n ) then we can choose a sequence (φ k ) k∈N ⊂ S (R n ) such that φ k → φ in H s (R n ). By (19) we have (−∆) s/2 (γ 1/2 1 m), (−∆) s/2 (γ 1/2 1 φ k ) L 2 (R n ) + q 1 (γ 1/2 1 m), (γ 1/2 1 φ k ) L 2 (R n ) = γ 1/2 1 γ 1/2 2 (q 2 − q 1 ), φ k L 2 (R n )
for all k ∈ N. Since γ 1/2 1 γ 1/2 2 (q 2 − q 1 ) ∈ L 2 (R n ), there holds γ 1/2 1 γ 1/2 2 (q 2 − q 1 ), φ k L 2 (R n ) → γ 1/2 1 γ 1/2 2 (q 2 − q 1 ), φ L 2 (R n ) as k → ∞. Again by [RZ22b, Lemma A.10], Hölder's inequality and the Sobolev embedding we see that
q 1 (γ 1/2 1 m), (γ 1/2 1 φ k ) L 2 (R n ) → q 1 (γ 1/2 1 m), (γ 1/2 1 φ) L 2 (R n )
as k → ∞. Finally, by [RZ22b,Corollary A.7] it follows that γ 1/2 Hence, (19) holds for all φ ∈ H s (R n ). Therefore, by the fractional Liouville reduction (Lemma 1.4), we see that m ∈ H s (R n ) satisfies div s (Θ γ 1 ∇ s m) = γ 1/2 1 γ 1/2 2 (q 2 − q 1 ) in R n as claimed.
1 φ k → γ 1/2 1 φ in H s (R n ) and hence (−∆) s/2 (γ 1/2 1 φ k ) → (−∆) s/2 (γ 1/2 1 φ) in L 2 (R n ),
but then by the Cauchy-Schwartz inequality it follows that
(−∆) s/2 (γ 1/2 1 m), (−∆) s/2 (γ 1/2 1 φ k ) L 2 (R n ) → (−∆) s/2 (γ 1/2 1 m), (−∆) s/2 (γ 1/2 1 φ) L 2 (R n ) as k → ∞.
Next we show that the assumptions of Theorem 1.5 imply the required regularity and a priori bounds for the potentials q i , allowing us to apply Theorem 1.3.
Lemma 4.5. Let 0 < s < min(1, n/2), > 0. Suppose that the the conductivities γ 1 , γ 2 ∈ L ∞ (R n ) with background deviations m 1 , m 2 fulfill the following conditions:
(i) γ 0 ≤ γ 1 (x), γ 2 (x) ≤ γ −1 0 for some 0 < γ 0 < 1, (ii) m 1 , m 2 ∈ H 4s+2 , n 2s (R n ) and there exists C 1 > 0 such that m i H 4s+2 , n 2s (R n ) ≤ C 1 for i = 1, 2, Then q i ∈ H δ, n 2s (R n ) for δ = 2 /3 with q i H δ, n 2s (R n ) ≤ M,
where M > 0 depends only on γ 0 , C 1 , n, s and .
Proof. First observe that we can write
q i = − (−∆) s m i γ 1/2 i = −(−∆) s m i 1 − m i m i + 1 = −(−∆) s m i + (−∆) s m i m i m i + 1
for i = 1, 2. By the assumption (ii) the first term belongs to H 2s+2 , n 2s (R n ) and hence it is sufficient to show that the second term is in H δ, n 2s (R n ) for some δ > 0. Now using m i ∈ H 4s+2 , n 2s (R n ) ∩ L ∞ (R n ) ⊂ H 2s+ , n s (R n ) (see (18)), the mapping properties of the fractional Laplacian and the Sobolev embedding
H 2s+2 , n 2s (R n ) → L ∞ (R n ) we see that (−∆) s m i ∈ H , n s (R n ) ∩ L ∞ (R n ). Now we claim that m i m i +1 ∈ H 2 3 , n 2s (R n )∩L q (R n ) for all n 2s ≤ q ≤ ∞. That m i m i +1 ∈ L q (R n )
follows from the uniform ellipticity of γ i , m i ∈ L n 2s (R n ) and interpolation in L p spaces. Next define Γ 0 := min(0, γ 1/2 0 −1) and choose Γ ∈ C ∞ b (R) such that Γ(t) = t t+1 for t ≥ Γ 0 . By [AF92,p. 156] and m i ∈ H 4s+2 , n 2s (R n )∩L ∞ (R n ), we deduce for i = 1, 2 that Γ(m i ) ∈ H 4s+2 , n 2s (R n ), but since m ≥ γ 1/2 0 −1 > −1 it follows that m i m i +1 ∈ H 4s+2 , n 2s (R n )∩L ∞ (R n ). Moreover, [AF92,p. 156] gives the estimate
m i m i + 1 H 4s+2 , n 2s (R n ) ≤ C( m i H 4s+2 , n 2s (R n ) + α m i 4s+2 H 4s+2 , n 2s (R n ) ) ≤ C,(20)
where α = 0 when 4s + 2 ≤ 1 and otherwise α = 1. Hence, the claim is proved. Next define p 1 := n s , p 2 := n 2s , r 2 := 3n 4s , s 1 := and θ := 2 3 .
Then there holds 1 < p 1 , p 2 , r 2 < ∞ and 1
p 2 = θ p 1 + 1 r 2 . Moreover, since (−∆) s m i ∈ H s 1 ,p 1 (R n ) ∩ L ∞ (R n ), m i m i +1 ∈ H θs 1 ,p 2 (R n ) ∩ L r 2 (R n ) we deduce from [RZ22b, Lemma A.6] that there holds (−∆) s m i m i m i + 1 ∈ H θs 1 ,p 2 (R n ) = H 2 3 , n 2s (R n ).
Hence, we have shown q i ∈ H 2 3 , n 2s (R n ) for i = 1, 2 as previously asserted. Moreover, [RZ22b, Lemma A.6 (i)] yields the estimate
(−∆) s m i m i m i + 1 H 2 3 , n 2s (R n ) ≤ C (−∆) s m i L ∞ (R n ) m i m i + 1 H 2 3 , n 2s (R n ) + m i m i + 1 L 3n 4s (R n ) (−∆) s m i θ H , n s (R n ) (−∆) s m i 1−θ L ∞ (R n ) ≤ C
where in the second inequality we used (20) and the assumption m i ∈ H 4s+2 , n 2s (R n ).
Our final technical lemma is an interpolation statement, required for the use of Theorem 4.3.
Lemma 4.6. Let 0 < s < min(1, n/2), > 0 and assume that Ω ⊂ R n is a smooth bounded domain. Assume that θ 0 ∈ (max(1/2, 2s/n), 1). Suppose that the the conductivities γ 1 , γ 2 ∈ L ∞ (R n ) with background deviations m 1 , m 2 fulfill the following conditions:
(i) γ 0 ≤ γ 1 (x), γ 2 (x) ≤ γ −1 0 for some 0 < γ 0 < 1, (ii) m 1 , m 2 ∈ H 4s+2 , n
2s (R n ) and there exists C 1 > 0 such that
m i H 4s+2 , n 2s (R n ) ≤ C 1 for i = 1, 2.
Then there holds
m i W 2s+
θ 0 ,θ 0 n/s (Ωe) ≤ C for i = 1, 2 and some constant C > 0 depending only on n, s, , θ 0 and C 1 .
Proof. Define s 1 := 2s + , s 2 := 4s + 2 , p 1 := n s , p 2 := n 2s and θ := 2 − 1/θ 0 .
Since, 1/2 < θ 0 < 1 we have 0 < θ < 1. Moreover, there holds 0 < s 1 < s 2 < ∞, 1 < p 1 , p 2 < ∞ and 2s + θ 0 = θs 1 + (1 − θ)s 2 and 1 θ 0
n s = θ p 1 + 1 − θ p 2 . Therefore, [RZ22b, Corollary A.3] implies u H 2s+ θ 0 ,θ 0 n s (R n ) ≤ C u θ H 2s+ ,n/s (R n ) u 1−θ H 4s+2 , n 2s (R n )
for all u ∈ H 2s+ ,n/s (R n ) ∩ H 4s+2 , n 2s (R n ). By the assumptions (i)-(ii) and the uniform estimate (18) this ensures
m i H 2s+ θ 0 ,θ 0 n s (R n ) ≤ C for i = 1, 2.
On the other hand the condition θ 0 > 2s n ensures θ 0 n s > 2 and therefore Theorem 2.2 shows
m i W 2s+ θ 0 ,θ 0 n/s (Ωe) ≤ m i W 2s+ θ 0 ,θ 0 n/s (R n ) ≤ m i H 2s+ θ 0 ,θ 0 n/s (R n ) ≤ C for i = 1, 2.
We are finally ready to complete the proof of Theorem 1.5:
Proof of Theorem 1.5. We start by recalling that the homogeneous Sobolev spaceḢ s (R n ) is defined as the space of tempered distributions whose Fourier transform belongs to L 1 loc (R n ) and satisfies
f 2Ḣ s (R n ) :=ˆR n |ξ| 2s | f (ξ)| 2 dξ < ∞,m L q (Ω) ≤ C m L 2n n−2s (Ω) ≤ C m L 2n n−2s (R n ) ≤ C m Ḣs (R n ) ≤ C Θ γ 1 ∇ s m, ∇ s m .
Testing (16) with m ∈ H s (R n ) and applying Lemma 4.4 we can estimate Observe that we have q i ∈ L n 2s (R n ) for i = 1, 2, since m i ∈ H 2s, n 2s (R n ) and the conductivities are uniformly elliptic. Thus, using Hölder's inequality and then the Cauchy-Schwarz inequality, we get
m L q (Ω) ≤ C Θ γ 1 ∇ s m, ∇ s m ≤ C ˆR n γ 1/2 1 γ 1/2 2 (q 2 − q 1 ) m dxI 1 ≤ C q 1 − q 2 L n 2s (Ω) γ 1/2 2 m L n n−2s (Ω) ≤ C q 1 − q 2 L n 2s (Ω) γ 1/2 2 L 2n n−2s (Ω) m L 2n n−2s (Ω) ≤ C q 1 − q 2 L n 2s (Ω) ,
where in the last estimate we have used that Ω is bounded and the assumption (i). On the other hand the second integral can be estimated by
I 2 ≤ C mγ 1/2 2 L ∞ (Ωe) q 1 − q 2 L 1 (Ωe) ≤ C γ 1/2 2 L ∞ (Ωe) q 1 − q 2 L 1 (Ωe) m L ∞ (Ωe) ≤ C γ 2 1/2 L ∞ (Ωe) γ 1/2 0 (−∆) s m 1 L 1 (Ωe) + (−∆) s m 2 L 1 (Ωe) m L ∞ (Ωe) ≤ C m L ∞ (Ωe) ,
where we have used the assumptions (i) and (iii). Thus, for all 1 ≤ q ≤ 2n/(n − 2s) there holds
m L q (Ω) ≤ C m L 2n n−2s (Ω) ≤ C( q 1 − q 2 L n 2s (Ω) + m L ∞ (Ωe) ).(21)
By Lemma 4.5 the assumptions in Theorem 1.3 on the potentials q i , i = 1, 2, namely that q i ∈ H δ, n 2s (R n ) with an a priori bound q i H δ, n 2s (R n ) ≤ M , are satisfied and we can estimate the first term in the right-hand side of (21) as q 1 − q 2 L n 2s (Ω) ≤ Cω( Λ q 1 − Λ q 2 * ), for some logarithmic modulus of continuity satisfying ω(x) ≤ C| log(x)| −σ for all 0 < x ≤ 1, where C, σ > 0. The second term of (21) can be estimated by first using Lemma 4.1 as
m L ∞ (Ωe) = γ 1/2 1 − γ 1/2 2 L ∞ (Ωe) ≤ C γ 1 − γ 2 1/2 L ∞ (Ωe)
and then applying the exterior stability result of Theorem 1.2 to obtain
γ 1 − γ 2 1/2 L ∞ (Ωe) ≤ C Λ γ 1 − Λ γ 2 1/2 * .
In summary, we have shown that there holds
m 1 − m 2 L q (Ω) ≤ C ω( Λ q 1 − Λ q 2 * ) + Λ γ 1 − Λ γ 2 1/2 * .
Next, we show that m 1 , m 2 satisfy the conditions in Theorem 4.3. Similarly, as for the estimate (18), by the Gagliardo-Nirenberg inequality in Bessel potential spaces (cf. [RZ22b, Corollary A.3,(iii)]) and the monotonicity of Bessel potential spaces, we have
m i H s,n/s (R n ) ≤ m i H s+ ,n/s (R n ) ≤ m i 1/2 H 2s+2 , n 2s (R n ) m i 1/2 L ∞ (R n ) ≤ m i 1/2 H 4s+2 , n 2s (R n ) m i 1/2 L ∞ (R n ) ≤ C m i 1/2 L ∞ (R n )
for i = 1, 2, where in the last step we used the assumption (ii). Moreover, by the fact that n/s > 2, Theorem 2.2, (ii) and Lemma 3.4, we have
m i W 2s+ , n s (Ωe) ≤ m i W 2s+ , n s (R n ) ≤ m i H 2s+ , n s (R n ) ≤ C m i 1/2 H 4s+2 , n 2s (R n ) m i 1/2 L ∞ (R n )
for i = 1, 2. Now, using the uniform bound (3) and the uniform ellipticity of γ i we deduce m W 2s+ ,n/s (Ωe) ≤ C. Applying Lemma 4.6 we also see that
m i W 2s+ θ 0 ,θ 0 n/s (Ωe) ≤ C.
This now demonstrates that m 1 − m 2 satisfies the condition (11) in Theorem 4.3. Therefore, we can apply Theorem 4.3 to deduce the estimate
m 1 − m 2 L q (Ω) ≤ C ω Λ γ 1 − Λ γ 2 * + Λ γ 1 − Λ γ 2 1 2 * + Λ γ 1 − Λ γ 2 1−θ 0 2 * + Λ γ 1 − Λ γ 2 1 2
for all 1 ≤ q ≤ 2n n−2s . Since θ 0 ∈ (1/2, 1) we have 1−θ 0 2 ∈ (0, 1/4). Hence, there holds x ≤ x 1/2 ≤ x 1−θ 0 2 for all 0 < x ≤ 1. Therefore, we obtain
ω Λ γ 1 − Λ γ 2 * + Λ γ 1 − Λ γ 2 1 2 * + Λ γ 1 − Λ γ 2 1−θ 0 2 * ≤ ω 3 Λ γ 1 − Λ γ 2 1−θ 0 2 * .
By the assumptions Λ γ 1 − Λ γ 2 * ≤ 3 −1/δ , 0 < δ < 1−θ 0 2 , we have
3 Λ γ 1 − Λ γ 2 1−θ 0 2 * ≤ Λ γ 1 − Λ γ 2 1−θ 0 2 −δ * .
Using the fact that ω(x) ≤ C| log x| −σ for 0 < x ≤ 1, we deduce
ω Λ γ 1 − Λ γ 2 * + Λ γ 1 − Λ γ 2 1 2 * + Λ γ 1 − Λ γ 2 1−θ 0 2 * ≤ Cω ( Λ γ 1 − Λ γ 2 * ) .
Next we observe that there holds x 1/2 ≤ C| log x| −σ for all 0 < x ≤ 1 and some C > 0. To see this, observe that this estimate is equivalent to Cx −1/2 ≥ 2 −σ | log x −1/2 | σ . Since 0 < x ≤ 1, this is the same as log x −1/2 ≤ C 1/σ 2 (x −1/2 ) 1/σ , but it is well-known that there holds log(y) ≤ y r /r for all y > 0 and r > 0. In fact, the last assertion is a straightforward consequence of the inequality log(z) ≤ z − 1 for all z > 0 by applying it to z = y r with y > 0, r > 0. Hence, the above estimate holds with C = (2σ) σ . This finally shows
Λ γ 1 − Λ γ 2 * ≤ C| log( Λ γ 1 − Λ γ 2 * )| −σ = Cω( Λ γ 1 − Λ γ 2 * ),
and we can conclude the proof.
Partial data reduction with minimal regularity assumptions on the domain
For the sake of completeness and possible future work on low regularity settings, we record a proposition considering a quantitative partial data reduction to the Schrödinger case with minimal assumptions on the domain Ω. These improvements come with the cost of having certain restrictions on the considered sets of measurements but on the other hand also allow a simpler proof. Furthermore, Proposition 5.1 is strong enough to conclude Theorem 1.5 after minor changes to the proof and using the partial data stability result for the Schrödinger case in [RS20]. This argument however does not establish the quantitative reduction "up to the boundary" like the proof of Theorem 4.3. In particular, this approach avoids using W s,p spaces and the explicit extension operators, which in part explains why the boundary regularity questions are not encountered and the proof is considerably simpler.
Finally, we emphasize that the partial data reduction to the Schrödinger case does not directly imply partial data stability for the conductivity case as the partial data uniqueness result is based on an additional unique continuation argument for the conductivities (see [CRZ22,RZ22c]) and the authors are not aware of quantitative unique continuation results of the following type: Let 1 ≤ p, q, r ≤ ∞, s, t ≥ 0 and W ⊂ R n be a nonempty open set. For all u ∈ X ⊂ H t,q (R n ) there holds
(22) u L p (R n ) ≤ F ( (−∆) s u L r (W ) , u L ∞ (W ) )
for some continuous function F : [0, ∞) × [0, ∞) → [0, ∞) with F (0, 0) = 0 and independent of u ∈ X where the set X encodes possible a priori assumptions. For the relevant regularity assumptions and choices of p, q, r, s, t, see [CRZ22,RZ22c] where u = m 1 −m 2 is the choice of u in our possible application. Such estimates for the special case u| W = 0, i.e. u L ∞ (W ) = 0, could already give new results towards the stability of the partial data problems for the fractional conductivity equation. In general, estimates of the type (22) would be interesting also in other function spaces and norms.
Proposition 5.1. Let 0 < s < min(1, n/2), s/n < θ 0 < 1 and > 0.
Let Ω ⊂ R n be a nonempty open set bounded in one direction. Suppose that the conductivities γ 1 , γ 2 ∈ L ∞ (R n ) with background deviations m 1 , m 2 and potentials q 1 , q 2 fulfill the following conditions:
(i) γ 0 ≤ γ 1 (x), γ 2 (x) ≤ γ −1 0 for some 0 < γ 0 < 1, (ii) There exists C 0 > 0 such that m i H 2s+ θ 0 ,θ 0 n/s (R n ) ≤ C 0 . for i = 1, 2. Let W 1 , W 2 , W Ω e be nonempty open sets such that W 1 ∪ W 2 W . Then there holds Λ q 1 − Λ q 2 H s (W 1 )→( H s (W 2 )) * ≤ C( Λ γ 1 − Λ γ 2 H s (W )→( H s (W )) * + Λ γ 1 − Λ γ 2 1−θ 0 2 H s (W )→( H s (W )) * )
where C > 0 depending only on s, , n, Ω, C 0 , θ 0 , W 1 , W 2 , W and γ 0 .
Proof. Suppose that f ∈ C ∞ c (W 1 ) and g ∈ C ∞ c (W 2 ). Choose a smooth cutoff function η| W 1 ∪W 2 = 1, 0 ≤ η ≤ 1 and supp(η) ⊂ W . We explain next how to modify the proof of Theorem 4.3, in order to obtain the quantitative partial data reduction result. To do so, we will establish sufficient estimates next. We first note that m 1 , m 2 ∈ H 2s+ ,n/s (R n ) ⊃ H s,n/s (R n ) with explicit bounds for the norms by (i), (ii), the Gagliardo-Nirenberg inequality in Bessel potential spaces and the monotonicity of Bessel potential spaces. Therefore we may continue the proof of Theorem 4.3 up to the point where we have the terms I 1 , I 2 , I 3 as in the proof of Theorem 4.3.
Since W 1 , W 2 ⊂ W , we have
|I 3 | ≤ C Λ γ 1 − Λ γ 2 H s (W )→( H s (W )) * f H s (R n ) g H s (R n ) .
as in the earlier proof. We may suppose, by taking smaller if necessary, that 0 < < 1 − s and 2s + is not an integer by the monotonicity of Bessel potential spaces. For the term I 1 , we may calculate that
|I 1 | ≤ Λ γ 1 H s (W )→( H s (W )) * γ −1/2 1 f H s (R n ) (γ −1/2 1 − γ −1/2 2 )g H s (R n ) ≤ C Λ γ 1 H s (W )→( H s (W )) * f H s (R n ) η(γ −1/2 1 − γ −1/2 2 ) C 0,s+ (R n ) g H s (R n ) since g = ηg.
We may then estimate using the embeddings to Hölder spaces, Gagliardo-Nirenberg inequality in Bessel potential spaces, the formula (5), boundedness of the multiplication with η and support conditions, that
η(γ −1/2 1 − γ −1/2 2 ) C 0,s+ (R n ) ≤ C η(γ −1/2 1 − γ −1/2 2 ) H 2s+ ,n/s (R n ) ≤ C η(γ −1/2 1 − γ −1/2 2 ) θ 0 H 2s+ θ 0 ,θ 0 n/s (R n ) η(γ −1/2 1 − γ −1/2 2 ) 1−θ 0 L ∞ (R n ) ≤ C γ −1/2 1 − γ −1/2 2 θ 0 H 2s+ θ 0 ,θ 0 n/s (R n ) γ −1/2 1 − γ −1/2 2 1−θ 0 L ∞ (W ) ≤ C m 1 m 1 + 1 − m 2 m 2 + 1 θ 0 H 2s+ θ 0 ,θ 0 n/s (R n ) γ 1 − γ 2 1−θ 0 2 L ∞ (W ) ≤ C γ 1 − γ 2 1−θ 0 2 L ∞ (W )
where in the last step we used the triangle inequality and a composition estimate for functions in Bessel potential spaces (see e.g. [AF92] and references therein). In fact, for any t > 0, 1 < p < ∞, δ > 0, there exists a polynomial function P : R 2 → R (of degree at most t + 1 and with nonnegative coefficients) such that
m m + 1 H t,p (R n ) ≤ P ( m H t,p (R n ) , m L ∞ (R n ) )
for all m ∈ H t,p (R n ) ∩ L ∞ (R n ) with m + 1 ≥ δ. This can be argued similarly as (20) in the proof of Lemma 4.5 but we decided to recall this alternative estimate here. This let us conclude that
|I 1 | ≤ C f H s (R n ) g H s (R n ) Λ γ 1 − Λ γ 2 1−θ 0 2 H s (W )→( H s (W )) *
by the exterior stability estimate (Theorem 1.2) since γ 1 , γ 2 are continuous by (ii). This is possible since the exterior stability estimate also holds for the considered partial data, i.e. γ 1 −γ 2 L ∞ (W ) ≤ C Λ γ 1 −Λ γ 2 H s (W )→( H s (W )) * . We may argue similarly with I 2 , which completes the proof.
Remark 5.2. Theorem 4.3 could be also adapted into the partial data setting as in Proposition 5.1. We omit presenting these details.
Exponential instability
In this Section we complement the above considerations about stability with an instability result in the flavour of [KRS21]. We start by recalling the definitions of -discrete sets and δ-nets. These can be given in the setting of a generic metric space (X, d).
Definition 6.1. Let (X, d) be a metric space, and assume , δ > 0. A set Y ⊂ X is said to be -discrete if for all y 1 , y 2 ∈ Y with y 1 = y 2 it holds d(y 1 , y 2 ) ≥ . A set Z ⊂ X is said to be a δ-net for a set X 1 ⊂ X if for all x ∈ X 1 there exists z ∈ Z such that d(x, z) ≤ δ.
We can now prove Theorem 1.7, which shows that the exponential stability obtained in Section 4 can not be improved. For this result it will suffice to consider conductivities whose exterior value is the constant 1.
Proof of Theorem 1.7. Define the set
X β := {f ∈ C c (B 1 ) ; f L ∞ ≤ , f C ≤ β},
and let X β := 1+X β . By [Man01, Lemma 2] (see also [RS18,Lemma 3.4]) we deduce the existence of an -discrete set Z ⊂ X β of cardinality
| Z| ≥ exp C(β/ ) n/ ,
with C = C( , n) > 0, where X β is seen as a metric space with respect to the L ∞ norm. By careful construction, it is also possible to ensure that 1 ≤ γ ≤ 2 for all γ ∈ Z (see [Man01, Proof of Corollary 1]). Let now γ ∈ Z, and let q be the corresponding transformed potential. Since γ ≥ 1, for all v ∈H s (B 1 ) we get
div s Θ γ ∇ s v, v H −s (R n )×H s (R n ) = B γ (v, v) ≥ (−∆) s/2 v 2 L 2 (R n ) ≥ λ 1,s v 2 L 2 (B 1 )
, where λ 1,s is the first Dirichlet eigenvalue of (−∆) s in B 1 (see e.g. [RS18, Proof of Lemma 3.2]). Therefore, by the fractional Liouville reduction
((−∆) s + q)v, v H −s (R n )×H s (R n ) = div s Θ γ ∇ s (γ −1/2 v), γ −1/2 v H −s (R n )×H s (R n ) ≥ λ 1,s γ −1/2 v 2 L 2 (B 1 ) ≥ λ 1,s 2 v 2 L 2 (B 1 ) ,
and eventually ((−∆) s + q) −1 L 2 (B 1 )→L 2 (B 1 ) ≤ 2 λ 1,s .
Since m := γ 1/2 − 1 ≥ 0, for the potential we compute
q L ∞ (B 1 ) = (−∆) s m 1 + m L ∞ (B 1 ) (−∆) s m L ∞ (B 1 ) ≤ (−∆) s m C −2s (R n ) .
Observe now that the fractional Laplacian (−∆) s acts as a bounded operator between C r and C r−2s for any r, r − 2s ∈ R + \ N. In order to see this, write the symbol as |ξ| 2s = ψ(ξ)|ξ| 2s + (1 − ψ(ξ))|ξ| 2s , where ψ ∈ C ∞ c (R n ) is 1 near the origin. The second term on the right hand side belongs to Hörmander's class S 2s 1,0 , and thus has the correct mapping properties (see e.g. [Tay11]). The first term on the right hand side corresponds to a convolution operator with kernel k = F −1 (ψ) * F −1 (|ξ| 2s ), which is L 1 as a convolution of a Schwartz function with a homogeneous function of order −n − 2s. Therefore u → k * u is bounded between any two Hölder spaces by the Fourier characterization of Hölder spaces ([Tri83, Section 2.3.7]). Thus
q L ∞ (B 1 ) (−∆) s m C −2s (R n ) m C (R n ) .
Moreover, γ ∈ Z ⊂ X β implies
m 2 + 2m = (m + 1) 2 − 1 = γ − 1 ∈ X β ,
and thus in particular m 2 + 2m ∈ C c (B 1 ). Define the function F : x → √ 1 + x − 1, which is smooth for non-negative x and has the property that F (m 2 + 2m) = m. Since C c is closed under composition with smooth functions, we have m ∈ C c (B 1 ) as well, with m C (B 1 ) m 2 + 2m C (B 1 ) . Eventually
q L ∞ (B 1 ) m C (R n ) γ − 1 C (B 1 ) ≤ β.
We shall now apply [RS18, Proposition 2.4]. Observe that the eigenvalue condition of the said proposition is not needed here, because in this case the potential q is comes from a fractional Liouville reduction, and so the Dirichlet problem for the transformed operator is already known to be wellposed.
Recall that {f h,k,l } is the basis of L 2 (B 3 \ B 2 ) constructed in [RS18, Lemma 2.1]. The operator Γ(q) := Λ q − Λ 0 mapping L 2 (B 3 \ B 2 ) to itself is completely characterized by the quantities a h 2 ,k 2 ,l 2 h 1 ,k 1 ,l 1 (q) := Γ(q)f h 1 ,k 1 ,l 1 , f h 2 ,k 2 ,l 2 L 2 (B 3 \B 2 ) . Let X := { Γ(q) ; q ∈ Q, Γ(q) X < ∞ },
where Q is the class of all transformed potentials, and Γ(q) X := sup h i ,k i ,l i
(1 + max{h 1 + k 1 , h 2 + k 2 }) n+2 |a h 2 ,k 2 ,l 2 h 1 ,k 1 ,l 1 (q)|.
By [RS18, Proposition 2.4] we obtain |a h 2 ,k 2 ,l 2 h 1 ,k 1 ,l 1 (q)| ≤ C n,s e −c max{h 1 +k 1 ,h 2 +k 2 } q L ∞ (B 1 ) ((−∆) s + q) −1 L 2 (B 1 )→L 2 (B 1 ) ≤ C n,s βe −c max{h 1 +k 1 ,h 2 +k 2 } , which means that if γ ∈ X β then Γ(q) ∈ X.
With this in mind, we can follow the proof of Lemma 3 in [Man01] (see also [RS18, Lemma 3.2]) to construct a δ-net Y for the image under Γ : q → Λ q −Λ 0 of the set of potentials corresponding to conductivities in X β . Here
δ := exp − − n (2n+3)
, and the cardinality of Y is
|Y | ≤ β exp C −n/ .
It is clear that for β large enough it must hold that | Z| > |Y |, which means that there exists two conductivities γ 1 , γ 2 ∈ X β with γ 1 − γ 2 L ∞ (B 1 ) ≥ and Λ q 1 − Λ q 2 L 2 (B 3 \B 2 )→L 2 (B 3 \B 2 ) Γ(q 1 ) − Γ(q 2 ) X δ, in light of the fact that Λ q is a bounded operator L 2 (B 3 \ B 2 ) → L 2 (B 3 \ B 2 ) (see [RS18, Remarks 2.2, 2.5]) and the related estimate [RS18, eq. (21)].
Since γ 1 = γ 2 = 1 in R n \ B 1 , by [CRZ22,Lemma 4.1] we deduce Λ q = Λ γ as operators on H s (R n \ B 1 ) → (H s (R n \ B 1 )) * . This lets us conclude that Λ γ 1 − Λ γ 2 H s (B 3 \B 2 )→(H s (B 3 \B 2 )) * ≤ Λ q 1 − Λ q 2 H s (B 3 \B 2 )→(H s (B 3 \B 2 )) * ≤ Λ q 1 − Λ q 2 L 2 (B 3 \B 2 )→L 2 (B 3 \B 2 ) δ.
that Eu| Ω = u and Eu W s,p (R n ) ≤ C u W s,p (Ω) for all u ∈ W s,p (Ω) and some C > 0. Using (23) we have Eu H s,p (R n ) ≤ C Eu W s,p (R n ) ≤ C u W s,p (Ω) .
By the definition of the · H s,p (Ω) norm this implies u H s,p (Ω) ≤ C u W s,p (Ω) and we can conclude the proof of Theorem 2.2, (i).
Proof of Lemma 3.1. Statement (i) has been essentially proved in [DNPV12, Lemma 5.3] but under the assumptions 0 ≤ φ ≤ 1 and µ = 1. We give a proof of this slightly more general result here for the sake of completeness. Since C 0,µ (Ω) ⊂ L ∞ (Ω), we have φu L p (Ω) ≤ φ L ∞ (Ω) u L p (Ω) and hence it remains to control the Gagliardo seminorm. We have
[φu] p W s,p (Ω) =ˆΩˆΩ |(φu)(x) − (φu)(y)| p |x − y| n+sp dxdy =ˆΩˆΩ |φ(x)(u(x) − u(y)) + (φ(x) − φ(y))u(y)| p |x − y| n+sp dxdy ≤ 2 p−1 ˆΩˆΩ |φ(x)| p |u(x) − u(y)| p |x − y| n+sp dxdy +ˆΩˆΩ |φ(x) − φ(y)| p |u(y)| p |x − y| n+sp dxdy ≤ 2 p−1 φ p L ∞ (Ω) [u] p W s,p (Ω) +ˆΩˆΩ
|φ(x) − φ(y)| p |u(y)| p |x − y| n+sp dxdy .
Now we writê
ΩˆΩ |φ(x) − φ(y)| p |u(y)| p |x − y| n+sp dxdy =ˆΩˆΩ |φ(x) − φ(y)| p |u(y)| p |x − y| n+sp χ B 1 (y) (x) dxdy +ˆΩˆΩ |φ(x) − φ(y)| p |u(y)| p |x − y| n+sp χ B 1 (y) c (x) dxdy =: I 1 + I 2 ,
where χ A denotes the characteristic function of the set A ⊂ R n . Next we estimate I 1 , I 2 . We have where ω n is the area of the unit sphere. Therefore, we get
I 1 ≤ [φ] p C 0,
[φu] p W s,p (Ω) ≤ 2 p−1 (1 + 2 p ω n sp φ p L ∞ (Ω) + ω n (µ − s)p
[φ] p C 0,µ (Ω) ) u p W s,p (Ω)
≤ C 1 + µ s(µ − s) φ p C 0,µ (Ω) u p W s,p (Ω) ,
where C only depends on n and p. This establishes the assertion (i).
Now let s = k + σ with k ∈ N and σ ∈ (0, 1). By classical results we have φu ∈ W k,p (Ω) with φu W k,p (Ω) ≤ C φ C k (Ω) u W k,p (Ω) for some C > 0 only depending on n, k and p. Thus it remains to estimate the Gagliardo seminorm of ∂ α (φu) for all multi-indices α of order k. By the Leibniz rule we have
[∂ α (φu)] W σ,p (Ω) ≤ C β≤α [∂ α−β φ∂ β u] W σ,p (Ω) ≤ C β≤α: α =β [∂ α−β φ∂ β u] W σ,p (Ω) + C[φ∂ α u] W σ,p (Ω) ≤ C β≤α: α =β 1 + µ σ(µ − σ) 1/p ∂ β−α φ C 0,µ (Ω) ∂ β u W σ,p (Ω) + C 1 + µ σ(µ − σ) 1/p φ C 0,µ (Ω) ∂ α u W σ,p (Ω) ≤ CC 0 1 + µ σ(µ − σ) 1/p β≤α: α =β ∂ β−α φ C 0,µ (Ω) ∂ β u W 1,p (Ω) + C 1 + µ σ(µ − σ) 1/p φ C 0,µ (Ω) ∂ α u W σ,p (Ω) ≤ C(1 + C 0 ) 1 + µ σ(µ − σ) 1/p k =0 ∇ φ C 0,µ (Ω) u W s,p (Ω)
for all α ∈ N n 0 with |α| = k and some constant C > 0 only depending on n, k. In the last estimate we used the bound from the case 0 < s < 1 and [DNPV12, Proposition 2.2], where C 0 > 0 is the norm of the extension operator E : W 1,p (Ω) → W 1,p (R n ) (see [Bre11,Theorem 9.7]).
Proof of Lemma 3.2. First note that dist(supp(u), Ω c ) ≥ d > 0. In fact, for x ∈ supp(u) ⊂ Ω, y ∈ Ω e consider the curve γ : [0, 1] → R n with γ(t) := x + t(y − x). Then there exists t 0 ∈ (0, 1) such that γ(t 0 ) ∈ ∂Ω because otherwise one would have [0, 1] = γ −1 (Ω) ∪ γ −1 (Ω e ) which is not possible. But then |x − y| ≥ |x − γ(t 0 )| ≥ d > 0. Next we distinguish the cases 0 < s < 1, s = k ∈ N and s = k + σ with k ∈ N, 0 < σ < 1.
Case 0 < s < 1: Clearly, we have ū L p (R n ) = u L p (Ω) and thus it remains to show [ū] W s,p (R n ) ≤ C u W s,p (Ω) . By symmetry we can split [ū] W s,p (R n ) aŝ
where [u] W σ,p (Ω) := ˆΩˆΩ |u(x) − u(y)| p |x − y| n+σp dxdy 1/p is the so-called Gagliardo seminorm. The Slobodeckij spaces are naturally endowed with the norm
see [ BCD11 ,
BCD11Section 1.3.1]. SinceḢ s (R n ) continuously embeds into L 2n/(n−2s) (R n ) for all 0 ≤ s < n/2 (see [BCD11, Theorem 1.38]), we find for any 1 ≤ q ≤ 2n n−2s
2
(q 1 − q 2 ) m dx =: I 1 + I 2 .
|x − y| n+sp dxdy =ˆΩˆΩ |u(x) − u(y)| p |x − y| n+sp dxdy +ˆΩ cˆΩc |ū(x) −ū(y)| p |x − y| n+sp dxdy + 2ˆΩˆΩ c |u(x)| p |x − y| n+sp dxdy = [u] p W s,p (Ω) + 2ˆs upp(u)|u(x)| p ˆΩ c dy |x − y| n+sp dy dx.
RZ22c]. This uses a general UCP result for the fractional Laplacians in[KRZ22].• Counterexamples for disjoint measurement sets. For any nonempty open disjoint sets W 1 , W 2 ⊂ Ω e with dist(W 1 ∪ W 2 , Ω) > 0 there exist two different conductivities γ 1
Given a bounded linear mapping A : X → Y between two Banach spaces, we denote its operator norm by A X→Y .
Here and in the rest of paper we use the notation A * := A H s (Ωe)→(H s (Ωe)) * .
Note A H s (W 1 )→( H s (W 2 )) * = sup{ | Au1, u2 | ; uj H s (R n ) = 1, uj ∈ C ∞ c (Wj) }.
Appendix A. Proofs of auxiliary resultsProof of Theorem 2.2. Let us first consider the case Ω = R n . In fact, this follows from embeddings between different function spaces:(i) If s ∈ R, 0 < p < ∞ and 0 < q 0 ≤ q 1 ≤ ∞ then F s p,q 0 (R n ) → F s p,q 1 (R n ) (cf. [Tri83, Section 2.3.2, Proposition 2]).Above F s p,q (R n ) denotes the Triebel-Lizorkin spaces, B s p,q (R n ) the Besov spaces and Λ s p,q (R n ) the Lipschitz spaces. For their definition and more details we refer to the monograph[Tri83]. These embeddings implyNow assume that 2 ≤ p < ∞ and Ω ⊂ R n is an arbitrary open set. Let u ∈ H s,p (Ω) and assume v ∈ H s,p (R n ) satisfies u = v| Ω . Then by(23)therefor some C > 0. By defintion of the Slobodeckij spaces and v| Ω = u we haveTaking the infimum over all extensions of u and recalling the definition of the norm · H s,p (Ω) we deduceThis proves Theorem 2.2, (ii). Next let 1 < p ≤ 2, s = k + σ with k ∈ N, 0 < σ < 1 and assume Ω is a C k,1 domain with bounded boundary. By Lemma 3.3 there is an extension operator E : W s,p (Ω) → W s,p (R n ) suchFor any x ∈ supp(u) there holds B d/2 (x) ⊂ Ω and hence we havêTherefore we getThis showsfor all 1 ≤ i ≤ n. This identity clearly also holds if the intersection is empty. Thus ∂ iū = ∂ i u ∈ L p (R n ) for all 1 ≤ i ≤ n. Hence, if u ∈ W 1,p (Ω) thenū ∈ W 1,p (R n ) and by the previous case there holds ū W 1,p (R n ) = u W 1,p (Ω) . By induction we see that for any k ∈ N we haveū ∈ W k,p (R n ) whenever u ∈ W k,p (Ω). Moreover, there holds ū W k,p (R n ) = u W k,p (Ω) .Case s = k + σ with k ∈ N, 0 < σ < 1: This follows immediately from the previous two cases.
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On continuous dependence for an inverse initial-boundary value problem for the wave equation. Zi Qi Sun, J. Math. Anal. Appl. 1501Zi Qi Sun. On continuous dependence for an inverse initial-boundary value problem for the wave equation. J. Math. Anal. Appl., 150(1):188-204, 1990.
Partial differential equations III. Nonlinear equations. Michael E Taylor, Applied Mathematical Sciences. 117Springersecond editionMichael E. Taylor. Partial differential equations III. Nonlinear equations, volume 117 of Applied Mathematical Sciences. Springer, New York, second edition, 2011.
Theory of function spaces. H Triebel, of Mathematik und ihre Anwendungen in Physik und Technik [Mathematics and its Applications in Physics and Technology. 38H. Triebel. Theory of function spaces, volume 38 of Mathematik und ihre Anwendungen in Physik und Technik [Mathematics and its Applications in Physics and Technology].
. Akademische Verlagsgesellschaft Geest & Portig, K.-G , LeipzigAkademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1983.
30 years of Calderón's problem. Gunther Uhlmann, Séminaire Laurent Schwartz-Équations aux dérivées partielles et applications. Année 2012-2013, Sémin.Équ. Dériv. Partielles, pages Exp. No. XIII, 25.École Polytech. PalaiseauGunther Uhlmann. 30 years of Calderón's problem. In Séminaire Laurent Schwartz-Équations aux dérivées partielles et applications. Année 2012- 2013, Sémin.Équ. Dériv. Partielles, pages Exp. No. XIII, 25.École Poly- tech., Palaiseau, 2014.
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[] |
[
"Noun-Phrase Analysis in Unrestricted Text for Information Retrieval",
"Noun-Phrase Analysis in Unrestricted Text for Information Retrieval"
] |
[
"David A Evans \nLaboratory for Computational Linguistics Carnegie Mellon Univeristy Pittsburgh\n15213PA\n",
"Chengxiang Zhai \nLaboratory for Computational Linguistics Carnegie Mellon Univeristy Pittsburgh\n15213PA\n"
] |
[
"Laboratory for Computational Linguistics Carnegie Mellon Univeristy Pittsburgh\n15213PA",
"Laboratory for Computational Linguistics Carnegie Mellon Univeristy Pittsburgh\n15213PA"
] |
[] |
Information retrieval is an important application area of natural-language processing where one encounters the genuine challenge of processing large quantities of unrestricted natural-language text. This paper reports on the application of a few simple, yet robust and efficient nounphrase analysis techniques to create better indexing phrases for information retrieval. In particular, we describe a hybrid approach to the extraction of meaningful (continuous or discontinuous) subcompounds from complex noun phrases using both corpus statistics and linguistic heuristics. Results of experiments show that indexing based on such extracted subcompounds improves both recall and precision in an information retrieval system. The noun-phrase analysis techniques are also potentially useful for book indexing and automatic thesaurus extraction.
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10.3115/981863.981866
| null | 3,876,333 |
cmp-lg/9605019
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1b04cc85965ce3e365e9d1522810b4b69b07f479
|
Noun-Phrase Analysis in Unrestricted Text for Information Retrieval
David A Evans
Laboratory for Computational Linguistics Carnegie Mellon Univeristy Pittsburgh
15213PA
Chengxiang Zhai
Laboratory for Computational Linguistics Carnegie Mellon Univeristy Pittsburgh
15213PA
Noun-Phrase Analysis in Unrestricted Text for Information Retrieval
Information retrieval is an important application area of natural-language processing where one encounters the genuine challenge of processing large quantities of unrestricted natural-language text. This paper reports on the application of a few simple, yet robust and efficient nounphrase analysis techniques to create better indexing phrases for information retrieval. In particular, we describe a hybrid approach to the extraction of meaningful (continuous or discontinuous) subcompounds from complex noun phrases using both corpus statistics and linguistic heuristics. Results of experiments show that indexing based on such extracted subcompounds improves both recall and precision in an information retrieval system. The noun-phrase analysis techniques are also potentially useful for book indexing and automatic thesaurus extraction.
Introduction
Information Retrieval
Information retrieval (IR) is an important application area of naturaManguage processing (NLP). 1 The IR (or perhaps more accurately "text retrieval") task may be characterized as the problem of selecting a subset of documents (from a document collection) whose content is relevant to the information need of a user as expressed by a query. The document collections involved in IR are often gigabytes of unrestricted natural-language text. A user's query may be expressed in a controlled language (e.g., a boolean expression of keywords) or, more desirably, a natural language, such as English.
A typical IR system works as follows. The documents to be retrieved are processed to extract indexing terms or content carriers, which are usually (Evans, 1990;Smeaton, 1992;Lewis & Sparck Jones, 1996) single words or (less typically) phrases. The indexing terms provide a description of the document's content. Weights are often assigned to terms to indicate how well they describe the document. A (natural-language) query is processed in a similar way to extract query terms. Query terms are then matched against the indexing terms of a document to determine the relevance of each document to the quer3a
The ultimate goal of an IR system is to increase both precision, the proportion of retrieved documents that are relevant, as well as recall, the proportion of relevant document that are retrieved. However, the real challenge is to understand and represent appropriately the content of a document and quer~ so that the relevance decision can be made efficiently, without degrading precision and recall. A typical solution to the problem of making relevance decisions efficient is to require exact matching of indexing terms and query terms, with an evaluation of the 'hits' based on a scoring metric. Thus, for instance, in vector-space models of relevance ranking, both the indexing terms of a document and the query terms are treated as vectors (with individual term weights) and the similarity between the two vectors is given by a cosine-distance measure, essentially the angle between any two vectors?
Natural-Language Processing for IR
One can regard almost any IR system as performing an NLP task: text is 'parsed" for terms and terms are used to express 'meaning'--to capture document content. Clearly, most traditional IR systems do not attempt to find structure in the naturallanguage text in the 'parsing' process; they merely extract word-like strings to use in indexing. Ideally, however, extracted structure would directly reflect the encoded linguistic relations among terms-captuing the conceptual content of the text better than simple word-strings.
There are several prerequisites for effective NLP in an IR application, including the following.
2 (Salton & McGill, 1983) 1. Ability to process large amounts of text
The amount of text in the databases accessed by modem IR systems is typically measured in gigabytes. This requires that the NLP used must be extraordinarily efficient in both its time and space requirements. It would be impractical to use a parser with the speed of one or two sentences per second.
2. Ability to process unrestricted text The text database for an IR task is generally unrestricted natural-language text possibly encompassing many different domains and topics. A parser must be able to manage the many kinds of problems one sees in natural-language corpora, including the processing of unknown words, proper names, and unrecognized structures. Often more is required, as when spelling, transcription, or OCR errors occur. Thus, the NLP used must be especially robust.
3. Need for shallow understanding While the large amount of unrestricted text makes NLP more difficult for IR, the fact that a deep and complete understanding of the text may not be necessary for IR makes NLP for IR relatively easier than other NLP tasks such as machine translation. The goal of an IR system is essentially to classify documents (as relevant or irrelevant) vis-a-vis a query. Thus, it may suffice to have a shallow and partial representation of the content of documents.
Information retrieval thus poses the genuine challenge of processing large volumes of unrestricted natural-language text but not necessarily at a deep level.
Our Work
This paper reports on our evaluation of the use of simple, yet robust and efficient noun-phrase analysis techniques to enhance phrase-based IR. In particular, we explored an extension of the ~phrasebased indexing in the CLARIT TM system ° using a hybrid approach to the extraction of meaningful (continuous or discontinuous) subcompounds from complex noun phrases exploiting both corpusstatistics and linguistic heuristics. Using such subcompounds rather than whole noun phrases as indexing terms helps a phrase-based IR system solve the phrase normalization problem, that is, the problem of matching syntactically different, but semantically similar phrases. The results of our experiments show that both recall and precision are improved by using extracted subcompounds for indexing.
Phrase-Based Indexing
The selection of appropriate indexing terms is critical to the improvement of both precision and recall in an IR task. The ideal indexing terms would directly represent the concepts in a document. Since 'concepts' are difficult to represent and extract (as well as to define), concept-based indexing is an elusive goal. Virtually all commercial IR systems (with the exception of the CLARIT system) index only on "words', since the identification of words in texts is typically easier and more efficient than the identification of more complex structures. However, single words are rarely specific enough to support accurate discrimination and their groupings are often accidental. An often cited example is the contrast between "junior college" and "college junior". Word-based indexing cannot distinguish the phrases, though their meanings are quite different. Phrase-based indexing, on the other hand, as a step toward the ideal of concept-based indexing, can address such a case directly.
Indeed, it is interesting to note that the use of phrases as index terms has increased dramatically among the systems that participate in the TREC evaluations. ~ Even relatively traditional word-based systems are exploring the use of multiword terms by supplementing words with statistical phrases--selected high frequency adjacent word pairs (bigrams). And a few systems, such as CLARIT--which uses simplex noun phrases, attested subphrases, and contained words as index terms--and New York University's TREC systemS--which uses "head-modifier pairs" derived from identified noun phrases--have demonstrated the practicality and effectiveness of thorough NLP in IR tasks.
The experiences of the CLAR1T system are instructive. By using selective NLP to identify simplex NPs, CLARIT generates phrases, subphrases, and individual words to use in indexing documents and queries. Such a first-order analysis of the linguistic structures in texts approximates concepts and affords us alternative methods for calculating the fit between documents and queries. In particular, we can choose to treat some phrasal structures as atomic units and others as additional information about (or representations of) content. There are immediate effects in improving precision:
1. Phrases can replace individual indexing words.
For example, if both "dog" and "hot" are used for indexing, they will match any query in which both words occur. But if only the phrase "hot dog" is used as an index term, then it will only match the same phrase, not any of the individual words.
3 (Evans et al., 1991;Evans et al., 1996) 4 (Harman, 1995;Harman, 1996) 5 (Strzalkowski, 1994) 2. Phrases can supplement word-level matches.
For example, if only the individual words "junior" and "college" are used for indexing, both "junior college" and "college junior" will match a query with the phrase "junior college" equally well. But if we also use the phrase "junior college" for indexing, then "junior college" will match better than "college junior", even though the latter also will receive some credit as a match at the word level.
We can see, then, that it is desirable to distinquish-and, if possible, extract--two kinds of phrases: those that behave as lexical atoms and those that reflect more general linguistic relations.
Lexical atoms help us by obviating the possibility of extraneous word matches that have nothing to do with true relevance. We do not want "hot" or "dog" to match on "hot dog". In essence, we want to eliminate the effect of the independence assumption at the word level by creating new words--the lexical atoms--in which the individual word dependencies are explicit (structural).
More general phrases help us by adding detail. Indeed, all possible phrases (or paraphrases) of actual content in a document are potentially valuable in indexing. In practice, of course, the indexing term space has to be limited, so it is necessary to select a subset of phrases for indexing. Short phrases (often nominal compounds) are preferred over long complex phrases, because short phrases have better chances for matching short phrases in queries and will still match longer phrases owing to the short phrases they have in common. Using only short phrases also helps solve the phrase normalization problem of matching syntactically different long phrases (when they share similar meaning). 6
Thus, lexical atoms and small nominal compounds should make good indexing phrases. While the CLARIT system does index at the level of phrases and subphrases, it does not currently index on lexical atoms or on the small compounds that can be derived from complex NPs, in particular, reflecting cross-simplex NP dependency relations. Thus, for example, under normal CLARIT processing the phrase "the quality of surface of treated stainless steel strip "7 would yield index terms such as "treated stainless steel strip", "treated stainless steel", "stainless steel strip", and "stainless steel" (as a phrase, not lexical atom), along with all the relevant single-word terms in the phrase. But the process would not identify "stainless steel" as a potential lexical atom or find terms such as "surface quality", "strip surface", and "treated strip".
To achieve more complete (and accurate) phrasebased indexing, we propose to use the following 6 (Smeaton, 1992) ZThis is an actual example from a U.S. patent document.
four kinds of phrases as indexing terms:
1. Lexical atoms (e.g., "hot dog" or 2.
3.
4.
perhaps "stainless steel" in the example above) Head modifier pairs (e.g., "treated strip" and "steel strip" in the example above)
Subcompounds (e.g., "stainless steel strip" in the example above)
Cross-preposition modification pairs (e.g., "surface quality" in the example above)
In effect, we aim to augment CLARIT indexing with lexical atoms and phrases capturing additional (discontinuous) modification relations than those that can be found within simplex NPs.
It is clear that a certain level of robust and efficient noun-phrase analysis is needed to extract the above four kinds of small compounds from a large unrestricted corpus. In fact, the set of small compounds extracted from a noun phrase can be regarded as a weak representation of the meaning of the noun phrase, since each meaningful small compound captures a part of the meaning of the noun phrase. In this sense, extraction of such small compounds is a step toward a shallow interpretation of noun phrases. Such weak interpretation is useful for tasks like information retrieval, document classification, and thesaurus extraction, and indeed forms the basis in the CLARIT system for automated thesaurus discovery.
Methodology
Our task is to parse text into NPs, analyze the noun phrases, and extract the four kinds of small compounds given above. Our emphasis is on robust and efficient NLP techniques to support large-scale applications.
For our purposes, we need to be able to identify all simplex and complex NPs in a text. Complex NPs are defined as a sequence of simplex NPs that are associated with one another via prepositional phrases. We do not consider simplex NPs joined by relative clauses.
Our approach to NLP involves a hybrid use of corpus statistics supplemented by linguistic heuristics. We assume that there is no training data (making the approach more practically useful) and, thus, rely only on statistical information in the document database itself. This is different from many current statistical NLP techniques that require a training corpus. The volume of data we see in IR tasks also makes it impractical to use sophisticated statistical computations.
The use of linguistic heuristics can assist statistical analysis in several ways. First, it can focus the use of statistics by helping to eliminate irrelevant structures from consideration. For example, syntactic category analysis can filter out impossible word modification pairs, such as [adjective, adjective] and [noun, adjective]. Second, it may improve the reliability of statistical decisions. For example, the counting ofbigrams that occur only within noun phrases is more reliable for lexical atom discovery than the counting of all possible bigrams that occur in the corpus. In addition, syntactic category analysis is also helpful in adjusting cutoff parameters for statistics. For example, one useful heuristic is that we should use a higher threshold of reliability (evidence) for accepting the pair [adjective, noun] as a lexical atom than for the pair [noun, noun]: a noun-noun pair is much more likely to be a lexical atom than an adjective-noun one.
The general process of phrase generation is illustrated in Figure 1. We used the CLARIT NLP module as a preprocessor to produce NPs with syntactic categories attached to words. We did not attempt to utilize CLARIT complex-NP generation or subphrase analysis, since we wanted to focus on the specific techniques for subphrase discovery that we describe in this paper. After preprocessing, the system works in two stages--parsing and generation. In the parsing stage, each simplex noun phrase in the corpus is parsed. In the generation stage, the structured noun phrase is used to generate candidates for all four kinds of small compounds, which are further tested for occurrence (validity) in the corpus.
Parsing of simplex noun phrases is done in multiple phases. At each phase, noun phrases are partially parsed, then the partially parsed structures are used as input to start another phase of partial pars-ing. Each phase of partial parsing is completed by concatenating those most reliable modification pairs together to form a single unit. The reliability of a modification pair is determined by a score based on frequency statistics and category analysis and is further tested via local optimum phrase analysis (described below). Lexical atoms are discovered at the same time, during simplex noun phrase parsing.
Phrase generation is quite simple. Once the structure of a noun phrase (with marked lexical atoms) is known, the four kinds of small compounds can be easily produced. Lexical atoms are already available. Head-modifier pairs can be extracted based on the modification relations implied by the structure. Subcompounds are just the substructures of the NP. Cross-preposition pairs are generated by enumerating all possible pairs of the heads of each simplex NP within a complex NP in backward order. 8
To validate discontinuous compounds such as non-sequential head-modifier pairs and crosspreposition pairs, we use a standard technique of CLARIT processing, viz., we test any nominated compounds against the corpus itself. If we find independently attested (whole) simplex NPs that match the candidate compounds, we accept the candidates as index terms. Thus for the NP "the quality of surface of treated stainless steel strip", the head-modifier pairs "treated strip", "stainless steel", "stainless strip", and "steel strip", and the cross-preposition pairs "strip surface", "surface quality", and "strip quality", would be generated as index terms only if we found independent evidence of such phrases in the corpus in the form of free-standing simplex NPs.
Lexical Atom Discovery
A lexical atom is a semantically coherent phrase unit. Lexical atoms may be found among proper names, idioms, and many noun-noun compounds. Usually they are two-word phrases, but sometimes they can consist of three or even more words, as in the case of proper names and technical terms. Examples of lexical atoms (in general English) are "hot dog", "tear gas", "part of speech", and "yon Neumann".
However, recognition of lexical atoms in free text is difficult. In particular, the relevant lexical atoms for a corpus of text will reflect the various discourse domains encompassed by the text. In a collection of medical documents, for example, "Wilson's disease" (an actual rheumatological disorder) may be used as a lexical atom, whereas in a collection of general news stories, "Wilson's disease" (reference to the disease that Wilson has) may not be a lexical atom. Note that in the case of the medical usage, we would commonly find "Wilson's disease" as a bigram and we would not find, for example, 8 (Schwarz, 1990) reports a similar strategy. "Wilson's severe disease" as a phrase, though the latter might well occur in the general news corpus. This example serves to illustrate the essential observation that motivates our heuristics for identitying lexical atoms in a corpus: (1) words in lexical atoms have strong association, and thus tend to co-occur as a phrase and (2) when the words in a lexical atom co-occur in a noun phrase, they are never or rarely separated.
The detection of lexical atoms, like the parsing of simplex noun phrases, is also done in multiple phases. At each phase, only two adjacent units are considered. So, initiall~ only two-word lexical atoms can be detected. But, once a pair is determined to be a lexical atom, it will behave exactly like a single word in subsequent processing, so, in later phases, atoms with more than two words can be detected.
Suppose the pair to test is [W1, W2]. The first heuristic is implemented by requiring the frequency of the pair to be higher than the frequency of any other pair that is formed by either word with other words in common contexts (within a simplex noun phrase). The intuition behind the test is that (1) in general, the high frequency of a bigram in a simple noun phrase indicates strong association and (2) The second heuristic requires that we record all cases where two words occur in simplex NPs and compare the number of times the words occur as a strictly adjacent pair with the number of times they are separated. The second heuristic is simply implemented by requiring that F(W1, W2) be much higher than DF(W1, W2) (where 'higher' is determined by some threshold).
Syntactic category analysis also helps filter out impossible lexical atoms and establish the thresh-21 old for passing the second test. Only the following category combinations are allowed for lexical atoms: [noun, noun], [noun, lexatom], [lexatom, noun], [adjective, noun], and [adjective, lexatom], where "lexatom" is the category for a detected lexical atom. For combinations other than [noun, noun], the threshold for passing the second test is high.
In practice, the process effectively nominates phrases that are true atomic concepts (in a particular domain of discourse) or are being used so consistently as unit concepts that they can be safely taken to be lexical atoms. For example, the lexical atoms extracted by this process from the CACM corpus (about 1 MB) include "operating system", "data structure", "decision table", "data base", "real time", "natural language", "on line", "least squares", "numerical integration", and "finite state automaton", among others.
Bottom-Up Association-Based Parsing
Extended simplex noun-phrase parsing as developed in the CLARIT system, which we exploit in our process, works in multiple phases. At each phase, the corpus is parsed using the most specific (i.e., recently created) lexicon of lexical atoms. New lexical atoms (results) are added to the lexicon and are reused as input to start another phase of parsing until a complete parse is obtained for all the noun phrases.
The idea of association-based parsing is that by grouping words together (based on association) many times, we will eventually discover the most restrictive (and informative) structure of a noun phrase. For example, if we have evidence from the corpus that "high performance" is a more reliable association and "general purpose" a less reliable one, then the noun phrase "general purpose high performance computer" (an actual example from the CACM corpus) would undergo the following grouping process: Word pairs are given an association score (S) according to the following rules. Scores provide evidence for groupings in our parsing process. Note that a smaller score means a stronger association.
1. Lexical atoms are given score 0. This gives the highest priority to lexical atoms.
2. The combination of an adverb with an adjective, past participle, or progressive verb is given score 0. 3. Syntactically impossible pairs are given score 100. This assigns the lowest priority to those pairs filtered out by syntactic category analysis. The association score (based principally on frequency) can sometimes be unreliable. For example, if the phrase "computer aided design" occurs frequently in a corpus, "aided design" may be judged a good association pair, even though "computer aided" might be a better pair. A problem may arise when processing a phrase such as "program aided design": if "program aided" does not occur frequently in the corpus and we use frequency as the principal statistic, we may (incorrectly) be led to parse the phrase as "[program (aided design)]".
One solution to such a problem is to recompute the bigram occurrence statistics after making each round of preferred associations. Thus, using the example above, if we first make the association "computer aided" everywhere it occurs, many instances of "aided design" will be removed from the corpus. Upon recalculation of the (free) bigram statistics, "aided design" will be demoted in value and the false evidence for "aided design" as a preferred association in some contexts will be eliminated.
The actual implementation of such a scheme requires multiple passes over the corpus to generate phrases. The first phrases chosen must always be the most reliable. To aid us in making such decisions we have developed a metric for scoring preferred associations in their local NP contexts.
To establish a preference metric, we use two statistics: (1) the frequency of the pair in the corpus, F(W1, W2), and (2) the number of the times that the pair is locally dominant in any NP in which the pair occurs. A pair is locally dominant in an NP iff it has a higher association score than either of the pairs that can be formed from contiguous other words in the NP. For example, in an NP with the sequence [X, Y, g], we compare S(X, Y) with S(Y, g); whichever is higher is locally dominant. The preference score (PS) for a pair is determined by the ratio of its local dominance count (LDC)--the total number of cases in which the pair is locally dominant--to its frequency:
LDC(WI 1W2)
PS(W1, W2) = r(Wl,W~)
By definition all two-word NIPs score their pairs as locally dominant.
In general, in each processing phase we make only those associations in the corpus where a pair's PS is above a specified threshold. If more than one association is possible (above theshold) in a particular NP, we make all possible associations, but in order of PS: the first grouping goes to the pair with highest PS, and so on. In practice, we have used 0.7 as the threshold for most processing phases. 9
Experiment
We tested the phrase extraction system (PES) by using it to index documents in an actual retrieval task. In particular, we substituted the PES for the default NLP module in the CLARIT system and then indexed a large corpus using the terms nominated by the PES, essentially the extracted small compounds and single words (but not words within a lexical atom). All other normal CLARIT processing-weighting of terms, division of documents into subdocuments (passages), vector-space modeling, etc.--was used in its default mode. As a baseline °When the phrase data becomes sparse, e.g., after six or seven iterations of processing, it is desirable to reduce the threshold. for comparison, we used standard CLARIT processing of the same corpus, with the NLP module set to return full NPs and their contained words (and no further subphrase analysis).l 0
The corpus used is a 240-megabyte collection of Associated Press newswire stories from 1989 (AP89), taken from the set of TREC corpora. There are about 3-million simplex NPs in the corpus and about 1.5-million complex NPs. For evaluation, we used TREC queries 51-100, ll each of which is a relatively long description of an information need. Queries were processed by the PES and normal CLARIT NLP modules, respectively, to generate query terms, which were then used for CLARIT retrieval.
To quantify the effects of PES processing, we used the standard IR evaluation measures of recall and precision. Recall measures how many of the relevant documents have been actually retrieved. Precision measures how many of the retrieved documents are indeed relevant. For example, if the total number of relevant documents is N and the system returns M documents of which K are relevant, then,
Recall = K IV
and Precision = ~-.
We used the judged-relevant documents from the TREC evaluations as the gold standard in scoring the performance of the two processes.
suggests that the PES could be used to support other IR enhancements, such as automatic feedback of the top-returned documents to expand the initial query for a second retrieval step) 2
Results
The results of the experiment are given in Tables 1, 2, and 3. In general, we see improvement in both recall and precision.
Recall improves slightly (about 1%), as shown in Table 1. While the actual improvement is not significant for the run of fifty queries, the increase in absolute numbers of relevant documents returned indicates that the small compounds supported better matches in some cases.
Interpolated precision improves significantly5 as shown in Table 2. The general improvement in precision indicates that small compounds provide more accurate (and effective) indexing terms than full NPs.
Precision improves at various returned-document levels, as well, as shown in Table 3. Initial precision, in particular, improves significantly. This 1°Note that the CLARIT process used as a baseline does not reflect optimum CLARIT performance, e.g., as obtained in actual TREC evaluations, since we did not use a variety of standard CLARIT techniques that significantly improve performance, such as automatic query expansion, distractor space generation, subterm indexing, or differential query-term weighting. Cf. (Evans et al., 1996) for details.
1 ~ (Harman, 1993) The PES, which was not optimized for processing, required approximately 3.5 hours per 20megabyte subset of AP89 on a 133-MHz DEC alpha processor) 3 Most processing time (more than 2 of every 3.5 hours) was spent on simplex NP parsing. Such speed might be acceptable in some, smallerscale IR applications, but it is considerably slower than the baseline speed of CLARIT noun-phrase identification (viz., 200 megabytes per hour on a 100-MIPS processor).
l~ Evans et al., 1996) 13Note that the machine was not dedicated to the PES processing; other processes were running simultaneously.
Conclusions
The notion of association-based parsing dates at least from (Marcus, 1980) and has been explored again recently by a number of researchers. TM The method we have developed differs from previous work in that it uses linguistic heuristics and locality scoring along with corpus statistics to generate phrase associations.
The experiment contrasting the PES with baseline processing in a commercial IR system demonstrates a direct, positive effect of the use of lexical atoms, subphrases, and other pharase associations across simplex NPs. We believe the use of N-P-substructure analysis can lead to more effective information management, including more precise IR, text summarization, and concept clustering. Our future work will explore such applications of the techniques we have described in this paper.
Figure 1 :
1General Processing for Phrase Generation
F
(W~, W2) > Maa:LDF(W~, W2) and F(W~, W2) > Ma3:RDF(W1, W2) Where, MaxLDF(W1, W2) = Maxw( U in( F(W, W1), DF(W, W2))) and MaxRDF(W1, W2) = Maxw( U in( DF(W1, W), F(W2, W) ) ) W is any context word in a noun phrase and F(X, Y) and DF(X, Y) are the continuous and discontinuous frequencies of [X, Y], respective135 within a simple noun phrase, i.e., the frequency of patterns [...X, Y...] and patterns [...X, ..., Y...], respectively.
4 .Figure 2 :
42Other pairs are scored according to the formulas given inFigure 2. Note the following effects of the formulas:When /;'(W1,W2) increases, S(W1,W2) decreases; When DF(W1, W2) increases, S(Wx, W2) decreases; When AvgLDF(W~, W2) or AvgRDF(W~, W2) increases, S(W1, W2) increases; and When F(Wx)-F(W1,W2) or F(W2)-F(W1, W2) increases, S(W1, W2) decreases. S(W1 W2)= I+LDF(W,,W2)+RDF(W1,W=) A(W1,W2) XlxF(W1,W2)+DF(W1,W,~) X Min(F(W, W1),DF(W,W',)) AvgLDF(Wa, W2) = ~-..Min( F( W2,W),D F( W1,W)) AvgRDF(W1, W2) = ~-..,WCRD IRDI A(W1, W2 ) = ~ F(W1)+F(W2)--2×F(WI,W2)+X2 Where • F(W) isfrequency of word W • F(W1, W2) is frequency of adjacent bigram [W1,W2] (i.e ..... W1 W2 ...) • DF(W1, W2) is frequency of discontinuous bigram [W1,W21 (i.e ..... W1...W2...) • LD is all left dependents, i.e., {W]min(F(W, Wl), DF(W, W2)) ~ 0} • RD is all right dependents, i.e., {WJmin( D F(W1, W), F(W2, W) ) ¢ 0} • ),1 is the parameter indicating the relative contribution of F(W1,W2) to the score (e.g., 5 in the actual experiment) • A2 is the parameter to control the contribution of word frequency (e.g., 1000 in the actual experiment) Formulas for Scoring
we want to avoid the case where [W1, W2] has a high frequency, but [W1, W2, W] (or [W, W1, W2]) has aneven higher frequency, which implies that W2 (or W1) has a stronger association with W than with W1 (or W2, respectively). More precisely, we require the following:
Table 2 :
2Interpolated Precision Results
Table 3 :
3Precision at Various Document Levels
AcknowledgementsWe received helpful comments from Bob Carpenter, Christopher Manning, Xiang Tong, and SteveHanderson, who also provided us with a hash table manager that made the implementation easier. The evaluation of the experimental results would have been impossible without the help of Robert Lefferts and Nata~a Mili4-Frayling at CLARITECH Corporation. Finally, we thank the anonymous reviewers for their useful comments.
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Recent developments in natural language text retrieval. T Strzalkowski, J Carballo, The Second Text REtrieval Conference (TREC-2). NIST Special Publication 500-215. Washington. Donna K. HarmanDC: U.S. Government Printing OfficeT. Strzalkowski and J. Carballo. 1994. Recent develop- ments in natural language text retrieval. In: Donna K. Harman (ed.), The Second Text REtrieval Conference (TREC-2). NIST Special Publication 500-215. Washing- ton, DC: U.S. Government Printing Office, pp. 123-136.
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"Th Wolf \nInstitute of Solid State Physics\nKarlsruhe Institute of Technology\n76131Karlsruhe, Germany\n",
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"L Castiglioni \nPhysik-Institut\nUniversität Zürich\nWinterthurerstrasse 1908057ZürichSwitzerland\n",
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"Département de Physique and Fribourg Center for Nanomaterials\nUniversité de Fribourg\n1700FribourgSwitzerland",
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"Institute of Solid State Physics\nKarlsruhe Institute of Technology\n76131Karlsruhe, Germany",
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We present a time-resolved angle-resolved photoelectron spectroscopy study of IrTe2, which undergoes two first-order structural and charge-ordered phase transitions on cooling below 270 K and below 180 K. The possibility of inducing a phase transition by photoexcitation with near-infrared femtosecond pulses is investigated in the charge-ordered phases. We observe changes of the spectral function occuring within a few hundreds of femtoseconds and persisting up to several picoseconds, which we interpret as a partial photoinduced phase transition (PIPT). The necessary time for photoinducing these spectral changes increases with increasing photoexcitation density and reaches timescales longer than the rise time of the transient electronic temperature. We conclude that the PIPT is driven by a transient increase of the lattice temperature following the energy transfer from the electrons. However, the photoinduced changes of the spectral function are small, which indicates that the low temperature phase is particularly robust against photoexcitation. We suggest that the system might be trapped in an out-of-equilibrium state, for which only a partial structural transition is achieved. arXiv:1801.07979v1 [cond-mat.str-el]
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10.1103/physrevb.97.075110
|
[
"https://arxiv.org/pdf/1801.07979v1.pdf"
] | 73,600,964 |
1801.07979
|
19246073815fe8c39694047ce0d5f2ecb9821dbd
|
Robustness of the charge-ordered phases in IrTe 2 against photoexcitation
C Monney
Physik-Institut
Universität Zürich
Winterthurerstrasse 1908057ZürichSwitzerland
Département de Physique and Fribourg Center for Nanomaterials
Université de Fribourg
1700FribourgSwitzerland
A Schuler
Physik-Institut
Universität Zürich
Winterthurerstrasse 1908057ZürichSwitzerland
T Jaouen
Département de Physique and Fribourg Center for Nanomaterials
Université de Fribourg
1700FribourgSwitzerland
M.-L Mottas
Département de Physique and Fribourg Center for Nanomaterials
Université de Fribourg
1700FribourgSwitzerland
Th Wolf
Institute of Solid State Physics
Karlsruhe Institute of Technology
76131Karlsruhe, Germany
M Merz
Institute of Solid State Physics
Karlsruhe Institute of Technology
76131Karlsruhe, Germany
M Muntwiler
Paul Scherrer Institut
5232Villigen PSISwitzerland
L Castiglioni
Physik-Institut
Universität Zürich
Winterthurerstrasse 1908057ZürichSwitzerland
P Aebi
Département de Physique and Fribourg Center for Nanomaterials
Université de Fribourg
1700FribourgSwitzerland
F Weber
Institute of Solid State Physics
Karlsruhe Institute of Technology
76131Karlsruhe, Germany
M Hengsberger
Physik-Institut
Universität Zürich
Winterthurerstrasse 1908057ZürichSwitzerland
Robustness of the charge-ordered phases in IrTe 2 against photoexcitation
(Dated: January 25, 2018)
We present a time-resolved angle-resolved photoelectron spectroscopy study of IrTe2, which undergoes two first-order structural and charge-ordered phase transitions on cooling below 270 K and below 180 K. The possibility of inducing a phase transition by photoexcitation with near-infrared femtosecond pulses is investigated in the charge-ordered phases. We observe changes of the spectral function occuring within a few hundreds of femtoseconds and persisting up to several picoseconds, which we interpret as a partial photoinduced phase transition (PIPT). The necessary time for photoinducing these spectral changes increases with increasing photoexcitation density and reaches timescales longer than the rise time of the transient electronic temperature. We conclude that the PIPT is driven by a transient increase of the lattice temperature following the energy transfer from the electrons. However, the photoinduced changes of the spectral function are small, which indicates that the low temperature phase is particularly robust against photoexcitation. We suggest that the system might be trapped in an out-of-equilibrium state, for which only a partial structural transition is achieved. arXiv:1801.07979v1 [cond-mat.str-el]
I. INTRODUCTION
Transition metal dichalcogenides (TMDCs) are layered quasi-two-dimensional materials which provoked tremendous attention in the last years, because of the possibility of easily reducing their size down to the monolayer and of their interesting properties despite their relative chemical simplicity. This makes them attractive candidates for modern devices. TMDCs were intensively studied in the 1970s as they are hosting charge density wave (CDW) phases at low temperature, involving a phase transition which is usually enabled by an enhancement in the electronic susceptibility due to nesting of the low-dimensional Fermi surface. Furthermore, recent research on materials with 4d and 5d electrons has highlighted the importance of strong spin-orbit coupling and a significant Hund's coupling for the physics of correlated materials 1,2 . In this framework, IrTe 2 with 5d valence electrons in dispersing valence states and showing charge ordering at low temperature is intriguing and raises the question of whether Mott physics develops at low temperature 3 . It offers the interesting case of a TMDC with potentially strong electronic correlations in spin-orbit coupled bands.
IrTe 2 undergoes a first-order structural phase transition at about T c1 = 270 K to a phase with a chargeordered state characterized by a wave vector q 1 = (1/5, 0, 1/5) with respect to the room temperature unit cell vectors. The first-order phase transition is accompanied by a change of its unit cell symmetry from trigonal to monoclinic 4 . It involves a large jump in transport and magnetic properties 4 as well as in heat capacity 5 . We call this phase LT1. This phase transition displays also hysteretic magnetic and electrical behaviors. At about T c2 = 180 K, a second phase transition occurs and the charge ordering wave vector changes to q 2 = (1/8, 0, 1/8). We call it LT2. These phases with different charge ordering patterns have stimulated many scanning tunneling microscopy studies which evidenced the occurence of additional ordering patterns at the surface 3,6-11 . In parallel, motivated by the TMDC structure of IrTe 2 and the possibility of finding a new CDW phase, many angle-resolved photoelectron spectroscopy (ARPES) investigations have targeted this material 3,[12][13][14][15][16] . All of these ARPES studies have provided evidence for spectral changes in the low-energy electronic structure down to 3 eV below the Fermi level E F . These changes appear abruptly across T c1 , while more subtle differences occur across T c2 . However, no CDW gap has been observed, nor any significant possibility for nesting of the Fermi surface, although the inner bands (nearΓ) disappear below T c1 and might therefore be gapped. Interestingly, a new band has been found in the LT1 phase at about 0.5 eV below E F 3,16 . The complexity of the band structure of IrTe 2 , together with its rich phase diagram involving many ordering patterns, makes it difficult to determine the mechanism behind its phase transitions. It has been put forward that the lattice reconstruction induces the lifting of an orbital degeneracy which stabilizes the LT1 phase. This can be viewed as a Jahn-Teller effect 12,13 . Another study proposed a Mott phase involving spin-orbit coupled states 3 to be at the origin of the phase transitions in IrTe 2 . In this framework, time-resolved techniques can provide new information about the nature of these phase transitions by discriminating such different mechanisms on the femtosecond to picosend timescales. The suppression of the Mott phase in TaS 2 has been shown to be as fast as the photoexcitation pulse [17][18][19] , while the suppression of phases involving a structural reconstruction requires often longer times [19][20][21] .
Here we perform a systematic time-resolved ARPES study of IrTe 2 in its low-temperature charge-ordered phases LT1 and LT2 and investigate the possibility of photoinducing the phase transition to the room temperature (RT) phase. Using 6 eV photons, we single out a specific band in its electronic structure, which we relate to these charge-ordered states. After photoexciting the material, a small transient shift of this band is observed and interpreted as the result of a partial PIPT. It is shown that the necessary time for photoinducing this spectral change increases with increasing photoexcitation density. The timescale of this spectral change turns out to be even slower than the rise time of the transient electronic temperature. We conclude that this partial PIPT is driven by the transient increase of the lattice temperature, after the energy transfer from the electrons. However, despite the high photoexcitation densities used here, the photoinduced changes are small, indicating that IrTe 2 is very robust against photoexcitation with near-infrared pulses. We propose that the system is not transiting to the RT phase, but is trapped in an out-of-equilibrium state, for which only a partial structural transition is achieved.
II. EXPERIMENTAL DETAILS
Single crystals of IrTe 2 were grown using the self-flux method. Resulting single crystals were characterized by magnetization measurements and single crystal x-ray diffraction studies, which confirms that T c1 = 270 K and T c2 = 180 K. The static ARPES data were acquired at the PEARL beamline 22 of the Swiss Light Source at the Paul Scherrer Institute (Switzerland).At 77 eV photon energy, the total energy resolution was 60 meV and the momentum resolution was 0.03Å −1 . The time-resolved ARPES data were obtained using light pulses produced by means of a commercial femtosecond oscillator (Coherent Mira Seed) and amplified in a high repetition rate (30 kHz) regenerative pulse amplifier (RegA 9050). IrTe 2 samples were cleaved in a base pressure of 10 −10 mbar and normal emission of photoelectrons was lying along (0001). The samples were excited by p-polarized laser pulses at 825 nm (1.5 eV). The samples were probed by p-polarized laser pulses of 6.0 eV photons, generated by frequency-doubling the fundamental 825 nm pulses twice in β−barium borate crystals. At 6.0 eV photon energy, the total energy resolution was 50 meV, the momentum resolution was 0.01Å −1 and the time cross-correlation was 250 fs (from the same sample). The absorbed fluences (absorbed excitation energy per unit area) appearing in this study are obtained by considering the mea-sured incident fluences and a sample reflectivity 5 of 50% at 825 nm. During the measurements with 6 eV photons, a bias voltage of −3 V with respect to ground in order to avoid the energy range of very low kinetic energies where the transmission of the electron detector almost vanishes. Furthermore this bias voltage helps to obtain sharper secondary electron edges for the determination of the sample work function. Sample temperature was measured with an accuracy of about ±10 K.
III. RESULTS
In Fig. 1, we show static ARPES data of IrTe 2 taken with high and low photon energies at three different temperatures corresponding to the three different phases of IrTe 2 . With a photon energy of 77 eV, the full Brillouin zone (displayed in Fig. 1(a)) is easily accessible with ARPES and at normal emission photoemission probes the region near Γ (the wave vector k z perpendicular to the surface corresponds to about 4.1c * with an inner potential 3 of 13 eV, c * being the Brillouin zone size perpendicular to the surface). From now on, we adopt the surface Brillouin zone notation as marked by bars. At room temperature, we observe a few sharp and dispersive bands along theΓM direction, as shown in Fig. 1 (b). After cooling the sample to 215 K ( Fig. 1 (c)), in the LT1 phase, the bands become suddenly broad in agreement with other ARPES studies 3,13-16 , so that it is difficult to distinguish the different contributions to the low energy electronic structure of IrTe 2 . In Fig. 1 (d), we show ARPES data measured at 80 K in LT2, which are similar to what has been measured at 215 K ( Fig. 1 (c)). Energy distribution curves (EDCs) integrated over ±0.3Å −1 aroundΓ are shown in Fig. 1 (e) for these three temperatures. It clearly demonstrates the strong spectral changes occuring in photoemission after cooling through the first phase transition. Notice especially the peak at about E − E F = −0.9 eV (labelled CO) which gains intensity in comparison to the peak at about -1.5 eV. Only small changes are visible in the spectra taken across the LT1-LT2 phase transition.
On the lower graphs of Fig. 1, we show the static ARPES data taken with a photon energy of 6 eV. With such an excitation energy, the photoelectrons have typical kinetic energies < 1 eV, meaning that only the center of the surface Brillouin zone is probed, with an excellent momentum resolution. Globally, the photoemission intensity distribution over the whole ARPES maps is significantly different than what is measured with 77 eV photons. This is expected because of the different position of the probed initial states along the k z -direction and the electronic structure of IrTe 2 displaying a significant dispersion along k z , despite its atomic structure being similar to that of many TMDCs 5 . In Fig. 1 (f), room temperature ARPES data indicate two bands dispersing close to each to other at the zone center. In comparison to Fig. 1 the zone outlined by the red dashed line is probed. By comparing to the data of Ootsuki et al., it appears that 6 eV photons probe the vicinity of the A point along the k z direction 15 . In addition, the final states of photoemission with 6 eV photons are expected to be significantly deviating from free electron final states 23,24 , which might affect strongly the photoemission matrix elements, and thus the photoemission intensity distribution. At 220 K in the LT1 phase, the ARPES spectrum is completely different, see Fig. 1 (g). One can now distinguish three flat bands (see dashed lines in graph (g)) instead of the two dispersing ones. When further decreasing the temperature to 150 K into the LT2 phase ( Fig. 1 (h)), there are little changes. Fig. 1 (i) shows the comparison of the angle-integrated EDCs for these three temperatures for the data obtained with 6 eV photons. This highlights a few important differences: (i) In addition to the drastic changes between room temperature and low temperature, there is a significant difference in intensity for the peak at -0.6 eV relative to the peak at about -0.2 eV between LT1 and LT2. From now on, we will consider the peak at -0.6 eV as the spectral signature of the LT phases in IrTe 2 and we will name it the charge order (CO) peak for convenience. (ii) The work function is different in the three phases: 5.3 eV at RT, 5.15 eV in LT1 and 5.1 eV in LT2. We attribute these different values to the changes in the electronic and atomic structure of IrTe 2 in the charge-ordered phases LT1 and LT2.
(b) (obtained with 77 eV photons), only (e) CO DF CO (i) (c) (d) (g) (h) LT1 LT2 (b) (f) RT(
Having established the spectral changes in static ARPES with 6 eV probe photons, we now move on to the time-resolved data. For this purpose, the samples are photoexcited with 1.5 eV photons. Given the complex electronic structure of IrTe 2 and the absence of a large band gap at low temperature, such an optical excitation can occur a priori via many direct transitions. In Fig. 2 (a), we show time-resolved ARPES data taken at 220 fs for IrTe 2 in the LT1 phase at 230 K for an absorbed excitation fluence of 1.9 mJ/cm 2 . To emphasize the dynamical changes induced by the photoexcitation, we subtract from these ARPES data taken at 220 fs an average of the data taken at delays before time 0. This difference map at 220 fs is shown in Fig. 2 (b). It highlights the momentum-resolved changes in the photoemission intensity occuring at different energies. Excited electrons transiently populate states above E F (positive intensity change) and intensity is lost by the appearance of holes (negative intensity change) just below E F . The excited electron distribution follows a non-trivial distribution with a shoulder at about 0.12 eV. This is better seen in the transient EDCs shown on Fig. 2 (c) for different times. At 220 fs, one distinguishes a shoulder above E F which is transiently populated and gives evidence for a band just above E F (see red arrow in 2 (c)). In addition to this population dynamics and corresponding hole dynamics, other transient changes occur at higher binding energies in the occupied states, below −0.3 eV. Interestingly, it affects also the CO peak, the spectral signature of the LT phases, which shifts towards E F (see the black arrow in the inset in Fig. 2 (c)).
In Fig. 2(d), we show the time-dependent photoemis- sion intensity of excited electrons at about 0.4 eV above E F (integrated in box 1 on Fig. 2 (b)), together with the time-dependent intensity in the upper edge of the CO peak (integrated in box 2). These two curves show very different behaviors: the green curve (box 1) displays the typical population dynamics for excited electrons near the Fermi level, while the blue curve (box 2) follows a step-like increase, with a slower rise than the green curve. The energy position of box 2 was chosen to minimize effects of the transient changes in the Fermi-Dirac distribution, while integrating the small transient changes near the CO peak. For this reason together with its unusal time-dependence, we infer that the photoemission changes below −0.3 eV are mainly photoinduced changes of the spectral function of IrTe 2 , rather than changes in the Fermi-Dirac distribution. Furthermore, given that the CO peak is the spectral signature of the LT1 phase, we conclude that it must be due to a (partial) photoinduced phase transition (PIPT) from the LT1 phase to the RT phase. This conclusion is further supported by Fig. 3: here we add about 5% of the static RT photoemission spectrum to 95% of the static LT1 spectrum (at 230 K), in order to visualize what would be the spectral signature of a partial PIPT. This simple construction simulates a situation for which only 5% of the probed volume would transit into the RT phase under the action of the pump pulse, e.g. as a consequence of sample inhomogeneity. The spectrum resulting from this combination is shown in green in Fig. 3. It compares well to the EDCs recorded for delays greater than 1000 fs when the new transient state is established like shown in Fig. 2 (c). In particular, it should be compared to the EDC of Fig. 2 (c) measured at 4500 fs, since the transient change of the CO peak in Fig. 2 (d) indicates that the PIPT effects remain over a long timescale of more than 4 ps. This establishes the observation of a partial PIPT from LT1 to RT in IrTe 2 upon excitation with strong 825 nm pump pulses. Additionnally, our data reveal a persistant shift of the leading edge near E F of about 4 meV (see the inset in Fig. 2 (c)). This might be due to a transient change of the spectral function just above E F , related to the band situated at about 0.12 eV (see red arrow in Fig. 2 (c)).
In Fig. 4, we show time-resolved ARPES data obtained in the LT2 phase. These data were acquired at 140 K with an absorbed excitation fluence of 2.3 mJ/cm 2 . Fig. 4 (a) shows the raw data at 170 fs and Fig. 4 (b) the corresponding difference map. In Fig. 4 (c) has been observed in LT1 (see Fig. 2), namely excited electrons above E F in a previously unoccupied band (see red arrow in Fig. 4 (c)), excited holes just below E F and photoinduced changes of the spectral function at higher binding energies, close to the CO peak (see black arrow in Fig. 4 (c)). In Fig. 4 (d), we show the time-dependent photoemission intensity integrated in the boxes 1 and 2 of Fig. 4 (b), following the dynamics of the excited electrons (green curve) in comparison to the spectral changes of the CO peak (blue curve) monitoring the PIPT. Similarly to the case of LT1, the dynamics of the PIPT are slower than the dynamics of the excited electrons. The CO peak intensity change, after having reached its maximum, relaxes back to its initial value on a very long timescale far above 3 ps. We now apply a similar reasoning as used to construct Fig. 3 in order to figure out to which phases the PIPT occurs when starting from LT2 phase. From Fig. 1 (i), we know that the IrTe 2 spectral function changes very little between the LT1 and LT2 phases. Therefore mixing 5% of the LT1 spectral function to 95% of the LT2 spectral function would result in a spectrum that is hardly different from the LT2 phase spectrum and such a difference would not be visible in photoemission. We therefore conclude that the transient photoinduced changes measured in the LT2 phase essentially originate from a small volume fraction transiting to the RT phase.
In order to investigate the partial PIPT occuring in IrTe 2 , we have performed time-resolved ARPES for different excitation fluences in both LT1 and LT2 phases. For each fluence, we have fitted the time-dependent photoemission intensity of the CO peak with a broadened step function to extract its rise-time and maximum intensity changes as shown by black curves in Fig. 2 (d) and in Fig. 4 (d). In Fig. 5 (a) and (b), we plot the rise times and maximum intensity changes (relative to the maximum CO peak intensity) as a function of fluence, respectively. Globally, we see that the rise time and maximum intensity change increase with fluence. At the highest absorbed fluence achieved here (about 2.7 mJ/cm 2 ), the rise time, i.e. the time necessary to achieve the partial PIPT, reaches a duration as long as 500 fs, both in LT1 and LT2 phases. This is clearly a very slow process on the timescale of electron motion. Surprisingly, in Fig. 5 (a), the rise time of the lowest fluence used here (about 0.4 mJ/cm 2 ) for LT1 phase, is completely out of the trend and shoots up to 500 fs. Despite the low intensity of the corresponding photoinduced changes, this behavior is observed in the raw data (not shown here). To understand this exception, we analyze the slope of the electronic distributions just below E F for all data obtained in LT1 phase and extract an electronic temperature from fits of the Fermi-Dirac distribution between about -0.05 and 0.05 eV. The extracted values are shown in Fig. 5 (c) and we see that the maximum of each curve scales with the absorbed fluence. Interestingly, the electronic temperatures obtained for the lowest fluence are significantly lower than for the other fluences and hardly go over T c1 . This might indicate that different dynamics is at play here, because a threshold energy cannot be reached at this fluence. We compare in Fig. 5 (d) the early evolution of the electronic temperature and the dynamics of the CO peak intensity change and of the mean excited electron energy E . Here E is calculated by integrating the energy of excited electrons, E − E F , weigthed with their photoemission intensity, I(t) − I(t < 0). In Fig. 5 (d), we see that the photoexcitation energy is quickly distributed into the excited electrons as observed within the detector window chosen here. The electronic temperature rises more slowly and reaches its maximum about 70 fs later. The main observation is that the increase in the intensity change of the CO peak is even slower and reaches its maximum at even longer times.
IV. DISCUSSION
The partial PIPT evidenced in this work involves small changes of the spectral function, which reach only a few percents of the intensity of the CO peak at the highest fluence used here (Fig. 5 (b)). This indicates that the LT phases of IrTe 2 are very robust against photoexcitation, at least as far as the electronic structure is concerned. It is tempting to emphasize the similarity with the PIPT observed in VO 2 , for which incident fluences as high as 25 mJ/cm 2 were necessary to achieve the complete PIPT in a material having an absorption length of 180 nm at 800 nm wavelength 25,26 . In time-resolved ARPES studies 27,28 , photoinduced changes of only a few percents have been obtained for incident fluences above 6 mJ/cm 2 . In VO 2 , the structural phase transition is of first order like in IrTe 2 . However, the similarity stops here, since the PIPT has been shown to occur on a very fast timescale 29 in VO 2 , being as fast as 80 fs. In the case of IrTe 2 , we observe a slow (partial) PIPT, with durations of the order of several hundreds of fs. Furthermore, we see that this PIPT rise time (see Fig. 5 (a)) is increasing for increasing fluences. This tells us that the time necessary to achieve the complete spectral changes observed here is dictated by a slow process governed by the pump pulse energy flow. We propose naturally that it is due to the energy flow from the excited electrons into the lattice, as also indicated by the hierarchy of timescales seen in Fig. 5 (d): first the pump energy is absorbed by the electrons, which are excited more than 1 eV above E F . These electrons then scatter down towards E F and thermalize, and the maximum electronic temperature is reached only ∼ 100 fs later. We conjecture that this remarkable delay is due to the presence of a presumably electron-like band in the unoccupied states, about 0.12 eV above E F (see red arrow in Fig. 2 (c) and Fig. 4 (c)). A central observation here is that the dynamical spectral signature of the PIPT is even slower than the electronic temperature. This is again compatible with our proposition that the PIPT is driven by the transient lattice temperature, which rises more slowly than the electronic temperature.
The fluence dependence of the rise time of the CO peak change (Fig. 5 (a)) can be understood qualitatively with a simple two temperature model analysis, including the transient electronic and lattice temperatures. For this purpose, we compute the electronic and lattice temperatures for IrTe 2 , according to the following differential equations
∂T e ∂t = −γ(T e ) · (T e − T l ) + P C e , ∂T l ∂t = C e C l γ(T e ) · (T e − T l ).
The coupling function, γ(T ) = 3λΩ 2 0 /( 2 πT ), is inversely proportional to temperature 30 . The differential equations can be derived from the model used by Perfetti et al. 31 but for only one phonon subsystem. We take the electronic specific heat C e as linear in the electronic temperature T e and a temperature dependent lattice specific heat C l fitted to experimental values 5 . The latent heat at the phase transition at T c1 is much smaller than the energy density deposited by the pump pulse (see below), and is therefore neglected. From the optical data of Fang et al. 5,32 , the absorption length in IrTe 2 at 825 nm is about 20 nm, which means that 1 mJ/cm 2 of absorbed fluence corresponds to an absorbed energy of 210 J/cm 3 . In our example here 36 , we first use an absorbed energy density of 210 J/cm 3 . The calculated temperatures are shown in Fig. 5 (e) and are typical for the output of the two temperature model: the electronic temperature rises as fast as the pump pulse intensity and reaches very high temperatures, while the lattice temperature rises more slowly due to the finite energy exchange rate between the electrons and the lattice and attains a lower temperature, 300 K in our example. On Fig. 5 (f), the calculated lattice temperatures are shown for three different absorbed energy densities and normalized to their maximum. One sees that the higher the absorbed energy density, the more time it takes for the lattice temperature to reach 95% of its maximum. This behavior is due to the complex temperature dependencies in the two temperature model and can be mainly traced back to the coupling function γ(T ) ∝ 1/T , which indicates that for increasing electronic temperatures, the energy transfer from electrons to lattice becomes less efficient. As a consequence, the timescale of the CO peak change, which represents the transient lattice temperature, increases with fluence ( Fig. 5 (a)) because for higher fluences (or absorbed energy densities), it will take more time for the CO peak change to reach its maximum transient value. In this interpretation, the PIPT timescale is given by the transient lattice temperature and can occur only as fast as the transient heating of the lattice.
The photoinduced spectral changes after a few ps do not exactly coincide with the simple superposition of static ARPES data shown in Fig. 3. In particular, we see significant photoinduced spectral changes on the peak at -0.2 eV (see e.g. Fig. 2(c)), in addition to a persistent shift of the leading edge at E F and no modification of the sample work function, which changes drastically through the phase transition in the static data. It is also very surprising that we can induce only small changes of the spectral function in our experiment, given the high fluences used here and the short absorption length of IrTe 2 . As written above, 1 mJ/cm 2 of absorbed fluence corresponds to an absorbed energy of 210 J/cm 3 . This represents a large energy density, which should result in a transient electronic temperature raising up to more than 1000 K and subsequently a lattice temperature raising above 300 K. From this estimation, and even from the observation that electronic temperatures of several 100s K above RT are reached at E F (Fig. 5(c)), it is clear that the excitation of electrons in IrTe 2 is not sufficient to trigger efficiently the PIPT. The latent heat of the first order structural phase transition (about 20 J/cm 3 , from Ref. 3) cannot explain this fact, since it is very small compared to the absorbed energy densities used here. This might be an indication that the system is in fact trapped in a metastable out-of-equilibrium state between LT1 and RT phases. It is therefore interesting to recall that the equilibrium phase transition taking place in IrTe 2 involves both a change of its unit cell symmetry and of the charge order, which breaks the translational symmetry. In a recent paper, Ivashko et al. have shown that, upon pressure, these two transitions occur at dif-ferent temperatures for samples with a few percents of Pt doping 33 . We propose here that a metastable out-ofequilibrium state involving only one of these two structural transitions could be achieved in IrTe 2 upon photoexcitation. This is further supported by the fact that no transient change of the sample work function is observed here. We hope that this will trigger interest for further time-resolved studies.
It has been proposed that the first-order CO phase transitions taking place in IrTe 2 are due to a Mott phase transition involving spin-orbit coupled states 3 . The phenomenology of successive first-order Mott phase transitions in IrTe 2 is reminiscent of the case of TaS 2 34,35 . TaS 2 has been intensively studied by time-resolved techniques and it has been shown that the Mott gap at Γ collapses quasi-instantaneously upon photoexcitation, what has been understood as evidence for its electronic origin [17][18][19] . Interpreting the CO peak in IrTe 2 as the lower Hubbard band of a Mott system, one would expect its response to photoexcitation to be ultrafast, well below the timeresolution of our experiment. The small shift of the CO peak observed in our study, which could be viewed as the precursor of a Mott gap collapse, is by far too slow to be related to the ultrafast physics of photoexcited Mott insulators. Our time-resolved ARPES study of IrTe 2 does therefore not give any evidence for Mott physics in this material.
V. CONCLUSION
In conclusion, we have studied the ultrafast dynamics of the first-order structural phase transitions in IrTe 2 with time-resolved ARPES. We observe that a partial phase transition can be photoinduced in this material using strong near-infrared pump pulses. The time necessary for this PIPT is increasing for increasing pump fluence and eventually becoming as high as 500 fs. Furthermore, this characteristic time is slower than the timescales necessary for depositing energy in the electronic subsystem and raising its temperature. We deduce that the partial PIPT is driven by the energy transfer from excited electrons into the lattice. However, the observed photoinduced changes in ARPES are small despite the large absorbed energy. We conjecture that this material is actually trapped in an out-of-equilibrium state. From the above we conclude that both the phase transitions in thermal equilibrium and those following ultrafast excitation are mainly driven by a lattice instability rather than by a Mott instability.
VI. ACKNOWLEDGEMENTS
Part of this work was performed at the PEARL beamline of the Swiss Light Source of the Paul Scherrer Institut in Switzerland. We thank J. Osterwalder for fruitful discussions. C.M. acknowledges the support by the SNSF
FIG. 1 :
1(a) Bulk and surface Brillouin Zone of IrTe2 in its room temperature phase. (b,c,d) Static ARPES data acquired with 77 eV photons alongΓM at (b) 300 K, (c) 215 K and (d) 80 K, corresponding to the RT, LT1 and LT2 phases, respectively. (e) Angle-integrated energy distribution curves (normalized to their maximum) integrated ±0.3Å −1 aroundΓ in the data of graphs (b,c,d). Static ARPES maps acquired with 6 eV photons alongΓM at (f) 298 K, (g) 230 K and (h) 160 K. The white dashed lines in graph (g) indicate the three flat bands mentioned in the main text. (i) Angle-integrated energy distribution curves (normalized to their maximum) from the data of graphs (f,g,h). A significant shift of the work function ∆Φ ∼ = 0.15 eV is observed between the RT and LT1/LT2 phases. Note that the ARPES maps in graphs (f,g) are not centered atΓ.
FIG. 2 :
2Time-resolved ARPES data in LT1 phase nearΓ, at 235 K. (a) Raw ARPES intensity map taken at + 220 fs and (b) the corresponding difference map (photoemission intensity difference between averaged data taken before 0 and data taken at 220 fs) for an absorbed fluence of 1.9 mJ/cm 2 . (c) Corresponding (angle-integrated) EDCs at different delay times. (d) Transient intensities integrated in the box 1 (excited electrons) and 2 (CO peak) of graph (b), together with a step-like fit.
FIG. 3 :
3Static (angle-integrated) EDCs taken at 300 K (in red, RT phase) and at 230 K (in black, LT1 phase). Their weighted sum (green thick line) is obtained as the addition of 5% of the static RT spectrum to 95% of the static LT1 spectrum.
FIG. 4 :
4, EDCs are displayed before 0 fs, at 170 fs and at 2500 fs. The photoinduced changes observed here are similar to what Time-resolved ARPES data in LT2 phase nearΓ, at 140 K. (a) Raw ARPES intensity map taken at + 170 fs and (b) the corresponding difference map (photoemission intensity difference between averaged data taken before 0 and data taken at 170 fs) for an absorbed fluence of 2.3 mJ/cm 2 . (c) Corresponding (angle-integrated) EDCs at different delay times. (d) Transient intensities integrated in the box 1 (excited electrons) and 2 (CO peak) of graph (b), together with a step-like fit.
FIG. 5 :
5(a) Rise time of the PIPT obtained from fits to the photoemission intensity changes of the CO peak in LT1 and LT2 phases, as a function of absorbed pump fluence. (b) Maximum intensity change of the CO peak (relative to the maximum CO peak intensity) as a function of absorbed fluence. (c) Transient electronic temperature for different absorbed pump fluences in LT1. (d) Comparison between the transient mean excited electronic energy, electronic temperature and CO peak intensity in LT1 for an absorbed fluence of 2.7 mJ/cm 2 . (e) Calculated transient electronic and lattice temperature for an absorbed energy density of 210 J/cm 3 using a two temperature model (see text for details). (f) Calculated transient lattice temperatures for three different absorbed energy densities and normalized to their maximum.
We are grateful to Cephise Cacho for sharing the time-resolved ARPES analysis software. This work was supported by the Swiss National Science Foundation through Div. P Z00p, II. F.W. was supported by the Helmholtz Society under contract VH-NG-840grant No. P Z00P 2 154867. We are grateful to Cephise Cacho for sharing the time-resolved ARPES analysis soft- ware. This work was supported by the Swiss National Science Foundation through Div. II. F.W. was supported by the Helmholtz Society under contract VH-NG-840.
was supported by the Karlsruhe Nano-Micro Facility (KNMF). M , M.M. was supported by the Karlsruhe Nano-Micro Facil- ity (KNMF).
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. B Sipos, A F Kusmatseva, A Akrap, H Berger, L Forró, A Tutis, Nature Mater. 7960B. Sipos, A.F. Kusmatseva, A. Akrap, H. Berger, L. Forró and A. Tutis, Nature Mater. 7, 960 (2008).
We also use a phonon energy of 30 meV, an electronphonon coupling constant λ of 0.2, and the linear term of the electronic specific heat is 1.5 · 10 −4 J/(cm 3 K 2 ). However the choice of these parameters does not influence our qualitative argument. We also use a phonon energy of 30 meV, an electron- phonon coupling constant λ of 0.2, and the linear term of the electronic specific heat is 1.5 · 10 −4 J/(cm 3 K 2 ). How- ever the choice of these parameters does not influence our qualitative argument.
|
[] |
[
"Bolometric Night Sky Temperature and Subcooling of Telescope Structures",
"Bolometric Night Sky Temperature and Subcooling of Telescope Structures"
] |
[
"R Holzlöhner \nEuropean Southern Observatory (ESO)\nKarl-Schwarzschild-Str. 285748GarchingGermany\n",
"S Kimeswenger \nInstitut für Astro-und Teilchenphysik\nLeopold-Franzens Universität Innsbruck\nTechnikerstr. 256020InnsbruckAustria\n\nInstituto de Astronomía\nUniversidad Católica del Norte\nAv. Angamos 0610AntofagastaChile\n",
"W Kausch \nInstitut für Astro-und Teilchenphysik\nLeopold-Franzens Universität Innsbruck\nTechnikerstr. 256020InnsbruckAustria\n",
"S Noll \nInstitut für Physik\nUniversität Augsburg\nUniversitätsstraße 186159AugsburgGermany\n\nDeutsches Fernerkundungsdatenzentrum\nDeutsches Zentrum für Luft-und Raumfahrt (DLR)\nMünchener Straße 20, Weßling-Oberpfaffenhofen 82234Germany\n"
] |
[
"European Southern Observatory (ESO)\nKarl-Schwarzschild-Str. 285748GarchingGermany",
"Institut für Astro-und Teilchenphysik\nLeopold-Franzens Universität Innsbruck\nTechnikerstr. 256020InnsbruckAustria",
"Instituto de Astronomía\nUniversidad Católica del Norte\nAv. Angamos 0610AntofagastaChile",
"Institut für Astro-und Teilchenphysik\nLeopold-Franzens Universität Innsbruck\nTechnikerstr. 256020InnsbruckAustria",
"Institut für Physik\nUniversität Augsburg\nUniversitätsstraße 186159AugsburgGermany",
"Deutsches Fernerkundungsdatenzentrum\nDeutsches Zentrum für Luft-und Raumfahrt (DLR)\nMünchener Straße 20, Weßling-Oberpfaffenhofen 82234Germany"
] |
[] |
Context. The term sky temperature is used in the literature in different contexts which often leads to confusion. In this work, we study T sky , the effective bolometric sky temperature at which a hemispherical black body would radiate the same power onto a flat horizontal structure on the ground as the night sky, integrated over the entire thermal wavelength range of 1-100 µm. We then analyze the thermal physics of radiative cooling with special focus on telescopes and discuss mitigation strategies. Aims. The quantity T sky is useful to quantify the subcooling in telescopes which can deteriorate the image quality by introducing an Optical Path Difference (OPD) and induce thermal stress and mechanical deflections on structures. Methods. We employ the Cerro Paranal Sky Model of the European Southern Observatory to derive a simple formula of T sky as a function of atmospheric parameters. The structural subcooling and the induced OPD are then expressed as a function of surface emissivity, sky view factor, local air speed and structure dimensions. Results. At Cerro Paranal (2 600 m) and Cerro Armazones (3 060 m) in the Atacama desert, T sky towards the zenith mostly lies 25-50 Kelvin below the ambient temperature near the ground, depending strongly on the precipitable water vapor (PWV) column in the atmosphere. The temperature difference can decrease by several Kelvin for higher zenith distances. The subcooling OPD scales linearly to quadratically with the telescope diameter and is inversely proportional to the local air speed near the telescope structure.
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10.1051/0004-6361/202038853
|
[
"https://arxiv.org/pdf/2010.01978v1.pdf"
] | 222,134,102 |
2010.01978
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5d9cc3ef9cf88e607455bfdc746d56a0665a0722
|
Bolometric Night Sky Temperature and Subcooling of Telescope Structures
October 6, 2020
R Holzlöhner
European Southern Observatory (ESO)
Karl-Schwarzschild-Str. 285748GarchingGermany
S Kimeswenger
Institut für Astro-und Teilchenphysik
Leopold-Franzens Universität Innsbruck
Technikerstr. 256020InnsbruckAustria
Instituto de Astronomía
Universidad Católica del Norte
Av. Angamos 0610AntofagastaChile
W Kausch
Institut für Astro-und Teilchenphysik
Leopold-Franzens Universität Innsbruck
Technikerstr. 256020InnsbruckAustria
S Noll
Institut für Physik
Universität Augsburg
Universitätsstraße 186159AugsburgGermany
Deutsches Fernerkundungsdatenzentrum
Deutsches Zentrum für Luft-und Raumfahrt (DLR)
Münchener Straße 20, Weßling-Oberpfaffenhofen 82234Germany
Bolometric Night Sky Temperature and Subcooling of Telescope Structures
October 6, 2020Received 06 July 2020; accepted 02 October 2020Astronomy & Astrophysics manuscript no. mainTelescopes -Instrumentation: miscellaneous -Site testing -Atmospheric effects -Radiative transfer -Convection
Context. The term sky temperature is used in the literature in different contexts which often leads to confusion. In this work, we study T sky , the effective bolometric sky temperature at which a hemispherical black body would radiate the same power onto a flat horizontal structure on the ground as the night sky, integrated over the entire thermal wavelength range of 1-100 µm. We then analyze the thermal physics of radiative cooling with special focus on telescopes and discuss mitigation strategies. Aims. The quantity T sky is useful to quantify the subcooling in telescopes which can deteriorate the image quality by introducing an Optical Path Difference (OPD) and induce thermal stress and mechanical deflections on structures. Methods. We employ the Cerro Paranal Sky Model of the European Southern Observatory to derive a simple formula of T sky as a function of atmospheric parameters. The structural subcooling and the induced OPD are then expressed as a function of surface emissivity, sky view factor, local air speed and structure dimensions. Results. At Cerro Paranal (2 600 m) and Cerro Armazones (3 060 m) in the Atacama desert, T sky towards the zenith mostly lies 25-50 Kelvin below the ambient temperature near the ground, depending strongly on the precipitable water vapor (PWV) column in the atmosphere. The temperature difference can decrease by several Kelvin for higher zenith distances. The subcooling OPD scales linearly to quadratically with the telescope diameter and is inversely proportional to the local air speed near the telescope structure.
Introduction
Any structure on the ground that is exposed to the night sky experiences a radiation imbalance and thus typically cools below the ambient air temperature (see Zhao et al. (2019) for an overview). Sky subcooling is strongest at sites with low water vapor column, hence under conditions commonly present at telescope sites (high and dry). As a consequence, the air surrounding for example a steel truss is cooled, causing a wake of cool air. Light rays passing the cooled air experience an optical path difference which induces a transient wavefront error in a telescope (Wilson 2013). This situation is most severe at small wind speeds and thus became known as the "Low-Wind Effect" that has recently been studied (Sauvage et al. 2015;Milli et al. 2018). A secondary detrimental consequence of subcooling is structural deflection due to thermal contraction, inducing mechanical stress and optical misalignment. Quantifying the effective sky temperature is thus required when designing advanced ground-based telescopes and analyzing their image quality.
Previous studies (Berdahl & Fromberg 1982;Gliah et al. 2011;Sun et al. 2017;Li et al. 2017Li et al. , 2018 used wide angle integrating detectors subtending nearly the whole sky. Moreover, these detectors are tested and calibrated at sites with lower altitudes and higher humidity. To derive the effects of the sky temperature on a telescope structure and the optical elements like the main mirror, a detailed knowledge of the sky irradiance as function of the direction on the sky is required as well. To our knowledge, such a study has not been presented before in the literature.
In Section 2, we discuss the night sky emissivity and define the effective bolometric sky temperature T sky . Section 3 then sets up a thermal balance model of a structure on the ground exposed to sky radiation and forced convection. Furthermore, we discuss the effects of heat conduction and thermal inertia. Finally, we conclude in Section 4.
Sky radiance and sky temperature
The spectrum of the radiance L sky for a given area on the sky irradiating a surface strongly depends on wavelength λ, zenith distance θ and weather conditions. Its total bolometric radiance E BOL , defined as integral of the radiance spectrum over all wavelengths, is dominated by those fractions of the wavelength domain where the atmosphere is completely opaque. In these regions, the mean free path of the light is often a few hundred meters only, in optically very thick lines it even may be only a few meters. Thus the main emission is strongly bound to the local temperature near the ground. Those regions where the atmosphere is nearly transparent do not contribute to the integrated radiance significantly. We use here the common definition of a sky temperature T sky as the equivalent temperature of a black body radiation curve of the same bolometric radiance E BOL (see Article number, page 1 of 14 arXiv:2010.01978v1 [astro-ph.IM] 5 Oct 2020 A&A proofs: manuscript no. main Fig. 1. The sky radiance L sky (λ) for Cerro Paranal at different PWV conditions typical for observations in direction towards zenith. The astronomical observation bands and the main ozone absorption are marked. For further details on species which are not important for us here (e.g. CO 2 , N 2 O and CH 4 ) see Fig. 1 and Table 1 in Smette et al. (2015). Li et al. 2017Li et al. , 2018, and references therein)
E BOL = ∞ 0 L sky (λ) dλ := σ π T 4 sky ,(1)
where the unit is W m −2 sr −1 , with the Stefan-Boltzmann constant σ = 5.670367 × 10 −8 W m −2 K −4 .
General trend and climatology of the sky temperature
With our European Southern Observatory (ESO) Sky Model 1 (Noll et al. 2012;Jones et al. 2013) we are able to calculate detailed spectral dependencies and integrate them to derive the bolometric flux and the sky temperature T sky . Although this sophisticated model contains a wide variety of components of illumination like scattered moonlight, zodiacal light, scattered starlight, atomic and molecular emission lines in the upper atmosphere (airglow), the vast majority of the bolometric flux originates from about 4 to 7 µm and longwards of 13.5 µm and is dominated by the emission of the lower atmosphere. Our model uses a spectral model of the Earth's lower atmosphere in local thermal equilibrium calculated by means of the radiative transfer code LNFL/LBLRTM (Clough et al. 2005). This code is used with the spectral line parameter database aer, which is based on HITRAN (Rothman et al. 2009 We show the dependency of the thermal sky radiance as function of the Precipitable Water Vapor (PWV) in Fig. 1. This quan-tity equals the integrated column density across the whole atmosphere, including all layers contributing to the opaqueness and is given in units of equivalent thickness in mm of liquid water. The opaque fraction of the spectral energy distribution resembles a black body radiation curve. While the transparency in the window around λ = 10 µm is only marginally affected by the H 2 O molecular bands, it varies strongly in the region from 15 to 28 µm (astronomical Q-band) with the water vapor content. Thus, the weather-dependent variations and uncertainties mainly originate from this wavelength band. There are marginal transparency windows beyond these wavelengths (astronomical midinfrared Z-band with 28 < λ < 40 µm). As Yoshii et al. (2010) showed this assumption still holds at sites of even higher altitude than Mauna Kea (e.g. as planned for the TAO 6.5 m telescope project at 5 640 m). Thus, an approximation with a black body curve is normally sufficient. The ozone absorption in the center of the N-band does not follow the trend of the other opaque regions due to the high altitude of the absorbing molecules.
The sky radiance (expressed as sky temperature as defined above) thus depends strongly on the PWV. This circumstance was already found by Berdahl & Fromberg (1982), but they characterized the measured radiation as a function of the relative humidity near the ground -respectively the dew point temperature T DP calculated from the humidity. Although radiation from low altitudes prevails, the variation is dominated by semi-transparent regions in the Q-band. Here we see radiation from higher layers, having different temperatures and a water vapor content which is not necessarily linked to the relative humidity measured near the ground.
We find a phenomenological fit function of the sky temperature with the water vapor content for dry conditions (Fig. 2) which is precisely passing the points with deviations less than 0.1 K. At more humid conditions (PWV > 15 mm) the relation still fits decently, but has some larger uncertainties in the order of 2 K. Although the fit is based on model data for the southern winter months June and July, it is also valid for the other months of the year. The small differences in the bimonthly mean profiles of pressure and temperature as well as the relative vertical distribution of water vapor at Cerro Paranal appear to have a negligible effect on the integrated radiance spectrum.
Moreover, the previous studies (Berdahl & Fromberg 1982;Gliah et al. 2011;Sun et al. 2017;Li et al. 2017Li et al. , 2018 used wide angle integrating detectors subtending nearly the whole Fig. 2. The correlation of the sky temperature T sky as a function of PWV at zenith (θ = 0). Only the range of conditions with PWV < 15 mm relevant for typical astronomical observations has been used for the final fit function. Fig. 3. The correlation of the sky temperature T sky as a function of the zenith distance at constant level of PWV sky. However, in order to calculate a radiation model for a structure, the irradiation of this structure as a function of its "exposed" direction to the sky has to be taken into account (see also Section 3.2). This effect is not so important for a structure like a whole building or a solar cell, as studied in e.g. Berdahl & Fromberg (1982) or Li et al. (2017). They thus needed only integrated values for the entire sky. However, structures of a telescope inside a dome are exposed in a more complex way to different fractions of the sky. Thus, the exact spatial impacts are essential. When tilting the telescope to higher zenith distances, the radiation is more and more dominated by low heights. Thus, we derived the correlation of the irradiance with zenith distance for a wide variety of parameters (Fig. 3). A comparison of the summer and winter models shows a very good match of the data points. There are only marginal differences for θ = 60 • and 70 • . Thus again, a general season-independent fit can be derived. We find that this correlation does not depend on the humidity after accounting for the zenith dependency mentioned above. Thus, a direct empirical fit function with both parameters for good weather conditions (PWV < 15 mm H 2 O and no clouds) is found:
T sky (T amb , PWV, θ) = T amb −45.75+6.52 ln(PWV)+0.00025 θ 2.5 , where T amb is the ambient temperature near the ground in Kelvin, PWV is in mm H 2 O and the zenith distance θ is measured in degrees.
Individual solutions and local estimators
Our software molecfit Kausch et al. 2015) was originally designed to correct astronomical spectra for telluric absorption. However, it can also be used to simulate individual thermal emission spectra for a certain date and time at a site. The code automatically derives the GDAS profile of that very day and time from the servers. In order to test extreme weather conditions which are not well represented by the sky model, we calculated emission spectra for the warmest and the coldest nights with respect to local midnight at Cerro Paranal in July 2018 (Fig. 4). The reference temperatures were measured 30 meters above ground at the Ambient Monitoring Station (ASM). They agree well with the temperatures that are derived from a fit of a black body to only those regions in the spectrum where the atmosphere is very opaque. For comparison, we also considered the warmest and coldest nights in 2018 at the high altitude site at Mauna Kea observatory in Hawaii, as measured at the weather monitor of the Canada-France-Hawaii Telescope (CFHT). This observatory site has significantly lower air temperatures due to its high altitude of 4 200 m (Fig. 5). Therefore, the relative humidity at this altitude tends to be lower. However, the latter and the dew point temperature calculated from it (Buck 1981) are only loosely correlated with the integrated water vapor column, i.e., the scatter is large (Otárola et al. 2010). Thus, we have to use PWV (indicating the impact of the most important and highly variable trace gas H 2 O) to optimally parameterize the atmospheric radiation (Otárola et al. 2010;Lakićević et al. 2016).
While we were able to take the PWV measured from the L-HATPRO microwave radiometer at Cerro Paranal (Kerber et al. 2012, see Section 2.3.1) for Fig. 4, the PWV values for Hawaii are only based on the (less accurate) GDAS profiles for the given nights. The PWV values and the resulting T sky for the integrated sky spectrum and using Eq. (2) are provided in Table 1. The fit formula causes deviations in T sky between −3.6 and +4.8 K for these extreme conditions, which suggests that its general uncertainty is not larger than a few Kelvin.
In case that a sky model is not available at a given site, we tested the possibility of estimating the values of T sky from the Article number, page 3 of 14 A&A proofs: manuscript no. main transmission curve. The latter essentially depends on the site altitude and the PWV (which can be obtained by means of GDAS data but might also be derived by various multi-wavelength observations without use of a model), and can therefore be easily tabulated for any site. Multiplying this transmission curve with a black body spectrum only depending on the ambient temperature T amb results in the desired estimator T est sky after integration over all wavelengths (Eq. (1)). As indicated in Figs. 4 and 5, such an estimate can match almost perfectly the accurate radiative transfer model result and confirms the parameterization in Eq. (2). Only the ozone band from 9.6 to 10.1 µm is strongly overestimated because most of the ozone emission originates from a higher layer in the atmosphere with lower temperature. This kind of emission might be different for very low altitude sites near human civilization, however, the latter is beyond our study. Thus, the ozone-related wavelength range was excluded from the integration. Other regions with special molecular features do not seem to be dominant enough to contribute to our result significantly. The resulting estimated sky temperature T est sky in Table 1 does not deviate by more than 2.1 K from T sky . For the more typical cases represented by the same parameters as used in Figs. 2 and 3, T est sky lies about 1.5 K above the value of the full model for PWV < 15mm (Fig. 6). Date The ESO Sky Model yields radiation spectra only up to a wavelength of 30 µm. Above this region, a pure black body emission is typically assumed in the standard online implementation (SkyCalc). The integration of that emission using Eq. (1) Fig. 6. Comparison of the sky temperature T full sky derived from the sky model for the full wavelength range, the temperature T sky derived from sky models assuming the approximation of pure black body radiation above 30 µm, and the estimated T est sky with the ambient temperature black body radiation and the atmospheric transmission curve only. leads to T sky used in the rest of the paper. This simplification slightly overestimates the radiation for very dry conditions and high sites. However, a complete model by us, using a custom patched version of the radiative transfer model (T full sky ), shows that the correction to the formula given above is marginal for PWV > 1.0 mm and even below. The discrepancies are of the order of only 0.2 K (Fig. 6).
T amb PWV T sky T Eq. (2) sky T est sky [ • C] [mm] [ • C] [ • C] [ • C]
Ambient meteorological conditions at Cerro Paranal
Methods and data selection
At Cerro Paranal there are several atmospheric site monitoring (ASM) systems installed which permanently record the atmospheric ambient conditions. We investigate data from a period of 4.5 years (2015-07-01T00:00:00 to 2019-12-31T23:59:59) based on the ESO Meteo Monitor, the L-HATPRO radiometer (Kerber et al. 2012), and the Differential Image Motion Monitor (DIMM). An overview on these data is given in Sarazin (2017). From the Meteo Monitor we extracted the wind direction, wind speed, and the ambient temperature T amb measured at a height of 30 meters (as in Section 2.2) since the telescope structure is more exposed to this part of the atmosphere than towards the ground (c.f. Section 2.1).
The L-HATPRO radiometer database provides integrated PWV measurements at zenith (θ = 0 • ) and measurements of the infrared sky temperature T IR achieved by a specific infrared radiometer mounted at the L-HATPRO, which operates in the 9.6 to 11.5 µm band. As mentioned in Section 2.1, the Earth's atmosphere is widely transparent in this wavelength range. Therefore, T IR measurements are very sensitive to (comparably warm) clouds at various heights. In particular, ice clouds (cirrus) can be detected, which is not possible with the microwave radiometer. The DIMM incorporates data on the seeing conditions and the sky transparency, the latter being measured by the relative rootmean-square (rms) of flux variations of reference stars along the line of sight during nighttime (normalized by the average flux).
We merged the data points of these three databases via their timestamps by finding the closest Meteo Monitor and DIMM neighbor to each individual radiometer measurement within ∆t ≤ ±300 s. By skipping those measurements with ∆t > 300 s, we automatically restrict our merged data set to nighttime because the DIMM only provides data taken between dusk and dawn. We also omitted data points with invalid values and those with PWV ≤ 0.02 mm.
The data set should represent typical observing conditions at Cerro Paranal, i.e., skipping bad weather conditions, which is important since the subcooling of the telescope structure (Section 3) is strongest at the best (= photometric) conditions. In addition, the ESO Sky Model does not cover cloudy conditions, thus Eq. (2) is not applicable. In order to exclude cloudy periods, we therefore apply a DIMM rms cut of 2% (which is consistent with the criterion for possible clouds of the online ASM tool at ESO). This cut reduces our sample by only about 10%, i.e., photometric conditions prevail at Cerro Paranal. In total we select 276 958 data points.
Statistics on the meteorological data
Investigations of the overall yearly variability reveal a strong trend towards higher PWV values in the southern summer months with an annual median value of PWV≈ 2.47 mm (10% percentile: 1.06 mm; 25% percentile: 1.56 mm; 75% percentile: 4.08 mm; 90% percentile: 6.39 mm). In addition, weather situations with winds coming from a 90 • cone centered on north are the most frequent ones (56% of the overall time), with a tendency towards higher wind speeds, higher PWV values, and more clouds. Since our data selection criteria exclude most of the bad conditions, the sensitivity of our investigations with respect to the wind direction is minimized.
Our empirical equation for T sky (Eq. 2) depends on PWV and the ambient temperature T amb . We assume these quantities to be independent. Our data set can be used to test this assumption. Figure 7 does not show a clear correlation. This finding can be tested by the Pearson coefficient r, which indicates whether a data set shows a linear correlation (r = ±1 means perfect linear correlation, r = 0 no linear correlation). Since the absolute value is small in our case (r = 0.16), we conclude that our parameterization of T sky is justified. The distribution of T sky , given in Fig. 8, Fig. 8. Histogram of the calculated T sky values. Its mean value is 246.2 K, the standard deviation σ = 5.7 K. reveals a median value of T sky = 246.2K and a standard deviation of σ = 5.7 K for the selected clear observing conditions. Figure 9 shows the infrared temperature T IR versus the sky temperature T sky determined with Eq. (2) (calculated by means of the T amb values from the ESO Meteo Monitor at θ = 0 • ). Since T IR has a relatively weak dependency on PWV but is strongly sensitive to even thin cirrus clouds (due to the small wavelength range used for the measurements), the infrared radiometer temperature is related to the sky temperature T sky only in case of photometric observing conditions and T sky 240 K. It is therefore not a good estimator for the sky temperature and its effect on the subcooling of telescope structures. In contrast, T sky derived by Eq.
(2) provides a robust estimate for that purpose only based on measurements of the PWV, the ambient temperature, and the zenith distance θ.
Ambient meteorological conditions at Cerro Armazones
In Section 3, we compute numerical quantities using typical environmental conditions at Cerro Armazones (3 060 m), the site of the ELT (Extremely Large Telescope) (Koehler 2015;Otárola et al. 2019). The median nighttime air temperature is T amb = 9.1 • C = 282.25 K and the median air pressure equals p = 712 hPa. PWV measurements like those with L-HATPRO at Cerro Paranal (Section 2.3.1) are not available for Cerro Armazones. Nevertheless, estimates by Otárola et al. (2010) and Lakićević et al. (2016) suggest that the median value for clear sky conditions at Cerro Paranal, i.e. 2.47 mm (Section 2.3.2), should be relatively similar. Possible systematic deviations of the order of half a millimeter are not critical for our purposes. Applying Eq. (2) for the selected PWV value, we obtain a difference between T sky and T amb of −39.9 K at zenith and −37.8 K at the typical observing zenith angle θ = 37 • . With the ELT median T amb , this results in T sky of 242.4 K and 244.5 K, respectively.
Otárola et al. (2019) presented a cumulative wind statistic for Cerro Armazones. However, that study was focused on radio astronomy and thus included daytime data. Fortunately, the complete weather data are publicly available 4 and we selected the nighttime wind data between 22:30 and 9:00 h UTC for the years 2006-2009. We find a nocturnal median wind speed at 7 meters above the ground of 7.0 m/s, 33% of the time the wind speed is below 5.2 m/s, 25% of the time below 4.3 m/s and 15% of the time below 3.0 m/s. From a related data set covering the years 2004-2008, we find that the median nocturnal temporal gradient of T amb equals −0.34 K/h when the temperature is falling and +0.25 K/h when it is rising (which is relevant for significantly shorter time periods). The corresponding quartiles are −0.17 and −0.60 K/h (falling) vs. +0.11 and +0.51 K/h (rising), respectively.
Thermal modeling of telescope structures
Radiative heat flux
In this section, we derive the steady-state temperature of a structure that radiatively cools against the night sky and is heated by forced convection of the ambient air. We consider a small surface of area A that has a spectral emissivity of (λ, γ), where γ is the angle between the surface normal n and the observing direction. The radiative heat flux from the sky that is absorbed by the surface can be written in spherical coordinates (centered at zenith) according to Lambert's Cosine Law (Pedrotti & Pedrotti 1993) aṡ
Q sky = A π/2 0 dθ sin(θ) 2π 0 dφ κ ∞ 0 dλ L sky (λ, θ, φ) (λ, γ),(3)
κ := max cos(γ), 0 ,
where L sky is the spectral radiance from the sky (unit W m −2 sr −1 m −1 ), γ = γ(θ, φ) and φ is the azimuth. Both θ and φ are measured in radians, in contrast to the previous section. If the surface is oriented horizontally (n points towards zenith), we find γ = θ and κ = cos(γ). Conversely, to calculate the emitted fluxQ surf from the surface to the sky, one simply replaces L sky (λ, θ, φ) in Eq. (3) by the black body spectral radiance
L bb (λ, T surf ) = 2 h Pl c 2 λ 5 exp h Pl c λ k B T surf − 1 −1 ,(5)
where T surf is the surface temperature, h Pl = 6.62618 × 10 −34 J s is Planck's constant, k B = 1.38065 × 10 −23 J/K is Boltzmann's constant and c = 2.99792 × 10 8 m/s is the speed of light in vacuum.
As shown in Figs. 4 and 5, the sky spectrum agrees with a black body fit spectrum within several wavelength ranges (e.g., near 7 µm or near 15 µm). The temperature of this fit spectrum is close to the ambient air temperature at high sites. The radiation imbalanceQ sky −Q surf is thus dominated by the wavelength bands where L sky (λ) − L bb (λ) has a large magnitude, specifically in the astronomical N-band (7 -13.5 µm) and the Q-band (17 -25 µm). The area between the spectra and their respective fits in Figs. 4 and 5 gives a graphical impression of this relationship.
The wavelength and angle resolved emissivity (λ, γ) is often unknown. Working with an average emissivity is possible, but it should be noted that its value is defined by the requirement to preserve the valueQ sky of the wavelength integral in Eq. (3). Any emissivity weighted by the full black body spectrum, as is sometimes provided by commercial measurement devices or listed in spec sheets, is not well suited to quantify subcooling. Instead, it is a better approximation to only consider the emissivity around λ = 11 µm and, if available, around 18 µm.
Sky view factor
The relation between spectral radiance and T sky is (Zhao et al. 2019)
σ T 4 sky = π/2 0 dθ sin(θ) cos(θ) 2π 0 dφ ∞ 0 dλ L sky (λ, θ, φ)(6)
= π E BOL , in accordance with Eq. (1), where the angular brackets denote the geometrical averaging over the sky hemisphere. Surfaces that are located inside buildings, such as a telescopes in a dome with open doors, can typically only see a solid angle Ω < 2π of the sky hemisphere. The sky view factor for a point q on the surface is defined by
F sky = 1 π π/2 0 dθ sin(θ) κ(θ, φ) 2π 0 dφ T (θ, φ),(7)
where the telescope transmission function T (θ, φ) equals 1 at directions θ, φ for which an unobstructed path from q to the sky exists and zero if the path is blocked by any warm black body. However, we also allow for partly obscured ray paths like propagation through a windscreen mesh or indirect ray paths to the sky through reflection on a mirror (often the primary mirror M1), or any partially reflective surface like a bare metal for which 0 < T < 1. The normalization factor π −1 ensures that 0 ≤ F sky ≤ 1. The sky view factor for a horizontal surface inside a building with a circular aperture centered at n that is seen from q under the one-sided angle θ a equals sin 2 (θ a ). This term assumes the value 1/2 at θ a = 45 • , although the solid angle of this aperture subtends only 29.3% of the hemisphere. If, on the other hand, the aperture is not centered at n, F sky will be smaller. This reasoning shows that the orientation of surfaces inside a telescope dome matters, not just the distance from the dome slit. The difference is particularly large when comparing the sky view factors for the top side of the spider trusses (with rectangular cross section) or the secondary mirror housing (F sky ≈ 0.8 -1) with its side faces where n is perpendicular to the telescope axis. Under no circumstances can the side faces see more than half of the sky hemisphere (F sky ≤ 1/2). Furthermore, the zenith angles θ at which the cosine weighting term κ is large are nearly perpendicular to the telescope axis; in fact, these rays may often point near the horizon (the side faces may even see part of the ground outside the telescope building). The average sky temperature is higher in these directions due to the θ 2.5 dependence; see Eq. (2). Fig. 10. Evaluation of sky view factors by projection. Red arrow: normal vector n above the point q on a surface. Blue quadrilaterals: two windows of solid angle Ω ≈ 0.16 sr through which the sky can be seen from q (i.e., T = 1); the right window is tilted versus n by γ = 60 • . Green rectangles: Projection of the windows onto the tangential plane with areas π F sky ≈ Ω and Ω × cos(60 • ) = Ω/2, respectively.
When dealing with non-circular apertures to the sky (e.g., the open door slit of a telescope dome) as sketched in Fig. 10, it can be more convenient to work with Cartesian coordinates x, y, z. The effect of κ in Eq. (7) is simply to project the area of the sky hemisphere that can be seen from the point q onto the unit disk on the tangential surface (perpendicular to n). If we let ρ = (x, y, z) be a vector of unit length that points from q to the sky and x, y, z are local coordinates so that any vector (x, y, 0) lies on the tangential surface and n = (0, 0, 1), we can write F sky = 1 π
x 2 +y 2 <1 dx dy T (x, y).
In buildings, the sky windows often have much more complicated shapes than in Fig. 10. One can numerically evaluate F sky in this case by describing the edges of the sky windows by a closed set of straight lines. The projected window then forms an (irregular) polygon on the unit disk whose area can be computed easily by the "shoelace formula" (Braden 1986). Dividing this area by π yields F sky . Any window edges whose rays ρ would point below the tangential plane, hence if γ ≥ 90 • and thus κ = 0, generate projected points that lie on the unit circle in this scheme. A collection of formulas to calculate F sky in special geometries is found in Chapter 10 of Lienhard (2011).
If a structure sees some part of the sky through a mirror, or in general through a series of reflective optics, F sky is defined by the ray bundle emanating from q that ultimately reach the sky, hence with T > 0. In other words, the effective aperture is the image of the real aperture (say, the rim of the dome slit) as seen from q. One can compute F sky by evaluating the angles γ of the ray segments between q and the first reflective surface. This method is correct even if the mirrors are powered (i.e., have nonzero curvature) by the principle of étendue conservation (Chaves 2017) and assuming that the sky is a Lambertian radiator (setting aside the dependence of T sky on θ).
Thermal balance equation
With the help of Eqs. (3 -7), we can write the radiative heat loss due to sky subcooling as Lienhard (2011)
Q sky −Q surf = A F sky σ T 4 sky T − T 4 surf < 0,(9)T surf = T amb + ∆T surf ,(10)
where T amb is the ambient air temperature inside the dome on the upwind side of the telescope structure and ∆T surf quantifies the surface subcooling. The expression T 4 sky T denotes the average sky temperature with the same transmission T as in Eqs. (7,8). As discussed in the previous subsection, this average geometrically depends on the patch of sky seen from q and the telescope pointing.
On the other hand, convection heats the surface at a rate oḟ
Q conv = A h (T amb − T surf ) = −A h ∆T surf > 0,(11)
where h is the interface heat transfer coefficient (in units of W K −1 m −2 ). In thermal equilibrium and neglecting thermal conduction, we require thaṫ
Q sky −Q surf +Q conv = 0,(12)
hence the heat fluxes are balanced. Our goal is to quantify the steady-state plate subcooling ∆T surf < 0 which satisfies
∆T surf = σ ω T 4 sky T − T amb + ∆T surf 4 ,(13)
ω := F sky h .
Expression (13) is a quartic equation with two real and two complex roots. One of the two real roots lies below −1300 K, so there is only one meaningful solution. We have calculated the latter in closed form, but the expression is rather complicated. Instead, we carry out a Taylor expansion to first order in ∆T surf about ∆T surf = δT . Solving for ∆T surf , we obtain
∆T surf = σ ω 1 + 4 T 3 0 σ ω T 4 sky T − T 3 0 T amb − 3 δT ,(14)
T 0 := T amb + δT.
In the remainder of this paper, we choose δT = −3 K so that we cover the typical range of subcooling in telescopes of about −6 K ≤ ∆T surf ≤ 0 with good accuracy. Following Lienhard (2011), we define the radiation heat transfer coefficient
h rad := 4 T 3 0 σ F sky ≈ 4.9 W K −1 m −2 × F sky ,(15)
and, rewriting Eq. (14), finally express ∆T surf as
∆T surf = η 1 + η ∆T D ,(16)∆T D := T 4 sky T 4 T 3 0 − T 0 − 4 δT 4 ≈ 0.89 ∆T sky ,(17)η := h rad h ,(18)
where the number η quantifies the subcooling efficiency. We thus find that ∆T surf scales linearly with η as long as η 1. We highlight that ∆T surf ∝ 1/h in this unsaturated regime, which applies to most cases of interest in telescope subcooling. At ELT median conditions (Section 2.4), η is of the order of 0.1 and the subcooling is driven by the temperature offset ∆T D that equals −33.9 K for θ = 37 • . In the opposite limit of η → ∞, we obtain ∆T surf → ∆T sky := T sky − T amb . In practice, this extreme can probably only be reached on a black surface facing the sky inside a vacuum vessel with a window that transmits infrared radiation.
Convective heat transfer
The thermal coupling of a structure to the ambient air through forced convection gives rise to a boundary flow problem (Martinez 2020b). The convective heat transfer coefficient h depends on the air speed v near the surface (which is often much lower than the wind speed outside the dome), the air pressure p and, to a large extent, on the shape of the structure. The structures closest to, or even inside, the telescope beam are usually steel trusses supporting the secondary and, possibly, the tertiary/quaternary mirrors. The simplest truss shape is cylindrical with diameter d. An approximate expression for h cyl , modeling the net heat exchange of airflow perpendicular to a cylinder, has been derived by Churchill & Bernstein (1977). Their original formula is expressed in terms of the Reynolds, Nusselt and Prandtl numbers (Martinez 2020a) and we have checked that the values of these numbers for typical telescope structures are within the validity range of the formula. We do not reproduce the general formula here but instead simplify it to
h cyl = c 1 √ β s 1 + c 2 β 5/8 4/5 ,(19)β := m s p v,
where c 1 = 0.0179 W K −1 m −1 , c 2 = 1.5 × 10 −4 for airflow at T amb = 282 K and s = π d is the cylinder circumference. We have introduced the Reynolds number scaling quantity m that equals 1.0 m W −1 if the incoming airflow is perfectly laminar and the cylinder has a smooth surface and m ≈ 1.5 m W −1 for partly turbulent inflow. However, if a precision of more than about 20% in h cyl is demanded, one has to resort to Computational Fluid Dynamics (CFD) simulations.
At v = 7.6 m/s and s = 0.6 π = 1.88 m, one reaches c 2 β 5/8 = 1. For much lower air speeds (more precisely, if β 1.3 × 10 6 ), the scaling law h cyl ∝ (m p v/s) 1/2 applies. We have to note that, although v = 7.6 m/s is close to the median wind speed on Cerro Armazones (cf. Section 2.4), the air speed inside the dome is often much lower. This holds in particular for structures below the M2 level, hence this regime is normally valid.
In the opposite limit of large β, e.g. for high air speed and/or large structures, h cyl ∝ β/s = m p v. The Reynolds number then becomes so large that the boundary layer is turbulent throughout, and thus the structure size is no longer relevant for h cyl . An example for this limit is subcooling of the outer dome cladding, where v approaches the free air wind speed and the interface length s can amount to tens of meters for streamlines that follow the dome surface. Figure 11 shows a plot of h cyl for three different values of s representative of structures in the VLT (Very Large Telescope, Paranal) and ELT telescopes and the ELT dome door (all for the ELT environmental conditions and m = 1.25 m W −1 ). Sundén et al. (2010) contains an overview of h for cylinders with non-circular cross sections. Figure 12 shows a plot of ∆T surf as a function of F sky according to Eq. (16), where the colors indicate four different local air speeds v. For these calculations and the remainder of this paper, we assume ELT median conditions as defined in Section 2.4 for the zenith angle θ = 37 • . The colored fans in Fig. 12 are meant to give an impression of the influence of the interface length s: As s grows from s = 0.4 π m = 1.26 m to s = 4.3 m, h cyl diminishes and thus η grows, hence the subcooling intensifies. Consequently, subcooling becomes more severe as telescopes grow. Figure 13 is a two-dimensional plot of T surf (F sky , v) for s = 4.3 m (ELT top ring truss circumference) and = 0.2, which is typical for dedicated low-emissivity paints; see Yarbrough (2010) for an overview of some respective commercial products. The tube structures of several large telescopes such as GTC (Gran Telescopio Canarias, La Palma), Keck I+II (Mauna Kea, Hawaii) and the two Gemini telescopes (Cerro Pachón, Chile and Mauna Kea) are covered with LO/MIT 5 paint by the company Solec (a silvery paint with small aluminum particles). On surfaces that scatter visible stray light, one may apply selective foil such as Acktar NanoBlack ®6 or Maxorb (Mason & Brendel 1982). Those are black at visible wavelengths but highly reflective and thus low in emissivity ( < 10%) in the thermal infrared. Combining Eq. (19) with Eq. (16), we find a simplified fit function near = 0.2 and F sky = 0.5 given by
∆T surf ≈ 0.636 ∆T sky F sky s 0.355 v −0.551 ,(20)
where s is entered in meters and v is in m/s and again ∆T sky = T sky − T amb . We found the power laws of s and v using the rule b = x f / f for a function f (x) = a x b with derivative f , applied at s = 0.6 π m = 1.88 m and v = 1 m/s.
Optical path difference
Temperature differences ∆T in an air volume cause the optical path length difference (OPD) of
OPD = dn dT ∆T q(l) dl,(21)
where q is a point in the air volume and l measures geometrical distance along the ray path. The refractive index of air n(λ, T, p) as a function of temperature T and pressure p is given by Edlen's modified formula (Bönsch & Potulski 1998). Its derivative by T can be written in simplified form as
dn dT = 1 ρ 0 dρ dT = − p ρ 0 R air T 2 ,(22)
where ρ = p/(R air T ) denotes the air density (R air = 287.06 J kg −1 K −1 is the gas constant of dry air, yielding ρ = 0.879 kg m −3 at ELT median conditions) and the constant ρ 0 = 4450 kg m −3 in the wavelength range 0.4 µm ≤ λ ≤ 3 µm. At ELT median conditions, we obtain dn/dT = −9.828 × 10 −12 K −1 Pa −1 × p that equals −7.0 × 10 −7 K −1 . For the ray path from the top of the ELT secondary mirror (M2) crown level to M1, then back to M2 and finally to the prime focus of total length 36 m + 31 m + 13 m = 80 m, an air cooling as small as ∆T = 25 mK causes an accumulated OPD of 1.4 µm, equivalent to about 0.64 λ in K-band (2 -2.4 µm). The thermal OPD is thus significant and may actually become the leading wavefront error contribution in 30+ meter class telescopes.
In the following, we establish the link between the subcooling temperature offset ∆T surf and the OPD. We rewrite the surface area in Eq. (11) as A = s L, where L is the length of a cylindrical truss that we assume to be oriented perpendicular to the local airflow vector v and perpendicular to the telescope optical axis, as would be the case e.g. for part of the top ring, as shown in Fig. 14. On the downwind side of the truss, a wake of cooled air is generated. We consider now a ray of light in a thin air slice perpendicular to v and assume for simplicity that the wake has a thickness along the ray of δ with constant temperature offset ∆T wake < 0. The total volume of cooled air flowing away Fig. 14. Slice through a cylindrical truss of diameter d = s/π running perpendicular to v and to the telescope axis. Red arrow: Ray of light traversing the cool wake of thickness δ and temperature difference ∆T wake . from the truss per time ∆t equals then V/∆t = δ L v and transports the heat flow
−Q conv = V ∆t c p ρ ∆T wake = c p ρ v L δ ∆T wake ,(23)
where c p = 1005 J kg −1 K −1 is the specific heat capacity of air at constant pressure. We can then derive the OPD imposed on the ray by the cooled air generated by a single truss of circumference s in a single ray pass by combining Eq. (23) with Eq. (11)
OPD = dn dT δ ∆T wake = dn dT s h ∆T surf c p ρ v .(24)
In reality, the wake is of course not isothermal, but instead the temperature distribution depends on the aerodynamics and is non-stationary. However, the actual temperature distribution along the ray is immaterial for the problem at hand since the path integral in Eq. (21) is a conserved quantity equal to δ ∆T wake when averaged over time and along the truss length L. In the absence of this spatial and temporal averaging, transient vortexes can distribute air pockets of different temperature in the plane orthogonal to the rays (Martinez 2020a) generating a transient wavefront error; however, this aspect is beyond the scope of this work.
When h rad h (e.g., with low-emissivity coatings and under most airflow conditions inside the dome) so that that ∆T surf is proportional to ∆T D /h, and since dn/dT is proportional to p and thus to ρ, we can simplify Eq. (24) to
OPD ≈ v 0 F sky s v .(25)
For ELT median conditions, v 0 ≈ 10 −7 m/s. We point out that the OPD does not depend on h: In steady state, the heat power lost by the radiation imbalance is exactly compensated by convective heat transfer from the inflowing air so that h cancels. Because the average air cooling is inversely proportional to the air mass passing the telescope structure per time, OPD ∝ v −1 . This fundamental relationship holds independently of the actual values of the structural subcooling ∆T surf that various parts of the telescope converge to in steady state and explains the name "Low Wind Effect" (Sauvage et al. 2015;Milli et al. 2018).
By the same argument, the OPD does not depend on air pressure either, thus telescopes at high altitude are in principle afflicted by the same magnitude of thermal OPD as at the ground level (actually, at high altitude the PWV column tends to be smaller, hence ∆T sky grows in magnitude and thus v 0 increases).
The structures in a telescope causing large OPD are those with high emissivity coatings (e.g., any regular industrial paint, irrespective of color) and those with large dimensions and/or high ratios F sky /v. The sky view factor F sky tends to be larger for structures closer to the dome slit than those near the primary mirror, but the same holds for v (Cullum & Spyromilio 2000). Moreover, we remind the reader that structures facing M1 may see a significant fraction of the sky in reflection. Therefore, it is not per se clear which telescope structures are the most critical in terms of causing subcooling OPD. Figure 15 shows a two-dimensional plot of the OPD as a function of F sky and v for a single ray pass through the wake of a single truss of circumference s = 4.3 m (ELT top ring) and = 0.2, again at ELT median conditions. The OPD from Fig. 15. Steady-state OPD from subcooling as a function of F sky and v for a single ray pass through the wake of a single truss of circumference s = 4.3 m and = 0.2 (with ELT median conditions). Contours: OPD in nanometers in increments of factors of 10 1/5 ≈ 1.585, starting at 10 nm. a single truss of typical size thus amounts to only a few tens of nanometers. Moreover, the OPD caused by e.g. the top ring may tend to spread rather uniformly across the pupil, in which case it would not strongly distort the wavefront of starlight. However, the situation is different for vertically oriented structures such as vertical trusses in the telescope tube or a tertiary/quaternary mirror tower. Assuming that the cold wake thickness is similar to the truss diameter δ ≈ d, the OPD for structures running parallel to the optical axis must be scaled by a factor up to the truss aspect ratio, for instance L/d = 22/0.6 ≈ 37 for one of the vertical ELT telescope tube trusses.
To estimate how the OPD from subcooling scales with telescope size in general, we assume that telescopes of increasing sizes are supported by trusses whose circumference s and length L in the direction parallel to the telescope axis both scale linearly with the primary mirror diameter D, causing the OPD from the wake of each truss to scale like D 2 . This relationship holds for the structure above the M1 level only, as long as we assume a roughly identical focal ratio of the primary mirrors. However, the tube structure of the upcoming generation of 30+ meterclass telescopes is becoming more delicate than that of the preceding 8-10 meter-class telescopes: The number of trusses increases, while each truss circumference grows only modestly. In the somewhat unrealistic limit that s does not grow at all and L rises proportionally with D, the OPD scales only linearly with D (but spreads more widely across the pupil). In practice, the scaling power law will lie somewhere in between these extremes.
The rays from a star at center field propagating through a rotationally symmetric telescope travel in a plane defined by the entry point of the ray in the pupil and the optical axis. The closer the entry point is to the central obscuration (CO), the closer the ray segments between reflections are to each other. The air volume that, in vertical projection, is close to the rim of the CO is thus traversed at least three times, boosting the OPD on the downwind side of the CO. Unfortunately, the M2 crown and the tertiary mirror tower introduce a lot of vertical surface area in this region with high F sky . Moreover, any vertically extended tertiary mirror tower and the M2 crown are optically conjugated to a large range of heights above the telescope, which can induce field-dependent errors in the adaptive optics correction. A quantitative analysis requires extensive CFD simulations of the entire dome volume with temperatures (Cho et al. 2011;Vogiatzis et al. 2014;Ladd et al. 2016;Oser et al. 2018;Vogiatzis et al. 2018).
Heat conduction
So far, we have neglected heat conduction within the structures, which will tend to equalize the structure temperature and thus spread the subcooling effect. In Appendix A, we derive the characteristic length scale due to heat conduction ξ with typical values of a few tens of centimeters in telescope trusses (see Table A.1).
The OPD is approximately proportional to F sky s as Eq. (25) shows. Therefore, it is fortunately equivalent to average the temperature around the perimeter of trusses when estimating the OPD in its wake, or, e.g. for rectangular cross sections, to sum the OPD contributions of the four side faces. However, one should not assume that the subcooling spreads evenly along the length of a truss, which typically far exceeds ξ.
Transient effects
Telescope structures have a significant thermal inertia that delays temperature changes. In Appendix B, we derive the characteristic thermal relaxation time scale τ with typical values of a few hours (see Table A.1).
Sky subcooling is thus a slow process that, in general, aggravates during the night. Note that any incorrect temperature conditioning of the telescope at sunset, including overheating (∆T surf > 0), will generate additional OPD, starting already at the beginning of the observations (Wilson 2013).
We point out that in contrast to the characteristic length ξ, the relaxation time τ is linearly dependent on w/h, where w is the truss wall thickness, and thus varies more strongly among different structures in a telescope, even if they are made of the same material. Additionally, the wind veers and the sky temperature drifts during the night, there is variable cloud coverage, the telescope pointing zenith angle varies etc., therefore sky sub-cooling is an inherently transient process and any steady-state computation is only an approximation.
A serious source of wavefront errors in some telescopes is "mirror seeing" (Wilson 2013), caused by ∆T surf on the optical mirror surface, which can sometimes become even more detrimental than "dome seeing" (Racine et al. 1991). However, mirror seeing is normally not linked to radiative cooling since typical mirrors have broadband coatings, hence very low emissivity such as < 2%. However, subcooling can become a problem if the uncoated backside of a secondary mirror is (partly) exposed to the night sky (the emissivity of typical mirror substrates made of Zerodur ® is ≈ 0.9).
To compound the complexity, the ambient air temperature T amb also fluctuates. During a typical night, T amb drops quickly after sunset and decreases later more slowly so that the radiation imbalance from the cold sky initially accelerates the thermalization of the telescope. Conversely, a rise in T amb exacerbates the subcooling. The median temporal gradients of T amb given in Section 2.4 are smaller than typical subcooling gradients of the order ofṪ amb ≈ −1.5 K/τ ≈ −1 K/h, but not negligible.
One can shorten the subcooling relaxation time, while lowering the emissivity and slowing down the structural deflection of the telescope structure, by covering the truss surfaces with a thermal insulation layer topped by aluminum foil or thin sheet metal; a strategy chosen at the Subaru Telescope on Mauna Kea, Hawaii (Bely 2006). Note, however, that applying an insulation layer on top of a large thermal mass does not diminish the subcooling (i.e., does not mitigate ∆T surf in the steady state) since the radiation imbalance on its surface is not reduced.
Conclusions
Subcooling of structures on the ground is caused by a thermal radiation imbalance between the night sky and a structure on the ground at ambient temperature which can reach 150 W m −2 and more. Structural elements in telescopes thus cool by a few Kelvin (see Eqs. (16,20)) and in turn cool the surrounding air. Wakes of cooled air flowing from elements in, or near, the telescope beam such as spider trusses, the top ring, the tertiary mirror tower etc. cause an Optical Path length Difference (OPD), introducing a transient wavefront error. While the size of modern telescopes keeps growing, the budgets of acceptable wavefront error are diminishing for diffraction-limited science cases. Therefore, subcooling poses a serious challenge to the generation of 30+ meter telescopes and second-generation instruments with adaptive optics correction in existing 10-meter class telescopes.
In summary, we find the following: • The integrated thermal sky radiation can be characterized by the effective bolometric sky temperature T sky . It is dominated by the lower atmosphere and thus T sky floats with the ambient air temperature T amb near the ground. Based on spectra from the Cerro Paranal Sky Model, we have derived a simple fit formula for T sky as a function of T amb , the Precipitable Water Vapor (PWV) column height, and the zenith angle θ; see Eq.
(2). The formula is valid for clear sky conditions, which are present at Cerro Paranal for about 90% of the time, and can be applied to any relatively dry site. For the investigated range of PWV values from 0.5 to 20 mm, the difference between T sky and T amb varies from −50 K to −25 K for θ = 0 • . The discrepancy can decrease by several Kelvin for higher zenith angles. • For Cerro Armazones, the site of the Extremely Large Telescope (ELT), we estimate a median T sky of about 244 K at a zenith angle of 37 • , which lies about 38 K below T amb .
• The OPD due to subcooling of a single truss of circumference s is proportional to F sky s/v, where denotes surface emissivity, F sky is the sky view factor averaged around the truss cross section and v is the local air speed; see Eqs. (24,25). Since large telescopes are generally supported by trusses with both, larger circumference and higher (vertical) length, the OPD scales between linearly and quadratically with the primary mirror diameter. • It is crucial to apply low-emissivity coatings to all telescope structures above the primary mirror level that are exposed to a large fraction of the sky. Good choices are either bare aluminum or paints with embedded aluminum particles. On surfaces that may scatter visible stray light, one can apply selective foil that is black in the visible but reflective in the thermal infrared such as Acktar NanoBlack ® or Maxorb. Conversely, regular industrial paint usually has high emissivity ( > 0.9), even if it is white. • One should maintain sufficient airflow in the telescope dome from sunset until the end of the night, e.g. by opening louvers. Because OPD ∝ F sky /v, the wavefront error from subcooling is not just caused by surfaces near the level of the dome slit, but also by any structures closer to the primary mirror if they are poorly ventilated and/or have high emissivity. • Special attention should be devoted to the outer dome cladding because of the combination of high sky view factor (F sky ≈ 1) and large surface area. It is advisable to use a low-emissivity coating, additionally with low solar absorptivity to limit daytime heating. Again, aluminum alloys or paints with aluminum pigments are good choices, while any color paints should be avoided. • The most critical areas in a telescope structure in terms of wavefront distortion due to subcooling tend to be the vertical ones (i.e., running parallel to the optical axis), in particular surfaces near the projected rim of the central obscuration and the lateral spider truss faces. • The ratio of heat capacity to convective efficiency of any telescope structure sets the timescale of thermal relaxation, but these quantities do not influence the steady-state subcooling temperature, as discussed in Section 3.7. The time to reach thermal equilibrium can be hours, hence subcooling often becomes more serious in the second half of the night. • Handheld commercial thermal cameras are not suitable to measure the bolometric sky temperature. These cameras normally use microbolometer detectors (Rogalski 2019) that are only sensitive in the wavelength range of about 8 -14 µm which largely overlaps with the astronomical N-band, thus a region with particularly high atmospheric transmission. On the other hand, if such cameras are employed to estimate subcooling of telescope structures, it is best to take the measurement at the end of the night directly after closing the dome to avoid measuring spuriously low temperatures due to partial sky reflection.
Fig. 4 .
4The sky radiance at zenith on the coldest and the warmest days at Cerro Paranal (2 600 m) in July 2018. The fit of a black body is based on the marked points only, defining an upper boundary.
Fig. 5 .
5Same as Fig. 4, but for Mauna Kea, Hawaii (4 200 m)
Fig. 7 .
7Two-dimensional histogram of the ambient temperature T amb as function of PWV for clear observing conditions (see text for more details). The bin widths of the histogram are 0.1 mm in PWV and 0.5 K in T amb . The resulting Pearson coefficient r is indicated (see text for more details).
Fig. 9 .
9T sky versus T IR at θ = 0 • as two-dimensional histogram (bin width of 0.5 K) for clear observing conditions.
Fig. 11 .
11Convective heat transfer coefficient h cyl versus local air speed v for three different interface lengths s representative of structures in the VLT and ELT telescopes, plus the ELT dome door.
Fig. 12 .
12Steady-state surface subcooling ∆T surf as a function of F sky for four different local air speeds v. The upper edges of the colored fans correspond to s = 1.26 m, the dashed lines to s = 1.88 m and the lower edges to s = 4.3 m, respectively.
Fig. 13 .
13Steady-state surface subcooling ∆T surf as a function of F sky and v for s = 4.3 m and = 0.2. Contours: Increments of −1 K; thick contour: Commonly accepted surface subcooling limit in telescopes of ∆T surf = −1.5 K.
Table 1 .
1Test of the extreme conditions at Cerro Paranal and at Mauna Kea: T sky obtained using Eqs. (1) and (2) as well as T est sky for the individual sky models from Figs. 4 and 5.
https://www.eso.org/observing/etc/skycalc/ 2 http://www.atm.ox.ac.uk/RFM/atm/ 3 https://www.ready.noaa.gov/gdas1.php
https://sitedata.tmt.org
https://web.archive.org/web/20170302132802/http://www.solec.org/ lomit-radiant-barrier-coating/lomit-technical-specifications/ 6 http://www.acktar.com/product/nano-black/ Article number, page 8 of 14 Holzlöhner, Kimeswenger, Kausch and Noll: Bolometric Night Sky Temperature and Subcooling. . .
Acknowledgements. W. Kausch is funded by the Hochschulraumstrukturmittel provided by the Austrian Federal Ministry of Education, Science and Research (BMBWF). S. Noll was financed by the project NO 1328/1-1 of the German Research Foundation (DFG). R. Holzlöhner wishes to thank Martin Brinkmann (ESO) for stimulating discussions on fluid dynamics and A. Otárola for help with the Cerro Armazones environmental data statistics. The authors further thank Jakob Vinther (ESO) for his continued support with SkyCalc.Appendix A: Heat conduction length scaleWe consider a plate and a pipe-shaped truss, both with wall thickness w and let y be the local surface tangent coordinate along the truss length L and x run along the orthogonal tangential direction. We can write the two-dimensional heat equation for the temperature field u(x, y) := ∆T surf (x, y) as − k m u xx + u yy = q sky − q surf + q conv (A.1)where u xx , u yy denote the respective second partial derivatives of u by x and y, q =Q/(A w) is the specific lateral heat flow per bulk wall volume (unit W m −3 ) and we have inserted the heat flow terms on the left-hand side of Eq. (12). In Eq. (A.1), we use Eq.(15)where now h rad = h rad (x, y), k m is the thermal conductivity of the truss wall material and we assume that the heat conduction normal to the surface is so efficient that the wall becomes isothermal along z, thus u(z) = const and k m /w h. We define the characteristic length ξ that quantifies the ratio of lateral heat conduction in the truss wall to convection and rewrite Eq. (A.1) aswhere we have assumed again h rad h as discussed after Eq. (15) and the radiative forcing term U rad (x, y) equals η ∆T D .We now consider two different simple structures: The first is a rectangular plate of dimensions L × s × w with L s w. We require that there be no heat flux across the plate edges, hence the temperature becomes stationary there. Further, the radiative forcing term contains a step function centered at x = 0, e.g. because the plate is bent into a L-shaped form with a 90 • corner running parallel to the long axis y where the sky view factor changes abruptly:where u x is the partial derivative of u by x and sgn(x) = 1 for x > 0, sgn(x) = −1 for x < 0 and 0 otherwise. The solution of Eq. (A.2) with the boundary conditions in Eq. (A.4) iswhere tanh, sinh and cosh denote the hyperbolic equivalents of the tan, sin and cos functions, respectively, and we have introduced the reduced variables x := x/ξ and s := s/(2ξ).The blue and red curves inFig. A.1show u(x), where s = π 0.6 m ≈ 1.88 m, which has the shape of an inverted letter S with its inflection point at (x = 0, u = U 0 ). For this plot we used the material properties of construction steel S355 as listed in Table A.1 and chose w = 10 mm, ∆U = −1 K and U 0 = −2 K. We assumed the two different conductive heat transfer coefficients h 1 = 6 W K −1 m −2 and h 2 = 3 W K −1 m −2 , corresponding to the scaling lengths ξ 1 = 0.27 m and ξ 2 = 0.39 m (blue and red curves, respectively).We can define a transition region width ∆x satisfying u x (0) ∆x/2 = ∆U, as indicated by the blue dotted line inFigBanyal & Ravindra (2011)where the last term is the Taylor expansion about the value s = 3 relevant for the presented examples. We conclude that the transition region can cover a significant fraction of a typical steel plate.The second structure is a pipe-shaped truss of diameter d = s/π with sinusoidal radiative forcing around its circumference instead of the step-like forcing, which could be a model of a vertical truss in the telescope tube, where F sky peaks on the side facing the telescope axis and becomes minimal on the opposite side oriented towards the inside of the dome. We modify the plate boundary conditions so that the problem becomes periodic in the circumferential coordinate x, as already analyzed by Joseph Fourier(1822)u(s/2) = u(−s/2), u x (s/2) = 0, U rad (x, y) = ∆U cos 2π j x/s + U 0 , (A.7)where j is any integer. As in the plate problem, U rad (x, y) lies between the extremes U 0 + ∆U and U 0 − ∆U, which are assumed A&A proofs: manuscript no. main now at x = 0 and x = ±s/2, respectively. The solution iswhich is plotted inFig. A.1in the turquoise and orange curves for j = 1 and s = 0.6 π m = 1.88 m and again setting ξ 1 = 0.27 m and ξ 2 = 0.39 m, respectively (since u(x ) is periodic in the pipe case, we have chosen to shift it along x so that its inflection point lies at x = 0 in the plot, as for the plate solution). The factor Ψ j compresses the amplitude of the sinusoidal temperature variation around the pipe, in particular if s < 2π j ξ. Note that we can approximate any periodic boundary condition by a Fourier series, hence a linear combination of solutions in Eq. (A.9) with different spatial frequencies j.If a plate is thermally insulated on one side, such as can be the case for the outer dome cladding, the convective heat exchange on the insulated side is basically turned off. Conversely, for plates whose two side faces are exposed to the ambient air, the interface surface area A is twice that of the insulated case and F sky should be averaged over the two sides. However, in such a case the average F sky cannot exceed the value 1/2 due to geometric constraints, hence the product F sky A may in some cases not differ strongly from the insulated plate case.Inside a sealed air-filled structure such as a pipe truss, the enclosed air exchanges heat energy with the internal walls and there is also internal thermal radiation. However, since the internal airflow is usually not forced and since the wall temperature variation is normally too small to induce a significant radiation imbalance, we can neglect these factors.Summarizing the plate and the pipe examples, we find that u(x) becomes compressed if the characteristic length ξ = (w k m /h) 1/2 is large compared to s, hence if lateral conduction dominates over convection. Irrespective of this condition, the average of u(x) around the truss circumference always equals the average of U rad , thus U 0 = −2 K in our two examples. The limit s 2π ξ may be called the isothermal regime, while the opposite limit s 2π ξ may be thought of as thermally disconnected. For a typical telescope pipe steel truss with a wall thickness of w = 10 mm, neither limit clearly applies around its circumference. However, the thermally disconnected regime applies along a typical length L 2π ξ.Appendix B: Thermal inertiaUntil now, we have discussed only the steady-state solution. We now finally consider the full time-dependent heat equation for a three-dimensional body in air with temperature offset u = u(x, y, z, t) := T body (x, y, z, t) − T amb (t) with time t c m ρ m u t +Ṫ amb −k m u xx + u yy + u zz = q sky −q surf +q conv , (B.1) where c m and ρ m are the specific heat capacity and the density of the body material, respectively.We can consider the pipe problem of the previous section with j = 1 and the initial condition u(x, y, z, 0) = 0 and, setting u zz =Ṫ amb = 0, find the solutionas a function of the reduced time t = t/τ with the lateral thermal relaxation timeWe now evaluate the time t = τ after which the term (1 − exp(−t/τ)) has grown from zero to 1 − 1/e ≈ 0.63. We assume again the material properties of construction steel S355 listed inTable A.1 and w = 10 mm to arrive at τ 1 = 1.64 hours and τ 2 = 3.27 hours for h 1 = 6 W K −1 m −2 and h 2 = 3 W K −1 m −2 , respectively.Table A.1 shows an overview of some material properties applicable to telescopes. We do not provide an emissivity value for S355 steel since it depends strongly on the surface preparation and regular steel is normally anyway coated to protect from rust. The aluminum alloy 5754 is a candidate for the ELT outer dome cladding ( = 6% corresponds to a clean and 20% to a contaminated surface, respectively).Zerodur is a low-expansion glass ceramic used for various telescope mirrors. The thermal conductivity of Zerodur and other glasses and ceramics is often so low that the relation k m /w h no longer holds for mirrors with typical thickness values of w = 50 mm (ELT M1 segments) to hundreds of millimeters. As a consequence, transient thermal phenomena become threedimensional problems(Banyal & Ravindra 2011)with the relaxation time along z of τ z = w 2 c m ρ m /k m .
. R K Banyal, B Ravindra, New Astron. 16328Banyal, R. K. & Ravindra, B. 2011, New Astron., 16, 328
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[] |
[
"In-situ observation of a soap film catenoid -a simple educational physics experiment",
"In-situ observation of a soap film catenoid -a simple educational physics experiment"
] |
[
"Masato Ito \nDepartment of Physics\nAichi University of Education\n448-8542KariyaJAPAN\n",
"Taku Sato \nDepartment of Physics\nAichi University of Education\n448-8542KariyaJAPAN\n"
] |
[
"Department of Physics\nAichi University of Education\n448-8542KariyaJAPAN",
"Department of Physics\nAichi University of Education\n448-8542KariyaJAPAN"
] |
[] |
The solution to the Euler-Lagrange equation is an extremal functional. To understand that the functional is stationary at local extrema (maxima or minima), we propose a physics experiment that involves using soap film to form a catenoid.A catenoid is a surface that is formed between two coaxial circular rings and is classified mathematically as a minimal surface. Using soap film, we create catenoids between two rings and characterize the catenoid in-situ while varying distance between rings. The shape of the soap film is very interesting and can be explained using dynamic mechanics. By observing catenoid, physics students can observe local extrema phenomena. We stress that in-situ observation of soap film catenoids is an appropriate physics experiment that combines theory and experimentation.
|
10.1088/0143-0807/31/2/013
|
[
"https://arxiv.org/pdf/0711.3256v5.pdf"
] | 53,681,173 |
0711.3256
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bb7d1f7053187a9fda7a1b013736019848ad9b33
|
In-situ observation of a soap film catenoid -a simple educational physics experiment
22 Dec 2009
Masato Ito
Department of Physics
Aichi University of Education
448-8542KariyaJAPAN
Taku Sato
Department of Physics
Aichi University of Education
448-8542KariyaJAPAN
In-situ observation of a soap film catenoid -a simple educational physics experiment
22 Dec 2009arXiv:0711.3256v5 [physics.class-ph]
The solution to the Euler-Lagrange equation is an extremal functional. To understand that the functional is stationary at local extrema (maxima or minima), we propose a physics experiment that involves using soap film to form a catenoid.A catenoid is a surface that is formed between two coaxial circular rings and is classified mathematically as a minimal surface. Using soap film, we create catenoids between two rings and characterize the catenoid in-situ while varying distance between rings. The shape of the soap film is very interesting and can be explained using dynamic mechanics. By observing catenoid, physics students can observe local extrema phenomena. We stress that in-situ observation of soap film catenoids is an appropriate physics experiment that combines theory and experimentation.
Introduction
The solution to the Euler-Lagrange equation is an extremal functional, which is stationary at local extrema (maxima or minima). For example, in classical mechanics [1], the functional (Lagrangian) for any given system leads to Newton's law of motion. However, physics students may have difficulty recognizing the behaviour of extremal functionals in physics experiments. Although we can observe some physical phenomena that are solutions to the Euler-Lagrange equation, the observation of local extrema is not easy. We therefore propose an accessible experiment that allows students to observe local extrema of a minimal surface, which is a solution to the Euler-Lagrange equation when choosing surface area as the functionals [2,3,4,5,6,7].
We stress that experimenting with the minimal surface formed by soap film is very interesting. In particular, the shape of a soap film inside a wire frame minimizes the surface area. The surface is classified mathematically as a minimal surface and, from the view point of dynamics and according to the principle of least action, its shape results from minimizing the surface energy of the soap film.
In the proposed experiment, we observe the dynamic behaviour of a catenoid, which is the shape a soap film takes between two coaxial rings. The catenoid can be explained using dynamic mechanics and differential geometry. We propose that in-situ observation and mathematical analysis of a catenoid soap film is suitable for an educational curriculum in experimental physics.
This article is organized as follows. In Sec.II, we describe the mathematical properties of the minimal surface and analyse the the shape of the catenoid. In Sec.III, we explain how to observe the catenoid and present some results of in-situ observation of a catenoid. In Sec.IV, we interpret the behaviour of the catenoid using dynamical mechanics. Finally, we give a brief summary and conclude in Sec.V.
Mathematics of Catenoid
In this section, we explain the mathematics of a minimal surface. For a given boundary, the minimal surface is that which has an extremal area [2,3,5]. If a minimal surface is parametrised in a three-dimensional orthogonal coordinate system (x, y, f (x, y)), the Euler-Lagrange equation leads to
1 + f 2 y f xx + 1 + f 2 x f yy − 2f x f y f xy = 0 ,(1)
where f i denotes the partial derivative with respect to index i. Because the partial differential Eq. (1) is rather complicated, it is difficult to solve. However it can be easily solved by imposing some assumptions (e.g., symmetry or boundary condition). Some well-known minimal surfaces include the catenoid, Helicoid, Scherk's surface, Enneper's surface, etc [2,3,8,9,10]. In this study, we focus on the catenoid, which is the minimal surface between two coaxial circular rings. If the minimal surface described by Eq. (1) is restricted to a surface of revolution with axial symmetry, we obtain the equation of a catenoid [2,3,8,9,10]. As shown in Fig. 1, a catenoid is a surface of revolution generated from a catenary curve with radius
r(z) = a cosh z a ,(2)
where a is the minimum value of r at z = 0. In the appendix, we derive Eq. (2) using the Euler-Lagrange equation.
In Fig. 2, the catenoid is described by three parameters h, R and a, where h is the distance between two coaxial circular rings, R is the radius of the equal-sized rings and a is the neck radius. According to Eq. (2), three parameters h, R and a are related by
h = 2a cosh −1 R a .(3)
As indicated by the right-handed graph of Fig. 2, the shape of a catenoid depends on the ratio h/R. For h/R > 1.33, it is not possible to form a catenoid, so the critical distance h c between rings is h c ≃ 1.33R. For any given h < h c , two catenoids are possible. Figure 2 shows that the neck radius of one catenoid (thick neck) is larger than the neck radius of other catenoid (thin neck). However, only one catenoid can physically exist, because the two catenoids correspond to extremal surfaces with a local maximum or a local minimum, we must compare the surface areas of two catenoids to determine which can exist. Using differential geometry, we obtain the following expression for the surface area S of the catenoid as a function of a:
S = 2πa 2 cosh −1 R a + 1 2 sinh 2 cosh −1 R a .(4)
Using Eqs. (3) and (4), the graph of S and h are depicted in Fig. 3 by Mathematica 5.1. As shown in Fig. 3, the surface area of the thick-neck catenoid is smaller everywhere than that of the thin-neck catenoid, so the surface area of the thick-neck catenoid is the absolute minimum for all values of h. Thus, the thick-neck catenoid is dynamically stable and the thinneck catenoid is unstable. In addition, we must consider the area of the disk inside the ring. Thus, we added a horizontal dashed line in Fig. 3, which corresponds to the surface area of the two disks. When the surface area of a stable catenoid equals the surface area of the two disks, the rings are separated by h 0 ≃ 1.05R. We are interested how the shape of the catenoid behaves as a function of the ring-separation h. Since the absolute minimum surface area is selected, the surface area of catenoid must always have the absolute minimum value. It is likely that a majority of physics students agree with this idea. From the viewpoint of classical mechanics, it seems that some physical phenomena in the natural world are described by functionals with absolute minimum.
From the viewpoint of the absolute minimum, we can predict the catenoid shape while gradually increasing h. As shown in Fig. 3, for 0 < h < h 0 , a stable thick-neck catenoid exists. For h 0 < h < h c , two disks exist. At h = h 0 , the stable catenoid jumps to the two disks. Thus, we expect that the stable catenoid separates in two and transfers to the two rings.
To confirm if the idea is correct, we performed the soap-film experiment. By increasing the distance between the two support rings, we can determine if the soap film changes from a stable catenoid to two rings at h = h 0 . In the next section, we show the results of this experiment.
in-situ observation of a catenoid
In the middle of the 19th century, physicist J. Plateau proposed that minimal surfaces could be visualized using soap film. To see the minimal surface for a given arbitrary boundary, it sufficed to soak a wire frame with its boundary in soapy water [3]. The soap film occupies the minimal surface to help minimize the surface tension. This demonstration, which combined mathematics and physics, was a great achievement by J. Plateau.
To observe the shape of the soap film, we used a familiar slide caliper with circular rings attached to the tips of the caliper (Fig. 4) The device enabled us to measure simultaneously both the distance h between the two rings and the neck radius a.
The experimental procedure is as follows. As indicated in caption of Fig. 5, when h = 0, soapy water is introduced between the rings. Next, by pulling a string attached to the movable part of the slide caliper, we can observe the formation of the soap-film catenoid. While pulling the string, the detailed behaviour of catenoid is recorded by a video camera with motion capture (1/120 sec per frame) mounted above the caliper (Fig. 6). Using this apparatus, we observe the catenoid in-situ.
While increasing the distance h, we recorded sequential images from the formation to the rupture of the catenoid, which are shown in Fig. 6. By analysing these photographs, we can measure both the distance h and neck radius a to generate the graph shown in Fig. 7.
By analyzing photographs, the graph as shown in Fig. 7 can be depicted. Starting from h = 0 (a = R), the shape of the soap film varies along the theoretical curve of a stable catenoid, as shown in Fig. 2. As mentioned in Sec. II, the graph of Fig. 3 predicts that the stable catenoid transforms into two disks at h = h 0 . However, in the experiment, the jump from the stable catenoid to two disks does not occur. Furthermore, as the distance h approaches the critical distance h c = 1.33R, the catenoid exhibits strange behaviour. Just before the critical point, we observe for an instant the unstable catenoid. The photograph of the unstable catenoid is shown in panel II of Fig. 7. Immediately, the unstable catenoid collapses, following which the soap film transforms into two disks.
A plot of the actual transition of the soap film is shown in Fig. 8. For 0 < h < h 0 , soap film occupies the stable catenoid, which is the absolute minimum for the surface area. When h = h 0 , the surface area of the stable catenoid is equal to that of the two disks. For h 0 < h, the surface area of the stable catenoid is greater than that of the two disks, which implies that the stable catenoid corresponds to a local minimum. For h 0 h h c , the stable catenoid corresponding to a local minimum exists. Finally, in the neighbourhood of h ∼ h c , the stable catenoid transforms into the unstable catenoid, then immediately collapses and transforms into two disks.
To summarize, the actual sequence followed by the shape of the soap-film is, stable catenoid (absolute minimum)
(i) → stable catenoid (local minimum) (ii) → unstable catenoid (local maximum) (iii) → rupture (iv)
→ two disks (absolute minimum).
Using a high quality video camera, we have observed the interesting behaviours of the soap film. In the next section, we interpret the behaviour not by kinematic analysis but by dynamic analysis.
Dynamic interpretation of catenoid
To explain the steps (i)-(iv) of (5), we must consider the surface energy V (x) of a soap film suspended two rings. As shown in Fig. 9, the coordinate x is measured mid-way between the two rings when h is fixed. By taking into account (5), we can imagine the form of V (x). Figure 8 indicates that the surface potential energy V (x) has three extremal points. If the soap film occupies a stable catenoid or the two disks, then the surface energy has a local minimal value [8]. The soap film of the unstable catenoid corresponds to a surface potential energy with the local maximum value. The illustration of surface energy V (x) is depicted in Fig. 9. Steps (i)-(iv) of (5) correspond to the transitions shown in Fig. 9 between surface energy curves.
Because the dynamic behaviour depends on the form of the surface potential energy, we can see the form of V (x) by observing the soap film. The circles in Fig. 9 correspond to the position x of the soap film (i.e. x corresponds to the neck radius a). The transitions (i)-(iv) of Fig. 9 are interpreted below.
(i) When h = 0, V (x) = 0. At this point, the soap film cannot form a membrane. Because the neck radius a = R, the circle in Fig. 9 is located at x = R, which corresponds to an absolute minimum of V (x). Upon increasing h, the catenoid forms and the absolute minimum point is raised because the formation of the soap film increases the surface energy V . The soap film shape is slightly perturbed because the string is pulled by hand to increase h, and this perturbation gives rise to fluctuations around the extremal point. However, despite of these fluctuations, the stable catenoid is in a dynamically stable state because it occupies an absolute minimum point on the surface energy curve. As h approaches h 0 , the surface area of stable catenoid approaches that of the surface area of the two disks. When h = h 0 , the surface energy of the stable catenoid and the two disks is the same, and both are absolute minimum values.
(ii) Increasing h increases the absolute minimum point corresponding to the stable catenoid. For h 0 < h < h c , the corresponding circle is in a local minimum point. In spite of the perturbations, the jump from stable catenoid to two rings does not occur because the potential barrier separating the local minimum from the absolute minimum at x = 0 prevents the soap film from transferring to the absolute minimum point which corresponds to two disks. This analysis reveals that kinematic analysis of Sec. II is not correct.
(iii) As h approaches h c , the local minimum point in V (x) attains the height necessary to overcome the potential barrier. Because of the perturbation, at h h c the circle can cross over the local maximum point corresponding to the unstable catenoid, which we succeeded in photographing (Fig. 7). In Fig. 9, this situation corresponds to the circle rolling over the top of a hill. Using a high quality video camera, it is possible to record the catenoid going over this local maximum in the surface energy.
(iv) The catenoid collapses after crossing over the maximum point, and the soap film transfers to the two rings corresponding to the absolute minimum point a x = 0. Referring to Fig. 9, the interpretation is that the circle rolls down a steep potential-energy hill.
Thus, the in-situ observations of a catenoid formed by soap film are completely explained by this dynamic interpretation.
At this point, two comments are in order. First, the concentration of soapsuds is not accounted for in the analysis. Secondly, the speed at which the rings are separated is not taken into consideration. In this experiment, a precise measurement of separation speed was difficult. Because we took care to keep the separation speed slower than competing processes (e.g., catenoid collapse), we consider that the speed does not change the essentials of the interpretation of the phenomenon.
Summary
We demonstrate in this article that the in-situ observation of soap film is suitable as an educational physics experiment. In the experiment proposed here, students can observe various interesting phenomena connected with local extrema. The film behaviour is interpreted using dynamic mechanics. Through the experiment and the analysis, students have the opportunity to study many fields, including differential geometry, minimal surfaces, and dynamical mechanics. The experimental apparatus is inexpensive and only common materials are required. We thus recommend the in-situ observation of soap-film catenoids as an educational physics experiment.
By integrating it, we get
r √ 1 + r ′2 = a ,(9)
where a is constant. Thus, we can obtain catenoid equation
r(z) = a cosh z − C a ,(10)
where C is constant. The case of C = 0 is Eq. (2).
Figure 1 :
1A catenary curve with the radius r(z) and the minimum radius a labelled (left panel). A catenoid is made by rotating the catenary curve around the z axis (centre panel). A three-dimensional schematic of a catenoid (right panel, created by using Mathematica 5.1).
Figure 2 :
2Drawing of a catenoid showing the three parameters (the distance h between the two circular rings, the radius R and the neck radius a) on which the catenoid shape depends (left panel). The graph of Eq. (3) is shown (right panel). The point C corresponds to the maximum inter-ring separation.
Figure 3 :
3The surface area S plotted as a function of the distance h between the two rings. The distance corresponding to point C is h c ≃ 1.33R. The intersection between the stable catenoid curve and the horizontal dashed line corresponding to two disks occurs at h 0 ≃ 1.05R.
Figure 4 :
4Photo of a slide caliper with two circular rings making a soap-film catenoid.
Figure 5 :
5Schematic of catenoid formation process. Soap water is introduced between the rings when h = 0 (left panel). The soap-film catenoid forms by pulling a string attached to the movable part of the caliper (right panel). The catenoid shape is recorded by a video camera mounted over the slide caliper.
Figure 6 :
6Sequential photographs showing the disappearance of the catenoid. The distance between two rings increases from left to right.
Figure 7 :
7The dashed curve and the solid curve in the left panel correspond to the unstable catenoid and the stable catenoid, respectively. The dots indicates data acquired from the video images. In the right panel, photographs corresponding to points I,II,III from the left panel are shown.
Figure 8 :
8Trace of the actual transition of soap film (bold line with arrow in left panel and dotted lines in insect). In the neighbourhood of the critical point, the stable catenoid transforms for a moment into the unstable catenoid, following which it vanishes and the soap film forms two disks. The insect shows a magnified view of this transformation process.
Figure 9 :
9The surface energy V (x) of soap film for various values of the parameter h. Here x represents the minimum catenoid radius. The circles indicate the positions occupied by the film. Equation (5) corresponds to the transitions (i)→(ii)→(iii)→(iv).
AcknowledgementsThe authors gratefully acknowledge the financial support of Aichi University of Education.Appendix: Catenoid equationWe derive the catenoid equation of Eq. (2). The cylindrical coordinate in z-axial symmetry is represented by r = (r(z) cos φ, r(z) sin φ, z) inFig. 1, where φ is the angle around z-axis. The surface area of revolution around z-axis is given bywhere ′ = d dz . We can read off the Lagrangian L = r √ 1 + r ′2 from Eq. (6), then the Euler-Lagrange equation leads to d dzFurthermore, Eq. (7) can be simplified as follows
H Goldstein, Classical Mechanics. Addison-Wesley Series in PhysicsH. Goldstein, Classical Mechanics (Addison-Wesley Series in Physics, 1980)
A Survey of Minimal Surface. Robert Osserman, Dover phoenix editionsRobert Osserman, A Survey of Minimal Surface (Dover phoenix editions, 1986).
John Opera, The Mathematics of Soap Films: Explorations with Maple. American Mathematical SocietyJohn Opera, The Mathematics of Soap Films: Explorations with Maple (American Mathematical Society, 2000)
Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces. Richard Courant, INC. Dover PublicationsRichard Courant, Dirichlet's Principle, Conformal Mapping, and Minimal Sur- faces (Dover Publications, INC., 1950)
S Kobayashi, K Nomizu, Foundations of Differential Geometry. 1S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol.1 and 2 (Interscience Tracts in Pure and Applied Mathematics, 1969)
Lectures of the Calculus of Variations (READ BOOKS. O Bolza, O. Bolza, Lectures of the Calculus of Variations (READ BOOKS, 2007)
Bruce Van Brunt, The calculus of variations. New YorkSpringer-VerlagBruce Van Brunt, The calculus of variations (Springer-Verlag, New York, 2004)
Stability and oscillations of a soap film: Analytic treatment. Loyal Durand, American Journal of Physics. 49Loyal Durand, "Stability and oscillations of a soap film: Analytic treatment", American Journal of Physics, 49, 334-343(1981).
Computing minimal surfaces via level set curvature flow. David L Chopp, Journal of computational Physics. 106David L. Chopp, "Computing minimal surfaces via level set curvature flow", Journal of computational Physics, 106, 77-91(1993).
Shapes of embedd minimal surfaces. T H Colding, W P Minicozzi, I I , Proceedings of the National Academy of Sciences. the National Academy of Sciences103T. H. Colding and W. P. Minicozzi II, "Shapes of embedd minimal surfaces", Proceedings of the National Academy of Sciences, 103, 11106-1111(2006).
|
[] |
[
"Robust estimation of SARS-CoV-2 epidemic at US counties",
"Robust estimation of SARS-CoV-2 epidemic at US counties",
"Robust estimation of SARS-CoV-2 epidemic at US counties",
"Robust estimation of SARS-CoV-2 epidemic at US counties"
] |
[
"Hanmo Li ",
"Mengyang Gu ",
"Hanmo Li ",
"Mengyang Gu "
] |
[] |
[] |
The COVID-19 outbreak is asynchronous at US counties. Mitigating the COVID-19 transmission requires not only the state and federal level order of protection measures such as social distancing and testing, but also public's awareness of the timedependent risk and reactions at county and community levels. We propose a robust approach to estimate the heterogeneous progression of SARS-CoV-2 at all US counties having no less than 2 COVID-19 associated deaths, and we develop the daily probability of contracting (PoC) SARS-CoV-2 for a susceptible individual to quantify the risk of SARS-CoV-2 transmission in community. We found that shortening only 5% of the infectious period of SARS-CoV-2 can reduce around 39% (or 78K, 95% CI: [66K , 89K ]) of the COVID-19 associated deaths in the US as of 20 September 2020. Our findings also indicate that the reduction of infection and deaths by shortened infectious period is more pronounced for areas with the effective reproduction number close to 1, suggesting that testing should be used along with other mitigation measures, such as social distancing and facial mask wearing, to reduce the transmission rate. Our deliverable includes a dynamic county-level map for local officials to determine optimal policy responses, and for public to better understand the risk of contracting SARS-CoV-2 on each day.
| null |
[
"https://arxiv.org/pdf/2010.11514v1.pdf"
] | 233,878,298 |
2010.11514
|
b86322fc82b52ee876bcf7347c6e410696fae202
|
Robust estimation of SARS-CoV-2 epidemic at US counties
Hanmo Li
Mengyang Gu
Robust estimation of SARS-CoV-2 epidemic at US counties
The COVID-19 outbreak is asynchronous at US counties. Mitigating the COVID-19 transmission requires not only the state and federal level order of protection measures such as social distancing and testing, but also public's awareness of the timedependent risk and reactions at county and community levels. We propose a robust approach to estimate the heterogeneous progression of SARS-CoV-2 at all US counties having no less than 2 COVID-19 associated deaths, and we develop the daily probability of contracting (PoC) SARS-CoV-2 for a susceptible individual to quantify the risk of SARS-CoV-2 transmission in community. We found that shortening only 5% of the infectious period of SARS-CoV-2 can reduce around 39% (or 78K, 95% CI: [66K , 89K ]) of the COVID-19 associated deaths in the US as of 20 September 2020. Our findings also indicate that the reduction of infection and deaths by shortened infectious period is more pronounced for areas with the effective reproduction number close to 1, suggesting that testing should be used along with other mitigation measures, such as social distancing and facial mask wearing, to reduce the transmission rate. Our deliverable includes a dynamic county-level map for local officials to determine optimal policy responses, and for public to better understand the risk of contracting SARS-CoV-2 on each day.
Introduction
The outbreak of new coronavirus 2019 (COVID-19) has caused nearly 200,000 deaths in the US, and among those, there are 2,277 counties with no less than 2 associated deaths as of 20 September 2020 1 . The ongoing COVID-19 pandemic has led to unprecedented non-pharmaceutical interventions (NPIs), including travel restrictions, lockdowns, social distancing, facial masks wearing, and quarantine to reduce the spread of SARS-CoV-2 in the US. The COVID-19 outbreak is prolonged and asynchronous across regions. It is thus of critical importance to estimate the dynamics of COVID-19 to determine appropriate protective measures before the availability of effective vaccine.
A non-negligible proportion of SARS-CoV-2 infectious individuals is asymptomatic or have mild symptoms 2 . We term the individuals the active infectious individuals who can transmit the disease to others, but may not be diagnosed yet. Identifying the number of active infectious individuals is crucial to monitor the transmission in a community. Another important timedependent quantity is the expected number of secondary cases resulted from each active infectious individual, or effective reproduction number. In this article, we estimate these two time-dependent quantities for all US counties with no less than 2 COVID-19 associated deaths; the population of some counties that falls within this category is even less than ten thousand. Furthermore, based on these two time-dependent quantities, we develop a more interpretable measure, called the daily probability of contracting (PoC) SARS-CoV-2 for an individual at county-level. The fine-grain estimation of disease progression characteristics allows public to understand the risk of contracting COVID-19 on a daily basis.
Predictive mathematical models are useful for analyzing an epidemic to guide policy responses 3 . The epidemiology compartmental models such as SIR, SEIR, SIRD, and their extensions 4-7 , stochastic agent based models 8,9 , branching processes 10 , and network analysis 11 have advanced our understanding of transmission rates and incubation period of SARS-CoV-2, which are connected to the traffic flow and mobility during the COVID-19 outbreaks at different regions 12,13 . The disease progression characteristics, such as the transmission rate, are often estimated based on the daily death toll 4, 7-9 . The increase of death associated with COVID-19 in small regions, however, is often temporally sparse, making it challenging to robustly estimate the progression of the epidemic in most US counties. Meanwhile, the COVID-19 observed confirmed cases (henceforth, observed confirmed cases) from the test data may significantly underestimate the population that have contracted the SARS-CoV-2. It was found in 14 that around 9.3% of the US individuals (or roughly 30 million) may have contracted the COVID by July 2020 based on serology tests, whereas less than 4.8 million COVID-19 positive cases have been confirmed in the US prior to August 2020 1 . The significant difference between the estimated prevalence and the number of observed confirmed cases in the US thus requires an estimate of the number of individuals who contracted COVID-19 but have not been tested positive. The focus herein is on integrating the death toll and test data to obtain a robust estimation of the disease Figure 1. a, The SIRDC model and the data used for analysis. b, c 7-day death toll forecast and 21-day death toll forecast against the held-out truth in 2,277 US counties with no less than 2 deaths as of 20 September 2020. Each dot is a cumulative death toll for one county at one held-out day. The counties from the same state are graphed with the same color.
progression characteristics of COVID-19 at county and community levels.
One critical quantity to evaluate an infectious disease outbreak is the time-dependent transmission rate, based on which one can compute the basic reproduction number and the effective reproduction number of the disease. Various approaches were proposed to estimate this parameter. The transmission rates are modeled as a decreasing function of the time in 4 , a function of NPIs in 8 and a geometric Brownian motion in 15 . Unlike the outbreak in China or other northeastern countries in Asia, the transmission rates of the COVID-19 progression in the US does not monotonically decrease due to the prolonged duration of the outbreak, and it is challenging to determine a suitable parametric form of this parameter in terms of time. In 7 , the transmission rate parameter was related to the initial values of infectious cases, resolving cases and up to two derivatives of the daily death toll. This method provides a flexible way to estimate the time-dependent transmission rate from the death toll and its derivatives; however, the method is unstable for a county with a moderate or small population size, as the numerical estimation of the derivatives of daily death toll is often unstable.
In this work, we propose a robust approach of integrating test data and death toll to estimate COVID-19 transmission characteristics by a Susceptible, Infectious, Resolving (but not infectious), Deceased and reCovered (SIRDC) model initially studied in 7 . We illustrate that the transition between different stages of disease progression in the SIRDC model in part a of the Figure 1. First, a part of the population are infected by the active infectious individuals each day, depending on the transmission rate parameter (β t ). After γ −1 day, an active infectious individual is expected to be no longer infectious, denoted by the resolving compartment, meaning that this individual will not transmit COVID-19 to others as a result of hospitalization or self-quarantine. We term the average length of an active infectious individual the infectious period. A resolving case is expected to be resolved (either recovered or deceased) after θ −1 day. The proportion of deaths from the number of resolved cases is controlled by the fatality rate parameter δ .
The innovation of our approach is on solving the compartmental models using a midpoint rule with a step size of 1 day, discussed in the method section, as the confirmed cases and death toll are updated daily in most US counties. Our estimate of transmission rates and reproduction numbers is robust and accurate to reproduce the number of death toll and other compartments for counties with medium to small population sizes (Figure 3 and Extended data Figure 1). The simulated studies also suggest that our approach is more robust than the solution in 7 (Extended Data Figure 2), as our solution does not require estimating derivatives of the daily death toll. Instead, the test data and death toll are used for estimation. Only two parameters, the initial values of the number of active infectious individuals and the number of resolving cases, are needed to be estimated numerically for each county, whereas the time-dependent transmission rates and all other compartments can be solved subsequently. Since only two parameters are estimated for each county, our estimation rarely depends on the initial values we choose for the optimization. A summary of the main findings, limitations and policy implication are given in Table 1. The transmission of SARS-CoV-2 is heterogeneous and asynchronous in US counties. It is thus important to assess the risk before lifting or replacing any mitigation measure in the community. We have developed a novel approach to integrate test data and death toll to estimate probability of contracting COVID-19, as well as the time-dependent transmission rate and the number of active infectious individuals at the county level in the US. Main findings and limitations National level order of protection measures reduce the transmission rate and active number of infectious individuals for most US counties in April, whereas the risk of contracting SARS-CoV-2 rebounded between late June and early July, as the protection measures were relaxed. We found that when the infectious period of SARS-CoV-2 is shortened by 5% and 10%, the number of deaths can be reduced from 199K to 120K (95% CI: [109K, 132K ]) and 80K (95% CI: [72K, 89K]) as of 20 September 2020, respectively, when other protection measures were kept the same. The reduction of infectious period can be achieved by extra testing in addition to the ongoing protection measures. Our model relies on the existing knowledge of the COVID-19 and model assumptions. Other information, such as demographic profiles, mobility and serology test data can be used to calibrate the model parameters and assumptions at the community level.
Policy implications
Our model indicates that extra testings, along with the current NPIs, can significantly reduce the number of deaths associated with COVID-19. The estimated probability of contracting COVID-19 can be used as an interpretable risk factor to guide policy responses in community.
Results
We first verify our model performance by forecast at the county level. The 7-day and 21-day death projections for 2, 277 US counties using data by 20 September 2020, for instance, are close to the held out test death toll in these counties, shown in part b and part c of Figure 1. The R 2 and Pearson correlation coefficient (ρ) are larger than 0.999 for both 7-day and 21-day forecast. The 21 day forecast of each considered county in Florida and California using observations by 20 September 2020 is provided in Extended Data Figure 6 and 7, respectively. The forecast of the death toll based on our model is accurate for most US counties, and around 95% of the held out test data is covered by nominal 95% predictive interval (Table S1 in supplementary materials), indicating that the uncertainty assessment is accurate. Based on the robust estimation of transmission rates, we derived the county-level estimation of daily PoC SARS-CoV-2 on each day. We classify the daily PoC SARS-CoV-2 in a community into 5 levels listed in Table 2. On 20 September, out of the 2,277 US counties, only 60 counties were at the controllable level and 311 counties were at the moderate level, whereas 1906 counties were at the either alarming, strongly alarming or hazardous level. The daily PoC SARS-CoV-2 measures the average probability to contract SARS-CoV-2 for a susceptible individual in a community, and the risk varies from individuals to individuals. Nonetheless, the PoC SARS-CoV-2 is an interpretable measure for public understanding of the average risk of contracting SARS-CoV-2 in a community on a given day.
We graph the estimated PoC SARS-CoV-2 of an individual at US counties on 20 April and 20 September in Figure 2. On 20 April, the PoC SARS-CoV-2 is large in northeastern regions, and in some southern states such as Arizona, New Mexico and New Orleans. On 20 September, the PoC SARS-CoV-2 is large in many inland states, for instance, Montana, North Dakota, Mississippi and Alabama. Although the PoC SARS-CoV-2 on 20 September in northeastern regions is substantially lower than that on 20 April, the probability of contracting SARS-CoV-2 for an individual is large in most other states on 20 September, suggesting that the relaxation of protection measures can lead to more population contracting COVID-19, and consequently more deaths at a rate no slower than that in late April.
The daily PoC SARS-CoV-2 can be used by officials to determine where mitigation policies can be lifted or replaced by other measures for different regions. The probability of contracting COVID-19 in many counties in Texas on 20 September, for example, is larger than those in Washington (part (a) and (d) in Figure 3), indicating that more protective measures should be undertaken in Texas to reduce the risk. The nationwide lockdown order and social distancing in spring effectively reduced the PoC SARS-CoV-2 in 4 out of 5 counties in Washington, while the PoC SARS-CoV-2 of all counties increases in late June and early July, as some of the nonpharmaceutical interventions (NPIs) were lifted (part b in Figure 3). Part (c) shows that the model fit to death toll. With only two parameters estimated numerically for each county, the fit is reasonably good for these counties at a wide range of dates. In comparison, though the outbreak of 5 counties in Texas started in early summer, the PoC SARS-CoV-2 in these Texas counties is much higher on 20 September than the one in Washington (part (e) in Figure 3). Our model also fits the death toll of the counties in Texas relatively well (part (f) in Figure 3).
The effectiveness of protection measures were studied to reduce the transmission rate 5, 6,8,9,11,16 , whereas the efficacy of these measures depends on the reactions from the public, which is likely to vary from region to region. Another simultaneous effort to mitigate the spread of COVID-19 outbreak is through testing and contacting tracing, which reduces the infectious period, and consequently, the number of active infectious individuals. For Washington and Texas, we simulate the model output with infectious period reduced by 5% (or equivalently 4.75 days in total), while the transmission rate (β t in SIRDC model) is March 2020 to 20 September 2020 with infectious period assumed to be 5 days (blue), 4.75 days (green) and 4.5 days (red). c, the estimate overall death toll in the US. The time period and interpretation of c are aligned with a and b, except that the black dots in c stand for the observed death toll in the US. held the same. We found that the PoC SARS-CoV-2 is reduced by 5 times for 12 counties out of 28 considered counties in Washington and 6 counties out of 209 considered counties in Texas, as shown in the Extended Data Figure 3. Furthermore, when we reduce the infectious period by 10% (or equivalently 4.5 days in total), while the transmission rate (β t in SIRDC model) is held the same, the PoC SARS-CoV-2 is reduced by 5 times for 26 counties in Washington and 146 counties in Texas, shown in Extended Data Figure 4.
We graph the estimated effective reproduction number, the number of active infectious individuals and cumulative death toll in the US, along with the simulated values when the average infectious period is reduced to 4.75 days and 4.5 days, from 5 days in Figure 4. First we found that mitigation measures in March effectively reduce the effective reproduction number to below 1, whereas the value rebounded in summer after some of these measures were relaxed in different regions. Consequently, US has experienced two waves of the outbreak in terms of the number of active infectious individuals (part (b) in Figure 4). The high test positive rate at the beginning of the epidemic (Extended Data Figure 5) indicates that a substantial number of active infectious individuals were not diagnosed in April due to the lack of diagnostic tests. According to our estimates, the peak of the first wave in April is larger than that of the second wave in July in terms of the number of active infectious individuals, whereas the peak of the daily observed confirmed cases in April is smaller than that of the second wave in July (Extended Data Figure 5).
Second, the simulated results suggest that a shortened infectious period of SARS-CoV-2 by 5% and 10% can reduce the total deaths from 199K to 120K (95% CI: [109K, 132K]) and 80K (95% CI: [72K, 89K]), respectively, as of 20 September 2020, when other protection measures were held as the same (part c in Figure 4). Note that since we held the transmission rate parameter (β t ) to be the same (a scenario that the public adhere to the protection measure same as the reality), the effective reproduction number changes very little (part a in Figure 4). However, the slightly shortened infectious periods of SARS-CoV-2 can reduce the active infectious individuals substantially (part b in Figure 4), as the number of active infectious individuals decreases.
We found that a shortened infectious period substantially reduces the number of active infectious individuals and fatality in the second wave, whereas the change is smaller in the first wave, since the effective reproduction number in the second wave is smaller than that in the first wave ( Figure 4). The county level estimation also validates this point (Extended Data Figure 3 and 4). This finding indicates that the efforts of shortening the infectious period of SARS-CoV-2 should not replace the other protection measures, such as social distancing and facial mask wearing to reduce the transmission rate.
Diagnostic tests can be used to shorten the length of infectious period of an active infectious individual. Drastically reducing the infectious period may not be possible without contact tracing, which is challenging when a large number of infection pairs exist. Reducing infectious period by around 5%, in comparison, can be achieved by periodically diagnostic tests every 20 day for each susceptible individual. A more efficient way is to test susceptible individuals with a high risk of contracting or spreading SARS-CoV-2, such as individuals with more daily contacts or have contacts with vulnerable population, e.g. workers from senior living facilities. Our estimation of contracting SARS-CoV-2 can be used as a response to develop regression models using covariates including demographic information and mobility to elicit personalized risk of contracting SARS-CoV-2 for susceptible individuals. Furthermore, efforts on reducing the length of infectious period should not replace other protection measures for reducing transmission rates of SARS-CoV-2, as the number of active infectious individuals and death toll can only 5/18 be effectively reduced, if the effective reproduction number is not substantially larger than 1.
Discussion
Our study have several limitations. First, our findings are based on available knowledge and model assumptions, as with all other studies. One critical parameter is the death rate, assumed to be 0.66% on average 17 , whereas this parameter can vary across regions due to the demographic profile of the population and available medical resources. The studies of prevalence of SARS-CoV-2 antibodies based on serology tests 14 can be used to determine the size of population that have contracted SARS-CoV-2, and thus provides estimates on death rate, as the death toll is observed. Besides, we assume the infected population can develop immunity since recovery at least for a few months, which is commonly used in other models. The exactly duration of post-immunity, however, remains unverified scientifically. Third, we assume that the number of susceptible individuals, and, consequently, the number of individuals that contracted SARS-CoV-2 can be written as a function of the number of observed confirmed cases and test positive rates, calibrated based on the death toll. More information such as the proportion of population adhere to the mitigation measures, mobility and demographic profile can be used to improve the estimation of susceptible individuals in a region.
Our results can be used to mitigate the ongoing pandemic by SARS-COV-2 and other infectious disease outbreak in future. The estimated daily PoC SARS-CoV-2 at the county level, for example, is an interpretable measure to understand the risk of contracting COVID-19 on a daily basis, and a surveillance marker to determine appropriate policy responses. Further studies of this measure relative to different mobility, demographic information and social economic status of individuals can provide more precise guidance for local officials to protect vulnerable population from contracting SARS-CoV-2, when an effective vaccine is not available.
Methods
SIRDC compartmental models
The SIRDC model for the jth county in the ith state in the US is described below:
S i, j (t) = −β i, j (t)S i, j (t)I i, j (t) N i, j , I i, j (t) = β i, j (t)S i, j (t)I i, j (t) N i, j − γI i, j (t), R i, j (t) = γI i, j (t) − θ R i, j (t), D i, j (t) = δ θ R i, j (t), C i, j (t) = (1 − δ )θ R i, j (t),(1)
where S i, j (t), I i, j (t), R i, j (t), D i, j (t) and C i, j (t) denote the number of individuals at these 5 compartmental groups on day t, respectively, and N i, j denotes the number of individuals in the j county at the ith state for i = 1, 2, ..., k, j = 1, 2, ..., n i with n i being the number of counties of the ith state considered in the analysis and t = 1, 2, ..., T i, j . The time-dependent transmission rate parameter is denoted by β i, j (t) and the inverse of average number of days an infectious individual can transmit the COVID-19 is denoted by γ. The inverse of the average number of dates for a case to get resolved (i.e. deceased or recovered) is denoted by θ and the proportion of deceased cases (i.e. death rate) is denoted by δ . The parameters (γ, θ , δ ) were invariant over time and held fixed in this study. Following 16 , we assume the infectious period to be 5 day on average and a case is expected to resolve after 10 days. The average death rate is assumed to be 0.66% 17 . Additional verification of these assumptions and a sensitivity analysis of these parameters are provided in the supplementary materials.
To determine the characteristics of the SARS-CoV-2 epidemic at US counties, we define the time-dependent effective reproduction number, i.e. the average number of secondary cases per primary cases as R i, j e f f (t) = R i, j 0 (t)S i, j (t)/N i, j , where the R i, j 0 (t) = β i, j (t)/γ denotes the basic reproduction number on day t. When R i, j e f f (t) < 1, it means that the number of the active infectious individuals will decrease (and vice versa, if R i, j e f f (t) > 1). The effective reproduction number was often used to quantify whether or not the disease is under control 18 . The effective reproduction number, however, does not directly quantify risk of contracting SARS-COV-2 for a susceptible individual, as the number of active infectious individuals in a region was not taken into consideration. We compute the average probability of contracting (PoC) SARS-CoV-2, denoted as P i, j (t) = R i, j e f f (t)I i, j (t)γ/(S i, j (t)) = β i, j (t)I i, j (t)/N i, j , which quantifies the risk of a susceptible individual in county j from state i to catch SARS-CoV-2 on day t. Here the risk is in an average sense among all susceptible individuals in a region.
The most critical parameter of the SIRDC is the transmission rate parameter, β i, j (t), as a function of time, based on which we have reproduction numbers on day t. To estimate the time dependent transmission rate for communities with small 6/18 population sizes, we derive a more robust estimation of transmission rate of each county based on death toll and testing data, discussed below.
Closed-form expressions of time-dependent transmission rates. Since the observations such as death toll and confirmed cases are typically updated daily, we approximate the solution of the ODEs of the SIRDC model in (1) by the midpoint rule of the integral with a step size of 1 day. For day t ∈ N + , the approximation is described below:
S i, j (t + 1) S i, j (t) . = exp − β i, j (t + 0.5) 2N i, j (I i, j (t) + I i, j (t + 1)) ,(2)I i, j (t + 1) I i, j (t) . = exp β i, j (t + 0.5) 2N i, j (S i, j (t) + S i, j (t + 1)) − γ ,(3)R i, j (t + 1) − R i, j (t) . = γ I i, j (t) + I i, j (t + 1) 2 − θ R i, j (t) + R i, j (t + 1) 2 ,(4)D i, j (t + 1) − D i, j (t) . = δ θ R i, j (t) + R i, j (t + 1) 2 ,(5)C i, j (t + 1) −C i, j (t) . = (1 − δ )θ R i, j (t) + R i, j (t + 1) 2 .(6)
Further by assuming the transmission rate parameter β i, j (t) is day-to-day invariant (i.e. a step function with step size 1). Based on equations (2) and (3), we obtain β i, j (t + 0.5) from t = 1 to T i, j − 1, iteratively, based on the sequence of susceptible individuals {S i, j (t)} T i, j t=1 and the initial number of active infectious individuals I i, j (1) described in algorithm 1.
Data: {S i, j (t)} T i, j t=1 , I i, j (1) Result: {β i, j (t + 0.5)} T i, j −1 t=1 , {I i, j (t)} T i, j t=1 S 1 = S i, j (1); S 2 = S i, j (2); I 1 = I i, j (1); for t = 1 to (T i, j − 1) do β i, j (t + 0.5) = β : S 2 S 1 − exp − β I 1 2N i, j (1 + exp{ β 2N i, j (S 1 + S 2 ) − γ}) = 0 ; I i, j (t + 1) = I 1 exp β i, j (t+0.5) 2N i, j (S 1 + S 2 ) − γ ;
S 1 = S 2 ; S 2 = S i, j (t + 1); I 1 = I i, j (t + 1); end Algorithm 1: Iterative approach for estimating transmission rate β i, j (t + 0.5). After we get the number of active infective individual (I i, j (t)) on each day, the sequences of the resolving, deceased and recovered compartments can be solved subsequently, following the same manner using equation (4)-(6), after specifying their initial values. Expressing the time-dependent transmission rate by the number of susceptive and infective cases is the key to integrate death toll and testing data for estimation.
In Extended Data Figure 1 and 2, we demonstrated that the solution by our approach is more accurate and more robust to solve the ODEs of SIRDC model than the method in 7 for both simulated and real scenarios. Other more accurate methods (such as the Runge-Kutta method) exist to solve the ODEs of SIRDC model, whereas the time-dependent transmission rates are not easily expressed as a function of the death toll and the number of active infectious individuals as the way they are in our solution.
Estimation of the number of susceptible individuals. Note that we have S i, j (t) + c o i, j (t) + c u i, j (t) = N i, j for any t, where c o i, j (t) and c u i, j (t) are the number of cumulative observed confirmed cases and unobserved confirmed cases, respectively. Estimating the number of susceptible individuals is equivalent to estimating the number of unobserved confirmed cases c u i, j (t), because the number of observed confirmed cases c o i, j (t) and the population N i, j are known. Here we combine them with the positive test rates to estimate c u i, j (t), as large positive test rates typically indicate a large number of unobserved confirmed cases. We assume that the total number of confirmed cases is equal to the observed confirmed cases, adjusted by the state level test positive rate p i (t), a power parameter α i and a weight parameter ω i , leading to the following formula of the susceptible population:
S i, j (t) = N i, j − c o i, j (t) − c u i, j (t) = N i, j − 1 ω i, j 1 {t≥2} t ∑ s=2 (p i (s)) α i ∆c o i, j (s) + (p i (1)) α i c o i, j (1) ,(7)
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where ∆c o i, j (t) is the observed daily confirmed cases on day t, for t = 1, 2, ..., T i, j , i = 1, 2, ..., k and j = 1, 2, ..., n i . Since the positive test rates are only available at the state level, the power parameter α i ∈ [0, 2] is estimated by the state-level observations. According to equation (7), the time-invariant weight ω i, j can be expressed below:
ω i, j = (p i (1)) α i c o i, j (1) I i, j (1) + R i, j (1) + D i, j (1) +C i, j (1) ,(8)
where I i, j (1), R i, j (1), D i, j (1) and C i, j (1) are the number of active infectious, resolving, deceased and recovered cases on day 1, respectively.
Estimation of initial values of infectious and resolving cases. We define day 1 of a county as the more recent date between 21 March 2020 and the date that the county has 5 observed confirmed cases for the first time. Since all counties were at an early stage of the epidemic on the starting day, we let the initial value of death toll D i, j (1) to be the observed death toll on day 1, and the initial value of the recovered cases to be 0. This assumption is not likely going to influence our analysis, as the number of recovered cases is only a negligible proportion of the susceptible individual on the starting day, if not zero. The only parameters to estimate are the number of infectious individuals I i, j (1) and the number of resolving cases R i, j (1) on day 1 for the jth county in the ith state, after the power parameter α i is estimated using the state-level observations to minimize the same loss function below:
(Î i, j (1),R i, j (1)) = arg min T i, j ∑ t=1 D i, j (t) −D i, j (t | I i, j (1), R i, j (1)) T i, j − t + 1 2 , s.t. 0 ≤ I i, j (1) + R i, j (1) ≤ U i, j , I i, j (1) ≥ 0, and R i, j (1) ≥ 0,(9)
where the upper bound U i, j is chosen to guarantee the estimated number of the susceptible cases S i, j (t) to be larger than 0:
U i, j = N i, j (p i (1)) α i c o i, j (1) 1 {T i, j ≥2} ∑ T i, j s=2 (p i (s)) α i ∆c o i, j (s) + (p i (1)) α i c o i, j (1) − (D i, j (1) +C i, j (1)), for t = 1, 2, . . . , T i, j .
Iterative updates for the fitted compartments and reproduction rates. After the initial values of infectious and resolving cases are estimated, we obtain the estimation of the susceptible cases from equation (7), and the infectious cases and transmission rates on each date for each county from Algorithm 1. The resolving cases, deaths and recovered cased can be derived subsequently from equation (4)-(6), respectively. The estimated basic and effective reproduction rates can be derived by the fitted time-dependent transmission rate, and the estimated probability of contracting for an individual COVID-19 can be computed based on transmission rate and number of infectious individuals for each county on each day.
Forecast. Our method can also be used as a tool for forecasting compartments (e.g. death toll), reproduction numbers and the probability of contracting COVID-19 at each county for a short time period. We extrapolate the transmission rate based on Gaussian processes implemented in RobustGaSP R package 19 with robust parameter estimation 20,21 . Based on the extrapolated transmission rates, the compartments can be solved iteratively based on equation (2)- (6).
We also found the forecast will generally be improved by assuming the cumulative deaths are modeled as D i, j (t) = D i, j (t) + z i, j (t), where z i, j is a zero-mean Gaussian process. The details of the forecast method, the comparison and uncertainty assessment of the forecast are presented in the supplementary materials.
Supplementary materials for robust estimation of SARS-CoV-2 epidemic at US counties The supplementary materials contain two parts. In the first part, We discuss the details of model parameter specification and conduct a sensitive analysis. The forecast algorithm and uncertain assessment are introduced in the second part.
Model parameter specification and sensitive analysis
We discuss the choice of model parameter and its sensitivity analysis. The following parameters of the SIRDC model were specified based on previous studies.
• The death rate or the infection fatality ratio (δ ) measures the proportion of death among all infected individuals. We assume δ = 0.66% following 17 .
• The inverse of the number of days an infectious individual can transmit the COVID-19 (γ). The average time of a COVID-19 patient to transmit disease is assumed to be 5 days in 16 , indicating that γ = 0.2. Another evidence comes from the study of incubation period. The latent period (exposed but not contagious) for COVID-19 is found to be 3.69 days on average 2 and the mean incubation period (time from infection to onset of symptoms) is 5.2 days 22 , meaning that infectious period being around 1.5 days before the onset of symptom. The diagnosis test could take less than one day to up to week. Assuming 3.5 days to get the result of a diagnosis test on average. The total infectious period is around 5 day.
• The inverse of the number of dates for resolving case to get resolved (θ ). According to the CDC report 23 , for mild and moderate symptom, the replication-competent virus has not been recovered after 10 days following symptom onset, indicating the individuals remains infectious no longer than 10 days after symptom onset. The infectious period could be as long as 20 days for patients with more severe illness from COVID-19 infection. Since the majority of the COVID-19 infection is mild to moderate, we assume the infectious period to be 13.5 days and after reducing 3.5 days from onset of the symptom to become resolving (after quarantine or hospitalized), it takes around 10 days for a resolving case to resolved on average.
We conduct a sensitive analysis to examine the change of the estimation in 4 different configuration.
• (Configuration 1) (γ, θ , δ ) = (0.2, 0.1, 0.0066), the default parameter set.
• (Configuration 2) (γ, θ , δ ) = (0.14, 0.1, 0.0066). The average length of infectious period changes from 5 days to 1 0.14 ≈ 7 days, whereas other parameters are held unchanged.
• (Configuration 3) (γ, θ , δ ) = (0.2, 0.067, 0.0066). The average length of resolving period changes from 10 days to 1 0.067 = 15 days, whereas other parameters are held unchanged.
• (Configuration 4) (γ, θ , δ ) = (0.2, 0.1, 0.0075). The infection fatality ratio changes from 0.66% to 0.75%, whereas other parameters are held unchanged.
After specifying the parameters (γ, θ , δ ), the transmission rate β (t) can be obtained from algorithm 1. Figure S1 gives result of the sensitive analysis. First we found the estimated death toll for 4 scenarios is almost the same (part d in Figure S1). Extending the infectious period from 5 to 7 days (Configuration 2) increases the number of active infectious individuals and effective reproduction number shown in part a and part c in Figure S1, respectively. Consequently, the peak of average daily PoC SARS-CoV-2 slightly increases in the first wave, whereas the scale of increase is smaller than the change in the effective reproduction number and active infectious individuals. The average daily PoC SARS-CoV-2 has almost no change for other periods, indicating that the length of the average infectious period has almost no influence of our estimation of PoC SARS-CoV-2 for most of the days.
Second, when the average length of resolving period changes from 10 to 15 days, the peak of PoC SARS-CoV-2, effective reproduction number and the number of active infectious individuals increases in the first wave, whereas these quantities remain largely unchanged for the rest of the days (part a-c in Figure S1). The result indicates that the average length of resolving period also barely affects the estimated characteristics of COVID-19 progression for most of the days.
When the death rate increases from 0.66% to 0.75%, the effective reproduction number seems to have almost no change (part a in Figure S1), whereas the PoC SARS-CoV-2 and the number of active infectious individuals (figure S1 part b-c) both reduce. This is because when the death rates increases, the number of individuals infected decreases, as the death toll is observed (and thus fixed). The death rate is a key parameter to calibrate and the studies of prevalence of SARS-CoV-2 antibodies based on serology tests 14 can be used to estimate the death rate in each state.
In conclusion, the parameter values of the average length of infectious period and the average length of resolving period do not change the COVID-19 progression characteristics for most of the days, including the fitted death toll. On the other hand, even the effective reproduction number and the fitted death toll has almost no change when the death rate changes, the number of active infectious individuals and the daily PoC SARS-CoV-2 depend critically on the death rate parameter. Further studies of prevalence would be useful for estimating the death rate parameter in different regions.
Forecast and uncertainty assessment
The epidemiological SIRDC model lead to a robust fit for most counties (considering only two parameters to estimate for each county in the model). The performance of forecast can be improved by adding a Gaussian Process (GP) to model the residual between the death toll and the estimate from the SIRDC model. GP is widely used to model temporal or spatio-temporal correlated observations. One advantage of a GP model is the internal assessment of the uncertainty of the forecast from the predictive distribution, which is of crucial importance. The aggregated model that combines the SIRDC model and the GP model for the jth county in the ith state in the US is described as follows.
D i, j (t) = F i, j (t) + z i, j (t) + ε i, j,t(S1)
where D i, j (t) and F i, j (t) denote the observed death toll and estimated death toll via the SIRDC model, respectively; The noise follows independently as a Gaussian distribution ε i, j,t ∼ N(0, σ 2 i, j,0 ) with variance parameter σ 2 i, j,0 . The latent temporal process z i, j (t) is modeled by a zero-mean GP, meaning that for time points {1, 2, . . . , T i, j }, z i, j = (z i, j (1), . . . , z i, j (T i, j )) T follows a multivariate normal distribution:
z i, j ∼ MN(0, Σ Σ Σ i, j )
where the (l, m) entry of Σ Σ Σ i, j is parameterized by a covariance function σ 2 i, j K i, j (l, m) for 1 ≤ l, m ≤ T i, j . Here σ 2 i, j is the variance parameter and K i, j (·, ·) is a one-dimensional correlation function. We use the power exponential correlation function:
K i, j (l, m) = exp − | l − m | b i, j a ,
where a is the roughness parameter fixed to be 1.9 as in other studies 24,25 , to avoid possible singularity in inversion of the covariance matrix using the Gaussian correlation (a = 2), and b i, j is a range parameter for each county estimated from the data.
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Data: c o i, j , D i, j , p i , c o i , and D i . Result: Estimates of county-level epidemiological compartmentsβ β β i, j ,Ŝ i, j ,Î i, j ,R i, j ,Ĉ i, j , forecastD * i, j , wherê D * i, j := D i, j (T i, j + 1), . . . ,D i, j (T i, j + T * ) T , and the uncertainty assessment of the compartments.
Step 1 Conduct a three-parameter constrained optimization treating state-level power parameter α i unknown to minimize the loss function in equation (9) using p i , c o i and D i .
Step 2 For each county, set initial values I i, j (1) = R i, j (1) = 1, 000, C i, j (1) = 0 and D i, j (1) to be the observed death toll on day 1. Find the optimized values of I i, j (1) and R i, j (1) to minimize equation (9).
Step 3 Simulate S = 500 samples of the observed confirmed cases sampled from the predictive distribution of a GP model of the observed confirmed cases. For each sample, obtain the other compartments and time-dependent transmission rate by equation (1)-(5) and algorithm 1 using the estimate of the initial values.
Step 4 Extrapolate the time-dependent transmission rate parameters from a GP model for each sample and obtain S = 500 samples of output death toll of the SIRDC at the forecast period.
Step 5 Sample the residuals from the predictive distribution (S2) at the forecast period and obtain S = 500 samples of the ensemble forecast for the death toll. Compute the mean for forecast and 95% predictive interval to quantify the uncertainty.
Algorithm S1: Ensemble forecast and uncertainty assessment.
We define the nugget parameter η i, j = σ 2 i, j,0 /σ 2 i, j . The range parameter b i, j , and the nugget parameter η i, j in equation (S1) are estimated based on the marginal posterior mode estimation using the rgasp function in the package RobustGaSP available on CRAN 20 .
Denote D i, j = (D i, j (1), ..., D i, j (T i, j )) T and F i, j = (F i, j (1), ..., F i, j (T i, j )) T .
After marginalizing out the variance parameter by the reference prior p(σ 2 i, j ) ∝ 1/σ 2 i, j , for any t * , the predictive distribution of z i, j (t * ), conditional on the observations, range parameter b i, j and nugget parameter η i, j , follows a non-central Student's t-distribution with degrees of freedom T i, j
20 z i, j (t * ) | D i, j , F i, j , b i, j , η i, j ∼ T ẑ i, j (t * ) ,σ 2 i, jK * i, j , T i, j (S2) whereẑ i, j (t * ) =F i, j (t * ) + r T i, j (t * )R −1 i, j (D i, j − F i, j ) σ 2 i, j = (D i, j − F i, j ) TR−1 i, j (D i, j − F i, j ) T i, j K * i, j =K i, j (t * ,t * ) + η i, j − r T i, j (t * )R −1 i, j r i, j (t * ) withR i, j = R i, j +η i, j I T i, j , the (l, m)th term of R i, j being K i, j (l, m) for 1 ≤ l, m ≤ T i, j , and r i, j (t * ) = (K i, j (t * , 1), . . . , K i, j (t * , T i, j )) T ,
by plugging in the estimated range parameter b i, j and nugget η i, j . The predictive meanẑ i, j (t * ) for forecast the death toll of the jth county in the ith state at a future day t * , and the predictive interval can be computed based on the quantiles and percentiles of the Student's t distribution.
An overview of our algorithm for the forecast the uncertainty assessment is given in algorithm S1, where the inputs are the county-level observed cumulative number of confirmed cases To evaluate the performance of our approach, we implement 7-day and 21-day forecasts using our approach and other approaches on 2,277 US counties with training period from 21 March 2020 to 20 September 2020, and with the forecast period starting from 21 September 2020. To compare the prediction performance of different methods, we computed the rooted mean square error (RMSE), proportion of the observations covered in the 95% predictive interval (P CI (95%)) and length of the 95% Table S1. 7-day and 21-day forecast for 2,277 US counties with training period from 21 March 2020 to 20 September 2020 and with prediction period starting from 21 September 2020. Four methods are compared. Our proposed approach that combines the SIRDC model and a zero-mean GP to model the residuals is denoted as SIRDC+GP. Second the death forecast by SIRDC model is denoted as SIRDC, which contains Step 1 and 2 in the algorithm S1 and provides point projection of the death toll. Third a GP with constant mean function is denoted as GP without linear trend, which equivalently replaces the SIRDC model of a constant mean parameter estimated by the data for each county. The fourth model is the same as the third method, except that the mean of GP contains constant mean and a linear trend of time with two linear coefficient parameters estimated from the data (denoted as GP with linear trend). The method of the best performance by each criterion is highlighted.
confidence interval (L CI (95%)), defined as follows:
RMSE = ∑ n i=1 ∑ n i j=1 ∑ s∈t * (D i, j (s) − D i, j (s)) 2 ∑ n i=1 n i T * P CI (95%) = 1 ∑ n i=1 n i T * n ∑ i=1 n i ∑ j=1 ∑ s∈t * 1 {D i, j (s)∈CI i, j,s (95%)} L CI (95%) = 1 ∑ n i=1 n i T * n ∑ i=1 n i ∑ j=1 ∑ s∈t * length{CI i, j,s (95%)}
where t * := (T i, j + 1, . . . , T i, j + T * ), T * = 7 and T * = 21 for the 7-day forecast and 21-day forecast, respectively. A model with small RMSE, P CI (95%) close to the nominal 95% and small L CI (95%) are precise for forecast and uncertainty assessment. The comparison between our approach and the other three approaches are recorded in Table S1. Our approach (denoted in SIRDC+GP) has the lowest RMSE among 4 methods considered herein. Approximately 95% of the held-out death toll are covered by the 95% predictive interval by our approach, indicating our uncertainty assessment is accurate. Although other approaches produce shorter length of the predictive interval, the number of held-out observations contained in the 95% predictive interval is smaller than ours. Therefore, combining the SIRDC model and GP for modeling the residuals can improve the predictive accuracy for forecasting COVID-19 associated death toll at US counties, compared to the one using the SIRDC model or the GP model alone.
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Extended data figures for robust estimation of SARS-CoV-2 epidemic at US counties Extended Data Figure 1. a-c, Comparisons between the estimation COVID-19 progression characteristics for Santa Barbara, CA as of 20 September 2020 by our algorithm 1 (blue solid curves) and the method F&J 7 (red dash curves) . The shaded area represents 95% confidence intervals. The black solid curve in part c is the observed cumulative death toll in Santa Barbara. d-f, Results for Imperial, CA as of 20 September 2020, which have the same interpretation as a-c. The transmission rate estimated from the method F&J is truncated to be within [0,10]. Figure 2. a-c, Simulated comparison with noise-free observations. The black circles are the solution of the ODEs of the SIRDC model via the default numerical solver Isoda in the function ode in deSolve R package. The green solid and dash curves are the numerical solutions from Runge-Kutta method with the 4th order integration and step size being 1 and 0.1, respectively. The Blue solid curves are the robust estimation from algorithm 1 and red dash curves are the estimation in 7 . In the simulation with noise-free observations, we let time duration be T = 100 days, the population size N = 10 7 , the initial values of 5 compartments chosen as (S(1), I(1), R(1), D(1),C(1)) = (N − 2000, 1000, 1000, 0, 0) and the transmission rate
Extended Data
β (t) = exp −0.7( 9 T −1 (t − 1) + 1) , for 1 ≤ t ≤ T . d-f
, results of the simulation with noisy observations, which have the same interpretation as a-c. In this simulation, we set the transmission rate β (t) = exp −0.7( 9 T −1 (t − 1) + 1) + ε, for 1 ≤ t ≤ T and ε ∼ N(0, 0.04), and the other parameters are held the same as in the noise-free simulation. The transmission rates estimated from the method F&J are truncated to be within [0,10]. The solution from our robust estimation approach, the Isoda, the Runge-Kutta method with the 4th order and step size being 0.1 overlap for both scenarios.
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Extended Data Figure 3. a-f, the simulated results of COVID-19 progression in Washington (the first row) and in Texas (the second row) that have the same interpretation as a-f in Figure 3 with the infection period changed from 5 days, to 4.75 days, whereas other parameters are held the same. Figure 4. a-f, the simulated results of COVID-19 progression characteristics in Washington (the first row) and in Texas (the second row) that have the same interpretation as a-f in Figure 3 with the infection period changed from 5 days to 4.5 days, whereas other parameters are held the same. 9.20310330840854, 9.34076484197512, 9.47495674925274, 9.60598265936292, 9.73441266884615 Apr May Jun Jul Aug Sep Oct 18384, 18385, 18386, 18387, 18388, 18389, 18390, 18391, 18392, 18393, 18394, 18395, 18396, 18397, 18398, 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 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10.257975959 Apr May Jun Jul Aug Sep Oct 18.1256062258264, 18.1643037285639, 18.2053706350613, 18.248516331611, 18.2935193943954, 18.3402007696657, 18.3884081702236, 18.4380066019006, 18.4888724605496, 18.5408898071103, 18.5939480172543, 18.6479403200922, 18.7027629216853, 18.7583145169991, 18.8144960603796, 18.8712107070915, 18.9283638659032, 18.9858633213106) Extended Data Figure 6. The 21-day prediction in 67 Florida counties with death toll no less than 2 as of 20 September 2020. The training period is from 21 March 2020 to 20 September 2020, whereas the prediction starts from 21 September 2020. The red curves are the cumulative observed death toll from 21 September 2020 to 11 October 2020 and the blue line indicates the forecast for the same period. The shaded area represents the 95% predictive intervals of the forecast for each analyzed county in Florida.
Extended Data
17/18
Apr May Jun Jul Aug Sep Oct 0 200 500 Alameda, population=1.67M 8, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18480, 18481, 18482, 18483, 18484, 18485, 18486, 18487, 18488, 18489, 18490, 18491, 18492, 18493, 18494, 18495, 18496, 18497, 18498, 18499, 18500, 18501, 18502, 18503, 18504, 18505, 18506, 18507, 18508, 18509, 18510, 18511, 18512, 18513, 18514, 18515, 18516, 18517, 18518, 18519, 18520 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 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18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18480, 18481, 18482, 18483, 18484, 18485, 18486, 18487, 18488, 18489 18384, 18385, 18386, 18387, 18388, 18389, 18390, 18391, 18392, 18393, 18394, 18395, 18396, 18397, 18398, 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474 18369, 18370, 18371, 18372, 18373, 18374, 18375, 18376, 18377, 18378, 18379, 18380, 18381, 18382, 18383, 18384, 18385, 18386, 18387, 18388, 18389, 18390, 18391, 18392, 18393, 18394, 18395, 18396, 18397, 18398, 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451 , 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18480, 18481, 18482, 18483, 18484, 18485, 18486, 18487, 18488, 18489, 18490, 18491, 18492, 18493, 18494, 18495, 18496, 18497, 18498, 18499, 18500, 18501, 18502, 18503, 18504, 18505, 18506, 18507, 18508, 18509, 18510, 18511, 18512, 18513, 18514, 18515, 18516, 18517, 18518, 18519, 18520, 18 Apr May Jun Jul Aug Sep Oct 0 40 80 Madera, population=0.16M 3,18414,18415,18416,18417,18418,18419,18420,18421,18422,18423,18424,18425,18426,18427,18428,18429,18430,18431,18432,18433,18434,18435,18436,18437,18438,18439,18440,18441,18442,18443,18444,18445,18446,18447,18448,18449,18450,18451,18452,18453,18454,18455,18456,18457,18458,18459,18460,18461,18462,18463,18464,18465,18466,18467,18468,18469,18470,18471,18472,18473,18474,18475,18476,18477,18478,18479,18480,18481,18482,18483,18484,18485,18486,18487,18488,18489,18490,18491,18492,18493,18494,18495,18496,18497,18498,18499,18500,18501,18502,18503,18504 98,18399,18400,18401,18402,18403,18404,18405,18406,18407,18408,18409,18410,18411,18412,18413,18414,18415,18416,18417,18418,18419,18420,18421,18422,18423,18424,18425,18426,18427,18428,18429,18430,18431,18432,18433,18434,18435,18436,18437,18438,18439,18440,18441,18442,18443,18444,18445,18446,18447,18448,18449,18450,18451,18452,18453,18454,18455,18456,18457,18458,18459,18460,18461,18462,18463,18464,18465,18466,18467,18468,18469,18470,18471,18472,18473,18474,18475,18476,18477,18478,18479,18480,18481,18482,18483,18484,18485,18486,18487,18488,18489 Mariposa, population=0.02M 83,18384,18385,18386,18387,18388,18389,18390,18391,18392,18393,18394,18395,18396,18397,18398,18399,18400,18401,18402,18403,18404,18405,18406,18407,18408,18409,18410,18411,18412,18413,18414,18415,18416,18417,18418,18419,18420,18421,18422,18423,18424,18425,18426,18427,18428,18429,18430,18431,18432,18433,18434,18435,18436,18437,18438,18439,18440,18441,18442,18443,18444,18445,18446,18447,18448,18449,18450,18451,18452,18453,18454,18455,18456,18457,18458,18459,18460,18461,18462,18463,18464,18465,18466,18467,18468,18469,18470,18471,18472,18473,18474 Mendocino, population=0.09M 368,18369,18370,18371,18372,18373,18374,18375,18376,18377,18378,18379,18380,18381,18382,18383,18384,18385,18386,18387,18388,18389,18390,18391,18392,18393,18394,18395,18396,18397,18398,18399,18400,18401,18402,18403,18404,18405,18406,18407,18408,18409,18410,18411,18412,18413,18414,18415,18416,18417,18418,18419,18420,18421,18422,18423,18424,18425,18426,18427,18428,18429,18430,18431,18432,18433,18434,18435,18436,18437,18438,18439,18440,18441,18442,18443,18444,18445,18446,18447,18448,18449,18450,18451,18452,18453,18454,18455,18456,18457,18458,18459 Merced, population=0.28M 8,18429,18430,18431,18432,18433,18434,18435,18436,18437,18438,18439,18440,18441,18442,18443,18444,18445,18446,18447,18448,18449,18450,18451,18452,18453,18454,18455,18456,18457,18458,18459,18460,18461,18462,18463,18464,18465,18466,18467,18468,18469,18470,18471,18472,18473,18474,18475,18476,18477,18478,18479,18480,18481,18482,18483,18484,18485,18486,18487,18488,18489,18490,18491,18492,18493,18494,18495,18496,18497,18498,18499,18500,18501,18502,18503,18504,18505,18506,18507,18508,18509,18510,18511,18512,18513,18514,18515,18516,18517,18518,18519,18520,18 Apr May Jun Jul Aug Sep Oct 0 2 4 Mono, population=0.01M 3,18414,18415,18416,18417,18418,18419,18420,18421,18422,18423,18424,18425,18426,18427,18428,18429,18430,18431,18432,18433,18434,18435,18436,18437,18438,18439,18440,18441,18442,18443,18444,18445,18446,18447,18448,18449,18450,18451,18452,18453,18454,18455,18456,18457,18458,18459,18460,18461,18462,18463,18464,18465,18466,18467,18468,18469,18470,18471,18472,18473,18474,18475,18476,18477,18478,18479,18480,18481,18482,18483,18484,18485,18486,18487,18488,18489,18490,18491,18492,18493,18494,18495,18496,18497,18498,18499,18500,18501,18502,18503,18504 Monterey, population=0.43M 98,18399,18400,18401,18402,18403,18404,18405,18406,18407,18408,18409,18410,18411,18412,18413,18414,18415,18416,18417,18418,18419,18420,18421,18422,18423,18424,18425,18426,18427,18428,18429,18430,18431,18432,18433,18434,18435,18436,18437,18438,18439,18440,18441,18442,18443,18444,18445,18446,18447,18448,18449,18450,18451,18452,18453,18454,18455,18456,18457,18458,18459,18460,18461,18462,18463,18464,18465,18466,18467,18468,18469,18470,18471,18472,18473,18474,18475,18476,18477,18478,18479,18480,18481,18482,18483,18484,18485,18486,18487,18488,18489 83,18384,18385,18386,18387,18388,18389,18390,18391,18392,18393,18394,18395,18396,18397,18398,18399,18400,18401,18402,18403,18404,18405,18406,18407,18408,18409,18410,18411,18412,18413,18414,18415,18416,18417,18418,18419,18420,18421,18422,18423,18424,18425,18426,18427,18428,18429,18430,18431,18432,18433,18434,18435,18436,18437,18438,18439,18440,18441,18442,18443,18444,18445,18446,18447,18448,18449,18450,18451,18452,18453,18454,18455,18456,18457,18458,18459,18460,18461,18462,18463,18464,18465,18466,18467,18468,18469,18470,18471,18472,18473,18474 Nevada, population=0.1M 368,18369,18370,18371,18372,18373,18374,18375,18376,18377,18378,18379,18380,18381,18382,18383,18384,18385,18386,18387,18388,18389,18390,18391,18392,18393,18394,18395,18396,18397,18398,18399,18400,18401,18402,18403,18404,18405,18406,18407,18408,18409,18410,18411,18412,18413,18414,18415,18416,18417,18418,18419,18420,18421,18422,18423,18424,18425,18426,18427,18428,18429,18430,18431,18432,18433,18434,18435,18436,18437,18438,18439,18440,18441,18442,18443,18444,18445,18446,18447,18448,18449,18450,18451,18452,18453,18454,18455,18456,18457,18458,18459 Orange, population=3.18M 8,18429,18430,18431,18432,18433,18434,18435,18436,18437,18438,18439,18440,18441,18442,18443,18444,18445,18446,18447,18448,18449,18450,18451,18452,18453,18454,18455,18456,18457,18458,18459,18460,18461,18462,18463,18464,18465,18466,18467,18468,18469,18470,18471,18472,18473,18474,18475,18476,18477,18478,18479,18480,18481,18482,18483,18484,18485,18486,18487,18488,18489,18490,18491,18492,18493,18494,18495,18496,18497,18498,18499,18500,18501,18502,18503,18504,18505,18506,18507,18508,18509,18510,18511,18512,18513,18514,18515,18516,18517,18518,18519,18520,18 Apr May Jun Jul Aug Sep Oct 0 20 40 60 Placer, population=0.4M 3,18414,18415,18416,18417,18418,18419,18420,18421,18422,18423,18424,18425,18426,18427,18428,18429,18430,18431,18432,18433,18434,18435,18436,18437,18438,18439,18440,18441,18442,18443,18444,18445,18446,18447,18448,18449,18450,18451,18452,18453,18454,18455,18456,18457,18458,18459,18460,18461,18462,18463,18464,18465,18466,18467,18468,18469,18470,18471,18472,18473,18474,18475,18476,18477,18478,18479,18480,18481,18482,18483,18484,18485,18486,18487,18488,18489,18490,18491,18492,18493,18494,18495,18496,18497,18498 11.1733114616063, 11.3027929392121, 11.4330536694415, 11.5623268064242, 11.6902230219606, 11.816721343794, 11.9419130119795, 12. 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18480, 18481, 18482, 18483, 18484, 18485, 18486, 18487, 18488, 18489, 18490, 18491, 18492, 18493, 18494, 18495, 18496, 18497, 18498, 18499, 18500, 18501, 18502, 18503, 18504, 18505, 18506, 18507, 18508, 18509, 18510, 18511, 18512, 18513, 18514, 18515, 18516, 18517, 18518 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18480, 18481, 18482, 18483, 18484, 18485, 18486, 18487, 18488, 18489, 18490, 18491, 18492, 18493, 18494, 18495, 18496, 18497, 18498, 18499, 18500, 18501, 18502, 18503, 18504 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18480, 18481, 18482, 18483, 18484, 18485, 18486, 18487, 18488, 18489, 18490 18384, 18385, 18386, 18387, 18388, 18389, 18390, 18391, 18392, 18393, 18394, 18395, 18396, 18397, 18398, 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475 8.96665339614193, 9.07965619843174, 9.19477525124454, 9.31205406562166, 9.43153360293837, 9.5532519807012, 9.67724452498403, 9.80354397127564 Apr May Jun Jul Aug Sep Oct 0 10 20 Shasta, population=0. 18M 368, 18369, 18370, 18371, 18372, 18373, 18374, 18375, 18376, 18377, 18378, 18379, 18380, 18381, 18382, 18383, 18384, 18385, 18386, 18387, 18388, 18389, 18390, 18391, 18392, 18393, 18394, 18395, 18396, 18397, 18398, 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18480, 18481, 18482, 18483, 18484, 18485, 18486, 18487, 18488, 18489, 18490, 18491, 18492, 18493, 18494, 18495, 18496, 18497, 18498, 18499, 18500, 18501, 18502, 18503, 18504, 18505, 18506, 18507, 18508, 18509, 18510, 18511, 18512, 18513, 18514, 18515, 18516, 18517, 18518, 18519 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18480, 18481, 18482, 18483, 18484, 18485, 18486, 18487, 18488, 18489, 18490, 18491, 18492, 18493, 18494, 18495, 18496, 18497, 18498, 18499, 18500, 18501, 18502, 18503, 18504, 18505 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18480, 18481, 18482, 18483, 18484, 18485, 18486, 18487, 18488, 18489, 18490 18384, 18385, 18386, 18387, 18388, 18389, 18390, 18391, 18392, 18393, 18394, 18395, 18396, 18397, 18398, 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474 18369, 18370, 18371, 18372, 18373, 18374, 18375, 18376, 18377, 18378, 18379, 18380, 18381, 18382, 18383, 18384, 18385, 18386, 18387, 18388, 18389, 18390, 18391, 18392, 18393, 18394, 18395, 18396, 18397, 18398, 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 8.30557817245775, 8.55149664314108, 8.78962166393395, 9.01966373587258, 9.24134886748988, 9.45442213618 Apr May Jun Jul Aug Sep Oct Extended Data Figure 7. The 21-day prediction in 50 California counties with death toll no less than 2 as of 20 September 2020. The training period is from 21 March 2020 to 20 September 2020, whereas the prediction starts from 21 September 2020. The red curves are the cumulative observed death toll from 21 September 2020 to 11 October 2020 and the blue line indicates the forecast for the same period. The shaded area represents the 95% predictive intervals of the forecast for each analyzed county in California.
Figure 2 .
2a, the estimated probability of contracting COVID-19 at 1,856 counties on 2020-04-20, and b, at 2,277 counties on 20 September 2020. The latitude and longitude of Hawaii are changed to make it fit into the map. The probability of contracting COVID-19 are truncated at 10 −6 , whereas only 78 counties on 20 April and 45 counties on 20 September are below this level, respectively.
Figure 3 .
3a, the estimated PoC SARS-CoV-2 in Washington state on 20 September 2020. b, the time-dependent PoC SARS-CoV-2 from 5 counties with the largest values on 20 September 2020 in Washington state. c, the observed (dots) and fitted (solid line) cumulative death toll in the 5 counties in figure b from the same time period. d-f, the results in Texas that have the same interpretation as a-c. Part e and f have different scale than part b and c, respectively.
Figure 4 .
4a and b, the estimate reproduction number and overall number of active cases in the US, including 50 states and Washington D.C., from 21
Figure S1 .
S1Sensitive analysis for 4 configurations of the SIRDC model parameters. Part a-d show the estimated effective reproduction number, PoC SARS-CoV-2, the number of active infectious individuals and cumulative death toll, respectively.
c o i, j = (c o i, j (1), ..., c o i, j (T i, j )) T , the county-level observed cumulative death toll D i, j , the state-level test positive rate p i = (p i (1), ..., p i (T i )) T , state-level confirmed cases c o i = (c o i (1), ..., c o i (T i )) T and state-level death toll D i = (D i (1), ..., D i (T i )) T .
Extended Data Figure 5 .
5a, the 7-day averaged daily confirmed cases in the US from 21 March 2020 to 20 September 2020. b, the 7-day averaged test positive rate in the US from 21 March 2020 to 20 September 2020.
1 c
1(NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA,
1 c
1(NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA,
1 c
1(NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA,
c(NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA,
1 c
1(NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA
, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA
, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA
c(NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA
1 A
1, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA
1 A
1, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA,
1 A 7
17, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA,
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 18.
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 277.
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 132.583671280803, 132.
, 1 ,
1NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 3071.
, 1 ,
1NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 22.
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 38.4382273220914, 39.
, 1 ,
1NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 23.3891051579308, 23.724986206175, 24.0675542524687, 24.
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 433.822926105194, 435.741882246948, 437.808328208961, 439.
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 137.115031363769, 137.225179281205, 137.350978072428, 137.
,
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 1284.23996955311, 1290.80771603291, 1297.74364799915, 1304.
1 ,
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1 A
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NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 67.
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1 ,
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1 ,
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1 A
1, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 417.
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NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 4.5105561792112, 4.84099767412334, 5.16366767709022, 5.47950236784872, 5.78913140647007, 6.09289884483634, 6.39093811135667, 6.6832329559027, 6.96966177404625, 7.25002956171186, 7.5240912903202, 7.79156936055411, 8.05216691200944,
, 4 8
48NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 144.24519944704, 145.299584286529, 146.406287787087, 147.536927895139, 148.679556964192, 149.828419751509, 150.980579229036, 152.134522022647, 153.289500491927, 154.4451938168, 155.601522253397, 156.758540698164, 157.916375740841, 159.075187663571, 160.NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 54.2101777444048, 54.385928669065, 54.5423352497078, 54.6858201988375, 54.8203549986479, 54.9486898935558, 55.0728726561426, 55.1945046885908, 55.3148817418983, 55.4350778010351, 55.5559984645869, 55.6784167998658, 55.8029985508726, 55.9303205635321, 56.NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA,
Table 1 .
1Policy summaryBackground
Table 2 .
2Interpretation of the daily PoC SARS-CoV-2 in a community.Daily PoC SARS-CoV-2 < 0.001%
0.001% to 0.01% 0.01% to 0.1% 0.1% to 1%
> 1%
Risk
controllable moderate
alarming
strongly alarming hazardous
Glades, population=0.01M8, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18480, 18481, 18482, 18483, 18484, 18485, 18486, 18487, 18488, 18489, 18490, 18491, 18492, 18493, 18494, 18495, 18496, 18497, 18498, 18499, 18500, 18501, 18502, 18503, 18504, 18505, 18506, 18507, 18508, 18509, 18510, 18511, 18512, 18513, 18514, 18515, 18516, 18517, 18518, 18519, 18520, 18Gulf, population=0.01M 3, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18480, 18481, 18482, 18483, 18484, 18485, 18486, 18487, 18488, 18489, 18490, 18491, 18492, 18493, 18494, 18495, 18496, 18497, 18498, 18499, 18500, 18501, 18502, 18503, 18504, 18505,Apr May Jun Jul Aug Sep Oct
0 5
15
Hamilton, population=0.01M98, 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18480, 18481, 18482, 18483, 18484, 18485, 18486, 18487, 18488, 18489, 18490,
Hardee, population=0.03M83, 18384, 18385, 18386, 18387, 18388, 18389, 18390, 18391, 18392, 18393, 18394, 18395, 18396, 18397, 18398, 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475,
Hendry, population=0.04M368, 18369, 18370, 18371, 18372, 18373, 18374, 18375, 18376, 18377, 18378, 18379, 18380, 18381, 18382, 18383, 18384, 18385, 18386, 18387, 18388, 18389, 18390, 18391, 18392, 18393, 18394, 18395, 18396, 18397, 18398, 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460,
Hillsborough, population=1.47M 98, 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433,18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18480, 18481, 18482, 18483, 18484, 18485, 18486, 18487, 18488, 18489, 18490,
Holmes, population=0.02M83, 18384, 18385, 18386, 18387, 18388, 18389, 18390, 18391, 18392, 18393, 18394, 18395, 18396, 18397, 18398, 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475,
Indian River, population=0.16M 368, 18369, 18370, 18371, 18372, 18373, 18374, 18375, 18376, 18377, 18378, 18379, 18380, 18381, 18382, 18383, 18384, 18385, 18386, 18387, 18388, 18389, 18390, 18391, 18392, 18393, 18394, 18395, 18396, 18397, 18398, 18399, 18400, 18401, 18402, 18403,18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460,
Jackson, population=0.05M8, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18480, 18481, 18482, 18483, 18484, 18485, 18486, 18487, 18488, 18489, 18490, 18491, 18492, 18493, 18494, 18495, 18496, 18497, 18498, 18499, 18500, 18501, 18502, 18503, 18504, 18505, 18506, 18507, 18508, 18509, 18510, 18511, 18512, 18513, 18514, 18515, 18516, 18517, 18518, 18519, 18520, 18 Apr May Jun Jul Aug Sep Oct Jefferson, population=0.01M 3, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18480, 18481, 18482, 18483, 18484, 18485, 18486, 18487, 18488, 18489, 18490, 18491, 18492, 18493, 18494, 18495, 18496, 18497, 18498, 18499, 18500, 18501, 18502, 18503, 18504, 18505, 10
5 10 15
Lafayette, population=0.01M 98, 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18480, 18481, 18482, 18483, 18484, 18485, 18486, 18487, 18488, 18489, 18490,
Lake, population=0.37M 83, 18384, 18385, 18386, 18387, 18388, 18389, 18390, 18391, 18392, 18393, 18394, 18395, 18396, 18397, 18398, 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 1
Lee, population=0.77M 368, 18369, 18370, 18371, 18372, 18373, 18374, 18375, 18376, 18377, 18378, 18379, 18380, 18381, 18382, 18383, 18384, 18385, 18386, 18387, 18388, 18389, 18390, 18391, 18392, 18393, 18394, 18395, 18396, 18397, 18398, 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460,
Leon, population=0.29M8, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18480, 18481, 18482, 18483, 18484, 18485, 18486, 18487, 18488, 18489, 18490, 18491, 18492, 18493, 18494, 18495, 18496, 18497, 18498, 18499, 18500, 18501, 18502, 18503, 18504, 18505, 18506, 18507, 18508, 18509, 18510, 18511, 18512, 18513, 18514, 18515, 18516, 18517, 18518, 18519, 18520, 18 Apr May Jun Jul Aug Sep Oct Levy, population=0.04M 3,18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18480, 18481, 18482, 18483, 18484, 18485, 18486, 18487, 18488, 18489, 18490, 18491, 18492, 18493, 18494, 18495, 18496, 18497, 18498, 18499, 18500, 18501, 18502, 18503, 18504, 18505,0 5 10
Liberty, population=0.01M 98, 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434,18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18480, 18481, 18482, 18483, 18484, 18485, 18486, 18487, 18488, 18489, 18490,0
4
8 12
, 18520, 18 Apr May Jun Jul Aug Sep Oct0 50
150
Martin, population=0.16M
3,
8, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18480, 18481, 18482, 18483, 18484, 18485, 18486, 18487, 18488, 18489, 18490, 18491, 18492, 18493, 18494, 18495, 18496, 18497, 18498, 18499, 18500, 18501, 18502, 18503, 18504, 18505, 18506, 18507, 18508, 18509, 18510, 18511, 18512, 18513, 18514, 18515, 18516, 18517, 18518, 18519, 18520, 18 Apr May Jun Jul Aug Sep Oct Pinellas, population=0.97M 3, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424,18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18480, 18481, 18482, 18483, 18484, 18485, 18486, 18487, 18488, 18489, 18490, 18491, 18492, 18493, 18494, 18495, 18496, 18497, 18498, 18499, 18500, 18501, 18502, 18503, 18504, 18505, 0
400 800
Polk, population=0.72M98, 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18480, 18481, 18482, 18483, 18484, 18485, 18486, 18487, 18488, 18489, 18490,1 355582724939, 725.515748171561, 727.856222133622, 730.308159858651, 732.838260535847, 735.427415898707, 738.063502166284, 740.73828179257, 743.445865746531, 746.181871700547, 748.94292885502, 751.726370135731, 754.530032321146, 757
Apr May Jun Jul Aug Sep Oct
0
300 600
Putnam, population=0.07M83, 18384, 18385, 18386, 18387, 18388, 18389, 18390, 18391, 18392, 18393, 18394, 18395, 18396, 18397, 18398, 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 18461, 18462, 18463, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 427327677173, 514.101683980298, 516.80373830906, 519.52717361383, 522.271160407293, 525.036814585745, 527.826025130745, 530.640966227729, 533.483865514584, 536.356884530771, 539.262053920819, 542.201237558104, 545.176112897103, 548.
Apr May Jun Jul Aug Sep Oct
0
20 40
Santa Rosa, population=0.18M368, 18369, 18370, 18371, 18372, 18373, 18374, 18375, 18376, 18377, 18378, 18379, 18380, 18381, 18382, 18383, 18384, 18385, 18386, 18387, 18388, 18389, 18390, 18391, 18392, 18393, 18394, 18395, 18396, 18397, 18398, 18399, 18400, 18401, 18402, 18403, 18404, 18405, 18406, 18407, 18408, 18409, 18410, 18411, 18412, 18413, 18414, 18415, 18416, 18417, 18418, 18419, 18420, 18421, 18422, 18423, 18424, 18425, 18426, 18427, 18428, 18429, 18430, 18431, 18432, 18433, 18434, 18435, 18436, 18437, 18438, 18439, 18440, 18441, 18442, 18443, 18444, 18445, 18446, 18447, 18448, 18449, 18450, 18451, 18452, 18453, 18454, 18455, 18456, 18457, 18458, 18459, 18460, 6788403928561, 39.8745445298696, 40.0769180182343, 40.2850048391262, 40.498092668356, 40.7156138027191, 40.9370979458802, 41.162145304188, 41.3904095743427, 41.6215864156586, 41.855405241066, 42.0916231478082, 42.3300202966118, 42.5
Apr May Jun Jul Aug Sep Oct
0
40 80
, 18520, 18 Apr May Jun Jul Aug Sep Oct0 100
250
Seminole, population=0.47M
3,
,18 Apr May Jun Jul Aug Sep Oct
0
20
40
Amador, population=0.04M
3,
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20
40
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150
8,
, 18519, 18520, 18 Apr May Jun Jul Aug Sep Oct Santa Barbara, population=0.45M0 50
150
3,
753755132334, 111.338202143676, 111.931306682137, 112.524216323358, 113.113425827223, 113.697416583834, 114.275563305737, 114.847683343052, 115.413823781839, 115.974151662504, 116.528893975075, 117.078303603022, 117.622639594609, 118.162155692848, 118.697093783182, 119.22768033818 Apr May Jun Jul Aug Sep Oct Santa Clara, population=1.93M0
200
98,
892426470318, 299.279856367286, 300.744344008124, 302.24942794278, 303.778378729182, 305.322635039809, 306.877643620095, 308.44100798183, 310.011556235422, 311.588831952981, 313.172799842154, 314.763668584273, 316.361781137717, 317.96754555988, 319.581390984978, 321.203739630123, Apr May Jun Jul Aug Sep Oct Santa Cruz, population=0.27M0 5
15
83,
3077135052886, 14.4963641007691, 14.687878979021, 14.8802012031488, 15.0728414877222, 15.2657492099915, 15.4590050355609, 15.6527205320469, 15.8470034052703, 16.041946153466, 16.2376235479312, 16.4340934210509, 16.6313985648228, 16.8295688529782, 17.0286232411708, 17.2285715285906 Apr May Jun Jul Aug Sep Oct0
40
80
Solano, population=0.45M
8,
, 18520, 18 Apr May Jun Jul Aug Sep Oct0 50
150
Sonoma, population=0.49M
3,
82257673615, 115.478763834034, 116.185438440129, 116.918341537021, 117.666087595771, 118.422593717617, 119.184404800464, 119.949511876249, 120.716756138084, 121.485500422287, 122.255436285251, 123.026465315484, 123.798623543393, 124.572032053913, 125.346864121125, 126.123323073492, 126.901627296514, 127.682000066577, 128.4646627 Apr May Jun Jul Aug Sep Oct Stanislaus, population=0.55M0
200 400
98,
167963401355, 341.09558049463, 343.907315555891, 346.620338325922, 349.247763390557, 351.79991162663, 354.285138754164, 356.710358735545, 359.081385564517, 361.403166305346, 363.679946707349, 365.915393349932, 368.112686682146, 370.274593879324, 372.403527246404, 374.501591965872, 376.570625780091, 378.612232426377, 380.62781012 Apr May Jun Jul Aug Sep Oct Sutter, population=0.1M0
5 10
83,
AcknowledgementsThis research is supported by the UCSB Office of Research COVID-19 seed grant program.Author contributions statementH.L. analyzed data, developed the model, derived mathematical results, wrote computer code, collected results and participated in manuscript writing. M.G. conceptualized the project, analyzed data, developed the model, derived mathematical results, wrote computer code, analyzed results, led manuscript writing.
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Association between mobility patterns and COVID-19 transmission in the USA: a mathematical modelling study. H S Badr, The Lancet Infect. Dis. Badr, H. S. et al. Association between mobility patterns and COVID-19 transmission in the USA: a mathematical modelling study. The Lancet Infect. Dis. (2020).
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Early dynamics of transmission and control of COVID-19: a mathematical modelling study. The lancet infectious diseases. A J Kucharski, Kucharski, A. J. et al. Early dynamics of transmission and control of COVID-19: a mathematical modelling study. The lancet infectious diseases (2020).
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|
[] |
[
"The fermion bag approach to lattice field theories",
"The fermion bag approach to lattice field theories"
] |
[
"Shailesh Chandrasekharan \nDepartment of Physics\nDepartment of Theoretical Physics\nDuke University\nBox 9030527708DurhamNorth CarolinaUSA\n\nTata Institute of Fundamental Research\nHomi Bhaba Road400005MumbaiIndia\n"
] |
[
"Department of Physics\nDepartment of Theoretical Physics\nDuke University\nBox 9030527708DurhamNorth CarolinaUSA",
"Tata Institute of Fundamental Research\nHomi Bhaba Road400005MumbaiIndia"
] |
[] |
We propose a new approach to the fermion sign problem in systems where there is a coupling U such that when it is infinite the fermions are paired into bosons and there is no fermion permutation sign to worry about. We argue that as U becomes finite fermions are liberated but are naturally confined to regions which we refer to as fermion bags. The fermion sign problem is then confined to these bags and may be solved using the determinantal trick. In the parameter regime where the fermion bags are small and their typical size does not grow with the system size, construction of Monte Carlo methods that are far more efficient than conventional algorithms should be possible. In the region where the fermion bags grow with system size, the fermion bag approach continues to provide an alternative approach to the problem but may lose its main advantage in terms of efficiency. The fermion bag approach also provides new insights and solutions to sign problems. A natural solution to the "silver blaze problem" also emerges. Using the three dimensional massless lattice Thirring model as an example we introduce the fermion bag approach and demonstrate some of these features. We compute the critical exponents at the quantum phase transition and find ν = 0.87(2) and η = 0.62(2).
|
10.1103/physrevd.82.025007
|
[
"https://arxiv.org/pdf/0910.5736v1.pdf"
] | 17,818,692 |
0910.5736
|
9c39a133bae5ba51313e2d0c9a6b9d8e6b4d91e8
|
The fermion bag approach to lattice field theories
29 Oct 2009
Shailesh Chandrasekharan
Department of Physics
Department of Theoretical Physics
Duke University
Box 9030527708DurhamNorth CarolinaUSA
Tata Institute of Fundamental Research
Homi Bhaba Road400005MumbaiIndia
The fermion bag approach to lattice field theories
29 Oct 2009numbers: 7110Fd0270Ss1130Rd0530Rt
We propose a new approach to the fermion sign problem in systems where there is a coupling U such that when it is infinite the fermions are paired into bosons and there is no fermion permutation sign to worry about. We argue that as U becomes finite fermions are liberated but are naturally confined to regions which we refer to as fermion bags. The fermion sign problem is then confined to these bags and may be solved using the determinantal trick. In the parameter regime where the fermion bags are small and their typical size does not grow with the system size, construction of Monte Carlo methods that are far more efficient than conventional algorithms should be possible. In the region where the fermion bags grow with system size, the fermion bag approach continues to provide an alternative approach to the problem but may lose its main advantage in terms of efficiency. The fermion bag approach also provides new insights and solutions to sign problems. A natural solution to the "silver blaze problem" also emerges. Using the three dimensional massless lattice Thirring model as an example we introduce the fermion bag approach and demonstrate some of these features. We compute the critical exponents at the quantum phase transition and find ν = 0.87(2) and η = 0.62(2).
matrix. If this determinant can be shown to be always positive the sign problem is solved. We refer to Monte Carlo algorithms based on of this approach as conventional algorithms. Even when the sign problem is solved, these conventional algorithms can be inefficient since the problem becomes completely non-local in the system size. One well known problem is that often the fermion matrix develops a large number of small eigenvalues. In these cases the algorithms slow down substantially with system size. In practical calculations, the small eigenvalues of the fermion matrix are controlled by the addition of new couplings to the theory which are then extrapolated to zero to extract physical answers. This introduces systematic errors which cannot easily be controlled. Finally, and most importantly, when the determinant is not positive, little insight can be gained about the fermion sign problem itself. In contrast to the auxiliary field method, the meron cluster method is based on cleverly rewriting the partition function as a sum over configurations that naturally divide the physical system into clusters or regions so that the sign problem is solved by re-summing configurations within each region. Due to the cleverness involved, the method is not widely applicable. On the other hand whenever it works, large system sizes can be studied more easily since the problem breaks up the system into smaller regions and one does not have to consider the entire system size to solve the fermion sign problem. In particular lattices of the order of 128 × 128 have been solved using this method [13]. Additional couplings to control the efficiency of the algorithm become unnecessary.
In this work we propose a more general approach to the fermion sign problem based on the underlying physics. In a sense we extend the meron cluster idea by combining it with the determinantal trick to solve the fermion sign problem in a wider class of theories. The essential idea is that many fermionic theories contain a coupling, which we will call U, such that when U = ∞ the fermions become paired into bosons and the partition function is naturally written with positive definite Boltzmann weights. In other words there is no fermion sign to worry about. When the coupling is large but not infinite, fermions become unpaired but remain confined to small regions which we refer to as fermion bags. The fermion sign problem is then confined to these bags and can sometimes be solved using the usual determinantal trick. When the bags remain small, the computational effort to solve the sign problem does not grow with the system size just like the meron cluster approach. Thus, Monte Carlo methods for these problems can be far more efficient than algorithms which do not take this physics into account. As the coupling reduces further the fermion bags merge and begin to grow with the volume. In this region the fermion bag approach loses its main advantage and suffers form similar slowing down as the auxiliary field methods.
However, it is useful to remember that at small couplings perturbation theory is usually a good approach and the recently proposed diagrammatic Monte Carlo method may be a better approach for small and moderate values of the couplings [14].
The main message behind the fermion bag idea is the following: When fermions are delocalized over the whole system, the increased computational cost associated to dealing with fermionic degrees of freedom is natural. But it is definitely unnatural in the regime where fermions are confined to small regions. The auxiliary field method to the fermion sign problem does not make use of this underlying physical picture. The similarity of the fermion bag approach to the meron cluster approach is striking: The bags, like the clusters, do not occupy the whole volume and makes the computational effort somewhat reduced. In addition, as we will discuss in this work, new insights and solutions to the fermion sign problems emerge. The fermion bag idea was first discussed in [9].
Our article is organized as follows. In section 2 we illustrate the ideas outlined above concretely using a simple but relatively less studied example of the massless lattice Thirring model constructed with a single flavor of staggered fermions. In particular we contrast the fermion bag approach with the auxiliary field method. In section 3, we introduce a fermion chemical potential and discuss how the silver-blaze problem [15], present in the auxiliary field method, is naturally solved in the fermion bag approach. In section 4, we given an example of a sign problem which seems unsolvable in the auxiliary field formulation but is solvable in the fermion bag approach. In Section 5 we discuss update algorithms for the massless Thirring model in the bag formulation.
In Section 6, we discuss the fermion bag distribution in the massless Thirring model. In particular we show that the typical fermion bag size does not grow with system size for U 1.2. In section 7, we discuss some results obtained using the bag approach in the massless Thirring model and in section 8, we discuss the quantum critical behavior. Section 9 contains our conclusions where we argue that the fermion bag approach is rather general and must be applicable to many lattice field theories. In particular we show how similar ideas may be adapted to the physics of the BCS-BEC crossover.
II. THE FERMION BAG APPROACH
Although the fermion bag approach is applicable to a wide class of problems in any dimension, it is useful to understand the details in the context of a simple model. Here we introduce the 4 fermion bag approach using the example of the massless lattice Thirring model with one flavor of staggered fermions on a three dimensional cubic lattice. The action is given by
S = − x,y ψ x D x,y ψ y − U x,α ψ x+α ψ x+α ψ x ψ x (1)
where the matrix D is the free staggered Dirac operator given by [16]
D x,y = η x,α 2 [δ x+α,y − δ x,y+α ].(2)
In our notation x ≡ (x 1 , x 2 , x 3 ) denotes a lattice site on a 3 dimensional cubic lattice of size L 3 , ψ x and ψ x , are Grassmann valued fields and α = 1, 2, 3 runs over the three positive directions.
The staggered fermion phase factors η x,α = exp(iπζ α · x) are defined through the 3-vectors ζ 1 = (0, 0, 0), ζ 2 = (1, 0, 0) and ζ 3 = (1, 1, 0). We also define the phase ε x = (−1) x 1 +x 2 +x 3 for later convenience.
The main feature of the model is that it contains massless fermions interacting with each other
with a U f (1) × U χ (1) chirally invariant interaction. Indeed it is easy to check that the action is invariant under the usual fermion number U f (1) transformations: ψ x → exp(iθ)ψ x and ψ x → exp(−iθ)ψ x , and the chiral U χ (1) transformations: ψ x → exp(iε x θ)ψ x and ψ x → exp(iε x θ)ψ x .
When U = 0 the model describes free massless Dirac fermions. At infinite U, all fermions are confined and the model reduces to a hardcore dimer model made up of paired fermions and the low energy physics is in the same universality class as the XY model in its broken phase [17]. Hence at some critical value U c the model undergoes a quantum phase transition. This model and its variants have been studied earlier with the auxiliary field method [18,19,20,21,22,23,24,25,26].
However, none of the earlier calculations were performed in the massless limit due to algorithmic difficulties. Here we use the fermion bag approach to tackle the massless limit for the first time.
The partition function of the model is given by
Z = x [dψ x dψ x ] exp(−S)(3)
where the integration is over the Grassmann fields. In the determinantal approach one uses the Hubbard-Stratanovich transformation to convert the four fermion coupling into a fermion bi-linear at the cost of introducing an integral over an auxiliary bosonic field. It is easy to verify that
Z = dφ [dψdψ] exp x,y ψ x (M[φ]) x,y ψ y(4)
where M[φ] is given by
M([φ]) = η µ (x) δ x+µ,y ( 1 2 + √ Ue iφµ(x) ) − δ x,y+µ ( 1 2 + √ Ue −iφµ(x) ) .(5)
The auxiliary field φ α (x) is integrated over the angles 0 ≤ φ µ (x) < 2π. Integrating over the Grassmann variables first we can obtain
Z = [dφ] Det(M([φ])).(6)
The matrix M is anti-Hermitian and so its eigenvalues are purely imaginary. Further, it anticommutes with the matrix Ξ x,y = ǫ x δ x,y which means that if λ is an eigenvalue then so is −λ. How is the fermion bag approach different? Instead of introducing an auxiliary field to rewrite the four-fermion term as a fermion bi-linear, we begin with the partition function given by
Z = [dψdψ] exp x,y ψ x D x,y ψ y + x,α Uψ x ψ x ψ x+α ψ x+α ,(7)
and expand it in powers of U using
exp(Uψ x ψ x ψ x+α ψ x+α ) = 1 + Uψ x ψ x ψ x+α ψ x+α .(8)
The Grassmann integration then gives
Z = nx,α=0,1 x,α U nx,α Det(W [n])(9)
where n x,α = 0, 1. The n x,α = 1 bonds are referred to as dimers. Note that in this approach the
Det(W [n]) = i Det(W [B i ])(10)
Thus, we see that fermions have become classical objects when they are considered as non-local objects in the form of bags. For this reason we call our method as the fermion-bag approach. The size, the shape and other properties of these bags encode the fermion physics.
Let us now discuss the effort required to generate a statistically independent configuration in the fermion bag approach using a specific algorithm discussed later in section V. In our algorithm, effort of obtaining a statistically independent configuration is then of the order N B N CG L 3 . Here we assume that one sweep through the lattice is sufficient to generate such a configuration. This is almost true due to the availability of a directed path update (see section V). When U is large, the bags are small comprising of a few neighboring bonds and thus independent of the volume.
Although each local update can be difficult the computational cost of a local update (N B N CG )
does not grow with the volume. This makes the fermion bag approach far more efficient for large system sizes compared to the determinantal approach. The former scales as N B N CG L 3 while the latter scales as L 7 as discussed earlier.
When U is small the bags can percolate and become as big as the system size. Here we expect
N B ∼ L 3 .
On the other hand since the fermions are almost free, the matrix inversion using the conjugate gradient algorithm becomes easy. we find that N CG ∼ L and hence the overall effort now grows as L 7 . On the other hand the auxiliary field method based on the HMC algorithm scales as L 5 and so is clearly superior. It may be interesting to explore a HMC type algorithm in the fermion bag approach if possible so as to combine the good features of both. But this is not the focus of the current work. Further, as mentioned earlier, at small U the diagrammatic Monte Carlo algorithm may be the best option [14].
At intermediate values of U, especially close to the phase transition, the HMC most likely continues to scale as L 7 or worse due to critical slowing down. On the other hand the scaling of 8 the bag algorithm is more tricky and needs to be studied carefully. We find three reasons to remain optimistic: (1) The bags of all sizes exist in the simulation so some updates are much faster, (2) The matrix W B x is the free matrix except for mesoscopic fluctuations coming from the boundaries of the bag. Hence N CG may scale favorably in the bag approach, (3) The existence of the directed path algorithm to update variables outside the bag may eliminate a lot of the critical slowing down.
More research is necessary to compare the two algorithms in the intermediate U region.
III. SOLUTION OF THE SILVER BLAZE PROBLEM
The fermion bag approach also offers new insights into sign problems. Here we discuss a simple resolution of the so called silver blaze problem, a general paradox related to sign problems that arises in the auxiliary field method in the presence of a fermion chemical potential [15]. Before we discuss how the fermion bag approach solves this problem, let us first review its origin in the current context of the massless lattice Thirring model.
In the presence of a chemical potential µ, the Dirac operator given in Eq.
(2) changes to
D(µ) x,y = η x,α 2 [δ x+α,y e µδα,t − δ x,y+α e −µδα,t ].(11)
In the auxiliary field method the four-fermion term is again converted to a fermion bi-linear using the Hubbard Stratanovich transformation and the partition function is again given by Eq. (6),
except that the matrix M[φ] is now given by M([φ]) = η α (x) δ x+µ,y ( 1 2 + √ U e iφα(x)+µδα,t ) − δ x−µ,y ( 1 2 + √ U e −iφµ(x)−µδα,t ) .(12)
Unfortunately, the properties that we used to argue that Det(M[φ]) ≥ 0 are no longer valid when µ = 0. Indeed the determinant can be negative as soon as µ = 0 for all values of U. This is the well known sign problem in the presence of a chemical potential.
Consider large values of U where the fermions become massive due to chiral symmetry breaking. In this phase the chemical potential should have no effect on the ground state of the system until a critical chemical potential is reached. This means, for low temperatures a small chemical potential must have little effect on the physics. However, the sign problem does not respect this mild behavior with respect to the chemical potential. The sign problem becomes severe as soon as the chemical potential is non-zero at small temperatures. This paradox has been called the silverblaze problem [15]. The auxiliary field method offers almost no explanation for this paradox, except the fact that the cancellations due to the sign problem are crucial to get the right physics. For small U when the fermions are massless, temporal winding bags proliferate and the fermion bag approach also suffers from a severe sign problem in the presence of a chemical potential. We do claim to have a solution to this sign problem in the case of a single flavor of staggered fermion.
However, in the next section we argue that the fermion bag approach allows us to solve this sign problem with an even number of flavors.
IV. NEW SOLUTIONS TO SIGN PROBLEMS
The fermion bag approach also offers new solutions to some seemingly unsolvable sign problems. In order to appreciate this consider the action of the N flavor model given by the action
S = − x,y ψ x D(µ) x,y ψ y + U x,α (ψ x+α ψ x ) (ψ x ψ x+α )(13)
where ψ x is an N-component column vector and ψ x is an N-component row vector. This action is invariant under a U(N) × U(N) symmetry. The partition function in the auxiliary field method turns out to be
Z = [dφ] Det(M([φ])) N(14)
where the matrix M[φ] is the same as the one-flavor model given in Eq. (12).
Since the Det(M[φ])
is a general complex number in the presence of a chemical potential, its Nth power remains complex for all N. Unfortunately, this sign problem remains unsolved within the fermion bag approach as well. On the other hand consider the model given by the action
S = − x,y ψ x D(µ) x,y ψ y − U (N!) 2 x,α (ψ x+α ψ x ) (ψ x ψ x+α ) N .(15)
This action is again invariant under the same U(N) × U(N) chiral symmetry. In the auxiliary field method one will need many auxiliary fields to convert the 4N-fermion term to a bi-linear. Further, even with these additional fields, it is difficult to see why the determinant of the fermion matrix that will arise will be positive for any value of N for the same reasons outlined above. On the other hand in the fermion bag approach this modified model is described by the partition function
Z = nx,α=0,1 x,α U nx,α Det(W [n]) N .(16)
Since Det(W [n]) is real, there is no sign problem with even N. Thus, the fermion bag approach is able to solve a sign problem that seems unsolvable with the auxiliary field method. In this context we must point out that there are indeed other actions that are invariant under U(N) × U(N) symmetry whose partition functions can be written without a sign problem using the auxiliary field method for even values of N.
11
V. THE MONTE CARLO METHOD
In order to solve the massless Thirring model using the fermion bag approach, in this section we discuss two update algorithms: (1) Local Heal Bath, and (2) Directed Path algorithm. For generality we discuss these algorithms for any dimension d, although the current work is focused on d = 3. We will argue that these two update algorithms together provide an efficient way to solve the problem for U 1.2. When 0.2 < U < 1.2 these algorithms do slow down dramatically, however they continue to provide a useful way to solve the problem.
A. Local Heat Bath
The first update we discuss creates and destroys dimers. This is accomplished with a local heat update. The exact update is as follows:
1. Pick a site x at random on the lattice.
We pick α with probability
P α = ω α α ω α(17)
5. If α = 0 we set n x,α = 0 for all values of α and stop. Otherwise we set n x+α = 1 and others to zero then stop.
We define a sweep as consisting of (L/2) 3 local heat updates updates.
The most time consuming step of this local heat bath update is the computation of This is the reason for the efficiency of the Monte Carlo method in the fermion bag method. We will show that this is indeed the case when U 1.0.
(W −1 [B x ]) x,
In order to compute the inverse we set the vector b y = δ x,y and then solve the equation
W v = b.
Practically we solve (−W 2 )v = (−W b) and since (−W 2 ) is a positive definite matrix, we can use the conjugate gradient method. The convergence of the answer is checked by the parameter γ = |W v−b| 2 . If this norm is less than 10 −20 we assume that the solution has been found. Another useful norm is γ ′ = |(−W 2 )v + W b| 2 and can be used to detect exact zero modes of W using the conjugate gradient method. Note that (−W b) eliminates the zero mode subspace from the source vector and the space on which conjugate gradient acts. Thus the conjugate gradient method can always make γ ′ arbitrarily small. If γ cannot be made smaller than 10 −20 even when γ ′ < 10 −30 , we declare that configuration to have an exact zero mode. This method appears to work reliably.
B. Directed Path Update
The second update preserves the number of dimers but moves them around. This update is similar to the directed path update discussed in [28] and reduces to it in the limit of large U. The philosophy behind it is similar to the worm algorithm [29]. The update is as follows:
1. Pick a site x at random. (c) If x is not the first site such that n x,α = 0 for all the values of α and we just came to the site from the previous site x +β, then we pick a direction γ with probability P γ (x)
to be discussed below. We set m x+β = 0 and m x+γ = 1. In other words we move the monomer from the site x +β to x +γ. Let us now discuss the probability P γ (x) on an active site x such that n x,α = 0 for all values of α and such that m y = 1 where y = x +β. The site x is associated to a fermion bag say B x . Note that the passive site y is not in the bag. Let x0 be another active site which is not in the bag, but contains a neighboring site which is in the bag. Thus, both x0 and y "touch" the bag B x but are not a part of it. Let us extend the bag to include both y and x0 call the extended bag B x,x0,y . The probability P γ is then given by
P γ (x) = ω x+γ γ ω x+γ(18)
where
ω z = |[(W [B x,x0,y ) −1 ] x 0 ,z | 2 .
It is possible to show that while ω z depends on x 0 , P γ does not.
Further, if x +γ does not belong to the bag B x,x0,y P γ (x) = 0. During the passive update on there is a probability to remain on the same site. This can be shown to be the correct probability to create a monomer at that site along with another monomer at the first site. Hence it contributes to the two point correlation function
G(x, y) = ψ x ψ x ψ y ψ y(19)
Thus, we can compute this correlation function during this update. Here we use this to compute the susceptibility. We have tested the algorithm against exact calculations on a small lattice and the results are given in the appendix. that the critical point where the fermion becomes massless is roughly around U ∼ 0.25 [22] we think that the scale M is not the mass of the fermion. We postpone the study of its origin to a later publication.
VI. DISTRIBUTION OF FERMION BAGS
In Fig. 4 the bag distribution is shown for many values of U at L = 8. We see that distribution changes qualitatively when U becomes small. Instead of decaying exponentially with size, a bump develops in the bag distribution at a value of S of about half the system size. The position of the bump then begins to grow till it becomes as big as the system size for very small U. This is easily understandable. As U reduces the bags merge so that now the whole lattice becomes one large bag with small regions of confined (or paired) fermions which in a sense form "dual bags".
It is useful to define a typical size of a fermion bag as the size of the bag that one encounters on an average during the update. More precisely we pick a site at random and define S τ as the size of the bag associated to that site. We can then average it over the ensemble of the configurations generated. It is easy to argue that
S τ = 1 L 3 B S 2 B(21)
where S B is the size of the bag B and the sum is over all the bags in a given configuration. In
VII. RESULTS IN THE MASSLESS THIRRING MODEL
In this section we present some results obtained using the fermion bag approach in the massless Thirring model with one flavor of staggered fermions in three dimensions. We focus on the following three observables:
1. Chiral condensate susceptibility:
χ = U L 3 x,y ψ x ψ x ψ y ψ y .(22)
Here the factor U ensures that in the U = ∞ limit the chiral condensate Σ is finite.
Fermion charge susceptibility
:
Q 2 f = x∈S,y∈S ′ J f α,x J f α,y .(23)
Here S and S ′ are two different two-dimensional surfaces perpendicular to the direction α and
J f α,x = η x,α 2 ψ x ψ x+α + ψ x+α ψ x(24)
is the conserved fermion current. In the bag formulation one can show that
Q 2 f = 1 2 x∈S,y∈S ′ η x,α η x,α (D −1 ) x,y+α (D −1 ) x+α,y + (D −1 ) x,y (D −1 ) x+α,y+α(25)
where D −1 is the inverse of the free Dirac operator inside the bag containing all the four points x, y, x + α, y + α. If any of these points is not part of the bag then the corresponding contribution is set to zero. It is easy to check that Q 2 f defined here is independent of the surfaces chosen on every bag configuration.
3. Chiral charge susceptibility:
Q 2 χ = x∈S,y∈S ′ J χ α,x J χ α,y(26)
where the notation is the same as for the fermion charge susceptibility, except the conserved chiral current is given by
J χ α,x = ε x η x,α 2 ψ x ψ x+α − ψ x+α ψ x .(27)
On every bag configuration let us define
q χ = x∈S(B) ε x η x,α (D −1 ) x,x+α + x∈S(C) 2ε x(28)
where S(B) is the set of sites on the two-dimensional surface S which belongs to some fermion bag while S(C) are the remaining set of sites on the surface. For a given bag configuration we find that q χ is also independent of the surface. Further
Q 2 χ = Q 2 f + q 2 χ ,(29)
where the average is over the fermion bag configurations generated.
For large values of U chiral symmetry is broken spontaneously and fermions acquire a mass.
Chiral perturbation theory predicts that [30],
q 2 χ = 4ρ s L[1 + 0.224 ρ s L + a (ρ s L) 2 ] (30a) χ = L 3 Σ 2 2 1 + 0.224 ρ s L + b (ρ s L) 2(30b)
and that Q 2 f goes to zero exponentially. The parameters ρ s , Σ, a and b are the low energy constants and can be determined from fitting the data. The constant ρ s is a mass scale and plays the role of F π in four dimensional chiral perturbation theory. The chiral condensate in the chiral limit is given by Σ. The factor 4 in the expression for q 2 χ is not standard. However in our definition the charge q χ takes only even values at U = ∞ and so ρ s will not be properly normalized without this factor. For small values of U chiral symmetry is restored, but due to the presence of massless fermions we expect
χ = U(χ 0 + χ 1 /L + χ 2 /L 2 + ...) (31a) q 2 χ = q 1 /L + q 2 /L 2 + q 3 /L 3 .... (31b) Q 2 f = γ 0 + γ 1 /L + γ 2 /L 2 + ...(31c)
For free staggered fermions (i.e., when U = 0 ) we find χ 0 ≈ 1.01093 and γ 0 ≈ 0.37085 while
q 2 χ = 0.
Our data are consistent with these expectations qualitatively for both large and small values of U. In Fig. 6 we plot the three observables scaled appropriately as a function of L for different values of U (left plots) and for L = 8, 12 as a function of U (right plots). We see that there the finite size scaling changes abruptly between U = 0.2 (symmetric phase) and U = 0.3 (broken phase). In particular Σ and ρ s are non-zero for U ≥ 0.3, but vanish for U ≤ 0.2. On the other hand Q 2 f remains non-zero when U ≤ 0.2 and begins to drop significantly when U ≥ 0.3. Thus, there is a clear phase transition in the model somewhere between these two couplings. We will study this quantum phase transition quantitatively in the next section.
Quantitatively, we can fit our data to expectations from chiral perturbation theory only for U ≥ 1.2 where the fermion bag approach allows us to go to large volumes. Typically we have found that the fits to the finite size scaling forms given in Eq. (30) are good in the range 16 ≤ L ≤ 32.
The values of the low energy constants that give good fits for U ≥ 1.2 are given in table I. We observe that a change from U = ∞ to U = 1.2 leads to only about 7% change in the mass scale ρ s and about 10% change in the chiral condensate. This again shows that the effort of conventional determinantal methods is unnecessary and the fermion bag approach is the better method in this range of couplings. For small values of U, in the symmetric phase, the fermion bag approach slows down considerably. Hence we have been able to perform calculations only up to L = 20. In While our data fits to the expected finite size scaling forms even for small U, due to the small system sizes used in the calculations we do not claim to be able to extract the L dependence reliably except for the free theory. In particular we expect large systematic errors in the determination of the coefficients shown in table II. In fact we have used the free theory to guide our fits. For example it is tempting to associate γ 0 = 0.370840.. to a universal constant for free massless fermions. It is likely that this does not change with U since we expect that the infrared physics to be that of free massless fermions. The small change that we observe in our fits perhaps is due to systematic errors associated to fitting the data at smaller lattice sizes. Further work is clearly necessary to reliably understand the small U regime.
VIII. QUANTUM PHASE TRANSITION
We next focus on the quantum phase transition in the model. This transition has already been studied earlier using mean field theory [18], auxiliary field method [21] and through a formulation as a strongly coupled U(1) lattice gauge theory with scalar fields [18,23]. In the latter two studies, algorithmic difficulties forced the use of a non-zero fermion mass. This usually makes the study of finite size scaling close to a second order transition more tricky since the fermion mass introduces a new length scale. However, by making a judicious choice of the scaling relations, both these studies concluded that the transition was consistent with a second order transition. The critical coupling was estimated to be U c = 0.25(1) [21] and U c = 0.28(1) [23], while the critical exponents were determined to be δ ≈ 2.5(4), β ≈ 0.71 (9) [21] and δ ≈ 3.1(3), ν ≈ 0.88(9) [23]. For comparison, mean field theory in d = 3 predicts U c = 0.177(6), δ = 2 and β = 1 [18,31].
The errors we quote here include systematic errors estimated from the different fitting procedures used in the previous work. In the present work we estimate these parameters again in the fermion bag approach.
One main difference as compared to the previous work is that we work with exactly massless fermions which helps with a clean finite size scaling analysis. Assuming the transition to be a conventional second order phase transition we expect
q 2 χ = κ 0 + κ 1 (U − U c )L 1 ν + κ 2 (U − U c ) 2 L 2 ν + ... (32a) χ/L 2−η = f 0 + f 1 (U − U c )L 1 ν + f 2 (U − U c ) 2 L 2 ν + ...(32b)
A combined fit of our data to these relations give the following results: we estimate β ≈ 0.71(4) and δ ≈ 2.70 (4). While, these results are in complete agreement with the earlier work, our values seem more accurate. One obvious caveat is that our results are also obtained on rather small lattices compared to what is available in bosonic models. Thus, we may be underestimating some systematic errors. On the other hand we have exactly massless fermions so some of the fitting procedures are cleaner. In any case accurate results on larger lattices are desirable to confirm these findings.
η ν U c f 0 f 1 f 2 f 3 κ 0 κ 1 κ 2 κ
It was pointed out in [32] that the Thirring model is different from the Gross-Nevue (GN) model in many ways. However, recent work suggests that the two models may be equivalent close to the quantum critical point and that studies in both these models are relevant to the physics of graphene [33,34]. The critical exponents in a lattice GN model with a U f (1) × Z 2 symmetry with staggered fermions have been computed and it was found that ν = 1.00(4) and η = 0.754(8) [35].
These critical exponents have also been obtained from other techniques [36,37] and they clearly appear to be different from what we find above. However, since the lattice symmetries are different between the two models, the critical behavior may fall under different universality classes. The critical behavior with a continuous U f (1) × U χ (1) GN model was studied in [38]. It was found that ν = 1.02 (8) and β = 0.89(10) which also seem to be different from our results but only at the 2-σ level. More work is necessary to clarify the issue of whether the GN model and the Thirring 25 model have the same critical exponents.
IX. DISCUSSION AND CONCLUSIONS
In this work we have introduced a new approach to the fermion sign problem which we call the fermion bag approach. The essential idea behind the new approach is to recognize that fermionic degrees of freedom are usually contained inside dynamically determined space-time regions (bags). Outside these regions they are hidden since they become paired into bosonic objects. Then it is likely that the sign problem is localized to the regions inside these bags and may be solved using determinantal tricks. Such a scenario is clearly natural in systems where there is a coupling U such that when U is infinite all the fermions are paired up and there is no sign problem.
Then as U becomes finite, the the fermions are liberated out but are confined to bags. In such systems, if the sign problem is contained within the bags and can be solved, then there will be regions in parameter space where the bags do not grow with the volume. In these regions of parameter space it should be possible to construct Monte Carlo algorithms which are far more efficient than conventional algorithms. In this work we showed an explicit example of a lattice field theory, namely the massless lattice Thirring model, where all these features are realized. In particular we developed a Monte Carlo algorithm and showed that it is efficient for large couplings as expected.
For smaller couplings the efficiency of the algorithm was lost, but still the fermion-bag approach continued to provide an alternative approach to the problem. In particular we could determine the critical exponents near the quantum phase transition present in the model with reasonable effort.
While the fermion bag approach loses its main advantage when the bags begin to merge and the fermions are delocalized in the entire space-time volume, it still provides new insights into the sign problem itself. For example we explained in this work, how a natural solution to the silver blaze problem emerges within this approach. Further, we also showed that new solutions to sign problems emerge. Although the discussion in this work focused on a single model, the idea behind the fermion bag approach is more general and must be applicable to other lattice field theories when formulated cleverly. As an example below we sketch how one can formulate a model to study the physics of the BCS-BEC cross over in the fermion bag approach.
Let
S 1 = − i,t(33a)
e 2µε ψ ↓,i,t+1 ψ ↑,i,t+1 ψ ↑,i,t ψ ↓,i,t + ψ ↓,i,t ψ ↑,i,t ψ ↑,i,t ψ ↓,i,t + ε e 2µε e i ψ ↓,e i ,t+1 ψ ↑,e i ,t+1 ψ ↑,i,t ψ ↓,i,t .
Note that the action is invariant under SU(2) spin symmetry and U(1) fermion number symmetry as required.
The partition function is given by Z =
x,t,s
[dψ x,t,s dψ x,t,s ] exp(−S). A sketch of such a configuration is shown in Fig. 9. The regions where the fermions are free are shown in as a bag in the figure. Further, it is easy to argue that the Boltzmann weight is always positive due to two flavor nature of the problem.
Based on the above reasoning we believe we have uncovered a somewhat unconventional approach to fermionic field theories which may prove to be a powerful alternative, especially when the coupling strengths are large and where perturbation theory is expected to fail.
APPENDIX A: ALGORITHM VERSUS EXACT RESULTS
In this section we present some exact results on a 2 3 lattice and compare them with the results from the algorithm in order to test the algorithm. Table III gives the various possible configurations (their degeneracy factors (Deg), the corresponding bag determinants (Bdet) and the Boltzmann weights (BWt)). We find that the partition function is given by
The average number of bonds is given by
N B = 1 4Z (216U + 624U 2 + 864U 3 + 576U 4 )(A2)
where we have normalized it so that for large U it approaches one. The typical bag size is given by S τ = 1 Z (648 + 972U + 600U 2 + 144U 3 )
The condensate susceptibility is obtained from configurations with two monomers. These along with their degeneracy factors, bag determinants and Boltzmann weights are given in
N
MD be the number of molecular dynamics steps necessary to generate a statistically independent configuration. Each step of the molecular dynamics update requires the computation of the force which requires a particular matrix element of (M[φ]) −1 . Typically this requires N CG L 3 operations where N CG is the number of conjugate gradient steps in the inversion process. Thus, the cost of generating an independent configuration in an HMC is given by L 3 N CG N MD . Both N CG and N MD are dependent on the physics and the model. In the current context, for large and intermediate values of U, the matrix M[φ] contains a non-zero density of small eigenvalues due to chiral symmetry breaking. Hence one expects N CG ∼ L 3 . On the other hand N MD grows with the largest correlation length in the problem and for the moment we will assume this to be L which is the best case scenario since the theory contains massless particles. Thus, the HMC effort scales at least as L 7 . One can reduce N CG drastically if we can can control the small eigenvalues of the matrix M[φ]. This is usually accomplished by adding a fermion mass term. This is the reason why all previous calculations of the Thirring model always used a non-zero fermion mass. No calculations of the massless Thirring model at large U have been attempted using the HMC method. At small values of U experience shows that N CG ∼ L since the fermions are almost free. Assuming that again N MD ∼ L, the HMC effort now scales as L 5 .
Grassmann integration leads to a determinant of a different matrix W [n], which is just the free fermion matrix where the sites connected to n x,α = 1 are dropped. It is easy to verify that W[n] is also anti-Hermitian and anti-commutes with Ξ and so Det(W [n]) ≥ 0. Thus the sign problem is again solved. Let us now show that we have captured important physics in this new formulation. Note that the configuration [n] divides the lattice into disconnected regions or "bags" B i , i = 1, 2, .... Each bag consists of sites connected with only n x,µ = 0 bonds. Inside each bag the fermions hop freely while outside they are confined in the form of dimers. The size and shape of the bags are dynamically determined by the value of U. One such configuration is illustrated in Fig. 1. Note that a single world line configuration of fermions inside the bag can give negative weights due to quantum mechanics. However, we can resum all the possible fermion world lines within the bag exactly. Indeed the quantum interference of all the fermion paths inside the bag B i is simply Det(W [B i ]) ≥ 0 and so
FIG. 1 :
1a local update requires the computation of a single matrix element of (W [B x ]) −1 where B x is the bag associated to the site x. The effort associated with this step is equal to N B N CG . An illustration of a "fermion-bag" configuration as discussed in the text.
FIG. 2 :
2An illustration of a "fermion-bag" configuration with one temporal winding bag and two nontemporal winding bags.The fermion bag approach also suffers from a sign problem in the presence of a chemical potential. But the sign problem tracks the physics of the model very closely. To see this note that the partition function is again given by Eq.(9) except that now W [n] is the free Dirac operator operator with the chemical potential given in Eq.(11) where the sites connected to n x,µ = 1 are dropped. Again it is no longer possible to argue that Det(W [n]) ≥ 0 when µ = 0. However, this sign problem is qualitatively different. Since the chemical potential only enters through fermion world lines that wrap around the temporal direction, the chemical potential completely drops out of the determinant of a fermion bag which lives completely inside the space time volume. We call these non-temporal winding bags and two such bags are shown inFig. 2. The fermions hopping within this bag will never have a fermion world line with a non-zero temporal winding. Only bags with a non-zero temporal winding are sensitive to the chemical potential. One such bag is also shown inFig. 2. At large U and small temperatures (large temporal direction), such bags are exponentially suppressed. Thus, the sign problem is naturally absent for small chemical potentials and low temperatures at large U in the fermion bag approach as dictated by physics.
The configurations are described by n x,α = 0, 1 bond variables. A dimer is represented by n x,α = 1. We will assume that α can take any of the 2d values: α = ±1, ±2, ... ± d, where the negative signs indicate negative directions. This means n x,α ≡ n x+α,−α . For convenience we also define site variables m x = 0, 1. A monomer is represented by m x = 1. To begin with we set m x = 0 at all sites. It is useful to remember that a site x that belongs to a fermion bag should have both m x = 0 and n x,α = 0, ∀α. The parity of a site x is defined as ε x = (−1) x 1 +x 2 +...+x d .
2 .
2There are 2d + 1 possible values for {n x,α }: n x,α = 1 for one of the 2d values of α or n x,α = 0, ∀α. In this latter configuration let us label the fermion bag that is connected to the site x as B x . Let W [B x ] be the free Dirac matrix inside this bag. If Det(W [B x ]) = 0, the update ends without changing the original configuration. Otherwise the update proceeds to the next step. 3. Let ω α = U|((W [B x ]) −1 ) x,x+α | 2 for the 2d values of α. We set ω 0 = 1.
x+α . It clearly depends on the size of the bag B x . When the typical bag size does not scale with the volume the time to compute the inverse also does not scale with the volume.
2 .
2If n x,α = 0, ∀α the update stops. If not we label x and all sites with the same parity as active sites. The sites with the opposite parity are labeled as passive sites. We then perform either an active or a passive update depending on our current site as discussed below. After each update we move through the lattice according to the rules of the update until we return to the first site, where the update ends.Active Update: If we are on an active site x, we do one of four things depending on the configuration on the site.13(a) If x is the first site such that m x = 0 and n x,α = 1, then we set n x,α = 0 and m x = 1 and m x+α = 1. In other words we break a dimer into two monomers. The update then moves to the site x +α.(b) If x is not the first site such that n x,α = 1 and we just came to the site from the previous site x +β, then we set n x,β = 1, m x+β = 0 and m x+α = 1. The update then moves to the site x +α.
( d )
dIf x is the first site such that m x = 1 and n x,α = 0, then we would have returned to it from the neighboring site x +β such that m x+β = 1. We then set m x = 0, m x+β = 0and n x,β = 1. The update then ends.Passive Update: If we are on a passive site x then we must have m x = 1. We pick one of the 2d + 1 directions α including 0 at random. If α = 0 the update remains on the same site, we get a contribution to the two-point correlation function discussed below. If α = 0 the update moves to the neighboring active site x +α.
The most time consuming step in this update is the computation of the probabilities P γ (x) on an active site x. Fortunately, as long as the fermion bags are not disturbed ω y does not change on any site y inside the bag. So P γ (x) is then simple to compute. However, if the fermion bag is disturbed the extra effort in computing the inverse is necessary. For large values of the U the bags are small and the effort does not grow with the volume.
FermionFIG. 3 :
3bags encode the fermionic physics, understanding their properties is an important research problem in itself. For example the eigenvalue distribution of the corresponding Dirac operator could be interesting. What role do the low eigenvalues play? Can their distribution be described by some simplified theory like random matrix theory? However, we postpone such studies to the future. Here we focus on computing a much simpler quantity, namely the size distribution of the fermion bags as a function of the coupling U. This quantity helps us understand the efficiency of the fermion bag approach and identify the range of U where the approach is clearly superior. Let N B (S) be the number of bags of size S in a single configuration. In Fig. 2 we plot the average of N B over the ensemble of configurations generated by the algorithm at U = 1.3 for L = 8, 12, 16 and 20. The figure shows that the number of bags of a given size increases with the volume but the density of the bags of a given size remains constant. Indeed the three data points for L = 12, 16, 20 collapse on a single curve once the density is plotted as shown in the inset of the figure. We find that N B (S) drops like a power for small values of S, but somewhere around S 100, a sudden drop in N B is observed. This behavior is similar for all values of L except that the sudden drop moves slightly. This we attribute to a finite size effect. For large values of S, we find N B (S) decays exponentially as a function of S. Assuming that the bag size represents a three dimensional lattice volume then we naturally expect N B (S) = A exp(−M 3 S) Plot of the average number of fermion bags N B of size S as a function of the S for L = 8, 12, 16 and 20 at U = 1.3. The solid line is an exponential fit as discussed in the text. The inset shows the same plot (L = 8 data has been omitted) scaled with the volume. The data collapse shows that the density of fermion bags of a given size does not depend on the volume. where M is a lattice mass scale. The data for L = 20 fits well to this form when S ≥ 84, we get A = 0.61(2), M = 0.286(1) and a χ 2 /DOF = 0.8. This fit is shown as a solid line in the plot of Fig. 3. It is tempting to relate the scale M with the mass of the fermion. However, given
FIG. 4 :
4Plot of the average distribution of fermion bags N B of size S as a function of the S for U = 0.1, 0.25, 0.5, 0.8, 1.3 and 1.5 for L = 8. We note that the distribution changes qualitatively between large and small U .
Fig. 5 FIG. 5 :
55we plot S τ as a function of L for U = 1.3, 1.2 and 0.8. The solid line is the fit given by 0.0806(2)L 3 . The figure shows clearly that for U = 1.2 and 1.3 the typical fermion bag size begins to saturate, indicating that the Dirac matrix used in the conjugate gradient has a typical size independent of the lattice size for large volumes. When U = 0.8 this advantage is no longer valid since the bags begin to grow with the spatial volume. Thus, the fermion bag approach is guaranteed to be efficient only when U 1.2. For smaller U the bag approach continues to be an alternative approach but becomes less attractive. However note that at U = 0.8 the bags only occupy roughly 1/10 the size of the system. Thus, the bag approach may continue to be competitive at intermediate values of U and moderate values of L. Plot of the typical bag size as a function of L for three different values of U . The solid line is a fit to the form AL 3 . Note the L axis is shown on a logarithmic scale.
Fig. 7
7we show results for U = 0.0, 0.1 and U = 0.2. In table II we show the results of the fits to Eq. (31).
FIG. 6 :FIG. 7 :
67Plots of 2χ/L 3 , q 2 χ /4L and Q 2 f as a function of the lattice size for various values of U is shown in the left figures. The same quantity is plotted as a function of U for L = 8, 12 on the right. Plots of χ, q 2 χ and Q 2 f as a function of the lattice size for small values of U . The solid lines are fits to the expected forms in The coefficients in Eq. (30) obtained by fitting the data. In the fits of χ the coefficient b was set to zero and ρ s was fixed from the second column. The data in the range 16 ≤ L ≤ 32 were used for fits of q 2 χ while the range 8 ≤ L ≤ 32 was used for χ. : The coefficients in Eq. (31) obtained by fitting the data. The missing coefficients have been assumed to be zero in the fit. For U = 0.0 we computed the coefficients exactly but assigned an error of 10 −6 uniformly. The results were then fit for L > 10. For U = 0.1 and 0.2 data from 8 ≤ L ≤ 20 were used in the fit.
3 0FIG. 8 :
38.62(2) 0.87(2) 0.2604(5) 0.074(4) 0.11(1) 0.11(2) 0.04(1) 0.354(6) 0.68(4) 0.65(9) 0.24(5) with a χ 2 /DOF of 1.6. The data used in the combined fit and the fits themselves are shown inFig. 8. Using hyper-scaling relations 2β = ν(d − 2 + η) and δ = (d + 2 − η)/(d − 2 + η) Plots of q 2 χ and χ/L 2−η as a function of U for L = 8, 12, 16 and 20. The solid lines show the combined fit of all the data to Eq. (32) as discussed in the text. Based on the fits we find the critical point to be U = 0.2604(5), ν = 0.87(2) and η = 0.62(2)
(i, t) be the coordinates of a cubic space-time lattice where i ≡ (i x , i y ), i x , i y = 0, 1, 2, ..., (L − 1) is the two dimensional spatial coordinate and t = 0, 1, 2.., (L t − 1) is the temporal coordinate. Let e i represent the four neighboring spatial sites of the site i. of a fermion bag configuration in a model described in the text that contains the physics of BEC-BCS cross over. The solid lines represent spin-up and spin-down fermions strongly paired into a spin-zero boson. The blobs represent regions where fermions can be liberated and become essentially free. be represented with s =↑, ↓. The fermion fields are represented by four independent Grassmann variables ψ s,i,t and ψ s,i,t per site which satisfy anti-periodic boundary conditions in time but periodic boundary conditions in space. Using this notation, consider the lattice field theory model described by the action S = S 0 + US 1 , where S 0 = − i,t,s ψ s,i,t+1 e µε − ψ s,i,t ψ s,i,t + ε t e µε e i ψ s,e i ,t+1 ψ s,i,t ,
( 34 )
34When U = ∞ the model describes the physics of paired fermions (hard-core bosons) hopping on the lattice, similar to the quantum XY model. On the other hand when U = 0, the fermions are free. Hence, as we tune U, the model should describe the physics of BEC-BCS crossover. For intermediate values of U we expect regions where fermions are paired and regions where they are free.
: Comparison between exact results and results from the Monte Carlo algorithm for the three observables N B , S τ and χ.
Thus, Det(M[φ]) ≥ 0 for every[φ] and the sign problem is solved. While different Monte Carlo algorithms exist to solve the remaining problem, the most popular is the Hybrid Monte Carlo (HMC) method due to its favorable scaling with the volume[27].Let us now briefly discuss the cost of the HMC algorithm. The HMC method is based on generating a new independent configuration [φ] based on a series of molecular dynamics update.The new configuration is then accepted or rejected based on a Metropolis accept reject step. Let
The efficiency may be comparable if not superior to the HMC method. For values of U < 0.2 the HMC algorithm will be a better approach. However, it is likely that the diagrammatic Monte Carlo provides a better algorithm for small and intermediate values of U [14].
TABLE III :
IIIContributions to the partition function for 2 3 lattice.Z = 81 + 216U + 312U 2 + 288U 3 + 144U 4
table IV .
IV30
Config.
Deg. Bdet. BWt
Config.
Deg. Bdet. BWt
3
9
27
24
1 24U
12
4
48U
6
1
6U
108 1 108U 2
72
1 72U 3
1
0
0
12
1 12U
24
1 24U 2
16
1 16U 3
TABLE IV :
IVContributions to the condensate susceptibility. The above exact expressions have been tested against our Monte Carlo method. The results for a few values of U are shown in table V.Based on this table we find that
χ =
U
Z
(27 + 90U + 132U 2 + 88U 3 )
(A4)
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|
[] |
[
"Efficient Computational Algorithm for Optimal Allocation in Regression Models",
"Efficient Computational Algorithm for Optimal Allocation in Regression Models"
] |
[
"Wei Gao \nSchool of Mathematics and Statistics\nKey Laboratory for Applied Statistics of MOE\nNortheast Normal University\n130024Changchun, JilinChina\n",
"Ping Shing Chan \nDepartment of Statistics\nThe Chinese University of Hong Kong\nShatin, Hong KongN. T\n",
"Hon Keung \nDepartment of Statistical Science\nSouthern Methodist University\n75275DallasTexasU.S.A\n",
"Tony Ng ",
"Xiaolei Lu \nSchool of Mathematics and Statistics\nKey Laboratory for Applied Statistics of MOE\nNortheast Normal University\n130024Changchun, JilinChina\n"
] |
[
"School of Mathematics and Statistics\nKey Laboratory for Applied Statistics of MOE\nNortheast Normal University\n130024Changchun, JilinChina",
"Department of Statistics\nThe Chinese University of Hong Kong\nShatin, Hong KongN. T",
"Department of Statistical Science\nSouthern Methodist University\n75275DallasTexasU.S.A",
"School of Mathematics and Statistics\nKey Laboratory for Applied Statistics of MOE\nNortheast Normal University\n130024Changchun, JilinChina"
] |
[] |
In this article, we discuss the optimal allocation problem in an experiment when a regression model is used for statistical analysis. Monotonic convergence for a general class of multiplicative algorithms for D-optimality has been discussed in the literature. Here, we provide an alternate proof of the monotonic convergence for D-criterion with a simple computational algorithm and furthermore show it converges to the D-optimality. We also discuss an algorithm as well as a conjecture of the monotonic convergence for A-criterion. Monte Carlo simulations are used to demonstrate the reliability, efficiency and usefulness of the proposed algorithms.
|
10.1016/j.cam.2013.10.040
|
[
"https://arxiv.org/pdf/1301.0877v2.pdf"
] | 2,048,886 |
1301.0877
|
6fea1fcce3b0bda54dd1d03319fccb93528ee183
|
Efficient Computational Algorithm for Optimal Allocation in Regression Models
25 Oct 2013
Wei Gao
School of Mathematics and Statistics
Key Laboratory for Applied Statistics of MOE
Northeast Normal University
130024Changchun, JilinChina
Ping Shing Chan
Department of Statistics
The Chinese University of Hong Kong
Shatin, Hong KongN. T
Hon Keung
Department of Statistical Science
Southern Methodist University
75275DallasTexasU.S.A
Tony Ng
Xiaolei Lu
School of Mathematics and Statistics
Key Laboratory for Applied Statistics of MOE
Northeast Normal University
130024Changchun, JilinChina
Efficient Computational Algorithm for Optimal Allocation in Regression Models
25 Oct 2013arXiv:1301.0877v2 [stat.CO]D-optimalityA-optimalityMaximum likelihood estimatorsAccelerated Life-testingMonte Carlo method
In this article, we discuss the optimal allocation problem in an experiment when a regression model is used for statistical analysis. Monotonic convergence for a general class of multiplicative algorithms for D-optimality has been discussed in the literature. Here, we provide an alternate proof of the monotonic convergence for D-criterion with a simple computational algorithm and furthermore show it converges to the D-optimality. We also discuss an algorithm as well as a conjecture of the monotonic convergence for A-criterion. Monte Carlo simulations are used to demonstrate the reliability, efficiency and usefulness of the proposed algorithms.
Introduction
Regression analysis is an useful technique in modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables. It is widely used in different fields of study. For instance, in reliability and life-testing experiments, often one of the primary purposes is to study the effect of covariates on the failure time distribution and to develop inference on the survival probability or some other reliability characteristic of an equipment. For this purpose, a regression model is used to incorporate these covariates in the statistical analysis.
Consider a general regression model
Y = µ(x, β) + σǫ (1)
where Y is the response variable, µ(x, β) is a known function which depends on the unknown parameters β ∈ ℜ p and the p covariates x = (x 1 , . . . , x p ) ′ ; and ǫ is a random variable with E(ǫ) = 0 and V ar(ǫ) = 1. We can rewrite the combinations of different levels in different covariates into k experimental conditions (or design points) represent by x l = (x 1l x 2l . . . x pl ), where x il is one of the levels of the i-th covariate. Note that when the intercept term present in the regression model, x 1l ≡ 1, l = 1, . . . , k.
Suppose that in an experiment, we have N items available for the test at k experimental conditions. We assign n l items for testing at experimental condition x l (l = 1, 2, . . . , k) with k l=1 n l = N, and observe the corresponding observations for estimation of parameters and/or prediction. In planning such an experiment, we have the flexibility in the choice of (n 1 , n 2 , . . . , n k ) for a given value of N and the experimental conditions x l , l = 1, 2, . . . , k.
Here, we consider the problem of optimal allocation of n 1 , n 2 , . . . , n k for a general regression model. This optimal allocation problem is usually referred as optimal design problem in the literature.
The optimal design problem in regression setting has long been studied in the literature, for example, Elfving (1952), Fedorov (1972), Silvey (1980). For extensive developments in optimal design, one may refer to Silvey (1980), Box and Draper (1987), Atkinson and Donev (1992), Liski et al. (2002), Seber and Wild (2003) and a concise introduction by O' Brien and Funk (2003). Besides the rich development in optimal design theory, different numerical computational algorithms have been proposed to obtain optimal designs under different scenarios. For instance, when µ(x, β) is linear functions of β, Wynn (1970) proposed a W -algorithm and Fedorov (1972) proposed a Valgorithm to search for the optimal design. Following the ideas in Wynn (1970) and Fedorov (1972), Mitchell (1974) proposed an algorithm for the maximization of |X T X|, where X is the design matrix. Then, Cook and Nachtsheim (1980) provided an empirical comparison of existing algorithms for the computer generation of exact D-optimal experimental including those due to Wynn (1970), Fedorov (1972) and Mitchell (1974) and proposed a modification of the Fedorov (1972) algorithm. However, as pointed out by Silvey (1980, p.34), these early algorithms have been criticized on the grounds of their slow convergence and some algorithms have been suggested to increase the speed of convergence (see, for example, Atwood, 1973, Silvey and Titterington, 1973and Wu, 1978. Meyer and Nachtsheim (1995) proposed a cyclic coordinate-exchange algorithm for constructing of D-optimal designs mainly for continuous design space. As they stated in Section 2.4, "For finite design spaces, the procedure is conceptually simple, although the computational demands can be prohibitive when q (dimensions of covariates) is large." Vandenberghe, Boyd and Wu (1998) have proposed the interior-point method to deal with more general problems but it also suffer from the slow convergence problem. Recent papers of Torsney and Mandal (2006), Harman and Pronzato (2007), Dette, Pepelyshev and Zhigljavsky (2008) and Torsney and Martin-Martin (2009) developed numerical computational algorithms for D-optimal designs. Yu (2010) discussed the monotonic convergence for a general class of computational algorithms for D-optimal design. In general, it is desirable to have a numerical computational algorithm to obtain optimal designs which (i) is simple and reliable; (ii) can be applied in general situations; and (iii) has a high convergence rate.
In this paper, we aim to develop efficient computational algorithms to obtain optimal allocation for a general regression model subject to the Doptimality and A-optimality criteria. Mathematical results related to the convergence and monotonicity of the proposed algorithms are developed. An extensive simulation study is performed to show the reliability of these algorithms. In Section 2, we consider the likelihood inference based on a general regression model and present the forms of the expected Fisher information matrix and the asymptotic variance-covariance matrix. The two optimal criteria are also discussed in Section 2. Then, the proposed computational algorithms for D-optimality and A-optimality criteria and their related mathematical properties are discussed in Section 3. Concluding remarks are provided in Section 4. The related proofs of the results are given in Appendix.
Model and Optimal Criteria
Model and Notations
One of the commonly used approaches to estimate the unknown parameters in a regression model in (1) is the maximum likelihood method. The maximum likelihood estimates (MLEs) are obtained by maximizing the likelihood function subject to the unknown parameters, β. The properties of the MLEs of the parameters in a regression model are then evaluated based on the asymptotic theory of MLEs. When µ(x, β) is a linear function of β, the expected Fisher information matrix of the MLE of β can be expressed as a function of n 1 , . . . , n k ,
Σ(n 1 , . . . , n k ) = 1 σ 2 n 1 x 1 x ′ 1 + · · · + n k x k x ′ k ,(2)
where n l is the number of repeated observations or measurements under the experimental condition x l . Thus, the asymptotic variance-covariance matrix of the MLE of β, which is the inverse of the expected Fisher information matrix, can also be expressed as a function of n 1 , . . . , n k . We can also write the expected Fisher information in terms of w l = n l /N, where w l = n l /N is the proportion of units (of a total of N units under test) to be assigned to the experimental condition x l , l = 1, 2, . . . , k,
Σ(w) = Σ(w 1 , . . . , w k ) = N(w 1 A 1 + w 2 A 2 + · · · + w k A k )(3)
where A 1 , . . . , A k are known nonnegative definite matrices which are functions of x 1 , . . . , x k and w l = n l /N is the proportion of units (of a total of N units under test) to be assigned to the experimental condition x l , l = 1, 2, . . . , k. The related applications that the inverse of covariance is decomposed into sums of linear nonnegative definite matrices has been considered by Vandenberghe, Boyd and Wu (1998), and Qu, Lindsay and Li (2000). The optimal allocation problem is equivalent to obtaining the values of w = (w 1 , w 2 , . . . , w k ) which optimized a specific objective function subject to the constraints w l ≥ 0 (l = 1, 2, . . . , k) and k l=1 w l = 1. It is noteworthy that if the expected Fisher information of the MLEs can be expressed in the form of (2) or (3), then the algorithms proposed in this manuscript are applicable. We can show that many of the commonly used regression model, such as multiple linear regression model with normal distributed errors, Weibull (extreme-value) regression model and Birnbaum-Saunders regression model, which have expected Fisher information of the MLEs in the form of (2) or (3). Thus, the proposed algorithms are applicable in those situations.
Optimal Criteria
The goal here is to determine the optimal planning of an experiment when regression analysis is used. We can determine the optimal allocation subject to different optimality criteria. If we are interested in the estimation of the model parameters, we may consider optimality in terms of:
[C1] Maximization of the determinant of the Fisher information matrix Σ:
This criterion is D-optimality, wherein the determinant of the Fisher information matrix is maximized, which results in minimum volume for the Wald-type joint confidence region for the model parameters (β, σ). w * is a D-optimal allocation for (3)
Proposed Computational Algorithm
In this section, we propose the computational algorithms to obtain the D-optimal and A-optimal choices of w. The properties of these algorithms are discussed.
Algorithm for D-optimal allocation
Theorem 1. w * is the D-optimal choice for (3) if and only if trace(A l Σ −1 (w * )) = p for w * l = 0(6)
and
trace(A l Σ −1 (w * )) ≤ p for w * l = 0.(7)
Proof: Let S(w) = log(|Σ(w)|), it is easy to check that S(w) is convex in w. By Kuhn-Tucker conditions (Kuhn and Tucker, 1951), w * is the optimal solution of (4) if and only if for all w(w j ≥ 0 and
w j = 1), 0 ≤ ∂S(w * ) ∂w (w − w * ) = − k l=1 trace(A l Σ −1 (w * ))(w l − w * l ) = − k l=1 trace(A l Σ −1 (w * ))w l + p,
which implies (6) and (7). Note that Eq. (6) given in Theorem 1 is consistent with the General Equivalence Theorem (Kiefer and Wolfowitz, 1960). Based on (6) and (7), the following iterative algorithm to obtain the Doptimal allocation in (4) is proposed.
Algorithm for D-optimal allocation 1. Set the initial value of w as w (0) = (1/k, · · · , 1/k) ′ .
2.
In the h-th step, update the value of w as
w (h) l = w (h−1) l trace(A l Σ −1 (w (h−1) )) p ,(8)
for l = 1, · · · , k. Note that when
A i = x i x ′ i , (8) can be expressed as w (h) l = w (h−1) l x ′ l Σ −1 (w (h−1) )x l p .
3. Repeat step 2 until the algorithm converge. One of the stopping rule based on absolute difference is stop when max w
(h) j − w (h−1) j < ζ,
where ζ is a small number specified by the user.
The monotonic convergence for multiplicative algorithms for D-optimality has been established in the literature (see, for example, Yu, 2010). Here, we provide an alternate proof of the monotonic convergence. Theorem 2. Let {w (h) } be given by (8), and then
log |Σ(w (h) )|−log |Σ(w (h−1) )| ≥ p k l=1 w (h) l log w (h) l w (h−1) l ≥ p 2 k l=1 |w (h) l − w (h−1) l | 2 and w (h) − w (h−1) → 0.
Now we consider the convergence of the proposed algorithm under the condition
a 1 A 1 + a 2 A 2 + · · · + a k A k = 0 ⇐⇒ a 1 = 0, a 2 = 0, · · · , a k = 0,(9)
that is, A 1 , · · · , A k are linearly independent, and this condition is a natural one in order for the models being identifiable. Theorem 3. Under the condition of (9), {w (h) } given by (8) is convergent and converges to the D-optimal solution of (4).
If A 1 , · · · , A k are linearly dependent, for the optimal allocation (4), its solution may not be unique and we can choose different initial values and get different optimal solutions. Compared with the W-algorithm proposed by Wynn (1970) and the V -algorithm proposed by Fedorov (1972), the value of w in the current step in our proposed algorithm is an explicit function of the value in the previous step which involves simple matrix manipulation while the value of w in each step of the W -and V -algorithms involve maximizations which required numerical procedures in computation. Therefore, the algorithms proposed here converge quicker and they are easy to program.
In order to study the convergent rate of the proposed algorithm, an extensive simulation study is performed. We generate the form of the Fisher information matrix in (3) with the elements of x 1 , x 2 , · · · , x k being independent identically uniform distributed in between −1 and 1, i.e., U(−1, 1), for number of design points k = 10, 20, 30, 40 and number of covariates p = 4, 5, 8, 10, 15, 20, 25, 30 with p < k. The stopping criteria of the algorithm are set to be max |w Table 1.
Algorithm for A-optimal allocation
Theorem 4. w * is the A-optimal choice for (3) if and only if
trace(Σ −1 (w * )A l Σ −1 (w * )) trace(Σ −1 (w * )) = 1 for w * l = 0(10)
and
trace(Σ −1 (w * )A l Σ −1 (w * )) trace(Σ −1 (w * )) ≤ 1 for w * l = 0.(11)
Based on (10) and (11), the following iterative algorithm to obtain the A-optimal allocation in (5) is proposed.
Algorithm for A-optimal allocation 1. Set the initial value of w as w (0) = (1/k, · · · , 1/k) ′ .
2.
In the h-th step, update the value of w as
w (h) l = w (h−1) l p trace(Σ −1 (w (h−1) )A l Σ −1 (w (h−1) )) trace(Σ −1 (w (h−1) )) + p − 1 ,(12)
for l = 1, · · · , k. Note that when (12) can be expressed as
A i = x i x ′ i ,w (h) l = w (h−1) l p x ′ l Σ −1 (w (h−1) )Σ −1 (w (h−1) )x l trace(Σ −1 (w (h−1) )) + p − 1 .
3. Repeat step 2 until the algorithm converge. One of the stopping rule based on absolute difference is stop when max |w
(h) j − w (h−1) j | < ζ,
where ζ is a small number specified by the user.
Although a theoretical justification of convergence of the proposed computational algorithm for A-optimality similar to Theorem 2 is not yet available, simulation results strongly support the validity and reliability of the algorithm. An extensive simulation study with settings presented in Section 3.1 is performed to study the properties of A-optimality. We have generated a wide range of settings and use our algorithm to compute the A-optimal allocation and we found that the algorithm converge in all these cases. The number of iterations and the elapse time (in unit of second) required to obtain the A-optimal allocation are recorded and their average values (with standard deviations in parenthesis) are presented in Table 2. We conjecture the monotonic convergence of the algorithm for A-optimality and the theoretically prove of convergence of the proposed computational algorithm for A-optimality seems to be an interesting open problem.
Concluding Remarks
We have proposed simple and efficient iterative algorithms to obtain the D-optimal and A-optimal allocations for general regression model. We have provided an alternate proof of the monotonic convergence of the proposed algorithm for D-optimality and demonstrate it converges to converges to optimal allocation. We have also shown that the proposed algorithm for A-optimality converges via an extensive Monte Carlo simulation study and conjecture the the monotonic convergence of the proposed algorithm for Aoptimality. The proposed computational algorithms converges fast and they are easy to program. These algorithms are programmed in R (R Development Core Team, 2012) and the programs are available from the authors upon request.
Appendix: Proof of Theorem 2 and Theorem 3 Lemma 1. π = (π 1 , π 2 , . . . , π k ) and θ = (θ 1 , θ 2 , . . . , θ k ) are two probability vectors in ℜ k , and then 2 k l=1 π l log(π l /θ l )
1/2 ≥ k l=1 |π l − θ l |.(13)
Proof: See the proof given by Kullback (1967), Csiszar (1967) or Kemperman (1969).
Lemma 2. For d 1 ≥ 0, · · · , d k ≥ 0, let Σ(w, d) = k l=1
w l d l A l , and then Σ(w) = Σ(w, 1) and
1 p log[|Σ(w, d)|] − 1 p log[|Σ(w)|] − k l=1w l log d l ≥ 0.(14)
wherew
l = w l trace(A l Σ −1 (w)) p .
Proof. Without loss of generality, suppose that d 1 > 0, · · · , d k > 0, let t 1 = log d 1 , · · · , t k = log d k , and
g(t 1 , . . . , t k ) = 1 p log[|Σ(w, exp{t})|] − k l=1w l t l ,
and then
∂g(t 1 , . . . , t k ) ∂t l = w l trace(A l Σ −1 (w, exp{t})) p exp{t l } −w l and ∂ 2 g(t 1 , . . . , t k ) ∂t l ∂t j = −w l w j trace(A l Σ −1 (w, exp{t})A j Σ −1 (w, exp{t})) p exp{t l + t j } +w l trace(A l Σ −1 (w, exp{t})) p exp{t l }I {l=j}. Γ = ∂ 2 g ∂t 2 1 ∂ 2 g ∂t 1 ∂t 2 · · · ∂ 2 g ∂t 1 ∂t k ∂ 2 g ∂t 2 ∂t 1 ∂ 2 g ∂t 2 2 · · · ∂ 2 g ∂t 2 ∂t k . . . . . . . . . . . . ∂ 2 g ∂t k ∂t 1 ∂ 2 g ∂t k ∂t 2 · · · ∂ 2 g ∂t 2 k ≥ 0
and ∂g(0, . . . , 0)/∂t l = 0, and then (0, . . . , 0) ′ is the global minimum point and so g(t 1 , . . . , t k ) ≥ g(0, . . . , 0) which implies that the Lemma holds.
Lemma 4.
A is a nonnegative matrix and then
|A| ≤ p i=1 a ii ,(15)
where a ij is the (i, j)-th element in A.
Proof: See the results given by Anderson (2003).
Proof of Theorem 2: Let a (l)
ii be the i−th diagonal of A l and by Lemma 4,
|Σ(w (h) )| ≤ p i=1 k l=1 w (h) l a (l) ii ≤ p i=1 max{a (l)
ii ; l = 1, · · · , k}, that is, the sequence {log |Σ(w (h) )|} is uniformly bounded. By Lemma 3, we also know the sequence {log |Σ(w (h) )|} increasing in n, and thus it converges.
We have
0 = lim h→∞ log |Σ(w (h) )| − log |Σ(w (h−1) )| ≥ lim h→∞ p × k l=1 w (h−1) l log trace(A l Σ −1 (w (h−1) )) p ≥ 0 which implies 0 = lim h→∞ k l=1 w (h−1) l log trace(A l Σ −1 (w (h−1) )) p ≥ k l=1 |w (h) l − w (h−1) l | and w (h) − w (h−1) → 0.
Let W be the set of accumulation points of {w (h) }, that is, for any w * ∈ W , there exist subsequence {w (hs) } which satisfies lim s→∞ w (hs) = w * .
For 1 ≤ l ≤ k and 1 ≤ i 1 < i 2 < · · · < i l ≤ k, define D(l : i 1 , · · · , i l ) = (x 1 , · · · , x k ) ′ k j=1 x j = 1, x j > 0, j ∈ {i 1 , · · · , i l }; x j = 0 otherwise , and D(l) = i 1 ,··· ,i l D(l : i 1 , · · · , i l ).
Thus we have
W ⊂ k l=1 D(l).(16)
Lemma 5.
If W D(l : i 1 , · · · , i l ) is not an empty set, w * ∈ W D(l : i 1 , · · · , i l ) satisfies log |Σ(w * )| = max w∈D(l: i 1 ,··· ,i l ) log |Σ(w)|.
Proof. By the definition W , there exists a subsequence {w (hs) } which is lim s→∞ w (hs) = w * .
By Theorem 2, lim s→∞ w (hs) − w (hs−1) = 0, have
w * j = lim s→∞ w (hs) j = lim s→∞ w (hs−1) j trace(A j Σ −1 (w (hs−1) )) p = w * j trace(A j Σ −1 (w * )) p which implies trace(A j Σ −1 (w * )) p = 1,
for j = i 1 , · · · , i l , and the conclusion holds by Theorem 1. Corollary A. Under (9), W D(l : i 1 , · · · , i l ) has no more than one element.
Proof. The function log|Σ(w)| is strictly concave under (9) and so if there have w * , w * * ∈ W D(l : i 1 , · · · , i l ) which satisfy log |Σ(w * )| = log |Σ(w * * )| = max w∈D(l:
i 1 ,··· ,i l ) log |Σ(w)|,
we have w * = w * * . So W D(l : i 1 , · · · , i l ) has no more than one element. Lemma 6. Let {y m } be a uniformly bounded sequence in R k . If y m − y m−1 → 0 k , as m → ∞, and the sequence is not convergent, then there are infinitely many accumulation points of the sequence, where 0 k denotes the k-dimensional zero vector.
Proof: See the Lemma A.1. given by Shi and Jiang (1998).
Proof of Theorem 3: Suppose the sequence {w (h) } does not converge under (9), and sequence {w (h) } has infinitely many accumulation points by Lemma 6. By (16),
W = W k l=1 D(l) = k l=1 i 1 ,··· ,i l (D(l : i 1 , · · · , i l ) ∩ W ) ,
and then the number elements W is less than 2 k − 1 which contradicts. So under (9), the sequence {w (h) } is convergent. Let w * = lim h→∞ w (h) , then
w * j = lim s→∞ w (h) j = lim s→∞ w (h−1) j trace(A j Σ −1 (w (h−1) )) p = w * j trace(A j Σ −1 (w * )) p which implies trace(A j Σ −1 (w * )) p = 1, w * j = 0 and trace(A j Σ −1 (w * )) p ≤ 1, w * j = 0.
So the sequence {w (h) } is convergent and converges to the D-optimal solution of (4) by Theorem 1.
if and only if w * = arg min w − log(|Σ(w)|) : subject to w l
]
Minimization of the trace of the variance-covariance matrix (Σ −1 ) of the MLE's: This criterion is A-optimality which minimizes the sum of the variances of the parameter estimates and provides an overall measure of variability from the marginal variabilities. w * is a A-optimal allocation for (3) if and only if w * = arg min w log(trace(Σ −1 (w)) : subject to w l
|
< 0.0001. For each combination of p and k, 50 replications are simulated and their corresponding D-optimal allocations are found by using the proposed algorithm. The number of iterations and the elapse time (in unit of second) required to obtain the D-optimal allocation are recorded and their average values (with standard deviations in parenthesis) are presented in
Table 1 :
1Simulated results for D-optimalityAverage no. of Average elapsed
k
p
iterations (s.d.) time in sec. (s.d.)
10 4
56.8 (27.9)
0.192 (0.094)
5
41.9 (16.4)
0.143 (0.058)
8
19.5 (10.6)
0.067 (0.040)
20 4
96.1 (69.0)
0.648 (0.467)
5
77.9 (29.6)
0.534 (0.203)
8
51.3 (11.5)
0.370 (0.085)
10
36.0 (9.5)
0.267 (0.073)
15
15.8 (4.7)
0.129 (0.040)
30 4
115.1 (94.8)
1.181 (0.982)
5
99.3 (35.9)
1.092 (0.539)
8
60.1 (16.7)
0.649 (0.179)
10
50.2 (14.7)
0.565 (0.162)
15
29.0 (4.5)
0.354 (0.057)
20
17.9 (3.9)
0.255 (0.054)
25
9.7 (2.8)
0.155 (0.046)
40 4
124.4 (63.9)
1.673 (0.853)
5
104.7 (38.6)
1.431 (0.530)
8
72.8 (28.5)
1.053 (0.419)
10
52.2 (10.2)
0.791 (0.153)
15
36.7 (6.2)
0.608 (0.104)
20
26.0 (6.0)
0.497 (0.119)
25
16.7 (3.5)
0.362 (0.077)
30
10.7 (2.1)
0.273 (0.055)
Table 2 :
2Simulated results for A-optimalityAverage no. of
Average elapsed
k
p iterations (s.d.) time in sec. (s.d.)
10 4
126.2 (77.0)
1.017 (0.618)
5
121.5 (67.3)
0.994 (0.554)
8
67.6 (20.7)
0.580 (0.180)
20 4
188.0 (79.3)
3.033 (1.283)
5
169.0 (73.9)
2.784 (1.218)
8
119.5 (30.0)
2.062 (0.514)
10
97.5 (18.0)
1.739 (0.318)
15
72.6 (20.2)
1.427 (0.399)
30 4
218.3 (84.6)
5.284 (2.046)
5
200.0 (68.0)
4.933 (1.682)
8
155.8 (38.6)
4.039 (1.004)
10
128.7 (30.0)
3.444 (0.809)
15
92.1 (16.2)
2.715 (0.475)
20
76.1 (12.7)
2.565 (0.432)
25
64.5 (12.0)
2.527 (0.472)
40 4
237.0 (108.6)
7.635 (3.492)
5
219.9 (78.0)
7.416 (2.936)
8
176.2 (45.6)
6.078 (1.571)
10
142.1 (29.3)
5.072 (1.059)
15
104.3 (18.8)
4.114 (0.744)
20
90.8 (18.8)
4.075 (0.842)
25
77.9 (10.1)
4.055 (0.528)
30
66.4 (8.3)
4.056 (0.502)
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|
[] |
[
"Gap Amplification for Small-Set Expansion via Random Walks",
"Gap Amplification for Small-Set Expansion via Random Walks"
] |
[
"Prasad Raghavendra [email protected] \nUniversity of California\nUniversity of California\nBerkeleyBerkeley\n",
"Tselil Schramm [email protected] \nUniversity of California\nUniversity of California\nBerkeleyBerkeley\n"
] |
[
"University of California\nUniversity of California\nBerkeleyBerkeley",
"University of California\nUniversity of California\nBerkeleyBerkeley"
] |
[] |
In this work, we achieve gap amplification for the Small-Set Expansion problem. Specifically, we show that an instance of the Small-Set Expansion Problem with completeness ε and soundness 1 2 is at least as difficult as Small-Set Expansion with completeness ε and soundness f (ε), for any function f (ε) which grows faster than √ ε. We achieve this amplification via random walks -the output graph corresponds to taking random walks on the original graph. An interesting feature of our reduction is that unlike gap amplification via parallel repetition, the size of the instances (number of vertices) produced by the reduction remains the same.
|
10.4230/lipics.approx-random.2014.381
|
[
"https://arxiv.org/pdf/1310.1493v3.pdf"
] | 8,717,677 |
1310.1493
|
2348617470a142fe7afbfc252445bc554c2edd03
|
Gap Amplification for Small-Set Expansion via Random Walks
2 Jul 2014
Prasad Raghavendra [email protected]
University of California
University of California
BerkeleyBerkeley
Tselil Schramm [email protected]
University of California
University of California
BerkeleyBerkeley
Gap Amplification for Small-Set Expansion via Random Walks
2 Jul 20141
In this work, we achieve gap amplification for the Small-Set Expansion problem. Specifically, we show that an instance of the Small-Set Expansion Problem with completeness ε and soundness 1 2 is at least as difficult as Small-Set Expansion with completeness ε and soundness f (ε), for any function f (ε) which grows faster than √ ε. We achieve this amplification via random walks -the output graph corresponds to taking random walks on the original graph. An interesting feature of our reduction is that unlike gap amplification via parallel repetition, the size of the instances (number of vertices) produced by the reduction remains the same.
Introduction
The small-set expansion problem refers to the problem of approximating the edge expansion of small sets in a graph. Formally, given a graph G = (V, E) and a subset of vertices S ⊆ V with |S| |V|/2, the edge expansion of S is
φ(S) = E(S,S) vol(S) ,
where vol(S) refers to the fraction of all edges of the graph that are incident on the subset S. The edge expansion of the graph G is given by φ G = min S⊆V,vol(S) 1 /2 φ(S). The problem of approximating the value of φ G is the well-studied uniform sparsest cut problem [LR99,ARV04,ALN08].
In the small-set expansion problem, the goal is to approximate the edge expansion of the graph at a much finer granularity. Specifically, for δ > 0 define the parameter φ G (δ) as follows:
φ G (δ) = min S⊆V,vol(S) δ φ(S).
The problem of approximating φ G (δ) for all δ > 0 is the small-set expansion problem.
The small-set expansion problem has received considerable attention in recent years due to its close connections to the unique games conjecture. To describe this connection, we will define a gap version of the problem.
Definition 1. For constants 0 < s < c < 1 and δ > 0, the SSE δ (c, s) problem is defined as follows: Given a graph G = (V, E) distinguish between the following two cases:
-G has a set S with vol(S) ∈ [δ/2, δ] with expansion less than 1 − c -All sets S with vol(S) δ in G have expansion at least 1 − s.
We will omit the subscript δ and write SSE(c, s) when we refer to the SSE δ (c, s) problem for all constant δ > 0.
Recent work by Raghavendra and Steurer [RS10] introduced the following hardness assumption and showed that it implies the unique games conjecture. Hypothesis 1.1. For all ε > 0, there exists δ > 0 such that SSE δ (1 − ε, ε) is NP-hard.
Theorem 1.2. [RS10] The small set expansion hypothesis implies the unique games conjecture.
Moreover, the small set expansion hypothesis is shown to be equivalent to a variant of the Unique Games Conjecture wherein the input instance is promised to be a small-set expander [RST12]. Assuming the small-set expansion hypothesis, hardness results have been obtained for several problems including Balanced Separator, Minimum Linear Arrangment [RST12] and the problem of approximating vertex expansion [LRV13].
In this work, we will be concerned with gap amplification for the small set expansion problem. Gap amplification refers to an efficient reduction that takes a weak hardness result for a problem Π with a small gap between the completeness and soundness and produces a strong hardness with a much larger gap. Formally, this is achieved via an efficient reduction from instances of problem Π to harder instances of the same problem Π. Gap amplification is a crucial step in proving hardness of approximation results. An important example of gap amplification is the parallel repetition of 2-prover 1-round games or Label Cover. Label cover is a constraint satisfaction problem which is the starting point for a large number of reductions in hardness of approximation [H01]. Starting with the PCP theorem, one obtains a weak hardness for label cover with a gap of 1 vs 1 − β 0 for some tiny absolute constant β 0 [ALM + 98]. Almost all label-cover based hardness results rely on the much stronger 1 vs ε hardness for label cover obtained by gap amplification via the parallel repetition theorem of Raz [Raz98]. More recently, there have been significant improvements and simplifications to the parallel repetition theorem [Rao08,Hol07,DS13].
It is unclear if parallel repetition could be used for gap amplification for small set expansion. Given a graph G, the parallel repetition of G would consist of the product graph G R for some large constant R. Unfortunately, the product graph G R can have small non-expanding sets even if G has no small non-expanding sets. For instance, if G has a balanced cut then G R could have a non-expanding set of volume 1 2 R . In this work, we show that random walks can be used to achieve gap amplification for small set expansion. Specifically, given a graph G the gap amplification procedure constructs G t on the same set of vertices as G, but with edges corresponding to t-step lazy random walks in G. Using this approach, we are able to achieve the following gap amplification.
Theorem 1. Let f be any function such that lim ε→0
f (ε) √ ε → ∞. Then If for all ε > 0, SSE ′ (1 − ε, 1 − f (ε)) is NP-hard then for all η > 0, SSE(1 − η, 1 /2) is NP-hard.
We remark here that the result has some discrepancy in the set sizes between the original instance and the instance produced by the reduction. For this reason, the reduction has to start with a slightly different version of the Small set expansion problem SSE ′ (See Definition 2).
The above result nicely complements the gap amplification result for the closely related problem of Unique Games obtained via parallel repetition [Rao08]. For the sake of completeness we state the result below. Theorem 1.3. [Rao08] Let f be any function such that lim ε→0
f (ε) √ ε → ∞. Then If for all ε > 0 if UG(1 − ε, 1 − f (ε)) is NP-hard then for all η > 0, UG(1 − η, 1 /2) is NP-hard.
Note that the size of the instance produced by our reduction remains bounded by O(n 2 ). In fact, the instance produced has the same number of vertices but possibly many more edges. This is in contrast to parallel repetition wherein the size of the instance grows exponentially in the number of repetitions used.
Technically, the proof of the result is very similar to an argument in the work of Arora, Barak and Steurer [ABS10] to show that graphs with sufficiently high threshold rank cannot be small-set expanders (see Steurer's thesis [Ste10] for an improved version of the result). The work of O'Donnell and Wright [OW12] recast these arguments using continous-time random walks instead of lazy-random walks, yielding cleaner and more general proofs. In this work, we will reuse the proof technique and obtain upper and lower bounds for the expansion profile of lazy random walks (see Theorem 3.1). These upper and lower bounds immediately imply the desired gap amplification result for small-set expansion.
Subsequent to our work, Kwok and Lau [KL14] have obtained a stronger analysis of our gap amplification theorem, yielding almost tight bounds.
Preliminaries
Unless otherwise specified, we will be concerned with an undirected graph G = (V, E) with n vertices and associated edge weights w : E → R + . The degree of vertex i denoted by d(i) = (i, j)∈E w(i, j). The volume of a set S ⊆ V is defined to be vol(S) = i∈S d(i). Henceforth, we will assume that the total volume is 1, i.e., i∈V d(i) = 1. The adjacency matrix A of the graph G has entries A i j = w(i, j). The degree matrix D is a n × n diagonal matrix with D ii = d(i).
Expansion profile. The expansion profile of a graph is defined as follows.
Definition 2.1. For a graph G, define the expansion profile φ G : R + → [0, 1] as φ G (δ) = min S⊆V,vol(S) δ φ(S) where φ(S) = E(S,S) vol(S) .
Lazy Random Walks. The transition matrix for a lazy random walk on G is given by
M = 1 2 (I + D −1 A)
The lazy random walk corresponds to staying at the same vertex with probability 1 2 , and moving to a random neighbor with probability 1 2 . We will let G t denote the graph corresponding to the t-step lazy random walk. The adjacency matrix of G t is given by DM t .
We recall a few standard facts about lazy random walks here.
Fact 2.2. If G is a graph with adjacency matrix A, then G's lazy random walk operator M = 1 2 (I + D −1 A) has the property that D 1 /2 Mv 2 2 = v T DM 2 v for any vector v. Proof. We use the fact that M = 1 2 D − 1 /2 (I + D − 1 /2 AD − 1 /2 ))D 1 /2 : D 1 /2 Mv 2 2 = 1 4 v T M T DMv = 1 4 v T D 1 /2 (I + D − 1 /2 AD − 1 /2 )D − 1 /2 DD − 1 /2 (I + D − 1 /2 AD − 1 /2 )D 1 /2 v = 1 4 v T D 1 /2 (I + D − 1 /2 AD − 1 /2 ) 2 D 1 /2 v = v T DM 2 v,
as desired.
Fact 2.3. If G is a graph with adjacency matrix A, then for the lazy random walk operator
M = 1 2 (I + D −1 A), we have ||D 1 /2 v|| 2 2 = v T Dv v T DMv v T DM 2 v = ||D 1 /2 Mv|| 2 2 .
Proof. Since the eigenvalues λ i of D − 1 /2 AD − 1 /2 are between [−1, 1], the eigenvalues of M ′ = 1 2 (
I + D − 1 /2 AD − 1 /2 ) are µ i = 1 2 (1 + λ i ), and so µ i ∈ [0, 1]. Let D 1 /2 v = α i u i be the decomposition of D 1 /2 v in terms of the eigenvectors of M ′ . Then we have D 1 /2 Mv = M ′ D 1 /2 v = α i µ i u i , and so v T Dv = α 2 i , v T DMv = α 2 i µ i , and v T DM 2 v = α 2 i µ 2 i . Since µ i ∈ [0, 1], we have v T Dv v T DMv v T DM 2 v, as desired.v ∈ R V , v 0 we have ||Dv|| 1 = ||DMv|| 1 . Proof. Let v ∈ R V . We have DMv 1 = 1 T D( I+D −1 A 2 )v = 1 2 ((1 T D)v + (1 T A)v) = Dv 1 ,
where the last inequality follows because 1 T D = 1 T A.
Small-Set Expansion Problem.
The formal statement of the SSE' problem is as follows.
Definition 2. For constants 0 < s < c < 1 and δ > 0, the Small-Set Expansion problem SSE ′ δ (c, s) is defined as follows: Given a graph G = (V, E), distinguish between the following two cases:
-G contains a set S such that vol(S) ∈ [δ/2, δ] and φ(S) 1 − c -All sets S with vol(S) 8δ in G have expansion φ(S) 1 − s.
The key difference from SSE δ (c, s) is that the soundness is slightly stronger in that even sets of size 8δ have expansion at least 1 − s.
Organization.
In Section 3, we will obtain upper and lower bounds (Theorem 3.1) for expansion profile of lazy random walks. Subsequently, we use these bounds to conclude the main result of the paper in Section 4. In Appendix A, we give a reduction that establishes the equivalence of the search versions of two different notions of Small-Set Expansion. We also present a reduction from SSE on irregular graphs to SSE on regular graphs in Appendix B. Finally, in Appendix C, we discuss some obstacles encountered in applying this reduction to the Unique Games problem.
Expansion profile of lazy random walks
Let G = (V, E) be a graph with adjacency matrix A, and diagonal degree matrix D. The transition matrix for a lazy random walk on G is M = 1 2 (I + D −1 A) = 1 2 D − 1 /2 (I + D − 1 /2 AD − 1 /2 )D 1 /2 . For every t ∈ N, let G t denote the graph corresponding to the t-step lazy random walk whose adjacency matrix is given by DM t . We will prove the following theorem about the expansion profile of G t .
Theorem 3.1. For all t ∈ N and η, δ ∈ (0, 1], if G t denotes the graph corresponding to the t-step lazy random walk in a graph G = (V, E) then,
min 1 − 1 − φ 2 G ( 4δ η ) 32 t , 1 − η φ G t (δ) t 2 · φ G (δ)
We will split the proof of the above theorem in to two parts: Lemma 3.2 and Lemma 3.3
Lemma 3.2. For every subset S ⊆ V, GT12], we have that the probability p(t) that a lazy random walk stays entirely in S for t steps is bounded below by
φ G t (S) t 2 · φ G (S), and therefore φ G t (δ) t 2 · φ G (δ). Proof. Fix a subset S ⊂ V. From [p(t) 1 − 1 2 φ(S) t .
Now, the expansion of S in G t is the probability of leaving the set on the tth step of the random walk, which is at most 1 − p(t). Hence,
φ G t (S) 1 − p(t) 1 − 1 − 1 2 φ(S) t t 2 φ(S),
as desired. The result immediately follows for all sets of volume δ.
Lemma 3.3. For all t, η,
φ G t (δ) min 1 − 1 − φ 2 G ( 4δ η ) 32 t , 1 − η
We prove this lemma by contradiction, by showing that if the expansion in the final graph is not large enough then there exists a vector with bounded Rayleigh quotient with respect to the original graph, from which we can extract a non-expanding set. The intuition is that the expansion of a set in the final graph DM t corresponds to the neighborhood of the random walk after t steps, and if the neighborhood is not large enough after t steps, there must be at least one step (or application of M) during which it did not grow.
Proof. Suppose by way of contradiction that this is not the case. Let β = φ G ( 4δ η ) and let δ ′ = 4δ η . Further, letβ = 1 2 β. Let S be a set of volume at most δ · vol(V) such that We first lower-bound ||w t 2 || 2 . By definition of expansion,
φ G t (S) min 1 − 1 −β 2 8 t , 1 − η .φ G t (S) = 1 − v T 0 DM t v 0 v T 0 Dv 0 which by Fact 2.2 implies that ||D 1 /2 M t 2 v 0 || 2 2 = vol(S)(1 − φ G t (S))
. Now, using (3.1) we get
||w t 2 || 2 2 = ||D 1 /2 M t 2 v 0 || 2 2 = vol(S)(1 − φ G t (S)) vol(S) · max η, (1 − 1 8β 2 ) t (3.2)
By Fact 2.3, we have ||w i || 2 ||w i+1 || 2 0 for all i, and (3.2) holds for all i t 2 . We now assert that there must be some i for which
||w i+1 || 2 2 ||w i || 2 2 > 1 − 1 4β 2 .
To see this, consider the product of all such terms for i < t 2 . Some algebraic simplification shows that
t 2 −1 i=0 w i+1 2 2 w i 2 2 = ||w t 2 || 2 2 ||w 0 || 2 2 > (1 − 1 8β 2 ) t · vol(S) vol(S) = 1 − 1 8β 2 t ,
where the second-to-last inequality follows from (3.2). Thus for some i < t 2 we have
w i+1 2 2 w i 2 2 > (1 − 1 8β 2 ) t 2 t > 1 − 1 4β 2 .
Then let w i be the vector corresponding to the first i for which ||w i+1 || 2 2 (1 − 1 4β 2 )||w i || 2 2 .
Since w i+1 is obtained from v i via one step of a lazy random walk and a normalization, we can bound the Rayleigh quotient of v i with respect to the Laplacian of DM = 1 2 (D + A):
v T i D(I − M)v i v T i Dv i = 1 − v T i DMv i v T i Dv i , by Fact 2.3, 1 − v T i DM 2 v i v T i Dv i
and by Fact 2.2,
= 1 − w i+1 2 2 w i 2 2 1 4β 2 . (3.3)
We now truncate the vector v i , then run Cheeger's algorithm on the truncated vector in order to find a non-expanding small set, and thus obtain a contradiction. Let θ = η 4 . We take the truncated vector
z i ( j) = v i ( j) − θ v i ( j) θ 0 otherwise .
By Fact 2.4, Dv i has L 1 mass vol(S). Thus, the total volume of the set S z of vertices with nonzero support in z i is
vol(S z ) = v i (j)>θ d( j) v i (j)>θ 1 θ d( j)v i ( j) 1 θ · Dv i 1 = 4 vol(S) η
Hence any subset of S z has volume at most 4 vol(S) η . For the vector v i , we know that Dv i 1 = vol(S). Moreover using (3.2),
D 1/2 v i 2 2 = w i 2 2 w t/2 2 2 η vol(S) .
Applying Lemma 1 to v i and z i to conclude,
z T i D(I − M)z i z T i Dz i 2 v T i D(I − M)v i v T i Dv i .
Using (3.3), this implies the following bound on the Rayleigh quotient of z i ,
z T i D(I − M)z i z T i Dz i 1 2β 2 .
Thus, when we run Cheeger's algorithm on z i , we get a set of volume at most 4 vol(S) η and of expansion less thanβ in DM, and therefore less than β in G. Since β = φ G ( 4δ η ), this is a contradiction. This completes the proof of Lemma 3.3.
The following lemma, which gives an upper bound on the Rayleigh quotient of a truncated vector, is a slight generalization of Lemma 3.4 of [ABS10].
Lemma 1. Let x ∈ R V be non-negative, let L be the weighted Laplacian of a graph G = (V, E) with weights w(i, j) and degree matrix D. Suppose that
4θ Dx 1 D 1/2 x 2 2 (3.4)
Then for the threshold vector y defined by
y(i) = x(i) − θ x(i) > θ 0 otherwise , we have y T Ly y T Dy 2 · x T Lx x T Dx .
Proof. First, we show y T Ly x T Lx.
y T Ly = (i, j)∈E w(i, j)(y(i) − y( j)) 2 = (i, j)∈E y(i),y(j) 0 w(i, j)(x(i) − x( j)) 2 + (i, j)∈E y(i) 0,y(j)=0 w(i, j)(x(i) − θ) 2 (i, j)∈E w(i, j)(x(i) − x( j)) 2 = x T Lx,
where the second-to-last inequality follows from the fact that if y(i) = 0, then x(i) θ. Now, we show that y T Dy 1 2 x T Dx. First, we note that
d(i)y(i) 2 d(i)x(i) 2 − 2θd(i)x(i) for all k. Thus, i∈V d(i)y(i) 2 i∈V d(i)x(i) 2 − 2θd(i)x(i) = i∈V d(i)x(i) 2 − 2θ i∈V d(i)x(i) 1 2 i∈V d(i)x(i) 2
Where the the last inequality follows by assumption (3.4).
Thus, we have y T Ly y T Dy 2 · x T Lx x T Dx , as desired.
Gap Amplification
In this section, we will prove Theorem 1 which we restate here for convenience.
Theorem 2. (Restatement of Theorem 1) Let f be any function such that lim ε→0
f (ε) √ ε → ∞. Then If for all ε > 0, SSE ′ (1 − ε, 1 − f (ε)) is NP-hard then for all η > 0 SSE(1 − η, 1 2 ) is NP-hard.
Proof. Fix ε small enough so that 64ε f (ε) 2 η. There exists such an ε since lim ε→0
f (ε) √ ε → ∞. Fix t = 64 f (ε) 2 .
Given an instance G of SSE ′ (1 − ε, 1 − f (ε)), the reduction just outputs the graph G t obtained via t-step lazy random walks on G. Since the adjacency matrix of G ′ can be calculated with log t matrix multiplications, this reduction clearly runs in time O(n 3 log t).
Completeness. If there exists a set of S with vol(S) ∈ [δ/2, δ] and φ G (S) ε then by Lemma 3.2 the same set S satisfies,
φ G t (S) t 2 φ G (S) = Θ ε f (ε) 2 η Soundness. If φ G (8δ) f (ε) then by applying Lemma 3.3 φ G t (δ) min 1 − 1 − 1 32 f (ε) 2 t , 1 /2 1 2
B Reduction from Irregular Graphs to Regular Graphs
In this section, we present a reduction from small set expansion on irregular graphs to small set expansion on regular graphs. Specifically, we prove the following theorem.
Theorem 3. There exists an absolute constant C such that for all γ, β ∈ (0, 1) there is a polynomial time
reduction from SSE δ (1 − γ, 1 − β) on a irregular graph G = (V, E) to SSE δ (1 − γ, 1 − β /C) on a 4-regular graph G ′ = (V ′ , E ′ )
Proof. The reduction is as follows: we replace each vertex v ∈ V with a 3-regular expander A v on deg(v) vertices. Using standard constructions of 3-regular expanders, we can assume that the graphs A v have edge expansion at least κ = 0.01. Now, for each edge (v, w) ∈ E, we add an edge between a particular vertex in A v and A w . The resulting graph on the expanders is
G ′ , with V ′ = ∪ v∈V A v . Note that G ′ is d-regular, and that |V ′ | = v∈V deg(v) = vol(V), as desired.
For the completeness, we note that if a set S ⊂ V with volume at most δ|V| has φ G (S) < γ, then the set S ′ = ∪ v∈S A v has the same number of edges leaving the set as S, and the number of vertices in the set is equal to vol(S). Thus, φ G ′ (S ′ ) < γ/4, as desired.
For soundness, suppose there is a set S ′ ⊂ V ′ with |S ′ | δ|V ′ | and φ G ′ (S ′ ) < β. Then we can partition S ′ into sets corresponding to each A v ; let B v = S ′ ∩ A v . Then consider the set
S * = ∪ |B v | 1 2 |A v | A v
, the set of A v that overlap with S ′ by at least half. We will argue that S * has expansion at most 10 κ β in G ′ . First, by definition of expansion we have
β φ G ′ (S ′ ) = v E(B v ,S ′ ) 4 v∈V |B v | = v E(B v , A v \ B v ) + E(B v ,S ′ \ A v ) 4 v∈V |B v | ,
where we distinguish between boundary edges of S ′ inside and outside of the A v . In particular, we have 4β
v∈V |B v | v∈V E(B v , A v \ B v ).
Now, we bound from below the number of boundary edges within A v . Since A v is an expander with expansion κ, we have
E(B v , A v \ B v ) κ · min(|B v |, |A v \ B v |).
Hence we will have,
S ′ ∆S * = v min(|B v |, |A v \ B v |) 1 κ v E(B v , A v \ B v ) 4β κ v∈V |B v | = 4β κ |S ′ |
Since G ′ is a 4-regular graph, we can upper bound the expansion of S * by
φ G ′ (S * ) E[S ′ ,S ′ ] + 4|S ′ ∆S * | 4|S ′ | − 4|S ′ ∆S * | 4β|S ′ | + 16 β /κ|S ′ | 4|S ′ | − 16 β /κ|S ′ | β (1 + 4 /κ) 1 − 4β /κ
Thus, in G the set S = {v | A v ∈ S * } has expansion at most 10 κ β, and vol(S) ∈ [ 1 2 δ vol(V), 2δ vol(V)], as desired.
C Discussion of Unique Games
The following simple example illustrates why a similar reduction will not work for Unique Games. Consider a unique games instance with two disconnected components, one of size s · n in which only 1 q of the constraints are satisfiable, and another component of size (1 − s)n in which all constraints are perfectly satisfiable. In this case, running a random walk on the label-extended graph will not alter the number of satisfiable constraints.
A slight modification of this approach would at first seem to be a promising avenue for overcoming this example at first: reweight the graph of constraints by (1 − w), and add to it an expander of weight w with arbitrary constraints. Now, in the previous example, following the proof of our completess case we observe that after t steps of the random walk, the formerly perfectly satisfiable component has expansion at most tw, the soundness is at most 1 − t(w + rs). Similarly, in the completeness case, our reduction goes from completeness 1 − c to soundness 1 − t(c + w). Thus, because the added constraints contribute equally to the soundness of the bad example and the completeness in general, this approach does not overcome the problem of isolated components.
Fact 2. 4 .
4For the lazy random walk operator M = 1 2 (I + D −1 A) and any vector
0 = 1 S be the vector corresponding to the indicator function of the set S. Define v i = M i v 0 , and for the diagonal degree matrix D of A, define w i = D 1 /2 v i . Note that ||w 0 || 2 2 = vol(S), and Dv 0 1 = vol(S). By Fact 2.4 we also have Dv i 1 = vol(S) for all i.
A Equivalence of Two Notions of the Small-Set Expansion ProblemThere is a slightly different version of the Small-Set expansion decision problem that differs from Definition 1 in the soundness case.Definition 3. For constants 0 < s < c < 1, and δ > 0, the Small-Set Expansion problem SSE = δ (c, s) is defined as follows: Given a graph G = (V, E) with vol(V) = N, distinguish between the following two cases:-G has a set of volume in the range [ 1 2 δN, δN] with expansion less than 1 − c -All sets in G of volume in the range [ 1 4 δN, δN] have expansion at least 1 − s. Clearly SSE = δ (c, s) is a harder decision problem than SSE δ (c, s) since the soundness assumption is weaker. There is no known reduction from SSE = δ (c, s) to SSE δ (c, s) that establishes the equivalence of the two versions. Here we observe that the search versions of these two problems are equivalent.Proof. Suppose we are given an algorithm A that finds a set S ′ of volume at most δN and expansion less than 1 − s whenever there exists a set S with vol(S) ∈ [ 1 4 δN, δN] and Φ(S) 2 − 2c. We construct a set S ⊆ V such that vol(S) ∈ [ 1 4 δN, δN] and φ(S) < 1 − s. We proceed iteratively, as follows.We start with an empy initial set, S out , and with the full graph, G 0 = G. If vol(S out ) ∈ [ 1 4 δN, δN], we terminate and return S out . Otherwise, at the ith step, we apply A to G i−1 to obtain a set S i of expansion less than 1 − s. If vol(S i ) ∈ [ 1 4 δN, δN] return S i , otherwise add the vertices in S i to S out . We then set G i = G i−1 \ S i . If no such set can be found, then we terminate and return no.Clearly, this algorithm terminates and runs in polynomial time. Suppose S ′ is a nonexpanding set with vol(S ′ ) ∈ [ 1 2 δN, δN]. As long as S out has volume smaller than 1 4 δN, S ′ − S out will have volume at least vol(S ′ )/2 and has expansion at most 2φ(S ′ ) 2 − 2c. Hence by the assumption about algorithm A, it will return a set S i of expansion at most 1 − s. The check of the volume of S i ensures that S out will never go from below the allowable volume range to above in a single step. Finally if S i was never returned for any step i, the union of all the sets S i has expansion at most 1 − s and volume in the range [δN/4, δN].
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[
"Multidimensional entropic uncertainty relation based on a commutator matrix in position and momentum spaces",
"Multidimensional entropic uncertainty relation based on a commutator matrix in position and momentum spaces"
] |
[
"Anaelle Hertz \nCentre for Quantum Information and Communication\nEcole polytechnique de Bruxelles\nUniversité libre de Bruxelles\n1050BrusselsBelgium\n",
"Luc Vanbever \nCentre for Quantum Information and Communication\nEcole polytechnique de Bruxelles\nUniversité libre de Bruxelles\n1050BrusselsBelgium\n",
"Nicolas J Cerf \nCentre for Quantum Information and Communication\nEcole polytechnique de Bruxelles\nUniversité libre de Bruxelles\n1050BrusselsBelgium\n"
] |
[
"Centre for Quantum Information and Communication\nEcole polytechnique de Bruxelles\nUniversité libre de Bruxelles\n1050BrusselsBelgium",
"Centre for Quantum Information and Communication\nEcole polytechnique de Bruxelles\nUniversité libre de Bruxelles\n1050BrusselsBelgium",
"Centre for Quantum Information and Communication\nEcole polytechnique de Bruxelles\nUniversité libre de Bruxelles\n1050BrusselsBelgium"
] |
[] |
The uncertainty relation for continuous variables due to Byalinicki-Birula and Mycielski expresses the complementarity between two n-tuples of canonically conjugate variables (x1, x2, · · · xn) and (p1, p2, · · · pn) in terms of Shannon differential entropy. Here, we consider the generalization to variables that are not canonically conjugate and derive an entropic uncertainty relation expressing the balance between any two n-variable Gaussian projective measurements. The bound on entropies is expressed in terms of the determinant of a matrix of commutators between the measured variables. This uncertainty relation also captures the complementarity between any two incompatible linear canonical transforms, the bound being written in terms of the corresponding symplectic matrices in phase space. Finally, we extend this uncertainty relation to Rényi entropies and also prove a covariance-based uncertainty relation which generalizes Robertson relation.
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10.1103/physreva.97.012111
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"https://arxiv.org/pdf/1711.04566v2.pdf"
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1711.04566
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dc8c0974909123ccdf143513fe127f9c1219bc5d
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Multidimensional entropic uncertainty relation based on a commutator matrix in position and momentum spaces
Anaelle Hertz
Centre for Quantum Information and Communication
Ecole polytechnique de Bruxelles
Université libre de Bruxelles
1050BrusselsBelgium
Luc Vanbever
Centre for Quantum Information and Communication
Ecole polytechnique de Bruxelles
Université libre de Bruxelles
1050BrusselsBelgium
Nicolas J Cerf
Centre for Quantum Information and Communication
Ecole polytechnique de Bruxelles
Université libre de Bruxelles
1050BrusselsBelgium
Multidimensional entropic uncertainty relation based on a commutator matrix in position and momentum spaces
The uncertainty relation for continuous variables due to Byalinicki-Birula and Mycielski expresses the complementarity between two n-tuples of canonically conjugate variables (x1, x2, · · · xn) and (p1, p2, · · · pn) in terms of Shannon differential entropy. Here, we consider the generalization to variables that are not canonically conjugate and derive an entropic uncertainty relation expressing the balance between any two n-variable Gaussian projective measurements. The bound on entropies is expressed in terms of the determinant of a matrix of commutators between the measured variables. This uncertainty relation also captures the complementarity between any two incompatible linear canonical transforms, the bound being written in terms of the corresponding symplectic matrices in phase space. Finally, we extend this uncertainty relation to Rényi entropies and also prove a covariance-based uncertainty relation which generalizes Robertson relation.
I. INTRODUCTION
At the heart of quantum mechanics, uncertainty relations reflect the impossibility to define -exactly and simultaneously -the value of two observables that do not commute, such as the positionx and momentum p of a particle. Uncertainty relations, expressed in terms of variances of observables, were first introduced by Heisenberg [1] and Kennard [2], and then generalized by Schrödinger [3] and Robertson [4]. Later on, it was shown by Hirschman [5] that uncertainty relations may also be formulated in terms of Shannon entropies instead of variances, leading to the first entropic uncertainty relation for canonically conjugate variablesx andp proven by Bialynicki-Birula and Mycielski [6] and Beckner [7]. Entropic uncertainty relations have also been developed for discrete observables in finite-dimensional spaces, see [8] for a review, but here we focus on continuous-spectrum observables in an infinite-dimensional space. Specifically, we use the notations of quantum optics and view variablesx andp as canonically conjugate quadrature components of a bosonic mode. Then, the n-modal version of the entropic uncertainty relation for the n-tuples x = (x 1 , x 2 , · · · x n ) and p = (p 1 , p 2 , · · · p n ) is expressed as 1 [6] h( x) + h( p) ≥ n ln(πe) (1) where h( x) and h( p) are the Shannon differential entropies of x and p, namely
h( x) ≡ h(|ψ( x)| 2 ) = − d x |ψ( x)| 2 ln |ψ( x)| 2 ,
h( p) ≡ h(|φ( p)| 2 ) = − d p |φ( p)| 2 ln |φ( p)| 2 , (2) * Electronic address: [email protected] 1 We set = 1 throughout this paper.
with |ψ( x)| 2 = | x 1 , x 2 , · · · x n |ψ | 2 and |φ( p)| 2 = | p 1 , p 2 , · · · p n |ψ | 2 being the probability distributions of x and p in the pure state |ψ . Of course, φ( p) is the Fourier transform of ψ( x), which is at the origin of the complementarity between x and p expressed by Eq. (1). Lately, this entropic uncertainty relation has been extended by taking x-p correlations into account [9], the significant advantage being that the resulting uncertainty relation is saturated by any pure n-modal Gaussian state.
In 2011, Huang [10] generalized the entropic uncertainty relation to a pair of observables that are not canonically conjugate. More precisely, defining the observableŝ
A = n i=1 (a ixi + a ipi ),B = n i=1 (b ixi + b ipi ),(3)
he showed that
h(Â) + h(B) ≥ ln(πe|[Â,B]|)(4)
where [Â,B] (which is a scalar) is the commutator between both observables. Obviously, if =x andB =p, this inequality reduces to Eq. (1). In addition, a similar result had earlier been obtained by Guanlei et al. [11] in the special case where n = 1, namely
h(x θ ) + h(x φ ) ≥ ln(πe| sin(θ − φ)|)(5)
wherex θ =x cos θ +p sin θ andx φ =x cos φ +p sin φ are two rotated quadratures. In this paper, we introduce a generalization of the uncertainty relation of Byalinicki-Birula and Mycielski, which is stated in the form of our Theorem 1. It addresses the situation where n arbitrary quadratures are jointly measured on n modes, expressing the balance between two such joint measurements (see Fig. 1). In other words, we state an entropic uncertainty relation between two arbitrary n-modal Gaussian projective measurements (or, equivalently, two n-mode Gaussian unitaries U A and U B ). The lower bound of our uncertainty relation, Eq. (32), depends on the determinant of a n × n matrix formed arXiv:1711.04566v2 [quant-ph] 12 Jan 2018 Figure 1: Schematic of two n-modal Gaussian projective measurements applied onto state |ψ , resulting in the n quadraturesŷ i 's orẑ i 's. These measurements can be implemented by applying a Gaussian unitary (U A or U B ) onto |ψ and measuring thex-quadratures of the n modes. Our uncertainty relation, Eq. (32), expresses the complementarity between theŷ i 's andẑ i 's, or equivalently between the linear canonical transforms associated with U A and U B .
| ⟩ x '1 '2 '3 'n x x x | ⟩ ℬ x 1 2 3 n x x x x
with the commutators between the n measured quadratures in both cases. In contrast, Eq. (1) is restricted to the case of measuring either all x quadratures or all p quadratures on n modes, while Refs. [10,11] treat the balance between two single-mode measurements only.
Interestingly, the probability distribution of the measured quadratures is given by the squared modulus of the linear canonical transform (LCT) associated with U A or U B , so that our entropic uncertainty relation also captures the complementarity between two incompatible n-dimensional LCTs as expressed by our Lemma 1. It simply reduces to Eq. (1) when the two LCTs are connected by a n-dimensional Fourier transform, mapping x = (x 1 , x 2 , · · · x n ) T onto p = (p 1 , p 2 , · · · p n ) T .
In Section II, we define general n-dimensional LCT's and give some useful properties. Section III presents our results on uncertainty relations for n modes. First, we derive a generalized entropic uncertainty relation based on differential Shannon entropies (our Theorem 1, with its extension to a larger-dimensional space), then we extend it to Rényi entropies (our Theorem 2), and finally we exhibit a covariance-based uncertainty relation (our Theorem 3). In Section IV, we conclude and suggest a conjecture for a generalized entropic uncertainty relation in the case where the commutators differ from scalars.
II. LINEAR CANONICAL TRANSFORMS
Before deriving our uncertainty relations, we need to properly define fractional Fourier transforms (FRFTs) along with their generalization to LCTs. Some early papers on FRFTs appeared in the 1920's, but this topic became investigated in depth only more recently in the fields of signal processing and quantum optics (see, e.g., [12][13][14][15][16] for more details). In one dimension, the FRFT of a wave function f (x) can be understood as the new wave function obtained when the Wigner function corresponding to f (x) undergoes a rotation of angle α in phase space. If α = π/2, then the FRFT simply coincides with the usual Fourier transform, connecting the time and frequency domains in the field of signal processing or the canonically conjugate x-and p-quadratures in quantum optics. Mathematically, the one-dimensional FRFT of function f (x) is defined as
F α (y) = 1 − i cot α 2π e i 2 y 2 cot α e −iyx sin α e i 2 x 2 cot α f (x) dx(6)
The one-dimensional FRFT can be generalized to onedimensional LCTs by including all affine linear transformations in phase space (x, p), going beyond rotations. Accordingly, the LCT of wave function f (x) is the new wave function obtained when the corresponding Wigner function undergoes a symplectic transformation S. The one-dimensional LCT of f (x) is defined as
F S (y) = 1 2πib e id 2b y 2 e −iyx b e ia 2b x 2 f (x) dx(7)
where S = a b c d is a symplectic matrix with a, b, c, and d being real parameters, and b = 0. The notion of LCT can readily be extended to n dimensions, the resulting transformation being also sometimes called n-dimensional FRFT. The physical interpretation is straightforward, namely a LCT is the transformation of a n-dimensional wave function f ( x) that is effected by any symplectic transformation in the 2ndimensional phase space of variables (x 1 , x 2 , · · · x n ) and (p 1 , p 2 , · · · p n ). We write the symplectic matrix S as
S = a b c d = 1 0 db −1 1 b 0 0 b −1 0 1 −1 0 1 0 b −1 a 1(8)
where a, b, c, and d are n × n real matrices, 1 1 is the n × n identity matrix, and det(b) = 0. Since S is symplectic, it obeys the constraint
SJS T = J with J = 0 1 −1 0 (9)
being the symplectic form, so that det(S) = 1. This also implies that ab T and cd T are symmetric matrices, and ad T − bc T = 1. The corresponding symplectic transformation in phase space is
y q = S x p ,(10)
where y = (y 1 , y 2 , · · · y n ) T and q = (q 1 , q 2 , · · · q n ) T form a new pair of canonically conjugate n-tuples. In state space, the LCT of f ( x) can be written as
F S [f ( x)]( y) = 1 (2π) n | det(b)| d xf ( x) e −i b −1 y· x × e i 2 [ x T (b −1 a) x+ y T (db −1 ) y] = C db −1 D b −1 FC b −1 a [f ( x)]( y)(11)
with
C r [f ]( x) = e i 2 x T r x f ( x) D b [f ]( x) = | det(b)|f (b x) F[f ]( x) = 1 (2π) n/2 d yf ( y)e −i x· y(12)
where C r represents the chirp multiplication, D b the squeezing (or dilation) operator, and F the usual Fourier transform. These operators are directly related to the decomposition of S in Eq. (8). Note also that the chirp multiplication (in one dimension) can be expressed as a product of the other two operators, namely C r = R θ−π/2 · D tan θ · R θ where R θ represent the rotation and r = tan θ − cot θ. Finally, note that the set of LCTs in phase space is in one-to-one correspondence with the set of Gaussian unitaries in state space [17], which can indeed be decomposed into passive linear-optics operations (phase shifters and beam splitters, i.e., rotations in phase space) and active squeezing operations (i.e., areapreserving dilations in phase space).
Here are some properties of LCTs that will be useful to prove our results in Section III.
Properties.
F
A F B = F AB 2. D b1 D b2 = D b1b2 3. F −1 = D −1 F where F −1 is the inverse Fourier transform. 4. |C r f | = |f | Proof.
1. Using Eq. (11) and the corresponding representation in phase space, Eq. (8), we see that F A F B is represented by the matrix AB. Since the symplectic matrices form a group, the product of two symplectic matrices is a symplectic matrix, which also admits decomposition (8) and thus represents the linear canonical transform F AB .
Proofs of 2, 3 and 4 are straightforward.
III. MULTIDIMENSIONAL UNCERTAINTY RELATIONS
A. Entropic uncertainty relation between two linear canonical transforms
Let |ψ be an arbitrary n-mode state. We wish to express the complementarity between two incompatible LCTs corresponding to two Gaussian unitaries (U A or U B ) applied onto |ψ . As shown in Fig. 1, we measure in both cases the n-tuple of output x-quadratures, which corresponds to applying two possible n-modal Gaussian projective measurements on |ψ . The vectors of measurement outcomes are noted, respectively, y = (y 1 , · · · , y n ) T or z = (z 1 , · · · , z n ) T . Denoting as A and B the symplectic transformations associated with U A and U B , and writing asr = (x 1 , · · · ,x n ,p 1 , · · · ,p n ) T the 2n-dimensional vector of input quadratures, we may express the corresponding vectors of output quadratures aŝ
r A = Ar ≡ y q ,r B = Br ≡ z o .(13)
where q (resp. o) is the vector of quadratures that are canonically conjugate with y (resp. z). The probability distributions for y and z are thus given by the squared modulus of the LCTs associated with U A and U B , namely
|F A [ψ( x)]( y)| 2 and |F B [ψ( x)]( z)| 2 .
In order to find an entropic uncertainty relation for y and z, we first express the complementarity between F A and F B in the following Lemma.
Lemma 1. Let F A and F B be two LCTs of a function f ( x) with x = (x 1 , · · · x n ).
Then, their squared moduli satisfy the entropic uncertainty relation
h(|F A | 2 ) + h(|F B | 2 ) ≥ ln (πe) n | det(B b A T a − B a A T b )| (14) where A = A a A b A c A d and B = B a B b B c B d(15)
are the symplectic matrices associated with F A and F B [acting on the quadrature operators as in Eq. (13)] and h(·) denotes Shannon differential entropy.
Proof. Let us define the function
G ( x) = C −b −1 a F A ( x) = e − i 2 x T (b −1 a) x F A ( x) . (16)
The inverse Fourier transform of G( x) is
g ( p) = F −1 [G ( x)]( p) = [D −1 F G] ( p) ,(17)
where the second equality results from property 3. Since
|G ( x)| 2 = |F A ( x)| 2 , the probability distributions are equal, so that h |G ( x)| 2 = h |F A ( x)| 2 .
Then, we may apply Eq. (1) to G and g, which gives
h |F A ( x)| 2 + h |g ( p)| 2 ≥ n ln(πe).(18)
With the change of variables p → b −1 p, we have
h |g ( p)| 2 = − |g ( p)| 2 ln |g ( p)| 2 d p = − g b −1 p 2 ln g b −1 p 2 d p | det(b)| .(19)
By using the above properties of LCTs, we have
g b −1 p 2 = det(b) D b −1 [g] ( p) 2 = | det(b)| [D b −1 D −1 F G] ( p) 2 = | det(b)| [D b −1 D −1 F C −b −1 a F A ] ( p) 2 = | det(b)| [C −db −1 D −b −1 F C −b −1 a F A ] ( p) 2 = | det(b)| F S F A ( p) 2 = | det(b)| F S A ( p) 2 ,(20)
where
we have defined S = a −b −c d , so by plugging it into Eq. (19), we get h |g ( p) | 2 = h |F S A ( p) | 2 − ln | det(b)| (21) since |F S A | 2 is a normalized function. Now, replacing h |g ( p) | 2 in Eq. (18), we obtain h |F A ( x)| 2 + h |F S A ( p) | 2 ≥ ln((πe) n | det(b)|).
(22) The last step is simply to define B = S A or equivalently S = BA −1 . Since symplectic matrices form a group and A and B are symplectic, S is necessarily symplectic too. The property that A is symplectic translates into
A −1 = JA T J T = A T d −A T b −A T c A T a ,(23)hence b = B a A T b − B b A T a .
Replacing b into Eq. (22) completes the proof of Eq. (14), which thus provides a n-dimensional entropic uncertainty relation for any two incompatible LCTs.
Note that in the special case of one mode (n = 1), we recover the result obtained by Guanlei et al. [18] for one-dimensional LCTs, namely [11]). Now, back to the nmode case, if we choose A = 1 and B being the direct sum of π/2 rotations on each modes (i.e., the usual ndimensional Fourier transform), then A a = 1,
h(|F A | 2 ) + h(|F B | 2 ) ≥ ln (πe|ab − a b|)(24)A b = 0, B a = 0, and B b = 1, so that B b A T a − B a A T b = 1.
Hence, we get back to the original entropic uncertainty relation of Bialynicki-Birula and Mycielski, Eq. (1). Finally, if we consider twice the same measurement, i.e., A = B, then
S = AA −1 = A a A b A c A d A T d −A T b −A T c A T a = A a A T d − A b A T c −A a A T b + A b A T a A c A T d − A d A T c −A c A T b + A d A T a .(25)
But since S = 1, we have A b A T a − A a A T b = 0, so that the lower bound in Eq. (14) is −∞. This means that we have no lower limit on the entropy h(|F A | 2 ) so the probability distribution |F A | 2 can be arbitrarily narrow, as expected.
Interestingly, in the special case where A = 1 1, it is possible to find a simpler alternative proof of Lemma 1. We define
S = B a B b −(B −1 b ) T 0 .(26)
and may easily check that S is a symplectic matrix by verifying that SJS T = J. Indeed
SJS T = −B b B T a + B a B T b B b B −1 b −(B −1 b ) T B T b 0 ,(27)
is equal to J since B a B T b is a symmetric matrix and det(B b ) = 0 (as B is also a symplectic matrix). Thus, S transformsr into a new vector of quadratures,
Sr = z −(B −1 b ) T x(28)
where z is the vector of position quadratures inr B [see Eq. (13)]. Since S is symplectic, z and −(B −1 b ) T x are two vectors of canonically conjugate quadratures, which we may plug into Eq. (1), giving
h −(B −1 b ) T x + h( z) ≥ n ln(πe).(29)
By using the scaling property of the differential entropy,
we have h(−(B −1 b ) T x) = h( x) + ln(| det((B −1 b ) T )|), so that Eq. (29) becomes h( x) + h( z) ≥ ln ((πe) n | det(B b )|) .(30)
Since the probability distribution of x is |F 1 1 | 2 and that of z is |F B | 2 , we recover Lemma 1 when A = 1.
B. Entropic uncertainty relation based on a matrix of commutators
Lemma 1 provides an entropic uncertainty relation for any two n-dimensional LCTs, F A and F B . As we show in the following theorem, this uncertainty relation can also be expressed in terms of a matrix of commutators between the measured variables. This is our main result. Theorem 1. Let y = (ŷ 1 , · · ·ŷ n ) T be a vector of commuting quadratures and z = (ẑ 1 , · · ·ẑ n ) T be another vector of commuting quadratures. Let the components of y and z be written each as a linear combination of the (x,p) quadratures of a n-modal system, namelŷ Then, the probability distributions of the vectors of jointly measured quadraturesŷ i 's orẑ j 's satisfy the entropic uncertainty relation
y i =h( y) + h( z) ≥ ln ((πe) n | det K|)(32)
where K ij = [ŷ i ,ẑ j ] denotes the n × n matrix of commutators (which are scalars) and h(·) denotes Shannon differential entropy.
Proof. Since the quadraturesŷ i commute, [ŷ i ,ŷ j ] = 0, ∀i, j, they can be jointly measured, and similarly for thê z j 's. Thus, the n measured quadratures correspond here to the output of F A or F B described by the symplectic matrix A or B, as defined in Eq. (15). We simply have to compute the commutator between quadratureŷ i (at the output of F A ) andẑ j (at the output of F B ):
K ji = [ŷ j ,ẑ i ] = 2n k=1 2n m=1 A jk B im [r k ,r m ] = i 2n m=1 n k=1 A jk B im δ m,k+n − 2n k=n+1 A jk B im δ m,k−n = i n k=1 A jk B i,k+n − 2n k=n+1 A jk B i,k−n = i n k=1 A jk B i,k+n − A j,k+n B ik = i n k=1 (A a ) jk (B b ) ik − (A b ) jk (B a ) ik = i B b A T a − B a A T b ij(33)
Using Lemma 1, we know that the probability distributions |F A ( y)| 2 and |F B ( z)| 2 satisfy the entropic uncertainty relation, Eq. (14). We now show that this result holds even if we jointly measure n quadratures on a larger-dimensional system. Theorem 1. (Extended version.) Let y n = (ŷ 1 , · · ·ŷ n ) T be a vector of commuting quadratures and z n = (ẑ 1 , · · ·ẑ n ) T be another vector of commuting quadratures. Let each components of the y n and z n be written as a linear combination of the (x,p) quadratures of a N -modal system with N > n, namelŷ
Since B b A T a − B a A T b = −iK T , we conclude that | det(B b A T a − B a A T b )| = | det(K)|,y i = N k=1 a i,kxk + N k=1 a i,kpk (i = 1, · · · n) z j = N k=1 b j,kxk + N k=1 b j,kpk (j = 1, · · · n) (34)
Then, the probability distributions of the vectors of jointly measured quadraturesŷ i 's orẑ j 's satisfy the entropic uncertainty relation
h( y n ) + h( z n ) ≥ ln ((πe) n | det K|) (35)
where K ij = [ŷ i ,ẑ j ] denotes the n × n matrix of commutators (which are scalars) and h(·) denotes Shannon differential entropy.
Proof. The N -dimensional vectors y and z can be decomposed as
y = y n y > , z = z n z > ,(36)
with y n = (y 1 , · · · y n ) T or z n = (z 1 , · · · z n ) T being the n measured quadratures, while y > = (y n+1 , · · · y N ) T or z > = (z n+1 , · · · z N ) T are being traced over. We writê
y i = 2N k=1 A i,krkẑj = 2N k=1 B j,krk(37)
with i, j = 1, · · · n, which generalizes Eq. (13) in the case wherer is a 2N -dimensional vector and A and B are 2N × 2N symplectic matrices (we only need to specify the upper block of size n × 2N of A and B, which defineŝ y i orẑ j , and complete the matrices by ensuring that they remain symplectic). We first note that the right-hand side term of Eq. (35) is invariant under symplectic transformations (if both symplectic matrices A or B are multiplied by a same symplectic matrix). Indeed, the commutation relations are preserved along symplectic transformations and the determinant is invariant under permutations (the order of the quadratures is irrelevant), hence det(K) is invariant. Thus, we may always apply some symplectic transformation on the N modes so that the measured quadratures in the first case are y i = x i , with i = 1, · · · n. The two upper blocks of matrix A are then given by
A a = 1 n×n 0 n×(N −n) · · · · · · N ×N(38)
and
A b = 0 n×n 0 n×(N −n) · · · · · · N ×N(39)
where we do not need to specify the matrix elements denoted with a dot. Next, we may assume with no loss of generality that the two upper blocks of B are given by
B a = B n×n C n×(N −n) D (N −n)×n 1 (N −n)×(N −n) N ×N(40)
and
B b = B n×n C n×(N −n) 0 (N −n)×n 1 (N −n)×(N −n) N ×N ,(41)
with B and C containing all b j,k entries for j = 1, · · · , n and k = 1, · · · , N , and B and C containing all b j,k entries for j = 1, · · · , n and k = 1, · · · , N . This is the case because the last N − n quadratures z > are traced over, so they may be chosen arbitrarily as long as B remains symplectic. This means that we must check that B a B T b is symmetric, which implies that
D = (C − C ) T (B ) −T(42)
where (·) −T stands for the inverse of the transpose of the matrix. Thus, the matrix D can always be chosen in order to ensure that B is symplectic. It is easy to write the n × n restricted matrix of commutators of the measured quadratures y n and z n (the first n quadratures of y and z), giving | det K| = | det B |, so the inverse of B is well defined as long as det K = 0. At this point, we only need to prove Eq. (35) in case the upper blocks of A and B are defined as above and | det K| is replaced by | det B |. As before, we define the symplectic matrix
S = B a B b −(B b ) −T 0 .(43)
It transforms the vector of quadraturesr into
Sr = B a x + B b p −(B b ) −T x = z n z > −(B ) −T x n (C ) T (B ) −T x n − x > (44) where x n = (x 1 , · · · x n ) T , x > = (x n+1 , · · · x N ) T .
This implies that z n and −(B ) −T x n are canonically conjugate n-tuples, so that we may apply Eq. (1) on the reduced state of the first n modes, giving
h −(B ) −T x n + h( z n ) ≥ n ln(πe).(45)
Using the scaling property of the differential entropy
h(−(B ) −T x n ) = h( x n ) + ln(| det(B ) −T |), we obtain h( x n ) + h( z n ) ≥ ln((πe) n | det B |)(46)
This implies Eq. (35), thus completing the proof of the extended version of Theorem 1.
Interestingly, Eq. (35) coincides with Eq. (4) in the special case n = 1. Thus, our Theorem 1 can be viewed as an extension of the result by Huang [10] when we measure more than one mode (n > 1). As already mentioned, we can check that if A = 1 and B is a direct sum of π/2rotations on each modes (i.e., the usual Fourier transform), then K = −i1 and we recover Bialynicki-Birula and Mycielski relation, Eq. (1).
C. Extension to Rényi entropies
The Shannon differential entropy is a special case of the family of Rényi differential entropies defined as
h α (|f ( x)| 2 ) = 1 1 − α ln d x (|f ( x)| 2 ) α(47)
when α → 1. Let us now derive generalized entropic uncertainty relations for these entropies.
Theorem 2. Let y = (ŷ 1 , · · ·ŷ n ) T be a vector of commuting quadratures, z = (ẑ 1 , · · ·ẑ n ) T be another vector of commuting quadratures, and let the components of these vectors be written each as a linear combination of the (x,p) quadratures of a N -modal system (N ≥ n). Then, the probability distributions of the vectors of jointly measured quadraturesŷ i 's orẑ j 's satisfy the Rényi entropic uncertainty relation
h α ( y) + h β ( z) ≥ n ln(α) 2 (α − 1) + n ln(β) 2 (β − 1) +n ln(π) + ln |det K| . (48) where 1 α + 1 β = 2, α > 0, β > 0,(49)K ij = [ŷ i ,ẑ j ]
is the matrix of commutators (which are scalars), and h α (·) is the Rényi differential entropy as defined in Eq. (47).
Proof. The proof follows exactly the same steps as the proof of Lemma 1 and Theorem 1 (both versions) except that Eq. (1) is replaced by its counterpart for Rényi entropies [19] h
α ( x) + h β ( p) ≥ n ln(π) + n ln(α) 2 (α − 1) + n ln(β) 2 (β − 1)(50)
for (α, β) satisfying Eq. (49). Note that Rényi entropies also verify the scaling property h α (S x) = h α ( x) + ln |S|.
As expected, in the limit where α → 1 and β → 1, we recover our uncertainty relations for Shannon differential entropies. Also, in a one-dimensional case (N = n = 1), Eq. (48) coincides with the result found in [18].
D. Covariance-based uncertainty relation
Finally, by exploiting Theorem 1, it is also possible to derive an uncertainty relation in terms of covariance matrices. This can been viewed as a n-dimensional extension of the usual Robertson uncertainty relation in position and momentum spaces where, instead of expressing the complementarity between observables andB (which are linear combinations of quadratures), namely
∆ ∆B ≥ |[Â,B]|/2(51)
with [Â,B] being a scalar, we consider the complementarity between two n-tuples of commuting observables.
Theorem 3. Let y = (ŷ 1 , · · ·ŷ n ) T be a vector of commuting quadratures, z = (ẑ 1 , · · ·ẑ n ) T be another vector of commuting quadratures, and let the components of these vectors be written each as a linear combination of the (x,p) quadratures of a N -modal system (N ≥ n). Let γ A ij = {ŷ i ,ŷ j } /2 − ŷ i ŷ j and γ B ij = {ẑ i ,ẑ j } /2 − ẑ i ẑ j be the (reduced) covariance matrices of theŷ i andẑ i quadratures. Then
det γ A 1 2 det γ B 1 2 ≥ | det K| 2 n (52)
where K ij = [ŷ i ,ẑ j ] denotes the commutator matrix.
Proof. Let us define the entropy powers of y and z as
N A = 1 2πe e 2 n h( y) , N B = 1 2πe e 2 n h( z) ,(53)
which allows us to rewrite Eq. (32) as an entropy-power uncertainty relation (see [9])
N A N B ≥ | det K| 2/n 4 .(54)
Since the maximum entropy for a fixed covariance matrix is reached by the Gaussian distribution, we have that N A ≤ (det γ A ) 1/n and N B ≤ (det γ B ) 1/n . Combining these inequalities with Eq. (54), we prove our theorem.
In the one-mode case, we obtain ∆ŷ 1 ∆ẑ 1 ≥ |[ŷ 1 ,ẑ 1 ]|/2 which is Robertson uncertainty relation applied to the two quadraturesŷ 1 andẑ 1 , as already mentioned. Thus, Theorem 3 extends this relation to two joint measurements of n modes and accounts for the correlations between the y i 's via the term det γ A (as well as between the z j 's via the term det γ B ). Note, however, that this covariance-based uncertainty relation is less strong than the entropic uncertainty relation since Theorem 3 follows from Theorem 1.
IV. CONCLUSIONS
We have derived an entropic uncertainty relation which applies to any two n-dimensional LCTs F A ( y) and F B ( z) or any two n-modal Gaussian projective measurements resulting in outcomes y and z. As implied by our Theorem 1, the sum of the entropy of the probability distributions for y and z is lower bounded by a quantity that depends on the determinant of the matrix of commutators [ŷ i ,ẑ j ], a quantity that is invariant under symplectic transformations. This is a generalization of the usual entropic uncertainty relation (1) due to Bialynicki-Birula and Mycielski in the case of any two n-dimensional observables that are not canonically conjugate but are connected by an arbitrary LCT.
Theorem 1 can also be viewed as a natural extension of the uncertainty relation (4) due to Huang [10]. As shown in Figure 1, the two considered measurements can be realized by applying a Gaussian unitary U A or U B before measuring thex quadratures. If we restrict ourselves to measuring thex quadrature of the first mode only, then the resulting quadrature is orB as defined in Eq. (3). Thus, our entropic uncertainty relation generalizes Huang's setup by including the measurement of any number of modes instead of the first one only. It naturally accounts for the correlations between the measured y i 's (as well as z j 's) via the use of joint entropies. Following the same scheme, we also recover the usual entropic uncertainty relation (1) by applying either the identity (U A = 1) or a tensor product of π/2 rotations on all mode (U B = R ⊗n π/2 ) before measuring all x quadratures. Our results still hold true (with some adaptations) when Shannon entropies are replaced by Rényi entropies, as proven in Theorem 2. They also imply a generalized version of Robertson uncertainty relation expressing the complementarity between two n-tuples of quadrature observables in terms of the determinant of a commutator matrix, see Theorem 3.
As a final note, it must be stressed that we have restricted ourselves to observables that are linear combinations of thex andp quadratures throughout this work, which implies that all commutators [ŷ i ,ẑ j ] are scalars, as well as det K. However, we believe that it should be possible to extend Theorem 1 to general vectors of commuting Hermitian operators A and B. Then, all commutators would be replaced by their mean values, in analogy with the usual Robertson relation. We therefore suggest the following conjecture: Conjecture 1. Let A = (A 1 , · · · A n ) be a vector of commuting observables, B = (B 1 , · · · B n ) be another vector of commuting observables and |ψ be the state of the system. The probability distributions of the jointly measured observables A i 's or B j 's in state |ψ satisfy the entropic uncertainty relation h( A) + h( B) ≥ ln ((πe) n | det ψ|K|ψ |)
where
K ij = [A i , B j ]
This would be a further generalization of the entropic uncertainty relation, also implying an extended Robertson relation involving a matrix of mean values of commutators instead of Eq. (52). Investigating this conjecture is an interesting topic of future work.
j,kpk (j = 1, · · · n). (31)
which concludes the proof of Eq. (32).
. W Heisenberg, Z. Phys. 43172W. Heisenberg, Z. Phys. 43, 172 (1927).
. E H Kennard, Z. Physik. 44326E. H. Kennard, Z. Physik 44, 326 (1927).
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. H P Robertson, Phys. Rev. 35667H. P. Robertson, Phys. Rev. 35, 667A (1930).
. I I Hirschman, Am. J. Math. 79152I. I. Hirschman, Am. J. Math. 79, 152 (1957).
. I Bialynicki-Birula, J Mycielski, Commun. Math. Phys. 44129I. Bialynicki-Birula and J. Mycielski, Commun. Math. Phys. 44, 129 (1975).
. W Beckner, Ann. Math. 102159W. Beckner, Ann. Math. 102, 159 (1975).
. P J Coles, M Berta, M Tomamichel, S Wehner, Rev. Mod. Phys. 8915002P. J. Coles, M. Berta, M. Tomamichel, and S. Wehner, Rev. Mod. Phys. 89, 15002 (2017).
. A Hertz, M G Jabbour, N J Cerf, J. Phys. A. 50385301A. Hertz, M. G. Jabbour, and N. J. Cerf, J. Phys. A 50, 385301 (2017).
. Y Huang, Phys. Rev. A. 8352124Y. Huang, Phys. Rev. A 83 052124 (2011).
Signal Process. X Guanlei, W Xiaotong, X Xiaotong, 892692X. Guanlei, W. Xiaotong and X. Xiaotong, Signal Pro- cess. 89 2692 (2009).
. A Bultheel, H Martinez-Sulbaran, Bull. Belg. Math. Soc. 13971A. Bultheel and H. Martinez-Sulbaran, Bull. Belg. Math. Soc. 13 971 (2006).
. S C Pei, J Ding, IEEE Trans. Sig. Proc. 494878S.C. Pei and J.J Ding, IEEE Trans. Sig. Proc. 49 (4), 878 (2001).
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The fractional Fourier tansform. H M Ozaktas, Z Zalevsky, M A Kutay, WileyChichesterH.M. Ozaktas, Z. Zalevsky and M.A. Kutay. The frac- tional Fourier tansform. Wiley, Chichester (2001).
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. C Weedbrook, S Pirandola, R Garcia-Patron, N J Cerf, T C Ralph, J H Shapiro, S Lloyd, Rev. Mod. Phys. 84621C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Rev. Mod. Phys. 84, 621 (2012).
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. I Bialynicki-Birula, Phys. Rev. A. 7452101I. Bialynicki-Birula, Phys. Rev. A 74 052101 (2006).
|
[] |
[
"Commissioning Run of the CRESST-II Dark Matter Search",
"Commissioning Run of the CRESST-II Dark Matter Search"
] |
[
"G Angloher \nMax-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany\n",
"M Bauer \nEberhard-Karls-Universität Tübingen\nD-72076TübingenGermany\n",
"I Bavykina \nMax-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany\n",
"A Bento \nMax-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany\n",
"A Brown \nDepartment of Physics\nUniversity of Oxford\nOX1 3RHOxfordUnited Kingdom\n",
"C Bucci \nMax-Planck-Institut für Physik\nDepartamento de Fisica\nINFN\nLaboratori Nazionali del Gran Sasso\nFöhringer Ring 6I-67010, D-80805Assergi, MünchenItaly, Germany\n\nUniversidade de Coimbra\nP3004 516CoimbraPortugal\n",
"C Ciemniak \nPhysik-Department E15\nTechnische Universität München\nD-85747GarchingGermany\n",
"C Coppi \nPhysik-Department E15\nTechnische Universität München\nD-85747GarchingGermany\n",
"G Deuter \nEberhard-Karls-Universität Tübingen\nD-72076TübingenGermany\n",
"F Von Feilitzsch \nPhysik-Department E15\nTechnische Universität München\nD-85747GarchingGermany\n",
"D Hauff \nMax-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany\n",
"S Henry \nDepartment of Physics\nUniversity of Oxford\nOX1 3RHOxfordUnited Kingdom\n",
"P Huff \nMax-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany\n",
"J Imber \nDepartment of Physics\nUniversity of Oxford\nOX1 3RHOxfordUnited Kingdom\n",
"S Ingleby \nDepartment of Physics\nUniversity of Oxford\nOX1 3RHOxfordUnited Kingdom\n",
"C Isaila \nPhysik-Department E15\nTechnische Universität München\nD-85747GarchingGermany\n",
"J Jochum \nEberhard-Karls-Universität Tübingen\nD-72076TübingenGermany\n",
"M Kiefer \nMax-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany\n",
"M Kimmerle \nEberhard-Karls-Universität Tübingen\nD-72076TübingenGermany\n",
"H Kraus \nDepartment of Physics\nUniversity of Oxford\nOX1 3RHOxfordUnited Kingdom\n",
"J.-C Lanfranchi \nPhysik-Department E15\nTechnische Universität München\nD-85747GarchingGermany\n",
"R F Lang \nMax-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany\n",
"B Majorovits \nDepartment of Physics\nUniversity of Oxford\nOX1 3RHOxfordUnited Kingdom\n",
"M Malek \nDepartment of Physics\nUniversity of Oxford\nOX1 3RHOxfordUnited Kingdom\n",
"R Mcgowan \nDepartment of Physics\nUniversity of Oxford\nOX1 3RHOxfordUnited Kingdom\n",
"V B Mikhailik \nDepartment of Physics\nUniversity of Oxford\nOX1 3RHOxfordUnited Kingdom\n",
"E Pantic \nMax-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany\n",
"F Petricca \nMax-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany\n",
"S Pfister \nPhysik-Department E15\nTechnische Universität München\nD-85747GarchingGermany\n",
"W Potzel \nPhysik-Department E15\nTechnische Universität München\nD-85747GarchingGermany\n",
"F Pröbst \nMax-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany\n",
"W Rau \nPhysik-Department E15\nTechnische Universität München\nD-85747GarchingGermany\n\nDepartment of Physics, Queen's University\nK7L 3N6OntarioCanada i Deceased\n",
"S Roth \nPhysik-Department E15\nTechnische Universität München\nD-85747GarchingGermany\n",
"K Rottler \nEberhard-Karls-Universität Tübingen\nD-72076TübingenGermany\n",
"C Sailer \nEberhard-Karls-Universität Tübingen\nD-72076TübingenGermany\n",
"K Schäffner \nMax-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany\n",
"J Schmaler \nMax-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany\n",
"S Scholl \nEberhard-Karls-Universität Tübingen\nD-72076TübingenGermany\n",
"W Seidel \nMax-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany\n",
"L Stodolsky \nMax-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany\n",
"A J B Tolhurst \nDepartment of Physics\nUniversity of Oxford\nOX1 3RHOxfordUnited Kingdom\n",
"I Usherov \nEberhard-Karls-Universität Tübingen\nD-72076TübingenGermany\n",
"W Westphal \nPhysik-Department E15\nTechnische Universität München\nD-85747GarchingGermany\n"
] |
[
"Max-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany",
"Eberhard-Karls-Universität Tübingen\nD-72076TübingenGermany",
"Max-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany",
"Max-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany",
"Department of Physics\nUniversity of Oxford\nOX1 3RHOxfordUnited Kingdom",
"Max-Planck-Institut für Physik\nDepartamento de Fisica\nINFN\nLaboratori Nazionali del Gran Sasso\nFöhringer Ring 6I-67010, D-80805Assergi, MünchenItaly, Germany",
"Universidade de Coimbra\nP3004 516CoimbraPortugal",
"Physik-Department E15\nTechnische Universität München\nD-85747GarchingGermany",
"Physik-Department E15\nTechnische Universität München\nD-85747GarchingGermany",
"Eberhard-Karls-Universität Tübingen\nD-72076TübingenGermany",
"Physik-Department E15\nTechnische Universität München\nD-85747GarchingGermany",
"Max-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany",
"Department of Physics\nUniversity of Oxford\nOX1 3RHOxfordUnited Kingdom",
"Max-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany",
"Department of Physics\nUniversity of Oxford\nOX1 3RHOxfordUnited Kingdom",
"Department of Physics\nUniversity of Oxford\nOX1 3RHOxfordUnited Kingdom",
"Physik-Department E15\nTechnische Universität München\nD-85747GarchingGermany",
"Eberhard-Karls-Universität Tübingen\nD-72076TübingenGermany",
"Max-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany",
"Eberhard-Karls-Universität Tübingen\nD-72076TübingenGermany",
"Department of Physics\nUniversity of Oxford\nOX1 3RHOxfordUnited Kingdom",
"Physik-Department E15\nTechnische Universität München\nD-85747GarchingGermany",
"Max-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany",
"Department of Physics\nUniversity of Oxford\nOX1 3RHOxfordUnited Kingdom",
"Department of Physics\nUniversity of Oxford\nOX1 3RHOxfordUnited Kingdom",
"Department of Physics\nUniversity of Oxford\nOX1 3RHOxfordUnited Kingdom",
"Department of Physics\nUniversity of Oxford\nOX1 3RHOxfordUnited Kingdom",
"Max-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany",
"Max-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany",
"Physik-Department E15\nTechnische Universität München\nD-85747GarchingGermany",
"Physik-Department E15\nTechnische Universität München\nD-85747GarchingGermany",
"Max-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany",
"Physik-Department E15\nTechnische Universität München\nD-85747GarchingGermany",
"Department of Physics, Queen's University\nK7L 3N6OntarioCanada i Deceased",
"Physik-Department E15\nTechnische Universität München\nD-85747GarchingGermany",
"Eberhard-Karls-Universität Tübingen\nD-72076TübingenGermany",
"Eberhard-Karls-Universität Tübingen\nD-72076TübingenGermany",
"Max-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany",
"Max-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany",
"Eberhard-Karls-Universität Tübingen\nD-72076TübingenGermany",
"Max-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany",
"Max-Planck-Institut für Physik\nFöhringer Ring 6D-80805MünchenGermany",
"Department of Physics\nUniversity of Oxford\nOX1 3RHOxfordUnited Kingdom",
"Eberhard-Karls-Universität Tübingen\nD-72076TübingenGermany",
"Physik-Department E15\nTechnische Universität München\nD-85747GarchingGermany"
] |
[] |
The CRESST cryogenic direct dark matter search at Gran Sasso, searching for WIMPs via nuclear recoil, has been upgraded to CRESST-II by several changes and improvements. The upgrade includes a new detector support structure capable of accommodating 33 modules, the associated multichannel readout with 66 SQUID channels, a neutron shield, a calibration source lift, and the installation of a muon veto. We present the results of a commissioning run carried out in 2007.The basic element of CRESST-II is a detector module consisting of a large (∼300 g) CaWO4 crystal and a very sensitive smaller (∼ 2 g) light detector to detect the scintillation light from the CaWO4. The large crystal gives an accurate total energy measurement. The light detector permits a determination of the light yield for an event, allowing an effective separation of nuclear recoils from electron-photon backgrounds. Furthermore, information from lightquenching factor studies allows the definition of a region of the energy-light yield plane which corresponds to tungsten recoils. A neutron test is reported which supports the principle of using the light yield to identify the recoiling nucleus.Data obtained with two detector modules for a total exposure of 48 kg-days are presented. Judging by the rate of events in the "all nuclear recoils" acceptance region the apparatus shows a factor ∼ten improvement with respect to previous results, which we attribute principally to the presence of the neutron shield. In the "tungsten recoils" acceptance region three events are found, corresponding to a rate of 0.063 per kg-day. Standard assumptions on the dark matter flux, coherent or spin independent interactions, then yield a limit for WIMP-nucleon scattering of 4.8 × 10 −7 pb, at M WIMP ∼ 50 GeV.
|
10.1016/j.astropartphys.2009.02.007
|
[
"https://arxiv.org/pdf/0809.1829v2.pdf"
] | 18,560,327 |
0809.1829
|
d25afffd28bf3a42a7fc71bc2777548f8a8e6eb1
|
Commissioning Run of the CRESST-II Dark Matter Search
16 February 2009 16 Feb 2009
G Angloher
Max-Planck-Institut für Physik
Föhringer Ring 6D-80805MünchenGermany
M Bauer
Eberhard-Karls-Universität Tübingen
D-72076TübingenGermany
I Bavykina
Max-Planck-Institut für Physik
Föhringer Ring 6D-80805MünchenGermany
A Bento
Max-Planck-Institut für Physik
Föhringer Ring 6D-80805MünchenGermany
A Brown
Department of Physics
University of Oxford
OX1 3RHOxfordUnited Kingdom
C Bucci
Max-Planck-Institut für Physik
Departamento de Fisica
INFN
Laboratori Nazionali del Gran Sasso
Föhringer Ring 6I-67010, D-80805Assergi, MünchenItaly, Germany
Universidade de Coimbra
P3004 516CoimbraPortugal
C Ciemniak
Physik-Department E15
Technische Universität München
D-85747GarchingGermany
C Coppi
Physik-Department E15
Technische Universität München
D-85747GarchingGermany
G Deuter
Eberhard-Karls-Universität Tübingen
D-72076TübingenGermany
F Von Feilitzsch
Physik-Department E15
Technische Universität München
D-85747GarchingGermany
D Hauff
Max-Planck-Institut für Physik
Föhringer Ring 6D-80805MünchenGermany
S Henry
Department of Physics
University of Oxford
OX1 3RHOxfordUnited Kingdom
P Huff
Max-Planck-Institut für Physik
Föhringer Ring 6D-80805MünchenGermany
J Imber
Department of Physics
University of Oxford
OX1 3RHOxfordUnited Kingdom
S Ingleby
Department of Physics
University of Oxford
OX1 3RHOxfordUnited Kingdom
C Isaila
Physik-Department E15
Technische Universität München
D-85747GarchingGermany
J Jochum
Eberhard-Karls-Universität Tübingen
D-72076TübingenGermany
M Kiefer
Max-Planck-Institut für Physik
Föhringer Ring 6D-80805MünchenGermany
M Kimmerle
Eberhard-Karls-Universität Tübingen
D-72076TübingenGermany
H Kraus
Department of Physics
University of Oxford
OX1 3RHOxfordUnited Kingdom
J.-C Lanfranchi
Physik-Department E15
Technische Universität München
D-85747GarchingGermany
R F Lang
Max-Planck-Institut für Physik
Föhringer Ring 6D-80805MünchenGermany
B Majorovits
Department of Physics
University of Oxford
OX1 3RHOxfordUnited Kingdom
M Malek
Department of Physics
University of Oxford
OX1 3RHOxfordUnited Kingdom
R Mcgowan
Department of Physics
University of Oxford
OX1 3RHOxfordUnited Kingdom
V B Mikhailik
Department of Physics
University of Oxford
OX1 3RHOxfordUnited Kingdom
E Pantic
Max-Planck-Institut für Physik
Föhringer Ring 6D-80805MünchenGermany
F Petricca
Max-Planck-Institut für Physik
Föhringer Ring 6D-80805MünchenGermany
S Pfister
Physik-Department E15
Technische Universität München
D-85747GarchingGermany
W Potzel
Physik-Department E15
Technische Universität München
D-85747GarchingGermany
F Pröbst
Max-Planck-Institut für Physik
Föhringer Ring 6D-80805MünchenGermany
W Rau
Physik-Department E15
Technische Universität München
D-85747GarchingGermany
Department of Physics, Queen's University
K7L 3N6OntarioCanada i Deceased
S Roth
Physik-Department E15
Technische Universität München
D-85747GarchingGermany
K Rottler
Eberhard-Karls-Universität Tübingen
D-72076TübingenGermany
C Sailer
Eberhard-Karls-Universität Tübingen
D-72076TübingenGermany
K Schäffner
Max-Planck-Institut für Physik
Föhringer Ring 6D-80805MünchenGermany
J Schmaler
Max-Planck-Institut für Physik
Föhringer Ring 6D-80805MünchenGermany
S Scholl
Eberhard-Karls-Universität Tübingen
D-72076TübingenGermany
W Seidel
Max-Planck-Institut für Physik
Föhringer Ring 6D-80805MünchenGermany
L Stodolsky
Max-Planck-Institut für Physik
Föhringer Ring 6D-80805MünchenGermany
A J B Tolhurst
Department of Physics
University of Oxford
OX1 3RHOxfordUnited Kingdom
I Usherov
Eberhard-Karls-Universität Tübingen
D-72076TübingenGermany
W Westphal
Physik-Department E15
Technische Universität München
D-85747GarchingGermany
Commissioning Run of the CRESST-II Dark Matter Search
16 February 2009 16 Feb 2009Preprint submitted to ElsevierDark MatterWIMPLow-temperature detectorsCaWO 4
The CRESST cryogenic direct dark matter search at Gran Sasso, searching for WIMPs via nuclear recoil, has been upgraded to CRESST-II by several changes and improvements. The upgrade includes a new detector support structure capable of accommodating 33 modules, the associated multichannel readout with 66 SQUID channels, a neutron shield, a calibration source lift, and the installation of a muon veto. We present the results of a commissioning run carried out in 2007.The basic element of CRESST-II is a detector module consisting of a large (∼300 g) CaWO4 crystal and a very sensitive smaller (∼ 2 g) light detector to detect the scintillation light from the CaWO4. The large crystal gives an accurate total energy measurement. The light detector permits a determination of the light yield for an event, allowing an effective separation of nuclear recoils from electron-photon backgrounds. Furthermore, information from lightquenching factor studies allows the definition of a region of the energy-light yield plane which corresponds to tungsten recoils. A neutron test is reported which supports the principle of using the light yield to identify the recoiling nucleus.Data obtained with two detector modules for a total exposure of 48 kg-days are presented. Judging by the rate of events in the "all nuclear recoils" acceptance region the apparatus shows a factor ∼ten improvement with respect to previous results, which we attribute principally to the presence of the neutron shield. In the "tungsten recoils" acceptance region three events are found, corresponding to a rate of 0.063 per kg-day. Standard assumptions on the dark matter flux, coherent or spin independent interactions, then yield a limit for WIMP-nucleon scattering of 4.8 × 10 −7 pb, at M WIMP ∼ 50 GeV.
Introduction
Evidence continues to grow that the majority of the mass of our galaxy and in the universe is made up of non-baryonic dark matter [1]. Precision measurements of the cosmic microwave background have given accurate figures for the density of matter and energy in the universe as a whole, suggesting that about a fourth of the energy density of the universe is in the form of dark matter [2]. Measurements of the rotation curves of other spiral galaxies, indicate large amounts of dark matter, which would suggest it also dominates our own galaxy. Gravitational lensing of light by galactic clusters indicates large amounts of dark matter, and dark matter is necessary for a reasonable description of the formation of galaxies.
Although some of these observations might also be explained by alternative theories such as modified Newtonian dynamics (MOND) [3], recent studies of the Bullet Cluster appear to favour the dark matter hypothesis [4]. However the need for direct detection of dark matter evidently remains strong.
A plausible origin for dark matter comes from particle physics in the form of WIMPs (weakly interacting massive particles). These would be stable massive particles with an interaction cross section typical for weak interaction processes. A relic density sufficient to make a significant contribution to the energy density of the universe arises naturally in this way. Supersymmetry provides a well motivated candidate in the form of the neutralino and this has been extensively studied theoretically.
According to the WIMP hypothesis a galaxy has a large gravitationally bound halo of such particles, making up most of the mass. The local mass density may be estimated from halo models and is found to be on the order of 0.3 GeVcm −3 . The range of the coherent or spin-independent WIMP-nucleon scattering cross-sections predicted by minimal supersymmetric models extends from below 10 −10 to above 10 −7 pb [5,6]; this makes the direct detection of such particles a difficult but in principle achievable task.
The CRESST project is a dark matter search aiming to detect the nuclear scattering of WIMPs by the use of cryogenic detectors. Since only small recoil energies (∼ keV's) are anticipated in WIMP-nucleus scattering, cryogenic detectors with their high sensitivity are well suited to the problem.
CRESST-II is an upgrade of the original CRESST appartus where the detectors are arranged in mod-ules, each consisting of two detectors, a large detector comprising the target mass and a similar but much smaller light detector measuring the light yield. While the large detector gives an accurate total energy measurement, the light yield measurement permits an effective rejection of events which are not nuclear recoils. This arises from the property of electron-photon events, constituting the dominant background, to give a much higher light output than nuclear recoils.
The large detector consists of a CaWO 4 crystal of ∼ 300 g with a tungsten superconducting-to-normal thermometer deposited on the surface. The energy of an event is measured in this detector. The energy deposited in the crystal is quickly converted into a gas of phonons which are then absorbed in the superconducting thermometer. The separate light detector, based on the same principle, measures the simultaneously emitted scintillation light. A comparison of these two signals then allows an effective background discrimination.
Results from an earlier prototype run with two modules were presented in Ref. [7], where taking the better of the two modules and assuming coherent WIMP scattering on the tungsten led to the limit 1.6 × 10 −6 pb. The improvements discussed here are aimed at ultimately increasing this sensitivity by up to two orders of magnitude. Such an increase would result, for example, with background free data and a threshold of 12 keV from 10 kg for 100 days giving 1000 kg-days. Or if the threshold could be reduced to 5 keV, with 300 kg days.
During 2007 an extended commissioning run of the new setup was carried out. Although this was primarily for optimization purposes, the results also represent an interesting further improvement. We report on them and the modifications for CRESST-II in the present paper.
Experimental Setup
The detector volume of the CRESST installation at Gran Sasso is surrounded by passive shielding consisting of 14 cm thick low background copper and 20 cm of lead. This inner shielding is entirely enclosed within a gas-tight radon box which is continuously flushed with N 2 gas. This part of the setup is as used in previous phases of the experiment and is described in Ref. [8]. The changes made in the upgrade to CRESST-II include the installation of a neutron shield and a muon veto. Figure 1 shows the He-4 He dilution refrigerator, low background detector environment, and the shielding as upgraded for CRESST-II. The detector carousel (CA) is connected to the mixing chamber of the cryostat (CR) by a long copper cold finger (CF) in order to reduce background originating from the dilution refrigerator. The gas-tight radon box (RB) encloses the low background copper (CU) and low background lead shielding (PB). It is covered by a plastic scintillator muon-veto (MV) and a 45 cm thick polyethylene neutron moderator (PE). Additional granular PE is placed between the baffles in the upper part of the cryostat to close the line of sight for neutrons coming from above. setup after the upgrade.
In the detector volume itself, a new detector support structure ("carousel") capable of holding the 33 modules has been mounted and a new 66-channel SQUID readout with associated wiring and data acquisition electronics has been installed. The various changes will be discussed in this section.
Detector Support Structure ("Carousel")
The new support structure for holding the detector modules is made from copper suitable for mil-liKelvin (mK) operation and electropolished to reduce surface contamination. The structure, which is depicted in Fig. 2, can accommodate a total of 33 detector modules, thus allowing an increase of the target mass by an order of magnitude (to ∼ 10 kg) relative to CRESST-I. Individual handling and dismounting of the modules is possible. The structure rests on custom-made CuSn6 springs [9] to reduce interference from vibrations. CuSn6 was chosen since it exhibits both intrinsic radiopurity and elasticity at mK temperatures. The thermal coupling of the carousel to the cold finger is intentionally set to be weak, with a relaxation time of about half an hour. This filters out high-frequency temperature variations of the cryostat, giving a high degree of temperature stability at the detectors.
Sixty-six channel SQUID readout
With the full complement of 33 modules, each with two superconducting thermometers, there will be 66 readout channels in CRESST-II. In the basic CRESST design these channels are read out with SQUID's. Figure 3 shows one such readout circuit, simplified for clarity. The actual implementation is discussed in detail in Ref. [10]. A constant bias current is shared between two branches, one containing the tungsten thermometer, the other the input coil for a SQUID in series with two reference resistors. The reference resistors assure proper branching of the current; two are used here to make the circuit more balanced and to minimize the level of crosstalk. Any change in the resistance of the thermometer changes the current through the SQUID input coil. The SQUID system then transforms this signal to a voltage pulse, which is processed by conventional follow-on electronics. The SQUIDs are DC SQUIDs using AC modulated flux-locked loops. Full details of the system are given in [10].
To handle the large number of detectors, a multichannel system was installed, consisting of 66 SQUID units. These are located at the bottom of the main helium bath, at 4 K. Woven twisted pair cables are used to connect the SQUIDs to the detectors in the mK environment, while another set of cables leads to connector boxes at room temperature, which then link to the control electronics and the data aquisition system.
In addition to the readout-line pair for the SQUID, each thermometer has a heater-line pair for temperature control and calibration pulses. Thus there are two pairs of lines for each thermometer and so a total of eight lines per module. With the many lines into the mK environment close attention must be paid in the design and fabrication of flanges and feed-throughs [10][11].
Neutron Shield
A significant new element in the setup is a neutron shield, consisting of a 45 cm thickness of polyethylene (PE) plates. The shield is mounted on rails to allow access to the inner parts of the experiment. The polyethylene moderates ambient neutrons to thermal energies where they are then captured on the hydrogen. The neutron capture range in CH 2 materials is 4.3 cm for MeV energies [12], so the shield should eliminate essentially all incoming neutrons.
Calibration Source Lift
During a run, CRESST detectors are regularly calibrated with a radioactive source supplying gamma quanta of known energy. In view of the many and closely spaced detectors, in CRESST-II it is necessary to be able to move and position the radioactive source during calibration runs.
Therefore the apparatus has been equipped with a calibration source lift. The lift consists of a plastic tube through which the source is moved by compressed air, the position being determined by strings attached from both ends of the tube. It spirals around the cold box inside the shielding and allows calibration of individual detector modules without opening the shielding. It is designed so that a rate of ∼ 0.5 Hz can be generated at any detector module with a 0.7 MBq 57 Co source. The system was used successfully several times during the commissioning run.
Muon Veto
With the aim of reducing the effects of muons entering the apparatus a muon veto has been installed. It consists of 20 plastic scintillator panels of area about 1 − 1.2 m 2 surrounding the radon box and the Cu/Pb shielding. By using three different types of panels, a total solid angle coverage of 98.7 % is achieved. A circular opening of 0.27 m 2 on top was unavoidable due to the entry for the cryostat (see Fig. 1). In a dedicated setup [13], the characteristics of each panel type were measured and calibrated. The photomultiplier signals are sent out of the Faraday cage optically to avoid interference with the cryogenic detectors. The veto triggered at ∼ 5 Hz during the run.
Detectors
Module housing
An important feature of the CRESST detector modules is their ability to reject events which are not nuclear recoils via a simultaneous measurement of a heat/phonon signal in the large detector and a scintillation signal [14] in the light detector.
The crystal and the light detector are enclosed together in a reflective and scintillating housing ( Fig. 4) of 3M foil, as described in Ref. [7]. This serves two purposes. Firstly, it increases the amount of light collected by the light detector. Secondly, its scintillation permits rejection of a potentially dangerous background from surface radioactivity on the crystal [15]. An important such possibility is polonium, a daughter of the radon decay chain with an alpha decay. If the alpha is lost and the recoiling lead nucleus stays in the crystal, this can mimic a tungsten recoil, i.e. a WIMP interaction. However, it was found that the scintillation of the alpha particle on the reflecting foil gives a light signal which discriminates events of this type and allows for their separation.
Thermometers (TES's)
The basic sensor in CRESST is a superconducting thin film thermometer, with the temperature sta-bilized at a suitable operating point in the middle of the superconducting-to-normal transition of the film . A small variation of the film resistance in such an arrangment gives a very sensitive temperature measurement; this system is often called a Transition Edge Sensor (TES) .
In the work described here tungsten thin films are used. Tungsten in its crystalline α phase becomes superconducting at T c ≈ 15 mK. The films are equipped with heaters with the dual function of stabilizing the detectors' operating points and the injection of heat pulses for calibration. The thermometer layouts for the CaWO 4 target and the light detector are similar. However, the light detector has additional Al/W phonon collectors for better energy collection. This is as described in ref. [7], but here the light detectors use a small portion of the thermal-link gold structure as the heater. This light detector layout is shown in Fig. 5.
For temperature regulation heater pulses of a fixed amplitude, so-called control pulses are injected through the heater lines every three seconds. Online pulse height evaluation of the resulting signal is used for regulation of the heater current which keeps the thermometer's temperature at the operating point. This is somewhat different from older work where temperature stabilization was on the baseline, and is found to provide excellent long term stability of operation. In addition to these control pulses for temperature stabilization, calibration pulses delivering a known energy are injected every 30 seconds. These are used in determining the energy to be identified with a pulse, as will be explained in section 4.2. The calibration and stabilization procedures are the same for both target and light detector thermometers.
Target Detector
The CRESST-II detectors use large CaWO 4 single crystals as the target or "absorber" mass. CaWO 4 has been selected for its high light output and the presence of the heavy nucleus tungsten, which gives a large factor ∼ A 2 in the presumed coherent (spin-independent) scattering of the WIMP. The crystals are cylindrical, of 4 cm height and 4 cm diameter, and about 300 g in weight.
Ten such crystals were installed for the commissioning run; two are used here for the dark matter analysis.
Light Detector
The scintillation light is measured by a separate cryogenic detector. This is made from a sapphire wafer of 40 mm diameter and 0.4 mm thickness, with an epitaxially grown silicon layer on one side for photon absorption. The weight is 2.3 g. This siliconon-sapphire (SOS) material exploits the excellent phonon transport properties of sapphire. The resulting detector with the sensor as in Fig. 5 is sensitive down to about 20 eV, or about seven optical photons.
Further progress in using the light yield to identify which nucleus is recoiling, as discussed below, depends largely on the improvement of the resolution of the light detector.
Data Taking and Analysis
Only very basic quality cuts are performed on the data. Pulses are rejected if they occur in one of the rare periods when the temperature of the cryostat is not stable, as inferred from the pulse height of the control pulses. Pileup is rejected by simple cuts on the baseline of the pulse. Due to the low trigger rate this does not introduce a significant dead time into the analysis.
Signal Processing and Pulse Fitting
For recording thermometer pulses, the output voltage of the SQUID electronics is split into two branches. In one branch the pulse is shaped and AC-coupled to a trigger unit, while in the other branch the signal is passed through an 8-pole antialiasing low-pass filter and then DC-coupled to a 16-bit transient digitizer. The time bin of the transient digitizer was chosen to be 40 µs, providing about 30 samples for the rising part of the pulse in the phonon channel. The record length of 4096 time bins includes a pre-trigger region of 1024 samples to record the baseline before the event, while the remaining 3072 post-trigger samples contain the pulse itself. The phonon and light channels of each module are read out together, whenever one or both trigger.
The pulses are fitted offline using a template fit, with a template constructed from pulses collected during a gamma calibration run. There is an individual template for each thermometer. Both pulses from a module are fitted simultaneously. Free parameters in the fit are the two baselines, the two pulse height amplitudes, and a common time shift relative to the trigger.
Energy Determination
Finally, the assignment of an energy to a pulse is made on the basis of the resulting amplitude or pulse height parameter. Since the pulses from the 57 Co calilbration with ∼ 122 keV and ∼ 136 keV photons establish a relation between pulse height and energy for each detector, a first simple determination of the pulse height-energy relation for a detector would be provided by a linear extrapolation from 122 keV to lower energies, and in early work this gave good results. Direct calibration with lower energy gammas is not possible since the calibration source is outside the coldbox and the photon must penetrate about 12 mm of copper.
A finer and more detailed determination of the pulse height-energy relation for a given detector is possible, however, by using the heater calibration pulses mentioned in section 3.2. For each thermometer, heater pulses delivering a known energy via ohmic heating are injected every 30 seconds. These are applied over the whole energy range of interest, so the thermometer response can be mapped down to low energies and the assumption of a simple lin- Fig. 6. Energy spectrum of the 300g detector "Verena" in the commissioning run (48 days). The prominent feature at 46.5 keV is due to a gamma from 210 P b decay in the crystal with its associated 17 keV β spectrum. The Cu fluorescence line at 8.05 keV is also clearly visible, and has resolution ∼ 300 eV. The close agreement of these energies with known values confirms the excellent accuracy of the heater pulse calibration method. ear extrapolation can be avoided. Both target and light detectors are calibrated by this procedure.
The accuracy and resolution of this method is illustrated in Fig. 6, which shows the energy spectrum of the target detector "Verena" in the commissioning run. The energy resolution may be gauged from the prominent feature starting around 46 keV and the line near 8 keV, which can be identified as a copper fluorescence line. The large feature is due to the gamma transition of a 210 Pb impurity in the crystal, with a broadened right shoulder from an accompanying 17 keV β spectrum. The steep left edge appears precisely at 46.5 keV as expected. The peak for the copper fluorescence line is at 8.05 keV, also as expected, and the energy resolution is ∼ 300 eV. These two features confirm the excellent accuracy of the calibration method with heater pulses. Further details of the energy spectrum can also be understood and will be discussed in a forthcoming publication.
Energy-Light Plots and Quenching Factor
Events from a given module are plotted as points in the energy-light yield plane, as in the following Figs. 7 and 8. An energy is assigned to an event as described in the previous section. The light yield is given by the ratio of the energy or pulse height in the light detector to that in the target detector, normalized such that it is 1 for a 122 keV calibration photon absorbed in the large detector. While the light yield for a 122 keV photon is therefore one by definition, the fact that the bands are roughly flat on the plots, and the average light yield is always near one, is not a matter of definition and reflects the accuracy of the thermometer calibrations and the approximate linearity 1 of the light production process [17].
It will be seen that when a source inducing nuclear recoils is present, as with neutrons in Fig. 7, there are two distinct bands, one of lower light yield from nuclear recoils and one of higher light yield from electron-photon events. The factor of reduction for the light yield of a nuclear recoil relative to that of the electron-photon event of the same energy is called (in our usage) the quenching factor. With the electron-photon band centered on 1, the center of the nuclear recoil band on Fig. 7 is simply at the inverse of the quenching factor.
Since the quenching factor varies widely, up to ∼ 40 for heavy elements, the level of light output can be used not only to distinguish nuclear recoils from electron-photon events, but also to distinguish tungsten recoils from those of the lighter nuclei. It is hoped that it may be ultimately possible to determine which particular nucleus, in a compound material like CaWO 4 , is recoiling. This would be very useful in verifying a positive dark matter signal, since a simple behavior for the signal with respect to changing the target nucleus is expected.
Since an understanding of the quenching factor is thus of great interest, a number of investigations have been carried out within the CRESST collaboration [15], [18], [19], [20], [21]. The results are summarized and discussed in [17].
Neutron Test
To directly test the response of the system to nuclear recoils a neutron test was carried out during the commissioning run. The test can be used to check the principle of identifying the recoiling nucleus via the light yield and to also determine the dispersion in the light output, which is important in applying the method.
Above ∼ 10 keV recoil energy, calculations show that the spectrum for neutron scattering on CaWO 4 Fig. 7. Low energy event distribution in the (Verena/SOS21) module in the presence of a neutron source during the commissioning run. Below the red curve 90 % of the oxygen recoils are expected, as calculated from the known quenching factor for oxygen and the energy resolution of the light detector. The good agreement with the prediction supports the use of this method to identify tungsten recoils in the dark matter analysis. The events above 47 keV between the electron and nuclear recoil bands can be understood as inelastic neutron scatterings where the tungsten nucleus is excited to a gamma-emitting level.
is totally dominated by oxygen recoils [22]. Therefore one anticipates that the events of the nuclear recoil band in this test should lie roughly on or below the line corresponding to the oxygen quenching factor ∼ 9 known from the previous quenching factor studies.
A neutron source was placed inside the polyethylene shield and Fig. 7 shows the results for one of the two modules (Verena/SOS21). Using the quenching factor of 9 and the measured energy-dependent resolution of the light detector, 90 % of the neutroninduced recoils should be below the red line.
The neutron test data appears to agree well with this prediction, and so supports our arguments in the dark matter analysis that an acceptance region for nuclear or tungsten recoils based on the same principle can be defined.
Dark Matter Limits
For the dark matter analysis we use data taken between March 27th and July 23rd 2007 with the two detector modules (Zora/SOS23) and (Verena/SOS21). The data are shown in Fig. 8. The cumulative exposure was 47.9 kg − days. For the tungsten this corresponds to an exposure of 30.6 kg − days.
We perform an analysis on the assumption of coherent or spin-independent scattering for the WIMP. This process should strongly favor tungsten recoils due to the ∼ A 2 factor in the WIMP-nucleus cross section. We define an acceptance region on the plots based on a) the quenching factor for the light yield and b) the maximum energy expected for tungsten recoil. A similar region for "all nuclear recoils" can be defined using the quenching factors for Ca and O. The quenching factor boundaries are shown on the plots of Fig. 8 by the curves. Below the upper curve 90 % of all nuclear recoils are expected, and below the lower curve 90 % of the tungsten recoils are expected. The energy boundaries are shown by Fig. 9. Coherent or spin-independent scattering cross section exclusion limit derived from the data of Fig. 8 using the maximum energy gap method. For comparison the limits from other experiments [27], [28], [29], [30], [31] and the range predicted by some supersymmetry models [32] are also shown.
the vertical lines. The upper limit at 40 keV is set by form-factor [23] effects, which effectively limit the energy transfer to the tungsten nucleus. The lower limit is set at 10 keV, where "leakage" from the electron/photon events becomes evident and so recoil discrimination becomes inefficient. Three candidate events (heavy black dots) are observed in Fig. 8 for the "tungsten recoils" acceptance region. The individual events are at 16.89 keV (Verena/SOS21) and at 18.03 keV and 33.09 keV (Zora/SOS23). The corresponding rate is 0.063 per kg-day. From this rate we can derive a limit on the coherent spin-independent WIMP-nucleon scattering cross section. Using standard assumptions on the dark matter halo [24,25] (WIMP mass density of 0.3 GeVcm −3 ), an upper limit for the coherent or spin-independent WIMP-nucleon scattering crosssection may be obtained using the maximum energy gap method [26]. This limit is plotted as the red curve in Fig. 9. The minimum of the curve, for a WIMP mass of ∼50 GeV, is at 4.8 × 10 −7 pb.
Backgrounds
To judge the improvement with respect to our previous setup of Ref. [7] we take the "all nuclear recoils" region and reset the low energy boundary of the acceptance region to 12 keV as it was there. In this acceptance region the earlier work had 16 events, for a rate of 0.87 per kg-day. As noted in Ref. [7], this rate was compatible with the estimates of [33] for neutron background. The present commissioning run has 4 events in the same acceptance region, for a rate of 0.083 per kg-day, so we may say there is a factor ∼ 10 improvement. We attribute this to the presence of the neutron shield.
On the other hand, according to the arguments in section 5 the "tungsten recoils" should be insensitive to neutron background. Indeed, the three events here from 48 kg-days are quite compatible with the zero events from 10 kg-days in Ref. [7]; the present rate of 0.063 per kg-day predicts only 0.6 events for 10 kg-days. The fact that the neutron shield appears to have a great effect on "all nuclear recoils" but little effect on "tungsten recoils" supports our argument that the light yield successfully distinguishes tungsten and other nuclear recoils.
The question arises, however, as to the nature of the few observed tungsten or nuclear recoil candidates. One possibility is evidently remaining neutrons. However with many absorption lengths for the shield very few should penetrate and indeed simulations [34] would give a rate of only ∼ 10 −5 per kgday, much less than the few events. A possible point here is that during the run a weak spot in the neutron shielding above the muon veto was identified and patched only after data taking was completed.
Another possible source for neutrons are some non-operational modules which were present during the run. These can act as non-vetoable neutron sources for the operational modules. However an estimate for the background events from an inactive detector module is well below 1.4 × 10 −5 per kg-day.
Apart from neutrons, another possibility arises from incomplete coverage of the inner surfaces of the detector module with scintillator, which could lead to unvetoed nuclear recoils from surface α-decays as discussed in section 3.1. In further work it is intended to paint some possibly uncovered areas with scintillating material.
Concerning muons, estimates [34] of muoninduced neutrons in the setup result in only 2.8 × 10 −3 per kg-day. No muon signals were found in coincidence with nuclear recoil events.
There thus appears to be no conclusive explanation for the few candidate events from conventional radioactive or particle sources. We hope to clarify some of these points in further work.
Conclusions
CRESST-II successfully completed its commissioning run in 2007. New elements of the apparatus were successfully installed and operated. A neutron test demonstrated the ability to detect nuclear recoils with a light yield consistent with that found from quenching factor studies, supporting the principle of identifying the recoil nucleus via the light yield.
Data were taken with two detector modules for a total of 48 kg-days. Three candidate events of uncertain origin are present in the acceptance region for tungsten recoils, yielding a rate of 0.063 per kg-day. A factor ∼ 10 improved performance is found with respect to previous work for the "all nuclear recoils" acceptance region.
A limit on coherent WIMP-nucleon scattering, is obtained, which at its lowest value, for M WIMP ≈ 50GeV, is 4.8 × 10 −7 pb.
Acknowledgements
Fig. 1 .
13
Fig. 2 .
2Detector carousel of ultrapure copper, electropolished to reduce surface contamination. The structure can accommodate 33 detector modules (i.e. 10 kg of target mass) which can be mounted or dismounted individually. The carousel is mounted at the lower end of the cold finger. The whole structure is cooled to ∼ 10 mK.
Fig. 3 .
3CRESST-II readout circuit. The tungsten film thermometer is in parallel with two reference resistors and the input coil of a DC SQUID. The circuit is biased by a constant current.
Fig. 4 .
4Schematic arrangement of a CRESST detector module. The module consists of two independent detectors: one with the CaWO 4 target crystal, providing a total energy measurement, and one with a silicon-on-sapphire (SOS) wafer for measuring the scintillation light from the target crystal. Both detectors are enclosed in a reflective, scintillating housing.
Fig. 5 .
5Layout and connection scheme of a light detector used in the commissioning run. Aluminum/tungsten phonon collectors surround the tungsten thermometer or TES. A small portion of the gold thermal link is used as the heater. Connections shown are the aluminum bond wires for the SQUID and heater circuits, and a gold bond wire for thermal coupling, leading to an electrically insulated pad on the detector holder.
Fig. 8 .
8Low-energy event distribution measured with two 300 g CaWO 4 detector modules during the commissioning run. The vertical axis represents the light yield (see text), and the horizontal axis the total energy, as measured by the phonon channel. Below the dashed black curve 90 % of all nuclear recoils, and below the solid red curve 90 % of the tungsten recoils are expected. The heavy black dots show the events in the "tungsten recoils" acceptance region. The intense regions of the electron recoil bands correspond to the lines ofFig. 6.
Upon closer examination it appears that the centroid of the electron-photon band falls slightly at lower energies. Such a decrease for γ rays in CaWO 4 has been seen in Ref.[16].
This article is dedicated to the memory of our friend and colleague Wolfgang Westphal, who significantly contributed to the success of CRESST.This work was partially supported by funds of the DFG (SFB 375, Transregio 27:"Neutrinos and Beyond"), the Munich Cluster of Excellence ("Origin and Structure of the Universe"), the EU networks for Cryogenic Detectors (ERB-FMRXCT980167) and for Applied Cryogenic Detectors (HPRN-CT2002-00322), and the Maier-Leibnitz-Laboratorium (Garching). We also thank Hans Ulrich Friebel for valuable technical support. Support was provided by the Science and Technology Facilities Council.
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[
"A Nearly Optimal Algorithm for the Geodesic Voronoi Diagram of Points in a Simple Polygon",
"A Nearly Optimal Algorithm for the Geodesic Voronoi Diagram of Points in a Simple Polygon"
] |
[
"Chih-Hung Liu [email protected] \nDepartment of Computer Science\nETH Zürich\nZürichSwitzerland\n"
] |
[
"Department of Computer Science\nETH Zürich\nZürichSwitzerland"
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The geodesic Voronoi diagram of m point sites inside a simple polygon of n vertices is a subdivision of the polygon into m cells, one to each site, such that all points in a cell share the same nearest site under the geodesic distance. The best known lower bound for the construction time is Ω(n + m log m), and a matching upper bound is a long-standing open question. The state-of-theart construction algorithms achieve O (n + m) log(n + m) and O(n + m log m log 2 n) time, which are optimal for m = Ω(n) and m = O( n log 3 n ), respectively. In this paper, we give a construction algorithm with O n + m(log m + log 2 n) time, and it is nearly optimal in the sense that if a single Voronoi vertex can be computed in O(log n) time, then the construction time will become the optimal O(n + m log m). In other words, we reduce the problem of constructing the diagram in the optimal time to the problem of computing a single Voronoi vertex in O(log n) time.ACM Subject ClassificationTheory of computation → Randomness, geometry and discrete structures tree and each pair of adjacent triangles have a parent-child relation; seeFigure 1(b). For each triangle, its diagonal shared with its parent partitions the polygon into two sub-polygons: the "lower" one contains it, and the "upper" one contains its parent. Then, the triangles are swept by the post-order and pre-order traversals of the rooted tree to respectively build, inside each triangle, the two diagrams with respect to sites in its lower and upper subpolygons. Finally, the two diagrams inside each triangle are merged into the final diagram.Very recently, Oh and Ahn [10] generalized the notion of plane sweep to a simple polygon. To supplant the scan line, one point is fixed on the polygon boundary, and another point is moved from the fixed point along the polygon boundary counterclockwise, so that the shortest path between the two points will sweep the whole polygon. Moreover, Guibas and Hershberger's data structure for shortest path queries[5,7]is extended to compute a Voronoi vertex among three sites or between two sites in O(log 2 n) time. This technique enables handling an event in O(log m log 2 n) time, leading to a total time of O(n + m log m log 2 n).Papadopoulou and Lee's method[11]has two issues inducing the n log(n+m) time-factor. First, while sweeping the polygon, an intermediate diagram is represented by a "wavefront" in which a "wavelet" is associated with a "subcell." Although this representation enables computing the common vertex among three subcells in O(1) time, since it takes Ω log(n + m) time to update such a wavefront, the Ω(n) vertices lead to the n log(n + m) factor. Second, when a wavefront enters a triangle from one diagonal and leaves from the other two diagonals, it will split into two. Since there are Ω(n) triangles, there are Ω(n) split events, and since a split event takes Ω log(n + m) time, the n log(n + m) factor arises again.Chih-Hung Liu 58:3
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A Nearly Optimal Algorithm for the Geodesic Voronoi Diagram of Points in a Simple Polygon
9 Mar 2018
Chih-Hung Liu [email protected]
Department of Computer Science
ETH Zürich
ZürichSwitzerland
A Nearly Optimal Algorithm for the Geodesic Voronoi Diagram of Points in a Simple Polygon
9 Mar 2018and phrases Simple polygonsVoronoi diagramsgeodesic distance
The geodesic Voronoi diagram of m point sites inside a simple polygon of n vertices is a subdivision of the polygon into m cells, one to each site, such that all points in a cell share the same nearest site under the geodesic distance. The best known lower bound for the construction time is Ω(n + m log m), and a matching upper bound is a long-standing open question. The state-of-theart construction algorithms achieve O (n + m) log(n + m) and O(n + m log m log 2 n) time, which are optimal for m = Ω(n) and m = O( n log 3 n ), respectively. In this paper, we give a construction algorithm with O n + m(log m + log 2 n) time, and it is nearly optimal in the sense that if a single Voronoi vertex can be computed in O(log n) time, then the construction time will become the optimal O(n + m log m). In other words, we reduce the problem of constructing the diagram in the optimal time to the problem of computing a single Voronoi vertex in O(log n) time.ACM Subject ClassificationTheory of computation → Randomness, geometry and discrete structures tree and each pair of adjacent triangles have a parent-child relation; seeFigure 1(b). For each triangle, its diagonal shared with its parent partitions the polygon into two sub-polygons: the "lower" one contains it, and the "upper" one contains its parent. Then, the triangles are swept by the post-order and pre-order traversals of the rooted tree to respectively build, inside each triangle, the two diagrams with respect to sites in its lower and upper subpolygons. Finally, the two diagrams inside each triangle are merged into the final diagram.Very recently, Oh and Ahn [10] generalized the notion of plane sweep to a simple polygon. To supplant the scan line, one point is fixed on the polygon boundary, and another point is moved from the fixed point along the polygon boundary counterclockwise, so that the shortest path between the two points will sweep the whole polygon. Moreover, Guibas and Hershberger's data structure for shortest path queries[5,7]is extended to compute a Voronoi vertex among three sites or between two sites in O(log 2 n) time. This technique enables handling an event in O(log m log 2 n) time, leading to a total time of O(n + m log m log 2 n).Papadopoulou and Lee's method[11]has two issues inducing the n log(n+m) time-factor. First, while sweeping the polygon, an intermediate diagram is represented by a "wavefront" in which a "wavelet" is associated with a "subcell." Although this representation enables computing the common vertex among three subcells in O(1) time, since it takes Ω log(n + m) time to update such a wavefront, the Ω(n) vertices lead to the n log(n + m) factor. Second, when a wavefront enters a triangle from one diagonal and leaves from the other two diagonals, it will split into two. Since there are Ω(n) triangles, there are Ω(n) split events, and since a split event takes Ω log(n + m) time, the n log(n + m) factor arises again.Chih-Hung Liu 58:3
Introduction
The geodesic Voronoi diagram of m point sites inside a simple polygon of n vertices is a subdivision of the polygon into m cells, one to each site, such that all points in a cell share the same nearest site where the distance between two points is the length of the shortest path between them inside the polygon. The common boundary between two cells is a Voronoi edge, and the endpoints of a Voronoi edge are Voronoi vertices. A cell can be augmented into subcells such that all points in a subcell share the same anchor, where the anchor of a point in the cell is the vertex of the shortest path from the point to the associated site that immediately precedes the point. An anchor is either a point site or a reflex polygon vertex. Figure 1(a) illustrates an augmented diagram. The size of the (augmented) diagram is Θ(n + m) [1]. The best known construction time is O (n + m) log(n + m) [11] and O(n + m log m log 2 n) [10]. They are optimal for m = Ω(n) and for m = O( n log 3 n ), respectively, since the best known lower bound is Ω(n + m log m). The existence of a matching upper bound is a long-standing open question by Mitchell [9].
Aronov [1] first proved fundamental properties: a bisector between two sites is a simple curve consisting of Θ(n) straight and hyperbolic arcs and ending on the polygon boundary; the diagram has Θ(n+m) vertices, Θ(m) of which are Voronoi vertices. Then, he developed a divide-and-conquer algorithm that recursively partitions the polygon into two roughly equalsize sub-polygons. Since each recursion level takes O (n + m) log(n + m) time to extend the diagrams between every pair of sub-polygons, the total time is O (n + m) log(n + m) log n .
Papadopoulou and Lee [11] combined the divide-and-conquer and plane-sweep paradigms to improve the construction time to O (n+m) log(n+m) . First, the polygon is triangulated and one resultant triangle is selected as the root such that the dual graph is a rooted binary
Our contribution
We devise a construction algorithm with O n + m(log m + log 2 n) time, which is slightly faster than Oh and Ahn's method [10] and is optimal for m = O( n log 2 n ). More importantly, our algorithm is, to some extent, nearly optimal since the log 2 n factor solely comes from computing a single Voronoi vertex. If the computation time can be improved to O(log n), the total construction time will become O n + m(log m + log n) , which equals the optimal O(n+m log m) since m log n = O(n) for m = O( n log n ) and log n = O(log m) for m = Ω( n log n ). In other words, we reduce the problem of constructing the diagram in the optimal time by Mitchell [9] to the problem of computing a single Voronoi vertex in O(log n) time.
At a high level, our algorithm is a new implementation of Papadopoulou and Lee's concept [11] using a different data structure of a wavefront, symbolic maintenance of incomplete Voronoi edges, tailor-made wavefront operations, and appropriate amortized time analysis.
First, in our wavefront, each wavelet is directly associated with a cell rather than a subcell. This representation makes use of Oh and Ahn's [10] O(log 2 n)-time technique of computing a Voronoi vertex. Each wavelet also stores the anchors of incomplete subcells in its associated cell in order to enable locating a point in a subcell along the wavefront.
Second, if each change of a wavefront needs to be updated immediately, a priority queue for events would be necessary, and since the diagram has Θ(m+n) vertices, an (m+n) log(m+ n) time-factor would be inevitable. To overcome this issue, we maintain incomplete Voronoi edges symbolically, and update them only when necessary. For example, during a binary search along a wavefront, each incomplete Voronoi edge of a tested wavelet will be updated.
Third, to avoid Ω(n) split operations, we design two tailor-made operations. If a wavefront will separate into two but one part will not be involved in the follow-up "sweep", then, instead of using a binary search, we "divide" the wavefront in a seemingly brute-force way in which we traverse the wavefront from the uninvolved part until the "division" point, remove all visited subcells, and build another wavefront from those subcells. If a wavefront propa- gates into a sub-polygon that contains no point site, then we adopt a two-phase process to build the diagram inside the sub-polygon instead of splitting a wavefront many times. Finally, when deleting or inserting a subcell (anchor), its position in a wavelet is known. Since re-balancing a red-black tree (RB-tree) after an insertion or a deletion takes amortized O(1) time [8,12,13], by augmenting each tree node with pointers to its predecessor and successor, an insertion or a deletion with a known position takes amortized O(1) time.
s 1 v root d (d) (d) (a) (b)
This paper is organized as follows. Section 2 formulates the geodesic Voronoi diagram, defines a rooted partition tree, and introduces Papadopoulou and Lee's two subdivisions [11]; Section 3 summarizes our algorithm; Section 4 designs the data structure of a wavefront; Section 5 presents wavefront operations; Section 6 implements the algorithm with those operations.
Preliminary
Geodesic Voronoi diagrams
Let P be a simple polygon of n vertices, let ∂P denote the boundary of P , and let S be a set of m point sites inside P . For any two points p, q in P , the geodesic distance between them, denoted by d(p, q), is the length of the shortest path between them that fully lies in P , the anchor of q with respect to p is the last vertex on the shortest path from p to q before q, and the shortest path map (SPM) from p in P is a subdivision of P such that all points in a region share the same anchor with respect to p. Each edge in the SPM from p is a line segment from a reflex polygon vertex v of P to ∂P along the direction from the anchor of v (with respect to p) to v, and this line segment is called the SPM edge of v (from p).
The geodesic Voronoi diagram of S in P , denoted by Vor P (S), partitions P into m cells, one to each site, such that all points in a cell share the same nearest site in S under the geodesic distance. The cell of a site s can be augmented by partitioning the cell with the SPM from s into subcells such that all points in a subcell share the same anchor with respect to s. The augmented version of Vor P (S) is denoted by Vor * P (S). With a slight abuse of terminology, a cell in Vor * P (S) indicates a cell in Vor P (S) together with its subcells in Vor * P (S). Then, each cell is associated with a site, and each subcell is associated with an achor. As shown in Figure 1(a), v is the anchor of the shaded subcell (in s 1 's cell), and the last vertex on the shortest path from s 1 to any point x in the shaded subcell before x is v.
A Voronoi edge is the common boundary between two adjacent cells, and the endpoints of a Voronoi edge are called Voronoi vertices. A Voronoi edge is a part of the geodesic bisector between the two associated sites, and consists of straight and hyperbolic arcs. Endpoints of these arcs except Voronoi vertices are called breakpoints, and a breakpoint is incident to an SPM edge in the SPM from one of the two associated sites, indicating a change of the corresponding anchor. There are Θ(m) Voronoi vertices and Θ(n) breakpoints [1].
In our algorithm, each anchor u refers to either a reflex polygon vertex of P or a point site in S; we store its associated site s, its geodesic distance from s, and its anchor with respect to s. The weighted distance from u to a point x is d(s, u) + |ux|.
Throughout the paper, we make a general position assumption that no polygon vertex is equidistant from two sites in S and no point inside P is equidistant from four sites in S. The former avoids nontrivial overlapping among cells [1], and the latter ensures that the degree of each Voronoi vertex with respect to Voronoi edges is either 1 (on ∂P ) or 3 (among three cells).
The boundary of a cell except Voronoi edges are polygonal chains on ∂P . For convenience, these polygonal chains are referred to as polygonal edges of the cell, the incidence of an SPM edge onto a polygonal edge is also a breakpoint, and the polygonal edges including their polygon vertices and breakpoints also count for the size of the cell.
A rooted partition tree
Following Papadopoulou and Lee [11], a rooted partition tree T for P and S is built as follows: First, P is triangulated using Chazelle's algorithm [3] in O(n) time, and all sites in S are located in the resulting triangles by Edelsbrunner et al's approach [4] in O(n+m log n) time. The dual graph of the triangulation is a tree in which each node corresponds to a triangle and an edge connects two nodes if and only if their corresponding triangles share a diagonal. Then, an arbitrary triangle whose two diagonals are polygon sides, i.e., a node with degree 1, is selected as the root, so that there is a parent-child relation between each pair of adjacent triangles. Figure 1(b) illustrates a rooted partition tree T .
For a diagonal d, let (d) and (d) be the two triangles adjacent to d such that (d) is the parent of (d), and call d the root diagonal of (d); also see Figure 1(b). d partitions P into two sub-polygons: P (d) contains (d) and P (d) contains (d). P (d) and P (d) are said to be "below" and "above" d, respectively. Assume that d ⊆ P (d); let S(d) = S ∩ P (d) and S (d) = S \ S(d), which indicate the two respective subsets of sites below and above d.
In this paper, we adopt the following convention: for each triangle , let d = v 1 v 2 be its root diagonal, let be its parent triangle, let d 1 , d 2 be the other two diagonals of , and let d 3 , d 4 be the other two diagonals of . Assume that d 4 is the diagonal between and its parent, and that
Subdivisions
Papadopoulou and Lee [11] introduced two subdivisions, SD and SD , of P , which can be merged into Vor * P (S). For each triangle with a root diagonal d, SD and SD respectively contain Vor * P S(d) ∩ and Vor * P S (d) ∩ ; SD also contains Vor * P (S) ∩ . Since S(d) and S(d 4 ) (resp. S (d) and S (d 4 )) may differ, a border forms along d in SD (resp. SD ) to "remedy" the conflicting proximity information. Figure 2(b)-(c) illustrate SD and SD . The incidence of a Voronoi edge or an SPM edge in Vor * P (S) onto a border in SD or SD is called a border vertex, and the border vertices partition a border into border edges. Both SD and SD have O(n + m) border vertices [11], O(m) of which are induced by Voronoi edges. Hereafter, a diagram vertex means a Voronoi vertex, a breakpoint, a border vertex, or a polygon vertex.
d d 1 d 2 d 3 d 4 v 1 v 2 v 1,2 v 3,4 (a) (b) (c)
Overview of the algorithm
We compute Vor * P (S) in the following three steps: 1. Build the rooted partition tree T for P and S in O(n + m log n) time. (Section 2.2) 2. Construct SD and SD in O n + m(log m + log 2 n) time by sweeping the polygon using the post-order and pre-order traversals of T , respectively. (Section 6.1 and Section 6.2) 3. Merge SD and SD into Vor * P (S) in O(n + m) time using Papadopoulou and Lee's method [11,Section 7].
By the above-mentioned running times, we conclude the total running time as follows.
Wavefront structure
A wavefront represents the "incomplete" boundary of "incomplete" Voronoi cells during the execution of our algorithm, and wavefronts will "sweep" the simple polygon P triangle by triangle to construct SD and SD . To avoid excessive updates, each "incomplete" Voronoi edge, which is a part of a Voronoi edge and will be completed during the sweep, is maintained symbolically, preventing an extra log n time-factor. During the sweep, candidates for Voronoi vertices in SD and SD called potential vertices will be generated in the unswept part of P .
Formal definition and data structure
Let η be a diagonal or a pair of diagonals sharing a common polygon vertex, and let S be a subset of S lying on the same side of η. A wavefront W η (S ) represents the sequence of Voronoi cells in Vor * P (S ) appearing along η, and each appearance of a cell induces a wavelet in W η (S ). The unswept area of W η (S ) is the part of P on the opposite side of η from S . Since Vor * P (S ) in the unswept area has not yet been constructed, Voronoi and polygonal edges incident to η are called incomplete. Each wavelet is bounded by two incomplete Voronoi or polygonal edges along η, and its incomplete boundary comprises its two incomplete edges and the portion of η between them. When the context is clear, a wavelet may indicate its associated cell. W η (S ) is stored in an RB-tree in which each node refers to one wavelet and the ordering of nodes follows their appearances along η. The RB-tree is augmented such that each node has pointers to its predecessor and successor, and the root has pointers to the first and last nodes, enabling efficiently traversing wavelets along η and accessing the two ending wavelets.
The subcells of a wavelet are the subcells in its associated cell incident to its incomplete boundary. The list of their anchors is also maintained by an augmented RB-tree in which their ordering follows their appearances along the incomplete boundary. Due to the visibility of a subcell, each subcell appears exactly once along the incomplete boundary. Since the rebalancing after an insertion or a deletion takes amortized O(1) time [12,13,8], inserting or deleting an anchor at a known position in an RB-tree, i.e., without searching, takes amortized O(1) time.
Incomplete Voronoi and polygonal edges
When a wavefront moves into its unswept area, incomplete Voronoi edges will extend, generating new breakpoints. If each breakpoint needs to be created immediately, all candidates for breakpoints should be maintained in a priority queue, leading to an Ω(n log n) running time due to Ω(n) breakpoints. To avoid these excessive updates, we maintain each incomplete Voronoi edge symbolically, and update it only when necessary. For example, when searching along the wavefront or merging two wavefronts, the incomplete Voronoi edges of each involved wavelet (cell) will be updated until the diagonal or the pair of diagonals.
Since a breakpoint indicates the change of a corresponding anchor, for a Voronoi edge, if the anchors of its incident subcells on "each side" are stored in a sequence, the Voronoi edge can be computed in time proportional to the number of breakpoints by scanning the two sequences [10, Section 4]. Following this concept, for each incomplete Voronoi edge, we store its fixed Voronoi vertex (in the swept area), its last created breakpoint, and its last used anchors on its two sides, so that we can update an incomplete Voronoi edge by scanning the two lists of anchors from the last used anchors. When creating a breakpoint, we also build a corresponding SPM edge, and then remove the corresponding anchor from the anchor list.
Each polygonal edge is also maintained symbolically in a similar way; in particular, it will also be updated when a polygon vertex is inserted as an anchor. Meanwhile, the SPM edges incident to a polygonal edge will also be created using its corresponding anchor list.
Potential vertices
We process incomplete Voronoi edges to generate candidates for Voronoi vertices called potential vertices. For each incomplete Voronoi edge, since its two associated sites lie in the swept area, one endpoint of the corresponding bisector lies in the unswept area and is a degree-1 potential vertex. For each two adjacent Voronoi edges along the wavefront, their respective bisectors may intersect in the unswept area, and the intersection is a degree-3 potential vertex. By Lemma 1, a potential vertex can be computed in O(log 2 n) time.
Potential vertices are stored in their located triangles; each diagonal of a triangle is equipped with a priority queue to store the potential vertices in the triangle associated with sites on the opposite side of the diagonal, where the key is the distance to the diagonal.
Wavefront operations
We summarize the eight wavefront operations, where K inv , A vis and I new are the numbers of involved (i.e., processed and newly created) potential vertices, visited anchors, and created diagram vertices, respectively:
Initiate: Compute Vor (S ) and initialize W (d,d2) (S ) in O |S |(log m + log 2 n) timeInsert: Insert S into W d1 S(d 2 ) to be W d1 S(d 2 ) ∪ S in O |S |(log m + log 2 n) time. Propagate: Propagate W d S (d) into P (d), provided that P (d) ∩ S = S(d) = ∅, to build SD ∩ P (d) = Vor * P (S) ∩ P (d) in O K inv (log m + log 2 n) + |SD ∩ P (d)| time.
Merge and Join operations differ in that the former also merges the two corresponding Voronoi diagrams in the underlying triangle, and the latter does not.
Readers could directly read the algorithm in Section 6 without knowing the detailed implementation for wavefront operations. For the operation times, we have three remarks.
Remark. During Extend and Propagate operations and at the end of the other operations, potential vertices will be generated according to new incomplete Voronoi edges. It will be clear in Section 6 that the total number of potential vertices in the construction of SD and SD is O(m), so a priority queue takes O(log m) time for an insertion or an extraction.
Remark. As stated in Section 4.2, we maintain incomplete Voronoi/polygonal edges symbolically. For the sake of simplicity, we charge the time to update an incomplete edge to the created breakpoints, and assign the corresponding costs to their located triangles.
Remark. Since a wavelet (resp. anchor) to remove from a wavefront (resp. wavelet) must be inserted beforehand, we charge its removal cost at its insertion time. For a wavelet, the cost is O(log m), and for an anchor, since the position is always known, the cost is amortized O (1). Similarly, we charge the cost to delete a diagram vertex at its creation time.
Initiate operation
An Initiate operation processes a triangle to compute Vor (S ) and initiate W (d,d2) (S ). Since no polygon vertex lies in a triangle, each resulting Voronoi cell has only one subcell. For the sake of simplicity, we assume that no diagonal of is a polygon side; the other cases are similar.
An Initiate operation consists of two simple steps: First, Vor (S ) is computed by constructing the Voronoi diagram of S without considering P and trimming the diagram by . Second, by tracing Vor (S ) along the pair (d, d 2 ) of diagonals, W (d,d2) (S ) is built as the aforementioned data structure (Section 4.1). During the tracing, if v 1,2 (resp. v 1 ) is a reflex polygon vertex, then it will be inserted into the anchor list of its located wavelet (cells) in W (d,d2) (S ). Note that if v 2 is a reflex polygon vertex, then v 2 will be inserted as an anchor after W (d,d2) (S ) has been divided or split at v 2 .
The operation time is O |S | · (log m + log 2 n) . The first step takes O(|S | log |S |) time to construct the Voronoi diagram using a standard divide-and-conquer or a plane-sweep algorithm [
Extend operation
An Extend operation extends a wavefront Wd(Q) from one diagonald of a triangle˜ to the opposite pair of diagonals (d 1 ,d 2 ) to construct W (d1,d2) (Q) and Vor * P (Q) ∩˜ , wherẽ d =ṽ 1ṽ2 ,d 1 =ṽ 1ṽ1,2 , andd 2 =ṽ 2ṽ1,2 , and Q lies on the opposite side ofd from˜ .
This operation is equivalent to sweeping the triangle with a scan line parallel tod and processing each hit potential vertex. The next hit potential vertex is provided from the priority queue associated withd (defined in Section 4.3), and will be processed in three phases: First, its validity is verified: for a degree-1 potential vertex, its incomplete Voronoi edge should be alive in the wavefront, and for a degree-3 one, its two incomplete Voronoi edges should be still adjacent in the wavefront. Second, the one or two incomplete Voronoi edges are updated up to the potential vertex. Since a degree-1 potential vertex is incident to a polygonal edge, the polygonal edge is also updated.
Finally, when a potential vertex becomes a Voronoi vertex, a wavelet will be removed from the wavefront. For a degree-1 potential vertex, since the wavelet lies at one end of the wavefront, a polygonal edge is added to its adjacent wavelet; for a degree-3 potential vertex, since the removal makes two wavelets adjacent, a new incomplete Voronoi edge is created, and one degree-1 and two degree-3 potential vertices are computed accordingly.
After the extension, ifṽ 1,2 is a reflex polygon vertex, the wavefront is further processed by three cases. If neitherd 1 nord 2 is a polygon side,ṽ 1,2 will later be inserted as an anchor while dividing or splitting W (d1,d2) (Q) atṽ 1,2 . If bothd 1 andd 2 are polygon sides, all the subcells will be completed along (d 1 ,d 2 ) and the wavefront will disappear. If onlyd 1 (resp.d 2 ) is a polygon side, all the subcells alongd 1 (resp.d 2 ) excluding the one containing v 1,2 will be completed, andṽ 1,2 will be inserted into the corresponding wavelet as the last or first anchor.
Merge operation
A Merge operation merges W η (Q) and W η (Q ) into W η (Q ∪ Q ) together with Vor * P (Q) ∩ and Vor * P (Q )∩ into Vor * P (Q∪Q )∩ where is the underlying triangle. In our algorithm, either Q = S , Q = S(d 1 ), and η = (d, d 2 ) or Q = S ∪ S(d 1 ), Q = S(d 2 ), and η = d; in both cases, S ⊆ Q. The border will form on ∂ \ η, i.e., d 1 for the former case and (d 1 , d 2 ) for the latter case. Although a wavefront only stores incomplete Voronoi cells, its associated diagram can still be accessed through the Voronoi edges of the stored cells. After the merge, new incomplete Voronoi edges will form, and their potential vertices will be created.
The Merge operation consists of two phases: (1) merge Vor * P (Q) ∩ and Vor * P (Q ) ∩ into Vor * P (Q ∪ Q ) ∩ and (2) merge W η (Q) and W η (Q ) into W η (Q ∪ Q ). The first phase is to construct so-called merge curves. A merge curve is a connected component consisting of border edges along ∂ \ η and Voronoi edges in Vor * P (Q ∪ Q ) ∩ associated with one site in Q and one site in Q ; the ordering of mergve curves is the ordering of their appearances along η. This phase is almost identical to the merge process by Papadopoulou and Lee [11, Section 5], but since our data structure for a wavefront is different from theirs, a binary search along a wavefront to find a starting endpoint for a merge curve requires a different implementation. We first state this difference here, and will describe the details of tracing a merge curve in Section 5.3.1.
Assume η to be oriented from v 1 to v 1,2 for η = (d, d 2 ) and from v 1 to v 2 for η = d. By [11, Lemma 4-6], a merge curve called initial starts from v 1,2 for the latter case (η = d), but all other merge curves have both endpoints on η, and those endpoints are associated with one site in S . Let Q be the set of sites in S that have a wavelet in W η (Q). If η = (d, d 2 ), a site in Q can have two wavelets in W η (Q), and with an abuse of terminology, such a site is imagined to have two copies, each to one wavelet. Since Q ⊆ Q, finding a starting endpoint for each merge curve except the initial one is to test sites in Q following the ordering of their wavelets in W η (Q) along η. After finding a starting endpoint, the corresponding merge curve will be traced; when the tracing reaches η again, a stopping endpoint forms, and the first site in Q lying after the site inducing the stopping endpoint will be tested.
Let x be the next starting endpoint, which is unknown, and let s be the next site in Q to test. A two-level binary search on W η (Q ) determines if s induces x, and if so, further determines the site t ∈ Q that induces x with s as well as the corresponding anchor.
The first-level binary search executes on the RB-tree for the wavelets in W η (Q ), and each step determines for a site q ∈ Q if its cell lies before or after t's cell along η or if q = t. Let y 1 and y 2 denote the two intersection points between η and the two incomplete edges of s (in W η (Q)), where y 1 lies before y 2 along η, and let z 1 and z 2 be the two points defined similarly for q (in W η (Q )). Since s lies in , the distance between s and any point in η can be computed in O(1) time. The two incomplete edges of q will be updated until z 1 and z 2 , so that the distance from q to z 1 (resp. to z 2 ) can be computed from the corresponding anchor. For example, if u is the anchor of the subcell that contains z 1 , d(z 1 , q) = |z 1 u| + d(u, q).
The determination considers four cases. (Assume s and t induce the "starting" endpoint.) z 2 lies before y 1 (resp. z 1 lies after y 2 ): t's cell lies after (resp. before) q's cell. z 1 lies before y 1 and z 2 lies between y 1 and y 2 (resp. z 2 lies after y 2 and z 1 lies between y 1 and y 2 ): if z 2 is closer to q than to s (resp. z 1 is closer to q than to s), then t's cell lies after (resp. before) q's cell; otherwise, t is q. Both y 1 and y 2 lie between z 1 and z 2 : t is q. Both z 1 and z 2 lie between y 1 and y 2 : let x s be the projection point of s onto η. If x s lies before z 1 , then t's cell lies before q's cell. If x s lies after z 2 : if z 2 is closer to q than to s, t's cell lies after q's cell; if both z 1 and z 2 are closer to s than to q, t's cell lies before q's cell; otherwise, t = q. If x s lies between z 1 and z 2 : if z 1 is closer to s than to q, t's cell lies before q's cell; otherwise, t = q. If the first-level search does not find t, then s does not induce the next starting endpoint x.
The second-level binary search executes on the RB-tree for t's anchor list to either determine the next starting endpoint x and t's corresponding anchor or report that s does not induce x. Let u be the current anchor of t to test, and let x s be the projection point of s onto η. u's "interval" on η can be decided by checking u's two neighboring anchors. If u's interval lies after x s , x lies before u's interval; otherwise, if both endpoints of u's interval are closer to (resp. farther from) t than to (resp. from) s, x lies after (resp. before) u's interval, and if one endpoint is closer to t than to s but the other is not, then x lies in u's interval and can be computed in O(1) time since d(x, s) = |xu| + d(u, t). If the second-level binary search does not find such an interval, s does not induce the next starting endpoint x.
The second phase (i.e., merging W η (Q) and W η (Q ) into W η (Q ∪ Q )) splits W η (Q) and W η (Q ) at the endpoints of merge curves, and concatenates active parts at these endpoints where a part is called active if it contributes to W η (Q ∪ Q ). In fact, the active parts along η alternately come from W η (Q) and W η (Q ). At each merging endpoint, the two cells become adjacent, generating a new incomplete Voronoi edge. Potential vertices of these incomplete Voronoi edges will be computed and inserted into the corresponding priority queues. For each ending polygon vertex of η, if it is reflex but has not yet been an anchor of W η (Q ∪ Q ), it will be inserted into its located wavelet as the first or the last anchor.
Tracing merge Curves in Merge Operation
We make some further definitions. The bisector B(Q, Q ) between Q and Q is the collection of points with the same minimum geodesic distance from both Q and Q , namely B(Q, Q ) = {x ∈ P | min q∈Q d(x, q) = min q ∈Q d(x, q )}. In our algorithm, each anchor u refers to either a reflex polygon vertex of P or a point site in S; we store its associated site s(u), its geodesic distance w(u) from s(u), and its anchor a(u) with respect to s(u). The weighted distance d w (x, u) from a point x to u is |xu| + w(u), and the weighted bisector between two anchors u 1 and u 2 is {x ∈ P | d w (x, u 1 ) = d w (x, u 2 )}.
The process to trace a merge curve from the starting endpoint is identical to that presented by Papadopolou and Lee. Note that when a wavelet (incomplete cell) is visited, its incomplete edges will be updated to η, enabling the tracing between (incomplete) subcells. The merge curve begins with the weighted bisector between the two anchors that induce the starting endpoint. When the merge curve changes the underlying subcell in one of the two diagrams, it continues with the weighted bisector between the new anchor and the other original anchor. When the merge curve hits η , the tracing continues along η in the direction along which the next point is closer to the current site in Q than to the current site in Q ; until reaching an equidistant point, it turns into the interior of again following the corresponding weighted bisector. The merge curve may visit η several times. Finally, it reaches η at the stopping endpoint. Border vertices form on η when the merge curve enters or leaves η and when the merge curve changes the underlying subcell in one side of η.
Since tracing a merge curve takes time proportional to the number of deleted and created vertices but the time to delete vertices has been charged at their creation, tracing all the merge curves takes O(I new ) time in total, where I new is the number of newly created diagram vertices.
Join operation
A Join operation joins W d S(d 3 ) ∪ S and W d S (d 4 ) into W d S (d) with building the border on d but without merging Vor * P S(d 3 ) ∪ S ∩ and Vor * P S (d 4 ) ∩ into Vor * P S (d) ∩ .
In the construction of SD (Section 6.2), our algorithm will actually join W d1 S(d 2 ) ∪ S and W d1 (S (d)) into W d1 (S (d 1 )), and join W d2 S(d 1 ) ∪ S and W d2 (S (d)) into W d2 (S (d 2 )). To interpret a Join operation, for the former case, one would replace d, d 3
) ∪ S than to S (d 4 ), namely {x ∈ d | min s∈S(d3)∪S d(x, s) ≤ min s ∈S (d4) d(x, s )}.
At a high level, a Join operation first builds the border on d, and then joins the two wavefronts according to the border. This operation is conceptually identical to the process in [11,Section 6] except that our data structure of a wavefront is different.
With a slight abuse of terminology, let b (d) denote the collection of connected components of border edges on d in SD . 1 These components in b (d) are called joint segments and are ordered from v 2 to v 1 . By [11,, b (d) satisfies the following conditions:
At most one joint segment starts at v 2 , and such a joint segment is called initial. Except the initial one, both endpoints of each joint segment are associated with a site in S . The first endpoint is called starting, and the second endpoint is called stopping. The second condition results from the fact that the Voronoi cells associated with S(d 3 ) and S(d 4 ) do not "cross" each other along d, and this desirable condition enables locating a starting endpoint of a non-initial joint segment by searching W d S (d 4 ) with sites in S .
Assume that d is oriented from v 2 to v 1 . Joint segments will be constructed one by one from v 2 to v 1 . For each joint segment (except the initial one), its starting endpoint is first located on d by searching W d S (d 4 ) and then it is traced from its starting endpoint along d until reaching its stopping endpoint. We remark that the initial joint segment will be directly traced from v 2 .
Let S |d be the set of sites in S that have a wavelet in W d S(d 3 )∪S . By the second condition, finding the starting endpoint of a joint segment is to "locate" sites of S |d in W d S (d 4 ) since each endpoint of a joint segment (except the initial one) is associated with a site in S |d . Therefore, sites in S |d will be tested following the ordering of their wavelets
in W d S(d 3 ) ∪ S
from v 2 to v 1 ; after tracing a joint segment, the test will continue on the next site in S |d , which lies after the site in S |d associated the last traced stopping endpoint. Note that each site in S |d has exactly one wavelet in
W d S(d 3 ) ∪ S .
Let x be the next starting endpoint, which is unknown, and let s be the site in S |d to test. A two-level binary search on W d S (d 4 ) , which is identical to the one in Section 5.3, can determine if s induces x, and if so, determine the site t ∈ S (d 4 ) that induces x with s together with the corresponding anchor. The total time to find starting points is trivially O |S |(log n + log m) .
The process to trace a joint segment from the starting endpoint is simpler than tracing a merge curve in Section 5. After constructing joint segments, we split each of W d S(d 3 ) ∪ S and W d S (d 4 ) at the endpoints of these joint segments, and concatenate the active parts at these endpoints into W d S (d) , where "active" means having a wavelet in W d S (d) . For each ending polygon vertex of d, i.e., v 1 or v 2 , if it is reflex but has not been an anchor of W d S (d) , it will be inserted into its located wavelet as the first or the last anchor.
The
Split operation
A Split operation splits a wavefront associated with a pair of diagonals at the common polygon vertex using a binary search. First, the operation locates the wavelet that contains the common polygon vertex and the corresponding anchor. Second, the operation splits the wavelet, i.e., the corresponding list of anchors, at the located anchor, and duplicates the located anchor since it appears in both resulting wavelets. Third, the operation splits the wavefront between the two resulting wavelets. Finally, if the common polygon vertex is reflex, the operation inserts it into both of the duplicate wavelets as an anchor.
The operation time is O(log n + log m). Since a wavefront has O(m) wavelets and a wavelet has O(n) anchors, both locating the common polygon vertex and spliting the wavelet and wavefront take O(log n + log m) time. Although inserting the common polygon vertex takes amortized O(1) time, since the worst-case time to insert an anchor is O(log n), the time to insert the common polygon vertex is dominated by the time-factor O(log n + log m). Since there is no new site, there is no new incomplete Voronoi edge and no new potential vertex.
Divide operation
A Divide operation divides a wavefront associated with a pair of diagonals at the common polygon vertex by traversing one diagonal instead of using a binary search. Although a Divide operation seems a brute-force way compared to a Split operation, since a Split operation takes Ω(log n + log m) time and there are Ω(n) events to separate a wavefront, if only Split operations are adopted, the total construction time would be Ω n(log n + log m) .
First, the wavefront is traversed from the end of the selected diagonal subcell by subcell, i.e., anchor by anchor, until reaching the common vertex. Then, the wavefront is separated at the common polygon vertex by removing all the visited anchors except the last one, duplicating the last one, and building a new wavefront for these "removed" anchors and the duplicate anchor from scratch. Finally, if the common polygon vertex is reflex, it is inserted into its located wavelets in both resultant wavefronts as an anchor without a binary search (since it is the first or last anchor of its located wavelets).
The total operation time is amortized O(A vis + 1), where A vis is the number of visited anchors. Since each wavelet (resp. anchor) records its two neighboring wavelets (resp. anchors) in the augmented RB-tree, the time to locate the common polygon vertex is O(A vis ). Recall that a cell must have one subcell, and the time to remove a wavelet or an anchor has been charged when it was inserted. Finally, building the new wavefront from scratch takes amortized O(A vis ) time, and inserting the common vertex takes amortized O(1) time.
Insert operation
For a triangle , an Insert operation inserts S into W d1 S(d 2 ) to form W d1 S(d 2 ) ∪ S . An Insert operation executes in the construction of SD, but its outcome will be used to construct SD . Let T denote the intermediate site set during the Insert operation, so that T = S(d 2 ) at the beginning.
For each site s ∈ S , let x s be its vertical projection point onto d 1 , and conduct a binary search on the wavefront W d1 (T ) to locate the anchor whose subcell contains x s . During the binary search, the two incomplete Voronoi/polygonal edges of each visited wavelet will be updated up to d 1 . Recall that the weighted distance between a point y and an anchor u associated with a site t is |yu| + d(u, t).
If x s is closer to s than to the located anchor under the weighted distance, traverse anchors in W d1 (T ) from x s along each direction of d 1 until either touching a polygon vertex of d 1 or reaching a point equidistant from s and the current visited anchor under the weighted distance. If all points in the "interval" of a visited anchor on d 1 are fully closer to s than to the anchor, remove the anchor from its associated wavelet, and if all the anchors of a wavelet have been removed, remove the wavelet from W d1 (T ). After the traversal, insert s into W d1 (T ), namely create and insert its wavelet into W d1 (T ). Since S(d 2 ) and S lie on the same side of d 1 and all sites in S(d 2 ) lie outside , each cell of a site in S(d 2 ) will not be separated by a cell of a site in S "along d 1 ", supporting the correctness of the above process.
After processing all the sites in S , check the two polygon vertices, v 1 and v 1,2 , of d 1 , and if v 1 (resp. v 1,2 ) should belong to a wavelet associated a site in S , insert v 1 (resp. v 1,2 ) into the wavelet as the first or last anchor. For each inserted wavelet, generate its incomplete Voronoi or polygonal edges, and compute the corresponding potential vertices.
The
Propagate operation
A Propagate operation propagates a wavefront W d S (d) into P (d), i.e., from the upside to the downside of d,
to build SD ∩ P (d) provided that S ∩ P (d) = S(d) = ∅. Since S(d) = ∅, then SD ∩ P (d) is exactly Vor * P (S) ∩ P (d).
To some extent, a Propagate operation is a generalized version of an Extend operation underlying a sub-polygon instead of a triangle.
The operation consists of two phases. The first phase constructs Vor P (S) ∩ P (d), and the second phase refines each cell into subcells to obtain Vor * P (S) ∩ P (d). The first phase "sweeps" the triangles in P (d) by a preorder traversal of the subtree of T rooted at (d). Similar to the Extend operation, this sweep processes potential vertices inside each triangle to construct Voronoi edges and to update the wavefront accordingly. However, this sweep will not process the polygon vertices of P (d), so that the anchor lists will not be updated, preventing constructing a Voronoi edge using the two corresponding anchor lists. Fortunately, Oh and Ahn [10] gave another technique that obtains in O(log n) time the two anchor lists of a Voronoi edge provided that the two Voronoi vertices are given, so a Voronoi edge can still be built in time proportional to log n plus the number of its breakpoints.
Therefore, the first phase takes O K d (log m+log 2 n)+|Vor P (S)∩P (d)| time, where K d is the number of involved potential vertices. Note that for the Voronoi edges that intersect d, their breakpoints outside P (d) will be counted in the respective triangles in P (d).
The second phase triangulates each cell in Vor P (S) ∩ P (d) and constructs the SPM from the associated site in each triangulated cell. Since Chazelle's algorithm [3] takes linear time to triangulate a simple polygon and Guibas et al's algorithm [6] takes linear time to build the SPM in a triangulated polygon, the second phase takes O(|SD ∩ P (d)|) time.
Remark. Although all the sites lie outside P (d), the information stored in W d S (d) allows us not to conduct Guibas et al's algorithm from scratch. For example, for each anchor u, its anchor a(u) is also stored, and the SPM edge of u is the line segment from u to the polygon boundary along the direction from a(u) to u.
58:15
To sum up, a Propagate operation takes O K d (log m + log 2 n) + |SD ∩ P (d)| time.
6
Subdivision construction
Construction of SD
To construct SD, we process each triangle by the postorder traversal of the rooted partition tree T and build SD ∩ . We first assume that no diagonal of is a polygon side, and we will discuss the other cases later. Let d be the root diagonal of and adopt the convention in Section 2.2 and Section 2.3. When processing , since its two children, (d 1 ) and (d 2 ), have been processed, W d1 S(d 1 ) and W d2 S(d 2 ) are available.
The processing of each triangle consists of 8 steps:
S(d 1 ) ∪ S ∩ . 4. Divide W (d,d2) S(d 1 ) ∪ S into W d S(d 1 ) ∪ S and W d2 S(d 1 ) ∪ S along d 2 . 5. Extend W d2 S(d 2 ) into to generate W (d1,d) S(d 2 ) and construct Vor * P S(d 2 ) ∩ . 6. Divide W (d1,d) S(d 2 ) into W d1 S(d 2 ) and W d S(d 2 ) along d 1 . 7. Insert S into W d1 S(d 2 ) to obtain W d1 S(d 2 ) ∪ S . 8. Merge W d S(d 1 ) ∪ S and W d S(d 2 ) into W d S(d) by which Vor * P S(d 1 ) ∪ S ∩ and Vor * P S(d 2 ) ∩ are merged into Vor * P S(d) ∩ = SD ∩ . We remark that W d1 S(d 2 ) ∪ S
and W d2 S(d 1 ) ∪ S will be used to construct SD . If exactly one diagonal d of is not a polygon side, it is either the root triangle or a leaf triangle. For the former, compute Vor (S ), extend W d (S(d)) into to build Vor * P (S(d)) ∩ , and merge Vor (S ) and Vor * P (S(d)) ∩ into Vor * P (S) ∩ = SD ∩ ; for the latter, compute Vor (S ) and initiate W d (S To apply Lemma 3, we bound K , I and A by the following two lemmas. Proof. We consider Step 4 (divide W (d,d2) S(d 1 ) ∪ S along d 2 ), which is similar to Step 6. Since there are O(n + m) anchors, it is sufficient to bound the number of anchors that are visited by Step 4 but still involved in the future construction of SD, namely SD ∩ P (d).
Since the subcell of each visited anchor intersects d 2 , if the subcell does not intersect d, its anchor will not be involved in constructing SD ∩ P (d). A subcell of a visited anchor intersects d in two cases. In the first case, the subcell contains v 2 and thus intersects both d and d 2 . Since there are O(n) triangles, the total number for the first case is O(n). In the second case, the subcell intersects both d and d 2 but does not contain v 2 . By the definition of W (d,d2) S(d 1 ) ∪ S , its associated "site" belongs to either S or S(d 1 ). For the former, since its anchor must be the site itself, the total number is S = m. For the latter, since all the sites in S(d 1 ) lie outside , only one site in S(d 1 ) can own a cell intersecting both d and d 2 . Moreover, due to the visibility of a subcell, only one subcell in such a cell can intersect both d and d 2 . Since there are O(n) triangles, the total number is O(n).
By Lemma 3, 4, and 5, we conclude the construction time of SD as follows.
Construction of SD
To construct SD , we processes each triangle by the preorder traversal of the partition tree T . For the root triangle , letd be its diagonal that is not a polygon side, build Wd(S ) = Wd S d )), and if S = S, further propagate Wd(S ) into P (d). For other triangles , we assume that neither nor its parent has a polygon side; the other cases can be processed in a similar way. Since has been processed, W d S (d) is available, and by the construction of SD, W d2 S(d 1 ) ∪ S and W d1 S(d 2 ) ∪ S have been generated. If S(d) = ∅, is processed by the following 4 steps:
1. Extend W d S (d) into
to obtain W (d1,d2) (S (d)) and Vor * P S (d) ∩ = SD ∩ . 2. If neither S(d 1 ) nor S(d 2 ) is empty, split W (d1,d2) (S (d)) into W d1 (S (d)) and W d2 (S (d)); otherwise, if S(d 1 ) (resp. S(d 2 )) is empty, divide W (d1,d2) (S (d)) into W d1 (S (d)) and W d2 (S (d)) along d 1 (resp. d 2 ). 3. Join W d1 S(d 2 ) ∪ S and W d1 (S (d)) into W d1 (S (d 1 )), and join W d2 S(d 1 ) ∪ S and W d2 (S (d)) into W d2 (S (d 2 )). 4. If S(d 1 ) (resp. S(d 2 )) is empty, propagate W d1 (S (d 1 )) (resp. W d2 (S (d 2 ))) into P (d 1 ) (resp. P (d 2 )) to build SD ∩ P (d 1 ) (resp. SD ∩ P (d 2 )).
We first analyze Split and Propagate operations. Step 2), although each of the two resultant wavefronts could be joined with another wavefront in Step 3, since the wavefront along the traversed diagonal will propagate into the corresponding sub-polygon in Step 4 and will not separate anymore, the reasoning of Lemma 5 implies that the total time for all the Divide operations is O(n + m). Finally, by Lemma 8, the total time for all the Propagate operations (Step 4) is O n + m(log m + log 2 n) , leading to the statement.
Figure 1
1(a) Augmented geodesic Voronoi diagram Vor * P (S). (b) Rooted partition tree T .
Lemma 1 .
1([10, Lemma 5 and 14]) It takes O(log 2 n) time to compute the degree-1 or degree-3 Voronoi vertex between two sites or among three sites after O(n)-time preprocessing.
d 1 , d, and d 4 are incident to v 1 , and d 2 , d, and d 3 are incident to v 2 . Let v 1,2 be the vertex shared by d 1 and d 2 , and let v 3,4 be the vertex shared by d 3 and d 4 . Denote the set of sites in as S = S(d) − S(d 1 ) − S(d 2 ). Figure 2(a) shows an illustration.
Figure 2
2(a) and . (b) SD ∩ . (c) SD ∩ . (Borders on d are indicated by arrows.)
Theorem 2 .
2Vor * P (S) can be constructed in O n + m(log m + log 2 n) time.
.
Extend: Extend one wavefront into a triangle from one diagonal to the opposite pair of diagonals to build the diagram inside the triangle in O K inv (log m + log 2 n) + I new plus amortized O(1) time. Merge: Merge two wavefronts sharing the same diagonal or the same pair of diagonals together with merging the two corresponding diagrams in O |S |(log m+log n)+K inv (log m+ log 2 n) + I new plus amortized O(1) time where is the underlying triangle. Join: Join two wavefronts sharing the same diagonal with building the border on the diagonal but without merging the two corresponding diagrams inside the underlying triangle in O |S |(log m + log n) + K inv (log m + log 2 n) + I new plus amortized O(1) time. Split: Split a wavefront using a binary search in O(log m + log n) time. Divide: Divide a wavefront by traversing one diagonal in amortized O(A vis + 1) time.
2 ,
2Section 3], and takes O(|S |) time to trim the diagram. The tracing in the second step simply takes O(|S |) time by first locating v 1,2 in V (S ), and then repeatedly traversing the boundary of the located Voronoi cell to find the next cell along (d, d 2 ) until reaching v 1 . The data structure can also be built from scratch in O(|S |) time. Since a cell in Vor (S ) has only one subcell, whose anchor is exactly the associated site, the position to insert v 1 or v 1,2 into a wavelet as an anchor can be computed in O(1) time, and although inserting an anchor at a known position takes amortized O(1) time, since the worst-case time to insert an anchor is O(log n), the time to insert v 1 and v 1,2 is dominated by the time-factor O(|S | · (log m + log 2 n) . Finally, since there are O(|S |) incomplete Voronoi edges, there are O(|S |) potential vertices, leading to the time factor |S | · (log m + log 2 n). (Computing a potential vertex takes O(log 2 n) time (Lemma 1), locating it in a triangle of T takes O(log n) time, and inserting it in the priority queue takes O(log m) time.)
The operation time is O K inv (log m + log 2 n) + I new plus amortized O(1), where K inv is the number of involved (i.e., processed and newly created) potential vertices and I new is the number of created diagram vertices. First, extracting a potential vertex from the priority queue takes O(log m) time, and verifying its validity takes O(1) time. Second, updating incomplete Voronoi and polygonal edges takes time linear in the number of created breakpoints (Section 4.2), i.e., O(I new ) time in total. Third, since computing a potential vertex takes O(log 2 n) time, locating it takes O(log n) time, and inserting it into a priority queue takes O(log m) time, creating a new potential vertex takes O(log 2 n + log m) time. Finally, completing subcells takes O(I new ) time, and insertingṽ 1,2 takes amortized O(1) time since its position in the anchor list is known.
The total operation time is O |S |(log n + log m) + K new (log m + log 2 n) + I new plus amortized O(1), where K new is the number of created potential vertices and I new is the number of created diagram vertices while merging the two diagrams. First, since each twolevel binary search takes O(log m + log n) time, finding starting points takes O |S |(log m + log n) time. Second, by Section 5.3.1 or by [11, Section 5], tracing a merge curve takes time linear in the number of deleted and created vertices, but the time to delete vertices has been charged at their creation, implying that tracing all the merge curves takes O(I new ) time. Third, an incomplete Voronoi edge generates at least one potential vertex, so the number of new incomplete Voronoi edges is O(K new ). Since an endpoint of a merge curve corresponds to a new incomplete Voronoi edge, there are O(K new ) split and O(K new ) concatenation operations, and since each operation takes O(log m + log n) time, it takes O K new (log m + log n) time to merge the two wavefronts. By the same analysis in Section 5.2, creating K new potential vertices takes O K new (log m + log 2 n) time. Finally, inserting an ending polygon vertex of η as an anchor takes amortized O(1) time.
3 .
3The process walks along d from the starting endpoint. Every time when one of the two corresponding anchors in W d S(d 3 ) ∪ S and W d S (d 4 ) changes, a border vertex is created accordingly. The process ends when reaching a point equidistant from the site of its located wavelet in W d S(d 3 ) ∪ S and the site of its located in W d S (d 4 ) ; this point is a border vertex and also the second endpoint of the joint segment. Note that the change of an anchor can be determined by the anchor lists, and the distance from a site to a point can be determined through the corresponding anchor. The time to trace all the joint segments is proportional to the number of created border vertices, i.e., O(I new ) time in total, where I new is the number of newly created diagram vertices.
time to update the wavefronts is O K new (log m + log 2 n) plus amortized O(1), where K new is the number of created potential vertices. First, an incomplete Voronoi edge generates at least one potential vertex, so the number of new incomplete Voronoi edges is O(K new ). Since an endpoint of a joint segment corresponds to a new incomplete Voronoi edge, there are O(K new ) split and O(K new ) concatenation operations, and since each operation takes O(log m + log n) time, it takes O K new (log m + log n) time to join the two wavefronts. Moreover, by the same analysis in Section 5.2, creating K new potential vertices takes O K new (log m + log 2 n) time. Finally, inserting v 1 or v 2 as an anchor takes amortized O(1) time. To conclude, a Joint operation takes O (|S |(log n + log m) + K new (log m + log 2 n) + I new plus amortized O(1) time.
total operation time is O(|S |(log m + log 2 n)). First, since each binary search takes O(log n + log m) time, the total time for binary searches is O |S |(log n + log m) . Second, inserting a wavelet takes O(log m) time, and the time to remove anchors and wavelets has been already charged at their insertion, leading to O(|S | log m) time. Third, although inserting a polygon vertex at a known position takes amortized O(1) time, since d 1 has only two polygon vertices, the amortized time to insert them is dominated by the time-factor |S | log m. Note that an Insertion operation occurs only if S = ∅. Finally, since there are at most |S | new wavelets in W d1 S(d 2 ) ∪ S , there are O(|S |) new incomplete Voronoi edges, implying that the time to create new potential vertices is O(|S |(log m + log 2 n)).
Lemma 3 .
3It takes O (|S | + K )(log m + log 2 n) + I plus amortized O(A + 1) time to process , where K is the number of involved potential vertices, I is the number of created diagram vertices, and A is the number of visited anchors in Steps 4 and 6. Proof. Step 1 takes O |S |(log m + log 2 n) time; Step 2 and Step 5 take O K (log m + log 2 n) + I ) plus amortized O(1) time; Step 3 and Step 8 take O |S |(log n + log m) + K (log m + log 2 n) + I plus amortized O(1) time; Step 7 takes O |S |(log m + log 2 n) time; Step 4 and Step 6 take amortized O(A + 1) time.
Lemma 4 .
4In the construction of SD, K = O(m) and I = O(n + m). Proof. There are 6 intermediate diagrams: Vor (S ) (step 1), Vor * P S(d 1 ) ∩ (step 2), Vor * P S(d 1 )∪S ∩ (step 3), Vor * P S(d 2 ) ∩ (step 5), Vor * P S(d 2 )∪S ∩ (step 7), and Vor * P S(d) ∩ = SD∩ (step 8). First, a potential vertex arises due to the formation of an incomplete Voronoi edge, and an incomplete Voronoi edge generates O(1) potential vertices. For the first diagram, the total number of Voronoi edges is O( |S |) = O(|S|) = O(m). For each of the other 5 diagrams, we can define borders in a similar way to SD and thus obtain a subdivision of P . Since each site has at most one cell in each resultant subdivision, Euler's formula implies that each subdivision contains O(m) Voronoi edges among cells, leading to the conclusion that K = O(m). Second, a created diagram vertex must be a vertex of the first diagram or the other 5 subdivisions. Since there are m sites, the first diagram results in O(m) vertices for all the triangles, and by the same reasoning of [11, Lemma 3], each subdivision has O(n + m) vertice, leading to that I = O(n + m). Lemma 5. In the construction of SD, A = O(n + m).
Theorem 6 .
6SD can be constructed in O n + m(log m + log 2 n) time. Proof. By Lemma 3, we need to bound (|S | + K ) · (log m + log 2 n) + I + A + 1 . It is trivial that |S | = |S| = m. By Lemma 4 and Lemma 5, K = O(m), I = O(n + m), and A = O(n + m), leading to the statement.
Lemma 7 .
7The total number of Split operations is O(m). . A Split operation occurs only if neither S(d 1 ) nor S(d 2 ) is empty. We recursively remove leaf nodes (triangles) containing no site from T , so that each leaf node in the resulting tree T contains a site. Then a Split operation occurs in an internal node of T with two children. Since there are m sites, T has O(m) leaf nodes, and since T is a binary tree, the number of internal nodes with two children is O(m), leading to O(m) Split operations.Lemma 8. The total time for all the Propagate operations is O n + m(log m + log 2 n) . Proof. For a sub-polygon to propagate, since each edge of the resultant subdivision has a vertex in the sub-polygon, the number of created edges is bounded by the number of created vertices. Therefore, by Section 5.8, the total time is O K(log m + log 2 n) + I , where K is the number of involved potential vertices and I is the number of created diagram vertices. Moreover, by the same reasoning of Lemma 4, K = O(m), and since all the sub-polygons to propagate are disjoint, I = O(|SD |) = O(m + n), leading to the statement. By Lemma 7, Lemma 8 and the reasoning of Theorem 6, we conclude the construction time of SD as follows. Theorem 9. SD can be constructed in O n + m(log m + log 2 n) time. Proof. The analysis for Extend operations (Step 1) is identical to that in the construction of SD, and the analysis for Join operations (Step 3) is similar to the analysis for Merge operations in the construction of SD, both of which yield O n + m(log m + log 2 n) time in total. For Split operations (Step 2), by Lemma 7, there are O(m) Split operations in total, and by Appendix 5.5, each Split operation take O(log m + log n) time, leading to O m(log m + log n) time in total. For a Divide operation (
) ∩ . The border on d while merging W d S(d 3 ) ∪ S and W d S (d 4 ) into W d S (d)is the collection of points in d that are closer to S(d 3, d 4 , and
with d 1 , d 2 , d, and , respectively, and for the latter case, one
would replace d, d 3 , d 4 , and
with d 2 , d 1 , d, and , respectively.
The reason for not merging Vor *
P S(d 3 )∪S ∩ and Vor *
P S (d 4 ) ∩ into Vor *
P S (d) ∩
is that S(d 3 ) ∪ S contributes nothing to SD ∩
= Vor *
P S (d 4
1 .
1Initiate Vor (S ) and W (d,d2) (S ).2. Extend W d1 S(d 1 ) into to generate W (d,d2) S(d 1 ) and construct Vor * P S(d 1 ) ∩ . 3. Merge W (d,d2) (S ) and W (d,d2) S(d 1 ) into W (d,d2) S(d 1 )∪S by which Vor (S ) andVor * P S(d 1 ) ∩ are merged into Vor * P
). If exactly two diagonals, d and d , of are not polygon sides, where d is the root diagonal, then compute Vor (S ) to initiate W d (S ) and W d (S ), extend W d S(d ) into to obtain W d S(d ) and Vor * P S(d ) ∩ , and merge W d (S ) and W d S(d ) to obtain W d S(d) and Vor * P S(d) ∩ = SD ∩ . By the operation times in Section 5, the time to process is summarized as follows.
In[11, Section 6], the authors define σ (d) as the Voronoi edges in SD ∩ that are associated with one site in S (d4) and one site in S(d3) ∪ S , but only compute σ (d) ∩ d, which is exactly b (d).
On the geodesic Voronoi diagram of point sites in a simple polygon. Boris Aronov, Algorithmica. 41-4Boris Aronov. On the geodesic Voronoi diagram of point sites in a simple polygon. Algo- rithmica, 4(1-4):109-140, 1989.
Voronoi Diagrams and Delaunay Triangulations. Franz Aurenhammer, Rolf Klein, Der-Tsai Lee, World ScientificFranz Aurenhammer, Rolf Klein, and Der-Tsai Lee. Voronoi Diagrams and Delaunay Tri- angulations. World Scientific, 2013.
Triangulating a simple polygon in linear time. Bernard Chazelle, Discrete & Computational Geometry. 6Bernard Chazelle. Triangulating a simple polygon in linear time. Discrete & Computational Geometry, 6:485-524, 1991.
Optimal point location in a monotone subdivision. Herbert Edelsbrunner, Leonidas J Guibas, Jorge Stolfi, SIAM J. Comput. 152Herbert Edelsbrunner, Leonidas J. Guibas, and Jorge Stolfi. Optimal point location in a monotone subdivision. SIAM J. Comput., 15(2):317-340, 1986.
Optimal shortest path queries in a simple polygon. Leonidas J Guibas, John Hershberger, Journal of Computer and System Sciences. 392Leonidas J. Guibas and John Hershberger. Optimal shortest path queries in a simple polygon. Journal of Computer and System Sciences, 39(2):126-152, 1989.
Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Leonidas J Guibas, John Hershberger, Daniel Leven, Micha Sharir, Robert E Tarjan, Algorithmica. 21-4Leonidas J. Guibas, John Hershberger, Daniel Leven, Micha Sharir, and Robert E. Tarjan. Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica, 2(1-4):209-233, 1987.
A new data structure for shortest path queries in a simple polygon. John Hershberger, Information Processing Letters. 385John Hershberger. A new data structure for shortest path queries in a simple polygon. Information Processing Letters, 38(5):231-235, 1991.
Sorted sequences. In Algorithms and Data Structures: The Basic Toolbox. Kurt Mehlhorn, Peter Sanders, Springer-VerlagBerlin HeidelbergKurt Mehlhorn and Peter Sanders. Sorted sequences. In Algorithms and Data Structures: The Basic Toolbox. Springer-Verlag Berlin Heidelberg, 2008.
Geometric shortest paths and network optimization. S B Joseph, Mitchell, Handbook of Computational Geometry. ElsevierJoseph S. B. Mitchell. Geometric shortest paths and network optimization. In Handbook of Computational Geometry, pages 633-701. Elsevier, 2000.
Voronoi diagrams for a moderate-sized point-set in a simple polygon. Eunjin Oh, Hee-Kap Ahn, 33rd International Symposium on Computational Geometry. Brisbane, Australia5215Eunjin Oh and Hee-Kap Ahn. Voronoi diagrams for a moderate-sized point-set in a simple polygon. In 33rd International Symposium on Computational Geometry, SoCG 2017, July 4-7, 2017, Brisbane, Australia, pages 52:1-52:15, 2017.
A new approach for the geodesic Voronoi diagram of points in a simple polygon and other restricted polygonal domains. Evanthia Papadopoulou, D T Lee, Algorithmica. 204Evanthia Papadopoulou and D. T. Lee. A new approach for the geodesic Voronoi diagram of points in a simple polygon and other restricted polygonal domains. Algorithmica, 20(4):319- 352, 1998.
Updating a balanced search tree in O(1) rotations. Robert Endre Tarjan, Information Processing Letters. 165Robert Endre Tarjan. Updating a balanced search tree in O(1) rotations. Information Processing Letters, 16(5):253-257, 1983.
Efficient Top-Down Updating of Red-Black Trees. Robert Endre Tarjan, TR-006-85Dapartment of Computer Science, Princeton UniversityTechnical ReportRobert Endre Tarjan. Efficient Top-Down Updating of Red-Black Trees. Technical report, Technical Report TR-006-85. Dapartment of Computer Science, Princeton University, 1985.
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[
"UNKNOTTING SUBMANIFOLDS OF THE 3-SPHERE BY TWISTINGS",
"UNKNOTTING SUBMANIFOLDS OF THE 3-SPHERE BY TWISTINGS"
] |
[
"Makoto Ozawa "
] |
[] |
[] |
By the Fox's re-embedding theorem, any compact submanifold of the 3-sphere can be re-embedded in the 3-sphere so that it is unknotted. It is unknown whether the Fox's re-embedding can be replaced with twistings. In this paper, we will show that any closed 2-manifold embedded in the 3-sphere can be unknotted by twistings. In spite of this phenomenon, we show that there exists a compact 3-submanifold of the 3-sphere which cannot be unknotted by twistings. This shows that the Fox's re-embedding cannot always be replaced with twistings.
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10.2422/2036-2145.201612_007
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[
"https://arxiv.org/pdf/1609.06573v3.pdf"
] | 119,279,905 |
1609.06573
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ec7950d058931fa5b2e8e1a092606653809d2de2
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UNKNOTTING SUBMANIFOLDS OF THE 3-SPHERE BY TWISTINGS
21 Sep 2016
Makoto Ozawa
UNKNOTTING SUBMANIFOLDS OF THE 3-SPHERE BY TWISTINGS
21 Sep 2016arXiv:1609.06573v1 [math.GT]
By the Fox's re-embedding theorem, any compact submanifold of the 3-sphere can be re-embedded in the 3-sphere so that it is unknotted. It is unknown whether the Fox's re-embedding can be replaced with twistings. In this paper, we will show that any closed 2-manifold embedded in the 3-sphere can be unknotted by twistings. In spite of this phenomenon, we show that there exists a compact 3-submanifold of the 3-sphere which cannot be unknotted by twistings. This shows that the Fox's re-embedding cannot always be replaced with twistings.
Introduction
Throughout this paper, we will work in the piecewise linear category. We assume that a surface is a compact, connected 2-manifold and that a 2-manifold is possibly disconnected.
Definition 1.1. Let X be a compact submanifold of the 3-sphere S 3 . Take a loop C in S 3 − X which is the trivial knot in S 3 . Then C bounds a disk D in S 3 , which may intersect X in its interior. Cut open S 3 along by D, rotate one copy of D by ±2π, and glue again two copies of D. Then we obtain another submanifold X ′ of S 3 and call this operation a twisting along C, which is denoted by (S 3 , X) C −→ (S 3 , X ′ ).
Remark 1.2.
A twisting along C is not a homeomorphism of S 3 , but it gives a homeomorphism of S 3 − C. We note that a twisting along C is also obtained by ±1-Dehn surgery along C. Definition 1.3. Let X be a compact submanifold of S 3 which has an n connected components X 1 , . . . , X n .
We say that X = X 1 ∪ · · · ∪ X n is completely splittable in S 3 if there exist n − 1 mutually disjoint 2-spheres S 1 , . . . , S n−1 in S 3 − X such that if we cut open S 3 along S 1 ∪ · · · ∪ S n−1 and glue 2(n − 1) 3-balls along their boundaries, then we obtain n pairs of the 3-sphere and the submanifold (S 3 , X 1 ), . . . , (S 3 , X n ).
For a connected component X i of X, we say that a pair (S 3 , X i ) is unknotted in S 3 if the exterior E(X i ) = S 3 − intN (X i ) consists of handlebodies. We say that X is unknotted if X is completely splittable and for every pair (S 3 , X i ), X i is unknotted in S 3 .
2010 Mathematics Subject Classification. 57Q35 (Primary), 57N35 (Secondary). Key words and phrases. embedding, closed surface, 3-sphere, twisting, crossing changes, unknotting, Fox's re-embedding.
The author is partially supported by Grant-in-Aid for Scientific Research (C) (No. 26400097), The Ministry of Education, Culture, Sports, Science and Technology, Japan. Remark 1.4. We remark that by the Fox's re-embedding theorem [2], any compact submanifold M of S 3 can be re-embedded in S 3 so that M is unknotted.
The following is the main subject of this paper. Problem 1.5. Can any Fox's re-embedding be replaced with twistings?
It is well-known that Problem 1.5 is true for any closed 1-manifold and for any closed 2-manifold which bounds handlebodies. Definition 1.6. Let F be a closed 2-manifold and α be a loop, namely, simple closed curve in F . We say that α is inessential in F if it bounds a disk in F . Otherwise, α is essential. We define the breadth b(F ) of F as the maximal number of mutually disjoint, mutually non-parallel essential loops in F .
Let F be a closed 2-manifold embedded in S 3 with b(F ) > 0. We say that F is compressible in S 3 if there exists a disk D embedded in S 3 such that D ∩ F = ∂D
and ∂D is essential in F . Such a disk is called a compressing disk for F . Then by cutting F along ∂D, and pasting two parallel copies of D to its boundaries, we obtain another closed 2-manifold F ′ with b(F ′ ) < b(F ). Such an operation is called a compression along D. Conversely, if F ′ is obtained from F by a compression along D, then there exists a dual arc α with respect to D, that is, α intersects D in one point and α ∩ F ′ = ∂α such that F can be recovered from F ′ by tubing along α. See Figure 1. We remark that if b(F ) = 0, then F consists of only 2-spheres and by the Alexander's theorem [1], F is unknotted in S 3 . On the other hand, we remark that if b(F ) > 0, then by [2] or [3], F is compressible in S 3 . Theorem 1.8. Any closed 2-manifold embedded in the 3-sphere can be unknotted by twistings.
Proof. Let F be a closed 2-manifold consisting of n closed surfaces F 1 , . . . , F n embedded in S 3 . We will prove Theorem 1.8 by induction on the breadth b(F ).
In the case of b(F ) = 0, by Remark 1.7, F is unknotted. Next suppose that when b(F ) < b, Theorem 1.8 holds, and assume that b(F ) = b. Then by Remark 1.7, there exists a compressing disk D for F . Let F ′ be the closed 2-manifold obtained from F by a compression along D. Then there exists an arc α such that α intersects D in one point, α ∩ F ′ = ∂α, and F can be obtained from F ′ by tubing along α. Since b(F ′ ) < b, by the assumption of induction, F ′ can be unknotted by twistings. Thus there exists a sequence of twistings
(S 3 , F ′ ) C1 −→ (S 3 , F ′ (1)) C2 −→ · · · Cm −→ (S 3 , F ′ (m)),
where F ′ (m) is unknotted. In each stage, we may assume that C i ∩ α = ∅ for i = 1, . . . , m. Therefore, this sequence extends to a sequence of twistings
(S 3 , F ) C1 −→ (S 3 , F (1)) C2 −→ · · · Cm −→ (S 3 , F (m)).
Let R be the closure of a connected component of S 3 − F ′ which contains α, and
put ∂R = F ′ 1 ∪· · ·∪F ′ k , where F ′ 1 , . . . , F ′ k are connected components of F ′ (m). Since F ′ (m) is unknotted, F ′ 1 ∪ · · · ∪ F ′ k bounds k handlebodies V 1 , .
. . , V k in S 3 − intR, and V 1 ∪ · · · ∪ V k is unknotted in S 3 , namely, V 1 ∪ · · · ∪ V k is ambient isotopic to a regular neighborhood of a plane graph G on the 2-sphere S. Then by crossing changes on α and crossing changes between α and V i , α can be unknotted, that is, α is isotopic to an arc on S. Since these crossing changes are obtained by twistings, there is a sequence of twistings
(S 3 , F ′ (m)) Cm+1 −→ (S 3 , F ′ (m + 1)) Cm+2 −→ · · · C m+l −→ (S 3 , F ′ (m + l)),
where F ′ (m), F ′ (m + 1), F ′ (m + 2), . . . , F ′ (m + l) are equivalent and C m+i ∩ α = ∅ for i = 1, . . . , l. Therefore, this sequence extends to a sequence of twistings
(S 3 , F (m)) Cm+1 −→ (S 3 , F (m + 1)) Cm+2 −→ · · · C m+l −→ (S 3 , F (m + l)).
Hence, by tubing F ′ along α, we obtain a sequence of twistings
(S 3 , F ) C1 −→ (S 3 , F (1)) C2 −→ · · · Cm −→ (S 3 , F (m)) Cm+1 −→ (S 3 , F (m + 1)) Cm+2 −→ · · · C m+l −→ (S 3 , F (m + l)), where F (m + l) is unknotted.
By the Waldhausen's theorem [9], any unknotted closed surface in S 3 is unique up to isotopy. Therefore, by Theorem 1.8, we have the following. Corollary 1.9. Any two closed 2-manifolds F = F 1 ∪· · ·∪F n and F ′ = F ′ 1 ∪· · ·∪F ′ n in the 3-sphere can be mutually related by twistings if n ≤ 2 and F is homeomorphic to F ′ . Example 1.10. We recall an example of closed surface H of genus 2 given by Homma [3], see also [6, 4.1 Theorem] as shown in Figure 2. The surface H separates S 3 into two components W 1 and W 2 , where W 1 is homeomorphic to the exterior of the 4-crossing Handcuff graph 4 1 in the table of [4], and W 2 is a boundary connected sum of two trefoil knot exteriors. It is remarkable that H is incompressible in W 1 , whereas H has only one compressing disk D in W 2 up to isotopy by [8], [7].
In spite of Theorem 1.8, there is a following phenomenon.
Theorem 1.11. The Homma's surface H cannot be unknotted by twistings in W 1 .
Proof. The key of proof is that any twistings in W 1 can take effect only crossing changes on the tube at D. Then one can show that the Handcuff graph 4 1 cannot be unknotted by crossing changes only on α.
Let C be a loop in W 1 which is trivial in S 3 . Let D be a compressing disk for H in W 2 which defines the boundary connected sum of two trefoil knot exteriors as shown in Figure 2. By compressing H along D, we obtain two tori T 1 and T 2 which bound two trefoil knot exteriors E 1 and E 2 in W 2 . Let α be an arc in W 2 which intersects D in one point and connects T 1 and T 2 such that H is obtained from T 1 ∪ T 2 by tubing along α. Since C is the trivial knot in S 3 , both of T 1
− int(E 1 ∪ E 2 ) − C. Hence C bounds a disk ∆ in S 3 − int(E 1 ∪ E 2 )
. Then we may assume that α intersects ∆ transversely and conclude that any twisting along C takes effect only on α.
Let K 1 ∪α∪K 2 be the Handcuff graph 4 1 , whose exterior is homeomorphic to W 1 . We take a double branched cover of S 3 along the trivial link K 1 ∪K 2 as follows. Let D i be a disk bounded by K i which intersects α in one point (i = 1, 2). We cut open S 3 along D 1 ∪ D 2 and take a copy of it. Those 3-manifolds are both homeomorphic to S 2 × I and whose boundary consists of 2-spheres D + we obtain S 2 × S 1 and a knotα obtained from α and α ′ as shown in Figure 4. By the Fox's re-embedding theorem [2], there exists a re-embedding of W 2 into S 3 such that W 2 is unknotted. However, this Fox's re-embedding cannot be obtained by twistings. Corollary 1.12. There exists a 3-submanifold of S 3 which cannot be unknotted by twistings.
1 ∪ D − 1 , D + 2 ∪ D − 2 , D ′ + 1 ∪ D ′ − 1 , D ′ + 2 ∪ D ′
Proof. Take W 2 as a 3-submanifold of S 3 .
However, we remark that by [5,Theorem 1.6], there exists a null-homologous link L in W 1 , which is reflexive in S 3 , such that a handlebody can be obtained from W 1 by a 1/Z-Dehn surgery along L, that is, [L] = 0 in H 1 (W 1 ; Z) and there exists a surgery slope 1/n i for each component L i of L such that a pair of S 3 and a handlebody is obtained from (S 3 , W 1 ) by a Dehn surgery along L. Therefore, the Fox's re-embedding can be replaced with a Dehn surgery along a link. At the time of writing of [5], it was unknown whether this Dehn surgery can be replaced with twistings. Corollary 1.12 shows that it is not always true.
Figure 1 .
1Compression along D Remark 1.7.
Figure 2 .
2The Homma's closed surface
Figure 3 .
3The 4-crossing Handcuff graph 4 1 and T 2 are compressible in S 3
− 2 .
2Then by gluing D + 1 and D ′ − 1 , D − 1 and D ′ + 1 , D + 2 and D ′ − 2 , D − 2 and D ′ + 2
Figure 4 .
4The double branched cover of S 3 along K 1 ∪ K 2
We note that [α] = 3[γ] in H 1 (S 2 ×S 1 ; Z) ∼ = Z, where γ is a generator of H 1 (S 2 ×S 1 ; Z). Suppose that K 1 ∪ α ∪ K 2 isunknotted by crossing changes on α. Then the homology class [α] in H 1 (S 2 × S 1 ; Z) does not change by the crossing changes, and we have [α] = 3[γ]. However, since K 1 ∪ α ∪ K 2 is trivial, we have [α] = [γ]. This is a contradiction.
On the subdivision of a 3-space by a polyhedron. J W Alexander, Proc. Nat. Acad. Sci. USA. 10J. W. Alexander, On the subdivision of a 3-space by a polyhedron, Proc. Nat. Acad. Sci. USA 10 (1924), 6-8.
On the imbedding of polyhedra in 3-space. R H Fox, Ann. of Math. 49R. H. Fox, On the imbedding of polyhedra in 3-space, Ann. of Math. 49 (1948), 462-470.
On the existence of unknotted polygons on 2-manifolds in E 3. T Homma, Osaka Math. J. 6T. Homma, On the existence of unknotted polygons on 2-manifolds in E 3 , Osaka Math. J. 6 (1954), 129-134.
A table of genus two handlebody-knots up to six crossings. A Ishii, K Kishimoto, H Moriuchi, M Suzuki, J. Knot Theory Ramifications. 211250035A. Ishii, K. Kishimoto, H. Moriuchi and M. Suzuki, A table of genus two handlebody-knots up to six crossings, J. Knot Theory Ramifications 21, 1250035 (2012).
Dehn surgery and Seifert surface systems. M Ozawa, K Shimokawa, Scuola Norm-Sci. to appear in AnnM. Ozawa, K. Shimokawa, Dehn surgery and Seifert surface systems, to appear in Ann. Scuola Norm-Sci.
On a complexity of a surface in 3-sphere. S Suzuki, Osaka J. Math. 11S. Suzuki, On a complexity of a surface in 3-sphere, Osaka J. Math. 11 (1974), 113-127.
On surfaces in 3-sphere: prime decomposition. S Suzuki, Hokkaido Math. J. 4S. Suzuki, On surfaces in 3-sphere: prime decomposition, Hokkaido Math. J. 4 (1975), 179- 195.
On surfaces in 3-space. Y Tsukui, Yokohama Math. J. 18Y. Tsukui, On surfaces in 3-space, Yokohama Math. J. 18 (1970), 93-104.
Department of Natural Sciences, Faculty of Arts and Sciences, Komazawa University, 1-23-1 Komazawa. F Waldhausen, Heegaard-Zerlegungen der 3-Sphäre. Setagaya-ku, Tokyo7Japan E-mail address: [email protected]. Waldhausen, Heegaard-Zerlegungen der 3-Sphäre, Topology 7 (1968), 195-203. Department of Natural Sciences, Faculty of Arts and Sciences, Komazawa Univer- sity, 1-23-1 Komazawa, Setagaya-ku, Tokyo, 154-8525, Japan E-mail address: [email protected]
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The classic Edgar Allan Poe story The Gold-Bug involves digging for pirate treasure. Locating the digging sites requires some simple trigonometry.
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"https://arxiv.org/pdf/1206.1761v1.pdf"
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1206.1761
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72c6758f4c2fb6765bbc64c6e36f011794a446fc
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May 2012
May 2012TRIGONOMETRY OF THE GOLD-BUG ERIK TALVILA
The classic Edgar Allan Poe story The Gold-Bug involves digging for pirate treasure. Locating the digging sites requires some simple trigonometry.
Introduction
The lanterns having been lit, we all fell to work with a zeal worthy a more rational cause; and, as the glare fell upon our persons and implements, I could not help thinking how picturesque a group we composed, and how strange and suspicious our labors must have appeared to any interloper who, by chance, might have stumbled upon our whereabouts.
We dug very steadily for two hours. This is an excerpt from the Edgar Allan Poe story The Gold-Bug [3]. Published in 1843, it is a gripping tale about Captain Kidd's lost pirate treasure map and digging for booty on the Carolina shore. Locating the digging spots requires a little trigonometry. Poe has our heroes dig two holes but we will show that they must really overlap.
The main character of the story is William Legrand, who is a nobleman in impoverished circumstances. He lives on tiny, mostly unpopulated, Charleston Island off the coast of South Carolina. With him is the old, freed, slave Jupiter. It is widely believed that Legrand is slightly unbalanced and Jupiter takes it upon himself to look after him. The third character is the narrator, an old friend, who accompanies them on the treasure hunt.
Legrand has found a parchment half buried in the sand beside an old ship's boat. It is written with peculiar symbols and turns out to be a cryptogram. He translates it and finds it gives veiled directions to a secluded spot in the woods on the mainland. (Poe gives a lovely explanation of how to crack a cryptogram.) Here the three companions dig by lantern light for what they hope is treasure.
The gold-bug itself is a living beetle that Legrand has found. It seems to be made of real gold. This gives the story a chilling air of mysticism. [H]e made furious resistance, and, leaping into the hole, tore up the mould frantically with his claws. In a few seconds he had uncovered a mass of human bones. . . Some simple trigonometry shows that, under reasonable assumptions for the radius of the tree and the distance the skull is from the trunk, the two holes must have overlapped, even though the narrator says they were several yards apart.
In Figure 1 The condition that the holes do not overlap is that AB > r A + r B . With (3.1) this reduces to (3.2) r + L < 250 24(r A + r B ) − 5 .
We know from the story that r A = 2 ′ . The narrator says that r B is slightly larger. Even if we say r B = 2 ′ , for non-overlapping holes we must have r + L < 250/91 . = 2 ′ 9 ′′ . This is clearly impossible. The tree is an enormously tall tulip tree, which stood, with some eight or ten oaks, upon the level, and far surpassed them all, and all other trees which I had then ever seen, in the beauty of its foliage and form, in the wide spread of its branches, and in the general majesty of its appearance. And Poe states that Jupiter climbs 60 or 70 feet up to the branch with the skull. We should take r to be the radius of the trunk at the height of the skull branch. For there to be a branch at that height substantial enough to hold a man's weight, r must be at least 2 inches and could easily be 6 inches or more. (A tulip tree on my campus was recently cut down. The stump measures 25 ′′ across. It was not a particularly old tree.) But that means L < 2 ′ 7 ′′ . This puts the skull too close to the tree trunk. Jupiter crawls out to the end of the branch so this must be more than 3 feet. Taking r B > 2 ′ only makes things worse. Hence, although it's a great story, Poe neglected to do his trigonometry homework and the two treasure holes must have overlapped.
From the wording in the story, it is not entirely clear whether the 50 feet to the centre of the two holes is measured from the eye socket drop points a and b, as we have taken it, or from the tree trunk. If from the trunk then the same assumptions on r A , r B and r as above require L be even smaller for the holes to not overlap.
Background
The Gold-Bug has been in the public domain for many years so there are a great number of editions available, both print and electronic. Excerpts here are taken from [3]. For more on fiction that contains mathematics, see Alex Kasman's web page [2].
In the story, Poe hints the treasure is Captain Kidd's. William Kidd, born circa 1654, was hanged for murder and piracy in England in 1701. He was certainly a privateer, which was legal in those times, and he held a Letter of Marque signed by King William III authorising him to attack French and certain other vessels. It is uncertain whether he was a pirate or not. After his death a legend grew that he had left a treasure buried on the North American Atlantic coast. For example, see [1].
2 .
2The digging directions . . . main branch seventh limb east side shoot from the left eye of the death'shead a bee line from the tree through the shot fifty feet out. . . At the digging site in the woods there is an enormous old tulip tree. Legrand has Jupiter climb it. Going up 70 feet, he climbs onto the seventh limb up the east side of the tree. Out on the limb a human skull has been fastened by the pirates. He drops the gold-bug through the right eye socket and it falls to the ground. They then draw a line a further 50 feet out from the tree in this direction and dig a hole 4 feet in diameter and 5 Date: Preprint May 31, 2012. To appear in Mathematical Gazette.
feet deep. Alas, no treasure. They then realise Jupiter has dropped the bug through the wrong eye socket. Legrand moves the mark under the skull 2 1/2 inches, as if it had fallen through the other eye socket. A line is then extended 50 feet through this new point away from the trunk. Here they dig a second time.3. TrigonometryI dug eagerly, and now and then caught myself actually looking, with something that very much resembled expectation, for the fancied treasure. . . [W]e were again interrupted by the violent howlings of the dog. . . .
we have the set up. Points labeled in the figure are O (centre of tree), a (drop point through left eye socket), b (drop point through right eye socket), A (first digging site), B (second digging site). Distances are r (radius of tree trunk), L (distance of each eye socket from tree trunk), r A (radius of first hole), r B (radius of second hole). All distances are in feet, as they are in the story. The angle between OA and OB is 2θ. Legrand states that points a and b are 2.5 ′′ = 5/24 ′ apart. The distances aA and bB are 50 ′ . Note that sin θ = 5/[48(r + L)]. The distance between the centres of the holes is (3.1) AB = 2(r + L + 50) sin θ = 5(r + L + 50) 24(r + L) .
Figure 1 .
1The digging site
Treasure and intrigue: The legacy of Captain Kidd. G Harris, Toronto, DundurnG. Harris, Treasure and intrigue: The legacy of Captain Kidd, Toronto, Dundurn, 2002.
Abbotsford, BC Canada V2S 7M8 E-mail address. E A Poe, The Gold-Bug, [email protected] of Mathematics & Statistics, University of the Fraser ValleyE.A. Poe, The Gold-Bug, http://www.online-literature.com/poe/32/ Department of Mathematics & Statistics, University of the Fraser Valley, Abbots- ford, BC Canada V2S 7M8 E-mail address: [email protected]
|
[] |
[
"Direct observation of enhanced magnetism in individual size-and shape-selected 3d transition metal nanoparticles",
"Direct observation of enhanced magnetism in individual size-and shape-selected 3d transition metal nanoparticles"
] |
[
"Armin Kleibert [email protected] \nSwiss Light Source\nPaul Scherrer Institut\n5232Villigen PSISwitzerland\n",
"Ana Balan \nSwiss Light Source\nPaul Scherrer Institut\n5232Villigen PSISwitzerland\n",
"Rocio Yanes \nDepartment of Physics\nUniversity of Konstanz\n78457KonstanzGermany\n",
"Peter M Derlet \nCondensed Matter Theory Group\nNUM\nPaul Scherrer Institut\n5232Villigen PSISwitzerland\n",
"C A F Vaz \nSwiss Light Source\nPaul Scherrer Institut\n5232Villigen PSISwitzerland\n",
"Martin Timm \nSwiss Light Source\nPaul Scherrer Institut\n5232Villigen PSISwitzerland\n",
"Arantxa Fraile Rodríguez \nDepartament de Física de la Matèria Condensada and Institut de Nanociència i Nanotecnologia (IN2UB)\nUniversitat de Barcelona\n08028BarcelonaSpain\n",
"Armand Béché \nEMAT\nUniversity of Antwerp\n2020AntwerpenBelgium\n",
"Jo Verbeeck \nEMAT\nUniversity of Antwerp\n2020AntwerpenBelgium\n",
"Rajen S Dhaka \nSwiss Light Source\nPaul Scherrer Institut\n5232Villigen PSISwitzerland\n\nDepartment of Physics\nIndian Institute of Technology Delhi\nHauz Khas, New Delhi-110016India\n\nInstitute of Condensed Matter Physics\nEcole Polytechnique Fédérale de Lausanne (EPFL)\n1015LausanneSwitzerland\n",
"Milan Radovic \nSwiss Light Source\nPaul Scherrer Institut\n5232Villigen PSISwitzerland\n\nInstitute of Condensed Matter Physics\nEcole Polytechnique Fédérale de Lausanne (EPFL)\n1015LausanneSwitzerland\n\nSwissFEL\nPaul Scherrer Institut\n5232Villigen PSISwitzerland\n",
"Ulrich Nowak \nDepartment of Physics\nUniversity of Konstanz\n78457KonstanzGermany\n",
"Frithjof Nolting \nSwiss Light Source\nPaul Scherrer Institut\n5232Villigen PSISwitzerland\n"
] |
[
"Swiss Light Source\nPaul Scherrer Institut\n5232Villigen PSISwitzerland",
"Swiss Light Source\nPaul Scherrer Institut\n5232Villigen PSISwitzerland",
"Department of Physics\nUniversity of Konstanz\n78457KonstanzGermany",
"Condensed Matter Theory Group\nNUM\nPaul Scherrer Institut\n5232Villigen PSISwitzerland",
"Swiss Light Source\nPaul Scherrer Institut\n5232Villigen PSISwitzerland",
"Swiss Light Source\nPaul Scherrer Institut\n5232Villigen PSISwitzerland",
"Departament de Física de la Matèria Condensada and Institut de Nanociència i Nanotecnologia (IN2UB)\nUniversitat de Barcelona\n08028BarcelonaSpain",
"EMAT\nUniversity of Antwerp\n2020AntwerpenBelgium",
"EMAT\nUniversity of Antwerp\n2020AntwerpenBelgium",
"Swiss Light Source\nPaul Scherrer Institut\n5232Villigen PSISwitzerland",
"Department of Physics\nIndian Institute of Technology Delhi\nHauz Khas, New Delhi-110016India",
"Institute of Condensed Matter Physics\nEcole Polytechnique Fédérale de Lausanne (EPFL)\n1015LausanneSwitzerland",
"Swiss Light Source\nPaul Scherrer Institut\n5232Villigen PSISwitzerland",
"Institute of Condensed Matter Physics\nEcole Polytechnique Fédérale de Lausanne (EPFL)\n1015LausanneSwitzerland",
"SwissFEL\nPaul Scherrer Institut\n5232Villigen PSISwitzerland",
"Department of Physics\nUniversity of Konstanz\n78457KonstanzGermany",
"Swiss Light Source\nPaul Scherrer Institut\n5232Villigen PSISwitzerland"
] |
[] |
Magnetic nanoparticles are important building blocks for future technologies ranging from nano-medicine to spintronics. Many related applications require nanoparticles with tailored magnetic properties. However, despite significant efforts undertaken towards this goal, a broad and poorly-understood dispersion of magnetic properties is reported, even within monodisperse samples of the canonical ferromagnetic 3d transition metals. We address this issue by investigating the magnetism of a large number of size-and shape-selected, individual nanoparticles of Fe, Co, and Ni using a unique set of complementary characterization techniques. At room temperature only superparamagnetic behavior is observed in our experiments for all Ni nanoparticles within the investigated sizes, which range from 8 to 20 nm. However, Fe and Co nanoparticles can exist in two distinct magnetic states at any size in this range: (i) a superparamagnetic state as expected from the bulk and surface anisotropies known for the respective materials and as observed for Ni; and (ii) a state with unexpected stable magnetization at room temperature. This striking state is assigned to significant modifications of the magnetic properties arising from metastable lattice defects in the core of the nanoparticles as concluded by calculations and atomic structural characterization. Also related with the structural defects, we find that the magnetic state of Fe and Co nanoparticles can be tuned by thermal treatment enabling one to tailor their magnetic properties for applications. This work demonstrates the importance of complementary single particle investigations for a better understanding of nanoparticle magnetism and for full exploration of their potential for applications.
|
10.1103/physrevb.95.195404
|
[
"https://arxiv.org/pdf/1702.03710v1.pdf"
] | 4,688,970 |
1702.03710
|
1e747c56f3c2eb8f5b682902f2d64a0a12268074
|
Direct observation of enhanced magnetism in individual size-and shape-selected 3d transition metal nanoparticles
Armin Kleibert [email protected]
Swiss Light Source
Paul Scherrer Institut
5232Villigen PSISwitzerland
Ana Balan
Swiss Light Source
Paul Scherrer Institut
5232Villigen PSISwitzerland
Rocio Yanes
Department of Physics
University of Konstanz
78457KonstanzGermany
Peter M Derlet
Condensed Matter Theory Group
NUM
Paul Scherrer Institut
5232Villigen PSISwitzerland
C A F Vaz
Swiss Light Source
Paul Scherrer Institut
5232Villigen PSISwitzerland
Martin Timm
Swiss Light Source
Paul Scherrer Institut
5232Villigen PSISwitzerland
Arantxa Fraile Rodríguez
Departament de Física de la Matèria Condensada and Institut de Nanociència i Nanotecnologia (IN2UB)
Universitat de Barcelona
08028BarcelonaSpain
Armand Béché
EMAT
University of Antwerp
2020AntwerpenBelgium
Jo Verbeeck
EMAT
University of Antwerp
2020AntwerpenBelgium
Rajen S Dhaka
Swiss Light Source
Paul Scherrer Institut
5232Villigen PSISwitzerland
Department of Physics
Indian Institute of Technology Delhi
Hauz Khas, New Delhi-110016India
Institute of Condensed Matter Physics
Ecole Polytechnique Fédérale de Lausanne (EPFL)
1015LausanneSwitzerland
Milan Radovic
Swiss Light Source
Paul Scherrer Institut
5232Villigen PSISwitzerland
Institute of Condensed Matter Physics
Ecole Polytechnique Fédérale de Lausanne (EPFL)
1015LausanneSwitzerland
SwissFEL
Paul Scherrer Institut
5232Villigen PSISwitzerland
Ulrich Nowak
Department of Physics
University of Konstanz
78457KonstanzGermany
Frithjof Nolting
Swiss Light Source
Paul Scherrer Institut
5232Villigen PSISwitzerland
Direct observation of enhanced magnetism in individual size-and shape-selected 3d transition metal nanoparticles
1
Magnetic nanoparticles are important building blocks for future technologies ranging from nano-medicine to spintronics. Many related applications require nanoparticles with tailored magnetic properties. However, despite significant efforts undertaken towards this goal, a broad and poorly-understood dispersion of magnetic properties is reported, even within monodisperse samples of the canonical ferromagnetic 3d transition metals. We address this issue by investigating the magnetism of a large number of size-and shape-selected, individual nanoparticles of Fe, Co, and Ni using a unique set of complementary characterization techniques. At room temperature only superparamagnetic behavior is observed in our experiments for all Ni nanoparticles within the investigated sizes, which range from 8 to 20 nm. However, Fe and Co nanoparticles can exist in two distinct magnetic states at any size in this range: (i) a superparamagnetic state as expected from the bulk and surface anisotropies known for the respective materials and as observed for Ni; and (ii) a state with unexpected stable magnetization at room temperature. This striking state is assigned to significant modifications of the magnetic properties arising from metastable lattice defects in the core of the nanoparticles as concluded by calculations and atomic structural characterization. Also related with the structural defects, we find that the magnetic state of Fe and Co nanoparticles can be tuned by thermal treatment enabling one to tailor their magnetic properties for applications. This work demonstrates the importance of complementary single particle investigations for a better understanding of nanoparticle magnetism and for full exploration of their potential for applications.
I. INTRODUCTION
Magnetic nanoparticles attract a wide interest in many fields ranging from bio-medicine to energy, magnetic data storage, and spintronics [1][2][3][4]. This interest is driven by the unique magnetic phenomena which occur at the nanoscale, such as single domain states and superparamagnetism (SPM) [5]. Moreover, enhanced magnetic moments and magnetic anisotropy energies have been reported for atomic clusters and nanoparticles [6][7][8]. These features are of great interest for novel applications, but achieving control remains challenging and requires deeper understanding of the magnetic properties at the nanoscale. Extensive efforts have been undertaken to establish simple laws to predict size-dependent properties such as the magnetic anisotropy energy [9][10][11][12]. However, experimental validation of scalable regimes has not been achieved so far, even for the common ferromagnetic 3d transition metals, Fe, Co, and Ni. Instead, the available literature reveals a significant scatter of magnetic properties which cannot be assigned only to particle size or environment. For instance, the magnetic anisotropy energies of Fe nanoparticles are reported to range from bulk-like to strongly enhanced values in different experiments [13][14][15][16][17][18][19]. Similarly, for Co nanoparticles the experimentally observed values vary over several orders of magnitude [20][21][22][23][24][25]. For Ni, the situation seems even more complex, since not only does the magnetic anisotropy energy vary, but also the magnetic moment of the particles differs in various reports [26][27][28][29][30][31][32]. Such variability is often assigned to shape, surface or interface effects [18,22,33]. However, an unambiguous interpretation of experimental data is difficult, since most of the reported investigations have been carried out with experimental techniques that average over large distributions of particle sizes, morphologies, and orientations. The situation might be further complicated by additional inter-particle interactions, which can largely affect ensemble properties such as magnetization curves acquired with bulk SQUID and vibrating sample magnetometry, or integrated X-ray magnetic circular dichroism (XMCD) spectroscopy [15,34,35].
In the present work we overcome these difficulties by investigating the magnetism of a large number of individual Fe, Co, and Ni nanoparticles by means of X-ray photo-emission electron microscopy (X-PEEM) together with the XMCD effect under ultrahigh vacuum conditions [15,[36][37][38][39]. The magnetic properties are directly correlated with morphological information of the very same nanoparticles such as size and shape obtained by scanning electron microscopy (SEM) and atomic force microscopy (AFM). Using this unique approach, we have recently shown that as grown Fe nanoparticles can be found in two different states with distinct magnetic properties at any size in the range from 8 nm to 20 nm [40]. Notably, half of the particles were found in a state with strikingly high magnetic anisotropy, resulting in stable magnetism at room temperature even in the smallest investigated nanoparticles, which could be of great interest for applications where nanomagnets with high magnetic anisotropy energy and high saturation magnetisation are required. However, the high anisotropy state was found to be metastable and to relax towards a state with the (much smaller) magnetic anisotropy of bulk Fe upon thermal excitation. Further, the experiments allowed us to exclude that the unusual high magnetic anisotropy energy is due to possible surface or shape contributions to the effective magnetic energy barriers, but instead the data indicate that the enhanced energy barriers originate from metastable, structural modifications in the volume of the nanoparticles.
While these data suggest that part of the controversy in the literature on the magnetic properties of Fe nanoparticles could be due to the presence of such metastable magnetic properties, important questions about the origin and nature of these observations remain open.
These questions concern particularly the presence of different crystallographic order within the investigated particle ensembles as well as thermal stability of the particle structure.
Moreover, it remained unclear whether similar magnetic behavior can be found in other 3d transition metal nanoparticle systems as well. Finally, quantitative estimates on the impact of structural defects on the magnetic properties are needed. In this work we address these issues and demonstrate that also as grown Co nanoparticles exhibit a similar size-independent coexistence of nanoparticles with distinct magnetic anisotropy energies, showing that the presence of metastable states with anomalous high magnetic barrier energies is a more general phenomenon and not solely restricted to Fe. However, in contrast to Fe, the state with enhanced magnetic anisotropy in Co can be promoted by thermal annealing and thus might be of great relevance for applications. In Ni nanoparticles, uniform SPM behaviour is found at room temperature with a magnetic blocking temperature of 100 K, confirming ferromagnetic order. To address the role of the particle structure, the magnetic data are correlated with characterization obtained by means of reflection high energy electron diffraction (RHEED) and high resolution scanning transmission electron microscopy (HR-STEM). Quantitative comparison of the experimental data with theoretical model calculations, allows us to rule out that the observed variability in the magnetic anisotropy energy in Fe and Co is due to particle interactions, surface contributions or shape and size-variations. Instead, our data and quantitative estimates suggest that lattice defects within the particles are at the origin of the reported magnetic diversity and of the observed metastability. Finally, we discuss additional implications of structural defects on the magnetism of nanoparticles.
III. EXPERIMENTAL DETAILS
The samples for the in situ X-PEEM experiments are prepared in three steps: (i) Au markers for particle identification in complementary microscopy investigations are lithographically prepared on Si(100) wafer substrates passivated with a native SiO x layer, see [41,42]. (ii) Upon introduction into the ultrahigh vacuum (UHV) surface preparation system (SPS) (base pressure ≤ 5×10 -10 mbar), the substrates are treated to remove adsorbates such as water which originate from exposure to the ambient atmosphere. In the case of the Fe nanoparticles the substrates were cleaned by mild sputtering with argon ions (kinetic energy: 500 eV, argon pressure: 5×10 -5 mbar, duration: 20 min), while for the Co and Ni nanoparticles, the substrates were thermally annealed in situ for 30 mins at about 525 K in the SPS. The SPS is directly attached to the PEEM instrument. For the RHEED studies, plain Si(001) wafers with the native SiO x surface layer are used. The wafers are annealed in the UHV RHEED system (base pressure: ≤ 5×10 -9 mbar) at a temperature of about 525 K until the pressure in the chamber recovers (after an initial increase) and the recorded RHEED pattern indicate a clean and flat SiO x surface. (iii) Finally, the nanoparticles are deposited onto the prepared substrates using an arc cluster ion source (ACIS), which is attached to the SPS [43][44][45]. For RHEED and X-PEEM investigations all samples are transferred under UHV conditions. This approach allows us to study the pristine magnetic properties of the nanoparticles.
In the ACIS, the nanoparticles are formed by condensation of metal vapor in a carrier gas consisting of a He/Ar mixture [43]. The metal vapor is generated by means of arc erosion from respective metal targets with a purity of 99.8%. An electrostatic quadrupole deflector is used to deflect a beam of mass-filtered nanoparticles onto the previously prepared Si substrates which are held either directly in the SPS or in a vacuum suitcase (base pressure ≤ 5×10 -9 mbar) attached to the SPS. A gold mesh placed in the nanoparticle beam path is used to measure the flux of the electrically charged particles during deposition and to control the final particle density on the substrates. For the X-PEEM investigations we choose a low particle density (a few nanoparticles per µm 2 ) to avoid magnetic dipolar interactions between the nanoparticles and to enable single particle resolution in the X-PEEM experiments (the particle-particle distance should be larger than 200 nm) [38]. For the RHEED experiments we choose a higher particle density of about 30 nanoparticles per µm 2 in order to obtain a sufficient signal-to-noise ratio in the diffraction data. At this coverage, agglomeration of the particles on the substrate is still avoided as confirmed by subsequent SEM images, so that also the RHEED data reflects the properties of an ensemble of isolated nanoparticles. Finally, samples with a particle density of a few tens of particles per µm 2 for ex situ HR-STEM investigations were deposited in the SPS. Commercially available 10 nm SiN membranes were used as substrates as-received (TEMwindows.com). During nanoparticle deposition the pressure temporarily increases to about 5×10 -6 mbar due to the presence of the Ar/He carrier gas, but recovers to the respective base pressure within a few minutes after deposition. For the present work the cluster source operation parameters as well as the mass-filter settings are held constant for all samples. This ensures similar growth, selection, and landing conditions in all experiments, with the kinetic energy of the particles prior to the impact on the substrate smaller than 0.1 eV/atom [44]. With these settings, the deposition takes place under so-called soft landing conditions, where no fragmentation of the particles or damage to the substrate is expected [46,47].
The crystallographic structure, the orientation of the deposited nanoparticles with respect to the substrate, as well as the thermal stability of the particles and the substrate, are determined by RHEED measurements [48,49]. The RHEED experiments are carried out with electrons with a kinetic energy of 35 keV at grazing incidence. This geometry enables one to investigate the quality of the substrates and the deposited nanoparticles simultaneously [48,50]. Data is recorded using a charge coupled device camera attached to the phosphor screen of the instrument. The temperature is set by means of resistive heating of a Si wafer piece under the sample. The sample temperature is read by a pyrometer (Maurer GmbH, Typ:
KTR 1075-1).
The in situ magnetic characterization of the samples is carried out using the PEEM (Elmitec GmbH) at the Surface/Interface: Microscopy (SIM) beamline of the Swiss Light Source [51].
The base pressure in the PEEM chamber is < 5×10 -9 mbar for the Fe nanoparticle experiments and < 5×10 -10 mbar for the Co and Ni nanoparticle investigations. For X-PEEM imaging the samples are illuminated with polarized mono-chromatic synchrotron radiation. The nanoparticles are visualized by means of elemental contrast maps, which are obtained by recording two images at a given sample site: first, a so-called "edge"-image is recorded with the photon energy resonantly tuned to the respective element-specific L 3 X-ray absorption edge. Then, a second so-called "pre-edge"-image is recorded with the photon energy tuned a few eV below the L 3 X-ray absorption edge energy. Pixel-wise division of the "edge"-and "pre-edge"-images finally yields the elemental contrast map, which reveals the nanoparticles as bright spots on the image, cf. Figs. 1(a) -1(c) [41]. The photon energies used in the resonant excitation of the L 3 X-ray photo-absorption edges for the "edge"-images are 708 eV for Fe, 778 eV for Co, and 852 eV for Ni. The photon energies used for recording the "pre- The magnetic properties of the particles are probed using the XMCD effect [52]. The latter gives rise to a magnetization-and helicity-dependent X-ray absorption cross section when tuning the photon energy resonantly to the L 3 absorption edge of the nanoparticles [52].
Magnetic contrast maps are obtained by pixelwise division of two X-PEEM images recorded with circularly polarized light of opposite helicity, C ± . In these maps, particles will exhibit a gray tone contrast ranging from black to white, depending on the projection of their magnetic moments onto the propagation vector of the X-ray beam, see Therefore, AFM is used to determine the height of the particles which serves as a measure of their size [53]. Finally, the morphology of the particles (exposed to air) was investigated by means of a HR-STEM (FEI Titan 3 equipped with Cs probe corrector) with high-angle annular dark-field (HAADF) imaging.
III. RESULTS
A. In situ magnetic characterization with single particle sensitivity
B. Structural characterization by means of RHEED
The crystallographic structure impacts on both the magneto-crystalline anisotropy and the shape of the particles and needs to be addressed experimentally, in particular, since the atomic lattice structure of nanoparticles can vary depending on the preparation technique and growth conditions [54]. For instance, in the literature Co and Ni nanoparticles were reported to exist in various structures ranging from hexagonal closed packed (hcp), primitive cubic, to face centered cubic (fcc) [27,55,56], while Fe has been stabilized in body centered cubic (bcc) and fcc at the nanoscale [48,57,58]. RHEED data taken for as grown Fe, Co, and Ni nanoparticles are shown in Figs. 2(a), 2(b), and 2(c) and exhibit characteristic Laue ring patterns from which the lattice structure can be deduced. For the Fe nanoparticles the Laue pattern is consistent with that of the bcc lattice [48]. The presence of diffraction rings confirms a nearly random crystallographic orientation of the particles upon deposition, which agrees with the random orientation of the magnetic moments of the FM particles, cf. Fig. 1(d). A texture is found on top of the (200) and the (110) rings. This indicates that the Fe nanoparticles preferentially rest with (100) and (110) facets parallel to the substrate surface [48]. This observation agrees with the expected shape according to a Wulff construction of monocrystalline bcc Fe nanoparticles given by a truncated dodecahedron exhibiting 6 (100) and 12
(110) facets [59,60], see also the inset in Fig. 6(a). The RHEED data of the Co and Ni nanoparticles show that they crystallize in the fcc structure, in accordance with other reports on gas phase grown systems in the present size range [56,61]. Again, we find a texture in the two lower index rings, (111) and (200), which suggests a preferred resting on (111) and (100) surface facets being consistent with the Wulff shape of mono-crystalline fcc nanoparticles given by truncated cuboctahedra with 8 (111) and 6 (100) surface facets [59] and being schematically depicted in the insets of Figs. 6(b) and 6(c). RHEED data taken at larger scattering angles further reveal an intact and flat, amorphous SiO x surface layer after the deposition of the nanoparticles (not shown). When increasing the sample temperature to 800 K, no change is observed for all samples, cf.
Figs. 2(d) -2(f), suggesting a high structural and chemical stability of the nanoparticles and of their interface with the substrate. Only when approaching the thermal decomposition temperature of the SiO x surface layer of about 1050 K, do the diffraction rings disappear and discrete diffraction spots occur (not shown) [63]. This observation indicates a chemical reaction of the particles with the exposed Si(001) substrate at higher temperatures. In turn, the absence of such diffraction spots in the RHEED pattern in the as grown samples, further confirms the non-destructive nature of the present soft-landing nanoparticle deposition, which not only avoids fragmentation of the nanoparticles, but also preserves the ultrathin SiO x surface layer (~1.5 nm thick) [46,47].
C. Ex situ morphology characterization of the nanoparticles
3(d) -3(f).
In what follows we refer always to the AFM data without correction of the oxide shell thickness. The HR-STEM data further confirm that the Co particles exhibit a larger variety of shapes as suggested by SEM and further reveal a number of twinned or polycrystalline particles, as can be seen for instance in Fig. 3(h).
D. Correlation of magnetic properties and particle size
E. Thermal stability of the magnetic properties
Due to their different lattices, bcc Fe and fcc Co nanoparticles are expected to show significantly distinct magnetic properties. In particular, the magneto-crystalline anisotropy energy of bulk fcc Co is known to result in magnetic energy barriers that are about two times smaller than that of bcc Fe. Therefore, it is a remarkable observation that both systems exhibit the same size-independent coexistence of FM and SPM nanoparticles at RT. To further understand this behaviour we studied the effect of thermal annealing on the magnetic properties of the Co nanoparticles as well as on those of the Ni nanoparticles. For the case of Fe we had recently demonstrated by means of in situ X-PEEM investigations that the FM state is metastable and can relax towards the SPM state [40]. In particular, it was found that all Fe nanoparticles lose their magnetic contrast when raising the sample temperature to 420 K. When cooling the sample to RT, the initial magnetic contrast is not recovered, indicating that all initially FM Fe nanoparticles undergo an irreversible transition to the SPM state [40]. observations reveal a remarkable difference between the Fe and the Co nanoparticles. A similar procedure has no effect for the magnetic state of the Ni nanoparticles, i.e., the entire ensemble remains SPM before, during, and after thermal annealing (not shown).
IV. DISCUSSION
A. Atomic level simulation of magneto-crystalline and effective surface anisotropy contributions for spherical nanoparticles
In order to evaluate the experimental findings we have performed advanced atomistic model calculations of the magnetic energy barriers E m of defect-free bcc Fe, fcc Co and fcc Ni nanoparticles, which also include the effect of non-collinear surface spin configurations due to the Néel-type surface anisotropy K s [68,69]. The simulations consider classical spins distributed over the lattice sites of spherical model particles with diameter D. The magnetic properties of the particles are described by an anisotropic Heisenberg Hamiltonian [69]. The effective energy landscapes of the many-spin particles are then evaluated using the Lagrangian multiplier method as described in Refs. [68,70,71]. The exchange constants are chosen to reproduce the bulk Curie temperatures using the classical spectral density method [72]. The magneto-crystalline anisotropy energy density is given by e MCA = The effect of the additional surface anisotropy due to spin non-collinearities at the surface is shown in Figs. 6(d) -6(f) as a function of the ratio K s /K 1 . We note that the high symmetry of spherical (as well as of Wulff-shaped) nanoparticles leads to a cancellation of the first order Néel surface anisotropy, and thus no surface effect is expected if non-collinear surface spin configurations are neglected [68]. The present calculations allow us to address this issue and to evaluate the actual contribution of K s to the magnetic energy barriers of spherical nanoparticles. Since K s is not a priori known and can significantly differ between various experimental reports, we have chosen to consider the range 0 < |K s /K 1 | < 800 in order to cover a large range of experimentally determined surface anisotropies deduced from thin film studies [74][75][76][77][78][79]. Note that for calculating K s /K 1 also K s is considered on a per atom basis. For comparison, ensemble measurements on nanoparticles have suggested |K s /K 1 | ~ 300 for Fe and |K s /K 1 | ~ 600 for Co [14,22]. Calculations are carried out for two particle sizes, 8 (red symbols) and 12 nm (black symbols). The data reveal that a sizeable enhancement of E m is possible for Fe and Co nanoparticles when |K s /K 1 | > 500, while for Ni no enhancement is found for |K s /K 1 | < 800. Despite the enhancement for Fe and Co, FM states are not induced by the considered surface contributions. We may note that for Co and Ni the surface anisotropy actually counteracts the magneto-crystalline anisotropy and thus initially reduces the magnetic energy barrier before it becomes the dominant contribution for higher values of K s . For Co we find further that the magnetic energy landscape changes significantly for K s > 250K 1 due to the increasing surface anisotropy, cf. grey shades in Fig. 6(e). As a result the magnetic easy axes reorient and for sufficient large values of K s , the easy axes will change to <100>.
B. Shape anisotropy contributions
The calculations show that the experimentally observed FM states in Fe and Co nanoparticles are not due to magneto-crystalline and surface anisotropy contributions. However, another contribution to the magnetic energy barrier can arise from deviations of the particle shape from the ideal spherical or highly symmetrical Wulff construction. The resulting dipolar stray field energy can cause a sizeable magnetic shape anisotropy, which can dominate over the other contributions [16]. Figs. 6(g) -6(i) show the calculated magnetic energy barriers for three selected particle sizes (8,12, and 20 nm) as a function of the aspect ratio of prolate Fe, Co, and Ni ellipsoids, respectively. The calculations were performed as described in Ref. [80] using bulk values for the saturation magnetization M s . The particle volume is kept constant for the different aspect ratios. In Figs. 6(g) -6(i) the shape-induced energy barrier of the nanoparticles has been added to the respective magneto-crystalline contributions shown in among the materials studied here, even smaller deviations from spherical geometry result in a significant enhancement of the magnetic energy barrier [81]. The situation is similar for Co, but the energy barriers are somewhat reduced when compared to Fe due to the smaller M s of fcc Co (1428.6 emu/cm 3 ) [81]. The saturation magnetization of fcc Ni is 510.3 emu/cm 3 being almost less than a third of that of Fe and Co [81]. Accordingly, for the Ni nanoparticles the shape-related contributions result in much smaller magnetic energy barriers. For comparison with the experimental data the SEM investigations described above give an upper limit for the aspect ratio of 1.15 for the selected particles. According to Figs. 6(g) and 6(h), shape anisotropy-induced FM states might be therefore possible for Fe and Co nanoparticles with sizes of 12 nm and above. For Ni the shape anisotropy results in SPM behaviour at all sizes, as experimentally observed. These conclusions remain the same even when adding the surface contributions shown in Figs. 6(d) -6(f).
C. Role of structural defects
The calculations show that surface and shape anisotropy contributions can indeed result in a sizeable enhancement of the magnetic energy barriers of nanoparticles. It also follows that the sensitivity of the shape anisotropy to relatively small variations of the particle morphology can lead to a sizeable diversity of magnetic energy barriers even in mono-disperse nanoparticle samples. Taking into account the large shape distribution observed in many experiments, these effects will certainly contribute to the dispersion of the magnetic anisotropy energies reported in the literature. However, the calculations provide no explanation for the presently observed FM states in Co and Fe nanoparticles with size below 12 nm. In addition, for the Fe nanoparticles, recent experiments provide further evidence that the FM states even for the larger nanoparticles cannot be explained by a combination of shape, surface and magneto-crystalline anisotropy contributions [39,82]. Similarly, the metastability of the magnetic energy barriers of the Fe and Co nanoparticles hints at an additional, sizeable and variable contribution to the total magnetic energy barriers. In what follows, we argue that the origin of such phenomena lies in the presence of lattice defects such as dislocations or stacking faults. Such defects may arise from particle growth kinetics which can result in complex structures [83]. Experimental evidence for such defects is provided by the HR-STEM investigations of the Co nanoparticles in the present work as well as reported in the literature [56,84], suggesting that such structural defects are abundant in 3d transition metal nanoparticles. Lattice defects may thus contribute to the magnetic properties in many experiments, although their effects are rarely discussed [32,84,85]. In the following, states are related to other crystal structures such as fcc-type Fe nanoparticles [86]. Rather, the data yield that FM and SPM nanoparticles are structurally very similar. This is further supported by the observation that the RHEED pattern exhibit also no noticeable changes upon thermal annealing up to 800 K, while the FM nanoparticles clearly undergo an irreversible transition to SPM behaviour upon annealing to 470 K [40]. These findings show that the transition from FM to SPM is not related with a structural phase transition, but that the distinct magnetic properties of SPM and FM nanoparticles are associated with smaller, metastable modifications of the bcc crystal lattice such as structural defects, which are typically dislocations, twinning and stacking faults in metallic nanoparticles [83,[87][88][89]. As we show below each type of these defects is expected to have a significant and specific impact on the magnetic properties of nanoparticles.
Dislocations are common defects in bulk bcc Fe and known to cause sizeable inhomogeneous strain fields around the dislocation core. The strain fields give rise to local magneto-elastic anisotropy energy contributions and are a source of pinning sites to magnetisation reversal [90]. Continuum mechanical calculations for bcc Fe predict for instance for a single edge (screw) dislocation within a {112} slip plane a uniaxial magnetic anisotropy with energy barriers of about 43 (10) µeV/atom at a distance of 0.5 nm from the dislocation core [91].
These values are clearly much larger than the magnetic energy barrier of 0.8 µeV/atom due to the magneto-crystalline anisotropy energy of the Fe bcc lattice, and thus a dislocation locally increases the magnetic energy barriers in Fe. In the bulk, strain fields associated with dislocations extend usually over a distance of a few 100 nm. Thus, when present in a nanoparticle with a size between 8 and 20 nm, the strain field and the associated magnetoelastic anisotropy of a dislocation will likely affect almost the entire volume of the nanoparticle. Accordingly, a single screw or edge dislocation in a Fe nanoparticle is expected to provide a significant additional contribution to the effective magnetic energy barrier, which may eventually give rise to the energy barriers required for the observed FM states.
Dislocations can also explain the metastability of the FM states in Fe nanoparticles. The high mobility of dislocations as known from bulk Fe allow them to be ejected from the finite volume of nanoparticles, e.g., upon thermal excitations, as demonstrated in molecular dynamics simulations [92]. The removal of the dislocation simultaneously lowers the elastic energy stored in the lattice of the particle as well as the magneto-elastic anisotropy contribution and thus could account for a transition from FM to more bulk-like SPM behaviour.
Much less is known so far about the effect of stacking faults and twinning on the magnetic properties of Fe. Theoretical work for bcc Fe predicts an influence of the magnetism on the stacking fault energy, which suggests that there is also an effect of the defects on the magnetic properties [93]. Stacking fault energies in bcc Fe are comparably high and therefore such defects are likely metastable in nanoparticles as for dislocations. In fact, stacking faults or twinning have been observed thus far only in bulk-like systems under high mechanical stresses, but not in bcc Fe nanoparticles [16,60,67,94]. Similarly, there is currently no evidence for dislocations or other defects in bcc Fe nanoparticles, including the present HR-STEM results. Based on our experimental observations, we assign the lack of direct experimental evidence for the existence of lattice defects in Fe nanoparticles to their metastability. In particular, the FM states in Fe nanoparticles are only stabilized upon deposition onto substrates with a sufficiently low free surface energy as the present passivated
Si wafers and they can spontaneously relax over time even at room temperature [40,82]. Thus, direct observation of the associated metastable defects in bcc Fe nanoparticles poses a challenging task. A successful route could be to embed the nanoparticles into suitable matrix materials to stabilize the FM states, prevent oxidation, and still allow for detailed transmission electron microscopy investigations.
For Co nanoparticles, the RHEED data also show the presence of only one crystallographic structure, the fcc lattice. This is in agreement with other reports about gas phase grown Co nanoparticles in the present size range (8 to 20 nm) [56]. For edge (screw) dislocations within a {111} slip plane in bulk Co one can estimate uniaxial anisotropy energies of 117 (68) µeV/atom at a distance of 0.5 nm from the dislocation core, which can be compared to the magnetic energy barrier of 0.3 µeV/atom given by the magneto-crystalline anisotropy. Thus, similar to Fe, these defects can significantly contribute to the total magnetic energy barrier of a nanoparticle. In contrast to bcc Fe, stacking faults and twinning are frequently observed in fcc Co nanoparticles [56,84]. Also, the present HR-STEM data show grains or twin boundaries in a number of particles, stable even upon ambient air exposure, cf. zones "A" and "B" in Fig. 3(h). Stacking faults in fcc materials can yield local hcp stacking. Based on the properties of bulk Co, hcp stacking could give rise to uniaxial anisotropies along the local caxis with an energy barrier of 35 µeV/atom, which is also much larger when compared to the magneto-crystalline anisotropy of fcc Co. Theory shows further that a single stacking fault in Co has a long range effect on the electronic and magnetic properties of the adjacent atomic layers [95]. Thus stacking faults can also significantly contribute to the magnetic energy barriers in Co nanoparticles. Moreover, they are to first order not related with strain and the formation of local hcp stacking may even lower the cohesion energy for Co nanoparticles [56]. Stacking faults might therefore be more stable when compared to dislocations and might be even promoted by thermal annealing. If so, a growing proportion of hcp stacking in individual particles could be related with the increasing number of FM Co nanoparticles observed upon thermal annealing as shown in Fig. 5. A respective two-phase mixture of fcc and hcp stacking in individual cobalt nanoparticles was indeed reported in Ref. [96]. Their thermal behaviour may further indicate that the FM properties in the Co nanoparticles are not due to metastable dislocations, which would be ejected from the particle as discussed for bcc
Fe. In fact, in metallic fcc nanoparticles combinations of different defects have been observed.
For instance, in multiply twinned fcc platinum (Pt) nanoparticles a complex combination of stacking faults, screw and edge dislocations was reported [88]. In these cases it is assumed that the dislocations reduce strain which results from a geometrical mismatch of the tetrahedral building blocks and thus stabilize the total structure. The effective magnetic energy barriers in fcc Co nanoparticles would then be the result of a complex competition between different anisotropy contributions.
Also the Ni nanoparticles exhibit only fcc lattice in RHEED. Estimates of the magneto-elastic anisotropy energy of edge (screw) dislocations along the <111> direction yields 95 (55) µeV/atom at a distance of 0.5 nm from the core. This value is almost as high as for fcc Co.
Since we find no FM states at RT, our data suggest that such dislocations are not stable in these particles. For fcc Ni nanoparticles, some authors have observed stacking faults or multiple twinning [32,85]. If hcp stacking is created in Ni, the present literature suggests that not only the magnetic energy barriers might be modified, but also the magnetic order could be locally affected. For hcp Ni nanoparticles, antiferromagnetic or ferromagnetic order or paramagnetic properties have been reported [26,27,30,97]. For the present Ni nanoparticles we observe FM states at 100 K. Thus these particles possess at least partial ferromagnetic order. The fact that we observe no FM states at RT suggests that stacking faults or any possible combination of defects in these particles are either not present in the investigated Ni nanoparticles or they do not yield sufficiently large magnetic energy barriers for stable room temperature magnetism.
Thus, besides magneto-crystalline, surface and shape anisotropies, lattice defects can significantly contribute to the magnetic properties of nanoparticles. The present data as well as a number of reports in the literature suggest that lattice defects in 3d transition metal nanoparticles are abundant and thus important for the understanding of their magnetic properties. Lattice defects alter not only the magnetic anisotropy energy or exchange interaction as discussed above, but are also known to affect the local magnetic structure. For instance, in thin films it was shown that the strain fields associated with screw or edge dislocations give rise to local non-collinear spin arrangements such as vortex-or lobe-like structures, which extend up to a few nanometres around the dislocation core [98]. While such a perturbation presents only a local phenomenon in the magnetic structure of a thin film, a similar dislocation would likely modify the entire magnetic structure of a nanoparticle in the present size range. If so, defects could result in non-collinear spin structures at dimensions far below the critical sizes for which the formation of magnetic single domain states is so far expected based on the dipolar interactions [5]. This would lead to a significantly different magnetic behaviour with particular impact on the analysis of magnetization curves. In case of screw dislocations, the broken inversion symmetry of the lattice might in addition give rise to magnetic chirality effects, which could lead to novel and thus far unexplored phenomena in magnetic nanoparticles [99].
V. CONCLUSIONS
In summary, we have studied the magnetic properties of ensembles of size-and shapeselected bcc Fe, fcc Co and fcc Ni nanoparticles with single particle sensitivity.
Substrates with Au-Markers for Complementary Microscopy
To enable identification of the very same nanoparticles using complementary microscopy, gold markers are prepared on Si(001) wafers by means of electron beam lithography. The appearance of the markers (here: "C1") and the nanoparticles in the different microscopes are shown in Fig. S-1.
X-PEEM Experimental Geometry
In
Simulated Magnetic Contrast Distribution for Different Scenarios
In order to assess the experimentally observed flat distribution of the normalized asymmetries of the FM Fe and Co nanoparticles, we have simulated XMCD asymmetry distributions resulting from three different scenarios for the orientation distribution of the magnetic moments of the deposited nanoparticles. In these simulations, the expected XMCD asymmetry is calculated for each nanoparticle according to the experimental geometry and the orientation of its magnetic moment given by a unit vector in spherical coordinates ) , ( m m φ θ We have set K 2 = 0 erg/cm 3 in the calculations for bcc Fe and fcc Co, since they are small compared to the respective values of K 1 .
Parameters Used for the Calculations of the Magnetic Energy Barriers
Magnetic Energy Barriers Required for a Magnetically Blocked State at Room
Temperature
The temperature-depending switching rate s τ of a magnetic nanoparticle can be expressed by an Arrhenius law:
where 0 v is the attempt frequency, m E is the magnetic energy barrier and B k is the Boltzmann constant. A magnetically blocked state is observed when the switching rate s τ becomes equal or smaller than the measurement time x τ . To estimate m E , the attempt frequency has to be determined.
The attempt frequency in the case of cubic anisotropy with Here, γ is the gyromagnetic ratio of an isolated electron being 1.760×10 11 rad×s -1 ×T -1 and V is the volume of the particle. To evaluate these equations, we consider further the temperature-dependence of the saturation magnetization:
Fig. S-1 of the Supplementary Material (SM)
edge"-images are 703 eV for Fe, 773 eV for Co, and 847 eV for Ni. A typical measurement sequence consists of averaging 10 individual frames with 1 s integration time each per photon energy from which a sequence of 10 elemental contrast maps is obtained. This sequence is then corrected for possible sample drift and finally averaged to yield elemental contrast images such as shown in Figs. 1(a) -1(c).
Fig. S-2 of the SM [42]. The time resolution of the present X-PEEM investigations is limited to τ x = 20 s, corresponding to a single pair of averaged C + and Cimages with a total acquisition time of 10 s each. For the data in Figs. 1(d) -1(f) a sequence of 20 magnetic contrast maps was acquired, drift-corrected and averaged. The spatial resolution of the X-PEEM experiments is limited to 50 -100 nm, thus largely exceeding the size of the particles. The nanoparticle size and morphology is therefore investigated by means of complementary SEM (Zeiss Supra VP55) and AFM (Veeco di 3100) after the in situ magnetic characterization. During transfer to the SEM the samples are exposed to ambient air. The lithographic marker structures allow one to identify the very same particles in X-PEEM, SEM and AFM, see Fig. S-1 of the SM [42]. The SEM is used to investigate the lateral morphology of the individual particles and to exclude close-lying particles, agglomerates and irregular, dendritic structures from the analysis. Since the spatial resolution is limited to 2 -3 nm, an accurate size determination with SEM is not possible.
X-PEEM elemental and magnetic contrast maps of as grown Fe, Co, and Ni nanoparticle samples recorded at room temperature (RT) are shown in Fig. 1. Bright spots in the elemental contrast maps in Figs. 1(a) for Fe, 1(b) for Co, and 1(c) for Ni indicate individual nanoparticles within the extended samples. As a guide to the eye, a number of particles are highlighted in Fig. 1 with white solid and dashed circles, respectively. Magnetic contrast maps of the same sample areas in Figs. 1(d) and 1(e) reveal that about half of the Fe and Co nanoparticles exhibit magnetic contrast ranging from black to white, for instance the particles highlighted with the solid white circles. The actual magnetic contrast of an individual nanoparticle depends on the orientation of its magnetic moment relative to the propagation vector of the impinging X-rays (see II.C). The presence of magnetic contrast further indicates a magnetically blocked state, i.e. a stable magnetic moment orientation over time periods longer than the measurement acquisition time of ~20 s in the present experiments [40]. Such particles are referred to as FM in this work. A detailed analysis of the magnetic contrast intensity distribution of a large number of FM nanoparticles reveals a random orientation of their magnetic moments reflecting the stochastic nature of the nanoparticle deposition process [for details of the analysis cf. the SM [42], in particular Figs. S-3 and S-4]. The lack of magnetic contrast in the other half of the nanoparticles can be assigned to thermally excited magnetic moment reversals in these nanoparticles at a rate higher than the measurement acquisition time, i.e. to superparamagnetic behavior, see also Ref. [40]. Such particles are referred to as SPM in this work. A number of SPM particles are highlighted with dashed white circles in Fig. 1. The absence of magnetic contrast in the Ni nanoparticle sample, Fig. 1(f), shows that all Ni nanoparticles are SPM at RT, with magnetic contrast occurring at about 100 K (not shown).
FIG. 1 .
1(a)-(c) Elemental contrast maps of as grown (a) Fe, (b) Co, and (c) Ni nanoparticles at RT. (d)-(f) Respective magnetic contrast maps. Examples of particles in superparamagnetic and magnetically blocked states are highlighted with dashed and solid circles, respectively.
13 FIG. 2 .
132(Color online) RHEED patterns for as grown (a) Fe, (b) Co, and (c) Ni nanoparticles obtained at RT. (d) -(f) corresponding diffraction patterns recorded at 800 K. Dashed arcs are guides to the eye and highlight the detected diffraction rings. The corresponding Miller indices are indicated by the small arrows[48,49,62].
Ex situ scanning electron microscopy (SEM) images of the samples are presented inFigs.3(a) -3(c). All samples reveal a variety of shapes which range from highly symmetrical and compact particles to dendritic structures or agglomerates of particles. This variety is assigned to growth kinetics in the cluster source used here for the deposition of gas phase grown nanoparticles. The growth depends on a number of parameters, such as temperature, nucleation density, and cooling rates[47,64]. With the chosen cluster source and deposition settings, which are identical for all reported experiments here, we observe a relatively high fraction (about 2/3) of compact, i.e., nearly spherical or cubic, nanoparticles of Fe and Ni [cf.the insets in Figs. 3(a) -3(c)]. In contrast, the Co samples show a significantly lower fraction of regular particles (about 1/3). Using substrates with gold marker structures allows us to perform detailed SEM investigations on the very same nanoparticles, which were previously magnetically characterized by means of X-PEEM, see SM [42] in particular Fig. S-1. Specifically, the SEM data are used to select only nearly spherical or cubic nanoparticles for the analysis of their magnetic properties for all three investigated systems Fe, Co, and Ni. Typical examples are shown in the insets to Figs. 3(a) -3(c). Particles with more complex morphology are not further considered in the present work. This choice is made to facilitate the comparison with the model calculations discussed below. FIG. 3. (a)-(c) SEM micrographs of the (a) Fe, (b) Co, and (c) Ni nanoparticle samples recorded after the PEEM experiments. The insets show magnified images of representative particles with a regular shape selected for further analysis. (d)-(f) AFM height distribution obtained from the Fe (d), Co (e), and Ni (c) particles selected based on the SEM data. (g)-(i) HAADF HR-STEM images of Fe (g), Co (h), and Ni (i) nanoparticles. The arrows indicate the thickness of the oxide shell which has formed due to ambient air exposure. The black dotted lines in panel (h) indicate two zones "A" and "B", respectively, with different crystallographic orientations. The size of the selected nanoparticles is obtained by measuring their height using atomic force microscopy (AFM) [65]. Figs. 3(d) -3(f) show the actual size distributions for the considered Fe, Co, and Ni nanoparticles as obtained from AFM analysis. The sizes are nearly the same for all three samples and range from about 8 to about 20 nm as expected from the identical deposition conditions. The mean values are about 12 nm. However, all particle heights are affected by the formation of an oxide shell, which occurs upon ambient air exposure after the X-PEEM investigations and during transfer to the SEM and AFM instruments. The oxide shell thickness is characterized by means of HR-STEM investigations carried out on air exposed reference samples. The HR-STEM images confirm the presence of an oxide shell formed in all samples, cf. Figs. 3(g) -3(i). The data suggests that Ni nanoparticles possess the thinnest oxide shell with a thickness of about 1 nm, while Co and Fe nanoparticles develop an oxide shell of 2 to 3 nm in agreement with previous studies [60,66,67]. The actual particle sizes in the ultrahigh vacuum during the X-PEEM and RHEED investigations might therefore be smaller by about 1 -2 nm when compared to the ex situ AFM results presented in Figs.
Finally
, our unique complementary microscopy approach is used to correlate magnetism and size of a large number of individual, shape-selected nanoparticles. This is achieved by analyzing the size distribution of FM and SPM nanoparticles, separately. The results of this analysis are shown in Figs. 4(a) -4(c) for all three systems. Surprisingly, the data reveal the same signature for Fe and Co nanoparticles, i.e., a nearly equal coexistence of SPM and FM nanoparticles, irrespective of size. In contrast, Ni nanoparticles are only found in the SPM state at RT. [Note that summing up the FM and SPM nanoparticle contributions in Fig. 4 yields directly the magnetically unresolved size distributions obtained by AFM as shown in Figs. 3(d) -3(f).] FIG. 4. (Color online) Relative fraction of SPM (red) and FM (blue) nanoparticles as a function of the particle size for as grown (a) Fe, (b) Co, and (c) Ni nanoparticles at RT.
FIG. 5 .
5Elemental and magnetic contrast maps of as grown Co nanoparticles (a) and (d), at RT, (b) and (e) at 470 K, (c) and (f) after cooling back to RT. Three particles labelled A, B, and C are highlighted for discussion. The results of a similar study for Co are shown in Fig. 5. We find that only a relatively small number of the initially FM particles lose their magnetic contrast when rising the temperature to 470 K, for example, particle A in Figs. 5(d) and 5(e). Instead most initially FM particles remain in a magnetically blocked state. Moreover, some particles, initially without magnetic contrast at RT, surprisingly display magnetic contrast at 470 K, e.g., nanoparticle B in Figs. 5(d) -5(f). These particles remain in the FM state when cooling back to RT. In addition, a number of nanoparticles become FM at RT only after the thermal cycle, such as nanoparticle C in Figs. 5(d) -5(f). This behaviour suggests thermally induced irreversible transitions from SPM to FM states. Finally, some particles of type A do not change their properties and exhibit a reversible transition from FM behaviour at RT to SPM behaviour at 470 K. Hence, these
1 2 ) + K 2 α 1 2 α 2 2 α 3 2 , where α i are the direction cosines and K 1,2 the tabulated first and second order anisotropy constants of the respective bulk material at room temperature. For ease of discussion we use the constants on a per atom basis, which are for bcc Fe: K 1 = 3.8 µeV/atom, K 2 = 0.008 µeV/atom, for fcc Co: K 1 = -3.8 µeV/atom, K 2 = -0.77 µeV/atom, and for fcc Ni:K 1 = -0.28 µeV/atom and K 2 = -0.12 µeV/atom. All other parameters are listed in the table given in the SM [42]. Before we discuss the effect of the surface anisotropy, we consider first the magnetic energy barriers due to the magneto-crystalline anisotropy. For the discussion we refer to the schematic Wulff-shaped particles as shown in the insets of Figs. 6(a) -6(c), which provide a better visualization of the crystallographic directions when compared to spheres. With the given values of K 1,2 we find that for the bcc Fe (fcc Co and Ni) particles the easy axes are along <100> (<111>) and the hard axes are along <111> (<100>). In Figs. 6(a) -6(c) the easy (hard) axes are indicated by black (red) arrows next to the schematics. The lowest energy path for the magnetic moments to switch from one easy axis to another is along the semi-hard axis, which is along the <110> direction for all systems. The corresponding path for the magnetization is schematically indicated by the dotted arcs in the insets of Figs. 6(a) -6(c). The respective magnetic energy barriers amount to 0.8 µeV/atom for bcc Fe, to 0.3 µeV/atom for fcc Co, and to 0.03 µeV/atom for fcc Ni. Finally, the total magnetic energy barriers E m as a function of D are shown by the black lines in Figs. 6(a) -6(c). For comparison with the present experiments, the dashed horizontal lines in all panels of Fig. 6 indicate the energy barriers above which a FM state is observed in the present experiments (for the calculation see the SM [42] and Ref. [73]). Values of E m below that threshold result in SPM behavior. The data in Figs. 6(a) -6(c) clearly show that for all investigated materials and sizes the magnetocrystalline anisotropy alone would result in SPM states.
FIG. 6 .
6(Color online) (a)-(c): Size-dependent magnetic energy barriers as given by the magneto-crystalline anisotropy energy of bulk Fe (bcc), Co (fcc), and Ni (fcc). The insets display nanoparticle shapes as predicted by the Wulff theorem for mono-crystalline bcc Fe and fcc Co and Ni, respectively. The red (black) arrows indicate the magnetic hard (easy) axes of the particles as given by the magneto-crystalline anisotropy energy. The dotted arcs in the insets indicate the low energy barrier pathway to flip the magnetization coherently from one easy axis to another. (d)-(f) Magnetic energy barriers including surface anisotropy contributions as discussed in the text. The grey shaded areas in case of Co indicate regions where a spin reorientation occurs due to dominant surface anisotropy. (g)-(i) Magnetic energy barriers due to magneto-crystalline contributions and shape anisotropy as a function of aspect ratio. The dashed lines indicate the magnetic energy barriers above which the relaxation time is larger than 20 s. Particles with equal or higher E m would appear as FM in our experiments.
Figs. 6
6(a) -6(c). The shape anisotropy scales with the square of the magnetization and the corresponding energy barrier strongly increases with particle size as well as with the value of M s of the respective material. In case of bcc Fe, which has the highest M s (1702.6 emu/cm3 )
we will consider the most common lattice defects and demonstrate that such defects explain not only the metastable magnetism and the exceptionally high magnetic anisotropy energies found in Fe and Co nanoparticles, but can further result in unexpected magnetic order and spin structures in nanoparticles. Finally, some of these defects may even give rise to novel phenomena such as magnetic chirality effects in magnetic nanoparticles, which might be of interest for future applications.Structural characterization of the as grown Fe nanoparticles by means of RHEED [see Fig. 2(a)] indicates solely bcc lattice symmetry, while X-PEEM reveals an almost equal amount of SPM and FM nanoparticles [see Figs. 1(d) and 4(a)]. Thus, it can be ruled out that the FM
X-PEEM the samples are illuminated with polarized, monochromatic synchrotron radiation with a propagation vector k impinging at an angle of incidence = k θ 16° with respect to the sample surface as shown in Fig. S-2. A raw X-PEEM image is shown in Fig. S-1(a). Using circularly polarized synchrotron radiation the X-ray magnetic circular dichroism (XMCD) effect leads to a magnetization-dependent intensity of a nanoparticle with a is the isotropic (i. e. non-magnetic) intensity, γ is a material and photon energy dependent constant, and ± C denotes circular right-and left-handed polarization.
Figure S- 1 .
1Si wafer substrates with Au-marker structure and Fe nanoparticles in (a) raw X-PEEM image, (b) elemental contrast image obtained at the Fe L 3 edge, (c) SEM image, and (d) AFM image. To illustrate the particle identification, three nanoparticles are highlighted with circles.
Figure S- 2 .
2Experimental geometry in X-PEEM. Experimentally obtained histograms of the respective XMCD asymmetry for the different systems for the Fe, Co, and Ni nanoparticle samples in their as grown state are shown in Fig. S-3. A negative (positive) value of the asymmetry A corresponds to a dark (bright) grey contrast in the respective magnetic contrast maps, see Figs. and can be assigned to SPM nanoparticles, while the flat part of the distribution is due to magnetic blocked FM nanoparticles. Figs. S-3(a) and S-3(b) show that Fe and Co nanoparticles have almost the same contributions of SPM and FM fractions and similar form of histograms in their as-deposited state, while the Ni nanoparticles in Fig. S-3(c) exhibit only the SPM peak.
Figure S- 3 .
3Experimentally obtained histograms of XMCD asymmetry of (a) Fe, (b) Co, and (c) Ni nanoparticles in the as grown state.
, cf. Fig. S-2. The simulations show that an ensemble of nanoparticles with fully random orientation of their magnetic moments results in a flat distribution of the XMCD asymmetry, cf. Fig. S-4(a). An in-plane random orientation of the magnetic moments results in the distribution shown in Fig. S-4(b). Finally, Fig. S-4(c) shows the histogram for a nanoparticle ensemble with preferred outofplane magnetization, where we allowed a standard deviation of 45° from full out of plane orientation ( respectively) of the moment to mimic a sample with deviations from perfect out-of-plane alignment of the magnetic moments. Figs. S-4(a) -S-4(c) show that the three scenarios result in a distinct distribution of the XMCD asymmetry. It follows that only a fully random orientation of magnetic moments yields a flat histogram compatible with the experimental data for the FM portion of the Fe and Co nanoparticles as shown in Figs. S-3(a) and S-3(b).
Figure S- 4 .
4Simulated XMCD asymmetry distribution for (a) fully, (b) in-plane, and (c) outofplane randomly orientated magnetic moments according to the experimental geometry shown in Fig. S-2. Each simulation was performed for 10 6 particles.
Where c T is the Curie temperature and 0 M the saturation magnetization at T = 0 K. A lower limit for the attempt frequency is obtained using the parameters for bulk bcc Fe, fcc Co and fcc Ni considering spherical nanoparticles with a diameter of D = 20 nm. The damping coefficient is set to 1 = α . At room temperature we obtain 0 v = 6.3×10 9 s -1 for bcc Fe,
Using a complementary microscopy approach we have directly correlated size and magnetism of a large number of individual nanoparticles. For bcc Fe and fcc Co our results clearly demonstrate that non-interacting and chemically pure nanoparticles of the same size and shape can have significantly distinct magnetic properties. Specifically, we find that a large portion of Fe and Co nanoparticles are found in a state with strikingly enhanced magnetic energy barriers manifested in magnetically blocked states at room temperature at sizes as small as 8 nm. While this unique state is irreversibly lost after thermal annealing in case of Fe, it can be promoted in the case of Co, which is promising for applications where high saturation magnetization and high magnetic anisotropy at small sizes are required. No such state is observed for fcc Ni nanoparticles at room temperature, but magnetic blocking is found at 100 K. Atomic level simulations show that effective surface and shape anisotropy contributions can lead to a sizeable enhancement of the magnetic energy barriers in all nanoparticles, but that these contributions are not sufficient to account for the observed room temperature blocking in the smallest Fe and Co nanoparticles under investigation. Similarly, their temperature-dependent behaviour cannot be explained by surface or shape effects. Based on these findings and complementary structural data we assign the strongly enhanced magnetic energy barriers and the metastable magnetic properties of the nanoparticles to lattice defects such as dislocations and stacking faults. Another likely consequence of the presence of such defects might be the occurrence of unexpected magnetic order, altered spin structures or novel properties such as magnetic chirality effects. To reveal these phenomena and to achieve an improved understanding of the magnetic properties of nanoparticles, increased experimental and theoretical efforts are urgently needed. Finally, our work underlines the importance of complementary single particle investigations for improving the understanding and control over magnetic phenomena at the nanoscale.
ACKNOWLEDGEMENTSWe thank A. Weber, R. Schelldorfer and J. Krbanjevic (Paul Scherrer Institut) for technical assistance. This work was supported by the Swiss Nanoscience Institute, University of Basel.
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Departament de Física de la Matèria Condensada and Institut de Nanociència i Nanotecnologia (IN2UB). 8028Barcelona, SpainUniversitat de BarcelonaDepartament de Física de la Matèria Condensada and Institut de Nanociència i Nanotecnologia (IN2UB), Universitat de Barcelona, 08028 Barcelona, Spain
. Hauz Khas. EMAT, University of Antwerp, 2020 Antwerpen, Belgium 6 Department of Physics, Indian Institute of Technology DelhiEMAT, University of Antwerp, 2020 Antwerpen, Belgium 6 Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi- 110016, India
Paul Scherrer Swissfel, Institut, 5232 Villigen PSI, Switzerland *Corresponding author. SwissFEL, Paul Scherrer Institut, 5232 Villigen PSI, Switzerland *Corresponding author, e-mail: [email protected]
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|
[] |
[
"Bias-Reduction in Variational Regularization",
"Bias-Reduction in Variational Regularization"
] |
[
"Eva-Maria Brinkmann ",
"Martin Burger ",
"Julian Rasch ",
"Camille Sutour "
] |
[] |
[] |
The aim of this paper is to introduce and study a two-step debiasing method for variational regularization. After solving the standard variational problem, the key idea is to add a consecutive debiasing step minimizing the data fidelity on an appropriate set, the so-called model manifold. The latter is defined by Bregman distances or infimal convolutions thereof, using the (uniquely defined) subgradient appearing in the optimality condition of the variational method. For particular settings, such as anisotropic 1 and TV-type regularization, previously used debiasing techniques are shown to be special cases. The proposed approach is however easily applicable to a wider range of regularizations. The two-step debiasing is shown to be well-defined and to optimally reduce bias in a certain setting.In addition to visual and PSNR-based evaluations, different notions of bias and variance decompositions are investigated in numerical studies. The improvements offered by the proposed scheme are demonstrated and its performance is shown to be comparable to optimal results obtained with Bregman iterations.where ∂ u H is the derivative of H with respect to the first argument. Now we proceed to a second step, where we only keep the subgradient p α and minimizêObviously, this problem is only of interest if there is no one-to-one relation between subgradients and primal values u, otherwise we always obtainû α = u α . The most interesting case with respect to applications is the one of J being absolutely one-homogeneous, i.e. J(λu) = |λ|J(u) for all λ ∈ R, where the subdifferential can be multivalued at least at u = 0.The debiasing step can be reformulated in an equivalent way aswith the (generalized) Bregman distance given by. We remark that for absolutely one-homogeneous J this simplifies to. The reformulation in terms of a Bregman distance indicates a first connection to Bregman iterations, which we make more precise in the sequel of the paper.Summing up, we examine the following twostep method: 1) Compute the (biased) solution u α of (1.1) with optimality condition (1.2), 2) Compute the (debiased) solutionû α as the minimizer of (1.3) or equivalently (1.4).In order to relate further to the previous approaches of debiasing 1 -minimizers given only the support and not the sign, as well as the approach with linear model subspaces, we consider another debiasing approach being blind against the sign. The natural generalization in the case of an absolutely one-homogeneous functional J is to replace the second step bydenotes the infimal convolution between the Bregman distances D pα J (·, u α ) and D −pα J (·, −u α ), evaluated at u ∈ X .The infimal convolution of two functionals F and G on a Banach space X is defined asThe dual product can be expressed as a symmetric Bregman distanceHence all three terms are nonnegative and we find in particular Au α = Aũ α , w α =w α and thus p α =p α .ICB pα J (cu λ , u α ) ≤ D pα J (cu λ , u α ) = J(cu λ ) − p α , cu λ = 0.Consequently,û α is also a solution of (3.3).Theorem 4.12. The setProof. The map u → D p J (u, v) is convex and nonnegative, henceis convex as a sublevel set of a convex functional. Moreover, for each c ≥ 0 we haveThus we deduceThe assertion follows by the nonnegativity of the maps u → D p J (u, v) and u → D −p J (u, −v). Note that for p = 0 the subset is proper, since e.g.
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10.1007/s10851-017-0747-z
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"https://arxiv.org/pdf/1606.05113v4.pdf"
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1606.05113
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08298854a3cc6eaca0d1a651583bfc1daa0228d0
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Bias-Reduction in Variational Regularization
June 23, 2017
Eva-Maria Brinkmann
Martin Burger
Julian Rasch
Camille Sutour
Bias-Reduction in Variational Regularization
June 23, 2017
The aim of this paper is to introduce and study a two-step debiasing method for variational regularization. After solving the standard variational problem, the key idea is to add a consecutive debiasing step minimizing the data fidelity on an appropriate set, the so-called model manifold. The latter is defined by Bregman distances or infimal convolutions thereof, using the (uniquely defined) subgradient appearing in the optimality condition of the variational method. For particular settings, such as anisotropic 1 and TV-type regularization, previously used debiasing techniques are shown to be special cases. The proposed approach is however easily applicable to a wider range of regularizations. The two-step debiasing is shown to be well-defined and to optimally reduce bias in a certain setting.In addition to visual and PSNR-based evaluations, different notions of bias and variance decompositions are investigated in numerical studies. The improvements offered by the proposed scheme are demonstrated and its performance is shown to be comparable to optimal results obtained with Bregman iterations.where ∂ u H is the derivative of H with respect to the first argument. Now we proceed to a second step, where we only keep the subgradient p α and minimizêObviously, this problem is only of interest if there is no one-to-one relation between subgradients and primal values u, otherwise we always obtainû α = u α . The most interesting case with respect to applications is the one of J being absolutely one-homogeneous, i.e. J(λu) = |λ|J(u) for all λ ∈ R, where the subdifferential can be multivalued at least at u = 0.The debiasing step can be reformulated in an equivalent way aswith the (generalized) Bregman distance given by. We remark that for absolutely one-homogeneous J this simplifies to. The reformulation in terms of a Bregman distance indicates a first connection to Bregman iterations, which we make more precise in the sequel of the paper.Summing up, we examine the following twostep method: 1) Compute the (biased) solution u α of (1.1) with optimality condition (1.2), 2) Compute the (debiased) solutionû α as the minimizer of (1.3) or equivalently (1.4).In order to relate further to the previous approaches of debiasing 1 -minimizers given only the support and not the sign, as well as the approach with linear model subspaces, we consider another debiasing approach being blind against the sign. The natural generalization in the case of an absolutely one-homogeneous functional J is to replace the second step bydenotes the infimal convolution between the Bregman distances D pα J (·, u α ) and D −pα J (·, −u α ), evaluated at u ∈ X .The infimal convolution of two functionals F and G on a Banach space X is defined asThe dual product can be expressed as a symmetric Bregman distanceHence all three terms are nonnegative and we find in particular Au α = Aũ α , w α =w α and thus p α =p α .ICB pα J (cu λ , u α ) ≤ D pα J (cu λ , u α ) = J(cu λ ) − p α , cu λ = 0.Consequently,û α is also a solution of (3.3).Theorem 4.12. The setProof. The map u → D p J (u, v) is convex and nonnegative, henceis convex as a sublevel set of a convex functional. Moreover, for each c ≥ 0 we haveThus we deduceThe assertion follows by the nonnegativity of the maps u → D p J (u, v) and u → D −p J (u, −v). Note that for p = 0 the subset is proper, since e.g.
Introduction
Variational regularization methods with nonquadratic functionals such as total variation or 1 -norms have evolved to a standard tool in inverse problems [10,34], image processing [13], compressed sensing [12], and recently related fields such as learning theory [15]. The popularity of such approaches stems from superior structural properties compared to other regularization approaches. 1 or even exact reconstruction of the support of the true solution. On the other hand it is known that such methods suffer from a certain bias due to the necessary increased weighting of the regularization term with increasing noise. Two wellknown examples are the loss of contrast in total variation regularization [10,27] or shrinked peak values in 1 -regularization. Accordingly, quantitative values of the solutions have to be taken with care.
Several approaches to reduce or eliminate the bias of regularization methods have been considered in literature: For 1 -regularization and similar sparsity-enforcing techniques an ad-hoc approach is to determine the support of the solution by the standard variational methods in a first step, then use a second debiasing step that minimizes the residual (or a general data fidelity) restricted to that support, also known as refitting [20,22,23]. A slightly more advanced approach consists in adding a signconstraint derived from the solution of the variational regularization method in addition to the support condition. This means effectively that the solution of the debiasing step shares an 1 -subgradient with the solution of the variational regularization method. A different and more general approach is to iteratively reduce the bias via Bregman iterations [27] or similar approaches [7,35]. Recent results for the inverse scale space method in the case of 1regularization (respectively certain polyhedral regularization functionals [8,25,5]) show that the inverse scale space performs some kind of debiasing. Even more, under certain conditions, the variational regularization method and the inverse scale space method provide the same subgradient at corresponding settings of the regularization parameters [6]. Together with a characterization of the solution of the inverse scale space method as a minimizer of the residual on the set of elements with the same subgradient, this implies a surprising equivalence to the approach of performing a debiasing step with sign-constraints. Recently, bias and debiasing in image processing problems were discussed in a more systematic way by Deledalle et al. [16,17]. They distinguish two different types of bias, namely method bias and model bias. In particular they suggest a debiasing scheme to reduce the former, which can be applied to some polyhedral one-homogeneous regularizations.
The key idea of their approach is the definition of suitable spaces, called model subspaces, on which the method bias is minimized. The remaining model bias is considered as the unavoidable part of the bias, linked to the choice of regularization and hence the solution space of the variational method. The most popular example is the staircasing effect that occurs for total variation regularization due to the assumption of a piecewise constant solution. In the setting of 1 -regularization a natural model subspace is the set of signals with a given support, which yields consistency with the ad-hoc debiasing approach mentioned above. Based on this observation, the main motivation of this paper is to further develop the approach in the setting of variational regularization and unify it with the above-mentioned ideas of debiasing for 1 -regularization, Bregman iterations, and inverse scale space methods.
Let us fix the basic notations and give a more detailed discussion of the main idea. Given a bounded linear operator A : X → Y between Banach spaces, a convex regularization functional J : X → R ∪ {∞} and a differentiable data fidelity H : Y × Y → R, we consider the solution of the variational method u α ∈ arg min u∈X H(Au, f ) + αJ (u) (1.1) as a first step. Here α > 0 is a suitably chosen regularization parameter. This problem has a systematic bias, as we further elaborate on below. The optimality condition is given by
For the sake of simplicity we carry out all analysis and numerical experiments in this paper for a least-squares data fidelity (related to i.i.d. additive Gaussian noise)
H(Au, f ) = 1 2 Au − f 2 Y (1.5)
for some Hilbert space Y, but the basic idea does not seem to change for other data fidelities and noise models. We show that the sets characterized by the constraints D pα J (u, u α ) = 0 and ICB pα J (u, u α ) = 0 constitute a suitable extension of the model subspaces introduced in [16] to general variational regularization. In particular, we use those manifolds to provide a theoretical basis to define the bias of variational methods and investigate the above approach as a method to reduce it. Moreover, we discuss its relation to the statistical intuition of bias. At this point it is important to notice that choosing a smaller regularization parameter will also decrease bias, but on the other hand strongly increase variance. The best we can thus achieve is to reduce the bias at fixed α by the two-step scheme while introducing only a small amount of variance. The remainder of the paper is organized as follows: In Section 2 we motivate our approach by considering bias related to the well-known ROFmodel [31] and we review a recent approach on debiasing [16]. In the next section we introduce our debiasing technique supplemented by some first results. Starting with a discussion of the classical definition of bias in statistics, we consider a deterministic characterization of bias in Section 4. We reintroduce the notion of model and method bias as well as model subspaces as proposed in [16] and extend it to the infinitedimensional variational setting. We furthermore draw an experimental comparison between the bias we consider in this paper and the statistical notion of bias. Finally, we comment on the relation of the proposed debiasing to Bregman iterations [27] and inverse scale space methods [32,7]. We complete the paper with a description of the numerical implementation via a firstorder primal-dual method and show numerical results for signal deconvolution and image denoising.
Motivation
Let us start with an intuitive approach to bias and debiasing in order to further motivate our method. To do so, we recall a standard example for denoising, namely the well-known ROFmodel [31], and we rewrite a recent debiasing approach [16] in the setting of our method.
Bias of total variation regularization
As already mentioned in the introduction, variational regularization methods suffer from a certain bias. This systematic error becomes apparent when the regularization parameter is increased. Indeed this causes a shift of the overall energy towards the regularizer, and hence a deviation of the reconstruction from the data in terms of quantitative values. Intuitively, this can be observed from the discrete version of the classical ROF-model [31], i.e.
u α ∈ arg min u∈R n
1 2 u − f 2 2 + α Γu 1 ,(2.1)
with a discrete gradient operator Γ ∈ R m×n . It yields a piecewise constant signal u α reconstructed from an observation f ∈ R n , which has been corrupted by Gaussian noise (see Figure 1(a)). Figure 1(b) shows the solution of (2.1) together with the true, noiseless signal we aimed to reconstruct. Even though the structure of the true signal is recovered, the quantitative values of the reconstruction do not match the true signal. Instead, jumps in the signal have a smaller height, which is often referred to as a loss of contrast. Without any further definition, one could intuitively consider this effect as the bias (or one part of the bias) of the ROF model. Hence, the goal of a bias reduction method would be to restore the proper signal height while keeping the (regularized) structure. It has been shown in [27,1] that this can be achieved by the use of Bregman iterations, i.e. by iteratively calculating u k+1 α ∈ arg min u∈R n 1 2 u − f 2 2 + αD p k α J (u, u k α ), (2.2) where in our case J(u) = Γu 1 , and p k α ∈ ∂J(u k α ) is a subgradient of the last iterate u k α . Since for total variation regularization the subgradient p k α essentially encodes the edge information of the last iterate, its iterative inclusion allows to keep edges while restoring the correct height of jumps across edges. We further elaborate on that in Section 4. Indeed, the reconstruction via (2.2) in Figure 1(b) shows an almost perfect recovery of the true signal even in terms of quantitative values. This indicates that Bregman iterations are able to reduce or even eliminate our heuristically defined bias. However, a theoretical basis and justification is still missing, i.e. a proper definition of the bias of variational methods, a proof that Bregman iterations indeed reduce the bias in that sense, and in particular a link to the statistical definition and understanding of bias. With this paper we aim to define a proper basis for this link, and in particular further establish the connection between bias reduction techniques and Bregman distances.
Recent debiasing and Bregman distances
In order to further motivate the use of Bregman distances for bias reduction let us recall and review a very recent approach on debiasing and work out its relation to Bregman distances. In [16], Deledalle et al. introduce a debiasing algorithm for anisotropic TV-type regularized problems
u α ∈ arg min u∈R n 1 2 Au − f 2 2 + α Γu 1 ,
with a linear operator A ∈ R n×d , a discrete gradient operator Γ ∈ R n×m and noisy data f ∈ R d . In [16] the authors argued that the loss of contrast characteristic for this kind of regularization is indeed bias in their sense. In order to correct for that error, the proposed debiasing method in [16] consists in looking for a debiased solutionû α such that Γû α and Γu α share the same support, butû α features the right intensities. Mathematically, the solutionû α of their debiasing problem is given bŷ
u α ∈ arg min u∈R n sup z∈F I 1 2 Au − f 2 2 + Γu, z ,(2.3)
where F I = {z ∈ R m | z I = 0}, and I is the set of indices corresponding to nonzero entries of Γu α . We can explicitly compute the supremum (the convex conjugate of the indicator function of the set F I ), which is
sup z∈F I Γu, z = ∞, (Γu) i = 0 for some i / ∈ I, 0, else.
Hence,û α can only be a minimizer of (2
.3) if supp(Γû α ) ⊂ supp(Γu α ), thuŝ u α ∈ arg min u∈R n 1 2 Au − f 2 2 s.t. supp(Γû α ) ⊂ supp(Γu α ). (2.4)
We can also enforce this support condition using the infimal convolution of two 1 -Bregman distances. Defining J(u) = Γu 1 , the subdifferential of J at u α is given by
∂J(u α ) = {Γ T q α ∈ R n | q α ∞ ≤ 1, (q α ) i = sign((Γu α ) i ) for (Γu α ) i = 0}.
Original TV Debiased Figure 2: TV denoising of a one-dimensional noisy signal and debiasing using the proposed approach with zero Bregman distance.
In particular |(q α ) i | = 1 on the support of Γu α . Let q α be such a subgradient and consider the 1 -Bregman distances D qα · 1 (·, Γu α ) and D −qα · 1 (·, −Γu α ). According to [24], their infimal convolution evaluated at Γu is given by:
ICB qα · 1 (Γu, Γu α ) = [D qα · 1 (·, Γu α )2D −qα · 1 (·, −Γu α )](Γu) = m i=1 (1 − |(q α ) i |)|(Γu) i |.
We observe that this sum can only be zero if |(q α ) i | = 1 or (Γu) i = 0 for all i. Assuming that a qualification condition holds, i.e. p α = Γ T q α ∈ ∂J(u α ) with |(q α ) i | < 1 for i / ∈ I, i.e. |(q α ) i | = 1 ⇔ (Γu α ) i = 0, we can rewrite the above debiasing method (2.3) as min u∈R n
1 2 Au − f 2 2 s.t. ICB qα · 1 (Γu, Γu α ) = 0.
Note that the zero infimal convolution exactly enforces the support condition (2.4) only if |(q α ) i | < 1 for all i ∈ I. Intuitively, since the subdifferential is multivalued at (Γu α ) i = 0, this leads to the question of how to choose q α properly. However, our method does not depend on the choice of a particular q α , but instead we use a unique subgradient p α coming from the optimality condition of the problem. We further comment on this in Section 4.
Debiasing
Inspired by the above observations, let us define the following two-step-method for variational regularization on Banach spaces. At first we compute a solution u α of the standard variational method
1) u α ∈ arg min u∈X 1 2 Au − f 2 Y + αJ(u),(3.
1) where A : X → Y is a linear and bounded operator mapping from a Banach space X to a Hilbert space Y, J : X → R ∪ {∞} denotes a convex and one-homogeneous regularization functional and f ∈ Y. We point out that in the following, we will always make the standard identification Y * = Y without further notice.
The first-order optimality condition of (3.1) reads:
p α = 1 α A * (f − Au α ), p α ∈ ∂J(u α ), (3.2)
and it is easy to show that this p α is unique (cf. Section 3.2, Thm. 3.1). We use this subgradient to carry over information about u α to a second step. In the spirit of the previous paragraph the idea is to perform a constrained minimization of the data fidelity term only:
2 a)û α ∈ arg min u∈X 1 2 Au − f 2 Y s.t. ICB pα J (u, u α ) = 0. (3.3)
If we reconsider the ad-hoc idea of 1 or TVtype debiasing from the introduction, it can be beneficial to add a sign or direction constraint to the minimization, rather than a support condition only. This can be achieved by the use of a single Bregman distance. Hence it is self-evident to define the following alternative second step:
2 b)û α ∈ arg min u∈X 1 2 Au − f 2 Y s.t. D pα J (u, u α ) = 0. (3.4)
We would like to point out that until now we only argued heuristically that the above method actually performs some kind of debiasing for specific problems. But since we are able to recover the debiasing method of [16] for J(u) = Γu 1 as a special case, at least for this specific choice of regularization (and a finitedimensional setting) our method is provably a debiasing in their sense.
However, our method is much more general. Since in contrast to [16] it does not depend on a specific representation of u α , it can theoretically be carried out for any suitable regularizer J. In particular, the method does not even depend on Second row: TV reconstruction and debiasing using the Bregman distance and its infimal convolution, respectively. The TV reconstruction recovers well the structures of the images but suffers from a loss of contrast, while the debiased solutions allow for a more accurate dynamic. 1 The color image is provided in order to point out that it is indeed a giraffe and not a cow.
the specific choice of the data term. In order to obtain a unique subgradient p α from the optimality condition it is desirable e.g. to have a differentiable data fidelity, but if we drop that condition, the data term is theoretically arbitrary.
Since this generalization requires more technicalities, we focus on a squared Hilbert space norm in this paper in order to work out the basics of the approach. Before we actually lay a theoretical foundation for our framework and prove that our method indeed is a debiasing method, we show some motivating numerical results and prove the well-definedness of the method.
A first illustration
To give a first glimpse of the proposed method, we revisit the ROF-reconstruction model (2.1) from Section 2 and show some numerical results in one and two dimensions.
Taking the subgradient p α of the TV reconstruction u α of the one-dimensional signal and performing our debiasing method, we obtain the results in Figure 2. The second step restores the right height of the jumps and yields the same result as the Bregman iterations we performed in Section 2.
As a second example we perform denoising on a cartoon image corrupted by Gaussian noise. The first row of Figure 3 shows the original image and its noisy version. The left image in the second row is the denoising result obtained with the ROF-model (2.1). We observe that noise has been reduced substantially, but some part of the contrast is lost. The second step of our method restores the contrast while keeping the structure of the first solution, yielding the two results depicted in the middle and on the right of the second row.
Well-definedness of the method
The aim of this section is to show that the method defined above is well-defined, i.e. that there always exists at least one solution to the problem. We fix the setup by restricting ourselves to conditions ensuring that the original variational problem (1.1) with quadratic data fidelity has a solution. The following result can be established by standard arguments:
Theorem 3.1.
Let Y be a Hilbert space, X be the dual space of some Banach space Z, such that the weak-star convergence in X is metrizable on bounded sets. Moreover, let A : X → Y be the adjoint of a bounded linear operator B : Y → Z, J be the convex conjugate of a proper functional on the predual space Z, and let the map u → 1 2 Au 2 Y + J(u) be coercive in X . Then the variational problem (1.1) with data-fidelity (1.5) has a minimizer u α ∈ X and there exists a subgradient p α ∈ ∂J(u α ) such that the optimality condition
p α = 1 α A * (f − Au α ) = 1 α B(f − Au α ) (3.5)
holds. Moreover, if u α =ũ α are two minimizers, then Au α = Aũ α and the corresponding subgradient is unique, i.e.,
p α = 1 α B(f − Au α ) = 1 α B(f − Aũ α ) =p α .
Proof. Since the functional J is proper, there exists a nonempty sublevel set of the functional u → 1 2 Au − f 2 Y + αJ(u), and by the coercivity assumption this sublevel set is bounded. The Banach-Alaoglu theorem now implies precompactness of the sublevel set in the weakstar topology. Since the latter is metrizable on bounded sets, it suffices to show that the objective functional is sequentially weak-star lower semicontinuous in order to obtain existence of a minimizer. For the regularization functional J, this follows from a standard argument for convex conjugates of proper functionals along the lines of [18]. The assumption A = B * guarantees further that A is continuous from the weakstar topology in X to the weak topology of Y and the weak lower semicontinuity of the norm also implies the weak-star lower semicontinuity of the data fidelity. Those arguments together yield the existence of a minimizer.
The first equation of the optimality condition for the subgradient p α follows from the fact that the data fidelity is Fréchet-differentiable. From the argumentation in [4,Remark 3.2] we see that the assumption A = B * furthermore implies that A * indeed maps to the predual space Z (and not to the bigger space Z * * ), such that (3.5) holds true. More precisely, this special property of A * is derived from the fact that A is sequentially continuous from the weak-star topology of X to the weak(-star) topology of Y, which implies that it posseses an adjoint which maps Y into Z regarded as a closed subspace of Z * * (note that the weak and the weak-star topology coincide on the Hilbert space Y). Consequently p α ∈ Z.
Finally, assume that u α andũ α are two solutions, then we find
p α = Bw α , w α = 1 α (f − Au α ),
and an analogous identity forp α respectivelyũ α . Consequently, we have
(w α −w α ) + 1 α A(u α −ũ α ) = 0.
Computing the squared norm of the left-hand side, we find
w α −w α 2 Y + 2 α p α −p α , u α −ũ α + 1 α 2 A(u α −ũ α ) 2
By exploiting that p α lies in the range of B we can prove coercivity and subsequently existence for problem (3.4). In fact, we can give a more general result. 2
Au − f 2 Y s.t. J(u) − p, u = 0.
Proof. Let A = {u ∈ X | J(u) − p, u = 0} be the admissible set. Since 0 ∈ A we can look for a minimizer in the sublevel set
S = u ∈ A | Au − f Y ≤ f Y .
By the triangle inequality we have
Au Y ≤ 2 f Y and hence 1 2 Au 2 Y ≤ 2 f 2 Y on S. Ac- cordingly, u → 1 2 Au 2 Y is bounded on S.
From the definition of the convex conjugate we know that for all u ∈ X , r ∈ X * we have r, u ≤ J * (r) + J(u).
(3.6)
Hence for u ∈ S we find
J(u) = p, u = p − Bw, u + w, Au ≤ p − Bw τ , τ u + w Y Au Y ≤ J * p − Bw τ + J(τ u) + w Y Au Y
which implies by the one-homogeneity of J that
J(u) ≤ w Y Au Y 1 − τ .
Thus we obtain the boundedness of u → Note that, provided that the operator A fulfills the conditions of Theorem 3.1, the assumptions of Theorem 3.2 always hold for p = p α obtained from (3.2) with w = 1 α (f − Au α ) and τ arbitrarily small, hence we conclude the existence of a minimizerû α of (3.4).
The situation for (3.3) is less clear, since there is no similar way to obtain coercivity. As we shall see in Section 4, (3.3) consists in minimizing a quadratic functional over a linear subspace, which immediately implies the existence ofû α if X has finite dimensions. In an infinitedimensional setting we cannot provide an existence result in general, since there is neither a particular reason for the subspace to be closed nor for the quadratic functional to be coercive (in ill-posed problems we typically deal with an operator A with nonclosed range).
Optimal debiasing on singular vectors
In the following we work out the behavior of the debiasing method on singular vectors [1], which represent the extension of the concept of classical singular value decomposition to nonlinear regularization functionals. According to [1], u λ ∈ X is a singular vector if for some λ > 0
λA * Au λ ∈ ∂J(u λ )
holds. Without going too much into detail, singular vectors can be considered as generalized "eigenfunctions" of the regularization functional J. As such, they describe a class of exact solutions to problem (3.1) in the following sense: Let us consider a multiple cu λ of such a singular vector for c > λα. According to [1], the solution u α of the variational problem (3.1) for data f = cAu λ is given by
u α = (c − αλ)u λ ,
and the subgradient from the optimality condition is
p α = λA * Au λ ∈ ∂J(u α ).
Hence u α recovers cu λ up to a (known) scalar factor αλ and shares a subgradient with u λ . This means that the variational method leaves the singular vector basically untouched, which allows for its exact recovery. Intuitively, the quantity −λαu λ hence represents the bias of the variational method in this case, which should be removed by our debiasing method (3.4). And indeed we obtainû α = cu λ as a minimizer of (3.4), since
Aû α − f Y = A(û α − cu λ ) Y = 0
and sinceû α lies in the admissible set due to the shared subgradient. If A has trivial nullspace, u α is even unique. Hence, the debiasing strategy leads to the exact reconstruction of the solution and corrects the bias −λαu λ . Note that this is indeed an important result, since if the debiasing method failed for singular vectors it would be doubtful whether the method is reliable in general.
Since the infimal convolution of Bregman distances is nonnegative and less or equal than either of the Bregman distances, it also vanishes atû α = cu λ . In particular
Bias and Model Manifolds
In the following we provide a more fundamental discussion of bias and decompositions obtained by debiasing methods. An obvious point to start is the definition of bias, which is indeed not always coherent in the imaging literature with the one in statistics.
Definitions of bias
We first recall the classical definition of bias in statistics. Let f be a realization of a random variable modeling a random noise perturbation of clean data f * = Au * , such that E[f ] = f * . If we consider a general canonical estimatorÛ (f ), the standard definition of bias in this setup is given by
B stat (Û ) = E[u * −Û (f )] = u * − E[Û (f )].
(4.1)
Unfortunately, this bias is hard to manipulate for nonlinear estimators. Hence, we consider a deterministic definition of bias, which relies on the clean data f * :
B * (Û ) = E[u * −Û (f * )] = u * −Û (f * ) = u * −Û (E[f ]). (4.2)
We immediately note the equivalence of the two definitions in the case of linear estimators, but our computational experiments do not show a significant difference between B stat and B * even for highly nonlinear variational methods. In general, the purpose of debiasing is to reduce the quantitative bias B d , i.e. here the error between u * andÛ (f * ) in an appropriate distance measure d:
B d (Û (f * )) = d(Û (f * ), u * ).
Let us consider the specific estimator u α (f * ), i.e. the solution of problem (3.1) with clean data f * . As already argued in Section 2, it suffers from a certain bias due to the chosen regularization. Following [16], this bias can be decomposed into two parts. The first part is related to the regularization itself, and it occurs if the assumption made by the regularization does not match the true object that we seek to recover. For example, trying to recover a piecewise linear object using TV regularization leads to the staircasing effect due to the assumption of a piecewise constant solution. This part of the bias is unavoidable since it is inherent to the regularization, and it is referred to as model bias. In particular, we cannot hope to correct it.
However, even if the regularity assumption fits, the solution still suffers from a systematic error due to the weight on the regularization. For TV regularization for example, this is the loss of contrast observed in Section 2. This remaining part is referred to as method bias, and this is the part that we aim to correct. As we shall see in the remainder of the section, the estimator u α (f * ) provides the necessary information to correct this bias. Deledalle et al. [16] define an appropriate linear model subspace related to that estimator, on which the debiasing takes place. It allows to define the model bias as the difference between u * and its projection onto the model subspace. The remaining part of the difference between the reconstructed solution and u * is then the method bias. In the following we reintroduce the notion of model subspaces provided by [16] and further generalize it to the variational setting in infinite dimensions. The latter may imply the nonclosedness of the model subspace and hence nonexistence of the projection of u * onto it. Moreover, it seems apparent that in some nonlinear situations it might be more suitable to consider a model manifold instead of a linear space and we hence generalize the definition in this direction. We remark that the use of the term manifold is for technical reasons. As we shall see, the sets we consider in the course of the paper are for example (linear) subspaces or convex cones. The latter are not linear, but can be considered as manifolds with boundaries. Therefore we shall use the term model manifold in general, and be more precise for particular instances of model manifolds.
Let us first assume that we are already given an appropriate model manifold.
Definition 4.1. Let M f * be a given model manifold and d : X ×X → [0, ∞) a distance mea- sure. An estimatorÛ (f * ) of u * is a debiasing of u α (f * ) ifÛ (f * ) ∈ M f * and d(Û (f * ), u * ) < d(u α (f * ), u * ).
If there exists a minimizer
u α (f * ) ∈ arg min v∈M f * d(v, u * ),(4.3)
we call it an optimal debiasing. In any case, we define the magnitude of the model bias as
B d mod (M f * ) = inf v∈M f * d(v, u * ).
Obviously the model bias only depends on the model manifold and for a given u α (f * ) it is hence, as already indicated, a fixed quantity that we cannot manipulate. Instead we want to perform the debiasing on the manifold only, so we consider another bias for elements of M f * only. Since according to the above definition there might exist more than one optimal debiasing, we shall from here on assume that we are given one of them.
Definition 4.2. For a fixed optimal debiasinĝ u α (f * ) on M f * , we define the magnitude of the method bias of v ∈ M f * related toû α (f * ) as B d meth (v) = d(v,û α (f * )).
The optimal debiasingû α (f * ) obviously does not suffer from method bias. Note that if the minimizer in (4.3) does not exist, which can happen in particular in ill-posed problems in infinite dimensions, then the magnitude of the method bias is not well-defined or has to be set to +∞.
With these definitions at hand, we now aim to compute an optimal debiasing, i.e. the solution of (4.3). The remaining questions are how to choose an appropriate model manifold M f * and the distance measure d. We start with the latter. An easy choice for the distance measure d is a squared Hilbert space norm: If the minimizer of (4.3) exists, e.g. if M f * is nonempty, convex and closed, the optimal debiasingû α (f * ) is the (unique) projection of u * onto M f * . We obtain a decomposition of the bias of any estimator v ∈ M f * into method and (constant) model bias:
v − u * = v −û α (f * ) method bias +û α (f * ) − u * model bias .
In case M f * is a closed subspace of X , this decomposition is even orthogonal, i.e.
B d (v) = v − u * 2 = v −û α (f * ) 2 + û α (f * ) − u * 2 = B d meth (v) + B d mod (M f * ).
Unfortunately, for general inverse problems with a nontrivial operator we do not know u * and hence cannot compute its projection onto M f * .
Instead we have access to the data f * = Au * (or rather to one noisy realization f of f * in practice, which we discuss later). In order to make the bias (and the associated debiasing) accessible, we can consider bias through the operator A. Hence the optimal debiasing comes down to computing the minimizer of (4.3) with a dis-
tance defined over A(M f * ), i.e. u α (f * ) = arg min v∈M f * Au * − Av 2 = arg min v∈M f * f * − Av 2 . (4.4)
Correspondingly, if such a minimizerû α (f * ) exists, we measure the magnitude of model and method bias in the output space, rather than in image space, i.e.
B d mod (M f * ) = inf v∈M f * Av − f * 2 , B d meth (v) = Aû α (f * ) − Av 2 .
We can hence at least guarantee that the optimal debiasing has zero method bias in the output space. For denoising problems without any operator (A being the identity), or for A invertible on M f * we obtain the equivalence of both approaches. In ill-posed inverse problems it is usually rather problematic to measure errors in the output space, since noise can also be small in that norm. Notice however that we do not use the output space norm on the whole space, but on the rather small model manifold, on whichif chosen appropriately -the structural components dominate. On the latter the output space norm is reasonable. The main advantage of this formulation is that we are able to compute a minimizer of (4.4), since it is in fact a constrained leastsquares problem with the data fidelity of (3.1). Its solution of course requires a proper choice of the underlying model manifold M f * , which we discuss in the following.
Model manifolds
In general, a model manifold can be characterized as the space of possible solutions for the debiasing step following the first solution u α (f ) of the variational problem (3.1). As such it contains the properties of u α (f ) that we want to carry over to the debiased solution. In the context of sparsity-enforcing regularization this is basically a support condition on the debiased solution.
Differential model manifolds
Deledalle et al. [16] use the notion of Fréchet derivative to define their model subspace in a finite-dimensional setting. We naturally generalize this concept using the directional derivative instead, and further extend it to infinite dimensions. The following definitions can e.g. be found in [33].
Definition 4.3. Let V and W be Banach spaces. A mapping F : V → W is called Fréchet differentiable at x ∈ V if there exists a linear and bounded operator DF (x; ·) : V → W such that lim g V →0 F (x + g) − F (x) + DF (x; g) W g V = 0. Definition 4.4. A mapping F : V → W is called directionally differentiable in the sense of Gâteaux at x ∈ V if the limit dF (x; g) := lim t→0 + F (x + tg) − F (x) t exits for all g ∈ V.
We can immediately deduce from the definition that, if the directional derivative dF (x; ·) exits, it is positively one-homogeneous in g, i.e.
dF (x; λg) = λdF (x; g) for all λ ≥ 0 and g ∈ V. If it is linear in g, we call F Gâteaux differentiable at x.
Provided a unique and Fréchet differentiable map f → u α (f ), Deledalle et al. [16] introduce the tangent affine subspace
M F f = u α (f ) + Du α (f ; g) | g ∈ Y ,
where Du α (f ; ·) : Y → X is the Fréchet derivative of u α (f ) at f . To be less restrictive, the easiest generalization of M F f is to consider the directional derivative.
Definition 4.5. If the map f → u α (f ) is direc- tionally differentiable with derivative du α (f ; ·), we define M G f = u α (f ) + du α (f ; g) | g ∈ Y .
Note that if the map is Fréchet differentiable, M G f is a linear space and coincides with the model subspace M F f . We now derive a few illustrative examples that we use throughout the remainder of the paper. In order to keep it as simple as possible, the easiest transition from the finite-dimensional vector space setting to infinite dimensions are the p -spaces of p-summable sequences:
Definition 4.6. For 1 ≤ p < ∞ we define the spaces p of p-summable sequences with values in R d by p (R d ) = (x i ) i∈N , x i ∈ R d : i∈N |x i | p < ∞ , where | · | denotes the Euclidean norm on R d . For p = ∞ we define ∞ (R d ) = (x i ) i∈N , x i ∈ R d : sup i∈N |x i | < ∞ . It is easy to show that p (R d ) ⊂ q (R d ) for 1 ≤ p ≤ q ≤ ∞.
In particular for d = 1 we denote by 1 , 2 and ∞ the spaces of summable, square-summable and bounded scalar-valued sequences.
Example 4.7. Anisotropic shrinkage. Let f ∈ 2 be a square-summable sequence. The solution of u α (f ) ∈ arg min u∈ 1 1 2 u − f 2 2 + α u 1 (4.5)
for α > 0 is given by
[u α (f )] i = f i − α sign(f i ), |f i | ≥ α, 0, |f i | < α.
Its support is limited to where |f i | is above the threshold α. The directional derivative du α (f ; g) of u α (f ) into the direction g ∈ 2 is given by
[du α (f ; g)] i = g i , |f i | > α 0, |f i | < α g i , |f i | = α, sign(f i ) = sign(g i ) 0, |f i | = α, sign(f i ) = sign(g i ).
Proof. See Appendix 8.1.
First, if we exclude the case |f i | = α, the directional derivative is linear, hence it is a Gâteaux derivative. In fact it is even an infinitedimensional Fréchet derivative, and the resulting model manifold coincides with the model subspace defined in finite dimensions in [16]:
M F f = u ∈ 2 | supp(u) ⊂ supp(u α (f )) .
The model manifold carries over information about the support of the first solution u α (f ).
Note that M F f contains all elements of 2 which share the same support as u α (f ), but as well allows for zeros where u α (f ) = 0. In that sense u α (f ) defines the maximal support of all u ∈ M F f . If we allow |f i | to be equal to α, we obtain a larger set which allows for support changes in the direction of f i on the threshold:
u ∈ M G f ⇔ u i = λ ∈ R, |f i | > α, 0, |f i | < α, λ ≥ 0, f i = α, λ ≤ 0, f i = −α.
Note that the case |f i | > α reveals a remaining shortcoming of the definition via the directional derivative, e.g. if f i > α it is counterintuitive to allow for negative elements in M G f , while this is not the case for f i = α. The main reason appears to be the strong deviation of the linearization in such directions from the actual values of [u α (f )] i , which is not controlled by the definition. However, minimizing the data term over M G f for the debiasing in Eq. (4.4) forces the changes to have the right sign and the debiased solutionû α (f ) corresponds to hard-thresholding:
[û α (f )] i = f i , |f i | ≥ α, 0, |f i | < α.
Note that we as well maintain the signal directly on the threshold.
We obtain analogous results for isotropic shrinkage, i.e. if f ∈ 2 (R d ) for d > 1. Since the computation of the derivative requires a little more work, we provide the results in Appendix 8.1. A more interesting example is the model manifold related to anisotropic 1 -regularized general linear inverse problems.
Example 4.8. Anisotropic 1 -regularization. For r > 1 let A : r → 2 be a linear and bounded operator and f ∈ 2 . Consider the solution u α (f ) of the 1 -regularized problem
u α (f ) ∈ arg min u∈ 1 1 2 Au − f 2 2 + α u 1 , (4.6)
where we assume that the solution is unique for data in a neighborhood of f . Computing the directional derivative directly is a more tedious task in this case, but computing the model manifold M G f is actually easier via a slight detour.
Let u α (f ) be the solution for data f and u α (f ) the solution for dataf . First, we derive an estimate on the two subgradients from the optimality conditions
0 = A * (Au α (f ) − f ) + αp α , p α ∈ ∂ u α (f ) 1 , 0 = A * (Au α (f ) −f ) + αp α ,p α ∈ ∂ u α (f ) 1 .
Following the ideas of [11], we subtract the two equations and multiply by u α (f ) − u α (f ) to arrive at
Au α (f ) − Au α (f ) 2 2 + α p α −p α , u α (f ) − u α (f ) = f −f , Au α (f ) − Au α (f ) ≤ 1 2 f −f 2 2 + 1 2 Au α (f ) − Au α (f ) 2 2 .
The last line follows from the Fenchel-Young inequality, obtained by applying the inequality (3.6) to J = 1 2 · 2 2 . The second term on the left hand side is a symmetric Bregman distance, i.e. the sum of two Bregman distances (cf. [11]), hence positive. Leaving it out and rearranging then yields
Au α (f ) − Au α (f ) 2 ≤ f −f 2 .
(4.7)
Since A * : 2 → s , where s −1 + r −1 = 1, A * is also continuous to ∞ , hence we derive the following estimate from the optimality conditions:
p α −p α ∞ = 1 α A * (Au α (f ) − Au α (f )) − A * (f −f ) ∞ ≤ A * α Au α (f ) − Au α (f ) 2 + A * α f −f 2 ≤ C α f −f 2 ,
where we used (4.7) for the last inequality and · denotes the operator norm. Next, we note that since A * maps to s and p α andp α lie in its range, they necessarily have to converge to zero. This implies the existence of N ∈ N such that for all i ≥ N both |(p α ) i | and |(p α ) i | are strictly smaller than 1 and hence u α (f ) and u α (f ) vanish for all i ≥ N . As a consequence it is sufficient to consider a finite dimensional setting for the following reasoning.
In view of the subdifferential of the 1 -norm,
∂ u 1 = {p ∈ ∞ : p ∞ ≤ 1, p i = sign(u i ) for u i = 0}, we have to consider several cases. If [u α (f )] i = 0 and |(p α ) i | < 1, we derive from |(p α ) i | ≤ |(p α ) i − (p α ) i | + |(p α ) i | ≤ C α f −f 2 + |(p α ) i |, that if f −f 2 is sufficiently small, then |(p α ) i | < 1. Hence [u α (f )] i = 0
, and the derivative related to the perturbed dataf vanishes.
In case [u α (f )] i = 0 and (p α ) i = 1, by a similar argument (p α ) i = −1 and thus [u α (f )] i ≥ 0 and [du α (f ; g)] i ≥ 0. Analogously, [du α (f ; g)] i ≤ 0 if [u α (f )] i = 0 and (p α ) i = −1. If [u α (f )] i = 0,
the directional derivative is an arbitrary real number depending on the data perturbation. Summing up we now know that every directional derivative is an element
v ∈ 1 fulfilling v i = 0, |(p α ) i | < 1, λ ≥ 0, (u α ) i = 0, (p α ) i = 1, λ ≤ 0, (u α ) i = 0, (p α ) i = −1. (4.8)
Note again that v differs from 0 only for a finite number of indices. Hence, for v satisfying (4.8), we can pick t > 0 sufficiently small such that p α is a subgradient ofũ = u α (f ) + tv. Indeed, for example if (u α ) i = 0 and (p α ) i = 1, then v i ≥ 0, soũ ≥ 0, and hence (p α ) i fulfills the requirement of a subgradient ofũ. The other cases follow analogously. Then from the optimality condition of u α (f ) we get:
A * (Au α (f ) − f )) + αp α = 0 ⇔ A * (A(u α (f ) + tv ũ ) − (f + tAv)) + αp α = 0.
We then deduce thatũ is a minimizer of problem (4.6) with dataf = f + tAv. Hence, there exists a data perturbation such that v is the directional derivative of u α (f ). Putting these arguments together we now know that u ∈ M G f if and only if
u i = λ ∈ R, [u α (f )] i = 0, 0, [u α (f )] i = 0, |(p α ) i | < 1, λ ≥ 0, [u α (f )] i = 0, (p α ) i = 1, λ ≤ 0, [u α (f )] i = 0, (p α ) i = −1.
It is not surprising that M G f has a similar structure as the model manifold for the anisotropic shrinkage in Example 4.7. It allows for arbitrary changes on the support of u α (f ) and permits
only zero values if [u α (f )] i = 0 and |(p α ) i | < 1. If we exclude the case where |(p α ) i | = 1 even though [u α (f )] i vanishes, debiasing on M G
f effectively means solving a least-squares problem with a support constraint on the solution. But we again find an odd case where changes are allowed outside of the support of the initial solution u α (f ). It occurs when |(p α ) i | = 1 even though [u α (f )] i vanishes, which seems to be the indefinite case. However, it has been argued in [24] that a subgradient equal to ±1 is a good indicator of support, hence it is reasonable to trust the subgradient in that case.
Variational model manifolds
As we have shown so far, the appropriate use of a derivative can yield suitable spaces for the debiasing. However, for already supposedly easy problems such as the latter example the explicit computation of such spaces or of the derivatives can be difficult or impossible. And even if it is possible, there remains the question of how to effectively solve the debiasing on those spaces, both theoretically and numerically.
On the other hand, the latter example implies that a subgradient of the first solution rather than the solution itself can provide the necessary information for the debiasing. This naturally leads us to the idea of Bregman distances in order to use the subgradient in a variational debiasing method. And indeed we show that the associated manifolds are closely related, and that they link the concept of model manifolds to the already presented debiasing method from Section 3. Furthermore, this does not only provide a theoretical framework, but also numerical solutions to perform debiasing in practice, even for more challenging problems.
In the following we introduce related manifolds motivated by the variational problem itself. The optimality condition of the variational problem (3.1) defines a unique map f → p α ∈ ∂J(u α ), which allows us to consider the following manifolds. We drop the dependence of u α on f for the sake of readability. Definition 4.9. For p α ∈ ∂J(u α ) defined by
(3.2) we define M B f = u ∈ X | D pα J (u, u α ) = 0 , M IC f = u ∈ X | ICB pα J (u, u α ) = 0 .
In order to assess the idea of the above manifolds, we first revisit the anisotropic shrinkage problem of Example 4.7.
Example 4.10. Anisotropic shrinkage. The optimality condition of problem (4.5) yields the subgradient
(p α ) i = f i − (u α ) i α = sign(f i ), |f i | ≥ α, fi α , |f i | < α,(4.9)
and for J = · 1 the Bregman distance takes the following form:
D pα 1 (u, u α ) = u 1 − p α , u = i∈N |u i | − (p α ) i u i = i∈N (sign(u i ) − (p α ) i )u i .
A zero Bregman distance thus means that either u i = 0 or sign(u i ) = (p α ) i . Having a closer look at the subgradient (4.9), we observe that if |f i | < α, then |(p α ) i | < 1. Hence the latter condition cannot be fulfilled, so in this case u i has to be zero. We can thus characterize the model manifold related to a zero Bregman distance as:
u ∈ M B f ⇔ u i = λ sign(f i ), λ ≥ 0, |f i | ≥ α, 0, |f i | < α.
As for M G f , the model manifold M B f fixes the maximum support to where |f i | ≥ α. However, M B f only allows for values on the support sharing the same sign as f i (respectively (u α ) i ).
By adapting the proof of [24], we obtain a similar result for the infimal convolution of Bregman distances, without the restriction on the sign:
ICB pα 1 (u, u α ) = [D pα 1 (·, u α )2D −pα 1 (·, −u α )](u) = i∈N (1 − |(p α ) i |)|u i |.
For this infimal convolution to be zero we need either u i = 0 or |(p α ) i | = 1. By the structure of the subgradient p α we thus find
u ∈ M IC f ⇔ u i = λ ∈ R, |f i | ≥ α, 0, |f i | < α.
Hence we observe the following connection between the manifolds:
M B f ⊂ M G f ⊂ M IC f .
Note that the manifold M G f related to the directional derivative seems to be the odd one of the three. While allowing for arbitrary sign for |f | > α, it only allows for changes in the direction of f directly on the threshold. In that sense, M B
f and M IC f seem to be more suitable in order to either include or exclude the signconstraint. A closer inspection at the manifolds reveals that M IC f is a linear space, as we further elaborate on in the next subsection. In this case it is actually even the span of M B f , which is however not true in general. This can e.g. be seen from the next example of isotropic TV-type regularization.
Example 4.11. Isotropic TV-type regularization. Let A : 2 (R n ) → 2 (R d ) and Γ : 2 (R n ) → 1 (R m ) be linear and bounded operators and J(u) = Γu 1 (R m ) for d, m, n ∈ N. We aim to find the variational model manifolds for the debiasing of the solution u α ∈ arg min
u∈ 2 (R n ) 1 2 Au − f 2 (R d ) + α Γu 1 (R m ) .
Given the (unique) subgradient p α ∈ ∂J(u α ) from the optimality condition, the chain rule for subdifferentials [18, p. 27] implies the existence of a q α ∈ ∂ · 1 (R m ) (Γu α ) such that p α = Γ * q α and D pα J (u, u α ) = D qα 1 (R m ) (Γu, Γu α ).
If we denote the angle between (Γu) i and (q α ) i by ϕ i , the Bregman distance reads:
D pα J (u, u α ) = D qα 1 (R m ) (Γu, Γu α ) = i∈N |(Γu) i | − (q α ) i · (Γu) i = i∈N |(Γu) i | 1 − cos(ϕ i )|(q α ) i |
For a zero Bregman distance we can distinguish two cases: If |(q α ) i | < 1, then (Γu) i has to be zero. If |(q α ) i | = 1, then either (Γu) i = 0 or cos(ϕ i ) = 1, hence (Γu) i = λ(q α ) i for λ ≥ 0. Hence the model manifold M B f is given by
u ∈ M B f ⇔ (Γu) i = λ(q α ) i , λ ≥ 0, |(q α ) i | = 1, 0, |(q α ) i | < 1.
In particular, if (Γu α ) i = 0, then by the structure of the 1 (R m )-subdifferential we know that (q α ) i = (Γuα)i |(Γuα)i| and thus (Γu) i = µ(Γu α ) i for some µ ≥ 0. So provided that |(q α ) i | < 1 whenever (Γu α ) i = 0 we find
u ∈ M B f ⇔ (Γu) i = µ(Γu α ) i , µ ≥ 0, (Γu α ) i = 0, 0, (Γu α ) i = 0.
Performing the debiasing on the latter manifold hence means minimizing the data term with a support and direction constraint on the gradient of the solution. This in particular allows to restore the loss of contrast which we have observed for TV regularization in Section 2. Note that the condition |(q α ) i | < 1 ⇔ (Γu α ) i = 0 excludes the odd case where the subgradient seems to contain more information than the first solution, as already seen in Example 4.8.
In the above illustration of the model manifold, the debiasing seems to rely on the choice of q α , which is obviously not unique. However, in practice we still use the unique subgradient p α from the optimality condition which avoids the issue of the choice of a "good" q α .
The computation of M IC f is a little more difficult in this case, since we cannot access an explicit representation of the functional ICB pα J (·, u α ). However, since
ICB qα 1 (R m ) (Γu, Γu α ) ≤ ICB pα J (u, u α ) (ICB qα 1 (R m ) (Γu, Γu α ) = i∈N G((Γu) i , (q α ) i ) with G : R m × R m → R defined as G((Γu) i , (q α ) i ) = |(Γu) i |(1 − | cos(ϕ i )||(q α ) i |), if |(q α ) i | < | cos(ϕ i )|, |(Γu) i || sin(ϕ i )| 1 − |(q α ) i | 2 , if |(q α ) i | ≥ | cos(ϕ i )|.
For G to be zero we once again distinguish two situations. If |(q α ) i | < 1, in the first case G can only vanish if (Γu) i = 0. In the second case, since 1 > |(q α ) i | ≥ | cos(ϕ i )|, we infer ϕ i / ∈ {0, π}, and hence neither the sinus nor the square root can vanish. This means once again that (Γu) i = 0. If |(q α ) i | = 1, we can only be in the second case and G vanishes independently of (Γu) i . Thus (Γu) i can be arbitrary. Putting the arguments together, we find
u ∈ M IC f ⇒ ICB qα 1 (R m ) (Γu, Γu α ) = 0 ⇔ (Γu) i = λ ∈ R m , |(q α ) i | = 1, 0, |(q α ) i | < 1.
This is indeed not the span of M B f , but it instead allows for arbitrary elements if |(q α ) i | = 1. Note that we gain the same support condition on the gradient as for M B f , but allow for arbitrary gradient directions on the support, which intuitively does not seem restrictive enough. However, in practice for the debiasing the direction is not arbitrary, but the data term decides, so we can expect a similar result for debiasing in M B
f and M IC f . Indeed the numerical studies in Section 6 confirm these expectations.
Properties of variational model manifolds
In the following we discuss some properties of the variational manifolds M B f and M IC f . All results are general and do not depend on the particular choice of a subgradient, so we drop the dependence on f in the notation of the manifolds. Let v ∈ X and p ∈ ∂J(v). We start with a result on the structure of M B :
i.e. if u is an element of the set, then every positive multiple cu is an element, too. Hence it is a convex cone. Since D p J (v, v) = 0 it is not empty.
The structure of M IC is even simpler; as announced in a special example above it is indeed a linear space:
Theorem 4.13. The set M IC = {u ∈ X |[D p J (·, v)2D −p J (·, −v)](u) = 0} is a nonempty linear subspace of X .
Proof. By analogous arguments as above we deduce the convexity and since Due to its convexity and absolute onehomogeneity J is a seminorm and thus satisfies the triangle inequality:
ICB p J (0, v) = inf φ+ψ=0 D p J (φ, v) + D −p J (ψ, −v) ≤ D p J (v, v) + D −p J (−v, −v) = 0 the set is not empty. For arbitrary c ∈ R \ {0} we have ICB p J (cu, v) = inf z J(cu − z) + J(z) − p, cu − 2z = |c| inf w J(u − w) + J(w) − p, u − 2w ,ICB p J (u 1 + u 2 , v) = inf z J(u 1 + u 2 − z) + J(z) − p, u 1 + u 2 − 2z ≤ J(u 1 + u 2 − z n 1 − z n 2 ) + J(z n 1 + z n 2 ) − p, u 1 + u 2 − 2z n 1 − 2z n 2 ≤ J(u 1 − z n 1 ) + J(z n 1 ) − p, u 1 − 2z n 1 + J(u 2 − z n 2 ) + J(z n 2 ) − p, u 2 − 2z n 2 → 0, for n → ∞.
Hence u 1 + u 2 ∈ M IC and M IC is a linear subspace.
As one may expect from the fact that the infimal convolution is a weaker distance than the original Bregman distance, we obtain an immediate inclusion between the corresponding manifolds:
Lemma 4.14. M B ⊂ M IC .
We finally elaborate on the importance of absolute one-homogeneity of J for our approach (respectively also other debiasing approaches as in [16]), such that the subdifferential can be multivalued. Otherwise the model manifolds may just contain a single element and debiasing in this manifold cannot produce any other solution. This is e.g. the case for a strictly convex functional. However, one can easily exclude this case since our assumption of J being onehomogeneous guarantees that it is not strictly convex.
Bias-variance estimates
Another justification for the deterministic definition of bias as well as our choice for the distance measure in Section 4.1 can be found in the variational model itself. In order to derive quantitative bounds for bias and variance in a variational model, we start with the Tikhonov regularization (Ridge regression) model related to the functional J(u) = 1 2 u 2 X . The optimality condition for this problem is given by
A * (Au α (f ) − f ) + αu α (f ) = 0.
We easily see that there exists w α = 1 α (f − Au α (f )) such that u α (f ) = A * w α and
Au α (f ) − Au * + αw α = f − Au * .
Now let us assume that a source condition u * ∈ Im[A * ] holds, i.e. u * = A * w * for some w * . In this case we can subtract αw * on both sides and take a squared norm to arrive at
Au α (f ) − Au * 2 Y + α 2 w α − w * 2 Y + 2α Au α (f ) − Au * , w α − w * = f − Au * 2 Y + α 2 w * 2 Y − 2α f − Au * , w * .
Now taking the expectation on both sides and using
E[f ] = f * = Au * we find E[ Au α (f ) − Au * 2 Y ] + α 2 E[ w α − w * 2 Y ] + 2αE[ u α (f ) − u * 2 X ] = E[ f − Au * 2 Y ] + α 2 w * 2 Y . (4.10)
The left-hand side is the sum of three error terms for the solution measured in different norms: in the output space, the space of the source element, and the original space used for regularization. All of them can be decomposed in a bias and a variance term, e.g.
E[ u α (f ) − u * 2 X ] = E[u α (f )] − u * 2 X + E[ u α (f ) − E[u α (f )] 2 X ].
The term E[ f − Au * 2 Y ] in (4.10) is exactly the variance in the data. As a consequence α w * Y measures the bias in this case. Note that in particular for zero variance we obtain a direct estimate of the bias via α w * Y . In the case of the variational model (3.1) this can be generalized using recent approaches [5,9,11,29] using the source condition A * w * ∈ ∂J(u * ). Now completely analogous computations as above yield
E[ Au α (f ) − Au * 2 Y ] + α 2 E[ w α − w * 2 Y ] + 2αE[D sym J (u α (f ), u * )] = E[ f − Au * 2 Y ] + α 2 w * 2 Y ,
with the only difference that we now use the symmetric Bregman distance
D sym J (u α (f ), u * ) = A * w α − A * w * , u α (f ) − u * ,
with A * w α ∈ ∂J(u α (f )). The bias-variance decomposition on the right-hand side remains the same. In the noiseless case it is then natural to consider this (here, deterministic) estimate as a measure of bias:
Au α (f * ) − Au * 2 Y + α 2 w α − w * 2 Y + 2αD sym J (u α (f * ), u * ) = α 2 w * 2 Y ,
Here, as already discussed in Section 4.1, we again consider a difference between the exact solution u * and the estimator for E[f ] = f * , i.e. the expectation of the noise, rather than the expectations of the estimators u α (f ) over all realizations of f (which coincide if J is quadratic). We observe that there are three natural distances to quantify the error and thus also the bias: a quadratic one in the output space and a predual space (related to w), and the symmetric Bregman distance related to the functional J. The first term Au α (f * ) − Au * 2 Y is exactly the one we use as a measure of bias. The second term
α 2 E[ w α − w * 2 Y ]
is constant on the model manifold M B f * , since by definition of the manifold p α = A * w α is a subgradient of all the elements in M B f * . The third term D sym J (u α (f * ), u * ) is not easy to control; if the manifold is appropriate, meaning that p α ∈ ∂J(u * ), then the symmetric Bregman distance vanishes for every element in M B f * . In any other case, we do not have access to a subgradient p * ∈ ∂J(u * ), so we cannot control the Bregman distance for any element of the manifold. Hence, with our method we minimize the part of the bias that we can actually control. In fact, if the model manifold is right, we even minimize the whole bias.
Back to the proposed method
To sum up, the debiasing method we have introduced in Equations (3.3) and (3.4) comes down to debiasing over M IC f * and M B f * , respectively, while the results of Section 3 guarantee the existence of the optimal debiasingû α (f * ) at least on M B f * . However in practice, we do not have access to the clean data f * , but often only to one noisy realization f , which makes the regularization in (3.1) necessary in the first place. Instead of the true model manifold M f * , we hence use an approximation M f computed from the noisy data f to perform the debiasing of the reconstruction
u α (f * )û(f * ) u α (f * )û(f * )
Clean data f * Figure 4: TV denoising and debiasing of the Giraffe and the Parrot images for either noisy data f or clean data f * , with the same regularization parameter α = 0.3.
u α (f ) for noisy data. The following experiments show that M f is a good approximation of M f * in terms of the resulting bias and bias reduction. They also relate the different definitions of bias that we have considered. In particular, we distinguish between the statistical bias of Equation (4.1) which is the expectation over several noisy realizations f and the deterministic bias that we define in Equation (4.2), which instead considers the outcome given the noiseless data f * . Figure 4 displays the TV denoising and debiasing (using the Bregman distance model manifold) results obtained with noisy data f (first row) or clean data f * (second row) with the same regularization parameter α = 0.3. We have performed the experiments for both the cartoon Giraffe image and the natural Parrot image 2 . First, for the Giraffe image we observe that the TV denoised solution u α (f * ) for clean data suffers from a heavy loss of contrast, i.e. from method bias. The debiased solution u α (f * ) however is again close to the original data f * . This shows that if the noiseless data is well represented by the choice of regularization (and hence M f * ), i.e. if there is no or little model bias, the debiasing procedure allows to recover the original signal almost perfectly. On the other hand, the same experiments on the natural Parrot image show the problem of model 2 http://r0k.us/graphics/kodak/ bias since the choice of regularization does not entirely match the data f * . The debiasing allows to recover the lost contrast, but even the result for noiseless data still suffers from bias, i.e. the loss of small structures, which is model bias in that case.
Besides, if α is big enough to effectively remove noise during the denoising step, then the TV solutions u α (f ) and u α (f * ) are close to each other. This leads to comparable model manifolds and hence debiased solutions, which confirms that M f is indeed a good approximation to M f * . Furthermore, we can assess the bias for both manifolds. On M f * we can only use the deterministic definition (4.2) of bias whereas on M f we use the statistical definition (4.1). Figures 5 and 6 show the bias estimation on the Giraffe cartoon image and the natural Parrot image. The first row shows the estimations of the statistical bias B stat for the two estimators u α (f ) andû α (f ) for noisy data f . In the second row the bias B * for the two estimators u α (f * ) andû α (f * ) for clean data f * is displayed. This deterministic bias can also be decomposed into the associated model and method bias, whereas such a decomposition has not been defined for the statistical bias. The overall deterministic bias B * (u α (f * )) = u * − u α (f * ) for TV denoising appears to be really close to the statistical TV denoising,
Bregman debiasing, B stat (u α (f )) = u * − E[u α (f )] B stat (û α (f )) = u * − E[û α (f )]
With noisy data f TV denoising, Bregman debiasing, Model bias: Method bias:
B * (u α (f * )) = u * − u α (f * ) B * (û α (f * )) = u * −û α (f * )û α (f * ) − u α (f * )
With clean data f * Figure 5: Bias estimation. First row: Statistical bias computed on five hundred noisy realizations of the Giraffe cartoon image. Second row: Deterministic bias computed between the clean data and the recovered solution from clean data f * . In the first column, TV denoising leads to bias. In the second column, the debiasing that has been performed has reduced (or suppressed) the method bias. The remaining (small) model bias is due to the necessary regularization. In the third column, the difference betweenû α (f * ) and u α (f * ) shows the bias that has been reduced by the debiasing step, hence the method bias.
bias on noisy data in the first row. The same applies for the bias of the debiased solutions in the second column. This confirms that the estimation of the model manifold that we perform with noisy data is indeed a good approximation to the ideal model manifold for clean data, and that the resulting statistical and deterministic bias are closely related. Besides, the difference u * −û α (f * ) in the second row shows the remaining bias after the debiasing step, which is model bias. For the Giraffe image, this bias is small because the cartoon image is well approximated in the model manifold associated to TV regularization. The Parrot image however suffers from a heavier model bias, for example the loss of the small structures around the eye. Finally, in the third column, the differenceû α (f * ) − u α (f * ) shows the error that has been removed by the debiasing step, which corresponds to the method bias. It is particularly interesting for the Parrot image. Here one can see the piecewise constant areas which correspond to the re-establishment of the lost contrast within the piecewise constant model provided by the model manifold.
TV denoising,
Bregman debiasing, B stat (u α (f )) = u * − E[u α (f )] B stat (û α (f )) = u * − E[û α (f )]
With noisy data f TV denoising, Bregman debiasing, Model bias: Method bias:
B * (u α (f * )) = u * − u α (f * ) B * (û α (f * )) = u * −û α (f * )û α (f * ) − u α (f * )
With clean data f * Figure 6: Bias estimation. First row: Statistical bias computed on five hundred noisy realizations of the Parrot natural image. Second row: Deterministic bias computed between the clean data and the recovered solution from clean data f * . On the first column, TV denoising leads to both kinds of bias, model bias and method bias. On the second column, the debiasing that has been performed has reduced (or suppressed) the method bias, and the remaining bias is model bias. On the third column, the difference betweenû α (f * ) and u α (f * ) shows the bias that has been reduced by the debiasing step, hence the method bias.
Relation to inverse scale space methods
We finally comment on the relation of the debiasing approaches to Bregman iterations respectively inverse scale space methods, which are rather efficiently reducing bias as demonstrated in many examples [27,32,7]. The Bregman iteration is iteratively constructed by
u k+1 ∈ arg min u∈X 1 2 Au − f 2 Y + αD p k J (u, u k ), p k+1 = p k + 1 α A * (f − Au k+1 ) ∈ ∂J(u k+1 ).
In the limit α → ∞ we obtain the time continuous inverse scale-space method, which is the differential inclusion
∂ t p(t) = A * (f − Au(t)), p(t) ∈ ∂J(u(t)),
with initial values u(0) = 0, p(0) = 0. A strong relation to our debiasing approach comes from the characterization of the primal solution given p(t) [8,25,26]
u(t) ∈ arg min u∈X Au − f 2 Y s.t. p(t) ∈ ∂J(u(t)).
This reconstruction step is exactly the same as the variational debiasing step using the Bregman distance, however with a different preceding construction of the subgradient p(t) (noticing that t corresponds to 1 α for the variational method).
From the last observation it becomes apparent that the Bregman debiasing approach with (3.2) and (3.4) is exactly equivalent if the variational method yields the same subgradient as the inverse scale space method, i.e. p α = p( 1 α ).
This can indeed happen, as the results for singular vectors demonstrate [1]. Moreover, in some cases there is full equivalence for arbitrary data, e.g. in a finite-dimensional denoising setting investigated in [6]. It has been shown that for A being the identity and J(u) = Γu 1 with ΓΓ * being diagonally dominant the identity p α = p( 1 α ) holds, which implies that the Bregman debiasing approach and the inverse scale space method yield exactly the same solution. For other cases that do not yield a strict equivalence we include the Bregman iteration for comparison in numerical studies discussed below.
Numerical Implementation
In Section 3 we have introduced a two-stepmethod (cf. Eq. (3.1) -(3.4)) in order to compute a variationally regularized reconstruction with reduced method bias in the sense discussed in Section 4. Its solution requires the minimization of the data fidelity over the model manifold defined by a zero Bregman distance or a zero infimal convolution thereof, respectively. This constraint is difficult to realize numerically, but can be approximated by a rather standard variational problem. We can translate the hard constraint into a soft constraint such that for γ > 0 the reformulated problems read: a)û α ∈ arg min
u∈X 1 2 Au − f 2 Y + γD pα J (u, u α ), b)û α ∈ arg min u∈X 1 2 Au − f 2 Y + γICB pα J (u, u α ).
For γ → ∞ we obtain the equivalence of the hard and soft constrained formulations. However, for the numerical realization already a moderately large γ is enough to enforce the constraint up to a satisfactory level. For our simulations we chose γ = 1000, but our tests showed that already for γ ≥ 500 the value of the Bregman distance or its infimal convolution stays numerically zero. Of course the choice of the parameter γ depends on the specific problem we aim to solve and probably has to be adjusted slightly for different image sizes or involved operators.
Discretization
For our numerical experiments we choose the setting X = R n , Y = R d and J(u) = Γu 1 .
In general Γ ∈ R n×m denotes a discrete linear operator, for the experiments with total variation regularization we choose a discretization of the gradient with forward finite differences. For a general linear forward operator A ∈ R n×d we hence end up with the following discrete optimization problems:
1. u α ∈ arg min u∈R n 1 2 Au − f 2 2 + α Γu 1 , 2. a)û α ∈ arg min u∈R n 1 2 Au − f 2 2 + γ ( Γu 1 − p α , u ) , b)û α ∈ arg min u∈R n 1 2 Au − f 2 2 + γ min z∈R n Γ(u − z) 1 − p α , u − z + Γz 1 + p α , z ,
where we leave out the particular spaces for the primal (and dual) variables for the sake of simplicity in the following. Taking a closer look at these minimization problems, we observe that we can exactly recover the optimization problem in the first step by means of problem 2. a) if we choose γ = α and p α = 0. We therefore concentrate on the minimization problems in the second step.
Primal-dual and dual formulation
Using the notion of convex conjugates [30], the corresponding primal-dual and dual formulations of our problems are given by a) min u max y1,y2
y 1 , Au + y 2 , Γu − γ p α , u − 1 2 y 1 2 2 − y 1 , f − ι B ∞ γ (y 2 ) = max y1,y2 − 1 2 y 1 2 2 − y 1 , f − ι B ∞ γ (y 2 ) − ι γpα (A * y 1 + Γ * y 2 ), b) min u,z max y1,y2,y3 y 1 , Au + y 2 , Γu − Γz + y 3 , Γz − γ p α , u + 2γ p α , z − 1 2 y 1 2 2 − y 1 , f − ι B ∞ γ (y 2 ) − ι B ∞ γ (y 3 ) = max y1,y2,y3 − 1 2 y 1 2 2 − y 1 , f − ι B ∞ γ (y 2 ) − ι B ∞ γ (y 3 ) − ι γpα (A * y 1 + Γ * y 2 ) − ι −2γpα (−Γ * y 2 + Γ * y 3 ),
Algorithm 1 Primal-Dual Algorithm for Variational Regularization (Step 1) Input: f , α > 0 Initialization: σ, τ > 0, u 0 =ū 0 = 0, y 0 1 = y 0 2 = 0 while not converged do y k+1
1 = y k 1 +σAu k −σf 1+σ y k+1 2 = Π B ∞ α (y k 2 + σΓu k ) u k+1 = u k − τ (A * y k+1 1 + Γ * y k+1 2 ) u k+1 = 2u k+1 − u k end while return u α = u k+1 , p α = 1 α A * (f − Au α ) (c.f. (3.2))
Algorithm 2 Primal-Dual Algorithm for Bias-Reduction with M B f (Step 2 a)) Input: f , γ > 0, p α , which is obtained via Algorithm 1 Initialization: σ, τ > 0, u 0 =ū 0 = 0, y 0 1 = y 0 2 = 0 while not converged do y k+1
1 = y k 1 +σAu k −σf 1+σ y k+1 2 = Π B ∞ γ (y k 2 + σΓu k ) u k+1 = u k − τ (A * y k+1 1 + Γ * y k+1 2 − γp α ) u k+1 = 2u k+1 − u k end while returnû α = u k+1
Algorithm 3 Primal-Dual Algorithm for Bias-Reduction with M IC f (Step 2 b)) Input: f , γ > 0 and p α , which is obtained via Algorithm 1.
Initialization: σ, τ > 0, u 0 = z 0 =ū 0 =z 0 = 0, y 0 1 = y 0 2 = y 0 3 = 0 while not converged do y k+1 1 = y k 1 +σAu k −σf 1+σ y k+1 2 = Π B ∞ γ (y k 2 + σΓ(u k − z k )) y k+1 3 = Π B ∞ γ (y k 3 + σΓz k ) u k+1 = u k − τ (A * y k+1 1 + Γ * y k+1 2 − γp α ) z k+1 = z k − τ (−Γ * y k+1 2 + Γ * y k+1 3 + 2γp α ) u k+1 = 2u k+1 − u k z k+1 = 2z k+1 − z k end while returnû α = u k+1
Solution with a primal-dual algorithm
In order to find a saddle point of the primal-dual formulations, we apply a version of the popular first-order primal-dual algorithms [28,19,14]. The basic idea is to perform gradient descent on the primal and gradient ascent on the dual variables. Whenever the involved functionals are not differentiable, here the 1 -norm, this comes down to computing the corresponding proximal mappings. The specific updates needed for our method are summarized in Algorithm 1 for the first regularization problem, and Algorithm 2 and Algorithm 3 for the two different debiasing steps.
We comment on our choice of the stopping criterion. We consider the primal-dual gap of our saddle point problem, which is defined as the difference between the primal and the dual problem for the current values of variables. As in the course of iterations the algorithm is approaching the saddle point, this gap converges to zero. Hence we consider our algorithm converged if this gap is below a certain threshold 1 > 0. We point out that the indicator functions regarding the ∞ -balls are always zero due to the projection of the dual variables in every update. Since the constraints with respect to the other indicator functions, for example A * y 1 + Γ * y 2 − γp α = 0 in case a), are hard to satisfy exactly numerically, we instead control that the norm of the left-hand side is smaller than a certain threshold 2 (respectively 3 for case b)). All in all we stop the algorithm if the current iterates satisfy: a) P D(u, y 1 , y 2 ) = − γ p α , u
+ 1 2 Au − f 2 2 + γ Γu 1 + 1 2 y 1 2 2 + y 1 , f /n < 1
and A * y 1 + Γ * y 2 − γp α 1 /n < 2 b) P D(u, z, y 1 , y 2 ) = − γ p α , u + 2γ p α , z
A * y 1 + Γ * y 2 − γp α 1 /n < 2 , − Γ * y 2 + Γ * y 3 + 2γp α 1 /n < 3 .
Note that we normalize the primal-dual gap and the constraints by the number of primal pixels n in order to keep the thresholds 1 , 2 and 3 independent of varying image resolutions. We give an example for the specific choice of parameters for our total variation denoising problems in Table 5.
Numerical Results
This section provides further experiments and numerical results that illustrate the proposed debiasing method.
1 -deconvolution
The first application that we illustrate is the deconvolution of a one-dimensional signal using anisotropic shrinkage (4.6). Figure 7 displays the original signal, the blurry signal corrupted by additive Gaussian noise with standard deviation σ = 0.05, the 1 -reconstructed signal and the debiased signals computed over the Bregman manifold M B f and the infimal convolution subspace M IC f . The last two completely overlap on these two plots. One can see that provided that the 1 -reconstruction finds the right peak locations, the debiasing method is able to restore the amplitude of the original signal. Figure 8 displays the evolution of the average bias of the estimated signals as well as the standard deviation of the error. They were computed over one thousand noisy realizations for the noisy, 1 -reconstructed and debiased signals, as a function of the regularization parameter α. These curves illustrate several behaviors: As expected, the residual variance decreases when the regularization parameter increases. For a very low value of α, the debiasing reintroduces some noise so the average variance is higher than for the 1 -reconstructed signal, revealing the biasvariance trade-off that has to be settled. As α increases, the gap between the variance of the 1 -reconstructed and debiased signal vanishes. On the other hand, the average bias is indeed smaller for the debiased signal than for the 1reconstructed signal. Besides, for small values of the regularization parameter the average bias for the debiased signal is stable and close to zero, showing the effective reduction of the method bias. Then it increases by steps which correspond to the progressive vanishing of the peaks, related to model bias. All in all, these plots show the ability of the proposed approach to reduce the method bias (here, the loss of intensity on the peaks), hence allowing for more efficient noise reduction and reconstruction for a wider range of regularization parameters.
Anisotropic TV denoising
In this subsection we study debiasing by means of the discrete ROF-model [31] given by:
u α (f ) ∈ arg min u∈R n 1 2 u − f 2 2 + α Γu 1 , (6.1)
where the 1-norm is anisotropic, i.e.
Γu 1 = m/2 i=1 |(Γu) 1,i | + |(Γu) 2,i |,
Cartoon image
The Giraffe cartoon image has been designed not to have model bias; it is piecewise constant, which makes it suitable for TV denoising and allows us to study the reduction of the method bias only. It takes values in [0, 1] and has been artificially corrupted with additive Gaussian noise with zero mean and variance σ 2 = 0.05, reaching an initial PSNR of about 13dB. The original image and a noisy realization are already displayed on the first line of Fig. 3 in Section 3. Figure 10 displays the TV denoising result as well as the debiased solutions computed on the Bregman manifold M B f or the infimal convolution subspace M IC f for different values of the regularization parameter α. On the first line, α = 0.15 is the optimal regularization parameter for TV denoising (in terms of PSNR, see Fig. 9-(b)). However, when performing the debiasing, noise is strongly amplified. On the second line, α = 0.3 is the optimal regularization parameter for debiasing, and overall, (in terms of PSNR, see Fig. 9-(b)). On the third line α = 0.6 leads to an oversmoothed solution, but the debiasing step still allows to recover a lot of the lost contrast.
Since we expect the variational method to systematically underestimate the value of the regularization functional and overestimates the residual (see [1] for a precise computation on singular values), we compare the corresponding quantities when varying α in Figure 9-(a). We observe that for a very large range of values of α there appears to be an almost constant offset between the values for the solution u α (f ) and the debiased solutionû α (f ) (except for very small values of α, when noise dominates). This seems to be due to the fact that the debiasing step can correct the bias in the regularization functional (here total variation) and residual to a certain extent. This corresponds well to the plot of PSNR vs. α in Fig. 9-(b), which confirms that the PSNR after the debiasing step is significantly larger than the one in u α (f ) for a large range of values of α, which contains the ones relevant in practice. The fact that the PSNR is decreased by the debiasing step for very small α corresponds to the fact that indeed the noise is amplified in such a case, visible also in the plots for the smallest value of α in Figure 10.
Altogether, these results show that the proposed debiasing approach improves the denoising of the cartoon image both visually and quantitatively.
Natural image
The debiasing can also be evaluated on natural images such as the Parrot picture. TV denoising on such images leads to both method bias and model bias. We expect to reduce the former with the proposed method, while the latter is due to the piecewise constant approximation associated with the ROF-model. The Parrot image takes values in [0, 1] and has been artificially corrupted with additive Gaussian noise with zero mean and variance σ = 0.05, reaching an initial PSNR of about 13dB. The original image and a noisy realization are displayed on the first line of Figure 13.
Analogously to Figure 10, Figure 13 also displays the TV denoising result as well as the de- biased solutions computed on the Bregman subspace or the infimal convolution subspace for different values of the regularization parameter α. On the second line, α = 0.15 is the optimal regularization parameter for TV denoising (in terms of PSNR, see Fig. 9-(c)). However, when performing the debiasing, the remaining noise is strongly amplified. On the third line, α = 0.3 is the optimal regularization parameter for debiasing (in terms of PSNR, see Fig. 9-(c)). On the fourth line α = 0.6 leads to an oversmoothed solution but the debiasing step still allows to recover the lost contrast. Note that in the Parrot case, the optimal re-sult in terms of PSNR is obtained for the TV denoising, for α = 0.15. However, the debiasing obtained with α = 0.3 visually provides a smoother result on the background, while preserving the fine structures such as the stripes around the eye. Note also that in each case the artifacts of TV denoising such as staircasing remain and even become more apparent. This however seems natural as the contrast is increased. Since these issues are in fact model bias they are not dealt with by the debiasing method we perform here, but could be reduced by an appropriate choice of regularization such as total generalized vari-ation [3].
Statistical behavior
For both images, the statistical behavior of the proposed debiasing methods can be evaluated by computing the statistical bias E[u * −Û ] as well as the variance Var[u * −Û ] between the true image u * and an estimatorÛ . In our case this is either the solution of the ROF-model (6.1) or the corresponding debiased result. Figure 11 displays the evolution of the estimated statistical bias and standard deviation of the TV, Bregman debiased and infimal convolution debiased estimators for the cartoon Giraffe and natural Parrot images, as a function of the regularization parameter α. These curves reflect some interesting behaviors: As expected, the residual variance decreases as the regularization parameter increases. Besides, the variance is always slightly higher for the debiased solutions, which reflects the bias-variance compromise that has to be settled. However, as the regularization parameter increases, the gap between the denoised and debiased variance decreases. On the other hand, as the regularization parameter grows, the bias increases for each method, and it always remains higher for the denoised solutions than for the debiased solutions. One interesting fact is the behavior of the bias curve for the cartoon Giraffe image: for low values of the regularization parameter (up to α ≈ 0.3), the evolution of the bias for the debiased solutions is relatively stable. This means that for those values, one can increase the regularization parameter in order to reduce the variance without introducing too much (at this point, method) bias. Then, for higher regularization parameters the bias increases in a steeper way, parallel to the evolution of the original bias for the TV denoised image. This reflects the evolution of the model bias from this point on, when the high regularization parameter provides a model subspace whose elements are too smooth compared to the true image. For the natural Parrot image, the model bias occurs even for small values of the regularization parameter, because the model manifold provided by the TV regularization does not properly fit the image prior.
These curves also illustrate the optimal biasvariance balance that can be achieved with or without the debiasing procedure. Intuitively, one would expect the optimal bias-variance trade-off to be reached when the bias and the standard deviation curves intersect each other. This is indeed confirmed by the PSNR curves from Fig. 9-(b) and 9-(c). Looking at those intersection points on both curves for the TV denoised solution on the one hand and for the debiased solutions on the other hand, one can see that the optimal compromise for the debiasing is reached for a higher regularization parameter than for the denoising. This offers more denoising performance, and it leads to a smaller (for the Giraffe image) or equal (for the Parrot image) average bias and standard deviation.
Isotropic TV denoising
Finally, we extend the examples presented in [16] with a few numerical results for isotropic TV denoising:
Γu 1 = m/2 i=1 |(Γu) 1,i | 2 + |(Γu) 2,i | 2 .
We then compare the denoising result to the solutions provided by the two alternative second steps of our debiasing method. Moreover, we also compare them to the result obtained from Bregman iterations. Figure 12 displays the optimal (in terms of PSNR) denoising and debiasing for the Giraffe and Parrot images. The regularization parameter has been set to α = 0.2 for the denoising result and to α = 0.3 for the debiasing. Similarly to the anisotropic case, the debiasing both visually and quantitatively improves the quality of the cartoon Giraffe image. For the natural Parrot image, even though the PSNR is not improved by the debiasing process, one can still observe that the higher regularization parameter offers a better denoising of the background, while the debiasing guarantees that the fine structures around the eye are preserved with a good contrast. Besides, the proposed debiasing approach offers similar results to Bregman iterations, displayed in the fourth column. However, the interesting aspect of our debiasing approach is that we only apply a twostep procedure, while Bregman iterations have to be performed iteratively with a sufficiently high number of steps. Note that our numerical approach to debiasing (see Section 5) is actually equivalent to performing one Bregman iteration with zero initialization of the subgradient, then updating the subgradient and solving a second Bregman step with a sufficiently high regularization parameter.
Conclusion
We have introduced two variational debiasing schemes based on Bregman distances and their infimal convolution, which are applicable for nonsmooth convex regularizations and generalize known debiasing approaches for 1 and TV-type regularization. Based on a recent axiomatic approach to debiasing by Deledalle and coworkers [16], which we further generalized towards infinite-dimensional problems, we were able to provide a theoretical basis of our debi-asing approach and work out meaningful model manifolds for variational methods. Moreover, we were able to relate the approach to Bregman iterations and inverse scale space methods.
From the numerical experiments we observe that the debiasing scheme improves the results for a wide range of regularization parameters, which includes the ones providing optimal results. Surprisingly, we often find visually optimal choices of the regularization parameters in the range where bias and standard deviation of the debiased solution are approximately of the Various questions remain open for future studies: one might study the generalization to other regularization schemes such as total generalized variation [3], spatially adaptive methods that would further reduce the model bias [21] or nonlocal methods for improved results on natural images. As already indicated in the introduction, the method is theoretically not restricted to squared Hilbert-space norms. Instead, it can be carried out for any suitable data fidelity H and we expect it to improve the results. From a theoretical, and in particular from a statistical viewpoint, the question is then how to relate the method to actual bias reduction, and how to properly motivate and define bias in this setting.
Another further improvement might be achieved by only approximating the model manifold by tuning the parameter γ without letting it tend to infinity.
We acknowledge a very recent and related work on the topic from another perspective, which has been developed in parallel to this work [17]. It will be interesting to investigate the connections in future work.
Appendix
We have included some examples and proofs in the Appendix in order not to interrupt the flow of the paper. These are in particular the proof for shrinkage and the calculation of the corresponding derivatives for isotropic and anisotropic shrinkage in Example 4.7, and the calculation of the infimal convolution of two 1 -Bregman distances in Example 4.11.
Shrinkage
Let f ∈ 2 (R d ) be a vector-valued signal for d ∈ N. Then the solution of the isotropic shrinkage problem
u α (f ) ∈ arg min u∈ 1 (R d ) 1 2 u − f 2 2 (R d ) + α u 1 (R d )
is given by the isotropic soft-thresholding
[u α (f )] i = (1 − α |fi| )f i , |f i | > α, 0, |f i | ≤ α.
Proof. We first point out, that the objective allows to exploit strong duality. Following [2,
dom( · 1 (R d ) ) ∩ cont 1 2 · −f 2 2 (R d ) = ∅.
Since the 1 (R d )-norm has full domain and the 2 (R d )-norm is continuous everywhere, this is trivially fulfilled. Hence, by the dual definition of the 1 (R d )-norm we find min u∈ 1 (R d )
1 2 u − f 2 2 (R d ) + α u 1 (R d ) = min u∈ 1 (R d ) sup r∈ ∞ (R d ) r ∞ (R d ) ≤α 1 2 u − f 2 2 (R d ) + r, u = sup r ∞ (R d ) ≤α min u∈ 1 (R d ) 1 2 u − f 2 2 (R d ) + r, u ,
where we used strong duality to interchange the infimum and the supremum. We can explicitely compute the minimizer for u as u = f − r and hence
sup r ∞ (R d ) ≤α min u∈ 1 (R d ) 1 2 u − f 2 2 (R d ) + r, u = sup r ∞ (R d ) ≤α − 1 2 r 2 2 (R d ) + r, f .
This supremum can be computed explicitely pointwise with the corresponding Lagrangian
L(r i , λ) = − 1 2 |r i | 2 + r i · f i + λ(|r i | 2 − α 2 )
with λ ≤ 0. Note that both the objective function and the constraints are continuously differentiable and that Slater's condition holds. Optimality with respect to r i yields
f i − r i + 2λr i = 0
and hence
r i = f i 1 − 2λ .
We distinguish two cases:
If |r i | = α, then α(1 − 2λ) = |f i | and u i = f i − r i = f i − f i 1 − 2λ = (1 − α |f i | )f i .
The nonpositivity of λ implies that |f i | ≥ α. In case |r i | < α, we obtain that λ = 0 and hence r i = f i and u i = 0 when |f i | < α. Note that since f ∈ 2 (R d ) there exists a finite N such that |f i | ≤ α for all i > N . Hence trivially u α (f ) ∈ 1 (R d ) as i∈N | [u α (f )] i | is a finite sum. This yields the assertion.
Remark: For d = 1 and a square-summable sequence f ∈ 2 we immediately obtain the anisotropic case: The solution to u α ∈ arg min
u∈ 1 1 2 u − f 2 2 + α u 1 (8.1)
for α > 0 is given by
[u α (f )] i = f i − α sign(f i ), |f i | ≥ α 0, |f i | < α.
Directional derivative: The computation of the directional derivative requires a little more work. At first, let us compute the directional derivative of the function F :
R d \{0} → R, x → 1 |x| into the direction g ∈ R d . We define G : R d \{0} → R, x → 1 |x| 2 and calculate dG(x; g) = lim t→0 + G(x + tg) − G(x) t = lim t→0 + 1 t 1 |x + tg| 2 − 1 |x| 2 = lim t→0 + 1 t |x| 2 − |x + tg| 2 |x| 2 |x + tg| 2 = lim t→0 + 1 t −2tx · g − t 2 |g| 2 |x| 2 |x + tg| 2 = −2 x · g |x| 4 .
Then by the chain rule we obtain
dF (x; g) = d √ G(x; g) = dG(x; g) 2 G(x) = −2 x · g |x| 4 |x| 2 = − x · g |x| 3 .
Let us further define the projection of a vector x ∈ R d onto another vector y ∈ R d \{0} as Π y (x) = y · x |y| 2 y.
We now have to compute
[du α (f ; g)] i = lim t→0 + 1 t [u α (f + tg)] i − [u α (f )] i
and we can distinguish four cases: Let at first |f i | > α. Then for t small enough we have |f i + tg i | > α and hence
lim t→0 + 1 t [u α (f + tg)] i − [u α (f )] i = lim t→0 + 1 t 1 − α |f i + tg i | (f i + tg i ) − 1 − α |f i | f i = lim t→0 + 1 t f i + tg i − α f i + tg i |f i + tg i | − f i + α f i |f i | = lim t→0 + 1 t tg i − αtg i |f i + tg i | −αf i 1 |f i + tg i | − 1 |f i | = g i − α g i |f i | + αf i f i · g i |f i | 3 = g i + α |f i | (Π fi (g i ) − g i ) .
For |f i | < α and t small enough we easily find |f i + tg i | < α and hence
[du α (f ; g)] i = 0.
In case |f i | = α we need to distinguish whether |f i + tg i | > α or |f i + tg i | ≤ α for arbitrarily small t. We hence compute
|f i + tg i | > α ⇔ |f i + tg i | 2 > α 2 ⇔ |f i | 2 + 2tf i · g i + t 2 |g i | 2 > α 2 ⇔ 2f i · g i + t|g i | 2 > 0,
which for arbitrarily small t is true only if f i ·g i ≥ 0. Analogously we find that |f i + tg i | < α for small t is only true if f i · g i < 0.
Hence let now |f i | = α and f i · g i ≥ 0. Then we obtain
[du α (f ; g)] i = lim t→0 + 1 t [u α (f + tg)] i = lim t→0 + 1 t 1 − α |f i + tg i | (f i + tg i ) .
Using α = |f i |, we find
lim t→0 + |f i |f i t 1 |f i | - 1 |f i + tg i | + g i - |f i |g i |f i + tg i | = |f i |f i f i · g i |f i | 3 = Π fi (g i ).
In the last case |f i | = α and f i · g i < 0, we find
[du α (f ; g)] i = lim t→0 + 1 t [u α (f + tg)] i = 0.
Summing up we have
[du α (f ;g)] i = g i + α |fi| (Π fi (g i ) − g i ) , |f i | > α, 0, |f i | < α, Π fi (g i ), |f i | = α, f i · g i > 0, 0, |f i | = α, f i · g i ≤ 0.
It remains to show that
u α (f + tg) − u α (f ) t − du α (f ; g) 1 (R d ) → 0 for t → 0 + . Again, since f ∈ 2 (R d ), there exists N ∈ N such that |f i | < α and hence [du α (f ; g)] i = 0 for all i > N .
The difference quotient as well vanishes for all i > N , hence the above 1 norm is a finite sum and thus we trivially obtain convergence in 1 (R d ).
Remark: For d = 1 and f ∈ 2 we obtain the anisotropic result:
[du α (f ; g)] i = g i , |f i | > α 0, |f i | < α g i , |f i | = α, sign(f i ) = sign(g i ) 0, |f i | = α, sign(f i ) = sign(g i ),
where we mention that here Π fi (g i ) = g i .
Model manifold:
The corresponding (isotropic) model manifold is given by
u ∈ M G f ⇔ u i = v ∈ R d , |f i | > α, 0, |f i | < α, λf i , λ ≥ 0, |f i | = α.
Analogously to the anisotropic case discussed in Example 4.7, the model manifold allows for arbitrary elements, here even including the direction, if the magnitude |f i | of the signal is strictly above the threshold parameter α. As already discussed in Example 4.7, |f i | = α is the odd case of the three, since in contrast to |f i | > α it only allows for changes into the direction of the signal f i . If we exclude that case, we again find a linear derivative, hence a Gâteaux derivative and even a Fréchet derivative. Accordingly the isotropic shrinkage is the immediate generalization of the anisotropic shrinkage, which we can find as a special case for d = 1.
Summing up, the debiasing procedure on this manifold again yields the solution of hard thresholding:
[û(f )] i = f i , |f i | ≥ α, 0, |f i | < α.
Note that we again maintain the signal directly on the threshold.
Infimal convolution of 1 Bregman distances
Theorem 8.1. Let Γ : 2 (R n ) → 1 (R m ) be linear and bounded and J(u) = Γu 1 (R m ) for m, n ∈ N. Let further q α ∈ ∂ · 1 (R m ) (Γu α ) such that p α = Γ * q α . Then ICB qα 1 (R m ) (Γu, Γu α ) ≤ ICB pα J (u, u α ).
Proof.
ICB pα J (u, u α ) = inf Note that we get equality for surjective Γ in Theorem 8.1.
Theorem 8.2. Let v, u ∈ 1 (R m ) and q ∈ ∂ v 1 (R m ) . Then
ICB q 1 (R m ) (u, v) = i∈N G(u i , q i ) with G : R m × R m → R defined as G(u i , q i ) = |u i |(1 − | cos(ϕ i )||q i |), |q i | < | cos(ϕ i )|, |u i || sin(ϕ i )| 1 − |q i | 2 , |q i | ≥ | cos(ϕ i )|.
where ϕ i denotes the angle between u i and q i , i.e. cos(ϕ i )|u i ||q i | = u i · q i with ϕ i := 0 for q i = 0 or u i = 0.
Proof. Let
f 1 (u) = D q 1 (R m ) (u, v) = u 1 (R m ) − q, u , f 2 (u) = D −q 1 (R m ) (u, −v)= u 1 (R m ) + q, u .
Since (f 1 2f 2 ) * = f * 1 + f * 2 and by the definition of the biconjugate, we know that
f 1 2f 2 ≥ (f * 1 + f * 2 ) * .
(1) We shall first compute the right-hand side. We have
f * 1 (w) = ι B ∞ (1) (w + q), f * 2 (w) = ι B ∞ (1) (w − q),
where ι B ∞ (1) denotes the characteristic function of the ∞ (R m )-ball
B ∞ (1) = w ∈ ∞ (R m ) | w ∞ (R m ) ≤ 1 . Thus (f * 1 + f * 2 ) * (u) = sup w∈ ∞ (R m ) u, w s.t. w + q ∞ (R m ) ≤ 1, w − q ∞ (R m ) ≤ 1.
Taking into account the specific form of these constraints, we can carry out the computation pointwise, i.e. sup wi∈R m u i · w i s.t. |w i + q i | ≤ 1, |w i − q i | ≤ 1.
From now on we drop the dependence on i for simplicity.
• Let us first consider the case |q| = 1. We immediately deduce that w = 0 and u · w = 0.
• Hence we assume |q| < 1 from now on, and set up the corresponding Lagrangian L(w, λ, µ) = −w · u + λ(|w − q| 2 − 1) + µ(|w + q| 2 − 1). (8.2) Both the objective functional and the constraints are differentiable, so every optimal point of (8.2) has to fulfill the four Karush-Kuhn-Tucker conditions, namely ∂ ∂w L(w, λ, µ) = 0, λ(|w − q| 2 − 1) = 0, λ, µ ≥ 0, µ(|w + q| 2 − 1) = 0, Slater's condition implies the existence of Lagrange multipliers for a KKT-point of (8.2). The first KKT-condition yields −u + 2λ(w − q) + 2µ(w + q) = 0. (8.3) * Let us first remark that the case u = 0 causes the objective function to vanish anyway, hence in the following u = 0.
* Then let us address the case q = 0 in which (8.3) yields u = 2(λ + µ)w.
In case |w| = 1 we find that 2(λ+µ) = |u|, hence w = u |u| . We infer w · u = u · u |u| = |u|.
Note that for |w| < 1, we find that λ = µ = 0 and hence u = 0. * If q = 0, we can distinguish four cases: 1st case: |w − q| 2 < 1, |w + q| 2 = 1. Thus λ = 0 and (8.3) yields u = 2µ(w + q).
Since |w + q| 2 = 1, we deduce µ = |u|/2, so w = u |u| − q and finally for the value of the objective function
w · u = u |u| − q · u = |u| − q · u.
2nd case: |w + q| 2 < 1, |w − q| 2 = 1. We analogously find w · u = |u| + q · u.
The first two cases thus occur whenever (insert w into the conditions) u |u| − 2q < 1 or u |u| + 2q < 1.
We calculate u |u| − 2q 2 < 1 ⇔ |q| 2 < q · u |u| ⇔ |q| < cos(ϕ).
Hence q · u > 0 and |u| − q · u = |u| − |q · u|.
In the second case we analogously find |q| < − cos(ϕ), hence q · u < 0 and |u| + q · u = |u| − |q · u|, so we may summarize the first two cases as w · u = |u| − |q · u| = |u|(1 − | cos(ϕ)||q|), whenever |q| < | cos(ϕ)|.
3rd case: |w − q| 2 = 1, |w + q| 2 = 1. At first we observe that from |w + q| 2 = |w − q| 2 we may deduce that w · q = 0. Therefore we have |w + q| 2 = 1 ⇒ |w| = 1 − |q| 2 .
We multiply the optimality condition (8.3) by q and obtain u · q = 2λ(w − q) · q + 2µ(w + q) · q ⇔ u · q = 2(µ − λ) |q| 2 ⇔ (µ − λ) = u 2 · q |q| 2 .
Multiplying (8.3) by w yields
u · w = 2(λ + µ)|w| 2 and another multiplication of (8.3) by u yields |u| 2 = 2(λ + µ)w · u + 2(µ − λ)q · u = 4(λ + µ) 2 |w| 2 + u · q |q| 2 , where we inserted the previous results in the last two steps. We rearrange and find 2(λ + µ) = |u| 2 − u · q |q| 2 |w| −1 .
Note that |w| > 0 since |q| < 1. This finally leads us to u · w = 2(λ + µ)|w| 2
= |u| 2 − u · q |q| 2 |w| = |u| 1 − u |u| · q |q| 2 (1 − |q| 2 ) = |u| (1 − | cos(ϕ)| 2 ) (1 − |q| 2 )
= |u|| sin(ϕ)| (1 − |q| 2 ).
4th case: |w − q| 2 < 1, |w + q| 2 < 1.
Here the first KKT-condition yields u = 0, which can only occur if the objective function w · u vanishes anyway. Summing up, we have
(f * 1 + f * 2 ) * (u) = i∈N G(u i , q i ) ≤ u 1 (R m ) .
(2) It remains to show that
(f 1 2f 2 )(u) = inf z∈ 1 (R m ) i∈N g i (z i ) ≤ (f * 1 + f * 2 ) * (u), where g i (z i ) = |u i − z i | + |z i | − q i · (u i − 2z i ) ≥ 0.
Again we need to distinguish four cases.
1st case: If |q i | < cos(ϕ i ), we have q i · u i > 0 and we can choose z i = 0 to obtain
g i (z i ) = |u i | − q i · u i = |u i | − |q i · u i |.
2nd case: Analogously if |q i | < − cos(ϕ i ), we have q i · u i < 0 and choose z i = u i , thus
g i (z i ) = |u i | + q i · u i = |u i | − |q i · u i |.
3rd case: If |q i | = 1, we compute for z i = ui 2 − c 2 q i , c > 0,
g i (z i ) = u i 2 + c 2 q i + u i 2 − c 2 q i − c|q i | 2 = c 2 q i + u i c + q i − u i c − 2 .
Using a Taylor expansion around q we obtain
q i + u i c = |q i | + q i |q i | · u i c + O(c −2 ), q i − u i c = |q i | − q i |q i | · u i c + O(c −2 ).
Hence with |q i | = 1 we find
g i (z i ) = c 2 (2|q i | + O(c −2 ) − 2) = O(c −1 ) → 0
for c → ∞. Hence for every ε there exists a c i > 0 such that g i (z i ) ≤ ε/2 i .
4th case: Finally, if |q i | ≥ | cos(ϕ i )| and |q i | < 1, we pick z i = 2λ i (w i −q i ), with λ i and w i being the Lagrange multiplier and the dual variable from the above computation of (f * 1 + f * 2 ) * . It is easy to see that
g i (z i ) = |u i || sin(ϕ i )| 1 − |q i | 2 .
Hence we define z := (z i ) i such that
z i = 0, if |q i | < cos(ϕ i ), u i , if |q i | < − cos(ϕ i ), ui 2 − ci 2 q i , if |q i | = 1, λ i (w i − q i ) if |q i | ≥ | cos(ϕ i )|, |q i | < 1.
Let z N denote z truncated at index N ∈ N, i.e.
z N i = z i , if i ≤ N, 0, else.
Then trivially z N ∈ 1 (R m ) and we compute
(f 1 2f 2 )(u) ≤ i∈N g i (z N i ) ≤ N i=1 G(u i , q i ) + ε 2 i + ∞ i=N +1 g i (0) = ∞ i=1 G(u i , q i ) + N i=1 ε 2 i + ∞ i=N +1 |u i | − q i · u i − G(u i , q i ) ≤ ∞ i=1 G(u i , q i ) + N i=1 ε 2 i + 3 ∞ i=N +1 |u i | → ∞ i=1 G(u i , q i ) + ε
as N → ∞. This completes the proof.
Figure 1 :
1Illustration of the bias of the ROF model on a 1D signal. (a) Original signal, and noisy signal corrupted by additive Gaussian noise. (b) Restoration of the noisy signal with TV regularization and Bregman iterations. The TV reconstruction recovers the structure of the signal but suffers from a loss of contrast, which is however well recovered with Bregman iterations.
Figure 3 :
3Denoising of a cartoon image. First row: original image, noisy image corrupted by Gaussian noise.
Theorem 3 . 2 .
32Let the conditions of Theorem 3.1 hold and let p ∈ ∂J(0) ∩ Z ⊂ X * be such that there exists w with J * p − Bw τ = 0 for some 0 < τ < 1. Then there exists a minimizer of min u∈X 1
1 2
1Au 2 Y + J(u) on S. The remaining steps follow the proof of Theorem 3.1.
From this example, we cannot immediately state that M B f ⊂ M IC f , because so far we only know that M B f as well as M IC f are subsets of the set {u ∈ X | ICB qα 1 (R m ) (Γu, Γu α ) = 0}. However, in the next subsection we see that M B f ⊂ M IC f is indeed true and it is actually a general property of the variational model manifolds.
where we use the one-to-one transform z = cw for c > 0 and z = c(u − w) for c < 0. This implies that ICB pJ (cu, v) = 0 if ICB p J (u, v) = 0. Now let u 1 , u 2 ∈ M IC , i.e.ICB p J (u 1 , v) = 0 and ICB p J (u 2 , v) = 0. Then by definition of the infimum there exist sequences (z n 1 ) n∈N , (z n 2 ) n∈N such that lim n→∞ J(u 1 − z n 1 ) + J(z n 1 ) − p, u 1 − 2z n 1 = 0, lim n→∞ J(u 2 − z n 2 ) + J(z n 2 ) − p, u 2 − 2z n 2 = 0.
Lemma 4 . 15 .
415Let J be strictly convex. Then M B is a singleton. Proof. For strictly convex J the mapping u → D p J (u, v) is strictly convex as well, hence D p J (u, v) = 0 if and only if u = v and M B = {v}.
Γu − Γz 1 + γ Γz 1
Figure 7 :
71 -deconvolution of a 1D signal. Original and noisy convolved signals, and 1reconstruction, Bregman debiasing and Infimal convolution debiasing.
Figure 8 :
81 -deconvolution of a 1D signal. Average bias and variance computed over one thousand realizations of the noisy signal for the noisy, restored and debiased signals.
Figure 9 :
9with (Γu) 1 and (Γu) 2 denoting the discrete gradient images in horizontal and vertical direction, respectively. We compare the original denoising result of Problem (6.1) to the proposed debiased solutions obtained with the Bregman manifold M B f or the infimal convolution subspace M IC f . Evolution of (a) The total variation and the residual for the cartoon Giraffe image, (b) The average PSNR for TV denoising, Bregman debiasing and infimal convolution debiasing for the cartoon Giraffe image and (c) The average PSNR for TV denoising, Bregman debiasing and infimal convolution debiasing for the natural Parrot image as a function of the regularization parameter α.
Figure 10 :
10Denoising of the Giraffe cartoon image for different values of the regularization parameter α. First column: TV denoising. Second column: Debiasing on the Bregman manifold. Third column: Debiasing on the infimal convolution subspace.
Figure 11 :
11Evolution of the average residual bias and standard deviation computed over 500 noisy realizations of (a) Giraffe and (b) Parrot for TV denoising, Bregman debiasing and infimal convolution debiasing.
Figure 13 :
13Denoising of the Parrot image for different values of the regularization parameter α. First column: TV denoising. Second column: Debiasing on the Bregman manifold. Third column: Debiasing on the infimal convolution subspace. same size.
Γ
(u − z) 1 (R m ) − p α , u − z + Γz 1 (R m ) + p α , z = inf z∈ 2 (R n ) Γ(u − z) 1 (R m ) − q α , Γ(u − z) + Γz 1 (R m ) + q α , Γz = inf Γz∈ 1 (R m ) Γ(u − z) 1 (R m ) − q α , Γ(u − z) + Γz 1 (R m ) + q α , Γz ≥ inf w∈ 1 (R m ) Γu − w 1 (R m ) − q α , Γu − w + w 1 (R m ) + q α , w = inf w∈ 1 (R m ) D qα 1 (R m ) (Γu − w, Γu α ) + D −qα 1 (R m ) (w, −Γu α ) = ICB qα 1 (R m ) (Γu, Γu α ).
Color image 1Original image
Noisy image
TV denoising
Bregman debiasing
ICB debiasing
P SN R = 19.63
P SN R = 22.75
P SN R = 22.70
Table 1 :
1Choice of parameters for a total vari-
ation denoising problem of an image of size
256x256 with values in [0, 1], corrupted by Gaus-
sian noise with variance 0.05.
and
P SN R = 25.38 P SN R = 24.69 P SN R = 24.76 P SN R = 24.60 Figure 12: Isotropic TV denoising and debiasing of the cartoon Giraffe and natural Parrot images, and comparison to Bregman iterations.Isotropic TV
Bregman debiasing
ICB debiasing
Bregman iterations
P SN R = 22.14
P SN R = 22.49
P SN R = 22.58
P SN R = 22.97
P SN R = 25.07 P SN R = 18.82 P SN R = 18.44Original image
Noisy image
TV denoising
Bregman debiasing
ICB debiasing
α = 0.15
α = 0.3 (optimal)
P SN R = 23.57
P SN R = 24.19
P SN R = 23.95
α = 0.6
P SN R = 21.20
P SN R = 22.29
P SN R = 22.29
Theorem 4.4.3 and Lemma 4.3.1], strong duality holds if
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[
"Lisa Randall \nMassachusetts Institute of Technology Cambridge\n02139MA\n",
"Nuria Rius \nMassachusetts Institute of Technology Cambridge\n02139MA\n"
] |
[
"Massachusetts Institute of Technology Cambridge\n02139MA",
"Massachusetts Institute of Technology Cambridge\n02139MA"
] |
[] |
We investigate the question of whether an additional light neutral scalar can explain the l + l − γγ events with high invariant mass photon pairs recently observed by the L3 collaboration. We parameterize the low energy effects of the unknown dynamics in terms of higher dimensional effective operators. We show that operators which allow for the scalar to be produced and decay into photon pairs will allow other observable processes that should have been seen in current experiments.
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10.1016/0370-2693(93)90946-f
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[
"https://arxiv.org/pdf/hep-ph/9304226v1.pdf"
] | 15,779,000 |
hep-ph/9304226
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0e94dac3bdfff9fc0df88b9d19c082eb2fe178fa
|
March 1993
Lisa Randall
Massachusetts Institute of Technology Cambridge
02139MA
Nuria Rius
Massachusetts Institute of Technology Cambridge
02139MA
March 1993Submitted to: Physics Letters BarXiv:hep-ph/9304226v1 6 Apr 1993 Why a Scalar Explanation of the L3 Events is Implausible *
We investigate the question of whether an additional light neutral scalar can explain the l + l − γγ events with high invariant mass photon pairs recently observed by the L3 collaboration. We parameterize the low energy effects of the unknown dynamics in terms of higher dimensional effective operators. We show that operators which allow for the scalar to be produced and decay into photon pairs will allow other observable processes that should have been seen in current experiments.
Introduction
The L3 collaboration has observed recently an excess of l + l − γγ events with high invariant mass photon pairs, M γγ ≃ 60 GeV [1]. Both DEL-PHI and ALEPH have two similar events each, and OPAL seems to have one candidate [2]; however these experiments are not so strongly peaked. Naively, such an observation would look like evidence of the discovery of a new neutral particle. However, such an interpretation requires a detailed understanding of the standard model background. A recent calculation of the hard bremsstrahlung process e + e − → µ + µ − γγ [3] yields a significantly higher cross section at the Z peak than previous theoretical predictions had indicated, pointing to the likelihood of a standard model explanation of the L3 events. In this note, we address the question of the likelihood of the discovery of a new scalar from a different vantage point; we ask whether a nonstandard model with such a scalar is consistent with other observational constraints. We systematically investigate the possible scalar couplings which could give rise to the L3 events and show they are almost all excluded.
We analyze the possibility of explaining the L3 events 1 by assuming the existence of a light neutral scalar (φ) of mass m φ ≃ 60 GeV. We assume the scalar φ is a gauge singlet. We parameterize low energy effects of unknown dynamics at a scale M in terms of higher dimensional effective operators, constructed out of the Standard Model (SM) fields and the extra neutral scalar. These operators are suppressed by powers of M 4−d , where d is the dimension of the corresponding operator. We impose SU (2) L × U (1) Y gauge invariance, which relates the process of interest with other observable effects which in general are measurable and allows us to confirm or rule out the models. We consider the most general possible low dimension operators which can explain the observed events. Our results do not rely on assumptions on the expected size of the coefficients of nonrenormalizable operators. We simply analyze other experimental consequences of the operators which could produce the observed events. In almost all cases this alone is sufficient to rule out the operator.
In section 2 we briefly discuss the possibility that the new particle couples only to gauge bosons. In section 3 we assume that the scalar couples to leptons and gauge bosons through lower dimensional gauge invariant effective operators. In section 4 we analyze higher dimensional four body operators and in section 5 we summarize our conclusions.
Scalar Coupled to Gauge Bosons
One can consider a model in which the scalar φ couples to the Z boson and is produced via the reaction
Z → Z * φ .(1)
The case of φ being the lightest CP-even neutral Higgs field (h) in models with a non-minimal Higgs sector, has been analyzed in ref. [5]. To account for the fact that h decays dominantly to 2γ, one can assume that h does not couple to fermions, but has essentially SM-type couplings to the Z and W bosons. This can be achieved within the kind of models referred to as model I in the literature [4], i.e., models with two Higgs doublets of which only one couples to fermions.
However, this simplest possibility encounters immediately an unavoidable problem which is independent of the particular model for the scalar φ. The decays of the Z boson are well known and the process (1) would yield also final states ννγγ, with a branching ratio determined by the well tested SM couplings of the Z to neutrinos. This implies that if we assume BR(Z → Z * φ → l + l − γγ) = 4 × 10 −6 to account for the L3 events, we will immediately obtain BR(Z → Z * φ → ννγγ) = 8 × 10 −6 . The null results of searches for events with high invariant mass photon pairs and missing energy [1] translate into the upper limit BR(Z → ννγγ) < 3 × 10 −6 at 95% CL, if we assume (as we have done in all our estimates) that the detection efficiency is one. Therefore we conclude that the l + l − γγ events can not be explained by the process (1).
Lowest Dimensional Operators
The next model we consider is one in which φ couples to leptons, and is therefore produced at LEP via
e + e − → Z → l + l − φ .(2)
The lowest dimensional gauge invariant operators involving the scalar and two charged leptons are of dimension d=5, namely
O a = 1 MĒ L He R φ (3) O b = 1 Mē R γ µ D µ e R φ , 1 MĒ L γ µ D µ E L φ(4)
where e R refers to the right handed charged lepton, E L is the SU (2) L doublet consisting of the charged left handed lepton and the neutrino and H is the standard Higgs doublet. We first consider O a . Notice that because φ is an SU (2) L singlet, it has to include the standard Higgs doublet. When the standard Higgs gets a vev, v, this operator reduces to a Yukawa coupling v Mē eφ .
We present two arguments against this model. We first assume that the decay of φ into two photons is induced by the gauge invariant effective operator
O = 1 M ′ φ(aB µν B µν + bW i µν W iµν ) (6) where B µν = ∂ µ B ν − ∂ ν B µ , W i µν = ∂ µ W i ν − ∂ ν W i µ + gf ijk W j µ W k ν
and a, b are arbitrary coefficients. When we write this operator in terms of the physical gauge fields, there are three pieces involving the neutral gauge bosons Z and γ:
O γ = d M ′ φF µν F µν , d ≡ ac 2 w + bs 2 w (7) O Z = h M ′ φZ µν Z µν , h ≡ as 2 w + bc 2 w (8) O γZ = 2k M ′ φF µν Z µν , k ≡ (b − a)c w s w (9)
where s w (c w ) denotes the sine (cosine) of the electroweak mixing angle.
The two Feynman diagrams that contribute to the process Z → l + l − φ are depicted in Fig. 1. A tedious but straightforward calculation leads to the following result for the partial decay width:
Γ(Z → l + l − φ) = α s 2 w c 2 w 1 192π 2 v M 2 M Z [v 2 C V (r) + a 2 C A (r)] (10)
where v (a) is the vector (axial) coupling of the lepton to the Z, given by
v = − 1 2 + 2s 2 w a = − 1 2 (11)
and r ≡ (
m φ M Z ) 2 .
The functions C V (r) and C A (r) take the form
C V (r) = 2r 2 F (r) − (1 − 2r − 3r 2 ) log r − 2 + 8r − 6r 2 (12) C A (r) = −2r 2 F (r) − (1 + 8r + 3r 2 ) log r (13) − 11 3 − 5r + 9r 2 − r 3 3(14)
with
F (r) = 2L 2 r 1 + r + log 2 r 1 + r − 1 2 log 2 r − π 2 6 ,(15)
and L 2 is the dilogarithm or Spence function. L3 has a total of ∼ 10 6 Z events and 4 l + l − γγ events have been observed, therefore we assume
BR(Z → l + l − φ) × BR(φ → γγ) ∼ 4 × 10 −6 ,(16)
which yields a lower bound on v/M . We obtain v/M ≥ 3.3, and therefore M ∼ 75 GeV for v = 250 GeV. This scale is so low that the validity of treating this as an effective operator might be questioned. This would be true particularly if one were to assume the operator arose from a loop diagram in a more fundamental theory. Furthermore, this particle would have quite a large width, on the order of its mass. Presumably we should not consider this operator further. We nevertheless show that such an operator is decisively ruled out in any case by current experimental data.
The experimental results on four fermion events at LEP constrain the partial decay width of φ into two photons to be at least of the same order of magnitude as the corresponding decay width into leptons 2 . In our model, these partial widths are, respectively:
Γ(φ → γγ) = d M ′ 2 m 3 φ 2π (17) Γ(φ → l + l − ) = 3 v M 2 m φ 8π (18)
where we have incorporated three lepton flavors. We then impose BR(φ → γγ) ≥ 0.5. This experimental constraint can be satisfied only if d 2 ∼ 13 when M ′ = M . Notice that taking M ′ > M makes d even larger, well beyond the realm of perturbation theory. We therefore take M ′ = M below. Let us analyze now the remaining terms in eq. (9). The operator O γZ leads to the process Z → φγ and the width is easily found to be
Γ(Z → φγ) = s 2 w c 2 w 6π (b − a) 2 M 2 M 2 Z − m 2 φ M Z 3(19)
Since M can be no larger than determined by eq. (16), the difference |b − a| must be less than 5 · 10 −2 . This is determined since BR(Z → 3γ) = BR(Z → φγ) × BR(φ → γγ), and BR(φ → γγ) ≥ 0.5, so we obtain BR(Z → 3γ) ≥ 5× 10 −2 (b− a) 2 . The experimental upper limit for this process, at 95% confidence level, is BR(Z → 3γ) < 1.4 × 10 −4 [6] and thus we conclude that |b − a| < 5 × 10 −2 . This means that in order to be consistent with the experimental data we have to assume b ∼ a and therefore d ∼ h in eq. (9). Finally, the last piece in eq. (9), O Z , contributes to the process Z → Z * φ, which produces final states of two fermions and the scalar when the Z * decays. The expression for the branching ratio of the process Z →f f φ, after the phase space integration, is rather cumbersome, so we only give here the numerical result for the neutrino decay channel:
BR(Z → Z * φ →ννφ) = h M 2 · 3.5 · 10 −2 GeV 2(20)
Since h ∼ d, this branching ratio is entirely determined. Together with the constraint BR(φ → γγ) ≥ 0.5 it implies that BR(Z →ννγγ) ∼ 4 × 10 −5 , which is excluded by LEP data [1].
One can also rule out this model by considering the rate for the scalar to be produced at TRISTAN. The interaction (5) would also yield the direct production e + e − → φ. The cross section for the process e + e − → φ → γγ is easily found to be
σ(s) = 1 4π dv M 2 2 s 2 |s − m 2 φ + im φ Γ φ | 2(21)
where Γ φ is the total width of the scalar and √ s is the center of mass energy.
Using that BR(φ → γγ) ∼ 0.5 we obtain σ = 120 nb, which is inconsistent with current experimental data (σ ∼ 50 nb for √ s = 60
GeV) [7]. This number was obtained within the framework of this model, in which the scalar is very broad. The width determined at LEP would make the situation even worse. We now consider the possibility that the Z decays into φ and two leptons directly through the contact term O b in eq. (4). Since the second of these operators would yield Z → ννφ with a branching ratio of the same order of magnitude as Z → l + l − φ, it is excluded. We therefore assume this operator is suppressed and restrict our attention to the first one.
In terms of the physical gauge fields, O b is written as
O b = 1 M φē R γ µ (∂ µ + ieA µ − ie s w c w Z µ )e R ,(22)
The last term yields the following partial decay width for the process Z → l + l − φ:
Γ(Z → l + l − φ) = α 24π 2 s 2 w c 2 w M 3 Z M 2 H(r)(23)
where H(r) = 3r 2 16 + r 4 log r + 1 8
3 8 + 8r 3 − 3r 2 − r 4 24(24)
and r = (m φ /M Z ) 2 .
Using again equation (16) derived from the L3 events, we obtain
1 M 2 ∼ 2. × 10 −3 .(25)
Although this result implies a very light mass scale (M ∼ 22 GeV), it depends also on the assumptions about unknown coefficient in front of O b . So we choose to study the further consequences of the operator and we will show that it can be excluded solely on an experimental basis. The first term in O b yields also a derivative coupling φll. It is straightforward to compute the width for the decay φ → l + l − induced by this coupling,
Γ(φ → l + l − ) = m φ 16π m l M 2(26)
It is suppressed by the mass of the corresponding lepton, m l , and with the mass scale given by (25) it turns out to be 5.9 × 10 −10 GeV, 2.6 × 10 −5 GeV and 7.5 × 10 −3 GeV for e, µ and τ respectively. The analysis done for the previous operators would not apply here because of the helicity suppression in the operator O b . In this case, the dominant φ decay model is naturally to photons. Furthermore, the production cross section at TRISTAN would be quite small. It is therefore best to rule out the operator directly, by calculating the production cross section for three photons with the mass scale M we have already determined.
We look now at the piece of O b involving the photon field in eq. (22). This term of O b yields e + e − → φγ → 3γ, which can also be measured at LEP. We obtain
σ(e + e − → φγ) = α 8M 2 s − m 2 φ s (27)
As the branching ratio of φ → 2γ is one in very good approximation, we just have to plug in eq.(27) the value of M determined by the L3 experiments to find σ(e + e − → 3γ) ∼ 0.4 nb. On the Z pole, the peak cross-section for Z production is roughly 55 nb. Combined with the experimental upper limit on the branching ratio for Z → 3γ [6], we get σ(e + e − → 3γ) < 8 × 10 −3 nb and thus we conclude that the L3 events can not be due to O b .
Higher Dimension Four Body Operators
If the neutral scalar is a singlet, the available higher dimensional gauge invariant operators have dimension d=7 and there are three kinds of which we present three representatives
O 2 = 1 M 3Ē L (D µ H)e R D µ φ (28) O 3 = 1 M 3Ē L H(D µ e R )D µ φ(29)O 4 = 1 M 3 B µνēR γ µ e R ∂ ν φ(30)
It is worth pointing out that these higher dimensional operators do not contain any vertex involving only two fermions and the scalar. Thus, in all these models φ decays dominantly into two photons, through the effective operator O introduced in section 3 (eq. (6)). Furthermore, unlike the lower dimensional operators considered in section 3, they can not be probed by direct production of the scalar at TRISTAN. Recall that in principle there are also operators analogous to O 4 but involving the left handed SU (2) L doublets; however those would once again yield the unobserved ννγγ events at LEP.
Notice that there can not be dimension 6 operators involving the Higgs field because, as it is an SU (2) L doublet, both E L and e R are necessary to make an SU (2) L × U (1) Y invariant and by Lorentz invariance this implies that two covariant derivatives are needed and therefore the lowest dimensional operator has dimension d=7.
There are many other operators involving two covariant derivatives as they can act on any couple of the four fields involved, as well as both on the same field. However, there is an essential difference between O 2 -type operators, in which one of the covariant derivatives acts on the Higgs doublet, and operators of type O 3 , in which no covariant derivative acts on the Higgs. The reason is that O 2 -type operators only contain Z and W gauge bosons and we will show that this fact prevents us from ruling them out with current experimental data. Operators of the O 3 kind involve also the photon and thus they yield to 3γ events at LEP with a cross section directly dictated by the related Z → l + l − γγ branching ratio, as we have shown for the lower dimensional operator O b in section 3. The same applies to O 4 .
In particular, the operator O 4 defined in eq. (30) has qualitatively the same consequences as the operator O b . In terms of the physical fields we have
O 4 = 1 M 3ē R γ µ e R ∂ ν φ(c w F µν − s w Z µν )(31)
The second piece leads to the process Z → l + l − φ with a branching ratio which agrees with the L3 results (eq. (16)) for M ∼ 74 GeV. Then, we compute the cross section for the process e + e − → φγ → 3γ, induced by the first term in eq. (31), using the same mass scale. We obtain σ(e + e − → 3γ) ∼ 8 · 10 −2 nb, which is excluded by LEP data [6]. Similar conclusions will hold for other operators of this sort. Of course there is a possibility of a fine tuned cancellation among the many operators but this is even sillier than the model already is.
Finally, we consider the operators of type O 2 . Since the results for all the operators of this kind are similar, we present here only the detailed calculation for the operator in eq. (29). When the Higgs field acquires a vacuum expectation value, O 2 contains the piece
O nc 2 = v M 3 ie 2c w s w Z µēL e R ∂ µ φ ,(32)
which leads to the following partial decay width for the process Z → l + l − φ:
Γ = 1 (4π) 3 e 2c w s w 2 v M 3 2 M 5 Z G(r)(33)
where
G(r) = − 4 15 1 − r 2 5(34)+ 1 + r 6 (1 − r) 3 (1 + r) 16 − 3r(1 − r 2 ) 8 − 3r 2 4 log r and r ≡ m 2 φ M 2 Z
. As in the previous models considered, the branching ratio inferred from the L3 events (eq. (16)) provides an upper limit for the coupling,
e 2c w s w v M 3 ∼ 5.2 · 10 −4 ,(35)
which implies M ∼ 56 GeV. The operator O 2 also induces e + e − → φZ * , which would produce photons and missing energy. However, the rate is too small to be observable. The operator also contains a charged current piece
O cc 2 = − v M 3 ie √ 2s w W µνL e R ∂ µ φ(36)
which induces the W decay W + → l + ν l φ. It is straightforward to obtain that the width for this decay channel is given by
Γ(W + → l + ν l φ) = 1 (4π) 3 e 2s w 2 v M 3 2 M 5 W G(r ′ )(37)
where r ′ ≡ m 2 φ M 2 W and the function G was defined in eq. (35). When we incorporate in this expression the result (35) we get Γ(W + → l + ν l φ) ∼ 2 · 10 −3 MeV, which is not measurable in current and projected experiments (it is expected that the total width of the W boson will be measured with a precision of 200 MeV at LEP II [8].).
We conclude that an operator of O 2 type cannot be ruled out as decisively as the others we have considered. It is however extremely unlikely that it is responsible for the observed events. First of all, the scale of mass suppression is once again too low to be really believable. Furthermore, the operator, if it existed, would most likely be chirally suppressed. And finally, it would be hard to understand why this operator should be induced and not the others which we have successfully excluded. We conclude that it is possible that a scalar could be produced through this direct contact term at the rate required, but it is extremely unlikely.
Conclusions
In this paper, we have considered the possibility that there is a singlet scalar responsible for the observed two photon invariant mass peak observed at LEP. Of course, there are more general possibilities one can consider. For example, φ might have been part of an SU(2) gauge multiplet. Presumably since the scale of the operators is always very low, this will not matter since one can insert the Higgs field (VEV) to make gauge invariant operators and pursue an analysis identical to this one. We suspect methods similar to these will rule out most particle models.
It might be objected that the scale of the operators is always so low that we were not justified in only considering the lowest dimension operators. Again, with a more complete model of what induced these operators one can mimic our analysis. Given the full operator contributing to φ production and decay, gauge invariance will ensure that there are related operators which lead to three photon production at LEP or excess two photon production at TRISTAN. Therefore, despite the limitations of this approach, we anticipate that the conclusion will be quite robust.
We conclude that it is very unlikely that the L3 events represent the discovery of a new particle. Even without information on the angular distribution or the standard model background, we see that the events are not easily explained in a particle physics model.
We will use only the L3 results (four events), but our conclusions are not essentially changed when we include the four LEP experiments.
We thank B. Wyslouch for private communication.
AcknowledgementsWe thank Bolek Wyslouch for motivating this investigation and for useful discusssions. This work is supported in part by funds provided by the U.
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[
"Inverse period mappings of K3 surfaces and a construction of modular forms for a lattice with the Kneser conditions",
"Inverse period mappings of K3 surfaces and a construction of modular forms for a lattice with the Kneser conditions"
] |
[
"Atsuhira Nagano "
] |
[] |
[] |
We construct modular forms on a 4-dimensional bounded symmetric domain of type IV via period mappings of a family of K3 surfaces. On the basis of moduli of K3 surfaces, we study the ring of our modular forms. Because of the Kneser conditions of the transcendental lattice, our modular group has a good arithmetic properties. Also, our results can be regarded as natural extensions of classical Siegel modular forms from the viewpoint of K3 surfaces.
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10.1016/j.jalgebra.2020.07.027
|
[
"https://arxiv.org/pdf/1903.01282v1.pdf"
] | 119,133,550 |
1903.01282
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f3ca47881bc245e405cc2531ac4db29a390f133a
|
Inverse period mappings of K3 surfaces and a construction of modular forms for a lattice with the Kneser conditions
1 Mar 2019 March 5, 2019
Atsuhira Nagano
Inverse period mappings of K3 surfaces and a construction of modular forms for a lattice with the Kneser conditions
1 Mar 2019 March 5, 2019
We construct modular forms on a 4-dimensional bounded symmetric domain of type IV via period mappings of a family of K3 surfaces. On the basis of moduli of K3 surfaces, we study the ring of our modular forms. Because of the Kneser conditions of the transcendental lattice, our modular group has a good arithmetic properties. Also, our results can be regarded as natural extensions of classical Siegel modular forms from the viewpoint of K3 surfaces.
Introduction
Let L be the even unimodular lattice of signature (3,19)
: L = U ⊕ U ⊕ U ⊕ E 8 (−1) ⊕ E 8 (−1)
. This lattice is called the K3 lattice, since it is isomorphic to the lattice of the second homology group of K3 surfaces. Let M = U ⊕ E 8 (−1) ⊕ E 6 (−1) be a lattice of signature (1, 15). We have the orthogonal complement We can consider the action of the group C * × Γ on D * . Then, we define the modular forms for Γ as follows.
Definition 0.1. If a holomorphic function f : D * → C given by Z → f (Z) satisfies the conditions
(i) f (λZ) = λ −k f (Z) (for all λ ∈ C * ), (ii) f (γZ) = χ(γ)f (Z) (for all γ ∈ Γ),
where k ∈ Z and χ ∈ Char(Γ) = Hom(Γ, C * ), then f is called a modular form of weight k and character χ for the group Γ.
Let A k (Γ, χ) be the vector space of the modular forms of Definition 0.1. We obtain the graded ring
A(Γ) = ∞ k=0 χ∈Char(Γ) A k (Γ, χ)
of modular forms. In this paper, we shall determine the structure of this ring.
Our lattice A of (0.1) not only can be regarded as a natural extension of the lattice A CD , but also has another good feature. This lattice is the simplest lattice with the Kneser conditions. These conditions induce good arithmetic properties (see Section 3.1). From that, the modular group Γ is generated by reflections for vectors with self-intersection −2. Moreover, the character group of Γ is very simple:
Char(Γ) = {id, det}.
So, the ring of our modular forms should be in the form
A(Γ) = ∞ k=0 A k (Γ, id) ⊕ ∞ k=0 A k (Γ, det).
We shall determine the structure of this ring. The following theorem is the main theorem of this paper.
Theorem 0.1. (see Corollary 4.1, Theorem 5.1 and Theorem 5.2) (1) There exists a subring of A(Γ) isomorphic to the polynomial ring C[t 4 , t 6 , t 10 , t 12 , t 18 ]. Here, t k gives a modular form of weight k and character id.
(2) There is a modular form s 9 (s 45 , resp.) of weight 9 (45, resp.) and character det. There exist relations s 2 9 = t 18 , s 2 45 = (an irreducible homogeneous polynomial in t 4 , t 6 , t 10 , t 12 and t 18 of weight 90).
These relations determine the structure of the ring A(Γ).
(3) The ring A(Γ) is given by a direct sum of four vector spaces:
A(Γ)
= C[t 4 , t 6 , t 10 , t 12 , t 18 ] ⊕ s 9 C[t 4 , t 6 , t 10 , t 12 , t 18 ] ⊕ s 45 C[t 4 , t 6 , t 10 , t 12 , t 18 ] ⊕ s 9 s 45 C[t 4 , t 6 , t 10 , t 12 , t 18 ].
Especially,
A(Γ, id) = ∞ k=0 A k (Γ, id) = C[t 4 , t 6 , t 10 , t 12 , t 18 ] ⊕ s 9 s 45 C[t 4 , t 6 , t 10 , t 12 , t 18 ], A(Γ, det) = ∞ k=0 A k (Γ, det) = s 9 C[t 4 , t 6 , t 10 , t 12 , t 18 ] ⊕ s 45 C[t 4 , t 6 , t 10 , t 12 , t 18 ].
In this paper, this main theorem will be proved via period mappings of a family of K3 surfaces. This idea is natural, because moduli spaces of polarized K3 surfaces are related to bounded symmetric domains of type IV . Also, there is the following good prototype.
In the work of Clingher-Doran [CD] (see also [NS]), they studied a family of explicit elliptic K3 surfaces S CD (α, β, γ, δ) (see (2.13)). The singular fibres of the elliptic fibration for their family are corresponding to the root systems of type E 8 and E 7 . This structure gives a polarization for their K3 surfaces. Also, they explicitly studied the Shioda-Inose structure on their K3 surfaces. They constructed Siegel modular forms of degree 2 by inverse period mappings for their K3 surfaces. Now, there exists a Siegel modular form of weight 35. According to Igusa [I1], this modular form satisfies a relation, which determines the structure of the ring of Siegel modular forms. It is remarkable that this modular form can be obtained by calculating the "discriminant" of the K3 surfaces of [CD] (see Section 2.3). Thus, the ring of Siegel modular forms can be studied via the geometric structure of appropriate K3 surfaces.
So, it is meaningful to construct new modular forms via inverse period mappings of K3 surfaces. In this paper, we will obtain the modular forms for our lattice A of (0.1) via a family of K3 surfaces. Our K3 surfaces are given by the equation of elliptic surface z 2 0 = y 3 0 + (t 4 x 4 0 + t 10 x 3 0 )y 0 + (x 7 0 + t 6 x 6 0 + t 12 x 5 0 + t 18 x 4 0 ), where x 0 , y 0 and z 0 are affine complex coordinates. The singular fibres of this elliptic fibration are corresponding to the root systems of type E 8 and E 6 . The family of [CD] is a subfamily of our family (see Proposition 2.3). So, Theorem 0.1 gives a natural and non-trivial extension of the work [CD] from the viewpoint of K3 surfaces. Especially, the pair of the relations of Theorem 0.1 (2) gives a counterpart of above-mentioned Igusa's relation. We remark that by the result of [Mo], our K3 surfaces do not admit the Shioda-Inose structure and the moduli space for our K3 surfaces is beyond the moduli space of principally polarized abelian surfaces.
Results of [CD] Our Results
Root Systems for Elliptic Surfaces
E 8 and E 7 E 8 and E 6 Transcendental Lattice U ⊕ U ⊕ A 1 (−1) U ⊕ U ⊕ A 2 (−1) Lie Group for Symmetric Domain SO 0 (2, 3) SO 0 (2, 4) ≃ SU (2, 2) Inverse Period Mapping
Classical Siegel Modular Forms Modular Forms for Γ Weights for χ = id 4, 6, 10, 12 4, 6, 10, 12, 18 Weights for χ = det 5, 30 9, 45 Table 1: The results of [CD] and our results
In Section 1 and Section 2, we will study the period mapping for our K3 surfaces precisely. In Section 1, we will investigate the structure of elliptic surfaces for our family. Here, we will determine the structure of the Néron-Severi lattice and the transcendental lattice of a generic member of our family. In Section 2, we will obtain an explicit expression of our period mapping. The Torelli type theorem and the surjectivity of period mappings for K3 surfaces are very important for our study. According to Corollary 2.1, our period mapping gives an isomorphism between the parameter space of our family of K3 surfaces and the quotient space D/Γ for the symmetric domain D. This guarantees that the inverse period mappings give modular forms for Γ.
In Section 3, we will see the properties of latices with the Kneser conditions. These properties enable us to study our period mapping precisely. Especially, we can study the structure of the branch loci coming from the action of the group Γ on D. In Section 4, we show that the polynomial ring C[t 4 , t 6 , t 10 , t 12 , t 18 ] in the parameters of our K3 surfaces gives a subring of modular forms of the trivial character. In the proof, we will consider the structure of line bundles over the modular variety. In Section 5, we will show that there exist the modular forms s 9 and s 45 of character det. The structure of the ring A(Γ) is determined by the equations for these two modular forms. In this argument, we will use some techniques of canonical orbibundles over our orbifolds, referring to the work of Hashimoto-Ueda [HU].
At the end of the introduction, we note that our modular forms not only give natural extensions of the classical Siegel modular forms, but also give modular forms on the bounded symmetric domain of type I 2,2 , which corresponds to the Hermitian form of signature (2, 2). This fact is coming from the isomorphism SO 0 (2, 4) ≃ SU (2, 2) of Lie groups. Here, we need to recall the pioneering work of Matsumoto-Sasaki-Yoshida [MSY] and [Ma]. They studied another family of K3 surfaces and obtained modular forms on I 2,2 . However, our motivation is different from their motivation and our modular forms are different from their modular forms. Each result has good features respectively (see Section 3.2). The author would like to point out that our modular group Γ is generated by reflections for vectors with self-intersection −2, whereas the modular group in [MSY] is not so. In fact, our argument of Section 4 and Section 5 is based on this property. Moreover, this characteristic of our modular group let us expect non-trivial applications of our modular forms (for detail, see the end of the last section).
Family of elliptic K3 surfaces S(t)
In this section, we define our K3 surfaces. They are given by hypersurfaces in a weighted projective space.
Hypersurfaces S(t)
Let t = (t 4 , t 6 , t 10 , t 12 , t 18 ) ∈ C 5 − {0}. We shall consider the hypersurfaces S(t) : z 2 = y 3 + (t 4 x 4 w 4 + t 10 x 3 w 10 )y + (x 7 + t 6 x 6 w 6 + t 12 x 5 w 12 + t 18 x 4 w 18 )
(1.1) of weight 42 in the weighted projective space P(6, 14, 21, 1) = Proj (C[x, y, z, w]). There is an action of the multiplicative group C * on P(6, 14, 21, 1) (C 5 − {0}, resp.) given by (x, y, z, w) → (x, y, z, λ −1 w) (t = (t 4 , t 6 , t 10 , t 12 , t 18 ) → λ · t = (λ 4 t 4 , λ 6 t 6 , λ 10 t 10 , λ 12 t 12 , λ 18 t 18 ), resp.) for λ ∈ C * . Then, the surface S(t) is invariant under the above action of C * . Hence, from the family
{S(t) | t ∈ C 5 − {0}} → C 5 − {0},
we naturally obtain the family
{S([t]) | [t]
∈ P(4, 6, 10, 12, 18)} → P(4, 6, 10, 12, 18).
Here, the point of P(4, 6, 10, 12, 18) corresponding to t = (t 4 , t 6 , t 10 , t 12 , t 18 ) ∈ C 5 − {0} is denoted by By an explicit calculation as in [NS] Section 3, we can see that P(4, 6, 10, 12, 18
) − T = {[t] | t 10 = t 12 = t 18 = 0}. (1.3)
Namely, T is an analytic subset of codimension 3 in the weighted projective spaceT = P(4, 6, 10, 12, 18). Anyway, we have the family
{S([t]) | [t] ∈ T } → T of elliptic K3 surfaces.
Here, let us consider the C * -bundle T * → T naturally coming from the above action t → λ · t of C * . We also have the family
{S(t) | t ∈ T * } → T * of K3 surfaces.
The above action of C * on (x, y, z, w) and t gives an isomorphism
λ : S(t) → S(λ · t) (1.4)
for any λ ∈ C * . There exists a unique holomorphic 2-form ω t on the K3 surface S(t) up to a constant factor. Then, we can take a holomorphic family {ω t } t∈T * , where ω t is the unique holomorphic 2-form on the K3 surface S(t). We can see that the above action of C * transforms ω t into λ −1 ω t . So, the isomorphism λ of (1.4) gives the correspondence
λ * ω λ·t = λ −1 ω t .
(1.5)
From (1.5), we have the family {ω [t] } [t]∈T of holomorphic 2-forms for the family {S([t]) | [t] ∈ T } of K3
surfaces.
Elliptic fibration and periods from S-marking
In this subsection, we will obtain a marking coming from the structure of elliptic K3 surfaces. Also, we will define the period mapping for our marked K3 surfaces. Set x 0 = x w 6 . Let R(x 0 , t) = 1 w 84 x 8 0 ( the discriminant of the right hand side of (1.1) in y) = 27x 6 0 + 54t 6 x 5 0 + (54t 12 + 4t 3 4 + 27t 2 6 )x 4 0 + (54t 18 + 12t 2 4 t 10 + 54t 6 t 12 )x 3 0 + (27t 2 12 + 12t 4 t 2 10 + 54t 6 t 18 )x 2 0 + (4t 3 10 + 54t 12 t 18 )x 0 + 27t 2 18 .
(1.6)
Set g ∨ 2 (x 0 , t) = 1 w 28 x 3 0 (t 4 x 4 w 4 + t 10 x 3 w 10 ) = t 4 x 0 + t 10 , g ∨ 3 (x 0 , t) = 1 w 42 x 4 0 (x 7 + t 6 x 6 w 6 + t 12 x 5 w 12 + t 18 x 4 w 18 ) = x 3 0 + t 6 x 2 0 + t 12 x 0 + t 18 .
Let r(t) be the resultant of the polynomials g ∨ 2 (x 0 , t) and g ∨ 3 (x 0 , t) in x 0 :
r(t) = t 3 10 + t 2 4 t 10 t 12 − t 18 t 3 4 − t 2 10 t 4 t 6 ,
Then, by an explicit calculation, we can see that the discriminant of the polynomial R(x 0 , t) in x 0 is given by r(t) 3 d(t), where d(t) is given by an irreducible homogeneous polynomial of weight 90 in t: d(t) =3125t 9 10 + 11664t 3 10 t 5 12 + 151875t 6 10 t 12 t 18 + 314928t 6 12 t 18 + 1968300t 3 10 t 2 12 t 2 18 + 4251528t 3 12 t 3 18 + 14348907t 5 18 + 16200t 4 t 5 10 t 3 12 + 472392t 4 t 2 10 t 4 12 t 18 − 273375t 4 t 5 10 t 2 18 − 5314410t 4 t 2 10 t 12 t 3 18 + 4125t 2 4 t 7 10 t 12 t 2 4 + 108135t 2 4 t 4 10 t 2 12 t 18 − 1259712t 2 4 t 10 t 3 12 t 2 18 + 4251528t 2 4 t 10 t 4 18 + 864t 3 4 t 3 10 t 4 12 − 3525t 3 4 t 6 10 t 18 + 23328t 3 4 t 5 12 t 18 − 378108t 3 4 t 3 10 t 12 t 2 18 + 1102248t 3 4 t 2 12 t 3 18 + 888t 4 4 t 5 10 t 2 12 + 26568t 4 4 t 2 10 t 3 12 t 18 + 227448t 4 4 t 2 10 t 3 18 + 16t 5 4 t 7 10 − 456t 5 4 t 4 10 t 12 t 18 − 85536t 5 4 t 10 t 2 12 t 2 18 + 16t 6 4 t 3 10 t 3 12 + 432t 6 4 t 4 12 t 18 − 1056t 6 4 t 3 10 t 2 18 + 62208t 6 4 t 12 t 3 18 + 16t 7 4 t 5 10 t 12 + 480t 7 4 t 2 10 t 2 12 t 18 − 16t 8 4 t 4 10 t 18 − 1536t 8 4 t 10 t 12 t 2 18 + 1024t 9 4 t 3 18 − 13500t 6 t 6 10 t 2 12 − 481140t 6 t 3 10 t 3 12 t 18 − 2834352t 6 t 4 12 t 2 18 − 1476225t 6 t 3 10 t 3 18 − 19131876t 6 t 12 t 4 18 − 5625t 4 t 6 t 8 10 − 200475t 4 t 6 t 5 10 t 12 t 18 − 236196t 4 t 6 t 2 10 t 2 12 t 2 18 − 2592t 2 4 t 6 t 4 10 t 3 12 − 69984t 2 4 t 6 t 10 t 4 12 t 18 + 422820t 2 4 t 6 t 4 10 t 2 18 + 944784t 2 4 t 6 t 10 t 12 t 3 18 − 3420t 3 4 t 6 t 6 10 t 12 − 107460t 3 4 t 6 t 3 10 t 2 12 t 18 − 174960t 3 4 t 6 t 3 12 t 2 18 − 1889568t 3 4 t 6 t 4 18 + 2772t 4 4 t 6 t 5 10 t 18 + 314928t 4 4 t 6 t 2 10 t 12 t 2 18 − 186624t 5 4 t 6 t 10 t 3 18 − 16t 6 4 t 6 t 6 10 − 576t 3 10 t 12 t 18 t 6 4 t 6 − 3456t 2 12 t 2 18 t 6 4 t 6 + 1152t 7 4 t 6 t 2 10 t 2 18 − 5832t 2 6 t 3 10 t 4 12 − 10125t 2 6 t 6 10 t 18 − 157464t 2 6 t 5 12 t 18 − 295245t 2 6 t 3 10 t 12 t 2 18 + 5314410t 2 6 t 2 12 t 3 18 − 5670t 4 t 2 6 t 5 10 t 2 12 − 170586t 4 t 2 6 t 2 10 t 3 12 t 18 + 3188646t 4 t 2 6 t 2 10 t 3 18 + 2700t 2 4 t 2 6 t 7 10 + 101898t 2 4 t 2 6 t 4 10 t 12 t 18 + 1102248t 2 4 t 2 6 t 10 t 2 12 t 2 18 + 216t 3 4 t 2 6 t 3 10 t 3 12 + 5832t 3 4 t 2 6 t 4 12 t 18 − 195048t 3 4 t 2 6 t 3 10 t 2 18 + 216t 4 4 t 2 6 t 5 10 t 12 + 6480t 4 4 t 2 6 t 2 10 t 2 12 t 18 − 216t 5 4 t 2 6 t 4 10 t 18 − 20736t 5 4 t 2 6 t 10 t 12 t 2 18 + 20736t 6 4 t 2 6 t 3 18 + 6075t 3 6 t 6 10 t 12 + 219429t 3 6 t 3 10 t 2 12 t 18 + 1338444t 3 6 t 3 12 t 2 18 + 4251528t 3 6 t 4 18 + 1215t 4 t 3 6 t 5 10 t 18 − 393660t 4 t 3 6 t 2 10 t 12 t 2 18 − 1259712t 2 4 t 3 6 t 10 t 3 18 − 216t 3 4 t 3 6 t 6 10 − 7776t 3 4 t 3 6 t 3 10 t 12 t 18 − 46656t 3 4 t 3 6 t 2 12 t 2 18 + 15552t 4 4 t 3 6 t 2 10 t 2 18 + 729t 4 6 t 3 10 t 3 12 + 19683t 4 6 t 4 12 t 18 − 8748t 4 6 t 3 10 t 2 18 − 2834352t 4 6 t 12 t 3 18 + 729t 4 t 4 6 t 5 10 t 12 + 21870t 4 t 4 6 t 2 10 t 2 12 t 18 − 729t 2 4 t 4 6 t 4 10 t 18 − 69984t 2 4 t 4 6 t 10 t 12 t 2 18 + 139968t 3 4 t 4 6 t 3 18 − 729t 5 6 t 6 10 − 26244t 5 6 t 3 10 t 12 t 18 − 157464t 5 6 t 2 12 t 2 18 + 52488t 4 t 5 6 t 2 10 t 2 18 + 314928t 6 6 t 3 18 .
(1.7)
Here, we can compute the singular fibres on these divisors corresponding to the discriminant of R(x 0 , t) (for detail, see [NS] Section 3.1 and [HU] Section 6). If t is on the divisor {t ∈ T | r(t) = 0}, then both g ∨ 2 (x 0 , t) and g ∨ 3 (x 0 , t) is equal to 0 for some x 0 . On such a point, the singular fibre for the elliptic surface S(t) is of Kodaira type II. We remark that this type of singularity does not acquire any new singularity.
On the other hand, for a generic point t of the divisor {t ∈ T | d(t) = 0}, two of singular fibres of type I 1 of (1.8) collapse into a singular fibre of Kodaira type I 2 . We note that this type of singularity acquires an A 1 -singularity.
Set
T = T − {[t] ∈ T | d(t) = 0 or t 18 = 0}. For a generic point [t] ∈ T , S([t])
gives an elliptic K3 surface π : S([t]) → P 1 (C) whose singular fibres are illustrated in Figure 1. Namely, singular fibres are of Kodaira type
II * + IV * + 6I 1 .
(1.8)
Here, π −1 (∞) is of type II * and π −1 (0) is of type IV * . The general fibre F satisfies the following:
F is linearly equivalent to a 0 + 2a 1 + 3a 2 + 4a 3 + 5a 4 + 6a 5 + 3a 6 + 4a 7 + 2a 8 .
(1.9)
• • • • • • • • • • • • • • • • • a H a a 2 a 3 a 4 a 5 a 6 a 7 a 8 O F b 0 b b 2 b 3 b 4 b 5 b 6 Figure 1: Elliptic fibres of S(t) Take [t] ∈ T . Let us consider the sublattice of H 2 (S([t]), Z) generated by F, O, a 1 , · · · , a 8 , b 1 , · · · , b 6 .
(1.10)
This lattice is a sublattice of NS(S([t])) which is isomorphic to the lattice
M = U ⊕ E 8 (−1) ⊕ E 6 (−1) (1.11)
of rank 16. So, we have an isometry
ψ : H 2 (S([t]), Z) → L such that ψ −1 (M ) ⊂ NS(S([t])
). We take γ 7 , · · · , γ 22 ∈ L as
γ 7 = ψ(F ), γ 8 = ψ(O), γ 8+j = ψ(a j ), γ 16+k = ψ(b k ) (j ∈ {1, · · · , 8}, k ∈ {1, · · · , 6}).
Then, γ 7 , · · · , γ 22 generate the lattice M . Here, since | det(M ) | = 3 is a prime number, M is a primitive sublattice of L. Therefore, there exist γ 1 , · · · , γ 6 ∈ L such that {γ 1 , · · · , γ 6 , γ 7 , · · · , γ 22 } (1.12)
gives a basis of L. Let δ 1 , · · · , δ 22 be the dual basis of the basis in (1.12) with respect to the intersection form given by the unimodular lattice L. Here, we note that δ 1 , · · · , δ 6 generate the lattice A of (0.1).
Let S 0 = S([t 0 ]) for [t 0 ] ∈ T be a reference surface. Take a sufficiently small neighborhood U of [t 0 ] in T such that there exists a topological trivialization τ : {S([t]) | [t] ∈ U } → S 0 × U . Let β : S 0 × U → S 0 be the canonical projection. Put r = β • τ. Then, r ′ [t] = r| S([t]) gives a C ∞ -isomorphism of complex surfaces. For any [t] ∈ U, we have an isometry ψ [t] : H 2 (S([t]), Z) → L given by ψ [t] = (r ′ [t] ) * .
We call this isometry the S-marking on U . By an analytic continuation along an arc α ⊂ T , we can define the S-marking on T . We remark that this depends on the choice of α. The S-marking preserves the Néron-Severi lattice. So, we obtain the local period mapping
Φ 1 : T → P 5 (C) (1.13)
given by
[t] → ξ = ψ −1 [t] (γ1) ω [t] : · · · : ψ −1 [t] (γ6) ω [t] ,
(1.14)
where ω [t] is the unique holomorphic 2-form on S([t]) up to a constant factor and γ 1 , · · · , γ 6 ∈ L are given by (1.12). We call the pair (S([t]), ψ [t] ) an S-marked K3 surface.
Definition 1.1. Suppose there are two S-marked K3 surfaces (S([t 1 ]), ψ 1 ) and (S([t 2 ]), ψ 2 ). We will say that (S([t 1 ]), ψ 1 ) and (S([t 2 ]), ψ 2 ) are equivalent (isomorphic, resp.) if there exists a biholomorphic mapping f : S([t 1 ]) → S([t 2 ]) such that (ψ 2 • f * • ψ −1 1 ) | M = id M (ψ 2 • f * • ψ −1 1 = id L , resp.
). Since the image of Φ 1 satisfies the Riemann-Hodge relation
ξA t ξ = 0, ξA t ξ > 0, (1.15) the image of Φ 1 should be contained in the 4-dimensional space D M = {ξ ∈ P 5 (C) | ξ satisfies (1.15)}.
( 1.16) This space has two connected components. Let D be one of the connected components. This is a bounded symmetric domain of type IV .
Techniques of elliptic surfaces and S-marked K3 surfaces
In this subsection, we shall see a relation between elliptic surfaces and S-marked K3 surfaces. Let π : S → P 1 (C) be an elliptic surface with a general fibre F . Then, by virtue of the Riemann-Roch theorem, we can see that π is the unique elliptic fibration up to Aut(P 1 (C)) such that F is a general fibre. So, we have the following definition.
Definition 1.2. Let (S 1 , π 1 , P 1 (C)) and (S 2 , π 2 , P 1 (C)) are two elliptic surfaces. If there exist a biholomorphic mapping f : S 1 → S 2 and ϕ ∈ Aut(P 1 (C)) such that ϕ • π 1 = π 2 • f , then we say that (S 1 , π 1 , P 1 (C)) and (S 2 , π 2 , P 1 (C)) are isomorphic as elliptic surfaces.
For an elliptic curve given by the Weierstrass form
z 2 0 = y 3 0 − g 2 (x 0 )y 0 − g 3 (x 0 ), the quotient j(x 0 ) = g 3 2 (x0) 4g 3 2 (x0)−27g 2
3 (x0) of the coefficients of the elliptic surface is called the j-invariant. Let (S 1 , π 1 , P 1 (C)) ((S 2 , π 2 , P 1 (C)), resp.) is an elliptic surface given by the Weierstrass form with the j-invariant j 1 (x 0 ) (j 2 (x 0 ), resp.), then there exists ϕ ∈ Aut(P 1 (C)) such that π −1 1 (x 0 ) and π −1 2 (ϕ(x 0 )) are singular fibres of the same type and j 2 • ϕ = j 1 holds.
For our K3 surface S([t]) of (1.1), π : (x, y, z, w) → (x, w) gives a natural elliptic fibration. Let x 0 = x w 6 be an affine coordinate.
Lemma 1.1. For [t 1 ] and [t 2 ] ∈ T , we suppose that (S([t 1 ]), π 1 , P 1 (C)) is isomorphic to (S([t 2 ]), π 2 , P 1 (C)) as elliptic curves. Then, it holds that [t 1 ] = [t 2 ].
Proof. We suppose that a biholomorphic mapping f : S([t 1 ]) → S([t 2 ]) gives an equivalence of elliptic surfaces. Then, there exists ϕ ∈ Aut(P 1 (C)) such that ϕ • π 1 = π 2 • f. Here, for j = 1, 2, π −1 j (0) (π −1 j (∞), resp.) is a singular fibre of Kodaira type II * (IV * , resp.). This implies that ϕ ∈ Aut(P 1 (C)) should be given by the mapping x 0 → λx 0 for some λ ∈ C * . Let us consider the discriminant R j (x 0 , t) (j = 1, 2) of the right hand side of the elliptic surface S([t j ]). They are given in the form of (1.6), that are polynomials in x 0 of degree 6. The six roots of each polynomial give the six images of the singular fibres of type I 1 on
P 1 (C) = x 0 -sphere. The roots of R 1 (x 0 , t) should be sent to those of R 2 (x 0 , t) by ϕ. Hence, by observing the coefficients of R 1 (x 0 , t) and R 2 (x 0 , t), we can see that [t 1 ] = [t 2 ]. Lemma 1.2. For [t 1 ] and [t 2 ] ∈ T , two S-marked K3 surfaces (S([t 1 ]), ψ 1 ) and (S([t 2 ]), ψ 2 ) are equivalent if and only if [t 1 ] = [t 2 ].
Proof. For [t 1 ] and [t 2 ] ∈ T , we suppose that there exists a biholomorphic mapping f :
S([t 1 ]) → S([t 2 ]) which gives an equivalence of S-marked K3 surfaces. So, it follows that (ψ 2 • f * • ψ −1 1 ) | M = id | M .
This implies that f * (F 1 ) = F 2 holds, where F j is a general fibre for the elliptic surface S([t j ]) (j = 1, 2). Then, F 2 is a general fibre not only for the elliptic fibration π 2 but also for another elliptic fibration π 1 • f −1 . Therefore, we have π 2 = π 1 • f −1 holds up to Aut(P 1 (C)). Hence, we proved that two S-marked K3 surfaces (S([t 1 ]), ψ 1 ) and (S([t 2 ]), ψ 2 ) are equivalent if and only if there exists an isomorphism of elliptic surfaces between (S([t 1 ]), π 1 , P 1 (C)) and (S([t 2 ]), π 2 , P 1 (C)). Due to Lemma 1.1, the assertion is proved. Proof. Since M of (1.11) is isomorphic to a sublattice of NS(S([t])) for [t] ∈ T , it is apparent that rank (NS(S([t]))) ≥ 16. Take a sufficiently small neighborhood U in T around a point [t 0 ] ∈ T . Here, we can apply the local Torelli theorem for K3 surfaces to our local period mapping Φ 1 of (1.13). Then, this theorem guarantees that there exists an isomorphism between (S([t 1 ]),
Picard number
ψ 1 ) and (S([t 2 ]), ψ 2 ) for [t 1 ], [t 2 ] ∈ U , if Φ 1 ([t 1 ]) = Φ 1 ([t 2 ]). Hence, by virtue of Lemma 1.2, the local period mapping Φ 1 is injective on the open set U (⊂ T ).
If we suppose that rank (NS(S([t]))) is not equal to 16 for generic point [t] ∈ T , we have a contradiction to the injectivity of Φ 1 on U . This prove the theorem.
Corollary 1.1. For a generic point [t] ∈ T , the lattice NS(S([t])) (Tr(S([t])), resp.) is given by the intersection matrix
M = U ⊕ E 8 (−1) ⊕ E 6 (−1) (A = U ⊕ U ⊕ A 2 (−1), resp.).
Remark 1.1. Reid listed weighted projective K3 hypersurfaces with Gorenstein singularities. They are often called 'famous 95' K3 surfaces. We note that our Néron-Severi lattice M = U ⊕ E 8 (−1) ⊕ E 6 (−1) is equal to that of No.88 of that list (see [B]).
Period mapping
In this section, we consider the period mapping for our K3 surfaces precisely.
Pseudo ample marked lattice polarized K3 surfaces with an elliptic fibration
Let S be a K3 surface. Let ω be the unique non-zero holomorphic 2-form on S up to a constant factor.
Regarding ω as a homomorphism H 2 (S, Z) → C, the Néron-Severi lattice NS(S) is equal to its kernel. Letting ρ be the rank of NS(S), then NS(S) is a non-degenerated lattice of signature (1, ρ − 1). We note that we can identify H 2 (S, Z) with H 2 (S, Z) by the Poincaré duality. Elements of V (S) + are effective divisor classes with positive self-intersection. The fundamental domain for this action is given by
C(S) = {x ∈ V (S) + | (x, δ) ≥ 0 for all δ ∈ ∆(S) + }.
We call We say that (S, j) is an ample M -polarized K3 surface if
C(S) + = {x ∈ V (S) + | (x, δ) > 0 for all δ ∈ ∆(S) + }thej(C(M ) + ) ∩ NS(S) ++ = φ.
Two M -polarized K3 surfaces (S 1 , j 1 ) and (S 2 , j 2 ) are isomorphic if there exists an isomorphism of K3 surfaces f : S 1 → S 2 such that j 2 = f * • j 1 .
P -marking and period mapping
Let us recall some basic properties of K3 surfaces (for detail, see [LP] or [L]). Letting S be a K3 surface, if x ∈ NS(S) satisfies (x, x) ≥ −2, then one of x or −x is representable by an effective divisor. An irreducible curve C is called a nodal curve if (C, C) = −2. We can see that a nodal curve C is smooth and rational. Let B(S) ⊂ ∆(S) be the set of classes of nodal curves. The set B(S) gives a root basis for ∆(S). This implies that (x, y) ≥ 0 (for all distinct x, y ∈ B(S)). Now, let us introduce P -marked K3 surfaces to study the period mapping for our elliptic K3 surface S([t]) of (1.1) precisely. Take a reference surface S 0 = S([t 0 ]) for a fixed [t 0 ] ∈ T . Take an S-marking ψ 0 : H 2 (S 0 , Z) → L as in Section 1.2. Let S be an algebraic K3 surface. An isometry ϕ : H 2 (S, Z) → L is called a P -marking of S if it satisfies the following conditions:
(i) ϕ −1 (M ) ⊂ NS(S) for the lattice M of (1.11), (ii) For a j , b j , O, F in (1.10), ϕ −1 • ψ 0 (a j ), ϕ −1 • ψ 0 (b j ), ϕ −1 • ψ 0 (O) and ϕ −1 • ψ 0 (F ) give effective divisors on S, (iii) ϕ −1 • ψ 0 (F ) is a nef divisor.
The pair (S, ψ) is called a P -marked K3 surface. Also, we define the equivalent classes and isomorphic classes as in Definition 1.1. We can define the period
Φ((S, ψ)) = ψ −1 •ψ0(γ1) ω : · · · : ψ −1 •ψ0(γ6) ω (2.7)
of a P -marked K3 surface (S, ψ), where ω is the unique holomorphic 2-form on S up to a constant factor. Proof. We note that a divisor D is nef if and only if (D, C) ≥ 0 for any effective divisor C on S. By the Riemann-Roch Theorem, we can see that there exists a unique elliptic fibration π : S → P 1 (C) = x-sphere such that ψ −1 • ψ 0 (F ) is a general fibre of π and ψ −1 • ψ 0 (O) is the zero-section of π. By considering the intersections of ψ −1 •ψ 0 (F ), ψ −1 •ψ 0 (a j ) and ψ −1 •ψ 0 (b j ), we can suppose that ψ −1 •ψ 0 (a j ) (ψ −1 •ψ 0 (b j ), resp.) is a component of the singular fibre π −1 (∞) (π −1 (0), resp.). It follows that π −1 (∞) (π −1 (0), resp.) should contain the subgraph of Kodaira type II * (IV * , resp.). According to the classification of singular fibres of elliptic fibration, π −1 (∞) (π −1 (0), resp.) should be of type II * (IV * , III * or II * , resp.). Then, by considering the Weierstrass models of elliptic surfaces, we can see that such an elliptic K3 surface (S, π, P 1 (C)) can be given by S([t]) ([t] ∈ T ) of (1.1). Moreover, due to Lemma 1.1 and a similar argument to the proof of Lemma 1.2, we can prove that {S([t]) | [t] ∈ T } gives the set of equivalent classes of P -marked K3 surfaces.
Lemma 2.2. Let (S 0 , ψ 0 ) be an S-marked K3 surface. Let (S, ψ) be a marked pseudo-ample M -polarized K3 surface. If C ∈ NS(S 0 ) be a nodal curve, then ψ −1 • ψ 0 (C) ∈ NS(S) is a nodal curve.
Proof. Since (ψ −1 • ψ 0 (C), ψ −1 • ψ 0 (C)) = (C, C) = −2, one of ψ −1 • ψ 0 (C) or −ψ −1 • ψ 0 (C) is effective.
From the definition, we can see that there exists κ ∈ C(M ) + such that ψ −1 (κ) is in the closure of the Kähler cone. Due to the argument in Section 2.1, ψ −1 0 (κ) ∈ H 2 (S 0 , Z) gives an ample divisor class. According to Nakai's criterion, we have
(ψ −1 (κ), ψ −1 • ψ 0 (C)) = (ψ −1 0 (κ), C) > 0.
So, it follows that ψ −1 • ψ 0 (C) is effective. Proof. Since a j (j ∈ {0, · · · , 8}) is a nodal curve on S 0 , due to Lemma 2.2, ψ −1 • ψ 0 (a j ) is a nodal curve on S. Take an arbitrary curve C on S. If C is linearly equivalent to ψ −1 • ψ 0 (a j ) for some j ∈ {0, · · · , 8}, C is a nodal curve in an elliptic fibre. So, we have (ψ −1 • ψ 0 (F ), C) = 0. If C is not linearly equivalent to ψ −1 • ψ 0 (a j ) (j ∈ {0, · · · , 8}), then
(ψ −1 • ψ 0 (a j ), C) ≥ 0, (2.8)
since B(S) gives a root system (recall (2.6)). Due to (1.9), we have
ψ −1 • ψ 0 (F ) =ψ −1 • ψ 0 (a 0 ) + 2ψ −1 • ψ 0 (a 1 ) + 3ψ −1 • ψ 0 (a 2 ) + 4ψ −1 • ψ 0 (a 3 ) + 5ψ −1 • ψ 0 (a 4 ) + 6ψ −1 • ψ 0 (a 5 ) + 3ψ −1 • ψ 0 (a 6 ) + 4ψ −1 • ψ 0 (a 7 ) + 2ψ −1 • ψ 0 (a 8 ) (2.9)
in NS(S). According to (2.8) and (2.9), we have (ψ −1 • ψ 0 (F ), C) ≥ 0. Therefore, ψ −1 • ψ 0 (F ) has a non-negative intersection number with for any nodal curve on C. This means that ψ −1 • ψ 0 (F ) is nef. Due to the above theorem, Theorem 2.2 and the definitions of period mappings, we have the following corollary.
Corollary 2.1. The period mapping (2.7) is explicitly given by
Φ : T ∋ [t] → ψ −1 •ψ0(γ1) ω [t] : · · · : ψ −1 •ψ0(γ6) ω [t] ∈ D.
(2.11)
This gives an extension of (1.14) defined on T . Also, for the group Γ of (2.10), the mapping (2.11) induces the isomorphism
Φ : T ≃ D/Γ = Q.
(2.12)
Family S CD (α, β, γ, δ)
In [CD], the family S CD (α, β, γ, δ) : z 2 1 = y 3 1 + (−3αx 4 1 − γx 5 1 )y 1 + (x 5 1 − 2βx 6 1 + δx 7 1 ) (2.13) of K3 surfaces was studied. This family is closely related to famous Siegel modular forms of degree 2. First, we recall the following result.
Proposition 2.1. ( [CD] or [NS]) For a generic point (α : β : γ : δ) ∈ P(4, 6, 10, 12), the Néron-Severi lattice NS(S CD (α, β, γ, δ)) (transcendental lattice Tr(S CD (α, β, γ, δ)), resp.) of the K3 surface S CD (α, β, γ, δ) is given by the intersection matrix U ⊕ E 8 (−1) ⊕ E 7 (−1) (U ⊕ U ⊕ A 1 (−1), resp.).
Let us recall the meaning and the importance of this family. By taking the subspace T 0 = {(α : β : γ : δ) ∈ P(4, 6, 10, 12) | γ = δ = 0}, we can obtain the period mapping
T 0 ∋ (α : β : γ : δ) → ξ 0 ∈ S 2 /Sp(4, Z), (2.14)
where S 2 is the Siegel upper half plane of degree 2 and Sp(4, Z) is the symplectic group. By the inverse period mapping (2.14), α (β, γ, δ, resp.) gives a Siegel modular form of weight 4 (6, 10, 12, resp.). More precisely, we have the following result.
Proposition 2.2. ( [I1], [I2], [CD] and [NS]) (1) The ring of Siegel modular forms on S 2 for the group Sp(4, Z) of even weight and the trivial character is generated by modular forms α, β, γ and δ of weight 4, 6, 10 and 12, respectively. Also, there exists a modular form χ 5 (χ 30 , resp.) of weight 5 (30, resp.) and non-trivial character. Moreover, setting χ 35 = χ 5 χ 30 , the space of Siegel modular forms on S 2 for the group Sp(4, Z) of odd weight and non-trivial character is equal to χ 35 C[α, β, γ, δ].
(2) More precisely, α, β, γ and δ have explicit expressions by the Igusa invariants: α = 1 9 I 4 , β = 1 27 (−I 2 I 4 + 3I 6 ), γ = 8I 10 , δ = 2 3 I 2 I 10 .
( 2.15) There exist relations which determine the structure of the ring of Siegel modular forms of degree 2. Namely, χ 2 5 and χ 2 30 are given by the polynomials in α, β, γ and δ. These relations are calculated by the explicit defining equation (2.13) of K3 surfaces as follows. Let R 0 (x 1 ) = 1
x 10 1 ( the discriminant of the right hand side of (2.13) in y 1 ).
Then, the discriminant of the polynomial R 0 (x 1 ) in x 1 is given by the product of γ 3 , r 3 0 and d 0 up to a constant factor. Here, r 0 is equal to the resultant of g2(x1)
x 4 1 and that of g3(x1)
x 5 1 , where we express (2.13) as z 2 1 = y 3 1 − g 2 (x 1 )y 1 − g 3 (x 1 ). Also, d 0 is a polynomial of weight 60 in α, β, γ and δ. The factor γ 3 implies that the square of χ 5 is equal to γ up to a constant factor. Also, via (2.15), we can see that d 0 is essentially equal to a factor of weight 60 in the famous relation for χ 2 30 by Igusa (see [I1] p.849) up to a constant factor.
Thus, the famous Siegel modular forms of degree 2 can be calculated by inverse period mappings for this family of K3 surfaces (2.13).
Our
M -polarized K3 surfaces S([t]) ([t] ∈ T )
is closely related to the Clingher-Doran family as follows. Proof. By putting x 1 = 1 x0 , y 1 = y0
x 4 0 and z 1 = z0
x 6 0 , (2.13) is transformed to
z 2 0 = y 3 0 + (−3αx 4 0 − γx 3 0 )y 0 + (x 7 0 − 2βx 6 0 + δx 5 0 ). (2.16)
This is equal to the form which is coming from (1.1) by the substitution x = x0 w 6 , y = y0 w 14 , z = z0 w 21 , t 4 = −3α, t 6 = −2β, t 10 = −γ, t 12 = δ and t 18 = 0.
We shall use Proposition 2.3 to prove the main theorem at end of this paper.
Lattice with the Kneser conditions
The Kneser conditions
In this section, we survey arithmetic properties of lattices. They are useful to study our period mapping of K3 surfaces.
Let K be an even integral lattice of signature (2, n). Namely, K is a free Z-module with a bilinear form K × K → Z such that (x, x) ∈ 2Z for any x ∈ K. Let K ∨ be the dual lattice. As in Section 2, we define the stable orthogonal group:
Õ(K) = Ker(O(K) → O(K ∨ /K)), where O(K) is the orthogonal group for the lattice K. Also, O(K) acts on D + ⊔ D − , where D + and D − are two connected components of {x ∈ P(K ⊗ C) | (x, x) = 0, (x, x) > 0}. We take a subgroup O + (K) = {γ ∈ O(K) | γ(D + ) = D + }. We set Γ =Õ + (K) =Õ(K) ∩ O + (K).
(3.1)
Definition 3.1. If an even integral lattice K satisfies the following conditions, we way that K satisfies the Kneser conditions:
(i) For the signature (s + , s − ) of K, min(s + , s − ) ≥ 2, (ii) There exists x ∈ K such that (x, x) = −2, (iii) rank 2 (K) ≥ 6, (iv) rank 3 (K) ≥ 5.
Here, rank p (K) for a prime number p means the rank of the mod p reduction of K.
This terminology is due to Kneser [K]. We have the following result:
Theorem 3.1. ) (1) If K satisfies the Kneser conditions, then Γ of (3.1) is generated by reflections σ δ : z → z + (z, δ)δ for δ ∈ K with (δ, δ) = −2.
(2) If K satisfies Kneser conditions and contains at least two hyperbolic planes U , then
Char(Γ) = {id, det}.
We note that the lattice A of (0.1) satisfies the conditions of the above theorem. In fact, our lattice A is the simplest lattice satisfying these conditions. [CD] or [MSY] In this subsection, let us see features of our family of K3 surfaces by comparing other famous families of polarized K3 surfaces.
Comparison with the results of
First, let us recall the family of K3 surfaces in Section 2.3 due to [CD]. The lattice
A CD = U ⊕ U ⊕ A 1 (−1).
does not satisfy the Kneser conditions. Nevertheless, this lattice also has good properties. We can see that O(A CD ) is equal to the stable orthogonal groupÕ(A CD ). According to Proposition 2.2, the classical Siegel modular forms of degree 2 is given by the modular forms for the lattice O(A CD ). Moreover, in [GN], it is proved that the group O(A CD ) is generated by reflections for vectors with self-intersection −2 and its character group is isomorphic to Z/2Z. Thus, the lattice A CD has good arithmetic properties.
Let us see another example in a famous work due to Matsumoto-Sasaki-Yoshida [MSY]. They studied a family of K3 surfaces with four complex parameters. From the viewpoint of hyperplane arrangements and differential equations, their family is very natural and interesting. Their K3 surfaces are given by the double covering of P 2 (C) blanched along six lines. The configuration of these lines are coming from the Grassmannian manifold Gr(3, 6). The periods for their K3 surfaces satisfy a system of partial differential equations, which is called a hypergeometric equation of type (3, 6). This system gives a natural extension of the classical Gauss hypergeometric equation. Here, a generic member of their family is a lattice polarized K3 surface with the transcendental lattice
A MSY = U (2) ⊕ U (2) ⊕ (−I 2 (2))
of signature (2, 4). We note that this lattice does not satisfy the Kneser conditions. We note that Matsumoto [Ma] studied modular forms for the group Γ AMSY , which is given by the lattice A MSY , coming from the inverse period mapping of a family of K3 surfaces with four parameters. However, the modular group Γ AMSY has more complicated structure than our modular group. Namely, it is generated by reflections for vectors with self-intersection not only −2 but also −4.
In the following contents of this paper, we will obtain modular forms for Γ =Õ + (A) by the inverse period mapping of our K3 surfaces. Our modular forms are defined on a 4-dimensional bounded symmetric domain of type IV , which corresponds to the Lie group SO 0 (2, 4) ≃ SU (2, 2). Also, the modular forms of [Ma] are also defined on the symmetric domain corresponding to SU (2, 2). However, our modular forms are different from their modular forms, since our modular group Γ =Õ + (A) is different from Γ AMSY . Their modular forms and our modular forms have different meaning and features. Modular forms from our K3 surfaces give natural extension of the classical Siegel modular forms, via [CD]. Especially, from the viewpoint of Dynkin diagrams, our elliptic K3 surfaces correspond to E 8 and E 6 , while elliptic K3 surfaces of [CD] correspond to E 8 and E 7 . Moreover, our modular group Γ has very good arithmetic properties surveyed in Theorem 3.1.
Structure of branches
We can apply Theorem 3.1 to our lattice A. So, the group Γ =Õ + (A) is generated by the reflections σ δ such that (δ, δ) = −2. Let ∆(A) be the set of vectors in A with self-intersection −2.
For g ∈ Γ, let us consider the linear subspace (D * ) g = {Z ∈ D * | g(Z) = Z}. If (D * ) g is a hyperplane, from Theorem 3.1 (1), it is equal to a reflection hyperplane δ ⊥ = {Z ∈ D * | (Z, δ) = 0} for some root δ ∈ ∆(A). We set
H D * = δ∈∆(A) δ ⊥ .
(3.2) Also, set
S D * = {Z ∈ D * | the stabilizer of Z is neither {id} nor {id, σ δ } ≃ Z/2Z for δ ∈ ∆(A)}. (3.3)
Then, H D * (S D * , resp.) is a countable union of hyperplanes (linear subspaces of codimension at least 2, resp.), since Γ is a countable group. Any points with the non-trivial stabilizer group are contained in H D * ∪ S D * . Therefore, the action of Γ on D * − (H D * ∪ S D * ) is free. Also, the stabilizer group of a point of H D * − (H D * ∪ S D * ) is given by {id, σ δ } for some δ ∈ ∆(A). Let H D and S D be the subsets in D given by the images of H D * and S D * , respectively. Similarly, we define the subsets H Q and S Q in Q.
Set Γ ′ = {γ ∈ Γ | det(γ) = 1}. This is a normal subgroup of Γ. From Theorem 3.1, Γ ′ = {γ ∈ Γ | γ can be given by a product of reflections of even numbers }.
Set Q 1 = D/Γ ′ . We naturally define the subsets H Q1 and S Q1 in Q 1 . We note that Q 1 − S Q1 → Q − S Q is a double covering branched along H Q − S Q . Since the action of Γ on D − (H D ∪ S D ) is free, any points whose stabilizer group is {id, σ δ } for some δ ∈ ∆(A) are contained in H D . Moreover, H D coincides with the set of fixed points of Γ/Γ ′ ≃ Z/2Z. Hence, it follows that the action of Γ ′ on D − S D is free.
Here, let us recall the period mapping Φ : T ≃ Q of (2.12), which gives an isomorphism. We set H T = Φ −1 (H Q ) and S T = Φ −1 (S Q ). Since T is a Zariski open set of the weighted projective spaceT = P(4, 6, 10, 12, 18) and H T is an analytic subgroup of codimension 1, there exists a weighted homogeneous polynomial ∆ T (t) ∈ C[t 4 , t 6 , t 10 , t 12 , t 18 ] such that H T = {[t] = (t 4 : t 6 : t 10 : t 12 : t 18 ) ∈ P(4, 6, 10, 12, 18) | ∆ T (t) = 0}.
(3.4)
Let us consider the double covering T 1 of T − S T branched along H T − S T :
T 1 = {([t], s) ∈ (T − S T ) × C | s 2 = ∆ T (t)}. (3.5) Since T 1 (Q 1 − S Q1 , resp.) is the double covering of T − S T (Q − S Q , resp.) branched along H T − S T (H Q − S Q ,resp
.) and the divisors H T is identified with H Q under the period mapping Φ of (2.12), we can obtain the lift Φ Q1 : T 1 ≃ Q 1 − S Q1 of Φ. Next, let us consider the universal covering T D of T 1 . Recalling that the action of Γ ′ on D − S D is free, we can see that Φ Q1 is lifted to Φ D : T D ≃ D − S D . So, we have the following diagram.
T D ΦD − −−− → D − S D free pD free T 1 ΦQ 1 − −−− → Q 1 − S Q1 Z/2Z p1 Z/2Z T − S T Φ| T −S T −−−−−→ Q − S Q inclusion inclusion T Φ − −−− → Q (3.6)
Let us consider the pull-back T * D → T D of the principal C * -bundle T * → T by the composition of the mapping
T D → T 1 → T − S T ֒→ T. The isomorphism Φ D : T D ≃ D − S D can be lifted to Φ D * : T D * ≃ D * − S D * .
(3.7)
Lemma 3.1. The mapping Φ Q1 (Φ D , Φ D * , resp.) is equivalent under the action of Γ/Γ ′ ≃ Z/2Z (Γ/{±1}, Γ, resp.).
Proof. We can see these properties by the above construction of the lifts of Φ. Here, we note the following. Since D is a subset of a projective space, Φ D should be equivalent under the action of the group Γ/{±id}. On the other hand, if we think about the action on D * , we can distinguish the action of γ ∈ Γ from that of (−γ) ∈ Γ. This is why Φ D * should be Γ-equivalent.
The above structure of the branches defines orbifolds that will be useful in the last section.
Sections of line bundles and polynomial ring in parameters
The natural projection ̟ : D * → D
gives a principal C * -bundle. By the period mapping Φ of (2.12), Q = D/Γ is isomorphic to a Zariski open set T of the weighted projective spaceT = P(4, 6, 10, 12, 18). Especially, Q gives a smooth variety. Since ̟ is equivalent under the action of Γ, we have a principal C * -bundle
̟ : D * /Γ → Q.
In this paper, let O Q (1) be the line bundle associated with ̟. By the definition of associated bundles, a section of O Q (1) corresponds to a holomorphic function η → s(η) (η ∈ D * /Γ) satisfying
s(λη) = λ −1 s(η) (λ ∈ C * ). (4.1) For k ∈ Z, the line bundle O Q (1) ⊗k is denoted by O Q (k).
Proposition 4.1. There is a commutative diagram t = (t 4 , t 6 , t 10 , t 12 , t 18 )
period mapping − −−−−−−−−− → η C * -action C * -action λ −1 · t = (λ −4 t 4 , λ −6 t 6 , λ −10 t 10 , λ −12 t 12 , λ −18 t 18 ) period mapping − −−−−−−−−− → λη
Especially, the correspondence D * /Γ ∋ η → t j (η) ∈ C, via the inverse of the period mapping Φ, defines a global section of the line bundle O Q (k).
Proof. In Corollary 2.1, the period mapping Φ of (2.11) is given by integrals of the holomorphic 2-
form ω [t] of S([t]) ([t] ∈ T )
. By virtue of the property (1.5) of ω [t] , we obtain the above commutative diagram. Together with Theorem 2.2 and Theorem 2.3, we can consider the inverse correspondence D * /Γ ∋ η → t j (η) ∈ C of the period mapping. Considering the diagram, η → t k (η) defines a global section of the line bundle O Q (k) in the sense of (4.1).
Moreover, we have the following theorem.
Theorem 4.1. The total coordinate ring on the smooth variety Q is isomorphic to the polynomial ring of t 4 , t 6 , t 10 , t 12 and t 18 :
L∈Pic(Q)
H 0 (Q, L) ≃ C[t 4 , t 6 , t 10 , t 12 , t 18 ].
Here, t k gives a global section of the line bundle O Q (k).
Proof. Since T of (1.3) is an analytic subset ofT = P(4, 6, 10, 12, 18) of codimension at least 2, by Hartogs's phenomenon, we have an isomorphism
ι * T : Pic(T ) ≃ Pic(T ) (4.2)
from the inclusion ι T : T ֒→T . By the isomorphism Φ : T → Q of (2.12), we obtain
Φ * : Pic(Q) ≃ Pic(T ). (4.3)
SinceT is a weighted projective space, Pic(T ) ≃ Z holds. Moreover, it holds Remark 4.1. We have the Satake-Baily-Borel compactificationQ of Q = D/Γ. We note thatQ−Q is an analytic subset ofQ of codimension at least 2 (see [BB] Proposition 3.15). So, by Hartogs's phenomenon again, the inclusion ι Q : Q ֒→Q induces an isomorphism ι * Q : Pic(Q) ≃ Pic(Q) of Picard groups and the total coordinate ring onQ is given by C[t 4 , t 6 , t 10 , t 12 , t 18 ] also.
L∈Pic(T ) H 0 (T , OT (L)) ≃ k∈Z C (k) [t],(4.
From the definition (4.1) of the associated line bundles, a global section s of O Q (k) induces a holomorphic mapping Z →s(Z) (Z ∈ D * ) satisfying the following functional equations s(λZ) = λ −ks (Z) (λ ∈ C * ), s(γZ) =s(Z) (γ ∈ Γ). Therefore, by recalling Definition 0.1, a global section s of O Q (k) defines a modular form of weight k and character id for the group Γ =Õ + (A). Hence, we have the following corollary.
Corollary 4.1. Let Γ =Õ + (A). The inverse of the period mapping Φ of (2.11) induces a subring C[t 4 , t 6 , t 10 , t 12 , t 18 ] of the ring A(Γ, id) = ∞ k=0 A k (Γ, id) of modular forms of character id. The above Corollary does not mean that the ring A(Γ, id) is equal to C[t 4 , t 6 , t 10 , t 12 , t 18 ]. Since the argument in this section is based on the structure of the line bundles over the smooth variety Q = D/Γ, we does not consider the branch loci coming from the action of the group Γ in the above argument. However, as we considered in Section 3, the action of Γ on D is not free. In fact, the branch loci coming from this action affect the structure of the ring of modular forms. In the next section, we will consider the orbifolds coming from the action of Γ. By considering orbibundles over the orbifolds, we will determine the precise structure of the ring of modular forms for Γ.
Theorem 5.1. There is a modular form s 9 (s 45 , resp.) of weight 9 (45, resp.) and character det. These forms satisfy the the following relations: (1.7).
s 2 9 = t 18 , s 2 45 = d(t) in
These relations determine the structure of the ring A(Γ).
Proof. Recall that two fibres of Kodaira type I 1 of (1.8) of elliptic surface S([t]) collapse into a singular fibre of type I 2 on the divisor {t ∈ T | d(t) = 0} of (1.7). Here, a singular fibre of type I 2 gives an A 1singularity. An A 1 -singularity acquires a (−2)-curve. (recall the argument in Section 1.2). So, according to the constructions of reflection hyperplanes H D * and the diagram (3.6), it follows that {t ∈ T | d(t) = 0} ⊂ {t ∈ T | ∆ T (t) = 0}. (5.8) Moreover, according to Proposition 2.1 and Proposition 2.3, the K3 surfaces S(t) with the generic Néron-Severi lattice U ⊕ E 8 (−1) ⊕ E 6 (−1) of S(t) are generically degenerated to the K3 surfaces with the Néron-Severi lattice U ⊕ E 8 (−1) ⊕ E 7 (−1) on the divisor {t ∈ T | t 18 = 0}. This implies that the divisor {t ∈ T | t 18 = 0} should be contained in the image of reflection hyperplanes H D * . Namely, {t ∈ T | t 18 = 0} ⊂ {t ∈ T | ∆ T (t) = 0}.
(5.9) From (5.8) and (5.9), the polynomial t 18 d(t) should divide the polynomial ∆ T (t). By virtue of Proposition 5.1, we conclude that ∆ T (t) = t 18 d(t) (5.10) up to a constant factor, because the polynomial t 18 d(t) is of weight 108. Recall the total coordinate ring on Q is C[t 4 , t 6 , t 10 , t 12 , t 18 ] (Theorem 4.1). We had the relationship between the character det for Γ and the double covering p 1 : T 1 → T branched along the divisor {t ∈ T | ∆ T (t) = 0} (recall Section 3.3). This implies that there exist modular forms of character det corresponding to the branch loci of the double covering p 1 and they determine the structure of the ring A(Γ). In our case, since we have the irreducible decomposition (5.10) of the polynomial ∆ T (t), there is a modular form s 9 (s 45 , resp.) of weight 9 (45, resp.) and character det satisfying the relation s 2 9 = t 18 (s 2 45 = d(t), resp.).
Remark 5.1. The classical elliptic discriminant form, which is of weight 12 and generates the vector spaces of cusp forms for SL(2, Z), is originally given by the discriminant of the right hand side of the Weierstrass equation of an elliptic curve. Also, as in Section 2.3, from the discriminant of the right hand side of the defining equation (2.13) of the K3 surface S CD , we can compute Siegel modular forms of nontrivial character, which determine the structure of the ring of Siegel modular forms. Our constructions of s 9 and s 45 is a counterparts of such constructions from discriminants of Weierstrass forms.
From Theorem 4.1 and Theorem 5.1, we obtain the structure of the ring of modular forms coming from the lattice A as follows.
Theorem 5.2.
A(Γ, id) = C[t 4 , t 6 , t 10 , t 12 , t 18 ] ⊕ s 9 s 45 C[t 4 , t 6 , t 10 , t 12 , t 18 ], A(Γ, det) = s 9 C[t 4 , t 6 , t 10 , t 12 , t 18 ] ⊕ s 45 C[t 4 , t 6 , t 10 , t 12 , t 18 ].
Due to Proposition 2.2, Proposition 2.3 and Remark 5.1, our family of K3 surfaces can be regarded as a natural extension of the family of (2.13) and our modular forms for the lattice A give natural counterparts of the classical Siegel modular forms from based on inverse period mappings of K3 surfaces. The author expects that we can study our modular forms more thoroughly as follows. Table VII in [ST] is {4, 6, 10, 12, 18}, which is equal to the set of weights of independent modular forms for the trivial character. Here, the group of No.33 is the unique group in the exceptional cases such that the rank is 5. The author expects that there exist closed relationship between our modular forms and the reflection group of No.33 in [ST]. More precisely, the author conjectures that we can study our modular forms via the invariants of that group, as Hirzebruch [H] studied Hilbert modular forms via Klein's invariants for the icosahedral group and so on.
• Gritsenko and Nikulin [GN] gave the Borcherds product expansion of the Siegel modular form χ 35 of weight 35, which we mentioned in Section 2.3. In their study, it is important that the orthogonal group for the lattice A CD is generated by reflections for vectors with self-intersection −2. Now, our modular group Γ has a similar property and s 9 s 45 gives a counterpart of χ 35 . The author expects that we can obtain the Borcherds product expansion of s 9 s 45 .
• Recently, among researchers studying the Langlands program, it is more and more important to study modular forms on the real 3-dimensional hyperbolic space H 3 R . Since they are real analytic functions, it is highly non-trivial to construct and study them explicitly. Matsumoto, Nishi and Yoshida [MNY] succeeded to obtain modular forms on H 3 R via the period mapping for K3 surfaces of [MSY]. However, their results are much complicated. Here, our modular forms are also defined on the 4-dimensional symmetric domain of type IV . Recall that we have a merit that our modular group is simpler than that of [MSY]. So, the author expects that we can apply our results to construct modular forms on H 3 R effectively.
A
= U ⊕ U ⊕ A 2 (−1) (0.1) of M in L. This is the lattice of signature (2, 4). From this lattice A, we define the 4-dimensional space D M = {[ξ] ∈ P(A ⊗ C) | (ξ, ξ) = 0, (ξ, ξ) > 0}. Let D be a connected component of D M . Namely, D is a bounded symmetric domain of type IV . Let Γ =Õ + (A) be the subgroup of the stable orthogonal group O(A) for the lattice A which preserves the connected component D (for detail, see Section 2.2). Then, Γ acts on D discontinuously. Let D * be the connected component of the set {ξ ∈ A⊗C | (ξ, ξ) = 0, (ξ, ξ) > 0} which projects to D.
[t] = (t 4 : t 6 : t 10 : t 12 : t 18 ). Set T = {[t] ∈ P(4, 6, 10, 12, 18) | S([t]) gives an elliptic K3 surface}.(1.2)
Theorem 1 . 1 .
11For a generic point [t] ∈ T , the Picard number is equal to 16.
Also, H 1,1 R (S) = H 1,1 (S) ∩ H 2 (S, R) has signature (1, 19). Let V (S) + be the component of V (S) = {x ∈ H 1,1 R (S) | (x, x) > 0} which contains the class of a Kähler form on S. This is called the positive cone. Set ∆(S) = {δ ∈ NS(S) | (δ, δ) = −2}. (2.1) Let ∆(S) + be the subset of effective classes of ∆(S) and ∆(S) − = ∆(S) + . Due to the Riemann-Roch theorem, we can see that ∆(S) = ∆(S) + ∆(S) − . Let W (S) be the subgroup of the orthogonal group of H 2 (S, Z) generated by reflections for elements of ∆(S). This is called the Weyl group. It acts on V (S) + .
Kähler cone for S. We set NS(S) + = C(S) ∩ H 2 (S, Z), NS(S) ++ = C(S) + ∩ H 2 (S, Z).Then, x ∈ N S(S) + gives a numerically effective divisor with (x, x) > 0. Also, the set N S(S) ++ consists of ample divisor classes. Let M be a lattice in (1.11) of signature (1, 15), which is embedded in the K3 lattice L. By fixing a generic point [t 0 ] ∈ T , we take a reference surface S 0 = S([t 0 ]) as in Section 1.2. We obtain the corresponding S-marking ψ 0 :H 2 (S 0 , Z) → L. By Corollary 1.1, we can suppose that ψ −1 0 (M ) = NS(S 0 ). Letting ∆(M ) = {δ ∈ M | (δ, δ) = −2}, we set ∆(M ) + = {δ ∈ ∆(M ) | ψ −1 0 (δ) ∈ NS(S 0 )gives an effective class}. Then, letting ∆(M ) − = {−δ | δ ∈ ∆(M ) + }, we have the decomposition ∆(M ) = ∆(M ) + ∆(M ) − . Set V (M ) = {y ∈ M R | (y, y) > 0}. Then, V (M ) has two connected components. We can take the connected component V (M ) + containing ψ 0 (x) for x ∈ V (S 0 ) + . Set C(M ) + = {y ∈ V (M ) + | (y, δ) > 0 for all δ ∈ ∆(M ) + }. Definition 2.1. An M -polarized K3 surface is a pair (S, j) where S is a K3 surface and j : M ֒→ NS(S) is a primitive lattice embedding. We say that (S, j) is a pseudo-ample M -polarized K3 surface if j(C(M ) + ) ∩ NS(S) + = φ.
Definition 2. 2 .
2Let S be a K3 surface and ψ : H 2 (S, Z) → L be an isometry of lattices satisfying ψ −1 (M ) ⊂ NS(S). Then, the pair (S, ψ) is called a marked M -polarized K3 surface. Here, the pair (S, ψ −1 | M ) is a M -polarized K3 surface in the sense of Definition 2.1. If (S, ψ −1 | M ) is a pseudo-ample M -polarized K3 surface, then we call (S, ψ) a pseudo-ample marked M -polarized K3 surface.Definition 2.3. Suppose (S 1 , ψ 1 ) and (S 2 , ψ 2 ) are two marked pseudo-ample M -polarized K3 surfaces. If (S 1 , ψ −1 1 | M ) is isomorphic to (S 2 , ψ −1 2 | M ) as M -polarized K3 surfaces, we say that (S 1 , ψ 1 ) and (S 2 , ψ 2 ) are isomorphic as pseudo-ample M -polarized K3 surfaces. Moreover, if there exists an isomorphism f : S 1 → S 2 such that ψ 1 = ψ 2 • f * , then we say that (S 1 , ψ 1 ) and (S 2 , ψ 2 ) are isomorphic as marked pseudo-ample M -polarized K3 surfaces.Let M M be the fine moduli space of marked M -polarized K3 surfaces. This can be obtained by gluing local moduli space of marked M -polarized K3 surfaces. The local period mappings are glued together to give a holomorphic mappingper : M M → D M .(2.2)Here, D M is the symmetric space in (1.16). The mapping (2.2) isétale and surjective.Theorem 2.1. (Dolgachev[D] Section3) The restriction of the period mapping (2.2) to the subset M pa M of isomorphism classes of marked pseudo-ample M -polarized K3 surfaces gives a surjective mapping per ′ : M pa M → D M . (2.3) Remark 2.1. Let M a M be the set of isomorphism classes of marked ample isomorphism classes. Set D • M = D M − δ∈∆(A) (H δ ∩ D M ), where ∆(A) = {δ ∈ A | (δ, δ) = −2}. Then, the period mapping (2.3) induces a bijection between M a M and D • M (see [D]). In this paper, the orthogonal group for the lattice M is denoted by O(M ). Let us consider the group Γ(M ) = {σ ∈ O(L) | σ(m) = m for any m ∈ M }, which acts on the moduli space M M by (S, ψ) → (S, ψ • σ). This action does not change the isomorphism class of the M -polarized K3 surface (S, ψ −1 | M ). So, M pa M /Γ(M ) gives the isomorphism classes of pseudo ample M -polarized K3 surfaces. Here, note that there exists an injective homomorphism Γ(M ) → O(A). Let Γ M be the image of this injection. We can see that Γ M is equal tõ O(A) = Ker(O(A) → Aut(A ∨ /A)). (2.4) Here, A ∨ = Hom(A, Z), A ∨ /A is the discriminant group of A and O(A) → Aut(A ∨ /A) is the natural homomorphism. The groupÕ(A) is called the stable orthogonal group of A. We remark thatÕ(A) is a subgroup of finite index in O(A). Theorem 2.2. (Dolgachev [D] Section 3) The period mapping (2.3) gives the following bijection M pa M /Γ(M ) ≃ D M /Õ(A). (2.5) Namely, the quotient space D M /Õ(A) gives the set of isomorphism classes of pseudo-ample M -polarized K3 surfaces. Remark 2.2. Due to [GHS1] Lemma 3.2, the index [O(A) :Õ(A)] is equal to the order of the finite group O(q A ). Here, q A is the discriminant quadratic form. In our case of the lattice A of (0.1), we can see that O(q A ) ≃ Z/2Z. Hence, the stable orthogonal group Γ M =Õ(A) does not coincides with O(A).
( 2 . 6 )
26If a primitive element D ∈ NS(S) − {0} is on the boundary of the positive cone and (D, α) ≥ 0 for all α ∈ B(S), then the linear system | D | defines an elliptic fibration.
Lemma 2. 1 .
1Any equivalent classes of P -marked K3 surfaces (S, ψ) are given by explicit elliptic K3 surfaces S([t]) ([t] ∈ T ) defined by the equation (1.1).
Theorem 2. 3 .
3Let (S, ψ) be a marked pseudo-ample M -polarized K3 surface. Then, ψ : H 2 (S, Z) → L gives a P -marking for S.
The orthogonal group O(A) for the lattice A acts on the space D M . Let D be a connected component of D M . Set O + (A) = {γ ∈ O(A) | γ(D) = D}. We set Γ =Õ + (A) =Õ(A) ∩ O + (A), (2.10) whereÕ(A) is the stable orthogonal group of (2.4). We note that [Õ(A) :Õ + (A)] = 2. Especially, we have D M /Õ(A) = D/Õ + (A).
Proposition 2. 3 .
3The family {S CD (α, β, γ, δ) | (α : β : γ : δ) ∈ T 0 } can be regarded as a subfamily of {S([t]) | [t] ∈ T } corresponding to t 18 = 0.
4)where C (k) [t] = C (k) [t 4 , t 6 , t 10 , t 12 , t 18 ] is the vector space of polynomials of weight k. Due to (4.2), Pic(Q) is generated by O Q (1). Together with (4.3) and (4.4), we havek∈Z C (k) [t] ≃ L∈Pic(T ) H 0 (T, O T (L)) ≃ k∈Z H 0 (Q, O Q (k)).(4.5)Especially, C (k) [t]gives the space H 0 (Q, O Q (k)) of sections of the line bundle O Q (k) via the period mapping.
•
Finite unitary reflection groups give important examples in invariant theory. Shepherd and Todd gave the classification of them. The set of degrees for the reflection group of No.33 in
Structure of the ring of modular forms for ΓIn this section, we will consider the structure of the canonical orbibundles on the orbifolds coming from the results in Section 3.3. For the definitions and properties of orbibundles, for example, see [ALR]. Also, we will use the techniques of orbifolds referring to the work [HU].We can see that the total space D * of the C * -bundle on D is a Stein manifold, because D is a Stein manifold. Also, we can see thatAcknowledgmentThe author would like to thank Professor Hironori Shiga for helpful advises and valuable suggestions. He thanks Professor Manabu Oura for suggesting me the results of [ST]. Moreover, he would like to thank Professor Mehmet Haluk Sengün for letting me know the paper [MNY]. This work is supported by JSPS Grant-in-Aid for Scientific Research (18K13383) and MEXT Leading Initiative for Excellent Young Researchers.Proposition 5.1. The degree of the polynomial ∆ T (t) in (3.4) is 108.Proof. Let us consider the canonical bundle Ω T1 on T 1 . Since T is an open subset of the weighted projective spaceT = P(4, 6, 10, 12, 18), we haveSince the double covering p 1 : T 1 → T is branched along H T1 , by considering the holomorphic differential forms, we haveHere, the double covering induces a morphismFrom (5.3) and (5.4), we obtainWe remark that the Picard group Pic(T 1 ) is generated by [p 1 ] * O T (1) and O T1 (H T1 ). Letting d be the degree of ∆ T (t), due to (3.5), we obtainHere, (5.6) gives the relation which determines the structure of Pic(T 1 ). Hence, we havewhere Z is generated by [p 1 ] * O T (1) and Z/2Z is generated byHere, we note that d ∈ 2Z because all weights for T are even. Also, according to the construction of (3.6), the torsion part (Z/2Z) in (5.7) should be generated by det ∈ Char(Γ).Since p D : T D → T 1 in (3.6) gives the universal covering, the orbifold [T D /Γ] is equivalent to [T 1 /(Z/2Z)] = T 1 . Also, recall that the orbifold [T D /Γ] is equivalent to O. So, the orbifold T 1 is equivalent to O. Therefore, by Lemma 5.1, Lemma 5.2 and (5.5), we haveThis implies that 4 = −50 + d 2 . So, we obtain d = 108.
Pic(O) = Char(C * × Γ). Pic(O) = Char(C * × Γ).
) and (5.1), there exists a line orbibundle corresponding to a character of C * × Γ given by (λ, γ) → λ −k det(γ) l (l ∈ Z/2Z). In the following argument. According to Theorem 3.1 (2. this bundle is denoted by O O (k) ⊗ det lAccording to Theorem 3.1 (2) and (5.1), there exists a line orbibundle corresponding to a character of C * × Γ given by (λ, γ) → λ −k det(γ) l (l ∈ Z/2Z). In the following argument, this bundle is denoted by O O (k) ⊗ det l .
Lemma 5.2. The lifted period mapping Φ D * in (3.7) gives an isomorphism of orbifolds between. T D * /(C * × Γ)] and [(D * − S D * )/(C * × Γsuch that [Φ D * ] * O [(D * −S D * )/(C * ×Γ)] (1) ≃ O [T D * /(C * ×Γ)Lemma 5.2. The lifted period mapping Φ D * in (3.7) gives an isomorphism of orbifolds between [T D * /(C * × Γ)] and [(D * − S D * )/(C * × Γ)] such that [Φ D * ] * O [(D * −S D * )/(C * ×Γ)] (1) ≃ O [T D * /(C * ×Γ)] (1).
Φ| T − Φ D : T D → D − S D Gives The Monodromy Covering Of, St : T − S T → Q − S Q . Therefore, Theorem 3.1 (1) and the argument of Section 3.3, the lifted period mapping. together with the proof of Proposition 4.1, we have the assertion for the mapping Φ D * . As far as we consider holomorphic sections of line bundles, by virtue of Hartogs's phenomenon, weProof. From Theorem 3.1 (1) and the argument of Section 3.3, the lifted period mapping Φ D : T D → D − S D gives the monodromy covering of Φ| T −ST : T − S T → Q − S Q . Therefore, together with the proof of Proposition 4.1, we have the assertion for the mapping Φ D * . As far as we consider holomorphic sections of line bundles, by virtue of Hartogs's phenomenon, we
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Abelianisation of orthogonal groups and the fundamental group of modular varieties. V Gritsenko, K Hulek, G K Sankaran, J. Alg. 322V. Gritsenko, K. Hulek and G. K. Sankaran, Abelianisation of orthogonal groups and the funda- mental group of modular varieties, J. Alg., 322, 2009, 463-478
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|
[] |
[
"Electric Dipole Moments in U(1) Models",
"Electric Dipole Moments in U(1) Models"
] |
[
"Alper Hayreter \nDepartment of Physics\nIzmir Institute of Technology\nIZTECH\nTR35430Turkey\n\nDepartment of Physics\nConcordia University\n7141 Sherbrooke WestH4B 1R6MontrealQuebecCanada\n",
"Aslı Sabancı \nDepartment of Physics\nIzmir Institute of Technology\nIZTECH\nTR35430Turkey\n",
"Levent Solmaz \nDepartment of Physics\nBalıkesir University\nTR10145BalıkesirTurkey\n",
"Saime Solmaz \nDepartment of Physics\nBalıkesir University\nTR10145BalıkesirTurkey\n"
] |
[
"Department of Physics\nIzmir Institute of Technology\nIZTECH\nTR35430Turkey",
"Department of Physics\nConcordia University\n7141 Sherbrooke WestH4B 1R6MontrealQuebecCanada",
"Department of Physics\nIzmir Institute of Technology\nIZTECH\nTR35430Turkey",
"Department of Physics\nBalıkesir University\nTR10145BalıkesirTurkey",
"Department of Physics\nBalıkesir University\nTR10145BalıkesirTurkey"
] |
[] |
We study electric dipole moments (EDM) of electron and proton in E(6)-inspired supersymmetric models with an extra U(1) invariance. Compared to the Minimal Supersymmetric Standard Model (MSSM), in addition to offering a natural solution to the µ problem and predicting a larger mass for the lightest Higgs boson, these models are found to yield suppressed EDMs.
|
10.1103/physrevd.78.055011
|
[
"https://arxiv.org/pdf/0807.1096v2.pdf"
] | 119,196,092 |
0807.1096
|
e591649a78181bad8fcf2eb16aea3d6e09aa4231
|
Electric Dipole Moments in U(1) Models
(Dated: August 19, 2008) 19 Aug 2008
Alper Hayreter
Department of Physics
Izmir Institute of Technology
IZTECH
TR35430Turkey
Department of Physics
Concordia University
7141 Sherbrooke WestH4B 1R6MontrealQuebecCanada
Aslı Sabancı
Department of Physics
Izmir Institute of Technology
IZTECH
TR35430Turkey
Levent Solmaz
Department of Physics
Balıkesir University
TR10145BalıkesirTurkey
Saime Solmaz
Department of Physics
Balıkesir University
TR10145BalıkesirTurkey
Electric Dipole Moments in U(1) Models
(Dated: August 19, 2008) 19 Aug 20081
We study electric dipole moments (EDM) of electron and proton in E(6)-inspired supersymmetric models with an extra U(1) invariance. Compared to the Minimal Supersymmetric Standard Model (MSSM), in addition to offering a natural solution to the µ problem and predicting a larger mass for the lightest Higgs boson, these models are found to yield suppressed EDMs.
I. INTRODUCTION
While solving the quadratic divergence of radiative corrections to the Higgs boson mass, the supersymmetrization of the Standard Model with minimal matter content brings a µ parameter with a completely unknown scale. On the other hand, extending the gauge structure SU (3) C × SU (2) L × U (1) Y of the Minimal Supersymmetric Model by a new U (1)
Abelian group provides an effective µ term related with the VEV of some extra singlet scalar field; thus a scale (∼ T eV ) can be dynamically generated for the µ parameter. The supersymmetric U (1) models have been intensely studied in the literature. While such models can be motivated by low-energy arguments like µ problem [1] of the MSSM they also arise at low-energies as remnants of GUTs such as SO (10) and E(6) [2,3,4]. These models necessarily involve an extra neutral vector boson [5,6] whose absence/presence to be established at the LHC.
The particle spectrum of U (1) models involve bosonic fields Z µ and S as well as their superpartners Z and S in addition to those in the MSSM. Therefore, such models can be tested in various observables ranging from electroweak precision observables to Z µ effects at the LHC. As a matter of fact, analysis of Higgs sector along with CP violation potential [7] as well as structure of EDMs [8] suggest several interesting signatures also at collider experiments [9]. One of the most important spots of these models is that the lower bound of the lightest Higgs boson mass (m h ≥114 GeV) can be satisfied already at the tree level, and radiative corrections (dominantly the top-stop mass splitting) is not needed to be as large as in the MSSM. This feature can have important implications also for the little hierarchy problem [10].
In this work we will study EDMs of electron and neutron in U (1) models stemming from E(6) GUT. Our main interest is to look at the reaction of EDMs to gauge extensions in comparison to the MSSM. The paper is organized as follows. In the next section we introduce the models. Section III is devoted to EDM predictions and their numerical analysis. In Section IV we conclude.
II. THE U (1) MODELS
The model is characterized by the gauge structure
SU (3) C × SU (2) L × U (1) Y × U (1) Y(1)
where g 3 , g 2 , g Y and g Y are gauge coupling constants respectively. Here the extra U
ing E6 → SO(10) × U (1) ψ followed by SO(10) → SU (5) × U (1) χ where U (1) Y is a linear
combination of ψ and χ symmetries:
U (1) Y = cos θ E6 U (1) χ − sin θ E6 U (1) ψ(2)
which, supposedly, is broken spontaneously at a TeV. There arises, in fact, a continuum of U (1) models depending on the value of mixing angle θ E 6 . However, for convenience and traditional reasons, one can pick up specific values of θ E 6 to form a set of models serving a testing ground. We thus collected some well-known models in Table I with the relevant normalization factors and a common gauge coupling constant
g Y = 5 3 g 2 tan θ W(3)
In theories involving more than one U (1) factor the kinetic terms can mix since for such symmetries the field strength tensor itself is invariant. In U (1) model, involving hypercharge U (1) Y and U (1) Y , the gauge part of the Lagrangian takes the form
− L gauge = 1 4 F µν Y F Y µν + 1 4 F µν Y F Y µν + sin χ 2 F µν Y F Y µν(4)
where F µν = ∂ µ Z ν − ∂ ν Z µ is the field strength tensor of the corresponding U (1) symmetry.
Kinetic part of Lagrangian can be brought into canonical form by a non-unitary transfor-
2 √ 15 Q η 2 Q I 2 √ 6 Q ψ 2 √ 10 Q N 2 √ 15 Q S u L , d L -2 0 1 1 -1/2 u R 2 0 -1 -1 1/2 d R -1 1 -1 -2 -4 e L 1 -1 1 2 4 e R 2 0 -1 -1 1/2 H u 4 0 -2 -2 1 H d 1 1 -2 -3 -7/2 S -5 -1 4 5 5/2mation Ŵ Ŷ W Y = 1 − tan χ 0 1/ cos χ Ŵ B W B (5)
whereŴ Y andŴ Y are the chiral superfields associated with the two U (1) gauge symmetries.
This transformation also acts on the gauge boson and gaugino components of the chiral superfields in the same form. The U (1) Y × U (1) Y part of covariant derivative in the case of no kinetic mixing is given by
D µ = ∂ µ + ig Y Y B µ + ig Y Q Y B µ(6)
however, with the presence of kinetic mixing this covariant derivative is changed to
D µ = ∂ µ + ig Y Y B µ + i −g Y Y tan χ + g Y cos χ Q Y B µ(7)
where g Y is gauge coupling constant and Q Y is fermion charges of U (1) Y symmetry. With a linear transformation of charges the covariant derivative takes the form [12]
D µ = ∂ µ + ig Y Y B µ + ig Y Q Y B µ(8)
in which the effective U (1) Y charges are shifted from its original value Q Y to
Q Y = Q Y cos χ − g Y g Y Y tan χ(9)
For the proper treatment of the models the most general superpotential should be considered [9], but for simplicity we parametrized U (1) models by the following superpotential
W = h u Q · H u U c + h d Q · H d D c + h e L · H d E c + h S S H u · H d(10)
where we discarded additional field (assuming that they are relatively heavy compared to this very spectrum) that are necessary for the unification of gauge couplings. Our conventions At this point, it is useful to explicitly state the soft breaking terms, the most general holomorphic structures are
are such that, for instance Q · H u ≡ Q T (iσ 2 ) H u = ij Q i H j u with 12 = − 21 = 1.− L sof t = ( i M i λ i λ i − A S h s SH d H u − A ij u h ij u U c j Q i H u − A ij d h ij d D c j Q i H d − A ij e h ij e E c j L i H d + h.c.) + m 2 Hu |H u | 2 + m 2 H d |H d | 2 + m 2 S |S| 2 + m 2 Q ij Q i Q * j + m 2 U ij U c i U c * j + m 2 D ij D c i D c * j + m 2 L ij L i L * j + m 2 E ij E c i E c * j + h.c.(11)
where the sfermion mass-squareds m 2 Q,...,E c and trilinear couplings A u,...,e are 3 × 3 matrices in flavor space. All these soft masses will be taken here to be diagonal. In general, all gaugino masses, trilinear couplings and flavor-violating entries of the sfermion mass-squared matrices are source of CP violation. However, for simplicity and definiteness we will assume a basis in which entire CP violating effects are confined into the gaugino mass M 1 (with M 1 = M 1 ), and the rest are all real (interested readers can chief to [13]).
These soft SUSY breaking parameters are generically nonuniversal at low energies. We will not address the origin of these low energy parameters as to how they follow via RG evolution from high energy boundary conditions, instead we will perform a general scan of the parameter space.
III. CONSTRAINTS AND IMPLICATIONS FOR EDMS
Due to the extra U (1) symmetry, associated Z boson can be expected to weigh around the electroweak bosons, and can exhibit significant mixing with the ordinary Z boson. The LEP data and other low-energy observables forbid Z-Z mixing to exceed one per mill level.
Indeed, precision measurements have shown that Z mass should not be less than ∼ 700
GeV for any of the models under concern (excluding leptophobic Z 's). Indeed, mixing of the Z and Z puts important restrictions on the mass and the mixing angle of the extra boson and this can be studied from the following Z − Z mixing matrix;
M 2 Z−Z = M 2 Z ∆ 2 ∆ 2 M 2 Z (12)
with M Z being the usual SM Z mass in the absence of mixing and
M 2 Z = 1 4 G 2 (|v u | 2 + |v d | 2 ) ∆ 2 = 1 2 Gg Y (Q Hu |v u | 2 − Q H d |v d | 2 )(13)M 2 Z = g 2 Y (Q 2 Hu |v u | 2 + Q 2 H d |v d | 2 + Q 2 s |v S | 2 )
where G 2 = g 2 Y + g 2 2 and g Y is the gauge coupling constant of the extra U (1). The mixing matrix can be diagonalized by an orthogonal transformation;
Z 1 Z 2 = cos φ sin φ − sin φ cos φ Z Z (14)
giving the mass eigenstates Z 1,2 with masses M Z 1 ,Z 2 where α is given by
tan 2α = 2∆ 2 M 2 Z − M 2 Z(15)
In the numerical analysis we considered α < 3 × 10 −3 and confined M Z > 700 GeV. Notice that when ∆ vanishes (tan β ∼ Q Hu /Q H d ) Z 1,2 can be identified with the ordinary Z and Z bosons; since we considered low tan β values, we will use the term Z for the heavy extra boson.
Besides this, the implication of the extra gauge boson can also be seen in sfermion sector, that is sfermion mass matrix is modified due to the presence of Z boson as;
M 2 f = M 2 f LL M 2 f LR M 2 f LR M 2 f RR (16) M 2 f LL = M 2 f L + h 2 f |H 0 f | 2 + 1 2 Y f L g 2 Y − T 3f g 2 2 |H 0 u | 2 − |H 0 d | 2 + g 2 Y Q f L |H 0 u | 2 Q Hu + |H 0 d | 2 Q H d + |S| 2 Q s M 2 f LR = h f A f H 0 f + h s SH 0 f (17) M 2 f RR = M 2 f R + h 2 f |H 0 f | 2 + 1 2 Y f R g 2 Y |H 0 u | 2 − |H 0 d | 2 + g 2 Y Q f R |H 0 u | 2 Q Hu + |H 0 d | 2 Q H d + |S| 2 Q s
in terms of shifted charge assignments. Sfermion mass matrix is hermitian and can be diagonalized by the unitary transformation
D † M 2 f D = diag(m 2 f 1 , m 2 f 2 )(18)
where D is the L − R mixing matrix for sfermions and is parametrized as
D = cos θ sin θ e −iφ sin θ e iφ cos θ (19)
It is worth to note that sfermion mass eigenvalues in U (1) models will be different than reference point [14], and additionally we assumed A s = A t . Notice that Q = 0 corresponds to MSSM prediction. This figure illustrates the difference between the MSSM and of the U (1) sfermion mass predictions, for the same input parameters. As should be inferred from this figure, opposite values of Q f L and Q f R can violate collider bounds for some of the U (1) models while this selection is current for the MSSM, that will be important in the numerical analysis and we will consider somewhat larger values of sfermion gauge eigenstates to overcome this issue.
In U (1) models compared to MSSM, there is an extra single scalar state in Higgs sector, an additional pair of higgsino and gaugino states are covered in neutralino sector and chargino sector is kept structurally unaltered though it is different than the MSSM due to the effective µ term. Now we will deal with these sectors.
A. Higgs Sector
The Higgs sector in U (1) models compared to MSSM is extended by a single scalar state S whose VEV breaks the U (1) symmetry and generates a dynamical µ ef f = h S S . For a detailed analysis of the Higgs sector with CP violating phases we refer to [15] and references therein. The tree level Higgs potential gets contributions from F terms, D terms and soft supersymmetry breaking terms:
V tree = V F + V D + V sof t ,(20)
in which
V F = |h s | 2 |H u · H d | 2 + |S| 2 (|H u | 2 + |H d | 2 ) ,(21)V D = G 2 8 |H u | 2 − |H d | 2 2 + g 2 2 2 (|H u | 2 |H d | 2 − |H u · H d | 2 ) + g 2 Y 2 Q Hu |H u | 2 + Q H d |H d | 2 + Q S |S| 2 2 ,(22)V sof t = m 2 u |H u | 2 + m 2 d |H d | 2 + m 2 s |S| 2 + (A s h s SH u · H d + h.c.).(23)
where G 2 = g 2 2 + g 2 Y and g Y = 3/5g 1 , g 1 is the GUT normalized hypercharge coupling.
At the minimum of the potential, the Higgs fields can be expanded as follows (see [16] for a detailed discussion):
H u = 1 √ 2 √ 2H + u v u + φ u + iϕ u , H d = 1 √ 2 v d + φ d + iϕ d √ 2H − d S = 1 √ 2 (v s + φ s + iϕ s ) ,(24)in which v 2 ≡ v 2 u + v 2 d = (246 GeV) 2 .
In the above expressions, a phase shift e iθ can be attached to S which can be fixed by true vacuum conditions considering loop effects (see [15] for details). Here it suffices to state that the spectrum of physical Higgs bosons consist of three neutral scalars (h, H, H ), one CP odd pseudoscalar (A) and a pair of charged Higgses H ± in the CP conserving case. In total, the spectrum differs from that of the MSSM by one extra CP-even scalar.
Notice that, the composition, mass and hence the couplings of the lightest Higgs boson of U (1) models can exhibit significant differences from the MSSM, and this could be an important source of signatures in the forthcoming experiments. It is necessary to emphasize that these models can predict larger values for m h , which hopefully will be probed in near future at the LHC. In the numerical analysis we considered m h > 90 GeV as the lower limit.
Besides this, as we will see, it is possible to obtain larger values such as m h ∼ 140 GeV within some of these E(6) based models.
B. Neutralino Sector
In U (1) models the neutralino sector of the MSSM gets enlarged by a pair of higgsino and gaugino states, namelyS (which we call as 'singlino') andB (which we call as bino-prime or zino-prime depending on the state under concern). The mass matrix for the six neutralinos in the (B,W 3 ,H 0 d ,H 0 u ,S,B ) basis is given by
M χ 0 = M 1 0 −m Z c β s W m Z s β s W 0 M K 0 M 2 m Z c β c W −m Z s β c W 0 0 −m Z c β s W m Z c β c W 0 −µ ef f −µ λ s β Q H d m v c β m Z s β s W −m Z s β c W −µ ef f 0 −µ λ c β Q Hu m v s β 0 0 −µ λ s β −µ λ c β 0 Q S m s M K 0 Q H d m v c β Q Hu m v s β Q S m s M 1 (25)
with gaugino mass parameters M 1 , M 2 , M 1 and M K [12] forB ,W 3 ,B andB −B mixing respectively. There arise two additional mixing parameters after electroweak breaking:
m v = g Y v and m s = g Y v s(26)
Moreover, supersymmetric higgsino mass and doublet-singlet higgsino mixing masses are generated to be
µ ef f = h S v S √ 2 , µ λ = h S v √ 2(27)
where
v = v 2 u + v 2 d .
The neutralino mass matrix can be diagonalized by a unitary matrix such that
N † M χ 0 N = diag(m χ 0 1 , ...,m χ 0 6 )(28)
The additional neutralino mass eigenstates due to new higgsino and gaugino fields encode effects of U (1) models wherever neutralinos play a role such as magnetic and electric dipole moments.
In fact, the neutralino-sfermion exchanges contribute to EDMs of quarks and leptons as follows:
d E f −χ 0 e = α EM 4π sin 2 θ W 2 k=1 6 i=1 Im(η f ik )m χ 0 i m 2 f k Qf B m 2 χ 0 i m 2 f k (29)
where the neutralino vertex is,
η f ik = − √ 2{tan θ W (Q f − T 3f )N 1i + g Y g 2 Q f L N 6i + T 3f N 2i }D f 1k − κ f N bi D f 2k × ( √ 2(tan θ W Q f N 1i + g Y g 2 Q f R N 6i )D f 2k − κ f N bi D f 1k )(30)
and
κ u = m u √ 2M W sin β , κ d,e = m d,e √ 2M W cos β(31)A(x) = 1 2(1 − x) 2 3 − x + 2 ln x 1 − x , B(x) = 1 2(x − 1) 2 1 + x + 2x ln x 1 − x(32)
Since H u and H d couple fermions differently due to their hypercharges, the b index in neutralino diagonalizing matrix must be carefully chosen in numerical analysis.
C. Chargino sector
Unlike the Higgs and Neutralino sectors, chargino sector is structurally unchanged in U (1) models compared to MSSM. However, chargino mass eigenstates become dependent upon U (1) breaking scale through µ ef f parameter in their mass matrix:
M χ ± = M 2 M W √ 2 sin β M W √ 2 cos β µ ef f (33)
which can be diagonalized by biunitary transformation
U M χ ± V −1 = diag(m χ + 1 ,m χ + 2 )(34)
where U and V are unitary mixing matrices. Since the chargino sector is structurally the same as with the MSSM, the fermion EDMs through fermion-sfermion-chargino interactions are given by
d E e−χ ± e = α EM 4π sin 2 θ W κ e m 2ν e 2 i=1m χ + i Im(U i2 V i1 )A m 2 χ + i mν e 2 (35) d E d−χ± e = − α EM 4π sin 2 θ W 2 k=1 2 i=1 Im(Γ dik )m χ + i m 2 u k QũB m 2 χ + i m 2 u k + (Q d − Qũ)A m 2 χ + i m 2 u k (36) d E u−χ ± e = − α EM 4π sin 2 θ W 2 k=1 2 i=1 Im(Γ uik )m χ + i m 2 d k QdB m 2 χ + i m 2 d k + (Q u − Qd)A m 2 χ + i m 2 d k(37)
where the chargino vertices are,
Γ uik = κ u V i2 D d1k (U i1 D d1k − κ d U i2 D d2k )(38)Γ dik = κ d U i2 D u1k (V i1 D u1k − κ u V i2 D u2k )(39)
D. Electron and Neutron EDMs
Total EDMs for electron and neutron is therefore the sum of all individual interactions, the electron EDM arises from CP-violating 1-loop diagrams with the neutralino and chargino
exchanges d E e = d E e−χ 0 + d E e−χ ±(40)
While studying neutron EDMs, besides neutralino and chargino diagrams, 1-loop gluino exchange contribution must also be taken into account, thus the EDM for quark-squarkgluino interaction can be written as;
d E q−g e = − 2α s 3π 2 k=1 Im(Γ 1k q ) mg m 2 q k QqB m 2 g m 2 q k(41)
with the gluino vertex,
Γ 1k q = D q2k D q1k(42)
However, for neutron EDM there are additionally two other contributions arising from quark chromoelectric dipole moment of quarks;
d C q−g = g s α s 4π 2 k=1 Im(Γ 1k q ) mg m 2 q k C m 2 g m 2 q k (43) d C q−χ 0 = g s g 2 16π 2 2 k=1 6 i=1 Im(η qik )m χ 0 i m 2 q k B m 2 χ 0 i m 2 q k (44) d C q−χ ± = −g s g 2 16π 2 2 k=1 2 i=1 Im(Γ qik )m χ ± i m 2 q k B m 2 χ ± i m 2 q k(45)
where,
C(x) = 1 6(x − 1) 2 10x − 26 + 2x ln x 1 − x − 18 ln x 1 − x(46)
and the CP violating dimension-six operator from 2-loop gluino-top-stop diagram is
d G = −3α s m t g s 4π 3 Im(Γ 12 t ) z 1 − z 2 m 3 g H(z 1 , z 2 , z t )(47)
with
z i = Mt i mg 2 , z t = m t mg 2(48)
and the 2-loop function is given by [17] H(z 1 , z 2 , z t ) = 1 2
1 0 dx 1 0 du 1 0 dy x (1 − x) u N 1 N 2 D 4(49)
with
N 1 = u (1 − x) + z t x (1 − x)(1 − u) − 2u x [z 1 y + z 2 (1 − y)], N 2 = (1 − x) 2 (1 − u) 2 + u 2 − 1 9 x 2 (1 − u) 2 , D = u (1 − x) + z t x (1 − x)(1 − u) + u x [z 1 y + z 2 (1 − y)](50)
Therefore total neutron EDM is written with the help of non-relativistic SU (6) coefficients of chiral quark model [18]
d n = 1 3 (4 d d − d u )(51)
in which all the contributions are gathered into u and d quark interactions
d E u = d E u−χ 0 + d E u−χ ± + d E u−g + d C u−χ 0 + d C u−χ ± + d C u−g + d G (52) d E d = d E d−χ 0 + d E d−χ ± + d E d−g + d C d−χ 0 + d C d−χ ± + d C d−g + d G (53)
The above analysis is at the electroweak scale and the evolution of d E,C,G 's down to hadronic scale is accomplished via Naivë Dimensional Analysis
d q = η E d E q + η C e 4π d C q + η G eΛ 4π d G(54)
where the QCD correction factors are η E = 1.53, η C 3.4 and Λ 1.19 GeV is the chiral symmetry breaking scale [19].
For the sake of generality, we give all the formulae which may contribute to electron and neutron EDM's, however, depending on the origin of CP violating phases, some of above equations may yield no contributions to the EDM's, as in our numerical analysis we considered only one CP-odd phase corresponding to complex bino (and bino-prime) mass, for simplicity. Therefore in our analysis contributions of gluinos for quark-squark-gluino interaction (d E q−g ), chromoelectric dipole moment of quarks (d C q−g ) and the CP violating dimension-six operator from the 2-loop gluino-top-stop diagram (d G ) will be missing. Care should be paid to the point that this phase can only provide a subleading contribution to the neutron EDM, for a complete treatment those missing contributions should be added too.
E. Numerical Analysis
In this part we will perform a detailed numerical study of various E(6)-based U (1) models in regard to their predictions for electron and neutron EDMs. We will compare the models given in Tab. I with each other and with the MSSM. In doing this, we consider bino (and bino-prime) mass to be complex and assume the rest of the parameters as real quantities (though this simplification might seem somewhat unrealistic we expect that results can still reveal certain salient features in such models).
During the analysis, to respect the collider bounds, we require the masses satisfy
m h > 90, m sf ermions > 100, m χ ± 1 > 105, M Z > 700(55)
(all in GeV) and the Z − Z mixing angle to be less than 3 × 10 −3 . Bounds from naturalness and perturbativity constraint are respected by considering 0.1 ≤ h s ≤ 0.75 [15,20,21].
Additionally, to make Z sufficiently heavy v s is scanned up to 10 TeV and low tan β regime is analyzed which is the preferred domain for the models and for which consideration of stop corrections suffice.
Imprints of different U (1) models related with electron and neutron EDM reactions are presented in Fig. 2. This figure depicts variations of EDMs with µ ef f in S, I, N , ψ and η models. In this figure and in the followings, since we did not take into consideration renormalization group running, we scanned the related parameters randomly. But we carefully used the same data points in each of the models. As can be seen from Fig. 2, with increasing µ ef f , eEDM (left panels) predictions start to raise from S to η model. Additionally, as the effective µ parameter deviates from the EW scale, eEDM predictions seem promising to bound the effective µ term in η and ψ models. But when it comes to nEDM (right panels)
as the µ ef f increases predictions for neutron EDM decreases from S to η model, respectively.
In other words, in terms of the difference between electron and neutron EDM predictions, the η model is the most striking one and the S model is the mildest model.
It is also useful to probe how EDM predictions vary with the mass of Z boson, which is given in Fig. 3. The left η panel of Fig 3 shows that it may be possible to bound Z mass from above once the eEDM predictions near the present experimental value (at least for certain range of parameters), whereas some models like S and I do not seem to react significantly to this variation. The most sensitive models to bound Z mass using the eEDM Straight lines in this and following figures denote corresponding eEDM and nEDM experimental constraints [27,28].
results are η, ψ and N models. On the other hand, it may also be possible to bound the mass of Z in S model using the nEDM measurements, as can be seen from the bottom S panel of Fig. 3. see the gray crosses in N and ψ panels).
As can be deduced from the previous figures there is a hierarchy among the models. This A rather interesting effect of the kinetic mixing can be investigated on the composition of the LSP candidate of the U (1) models. For the selected range of the parameters, all U (1) models share the same LSP candidate with the MSSM, which is bino. But also notice that singlino dominated neutralino can be a good candidate for the LSP [22,23], for this kind of models.
In our domain, without the kinetic mixing its composition can be expected to be very Our last figure is Fig. 9 where we present tan β dependencies of the electron and neutron
EDMs. Here tan β is scanned up to 10 and the most striking difference between the MSSM and U (1) models, for the models under concern, turns out to be the smallness of tan β (can be as small as 0.5), which is ruled out for the MSSM. Additionally, for most of the models eEDM and nEDM predictions decrease with decreasing tan β as in the MSSM. The only exception to this observation is found for η model where the sensitivity of eEDM predictions are very small. But, in general, this common tendency of U (1) models show that it is easier to evade EDM constraints in such models where tan β ∼ 1 is actually the natural value. As can be seen from the figures presented in this section, we did not try to constrain complex phases but instead we tried to demonstrate the general tendencies in U (1) models, and apparently all the examples given here are well below the experimental bounds.
IV. CONCLUSION
In this work we have performed a study of EDMs (of electron and neutron) in U (1) models descending from E(6) SUSY GUT. With anticipated increase in precision of EDM measurements, our results show that these models give rise to observable signatures not shared by the MSSM. Indeed, U (1) models generically possess different predictions for EDMs compared to MSSM (see Fig. 4). This very feature provides a way of determining nature of the supersymmetric model at the TeV scale via EDM measurements.
Apart from comparisons with the MSSM, different E(6)-based U (1) models are found to have different predictions for various observables studied in the text. Indeed, sensitivity of EDMs to µ parameter (see Fig. 2), to Z mass (see Fig. 3), and to tan β are different for different models. Furthermore, eEDM and nEDM are found to exhibit different dependencies in each case. These features establish the fact that, once precise measurements are attained (presumably at a high-energy linear collider) one can determine likely breaking directions for E(6) grand unified group down to that of the MSSM. Also interesting are the predictions of different U (1) models for m h (which is plotted against µ ef f in Fig. 5). Indeed, both range and shape of the allowed domain are different for different models, and this feature also helps determining the correct model (of E (6) origin) once precise measurements of associated quantities are available.
It is not surprising that these models can have important implications also for FCNC observables (including their CP asymmetries) [24]. Moreover, the EDMs discussed above can be correlated with the CP asymmetries (of B meson decays [25]) or with the Higgs sector itself [26] so as to further bound such models with the information available from B factories and Tevatron. This kind of analysis will be given elsewhere.
To conclude, the problem of CP violation (in particular EDMs) is a particularly important issue of U (1) models for various reasons, most notably, the approximate reality of the effective µ parameter. Analyses of various observables (including the FCNC ones) can shed
be a light (broken at a TeV) linear combination of a number of U(1) symmetries (in effective string models there are several U(1) factors whose at least one combination can survive down to the TeV scale). There are a number of U (1) models studied in literature, all of them offer a dynamical solution to the µ problem of the MSSM via spontaneous breaking of extra U (1) Abelian factor at the TeV scale depending on the model, and many of them respecting gauge couplings unification predicts extra fields in order to sort out gauge and gravitational anomalies from the theory. These models typically arise from SUSY GUTs and strings. From E(6) GUT, for example, two extra U (1) symmetries appear in the break-
The righthanded fermions are contained in the chiral superfields U , D, E via their charge-conjugates e.g. U = u R , (u R ) C . What a U (1) model does is basically to allow a dynamical effective µ ef f = h s S related to the scale of U (1) breaking instead of an elementary µ term which troubles supersymmetric Higgsino mass in the MSSM. Notice that a bare µ term cannot appear in the superpotential due to U (1) invariance.
FIG. 1 :
1in the MSSM due to the contribution of extra gauge boson and kinetic mixing. In general D U (1) = D M SSM and the MSSM results can be recovered by assuming no kinetic mixing (sin χ = 0) and no charges under U (1) at all. Impact of selectron U (1) charge Q on the selectron masses (In GeVs). But the existence of the U (1) charges have profound impact on the sfermion eigenvalues. To show this we present Fig. 1 in which selectron mass eigenvalues are plotted against U (1) charges for two different cases. In panel a) we assumed Q eL = − Q eR = Q to be compared with panel b) in which Q eL = Q eR = Q , with the following inputs: h s = 0.5, v s = 5 TeV, Q Hu = Q H d = −0.05 and the rest of the parameters are taken as in SPS1a
FIG. 2 :
2µ ef f versus eEDM (left panels) and nEDM (right panels) in U (1) models (top to bottom: S, I, N, ψ and η models). As inputs, all trilinears are scanned in -2 to 2 TeV, all sfermions are scanned in 0.5 to 1 TeV separately. The resulting data sets are used to obtain in every model with tan β = 3. Absolute value of EDM predictions are given in log 10 base, µ ef f values are given in GeVs.
FIG. 3 :
3M Z versus eEDM (left panels) and nEDM (right panels) in U (1) models, as in Fig. 2. Our next figure is Fig. 4 in which electron and neutron EDM predictions are presented for the MSSM and for the aforementioned U (1) models against variations in the phase of bino. In S and I models eEDM predictions are generally well below the MSSM predictions. On the other hand, in η model it is possible to get lower predictions for nEDM. Notice that while majority of the points obtained are above the MSSM predictions there are regions where it is possible to obtain smaller EDM values for both of the electron and neutron (i.e.
FIG. 4 :
4The phase of M 1 versus eEDM (left panels) and nEDM (right panels) in U (1) models.Here our shading convention is such that dark triangles correspond to MSSM and gray crosses are for U (1) models. Inputs are as inFig. 2.
FIG. 5 :
5situation is also shared by the mass of the lightest Higgs boson. We provideFig. 5 in which mass of the lightest Higgs boson is plotted against variations of µ ef f . Here again, predictions for the mass of the lightest Higgs boson are in an order increasing from S to η model. Notice that while the LEP2 bound on SM like Higgs boson confines its mass to be larger than 114 GeV it can not be used directly in U (1) models, so we accepted 90 GeV as the lower bound. But all of the models are capable of satisfying m h > 114 GeV. Additionally, compared to the MSSM, in these U (1) models it is possible to find larger m h predictions for m h i.e. see η or ψ panels. Effective µ versus m h in U (1) models (All in GeVs). Inputs are the same with Fig. 2. Another important issue worth noticing within these models is the possibility of kinetic mixing. As should be predicted it modifies EDM predictions (as well as many other properties of the models) in accordance with its magnitude. To give a concrete example of its impact, we selected N model for which eEDM and nEDM predictions are generally larger than the MSSM. So, we provide Fig. 6 for electron and neutron EDMs. As can be seen the very figure, even very small values of the kinetic mixing angle (i.e. χ=-0.1) can yield sizable variations for the EDM predictions of the electron, but, its impact on the neutron EDM is rather small. Meanwhile, nonzero choices of the mass terms M K (see the c panels) can also reduce both of the eEDM and nEDM predictions. When both of the χ and M K are in charge (see the d panels), we see that, both of the eEDM and nEDM predictions in the N model can be smaller than the MSSM predictions.
FIG. 6 :FIG. 7 :FIG. 8 :
678similar to the MSSM's lightest neutralino. This can be inferred from Fig. 7 where singlino (gray crosses) and Z -ino (dark triangles) compositions of the LSP candidate are plotted against varying M K with (left panel) and without (right panel) the kinetic mixing scanned randomly in [-0.3,0]. Notice that when M K ∼ 0 GeV, even if the kinetic mixing is turned on, the composition of the LSP candidate can not be expected to be very different from the MSSM. For a clear picture of this phenomena we support Figs. 6 and 7 with Fig. 8, where the mass eigenvalues of the N model neutralinos are plotted against varying M The eEDM (left panels) and the nEDM (right panels) versus argument of M 1 in N model (Dark triangles : MSSM, gray crosses : N model). Here we fixed tan β = 5, m sleptons = 400 GeV, m squarks = 750 GeV, all trilinars=-1500 GeV, M 2 = 190 GeV (M 1 = 0.56 M 2 , M 3 = 2.8 M 2 ) In panel a) mixing angle χ = 0, M Y X = 0, b) mixing angle χ = −0.3, −0.2, −0.1, 0 and M Y X = 0, c) mixing angle χ = 0 but M Y X scanned randomly in 0 to 0.5 TeV d) χ = −0.3, −0.2, −0.1, 0 and M Y X scanned randomly in 0 to 0.5 TeV. Notice that M Y X ∼ M K for small χ values as in our cases (see [12] for details) (panel b)) and without (panel a)) mixing angle. As can be seen from Fig 8, mass of the LSP candidate of the related model is sensitive to M K . This tendency reduces as we go away Singlino (gray crosses: |N 1,5 | 2 ) and Z -ino (dark tringles: |N 1,6 | 2 ) compositions of the lightest neutralino against M K in N model. Inputs are from c) and d) panels of Fig. 6. (for a) and b) panels they are of the order 10 −7 ). from the lightest neutralino up to 5th and 6th neutralinos. For those two heavy neutralinos impact of nonzero mixing angle can dominate the effect of M K if both of them are in charge (see panel b) of Fig 8). For the selected range of parameters lightest neutralino is very similar to the MSSM's neutralino as far as the mentioned variables are off; when they are on, their corresponding impact on the composition and on the mass of the lightest neutralino can be ∼ 10-20 % as can be seen from the very figures. Neutralino masses versus M K corresponding to the same panels of Fig. 7 (All in GeVs).
FIG. 9 :
9tan β versus eEDM (left panels) and nEDM (right panels) predictions in different U (1) models. We used the conventions ofFig. 3. Here again straight lines denotes the corresponding EDM bounds.
Fig. 6
6makes it clear that the soft-breaking mass that mix U (1) Y and U (1) gauginos is a sensitive source of EDMs. Indeed, as happens in models of paraphotons, entire matter can be neutral under U (1) symmetry yet such a kinetic mixing (that mix gauge bosons and gauginos ) can exist and can have important implications. These figures make it clear that EDMs vary significantly with this parameter.
TABLE I :
IGauge quantum numbers of several U (1) models[11]
We all would like to thank to D. A. DEMİR for his contributions with inspiring and illuminating discussions in various stages of this work. We all would like to thank to D. A. DEMİR for his contributions with inspiring and illuminating discussions in various stages of this work.
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|
[] |
[
"COLOURED QUIVERS FOR RIGID OBJECTS AND PARTIAL TRIANGULATIONS: THE UNPUNCTURED CASE",
"COLOURED QUIVERS FOR RIGID OBJECTS AND PARTIAL TRIANGULATIONS: THE UNPUNCTURED CASE"
] |
[
"Robert J Marsh ",
"Yann Palu "
] |
[] |
[] |
We associate a coloured quiver to a rigid object in a Hom-finite 2-Calabi-Yau triangulated category and to a partial triangulation on a marked (unpunctured) Riemann surface. We show that, in the case where the category is the generalised cluster category associated to a surface, the coloured quivers coincide. We also show that compatible notions of mutation can be defined and give an explicit description in the case of a disk. A partial description is given in the general 2-Calabi-Yau case. We show further that Iyama-Yoshino reduction can be interpreted as cutting along an arc in the surface.
|
10.1112/plms/pdt032
|
[
"https://arxiv.org/pdf/1012.5790v4.pdf"
] | 53,549,086 |
1012.5790
|
0a6610ce3da00b0ed021ded0608a75cd849e900f
|
COLOURED QUIVERS FOR RIGID OBJECTS AND PARTIAL TRIANGULATIONS: THE UNPUNCTURED CASE
20 Dec 2011
Robert J Marsh
Yann Palu
COLOURED QUIVERS FOR RIGID OBJECTS AND PARTIAL TRIANGULATIONS: THE UNPUNCTURED CASE
20 Dec 2011
We associate a coloured quiver to a rigid object in a Hom-finite 2-Calabi-Yau triangulated category and to a partial triangulation on a marked (unpunctured) Riemann surface. We show that, in the case where the category is the generalised cluster category associated to a surface, the coloured quivers coincide. We also show that compatible notions of mutation can be defined and give an explicit description in the case of a disk. A partial description is given in the general 2-Calabi-Yau case. We show further that Iyama-Yoshino reduction can be interpreted as cutting along an arc in the surface.
Introduction
Let (S, M ) be a pair consisting of an oriented Riemann surface S with non-empty boundary and a set M of marked points on the boundary of S, with at least one marked point on each component of the boundary. We further assume that (S, M ) has no component homeomorphic to a monogon, digon, or triangle. A partial triangulation R of (S, M ) is a set of noncrossing simple arcs between the points in M . We define a mutation of such triangulations, involving replacing an arc α of R with a new arc depending on the surface and the rest of the partial triangulation. This allows us to associate a coloured quiver to each partial triangulation of M in a natural way. The coloured quiver is a directed graph in which each edge has an associated colour which, in general, can be any integer.
Let C be a Hom-finite, 2-Calabi-Yau, Krull-Schmidt triangulated category over a field k. A rigid object in C is an object R with no self-extensions, i.e. satisfying Ext 1 C (R, R) = 0. Rigid objects in C can also be mutated. In this case the mutation involves replacing an indecomposable summand X of R with a new summand depending on the relationship between X and the rest of the summands of R. As above, this allows us to associate a coloured quiver to each rigid object of C in a natural way.
In [BZ] the authors study the generalised cluster category C (S,M) in the sense of Amiot [Ami09] associated to a surface (S, M ) as above. Such a category is triangulated and satisfies the above requirements. It is shown in [BZ] that, given a choice of (complete) triangulation of (S, M ), there is a bijection between the simple arcs in (S, M ) (joining two points in M ), up to homotopy, and the isomorphism classes of rigid indecomposable objects in C (S,M) . If X α denotes the object corresponding to an arc α then Ext 1 C (S,M ) (X α , X β ) = 0 if and only if α and β do not cross. It follows that there is a bijection between partial triangulations of (S, M ) and rigid objects in C (S,M) . Our main result is that the coloured quivers defined above coincide in this situation and that the two notions of mutation are compatible.
Suppose that α is a simple arc in (S, M ) as above. Let X α be the indecomposable rigid object corresponding to α. Iyama-Yoshino [IY08] have associated (in a more general context) a subquotient category (C (S,M) ) Xα to X α which we refer to as the Iyama-Yoshino reduction of C (S,M) at X α . The Iyama-Yoshino reduction is again triangulated. We show that (C (S,M) ) Xα is equivalent to C (S,M)/α where (S, M )/α denotes the new marked surface obtained from (S, M ) by cutting along α.
By studying the combinatorics, we are able to give an explicit description of the effect of mutation on coloured quivers associated to a disk with n marked points. The corresponding cluster category in this case was introduced independently in [CCS06] (in geometric terms) and in [BMR + 06] as the cluster category associated to a Dynkin quiver of type A n−3 . We also give a partial explicit description of coloured quiver mutation in the general (2-Calabi-Yau) case, together with a categorical proof. In general, there are quite interesting phenomena: we give an example to show that infinitely many colours can occur in one quiver, and also show that zero-coloured 2-cycles can occur (in contrast to the situation in [BT09]).
We remark that in the case of a cluster tilting object T in an acyclic cluster category the categorical mutation we define coincides with that considered in [BMR + 06]; also with that in the 2-Calabi-Yau case considered in [BIRSc09,Pal09]. It also coincides in the maximal rigid case considered in [GLS06,BIRSc09,IY08]. In this case, the coloured quiver we consider here encodes the same information as the matrix associated to T in [BMV10] provided there are no zero-coloured two-cycles. With this restriction, the mutation of this matrix coincides [BMV10, 1.1] with the mutation [FZ02] arising in the theory of cluster algebras. We note also that this fact for the cluster tilting case was shown in [BIRSc09] under the assumption that there are no two-cycles or loops in the quiver of the endomorphism algebra; the cluster category case was considered in [BMR08] and the stable module category over a preprojective algebra was considered in [GLS06]. See also [BIRSm] and [KY11], where mutation of quivers with potential [DWZ08] has been studied in a categorical context. There has been a lot of work on this subject: see the survey [Kel10] for more details.
The geometric mutation of partial triangulations mentioned above specialises to the usual flip of an arc in the triangulation case (see [FST08,Defn. 3.5]). Coloured quivers similar to those considered here have been associated to m-cluster tilting objects in an (m + 1)-Calabi-Yau category in [BT09] (in this case, the number of colours is fixed at m + 1). The geometric mutation we define here should also be compared with the geometric mutation for m-allowable arcs in a disk [BT09,Sect. 11]; see also the geometric model of the m-cluster category of type A in [BM08].
We also note that the 2- [BM] to the general rigid object case, using Gabriel-Zisman localisation. This result suggests that the mutation of general rigid objects should be considered.
We note that some of our definitions and results could be generalised to the punctured case, except for that fact that we rely on results in [BZ] which apply only to the unpunctured case and are not yet known in full generality (see the recent [CIL-F]). Hence we restrict here to the unpunctured case.
The paper is organised as follows. In Section 1 we set up notation and recall the results we need. In Section 2 we define the mutation and the coloured quiver of a rigid object in a triangulated category. In Section 3 we define mutation and the coloured quiver of a partial triangulation in a marked surface. In Section 4 we show that cutting along an arc corresponds categorically to Iyama-Yoshino reduction. In particular, the coloured quiver after cutting along an arc in a partial triangulation can be obtained from the coloured quiver of the partial triangulation by deleting a vertex. In Section 5 we show that, for a partial triangulation of a surface and the corresponding rigid object in the cluster category of the surface, the two notions of coloured quiver coincide. In Section 6 we show that mutation in the type A case can be described purely in terms of the coloured quiver and give an explicit description. We also give the example mentioned above in which the associated coloured quiver contains infinitely many colours. Finally, in Section 7, we give a partial explicit description and categorical interpretation of coloured quiver mutation. This last result holds in any Hom-finite, Krull-Schmidt, 2-Calabi-Yau triangulated category.
1. Preliminaries 1.1. Riemann surfaces. In this section, we recall some definitions and results from [FST08] and [LF09].
We consider a pair (S, M ) consisting of an oriented Riemann surface with boundary S and a finite set M of marked points on the boundary of S, with at least one marked point on each boundary component. We refer to such a pair as a marked surface. We fix, once and for all, an orientation of S, inducing the clockwise orientation on each boundary component.
Note that:
• We do not assume the surface to be connected.
• We only consider unpunctured marked surfaces. We will always assume that (S, M ) does not have any component homeomorphic to a monogon, a digon or a triangle.
Up to homeomorphism, each component of (S, M ) is determined by the following data:
• the genus g,
• the number of boundary components b and • the number of marked points on each boundary component {n 1 , . . . , n b }. An arc γ in (S, M ) is (the isotopy class relative to endpoints of) a curve in S whose endpoints belong to M , which does not intersect itself (except possibly at endpoints) and which is not contractible to a point. The marked points on a boundary component divide it into segments, and we say that an arc isotopic to an arc along one of these segments is a boundary arc. The term arc will usually refer to a non-boundary arc.
The set of all (non-boundary) arcs in (S, M ) is denoted by A 0 (S, M ). Two arcs are said to be non-crossing if their isotopy classes contain representatives which do not cross, i.e. their crossing number is zero. If R is a collection of non-crossing arcs in (S, M ), we will denote by A 0 R (S, M ) the set of arcs in (S, M ) which do not cross any arc in R and which do not belong to R.
A partial triangulation of (S, M ) is a collection of non-crossing arcs. A maximal collection of non-crossing arcs is called a triangulation. The number n of arcs in any triangulation of a connected marked surface is given by the formula:
n = 6g + 3b + c − 6,
where c is the number of marked points in M (see e.g. [FST08, Prop. 2.10]).
Let T be a triangulation. By [FST08,Sect. 4] and [LF09, Sect. 3] a quiver Q = Q T , together with a potential (a linear combination of cycles in Q T up to cyclic permutation) W T can be associated to T as follows. The vertices of Q are α β γ Figure 1. The quiver with potential associated to a triangulated surface. The potential W = αβγ + · · · . the arcs of the triangulation. There is an arrow from γ to γ ′ for each triangle in which γ ′ follows γ with respect to the orientation of S, and the potential W T is the sum of all the 3-cycles; see Figure 1, where part of a triangulated surface is shown.
For an arrow a ∈ Q 1 , the cyclic derivative ∂ a sends a cycle a 1 · · · a d to the sum d k=1 δ a k a a k+1 · · · a d a 1 · · · a k−1 . It is extended to potentials by linearity. The Jacobian algebra of the quiver with potential (Q T , W T ) is the quotient of the complete path algebra kQ T by the closure of the ideal generated by the cyclic derivatives ∂ a W T , for all a ∈ Q 1 . We note that, by [CIL-F, Thm. 5.5], the Jacobian algebra can be taken to be the quotient of the path algebra kQ T by the ideal generated by the corresponding cyclic derivatives in kQ T . 1.2. Cluster categories associated with Riemann surfaces. Let k be a field. If C is a triangulated category, we will usually denote its shift functor by Σ. All the triangulated k-categories under consideration in this paper are assumed to be Krull-Schmidt, Hom-finite (all morphism spaces are finite-dimensional k-vector spaces) and admit non-zero rigid objects (objects R such that C(R, ΣR) = 0). All rigid objects will be assumed basic (their summands are pairwise non-isomorphic). We will assume moreover that the triangulated categories are 2-Calabi-Yau, so that there are bifunctorial isomorphisms C(X, ΣY ) ≃ DC(Y, ΣX) for all objects X, Y , where D is the vector space duality D = Hom k (−, k). A rigid object T is called a cluster tilting object if, in addition, for all objects X in C, C(X, ΣT ) = 0 = C(T, ΣX) implies that X belongs to add T .
The main examples of such categories that we consider are the (generalised) cluster categories associated with marked surfaces, the definition of which is recalled in the following sections.
1.2.1. Ginzburg dg algebras. Let (Q, W ) be a quiver with potential (i.e. a QP). In this paper, we are mostly interested in QPs arising from triangulations of marked surfaces.
The Ginzburg dg-algebra Γ(Q, W ) is defined as follows: First define a graded quiver Q. The vertices of Q are the vertices of Q, and the arrows are given as follows:
• the arrows of Q, of degree 0;
• for each arrow α in Q from i to j, an arrow α * from j to i, of degree −1; • for each vertex i in Q, a loop e * i at i, of degree −2. The underlying graded algebra of Γ(Q, W ) is the path algebra of the graded quiver Q. It is equipped with the unique differential d sending
• the arrows of degree 0 and each e i to 0;
• the arrow α * to ∂ α W , for each α ∈ Q 1 , and • the loop e * i to e i ( α [α * , α]) e i , for i ∈ Q 0 . The cohomology of Γ(Q, W ) in degree zero is the Jacobian algebra J(Q, W ).
1.2.2.
Generalised cluster categories. The cluster categories associated with acyclic quivers were introduced in [CCS06] in the A n case and in [BMR + 06] in the acyclic case. Amiot defined, in [Ami09], the generalised cluster categories, associated with quivers with potentials whose Jacobian algebra is finite dimensional.
Let (Q, W ) be a quiver with potential such that the Jacobian algebra J(Q, W ) is finite dimensional, and let Γ = Γ(Q, W ) be the associated Ginzburg dg algebra.
Let DΓ be the derived category of Γ, and let D b Γ be the bounded derived category. The perfect derived category per Γ is the smallest triangulated subcategory of DΓ containing Γ and stable under taking direct summands.
Theorem This definition is motivated by the following:
Theorem [Ami09, Sect. 3]:
The cluster category C (Q,W ) is Hom-finite and 2-Calabi-Yau. Moreover, the image of Γ in C (Q,W ) is a cluster tilting object whose endomorphism algebra is isomorphic to the Jacobian algebra J(Q, W ). If Q is acyclic, then W = 0 and the triangulated categories C (Q,0) and C Q are equivalent.
We also recall the 2-Calabi-Yau tilting theorem which applies in this context:
Theorem [KR07, Prop. 2.1] Let C be a triangulated Hom-finite Krull-Schmidt 2-Calabi-Yau category over a field k. If T is a cluster tilting object in C, then the functor C(T, Σ −) induces an equivalence between the category C/T and the category of finite dimensional End C (T )-modules.
Note that the assumption in the paper that k be algebraically closed is not required for this result. We also note that this result has been generalised (see [IY08,Prop. 6.2], [KZ08, Sect. 5.1]).
1.2.3. Cluster categories from surfaces. Let (S, M ) be a marked surface, and let T be a triangulation of (S, M ). Let (Q, W ) be the quiver with potential associated with T . The following particular case of a theorem of Keller-Yang shows that the cluster category C (Q,W ) does not depend on the choice of a triangulation. Let T ′ be a triangulation of (S, M ) obtained from T by a flip. Denote by (Q ′ , W ′ ) the associated quiver with potential.
Theorem [KY11]: There is a triangle equivalence C (Q ′ ,W ′ ) ≃ C (Q,W ) .
Since any two triangulations of (S, M ) are related by a sequence of flips, the theorem above shows that the cluster category C (Q,W ) is independent of the choice of the triangulation T . The resulting category is denoted C (S,M) and is called the cluster category associated with the marked surface (S, M ). (We refer also to [BIRSm, Theorem 5.1]).
These categories have been studied by Brüstle-Zhang in [BZ]. We now recall those of their main results which will be used in the article.
Fix a triangulation T = {γ 1 , . . . , γ m } of (S, M ) with associated quiver with po-
tential (Q, W ). Let T = T 1 ⊕· · ·⊕T m be the image of Γ(Q, W ) under per Γ(Q, W ) → C (Q,W ) ≃ C (S,M) . Note that T is a cluster tilting object. With each arc γ not in T is associated [ABCJP10, Proposition 4.2] an indecom- posable J(Q, W )-module I(γ). Let X γ be the unique (up to isomorphism) inde- composable object in C (S,M) such that C (S,M) (T, ΣX γ ) ≃ I(γ). Define X γ k = T k , for k = 1, . . . , m.
Theorem [BZ]:
• The map γ → X γ is a bijection between the arcs of (S, M ) and the (isomorphism classes of ) exceptional (i.e. indecomposable rigid) objects of C (S,M) . • For any two exceptional objects X α and X β , we have Ext 1 C (S,M ) (X α , X β ) = 0 if and only if the arcs α and β do not cross.
• The shift functor of C (S,M) acts on the arcs of (S, M ) by moving both endpoints clockwise along the boundary to the next marked points.
Note that a bijection with these properties is not unique in general. We note that our choice of an orientation of the Riemann surface differs from that of [BZ], but coincides with that of [BT09, Section 11].
We extend the bijection in the first part of the previous Theorem to a bijection between partial triangulations of (S, M ) and rigid objects in C (S,M) in the obvious way.
1.3. Iyama-Yoshino reduction. For an object X in a triangulated category C, we write ⊥ X for the full subcategory of C whose objects are those objects Y of C such that Hom C (Y, X) = 0. The subcategory X ⊥ is similarly defined. For an additive subcategory D of C, we write C/[D] for the quotient category whose objects are the same as the objects of C with morphisms given by the morphisms of C modulo those morphisms factoring through D. If D is the additive closure of an object X in C then we just write C/X for C/[D].
Theorem: [IY08, 4.2, 4.7] Let C be a 2-Calabi-Yau triangulated category and R a rigid object in C. Then the subfactor category ⊥ (ΣR)/R of C is again a 2-Calabi-Yau triangulated category.
We refer to the subfactor category ⊥ (ΣR)/R as the Iyama-Yoshino reduction of C at R and denote it C R . We denote its shift by Σ R and the quotient functor
⊥ (ΣR) −→ C R by π R . See also [BIRSc09, II.2.1].
We recall a result of Keller:
Theorem: [Kel09, 7.4] Let Q, W be a quiver with potential whose Jacobian algebra is finite dimensional. Let i be a vertex of Q and let P i be the image of the indecomposable projective module over Γ(Q, W ) corresponding to i, under the quotient functor Π : per(Γ(Q, W )) → C Q,W . Then the Iyama-Yoshino reduction of C (Q,W ) at P i is triangle equivalent to C (Q ′ ,W ′ ) , where Q ′ is Q with vertex i (and all incident arrows) removed, and W ′ is W with all cycles passing through i deleted.
Coloured quivers for rigid objects
2.1. Mutation and coloured quivers of rigid objects. Let k be a field. Let C be a k-linear Hom-finite, Krull-Schmidt, 2-Calabi-Yau triangulated k-category. Let R = R 1 ⊕ · · · ⊕ R m be a basic rigid object in C and let X be an indecomposable rigid object in C Ext-orthogonal to R, i.e. such that C(X, ΣR) = 0 = C(R, ΣX). For k = 1, . . . , m and c ∈ Z, consider triangles
X (c) f c −→ B (c) g c −→ X (c+1) −→ ΣX (c)
where f c is a minimal left add R-approximation and g c is a minimal right add Rapproximation and where X (0) = X. These will be called the exchange triangles for X with respect to R. They can be constructed using induction on c. We will often write κ (c) R X for X (c) , and κ for κ (1) ; κ R X will be referred to as the twist of X with respect to R. Note that κκ (c) = κ (c+1) = κ (c) κ for all c.
These exchange triangles lift the triangles
X (c) −→ 0 −→ Σ R X (c) −→ Σ R X (c) in the Iyama-Yoshino reduction ⊥ (ΣR)/R canonically to C. Therefore, X (c) is
indecomposable, rigid and Ext-orthogonal to add R for all c. This justifies the following definition:
Definition: The mutation of R at R k is the rigid object µ R k R = R/R k ⊕ κ R/R k R k .
We will often write µ k for µ R k and call it the mutation at k.
We note that our use of Iyama-Yoshino to define the mutation above is similar to that of [BØO,Sect. 3] where cluster tilting objects are mutated at a nonindecomposable summand.
In [BT09], the authors associate coloured quivers to d-cluster-tilting objects in (d + 1)-Calabi-Yau categories. Here we use the same definition to associate a coloured quiver to R.
Definition: The coloured quiver Q = Q R associated with the rigid object R is defined as follows: The set of vertices is Q 0 = {1, . . . , m}. The set Q (c) 1 (i, j) of c-coloured arrows from i to j has cardinality given by the multiplicity of
R j in B (c) i , where R (c) i f c i −→ B (c) i g c i −→ R (c+1) i −→ ΣR (c) i
are the exchange triangles as above for R i with respect to R/R i .
When the category C is a (generalised) cluster category, it is often the case that a sequence of exchange triangles is periodic. With each vertex k of Q is thus associated an integer d k (possibly infinite): the periodicity of the sequence of exchange triangles for R k . In order to avoid keeping infinitely many arrows starting at each vertex when not necessary, the colours of the arrows starting at a vertex k are considered as elements in Z/d k . Note that the periodicity depends on the starting vertex.
Remark:
• Analogous definitions would apply to a functorially finite, strictly full rigid subcategory R of C closed under direct sums and direct summands, such that, for each indecomposable R ∈ R, the subcategory R \ R is again functorially finite. • Analogous definitions would also apply to rigid objects in a stably 2-Calabi-Yau Frobenius category.
Mutation of rigid objects and Iyama-Yoshino reductions.
The following lemma shows that the mutation of rigid objects is well-behaved with respect to Iyama-Yoshino reductions. This will turn out to be helpful in simplifying the proof of Theorem 15 in Section 7. Let R be a rigid object in C. Let C R = ⊥ (ΣR)/(R) be the Iyama-Yoshino reduction of C with respect to R, with shift Σ R .
Let T be a rigid object in C, containing R as a direct summand. Assume that T k is a summand of T but not of R, and consider the exchange triangle with respect to T /T k :
( * ) T k f −→ B (0) k g −→ T (1) k ε −→ ΣT k . Here B (0) k belongs to add T , where T = T k ⊕ T . Lemma 1. The induced morphism f is a minimal left π R (T )-approximation in C R . Proof. The triangle ( * ) in C induces a triangle T k f −→ B (0) k g −→ T (1) k −→ Σ R T k , in C R .
We have:
C R (Σ −1 R π R (T (1) k ), π R (T )) ≃ Ext 1 CR (π R (T (1) k ), π R (T )) ≃ Ext 1 C (T (1) k , T ) = 0, using [IY08, Lemma 4.8]. Hence, the morphism f is a left π R (T )-approximation. It is left minimal since T (1) k is indecomposable in C R . √ Remark: Write B (0) k = R (0) k ⊕ C (0) k , with C (0) k having no summands in common with R. Then the morphism T k f ′ −→ C (0) k is not a left T /R-approximation in C in general.
Let Q be the coloured quiver of T in C, and let Q be the coloured quiver of π R (T ) in C R . Lemma 1 has the following immediate corollary:
Corollary 2. The coloured quiver Q is the full subquiver of Q with vertices corresponding to the indecomposable summands of T /R.
Moreover, computing the minimal T -approximation f ∈ C in the triangle ( * ) amounts to computing the minimal add T j -approximation of T k in the Iyama-Yoshino reduction C T /Tj for all j = k. More precisely:
Lemma 3. Let R = R 1 ⊕ · · · ⊕ R m be a rigid object in C and let 1 ≤ k ≤ m. For each j = 1, . . . , m, let C j denote the Iyama-Yoshino reduction of C with respect to R/(R k ⊕R j ). For j = k, let f j : R k −→ R nj j be a map in C be such that R k fj −→ R nj j is a minimal left add R j -approximation in C j . Then the morphism: R k [fj ] −→ j =k R nj j is a minimal left add R/R k -approximation in C. Proof. Let i = k, and let R k f −→ R i be an arbitrary morphism in C. Since f i is an add R i -approximation in C i , there are morphisms j =k R nj j g1 −→ R i , R k β1 −→ j =i,k R a (1) j j and j =i,k R a (1) j j α1 −→ R i in C (for some a(1)
j ) such that f = g 1 [f j ] + α 1 β 1 . Note that α 1 must be a radical map, as no summand of its domain is isomorphic to R i . Reducing to C j for some j = i, k, we see that the component
⊕ j =k R a (r) j j αr G G · · · G G ⊕ j =k R a (2) j j α2 G G ⊕ j =i,k R a (1) j j α1 G G R i · · · R k [fj ] G G βr h h | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | β2 ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ β1 y y f d d ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ⊕ j =k R nj j g1,β 1,j of β 1 mapping to R a (1) j j factors through f j : R k → R nj j up to a map factoring through add ⊕ l =j,k R l . That is, we can write β 1,j as u j f j + w j v j for some u j : R nj j → R a (1) j j , v j : R k → X j , and w j : X j → R k , where X j ∈ add ⊕ l =j,k R l .
Note that w j is a radical map, since none of the summands in X j are isomorphic to R j .
Adding over all j for j = i, k, we obtain maps α 2 :
⊕ l =k R a (2) l l → ⊕ j =i,k R a (1) j j , β 2 : R k → ⊕ l =k R a (2) l l and γ 2 : ⊕ j =k R nj j → ⊕ j =i,k R a (1) j j (for some a (2) l ) such that β 1 = α 2 β 2 + γ 2 [f j ]. Setting g 2 = α 1 γ 2 we obtain α 1 β 1 = α 1 α 2 β 2 + g 2 [f j ], so f = g 1 [f j ] + α 1 β 1 = α 1 α 2 β 2 + (g 1 + g 2 )[f j ].
Here, α 2 is a radical map, since all of its summands, the w j , are radical. See Figure 2.
Iterating this step we construct, for all r ≥ 3, morphisms g r : l ), such that f = β n α n · · · α 1 + (g 1 + · · · + g n )[f j ] and each of the α i is a radical map. Since C is Hom-finite, the radical of End(R) is nilpotent and the composition β n α n · · · α 1 vanishes for n big enough. Therefore f factors through [f j ] and [f j ] is a left add R/R k -approximation in C. The left minimality of [f j ] follows from the left minimality of each f j . √
j =k R nj j −→ R i , α r : j =k R a (r) j j −→ j =k R a (r−1) j j , and β r : R k −→ j =k R a(
Coloured quivers for partial triangulations
Let (S, M ) be an unpunctured oriented Riemann surface with boundary and marked points. We will always assume that each boundary component contains at least one marked point and that no component of (S, M ) is a monogon, a digon or a triangle.
3.1. Composition of arcs. Let α and β be two oriented arcs in (S, M ) with β(1) = α(0). The composition αβ is the arc given by
t −→ β(2t) if 0 ≤ t ≤ 1/2 α(2t − 1) if 1/2 ≤ t ≤ 1.
See Figure 3.
Note that the composition only makes sense for oriented arcs.
3.2.
Twisting an arc with respect to a partial triangulation. In this section, our aim is to generalise the flip of triangulations to the twist of an arc with respect to a partial triangulation. Let R be a partial triangulation of (S, M ), i.e. a collection of non-crossing arcs γ 1 , . . . , γ m . Let α be an arc in (S, M ) which does not cross R and does not belong to R, i.e. α ∈ A 0 R (S, M ). We define the twist of α with respect to R as follows: Choose an orientation α of α. Consider the arcs of the partial triangulation R which admit α(0) as an endpoint. Restrict to a neighbourhood of α(0) small enough not to contain any loop. The orientation of the boundary containing α(0) induces an ordering on the parts of the arcs included in the neighbourhood (see Figure 4). Let α s be the arc, in R or boundary, which follows α in this ordering (note that it is not allowed to be α itself). Similarly, define α t with respect to the endpoint α(1). These will be called the arcs following α in R.
We give α s and α t the orientations α s and α t described in the local pictures of Figure 4. Note that this orientation coincides with the orientation of the boundary if α s or α t is a boundary arc.
For an oriented arc β, let [β] denote the underlying unoriented arc. Define the twist of the arc α with respect to R to be the underlying unoriented arc of the composition:
κ R (α) = [α t α α s −1 ].
See Figure 5 for an example of a twist. Note that the definition of the arc κ R (α) does not depend on the choice of an orientation for α. It is easy to check, using a case-by-case analysis depending on whether or not α, α s , α t are loops and the order in which they appear at their end-points, that κ R (α) does not cross any arc in R, i.e. that κ R (α) ∈ A 0 R (S, M ). The twist with respect to R is invertible with inverse κ −1 R , which can also be defined similarly.
µ γ R = R ⊔ κ R (γ)
. If R is a triangulation, then, for γ ∈ R, µ γ R is the usual flip of R at the arc γ ∈ R; see [FST08, Defn. 3.5].
3.4. Coloured quivers. Let α and β be two arcs in (S, M ), and let R be a partial triangulation not containing α. For all c ∈ Z, define the numbers q (c)
R (α, β) by: q R (α, β) = 2 if β = α s = α t 1 if β ∈ {α s , α t } and α s = α t 0 otherwise, q (c) R (τ, τ ′ ) = q R (κ c R (τ ), τ ′ )
. With a partial triangulation R = {γ 1 , . . . , γ m }, we associate a coloured quiver Q R :
Definition The coloured quiver Q R associated with the partial triangulation R is defined as follows: The set of vertices is Q 0 = {1, . . . , m}. The set Q (c) 1 (i, j) of c-coloured arrows from vertex i to vertex j has cardinality q (c)
Ri (γ i , γ j ), where R i = R \ {γ i }.
Here also, the colours can be thought of as elements of Z/d k where d k is the periodicity of twisting the arc corresponding to the starting vertex.
For an example of a coloured quiver associated to a partial triangulation of a torus and the effect of mutation on the quiver, see Section 6.2.
Cutting along an arc and CY reduction
Let (S, M ) be as in Section 3.
4.1.
Cutting along an arc. Let α be an arc on (S, M ) not homotopic to a point or a boundary arc. Fix a representative of α, also denoted by α, whose intersection with the boundary of S consists only of its endpoints. Then the marked surface obtained from (S, M ) by cutting along the arc α is the Riemann surface with boundary obtained by cutting along the arc α together with the image of the marked points M after cutting. Up to homeomorphism, it does not depend on the choice of representative of α. We will denote it by (S, M )/α. Note that if α is not a loop, then each endpoint of α gives rise to two distinct marked points in (S, M )/α. If α is a loop, its endpoint gives rise to three distinct marked points in (S, M )/α. The resulting marked surface cannot contain a monogon as a connected component, since α is not homotopic to a point. No connected component can be a bigon, since α is not a boundary arc. If a component homeomorphic to a triangle has been created, we remove it.
There is a natural bijection between the arcs on (S ′ , M ′ ) and the arcs of (S, M ) which do not cross the arc α. Moreover, the (partial) triangulations of (S ′ , M ′ ) correspond, through this bijection, to the (partial) triangulations of (S, M ) containing the arc α. Given a collection R of non-crossing arcs, one can cut successively along each arc. Whatever order is chosen yields the same new surface, by Remark 4. The corresponding surface will be called the reduction of (S, M ) with respect to R, and will be denoted by (S, M )/R. We will denote the natural bijection between √ Remark: Lemma 7 shows that the equivalence above is well-behaved with respect to well-chosen bijections between arcs and exceptional objects. Figure 6 shows the effect of cutting along an arc in a triangulation of a torus with a single boundary component containing two marked points. We cut along the red arc (numbered 3) and obtain a cylinder with four marked points as shown, with triangulation given by the remaining arcs. In the last step, the cylinder has been rotated around to get a simpler picture. The effect on the corresponding quiver with potential is shown in Figure 7.
Proposition 6. Let (S, M ) be a marked surface and R a partial triangulation of (S, M ). Let R ′ be a collection of arcs containing R. Then the coloured quiver associated to π R (R ′ \ R) in (S, M )/R coincides with the coloured quiver associated to R ′ in (S, M ) with the vertices corresponding to R and all arrows incident with them removed.
Proof. It is clear that the vertices of each coloured quiver correspond to the arcs in R ′ \ R. In the definition of the twist κ R (see Section 3.2), no distinction is made between arcs in R and boundary arcs. Then, looking at the definition of the Figure 7. The change in the quiver with potential from the cut in Figure 6. The potential in each case is given by the sum of the 3-cycles containing black dots. coloured quiver of a partial triangulation (see Section 3.4) we see that the arrows between arcs in R ′ \ R are the same when considered in either coloured quiver. The result follows. √ We now give an example. In Figure 8, we start with a partial triangulation of a torus with a single boundary component with two marked points. This has been obtained by removing arcs 4 and 5 from the triangulation considered in Figure 6. As before, we cut along the red arc (numbered 3) and obtain a cylinder with four marked points as shown, with a partial triangulation given by the remaining arcs. Figure 9 gives the corresponding coloured quiver associated to the partial triangulation in Figure 8, together with the new quiver obtained after cutting along the red arc (numbered 3), i.e. with vertex 3 and all arrows incident with it removed.
× × ✎ ✎ ✎ ✎ ✎ ✎ ✎ ✎ 2 o o # # ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ 4 2 # # ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ o o • • → • 3 G G 1 ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ q q ✎ ✎ ✎ ✎ ✎ ✎ ✎ ✎ 5 o o 1 ✴ ✴ ✴ ✴ ✴ ✴ ✴ ✴ q q ✎ ✎ ✎ ✎ ✎ ✎ ✎ ✎ 5 o o
Compatibility
5.1. Compatibility of the mutations. Let R be a partial triangulation of (S, M ). Complete R to a triangulation T of (S, M ), and let T be the associated cluster tilting object in C = C (S,M) . Let R be the direct summand of T corresponding to
R.
We thus obtain a map α → X α between the arcs of (S, M ) and the isomorphism classes of exceptional objects in C (S,M) (see section 1.2.3). We denote by π R the bijection A 0 R (S, M ) → A 0 (S, M )/R ; recall also that π R denotes the functor ⊥ (ΣR) → C R . Consider the partial triangulation π R (T \ R) of (S, M )/R. Note that T ′ := π R (T ) is cluster tilting in C (S,M)/R ≃ C R by [IY08,Theorem 4.9]. This cluster tilting object induces a bijection β → Y β between the arcs in (S, M )/R and the exceptional objects in C R .
Lemma 7. Let α be an arc in
A 0 R (S, M ). Then the image of X α under π R is isomorphic to Y π R α .
Proof. Using [JP, Proposition 3.5], the modules associated with π R X α and Y π R α are seen to be isomorphic. √
Let α be an arc in (S, M ) which is not in R and which does not cross R, i.e. α ∈ A 0 R (S, M ). Fix an orientation α of α and let α s and α t be the two (possibly boundary) arcs following α in R (as defined in section 3.2). Recall that κ R (α) is defined to be [α t αα s −1 ]. If γ is any arc in (S, M ) then recall we have (from [BZ]; see Section 1.2.3):
(1)
ΣX γ = X κ φ (γ) ,
where φ denotes the empty set of arcs in (S, M ). Thus κ φ (α) is obtained from the arc α by composition with the two boundary arcs which follow α (see Section 3.2).
The following corollary describes the twist of an arc in terms of the action of the shift functor of an Iyama-Yoshino reduction. In other words, we have a commutative diagram: M ). By (1), noting that R becomes part of the boundary of (S, M )/R, we have Σ R Y π R (α) ≃ Y π R κ R (α) . The result follows. √
A 0 (S, M )/R shift G G A 0 (S, M )/R A 0 R (S, M ) π R y y κ R G G A 0 R (S, M ). π R y y Proof. Let α ∈ A 0 R (S,
We now have the ingredients we need in order to show that the two mutations (of partial triangulations and rigid objects) are compatible.
Proposition 9. Let R be the rigid object in C (S,M) associated with the partial triangulation R as above. Let α be an arc in A 0 R (S, M ). Then we have:
κ R X α ≃ X κ R (α) and µ k R ≃ X µ k R .
Proof. Since α does not cross R, it follows from [BZ] (see Section 1.2.3) that
X α ∈ ⊥ (ΣR). Similarly, X κ R (α) ∈ ⊥ (ΣR), since κ R (α) does not cross R. By the description of the shift Σ R of C R in [IY08, 4.1], π R (κ R (X α )) ≃ Σ R (π R X α ) in C R . By Lemma 7, Σ R (π R X α ) ≃ Σ R Y π R (α) . By Corollary 8, we have Σ R Y π R (α) ≃ Y π R κ R (α) . By Lemma 7, Y π R κ R (α) ≃ π R X κ R (α). Hence π R (κ R X α ) ≃ π R (X κ R (α) ).
Note that κ R X α is an indecomposable object in ⊥ (ΣR) which is not in add R (see Section 2.1). Since κ R α does not cross R and does not lie in R, the same is true of X κ R (α) . It follows that κ R X α ≃ X κ R (α) , proving the first part of the Proposition. The second part follows. √
5.2.
Compatibility of the coloured quivers. As in the previous section, let α ∈ A 0 R (S, M ); we fix an orientation of α and let α s and α t be the two (possibly boundary) arcs following α in R (as defined in section 3.2). Note that it is possible that α s = α t . We choose a triangulation T of (S, M ) containing R and α and let T be the corresponding cluster tilting object, containing R and as a direct summand and X α as an indecomposable direct summand. Recall that X γ = 0 if γ is a boundary arc.
Lemma 10. There is a minimal left add R-approximation of X α in C (S,M) of the form X α −→ X αs ⊕ X αt .
Proof. By the 2-Calabi-Yau tilting theorem (see Section 1.2.2), the functor H = C(T, Σ −) induces an equivalence between C/T and mod J(Q, W ). Hence H induces an equivalence between Σ −1 add T and the category P of projective modules over J(Q, W ). Let P α = H(Σ −1 X α ) for each arc α in T and let P R = H(Σ −1 add R).
Then it is enough to show that there is a minimal left P R -approximation of P α in mod J(Q, W ) of the form P α −→ P αs ⊕ P αt . We recall that J(Q, W ) is gentle (see Section 1.1). In particular, the defining relations are all zero-relations. Let γ 1 , γ 2 , . . . , γ j be the arcs in T incident with α(0) which are after α in the order induced by the orientation of the boundary at α(0) (and listed in that order); see Section 3.2. Similarly, let δ 1 , δ 2 , . . . , δ k be the arcs in T around α(1) which are after α in the order induced by the orientation of the boundary at α(1).
Because of the zero-relations in J(Q, W ), the only non-zero paths in Q starting at α are paths:
α −→ γ 1 −→ γ 2 −→ · · · −→ γ j and α −→ δ 1 −→ δ 2 −→ · · · −→ δ k . Thus the only non-zero morphisms from P α to some indecomposable projective module lie in the composition chains: P α −→ P γ1 −→ P γ2 −→ · · · −→ P γj and P α −→ P δ1 −→ P δ2 −→ · · · −→ P δ k , or are linear combinations of these (noting that the chains may overlap).
If α t is a boundary arc, but α s is not, then α s occurs in the first chain above. It is easy to see that the non-zero map P α −→ P αs coming from the chain of compositions is a left minimal P R -approximation and we are done. The argument is similar if α s is a boundary arc but α t is not. If both α s and α t are boundary arcs then the zero map is a left minimal P R -approximation.
We are left with the case where neither α s nor α t is a boundary arc. Thus α s = γ i for some i while α t = δ i ′ for some i ′ . Let f s and f t be the non-zero morphisms arising from the above chains of compositions and let f : P α −→ P αs ⊕ P αt be the map with components f s , f t . It follows from the above that f is a left P Rapproximation of P α . It remains to check that f is left minimal.
We note that if we had f s = kh for some h : P α −→ P β and k : P β −→ P αs for some β ∈ R then k would have to be an isomorphism since the path in Q from α to α s is not equal to any other path in Q from α to α s , and α s is the first arc in R appearing along this path. A similar statement holds for f t .
If f were not left minimal, a summand of form 0 −→ P αs (respectively, 0 −→ P αt ) would split off and we would have a left P R -approximation of the form g s : P α −→ P αs (respectively, g t : P α −→ P αt ). We consider only the first case (the second case requires a similar argument). In this case, f t factors through g s , i.e. f t = vg s for some map v : P αs → P αt . By the above, v is an isomorphism and g s = v −1 f t .
Again, since g s is a left P R -approximation, we also have that f s factors through g s , i.e. f s = wg s for some w : P αs −→ P αt . By the above, w is an isomorphism.
Hence we have f s = wv −1 f t where wv −1 is an isomorphism. This is a contradiction since f s and f t arise from two different paths starting at α. The result is proved. √ Theorem 11. Let R be the rigid object in C (S,M) associated with the partial triangulation R. Then the coloured quivers Q R and Q R coincide.
Proof. By Proposition 9, it is enough to prove that the sets of 0-coloured arrows coincide. This follows from Lemma 10. √
Some examples
6.1. The A n case. In this section, we assume that the category C is the cluster category of type A n . Suppose that R is a basic rigid object in C. In Section 2.1 we have associated a coloured quiver Q with R. If R k is an indecomposable direct summand of R then the rigid object µ k R also has a coloured quiver, Q, associated with it, and we can ask if Q can be computed from Q. This is known in the d-cluster-tilting object case of a d+1-Calabi-Yau category [BT09, Thm. 2.1] but is not known for a general rigid object. In Section 7 we will indicate some results in this direction with a categorical proof, but here we give a complete answer for the cluster category of type A using a combinatorial (geometric) proof. In this case, the corresponding surface is a disk with n + 3 marked points (see [CCS06]), which we shall denote (S, M ); as usual, we denote by R the set of noncrossing arcs in (S, M ) corresponding to the indecomposable direct summands of R. α β Figure 10. The arc α has order 3 under twisting and lies in a hexagon, while the arc β has order 5 and lies in a pentagon.
The following is easy to check (d i was defined in section 2.1):
Lemma 12. Let i ∈ I. Then (a) d i = max{c ≥ 0 : q (c) ij = 0} + min{c ≥ 0 : q (c) ji = 0} + 1. (b) For all c ∈ Z, q (c) ii = 0.
Let R i be a summand of R and let α i denote the corresponding arc in (S, M ). If the arc corresponding to R i in (S, M )/(R \ α i ) is a diameter of a 2d-sided polygon for some d then twisting α i with respect to R \ R i has order d, while in all other cases the order of this twisting coincides with the number of sides of the polygon. See Figure 10 for an example.
It is easy to see that this situation occurs if and only if for any j such that i and j are ends of a common arrow in Q, there is a unique colour c such that q (c) ij = 0, and thus this can be detected in Q. This irregularity makes it difficult to compute the new coloured quiver after mutation of R at an arc. But, since it is detectable in Q, we can make a first pass and add second arrows between i and j to Q in such cases. We do this when computing the coloured quiver Q of R by continuing to mutate R at α i a further d times and adding the new arrows arising to Q; see step (ii) below. At the end, this procedure must be reversed.
Thus, given a coloured quiver Q of a rigid object and vertex k as above, we consider the following algorithm to define a new quiver Q. The letters i, j always refer to distinct vertices. We note that after this step, for any two vertices i, j, there will either be no arrows between i and j or exactly two arrows in each direction. In the latter case, we indicate the two colours in a given direction by l ≤ l ′ for some letter l and we will write the colours as a pair labelling a single arrow.
(iii) Suppose we have the following arrows:
i (a,a ′ ) G G k (0,b ′ ) o o j (c,c ′ ) G G k (d,d ′ ) o o
where d = 0. Add the following arrows:
i (d,d+a ′ −a) G G j (c,c ′ ) o o
and cancel any pairs of arrows between i and j in the same direction whose colours differ by 1. (iv) Suppose we have the following full subquiver of Q where j = k:
k (0,b ′ ) G G i (a,a ′ ) o o (d,d ′ ) G G j (c,c ′ ) o o .
Then change the arrows between i and j to:
i (d,d+b ′ ) G G j (c,c ′ ) o o .
(v) Apply the following rule to all vertices i with an arrow to or from k:
i (a,a ′ ) G G k (b,b ′ ) o o → i (a+1,a ′ +1) G G k (b−1,b ′ −1) o o if b = 0; i (0,a ′ −a) G G k (b ′ −1,a+b ′ ) o o if b = 0. If b = 0 then d i is replaced with d i + b ′ − a = b ′ + a ′ − a.
(vi) Whenever we have arrows:
i (c,c+ 1 2 di) G G j (d,d ′ ) o o ,
remove the arrow with colour c + 1 2 d i and replace d i with 1 2 d i . Proposition 13. Let R = i∈I R i be a rigid object in the cluster category of type A n , with associated quiver Q. Let R k be a summand of R and let Q denote the quiver of µ k (R). Then Q can be computed using the above algorithm.
Proof. Let Γ be the quiver obtained from Q after Step (ii) of the algorithm has been applied and let Γ be the quiver so obtained from Q. Since Step (vi) takes Γ to Q it is enough to show that Steps (iii) to (v) take Γ to Γ.
We consider each possible configuration of the arcs corresponding to the summands of R in the disc and check that in each case, the above rule gives the correct answer. It is easy to check that the algorithm works in the case where there are no arrows of colour zero starting at k. The other possible configurations are given in Figures 11,12,13 and 14, together with the corresponding quivers Γ and Γ. In each case, the label on part of the boundary indicates the number of boundary segments between the two nearest arc ends on the boundary. (The black dot indicates the end of arc k to indicate that this has changed after the mutation). Case I can be regarded as an instance of Step (iii) with a = r, a ′ = r + s, b = 0, b ′ = q + t + 1, c = t, c ′ = p + t, d = q and d ′ = q + r + 1, followed by Step (v). Case II can be regarded as an instance of Step (iii) with a = q + t + 1, a ′ = q + r + t + 1, b = 0, b ′ = s, c = t, c ′ = p + t, d = q + 1 and d ′ = q + s + 1, followed by Step (v). Case III can be regarded as instance of Step (iv) with a = s, a ′ = q + s + t + 1, b = 0, b ′ = r, c = t, c ′ = p + t, d = q and d ′ = q + s + 1, followed by Step (v). In Case IV, only Step (v) is applied. The result follows. Figure 11. Case I. Here we have p ≥ 2, q ≥ 1, r ≥ 1, s ≥ 2, t ≥ 0. Note that d i changes from r + s + 1 to q + t + s + 1. Figure 12. Case II. Here we have p ≥ 2, q ≥ 0, r ≥ 2, s ≥ 2, t ≥ 0. Note that d i changes from q + r + t + 2 to r + s.
6.
2. An example with infinitely many colours. We consider again the example from Figure 8, i.e. a torus with a single boundary component with two marked points. We show again the partial triangulation of this surface in Figure 15. Several copies are drawn to make it easier to see mutations at each of the arcs. The corresponding coloured quiver is given below the surface: note that removing any of the three arcs leaves a hexagon; it follows that mutation at any of the arcs has order 3, and we get finitely many colours: 0, 1 and 2, appearing as labels on the arrows. Now suppose we mutate at arc 1. We obtain the partial triangulation in Figure 16; the corresponding quiver is given below the picture of the surface. Here, an Figure 13. Case III. Here we have p ≥ 2, q ≥ 0, r ≥ 2, s ≥ 0, t ≥ 0, q + t ≥ 1. Note that d i changes from q + s + t + 2 to q + r + t + 1. Figure 14. Case IV. Here we have p ≥ 2, q ≥ 1, r ≥ 2, s ≥ 1. Note that d i1 changes from q + r + 1 to r + s + 1 and d i2 changes from p + s + 1 to p + q + 1. arrow is labelled with Z to represent an infinite number of arrows, one coloured n for each integer n. This infinity of arrows comes from mutating at arc number 2. If we cut along the remaining arcs in the partial triangulation, we obtain a cylinder. Then, after each mutation a small neighbourhood of the triangulation is the same at each end (which explains the regularity), but as more and more mutations are made the arc wraps itself more and more around the cylinder. Thus we see that, even if the quiver is locally finite to start with, after a mutation it might not be. Figure 16. The result of mutating the partial triangulation in Figure 15 at arc 1 and the corresponding coloured quiver.
i 1 i 1 i 1 i 1 i 2 i 2 i 2 i 2 k k
Remark 14. We note that in the example in Figure 16, the coloured quiver contains a two-cycle of arrows both coloured zero, a situation that does not arise in the coloured quivers arising in m-cluster categories [BT09, Sect. 2].
Partial categorical interpretation
In this section, we prove the following result, which gives a partial description of the mutation of a coloured quiver associated to a rigid object in a categorical context. Note that the result applies to any 2-Calabi-Yau triangulated category (under mild assumptions), and is not restricted to the surface case.
Theorem 15. Let C be a Hom-finite, Krull-Schmidt, 2-Calabi-Yau triangulated k-category. Let Q be the coloured quiver associated with a rigid object R ∈ C and let Q be the coloured quiver associated with µ k R, for some vertex k of Q. Denote the periodicity associated with vertex i of Q (resp. Q) by d i (resp. d ′ i ). (i) We have:
• d k = d ′ k and • for any j ∈ Q 0 and any c ∈ Z/d k , q
(c) k,j = q (c+1) k,j . (ii) Let i, j ∈ Q 0 be such that q (0) k,j = 0 = q (0)
k,i . Then we have:
• d ′ i = d i , d ′ j = d j ; • for any c ∈ Z/d j , q (c) j,k = q (c−1) j,k ; • for any c ∈ Z/d i , q (c) i,j = q (c) i,j .
We note that, for the cases covered by the theorem, the new coloured quiver depends only on the old coloured quiver and not on the particular choice of rigid object or category C. 7.1. Proof of Theorem 15. We break the proof of Theorem 15 down into smaller steps, which we present as individual lemmas.
Let R ∈ C be rigid and let Q be the associated coloured quiver.
Lemma 16. We have d ′ k = d k and, for any c ∈ Z/d k and any j ∈ Q 0 , q
(c) k,j = q (c+1) k,j .
Proof. The exchange triangles for R
(1) k can be deduced from those for R k , so that we have (R
(1) k ) (c) = R (c+1) k . √
Lemma 17. Let j ∈ Q 0 be such that q (0) k,j = 0. Then d ′ j = d j and for any c ∈ Z/d j we have: q
(c) j,k = q (c−1) j,k .
Proof. By Corollary 2, we may assume that R = R j ⊕ R k . Since q (0) k,j = 0, the first exchange triangle for R k with respect to R j is:
R k −→ 0 −→ R (1) k = −→ ΣR k . Let . . . , R (−1) j −→ R s−1 k −→ R j −→ ΣR (−1) j , R j −→ R s0 k −→ R (1) j −→ ΣR j , .
. . be the exchange triangles for R j with respect to R k . Since q (0) k,j = 0, we have s −1 = 0 and R j is isomorphic to ΣR (−1) j . The exchange triangles for R j with respect to R (1) k = ΣR k are thus obtained from those with respect to R k by applying the shift functor:
. . .
ΣR (−3) j −→ ΣR s−3 k −→ ΣR (−2) j −→ ΣR (−2) j −→ ΣR s−2 k −→ R j −→ R j −→ 0 −→ ΣR j −→ ΣR j −→ ΣR s0 k −→ ΣR (1) j −→ ΣR (1) j −→ ΣR s1 k −→ ΣR
(2) j −→ . . .
√
Lemma 18. Let i, j ∈ Q 0 be such that q Proof. By Corollary 2, we may assume that R = R i ⊕ R j ⊕ R k . Let . . . , R
(−1) i → R t−1 j → R i → ΣR (−1) i , R i → R s0 k ⊕ R t0 j → R (1) i → ΣR i , . . .
be the exchange triangles in C for R i with respect to R j ⊕ R k . We denote by R (c) * i the twists of R i with respect to µ k R/R i . Our assumptions have the following consequences:
(i) R
(1) k = ΣR k and (ii) the spaces C(R k , R j ) and C(R k , R i ) vanish.
Let C be the Iyama-Yoshino reduction of C with respect to ΣR k = R The result then follows from (a) and (a') by Corollary 2.
Let us first prove that (a) and (b) hold for c = 0. Note that, by (ii), both R i and R j belong to (Σ −1 R
(1) k ) ⊥ , so that (a) makes sense. Let us denote by f ′ f the map R i → R s0 k ⊕ R t0 j . Since C(R k , R j ) = 0, the map R i f −→ R t0 j is a left add R japproximation in C, thus so is f in C. Let g ∈ End C (R t0 j ) be such that g f = f . = f ′ f . By minimality, g is an isomorphism. Thus f is leftminimal. By Lemma 3 and Lemma 17, the first exchange triangle with respect to µ k R for R i is
R i −→ R t0 j −→ R (1) * i −→ ΣR i .
The triangle (b) is easily constructed by applying the octahedral axiom to the composition R i −→ R s0 k ⊕ R t0 j proj −→ R t0 j as follows:
R s0 k R s0 k R i G G R s0 k ⊕ R t0 j G G R (1) i G G ΣR i R i G G R t0 j G G 0 R (1) * i G G ΣR i ΣR s0 k ΣR s0 k .
Assume that (a) and (b) hold for some c, and let us first prove that (a) holds for c + 1. Note that, by construction, R (c+1) * i belongs to R ⊥ k = Σ −1 (R (1) k ) ⊥ and so does R j , by (ii), so (a) makes sense. Write X for X c+1 . Since X belongs to add R k , the space C(X, R j ) vanishes and the morphism R
(c+1) i → R sc+1 k ⊕ R tc+1 j
induces a morphism of triangles:
X G G ✤ ✤ ✤ R (c+1) i G G R (c+1) * i G G m ✤ ✤ ✤ ΣX ✤ ✤ ✤ R sc+1 k G G R sc+1 k ⊕ R tc+1 j G G R tc+1 j 0 G G ΣR sc+1 k .
We claim that m is a minimal left add R j -approximation in C. Let f belong to C(R (c+1) * i , R j ). The following diagram illustrates the proof:
Σ −1 R (c+2) i v R (c+1) i q G G u R (c+1) * i p G G m f % % ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ΣX R sc+1 k ⊕ R tc+1 j π 2 G G a I I ❉ • ❏ ❖ ❚ ( ( ❴ ❛ ❜ R tc+1 j b D D ✸ ❊ ❚ ( R (c+2) i R j
where π 2 denotes the second projection. Since the space C(R (c+2) i , ΣR j ) vanishes, we have f qv = 0 and there exists a morphism a such that f q = au. By (ii), the morphism a factors through π 2 . Let b be such that a = bπ 2 . We then have f q = bπ 2 u = bmq, and the morphism f − bm factors through p. Since the object X belongs to add R k , this implies that f − bm lies in the ideal (ΣR k ). That is m is a left add R j -approximation in C. Let g ∈ End C (R tc+1 j ) be such that g m = m, that is gm − m belongs to the ideal (ΣR k ). This implies that the composition (m − gm)q vanishes since C(R (c+1) i , ΣR k ) = 0. Let h ∈ C(ΣX, R tc+1 j ) be such that gm = m + hp. We have: gmq = mq + hpq = mq. Since mq = π 2 u is left minimal, the morphism g is an isomorphism in C, thus so is g in C. Hence (a) holds for c + 1.
Let us now prove that (b) holds for c + 1. By Lemma 3, Lemma 17 and (a) for c + 1, we have a minimal left add R j ⊕ ΣR k approximation of R (c+1) * i in C of the form [ m r ] for some r : R (c+1) * i −→ ΣR sc k , which we complete to an exchange triangle
R (c+1) * i [ m r ] −→ R tc+1 j ⊕ ΣR sc k −→ R (c+2) * i −→ ΣR (c+1) * i .
Complete the commutative square
R (c+1) i u G G q R tc+1 j ⊕ R sc+1 k [ 1 0 0 0 ] R (c+1) * i [ m r ] G G R tc+1 j ⊕ ΣR sc k to a commutative diagram R (c+1) i u G G q R tc+1 j ⊕ R sc+1 k
Theorem: Let T be a triangulation of a marked surface (S, M ), and let T ′ be the triangulation obtained by flipping T at an arc γ. Then: (a) [LF09, Thm. 36] The Jacobian algebra J(Q T , W T ) is finite dimensional. (b) [ABCJP10, Thm. 2.7] The Jacobian algebra J(Q T , W T ) is gentle and Gorenstein of Gorenstein dimension 1. (c) [FST08, Prop. 4.8] The quiver Q T ′ is given by the Fomin-Zelevinsky mutation of Q T at the vertex corresponding to γ. (d) [LF09, Thm. 30] The quiver with potential (Q T ′ , W T ′ ) is given by the QP mutation (see [DWZ08, Sect. 5]) of (Q T , W T ) at the vertex corresponding to γ.
[Kel09, Sect. 6]: The Ginzburg dg algebra Γ is homologically smooth and 3-Calabi-Yau as a bimodule. In particular, there is an inclusion D b Γ ⊂ per Γ. Definition [Ami09, Sect. 3]: The (generalised) cluster category C (Q,W ) associated with the quiver with potential (Q, W ) is the Verdier localisation per Γ/D b Γ.
Figure 2 .
2Proof of Lemma 3
Figure 3 .
3Composition
Figure 4 .
4Orientation of α s and α t
Figure 5 .
5An example of a twist 3.3. Mutation of partial triangulations. Let R be a partial triangulation of (S, M ), and let γ ∈ R. Write R = R ⊔ {γ}. The mutation of R at γ is the partial triangulation
Remark 4 .
4The surface (S, M )/α can also be constructed as follows. Let T be a triangulation of (S, M ) containing α. The surface (S, M ) is then obtained from the triangles of the triangulation by gluing matching sides of triangles in a prescribed orientation. The surface (S, M )/α is obtained from the same triangles by respecting the same gluings except for the sides which correspond to α, which are not glued together anymore.
A 0 R
0(S, M ) and A 0 ((S, M )/R) by π R . 4.2. Compatibility with CY reduction. Let R be a basic rigid object in C (S,M) , and let R be the associated partial triangulation. We denote by C R = ⊥ (ΣR)/(R) the Calabi-Yau reduction of C (S,M) with respect to R, and by (S, M )/R the marked surface obtained from (S, M ) by cutting along the arcs of R. Proposition 5. The triangulated categories C (S,M)/R and C R are equivalent. Proof. Complete the collection of arcs R to a triangulation T . Let (Q, W ) be the QP associated with T . By definition, there is an equivalence of triangulated categories C (S,M) ≃ C (Q,W ) . By [Kel09, Theorem 7.4] (see section 1.3), the category C R is triangle equivalent to the cluster category C (Q ′ ,W ′ ) , where (Q ′ , W ′ ) is obtained from (Q, W ) by deleting the vertices of Q which correspond to arcs in R, and all adjacent arrows. On the other hand, the arcs in T not in R induce a triangulation of the surface (S, M )/R. It follows from Remark 4 that (Q ′ , W ′ ) is the QP associated with this triangulation. Thus C (S,M)/R is equivalent to C (Q ′ ,W ′ ) .
Figure 6 .
6Cutting along an arc, numbered 3, in a torus to get a cylinder: triangulation case.
4
Figure 8 .Figure 9 .
89Cutting along an arc in a torus to get a cylinder: partial triangulation case. The effect of cutting along arc 3 on the coloured quiver of the partial triangulation inFigure 8.
Corollary 8 .
8Under the bijection A 0 R (S, M ) ↔ A 0 (S, M )/R , the induced action of the shift functor of C (S,M)/R on A 0 R (S, M ) coincides with that of the twist κ R .
Compute d i for all i ∈ I using the formula in Lemma 12. (ii) For any pair of vertices i, j for which there is a unique c with q For all such i arising in this way, replace d i with 2d i .
Figure 15 .
15A partial triangulation of a torus with a single boundary component and two marked points and the corresponding coloured quiver.
i . Then, for any c ∈ Z/d i , we have:
→→
in C of a morphism f ∈ C is denoted f . By induction on c ≥ 0, we are going to construct:(a) A minimal left add R j -approximation R ΣX c+1 in C, with X c+1 in add R k ; (a') a minimal right add R j -approximation R ΣX −c−1 in C, with X −c−1 in add R k .
add ΣR k . Since R i belongs to ⊥ (ΣR k )
Calabi-Yau tilting theorem of Keller-Reiten [KR07, Prop. 2.1] (see also Koenig-Zhu [KZ08, Cor. 4.4] and Iyama-Yoshino [IY08, Prop. 6.2]) was recently generalised
The use of Iyama-Yoshino reduction would be replaced by [BIRSc09, Theorem I.2.6] (see also [GLS06, Lemma 5.1] and [AO, Sect. 4]).
Acknowledgements Robert Marsh would like to thank Aslak Bakke Buan for some helpful discussions. This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/G007497/1].(c+1) i ΣR tc+1 j ⊕ ΣR sc+1 k ΣR (c+2) i whose rows and columns are triangles. By construction, R, ΣR k ) = 0. Thus ΣY also belongs to the extension-closed subcategory ⊥ (Σ 2 R k ), and the morphism η vanishes, since X ∈ add R k . As a consequence, the triangle in the third row splits and Y belongs to add R k . Define X c+2 to be Y . Then we see that (b) has been shown.The statements (a') and (b') can be deduced from (a) and (b) by duality: Consider in the category C op the object R i ⊕ R j ⊕ ΣR k . It is a rigid object:and similarly, C op (R j , Σ op ΣR k ) = 0. Moreover, it satisfies the assumptions we made to prove (a) and (b):with Y c+1 in add ΣR k . Let X −c−1 = Σ −1 Y c+1 , to get the triangles (b').By (a) applied to C op , there are minimal right add R j approximations Rin C R k = ⊥ (ΣR k )/(R k ). This proves that we have t op c = t −c−1 . Written in C,
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[] |
[
"Generalized Reverse Young and Heinz Inequalities",
"Generalized Reverse Young and Heinz Inequalities"
] |
[
"Shigeru Furuichi ",
"· Mohammad ",
"Bagher Ghaemi ",
"· Nahid ",
"Gharakhanlu "
] |
[] |
[] |
In this paper, we study the further improvements of the reverse Young and Heinz inequalities for the wider range of v, namely v ∈ R. These modified inequalities are used to establish corresponding operator inequalities on a Hilbert space.
|
10.1007/s40840-017-0483-y
|
[
"https://arxiv.org/pdf/1801.10389v1.pdf"
] | 119,133,337 |
1801.10389
|
f9d10a169e7448018172993b1fd44811a24ba2a6
|
Generalized Reverse Young and Heinz Inequalities
31 Jan 2018
Shigeru Furuichi
· Mohammad
Bagher Ghaemi
· Nahid
Gharakhanlu
Generalized Reverse Young and Heinz Inequalities
31 Jan 2018Received: date / Accepted: dateBulletin of the Malaysian Mathematical Sciences Society manuscript No. (will be inserted by the editor)Young's inequality · Heinz inequality · Operator inequality Mathematics Subject Classification (2010) 15A39 · 47A63 · 47A60 · 47A64
In this paper, we study the further improvements of the reverse Young and Heinz inequalities for the wider range of v, namely v ∈ R. These modified inequalities are used to establish corresponding operator inequalities on a Hilbert space.
for a, b ≥ 0 and v ∈ [0, 1]. If v = 1 2 , we obtain the arithmetic-geometric mean inequality a + b 2 ≥ √ ab.
The Heinz means, introduced in [3], are defined by
H v (a, b) = a 1−v b v + a v b 1−v 2
for a, b ≥ 0 and v ∈ [0, 1]. It is easy to see that
√ ab ≤ H v (a, b) ≤ a + b 2 v ∈ [0, 1],
which is called Heinz inequality.
A♯ v B = A 1 2 A − 1 2 BA − 1 2 v A 1 2
and the v-weighted operator arithmetic mean of A and B, denoted by A∇ v B, is
A∇ v B = (1 − v)A + vB.
When v = 1 2 , A♯ 1 2 B and A∇ 1 2 B are called operator geometric mean and operator arithmetic mean, and denoted by A♯B and A∇B, respectively [19].
A♯ v B = B♯ 1−v A.
It is well known that if A, B ∈ B ++ (H) and v ∈ [0, 1], then [8,9] A∇ v B ≥ A♯ v B, which is the operator version of the scalar Young's inequality. An operator version of Heinz means was introduced in [3] by
H v (A, B) = A♯ v B + A♯ 1−v B 2 ,
where v ∈ [0, 1]. In particular H 1 (A, B) = H 0 (A, B) = A∇B. It is easy to see that the Heinz operator means interpolate between the arithmetic and geometric operator means
A♯B ≤ H v (A, B) ≤ A∇B,
Generalized Reverse Young and Heinz Inequalities 3 which is called the Heinz operator inequality [14,15]. We note that we use in Sect. 3 the following notations
A♮ v B ≡ A 1 2 A − 1 2 BA − 1 2 v A 1 2 ,Ĥ v (A, B) ≡ A♮ v B + A♮ 1−v B 2 ,
for all v ∈ R including the range v / ∈ [0, 1]. Improvements of Young and Heinz inequalities and their reverses have been done for the weight v ∈ [0, 1] by many researchers. We refer the reader to [1,2,4,5,6,7,11,12,13,14,15,16,17,18,20,21,22,23,24] as a sample of the extensive use of Young and Heinz inequalities. One of the first refinements is as follows in Lemma 1 which one positive term was added to the right-hand side of the Young's inequality.
Lemma 1 ([16]) Let a, b ≥ 0 and v ∈ [0, 1]. Then (1 − v)a + vb ≥ a 1−v b v + r 0 √ a − √ b 2 where r 0 = min{v, 1 − v}.
However, in the recent paper [24], Zhao and Wu provided two refining terms of Young's inequality in the following way.
Lemma 2 ([24]) Let a, b ≥ 0 and v ∈ [0, 1]. (i) If v ∈ [0, 1 2 ], then (1 − v)a + vb ≥ a 1−v b v + v( √ a − √ b) 2 + r 0 ( √ a − 4 √ ab) 2 , (ii) If v ∈ [ 1 2 , 1], then (1 − v)a + vb ≥ a 1−v b v + (1 − v)( √ a − √ b) 2 + r 0 ( √ b − 4 √ ab) 2 ,
where r = min{v, 1 − v} and r 0 = min{2r, 1 − 2r}.
In the same paper, the following refined reverse versions have been proved too.
Lemma 3 ([24, Lemma 2]) Let a, b ≥ 0 and v ∈ [0, 1]. (i) If v ∈ [0, 1 2 ], then (1 − v)a + vb ≤ a 1−v b v + (1 − v)( √ a − √ b) 2 − r 0 ( √ b − 4 √ ab) 2 , (ii) If v ∈ [ 1 2 , 1], then (1 − v)a + vb ≤ a 1−v b v + v( √ a − √ b) 2 − r 0 ( √ a − 4 √ ab) 2 ,
where r = min{v, 1 − v} and r 0 = min{2r, 1 − 2r}.
Sababheh and Choi [21] obtained a complete refinement of the Young's inequality by adding as many refining terms as we like. For a, b > 0, n ∈ N and v ∈ [0, 1]
(1 − v)a + vb ≥ a 1−v b v + n k=1 s k (v) 2 k a 2 k−1 −j k (v) b j k (v) − 2 k a 2 k−1 −j k (v)−1 b j k (v)+1 2 ,
where [x] is the greatest integer less than or equal to x and
j k (v) = [2 k−1 v], r k (v) = [2 k v], s k (v) = (−1) r k (v) 2 k−1 v + (−1) r k (v)+1 r k (v) + 1 2 .
Quite recently, Sababheh and Moslehian [22] gave a full description of all other refinements of the reverse Young's inequality in the literature as follows.
Lemma 4 ( [22, Theorem 2.1]) Let a, b > 0 and v ∈ [0, 1]. (i) If v ∈ [0, 1 2 ], then (1 − v)a + vb ≤ a 1−v b v + (1 − v)( √ a − √ b) 2 − S n (2v, √ ab, b). (ii) If v ∈ [ 1 2 , 1], then (1 − v)a + vb ≤ a 1−v b v + v( √ a − √ b) 2 − S n (2(1 − v), √ ab, a). Where S n (v, a, b) = n k=1 s k (v) 2 k b 2 k−1 −j k (v) a j k (v) − 2 k a j k (v)+1 b 2 k−1 −j k (v)−1 2 , j k (v) = [2 k−1 v], r k (v) = [2 k v], s k (v) = (−1) r k (v) 2 k−1 v + (−1) r k (v)+1 r k (v) + 1 2 .
In the study of Young's inequalities, supplemental Young's inequality
a v v 1−v ≥ va + (1 − v)b
for a, b > 0 and v / ∈ [0, 1] is often discussed. Our main idea in this paper is to extend the range of v and to give the tighter bounds of the reverse Young's inequalities proved in [22] and [24]. In Theorem 1, we will obtain a new generalization of the reverse Young's inequality which is stronger than the reverse Young's inequalities shown in [22, Theorem 2.1] and [24, Lemma 2]. Theorem 3 is another refinement of [21, Theorem 2.9] which extend the range of v. In Sect. 3, these modified inequalities are used to establish corresponding operator inequalities on a Hilbert space. We emphasize that the significance of the inequalities in this paper is to have the wider range, namely v ∈ R, and tighter bounds. In this section, we present the numerical inequalities needed to prove the operator versions. We start from the following lemma to prove our main result. 1 2 ]. So by changing two elements a, b and two weights v, 1 − v in (i), the desired inequality is obtained.
Lemma 5 ([2, Lemma 2.1]) Let a, b > 0 and v / ∈ [0, 1]. Then (1 − v)a + vb ≤ a 1−v b v . Corollary 1 Let a, b > 0 and 1 2 = v ∈ R. (i) If v / ∈ [0, 1 2 ], then (1 − v)a + vb ≤ a 1−v b v + v( √ a − √ b) 2 . (ii) If v / ∈ [ 1 2 , 1], then (1 − v)a + vb ≤ a 1−v b v + (1 − v)( √ a − √ b) 2 . Proof (i) If v / ∈ [0, 1 2 ], then (1 − v)a + vb − v( √ a − √ b) 2 = (1 − 2v)a + (2v) √ ab ≤ a (1−2v) ( √ ab) 2v ( by Lemma 5) = a 1−v b v . (ii) If v / ∈ [ 1 2 , 1], then (1 − v) / ∈ [0,
⊓ ⊔ Next, we represent our main result which is the reverse Young's inequality for v ∈ R.
Theorem 1 Let a, b > 0, n ∈ N such that n ≥ 2 and 1 2 = v ∈ R. Then,
(i) If v / ∈ [ 1 2 , 2 n−1 +1 2 n ], then (1 − v)a + vb ≤ a 1−v b v + (1 − v)( √ a − √ b) 2 + (2v − 1) √ ab n k=2 2 k−2 2 k b a − 1 2 .
(1)
(ii) If v / ∈ [ 2 n−1 −1 2 n , 1 2 ], then (1 − v)a + vb ≤ a 1−v b v + v( √ a − √ b) 2 + (1 − 2v) √ ab n k=2 2 k−2 2 k a b − 1 2 . (2) Proof (i) If v / ∈ [ 1 2 , 2 n−1 +1 2 n ], we have (2 n v−2 n−1 ) / ∈ [0, 1] and (2 n−1 −2 n v+1) / ∈ [0, 1]. Now compute (1 − v)a + vb − (1 − v)( √ a − √ b) 2 − (2v − 1) √ ab n k=2 2 k−2 2 k b a − 1 2 = (1 − v)a + vb − (1 − v)( √ a − √ b) 2 − (2v − 1) √ ab 4 b a − 1 2 + 2 8 b a − 1 2 + 4 16 b a − 1 2 − · · · − 2 n−4 (2v − 1) √ ab 2 n−2 b a − 1 2 − 2 n−3 (2v − 1) √ ab 2 n−1 b a − 1 2 − 2 n−2 (2v − 1) √ ab 2 n b a − 1 2 = (1 − v)a + vb − (1 − v)(a − 2 √ ab + b) − (2v − 1) √ ab b a − 2 4 b a + 1 − 2(2v − 1) √ ab 4 b a − 2 8 b a + 1 − 4(2v − 1) √ ab 8 b a − 2 16 b a + 1 − · · · − 2 n−4 (2v − 1) √ ab 2 n−3 b a − 2 2 n−2 b a + 1 − 2 n−3 (2v − 1) √ ab 2 n−2 b a − 2 2 n−1 b a + 1 − 2 n−2 (2v − 1) √ ab 2 n−1 b a − 2 2 n b a + 1 = 2(1 − v) √ ab + (1 − 2v) √ ab n−2 l=0 2 l + 2 n−1 (2v − 1) √ ab 2 n b a = 2(1 − v) + (1 − 2v) n−2 l=0 2 l √ ab + 2 n−1 (2v − 1) √ ab 2 n b a = 2(1 − v) + (1 − 2v)(2 n−1 − 1) √ ab + 2 n−1 (2v − 1) √ ab 2 n b a = (2 n−1 − 2 n v + 1) √ ab + (2 n v − 2 n−1 ) √ ab 2 n b a ≤ √ ab (2 n−1 −2 n v+1) √ ab 2 n b a (2 n v−2 n−1 ) ( by Lemma 5) = a 1−v v v .
So we get the following inequality
(1 − v)a + vb − (1 − v)( √ a − √ b) 2 − (2v − 1) √ ab n k=2 2 k−2 2 k b a − 1 2 ≤ a 1−v v v which is equivalent to (1). (ii) If v / ∈ [ 2 n−1 −1 2 n , 1 2 ], then (1 − v) / ∈ [ 1 2 , 2 n−1 +1 2 n
]. Now by changing two elements a, b and replacing the weight v with (1 − v) in (i), the desired inequality (2) is deduced.
⊓ ⊔
Remark 1
We would remark that if we rewrite Theorem 1 for n = 1, then we get Corollary 1.
Remark 2 From the equality of the proof in Theorem 1, the inequality (1) is equivalent to
a 1−v b v ≥ √ ab + 2 n v − 1 2 √ ab 2 n b a − 1 ,(3)
which gives the following inequality
b a v− 1 2 ≥ 1 + 2 n v − 1 2 2 n b a − 1 .
Since lim r→0
t r −1 r = log t, by putting r = 1 2 n , we have lim n→∞ 2 n 2 n b a − 1 = log b a .
Thus, we have the following inequality in the limit of n → ∞ for the inequality (1) in Theorem 1:
log b a v− 1 2 ≤ b a v− 1 2 − 1,(4)
for a, b > 0 and 1 2 = v ∈ R, which comes from the condition v / ∈ 1 2 , 2 n−1 +1 2 n in the limit of n → ∞. The above inequality recover the equality in the case v = 1 2 . Therefore, we have the inequality (4) for all v ∈ R. We notice that the inequality (4) can be proven directly by putting
x = b a v−1/2 in the inequality log x ≤ x − 1, (x > 0).(5)
Similarly in the limit of n → ∞ for the inequality (2) in Theorem 1, we get the inequality
log a b 1 2 −v ≤ a b 1 2 −v − 1,
by changing two elements a, b and replacing the weight v with (1 − v) in the inequality (4). Next, in Remarks 3 and 4, we show that Theorem 1 recover the inequalities in Lemma 3. To achieve this, we compare Lemma 3 with Theorem 1 in the cases such as n = 2 and n = 3 where v ∈ [0, 1]. First, we notice that Lemma 3 is equivalent to the following proposition.
Proposition 2 Let a, b ≥ 0 and v ∈ [0, 1]. (i) If v ∈ [0, 1 4 ], then (1 − v)a + vb ≤ a 1−v b v + (1 − v)( √ a − √ b) 2 − 2v( √ b − 4 √ ab) 2 . (ii) If v ∈ [ 1 4 , 1 2 ], then (1 − v)a + vb ≤ a 1−v b v + (1 − v)( √ a − √ b) 2 + (2v − 1)( √ b − 4 √ ab) 2 . (iii) If v ∈ [ 1 2 , 3 4 ], then (1 − v)a + vb ≤ a 1−v b v + v( √ a − √ b) 2 − (2v − 1)( √ a − 4 √ ab) 2 . (iv) If v ∈ [ 3 4 , 1], then (1 − v)a + vb ≤ a 1−v b v + v( √ a − √ b) 2 + (2v − 2)( √ a − 4 √ ab) 2 .
Remark 3 Consider Theorem 1 in the case n = 2 with v ∈ [0, 1]. For a, b > 0, we have the following inequalities
(i) If v / ∈ [ 1 2 , 3 4 ], then (1 − v)a + vb ≤ a 1−v b v + (1 − v)( √ a − √ b) 2 + (2v − 1)( √ b − 4 √ ab) 2 .(6)(i) If v / ∈ [ 1 4 , 1 2 ], then (1 − v)a + vb ≤ a 1−v b v + v( √ a − √ b) 2 − (2v − 1)( √ a − 4 √ ab) 2 .(7)
In our recent paper [11], we showed that the right-hand sides of both inequalities (6) and (7) give tighter upper bounds of the v-weighted arithmetic mean than those in Proposition 2.
Remark 4 As a direct consequence of Theorem 1 in the case n = 3 with restricted range v ∈ [0, 1], we have
(i) If v / ∈ [ 1 2 , 5 8 ], then (1 − v)a + vb ≤ a 1−v b v + (1 − v)( √ a − √ b) 2 + (2v − 1)( √ b − 4 √ ab) 2 + (4v − 2)( 8 √ ab 3 − 4 √ ab) 2 .(8)(i) If v / ∈ [ 3 8 , 1 2 ], then (1 − v)a + vb ≤ a 1−v b v + v( √ a − √ b) 2 − (2v − 1)( √ a − 4 √ ab) 2 − (4v − 2)( 8 √ a 3 b − 4 √ ab) 2 .(9)
We here give advantages of inequalities (8) and (9) in comparison with Proposition 2.
(a) Firstly, we compare Proposition 2 with the inequality (8) which holds in the cases v ∈ [0, 1 4 3 4 ] and v ∈ [ 3 4 , 1] . (a1) In the case v ∈ [0, 1 4 ], we have (2v−1) < (−2v) and (4v−2) < 0. Indeed, the right-hand side of inequality (8) is less than the right-hand side of (i) in Proposition 2. For the case of v ∈ [ 1 4 , 1 2 ], we have (4v − 2) < 0. So we easily find that the right-hand side of inequality (8) is less than the right-hand side of (ii) in Proposition 2. (a2) For the case of v ∈ [ 5 8 , 3 4 ], we claim that the right-hand side of inequality (8) is less than or equal to the right-hand side of (iii) in Proposition 2. To prove our claim, we show that the following inequality holds:
], v ∈ [ 1 4 , 1 2 ], v ∈ [ 5 8 ,(1 − v)( √ a − √ b) 2 + (2v − 1)( √ b − 4 √ ab) 2 + (4v − 2)( 8 √ ab 3 − 4 √ ab) 2 ≤ v( √ a − √ b) 2 − (2v − 1)( √ a − 4 √ ab) 2 ,(10)
which is equivalent to the inequality
2(1 − 2v)t 1/4 3t 1/4 − 1 − 2t 3/8 ≥ 0,
for t > 0 and v ∈ [ 5 8 , 3 4 ]. To obtain this, it is enough to prove (3t 3 4 , 1], we claim that the right-hand side of inequality (8) is less than or equal to the right-hand side of (iv) in Proposition 2. To prove our claim, we show that the following inequality holds:
1/4 − 1 − 2t 3/8 ) ≤ 0. If t 1/8 = x, then we easily find that f (x) = 3x 2 − 2x 3 − 1 is increasing for 0 < x < 1 and decreasing where x > 1. Indeed, f (x) = 3x 2 − 2x 3 − 1 ≤ 0 where x > 0 and so 3t 1/4 − 1 − 2t 3/8 ≤ 0. (a3) For the case of v ∈ [(1 − v)( √ a − √ b) 2 + (2v − 1)( √ b − 4 √ ab) 2 + (4v − 2)( 8 √ ab 3 − 4 √ ab) 2 ≤ v( √ a − √ b) 2 − 2(1 − v)( √ a − 4 √ ab) 2 ,(11)
which is equivalent to the inequality
(4v − 3) + (3 − 8v)t 1/2 + (4 − 4v)t 1/4 + (8v − 4)t 5/8 ≥ 0,(12)
for t > 0 and v ∈ [ 3 4 , 1]. To obtain the inequality (12), it is sufficient to
prove f (x, v) ≥ 0 where x = t 1/8 > 0 and f (x, v) ≡ (8v − 4)x 5 + (3 − 8v)x 4 + (4 − 4v)x 2 + (4v − 3). Since df (x,v) dv = 4(x − 1) 2 (2x 3 + 2x 2 + 2x + 1) ≥ 0, we have f (x, v) ≥ f (x, 3 4 ) = x 2 (x − 1) 2 (2x + 1) ≥ 0 for x > 0.
(b) Secondly, we compare Proposition 2 with the inequality (9) which holds in the cases v ∈ [0, 1 4 ], v ∈ [ 1 4 , 3 8 ], v ∈ [ 1 2 , 3 4 ] and v ∈ [ 3 4 , 1]. (b1) For the case of v ∈ [0, 1 4 ], we claim that the right-hand side of inequality (9) is less than or equal to the right-hand side of (i) in Proposition 2. To prove our claim, we give the following inequality
v( √ a − √ b) 2 − (2v − 1)( √ a − 4 √ ab) 2 − (4v − 2)( 8 √ a 3 b − 4 √ ab) 2 ≤ (1 − v)( √ a − √ b) 2 − 2v( √ b − 4 √ ab) 2 ,
which we get it by replacing v with (1 − v) and changing the elements a, b in the inequality (11) . (b2) For the case of v ∈ [ 1 4 , 3 8 ], we claim that the right-hand side of inequality (9) is less than or equal to the right-hand side of (ii) in Proposition 2. To prove our claim, we give the following inequality
v( √ a − √ b) 2 − (2v − 1)( √ a − 4 √ ab) 2 − (4v − 2)( 8 √ a 3 b − 4 √ ab) 2 ≤ (1 − v)( √ a − √ b) 2 − (1 − 2v)( √ b − 4 √ ab) 2 ,
which is deduced by replacing v with (1 − v) and changing the elements a, b in the inequality (10) . (b3) For the case of v ∈ [ 1 2 , 3 4 ], we have −(4v − 2) < 0. So we easily find that the right-hand side of inequality (9) is less than the right-hand side of (iii) in Proposition 2. In the case v ∈ [ 3 4 , 1] , we have −(2v−1) < (2v−2) and −(4v − 2) < 0. That is the right-hand side of inequality (9) is less than the right-hand side of (iv) in Proposition 2.
Thus, according to Remark 3 and Remark 4, Theorem 1 recover Lemma 3. We notice that the range of the reverse Young's inequalities in Theorem 1 is wider than Lemma 3, namely Theorem 1 holds in the case v ∈ R and Lemma 3 holds for v ∈ [0, 1].
Next, we compare Theorem 1 with Lemma 4. In Theorem 1, we have (
I) v ≤ 0, (II) 0 ≤ v < 2 n−1 −1 2 n , (III) 2 n−1 −1 2 n ≤ v ≤ 1 2 , (IV) 1 2 ≤ v ≤ 2 n−1 +1 2 n , (V) 2 n−1 +1 2 n
< v ≤ 1 and (VI) v ≥ 1. For the cases of (II) and (V), we claim that Theorem 1 has tighter upper bounds than those in Lemma 4, while Lemma 4 recover Theorem 1 in the cases (III) and (IV). Therefore we conclude that Theorem 1 and Lemma 4 are different refinements of the reverse Young's inequality which both of them recover Lemma 3. However, we emphasize that Theorem 1 gives the reverse Young's inequality in the wider range than Lemma 4, namely in the case v ∈ R. This justify why our refinement in Theorem 1 is better than Lemma 4. To prove our claims, we compare Theorem 1 with Lemma 4 in the same steps such as n = 2. For this purpose, we list up the following corollaries which are deduced directly from Theorem 1 and Lemma 4, respectively.
Corollary 2 Let a, b > 0 and v ∈ R.
(i) If v / ∈ [ 1 2 , 3 4 ], then (1 − v)a + vb − a 1−v b v ≤ (1 − v)( √ a − √ b) 2 + (2v − 1)( √ b − 4 √ ab) 2 .(13)(ii) If v / ∈ [ 1 4 , 1 2 ], then (1 − v)a + vb − a 1−v b v ≤ v( √ a − √ b) 2 − (2v − 1)( √ a − 4 √ ab) 2 . (14)
Corollary 3 Let a, b > 0 and v ∈ [0, 1]. 1 4 ], we can find examples such that the right-hand side of (13) is tighter than that of (15) in Corollary 3. Actually, take a = 1, b = 16 and v = 1/8, then the right-hand side of (13) is equal to 4.875, while the right-hand side of (15) is nearly equal to 6.2892. Indeed, the inequality (13) can recover Corollary 3 where v ∈ [0, 1 4 ]. In the case v ∈ [ 3 4 , 1], we claim that the right-hand side of the inequality (13) is less than or equal to the right-hand side of (18). According to (11), we have
(i) If v ∈ [0, 1 4 ], then (1 − v)a + vb − a 1−v b v ≤ (1 − v)( √ a − √ b) 2 − 2v( √ b − 4 √ ab) 2 − 4v( √ b − 8 √ ab 3 ) 2 .(15)(ii) If v ∈ [ 1 4 , 1 2 ], then (1 − v)a + vb − a 1−v b v ≤ (1 − v)( √ a − √ b) 2 + (2v − 1)( √ b − 4 √ ab) 2 − (4v − 1)( 8 √ ab 3 − 4 √ ab) 2 . (16) (iii) If v ∈ [ 1 2 , 3 4 ], then (1 − v)a + vb − a 1−v b v ≤ v( √ a − √ b) 2 − (2v − 1)( √ a − 4 √ ab) 2 + (4v − 3)( 8 √ ab 3 − 4 √ ab) 2 .(17)(iv) If v ∈ [ 3 4 , 1], then (1 − v)a + vb − a 1−v b v ≤ v( √ a − √ b) 2 + (2v − 2)( √ a − 4 √ ab) 2 − 4(1 − v)( 8 √ ab 3 − √ a) 2 .(18)(1 − v)( √ a − √ b) 2 + (2v − 1)( √ b − 4 √ ab) 2 ≤ v( √ a − √ b) 2 + (2v − 2)( √ a − 4 √ ab) 2 − (4v − 2)( 8 √ ab 3 − 4 √ ab) 2 .
So to prove our claim, we show that the following inequality holds
(2 − 4v)( 8 √ ab 3 − 4 √ ab) 2 ≤ (4v − 4)( 8 √ ab 3 − √ a) 2 ,
which is equivalent to
g(x, v) = (4v − 2)x 6 + (2 − 4v)x 5 + (2v − 1)x 4 + (2 − 4v)x 3 + (2v − 1) ≥ 0, where x = t 1 8 > 0 and v ∈ [ 3 4 , 1]. Since dg(x,v) dv = 2(x − 1) 2 (2x 4 + 2x 3 + 3x 2 + 2x + 1) ≥ 0, we have g(x, v) ≥ g(x, 3 4 ) = (x − 1) 2 (x 4 + x 3 + 3 2 x 2 + x + 1 2 ) ≥ 0.
This justify that the inequality (13) also recover Corollary 3 where v ∈ [ 3 4 , 1]. However, in the case v ∈ [ 1 4 , 1 2 ], we easily find that the right-hand side of (16) is less than or equal to the right-hand side of (13). That is Corollary 3 is a refinement of (13) where v ∈ [ 1 4 , 1 2 ]. Secondly, we compare Corollary 3 with the inequality (14) which holds in the cases v ∈ [0, 1 4 ], v ∈ [ 1 2 , 3 4 ] and v ∈ [ 3 4 , 1]. The comparison is done by the same way as in the first step, and we omit it.
Sababheh and Choi gave the following refinement of Lemma 5 which it's complete proof can be found in [22,Theorem 2.2].
Lemma 6 ( [21, Theorem 2.9]) Let a, b > 0. Then, we have
(i) If v ≤ 0, then (1 − v)a + vb ≤ a 1−v b v + v n k=1 2 k−1 √ a − 2 k a 2 k−1 −1 b 2 . (19) (ii) If v ≥ 1, then (1 − v)a + vb ≤ a 1−v b v + (1 − v) n k=1 2 k−1 √ b − 2 k ab 2 k−1 −1 2 .(20)
We can extend the ranges of v in Lemma 6 to those in following theorem, by the similar way to the line of proof of Theorem 1. As we discussed the case of n → ∞ in Remark 2, we also give the following remark. for v = 0 in the limit of n → ∞. The above inequality trivially holds for all v ∈ R. It can be also obtained by the inequality (5). Similarly, the inequality (20) for v = 1 in the limit of n → ∞. The above inequality trivially holds for all v ∈ R. It can be also obtained by the inequality (5).
As a direct consequence of Theorems 1 and 3, we have the following reverse inequalities with respect to the Heinz means.
Corollary 4 Let a, b > 0, n ∈ N such that n ≥ 2 and 1 2 = v ∈ R.
(i) If v / ∈ [ 1 2 , 2 n−1 +1 2 n ], then a + b 2 ≤ H v (a, b) + (1 − v)( √ a − √ b) 2 + v − 1 2 √ ab n k=2 2 k−2 2 k a b − 1 2 + 2 k b a − 1 2 (ii) If v / ∈ [ 2 n−1 −1 2 n , 1 2 ], then a + b 2 ≤ H v (a, b) + v( √ a − √ b) 2 + 1 2 − v √ ab n k=2 2 k−2 2 k a b − 1 2 + 2 k b a − 1 2 .
Corollary 5 Let a, b > 0, n ∈ N and v ∈ R.
(i) If v / ∈ [0, 1 2 n ], then a + b 2 ≤ H v (a, b) + v n k=1 2 k−2 √ a − 2 k a 2 k−1 −1 b 2 + √ b − 2 k ab 2 k−1 −1 2 . (ii) If v / ∈ [ 2 n −1 2 n , 1], then a + b 2 ≤ H v (a, b) + (1 − v) n k=1 2 k−2 √ a − 2 k a 2 k−1 −1 b 2 + √ b − 2 k ab 2 k−1 −1 2 .
Let B(H) denote the C * -algebra of all bounded linear operators on a complex Hilbert space H. A self-adjoint operator A ∈ B(H) is called positive, and we write A ≥ 0 if Ax, x ≥ 0 for all x ∈ H. The set of all positive operators is denoted by B + (H). The set of all invertible operators in B + (H) is denoted by B ++ (H). We say A ≥ B if A − B ≥ 0. Let A, B ∈ B ++ (H) and v ∈ [0, 1]. The v-weighted operator geometric mean of A and B, denoted by A♯ v B, is defined as
For positive operators A, B and v ∈ [0, 1], we have [10, Definition 5.2]
Theorem 3
3Let a, b > 0, n ∈ N and v ∈ R.
], then the inequality (19) holds. (ii) If v / ∈ [ 2 n −1 2 n , 1], then the inequality (20) holds.
Acknowledgements The authors express their gratitude to the editor-in-chief Prof. Rosihan M. Ali and the anonymous referees for their careful reading and detailed comments which have considerably improved the paper.In this section by applying Kubo-Ando theory[19]and thanks to Theorems 1 and 3, we have the following operator inequalities.Theorem 4 Let A, B ∈ B ++ (H), n ∈ N such that n ≥ 2 and 1 2 = v ∈ R. Then, we have the following inequalities.Proof (i) According to the inequality (1), the following inequality holds for t ≥ 0By functional calculus, if we replace t with A − 1 2 BA − 1 2 and then multiplying both sides of the inequality by A Theorem 5 Let A, B ∈ B ++ (H), n ∈ N and v ∈ R. Then, we have the following inequalities.Generalized Reverse Young and Heinz Inequalities 15 Proof (i) According to Theorem 3 in the case v / ∈ [0, 1 2 n ], the following inequality holds for t ≥ 0By functional calculus, if we replace t with A − 1 2 BA − 1 2 and then multiplying both sides of the inequality by A 1 2 , the desired inequality is deduced.(ii) By applying Theorem 3 for the case of v / ∈ [ 2 n −1 2 n , 1], we get the desired inequality in the same way as in (i).⊓ ⊔As a direct consequence of Theorems 4 and 5, we get the generalized reverse Heinz operator inequalities as follows. That is Corollaries 6 and 7 are operator versions of Corollaries 4 and 5 respectively.Corollary 6 Let A, B ∈ B ++ (H), n ∈ N such that n ≥ 2 and 1 2 = v ∈ R. Then we have the following inequalities..Remark 8 Putting n = 2 in Corollary 6 with v ∈ [0, 1] gives the inequalities obtained in[11,Corollary 6].Corollary 7 Let A, B ∈ B ++ (H), n ∈ N and v ∈ R. Then, we have the following inequalities.
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[
"A representation theorem for generators of BSDEs with general growth generators in y and its applications ✩",
"A representation theorem for generators of BSDEs with general growth generators in y and its applications ✩"
] |
[
"Lishun Xiao \nSchool of Mathematics\nChina University of Mining and Technology\n221116XuzhouJiangsuP.R. China\n",
"Shengjun Fan \nSchool of Mathematics\nChina University of Mining and Technology\n221116XuzhouJiangsuP.R. China\n"
] |
[
"School of Mathematics\nChina University of Mining and Technology\n221116XuzhouJiangsuP.R. China",
"School of Mathematics\nChina University of Mining and Technology\n221116XuzhouJiangsuP.R. China"
] |
[] |
In this paper we first prove a general representation theorem for generators of backward stochastic differential equations (BSDEs for short) by utilizing a localization method involved with stopping time tools and approximation techniques, where the generators only need to satisfy a weak monotonicity condition and a general growth condition in y and a Lipschitz condition in z. This result basically solves the problem of representation theorems for generators of BSDEs with general growth generators in y. Then, such representation theorem is adopted to prove a probabilistic formula, in viscosity sense, of semilinear parabolic PDEs of second order. The representation theorem approach seems to be a potential tool to the research of viscosity solutions of PDEs.
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10.1016/j.spl.2017.06.014
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[
"https://arxiv.org/pdf/1701.03870v1.pdf"
] | 119,642,492 |
1701.03870
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fe0eddfcec05fe6ef38cd192f9bcbad0de2d88f2
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A representation theorem for generators of BSDEs with general growth generators in y and its applications ✩
14 Jan 2017
Lishun Xiao
School of Mathematics
China University of Mining and Technology
221116XuzhouJiangsuP.R. China
Shengjun Fan
School of Mathematics
China University of Mining and Technology
221116XuzhouJiangsuP.R. China
A representation theorem for generators of BSDEs with general growth generators in y and its applications ✩
14 Jan 2017Backward stochastic differential equationRepresentation theoremGeneral growthWeak monotonicityViscosity solution 2010 MSC: 60H1035K58
In this paper we first prove a general representation theorem for generators of backward stochastic differential equations (BSDEs for short) by utilizing a localization method involved with stopping time tools and approximation techniques, where the generators only need to satisfy a weak monotonicity condition and a general growth condition in y and a Lipschitz condition in z. This result basically solves the problem of representation theorems for generators of BSDEs with general growth generators in y. Then, such representation theorem is adopted to prove a probabilistic formula, in viscosity sense, of semilinear parabolic PDEs of second order. The representation theorem approach seems to be a potential tool to the research of viscosity solutions of PDEs.
Introduction
By Pardoux and Peng [1990] we know that the following one dimensional backward stochastic differential equation (BSDE for short):
Y t = ξ + T t g(s, Y s , Z s ) ds − T t Z s , dB s , t ∈ [0, T ],(1)
admits a unique adapted and square-integrable solution (Y t (g, T, ξ), Z t (g, T, ξ)) t∈[0,T ] , provided that g is Lipschitz continuous with respect to (y, z), and that ξ and (g(t, 0, 0)) t∈[0,T ] are square-integrable. We call T with 0 ≤ T < ∞ the terminal time, ξ the terminal condition and g the generator. BSDE (1) is also denoted by BSDE (g, T, ξ). Briand, Coquet, Hu, Mémin, and Peng [2000] constructed a representation theorem for the generator of BSDE (1): for each (y, z) ∈ R × R d ,
g(t, y, z) = L 2 − lim ε→0 + 1 ε [Y t (g, t + ε, y + z · (B t+ε − B t )) − y] , ∀t ∈ [0, T ),(2)
where the generator g is Lipschitz continuous in (y, z) and satisfies two additional conditions, i.e., E sup t∈[0,T ] |g(t, 0, 0)| 2 < ∞ and g(t, y, z) is continuous with respect to t. It has been widely recognized that the representation theorems provide a crucial tool for investigating properties of the generator and g-expectations by virtue of solutions of the BSDEs. Note that g-expectations were first introduced by Peng [1997] to explain the nonlinear phenomena in mathematical finance. More details on this direction are referred to Jiang [2005], Jiang [2008], Jia [2010] and Ma and Yao [2010], etc.
Particularly, many attempts have been made to weaken the conditions required by the generator g. For instance, Jiang [2006] and Jiang [2008] eliminated the above two additional assumptions and proved that (2) remains valid for dt -a.e. t ∈ [0, T ] in L p (1 ≤ p < 2) sense; Jia [2010] proved a representation theorem where the generator g is Lipschitz continuous in y and uniformly continuous in z; Fan and Jiang [2010] presented a representation theorem where g is only continuous and of linear growth in (y, z). However, all these works only handled the case of the generator g with a linear growth in (y, z).
In subsequent works on this direction, researchers were devoted to the case that the generator g has a nonlinear growth in (y, z). For example, Ma and Yao [2010] investigated the representation theorem when the generator g has a quadratic growth in z. They applied a stopping time technique to truncate the terminal condition |B s − B t |, which ensures the first part of the solution is essentially bounded. This idea will be borrowed to deal with our problems. Furthermore, by adopting an approximation technique, Fan, Jiang, and Xu [2011] weakened the conditions required by the generator g in y to a monotonicity condition with a polynomial growth. Recently, Zheng and Li [2015] further studied a representation theorem when the generator g is monotonic with a convex growth in y, and has a quadratic growth in z, where a vital property of convex functions, superadditivity, plays a key role. We mention that up to now the problem of representation theorems for generators of BSDEs with general growth generators in y has not been solved completely.
In the present paper, we first prove a general representation theorem for generators of BSDEs where the generator g only needs to satisfy a weak monotonicity condition and a general growth condition in y, and a Lipschitz condition in z. This basically answers the problem of representation theorems for generators of BSDEs with general growth generators in y, which has been standing for more than decade.
Then, utilizing such representation theorem we establish a converse comparison theorem for solutions of BSDEs and a probabilistic formula for viscosity solutions of systems of second order semilinear parabolic partial differential equations (PDEs for short).
For the representation theorem, the difficulty mainly comes from the general growth conditions of the generator g in y. The whole idea to handle it is that we first use a stopping time to truncate the terminal condition |B s − B t | and s t |g(r, 0, 0)| 2 dr such that the first part of the solution, (Y t ) t∈[0,T ] , is essentially bounded by a constant K > 0, and then treat g(t, q K (y), z) with q K (y) = Ky/(|y| ∨ K), which has a linear growth in y, instead of g(t, y, z), by virtue of an approximation technique. We also mention that the proof procedure of our representation theorem is both simpler and clearer than those in the literature.
For the probabilistic formula for solutions of semilinear parabolic PDEs, which is actually a nonlinear Feynman-Kac formula, its connection with the representation theorem for generators of BSDEs is dis-covered for the first time. More precisely, the probabilistic formula for semilinear PDEs holds as soon as such representation theorem holds. This can also be regarded as a new application of the representation theorem. We believe that the representation theorem will be a potential tool to investigate the viscosity solutions of PDEs.
The rest of this paper is organized as follows. Section 2 contains some notations and preliminaries.
Section 3 presents the representation theorem and a converse comparison theorem. Section 4 proves the probabilistic formula for viscosity solutions of second order semilinear parabolic PDEs.
Preliminaries
Let T > 0 be a given real number, (Ω, F , P) a probability space carrying a standard d-dimensional
Brownian motion (B t ) t≥0 and (F t ) t≥0 the natural σ-algebra filtration generated by (B t ) t≥0 . We assume that F T = F and (F t ) t≥0 is right-continuous and complete. We denote the Euclidean norm and dot product by | · | and ·, · , respectively. For each real number p > 1, let L p (F t ; R) be the set of real-valued and (F t )-measurable random variables ξ such that E[|ξ| p ] < ∞; and S 2 (0, T ; R) (or S 2 for simplicity) the set of real-valued, (F t )-adapted and continuous processes (
Y t ) t∈[0,T ] such that E sup t∈[0,T ] |Y t | 2 < ∞. Moreover, let H 2 (0, T ; R d ) (or H 2 ) denote the set of (equivalent classes of) (F t )-progressively measurable R d -valued processes (Z t ) t∈[0,T ] such that E T 0 |Z t | 2 dt < ∞.
Finally, let K denote the set of nondecreasing and concave continuous function κ : R + → R + satisfying κ(0) = 0, κ(u) > 0 for each u > 0 and 0 + du/κ(u) = ∞. Since κ(·) increases at most linearly, we denote the linear growth constant of κ ∈ K by A > 0, i.e., κ(u) ≤ A + Au for all u ∈ R + .
We will deal with the one dimensional BSDE which is an equation of type (1), where the terminal condition ξ is F T -measurable and the generator g(t, y, z) :
Ω × [0, T ] × R × R d → R is (F t )-progressively
measurable for each (y, z), where and hereafter we suppress ω for brevity. In this paper, a solution of BSDE (1) is a pair of processes Y t (g, T, ξ), Z t (g, T, ξ) t∈[0,T ] in S 2 × H 2 which satisfies BSDE (1). Next we list some lemmas which will be used in the proof of our main result.
Lemma 1 (Proposition 2.2 in Jiang [2008]). Let 1 ≤ p < 2. For any (φ t ) t∈[0,T ] ∈ H 2 (0, T ; R), we have φ t = L p − lim ε→0 + 1 ε t+ε t φ s ds, dt -a.e. t ∈ [0, T ).
The following lemma presents an a priori estimate for solutions of BSDE (1), which comes from Proposition 3.1 in Fan and Jiang [2013]. To state it, we need the following assumption.
(A) There exists a positive constant µ such that dP×dt -a.e., for each y ∈ R and z ∈ R d ,
yg(t, y, z) ≤ κ(|y| 2 ) + µ|y||z| + |y|f t , where κ ∈ K and (f t ) t∈[0,T ] is a non-negative, real-valued and (F t )-progressively measurable process with E T 0 f t dt 2 < ∞.
Lemma 2. Assume that g satisfies (A) and (Y t , Z t ) t∈[0,T ] is a solution of BSDE (1). Then there exists a constant C > 0 depending on µ and T such that
for each 0 ≤ u ≤ t ≤ T , E sup r∈[t,T ] |Y r | 2 F u + E T t |Z r | 2 dr F u ≤ C E |ξ| 2 F u + T t κ E |Y r | 2 |F u dr + E T t f r dr 2 F u .
Next we introduce an existence and uniqueness result for solutions of BSDE (1), which is actually a corollary of Theorem 1 in Fan [2015]. The necessary assumptions are given as follows.
(H1) E T 0 |g(t, 0, 0)| 2 dt < ∞;
(H2) dP×dt -a.e., for each z ∈ R d , y → g(t, y, z) is continuous;
(H3) g satisfies a weak monotonicity condition in y, i.e., there exists a function ρ ∈ K such that dP× dt -a.e., for each y 1 , y 2 ∈ R and z ∈ R d ,
(y 1 − y 2 ) g(t, y 1 , z) − g(t, y 2 , z) ≤ ρ(|y 1 − y 2 | 2 ); (H4) g has a general growth in y, i.e., ψ α (t) := sup |y|≤α |g(t, y, 0)−g(t, 0, 0)| ∈ H 2 (0, T ; R) for each α ≥ 0;
(H5) There exists a constant λ ≥ 0 such that dP×dt -a.e., for each y ∈ R and z 1 ,
z 2 ∈ R d , |g(t, y, z 1 ) − g(t, y, z 2 )| ≤ λ|z 1 − z 2 |.
We also need the following growth condition introduced by Pardoux [1999] to compare with our result.
(H4') There exists a continuous increasing function ϕ : R + → R + such that dP × dt -a.e., for each y ∈ R, |g(t, y, 0)| ≤ |g(t, 0, 0)| + ϕ(|y|).
Remark 3. (i) It is clear that (H4) is weaker than (H4') under the condition (H1). If the increasing function ϕ(·) is also convex, (H4') becomes the convex growth condition adopted by Zheng and Li [2015].
Hence, the general growth condition (H4) is also weaker than the convex growth condition. We mention that the supperadditivity of convex function ϕ(·) plays a key role in the proof of Zheng and Li [2015]. Yet ψ α (t) in (H4) enjoys no other more explicit expressions except for the integrable property. This is the main difficulty to prove the corresponding representation theorem for generators under (H4).
(ii) Originally, it is required that ψ α (t) in (H4) belongs to L 1 ([0, T ] × Ω) in the existence and uniqueness theorem of the solution established in Briand, Delyon, Hu, Pardoux, and Stoica [2003]. However, to obtain our representation theorem the integrability of ψ α (t) needs to be strengthened to the case in (H4).
Proposition 4. Let g satisfy (H1) -(H5). Then for each ξ ∈ L 2 (F T ; R), BSDE (1) admits a unique
solution Y t (g, T, ξ), Z t (g, T, ξ) t∈[0,T ] in S 2 × H 2 .
The following Proposition 5 can be seen as the starting point of this paper. The crucial steps to prove Proposition 5 are the approximation using the convolution technique proposed by Lepeltier and San Martin [1997]. For readers' convenience, the proof procedures are given to adapt our settings.
Proposition 5. Assume that the generator g satisfies (H1), (H2) and (H4). Then for each α ≥ 0, there exists a (F t )-progressively measurable process sequence {(g n α (t)) t∈[0,T ] } ∞ n=1 such that lim n→∞ E[|g n α (t)| 2 ] = 0 for dt -a.e. t ∈ [0, T ], and dP×dt -a.e., for each n ≥ 1 and y ∈ R, we have |g n α (t)| ≤ 2ψ α (t) + 4|g(t, 0, 0)| and |g(t, q α (y), 0) − g(t, 0, 0)| ≤ |g n α (t)| + 2n|y|,
where and hereafter q α (y) := αy/(|y| ∨ α) for each α ≥ 0 and y ∈ R.
Proof. Let the assumptions hold and fix a real number α ≥ 0. Note that |q α (y)| ≤ α ∧ |y|. By (H4) we know that dP×dt -a.e., for each y ∈ R,
|g(t, q α (y), 0)| ≤ ψ α (t) + |g(t, 0, 0)|.(3)
Then, for each n ≥ 1, we can define the following two (F t )-progressively measurable processes:
g n α (t) := inf u∈R {g(t, q α (u), 0) + n|u|},ḡ n α (t) := sup u∈R {g(t, q α (u), 0) − n|u|}.
Now we show some properties of g n α (t). Firstly, it follows from (3) that dP×dt -a.e., for each u ∈ R, g(t, q α (u), 0) ≥ −ψ α (t) − |g(t, 0, 0)|, which implies that g n α (t) is well defined and for each n ≥ 1,
g n α (t) ≥ inf u∈R {−ψ α (t) − |g(t, 0, 0)| + n|u|} ≥ −ψ α (t) − |g(t, 0, 0)|.
On the other hand, it is clear from the definition that g n α (t) is nondecreasing in n and dP× dt -a.e., g n α (t) ≤ g(t, q α (0), 0) = g(t, 0, 0). Thus, we know that dP×dt -a.e., lim sup n→∞ g n α (t) ≤ g(t, 0, 0) and for each n ≥ 1,
|g n α (t)| ≤ ψ α (t) + |g(t, 0, 0)|.(4)
Moreover, it follows from the definition of g n α (t) and (3) that there exists a sequence {u n } ∞ n=1 such that dP×dt -a.e., g n α (t) ≥ g(t, q α (u n ), 0) + n|u n | − 1 n ≥ −ψ α (t) − |g(t, 0, 0)| + n|u n | − 1 n , which together with (4) implies that u n → 0 as n → ∞, and that dP × dt -a.e., lim inf n→∞ g n α (t) ≥ lim n→∞ g(t, q α (u n )) = g(t, 0, 0). Therefore, dP×dt -a.e., lim n→∞ g n α (t) ↑= g(t, 0, 0). Besides, assumptions (H1) and (H4) indicate that E[|g(t, 0, 0)| 2 ] < ∞ and E[|ψ α (t)| 2 ] < ∞ for dt -a.e. t ∈ [0, T ]. Thus, Lebesgue's dominated convergence theorem together with (4) yields that for dt -a.e. t ∈ [0, T ],
lim n→∞ E[|g n α (t) − g(t, 0, 0)| 2 ] = 0.(5)
In the sequel, using a similar argument to that above we can prove that for dt -a.e. t ∈ [0, T ], P -a.s., |g n α (t)| ≤ ψ α (t) + |g(t, 0, 0)|, ∀n ≥ 1 and lim n→∞ E[|g n α (t) − g(t, 0, 0)| 2 ] = 0.
Finally, by the definitions of g n α (t) and g n α (t) we easily get that dP×dt -a.e., for each n ≥ 1 and y ∈ R, g(t, q α (y), 0) − g(t, 0, 0) ≥ g n α (t) − g(t, 0, 0) − n|y|, g(t, q α (y), 0) − g(t, 0, 0) ≤ḡ n α (t) − g(t, 0, 0) + n|y|. (7) Hence, we can pick the process sequence {(g n α (t)) t∈[0,T ] } ∞ n=1 as follows:
g n α (t) := |g n α (t) − g(t, 0, 0)| + |ḡ n α (t) − g(t, 0, 0)|, and the desired conclusion follows from (4) -(7).
A general representation theorem for generators of BSDEs
In this section we present the representation theorem and its detailed proof. Our proof is partially motivated by Ma and Yao [2010] and Fan, Jiang, and Xu [2011]. An example is also provided to illustrate the novelty of our representation theorem. At the end of this section, a converse comparison theorem for solutions of BSDEs is established.
Theorem 6 (Representation Theorem). Assume that the generator g satisfies (H1) -(H5). Then for each y ∈ R, z ∈ R d , 1 ≤ p < 2 and dt -a.e. t ∈ [0, T ),
g(t, y, z) = L p − lim ε→0 + 1 ε Y t (g, (t + ε) ∧ τ, y + z, B (t+ε)∧τ − B t ) − y , where τ := inf s ≥ t : |B s − B t | + s t |g(r, 0, 0)| 2 dr > 1 ∧ T .
Corollary 7. Assume that the generator g is deterministic and satisfies (H1) -(H5), and τ is defined in Theorem 6. Then for each y ∈ R, z ∈ R d and dt -a.e. t ∈ [0, T ),
g(t, y, z) = lim ε→0 + 1 ε Y t (g, (t + ε) ∧ τ, y + z, B (t+ε)∧τ − B t ) − y .
Remark 8. If for each y ∈ R and z ∈ R d , t → g(t, y, z) is continuous, then the corresponding identities in Theorem 6 and Corollary 7 remain valid for all t ∈ [0, T ).
Next we provide an example of a generator which satisfies the general growth condition (H4) but does not meet (H4').
Example 9. Let T > 0 be a fixed real number. For each (ω, t, y, z) ∈ Ω × [0, T ] × R × R d , define the generator g as follows:
g(t, y, z) := −e y|Bt| + h(|y|) + |z|,
where h(x) := (−x ln x)1 0≤x≤δ + h ′ (δ−)(x − δ) + h(δ) 1 x>δ with δ small enough. It is not very hard to verify that the generator g fulfills assumptions (H1) -(H5) with ρ(·) = h(·), and that it does not satisfy the growth condition (H4'). Hence, the existing works including Jiang [2006], Jia [2010], Fan, Jiang, and Xu [2011] and Zheng and Li [2015] can not imply the conclusion of Theorem 6.
Proof of Theorem 6. Assume that g satisfies (H1), (H2), (H3) with ρ(·), (H4) and (H5) with λ. Fix
(t, y, z) ∈ [0, T ) × R × R d and choose ε > 0 such that ε ≤ T − t.
We define the following stopping time,
τ := inf s ≥ t : |B s − B t | + s t |g(r, 0, 0)| 2 dr > 1 ∧ T.
Proposition 4 indicates that the following BSDE
Y ε s = y + z, B (t+ε)∧τ − B t + t+ε s 1 r<τ g(r, Y ε r , Z ε r ) dr − t+ε s Z ε r , dB r , s ∈ [t, t + ε],
admits a unique solution in S 2 × H 2 , Y s (g, (t + ε) ∧ τ, y + z, B (t+ε)∧τ − B t ), Z s (g, (t + ε) ∧ τ, y + z, B (t+ε)∧τ − B t ) s∈ [t,t+ε] .
For notational simplicity, it is denoted by (Y ε s , Z ε s ) s∈ [t,t+ε] . Next we first show that Y ε · is bounded. It follows from (H3) and (H5) for g that dP×ds -a.e., for each y ∈ R and z ∈ R d , y · 1 s<τ g(s, y, z) ≤ ρ(| y| 2 ) + | y|1 s<τ |g(s, 0, z)| ≤ ρ(| y| 2 ) + λ| y|| z| + | y|1 s<τ |g(s, 0, 0)|, which means that (A) is satisfied by 1 s<τ g with κ(·) = ρ(·), µ = λ and f s = 1 s<τ |g(s, 0, 0)|. Then, according to Lemma 2, Hölder's inequality and the definition of τ we obtain that there exists a constant C > 0 depending on λ and T such that for each t ≤ u ≤ s ≤ t + ε,
E Y ε s 2 F u ≤ 2C(|y| 2 + |z| 2 ) + C t+ε s ρ E Y ε r 2 F u dr + CT E t+ε s 1 r<τ |g(r, 0, 0)| 2 dr F u . ≤ C t+ε s ρ E Y ε r 2 F u dr + 2C(|y| 2 + |z| 2 ) + CT.
Furthermore, since ρ(x) ≤ Ax+A for all x ≥ 0, Gronwall's inequality yields that for each t ≤ u ≤ s ≤ t+ε,
E Y ε s 2 F u ≤ 2C(|y| 2 + |z| 2 ) + CT + CAT e CAT =: M 2 .
Then, substituting u = s in the previous inequality yields that P -a.s., |Y ε s | ≤ M for each s ∈ [t, t + ε]. In what follows, we set, for each s ∈ [t, t + ε],
Y ε s := Y ε s − y − z, B s∧τ − B t , Z ε s := Z ε s − 1 s<τ z.
Then, it is not hard to verify that ( Y ε s , Z ε s ) s∈[t,t+ε] ∈ S 2 × H 2 is a solution of the following BSDE:
Y ε s = t+ε s g(r, Y ε r , Z ε r ) dr − t+ε s Z ε r , dB r , s ∈ [t, t + ε],(8)
where g(s, y, z) := 1 s<τ g(s, y + y + z, B s∧τ − B t , z + z) for each y ∈ R and z ∈ R d . And by the definition of τ we know that P -a.s., for each s ∈ [t, t + ε], | Y ε s | ≤ |Y ε s | + |y| + |z| ≤ M + |y| + |z| =: K.
From the definition of g, it is straightforward to check that g fulfills (H2), (H3) and (H5) with λ.
Moreover, by (H4) and (H5) for g we have that dP×ds -a.e., for each α ≥ 0 and y ∈ R with | y| ≤ α, | g(s, y, 0) − g(s, 0, 0)| ≤ 2λ|z| + |g(s, y + y + z, B s∧τ − B t , 0) − g(s, 0, 0)|
+ |g(s, y + z, B s∧τ − B t , 0) − g(s, 0, 0)| ≤ 2λ|z| + ψ α+|y|+|z| (s) + ψ |y|+|z| (s).(10)
Set ψ α (s) := sup | y|≤ α | g(s, y, 0) − g(s, 0, 0)|. Then
E T 0 | ψ α (r)| 2 dr ≤ 12λ 2 |z| 2 T + 3E T 0 |ψ α+|y|+|z| (r)| 2 + |ψ |y|+|z| (r)| 2 dr < ∞,
which implies that g satisfies (H4) with ψ α (t). Similar arguments as (10) yield that dP×ds -a.e., | g(s, 0, 0)| ≤ λ|z| + ψ |y|+|z| (s) + 1 s<τ |g(s, 0, 0)|, from which it follows that
E T 0 | g(s, 0, 0)| 2 ds ≤ 3λ 2 |z| 2 T + 3E T 0 |ψ |y|+|z| (s)| 2 ds + 3E T 0 |g(s, 0, 0)| 2 ds < ∞.
Hence, (H1) holds true for g. Thus, we have shown that g satisfies (H1) -(H5). Now we can demonstrate the following Lemma 10.
Lemma 10. For dt -a.e. t ∈ [0, T ), we have that
lim ε→0 + 1 ε E sup r∈[t,t+ε] | Y ε r | 2 + t+ε t | Z ε r | 2 dr = 0.
Proof. It follows from (H4) and (H5) for g together with (9) that dP×ds -a.e.,
Y ε s · g(s, Y ε s , Z ε s ) ≤ λ| Y ε s || Z ε s | + | Y ε s | | g(s, Y ε s , 0) − g(s, 0, 0)| + | g(s, 0, 0)| ≤ λ| Y ε s || Z ε s | + | Y ε s | ψ K (s) + | g(s, 0, 0)| ,
which indicates that (A) is fulfilled by g with κ(·) ≡ 0, µ = λ and f s = ψ K (s) + | g(s, 0, 0)|. Then
Lemma 2 and Hölder's inequality contribute to the existence of a constant C ′ > 0 such that
The first term on the right hand side of the previous inequality tends to 0 as ε → 0 + because of Lemma 10.
Concerning the second term, we can deduce from (11) that for each n ≥ 1,
E T 0 | g n K (r)| 2 dr ≤ 32E T 0 | ψ K (r)| 2 + | g(r, 0, 0)| 2 dr < ∞.
Thus, Lemma 1 indicates that for dt -a.e. t ∈ [0, T ), each n ≥ 1 and 1 ≤ p < 2,
lim ε→0 + E 1 ε t+ε t g n K (r) dr p = E [| g n K (t)| p ] .
Note that the right hand side in the previous identity tends to 0 as n → ∞. By sending ε → 0 + and then n → ∞ in (12), we get that for dt -a.e. t ∈ [0, T ), lim ε→0 + E [|M ε t − N ε t | p ] = 0. Now we consider the term (N ε t − g(t, 0, 0)). Taking into account that (H1) holds for g, we can derive from Jensen's inequality and Lemma 1 that for dt -a.e. t ∈ [0, T ) and each 1 ≤ p < 2, as ε → 0 + ,
E [|N ε t − g(t, 0, 0)| p ] ≤ E 1 ε t+ε t | g(r, 0, 0) − g(t, 0, 0)| dr p → 0.
The proof of Theorem 6 is complete.
Remark 11. With analogous techniques and arguments in the proof of Theorem 6, the representation theorem via coupled forward and backward stochastic differential equations (like Jiang [2005]) and in stochastic process spaces (like Fan, Jiang, and Xu [2011]) can also be proved.
With the help of Theorem 6, we can obtain a corresponding converse comparison theorem for solutions of one dimensional BSDEs with weak monotonicity and general growth generators in y. Its proof is omitted since it is classical. Detailed arguments are referred to Jiang [2006].
Theorem 12 (Converse comparison theorem). Let the generators g i (i = 1, 2) satisfy (H1) -(H5). If for each t ∈ [0, T ] and ξ ∈ L 2 (F t ; R), the solutions Y s (g i , t, ξ), Z s (g i , t, ξ) s∈[0,t] of BSDE (g i , t, ξ) satisfy that P -a.s., Y s (g 1 , t, ξ) ≥ Y s (g 2 , t, ξ) for each s ∈ [0, t], then for each y ∈ R and z ∈ R d , g 1 (t, y, z) ≥ g 2 (t, y, z), dP×dt -a.e..
Applications to viscosity soultions of semilinear parabolic PDEs
This section will demonstrate a new application of the representation theorem for generators of BS-DEs. Indeed, utilizing the representation theorem obtained in Theorem 6, we can construct a probabilistic formula for viscosity solutions of a semilinear parabolic PDE of second order.
Let b(t,x) : [0, T ] × R n → R n , σ(t,x) : [0, T ] × R n → RY t,x s = Φ(X t,x T ) + T s g(r, X t,x r , Y t,x r , Z t,x r ) dr − T s Z t,x r , dB r , s ∈ [t, T ],
where Φ : R n → R and g : [0, T ] × R n × R × R d → R are continuous. Assume that for eachx ∈ R n , g(t,x, y, z) satisfies (H3) -(H5), and Φ(·) and g(t, ·, 0, 0) are polynomial growth, i.e., there exist two constants L > 0 and p > 0 such that for each t ∈ [0, T ],x ∈ R n , y ∈ R and z ∈ R d , |Φ(x)| + |g(t,x, 0, 0)| ≤ L(1 + |x| p ).
Note that (H1) -(H2) are fulfilled trivially for g(·, x, ·, ·). Then the previous BSDE admits a unique
solution (Y t,x s , Z t,x s ) s∈[t,T ] ∈ S 2 × H 2 . It is obvious that for each s ∈ [t, T ], Y t,x s is F t s = σ{B r − B t , t ≤ r ≤ s} ∨ N measurable,
where N is the class of the P-null sets of F . In particular, Y t,x t is deterministic.
From the uniqueness for solutions of BSDEs, we also have that Y t,x t+ε = Y t+ε,X t,x t+ε t+ε for any ε > 0.
We define the infinitesimal generator of the Markov process (X t,x s ) s∈[t,T ] as follows,
L s := 1 2 i,j (σσ * )(s,x) ∂ 2 ∂x i ∂x j + i b i (s,x) ∂ ∂x i , s ∈ [0, T ],x ∈ R n , 1 ≤ i, j ≤ n,
where and hereafter σ * represents the transpose of σ. Next we will prove that the function u(t, x) := Y t,x t , (t, x) ∈ [0, T ] × R n , is a viscosity solution of the following system of backward semilinear parabolic PDE of second order:
∂ t u(t, x) + L t u(t, x) + g(t,
x, u(t, x), (σ * ∇u)(t, x)) = 0, (t, x) ∈ [0, T ) × R n ;
u(T, x) = Φ(x), x ∈ R n .(13)
Now we adapt the notion of viscosity solution of PDE (13) from Crandall, Ishii, and Lions [1992] and Pardoux [1999].
Definition 13. Let u ∈ C([0, T ] × R n ; R) satisfy u(T, x) = Φ(x) for x ∈ R n . u is called a viscosity subsolution (resp. supersolution) of PDE (13) if, whenever ϕ ∈ C 1,2 b ([0, T ] × R n ; R), and (t, x) ∈ [0, T ) × R n is a global maximum (resp. minimum) point of u − ϕ, we have ∂ t ϕ(t, x) + L t ϕ(t, x) + g(t, x, u(t, x), (σ * ∇ϕ)(t, x)) ≥ (resp. ≤) 0.
And u is called a viscosity solution of PDE (13) if it is both a viscosity subsolution and supersolution.
Theorem 14. Under the above assumptions, u(t, x) := Y t,x t , (t, x) ∈ [0, T ]×R n , is a continuous function of (t, x) which grows at most polynomially and it is a viscosity solution of PDE (13).
Proof. The continuity is a consequence of the continuity of (X t,x s ) s∈[t,T ] with respect to (t, x) and Lemma 2. The polynomial growth follows from classical moment estimates for (X t,x s ) s∈[t,T ] and the assumptions on the growth of Φ and g. Note that u(T, x) = Φ(x) is trivially satisfied. We only prove that u is a viscosity subsolution since the proof of the other assertion is symmetric.
Take any ϕ ∈ C 1,2 b ([0, T ] × R n ; R) and (t, x) ∈ [0, T ) × R n such that u − ϕ achieves a global maximum at (t, x). Without loss of generality, we set u(t, x) = ϕ(t, x). For 0 ≤ t ≤ t + ε ≤ T with ε small enough and x ∈ R n , Proposition 4 shows that u(t, x) = u(t + ε, X t,x t+ε ) + t+ε t g(r, X t,x r , Y t,x r , Z t,x r ) dr − t+ε t Z t,x r , dB r .
Let (Y ε s , Z ε s ) s∈[t,t+ε] ∈ S 2 × H 2 denote the unique solution of the following BSDE:
n×d be two measurable continuous functions which are Lipschitz continuous inx uniformly with respect to t. Then for each given t ∈ [0, T ] andx ∈ R n , the following SDE admits a unique solution (X t,x s ) s∈[t,T ] in S 2 , ) dB r , s ∈ [t, T ]. Now we consider the following BSDE:X t,x
s = x +
s
t
b(r, X t,x
r ) dr +
s
t
σ(r, X t,x
r
Thus, (H1) and (H4) for g and the absolute continuity of integrals yield the desired result.Come back to the proof of Theorem 6. Taking s = t and then the conditional expectation with respect to F t in both sides of BSDE (8) result in the following identity, P -a.s.,Next we set,Thus, it holds thatThen it is sufficient to prove that (M ε t − N ε t ) and (N ε t − g(t, 0, 0)) tend to 0 in L p (1 ≤ p < 2) sense for dt -a.e. t ∈ [0, T ) as ε → 0 + , respectively.We first treat the term (M ε t − N ε t ). Since g satisfies (H1), (H2) and (H4), we can deduce from Proposition 5 and (9) that there exists a process sequence {( g n K (t)) t∈[0,T ] } ∞ n=1 such that for dt -a.e. t ∈ [0, T ], lim n→∞ E[| g n K (t)| 2 ] = 0, and dP×ds -a.e., for each n ≥ 1,Then Jesen's and Hölder's inequalities yield that for dt -a.e. t ∈ [0, T ), n ≥ 1 and 1 ≤ p < 2,Then comparison theorem (Theorem 3 inFan [2015]) provides that u(t, x) = ϕ(t, x) ≤ Y ε t . Moreover, Itô's formula to ϕ(r, X t,x r ) arrives atThus, we obtain that, settingŶ εwhere the generator is defined as follows, for each y ∈ R and z ∈ R d , G(r, X t,x r , y, z) := (∂ r ϕ + L r ϕ) (r, X t,x r ) + g r, X t,x r , y + ϕ(r, X t,x r ), z + (σ * ∇ϕ)(r, X t,x r ) .Note that G(t, X t,x t , y, z) = G(t, x, y, z) is deterministic and continuous in t. Theorem 6, Corollary 7 and Remark 8 yield thatwhich together with the factŶ ε t ≥ 0 implies the desired result.Remark 15. (i) The required conditions of g in Theorem 14 are weaker than some existing results includingPeng [1991],Pardoux and Peng [1992]andPardoux [1999]. And, the representation theorem approach used in Theorem 14 overcomes the difficulty that the strict comparison theorem for solution of BSDEs employed in Theorem 3.2 ofPardoux [1999]does not hold in our framework.(ii) From the proof procedure of Theorem 14 we can observe that the probabilistic formula for semilinear PDEs holds as soon as the representation theorem for generators holds, where the latter one is determined by the conditions of the original generator g. From this point of view, the problem of a probabilistic formula for viscosity solutions of semilinear PDEs can be transformed to a representation problem for the generator of a BSDE.(iii) Analogous arguments to that in Theorem 14 can be extended easily to systems of multidimensional semilinear second order PDEs of both parabolic and elliptic types, more detailed proof procedures are referred toPardoux [1999]. However, for the notion of viscosity solutions to make sense, an extra assumption on g is needed, i.e., the ith coordinate of g depends only on the ith row of z.
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|
[] |
[
"epl draft η-mesic nuclei in relativistic mean-field theory",
"epl draft η-mesic nuclei in relativistic mean-field theory"
] |
[
"C Y Song \nDepartment of Physics\nNankai University\n300071TianjinChina\n",
"X H Zhong \nInstitute of High Energy Physics\nChinese Academy of Sciences\n100039BeijingChina\n",
"L Li \nDepartment of Physics\nNankai University\n300071TianjinChina\n",
"P Z Ning \nDepartment of Physics\nNankai University\n300071TianjinChina\n"
] |
[
"Department of Physics\nNankai University\n300071TianjinChina",
"Institute of High Energy Physics\nChinese Academy of Sciences\n100039BeijingChina",
"Department of Physics\nNankai University\n300071TianjinChina",
"Department of Physics\nNankai University\n300071TianjinChina"
] |
[] |
With the η-nucleon (ηN ) interaction Lagrangian deduced from chiral perturbation theory, we study the possible η-mesic nuclei in the framework of relativistic mean-field theory. The η single-particle energies are sensitive to the ηN scattering length, and increase monotonically with the nucleon number A. If the scattering length is in the range of a ηN = 0.75 ∼ 1.05 fm and the imaginary potential V0 ∼ 15 MeV, some discrete states of 12 η C, 16 η O and 20 η Ne should be identified in experiments. However, when the scattering length a ηN < 0.5 fm, or the imaginary potential V0 > 30 MeV, no discrete η meson bound states could be observed in experiments.
|
10.1209/0295-5075/81/42002
|
[
"https://arxiv.org/pdf/0802.3738v2.pdf"
] | 15,977,679 |
0802.3738
|
c96a6c5919d855dc2895f1180487085243760d9f
|
epl draft η-mesic nuclei in relativistic mean-field theory
27 Feb 2008
C Y Song
Department of Physics
Nankai University
300071TianjinChina
X H Zhong
Institute of High Energy Physics
Chinese Academy of Sciences
100039BeijingChina
L Li
Department of Physics
Nankai University
300071TianjinChina
P Z Ning
Department of Physics
Nankai University
300071TianjinChina
epl draft η-mesic nuclei in relativistic mean-field theory
27 Feb 2008PACS 21.10.Pc -First pacs description PACS 21.30.Fe -Second pacs description
With the η-nucleon (ηN ) interaction Lagrangian deduced from chiral perturbation theory, we study the possible η-mesic nuclei in the framework of relativistic mean-field theory. The η single-particle energies are sensitive to the ηN scattering length, and increase monotonically with the nucleon number A. If the scattering length is in the range of a ηN = 0.75 ∼ 1.05 fm and the imaginary potential V0 ∼ 15 MeV, some discrete states of 12 η C, 16 η O and 20 η Ne should be identified in experiments. However, when the scattering length a ηN < 0.5 fm, or the imaginary potential V0 > 30 MeV, no discrete η meson bound states could be observed in experiments.
Introduction. -Since the η-mesic nuclei were predicted by Haider et al., [1,2] the topics on the ηN interactions and η-mesic nuclei are studied extensively. Although all of the theory models predict that the interaction between η-meson and nucleon is attractive, its strength (i.e. the predicted η nuclear potential ) has strong model dependence, spans from about -20 MeV to -100 MeV [3][4][5][6].
Because of the uncertainties of the η nuclear potentials, the predictions of the η-mesic nuclei are very different in different models [1,[7][8][9][10][11][12][13][14][15][16][17][18][19]. For example, some models predicted that η-mesic nuclei could be found in the nuclei with nucleon number A > 10 [1], while some other models predicted that they could be found in very light nuclei with A ≥ 2 [15][16][17].
Experimentally, several experiments bad been performed [20,21], but no evidence of η-mesic nuclei was found. Recently, Sokol et al. [22] claimed that they observed a η-mesic nucleus, 11 η C, by measuring the invariant mass of correlated π + n pairs in a photo-mesonic reaction. And more recently, M. Pfeiffer et al. [23] also claimed they observed some information of a η-mesic nucleus, 3 η He. To get a further understanding on η-mesic nuclei, more studies, both in theory and experiments, are needed.
In our previous work, the ηN interaction Lagrangian had been derived from the chiral perturbation theory (ChPT) [24], in which the off shell term has been related with the ηN scattering length by a off-shell term parameter κ. Combining this ηN Lagrangian with the Lagrangian for nucleons in relativistic mean field theory (RMF), we have obtained the equations of motion for nu-cleons and mesons. By solving the these equations selfconsistently in RMF, the static properties of η-mesic nuclei, such as the single-particle energy spectra, are gotten. Similar method can be found in the study of kaonic nuclei as well [25,26]. In the RMF calculations, with the existing data of the scattering lengths, the lower limits of the 1s state single-particle η binding energies are 9 ± 7 MeV, and the upper limits are 70 ± 10 MeV. With large scattering length a ηN = 0.75 ∼ 1.05 fm and small imaginary potential V 0 ∼ 15 MeV, the discrete bound states of 12 η C, 16 η O and 20 η Ne may be identified in experiments. This work is organized as follows. In the subsequent section, the Lagrangian density is given, the equations of motion for nucleons and the meson fields σ, ω, ρ, and η are deduced, the imaginary part of the self-energies are introduced. We then present our results and discussions in Sec. III. Finally a summary is given in Sec. IV.
Framework. -Lagrangian and equations of motion.
In relativistic mean field theory, the standard Lagrangian density for an ordinary nucleus can be written as [27,28]
L 0 = L Dirac + L σ + L ω + L ρ + L A ,(1)
where
L Dirac =Ψ N (iγ µ ∂ µ − M N )Ψ N ,(2)L σ = 1 2 ∂ µ σ∂ µ σ − 1 2 m 2 σ σ 2 − g σNΨN σΨ N − 1 3 g 2 σ 3 − 1 4 g 3 σ 4 ,(3)
p-1 C. Y. Song et al.
L ω = − 1 4 F µν F µν + 1 2 m 2 ω ω µ ω µ −g ωNΨN γ µ Ψ N ω µ ,(4)L ρ = − 1 4 G µν G µν + 1 2 m 2 ρ ρ µ · ρ µ −g ρNΨN ρ µ · IΨ N ,(5)L A = − 1 4 H µν H µν − eΨ N γ µ I c A µ Ψ N ,(6)
with
F µν = ∂ ν ω µ − ∂ µ ω ν ,(7)G µν = ∂ ν ρ µ − ∂ µ ρ ν ,(8)H µν = ∂ ν A µ − ∂ µ A ν .(9)
In the above equations, the meson fields are denoted by σ, ω µ , and ρ µ , with masses m σ , m ω , m ρ , respectively. Ψ N is the nucleon field with corresponding mass M N . A µ is the electromagnetic field. g σN , g ωN , and g ρN are, respectively, the σ-N , ω-N , and ρ-N coupling constants. I c = (1 + τ 3 )/2 is the Coulomb interaction operator with τ 3 being the third component of the isospin Pauli matrices for nucleons. I is the nucleon isospin operator. In the calculations, we adopt the NL-SH parameter set (see Tab. 1) [29], which describes the properties of finite nuclei reasonably.
For an η-nucleus system, another Lagrangian density L η describing the ηN interactions should be added to L 0 . In this work, the Lagrangian density L η is adopted the one deduced from the heavy baryon chiral perturbation theory up to the next-to-leading-order terms [24], which is given by
L η = 1 2 ∂ µ η∂ µ η − 1 2 m 2 η − Σ ηN f 2 πΨ N Ψ N η 2 + 1 2 · κ f 2 πΨ N Ψ N ∂ µ η∂ µ η,(10)
where m η = 547.311 MeV corresponds to the mass of ηmeson, Σ ηN is the ηN sigma term, κ is a parameter of the "off-shell" term. f π ≃ 93 MeV is the pseudoscalar meson decay constants. According to our previous work [24], we set Σ ηN = 280 MeV. The "off-shell" term parameter κ was determined by the ηN scattering length a ηN ,
κ = 4πf 2 π 1 m 2 η + 1 m η M N a ηN − Σ ηN m 2 η .(11)
The scattering length has large uncertainties, which scatters in a large range a ηN = 0.2 ∼ 1.1 fm [30][31][32][33][34]. Thus, the corresponding value of κ is in the range of (−0.13 ∼ 0.40) fm. It should be emphasized that in the ChPT the contributions of N * (1535) can not be seen directly, however, its contributions are included by the scattering length, which relates to the resonance N * (1535) directly.
In the mean field approximation, the meson-fields σ, ω µ , and ρ µ , and the photons A µ are replaced with their mean values, σ , ω µ , ρ µ and A µ , respectively. For a spherical nucleus, only the mean values of the time components ω 0 , ρ 0 and A 0 remain, which are denoted by ω 0 , and ρ 0 , and A 0 respectively. From the Lagrangian for the ηnucleus system, the equations of motion for nucleons, ω, σ, ρ, and photons are deduced, which are given by
α · P + β[M N + S(r)] + V (r) Ψ N = EΨ N , (12) −∇ 2 + m 2 σ σ 0 = −g σNΨN Ψ N − g 2 σ 2 0 − g 3 σ 3 0 , (13) −∇ 2 + m 2 ω ω 0 = g ωNΨN γ 0 Ψ N ,(14)−∇ 2 + m 2 ρ ρ 0 = g ρNΨN γ 0 IΨ N ,(15)−∇ 2 A 0 = eΨ N γ 0 I c Ψ N ,(16)
with
S(r) = g σN σ 0 − 1 2 · Σ ηN f 2 π η 2 − 1 2 · κ f 2 π ∂ µ η∂ µ η, (17) V (r) = g ωN ω 0 + g ρN τ 3 ρ 0 + eI c A 0 .(18)
In the calculation the spacial terms of the last term in Eq.(17) are neglected for a simplicity. And the equation of motion for η meson is derived as
− ∇ 2 + (m 2 η − E 2 ) + Π η = 0,(19)
with
Π = − 1 f 2 π (1 + κ f 2 π ̺ s ) (κm 2 η + Σ ηN )̺ s .(20)
In the above equations, E is the nucleon single-particle energy, E is the single-particle energy for η meson, ̺ s = Ψ N Ψ N is the scalar density of nucleons, and Π is the selfenergy of the η meson in the nucleus.
Imaginary potential. Within the framework of RMF model, there is only a real part for the self-energy of the η meson in the nucleus. Considering there are strong absorption for the η-mesons in a nucleus, in the realistic calculations the imaginary part of the self-energy should be included. Thus, as done in Refs. [25,26], we assume a specific form for the self-energy:
Π = Π + i −2(ReE)f V 0 ̺ ̺ 0 .(21)
The imaginary part of the potential ImU is adopted the simple "t̺" form, namely, ImU = −f V 0 ̺/̺ 0 . f is a suppression factor, which will be discussed later. V 0 is the p-2 η mesic nuclei in relativistic mean-field theory imaginary potential depth at normal nuclear density ̺ 0 , which has strong model dependence. The shallowest value of V 0 ∼ 10 MeV is given by fitting larger scattering length using the "t̺" form [17]. While Waas and Weise studied the s-wave interactions of η-meson in nuclear medium, and got V 0 ≃ 22 MeV [3]. Inoue and Oset also obtained V 0 ≃ 29 MeV with chiral unitary approach [6]. Using the chiral doublet model to incorporate the medium effects of the N * (1535) resonance, Jido and Nagahiro et al. predicted the largest imaginary potential depth V 0 ≃ 50 MeV [8,14,19]. Chiang et al. [4] suggested the imaginary potential depth in the range of (12 ∼ 49) MeV by assuming that the mass of the N * (1535) did not change in the medium. Thus, in the present work, we set the imaginary potential depth V 0 in the range of 10 ∼ 50 MeV to cover all the possible ranges.
Considering the decay channels should be reduced for the η-meson being bound in a nucleus, the suppression factor, f , is introduced to multiply the imaginary part to decrease the imaginary potentials (widths) 1 . This method has been used to calculate the width of kaonic nuclei [25,26,35]. There are two main decay channels for η-mesic nuclei. One is the mesonic decay channel, ηN → πN . The corresponding suppression factor is given by [25,26,35]
f 1 = M 3 01 M 3 1 [M 2 1 − M 2 + ][M 2 1 − M 2 − ] [M 2 01 − M 2 + ][M 2 01 − M 2 − ] Θ(M 1 − M + ),(22)f 2 = M 3 02 M 3 2 [M 2 2 − 4M 2 N ]M 2 2 [M 2 02 − 4M 2 N ]M 2 02 Θ(M 2 − 2M N ),(23)
where M 02 = m η + 2M N , M 2 = ReE + 2M N correspond to the energies of the free system and the bound system of ηN N , respectively. The mesonic decay and non-mesonic decay are studied in Ref. [36], the ratio for the two decay modes are about 90% and 10%, respectively. Thus the suppression factor f can be written as
f = 0.9f 1 + 0.1f 2 .(24)
Single-particle η binding energy and width. Then the modified Klein-Gordon equation can be expressed as,
− ∇ 2 + (m 2 η − E 2 ) + Π η = 0.(25)
The complex eigenenergy is
E = −B s,p η + m η − iΓ/2,(26)
1 The energy of a free system is larger than the energy of a bound system, thus, for a decay channel its phase space should be suppressed for a bound system. As an example, we can see Eq. (22) and Eq.(23). where the real part corresponds to the single-particle η binding energy, which is defined as
B s,p η = m η − ReE,(27)
and the imaginary part of the complex eigenenergy corresponds to the width
Γ = −2ImE.(28)
Solving the equations (12) - (16) and Eq.(25) selfconsistently, we can obtain the single-particle energy spectra and widths of η mesic nuclei.
Results and discussions. -In this section, the single-particle energy spectra and the widths of the possible η-mesic nuclei, such as 12 For a ηN = 0.20 fm (see Tab. 2), it is found that the imaginary potential depth V 0 has effects on the lighter nuclei to form η quasi-bound states. For example, with V 0 = 15 MeV, quasi-bound states can be found with nucleon number A ≥ 20, however, they are only found in the A ≥ 36 nuclei with V 0 = 50 MeV. The 1s state singleparticle binding energies are (9 ± 7) MeV, increasing with the nucleon number. The widths are much larger than the single-particle binding energies even we use the smallest V 0 = 15 MeV. Thus, no η-mesic nuclei can be observed in experiments.
η C, 16 η O, 20 η Ne, 24 η Mg, 28 η Si,
For a ηN = 0.50 fm (see Tab. 3) the ground state singleparticle binding energies are (26 ± 10) MeV. If the imaginary part V 0 = 15 MeV, the decay widths are comparable with the the binding energies, thus, in this case the η-mesic nuclei maybe observed in the light nuclei when a ηN ≥ 0.50 fm. On the contrary, when a ηN < 0.50 fm, no η-mesic nuclei can be observed in experiments.
For a ηN = (0.75 ∼ 1.05) fm (see Tab. 4), the 1s state single-particle binding energies are in the range of (48 ± 10 ∼ 70 ± 10) MeV, and those of 1p states are in the region of (15 ± 12 ∼ 38 ± 21) MeV, increasing monotonically with the nucleon number A. The separations of the single-particle η binding energies between the 1p and 1s states are on the magnitude of (30 ± 10 ∼ 35 ± 13) MeV, decreasing with the increment of the nucleon number in general. When V 0 ∼ 15 MeV, the sum of the half widths of the 1s and 1p states are narrower than the separations of the single-particle η binding energies between 1s and 1p states for C, O, Ne, which implies that some discrete states should be identified in experiments for these nuclei. However, if V 0 > 30 MeV no η mesic nuclei could be observed in experiments according to our calculations. From Tab. 3 and Tab. 4, it is found that the widths of the 1s states are in the ranges of (28 ± 7 ∼ 104 ± 14) MeV and those of 1p states are (26 ± 3 ∼ 89 ± 7) MeV, respectively, for V 0 = (15 ∼ 50) MeV. The imaginary potential depth V 0 has slight effects on the values of the single-particle energy B s,p η , the effects decrease with the increment of V 0 . For example, if we change V 0 from 15 MeV to 50 MeV, the single-particle energies decrease about (3 ∼ 8) MeV for both 1s and 1p states .
Summary. -Some possible η mesic nuclei from 12 η C to 44 η Ti have been studied in RMF. The η single-particle energy is sensitive to the ηN scattering length (i.e. "offshell" term parameter κ). In the whole possible range for the scattering length, the lower limits of the 1s state single-particle η binding energies are (9 ± 7) MeV, and the upper limits are (70 ± 10) MeV. The widths of 1s states are in the ranges of (28 ± 7 ∼ 104 ± 14) MeV and those of 1p states are (26 ± 3 ∼ 89 ± 7) MeV.
When the scattering length a ηN = (0.75 ∼ 1.05) fm, and the imaginary potential V 0 ≤ 15 MeV, the sum of the half widths of the 1s and 1p states for 12 η C, 16 η O and 20 η Ne are smaller than the separations of the single-particle binding energies between the two low-lying states of these η-mesic nuclei, which implies that discrete η meson bound states may be identified in experiments in these nuclei. However, when the scattering length a ηN < 0.5 fm, or the imaginary potential V 0 > 30 MeV, no discrete η meson bound states could be identified in experiments. Finally, we should point out that it is an attempt to study the η mesic nuclei with the ηN interaction deduced from ChPT. In our method the contributions of resonances, such as N * (1535), are only included indirectly by the ηN scattering length, which relates to the resonances directly. The imaginary potential is phenomenologically introduced in this paper, which has a large uncertainty. Thus, more realistic ηN interaction which introduces the resonances naturally, and more fundamental imaginary potential should be pursued in the future work. * * *
The authors thank N. Auerbach for many good suggestions. This work was supported, in part, by the Natural Science Foundation of China (grants 10575054 and 10775145), China Postdoctoral Science Foundation, and K. C. Wong Education Foundation, Hong Kong. p-4 η mesic nuclei in relativistic mean-field theory
where M 01 = m η +M N , M + = m π +M N , M − = M N −m π and M 1 = ReE + M N is the energy of the bound system ηN . The other channel is the non-mesonic decay channel, ηN N → N N , and the corresponding suppression factor is[25,26,35]
32 η S, 36 η
3236Ar, 40 η Ca and 44 η Ti are calculated in RMF. For the uncertainties of the parameter κ (i.e. the scattering length a ηN ), which give large uncertainties for the η nuclear potentials, we choose four values of κ (−0.13, 0.04, 0.19 and 0.40 fm corresponding to a ηN = 0.20, 0.50, 0.75, 1.05 fm) to cover all the possible scattering lengths. In each case, we also suppose V 0 = 15, 30 and 50 MeV, respectively, which can cover all the possible ranges of the imaginary potential. The results, including the single-particle η binding energies (B s,p η ) and the widths (Γ), for κ = −0.13 fm (a ηN = 0.20 fm) and κ = 0.04 fm (a ηN = 0.50 fm) are shown in Tab. 2 and Tab. 3, respectively. And the results for κ = 0.19 , 0.40 fm (a ηN = 0.75 , 1.05 fm) are listed in Tab. 4.
η 44 η
44Ca 1s 35.1 28.6 33.8 57.5 31.1 96.8 1p 16.8 24.8 14.7 50.7 10.9 87.3 Ti 1s 36.0 28.0 34.8 56.9 32.4 95.5 1p 18.8 25.7 16.9 51.7 13.4 88.7
20 η 24 η 36 η 40 η 44 η
202436404443.2 29.6 41.6 59.8 38.2 100.8 65.4 31.4 64.1 62.9 61.0 104.9 1p 13.2 21.0 10.8 43.0 6.0 75.0 31.1 24.4 29.3 49.2 25.6 83.1 Ne 1s 46.3 27.6 45.1 55.3 42.4 92.7 69.1 26.9 68.1 53.8 66.0 89.8 1p 18.6 22.5 16.6 45.7 12.2 80.4 37.8 23.5 36.5 47.2 33.4 81.4 Mg 1s 50.9 28.9 49.8 57.9 47.2 96.9 74.7 28.5 73.7 57.0 71.5 95.0 1p 25.2 24.8 23.4 50.7 19.5 87.4 46.0 26.2 44.7 52.6 41.8 88.3 28 η Si 1s 55.1 30.5 53.9 61.2 51.3 102.2 79.7 30.1 78.7 60.3 76.4 100.3 1p 31.0 27.4 29.3 55.4 25.6 93.7 53.1 28.7 51.8 57.5 48.8 96.3 32 η S 1s 57.4 32.2 56.1 63.7 53.0 106.4 82.9 31.9 81.7 63.8 78.8 106.1 1p 30.9 26.3 29.2 53.0 25.6 89.7 52.8 27.2 51.5 54.6 48.6 91.3 Ar 1s 56.2 30.7 55.0 61.6 52.2 102.8 81.2 29.6 80.1 59.1 77.8 98.3 1p 32.5 26.2 30.9 52.7 27.6 89.0 54.3 26.6 53.1 53.3 50.5 89.1 Ca 1s 55.4 29.5 54.3 59.0 51.8 98.5 79.8 28.3 78.9 56.5 76.8 94.0 1p 34.3 26.1 33.0 52.6 30.1 88.7 56.2 26.4 55.2 52.8 52.9 88.3 Ti 1s 56.3 28.7 55.3 58.1 53.1 97.0 80.7 28.1 79.9 56.3 78.0 93.6 1p 36.8 26.8 35.5 53.3 32.8 89.6 59.0 26.9 58.1 54.0 55.8 90.1
Table 1 :
1Parameters used in the present calculations.M N
m σ
m ω
m ρ
939.0 526.059 783.0
763.0
g σN
g ωN
g ρN
g 3
g 2
10.444 12.945 8.766 -15.8337 -6.9099 f m −1
Table 2 :
2The single-particle η binding energies, B s,p η = mη − ReE and the widths, Γ, (both in MeV), in various nuclei for κ=-0.13 fm (a ηN = 0.20 fm), where the complex eigenenergies are, E = −B s,p η + mη − iΓ/2.V 0 = 15
V 0 = 30
V 0 = 50
B s,p
η
Γ
B s,p
η
Γ
B s,p
η
Γ
16
η O
1s
-
-
-
20
η Ne 1s 4.1
21.2
-
-
24
η Mg 1s 6.1
23.8 2.8
51.3
-
28
η Si
1s 7.9
26.0 4.9
55.3
-
32
η S
1s 8.5
26.5 5.3
56.0
-
36
η Ar 1s 8.9
25.8 6.1
54.3 1.8
95.3
40
η Ca 1s 9.2
25.4 6.8
53.1 3.2
92.7
44
η Ti
1s 10.0 25.8 7.7
53.7 4.4
93.2
132
η Xe 1s 15.2 27.8 14.2 55.9 12.1 94.0
1p
7.3
26.6
5.8
54.1
3.0
92.1
208
η Pb 1s 16.3 28.4 15.6 56.8 13.9 94.6
1p
9.7
28.4
8.9
56.9
7.1
95.0
Table 3 :
3The single-particle η binding energies, B s,p η = mη − ReE and the widths, Γ, (both in MeV), in various nuclei for κ=0.04 fm (a ηN = 0.50 fm), where the complex eigenenergies are, E = −B s,p η + mη − iΓ/2.V 0 = 15
V 0 = 30
V 0 = 50
B s,p
η
Γ
B s,p
η
Γ
B s,p
η
Γ
12
η C
1s 26.2 30.7 23.2 64.0 17.2 109.9
1p
-
-
-
16
η O
1s 24.8 27.2 22.9 55.3 18.9 94.5
1p
-
-
-
20
η Ne 1s 27.5 26.6 25.8 53.8 22.6 91.5
1p
-
-
-
24
η Mg 1s 31.2 28.5 29.6 57.4 26.5 97.1
1p 8.8
22.2 5.8
46.7
-
28
η Si
1s 34.5 29.5 33.1 59.4 30.1 100.1
1p 13.1 25.4 10.4 52.3 5.4
92.2
32
η S
1s 36.1 30.6 34.4 61.6 30.9 103.7
1p 13.4 24.3 10.8 49.9 5.9
86.9
Table 4 :
4The single-particle η binding energies, B s,p η = mη − ReE and the widths, Γ, (both in MeV), in various nuclei for κ=0.19 fm (a ηN = 0.75 fm) and κ=0.40 fm (a ηN = 1.05 fm), where the complex eigenenergies are, E = −B s,p η + mη − iΓ/2.κ=0.19 fm(a ηN = 0.75 fm)
κ=0.40 fm(a ηN = 1.05 fm)
V 0 = 15
V 0 = 30
V 0 = 50
V 0 = 15
V 0 = 30
V 0 = 50
B s,p
η
Γ
B s,p
η
Γ
B s,p
η
Γ
B s,p
η
Γ
B s,p
η
Γ
B s,p
η
Γ
12
η C
1s 46.2 34.9 43.8 70.3 38.6 118.2 69.8 35.5 67.8 70.9 63.3 118.2
1p 6.6
19.4 3.2
40.5
-
-
23.4 23.7 21.1 47.8 15.7 83.3
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|
[] |
[] |
[
"Kurt Hinterbichler \nCenter for Particle Cosmology\nDepartment of Physics and Astronomy\nUniversity of Pennsylvania\n19104PhiladelphiaPennsylvaniaUSA\n",
"James Stokes \nCenter for Particle Cosmology\nDepartment of Physics and Astronomy\nUniversity of Pennsylvania\n19104PhiladelphiaPennsylvaniaUSA\n",
"Mark Trodden \nCenter for Particle Cosmology\nDepartment of Physics and Astronomy\nUniversity of Pennsylvania\n19104PhiladelphiaPennsylvaniaUSA\n"
] |
[
"Center for Particle Cosmology\nDepartment of Physics and Astronomy\nUniversity of Pennsylvania\n19104PhiladelphiaPennsylvaniaUSA",
"Center for Particle Cosmology\nDepartment of Physics and Astronomy\nUniversity of Pennsylvania\n19104PhiladelphiaPennsylvaniaUSA",
"Center for Particle Cosmology\nDepartment of Physics and Astronomy\nUniversity of Pennsylvania\n19104PhiladelphiaPennsylvaniaUSA"
] |
[] |
We study the background cosmology of two extensions of dRGT massive gravity. The first is variable mass massive gravity, where the fixed graviton mass of dRGT is replaced by the expectation value of a scalar field. We ask whether self-inflation can be driven by the self-accelerated branch of this theory, and we find that, while such solutions can exist for a short period, they cannot be sustained for a cosmologically useful time. Furthermore, we demonstrate that there generally exist future curvature singularities of the "big brake" form in cosmological solutions to these theories. The second extension is the covariant coupling of galileons to massive gravity. We find that, as in pure dRGT gravity, flat FRW solutions do not exist. Open FRW solutions do exist -they consist of a branch of self-accelerating solutions that are identical to those of dRGT, and a new second branch of solutions which do not appear in dRGT.
|
10.1016/j.physletb.2013.07.009
|
[
"https://arxiv.org/pdf/1301.4993v3.pdf"
] | 118,504,427 |
1301.4993
|
0bf551695a043626cf78bc0e1be3b78b6d4b4b5d
|
16 Dec 2013 (Dated: December 17, 2013)
Kurt Hinterbichler
Center for Particle Cosmology
Department of Physics and Astronomy
University of Pennsylvania
19104PhiladelphiaPennsylvaniaUSA
James Stokes
Center for Particle Cosmology
Department of Physics and Astronomy
University of Pennsylvania
19104PhiladelphiaPennsylvaniaUSA
Mark Trodden
Center for Particle Cosmology
Department of Physics and Astronomy
University of Pennsylvania
19104PhiladelphiaPennsylvaniaUSA
16 Dec 2013 (Dated: December 17, 2013)Cosmologies of extended massive gravity
We study the background cosmology of two extensions of dRGT massive gravity. The first is variable mass massive gravity, where the fixed graviton mass of dRGT is replaced by the expectation value of a scalar field. We ask whether self-inflation can be driven by the self-accelerated branch of this theory, and we find that, while such solutions can exist for a short period, they cannot be sustained for a cosmologically useful time. Furthermore, we demonstrate that there generally exist future curvature singularities of the "big brake" form in cosmological solutions to these theories. The second extension is the covariant coupling of galileons to massive gravity. We find that, as in pure dRGT gravity, flat FRW solutions do not exist. Open FRW solutions do exist -they consist of a branch of self-accelerating solutions that are identical to those of dRGT, and a new second branch of solutions which do not appear in dRGT.
INTRODUCTION AND OUTLINE
An interacting theory of a massive graviton, free of the Boulware-Deser mode [1], has recently been discovered [2,3] (the dRGT theory, see [4] for a review), allowing for the possibility of addressing questions of interest in cosmology. Pure dRGT massive gravity admits self-accelerating solutions [5][6][7][8][9][10][11], in which the de Sitter Hubble factor is of order the mass of the graviton. Since having a light graviton is technically natural [12,13], such a solution is of great interest in the late-time universe to account for cosmic acceleration.
A natural question is whether a similar phenomenon might drive inflation in the early universe. To use the self-accelerating solution of massive gravity for inflation (i.e. "self-inflation"), the graviton mass would have to be of order the Hubble scale during inflation. Yet, we know that the current graviton mass cannot be much larger than the Hubble scale today [14].
Thus, for self-inflation to be possible, the graviton mass must change in time. One idea of how to realize this is to promote the graviton mass to a scalar field, Φ, which has its own dynamics and can roll [9,17]. The expectation value (VEV) of Φ then sets the mass of the graviton. We can imagine that at early times Φ has a large VEV, so that the graviton is very massive, and the universe self-inflates with a large Hubble constant. Then, at late times, Φ rolls to a smaller VEV, self-inflation ends and the graviton mass attains a small value consistent with present day measurements.
In this letter, we will see that, in practice, such an inflation-like implementation of massive gravity is difficult to achieve in this model. Pure dRGT theory has a constraint, stemming from the Bianchi identity, which forbids standard FRW evolution in the flat slicing [9] (the self-accelerating solutions are found in other slicings). There appears an analogous constraint in the variable mass theory, and this constraint, while it no longer forbids flat FRW solutions, implies that self-inflation cannot be sustained for a cosmologically relevant length of time. In addition, we show that non-inflationary cosmological solutions to this theory may exhibit future curvature singularities of the "big brake" type.
In the second half of this letter (which can be read independently from the first), we consider the covariant galileon extension of massive gravity introduced in [15]. This theory has a new scalar degree of freedom π, which describes brane bending in an additional spatial dimension. (Unlike the variable mass theory, π here does not act to set the graviton mass, which is fixed. So we are not interested in self-inflation here, but are just exploring the basic cosmological equations.)
We derive the background cosmological equations for this theory, and find that the presence of the scalar leads to a more complicated constraint than in pure dRGT. We discuss the possible solutions in the case of zero and negative spatial curvature. We find that, as in pure dRGT theory, this constraint forbids flat FRW solutions. For an open FRW ansatz, however, solutions can exist and they come in two branches. The first branch consists of self-accelerating solutions that are identical to the selfaccelerating solutions of pure dRGT theory. The second branch consists of novel solutions which are not found in pure massive gravity.
VARIABLE MASS MASSIVE GRAVITY
We start with variable mass massive gravity. This is dRGT theory in which the graviton mass squared is promoted to a scalar field Φ,
S = S EH + S mass + S Φ ,(1)
where
S EH = 1 2 M 2 P d 4 x √ −g R ,(2)S mass = M 2 P d 4 x √ −g Φ (L 2 + α 3 L 3 + α 4 L 4 ) , (3) S Φ = − d 4 x √ −g 1 2 g(Φ)(∂Φ) 2 + V (Φ) . (4)
Here α 3 , α 4 are the two free parameters of dRGT theory. We have allowed for an arbitrary kinetic function g(Φ) and potential V (Φ), so that there is no loss of generality in the scalar sector. The mass term consists of the ghostfree combinations [3],
L 2 = 1 2 [K] 2 − [K 2 ] , L 3 = 1 6 [K] 3 − 3[K][K 2 ] + 2[K 3 ] , L 4 = 1 24 [K] 4 − 6[K] 2 [K 2 ] + 3[K 2 ] 2 + 8[K][K 3 ] − 6[K 4 ] ,(5)
where K µ ν = δ µ ν − √ g µσ η σν , η µν is the non-dynamical fiducial metric which we have taken to be Minkowski, and the square brackets are traces. To work in the gauge invariant formalism, we introduce four Stückelberg fields φ a through the replacement η µν → ∂ µ φ a ∂ µ φ b η ab . Variable mass massive gravity was first considered in [9], and further studied in [17][18][19][20] (see also [16] for a more symmetric scalar extension of dRGT). dRGT gravity has been demonstrated to be ghost-free through a variety of different approaches [21][22][23][24][25], and the introduction of the scalar field does not introduce any new Boulware-Deser like ghost degrees of freedom into the system [17].
For cosmological applications we take a Friedmann, Robertson-Walker (FRW) ansatz for the metric, so that
ds 2 = −N 2 (t)dt 2 + a 2 (t)Ω ij dx i dx j ,(6)
where
Ω ij = δ ij + κ 1 − κr 2 x i x j(7)
is the line element for a maximally symmetric 3-space of curvature κ and r 2 = x 2 + y 2 + z 2 . We also take the assumptions of homogeneity and isotropy for the scalar field,
Φ = Φ(t).(8)
Consider first the case of flat Euclidean sections (κ = 0). We work in the gauge invariant formulation, and the Stueckelberg degrees of freedom take the ansatz [9,10].
φ i = x i , φ 0 = f (t),(9)
where f (t), like a(t), is a monotonically increasing function of t.
Inserting (6) and (9) into the action, we obtain the mini-superspace action
S EH = 3M 2 P dt −ȧ 2 a N ,(10)S mass = 3M 2 P dt Φ N F (a) −ḟ G(a) ,(11)S Φ = dt a 3 1 2 N −1 g(Φ)Φ 2 − N V (Φ) .(12)
where
F (a) = a(a − 1)(2a − 1) + α 3 3 (a − 1) 2 (4a − 1) + α 4 3 (a − 1) 3 ,(13)G(a) = a 2 (a − 1) + α 3 a(a − 1) 2 + α 4 3 (a − 1) 3 . (14)
This mini-superspace action is invariant under time reparametrizations, under which f transforms like a scalar.
There are four equations of motion, obtained by varying with respect to F, N, Φ and a. As in GR, the Noether identity for time reparametrization invariance tells us that the acceleration equation obtained by varying with respect to a is a consequence of the other equations, so we may ignore it. After deriving the equations, we will fix the gauge N = 1 (this cannot be done directly in the action without losing equations).
Varying with respect to f we obtain the constraint pointed out in [18],
Φ = C G(a) ,(15)
where C is an arbitrary integration constant. (Note that the analogous equation in the fixed mass theory implies that a = constant, so there are no evolving flat FRW solutions in that case [9].) Varying with respect to N , we obtain the Friedmann equation,
3M 2 P H 2 + ΦF (a) a 3 = 1 2 g(Φ)Φ 2 + V,(16)
and varying with respect to Φ we obtain the scalar field equation
g(Φ) Φ + 3HΦ + 1 2 g ′ (Φ)Φ 2 + V ′ (Φ) = 3M 2 P F (a) a 3 −ḟ G(a) a 3 .(17)
Rather than solving the coupled second-order Einsteinscalar equations of motion, one can instead reduce the system to a single first-order Friedmann equation. The relation (15) can be used to eliminate Φ and its first derivative from (16), which then becomes a first-order differential equation in a which determines the scale factor,
H 2 = V C G(a) − 3M 2 P C F (a) a 3 G(a) 3M 2 P − 1 2 C 2 g C G(a) G ′ (a) 2 a 2 G(a) 4 .(18)
Once we have solved for the scale factor, the scalar Φ is determined from (15) and the Stueckelberg field f is determined by solving (17) 1 .
Singularities
One feature of this model that has not been noticed previously is that it allows for the possibility of curvature singularities at finite values of a. These can happen when the evolution attempts to pass through values of a for which the denominator of the right hand side of (18) goes to zero.
When this happens a is finite, but the Hubble parameter, and henceȧ, is blowing up. The scalar curvature is also blowing up, so this is a genuine curvature singularity; a "big brake" where the universe comes to some finite scale factor and gets stuck [26,27]. Similar types of singularities also occur in DGP [28].
For example, Taylor expanding the denominator of (18) for large a we obtain a critical value of the scale factor at which the Hubble parameter diverges,
a cr = √ 3 g(0) 2M 2 P 1/6 C 3 + 3α 3 + α 4 1/3 .(19)
Self-Inflation
We now consider the possibility that the graviton has a large mass in the early universe, through some natural displacement (and resulting VEV) of the scalar field from its true minimum near zero. We seek dynamics such that the scalar field slowly rolls down its potential, during which time the graviton remains massive, resulting in a large self-acceleration of the universe. This selfacceleration comes from the second term on the left-hand side of the Friedmann equation (16). For this to be true self-acceleration this term should be much larger than both the scalar kinetic energy and potential on the right hand side, so that the acceleration is primarily driven by the modification to gravity and not by the scalar field. After many e-folds, Φ should roll towards zero, 1 Note that in general the Stueckelberg field cannot be chosen arbitrarily as in [18] but is non-trivially constrained by the choice of mass term, or in this case, kinetic function g(Φ).
self-inflation should end, and the graviton mass should become small at late times. Thus, assume we have an inflationary solution a ∼ e Ht , with H ∼ constant. The scale factor is growing exponentially, so we Taylor expand the entire right hand side of (18) for large a, as
H 2 = V (0) 3M 2 P +C V ′ (0) M 2 P − (6 + 4α 3 + α 4 ) 3 + 3α 3 + α 4 1 a 3 +O 1 a 4 .(20)
We see that the dependence on all of the massive gravity modifications redshifts away exponentially, at least as fast as a −3 , and we are left with inflation driven only by the value of the potential at Φ = 0. (In particular, contributions sensitive to the function g only start to enter at O (1/a 7 ).)
Said another way, we only have self-inflation if the quantity φF (a) a 3
in (16) is approximately constant when a ∼ e Ht . But the constraint equation (15) makes this impossible: we see from (15) that Φ ∼ 1 a 3 , since G(a) ∼ a 3 for large a. Since F (a) ∼ a 3 , the quantity ΦF (a) a 3 behaves like ∼ Φ ∼ a −3 , so it goes to zero exponentially fast and we cannot sustain self-inflation.
Having encountered an obstacle to the possibility of self-inflation in the flat slicing, we now investigate the possibility in the open slicing (κ < 0). Following [10] we take the Stueckelberg ansatz to be
φ 0 = f (t) 1 − κr 2 , φ i = √ −κf (t)x i .(21)
The mini-superspace action then becomes
S EH = 3M 2 P dt −ȧ 2 a N + κN a ,(22)S mass = 3M 2 P dt Φ N F (a, f ) −ḟ G(a, f ) ,(23)S Φ = dt a 3 1 2 N −1 g(Φ)Φ 2 − N V (Φ) ,(24)
where
F (a, f ) = a(a − √ −κf )(2a − √ −κf ) + α 3 3 (a − √ −κf ) 2 (4a − √ −κf ) + α 4 3 (a − √ −κf ) 3 ,(25)G(a, f ) = a 2 (a − √ −κf ) + α 3 a(a − √ −κf ) 2 + α 4 3 (a − √ −κf ) 3 .(26)
Again, we have time reparametrization invariance so we can ignore the acceleration equation, and we will fix the gauge N = 1 after deriving the equations of motion. The constraint equation arising from varying with respect to f is
(ȧ − √ −κ)Φ ∂G(a, f ) ∂a + G(a, f )Φ = 0 .(27)
The Friedmann equation obtained by varying with respect to N is (28) and the equation of motion for Φ is
3M 2 P H 2 + κ a 2 + ΦF (a, f ) a 3 = 1 2 g(Φ)φ 2 + V (Φ),g(Φ) Φ + 3HΦ + 1 2 g ′ (Φ)Φ 2 + V ′ (Φ) = 3M 2 P F (a, f ) a 3 −ḟ G(a, f ) a 3 .(29)
In order to obtain inflation driven by the graviton mass, the term ΦF (a, f )/a 3 in the Friedmann equation (28) must be approximately constant when a ∼ e Ht . Rearranging the constraint equation (27) to isolate Φ giveṡ
Φ Φ = − H − √ −κ a a ∂G(a,f ) ∂a G(a, f ) .(30)
Now there are three possibilities: the first is that a(t) ∼ e Ht grows faster than f (t). In this case, the righthand side of (30) approaches a constant at late times, Φ/Φ ∼ −3H + O(1/a), which tells us that Φ decreases exponentially at late times, Φ(t) ∼ e −3Ht . This, in turn, implies that the self-acceleration quantity ΦF (a, f )/a 3 in (28) decreases exponentially like Φ, since F (a, f )/a 3 approaches a constant. So once again, we cannot sustain inflation in this case. The second possibility is that f (t) grows faster than a(t) ∼ e Ht . In this case, the righthand side of (30) goes to zero at late times, so Φ becomes constant. The self-acceleration quantity ΦF (a, f )/a 3 in (28) now grows without bound as ∼ f 3 (−κ) 3/2 /a 3 at late times, so again we do not achieve sustained inflation. Finally, there is the possibility that f (t) ∼ e −Ht , growing at the same rate as a(t). This case follows the same pattern as the first possibility -the right-hand side of (30) approaches a constant at late times, and the selfacceleration quantity decreases exponentially.
In summary, flat and open FRW solutions exist in mass-varying massive gravity, but the constraint equation (the one which forbids flat FRW solutions in pure dRGT theory) does not allow for long-lasting selfinflation. Finally, homogeneous and isotropic closed FRW solutions are not possible for the same reason they are not in dRGT -the fiducial Minkowski metric cannot be foliated by closed slices.
THE DBI MODEL
We now turn to the cosmology of the coupled galileonmassive gravity model introduced in [15]. One way to construct the theory is to consider a dynamical metric, g µν , on the worldvolume of a 3-brane in a five dimensional flat bulk, with embedding coordinates X A (x), coupled to the induced metric through dRGT interactions. The worldvolume theory is
S[g,ḡ] = S R [g] + S mass [g,ḡ] + S π [ḡ],(31)
where S R and S mass are the usual Einstein-Hilbert and dRGT mass terms except that now
g µν = ∂ µ X A ∂ ν X B η AB(32)
is the pull-back of the 5D Minkowski metric to the 4D brane. The action S π consists of ghost-free scalars constructed fromḡ µν and the extrinsic curvature [29]. The only term which will be relevant for us is the DBI term
S π = −Λ 4 d 4 x √ −ḡ,(33)
where Λ is some characteristic mass scale. In the flat slicing, the other possible higher order terms in S π all vanish on the cosmological backgrounds we consider, so we may exclude them without loss of generality. They do contribute in the curved slicing, however the final constraint equation (48) and the conclusions drawn from it are not modified in their presence, so we ignore them in what follows. The embedding functions are scalar fields; the first four become the Stueckelberg fields, and the fifth becomes an extra physical scalar, the brane bending mode π,
X µ = φ µ , X 5 ≡ π.(34)
Consider first the case of flat Euclidean space (κ = 0). We take the FRW ansatz (6) and (9), along with homogeneity for the scalar π = π(t). The mini-superspace action is
S R = 3M 2 P dt −ȧ 2 a N ,(35)S mass = 3M 2 P dt m 2 N F (a) − G(a) ḟ 2 −π 2 ,(36)S π = −Λ 4 dt ḟ 2 −π 2 ,(37)
where F (a) and G(a) are as in (13) and (14). The mini-superspace action is invariant under time reparametrizations, under which both f and π transform like scalars. Thus, as before we may ignore the acceleration equation of motion obtained by varying with respect to a, since it is redundant. We fix the gauge N = 1 after deriving the equations.
Varying with respect to N we obtain the Friedmann equation
H 2 + m 2 F (a) a 3 = 0.(38)
The f and π equations are, respectively
δS δf = d dt 3M 2 P m 2 G(a) + Λ 4 ḟ ḟ 2 −π 2 = 0, (39) δS δπ = − d dt 3M 2 P m 2 G(a) + Λ 4 π ḟ 2 −π 2 = 0. (40)
Taking the combinationḟ δS δf +π δS δπ of the two equations of motion (39) and (40) , we arrive at the following constraint equation,
ḟ 2 −π 2 d dt [G(a)] = 0.(41)
The square root cannot be zero because it appears in the denominator of (39) and (40), so we must have d dt [G(a)] = 0, which implies that a must be constant and thus there are no evolving flat FRW solutions. This is the same phenomenon as in pure dRGT theory [9], and we see that the DBI extended theory has a similar structure. The inclusion of matter minimally coupled to g µν does not affect this conclusion because it does not contribute to the equations (39), (40) and hence the constraint (41) is unchanged.
We now turn to the case of nonzero spatial curvature κ < 0. Using again the ansatz (21) we obtain
S R = 3M 2 P dt −ȧ 2 a N + κN a ,(42)S mass = 3M 2 P dt m 2 N F (a, f ) − G(a, f ) ḟ 2 −π 2 ,(43)S π = −Λ 4 dt ( √ −κf ) 3 ḟ 2 −π 2 ,(44)
with F (a, f ) and G(a, f ) as in (25) and (26). Varying with respect to N and setting N = 1, we obtain the Friedmann equation
H 2 + κ a 2 + m 2 F (a, f ) a 3 = 0.(45)
The f and π equations are, respectively
δS δf = 3M 2 P m 2 ∂F (a, f ) ∂f − ∂G(a, f ) ∂f ḟ 2 −π 2 − 3Λ 4 ( √ −κ) 3 f 2 ḟ 2 −π 2 + + d dt 3M 2 P m 2 G(a, f ) + Λ 4 ( √ −κf ) 3 ḟ ḟ 2 −π 2 = 0,(46)δS δπ = − d dt 3M 2 P m 2 G(a, f ) + Λ 4 ( √ −κf ) 3 π ḟ 2 −π 2 = 0.(47)
Taking the combinationḟ δS δf +π δS δπ of the two equations of motion (46) and (47), and using the relation
∂F (a,f ) ∂f = − √ −κ ∂G(a,f ) ∂a
, which can be checked straightforwardly from the definitions (25) and (26), we arrive at the following constraint equation,
∂G(a, f ) ∂a ȧ ḟ 2 −π 2 − √ −κḟ = 0.(48)
There are now two possible branches of solutions. The first consists of the solutions for which ∂G ∂a = 0. Solving this algebraically for f , we find f = C ± aR for some constant C ± depending only on the coefficients α 2 , α 3 (there can be two branches here, since we must solve a quadratic equation for f ). Then, reinserting this into (45), we obtain a modified Friedmann equation. The modification
F (a,C±aR) a 3
is a constant, depending only on α 2 , α 3 , so that when this constant is negative, we have self-acceleration with ρ self ∼ M 2 P m 2 . These solutions are exactly the same self-accelerated solutions that exist for the pure dRGT theory [10]. The solution for π can then be determined by solving (47).
However, for the theory at hand, there exists a new possibility. This second branch consists of solutions for whichȧ
ḟ 2 −π 2 = √ −κḟ .(49)
In the case of the pure dRGT theory where π = 0, this branch gives only solutions for which a = √ −κt, which is just Minkowski space in Milne coordinates. Here we have the possibility of non-trivial solutions on this branch. Solving (49) forπ givesπ = ±ḟ 1 + κ a 2 , and substituting this into the π equation of motion (47) we see thatḟ cancels and we are left with an algebraic equation in f , namely
3M 2 P m 2 G(a, f ) + Λ 4 ( √ −κf ) 3 ȧ 2 + κ −κ = C (50)
where C is the integration constant from integrating (47). This is a cubic equation which can be solved for f . Eliminating f from the Friedmann equation (45) yields a separable equation of motion for the scale factor which can have solutions with non-trivial evolution.
SUMMARY AND CONCLUSIONS
In this letter we have explored the cosmology of two different extensions of the dRGT massive gravity theory.
The first model is variable mass massive gravity, which results from promoting the fixed mass term of the dRGT model to the vacuum expectation value of a dynamical scalar field, as suggested in [9]. In dRGT theory, there is a constraint equation that forbids non-trivial flat FRW solutions, though self-accelerating open solutions exist. In the variable mass theory, the form of the constraint is different, and it no longer forbids flat FRW solutions. The constraint, however, makes it difficult to realize the idea of self-inflation, i.e. using the self-acceleration properties of massive gravity in the early universe to drive inflation. Furthermore, we have demonstrated for the first time that a large class of cosmologies within these models exhibit a future curvature singularity of the "big brake" form.
The second extension is the coupled galileon-massive gravity scalar-tensor theory of [15]. In the flat slicing, the constraint equation takes a similar form to that of the pure dRGT theory, and it similarly rules out nontrivial FRW solutions. In the open slicing, the theory reproduces the self-accelerating branch discovered in pure massive gravity, and in addition, provides a new branch of evolving cosmological solutions, the detailed general properties of which will be the topic of future work.
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[
"Simulating Hamiltonian dynamics using many-qudit Hamiltonians and local unitary control",
"Simulating Hamiltonian dynamics using many-qudit Hamiltonians and local unitary control"
] |
[
"Michael J Bremner \nSchool of Physical Sciences\nThe University of Queensland\n4072QueenslandAustralia\n\nInstitute for Quantum Information\nCalifornia Institute of Technology\n91125PasadenaCAUSA\n",
"Dave Bacon \nInstitute for Quantum Information\nCalifornia Institute of Technology\n91125PasadenaCAUSA\n\nDepartment of Physics\nCalifornia Institute of Technology\n91125PasadenaCAUSA\n",
"Michael A Nielsen \nSchool of Physical Sciences\nThe University of Queensland\n4072QueenslandAustralia\n\nInstitute for Quantum Information\nCalifornia Institute of Technology\n91125PasadenaCAUSA\n\nSchool of Information Technology and Electrical Engineering\nThe University of Queensland\n4072QueenslandAustralia\n"
] |
[
"School of Physical Sciences\nThe University of Queensland\n4072QueenslandAustralia",
"Institute for Quantum Information\nCalifornia Institute of Technology\n91125PasadenaCAUSA",
"Institute for Quantum Information\nCalifornia Institute of Technology\n91125PasadenaCAUSA",
"Department of Physics\nCalifornia Institute of Technology\n91125PasadenaCAUSA",
"School of Physical Sciences\nThe University of Queensland\n4072QueenslandAustralia",
"Institute for Quantum Information\nCalifornia Institute of Technology\n91125PasadenaCAUSA",
"School of Information Technology and Electrical Engineering\nThe University of Queensland\n4072QueenslandAustralia"
] |
[] |
When can a quantum system of finite dimension be used to simulate another quantum system of finite dimension? What restricts the capacity of one system to simulate another? In this paper we complete the program of studying what simulations can be done with entangling many-qudit Hamiltonians and local unitary control. By entangling we mean that every qudit is coupled to every other qudit, at least indirectly. We demonstrate that the only class of finite-dimensional entangling Hamiltonians that aren't universal for simulation is the class of entangling Hamiltonians on qubits whose Pauli operator expansion contains only terms coupling an odd number of systems, as identified by Bremner et. al. [Phys. Rev. A, 69, 012313 (2004)]. We show that in all other cases entangling many-qudit Hamiltonians are universal for simulation.
|
10.1103/physreva.71.052312
|
[
"https://arxiv.org/pdf/quant-ph/0405115v1.pdf"
] | 31,586,090 |
quant-ph/0405115
|
95235200b19dbca964f8e532058eaba9de14ba41
|
Simulating Hamiltonian dynamics using many-qudit Hamiltonians and local unitary control
20 May 2004
Michael J Bremner
School of Physical Sciences
The University of Queensland
4072QueenslandAustralia
Institute for Quantum Information
California Institute of Technology
91125PasadenaCAUSA
Dave Bacon
Institute for Quantum Information
California Institute of Technology
91125PasadenaCAUSA
Department of Physics
California Institute of Technology
91125PasadenaCAUSA
Michael A Nielsen
School of Physical Sciences
The University of Queensland
4072QueenslandAustralia
Institute for Quantum Information
California Institute of Technology
91125PasadenaCAUSA
School of Information Technology and Electrical Engineering
The University of Queensland
4072QueenslandAustralia
Simulating Hamiltonian dynamics using many-qudit Hamiltonians and local unitary control
20 May 2004PACS numbers: 0367-a, 0365-w
When can a quantum system of finite dimension be used to simulate another quantum system of finite dimension? What restricts the capacity of one system to simulate another? In this paper we complete the program of studying what simulations can be done with entangling many-qudit Hamiltonians and local unitary control. By entangling we mean that every qudit is coupled to every other qudit, at least indirectly. We demonstrate that the only class of finite-dimensional entangling Hamiltonians that aren't universal for simulation is the class of entangling Hamiltonians on qubits whose Pauli operator expansion contains only terms coupling an odd number of systems, as identified by Bremner et. al. [Phys. Rev. A, 69, 012313 (2004)]. We show that in all other cases entangling many-qudit Hamiltonians are universal for simulation.
I. INTRODUCTION
A. Overview
One remarkable aspect of Nature is that it can be modeled by equations whose solution may be obtained by algorithmic means. This empirically observed fact allows us to construct physical theories that make predictions as to how Nature will behave. Of course, while we can simulate Nature, our capacity to do so is limited by the way we choose to perform the simulation and the complexity of the system to be simulated. Feynman's landmark paper on quantum computation [1] discussed the apparent inability of classical computers to efficiently simulate quantum systems and suggested that a quantum computer might succeed where classical computers fail. In this paper we study a class of simulation protocols motivated by the example of quantum computation. In particular, we examine the following question: given a composite system with a finite-dimensional Hamiltonian and the ability to perform arbitrary local unitary operations, what other Hamiltonians can we simulate?
The simulation of quantum systems by quantum computers is a topic that has attracted considerable attention. A considerable literature (see [2] and references therein) addresses the question of how to adequately simulate physically interesting closed quantum systems. Issues of particular interest include the complexity of protocols for simulating initial states, simulating evolutions, and for extracting physically important information from the final state of the computer. Each of these issues must be addressed in any comparative study of quantum and classical computers, and their capacity to simulate Nature.
While state preparation and measurement are vital elements of any simulation of a quantum system, we focus in this paper on the simulation of evolutions of systems. Hamiltonian simulation protocols using singlequdit unitary operations as an additional resource have received considerable attention in recent years due to their relationship with various models of quantum computation. One of the more noteworthy advances was the discovery that all two-body Hamiltonians can simulate all other Hamiltonians on the set of qudits that they entangle, when combined with single-qudit unitary operations [3,4,5,6,7,8,9,10,11,12]. This body of work also demonstrated that these Hamiltonians could be used to efficiently simulate any other two-body Hamiltonian that acts on the network of qudits they entangle. This includes a Hamiltonian that can implement the cnot operation, thus implying that all entangling two-body Hamiltonians and single-qubit unitary operations are universal for quantum computation.
The results and tools used to study two-body Hamiltonian simulation have been applied fruitfully to several related problems. There is now a considerable literature on time-optimal strategies for simulating two-qubit Hamiltonians and quantum gates; see, for example, [5,13,14,15,16,17,18,19,20,21,22,23,24,25,25,26,27], and references therein. This body of investigation has led to interest in applying these theoretical results to practical proposals for quantum computation [28].
More recently, studies have focused on using systems with many-qubit interactions for Hamiltonian simulation and gate-synthesis [21,27,29,30,31]. A number of these papers have investigated the structure of systems with many-body interactions for the purposes of gate synthesis and algorithm design [21,27,30,31]. Several authors have recently examined the effects of many-body interactions in quantum dot [32] and optical lattice [33,34] systems.
For the purposes of this paper we are most concerned with the work in [29], where the authors established which Hamiltonians with many-qubit interactions are universal when combined with the ability to perform arbitrary single-qubit unitary operations. In a similar vein we examine which Hamiltonians with many-qudit interactions are universal when combined with arbitrary single-qudit unitary operations. Our final result is a striking generalization of the conclusion in [29]. [29] showed that the only class of non-universal entangling Hamiltonians on qubits are the odd entangling Hamiltonians, i.e., those Hamiltonians whose Pauli operator expansion contains only terms coupling an odd number of qubits. Furthermore, [29] showed that the odd entangling Hamiltonians can all simulate one another, so there is a sense in which there are only two types of many-qubit entangling Hamiltonian. Remarkably, in this paper we will see that when the systems involved are not all qubits, this structure actually simplifies, with all entangling Hamiltonians capable of simulating all other entangling Hamiltonians, i.e., we show that apart from the many-qubit case, there is only one type of many-body entangling Hamiltonian.
Our primary concern in this paper is with questions of universality in many-qudit systems, without regard to the issue of complexity. Thus, when we say a set of resources is universal on a set of qudits, we are stating that these resources can be used to simulate any Hamiltonian on those qudits, without implication that this simulation is efficient or inefficient. This is in contrast to the notion of universality for quantum computation which requires that any universal set of resources can simulate a standard gate set with a polynomial overhead in the number of qubits used. That said, it is often possible to exploit the structure of certain classes of many-body Hamiltonians to develop efficient simulation algorithms. For instance much headway can be made in developing efficient Hamiltonian simulation protocols that use k-local Hamiltonians by adapting the methods developed for systems of qubits in [29] to systems of qudits.
B. Terminology and statement of results
Before turning to the discussion and proof of the main results of this paper it is helpful to introduce some terminology. Generally, we will use the term qudit to describe any quantum system with a finite-dimensional state space. As an example of our usage, a threequdit system might contain a two-dimensional system (a qubit), a five-dimensional system, and a four-dimensional system.
We are interested in the properties of the Hamiltonian dynamics of an n-qudit system. As we will see, a great deal can be said about the properties of a Hamiltonian simply by examining its structure in a suitable representation. In [29], the authors found that the universality properties of a many-body Hamiltonian acting on qubits could be identified by expanding it in the Paulioperator basis, i.e., tensor products of X, Y , Z and I. In this paper we expand upon this analysis by examining the properties of an n-qudit Hamiltonian written in a d-dimensional generalization of the Pauli basis.
An arbitrary Hamiltonian on n qudits can be uniquely written as
H = α h α H α ,(1)
where h α are real coefficients and each H α in the expansion is a tensor product of Hermitian operators acting on the individual quits,
H α = n j=1 H (j) α ,(2)
where H (j) α acts on qudit j, and is either the identity operator, or one of a set of traceless Hermitian matrices known as the Gell-Mann matrices. The Gell-Mann matrices generalize the Pauli matrices, and thus this expansion is a generalization of the expansion for qubits used in [29]. The Gell-Mann matrices for a d-dimensional quantum system consist of: (a) d−1 matrices of the form
W m = 1 m(m − 1) m−1 b=1 |b b| − (m − 1)|m m| ,(3)
where 2 ≤ m ≤ d; and (b) the Pauli-like matrices:
X ab = 1 √ 2 (|a b| + |b a|) (4) Y ab = −i √ 2 (|a b| − |b a|)(5)
where 1 ≤ a < b ≤ d. These act as the Pauli X and Y on the two-dimensional subspace spanned by the vectors |a and |b . We sometimes refer to the W m matrices as Cartan subalgebra elements of the Gell-Mann matrices, since they span a Cartan subalgebra of the Lie algebra su(d) generated by the Gell-Mann matrices. However, it is worth emphasizing that we do not use any special properties of Cartan subalgebras, and the reader does not need to be familiar with the properties of Cartan subalgebras to follow the details of the paper; our use of the term is a convenience of nomenclature only. Note that the Gell-Mann matrices are traceless and Hermitian, and form a complete basis for traceless Hermitian matrices. The representation Eq. (1) is useful as it highlights which qudits interact and which do not. In particular, given a term H α let S α be the set of qudits upon which H α acts non-trivially, that is, the set of qudits for which H (j) α is traceless. We say that the qudits in S α are coupled by H α and refer to H α as a coupling term. We also say that H α is entangling on the set S α . More generally, we say that a Hamiltonian H is entangling on some set of qudits if it is not possible to partition this set of qudits into two non-trivial sets S andS such that every term H α in the expansion of H couples either a subset of S or ofS. In graph-theoretic language, if the qudits corresponded to vertices on a hypergraph and the couplings corresponded to hyperedges, the condition that the Hamiltonian is entangling on a set of qudits is simply that the hypergraph connects this set. As such we say that a Hamiltonian connects the set of qudits it entangles.
Our strategy for demonstrating universality in this paper is to show that some set of resources is capable of simulating another set already known to be universal. In particular, we make reference to two theorems that categorize large classes of Hamiltonians as universal up to single-qudit unitary operations. The first was mentioned in the introduction: two-body entangling Hamiltonians are universal for quantum computation [10]. Using the terminology just introduced, this theorem may be stated as follows [10]:
Theorem 1.
Suppose H is a two-body Hamiltonian, that is, every coupling term in the Gell-Mann expansion of H couples at most two qudits. If H is entangling on a set of n qudits (that is, the coupling terms in H connect these qudits) then evolutions of H together with single-qudit unitary operations are universal for quantum computation on these n qudits.
The second universality theorem that we use involves Hamiltonians acting on sets of qubits that have coupling terms that may couple more than two qubits. This theorem is stated [29]:
Theorem 2.
Suppose H is an arbitary entangling Hamiltonian on a set of n qubits. Evolutions of H and single-qubit unitary operations are universal on those n qubits if and only if the Gell-Mann (i.e., Pauli) expansion of H contains at least one coupling term that couples an even number of qubits.
Theorem 2 tells us that for a Hamiltonian acting on qubits alone to be universal, it must have a coupling term acting on an even number of qubits. If H does not contain such a coupling term then we shall call it an odd Hamiltonian, since all its terms couple an odd number of qubits. What can the odd Hamiltonians simulate? This question was also answered in [29]:
Theorem 3.
Let H be an odd entangling Hamiltonian, that is, every term in the Gell-Mann expansion of H couples an odd number of qubits. Then H and single-qubit unitaries can simulate any other odd Hamiltonian on the n qubits.
[29] also demonstrated that the Lie algebra generated by the odd entangling Hamiltonians on n qubits (and local unitaries) corresponds to the Lie algebras so(2 n ) and sp(2 n ), for even or odd n respectively. Furthermore, [29] showed that the odd Hamiltonians can be made universal with appropriate encodings.
In this paper we demonstrate that if a Hamiltonian is entangling on a set of qudits, then this Hamiltonian is universal on those qubits, when assisted by local unitary operations. The only exception to this result is the special case when the Hamiltonian is an odd Hamiltonian acting on qubits only.
C. Outline
Theorem 1 shows that if a Hamiltonian connects a set of qudits with two-qudit couplings then this Hamiltonian is universal with single-qudit unitaries. Our strategy in this paper is to show that a many-body Hamiltonian (that isn't one of the odd qubit-only Hamiltonians) connecting a set of n qudits can simulate a two-body Hamiltonian connecting the same set of qudits. This is done by defining a series of simulation protocols, each identifying broad classes of Hamiltonians that any entangling Hamiltonian can simulate, until we arrive at the eventual result.
The structure of the paper is as follows. In Section II we introduce some simple general simulation techniques that are used often in this paper. Section III introduces a simulation technique known as term isolation. This simulation technique allows us to simulate any particular coupling term, H α , that is present in the Gell-Mann expansion of H, thus isolating the term. In Section IV we show that given some term coupling k qudits, we can simulate new coupling terms that couple fewer than k qudits. We also discuss the limitations on this type of simulation. Section V examines how we can use a term that couples k qudits to simulate a coupling between two qudits. Finally we prove the main result of the paper: that the only non-universal class of entangling Hamiltonians is the class of odd Hamiltonians. This is argued through an exhaustive demonstration that all n-qudit entangling Hamiltonians other than the odd many-qubit Hamiltonians are indeed universal.
II. SIMPLE SIMULATIONS
In this section, we review some simple Hamiltonian simulation techniques studied in previous papers [3,4,5,6,7,8,9,10,11,12], and that will form the basis for our later results. By a Hamiltonian simulation we mean a sequence of evolutions due to our system Hamiltonian, H, which is assumed fixed, interleaved with single-qudit unitary operations. The goal is to approximate (to arbitrary accuracy) evolution according to some other Hamiltonian. If that is possible for some desired Hamiltonian we say that Hamiltonian can be simulated. The theory of Lie algebras and Lie groups ensures that the techniques decribed in this section exhaust the set of possible simulations that can be performed given some Hamiltonian and single-qudit unitaries.
A. Conjugation by a unitary operator
A quantum system with Hamiltonian H evolves in time via the unitary operation e −iHt . Say we are also given the ability to perform some unitary operation, U , and its inverse, U † . Then performing the sequence of unitary operations U e −iHt U † = e −iUHU † t , we see that we can simulate an evolution according to the conjugated Hamiltonian U HU † . In this paper, as we have given ourselves the ability to perform arbitrary single-qudit unitaries, we will often conjugate a Hamiltonian by unitaries of the
form U = U 1 ⊗ U 2 ⊗ . . . ⊗ U n .
B. Simulating linear combinations
Suppose we can simulate two different Hamiltonians, H 1 and H 2 . Then we can simulate the sum of these Hamiltonians, since e −iH1∆ e −iH2∆ ≈ e −i(H1+H2)∆ for small ∆, and with successive evolutions we can simulate the Hamiltonian H 1 + H 2 for an arbitrary time t. Imagine that we could evolve our system by a whole set of Hamiltonians, H, and their negatives 1 . It follows that we can simulate arbitrary linear combinations of any of the elements of H.
C. Simulating commutators of Hamiltonians
Another simple simulation protocol that can be performed is the simulation of a commutator of two different Hamiltonians. This is possible as e −iH1∆ e iH2∆ e iH1∆ e −iH2∆ ≈ e −i(i[H1,H2])∆ 2 . So if we can simulate H 1 , H 2 and their negations we can simulate the commutator of these Hamiltonians.
D. Simulating Hamiltonians that couple the same qudits
Consider the general expression for a Hamiltonian acting on a system of qudits in Equations (1) and (2), and recall that H α couples a set of qudits S α . We now introduce a theorem from [8] to show it is possible to use H α and single-qudit unitaries to exactly simulate any other coupling term that couples the set of qudits S α :
1 Given that we can simulate H, it turns out always to be possible to simulate −H, using single-qudit unitary operations. This follows from Equation (11), later in the paper, which shows how to express −H as a sum of terms of the form U HU † , where U are local unitary operations. By the methods of simulation we've already introduced, it follows that −H can be simulated.
Theorem 4. Let A and B be any two traceless Hermitian operators in d dimensions and assume that B = 0. There is an algorithm to find a set of at most d 2 unitary operators, U n , and constants c n > 0 such that:
A = n c n U n BU † n .(6)
Key to proving this theorem is a result from the theory of operator majorization, Uhlmann's theorem [35]. Although we do not need the theory of majorization in this paper, for the benefit of readers familiar with majorization, we make the following summary remarks. Recall that Uhlmann's theorem tells us that if P ≺ Q (that is, P is majorized by Q) then P = n p n U n QU † n , for some unitary operators U n and some p n that form a probability distribution. The proof of Theorem 4 in [8] follows by showing that A ≺ cB for some positive constant c.
Any coupling term H α in H is a tensor product of traceless terms acting on S α . If we replace B in Theorem 4 by the individual tensor factors appearing in H α , then we see that we can simulate any A that is a tensor product of traceless Hermitian operators acting on the same set S α . This result will be extremely useful in the remainder of this paper. It tells us that if we can simulate some coupling H α , we can simulate every other coupling on the same set of qudits.
III. TERM ISOLATION
In Section II D, we saw that any coupling term, H α , in the expansion H = α h α H α (Equation (1)), could be used to simulate any other coupling term that entangles the same set of qudits. If we have a Hamiltonian that is simply a coupling term on a given set of qudits, we can immediately say a great deal about what can be simulated with that Hamiltonian. In general we do not have this luxury of interpretation. Instead, some general Hamiltonian, H = α H α , has many different coupling terms that couple many different sets of qudits. Term isolation is a simulation technique that uses H and singlequdit unitaries to simulate any particular term H α in the expansion of H alone.
Term isolation allows us to think about H in a different way, showing that the ability to simulate H is equivalent to the ability to simulate the coupling terms {H α } individually. Thus, we can perform our analysis entirely in terms of the set {H α } and still encapsulate all of the Hamiltonian simulation properties of H. Given that the elements of the set {H α } have a much simpler structure than a general H, term isolation is a powerful tool for analysis.
We now show that term isolation can always be performed. If we demonstrate that we can use H and singlequdit unitaries to simulate some H α coupling an arbitrarily chosen set of qudits, then we know from Section II D that it can be used to simulate any other term coupling the same qudits.
Without loss of generality we may assume that the term being isolated is of the form
H α = k j=1 W (j) bj ⊗ I ⊗n−k ,(7)
where k is the number of qudits in the set S α . To see that there is no loss of generality in assuming this form, note that we can always relabel the qudits in S α so that they are the first k qudits in the system, and any operators X ab or Y ab in H α are equivalent under local unitaries to W 2 . Any term in the expansion of H, H β , that isn't the term H α that we wish to keep, is different from H α in at least one of three ways. Either:
Case 1: H β has terms acting non-trivially on qudits outside of S α , the set of qudits upon which H α acts.
Case 2: H β acts on a strict subset of S α .
Case 3: H β acts on the same qudits as H α but is a tensor product of different elements of the Gell-Mann basis. That is, H α = H β , even though H β couples the set S α .
Each of these cases identifies a special difference between H α and H β . In the following sections these differences are exploited to define simulations that remove undesirable terms.
As we have previously stated, every simulation in this section may be represented as a sequence of linear combinations, commutators and conjugations by local unitaries. We often denote a sequence of operations of this type on a Hamiltonian, H, by a scripted letter. For example, in Section III A we define the depolarizing channel, which is a linear combination of conjugations by local unitaries, and write D[H] = H D to symbolize the depolarizing channel acting on H, resulting in the simulated Hamiltonian H D . The action of D on H defines a simulation. We can also compose simulation techniques, so, for example, in Section III B we define a simulation T [H D ] = H T .
A. Case 1
We begin by noting the identity
Up U p JU † p = d tr(J)I,(8)
where J is an operator acting on some qudit of dimension d and the sum is over all d 2 elements of the d-dimensional Pauli group 2 , where we omit repeated summation when two elements in the Pauli group differ merely by a phase factor. We note there is a simple extension of Equation (8) for multiple-qudit systems,
U (j) p (U (1) p ⊗ . . . ⊗ U (n) p )J(U (1) p ⊗ . . . ⊗ U (n) p ) † = D tr(J)I ⊗n ,(9)
where the superscripts indicate the different qudit systems, of respective dimension d (j) , D = d (1) ...d (n) is the dimension of the combined system, I represents the appropriate identity operator for each subsystem, and the sum is over conjugations by all elements of the Pauli group for each qudit, again omitting repeated sums over elements that are the same up to a phase factor. We define the simulation D[H] = H D to be the multiple-qudit depolarizing channel acting on the n − k qudits that aren't coupled by H α ,
D[H] = U (j) p (U (k+1) p ⊗ ... ⊗ U (n) p )H(U (k+1) p ⊗ ... ⊗ U (n) p ) † = H D .(10)
H α acts on the first k qudits of an n-qudit system, that is, the set S α . If we examine the simulated Hamiltonian, H D , we find from Equation (9) that any terms H β in H that act non-trivially on qudits outside the set S α are eliminated. The simulation leaves the coupling term H α unchanged except for an unimportant positive scaling factor. Thus we have removed all the Case 1 terms H β from the Hamiltonian, and need only consider the remaining Case 2 and Case 3 terms.
B. Case 2
The Hamiltonian H D is a linear combination of terms that couple the set of qudits S α or some subset of S α . It turns out that we can use another extension of Equation (8) to simulate a Hamiltonian, H T , that only has terms that couple the set S α . In Equation (8), if J is a traceless operator we find that the right hand side of the equation is zero. Noting that I is an element of the Pauli group, we find
Up =I U p JU † p = −J,(11)
which always holds for traceless J. Using single-qudit unitaries from the Pauli group we consider the following summation,
U (1) p =I,U(2)p =I (U (1) p ⊗U (2) p )(J (1) ⊗J (2) )(U (1) p ⊗U (2) p ) † .(12)
If J (1) and J (2) are traceless, this expression is equal to J (1) ⊗ J (2) . If J (2) is traceless and J (1) is the identity, this expression is equal to −[(d (1) ) 2 − 1]I ⊗ J (2) . With this in mind we define a simulation:
T (j) [H] ≡ ((d (j) ) 2 − 1)H + U (1) p ,U (j) p =I (U (1) p ⊗ U (j) p )H(U (1) p ⊗ U (j) p ) † .(13)
Performing T (j) for j = 2, ..., k, only terms that couple the same qudits as H α are not eliminated. So performing the following sequence of simulations,
T [H D ] = T (k) [T (k−1) [...[T (2) [H D ]]...]] = H T(14)
the simulated Hamiltonian, H T , is a linear combination of terms that couple the same qudits as H α .
C. Case 3
We have shown how to simulate a Hamiltonian H T that only contains terms which couple the same qudits as H α . To eliminate the remaining terms we define the following operators that are both unitary and Hermitian,
Z a ≡ I − 2|a a| = d j=1 |j j| − 2|a a|.(15)
Notice that the Z a operators commute with each of the Cartan subalgebra elements, W m , in Equation (3). Hence, each of the Z a will also commute with H α as it is a tensor product of elements of the Cartan subalgebra. Further notice that Z a anti-commutes with X lm and Y lm if a = l or a = m and commutes otherwise. We can use this fact to define a simulation that eliminates terms with X lm and Y lm operators present in H T . We define a simulation
Z (j) a [H] = H + Z (j) a HZ (j) a ,(16)
where the superscript j indicates a Z a operator acting on the jth qudit, with identities acting elsewhere. If there exists any term with an X lm or Y lm operator on the jth qudit, and such that a = l or a = m, then this term will be eliminated from H T by the simulation Z (j)
a [H T ].
Expanding on this idea we can eliminate every term on the jth qudit that has the form X lm or Y lm by performing the following simulation:
Z (j) [H T ] ≡ Z (j) d [Z (j) (d−1) [...[Z (j) 1 [H T ]]...]](17)
where d is the dimension of the jth qudit. The effect of this simulation on H α is simply to rescale it. Now, if we perform the simulation Z (j) for each qudit in S α ,
Z[H T ] = Z (k) [Z (k−1) [...[Z (1) [H T ]]...]] = H Z ,(18)
all that remains in the newly simulated Hamiltonian, H Z , is a linear combination of terms that commute with the Cartan subalgebra elements. We have now simulated a Hamiltonian with no X-and Y -type terms. H Z is a linear combination of terms that are tensor products of operators from the Cartan subalgebra. Consider the unitary representation, P (j) (π), of the permutation group S bj −1 that permutes the elements of the diagonal basis of the Cartan subalgebra, |a , for a = 1, . . . , b j − 1 on the jth qudit. When a ≥ b j we have P (j) (π)W a P (j) † (π) = W a . When a < b j , we find that the effect of conjugating W a by a permutation operation is to shift around the diagonal elements of W a . Now, we can eliminate any terms in H Z that contain an operator W (j) a with a < b j by performing the simulation
P (j) [H Z ] = π∈S b−1 P (j) H Z P (j) † .(19)
This works because W (j) a is a diagonal, traceless operator and the permutation, P (j) , distributes each of the diagonal elements of W (j) a equally. The effect of P (j) on terms W (j) a acting on the jth qudit and with a ≥ b j is to simply scale them by a factor of (b j − 1)!. Performing the following simulation,
P[H Z ] = P (k) [P (k−1) [...[P (1) [H Z ]]...]] = H P(20)
we produce a Hamiltonian H P that is a linear combination of terms that couple the same qudits as H α and are tensor products of operators W a with a ≥ b j .
In Section II C we pointed out that it is possible to simulate a Hamiltonian proportional to the commutator of two Hamiltonians that are both simulatable. Now, we note that the commutator −i[W
(j) a , X bj −1 bj ] = 0 if a > b j . If a = b j we find −i[W (j) bj , X bj −1 bj ] = √ bj √ bj −1 Y bj −1 bj .
We can make use of this distinction to find a way to remove the unwanted terms from H P . We define the simulation
X (j) [H] ≡ −i[H, X (j) bj −1 bj ].(21)
Then if we perform the following sequence of simulations,
X [H P ] = X (k) [X (k−1) [...[X (1) [H P ]]...]] = H X(22)
we find that H X = ⊗ k j=1 Y bj −1,bj ⊗ I ⊗n−k , up to some unimportant but non-zero constant multiple. We have now simulated a single coupling term that couples the same qudits as H α . Recall in Section II D we noted that a coupling term can be used with single-qudit unitaries to simulate any other term coupling the same set of qudits. So, we can use H X and single-qudit unitaries to simulate H α , the desired term. Thus we have demonstrated that it is possible to isolate H α from H.
IV. SIMULATING NEW COUPLING TERMS
Term isolation shows that the ability to simulate a Hamiltonian H = α h α H α is equivalent to the ability to simulate the set of coupling Hamiltonians, {H α }, given single-qudit unitary operations. Additionally, we learnt in Section II D that given H α and single-qudit unitaries we can simulate any coupling term that couples the same qudits as H α . So far we have not presented any way of simulating some coupling term that couples a different set of qudits than any of the terms in the set {H α }. In this section we will take a key step towards a proof of universality, showing how to use single-qudit unitaries and a term H α coupling k qudits in order to simulate a term that couples k − 1 qudits.
A. Evaluation of commutators
In [29] it was shown that if H α coupled qubits, its capacity to simulate other coupling terms depended on the number of qubits that it coupled. More specifically, it was shown that if H α coupled k qubits and k was an odd number, then H α couldn't be used with singlequbit unitaries to simulate a coupling term that coupled k − 1 qubits. One way of seeing why this is true is to examine the commutator of two Hamiltonians, [H α , H β ], that couple the same set of qubits S α . It is easy to show that the commutator [H α , H β ] = 0 if and only if there are an odd number of locations in S α where H α and H β differ. From this restriction it is possible to prove, as was done in [29], that coupling terms coupling an odd number of qubits can only ever simulate other Hamiltonians that have odd couplings.
What is different when not all the systems are qubits? The purpose of this subsection is to investigate the commutator of two specially chosen couplings H α and H β that couple the same set of qudits, S α . In the case of qubits, it is not difficult to convince oneself that when S α contains an even number of qubits, the commutator [H α , H β ] is either zero, or else couples a set of qubits that is a strict subset of the original set S α . We will show by an explicit calculation that when one or more of the systems is not a qubit, it is possible to choose H α and H β so that the commutator [H α , H β ] contains terms coupling the entire set S α . Remarkably, we will see in the remainder of the paper that this is the key fact that simplifies the study of universality when not all the systems are qubits.
We begin by choosing H α = k j=1 X
[H α , H β ] = k j=1 1 2 √ 2 (X (j) bb ′ + iY (j) bb ′ ) − k j=1 1 2 √ 2 (X (j) bb ′ − iY (j) bb ′ ).(23)
This expression contains Hermitian and skew-Hermitian terms. Upon expansion of the above expression we find that all of the Hermitian terms sum to zero, leaving only a sum of skew-Hermitian terms remaining. These terms correspond to a sum of tensor product terms containing odd numbers of Y bb ′ terms. All of the terms couple the entire set S α . It is easy to verify that this sum is always non-zero, simply by inspection of the coefficients of the relevant terms. So far we have only considered the case where we could choose to simulate H α and H β for X (j) ab and X (j)
ab ′ , b = b ′ .
We can only do this when each subsystem has dimension d > 2. If we have subsystems where d = 2, the situation changes slightly, but the results are similar, provided not all of the subsystems are qubits.
For every j where the qudit has dimension d > 2 we choose H
(j) α = X (j) ab and H (j) β = X (j) ab ′ with b = b ′ . For every j where the qudit has dimension d = 2, we choose H (j) α = X, and H (j) β = Y .
Provided H α and H β do not couple qubits exclusively, a straightforward calculation along lines similar to that already done shows that [H α , H β ] is a non-zero sum of terms, each of which is skew-Hermitian and couples all k qudits. The only subtlety in the calculation is the need to analyse separately the cases where there are an even number of qubits in the set S α , which gives rise to a commutator which is a non-zero sum of tensor product terms containing an odd number of Y bb ′ terms, and the case where there are an odd number of qubits in the set S α , which gives rise to a commutator which is a non-zero sum of tensor product terms containing an even number of Y bb ′ terms.
B. Simulating identity operators
Given some term, H α , coupling a set of qudits S α , we show how the results on commutators just obtained allow us to simulate other coupling term that couples a subset of S α with just one qudit removed. More precisely:
H ′ = I ⊗ H γ ,(24)
provided H α does not couple qubits exclusively. The coupling term H γ may couple any k − 1 qudit subset of S α , subject to the constraint that the subset not be qubits exclusively.
Proof: Given H α we can simulate any other coupling term, H β = ⊗ n j=1 H (j) β , that acts non-trivially on the same set of k qudits, S α . We label the qudits so that S α consists of qudits 1, . . . , k, and so that our goal is to simulate a coupling on qudits 2, . . . , k, i.e., the goal is to remove qudit 1. To this end, we choose H
(1) β so that H (1) α = H (1)
β . Note that, by assumption, the set 2, . . . , k does not contain qubits exclusively. Evaluating the commutator, we find:
i[H α , H β ] = i(H (1) α ) 2 ⊗ n j=2 H (j) α , n j=2 H (j) β .(25)Setting N ≡ n j=2 H (j) α , N ′ ≡ n j=2 H (j)
β , and applying Equation (8) to the first qudit, we see that it is possible to simulate Theorem 1 stated that if a set of qudits is connected by a Hamiltonian, H, with two-body interactions, then evolutions by H and single-qudit unitaries form a universal set of operations on that set of qudits [10]. A set of 2-qudit coupling terms connecting the same set of qudits is also universal as they can simulate a two-body Hamiltonian on the set of qudits. We prove in this section the main result of this paper: that a generic Hamiltonian, H, entangling a set of qudits can simulate a set of 2-qudit coupling terms connecting the qudits, and is thus universal. The only exception to this rule is the case where H is a sum of odd coupling terms, as discussed in [29], and summarized in Theorems 2 and 3 in the present paper.
H ′ = iI ⊗ [N, N ′ ].(26)
We begin by proving Theorem 5, which shows that a coupling term, H α , that couples a set of k qudits, S α , can be used to simulate a set of 2-qudit couplings that connect the set S α . This implies that H α and single-qudit unitaries are a universal set on the qudits S α . We conclude with Theorem 6, showing that an arbitrary entangling Hamiltonian on n qudits is universal for the qudits it entangles.
A. Theorem 5: Using a term coupling many qudits to simulate a term coupling two qudits.
Theorem 5. Suppose H α = n j=1 H (j) α
couples k qudits. Then H α and single-qudit unitary operations can be used to simulate a set of two-qudit couplings connecting every qudit coupled by H α , provided H α does not couple qubits exclusively, and k > 1. Thus H α and single-qudit unitaries are universal on the set of qudits coupled by H α .
Proof: Without loss of generality we may label the systems so that H α couples systems 1 through k, and system 1 is not a qubit. Fix j in the range 2 through k. Applying Lemma 1 repeatedly, we see that we can simulate a Hamiltonian coupling system 1 and system j. It follows that H α and single-qudit unitaries are universal on the set of qudits coupled by H α . ✷ B. Theorem 6: Which Hamiltonians are universal?
With Theorem 5 in mind, we now prove that the only non-universal set of entangling Hamiltonians is the set of odd Hamiltonians acting on qubits alone. Proof: The forward implication follows from Theorem 2, as does the reverse implication when all systems are qubits. Thus, all that needs proof is the reverse implication in the case when H is an entangling Hamiltonian that does not act exclusively on qubits. We will show how to construct a set of two-body couplings that connect all n qudits.
To construct this set, begin by picking a system that is not a qubit, and label it system 1. We will explain how to construct a set, S, of systems to which 1 can be coupled via a two-body interaction. We begin by setting S = {1}, and aim to add in other systems that can be coupled to 1 via two-body interactions. Our strategy is to show that provided S is not yet maximal, i.e., does not yet contain all n qudits, then it is always possible to add an extra qudit into S.
To see this, suppose S is not yet maximal. Then it is always possible to pick a qudit j inside S and a qudit k outside of S such that H contains a coupling term H jk which couples systems j and k. (Other systems may also be coupled by H jk .) In the case when either j or k is not a qubit, Theorem 5 shows that a term coupling just j and k may be simulated. Theorem 1 implies that we can also simulate a term coupling system 1 and k, and so system k may be added to S.
The other possible case is when j and k are both qubits. In this case, suppose without loss of generality that H jk has the form X (j) ⊗ X (k) ⊗ . . ., where the superscripts label the systems. We may also simulate the coupling X
12 ⊗ Z (j) , since system j is in S. Taking the commutator of these two couplings, we see that we may simulate couplings of the form X
12 ⊗ Y (j) ⊗ X (k) ⊗ . . .. Applying Theorem 5, we see that it is possible to simulate a twobody coupling between system 1 and k, and thus system k may be added to S. ✷
VI. CONCLUSION
We have demonstrated that many-qudit Hamiltonians combined with local unitary operations are always universal for simulation on any connected set of subsystems upon which the interactions act nontrivially, provided that Hamiltonian is not an odd Hamiltonian acting on qubits. This result is rather intriguing and elegant, especially in the light of the general lack of broad results for many-body (as opposed to two-body) problems in quantum information science. In the study of pure state bipartite entangled states, for example, a single unit of currency, the maximally entangled state, has been identified and the fungible nature of this currency has been established. On the other hand, a similar currency and set of fungible transformations has not been identified for systems consisting of more than two parties. Given this difficulty in understanding the structure of quantum states, it is quite remarkable that, with the exception of odd entangling Hamiltonians, all of the different manyqudit interactions are equivalent. Even in the case of odd entangling Hamiltonians, universal simulation can be achieved using an encoding which wastes only a single extra qubit of space [29]. Thus there is a real sense in which, for simulation, all interactions have been created equal.
Part of the simplicity of our result stems from our focus on universality for simulation as opposed to universality for quantum computation, which requires that issues of efficiency be taken into account. When one adds the requirement of efficiency of simulation, then problems of universality become much more difficult: indeed this is perhaps one of the fundamental problems in the study of the computational complexity of quantum circuits. A well-developed theory of efficient simulation is a task of great importance and, judging from the difficulties encountered in proving lower bounds for problems in classical circuit complexity, this task is probably an immensely difficult problem. This paper can be seen, however, as a necessary precursor to any attempt to advance this program.
ab ′ where for all j we set b = b ′ .(We assume initially that all systems are of dimension 3 or greater.) Given these forms for H α and H β , what does [H α , H β ] look like? We find
Lemma 1 .
1Given the ability to evolve via H α which couples k qudits, and local unitary operations, it is possible to simulate H ′ such that
Theorem 6 .
6Single-qudit unitary operations, and evolutions via a Hamiltonian, H, that connects a set of n qudits, is a universal set of operations on those n qudits if and only if H is not an odd Hamiltonian acting on qubits alone.
Finally, we note that as N and N ′ don't act exclusively on qubits, our earlier results on commutators show that we can ensure that [N, N ′ ] is a non-zero linear combination of terms that couple S α , less the first qudit. Term isolation allows us to simulate one of the coupling terms in [N, N ′ ] alone, i.e., H ′′ = I ⊗ H γ , as required. ✷ V. UNIVERSALITY
The properties of the d-dimensional Pauli group were extensively studied in[36]. We will not use any further special properties of this group and refer the interested reader to[36] for further information.
AcknowledgmentsWe thank Jennifer Dodd, Henry Haselgrove and Andrew Hines checking this manuscript and for helpful discussions. This work was supported in part by the National Science Foundation under Grant. No. EIA-0086038.
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"THE DIAMETER OF THE GENERATING GRAPH OF A FINITE SOLUBLE GROUP",
"THE DIAMETER OF THE GENERATING GRAPH OF A FINITE SOLUBLE GROUP"
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[
"Andrea Lucchini "
] |
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Let G be a finite 2-generated soluble group and suppose thatWe construct a soluble 2-generated group G of order 2 10 · 3 2 for which the previous result does not hold. However a weaker result is true for every finite soluble group: if a 1 , b 1 = a 2 , b 2 = G, then there exist c 1 , c 2 such that a 1 , c 1 = c 1 , c 2 = c 2 , a 2 = G.1991 Mathematics Subject Classification. 20P05, 20D10, 20E18.
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10.1016/j.jalgebra.2017.08.020
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"https://arxiv.org/pdf/1701.03346v1.pdf"
] | 55,638,250 |
1701.03346
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93bcf43f623f648f322465629cad018f124b0c1c
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THE DIAMETER OF THE GENERATING GRAPH OF A FINITE SOLUBLE GROUP
12 Jan 2017
Andrea Lucchini
THE DIAMETER OF THE GENERATING GRAPH OF A FINITE SOLUBLE GROUP
12 Jan 2017
Let G be a finite 2-generated soluble group and suppose thatWe construct a soluble 2-generated group G of order 2 10 · 3 2 for which the previous result does not hold. However a weaker result is true for every finite soluble group: if a 1 , b 1 = a 2 , b 2 = G, then there exist c 1 , c 2 such that a 1 , c 1 = c 1 , c 2 = c 2 , a 2 = G.1991 Mathematics Subject Classification. 20P05, 20D10, 20E18.
Introduction
Let G be a finite group. The generating graph for G, written Γ(G), is the graph where the vertices are the nonidentity elements of G and there is an edge between g 1 and g 2 if G is generated by g 1 and g 2 . If G is not 2-generated, then there will be no edges in this graph. Thus, it is natural to assume that G is 2-generated when looking at this graph. There could be many isolated vertices in this graph. For example, all of the elements in the Frattini subgroup will be isolated vertices. We can also find isolated vertices outside the Frattini subgroup (for example the nontrivial elements of the Klein subgroup are isolated vertices in Γ(Sym(4)). Let ∆(G) be the subgraph of Γ(G) that is induced by all of the vertices that are not isolated. In [4] it is proved that if G is a 2-generated soluble group, then ∆(G) is connected. In this paper we investigate the diameter diam(∆(G)) of this graph. Theorem 1. If G is a 2-generated finite soluble group, then ∆(G) is connected and diam(∆(G)) ≤ 3.
The situation is completely different if the solubility assumption is dropped. It is an open problem whether or not ∆(G) is connected, but even when ∆(G) is connected, its diameter can be arbitrarily large. For example if G is the largest 2-generated direct power of SL(2, 2 p ) and p is a sufficiently large odd prime, then ∆(G) is connected but diam(∆(G)) ≥ 2 p−2 − 1 (see [2,Theorem 5.4]).
For soluble groups, the bound diam(∆(G)) ≤ 3 given in Theorem 1 is best possible. In Section 3 we construct a soluble 2-generated group G of order 2 10 · 3 2 with diam(∆(G)) = 3. However we prove that diam(∆(G)) ≤ 2 in some relevant cases.
Theorem 2. Suppose that a finite 2-generated soluble group G has property that | End G (V )| > 2 for every nontrivial irreducible G-module which is G-isomorphic to a complemented chief factor of G. Then diam(∆(G)) ≤ 2, i.e. if a 1 , b 1 = a 2 , b 2 = G, then there exists b ∈ G with a 1 , b = a 2 , b = G.
It follows from the proof of [4,Lemma 6] that the number, say φ G,N (X, k), of k-tuples (g 1 n 1 , . . . , g k n k ) generating G with X is independent of the choice of (g 1 , . . . , g k ). In particular
φ G,N (X, k) = |N | k P G,N (X, k)
where P G,N (X, k) is the conditional probability that k elements of G generate G with X, given that they generate G with XN . Proposition 6 ([9] Proposition 16). If N is a normal subgroup of a finite group G and k is a positive integer, then
P G,N (X, k) = X⊆H≤G HN =G µ(H, G) |G : H| k .
where µ is the Möbius function associated with the subgroup lattice of G. Corollary 7. Let N be a minimal normal subgroup of a finite group G. Assume that N is abelian and let q = | End G (N )|. For every X ⊆ G, if k ≥ d X (G) and P G,N (X, k) = 0, then P G,N (X, k) ≥ q−1 q . Proof. We may assume that N is not contained in the Frattini subgroup of G (otherwise P G,N (X, k) = 1). In this case, if H is a proper supplement of N in G, then H is a maximal subgroup of G and complements N . Therefore µ(H, G) = −1 and |G : H| = |N |. It follows from Proposition 6 that
P G,N (X, k) = 1 − c |N | k ,
where c is the number of complements of N in G containing X. If c = 0, then P G,N (X, k) = 1. Assume c = 0 and fix a complement H of N in G containing X. Let Der X (H, N ) be the set of derivations δ from H to N with the property that x δ = 1 for every x ∈ X. The complements of N in G containing X are precisely the subgroups of G of the kind H δ = {hh δ | h ∈ H} with δ ∈ Der X (H, N ), hence
P G,N (X, k) = 1 − | Der X (H, N )| |N | k .
Now let F q = End G (N ). Both N and Der X (H, N ) can be viewed as vector spaces over F q . Let n = dim Fq N, a = dim Fq Der X (H, N ). We have P G,N (X, k) = 1 − q a q nk . Since P G,N (X, k) = 0, we have a < kq and
P G,N (X, k) = 1 − q a q nk ≥ 1 − 1 q = q − 1 q .
Proof of Theorem 2. We prove the theorem by induction on the order of G. We may assume that G is not cyclic and that the Frattini subgroup of G is trivial. We distinguish two cases: a) All the minimal normal subgroups of G have order 2. In this case G is an elementary abelian group of order 4 and a 1 and a 2 are nontrivial elements of G.
If b / ∈ {1, a 1 , a 2 }, then a 1 , b = a 2 , b = G. b) G contains a minimal normal subgroup N with |N | ≥ 3. By assumption q = | End G (N )| ≥ 3. Assume a 1 , b 1 = a 2 , b 2 = G. By induction there exists g ∈ G such that a 1 , g N = a 2 , g N = G. For i ∈ {1, 2}, let Ω i = {n ∈ N | a i , gn = G}.
Since a i , b i = G, we have d {ai} (G) ≤ 1, hence, by Lemma 5, P G,N ({a i }, 1) = 0 and consequently we deduce from Corollary 7 that
|Ω i | = |N |P G,N ({a i }, 1) ≥ |N | q − 1 q ≥ 2|N | 3 .
But then Ω 1 ∩Ω 2 = ∅. Let b = gn with n ∈ Ω 1 ∩Ω 2 . Then G = a 1 , b = a 2 , b .
Proof of Corollary 3. Let G ′ be the derived subgroup of G. If |G ′ | is odd, then G ′ is soluble by the Feit-Thompson Theorem, and consequently G is also soluble. Moreover if X and Y are normal subgroups of G such that A = X/Y is a nontrivial irreducible G-module then |A| is a power of a prime divisor p of |G ′ | and F = End G (A) is a finite field of characteristic p. Hence |F | ≥ p ≥ 3 and we may apply Theorem 2.
Proof of Corollary 4. We may assume Frat(G) = 1. This means that
G = M ⋊ H where H is abelian and M = V 1 × · · · × V u is the direct product of u irreducible non trivial H-modules V 1 , . . . , V u . Let F i = End H (V i ) = End G (V i ): for each i ∈ {1, . . . , u}, V i is an absolutely irreducible F i H-module so dim Fi V i = 1. Now assume that A is a nontrivial irreducible G-module G-isomorphic to a complemented chief factor of G: it must be A ∼ =G V i for some i, so | End G (A)| = |F i | = |V i | = |A|.if V is a nontrivial irreducible G-module with End G (V ) = F 2 , then a chief series of G does not contain dim F2 V different complemented factors G-isomorphic to V. Then diam(∆(G)) ≤ 2. 3. A finite soluble group G with diam(∆(G)) > 2
Let first recall some results that we will be applied in the discussion of our example. Let G be a finite soluble group, and let V G be a set of representatives for the irreducible G-groups that are G-isomorphic to a complemented chief factor of G.
For V ∈ V G let R G (V ) be the smallest normal subgroup contained in C G (V ) with the property that C G (V )/R G (V ) is G-isomorphic to a direct product of copies of V and it has a complement in G/R G (V ). The factor group C G (V )/R G (V ) is called the V -crown of G. The non-negative integer δ G (V ) defined by C G (V )/R G (V ) ∼ =G V δG(V )
is called the V -rank of G and it coincides with the number of complemented factors in any chief series of G that are G-isomorphic to V . If δ G (V ) = 0, then the V -crown is the socle of G/R G (V ). The notion of crown was introduced by Gaschütz in [8]. We have (see for example [10, Proposition 2.4]):
Proposition 9. Let G and V G be as above. Let x 1 , . . . , x u be elements of G such that x 1 , . . . , x u , R G (V ) = G for any V ∈ V G . Then x 1 , . . . , x u = G.
Now let V be a finite dimensional vector space over a finite field of prime order. Let K be a d-generated linear soluble group acting irreducibly and faithfully on V and fix a generating d-tuple (k 1 , . . . , k d ) of K. For a positive integer u we consider the semidirect product G u = V u ⋊ K where K acts in the same way on each of the u direct factors. Put F = End K (V ). Let n be the dimension of V over F . We may identify K = k 1 , . . . , k d with a subgroup of the general linear group GL(n, F ). In this identification k i becomes an n × n matrix X i with coefficients in F ; denote by
A i the matrix I n − X i . Let w i = (v i,1 , . . . , v i,u ) ∈ V u .
Then every v i,j can be viewed as a 1 × n matrix. Denote the u × n matrix with rows v i,1 , . . . , v i,u by B i . The following result is proved in [3,Section 4].
Proposition 10. The group G u = V u ⋊ K can be generated by d elements if and only if u ≤ n(d − 1). Moreover (1) rank A 1 . . . A d = n. (2) k 1 w 1 , . . . , k d w d = V u ⋊ K if and only if rank A 1 · · · A d B 1 · · · B d = n + u.
In this section we will use in particular the following corollary of the previous proposition:
Corollary 11. Let V = F 2 × F 2 , where F 2 is the field with 2 elements and let Γ = GL(2, 2) ⋉ V 2 . Assume that k 1 , k 2 = GL(2, 2) and let γ 1 = k 1 (v 1 , v 2 ), γ 2 = k 2 (v 3 , v 4 ) in Γ. We have that Γ = γ 1 , γ 2 if and only if 1 − k 1 1 − k 2 v 1 v 3 v 2 v 4 = 0.
Now we are ready to start the construction of a finite 2-generated soluble G with diam(∆(G)) > 2. Let H = GL(2, 2) × GL(2, 2) and let W = V 1 × V 2 × V 3 × V 4 be the direct product of four 2-dimensional vector spaces over the field F 2 with two elements. We define an action of H on W by setting
(v 1 , v 2 , v 3 , v 4 ) (x,y) = (v x 1 , v x 2 , v y 3 , v y 4 )
and we consider the semidirect product
G = H ⋉ W. Let N 1 :=C G (V 3 ) = C G (V 4 ) = {(k, 1) | k ∈ GL(2, 2)}, N 2 :=C G (V 1 ) = C G (V 2 ) = {(1, k) | k ∈ GL(2, 2)}.
A set of representatives for the G-isomorphism classes of the complemented chief factor of G contains precisely 5 elements:
• Z, a central G-module of order 2, with R G (Z) = G ′ = SL(2, 2) 2 ⋉ W. • U 1 , a non central G-module of order 3, with R G (U 1 ) = N 2 ⋉ W. • U 2 , a non central G-module of order 3, with R G (U 2 ) = N 1 ⋉ W. • V 1 , with R G (V 1 ) = V 3 × V 4 × N 2 . • V 3 , with R G (V 3 ) = V 1 × V 2 × N 1 . Let (x 1 , y 1 )(v 11 , v 12 , v 13 , v 14 ) = g 1 , (x 2 , y 2 )(v 21 , v 22 , v 23 , v 24 ) = g 2 .
We want to apply Proposition 9 to check whether g 1 , g 2 = G. The three conditions
g 1 , g 2 R G (Z) = G, g 1 , g 2 R G (U 1 ) = G, g 1 , g 2 R G (U 2 ) = G are equivalent to g 1 , g 2 W = G, i.e. to (x 1 , y 1 ), (x 2 , y 2 ) = H. Moreover g 1 , g 2 R G (V 1 ) = G if and only if x 1 (v 11 , v 12 ), x 2 (v 21 , v 22 ) = (V 1 × V 2 ) ⋊ GL(2, 2), g 1 , g 2 R G (V 3 ) = G if and only if y 1 (v 31 , v 32 ), y 2 (v 41 , v 42 ) = (V 3 × V 4 ) ⋊ GL(2, 2).
Applying Corollary 11 we conclude that
g 1 , g 2 = G
if and only if the following conditions are satisfied:
(1) (x 1 , y 1 ), (x 2 , y 2 ) = H = GL(2, 2) × GL(2, 2),
(2) det 1 − x 1 1 − x 2 v 11 v 21 v 12 v 22 = 0,(3)det 1 − y 1 1 − y 2 v 13 v 23 v 14 v 24 = 0.
Consider the following elements of GL(2, 2):
x := 1 0 1 1 , y := 1 1 1 0 , z := 1 1 0 1 , and the following elements of
= 1, det 1 − x 1 − z 0 e 1 e 2 0 = det 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 = 1, det 1 − x 1 − y e 1 0 e 2 0 = det 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 0 = 1, det 1 − x 1 − z e 1 e 1 e 2 0 = det 0 0 0 1 1 0 0 0 1 0 1 0 0 1 0 0 = 1,
so either a 1 , b 1 as a 2 , b 2 satisfy the three conditions (1), (2) (3) and therefore
a 1 , b 1 = a 2 , b 2 = G. Now we want to prove that there is no b ∈ G with a 1 , b = a 2 , b = G. Let b = (h 1 , h 2 )(v 1 , v 2 , v 3 , v 4 )
, and assume by contradiction that a 1 , b = a 2 , b = G. We must have in particular that condition (1) holds, i.e. (x, x), (h 1 , h 2 ) = H.
Since (x, x) has order 2 and H cannot be generated by two involutions (otherwise it would be a dihedral group) at least one of the two elements h 1 , h 2 must have order 3: it is not restrictive to assume h 1 = y. Let v 1 = (α, β), v 2 = (γ, δ). Conditions (2) and (3) must be satisfied, hence we must have
det 1 − x 1 − y 0 v 1 e 2 v 2 = det 1 − x 1 − y e 1 v 1 e 2 v 2 = 1. However det 1 − x 1 − y 0 v 1 e 2 v 2 = det 0 0 0 1 1 0 1 1 0 0 α β 0 1 γ δ = α, det 1 − x 1 − y e 1 v 1 e 2 v 2 = det 0 0 0 1 1 0 1 1 1 0 α β 0 1 γ δ = α + 1.
However, since α ∈ F 2 either α = 0 or α + 1 = 0, so there is no b ∈ G with a 1 , b = a 2 , b = G.
A problem in linear algebra
Before to prove Theorem 1, we need to collect a series of results in linear algebra. Denote by M r×s (F ) the set of the r × s matrices with coefficients over the field F.
V with dim W 1 = dim W 2 , then V contains a subspace U such that V = W 1 ⊕ U = W 2 ⊕ U.
Lemma 13. Let v 1 , . . . , v n , w 1 , . . . w n ∈ F n , where F is a finite field and either |F | > 2 or n = 1. There exist z 1 , . . . , z n ∈ F n so that the two sequences v 1 + z 1 , . . . , v n + z n , w 1 + z 1 , . . . , w n + z n are both basis of F n .
Proof. Equivalently, we want to prove that for every pair of matrices A, B ∈ M n×n (F ), there exists C ∈ M n×n (F ), such that det(A+C) = 0 and det(B+C) = 0. Since either |F | > 2 or n = 1, every element of M n×n (F ) can be expressed as the sum of two units [11]. In particular
A − B = U − V with U, V ∈ GL(n, F ). We may take C = U − A = B − V.
Lemma 14. Let F be a finite field and assume r ≤ n. Given R ∈ M r×n (F ) and S ∈ M r×r (F ) consider the matrix R S ∈ M r×(n+r) . Assume rank R S = r and let π R,S be the probability that a matrix Z ∈ M r×n (F ) satisfies the condition rank(R + SZ) = r. Then π R,S > 1 − q r q n (q − 1) .
Proof. There exist m ≤ r, X ∈ GL(r, F ) and Y ∈ GL(r, F ) such that
XSY = I m 0 0 0 ,
where I m is the identity element in M m×m (F ). Since r = rank R S = rank X R S I n 0 0 Y = rank XR XSY and rank(R + SZ) = rank(X(R + SZ)) = rank(XR + XSZ)
= rank(XR + XSY (Y −1 Z)),
it is not restrictive (replacing R by XR, S by XSY and Z by Y −1 Z) to assume
S = I m 0 0 0 .
Denote by v 1 , . . . , v r the rows of R and by z 1 , . . . , z r the rows of Z. The fact that the rows of (R S) are linearly independent implies that v m+1 , . . . , v r are linearly independent vectors of F n . The condition rank(R + SZ) = r is equivalent to ask that v 1 + z 1 , . . . , v m + z m , v m+1 , . . . , v r are linearly independent. The probability that z 1 , . . . , z m satisfy this condition is
1 − q r−m q n 1 − q r−m+1 q n · · · 1 − q r−m+(m−1) q n .
Hence
π R,S = 1 − q r−m q n 1 − q r−m+1 q n · · · 1 − q r−m+(m−1) q n ≥ 1 − q r−m (1 + q + · · · + q m−1 ) q n = 1 − q r−m (q m − 1) q n (q − 1) > 1 − q r q n (q − 1)
.
Lemma 15. Assume that F is a finite field and that A, B 1 , B 2 , D 1 and D 2 are elements of M n×n (F ) with the property that
rank A B 1 = rank A B 2 = n, rank B 1 D 1 = rank B 2 D 2 = n.
Moreover assume that at least one of the following conditions holds:
(1) |F | > 2; (2) det A = 0; (3) n ≥ 2 and (det B 1 , det B 2 ) = (0, 0). Then there exists C ∈ M n×n (F ) such that det A B 1 C D 1 = 0 and det A B 2 C D 2 = 0.
Proof. Let r = rank(A). There exist X, Y ∈ GL(n, F ) such that XAY = I r 0 0 0 where I r is the identity element in M r×r (F ). Let B 11 , B 21 ∈ M r×n (F ) and B 12 , B 22 ∈ M (n−r)×n (F ) such that
XB 1 = B 11 B 12 , XB 2 = B 21 B 22 .
For i ∈ {1, 2}, since
n = rank A B i = rank X A B i Y 0 0 I n = rank I r 0 B i1 0 0 B i2 ,
it must be rank(B i2 ) = n − r. In particular there exists Z i ∈ GL(n, F ) such that
XB i Z i = B i1 B i2 Z i = B * i1 B * i2 0 I n−r with B * i1 ∈ M r×r (F ), B * i2 ∈ M r×(n−r) (F ). Notice that det XAY XB i Z i CY D i Z i = det X 0 0 I n A B i C D i Y 0 0 Z i = det(X) det(Y ) det(Z i ) det A B i C D i .
This means that it is not restrictive to assume
A = I r 0 0 0 , B i = B * i1 B * i2 0 I n−r with B * i1 ∈ M r×r (F ), B * i2 ∈ M r×(n−r) (F ). Let C 1 , D i1 ∈ M n×r (F ) and C 2 , D i2 ∈ M n×(n−r) (F ) such that C 1 C 2 = C and D i1 D i2 = D. Notice that det A B i C D i = det I r 0 B * i1 B * i2 0 0 0 I n−r C 1 C 2 D i1 D i2 = (−1) n det I r 0 B * i1 C 1 C 2 D i1 = (−1) n det I r 0 B * i1 C 1 C 2 D i1 I r 0 −B * i1 0 I n−r 0 0 0 I r = (−1) n det I r 0 0 C 1 C 2 D i1 − C 1 B * i1 = (−1) n det C 2 D i1 − C 1 B * i1
. Assume that we can find C 1 such that rank(D 11 − C 1 B * 11 ) = rank(D 21 − C 1 B * 21 ) = r and let W 1 , W 2 be the subspaces of F n spanned, respectively, by the columns of the two matrices D 11 − C 1 B * 11 and D 21 − C 1 B * 21 . By Lemma 12, there exists a subspace U of F n such that F n = W 1 ⊕ U = W 2 ⊕ U. If C 2 is a matrix whose columns are a basis for U, then det C 2 D 11 − C 1 B * 11 = 0 and det C 2 D 21 − C 1 B * 21 = 0 and C = (C 1 C 2 ) is a matrix with the request property. Set rank(R 1 + S 1 Z) = rank(R 2 + S 2 Z) = r.
Notice that R 1 , R 2 ∈ M r×n (F ), S 1 , S 2 ∈ M r×r (F ) have the property that rank R 1 S 1 = rank R 2 S 2 = r.
First assume that either |F | = q > 2 or r < n : by Lemma 14, we have π R1,S1 > 1 2 and π R2,S2 > 1 2 and this is sufficient to ensure that a matrix Z with the requested property exists. Therefore we may assume r = n (i.e. det A = 0) and q = 2. In this case, we assume also that at least one of the two matrices B 1 and B 2 is invertible. Let for example det B 1 = 0. This implies det S 1 = 0. There exist m ≤ n and X, Y ∈ GL(n, F ) such that
XS 2 Y = I m 0 0 0 . Notice that R 2 +S 2 Z is invertible if and only if XR 2 Y +XS 2 Y Y −1 ZY is invertible. Moreover R 1 + S 1 Z is invertible if and only if R 1 Y + S 1 Y Y −1 ZY is invertible, if and only if (S 1 Y ) −1 R 1 Y + Y −1 ZY is invertible.
This means that (replacing R 1 by (S 1 Y ) −1 R 1 Y , R 2 by XR 2 Y and Z by Y −1 ZY ) we may assume S 1 = I n and S 2 = I m 0 0 0 .
Let v 1 , . . . , v n be the rows of R 1 , w 1 , . . . , w n the rows of R 2 and z 1 , . . . , z n the rows of Z. Our request on Z is equivalent to ask that the sequences v 1 + z 1 , . . . , v m + z m , v m+1 + z m+1 , . . . , v n + z n , w 1 + z 1 , . . . , w m + z m , w m+1 , . . . , w n , are both linearly independent. Notice that the condition rank(R 2 S 2 ) = n implies in particular that w m+1 , . . . , w n are linearly independent. First assume m = 1. For j > m, let z j = v j + w j so that z j + v j = w j and let W = w m+1 , . . . , w n . We then work in the vector space F n /W of dimension m and our request is that the vectors v 1 + z 1 + W, . . . , v m + z m + W and the vectors w 1 + z 1 + W, . . . , w m + z m + W are linearly independent: Lemma 13 ensures that this request is fulfilled for a suitable choice of z 1 , . . . , z m . Finally assume m = 1. As before for j > 2, let z j = v j +w j and let W = w 3 , . . . , w n . We want to find z 1 and z 2 so that the two vectors v 1 + z 1 + W, v 2 +z 2 +W and the two vectors w 1 +z 1 +W, w 2 +W are linearly independent. This is always possible. First choose z 1 so that w 1 +z 1 +W / ∈ w 2 +W and v 1 +z 1 / ∈ W. Once z 1 has been fixed, choose z 2 so that v 2 + z 2 + W / ∈ v 1 + z 1 + W .
Remark 16. Notice that when |F | = 2 and det A = 0, we cannot drop the assumption (det B 1 , det B 2 ) = (0, 0). Consider for example
A = 0 1 1 1 , B 1 = B 2 = 0 0 1 0 , D 1 = 0 0 0 1 , D 2 = 1 0 0 1 .
Then, as we noticed at the end of Section 3, there is no C ∈ M 2×2 (F ) with
det A B 1 C D 1 = 0 and det A B 2 C D 2 = 0.
This restriction in the statement of Lemma 15 is indeed the reason why we cannot have diam(∆(G)) = 2 for every 2-generated finite soluble group G.
Proposition 17. Let F be a finite field and let n be a positive integer. Assume that either n ≥ 2 or |F | > 2. Assume that A 0 , A 1 , A 2 , A 3 , B 0 and B 3 are elements of M n×n (F ) with the property that
rank(A 0 A 1 ) = rank(A 1 A 2 ) = rank(A 2 A 3 )
and
rank A 0 B 0 = rank A 3 B 3 = n.
Then there exist B 1 , B 2 ∈ M n×n (F ) such that
det A 0 A 1 B 0 B 1 = 0, det A 1 A 2 B 1 B 2 = 0, det A 2 A 3 B 2 B 3 = 0.
Proof. Set i = 1 if either |F | > 2 or |F | = 2 and det A 1 = 0, i = 2 otherwise. a) Assume i = 1. First choose B 2 so that det
A 2 A 3 B 2 B 3 = 0.
Then, since either det A 1 = 0 or |F | > 2, by Lemma 15 there exists B 1 such that
det A 0 A 1 B 0 B 1 = 0, det A 1 A 2 B 1 B 2 = 0. b) Assume i = 2. First choose B 1 so that det A 0 A 1 B 0 B 1 = 0.
Then, since det A 1 = 0 or |F | > 2, by Lemma 15 there exists B 2 such that det
A 1 A 2 B 1 B 2 = 0, det A 2 A 3 B 2 B 3 = 0.
Corollary 18. Let K be a non-trivial 2-generated linear soluble group acting irreducibly and faithfully on V and consider the semidirect product G = V δ ⋊ K with δ ≤ n = dim EndG(V ) V. Assume that there exists x 0 , x 1 , x 2 , x 3 , in K such that (1) x 0 w 0 and x 3 w 3 are non isolated vertices in the generating graph of G,
(2) x 0 , x 1 = x 1 , x 2 = x 2 , x 3 = K. Then there exist w 1 , w 2 ∈ G with x 0 , x 1 w 1 = x 1 w 1 , x 2 w 2 = x 2 w 2 , x 3 = G.
Proof. Since V δ ⋊ K is an epimorphic image of V n ⋊ K, it suffices to prove the statement in the particular case G = V n × K. We may identify x 0 , x 1 , x 2 , x 3 with X 0 , X 1 , X 2 , X 3 ∈ GL(n, F ), where F = End G (V ) and w 0 , w 1 , w 2 , w 3 ∈ V n with four matrices B 0 , B 1 , B 2 , B 3 in M n×n (F ). We now apply Proposition 10. Let
A 0 = I n − X 0 , A 1 = I n − X 1 , A 2 = I n − X 2 , A 3 = I n − X 3 .
Conditions (1) and (2) implies that
rank(A 0 A 1 ) = rank(A 1 A 2 ) = rank(A 2 A 3 ) and rank A 0 B 0 = rank A 3 B 3 = n.
Moreover the statement is equivalent to say that there exist B 1 , B 2 ∈ M n×n (F ) with
det A 0 A 1 B 0 B 1 = 0, det A 1 A 2 B 1 B 2 = 0, det A 2 A 3 B 2 B 3 = 0.
The existence of B 1 and B 2 is ensured by Lemma 15 (notice that the fact that K is a non-trivial subgroup of GL(n, F ) implies that n ≥ 2 if |F | = 2).
Proof of Theorem 1
At the beginning of Section 3 we recalled some properties of the crowns of a finite soluble group. In the proof of Theorem 1, we will use other two related results.
Lemma 19. [1, Lemma 1.3.6] Let G be a finite solvable group with trivial Frattini subgroup. There exists a crown C/R and a non trivial normal subgroup U of G such that C = R × U.
Lemma 20. [6,Proposition 11] Assume that G is a finite soluble group with trivial Frattini subgroup and let C, R, U as in the statement of Lemma 19. If HU = HR = G, then H = G.
Proof of Theorem 1. We prove the theorem making induction on the order of G. Choose two non-isolated vertices x and y in the generating graph of G. Let F = Frat(G) be the Frattini subgroup of G. Clearly xF and yF are non-isolated vertices of the generating graph of G/F. If F = 1, then by induction there exists a path xF = g 0 F, . . . , g n F = yF in the graph Γ(G/F ), with n ≤ 3. For every 0 ≤ i ≤ n−1, we have G = g i , g i+1 F = g i , g i+1 , hence x = g 0 , . . . , g n = y is a path in Γ(G). Therefore we may assume F = 1. In this case, by Lemma 19, there exists a crown C/R of G and a normal subgroup U of G such that C = R × U. We have R = R G (A) where A is an irreducible G-module and U ∼ =G A δ for δ = δ G (A). By induction the graph Γ(G/U ) contains a path xU = g 0 U, g 1 U, . . . , g n−1 U, g n U = yU with n ≤ 3. We may assume n = 3 : indeed if n = 1 we may consider the path g 0 U, g 1 U, g 0 U, g 1 U and if n = 2 we may consider the path g 0 U, g 0 g 1 U, g 1 U, g 2 U. So we are assuming (5.1)
x, g 1 U = g 1 , g 2 U = g 2 , y U = G.
We work in the factor groupḠ = G/R. We haveC = C/R = U R/R ∼ = U ∼ = A δ and either A ∼ = C p is a trivial G-module andḠ ∼ = (C p ) δ orḠ =Ū ⋊H ∼ = A δ ⋊ K where K ∼ =H acts in the same say on each of the δ factors of A δ and this action is faithful and irreducible. SinceḠ is 2-generated, we have δ ≤ 2 if A is a trivial G-module, δ ≤ n := dim EndG(A) A otherwise. By Theorem 2 in the first case (we are working in the nilpotent group A δ ) and by Proposition 18 in the second case, there exist u 1 , u 2 ∈ U with x,ḡ 1ū1 = ḡ 1ū1 ,ḡ 2ū2 = ḡ 2ū2 ,ȳ =Ḡ. i.e.
(5.2) x, g 1 u 1 R = g 1 u 1 , g 2 u 2 R = g 2 u 2 , y R = G.
By Lemma 20, from (5.1) and (5.2), we deduce
x, g 1 u 1 = g 1 u 1 , g 2 u 2 = g 2 u 2 , y = G.
1 :=(x, x)(0, e 2 , 0, e 2 ), a 2 := (x, x)(e 1 , e 2 , e 1 , e 2 ) b 1 :=(y, z)(e 1 , 0, e 1 , 0), b 2 := (y, z)(0, 0, e 1 , 0). It can be easily checked that (x, x), (y, z) = H.
Lemma 12 .
12[4, Lemma 3] Let V be a finite dimensional vector space over the field F . If W 1 and W 2 are subspaces of
R 1 =
1D T 11 , R 2 = D T 21 , S 1 = B * T 11 , S 2 = B * T 21 , Z = −C T 1 . The previous observation implies that a matrix C with the requested properties exists if and only if there exists Z ∈ M r×n (F ) such that (4.1)
Andrea LucchiniUniversità degli Studi di Padova Dipartimento di Matematica "Tullio Levi-Civita" Via Trieste 63, 35121 Padova, Italy email: [email protected]
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Crowns and factorization of the probabilistic zeta function of a finite group. E Detomi, A Lucchini, J. Algebra. 2652E. Detomi and A. Lucchini, Crowns and factorization of the probabilistic zeta function of a finite group, J. Algebra, 265 (2003), no. 2, 651-668.
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[] |
[
"A Machine Learning-based Recommendation System for Swaptions Strategies",
"A Machine Learning-based Recommendation System for Swaptions Strategies"
] |
[
"Adriano Soares Koshiyama \nDepartment of Computer Science\nUniversity College London London\nUnited Kingdom\n",
"Nick Firoozye \nDepartment of Computer Science\nUniversity College London London\nUnited Kingdom\n",
"Philip Treleaven \nDepartment of Computer Science\nUniversity College London London\nUnited Kingdom\n"
] |
[
"Department of Computer Science\nUniversity College London London\nUnited Kingdom",
"Department of Computer Science\nUniversity College London London\nUnited Kingdom",
"Department of Computer Science\nUniversity College London London\nUnited Kingdom"
] |
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Derivative traders are usually required to scan through hundreds, even thousands of possible trades on a daily basis. Up to now, not a single solution is available to aid in their job. Hence, this work aims to develop a trading recommendation system, and apply this system to the so-called Mid-Curve Calendar Spread (MCCS), an exotic swaption-based derivatives package. In summary, our trading recommendation system follows this pipeline: (i) on a certain trade date, we compute metrics and sensitivities related to an MCCS; (ii) these metrics are feed in a model that can predict its expected return for a given holding period; and after repeating (i) and (ii) for all trades we (iii) rank the trades using some dominance criteria. To suggest that such approach is feasible, we used a list of 35 different types of MCCS; a total of 11 predictive models; and 4 benchmark models. Our results suggest that in general linear regression with lasso regularisation compared favourably to other approaches from a predictive and interpretability perspective.
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[
"https://arxiv.org/pdf/1810.02125v1.pdf"
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1810.02125
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96af77c627fabcec662622a634dea07d8e82d4f2
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A Machine Learning-based Recommendation System for Swaptions Strategies
October 5, 2018
Adriano Soares Koshiyama
Department of Computer Science
University College London London
United Kingdom
Nick Firoozye
Department of Computer Science
University College London London
United Kingdom
Philip Treleaven
Department of Computer Science
University College London London
United Kingdom
A Machine Learning-based Recommendation System for Swaptions Strategies
October 5, 2018
Derivative traders are usually required to scan through hundreds, even thousands of possible trades on a daily basis. Up to now, not a single solution is available to aid in their job. Hence, this work aims to develop a trading recommendation system, and apply this system to the so-called Mid-Curve Calendar Spread (MCCS), an exotic swaption-based derivatives package. In summary, our trading recommendation system follows this pipeline: (i) on a certain trade date, we compute metrics and sensitivities related to an MCCS; (ii) these metrics are feed in a model that can predict its expected return for a given holding period; and after repeating (i) and (ii) for all trades we (iii) rank the trades using some dominance criteria. To suggest that such approach is feasible, we used a list of 35 different types of MCCS; a total of 11 predictive models; and 4 benchmark models. Our results suggest that in general linear regression with lasso regularisation compared favourably to other approaches from a predictive and interpretability perspective.
Introduction
Derivative traders are usually required to scan through hundreds, even thousands of possible trades on a daily basis. A concrete case is the so-called Mid-Curve Calendar Spread (MCCS), a derivatives package that involves selling an option on a forward-starting swap and buying an option on a spot-starting swap with longer expiration [8,26]. In such a package, traders look for the historical carry and the breakeven width levels, metrics that can be easily inferred from the terminal or aged payoff profile of the MCCS, shown in several heatmaps made by the research team. After that, they rank the most prominent ones to offer a client or to proceed in some proprietary trading. In general, the straightforwardness and swiftness that the decisions are made is the main upside of this framework.
However, one might notice that the main downsides of such approach are: (i) substantial information on the underlying like sensitivities, implied volatility, etc. are usually not taken into account; (ii) using the previous example, high historical values for carry and breakeven widths are more necessary rather than sufficient conditions for a profitable MCCS trade, being such argument extensible to other trades as well; (iii) a trader can quickly judge if an individual trade is worthwhile to invest, but may take some time to find it; and (iv) after a given period, traders tends to only look at a small subset of possible trades (small area on the heatmap), rather than the all available selection. Hence, a systematic approach where more information at hand is crossed and aggregated to find good trading picks and undoubtedly increase the trader's productivity.
Therefore, the objective of this work is to develop a trading recommendation system that can aid derivatives traders in their day-to-day routine. Being more specific, our solution is based on the following pipeline: (i) on a certain trade date, we compute metrics and sensitivities related to MCCS; (ii) these metrics are feed in a model that can predict its expected return for a given holding period; and after repeating (i) and (ii) for all trades we (iii) rank the trades using some dominance criteria. Our final solution is a model-based heatmap with the attractiveness scores for each trade, which can be offered to the traders and salespeople on a daily basis.
In this sense, we organised this work as follows: next section presents a literature review on existing approaches to return/price prediction/estimation in different areas and instruments, as well as a brief description on MCCS trades. The third section displays the dataset that comports the MCCS trades, showing how the information is computed and gathered, which variables are the input and outputs, and the main assumptions that are embedded in it. Then, we move to modelling strategy, highlighting the main models that are going to be used as candidates for the recommendation system, how they are tested and have their performance assessed. Finally, we exhibit the results and discussions, closing this work with some concluding remarks and future directions for research.
Background
Related Works
Literature provides a growing body of evidence that price changes can be predicted, that is, in particular circumstances and periods securities violate the Efficient Market Hypothesis [5,24]. In this sense, researchers have employed different modelling approaches and information sets to predict price changes across a range of assets. When we scan the literature for cash instruments (equities, bonds, foreign exchange, etc.) focused only in using past returns as the main source for prediction, we can find works that tap into Bayesian forecasting [33], Nonparametric Predictive Inference [2], Forecasting Combination [12], Generalized Exponential Weighted Moving Average [25], Support Vector Machines (SVM) [22], Shallow and Deep Neural Networks architectures [7,9,18,32], Random Forest and Gradient Boosting Trees [23], and so forth. The list of proposed methodologies keeps growing, in which equities or indices appears as the dominant asset class to apply these algorithms. Collectively, they provide evidence that some forecastability over returns can be achieved by putting in place complex models with a suitable training scheme.
Contrasting with the emphasis that researchers in cash instruments put on return predictability, when we devote our attention to research in derivatives instruments (options, swaps, swaptions, etc.) it is clear that most of the effort is concentrated on pricing these contracts. In parallel to the traditional framework, alternative ways of pricing and trading started to emanate relying on fewer assumptions and more data-driven. We can pinpoint approaches that use Neural Networks for option pricing and hedging with daily S&P 500 index daily call options [17] as well as for real-time pricing and hedging options on currency futures of EUR/USD at tick level [31]. It is worth to mention other approaches in the derivatives realm, such that the prediction of pricing and hedging errors for equity-linked warrants with Gaussian Process models [19], building machine learning models for predicting option prices over KOSPI 200 Index options [27] and a general study on forecasting option price distributions using Bayesian kernel methods [28].
When we devote our attention to the asset type that this work is dedicated, interest rate swaptions, a similar pattern persists: most of the research is related to pricing and not to return prediction. Regarding pricing, the same tradition of relying on stochastic calculus techniques is followed [4,29]. Regarding potential alternatives using more data-driven approaches as we saw with currency, indices and equities options, we can only mention the work of Souza et al. [30] which calibrates the Vasicek interest rate model under the risk neutral measure by learning the model parameters using Gaussian processes for regression. Considering trading strategies and return prediction, we can find even less academic research, being perhaps most of the research residing inside the counterparts that exchange such products (banks, hedge funds, etc.). This shortage of published research might be linked with the absence of ready to use and publicly available datasets, similar to the ones found in cash products since these instruments are traded off-exchange.
Based on this review of existing approaches to return/price prediction/estimation in different areas and instruments, to the best of our knowledge, our work is the first attempt to build a trading recommendation system in the context of derivatives. Our approach is not the only novel from a modelling perspective, but instead of trading the vanilla product (receiver/payer interest rate swaption), we prefer to focus on options strategies (calendar spreads, straddles, etc.) which in many cases is the package that is in practice traded. By thinking in terms of the package, in this case, a Mid-Curve Calendar Spread, rather than the individual constituents we unlock some features that can only be computed in this situation, like the carry at expiry, breakeven width and so on. Therefore, we can train our models not only using past returns but also using sensitivities as well as information derived from the package payoff function. By portraying in this manner our investment strategy, we have a large information set that can substantially add information to aid forecasting returns. But as a counter-effect, this poses a new challenge on separating relevant features in a dynamic context. In this respect, the combination of temporal cross-validation, a diverse set of models and regularisation/feature selection can provide a robust framework for trading strategies backtesting and assessment. But before presenting such framework, next section gives a brief view on MCCSs trades.
Mid-Curve Calendar Spreads
Mid-Curve Calendar Spread (MCCS) is a package involving short selling an option on a forward-starting swap and going long a longer-expiry swaption on the same underlying swap [8]. There is a counterpart with many similarities for equities -check in [26] for more information. Investors typically use MCCS to take a view on forwarding volatility. This comes from the fact that, conceptually, spot volatility can be decomposed into forward volatility and mid-curve volatility. Taking 10y10y 1 for example, Figure 1 illustrate the time periods covered by different interest rate volatilities and their instruments. The red lines indicate the time over which interest rate volatility exposure is taken, and the grey line indicates the underlying forward swap rate. Figure 1: Mid-curve Swaption: 5y mid-curve on 5y10y swap rate -the volatility of a forward-starting swaption, called mid-curve, whose strike is set at inception and but the underlying swap starts several years following the option expiry date. We plot the payoff profiles for current volatility and up and down volatility scenarios, noting that the long vega position means that the payoff profile shifts up in a rising volatility environment and correspondingly shifts down in a falling vol environment. We calculate the (volatility adjusted) breakevens as being 0.41% − 0.47%, giving little protection against selloffs. We note that forwards in a ±1 volatility band leave them at 0.40% − 0.48%, a range just marginally larger than our breakeven range (i.e., the trade should pay off just slightly less than 66% of the time). MCCS can result in what we think of as turbocharged carry, primarily because of the risks that they have (which fortunately can be balanced with the returns in a way which results in relatively attractive trades). Based on these characteristics of MCCS, next section presents how we elaborated the methodology to build this trading recommendation system.
Methodology
In summary, our solution develops the following roadmap (also schematically described in Figure 2. Modelling: These metrics are feed in a predictive model that outputs its expected return for a given holding period (e.g., one year); 3. Recommendation: After repeating (i) and (ii) for all MCCS we (iii) rank them based on the expected returns using some criteria.
Following this outlined structure, the next three subsections describe in more details when and which MCCS trades were recorded (Dataset), which predictive models were trained and how they were assessed (Modelling) and how the long/short trading signal is computed for each MCCS (Recommendation). Finally, last subsection presents which metrics were used to evaluate the recommendation system performance when a certain predictive model candidate is underpinning it.
Dataset
During our experiments, we opted to use the trades displayed in Table 1. Although many other configurations are available in practice, these are the ones with longest historical data available, which is important when it is necessary to fit a predictive model. As it can be seen, all trades are in Euro, ranging from different expiries (1y-5y), forwards (1y-5y) and swap tenures (1y-5y and 8y). For each configuration, at time t we agree with a counterpart to trade this package using the At the Money Forward (ATMF) rate as the strike, paying or receiving the present value P V t . The P V t is computed via SABR model [29], using information and parameters (e.g., spot, forward rates and rate-rate correlation) calibrated using market data on a daily basis. From the same model that computed the P V t , we can also obtain other metrics and sensitivities as those displayed in Table 2. Carry and BE Width are those obtained looking at the payoff profile at expiry. The Aged 1y Carry is produced by ageing the trade by one year (moving closer to the expiration) and estimate the payoff profile computing the carry. Theta, Vega and Gamma are the sensitivities of the instruments by a change in time, volatility and a wider range of underlying rate movements, respectively. These and the ATMF Implied Vol are backed by the SABR model too. Curve, Time and Volatility Carry are the amount of Aged 1y Carry that can be attributed to the changes in certain sensitivities from spot to forward, such as the Delta (Curve), Theta (Time) and Vega (Volatility). These can also be used as tools to understand which factors most influence the instrument value over time.
After computing all these metrics at time t, we hold the trade until t + h where h can be two weeks, one month, one year, and so on, as long as t + h is before or at expiration. In time t + h we compute the P V t+h of the same trade again, using the new economic scenario available (e.g. rates, change in model parameters). By agreeing on buying back or selling the current trade for P V t+h we can compute the Holding k-period Return of the trade started at time t by:
R (h) t = P V t+h − P V t P V t(1)
In summary, Table 3 presents an example of information in a wide format that is available when we combine the data from time t and t + h. Table 3: Example of information available at time t and t + h for the MCCS.
Instant (t) P Vt R (h) t R (h) t−h−1 Strike Features 1 300 0.3 - 3.4 ... 2 320 0.2 - 3.2 ... ... ... ... ... ... ... t + h − 1 250 -0.1 - 1.8 ... t + h 260 -0.05 - 1.9 ... t + h + 1 270 -0.2 0.2 2.0 ... ... ... ... ... ... ... T 250 - 0.1 2.2 ...
Note that in the most contemporaneous period (close to T ) we do not have the P V T +h and so, we cannot compute R
(h) T . Conversely, if we want to use lagged returns R (k) t−h−1 as explanatory forces for R (h)
t , then in the beginning this information is also not available. Therefore, our dataset is trimmed at the beginning and the end mainly by the value of h. If h is small, such as two weeks or one month, the trimming is imperceptible and, therefore, may not affect the model fitting and validation. However, if h is large such as two or three years, this might reduce the samples available substantially, decreasing the range of models and cross-validation schemes that might be employed for this task. Based on these procedures, metrics and observations, Table 4 express other details that we used during our experiments to generate the dataset. Therefore, we gathered data from trades entered on a weekly basis from September 2006 to September 2016. These trades are struck ATMF, using the P V t computed from the Middle Rate (in practice, some bid-ask spread would be imbued proportional to the Vega). After holding for one year (h = 1y) the trade, we compute the arithmetical returns that are, therefore by definition, automatically annualised. These returns are gross, and so we need to take into account the transaction costs (hedging costs and fixed fees charged by the derivatives desk) as well as some future funding rate. These values are also outlined in Table 4, where the transaction costs of 0.75 as a fraction of Vega were chosen not only to taken into account the transaction cost, but also some potential bid-ask spread on the start/unwind of the trade. The 3-month London Interbank Overnight Rate (LIBOR) was chosen as the funding cost/benchmark rate to compute excess returns.
Using these assumptions, next subsection presents the modelling strategy that taps into this dataset to create the recommendation system for the MCCS trades.
Modelling
In relation to modelling, our general model is a system of uncoupled equations:
R (1y) t,1 = f 1 (f eatures t,1 ) + ε t,1 =R (1y) t,1 + ε t,1 (2) R (1y) t,2 = f 2 (f eatures t,2 ) + ε t,2 =R (1y) t,2 + ε t,2(3)
...
R (1y) t,n = f n (f eatures t,n ) + ε t,n =R (1y) t,n + ε t,n(4)
where for each MCCS trade (i = 1, ..., n) there is an i-th predictive model f i that is feed with a set of pre-calculated features (BE Width, Carry, etc.) and returns an estimate of the holding 1y-period returnR (1y) t,i . As the model is an approximation, some noise/error is expected, and in the modelling aspect, this is expressed as the ε t,i component. After defining which variable is intended to be predicted, the remaining points are: which models are available to embody f i and how the fitting, validation and selection of these models are going to be made.
About the first point, in the first rows of Table 5 we display the models that we used during our experiments, with their mathematical descriptions and usage found in the following references [3,11,16,20]. In Table 5 Model column presents a plethora of models that this work has fitted for this prediction purpose: we started from simple predictive models such as Classical Linear Regression, k-Nearest Neighbours and Classification and Regression Tree, towards those that can seamlessly exhibit nonlinear behaviours, like Random Forest, Kernel Ridge Regression, Multi-Layer Perceptron and Support Vector Regression. Some of these methods had their hyperparameters held constant across all experiments (Fixed Hyperparameters column), or because we wanted to apply a particular form of a method (RBF kernel, single hidden layer, etc.) or because during a warm-up phase we noticed that they did not affect substantially the results (hyperbolic tangent, increasing number of trees, etc.).
For certain models, the Cross-Validated Parameters column shows which hyperparameters were optimised before the prediction step. For instance, suppose the case of Ridge Regression and the need to define the regularisation value (λ) appropriately. Consider that we have a set of training pairs (f eatures t , R (1y) t ) L t=1 of size L, and for this sample we subset it in k-rolling-cross-validation (k-rollingcv) folders (better explained later in this subsection). Then, we train and test using this scheme the Ridge Regression model with one of the predefined λ, say λ = 10 0 . We compute some performance function on the test set (Mean Squared Error -MSE) and repeat this process for all λ values available. We use in the final model the λ that on average had the lowest MSE.
We fitted usual benchmarks found in the literature for regression and forecasting modelling: the Average and Naive models [16,21]. We also implemented the benchmarks that traders use to assess whether a particular MCCS is worth to be pitched or traded: BE Width and Carry at Expiry. We replicated the way traders look to these features, by computing z-scores 3 based on average and standard deviation on rolling window of size equal to 1 year. The signal for going long/short is done by a thumb rule with a simple rationale: if a certain metric has a z-score above or equal to ±3, the trader goes fully long (+)/short(-) in the trade, since it is a very extreme event. Otherwise, it reduces the leverage on it, until it below one standard deviation of distance from the rolling average.
We removed any missing data, and clipped extremes values, mainly in returns above the 95% percentiles (in our case it can be due to some numerical problems, or some extreme scenarios related to 2008-2009 financial crisis period). Next subsection presents the final component of our roadmap: recommendation system.
Recommendation
The recommendation of a certain trade can be made solely on some normalised version of the expected return for holding 1y-period the i-th trade (R (1y) t,i ). Given that each model will be providing individual forecasts for each MCCS and after that their performance will be assessed locally and globally, a more suitable manner to proceed would be to assign a credit based on the tracking record of a model to predict a particular MCCS trade. Hence, we will be weighted up or down a signal not only based on the magnitude of a model prediction but also by its quality. Then, consider asR (1y) t,i the expected return for holding 1y-period 3 a z-score is defined by: Z − score = X−µ σ where X represent the actual value of a certain variable, µ and σ the average and standard deviation of X in a period. the i-th trade. Now, define the new signal function S t,i by:
S t,i =R (1y) t,i × RhoR(1y) t,i ,R (1y) t,i max(|R (1y) t,i × RhoR(1y) t,i ,R (1y) t,i |, ..., |R (1y) t−h,i × RhoR(1y) t−h,i ,R (1y) t−h,i |, 0)
where the strength of the i-th long/short signal is given by its expected return, scaled by the maximum weighted return that a long/short position on the same trade (that is why the returns are in absolute terms) was expected to yield in the previous h-period (in this case 1 year). Therefore, the trade with the maximum weighted return in absolute terms will have |S t,i | = 1 as well as those close to zero will yield S t,i ≈ 0. The weight/credit of a certain prediction is based on the historical Pearson correlation coefficient, that is, adherence between the actual and predicted values.
Evaluation Metrics
Below we outline two types of metrics: one that focuses on the predictive performance that the model provided, and other three that are based on the profit/loss that its application harvested during the backtest. Set by R
(S) t = R (1y) t × S t (R (1y) t
) the strategy return (combination of the realized/observed excess returns and the signal -function of a model prediction) 4 , we can compute the following metrics:
• Pearson Correlation Coefficient (Rho): it is a dimensionless measure of the linear dependence between the actual and predicted values:
Rho = Cov[R (1y) t ,R (1y) t ] V ar[R (1y) t ]V ar[R (1y) t ](5)
where Cov and V ar are the covariance and variance operators. It ranges from [−1, +1], with −1 representing a perfect inverse linear association, and +1 the opposite. In our case, we benefit more when Rho is close to +1. In the context of linear models, a higher predictive power is a necessary condition for profitable trades (see [1]), hence by minimising the predictive error we are somewhat trailing a path for profits maximisation, albeit such causation is not very clear since this is not a sufficient condition.
• Average Return (Avg Return): is the arithmetic average of the strategy returns:R
(S) = T t=1 R (S) t T(6)
• Standard Deviation: is the estimator of the dispersion around the strategy average returns (a risk measure in certain sense):
σ R (S) = T t=1 (R (S) t −R (S) ) 2 T(7)
• Information Ratio: is the average annualized return of a strategy earned in excess of a particular benchmark per unit of risk (measured in terms of standard deviation):
IR =R (S) −B σ R (S)(8)
whereB is the average return of the benchmark (e.g., treasury bond, equity index). In our case, it was already set to the 3-month LIBOR rate (Table 4). It should be mentioned that Information Ratio makes each strategy performance comparable: since we are adjusting average returns by the risk assumed for each strategy, it removes the leverage component that is magnifying/shrinking the returns provided by a certain strategy. Similarly, different remarks can be made over the global picture: (i) the Naive and Mean Pred models underperformed, but the traders benchmarks did perform reasonably well, surpassing the predictive models in many occasions; (ii) from the linear regression family, Lasso Regression followed by Ridge Regression are the ones that performed better; (iii) most nonlinear models failed to provide a decent average return; and (iv) MLP fared well for the trades in the EUR Xy1y1y range, but did not repeat a more stable across other trades. When we take into account the variability seen in the stream of returns generated by the recommendation system, we may encounter a different picture. Figure 5 shows a heatmap with the Information Ratio for all the available combinations of models and trades.
Results and Discussions
In general, the models kept their positions unaltered in comparison to the Average Return (%) -linear models still fared better than the nonlinear ones -, but now all are standing on a similar scale. Based on these Information Ratio results, Table 6 presents a statistical analysis using the average ranks 5 , Friedman test and Holm posthoc procedure [10].
When we look at the average rank, Lasso Regression was the top positioned (3.23) while Mean Pred remained most of the time as the worst choice (12.86). The trader's benchmarks performed pretty well, being placed in the third and fourth places. When we compare whether such result fared by Lasso Regression was substantially different from Ridge Regression (4.86), we arrive with a Z-score equal to 1.63 and a p-value of 0.0517. If we set our initial significance level as 0.05 and correct using the Holm procedure (last column) we can assert that Lasso did not perform significantly different from Ridge Regression, but way better than the other models. Therefore, Lasso Regression is capturing some information beyond that is being spanned by the trader's benchmarks, as well as beating almost all other predictive models for this particular task. Our throughout analysis suggests that Lasso Regression seems to provide in general the best results across a range of metrics and criteria. Figure 6 uses boxplots as a visualisation tool to decompose the aggregated results shown before, by informing per trade the returns obtained from using the Lasso Regression trading recommendation system.
Overall, some patterns can be spotted from the boxplots: (i) in general the medians are located above zero, meaning that more than half of the trades tended to yield positive returns; (ii) Lasso Regression is exploring long/short position regardless of the most frequent outcome for each MCCS tenure; (iii) it tended to perform well for EUR 3yXyXy trades and since these tended to be historically a challenging pick (medians are centred to zero in these trades), it means that the model is actually capturing some signal from the data and not naively guessing long/short positions; and (iv) the returns distribution, mostly with higher forward and swap tenure (second and third row), tended to be right- skewed -because the third quartile is far from the median, whereas the first is squeezed towards the median. This last fact denounces that Lasso Regression frequently generates small negative outcomes, and dangerous scenarios are not as likely, which tends to be a desired property for quantitative strategies in general. Given that, we now look at the aggregated returns harvested by Lasso Regression during its test phase. These results are consolidated in Figure 7, where: (i) the top plot shows the average return with standard deviations obtained across all MCCS per trading week; (ii) the middle reveal histograms, where the left one represents the returns obtained across all MCCS regardless of the trading date, while the right ones displayed the same data but conditioned per position; and (iii) finally the bottom image presents the trading success rate for each long/short position suggested by the model (left), with the break down by long/short position displayed as well (right). To clarify, trading success in this context means being long/short when the returns of trade were positive/negative regardless of its magnitude. In respect to the top image, we can see that Lasso Regression started well during the first year but suffered a drawdown in the second to third year. This period was marked by higher volatility, mainly due to the final developments of the Euro Crisis period (2010-2012). However, from the third year onwards the average returns always scored positive values, usually ranging from 10% to 20% in average. Such performance can be seen stamped on the middle left histogram, where the bulk of returns lies above zero, and not only that but concentrated close to 15%. This performance was largely generated by Lasso Regression suggesting short positions (middle right), while the long positions were not so successful. Such pattern can be better seen in the histogram located at the bottom, where a bimodal distribution for trading recommendation success rate is depicted. Probably the verified outperformance coming from taking short positions in the MCCS is linked with betting against the volatility/variance risk-premium trade [6]. Roughly this strategy harvest the premium paid by a counterpart for the insurance on large swings in the market (almost the same as selling a put for equities options). Since in general, the market tends to remain range-bounded, the investor shorting the trade can repurchase it later for a smaller premium, profiting from this differential. Lasso Regression did dynamically the opposite and profited from it, largely because in this last 5-6 years was populated of higher volatility periods and tail events. Figure 8 help us to analyse which features are being most significant by Lasso Regression for each particular trade.
Each cell corresponds to a normalised t-stats 6 from the model coefficients built in the last step from k-rolling-cv. Implied Vol was the most significant feature pointed out by Lasso Regression, is negatively related with the MCCS returns. Other important features were the BE Width -slightly positively correlated with returns -and the Carry at Expiry -negatively related, but probably due to the depressed levels of carry that has been seen in the last batch of data. Lasso Regression promoted in general very sparse models, being Figure 6: Boxplot with the returns obtained from Lasso Regression for each instrument. All y-axes were fixed between 0.3 and -0.3 (30% and -30% annualised return) to facilitate their visualisations and comparisons. the other features playing specific roles for some trades like Time Carry for short-dated trades. Finally, the lagged returns were just relevant for few trades, and perhaps could be omitted for certain trades in the future to guarantee a broader dataset. We close this section showing results of Lasso Regression for a specific trade: EUR 4y5y5y - Figure 9.
Starting from the top, it can be seen that the Lasso Regression was able to predict reasonably well the observed returns after 1 year holding this trade. It is not perceived any over/underestimation of values, with the mirror-shape format indicating a good fit. This observation is reflected in the middle image, where the long/short positions track well the observed returns of the EUR 4y5y5y. The main mistakes are due to events that were not incorporated into the model and perhaps were also unforecastable: (i) March to April 2012, possibly due to the Greek Debt Restructuring Agreement; (ii) May to November 2013, linked with the Taper Tantrum event in the US and its effect on Europe rates. Although such events have influenced in the strategies return, last plot shows that the trading success of Lasso Regression has attained a historical Area Under the Curve (AUC) of 0.85, with the capacity to control the false alarms to 5% and still recommend trades accurately 40% of the time. This is a good indicator since in the onset of the recommendation system it is better to reduce the chances of suggesting a bad trade than missing a good pick.
Conclusions
This work proposed a trading recommendation system for Mid-Curve Calendar Spread Trades (MCCS). We proposed a recommendation system that could analyse and rank a set of fixed income derivatives trades. Our first experiment is designing and applying this method for Mid-Curve Calendar Spread trades. Therefore, we started the methodology by showing the dataset: it comprised of 35 MCCS trades, ranging from September 2006 to September 2016, with different expirations, forward and swap tenures. For each particular trade, we described how the sampling of inputs (metrics, sensitivities and lagged returns) and outputs (returns from unwinding the trade after one year of its start) were computed on a weekly basis. Then, we displayed the modelling strategy by highlighting the models that were trained as well as which hyperparameters were investigated during the nested resampling step. Before entering the results section, we presented the backtesting setting with the performance measures used to compare different methodologies.
Most models provided results better than the modelling benchmarks (Mean and Naive), yet very few were able to outperform the trader's benchmarks. Our results suggested that linear models with shrinkage procedures (e.g., Ridge and Lasso) tended to perform better than their nonlinear counterparts (like Kernel Ridge Regression, SVR and MLP). Also, regarding interpretability, they tend to be easier to convey to the traders, since most are versed in linear models. When we delved into Lasso Regression results, we found out that this model wielded some interesting features like: (i) it learned a type of volatility buying/selling strategy without being programmed to do so; (ii) its returns distribution across all MCCS tended to be right-skewed, meaning that we are more hedged towards dangerous scenarios with greater chances of upsides; (iii) it matched traders view on selecting good trades, but adding some dynamic view on it since Carry at Expiry is now negatively linked with returns, rather than the original view from the traders. We believe that Lasso Regression will be our choice for a first version of the trading recommendation system, with future developments giving space to different models and mixed approaches.
Figure 2
2presents the payoff profile for an EUR 1m1y2y 2 .
Figure 2 :
2Payoff profile for an EUR 1m1y2y.
Figure 3 :
3Flowchart describing the input-output schemes from the proposed trading recommendation system for MCCS trades.1. Data: On a certain trade date, we calculate metrics and sensitivities related to an MCCS package;
Abbreviation
Figure 4
4displays a heatmap with the results of all models for each trade regarding Average Return (%).
Figure 4 :
4Heatmap with the historical Average Return (%) during test set.
Figure 5 :
5Heatmap with the Information Ratio during the test set.
Figure 7 :
7Aggregated returns over the period and histogram of aggregated returns and success rate from trades suggested using Lasso Regression.
Figure 8 :
8Heatmap with the Feature Significance (%) obtained from normalised t-stats of Lasso Regression coefficients.
Figure 9 :
9Lasso Regression results for EUR 4y5y5y: predicted versus observed values, returns and long/short signals over the period and receiving operating characteristic curve based on the success rate.
Table 1 :
1Configuration of the MCCS trades used.Currency
Expiry
Forward
Swap
Currency
Expiry
Forward
Swap
EUR
1y
1y
1y
EUR
3y
3y
2y
EUR
1y
1y
4y
EUR
3y
4y
1y
EUR
1y
2y
3y
EUR
3y
5y
5y
EUR
1y
2y
8y
EUR
4y
1y
1y
EUR
1y
3y
2y
EUR
4y
1y
4y
EUR
1y
4y
1y
EUR
4y
2y
3y
EUR
1y
5y
5y
EUR
4y
2y
8y
EUR
2y
1y
1y
EUR
4y
3y
2y
EUR
2y
1y
4y
EUR
4y
4y
1y
EUR
2y
2y
3y
EUR
4y
5y
5y
EUR
2y
2y
8y
EUR
5y
1y
1y
EUR
2y
3y
2y
EUR
5y
1y
4y
EUR
2y
4y
1y
EUR
5y
2y
3y
EUR
2y
5y
5y
EUR
5y
2y
8y
EUR
3y
1y
1y
EUR
5y
3y
2y
EUR
3y
1y
4y
EUR
5y
4y
1y
EUR
3y
2y
3y
EUR
5y
5y
5y
EUR
3y
2y
8y
Table 2 :
2Metrics and sensitivities computed for each available package at time t.Features
P V
Strike
Carry at Expiry (Carry)
Breakeven Width (BE Width)
Aged 1y Carry
Theta
ATMF Implied Volatility (Implied Vol)
Gamma
Vega
Curve Carry (Aged 1y)
Time Carry (Aged 1y)
Volatility Carry (Aged 1y) (Vol Carry)
Table 4 :
4Details used to generate the MCCS trade dataset.Detail
Value
Period
September 2006 to September 2016
Holding Period (h)
1 year
Trade Frequency
Weekly (usually on Wednesday)
Strike
At the Money Forward (ATMF)
Lagged data (h − p)
p = 1, 2 and 3 lagged returns
Assumption
Characteristic
Bid-Ask
Middle Rate
Transaction Costs
Entry and Unwind = 0.75 × V egat
Funding Rate
Libor 3 month rate
Table 5 :
5Parameters used to model the MCCS trade dataset.
Table 6 :
6Average ranks, Friedman and Holm post-hoc statistical tests and analysis for Information Ratio.Model
Avg Rank
Z-score
p-value
Holm Correction
Mean Pred
12.86
9.63
<0.0001
0.0036
Naive
11.89
8.66
<0.0001
0.0038
SVR-RBF
10.60
7.37
<0.0001
0.0042
CART
10.11
6.89
<0.0001
0.0045
Random Forest
9.31
6.09
<0.0001
0.0050
kNN
8.23
5.00
<0.0001
0.0056
Classic Regression
8.17
4.94
<0.0001
0.0063
Grad Boost Reg
7.69
4.46
<0.0001
0.0071
KRR-RBF
7.49
4.26
<0.0001
0.0083
MLP
7.20
3.97
<0.0001
0.0100
BackSel Regression
6.37
3.14
0.0008
0.0125
Z-Score: CarryAtExpiry
6.09
2.86
0.0021
0.0167
Z-Score: BE-Width
5.91
2.69
0.0036
0.0250
Ridge Regression
4.86
1.63
0.0517
0.0500
Lasso Regression
3.23
-
-
-
Friedman Chi-Square
117.22
<0.0001
This notation is extensively used during this work. In this case, the first 10y means a spot swaption with 10 year of expiration, while the second 10y refers to the swap tenure.2 Short selling a 1m1y2y mid-curve swaption and going long a longer-expiry 13m2y spot swaption.
For the sake of brevity we dropped the subscript that refers to a particular trade (i).
When we rank the models for a single MCCS, it means that we sort all them in such way that the best performer is in the first place (receive value equal to 1), the second best is positioned in the second rank (receive value equal to 2), and so on. We can repeat this process for all trades and compute metrics, like the average rank (e.g., 1.35 means that a particular model was placed mostly near to the first place).
By normalised t-stats we mean dividing each coefficient t-stat by the sum of the absolute values of all t-stats in the model. The result is a number between -1 and +1, indicating the significant magnitude in comparison to other variables, as well as the direction in which it affects the model predictions. We multiplied it by a one hundred just to work on a more convenient scale of -100% and 100%.
AcknowledgmentAdriano Soares Koshiyama wants to acknowledge the funding for its PhD studies provided by the Brazilian Research Council (CNPq) through the Science Without Borders program. Also, the authors would like to thanks, Guillaume Andrieux, Tomoya Horiuchi, Gerald Rushton, Tam Rajendran, and Anthony Morris for all the comments and support during this research.
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|
[] |
[
"Complex Engel Structures",
"Complex Engel Structures"
] |
[
"Zhiyong Zhao [email protected] \nMathematics Department\nDuke University\n27708-0230DurhamNCUSA\n"
] |
[
"Mathematics Department\nDuke University\n27708-0230DurhamNCUSA"
] |
[] |
We study the geometry of Engel structures, which are 2-plane fields on 4-manifolds satisfying a generic condition, that are compatible with other geometric structures.A complex Engel structure is an Engel 2-plane field on a complex surface for which the 2-planes are complex lines. We solve the equivalence problems for complex Engel structures and use the resulting structure equations to classify homogeneous complex Engel structures. This allows us to determine all compact, homogeneous examples.Compact manifolds that support homogeneous complex Engel structures are diffeomorphic to S 1ˆS U p2q or quotients of C 2 , S 1ˆS U p2q, S 1ˆG or H by co-compact lattices, where G is the connected and simply-connected Lie group with Lie algebra sl 2 pRq and H is a solvable Lie group.
| null |
[
"https://arxiv.org/pdf/1805.07660v1.pdf"
] | 119,270,287 |
1805.07660
|
f218dce3a2dac81b7ecd381303419a1c09962ae6
|
Complex Engel Structures
19 May 2018
Zhiyong Zhao [email protected]
Mathematics Department
Duke University
27708-0230DurhamNCUSA
Complex Engel Structures
19 May 2018and phrases: complex Engel structurestructure equationhomoge- neous manifold 1991 Mathematics Subject Classification: 58H99
We study the geometry of Engel structures, which are 2-plane fields on 4-manifolds satisfying a generic condition, that are compatible with other geometric structures.A complex Engel structure is an Engel 2-plane field on a complex surface for which the 2-planes are complex lines. We solve the equivalence problems for complex Engel structures and use the resulting structure equations to classify homogeneous complex Engel structures. This allows us to determine all compact, homogeneous examples.Compact manifolds that support homogeneous complex Engel structures are diffeomorphic to S 1ˆS U p2q or quotients of C 2 , S 1ˆS U p2q, S 1ˆG or H by co-compact lattices, where G is the connected and simply-connected Lie group with Lie algebra sl 2 pRq and H is a solvable Lie group.
Introduction
A distribution is a subbundle D Ă T M of the tangent bundle of a manifold M . We will consider certain distributions with special properties, for example, distributions with some integrability conditions. Given a distribution of rank n on a manifold M m , if, for each point of M , there exists a coordinate neighborhood U and local coordinates x 1 , x 2 ,¨¨¨, x m such that B Bxi , i " 1,¨¨¨, n forms a local basis for the distribution on U , then the distribution is said to be completely integrable.
Besides complete integrability, one can consider partially integrable distributions. An extreme condition is to be nowhere integrable, i.e., for every p P M , there exist X, Y , sections of D,
such that rX, Y s p is not a section of D. For example, contact structures are nowhere integrable distributions on odd dimensional manifolds. The study of contact structures [8] usually involves interplay between geometry, topology and dynamics. Contact structures play an important role in the study of low-dimensional topology.
We will study Engel structures, which are certain non-integrable distributions defined on 4-manifolds.
We will see that, locally, all Engel structures are isomorphic but the global theory of Engel structures is not trivial. A 4-manifold can carry many nonisomorphic Engel structures [13]. There are relations between contact structures on 3-dimensional manfiolds and Engel structures. For example, V.
Gershkovich [13] proved that each Engel manifold carries a canonical one-dimensional foliation and an Engel structure defines a contact distribution on any three-dimensional submanifold transversal to the canonical foliation.
We will solve the equivalence problem for complex Engel structures and present the classification of compact quotients of homogeneous complex Engel structures. Engel structures (to be defined below) can be characterized in terms of the derived system construction.
Proposition 1.1.
[2] Given a Pfaffian system I, there exists a bundle map δ : I Ñ Λ 2 pT˚M {Iq that satisfies δω " dω mod pIq for all ω P ΓpIq.
Definition 1.1. By Proposition 1.1, we have a bundle map δ. Set I p1q " ker δ and call I p1q the first derived system. Continuing with this construction, we can get a filtration I pkq è¨¨Ă I p2q Ă I p1q Ă I p0q " I, defined inductively by I pk`1q " pI pkq q p1q .
I pkq is called the kth derived system. Now we present the definition and characterization of Engel structures.
Definition 1.2 (Engel Structure). Given a 4-manifold M and a Pfaffian system I Ă T˚M , an
Engel structure is a sub-bundle D " I K of the tangent bundle of M that satisfies: p1q I is of rank 2, 2 I p1q is of rank 1 and 3 I p2q " 0. A manifold endowed with an Engel structure D " I K is called an Engel manifold. Definition 1.3. Given an ideal I generated by a Pffafian system I, a vector field ξ is called a Cauchy characteristic vector field of I if ξ ⌟ I Ă I. At a point x P M , the set of Cauchy characteristic vector fields is ApIq x " tξ x P T x M |ξ x ⌟ I x Ă I x u Ă I K and the retracting space or Cartan system is defined to be
CpIq x " ApIq K x Ă Tx M .
By the definition of Engel structure I K , there is a canonical flag of sub-bundles 0 Ă I p1q Ă I Ă CpI p1q q Ă T˚M .
V. Gershkovich [13] proved the following theorem which can also be found in [6].
Theorem 1.2. If an orientable 4-manifold admits an orientable Engel structure, then it has trivial tangent bundle.
T. Vogel [1] proved the converse of the above theorem: Theorem 1.3. Every parallelizable 4-manifold admits an orientable Engel structure.
Thus for an orientable 4-manifold, parallelizability is equivalent to the existence of an orientable Engel structure. This is a global characterization of manifolds that support orientable Engel structures. Locally, we have the following Engel normal form [2], which implies that there is no local invariant for Engel structures, i.e., all Engel structures are locally equivalent. coordinates px, y 0 , y 1 , y 2 q : U Ñ R 4 such that I| U " tdy 0´y1 dx, dy 1´y2 dxu .
In this paper, we will consider complex Engel structures.
Definition 1.4. Given a complex manifold pM, J, Iq with an Engel structure D " I K , if I K Ă T M is a complex line field, the Engel structure is called a complex Engel structure.
Geometry of Complex Engel Structures
Let J be the complex structure on the underlying manifold M . Choose a local J-complex coframing
pω 1 , ω 2 q on an open set U Ă M such that 1. ω 1 , ω 2 are of J-type (1,0) 2. ω 2 " 0 defines I K on U
Then pω 1 , ω 2 q is called a 0-adapted coframing for the Engel structure.
Lemma 2.1. The 0-adapted coframings are the sections of a G-structure on M , where G Ă GLp2, Cq is the 3-dimensional complex subgroup G " #˜a b 0 c¸ˇˇˇˇa , b, c P C and a, c ‰ 0 + .
In the following analysis, we will denote the conjugate of ω 1 , ω 2 byω 1 ,ω 2 instead of ω 1 , ω 2 .
Theorem 2.2. A complex Engel structure has a canonical coframing pω 1 , ω 2 q (i.e., an e-structure)
such that
dω 2 " ω 1^ω1 mod ω 2 , dpω 2`ω2 q "´1 2 pω 1´ω1 q^pω 2´ω2 q mod ω 2`ω2 .(1)
Proof. Since ω 2 " 0 defines the complex Engel structure, a 0-adapted coframing pω 1 , ω 2 q is defined on an open set U Ă M up to the following change of coframing:ω
1 ω 2¸"˜a b 0 c¸˜ω 1 ω 2¸,
where a, b, c are complex functions on U and ac ‰ 0 on U .
The Engel condition implies that
dω 2 " A ω 1^ω1 mod ω 2 ,ω 2 , where A ‰ 0. Defineω 2 " 1 A ω 2 , then dω 2 " d`1 A˘^ω 2`1 A dω 2 " 1 A Aω 1^ω1 mod ω 2 ,ω 2 " ω 1^ω1 mod ω 2 ,ω 2 .(2)
Thus, after rescaling ω 2 , we can arrange that A " 1. Then
dω 2 " ω 1^ω1 mod ω 2 ,ω 2 .(3)
Such coframings will be said to be 1-adapted. They are the sections of a G 1 -structure, where
G 1 Ă G is defined by c " aā.
The change of the coframing that preserves (3) is reduced tõω
1 ω 2¸"˜a b 0 aā¸˜ω 1 ω 2¸.(4)
Now ω 2 is defined up to a real multiple, so the real and imaginary parts of ω 2 are uniquely defined up to a real multiple. Suppose the real part of ω 2 spans I p1q , the first derived system.
By (3), the real part of ω 2 spans I p1q , the first derived system, i.e., dpω 2`ω2 q " 0 mod ω 2 ,ω 2 .
Since dpω 2`ω2 q is a real 2-form, there must be a complex function p 1 such that dpω 2`ω2 q " pp 1 ω 1´p1ω1 q^pω 2´ω2 q mod ω 2`ω2 .
Because I p2q " p0q, dpω 2`ω2 q^pω 2`ω2 q ı 0 implies that p 1 ‰ 0. By replacing pω 1 , ω 2 q bý´1 2p1 ω 1 , 1 4p1p1 ω 2¯, we can arrange p 1 "´1 2 . After this arrangement, a is fixed to be 1 in the transformation. Now the coframing satisfies
dpω 2`ω2 q "´1 2 pω 1´ω1 q^pω 2´ω2 q mod ω 2`ω2 .(5)
Such coframings will be said to be 2-adapted. They are the sections of a G 2 -structure, where G 2 Ă G 1 is defined by a " 1. After setting a " 1, by (4), ω 2 is unique and ω 1 is unique modulo ω 2 , i.e. a change of coframing that preserves (3) and (5) is reduced tõω
1 ω 2¸"˜1 b 0 1¸˜ω 1 ω 2¸.
Because ω 2 is uniquely defined now, we can write
dω 2 " ω 1^ω1`f ω 1^ω2 mod ω 2
for some function f . Note that there is noω 1^ω2 term since we assumed that pω 1 , ω 2 q be of type (1,0), and the underlying almost complex structure is integrable.
By adding a multiple of ω 2 to ω 1 , we can arrange f " 0.
dω 2 " ω 1^ω1 mod ω 2 .(6)
After arranging this modification, the coframing satisfying (5) and (6)
dω 1 "´pp 1 ω 1`p2 ω 2`q1ω1`q2ω2 q^ω 1´p q 2 ω 1`r1ω1`r2ω2 q^ω 2 dω 2 " pω 1´ω2 q^ω 1´p p 1 ω 1`p2 ω 2`p1ω1`p2ω2 q^ω 2(7)
where p 1 , p 2 , q 1 , q 2 , r 1 , r 2 are complex functions. Thus, a complex Engel structure has 6 fundamental functional invariants.
Proof. From Theorem 2.2, the structure equation can be writen as
dω 2 " ω 1^ω1`p pω 1`qω1`rω2 q^ω 2
for some functions p, q, r. Therefore dpω 2`ω2 q " ppω 1`qω1`rω2 q^ω 2`ppω1`q ω 1`r ω 2 q^ω 2 " ppp´qqω 1`p q´pqω 1 q^ω 2 mod pω 2`ω2 q .
But according to (5) dpω 2`ω2 q "´1 2 pω 1´ω1 q^pω 2´ω2 q "´pω 1´ω1 q^ω 2 mod pω 2`ω2 q .
By comparing (8) and (9), we find q "p`1. Thus
dω 2 " ω 1^ω1`ω1^ω2`p pω 1`pω1`rω2 q^ω 2 .
Let α "´ppω 1`r ω 2 q, then
dω 2 " ω 1^ω1`ω1^ω2´p α`ᾱq^ω 2 ,(10)
where α is a p1, 0q-form, uniquely defined by (10). Let γ be a p1, 0q´form and β be any 1-form.
The structure equation can be written as
dω 1 "´pα`γq^ω 1´β^ω2 , dω 2 " ω 1^ω1`ω1^ω2´p α`ᾱq^ω 2 .(11)
Taking the exterior derivative of dω 2 then yieldśγ^ω 1^ω1`ω1^β^ω2`ω1^γ^ω1 " 0 mod ω 2 .
Recall that γ is a p1, 0q-form, sō
γ^ω 1`β^ω2 " 0 mod ω 1 , ω 2 .
Let γ " q 1 ω 1`q2 ω 2 , then
β " q 2 ω 1 modω 1 , ω 2 ,ω 2 .
Thus, the final structure equation of a complex Engel structure is
dω 1 "´pp 1 ω 1`p2 ω 2`q1ω1`q2ω2 q^ω 1´p q 2 ω 1`r1ω1`r2ω2 q^ω 2 , dω 2 " pω 1´ω2 q^ω 1´p p 1 ω 1`p2 ω 2`p1ω1`p2ω2 q^ω 2 .(12)
Remark 2.1. We can take exterior derivatives of (12), and see that there are no further relations on p 1 , p 2 , q 1 , q 2 , r 1 , r 2 . All differential invariants of complex Engel structures are these six or their derivatives with respective to the canonical coframing.
Homogeneous Complex Engel Structures
In this section, we will classify homogeneous complex Engel structures. The group of diffeomorphisms preserving a complex Engel structure also preserves its canonical coframing and hence preserves its fundamental invariants. Thus, if it is homogeneous, then the invariants must be constant. Assume that the functions be constant and take exterior derivatives of dω 1 , dω 1 , dω 2 and dω 2 , and set all of these to be zero. This will yield quadratic equations on p, q, r. After solving these equations, which was done with the help of MAPLE, we arrive at the following theorem:
Theorem 3.1 (Classification of Homogeneous Complex Engel Structures). There are six distinct two-parameter families of homogeneous structure equations of complex Engel structures. The constants pp 1 , p 2 , q 1 , q 2 , r 1 , r 2 q in equation (7) are listed as follows for the six cases:
• Case C1: pa`ib, 0, 0, 0, 0, 0q
• Case C2: p 1 2`i b, 0, 2ia, 0, 0, 0q
• Case C3: p 1 2´i b, 0, 2ib, 1 2 p2b`iqp2ia´bq, 2b 2´i b, pb 2`1 4 qp2a`ibqq
• Case C4: p0, a´ib, 2ib, a´ib,´a´ib, 0q
• Case C5: pap1´2ibq, 1 4 p2a´1qp2b`iq 2 , 2ib,´1 4 p2b`iq 2 , 1 4 p2b`4ab`ip2a´1qqp´2b`iq, 1 4 ap´1`2biqp1`4b 2 qq
• Case C6: p 1 2 pcos a`i sin aqp2b`iq, 1 4 p´sin a`2b cos a´1qp2b`iq 2 , 2ib, 1 8 p2ib sin a`i cos a´2b cos a´2ib`sin a`1qp2b`iq 2 , 1 4 p1`2biqp2ib sin a`i cos a`2b cos a´2ib´sin a´1q, 1 16 p1`4b 2 qp´1`2ibqp2ib sin a`i cos a`2b cos a´2ib´sin a´1qq
where a and b are real constants.
Proof. First take the exterior derivatives of dω 2 and dω 2 , which yield
r 2 " p 1 q 2`p2´q2 , r 2 "p 1q2`p2´q2 .(13)
Substituting these into exterior derivatives of dω 2 and dω 2 yields q 1`q1 " 0 So q 1 is pure imaginary.
Set
q 1 " 2iq 0 ,q 1 "´2iq 0 .(14)
Substituting these relations into dω 2 and dω 2 yields
ℑp 2 " q 0 pp 1`p1´1 q ,
where ℑp 2 means the imaginary part of p 2 .
Now take the exterior derivatives of dω 1 and dω 1 and set these to be zero. The equations are quadratic expressions in the coefficients of dω 1 , dω 2 . Then solve these quadratic equations. We get the six different 2-parameter solutions, listed in the Theorem.
Compact Homogeneous Complex Engel Structures
We have proved the classification result for homogeneous complex Engel structures. Now we can classify compact homogeneous complex Engel structures. We will prove the following theorem: • Case C1: If a " 1 2 and b " 0, the Lie algebra is a 4-dimensional solvable Lie algebra. There exists a compact quotient that supports a homogeneous complex Engel structure if and only if a " 1 2 and b " 0.
• Case C2: If there exists a complex number λ and a matrix A P SL 3 pZq such that
1. b "´a 2. |λ| ‰ 1
3. the eigenvalues of A are pλλq´1, λ,λ 4. there exists k P Z such that´1 2a logp|λ|q " arg λ`2kπ
then there exists a co-compact lattice Γ such that G{Γ supports a homogeneous complex Engel structure.
• Case C3:
1. If a "´1 4 , the Lie algebra is a solvable Lie algebra, and there exists a compact quotient.
2. If a " b 2 ‰ 0, the Lie algebra is a solvable Lie algebra, but there does not exist a compact quotient.
3. If pa ă´1 4 q or p0 ď a ă b 2 q or pb " 0 and a ă 0q, the Lie algebra is Rˆslp2, Rq.
There exists a compact quotient.
4. If p´1 4 ă a ă 0q or pa ą b 2 q or pb " 0 and a ą 0q, the Lie algebra is Rˆsup2q. There exists a compact quotient.
• Case C4: There is no compact quotient that supports a homogeneous complex Engel structure.
• Case C5: If a " 1 2 , there exists a co-compact lattice Γ of a solvable Lie group G that G{Γ supports a homogeneous complex Engel structure.
• Case C6: There is no compact quotients that supports a homogeneous complex Engel structure unless a "´π 2`2 kπ, k P Z and b " 0. Under this condition, this is a special case of case
C1.
In summary, compact quotients that support homogeneous complex Engel structures can occur in case C1, case C2, case C3, and case C5.
We will prove the theorem in the following section by analyzing the structure equation for each case. If the structure equation is solvable, we will also provide a local coordinate system expression of the coframing. 5 Proof of Theorem 4.1
Homogeneous Case C1
In this case, pp 1 , p 2 , q 1 , q 2 , r 1 , r 2 q " pa`ib, 0, 0, 0, 0, 0q. The structure equation is
dω 1 " 0 , dω 2 " pω 1´ω2 q^ω 1´p pa`ibq ω 1`p a´ibqω 1 q^ω 2 .(15)
By (15),
dpω 1^ω2^ω2 q "´p1´2a`2biqω 1^ω1^ω2^ω2 .(16)
By Stokes' Theorem, there is no compact example unless a " 1 2 and b " 0. By (15), dω 1 " 0. By the complex Poincaré Lemma, there exists a holomorphic function z locally on the manifold such that
ω 1 " dz .(17)
Thus
dω 2 " dz^dz`r´pa`ibqdz`p1´a`ibqdzs^ω 2 .
We will find a local coordinate system for the coframing pω 1 , ω 2 q in order to explicitly describe its group of symmetries. Since the groups of symmetries are different for different a and b, we will consider two cases: a`ib " 1 (special case) and generic case.
Special Case
If a`ib " 1, i.e. a " 1 and b " 0,
dpω 2`z dzq "´dz^pω 2`z dzq .(18)
By the complex Frobenius Theorem, there exists a complex function f and a holomorphic function w locally on the manifold such that
ω 2`z dz " f dw .(19)
Since pω 1 , ω 2 q is a coframing, ω 1 and ω 2 are linearly independent. By comparing the local coordinate expressions (17) and (19), we know that f is nowhere zero on its defining domain. Substituting
(19) into (18) yields df^dw "´f dz^dw .
Setting f " e´zg for some function g ‰ 0, we have dg^dw " 0. So the function g is a function of w only.
ω 2`z dz " e´zgpwqdw .
By definingw " ş gpwqdw and dropping the tilde in the local coordinate, we get
ω 2`z dz " e´zdw .
Since pω 1 , ω 2 q is a p1, 0q-coframing, pz, wq can serve as a local holomorphic coordinate system in a neighborhood of the manifold. In the coordinate system of pz, wq, the coframing can be expressed as ω 1 " dz .
ω 2 "´zdz`e´zdw .(20)
We consider the symmetry group of the coframing in these local coordinate. Let
Γ " tk 1`i k 2 |pz, wq Ñ pz, w`k 1`i k 2 q, where k 1 , k 2 P Zu .
Γ acts freely and discontinuously on the coframing in the local coordinate. So we can take a global model for this complex Engel structure
M 4 " C 2 {Γ -R 2ˆT 2 .
Generic Case
If a`ib ‰ 1, i.e. a ‰ 1 or b ‰ 0 , by (15), we have
dˆω 2´d z 1´a`ib˙" r´pa`ibqdz`p1´a`ibqdzs^ˆω 2´d z 1´a`ib˙.(21)
By the complex Frobenius Theorem, there exists a complex function f and a holomorphic function w locally on the manifold such that
ω 2´d z 1´a`ib " f dw .(22)
Substituting (22) into (21) yields
df^dw " f r´pa`ibqdz`p1´a`ibqdzs^dw .
By defining f " e´p a`ibqz`p1´a`ibqz g for some function g ‰ 0, we get dg^dw " 0. Thus
ω 2´d z 1´a`ib " e´p a`ibqz`p1´a`ibqz gpwqdw .
By definingw " ş gpwqdw and dropping the tilde, we have
ω 2´d z 1´a`ib " e´p a`ibqz`p1´a`ibqz dw .
Since pω 1 , ω 2 q is a coframing, pz, wq can serve as a local holomorphic coordinate system on a open set. The coframing can be written as
ω 1 " dz , ω 2 " dz 1´a`ib`e´p a`ibqz`p1´a`ibqz dw .
We will analyze the symmetry group of the coframing. Define
G a,b " tpα, βq|´pa`ibqα`p1´a`ibqᾱ " 2kπi, where α, β P C and k P Zu ,
where G a,b acts on the local coordinate as pz, wq Ñ pz`α, w`β q. We will analyze the elements
of G a,b . Let α " α 0`i α 1 , where α 0 , α 1 P R. We have p1´2aqα 0`2 bα 1 " 0 , α 1 " 2kπ ,(23)
where k P Z.
For different a, b, there exist three families of solution for α:
1. if a ‰ 1 2 , then α 0 "´4 bkπ 1´2a , α 1 " 2kπ. Define Γ 1 " "ˆˆ´4 bπ 1´2a`i 2π˙k, β 0`i β 1˙ˇk , β 0 , β 1 P Z * .
We can get a non-compact quotient
C 2 {Γ 1 -RˆS 1ˆT 2
that supports a homogeneous complex Engel structure.
2. if a " 1 2 , b " 0, then α 1 " 2kπ. Define Γ 2 " t pα 0`i 2kπ, β 0`i β 1 q| k, α 0 , β 0 , β 1 P Zu .
We get a compact quotient C 2 {Γ 2 that supports a homogeneous complex Engel structure.
In this case, we can define θ 1 "´ω 1 and θ 2 " ω 1´1 2 ω 2 . The structure equation is
dθ 1 " 0 , dθ 2 " 1 2 pθ 1´θ1 q^θ 2 .(24)
Define θ 1 " α`iβ and θ 2 " γ`iδ for real parts and imaginary parts decomposition. We
have dα " 0 , dβ " 0 , dγ "´β^δ , dδ " β^γ .(25)
Thus the Lie algebra is a 4-dimensional solvable Lie algebra g with nontrivial brackets:
rX, Y s " Z, rX, Zs "´Y,
where X, Y, Z P g. By the classification results in [15], the corresponding connected and simply-connected Lie group has a co-compact lattice.
3. if a " 1 2 , b ‰ 0, then α 1 " 0. Define Γ 3 " t pα 0 , β 0`i β 1 q| α 0 , β 0 , β 1 P Zu .
Then we can get a non-compact quotient
C 2 {Γ 3 -R 1ˆS1ˆT 2 .
In summary, there exists a compact quotient of type C1 that can support a homogeneous complex
Engel structure if and only if a " 1 2 and b " 0.
Homogeneous Case C2
Now, pp 1 , p 2 , q 1 , q 2 , r 1 , r 2 q " p 1 2`i b, 0, 2ia, 0, 0, 0q. Assume a ‰ 0, otherwise, it is a special case of C1. The structure equation is
dω 1 " 2iaω 1^ω1 , dω 2 " pω 1´ω2 q^ω 1´"`1 2`i b˘ω 1``1 2´i b˘ω 1 ‰^ω 2 .(26)By (26) d`ω 1``´1 2`i p2a´bq˘ω 2˘"`1 2`i b˘pω 1´ω1 q^`ω 1``´1 2`i p2a´bq˘ω 2˘.
By the complex Frobenius Theorem, there exist complex functions p and q, and holomorphic functions z and w such that
ω 1 " pdz , ω 1`´´1 2`i p2a´bq¯ω 2 " qdw .
To calculate the function p, write ω 1 " α`iβ, where α and β are the real and imaginary part of ω 1 , respectively. From the structure equation
dα "´4aα^β , dβ " 0 .
Thus there exist functions f, x, y such that α " f dy, β " dx and df " 4af dx mod dy. After redefining y, we can write α " e 4ax dy. Thus
ω 1 " e 4ax d`y´i 4a e´4 ax˘.(27)
Define z " y´i 4a e´4 ax as a local holomorphic coordinate ( Note: ℑpzq ‰ 0. So, according to the sign of a, we can restrict the definition of z to half of the complex plane), then
ω 1 "´i 2apz´zq dz .
From the structure equation, we get
dq^dw "´1 2`i b¯i 2apz´zq dpz´zq^qdw Thus dq "´1 2`i b¯i 2apz´zq dpz´zqq mod dw
After redefining w, we can take q " pz´zq´2 b`i 4a . So in the local holomorphic coordinate system z, w,
ω 1 "´i 2apz´zq dz , ω 2 " 2 1`2p2a´bqiˆpz´z q´2 b`i 4a dw`i 2apz´zq dz˙.
Compact case of case C2
By (26),
dpω 1^ω2^ω2 q " 2ipa`bqω 1^ω1^ω2^ω2 .(28)
If a`b ‰ 0, the volume form is exact. By Stokes' Theorem, there cannot be a compact quotient that supports a homogeneous complex Engel structure when a`b ‰ 0. In the following, only consider a`b " 0. Define θ 1 " ω 1 . The structure equation is
dθ 1 " 2iaθ 1^θ1 , dθ 2 "ˆ1 2´i a˙pθ 1´θ1 q^θ 2 .(29)
Let θ 1 " α`iβ , θ 2 " γ`iδ be real part and imaginary part decompositions. Then (29) is equivalent
to dα "´4aα^β , dβ " 0 , dγ " β^δ´2aβ^γ , dδ "´β^γ´2aβ^δ .
Let X 1 , X 2 , X 3 , X 4 be left-invariant vector fields dual to the left-invariant forms α, β, γ, δ, respectively. Then the nontrivial brackets are
rX 2 , X 1 s " 4aX 1 , rX 2 , X 4 s " X 3´2 aX 4 , rX 2 , X 3 s "´2aX 3´X4 .
By [15], there exists a co-compact lattice for some a. We will calculate the conditions for the existence of a co-compact lattice.
Let f "´2a`i, V " X 3`i X 4 and W " X 3´i X 4 . The nontrivial brackets are
rX 2 , X 1 s "´pf`f qX 1 , rX 2 , V s " f V , rX 2 , W s "f W .
Since center of the Lie algebra g is trivial, we have an exact sequence
0 Ñ g ad ÝÑ Endpgq,
where g ad ÝÑ Endpgq is the adjoint representation. Let X " xX 1`y X 2`z V`zW be an element of g. Then
adpXqpX 1 , V, W, X 2 q " pX 1 , V, W, X 2 q » - - - - -´p f`f qy 0 0 pf`f qx 0 f y 0´f z 0 0f y´fz 0 0 0 0 fi ffi ffi ffi ffi fl .
The connected and simply-connected Lie group corresponding to the Lie algebra g is
G - $ ' ' ' ' & ' ' ' ' % » - - - - - y´p f`fq 0 0 x 0 y f 0 z 0 0 yfz 0 0 0 1 fi ffi ffi ffi ffi flˇˇˇˇˇˇˇˇˇx , y P R, y ą 0, z P C , / / / / . / / / / - . Proposition 5.1.
If there exists a complex number λ and a matrix A P SL 3 pZq such that
1. |λ| ‰ 1 2. the eigenvalues of A are pλλq´1, λ,λ 3. there exists k P Z such that´1 2a logp|λ|q " arg λ`2kπ
then there exists a co-compact lattice Γ such that G{Γ supports a homogeneous complex Engel structure.
Proof. Let N Ă G be the subgroup
N " $ ' ' ' ' & ' ' ' ' % » - - - - - 1 0 0 r 0 1 0 z 0 0 1z 0 0 0 1 fi ffi ffi ffi ffi flˇˇˇˇˇˇˇˇˇr P R, z P C , / / / / . / / / / - .(30)
It is easy to verify that N is a normal subgroup of G. Thus
G{N - $ ' ' ' ' & ' ' ' ' % » - - - - - y´p f`f q 0 0 0 0 y f 0 0 0 0 yf 0 0 0 0 1 fi ffi ffi ffi ffi flˇˇˇˇˇˇˇˇˇy P R, y ą 0 , / / / / . / / / / - is a quotient group. Let L 1 " x v 1 , v 2 , v 3
y be a lattice of the normal subgroup N , to be determined later. We need to find a lattice L 2 of G{N such that the lattice of the group G is L "
# « γ v 0 1 ffˇˇˇˇˇw here γ P L 2 , v P L 1 + .
By the multiplication rule of the group G, this is equivalent to γ v P L 1 for any γ P L 2 and any v "
» - - - v 1 v 2 v 3
fi ffi ffi fl P L 1 . Hence we need to find a ij P Z and c ą 0 and c ‰ 1 such that
γ c v 1 " a 11 v 1`a21 v 2`a31 v 3 , γ c v 2 " a 12 v 1`a22 v 2`a32 v 3 ,(31)γ c v 3 " a 13 v 1`a23 v 2`a33 v 3 ,
where γ c is the linear transform with transformation matrix
The eigenvalues of the matrix A should be c´p f`fq , c f and cf for some c ą 0. Assume the eigenvalues are pλλq´1, λ,λ and fix c´2 a`i " λ .
Thus c " |λ|´1 2a .
By (33),
|λ|ˆ|λ|´1 2a i " |λ|ˆe i arg λ .(35)
Thus the eigenvalue λ and the parameter a satisfý
1 2a logp|λ|q " arg λ`2kπ(36)
for certain k P Z.
We will calculate an explicit condition on the existence of co-compact lattice. Since the eigenvalues are pλλq´1, λ,λ, the characteristic polynomial of the matrix A is
px´pλλq´1qpx´λqpx´λq " x 3´´`λλ˘´1`λ`λ¯x2`´λ`λλ˘´1`λ`λλ˘´1`λλ¯x´1 .(37)
Since A P SL 3 pZq, there exist m, n P Z such that λλ˘´1`λ`λ " m , λ`λλ˘´1`λ`λλ˘´1`λλ " n . Define p " λ`λ and q "`λλ˘´1. It is easy to verify that p and q are real numbers and 4 q ě p 2 . To be a co-compact lattice, 0 ă q ă 1. Then by (38),
p`q " m , pq`1 q " n .(39)
Then the eigenvalue is λ " p
2`i b 1 q´p 2 4 .
Remark 5.1. There exist countably infinite families of solutions for p and q, that yield infinitely many families of co-compact lattices. The co-compact lattices can be derived from solutions of (39). Now we give an example for some a such that there exists a compact quotient. λ "´s 12´1 s`?
3`s 6´2 s2 i .
We can calculate a by (36) and c by (34).
Homogeneous Case C3
Now pp 1 , p 2 , q 1 , q 2 , r 1 , r 2 q "´1 2´i b, 0, 2ib, 1 2 p2b`iqp2ia´bq, 2b 2´i b, pb 2`1 4 qp2a`ibqĀ ssume at least one of a or b is nonzero. Otherwise, it will be a special case of C1 with a " 1 2 and b " 0. The structure equation is
dω 1 "´r´2ibω 1`1 2 p2b´iqp´2ia´bqω 2 s^ω 1 " 1 2 p2b`iqp2ia´bqω 1`p 2b 2`i bqω 1`ˆb 2`1 4˙p 2a´ibqω 2 ^ω 2 , dω 2 " ω 1^ω1´ˆ1 2´i b˙pω 1´ω1 q^ω 2 .(40)
Define θ " ω 1``´1 2`i b˘ω 2 . By (40), we have
dθ " "`1 2`i b˘pω 1`p 2a´ibqω 2 q´`1 2`2 a˘ω 1 ‰^θ .(41)
Since the symmetry groups are different for different parameters, we will consider the structure equation with different parameters:
1. a "´1 4 2. a " b 2 ‰ 0 3. a ‰´1 4 and a ‰ b 2 5.3.1 a "´1 4
The structure equation (41) reduces to
d´ω 1`´´1 2`i b¯ω 2¯"´1 2`i b¯´ω 1`´´1 2`i b¯ω 2¯( 42)´ω 1`´´1 2`i b¯ω 2¯.
Let ω 1`´´1 2`i b¯ω 2 " α`iβ, where α and β are real 1-forms. Then
dα "´2bα^β , dβ " α^β .(43)
Since dpα`2bβq " 0, there exists function x such that α`2bβ " dx. By the structure equation, we have dβ " dx^β, that implies the existence of a function y such that β " e x dy. Thus in terms of local coordinate x, y, α " dx´2be x dy , β " e x dy .
So
ω 1`ˆ´1 2`i b˙ω 2 " e x dp´e´x´2by`iyq .
Let z "´e´x´2by`iy.
Then
ω 1`´´1 2`i b¯ω 2 " dź´1 2`i b¯z`´´1 2`i b¯z .
By (40), we have
dω 2 " ω 1^ω1´ˆ1 2´i b˙pω 1´ω1 q^ω 2 "´´1 2`i b¯dz^ω 2´1 2`i b¯z`´´1 2`i b¯z`´´´1 2`i b¯dz^ω 2´1 2`i b¯z`´´1 2`i b¯z dz^dz´´1 2`i b¯z`´´1 2`i b¯z¸2 .(44)
Lemma 5.2. Assume xdz, ω 2 y forms a Frobenius system. Then there exists a function f and a holomorphic function w such that
ω 2 " dw`f dz
Proof. Since xdz, ω 2 y forms a Frobenius system of rank 2, there exist functions u, v, a, b such that dz " adu`bdv and du, dv are linearly independent . Since dz ‰ 0, at least one of a, b is nonzero.
Without loss of generality, assume b ‰ 0. So dv " 1 b pdz´aduq. So v is a function of z and x and locally we take z and u, instead of v and u, as local coordinates.
By the complex Frobenius Theorem, there exist functions rpz, uq and spz, uq such that ω 2 " rpz, uqdu`spz, uqdz " d´ż rpz, uq du¯´´ż Brpz, uq Bz du¯dz`spz, uqdz " d´ż rpz, uq du¯`"spz, uq´´ż Brpz, uq Bz du¯ıdz .
Since ω 2 and dz are linearly independent, d´ş rpz, uq du¯‰ 0. Let w "´ş rpz, uq du¯and f " spz, uq´´ş Brpz,uq Bz du¯, then ω 2 " dw`f dz .
Let D "´1 2`i b. By (44) and Lemma 5.2, we get df^dz " dz Dz`Dz^ˆD dw´Ddw`dz Dz`Dz´Df dz˙.
Thus Bf Bw "´D Dz`Dz , Bf Bw "D Dz`Dz , Bf Bz "Df´1 Dz`Dz Dz`Dz .(45)
Let f " f 1`i f 2 , where f 1 and f 2 are real and imaginary parts of f , respectively. Let z " x`iy and w " u`iv, then (45) is equivalent to
Bf 1 Bu " 0 , Bf 1 Bv " 0 , Bf 2 Bu " 2b x`2by , Bf 2 Bv "´1 x`2by ,
Bf 1 Bx´B
f 2 By " f 1`2 bf 2´2 x`2by
x`2by ,
Bf 1 By`B f 2 Bx " 2bf 1´f2 x`2by .(46)
So f 1 " f 1 px, yq, f 2 " gpx, yq`2 bu x`2by´v x`2by . The equation is equivalent to
Bf 1 Bx´B g By " f 1`2 bg´2 x`2by
x`2by ,
Bf 1 By`B g Bx " 2bf 1´g x`2by .(47)
Consider the differential ideal I " xθ 1 , θ 2 y, where θ 1 " df 1´p dx´qdy ,
θ 2 " dg`ˆq´2 bf 1´g x`2by˙d x´˜p´f 1`2 bg´2 x`2by x`2by¸d y .(48)
and dθ 1 "´π 1^d x´π 2^d y ,
dθ 2 " π 2^d x´π 1^d y ,(49)
where π 1 " dp mod pdxq and π 2 " dq mod pdyq. This system is involutive and its Cartan characters are ps 1 , s 2 q " p2, 0q. So the solution depends on 2 functions of 1 variable.
We will calculate the coframing in local coordinate for b " 0. Let F 1 " F p´izq, F 2 " Gpizq be two functions of one variable z and C be a constant. The general solution is of the following form
f 1 " F 1 1`F 1 2`ˆ4 pz`zq 2`C˙z`z 2 , f 2 " iF 1 2´i F 1 1´2 z`zˆF 1`F2`w´w 2i˙.
Thus in the case b " 0, the coframing is
ω 1 " 1 2 pdw`f dzq´2 dz z`z , ω 2 " dw`f dz .
In our original parametrization, z "´e´x´2by`iy. So z`z ă 0. Take a special form F 1 " 0 and F 2 " 0. Then f pz, wq " 2´pw´wq z`z .
The coframing can be written as
ω 1 " 1 2ˆd w´2`p w´wq z`z dz˙, ω 2 " dw`2´p w´wq z`z dz .
We will prove that there exist compact quotients that support homogeneous complex Engel structures. Before proving this, we need to know the Lie algebra of the homogeneous complex Engel structures.
Proposition 5.3. There exists a basis pX 1 , X 2 , X 3 , X 4 q such that the nontrivial brackets of the Lie algebra are rX 2 , X 3 s " X 1 , rX 2 , X 4 s " X 2 , rX 3 , X 4 s "´X 3 .
Proof. Recall that dα "´2bα^β , dβ " α^β .
Define
ω 1 " γ`iδ. dγ "´2bα^β´δ^β`2bδ^α , dδ " α^β´α^δ´2bβ^δ .(52)
Let e 1 , e 2 , e 3 , e 4 be left-invariant vector fields dual to the left-invariant forms α, β, γ, δ, respectively.
Then by (52), the nontrivial brackets are re 1 , e 2 s "´2be 1`e2´2 be 3`e4 re 1 , e 4 s "´2be 3´e4 (53) re 2 , e 4 s " e 3´2 be 4 .
We will consider 2 separate cases:
1. b " 0 2. b ‰ 0
If b " 0, the nontrivial brackets are re 1 , e 2 s " e 2`e4 re 1 , e 4 s "´e 4 re 2 , e 4 s " e 3 .
Define X 1 "´2e 3 , X 2 " e 4 , X 3 " 2e 2`e4 , X 4 " e 1 . Then rX 2 , X 3 s " X 1 , rX 2 , X 4 s " X 2 , rX 3 , X 4 s "´X 3 .
If b ‰ 0, defineẽ 1 " e 1´1 2b e 2´1 2b`e 4´1 2b e 3˘,ẽ4 " e 4´1 2b e 3 . The nontrivial brackets are rẽ 1 , e 2 s "´2bẽ 1`ẽ4``1 2b´2 b˘e 3 rẽ 1 ,ẽ 4 s "´`2b`1 2b˘e 3 re 2 ,ẽ 4 s "´2bẽ 4
Defineẽ 1 "ẽ 1´1 4bẽ 4 . Then the nontrivial brackets are rẽ 1 , e 2 s "´2bẽ 1``1 2b´2 b˘e 3 rẽ 1 ,ẽ 4 s "´`2b`1 2b˘e 3
re 2 ,ẽ 4 s "´2bẽ 4 Define X 1 "´4 b 2`1 2b e 3 , X 2 "ẽ 1 , X 3 "ẽ 4 , X 4 " 1 4b 2`1``´2 b´1 2b˘e 2´`´2 b`1 2b˘e 4˘. Then rX 2 , X 3 s " X 1 , rX 2 , X 4 s " X 2 , rX 3 , X 4 s "´X 3 .(55)
Thus the theorem is true for both cases.
By [15], there exists a co-compact lattice when a "´1 4 . Thus there exists a compact quotient of type C3 when a "´1 4 .
a
" b 2 ‰ 0 Define θ 1 " ω 1`p 2b 2`i bqω 2`1 4b 2`1 pp4b 2´1 q´4biqω 1`b 4b 2`1 p2bp4b 2´3 q´p12b 2´1 qiqω 2 (56)
Then the coframing pθ, θ 1 q satisfies
dθ 1 " 0 , dθ "`´1 2`i b˘θ 1^θ .
Thus there exists a holomorphic coordinate system pz, wq such that θ 1 and θ can be written as linear combinations of pdw, dwq and dz, respectively. There exists a function f pw,wq such that
θ 1 " df ,
θ " e p´1 2`i bqf dz .
Thus
ω 1 " e´´1 2`i b¯f dz´`´1 2`i b˘ω 2 . By (56), we have´2 b 2´1 2´2 ib¯ω 2`´2 b 2`1 2¯ω 2 " df´e p´1 2`i bqf dz´1 4b 2`1´p 4b 2´1 q´4bi¯e p´1 2´i bqf dz .
Let ω 2 " hdz`gdw, where h and g are functions. By scaling, fix g " 1.
Take f "´2b 2`1 2¯w2 b 2´1 2´2 ib¯w, then h "´2 e p´1 2`i bq «´2 b 2`1 2¯w`´2 b 2´1 2´2 ib¯w ff 4b 2`1 .
So
ω 2 "´2 e p´1 2`i bqrp2b 2`1 2 qw`p2b 2´1 2´2 ibqws 4b 2`1 dz`dw and ω 1 " e p´1 2`i bqrp2b 2`1 2 qw`p2b 2´1 2´2 ibqws dz´ˆ´1 2`i b˙ω 2 .
Proposition 5.4. There does not exist a compact quotient that supports a homogeneous complex Engel structure when a " b 2 ‰ 0.
Proof. After changing θ 1 to 1´1 2`i b¯θ 1 , the structure equation is
dθ 1 " 0 , dθ " θ 1^θ .(57)
Define θ 1 " α`iβ, θ " γ`iδ as the real and imaginary parts decompositions. Then (57) is
equivalent to dα " 0 , dβ " 0 , dγ " α^γ´β^δ , dδ " α^δ`β^γ .(58)
Let´X 3 , X 4 , X 1 , X 2 be left-invariant vector fields dual to the left-invariant forms α, β, γ, δ, respectively. Then the nontrivial brackets are
rX 1 , X 3 s " X 1 , rX 2 , X 3 s " X 2 , rX 1 , X 4 s "´X 2 , rX 2 , X 4 s " X 1 .
The Lie algebra is a solvable Lie algebra, denoted by g 4,10 in [17]. According to the classification results of the existence of co-compact lattices for 4-dimensional solvable Lie groups in [15], the connected and simply-connected Lie group corresponding to g 4,10 does not have a co-compact lattice. Therefore, there does not exist a compact quotient that supports a homogeneous complex Engel structure in this case.
5.3.3
a ‰´1 4 and a ‰ b 2
Let ω 1 " α`iβ and ω 2 " γ`iδ. From the structure equation (40), we have dα "´4bα^β´2b 2 α^γ`bα^δ`p4ab´2bqβ^γ´p2a`4b 2 qβ^δ
2bˆ1 4`b 2˙γ^δ , dβ "´4abα^γ`2aα^δ`2b 2 β^γ´bβ^δ´4aˆ1 4`b 2˙γ^δ , dγ " β^pδ´2bγq , dδ " β^p2α´γ´2bδq .(59)
Defineα "´2aα`bβ´p4a 2´b2 qγ`4abδ. Since at least one of a or b is not zero,α ‰ 0 and dα " 0. Let e 1 , e 2 , e 3 , e 4 be the dual vector fields of the 1-forms α, β, γ, δ, respectively.
Remark 5.2. From Lie theory, there is a split exact sequence [14] 0
Ñ Radpgq Ñ g Ñ g{Radpgq Ñ 0 ,
where Radpgq is the radical ideal of g. Thus g -Radpgq ' g{Radpgq .
Denote g 1 " g{Radpgq. We will prove the following theorem:
Theorem 5.5. For the Lie algebra g corresponding to the structure equation(59), Radpgq is 1dimensional and g 1 is simple 3-dimensional Lie algebra. Specially,
• pa ă´1 4 q or p0 ď a ă b 2 q or pb " 0 and a ă 0q, the Lie algebra is Rˆslp2, Rq.
• p´1 4 ă a ă 0q or pa ą b 2 q or pb " 0 and a ą 0q, the Lie algebra is Rˆsup2q.
Proof. We will prove this theorem by analyzing the result for the following 3 cases:
• 1. b " 0
In this caseα " α`2aγ. Defineβ " β,γ "´2α`γ,δ " δ, then dβ "´aγ^δ , dγ "´p1`4aqδ^β , dδ "´β^γ .
(60)
If a ą 0, we defineβ " 1
? 1`4aβ ,γ " 1 ? aγ ,δ " 1 ? ap1`4aqδ . (60) is equivalent to dβ "γ^δ , dγ "δ^β , dδ "β^γ .
Thusβ,γ,δ are left-invariant forms of the Lie group SU p2q. So if a ą 0, the manifold can be taken as S 1ˆS U p2q.
If a ă 0, Radpgq " te 1`2 e 3 u and g{Radpgq " tu " e 1´1 2a e 3 , v " e 2 , w " e 4 u. Define
H "´4 ?´a 1`4a u , X " ?´a v`w , Y " ?´a ap1`4aq v´1 ap1`4aq w .
The nontrivial brackets are
rH, Xs " 2X , rH, Y s "´2Y , rX, Y s " H .
pH, X, Y q forms a canonical basis for the Lie algebra slp2, Rq. Since SL 2 pRq has co-compact lattices [12], in this case, there exists a compact quotient that supports homogeneous complex Engel structure.
• 2. a " 0 By our assumption, b ‰ 0. The radical ideal is Radpgq "
! e 1`2 e3`4be4 1`4b 2 )
and g{Radpgq "
tu " e 1 , v " e 2´e 3 b , w " e 4 u. The nontrivial brackets are ru, vs " 2bu`2w
ru, ws "´bu rv, ws "ˆ2b 2´1 2˙u`b v`2bw Define $ & % A " 2b´1 4b 2 , B " 1 2b , C " 1 2b 2 , D " 2b`1, E "´2b, F " 2, s " 2, if b ‰´1 2 , A " 2b`1 4b 2 , B "´1 2b , C " 1 2b 2 , D " 2b´1, E " 2b, F " 2, s "´2, if b ‰ 1 2 .
and H " sv ,
X " Au`Bv`Cw , Y " Du`Ev`F w .
Then pH, X, Y q forms a canonical basis for the Lie algebra slp2, Rq such that
rH, Xs " 2X , rH, Y s "´2Y ,
rX, Y s " H .
Since SL 2 pRq has co-compact lattices [12], in this case there exists a compact quotient that supports a homogeneous complex Engel structure.
• 3. a ‰ 0, a ‰´1 4 and b ‰ 0
The radical ideal is Radpgq "
! e 1`2 e3`4be4 1`4b 2 )
and
g{Radpgq "
" u " e 1`e 4 2b , v " e 2´e 4 4a , w " e 3`4 a 2´b2 4ab e 4 * .
The nontrivial brackets are
ru, vs "´´8 ab 2`4 a 2´b2 4ab u`1 2b w , ru, ws "´p´4ab 2`4 a 2´b2`a qˆ1 4a u`1 2b v˙, rv, ws "´4 b 4`1 6a 3`1 2ab 2`b2 8ab u`´4 ab 2`4 a 2´b2`a 4a v´8 ab 2`4 a 2´b2 4ab w .
After the proof of the following proposition, we will finish the proof of the theorem.
Proposition 5.6. If a ‰ 0, a ‰´1 4 and b ‰ 0, there exists a compact quotient that supports a homogeneous complex Engel structure.
Proof. 1. a ą b 2 Define U " c 1 4ab 2`4 a 2´b2`a , V " c a 2p´4ab 2`4 a 2´b2`a q and A " V 2a p´8ab 2 U`4a 2 U´b 2 U`bq, B " V, C "´U V, D "´V 2a p´8ab 2 U`4a 2 U´b 2 U´bq, E " V, F " U V, s " 2bU .
Then define H " sv ,
X " Au`Bv`Cw ,(61)
Y " Du`Ev`F w .
Then pH, X, Y q forms a canonical basis for the Lie algebra sup2q with the following nontrivial brackets rH, Xs " Y , rH, Y s "´X , rX, Y s " H .
Since SU p2q has co-compact lattices, there exists a compact quotient that supports a homogeneous complex Engel structure.
2. 0 ă a ă b 2 Define U " c´1 4ab 2`4 a 2´b2`a , V " c´á 4ab 2`4 a 2´b2`a and A "´V 2a p´8ab 2 U`4a 2 U´b 2 U´bq, B " V, C " U V, D " V 2a p´8ab 2 U`4a 2 U´b 2 U`bq, E " V, F "´U V, s " 4bU.
Then define H " sv ,
X " Au`Bv`Cw , Y " Du`Ev`F w .
Then pH, X, Y q forms a canonical basis for the Lie algebra slp2, Rq with nontrivial brackets
rH, Xs " 2X , rH, Y s "´2Y ,
rX, Y s " H .
Since SL 2 pRq has co-compact lattices [12], there exists a compact quotient that supports a homogeneous complex Engel structure.
3.´1 4 ă a ă 0 Define U " c 1 4ab 2`4 a 2´b2`a , V " c a 2p´4ab 2`4 a 2´b2`a q and A " V 2a p´8ab 2 U`4a 2 U´b 2 U`bq, B " V, C "´U V, D "´V 2a p´8ab 2 U`4a 2 U´b 2 U´bq, E " V, F " U V, s " 2bU .
Then define H " sv ,
X " Au`Bv`Cw , Y " Du`Ev`F w .
Note Y " X. pH, X, Y q forms a canonical basis of the Lie algebra sup2q with nontrivial brackets rH, Xs " X , rH, Xs "´X , rX, Xs " H .
Since SU p2q has co-compact lattices, there exists a compact quotient that supports a homogeneous complex Engel structure.
a ă´1 4
Define U "
c 1 4ab 2`4 a 2´b2`a , V " c´a 2p´4ab 2`4 a 2´b2`a q and A " V 2a p´8ab 2 U`4a 2 U´b 2 U`bq, B " V, C "´U V, D "´V 2a p´8ab 2 U`4a 2 U´b 2 U´bq, E " V, F " U V, s " 2bU.
Then define H " sv ,
X " Au`Bv`Cw , Y " Du`Ev`F w .
pH, X, Y q forms a canonical basis of the Lie algebra slp2, Rq with nontrivial brackets
rH, Xs " Y , rH, Y s "´X ,
rX, Y s "´H .
Since SL 2 pRq has co-compact lattices [12]), there exists a compact quotient that supports a homogeneous complex Engel structure.
Since there exists a co-compact lattice for each case, we have proved the theorem.
Homogeneous Case C4
Now pp 1 , p 2 , q 1 , q 2 , r 1 , r 2 q " p0, a´ib, 2ib, a´ib,´a´ib, 0q. The structure equation is
dω 1 " 2ibω 1^ω1´p a`ibqω 2^ω1`p a´ibqω 1^ω2 , dω 2 " pω 1´ω2 q^ω 1´p a`ibqω 2^ω2 .(62)
It is easy to verify that dpω 1^ω2^ω2 q " p1`2biqω 1^ω1^ω2^ω2 ‰ 0.
By Stokes' Theorem, there is no compact quotient of type C4 that supports a homogeneous complex Engel structure.
We will find local coordinate representations of ω 1 and ω 2 under different conditions for a and b.
The symmetry groups of the coframing are different for different a and b. We define A " 1 and
B " $ & %´1 2˘c`i b, where c " b 1 4´p b 2`a q, if b 2`a ď 1 4 1 2˘i c`ib, where c " b pb 2`a q´1 4 , if b 2`a ą 1 4(63)
By (62),
dpAω 1`B ω 2 q^pAω 1`B ω 2 q " 0 . 5.4.1 a " b " 0
The structure equation (62) reduces to dω 1 " 0 , dω 2 " pω 1´ω2 q^ω 1 .
Since dω 1 " 0, by the complex Poincaré Lemma, there exists a holomorphic function z such that ω 1 " dz. So dω 2 " pdz´ω 2 q^dz, that is equivalent to dpdz´ω 2 q "´pdz´ω 2 q^dz. By the complex Frobenius Theorem, there exists a function f and a holomorphic function w such that dz´ω 2 " f dw and df^dw " f dz^dw. By modifying w, we can write dz´ω 2 " ez dw. So in the local coordinate system pz, wq, the coframing is
ω 1 " dz , ω 2 "´ez dw`dz .
Take a discrete symmetry group of the coframing
Γ " tp2α 0 πi, β 0`i β 1 q|α 0 , β 0 , β 1 P Zu -Z 3 .
Γ acts on the local coordinate as pz, wq Þ Ñ pz`2α 0 πi, w`β 0`i β 1 q. This action keeps the coframing invariant. The quotient can be taken as
C 2 {Γ -R 1ˆS1ˆT 2 . 5.4.2 b 2`a ă 1 4 or b 2`a ą 1 4
Since the calculations are similar for these two cases, without loss of generality, assume b 2`a ă 1 4 . Define U "´1 2`c`i b, V "´1 2´c`i b and θ 1 " ω 1`U ω 2 , θ 2 " ω 1`V ω 2 .
Then the structure equation is
dθ 1 " p 1 2´c`i bqθ 2^θ1 , dθ 2 " p 1 2`c`i bqθ 1^θ2 .
By the complex Frobenius Theorem, there exist coordinates z and w and functions f and g, such that θ 1 " f dz, θ 2 "ḡdw. Since dpθ 1^θ2 q " dpf gdz^dwq " 0, so there exists a function F pz, wq such that f g " F pz, wq. Since
dθ 1 " df^dz "´1 2´c`i b¯F pz, wqdw^dz , dθ 2 " dg^dw "´1 2`c`i b¯F pz, wqdz^dw ,
f and g are both functions of z, w and
f w "´1 2´c`i b¯f g , g z "´1 2`c`i b¯f g .
So there exist functions Apzq and Bpwq such that f pz, wq gpz, wq " Apzq Bpwq .
So there exists a function Gpz, wq such that θ 1 " ApzqGpz, wqdz, θ 2 " BpwqGpz, wqdw.
After redefining z and w and the function f pz, wq, we can write
θ 1 " f pz, wqdz, θ 2 " f pz, wqdw $ & % f w "´1 2´c`i b¯f 2 f z "´1 2`c`i b¯f 2
Thus there exists a constant C such that
f pz, wq " 1´1 2`c`i b¯z`´1 2´c`i b¯w`C .
After translating z or w,
θ 1 " dź´1 2`c`i b¯z`´1 2´c`i b¯w , θ 2 " dẃ´1
2`c`i b¯z`´1 2´c`i b¯w . Now change the notation from w tow, then
θ 1 " dź´1 2`c`i b¯z`´1 2´c`i b¯w , θ 2 " dẃ´1 2`c`i b¯z`´1 2´c`i b¯w .
So the Engel structure can be defined on the complex 2-plane except two lineś´1
2`c`i b¯z`´1 2´c`i b¯w " 0,´1 2`c`i b¯z`´1 2´c`i b¯w " 0.(64)
Now we can get the local coordinate representation of ω 1 and ω 2
ω 1 " 1 2c " p 1 2`c`i bqθ 1`p´1 2`c`i bqθ 2 ı , ω 2 " 1 2c pθ 1´θ2 q.
Let λ and µ be two complex constants. The symmetry group of the coframing is pz, wq Ñ pλz,λwq and pz, wq шz`ˆ1 2´c`i b˙µ, w`ˆ1 2`c`i b˙μ˙.
5.4.3 b 2`a " 1 4
The structure equation is
dω 1 " 2ibω 1^ω1´ˆ1 4´b 2`i b˙ω 2^ω1`ˆ1 4´b 2´i b˙ω 1^ω2 , dω 2 " pω 1´ω2 q^ω 1´ˆ1 4´b 2`i b˙ω 2^ω2 .
Define θ " ω 1``´1 2`i b˘ω 2 . Then dθ "´1 2`i b¯θ^θ.
Let θ " α`iβ be the real part and imaginary part decomposition. Then dα "´2bα^β , dβ " α^β, so dpα`2bβq " 0. Thus there exists a real function x such that α`2bβ " dx. Thus, dβ " dx^β.
This is equivalent to dpe´xβq " 0, that implies the existence of a real function y such that β " e x dy.
So θ " dx´2be x dy`ie x dy.
Let z "´e´x´2by`iy. Then
θ " dz p´1 2`i bqz`p´1 2´i bqz .
Recall that the structure equation is
dω 1 "ˆ1 2`i b˙θ^ω 1`ˆ´1 2`i b˙ω 1^θ , dω 2 " θ^θ`ˆ1 2`i b˙`θ^ω 2`θ^ω2˘.
Let ω 1 " γ`iδ be the real part and imaginary part decomposition. Since θ " α`iβ, the exterior derivative of ω 1 can be written as dγ " α^γ`β^δ´2bα^δ`2bβ^γ, dδ " 0.
So there exists a real function u such that δ " du. Since xγ, duy forms a Frobenius system, there exist real functions p, q, v such that γ " pdv`qdu. Write dp " p x dx`p y dy`p u du`p v dv , dq " q x dx`q y dy`q u du`q v dv .
Then from the structure equation, the functions p and q satisfy p x " ṕ x´2by ,
p y " 2bṕ x´2by , p u " q v , q x " q´2b x´2by , q y " 1`2bq x´2by .
From the first two equations, we know that there exists a function C 1 pu, vq such that p " C1pu,vq x`2by . From the last two equations, we know that there exists a function C 2 pu, vq such that q " p2bx´yq`C2pu,vq
x`2by .
From the third equation we know that BC1 Bu " BC2 Bv . Thus d´ş C 1 pu, vqdv¯" C 1 pu, vqdv`C 2 pu, vqdu. Thus
γ " C 1 pu, vq x`2by dv`p 2bx´yq`C 2 pu, vq x`2by du " C 1 pu, vqdv`C 2 pu, vqdu x`2by`2 bx´y x`2by du " dp ş C 1 pu, vqdvq x`2by`2 bx´y x`2by du.
Now define p ş C 1 pu, vqdvq as new v, then γ " dv x`2by`2 bx´y x`2by du.
So
ω 1 " γ`iδ " dv x`2by`2 bx´y x`2by du`idu " 1 x`2by rdv`p2b`iqpx`iyqdus " 1 x`2by rdpv`p2b`iqpx`iyquq´p2b`iqu dpx`iyqs .
Define w " v`p2b`iqpx`iyqu and redefine z " x`iy. The coframing can be written as
ω 1 " 1 x`2by rdw´p2b`iqu dzs "´1 p´1 2`i bqz`p´1 2´i bqẑ " dw`ˆ1 2´i b˙w´w p´1 2`i bqz`p´1 2´i bqz dz , ω 2 " 1 1 2`i b pθ´ω 1 q " 1 p´1 2`i bqp´1 2`i bqz`p´1 2´i bqẑ " dz`dw`ˆ1 2´i b˙w´w p´1 2`i bqz`p´1 2´i bqz dz ,
where pz, wq is a local holomorphic coordinate system on the manifold. And ω 2 " 0 defines the complex Engel structure. Let λ, s, t be any real constants. The coframing is invariant under the following local transformation pz, wq шλz`ˆb´1 2 i˙t, λw`s˙.
We can take a discrete subgroup Γ such that M is locally biholomorphic to C 2 {Γ -R 2ˆT 2 .
Homogeneous Case C5
Assume a ‰ 0. Otherwise, this is a special case of homogeneous case C4, with a`b 2 " 1 4 . The structure equation is
dω 1 "ˆb 2´1 4´i b˙ω 2^ω1`ˆ1 4 a`ab 2`iˆ1 2 ab`2ab 3˙˙ω 2^ω2`2 ibω 1^ω1ˆ1 4´1 2 a´2ab 2´b2´i b˙ω 1^ω2`ˆ1 2 a´2ab 2´2 iab˙ω 2^ω1 , dω 2 "ˆ´1 4`1 2 a´2ab 2`b2`i p2ab´bq˙ω 2^ω2`ω1^ω1
p´a`2iabq ω 1^ω2`p´1`a`2 iabq ω 2^ω1 .
Define θ " ω 1``´1 2`i b˘ω 2 , that satisfies
dθ "ˆ1 2`i b˙θ^θ.(65)
This structure equation is same as that of the case C4, with b 2`a " 1 4 . But in case C5, a and b do not have to satisfy this relation.
Let θ " α`iβ be the real and imaginary part decomposition. By (65), dα "´2bα^β, dβ " α^β.
Since dpα`2bβq " 0, there exists a real function x such that α`2bβ " dx. Thus dβ " dx^β. So there exists a real function y such that β " e x dy, that yields θ " dx´2be x dy`ie x dy .
Let z "´e´x´2by`iy. Then
θ " dz p´1 2`i bqz`p´1 2´i bqz .
The symmetry groups of the coframing are different for different parameters a and b. In the following sections, we will consider the following cases:
1. a ‰˘1 2 2. a " 1 2 , b " 0 3. a " 1 2 , b ‰ 0 4. a "´1 2 5.5.1 a ‰˘1 2 Define A " r`is, where r " 1 1 2`a¯p 1`4b 2 q s " 2b 1 2´a¯p 1`4b 2 q .
After defining θ 2 " ω 2`A θ, the structure equation reduces to
dθ 2 " 2a´´1 2`i b¯θ^θ 2`p 2a´1q´1 2`i b¯θ 2^θ`´1 2`i b¯θ^θ 2 .
Let θ 2 " γ`iδ and θ " dx`idy Apzq , where Apzq " p´1 2`i bqz`p´1 2´i bqz "´x´2by is a real function.
By (66), xγ, dyy and xδ, dxy are two Frobenius systems, that implies the existence of functions p, q, r, s and u, v such that γ " pdu`qdy, δ " rdv`sdx.
Write dp " p x dx`p y dy`p u du`p v dv, dq " q x dx`q y dy`q u du`q v dv, dr " r x dx`r y dy`r u du`r v dv, ds " s x dx`s y dy`s u du`s v dv.
By (66),
dγ " dp^du`dq^dy " p x dx^du`p y dy^du`p v dv^du`q x dx^dy`q u du^dy`q v dv^dy " 1 x´2by rpp1´2aqdx^du`pqp1´2aq´sqdx^dỳ
rdy^dv´4abpdy^dus.
Then the functions p and q must satisfy
p y´qu "´4 abṕ x´2by
,
p x " 1´2á x´2by p, p v " 0, q v "´ŕ x´2by , q x " qp1´2aq´ś x´2by .
Thus there exists a function C 1 pu, yq such that p " px`2byq 2a´1 C 1 pu, yq. After redefining u and q, we can assume C 1 pu, yq " 1. Then p " px`2byq 2a´1 .
By (66),
dδ " dr^dv`ds^dx
" r x dx^dv`r y dy^dv`r u du^dv`s y dy^dx`s u du^dx`s v dv^dx
" 1 x´2by " 2bpdx^du`p2bq`p4ab´2bqsqdx^dỳ p´4ab`2bqrdy^dv´2ardx^dv ı .
Then the functions r and s must satisfy r x´sv "´2 aŕ x´2by , r y "´4 ab`2b x´2by r,
r u " 0, s y "´2 bq`p´4ab`2bqś x´2by , s u "´2 bṕ x´2by .(67)
So there exists a function C 2 px, vq such that r " px`2byq 2a´1 C 2 px, vq. After redefining v and s,
we can assume C 2 px, vq " 1. Then
r " px`2byq 2a´1 .
Substituting this equation into (67), the equations are as follows:
q u "´2bpx`2byq 2a´2 , q v " px`2byq 2a´2 , q x " p2a´1qq`s x`2by , s v "´px`2byq 2a´2 , s u " 2bpx`2byq 2a´2 , s y " 2bq`p4ab´2bqs x`2by .
This yields
θ 2 " γ`iδ " px`2byq 2a´1 pdu`idvq`is´dx´i q s dy¯.(68)
To get a local holomorphic coordinate system, let q "´s. Then
q u "´2bpx`2byq 2a´2 , q v " px`2byq 2a´2 , q x " p2a´2qq
x`2by , q y "´p´4 ab`4bqq x`2by .
From these equations, we get
q " p1´2bu`vqpx`2byq 2a´2
Substituting this equation into (68) yields
θ 2 " γ`iδ " ppdu`qdyq`iprdv`sdxq " ppx`2byq 2a´1 du`p1´2bu`vqpx`2byq 2a´2 dyq ippx`2byq 2a´1 dv´p1´2bu`vqpx`2byq 2a´2 dxq " px`2byq 2a´1 dpu`ivq´ip1´2bu`vqpx`2byq 2a´2 dpx`iyq.
Define w " u`iv. We get the local coordinate representation of θ 2 :
θ 2 " p´Apzqq 2a´1 " dw`"i`ˆ1 2´b i˙w´ˆ1 2`b i˙w dz Apzq .
Recall that
θ " ω 1`ˆ´1 2`i b˙ω 2 , θ 2 " ω 2`« 1 1 2`a˘p 1`4b 2 q`i 2b 1 2´a˘p 1`4b 2 q ff θ.
Thus in the local holomorphic coordinate system pz, wq, the coframing can be written as
ω 1 " # 1`ˆ´1 2`i b˙« 1 1 2`a˘p 1`4b 2 q`i 2b 1 2´a˘p 1`4b 2 q ff+ θ´ˆ´1 2`i b˙θ 2 ω 2 " θ 2´« 1 1 2`a˘p 1`4b 2 q`i 2b 1 2´a˘p 1`4b 2 q ff θ. 5.5.2 a " 1 2 , b " 0
In this case, we can choose pa, bq such that
lim aÑ 1 2 bÑ0 2b 1 2´a¯p 1`4b 2 q " 0
Then all the formula in the case a ‰ 1 2 applies to this case a " 1 2 , b " 0. The local coordinate representation of the coframing is
ω 1 "ˆ1 2`i b˙θ´ˆ´1 2`i b˙θ 2 ω 2 " θ 2´θ . 5.5.3 a " 1 2 , b ‰ 0
The structure equation is
dω 2 " ω 1^ω1`ˆ´1 2`i b˙ω 1^ω2`ˆ´1 2`i b˙ω 2^ω1 .(69)
Substituting θ " ω 1`p´1 2`i bqω 2 into (69) yields
dω 2 " " θ´´´1 2`i b¯ω 2 ı^"θ´´´1 2´i b¯ω 2 ì´1 2`i b¯θ^ω 2`´´1 2`i b¯ω 2^"θ´´´1 2´i b¯ω 2 ı " θ^θ`´1 2`i b¯θ^ω 2`´´1 2`i b¯θ^ω 2 .
From the structure equation,
dδ " dr^dv`ds^dx
" r x dx^dv`r y dy^dv`r u du^dv`s y dy^dx`s u du^dx`s v dv^dx "´2 Apzq 2 dx^dy`1 Apzq dx^p´rdv`2bpdu`2bqdyq.
Then the functions r and s must satisfy r x´sv "´ŕ x´2by , r y " 0, r u " 0,
s y " 2 px`2byq 2`2 b
x`2by q, s u "´2 bṕ x´2by .
Since there is ambiguity for choosing r and v, by modifying v we can arrange r " 1.
Thus (70) and (71) reduces to
q u "´2 b x`2by , q v " 1 x`2by , q x " s x`2by , s v "´1
x`2by , s y " 2 px`2byq 2`2 b
x`2by q, s u " 2b x`2by .
Thus there exist functions C 1 px, yq and C 2 px, yq such that q " 1 Apzq p2bu´vq`C 1 px, yq and s "´1 Apzq p2bu´vq`C 2 px, yq.
The functions C 1 px, yq and C 2 px, yq satisfy
C 1x " 1 x`2by C 2 , C 2y " 2b x`2by C 1`2 px`2byq 2 .
Choose C 2 px, yq "´C 1 px, yq such that z is a local holomorphic coordinate. Then C 1x "´1
x`2by C 1 ,
C 1y "´2 b x`2by C 1´2 px`2byq 2 .
Then C 1 px, yq "´l npx`2byq bpx`2byq .
So ω 2 " γ`iδ " ppdu`qdyq`iprdv`sdxq " dpu`ivq`ˆ1 Apzq p2bu´vq´l npx`2byq bpx`2byq˙d py´ixq.
Let w " u`iv. Then
ω 2 " dw´i 1 Apzq "´b`i 2¯w`´b´i 2¯w`1 b lnp´Apzqq dz.
Recall that θ " ω 1`ˆ´1 2`i b˙ω 2 .
So we can write ω 1 as
ω 1 " dz Apzq´ˆ´1 2`i b˙ω 2 .
Then there are functions C 1 px, yq and C 2 px, yq such that q " 1 Apzq 3 p2bu´vq`C 1 px, yq
and s "´1 Apzq 3 p2bu´vq`C 2 px, yq.
The functions C 1 px, yq and C 2 px, yq satisfy C 1x " C 2´2 C 1 x`2by , C 2y " 2bC 1´4 bC 2 x`2by`2 px`2byq 2 .
Choose C 2 px, yq "´C 1 px, yq such that z is a local holomorphic coordinate. Then C 1x "´3
x`2by C 1 ,
C 1y "´6 b x`2by C 1´2 px`2byq 2 .
We can solve C 1 px, yq C 1 px, yq "´2 ypx`byq px`2byq 3 .
By (74), we get q " 1 Apzq 3 p2bu´vq´2 ypx`byq px`2byq 3 .
By (75), we get s "´1 Apzq 3 p2bu´vq`2 ypx`byq px`2byq 3 .
So ω 2 " γ`iδ " ppdu`qdyq`iprdv`sdxq " 1 Apzq 2 dpu`ivq`´1 Apzq 3 p2bu´vq´2
ypx`byq px`2byq 3¯d py´ixq.
Let w " u`iv. Then
ω 2 " 1 Apzq 2 dw´i 1 Apzq 3 "´b`i 2¯w`´b´i 2¯w´i 2 pz´zqpp1´ibqz`p1`ibqzq ı dz.
Recall that θ " ω 1`ˆ´1 2`i b˙ω 2 , so we can write ω 1 such that ω 1 " dz Apzq´p´1 2`i bqω 2
Compact case of type C5
From the structure equation, dpω 1^ω2^ω2 q " p1´2aqp1`2biqω 1^ω1^ω2^ω2 ‰ 0.
If a ‰ 1 2 , there is no compact quotient of type C5. In the following analysis, we assume a " 1 2 . Recall that dθ "´1 2`i b¯θ^θ,
dω 2 " θ^θ`ˆ1 2`i b˙θ^ω 2`ˆ´1 2`i b˙θ^ω 2 .(76)
Let θ " α`iβ and ω 2 " γ`iδ be the real and imaginary part decompositions. By (76), dα "´2bα^β, dβ " α^β, dγ " β^δ´2bβ^γ, dδ "´2α^β´α^δ`2bα^γ.
Let e 1 , e 2 , e 3 , e 4 be left-invariant vector fields dual to the left-invariant forms α, β, γ, δ, respectively.
The nontrivial brackets are re 1 , e 2 s "´2be 1`e2´2 e 4 , re 1 , e 3 s " 2be 4 , re 1 , e 4 s "´e 4 , re 2 , e 3 s "´2be 3 , re 2 , e 4 s " e 3 .
Make a basis change:ẽ 3 " e 3`2 be 4 ,ẽ 2 " e 2´2 be 1 . Then re 1 ,ẽ 3 s " 0, re 2 ,ẽ 3 s " 0.
After dropping the tildes, the nontrivial brackets are re 1 , e 2 s " e 2´2 e 4 , re 1 , e 4 s "´e 4 , re 2 , e 4 s " e 3 .
Defineẽ 2 " e 2´e4 . Then re 1 ,ẽ 2 s "ẽ 2 , re 1 , e 4 s "´e 4 , rẽ 2 , e 4 s " e 3 .
Then define X 1 "´e 3 , X 2 " e 4 , X 3 "ẽ 2 , X 4 " e 1 . The nontrivial brackets are rX 2 , X 3 s " X 1 , rX 2 , X 4 s " X 2 , rX 3 , X 4 s "´X 3 .
By the classification result of solvmanifolds [15], there exists a co-compact lattice Γ such that G{Γ is compact and supports a homogeneous complex Engel structure.
Homogeneous Case C6
In this case, the symmetry groups of the coframing are different for different parameters a and b.
We will consider the following 2 cases:
1. a "´π 2`2 kπ, k P Z.
In this case, the constants pp 1 , p 2 , q 1 , q 2 , r 1 , r 2 q are p 1 2´i b, 0, 2bi, 1 2 ibp2b`iq 2 ,´bp´2bì q,´1 4 bp´2b`iqp2b`iq 2 q. This is a special case of homogeneous case C3, with a " b 2 . There is no compact quotient that can support a homogeneous complex Engel structure unless b " 0.
Under this condition, this is a special case of case C1.
2. a ‰´π 2`2 kπ, k P Z.
By the structure equation, we get dpω 1^ω2^ω2 q "´2``b`1 2 i˘cos a``´1 2`b i˘psin a`1q˘ω 1^ω1^ω2^ω2 , dpω 1^ω1^ω2 q "´1 2 p´sin a`2b cos a´1q`4b 2´1`4 bi˘ω 1^ω1^ω2^ω2 .
It is easy to verify that dpω 1^ω2^ω2 q " dpω 1^ω1^ω2 q " 0 if and only if a "´π 2`2 kπ, k P Z.
But this is contradictory to our assumption. Thus the volume form is exact in this case. By Stokes' Theorem, there does not exist compact quotient that supports a homogeneous complex Engel structure.
Theorem 1. 4 (
4Engel normal form). Let I be a Pfaffian system on M 4 such that I K Ă T M is an Engel structure. Then every point of M has an open neighborhood U on which there exists local
Theorem 4. 1 (
1Classification of Compact Homogeneous Complex Engel Structures). Let g be the 4-dimensional Lie algebra of symmetry vector fields of a complex Engel structure. For the six distinct 2-parameter families of homogeneous complex Engel structures listed in Theorem 3.1, the results about compactness in each case are listed as follows:
fl . Since xγ c v 1 , γ c v 2 , γ c v 3 y will be a new basis for the lattice L 1
fl P SL 3 pZq.
fl P SL 3 pZq. Define s "
is completely determined: the original G-structure defines a canonical sub e-structure.Thus by a Theorem of Kobayashi [16],
Corollary 2.3. The symmetry group of a complex Engel structure acts freely on the underlying
connected manifold.
Theorem 2.4. The canonical coframing of a complex Engel structure satisfies
Thendγ "´´2
a
Apzq`1 Apzq¯d
x^γ`1
Apzq
dy^δ´4
ab
Apzq
dy^γ
dδ "
2b
Apzq
dx^γ`´´4
ab
Apzq`2
b
Apzq¯d
y^δ´2
a
Apzq
dx^δ.
Since plog f gq wz " plog f gq zw ,´f z f¯w "´g w g¯z .Since plog f q zw "´f z f¯w and plog gq zw "´g w g¯z ,So xγ, dyy and xδ, dxy are two Frobenius systems. There exist functions p, q, r, s and u, v such that γ " pdu`qdy δ " rdv`sdx.From the structure equation,Since there are ambiguities for choosing p and u, by modifying u we can arrange p " 1.a "´1 2The structure equation isLet ω 2 " γ`iδ be the real and imaginary part decomposition. ThenSo xγ, dyy and xδ, dxy are two Frobenius systems. So there exist functions p, q, r, s and u, v such that γ " pdu`qdy, δ " rdv`sdx.From the structure equation,r2dx^ppdu`qdyq`dy^prdv`sdxq`2bdy^ppdu`qdyqs.This yieldsApzq .(72)The functions r and s must satisfySince there is ambiguity for choosing r and v, by modifying v we arrange r " 1 Apzq 2 . (72) and (74) reduce to2q´s Apzq ,s v "´1 px`2byq 3 , s y " 2 px`2byq 2`´2 bq`4bś px`2byq , s u " 2b px`2byq 3 .
Existence of Engel structures. Thomas Vogel, Ann. of Math. 2Vogel, Thomas, Existence of Engel structures, Ann. of Math. (2), 169 (2009), 79-137.
Exterior differential systems. R L Bryant, S S Chern, R B Gardner, H L Goldschmidt, P A Griffiths, Mathematical Sciences Research Institute Publications. 18475Springer-VerlagBryant, R. L. and Chern, S. S. and Gardner, R. B. and Goldschmidt, H. L. and Griffiths, P. A., Exterior differential systems, Mathematical Sciences Research Institute Publications, 18, Springer-Verlag, New York, 1991, viii+475.
The method of equivalence and its applications. Robert B Gardner, CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics58127Gardner, Robert B., The method of equivalence and its applications, CBMS-NSF Regional Conference Series in Applied Mathematics, 58, Society for Industrial and Applied Mathematics (SIAM), 1989, viii+127.
Shing Yau, Tung, Parallelizable manifolds without complex structure. 15Yau, Shing Tung, Parallelizable manifolds without complex structure, Topology, 15, 1976, 51- 53.
Holomorphic Engel structures. Francisco Presas, Solá Conde, Luis E , Revista Matemática Complutense. 27Rev. Mat. Complut.Presas, Francisco and Solá Conde , Luis E., Holomorphic Engel structures, Rev. Mat. Complut., Revista Matemática Complutense, 27, 2014, 327-344.
Characteristic classes for the degenerations of two-plane fields in four dimensions. Maxim Kazarian, Richard Montgomery, Boris Shapiro, Pacific J. Math. 179Kazarian, Maxim and Montgomery, Richard and Shapiro, Boris, Characteristic classes for the degenerations of two-plane fields in four dimensions, Pacific J. Math., 179, 1997, 355-370.
An introduction to differentiable manifolds and Riemannian geometry. William M Boothby, Pure and Applied Mathematics. 120430Boothby, William M., An introduction to differentiable manifolds and Riemannian geometry, Pure and Applied Mathematics, 120, 1986, xvi+430.
Contact geometry, Handbook of differential geometry. Hansjörg Geiges, IIGeiges, Hansjörg, Contact geometry, Handbook of differential geometry. Vol. II, 315-382, 2006.
Arithmetic groups, Topics in the theory of algebraic groups. James E Humphreys, Notre Dame Math. Lectures. 10Humphreys, James E., Arithmetic groups, Topics in the theory of algebraic groups, Notre Dame Math. Lectures, 10, 73-99.
Discrete subgroups of Lie groups. M S Raghunathan, Ergebnisse der Mathematik und ihrer Grenzgebiete. 68227Springer-VerlagBandRaghunathan, M. S., Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972, ix+227.
Symplectic homogeneous spaces. Bon Chu, Yao, Trans. Amer. Math. Soc. 197Chu, Bon Yao, Symplectic homogeneous spaces, Trans. Amer. Math. Soc., 197, 1974, 145-159.
Five lectures on lattices in semisimple Lie groups. Yves Benoist, Géométriesà courbure négative ou nulle, groupes discrets et rigidités. 18Benoist, Yves, Five lectures on lattices in semisimple Lie groups, Géométriesà courbure négative ou nulle, groupes discrets et rigidités, Sémin. Congr., 18, 117-176.
Exotic Engel structures on R 4. Vladimir Gershkovich, Russian J. Math. Phys. 3Gershkovich, Vladimir, Exotic Engel structures on R 4 , Russian J. Math. Phys., 3, 1995, 207- 226.
Representation theory. William Fulton, Joe Harris, Graduate Texts in Mathematics. 129551Springer-VerlagFulton, William and Harris, Joe, Representation theory, Graduate Texts in Mathematics, 129, Springer-Verlag, New York, 1991, xvi+551.
On low-dimensional solvmanifolds. Christoph Bock, Asian J. Math. 20Bock, Christoph, On low-dimensional solvmanifolds, Asian J. Math., 20, 2016, 199-262.
Le groupe des transformations qui laissent invariant une parallélisme. Shôshichi Kobayashi, Colloque de topologie de Strasbourg ; Institut de Mathématique, Université de Strasbourg.Kobayashi, Shôshichi, Le groupe des transformations qui laissent invariant une parallélisme, Colloque de topologie de Strasbourg, 1954-1955, 5, Institut de Mathématique, Université de Strasbourg., 1954.
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Complex and Lagrangian Engel structures. Zhiyong Zhao, PhD dissertationZhao, Zhiyong, Complex and Lagrangian Engel structures, PhD dissertation, 2018.
|
[] |
[
"Neutron-Mirror-Neutron Oscillations in a Trap",
"Neutron-Mirror-Neutron Oscillations in a Trap"
] |
[
"B Kerbikov \nState Research Center Institute for Theoretical and Experimental Physics\nMoscowRussia\n",
"O Lychkovskiy \nState Research Center Institute for Theoretical and Experimental Physics\nMoscowRussia\n"
] |
[
"State Research Center Institute for Theoretical and Experimental Physics\nMoscowRussia",
"State Research Center Institute for Theoretical and Experimental Physics\nMoscowRussia"
] |
[] |
We calculate the rate of neutron-mirror-neutron oscillations for ultracold neutrons trapped in a storage vessel. Recent experimental bounds on the oscillation time are discussed. *
|
10.1103/physrevc.77.065504
|
[
"https://arxiv.org/pdf/0804.0559v4.pdf"
] | 118,522,622 |
0804.0559
|
242f7b24acd936aa76510eb6ec610e85d53f3184
|
Neutron-Mirror-Neutron Oscillations in a Trap
1 Jun 2008
B Kerbikov
State Research Center Institute for Theoretical and Experimental Physics
MoscowRussia
O Lychkovskiy
State Research Center Institute for Theoretical and Experimental Physics
MoscowRussia
Neutron-Mirror-Neutron Oscillations in a Trap
1 Jun 2008
We calculate the rate of neutron-mirror-neutron oscillations for ultracold neutrons trapped in a storage vessel. Recent experimental bounds on the oscillation time are discussed. *
Introduction
During the last couple of years we are witnessing the revival of the interest to the "mirror particles", "mirror matter" and "mirror world". The idea of the existence of the hypothetical hidden sector to compensate mirror asymmetry was first explicitly formulated in [1]. The subject has a rich history -see the review paper [2]. The present wave of interest to mirror particles has been to a great extent initiated by the quest for neutron-mirror-neutron oscillations (n−n ′ ). It was conjectured that n−n ′ oscillations may play an important role in the propagation of ultra high energy cosmic rays and that the oscillation time τ osc may be as small as τ osc ∼ 1s [3]. Implications of mirror particles for cosmology and astrophysics were discussed in a number of papers e.g., [4]. Last year the first experimental data on n − n ′ transitions were published with the results τ osc ≥ 103s [5] and τ osc ≥ 414s [6]. Possible laboratory experiments to search for n − n ′ oscillations were discussed in [7].
Experimental results [5,6] were obtained using the ultracold neutrons (UCN), i.e., neutrons with the energy E < 10 −7 eV storaged in a trap.
Previously similar experimental setup was used in the search for neutron-antineutron oscillations (see [8] and references therein). The crucial difference between n − n ′ and n −n oscillations is that n ′ freely escapes from the trap whilen either annihilates on the trap walls or gets reflected. Therefore formalism developed for the n −n oscillations cannot be adjusted to treat n − n ′ transitions. Still the two processes have a common point. This is a problem of a correct quantum mechanical description of the UCN wave function (w.f.). Most often it is assumed that the w.f. of the bottled UCN corresponds to a stationary state of a particle inside a potential well [9,10]. Alternatively, other authors [11] describe oscillations of the trapped neutrons in the basis of the free plane waves. Both pictures do not correspond to the physics of real experiments. The process proceeds in time in three stages.
At the first stage the filling of the trap takes place, then the neutrons are kept inside the trap during the storage time (hundreds of seconds), and finally neutrons leave the trap to the detectors. Therefore the w.f. undergoes a complicated evolution which hardly can be described without resorting to approximations. We shall first evaluate the neutron-mirror-neutron oscillations using a stationary w.f. as the initial state w.f. Then we shall do the same using wave packet instead of a stationary w.f. The paper is organized as follows. We start in Section 2 with the analysis of the oscillations in the stationary w.f. approach. Transitions take place from one of the trap eigenstates. In Section 3 transitions are considered in presence of a superimposed magnetic field. A general equation for the transition rate is derived and the limits of weak and strong field are considered. Section 4 is devoted to the wave packet formalism. The evolution of the UCN wave packet (w.p.) is encoded using the trap Green's function. Neutron-mirror-neutron transition rate is calculated. In Section 5 the main conclusions are presented and open problems are formulated. Appendix contains comparison between the infinite and finite well models.
Stationary Wave Function Approach
The problem of neutron-mirror-neutron oscillations in free space can be solved by diagonalization of the time-dependent two-channel Schrodinger equation with the result [12] |ψ n ′ (t)| 2 = 4ε 2
ω 2 + 4ε 2 exp(−Γ β t) sin 2 1 2 √ ω 2 + 4ε 2 t ,(1)
where ω = E n − E n ′ = |µ n |B is the energy difference between neutron and mirror neutron due to superimposed magnetic field (mirror neutron does not feel "our" magnetic field), ε = τ −1 osc is the mixing parameter, Γ β is the neutron β-decay width. In arriving at (1) the spatial part of the w.f. was factored out making use of the fact that in free space the w.f.-s of n and n ′ are of the same form. In the trap, however, the situation is different: the neutron is confined while for the mirror neutron the trap walls do not exist. As already mentioned in the Introduction, the description of the trapped UCN is a nontrivial problem. The naive guess would be that inside the trap the neutron w.f. corresponds to a discrete eigenstate. Here we assume that the neutron w.f. is that of a particle in a potential well with the boundary conditions corresponding (in the first approximation) to a complete reflection.
In order to make calculations tracktable and transparent we shall consider the following simple model of a trap. Let it be a one-dimensional square well of width L = 1m with walls at x = 0 and x = L, i.e., the potential of the form
U(x) = V, x < 0 0, 0 < x < L V, x > L .
(
The height of the potential well depends on the material with the typical value V = 2 · 10 −7 eV which will be used in what follows. For such a well the limit for stored UCN velocity is 6.2m/s. The number of discrete levels in such a trap may be estimated as
M ≃ L √ 2mV π ≃ 10 8 π .(3)
We choose the UCN energy to be E = 0.8 · 10 −7 eV. This energy corresponds to a level with quantum number j ≃ 2 · 10 7 . Positions and eigenfunctions of such highly excited states in a finite-depth potential are very close to the same quantities in the infinite well (except for the levels close to the upper edge of the well; we not consider such levels). The finite-depth corrections are considered in the Appendix. The eigenvalues and eigenfunctions for the infinite well are
E j = π 2 j 2 2mL 2 , k j = πj L , j = 1, 2, 3.. (4) ϕ j (x) = 2 L sin k j x.(5)
Another important quantity characterizing highly excited states is the classical frequency ω cl [11]
ω cl = π 2 mL 2 j = 2π τ cl = δE j ,(6)
where τ cl is the time of the classical period and
δE j = E j+1 − E j ≃ 0.8 · 10 −14 eV
is the level spacing. Levels with j ≫ 1 are almost equidistant. In the semiclasical limit we may also define the trap crossing time τ
τ = τ cl 2 = mL k j ≃ 0.26s(7)
for j = 2 · 10 7 . Next we calculate the rate of (n − n ′ ) oscillations for the neutron at the j − th discrete level. The neutron and mirror neutron w.f.-s in a two-component basis arẽ
ϕ j (x) = 2 L sin k j x 1 0 ≡ ϕ j (x) 1 0 ,(8)f p (x) = 1 √ 2π e ipx 0 1 ≡ f p (x) 0 1 ,(9)
where −∞ < p < +∞. The (n − n ′ ) system is described by the Hamiltonian
H =Ĥ 0 +Ŵ = k 2 2m + U 0 0 p 2 2m + 0 ε ε 0 .(10)
The states (8) and (9) are the eigenstates ofĤ 0 , therefore it is convenient to use the interaction representation. The probability to find at time t a mirror neutron instead of a neutron reads
P nn ′ = dp| f p | exp −i t 0 dt ′Ŵ int (t ′ ) |φ j | 2 ,(11)
whereŴ int (t) = e iĤ 0 tŴ e −iĤ 0 t . In the first order of perturbation theory we get
P nn ′ = dp| f p | t 0 dt ′Ŵ int (t ′ )|φ j | 2 = = ε 2 dp| t 0 dt ′ e −i(E j −Ep)t ′ | 2 | f p |ϕ j | 2 ,(12)
where
E j = k 2 j 2m , E p = p 2 2m . The time-dependent integral is a standard one w(E p ) = | t 0 dt ′ e −i(E j −Ep)t ′ | 2 = 4 sin 2 (Ep−E j )t 2 (E p − E j ) 2 .(13)
The overlap of the w.f.-s reads
g j (p) = | f p |ϕ j | 2 = 4k 2 j πL(p 2 − k 2 j ) 2 sin 2 pL + πj 2 , j = 1, 2, ...(14)
From (12), (13) and (14) we obtain
P nn ′ = ε 2 +∞ −∞ dpg j (p)w(E p ).(15)
It is convenient to change integration from dp to dE p taking into account that g(p) = g(−p). Then
P nn ′ = 2mε 2 dE p g(E p )w(E p ) p ,(16)
where the factor 2 comes from the fact that two plane waves e ±ipx correspond to the same energy E p . Both functions g(E p ) and w(E p ) are peaked at E p = E j . According to (14) and (13) the widths ∆E g p and ∆E w p of the corresponding maxima are
∆E g p ≃ π/τ, ∆E w p ≃ 4π/t,(17)
with τ being the trap crossing time. At times t ≫ τ we may substitute g(E p )/p by its value at p = k j and take it out of the integral (16). From (14) one gets g(E j ) = L/4π. The remaining integration in (16) can be extended to (−∞ < E p < +∞) yielding 2πt. Collecting all pieces together we obtain
P nn ′ = ε 2 τ t.(18)
At very short times t ≪ τ the function w(E p ) becomes smoother than g(E p ). Hence w(E p ) can be taken out of the integral (16). The remaining integral is time-independent while w(E p ) ∼ t 2 . As a result P nn ′ ∼ ε 2 t 2 and we can not define the transition probability per unit time [12]. On the other hand, Eq.(18) is valid only for times shorter than the neutron β -decay time t β since we have define the eigenstate (8) neglecting the β-decay. The condition τ ≪ t ≪ t β was with a fair accuracy satisfied in experiments [5,6].
Stationary approach with magnetic field included
The search for n − n ′ oscillations is experiments with bottled UCN is based on the comparison of UCN storage with and without superimposed magnetic field [5,6]. It is assumed that there is no mirror magnetic field in the laboratory and therefore the interaction of the neutron with magnetic field lifts the degeneracy and thus suppresses the oscillations.
In magnetic field B the energy of the trapped neutron becomes equal to
E j = k 2 j 2m + µB,(19)
where µ = −µ n = 1.91µ N (µ N = e/2m p ). Inclusion of the magnetic field does not alter the functions w(E p ) and g(p) given by Eq.-s (13) and (14). There is, however, an important difference between our present considerations and the previous section. As we see from (19) w(E p ) now peaks at p = ± k 2 j + 2mµB while the maximum of g(p) is as before at p = ±k j . As a result instead of (16) we obtain
P nn ′ = 4ε 2 t (µB) 2 τ 1 + 2mµB k 2 j cos 2 k j L 2 1 + 2mµB k 2 j , j = 1, 3, ... sin 2 k j L 2 1 + 2mµB k 2 j , j = 2, 4...(20)
This equation can be simplified taking into account that the quantities (µB) and k 2 j /2m differ by many orders of magnihides. In experiments [5,6] the value of the magnetic field varied in the interval (1−2)nT ≤ B ≤ (few)µT which corresponds to 10 −16 eV < ∼ µB < ∼ 10 −13 eV, while k 2 j /2m ≃ 10 −7 eV (note that the unshielded Earth magnetic field corresponds to µB ≃ 3 · 10 −12 eV≪ k 2 j 2m ). Therefore Eq. (20) easily reduces to
P nn ′ ≃ 4ε 2 t τ sin 2 ( 1 2 µBτ ) (µB) 2 ,(21)
where τ is the trap crossing time. For our model of the trap described in section 2 we have τ /2 ≃ 2·10 14 eV −1 . Therefore in the limit of weak magnetic field B ≃ nT Eq. (21) yields
P nn ′ ≃ ε 2 τ t,(22)
as expected (see (18)). In the opposite limit of strong magnetic field B ≃ (few) µT we have to take into account that the quantities τ and B in (21) experience fluctuations leading to rapid oscillations of the function sin 2 ( 1 2 µBτ ). In particular, the crossing time τ may vary either due to changes of L at each cossision, or due to variations of the neutron velocity. Substituting the rapidly oscillating quantity in (21) by its mean value equal to 1/2 we obtain the equation describing the neutron-mirror-neutron transitions in strong magnetic field
P nn ′ = ε 2 2t (µB) 2 τ .(23)
The wave packet approach
We now turn to the question formulated in the Introduction, namely to the problem of the UCN w.f. evolution and to the calculation of the oscillations in the wave packet approach. In order to get physically transparent results and to avoid numerical calculations suited to a concrete experiment we assume that UCN coming to the trap from the source are described by the Gaussian wave packets (w.p.) [13]. The w.p. moving from the left and for t = 0 centered at x = 0 is given by the following expression
Ψ k (x, t = 0) = (πa 2 ) −1/4 exp − (x − x 0 ) 2 2a 2 + ikx ,(24)
where a is the width of the w.p. and k is its central momentum. The normalization of the w.p. (23) corresponds to one particle in the entire onedimensional space,
+∞ −∞ dx |Ψ k (x, t = 0)| 2 = 1.(25)
Let the UCN energy be equal to the value chosen in Section 2, E = 0.8 · 10 −7 eV, and let the beam resolution be equal to ∆E/E = 10 −3 . Thus the set of parameters to be used is 1
E = 0.8 · 10 −7 eV, λ = 2π k ≃ 10 −5 cm, a ≃ 3.2 · 10 −3 cm.(26)
The condition a ≫ λ ensures the localization of the w.p. We remind that the above value of E corresponds to the level E j with a very high quantum number j ≃ 2 · 10 7 . Next we estimate the number of levels within ∆E. One has
∆j = ∆E ω cl = v(∆k) ω cl = L πa ≃ 10 4 .(27)
The large number of levels forming the w.p. is a necessary condition for the trapped w.p. to be localized (in free space this condition reads a ≫ λ, see above). The time evolution of the initial w.p. (24) proceeds according to the following law
Ψ k (x, t) = dx ′ G(x, t; x ′ , 0)Ψ k (x ′ , 0),(28)
where G(x, t; x ′ , t ′ ) is the trap Green's function. In the infinite well approximation we may use the spectral decomposition of the Green's function over the set of eigenfunctions (5) and write
Ψ k (x, t) = ∞ j=1 e −iE j t ϕ j (x) L 0 dx ′ ϕ * j (x ′ )Ψ k (x ′ , 0).(29)
The width of the w.p. (29) increases with time according to
a ′ = a 1 + t ma 2 2 1/2 ≃ a t ma 2 ,
where for our model the spreading time is ma 2 ≃ 1.7 · 10 −2 s and t/ma 2 ≃ 60t(s). A so-called collapse time t c [14] corresponds to a ′ = L and is equal to t c ≃ 500s. At t = t c the w.p. spreads uniformly over the entire well and the stationary regime considered in Section 2 sets in [15]. We note in passing that there is another time scale in the problem, the so-called revival time t rev = 4mL 2 /π ≃ 2.10 7 s when the w.p. regains its initial shape -see [14] and references therein.
The initial w.p Ψ k (x, 0) contains only right running wave -see (24). The trapped w.p. (29) contains both right and left running waves, i.e., it correctly describes reflections from the trap walls. We assume that the point x 0 (see (24) is not in the immediate vicinity of the trap walls, i.e., x 0 is at least few times of a away from the walls. Then the integration in (29) may be extended to the entire one-dimensional space . This yields
F (k, k j ; L, a, x 0 ) ≡ +∞ −∞ dx ′ ϕ * j (x ′ )Ψ k (x ′ , 0) = i a √ π L 1/2 × × exp − a 2 (k − k j ) 2 2 + i(k − k j )x 0 − exp [...k j → −k j ...] .(30)
Then we can calculate the transition probability P nn ′ following the procedure described in Section 2. Instead of the w.f. (8) we now have
Ψ k (x, t) = ∞ j=1 e −iE j t ϕ j (x)F j (k),(31)
with F j (k) being the shorthand notation for the function F j (k, k j ; L, a, x 0 ) defined by (30). The normalization condition for F j (k) reads
j |F j (k)| 2 = 1.(32)
In line with (15) and following the arguments presented after (16) we write
P nn ′ = ε 2 j,l F j (k)F * l (k)e i 2 (E l −E j )t × × +∞ −∞ dp 2 sin (Ep−E j )t 2 (E p − E j ) 2 sin (Ep−E l )t 2 (E p − E l ) f p |ϕ j ϕ l |f p .(33)
Consider first the contribution P (1) nn ′ of the diagonal terms with j = l. We have
P (1) nn ′ = ε j |F j (k)| 2 +∞ −∞ dp 4 sin (Ep−E j )t 2 (E p − E j ) 2 f p |ϕ j ϕ j |f p = = ε 2 j |F j (k)| 2 2m k j 2πt L 4π = ε 2 τ t,(34)
with τ being the weighted crossing time
τ = j |F j (k)| 2 τ (k j ),(35)
and τ (k j ) = mL/k j . Next we turn to the contribution P (2) nn" of the nondiagonal terms in (32). In this case we are dealing with a two-hump function with maxima at E p = E j and E p = E l . Therefore we may write
P (2) nn ′ = 4πmε 2 j F j (k) k j l F * l (k)e 1 2 (E l −E j )t 2 sin (E j −E l )t 2 (E j − E l ) × × f j |ϕ j ϕ l |f j + (j ↔ l)} .(36)
Replacing summation over l by integration we obtain
P (2) nn ′ ≃ 8ε 2 j |F j | 2 m 2 L 2 k 2 j = 8ε 2 τ 2(37)
where
τ 2 = j |F j | 2 τ 2 j .(38)
Collecting the two contributions (34) and (37) together we get the final result
P nn ′ = ε 2 τ t 1 + 8 τ 2 τ t .(39)
Conclusions
We have calculated the rate of neutron-mirror-neutron oscillations for trapped UCN. Two types of the UCN w.f.-s were used: the stationary solution for a particle inside a potential well and the Gaussian w.p. Calculations were performed in the first-order perturbation theory. This approximation is legitimate provided P nn ′ ≪ 1. From (22) and (23) it follows that this condition holds for τ nn ′ ≫ 7s and τ nn ′ ≫ 0.1s in the weak and strong magnetic fields correspondingly. Obviously the first order perturbation theory describes the transition of the neutron into mirror neutron. The inverse process appears only in the second order in ε. For the analysis of the experiments [5,6] first order perturbation theory is a fair approximation. The above analysis has been performed for a simple one-dimensional trap. We think that such a model correctly describes the principal features of the process. Generalization to the three-dimensional rectangular trap is trivial. The simplest way to generalize our result to the trap with arbitrary geometry is to substitute the crossing time τ by the effective crossing time corresponding to a given trap geometry.
Experimental data [5,6] were analyzed using the free space equation (1) with the time t being limited by the crossing time τ .
Equation (1) contains only time dependence since the spatial parts of n and n ′ w.f.-s were factored out using the fact that in free space the coordinate w.f.-s of n and n ′ have the same form. For bottled UCN the situation is different. Neutron is confined inside the trap while mirror neutron freely crosses the trap walls. Therefore the use of the (1) to describe oscillations of trapped UCN seems questionable 2 . However our accurate approach justifies the analysis of the experimental data based on Eq.(1) [5,6]. This can be explained by the semiclassical character of the UCN motion inside the trap of macroscopic size. In the stationary approach the typical UCN energy corresponds to the states with j ≫ 1, i.e., to the semiclassical part of the spectrum. In the w.p. formalism the classical limit corresponds to (∆j) ∼ (j) 1/2 → ∞ [16]. The w.p. (24),(26) is close to this limit. For UCN with extremely low energy, E < ∼ 10 −16 eV, the oscillation pattern changes. We shall consider this question elsewhere. The fraction of UCN with such energies in experiments is negligible. Another point which deserves a dedicated study is decoherence of the UCN w.f. and subsequent randomization of the oscillation process. This might occur due to dephasing of the w.f. caused by collisions with the trap walls and with the residual gas.
Appendix
Calculations presented above were performed for the infinite well model of a trap. Here we consider the finite potential and show that there is only a minor difference between the two models. Consider the potential well defined by Eq. (2). Matching the logarithmic derivatives of the w.f.-s at x = 0 and x = L we obtain the eigenvalue equation
k ′ j L = πj − 2 arcsin k ′ j √ 2mV
, (A.1) (the notation k j is kept for k j = πj/L). The small parameter in the problem is δ = 2 mV L 2 1/2 ≃ 2 · 10 −8 .
(A.2)
Expanding (A.1) with respect to δ we obtain
k ′ j ≃ πj L (1 − δ), E ′ j ≃ π 2 j 2 2mL 2 (1 − 2δ). (A.3)
Therefore the levels in the finite well are shifted relative to the infinite well levels by In the finite well the w.f. penetrates into classically forbidden regions inside the trap walls. However neutron-mirror-neutron transitions inside the walls may be neglected since both the penetration depth d and the collision time τ coll are small: d ∼ 10 −6 cm, τ coll ∼ 10 −8 s.
E j − E ′ j ≃ 4 · 10 −15 eV. (A.4) From (A.3) it follows that the spectrum in the finite well (2) is the same as in somewhat wider infinite well L ′ = L(1 + δ). (A.5)
The problem of the choice of the w.p. parameters will be addressed in the next Section.
To describe the free space experiments by Eq.(1) one still has to supplement it by a boundary condition at x = 0 where the reactor is placed. Otherwise at t = πτ nn ′ /2 the reactor becomes a source of mirror neutrons. We are grateful to L.B.Okun and M.I.Vysotsky for drawing our attention to this.
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|
[] |
[
"Deep Impression: Audiovisual Deep Residual Networks for Multimodal Apparent Personality Trait Recognition",
"Deep Impression: Audiovisual Deep Residual Networks for Multimodal Apparent Personality Trait Recognition"
] |
[
"Yagmur Güçlütürk \nDonders Institute for Brain\nCognition and Behaviour\nRadboud University\nNijmegenthe Netherlands\n",
"Umut Güçlü \nDonders Institute for Brain\nCognition and Behaviour\nRadboud University\nNijmegenthe Netherlands\n",
"Marcel A J Van Gerven [email protected] \nDonders Institute for Brain\nCognition and Behaviour\nRadboud University\nNijmegenthe Netherlands\n",
"Rob Van Lier [email protected] \nDonders Institute for Brain\nCognition and Behaviour\nRadboud University\nNijmegenthe Netherlands\n"
] |
[
"Donders Institute for Brain\nCognition and Behaviour\nRadboud University\nNijmegenthe Netherlands",
"Donders Institute for Brain\nCognition and Behaviour\nRadboud University\nNijmegenthe Netherlands",
"Donders Institute for Brain\nCognition and Behaviour\nRadboud University\nNijmegenthe Netherlands",
"Donders Institute for Brain\nCognition and Behaviour\nRadboud University\nNijmegenthe Netherlands"
] |
[] |
Here, we develop an audiovisual deep residual network for multimodal apparent personality trait recognition. The network is trained end-to-end for predicting the Big Five personality traits of people from their videos. That is, the network does not require any feature engineering or visual analysis such as face detection, face landmark alignment or facial expression recognition. Recently, the network won the third place in the ChaLearn First Impressions Challenge with a test accuracy of 0.9109.
|
10.1007/978-3-319-49409-8_28
|
[
"https://arxiv.org/pdf/1609.05119v1.pdf"
] | 12,693,562 |
1609.05119
|
56ba40849acfe4f6a8a2338f5bb31be2ce6df712
|
Deep Impression: Audiovisual Deep Residual Networks for Multimodal Apparent Personality Trait Recognition
Yagmur Güçlütürk
Donders Institute for Brain
Cognition and Behaviour
Radboud University
Nijmegenthe Netherlands
Umut Güçlü
Donders Institute for Brain
Cognition and Behaviour
Radboud University
Nijmegenthe Netherlands
Marcel A J Van Gerven [email protected]
Donders Institute for Brain
Cognition and Behaviour
Radboud University
Nijmegenthe Netherlands
Rob Van Lier [email protected]
Donders Institute for Brain
Cognition and Behaviour
Radboud University
Nijmegenthe Netherlands
Deep Impression: Audiovisual Deep Residual Networks for Multimodal Apparent Personality Trait Recognition
Big Five personality traitsaudiovisualdeep neural net- workdeep residual networkmultimodal
Here, we develop an audiovisual deep residual network for multimodal apparent personality trait recognition. The network is trained end-to-end for predicting the Big Five personality traits of people from their videos. That is, the network does not require any feature engineering or visual analysis such as face detection, face landmark alignment or facial expression recognition. Recently, the network won the third place in the ChaLearn First Impressions Challenge with a test accuracy of 0.9109.
Introduction
Appearances influence what people think about the personality of other people, even without having any interaction with them. These judgments can be made very quickly -already after 100 ms [35]. Although some studies have shown that people are good at forming accurate first impressions about the personality traits of people after viewing their photographs or videos [4,21], it has also been shown that simply relying on the appearance does not always result in correct first impression judgments [22].
Several characteristics of people varying from clothing to facial expressions, contribute to the first impression judgments about personality [29]. For example, [30] has shown that the photographs of the same person taken with a different facial expression changes the judgments about the person's personality traits such as trustworthiness and extravertedness as well as other perceived characteristics such as attractiveness and intelligence. Furthermore, people are better at guessing other's personality traits if they find them attractive after short encounters with them [18]. The same study also showed that people form more positive first impressions about more attractive people.
Studies of personality prediction generally either deal with correctly recognizing the actual personality traits of people, which can be measured as self-or acquaintance-reports or apparent personality traits, which are the impressions about the personality of an unfamiliar individual [34]. Below we review the recent work in apparent personality prediction.
Most of the previous work on apparent personality modeling and prediction have been in the domain of paralanguage, i.e. speech, text, prosody, other vocalizations and fillers [34]. Conversations (both text and audio) [19] and speech clips [23,20] were the materials that were most commonly analyzed. In this domain, INTERSPEECH 2012 Speaker Trait Challenge [25] enabled a systematic comparison of computational methods by providing a dataset comprising audio data and extracted features. The competition had three sub-challenges for predicting the Big Five personality traits, likability and pathology of speakers, .
Recently, prediction of apparent personality traits from social media content has become a challenge that attracted much attention in the field. For example [6,26] demonstrated that the images that the users "favorite" on Flickr enabled the prediction of both apparent and actual (self-assessed) personality traits of Flickr users. [32] looked at the influence of a large number of physical attributes (e.g. chin length, head size, posture) on people's impressions regarding approachability, youthful-attractiveness and dominance of them. They studied these influences based on people's impressions formed after looking at face photographs. They performed factor analysis to quantify the contribution of physical attributes and used these factors as inputs to a linear neural network to predict impressions. Their predictions were significantly correlated to the actual impression data.
Given that the exact facial expression [30] and the posture [32] of the person in a photograph influences the first impression judgments about that person, as well as the importance of paralinguistic information in impression formation [19], continuous audio-visual data seems to be a suitable medium to study first impressions. In a series of studies using YouTube video blogs (vlogs) [1,3,2,29], researchers showed that this is indeed the case. Furthermore, [5] showed that audiovisual annotations along with audiovisual cues enabled the best prediction performance for their regression models compared to either using only either one of them.
At the same time, deep neural networks [24,16] in general and deep residual networks [11] in particular have achieved state-of-the-art results in many computer vision tasks. For example, [11] won the first places in the object detection task and the object localization task at the ImageNet Large Scale Visual Recognition Challenge 2015 1 with their seminal work that introduced deep residual networks. Furthermore, deep residual networks have been successfully used in a variety of other computer vision tasks ranging from style transfer [14] and image super-resolution [14] to semantic segmentation [7] and face hallucination [9].
Recently, [33] suggested that deep neural networks can be used for personality trait recognition because of the hierarchical organization of the personality traits [36]. Following this line of reasoning as well as the recent success of deep residual networks, we develop an audiovisual deep residual network for multimodal personality trait recognition. The network is trained end-to-end for predicting the apparent Big Five personality traits of people from their videos. That is, the network does not require any feature engineering or visual analysis such as face detection, face landmark alignment or facial expression recognition. Figure 1 shows an illustration of the network architecture. The network comprises an auditory stream of a 17 layer deep residual network, a visual stream of another 17 layer deep residual network and an audiovisual stream of a fullyconnected layer.
Methods
Architecture
The auditory stream and the visual stream are similar to the first 17 layers of the 18 layer deep residual network in [11]. That is, each stream comprises one convolutional layer and eight residual blocks of two convolutional layers. The convolutional layers are followed by batch normalization [13] (all layers), rectified linear units (all layers), max pooling (first layer) and global average pooling (last layer). In the residual blocks that do not change the dimensionality of their inputs, identity shortcut connections are used. In the remaining residual blocks, convolutional shortcut connections are used. In contrast to [11], the number of convolutional kernels are halved.
Similar to [8], the difference between the auditory stream and the visual stream is that inputs, convolutional/pooling kernels and strides of the auditory stream are one-dimensional whereas those of the visual stream are twodimensional if the number of channels are ignored. That is:
-An n 2 × 1 × 1 input of the auditory stream corresponds to an n × n × m input of the visual stream. -An n 2 × 1 × m/n 2 × 1 convolutional/pooling kernel of the auditory stream corresponds to an n × n × m/n × n convolutional/pooling kernel of the visual stream. -An n 2 × 1 stride of the auditory stream corresponds to an n × n stride of the visual stream.
where m is the number of channels. Outputs of the auditory stream and the visual stream are merged in an audiovisual stream. The audiovisual stream comprises a fully-connected layer. The fully-connected layer is followed by hyperbolic tangent units. Outputs of the audiovisual stream are scaled to [0, 1].
Training
We used Adam [15] with initial α = 0.0002, β 1 = 0.5, β 2 = 0.999, = 10 −8 and mini-batch size = 32 to train the network by iteratively minimizing the mean absolute error loss function between the target traits and the predicted traits for 900 epochs. We initialized the biases/weights as in [10] and reduced α by a factor of 10 after every 300 epochs. Each training video clip was processed as follows:
-The audio data and the visual data of the video clip are extracted.
-A random 50176 sample temporal crop of the audio data is fed into the auditory stream. The activities of the penultimate layer of the auditory stream are temporally pooled.
Validation/Test
Each validation/test video clip was processed as follows:
-The audio data and the visual data of the video clip are extracted.
-The entire audio data are fed into the auditory stream. The activities of the penultimate layer of the auditory stream are temporally pooled (see below note). It should be noted that the network can process video clips of arbitrary sizes since the penultimate layers of the auditory stream and the visual stream are followed by global average pooling.
Results
We evaluated the network on the dataset that was released as part of the ChaLearn First Impressions Challenge 2 [17]. The dataset consists of 10000 15second-long video clips that were drawn from YouTube 3 , of which 6000 were used for training, 2000 were used for validation and 2000 were used for test. The video clips were annotated with the Big Five personality traits (i.e. openness to experience, conscientiousness, extraversion, agreeableness, and neuroticism) by Amazon Mechanical Turk 4 workers. Each trait was represented with a value between the range [0, 1].
The video clips were preprocessed by temporally resampling the audio data to 16000 Hz as well as spatiotemporally the video data to 456 pixels × 256 pixels and 25 frames per second.
We implemented the network in Chainer [31] with CUDA and cuDNN. Most of the processing took place on a single chip of an Nvidia Tesla K80 GPU accelerator 5 . Processing took approximately 50 milliseconds per training example and 2.7 seconds per validation/test example on a single chip of an Nvidia Tesla K80 GPU accelerator. Figure 2 shows five validation examples and the corresponding predictions.
Accuracy was defined as 1 -mean absolute error. We report the validation accuracy of the network after 300, 600 and 900 epochs of training (Table 1). Average validation accuracy of the network increased as a function of number of epochs of training with the highest average validation accuracy of 0.9121. We report also the test accuracy of the network after 900 epochs of training, which won the third place in the challenge, and compare it with those of the models that won the first two places in the challenge (Table 2).
Post challenge models
For completeness, we briefly report our preliminary work on two models that we have evaluated after the end of the challenge. First, we separately fine-tuned the original DNN after 300 epochs of training for each trait. Everything about the fine-tuned DNNs (i.e. architecture, training and validation/test) were the same with the original DNN except for their fullyconnected layers that output one value rather than five values.
Second, we trained a recurrent neural network (RNN) on top of the original network. The RNN comprised two layers of 512 long short-term memory units [12] and one layer of five linear units. At each time point, the RNN took as input the layer 5 features of a second-long video clip and the output of the RNN was the predicted traits. Dropout [27] was used to regularize the hidden layers.
We used Adam to train the model by iteratively minimizing the mean absolute error loss function between the target traits and the predicted traits at each time point. Backpropagation was truncated after every 15 time points. Once the model was trained, the predicted traits were averaged over the entire video clip. Table 3 shows the validation accuracy of the post challenge models. While the post challenge models failed to outperform the challenge model to a large extent, we strongly believe that variants thereof have the potential to do so and will be the subject matter of future work.
Conclusion
In this study, we presented our approach and results that won the third place in the ChaLearn First Impressions Challenge. Summarizing, we developed and trained an audiovisual deep residual network for predicting the apparent personality traits of people in an end-to-end manner. This approach enabled us to obtain very high performance for all traits while exploiting the similarities between the organization of the personality traits and the deep neural networks in terms of the hierarchical organization and circumventing extensive analyses for identifying/designing relevant features for the task of apparent personality traits prediction. Our results demonstrate the potential of deep neural networks in the field of automatic (perceived) personality prediction. Future work will focus on the extensions of the current work with recurrent neural networks and language models as well as identifying the factors that drive first impressions.
Fig. 1 .
1Illustration of the network architecture.
-A random 224 pixels × 224 pixels spatial crop of a random frame of the visual data is randomly flipped in the left/right direction and fed into the visual stream. The activities of the penultimate layer of the visual stream are spatially pooled. -The pooled activities of the auditory stream and the visual stream are concatenated and fed into the fully-connected layer. -The fully-connected layer outputs five continuous prediction values between the range [0, 1] corresponding to each trait for the video clip.
-
The entire visual data are fed into the visual stream one frame at a time. The activities of the penultimate layer of the visual stream are spatiotemporally pooled (see below note).-The pooled activities of the auditory stream and the visual stream are concatenated and fed into the fully-connected layer. -The fully-connected layer outputs five continuous prediction values between the range [0, 1] corresponding to each trait for the video clip.
Table 1 .
1Validationaccuracies of the challenge model after 300, 600 and 900 epochs
of training.
Epoch
Validation accuracy
Average Openness Agreeableness Conscientiousness Neuroticism Extraversion
300 0.906461 0.905451
0.911128
0.902121
0.907886
0.905721
600 0.911929 0.911924
0.915610
0.911717
0.909891 0.910503
900 0.912132 0.911983
0.915466
0.913077
0.909705
0.910429
Table 2. Test accuracies of the models that won the first three places in the challenge.
Rank
Test accuracy
Average
Openness Agreeableness Conscientiousness Neuroticism Extraversion
1 [37] 0.912968037541 0.91237757 0.91257098
0.91663743
0.9099631 0.91329111
2 [28] 0.912062557634 0.91167725 0.91186694
0.91185413
0.90991703 0.91499745
3 (ours) 0.910932616931 0.91108539 0.91019226
0.91377735
0.90890031 0.91070778
. . .
10
0.875888740066 0.87026111 0.88423626
0.87270874
0.87526563 0.87697196
Fig. 2. Example thumbnails of the videos of five people and the corresponding predicted
personality traits. Each trait takes a value between [0, 1]. Each color represents a
trait. From left to right: Openness, agreeableness, conscientiousness, neuroticism and
extraversion.
Table 3 .
3Validation accuracies of the challenge model and the post challenge models.Model
Validation accuracy
Average Openness Agreeableness Conscientiousness Neuroticism Extraversion
DNN
0.912132 0.911983
0.915466
0.913077
0.909705
0.910429
5 × DNN 0.911987 0.911522
0.915413
0.913211
0.909062
0.910727
DNN + RNN 0.912158 0.911676
0.915761
0.913300
0.909056
0.910996
http://image-net.org/.
http://gesture.chalearn.org. 3 http://www.youtube.com/. 4 http://www.mturk.com/. 5 The implementation is available at https://github.com/yagguc/deep_impression.
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|
[
"https://github.com/yagguc/deep_impression."
] |
[
"Tailoring steep density profile with unstable points",
"Tailoring steep density profile with unstable points"
] |
[
"Shun Ogawa ",
"Xavier Leoncini ",
"Alexei Vasiliev ",
"Xavier Garbet ",
"\nRIKEN Center for Brain Science\nLaboratory for Neural Computation and Adaptation\nAix Marseille Univ\n2-1 Hirosawa Wako351-0198SaitamaJapan\n",
"\nUniversité de Toulon\nCNRS\nMarseilleCPTFrance\n",
"\nSpace Research Institute\nCEA\nProfsoyuznaya 84/32117997MoscowRussia\n",
"\nIRFM\nF-13108St. Paul-lez-Durance cedexFrance\n"
] |
[
"RIKEN Center for Brain Science\nLaboratory for Neural Computation and Adaptation\nAix Marseille Univ\n2-1 Hirosawa Wako351-0198SaitamaJapan",
"Université de Toulon\nCNRS\nMarseilleCPTFrance",
"Space Research Institute\nCEA\nProfsoyuznaya 84/32117997MoscowRussia",
"IRFM\nF-13108St. Paul-lez-Durance cedexFrance"
] |
[] |
The mesoscopic properties of a plasma in a cylindrical magnetic field are investigated from the view point of test-particle dynamics. When the system has enough time and spatial symmetries, a Hamiltonian of a test particle is completely integrable and can be reduced to a single degree of freedom Hamiltonian for each initial state. The reduced Hamiltonian sometimes has unstable fixed points (saddle points) and associated separatrices. To choose among available dynamically compatible equilibrium states of the one particle density function of these systems we use a maximum entropy principle and discuss how the unstable fixed points affect the density profile or a local pressure gradient, and are able to create a steep profile that improves plasma confinement.Being able to sustain a steep density profile in hot magnetized plasma is one of the major key points to achieve magnetically confined fusion devices. These steep profiles are typically associated with the emergence in the plasma of so-called internal transport barriers (ITB) [1, 2]. Both the creation and study of these barriers have generated numerous investigations mostly numerical using either a fluid, or magnetic field or kinetic perspective or combining some of these. In this paper starting from the direct study of particle motion, we propose a simple mechanism to set up a steep profile which may not have been fully considered yet. Indeed, charged particle motion in a non-uniform magnetic field [3-9] is one of main classical issues of physics of plasmas in space or in fusion reactors. To tackle this problem the guiding center [7] and the gyrokinetic [8] theories are developed to trace the particle's slower motion by averaging the faster cyclotron motion. These reductions suppress computational cost and they are widely used to simulate the magnetically confined plasmas in fusion reactors[6]. These reduction theories assume existence of an invariant or an adiabatic invariant of motion associated with the magnetic moment. Meanwhile, this assumption does not always hold true. Then, recently, studies on full particle orbits without any reductions are done to look into phenomena ignored by these reductions and to interpolate the guiding center orbit. There exists a case that a guiding center trajectory and a full trajectory are completely different[10]. Further, it is found that the assumption of the invariant magnetic moment breaks[11,12]due to the chaotic motion of the test particles. * [email protected] † [email protected] Let us quickly review the single particle motion and adiabatic chaos. We consider a model of charged particle moving in a non-uniform cylindrical magnetic field B(r ) = ∇ ∧ A 0 (r ). The vector potential A 0 (r ) is given bywhere the cylinder is parametrized with the coordinate (r, θ, z), B 0 is strength of the magnetic field, z has 2πR perperiodicity, e θ and e z are basic units for each direction, and q(r ) is a winding number called a safety factor of magnetic field lines. The Hamiltonian of the particle H = v 2 /2, where v denotes particle's velocity, has three constants of motion, the energy, the angular momentum, and the momentum, associated with time, rotational, and translational symmetry of the system respectively, so that the Hamiltonian H on the six dimensional phase space is reduced into the single-degree-of-freedom Hamiltonian on the two dimensional phase space, (r, p r )-plane[11,12],where p i stands for the conjugate momentum for i = r , z, and θ respectively, and where v θ = rθ and v z =ż. The upper dot denotes d /d t . Invariants p z and p θ are fixed by the initial condition of the particle in the 6-dimensional phase space. Appropriately setting the safety factor q and choosing initial condition, we can find an unstable fixed point in (r, p r ) phase plane, which can induce the adiabatic chaos[13][14][15][16][17]when the weak magnetic perturbation or the curvature effect added to the flat torus (cylinder) exist[11,12].
|
10.1016/j.physleta.2018.09.014
|
[
"https://arxiv.org/pdf/1611.00063v3.pdf"
] | 119,192,859 |
1611.00063
|
308164342edbae8821d7d3e0d7028f8ef90fabea
|
Tailoring steep density profile with unstable points
5 Apr 2018
Shun Ogawa
Xavier Leoncini
Alexei Vasiliev
Xavier Garbet
RIKEN Center for Brain Science
Laboratory for Neural Computation and Adaptation
Aix Marseille Univ
2-1 Hirosawa Wako351-0198SaitamaJapan
Université de Toulon
CNRS
MarseilleCPTFrance
Space Research Institute
CEA
Profsoyuznaya 84/32117997MoscowRussia
IRFM
F-13108St. Paul-lez-Durance cedexFrance
Tailoring steep density profile with unstable points
5 Apr 2018
The mesoscopic properties of a plasma in a cylindrical magnetic field are investigated from the view point of test-particle dynamics. When the system has enough time and spatial symmetries, a Hamiltonian of a test particle is completely integrable and can be reduced to a single degree of freedom Hamiltonian for each initial state. The reduced Hamiltonian sometimes has unstable fixed points (saddle points) and associated separatrices. To choose among available dynamically compatible equilibrium states of the one particle density function of these systems we use a maximum entropy principle and discuss how the unstable fixed points affect the density profile or a local pressure gradient, and are able to create a steep profile that improves plasma confinement.Being able to sustain a steep density profile in hot magnetized plasma is one of the major key points to achieve magnetically confined fusion devices. These steep profiles are typically associated with the emergence in the plasma of so-called internal transport barriers (ITB) [1, 2]. Both the creation and study of these barriers have generated numerous investigations mostly numerical using either a fluid, or magnetic field or kinetic perspective or combining some of these. In this paper starting from the direct study of particle motion, we propose a simple mechanism to set up a steep profile which may not have been fully considered yet. Indeed, charged particle motion in a non-uniform magnetic field [3-9] is one of main classical issues of physics of plasmas in space or in fusion reactors. To tackle this problem the guiding center [7] and the gyrokinetic [8] theories are developed to trace the particle's slower motion by averaging the faster cyclotron motion. These reductions suppress computational cost and they are widely used to simulate the magnetically confined plasmas in fusion reactors[6]. These reduction theories assume existence of an invariant or an adiabatic invariant of motion associated with the magnetic moment. Meanwhile, this assumption does not always hold true. Then, recently, studies on full particle orbits without any reductions are done to look into phenomena ignored by these reductions and to interpolate the guiding center orbit. There exists a case that a guiding center trajectory and a full trajectory are completely different[10]. Further, it is found that the assumption of the invariant magnetic moment breaks[11,12]due to the chaotic motion of the test particles. * [email protected] † [email protected] Let us quickly review the single particle motion and adiabatic chaos. We consider a model of charged particle moving in a non-uniform cylindrical magnetic field B(r ) = ∇ ∧ A 0 (r ). The vector potential A 0 (r ) is given bywhere the cylinder is parametrized with the coordinate (r, θ, z), B 0 is strength of the magnetic field, z has 2πR perperiodicity, e θ and e z are basic units for each direction, and q(r ) is a winding number called a safety factor of magnetic field lines. The Hamiltonian of the particle H = v 2 /2, where v denotes particle's velocity, has three constants of motion, the energy, the angular momentum, and the momentum, associated with time, rotational, and translational symmetry of the system respectively, so that the Hamiltonian H on the six dimensional phase space is reduced into the single-degree-of-freedom Hamiltonian on the two dimensional phase space, (r, p r )-plane[11,12],where p i stands for the conjugate momentum for i = r , z, and θ respectively, and where v θ = rθ and v z =ż. The upper dot denotes d /d t . Invariants p z and p θ are fixed by the initial condition of the particle in the 6-dimensional phase space. Appropriately setting the safety factor q and choosing initial condition, we can find an unstable fixed point in (r, p r ) phase plane, which can induce the adiabatic chaos[13][14][15][16][17]when the weak magnetic perturbation or the curvature effect added to the flat torus (cylinder) exist[11,12].
Xavier Garbet
CEA, IRFM, F-13108 St. Paul-lez-Durance cedex, France
The mesoscopic properties of a plasma in a cylindrical magnetic field are investigated from the view point of test-particle dynamics. When the system has enough time and spatial symmetries, a Hamiltonian of a test particle is completely integrable and can be reduced to a single degree of freedom Hamiltonian for each initial state. The reduced Hamiltonian sometimes has unstable fixed points (saddle points) and associated separatrices. To choose among available dynamically compatible equilibrium states of the one particle density function of these systems we use a maximum entropy principle and discuss how the unstable fixed points affect the density profile or a local pressure gradient, and are able to create a steep profile that improves plasma confinement.
Being able to sustain a steep density profile in hot magnetized plasma is one of the major key points to achieve magnetically confined fusion devices. These steep profiles are typically associated with the emergence in the plasma of so-called internal transport barriers (ITB) [1,2]. Both the creation and study of these barriers have generated numerous investigations mostly numerical using either a fluid, or magnetic field or kinetic perspective or combining some of these. In this paper starting from the direct study of particle motion, we propose a simple mechanism to set up a steep profile which may not have been fully considered yet. Indeed, charged particle motion in a non-uniform magnetic field [3][4][5][6][7][8][9] is one of main classical issues of physics of plasmas in space or in fusion reactors. To tackle this problem the guiding center [7] and the gyrokinetic [8] theories are developed to trace the particle's slower motion by averaging the faster cyclotron motion. These reductions suppress computational cost and they are widely used to simulate the magnetically confined plasmas in fusion reactors [6]. These reduction theories assume existence of an invariant or an adiabatic invariant of motion associated with the magnetic moment. Meanwhile, this assumption does not always hold true. Then, recently, studies on full particle orbits without any reductions are done to look into phenomena ignored by these reductions and to interpolate the guiding center orbit. There exists a case that a guiding center trajectory and a full trajectory are completely different [10]. Further, it is found that the assumption of the invariant magnetic moment breaks [11,12] due to the chaotic motion of the test particles.
Let us quickly review the single particle motion and adiabatic chaos. We consider a model of charged particle moving in a non-uniform cylindrical magnetic field B(r ) = ∇ ∧ A 0 (r ). The vector potential A 0 (r ) is given by
A 0 (r ) = B 0 r 2 e θ − B 0 F (r )e z , F (r ) = r 0 r d r R per q(r ) ,(1)
where the cylinder is parametrized with the coordinate (r, θ, z), B 0 is strength of the magnetic field, z has 2πR perperiodicity, e θ and e z are basic units for each direction, and q(r ) is a winding number called a safety factor of magnetic field lines. The Hamiltonian of the particle H = v 2 /2, where v denotes particle's velocity, has three constants of motion, the energy, the angular momentum, and the momentum, associated with time, rotational, and translational symmetry of the system respectively, so that the Hamiltonian H on the six dimensional phase space is reduced into the single-degree-of-freedom Hamiltonian on the two dimensional phase space, (r, p r )-plane [11,12],
H eff (r, p r ) = p 2 r /2 + V eff (r ), V eff (r ) = v 2 θ + v 2 z 2 = p θ r −1 − B 0 r /2 2 2 + (p z + B 0 F (r )) 2 2 .
(2)
where p i stands for the conjugate momentum for i = r , z, and θ respectively, and where v θ = rθ and v z =ż. The upper dot denotes d /d t . Invariants p z and p θ are fixed by the initial condition of the particle in the 6-dimensional phase space. Appropriately setting the safety factor q and choosing initial condition, we can find an unstable fixed point in (r, p r ) phase plane, which can induce the adiabatic chaos [13][14][15][16][17] when the weak magnetic perturbation or the curvature effect added to the flat torus (cylinder) exist [11,12]. This Letter aims to exhibit one possibility that the unstable fixed point inducing chaotic motion modifies mesoscopic properties of plasmas, local density and pressure gradients which are believed to be associated with the internal transport barriers (ITBs) [1, 2] a feature missed by gyrokinetics, or a pure magneto-hydrodynamic approach. For this purpose, we shall compute an equilibrium radial density function ρ(r ) from stationary kinetic distribution. When neglecting the feedback on the fields of the motion of the particles governed by the Hamiltonian (2) computing a stationary state of an ensemble of particles, i.e. a stationary one-particle density function, resumes to find a density function f 0 on the phase space, which commutes with Hamiltonian (2). Since the motion is integrable, these solutions correspond after a local change to action-angle variables to functions depending only on the actions of the Hamiltonian with a uniform distribution of the associated angles (see for instance for a similar situation [23]). As a consequence, there exist infinitely many steady states. In order to choose one, we may assume a vanishing collisionality, and consider that the one maximizing the information entropy under suitable constraint conditions is picked out [22,24]. In principle, we should consider a Vlasov-Maxwell system consisting of the collisionless Boltzmann equation describing a temporal evolution of single particle density functions of ions and electrons, coupled with the Maxwell equation determining a self-consistent electro-magnetic field [20,21], when neglecting the selfconsistency we notably neglect electrons (their presence insures a neutralizing background, and the current to get the right poloidal component of magnetic field), inter-particle interactions, radiation from moving charged particles and back reaction from the electro-magnetic field. As such we are looking for a steady state of a truncated Vlasov-Maxwell system with an ion moving in a static magnetic field. In this setting we qualitatively discuss which kind of vector potentials F (r ) or q-profiles are likely to bring about the unstable fixed point for the test particle motion. Then, we look into the effect of the unstable fixed point for the density profile and the local-pressure gradient. We shall end this Letter by remarking the relation between the steep density profile (particle's ITBs) and the magnetic ITBs [18,19].
Among the different stationary distributions, let us now compute the general form of f 0 which maximizes the information entropy (also called a density of the Boltzmann Gibbs. The Boltzmann's constant k B is set as unity. We thus have to maximize the functional
S [ f ] = − µ f ln f d 3 pd 3 q,(3)
subject to the normalization condition (conservation of the number of particles) and energy, momentum, and angular momentum conservations, which are respectively
N [ f ] = µ f d 3 pd 3 q, E [ f ] = µ H eff f d 3 pd 3 q, P [ f ] = µ p z f d 3 pd 3 q, L [ f ] = µ p θ f d 3 pd 3 q,(4)
where the integral µ •d 3 pd 3 q means average over the six dimensional single particle phase space, noted here µspace. The solution to this variational problem is
f 0 = e −βH eff −γ 1 −γ θ p θ −γ z p z (5)
where β, γ 1 , γ θ , and γ z are the Lagrangian multipliers, associated with energy conservation, normalization, momentum and angular momentum conditions respectively. The parameter β corresponds to the thermodynamical temperature as
T −1 th ≡ β = δS /δE ,(6)
and it can be safely assumed positive. It should be noted that −γ θ and −γ z are proportional to the ensemble averages of v θ and v z respectively. In the literature is has been admitted that when an ITB exists plasma rotation exists. We then expect that in such state the averages of v θ and v z are not 0 and so are γ θ and γ z . The spatial density function n(q) is deduced from this result as
n(q) ≡ f 0 d 3 p = f 0 r −1 d p θ d p z d p r .(7)
Thus, the density n(r )d 3 q is proportional to (8) and is independent of θ and z. We then obtain a radial density function ρ(r ) given by
n(q)r d r d θd z ∝ e γ θ 2 −B 0 − γ θ β r 2 +γ z B 0 F (r ) r d r d θd z,ρ(r ) = n(q)r d θd z r d θd z = 1 4π 2 r R per n(q)r d θd z,(9)
as
ρ(r ) = exp −ar 2 − bF (r ) ∞ 0 exp −ar 2 − bF (r ) d r , a = γ θ 2 B 0 − γ θ β , b = −γ z B 0 .(10)
We can notice from Eq.(10) that the equilibrium profile is not flat as soon as γ θ is not zero and that it depends on the poloidal magnetic field configuration when γ z = 0. Given the definitions, this means as soon as the plasma moves the profiles are not flat. Since we are considering an equilibrium configuration, we may as well end up with a non-flat temperature profile, but here we have to consider the local radial kinetic temperature of the particles rather than the thermodynamic one (6), so this would correspond to the average of the energy at constant radius. In the same spirit as for the density we can compute the spatial energy density function ε(q) is deduced from this result as
ε(q) ≡ f 0 H eff d 3 p = f 0 H eff r −1 d p θ d p z d p r .(11)
Thus, we notice that
ε(q) = − ∂n(q) ∂β(12)
so the kinetic temperature profile T (r ) is proportional to ρ(r ). In the same spirit it should be noted that the local pressure P (r ) is proportional to the radial density ρ(r ), because we assume the equation of state P (r ) = N ρ(r )T th holds locally true, where N is the number of particles and here we consider the equilibrium temperature T th . We now move on and consider how the existence of the unstable fixed points with relevant energy level affects the obtained equilibrium density profile. For this purpose we have to discuss how the safety factor is chosen, in other words which function F in Eq. (1) leads to the emergence of "practical" unstable points in the effective potential V eff . Indeed, as a first point to pin out, if the amplitude of F is large, we can expect that the term v 2 z /2 = (p z +F ) 2 /2 in the Hamiltonian (2) becomes also large, then the unstable points appear in the phase space at so high energy level that they become physically irrelevant. Therefore, the amplitude of F should be small.
Moreover if the variations of F (r ) are smooth and "gentle" with r , so does again (p z + F ) 2 /2 in V eff , then V eff has only one minimum point that is essentially governed by the term (p θ /r − B 0 r /2) 2 /2. Thus, enough concavity of v 2 z /2 near but not at the minimum point of (p θ /r − B 0 r /2) 2 /2, r = 2p θ /B 0 is necessary so that V eff has unstable points. These considerations are illustrated in Fig. 1. In the panel (d), we assume that there exists an r such that p z + F (r ) = 0. We stress out as well that if |p z | is sufficiently large, it is also possible to create an unstable point, but then again the energy level is so high that it is irrelevant for the mesoscopic profiles in considered plasmas.
Given the obtained density profile (10), we can notice that a sudden fast variation of the function F , will lead to strong variations of the profile, as long as r is not too large, for instance a step like profile should translate in a steep profile. With this in mind, since this effect is present if γ z , related to the average velocity along the cylinder axis, we may expect that in the context of magnetic fusion with machines with large aspect ratios the presence of zonal flows along the toroidal direction is important to increase confinement. Going back to our simple model, since the safety factor q(r ) can be directly associated with the function F (r ) at the origin of the unstable fixed points, and q(r ) is a crucial parameter for the operation of magnetized fusion machine, let us discuss more how the constraints discussed previously translate on the q-profile. For instance let us consider a situation with a non-monotonous profile such that q(r ) has a minimum q 0 at r = α and the spatial scale is characterized with λ. Then locally q(r ) can be expressed as
q(r ) = q 0 1 + λ 2 (r − α) 2 , r ∼ α.(13)
Recalling Eq. (1), the function F is scaled as q −1 0 λ −1 , and v 2 z /2 = (p z + F ) 2 /2 ∼ q −2 0 λ −2 , so that this provides a typical energy level of the particles located near a separatrix. It should be noted that the width of the well of v 2 z /2 scales as λ −1 (see Fig. 1). For a fixed value of q 0 , a large value of λ creates unstable fixed points with relevant energy levels for the particle whose angular momentum is p θ ∼ B 0 α 2 /2. As λ gets to be larger, the number of the particles with un- stable fixed point increase. This is because, roughly speaking, the energy levels of unstable points get to be lower, and the one-particle density (5) is proportional to e −βH eff . As a consequence of these considerations we illustrate on Fig. 2 how to adjust a given q-profile in order to create unstable fixed points whose location is r ∼ α. One can set up unstable fixed point around r ∼ α by modifying the q-profile so that it has concavity around r = α.
F O(1/q 0 2 2 ) O(1/q 0 ) O(1/ ) O(1/ ) r r v z 2 /2 V eff r v 2 /2 r (2p /B 0 ) 1/2 (a) (b) (c) (d)
We then have a form of density profile (10) and a condition for q-profile exhibiting unstable fixed points. We next consider where the unstable points appear, and we shall exhibit that the emergence of unstable fixed points induces the presence of a local steep profile in their vicinity, i.e. their radial positions are inducing the existence of locally strong density gradients. For this purpose we simply consider the q-profile given by Eq. (13) with parameters q 0 = 0.12, λ = 55, and α = 0.18 0.4243, in Figs. 3. When including perturbations, we point out that the adiabatic chaos due to separatrix crossing in this magnetic field has been discussed in Ref. [11]. The results are displayed in Fig. 3, where two density profiles (10) obtained for two q-profiles with and without unstable fixed points are shown. The parameter a is changed so that they have same density in the center of cylinder. We note a can be changed keeping the thermodynamical temperature β −1 and changing the average of angle velocity v θ . We find the steep region which corresponds to the local steep density gradient around r = α on which q(r ) satisfies q (r ) = 0 for the q-profile with unstable points. Going further on, reasoning directly with a q-profile The solid and dotted curves express the density profiles with q-profiles with unstable points (shaded region) and without unstable points respectively. The parameters in a local q-profile [Eq. (13)] are determined as q 0 = 0.12, and α = 0.18 0.4243, and λ = 55 for the solid curve and λ = 2 for the dotted one. In both profiles, γ z and β are same, but γ θ is different. may lead to some physical problems. For a given magnetic configuration inspired from a tokamak, the toroidal component of the magnetic field is generated by the current within the plasma. In the large aspect ratio (cylindrical) limit, this readily gives a r dependence of the current When looking at the current profile given by the q-profile giving rise to Fig. 3, we end up with two different region one, with a negative current for r sufficiently large, and one with a positive one for small r . This is likely to be physically not realistic. Eventhough it may have appeared as possible, the actual stability of these configuration is doubtful, see for instance [29][30][31] .In order to generate a profile that could be more physically relevant, we may have again a look at Eq. (10). We can notice that a "steep" variation of the function F (r ) will likely trigger a steepness in the density profile. Taking into account Eq. (14), we can look for profiles that can achieve this variation while keeping a positive current. Given the form of Eq. (14), it appears as simpler to look for just one Bessel function of the first kind F = F 0 J ν (λr ) and to create only one separatrix, we consider the case when no plasma current is present, then the minimum of the effective potential (2) is located at r 0 such that p θ = B 0 r 2 0 /2. Given the expression (14), and our choice of F , expecting some current in r = 0, implies that ν = 2, we adjust λ to keep a positive current up to R = 1, which leads to λ = 1/r 1 , with r 1 being the first zero of J 2 (r ). The shape of J 2 , leads to expect a maximum near r = 1/2 for F , so we can expect that by tuning the parameters a and b, we will capture many r 0 's (for the different p θ 's), giving rise to regions in phase space with separatices that have non-negligible statistical weights. An illustration of possible obtained profiles are depicted in Fig. (5).
j (r ) = 1 r ∂ ∂r r ∂F ∂r .(14)
Before concluding this letter we would like to make some remarks on the observed profiles which indicate the presence of what we may call an ITB although the underlying physical mechanisms are quite different. We would like to stress out the similarities our observations and the magnetic ITBs discussed for instance in [19]. One of the main difference is that if the plateau of q(r ) appears the transport barrier emerges even if the value of q is far from rational number m/n with small integers m and n, in that sense the creation of the barrier with the q-profile (13) is not correlated to the existence of a resonant surface, on the other hand for the considered example, we can notice that the location of the magnetic ITB coincides with the place on which the local density gradient exists. Another study between the particle's motion and the existence of an ITB using a pure field line approach and the existence of a stable magnetic tori has been performed from the view point of the difference between the magnetic winding number q(r ) and the effective one q eff (r ) for the guiding venter orbit of the energetic particles [25]. In a recent study [26], it is shown that the resonance shift due to the grad B drift and its disappearance due to the curvature drift effect can create an invariant tori in the particle dynamics while there are none for magnetic field lines, and this is confirmed both analytically and numerically with the full particle orbits around the resonance points. It should be remarked that the guiding center theory is useless to clarify it unlike Ref. [26].
The above consideration provides de facto another difference between the magnetic ITB and the effective ITB induced by the separatrices. Moreover, we can stress out that the magnetic ITB is present in both situations described in Fig. 3, while the steep profile occurs only when the hyperbolic points are present. When considering the degenerate q-profile we also note that the two unstable fixed points appear around the magnetic ITB and when the parameter λ in Eq. (13) gets to be large, the steep region of ρ(r ) gets to be strong as the influence of the separatrix grows because the gap of F (r ) between r < α and r > α becomes smaller (see Fig. 1).
To conclude, we have shown in this letter that steep equilibrium density profiles can emerge due to the presence of a separatrix in the passive particle orbits, this phenomenon is not related to the existence of a local resonant surface and the observed phenomenon is reminiscent of the presence of an ITB although the physical mechanisms inducing it are a priori quite different in interpretation. We also discussed how the q-profile can be tuned in order to generate such barriers.
We finally remark on what happens if we consider a toroidal configuration. Let us imagine our cylindrical system is an infinite toroidal radius limit of the toroidal system as Ref. [12]. The finite toroidal radius effect breaks the integrability and it thus can induces adiabatic chaos. It has been known for a while that the presence of chaos affects sometimes the density profile locally. For instance, the averaging effect in plasmas from the global chaos induced by the resonance overlapping [27] has been found and it modifies the density profile [28]. Even though we are directly tackling the passive particle motions of ions and thus a different type of localized chaos in this letter, similar things can be expected. In the present case the unstable fixed points are located around the place in which the steepness of density (10) and the local pressure gradient exist as shown in Fig. 3. Therefore the flattening effect makes them steeper locally. As a result we can expect that this steepening effect observed in the cylindrical configuration can be robust at least as long as strongly chaotic motion remains localized near each hyperbolic point.
FIG. 1 .
1Schematic picture showing how the saddle point appears around the bottom of safety factors. A Panel (a) represents a part of v2 θ /2 = (p θ /r − B 0 r /2) 2 /2, (b) V eff , (c) F (r ), and (d) v 2 z /2 = (p z + F (r )) 2 /2. In the panel (d), we consider that there exists an r such that p z +F (r ) = 0. The parameters q 0 , λ, and α are associated with Eq.(13).
FIG. 2 .,
2Schematic picture exhibiting how to create unstable fixed point by modifying q-profile. The solid curve represents the given q-profile. Dotted and bold curves are the graph of q 0 (1+λ 2 (r −α) 2 ) for q 0 = 0.12, α = 0.18 and λ = 2, 55 respectively. The q(r ) with λ = 55 induces the unstable fixed points, so does not one with λ = 2. (a) and (b) correspond respectively to the q-profiles without and with unstable fixed points. =0.4, b=1 λ=55, =4, b=1FIG. 3. (Color online)
online) Effective potential for F (r ) = F 0 J 2 (λr ). An Unstable point appears as expected around r 0 = 0.6. Here we choose the parameters as F 0 = 1, p θ = 0.5, p z = 1, B 0 = 4.
FIG. 5 .
5(Color online) Top: The gray(red) and black curves express the density profiles with only one unstable point. Bottom: same current profile for both configurations. The parameters are F 0 = 1, a = b = 10 for the gray(red) curve F 0 = 1, a = 10, b = 0 for the black.
ACKNOWLEDGMENTSS. O. and X. L. thank G. Dif-Pradalier for useful and encouraging discussions. X. L. thanks E. Laribi for careful reading of the manuscript. This work has been carried out within the framework of the French Research Federation for Magnetic Fusion Studies. The project leading to this publication has received funding from Excellence Initiative of Aix-Marseille University -AMIDEX, a French "Investissements d'Avenir" programme.
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|
[] |
[
"Many-body localization in a tilted potential in two dimensions",
"Many-body localization in a tilted potential in two dimensions"
] |
[
"Elmer V H Doggen \nInstitute for Quantum Materials and Technologies\nKarlsruhe Institute of Technology\n76021KarlsruheGermany\n\nInstitut für Theorie der Kondensierten Materie\nKarlsruhe Institute of Technology\n76128KarlsruheGermany\n",
"Igor V Gornyi \nInstitute for Quantum Materials and Technologies\nKarlsruhe Institute of Technology\n76021KarlsruheGermany\n\nInstitut für Theorie der Kondensierten Materie\nKarlsruhe Institute of Technology\n76128KarlsruheGermany\n\nIoffe Institute\n194021St. PetersburgRussia\n",
"Dmitry G Polyakov \nInstitute for Quantum Materials and Technologies\nKarlsruhe Institute of Technology\n76021KarlsruheGermany\n"
] |
[
"Institute for Quantum Materials and Technologies\nKarlsruhe Institute of Technology\n76021KarlsruheGermany",
"Institut für Theorie der Kondensierten Materie\nKarlsruhe Institute of Technology\n76128KarlsruheGermany",
"Institute for Quantum Materials and Technologies\nKarlsruhe Institute of Technology\n76021KarlsruheGermany",
"Institut für Theorie der Kondensierten Materie\nKarlsruhe Institute of Technology\n76128KarlsruheGermany",
"Ioffe Institute\n194021St. PetersburgRussia",
"Institute for Quantum Materials and Technologies\nKarlsruhe Institute of Technology\n76021KarlsruheGermany"
] |
[] |
Thermalization in many-body systems can be inhibited by the application of a linearly increasing potential, which is known as Stark many-body localization. Here we investigate the fate of this phenomenon on a two-dimensional disorder-free lattice with up to 24 × 6 sites. Similar to the one-dimensional case, "density-polarized" regions can act as bottlenecks for transport and thermalization on laboratory timescales. However, compared to the one-dimensional case, a substantially stronger potential gradient is needed to prevent thermalization when an extra spatial dimension is involved. The origin of this difference and implications for experiments are discussed. We argue that delocalization is generally favored for typical states in two-dimensional Stark many-body systems, although nonergodicity can still be observed for a specific choice of initial states, such as those probed in experiments. arXiv:2202.04948v2 [cond-mat.str-el]
| null |
[
"https://arxiv.org/pdf/2202.04948v2.pdf"
] | 246,706,179 |
2202.04948
|
4183bc527e0c5185c65878046b21412eba120494
|
Many-body localization in a tilted potential in two dimensions
Elmer V H Doggen
Institute for Quantum Materials and Technologies
Karlsruhe Institute of Technology
76021KarlsruheGermany
Institut für Theorie der Kondensierten Materie
Karlsruhe Institute of Technology
76128KarlsruheGermany
Igor V Gornyi
Institute for Quantum Materials and Technologies
Karlsruhe Institute of Technology
76021KarlsruheGermany
Institut für Theorie der Kondensierten Materie
Karlsruhe Institute of Technology
76128KarlsruheGermany
Ioffe Institute
194021St. PetersburgRussia
Dmitry G Polyakov
Institute for Quantum Materials and Technologies
Karlsruhe Institute of Technology
76021KarlsruheGermany
Many-body localization in a tilted potential in two dimensions
(Dated: April 14, 2022)
Thermalization in many-body systems can be inhibited by the application of a linearly increasing potential, which is known as Stark many-body localization. Here we investigate the fate of this phenomenon on a two-dimensional disorder-free lattice with up to 24 × 6 sites. Similar to the one-dimensional case, "density-polarized" regions can act as bottlenecks for transport and thermalization on laboratory timescales. However, compared to the one-dimensional case, a substantially stronger potential gradient is needed to prevent thermalization when an extra spatial dimension is involved. The origin of this difference and implications for experiments are discussed. We argue that delocalization is generally favored for typical states in two-dimensional Stark many-body systems, although nonergodicity can still be observed for a specific choice of initial states, such as those probed in experiments. arXiv:2202.04948v2 [cond-mat.str-el]
I. INTRODUCTION
Many systems encountered in nature obey the ergodic hypothesis, that is, each microstate consistent with fixed macroscopic thermodynamic variables according to the appropriate statistical ensemble is as likely as the other. However, some systems are nonergodic. Understanding the origin of nonergodicity in quantum systems [1] is of particular relevance to describing decoherence and the crossover from quantum to classical behavior. A paradigmatic example of nonergodicity in quantum many-body systems is many-body localization (MBL), which occurs at nonzero density of excitations, driven by the interplay between interactions and disorder [2][3][4][5][6][7].
Recently, interest has increased in studying many-body systems that exhibit nonergodicity even without the presence of any disorder [8,9]. One such system consists of interacting particles under the influence of a linear potential, i.e., the interacting many-body analog of Wannier-Stark localization. Experimental realizations in one dimension have shown that localization can also persist in this case [10][11][12].
On the other hand, it has been observed experimentally [10][11][12][13] that "Stark-MBL" systems can exhibit features of ergodic systems. This suggests a transition in such systems from a delocalized phase to a localized one at a certain critical value of the gradient of the linear potential. Numerical studies in one dimension have indicated that such a transition does indeed occur [14,15], similarly to the predicted transition in disorder-driven MBL systems. However, as we have previously shown [16], within the purported delocalized region some states are, in fact, anomalously long-lived (see also Refs. [17,18]).
One may understand this phenomenon through the Hilbert-space shattering (fragmentation) [19,20] that oc- * Corresponding author: [email protected] curs in the limit of an infinitely large gradient of the potential, permitting a mapping to fractonic or constrained systems. Hilbert-space shattering implies that the Hilbert space of the system is divided into an exponentially large [20] (in the system size) number of disconnected sectors, preventing thermalization. At a finite value of the potential gradient these sectors are connected, but in a sufficiently weak manner such that thermalization can be strongly suppressed on laboratory timescales. Importantly, in the one-dimensional (1D) case, the probability for the transport-blocking regions of an arbitrarily large length λ to occur is unity in the thermodynamic limit, giving rise to a finite density of such regions. The thermodynamic limit is established in systems whose length is exponentially large in λ.
A question of interest is to what degree nonergodicity in the various systems that exhibit it shares a similar origin and features. In this context, it is worthwhile to investigate the influence of geometry and dimensionality on localization-related phenomena. Disorder-driven MBL is only expected to be stable in one dimension, according to the avalanche theory of the MBL transition [21][22][23][24]. Within this theory, a vital role is played by rare weakly disordered regions, which lead to the emergence of growing "ergodic spots" that can eventually thermalize the whole system. Such regions are far more likely to occur in dimensions higher than one, destroying, in particular, MBL in two-dimensional (2D) systems in the thermodynamic limit. However, in the case of Stark MBL there are no possible rare configurations of the potential. From this perspective, Stark-MBL systems are more prone to localization, and the above distinction between 1D and 2D geometries is less prominent.
On the other hand, despite the differences behind the physics of localization, some features familiar from disorder-driven MBL systems have been reported also in Stark-MBL systems, such as logarithmic growth of entanglement and Poissonian energy level statistics in the localized phase [14,15]. Furthermore, as we show below, a distinction between one-and higher-dimensional 1. (a,b): Schematics of the geometry and initial conditions. (a): Charge-density wave (CDW) initial condition with wavelength 2λ, where sites are either occupied (red dots) or unoccupied (light blue dots). The initial state is periodic in the i-direction, in the same direction as the potential gradient. systems, as in disorder-driven MBL, is also present in the case of Stark MBL. This distinction is present because the suppression of transport in one dimension is related to blocking (polarized) regions, which occur with unit probability in the limit of large system sizes [16]. However, this probability vanishes in the thermodynamic limit for 2D systems. It is the goal of this paper to sort out the similarities and differences between Stark MBL and conventional MBL with regard to the role of dimensionality.
From a technical point of view, a major obstacle is that the numerical complexity of a generic, unconstrained many-body quantum system on a lattice scales as f N , where N is the number of sites on the lattice and f the number of local degrees of freedom. Exact algorithms can only handle system sizes up to N ≈ 25, even in the simplest case f = 2 (e.g., a spin-1/2 system). This means that, in order to access meaningfully large systems, approximate methods need to be used. Recently, one such method-the time-dependent variational principle (TDVP) [25]-has proven to be exceptionally powerful, yielding reliable results for (almost) localized systems [16,24,[26][27][28][29][30][31]. This is true even up to relatively large times and system sizes, comparable to those of the experiment. Here, we use the TDVP to elucidate the physics of Stark MBL in two dimensions.
II. MODEL AND METHOD
We consider hard-core bosons on a 2D square lattice as described by the Hamiltonian:
H = ij;i j − J 2 b † ij b i j + H.c. + Un ijni j + ij inij ,(1)where b ij (b † ij )
is the annihilation (creation) operator for a boson on the site with indices i ∈ [1, L], j ∈ [1, d] and n ij = b † ij b ij . The summation over ij; i j is restricted to nearest neighbors, with open (periodic) boundary conditions in the i (j)-direction. In the following, we use units with = 1 and choose J = 1 for the energy and time scales. Moreover, we set the interaction strength U = 1. The on-site potential varies only in the idirection, namely i = W i, where W is the potential gradient. This model is similar to the one studied in a recent experiment [13], except that we consider the simpler case of hard-core bosons instead of two-component fermions. Dynamics governed by Eq. (1) is computed, using the TDVP, up to time t = 500. The TDVP belongs to the matrix-product-state (MPS) class of algorithms [32], a type of variational representation of the many-body wave function with a controllable error. The TDVP dynamics obeys the Schrödinger-like equation:
d dt |ψ = −iP MPS H|ψ ,(2)
where P MPS projects the dynamics onto the variational manifold. The number of independent parameters in the manifold scales with the bond dimension χ, which is the main control parameter used to verify convergence of the algorithm. A major appeal of this method, compared to other MPS algorithms, is that globally conserved quantities are conserved in the numerical procedure, enhancing accuracy. We employ the hybrid one-site and two-site implementation of the TDVP [16,24] in a parallelized fashion (details and benchmarks are presented in Appendix).
III. RESULTS
With the TDVP, we compute dynamics starting from an initial product state, using a sufficiently large bond dimension. We consider two different initial states |ψ , which are product states in the particle occupation basis. The first, depicted schematically in Fig. 1a, is a steplike charge-density wave (CDW) configuration with wavelength 2λ and dimensions L in the i-direction and d in the j-direction. This choice corresponds to the one employed in the experimental realization of Ref. [13]. We furthermore consider an additional initial state, depicted in Fig. 1b, where the aforementioned state is perturbed, breaking the translational invariance in the j-direction. We approach the two-dimensional limit by considering a width of up to 6 lattice sites.
Choosing the potential to be translationally invariant perpendicular to the axis of the CDW is appealing from the perspective of investigating localization properties, because it is expected that this setup is the most amenable to delocalization. Furthermore, with this arrangement, we can directly investigate the fate of the long-lived 1D states studied in Ref. [16] upon increasing the width of the system, thus going towards the 2D case.
Aside from considering the expectation values of particle densities n ij ≡ b † ij b ij , we consider the memory of the initial state, as quantified using the imbalance I, an experimentally accessible quantity [33]:
I(t) = 4 Ld ij n ij (t) − 1/2 n ij (t = 0) − 1/2 . (3)
A state that is unchanged from the initial state obeys I = 1, while for a delocalized state at half filling I = 0. We further consider the bipartite von Neumann entropy of entanglement S [34]:
S(t) = max A [−Tr(ρ A ln ρ A )], ρ A ≡ Tr B |ψ(t) ψ(t)|.(4)
Here Tr B traces out the degrees of freedom corresponding to part B. We choose the bipartition between subsystems A and B such that the entropy is the maximum (this is the meaning of max A above) of all the possible bipartitions, which turns out to correspond to the position of a domain wall in the setups in Figs. 1a and b (e.g., for the parameters in Fig. 1c, the maximum of S is achieved at i = 8 and 16). Let us first consider dynamics for a particular choice of parameters L = 24, d = 6, λ = 4, and W = 4, as shown in Fig. 1a. After a brief initial evolution (see Appendix), dynamics is frozen up to t = 500 hopping times, without any appreciable change in the state, apart from regular oscillations, see Fig. 1c. The possibility of transverse dynamics, therefore, does not appear to lead to delocalization in the i-direction.
In Fig. 2, we show the imbalance as a function of time. Because the initial density pattern mostly survives (see Fig. 1c), I remains close to 1. Strong oscillations in time are present, which are more pronounced for smaller widths d. This is due to the dynamics being constrained to only a few sites. The average value (solid lines in Fig. 2, obtained through a Fourier transform, analogous to Ref. [17]) is only weakly dependent on system size, suggesting convergence to an asymptotic value for larger system sizes. This implies that localization survives up to long times, similar to the 1D case. We now consider the von Neumann entropy of entanglement (4). Similar to the imbalance studied above, the entropy shows an initial rapid change and then saturates at a fixed value. This saturation value of the entropy scales with the system width d, roughly as S ∝ d. Such behavior is expected, because the dynamics is constrained only in the i-direction. Indeed, the number of involved sites scales linearly with d, which leads to the same scaling for the entropy.
Perturbing the initial state in the manner depicted in Fig. 1b breaks translational invariance in the j-direction and introduces additional broadening of the left domain walls (at i = 4, 12, and 20; we choose the same parameters L, d, λ, and W as for Fig. 1a). Comparing the two cases (Figs. 1c and d), we observe highly similar dynamics at the right (unperturbed) domain walls, whereas the dynamics is different for the left (perturbed) boundaries. Nonetheless, the dynamics in Fig. 1d appears frozen over the observed timescales, with no sign of macroscopic ther- Let us now inspect the dynamics of the imbalance and entropy in the perturbed and unperturbed cases in more detail. In Fig. 4, a comparison between the CDW initial condition and the perturbed CDW is shown. We note two essential differences. Firstly, in the perturbed case, the imbalance I initially drops to a lower value, 1 − I ∼ 1/d. After the drop, similarly to the unperturbed case, no significant decay of the imbalance is observed. Secondly, the time dependence of the entropy is markedly different. In the unperturbed case, the entropy quickly reaches a plateau at S ≈ 2 from the initial value S = 0, after which there is a barely noticeable increase in time. In the perturbed case, a substantially faster increase is visible at the second stage. The maximum value of the entropy is reached at the domain walls with the |10 configuration found at column indices i = 8 and 16 (right domain walls), while the perturbation is at the left domain walls. Hence, correlations due to the perturbation do penetrate through the domain wall. The behavior is reasonably well fitted by a linear dependence, depicted in Fig. 4, which provides a better fit than a logarithmic dependence over the depicted time window. Despite this steady growth of entanglement, no significant decay of the imbalance is observed, suggesting long-lived stability of such localized states. Such behavior is not contradictory: in the case of disordered MBL in one dimension, entropy growth is thought to be logarithmic, while transport remains frozen [35].
If the period of the CDW is reduced, the polarized striped regions are less effective at inhibiting transport. In Fig. 5, we show the case where λ = 1 (and no perturbation to the CDW), with L = 24 and d = 3. Even though the width of the system is limited, we can observe (Fig. 1a) and perturbed charge density wave (Fig. 1b). The dashed red lines indicate a linear fit to the curve S(t) = at + b, yielding the values aCDW = (11 ± 5) · 10 −5 and apCDW = (59 ± 4) · 10 −5 for the CDW and perturbed CDW respectively (95% confidence intervals). The thick lines represent filtered data using a Savitzky-Golay polynomial fitting procedure of third order [36]. a dramatic quantitative difference compared to dynamics for the 1D case [14][15][16]. Namely, while in the 1D case, already a modest value W 1 is sufficient to observe clear saturation on these timescales, we do not observe such saturation in the quasi-1D case even at a much stronger value of the tilt, W = 10.
This difference is understandable as follows. In the 1D case, violations of the eigenstate thermalization hypothesis [1] result from specific nonergodic states identified in Ref. [16]. These states with a blocking region of length exceeding a given λ (which is a decreasing function of the field W ) have measure zero in the whole Hilbert space in the limit L → ∞, but still occur with a unit probability in the subspace of random product states. More specif-ically, they occur with a spatial density that scales as 2 −λ .
For a quasi-1D system with width d, however, these blocking regions require polarized regions of area dλ. The probability of finding such a state scales as 2 −dλ . In the picture presented in Ref. [16], this means that localization is less enduring with respect to tuning the potential gradient away from W → ∞ (recall that in this limit there is a mapping to a constrained, nonergodic system [20]). In the fully 2D limit, d = L → ∞, the appearance of a fully blocking region then becomes vanishingly unlikely, in stark contrast to the 1D case.
IV. CONCLUSION AND OUTLOOK
In one dimension, the application of a linearly increasing potential induces localization, which survives the introduction of interactions (Stark many-body localization). Here, we have shown that long-lived localized states exist also in higher dimensions, namely, on a 2D lattice. This behavior is in line with the notion of Hilbertspace shattering, following a mapping to a constrained system that becomes exact at infinitely large potential gradient.
However, it is noteworthy that the values of the field W required to observe localization in higher dimensions are substantially larger than in the 1D case [16], where we observed long-lived localization in the 1D analog of the model (1) even for W = 0.3. The additional dimension, therefore, aids thermalization of the system. We can explain this dependence on dimensionality by noting that the blocking polarized regions represent an exponentially smaller part of the whole Hilbert space as a function of width d, compared to the 1D case. Therefore, there is a parallel to the "standard" type of MBL in a disordered potential, in which dimensionality is argued to play a crucial role [21,23,24] in that it determines the importance of rare fluctuations of disorder responsible for delocalization. In the Stark-MBL case, no disorder is present, but "rare events"-rare states with local constraintsalso play a crucial role, favoring, in contrast to the rare ergodic spots in the disordered case, localization. (In this sense, they are similar to transport-hindering rare events of the Griffiths type [4,5].) The relative number of such states as a fraction of the Hilbert space is greatly suppressed for higher dimensions, again in contrast to the disordered case, where the ergodic spots are more probable with increasing dimensionality. Remarkably, the proliferation of rare ergodic regions in conventional 2D MBL and the suppression of blocking regions in 2D Stark MBL both have a delocalizing effect. The key difference between 1D and 2D Stark MBL is that, in two dimensions, the polarized blocking regions are expected to be irrelevant for the thermalization of typical (product) states, in contrast to the 1D case [16].
At smaller values of the gradient W , an analytical description, assuming an incoherent (hydrodynamic) pic-ture, predicts subdiffusive transport [13,37]. This leads to strong growth of the entanglement, and is, therefore, extremely demanding for the numerical simulations based on matrix product states. The experiment of Ref. [13] does find, however, an exponential decrease of the delocalization rate in the observed subdiffusive regime as a function of the size of polarized regions, suggesting a trend toward localization for the CDW initial states. It may nonetheless be difficult to observe robust localization in a 2D system, as it is more challenging to prepare a cleanly polarized plaquette as opposed to a 1D charge-density wave. Indeed, as we have seen, a single (hole) defect in any one site at the boundary of the plaquette region quickly destroys polarization in the direction perpendicular to the field gradient (see Appendix).
An intriguing open question is whether the numerically observed localization represents a genuine long-lived localized phase, or a transient "prethermal" state [38]. Contrary to the one-dimensional case [16], we observe growth of entanglement with time, while transport remains frozen. This is potentially a signature of such a prethermal regime at timescales far beyond the numerically accessible range. Note, however, in disordered MBL systems the growth of entanglement is characteristic of both the ergodic and nonergodic states [35]. We stress that, in the context of this work, by "MBL" we mean long-lived localized states -the weakest possible criterion for MBL. The stability criteria for the MBL phase and the properties of the transition in the thermodynamic limit are still debated even for 1D disordered MBL [23,[39][40][41][42][43][44][45][46][47]. It is difficult to address questions pertaining to the thermodynamic limit through experimental or numerical means, especially in two dimensions, so that further analytical work in this direction is needed.
ACKNOWLEDGMENTS
We thank F. Pollmann and P. Sala for useful discussions. Simulations were performed using the TeNPy library [48], version 0.7.2.
Appendix A: Numerical details
In this Appendix, we discuss technical details of the numerical simulations, as well as provide several benchmarks and a comparison to a different algorithm. In this work, we have employed matrix product states (MPS) simulations, a class of variational algorithms for solving many-body problems [32], in which the number of variational parameters is controlled by the bond dimension χ.
We employ the time-dependent variational principle (TDVP) [25] in a hybrid implementation (the same as used in Refs. [16,24]), where the two-site algorithm, which allows expansion of χ at every time step, is used up to τ = max(2, τ χ ), where τ χ is the time in which χ has reached the maximum set value. After the time t = τ , the remainder of the dynamics is computed using the single-site algorithm. This algorithm does not allow further expansion (or reduction) of the bond dimension, but has the benefit that the energy is globally conserved by the dynamics. This is opposed to other MPS-based algorithms, in which the various truncation procedures leads to violations of energy conservation, even for a timeindependent Hamiltonian, which results in accumulating errors. Loosely speaking, one can identify this difference as the difference between implicit and explicit numerical integration schemes for solving partial differential equations. Instead of the parallel implementation used for disorder [24], here we employ parallelization of the Intel Math Kernel Library (MKL) routines for an additional speedup.
Bond dimension
Let us first consider the dependence of the result on the bond dimension χ of the MPS. Recall that χ controls the number of variational parameters in the MPS, allowing for stronger entanglement throughout the sys- tem as χ increases. We should then expect the result of the simulation to converge for sufficiently high χ. A comparison for the choices χ = {256, 384} is shown in Fig. 6. The results are in good agreement, with only a small discrepancy visible at late times.
Influence of the wavelength λ
In the following, we consider the effect of changing the wavelength λ of the initial charge-density wave. As long as we are in the localized phase, we should expect localization to remain robust upon increasing λ; this increases the length of polarized regions in the system. Results for the parameters L = 24, d = 3, W = 4, and λ = 6 are shown in Fig. 7. Comparison to the case λ = 4 in the main text indeed reveals the results are mostly unchanged: the maximum bipartite entropy is approximately the same. The imbalance also appears to saturate at a finite value, which in this case is slightly closer to 1, corresponding to the reduced number of domain wall borders.
Of interest is also the case where the wavelength is relatively small. If λ = 1, we obtain the so-called (columnar) Néel state as an initial condition, where columns are initially occupied and unoccupied in an alternating fashion. This state should be among the most susceptible to delocalization (out of the possible states with translational invariance in the j-direction), as is indeed observed numerically (see Fig. 8 and the main text). Even for a much larger choice of the tilt W = 10, the system tends to delocalize over time, and convergence with bond dimension is lost around t ≈ 200 due to the growth of entanglement.
3. Influence of system size L Similarly to the above section, we can investigate the effect of changing system size. As per the same reasoning as for what happens in the case of increasing λ, little should change upon increasing the system size in the localized regime. The results for the choice L = 32, d = 3, W = 4, and λ = 4 are shown in Fig. 9. Again, the results are highly similar to the choice L = 24 shown in the main text.
Dynamics in the transverse direction
It is instructive, also for the purposes of benchmarking, to investigate the dynamics of individual sites. Let us consider the case of the perturbed CDW as discussed in the main text. In the transverse direction, no potential gradient is present. Hence, particles are allowed to move freely in this direction, and we should expect rapid "thermalization" -albeit restricted only to this transverse dimension. This is precisely what is found, as depicted in Figs. 10 and 11. On a relatively modest timescale O(d) the site densities reach a value 1− n ∼ 1/d, corresponding to the average density in the initial state. Around this average value, there are oscillations that are not damped because the dynamics is unitary and the system is closed.
Note that due to symmetry in the initial condition, the dynamics for the site pairs j = {1, 5} and j = {2, 4} is identical. Since this symmetry is not explicitly imposed by the algorithm (in fact, the symmetry is broken through the mapping from a 2D lattice to a 1D chain, which breaks translational symmetries in the transverse direction), this provides another benchmark for the method. Only a very small difference in the densities is found, barely visible on the scale of the plot, even at the latest time window t ∈ [475, 500] (Fig. 11).
Quadratic modulation of the potential
In Refs. [14,49] it has been argued that adding a small perturbation to the potential, taken to be in the form of a small quadratic term, can dramatically influence the localization properties of the model. The reason put forward is that the unperturbed linear potential permits resonant processes, enhancing delocalization. However, in Ref. [50] the effect of this type of perturbation on the melting of domain walls has been considered, and the authors have not found crucial differences caused by the quadratic modulation. Here we show the effect of such a quadratic modulation in the 2D case, as considered in the main text. To wit, the full potential is now given by:
i,quad = W i − αi 2 ,(A1)
where α = 0.01. Hence, the gradient of the field as a whole is approximately W close to the left end of the system, but for W = 4 the potential reaches zero at L = 20, and then becomes negative. This is, thus, a rather significant perturbation of the potential.
The results for the dynamics in both cases α = 0 and α = 0.01 are shown in Fig. 12. Here the initial condition with a perturbed CDW is chosen, corresponding to the green line in Fig. 4 of the main text. The addition of the quadratic potential reduces the magnitude of oscillations in the imbalance, which can be attributed to the breaking of the aforementioned resonances. However, the qualitative behavior appears unchanged, with only a slightly lower imbalance and slightly higher entropy in the case of the quadratic modulation. The increased delocalization can be attributed to the effectively weaker gradient for the rightmost domain wall, enhancing the melting thereof [50].
Comparison to the WII method
We now compare the results of the TDVP algorithm, used in the main text, to a different method also belonging to the class of MPS algorithms: the W II method [51] (we again use the TeNPy library [48] to implement it). This method shares some features of the TDVP algorithm, such as the ability to handle long-range terms. The latter is an essential ingredient used for the mapping of the two-dimensional square lattice to the 1D structure of the MPS. A key difference between the methods is that the W II method suffers from a truncation error at each time step, associated with truncated singular value decompositions. This type of error does not appear in single-site TDVP; however, a projection error, induced by the projector P MPS takes its place. Moreover, single-site TDVP is an implicit integration method; such methods tend to be suitable for oscillatory problems. The reader is referred to the review [52] for a detailed discussion of the differences between these two algorithms, and to the review [31] for an in-depth discussion of the application of MPS-type algorithms to the MBL problem. In the particular case of our model, we find that the TDVP performs better in terms of computational time, primarily because a smaller time step of the integrator is required for the W II method, δt = 0.01 as opposed to δt = 0.05 for the TDVP.
The result is shown in Fig. 13. At short times, the two methods are in good agreement, but the agreement deteriorates over time; the maximum bond dimension χ = 256 is reached at t ≈ 1, and repeated truncation errors cause the result to diverge from the TDVP method.
Exact results
For sufficiently small system sizes, we can compare to numerically exact results, where there is no truncation of the Hilbert space through χ. In Fig. 14, we show such results, where we take L = 8, d = 2, W = 4, and λ = 4. The dynamics is almost entirely frozen, with the imbalance (not shown) saturating at I ≈ 0.993. This is because the orientation of the single domain wall is from unoccupied sites on the left side to occupied sites on the right side. As seen in the main text, it is the opposite orientation that allows for a greater degree of domain wall melting. The symmetry is broken through the sign of the hopping J.
FIG. 1. (a,b): Schematics of the geometry and initial conditions. (a): Charge-density wave (CDW) initial condition with wavelength 2λ, where sites are either occupied (red dots) or unoccupied (light blue dots). The initial state is periodic in the i-direction, in the same direction as the potential gradient. (b): As in panel (a), but with a perturbation that breaks the translational invariance in the j-direction. (c): Particle density n, averaged over sites within a given column (fixed i), as a function of time. The initial state is a unidirectional CDW as depicted in panel (a), with λ = 4. (d): As in panel (c), but starting from the perturbed CDW initial state depicted in panel (b).
(b): As in panel (a), but with a perturbation that breaks the translational invariance in the j-direction. (c): Particle density n, averaged over sites within a given column (fixed i), as a function of time. The initial state is a unidirectional CDW as depicted in panel (a), with λ = 4. (d): As in panel (c), but starting from the perturbed CDW initial state depicted in panel (b).
FIG. 2 .
2Imbalance (3) as a function of time for various widths of the system, d ∈ [3, 6]. Other parameters are the same as in Fig. 1. Top panels show individual imbalance curves for d ∈ [3, 6]. The thick lines in the bottom panel show the leading behavior of the imbalance without oscillations, using a low-pass filter.
FIG. 3 .
3Entanglement entropy (4) as a function of time, for the same parameters as inFig. 2.malization. Notably, in both Figs. 1c and d, the spatial range of the domain-wall melting in the i-direction does not exceed the width of the system d = 6. In other words, any 2D subblocks of size 6 × 6 (seeFigs. 1aand b) can be considered as non-thermalized on the timescale of observation.
FIG. 4 .
4Comparison of the imbalance (left panel) and entropy (right panel) dynamics for the same choice of parameters, L = 24, d = 6, W = 4, λ = 4, and χ = 384, but different choices of initial condition: the charge density wave
L
= 24, d = 3, W = 10, λ = 1 FIG. 5. Imbalance dynamics in the case of a short-wavelength charge-density wave, with λ = 1. Note the choice of a stronger gradient W = 10.
FIG. 6 .
6Comparison of the dynamics in the case of the CDW initial condition for different choices of the bond dimension χ = {256, 384}, for both the imbalance (top panel) and the entropy (bottom panel).
FIG. 7 .
7Imbalance dynamics (top) and entropy (bottom) for a choice of the initial condition corresponding to λ = 6.
FIG. 8 .
8Imbalance dynamics (top) and entropy (bottom) for λ = 1, with a larger value of the tilt (W = 10) compared to the main text. Convergence with χ is lost at t ≈ 200.
FIG. 9 .
9Imbalance dynamics (top) and entropy (bottom) for a different choice of system length compared to the main text, L = 32.
FIG. 10 .FIG. 11 .
1011Dynamics for the on-site particle density n j for a fixed row i = 5 in the case of the perturbed CDW initial condition. Parameters are L = 24, d = 6, W = 4, λ = 4, and χ = 384. Shown is the early time window t ∈ [0, 25]. AsFig. 10, but at a late time window t ∈ [475, 500].
FIG. 12 .
12Comparison of dynamics in the case where a small quadratic perturbation is present, compared to the case where it is absent. The initial condition is chosen to be the perturbed CDW.
FIG. 13 .
13Comparison of the TDVP and WII algorithms over the time window t ∈ [0, 10].
FIG. 14 .
14Numerically exact dynamics of the density, for L = 8, d = 2, W = 4, and λ = 4.
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"A Note on Systematic Conflict Generation in CA-EN-type Causal Structures",
"A Note on Systematic Conflict Generation in CA-EN-type Causal Structures"
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"Antoni Lige ' Za [email protected] \nLAAS du CNRS\n7, Av. du Colonel Roche31077Toulouse CedexFrance\n"
] |
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"LAAS du CNRS\n7, Av. du Colonel Roche31077Toulouse CedexFrance"
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This paper is aimed at providing a very first, more "global", systematic point of view with respect to possible conflict generation in CA-EN-like causal structures. For simplicity, only the outermost level of graphs is taken into account. Localization of the "conflict area ", diagnostic preferences, and bases for systematic conflict generation are considered. A notion of Potential Conflict Structure (PCS) constituting a basic tool for identification of possible conflicts is proposed and its use is discussed.
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"https://arxiv.org/pdf/1411.4616v1.pdf"
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1411.4616
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4f4769be0778287b108032558f85da3a5794b8b5
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A Note on Systematic Conflict Generation in CA-EN-type Causal Structures
17 Nov 2014
Antoni Lige ' Za [email protected]
LAAS du CNRS
7, Av. du Colonel Roche31077Toulouse CedexFrance
A Note on Systematic Conflict Generation in CA-EN-type Causal Structures
17 Nov 2014
This paper is aimed at providing a very first, more "global", systematic point of view with respect to possible conflict generation in CA-EN-like causal structures. For simplicity, only the outermost level of graphs is taken into account. Localization of the "conflict area ", diagnostic preferences, and bases for systematic conflict generation are considered. A notion of Potential Conflict Structure (PCS) constituting a basic tool for identification of possible conflicts is proposed and its use is discussed.
Introduction
Diagnostic reasoning is an activity oriented towards detection of faulty behaviour and its explanation, i.e. isolation of faulty components responsible for the observed misbehaviour of the analyzed system. Model-based diagnosis is based on explicit system model applied for diagnostic inference. A widely accepted approach consists in consistency-based reasoning where the analysis is aimed at regaining consistency of the predicted model output with current observations by retracting some of the assumptions about correct behaviour of certain components. The sets of elements suspected to contain at least one faulty component are identified by detecting inconsistency between the observed and predicted behaviour. Such sets, called conflict sets are basic products for generating diagnoses.
A complete diagnostic procedure following the consistency-based approach should cover:
• detection and localization of misbehaviour,
• restriction of the search area (hierarchical fault diagnosis),
• systematic conflict generation, taking into account that:
-all conflicts should be found, but -only minimal conflicts should be generated, * On leave from: Institute of Automatics AGH, al. Mickiewicza 30, 30-059 Kraków, Poland; e-mail: [email protected]. The author's stay in LAAS du CNRS was supported by a MENESRIP-DAEIF scholarship No.: 174755 K through CIES.
• diagnoses generation,
• verification and repair. This paper is mostly concerned with systematic conflict generation. The problem of conflict generation appears to be one of the most important problems in automated diagnosis of dynamic systems based on domain model of correct system behaviour.
Conflict sets [22] (or conflicts, for short) are the sets of components of the system such that under the assumed model and observed output not all the components of any conflict set cannot be claimed to work correctly. Such sets of "suspected" elements are used then for potential diagnoses generation in the form of hitting sets for all the conflicts (i.e. a diagnosis is any set having nonempty intersection with any conflict set, and build from the elements of conflict sets only). This kind of diagnostic approach is based on Reiter's theory [22] of diagnosis from first principles, and DeKleer's work on diagnostic systems [5]. In application to dynamic systems the theory describing system behaviour is constituted by a causal qualitative model of correct system behaviour in the form of CA-EN causal graphs incorporating qualitative calculus [2].
When considering the problem of conflict set generation, the following simplifying assumptions will be considered to hold:
• the causal qualitative model is complete in the sense that the behaviour of all variables of interest can be effectively calculated (simulated) under the assumption of correct behaviour of system components,
• the possibly observed incorrect behaviour of certain variables is due to one or more faults of components only; no misbehaviour caused by incorrect design or implementation, closed-loops feedback effects, wrong control actions (input variables), external noise, impreciseness of the model or measurements are taken into account,
• the potential faults can be caused only by elements "assigned" to influence relations represented by edges of the graph; for simplicity one can assume that one influence is represented by one "identifiable component" assigned to it,
• quasi-static faults are considered only, i.e. faults that can be observed for some time period, causing steady-state-like misbehaviour observed during some time interval; no temporal misbehaviour is considered here, i.e. faults are assumed to be of permanent nature,
• the structure of the subgraph of interest does not change during its analysis,
• the considered graph has no loops,
• no time (dynamics) is taken into account,
• all the influences are "calculable", i.e. the equations describing the signal propagation are solvable both in the forward and in the backward direction,
• all the misbehaving variables can be detected.
Further, no knowledge about potential misbehaviour modes of the components will be used. The assumed goal is to find, possibly all, explanations, i.e. faults, responsible for the observed misbehaviour.
Graphical notation
For the sake of representing graphically various features concerning analyzed cases a simple extension of the causal graphs symbolics is proposed. Throughout the paper the following extension of the basic notation of [2] will be used:
• P , Q, R -input/control variables, • U, V , W -intermediate variables,
• X, Y , Z -output variables; X is also used as any variable (without making precise its position),
• [U], [ ] -a variable not measurable,
• X, X, • -a measured variable,
• X , X * , * -a variable observed to misbehave,
• X+, X + , X− , X − -extended notions of misbehaviour, i.e. providing the information that the value is too big or too small, respectively (reserved for future use).
As usually, an arrow (−→) means causality. Families of variables are to be denoted with boldface characters. e.g X = {X 1 , X 2 , . . . , X k }. Influences (equations) are denoted by I, e.g. P I −→ X means that P influences X through I; a component responsible for the correct work of I is to be denoted by c. Faulty components or influences will be also denoted by c * and c * , respectively. For simplicity, assuming that one component c is responsible for the correct behaviour of influence I, we can interchange components with influences and vice versa.
In case of dynamic equations a "time-flattening" procedure may be applied. This means that a differential equation can be replaced with an appropriate set of algebraic equations describing the relationship among the variable values in the subsequent time instants, as it is done for numerical solution of differential equations.
Causal graph
The class of considered causal graphs is quite a general one. By a causal graph we understand a set of variables (taking either numerical or symbolic values) and represented by the graph nodes, and a set of causal influences defined with appropriate equations, and represented by the arcs of the causal graph. Thus any causal graph is assumed to be a structure of the form G = (X, Ψ), where X is the set of all the variables and Ψ is the set of influences/equations allowing for calculation of certain non-input variables on the base of the input ones. It is assumed that the equations defined by Ψ are forward and backward calculable, i.e. having all input variables for one influence the output can be calculated, and having the all but one input variable values and the value of the output variable, the single undefined input variable can be calculated. It is no matter here if the calculation are analytical or numerical ones.
Basic problem formulation
A reasonable assumption is to start the diagnostic procedure at discovering that at least one of the observed (i.e. measurable and measured) variables misbehaves. This can be done basically in two ways:
• the detection can be a model-based one, i.e. the value of the variable is predicted (by simulation, under assumption of correct work of all the components) and compared with the observed value. In case significant discrepancy is observed, the variable is classified as misbehaving or non O.K. This method is in fact applied in CA-EN [2]; some further details considered there cover the problem of minimal time period of the discrepancy being observed (to avoid false alarms caused by fluctuations, etc.), and problems following from qualitative character of the values of the considered variables.
• the detection can be based on expected behaviour approach [12,11], i.e. the definition of expected normal behaviour or expected failure behaviour can be stated explicitly. The definitions can be based both on the analysis of the model and on the expert domain knowledge and former experience. The advantage of the latter approach is that the detection can be quicker, as it does not require simulation with use of the system model.
Moreover, any combination of the above two approaches can be applied. In case of complex systems with qualitative models and big degree of uncertainty and vagueness in the domain knowledge, such a combination may be necessary in order to reasonably cover most of the real failures and avoid covering the false ones. Further, general "integrity constraint", e.g. in the form of logical formulae can be provided to describe consistent and inconsistent patterns of variable/values combinations [14].
The starting point for diagnostic procedure consists in determining a nonempty set of misbehaving variables X * = {X * 1 , X * 2 , . . . , X * k }. We assume that during the operation of diagnostic procedure this set remains unchanged, i.e. quasi-static faults are to be diagnosed. Our goal here is to determine possibly all minimal conflict sets for the observed set X * . This conflict sets may be used then for diagnoses generation.
It seems reasonable to distinguish the following stages to be carried out in course of a systematic conflict generation procedure:
1. Domain restriction: Restriction (possibly maximal) of the initial graph to a subgraph containing only the variables and components "involved" in the creation of observed misbehaviour; however, certain boundary variables may also be useful, e.g. to eliminate certain suspected components and/or to further structure the potential conflict sets. Note that in a more complex system several independent faults may occur at a time in "geographically separated" areas of the system; it seems reasonable to perform diagnostic reasoning then independently for any such area, 2. Strategy selection: Establishing a strategy for conflict generation, e.g. "hot" starting points, order of generation, restrictions, etc. One of the key issues there is that the generation of conflicts should be efficient both with respect to time of generation and with respect to their "parameters" (conflict sets should be as precise as possible, i.e. minimal), 3. Efficient conflict generation: Systematic conflict generation, usually from from the "smallest" to the "biggest" ones with deleting non-minimal conflicts and efficient elimination of potential conflicts sets which are not real conflicts. This point is crucial for the diagnostic efficiency -if not all the conflicts are generated, some faulty elements may be missing in final diagnoses; if conflicts are not minimal, too many diagnoses are likely to be obtained.
The conflict generation stage can be interleaved with diagnoses generation. A post-analysis of conflicts after generation stage may be useful as well; elimination of certain conflicts -if possible -leads to smaller diagnoses, while considering minimal conflicts leads to smaller number of potential diagnoses. In order to establish efficient strategy both preferences and current limitations should be taken into account. Further, expert domain knowledge and heuristics may be useful (e.g. in the form of pre-schedules or plans for ordering conflict generation).
Below we consider the three above stages as separate problems.
Graph restriction
Let us consider the problem of restricting the area of focus in order to minimize the domain for potential conflict calculation. This can be done by restricting the initial graph G to a subgraph, say G', sufficient for the diagnostic task. Intuitively, the goal is to rule out most of the variables and influences not taking part in the formation of the observed misbehaviour. We discuss below some of the most straightforward possibilities. Consider the most simple and intuitive restriction consisting in limiting the area of interest to elements from which there is a signal flow to the misbehaving variables (as in [2]); in other words, the idea of causality is applied to elimination of items not having the causal influence on the observed misbehaviour. Let ANT (X) denote the subgraph composed of components (influences) such that through any c of ANT (X) there is a directed path to X. Similarly, let DESC(X) denote the subgraph composed of all components (influences) such that there is a directed path from X through any c ∈ DESC(X). Similar notation can be applied to sets of variables. Only the elements involved in ANT (X) can have influence on the behaviour of X.
It would seem natural to limit the area of interest to ANT (X * ) but the simplest example concerning back-propagation shows that in some cases it may be not sufficient (see Fig.1).
Note that, in such a case at least two problems following from missing of some auxiliary information occurs:
• lack of intermediate point checking,
• compensation phenomenon for multiple faults in line.
In case c 1 is faulty c 3 may compensate for its misbehaviour so that Y behaves correctly. In such a case even if c 3 ∈ ANT (X * ), it should certainly be taken into account. The use of it may be twofold: if {c 2 , c 3 } is a conflict set (apart from {c 1 , c 2 }), then probably c 2 is faulty; however the explanation that c 1 and c 3 are faulty and c 3 compensates for the fault of c 1 at Y is also possible (the compensation phenomenon takes place). In case {c 2 , c 3 } is not a conflict set, then most probably c 1 is the only faulty element there. Thus the part with c 3 provides new discrimination information. We shall refer to this type of structures as "fork-like". The next approximation may be to limit the area of analysis to the set defined as DESC(ANT (X * )) and it seems to be quite reasonable 1 . However, again, extension of the former recent example shows that in certain cases this heuristic simplification may lead to incomplete conflict generation possibilities (see Fig. 2).
[U] Y X* P c1 c2 c3 ANT(X*) {c1,c2} {c2,c3} {c1,c3}
One of the possible conflict sets is {c 3 , c 8 , c 7 , c 6 , c 5 , c 4 }, and in order to calculate it one has to consider a complex combination of ascending/descending elements. Of course this example may be extended horizontally on any arbitrary number of variables; we shall refer to it as "side-
C = {c4,c5,c6,c7,c8,c3} [U] [V] [W] P Q R Y Z X* c1
c2 c3 c5 c6 c7 c8 c4 DESC(ANT(X*)) wave" or "side-effect" example.
The next obvious possibility is an arbitrary combination of the form
DESC(ANT (. . . DESC(ANT (X * )) . . .))
or similar; the main problem is where to stop. Of course, an obvious solution is to cover all the connected graph, i.e. break the procedure of subgraph growing on the "natural boundaries". Below a more restrictive proposal, still satisfying the requirement of generating all possible conflicts is outlined.
Let us consider an arbitrary subgraph of the initial causal graph containing at least one unmeasured variable. By extending the graph we shall understand adding subsequent links (with assigned to them variables). Extending on a certain path (paths) to or from an unmeasured variable leads to a closure if all the "boundary" variables are measured ones, and the values of the unmeasured variables incorporated in the subgraph can be calculated (either by for-or by back-propagation) from the boundary variables. The smallest set of measured boundary variables defining a closure for the variables of the initial graph on all the paths to or from it "cuts out" the subgraph of interest. It is to be denoted by CLO(X) for the initial set of variables being X. In other words, the CLO(X) is a minimal subgraph covering X, "cut off" from the basic causal graph at measured variables only (and including all of them). Below we propose a formal definition following the above intuitions. Definition 1. Let X be an arbitrary set of variables (both measured and unmeasurable ones). The closure of X, to be denoted as CLO(X) is a subgraph of the causal graph satisfying the following conditions:
• CLO(X) incorporates all the variables of X,
• all the input and output variables of CLO(X) are measured ones,
• all the input and output variables are the O.K. ones,
• CLO(X) is minimal with respect to set inclusion of the set of subgraph nodes.
The meaning of input and output variables is straightforward. An input variable is one from which the signal is directed inside the structure and taking it value from outside the structure. An output variable is one taking its value from inside the graph and, if some links point from this variable they must all go outside the closure. Thus a variable to which several links point can be a boundary variable only if all the links are pointing inside or outside the closure. For example, on Fig. 2 variable Y is not a boundary variable for the structure incorporating elements c 1 , c 4 , and c 5 .
For intuition, the construction of the CLO(X) cuts out a specific subgraph from the initial graph, but one can cut only through measured and correct variables and not through links (see Fig. 3).
This means that one should try to isolate the subgraph with respect to information coming in or out of the subgraph; one tries to isolate it from information theory point of view. Of course, the selected subgraph should be the smallest one, covering the misbehaving variables. Further, the variables through which we cut should behave correctly -the goal is to isolate misbehaving part of the graph. This is also like growing the misbehaving area until correctly behaving measured variables are reached on all paths going outside.
The input and output variables will be referred to as boundary variables; in fact, they constitute a kind of a frontier such that all the information coming in or out goes through the boundary variables. Intuitively, if they behave O.K. no information about misbehaviour inside CLO(X) can go out from the structure and thus be manifested outside; similarly, no information about possible faults outside can go inside the structure and thus be a cause for the observed misbehaviour of the incorporated variables.
Note that the above operation separates in fact a subgraph closed from the point of view of information theory; assuming the "Markow-like " character of the causal graphs, no information can be transferred inside or outside the CLO-sured subgraph in a way different than through the boundary variables. Thus, no information from the outside can have influence on the diagnostic process, provided that the goal is to explain the misbehaviour of the variables inside the CLOsure.
Now the natural consequence of the above idea is to restrict the subgraph for conflict generation to be CLO(X * ) (see Fig. 3 consisting closures for the misbehaving variables can be constructed, the diagnostic process can be performed for any identified sub-area independently from each other. Roughly speaking, in such a case the closure would constitute "islands"of isolated misbehaviour on the area of the initial causal graph. The idea of the algorithm for constructing CLO(X * ) may be as follows. First construct a CLO({X * }) for any variable X * ∈ X * . This can be done by following any path to and from X * until a measurable and correct variable is spotted. Then any closures of single variables sets having some elements in common should be composed into connected subgraphs. The process may result with one or more connected subgraphs.
Note that the isolation process can be done for both misbehaving and O.K. variables, i.e. one can perform a construction like the above closure for a set of correct variables. In this case, such a closure should be excluded from the diagnostic process (maybe without the boundary variables).
As it can be seen, the above choice is quite a natural one, and it follows also from the approach to conflict calculation presented later.
Strategy for conflict calculation
In order to consider the problems of strategy of conflicts generation, one should first answer the questions concerning preferences among diagnoses to be generated and sets of diagnoses viewed as "final solutions ".
Recall that any diagnosis D is just a set of components, D = {c 1 , c 2 , . . . , c m } such that assuming them faulty is sufficient for restoring the consistency of the domain theory with the observations. The sets of diagnoses are denoted as D = {D 1 , D 2 , . . . , D n }..
Considering preferences among diagnoses we must take into account the risk of basing our diagnostic procedure on incomplete set of conflicts, i.e. a case when not all the conflicts are calculated. Intuitively, the diagnoses calculated in such a case will be "incomplete" or "partial". This observation is supported by the following proposition.
Proposition 1. Let C i denote sets of conflict sets, and D i sets of diagnoses calculable from
C i , i = 1, 2.
Then, if C 1 ⊆ C 2 , then also:
1. ||D 1 || ≤ ||D 2 ||, and 2. ∀D 1 ∈ D 1 ∃D 2 ∈ D 2 such that D 2 ⊇ D 1 .
According to the above proposition, in case of more conflict sets accessible (i.e. known; in our case the problem is to discover the conflicts since they are "hidden" in the graph structure), the diagnoses are possibly more precise 2 . Another observation is that simultaneously, having more conflicts we can expect generation of more diagnoses, and this is what we would like to avoid. As the diagnoses are only potential explanations of the observed misbehaviour, they require further verification, thus the number of diagnoses generated should be relatively small. The intuition is that generation of the conflicts should be limited to the minimal ones (or at least as small as possible). The following proposition is related to the problem of limiting the number of possible diagnoses.
Proposition 2.
Let C i denote sets of conflict sets, and D i sets of diagnoses calculable from C i , i = 1, 2. Further, assume that C 1 and C 2 have the same number of elements. Then, if for any C 1 ∈ C 1 there exists C 2 ∈ C 2 such that C 1 ⊆ C 2 , then there is also ||D 1 || ≤ ||D 2 ||.
The above proposition allows for comparison of two sets of conflicts with the same number of elements, but different "degree of preciseness"; the more precise conflicts are better, they lead to generating less possible diagnoses.
Finally, let us consider the problem of adding new conflict sets to an existing set of conflicts. If in the already found conflict set there is one smaller than the one to be added, then adding the new one is not necessary; either no new diagnoses will be generated or the diagnoses will not be minimal. The following proposition states it more precisely.
Proposition 3.
Let C i denote sets of conflict sets, and D i sets of diagnoses calculable from C i , i = 1, 2. Let C 2 contain all the conflicts of C 1 and some other not minimal ones, i.e.
C 2 = C 1 ∪ C ′ 1 , where for any C ′ 1 ∈ C ′ 1 there exists C 1 ∈ C 1 such that C 1 ⊆ C ′ 1 . Then for any diagnosis D 2 ∈ D 2 there is a diagnosis D 1 ∈ D 1 such that D 1 ⊆ D 2 . Further, there is also ||D 1 || ≤ ||D 2 ||.
This proposition justifies the intuition that non-minimal conflicts are useless for diagnostic efficiency -adding non minimal conflicts not only may lead to more diagnoses, but also to generation of non minimal ones as well.
The above considerations seems to justify the following assumptions concerning the strategy of conflict generation:
1. Conflicts should be generated in a systematic way, so that all the necessary conflicts are obtained; if a conflict set is missing, there is a risk of generating partial (incomplete) diagnoses.
2. Conflicts should be generated from i = 1 towards i = k, where i is the number of components in a conflict set and k is the maximal number of components in the analyzed subgraph; this assures that more precise conflicts are generated first.
3. All conflicts comprising i elements should be calculated before ones comprising i + 1 elements; a conflict which is a superset of some previously generated conflict is not to be considered.
4. The procedure can be stopped when either no new conflicts can be generated, or for any new conflict to be generated a subset conflict has already been generated (minimality requirement).
Note that the following further auxiliary rules may be proposed for enhancing he diagnostic process:
• the conflict generation procedure may be stopped for some number of conflicts generated arbitrarily; this may be the case when diagnoses containing only some limited number of faulty components are probable,
• the conflict generation procedure may also be stopped for some i arbitrarily; this may be the case when too complex conflicts are too costly to calculate, etc.,
• more data (measurements, tests) may become available cutting down in a natural way the size of conflict sets,
• conflict generation may be interleaved with diagnoses generation (see also [22]); some diagnosis found valid may stop the process,
• expert-designed schedules for conflict generation finding the most probable conflicts "around" elements most likely exhibiting faulty behaviour can be used to speed-up the diagnostic procedure by turning it into routine procedures,
• if accessible, the knowledge about modes of faulty behaviour of the components and its influence on the behaviour of variables can be used for further selection.
An approach to systematic conflict generation
The basic assumption here is that both propagation and back-propagation can be regarded as mathematical constraints, i.e. they provide some In case of back-propagation there can be some difficulties with "inverting" the calculations so as to obtain the values for one of the arguments. However, from theoretical point of view it seems that in any case one can solve the equation numerically, and so inverting it is not a necessary procedure; of course, if applicable, it can contribute to computational efficiency.
From purely mathematical point of view, in order to generate conflicts one must have more equations than variables; in our case they are unmeasured variables. Thus if n denotes the number of equations (both for back-and for-propagation) defined for some subgraph, and m is the number of unmeasured variables involved in the computation, then the condition necessary for potential conflict generation is that n ≥ m + 1. Further, for any such substructure there exists a potential possibility of generating no more than n m+1 possible conflicts. This can be illustrated with Fig.4, where n = 4, m = 1, and we have 4 2 potential conflicts.
[U] Usually, there are less conflicts, since not all the structures described by n equations and containing m variables allow for calculation of a "full-size" conflict; examples include chains with pending unmeasured variables. Further, all the conflicts are only potential -if a conflict is really observed or not depends on actual computations, and, of course, on the existence of faulty elements (it is assumed that no conflicts are generated due to inadequate calculations or inadequate model).
Q P R X Y Z c1
Taking into account the above considerations and in order to achieve better efficiency of conflict generation, the following, two-stage, transparent procedure of conflict generation is proposed;
• identification of potential conflict structures, i.e. sets of influences assuring necessary conditions for conflict existence, and then
• verification for any such a structure and selected set of equations if a conflict exists; this is to be done by an attempt at "solving" these equations.
Splitting the procedure of conflict generation into these two stages seems to advantageous for the sake of transparency and systematic conflict generation. Moreover, identification of a conflict structure is equivalent to having the knowledge about it components. Thus exploration of non-minimal potential conflicts can be abandoned without performing the real calculations. Moreover, certain heuristics can be applied to preselect the potential conflict structures for further investigation, leaving a large part of them without performing costly mathematical calculations.
The key issue for carrying on is to introduce a definition of a Potential Conflict Structure, shortly PCS. This notion denotes a subgraph of the causal graphs, for which there is a possibility (always potential) of calculating a conflict via obtaining two different values for the same variable. A P CS comprising m unmeasured variables and leading to detection of potential conflict at a variable X will be denoted as P CS m (X). The number of unmeasured variables m will be referred to as the order of a conflict structure. Variable X can be measured one or an unmeasured one.
Note that some most interesting are potential conflict structures having no unmeasured variables, i.e. P CS 0 -they are always of the form P 1 , P 2 , . . . , P j c −→ X, where all the variables are measured; if such a structure provides a real conflict, then the conflict consists of one element c and in fact is a partial diagnosis. In other words, component c is faulty and must be an element of any valid diagnosis; further the fault of c is a cause of the observed misbehaviour of X. Therefore conflict structures of zero order should always be explored first (if existing). Now let us pass to potential conflict structures of larger size. First we put forward the following definition.
Definition 2. A variable X is well-defined (defined, for short) iff:
• either it is a measured variable, or • its value can be calculated on the base of some other variables, which are well-defined.
If there are two or more, e.g. k independent ways of calculating the value of X, then X is said to be k-defined.
The independent ways of calculating the variable may consist in measuring the value and calculating it with different sets of equations. The calculation of a well-defined variable can be done no matter how -forward propagation is as good as backward one (at least from purely mathematical point of view). Now we can define a potential conflict structure on m unmeasured variables.
Definition 3. A Potential Conflict Structure for variable X defined on m unmeasured variables is any subgraph of the causal graph, such that:
• it comprises exactly m unmeasured variables (including X, if unmeasured),
• all the variables are well-defined, and X is double-defined;
• for the P CS being defined on m unmeasured variables it is necessary that all the values of the m variables are necessary for making X double-defined.
Variable X will be called the head of the PCS. [U]
[U]
P X P X Y P Q R X [U] [V] [U] [V]
[U]
[V] P X P X Z P Q X Figure 5: Examples -potential conflict structures.
Note that any of the graphs represent at least two potential conflict structures, i.e. ones for different head variable; in these cases the P CS constitute the same graphs and the same conflicts will eventually be generated. Thus for any such P CS the calculation is to be performed only once, and the selection of the head variable is to be done arbitrarily, e.g. with respect to making easier the problem of equations solving.
Potential conflict structures can take arbitrary "shape" and it is, in general, difficult to say if some structure is a P CS at the first sight. The definition is in fact recursive, and so the algorithm for detection of P CS must be. But there are at least three typical conflict structures with some nice properties and interpretation. They are: a "chain" of calculable variables, a "pyramid" and various types of "forks ", where back-propagation plays important role.
The algorithm for detecting a P CS on m unmeasured variables is a recursive one; the starting point is a selected variable, the head of the structure; then, it must search for exactly m unmeasured variables connected to the selected one if it is a measured variable (a kind of path tracing), or m − 1 unmeasured variables if the head is an unmeasured variable itself. When traversing the graph a check if all the unmeasured variables are well defined and if having them the head variable is double defined is to be performed. The basic procedure should be repeated for any selected variable for m changing from 0 to the maximal number of unmeasured variables.
Note that the basic procedure does not guarantee that the conflict sets are generated according to growing number of elements; this is so, because there can be many influences "pointing to" a single unmeasured variable and thus a potential conflict set may contain many elements assigned to these influences. However, for any P CS m it is possible to assign a number j of elements assigned to the influences necessary for conflict calculation; this can be done before the calculation of the appropriate conflict; there is also always j ≥ m+1. Thus there is a simple way of defining a second characteristics of any P CS, i.e. a number j of elements of the conflict set to be possibly determined; this may be noted as P CS j m . And finally, during calculation of conflicts for the same m for any unmeasured variable, the order of calculations can be done with respect to the value of j.
An outline of algorithmic approach
To summarize, an outline of an algorithm for systematic conflict generation can be as follows:
1. Detect the set of misbehaving variables X * , 2. Define the restricted subgraph being the object of analysis to be CLO(X * ); any other choice (e.g. a heuristic one) is possible, but the completeness can be violated, 3. For any measured variable X ∈ CLO(X * ) detect all the conflicts calculable without the use of the values of unmeasured variables; the calculation of conflicts can be ordered with respect to the number of elements in conflict sets; this step refers to the zero-order conflict generation, 4. For any variable X ∈ CLO(X * ) detect sequentially all P CS j m ; the order of generation is from m = 1 to the number of variables in the subgraph of interest. For any variable and m established order the P CS j m according to increasing values of j, 5. Repeated P CS, i.e. ones different only with respect to the head variable should be abandoned (leaving exactly one of them for investigation); further, P CS leading to nonminimal conflicts can be abandoned before numerical investigation, 6. Stop the procedure when there are no more P CS to be generated (or earlier, according to some heuristics or when an appropriated diagnosis is generated).
An Example
A simple test program for support of direct determining potential conflict structures was implemented in PROLOG. The program is a meta-interpreter using several simple recursive rules. It calculates the potential conflict structures for a specified unmeasured variable. An example subgraph CLO-sured with measured variables is shown in Fig. 6; potential conflicts calculated with use of the program are listed there as well.
During calculation of the conflicts the following rules may be useful in order to avoid repeated computations and improve the overall efficiency:
• head variables preselection: this seems to be a most important one heuristic rule for achieving reasonable efficiency; the candidate variables for heads of the P CS should be preselected. A reasonable heuristic may consists in selecting all the misbehaving variables and the unmeasured ones (other variables, i.e. the measured O.K. ones, will be incorporated in the calculated P CS or simply are not necessary; a strong simplification may consist in selecting the non O.K. variables as heads for P CS-s,
• eliminating repetitions: the concept of P CS allows for identification of conflict set elements before calculation of the potential conflict; thus, whenever a P CS identical to another already generated appears, there is no need to calculate the conflict once more. This is important, since starting from different variables and extending the P CS-s it seem unavoidable to generate the same P CS several times.
• limiting the size of generated conflicts: again, if a P CS has been generated such that its elements constitute a superset of a conflict set which has already been generated the there is no need to investigate this P CS,
• further decomposition: generation of conflicts can be stopped at a boundary composed of measured variables; this is similar to considering the CLO-sure, but this time for a substructure of the selected subgraph,
• re-use of calculations: once calculated, influences (values of the variables) can be reused in calculation of several conflicts; they need just to be stored,
• user-defined scenarios: some schedules or expert-defined scenarios providing guidance for calculating conflicts in specific cases defined by the selection of misbehaving variables and type of their misbehaviour can be used to guide the procedure of conflict generation,
• measurement introduction: for certain P CS seeming too large, suggestions of measurement points can be done so as to structure them down into manageable objects.
• pre-elimination of potential faulty elements: the observed nature of faults may indicate that certain types of faults are not to be taken into account; this observation may lead to eliminate certain components from further considerations. Hence, even if the set of influences used for conflict generation are large, the generated conflicts may turn out to be quite small.
Closing remarks
From the above considerations one can expect that the final efficiency of conflict (not diagnoses!) generation is bound to depend on a variety of factors. Further, one can expect that efficient conflicts, i.e. ones leading to a small number of well-localized diagnoses can be obtained only if the unmeasured variables are relatively sparse. In case there are only few measurements one cannot expect that the isolation of faulty components will be effective. With respect to this problem, the back-propagation seems to appear to be a crucial issueroughly speaking, it may play a role similar to introducing measurements and thus contribute to limiting the size of conflicts.
Another aspect which seems well worth investigating is the problem of modes of faulty behaviour of the components and their influence on the observed behaviour of variables; an a priori knowledge about possible faults of components can be used to model the faulty behaviour and thus to select out certain possible faults if the modeled behaviour is not observed. Further, certain conflict sets can be eliminated during generation.
But the most challenging issue seems to be formalization of an approach based on combination of direct search for faulty components (as in [15]) with procedures based on conflict generation; Let us recall that the approach based on conflict generation according to Reiter's theory [22] is justified only under several relatively strong assumptions; further, after generating conflicts a next stage for generating diagnoses is necessary. The overall procedure seems to be hardly fitting on-line systems requirements (not to say about the real-time ones). On the other hand, the idea of such a backward-search procedure seems to follow from the general search principles and abductive reasoning: by appropriate use of functional element descriptions and information about measured values one should construct the hypotheses explaining the observed behaviour. In order to consider some more specific bases for such an approach, an "axiomatization" of the domain seem to be necessary. Then the basic step should consist in generation of a search space for abductive diagnostic reasoning, providing a model fitting the diagnostic purposes.
Figure 1 :
1Example -restriction of the domain to ANT (X * ).
Figure 2 :
2Example -Restriction of the domain to DESC(ANT (X * )).
Figure 3 :
3for and intuitive idea). Moreover, if several separated subgraphs An information closure.
equations determining the relations among variable values. An equation with one unknown value of a variable defines this value; if all the values are known, then such an equation provides a possibility of conflict generationthe components responsible for holding of this equation may not be all working correct if the equation is not satisfied by the observed variable values.
Figure 4 :
4Example -potential conflict structures selection.
Examples of P CS m for m ∈ {0, 1, 2} are shown onFig.
Figure 6 :
6Example -generation of PCS-s.
This, in fact, is the core of the approach equivalent to assuming that back-propagation is not combined with propagation forward.
At this stage we assume to prefer as precise (complete) diagnoses as possible. By a more precise diagnosis we understand here one explaining more of the observed failures; contrary to precise (complete) diagnoses, the imprecise (incomplete) ones explain only part of the observed misbehaviour. This may seem contrary to the usual preference of minimal diagnoses. Our standpoint is, however, following from the risk that not all the conflicts are likely to be generated. For an established set of conflicts preferring minimal diagnoses is still an obvious choice.
Acknowledgment:The Author thanks Dr. Louise Travé-Massuyès for many comments and discussions helpful in improving this work.
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Normality and faults in logic-based diagnosis. D Poole, Proceedings of IJCAI'89. IJCAI'89D. Poole, 'Normality and faults in logic-based diagnosis', Proceedings of IJCAI'89, 1304-1310.
A theory of diagnosis from first principles. R Reiter, Artificial Intelligence. 32R. Reiter, 'A theory of diagnosis from first principles' Artificial Intelligence, 32, 57-95, 1987.
Artificial Intelligence. A modern approach, Prentice Hall series in Artificial Intelligence. S T Russell, P Norvig, S.T. Russell and P. Norvig, Artificial Intelligence. A modern approach, Prentice Hall series in Artificial Intelligence, 1995.
Knowledge Based Systems for Test and Diagnosis. G Saucier, M.A. BreuerNorth-Holland, AmsterdamG. Saucier, A. Ambler and M.A. Breuer (Eds.), Knowledge Based Systems for Test and Diagnosis, North-Holland, Amsterdam, 1989.
Knowledge-based diagnosis -an important challenge and touchstone for AI. P Struss, Proceedings of ECAI'92. ECAI'92P. Struss, Knowledge-based diagnosis -an important challenge and touchstone for AI, Proceedings of ECAI'92, 1992.
Diagnostic Problem Solving. Combining heuristic Approximate and Causal Reasoning, North Oxford Academic, A Division of Kogan Page. P Torasso, L Console, LondonP. Torasso and L. Console, Diagnostic Problem Solving. Combining heuristic Approxi- mate and Causal Reasoning, North Oxford Academic, A Division of Kogan Page, Lon- don, 1989.
|
[] |
[
"Photoassociation to the 2 1 Σ + g state in ultracold 85 Rb 2 in the presence of a shape resonance",
"Photoassociation to the 2 1 Σ + g state in ultracold 85 Rb 2 in the presence of a shape resonance"
] |
[
"M A Bellos \nDepartment of Physics\nUniversity of Connecticut\n06269StorrsConnecticutUSA\n",
"R Carollo \nDepartment of Physics\nUniversity of Connecticut\n06269StorrsConnecticutUSA\n",
"D Rahmlow \nDepartment of Physics\nUniversity of Connecticut\n06269StorrsConnecticutUSA\n",
"J Banerjee \nDepartment of Physics\nUniversity of Connecticut\n06269StorrsConnecticutUSA\n",
"E E Eyler \nDepartment of Physics\nUniversity of Connecticut\n06269StorrsConnecticutUSA\n",
"P L Gould \nDepartment of Physics\nUniversity of Connecticut\n06269StorrsConnecticutUSA\n",
"W C Stwalley \nDepartment of Physics\nUniversity of Connecticut\n06269StorrsConnecticutUSA\n"
] |
[
"Department of Physics\nUniversity of Connecticut\n06269StorrsConnecticutUSA",
"Department of Physics\nUniversity of Connecticut\n06269StorrsConnecticutUSA",
"Department of Physics\nUniversity of Connecticut\n06269StorrsConnecticutUSA",
"Department of Physics\nUniversity of Connecticut\n06269StorrsConnecticutUSA",
"Department of Physics\nUniversity of Connecticut\n06269StorrsConnecticutUSA",
"Department of Physics\nUniversity of Connecticut\n06269StorrsConnecticutUSA",
"Department of Physics\nUniversity of Connecticut\n06269StorrsConnecticutUSA"
] |
[] |
We report the first observation of photoassociation to the 2 1 Σ + g state of 85 Rb2. We have observed two vibrational levels (v ′ =98, 99) below the 5s 1/2 + 5p 1/2 atomic limit and eleven vibrational levels (v ′ =102-112) above it. The photoassociation-and subsequent spontaneous emission-occur predominantly between 15 and 20 Bohr in a region of internuclear distance best described as a transition between Hund's case (a) and Hund's case (c) coupling. The presence of a g-wave shape resonance in the collision of two ground-state atoms affects the photoassociation rate and lineshape of the J ′ = 3 and 5 rotational levels.
|
10.1103/physreva.86.033407
|
[
"https://arxiv.org/pdf/1208.3649v1.pdf"
] | 119,105,027 |
1208.3649
|
f53e8018855add1820757cebd558a7ca88e0d3b1
|
Photoassociation to the 2 1 Σ + g state in ultracold 85 Rb 2 in the presence of a shape resonance
17 Aug 2012 (Dated: May 2, 2014)
M A Bellos
Department of Physics
University of Connecticut
06269StorrsConnecticutUSA
R Carollo
Department of Physics
University of Connecticut
06269StorrsConnecticutUSA
D Rahmlow
Department of Physics
University of Connecticut
06269StorrsConnecticutUSA
J Banerjee
Department of Physics
University of Connecticut
06269StorrsConnecticutUSA
E E Eyler
Department of Physics
University of Connecticut
06269StorrsConnecticutUSA
P L Gould
Department of Physics
University of Connecticut
06269StorrsConnecticutUSA
W C Stwalley
Department of Physics
University of Connecticut
06269StorrsConnecticutUSA
Photoassociation to the 2 1 Σ + g state in ultracold 85 Rb 2 in the presence of a shape resonance
17 Aug 2012 (Dated: May 2, 2014)
We report the first observation of photoassociation to the 2 1 Σ + g state of 85 Rb2. We have observed two vibrational levels (v ′ =98, 99) below the 5s 1/2 + 5p 1/2 atomic limit and eleven vibrational levels (v ′ =102-112) above it. The photoassociation-and subsequent spontaneous emission-occur predominantly between 15 and 20 Bohr in a region of internuclear distance best described as a transition between Hund's case (a) and Hund's case (c) coupling. The presence of a g-wave shape resonance in the collision of two ground-state atoms affects the photoassociation rate and lineshape of the J ′ = 3 and 5 rotational levels.
I. INTRODUCTION
Photoassociation (PA) of ultracold atoms has become a standard tool for high-resolution spectroscopy [1], with a wide range of applications [2]. There is strong interest in producing cold and ultracold molecules in their lowest electronic and rovibrational state. Molecules in their lowest energy state are immune to inelastic collisions and can be used as a platform for quantum information [3] and cold chemistry [4]. A wide variety of experiments target the formation of stable molecules at low temperatures, such as buffer gas cooling [5], Stark deceleration [6], STI-RAP transfer of magnetoassociated atoms [7] and photoassociated atoms [8], direct laser cooling of molecules [9], and rotating supersonic sources [10], to name a few.
Traditionally, photoassociation has been limited to vibrational levels near the asymptotic limit of the excited electronic state, and consequently to large internuclear separations. Photoassociation at short range was believed to have very low rates due to the small amplitude of the ground-state wavefunction at short internuclear distances. However, photoassociation of LiCs [11] and more recently of Rb 2 [12], RbCs [13] and NaCs [14] has shown that short-range photoassociation is possible, allowing the formation of deeply-bound molecules, including the lowest rovibrational levels.
Several factors can increase the photoassociation rate at short range. The amplitude of the ground-state wavefunction can be increased at short range by scattering resonances, i.e., shape resonances or Feshbach resonances [15]. The amplitude of the excited-state wavefunction can be large if the excited state is constrained over a narrow range of internuclear distances, e.g. at the bottom of a potential well, or, in quasibound levels where the outer turning points are at small distances. Furthermore, transition dipole moments can increase at short range, thereby increasing the transition rate, although this is not the case in this work.
In previous work, we photoassociated to the 1 3 Π g state of Rb 2 and observed spontaneous decay to the lowest vibrational levels of the a 3 Σ + u state [12]. In some favorable cases, the spontaneous decay populated mostly a single vibrational level, the v ′′ =0 level.
In this work we photoassociate to the neighboring 2 1 Σ + g state and again observe that transitions to high rotational levels yield stronger signals than to low rotational levels. We now attribute this atypical rotational distribution to the presence of a shape resonance. The 2 1 Σ + g state is appealing because it contains both reddetuned (i.e. bound) levels and blue-detuned (i.e. quasibound) levels. Also, since these levels spontaneously decay to intermediate vibrational levels of the a 3 Σ + u state, one can now populate vibrational levels at the bottom, middle, and top [16] of the a 3 Σ + u potential well, with relatively narrow vibrational distributions.
II. EXPERIMENTAL SETUP
Our experiment is performed in a magneto-optical trap (MOT) of 85 Rb with a peak density of ∼ 1 × 10 11 cm −3 , a total atom number of ∼ 8 × 10 7 , and a temperature of ∼ 120 µK. We load the MOT from a getter source, with a loading time of ∼ 2 s and a non-alkali background pressure < 1 × 10 −10 torr. The MOT trapping laser is locked 14 MHz below the 5s 1/2 , F = 3 → 5p 3/2 , F ′ = 4 atomic transition, and a repumping laser is tuned to the 5s 1/2 , F = 2 → 5p 3/2 , F ′ = 3 transition to prevent buildup of atoms in the lower F = 2 hyperfine ground level. The energy splitting between the two hyperfine ground levels is one that routinely appears in our spectra as the presence of "hyperfine ghost" lines. These "hyperfine ghost" lines occur 0.10126 cm −1 above strong PA transitions (as shown in Fig. 4, for example) and may originate from short-range hyperfine-changing collisions [17,18].
We photoassociate with a single-mode cw Ti:Sapphire laser (Coherent 899-29) pumped by an argon ion laser (Coherent Innova 400). After fiber optic coupling we have over 1 W of usable power at the vacuum chamber. The PA laser is weakly focused to the size of the MOT, yielding a maximum PA intensity of about 100 W/cm 2 . The tuning range used for this experiment varied between 780 nm and 795 nm. Resonantly enhanced multi-photon ionization (REMPI) is performed by use of a pulsed dye laser (Continuum ND6000). It is operated between 625 nm and 675 nm using a DCM dye solution. The pulsed dye laser is pumped at 10 Hz by a 532 nm Nd:YAG laser (Spectra-Physics Lab 150). The REMPI pulse energy and linewidth are 5 mJ and 0.5 cm −1 , respectively.
After ionization, molecules are detected by a discrete dynode multiplier (ETP model 14150). Two electric field grids focus the ions through a long field-free tube to the detector, resulting in a small detection region centered at the location of the MOT. Molecular ions are distinguished from scattered photons and atomic ions by timeof-flight mass spectrometry, and the signal is integrated by a gated boxcar integrator (SRS model SR250). We turn off the MOT lasers starting 10 µs before each ionizing pulse and ending 10 µs after it to depopulate the excited 5p 3/2 state so as to decrease Rb + signals.
III. MOLECULE FORMATION PATHWAYS AND TRANSITION MOMENTS
The 2 1 Σ + g state has been previously observed by laserinduced fluorescence from the highly excited C (2) 1 Π u state [19]. To the best of our knowledge the present work is the first observation of the 2 1 Σ + g state through excitation rather than decay. This state has eluded many experiments because excitation from the ground X 1 Σ + g state is forbidden by single-photon electric dipole parity selection rules. Furthermore, single-photon excitation from deeply-bound levels of the a 3 Σ + u state is forbidden by spin selection rules. However, by photoassociating from the triplet state of colliding atoms at a range of internuclear distances where spin is not a good quantum number, the spin selection rule breaks down, allowing the transition.
The photoassociation laser thus converts a small fraction of colliding atom pairs in the MOT into molecules in the 2 1 Σ + g state. The subsequent decay of these excited molecules forms molecules in the X 1 Σ + g and a 3 Σ + u states through a variety of pathways, as discussed below. These X or a state molecules can then be detected by REMPI through any of several possible intermediate states.
In this experiment we use REMPI through the 2 3 Σ + g state to detect a 3 Σ + u molecules as shown in Fig. 1. The REMPI laser monitors the population of a single vibrational level in the a 3 Σ + u state, typically between v ′′ =18 and 24. We produce photoassociation spectra by scanning the PA laser while fixing the REMPI laser on resonance with an intermediate state.
It is useful to consider electric dipole (E1) selection rules for spontaneous emission from 2 1 Σ + g to lower states using first the Hund's case (a) basis and then Hund's case (c) basis. The 2 1 Σ + g state corresponds to the 2(0 + g ) state in Hund's case (c) notation, which further correlates to the 5s 1/2 +5p 1/2 atomic limit [22]. For Hund's case (a), single-photon decay from 2 1 Σ + g to X 1 Σ + g is forbidden by parity selection rules (g ↔ u). However, a two-step cascade decay through the A 1 Σ + u state is allowed, and has been observed in Cs 2 [23]. The first step of such Arrows indicate photoassociation (PA) to 2 1 Σ + g , spontaneous emission (SE) to a 3 Σ + u , and REMPI to Rb + 2 . Note that the 2 1 Σ + g state has a barrier of 250 cm −1 at R = 19 a0. Potential energy curves are from Refs. [20,21].
a cascade typically has a low transition rate due to the low transition frequency. This transition rate is given by summing over Einstein A coefficients,
A v ′ →v ′′ ∝ ν 3 ψ v ′′ | µ(R) | ψ v ′ 2 ,(1)
where ψ v ′ is the upper-state wavefunction, ψ v ′′ is the lower-state wavefunction, µ(R) is the transition dipole moment as a function of internuclear distance, and ν is the transition frequency. Decay to the a 3 Σ + u state is spin forbidden. As mentioned above, we nevertheless observe population in this state due to singlet-triplet mixing. This corresponds to Hund's case (c), where the spin quantum number is not well defined. Decay from 2(0 + g ) to the ground 1(0 + g ) state is still forbidden by the (g ↔ u) selection rule, just as in Hund's case (a). However decay from 2(0 + g ) to 1(1 u ) (one of the two components of the a 3 Σ + u state, also denoted as a 3 Σ + u (Ω = 1 u )) is now allowed. Furthermore, the decay to 1(1 u ) dominates over other states, as it has the largest ν 3 factor in Eq. 1. These decay pathways are summarized in Fig. 2.
Therefore at short range, in Hund's case (a), levels of the 2 1 Σ + g state are metastable but at long range, in Hund's case (c), they are not. This is evident in the transition dipole moment (TDM) shown in Fig. 3(b) taken from Allouche and Aubert-Frécon [20]. In the region in- [24,25] showing photoassociation (PA) to 2(0 + g ) and spontaneous emission (SE) to 1(1u). (b) transition dipole moment between the 2(0 + g ) and 1(1u) states, from Ref. [20].
Hund's case (c) Hund's case (a) 2( u ) 1( u ) b 3 u A 1 + u 1( + u ) 2( + u ) 2( u ) 1( u ) 1( + g ) 1( u ) 2( + g )
side 15 a 0 , the 2 1 Σ + g state is well described by Hund's case (a) and the TDM is zero. Levels with both turning points inside 15 a 0 should be metastable as the only decay path is through the slow transition to the A 1 Σ + u state.
The region between 15 and 20 a 0 is where most of the photoassociation and decay occurs in this experiment. The TDM connecting the 2(0 + g ) and 1(1 u ) states accounts for both the photoassociation step from free triplet colliding atoms and the decay back to 1(1 u ). This region is best described as an intermediate between Hund's case (a) and Hund's case (c) couplings, making all of the allowed Hund's case (a) and Hund's (c) decays shown in Fig. 2 possible. For example, the branching ratio for decay from v ′ = 101 to the A 1 Σ + u state compared to the 1(1 u ) state is calculated to be 1:10 based on the ratio of Einstein A coefficients as shown in Table I. The rapid transition to Hund's case (c) between 15 and 20 a 0 is due in part to an avoided crossing between the 2(0 + g ) and 3(0 + g ) states at 18 a 0 , as shown in Fig. 3(a), which increases the triplet character of the 2(0 + g ) state. The change in coupling to Hund's case (c) occurs roughly between 20 and 40 a 0 for most other electronic states in Rb 2 [20]. Generally speaking, the range at which this change in coupling occurs is inversely proportional to the strength of the fine structure splitting between the P 1/2 and P 3/2 asymptotes. The heavier the alkali dimer, the stronger the fine structure splitting, and therefore the smaller the distance at which the coupling changes from Hund's case (a) to Hund's case (c).
IV. PHOTOASSOCIATION SPECTROSCOPY
The energies of the observed rovibrational levels are listed in Table I and the spectrum of a single vibrational level is shown in Fig. 4. The vibrational assignments are determined by comparing the measured vibrational energy spacings (∆ G v+1/2 ) with those generated from abinitio potential energy curves from Dulieu and Gerdes (DG) [24] and Allouche and Aubert-Frécon (AA) [20]. The DG potential has v ′ =113 as the uppermost vibrational level, while the AA potential has v ′ =126. The experimental vibrational spacings match the DG potential very closely, so we adopt the corresponding vibrational numbering. Vibrational spacings from the bottom of the potential well [19], on the other hand, match the AA potential somewhat closer. The rotational assignments are verified by fitting the energies to B v [J(J + 1)], which also determines the rotational constants B v listed in Table I. We do not take into account small frequency shifts induced by the PA [27] and trapping lasers, which can shift the line position, typically by about 10 MHz. The rotational constants calculated for the DG potentials are larger than the measured rotational constants, while those calculated for the AA potential are smaller.
Levels above v ′ =99 are quasibound and can tunnel through the potential barrier and dissociate into 5s 1/2 TABLE I. Energies and assignment (E v ′ ,J ′ ) of observed levels, with respect to the ground state dissociation limit of two Rb 5s 1/2 , F = 3 atoms. A single v ′ =112 line is observed at 12811.86 cm −1 with an unknown rotational assignment. Systematic uncertainties are ±0.03 cm −1 , while random uncertainties are ±0.001 cm −1 . The experimental rotational constants (Bv) are given along with fitting uncertainties when known. Also shown are Einstein A coefficients for spontaneous emission from a single vibrational level v ′ of the 2 1 Σ + g state to all vibrational levels v ′′ of the a 3 Σ + u (Ω = 1u) and A 1 Σ + u states, denoted by A v ′ →a and A v ′ →A respectively. These A coefficients are calculated using LEVEL 8.0 [26]. and 5p 1/2 atoms. The tunneling probability competes with spontaneous emission only for the two highest vibrational levels. For v ′ =112, the rates are comparable, which decreases the observed signal size. The uppermost vibrational level, v ′ =113, is calculated to dissociate rapidly. Its predicted linewidth (20 GHz) is orders of magnitude broader than for lower vibrational levels, making it very difficult to observe experimentally.
E v ′ ,J ′ (cm −1 ) Bv (10 −4 cm −1 ) A v ′ →a (10 5 s −1 ) A v ′ →A (10 5 s −1 ) v ′ J ′ =0 J ′ =1 J ′ =2 J ′ =3 J ′ =4 J ′ =5
The first two quasibound levels above the atomic limit, v ′ = 100 and v ′ = 101, are not observed. One possibility is that the absence of these vibrational levels is due to a small FCF overlap for the PA transitions. However, the calculated FCFs for these levels are predicted to be higher than for most of the observed levels. Another possibility is that the PA laser coincidentally couples molecules to higher-lying states, through bound-to-bound or boundto-free transitions, pumping them away from the detection pathway. Although we cannot rule out accidental bound-bound transitions, it is unlikely for such coincidences to occur for all rotational levels of two successive vibrational levels. Bound-free transitions to a repulsive potential, on the other hand, may occur for a wide range of laser frequencies, but are limited by the wavefunction overlap. This makes it unlikely that bound-free transitions would dominate over the spontaneous decay from the 2 1 Σ + g state to the a 3 Σ + u state. Another explanation is that optical shielding [28] from the blue-detuned photoassociation beam does not allow atom pairs to reach small interatomic distances and photoassociate. Instead, atom pairs are excited to the outer region of the 2 1 Σ + g barrier before they can photoassociate at short range to levels inside the barrier. The optical shielding effect is strongest for levels just above the asymptote, namely for v ′ = 100, and decreases with energy above the asymptote. Modeling this optical shielding at short internuclear distances is beyond the scope of this work, but offers an avenue for future work. For example, one empirical test for the optical shielding is to monitor the PA signal of a given line and look for reductions in PA signal upon the introduction of an additional blue-detuned laser tuned below the PA line and above the atomic limit. Another test is to search for differences between the intensity dependence of red-detuned photoassociation [27,29] and the intensity dependence of blue-detuned photoassociation.
Other than these missing vibrational levels and the potential for optical shielding, we were unable to find differences between photoassociation to red-detuned and blue-detuned levels. We suspect that effects due to the trapping potentials of red-detuned PA beams-or the anti-trapping potentials of blue-detuned beams-should appear at higher intensities and smaller detunings than those used in the experiment.
We did not search for levels below v ′ =98. We expect their signal size to decrease due to a decreasing TDM as discussed in Sec. III.
A. Rotational levels, partial waves, and shape resonances
The distribution of rotational lines of a single vibrational level carries information about the partial wave content of the colliding ground-state atoms. The relative strength of rotational lines in the spectrum is related to the relative partial wave amplitudes. A single rotational line, J, is typically made up of several partial waves.
From conservation of angular momentum [2,34], we know that J = l+ j, where J is the total angular momentum of the molecule, l is the partial wave or "mechanical rotational quantum" of the ground-state collision, and j = j a + j b is the total atomic electronic angular momentum of both atoms at their asymptotic limit. For a potential curve converging to the 5s 1/2 + 5p 1/2 asymptote; j a = 1/2 and j b = 1/2, implying that j = 0, 1. Therefore J can take values J = l − 1, l, or l + 1. Furthermore, a symmetry consideration [2,35] requires that in states with (+) symmetry; odd J's come from even l's, and even J's come from odd l's. This additional requirement restricts the values of J to J = l ± 1. Therefore, for the 2 1 Σ + g state, s-wave collisions contribute only to the J ′ =1 line, p-wave collisions contribute to the J ′ =0 and J ′ =2 lines, and so forth. The g-wave collisions-the highest observed partial wave-contribute to the J ′ =3 and J ′ =5 lines, as shown in Fig. 4.
In our spectra, the strongest-and sometimes onlylines are the J ′ =3 and J ′ =5 lines. This implies that the l=4 partial wave is the strongest contribution to the PA transitions, clearly at odds with the common notion that s-wave scattering dominates processes at ultracold temperatures. This enhancement of the g-wave is caused by a shape resonance in the scattering of ground-state atom pairs. This shape resonance is due to the presence of a quasibound level inside the centrifugal potential barrier associated with non-zero angular momentum scattering as is shown in Fig. 5(a). This quasibound level enhances the continuum wavefunction amplitude inside the centrifugal barrier [18,35]. The population of this level depends strongly on the temperature of the system. At temperatures in the quantum degenerate regime (e.g., 1 µK), we would not expect any significant population of the quasibound level and the shape resonance should not be observable. The curves in Fig. 5(a) are generated by adding a centrifugal term to the highly accurate a 3 Σ + u potential from Strauss et al. [25]. The l=4 quasibound level was found by numerically solving for bound states with LEVEL 8.0 [26]. This rovibrational level (v ′′ =39, l=4) has a calculated resonance energy of E=+0.66 mK and a tunneling width of Γ/2π=0.1 MHz. This same triplet quasibound level has previously been observed for 85 Rb 2 by Boesten et al. [17] and a corresponding resonance has been observed in 87 Rb 2 [18,30,32,33,36]. Shape resonances have also been observed in other alkali dimers, for example, in K 2 [37,38], Li 2 [39], and NaCs [14]. In Fig. 5(b) we plot the Maxwell-Boltzmann temperature distribution at the average MOT temperature,
f (E) = E πkT exp −E kT ,(2)
showing the overlap of energies between the quasibound state and the energies of the atoms in the MOT. Many of the colliding atom pairs can tunnel through the centrifugal g-wave barrier barrier and populate the quasibound state. Since our calculated parameters (energy, width, and partial wave) of this triplet-state shape resonance in 85 Rb 2 match experimental values [17] so closely, we have extended the calculation to other combinations of scattering states and isotopologues of Rb 2 as shown in Table II. It is interesting to note that the singlet and triplet states of 85 Rb 2 both have g-wave shape resonances, albeit with different energies and widths. Similarly, the singlet and triplet states of 87 Rb 2 both have a d-wave shape resonance. This occurs because the last few vibrational levels of the singlet and triplet states are nearly degenerate [25], making the singlet and triplet quasibound levels also nearly degenerate.
Unusually high rotational levels, due to mechanisms other than a shape resonance, have been observed by other groups and explained in terms of attraction caused by the trapping laser [40], and attraction caused by dipole trapping from a highly focused PA beam [41]. We do not see evidence for these effects in our work, which as we describe, in this section and the following, is fully accounted for by the shape resonance.
B. Lineshapes
Another manifestation of shape resonances is through the photoassociation lineshape [31]. In Fig. 6 we compare the lineshapes arising from s-wave and g-wave collisions, i.e., the J ′ =1 and the J ′ =3 lines. The J ′ =1 lineshape shown in Fig. 6 (a) is distinctly asymmetric, with a tail on the red side. The asymmetry is due to the high energy tail in the energy distribution of atoms in the MOT, as shown in Fig. 5(b). These lineshapes were modeled with the "Wigner law" lineshape given by Jones et al. [34] (Eq. 3). Although the fit is good, the resulting fitting parameters are not physical in the case of Rb 2 at a temperature of 120 µK. This is due to a partial breakdown of the lineshape model, also discussed in Ref. [34]. For example, the temperature extracted from the fit is two to three times higher than 120±20 µK measured by ballistic expansion. The lineshapes of J ′ =3 lines are in contrast almost symmetric, implying that only a narrow range of collisional energies participates in the photoassociation process. Furthermore the linewidths for the J ′ =3 lines are generally narrower than for the J ′ =1 lines, again due a narrower range of collisional energies participating in the photoassociation. The ground-state g-wave collision in the MOT populates the quasibound level, making the photoassociation more resemble a bound-bound transition than a free-bound transition. Since there is negligible thermal broadening, one can extract the excited state natural linewidths by fitting a simple Lorentizian lineshape.
V. CONCLUSION
We have observed photoassociation to thirteen vibrational levels of the 2 1 Σ + g state in 85 Rb 2 . Most of these levels lie above the corresponding 5s 1/2 + 5p 1/2 asymptote. These levels spontaneously decay predominantly to the lowest triplet a 3 Σ + u state even though the transition is forbidden by spin selection rules. The presence of a shape resonance causes strong transitions to rotational levels J ′ =3 and 5, and also influences their lineshapes.
VI. ACKNOWLEDGEMENTS
We thank O. Dulieu and A. Gerdes for providing us with unpublished potential curves, and I. Simbotin for useful discussions. We gratefully acknowledge funding from NSF (PHY-0855613) and AFOSR MURI (FA 9550-09-1-0588) Lines are offset vertically for clarity and horizontally to match the peak position. The small deviation between experimental and fitted lineshape in (b) is not physical and is due to the PA laser scan rate.
FIG. 1 .
1(Color online) Potential energy curves with molecule formation and detection pathways.
online) Some spontaneous emission pathways in Hund's case (a) and Hund's case (c) basis sets. Solid arrows denote allowed transitions and dashed arrows denote forbidden transitions according to one-photon E1 selection rules. Some transitions that are forbidden in Hund's case (a) are allowed in Hund's case (c).
FIG
. 3. (Color online) Potential energy curves
FIG. 4 .
4PA spectrum of the 2 1 Σ + g v ′ =109 level showing the rotational assignment (J ′ ) and the possible partial wave (l) content of each rotational line. The line intensities reveal the presence of partial waves l = 0, 1, 2, 4 and exclude the presence of partial waves l = 3, 5, 6. Lines marked with an asterisk (*) are hyperfine ghost lines of the J ′ =3 and 5 transitions as described in the text.
FIG
. 5. (Color online) (a) Close-up of the a 3 Σ + u effective potential curves of 85 Rb2 around the region of the centrifugal barriers for s, p, d, f , and g-partial waves. The g-wave quasibound level responsible for the shape resonance is also shown. (b) The Maxwell-Boltzmann energy distribution of a MOT at 120 µK showing the high energy tail extending to, and past, the quasibound level.
FIG. 6 .
6(Color online) Experimental (•) and fitted (-) lineshapes for J ′ =1 (a) and J ′ =3 (b) at different PA powers.
TABLE II .
IICalculated and previously determined values for the lowest shape resonance in the ground and lowest triplet states of 85 Rb2 and 87 Rb2. The calculated values are obtained using the potentials of Ref.[25].Molecule State
Calculated
Previous work
Partial wave, Resonance Tunneling
Partial wave,
Resonance Tunneling Ref.
l
energy
width/2π
l
energy
width/2π
(mK)
(MHz)
(mK)
(MHz)
85 Rb2
a 3 Σ +
u
4
0.66
0.1
4
0.6-0.8
0.04-0.16 [17]
85 Rb2
X 1 Σ +
g
4
0.28
0.002
87 Rb2
a 3 Σ +
u
2
0.25
4.5
2
1.5-7.5
[30]
2
0.28
[31]
2
0.312(25)
2.4(0.5) [32]
2
0.300(70)
3
[33]
87 Rb2
X 1 Σ +
g
2
0.33
5.7
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"Software systems for operation, control, and monitoring of the EBEX instrument",
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"Michael Milligan \nUniversity of Minnesota School of Physics and Astronomy\n55455MinneapolisMN\n",
"Peter Ade \nCardiff University\nCF24 3AACardiffUnited Kingdom\n",
"François Aubin \nMcGill University\nH3A 2T8MontréalQuebecCanada\n",
"Carlo Baccigalupi \nScuola Internazionale Superiore di Studi Avanzati\n34151TriesteItaly\n",
"Chaoyun Bao \nUniversity of Minnesota School of Physics and Astronomy\n55455MinneapolisMN\n",
"Julian Borrill \nLawrence Berkeley National Laboratory\n94720BerkeleyCA\n",
"Christopher Cantalupo \nLawrence Berkeley National Laboratory\n94720BerkeleyCA\n",
"Daniel Chapman \nColumbia University\n10027New YorkNY\n",
"Joy Didier \nColumbia University\n10027New YorkNY\n",
"Matt Dobbs \nMcGill University\nH3A 2T8MontréalQuebecCanada\n",
"Will Grainger \nCardiff University\nCF24 3AACardiffUnited Kingdom\n",
"Shaul Hanany \nUniversity of Minnesota School of Physics and Astronomy\n55455MinneapolisMN\n",
"Seth Hillbrand \nColumbia University\n10027New YorkNY\n",
"Johannes Hubmayr \nNational Institute of Standards and Technology\n80303BoulderCO\n",
"Peter Hyland \nMcGill University\nH3A 2T8MontréalQuebecCanada\n",
"Andrew Jaffe \nImperial College\nSW7 2AZLondonEngland, United Kingdom\n",
"Bradley Johnson \nUniversity of California\n94720Berkeley, BerkeleyCA\n",
"Theodore Kisner \nLawrence Berkeley National Laboratory\n94720BerkeleyCA\n",
"Jeff Klein \nUniversity of Minnesota School of Physics and Astronomy\n55455MinneapolisMN\n",
"Andrei Korotkov \nBrown University\n02912ProvidenceRI\n",
"Sam Leach \nScuola Internazionale Superiore di Studi Avanzati\n34151TriesteItaly\n",
"Adrian Lee \nUniversity of California\n94720Berkeley, BerkeleyCA\n",
"Lorne Levinson \nWeizmann Institute of Science\n76100RehovotIsrael\n",
"Michele Limon \nColumbia University\n10027New YorkNY\n",
"Kevin Macdermid \nMcGill University\nH3A 2T8MontréalQuebecCanada\n",
"Tomotake Matsumura \nCalifornia Institute of Technology\n91125PasadenaCA\n",
"Amber Miller \nColumbia University\n10027New YorkNY\n",
"Enzo Pascale \nCardiff University\nCF24 3AACardiffUnited Kingdom\n",
"Daniel Polsgrove \nUniversity of Minnesota School of Physics and Astronomy\n55455MinneapolisMN\n",
"Nicolas Ponthieu \nInstitut d'Astrophysique Spatiale\nUniversite Paris-Sud\n91405OrsayFrance\n",
"Kate Raach \nUniversity of Minnesota School of Physics and Astronomy\n55455MinneapolisMN\n",
"Britt Reichborn-Kjennerud \nColumbia University\n10027New YorkNY\n",
"Ilan Sagiv \nUniversity of Minnesota School of Physics and Astronomy\n55455MinneapolisMN\n",
"Huan Tran \nUniversity of California\n94720Berkeley, BerkeleyCA\n",
"Gregory S Tucker \nBrown University\n02912ProvidenceRI\n",
"Yury Vinokurov \nBrown University\n02912ProvidenceRI\n",
"Amit Yadav \nInstitute for Advanced Study\n08540PrincetonNJ\n",
"Matias Zaldarriaga \nInstitute for Advanced Study\n08540PrincetonNJ\n",
"Kyle Zilic \nUniversity of Minnesota School of Physics and Astronomy\n55455MinneapolisMN\n"
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We present the hardware and software systems implementing autonomous operation, distributed real-time monitoring, and control for the EBEX instrument. EBEX is a NASA-funded balloon-borne microwave polarimeter designed for a 14 day Antarctic flight that circumnavigates the pole.To meet its science goals the EBEX instrument autonomously executes several tasks in parallel: it collects attitude data and maintains pointing control in order to adhere to an observing schedule; tunes and operates up to 1920 TES bolometers and 120 SQUID amplifiers controlled by as many as 30 embedded computers; coordinates and dispatches jobs across an onboard computer network to manage this detector readout system; logs over 3 GiB/hour of science and housekeeping data to an onboard disk storage array; responds to a variety of commands and exogenous events; and downlinks multiple heterogeneous data streams representing a selected subset of the total logged data. Most of the systems implementing these functions have been tested during a recent engineering flight of the payload, and have proven to meet the target requirements.The EBEX ground segment couples uplink and downlink hardware to a client-server software stack, enabling real-time monitoring and command responsibility to be distributed across the public internet or other standard computer networks. Using the emerging dirfile standard as a uniform intermediate data format, a variety of front end programs provide access to different components and views of the downlinked data products. This distributed architecture was demonstrated operating across multiple widely dispersed sites prior to and during the EBEX engineering flight.
|
10.1117/12.857583
|
[
"https://arxiv.org/pdf/1006.5256v2.pdf"
] | 27,886,645 |
1006.5256
|
7d8dafca703afa17b3be06b3924f9c993f9ee6a2
|
Software systems for operation, control, and monitoring of the EBEX instrument
5 Jul 2010
Michael Milligan
University of Minnesota School of Physics and Astronomy
55455MinneapolisMN
Peter Ade
Cardiff University
CF24 3AACardiffUnited Kingdom
François Aubin
McGill University
H3A 2T8MontréalQuebecCanada
Carlo Baccigalupi
Scuola Internazionale Superiore di Studi Avanzati
34151TriesteItaly
Chaoyun Bao
University of Minnesota School of Physics and Astronomy
55455MinneapolisMN
Julian Borrill
Lawrence Berkeley National Laboratory
94720BerkeleyCA
Christopher Cantalupo
Lawrence Berkeley National Laboratory
94720BerkeleyCA
Daniel Chapman
Columbia University
10027New YorkNY
Joy Didier
Columbia University
10027New YorkNY
Matt Dobbs
McGill University
H3A 2T8MontréalQuebecCanada
Will Grainger
Cardiff University
CF24 3AACardiffUnited Kingdom
Shaul Hanany
University of Minnesota School of Physics and Astronomy
55455MinneapolisMN
Seth Hillbrand
Columbia University
10027New YorkNY
Johannes Hubmayr
National Institute of Standards and Technology
80303BoulderCO
Peter Hyland
McGill University
H3A 2T8MontréalQuebecCanada
Andrew Jaffe
Imperial College
SW7 2AZLondonEngland, United Kingdom
Bradley Johnson
University of California
94720Berkeley, BerkeleyCA
Theodore Kisner
Lawrence Berkeley National Laboratory
94720BerkeleyCA
Jeff Klein
University of Minnesota School of Physics and Astronomy
55455MinneapolisMN
Andrei Korotkov
Brown University
02912ProvidenceRI
Sam Leach
Scuola Internazionale Superiore di Studi Avanzati
34151TriesteItaly
Adrian Lee
University of California
94720Berkeley, BerkeleyCA
Lorne Levinson
Weizmann Institute of Science
76100RehovotIsrael
Michele Limon
Columbia University
10027New YorkNY
Kevin Macdermid
McGill University
H3A 2T8MontréalQuebecCanada
Tomotake Matsumura
California Institute of Technology
91125PasadenaCA
Amber Miller
Columbia University
10027New YorkNY
Enzo Pascale
Cardiff University
CF24 3AACardiffUnited Kingdom
Daniel Polsgrove
University of Minnesota School of Physics and Astronomy
55455MinneapolisMN
Nicolas Ponthieu
Institut d'Astrophysique Spatiale
Universite Paris-Sud
91405OrsayFrance
Kate Raach
University of Minnesota School of Physics and Astronomy
55455MinneapolisMN
Britt Reichborn-Kjennerud
Columbia University
10027New YorkNY
Ilan Sagiv
University of Minnesota School of Physics and Astronomy
55455MinneapolisMN
Huan Tran
University of California
94720Berkeley, BerkeleyCA
Gregory S Tucker
Brown University
02912ProvidenceRI
Yury Vinokurov
Brown University
02912ProvidenceRI
Amit Yadav
Institute for Advanced Study
08540PrincetonNJ
Matias Zaldarriaga
Institute for Advanced Study
08540PrincetonNJ
Kyle Zilic
University of Minnesota School of Physics and Astronomy
55455MinneapolisMN
Software systems for operation, control, and monitoring of the EBEX instrument
5 Jul 2010CMBmillimeter-wave telescopesflight control systemsballooningdata handling
We present the hardware and software systems implementing autonomous operation, distributed real-time monitoring, and control for the EBEX instrument. EBEX is a NASA-funded balloon-borne microwave polarimeter designed for a 14 day Antarctic flight that circumnavigates the pole.To meet its science goals the EBEX instrument autonomously executes several tasks in parallel: it collects attitude data and maintains pointing control in order to adhere to an observing schedule; tunes and operates up to 1920 TES bolometers and 120 SQUID amplifiers controlled by as many as 30 embedded computers; coordinates and dispatches jobs across an onboard computer network to manage this detector readout system; logs over 3 GiB/hour of science and housekeeping data to an onboard disk storage array; responds to a variety of commands and exogenous events; and downlinks multiple heterogeneous data streams representing a selected subset of the total logged data. Most of the systems implementing these functions have been tested during a recent engineering flight of the payload, and have proven to meet the target requirements.The EBEX ground segment couples uplink and downlink hardware to a client-server software stack, enabling real-time monitoring and command responsibility to be distributed across the public internet or other standard computer networks. Using the emerging dirfile standard as a uniform intermediate data format, a variety of front end programs provide access to different components and views of the downlinked data products. This distributed architecture was demonstrated operating across multiple widely dispersed sites prior to and during the EBEX engineering flight.
INTRODUCTION
Science goals
The E and B EXperiment (EBEX) is balloon-borne microwave polarimeter designed to study the polarization of the cosmic microwave background (CMB) and the foreground emission of thermal dust in our galaxy. 1, 2 These measurements will: detect or constrain the primordial B-mode polarization of the CMB, a predicted signature of gravity waves produced by cosmic inflation; 3, 4 characterize the polarized foreground dust emission, which is a necessary step in determining the CMB B-mode signal; 5,6 and measure the predicted effect of gravitational lensing on the CMB. 7 The science goals of EBEX are described more fully in other publications. 1, 2, 8
Instrument description
The EBEX instrument consists of a 1.5 meter clear aperture Gregorian-type telescope that feeds a cryogenic receiver, all of which are mounted on the inner frame of the EBEX gondola. Pointing control is maintained by driving the inner frame in elevation, while a pivot and reaction wheel turn the outer frame azimuthally relative to the balloon flight line. Attitude sensors including a sun sensor, star cameras, differential GPS, gyroscopes, magnetometer, and clinometers are mounted as appropriate on the inner and outer frames. The flight computers, Attitude Control System (ACS) crate, and disk storage pressure vessels are mounted on the outer frame. Inside the cryostat reimaging optics focus the input radiation onto two focal planes each carrying up to 960 transition edge sensor (TES) bolometers, up to 1920 total bolometers. A polarimetric system, consisting of a half wave plate (HWP) 9 spinning on a superconducting magnetic bearing 10 and a wire grid, modulates polarization information into the phase and amplitude of the component of the radiation intensity at the focal plane corresponding to four times the HWP rotation frequency. 11 The TES are read out through SQUID amplifiers via a frequency domain multiplexing scheme that connects up to 16 TES to each SQUID. The SQUIDs in turn are connected in groups of four to digital frequency-domain multiplexing readout (DfMux) boards. 12 The design of the EBEX instrument is detailed elsewhere, 2,8,13 and the bolometer readout system is described in Hubmayr et al 14 and Aubin et al. 15 EBEX completed a 13 hour engineering flight from Ft. Sumner, New Mexico in June 2009. In this paper we describe the software and data flow architecture that make up the EBEX control and monitoring systems.
Computing and system control overview
In order to meet the science goals, EBEX autonomously executes several tasks in parallel.
The instrument maintains real-time pointing control to better than the 0.5 • requirement and logs sufficient data from the pointing sensors to allow post-flight pointing reconstruction to better than the 9 requirement. The pointing system can realize several predefined instrument scan modes, as well as drift, slew, and coordinate tracking motions. The two redundant flight computers (see Sec. 2.1) execute all pointing actions synchronously, with a watchdog card selecting the less-recently rebooted computer to control the instrument. The pointing system is discussed in detail by Reichborn-Kjennerud. 16 Both SQUIDs and TES bolometers periodically require active tuning, such as during cycling of the sub-Kelvin adsorption refrigerators. 17 This instrument reads out up to 1792 of the 1920 bolometers, multiplexed through 112 SQUIDs, operated by 28 DfMux boards. These setup and tuning operations are managed over the gondola Ethernet network by the flight computers, as discussed in Sec. 2.2.
Bolometers are read out at 190.73 Hz 16-bit samples. Depending on the multiplexing level each DfMux board reads out between 32 and 64 bolometers, producing a data stream of between 21 and 42 kilobytes/s, or 2.1 to 4.2 gigabytes per hour for the full complement of 28 boards. The ACS generates an additional data stream of approximately 20 KB/s (70 megabytes per hour), and the angular encoders on the rotating HWP produce a combined 21 KB/s (75 MB/h). This output data is transferred over the ethernet network to the flight computer and logged to disk. Consequently for a 14 day flight the onboard disk array must provide over 1.5 terabytes total storage per redundant copy written. The storage system is discussed in Sec. 2.3.
In addition to planned housekeeping operations, the possibility of unplanned events demands that EBEX possess the ability to respond to some exogenous contingencies, that sufficient operational data be downlinked to enable human diagnosis of unexpected conditions, and that the telecommanding interface be flexible enough to exercise the full range of recovery options available in the flying hardware. The necessary downlink (Sec. 2.4) is provided by a 1 Mbit/s line-of-sight (LOS) transmitter available for roughly the first day of flight, and a much slower TDRSS satellite relay afterwards. The telecommanding uplink relies on satellite relay or an HF-band LOS transmission, and in practice is limited to less than ten 15-bit command tokens per second.
All of the above activities can be triggered from the ground via uplinked commands, as well as scheduled via onboard schedule files. The scheduling system operates in local sidereal time, allowing planned observations to account for the motion of the balloon in longitude, which cannot be precisely known in advance. Within the limits of the underlying operating system, actions can be scheduled arbitrarily far in the future. Uplinked commands can select between alternative stored schedules.
The communications infrastructure of the Columbia Scientific Balloon Facility (CSBF) provides the LOS downlink signal at the launch site, and provides connections to satellite-based telemetry and telecommanding via the Operations Control Center in Palestine, Texas. 18 During a long duration balloon flight, many collaboration personnel will be positioned at the launch site, while other collaborators may be geographically dispersed. To support this scenario the EBEX ground segment couples uplink and downlink hardware to a client-server software stack (see Sec. 2.5 and Fig. 2). The full high rate LOS data stream is available at multiple client workstations at the launch site, and portions of this data can be made available via the public internet for remote real-time examination. Likewise telecommanding is forwarded over network links to the EBEX ground station and CSBF uplink.
To meet the reliability and development time requirements of this project we use commercially available hardware and existing software whenever practical. With the exception of the FPGA-based DfMux and ACS boards, onboard computers and networking hardware are available industrial embedded models which we have qualified in thermal and vacuum conditions approximating balloon flight. The ACS, many aspects of the gondola and pointing system design, and several components of the software chain described here originate with the BLAST project, 19 and are described by Wiebe. 20 The housekeeping system makes extensive use of embedded monitoring boards 21 originally developed for the ATLAS experiment at CERN. 22, 23
SYSTEMS
The EBEX gondola comprises several subsystems of networked components, with the flight computer crate acting as the point of intersection.
An Ethernet network of industrial ring switches 24 connects the flight computers, disk storage system, bolometer readout boards, HWP encoder readouts, sun sensor, and star camera. This network is shown in Fig. 1. The use of ring switches provides resilience to network breaks or failure of a single switch. Optical fiber connections are used where electrical isolation is necessary.
The GPS receiver, multiple actuators, and the CSBF support package (which includes the commanding uplink and low rate satellite telemetry) communicate directly with the flight computers via serial ports. Additional sensors and controls connect directly to hardware in the ACS crate. The ACS communicates with the flight computers via a custom bidirectional bus termed the "E-bus." 16,20 Housekeeping monitoring and control is handled by custom boards equipped with embedded monitoring boards, 23 which are connected by a Controller Area Network 25 bus (CANbus). The flight computers communicate with this network via Kvaser USB-CANbus adapters. 26 Because the housekeeping system, ACS, and bolometer readouts are asynchronous, all systems embed in their data streams a common timestamp using EBEX "ticks" which is recorded for post-flight alignment. The systems maintain a relative synchronization of ∼ 10 µs by resynchronizing to an onboard precision clock every 164 ms. The time servers broadcast synchronization messages onto the CANbus, and distribute timing data to the DfMux boards and ACS via an RS-485 serial link that does not connect to the flight computers. The housekeeping and timing subsystems are described in Sagiv et al. 13 EBEX ethernet network Uses ring switches to make self-healing rings: Withstands failure of one connection in each ring; Loss of a switch: loses only its clients. 7
Flight control program -fcp
The flight computer crate contains two Ampro single board computers, 27 each configured with a 1.0 GHz Celeron processor, 256 MiB RAM, and a 1 GB solid state flash disk module. The module stores the computer operating system, currently Debian GNU/Linux 4.0 28 with Linux kernel 2.6.18 and additional modular drivers for the ACS E-bus and USB-CANbus adapter. The flight control program fcp resides on the flash module as well, which the operating system is configured to run immediately after the computer boots.
fcp is a derivative of the BLAST experiment's mcp, 20 and preserves its overall architecture as a monolithic program running multiple concurrent, event-driven threads, with a main loop handling pointing, frame generation, and data logging clocked to the E-bus. We have added code modules implementing control and readout of the DfMux boards, housekeeping via the CANbus, storage to the networked disk storage array, and the downlink scheme discussed below. Other modules have been modified as needed.
Flight computer redundancy is implemented via a watchdog card connected to the IEEE 1284 parallel port of each computer. In nominal operation the fcp WatchDog thread toggles a pin on the parallel port at 25 Hz. If this action ceases for more than a configurable length of time, a fault is inferred. The watchdog card will power cycle the faulty computer and switch control to the other computer. Besides crashes in the software or hardware of the flight computer, fcp can programmatically trigger this sequence of events by terminating the WatchDog thread in response to certain error conditions. The identity of the computer in control is communicated to both flight computers via the E-bus, and recorded as the incharge variable. During the North American engineering flight dataset the value of this variable changes only once, at 8:19 UTC, due to an intentional pre-flight reboot of the then-active flight computer. This indicates that there were no such reboots of the in-charge flight computer between launch at 14:01 UTC and termination after 03:18 UTC.
Distributed networked bolometer readout architecture
Each DfMux readout board combines analog signal processing hardware with an FPGA implementing digital signal processing modules and a soft CPU running an embedded Linux distribution. described in detail by Dobbs et al. 12 Operations comprising the setup, tuning, and maintenance of the detectors and readout system are controlled by the flight computer via requests over the Ethernet network, and readout data are returned over the same network.
Low level operations are exposed via small programs in the DfMux firmware implementing the Common Gateway Interface. 29 More complex algorithms are invoked as jobs through an interface called "Algorithm Manager," which passes data using JavaScript Object Notation. 30 On each DfMux board a program, implemented by code in a subset of the Python language, 31 listens on a network port for requests to start, stop, or collect the output of jobs. Because of memory and CPU constraints in the embedded environment, no more than two jobs may run at a time on each board. In fcp the algMan module maintains queues of pending and running jobs and attempts to run all requested jobs as soon as possible, while ensuring that on a per-board basis all jobs are run in the order requested. To the rest of fcp, algMan exposes routines to trigger algorithm requests to a single board. It also provides a higher level interface based on stored parameter files. In these files sets of algorithm parameters are defined on a per-SQUID basis. After commanding fcp to parse one of the stored files, algMan will respond to these high-level commands by dispatching algorithm requests for the corresponding operation for each SQUID defined in the parameter file.
Regardless of the method of invocation, requested operations will produce output strings in the JavaScript Object Notation format which are returned to algMan. These strings, generically termed "algorithm results," are logged to disk and added to the file downlink system queue.
DfMux boards output data samples by broadcasting User Datagram Protocol 32 packets to a multicast address over the Ethernet network. Each packet is 1428 bytes and consists of a header and 13 frame structures. In the case of the bolometer readout boards in the configuration flown in the 2009 engineering flight, with 8 bolometer per SQUID multiplexing (32 total bolometer channels per board) these frames contain a timestamp and one 16-bit sample for each of the 32 channels recorded at the corresponding time. For a 190.73 Hz sample rate each board broadcasts packets at 14.67 Hz. Within each bolometer readout crate, the DfMux boards are synchronized to a common 25 MHz oscillator so that the bolometers for all boards in the crate are sampled at the same time.
In fcp the UDPS Listener packet reader thread listens on the multicast address. Each packet is inspected to determine its origin, and the pdump module writes it to disk in a packet dump (.pdump) file corresponding to the originating board. The .pdump files are rotated every 15 minutes to limit the maximum file size produced. Fig. 3 demonstrates the performance of this readout system for a typical readout board. Excluding a brief period around 17:35 UTC when the boards were commanded to reboot during a SQUID tuning procedure, no board is missing more than 65 packets from the logged packet data, for a loss rate of < 0.01%. Testing on the ground shows that under simulated load equivalent to the full planned complement of 28 boards, loss rates remain similarly low. 11 of the 12 bolometer readout boards were synchronized to the common oscillators in their respective crates for the entire flight. The twelfth board was left unsynchronized due to a misconfigured startup script.
Two DfMux boards are also used to read the optical angular encoder on the HWP. They each sample a single channel at 3.052 KHz. Each HWP encoder packet contains 416 samples, and thus each board broadcasts packets at 7.34 Hz. The structure of the bolometer readout packets is reused for the HWP encoder readout, so the same code processes both types of packet stream.
The code defining the packet format is written in portable C that is compiled into the packet streamer program onboard the DfMux CPU, UDPS Listener, and the standalone parser program used to extract data from packet streams and saved dumps.
ATAoE onboard storage
EBEX will fly with over 3 terabytes of hard disk storage. This allows the flight computers to write two redundant copies of all data produced in flight to separate disks. We use the ATA over Ethernet (ATAoE) protocol 33 in order to implement the onboard disk storage array. Ethernet has several attractive features. It provides a manyto-many topology so that redundant disks can be provided without foreknowledge of which flight computer will need one. It is physically straightforward to route signals from the flight computer crate in vacuum into the pressure vessels holding hard disks. Finally, Ethernet is already in use onboard so it avoids adding an additional networking technology. Drivers for the ATAoE protocol are a standard part of the Linux kernel.
As shown in Fig. 1, the disk drives are divided between two pressure vessels. Each vessel contains a ring switch, a passive backplane for power and signal distribution, and up to seven 2.5" laptop disk drives mounted on ATAoE blades. 34 Each blade is connected independently to the ethernet ring switch. In fcp the EBEX AOE module abstracts detection, setup and low-level management of the array. Disk usage is flagged in non-volatile memory present on each blade to ensure that the two flight computers do not attempt to simultaneously mount the same disk. This module will only present as available disks which are not already in use and which have sufficient free space remaining. The aoeMan module adds an additional layer of abstraction, allowing fcp code to request file operations without any detailed knowledge of the disk array. It mounts disks as needed to supply the requested free space, and translates filenames to correspond with the correct mount points in the filesystem namespace.
Downlink and data logging
fcp produces a 1 Mbit/s biphase encoded output data stream, suitable for transmission over the CSBF-provided line-of-sight downlink. This stream combines all output channels of the ACS and housekeeping systems, packet data streams from five selectable DfMux boards, and a file downlink system called filepig, used to retrieve algorithm results, diagnostic logs, and other irregularly formatted data. At the launch site the EBEX ground station uses a commercial bit synchronizer, custom decommutator card, and the decomd software to decode and store this data stream to disk.
As detailed in Fig. 4, the downlink stream is composed of 1248 byte frames generated at 100 Hz. These are grouped into superframes of 20 frames. Each frame begins with a sync word and counters, followed by channel data. Each 2-byte word of channel data can either contain samples of a "fast channel" at 100 Hz, or have 20 "slow channels" multiplexed over the superframe at 5 Hz. In the 2009 engineering flight, this channel data totalled 194 bytes per frame, encoding 59 fast channels and 480 slow channels. This channel data is also logged to disk onboard the gondola.
The remaining space in each frame (1048 bytes, after overhead, for the 2009 flight configuration) is aggregated across the superframe and used to transfer DfMux readout packets and filepig data blocks. In fcp this format is defined by the "Biphase marshaler" module, which accepts data from UDPS Listener and filepig and assembles the superframe data area. Every 200 ms the fcp downlink code queries the marshaler for an assembled data area to incorporate into the transmitted frames.
The marshaler uses fixed slots in the superframe to provision a deterministic bandwidth to each downlinked data stream, and to ensure that if one frame is lost or corrupted, data in the surrounding frames can still be correctly reassembled. UDPS Listener, described above, passes whole packets, and thus requires 1428-byte slots. In 200 ms a bolometer readout board produces on average 2.93 packets, and a HWP encoder board produces 1.47. Thus a group of three slots for bolometer readout or two slots for encoder readout yields a stream with adequate capacity to downlink the entire packet data output of a DfMux board. With 14 slots, streams are defined to downlink the output of four bolometer readout boards and one HWP encoder board. Uplinked commands select which five boards out of the total complement are allotted a downlink stream.
filepig, so named because it allows files to "piggyback" on a frame-based protocol, claims the odd-sized chunk of space at the end of the data area after packet streams have been allocated. It exposes an interface by which fcp code may queue the filenames of data objects already written to disk. Files are broken into chunks together with minimal header and error detection data and downlinked. Support exists, presently unused, to plug in transformations for more robust error correction or data compression, and to resend corrupted data in response to uplinked commands. For the engineering flight 968 bytes per superframe were left for the filepig data chunk, providing about 4.2 KB/s file downlink bandwidth. Over the 13 hour flight 10898 files totalling 61 MB were retrieved.
Ground tools and architecture
The BLAST telemetry chain 20 is employed largely unmodified on the ground. As illustrated in Fig. 2, the biphase encoded bitstream is converted back into data frames in the Ground Station computer and logged to disk. The interloquendi server permits clients to fetch streams of frames remotely via TCP/IP connections. defile then decodes the channel data in these frames into dirfile 35 -format data files. Front end programs such as palantir and KST 36 allow real-time display of the streamed channels.
To this EBEX adds support in the frame handling code for the superframe data area, and support in defile for extracting packet streams and downlinked files from those frames. These additional data products are written alongside the channel-based data on each connecting client workstation. Scripts employing the parser program automate the production of bolometer and HWP encoder time streams in dirfile format from extracted .pdump files.
Time streams can be displayed in real-time using either KST or Python tools that understand the dirfile format. The EBEX Alignment Tools is a suite of programs for further processing these streams, including interpolation and alignment to a common sample rate and timing, decoding the HWP angular encoder signal to HWP position, and template-based removal of the HWP rotation signal from bolometer timestreams.
We have also written a Python/TK front end to the Algorithm Manager system. By monitoring the names of the files downlinked through filepig, it is possible to select those corresponding to algorithm result strings. Parsing these files permits display on a board-by-board basis, in close to real time, of the job execution activity occurring in the readout system DfMux boards. A dashboard interface presents selected information from each board using labels and color coding, and the user can select individual boards or jobs for more detailed display. This front end provides immediate visual feedback on complex operations, such as detector system tuning, that entail parallel execution of a sequence of jobs on each bolometer readout board.
ebexcmd accepts fcp commands in textual format, which it can either relay to a listening ebexcmd over a network connection, or convert to the binary representation suitable for transmission over CSBF uplink hardware. Commands can therefore be generated on any host permitted to connect to the ground station, and those commands will then be uplinked. Commands are most commonly selected through the narsil front end, but are also generated by Python scripts and may even be entered manually from a command line.
This ground infrastructure provides network transparency in both data distribution and commanding, allowing flight operators to monitor and control the instrument from an arbitrary number of networked workstations. During the 2009 integration campaign and flight, this system routinely connected as many as ten client workstations over the private internal network at the New Mexico launch site. Late in the flight line-of-sight communications were only possible from the downrange station in Arizona, and commands were successfully relayed from the launch site through the downrange ground station ebexcmd. Streaming of frame data via interloquendi from the downrange station to the launch site, and from the launch site to collaborators at their home institutions, worked only intermittently due to bandwidth constraints at the launch site.
CONCLUSION
EBEX combines a large format bolometer array, and the correspondingly large data volume, with a complex readout system architecture. As a result, EBEX solves for a balloon flight environment problems in data handling, communications, and control that are typically associated with ground based observatories. The required 3 terabyte in-flight storage capacity is achieved using a high speed gondola ethernet network and networked disk storage arrays. The readout system is controlled from a central flight computer using a custom distributed job control scheduler, and it is monitored by extending a frame-oriented telemetry system to support asynchronous packet streams and event-driven downlink of arbitrary data in files. On the ground, a networked real-time data distribution and command relay architecture allows shared monitoring and control of the instrument.
Figure 1 .
1Configuration of the EBEX gondola ethernet network planned for the long duration Antarctic flight.
FCPFigure 2 .
2Schematic diagram of command and data flows in the EBEX flight and ground systems. Square corner boxes represent physical components, and rounded boxes generally represent software modules. The left side of the figure comprises flight systems, including the flight computer running fcp (Sec. 2.1) connected to DfMux boards in the Data Acquisition System (Sec. 2.2) and the disk storage system (Sec. 2.3). The center of the figure represents the ground station, containing the interface to CSBF downlink equipment (biphase, decom, decomd, Sec. 2.4), and the server portion of the data distribution software chain (interloquendi). Sample operator console configurations (Sec. 2.5) are shown on the right. A data monitoring terminal at top illustrates the client (defile) and display (KST, palantir ) portions of the data distribution chain. Below, a commanding station illustrates the command uplink chain via narsil and ebexcmd. The heavy dashed lines represent radio communications between the gondola on the left and ground on the right. In the interest of space data paths for satellite downlinks are omitted.
Figure 3 .
3Synchronization flags for a typical bolometer readout board during the 2009 flight. The flag values indicate: 0 -sample present and synchronized; 1 -padding at ends; 2 -missing data; 4 -wrong sample rate. The anomalous behavior around 17:35 UTC corresponds to a commanded reboot of the DfMux boards. Most of the isolated spikes to state 2 indicate single packets missing from the recorded data stream, 20 in total for this board. Otherwise for this board data samples were logged for the entire flight, and those samples were synchronized to the common oscillator.
Figure 4 .
4Schematic of the line-of-sight downlink superframe discussed in Sec. 2.4. This structure is repeated at 5 Hz over the 1 Mbit/s transmitter. The horizontal rows indicate the 20 individual 1248 byte frames, transmitted at 100 Hz. Each frame starts with 200 bytes of header and housekeeping channel data. The remaining 1048 bytes in each frame is aggregated across the superframe to form a 20960 byte data area. 14 slots of 1428 bytes each are allotted for DfMux packets and are grouped into five logical streams (denoted here by matching hatch patterns), accomodating the complete data output of four bolometer readout boards and one HWP encoder readout board. The final 968 bytes of the superframe is used by the filepig file downlink system.
The DfMux hardware isdfMUX boards
PCI card
USB
CAN
PCI card
serial I/O
serial I/O
other serial
devices
network
Ethernet
Gondola
bitsync
decom
LINUX kernel
decomd
hard disk
storage
interloquendi
CSBF
GSE
Operator Console
defile
storage
disk
palantir
KST
Operator Console
launcher script
ebexcmd
narsil
channels
serial I/O
serial I/O
TDRSS /
Iridium
DAS Crates
Flight Computer
Attitude
Control
System
Ground Station Computer
ebexcmd
Gondola CANbus
network + devices
ATAoE Disk Array
CSBF Support Package
biphase
E−bus
biphase
Ethernet
Ethernet
Ethernet
Ethernet
Ethernet
Ethernet
ACKNOWLEDGMENTSEBEX is supported by NASA through grants number NNG04GC03G, NNG05GE62G, NNX08AG40G, and NNX07AP36H. Additional support comes from the National Science Foundation through grant number AST 0705134, the French Centre national de la recherche scientifique (CNRS), and the UK Science and Technology We thank the Columbia Scientific Balloon Facility for their energetic support. We gratefully acknowledge the efforts of our colleagues in the BLAST project that produced the foundation for much of this work.
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ATA over Ethernet Specification. S Hopkins, B Coile, The Brantley Coile Company, IncHopkins, S. and Coile, B., ATA over Ethernet Specification. The Brantley Coile Company, Inc. (Feb. 2009).
Coraid Inc, EtherDrive storage solutions. CORAID Inc., http://www.coraid.com, EtherDrive storage solutions.
. D V Wiebe, Dirfile Standards, Wiebe, D. V., Dirfile Standards. http://getdata.sourceforge.net.
. B Netterfield, KST. KDE Project. Netterfield, B. et al., KST. KDE Project, http://kst.kde.org/.
T Damour, R T Jantzen, R Ruffini, Proceedings of the Twelfth Marcel Grossmann Meeting on General Relativity. the Twelfth Marcel Grossmann Meeting on General RelativityWorld ScientificDamour, T., Jantzen, R. T., and Ruffini, R., eds., [Proceedings of the Twelfth Marcel Grossmann Meeting on General Relativity ], World Scientific (2010).
|
[] |
[
"Prepared for submission to JHEP H → bbj at Next-to-Next-to-Leading Order Accuracy",
"Prepared for submission to JHEP H → bbj at Next-to-Next-to-Leading Order Accuracy",
"Prepared for submission to JHEP H → bbj at Next-to-Next-to-Leading Order Accuracy",
"Prepared for submission to JHEP H → bbj at Next-to-Next-to-Leading Order Accuracy"
] |
[
"Roberto Mondini [email protected] \nDepartment of Physics\nUniversity at Buffalo\nThe State University of New York\n14260BuffaloUSA\n",
"Ciaran Williams [email protected] \nDepartment of Physics\nUniversity at Buffalo\nThe State University of New York\n14260BuffaloUSA\n",
"Roberto Mondini [email protected] \nDepartment of Physics\nUniversity at Buffalo\nThe State University of New York\n14260BuffaloUSA\n",
"Ciaran Williams [email protected] \nDepartment of Physics\nUniversity at Buffalo\nThe State University of New York\n14260BuffaloUSA\n"
] |
[
"Department of Physics\nUniversity at Buffalo\nThe State University of New York\n14260BuffaloUSA",
"Department of Physics\nUniversity at Buffalo\nThe State University of New York\n14260BuffaloUSA",
"Department of Physics\nUniversity at Buffalo\nThe State University of New York\n14260BuffaloUSA",
"Department of Physics\nUniversity at Buffalo\nThe State University of New York\n14260BuffaloUSA"
] |
[] |
We present the calculation of the decay H → bbj at next-to-next-to-leading order (NNLO) accuracy. We consider contributions in which the Higgs boson couples directly to bottom quarks, i.e. our predictions are accurate to order O(α 3 s y 2 b ). We calculate the various components needed to construct the NNLO contribution, including an independent calculation of the two-loop amplitudes. We compare our results for the two-loop amplitudes to an existing calculation finding agreement. We present additional checks on our two-loop expression using the known infrared factorization properties as the emitted gluon becomes soft or collinear. We use our results to construct a Monte Carlo implementation of H → bbj and present jet rates and differential distributions in the Higgs rest frame using the Durham jet algorithm.
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[
"https://arxiv.org/pdf/1904.08961v2.pdf"
] | 126,166,894 |
1904.08961
|
846f75277c0687fec42aa45877756a85557abb3d
|
Prepared for submission to JHEP H → bbj at Next-to-Next-to-Leading Order Accuracy
29 Apr 2019
Roberto Mondini [email protected]
Department of Physics
University at Buffalo
The State University of New York
14260BuffaloUSA
Ciaran Williams [email protected]
Department of Physics
University at Buffalo
The State University of New York
14260BuffaloUSA
Prepared for submission to JHEP H → bbj at Next-to-Next-to-Leading Order Accuracy
29 Apr 2019
We present the calculation of the decay H → bbj at next-to-next-to-leading order (NNLO) accuracy. We consider contributions in which the Higgs boson couples directly to bottom quarks, i.e. our predictions are accurate to order O(α 3 s y 2 b ). We calculate the various components needed to construct the NNLO contribution, including an independent calculation of the two-loop amplitudes. We compare our results for the two-loop amplitudes to an existing calculation finding agreement. We present additional checks on our two-loop expression using the known infrared factorization properties as the emitted gluon becomes soft or collinear. We use our results to construct a Monte Carlo implementation of H → bbj and present jet rates and differential distributions in the Higgs rest frame using the Durham jet algorithm.
Introduction
The discovery of the Higgs boson [1,2] has set a large part of the agenda in high energy physics for the foreseeable future. Of primary concern is the need to determine the properties of the Higgs boson in relation to the predictions of the Standard Model (SM). This is mainly achieved through measurements of the couplings of the Higgs boson to the other SM particles and the Higgs coupling to itself. The Higgs self-coupling is of particular interest, since it is intimately linked to the electroweak symmetry breaking potential, the form of which is still unconstrained through measurements of the Higgs mass alone (although its remaining properties are predicted in the SM). Any additional physics beyond the Standard Model (BSM) could lead to significant changes in the shape of the electroweak symmetry breaking potential, and thus lead to deviations from the SM predictions.
Measuring the properties of the Higgs boson is an ongoing task. In regards to that, the LHC has already achieved a remarkable precision with existing Run II measurements and will significantly improve upon these results over the course of the next decade. Plans are afoot for future colliders beyond the LHC (FCs) and a particularly appealing prospect regarding Higgs precision physics is the construction of a lepton collider. Due to the clean experimental conditions, future lepton colliders should be able to probe the properties of the Higgs boson down to per-mille level accuracy [3][4][5].
The Higgs boson decays predominately to bottom quark pairs (bb), and therefore a large part of the experimental program at the LHC and putative FCs consists in measuring the properties of this decay. At the LHC the H → bb process can be accessed through associated production channels pp → V H followed by a subsequent H → bb decay [6,7] or directly, by using jet substructure techniques and by looking in the high-p T H + j channel [8], where the backgrounds can be controlled to such a level as to make this measurement a possibility.
In both situations precise predictions are mandatory to ensure that theoretical calculations have a similar or smaller uncertainty than the experimental counterparts. This will become even more pressing at an FC, for which historical measurements from LEP for Z/γ * → jets already show that the level of experimental uncertainty will be very small indeed.
Given its importance for LHC physics, the study of Higgs plus multi-parton production has received significant theoretical attention over the last couple of decades. Working within the effective field theory, in which the top quark is treated as infinitely heavy, the production of a Higgs through gluon fusion is known to N 3 LO in QCD [9,10], and recently, using the method of Q T subtraction [11], differential predictions at this order have been computed [12]. In order to compute pp → H differentially at N 3 LO, pp → H + j must be available at NNLO, pp → H +2j at NLO, and pp → H +3j at LO. These computations have all been performed [13][14][15] 1 . Of particular note for this work is the calculation of pp → H +j at NNLO, which requires the analytic computation of H → 3 partons in the EFT [17]. The related process in which the Higgs boson decays to three partons via a tree-level coupling to b-quarks has been less well-studied in the literature. Attention has naturally been focused on the H → bb process which has been studied at NLO [18] and NNLO [19][20][21], and inclusively is known to O(α 4 s ) [22]. No complete NNLO prediction for H → bbj is available, although a calculation of the two-loop amplitudes has been presented [23].
The aim of this paper is twofold. Firstly, we perform an independent computation of the two-loop amplitudes for H → bbg which have been presented in the literature in Ref. [23]. Secondly, we use these results to produce a NNLO Monte Carlo code for the H → bbj process. The primary goal is to establish whether we can effectively integrate out the additional jet at NNLO. By successfully doing so, we open up the possibility of studying H → bb decay at N 3 LO. We perform this calculation in a companion paper [24].
Our paper proceeds as follows. In Section 2 we give a general overview of the calculation, while a detailed discussion of our two-loop computation is presented in Section 3. We discuss the results of our Monte Carlo implementation of H → bbj in Section 4. After drawing our conclusions, we present the full analytic results of our two-loop amplitudes in the appendix.
2 Overview of the calculation
General overview
In this paper we consider the decay of a Higgs boson to a bottom quark pair and an additional jet at NNLO in QCD. In perturbation theory up to NNLO the partial decay width is expanded as follows:
Γ NNLO H→bbj = Γ LO H→bbj + ∆Γ NLO H→bbj + ∆Γ NNLO H→bbj .
(2.1)
The above formula introduces the notation we will use in this paper: Γ X H→bbj defines the partial width at order X in perturbation theory, while ∆Γ X H→bbj defines the coefficient which enters the expansion for the first time at this order. Representative Feynman diagrams for our NNLO calculation are shown in Fig. 1. Specifically, at NNLO we need to compute twoloop amplitudes for H → bbg, one-loop amplitudes for H → bbgg and H → bbqq (including identical-quark terms H → bbbb), and tree-level amplitudes for H → bbggg, H → bbqqg, and H → bbbbg.
Radiative corrections to the H → bb decay were first studied nearly forty years ago [18], when it was shown that there are sizable differences between calculations in the "massless theory", in which the b-quark mass is dropped in the phase space and kinematics but kept in the b-quark Yukawa coupling, and in the full theory, in which the b-quark mass is retained throughout. These differences were shown to be primarily due to logarithms of the form log (m 2 b /m 2 H ). It was also discussed how these effects can be reinstated in the massless theory by running the b-quark mass in the Yukawa coupling. Using the b-quark mass evolved to the Higgs scale in the massless theory results in much smaller differences between the two theories. This is advantageous since it is theoretically convenient to work in the massless limit, due to the reduced complexity of higher-order Feynman diagrams. In the massless theory the inclusive partial width for the H → bb decay channel is known to an impressive O(α 4 s ) accuracy [22]. The form factor for H → bb at three loops is also In this paper we will therefore work in the massless theory in which the b-quark mass is dropped from the phase space and kinematics, but kept in the Yukawa coupling with the b-quark mass run to the Higgs scale. As mentioned above, a result of the massless theory assumption is that it simplifies the calculation by reducing the number of Feynman diagrams which must be included at one and two loops. We refer in particular to diagrams in which the Higgs boson couples indirectly to the b-quarks, for which example topologies are shown in Fig. 2. At O(α 3 s ) these diagrams interfere with the respective tree-level amplitudes for H → bbg and H → bbgg for the two-loop and one-loop calculations respectively. A simple helicity argument indicates that these interference terms are zero. In the H → bbg and H → bbgg tree-level amplitudes the scalar Higgs boson couples directly to the two (massless) quarks, which therefore must have identical helicity assignments (both positive or negative). On the other hand, the diagrams in which the Higgs couples implicitly to the b quarks as shown in Fig. 2 always result in the final-state bb pair coupling directly to a gluon. This vertex requires that the fermions have opposite helicities, and therefore there is no combination that allows non-zero interference terms to exist, resulting in no net contribution from these diagrams at NNLO (the H → bbgg box squared would first enter at O(α 4 s )). A slight subtlety arises when we consider the one-loop triangle diagram in which the Higgs boson couples indirectly to the bottom quarks (i.e the left diagram in Fig. 2 with no additional gluon exchanged in the loop). This diagram would self-interfere at O(α 3 s ) and is therefore not excluded from our NNLO calculation by the argument presented above. However, the trace over the fermion loop for this diagram contains five γ matrices and hence this term vanishes in the massless theory. In order for this diagram to give a nonzero contribution, the quark mass must be retained in the loop. This is the case when the loop particle is a top quark, and hence there exists a top Yukawa contribution which first enters at O(α 3 s ) in our calculation. Schematically, the perturbative expansion of the decay width Γ NNLO H→bbj in the full theory is of the form:
Γ NNLO H→bbj ∼ α s y 2 b A b + α 2 s y 2 b B b + y t y b B tb + α 3 s y 2 b C b + y 2 t C t + y t y b C tb + O(α 4 s ) ,(2.2)
where y b and y t are the bottom and top Yukawa couplings respectively. From the arguments given above it is clear that in the full theory the interference terms y t y b B tb and y t y b C tb are suppressed by the bottom-quark mass (since a helicity flip is needed to make a non-zero interference term). However, since the top Yukawa coupling is large, these mixed terms are of phenomenological relevance. Specifically, in an effective theory in which the top-quark loop is integrated out, the term y t y b B tb contributes to around 30% of the O(α 2 s ) coefficient [26]. For our theoretical setup, the mixed term B tb and C tb are exactly zero. In addition, at O(α 3 s ) the pure top contribution y 2 t C t mentioned above needs to be included. Indeed, while formally this term enters the perturbative expansion as a one-loop squared contribution, the higher-order corrections are known to be large (and well-studied in the EFT approach). This means that for a good phenomenological description higher-order terms proportional to y 2 t should be included as well. The IR properties of this piece are further complicated by the presence of collinear singularities as the bb pair becomes unresolved (in the massless theory) since this piece factors onto a different LO term (H → gg). In this paper we drop the y 2 t term for two reasons. Firstly, we are interested in the theoretical computation of the y 2 b terms (which is new), while the study of the y 2 t contribution has received significant attention in the literature through the various studies of H + j at the LHC. Secondly, we wish to use this computation to perform the N 3 LO calculation of the y 2 b terms for H → bb. We leave the inclusion of the top Yukawa contributions to a future study, while we remind the reader that these contributions should be included before a complete phenomenological study is performed.
N -jettiness slicing
In order to regulate the IR divergences present in our NNLO calculation we employ the N -jettiness slicing method [27,28]. Since there are three partons in the final state at LO we use the 3-jettiness variable τ 3 to separate our calculation into two pieces. For a parton-level event the 3-jettiness variable [29] is defined as follows:
τ 3 = j=1,m min i=1,2,3 2q i · p j Q i , (2.3)
where the index j runs over the m partons in the phase space (with momenta p j ), while q i represent the momenta of the three most energetic jets, clustered in our case with the Durham jet algorithm [30,31]. Q i are the hard scales in the process, which are typically taken to be Q i = 2E i with E i the energy of the i-th jet. We then introduce a variable τ cut 3 that separates the phase space into two regions. The region τ 3 <τ cut 3 contains all of the doubly-unresolved regions of phase space and here the partial width can be approximated with the following convolution, derived from SCET [29,32]:
Γ H→3j τ 3 < τ cut 3 ≈ 3 i=1 J i ⊗ S ⊗ H + O(τ cut 3 ) . (2.4)
In the above equation the terms J i correspond to the jet functions which describe collinear emissions, S denotes the soft function for three colored partons, and H is the processspecific hard function. The explicit expressions for the jet functions J i needed for our NNLO computation can be found in Ref. [33]. For the soft function, we use the results for the 1-jettiness soft function with arbitrary kinematics computed in Ref. [34] (see also Ref. [35]). The calculation of the hard function for this process is one of the primary aims of this paper and is discussed in Section 3. In order for the approximate form of the partial width in Eq. (2.4) to be accurate, τ cut 3 should be taken as small as possible to minimize the power corrections which vanish in the limit τ cut 3 → 0. Since any doubly-unresolved contribution resides in the region τ 3 < τ cut 3 , the region τ 3 > τ cut 3 corresponds to the NLO calculation of H → bbjj. The methods to compute one-loop expressions are by now well-established so we do not spend significant time on them here. In this section we limit ourselves to a brief description of the computation. One-loop amplitudes are computed analytically using the generalized unitarity approach [36]. Specifically, quadruple cuts are used to compute box coefficients [37], triple cuts are used to compute the triangle coefficients [38], double cuts are used to compute bubble coefficients [39], and the rational pieces are computed using d-dimensional unitarity techniques as outlined in Ref. [40]. Our calculation is checked numerically using the d-dimensional unitarity algorithm presented in Ref. [41]. The resulting expressions are rather compact, with a similar level of complexity to the H → gggg amplitudes presented in Ref. [42]. Tree-level amplitudes are computed using the BCFW recursion relations [43] and all tree-level amplitudes present in the calculation have been checked against Madgraph [44]. Finally, IR divergences in the NLO calculation are regulated using Catani-Seymour dipole subtraction [45].
Hard function for H → bbg at NNLO
In this section we describe the calculation of the hard function H of Eq. (2.4) for the process H → bbg at NNLO accuracy. We define the hard function as a perturbative series in powers of the renormalized strong coupling α s ≡ α s (µ) at the renormalization scale µ: where M ( ),ren is the MS-renormalized -loop amplitude in the notation of Ref. [46]. The calculation of M ( ),ren with = 0, 1, 2 is described in the following sections.
H = H LO + α s 2π H NLO + α s 2π 2 H NNLO + O(α 3 s ) .
Notation and kinematics
We consider the decay
H → b(p 1 )b(p 2 ) g(p 3 ) .
The Mandelstam invariants for this process are defined as
s = (p 1 + p 2 ) 2 > 0 t = (p 1 + p 3 ) 2 > 0 u = (p 2 + p 3 ) 2 > 0
and satisfy s + t + u = m 2 H with m H the mass of the Higgs boson. We also introduce the dimensionless quantities
x = s m 2 H y = t m 2 H z = u m 2 H (3.5)
which satisfy 0 < x < 1, 0 < y < 1, 0 < z < 1, and x + y + z = 1.
We follow the notation introduced in Ref. [23], in which the unrenormalized amplitude for H → bbg is written in terms of two tensor structures:
M = i α s 2π 1 2 y b m 2 H T a ij µ (p 3 ) [A 1 T µ 1 + A 2 T µ 2 ] ,(3.6)
where α s is the bare strong coupling constant, y b is the bare bottom Yukawa coupling, T a ij is the color matrix with gluon color index a and quark indices i and j, and µ (p 3 ) is the gluon polarization vector. Finally, the tensors T µ 1 and T µ 2 are defined as
T µ 1 =ū(p 1 ) p 3 γ µ v(p 2 ) T µ 2 = p µ 1 − t u p µ 2 ū(p 1 ) v(p 2 ) . (3.7)
The coefficients A m (m = 1, 2) have perturbative expansions in powers of α s :
A m = A (0) m + α s 2π A (1) m + α s 2π 2 A (2) m + O(α 3 s ) (3.8)
where the coefficients A
1 2(d − 3)tu T µ † 1 − 1 2(d − 3)st T µ † 2 P µ 2 = − 1 2(d − 3)st T µ † 1 − (d − 2)u 2(d − 3)s 2 t T µ † 2 (3.9)
to the appropriate amplitude, namely
A ( ) m = pol P ν m * ν (p 3 )M ( ) (3.10)
where M ( ) is the -loop amplitude written, for instance, as the sum of Feynman diagrams. The sum over the polarization states of the external gluon is performed as
pol µ (p 3 ) ν * (p 3 ) = −g µν + p µ 3 q ν + q µ p ν 3 q · p 3 (3.11)
where q is an auxiliary vector. In our calculation we choose q = p 1 .
Calculation
We now discuss the calculation of the coefficients A m to second order. We generate the tree-level, one-loop, and two-loop Feynman diagrams using FeynArts [47]. At tree level, by applying Eq. are written in terms of scalar one-loop and two-loop integrals respectively. We reduce them to an irreducible set of master integrals (MIs) using the programs Kira [48] and LiteRed [49]. The topologies needed to reduce all integrals appearing in the calculation are the same as those presented in Eqs. (3.2)-(3.5) of Ref. [23].
At the one-loop level, there are two master integrals, namely the bubble and the box integral. Their explicit results are presented in Appendix A of Ref. [50], where in particular the result for the box integral is given as a series in the regulator and in terms of HPLs [51] and two-dimensional HPLs (2dHPLs) [52,53].
At two loops, all required master integrals are known in the literature and can be divided into three groups: planar integrals, whose results are presented in Ref. [52], nonplanar integrals, computed in Ref. [53], and products of two one-loop integrals. As in the case of the one-loop box integral, the results for the two-loop planar and non-planar integrals are expressed as Laurent series in and in terms of HPLs and 2dHPLs. Furthermore, following the discussion in Section (3.3) of Ref. [50], we observe that in our calculation each master integral can be present in up to six kinematic configurations (i.e. with all possible permutations of the independent external momenta p 1 , p 2 , p 3 ). This means that, after substituting the explicit results of the MIs, our results for the coefficients A ( ) m initially contain HPLs with three arguments (x, y, or z) and 2dHPLs with six combinations (x, y, or z in the index vector and in the argument). In order to simplify our expressions, we can express all HPLs and 2dHPLs appearing in the calculation in terms of HPLs and 2dHPLs belonging to one unique kinematic configuration. Following Refs. [50,52,53], we choose 2dHPLs of argument y and index z and HPLs of arguments y and z as the unique set.
One way of obtaining the relations needed to convert all "spurious" HPLs to a unique set is by exploiting their integral representation and applying interchange of arguments formulae as described in Refs. [50,52]. In this work we proceed in a slightly different way, following the work on multiple polylogarithms (MPLs), of which HPLs and 2dHPLs are examples, of Ref. [54]. In Ref. [54] it is shown that MPLs form a Hopf algebra and that a coproduct on MPLs can be defined. The coproduct allows one to systematically decompose MPLs of any weight into MPLs of lower weights. Since at weight 1 it is trivial to convert HPLs and 2dHPLs of different arguments and/or indices to a unique set, we can apply the coproduct with a bottom-up approach to find relations between HPLs and 2dHPLs of different kinematic configurations at any weight. In our case we derive all the relations required to reduce HPLs and 2dHPLs of up to weight 4 to the chosen set using the coproduct method. We also use GiNaC to numerically evaluate the 2dHPLs for checking purposes.
MS-renormalized amplitudes
We now construct the MS-renormalized amplitudes M ( ),ren that are needed for the hard function computation at NNLO accuracy. Through Eq. (3.6) this is equivalent to constructing the MS-renormalized coefficients A ( ),ren m .
UV renormalization
We start by removing the UV divergences from the coefficients A ( ) m computed in the previous section. We renormalize the bare strong coupling constant and Yukawa coupling by performing the replacements
α s → α s S Z α (3.12) y b → y b Z y (3.13) with S = exp ( γ E ) (4π)
, α s ≡ α s (µ) and y b ≡ y b (µ) at the renormalization scale µ. The renormalization factors are given by
Z α = 1 + α s 2π r 1 + α s 2π 2 r 2 + O(α 3 s ) (3.14) Z y = 1 + α s 2π s 1 + α s 2π 2 s 2 + O(α 3 s ) (3.15)
with r 1 , r 2 , s 1 , s 2 explicitly defined in Appendix A. By inserting Eqs. :
A (0),UV-fin m = A (0) m (3.16) A (1),UV-fin m = S A (1) m + s 1 + r 1 2 A (0) m (3.17) A (2),UV-fin m = S 2 A (2) m + s 1 + 3r 1 2 S A (1) m + s 2 + r 1 s 1 2 + r 2 2 − r 2 1 8 A (0) m . (3.18)
IR subtraction and conversion to MS scheme
In order to obtain the hard function we remove the explicit soft and collinear divergences from the UV-renormalized coefficients. The IR structure of one-loop and two-loop QCD amplitudes is universally known [55] and can be written using Catani's subtraction operators I ( ) ( ). The finite coefficients A ( ),fin m are defined as
A (0),fin m = A (0),UV-fin m (3.19) A (1),fin m = A (1),UV-fin m − I (1) ( )A (0),UV-fin m (3.20) A (2),fin m = A (2),UV-fin m − I (1) ( )A (1),UV-fin m − I (2) ( )A (0),UV-fin m . (3.21)
The explicit expressions of the subtraction operators for H → bbg can be found in Appendix A. In Appendix B we show the complete results for the coefficients A ( ),fin m . Specifically, following the notation of Eq. m;n presented in Appendix B. Finally, following the discussion in Section (2.1) of Ref. [46], we obtain the MS-renormalized coefficients A ( ),ren m in the following way:
A (0),ren m = A (0),fin m (3.23) A (1),ren m = A (1),fin m + C 0 A (0),fin m (3.24) A (2),ren m = A (2),fin m + C 0 A (1),fin m + C 2 A (0),fin m (3.25)
where C 0 and C 2 are defined in Appendix A. By using Eqs. (3.2)-(3.4) and (3.6) we obtain the hard function at NNLO accuracy. Explicitly, the interferences are constructed as follows:
M (m),ren M (n),ren * = N LO 4yz A (m),ren 1 A (n),ren 1 * + 2x 2 y z A (m),ren 2 A (n),ren 2 * − 2xy A (m),ren 1 A (n),ren 2 * − 2xy A (m),ren 2 A (n),ren 1 * (3.26) where N LO = αs 2π y 2 b N c C F .
Comparison with existing results
We can compare our results for the coefficients A ( ),fin m up to = 2 with the existing results in the literature [23]. At tree level the agreement is trivial. Since we defined the tensors T µ 2;n to match our notation. We find complete agreement for all coefficients at one-loop level 2 and at two-loop level 3 . The agreement at two loops is explicitly shown in table 1 where we perform a numerical comparison between the two sets of results for a random phase-space point.
1; lit and A (0) 2; lit ) is A (0) 1; lit = 2i z A (0),fin 2 and A (0) 2; lit = i A (0),
Factorization properties of the two-loop amplitude
Although we established agreement between our two-loop amplitude and an existing result in the literature, both share certain similarities (namely an expansion in the same master integrals). We therefore initiate further testing of our calculation by investigating the analytic structure of our result in the limits in which one of the partons becomes unresolved. Such a check was not detailed previously. We do so by checking that our two-loop amplitude correctly reproduces the known IR factorization properties of QCD [56,57] when the external gluon becomes either soft or collinear to one of the quarks. We note that a further by-product of this check is a confirmation of the computed factorization limits of QCD for the soft [56] and collinear [57] limit.
Coefficient
Ref. [23] Our result B 3.26338843478 · 10 6 3.26338843480 · 10 6 0 6.52342650778 · 10 7 6.52342650793 · 10 7 Table 2. Numerical comparison of our two-loop results with the known soft limit for y = z = 10 −10 and µ 2 =
m 2 H
2 . An overall factor of α 3 s y 2 b has been extracted from both results.
Soft-gluon limit
In the limit of soft gluon, the momentum of the gluon vanishes, i.e. p 3 → 0 which implies that y, z → 0 simultaneously. The soft-gluon limit at two loops reads:
2 Re M (2) H→bbg M (0) * H→bbg → S (2) H→bbg = 2 Re S (0) (y, z)M (2) H→bb M (0) * H→bb + S (1) (y, z)M (1) H→bb M (0) * H→bb + S (2) (y, z)M (0) H→bb M (0) * H→bb ,(3.28)
where the relevant H → bb matrix elements and the soft currents S (0) (y, z), S (1) (y, z), S (2) (y, z) are presented in Appendix C. Using our results for the unrenormalized IR-
divergent coefficients A (2) m we construct the interference 2 Re M (2) H→bbg M (0) *
H→bbg as a series in in order to compare it with the known soft limit S
H→bbg defined above. Since the soft limit diverges as (yz) −1 , we multiply both expressions by a factor of y z. We show the obtained numerical results in table 2. The agreement between the known soft limit and our results is excellent.
Collinear limit
In the limit of the gluon becoming collinear to the outgoing quark, the invariant t vanishes which means y → 0 while z = 0. The collinear limit at two loops reads: 2.584146 · 10 6 2.584189 · 10 6 0 3.09852 · 10 7 3.09870 · 10 7 Table 3. Numerical comparison between our two-loop results and the known collinear limit for y = 10 −12 , z = 0.23 and µ 2 =
2 Re M (2) H→bbg M (0) * H→bbg → C (2) H→bbg = 2 Re C (0) (y, z)M(2)m 2 H
2 . An overall factor of α 3 s y 2 b has been extracted from both results.
+ C (1) (y, z)M (1) H→bb M (0) * H→bb + C (2) (y, z)M (0) H→bb M (0) * H→bb . (3.29)
The splitting functions C (0) (y, z), C (1) (y, z), C (2) (y, z) are given in Appendix C. We compare our result for 2 Re M
H→bbg . We multiply both expressions by a factor of y to remove the leading divergence. The numerical results are shown in table 3. We observe excellent agreement between our result and the known collinear limit.
Summary
In this section we have presented the computation of the hard function required to construct the τ 3 < τ cut 3 part of our NNLO calculation. We have compared our calculation to a similar existing result in the literature and found agreement. We have also verified that our expressions reproduce the known soft and collinear limits at this order and are therefore confident in using our results for the phenomenology presented in the subsequent sections of this paper.
Results
We have implemented the results discussed in the previous sections into a fully-flexible parton-level Monte Carlo code. Our code is based upon the existing structure of MCFM [58][59][60][61] and could be easily included in a future release of the code. Here we present phenomenological results for H → bbj . As outlined in Section 2, the b-quark mass is set to zero kinematically, but kept in the Yukawa coupling. In order to account for some of the effects of the missing b-mass terms we evolve the b-quark mass to the Higgs scale (m H = 125 GeV) using the two-loop running for NLO predictions, and three-loop running for NNLO predictions. This results in an effective b-quark mass of 2.94 GeV at NNLO (for our central scale choice µ = m H ). We also use G F = 0.116639 × 10 −4 GeV −2 and m W = 80.385 GeV. We take α s (m Z ) = 0.118 and we run the coupling at one, two, and three loops for LO, NLO, and NNLO calculations respectively. All results in this paper compute the width in units of MeV. In order to compute rates and distributions for H → bbj, a jet algorithm must be applied. In this paper we will present results using the Durham jet algorithm Figure 3. The τ cut 3 dependence of the NNLO coefficient for three different jet definitions. [30,31], which takes the variable y cut as an input variable. Starting at the parton level, the algorithm computes the following quantity for every possible pair of partons (i, j):
� ��� =��� ����� ����� ����� ����� ����� ����� ΔΓ �→� � ���� (τ � ��� ) � ��� =����� -���� -���� -���� -���� -���� -���� -���� ΔΓ �→� � ���� (τ � ��� ) � ��� =������ μ � =� � ����� ����� ����� ����� ����� ����� � -��� -��� -��� -��� -��� -��� -��� τ � ��� [���] ΔΓ �→� � ���� (τ � ��� )y ij = 2 min(E 2 i , E 2 j )(1 − cos θ ij ) Q 2 (4.1)
where E i is the energy of parton i, θ ij is the angle between partons i and j, and in our case Q = m H . If y ij < y cut the pairs are combined into a new object with momentum p i + p j . The algorithm then repeats until no further clusterings are possible and the remaining objects are classified as jets. These algorithms have been widely used at LEP to study e + e − → jets, which is the process most similar to our H → bbj calculation. Our results are presented in the Higgs rest frame. We first validate our calculation by studying the dependence of the NNLO coefficient on the unphysical slicing parameter τ cut 3 . To do so we focus on three representative clustering options corresponding to y cut = 0.1, 0.002 and 10 −4 . These choices span the various regions of interest theoretically and experimentally. The value y cut = 0.1 is within the perturbative regime, in which the higher-order corrections are expected to be small and agreement with future data should be good (assuming similarity to the NNLO calculations of e + e − → jets [62,63]). The second choice y cut = 0.002 corresponds to the region in which the threejet rate peaks. Finally, the choice y cut = 10 −4 is around the region in which the NNLO three-jet rate turns negative and becomes unphysical (the need for resummation of large y cut logarithms has set in long before this value is reached). The final choice is of particular relevance to this paper, since it corresponds to integrating the NNLO calculation with a very weak jet cut. Creating stable (and slicing-independent) results in this region allows us to test the code in phase-space configurations which correspond to two hard jets and one soft/collinear jet. Such configurations occur copiously in the calculation of H → bb at N 3 LO (where the soft jet is not required), and therefore establishing our code here is a prerequisite for this computation.
Our results for the three y cut values are presented in Fig. 3. Asymptotic behavior is established in each region, with the dependence on missing power corrections having, as expected, a notable dependence on y cut . For the larger choices the dependence on τ cut 3 is rather mild, as the result for the largest value of τ cut 3 is less than 10% different to that obtained in the asymptotic region (around τ cut 3 ≤ 0.05 GeV for y cut = 0.1 and τ cut 3 ≤ 0.01 for y cut = 0.002). The dependence on τ cut 3 for y cut = 10 −4 is greater and asymptotic behavior is found for τ cut 3 ≤ 0.005 GeV. We therefore conclude that the power corrections are under control and that our code can be used to make phenomenological predictions. We note in passing that an LHC jet would be clustered using a k T -style algorithm and a jet with around p T > 30 GeV would loosely scale like m 2 H y cut ∼ 30 GeV, so that the LHC case would look most like our results obtained when y cut ∼ 0.1. In this region we have established that the power corrections are small and under control, and therefore our code could readily be applied to LHC processes such as pp → V (H → 3j). We leave this study to future work.
In Fig. 4 we show the exclusive three-jet rate at LO, NLO, and NNLO as a function of y cut . We present results for the three-jet rate normalized to the N 3 LO H → bb inclusive rate [22]. In order to make each prediction we have set τ cut 3 = 10 −2 GeV, which is in the asymptotic region for nearly all of the phase space of interest. This choice is slightly too large for the smallest value of y cut studied as discussed in the previous paragraph. However, the error on the coefficient for this choice is around 5%, which corresponds to a phenomenologicallyacceptable ∼ 2% correction on the total fractional jet rate. Our figure can be compared to similar results obtained for e + e − → jets [62,63]. The pattern is broadly the same, with a small positive correction in the large y cut region (around 10%), which transitions to a decrease in the jet rate for y cut ≤ 0.01. The three-jet rate is maximum at around y cut = 0.002 and then turns over, becoming negative (and hence unphysical) in the region around 10 −4 . Along with the central scale choice of µ = m H we also provide predictions for jet rates obtained with renormalization scales µ = {2, 1/2} × m H . In addition to the implicit dependence in the loop integral expansion, the predictions depend on µ also through the running of α s and m b at two-and three-loop order for our NLO and NNLO predictions respectively. We observe that the inclusion of the NNLO corrections substantially improves the overall scale dependence. This is especially true in the perturbative region specified by y cut > 0.01 where we observe improvement of around a factor of two. For instance, at y cut = 0.1 the overall scale dependence of the jet rate at NNLO is {+3, −6}%, compared to {+11, −10}% for the same jet cut at NLO.
In Fig. 5 we turn our attention to differential distributions. We present the differential Figure 4. The three-jet rate at LO, NLO, and NNLO as a function of y cut for the Durham jet algorithm. The renormalization scale is set to µ = m H . Figure 5. The maximum energy of the jets (divided by the Higgs mass) for different jet-clustering options. The right-hand panel presents the ratio of the NNLO to NLO (with µ = m H ) predictions for each jet-clustering option.
������ �-��� Γ �→� � �� Γ �→� � ��� Γ �→� � ���� � � /� ⩽ μ ⩽ �� � ��� ��� ��� ��� ��� ��� Γ �→� � � /Γ �→� � _ ���� [μ=� � ] ����[μ]/���[μ=� � ] ���[μ]/���[μ=� � ] �� -� ����� ����� ����� � ��� ��� ��� ��� ��� � ��� ����������� �-��� � � /� ≤ μ ≤ �� � � ��� =����� � ��� =���� � ��� =��� ���� ���� ���� ���� ���� �� -� ����� ����� �� � � ��� /� � �Γ �→ � � ���� /�(� � ��� /� � ) � ��� =��� ����(μ)/���(μ=� � ) ��� ��� ��� ��� ����� � ��� =���� ����(μ)/���(μ=� � ) ��� ��� ��� ��� ��� ����� � ��� =����� ����(μ)/���(μ=��) ���� ���� ���� ���� ���� ��� ��� ��� � � ��� /� � �����
distribution for the energy component (rescaled by the Higgs mass) of the maximum-energy jet in three-jet events clustered with y cut = 0.2, 0.02, and 0.002. Comparing the three curves we observe that as y cut decreases new phase space opens up near what would correspond to a two-jet LO topology, which occurs around m H /2. These configurations correspond to two nearly back-to-back jets with a soft/collinear third jet. In the perturbative region of y cut = 0.2 the prediction is more physically sensible, the majority of jets having an energy close to m H /3 with the most energetic jet peaking slightly higher than this value. For the cases y cut =0.2 and 0.02 the ratio of NNLO to NLO is reasonably flat and small (between 5−10%) until E max /m H becomes large enough that there is no LO phase space configuration possible. In this region the NLO prediction is the first non-zero prediction and it is hence susceptible to large corrections at the next order. The scale variation mimics that of the total jet rate and is reasonably flat in the region in which the phase space is accessible to all of the contributing parton-level phase spaces. We have also computed differential distributions for smaller values of y cut = 2 × 10 −4 . They are not presented in Fig. 5 since, for such a small value of y cut , the differential prediction is negative over a large range of phase space. We mention these predictions here simply to note that the code can produce stable distributions with small MC uncertainties even in this region, which is relevant to the N 3 LO results obtained in our companion paper.
Conclusions
In this paper we have presented the calculation of H → bbj at NNLO. We have focused on the contributions in which the Higgs boson couples directly to bottom quarks. We have performed an independent computation of the two-loop amplitudes needed at this order and found agreement with a previous calculation in the literature. Additionally, we checked our result using the known IR factorization properties of QCD when the emitted gluon becomes soft or collinear to one of the fermions and found complete agreement with the predictions in both limits. We have presented the two-loop amplitudes for H → bbg in full in the Appendix. In order to regulate the IR divergences present at this order we used the N -jettiness slicing technique to separate the calculation into two components. In the region of small τ 3 we use SCET to construct an approximate form of the decay width. We used a computation of the 1-jettiness soft function, valid for arbitrary kinematics, coupled with the known jet functions and our computation of the hard function to construct the below-cut piece. The region τ 3 >τ cut 3 corresponds to the NLO computation of the H → bbjj process, for which we calculated all of the needed helicity amplitudes using on-shell techniques of generalized unitarity for the one-loop pieces and BCFW recursion relations for the H → bbjjj tree-level amplitudes.
We implemented our results into a Monte Carlo code, based upon the existing Njettiness slicing calculations of MCFM, and used it to produce differential distributions and jet rates for H → bbj at NNLO using the Durham jet algorithm. Our calculation neglected top quark-induced contributions, which are phenomenologically relevant. By combing our results with the available H + j EFT results we can produce predictions for H → bbj relevant for the LHC and FCs which include both top and bottom Yukawa contributions. Additionally, by performing the appropriate kinematic crossings of our results we can compute pp → H + b at NNLO for LHC kinematics. We leave these applications to future studies.
One of the main goals of this paper was to investigate whether a stable (slicingindependent) Monte Carlo code could be constructed for very small jet cuts. We have established this by presenting rates and differential distributions for a variety of values of the jet-clustering variable y cut . We are therefore able to effectively integrate out the jet at NNLO and use our results in a N 3 LO calculation. We pursue this approach in a companion paper to this article.
Note added
A previous version of this manuscript, submitted to the arXiv, incorrectly claimed a disagreement with the existing two-loop calculation presented in Ref. [23]. This was due to a miss-interpretation on our part of the results of Ref. [23]. Specifically, in two of the coefficients in Ref. [23] t and u are retained (the remaining coefficients are functions of y and z only), whereas our results are constructed in terms of y and z only. Our notation is different from Ref. [23] (t ↔ u) and the residual t and u dependence in the two coefficients was incorrectly evaluated in our initial comparison. Once this is altered accordingly our results are in perfect agreement.
A Formulae for renormalization and IR subtraction
The renormalization coefficients of Eqs. (3.14) and (3.15) are defined as
r 1 = − β 0 2 (A.1) r 2 = β 2 0 4 2 − β 1 8 (A.2) s 1 = − 3C F 2 (A.3) s 2 = 3 8 2 3C 2 F + β 0 C F − 1 8 3 2 C 2 F + 97 6 C F C A − 10 3 C F T R N f (A.4) with β 0 = 11 3 C A − 4 3 T R N f (A.5) β 1 = 34 3 C 2 A − 20 3 C A T R N f − 4C F T R N f (A.6) and T R = 1 2 , C A = N c , C F = N 2 c −1 2Nc .
The subtraction operators I ( ) ( ) for generic QCD processes can be found in Ref. [55]. For completeness, we show here the explicit expressions for the subtraction operators in CDR for the process H → bbg :
I (1) ( ) = e γ E 2 Γ(1 − ) − m 2 H µ 2 − 1 2 + 3 2 (C A − 2C F ) x − + − C A 2 − 5C A 3 + T R N f 3 y − + z − (A.7) I (2) ( ) = e − γ E Γ(1 − 2 ) Γ(1 − ) β 0 2 + K I (1) (2 ) + H (2) ( ) − 1 2 I (1) ( ) I (1) ( ) + β 0 (A.8) where K = 67 18 − ζ 2 C A − 10 9 T R N f (A.9) H (2) ( ) = 1 2H (2) q + H (2) g (A.10) with H (2) q = C F T R N f − 25 216 + ζ 2 8 + C F C A 245 864 − 23ζ 2 32 + 13ζ 3 8 + C 2 F − 3 32 + 3ζ 2 4 − 3ζ 3 2 (A.11) H (2) g = C 2 A 5 48 + 11ζ 2 96 + ζ 3 8 + C A T R N f − 29 54 − ζ 2 24 + 1 4 C F T R N f + 5 27 T 2 R N 2 f . (A.12)
Finally, we present the expressions for C 0 and C 2 in Eqs. (3.24) and (3.25). The coefficient C 0 corresponds to the 0 order of the series expansion of I (1) ( ). Explicitly:
C 0 = 1 4 (C A − 2C F ) L(x) 2 − 3L(x) − ζ 2 − 1 6 T R N f [L(y) + L(z)] + C A 12 10 (L(y) + L(z)) − 3 L(y) 2 + L(z) 2 + 6ζ 2 (A.13)
where for brevity L(a) = ln − m 2 H µ 2 + ln a. The coefficient C 2 is defined as
C 2 = 1 2 C 2 0 + γ cusp 1 8 C 0 + 3ζ 2 64 Γ 0 + β 0 2 C 1 (A.14) where γ cusp 1 = C A 268 9 − 8ζ 2 − 80 9 T R N f (A.15) Γ 0 = −4 (C A + 2C F ) (A.16) C 1 = − 1 48 (C A − 2C F ) 4L(x) 3 − 18L(x) 2 + 6ζ 2 L(x) − 6ζ 3 − 9ζ 2 + 1 12 T R N f L(y) 2 + L(z) 2 + ζ 2 + C A 24 2L(y) 3 − 10L(y) 2 + 3ζ 2 L(y) + 2L(z) 3 − 10L(z) 2 + 3ζ 2 L(z) − 6ζ 3 − 10ζ 2 . (A.17)
B One-loop and two-loop coefficients for H → bbg
We present the explicit results for the coefficients A
1 = 2 √ 2 π 1 y + 1 z A (0),fin 2 = A (0) 2 = 2 √ 2 π − 2 y . (B.1)
At one loop the coefficients read:
B (1) 1;1 = 3 4N c + N F 6 − 5N c 3 (B.2) B(1)1;0 − 1 4N c − N c 4 . (B.5)(1)
At two loops:
B (2) 1;2 = 9 32N 2 c + N F 4N c − 31 16 + N 2 F 24 − 17N F N c 24 + 35N 2 c 12 (B.6) B (2) 1;1 = − 3 16N+ 378H(1, 0; z) − 432ζ 3 + 477ζ 2 − 721 (B.7) B(2)1;0 = 1 N 2 c D (2) 1;0,a + N F N c D(2)1;0,b + D(2)1;0,c + N 2 F D(2)1;0,d + N F N c D(2)1;0,e + N 2 c D (2) 1;0,f (B.8) B (2) 2;2 = B (2) 1;2 (B.9) B (2) 2;1 = B (2) 1;1 − 3 16N 2 c − N F 8N c + 11 16 − N F N c 8 + 7N 2 c 8 (B.10) B(2)2;0 = 1 N 2 c D (2) 2;0,a + N F N c D(2)
2;0,b + D where S (0) (y, z) and S (1) (y, z) have been adapted from Eqs. (12), (13), and (26) of Ref. [64], while S (2) (y, z) is taken from Eq. (11) of Ref. [56].
− 12H(1, 1 − z, 0; y) − 8H(1; z)H(1, 1 − z, z; y) + 16H(1; z)H(1 − z, 0, 0; y) − 8H(0; z)H(1 − z, 0, 1 − z; y) + 24H(1 − z, 0, 1 − z; y) − 8H(1; z)H(1 − z, 1, 0; y) − 24H(1 − z, 1, 0; y) + 24H(1 − z, 1 − z, 0; y) + 32H(1; z)H(1 − z, 1 − z, z; y) − 24H(1 − z, z, 1 − z; y) −+ 16H(1 − z, 0, 0, 1 − z; y) − 8H(1 − z, 0, 1, 0; y) + 16H(1 − z, 0, 1 − z, 0; y) − 16H(1 − z, 1, 0, 0; y) − 8H(1 − z, 1, 0, 1 − z; y) + 16H(1 − z, 1, 1, 0; y) − 8H(1 − z, 1, 1 − z, 0; y) + 16H(1 − z, 1 − z, 0, 0; y) + 32H(1 − z, 1 − z, z, 1 − z; y) − 16H(z, 0, 1, 0; y) − 16H(z, 0, 1 − z, 1 − z; y) + 16H(z, 0, z, 1 − z; y) − 16H(z, 1 − z, 0, 1 − z; y) + 16H(z, 1 − z, 1, 0; y) − 16H(z, 1 − z, 1 − z, 0; y) + 32H(z, 1 − z, z, 1 − z; y) + 16H(z, z, 0, 1 − z; y) + 16H(z, z, 1 − z, 0; y) + 32H(z, z, 1 − z, 1 − z; y) − 32H(z, z, z, 1 − z; y) − 32ζ 3 H(1; y) − 16ζ 3 H(1; z) + 16ζ 3 H(1 − z; y) − 84ζ 3 + 19 − z 3 8 x (1 − y) 2 (y + z) ζ 2 H(0; z) − 3ζ 3 + z 2 8 x (1 − y) (y + z) − ζ 2 H(0; z) + H(1, 0; z) + 3ζ 3 + ζ 2 + z 8 (1 − y) (y + z) − zH+ H(0, 1, 0; y) + ζ 2 H(1; z) + (2 − z) z 8 (1 − z) (y + z) 2 − ζ 2 H(1 − z; y) − H(1 − z, 1, 0; y) − H(0, 1, 0; y) − ζ 2 H(1; z) + z (z + 2) 8 (1 − y) (y + z) 2 ζ 2 H(1 − z; y) + H(1, 0; z)H(1 − z; y) + z 2 (1 + z) 8 (1 − y) 2 (y + z) 2 ζ 2 H(1 − z; y) + H(1, 0; z)H(1 − z; y) − z 3 (1 + z) 8 x (1 − y) 2 (y + z) 2 ζ 2 H(1; z) + H(0, 1, 0; z) + H(1, 1, 0; z) − z 2 8 x (1 − y) (y + z) 2 ζ 2 H(1; z) + H(0, 1, 0; z) + H(1, 1, 0; z) (B.12) D(2)+ 31zH(0; z) − 38zH(1, 0; z) − 6zH(0, 1, 0; z) − 18zH(1, 1, 0; z) − 6H(1, 1, 0; z) − 12zζ 3 − 38zζ 2 + 6ζ 3 + 1 8 x (y + z) 2 z 2 H(0; y)H(1, 0; z)
The collinear functions in Eq. (3.29) are
C (0) (y, z) = 4πα s C F 2 n=1
Sp (0) n ( ) (C.7)
C (1) (y, z) = 1 2 1 (2π) 2 (4πα s ) 2 S C F 2 n=1
Sp (1) n ( ) Sp (0) n ( ) (C.8)
C (2) (y, z) = 1 2 1 (2π) 4 (4πα s ) 3 S 2 C F 2 n=1
Sp (2) n ( ) Sp (0) n ( ) .
s (1) ( ) = N c 2 2 1 − ∞ m=1 m Li m 1 − z −z − 1 N 2 c Li m −z 1 − z . (C.15)
The two-loop splitting functions Sp and Sp
(2),fin 2 correspond to Eqs. (4.24) and (4.25) of Ref. [57] respectively with the replacement w → z.
Figure 1 .
1Representative Feynman diagrams for the H → bbj process at NNLO.
, NLO, and NNLO coefficients of the hard function are H LO = M (0),ren M (0),ren * (3.2) H NLO = 2 Re M (1),ren M (0),ren * (3.3) H NNLO = M (1),ren M (1),ren * + 2 Re M (2),ren M (0),ren * (3.4)
≥ 1 contain UV and IR divergences which are regularized in d = 4 − 2 dimensions. At any order in α s , the coefficients A ( ) m are obtained by applying the projectors
( 3 .
310) and by carrying out the trace calculations in d dimensions we directly obtain A (0) m . At one loop and two loops, after using Eq. (3.10) the coefficients A
( 3 .
312) and (3.13) into Eqs. (3.6) and (3.8), we obtain the UV-finite coefficients A ( ),UV-fin m
and two loops we can compare the coefficients B ( ) m;n with those presented in Appendix B and C of Ref. [23]. Since the coefficients B ( ) m;n have been rescaled by the tree-level coefficients, we only need to swap B
−−
2H(0; y)H(1; z) + 4H(1; z)H(z; y) − 2H(0; z)H(1 − z; y) − 3H(1 − z; y) − 2H(0, 1 − z; y) − 2H(1 − z, 0; y) + 4H(z, 1 − z; y) + 2H(1, 0; y) − 3H(1; z) + 2H(0, 1; z) 6H(0; y)H(0; z) − 10H(0; y) − 6H(1, 0; y) − 10H(0; z)
−−
4H(1; z)H(z; y) + 2H(0, 1 − z; y) + 2H(1 − z, 0; y) − 4H(z, 1 − z; y) − 2H(1, 0; y) + 3H(1; z) − 2H(0, 1; z) − 8ζ 3 + 4ζ 2 y)H(1; z) + 756H(0; z)H(1 − z; y) + 1134H(1 − z; y) − 1512H(1; z)H(z; y) + 756H(0, 1 − z; y) + 756H(1 − z, 0; y) − 1512H(z, 1 − z; y) − 270H(0; y) − 918H(1, 0; y) − 270H(0; z) + 1134H(1; z) − 756H(0, 1; z) − 162H(1, 0; z) + 108ζ 3 − 135ζ 2 51H(0; y) − 18H(1, 0; y) − 51H(0; z) − 18H(1, 0; z) − 24ζ 2 y)H(0; z) + 630H(0; y) + 378H(1, 0; y) + 630H(0; z)
12H(0; y) + 36H(1; z) − 16H(1; z)H(1 − z; y) + 12H(1 − z; y) + 16H(0, 1; y) − 16H(0, 1 − z; y) + 16H(1, 1; y)+ 16H(1 − z, 1; y) − 32H(1 − z, 1 − z; y) + 28 − 24H(0; y)H(1; z) − 18H(1; z) − 12H(0; z)H(1 − z; y) + 18H(1; z)H(1 − z; y) − 18H(1 − z; y) + 40H(1; z)H(z; y) + 24H(z; y)H(0, 1; z) + 24H(0, 1; z) + 24H(0; z)H(0, 1 − z; y) + 24H(1; z)H(0, 1 − z; y) − 8H(0, 1; z)H(0, 1 − z; y) − 24H(0, 1 − z; y) − 24H(1; z)H(0, z; y) − 12H(0; z)H(1, 0; y) − 12H(1; z)H(1, 0; y) + 8H(0, 1; z)H(1, 0; y) + 24H(1, 0; y) + 12H(0; y)H(1, 0; z) − 24H(z; y)H(1, 0; z) − 16H(0, 1 − z; y)H(1, 0; z) − 16H(0, z; y)H(1, 0; z) + 24H(0; y)H(1, 1; z) − 48H(z; y)H(1, 1; z) + 16H(0, 0; y)H(1, 1; z) − 16H(0, z; y)H(1, 1; z) + 18H(1, 1; z)− 8H(0, 1; z)H(1, 1 − z; y) + 12H(0; z)H(1 − z, 0; y) + 24H(1; z)H(1 − z, 0; y) − 8H(0, 1; z)H(1 − z, 0; y) − 24H(1 − z, 0; y) + 24H(0; z)H(1 − z, 1 − z; y) z)H(z, 1 − z; y) + 40H(z, 1 − z; y) + 48H(1; z)H(z, z; y) − 16H(0, 1; z)H(z, z; y) + 16H(1, 0; z)H(z, z; y) + 32H(1, 1; z)H(z, z; y) − 8H(1; y)H(0, 0, 1; z) + 16H(1 − z; y)H(0, 0, 1; z) + 16H(0; z)H(0, 0, 1 − z; y) + 16H(1; z)H(0, 0, 1 − z; y) − 16H(1; z)H(0, 0, z; y) − 8H(1; z)H(0, 1, 0; y) + 12H(0, 1, 0; y) + 8H(1 − z; y)H(0, 1, 0; z) − 16H(0; y)H(0, 1, 1; z) + 16H(z; y)H(0, 1, 1; z) − 24H(0, 1, 1; z) + 16H(1; z)H(0, 1 − z, 0; y) + 24H(0, 1 − z, 1 − z; y) − 8H(1; z)H(0, 1 − z, z; y) − 16H(0; z)H(0, z, 1 − z; y) − 16H(1; z)H(0, z, 1 − z; y) − 24H(0, z, 1 − z; y) + 16H(1; z)H(0, z, z; y) − 16H(1; z)H(1, 0, 0; y) − 8H(0; y)H(1, 0, 1; z) − 8H(1; y)H(1, 0, 1; z) + 24H(1 − z; y)H(1, 0, 1; z) + 16H(z; y)H(1, 0, 1; z) − 12H(1, 0, 1; z) − 12H(1, 0, 1 − z; y) + 8H(1; z)H(1, 0, z; y) + 8H(1 − z; y)H(1, 1, 0; z) + 12H(1, 1, 0; z)
−
16H(0; z)H(z, 0, 1 − z; y) − 16H(1; z)H(z, 0, 1 − z; y) − 24H(z, 0, 1 − z; y) + 16H(1; z)H(z, 0, z; y) − 16H(1; z)H(z, 1 − z, 0; y) − 24H(z, 1 − z, 0; y) − 16H(0; z)H(z, 1 − z, 1 − z; y) − 48H(z, 1 − z, 1 − z; y) + 32H(1; z)H(z, 1 − z, z; y) + 16H(1; z)H(z, z, 0; y)+ 16H(0; z)H(z, z, 1 − z; y) + 32H(1; z)H(z, z, 1 − z; y) + 48H(z, z, 1 − z; y) − 32H(1; z)H(z, z, z; y) + 16H(0, 0, 1, 1; z) + 16H(0, 0, 1 − z, 1 − z; y)− 16H(0, 0, z, 1 − z; y) + 16H(0, 1, 0, 1; z) − 8H(0, 1, 0, 1 − z; y) + 16H(0, 1, 1, 0; y) + 16H(0, 1, 1, 0; z) − 8H(0, 1, 1 − z, 0; y)+ 16H(0, 1 − z, 0, 1 − z; y) − 8H(0, 1 − z, 1, 0; y) + 16H(0, 1 − z, 1 − z, 0; y) − 8H(0, 1 − z, z, 1 − z; y) − 16H(0, z, 1 − z, 1 − z; y) + 16H(0, z,z, 1 − z; y) + 8H(1, 0, 0, 1; z) − 16H(1, 0, 0, 1 − z; y) + 8H(1, 0, 1, 0; y) − 16H(1, 0, 1 − z, 0; y) + 8H(1, 0, z, 1 − z; y) 16H(1, 1 − z, 0, 0; y) − 8H(1, 1 − z, 1, 0; y) − 8H(1, 1 − z, z, 1 − z; y)
−+
2H(0, 1; z)H(1 − z; y) − 2H(0; z)H(0, 1 − z; y) + 2H(1; z)H(0, z; y) − 2H(1; z)H(1 − z, z; y) + 2H(0, z, 1 − z; y) − 2H(1 − z, z, 1 − z; y) + H(0, 1; z) − 2H(0, 0, 1; z) − 2H(1, 0, 1; z) + z 2 8 (1 − y) 2 (y + z) − H(0, 1; z)H(1 − z; y) − H(0; z)H(0, 1 − z; y) + H(1; z)H(0, z; y) − H(1; z)H(1 − z, z; y)+ H(0, z, 1 − z; y) − H(1 − z, z, 1 − z; y) − H(0, 0, 1; z) − H(1, 0, 1; z) + z 8 (1 − z) 2 (y + z) zζ 2 H(0; y) + 3zζ 2 H(1 − z; y) − 2ζ 2 H(1 − z; y) − zH(0; y)H(0, 1; z) − zH(0, 1; z)H(1 − z; y) − zH(1; z)H(0, z; y) − zH(1; z)H(1 − z, z; y) + 3zH(0, 1, 0; y) − zH(0, z, 1 − z; y) + 3zH(1 − z, 1, 0; y) − zH(1 − z, z, 1 − z; y) − 2H(1 − z,1, 0; y) − 2H(0, 1, 0; y) + 3zζ 2 H(1; z) − 2ζ 2 H(1; z) + zH(0, 0, 1; z) − zH(1, z) (y + z) − z 2 H(0; y)H(1; z) − z 2 H(0, 1 − z; y)+ z 2 H(1, 0; y) − z 2 H(1 − z, 0; y) − 2zζ 2 H(0; y) − 2zζ 2 H(1 − z; y) + 2ζ 2 H(1 − z; y) + 2zH(0; y)H(0, 1; z) + 2zH(0, 1; z)H(1 − z; y) + 2zH(1; z)H(0, z; y) + zH(1, 0; y) + 2zH(1; z)H(1 − z, z; y) − 2zH(0, 1, 0; y) + 2zH(0, z, 1 − z; y) − 2zH(1 − z, 1, 0; y) + 2zH(1 − z, z, 1 − z; y) + 2H(1 − z, 1, 0; y) + 2H(0, 1, 0; y) + z 2 H(0, 1; z) − 2zζ 2 H(1; z) + 2ζ 2 H(1; z) − zH(0, 1; z) − 2zH(0, 0, 1; z) + 2zH(1, 0, 1; z) + 6zζ 3 + zζ 2 + 1 8 x (y + z) − z 2 ζ 2 H(0; y) − 2z 2 ζ 2 H(1; y) + 6z 2 H(0; y)H(0; z) − z 2 H(1, 0; y)H(0; z) + 6z 2 H(1, 0; y) − 2z 2 H(1, 1, 0; y) + 2zζ 2 H(0; y) − 6zH(0; y)H(0; z) − 6zH(1, 0; y) − ζ 2 H(0; y) − z 2 ζ 2 H(0; z) + 6z 2 H(1, 0; z) + 2zζ 2 H(0; z) − 7zH(1, 0; z) + 4z 2 ζ 3 + 6z 2 ζ 2 − 10zζ 3 − 7zζ 2 + 2ζ 3 + 1 8 (1 − z)2 yζ 2 H(1; z) + yζ 2 H(1 − z; y) − zζ 2 H(0; y) − 2zζ 2 H(1 − z; y) (1 − z, z, 1 − z; y) + zH(0; y)H(0, 1; z) + zH(0, 1; z)H(1 − z; y) + zH(1; z)H(0, z; y) + zH(1; z)H(1 − z, z; y) − 2zH(0, 1, 0; y) + zH(0, z, 1 − z; y) − 2zH(1 − z, 1, 0; y) + H(1 − z, 1, 0; y) + zH(1 − z, z, 1 − z; y) + yζ 2 H(0; y) + yH(0, 1, 0; y) + H(0, 1, 0; y) − 2zζ 2 H(1; z) + ζ 2 H(1; z) − zH(0, 0, 1; z) + zH(1, 0, 1; z) − 3yζ 3 yH(1; z)H(z; y) + yH(0, 1; z) − 2H(0; y)H(0, 1; z) − 2H(0, 1; z)H(1 − z; y) − yH(0, 1 − z; y) + zH(0, 1 − z; y) − 2H(1; z)H(0, z; y) − zH(1, 0; y) − yH(1 − z, 0; y) + zH(1 − z, 0; y) − 2H(1; z)H(1 − z, z; y) + yH(z, 1 − z; y) − 2H(0, z, 1 − z; y) + H(1 − z, 1, 0; y) − 2H(1 − z, z, 1 − z; y) + 2ζ 2 H(0; y) + yH(1, 0; y) − H(1, 0; y) + H(0, 1, 0; y) + ζ 2 H(1; z) − zH(0, 1; z) + H(0, 1; z) + 2H(0, 0, 1; z) − 2H(1, 0, 1; z) − 6ζ 3 − ζ 2 + 1 8 (y + z) − 4zζ 2 H(0; y) + 4zζ 2 H(1; y) + 2zζ 2 H(1 − z; y) − 2ζ 2 H(1 − z; y) + 6zH(0; y)H(0; z) + 3zH(0; y)H(1; z) − 3zH(0; z)H(1 − z; y) + 3zH(0, 1 − z; y) + 2zH(1, 0; y)H(0; z) + 3zH(1, 0; y) − 2zH(0; y)H(1, 0; z) + 2zH(1, 0; z)H(1 − z; y) + 2zH(0; z)H(1 − z, 0; y) + 3zH(1 − z, 0; y) + 2zH(0, 1, 0; y) + 4zH(1, 1, 0; y) + 2zH(1 − z, 1, 0; y) + H(0; y)H(1, 0; z) − 2H(1 − z, 1, 0; y) + ζ 2 H(0; y) − 3H(0, 1, 0; y) + 2zζ 2 H(0; z) − 6zζ 2 H(1; z) − 3zH(0, 1; z) + 6zH(1, 0; z) − 4zH(0, 1, 0; z) − 6zH(1, 1, 0; z) (y + z) 2 − z 2 H(0; y)H(1, 0; z) + z 2 H(0, 1, 0; y) + 2zH(0; y)H(1, 0; z) − 2zH(0, 1, 0; y) − H(0; y)H(1, 0; z) + H(0, 1, 0; y) − z 2 ζ 2 H(1; z) + 2z 2 H(0, 1, 0; z) − z 2 H(1, 1, 0; z) + 6zζ 2 H(1; z) − 2ζ 2 H(z) 2 − 2z 2 ζ 2 H(1 − z; y) − z 2 H(0; y)H(1, 0; z) − 2z 2 H(1, 0; z)H(1 − z; y) − 2z 2 H(0; z)H(1 − z, 0; y) + z 2 H(0, 1, 0; y) − 2z 2 H(1 − z, 1, 0; y) − 2zH(0; y)H(1, 0; z) − 2zH(1, 0; z)H(1 − z; y) + 4zH(0, 1, 0; y) + 2zH(1 − z, 1, 0; y) + H(0; y)H(1, 0; z) − H(0, 1, 0; y) + z 2 H(0, 1, 0; z) − 4zζ 2 H(1; z) + 2ζ 2 H(1; z) − 2zH(0, 1, 0; z) − 6zH(1, 1, 0; z)
−+−−−−
324H(0, 0; z)H(1 − z; y) + 216H(0; y)H(0, 1; z) − 216H(0, 1; z)H(1 − z; y) − 108H(0, 1; z)H(z; y) − 108H(0; z)H(0, 1 − z; y) + 216H(1; z)H(0, 1 − z; y) + 279H(0, 1 − z; y) + 324H(1; z)H(0, z; y) + 54H(1, 0; y)H(0; z) − 54H(0; y)H(1, 0; z) + 216H(1, 0; z)H(1 − z; y) + 324H(1, 0; z)H(z; y) + 216H(0; y)H(1, 1; z) − 432H(1, 1; z)H(z; y) − 108H(0; z)H(1 − z, 0; y) + 216H(1; z)H(1 − z, 0; y) + 279H(1 − z, 0; y) + 216H(0; z)H(1 − z, 1 − z; y) + 324H(1 − z, 1 − z; y) − 432H(1; z)H(1 − z, z; y) + 324H(1; z)H(z, 0; y) + 324H(0; z)H(z, 1 − z; y) − 432H(1; z)H(z, 1 − z; y) − 720H(z, 1 − z; y) − 432H(1; z)H(z, z; y) − 324H(0, 0, 1 − z; y) − 324H(0, 1 − z, 0; y)+ 216H(0, 1 − z, 1 − z; y) + 324H(0, z, 1 − z; y) − 324H(1 − z, 0, 0; y) + 216H(1 − z, 0, 1 − z; y) + 216H(1 − z, 1 − z, 0; y) − 432H(1 − z, z, 1 − z; y) + 324H(z, 0, 1 − z; y) + 324H(z, 1 − z, 0; y) − 432H(z, 1 − z, 1 − z; y) − 432H(z, z, 1 − z; y) − 216ζ 2 H(1; y) + 81H(0; y) − 360H(1, 0; y) + 54H(0, 1, 0; y) + 324H(1, 0, 0; y) − 216H(1, 1, 0; y) + 54ζ 2 H(1; z) + 189H(0; z) + 204H(1; z) − 441H(0, 1; z) − 81H(1, 0; z) 1296H(1; z)H(1 − z; y) + 1917H(1 − z; y) − 1296H(0, 1; y) 486H(0; y)H(0; z) − 432H(0; z) − 1926H(0; y)H(1; z) − 951H(1; z) − 1440H(0; z)H(1 − z; y) − 1782H(1; z)H(1 − z; y) − 951H(1 − z; y) + 5148H(1; z)H(z; y) + 2268H(1; z)H(0, 0; y) + 2268H(1 − z; y)H(0, 0; z) −702H(0; y)H(0, 1; z) + 1188H(1 − z; y)H(0, 1; z) − 864H(z; 324H(0, 1; z)H(1, 0; y) + 2250H(1, 0; y) + 1026H(0; y)H(1, 0; z) + 270H(1 − z; y)H(1, 0; z) − 1296H(z; y)H(1, 0; z) + 648H(0, 0; y)H(1, 0; z) − 1296H(0, z; y)H(1, 0; z) − 324H(1, 0; y)H(1, 0; z) + 324H(1, 0; z) − 1188H(0; y)H(1, 1; z) + 2376H(z; y)H(1, 1; z) + 648H(0, z; y)H(1, 1; z) − 1782H(1, 1; z) − 324H(0, 1; z)H(1, 1 − z; y) + 324H(1, 0; z)H(1, 1 − z; y)+ 1566H(0; z)H(1 − z, 0; y) − 1188H(1; z)H(1 − z, 0; y) + 648H(0, 0; z)H(1 − z, 0; y) + 972H(0, 1; z)H(1 − z, 0; y) + 972H(1, 0; z)H(1 − z, 0; y) − 1926H(1 − z, 0; y) − 1188H(0; z)H(1 − z, 1 − z; y) + 1296H(1, 0; z)H(1 − z, 1 − z; y) − 1782H(1 − z, 1 − z; y) + 2376H(1; z)H(1 − z, z; y) − 1944H(0, 1; z)H(1 − z, z; y) + 648H(1, 0; z)H(1 − z, z; y) − 1296H(1; z)H(z, 0; y) + 648H(0, 1; z)H(z, 0; y) + 648H(1, 1; z)H(z, 0; y) − 1296H(0; z)H(z, 1 − z; y) + 2376H(1; z)H(z, 1 − z; y) − 648H(0, 1; z)H(z, 1 − z; y) + 648H(1, 0; z)H(z, 1 − z; y) + 5148H(z, 1 − z; y) + 432H(1; z)H(z, z; y) − 3240H(0, 1; z)H(z, z; y) + 648H(1, 0; z)H(z, z; y) − 1296H(1, 1; z)H(z, z; y) − 648H(0; y)H(0, 0, 1; z) − 324H(1; y)H(0, 0, 1; z) − 1296H(1 − z; y)H(0, 0, 1; z) − 2592H(z; y)H(0, 0, 1; z) + 108H(0, 0, 1; z) + 2268H(0, 0, 1 − z; y) + 648H(1; z)H(0, 0, z; y) − 324H(0; z)H(0, 1, 0; y) + 324H(1; z)H(0, 1, 0; y) − 2484H(0, 1, 0; y) − 324H(0; y)H(0, 1, 0; z) + 324H(1; y)H(0, 1, 0; z) + 1296H(1 − z; y)H(0, 1, 0; z) + 648H(z; y)H(0, 1, 0; z) − 1512H(0, 1, 0; z) − 648H(z; y)H(0, 1, 1; z) + 1188H(0, 1, 1; z) + 2268H(0, 1 − z, 0; y) − 648H(0; z)H(0, 1 − z, 1 − z; y) − 1188H(0, 1 − z, 1 − z; y) + 324H(1; z)H(0, 1 − z, z; y) + 1296H(1; z)H(0, z, 0; y) 324H(0; z)H(1, 1 − z, 0; y) + 486H(1, 1 − z, 0; y) − 324H(1; z)H(1, 1 − z, z; y) + 648H(0; z)H(1 − z, 0, 0; y) + 2268H(1 − z, 0, 0; y) + 324H(0; z)H(1 − z, 0, 1 − z; y) − 1188H(1 − z, 0, 1 − z; y) + 648H(1; z)H(1 − z, 0, z; y) + 324H(0; z)H(1 − z, 1, 0; y) + 972H(1; z)H(1 − z, 1, 0; y) + 1458H(1 − z, 1, 0; y) + 648H(0; z)H(1 − z, 1 − z, 0; y) − 1188H(1 − z, 1 − z, 0; y) + 648H(1; z)H(1 − z, z, 0; y) + 648H(0; z)H(1 − z, z, 1 − z; y) + 2376H(1 − z, z, 1 − z; y) − 2592H(1; z)H(1 − z, z, z; y) + 648H(1; z)H(z, 0, 1 − z; y) − 1296H(z, 0, 1 − z; y) + 648H(1; z)H(z, 0, z; y) + 648H(1; z)H(z, 1 − z, 0; y) − 1296H(z, 1 − z, 0; y) + 648H(0; z)H(z, 1 − z, 1 − z; y)+ 2376H(z, 1 − z, 1 − z; y) − 1296H(1; z)H(z, 1 − z, z; y) + 648H(1; z)H(z, z, 0; y) + 648H(0; z)H(z, z, 1 − z; y) − 1296H(1; z)H(z, z, 1 − z; y) + 432H(z, z, 1 − z; y) − 3888H(1; z)H(z, z, z; y) − 648H(0, 0, 1, 0; y) + 1296H(0, 0, 1, 0; z) + 648H(0, 0, z, 1 − z; y) + 324H(0, 1, 0, 1 − z; y) − 1296H(0, 1, 1, 0; y) + 648H(0, 1, 1, 0; z) + 324H(0, 1, 1 − z, 0; y) + 972H(0, 1 − z, 1, 0; y) + 324H(0, 1 − z, z, 1 − z; y) + 1296H(0, z, 0, 1 − z; y) + 1296H(0, z, 1 − z, 0; y) + 648H(0, z, 1 − z, 1 − z; y) + 648H(0, z, z, 1 − z; y) − 324H(1, 0, 0, 1; z) + 648H(1, 0, 0, 1 − z; y) − 1296H(1, 0, 1, 0; y) + 648H(1, 0, 1, 0; z) + 648H(1, 0, 1 − z, 0; y) + 324H(1, 0, z, 1 − z; y) − 1296H(1, 1, 0, 0; y) + 648H(1, 1, 0, 1; z) − 1296H(1, 1, 1, 0; y) + 972H(1, 1, 1, 0; z) + 648H(1, 1 − z, 0, 0; y) − 324H(1, 1 − z, z, 1 − z; y) + 648H(1 − z, 0, 1, 0; y) + 648H(1 − z, 0, z, 1 − z; y) + 648H(1 − z, 1, 0, 0; y) + 972H(1 − z, 1, 0, 1 − z; y) + 972H(1 − z, 1, 1 − z, 0; y) + 1296H(1 − z, 1 − z, 1, 0; y) + 648H(1 − z, z, 0, 1 − z; y) + 648H(1 − z, z, 1 − z, 0; y) − 2592H(1 − z, z, z, 1 − z; y) 2 H(1 − z; y) − yH(0; y)H(1; z) + zH(0; y)H(1; z) + yH(1; z)H(z; y) + yH(0, 1; z) − 2H(0; y)H(0, 1; z) − 2H(0, 1; z)H(1 − z; y) − yH(0, 1 − z; y) + zH(0, 1 − z; y) − 2H(1; z)H(0, z; y) − zH(1, 0; y) − yH(1 − z, 0; y) + zH(1 − z, 0; y) − 2H(1; z)H(1 − z, z; y) + yH(z, 1 − z; y) − 2H(0, z, 1 − z; y) + 2H(1 − z, 1, 0; y) − 2H(1 − z, z, 1 − z; y) + 2ζ 2 H(0; y) + yH(1, 0; y) − H(1, 0; y) + 2H(0, 1, 0; y) + 2ζ 2 H(1; z) − 6H(0; z) − zH(0, 1; z) + H(0, 1; z) + 2H(0, 0, 1; z) − 2H(1, 0, 1; z) − 6ζ 3 − ζ 2 + 1 24 (y + z) − 72ζ 2 H(1 − z; y) + 7zH(0; y) + 18zH(0; y)H(0; z) + 9zH(0; y)H(1; z) − 9zH(0; z)H(1 − z; y) + 36H(0, 1; z)H(1 − z; y) + 9zH(0, 1 − z; y) + 9zH(1, 0; y) − 18H(1, 0; y)H(1; z) − 18H(1, 0; z)H(1 − z; y) + 9zH(1 − z, 0; y) + 36H(1; z)H(1 − z, z; y) − 18H(1, 0, 1 − z; y) − 18H(1, 1 − z, 0; y) − 18H(1 − z, 1, 0; y) + 36H(1 − z, z, 1 − z; y) + 36ζ 2 H(1; y) + 36H(1, 1, 0; y) − 36ζ 2 H(1; z) − 7zH(0; z) − 9zH(0, 1; z) + 18zH(1, 0; z) + 18H(1, 0, 1; z) 2 H(1, 0; z) + y 2 H(1, 0; y) + 2yH(0; z) − 2yH(0; y) + 1 1296 x − 108yH(0; y)H(0; z) + 180H(0; y)H(0; z) − 324H(0; y)H(0, 0; z) − 270H(0; y)H(1, 0; z) − 216yH(0; z) − 324H(0, 0; y)H(0; z) − 54H(1, 0; y)H(0; z) − 108yH(1, 0; z) − 216H(1, 0; z)H(1 − z; y) − 216H(0; z)H(1 − z, 0; y) − 216H(1 − z, 1, 0; y) + 216yH(0; y) + 855H(0; y) − 738H(0, 0; y) − 108yH(1, 0; y) z)H(1; y) − 2376H(1; y) + 648H(1; y)H(1; z) + 2916H(1; z) − 648H(0; z)H(1 − z; y) − 1296H(1; z)H(1 − z; y) + 3348H(1 − z; y) − 1296H(1; z)H(z; y) + 1296H(0, 1; y) + 1296H(0, 1; z) + 1296H(0, 1 − z; y) + 1944H(1, 0; y) + 1296H(1, 0; z) + 1296H(1, 1; y) + 648H(1, 1; z) + 648H(1, 1 − z; y) − 648H(1 − z, 0; y) − 1296H(1 − z, 1; y) − 1296H(z, 1 − z; y) + 954 − 2652H(0; y) − 2700H(0; y)H(0; z) − 6000H(0; z) + 4536H(0; z)H(0, 0; y) + 5760H(0, 0; y) + 4536H(0; y)H(0, 0; z) + 1296H(0, 0; y)H(0, 0; z) + 5760H(0, 0; z) + 108H(0; z)H(1, 0; y) − 4500H(1, 0; y) + 4428H(0; y)H(1, 0; z) + 3348H(1 − z; y)H(1, 0; z) + 1296H(0, 1 − z; y)H(1, 0; z) + 648H(1, 0; y)H(1, 0; z) − 4500H(1, 0; z) + 648H(1, 0; z)H(1, 1 − z; y) + 3348H(0; z)H(1 − z, 0; y) − 648H(1, 0; z)H(1 − z, 0; y) − 1296H(1, 0; z)H(z, 0; y) − 1296H(1, 0; z)H(z, 1 − z; y) + 648H(0; z)H(0, 1, 0; y) + 2052H(0, 1, 0; y) + 648H(0; y)H(0, 1, 0; z) + 648H(1; y)H(0, 1, 0; z) + 648H(1 − z; y)H(0, 1, 0; z) + 1296H(z; y)H(0, 1, 0; z) + 3024H(0, 1, 0; z) + 1296H(0; z)H(0, 1 − z, 0; y) + 1296H(0; z)H(1, 0, 0; y) + 4536H(1, 0, 0; y) + 1296H(0; y)H(1, 0, 0; z) + 4536H(1, 0, 0; z) − 2376H(1, 1, 0; y) + 648H(1; y)H(1, 1, 0; z) − 1296H(1 − z; y)H(1, 1, 0; z) − 1296H(z; y)H(1, 1, 0; z) + 2916H(1, 1, 0; z) + 648H(0; z)H(1, 1 − z, 0; y) − 648H(0; z)H(1 − z, 1, 0; y) + 3348H(1 − z, 1, 0; y) − 1296H(0; z)H(z, 1 − z, 0; y) 38z 2 H(0; y)H(0; z) + 3z 2 H(1, 0; y)H(0; z) − 38z 2 H(1, 0; y) + 6z 2 H(1, 1, 0; y) − 6zζ 2 H(0; y) + 18zH(0; y) + 38zH(0; y)H(0; z) + 38zH(1, 0; y) + 3ζ 2 H(0; y) + 3z 2 ζ 2 H(0; z) − 38z 2 H(1, 0; z) + 38zH(1, 0; z) − 12z 2 ζ 3 − 38z 2 ζ 2 + 12zζ 3 + 38zζ 2 − 6ζ 3 2 H(0; y) + 24zζ 2 H(1; y) − 6zζ 2 H(1 − z; y) − 31zH(0; y) − 38zH(0; y)H(0; z) + 12zH(1, 0; y)H(0; z) − 38zH(1, 0; y) − 12zH(0; y)H(1, 0; z) − 6zH(1, 0; z)H(1 − z; y) − 6zH(0; z)H(1 − z, 0; y) + 12zH(0, 1, 0; y) + 24zH(1, 1, 0; y) − 6zH(1 − z, 1, 0; y) − 3H(0; y)H(1, 0; z) − 3ζ 2 H(0; y) + 3H(0, 1, 0; y) + 12zζ 2 H(0; z) − 18zζ 2 H(1; z) − 6ζ 2 H(1; z)
− z 2 −−
2H(0, 1, 0; y) − 2zH(0; y)H(1, 0; z) + 2zH(0, 1, 0; y) + H(0; y)H(1, 0; z) − H(0, 1, 0; y) + 2z 2 ζ 2 H(1; z) − z 2 H(0, 1, 0; z) + 2z 2 H(1, 1, 0; z) − 4zζ 2 H(1; z) + 2ζ 2 H(1; z) − 4zH(1, 1, 0; z) y + z) 2 2z 2 ζ 2 H(1 − z; y) + z 2 H(0; y)H(1, 0; z) + 2z 2 H(1, 0; z)H(1 − z; y) + 2z 2 H(0; z)H(1 − z, 0; y) − z 2 H(0, 1, 0; y) + 2z 2 H(1 − z, 1, 0; y) + 2zH(0; y)H(1, 0; z) − 2zH(0, 1, 0; y) − H(0; y)H(1, 0; z) + H(0, 1, 0;y) − z 2 H(0, 1, 0; z) + 4zζ 2 H(1; z) − 2ζ 2 H(1; z) + 4zH(1, 1, 0; z) − 2H(1y)H(1; z) + 4H(0; z)H(1 − z; y) − 18H(1 − z; y) − 24H(1; z)H(z; y) + 4H(0, 1 − z; y) + 4H(1 − z, 0; y) − 24H(z, 1 − z; y) − 24H(0; y) − 4H(1, 0; y) − 18H(1; z) − 4H(0, 1; z) − 7 + z 8 (1 − y) (y + z) − 2ζ 2 H(1 − z; y) + 4zH(0; z)H(1 − z; y) y) 2 (y + z) − ζ 2 H(1 − z; y) + H(0, 1; z)H(1 − z; y) − H(1, 0; z)H(1 − z; y) + H(0; z)H(0, 1 − z; y) − H(1; z)H(0, z; y) + H(1; z)H(1 − z, z; y) − H(0, z, 1 − z; y) + H(1 − z, z, 1 − z; y) + H(0, 0, 1; z) + H(1, 0, 1; z) + z 2 8 (1 − z) 2 (y + z) − ζ 2 H(1 − z; y)+ H(0; y)H(0, 1; z) + H(0, 1; z)H(1 − z; y) + H(1; z)H(0, z; y) + H(1; z)H(1 − z, z; y) + H(0, z, 1 − z; y) − H(1 − z, 1, 0; y)+ H(1 − z, z, 1 − z; y) − ζ 2 H(0; y) − H(0, 1, 0; y) − ζ 2 H(1; z) − H(0, 0, 1; z) + H(1, 0, 1; z) + 3ζ 3 + z 8 (1 − z) (y + z) 2ζ 2 H(1 − z; y) + 4zH(0; y)H(1; z) − 3H(0; y)H(1; z) − 2H(0; y)H(0, 1; z) − 2H(0, 1; z)H(1 − z; y) + 4zH(0, 1 − z; y) − 3H(0, 1 − z; y) − 2H(1; z)H(0, z; y) − 4zH(1, 0; y) + 4zH(1 − z, 0; y) − 3H(1 − z, 0; y) − 2H(1; z)H(1 − z, z; y) − 2H(0, z, 1 − z; y) + 2H(1 − z, 1, 0; y) − 2H(1 − z, z, 1 − z; y) + 2ζ 2 H(0; y) + 2H(1, 0; y) + 2H(0, 1, 0; y) + 2ζ 2 H(1; z) − 4zH(0, 1; z) + 4H(0, 1; z) + 2H(0, 0, 1; z) − 2H(1, 0, 1; z) − 6ζ 3 − ζ 2 + 1 8 x (y + z)2z 2 H(0; y)H(0; z) + 2z 2 H(1, 0; y) + 4zζ 2 H(0; y) − 4zζ 2 H(1; y) − 6zH(0; y) − 8zH(0; y)H(0; z) − 2zH(1, 0; y)H(0; z) − 6zH(1, 0; y) + 4zH(0; y)H(1, 0; z) − 4zH(0, 1, 0; y) − 4zH(1, 1, 0; y) − 4H(0; y)H(1, 0; z) − 4ζ 2 H(0; y) + 6H(0; y) − H(1, 0; y)+ 4H(0, 1, 0; y) + 2z 2 H(1, 0; z) − 4zζ 2 H(0; z) + 6zζ 2 H(1; z) − 8ζ 2 H(1; z) + 6zH(0; z) − 9zH(1, 0; z) + 6zH(1, 1, 0; z) + H(1, 0; z) − 8H(1, 1, 0; z) + 2z 2 ζ 2 + 2zζ 3 − 7zζ 2 + 8ζ 3 + ζ 2 + 1 8 (1 − z) 2 − yζ 2 H(1; z) − yζ 2 H(1 − z; y) + zζ 2 H(0; y) + zζ 2 H(1 − z; y)+ yH(0; y)H(0, 1; z) + yH(0, 1; z)H(1 − z; y) + yH(1; z)H(0, z; y) + yH(1; z)H(1 − z, z; y) − yH(0, 0, 1; z) + yH(0, z, 1 − z; y) + yH(1, 0, 1; z) − yH(1 − z, 1, 0; y) + yH(1 − z, z, 1 − z; y) − zH(0; y)H(0, 1; z) − zH(0, 1; z)H(1 − z; y) − zH(1; z)H(0, z; y) − zH(1; z)H(1 − z, z; y) + zH(0, 1, 0; y) − zH(0, z, 1 − z; y) + zH(1 − z, 1, 0; y) − zH(1 − z, z, 1 − z; y) + y(−ζ 2 )H(0; y) − yH(0, 1, 0; y) + zζ 2 H(1; z) + zH(0, 0, 1; z) − zH(1, 0, 1; z) + 3yζ 3 − 3zζ 3 1 − z, 0; y) + 1296H(1 − z, 1 − z; y) − 2133 − 540H(0; y) − 324H(0; y)H(0; z) + 432H(0; z) − 1278H(0; y)H(1; z) − 708H(1; z) − 1278H(0; z)H(1 − z; y) − 1782H(1; z)H(1 − z; y) − 708H(1 − z; y) + 4176H(1; z)H(z; y) + 2268H(1; z)H(0, 0; y) + 2268H(1 − z; y)H(0, 0; z) − 702H(0; y)H(0, 1; z) + 1188H(1 − z; y)H(0, 1; z) − 864H(z; 324H(1, 0; y)H(1, 0; z) + 486H(1, 0; z) − 1188H(0; y)H(1, 1; z) + 2376H(z; y)H(1, 1; z) + 648H(0, z; y)H(1, 1; z) − 1782H(1, 1; z) − 324H(0, 1; z)H(1, 1 − z; y) + 324H(1, 0; z)H(1, 1 − z; y) + 1566H(0; z)H(1 − z, 0; y) − 1188H(1; z)H(1 − z, 0; y) + 648H(0, 0; z)H(1 − z, 0; y) + 972H(0, 1; z)H(1 − z, 0; y) + 972H(1, 0; z)H(1 − z, 0; y) − 1278H(1 − z, 0; y) − 1188H(0; z)H(1 − z, 1 − z; y) + 1296H(1, 0; z)H(1 − z, 1 − z; y) − 1782H(1 − z, 1 − z; y) + 2376H(1; z)H(1 − z, z; y) − 1944H(0, 1; z)H(1 − z, z; y) + 648H(1, 0; z)H(1 − z, z; y) − 1296H(1; z)H(z, 0; y) + 648H(0, 1; z)H(z, 0; y) + 648H(1, 1; z)H(z, 0; y) − 1296H(0; z)H(z, 1 − z; y) + 2376H(1; z)H(z, 1 − z; y) − 648H(0, 1; z)H(z, 1 − z; y) + 648H(1, 0; z)H(z, 1 − z; y) + 4176H(z, 1 − z; y) + 432H(1; z)H(z, z; y) − 3240H(0, 1; z)H(z, z; y) + 648H(1, 0; z)H(z, z; y) − 1296H(1, 1; z)H(z, z; y) − 648H(0; y)H(0, 0, 1; z) − 324H(1; y)H(0, 0, 1; z) − 1296H(1 − z; y)H(0, 0, 1; z) − 2592H(z; y)H(0, 0, 1; z) (z; y)H(0, 1, 0; z) − 1512H(0, 1, 0; z) − 648H(z; y)H(0, 1, 1; z) 324H(0; z)H(1, 1 − z, 0; y) + 486H(1, 1 − z, 0; y) − 324H(1; z)H(1, 1 − z, z; y) + 648H(0; z)H(1 − z, 0, 0; y) + 2268H(1 − z, 0, 0; y) + 324H(0; z)H(1 − z, 0, 1 − z; y) − 1188H(1 − z, 0, 1 − z; y) + 648H(1; z)H(1 − z, 0, z; y) + 324H(0; z)H(1 − z, 1, 0; y) + 972H(1; z)H(1 − z, 1, 0; y) + 1458H(1 − z, 1, 0; y) + 648H(0; z)H(1 − z, 1 − z, 0; y) − 1188H(1 − z, 1 − z, 0; y) + 648H(1; z)H(1 − z, z, 0; y) + 648H(0; z)H(1 − z, z, 1 − z; y) + 2376H(1 − z, z, 1 − z; y) − 2592H(1; z)H(1 − z, z, z; y) + 648H(1; z)H(z, 0, 1 − z; y) − 1296H(z, 0, 1 − z; y) + 648H(1; z)H(z, 0, z; y) + 648H(1; z)H(z, 1 − z, 0; y) − 1296H(z, 1 − z, 0; y) + 648H(0; z)H(z, 1 − z, 1 − z; y) + 2376H(z, 1 − z, 1 − z; y) − 1296H(1; z)H(z, 1 − z, z; y) + 648H(1; z)H(z, z, 0; y) + 648H(0; z)H(z, z, 1 − z; y) − 1296H(1; z)H(z, z, 1 − z; y) + 432H(z, z, 1 − z; y) − 3888H(1; z)H(z, z, z; y) − 648H(0z, 1, 0; y) + 324H(0, 1 − z, z, 1 − z; y) + 1296H(0, z, 0, 1 − z; y) + 1296H(0, z, 1 − z, 0; y) + 648H(0, z, 1 − z, 1 − z; y) + 648H(0, z, z, 1 − z; y) − 324H(1
-level splitting functions in CDR are (see Eqs. (4.17) and (4.18) of Ref. [57]loop, the splitting functions in CDR are (see Eqs. (4.21) and (4.22) of Ref.
Figure 2. Examples of Feynman diagrams that do not enter our calculation at NNLO.known[25], so that, once a NNLO calculation of H → bbj is complete, all of the component pieces for H → bb at N 3 LO are available.
Table 1. Numerical comparison between our two-loop results and those of Ref.[23] for y = 0.19 and z = 0.67 after adjusting for an overall 1/4 factor.(2)
1;2
15.1770833333333
15.1770833333333
B
(2)
1;1
−61.1367801007938 −61.1367801007938
B
(2)
1;0
77.6770380202061
77.6770380202060
B
(2)
2;2
15.1770833333333
15.1770833333333
B
(2)
2;1
−54.6784467674605 −54.6784467674605
B
(2)
2;0
74.2152337563907
74.2152337563904
Coefficient
y z S
(2)
H→bbg
Our result
−4
81.7702729678
81.7702729678
−3
3818.49680411
3818.49680413
−2
130763.8079162
130763.8079168
−1
Indeed, H + 3j is also available at NLO in QCD[16].
After adjusting for a small typo (i.e. changing 12H(0, 2; y) → 2H(0, 2; y) in the last line of B(1) 1;0 ) and dividing the literature results by an overall factor of 2. 3 After taking into account the different definition of the Mandelstam invariants t and u of Ref. [23] and after adjusting the literature results by an overall factor of 4.
AcknowledgmentsWe thank Ulrich Schubert for suggesting the coproduct method to simplify the 2dHPLs in the two-loop amplitudes and for helping with its implementation. We are grateful to Taushif Ahmed for clarifying the results of Ref.[23]. CW and RM are supported by a National Science Foundation CAREER award number PHY-1652066. Support provided by the Center for Computational Research at the University at Buffalo.+ 648H(z, 0, 1 − z, 1 − z; y) + 648H(z, 0, z, 1 − z; y) + 648H(z, 1 − z, 0, 1 − z; y) + 648H(z, 1 − z, 1 − z, 0; y) − 1296H(z, 1 − z, z, 1 − z; y) + 648H(z, z, 0, 1 − z; y) + 648H(z, z, 1 − z, 0; y) − 1296H(z, z, 1 − z, 1 − z; y) − 3888H(z, z, z, 1 − z; y) − 972ζ 3 H(1; y) + 648ζ 3 H(1; z) + 1620ζ 3 H(1 − z; y) + 1845ζ 3 + 2464 + 3ζ 3 z 3 8 x (1 − y) 2 (y + z)2;0,d = D
Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC. G Aad, ATLAS CollaborationPhys. Lett. 7161207.7214ATLAS Collaboration, G. Aad et. al., Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B716 (2012) 1-29 [1207.7214].
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[] |
[
"Continuous-wave versus time-resolved measurements of Purcell-factors for quantum dots in semiconductor microcavities",
"Continuous-wave versus time-resolved measurements of Purcell-factors for quantum dots in semiconductor microcavities"
] |
[
"M Munsch ",
"A Mosset ",
"A Auffèves ",
"S Seidelin ",
"J P Poizat ",
"J.-M Gérard ",
"A Lemaître ",
"I Sagnes ",
"P Senellart ",
"\nCEA/CNRS/UJF Joint team \"Nanophysics and semiconductors\", CEA/INAC/SP2M\nCEA/CNRS\nUJF Joint team \"Nanophysics and semiconductors\"\nInstitut Néel-CNRS\nBP 166, 25, rue des Martyrs, 17 rue des Martyrs38042, Cedex 9, 38054Grenoble, GrenobleFrance, France\n",
"\nLaboratoire de Photonique et de Nanostructures, LPN/CNRS, Route de Nozay\n91460MarcoussisFrance\n"
] |
[
"CEA/CNRS/UJF Joint team \"Nanophysics and semiconductors\", CEA/INAC/SP2M\nCEA/CNRS\nUJF Joint team \"Nanophysics and semiconductors\"\nInstitut Néel-CNRS\nBP 166, 25, rue des Martyrs, 17 rue des Martyrs38042, Cedex 9, 38054Grenoble, GrenobleFrance, France",
"Laboratoire de Photonique et de Nanostructures, LPN/CNRS, Route de Nozay\n91460MarcoussisFrance"
] |
[] |
The light emission rate of a single quantum dot can be drastically enhanced by embedding it in a resonant semiconductor microcavity. This phenomenon is known as the Purcell effect, and the coupling strength between emitter and cavity can be quantified by the Purcell factor. The most natural way for probing the Purcell effect is a time-resolved measurement. However, this approach is not always the most convenient one, and alternative approaches based on a continuouswave measurement are often more appropriate. Various signatures of the Purcell effect can indeed be observed using continuous-wave measurements (increase of the pump rate needed to saturate the quantum dot emission, enhancement of its emission rate at saturation, change of its radiation pattern), signatures which are encountered when a quantum dot is put on-resonance with the cavity mode. All these observations potentially allow one to estimate the Purcell factor. In this paper, we carry out these different types of measurements for a single quantum dot in a pillar microcavity and we compare their reliability. We include in the data analysis the presence of independent, nonresonant emitters in the microcavity environment, which are responsible for a part of the observed fluorescence.
|
10.1103/physrevb.80.115312
|
[
"https://arxiv.org/pdf/0906.0750v1.pdf"
] | 31,161,544 |
0906.0750
|
6fc39f1e9298dadeda462f3fca98f50568c50446
|
Continuous-wave versus time-resolved measurements of Purcell-factors for quantum dots in semiconductor microcavities
3 Jun 2009
M Munsch
A Mosset
A Auffèves
S Seidelin
J P Poizat
J.-M Gérard
A Lemaître
I Sagnes
P Senellart
CEA/CNRS/UJF Joint team "Nanophysics and semiconductors", CEA/INAC/SP2M
CEA/CNRS
UJF Joint team "Nanophysics and semiconductors"
Institut Néel-CNRS
BP 166, 25, rue des Martyrs, 17 rue des Martyrs38042, Cedex 9, 38054Grenoble, GrenobleFrance, France
Laboratoire de Photonique et de Nanostructures, LPN/CNRS, Route de Nozay
91460MarcoussisFrance
Continuous-wave versus time-resolved measurements of Purcell-factors for quantum dots in semiconductor microcavities
3 Jun 2009(Dated: June 3, 2009)numbers: 4250Pq7867Hc7890+t
The light emission rate of a single quantum dot can be drastically enhanced by embedding it in a resonant semiconductor microcavity. This phenomenon is known as the Purcell effect, and the coupling strength between emitter and cavity can be quantified by the Purcell factor. The most natural way for probing the Purcell effect is a time-resolved measurement. However, this approach is not always the most convenient one, and alternative approaches based on a continuouswave measurement are often more appropriate. Various signatures of the Purcell effect can indeed be observed using continuous-wave measurements (increase of the pump rate needed to saturate the quantum dot emission, enhancement of its emission rate at saturation, change of its radiation pattern), signatures which are encountered when a quantum dot is put on-resonance with the cavity mode. All these observations potentially allow one to estimate the Purcell factor. In this paper, we carry out these different types of measurements for a single quantum dot in a pillar microcavity and we compare their reliability. We include in the data analysis the presence of independent, nonresonant emitters in the microcavity environment, which are responsible for a part of the observed fluorescence.
I. INTRODUCTION
Coupling an emitter to a cavity strongly modifies its radiative properties, giving rise to the observation of cavity quantum electrodynamics effects (CQED), which can be exploited in the field of quantum information and fundamental tests of quantum mechanics. A variety of systems allows one to implement different CQED schemes, ranging from Rydberg atoms 1 and alkaline atoms in optical cavities 2,3 , to superconducting devices 4 , as well for semi-conducting quantum dots (QDs) (for an early review, see 5 ) coupled to optical solid-state cavities. Thanks to impressive recent progress in nanoscale fabrication techniques, vacuum Rabi splitting 6,7 , giant optical nonlinearities at the single photon level 8,9 , and vacuum Rabi oscillation in the temporal domain 10 have been demonstrated for single InAs/GaAs QDs coupled to microcavities. Success in sophisticated CQED experiments requires first of all an efficient enhancement of the spontaneous emission (SE) of an emitter coupled to a resonant single mode cavity 11 . The dynamical role of the cavity is quantified by the so-called Purcell factor F , namely the ratio between the emitter's SE rate with and without the cavity. For an emitter perfectly coupled to the cavity 12 the Purcell-factor only depends on the cavity parameters and takes on the value denoted F P which is given by
F P = 3 4π 2 Q V λ n 3 .
(1)
where Q is the cavity quality factor, V the cavity volume, λ the wavelength for the given transition, and n the refractive index. The Purcell effect using QDs as emitters has first been observed when coupled to pillar type microcavities in the late 90ies 13 . Moreover, when its radiation pattern is directive, the cavity efficiently funnels the spontaneously emitted photons in a single direction of space. This geometrical property allows one to implement efficient sources of single photons 14,15,16 , or even single, indistinguishable photons 17,18 . A high Purcell factor also enhances the visibility of CQED based signals like QD induced reflexion 8,19 . Beyond its seminal role, the Purcell factor appears thus as an important parameter which measures the ability of a QD-cavity system to show CQED effects, and has therefore become a figure of merit for quantifying these effects. It is obviously important to develop reliable methods to measure accurately this figure of merit. Two types of measurements are possible. The first one is the most intuitive and simply consists in comparing the lifetime of a QD at and far from resonance with the cavity mode, using a time-resolved setup 13 . This is feasible only as long as the resonant QD lifetime is longer than the time resolution of the the detector, or more generally longer than any other time scales involved, such as the exciton creation time (capture and relaxation of electron and holes inside the QD). For a large Purcell factor, this might be a limiting condition. Instead, the Purcell effect can be estimated from measurements under continuouswave (CW) excitation 20 . When approaching QD-cavity resonance, the pump rate required to saturate the emission of the QD is higher due to the shortening of the exciton lifetime. The Purcell effect also produces a preferential funneling of the QD SE into the cavity mode, and thus increases the photon collection efficiency in the output cavity channel. Measuring either the saturation pump rate or the PL intensity as a function of detuning, enables thereby one to measure the Purcell factor. This paper first aims at evaluating the consistency of these different methods, and to compare their accuracy. Moreover, both methods suffer from the same problem, related to the fact that the cavity is illuminated by many other sources in addition to the particular QD being studied. Even far detuned QDs can efficiently emit photons at the cavity frequency. This feature has been observed by several groups worldwide 7,10,16,21 , leading to theoretical effort to understand this phenomenon 21,22,23,24 . All the models involve the decoherence induced broadening of the QDs combined with cavity filtering and enhancement. Even though one can easily isolate the contribution of the single QD when it is far detuned from the cavity mode, this becomes much more difficult near resonance when other sources emitting via the cavity have to be taken into account. With this aim, we have developed a model which includes these contributions and therefore enables us to fit the experimental data and to derive a correct value of the Purcell factor.
II. SAMPLE CHARACTERISTICS AND SETUP
To fabricate the samples, a layer of InAs self-assembled QDs is grown by molecular beam epitaxy and located at the center of a λ-GaAs microcavity surrounded by two planar Bragg mirrors, consisting of alternating layers of Al 0.1 Ga 0.9 As and Al 0.95 Ga 0.05 As. The top (bottom) mirror contains 28 (32) pairs of these layers. The quality factor of the planar cavity is 14000. In a subsequent step, the planar cavity is etched in order to form a micro-pillar containing the QDs. The specific micro-pillar discussed in the following has a diameter of 2.3 µm and the density of the quantum dots is approximately 2.5 × 10 −9 QDs/cm 2 .
The etching of the Bragg mirrors into a micro-pillar can deteriorate the quality factor of the cavity. To measure the micro-pillar quality factor, we perform a photoluminescence measurement at high power such that the ensemble of QDs act as a spectrally broad light source, which is used for probing the cavity 15,25 . From this measurement, we extract a quality factor of our spectra of cavity and QD when varying the temperature (here from 5 K to 30 K). C indicates the cavity mode, whereas Xa and X b corresponds to the two QDs spectrally closest to the cavity (see text). The white line indicates the temperature for which the spectrum shown in the top has been recorded. specific 2.3 µm diameter sample mentioned above of Q = λ/∆λ = 7500. This value agrees (to within 10%) with reflectivity measurement using white light. We will, in the following section, use the corresponding bare cavity linewidth κ 0 = λ/Q (in nanometers). Using equation 1 together with the measured value for the quality factor, we obtain F P = 18.6.
Our sample is located in a cryostat held at 4K. For the continuous-wave measurements, the QDs are excited using a standard laser diode emitting at 820 nm (off-resonant excitation in the GaAs barrier), while for the time resolved measurements, we use a pulsed Ti:Sa laser centered at 825 nm (80 MHz repetition rate and 1 ps pulse width). The emitted light is recollected after passing a spectrometer (1.5 m focal and 0.03 nm resolution). The spectrometer has two output channels: one channel leads to a CCD camera (for the CW measurements), the other to an avalanche photo diode (APD) with a 40 ps time resolution which, combined with a 5 ps resolution for the data acquisition card and 65 ps resolution due to the spectrometer, gives us an overall resolution of 80 ps.
In fig. 1 we give an overview of the different lines observed in a typical photoluminescence experiment for our particular micro-pillar to be studied in the following. Centered around 895 nm, we observe what is usually referred to as the inhomogeneous line, composed of hundreds of QDs. The micro-pillar has been processed such that the cavity resonance is located on the low energy wing of this inhomogeneous line, where the QD density is very low, allowing us to optically isolate one single QD (denoted X a ) to be studied, and in particular scanned through cavity resonance. We also note that its corresponding bi-exciton (XX a ) is blue shifted by about 1 nm, an amount which is larger than the cavity linewidth. For a given temperature, we can therefore make the biexciton off resonance with the cavity, while having the exciton centered at resonance. For this specific micropillar, this happens at 19.5 K. In this case, a second QD (X b ) appears about 3 cavity linewidths away (with its bi-exciton XX b even further away), and is therefore also minimally affected by the cavity. All other QDs are much further detuned.
In fig. 2 we show the temperature dependence of the cavity resonance frequency, as well as the two relevant QDs emission wavelengths. The cavity frequency varies due to a temperature dependent refractive index, while the QD exciton energy follows the expected temperature dependence of the GaAs bandgap. Due to this difference in temperature dependence, we can vary the cavity -QD detuning 6,7,26 .
III. CONTINUOUS-WAVE MEASUREMENTS
Even though the Purcell effect is a dynamical phenomenon, it can be measured without a time resolved setup. This can be understood as follows. As the emitter's lifetime decreases near resonance due to the Purcell effect, it becomes harder to saturate the optical transition. This can be quantified by measuring the increase in the pump rate required to saturate the emitter (see section III A), or by measuring the actual cycling rate in a PL measurement at saturation (section III B). So by comparing the on-and off-resonant saturation pump rate or PL intensity, the Purcell-factor can be measured.
More recently it has been demonstrated that one can also extract the Purcell-factor due to the change in the fraction of SE that is funneled into the cavity mode 25 . This is FIG. 3: a) The PL of the QD arriving at the detector can be separated into two channels: one part emitted into loss channels (γ) but redirected to the detector with a probability χ leak and the part emitted into the cavity Γ(∆), and detected with a probability χcav. b) Three-level scheme including the exciton | X and bi-exciton | XX . The notations are defined in the text. 2 . The open circles allows us to extract ǫ above and ǫ below as described in section III D. Filled black circles indicate the saturation intensity Isat, and the corresponding pump power needed to saturate the QD, denoted Psat.
0 x 1 0 5 1 . 5 1 . 0 0 . 5 0 . 0 I X , d e t ( a . u . ) 3 0 0 2 0 0 1 0 0 0 P ( µ W ) ( P s a t , I s a t ) D / k = 0 D / k = - 0 . 1 4 D / k = - 0 . 3 2 D / k = - 0 . 9 6 D / k = - 1 . 3 5 D / k = - 2 .
done by measuring the SE rate as a function of detuning for fixed pump power, as will be done in section III C. An illustration of the principles is given in fig. 3. A QD is embedded in a cavity whose fundamental mode is nearly resonant with the excitonic X a transition ( fig. 1a). We denote ∆ the detuning between the excitonic transition and the cavity mode. The QD is non-resonantly pumped with a rate r, and decays by emitting photons either in the cavity mode, or in other leaky modes with a rate which we suppose to be independent of the detuning ∆ and identical to that of the bulk material (which is a reasonable approximation for QDs in micro-pillar cavities 13 and microdisks). As suggested by the PL spectra shown in part II, the QD should be modeled by a threelevel system which includes the bi-exciton ( fig. 3b). In the following we will concentrate solely on X a , so for simplicity we will omit the subscript a. We denote γ and γ XX the coupling of the X and bi-excitonic (XX) transition with the leaky modes. In addition to the leaky modes, the X transition is coupled to the cavity mode with a rate Γ(∆) = γF L(∆) where F is the effective Purcell factor experienced by the QD, taking into account that it is not perfectly coupled to the cavity (in contrast to F P given in equation 1, which is only an upper bound for F ). Moreover, L(∆) = 1/(1+∆ 2 /κ 2 0 ) is a Lorentzian of width κ 0 corresponding to the empty cavity line shape. When pumping with a rate r, the average excitonic population is then given by
p X (∆, r) = 1 1 + r γXX + γ+Γ(∆) r .(2)
As it was mentioned in the first part of this paper, the role of the cavity is not only to enhance the cycling rate for the exciton (X), but also to efficiently funnel the emitted photons into the cavity mode. Provided its emission pattern is directional, which is the case for micropillars, the coupling with a conveniently positioned detector can be very efficient, whereas the coupling between leaky modes and detector remains poor. These geometrical efficiencies are respectively denoted χ cav and χ leak (see fig. 3 and ref 24 ). The PL intensity from our single QD collected by the detector can thus be written in the following way:
I X,det (∆, r) = I leak X (∆, r) + I cav X (∆, r),(3)
where
I leak X (∆, r) = χ leak γp X (∆, r)(4)
is the PL intensity emitted through the leaky modes and
I cav X (∆, r) = χ cav Γ(∆)p X (∆, r)(5)
the detected PL intensity emitted spatially into the cavity mode. Please note that the notation cav applies to geometrical considerations, but not to the emission frequency (this PL contribution is indeed emitted at the QD frequency). In our experiment, to separate I X,det from the PL intensity from all other light sources, we use of the spectrometer to focus on a window centered on our selected QD (see the inset in fig. 1) and we then fit the line shape corresponding to the single QD with a Lorentzian function. When the QD-cavity detuning is large, it is easy to separate the QD line shape from the cavity, but as the detuning decreases, they will partially overlap with each other. When this happens, to avoid a part of the cavity peak erroneously being included in the single QD line shape, we also do a Lorentzian fit on the cavity profile, which we then subtract from the QD line shape. Note that in doing this, we also involuntarily omit from I X,det the part of the QD PL which is emitted at the cavity frequency, but this part constitutes a small fraction of the total signal. An example of typical experimental data is pictured in fig. 4, where the PL intensities for different detunings ∆ are plotted. As we generally measure the pump power denoted P and not the pump rate r, we have chosen to plot the data as a function of the former (and we do the same in the graphs to follow). This also means that P sat is the pump power corresponding to the pump rate r sat .
For each detuning, the maximal intensity I sat X,det (∆, r sat (∆)) is reached when the X transition is saturated, where r sat (∆) denote the pump rate required to saturate the transition (saturation pump rate). Note that the highest values of I sat X,det and corresponding r sat are reached at resonance, which is coherent with the enhancement of the X transition rate induced by the cavity. In the following, we will analyze the curves presented in fig. 4 (and further equivalent curves not added to the graph for clarity), in four different ways (section III A-III D).
A. Saturation pump rate measurements
In the first method the Purcell factor is extracted from the dependence of the saturating pumping rate intensity with the detuning r sat (∆), corresponding to black filled circles in fig. 4. This method has been proposed as a substitute for the time-resolved measurements, and has been widely used for micropillars 20 , microdiscs 20 and photonic crystals 27 . The analytic expressions can be found by determining the pump rate corresponding to the maximum intensity of equation 3. We obtain r sat (∆) ∝ 1 + F L(∆). In fig. 5a we have plotted the data and the fit according to equation 6 where we have imposed the bare cavity linewidth based on independent measurements. From the first fit, we extract a Purcell-factor of F = 3.7 ± 1.0,
where the relatively large error is due to the uncertainty of r sat . The slope of the baseline in fig. 5a is due to the increase in temperature for increased detuning. As mentioned in section II, we use an optical excitation obtained through the pumping of the GaAs barrier material. The mean free path of the electrons and holes increase with temperature, so that the excitation rate of the QD tends to increase for a fixed pump rate. As a test, we have checked that the PL of another far detuned QD, X b , gives rise to an equivalent slope during the same experiment, see fig. 5b.
B. Saturation PL intensity measurements
Another similar approach again based on the black filled circles in fig. 4 has been used in recent papers 28,29 . This method corresponds to exploiting directly the maximum intensity of equation 3 given by
I sat X,det (∆) ∝ F L(∆) 1 + 2 + 2F L(∆) ,(8)
In fig. 6 we have plotted the data and the fit (the maximum normalized to one) according to equation 8, again with the bare cavity linewidth fixed. From the fit, we extract a Purcell-factor of F = 2.4 ± 1.2.
In this case, the intrinsic uncertainty in the PL measurement is quite small, but is amplified by the fitting procedure, resulting in the stated errorbar.
C. PL intensity with fixed pump rate
In the two previous sections, we have used the data corresponding to the saturation pump rate and intensity. Instead, we can also use the emitted PL intensity, not at saturation, but for a fixed pump rate 25 . This amounts to using the PL intensity corresponding to the intersection of the curves in fig. 4 with a straight vertical cut. In particular, we have plotted the PL intensity for powers below and above saturation in fig. 7. The fit corresponds again to equation 3, but this time with the pump rate fixed (r = 30µW and r = 300µW , for the two curves, respectively). From both curves we have subtracted a global offset corresponding to the PL intensity I X,det at ∆ = ∞.
Below saturation the change in the light intensity I cav X as the QD is scanned across the cavity resonance is due to the geometrical redirection of the emission alone (a modification in the emission pattern). What we detect is a projection of a fraction of the micro-pillar emission pattern onto the microscope aperture. More precisely, for low powers (well below saturation) p X (∆, r) = r γ+Γ(∆) , and we obtain
I cav X (∆) ∝ F L(∆) 1 + F L(∆) ≡ β(∆),(10)
where we have defined the function β(∆) which can be interpreted as the fraction of the emission pattern overlapping with the cavity mode. This function is broader than the Lorentzian profile of the cavity mode by a factor (F + 1). Above saturation, the geometrical redirection of the emission pattern is still present but the light intensity I cav X follows now the L(∆) profile of the cavity owing to the additional effect of the larger emission rate of the quantum dot caused by the shortening of its lifetime. More precisely, in the regime well above saturation we have p X (∆, r) ≈ γ XX /r, and we get
I cav X (∆, r) ∝ F L(∆).(11)
From the ratio of the two widths, we extract a Purcellfactor of
F = 3.2 ± 0.9,(12)
where the stated uncertainty arises from the intensity measurements, which is the dominant source of error in this case.
D. PL intensity ratio at low and high pump rate
This method also consists in comparing the light emitted by the single QD at resonance and far from resonance, below and above saturation, but only requires four of the measurements used above. Here we do not subtract the offset due to χ leak as done above, which has the advantage that it allows us to quantify χ cav /χ leak . We define as ǫ(∆, r) the following ratio ǫ(∆, r) = I X,det (0, r)
I X,det (∆, r) = p X (0, r) p X (∆, r) × χ leak + χ cav F χ leak + χ cav F L(∆) (13) ≡ p X (0, r) p X (∆, r) α(∆),(14)
where the parameter α(∆) depends on the cavity funneling properties. For pump rates below the pump rate required to saturate (where p X (∆, r) = r γ+Γ(∆) )
ǫ below (∆) = α(∆) × 1 + F L(∆) 1 + F ,(15)
whereas above the saturation pump rate (again using that p X (∆, r) ≈ γ XX /r)
ǫ above (∆) = α(∆).(16)
Taking the ratio between ǫ below and ǫ above , α(∆) cancels, and with an independent measurement of κ 0 (see section II), we obtain a Purcell-Factor of
F = 2.5 ± 0.5,(17)
where the error arises from the uncertainty on the intensity measurements. From the separate value of ǫ above (or ǫ below ) we get
χ cav /χ leak ∼ 15 ± 4.5,(18)
confirming that the cavity is much better coupled to the detector than the leaky modes. This ratio depends on the radiation pattern of the micro-pillar, and the numerical aperture of the collection objective (0.4 for the above stated ratio of χ cav /χ leak ).
Note that we have only included the presence of exciton and bi-exciton in all given formulas. We have, however, repeated the above analysis, allowing for all orders of exciton levels, without any significant change in final results within the range of used pump powers.
IV. TIME-RESOLVED MEASUREMENTS
As a way to confirm our continuous-wave measurements of the Purcell factor, we have performed a detailed study of the lifetime as a function of the detuning, using time resolved spectroscopy. This technique has been used extensively for many different systems since it was the first method to be used. In fact, the Purcell factor can be written as
F = τ (∆ = 0) τ (∆ = ∞) − 1,(19)
where τ is the lifetime of the QD, and ∆ again is the detuning. Opposite equation 1, this definition also applies to an emitter that is not perfectly coupled to the cavity (within the approximation where γ leak = γ bulk , the latter denoting the SE of the QD into the unprocessed, or bulk, material).
In fig. 8 we show the measured lifetime of our quantum dot for different pumping powers. In a) the QD is detuned from the cavity resonance, while in b) it is at resonance. In the first case (a), we show two different powers. When P ≤ P sat (lowest lying curve) the QD exhibits the typical mono-exponential decay. When P ≥ P sat (highest lying curve), the effect of the bi-exciton can be observed as a rounding off of the curve at short time, which corresponds to the delay in the radiation of the exciton. The data fit very well with a model including three levels (a ground state, the exciton and bi-exciton states) and we extract the exciton and bi-exciton lifetimes, which are the same for the two different powers: τ X = 0.80 ± 0.05 ns and τ XX = 0.40 ± 0.02 ns (20) As the biexciton is not influenced by the Purcell effect (for the detunings used in this experiment), the obtained value can be used as a fixed parameter when we then fit the data for the resonant case. Note that all our fits have been convoluted with the experimental system response time (80 ps time resolution). On the contrary, the resonant case (b), shows a power dependency that can not be explained by our simple three level model used above. We clearly observe in fig. 8b a change from a quasi monoexponential decay to a bi-exponential decay, when lowering the pump rate. We exclude a prominent role of dark exciton since a mono-exponential behavior is observed in the non-resonant case (a). In addition, the fact that the second lifetime of the exponential decay is fast (less than 1 ns) also tends to eliminate this hypothesis. We believe that this behavior is due to detuned emitters, which contribute to the collected intensity via the cavity emission. Recent experiments 7,10,16,21 show that QD's could emit photons in the cavity mode even at rather high detunings (several times the cavity linewidth). In contrast to CW measurements where we could separate the emission of our QD from the one of the cavity using appropriate lorentzian fits, in the present case we do not have access to the full spectra, and therefore cannot use the same technique. Instead we must select a frequency window around the QD line, for which we integrate all PL. This makes us unable to filter out the cavity component which overlaps in frequency with the chosen window (when close to resonance, as in fig. 8b). As a result we measure two different times: the shorter one is the lifetime of our single QD (undergoing Purcell effect), whereas the longer one corresponds to the lifetime of other detuned emitters. The higher the pump power, the more dominant is the signal due to the contribution of the detuned emitters. Therefore, at high powers, the light from other emitters tends to make the signal invisible for our single QD. This is illustrated in fig. 9, where we have shown the spectra corresponding to three different pump powers, ranging from high (a) to low (c), but for a fixed detuning. The fraction of light emitted via the cavity clearly dominates at high powers, but decrease while lowering the pump power. This is why, for high powers, only one lifetime can be observed (upper curves in fig. 8b), and this lifetime is obviously no longer the QD radiative lifetime, but corresponds to the light emitted via the cavity. Only for lower pump power, the true lifetime also becomes visible (lower curve) as seen by the bi-exponential decay. We therefore need to include these additional emitters that we can model (within our pumping range) with a two-level system whose lifetime corresponds to an average lifetime, which can be measured in an independent experiment, in which all QDs are far detuned. We obtain 0.8 ± 0.05 ns.
The exciton lifetime is the only free parameter in our fits (the bi-exciton lifetime is a fixed parameter). The excellent agreement between data and fit seems to vali- date our model. We find for the non resonant and resonant case, τ (∆ = ∞) = 0.80 ± 0.05 ns and τ (∆ = 0) = 0.2 ± 0.01 ns, which gives a Purcell factor of:
F = 3.0 ± 0.5(21)
where the stated uncertainty arises from the exponential fit. Based on the above discussion, we remark that the low power condition is a necessary but not sufficient criterium for measuring the correct lifetime. Indeed, though all shown powers in fig. 8 are below saturation only the complete model gives the right lifetime. In fig. 10 we have plotted the exciton lifetime obtained by measurements equivalent to those in fig. 8 for different values of the detuning. The shape of the curve should be the Lorentzian profile of the cavity, confirmed by the fit.
V. FINAL DISCUSSION
We have presented several ways to measure the Purcellfactor, which is an important figure of merit in cavity QED. All our CW measurements agree with each other, within the experimental uncertainty, for a Purcellfactor of 3.0 ± 0.4. We emphasize that in our evaluation of the error-bars, we have not taken into account the stated 10 % uncertainty for the bare cavity linewidth (see section II). A simple PL measurement of the cavity linewidth has negligible error-bars, but when probing the cavity by reflectivity measurements, this value turns out to be about 10 % different. We also point out that the value measured by reflectivity is systematically higher than the one measured in PL. We will here revisit the obtained results for the Purcell factor, in order to see how a 10% deviation on the quality factor would affect the values. While the first method (based on saturation pump power, in section III A) does not depend on this parameter, all the other CW methods here presented do. In particular, the second technique, which uses the saturation intensity (III B), drastically depends on this parameter. In our case, an uncertainty of 10% on the quality factor would make the measurement based on this method useless. Though we still can fit the data with a correct shape, the obtained Purcell-factor is absurd and exceeds the theoretical value. Finally, concerning the third method (III C), the modification of the Purcell factor induced by the 10% change in the quality factor amounts to 20 %, which is slightly below the stated error-bars due to the imprecision on the measurement. Therefore, these error-bars are not significantly increased when allowing the given deviation on the quality factor. The time resolved measurements also agree within the error-bars with the CW measurements. The fact that we clearly do not observe a single exponential decay at resonance, confirms the hypothesis that other light sources contribute to the light emitted into the cavity channel. In particular, for the time resolved measurement, if not including this light in our model, the lifetime appears to be pump power dependent, even when we pump way below saturation which is clearly non-physical. We thus underline that the commonly adopted criterium that the time resolved spectroscopy of an exciton has to be made below saturation, might not be sufficient for CQED experiment. Moreover, if additional emitters are present in the environment of the considered QD, it might be adequate to include their presence in the data analysis.
In conclusion, the agreement of the time resolved measurements with the CW measurements suggests that both methods are reliable. The dramatic influence of the cavity linewidth uncertainty on the Purcell-factor error-bars might be a reason for preferring Q-independent measurements such as time resolved spectroscopy. On the other hand, the time resolved measurements suffer from a lower signal-to-noise ratio, and for some systems (photonic crystals, in particular, where the radiation pattern is less favorable), this becomes a limiting factor, making the CW measurements more desirable. In that case, based on above considerations, we advise to use the method based on the saturation intensity with precaution, unless a very precise measurement of the cavity quality factor is available. If this is not the case, the other CW methods here presented seem more robust against an uncertainty on this parameter.
FIG. 1 :
1The full spectrum recorded at 4K showing the inhomogeneous line, with a zoom on the section of interest including two isolated quantum dots (Xa and X b ) and their respective bi-excitons (XXa and XX b ), and the cavity mode (C).
FIG. 2: PL spectra of cavity and QD when varying the temperature (here from 5 K to 30 K). C indicates the cavity mode, whereas Xa and X b corresponds to the two QDs spectrally closest to the cavity (see text). The white line indicates the temperature for which the spectrum shown in the top has been recorded.
7 2 FIG. 4 :
24Photoluminescence intensity (I) as a function of pump power (P) for different cavity versus quantum dot detunings.
pump power as a function of detuning for our single QD and b) saturation pump power for our single QD Xa and a "control" QD X b as a function of temperature. The QD Xa goes through the cavity resonance while X b remains detuned throughout the scan.
FIG. 6 :
6Saturation intensity for the single QD as a function of detuning. The size of the error-bars corresponds approximately to the extent of the data points and are therefore not shown.
FIG. 7 :
7Measurements of the PL intensity at fixed pump power (30 µW and 300 µW , respectively). The two curves can be thought of as the intersection of the curves infig. 4with vertical lines centered at P=30 µW and P=300 µW (with several more similar curves added).
FIG. 8 :
8Lifetime measurements at different pump powers of the quantum dot while a) far detuned from the cavity and b) close to resonance. In a) the solid line corresponds to P = 3Psat and dotted line to P = Psat. In b) we have P = Psat (solid line) and P = Psat/10 (dotted line) and P = Psat/30 (dashed line).
FIG. 9 :
9Three different line spectra, each corresponding to the QD (solid line) and the cavity (shaded area). The spectra are shown for the pump power decreasing from a) through c). For all three cases, the detuning is fixed (∆ = −0.2κ0). The square frame indicates for each spectrum the integration window.
FIG. 10 :
10Exciton lifetimes measured at low intensity as a function of QD-cavity detuning.
We thank B. Gayral for fruitful discussions and M. Rigault for his help and enthusiasm in the initial stages of the measurements, and we acknowledge QAP for financial support (Contract No. 15848).
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|
[] |
[
"Localization of Protein Aggregation in Escherichia coli Is Governed by Diffusion and Nucleoid Macromolecular Crowding Effect",
"Localization of Protein Aggregation in Escherichia coli Is Governed by Diffusion and Nucleoid Macromolecular Crowding Effect"
] |
[
"Anne-Sophie Coquel \nInstitut National de la Santé et de la Recherche Médicale\n1001Unité, ParisFrance\n\nEPI Beagle\nINRIA Rhone-Alpes\nVilleurbanneFrance\n\nUniversity of Lyon\nLIRIS UMR5205 CNRS\nVilleurbanneFrance\n",
"Jean-Pascal Jacob \nUMR 8145\nUniversity Paris Descartes\nMAP5 -CNRSParisFrance\n",
"Mael Primet \nUMR 8145\nUniversity Paris Descartes\nMAP5 -CNRSParisFrance\n",
"Alice Demarez \nInstitut National de la Santé et de la Recherche Médicale\n1001Unité, ParisFrance\n",
"Mariella Dimiccoli \nUMR 8145\nUniversity Paris Descartes\nMAP5 -CNRSParisFrance\n",
"Thomas Julou \nLaboratoire de Physique Statistique de l'École Normale Supérieure\nUMR 8550\nCNRS\nParisFrance\n",
"Lionel Moisan \nUMR 8145\nUniversity Paris Descartes\nMAP5 -CNRSParisFrance\n",
"Ariel B Lindner [email protected] \nInstitut National de la Santé et de la Recherche Médicale\n1001Unité, ParisFrance\n\nFaculty of Medicine\nParis Descartes University\nParisFrance\n",
"Hugues Berry [email protected] \nEPI Beagle\nINRIA Rhone-Alpes\nVilleurbanneFrance\n\nUniversity of Lyon\nLIRIS UMR5205 CNRS\nVilleurbanneFrance\n"
] |
[
"Institut National de la Santé et de la Recherche Médicale\n1001Unité, ParisFrance",
"EPI Beagle\nINRIA Rhone-Alpes\nVilleurbanneFrance",
"University of Lyon\nLIRIS UMR5205 CNRS\nVilleurbanneFrance",
"UMR 8145\nUniversity Paris Descartes\nMAP5 -CNRSParisFrance",
"UMR 8145\nUniversity Paris Descartes\nMAP5 -CNRSParisFrance",
"Institut National de la Santé et de la Recherche Médicale\n1001Unité, ParisFrance",
"UMR 8145\nUniversity Paris Descartes\nMAP5 -CNRSParisFrance",
"Laboratoire de Physique Statistique de l'École Normale Supérieure\nUMR 8550\nCNRS\nParisFrance",
"UMR 8145\nUniversity Paris Descartes\nMAP5 -CNRSParisFrance",
"Institut National de la Santé et de la Recherche Médicale\n1001Unité, ParisFrance",
"Faculty of Medicine\nParis Descartes University\nParisFrance",
"EPI Beagle\nINRIA Rhone-Alpes\nVilleurbanneFrance",
"University of Lyon\nLIRIS UMR5205 CNRS\nVilleurbanneFrance"
] |
[] |
Aggregates of misfolded proteins are a hallmark of many age-related diseases. Recently, they have been linked to aging of Escherichia coli (E. coli) where protein aggregates accumulate at the old pole region of the aging bacterium. Because of the potential of E. coli as a model organism, elucidating aging and protein aggregation in this bacterium may pave the way to significant advances in our global understanding of aging. A first obstacle along this path is to decipher the mechanisms by which protein aggregates are targeted to specific intercellular locations. Here, using an integrated approach based on individual-based modeling, time-lapse fluorescence microscopy and automated image analysis, we show that the movement of aging-related protein aggregates in E. coli is purely diffusive (Brownian). Using single-particle tracking of protein aggregates in live E. coli cells, we estimated the average size and diffusion constant of the aggregates. Our results provide evidence that the aggregates passively diffuse within the cell, with diffusion constants that depend on their size in agreement with the Stokes-Einstein law. However, the aggregate displacements along the cell long axis are confined to a region that roughly corresponds to the nucleoid-free space in the cell pole, thus confirming the importance of increased macromolecular crowding in the nucleoids. We thus used 3D individual-based modeling to show that these three ingredients (diffusion, aggregation and diffusion hindrance in the nucleoids) are sufficient and necessary to reproduce the available experimental data on aggregate localization in the cells. Taken together, our results strongly support the hypothesis that the localization of aging-related protein aggregates in the poles of E. coli results from the coupling of passive diffusion-aggregation with spatially non-homogeneous macromolecular crowding. They further support the importance of ''soft'' intracellular structuring (based on macromolecular crowding) in diffusion-based protein localization in E. coli. Citation: Coquel A-S, Jacob J-P, Primet M, Demarez A, Dimiccoli M, et al. (2013) Localization of Protein Aggregation in Escherichia coli Is Governed by Diffusion and Nucleoid Macromolecular Crowding Effect. PLoS Comput Biol 9(4): e1003038.
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10.1371/journal.pcbi.1003038
| null | 10,360,467 |
1303.1904
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8dcb490a12c4a84f83131b478fc973b4bfed63c8
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Localization of Protein Aggregation in Escherichia coli Is Governed by Diffusion and Nucleoid Macromolecular Crowding Effect
Anne-Sophie Coquel
Institut National de la Santé et de la Recherche Médicale
1001Unité, ParisFrance
EPI Beagle
INRIA Rhone-Alpes
VilleurbanneFrance
University of Lyon
LIRIS UMR5205 CNRS
VilleurbanneFrance
Jean-Pascal Jacob
UMR 8145
University Paris Descartes
MAP5 -CNRSParisFrance
Mael Primet
UMR 8145
University Paris Descartes
MAP5 -CNRSParisFrance
Alice Demarez
Institut National de la Santé et de la Recherche Médicale
1001Unité, ParisFrance
Mariella Dimiccoli
UMR 8145
University Paris Descartes
MAP5 -CNRSParisFrance
Thomas Julou
Laboratoire de Physique Statistique de l'École Normale Supérieure
UMR 8550
CNRS
ParisFrance
Lionel Moisan
UMR 8145
University Paris Descartes
MAP5 -CNRSParisFrance
Ariel B Lindner [email protected]
Institut National de la Santé et de la Recherche Médicale
1001Unité, ParisFrance
Faculty of Medicine
Paris Descartes University
ParisFrance
Hugues Berry [email protected]
EPI Beagle
INRIA Rhone-Alpes
VilleurbanneFrance
University of Lyon
LIRIS UMR5205 CNRS
VilleurbanneFrance
Localization of Protein Aggregation in Escherichia coli Is Governed by Diffusion and Nucleoid Macromolecular Crowding Effect
Received October 8, 2012; Accepted March 5, 2013; Published April 25, 2013Editor: Stanislav Shvartsman, Princeton University, United States of America Funding: This work was funded by the French National Institute for Research in Computer Science and Control, INRIA (grant AEN ColAge) and the French National Research Agency, ANR (grant PagDeg). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist. * (HB) . These authors contributed equally to this work.
Aggregates of misfolded proteins are a hallmark of many age-related diseases. Recently, they have been linked to aging of Escherichia coli (E. coli) where protein aggregates accumulate at the old pole region of the aging bacterium. Because of the potential of E. coli as a model organism, elucidating aging and protein aggregation in this bacterium may pave the way to significant advances in our global understanding of aging. A first obstacle along this path is to decipher the mechanisms by which protein aggregates are targeted to specific intercellular locations. Here, using an integrated approach based on individual-based modeling, time-lapse fluorescence microscopy and automated image analysis, we show that the movement of aging-related protein aggregates in E. coli is purely diffusive (Brownian). Using single-particle tracking of protein aggregates in live E. coli cells, we estimated the average size and diffusion constant of the aggregates. Our results provide evidence that the aggregates passively diffuse within the cell, with diffusion constants that depend on their size in agreement with the Stokes-Einstein law. However, the aggregate displacements along the cell long axis are confined to a region that roughly corresponds to the nucleoid-free space in the cell pole, thus confirming the importance of increased macromolecular crowding in the nucleoids. We thus used 3D individual-based modeling to show that these three ingredients (diffusion, aggregation and diffusion hindrance in the nucleoids) are sufficient and necessary to reproduce the available experimental data on aggregate localization in the cells. Taken together, our results strongly support the hypothesis that the localization of aging-related protein aggregates in the poles of E. coli results from the coupling of passive diffusion-aggregation with spatially non-homogeneous macromolecular crowding. They further support the importance of ''soft'' intracellular structuring (based on macromolecular crowding) in diffusion-based protein localization in E. coli. Citation: Coquel A-S, Jacob J-P, Primet M, Demarez A, Dimiccoli M, et al. (2013) Localization of Protein Aggregation in Escherichia coli Is Governed by Diffusion and Nucleoid Macromolecular Crowding Effect. PLoS Comput Biol 9(4): e1003038.
Introduction
While aging is a fundamental characteristic of living systems, its underlying principles are still to be fully deciphered. Recent observations of ageing in unicellular models, in absence of genetic or environmental variability, have paved way to new quantitative experimental systems to address ageing's underlying molecular mechanisms [1,2]. Further, the notion of aging was extended beyond asymmetrically dividing unicellular organisms such as the budding yeast Saccharomyces cerevisiae or the bacterium Caulobacter crescentus -where a clear morphological difference and existence of a juvenile phase distinguishes between the aging mother cell and its daughter cells [3,4] -to symmetrically dividing bacteria. This pushed aging definition to demand functional asymmetry as minimal requirement for a system to age [5]. Specifically, Escherichia coli and Bacillus subtilis were shown to age as observed by loss of fitness at small generation scale (,10) [6-8 (for B. subtilis), [9][10][11] and increased probability of death at larger generation scale (up to 250 generations) [12]. Age in this system was defined as the number of consecutive divisions a cell has inherited the older cellular pole [7]; the sibling that inherits the older cell pole was shown to grow slower than the newer pole sibling.
From a cellular viewpoint, aging is arguably due to the accumulation of damage over time that degenerates cellular functions, ultimately affecting the survival of the organism [1,2]. In the case of E. coli, a significant portion of the age-related fitness loss is accounted for by the presence of protein aggregates that accumulate in the bacterial older poles [7,9,10]. Such accumulation is reminiscent of many known age-related protein folding diseases [1]. Preferential sequestration of damaged proteins is also observed in S. cerevisiae between the bud and the mother cell [13][14][15] and between specific intracellular compartments in yeast and mammalian cell [16,17]. Therefore spatial localization, as nonhomogeneous distribution of damaged protein aggregates in the cytoplasm, has been postulated to be an optimized strategy allowing cell populations to maintain large growth rates in the face of the accumulation of damages that accompany metabolism during cell life [14,18,19]. These results suggest that spatial localization of damaged protein aggregates could present an ageing process conserved across different living kingdoms. Given the documented link between protein aggregation and ageing, the short life-span, ease of quantification of large number of individuals, molecular biology and genetics accessibility of E. coli may make this bacterium into a relevant model system to elucidate protein aggregation role in a ageing.
A first obstacle along this path is to understand the mechanisms by which cells can localize protein aggregates at specific locations within their intracellular space. Generally, thermal agitation and the resulting diffusion (Brownian movement) of proteins forbid localization in space on long timescale, since diffusion is a mixing process that will render every accessible position equiprobable. Inside eukaryotic cells, active mechanisms such as directed transport or sub-compartmentalization by internal membranes permit to counteract the uniforming effects of diffusion. It is however known since the 1952 seminal paper by Alan Turing [20] that subtle interactions between chemical reactions and diffusion can spontaneously lead to steady states with non-uniform spatial extension. This is also true for bacteria, as exemplified by the spatial oscillations in the minCDE system [21] or in the case of diffusion-trapping coupling [22]. Recently, the importance of precise sub-cellular localization of proteins within bacteria has become apparent [23][24][25]. In absence of a general cytoskeletonbased directed, active transport mechanism nor internal membranes, this would favor diffusion-reaction based localization within bacteria (see however [26][27][28]).
Specifically it is still unclear whether for single-cell organisms, preferential localization mechanism of damaged proteins is based on active directed transport or passive Brownian diffusion. In S. cerevisiae, initial reports incriminated a role for active directed transport (actin cytoskeleton) or sub-compartmentalization (membrane tethering) in the segregation of molecular damages (damaged proteins, episomal DNA) in the mother cell [13,29]. Yet, more recent reports contradict the need for directed transport, e.g. on the actin cable, and favor diffusion-based localization [16,30,31].
In E. coli, protein aggregates have consistently been reported to localize in the cell poles and in the middle of the cell [7,9,10]. The number of distinct aggregates per cell seems to depend on the cellular environment. In non-stressed conditions, at most one aggregate per cell is observed with rare cases (,4%) of two foci per cell detected [6]; under heat shock, most cells contain two or three aggregates [8,9]. In heat-shock conditions, Winkler et al. [9] concluded in favor of a Brownian passive motion of the protein aggregates. This study also pointed out that one simple possible passive aggregate localization mechanism may be based on spatially non-homogeneous macromolecular crowding. Indeed, in healthy cells, the bacterial chromosome spontaneously condensates [32] thus delineating a restricted sub-region of the cell called ''nucleoid'', where molecular crowding is much larger than in the rest of the cytoplasm [33]. Macromolecular crowding then alternates along the cell long axis between low intensity zones (cytosol) and large intensity ones (nucleoid). Monte-Carlo dynamics modeling suggests that such non-homogeneous spatial distribution of the molecular crowding may be sufficient to localize large proteins to the cell poles [34]. In line with this proposal are experimental reports that the observed aggregates preferentially localize in the nucleoid-free regions of the cell [7,9], i.e. precisely in the regions of alleged lower macromolecular crowding. In spite of these hints though, whether the transport of the aging-related protein aggregates in E. coli is of a directed active nature or purely passive Brownian origin remains elusive, since contradictory results indicate that this process would include ATP-dependent stages [9].
Here, our aim is to determine whether the movement of agingrelated protein aggregates in E. coli is purely diffusive (Brownian) or includes some active process (ATP-dependent, directed transport or membrane tethering). To this aim, we devised an integrated approach combining time-lapse fluorescence microscopy of E. coli cells in vivo, open-source automated image analysis, and individual-based modeling. Our results strongly indicate that purely diffusive pattern of aggregates mobility combined with nucleoid occlusion underlie their accumulation in polar and midcell positions.
Results
Trajectory analysis of single protein aggregates
In vivo analysis of individual trajectories of proteins of interest (or aggregates thereof) is a powerful method to determine whether the movement of the target protein is of Brownian nature or additionally exhibits further ingredients (active directed transport, caging or corralling effects, transient trapping, anomalous subdiffusion) [15,[35][36][37][38][39][40]. Here, we focused on naturally forming protein aggregates tethered with the small heat-shock protein IbpA in E. coli whose spatio-temporal dynamics have been implicated in aging of the bacteria [7,13].
To characterize the motion of IbpA-tethered aggregates in single E. coli cells, we monitored intracellular trajectories of single foci of IbpA-YFP fusion proteins [7,41] in non-stressed conditions (37uC, in LB medium, see Materials and Methods). For the automatic quantification of the resulting time-lapse fluorescence microscopy movies, we developed dedicated image analysis and tracking software tools (see Materials and Methods). This software
Author Summary
Localization of proteins to specific positions inside bacteria is crucial to several physiological processes, including chromosome organization, chemotaxis or cell division. Since bacterial cells do not possess internal sub-compartments (e.g., cell organelles) nor vesicle-based sorting systems, protein localization in bacteria must rely on alternative mechanisms. In many instances, the nature of these mechanisms remains to be elucidated. In Escherichia coli, the localization of aggregates of misfolded proteins at the poles or the center of the cell has recently been linked to aging. However, the molecular mechanisms governing this localization of the protein aggregates remain controversial. To identify these mechanisms, we have devised an integrated strategy combining innovative experimental and modeling approaches. Our results show the importance of the increased macromolecular crowding in the nucleoids, the regions within the cell where the bacterial chromosome preferentially condensates. They indicate that a purely diffusive pattern of aggregates mobility combined with nucleoid occlusion underlies their accumulation in polar and mid-cell positions. suite performs automatic segmentation and tracking of the cells (Fig. 1A). Moreover, it automatically detects the fluorescence aggregates foci and monitors their movements relative to the cell in which they are located, with sub-pixel resolution.
Localization of protein aggregates is non-homogeneous along the cells. Detectable protein aggregates (in the form of localized fluorescence foci) were observed in half of the cells monitored (54%; N cells = 1625 recorded in 72 independent movies), in agreement with previous experimental reports [7,13]. No further foci were detected by doubling exposure time (see materials and methods). This suggests that smaller undetected aggregates that may exist either diffuse faster than the acquisition time and are therefore not recorded as ''localized'', or alternatively, that they merge into bigger, detectable aggregates before full maturation of the fluorophore (#7.5 min). [42]. Cells in nonstressed conditions tend to exhibit smaller copy numbers of distinct protein aggregates than in heat-shocked conditions (compare e.g. [7] with [9,10]). Accordingly, in our hands, nearly all the focicontaining cells (98.8%) displayed a single fluorescence focus within the imaging time, while the remaining cells had at most two foci. The distribution of the (initial) spatial localization of the aggregates is displayed in Fig. 1. As a convention, we denote the long axis of the bacteria cell as its x-axis and the short one as its yaxis (Fig. 1B). In this figure, we express the aggregate position relative to the cell center of mass, and rescale to [21, +1] 2 range. Thus the two bacterial poles correspond to x = 21 and x = +1 in this relative scale.
The histogram of the location of the aggregate at the starting point of each trajectory is shown in Fig. 1C. The distribution is highly non-homogeneous, with most of the aggregates predominantly localized at one of the two poles, and the others mainly around the middle of the cell. This distribution is similar to the results obtained in [7] (Fig. 2-E-F in [7]), except that here, since we do not differentiate between old and new poles, the amplitude of the polar modes in Fig. 1C are roughly symmetrical. Similar distributions were also obtained in heat-shock conditions [9,10].
The marked localization of the aggregates suggests they might be tethered to the membrane at the poles (and center) at some point of the aggregation process, thus restricting their motion. Fig. 1D shows the location of the polar aggregates at first detection (both poles were pooled). Because these experimental results are two-dimensional projections of three-dimensional positions, one cannot directly determine whether the aggregates are bound to the cell membrane or spread in the three dimensional cytoplasmic bulk. To this aim, we generated 10 4 aggregate positions (uniformly) at random in a volume of the same shape and dimensions than the cell pole. Fig. 1E shows the two-dimensional projection of these positions when the proteins were randomly located in the three-dimensional bulk whereas Fig. 1F shows the two-dimensional projections when the proteins were randomly located on the cytoplasmic membrane enclosing the bulk.
To quantify these plots, we analyzed the local density of protein positions in the two-dimensional projections. Assuming the 2d projection of the pole is a semi-ellipse of radii a x and a y (green dashed shapes in Fig. 1 D-G), its area is 1/2pa x a y . The area of the elementary semi-elliptic crescent D s (gray in Fig. 1G) delimited by the semi-ellipse of radii sa x and sa y (0,s,1) and that of radii (s+ds)a x and (s+ds)a y is thus A(D s ) = pa x a y ds(s+ds/2). Varying s between 0 and 1, we counted the number of aggregate positions n s found within the semi-elliptic disk D s and computed the corresponding density (correlation function) r(s) = n s /A(D s ). Therefore, when s approaches 1, r(s) represents the local density of aggregate positions close to the external boundary of the pole (green dashed shapes in Fig. 1 D-F) whereas for small s values, r(s)
gives the local density of aggregate positions close to center of the pole (i.e. the center of the semi-ellipse). To validate the approach, we generated random aggregate positions directly in two dimensions (uniform distribution inside the 2D semi-ellipse) and checked that the density r(s) exhibits a constant value with s ( Fig. 1H, dotted black line).
When r(s) was used to quantify the data of Fig. 1E (aggregates in the bulk), the density was roughly constant up to s,0.3 then decayed smoothly for larger values (Fig. 1H, blue line), in agreement with the slight decay of the aggregate density close to the semi-ellipse boundary that is already visible in Fig. 1E. By contrast, with the data shown in Fig. 1F (aggregates in the membrane around the bulk), r(s) increased first slowly with increasing s, then very abruptly close to 1 (Fig. 1H, red line), in agreement with the accumulation of aggregate locations close to the semi-ellipse boundary that is already visible in Fig. 1F. Therefore, these simulated data show that the distortions due to the projection in two dimensions are expected to manifest as strongly increasing r(s) values close to s = 1 if the aggregates are tethered to the membrane versus smoothly decreasing r(s) if the aggregates are randomly spread in the 3d bulk. Finally, we used this approach to quantify the experimental data described above (Fig. 1I). The local density r(s) exhibits a non-monotonous behavior. Close to the pole boundary, r(s) clearly shows an almost linear decay, which is very similar to the behavior observed at large s for bulk simulations (Fig. 1H). For small s values however, r(s) increases with s, thus indicating a phenomenon that hinders aggregate location close to the cell center. To conclude, these results plead in favor of a 3d bulk distribution of the aggregates in the poles, thus rejecting the hypothesis that they would be tethered or attached at the cell membrane.
Polar protein aggregates exhibit Brownian motion. We next quantified the displacements of the polar aggregates. To this end we used two temporal regimes in our time-lapse fluorescence microscopy. Low sampling frequency (LF; 0.33 Hz) over long time scale (LT; 5 minutes) was used to assure that we monitor entirely the range of displacements (n = 1149), whereas high frequency sampling (HF; 1.67 Hz) was used to increase the temporal resolution of the linear initial regimes observed in mean-squared displacements. These conditions were optimized considering the trade-off between extensive exposure leading to bleaching of the foci and satisfactory temporal resolution.
The resulting mean displacements along the x or y-axis are shown in Fig. S1. For both sampling frequencies, the mean displacement along the y-axis is roughly stationary and fluctuates around a close-to-zero value. In order to analyze the fluctuations, we rescaled the mean displacement along the y-axis so that it fluctuates around exactly zero.
The mean displacement along the x-axis displays a close-tolinear increase with time ( Fig. S1) with a slope of around 0.05 mm/ min, a value that corresponds to the linear approximation of the cell elongation rate under our experimental conditions (doubling time around 30 minutes, during which the cell half-length grows from <1.0-2.0 to <2.0-3.5 mm). Therefore, the raw movement of the aggregates along the x-axis is dominated by ballistic displacement toward the pole under the effect of cell elongation. We corrected for this passive transport by subtracting the increase rate of the cell elongation.
The corrected mean displacements ( Fig. 2A) are stationary and slightly fluctuate around zero, for both HF and LF trajectories. This is a typical characteristic of Brownian motion ('random walk'). To confirm this hypothesis one has to study higher-order moments of the displacement, in particular the second one. The Fig. 2B. The LF and HF data here again are in very good agreement, with the HF data nicely aligned on the LF ones. This agreement is an important test of the coherence and quality of our measurement and analysis methodology. The inset of the figure shows a magnification of the HF data until time t = 30 sec. For the first 10 to 15 seconds, the HF data exhibits a clear linear behavior. As expected from an unbounded Brownian motion the same slope was observed for both the xand y-axis. The non-zero intercept with the y-axis is typically due to the noise in the experimental determination of the aggregate position [43]. Such a linear dependence of the mean-squared displacement (MSD) is a further indication that the movement of the aggregates is Brownian diffusion, as one expects h u t ð Þ{u 0 ð Þ ð Þ 2 i~2Dut in the case of a random walk (where D x or D y are the diffusion constant in the xor y-direction, respectively). Using the first 15 seconds of the HF data, our estimates yield D x <5.1610 24 mm 2 /s and D y <4.0610 24 mm 2 /s. Note that these values are at best rough estimates since the data are averaged over aggregates of very variable initial sizes (whose mobility is expected to vary accordingly; see below). Nevertheless, the fact that the values for the xand y-axes are similar is another indication of the isotropy of the Brownian motion that seems to govern the movement of the aggregates. These values are compatible with previous experimental reports of the diffusion constants of large multi-protein assemblies in bacteria, such the origin of replication in E. coli (around 10 24 mm 2 /s [40]) but are significantly smaller than the values reported for single fluorescent proteins such as mEos2 or GFP (1 to 10 mm 2 /s [39,44]).
resulting mean-squared displacement u t ð Þ{u 0 ð Þ ð Þ 2 D E is dis- played in
Altogether the analysis of the first 15 seconds of the HF data pleads in favor of the hypothesis that the aggregates' motion is due to diffusion, thus excluding directed transport due to some active Coordinates along the x and y-axis are shown in red and black, respectively. Low frequency sampling trajectories (LF) are displayed using full lines and high frequency ones (HF) using open symbols. Light red and black swaths indicate + and 295% confidence intervals for the x-and y-axis data, respectively (for clarity, 2 and + intervals for the x-and y-axis data, respectively, mechanisms. In order to further quantify the diffusive character of the aggregates at hand, we divided the LF data into 5 classes of increasing initial median fluorescence (table 1), so as to average data over aggregates of more homogeneous size. As depicted in Fig. 3A & B, the initial slope of the MSD vs. Time curves increases when the average intensity in the class decreases. Assuming that the initial median fluorescence is proportional to the initial size of the aggregate, this result further supports a diffusive behavior, for which the diffusion constant is expected to decrease as the molecular size increases. Plotting the data in log-log scale (Fig. 3C & D) evidences initial slopes (i.e. exponents of a possible power law) of 1.04+/20.14 and 0.83+/20.08 (on y and x direction respectively; mean slope, excluding the highest fluorescent class, see discussion). Note that the data for the y-direction are expected to be less noisy because they were not subjected to correction for cell growth, unlike the data for the x-direction. Moreover, in the xdirection, the aggregate movements appear to be restricted by a ''soft'' boundary (i.e. not a membrane, see below), which is expected to hinder interpretations of the MSD curves. Even so, the value of the exponent in the x-direction (0.83) is not significantly smaller than 1.0 (one-tailed t-test, 0.01 significance level). Therefore, we conclude from this data that the MSD at short times (before saturation) evolves linearly with time (MSD,t a with a = 1), as expected from a Fick-like normal diffusion. This is in contrast with recent reports where anomalous diffusion was recorded (a in [0.40,0.75]) for RNA-protein assemblies [45,46] and further supports our conclusion favouring passive pure diffusion mechanism of protein aggregates in E. coli. In our results, anomalous diffusion can be excluded except for the largest aggregate class (black circles in fig. 3A-D); however, the movement amplitude of these very large aggregates is too low to allow precise quantification. We further controlled for growth rate and cell-length effects on the diffusion pattern observed. Among the different cells we imaged, the variations in cell growth were very small, whereas cell length varied appreciably. However, the initial slopes showed no significant differences when the data were clustered into sub-classes of cell lengths (Fig. S2).
are omitted) (A) Corrected mean displacement h u t ð Þ{u 0 ð Þ{u c t ð Þ ð Þ 2 i where u c (t) is
Protein aggregates exhibit passive confinement within bacterial polar region. Following the short, approximately 15 seconds linear regime, the MSD change with time decelerates (inset, Fig. 2B) and reaches full saturation after about 40 seconds (LF data, Fig. 2B). MSD Saturation occurs in both x-and y-data at values of 0.030 and 0.040 mm 2 , respectively. This corresponds to a restriction of the movements of the aggregates in a sub-region of the bacterial interior, with characteristic size <400 nm. To understand these results, one has to take into account the size variability of the monitored aggregates. Indeed, when considering the 5 sub-classes of aggregates (Table 1), the observed saturation levels are not constant but clearly decrease when the initial total fluorescence increases. This suggests that the size of the large aggregates is of the same order than the cell dimension, so that the intracellular space available for an aggregate of size r is in fact L-r,,L (where L is the cell size). In turn, this enables us to estimate the actual size of the foci, independent of the microscope resolution.
To this end, we used the data in Fig. 3A and C to obtain reliable estimates of the sizes of the aggregates and that of the intracellular subregion in which they are confined. We used an automatized procedure based on parameter optimization by an evolutionary strategy of the parameters of an individual-based simulation for constrained diffusion (see Materials and Methods for full description). In short, our strategy can be seen as an automatized fit of the values of 12 parameters: the dimensions L X and L Y = L Z of the intracellular subregion in which the aggregates are confined and the average radius r i and diffusion constant D i of the aggregates belonging to class i (i = 1..5). We start by setting these parameters to initial guess values. We then simulate the confined diffusion of 5,000 spherical molecules per class that are endowed with the corresponding values of r i and D i (i = 1..5) and diffuse in a spatial domain which size is given by the initial guess values of L X and L Y = L Z . The values of the MSD (averaged over each class) in the x and y directions are then sampled during the simulation at the same frequency (0.33 Hz) as in the LF data of Fig. 3A and C. We then compute the distance between the resulting curves for the simulated data and those for the experimental LF data. This distance represents the least-square error between the MSD predicted by the simulations using the initial values of the parameters. To minimize this error automatically, we used an evolutionary strategy called CMA-ES (see [47] and Material and Methods). CMA-ES automatically finds the values of the parameters that yield the smallest distance between experimental and simulation data. Note that because the correction procedure (that removes the effect of cell elongation) for the x-data is strong (i.e. the aggregate motion in this direction is originally dominated by cell elongation), it is difficult to apply this fitting procedure on the experimental MSD data for the xand y-axis simultaneously. This obliged us to fit the parameters separately on the x and y-data. The numerical values indicated in table 1 represent the average of these two fits (the ''6'' values indicate the total variation range).
The class-averaged MSD corresponding to the optimized parameters are shown in Fig. 3A&C as lines and are in agreement with the experimental data. The best-fit values for the cell dimension parameters are L X = 650 nm and L Y = L Z = 750 nm. Considering our pixel size (64 nm), we conclude that the value of the space available to the aggregate in the y and z directions is close to the whole space available in this direction (around 900 nm). Therefore the aggregate motion in the y and z directions seems to be constrained only by the inner cell membrane. In strong contrast, while the total intracellular space in the x-direction ranges from 2 to 5 mm (depending on the cell elongation), our fitting procedure indicates that the aggregates do not move beyond 650 nm in this direction. As they are initially located in the cell pole, this shows that the aggregates remain in the pole, where a passive mechanism hinders their free movement and keep them from leaving a subregion spanning <1/4th to 1/9th of the total cell length. This is coherent with the positioning of the nucleoids, hindering the diffusion of the aggregates (see below).
The experimental relation between the aggregate radius and diffusion constant, as determined by the fitting procedure above (listed in table 1), shows a good fit to a Stokes-Einstein relation: D(r) = C 0 /r where one expects C 0 = kT/(6pg) (describing simple sphere diffusion in a liquid of viscosity g) (Fig. 3E). We obtain a remarkable fit to the experimental data with C 0 = 47230 nm 3 /s. Note that a power-law fitting D(r) = C 0 /r b gives b = 0.89+/20.14 (thus that does not exclude b = 1.0) with a similar quality of fit (chisquare values). In the inset of Fig. 3E, the experimentally determined values of D are plotted as a function of 1/r. The relation is linear except perhaps for very large values of r, thus emphasizing the very good agreement with the Stokes-Einstein law.
These results confirm that some passive mechanism hinders the free movement of the aggregates along the long axis. They also show that the aggregates follow the Stokes-Einstein law even for large sizes since the diffusion constant of the aggregates simply decays as the inverse of its radius.
Simulation of aggregation, Brownian diffusion, and molecular crowding reconstitutes the observed patterns
If the spatial dynamics of protein aggregates in E. coli is indeed based exclusively on Brownian motion and aggregation, we should be able to reproduce by computer simulations the observed spatial patterns of the aggregates described here and in previous experimental reports [7,9,10]. In particular, given the relative simplicity of these two ingredients (diffusion+aggregation), individual-based modeling is a very convenient tool in this framework. Here, we used an individual-based model in which proteins diffuse via lattice-free random walks in a 3D domain that has the size of a typical E. coli cell (see Material and Methods and Fig. 4A). When two molecules meet along their respective random walks, they form a single aggregate with probability p ag . The size of the resulting aggregate is updated according to the size of its constituting molecules and its diffusion constant is updated as a function of the aggregate size using a simple Stokes-Einstein law.
In turn, such aggregates can combine to form bigger aggregates.
The above experimental finding suggests that the aggregate's Brownian motion is restricted in the x-direction over a distance that is comprised between 1/4 th to 1/9 th of the cell long axis. These numbers are in agreement with the hypothesis that the physical structures that hinders aggregate diffusion are the nucleoids, i.e. the subregion of the bacterial cytoplasm where the chromosome condensates, thus increasing molecular crowding and diffusional hindrance [33]. In a typical E. coli culture in exponential phase, most cells contain two nucleoids [48] and the free space between each nucleoid and the closest pole end is around 500-600 nm (deduced from [49]). We thus tested the hypothesis (already pointed out in [7] or [10]) that the restriction to the free movement of the obstacles along the long axis of the bacteria is caused by an increased molecular crowding in the nucleoids. To simulate this increased molecular crowding in our individual-based model, we added immobile non-reactive obstacles to represent the densely packed nucleoids (Fig. 4A, see Materials and Methods).
Each simulation is initiated by positioning N p single proteins at random inside the cells. The simulation then proceeds by moving the proteins via random walks and letting them aggregate (with resulting update of the aggregate radius and diffusion constant) when they encounter. The initial number of proteins N p was varied in order to account for different experimental conditions. We used N p = 100 to emulate experiments in non-stressed conditions (as in this work) where one does not expect the presence of large numbers of proteins in the aggregates. For experiments where aggregation is intensified using for example heat shock treatment [9,10], one generally expects to recover more proteins in the aggregates. In such conditions, the number of proteins in the aggregates was evaluated to be between 2,000 and roughly 20,000 molecules [10]. Here, to account for these data, we used at most a total of N p = 7,000 molecules in the simulations.
In the experiments, protein aggregates are detected by fluorescence microscopy when their size is large enough; though we have currently no means to quantify this threshold size, we estimate our detection limit to contain about 10-50 YFP copies, based on the detection of slow mobile particles [50]. Therefore, in the simulations, we had to arbitrarily fix a threshold for the number of proteins per aggregate above which the aggregate is considered large enough to be detected. In non-stressed conditions (N p = 100), we used a detection threshold of 30 proteins per aggregate, whereas with N p = 7,000 total molecules per cell, we varied the detection threshold between 30 and 1750 monomeric proteins per aggregates.
We first focused on the distribution of the position of the aggregates at their first detection along the bacterial long axis. The corresponding distribution is shown in Fig. 4B. In non-stressed conditions (N p = 100), the distribution of the aggregate position is qualitatively very similar to our experimental results (see Fig. 1C): most of the aggregates are located in the poles, the rest being mainly located in the cell center. This result is largely robust to the value of the detection threshold of the aggregates. For very large values of the detection threshold, the distributions of the aggregate location tend to decay inside the nucleoids and to increase in the poles. However, outside such extreme ranges, the spatial distribution of the aggregates is qualitatively robust to the detection threshold. In Fig. 4B, for instance, the full lines correspond to aggregate detection thresholds ranging from to 5 to 50 proteins per aggregate. The resulting spatial distributions are however similar and match all well with the experimental result (dashed line histogram). This robustness is also observed when the initial locations of the proteins are taken inside or immediately around the nucleoids (Fig. S3A & C). Importantly, in agreement with experimental reports, the nucleoids in the simulations are not fully impermeable to the aggregates, since the probability to observe large aggregates in the nucleoids is not null. Rather, the presence of the largest molecular crowding in the nucleoids reduces the probability that large aggregates form in the nucleoids.
In addition, starting the simulation with a non-homogeneous distribution of proteins, such as confinement to the nucleoid area or just englobing it, does not change the aggregate number per cell and distribution outcome, though a short delay in the accumulation of aggregates in observed (Fig. S3, compare e.g. B & D with Fig. 4C & E, respectively). This is dictated by the crowded nucleoid model we adopted with realistic mesh size that does not prohibit diffusion of monomeric proteins within the nucleoid crowded area.
Another intriguing experimental result is that the number of aggregates simultaneously present in a cell depends on the experimental conditions. In non-stressed conditions only 1.2% of the observed cells presented more than a single aggregate within the observation time [7] whereas in heat-shocked conditions a majority of the cells present two distinct aggregates, and the rest is roughly equally distributed between one and three protein aggregates per cell. These observations are faithfully reproduced by diffusion-aggregation mechanisms. Indeed, in our simulations emulating non-stressed conditions (100 proteins), a vast majority of the simulations develop a unique detectable aggregate within the simulation time (Fig. 4C). Despite the increase with time of the probability for a simulation to feature two detectable aggregates per cell, the probability to observe only one aggregate remains largely higher, even at the end of the simulations.
In simulated heat-shock conditions (see Fig. 4D, total of 7,000 proteins) with a very low aggregate detection threshold, virtually all simulations display at least 4 distinct detectable protein aggregates per cell from the very beginning of the simulation. These results are comparable with the experimental data obtained in [7], where the addition of streptomycin was shown to strongly increase the size and number of detected aggregates per cell (to 5 or more). When we increased the aggregate detection threshold (illustrated by a detection threshold of 1750 aggregates in Fig. 4E) the simulations showed very different behavior. For short simulation times, we mainly observed cells with a unique detected aggregate. As simulation time increases, the probability to detect a unique aggregate decays in favor of the probability of detect 2 and 3 aggregates simultaneously in the cells. Eventually, most of the simulations (around 60%) display two aggregates, and the others are roughly evenly shared between 1 and 3 detected aggregates per cell, in agreement with the experimental results reported in [10] that employ heat-shock triggered aggregation. These simulations predict that the number of detected aggregates in the cell is crucially dependent on two main factors: the aggregate detection threshold and the total number of aggregation-prone proteins. Therefore they suggest that the discrepancy observed concerning the number of aggregates per cell between non-stressed and heat shock conditions is due to the larger quantity of aggregate-prone proteins resulting from the heat shock. Taken together, our results show that the three basic ingredients we considered in our simulations (passive Brownian motion, aggregation, increased molecular crowding in the nucleoids) are sufficient to reproduce several experimental observations on the spatial distribution and number of protein aggregates in E. coli. Therefore, they confirm the conclusion drawn from our experimental results above that the movement of the chaperone protein aggregates in E. coli is driven by passive diffusion (Brownian motion). They moreover indicate that the observed non-homogeneous spatial distribution is not due to active or directed aggregate movement but is a mere result of the interplay between Brownian diffusion and molecular crowding.
Discussion
Our objective in this work was to decipher the mechanisms by which protein aggregates in E. coli localize to specific intracellular regions, i.e., cellular poles.
Using single-particle tracking of protein aggregates marked with the small heat shock chaperone IbpA (inclusion-proteins Binding Protein A) translationally-fused to the yellow fluorescence protein (YFP), our results indicate that protein aggregate movements are purely diffusive, with coefficient constants of the order of 500 nm 2 /s, depending on their size. Noteworthy, recent quantification of the movements and polar accumulation in the poles of MS2 multimeric RNA-protein complexes and fluorescentlylabelled chromosomal loci concluded a high degree of anomalous diffusion, as reflected by slopes of 0.4-0.75 in log-log plots of time-MSD relationships [45,46]. This suggests that unlike pure protein aggregates, these complexes have further significant interactions with cellular components.
Applying evolutionary strategy for parameters estimation under the hypothesis of confined diffusion, we used our experimental data to estimate the average size and diffusion constant of the aggregates and the distances over which their movement is confined. As expected, the aggregate diffusion constant decreases with increasing aggregate sizes, but, more surprisingly, we find that the relation between the aggregate diffusion constant and their size is in very good agreement with the Stokes-Einstein law, thus strengthening the demonstration of pure Brownian motion. The agreement with the Stokes-Einstein law, that predicts a decay of the diffusion constant as the inverse of the radius, D,1/r, was found valid for all the estimated aggregate radii, even as large as 250-270 nm.
Recent experimental tests of the validity of this law in E. coli were more ambiguous. Kumar and coworkers [51] quantified the diffusion of a series of 30-250 kDa fusion proteins (some of which contained native cytoplasmic E. coli proteins) in E. coli cytoplasm and found very strong deviation from the Stokes-Einstein laweven for small proteins-with very sharp decay of the diffusion constant D,1/r 6 . However, using GFP multimers of increasing sizes, Nenninger et al. [52] found very good agreement with Stokes-Einstein law from 20 to 110 kDa, i.e. up to tetramers, while the diffusion constant for pentamers (138 kDa) was found smaller than Stokes-Einstein prediction. Moreover, deviations from the Stokes-Einstein law was suggested an indication of specific interactions of the diffusing protein with other cell components. A tentative interpretation of our observation that even large cytoplasmic protein aggregates in E. coli do follow Stokes-Einstein law, would be that these aggregates actually have limited interactions with other cell components. This hypothesis would match very well with the putative protective function of the aggregates as scavengers of harmful misfolded proteins, allowing their retention within large, stable objects [1].
A second major finding of our study is the demonstration that the Brownian motion of the aggregates is restricted by the cell membrane in the section plane of the cell, while, along the cell long axis, the aggregates are confined to a region that roughly corresponds to the nucleoid-free space in the pole, thus confirming the importance of hindered diffusion in the nucleoids. In further support to this hypothesis, we used 3D individual-based modeling to show that these three ingredients are sufficient to explain the most salient experimental observations. Our simulations exhibit spatial distributions of the aggregates that are similar to those observed in non-stressed as well as heat-shock conditions. They also explain the differences in the number of distinct aggregates per cell as a mere difference in the total number of aggregationprone (misfolded) proteins. Therefore, our results strongly support the hypothesis that the localization of aging-related protein aggregates in the center and poles of E. coli is due to the coupling of passive diffusion-aggregation with the spatially non-homogeneous macromolecular crowding resulting from the localization of the nucleoid(s).
Our computational approach can be further extended to address asymmetric division of cellular components in dividing cells. Due to computation time limitations inherent to individualbased models, a valid approach to pursue would be to derive a mean-field model of the diffusion-coagulation process, using e.g. integro partial differential equations with position-dependent properties for the diffusion constant or operator (Laplacian) combined with a coagulation operator [53]. This approach would allow to reach simulated times large enough to account for several cell generations and focus on the location of the larger aggregates along the lineage.
simulations with different detection thresholds (an aggregate must contain at least 5, 10, 20 or 50 monomeric proteins, respectively, to be detected). Total number of proteins in the simulations N p = 100. The dashed line is an histogram showing the distribution of the experimental data. (C-E) Timeevolution of the probabilities to observe exactly 1 (red), 2 (green), 3 (blue) or more than for 4 (brown) distinct aggregates simultaneously in the simulations. The simulations emulated non-stressed conditions (C), with N p = 100 total proteins and aggregate detection threshold = 30 or heat-shock triggered aggregation, with N p = 7,000 total proteins and aggregate detection threshold = 30 (D) or 1750 (E). doi:10.1371/journal.pcbi.1003038.g004
As a whole, our results emphasize the importance for diffusionbased protein localization of the ''soft'' intracellular structuring of E. coli along the large axis due to increased macromolecular crowding in the nucleoids. In addition to this implication in the localization of aging-related protein aggregates, the structuring effect of the nucleoids has very recently been evidenced in the accurate and robust positioning of the divisome proteins (that mediate bacterial cytokinesis) [54] or the non-homogeneous spatial distribution of the transcription factor LacI [55].
Wild-type E. coli cells (e.g., without any expression of fluorescent proteins) exhibit qualitatively and quantitatively the same ageing phenotype in terms of gradual fitness as those expressing fluorescent markers (as the IbpA-YFP fusion) [7]. Yet, in previous studies, limited to very few ageing generations (typically less than 8), visible aggregates in wild-type ageing cells were seldomly detected, probably because of the phase contrast detection threshold (typically 500 nm) [7]. To ensure that polar accumulation of aggregates is indeed a phenotype of wild-type cells, we used the recently developed 'mother machine' microfluidics system [12] to grow ageing wild-type E. coli through .150 generations. Under these conditions, many of the ageing cells indeed accumulate clearly visible aggregates (Fig. S4), pointing to the validity of our approach to use the IbpA-yfp system for better detection [7]. The mechanism described here for IbpA-yfp tethered aggregates can be generalized as ample evidence exist for polar localization of aggregates resulting from heterologous over-expression of proteins, streptomycin treatment [7 and ref therein], large protein assemblies of fluorescently-labelled protein fusions (due to avidity of low multimerization propensity of some fluorescent proteins and independent of the diffusive positioning of the native proteins studied [13]), large RNA-protein assemblies [45,46]. In all cases, given the non-specific nature of hydrophobic interactions governing aggregate assembly, it is unsurprising that co-localization may occur amongst different aggregated polypeptides and chaperones, based on the common diffusive mechanism of polar accumulation described here. Moreover, our recent work demonstrates that large engineered RNA assemblies accumulate as well in the cells' poles [56, (electron microscope images therein)]. Therefore, the polar localization pattern of low diffusive elements in bacteria is not limited to large purely protein assemblies. We propose that it might be a more general process concerning other cell constituents, such as nucleic acids.
Materials and Methods
Bacterial strains
The sequenced wild-type E. coli strain, MG1655 [57] was modified to express an improved version of the YFP fluorescent protein fused to the C terminus of IbpA [35] under the control of the endogenous chromosomal ibpA promoter resulting in the MGAY strain. E. coli strains were grown in Luria-Bertani (LB) broth medium half salt at 37uC. For more information about the cloning, see [7], S.I.
Fluorescence time-lapse microscopy setup
After an overnight growth at 37uC, MGAY cultures were diluted 200 times. When the cells reached an absorbance 0.2 (600 nm), they were placed on microscope slide that was layered with a 1.5% agarose pad containing LB half salt medium. The agarose pad was covered with a cover-slide, the boarder of which was then sealed with nail polish oil.
Cells were let to recover for 1 hour before observation using Nikon automated microscope (ECLIPSE Ti, Nikon INTENSI-LIGHT C-HGFIE, 1006 objective) and the Metamorph software (Molecular Devices, Roper Scientific), at 37uC. Phase contrast and fluorescence images (25% lamp energy, 1 second illumination LF movies and 600 milliseconds for HF movies) were sampled at two different time-scales. For low-frequency (LF) movies, images were taken every 3 seconds for a total of 5 minutes, while for highfrequency (HF) movies, fluorescence images were taken about every 0.60 seconds for a total of 2 min (and phase contrast images were sampled about every 7 fluorescence images). Fluorescence excitation light energy level used here is 5-fold higher than previously described [7] to allow proportional decrease of exposure time, enabling a higher temporal resolution. Under these conditions, doubling the exposure time did not result in further detection of fluorescent foci yet resulted in accelerated bleaching that prevented consecutive time lapse imaging of the observed foci.
Image analysis and aggregate tracking
Phase contrast images were analyzed by customized software ''Cellst'' [58] for cell segmentation and single cell lineage reconstruction. Phase contrast images were denoised using the flatten background filter of Metamorph software for long movies or a mixed denoising algorithm [58] for fast movies. The mixed denoising algorithm combines two famous image denoising methods: NL-means denoising [59], which is patch-based and Total Variation denoising [60,61], which is used as regularization. The Cellst software was used to automatically segment the cells on most of the images, albeit when necessary, manual corrections were applied. At the end of the whole segmentation and tracking process, Cellst also calculates the lineage of every cell in the movie.
The fluorescent protein aggregates were detected by another customized software. Detection of each spot was realized using the a contrario methodology based on a circular patch model with a central zone of detection and an external zone of context. The patch radius was then optimized to optimally match the spot. The energy of each spot was computed in the following way: the total image was modeled as a sum of a constant background and 2D circular Gaussian curves, centered on the maximal intensity pixel of the detected spots with a deviation of 3 pixels. The quadratic minimum deviation between the image and the model enabled to calculate the Gaussian coefficients. These coefficients were considered the energy values of each spot. The coordinates of the detected spots were then refined to subpixel resolution. This was achieved by computing a weighted average of the coordinates of the pixels in a circular neighborhood of the detected spot. The weights were given by the intensity values of the pixels to which the local background is subtracted. The local background was then computed as the median value of the pixel intensity in the neighborhood. Only pixels having intensity bigger than the median value were considered in the weighted average. This algorithm has been tested on both synthetic and real image and it shown a precision of 1/10 of a pixel on very poor contrasted spots.
After detection and localization, the movements of the fluorescent aggregates were tracked and quantified by a third customized software named ''aggtracker'' based on the cell lineage and the detected spots. The algorithm uses the lineage and cell information to ensure that an aggregate is consistently tracked through points with points that are inside the same cell.
The output of this software are time-series for the coordinates x and y (in pixels) of each fluorescence spot as well as the affiliation of the spot to the cell it is in. The last step consisted in the projection of the coordinates of the fluorescence spot from their initial absolute values in the image (in pixels) to their value along the long and short axes of the 2d image of the cell. To this aim, we used active skeletons. A skeleton represents an object by a median line (the center line in the case of a tubular bacteria). Here we used one active skeleton for each cell, providing the long axis of the cell image (the median line) and its short axis (along the skeleton width). Active skeletons were adapted to bacteria in order to optimize the position of the skeleton in the image of the cell. The coordinates of the fluorescence spots were then expressed as the coordinate of the center of the fluorescence spot in the basis composed of the active skeletons that localize the cell long and short axes. We exploited the simple shape of the skeleton to estimate the total cell width and length as that of the respective skeleton. As a convention, we refer below to the aggregate coordinate along the long axis as the x-coordinate and that along the short axis as the y-coordinate.
In order to improve precision, aggregate trajectories made of less than 10 successive images in the movies were not further taken into account. In total, we obtained 1644 aggregate trajectories.
Individual-based modeling of protein aggregation
To simulate the diffusion and aggregation process of proteins in a single cell, we used a 3d individual-based lattice-free model. Each protein p was explicitly modeled as a sphere of radius r p centered at coordinates (x p , y p , z p ) in the 3d intracellular space of the cell. We simulated protein diffusion in the cell and aggregation as they encounter using as realistic conditions as possible. In particular, the radius and diffusion coefficient of the protein aggregates explicitly increased as they grow. Moreover, we explicitly modeled the larger molecular crowding in the nucleoids. Details of the simulations are as follows.
The bacterial cell was simulated as a 3d square cylinder with width and depth 1.0 mm [62] and length 4.0 mm (chosen to correspond to a bacterial cell just before division) and reflective boundaries. Note that we also ran simulations with more realistic cell shapes (i.e. spherical caps at cell ends) and did not find significant differences compared to square cylinders (except for the much higher computation cost with spherical caps).
Within each cells, we also explicitly modeled the larger molecular crowding found in the nucleoids. Indeed, in healthy cells, the bacterial chromosome condensates into a restricted subregion of the cell called ''nucleoid'', where molecular crowding is much larger than in the rest of the cytoplasm [33]. To model this increased molecular crowding in the nucleoids, we placed at random (with uniform probability) 50,000 bulky immobile, impenetrable and unreactive obstacles (radius 10 nm) in the region of the cell where a nucleoid is expected. Because cell cycle and DNA replication in E. coli are not synchronized, roughly 75% of the cells in exponential phase contain two nucleoids [48]. We thus explicitly positioned two nucleoids within the cell. The location and size of the two nucleoids were estimated from DAPIstained inverted phase contrast images of the nucleoids found in [49]. Both nucleoids were 3d square cylinders of length 1220 nm (along the cell long axis) and width and height 532 nm. Each nucleoid started at 540 nm from each cell pole and was centered on the cell long axis. The volume occupied by the two nuclei area thus formed is about 12%, which is consistent with literature [63,64].
Each simulation was initialized by positioning N p individual IbpA-YFP proteins (monomers) at non-overlapping randomly chosen (with uniform probability) locations in the free intracellular space of the cell (i.e. the whole interior of the cell minus the space occupied by the obstacles in the nucleoids). At each time step, each molecule is independently allowed to diffuse over a distance d that depends on the protein diffusion constant D p , according to d = (6 D p Dt) 1/2 , where Dt is the time step, in agreement with basic Brownian motion. Note that D p itself depends on the aggregate size r p (see below). The new position of the protein (x9,y9,z9) was then computed by drawing two random real numbers, h and c, uniformly distributed in [0, 2p] and [21,1], respectively, and spherical coordinates: x9 = x(t)+d sin(acos(c)) cos(h); y9 = y(t)+d sin(acos(c)) sin(h) and z9 = z(t)+d c where (x(t),y(t),z(t)) is the initial position of the protein. If the protein in this new position (x9,y9,z9) overlaps with any of the immobile obstacles (i.e. if there exists at least one obstacle such that the distance between the obstacle center and (x9,y9,z9) is smaller than the sum of their radii) the attempted movement is rejected (x(t+Dt),y(t+Dt),z(t+Dt)) = (x(t),y(t),z(t)). This classical approximation of the aggregate reflection by the static obstacles is not expected to change the simulation results significantly, but it drastically reduces the computation load. If no obstacle overlaps, the movement is accepted, i.e. (x(t+Dt),y(t+Dt),z(t+Dt)) = (x9,y9,z9).
After each molecule has moved once, the algorithm searches for overlaps between proteins. Two proteins are overlapping whenever the distance between their centers is smaller than the sum of their radii. Each overlapping pair was allowed to aggregate with (uniform) probability p ag (irrespective of their size). In our simulations, p ag was varied between 0.1 and 1.0 (limited at the lower band by simulation time needed to score enough aggregation events). To model the aggregation from two overlapping proteins, we could not, for computation time reasons, keep track of the shape of the aggregates (i.e. the individual location of each protein in the aggregates). Instead, we used the simplifying hypothesis that all along the simulation, the aggregates maintain a spherical shape with constant internal density. It follows that the radius of an aggregate C, born out of the aggregation of two aggregates A and B of respective size r A and r B is r C = (r A 3 +r B 3 ) 1/3 . Upon aggregation, we thus remove the aggregates A and B from the cell, and add a new aggregate with size r C , centered at the center of mass of the two former aggregates A and B.
Finally, to set the diffusion constant of the aggregates, we used the classical Stokes-Einstein relation for a Newtonian fluid, where the diffusion constant is inversely proportional to its radius. In our case, this leads to D p = D 0 r 0 /r p where r 0 and D 0 are the radius and diffusion constant, respectively, of individual (monomeric) IbpA-YFP molecules. Note that this relation could be violated for large molecules in the cytoplasm of E. coli [52,53]. In a subset of simulations, we used power law relations, such as D p /r p 26 , as suggested in [51], without noticeable change in our results (except that the time needed to reach a given threshold aggregate size was increased). Note that aggregation was considered irreversible in our model (i.e. aggregates do never breakdown into smaller pieces). This is in agreement with our experimental observations, where we never measured decay of foci fluorescence.
The diffusion constant of the 26-kDa GFP (radius 2 nm) in E. coli cytoplasm is around 7.0 mm 2 /s and that of the GFP-MBP fusion (72 kDa) around 2.5 mm 2 /s [37]. Using this data and the Stokes-Einstein relation combined to our constant spherical hypothesis, led to estimates of the radius and diffusion coefficient of the individual (monomeric) 39 kDa IbpA-YFP fusion of r 0 = 3 nm and D 0 = 4.4 mm 2 /s. The value of the time step Dt has to be small enough so that proteins cannot jump over each other during a single time step, meaning that the distance diffused during a single time step is limited by d,4r 0 . Using the definition for d above, one then has Dt,8/3 r 0 2 /D 0 <5 ms. Here, we used Dt = 1 ms yielding d = 5 nm for monomeric proteins.
Every simulation was run for a total of 2610 6 time steps. The translation of this value into real time is hardly possible since we have no indication of the experimental value of the aggregation probability per encounter p ag (see above) even less so of its dependence on the aggregate size. A lower bound can be estimated to 2 seconds real time (for 2610 6 time steps) if the aggregation is always diffusion limited (i.e. p ag = 1). On general grounds however, the experimental value of p ag can be expected to be smaller, so that the 2610 6 simulation time steps would correspond to more than this 2 seconds real time minimal value. For the results to be statistically significant, we ran n run simulations for each parameter and condition, with different realization of the random processes (initial location, random choice of the positions or of the aggregation events) and averaged the results over these n run simulations. In the results presented here we used n run = 10 3 .
Fitting procedure for the aggregate radius, diffusion constant and cell dimensions
The data from the LF movies were partitioned into 5 classes based on the aggregate fluorescence intensity at the beginning of the measured trajectory, yielding 5 pairs of experimental curves for the mean-squared displacement, h x exp The aim of the fitting procedure is to minimize the distance between the experimental and theoretical curves, ie to minimize the cost function: where the indices j are over the N time steps. The formulation of this cost function corresponds to the traditional least squares, so that the optimization procedure actually looks for best fits in the least-square sense (minimization of the squared residuals between the theoretical predictions and experimental observations). To minimize automatically the cost function F, thus adjusting the theoretical to the experimental curves, we used the C++ implementation of the evolutionary strategy algorithm CMA-ES [39] with population size 12 and 400 generations. For each initialization type, the graph shows the distribution of the localization at first detection of the protein aggregates along the long axis (x-axis) (A,C) and the time-evolution of the probabilities to observe exactly 1 (red), 2 (green), 3 (blue) or more than for 4 (brown) distinct aggregates simultaneously in the simulations (B,D). In A and C, the different curves correspond to different detection thresholds. For both initial locations inside the nucleoids (A,B), the simulations corresponded to non-stressed conditions (100 proteins, detection threshold = 30) and aggregation probability p ag = 1. (EPS) Figure S4 Aggregate formation in ageing wild type E. coli cells. E. coli wild-type K12 MG1655 cells were inoculated into the microfluidics ''mother machine'' device (see [12]) and grown at 37uC in LB media. Briefly, the dead-end part of channels maintains the old pole cell (top of images, white arrows); at the opposite side the channels are open to flow that washes away the progeny. Cells were followed by time-lapse phase contrast microscopy for 32 hours. As can be seen, protein aggregates (yellow arrowheads) are formed within the old-pole of the ageing cells.
i (t){x exp i (0) À Á 2 i~fx (t) and
Supporting Information
(EPS)
Figure 1 .
1Localization of the detected aggregates in the cells. (A) In each image on the time-lapse fluorescence movies, the bacterial cells are automatically isolated (each individual cell is given a unique random color). The aggregates appearing during the movie are automatically detected and their trajectory within the cell quantified (internal trajectories). (B) By convention, we referred to the projection of the aggregate location on the long axis of the cell as the x-component and that along the short axis as the y-component. (C) Histogram of the x-component of the initial position of the trajectories (total of 1,644 trajectories). Since the cell length at the start of the trajectory is highly variable, the x-component was rescaled by division by the cell half-length. After this normalization, the cell poles are located at locations 21.0 and 1.0 respectively, for every trajectory. (D) Experimentally measured positions of the aggregates detected in the poles (both poles pooled, n = 9,242 points). The green-dashed curves in (D-F) locate the 2d projection of the 3d semi-ellipsoid that was used to approximate the cell pole. (E) Synthetic data for bulk positions: 10,000 3d positions were drawn uniformly at random in the 3d semi-ellipsoid pole. The figure shows the corresponding 2d projections. (F) Synthetic data of membranary positions: 10,000 3d positions were drawn uniformly at random in the external boundary (membrane) of the 3d semi-ellipsoid pole. The figure shows the corresponding 2d projections. (G) To quantify figures D-F, the correlation function r(s) was computed as the density of positions located within crescent D(s) (gray). See text for more detail. (H-I) Local density of aggregate positions r(s) in the synthetic (H) and experimental (I) data shown in E (bulk, blue), F (membranary, red) and D (experimental, orange). The dashed black line shows the local density computed for 10,000 synthetic 2d positions that were drawn uniformly at random in the 2d semi-ellipse resulting from the 2d projection of the 3d pole ellipsoid (green dashed curve in D-F). doi:10.1371/journal.pcbi.1003038.g001
Figure 2 .
2Single-aggregate tracking analysis inside E. coli cells.
the applied correction. For the y-component, the correction is the time-average of the y-coordinate. For the x-component, the applied correction is cell growth : u x (t)~L t ð Þ{L 0 ð Þ ð Þ Dt where L(t) is the cell halflength at time t and Dt is the time interval between two consecutive images. (B) Corresponding mean squared displacements MSD c~h u t ð Þ{u 0 ð Þ{u c t ð Þ ð Þ 2 i . The inset shows a magnification of the HF results and their close-to-linear behavior for the first 10-15 seconds (dashed line). doi:10.1371/journal.pcbi.1003038.g002
Figure 3 .
3Size-dependence of the diffusion constants. Trajectories from the LF movies(Fig. 2)were clustered into 5 classes of increasing initial median fluorescence(Table 1)and the corresponding MSD were averaged in each class. Symbols (open circles) show the MSD for the x-(A) and ydirections (B) for each class. Curve colors correspond to the classes from Table 1(with median fluorescence increasing from top to bottom). The corresponding full lines show the results of the fitting procedure for each class (see text and Material and Methods). Panels (C) and (D) show the corresponding log-log plots, to explore for possible anomalous diffusion. The straight lines are linear fits over the initial regimes (first 21 seconds), before movement restriction starts saturating the MSDs. The slopes of these lines are the anomalous exponents as defined by MSD(t),t a . Each panel indicates the average (+/2 s.d.) of the exponents determined for the 4 smallest aggregates classes (thus excluding the largest class, represented by black circles). The resulting values of the diffusion constant D are plotted against the radius r in (E), keeping the same color code as in (A-D). Full circles indicate the values determined from fitting the MSD in the x-direction, while full squares show the values from the fit in the y-direction. The full line is a fit to a Stock-Einstein law D(r) = C 0 /r, yielding C 0 = 47.23610 3 nm 3 /s. The inset replots these data as a function of 1/r. doi:10.1371/journal.pcbi.1003038.g003
Figure 4 .
4Individual-based models of chaperone protein diffusion and aggregation. (A) Geometry of the 3D model used in individualbased models for E. coli intracellular space. Numbers indicate distances in mm. The blue boxes inside the bacteria locate the nucleoids, where increased molecular crowding is modeled by the insertion of bulk immobile obstacles. (B) Comparison between simulations and experiments of the localization at first detection of the protein aggregates along the long axis (x-axis). The full lines show the spatial distributions extracted from the
i~fy(t) where i = {1,…,5} indexes the intensity class. Corresponding theoretical values were obtained by individual-based simulations of confined random walks similar to those described above but modified as follows: the cells, of dimensions LX (length), LY = LZ = LYZ (height and width) were devoid of nucleoids or aggregation (aggregation probability p ag = 0) and we used N = 5,000 IbpA-YFP proteins. Each 12-uplet of parameters {LX, LYZ, r i , D i } yields two theoretical curves
Figure S1
S1Mean displacements of single-aggregates. The figure shows the time evolution of the mean displacement, u t ð Þ{u 0 ð Þ h i , where u~x or y (brackets denote averaging over the trajectories). Coordinates along the x and y-axis are shown in red and black, respectively. Low frequency sampling trajectories (LF) are displayed using full lines and high frequency ones (HF) using open symbols. The inset schematizes the increase of the cell half-length during growth that dominates the movement along the x-axis. (EPS) Figure S2 Diffusion measurements clustered by cell length. Trajectories from the LF movies (Fig. 2) were clustered into 4 classes corresponding to the cell size at the time of measurement: L#3.4 mm (light blue), 3.4 mm,L#4.0 mm (red), 4.0 mm,L#4.8 mm (lilac) or L$4.8 mm (green). The corresponding MSD were averaged in each class for the x-(A) and y-directions (B). (EPS) Figure S3 Simulation of aggregate formation dynamics and location for protein initializations in or around the nucleoids. The positions of the proteins monomers were initialized (uniformly) at random inside the nucleoids (A-B) or around (i.e. within a layer of 20 nm) around them (C-D).
Table 1 .
1Comparison of the estimated radius and diffusion constants of protein aggregates depending on their initial median fluorescence.Intensity classes
# aggregates
r (nm)
D (nm 2 /s)
I,1459
128
5462
837638
1459,I,2015
115
105611
55062
2015,I,2905
128
141633
437626
2905,I,4727
129
174610
26164
I.4727
128
273612
92626
Data are averages of the fits on the MSDx-and MSDy-data and +/2 values
locate min-max.
doi:10.1371/journal.pcbi.1003038.t001
PLOS Computational Biology | www.ploscompbiol.org
April 2013 | Volume 9 | Issue 4 | e1003038
AcknowledgmentsWe thank N. Hansen, for providing the source code of CMA-ES for various programming languages (downloadable at http://www.lri.fr/ ,hansen/cmaes_inmatlab.html). We also thank the CNRS-IN2P3 Computing Center (cc.in2p3.fr) for providing computer resources.Author Contributions
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|
[] |
[
"Introducing SPAIN (SParse Audio INpainter)",
"Introducing SPAIN (SParse Audio INpainter)"
] |
[
"Ondřej Mokrý \nFaculty of Mechanical Engineering\nBrno University of Technology\nBrnoCzech Republic\n",
"Pavel Záviška ",
"Pavel Rajmic [email protected] ",
"Vítězslav Veselý [email protected] \nFaculty of Mechanical Engineering\nBrno University of Technology\nBrnoCzech Republic\n",
"\nSignal Processing Laboratory\nBrno University of Technology\nBrnoCzech Republic\n"
] |
[
"Faculty of Mechanical Engineering\nBrno University of Technology\nBrnoCzech Republic",
"Faculty of Mechanical Engineering\nBrno University of Technology\nBrnoCzech Republic",
"Signal Processing Laboratory\nBrno University of Technology\nBrnoCzech Republic"
] |
[] |
A novel sparsity-based algorithm for audio inpainting is proposed by adapting the SPADE algorithm by Kitić et al. for audio declipping into the task of audio inpainting. SPAIN (SParse Audio INpainter) comes in synthesis and analysis variants. Experiments show that both A-SPAIN and S-SPAIN outperform other sparsity-based inpainting algorithms. Moreover, A-SPAIN performs on a par with the state-of-the-art method based on linear prediction in terms of the SNR, and, for larger gaps, SPAIN is even slightly better in terms of the PEMO-Q psychoacoustic criterion.
|
10.23919/eusipco.2019.8902560
|
[
"https://arxiv.org/pdf/1810.13137v4.pdf"
] | 53,109,833 |
1810.13137
|
59e70ea5dcbc2b497d31212489e25607ec2a237d
|
Introducing SPAIN (SParse Audio INpainter)
Ondřej Mokrý
Faculty of Mechanical Engineering
Brno University of Technology
BrnoCzech Republic
Pavel Záviška
Pavel Rajmic [email protected]
Vítězslav Veselý [email protected]
Faculty of Mechanical Engineering
Brno University of Technology
BrnoCzech Republic
Signal Processing Laboratory
Brno University of Technology
BrnoCzech Republic
Introducing SPAIN (SParse Audio INpainter)
Index Terms-InpaintingSparseCosparseSynthesisAnaly- sis
A novel sparsity-based algorithm for audio inpainting is proposed by adapting the SPADE algorithm by Kitić et al. for audio declipping into the task of audio inpainting. SPAIN (SParse Audio INpainter) comes in synthesis and analysis variants. Experiments show that both A-SPAIN and S-SPAIN outperform other sparsity-based inpainting algorithms. Moreover, A-SPAIN performs on a par with the state-of-the-art method based on linear prediction in terms of the SNR, and, for larger gaps, SPAIN is even slightly better in terms of the PEMO-Q psychoacoustic criterion.
I. INTRODUCTION
The term "inpainting" has spread to the audio processing community from the image processing field, when a paper by Adler et al. was published, entitled simply "Audio inpainting" [1]. The goal of so termed restoration task is to fill degraded or missing parts of data, based on its reliable part, and preferably in an unobtrusive way. Although the term "inpainting" is rather new in this context, the problem itself is much older and different approaches have been proposed to deal with it in past decades.
One of the first successful approaches was the time-domain interpolation of the missing audio samples. An adaptive interpolation method was presented by Janssen et al. in [2], which was built on modeling the signal as an autoregressive (AR) process. The method can be roughly described as the minimization of a suitable functional, which includes as the variables both the estimates of the missing samples and the AR coefficients of the restored signal. Although the algorithm is designed as iterative, only a single iteration was considered in [2]. On today's hardware, nonetheless, hundreds of iterations can be computed in a reasonably short time. Doing this improves the results substantially and keeps the Janssen algorithm the state-of-the-art in audio inpainting in terms of the SNR. Note, however, that when applied to compact "gaps", the AR-modeling suggests that this method is successful for sounds that contain harmonics and these do not "evolve" within the gap. As such, this approach is beneficial for gaps of up to ca 45 milliseconds, which can be observed also from the experiments described below. Various prediction methods were also presented in [3].
The work was supported by the joint project of the FWF and the Czech Science Foundation (GAČR): numbers I 3067-N30 and 17-33798L, respectively. Research described in this paper was financed by the National Sustainability Program under grant LO1401. Infrastructure of the SIX Center was used.
A different approach is the sparsity-based audio inpainting, reported more recently in [1]. It benefits from the observation that the energy of an audio signal is often concentrated in a relatively small number of coefficients, with respect to a suitable, redundant time-frequency transform. The goal is to find a signal that has sparse representation under a transform such as the STFT (Short-Time Fourier Transform, aka the Discrete Gabor Transform, DGT), while it belongs to the set of feasible solutions; for audio inpainting, a solution is feasible if it is identical to the reliable parts of the signal.
The above requirements can be formulated as an optimization task. Its solution cannot be attained in practice due to the NP-hardness, but the solution can be reasonably approximated, for example using the Orthogonal Matching Pursuit (OMP) as in [1]. Another kind of approximation uses the 1 norm, which leads to a convenient convex minimization [4]. One of the modern approaches is also to search for sparsity in a nonlocal way [5].
Further methods have been developed to deal with long missing segments of audio signal. Stationary signals can be restored using sinusoidal modeling [6] or via model-based interpolation schemes [7]. In repetitive signals such as rock and pop music, there is for the distorted part usually an intact one that is sufficiently similar and that can be used for filling the missing data [8]. A different method, still based on self-similarity, was introduced in [9], aiming originally at the packet loss concealment. One of the latest methods employs a deep neural network for the task of audio inpainting [10].
The algorithm presented in the paper is inspired by the state-of-the-art among sparsity-based approaches to audio declipping, the so-called SParse Audio DEclipper (SPADE) [11], [12] and it is in line with the universal framework presented in [13]. We benefit from the close relation between the sparsitybased inpainting and declipping problems. We formulate the two variants of the inpainting problem as the optimization tasks
min x,z z 0 s. t. x ∈ Γ and Ax − z 2 ≤ ε, (1a) min x,z z 0 s. t. x ∈ Γ and x − Dz 2 ≤ ε,(1b)
where (1a) and (1b) present the formulation referred to as the analysis and the synthesis variant, respectively. In both formulations, Γ = Γ(y) ⊂ C N is the set of feasible solutions (i.e. the set of time-domain signals that are equal to the observed signal y in its reliable parts), A : C N → C P is the analysis operator of a Parseval tight frame and D = A * is its synthesis counterpart, with P ≥ N . The formulation (1) reflects the goal to find a signal from Γ with the highestsparsity representation, as was informally stated above.
The intention is to make use of the Alternating Direction Method of Multipliers (ADMM) for the optimization of the above non-convex problems, as was the case in the original SPADE paper [11]. For this purpose, a more appropriate formulation is needed. We introduce the (unknown) sparsity k, which is to be minimized, and indicator functions 1 of two sets: the set of feasible solutions Γ and the set of k-sparse vectors, the latter denoted 0 ≤ k for short. This leads to
min x,z,k {ι Γ (x) + ι 0≤k (z)} s.t. Ax − z = 0, x − Dz = 0.(2)
Note that in (2), a more conservative constraint binding the variables x and z is used than it was in (1). The reason is that such a constraint falls within the general ADMM scheme [15]. Such an alteration is nevertheless legitimate, since in the resulting iterative algorithm presented below, the stopping criterion will in effect relax the constraint into the form of (1).
II. SPAIN (SPARSE AUDIO INPAINTER)
In [11], the SPADE algorithm was introduced to tackle (2). It is built on the idea that for a fixed k, problem (2) can be approximately solved by the ADMM. The SPADE increases the value of k during the iterations of ADMM (starting from a sufficiently small value), until the condition Ax − z 2 ≤ ε (analysis approach, A-SPADE) or x − Dz 2 ≤ ε (synthesis approach, S-SPADE) is met for the chosen tolerance ε > 0. This ensures k-sparsity of the signal coefficients during iterations and the algorithm thus provides an approximation of the solution to (1).
Section II-A shows that A-SPAIN can be derived from A-SPADE. In Subsection II-B, on the other hand, S-SPAIN is introduced as a brand new inpainting algorithm. The derivation is in line with the new variant of S-SPADE for declipping, proposed in [16].
A. A-SPAIN derived from A-SPADE
The A-SPADE algorithm (Alg. 1) was originally designed for the declipping task, where the set of feasible solutions Γ is the (convex) set of time-domain signals whose samples are identical to the observed ones in the reliable part, while the restored samples are required to lie above or below the upper or the lower clipping thresholds, respectively.
The key observation is that the formulations of inpainting and declipping differ only by the definition of the set of feasible solutions Γ, which is less restrictive for inpainting. Alg. 1 is formulated general enough to cover both the A-SPADE and the new A-SPAIN. In effect, the actual task being solved is a matter of the projection on line 3 of the algorithm.
The operator H k used in the algorithm denotes hard thresholding, which sets all but k largest elements of the argument to zero. Note that in our implementation, the structures of D and A and the fact that a real signal is being processed are taken into account, leading to thresholding conjugate pairs of coefficients at a time instead of thresholding of a single component.
Although the analysis version of the algorithm was derived from ADMM in [11] based on the formulation (2), the synthesis variant was surprisingly built on a different basis (see the technical report [17] for further explanation). A more consistent approach is to derive the synthesis variant in analogy to the analysis variant. Thus, in the following subsection, the novel derivation of S-SPAIN from ADMM is described.
B. S-SPAIN derived from ADMM
This subsection walks through the main steps leading to the S-SPAIN; for further details, see [17].
The ADMM is a scheme to solve optimization problems of the form min
z f (z) + g(Dz),(3)
where D is a linear operator. The functions f, g are assumed to be real and convex. Problem (3) can be reformulated by introducing a slack variable x such that x = Dz, leading to
min z f (z) + g(x) s.t. x − Dz = 0,(4)
which corresponds to the synthesis variant of (2). Next, the Augmented Lagrangian is formed as
L ρ (x, λ, z) = f (x)+g(z)+λ (x−Dz)+ ρ 2 x−Dz 2 2 ,(5)
where ρ > 0 is called the penalty parameter and λ ∈ R N is the dual variable. The general scheme of ADMM then consists of three steps-minimization of L ρ over z, minimization of L ρ over x and the update of the dual variable λ.
For the purpose of audio inpainting, we set f (z) = ι 0≤k (z) and g(x) = ι Γ (x). Note that such a function f is not convex, therefore the conditions of ADMM are not met. Nevertheless, ADMM may converge even in such a case and the experiments show that it provides reasonable results for audio inpainting.
It is convenient to introduce the so-called scaled dual variable u = λ/ρ at this moment. After this, it is straightforward to arrive at the ADMM steps for S-SPAIN in the following form:
z (i+1) = arg min z Dz − x (i) + u (i) 2 2 s.t. z 0 ≤ k, (6a) x (i+1) = arg min x Dz (i+1) − x + u (i) 2 2 s.t. x ∈ Γ, (6b) u (i+1) = u (i) + Dz (i+1) − x (i+1) .(6c)
Note that in the convex case, the minimizations over x and over z can be switched without affecting the convergence of ADMM [15]. We assume that the convergence will not be violated in the non-convex case either. The solution to the subproblem (6b) is the projection of (Dz (i+1) + u (i) ) onto Γ, which is easy to compute in the time domain. The subproblem (6a) is more challenging-it Algorithm 1: A-SPADE / A-SPAIN, depending on the particular choice of Γ Require: A, y, Γ, s, r, ε
1x (0) = y, u (0) = 0, i = 0, k = s 2z (i+1) = H k Ax (i) + u (i) 3x (i+1) = arg min x Ax −z (i+1) + u (i) 2 2 s.t. x ∈ Γ 4 if Ax (i+1) −z (i+1) 2 ≤ ε then 5 terminate 6 else 7 u (i+1) = u (i) + Ax (i+1) −z (i+1) 8 i ← i + 1 9
if i mod r = 0 then 10 k ← k + s 11 end 12 go to 2 13 end 14 returnx =x (i+1) corresponds to the sparse synthesis approximation, where the goal is to get as close as possible to the signal (x (i) − u (i) ) by synthesis using D in such a way that the expansion coefficients are k-sparse. Such a problem is NP-hard due to the nonorthogonality of D in practice; therefore approximate solutions are enforced. One possibility is to utilize hard thresholding and compute
z (i+1) ≈ H k D * (x (i) − u (i) ) .(7)
In [17] we show that such an approximation is reasonably close to the solution of (6a). Note that in A-SPAIN, on the contrary, the thresholding step provides an exact solution of the corresponding ADMM subproblem. Alternatively, z (i+1) can be approximated using k iterations of OMP [18], which was not considered neither in the original SPADE [11] nor the novel synthesis version [16]. Such an approach is, however, computationally much more demanding than the thresholding (since each iteration of OMP employs one synthesis and one analysis 2 ) and the experiments show that although it provides a better approximation for the solution of (6a), it surprisingly does not lead to a better result of the whole S-SPAIN algorithm. The S-SPAIN algorithm is presented in Alg. 2. In the following section, the S-SPAIN variants using hard thresholding and OMP as an approximation of step 2 of the algorithm will be denoted as S-SPAIN H and S-SPAIN OMP, respectively.
III. EXPERIMENTS AND RESULTS
A. Evaluation based on signal-to-noise ratio (SNR)
For the experiment, ten music recordings sampled at 16 kHz or 44.1 kHz were used, covering different degrees of sparsity of the STFT coefficients. The main source was the signals that were examined in the papers [1], [11], [20].
Algorithm 2: S-SPAIN, task in step 2 is to be approximated either by the hard thresholding or OMP Require: D, y, Γ, s, r, ε
1x (0) = D * y, u (0) = 0, i = 0, k = s 2z (i+1) = arg min z Dz −x (i) + u (i) 2 2 s.t. z 0 ≤ k 3x (i+1) = arg min x Dz (i+1) − x + u (i) 2 2 s.t. x ∈ Γ 4 if Dz (i+1) −x (i+1) 2 ≤ ε then 5 terminate 6 else 7 u (i+1) = u (i) + Dz (i+1) −x (i+1) 8 i ← i + 1 9
if i mod r = 0 then 10 k ← k + s In each test instance, the objective was to recover a signal containing six gaps, starting at random points in the signal such that the gaps do not overlap and are not too close to affect the restoration [21]. The gap length was chosen from the set of 10 available lengths, distributed between 5 and 50 ms.
As the competitors of SPAIN, we used the Janssen algorithm, the OMP and both the synthesis and the analysis approaches of the 1 relaxation. 3 In OMP and SPAIN, we used the overcomplete DFT with redundancy of the transform set to 2 (meaning that the number of frequency channels is two times higher than the number of signal samples). All algorithms were applied frame-wise: the signal was windowed using the Hann window 64 ms long with a shift of 16 ms (i.e. 75% overlap), and the restored blocks were combined using the overlap-add scheme. For the 1 relaxation, the DGT was used with the same window parameters and with number of frequency channels corresponding to length of the transform window in samples. Besides this, weighted 1 relaxation was also employed. 4 As the performance measure, we used the signal-to-noise ratio (SNR), defined SNR (x,x) = 10 log 10
x 2 2 x −x 2 2 [dB],(8)
wherex stands for the recovered gap and x denotes the corresponding segment of the uncorrupted signal [1]. Fig. 1 shows the overall results; for each gap length and each algorithm, the average value of SNR was computed from all the signals and all positions of the gap. It is clear that for gaps of up to 40 ms, Janssen and A-SPAIN outperform all other methods and that these two algorithms behave almost identically in terms of the SNR (the biggest difference is for the longest gaps greater than 45 ms, where the performance of Janssen drops). Regarding S-SPAIN, it performs like the two do for shorter gaps (up to 25 ms), and it outperforms the OMP and the 1 -relaxation methods even for the longer gaps. Note that in Fig. 1, results of S-SPAIN OMP are not presented. The reason is the computational complexity of this approach (a single test instance takes up to a few days!).
The scatter plot in Fig. 2 shows detailed comparison of S-SPAIN H with S-SPAIN OMP. Due to the computational time consumed by the S-SPAIN OMP, a shortened experiment was performed with both the gap lengths and the window length shortened to their quarter compared to the first experiment (i.e. window length 16 ms, overlap 4 ms, gap lengths ranging from 1.25 to 12.5 ms). Each cross in Fig. 2 corresponds to one test instance and its coordinates are SNR for S-SPAIN H and S-SPAIN OMP. The diagonal line in the plot divides the areas in which S-SPAIN H (under the line) or S-SPAIN OMP (above the line) perform better. It is clear that except for a small area corresponding to a very low SNR, S-SPAIN H outperforms S-SPAIN OMP in majority of cases. This result is quite surprising since the OMP is supposed to solve the optimization subproblem (6a) better than the simpler hard thresholding does.
A similar comparison is in Fig. 3, this time for S-SPAIN H versus A-SPAIN (back again for the setting corresponding to Fig. 1). Here, the difference between the synthesis and the analysis approach is more apparent than it was in Fig. 1. A possible explanation of the superiority of A-SPAIN is that the ADMM subproblems in A-SPAIN are solved exactly, whereas in S-SPAIN, the thresholding operator provides just an approximation of (6a).
B. Evaluation based on PEMO-Q
For further comparison of SPAIN with Janssen, we performed a second experiment, aiming at a more profound eval- uation using PEMO-Q [22]. In order to correctly evaluate the restored signals with PEMO-Q, only sound excerpts sampled at 44.1 kHz from the previous experiment had to be used (which makes six test signals in total). We also considered only the gaps with lengths from the set of {20, 30, 40, 50} ms. The measured quantity was the objective difference grade (ODG) which simulates the human perception, comparing the restored signal to the reference (original, not degraded signal). The ODG ranges from 0 to −4 and rates the audio degradation as: The results in terms of ODG are presented in Fig. 4. Each plot shows average values over the six audio samples, given the gap length. As expected, the plots indicate that the longer the gap, the worse the reconstruction. Nevertheless, the remarkable result here is that besides the gap length of 20 ms, A-SPAIN slightly outperforms both S-SPAIN H and Janssen.
C. Reproducible research
The MATLAB codes for SPAIN and the sound excerpts are available at https://bit.ly/2zdhbpp.
IV. CONCLUSION
The paper presented a novel inpainting algorithm (SPAIN) developed by an adaptation of successful declipping method, SPADE, to the context of inpainting. It was shown that the analysis variant of SPAIN performs the best in terms of SNR among sparsity-based methods. Furthermore, A-SPAIN was demonstrated to reach results on a par with the state-of-theart Janssen algorithm for audio inpainting in terms of SNR. Finally, the objective test using PEMO-Q, which takes into account the human perception of sound, showed that A-SPAIN even slightly outperforms Janssen for larger gap sizes.
Fig. 1 :
1Inpainting results in terms of the SNR.
Fig. 2 :
2Comparison of S-SPAIN H and S-SPAIN OMP.
Fig. 3 :
3Comparison of S-SPAIN H and A-SPAIN.
Fig. 4 :
4Average ODG values. Showing results for the observed, degraded signal (anchor, An) and the reconstruction by A-SPAIN (A), S-SPAIN H (S) and Janssen (J).
The indicator function of a set Ω, denoted ι Ω (x), attains the value 0 for x ∈ Ω and ∞ otherwise[14].
It may additionally employ the pseudoinverse, which, however, is not necessary, as implementation using QR factorization was used[19].
Implementation of the Janssen algorithm was taken from the Audio Inpainting Toolbox by Adler et al.[1]. OMP was implemented using the Sparsify Toolbox by Thomas Blumensath[19]. For the synthesis and the analysis 1 relaxations, our own implementations of the Douglas-Rachford and the Chambolle-Pock algorithms were used, respectively.4 In weighted 1 relaxation, the objective function is W · 1 . The diagonal matrix W allows us to favor chosen coefficients of the STFT when searching for the restored signal. The same weighting, i.e. according to 2 norm of atoms, was proposed in[1] for OMP.
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[] |
[
"PT spectroscopy of the Rabi problem",
"PT spectroscopy of the Rabi problem"
] |
[
"Yogesh N Joglekar \nDepartment of Physics\nIndiana University Purdue University Indianapolis (IUPUI)\n46202IndianapolisIndianaUSA\n",
"Rahul Marathe \nDepartment of Physics\nUniversity of Pune\n411007Ganeshkhind, PuneIndia\n",
"P Durganandini \nDepartment of Physics\nUniversity of Pune\n411007Ganeshkhind, PuneIndia\n",
"Rajeev K Pathak \nDepartment of Physics\nUniversity of Pune\n411007Ganeshkhind, PuneIndia\n"
] |
[
"Department of Physics\nIndiana University Purdue University Indianapolis (IUPUI)\n46202IndianapolisIndianaUSA",
"Department of Physics\nUniversity of Pune\n411007Ganeshkhind, PuneIndia",
"Department of Physics\nUniversity of Pune\n411007Ganeshkhind, PuneIndia",
"Department of Physics\nUniversity of Pune\n411007Ganeshkhind, PuneIndia"
] |
[] |
We investigate the effects of a time-periodic, non-hermitian, PT -symmetric perturbation on a system with two (or few) levels, and obtain its phase diagram as a function of the perturbation strength and frequency. We demonstrate that when the perturbation frequency is close to one of the system resonances, even a vanishingly small perturbation leads to PT symmetry breaking. We also find a restored PT -symmetric phase at high frequencies, and at moderate perturbation strengths, we find multiple frequency windows where PT -symmetry is broken and restored. Our results imply that the PT -symmetric Rabi problem shows surprisingly rich phenomena absent in its hermitian or static counterparts.
|
10.1103/physreva.90.040101
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[
"https://arxiv.org/pdf/1407.4535v1.pdf"
] | 118,724,584 |
1407.4535
|
8cdc4e81864dc529c4f2f3dbe5287e1d638d2137
|
PT spectroscopy of the Rabi problem
Yogesh N Joglekar
Department of Physics
Indiana University Purdue University Indianapolis (IUPUI)
46202IndianapolisIndianaUSA
Rahul Marathe
Department of Physics
University of Pune
411007Ganeshkhind, PuneIndia
P Durganandini
Department of Physics
University of Pune
411007Ganeshkhind, PuneIndia
Rajeev K Pathak
Department of Physics
University of Pune
411007Ganeshkhind, PuneIndia
PT spectroscopy of the Rabi problem
(Dated: July 18, 2014)
We investigate the effects of a time-periodic, non-hermitian, PT -symmetric perturbation on a system with two (or few) levels, and obtain its phase diagram as a function of the perturbation strength and frequency. We demonstrate that when the perturbation frequency is close to one of the system resonances, even a vanishingly small perturbation leads to PT symmetry breaking. We also find a restored PT -symmetric phase at high frequencies, and at moderate perturbation strengths, we find multiple frequency windows where PT -symmetry is broken and restored. Our results imply that the PT -symmetric Rabi problem shows surprisingly rich phenomena absent in its hermitian or static counterparts.
Introduction.
A two-level system coupled to a sinusoidally varying potential is a prototypical example of a time-dependent, exactly solvable Hamiltonian, with profound implications to atomic, molecular, and optical physics [1]. When the frequency of perturbation ω is close to the characteristic frequency ∆ of the two-level system -near resonance -the system undergoes complete population inversion for an arbitrarily small strength γ of the potential [2]. The implications of this result to spin magnetic resonance, Rabi flopping [3], and its generalization, namely the Jaynes-Cummings model [4,5], have been extensively studied over the past half century [6,7]. Surprisingly, the quantum Rabi problem, where the full quantum nature of the perturbing bosonic field is taken into account, has only been recently solved [8].
The two-level model is useful because it is applicable to many-level systems when the perturbation frequency is close to or resonant with a single pair of levels. As the detuning away from resonance |∆ − ω| increases, the perturbation strength necessary for population inversion increases linearly with it; in a many-level system, with increased potential strength, transitions to other levels have to be taken into account and the resultant problem is not exactly solvable. Therefore, understanding the behavior of a system in the entire parameter space (γ, ω) requires analytical and numerical approaches. All of these studies are restricted to hermitian potentials.
In recent years, discrete Hamiltonians with a hermitian tunneling term H 0 and a non-hermitian perturbation V that are invariant under combined parity and timereversal (PT ) operations have been extensively investigated [9][10][11][12][13][14][15][16]. The spectrum λ of a PT -symmetric Hamiltonian is real when the strength γ of the non-hermitian perturbation is smaller than a threshold γ P T set by the hermitian tunneling term. Traditionally, the emergence of complex-conjugate eigenvalues that occurs when the threshold is exceeded, γ > γ P T , is called PT symmetry breaking [17][18][19]. It is now clear that PT -symmetric Hamiltonians represent open systems with balanced gain and loss, and PT symmetry breaking is a transition from a quasiequilibrium state (PT -symmetric state) to a state with broken reciprocity (PT -broken state) [20]. Recent experiments on optical waveguides [21][22][23] and resonators [24] with amplification and absorption have shown that PT systems display a wealth of novel phenomena [25,26] that are absent in closed or purely dissipative systems. We note that all experiments and most of the theoretical work, with few exceptions [27][28][29], have only explored systems with static gain and loss potentials.
And what of a system perturbed by a time-periodic gain-loss potential V (t)? What is the criterion for PTsymmetry breaking? What is the analog of Rabi flopping in such a case? We answer these questions by investigating small, N -site lattices perturbed by a pair of balanced gain-loss potentials ±iγ cos(ωt) located at parity symmetric sites. Such systems can be realized in coupled waveguides with a complex refractive index [22,23,28] that varies along the propagation direction, or in coupled resonators [24].
Our primary results are as follows: i) The system can be in "PT -broken phase" even if the spectrum λ (t) of the Hamiltonian H(t) is real at all times. ii) Near every resonance, the PT symmetric threshold is reduced from its static value γ P T to the detuning, γ P T (ω) ∝ |ω − ∆|; in particular, it vanishes at the resonance. iii) For any gain-loss strength including γ γ P T , the PT -symmetric phase is restored at high frequencies ω > ω c ∝ γ. iv) At intermediate strengths, γ ∼ γ P T , the PT symmetry is broken in multiple windows in the frequency domain. Thus, a harmonic, PT -symmetric perturbation provides a new spectroscopic tool for investigating the level structure of a system. PT phase diagram. The Hamiltonian for an N -site lattice with constant tunneling is is given by x →x = N + 1 − x and the antilinear timereversal operator acts as i → −i. The spectrum of H 0 is given by n = −2 J cos(k n ) = − n and the normalized eigenfunctions are ψ n (x) = x|n = sin(k n x)/ 1 + 1/N where k n = nπ/(N + 1) with 1 ≤ n ≤ N . The energy differences ∆ nm = n − m > 0 define the possible resonances for this N -level system. Motivated by the Rabi problem, here we will only consider N = {2, 3, 4}. This system is perturbed by a balanced gain-loss potential
H 0 = − J N −1 x=1 (|x x + 1| + |x + 1 x|) ,(1)V (t) = i γ cos(ωt) (|x 0 x 0 | − |x 0 x 0 |) = V † (t). (2)
Eq.
(2) implies that at time t = 0, x 0 is the gain or amplification site andx 0 is the loss or absorption site. The nonhermitian potential satisfies PT V (t)PT = V (t). The total Hamiltonian
H(t) = H 0 + V (t) is periodic in time, i.e. H(t + 2π/ω) = H(t)
, and its properties are best analyzed via its Floquet counterpart, H = −i ∂ t + H [30][31][32]. In the frequency domain, the non-Hermitian, PTsymmetric Floquet Hamiltonian is given by
H p,q x,x = −ip ωδ p,q δ x,x − δ p,q J(δ x,x +1 + δ x,x −1 ) + i γδ x,x (δ p,q+1 + δ p,q−1 )(δ x,x0 − δ x,x0 )(3)
where p, q ∈ Z denote the Floquet band indices. In practice, a truncated Floquet Hamiltonian with |p| ≤ N f is used, so that H is an
N (2N f + 1) × N (2N f + 1) matrix.
We define PT -symmetry breaking as the emergence of complex-conjugate eigenvalues for the Floquet Hamiltonian H [27]. As we will show below, this can occur even if the instantaneous eigenvalues λ (t) of the time-dependent Hamiltonian H(t) are purely real over the entire period T ω = 2π/ω. Figure 1 is the PT -symmetric phase diagram of a twolevel system in the (γ, ω) plane. Figure 1(a) shows the maximum imaginary part of the spectrum of the Hamiltonian H with 101 Floquet bands. The region with zero imaginary part (dark blue) is the PT -symmetric phase and the region with nonzero imaginary part (all other colors) is the PT -broken phase; the static, ω = 0, threshold is given by γ P T /J = 1. Figure. 1(a) has three universal features that appear in systems with larger N as well. The first is a vanishingly small PT -symmetric threshold γ P T (ω) that occurs when ω is close to the single resonance frequency for the system, ∆ 21 = 2J. The second is the the emergence of PT -symmetric phase that occurs at large frequencies [29] ω > ω c ∝ γ for any gain-loss strength including γ/J ≥ 1. The third feature is the presence of multiple windows along the frequency axis such that the PT -symmetry is broken within each window. Fig. 1(b) is a higher resolution close-up of the parameter space marked by the white rectangle in Fig. 1(a). It shows that for a fixed γ, as the frequency of the PT potential is changed, multiple PT -symmetric and PTbroken regions emerge. These regions are present both below and above the static threshold.
A complementary method to distinguish the PTbroken (PTB) region from the PT -symmetric (PTS) region is to track the net intensity I(t) = ψ(t)|ψ(t) = ψ(0)|G † (t)G(t)|ψ(0) of an initially normalized state |ψ(0) . We obtain the non-unitary time-evolution operator
G(t) = T e − i t 0 dt H(t ) ≈ t/δt k=1 e − i δtH(kδt) ,(4)
where the discretization time-step δt/T ω 1 is chosen to ensure that G(t) is independent of it. Figure 1(c) shows the maximum intensity I max reached before time t at cutoff times t = 5T ω (open red circles) and t = 10T ω (solid blue squares), as a function of the PT -perturbation frequency ω on the vertical axis. These results are for γ/J = 0.9, initial state localized on the first site, and a time-step δt/T ω = 10 −5 . In the PT -symmetric region, I(t) undergoes bounded oscillations and therefore I max is the same for the two time-cutoffs. In a sharp contrast, in the PT -broken region, I(t) increases exponentially with time and I max doubles on the logarithmic scale when the cutoff is doubled. Thus, both approaches show the existence of multiple frequency windows where PT symmetry is broken; note that the phase-boundaries in Fig. 1(c) do not exactly match those in Fig. 1(b) due to the finite time-cutoff.
We remind the reader that when γ/J ≤ 1, the 2 × 2 Hamiltonian H(t) = i γ cos(ωt)σ z − Jσ x has a purely real spectrum λ (t) = ± [J 2 −γ 2 cos 2 (ωt)] 1/2 at all times, and yet, the norm of the time-evolution operator G(t) is either bounded or exponential-in-time at different frequencies [27]. Figure 2 shows the PT phase diagram of a three-level system obtained by using 81 Floquet bands. We plot the base-10 logarithm of the largest imaginary part of eigenvalues of H to easily distinguish the PT -symmetric phase (blue) and PT -broken phase (red). For a threelevel system, the unperturbed spectrum is given by
n = {0, ± √
2J}, and has two resonance frequencies ∆ 21 = √ 2J = ∆ 32 and ∆ 31 = 2 √ 2J. Fig. 2 shows a linearly vanishing γ P T (ω) at the first resonance ω/J = √ 2, a PT -restored phase at high frequencies, and a number of frequency windows where PT symmetry is broken at moderate values of γ/J 1. There is no PT -broken region near the second resonance, ω/J = 2 √ 2 (not shown) because V (t) does not connect states with same parity. Thus, the phase diagram for a three-level system shares the fundamental characteristics of that for a two-level system. Lastly, we consider a four-level system, N = 4. In this case, the resonances between states with opposite parity occur at ∆ 21 /J = 1 = ∆ 34 /J, ∆ 23 /J = ( √ 5−1) ≈ 1.236, and ∆ 14 /J = ( √ 5 + 1) ≈ 3.236. There are two inequivalent locations for the gain-loss potential, x 0 = 1 and x 0 = 2, and both have the same static threshold γ P T /J = 1. In the first case, only center-two of the four eigenvalues become complex at the threshold, whereas in the second case all four simultaneously become complex [33]. Figure 3 shows the PT phase diagram obtained with 41 Floquet bands. Both panels, (a) and (b), demonstrate the three salient features discussed earlier for both two and three-level systems. Understanding the phase boundaries. We now derive the PT -phase boundaries at occur at high frequencies or vanishingly small γ near a resonance. The natural time-scale for an unperturbed system is proportional to 1/J and its static PT breaking threshold is also set by J. At high frequencies ω/J 1, the rapidly varying potential iγ cos(ωt) is replaced by its average over the characteristic time-scale, γ av ∝ (γ/ω)J. For any gain-loss strength γ, no matter how large, increasing the frequency reduces the effective strength γ av and thus restores the PT -symmetric phase. The slope of the linear phaseboundary in the region ω/J 1 will depend upon the number of levels N and the location x 0 of the gain-loss potential, but the linear behavior of the phase boundary is universal.
Next we derive the cone-shaped phase boundary that occurs at small γ in the neighborhood of a resonance ω ∼ ∆ nm . For a state |ψ(t) = n c n (t) exp(−i n t/ )|n , the interaction-picture equation of motion for the leveloccupation coefficients c n (t) is given by
i∂ t c n (t) = N m=1 V nm (t)e +i∆nmt c m (t),(5)V nm (t) = iγ cos(ωt)[1 − (−1) n+m ] n|x 0 x 0 |m . (6)
For a two-level system, when ω ≈ ∆ 21 = 2J, averaging over high-frequency terms simplifies Eq.(5) to (7) Eq. (7) implies that when |ω − 2J| > γ, the coefficients c 1 (t) and c 2 (t) oscillate in time and remain bounded, and the system is in the PT -symmetric phase. When |ω − 2J| < γ, c 1,2 (t) increase with time exponentially, and the system is in the PT -broken phase. Thus, PTsymmetric phase boundary is given by γ P T (ω) = |ω−2J|, and the threshold gain-loss strength vanishes as ω → ∆ 21 = 2J. A visual inspection of Fig. 1(a) shows that, indeed, the slope of the phase-boundary lines fanning away from ω = 2J is one. Eq.(7) also implies that along the cone-shaped phase boundary, the net intensity I(t) grows quadratically with time [20,21,24]. When N = 3 a similar analysis with three equidistant levels implies that near resonance c 2 (t) satisfies a third-order differential equation, ∂ 3 t c 2 (t) + (ω − √ 2J) 2 − (3γ/4) 2 ∂ t c 2 (t) = 0. Therefore, the phase-boundary separating the PT -broken region from the PT -symmetric region is given by γ P T (ω) = (4/3)|ω− √ 2J|. This, too, can be verified by a visual inspection of Fig. 2. Our analysis also predicts that along this phase boundary, the net intensity of an initially normalized state increases quartically with time, i.e. I(t) ∝ t 4 , because ∂ 3 t c 2 (t) = 0. In an N -level system, our analysis is applicable to a pair of levels m, n when the PTperturbation frequency is close to the resonance that connects those levels, ω ∼ ∆ nm .
∂ 2 t c 1,2 (t) + (ω/2 − J) 2 − (γ/2) 2 c 1,2 (t) = 0.
Thus, a vanishingly small PT perturbation induces PT -symmetry breaking when the frequency of the perturbation is close to a resonance. Near resonance, the spatial oscillations of a state match the gain-loss temporal oscillations; as a result, it spends most of the time on the gain-medium site, leading to an exponential growth in the net intensity. Conclusion. In this paper, we have proposed the PTsymmetric Rabi model. We have shown that a harmonic, gain-and-loss perturbation leads to a rich PT phase diagram with three salient features. Among them is the existence of multiple frequency windows in which PTsymmetry is broken and restored. Time-dependent PT potentials have been extensively investigated in continuum one-dimensional optical structures [25,26]. Our results show that such potentials are a surprising spectroscopic probe, where the phase of the system -PT -broken or PT -symmetric -denotes the proximity of the perturbation frequency to a resonance of the system.
FIG. 1 .
1where |x is a normalized state localized on site x, J is the tunneling rate, and = h/(2π) is the scaled Planck's constant. The action of the parity operator on the latticearXiv:1407.4535v1 [physics.optics] (color online) (a) PT phase diagram of a two-level system in the (γ, ω) plane; plotted is the largest imaginary part of the spectrum of H. The static threshold γP T /J = 1 is suppressed down to zero when the perturbation frequency matches a resonance, ω/J = 2. (b) A close-up of the area marked by the white rectangle in panel (a) shows that for moderate γ/J ∼ 1, multiple PT -symmetric (PTS) and PT -broken (PTB) regions occur when ω is varied. (c) Maximum of the net intensity I(t) = ψ(0)|G † (t)G(t)|ψ(0) at two cutoff times t = 10π/ω (open red circles) and t = 20π/ω (solid blue squares), on horizontal logarithmic axis, as a function of ω on the vertical axis, at γ/J = 0.9 (white dot-dashed line in panel (b)). PTB regions are distinguished by a clear dependence of Imax on the cutoff.
FIG. 2 .
2(color online) PT -symmetric phase diagram of a three-level system; shown is the base-10 log of the maximum imaginary part of the spectrum of H. Its three featuresa linearly vanishing PT -threshold at the resonance ω/J = √ 2 ≈ 1.41, a restored PT -symmetric phase at high frequencies, and multiple frequency windows with PT -symmetric and PT -broken phases at moderate γ/J -are universal.
FIG. 3 .
3(color online) Phase diagram of a 4-site lattice with gain-loss potential at the ends, x0 = 1 (a), or in the center, x0 = 2 (b). Both panels show a restored PT -symmetric phase at high ω, and multiple frequency windows with PTS and PTB regions at moderate γ/J. They also show a linearly vanishing threshold γP T (ω) at each of the three relevant resonances, marked by dashed (a) or solid (b) white lines.
Although we have focused only on few-level systems here, our results are applicable to larger lattices, particularly in the vicinity of a resonance. Deep in the PTbroken phase, at long times, nonlinear effects also become relevant, although they do not affect our findings.This work was supported by NSF DMR-1054020 (YJ), Department of Science and Technology, India (RM), BUCD University of Pune (PD), and University of Pune research grant BCUD/RG/9 (RP). YJ thanks the hospitality of University of Pune where this work began.
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[] |
[
"Comptes Rendus Mathématique On a congruence involving q-Catalan numbers \"Elementary\" Number Theory / Théorie \"élémentaire\" des nombres On a congruence involving q-Catalan numbers Sur une congruence impliquant des q-nombres de Catalan",
"Comptes Rendus Mathématique On a congruence involving q-Catalan numbers \"Elementary\" Number Theory / Théorie \"élémentaire\" des nombres On a congruence involving q-Catalan numbers Sur une congruence impliquant des q-nombres de Catalan"
] |
[
"Ji-Cai Liu \nDepartment of Mathematics\nWenzhou University\n325035WenzhouPR China\n",
"Ji-Cai Liu \nDepartment of Mathematics\nWenzhou University\n325035WenzhouPR China\n"
] |
[
"Department of Mathematics\nWenzhou University\n325035WenzhouPR China",
"Department of Mathematics\nWenzhou University\n325035WenzhouPR China"
] |
[] |
Based on a q-congruence of the author and Petrov, we set up a q-analogue of Sun-Tauraso's congruence for sums of Catalan numbers, which extends a q-congruence due to Tauraso.Résumé. À partir d'une q-congruence de l'auteur et Petrov, nous établissons un q-analogue de la congruence de Sun-Tauraso pour des sommes de nombres de Catalan, qui étend la q-congruence due à Tauraso.
|
10.5802/crmath.35
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[
"https://comptes-rendus.academie-sciences.fr/mathematique/article/CRMATH_2020__358_2_211_0.pdf"
] | 119,617,732 |
1902.09983
|
6d51fdaaea1ff689d635b6a3b65bbfb075cabc42
|
Comptes Rendus Mathématique On a congruence involving q-Catalan numbers "Elementary" Number Theory / Théorie "élémentaire" des nombres On a congruence involving q-Catalan numbers Sur une congruence impliquant des q-nombres de Catalan
Ji-Cai Liu
Department of Mathematics
Wenzhou University
325035WenzhouPR China
Ji-Cai Liu
Department of Mathematics
Wenzhou University
325035WenzhouPR China
Comptes Rendus Mathématique On a congruence involving q-Catalan numbers "Elementary" Number Theory / Théorie "élémentaire" des nombres On a congruence involving q-Catalan numbers Sur une congruence impliquant des q-nombres de Catalan
Manuscript received 7th January 2020, revised and accepted 10th March 2020.Volume 358, issue 2 (2020), p. 211-215. This article is licensed under the CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE. Comptes Rendus Mathématique 2020, 358, n 2, p. 211-215
Based on a q-congruence of the author and Petrov, we set up a q-analogue of Sun-Tauraso's congruence for sums of Catalan numbers, which extends a q-congruence due to Tauraso.Résumé. À partir d'une q-congruence de l'auteur et Petrov, nous établissons un q-analogue de la congruence de Sun-Tauraso pour des sommes de nombres de Catalan, qui étend la q-congruence due à Tauraso.
Introduction
In combinatorics, the Catalan numbers are a sequence of natural numbers, which play an important role in various counting problems. The nth Catalan number is given by the following binomial coefficient:
C n = 2n n 1 n + 1 = 2n n − 2n 2n + 1 .
Closely related numbers are the central binomial coefficients 2n n for n ≥ 0. Both Catalan numbers and central binomial coefficients satisfy many interesting congruences (see, for instance, [7,[9][10][11]). In 2011, Sun and Tauraso [11] proved that for primes p ≥ 5,
p−1 k=0 2k k ≡ p 3 (mod p 2 ),(1)p−1 k=0 C k ≡ 3 2 p 3 − 1 2 (mod p 2 ),(2)
where · p denotes the Legendre symbol. In the past few years, q-analogues of congruences (q-congruences) for indefinite sums of binomial coefficients as well as hypergeometric series attracted many experts' attention (see, for example, [2][3][4][5][6]8,12,13]). It is worth mentioning that Guo and Zudilin [6] developed an interesting microscoping method to prove many q-congruences.
In order to discuss q-congruences, we first recall some q-series notation. The q-binomial coefficients are defined as
n k = n k q = (q;q) n (q;q) k (q;q) n−k if 0 k n, 0 otherwise,
where the q-shifted factorial is given by (a; q) n = (1 − a)(1 − aq) · · · (1 − aq n−1 ) for n ≥ 1 and (a; q) 0 = 1. Moreover, the q-integers are defined by [n] q = (1− q n )/(1− q), and the nth cyclotomic polynomial is given by
Φ n (q) = 1≤k≤n (n,k)=1 (q − e 2kπi /n ).
Recently, the author and Petrov [8] established a q-analogue for (1) as follows:
n−1 k=0 q k 2k k ≡ n 3 q n 2 −1 3 (mod Φ n (q) 2 ),(3)
which was originally conjectured by Guo [2] and generalises a q-congruence of Tauraso [12].
There are several natural q-analogues of Catalan numbers (see [1]). Here and throughout the paper, we consider the following q-analogue of Catalan numbers:
C n (q) = 1 [n + 1] q 2n n = 2n n − q 2n n + 1 .(4)
In 2012, Tauraso [12] obtained a weak q-version of (2) as follows:
n−1 k=0 q k C k (q) ≡ q n/3 if n ≡ 0, 1 (mod 3) −1 − q (2n−1)/3 if n ≡ 2 (mod 3) (mod Φ n (q)),
where x denotes the integral part of real x. In this note, we aim to set up a q-analogue of (2) as well as another related q-congruence for sums of binomial coefficients.
Theorem 1.
For any positive integer n, the following holds modulo Φ n (q) 2 :
n−1 k=0 q k C k (q) ≡ −q n 2 −1 3 − q n(2n−1) 3 if n ≡ 2 (mod 3) q n 2 −1 3 − n−1 3 (q n − 1) if n ≡ 1 (mod 3).(5)
In order to prove (5), we shall establish the following q-congruence.
Theorem 2.
For any positive integer n, the following holds modulo Φ n (q) 2 :
n−1 k=0 q k+1 2k k + 1 ≡ q n(2n−1) 3 if n ≡ 2 (mod 3), n−1
An auxiliary result
Lemma 3. For any positive integer n, the following holds modulo Φ n (q):
n−1 k=1 k − 1 3 (−1) k q 1 3 2k 2 −k k−1 3 − k(k−1) 2 1 − q k ≡ 0 if n ≡ 2 (mod 3), n−1 6 if n ≡ 1 (mod 3).(7)
Proof. Note that
n−1 k=1 (−1) k k − 1 3 q 1 3 2k 2 −k k−1 3 − k(k−1) 2 1 − q k = n−3 3 k=0 (−1) k q (k+1)(3k+2) 2 1 − q 3k+2 − n−1 3 k=1 (−1) k q k(3k+5) 2 1 − q 3k .
We shall distinguish two cases to prove (7).
Case 1. n ≡ 2 (mod 3)
. This case is equivalent to
n−1 k=0 (−1) k q (k+1)(3k+2) 2 1 − q 3k+2 − n k=1 (−1) k q k(3k+5) 2 1 − q 3k ≡ 0 (mod Φ 3n+2 (q)).(8)
Let ω be a primitive (3n + 2)th root of unity. Letting k → n − k in the following sum gives
n−1 k=0 (−1) k ω (k+1)(3k+2) 2 1 − ω 3k+2 = n k=1 (−1) n−k ω (n−k+1)(3n−3k+2) 2 1 − ω 3n−3k+2 = n k=1 (−1) n−k ω k(3k−1) 2 + (3n+2)(n+1) 2 −(3n+2)k 1 − ω 3n−3k+2 = n k=1 (−1) k ω k(3k+5) 2 1 − ω 3k ,
where we have used the fact that ω (3n+2)(n+1) 2 = (−1) n+1 . Thus,
n−1 k=0 (−1) k ω (k+1)(3k+2) 2 1 − ω 3k+2 − n k=1 (−1) k ω k(3k+5) 2 1 − ω 3k = 0,
which is equivalent to (8).
Case 2. n ≡ 1 (mod 3). Let ζ be a primitive (3n + 1)th root of unity. It suffices to show that
n−1 k=0 (−1) k ζ (k+1)(3k+2) 2 1 − ζ 3k+2 − n k=1 (−1) k ζ k(3k+5) 2 1 − ζ 3k = n 2 .(9)
Note that
n−1 k=0 (−1) k ζ (k+1)(3k+2) 2 1 − ζ 3k+2 = 2n k=n+1 (−1) 2n−k ζ (2n−k+1)(6n−3k+2) 2 1 − ζ 6n−3k+2 = 2n k=n+1 (−1) k ζ k(3k−1) 2 +(3n+1)(2n−2k+1) 1 − ζ −3k = − 2n k=n+1 (−1) k ζ k(3k+5) 2 1 − ζ 3k ,
where we replace k by 2n − k in the first step. Thus,
n−1 k=0 (−1) k ζ (k+1)(3k+2) 2 1 − ζ 3k+2 − n k=1 (−1) k ζ k(3k+5) 2 1 − ζ 3k = − 2n k=1 (−1) k ζ k(3k+5) 2 1 − ζ 3k .(10)
Furthermore, letting k → 2n + 1 − k on the right-hand side of (10) gives
n−1 k=0 (−1) k ζ (k+1)(3k+2) 2 1 − ζ 3k+2 − n k=1 (−1) k ζ k(3k+5) 2 1 − ζ 3k = − 2n k=1 (−1) 2n+1−k ζ (2n+1−k)(6n−3k+8) 2 1 − ζ 3(2n+1−k) = − 2n k=1 (−1) 1−k ζ (3k−1)(k−2) 2 +(3n+1)(2n+3−2k) 1 − ζ 1−3k = − 2n k=1 (−1) k ζ k(3k−1) 2 1 − ζ 3k−1 .(11)
An identity due to the author and Petrov [8, (2.4
)] says 2n k=1 (−1) k ζ k(3k−1) 2 1 − ζ 3k−1 = − n 2 .(12)
Then the proof of (9) follows from (11) and (12).
Proof of Theorem 2
Now we are in a position to prove Theorem 2. We recall the following identity:
n−1 k=0 q k 2k k + 1 = n−1 k=0 n − k − 1 3 q 1 3 2(n−k) 2 −(n−k) n−k−1 3 −3 2n k ,(13)
which was proved by Tauraso in a more general form (see [12,Theorem 4.2]). Since 1 − q n ≡ 0 (mod Φ n (q)), we have
1 − q 2n = (1 + q n )(1 − q n ) ≡ 2(1 − q n ) (mod Φ n (q) 2 ).
It follows that for 1 ≤ k ≤ n − 1,
2n k = (1 − q 2n )(1 − q 2n−1 ) · · · (1 − q 2n−k+1 ) (1 − q)(1 − q 2 ) · · · (1 − q k ) ≡ 2(1 − q n ) (1 − q −1 ) · · · (1 − q −k+1 ) (1 − q)(1 − q 2 ) · · · (1 − q k ) (mod Φ n (q) 2 ) = 2(q n − 1) (−1) k q − k(k−1) 2 1 − q k .(14)
Multiplying both sides of (13) by q and substituting (14) into the right-hand side of (13), we arrive at We complete the proof of (6) by combining (7) and (16).
n−k) 2 −(n−k) n−k−1 3 − k(k−1) 2 1 − q k (mod Φ n (q) 2 ). q k (mod Φ n (q)),where we set k → n − k in the first step. Thus, q k (mod Φ n (q) 2 ).(16)
(q n − 1) if n ≡ 1 (mod 3).(6)It is clear that(5) can be directly deduced from(3),(4)and(6). The remainder of the paper is organized as follows. We first set up a preliminary result in the next section, and prove Theorem 2 in Section 3.C. R. Mathématique, 2020, 358, n 2, 211-215
q-Catalan numbers. J Fürlinger, J Hofbauer, J. Comb. Theory, Ser. A. 40J. Fürlinger, J. Hofbauer, "q-Catalan numbers", J. Comb. Theory, Ser. A 40 (1985), p. 248-264.
Proof of a q-congruence conjectured by Tauraso. V J W Guo, Int. J. Number Theory. 151V. J. W. Guo, "Proof of a q-congruence conjectured by Tauraso", Int. J. Number Theory 15 (2019), no. 1, p. 37-41.
q-Analogues of two Ramanujan-type formulas for 1/π. V J W Guo, J.-C Liu, J. Difference Equ. Appl. 248V. J. W. Guo, J.-C. Liu, "q-Analogues of two Ramanujan-type formulas for 1/π", J. Difference Equ. Appl. 24 (2018), no. 8, p. 1368-1373.
Some new q-congruences for truncated basic hypergeometric series. V J W Guo, M J Schlosser, ID 268Symmetry. 11212 pagesV. J. W. Guo, M. J. Schlosser, "Some new q-congruences for truncated basic hypergeometric series", Symmetry 11 (2019), no. 2, article ID 268 (12 pages).
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A q-microscope for supercongruences. V J W Guo, W Zudilin, Adv. Math. 346V. J. W. Guo, W. Zudilin, "A q-microscope for supercongruences", Adv. Math. 346 (2019), p. 329-358.
On two conjectural supercongruences of Apagodu and Zeilberger. J.-C Liu, J. Difference Equ. Appl. 2212J.-C. Liu, "On two conjectural supercongruences of Apagodu and Zeilberger", J. Difference Equ. Appl. 22 (2016), no. 12, p. 1791-1799.
Congruences on sums of q-binomial coefficients. J.-C Liu, F Petrov, ID 102003Adv. Appl. Math. 11611 pagesJ.-C. Liu, F. Petrov, "Congruences on sums of q-binomial coefficients", Adv. Appl. Math. 116 (2020), article ID 102003 (11 pages).
From generating series to polynomial congruences. S Mattarei, R Tauraso, J. Number Theory. 182S. Mattarei, R. Tauraso, "From generating series to polynomial congruences,", J. Number Theory 182 (2018), p. 179- 205.
New congruences for central binomial coefficients. Z.-W Sun, R Tauraso, Adv. Appl. Math. 451Z.-W. Sun, R. Tauraso, "New congruences for central binomial coefficients", Adv. Appl. Math. 45 (2010), no. 1, p. 125- 148.
On some new congruences for binomial coefficients. Int. J. Number Theory. 73---, "On some new congruences for binomial coefficients", Int. J. Number Theory 7 (2011), no. 3, p. 645-662.
q-Analogs of some congruences involving Catalan numbers. R Tauraso, Adv. Appl. Math. 485R. Tauraso, "q-Analogs of some congruences involving Catalan numbers", Adv. Appl. Math. 48 (2012), no. 5, p. 603- 614.
Some q-analogs of congruences for central binomial sums. Colloq. Math. 1331---, "Some q-analogs of congruences for central binomial sums", Colloq. Math. 133 (2013), no. 1, p. 133-143.
|
[] |
[
"Fossilized condensation lines in the Solar System protoplanetary disk",
"Fossilized condensation lines in the Solar System protoplanetary disk"
] |
[
"A Morbidelli \nLaboratoire Lagrange\nUMR7293\nUniversité Côte d'Azur\nCNRS\nObservatoire de la Côte d'Azur. Boulevard de l'Observatoire\n06304, Cedex 4NiceFrance\n",
"B Bitsch \nDepartment of Astronomy and Theoretical Physics\nLund University\nBox 4322100LundSweden (\n",
"A Crida \nLaboratoire Lagrange\nUMR7293\nUniversité Côte d'Azur\nCNRS\nObservatoire de la Côte d'Azur. Boulevard de l'Observatoire\n06304, Cedex 4NiceFrance\n",
"M Gounelle \n) IMPMC, Muséum national d'histoire naturelle, Sorbonne Universités, CNRS, UPMC & IRD\n57 rue Cuvier, France (4) Institut Universitaire de France 103 bd Saint Michel75005, 75005Paris, ParisFrance (\n",
"T Guillot \nLaboratoire Lagrange\nUMR7293\nUniversité Côte d'Azur\nCNRS\nObservatoire de la Côte d'Azur. Boulevard de l'Observatoire\n06304, Cedex 4NiceFrance\n",
"S Jacobson \nLaboratoire Lagrange\nUMR7293\nUniversité Côte d'Azur\nCNRS\nObservatoire de la Côte d'Azur. Boulevard de l'Observatoire\n06304, Cedex 4NiceFrance\n\nUniversity of Bayreuth\nD-95440BayreuthGermany\n",
"A Johansen \nDepartment of Astronomy and Theoretical Physics\nLund University\nBox 4322100LundSweden (\n",
"M Lambrechts \nLaboratoire Lagrange\nUMR7293\nUniversité Côte d'Azur\nCNRS\nObservatoire de la Côte d'Azur. Boulevard de l'Observatoire\n06304, Cedex 4NiceFrance\n",
"E Lega \nLaboratoire Lagrange\nUMR7293\nUniversité Côte d'Azur\nCNRS\nObservatoire de la Côte d'Azur. Boulevard de l'Observatoire\n06304, Cedex 4NiceFrance\n"
] |
[
"Laboratoire Lagrange\nUMR7293\nUniversité Côte d'Azur\nCNRS\nObservatoire de la Côte d'Azur. Boulevard de l'Observatoire\n06304, Cedex 4NiceFrance",
"Department of Astronomy and Theoretical Physics\nLund University\nBox 4322100LundSweden (",
"Laboratoire Lagrange\nUMR7293\nUniversité Côte d'Azur\nCNRS\nObservatoire de la Côte d'Azur. Boulevard de l'Observatoire\n06304, Cedex 4NiceFrance",
") IMPMC, Muséum national d'histoire naturelle, Sorbonne Universités, CNRS, UPMC & IRD\n57 rue Cuvier, France (4) Institut Universitaire de France 103 bd Saint Michel75005, 75005Paris, ParisFrance (",
"Laboratoire Lagrange\nUMR7293\nUniversité Côte d'Azur\nCNRS\nObservatoire de la Côte d'Azur. Boulevard de l'Observatoire\n06304, Cedex 4NiceFrance",
"Laboratoire Lagrange\nUMR7293\nUniversité Côte d'Azur\nCNRS\nObservatoire de la Côte d'Azur. Boulevard de l'Observatoire\n06304, Cedex 4NiceFrance",
"University of Bayreuth\nD-95440BayreuthGermany",
"Department of Astronomy and Theoretical Physics\nLund University\nBox 4322100LundSweden (",
"Laboratoire Lagrange\nUMR7293\nUniversité Côte d'Azur\nCNRS\nObservatoire de la Côte d'Azur. Boulevard de l'Observatoire\n06304, Cedex 4NiceFrance",
"Laboratoire Lagrange\nUMR7293\nUniversité Côte d'Azur\nCNRS\nObservatoire de la Côte d'Azur. Boulevard de l'Observatoire\n06304, Cedex 4NiceFrance"
] |
[] |
The terrestrial planets and the asteroids dominant in the inner asteroid belt are water poor. However, in the protoplanetary disk the temperature should have decreased below water-condensation level well before the disk was photo-evaporated. Thus, the global water depletion of the inner Solar System is puzling. We show that, even if the inner disk becomes cold, there cannot be direct condensation of water. This is because the snowline moves towards the Sun more slowly than the gas itself. Thus the gas in the vicinity of the snowline always comes from farther out, where it should have already condensed, and therefore it should be dry. The appearance of ice in a range of heliocentric distances swept by the snowline can only be due to the radial drift of icy particles from the outer disk. However, if a planet with a mass larger than 20 Earth mass is present, the radial drift of particles is interrupted, because such a planet gives the disk a super-Keplerian rotation just outside of its own orbit. From this result, we propose that the precursor of Jupiter achieved this threshold mass when the snowline was still around 3 AU. This effectively fossilized the snowline at that location. In fact, even if it cooled later, the disk inside of Jupiter's orbit remained ice-depleted because the flow of icy particles from the outer system was intercepted by the planet. This scenario predicts that planetary systems without giant planets should be much more rich in water in their inner regions than our system. We also show that the inner edge of the planetesimal disk at 0.7 AU, required in terrestrial planet formation models to explain the small mass of Mercury and the absence of planets inside of its orbit, could be due to the silicate condensation line, fossilized at the end of the phase of streaming instability that generated the planetesimal seeds. Thus, when the disk cooled, silicate particles started to drift inwards of 0.7 AU without being sublimated, but they could not be accreted by any pre-existing planetesimals.
|
10.1016/j.icarus.2015.11.027
|
[
"https://arxiv.org/pdf/1511.06556v1.pdf"
] | 54,642,403 |
1511.06556
|
73ac4167aaa3d5499b6187b964be85f9b842db61
|
Fossilized condensation lines in the Solar System protoplanetary disk
20 Nov 2015
A Morbidelli
Laboratoire Lagrange
UMR7293
Université Côte d'Azur
CNRS
Observatoire de la Côte d'Azur. Boulevard de l'Observatoire
06304, Cedex 4NiceFrance
B Bitsch
Department of Astronomy and Theoretical Physics
Lund University
Box 4322100LundSweden (
A Crida
Laboratoire Lagrange
UMR7293
Université Côte d'Azur
CNRS
Observatoire de la Côte d'Azur. Boulevard de l'Observatoire
06304, Cedex 4NiceFrance
M Gounelle
) IMPMC, Muséum national d'histoire naturelle, Sorbonne Universités, CNRS, UPMC & IRD
57 rue Cuvier, France (4) Institut Universitaire de France 103 bd Saint Michel75005, 75005Paris, ParisFrance (
T Guillot
Laboratoire Lagrange
UMR7293
Université Côte d'Azur
CNRS
Observatoire de la Côte d'Azur. Boulevard de l'Observatoire
06304, Cedex 4NiceFrance
S Jacobson
Laboratoire Lagrange
UMR7293
Université Côte d'Azur
CNRS
Observatoire de la Côte d'Azur. Boulevard de l'Observatoire
06304, Cedex 4NiceFrance
University of Bayreuth
D-95440BayreuthGermany
A Johansen
Department of Astronomy and Theoretical Physics
Lund University
Box 4322100LundSweden (
M Lambrechts
Laboratoire Lagrange
UMR7293
Université Côte d'Azur
CNRS
Observatoire de la Côte d'Azur. Boulevard de l'Observatoire
06304, Cedex 4NiceFrance
E Lega
Laboratoire Lagrange
UMR7293
Université Côte d'Azur
CNRS
Observatoire de la Côte d'Azur. Boulevard de l'Observatoire
06304, Cedex 4NiceFrance
Fossilized condensation lines in the Solar System protoplanetary disk
20 Nov 2015
The terrestrial planets and the asteroids dominant in the inner asteroid belt are water poor. However, in the protoplanetary disk the temperature should have decreased below water-condensation level well before the disk was photo-evaporated. Thus, the global water depletion of the inner Solar System is puzling. We show that, even if the inner disk becomes cold, there cannot be direct condensation of water. This is because the snowline moves towards the Sun more slowly than the gas itself. Thus the gas in the vicinity of the snowline always comes from farther out, where it should have already condensed, and therefore it should be dry. The appearance of ice in a range of heliocentric distances swept by the snowline can only be due to the radial drift of icy particles from the outer disk. However, if a planet with a mass larger than 20 Earth mass is present, the radial drift of particles is interrupted, because such a planet gives the disk a super-Keplerian rotation just outside of its own orbit. From this result, we propose that the precursor of Jupiter achieved this threshold mass when the snowline was still around 3 AU. This effectively fossilized the snowline at that location. In fact, even if it cooled later, the disk inside of Jupiter's orbit remained ice-depleted because the flow of icy particles from the outer system was intercepted by the planet. This scenario predicts that planetary systems without giant planets should be much more rich in water in their inner regions than our system. We also show that the inner edge of the planetesimal disk at 0.7 AU, required in terrestrial planet formation models to explain the small mass of Mercury and the absence of planets inside of its orbit, could be due to the silicate condensation line, fossilized at the end of the phase of streaming instability that generated the planetesimal seeds. Thus, when the disk cooled, silicate particles started to drift inwards of 0.7 AU without being sublimated, but they could not be accreted by any pre-existing planetesimals.
Introduction
The chemical structure of a protoplanetary disk is characterized by a condensation front for each chemical species. It marks the boundary beyond which the temperature is low enough to allow the condensation of the considered species, given its local partial pressure of gas. If one assumes that the disk is vertically isothermal and neglects pressure effects, the condensation front is a vertical straight-line in (r, z) space. This is the reason for the wide-spread use of the term "condensation line". However, the vertical isothermal approximation is in many cases a poor proxy for the thermal structure of the disk (see below), so that in reality the condensation "line" is a curve in (r, z) space, like any other isothermal curve (Isella and Natta, 2005).
Probably the most important condensation line is that for water, also called the ice-line or the snowline. In the Solar System water accounts for about 50% of the mass of all condensable species (Lodders, 2003). The fact that the inner Solar System objects (terrestrial planets, asteroids of the inner main belt) are water poor, whereas the outer Solar System objects (the primitive asteroids in the outer belt, most satellites of the giant planets and presumably the giant planets cores, the Kuiper belt objects and the comets) are water rich, argues for the importance of the snowline in dividing the protoplanetary disk in two chemically distinct regions.
Thus, modeling the thermal structure of the disk has been the subject of a number of papers. There are two major processes generating heat: viscous friction and stellar irradiation. Chiang and Goldreich (1997), Dullemond et al. (2001 and Dullemond (2002) neglected viscous heating and considered only stellar irradiation of passive disks. They also assumed a constant opacity (i.e. independent of temperature). Chiang and Goldreich demonstrated the flared structure of a protoplanetary disk while the Dullemond papers stressed the presence of a puffed-up rim due to the face illumination of the disk's inner edge. This rim casts a shadow onto the disk, until the flared structure brings the outer disk back into illumination. Hueso and Guillot (2005), Davis (2005), Garaud and Lin (2007), Oka et al. (2011), Bitsch et al. (2014a and Baillié et al. (2015) considered viscous heating also and introduced temperature dependent opacities with increasingly sophisticated prescriptions. They demonstrated that viscous heating dominates in the inner part of the disk forṀ > 10 −10 M /y (Oka et al., 2011), whereṀ is the radial mass-flux of gas (also known as the stellar accretion rate) sustained by the viscous transport in the disk. In the most sophisticated models, the aspect ratio of the disk is grossly independent of radius in the region where the viscous heating dominates, although bumps and dips exist (with the associated shadows) due to temperature-dependent transitions in the opacity law (Bitsch et al., 2014a(Bitsch et al., , 2015. The temperature first decreases with increasing distance from the midplane, then increases again due to the stellar irradiation of the surface layer. The outer part of the disk is dominated by stellar irradiation and is flared as predicted earlier; the temperature in that region is basically constant with height near the mid-plane and then increases approaching the disk's surface.
As a consequence of this complex disk structure, the snowline is a curve in the (r, z) plane (see for instance Fig. 4 of Oka et al., 2011). On the midplane, the location of the snowline is at about 3 AU when the accretion rate in the disk isṀ = 3-10 × 10 −8 M /y. When the accretion rate drops to 5-10 × 10 −9 M /y the snowline on the midplane has moved to 1 AU (Hueso and Guillot, 2005;Davis, 2005;Garaud and Lin, 2007;Oka et al., 2011;Baillié et al., 2015;Bitsch et al., 2015). Please notice that a disk should not disappear before that the accretion rate decreases toṀ 10 −9 M /y (Alexander et al., 2014). The exact value of the accretion rate for a given snowline location depends on the disk model (1+1D as in the first four references or 2D as in the last one) and on the assumed dust/gas ratio and viscosity but does not change dramatically from one case to the other for reasonable parameters, as we will see below (eq. 9).
The stellar accretion rate as a function of age can be inferred from observations. Hartmann et al. (1998) found that on averageṀ = 10 −8 M /y at 1 My andṀ = 1-5×10 −9 M /y at 3 My. The accretion rate data, however, appear dispersed by more than an order of magnitude for any given age (possibly because of uncertainties in the measurements of the accretion rates and in the estimates of the stellar ages, but nevertheless there should be a real dispersion of accretion rates in nature). In some cases, stars of 3-4 My may still have an accretion rate of 10 −8 M /y (Hartmann et al., 1998;Manara et al., 2013).
The Solar System objects provide important constraints on the evolution of the disk chemistry as a function of time. Chondritic asteroids are made of chondrules. The ages of chondrules span the ∼ 3 My period after the formation of the first solids, namely the calcium-aluminum inclusions (CAIs; Villeneuve et al., 2009;Connelly et al., 2012;Bollard et al., 2014;Luu et al., 2015). The measure of the age of individual chondrules can change depending on which radioactive clock is used, but the result that chondrule formation is protracted for ∼ 3 My seems robust. Obviously, the chondritic parent bodies could not form before the chondrules. Hence, we can conclude that they formed (or continued to accrete until; Johansen et al., 2015) 3-4 My after CAIs.
At 3 My (typicallyṀ = 1-5×10 −9 M /y) the snowline should have been much closer to the Sun than the inner edge of the asteroid belt (the main reservoir of chondritic parent bodies). Nevertheless, ordinary and enstatite chondrites contain very little water (Robert, 2003). Some water alteration can be found in ordinary chondrites (Baker et al., 2003) as well as clays produced by the effect of water (Alexander et al. 1989). Despite these observations, it seems very unlikely that the parent bodies of these meteorites ever contained ∼ 50% of water by mass, as expected for a condensed gas of solar composition (Lodders, 2003).
One could think that our protoplanetary disk was one of the exceptional cases still showing stellar accretion 10 −8 M /y at ∼ 3 My. However, this would not solve the problem. In this case the disk would have just lasted longer, while still decaying in mass and cooling. In fact, the photo-evaporation process is efficient in removing the disk only when the accretion rate drops at 10 −9 M /y (see Fig. 4 of Alexander et al., 2014). Thus, even if the chondritic parent bodies had formed in a warm disk, they should have accreted a significant amount of icy particles when, later on, the temperature decreased below the water condensation threshold, but before the disk disappeared.
The Earth provides a similar example. Before the disk disappears (Ṁ ∼ 10 −9 M /y), the snowline is well inside 1 AU (Oka et al., 2011). Thus, one could expect that plenty of ice-rich planetesimals formed in the terrestrial region and our planet accreted a substantial fraction of water by mass. Instead, the Earth contains no more than ∼ 0.1% of water by mass (Marty, 2012). The water budget of the Earth is perfectly consistent with the Earth accreting most of its mass from local, dry planetesimals and just a few percent of an Earth mass from primitive planetesimals coming from the outer asteroid belt, as shown by dynamical models (Morbidelli et al., 2000;Raymond et al., 2004Raymond et al., , 2006Raymond et al., , 2007O'Brien et al., 2006O'Brien et al., , 2014. Why water is not substantially more abundant on Earth is known as the snowline problem, first pointed out clearly by Oka et al. (2011). Water is not an isolated case in this respect. The Earth is depleted in all volatile elements (for lithophile volatile elements the depletion progressively increases with decreasing condensation temperature; McDonough and Sun, 1995). Albarede (2009), using isotopic arguments, demonstrated that this depletion was not caused by the loss of volatiles during the thermal evolution of the planet, but is due to their reduced accretion relative to solar abundances. Furthermore, a significant accretion of oxidized material would have led to an Earth with different chemical properties (Rubie et al., 2015). Mars is also a water-poor planet, with only 70-300ppm of water by mass (McCubbin et al., 2012).
Thus, it seems that the water and, more generally, the volatile budget of Solar System bodies reflects the location of the snowline at a time different from that at which the bodies formed. Interestingly and never pointed out before, the situation may be identical for refractory elements. In fact, a growing body of modeling work (Hansen, 2009;Walsh et al., 2011;Jacobson and Morbidelli, 2014) suggests that the disk of planetesimals that formed the terrestrial planets had an inner edge at about 0.7 AU. This edge is required in order to produce a planet of small mass like Mercury (Hansen, 2009). On the midplane, a distance of 0.7 AU corresponds to the condensation line for silicates (condensation temperature ∼ 1300 K) for a disk with accretion rateṀ ∼ 1.5 × 10 −7 M /y, typical of an early disk. Inside this location, it is therefore unlikely that objects could form near time zero. The inner edge of the planetesimal disk at 0.7 AU then seems to imply that, for some unknown reason, objects could not form there even later on, despite the local disk's temperature should have dropped well below the value for the condensation of silicates. Clearly, this argument is more speculative than those reported above for the snowline, but it is suggestive that the snowline problem is common to all chemical species. It seems to indicate that the structure of the inner Solar System carries the fossilized imprint of the location that the condensation lines had at an early stage of the disk, rather than at a later time, more characteristic of planetesimal and planet formation; hence the title of this Note. Interestingly, if this analogy between the silicate condensation line and the snowline is correct, the time of fossilization of these two lines would be different (the former corresponding to the time whenṀ ∼ 1.5 × 10 −7 M /y, the latter whenṀ ∼ 3 × 10 −8 M /y).
The goal of this Note is to discuss how this might be understood. This Note will not present new sophisticated calculations, but simply put together results already published in the literature and connect them to propose some considerations, to our knowledge never presented before, that may explain the fossilization of the condensation lines, with focus on the snowline and the silicate line.
Below, we start in Sect. 2 with a brief review of scenarios proposed so far to solve the snowline and the 0.7 AU disk edge problems. In Sect. 3 we discuss gas radial motion, the radial displacement of the condensation lines and the radial drift of solid particles. This will allow us to conclude that the direct condensation of gas is not the main process occurring when the temperature decreases, but instead it is the radial drift of particles from the outer disk that can repopulate the inner disk of condensed species. With these premises, in Sect. 4 we focus on the snowline, and discuss mechanisms for preventing or reducing the flow of icy particles, so to keep the Solar System deficient in ice inside ∼ 3 AU even when the temperature in that region dropped below the ice-condensation threshold. In Section 5 we link the inner edge of the planetesimal disk to the original location of the silicate condensation line and we attempt to explain why no planetesimals formed inside this distance when the temperature dropped. A wrap-up will follow in Sect. 6 and an Appendix on planet migration in Sect. 7.
Previous models
The condensation line problem is a subject only partially explored. For the snowline problem, Livio (2012, 2013) proposed that the dead zone of the protoplanetary disk piled up enough gas to become gravitationally unstable. The turbulence driven by self-gravity increased the temperature of the outer parts of the dead zone and thus the snowline could not come within 3 AU, i.e. it remained much farther from the star than it would in a normal viscously evolving disk. This model, however, has some drawbacks. First, it predicts an icy region inside of the Earth's orbit, so that Venus and Mercury should have formed as icy worlds. Second, from the modeling standpoint, the surface density ratio between the deadzone and the active zone of the disk is inversely proportional to the viscosity ratio only in 1D models of the disk. In 2D (r, z) models (Bitsch et al., 2014b) the relationship between density and viscosity is non-trivial because the gas can flow in the surface layer of the disk. Thus, the deadzone may not become gravitationally unstable.
Hubbard and Ebel (2014) addressed the deficiency of the Earth in lithophile volatile elements. They proposed that grains in the protoplanetary disk are originally very porous. Thus, they are well coupled with the gas and distributed quite uniformly along the vertical direction. The FU-Orionis events, that our Sun presumably experienced like most young stars, would have heated above sublimation temperature the grains at the surface of the disk. Then, the grains would have recondensed, losing the volatile counterpart and acquiring a much less porous structure and a higher density. These reprocessed grains would have preferentially sedimented onto the disk's midplane, featuring the major reservoir of solids for the accretion of planetesimals and the planets. Planetesimals and planets would therefore have accreted predominantly from volatile depleted dust, even though the midplane temperature was low. This model is appealing, but has the problem that the phase of FU-Orionis activity of a star lasts typically much less than the disk's lifetime. Thus eventually the devolatilazation of the grains would stop and the planetesimals and planets would keep growing from volatile-rich grains. Also, it neglects the radial drift of icy particles on the mid-plane from the outer disk.
Concerning the inner edge of the planetesimal disk at 0.7 AU, an explanation can be found in Ida and Lin (2008). The authors pointed out that the timescale for runaway growth of planetary embryos decreases with heliocentric distance. Because the radial migration speed of embryos is propotional to their mass (Tanaka et al., 2002), the innermost embryos are lost into the star and are not replaced at the same rate by embryos migrating inward from farther out. This produces an effective inner edge in the solid mass of the disk, that recedes from the Sun as time progresses (see Fig. 2 of Ida and Lin, 2008). The major issue here is whether planets and embryos can really be lost into the star. The observation of extrasolar planets has revealed the existence of many "hot" planets, with orbital periods of a few days. Clearly, these planets would be rare if there had existed no stopping mechanism to their inward migration, probably due to the existence of an inner edge of the protoplanetary disk where the Keplerian period is equal to the star's rotation period (Koenigl, 1991;Lin et al., 1996), acting like a planet-trap (Masset et al., 2006). The presence of planet-trap would change completely the picture presented in Ida and Lin (2008) (see for instance Cossou et al., 2014).
More recently, Batygin and Laughlin (2015) and Volk and Gladman (2015) proposed that the the Solar System formed super-Earths inside of 0.7 AU, but these planets were lost, leaving behind only the "edge" inferred by terrestrial planet formation models. In Batygin and Laughlin (2015), the super-Earths are pushed into the Sun by small planetesimals drifting by gas-drag towards the Sun and captured in mean motion resonances. Again, we are faced with the issue of the probable presence of a planet-trap at the inner edge of the protoplanetary disk. With a planet-trap, the super-Earths would probably not have been removed despite of the planetesimals push. In Volk and Gladman (2015), instead, the super-Earths become unstable and start to collide with each other at velocities large enough for these collisions to be erosive. There is no explicit modeling, however, of the evolution of the system under these erosive collisions. We expect that the debris generated in the first erosive collisions would exert dynamical friction on the planets and help them achive a new, stable configuration (see for instance Chambers, 2013). Thus, we think it is unlikely that a system of super-Earths might disappear in this way.
From this state-of-art literature analysis it appears that the condensation line problem is still open. Thus, we believe that it is interesting to resume the discussion and approach the problem globally, i.e. addressing the general issue of the "fossilization" of condensation lines at locations corresponding to some "early" times in the disk's life.
Relevant radial velocities
In this section we review the radial velocities of the gas, of the condensation lines and of solid particles. This will be important to understand how a portion of the disk gets enriched in condensed elements as the disk evolves and cools, and it will give hints on how a region could remain depleted in a chemical species even when the temperature drops beyond its corresponding condensation value.
The seminal work for the viscous evolution of a circumstellar disk is Lynden-Bell and Pringle (1974). We consider the disk described in their section 3.2, which can be considered as the archetype of any protoplanetary disk, which accretes onto the star while spreading in the radial direction under the effect of viscous transport. The viscosity ν is assumed to be constant with radius in Lynden-Bell and Pringle's work, but the results we will obtain below are general for a viscously evolving disk, even with more realistic prescriptions for the viscosity (whenever we need to evaluate the viscosity, we will then adopt the α prescription of Shakura and Sunyaev, 1973).
According to eq. (18') of Lynden-Bell and Pringle (1974), the radial velocity of the gas is
u r = − 3 2 ν r 1 − 4a(GM r) 2 τ ,(1)
where G is the gravitational constant, M is the mass of the central star, a is a parameter describing how sharp is initially the outer edge of the disk and τ is a normalized time, defined as
τ = βνt + 1 ,(2)
where t is the natural time and β = 12(GM ) 2 a. Still according to the same paper, the surface density of the gas evolves as:
Σ = Cτ −5/4 3πν exp − a(GM r) 2 τ ,(3)
where C is a parameter related to the peak value of Σ at r = 0 and t = 0. The disk described by this equation spreads with time (the term a(GM r) 2 /τ becoming smaller and smaller with time), while the peak density declines as τ −5/4 . The motion of the gas is inwards for r < r 0 ≡ τ /(4a)/(GM ) and outwards beyond r 0 , which itself moves outwards as r 0 ∝ √ t.
We now focus on the inner part of the disk, where r << r 0 . In this region we can approximate a(GM r) 2 /τ with 0 and therefore the equations for the radial velocity of the gas and the density become:
u r = − 3 2 ν r ,(4)Σ = Cτ −5/4 3πν .(5)
Thus, the stellar accretion rate iṡ
M = −2πrΣu r (r) = 3πνΣ = Cτ −5/4 .(6)
That is, the accretion rate in the inner part of an accretion disk is independent of radius. Eq. (4) gives the radial velocity of the gas, i.e. the first of the expressions we are interested in. Notice that Takeuchi and Lin (2002) found that, in a three dimensional disk, the radial motion of the gas can be outwards in the midplane and inwards at some height in the disk. Nevertheless the global flow of gas is inwards (the inward flow carries more mass than the outwards flow). The velocity u r in (4) can be considered as the radial speed averaged along the vertical direction and ponderated by the mass flow. For our considerations below we can consider this average speed, without worrying about the meridional circulation of the gas.
We now compute the speed at which a condensation line moves inwards in this evolving disk. Neglecting stellar irradiation (which is dominant only in the outer part of the disk; Oka et al., 2011, Bitsch et al., 2015, the temperature on the midplane of the disk can be obtained by equating viscous heating (Bitsch et al., 2013):
Q + = 2πrδr 9 8 ΣνΩ 2 ,(7)
with radiative cooling:
Q − = 4πrδrσT 4 /(κΣ) ,(8)
where Ω = GM/r 3 is the orbital frequency, σ is Boltzman constant, κ is the opacity (here assumed independent of radius and time, for simplicity), T is the temperature and δr is the radial width of the considered annulus. Thus, the expression for the temperature is:
T = A[κνΣ 2 ] 1/4 r −3/4 = A[κΣṀ /(3π)] 1/4 r −3/4 ,(9)
where A = [9GM/(16σ)] 1/4 . So, the temperature changes with time (through Σ andṀ ) and with radius. Eq. 9 also implies that, for a given value ofṀ , T is weakly dependent (i.e. to the 1/4 power) on the product κΣ, namely on the remaining disk parameters. This is why we can link the location of a given condensation line with the disk's accretion rate with small uncertainty.
The derivatives of the temperature with respect to radius and time are:
dT dr = ∂T ∂r = − 3 4 A[κνΣ 2 ] 1/4 r −7/4 ,(10)dT dt = ∂T ∂t = 1 2 A[κν] 1/4 r −3/4 Σ −1/2 dΣ dt = − 5 8 A[κν] 1/4 r −3/4 Σ −1/2 Cτ −9/4 3πν dτ dt = − 5 8 A[κν] 1/4 r −3/4 Σ −1/2 Cτ −9/4 β 3π(11)
(in the derivation of the equations above, please remember that we assumed that κ and ν are constant in time and space and we derived in Eq. 5 that, at equilibrium, Σ in the inner part of the disk is independent of radius, so that the total derivatives of T are equal to its partial derivatives).
Therefore, assuming that the location of a condensation line just depends on temperature (i.e. neglecting the effect of vapor partial pressure), the speed at which a condensation line moves inwards (which is the second expression we are looking for) is:
v cond r = − dr dT dT dt = − 5 6 r Σ Cτ −9/4 β 3π(12)
which, using (5) and approximating τ with βνt, gives:
v cond r = − 5 6 r t .(13)
By comparing (13) with (4) we find that u r > v cond r for t > 5 9
r 2 ν .(14)
The inequality (14) implies that, after approximately half a viscous timescale t ν ≡ r 2 /ν, the radial motion of the gas is faster than the displacement of a given condensation line. The lifetime of a disk is typically several viscous timescales for the inner region. In fact, Hartmann et al. (1998) found that the time-decay of the accretion rate on stars implies that, if one adopts an α-prescription for the viscosity (i.e. ν = αH 2 Ω; Shakura and Sunyaev, 1973), the value of the coefficient α is 0.001-0.01. At 3 AU, assuming a typical aspect ratio of 5%, the viscous timescale is t ν = 3 × 10 4 -3 × 10 5 y; at 0.7 AU t ν is about 10 times shorter. Both values are considerably shorter than the typical disk's lifetime of a few My. Thus, for the regions and timescales we are interested in (either the asteroid belt at t ∼ 1-3 My or the region around 0.7 AU at 0.1 My) the condition (14) is fulfilled.
This result has an important implication. For t 1/2t ν the condensation lines move very quickly. Thus there is the possibility that gas condenses out locally when the temperature drops. But for t 1/2t ν this process of direct condensation of gas loses importance. This can be understood from the sketch in the left part of Fig. 1. Consider the location r 0 of a condensation line at time t 0 ∼ 1/2t ν , where t ν ≡ r 2 0 /ν. The gas beyond the condensation line (r > r 0 ) is "dry", in the sense that the considered species is in condensed form; instead the gas at r < r 0 is "wet" in the sense that it is rich in the vapor of the considered species. Now, consider first the idealized case where the condensed particles are large enough to avoid radial drift. Because the radial drift of the gas is faster than the radial motion of the condensation line, the outer radial boundary of the wet region moves away from the condensation line, in the direction of the star. In reality the boundary between the two gases in wet and dry form is fuzzy, because it is smeared by turbulent diffusion. But it is clear, from the difference in radial velocities and a simple process of dilution, that the gas just inwards of the condensation lines has to become more and more depleted in the considered species as time progresses. Thus, as the condensation line advances towards the star, the amount of mass that can condense locally is very limited. Thus, the condensed material can be (mostly) found only beyond the original location r 0 of the condensation line. . The orange shaded region shows the "wet" gas, rich in the vapor of the considered species. The blue shaded region shows the "dry" gas, depleted in the vapor of considered species because the latter has condensed out. The outer boundary of the wet gas region also moves towards the star, with speed ur. Because ur > v cond r this boundary moves away from the condensation line. Thus, if there is no radial drift of particles (left part of the plot) the condensation line moves in a disk of dry gas, and therefore the amount of material that can condense out locally is negligible. Instead, in the case shown in the right part of the plot, the condensed particles repopulate, by radial drift, the disk down to the position of the condensation line. Also, the region (green) in between the "wet" and "dry" domains is resupplied in vapor of the considered species by particles sublimating when they pass through the condensation line (see the symbol for the sublimation front). Thus, the mass-flux of particles governs the abundance of the considered species in gaseous or solid form on both sides of the condensation line.
Does this mean that a region of the disk originally too hot for a species to condense will remain depleted in that species forever, even if the temperature eventually drops well below the condensation threshold? In principle no, because in a more realistic case (at least some of) the condensed particles are small enough to drift inwards by gas-drag, so that they can populate any region that has become cold enough to host them in solid form (see the right part of Fig. 1). Also, particles drifting through the condensation line can sublimate, thus resupplying the gas of the considered species in vapor form. Thus, particle drift is the key to understanding the condensation line problem.
A solid particle can be characterized by a dimensionless parameter called the Stokes number:
τ f = ρ b R ρ g c s Ω ,(15)
where ρ b is the bulk density of the particle, ρ g is the density of the gas, R is the size of the particle and c s is the sound speed. A particle feels a wind from the gas, which has two components. The azimuthal component is due to the fact that the gas rotates around the star at a sub-Keplerian speed due to the pressure gradient in the disk; the radial component is due to the fact that the gas flows towards the Star, due to its own viscous evolution. Both components cause the radial drift of the particles towards the star at the speed (Weidenschilling, 1977;Takeuchi and Lin, 2002):
v r = −2 τ f τ 2 f + 1 ηv K + u r τ 2 f + 1 ,(16)
where v K is the Keplerian velocity, u r is the radial velocity of the gas and η is a measure of the gas pressure support:
η = − 1 2 H r 2 d log P d log r .(17)
Eq. (16) is the final radial speed we looked for in this section. For mm-size particles or larger, typically the first term in (16) dominates over the second one.
The radial speed of particles is very fast. In fact, the typical value of η is ∼ 3 × 10 −3 so that the drift speed at 1 AU is ∼ 4τ f × 10 −2 AU/y. Even particles as small as a millimeter (τ f ∼ 10 −3 at ∼ 1 AU) would travel most of the radial extent of the disk within the disk's lifetime. Solid particles condensed in the outer disk are therefore expected to potentially be delivered in the inner disk.
In conclusion, solving the snowline problem, i.e. understanding why Solar System objects remained depleted in species that should have condensed locally before the removal of the gas-disk, requires finding mechanisms that either prevent the radial drift of particles or inhibit the accretion of these particles onto pre-existing objects. Below we investigate some mechanisms, focussing on the cases of the snowline and the silicate line.
The snowline
In this section we discuss several mechanisms that could have potentially prevented the drift or the accretion of icy particles in the asteroid belt and the terrestrial planet zone even after that the snowline had passed across these regions.
Fast growth
If icy particles had accreted each other quickly after their condensation, forming large objects (km-size or more) that were insensitive to gas drag, the inner Solar System would have received very little flux of icy material from the outer part of the disk (as in the example illustrated in the left part of Fig. 1).
We think that this scenario is unlikely. The growth of planetesimals should have been extremely efficient for the fraction of the leftover icy particles to be small enough to have a negligible effect on the chemistry of the inner Solar System bodies. Such an efficient accretion has never been demonstrated in any model.
Observational constraints suggest the same, by showing that disks are dusty throughout their lifetime (see Williams andCieza, 2011 or Testi et al., 2014 for reviews), with the exception of the inner part of transitional disks (Espaillat et al., 2014) that we will address in sect. 4.4. The Solar System offers its own constraints against this scenario. In chondrites, the ages of the individual chondrules inside the same meteorite span a few millions of years (Villeneuve et al., 2009;Connelly et al., 2012). Despite this variability, it is reasonable to assume that all particles (chondrules, CAIs,...) in the same rock got accreted at the same time. Thus, the spread in chondrule ages implies that particles were not trapped in planetesimals as soon as they formed; instead they circulated/survived in the disk for a long time before being incorporated into an object. Similarly, CAIs formed earlier than most chondrules (Connelly et al., 2012;Bollard et al., 2014), but they were incorporated in the meteorites with the chondrules; this means that the CAIs also spent significant time in the disk before being incorporated in macroscopic objects. Thus, it seems unlikely that virtually all icy particles had been accreted into planetesimals at early times, given that this did not happen for their refractory counterparts (CAIs and chondrules).
Inefficient accretion
A second possibility could be that the water-rich particles that drifted into the asteroid belt once the latter became cold enough, were very small. Small particles accrete inefficiently on preexisting planetesimals because they are too coupled with the gas (Lambrechts and Johansen, 2012; and they are also very inefficient in triggering the streaming instability (Youdin and Goodman, 2005;Bai and Stone, 2010a,b;Carrera et al., 2015). Also, small particles are not collected in vortices, but rather accumulate in the low-vorticity regions at the dissipation scale of the turbulent cascade (Cuzzi et al., 2001). The levels of concentration that can be reached, however, are unlikely to be large enough to allow the formation of planetesimals (Pan et al., 2011). Thus, if the flux of icy material through the asteroid belt and the terrestrial planet region was mostly carried by very small particles, very little of this material would have been incorporated into asteroids and terrestrial planets precursors. But how small is small? Again, chondrites give us important constraints. Chondrites are made of chondrules, which are 0.1-1 mm particles. Thus, particles this small could accrete into (or onto -Johansen et al., 2015) planetesimals. The ice-rich particles flowing from the outer disk are not expected to have been smaller than chondrules. Lambrechts and Johansen (2014) developed a model of accretion and radial drift of particles in the disk based on earlier work from Birnstiel et al. (2012). They found that the size of particles available in the disk decreased with time (the bigger particles being lost faster by radial drift). They estimated that, at 1 My, the particles at 2 AU were a couple of cm in size, so more than 10 times the chondrule size; the particles would have been chondrule-size at ∼ 10 My. Thus, we don't see any reason why the asteroids should have accreted chondrules but not ice-rich particles, if the latter had drifted through the inner part of the asteroid belt. Consequently, this scenario seems implausible as well.
Filtering by planetesimals
Particles, as they drifted radially, passed through a disk which presumably had already formed planetesimals of various sizes. Each planetesimal accreted a fraction of the drifting particles. If there were many planetesimals and they accreted drifting particles efficiently, the flow of icy material could have been decimated before reaching the inner Solar System region. There are a few exceptions to this, however, also illustrated in the Guillot et al. paper. If the disk hosted a population of km-size planetesimals with a total mass corresponding to the solid mass in the Minimum Mass Solar Nebula model (MMSN;Weidenschillig, 1977b;Hayashi, 1981) and the turbulent stirring in the disk was weak (α = 10 −4 in the prescription of Shakura and Sunyaev, 1973) particles smaller than a millimeter in size could have been filtered efficiently and failed to reach the inner Solar System (however, see Fig. S2 in Johansen et al., 2015, for a different result). We think that it is unlikely that these parameters are pertinent for the real protoplanetary disk. In fact, we have seen above that icy particles are expected to have had sizes of a few cm at 1 My at 2 AU . Moreover, we believe it is unlikely that the size of the planetesimals that carried most of the mass of the disk was about 1 km. The formation of km-size planetesimals presents unsolved problems (e.g. the m-size barrier -Weidenschilling, 1977and the bouncing barrier - Güttler et al., 2009). Instead, modern accretion models (e.g. Johansen et al., 2007Cuzzi et al., 2010) and the observed size distributions in the asteroid belt and the Kuiper belt suggest that planetesimals formed from self-gravitating clumps of small particles, with characteristics sizes of 100km or larger (Morbidelli et al., 2009). Therefore, more interesting is the other extreme of the parameter space identified by Guillot et al. (2014). If a MMSN mass was carried by "planetesimals" more massive than Mars and the turbulent stirring was small (again, α = 10 −4 ), particles larger than 1 cm in size would have been filtered efficiently and would have failed to reach the inner Solar System. The lower limit of the mass of the filtering planetesimals decreases to 1/10 of a Lunar mass if the "particles" were meter-size boulders. Clearly, this is an important result. It is unclear, however, whether the protoplanetary disk could host so many planetesimals so big in size. The mass in solids in the MMSN model between 1 and 35 AU is about 50 M ⊕ . Assuming that this mass was carried by Mars-mass bodies would require the existence of about 500 of these objects.
Filtering by proto-Jupiter
At some point in the history of the protoplanetary disk, Jupiter started to form. The formation of the giant planets is not yet very clearly understood, so it is difficult to use models to assert when and where Jupiter had a given mass.
However, it has been pointed out in Morbidelli and Nesvorny (2012) that when a planet reaches a mass of the order of 50 Earth masses it starts opening a partial gap in the disk. In an annulus just inside the outer edge of the gap the pressure gradient of the gas is reversed. Therefore, in this annulus the rotation of the gas around the Sun becomes faster than the Keplerian speed. Thus, the drag onto the particles is reversed. Particles do not spiral inwards, but instead spiral outwards. Consequently, particles drifting inwards from the outer disk have to stop near the outer edge of the gap. This process is often considered to be at the origin of the so-called "transitional disks" (Espaillat et al., 2014), which show a strong depletion in mm-sized dust inside of some radius, with no proportional depletion in gas content.
This mechanism for stopping the radial drift of solid particles has been revisited in Lambrechts et al. (2014), who used three dimensional hydro-dynamical simulations to improve the estimate of the planet's mass-threshold for reversing the gas pressure gradient. They found that the massthreshold scales with the cube of the aspect ratio h of the disk and is:
M iso = 20M ⊕ h 0.05 3 ,(18)
quite insensitive to viscosity (within realistic limits). Only particles very small and well-coupled with the gas (about 100µm or less; Paardekooper and Mellema, 2006) would pass through the gap opened by the planet and continue to drift through the inner part of the disk. However, these particles are difficult to accrete by planetesimals, because they are "blown in the wind" (Guillot et al., 2014). Thus, they are not very important for the hydration of inner Solar System bodies. The particles which would be potentially important are those mm-sized or larger, because for these sizes the pebble accretion process is efficient (Ormel and Klahr 2010;Lambrechts and Johansen, 2012); however, for these particles the gaps barrier is effective.
According to Bitsch et al. (2015), when the snowline was at 3 AU (disk's accretion ratė (i.e. beyond ∼ 3 AU) until all the asteroids formed (about 3 My after CAIs). We think that this scenario is reasonable and realistic given that (i) Jupiter exists and thus it should have exceeded 20 M ⊕ well within the lifetime of the disk and (ii) Jupiter is beyond 3 AU today. Nevertheless, there are important migration issues for Jupiter, that we will address in the Appendix.
In this scenario, the current chemical structure of the Solar System would reflect the position of the snowline fossilized at the time when Jupiter achieved ∼20 M ⊕ 1 . Therefore it makes sense that the fossilized snowline position corresponds to a disk already partially evolved (i.e. withṀ of a few 10 −8 M /y, instead of a few 10 −7 M /y, typical of an early disk), given that it may take considerable time (up to millions of years) to grow a planet of that mass.
We also notice that this scenario is consistent, at least at the qualitative level, with the fact that ordinary and enstatite chondrites contain some water (typically less than 1% by mass, well below solar relative abundance of ∼ 50%) and show secondary minerals indicative of water alteration (Baker et al., 2003;Alexander et al., 1989). In fact, it is conceivable that some particles managed to jump across the orbit of the proto-Jupiter (either because they were small enough to be entrained in the gas flow or by mutual scattering, once sufficiently piled up at the edge of Jupiter's gap). Because the entire asteroid region presumably had a temperature well below ice-sublimation by the time these chondrites formed, the icy particles that managed to pass through the planet's orbit would have been available in the asteroid belt region to be accreted. Of course, if most of the particles were retained beyond the orbit of Jupiter, the resulting abundance of water in these meteorites would have remained well below 50% by mass, as observed.
The barrier to particle radial drift induced by the presence of the proto-Jupiter would not just have cut off the flow of icy material. It would also have cut off the flow of silicates. Thus, once the local particles had drifted away, the accretion of planetesimals should have stopped everywhere inside the orbit of the planet. Thus, it seems natural to expect that the chondrites should have stopped accreting at about the time of fossilization of the snowline position. As we said in the introduction, chondrites accreted until ∼ 3 My after CAI formation. Thus the formation of a 20 M ⊕ proto-Jupiter should have occurred at about 3 My. Because the position of the snowline should have been at about 3 AU when this happened, this implies that the solar protoplanetary disk had an accretion rateṀ ∼ 3 × 10 −8 M /y (to sustain a snowline at 3 AU) at ∼3 My. Therefore, our disk was not typical (typical disks have a lower accretion rate at that age; Hartmann et al., 1998), although still within the distribution of observed accretion rates at this age (Manara et al., 2013).
There is, however, a second possibility. If there had been some mechanism recycling particles (i.e. sending the particles back out once they came close enough to the star), or producing new particles in situ, it is possible to envision that chondrules continued to form and chondrites continued to accrete them even after the flow of particles from the outer disk was cut off. In that case, the proto-Jupiter should still have formed when the snowline was at about 3 AU, but we would not have any chronological constraint on when this happened. It could have happened significantly earlier than 3 My. Thus, the solar protoplanetary disk might have had a typical accretion rate as a function of time.
A mechanism for producing small particles in situ is obviously collisions between planetesimals. Chondrules have been proposed to have formed as debris from collisions of differentiated planetesimals (Libourel and Krot, 2007;Asphaugh et al., 2011), although this is still debated (see e.g. Krot et al., 2009 for a reivew).
Several mechanisms leading to a recycling of particles have been proposed, such as x-winds (Shu et al. 1996(Shu et al. , 1997(Shu et al. , 2001, gas outflow on the midplane (Takeuchi and Lin 2002;Ciesla 2007;Bai and Stone, 2010a) and disk winds (Bai, 2014;Staff et al., 2014). Independently of the correct transport mechanism(s), the very detection of high temperature materials within comets (Brownlee et al. 2006;Nakamura et al. 2008;Bridges et al. 2012) demonstrates strong transport from the inner disk regions to the outer disk region. However, it is not clear at which stage of the disk's life this outward transport was active and whether it concerned also particles larger than the microscopic ones recovered in the Stardust samples.
If a mechanism for recycling/producing particles in the inner disk really existed, another implication would be that planetesimals on either side of Jupiter's orbit eventually accreted from distinct reservoirs. The planetesimals inside of the orbit of the planet accreted only material recycled from the inner disk; instead the planetesimals outside of the proto-Jupiter's orbit accreted outer disk material, although possibly contaminated by some inner-disk material transported into the outer disk. In this respect, it may not be a coincidence that ordinary and carbonaceous chondrites appear to represent distinct chemical and isotopic reservoirs (Jacquet et al. 2012) because, in addition to the water content, these meteorites show two very distinct trends in the ∆ 17 O-54 Cr * isotope space (Warren et al. 2011). Today the parent bodies of both classes of meteorites reside in the asteroid belt, i.e. inside of the orbit of Jupiter. But in the Grand Tack scenario (Walsh et al., 2011) the parent bodies of the carbonaceous chondrites formed beyond Jupiter's orbit and got implanted into the asteroid belt during the phase of Jupiter's migration.
The silicate line
According to terrestrial planet formation models, the small mass of Mercury and the absence of planets inside its orbit can be explained only by postulating that the disk of planetesimals and planetary embryos had an inner edge near 0.7 AU (Hansen, 2009). We argued in the introduction that this inner edge might reflect the location of the silicate condensation line at a very early age of the disk. But what could have prevented particles from drifting inside of 0.7 AU, once the disk there had cooled below the silicate condensation temperature? Obviously no giant planets formed near 0.7 AU, so the scenario invoked for the snowline cannot apply to this case.
A solution can be found in the results presented in Johansen et al., 2015. The authors pointed out that the very early disk is the most favorable environment for the production of planetesimal seeds via the streaming instability. This is because the streaming instability requires the presence of large particles, with Stokes numbers of the order of 0.1-1 (e.g. significantly larger than chondrulesize particles). These particles drift very quickly in the disk, so they are rapidly lost. As shown in Lambrechts and Johansen (2014) the mass ratio between solid particles and gas, as well as the size of the dominant particles, decrease with time. Thus, the streaming instability becomes more and more unlikely to happen as time progresses. Therefore, Johansen et al. argue that planetesimals formed in two stages. In the first stage planetesimal seeds formed by the streaming instability, triggered by large particles (decimeter across). This stage lasted for a short time only, due to the rapid loss of these large particles by radial drift. In the second stage the planetesimal seeds kept growing by accreting chondrule-sized particles, the only ones surviving and still drifting in the disk after a few My.
This model of planetesimal formation would provide a natural explanation for the fossilization of the silicate line. Imagine that, when the phase of streaming instability was over, the accretion rate in the protoplanetary disk wasṀ ∼ 1.5 × 10 −7 M /y. In this case, the silicate sublimation line was at ∼ 0.7 AU. Thus, presumably no planetesimal seeds could have formed within this radius, because until that time only the very refractory material would have been available in solid form there (a small fraction of the total mass, insufficient to trigger the streaming instability). After the streaming instability phase was over, planetesimals continued to grow by the accretion of smaller particles onto the planetesimal seeds . Meanwhile the disk cooled so that silicate particles could drift within 0.7 AU and remain solid. However, because of the absence of planetesimal seeds within this radius, they could not be accreted by anything, and thus they simply drifted towards the Sun. Clearly, in this scenario the final planetesimal disk would have an inner edge at 0.7 AU, fossilizing the early location of the silicate line that characterized the formation of the planetesimal seeds.
Admittedly, this scenario is still qualitative, but it is seducing in its simplicity. The possibility to explain location of the inner edge of the planetesimal disk by this mechanism is an additional argument in favor of the two-phases model for planetesimal growth, simulated in .
Conclusions
The chemical composition of the objects of the Solar System seems to reflect a condensation sequence set by a temperature gradient typical of an "early" disk, still significantly warm in its inner part. Particularly significant is the situation concerning water. The terrestrial planets and the asteroids predominant in the inner belt are water-poor. However, the disk's temperature should have decreased below the water-condensation level even at 1 AU before the disappearance of the gas. So, the question why terrestrial planets and asteroids are not all water rich is a crucial one (Oka et al., 2011). Interestingly, water is not the only chemical species that reveals this conundrum. Inner Solar System objects are in general more depleted in volatile element than they should be, given the temperatures expected in the disk at its late stages. Also, terrestrial planet formation models argue that, in order to explain the small mass of Mercury, the planetesimal disk had to have an inner edge at about 0.7 AU (Hansen, 2009;Walsh et al., 2011). This boundary could correspond to the location of the evaporation front for silicates, but again only for a massive (i.e. "early") disk.
In this Note we have discussed how the composition of Solar System objects could reflect the location of the condensation lines fossilized at some specific stages of the disk's evolution. Some mechanisms have been quickly dismissed, others look promising and we propose them to the community for further discussion and investigation.
First, we have demonstrated that the radial motion of gas towards the central star is faster than the inward motion of the condensation lines. This implies that there cannot be condensation of gas in a region swept by the motion of a condensation line, even if this may sound paradoxical. This is because the gas in the considered region comes from farther out and therefore it should have already condensed there (see Fig. 1, left side). Thus, the enrichment of a disk region in condensed elements when the temperature drops can only be due to the radial drift of solid particles from the more distant disk (see Fig. 1, right side).
With this consideration in mind, the scenario that we propose to explain the fossilized condensation lines is the one sketched in Fig. 2. Planetesimal formation occurred in two stages, as proposed in . In the very early disk, dust coagulation produced pebbles and boulders of sizes ranging from decimeters to, possibly, a meter. These objects were very effective in triggering the streaming instability (Youdin and Goodman, 2005;Johansen et al., 2007;Bai and Stone, 2010a,b) and areodynamically clumped together forming planetesimal seeds of about 100 km in size. This phase, however, could not last long, because these boulders drifted quickly through the disk and those that were not rapidly incorporated into a planetesimal seed got lost by drifting into the Sun. We propose that, when this stage ended, the silicate condensation line was at 0.7 AU. This would correspond to a disk with an accretion rate ofṀ ∼ 1.5 × 10 −7 M /y, for a nominal metallicity of 1%. Then, no planetesimal seeds could have formed within this radius, because of the lack of a sufficient amount of solids. In the second stage, the surviving solid particles were too small and their mass ratio with the gas too low to trigger streaming instability . Thus, these particles could only be accreted onto the already formed planetesimal seeds . If no seeds existed inside of 0.7 AU, this radius remained the inner edge of the planetesimal disk, a fossil trace of the silicate line at the end of the first stage of planetesimal growth.
Meanwhile the temperature in the disk continued to decrease. The snowline moved towards the Sun. The region swept by the snowline became increasingly enriched in icy material due to the radial drift of particles from the outer disk. When the mass of the proto-Jupiter reached 20 M ⊕ , however, this flux of icy particles stopped. The opening of a shallow gap by the protoplanet created a barrier to the inward drift of the particles by gas drag. Thus, the flux of icy material across Jupiter's orbit was interrupted, presumably making our Solar System look like a "transitional disk". We propose that the proto-Jupiter reached this critical mass when the snowline was at about 3 AU. This corresponds to a disk with a stellar accretion rate ofṀ ∼ 3 × 10 −8 M /y, assuming the canonical metallicity of 1%. Thus, the objects inside 3 AU, which could not accrete ice up to that time because the temperature was too high, could not accrete a significant amount of ice also after that the temperature dropped, because of the interrupted icy-particle flow. Instead, they could have continued to accrete refractory particles if the latter were recycled or continuously reproduced in the disk, thanks to the existence of outwards flows in the midplane (Takeuchi and Lin, 2002;Ciesla, 2007;Bai and Stone, 2010a), x-winds (Shu et al., 1996, disk winds (Bai, 2014;Staff et al., 2014) or collisions (Libourel and Krot, 2007;Asphaugh et al., 2011). Thus, the resulting chemistry of planetesimals reflect the location of the snowline fossilized at the time the proto-Jupiter reached 20 M ⊕ .
Notice that, because it takes more time to form the proto-Jupiter than the planetesimal seeds, it makes sense that the snowline appears fossilized at the location corresponding to a "later" disk than the silicate line (Ṁ ∼ 3 × 10 −8 M /y instead ofṀ ∼ 1.5 × 10 −7 M /y).
This scenario is much simpler than what has been proposed so far for the snowline problem or the origin of the inner edge of the planetesiamal disk (reviewed in Sect. 2). It is indeed appealing for its simplicity.
This scenario leads to a few predictions. For the Solar System it predicts that the condensation lines corresponding to species much more volatile than water (e.g. CO, with a condensation temperature of ∼ 25 K) should not have been fossilized because only Jupiter and Saturn are sufficiently massive to stop the flow of drifting particles and these planets should always have been too close to the Sun. Thus, the composition of outer Solar System bodies should reflect the location of these lines at the end of the disk's lifetime. Perhaps this can explain the compositions of Uranus and Neptune (Ali-Dib et al., 2014).
For extrasolar planetary systems, the scenario predicts that systems without giant planets should be much more volatile rich in their inner parts than a system like ours. This seems consistent with the observations, which show a large number of systems of low-density super-Earths in close-in orbits and no giant planets farther out (Fressin et al., 2013). In principle the low bulk-densities could be explained by the presence of extended H and He atmosphere around rocky planets. But Lopez and Fortney (2014) concluded that the observed size distribution of extrasolar planets, with a sharp drop-off above 3 R ⊕ , is diagnostic that most super-Earths are water-rich. In fact, refractory planets with extended atmospheres would have a more uniform size distribution.
Finally, the fossilization of the silicate line should be a generic process, although the location at which this condensation line is fossilized may change from disk to disk depending on the duration of the streaming instability stage and the evolution of the temperature in that timeframe. This suggests that "hot" extrasolar planets (with orbital radii significantly smaller than Mercury's) did not form in situ but migrated to their current orbits from some distance away.
Clearly, the scenario proposed in this Note remains speculative. However, with the improved understanding of disk evolution and its chronology, planetesimal accretion and giant planet growth, it will be possible in a hopefully not distant future to test it on more quantitative grounds against the available constraints.
Appendix: Planet migration issues
Planets are known to migrate in disks (see Baruteau et al., 2014 for a review). Thus, we discuss here possible scenarios that could explain how the proto-Jupiter remained beyond ∼ 3 AU until the chondrite formation time.
A first possibility is that Jupiter's core started to form sufficiently far in the disk, so that it could not reach 3 AU before ∼ 3 My. This is one of the approaches taken in Bitsch et al., 2015b. In their model, Jupiter started growing by pebble accretion at about 20 AU.
A second possibility is offered by the subtle action of the entropy-driven corotation torque (Paardekooper and Mellema, 2006b;Paardekooper et al., 2010Paardekooper et al., , 2011Casoli, 2009, 2010). This torque can reverse the migration of intermediate-mass planets (several Earth masses) in localized regions of the disk where the temperature gradient is steep. Bitsch et al. (2014aBitsch et al. ( , 2015 showed migration maps as a function of location and planet mass at different evolutionary stages (i.e. different values ofṀ ) of the disk. They found that the outward migration region is adjacent to the snowline. It typically ranges from the snowline location up to a few AUs beyond the snowline. For an early disk (Ṁ = 7 × 10 −8 M /y) outward migration concerns planets with masses between 5 and 40 Earth masses. A proto-Jupiter in this mass range would therefore be retained at 6-8 AU (depending on its mass), the snowline being at ∼ 4 AU (see Fig. 7 of Bitsch et al., 2015). When the disk loses mass and cools, the outward migration region shifts towards the Sun with the snowline position. But it also shrinks in the planet-mass parameter space. In a late disk withṀ = 8.75 × 10 −9 M /y (snowline at ∼ 1AU ) the outward migration region still extends to ∼ 3AU , but only for planet masses smaller than 10 M ⊕ . Thus, in principle, a pebble-stopping planet of 20 M ⊕ should have been released to free Type-I migration and should have penetrated into the inner Solar System, possibly too early relative to the chondrite formation time of ∼ 3 My.
We stress, however, that the exact location and shape of the outward migration region depends on the adopted parameters for the disk, as shown in Fig. 3. In particular, the upper limit in planet mass for outward migration is due to torque saturation. It can be increased if the disk is more viscous. The size of the coorbital region of a planet (the one characterized by horseshoe streamlines) has a width x s ∝ M p , where M p is the planet mass. The timescale for viscous transport across Each color corresponds to a different disk, whose parametersṀ , α and κ are reported on the plot. To help reading this plot, the red arrows show the direction of migration of planets of different masses and locations for the case withṀ = 10 −8 M /y, α = 0.01 and κ =1%, corresponding to the red contour. A planet of an appropriate mass (between 2.5 and 20 M E for the case of the red contour) migrating inwards from the outer disk would stop at the right-hand-side boundary of the outward migration region. The black and red contorus are too limited in planet-mass range to be able to trap a 20 M ⊕ planet, but the other disks could retain a proto-Jupiter of this mass beyond 2.5 or 3 AU. this region is therefore t ν ∝ x 2 s /ν ∝ M p /ν .
The libration timescale in the horseshoe region is
t lib ∝ 1 M p .(20)
Saturation is achieved when t lib << t ν . If we want saturation to occur for a planet N times bigger in mass, the scaling of (19) and (20) implies that ν has to be N 3/2 times bigger.
However, to have the outward migration region in the same radial range we need that the thermal structure of the disk does not change with the increase in viscosity. Because the viscous transport in the disk is proportional to νΣ, we need that Σ is N 3/2 smaller. And because the temperature in the disk is proportional to (κνΣ 2 ) 1/4 (see 9) we need that the opacity κ (basically the mass ratio between micron-sized dust and gas) scales as N 3/2 . In other words, a planet of 20 M ⊕ can be retained at 3 AU in a disk withṀ = 8.75 × 10 −9 M /y if the viscosity is ∼ 3 times higher and the opacity ∼ 3 times larger than assumed in Bitsch et al. (2015). Given the uncertainties on disk parameters, we cannot exclude this possibility. Fig. 3 indeed shows "late disks" (i.e. with a smallṀ -a few 10 −9 M /y) with an outward migration region capable of retaining a 20 M ⊕ planet beyond 2.5-3 AU.
Nevertheless, however we play with disk parameters, it is clear that a planet is eventually released to inward migration once it becomes massive enough. Thus, Jupiter should have eventually invaded the asteroid belt, (unless it started so far out in the disk that it was not able to reach the asteroid belt within the disk's lifetime; see some simulations in Bitsch et al., 2015b). The migration of Jupiter through the belt is contemplated in the so-called Grand Tack scenario (Walsh et al., 2011), in which Jupiter reached 1.5 AU before being pulled back to its current distance by the presence of Saturn. This scenario of inward-then-outward migration of Jupiter can explain the excitation and depletion of the asteroid belt and the abortion of the growth of Mars. Again, because chondritic planetesimals accrete until ∼ 3 My, what is important is that Jupiter did not invade the asteroid belt till that time.
Fig. 1 .
1-Sketch of the radial motion of a condensation line and of the gas, averaged over the vertical direction in the disk. The left/right parts of the figure differ by the assumption that condensed particles do not drift/drift, respectively. From top to bottom, each panel depicts the situation at different times, labeled on the left of each panel, with t0 < t1 < t2. The time t0 is defined as about half of the viscous timescale at the location of the condensation line r0. The main horizontal arrow indicates the radial direction. The vertical dashed line shows the location of the condensation line as a function of time, approaching the star with a speed v cond r
Guillot et al. (2014) developed a very complete analytic model of the process of filtering of drifting dust and pebbles by planetesimals. Unfortunately, the results are quite disappointing from our perspective. As shown by Figs. 21 and 22 of Guillot et al., in general only large boulders (about 10m in size) drifting from the outer disk (35 AU for the calculations illustrated in those figures, but the result is not very sensitive on this parameter) would have been accreted by planetesimals before coming within a few AU from the Sun.
Fig. 2 .
2-Sketch of the solution to the follisized condensation lines problem proposed in this paper. The top and central panels show the situation at the times when the silicate line, first, then the snowline, remain fossilized. The bottom panel sketches the situation after the fossilization of the snowline, under the assumption where accretion of planetesimals in the inner disk continues thanks to the recycling of small particles in outwards flows. If a recycling or particle-generation mechanism did not exist, all planetesimals inside of the orbit of proto-Jupiter should have stopped accreting at the time of fossilization of the snowline. See text for detailed description.
Fig. 3 .
3-Contours of the outward migration region in the parameter space heliocentric distance vs. planetary mass.
M ∼ 3 × 10 −8 M /y) the aspect ratio of the disk was around 0.05 up to ∼ 10 AU . So, the mass of 20 M ⊕ is the minimum value required for the proto-Jupiter in order to stop the drift of icy pebbles and large grains. Basically, the constraint that asteroids and the precursors of the terrestrial planets did not accrete much ice translates into a constraint on the mass and location of the proto-Jupiter. More specifically, the proto-Jupiter needs to have reached 20 M ⊕ before the disk dropped below an accretion rate ofṀ ∼ 3 × 10 −8 M /y and it needs to have remained beyond the asteroid belt
In reality, at this time there may still have been a reservoir of icy particles between the snowline location and the orbit of Jupiter. However, because of the typical drift rate of particles (10cm/s for mm-size dust, 1m/s for cm-size pebbles), if this reservoir was just a few AU wide (see Appendix), it should have emptied in 10 4 -10 5 y, i.e. before that the snowline could move substantially.
We acknowledge support by the French ANR, project number ANR-13-13-BS05-0003-01 projet MOJO (Modeling the Origin of JOvian planets). A.J. is grateful for the financial support from the European Research Council (ERC Starting Grant 278675-PEBBLE2PLANET) and the Swedish Research Council (grant 2014-5775). A.J. and B.B. thanks the Knut and Alice Wallenberg Foundation for their financial support. S. J. was supported by the European Research Council (ERC) Advanced Grant ACCRETE (contract number 290568). We are grateful to C. Dullemond and an anonymous reviewer for their constructive reviews.References
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[] |
[
"A DIGITAL BINOMIAL THEOREM FOR SHEFFER SEQUENCES",
"A DIGITAL BINOMIAL THEOREM FOR SHEFFER SEQUENCES"
] |
[
"Toufik Mansour ",
"Hieu D Nguyen "
] |
[] |
[] |
We extend the digital binomial theorem to Sheffer polynomial sequences by demonstrating that their corresponding Sierpiński matrices satisfy a multiplication property that is equivalent to the convolution identity for Sheffer sequences.
| null |
[
"https://arxiv.org/pdf/1510.08529v1.pdf"
] | 119,321,895 |
1510.08529
|
df57b9ffc182619f8563d5748d316a1ff70b5367
|
A DIGITAL BINOMIAL THEOREM FOR SHEFFER SEQUENCES
29 Oct 2015
Toufik Mansour
Hieu D Nguyen
A DIGITAL BINOMIAL THEOREM FOR SHEFFER SEQUENCES
29 Oct 2015
We extend the digital binomial theorem to Sheffer polynomial sequences by demonstrating that their corresponding Sierpiński matrices satisfy a multiplication property that is equivalent to the convolution identity for Sheffer sequences.
Introduction
The binomial theorem is a fundamental result in mathematics: (x + y) n = n k=0 n k
x k y n−k .
One generalization of the binomial theorem, due to Callan [2] (see also [6]), expresses the exponents appearing in (1) in terms of the binary sum-of-digits function s(m):
(x + y) s(n) = 0≤m≤n (m,n−m) carry-free
x s(m) y s(n−m) ,(2)
where a pair of non-negative integers (j, k) is said to be carry-free if their sum in binary involves no carries. We refer to (2) as the digital binomial theorem. Several extensions of the digital binomial theorem have recently been found. For example, a non-binary version is given in [7] for any integer base b > 2:
N −1 i=0 x + y + n i − 1 n i = 0≤m b n N −1 i=0 x + m i − 1 m i N −1 i=0 y + n i − m i − 1 n i − m i .(3)
Here, n i and m i denote the i-th digit in the base-b expansion of n and m, respectively. Also, m b n denotes the fact that m i ≤ n i for all i. Another example is a q-analog given in [4]:
N −1 i=0 x + q i y + n i − 1 n i = 0≤m≤n (m,n−m) carry-free q zn(m) x s(m) y s(n−m) .(4)
In this paper, we present a digital binomial theorem for Sheffer sequences and those of binomial type by considering a polynomial generalization of (1). Let f (t) be a delta series and g(t) be an invertible series, i.e.,
f (t) = ∞ k=0 a k t k (a 0 = 0, a 1 = 0), g(t) = ∞ k=0 b k t k (b 0 = 0).
The polynomial sequence s n (x), n = 0, 1, . . . , is said to be Sheffer for (g(t), f (t)) if it has generating function ([8, Theorem 2.3.4, p. 18])
1 g(f (t)) e xf (t) = ∞ n=0 s n (x) n! t n (5) wheref (t) is the compositional inverse of f (ts n (x + y) = n k=0 n k p k (x)s n−k (y).(6)
Here, p n (x) is the polynomial sequence associated to s n (x), i.e., p n (x) has generating function
e xf (t) = ∞ n=0 p n (x) n! t n .
It is known that p n (x) is of binomial type, i.e., p n (x) satisfies the binomial identity (([8, Theorem 2.4.7, p. 26]) p n (x + y) = n k=0 n k p k (x)p n−k (y).
In the special case wheref (t) = t in (5) so that p n (x) = x n , the Sheffer sequence s n (x) is then called an Appell sequence. There are many well-known examples of Sheffer sequences, e.g., the Bernoulli, Hermite (probabilistic version), and Laguerre polynomials defined by the generating functions
t e t − 1 exp(xt) = ∞ n=0 B n (x) n! t n , exp(xt − t 2 /2) = ∞ n=0 H n (x) n! t n , and 1 (1 − t) α+1 exp − xt 1 − t = ∞ n=0 L α n (x) n! t n ,
respectively. Observe that B n (x) and H n (x) are both Appell sequences. It is well known that all three Sheffer sequences satisfy the convolution identities
B n (x + y) = n k=0 n k x k B n−k (y), H n (x + y) = n k=0 n k x k H n−k (y), and L α n (x + y) = n k=0 L −1 k (x)L α n−k (y),
respectively. These polynomials also have extensions that are also Sheffer sequences, e.g., Bernoulli polynomials of higher order [5] and generalized Hermite polynomials [10]. If we renormalize a Sheffer sequence s n (x) and its associated sequence p n (x) by defininḡ s n (x) = s n (x)/n! andp n (x) = p n (x)/n!, then (6) and (7) are equivalent tō
s n (x + y) = n k=0p k (x)s n−k (y) (8) andp n (x + y) = n k=0p k (x)p n−k (y),(9)
respectively. Identities (8) and (9) form the basis for our main result.
Theorem 1. Let n be a non-negative integer with base b expansion n = n N −1 b N −1 +· · ·+n 0 b 0 and {s n (x)} be a Sheffer polynomial sequence with associated sequence {p n (x)}. Then N −1 i=0s n i (x i + y i ) = 0≤m b n N −1 i=0p m i (x i ) N −1 i=0s n i −m i (y i )(10)
and
N −1 i=0p n i (x i + y i ) = 0≤m b n N −1 i=0p m i (x i ) N −1 i=0p n i −m i (y i ) ,(11)where m = m N −1 b N −1 + · · · + m 0 b 0 .
In the case where x 0 = . . . = x N −1 and y 0 = . . . = y N −1 , we obtain as a corollary the following result:
Corollary 2. For n = b N − 1, we havē s b−1 (x + y) N = 0≤m≤n N −1 i=0p m i (x) N −1 i=0s b−1−m i (y) (12) andp b−1 (x + y) N = 0≤m≤n N −1 i=0p m i (x) N −1 i=0p b−1−m i (y)(13)
If we specialize to Bernoulli polynomials by setting s n (x) = B n (x) and p n (x) = x n , then (12) gives the following result.
Theorem 3. For n = b N − 1, we have B b−1 (x + y) N = 0≤m≤n x s b (m) N −1 i=0 B b−1−m i (y) (14) where s b (m) is the base-b sum-of-digits function.
Similar formulas can be obtained for other special polynomials such as Hermite and Laguerre polynomials. Also, we remark that setting y = 0 in (14) yields a formula for higher powers of Bernoulli polynomials in terms of Bernoulli numbers B n := B n (0) and the sumof-digits function:
B b−1 (x) N = 0≤m≤n x s b (m) N −1 i=0 B b−1−m i .
The proof of Theorem 1 will be given in the next section where we investigate a Sheffer sequence analog of the Sierpiński matrix and use its multiplicative property to derive (10) and (11).
Sierpinski Matrices of Sheffer Type
Throughout this paper we assume that b is an integer greater than 1. We begin by introducing the notion of digital dominance as defined in [1] (see also [7]).
Definition 4. Let m and n be non-negative integers with base b expansions m = m N −1 b N −1 + · · · + m 0 b 0 and n = n N −1 b N −1 + · · · + n 0 b 0 , respectively. We denote m b n to mean that m is digitally less than n in base b, i.e., m k ≤ n k for all k = 0, . . . , N − 1.
Next, we define a sequence of generalized Sierpiński matrices corresponding to the Sheffer sequence s n (x) and its associated sequence p n (x).
Definition 5.
Let N be a non-negative integer. Denote x N = (x 0 , . . . , x N −1 ). If N = 0, we set S b,0 (x 0 ) = P b,0 (x 0 ) = 1. For N > 0, we define the N-variable Sierpiński matrices
S b,N (x N ) = (α N (j, k, x N )) matrices P b,N (x N ) = (β N (j, k, x N )) of dimension b N × b N by α N (j, k, x N ) = N −1 i=0s d i (x i ) if 0 ≤ k ≤ j ≤ b N − 1 and k b j; 0, otherwise,(15)
and
β N (j, k, x N ) = N −1 i=0p d i (x i ) if 0 ≤ k ≤ j ≤ b N − 1 and k b j; 0, otherwise,(16)respectively, where j − k = d 0 b 0 + d 1 b 1 + . . . + d N −1 b N −1 is the base-b expansion of j − k.
The following lemma gives a recurrence for S b,N (x N ) and P b,N (x N ).
Lemma 6. The generalized Sierpiński matrices S b,N (x N ) and P b,N (x N ) satisfy the recurrence
S b,N +1 (x N +1 ) = S b,1 (x N ) ⊗ S b,N (x N ),(17)
and
P b,N +1 (x N +1 ) = P b,1 (x N ) ⊗ P b,N (x N ), (18) respectively, where we define S b,1 (x) = 1 0 0 · · · 0 s 1 (x) 1 0 · · · 0 s 2 (x)s 1 (x) 1 · · · 0 . . . . . . . . . . . . . . . s b−1 (x)s b−2 (x)s b−3 (x) · · · 1 = s j−k (x), if 0 ≤ k ≤ j ≤ b − 1; 0, otherwise and P b,1 (x) = 1 0 0 · · · 0 p 1 (x) 1 0 · · · 0 p 2 (x)p 1 (x) 1 · · · 0 . . . . . . . . . . . . . . . p b−1 (x)p b−2 (x)p b−3 (x) · · · 1 = p j−k (x), if 0 ≤ k ≤ j ≤ b − 1; 0, otherwise.
Proof. Following [4], we shall prove (17) by induction on N. It is clear that (17) holds for N = 0. Next, assume that (17) holds for N − 1. To prove that(17) holds for N, we express S N +1 (x N +1 ) a b × b matrix of blocks (A p,q ) 0≤p,q≤b−1 :
S N +1 (x N +1 ) = A 0,0 . . . A 0,b−1 . . . . . . . . . A b−1,0 . . . A b−1,b−1 ,
where each A p,q is a square matrix of size b N . We consider two cases depending on the position of A p,q : Case 1. p < q. Then by definition of S b,N +1 (x N +1 ) we have that α N +1 (p, q, x N +1 ) = 0, which implies A p,q = 0. N (x N , r N ). This proves (17). The proof for (18) is analogous and will be omitted.
Case 2. p ≥ q. Let α N +1 (j, k, x N +1 ) be an arbitrary entry of A p,q . Then pb N ≤ j ≤ (p+1)b N − 1 and qb N ≤ k ≤ (q +1)b N −1. Set j ′ = j −pb N and k ′ = k −qb N . If j < k, then by definition α N +1 (j, k, x N ) = 0. Therefore, we assume j ≥ k. Let j − k = d 0 b 0 + d 1 b 1 + · · · + d N b N , where d N = p − q. Then j ′ − k ′ = d 0 b 0 + d 1 b 1 + · · · + d N −1 b N −1 . Since k b j if and only if k ′ b j ′ ,we have α N +1 (j, k, x N +1 ) = N i=0s d i (x i ) if 0 ≤ k ≤ j ≤ b N +1 − 1 and k b j; 0 otherwise. =s d N (x N )α b,N (j ′ , k ′ , x N ).
It follows that
A p,q =s p−q (x N )S b,N (x N , r N ) and hence S b,N +1 (x N +1 ) = S b,1 (x N , r N ) ⊗ S b,
The generalized Sierpiński matrices S b,N (x N ) and P b,N (x N ) satisfy the following multiplicative property: Theorem 7. Let N be a non-negative integer. Then
P b,N (x N )S b,N (y N ) = S b,N (x N + y N )(19)
and
P b,N (x N )P b,N (y N ) = P b,N (x N + y N ), (20) where we define x N + y N = (x 0 + y 0 , x 1 + y 1 , . . . , x N −1 + y N −1 ).
Proof. We shall prove (19) using induction. To prove that (19) holds for the base case N = 1, let γ(j, k) denote the (j, k)-entry of T = P b,1 (x 1 )S b,1 (y 1 ). Since T is lower-triangular, it follows that γ(j, k) = 0 if j < k. Therefore, we assume j ≥ k. By definition of S b,1 (x 1 ) and P b,1 (y 1 ), we have
γ(j, k) = j i=kp j−i (x 0 )s i−k (y 0 ) = j−k i=0p i (x 0 )s j−k−i (y 0 ).
Since s n (x) is a Sheffer sequence, it follows from (6) that γ(j, k) =s j−k (x 0 + y 0 ), or equivalently, P b,1 (x 1 , r 1 )S b,1 (y 1 , r 1 ) = S b,1 (x 1 + y 1 , r 1 ). This proves (19) for N = 1.
Next, assume that (19) holds for arbitrary N. To prove that (19) holds for N + 1, we employ Lemma 6 and the mixed-property of a Kronecker product:
P b,N +1 (x N +1 )S b,N +1 (y N +1 ) = (P b,1 (x N ) ⊗ P b,N (x N )(S b,1 (y N ) ⊗ S b,N (y N )) = (P b,1 (x N )S b,1 (y N )) ⊗ (P b,N (x N )S b,N (y N )).
Moreover, by the induction hypothesis and Lemma 6 again, we obtain
P b,N +1 (x N +1 )S b,N +1 (y N +1 ) = S b,1 (x N + y N ) ⊗ S b,N (x N + y N ) = S b,N +1 (x N +1 + y N +1 ).
Hence, (19) holds for N + 1. The proof of (20) is similar and will be omitted.
Proof of Theorem 1. We equate the matrix entries at position (n, 0) on both sides of (19) to obtain
α N (n, 0, x N + y N ) = 0≤m b n β N (n, m, x N )α N (m, 0, y N ) = 0≤m b n β N (n, n − m, x N )α N (n − m, 0, y N ).
This yields (10) as desired. The derivation of (11) is similar and left for the reader to verify.
We conclude with two ideas on how to extend Theorem 1. The first is to iterate (10) and (11) to obtain a trinomial version:
N −1 i=0s n i (x i + y i + z i ) = 0≤m b n N −1 i=0p m i (x i + y i ) N −1 i=0s n i −m i (z i ) = 0≤m b n 0≤l b m N −1 i=0p l i (x i ) N −1 i=0p m i −l i (y i ) N −1 i=0s n i −m i (z i )
and similarly,
N −1 i=0p n i (x i + y i + z i ) = 0≤m b n 0≤l b m N −1 i=0p l i (x i ) N −1 i=0p m i −l i (y i ) N −1 i=0p n i −m i (z i ) where l = l N −1 b N −1 + · · · + l 0 b 0 .
Of course, we can also iterate repeatedly to obtain to the multinomial formulas
N −1 i=0 s n i (x (1) i + · · · + x (d) i ) = 0≤m (d−1) b n . . . 0≤m (1) b m (2) N −1 i=0p m (1) i (x (1) i ) · · · N −1 i=0p m (d−1) i −m (d−2) i (x (d−1) i ) N −1 i=0s n i −m (d−1) i (x (d) i ) and N −1 i=0 p n i (x (1) i + · · · + x (d) i ) = 0≤m (d−1) b n . . . 0≤m (1) b m (2) N −1 i=0p m (1) i (x (1) i ) · · · N −1 i=0p m (d−1) i −m (d−2) i (x (d−1) i ) N −1 i=0p n i −m (d−1) i (x (d) i )
where m (j) = m
The digital version of (22) and its multinomial generalization then becomes clear and extends . . .
0≤m (1) b m (2) N −1 i=0 L 1 |p m (1) i (x) · · · N −1 i=0 L d |p m (d−1) i −m (d−2) i (x) N −1 i=0 L d |p n i −m (d−1) i (x) .
idea is to view the binomial convolution identity for Sheffer sequences in the context of umbral calculus and linear functionals on the vector space of polynomials. Define the product of two such linear functionals L and M by and Rota [9, Prop. 3.3, p. 102] proved that for any polynomial sequence p n (x) of binomial type, we have LM|p n (x) = n k=0 n k L|p k (x) M|p n−k (x) .
Dominance orders, generalized binomial coefficients, and Kummer's theorem. T Ball, T Edgar, D Juda, Math. Mag. 87T. Ball, T. Edgar, and D. Juda, Dominance orders, generalized binomial coefficients, and Kummer's theorem, Math. Mag. 87 (2014), 135-143.
Sierpinski's triangle and the Prouhet-Thue-Morse word. D Callan, preprintD. Callan, Sierpinski's triangle and the Prouhet-Thue-Morse word, preprint, http://arxiv.org/abs/math/0610932.
Umbral calculus associated with Bernoulli polynomials. D S Kim, T Kim, J. Number Theory. 147D. S. Kim and T. Kim, Umbral calculus associated with Bernoulli polynomials, J. Number Theory 147 (2015), 871-882.
T Mansour, H D Nguyen, A q-digital binomial theorem. preprintT. Mansour and H. D. Nguyen, A q-digital binomial theorem, preprint, 2015. http://arxiv.org/abs/1506.07945.
Sheffer sequences of polynomials and their applications. D S Kim, T Kim, S.-H Rim, D V Dolgy, Adv. Diff. Eqs. 118D. S. Kim, T. Kim, S.-H. Rim, and D. V. Dolgy, Sheffer sequences of polynomials and their applications, Adv. Diff. Eqs. 2013:118.
H D Nguyen, A digital binomial theorem. preprintH. D. Nguyen, A digital binomial theorem, preprint, 2015. http://arxiv.org/abs/1412.3181.
A generalization of the digital binomial theorem. H D Nguyen, 15.5.7J. Integer Seq. 18H. D. Nguyen, A generalization of the digital binomial theorem, J. Integer Seq. 18 (2015), Article 15.5.7.
The umbral calculus. S Roman, Academic PressOrlando, FLS. Roman, The umbral calculus, Academic Press, Orlando, FL, 1984.
The umbral calculus. S Roman, G.-C Rota, Adv. Math. 272S. Roman and G.-C. Rota, The umbral calculus, Adv. Math. 27 (1978), No. 2, 95-188.
A Note on a Generating Function for the Generalized Hermite Polynomials, Nederl A/cad. H M Srivastava, Wetensch. Proc. Ser. A. 79H. M. Srivastava, A Note on a Generating Function for the Generalized Hermite Polynomials, Nederl A/cad. Wetensch. Proc. Ser. A 79 (1976), 457-61.
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[] |
[
"Tunnelling processes for Hadamard states through a 2+1 dimensional black hole and Hawking radiation",
"Tunnelling processes for Hadamard states through a 2+1 dimensional black hole and Hawking radiation"
] |
[
"Francesco Bussola [email protected] \nDipartimento di Fisica\nUniversità di Pavia\nVia Bassi 627100PaviaItaly\n\nINFN\nSezione di Pavia -Via Bassi 627100PaviaItaly\n",
"Claudio Dappiaggi [email protected] \nDipartimento di Fisica\nUniversità di Pavia\nVia Bassi 627100PaviaItaly\n\nINFN\nSezione di Pavia -Via Bassi 627100PaviaItaly\n"
] |
[
"Dipartimento di Fisica\nUniversità di Pavia\nVia Bassi 627100PaviaItaly",
"INFN\nSezione di Pavia -Via Bassi 627100PaviaItaly",
"Dipartimento di Fisica\nUniversità di Pavia\nVia Bassi 627100PaviaItaly",
"INFN\nSezione di Pavia -Via Bassi 627100PaviaItaly"
] |
[] |
We analyse the local behaviour of the two-point correlation function of a quantum state for a scalar field in a neighbourhood of a Killing horizon in a 2 + 1-dimensional spacetime, extending the work of Moretti and Pinamonti in a 3 + 1-dimensional scenario. In particular we show that, if the state is of Hadamard form in such neighbourhood, similarly to the 3 + 1-dimensional case, under a suitable scaling limit towards the horizon, the two-point correlation function exhibits a thermal behaviour at the Hawking temperature. Since the whole analysis rests on the assumption that a Hadamard state exists in a neighbourhood of the Killing horizon, we show that this is not an empty condition by verifying it for a massive, real scalar field subject to Robin boundary conditions in the prototypic example of a three dimensional black hole background: the non-extremal, rotating BTZ spacetime.arXiv:1806.00427v1 [gr-qc] 1 Jun 2018The Hadamard condition has been fist introduced as a selection criterion for physically admissible quantum states, since on the one hand it guarantee that the short-distance behaviour of the state coincides with that of the Poincaré vacuum, while, on the other hand, it ensures that the quantum fluctuations of all observables are finite and that there exists a covariant scheme to construct Wick polynomials,[8,9,10,11].Subsequently, in a very influential paper by Haag and Frendehagen[12], it has been shown that the Hadamard condition is deeply connected to the presence of Hawking radiation, although, in this paper, the focus was on the role of radiative modes. Only in [7] the attention has been switched towards the analysis of the interplay between Hadamard states and the thermal properties of field theoretical models in a neighbourhood of a Killing horizon. The key ingredient in this paper has been the integral kernel of a two-point function ω 2 (x, y). In particular, using the structural properties of a Killing horizon[11,13]and provided that ω 2 is of Hadamard form ina neighborhood of the horizon, it has been proven that, independently of the choice of such bi-distribution, ω 2 acquires a thermal spectrum with respect to the notion of time and energy associated with the Killing field, generating the horizon. This result is obtained by smearing ω 2 with two test functions whose support is close to the horizon. More precisely, if the points x and y of the integral kernel of ω 2 lie on the same side of the horizon, the Fourier transform of the twopoint function displays a Bose factor at the Hawking temperature. Conversely, if x and y are kept at the opposite sides of the horizon, the resulting spectrum agrees with the transition probability between two weakly coupled reservoirs which are in thermal equilibrium at the Hawking temperature, in perfect agreement with the predictions of [2].Yet a close scrutiny of [2] unveils that their analysis can be applied to any spacetime with a Killing horizon, regardless of the dimension of the background, while the results of [7] are based on the leading singular behaviour of the local form of the two-point function of a Hadamard state in four spacetime dimensions. This prompts the question whether the above conclusions can be extended to any dimension and in this paper we investigate the case of a three-dimensional manifolds with a Killing horizon, showing that the results of [7] are still valid regardless of the different singular structure of the Hadamard states. The interest towards this scenario is motivated also by the fact that the analysis of [7] rests on the assumption that there exists a state which is of Hadamard form in a neighbourhood of the Killing horizon. While this is not an obstruction when the underlying manifold is globally hyperbolic, since the existence of Hadamard sates is guaranteed a priori thanks to a deformation argument [14], the situation is completely different when the background does not enjoy such property. This is especially relevant in 2 + 1 dimensions, when considering the prototypic example of a 2 + 1-dimensional black hole spacetime with a Killing horizon: the BTZ solution of the Einstein's equations with a negative cosmological constant[15,16]. This is a stationary, axisymmetric manifold which is locally isometric to AdS 3 , the three-dimensional anti-de Sitter spacetime, and it is not globally hyperbolic. At the level of field theory this has far reaching consequences since there is no guarantee that one can identify a state for a free, massive, real scalar field theory on BTZ which is of Hadamard form in a neighborhood of the outer horizon. This entails that the generalization of the work of [7] might not be testable in the basic example of a three dimensional black-hole, albeit the tunnelling processes and Hawking radiation on BTZ have been investigated from different, complementary viewpoints by several research groups, see e.g.[17,18,19,20]. Hence, as a last step in this work, we argue that this is not the case since one can exploit that BTZ can be constructed directly from AdS 3 via a suitable periodic identification of points in order to individuate, starting from the ground state for a massive, real scalar field on the universal cover of AdS 3 with boundary conditions of Robin type [21], a Hadamard state in the whole BTZ spacetime. This paper is structured as follows. Section 2 offers a brief review of the method proposed by [7] and sets some notations and conventions. In addition we generalize the results of [7] to a 2 + 1-dimensional
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10.1088/1361-6382/aaf335
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[
"https://arxiv.org/pdf/1806.00427v1.pdf"
] | 56,229,505 |
1806.00427
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d55d16a1a3d955f3e49c4b731974f204520e35f2
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Tunnelling processes for Hadamard states through a 2+1 dimensional black hole and Hawking radiation
June 4, 2018
Francesco Bussola [email protected]
Dipartimento di Fisica
Università di Pavia
Via Bassi 627100PaviaItaly
INFN
Sezione di Pavia -Via Bassi 627100PaviaItaly
Claudio Dappiaggi [email protected]
Dipartimento di Fisica
Università di Pavia
Via Bassi 627100PaviaItaly
INFN
Sezione di Pavia -Via Bassi 627100PaviaItaly
Tunnelling processes for Hadamard states through a 2+1 dimensional black hole and Hawking radiation
June 4, 2018
We analyse the local behaviour of the two-point correlation function of a quantum state for a scalar field in a neighbourhood of a Killing horizon in a 2 + 1-dimensional spacetime, extending the work of Moretti and Pinamonti in a 3 + 1-dimensional scenario. In particular we show that, if the state is of Hadamard form in such neighbourhood, similarly to the 3 + 1-dimensional case, under a suitable scaling limit towards the horizon, the two-point correlation function exhibits a thermal behaviour at the Hawking temperature. Since the whole analysis rests on the assumption that a Hadamard state exists in a neighbourhood of the Killing horizon, we show that this is not an empty condition by verifying it for a massive, real scalar field subject to Robin boundary conditions in the prototypic example of a three dimensional black hole background: the non-extremal, rotating BTZ spacetime.arXiv:1806.00427v1 [gr-qc] 1 Jun 2018The Hadamard condition has been fist introduced as a selection criterion for physically admissible quantum states, since on the one hand it guarantee that the short-distance behaviour of the state coincides with that of the Poincaré vacuum, while, on the other hand, it ensures that the quantum fluctuations of all observables are finite and that there exists a covariant scheme to construct Wick polynomials,[8,9,10,11].Subsequently, in a very influential paper by Haag and Frendehagen[12], it has been shown that the Hadamard condition is deeply connected to the presence of Hawking radiation, although, in this paper, the focus was on the role of radiative modes. Only in [7] the attention has been switched towards the analysis of the interplay between Hadamard states and the thermal properties of field theoretical models in a neighbourhood of a Killing horizon. The key ingredient in this paper has been the integral kernel of a two-point function ω 2 (x, y). In particular, using the structural properties of a Killing horizon[11,13]and provided that ω 2 is of Hadamard form ina neighborhood of the horizon, it has been proven that, independently of the choice of such bi-distribution, ω 2 acquires a thermal spectrum with respect to the notion of time and energy associated with the Killing field, generating the horizon. This result is obtained by smearing ω 2 with two test functions whose support is close to the horizon. More precisely, if the points x and y of the integral kernel of ω 2 lie on the same side of the horizon, the Fourier transform of the twopoint function displays a Bose factor at the Hawking temperature. Conversely, if x and y are kept at the opposite sides of the horizon, the resulting spectrum agrees with the transition probability between two weakly coupled reservoirs which are in thermal equilibrium at the Hawking temperature, in perfect agreement with the predictions of [2].Yet a close scrutiny of [2] unveils that their analysis can be applied to any spacetime with a Killing horizon, regardless of the dimension of the background, while the results of [7] are based on the leading singular behaviour of the local form of the two-point function of a Hadamard state in four spacetime dimensions. This prompts the question whether the above conclusions can be extended to any dimension and in this paper we investigate the case of a three-dimensional manifolds with a Killing horizon, showing that the results of [7] are still valid regardless of the different singular structure of the Hadamard states. The interest towards this scenario is motivated also by the fact that the analysis of [7] rests on the assumption that there exists a state which is of Hadamard form in a neighbourhood of the Killing horizon. While this is not an obstruction when the underlying manifold is globally hyperbolic, since the existence of Hadamard sates is guaranteed a priori thanks to a deformation argument [14], the situation is completely different when the background does not enjoy such property. This is especially relevant in 2 + 1 dimensions, when considering the prototypic example of a 2 + 1-dimensional black hole spacetime with a Killing horizon: the BTZ solution of the Einstein's equations with a negative cosmological constant[15,16]. This is a stationary, axisymmetric manifold which is locally isometric to AdS 3 , the three-dimensional anti-de Sitter spacetime, and it is not globally hyperbolic. At the level of field theory this has far reaching consequences since there is no guarantee that one can identify a state for a free, massive, real scalar field theory on BTZ which is of Hadamard form in a neighborhood of the outer horizon. This entails that the generalization of the work of [7] might not be testable in the basic example of a three dimensional black-hole, albeit the tunnelling processes and Hawking radiation on BTZ have been investigated from different, complementary viewpoints by several research groups, see e.g.[17,18,19,20]. Hence, as a last step in this work, we argue that this is not the case since one can exploit that BTZ can be constructed directly from AdS 3 via a suitable periodic identification of points in order to individuate, starting from the ground state for a massive, real scalar field on the universal cover of AdS 3 with boundary conditions of Robin type [21], a Hadamard state in the whole BTZ spacetime. This paper is structured as follows. Section 2 offers a brief review of the method proposed by [7] and sets some notations and conventions. In addition we generalize the results of [7] to a 2 + 1-dimensional
Introduction
Quantum field theory on curved backgrounds aims at exploring the behaviour of quantum systems in the presence of a fixed, classical metric. In this setting, particular attention has been devoted to the case when the geometry of the spacetime describes a black hole. Aside from the intrinsic importance of these gravitating systems, they played a paramount role in revealing new unexpected, quantum phenomena, such as the emission of Hawking radiation from a black hole [1].
Since its first computation, many alternative derivations of such phenomenon have been proposed, though eventually, a good deal of attention has been devoted to studying the interplays between radiating black holes and the local properties of the underlying quantum field in a neighbourhood of a point on the event horizon. Such viewpoint culminated in the seminal paper by Parikh and Wilczek [2] in which they associate the arise of thermal effects to a tunnelling process through the horizon, see also [3,4,5,6].
We will not dwell into the details of these papers, rather we will focus on the viewpoint, first advocated and thoroughly analysed in [7]. Herein, the main message has been to show that the arise of a thermal behaviour from a tunnelling process, as proposed in [2], can be ascribed only to the leading singular structure of the two-point correlation function of a Hadamard state for the underlying field theory, which is taken to be for simplicity a real, massive scalar field. spacetime equipped with a Killing horizon. In Section 3 we test our general analysis in the distinguished case of a scalar field on BTZ spacetime. Subsection 3.1 lists some notable geometrical properties of such background, in particular that it possesses two bifurcate horizons, an inner and an outer one. Finally in Subsection 3.2 we consider a massive real scalar field theory on BTZ spacetime showing that, for boundary conditions of Robin type, it admits a Hadamard state in a neighbourhood of the outer horizon, which is the key prerequisite to apply the reasoning of Section 2. This statement combines two distinct results. The first, based on [22], is the explicit construction, via a mode decomposition, of the ground state for a massive real scalar field with Robin boundary conditions, living in the exterior region of the outer horizon of a BTZ spacetime. This allows in turn to construct explicitly states obeying the KMS condition at arbitrary temperature. The second result is based on [21] and it refers to the proof of the existence of a one parameter family of maximally symmetric states, locally of Hadamard form, for a real, massive scalar field on the universal cover of anti-de Sitter spacetime. Each of these states is characterized by the choice of a boundary condition of Robin type.
Tunnelling processes in + 1 dimensions
In this section we review the tunnelling process, first devised by Parikh and Wilczeck in [2], here presented with the local approach as proposed by Moretti and Pinamonti in [7]. As we will show, although developed in connection to four dimensional spacetimes, this viewpoint is well suited to be generalized to arbitrary spacetime dimensions, 2 + 1 in particular.
General setting
In the following, let (M, g) be a smooth, three dimensional, connected Lorentzian manifold. In addition, following [7, Def. Observe that the hypotheses above encompass the case of (M, g) being endowed with a Killing field K generating a bifurcate Killing horizon, see also Section 3. In this case K is vanishing on a two-dimensional acausal submanifold B and it is lightlike on two K-invariant null submanifolds, H + and H − such that B = H + ∩ H − . Most notably any neighbourhood of a point p ∈ H ± , not intersecting B, satisfies the geometric hypotheses. From now we shall assume to work in this setting. This is not at all a restriction since, whenever we consider O ⊂ M fulfilling the hypotheses above, it is possible to deform smoothly (M, g) so for a bifurcate Killing horizon to exist. On account of our interest in evaluating quantities only defined in O, this procedure is harmless.
In view of [11,13] we can consider in O a distinguished coordinate patch (V, U, x 3 ). Here U denotes an affine parameter along the null geodesics, which are the integral lines of K with the origin fixed to lie at B. At the same time V is the affine parameter, with origin at B, of the integral curves of the futurepointing lightlike vector field n H+ of H + . This is built as the parallel transport of n, the unique, future pointing, lightlike vector at B such that with g(n, − ∂ ∂U ) = − 1 2 . The remaining unknown, x 3 denotes any, but fixed coordinate defined on an open neighbourhood of a point lying in B. Every such chart defines per restriction a counterpart on O.
With respect to the coordinates (U, V, x 3 ), it turns out that the Killing vector field K in O reads
K = K 1 ∂ ∂V + K 2 ∂ ∂U + K 3 ∂ ∂x 3(1)
and, if there exists O ⊂ O such that its closure is compact and such that p ∈ O , then
K 1 (p) = −κV + V 2 R 1 (p), K 2 (p) = κU + V 2 R 2 (p), K 3 (p) = V R 3 (p). Here R 1 , R 2 , R 3 are bounded smooth functions on O , cf. [7, Prop. 2.1]
, while κ is the surface gravity. In addition the line element of the metric g reads
g H + = − 1 2 dU ⊗ dV − 1 2 dV ⊗ dU + h(x 3 )dx 3 ⊗ dx 3(2)
where H + indicates that the metric is defined in a neighbourhood of H + , while h is a strictly positive function depending only on x 3 . Combining together (1) with (2) it turns out that, at the leading order
in V , g H + (K, K) = κ 2 U V + O(V 2O = O s ∪ O 0 ∪ O t where O 0 = O ∩ H + , while O s . = {p ∈ O | V (p) < 0} and O t . = {p ∈ O | V (p) > 0},(3)
where the subscripts s and t indicate that the vector field K is spacelike and timelike in O s and O t respectively. Heuristically and comparing with standard scenarios such as Schwarzschild spacetime, we can divide a neighbourhood of a point at the horizon in an interior (s) and an exterior (t) region.
Having established (2), it is possible to study the properties of the geodesic distance, which will play a key role in the next sections. Since the outcome is a slavish adaptation to a three dimensional scenario of [7, Prop. 2.1], we will not dwell into the details of a systematic proof. More precisely, let p ∈ O be a point of coordinates (U, V, x 3 ), so that p ∈ H + if and only if V = 0. Then it holds that 1. Let O ⊂ O be any geodesically convex neighbourhood of H + and let p, q ∈ H + ∩ O. Then, the squared geodesic distance between these points is
σ(x(p), x(q)) ≡ (x 3 (p), x 3 (q)) . = x3(q) x3(p) dλ f (λ) 2 ,(4)
where x(p) (resp. x(q)) indicates the representation of the point p (resp. of the point q) in terms of the coordinates (U, V, x 3 ). Furthermore f 2 = h, h being the function appearing in (2), while x 3 (p) and x 3 (q) are respectively the evaluation of p and q along the coordinate x 3 .
2. Let p ∈ O and, for any but fixed, admissible value of the coordinates U , V , let S U ,V be the collection of points q lying in the cross section of O at constant V and U . For δ > 0, let
G δ (p, V , U ) = {q ∈ S U ,V | (x 3 (p), x 3 (q)) < δ 2 },(5)
being as in (4). Then δ can be chosen in such a way that the smooth map G δ (p, V , U ) q → σ(x(p), x(q)) has minimum in a unique point q(p, V , U ). As a consequence of (4) x 3 (q) = x 3 (p) if p ∈ H + ∩ O, whereas in general, there exists three bounded functions F i , i = 1, 2, 3, depending smoothly on x(p), U , V such that
σ(x(p), x(q)) = (x 3 (p), x 3 (q)) − (U − U )(V − V ) + R(x(p), V, V , U ) (6) where R(x(p), U , V ) = F 1 V 2 + F 2 V 2 + F 3 V V .
Two-point Correlation Functions and their Scaling Behaviour
On top of the spacetime (M, g) we consider for definiteness a real, scalar field Φ : M → R whose dynamics is ruled by the Klein-Gordon equation
P Φ = 2 − m 2 − ξR Φ = 0,(7)
where 2 is the D'Alembert wave operator built out of g, m 2 ≥ 0, R is the Ricci scalar while ξ ∈ R. This choice is motivated mainly for its connection with Section 3, although the following analysis is independent from the choice of a detailed free field theory relying only on the specific assumptions on the singular structure of the two-point correlation function. The details of the quantization of (7) are an overkilled topic which has been thoroughly discussed in the literature. Therefore we will not dwell in the details, referring an interested reader for example to [23]. For the sake of clarity, we outline only the main points. To start with we assume that (7) can be solved in terms of an advanced and a retarded fundamental solutions G ± :
C ∞ 0 (M) → C ∞ (M) such that P • G ± = G ± • P = id| C ∞ 0 (M) and such that supp(G ± (f )) ⊆ J ∓ (supp(f )) for all f ∈ C ∞ 0 (M).
Under such premise, we can associate to the scalar field Φ an algebra of observables A(M). This is constructed from the following basic structures: f
1. T (M) = ∞ n=0 (C ∞ 0 (M)) ⊗n , where C ∞ 0 (M) ⊗0 ≡ C be⊗ f − f ⊗ f − iG(f, f )I (CCR) where f, f ∈ C ∞ 0 (M) while I is the identity of T (M) and G . = G − − G + .
Hence we set A(M) = T (M) I(M) , which is also called algebra of observables. Expectation values of observables can be computed in terms of correlation functions ω n which are C-linear maps on C ∞ 0 (M) ⊗n compatible both with the relations defining the ideal I(M ) and with the positivity and normalization requirements needed to define a full-fledged state for the underlying quantum theory, see for example [10]. If one is interested in the local properties of the system in a submanifold O, one might define a local algebra of observable as the restriction of A(M) on O.
In between the plethora of admissible correlation functions, we will be focusing on the special subclass of those of Hadamard form, see [11]. More precisely, adopting the notation and conventions of [24], we consider a two-point correlation function which descends from a distribution ω 2 ∈ D (M × M) such that the restriction of its integral kernel to O × O reads
ω 2, (x, x ) = ∆(x, x ) 1/2 4π σ (x, x ) + w (x, x ) ,(8)
where x, x stand for the coordinates of two arbitrary points in O,
∆ ∈ C ∞ ( O × O) is the Van Vleck- Morette determinant, a quantity built only out of the metric, while w (x, x ) is a smooth function on O × O. The function σ (x, x ) := σ(x, x ) + 2i (T (x) − T (x )) + 2 , where σ(x, x ) is the squared geodesic
distance between x and x , > 0 while T is any, but fixed time function. Recall that O is a convex geodesic neighbourhood with non vanishing intersection with H + . Having introduced all ingredients needed, we can now focus on extending to the 2 + 1-dimensional scenario the analysis of [7]. Let us consider thus two one-parameter families of test functions f λ , f λ ∈ C ∞ ( O), λ ∈ R such that both obey to the constraints
f λ (V, U, x 3 ) = 1 λ f V λ , U, x 3 and f = ∂F ∂V , F ∈ C ∞ 0 ( O).(9)
In the following, taking into account (8), we shall evaluate
lim λ→0 + ω 2 (f λ , f λ ) = lim λ→0 + lim →0 + O× O ∆(x, x ) 1/2 4π σ (x, x ) + w (x, x ) f λ (x)f λ (x )dµ g (x)dµ g (x ) ,(10)
where
dµ g = h(x 3 )dx 3 dU dV is the metric induced volume form on O.
We observe that only the contribution to (2.2) of the singular part of the two-point correlation function is relevant since, being w smooth, the integral with respect to either V or V vanishes in view of the assumptions on the test functions. To this end we need to introduce an auxiliary cut-off function built as follows. Let δ > 0 and let G δ (p, V , U ) be as in (5).
We define G δ (p, V , U ) p → χ δ (x, x ) ≥ 0 where x indicates the coordinates of p, while x = (V , U , x 3 ) those of p ,
to be any smooth and compactly supported function subject to the constraint
χ δ (x, x ) = 1, for 0 ≤ (x 3 (p), x 3 (p )) ≤ δ 2 + 1 2 (x 3 (p), x 3 (q))
Here q refers to the unique point q(p, V , U ) ∈ O, minimizing the function G δ (p, V , U ) p → σ(x, x ). Hence, for any pair of points p, p ∈ O of coordinates x, x respectively, we rewrite the contribution to the two-point correlation function coming from the singular part of (8) as
O× O dµ g (x)dµ g (x ) ∆(x, x ) 1 2 f λ (x)f λ (x ) 4π σ (x, x ) χ δ (x, x )+ (11a) O× O dµ g (x)dµ g (x ) ∆(x, x ) 1 2 f λ (x)f λ (x ) 4π σ (x, x ) (1 − χ δ (x, x )) (11b)
where dµ g stands from the volume form induced by (2). In the following our strategy is to evaluate separately the limits of the above two terms as λ and tend to 0 + .
Evaluation of (11b): Following the same argument as in [7], by shrinking if necessary O, it turns out that the integrand is nowhere singular, being actually jointly smooth in all variables including if = 0. Hence we can apply Lesbegue dominated convergence theorem to exchange both limits with the integrals. The end point is an integrand, which is the derivative with respect to V and V of a compactly supported smooth function. Hence the overall integral vanishes.
Evaluation of (11a): In this case we need to deal with the singularity due to σ (p, p ) as → 0 + . To this avail, for every p ∈ O, let ρ :
G δ (p, V , U ) → [0, ∞) be such that σ(x, x ) = ρ(x ) 2 + σ(x, x(q)), .(12)
where G δ (p, V , U ) is defined in (5) while q is the point therein in which, for fixed p, the function σ attains its unique minimum. Furthermore, from now on, in each G δ (p, V , U ) we switch from the coordinates (U , V , x 3 ) to (U , V , ρ). Using (12) and the Taylor expansion (6), (11a) reduces to
O× O dµ g (x)dµ g (x ) ∆(x, x ) 1/2 χ δ (x, x )f λ (x)f λ (x ) 4π ρ 2 + (x 3 , x 3 ) − (U − U − i )(V − V − i ) + R(x, V , U ) .(13)
To write this integral in a more manageable form, we observe that the denominator of (13) is the derivative with respect to ρ of
Ξ(ρ, V, V , R) = ln ρ 2 + (x 3 (p), x 3 (q)) − (U − U − i )(V − V − i ) + R(x, V , U ) 1 2 + ρ ,
where we indicate only the explicit dependence of those variable which will play a role in the following discussion. Furthermore it is convenient to rescale (V, V ) to (λV, λV ) so that, calling ∆ λ , R λ , dµ g λ the quantities transformed accordingly and considering the hypothesis that
f = ∂ V F , f = ∂ V F , we get O λ × O λ dµ g λ (x)dµ g λ (x ) ∆ λ (x, x ) 1 2 4π χ δ (x, x ) ∂ V F (x) ∂ V F (x ) ∂ ρ Ξ (ρ, λV, λV , R λ ) .
Since ρ has a domain of definition of the form [0, ρ 0 ) we can integrate by parts in this variable so to deal with the contribution from ∂ ρ Ξ. Of the ensuing boundary terms, that due to ρ 0 vanishes being F compactly supported, while that due to ρ = 0 yields, up to an innocuous rescaling of as λ
− O λ × O λ dµ g λ (x)dU dV |g| ρ=0 ∆ λ (x, x ) 1 2 4π ∂ V F (x) ∂ V F (x )| ρ=0 Ξ 0, V, V , R λ λ + ln λ ,(14)
where we have written dµ g (x ) = |g|dρ dU dV and we have implicitly used that, when ρ = 0, (x 3 , x 3 ) = 0. Since we are interested in taking both the limit of (14) as λ and tend to 0 + , we observe that the term proportion to ln λ does not contribute. Since both F is smooth and compactly supported, one can integrate by parts in V . The boundary terms vanish while, due to the derivative of ∆ 1 2 λ , the remaining integral is proportional to λ ln λ.
At the same time, in order to evaluate the remaining term in (14), first of all we notice that, in view of Eq. (6), being R quadratic in V and V , there exists C ∈ R such that, for λ sufficiently close to 0 + , |λ −1 R λ | < Cλ. In addition, by direct inspection of (2) it turns out that, when either x or x tend to the horizon and λ → 0
+ , | det g λ | → 1 2 while dµ g (x) = dU dV 2 dµ(x 3 ) where dµ(x 3 ) = h(x 3 )dx 3
. As a last ingredient observe that, at a distributional level, letting z = w + iy, it holds lim y→0 + ln(z) = ln |w| + iπ(1 − Θ(w)), Θ being the Heaviside function. Gathering all these data together, the remaining contribution from (14) reads as in [7] − 1 32π lim
→0 + O× O ∂ V F (x)∂ V F (x ) ln(−(U − U )(V − V − i ))dU dV dU dV dµ(x 3 ) ,(15)
where we have restored the limit as → 0 + and where we used that ∆ 0 = 1 when x 3 = x 3 and V = V = 0. As a consequence we managed to rearrange (2.2) in the form 15 which is a useful expression to investigate the energy spectrum of the two-point correlation function, as seen by an observer moving along the curves generated by K with respect to the associated Killing time τ .
Thermal spectrum of the correlation functions
Having established the form (15) for the two-point correlation function of a real, massive scalar field in 2 + 1 dimensions, we can repeat verbatim the analysis of [7] aimed at computing the energy spectrum of (15) as seen by an observer the moves along the integral curves of K. We summarize the procedure. It calls for considering two test functions which are squeezed on the Killing horizon. From an operational point of view this consists of considering the leading behaviour of (15) as V is close to 0. To this end, a direct inspection of (1) unveils that, calling τ the affine parameter of the integral curves of K, it holds that, up to an irrelevant additive constant
V = −e −κτ for V < 0, V = e −κτ for V > 0.(16)ω 2 (Φ(f λ )Φ(f λ )) = lim →0 + − κ 2 128π R 4 ×B dτ dU dτ dU dx 3 F (τ, U, x 3 )F (τ , U , x 3 ) (sinh κ 2 (τ − τ ) + i ) 2 ,
where, in view of the compactness of both F and F we can extend the domain of integration for the variables τ, τ , U, U to the whole real axis, while B indicates a one-dimensional, connected, domain of integration, diffeomorphic to the bifurcation horizon. We can rewrite this last expression in Fourier space with respect to the variables τ and τ as
lim λ→0 ω 2 (Φ(f λ )Φ(f λ )) = 1 64 R 2 ×B dU dU dx 3 ∞ −∞ dEE F (E, U, x 3 ) F (E, U , x 3 ) 1 − e −β H E ,
where β H = 2π κ while both F and F indicate the Fourier transform of F and F respectively. This result can be interpreted as follows: Whenever a state for a real, massive Klein Gordon field, is such that its two-point function is of Hadamard form in a geodesic neighbourhood of a point of a Killing horizon, then the modes of the two-point correlation function, built with respect to (16), follow in the region external to the horizon a thermal distribution at the Hawking temperature β −1 H . Case 2) Let us consider now two 1-parameter families of test functions f λ and f λ obeying (9), up to the replacement of O with O t for f λ and with O s for f λ . Following the same calculation as in the previous case the result is left unchanged except for the hyperbolic sine being replaced by the hyperbolic cosine. Eventually, rewriting the result in Fourier space with respect to the variables τ and τ , we obtain
lim λ→0 ω 2 (Φ(f λ )Φ(f λ )) = 1 32 R 2 ×B dU dU dx 3 ∞ −∞ dEE F (E, U, x 3 ) F (E, U , x 3 ) sinh(β H E/2) ,(17)
where, once more, β H = 2π κ while both F and F indicate the Fourier transform of F and F respectively. Since the support of the test functions and hence of the observables is located both in the interior and in the exterior region with respect to the Killing horizon, once can interpret |ω 2 (f λ , f λ )| 2 as a tunnelling probability through the horizon. By considering wave packets which are peaked around E 0
1, (17) yields lim λ→0 |ω 2 (Φ(f λ ), Φ (f λ ))| 2 ≈ E 2 0 e −β H E0 ,
which is nothing but the original result of [2].
Hawking radiation from a BTZ black hole
The analysis of the previous section rests mainly on two assumptions, the existence both of a (local) Killing horizon H + and of a state for a real, massive scalar field which is of Hadamard form at least in a geodesic neighbourhood of a point at H + . While the first requirement indicates a constraint on the underlying geometry, pointing to considering black hole spacetimes or a generalization thereof, the second one is more subtle since it refers to the underlying quantum theory. The construction of Hadamard states is a thoroughly investigated problem and it is by now established their existence when the underlying manifold is globally hyperbolic [14]. In the presence of a black hole, global hyperbolicity of a domain which crosses the whole horizon is not at all an obvious feature, being satisfied in some specific scenarios such as for example the four-dimensional Schwarzschild spacetime and its Kruskal extension. In this case one can consider in a neighbourhood of the event horizon as Hadamard states for the Klein-Gordon field both the Unruh [25,26] and the Hartle-Hawking states [27,28]. In 2 + 1 dimensions, the situation is far more intricate if one is interested in black hole spacetimes. As a matter of fact, in order for the underlying geometry to possess a global Killing horizon, one is forced to considering only solutions to the Einstein's equation with negative cosmological constant, that is asymptotically AdS spacetimes. The prime example is the renown BTZ spacetime [16], which is not globally hyperbolic, as one can readily infer since it possesses a timelike, conformal boundary. Hence, even in the prototypical example of a 2 + 1-dimensional black hole spacetime, one cannot invoke a general result to conclude that, for a real, massive scalar field, a state which is Hadamard in a neighbourhood of the horizon exists. Therefore, one of the key assumptions at the heart of the analysis in Section 2.1 is not a priori verified and it prompts the question whether any quantum state, meeting the wanted hypotheses, exists. In this section we prove that such a pathological scenario does not occur.
BTZ geometry
In this Section, we recall the basic structural feature of the BTZ black-hole spacetime. This is a stationary, axisymmetric, (2+1)-dimensional solution of the vacuum Einstein field equations with a negative cosmological constant Λ = −1/ 2 [15,16]. Henceforth we set = 1. The line element reads
ds 2 = −N 2 dt 2 + N −2 dr 2 + r 2 dφ + N φ dt 2 ,(18)
where t ∈ R, φ ∈ (0, 2π), r ∈ (0, ∞), while
N 2 = −M + r 2 + J 2 4r 2 , N φ = − J 2r 2 .
Here the parameter M is interpreted as the mass of the black hole, while J as its angular momentum. In the range M > 0, |J| ≤ M , the values Figure 1: Penrose diagrams of the BTZ black hole for the rotating 0 < r − < r + (left) and the static 0 = r − < r + (right) cases.
r 2 ± = 1 2 M ± M 2 − J 2 .(19)r = r+ r = ∞ r = r− r = 0 r = r+ r = ∞ r = 0 1
represent an outer event horizon and an inner Cauchy horizon for the black hole. In particular the former is a a Killing horizon generated by the Killing field
K := ∂ t + Ω H ∂ φ ,(20)
where Ω H := N φ (r + ) = r− r+ can be interpreted as the angular velocity of the horizon itself. With reference to the analysis of section 2.1, we will identify O BT Z t as per (3) as the whole region r > r + , while O BT Z s as that for which r − < r < r + . A characteristic feature of BTZ spacetime, which distinguish it from other rotating black holes, such as for example Kerr, is that K is a timelike Killing vector field across the whole O BT Z t . A further, relevant, geometric quantity is the surface gravity [29,Ch. 12] which, in the case at hand, can be computed from the defining equation K µ ∇ µ K ν = κK ν , K as in (20), to be
κ = r 2 + − r 2 − r + .
We stress that the BTZ black hole spacetime is not globally hyperbolic. In fact, the timelike surface r = ∞ behaves as a conformal boundary for (M, g). This implies that the solutions of the equations of motion for a free field theory must be obtained by imposing suitable boundary conditions at the boundary. The Penrose diagrams of this spacetime are shown in Fig. 1.
As a final remark we stress that BTZ spacetime is a manifold, locally of constant, negative scalar curvature. The underlying reason is that it can be constructed via a suitable periodic identification of points, directly from AdS 3 , the three dimensional anti-de Sitter, maximally symmetric solution to the vacuum Einstein's equations with negative cosmological constant. Since this feature will play a distinguished role in our analysis, we review the constructive procedure of BTZ spacetime succinctly, following [30,Ch. 12].
The starting point is R 4 endowed with Cartesian coordinates X i , i = 0, ..., 3 and with the line elements ds 2 = −dX 2 0 − dX 2 1 + dX 2 2 + dX 2 3 . AdS 3 is realized thus as the hyperboloid
−X 2 0 − X 2 1 + X 2 2 + X 2 3 = −1,
where the cosmological constant Λ has been normalized to −1. Starting from this realization of anti-de Sitter spacetime, it is possible to cover the whole BTZ solution by means of three patches corresponding respectively to the regions i) r ≥ r + , ii) r − ≤ r ≤ r + and iii) r ≤ r − . For our later purposes only the first two are relevant, since we are interested in the behaviour of a real, massive, scalar field theory in a neighbourhood of r = r + . Hence we make explicit only the first two set of coordinate transformations which read as follows:
Region i) X 0 = α(r) cosh r + φ − r − t , X 1 = α(r) − 1 sinh r + t − r − φ X 2 = α(r) sinh r + φ − r − t , X 3 = α(r) − 1 cosh r + t − r − φ ,(21)
and
Region ii)
X 0 = α(r) cosh r + φ − r − t , X 1 = − α(r) − 1 sinh r + t − r − φ X 2 = α(r) sinh r + φ − r − t , X 3 = − α(r) − 1 cosh r + t − r − φ .(22)
Here (t, r) are the same coordinates used in (18) while φ ∈ R and α(r) =
r 2 −r 2 − r 2 + −r 2 −
, r ± being as in (19). In order to obtain the BTZ metric we need to switch from φ to φ which is obtained considering the former and imposing the periodic identification φ ≡ φ + 2π. As a side remark, we observe that the region i) could be also obtained via a similar, periodic identification procedure starting from the Poincaré patch of AdS 3 .
KMS state for a massive scalar field
We now consider a real, massive scalar field Φ : M → R satisfying the Klein-Gordon equation
(2 g − m 2 − ξR)Φ = 0 ,(23)
where 2 g is the D'Alembert wave operator built out of (18), R = −6 is the scalar curvature, while ξ ∈ R and m 2 ≥ 0 is the mass. Since the underlying background is not globally hyperbolic, solutions of (23) cannot be constructed only by assigning initial data on a two-dimensional, smooth, spacelike hypersurface, but one must impose also suitable boundary conditions as r → ∞. This problem has been studied thoroughly in [22] (see also [31] for an investigation of the static case) where (23) has been analysed in the stationary region r > r + by considering boundary conditions of Robin type. Observe that this corresponds to (21) in the previous section. The ensuing space of classical solutions of the equation of motion has been used subsequently to construct, for each admissible boundary condition, the two-point function of the associated ground state, while the ensuing algebra of observables can be constructed along the same lines used for the counterpart of a real, massive scalar field in the Poincaré patch of a (d + 1)−dimensional AdS spacetime, see [32].
Here we only report the final result, leaving an interested reader to [22] for all the technical details of the derivation. Most notably, it turns out that we need to distinguish two different scenarios, which are ruled by the parameter µ 2 .
= m 2 − 6ξ, which, according to our conventions, is dimensionless. Introducing for convenience the new coordinate z =
r 2 −r 2 + r 2 −r 2 − ∈ (0, 1),
1. If µ 2 ≥ 0, there exists only one admissible asymptotic behaviour at the conformal boundary of the solutions of (23). Hence, in this case, one does not need to impose boundary conditions as r → ∞ (equivalently z → 1) and the two-point function of the underlying ground state reads
ω 2 (x, x ) = lim →0 + k∈Z e ik( φ− φ ) ∞ 0 d ω (2π) 2 e −i ω( t− t −i ) A B − A B C Ψ 1 (z)Ψ 1 (z ) ,(24)
whereφ = φ − Ω H t,t = t, while we recall that Ω H is the angular velocity of the horizon. The remaining unknowns, namely A, B and C are constants while Ψ 1 is a function depending only on z constructed out of (7) via a mode expansion. For convenience we list their form explicitly in the appendix. Observe thatω = ω−kΩ H represents the Fourier parameter associated to the Killing field K as in (20). Since K is stationary across the whole region r > r + . Therefore, having constructed (24) only out of positive values of ω, we are justified in calling ω 2 the two-point correlation function of a ground state.
2. If −1 < µ 2 < 0, there exists a one-parameter family of admissible boundary conditions of Robin type which can be assigned at z = 1 and which admits an associated ground state. Calling the underlying parameter ζ, it turns out that, there exists a value ζ * ∈ (0, π 2 ) such that, whenever ζ ∈ [0, ζ * ], the two-point function of the underlying ground state reads
ω ζ 2 (x, x ) = lim →0 + k∈Z e ik( φ− φ ) ∞ 0 d ω (2π) 2 e −i ω( t− t −i ) AB − AB C |cos(ζ)B − sin(ζ)A| 2 Ψ ζ (z)Ψ ζ (z ) .(25)
where Ψ ζ (z) = cos(ζ)Ψ 1 (z) + sin(ζ)Ψ 2 (z), where Ψ 1 and Ψ 2 are listed in the appendix.
It is noteworthy that, since (24) and (25) identify ground states, they are all of Hadamard form as it is proven in full generality in [33]. As a next step, we show how to switch from (24) and (25) to an associated thermal equilibrium state, satisfying the KMS condition. To this end we apply a standard procedure, see for example [34]. The main ingredients are (24) and (25), the two-point correlation functions of a ground state, and the existence of a timelike Killing field K as in (20) with an associated action α t :
C ∞ 0 (M) → C ∞ 0 (M) such that, for all f ∈ C ∞ 0 (M) and for all t ∈ R, α t f (x) = f ( α − t (x)),
where α − t (x) indicates the flow of x ∈ M built out of the integral curves of K. Starting from this premise, we say that a two-point correlation function ω 2,β ∈ D (M × M) satisfies the KMS condition at the inverse temperature β > 0 with respect to α t if, for every f, f ∈ C ∞ 0 (M), it holds
R d t ω 2,β (f, α t (f ))e −i ω t = R d t ω 2,β (α t (f ), f )e −i ω( t+iβ) .(26)
Having already established the existence of a ground state in the exterior region O BT Z t , cf. (24) and (25), built out of the positive frequencies ω, one can construct straightforwardly an associated two-point correlation function obeying (26). If µ 2 ≥ 0, the integral kernel reads
ω 2,β (x, x ) = lim →0 + k∈Z e ik( φ− φ ) ∞ 0 d ω (2π) 2 A B − A B C e −i ω( t− t −i ) e β ω e β ω − 1 + e i ω( t− t +i ) e −β ω 1 − e −β ω Ψ 1 (z)Ψ 1 (z ), .(27)while, if −1 < µ 2 < 0 and if ζ ∈ [0, ζ * ], ω ζ 2,β (x, x ) = lim →0 + k∈Z e ik( φ− φ ) ∞ 0 d ω (2π) 2 AB − AB C |cos(ζ)B − sin(ζ)A| 2 e −i ω( t− t −i ) e β ω e β ω − 1 + e i ω( t− t +i ) e −β ω 1 − e −β ω Ψ ζ (z)Ψ ζ (z ) .(28)
We observe that, for all β > 0, both (27) and (28) identify two-point correlation functions satisfying the Hadamard condition as one can infer either by using the results of [33] or by observing that ω 2,β − ω 2 and ω ζ 2,β − ω ζ 2 are smooth functions. In order to conclude the analysis one should study if there exists a specific value of β for which both (27) and (28) can be seen as the restriction to r > r + of the two-point correlation function of a state, which is of Hadamard form also in a neighbourhood of the outer horizon.
In order to give an answer to this query, it suffices to outline the procedure described in [30,Ch. 12]. The starting point consists of recalling that BT Z is realized form CAdS 3 , the universal cover of AdS 3 via the identifications (21) and (22). As a consequence, starting from the integral kernel of any two-point function in CAdS 3 , one can build a counterpart on BT Z by means of the method of images which implements the periodic identification built in (21) and in (22). In [30] it is shown that the ensuing two-point correlation function is periodic with respect to the time variable t under the shift iβ H , where
β H = 2π T H = 2πr+ r 2 + −r 2 −
is proportional to the inverse Hawking temperature of the BTZ black hole. As a by-product the restriction of such two-point correlation function, restricted to the region r > r + enjoys the KMS property at the Hawking temperature.
In [30] this procedure is made explicit starting from the two-point correlation function in CAdS 3 for the ground state of a massless, conformally coupled real scalar field enjoying either the Dirichlet or the Neumann boundary condition. Most notably both these states are of (local) Hadamard form and such property is pertained by the counterpart built on BTZ spacetime.
Although the analysis is valid apparently for two rather special cases, the conclusion drawn can be applied also to all cases of our interest for the following reason:
1. In [21] it has been shown that one can construct the two-point correlation function associated to the ground state of any Klein-Gordon field with an arbitrary coupling to scalar curvature enjoying a Robin boundary condition. In addition, all these two-point functions are locally of Hadamard form thanks to the analysis in [33].
2. Since anti-de Sitter space and its universal cover are maximally symmetric solutions of the Einstein's equations with a negative cosmological constant, one can show that, since the two-point function of the ground state is also maximally symmetric regardless of the chosen Robin boundary condition, the associated integral kernel depends on the spacetime points x, x only via the geodesic distance σ AdS3 (x, x ) [35].
3. in [30] it is shown that, by applying the method of images, one obtains a two-point correlation function in BT Z spacetime which is periodic in t under a shift of iβ H . This conclusion is drawn as a consequence of the explicit form of σ AdS3 (x, x ) and thus it can be applied also to any state whose two-point correlation function depends only on the spacetime points via the geodesic distance. This is the case for the ground states built in [21] Hence, starting from the work in [21], we obtain states in BTZ spacetime which are of Hadamard form in a neighbourhood of the outer horizon. This proves that all the ingredients needed to apply the method described in Section 2.1 exist. In addition their restriction to the region r > r + must enjoy the KMS property at the Hawking temperature besides the Robin boundary condition. Thus they must coincide with (27) and with (28).
Conclusions
In this paper we have achieved two main results. On the one hand, in the first part of the manuscript, we generalised the method proposed by Moretti and Pinamonti [7] to the 2 + 1-dimensional case, showing that two-point correlation function of any state, which is of Hadamard form in a neighbourhood of a Killing horizon, exhibits a thermal behaviour in the exterior region. Moreover the spectrum of the energy modes obeys a Bose-Einstein distribution at the Hawking temperature. In addition this method also shows that the tunnelling probability through the horizon is in agreement with the results found in [19], and with the semiclassical method proposed in [2]. The main difference is that this new result adopts the point of view proper of quantum field theory in curved spacetime and it is applicable regardless of the underlying quantum state, provided that it is of Hadamard form in a neighbourhood of a bifurcate horizon.
On the other hand, in the second part we have shown that the work hypotheses used in the previous analysis are valid in concrete cases. Here we address this problem considering a massive, real scalar field with Robin boundary conditions on a BTZ spacetime -a 2 + 1-dimensional black hole solution of Einstein's field equations with a negative cosmological constant. This part takes advantage of the recent results in [21,22] combined with the classic ones in [30].
It is important to stress that our work seems to indicate that the analysis in [7] might only be sensible to the Hadamard condition and not to the specific form of the integral kernel of the two-point function of a Hadamard state in four dimensions. It would be interesting to investigate in the future whether the findings of Section 2 apply also for spacetime dimensions n > 4.
2.1] we assume that there exists an open subset O ⊂ M such that (a) there exists K ∈ Γ(T O) for which L K g = 0, (b) the orbits of K lying in O are diffeomorphic to an open interval of R, (c) there exists a two dimensional, connected submanifold H ⊂ O, dubbed local Killing horizon, invariant under the action of the group of local isometries generated by K, (d) K| H is lightlike and the intersection between H and the integral curves of K identifies a smooth 2-dimensional submanifold of H, (e) κ, the surface gravity of H is a non vanishing, positive constant.
the universal tensor algebra of testfunctions, endowed with the * -operation induced from complex conjugation, 2. I(M) be the * -ideal of T (M) generated by elements of the form P f (equation of motion) and
Two cases should be considered when analysing ω 2 (x, x ): both points x, x lie in the exterior region O t , cf.(3), or one lies in O t and the second in O s , the interior region.Case 1) Let us consider two 1-parameter families of test functions f λ and f λ obeying (9), up to the replacement of O with O t . Inserting this hypothesis in (15) we can first integrate by parts both in V and in V . On account of the compactness of the support of both F and F , no boundary term contributes to the result. Eventually, replacing V with (16), we obtain lim λ→0
). Since U is per construction of definite sign in O, say positive, it turns out that, by shrinking if necessary O, we can separate it as the union of three disjoint regions:
AcknowledgementsWe would like to thank Nicolò Drago for the precious and continuous discussions. We are also grateful to Igor Khavkine, Jorma Louko and Nicola Pinamonti for the useful comments. The work of F. B. was supported by a Ph.D. fellowship of the University of Pavia, that of C. D. was supported by the University of Pavia.A Notable QuantitiesIn the following we list the constants and the functions appearing in(24)and in(25), as calculated in[22]. With reference to the former, we setwhere ω = ω − kΩ H , while ω ∈ R and k ∈ Z. At the same time we definewhere F is the Gaussian hypergeometric function.
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|
[] |
[
"Ground state properties of a three-site Bose-Fermi ring with a small number of atoms",
"Ground state properties of a three-site Bose-Fermi ring with a small number of atoms"
] |
[
"Santiago F Caballero-Benítez \nARC Centre of Excellence for Quantum-Atom Optics and Nonlinear Physics Centre\nResearch School of Physics and Engineering\nAustralian National University\nACT 0200CanberraAustralia\n",
"Elena A Ostrovskaya \nARC Centre of Excellence for Quantum-Atom Optics and Nonlinear Physics Centre\nResearch School of Physics and Engineering\nAustralian National University\nACT 0200CanberraAustralia\n"
] |
[
"ARC Centre of Excellence for Quantum-Atom Optics and Nonlinear Physics Centre\nResearch School of Physics and Engineering\nAustralian National University\nACT 0200CanberraAustralia",
"ARC Centre of Excellence for Quantum-Atom Optics and Nonlinear Physics Centre\nResearch School of Physics and Engineering\nAustralian National University\nACT 0200CanberraAustralia"
] |
[] |
We investigate a three-site ring system with a small number of quantum degenerate bosons and fermions. By means of the exact diagonalization of the Bose-Fermi-Hubbard Hamiltonian, we show that the symmetry of the ground state configuration is a function of both the boson-boson and the inter-species interaction in the system. The phase diagram of the system, constructed by computing the exact two-body spatial correlations, reveals nontrivial insulating phases that exist even in the strong bosonic tunneling limit and for incommensurate filling of bosons. These insulating phases are due to the inter-species interactions in the system and are not necessarily accompanied by the suppression of the particle number fluctuations.PACS numbers:
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10.1088/0953-4075/44/13/135301
|
[
"https://arxiv.org/pdf/1011.5002v1.pdf"
] | 119,219,389 |
1011.5002
|
a83fe3ea7ed23cc6c5827d7fa0f713d8d9829f05
|
Ground state properties of a three-site Bose-Fermi ring with a small number of atoms
23 Nov 2010
Santiago F Caballero-Benítez
ARC Centre of Excellence for Quantum-Atom Optics and Nonlinear Physics Centre
Research School of Physics and Engineering
Australian National University
ACT 0200CanberraAustralia
Elena A Ostrovskaya
ARC Centre of Excellence for Quantum-Atom Optics and Nonlinear Physics Centre
Research School of Physics and Engineering
Australian National University
ACT 0200CanberraAustralia
Ground state properties of a three-site Bose-Fermi ring with a small number of atoms
23 Nov 2010arXiv:1011.5002v1 [cond-mat.quant-gas]PACS numbers:
We investigate a three-site ring system with a small number of quantum degenerate bosons and fermions. By means of the exact diagonalization of the Bose-Fermi-Hubbard Hamiltonian, we show that the symmetry of the ground state configuration is a function of both the boson-boson and the inter-species interaction in the system. The phase diagram of the system, constructed by computing the exact two-body spatial correlations, reveals nontrivial insulating phases that exist even in the strong bosonic tunneling limit and for incommensurate filling of bosons. These insulating phases are due to the inter-species interactions in the system and are not necessarily accompanied by the suppression of the particle number fluctuations.PACS numbers:
I. INTRODUCTION
The ability to load ultracold quantum degenerate gases into periodic or quasi-periodic potentials of different geometries, created by optical lattices [1] has opened the way to realization of well known condensed matter systems with atoms of fermionic [3] and bosonic character [4]. Such model systems allow us to gain deeper understanding of the fundamental problems in many body physics beyond the weak coupling, mean-field regime, such as transition between superfluid and Mott-insulator [5]. In the strongly interacting regime, new phenomena can emerge, such as quantum magnetism [6] and the supersolid phase [7]. In general, these emergent phenomena belong to the class of quantum phase transitions and their mechanisms are strongly dependent both on the geometry of the lattice and the interaction between the atoms loaded into the lattice. The theoretical and experimental studies of these effects in large many-sites lattice systems are important both from the fundamental and applied points of view. For example, the physics of ultracold lattice systems where frustration can occur, is fundamental for the development of fault tolerant systems and is important for quantum computing [8].
On the other hand, the possibility to address atoms in single sites [10] and to create mixtures of bosons and fermions [11] draws the attention to the study of finite, inhomogeneous systems [12] with complex interspecies interactions. The study of such two-or threedimensional systems enables understanding of basic physics of more complicated systems, such as interactiondependent properties of ground states (such as frustrations) and the role of symmetry breaking in transitions between different quantum phases. Small, strongly correlated systems may also lend themselves to the process of controlled quantum state preparation and manipulation making it potentially useful in quantum information, quantum-limited measurements, and atomtronics [15,16].
In this paper we consider a minimal finite twodimensional lattice model, namely a three-site ring of a Bose-Fermi ultracold mixture. Such a three-site sys-tem may be realized experimentally by engineering magnetic microtraps on an atomic chip, or by combining a harmonic potential with a triangular or Kagome lattice, as suggested in [13]. With the small number of atoms, this system lends itself to the Bose-Fermi-Hubbard model solvable by means of direct diagonalization. We consider the ground state of the system and investigate how the admixture of fermions leads to various phases, depending on the filling factor and inter-species interaction strength. In particular, we consider unusual insulating phases resulting from the inter-species interaction, which is connected to the existence of macroscopic self-trapping states in the mean-field regime [14].
II. THE MODEL AND THE GROUND STATE CONFIGURATION
Based on the standard Bose-Fermi Hubbard model (see, e.g., [12]), the Hamiltonian for the three-site ring can be written as follows:
H = H b + H f + H bb + H bf ,(1)
where,
H ξ = −T ξ <l,m> ξ † lξm +ξ † mξl ,(2)H bb = U bb 2 3 l=1n b l n b l − 1 ,(3)H bf = U bf 3 l=1n b ln f l .(4)
Hereb † (b ) are the creation (annihilation) operators for the bosons andf † (f ) the creation (annihilation) operators for the fermions; the number operators are: n ξ =ξ †ξ , ξ ∈ {b, f }. The nearest neighbour tunneling coefficient is T ξ , the intra-and inter-species interaction strengths are U bb > 0 and U bf , respectively. In general, one can write a state vector of the system as follows:
|Ψ = |Ψ f ⊗|Ψ b = |n f 1 , n f 2 , n f 3 ; f ⊗|n b 1 , n b 2 , n b 3 b , where, n f 1 + n f 2 + n f 3 = N f , and n b 1 + n b 2 + n b 3 = N b .
In the case of inter-species repulsion and N f = 1, the states corresponding to the lowest energy levels are the ones with the following on-site particle distributions:
|1 + → |1, 0, 0 f ⊗ |0, 3, 0 b , |2 + → |1, 0, 0 f ⊗ |0, 1, 2 b , |3 + → |0, 0, 1 f ⊗ |0, 2, 1 b , |4 + → |1, 0, 0 f ⊗ |1, 1, 1 b ,
where |1 + , |2 + and |3 + have six-fold degeneracy, while |4 + is a triplet. These states are schematically shown in Fig. 1 and the corresponding energies are plotted in Fig. 2
(a).
For attraction and N f = 1 (see Fig. 1 and Fig. 2 (b)), the lowest-lying energy states are as follows:
|1 − → |0, 0, 1 f ⊗ |0, 0, 3 b , |2 − → |1, 0, 0 f ⊗ |2, 0, 1 b , |3 − → |1, 0, 0 f ⊗ |1, 0, 2 b , |4 − → |0, 1, 0 f ⊗ |1, 1, 1 b ,
where the states with symmetry |1 − and |4 − are triplets and |2 − and |3 − have six-fold degeneracy. Similar ground state configurations occur at incommensurate filling, as shown in Fig. 3.
As the bosonic interaction strength changes, the ground state of the system is changing symmetry. The hierarchy of the lowest lying energy levels corresponding to different states is shown in Fig. 2 for the commensurate and in Fig. 4 for incommensurate filling of bosons. In the case of commensurate filling, for a fixed value of the inter-species interaction and growing U bb , the , and its magnitude is proportional to the interaction strength. It is reminiscent of the gap in the excitation spectrum of bosonic systems in lattices that indicates superfluid (SF) to Mott-insulator (MI) transition. Indeed, using the exact diagonalization of the Hamiltonian (1) and extracting characteristic behavior of tunneling correlations and particle number fluctuations, we can construct the phase diagram of the ground state and examine the transition to the insulating states in our small-scale Bose-Fermi system.
III. PHASE DIAGRAM
The typical characteristic quantity for different quantum phases of an ultracold quantum degenerate gas in a the lattice system is the spatial (tunneling) correlation for the bosonic component η b and the tunneling correlation of the fermionic component η f , given by
η ξ = | ξ † iξi+1 |/n ξ avg , ξ ∈ {b, f }, where n f avg = N f /3.
The boson tunneling correlation can be used to construct the phase diagram of the model, depending on the interactions. In a bosonic lattice this quantity tends to zero when the system is in the Mott-insulator regime, and approaches one when the system is in the superfluid regime. Similarly, in our small, finite, mixed-species system, this quantity can be used to delineate between an insulating and a superfluid state. The fermion tunneling correlation measures the mobility of the fermion in the system. When its value is close to one, the fermion is mobile, and when it is zero, the fermion is pinned to one of the sites.
A. Commensurate boson filling
As a start, we analyze the components of the ground state for the system with a commensurate number of bosons, N f = 1 (n f avg = 1/3), and for the fixed interspecies interaction strength |U bf | = 10. The boson tunneling correlation for N b = 3, 9 is shown in Fig. 6. Due to the small number of particles, there exists a distinct crossover region between the SF (η b = 1) and insulating (η b = 0) regimes [13], where 0 < η b < 1. When this quantity is closer to either of the two extreme values, we will refer to it as a SF-or MI-enhanced regime, respectively. It can be seen that the system exhibits a rich phase diagram with an asymmetric behavior depending on the sign of the inter-species interaction and the number of bosons.
For the case of attraction between bosons and fermions, U bf < 0, a new insulating phase corresponding to the ground state |1 − (Fig. 1, d) appears. It is strikingly different from the regular MI-like insulating state in the pure bosonic system, |4 ± (see Fig. 1), which appears at U bf = 0 and dominates the phase diagram for larger inter-species interaction strength. In this interaction-induced insulating phase the bosonic occupation of the ring sites is strongly unbalanced, in analogy with the mean-field macroscopic self-trapped states of a Bose-Fermi mixture [14]. As one increases the repulsion between bosons, a new SF enhancement region appears and then the ground state of the system is once again dominated by the regular insulating state |4 − . With the increasing inter-species interaction and the number of bosons, the SF enhancement region breaks into several filaments separated by the insulating states |2 − that appear due to interaction between bosons and fermions. This change in the structure of ground state can also be followed in Fig. 2 (b).
In the case of inter-species repulsion, U bf > 0, the system is superfluid for small |U bf |. In Fig. 2(a) one can clearly see that the ground state in this region is superposition of degenerate states |1 + and |2 + , which is a typical sign of frustration. Insulating regions appear for larger inter-species interaction and are dominated by the |2 + state [see Fig. 6 (b)]. For large U bb transition to the regular insulating state |4 + occurs.
For both signs of inter-species interaction, the behavior in the regions where U bb ∼ U bf is SF-enhanced. The effect of increasing the number of bosons is to scale up the regions of SF behaviour and introduce additional insulating regions due to inter-particle interactions. The appearance of the gap in the energy spectrum (Fig. 5) correlates exactly with the insulating regions in the phase diagram.
While the phase diagrams in Fig. 6 (a,b) are based on the behaviour of the bosonic fraction, it is useful to examine the tunneling correlation of the fermion in the system. As seen in Fig. 6(c), in the region corresponding to the regular bosonic insulating state, (i.e. for |U bb | ≫ 1) and for low inter-species coupling, the fermions are free to hop between the ring sites, which can also be deduced from the symmetry of the |4 ± state [see Fig. 1]. In the mean-field picture [14] this behavior reflects the fact that the effective interaction-induced potential seen by the fermion is weak and completely symmetric. As one increases |U bf |, the fermions localize and no tunneling is possible. In contrast to the regular insulating phase, the bosonic insulating phases that arise purely due to the inter-species interaction and correspond to symmetrybroken states (e.g., |1 − or |2 + ), naturally give rise to the regions of suppressed tunneling of the fermion.
To characterize the interaction-induced insulating phases further, we look at the boson number fluctuations in the system given by σ = n b in b i − n b i 2 /n b avg , see Fig.6(d). As expected from the MI behavior, the fluctuations in the regular insulating phase (with the |4 ± symmetry) approach zero as one increases the strength of repulsion between bosons. In contrast, for attractive inter-species interaction the interaction-induced insulating region at low values of U bb is dominated by fluctuations, which can be taken as a signature of the new insulating state.
B. Incommensurate boson filling
It has been noted (see, e.g., [13,18]) that for a fewparticle pure bosonic system at incommensurate filling a small superfluid fraction is always present, therefore no insulating state can occur. We find that this is indeed the case in our system with no fermions (or U bf = 0). The insulating phases in this case occur only in the pres- ence of fermions, with non-zero inter-species interaction strength. For the repulsive inter-species interaction the insulating phase at large U bb is suppressed, as shown in Fig. 7(a), which is typical for N b = 3m + 1 bosons with m being a positive integer. In contrary, the interactioninduced insulating region is very prominent for small N b and is broken up by regions of enhanced superfluidity for larger N b . For inter-species attraction both the regular MI-like insulating region at large |U bb | and the insulating region at lower values of U bb are retained due to the inter-species interaction. As in the case of commensurate filling, at large U bb the new insulating domains that appear due to the inter-species interaction are dominated by the ground state configuration (f ) in Fig. 3. In general, both for incommensurate and commensurate filling of bosons the boundaries of the insulating regions are pushed to larger interaction strength values as one increases the number of bosons.
In striking difference from the commensurate case, the tunneling of fermions is almost completely suppressed in the attractive inter-species interaction region, U bf < 0 [see Fig. 7(b)]. For repulsive inter-species interaction, U bf > 0, the fermion gains mobility as one increases the boson interaction strength, U bb . This is due to the fact that, for larger U bb and repulsive inter-species interaction, the insulating phase is dominated by the state (c) in Fig. 3 [see also Fig. Fig. 4 (a)]. In this case the fermion is always able to hop between the sites with lower bosonic occupation numbers. In contrast, in the case of inter-species attraction the insulating phase is dominated by the state (f ) in Fig. 3 [see also Fig. 4 (b)] and the fermions are pinned to a site with the highest occupation of bosons. For weaker inter-species interaction, U bf < U bb , the fermion is delocalized as in the commensurate case.
C. Role of the fermion filling factor
So far, we have considered one fermion interacting with N b bosons. Therefore, the results presented above are also applicable to a mixture of two bosonic species (see, e.g., [17]) with a single atom in one of the components. In the case of N f = 1, the Bose-Fermi-Hubbard model is still valid for the fermion filling factor less than 1 (the system with no fermions is equivalent to the system with three fermions), and the fermion statistics influences the ground state configuration. In particular, the system is particle-hole symmetric for 1/3 and 2/3 filling of fermions. This fact is reflected in the behavior of the phase diagram, so that the case of 1/3 fermion filling with repulsive inter-species interactions corresponds to the case of 2/3 fermion filling with attractive interspecies interaction. For example, in the case of 2/3 filling of fermions the interaction-induced insulating regions appear for repulsive rather then attractive interaction between species, as seen in Fig. 8. Other characteristic properties of the system, such as the behaviour of the fermion tunneling correlations and bosonic number fluctuations, are qualitatively the same as for N f = 1.
IV. CONCLUSIONS
In conclusion, we have analyzed the ground state of a small-scale system of quantum degenerate bosons and fermions in a three-site ring configuration. We have restricted the consideration to the fermion filling factor less or equal than one, which as allowed us to employ a standard Bose-Fermi-Hubbard Hamiltonian. By examining the tunneling correlations and particle fluctuations in the system, we have found that the system admits mobile and insulating states that are analogous to the superfluid and Mott-insulator states in infinite lattices. The novel insulating states identified in this small-scale system for both commensurate and incommensurate filling of bosons, are purely due to the inter-species interactions, and can be controlled by controlling the interaction strengths and the number of fermions injected into the system. This work is supported by the Australian Research Council (ARC). The authors acknowledge useful discussions with Dr. Chaohong Lee and Dr. Tristram Alexander.
FIG. 1 :
1(Color online) Schematics of the on-site state configurations of three bosons (blue circles) and one fermion (red circle), with the lowest energy. The figures correspond to the states: (a) |1 + , (b) |2 +, (c) |3 +, and (f) |4 + for repulsive inter-species interaction, and (d) |1 −, (e) |2 − , (c) |3 −, and (f) |4 − for inter-species attraction.
FIG. 2 :
2(Color online) The energy of the states with the largest contribution to the ground state as a function of U bb for inter-species (a) repulsion and (b) attraction with the fixed magnitude |U bf | = 10. States with different symmetries are marked with the points. The unmarked states in (a) and (b) are |3 + and |3 − , respectively. ground state structure evolves from a mixture of degenerate states (a) and (b) to the state (f ) (inFig 1)in the case of inter-species repulsion, and from (d) to (e) and (f ) in the case of inter-species attraction. Similarly, in the case of incommensurate filling, the ground state evolves from a mixture of states (a) and (b) to (c) and to the mixture of degenerate states (c) and (f ) (seeFig. 3) for inter-species repulsion, and from (d) to (e), to the mixture of states (c) and (f ) for the attractive inter-species interaction.The energy spectrum of this small scale system exhibits a gap, ∆, given by the energy difference between the ground state and first excited manifold [seeFig. 5 (a)]. For the commensurate filling of bosons this gap opens up at sufficiently strong boson repulsion for both repulsive and attractive inter-species interaction, as well as for U f b = 0 [see Fig. 5(b)]
FIG. 3 :
3(Color online) Schematics of the on-site state configurations of four bosons (blue circles) and one fermion (red circle), contributing to the lowest energy band. The panels (a), (b), (c), and (f) correspond to repulsive inter-species interaction and (d), (e), and (f) -to inter-species attraction.
online) The energy of the states with the largest contribution to the ground state as a function of U bb for inter-species (a) repulsion and (b) attraction with the fixed magnitude of inter-species interaction |U bf | = 10. Energies of states with different symmetries shown inFig. 3are marked with the corresponding letters.
FIG. 5 :
5(Color online) (a) Structure of the energy spectrum vs. U bf for commensurate filling of bosons (U bb = 1, N b = 3). (b) The dependence of the gap, ∆ = E1 − E0 on the intraand inter-species interaction strengths for N b = 3.
FIG. 6 :
6(Color online) (a,b) Density plots of the boson tunneling correlation, η b (U bf , U bb ), for (a) N b = 3 and (b) N b = 9. (c,d) Density plots of the (c) fermion tunneling correlation, η f (U bf , U bb ) and (d) boson number fluctuations, σ(U bf , U bb ) for N b = 3.
FIG. 7 :
7(Color online) Density plot of the (a) boson and (b) fermion tunneling correlation for N f = 1,N b = 4.FIG. 8: (Color online) Density plot of the (a) boson and (b) fermion tunneling correlation for N f = 2,N b = 10.
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[
"THE IRREDUCIBLE REPRESENTATIONS OF 3-DIMENSIONAL SKLYANIN ALGEBRAS",
"THE IRREDUCIBLE REPRESENTATIONS OF 3-DIMENSIONAL SKLYANIN ALGEBRAS"
] |
[
"Kevin De Laet "
] |
[] |
[] |
In this article a complete description is given of the simple representations of a 3-dimensional Sklyanin algebra associated to a torsion point. In order to determine these irreducible representations, a review is given of classical results regarding representation theory of graded rings with excellent homological and algebraic properties.
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[
"https://arxiv.org/pdf/1707.03813v1.pdf"
] | 119,633,447 |
1707.03813
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d8c0618f60450ff73956ef19747856634abc2223
|
THE IRREDUCIBLE REPRESENTATIONS OF 3-DIMENSIONAL SKLYANIN ALGEBRAS
12 Jul 2017
Kevin De Laet
THE IRREDUCIBLE REPRESENTATIONS OF 3-DIMENSIONAL SKLYANIN ALGEBRAS
12 Jul 2017
In this article a complete description is given of the simple representations of a 3-dimensional Sklyanin algebra associated to a torsion point. In order to determine these irreducible representations, a review is given of classical results regarding representation theory of graded rings with excellent homological and algebraic properties.
Introduction
The 3-dimensional Sklyanin algebras form the most important class of Artin-Schelter regular algebras of global dimension 3. These algebras are parametrized by the following geometric data: an elliptic curve E and a point τ ∈ E. In addition, they form a class of graded algebras with excellent ringtheoretical and homological properties, which was shown in the original papers by Artin, Tate and Van den Bergh, [5] and [3].
Recently, the representation theory of these algebras have gained attention, see for example [12], [8], [21] and [20]. In [18] it was shown that a Sklyanin algebra is a finite module over its center if and only if τ is a torsion point of E, in which case the Sklyanin algebra is a maximal order over its center. Consequently, in the case of a torsion point, it is useful to describe the simple representations.
The point of this article is to explain the connection between the study of fat point modules, PGL n × C * -stabilizers of simple representations and C * -families of (non-trivial) simple representations of a graded algebra A with good properties. This connection is known by the experts, but it is not well-known by the generic mathematician working in noncommutative algebraic geometry. As a consequence, we verify [21, Conjecture 1.5] using these methods. 1.2. Conventions. In this article, the following notations and conventions are used.
• We will work over C.
• The element ω ∈ C * will be a primitive third root of unity.
• The element ρ ∈ C * will be a primitive nth root of unity for a specified n ∈ N.
• For a graded algebra A, Proj(A) = qgr(A) is the category of finitely generated, graded A-modules, modulo torsion modules. Recall that there is an automorphism on Proj(A) by the shift functor
: if M = ⊕ i∈N M i ∈ Proj(A), then M [1]
is the graded module with degree k-part isomorphic to M k+1 . • For A any algebra, the set irrepA is the set of simple representations of A up to equivalence. With irrep k A for k ∈ N the subset of k-dimensional simple representations up to equivalence are denoted. • If A is a sheaf of coherent algebras over X with X a projective scheme, then irrepA denotes the sheaf of sets of simple representations up to equivalence over X, that is,
(irrepA)(U ) = irrepA| U .
With irrep k A we denote the subsheaf of sets of simple k-dimensional representations up to equivalence over X.
Noncommutative projective geometry
2.1. Cayley-Hamilton algebras. In this section a review of the results of [10], [9] and [11] is discussed. We will assume that
• A = C ⊕ A 1 ⊕ A 2 ⊕ . .
. is a graded algebra, finitely generated in degree one with quadratic relations,
• A is a domain, • A has global dimension d and H A (t) = (1 − t) −d for some d ∈ N,
and • A is a finite module over its center R = Z(A), which is a normal domain. The algebras of current interest, the Sklyanin algebras at torsion points, have all these properties by [19,Corollary 1.3].
Let n ∈ N be equal to n = max{m : ∃φ : A / / / / M m (C) algebra morphism}. Under these conditions, A is a Cayley-Hamilton of degree n, that is, there exists an R-linear map tr : A / / R such that • tr(1) = n, • tr(ab) = tr(ba) for each a, b, ∈ A, and • χ n,a (a) = 0 for each a ∈ A. The element χ n,a (a) is the nth Cayley-Hamilton polynomial of degree n, expressed in the sums of powers of a, as explained in [14,Section 2.3].
A grading on an algebra A is equivalent to an action of the torus group C * , by the rule a ∈ A n ⇔ t · a = t k a for each k ∈ N, t ∈ C * , a ∈ A. As such, tr is also a degree-preserving map, that is, tr(A n ) ⊂ A n . Example 1. Let n ∈ N be an integer strictly larger than 1 and take the algebra
A ρ = C ρ [x, y] = C x, y /(xy − ρyx),tr(x k y l ) = nx k y l , if (k, l) ∈ (nN) 2 , 0, otherwise
The couple (A ρ , tr) determines a Cayley-Hamilton algebra of degree n.
To A is associated the affine variety of trace-preserving n-dimensional representations, that is,
X A = trep n A := {φ : A / / M n (C) algebra morphism : tr •φ = φ • tr},
with the trace on M n (C) being the usual trace map. The projective special linear group of degree n, PGL n (C) acts on this variety by conjugating matrices, such that PGL n (C)-orbits correspond to isomorphism classes of n-dimensional representations. By a result of Artin [1, Section 12.6], one has
X A /PGL n (C) ∼ = Spec(R).
In addition, as A is graded, C * -acts on X A . The only closed orbit, as in the commutative case, is the trivial representation (A/A + ) ⊕n , which is also the only orbit with a non-trivial C * -stabilizer. From the fact that the actions of C * and PGL n (C) on X A commute, it follows that X A is a PGL n (C) × C * -variety. Let M be a simple n-dimensional A-representation, then its PGL n (C)-stabilizer is trivial by Schur's lemma. As mentioned before, the C * -stabilizer of M is also trivial, but the PGL n (C) × C * -stabilizer doesn't have to be trivial. From [6, lemma 4, theorem 2], it follows that this stabilizer is always finite and is conjugated to the group generated by (g ζ , ζ) ∈ PGL n (C) × C * , with ζ = ρ n e for some e ∈ N and
g ζ = diag(1, . . . , 1 m0 , ζ, . . . , ζ m1 , . . . , ζ e−1 , . . . , ζ e−1 me−1 )
If M is a k-dimensional simple A-representation with k < n, one can look at rep k A and calculate the PGL k (C) × C * -stabilizer.
. . , a k ) ∈ N k with 0 ≤ a 1 ≤ a 2 ≤ . . . ≤ a k < e such that A m g /m g ∼ = M k (C[t, t −1 ])(a 1 , a 2 , . . . , a k )
with deg(t) = e. If deg(t) = 1 then we will suppress the integers a 1 , . . . , a k . This isomorphism is a graded isomorphism. Recall that for a Z-graded ring S and a k-tuple (a 1 , . . . , a k ) ∈ Z k the matrix ring M k (S)(a 1 , a 2 , . . . , a k ) has degree m-part with m j > 0 and e−1 j=0 m j = k, which we will assume throughout this paper. By assumption, this means that if x ∈ A 1 , then
for m ∈ Z M k (S)(a 1 , a 2 , . . . , a k ) m = S m S m−a1+a2 . . . S m−a1+a k S m−a2+a1 S m . . . S m−a2+a k . . . . . . . . . . . . S m−a k +a1 S m−a k +a2 . . . S m .
By construction, there is a commutative diagram
A m φ / / / / M k (C) A m g ψ / / / / i O O M k (C[t, t −1 ])(a 1 , a 2 , . . . , a k ) t →1 O O By [10, Theorem 1], M k (C[t, t −1 ])(a 1 ,ψ(x) = 0 0 . . . M (x) 0 t M (x) 1 0 . . . 0 . . . . . . . . . . . . 0 . . . M (x) e−1 0 with M (x) l ∈ M m l ×m l−1 (C)
, indices taken in Z e . But then, by construction, φ has as PGL k (C) × C * -stabilizer the cyclic group (g ζ , ζ) .
Example 2. Continuing example 1 with ρ = −1, then the center of A = A −1 is C[x 2 , y 2 ].
Over m = (x 2 − 1, y 2 − 1) lies a unique two-dimensional simple representation. Then m g = (x 2 − y 2 ). As Z(A m g ) ⊂ ⊕ k∈2Z (A m g ) k and A m g is generated over its center by degree one elements, it follows that
A m g /m g ∼ = M 2 (C[t, t −1 ])(0, 1), deg(t) = 2.
By construction, a C * -family of simple two-dimensional representations of A is determined by
0 t 1 0 , 0 −t 1 0 ∈ M 2 (C[t, t −1 ])(0, 1).
For each t = 0 and associated algebra morphism φ t obtained by specialising these two matrices, the PGL 2 (C) × C * -stabilizer is determined by
1 0 0 −1 , −1 .
Periodic fat point modules. Associated to
A m g ψ / / / / M k (C[t, t −1 ])(0, . . . , 0 m0 , 1, . . . , 1 m1 , . . . , e − 1, . . . , e − 1 me−1 )
are e e-periodic fat point modules ([10, Section 6]). Recall that a fat point module is a simple object of Proj(A). For j ∈ N, let V j = C m l if j ≡ l mod Z e . Let M (ψ) = ⊕ j∈N V j and define an action of A by extending
x ∈ A 1 , v ∈ M (ψ) j = V j : xv := M (x) j+1 v as an element of M (ψ) j+1 = V j+1 . As a C[t]-module, this is just ⊕ e−1 i=0 (C[t])[i] ⊕mi .
Of course, this makes sense as t acts faithfully on this module and so
END(⊕ e−1 i=0 (C[t, t −1 ][i]) ⊕mi ) = M k (C[t, t −1 ])(0, . . . , 0 m0 , 1, . . . , 1 m1 , . . . , e − 1, . . . , e − 1 me−1 )
.
By surjectivity of ψ, M (ψ) is a fat point module of A. However, if e > 1, then M (ψ)[k] ∼ = M (ψ) for 1 ≤ k ≤ e−1, but M (ψ)[e] ∼ = M (ψ) in Proj(A). By definition, M (ψ)
is an e-periodic fat point module of multiplicity (m 0 , m 1 , . . . , m e−1 ). The other way round, suppose that M is an e-periodic fat point module of multiplicity (m 0 , m 1 , . . . , m e−1 ). Then to M is associated a e−1 i=0 m i = k-dimensional representation S by the following rule:
x ∈ A 1 , v ∈ M j : if xv = M (x) j+1 v as an element of M j+1 = V j+1
then one associates the algebra map
ψ M (x) = 0 0 . . . M (x) 0 M (x) 1 0 . . . 0 . . . . . . . . . . . . 0 . . . M (x) e−1 0 .
This algebra map corresponds to a simple representation of A, for if this was not simple, then this would imply that M is not a fat point module.
In short, we have Proof. Let M be an e-periodic fat point. As M has a finite projective resolution, it follows that
f (t) (1 − t) l = p(t) 1 − t e with (1 − t, f (t)) = 1, deg(p(t)) = e − 1.
But then it follows that (1 − t) l divides 1 − t e , which can only happen if l = 1. In this case,
f (t)(1 + t + . . . + t e−1 ) = p(t),
which can only happen if f (t) = m is constant. But then the Hilbert series of M is
m(1 − t) −1 .
The integer m of the proposition is called the multiplicity of M . Example 3. Continuing example 2, one sees that the algebra morphism
A/m g / / / / M 2 (C[t, t −1 ])(0, 1) , (x, y) → 0 t 1 0 , 0 −t 1 0
determines two point modules, the point module P with associated infinite quiver
1 1 1 1 . . .
Sheaf of algebras A.
In this subsection we follow [9]. Let g be a central element of A of degree strictly larger than 0. Localizing A at g one gets the graded ring Λ = A[g −1 ] with center Σ = R[g −1 ] and taking the degree 0 part Λ 0 , one gets an algebra over Σ 0 which is finitely generated as a Σ 0 -module. Gluing these algebras, one gets a sheaf of orders A, finite over R = Proj(R). We have Λ 0 ⊂ Q gr (A) 0 , which is a division algebra, say of PI-degree s. However, it is not always so that Z(Λ 0 ) = Σ 0 . for some maximal graded ideal m g of B, which we know from the previous discussions corresponds to a sum of e distinct but shift-equivalent fat point modules of multiplicity n e . As a fat point corresponds to a simple representation of B 0 , we find that the PI-degree of A is s = n e . As there are e fat points lying over the graded prime ideal m g ∩ R, it follows that the map
Z Φ / / / / R is generically e-to-one.
The shift functor on Proj(A) induces an automorphism on the fat points of a fixed multiplicity. As we now know that fat points correspond to simple representations of A, this shift functor also induces an automorphism α on Z. As Z = A/PGL n (C) and Spec(R) = A/PGL n (C), it follows that Φ is the quotient map by the group G = α . As Proj(C) = Proj(C (k) ) for any affine, commutative ring C,
Proj(C[x, y 3 , z 3 , yz]) ∼ = Proj(C[x, y 3 , z 3 , yz] (3) ) = Proj(C[x 3 , y 3 , z 3 , xyz]) ∼ = Proj(C[u, v, w, d]/(uvw − d 3 )).
Consequenty, the map
P 2 Φ / / / / P 2 /Z 3 is indeed just the quotient map by the automorphism α = diag(1, ω, ω 2 ).
Armed with this knowledge, we can now tackle the problem of describing all the simple representations of 3-dimensional Sklyanin algebras.
3-dimensional Sklyanin algebras
In this section, set p = 3. Let E be an elliptic curve embedded in P 2 and a torsion point τ ∈ E of order n > 1. The associated 3-dimensional Sklyanin algebra A = A(E, τ ) is a finite module over its center by [19,Theorem 1.2]. In this section we will give a complete description of the simple representations of A.
The first thing we have to notice is: if g is the unique central element of degree 3, then all fat points of A/(g) are point modules, which are parametrized by E. This follows from the fact that A/(g) = O τ (E) is the twisted coordinate of E associated to τ [5, Theorem 6.8]. In addition, the shift functor [1] on E ⊂ Proj(O τ (E)) is addition with τ , so that each point module is n-periodic. From this we deduce using Theorem 3:
Proposition 3. Each non-trivial simple representation of O τ (E) is n-dimensional with PGL n (C) × C * -stabilizer isomorphic to Z n .
This fact is independent of the conditions (n, 3) = 1 or (n, 3) = 3.
R = C[u, v, w, g]/(Ψ(u, v, w) − g n ),
with deg(u) = deg(v) = deg(w) = n, deg(g) = 3 and Ψ(u, v, w) a homogeneous polynomial of degree 3n which embeds the elliptic curve E ′ = E/ τ in P 2 [u:v:w] . Consequently, the central proj is P 2 ∼ = Proj(R).
For the sake of completeness, a review of the discussion in [8] will be given. By [2,Theorem 3.4], all the fat points that are not annihilated by g are of multiplicity n, which also follows from [4, Theorem 7.3] and the above discussion regarding simple representations of the sheaf A of dimension n and multiplicity n fat point modules. • If g ∈ m then S is a n-dimensional representation of A with trivial PGL n (C)× C * -stabilizer. The C * -orbit of S corresponds to a unique 1-periodic fat point module of multiplicity n, which in turn corresponds to a unique ndimensional representation of A. Consequently, one has
A m g /m g ∼ = M n (C[t, t −1 ]) with deg(t) = 1.
• If g ∈ m but dim(S) = 1 then S is n-dimensional with PGL n (C) × C *stabilizer isomorphic to Z n . The C * -orbit of S corresponds to n shiftequivalent point modules, who in turn correspond to n 1-dimensional representations of A. Consequently, on has
A m g /m g ∼ = M n (C[t, t −1 ])(0, 1, . . . , n − 1) with deg(t) = n.
• If dim(S) = 1 then S is the trivial representation A/A + .
Consequently, in the affine case, we have
irrep n A 1:1 / / / / irrepR \ V(u, v, w, g) , irrep 1 A 1:1 / / / / irrepV(u, v, w, g)
while in the projective case
irrep n A 1:1 / / / / Proj(R) \ E ′ = P 2 \ E ′ , irrep 1 A = E n:1 / / / / E ′ .
3.2. (n, 3) = 3. If (n, 3) = 3, then there will be a difference between R = Proj(R) and Z. Let E ′ = E/ 3τ and E ′′ = E ′ / τ . We will assume that n = 3.
Theorem 6. [18,Theorem 4.8] The center of A is isomorphic to
R = C[u, v, w, g]/(Ψ(u, v, w) + 3z 2 g n 3 + 3zg 2n 3 + g n ) with deg(u) = deg(v) = deg(w) = n, deg(g) = 3. The polynomial Ψ(u, v, w) of degree 3n corresponds to E ′′ embedded in P 2
[u:v:w] and z is a linear polynomial in u,v,w that vanishes on three inflection points of E ′′ .
By Proposition 2, the map Z Φ / / / / R will be a quotient map by an automorphism α of order 3, which is the greatest common divisor of the degrees of the homogeneous elements of R. By [18,Theorem 4.7], Z ∼ = P 2 and E ′ ⊂ Z. Let s = n 3 , then again by [2,Theorem 3.4], all the fat points not annihilated by g are of multiplicity s.
As E ′ is the ramification locus of A (that is, the closed subset of Z where A is not Azumaya), it follows that α has to induce an automorphism of order three on E ′ without fix points. This can only be if α is conjugated to diag(1, ω, ω 2 ) ⊂ PGL 3 (C). But then, α has three fixed points, which correspond to three fat points of multiplicity s which are 1-periodic. They in turn correspond to three C * -families of simple representations of dimension s, with trivial PGL s (C) × C * -stabilizer.
Theorem 7. The simple representations of A come in four types. Let S be a simple A-module with annihilator m = Ann(S).
• If g ∈ m and S is a quotient of a fat point F that is not fixed by the shift functor, then S is n-dimensional with PGL n (C) × C * -stabilizer isomorphic to Z 3 . The fat point F is 3-periodic and has multiplicity s. F and its shifts F • If g ∈ m and S is a quotient of a fat point F that is fixed by the shift functor, then S is s-dimensional with trivial PGL s (C)×C * -stabilizer. The multiplicity of F is s and F corresponds to a unique s-dimensional representation of A. Consequently, one has
A m g /m g ∼ = M s (C[t, t −1 ]) with deg(t) = 1.
• If g ∈ m and S is not one-dimensional, then S is n-dimensional with PGL n (C)×C * -stabilizer isomorphic to Z n . The C * -orbit of S corresponds to n shift-equivalent point modules, who in turn correspond to n 1-dimensional representations of A. Consequently, on has A m g /m g ∼ = M n (C[t, t −1 ])(0, 1, . . . , n − 1) with deg(t) = n.
• If dim(S) = 1 then S is the trivial representation A/A + .
Regarding the center R of A, we can now give a different description. In the affine case, we now find
Conjecture 1 .
1[21, Conjecture 1.5] The simple representations of a 3-dimensional Sklyanin algebra with τ a torsion point of order n with (n, 3) = 3 are of dimension n, n 3 or 1. The simple representations of dimension n 3 form a 3 : 1-cover over three lines intersecting in a unique point. 1.1. Statement. The author hereby acknowledges that his own contribution is mainly combining previous results by Artin, Tate, Van den Bergh, Le Bruyn, Smith and others.
classically graded by deg(x) = deg(y) = 1. Then Z(A ρ ) = C[x n , y n ]. The natural trace map on A ρ is the C-linear extension of:
2. 2 .
2Graded matrix rings. Let M be a simple A-module of dimension k with k ≤ n and let m = Ann A (M ). Then we have a map A m φ / / M k (C) Let m g = max{I ⊂ m : I homogeneous ideal}. Then m g is a maximal homogeneous ideal. By a graded version of Artin-Wedderburn [13, theorem I.5.8], there is an integer e ∈ N \ {0} and a k-tuple (a 1 , .
− 1 me− 1 )
11a 2 , . . . , a k ) is generated in degree one over its center if and only if (a 1 , a 2 , . . . , a k ) = (0, . . . , 0 m0 , 1, . . . , 1 m1 , . . . , e − 1, . . . , e
Theorem 2 .•
2[10, Proposition 3] There is a one-to-one correspondence between• simple k-dimensional representations with PGL k (C)×C * -stabilizer generated by (g ζ , ζ) with g ζ = diag(1, . . . , 1 m0 , ζ, . . . , ζ m1 , . . . , ζ e−1 , . . . , ζ shift-equivalence classes of fat point modules of period e and multiplicity (m 0 , m 1 , . . . , m e−1 ). If m 0 = m 1 = . . . = m e−1 = 1, then M is called a point module. In fact, one has Proposition 1. If A has finite global dimension and has Hilbert series (1 − t) −d , then the Hilbert series of a fat point M is 1-periodic, that is, H M (t) = m(1 − t) −1 for some m ∈ N.
Example 4 .
4For any quantum algebra A ρ from example 1, localizing at for example x, one finds that Λ 0 = C[ y x ] and Σ 0 = C[ y n x n ]. Consider now a fat point module M which is not annihilated by g. Then g acts faithfully on M and M [g −1 ] is a graded A[g −1 ]-module. The module M [g −1 ] becomes a gr-simple module (that is, it has no non-trivial graded submodules). By the equivalence of categories of Dade's Theorem [13, Theorem 3.1.1], N = M [g −1 ] 0 is a simple Λ 0 -module. Let Z = Spec(Z(A)) be the central proj of A. By a result of Artin [1, Section 12.6], Z parametrizes isomorphism classes of semi-simple trace-preserving sdimensional representations of A. There is a surjective map Z Φ / / / / R . This map will be finite and therefore generically e-to-one for some e ∈ N. Proposition 2. The integer e is equal to the least common multiple of the degrees of homogeneous elements of R. Proof. Localize A at a finite number of generators of R, say B = A[(x 1 · · · x t ) −1 ] with R = C[x 1 , . . . , x t ]/(I) for I some set of relations. Then any simple representation of dimension n of B comes from an algebra morphism B m g / / / / M n (C[t, t −1 ])(0, . . . , 0 m , 1, . . . , 1 m , . . . , e − 1, . . . , e − 1 m ) ,
Theorem 3 .A
3The following sets are in one-to-one correspondence:• e-periodic multiplicity m-fat points of A, • e m-dimensional simple representations of A such that the corresponding points in Z form an orbit under α, • graded maximal ideals m g of A such thatA/m g ∼ = M em (C[t, t −1 ])(0, . . . , 0 m , 1, . . . , 1 m , . . . , e − 1, . . . , e − 1 m ), deg(t) = e,• simple me-dimensional representations with PGL me (C) × C * -stabilizer generated by a subgroup conjugated to (g ζ , ζ) withg ζ = diag(1, . . . , = C x, y, z /(yz − ωzy, zx − ωxz, xy − ωyx) be an Artin-Schelter regular algebra of global dimension 3. Then the center of A is isomorphic to R ∼ = C[u, v, w, d]/(uvw − d 3 ). However, the central proj of A is isomorphic to P 2 , and in fact Proj(A) ∼ = coh(P 2 ). In fact, Proj(R) = P 2 /Z 3 , with Z 3 acting on P 2 = P(CZ 3 ). The last fact can be proved using invariant theory: if P 2 = Proj(C[x, y, z]) with Z 3 -weights 0, 1, 2, then C[x, y, z] Z3 = C[x, y 3 , z 3 , yz].
The first thing one has to determine is the center of A.
Theorem 4 .
4[18, Theorem 4.8] The center of A is isomorphic to
Theorem 5 .
5[8, Lemma 1, Theorem 4] The simple representations of A come in three types, let S be a simple A-module with annihilator m = Ann(S).
[ 1 ]
1and F [2] correspond to three different s-dimensional representations of A. Consequently, one has A m g /m g ∼ = M n (C[t, t −1 ])(0, . . . , 0 s , 1, . . . , 1 s , 2, . . . , 2 s ) with deg(t) = 3.
Proposition 4 .
4The center R is isomorphic to C[u, v, w, d, g]/(uvw − d 3 , g s − l) with l a linear form in u, v, w and d such that R/(g) is the homogeneous coordinate ring of E ′′ , generated in degree n.Proof. By [18, Theorem 3.7, Proposition 4.2], there are three normal elements u 0 , v 0 , w 0 of degree s such that Proj(C[u 0 , v 0 , w 0 ]) = P 2 = Z. By the fact that the shift functor corresponds to diag(1, ω, ω 2 ) for an open subset of fat points, it follows that each irreducible representation of dimension n restricted to C[u 0 , v 0 , w 0 ] is determined by the matrices for (λ, µ, η) ∈ A 3 . open subset of A 3 . But then, u = u 3 0 , v = v 3 0 , w = w 3 0 and d = u 0 v 0 w 0 are sent to scalar matrices for an open subset of Spec(R), so they are central in A. In addition, they are linearly independent, so they generate R (n) by Theorem 6. But then g s = l for some linear form in u, v, w and d. The fact that V(uvw − d 3 , g s − l) ∩ V(g) = E ′′ follows from the fact that the center of A/(g) is O(E/ τ ) = O(E ′′ ).
///
/ / / irrepR \ V(uv, vw, uw) , / / / V(uv, uw, uw) \ V(u, v, w, g) , / / / irrepZ \ E ′ , irrep 1 A = E s:1 / / / / E ′ , (irrepZ = P 2 ) \ V(u 0 v 0 , v 0 w 0 , u 0 w 0 ) 3:1 / / / / R \ V(uv, vw, wu, d) , V(u 0 v 0 , v 0 w 0 , u 0 w 0 ) 1:1 / / / / V(uv, vw, wu, d) .
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Some algebras associated to automorphisms of elliptic curves. M Artin, J Tate, M Van Den, Bergh, The Grothendieck Festschrift. SpringerM. Artin, J. Tate, and M. Van den Bergh. Some algebras associated to au- tomorphisms of elliptic curves. In The Grothendieck Festschrift, pages 33-85. Springer, 2007.
The local structure of graded representations. R Bocklandt, S Symens, Communications in Algebra. 3412R. Bocklandt and S. Symens. The local structure of graded representations. Communications in Algebra, 34(12):4401-4426, 2006.
A Cazzaniga, A Morrison, B Pym, B Szendroi, arXiv:1510.08116Motivic Donaldson-Thomas invariants of some quantized threefolds. arXiv preprintA. Cazzaniga, A. Morrison, B. Pym, and B. Szendroi. Motivic Donaldson-Thomas invariants of some quantized threefolds. arXiv preprint arXiv:1510.08116, 2015.
The Geometry of Representations of 3-dimensional Sklyanin algebras. Algebras and Representation Theory. K , De Laet, L. Le Bruyn, 18K. De Laet and L. Le Bruyn. The Geometry of Representations of 3-dimensional Sklyanin algebras. Algebras and Representation Theory, 18(3):761-776, 2015.
Central singularities of quantum spaces. L , Le Bruyn, Journal of Algebra. 1771L. Le Bruyn. Central singularities of quantum spaces. Journal of Algebra, 177(1):142-153, 1995.
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Noncommutative geometry and Cayley-smooth orders. L , Le Bruyn, CRC PressL. Le Bruyn. Noncommutative geometry and Cayley-smooth orders. CRC Press, 2007.
L , Le Bruyn, arXiv:1705.03243The superpotential xyz+xzy+c/3(xxx+yyy+zzz). arXiv preprintL. Le Bruyn. The superpotential xyz+xzy+c/3(xxx+yyy+zzz). arXiv preprint arXiv:1705.03243, 2017.
Graded ring theory. C Nastasescu, F Van Oystaeyen, ElsevierC. Nastasescu and F. Van Oystaeyen. Graded ring theory. Elsevier, 2011.
A formal inverse to the Cayley-Hamilton theorem. C Procesi, Journal of algebra. 1071C. Procesi. A formal inverse to the Cayley-Hamilton theorem. Journal of algebra, 107(1):63-74, 1987.
Explicit representations of 3-dimensional Sklyanin algebras associated to a point of order 2. D J Reich, C Walton, arXiv:1512.09167arXiv preprintD. J. Reich and C. Walton. Explicit representations of 3-dimensional Sklyanin algebras associated to a point of order 2. arXiv preprint arXiv:1512.09167, 2015.
Point modules over Sklyanin algebras. S P Smith, Mathematische Zeitschrift. 2151S. P. Smith. Point modules over Sklyanin algebras. Mathematische Zeitschrift, 215(1):169-177, 1994.
Degenerate 3-dimensional Sklyanin algebras are monomial algebras. S P Smith, Journal of Algebra. 358S. P. Smith. Degenerate 3-dimensional Sklyanin algebras are monomial alge- bras. Journal of Algebra, 358:74-86, 2012.
The center of the 3-dimensional and 4-dimensional Sklyanin algebras. K-theory. S P Smith, J Tate, 8S. P. Smith and J. Tate. The center of the 3-dimensional and 4-dimensional Sklyanin algebras. K-theory, 8(1):19-63, 1994.
J Tate, M Van Den, Bergh, Homological properties of Sklyanin algebras. Inventiones mathematicae. 124J. Tate and M. Van den Bergh. Homological properties of Sklyanin algebras. Inventiones mathematicae, 124(1):619-648, 1996.
Representation theory of three-dimensional Sklyanin algebras. C Walton, Nuclear Physics B. 8601C. Walton. Representation theory of three-dimensional Sklyanin algebras. Nu- clear Physics B, 860(1):167-185, 2012.
C Walton, X Wang, M Yakimov, arXiv:1704.04975The Poisson geometry of the 3-dimensional Sklyanin algebras. Middelheimlaan 1, B-2020 Antwerp (BelgiumDepartment of Mathematics, University of AntwerparXiv [email protected]. Walton, X. Wang, and M. Yakimov. The Poisson geometry of the 3- dimensional Sklyanin algebras. arXiv preprint arXiv:1704.04975, 2017. Department of Mathematics, University of Antwerp, Middelheimlaan 1, B-2020 Antwerp (Belgium), [email protected]
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"Magnetohydrodynamic Stability at a Separatrix: Part I",
"Magnetohydrodynamic Stability at a Separatrix: Part I"
] |
[
"\nCulham Science Centre\nOX14 3DB. *AbingdonOxfordshire\n"
] |
[
"Culham Science Centre\nOX14 3DB. *AbingdonOxfordshire"
] |
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The rapid deposition of energy by Edge Localised Modes (ELMs) onto plasma facing components, is a potentially serious issue for large Tokamaks such as ITER and DEMO. The trigger for ELMs is believed to be the ideal Magnetohydrodynamic Peeling-Ballooning instability, but recent numerical calculations have suggested that a plasma equilibrium with an X-point -as is found in all ITER-like Tokamaks, is stable to the Peeling mode. This contrasts with analytical calculations (G. Laval, R.Pellat, J. S. Soule, Phys Fluids, 17, 835, (1974)), that found the Peeling mode to be unstable in cylindrical plasmas with arbitrary cross-sectional shape. However the analytical calculation only applies to a Tokamak plasma in a cylindrical approximation. Here, we re-examine the assumptions made in cylindrical geometry calculations, and generalise the calculation to an arbitrary Tokamak geometry at marginal stability. The resulting equations solely describe the Peeling mode, and are not complicated by coupling to the ballooning mode, for example. We find that stability is determined by the value of a single parameter ∆ ′ that is the poloidal average of the normalised jump in the radial derivative of the perturbed magnetic field's normal component. We also find that near a separatrix it is possible for the energy principle's δW to be negative (that is usually taken to indicate that the mode is unstable, as in the cylindrical theory), but the growth rate to be arbitrarily small.
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"https://arxiv.org/pdf/0902.3585v1.pdf"
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0902.3585
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b039ebb14050b14d586fa7d260c928c33f7acd31
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Magnetohydrodynamic Stability at a Separatrix: Part I
20 Feb 2009 (Dated: February 20, 2009)
Culham Science Centre
OX14 3DB. *AbingdonOxfordshire
Magnetohydrodynamic Stability at a Separatrix: Part I
20 Feb 2009 (Dated: February 20, 2009)A.J. Webster & C.G. Gimblett Euratom/UKAEA Fusion Association,numbers: 5255Tn5230Cv5255Fa5235Py * Electronic address: anthonywebster@ukaeaorguk
The rapid deposition of energy by Edge Localised Modes (ELMs) onto plasma facing components, is a potentially serious issue for large Tokamaks such as ITER and DEMO. The trigger for ELMs is believed to be the ideal Magnetohydrodynamic Peeling-Ballooning instability, but recent numerical calculations have suggested that a plasma equilibrium with an X-point -as is found in all ITER-like Tokamaks, is stable to the Peeling mode. This contrasts with analytical calculations (G. Laval, R.Pellat, J. S. Soule, Phys Fluids, 17, 835, (1974)), that found the Peeling mode to be unstable in cylindrical plasmas with arbitrary cross-sectional shape. However the analytical calculation only applies to a Tokamak plasma in a cylindrical approximation. Here, we re-examine the assumptions made in cylindrical geometry calculations, and generalise the calculation to an arbitrary Tokamak geometry at marginal stability. The resulting equations solely describe the Peeling mode, and are not complicated by coupling to the ballooning mode, for example. We find that stability is determined by the value of a single parameter ∆ ′ that is the poloidal average of the normalised jump in the radial derivative of the perturbed magnetic field's normal component. We also find that near a separatrix it is possible for the energy principle's δW to be negative (that is usually taken to indicate that the mode is unstable, as in the cylindrical theory), but the growth rate to be arbitrarily small.
I. INTRODUCTION
Thermonuclear fusion requires plasmas with a pressure of at least an atmosphere, and temperatures in excess of 100 million degrees Kelvin. These conditions can be achieved in Tokamaks such as JET [1], but the plasmas are subject to a number of instabilities, the consequences of which range from benign to structurally damaging. By understanding the instabilities that can occur, they can be avoided or mitigated. A class of instabilities that are only partly understood are Edge Localised Modes (ELMs) [2]. ELMs can lead to a rapid deposition of energy onto plasma facing components, and this is a potentially serious issue for proposed large tokamak devices such as ITER [3].
Our present understanding of ELMs is based on the linear ideal Magnetohydrodynamic
Peeling-Ballooning instability (Wilson et al [4], Gimblett et al [5]), which is thought to trigger ELMs, that subsequently evolve non-linearly. The studies upon which this understanding were based considered Tokamak equilibria with a smoothly shaped magnetic flux-surface at the plasma-vacuum boundary. In contrast, modern Tokamak plasmas have a cross-section in which the outermost flux surface is redirected onto divertor plates, forming a separatrix with a sharp "X-point" where the magnetic topology changes from closed (confined plasma), to open field lines along which plasma can flow to the divertor plates [1].
The first numerical evidence for a stabilising effect from the separatrix was found by Medvedev et al [6]. More recently, numerical studies of the Peeling-Ballooning instability in these X-point plasmas (Huysmans[7]), have found that as the plasma's outermost flux surface is made increasingly close to that of a separatrix with an X-point, the Peeling mode becomes stabilised. Crucially the stabilisation appeared to happen before the plasma formed a separatrix with an X-point, shaping alone appeared to be sufficient to stabilise the mode. This appears to be contrary to theoretical work by Laval et al [8], which has indicated that the peeling mode is unstable in cylindrical plasmas with an arbitrarily shaped cross-section.
In addition the ELITE code (Wilson et al [9]) has recently been used to examine Peeling mode stability as the outermost flux surface approaches the separatrix. It was found that although the growth rate reduced in size as the boundary more closely approximated a separatrix, it did so increasingly slowly, and its asymptotic behaviour was uncertain (Saarelma [10]). To help understand and reconcile these results, here in the first part of this two part paper we re-examine the assumptions made in the derivation of the peeling mode stability criterion for a cylinder, and generalise the calculation so that it applies to a toroidal Tokamak plasma.
As with the original studies of the Peeling mode in a cylinder, for simplicity we will firstly consider marginal stability, and generalise the condition for Peeling mode stability in a straight cylinder to a condition for Peeling mode stability in an arbitrary cross-section Tokamak plasma. The resulting equations only describe Peeling mode stability, and are not complicated by coupling to the Ballooning mode instability, for example. This paper generalises previous analytic calculations in a number of ways. Firstly it applies to axisymmetric toroidal geometries, as opposed to the cylindrical geometry in which the Peeling mode has been extensively studied (for example see Laval et al [8], Lortz [15], Connor et al [14]). It allows for equilibrium poloidal currents at the plasma edge, in addition to the toroidal current that is solely included in previous analytic studies. The skin currents that are induced by a plasma perturbation are related to the difference between the magnetic field in the plasma and the vacuum, and it is found that at marginal stability a plasma perturbation induces a skin current that is parallel and proportional to the equilibrium edge current and proportional to the amplitude of the radial plasma displacement. The complicated-looking plasma-vacuum boundary condition that is usually found in association with the energy principle may be expressed as a simple relationship between the normal components of the plasma and vacuum magnetic fields. This is used to relate the generalised equations for Peeling mode stability at marginal stability to the energy principle, and this allows us to define the Peeling mode in terms of the energy principle's δW . With this energy principle for the Peeling mode we can consider the trial function used by Laval et al [8], finding that a single parameter ∆ ′ determines the sign and magnitude of δW . Finally the instability's growth rate is considered.
II. BACKGROUND
Laval et al [8] considered a large aspect ratio ordering that neglects toroidal effects, and also neglects any equilibrium poloidal current at the plasma edge. The work suggests that the Peeling mode will be unstable for a non-zero edge current, regardless of the plasma cross section. Later we will reconsider Peeling mode stability for arbitrary cross section Tokamak plasmas, but firstly we consider some properties of the trial function considered by Laval et al [8]. [8], that for the most unstable modes with m ∼ nq has ξ ∼ e inqθ , with θ the usual straight field line angle [11]. In the second part to this paper we will show that the length l along a flux surface in the poloidal cross section may be parameterised by α, with α = −π..π, as can θ. It is also shown that a physically reasonable model
for θ has θ(α) ≃ 1 q α −π dα " α 2 2 +ǫ 2 " and l(α)/l(π) ≃ α −π α 2 2 + ǫ 2 1/4 dα/ α 2 2 + ǫ 2 1/4 dα.
Therefore by parameterising both θ(α) and l(α), then the figure plots θ (vertical axis) versus l/l(π) (horizontal axis), for ǫ = 0.001 and n = 20.
Note that for an element of length dl along a flux surface in the poloidal plane, dl Bp = JχBp Bp dχ = νR 2 I dχ with J χ the Jacobian, B p the poloidal field, ν = IJχ R 2 is the local field-line pitch, and χ, φ, ψ an orthogonal toroidal co-ordinate system (for example, see Freidberg [12] for details). Thus the poloidal angle used in Laval et al,
2π l 0 dl Bp / dl Bp = 2π χ νdχ ′ / νdχ ′ = θ
is the same as the usual straight field line angle [11],
and q = 1 2π νdχ = 1 2π I R 2 dl
Bp . The perturbation they consider has a plasma displacement ξ ∼ e imθ , so if we plot ξ versus the length along a flux surface in the poloidal plane, then near the separatrix in an X-point equilibrium ξ will oscillate arbitrarily rapidly as we approach the X-point (see figure 1). The most unstable modes have m ≃ nq, so in the figure we plot a mode with m ≃ nq, for which e imθ ≃ e in R χ νdχ . Alternately, when m ≪ nq or m ≫ nq, then the mode localises poloidally in the vicinity of the X-point, and is approximately constant elsewhere. The rapid oscillation of ξ near the X-point makes it questionable whether it is physically acceptable, and other terms beyond ideal MHD need to be considered, but it cer-tainly means that the closer we approach the separatrix the greater the number of Fourier modes required (since m ≃ nq), and the smaller a computer code's mesh spacing would need to be to represent the mode. Therefore as we approach the separatrix it will be increasingly difficult for a numerical calculation to represent the mode.
III. CYLINDRICAL PLASMAS
Here we outline the derivation of the marginal stability condition for an arbitrarily large aspect ratio (cylindrical) equilibrium with the Tokamak ordering (Freidberg [12], Wesson [1]).
We start from the usual force balance equation J ∧ B = ∇p, and take the curl of both sides, expanding to give,
0 = B.∇ J − J.∇ B(1)
Linearising the equation then gives for the equilibrium quantities
0 = B 0 .∇ J 0 − J 0 .∇ B 0(2)
and for the perturbed quantities
0 = B 0 .∇ J 1 + B 1 .∇ J 0 − J 0 .∇ B 1 − J 1 .∇ B 0 + ξ.∇ B 0 .∇ J 0 − J 0 .∇ B 0(3)
Because Eq. 2 holds everywhere, with B 0 .∇ J 0 − J 0 .∇ B 0 having the constant value of zero, the last term in Eq. 3 that arises from the displacement of the plasma surface by ξ, is zero.
In the large aspect ratio approximation (Freidberg[12], Wesson [1]), Eq. 3 further simplifies
to 0 = B 0 .∇ J 1 + B 1 .∇ J 0(4)
Again, in the large aspect ratio (cylindrical) approximation we may write B 1 = e z ∧ ∇ψ,
for which J 1 = ∇ ∧ e z ∧ ∇ψ = e z ∇ 2ψ − e z .∇∇ψ(5)
With this same ordering (Freidberg [12], Wesson [1]), J 0 is taken to be parallel to e z . Therefore the e z component of Eq. 4 gives
0 = B 0 .∇ J 1 . e z + B 1 .∇J 0z(6)
Substituting for J 1 then gives
0 = B 0 .∇ ∇ 2ψ + B 1 . e r dJ 0z dr(7)
Symmetry with respect to both the axial and the poloidal coordinates, means that we need only consider a single mode,ψ = e ikz+imθψ m (r) (8) where r, θ, and z are the usual cylindrical coordinates, k is a dimensional mode number, and m is a non-dimensional poloidal mode number. Hence we have ∇ψ = e ikz+imθ ik e z + im r e θ ψ m (r) + e r dψ m dr (9) and because B 1 = e z ∧ ∇ψ, then B 1 .∇r = − e θ .∇ψ, giving
B 1 . e r = −e ikz+imθ im rψ m(10)
We also have that
∇ 2ψ = −k 2 − m 2 r 2 ψ m + 1 r ∂ ∂r r ∂ψ m ∂r e ikz+imθ(11)
Hence Eq. 7 gives
0 = ikB z + im B p r −k 2 − m 2 r 2 ψ m + 1 r ∂ ∂r r ∂ψ m ∂r + B 1r dJ 0φ dr(12)
∇. B = 0 requires the normal component of B to be continuous across the plasma-vacuum surface, that with B 1 = e z ∧ ∇ψ requiresψ m to be continuous across the surface. The perturbed field from dψ m /dr may be discontinuous however. Also J 0z = J a just inside the plasma, but J 0z = 0 in the vacuum outside the plasma, so if we integrate from an arbitrarily small distance inside the plasma surface to an arbitrarily small distance outside the surface, we get
0 = ikB z + im B p r dψ m dr − im rψ m [|J 0φ |](13)
where [|f |] indicates the difference between f evaluated in the vacuum just outside of the plasma, and f evaluated just inside the plasma ([|f |] does not equal the integral of f from just inside to just outside the plasma). Writing k = − n R and q = rB z /RB p , then gives
0 = m − nq m r dψm dr ψ m + rJ a B p(14)
as the condition for marginal stability.
IV. TOKAMAK PLASMAS
A. Assumptions
Here we re-examine the assumptions made in going from Eq. 12 to Eq. 13. To obtain Eq.
13, we allowed the perturbed fields to be discontinuous, but required the equilibrium fields to be continuous across the plasma-vacuum boundary. Because a discontinuous magnetic field requires a skin current [13], we will allow perturbed skin currents, but the continuous equilibrium fields imply zero equilibrium skin currents. Therefore we assume that: (a)
there are no equilibrium skin currents, but J 0 can be discontinuous at the plasma-vacuum interface, and (b) perturbations to the magnetic field induce surface skin currents. The first of these assumptions means that with the exception of J 0 , the equilibrium quantities will be continuous across the plasma-vacuum boundary. So if we integrate along the unit normal to the plasma surface, from a distance ǫ just inside the surface to a distance ǫ just outside the plasma, then we will get
l+ǫ l−ǫ f (η(l ′ ))dl ′ = ǫf (η(l)) → 0 as ǫ → 0(15)
where η(l ′ ) is a path parameterised by l ′ , that is parallel to the unit normal to the plasma, and where f is a continuous equlibrium quantity. We also get
l+ǫ l−ǫ df dl ′ dl ′ = f (l + ǫ) − f (l − ǫ) → 0 as ǫ → 0(16)
However, because the equlibrium current is discontinuous at the plasma surface being nonzero within the surface but zero in the vacuum, its derivatives act like delta functions, and
hence l+ǫ l−ǫ d J 0 dl ′ dl ′ = J 0 (l + ǫ) − J 0 (1 − ǫ) ≡ J 0(17)
and similarly
l+ǫ l−ǫ f d J 0 dl ′ dl ′ = f (l) J 0(18)
The second of these assumptions, that the perturbation will produce skin currents at the plasma surface (i.e. that the perturbed magnetic field can be discontinuous at the perturbed plasma surface), means that for a perturbed current J 1
l+ǫ l−ǫ J 1 (l ′ )dl ′ ≡ σ RB p = 0(19)
i.e. the skin current acts like a delta function at the plasma surface. The reason for including the factor of 1/RB p will be clear later. Similarly if f is a continuous function at the plasma surface,
l+ǫ l−ǫ f J 1 (l ′ )dl ′ = f (l) σ RB p(20)
A few other remarks are worthwhile. Firstly the terms in Eq. 3 are evaluated at the equilibrium surface positions, but give the value of the perturbed force balance equation at the perturbed surface. Secondly, the perturbed unit normal n = n 0 + n 1 , with n 1 ∼ |ξ| ≪ 1.
Hence we have that
l+ǫ l−ǫ f d J 0 dψ dl ′ = l+ǫ l−ǫ f ∇ψ.∇ J 0 R 2 B 2 p dl ′ = l+ǫ l−ǫ f RBp n.∇ J 0 − n 1 .∇ J 0 dl ′ = l+ǫ l−ǫ f RBp n.∇ J 0 + O(ξ) = l+ǫ l−ǫ f RBp d J 0 dl ′ dl ′ = f RBp J 0 (21) or equivalently, 1+ǫ l−ǫ f d J 0 dψ dl ′ = 1 RB p ψ + ψ − f d J 0 dψ dψ + O(ξ)(22)
where ψ + ψ − indicates an integral from just inside the last closed flux surface, to just outside it. Similarly because ∇ψ. n = ∇ψ.
n 0 + O(ξ) = RB p + O(ξ), then 1+ǫ l−ǫ f dl = 1 RB p ψ + ψ − f dψ + O(ξ)(23)
and
σ = ψ + ψ − J 1 dψ + O(ξ 2 )(24)
Bearing the above remarks in mind, we now integrate Eq. 3 across the plasma surface, distinguishing perturbed quantities by a subscript of 1 and the equilibrium currents and magnetic field by a subscript of 0, to get
0 = B 0 .∇ ψ + ψ − J 1 dψ + ψ + ψ − B 1 .∇ψ ∂ J 0 ∂ψ dψ − ψ + ψ − J 1 dψ .∇ B 0 + O ξ 2(25)
Note that because the terms in Eq. 3 are of order ξ (it describes a linearised perturbation to the plasma), the O(ξ) corrections that arise when integrating along the normal to the surface produce terms of order ξ 2 and are neglected. Therefore at leading order we have
0 = B 0 .∇ σ 1 + B ψ 1 J 0 − σ 1 .∇ B 0(26)
where B ψ 1 = B 1 .∇ψ. This is the generalised force balance equation for Peeling mode marginal stability, valid for an arbitrary cross-section Tokamak plasma. The last term is a new term that is not present in a circular cross-section cylindrical geometry.
The procedure of integrating across the plasma-vacuum boundary is clearer if we project out the components before integrating across the surface. We have done this as a check, but it is algebraically cumbersome, and obscures the physical arguments that are clearer in the presentation above.
B. Ampere's law at the surface Before proceeding it is worth examining some relationships between the magnetic fields and the skin currents. At the plasma-vacuum interface, ∇. B = 0 and Ampere's law imply that [13] n.
B = 0 (27) n ∧ B V − B = σ RB p(28)
with σ = ψ + ψ − J 1 dψ as before, n denotes the unit normal to the surface, and B V the magnetic field in the vacuum. Because we assume zero equilibrium skin currents
σ 0 RB p = n 0 ∧ B V 0 − B 0 = 0(29)
and therefore because Eq. 27 gives n 0 . B V 0 − B 0 = 0, it then follows that B V 0 − B 0 is not parallel to n 0 , and hence the only way to satisfy Eq. 29 is if B V 0 = B 0 . Considering the lowest order perturbation, we have
σ RBp = n 1 ∧ B V 0 − B 0 + n 0 ∧ B V 1 − B 1 +ξ.∇ n 0 ∧ B V 0 − B 0 (30) which using B V 0 = B 0 and ∇ψ = RB p n 0 , gives σ = ∇ψ ∧ B V 1 − B 1(31)
and hence
∇ψ. σ = 0(32)
In the ψ, χ, φ coordinate system it is possible to directly evaluate σ = l+ǫ l−ǫ ∇ ∧ B 1 dl ′ , giving
σ = R 2 ∇φ[| B p . B 1 |] − R 2 B p [|∇φ. B 1 |](33)
C. Skin currents at marginal stability Using ∇ψ. σ = 0, while projecting out components of Eq. 26, gives
0 = B 0 .∇ σ. B p + B 1 .∇ψ B p . J 0 − σ.∇B 2 p 0 = B 0 .∇ ( σ.∇φ) + B 1 .∇ψ ∇φ. J 0 + 2I σ.∇R R 3 (34) Because 2I σ.∇R R 3 = −σ.∇ I R 2 = − σ. Bp B 2 p B p .∇ I R 2 , then we can rewrite Eq. 34 as 0 = B 0 .∇ σ. Bp B 2 p + B 1 .∇ψ [| Bp. J 0|] B 2 p 0 = B 0 .∇ ( σ.∇φ) + B 1 .∇ψ ∇φ. J 0 − σ. Bp B 2 p B p .∇ I R 2(35)
Next we note that for B 0 = I(ψ)∇φ + ∇φ ∧ ∇ψ, then
Bp. J 0 B 2 p = −I ′ ∇φ. J 0 = −p ′ − II ′ R 2 (36) giving Bp. J 0 B 2 p = I ′ a ∇φ. J 0 = p ′ a + IaI ′ a R 2(37)
Using B 1 .∇ψ = B 0 .∇ξ ψ and also noting that I = I(ψ), Eqs. 35 now become
0 = B 0 .∇ σ. Bp B 2 p + I ′ a ξ ψ 0 = B 0 .∇ σ.∇φ + ξ ψ p ′ a + IaI ′ a R 2 − B 0 .∇ I R 2 σ. Bp B 2 p + I ′ a ξ ψ(38)
Hence we have that
σ. Bp B 2 p = −I ′ a ξ ψ + f φ − χ νdχ ′ σ.∇φ = − p ′ a + IaI ′ a R 2 ξ ψ + I R 2 f φ − χ νdχ ′(39)
where f is some function of φ − χ νdχ ′ , so that B 0 .∇f = 0. Therefore using Eq. 36 to write Eqs. 39 in terms of the equilibrium current J 0 , we have the skin currents at marginal stability given by
σ = J 0 ξ ψ + B 0 f φ − χ νdχ ′(40)
If we require that σ = 0 for ξ ψ = 0, then because B 0 .∇ξ ψ = 0, we simply have that
σ = J 0 ξ ψ(41)
i.e. that the skin current due to a perturbation at marginal stability is equal to the product of the equilibrium current at the edge and the radial displacement of the plasma.
Notice that at this point we have not used any expansion, e.g. in terms of straight field line co-ordinates, and we have not neglected any poloidal dependencies of the equilibrium quantities. We have implicitly assumed the existence of the χ, φ, ψ coordinate system, but it is possible to re-express the coordinates in terms of arc length along a flux surface, and the consequent relations expressed in Eqs. 33 for example, remain valid (except at the point of zero size that is the X-point). Hence the results appear valid at the separatrix.
D. Marginal stability
To relate this to the previous cylindrical condition for marginal stability, Eq. 14, we consider ∇. B, which may be written
∇. B = 1 J χ ∂ ∂ψ J χ ∇ψ. B + ∂ ∂χ J χ ∇χ. B + ∂ ∂φ J χ ∇φ. B(42)
with J χ the Jacobian. Then we evaluate [|∇. B|], the difference between ∇. B evaluated in the vacuum just outside the plasma and ∇. B evaluated just inside the plasma.
Noting that J χ ∇χ.
B 1 = J χ 1 JχBp ∇χ |∇χ| . B 1 = Bp. B 1 B 2 p , then [|∇. B 1 = 0|] requires that 0 = 1 J χ ∂ ∂ψ J χ ∇ψ. B 1 + 1 J χ ∂ ∂χ B p . B 1 B 2 p + ∂ ∂φ ∇φ. B 1(43)
Eq. 33 gives − σ. Bp
R 2 B 2 p = ∇φ. B 1 σ.∇φ = B p . B 1(44)
So we have
0 = 1 J χ ∂ ∂ψ J χ ∇ψ. B 1 + 1 J χ ∂ ∂χ σ.∇φ B 2 p + ∂ ∂φ − σ. B p R 2 B 2 p(45)
To simplify this we consider a limit of high toroidal mode number n and note that
1 Jχ ∂ ∂χ = B 0 .∇ − I R 2 ∂
∂φ . Then we note that B 0 .∇ is of order 1 to prevent a large stabilising contribution from field-line bending, but ∂ ∂φ is of order n. We also have ∂ ∂ψ ∇ψ. B 1 ∼ n. Because the system is axisymmetric, we need only consider a single Fourier mode in the toroidal angle, and take σ ∼ e −inφ . Then we have
0 = 1 J χ ∂ ∂ψ J χ ∇ψ. B 1 + in R 2 B 2 p I σ.∇φ + σ. B p + O(1)(46)
Later in Section V we will find that [|∇ψ. B 1 |] = 0, so there will be zero contribution from the terms involving ∂J χ /∂ψ. Rearranging the resulting equation leaves
R 2 B 2 p i n ∂ ∂ψ ∇ψ. B 1 = B 0 . σ + O 1 n(47)
Eq. 47 is our generalised criterion for marginal stability to the Peeling mode at high-n, with σ given by Eq. 41.
E. Cylindrical limit
Using Eq. 41 ( σ = ξ ψ J 0 = J 0 ∇ψ. ξ), then for large aspect ratio (cylindrical) geometry σ = J 0 (RB p )ξ r and Eq. 47 becomes
where we have also used that for a cylinder q = rB 0 R 0 Bp . Finally the plasma vacuum boundary conditions (e.g. see Freidberg [12]), of
n 0 . B V 1 = B 0 .∇ n 0 . ξ − n 0 . ξ n 0 . n 0 .∇ B 0(50)
for a cylinder simplify to give b V r = B 0 .∇ξ r = b r . So we regain the usual condition for marginal stability to the Peeling mode in cylindrical geometry, of
∆ ′ a ∆ a + J = 0(51)
with ∆ ′ a = r br dbr dr and ∆ a = 1 − nq m .
V. THE PLASMA-VACUUM BOUNDARY CONDITIONS
The question now arises: what are the equivalent terms in the energy principle that correspond to our ordering for the Peeling mode? Before addressing this question, we first
show that the plasma-vacuum boundary condition that is usually given in conjunction with the energy principle simply requires continuity of the normal component of the perturbed magnetic field, evaluated with the equilibrium normal to the surface at the equilibrium surface position. This is shown to agree with a simpler and more intuitive derivation that is given later.
We start from the boundary condition usually given in conjunction with the energy principle (Freidberg [12]), of
n 0 . B V 1 = B 0 .∇ n 0 . ξ − n 0 . ξ n 0 . ( n 0 .∇) B 0(52)
with n 0 the equilibrium unit normal. We will rewrite the expression in terms of ∇ψ = n 0 RB p , and then simplify the result. Firstly we multiply by RB p , and rearrange the equation to get
∇ψ. B V 1 = RB p B 0 .∇ n 0 . ξ − ∇ψ. ξ 1 R 2 B 2 p ∇ψ.∇ψ.∇ B 0 = B 0 .∇ RB p n 0 . ξ − n 0 . ξ B 0 .∇ (RB p ) − ∇ψ. ξ R 2 B 2 p ∇ψ.∇ B 0 .∇ψ − B 0 .∇ψ.∇ψ∇ψ = B 0 .∇ ∇ψ. ξ − ∇ψ. ξ RBp B 0 .∇ (RB p ) − ∇ψ. ξ R 2 B 2 p −∇ψ. B 0 .∇ψ∇ψ = B 0 .∇ ∇ψ. ξ − ∇ψ. ξ RBp B 0 .∇ (RB p ) − ∇ψ. ξ R 2 B 2 p − B 0 .∇ R 2 B 2 p 2 = B 0 .∇ ∇ψ. ξ = ∇ψ. B 1(53)
Alternately, ∇. B = 0 at the surface requires n. B = n. B V , with n = n 0 + n 1 , where n 0 is the unit normal to the equilibrium magnetic flux surfaces (and hence also the equilibrium plasma surface) and n 1 is the unit normal perturbation from the equilibrium unit normal n 0 (and hence also the perturbation from the equilibrium unit normal to the plasma surface).
VI. THE ENERGY PRINCIPLE
Now we return to the relationship between our equations for marginal stability of the Peeling mode, and the energy principle. We start from the high mode number formulation for δW = δW F + δW S + δW V , given in Connor et al [14]. Looking firstly at δW S , then later δW V , we have
δW S = π dχ ξ * n J χ Bk R 2 B 2 p J χ B 2 1 n ∂ ∂ψ J χ Bk ξ ψ − B. J B 2 ξ ψ(59)
with J χ Bk = iJ χ B.∇. Note that Connor et al [14] took ξ ∼ e +inφ , however here we have ξ ∼ e −inφ , which is reflected by n → −n in the expression for δW . We integrate by parts, noting that ξ ∼ e −inφ but ξ * ∼ e inφ , and also replace J χ Bk with iJ χ B.∇ to give
δW S = −π J χ dχ i n B.∇ξ * R 2 B 2 p J χ B 2 i n ∂ ∂ψ J χ B.∇ξ ψ − B. J B 2 ξ ψ(60)
Using B.∇ξ * ψ = ∇ψ. B * 1 , then we get
δW S = π J χ dχ i n ∇ψ. B * 1 − R 2 B 2 p B 2 i n 1 J χ ∂ ∂ψ J χ ∇ψ. B 1 + B. J B 2 ξ ψ(61)
To obtain the vacuum solution for B V 1 = ∇V , we need to solve ∇ 2 V = 0, which requires
0 = 1 J χ ∂ ∂ψ (J χ ∇ψ.∇V ) + ∂ ∂χ (J χ ∇χ.∇V ) + ∂ ∂φ (J χ ∇φ.∇V ) (62) Using 1 Jχ ∂ ∂χ = B.∇ − I R 2 ∂ ∂φ , |∇χ| 2 = 1 J 2 χ B 2 p
, and V ∼ e −inφ , we can rewrite this as
0 = 1 J χ ∂ ∂ψ (J χ ∇ψ.∇V ) + B.∇ + in I R 2 1 B 2 p B.∇ + in I R 2 V − n 2 R 2(63)
Then using B.∇ ∼ 1, and taking the high-n limit, gives
0 = 1 J χ ∂ ∂ψ (J χ ∇ψ.∇V ) − n 2 I 2 R 4 B 2 p V − n 2 1 R 2 V(64)
which using B V 1 = ∇V , rearranges to give the result that for n ≫ 1,
V = R 2 B 2 p B 2 1 n 2 1 J χ ∂ ∂ψ J χ ∇ψ. B V 1(65)
The vacuum contribution to δW is
δW V = 1 2 dr B V 1 2 (66)
Using ∇ 2 V = 0, B 1 = ∇V , and Gauss' theorem, this may be written as an integral over the plasma surface, with
δW V = − 1 2 V dS.∇V *(67)
Using dS = ∇ψ RBp (J χ B p dχ) (Rdφ), and integrating with respect to φ gives
δW V = −π J χ dχV ∇ψ.∇V *(68)
Using the expression Eq. 65 for V at high-n, and the boundary condition ∇ψ. B V 1 = ∇ψ. B 1 (that Section V shows to be equivalent to the boundary condition that is usually given in formulations of the energy principle), we get
δW V = −π J χ dχ ∇ψ. B 1 R 2 B 2 p B 2 1 n 2 1 J χ ∂ ∂ψ J χ ∇ψ. B V 1(69)
Inserting −1 = i 2 into δW V , we get
δW S + δW V = π J χ dχ i n ∇ψ. B * 1 R 2 B 2 p B 2 i n 1 Jχ ∂ ∂ψ J χ ∇ψ. B V 1 − R 2 B 2 p B 2 i n 1 Jχ ∂ ∂ψ J χ ∇ψ. B 1 + B. J B 2 ξ ψ(70)
Because ∇ψ. B V 1 = ∇ψ. B 1 , the terms involving ∂J χ /∂ψ will cancel. Then using the notation [|f |] to denote the difference between f evaluated just outside and just inside the plasma,
gives δW S + δW V = π J χ dχ i n ∇ψ. B * 1 B 2 R 2 B 2 p i n ∂ ∂ψ ∇ψ. B 1 + B. Jξ ψ (71)
The term in {} is exactly Eq. 47 with σ = J 0 ξ ψ as given by Eq. 41. Therefore marginal stability of the Peeling mode corresponds (at high-n) to taking the plasma's contribution to δW , δW F ≡ 0, and then solving δW S + δW V = 0. This suggests we should define the high-n Peeling mode as a mode (represented by a trial function in this analysis), that allows us to neglect δW F compared to δW S + δW V (by for example being sufficiently localised), and whose subsequent stability is determined by δW S and δW V .
VII. X-POINT PLASMAS
Now we consider the stability of Peeling modes to the trial function considered by Laval et al [8], that consists of a single Fourier mode with ξ = ξ m (ψ)e imθ−inφ , where θ = 1 q χ νdχ ′ is the usual straight field-line poloidal coordinate. After taking Eq. 71, and using ∇ψ.
B V 1 = ∇ψ. B 1 = B.∇ξ ψ , we have δW S + δW V = π J χ dχ i n B.∇ξ * ψ B 2 R 2 B 2 p ∂ ∂ψ ∇ψ. B 1 ∇ψ B 1 i n B.∇ξ ψ + B. Jξ ψ(72)
Substituting the trial function into Eq. 72 gives
δW = −2π 2 |ξ m | 2 R 0 ∆ ∆∆ ′ +Ĵ (73) where ∆ ≡ m − nq nq (74) J ≡ 1 2π dl IR 0 R 2 B p J. B B 2(75)∆ ′ ≡ 1 2π dlR 0 B p I 2 R 2 B 2 ∂ ∂ψ ∇ψ. B 1 ∇ψ. B 1 (76)
and with dl = J χ B p dχ an element of arc length in the poloidal cross-section, and R 0 a typical measure of the major radius such as it's average for example. Note that because ξ m is a Fourier component of ξ.∇ψ ∼ ξ(RB p ), the dimensions of |ξ m | 2 /R 0 are energy. Equation 73
may easily be minimised for ∆ (or equivalently, minimised with respect to choice of toroidal mode number), with
∆ = −Ĵ 2∆ ′ (77) giving δW = π 2 2 |ξ m | 2 R 0 2Ĵ 2 ∆ ′(78)
If ∆ is chosen to maximise the growth rate, then a similar but different value will be found.
Similarly, there is no reason why there should not be a more unstable mode than the trial function we have considered. However, our primary interest is stability to the trial function that was found to be unstable by Laval et al [8].
A calculation of δW requires the evaluation of ∆ ′ for a plasma equilibrium with a separatrix. The calculation of ∆ ′ is the main subject of the second part to this paper. As an introduction to this, we note that for a circular cross section plasma, an estimate for ∆ ′ may be found by approximating the perturbation to the magnetic field near the edge of the plasma as being the same as for a vacuum, then solving Laplace's equation both inside and outside the plasma, and matching the solutions at the plasma-vacuum boundary. Then for a circular cross-section ∆ ′ = −2m ≃ −2nq.
Observe that for J .∇φ = 0, B p . J = −I ′ B 2 p , for whicĥ
J = 1 2π dl IR 0 R 2 B p (−I ′ )B 2 p B 2 ∼ 1(79)
and as discussed above we will have δW → 0 as q → ∞. However, for J.∇φ = 0, then
B. J = −Ip ′ − B 2 I ′ and thereforeĴ = 1 2π dl IR 0 R 2 Bp (−Ip ′ −B 2 I ′ ) B 2 ≃ R 0 J. B B 2 I R 2 dl Bp = R 0 J. B B 2 q(80)
For which if ∆ ′ ∼ −nq as is the case for a circular cross-section, thenĴ 2 /∆ ′ would be of order −q, and δW < 0, suggesting that the mode would be unstable.
Although the sign of δW is usually taken to indicate whether a mode is unstable or not, the growth rate determines how unstable the mode is (i.e. how rapidly it develops). For example if our trial function ξ m (ψ)e imθ = ξ ψ = ∇ψ. ξ had been ξ m (ψ)e imθ = ∇ψ. ξ/RB p so that it had dimensions of length as opposed to dimensions of length times RB p , then we would no longer haveĴ ∼ q, despite our model only depending on the poloidal structure of the mode. The dependence of δW on the normalisation of the plasma perturbation does not affect the calculation of the growth rate however, for which the consequences of the normalisation of the plasma perturbation will cancel. The growth rate is discussed later.
Are there any reasons why a computer code might fail to find an unstable mode? One possibility is that the need for m ∼ nq will require very high poloidal mode numbers as q → ∞, and this could potentially prevent a numerical code from seeing the instability. Also important, is the need to consider the most unstable mode. Minimising δW with respect to the toroidal mode number
gave ∆ = −Ĵ /2∆ ′ . If ∆ ′ ≃ −2nq andĴ ≃ qR 0 ( J. B)/B 2 (for J.∇φ = 0), this would require ∆ ≃ −Ĵ 2∆ ′ = R 0 J. B B 2 1 4n (81)
that is independent of q and the poloidal mode number. A final possibility is that ∆ ′ might diverge more rapidly thanĴ as we approach the separatrix. The second part to this paper calculates ∆ ′ analytically, thereby avoiding the numerical problems associated with an X-point.
Another, less obvious reason why an unstable mode might not be found in computer calculations is that despite δW < 0 indicating that it is energetically favourable for the mode to be unstable, the growth rate can still be vanishingly small. This possibility is explored next.
VIII. THE GROWTH RATE
So far we have only considered δW , because its sign is usually presumed to be sufficient to indicate whether a mode is stable or not. The growth rate γ for a mode with ξ ∼ e γt is obtained from γ 2 = −δW/ 1 2 ρ 0 |ξ| 2 dr, where 1 2 ρ 0 |ξ| 2 dr is the kinetic energy term. Next we will estimate the kinetic energy term, so as to estimate the growth rate. The surprising result that we will find is that even if δW < 0, indicating it is energetically favourable for an instability, the kinetic energy term can diverge so strongly that although it may be energetically favourable for a mode to be unstable, its growth rate is vanishingly small. An alternative complementary calculation to the one given below, with the same conclusions, is given in Appendix A.
To estimate ρ 0 | ξ| 2 dr, we write
ξ = ξ ψ ∇ψ R 2 B 2 p + ξ B B B 2 + ξ ⊥ B ∧ ∇ψ R 2 B 2 p B 2(82)
for which
| ξ| 2 = |ξ ψ | 2 R 2 B 2 p + |ξ B | 2 B 2 + |ξ ⊥ | 2 R 2 B 2 p B 2(83)
It is convenient to write ξ in the form of Eq. 82 so that we can use the results from a high-n ordering (e.g. see Webster and Wilson [19]), that gives
ξ ⊥ = i n ∇ψ.∇ξ ψ = i n R 2 B 2 p ∂ξ ψ ∂ψ(84)
and
B.∇ξ B − ξ B B.∇B 2 B 2 = ξ ψ ∂ ∂ψ 2p + B 2 + Iξ ⊥ R 2 B 2 p B 2 B.∇B 2(85)
Before continuing further we make some observations on the high-n ordering that for
∇. ξ = 0 usually leads to ξ ⊥ = i n R 2 B 2 p ∂ξ ψ
∂ψ . This analysis is used in the derivation of δW used in Section VI onwards, and to derive the equations solved by ELITE [9]. The ordering implicitly assumes that ∂ψ . For the present we will continue to use the ordering employed by Connor [14], that is also used to derive the equations solved by ELITE.
Because the trial function that we consider consists of a single Fourier mode, Eq. 85 may be solved for ξ B , with
ξ B = ξ ψ ∂ ∂ψ (2p + B 2 ) + I B.∇B 2 B 2 i n ∂ξ ψ ∂ψ I qR 2 (im − inq) − B.∇B 2 B 2(86)
giving
|ξ B | 2 B 2 = 1 B 2 |ξ ψ | 2 ∂ ∂ψ (2p + B 2 ) 2 + 1 n 2 ∂ξ ψ ∂ψ 2 I 2 B.∇B 2 B 2 2 I 2 R 4 n 2 ∆ 2 + B.∇B 2 B 2 2(87)
with ∆ = (m − nq)/nq as before. Using Eq. 84 for ξ ⊥ , and substituting this and 87 into Eq.
83, gives
| ξ| 2 = |ξ ψ | 2 B 2 R 2 B 2 p + ∂ ∂ψ (2p + B 2 ) 2 I 2 R 4 n 2 ∆ 2 + B.∇B 2 B 2 2 + 1 n 2 ∂ξ ψ ∂ψ 2 R 2 B.∇B 2 B 2 2 + I 2 B 2 p R 2 B 2 n 2 ∆ 2 I 2 R 4 n 2 ∆ 2 + B.∇B 2 B 2 2 (88) Noting that R 2 B.∇B 2 B 2 ∼ B 2 p and I 2 B 2 p R 2 B 2 ∼ B 2 p , whereas I 2 R 4 n 2 ∆ 2 + B.∇B 2 B 2 2 ∼ B 2 /R 2 as
B p → 0, with ∆ ∼ 1/n as found previously, then we find
| ξ| 2 ∼ |ξ ψ | 2 R 2 B 2 p + R 2 B 2 p B 2 1 n 2 ∂ξ ψ ∂ψ 2(89)
In other words, we may neglect the term in |ξ B | 2 compared with the |ξ ψ | 2 and |ξ ⊥ | 2 terms.
We will consider each of these terms in turn. Firstly
ρ 0 |ξψ| 2 R 2 B 2 p dr = 2π dl Bp dψρ 0 |ξ ψ | 2 R 2 B 2 p ∼ ρ 0 R 2 ψs |ξ ψ | 2 dψ dl B 3 p ρ 0 I∇φ. J ψs q ′ |ξ ψ | 2 dψ(90)
with ψ = ψ s at the plasma surface, and where we used [18]
q ′ (ψ) ≃ 1 2π ν B 2 p ∇φ. J − ∂B 2 p ∂ψ dχ ∇φ. J 2π dl B 3 p(91)
with dl = (J χ B p )dχ, and at large aspect ratio ∇φ. J is approximately a function of the poloidal magnetic flux. Although we have not considered the radial structure of the mode in this paper, to estimate these terms we will adopt the ansatz that near the plasma's edge ξ m can be approximated by a power law, with
ξ m = ξ 0 (ψ a − ψ s ) p (ψ a − ψ) p(92)
where ψ = ψ a at the separatrix, and ψ = ψ s at the plasma surface. This is consistent with studies that do consider a mode's radial structure [14,15]. We also use the result found here [16] and elsewhere [20], that for a conventional X-point (as opposed to the Xpoint produced by a "snowflake" divertor [20]), near a separatrix we have
q ≃ −q 0 ln ψ a − ψ ψ a(93)
for some constant q 0 ∼ 1. Under these assumptions
q ′ |ξ ψ | 2 dψ ∼ |ξ ψ | 2 (94) giving ρ 0 |ξ ψ | 2 R 2 B 2 p dr ∼ ρ 0 r B B p |ξ ψ | 2 ψs(95)
where we took I∇φ J ∼ B B p /r, with r a measure of the plasma radius.
Next we consider,
ρ 0 R 2 B 2 p B 2 1 n 2 ∂ξ ψ ∂ψ 2 dr ∼ ρ 0 B 2 1 n 2 ψs R 2 B p dl ∂ξ ψ ∂ψ 2 dψ(96)
Finally, taking ∇φ. J = 0 and ∆ ′ ≃ −2nq (∆ ′ is calculated in the second part to this paper), then gives
γ 2 = −δW drρ 0 |ξ| 2 ∼ γ 2 A R 0 J . B B 2 2 q ψ s q ′(102)
with γ A ≡ B 2 /(ρ 0 R dl). Because q ′ → ∞ more rapidly than q as we approach the separatrix, then for an outermost flux surface that is made increasingly close to that of a separatrix with ψ s → ψ a , we have that γ 2 → 0 and
ln γ γ A = − 1 2 ln ψ s q ′ q(103)
with γ A the Alfven frequency, indicating that the growth rate γ → 0 as q ′ → ∞.
IX. SUMMARY
This paper re-explores the stability of the Peeling mode for toroidal Tokamak geometry.
It starts from a simple approach to Peeling mode stability at marginal stability in cylindrical geometry, then generalises this to toroidal Tokamak equilibrium. In the process of doing so we find a number of interesting results, namely 1. At marginal stability, a plasma perturbation induces a skin current that is parallel and proportional to the equlibrium current at the edge, and proportional to the radial plasma displacement.
2. For zero equilibrium skin current the usual plasma-vacuum boundary conditions (Freidberg[12]), are identical to the requirement that n 0 . B 1 = n 0 . B V 1 , with the quantities evaluated at the equilibrium position.
3. The equilibrium conditions (force balance) for the Peeling mode at marginal stability and high toroidal mode number n, are identical to requiring δW S + δW V = 0, where δW S and δW V are the surface and vacuum contributions to the energy principle's δW = δW F + δW S + δW V , with δW F the plasma's contribution to the energy principle (Freidberg[12]).
This suggests the Peeling mode be defined as a mode for which δW F ≪ δW S + δW V . For the trial function used by Laval et al [8], that consisted of a single Fourier mode in straight field line co-ordinates, we find that the most unstable choice of ∆ gives δW = π 2 2 |ξm| 2 R 0 2Ĵ 2 ∆ ′ . To evaluate δW for this model, it is necessary to know ∆ ′ for a plasma cross-section with a separatrix and X-point at the plasma-vacuum boundary. Doing this without making the usual approximations (i.e. with a discretisation of space), as are usually made in numerical calculations, is the subject of the second part to this paper [16].
Finally we considered the growth rate, and found that even with δW < 0, the growth rate can be vanishingly small. This is because the kinetic energy term was found to diverge like q ′ as the outermost flux surface becomes increasingly close to a separatrix. When this divergence is sufficiently rapid (as would be the case for ∆ ′ ≃ −2nq), then ln(γ/γ A ) asymptotes to ln(γ/γ A ) = − 1 with γ 2 A = B 2 /ρ 0 R 0 dl. Therefore as the outermost plasma surface more closely approximates a separatrix with ψ s → ψ a , we have that γ/γ A → 0. Note that because we have taken q ∼ q 0 ln ψa−ψ ψa
FIG. 1 :
1A plot of the trial function used by Laval et al
b r = B.∇ξ r , taking ξ ∼ e imθ−inφ , and using 1 nq = 1 m + m−nq nq with m ≃ nq,
Then n. B = n. B V requires ( n 0 + n 1 ) . B 0 + B 1 + ξ.∇ n 0 . B 0 = ( n 0 + n 1 ) .B V 0 + B V 1 + ξ.∇ n 0 . B V 0(54)where ξ is the displacement of the plasma from its equilibrium position, and where all quantities are evaluated at their equilibrium positions. Similarly at equilibrium it is the quantities evaluated at their equilibrium positions. Therefore because n 0 . B 0 = n 0 . B V 0 = 0 everywhere, then ξ.∇ n 0 . B 0 = ξ.∇ n 0 . are no skin currents at equilibrium, then as shown previously in the main text, we have B 0 = B V 0 , so Eq. 56 becomes n 0 . B 1 − B
∂ψ ∼ n ≫ 1. Whereas a sufficiently large n can always be found to ensure 1, and terms of this type will often be negligible order one contributions anyhow, future calculations would be improved by including them. For example ξ ⊥ would then become ξ ⊥ =
For the trial function ξ ψ = ξ m (ψ)e imθ with θ = 1 most strongly, so the largest contribution to the expression will be from either dξm Similarly, because νdχ = 2πq and θ = 1 q χ νdχ ′ , we have∼ n 2 q ′ 2 |ξ m | 2 ∼ n 2 |ξ 0 | 2 q ′ | ψ=ψsTherefore, because ψ s ∼ R 2 B p dl/ dl we have (ψ s q ′ ) |ξ ψ | 2q
χ νdχ ′ , we have
∂ξ ψ
∂ψ
2
dψ ∼
dξ m
dψ
2
+ |ξ m | 2 ∂θ
∂ψ
2
m 2 dψ
(97)
the last expression ignores terms in dξm
dψ
∂θ
∂ψ because either dξm
dψ
or ∂θ
∂ψ
will diverge
dψ
2
or
|ξ m | 2 ∂θ
∂ψ
2
m 2 . Taking ξ m as in Eq. 92, gives
dξ m
dψ
2
dψ ∼ |ξ 0 | 2 q ′
ψ=ψs
(98)
∂θ
∂ψ
2
= −
q ′
q
θ +
1
q
χ ∂ν
∂ψ
dχ ′
2
∼
q ′
q
2
(99)
that with m ≃ nq gives
m 2 |ξ m | 2 ∂θ
∂ψ
2
(100)
ρ 0
R 2 B 2
p
B 2
1
n 2
∂ξ ψ
∂ψ
2
dr ∼
ρ 0 dl
B 2 ψ=ψs
ln(ψ s q ′ /q), a result that may be compared with those from codes such as ELITE.
AcknowledgmentsThanks to Tim Hender for reading and commenting on an earlier draft of this paper. Here we provide an alternative derivation for the growth rate, to that given in Section VIII. In the following we will, 1. Continue to use the high-n ordering of Connor et al[14], for which ∇. ξ = 0 requires that2. We will use the arguments from Section VIII, that lead us to expect that ∂ξ ψ /∂ψ ∼ mq ′ q |ξ m | ∼ nq ′ |ξ m |, for the trial function of Laval et al[8]with ξ ψ = ξ m (ψ)e imθ .3. As in Section VIII we will continue to assume that near the separatrix,for some constant q 0 ∼ 1, with ψ a the value of ψ at the separatrix. We will also continue to assume that near the separatrix we can approximate |ξ m (ψ)| as a power law, withwhere ψ s < ψ a is the value of ψ at the plasma-vacuum surface.With the assumptions of 1, 2, and 3, we require thatHowever, we must have ξ ⊥ ≪ 1 as ψ → ψ s , and therefore we require ξ 0 =ξ 0 (ψa−ψs) ψa witĥ ξ 0 ≪ 1 a constant, so that as ψ → ψ a we have ξ ⊥ ∼ξ 0 ≪ 1. Therefore we also have as a consequence that ξ ψ ∼ξ 0In the limit where the plasma surface tends to a separatrix, with ψ s → ψ a , we then must have ξ ψ ∼ξ 0 (ψa−ψs) ψa → 0. Therefore δW , for which ξ m is evaluated at ψ = ψ s , hasNext we consider the growth rate.Using arguments from Section VIII, we expectNow with ξ ⊥ given by Eq. A4, we havewhere denotes a poloidal average, and in the last line we used ψ a ∼ B p R 2 . Hence using Eqs. A6 and A8 we find
J Wesson, Tokamaks Oxford. Oxford University PressWesson J. 1997 Tokamaks Oxford, Oxford University Press.
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S Medvedev, Yu, T C Hender, O Sauter, 28th EPS Conference on Contr. Fusion and Plasma Phys., ECA. 25Medvedev S.Yu, Hender T.C., Sauter O. et al, 28th EPS Conference on Contr. Fusion and Plasma Phys., ECA Vol 25 A (2001) 21-24.
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. S Saarelma, private communicationSaarelma S., 2007 private communication.
MHD Instabilities. G Bateman, MIT PressBateman G., 1980 MHD Instabilities, MIT Press.
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. J D Jackson, Classical Electrodynamics. John Wiley & SonsJackson J.D., 1975 Classical Electrodynamics New York, John Wiley & Sons.
. J W Connor, R J Hastie, H R Wilson, Phys. Plasmas. 572687Connor J.W., Hastie R.J., Wilson H.R., 1998 Phys. Plasmas 5, no. 7, 2687.
. D Lortz, Nucl. Fusion. 1549Lortz D. 1975 Nucl. Fusion 15, 49.
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A J Webster, H R Wilson, Proceedings of the joint Varenna-Lausanne International Workshop on the Theory of Fusion Plasmas. the joint Varenna-Lausanne International Workshop on the Theory of Fusion Plasmas417Webster A.J., Wilson H.R., 2002 Proceedings of the joint Varenna-Lausanne International Workshop on the Theory of Fusion Plasmas, 417.
. D D Ryutov, R H Cohen, T D Rognlien, M V Umansky, Phys. Plasmas. 1592501Ryutov D.D, Cohen R.H., Rognlien T.D., Umansky M.V., 2008 Phys. Plasmas, 15, 092501
|
[] |
[
"Genuine Counterfactual Communication with a Nanophotonic Processor",
"Genuine Counterfactual Communication with a Nanophotonic Processor"
] |
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"I Alonso Calafell \nVienna Center for Quantum Science and Technology (VCQ)\nFaculty of Physics\nUniversity of Vienna\nBoltzmanngasse 5A-1090ViennaAustria\n",
"T Strömberg \nVienna Center for Quantum Science and Technology (VCQ)\nFaculty of Physics\nUniversity of Vienna\nBoltzmanngasse 5A-1090ViennaAustria\n",
"D R M Arvidsson-Shukur \nDepartment of Physics\nCavendish Laboratory\nUniversity of Cambridge\nCB3 0HECambridgeUnited Kingdom\n",
"L A Rozema \nVienna Center for Quantum Science and Technology (VCQ)\nFaculty of Physics\nUniversity of Vienna\nBoltzmanngasse 5A-1090ViennaAustria\n",
"V Saggio \nVienna Center for Quantum Science and Technology (VCQ)\nFaculty of Physics\nUniversity of Vienna\nBoltzmanngasse 5A-1090ViennaAustria\n",
"C Greganti \nVienna Center for Quantum Science and Technology (VCQ)\nFaculty of Physics\nUniversity of Vienna\nBoltzmanngasse 5A-1090ViennaAustria\n",
"N C Harris \nQuantum Photonics Group, RLE\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUSA\n",
"M Prabhu \nQuantum Photonics Group, RLE\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUSA\n",
"J Carolan \nQuantum Photonics Group, RLE\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUSA\n",
"M Hochberg \nElenion Technologies\n10016New YorkNYUSA\n",
"T Baehr-Jones \nElenion Technologies\n10016New YorkNYUSA\n",
"D Englund \nQuantum Photonics Group, RLE\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUSA\n",
"C H W Barnes \nDepartment of Physics\nCavendish Laboratory\nUniversity of Cambridge\nCB3 0HECambridgeUnited Kingdom\n",
"P Walther \nVienna Center for Quantum Science and Technology (VCQ)\nFaculty of Physics\nUniversity of Vienna\nBoltzmanngasse 5A-1090ViennaAustria\n"
] |
[
"Vienna Center for Quantum Science and Technology (VCQ)\nFaculty of Physics\nUniversity of Vienna\nBoltzmanngasse 5A-1090ViennaAustria",
"Vienna Center for Quantum Science and Technology (VCQ)\nFaculty of Physics\nUniversity of Vienna\nBoltzmanngasse 5A-1090ViennaAustria",
"Department of Physics\nCavendish Laboratory\nUniversity of Cambridge\nCB3 0HECambridgeUnited Kingdom",
"Vienna Center for Quantum Science and Technology (VCQ)\nFaculty of Physics\nUniversity of Vienna\nBoltzmanngasse 5A-1090ViennaAustria",
"Vienna Center for Quantum Science and Technology (VCQ)\nFaculty of Physics\nUniversity of Vienna\nBoltzmanngasse 5A-1090ViennaAustria",
"Vienna Center for Quantum Science and Technology (VCQ)\nFaculty of Physics\nUniversity of Vienna\nBoltzmanngasse 5A-1090ViennaAustria",
"Quantum Photonics Group, RLE\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUSA",
"Quantum Photonics Group, RLE\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUSA",
"Quantum Photonics Group, RLE\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUSA",
"Elenion Technologies\n10016New YorkNYUSA",
"Elenion Technologies\n10016New YorkNYUSA",
"Quantum Photonics Group, RLE\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUSA",
"Department of Physics\nCavendish Laboratory\nUniversity of Cambridge\nCB3 0HECambridgeUnited Kingdom",
"Vienna Center for Quantum Science and Technology (VCQ)\nFaculty of Physics\nUniversity of Vienna\nBoltzmanngasse 5A-1090ViennaAustria"
] |
[] |
In standard communication information is carried by particles or waves 1-3 . Counterintuitively, in counterfactual communication particles and information can travel in opposite directions. The quantum Zeno effect allows Bob to transmit a message to Alice by encoding information in particles he never interacts with 4,5 . The first suggested protocol 6 not only required thousands of ideal optical components, but also resulted in a so-called "weak trace" 7 of the particles having travelled from Bob to Alice, calling the scalability and counterfactuality of previous proposals 8-12 and experiments 13,14 into question. Here we overcome these challenges, implementing a new protocol 15 in a programmable nanophotonic processor, based on reconfigurable silicon-on-insulator waveguides that operate at telecom wavelengths 16 . This, together with our telecom single-photon source and highly-efficient superconducting nanowire single-photon detectors, provides a versatile and stable platform for a high-fidelity implementation of genuinely trace-free counterfactual communication, allowing us to actively tune the number of steps in the Zeno measurement, and achieve a bit error probability below 1 %, with neither post-selection nor a weak trace. Our demonstration shows how our programmable nanophotonic processor could be applied to more complex counterfactual tasks and quantum information protocols 17-19 .Interaction-free measurements allow one to measure whether or not an object is present without ever interacting with it 20 . This is made clear in Elitzur and Vaidman's well-known bomb-testing gedanken experiment 21 . In this experiment, a single photon used in a Mach-Zehnder interferometer (MZI) sometimes reveals whether or not an absorbing object (e.g. a bomb) had been placed in one of the interferometer arms, without any interaction between the photon and the bomb. It was later shown that the quantum Zeno effect, wherein repeated observations prevent the system from evolving 4,5 , can be used to bring the success probability of this protocol arbitrarily close to unity 4,5,22,23 . Such protocols are often referred to as "counterfactual", and have now been applied to quantum computing 24 , quantum key distribution 17-19 and communication 6,15 . Here we experimentally implement a counterfactual communication (CFC) protocol where information can propagate without being carried by physical particles.The first suggested protocol for CFC was developed by Salih et al., and it is based on a chain of nested MZIs 6 . There are four main concerns with this scheme:(1) Achieving a high success probability (say > 95 %) requires thousands of optical elements. (2) An analysis of the Fisher information flow shows that to retain counterfactuality in Salih's protocol, absolutely perfect quantum channels are needed 12 . (3) Alice's particles leave a weak trace in Bob's laboratory, raising doubts about the * These authors contributed equally to this work.FIG. 1. Architecture of the chained MZI protocol. Alice inputs a photon into the transmission channel, consisting of a row of beamsplitters (BSs) and the lower row of mirrors (marked with an 'm'). a. If Bob intends to send a logic 0, he places mirrors in his laboratory to form MZIs that span his lab and the transmission channel, creating constructive interference in Bob's port (DB). b. If he intends to send a logic 1, he removes the mirrors, causing the photons to arrive back in Alice's laboratory (DA) with high probability.scheme's counterfactuality 8-10 . (4) This scheme requires post-selection to remove the CFC violations.To get around these issues, we implement a novel CFC protocol 15 that does not need post-selection and requires orders of magnitude fewer optical elements than nested MZI protocols. In this scheme photons travel from Alice to Bob but information from Bob to Alice. Note that arXiv:1808.04856v1 [quant-ph]
|
10.1364/cleo_qels.2019.fth4a.3
|
[
"https://arxiv.org/pdf/1808.04856v1.pdf"
] | 119,378,244 |
1808.04856
|
c2d831fcb5f143901e43812f21d98fa65e1c16f6
|
Genuine Counterfactual Communication with a Nanophotonic Processor
14 Aug 2018
I Alonso Calafell
Vienna Center for Quantum Science and Technology (VCQ)
Faculty of Physics
University of Vienna
Boltzmanngasse 5A-1090ViennaAustria
T Strömberg
Vienna Center for Quantum Science and Technology (VCQ)
Faculty of Physics
University of Vienna
Boltzmanngasse 5A-1090ViennaAustria
D R M Arvidsson-Shukur
Department of Physics
Cavendish Laboratory
University of Cambridge
CB3 0HECambridgeUnited Kingdom
L A Rozema
Vienna Center for Quantum Science and Technology (VCQ)
Faculty of Physics
University of Vienna
Boltzmanngasse 5A-1090ViennaAustria
V Saggio
Vienna Center for Quantum Science and Technology (VCQ)
Faculty of Physics
University of Vienna
Boltzmanngasse 5A-1090ViennaAustria
C Greganti
Vienna Center for Quantum Science and Technology (VCQ)
Faculty of Physics
University of Vienna
Boltzmanngasse 5A-1090ViennaAustria
N C Harris
Quantum Photonics Group, RLE
Massachusetts Institute of Technology
02139CambridgeMassachusettsUSA
M Prabhu
Quantum Photonics Group, RLE
Massachusetts Institute of Technology
02139CambridgeMassachusettsUSA
J Carolan
Quantum Photonics Group, RLE
Massachusetts Institute of Technology
02139CambridgeMassachusettsUSA
M Hochberg
Elenion Technologies
10016New YorkNYUSA
T Baehr-Jones
Elenion Technologies
10016New YorkNYUSA
D Englund
Quantum Photonics Group, RLE
Massachusetts Institute of Technology
02139CambridgeMassachusettsUSA
C H W Barnes
Department of Physics
Cavendish Laboratory
University of Cambridge
CB3 0HECambridgeUnited Kingdom
P Walther
Vienna Center for Quantum Science and Technology (VCQ)
Faculty of Physics
University of Vienna
Boltzmanngasse 5A-1090ViennaAustria
Genuine Counterfactual Communication with a Nanophotonic Processor
14 Aug 2018(Dated: August 16, 2018)
In standard communication information is carried by particles or waves 1-3 . Counterintuitively, in counterfactual communication particles and information can travel in opposite directions. The quantum Zeno effect allows Bob to transmit a message to Alice by encoding information in particles he never interacts with 4,5 . The first suggested protocol 6 not only required thousands of ideal optical components, but also resulted in a so-called "weak trace" 7 of the particles having travelled from Bob to Alice, calling the scalability and counterfactuality of previous proposals 8-12 and experiments 13,14 into question. Here we overcome these challenges, implementing a new protocol 15 in a programmable nanophotonic processor, based on reconfigurable silicon-on-insulator waveguides that operate at telecom wavelengths 16 . This, together with our telecom single-photon source and highly-efficient superconducting nanowire single-photon detectors, provides a versatile and stable platform for a high-fidelity implementation of genuinely trace-free counterfactual communication, allowing us to actively tune the number of steps in the Zeno measurement, and achieve a bit error probability below 1 %, with neither post-selection nor a weak trace. Our demonstration shows how our programmable nanophotonic processor could be applied to more complex counterfactual tasks and quantum information protocols 17-19 .Interaction-free measurements allow one to measure whether or not an object is present without ever interacting with it 20 . This is made clear in Elitzur and Vaidman's well-known bomb-testing gedanken experiment 21 . In this experiment, a single photon used in a Mach-Zehnder interferometer (MZI) sometimes reveals whether or not an absorbing object (e.g. a bomb) had been placed in one of the interferometer arms, without any interaction between the photon and the bomb. It was later shown that the quantum Zeno effect, wherein repeated observations prevent the system from evolving 4,5 , can be used to bring the success probability of this protocol arbitrarily close to unity 4,5,22,23 . Such protocols are often referred to as "counterfactual", and have now been applied to quantum computing 24 , quantum key distribution 17-19 and communication 6,15 . Here we experimentally implement a counterfactual communication (CFC) protocol where information can propagate without being carried by physical particles.The first suggested protocol for CFC was developed by Salih et al., and it is based on a chain of nested MZIs 6 . There are four main concerns with this scheme:(1) Achieving a high success probability (say > 95 %) requires thousands of optical elements. (2) An analysis of the Fisher information flow shows that to retain counterfactuality in Salih's protocol, absolutely perfect quantum channels are needed 12 . (3) Alice's particles leave a weak trace in Bob's laboratory, raising doubts about the * These authors contributed equally to this work.FIG. 1. Architecture of the chained MZI protocol. Alice inputs a photon into the transmission channel, consisting of a row of beamsplitters (BSs) and the lower row of mirrors (marked with an 'm'). a. If Bob intends to send a logic 0, he places mirrors in his laboratory to form MZIs that span his lab and the transmission channel, creating constructive interference in Bob's port (DB). b. If he intends to send a logic 1, he removes the mirrors, causing the photons to arrive back in Alice's laboratory (DA) with high probability.scheme's counterfactuality 8-10 . (4) This scheme requires post-selection to remove the CFC violations.To get around these issues, we implement a novel CFC protocol 15 that does not need post-selection and requires orders of magnitude fewer optical elements than nested MZI protocols. In this scheme photons travel from Alice to Bob but information from Bob to Alice. Note that arXiv:1808.04856v1 [quant-ph]
In standard communication information is carried by particles or waves [1][2][3] . Counterintuitively, in counterfactual communication particles and information can travel in opposite directions. The quantum Zeno effect allows Bob to transmit a message to Alice by encoding information in particles he never interacts with 4,5 . The first suggested protocol 6 not only required thousands of ideal optical components, but also resulted in a so-called "weak trace" 7 of the particles having travelled from Bob to Alice, calling the scalability and counterfactuality of previous proposals [8][9][10][11][12] and experiments 13,14 into question. Here we overcome these challenges, implementing a new protocol 15 in a programmable nanophotonic processor, based on reconfigurable silicon-on-insulator waveguides that operate at telecom wavelengths 16 . This, together with our telecom single-photon source and highly-efficient superconducting nanowire single-photon detectors, provides a versatile and stable platform for a high-fidelity implementation of genuinely trace-free counterfactual communication, allowing us to actively tune the number of steps in the Zeno measurement, and achieve a bit error probability below 1 %, with neither post-selection nor a weak trace. Our demonstration shows how our programmable nanophotonic processor could be applied to more complex counterfactual tasks and quantum information protocols [17][18][19] .
Interaction-free measurements allow one to measure whether or not an object is present without ever interacting with it 20 . This is made clear in Elitzur and Vaidman's well-known bomb-testing gedanken experiment 21 . In this experiment, a single photon used in a Mach-Zehnder interferometer (MZI) sometimes reveals whether or not an absorbing object (e.g. a bomb) had been placed in one of the interferometer arms, without any interaction between the photon and the bomb. It was later shown that the quantum Zeno effect, wherein repeated observations prevent the system from evolving 4,5 , can be used to bring the success probability of this protocol arbitrarily close to unity 4,5,22,23 . Such protocols are often referred to as "counterfactual", and have now been applied to quantum computing 24 , quantum key distribution [17][18][19] and communication 6,15 . Here we experimentally implement a counterfactual communication (CFC) protocol where information can propagate without being carried by physical particles.
The first suggested protocol for CFC was developed by Salih et al., and it is based on a chain of nested MZIs 6 . There are four main concerns with this scheme: (1) Achieving a high success probability (say > 95 %) requires thousands of optical elements. (2) An analysis of the Fisher information flow shows that to retain counterfactuality in Salih's protocol, absolutely perfect quantum channels are needed 12 . (3) Alice's particles leave a weak trace in Bob's laboratory, raising doubts about the FIG. 1. Architecture of the chained MZI protocol. Alice inputs a photon into the transmission channel, consisting of a row of beamsplitters (BSs) and the lower row of mirrors (marked with an 'm'). a. If Bob intends to send a logic 0, he places mirrors in his laboratory to form MZIs that span his lab and the transmission channel, creating constructive interference in Bob's port (DB). b. If he intends to send a logic 1, he removes the mirrors, causing the photons to arrive back in Alice's laboratory (DA) with high probability.
scheme's counterfactuality [8][9][10] . (4) This scheme requires post-selection to remove the CFC violations.
To get around these issues, we implement a novel CFC protocol 15 that does not need post-selection and requires orders of magnitude fewer optical elements than nested MZI protocols. In this scheme photons travel from Alice to Bob but information from Bob to Alice. Note that Each MZI is equipped with a pair of thermo-optic phase shifters, which allows us to treat them as beamsplitters with fully tunable reflectivities (set via θ ∈ [0, 2π]) and phases ((φ ∈ [0, 2π]). In our work, we set θ to π, 0 or π/2N , to implement mirrors (circles), SWAPs (triangles) and beamsplitters (squares), respectively. In Alice's laboratory (the pink shaded region) a spontaneous parametric down-conversion source creates a frequency non-degenerate photon pair at λH = 1563 nm, λT = 1565.8 nm. Detection of the λH photon in detector H heralds the λT photon that is injected into the transmission channel. This channel is comprised of the lower half of the PNP, in which MZIs are set to act as mirrors, as well as the MZIs that couple the upper and lower half of the waveguide. The latter of these MZIs are configured to act as beamsplitters, whose reflectivity varies with N (the number of beamsplitters used in the protocol) as R(N ) = cos 2 (π/2N ). Bob's laboratory consists of the upper set of MZIs (blue shaded area), which he can set as mode swaps to send a logic 1 or b. as mirrors to send a logic 0. The photons are detected by superconducting nanowire single-photon detectors with detection efficiencies of approximately 90 %. Coincident detection events are recorded with a custom-made Time Tagging Module (TTM). c. Micrograph of the PNP with dimensions 4.9 mm × 2.4 mm.
the very recent proposals 25,26 discussing means of making the Salih scheme trace-free still require the post-selected removal of non-counterfactual events as well as thousands of ideal optical components.
We perform our experiment using telecom singlephotons in a state-of-the-art programmable nanophotonic processor (PNP) 16 , which is orders of magnitude more precise and stable than previous bulk-optic approaches 22,23 . Our PNP also provides unprecedented tunability, which we use to investigate the scaling of the protocol by changing the number of chained interferometers. By combining the novel CFC protocol with our advanced photonic technology, we are able to implement counterfactual communication with a bit success probability above 99 %, without post-selection.
Our protocol uses a series of N beamsplitters with reflectivity R = cos 2 (π/2N ), which, together with mirrors, form a circuit of N −1 chained MZIs. As shown in Figure 1, the communication protocol begins with Alice injecting a single photon into her input port. If Bob wants to send a logic 0 he leaves his mirrors in place, causing the photon to self interfere such that it exits in D B with unit probability (Figure 1a). To send a logic 1 Bob locally modifies the circuit to have the upper paths open (Fig-ure 1b). In this case the photon will successfully reflect off of all the beamsplitters and exit in D A with probability R N . Removing the mirrors effectively collapses the wavefunction after every beamsplitter, suppressing interference and implementing the Zeno effect. The probability that the photon remains in the lower arm after N beamsplitters can be made arbitrarily high by increasing N (and changing the reflectivities accordingly).
Since any implementation is restricted to a finite number of beamsplitters, there will be a probability for a photon to exit the wrong port when Bob tries to send a logic 1. This error probability is a function that decreases with N as P 1,err = 1 − R(N ) N . In the non-ideal case, optical losses in the system will increase this probability further. The errors associated with Bob's attempt to transmit a logic 0 are of a different nature. In theory, he can always perfectly transmit a logic 0, independent of N ; that is, P 0,err = 0. In practice, however, imperfections in the interferometers will lead to cases in which the photon re-enters Alice's laboratory and she incorrectly records a logic 1. This leads to a counterfactual violation, as the wavefunction "leaks" from Bob's to Alice's laboratory 12 . Although they do not contribute to a counterfactual violation, dark counts in Alice's detector will also increase this error rate.
We can overcome the bit errors by encoding each logical bit into M single photons, at the cost of slightly increasing the CFC violation. If Alice sends M photons into the transmission channel without detecting any at D A , she will record a logic 0. On the other hand, if she detects one or more photons in her laboratory, she will record a logic 1. Assuming messages with an equal number of 0s and 1s, the average bit error probability is given by:
P err (M ) = 1 2 (P 1,err ) M + M P 0,err(1)
where the second term is an approximation of 1 − P M 0 valid for small values of M P 0,err . By increasing M we can thus decrease the contributions of P 1,err exponentially while only increasing those of P 0,err linearly. The counterfactual violation probability for a random bit is given by
P CF C (M ) = 1 2η M P 0,err ,(2)
where η is the detector efficiency. We can thus find an M that minimises the average bit error, while also maintaining a low counterfactual violation probability. In our experiment this expression slightly overestimates the violation probability, as it includes the detector dark counts. As illustrated in Figure 2, we implement a series of cascaded MZIs using a PNP. At the intersections of each of the modes shown in the figure there are smaller MZIs that act as beamsplitters with tunable reflectivities and phases. This allows us to vary our CFC protocol using two to six beamsplitters. Additionally, the high interferometric visibility of the PNP, which we measure to be 99.94 % on average, allows us to keep the rate of counterfactual violations low. The single photons are generated in a spontaneous parametric down conversion process and detected using superconducting nanowire single-photon detectors with detection efficiencies η ∼ 90 % (see Appendix).
To study the performance of this CFC protocol we measure the average bit error, as a function of the number of photons in which the bit is encoded, M, for five different values of N number of BSs. For the logic 0, we configure the MZIs in Bob's laboratory as mirrors (see Figure 2), while for the logic 1 we let the MZIs in Bob's laboratory act as SWAP gates, routing the light out of the interferometer chain. Since Alice cannot access detector D B , she assumes that a photon is injected in the transmission channel every time she detects a heralding photon in D H . We thus run the measurement until we have M recorded single-photon events in D H (typical rates were 1.1 MHz) and look for the coincidences that these events have with D A within a set coincidence window ∆τ = 2.5 ns that is shorter than the pulse separation. Our heralding efficiency was ∼ 3 % through the PNP. Figure 3a shows the experimental average error probability of our CFC protocol as a function of M for different N . We also include a theoretical calculation of the expected error probabilities, which considers the heralding efficiency of the single photons and the success probability of the interferometer that is in good agreement with the experimental data. Note that these are not fits to the data, but rather models with no free parameters. As theoretically predicted, the error rate of the logic 1 decreases exponentially with increasing M and the error rate of the {10, 50, 320, 500}. The white and black pixels are defined to correspond to logic 1 and logic 0, respectively. The success probability increases with increasing M , reaching 99 % for M = 320. The CFC violation probability (PCF C ) also increases with increasing M , but it remains as low as 0.6 % for M = 320. Note that this CFC violation comes only from the logic 0 errors, which we can directly measure; the total CFC violation is larger as discussed in the main text. Increasing M beyond 320 increases the success probability at the expense of increasing the CFC violation. As it can be observed, these probabilities are directly related to the transmission fidelity (F ) of the white pixels, which increases with M , and the transmission fidelity of the black pixels, which decreases with M .
logic 0 increases linearly with M . We observe that higher N requires smaller M , and also results in lower bit error probabilities.
The success probability of this CFC scheme is highly sensitive to the fidelity of the interferometers and the overall heralding efficiency, which depends on the singlephoton source and the coupling efficiency throughout the system. Hence, we optimized the setup for the N = 6 case. Figure 3b shows the corresponding error probability of the logic 1 and the logic 0. The inset in Figure 3b shows the average error probability, where we find a minimum of 1.5 % for M = 320, while the average counterfactual violation is kept at 2.4 %. Owing to on-chip backscattering in Bob's laboratory (i.e. imperfect SWAP operations) a small "amount" of wavefunction amplitude leaks back into the transmission line in the 1 bit process. Although these do not all lead to detection events in Alice's laboratory, the sum of their squares provides an upper bound on the probability of a counterfactual violation. We estimate that the probability for a photon to reflect off of a SWAP operation is at most 1 percent. Hence, in our experiment (Fig. 4) with M = 320 and N = 6, the contribution from the logic 1 to a CFC violation is less than 1.1 %. Note that this violation probability decreases with N , even if the errors remain the same.
To demonstrate the performance of the communication protocol we proceed to analyse the quality of a message in the form of a black and white image, sent from Bob to Alice, for N = 6 and M = {10, 50, 320, 500}. We arbitrarily define the white and black pixels of the image as logic 1 and logic 0, respectively. Figure 4 shows the message transmitted from Bob to Alice for different numbers of encoding photons. We de-fine the image fidelity as
F = T i=1 1 + (−1) Ai+Bi 2T (3)
where B i is the bit that Bob intended to send, A i is the bit that Alice recorded, and T is the total number of bits in the image. In this case we define the CFC violation probability as the number of incorrectly transmitted logic 0s (black pixels) over T . The encoding using M = 10 is clearly not enough to overcome the losses of the system. As we increase M , the success probability and legibility of the message increases, reaching 99 % for M = 320, while the CFC violation probability from 0 bit errors remains as low as 0.6 %. If the CFC violation of the 1 bit is accounted for, this value increases to 2.3 %. For larger M s the success probability increases, but so does the CFC violation. Note that these values are lower than the value in Figure 3b due to the unbalanced distribution of black and white pixels in the image. Our high-fidelity implementation of a trace-free counterfactual communication protocol without postselection was enabled by a programmable nano-photonic processor. The high (99.94 %) average visibility of the individual integrated interferometers allowed bit error probabilities as low as 1.5 %, while, at the same time, keeping the probability for the transmission of a single bit to result in a counterfactual violation below 2.4 %. By combining our state-of-the-art photonic technology with a novel theoretical proposal we contradicted a crucial premise of communication theory 1 : that a message is carried by physical particles or waves. In fact, our work shows that "interaction-free non-locality", first described by Elitzur and Vaidman 21 , can be utilised to send information that is not necessarily bound to the trajectory of a wavefunction or to a physical particle. In addition to enabling further high-fidelity demonstrations of counterfactual protocols, our work highlights the important role that technological advancements can play in experimental investigations of fundamentals of quantum mechanics and information theory. We thus anticipate nanophotonic processors, such as ours, to be central to future photonic quantum information experiments all the way from the foundational level to commercialized products. The authors would like to express their gratitude towards J. Zeuner for helpful discussions and T. Rögelsperger for the artistic input. They furthermore thank LioniX International BV for the manufacturing of the interposer waveguides used in the experiment.
APPENDIX
Telecom Photon Source-We use a pulsed Ti:Sapphire laser with a repetition rate of 76 MHz, an average power of 0.2 W, a central wavelength of 782.2 nm, and a pulse duration of 2.1 ps. The repetition rate is doubled via a passive temporal multiplexing stage 27,28 . This beam pumps a periodically-poled KTP crystal phase matched for collinear type-II spontaneous parametric down conversion, generating frequency non-degenerate photon pairs at λ H = 1563 nm, λ T = 1565.8 nm. Registering the shorter wavelength photon at the detector D H heralds the presence of the longer wavelength one, which is sent to the waveguide.
Programmable Nanophotonic Processor-Our cascaded Mach-Zehnder interferometers (MZIs) are implemented in a silicon-on-insulator (SOI) programmable waveguide, developed by the Quantum Photonics Laboratory at the Massachusetts Institute of Technology 16 . The device consists of 88 MZIs, each accompanied by a pair of thermo-optic phase shifters that facilitate full control over the internal and external phases of the MZIs. The phase shifters are controlled by a 240-channel, 16-bit precision voltage supply, allowing for a phase precision higher than 250 µrad. The coupling of the single photons in/out of the chip is performed using two Si 3 N 4 -SiO 2 waveguides manufactured by Lionix International, that adiabatically taper the 10 µm x 10 µm mode from the single mode fiber down to 2 µm x 2 µm, matching the mode field diameter of the programmable waveguide at the input facet. The total insertion loss per facet was measured as low as 3 dB.
Superconducting Nanowire Single-Photon Detectors-The photons are detected using superconducting nanowire single-photon detectors 29,30 . These detectors are produced by photonSpot and are optimized to reach detection efficiencies ∼ 90 % at telecom wavelengths.
FIG. 2 .
2Experimental setup. a. Our experiment is implemented in a programmable nanophotonic processor (PNP), which is composed of 26 interconnected waveguides. The waveguides are coupled by 88 Mach-Zehnder interferometers (MZIs), as indicated by the top-left inset.
FIG. 3 .
3Success probabilities of the CFC communication. The curves are theoretical models of our experiment with no free parameters, and the points are experimental data. a. Measured average bit error (as defined in the main text) of the protocol for different number of beamsplitters (N ) as a function of the number of photons (M ) used to encode each bit. For small M the cos 2N (π/2N ) dependence of the logic 1 error dominates the average error, making the latter decrease with M as expected. As M is increased more, the linearly growing error in the logic 0, caused by imperfect destructive interference in Alice's port (DA), starts to dominate. b. In the N = 6 case, the optimization of the interferometer fidelity and heralding efficiency leads to an average bit error rate of 1.5 % for M = 320, where the average CFC violation probability is 2.4 %.
FIG. 4 .
4Image sent from Bob to Alice by encoding bits in different numbers of single photons M =
ACKNOWLEDGMENTS I.A.C. and T.S. acknowledge support from the University of Vienna via the Vienna Doctoral School. L.A.R. acknowledges support from the Templeton World Charity Foundation (fellowship no. TWCF0194). P.W. acknowledges support from the European Commission through ErBeSta (No. 800942), the Austrian Research Promotion Agency (FFG) through the QuantERA ERA-NET Cofund project HiPhoP, from the Austrian Science Fund (FWF) through CoQuS (W1210-4) and NaMuG (P30067-N36), the U.S. Air Force Office of Scientific Research (FA2386-233 17-1-4011), and Red Bull GmbH. D.R.M.A.S. acknowledges support from the EPSRC, Hitachi Cambridge, Lars Hierta's Memorial Foundation and the Sweden-America Foundation. N.H. was supported in part by the Air Force Research Laboratory RITA program (FA8750-14-2-0120); Research program FA9550-16-1-0391, supervised by Gernot Pomrenke; and D.E. acknowledges partial support from the Office of Naval Research CONQUEST program.
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|
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[
"ON THE USE OF CONFORMAL MAPS FOR THE ACCELERATION OF CONVERGENCE OF THE TRAPEZOIDAL RULE AND SINC NUMERICAL METHODS",
"ON THE USE OF CONFORMAL MAPS FOR THE ACCELERATION OF CONVERGENCE OF THE TRAPEZOIDAL RULE AND SINC NUMERICAL METHODS"
] |
[
"Richard Mikael Slevinsky ",
"Sheehan Olver "
] |
[] |
[] |
We investigate the use of conformal maps for the acceleration of convergence of the trapezoidal rule and Sinc numerical methods. The conformal map is a polynomial adjustment to the sinh map, and allows the treatment of a finite number of singularities in the complex plane. In the case where locations are unknown, the so-called Sinc-Padé approximants are used to provide approximate results. This adaptive method is shown to have almost the same convergence properties. We use the conformal maps to generate high accuracy solutions to several challenging integrals, nonlinear waves, and multidimensional integrals.
|
10.1137/140978363
|
[
"https://arxiv.org/pdf/1406.3320v1.pdf"
] | 45,882,414 |
1406.3320
|
2fd26a01c66b8e75c4f032fdf6925ca18e04bb20
|
ON THE USE OF CONFORMAL MAPS FOR THE ACCELERATION OF CONVERGENCE OF THE TRAPEZOIDAL RULE AND SINC NUMERICAL METHODS
Richard Mikael Slevinsky
Sheehan Olver
ON THE USE OF CONFORMAL MAPS FOR THE ACCELERATION OF CONVERGENCE OF THE TRAPEZOIDAL RULE AND SINC NUMERICAL METHODS
Trapezoidal rule Sinc numerical methods Conformal maps AMS subject classifications 30C3041A3065D3065L10
We investigate the use of conformal maps for the acceleration of convergence of the trapezoidal rule and Sinc numerical methods. The conformal map is a polynomial adjustment to the sinh map, and allows the treatment of a finite number of singularities in the complex plane. In the case where locations are unknown, the so-called Sinc-Padé approximants are used to provide approximate results. This adaptive method is shown to have almost the same convergence properties. We use the conformal maps to generate high accuracy solutions to several challenging integrals, nonlinear waves, and multidimensional integrals.
1. Introduction. The trapezoidal rule is one of the most well-known methods in numerical integration. While the composite rule has geometric convergence for periodic functions, in other cases it has been used as the starting point of effective methods, such as Richardson extrapolation [36] and Romberg integration [37]. The geometric convergence breaks down with endpoint singularities, and this issue inspired a different approach to improve on the composite rule. From the Euler-Maclaurin summation formula, it was noted that some form of exponential convergence can be obtained for integrands which vanish at the endpoints, suggesting that undergoing a variable transformation may well induce this convergence [38,40,48]. After this observation, the race was on to determine exactly which variable transformation, and therefore which decay rate, is optimal. Numerical experiments showed the exceptional promise of rules such as the tanh substitution [9], the erf substitution [49], the IMT rule [19], and the tanh-sinh substitution [50], among others [22]. But exactly which one is optimal, and in which setting?
Using a functional analysis approach, this question was beautifully answered by establishing the optimality of a double exponential endpoint decay rate for the trapezoidal rule on the real line for approximating analytic integrands [44]. The domain of analyticity is described in terms of a strip of maximal width π centred on the real axis in the complex plane. This optimality also prescribed the optimal step size and a near-linear convergence rate O(e −kN/ log N ), where N is the number of sample points and k is a constant proportional to the strip width.
The results allowed for displays of strong performance for integrals with integrable endpoint singularities without changing the rule in any way [23][24][25]47]. The double exponential transformation was also adapted to Fourier and general oscillatory integrals in [34,35]. Recognizing the trapezoidal rule as the integration of a Sinc expansion of the integrand, the double exponential advocates adapted their analysis to Sinc approximations [46], and also to all the numerical methods therewith derived, such as Sinc-Galerkin and Sinc-collocation methods [17,28,45] for initial and boundary value problems, Sinc indefinite integration [52], iterated integration [27], and Sinccollocation for integral equations [26,29], all obtaining the near-linear convergence rate O(e −kN/ log N ). More recently, the researchers then focused on improving the original convergence results by developing more precise upper estimates on the error given bounds on the function [30,31,51,53,54].
In this work, we report an improvement of the trapezoidal rule in the context of a finite number of singularities -of any kind -near the contour of integration. This problem has been considered before, in Gaussian quadrature and in Sinc quadrature [7,11,21,39]. The prevailing philosophy seems to be to characterize singularities as specifically as possible, then account for them by either adding terms from Cauchy's residue theorem to the approximation or by modifying the weights and abscissas.
While the examples and applications show exceptional performance of the algorithms, the case of general singularities is still untreated. In the optimal double exponential framework, singularities near the integration contour may reduce the width of the strip of analyticity about the real axis. As the double exponential decay rate is typically induced by a variable transformation, we seek to find variable transformations which place the threatening singularities on the upper and lower edges of the maximal strip of width π. In this work, such variable transformations are conformal maps which maximize the convergence rate despite the presence of the singularities.
The idea of using conformal maps to speed up numerical computations is not new. In fact, it was recently pioneered by Tee and Trefethen [55], Hale and Trefethen [15], Hale and Tee [14], and Hale's so-titled Ph. D. thesis [13]. Inspired by this work, we investigate the use of the Schwarz-Christoffel map from the strip of width π to a polygonally bounded region, with the possibility of some sides being infinite. In this way, the edges of the strip of width π contain the pre-images of the function's singularities, their possible branches, and other objects limiting analyticity. In the case dealt with in [14], an algorithm is constructed to solve the Schwarz-Christoffel parameter problem, and the integral definition of the Schwarz-Christoffel map is simplified by partial fraction decomposition.
For the strip map variation of the Schwarz-Christoffel map, the explicit integration of the Schwarz-Christoffel map could not be done [14], so in this work, an approximate map is constructed based on polynomial adjustments to the sinh map. We choose the sinh map because it appears in every double exponential map of the canonical finite, infinite, and semi-infinite domains. The polynomial adjustments add exactly enough parameters to locate a finite number of pre-images of singularities on the edges of the maximal strip of width π, and the parameter problem -the determination of such polynomials adjustments -is in complete analogy with the Schwarz-Christoffel parameter problem. However, the approximate map is significantly less expensive to evaluate, and therefore well-paired with the double exponential transformation for high precision numerical experiments.
The problem of poles limiting analyticity has been noted in [47], where Sugihara and Matsuo show the double exponential Sinc expansion of the function:
f (x) = x (1 − x) e −x (1/2) 2 + (x − 1/2) 2 , x ∈ [0, 1],(1.1)
is not as efficient as methods of polynomial interpolation. To demonstrate how simple our nonlinear program can be, we note that while the original double exponential transformation for the problem is φ(t) = 1 2 tanh( π 2 sinh t) + 1 2 , the optimized map is φ(t) = 1 2 tanh( π 4 sinh t) + 1 2 , and the convergence rate is approximately tripled.
We demonstrate the merits of our algorithm on four integrals, each with its own combination of singularities. In these cases, the algorithm obtains approximately 2.5-4 times as many correct digits as a naïve double exponential transformation for the same number of function evaluations. The algorithm is applied to obtain solutions of the forced Benjamin-Ono equation describing nonlinear waves. Then, the algorithm also shows its merit in the evaluation of multidimensional integrals.
High accuracy in scientific computing is important in many applications. As an example, high accuracy results are required to confidently obtain results from an integer relation algorithm such as PSLQ [10] for finding closed forms for integrals. As well, it is also important to have a rapidly convergent algorithm so that problems are solved in acceptable computational times.
2. Quadrature and Sinc methods by variable transformation. Using a variable transformation to induce exponential decay at the endpoints is first performed in [49] and it is extended to double exponential decay in [50]. They find that variable transformations that induce double exponential decay at the endpoints perform better than single exponential transformations.
For infinite and semi-infinite integrals with or without pre-existing exponential decay, various other transformations have been proposed to induce double exponential decay. Examples from [54] are included in Table 2.1.
Interval
Single Exponential Double Exponential
[−1, 1] tanh(t/2) tanh( π 2 sinh t) (−∞, +∞) sinh(t) sinh( π 2 sinh t) [0, +∞) log(exp(t) + 1) log(exp( π 2 sinh t) + 1) [0, +∞) exp(t) exp( π 2 sinh t)
We follow closely the rigorous derivations in [44,46] of the optimality of the trapezoidal rule and Sinc numerical methods for functions with double exponential decay as x → ±∞. Let d be a positive number and let D d denote the strip region of width 2d about the real axis:
D d = {z ∈ C : |Im z| < d}.
(2.1)
Let ω(z) be a non-vanishing function defined on the region D d , and define the Hardy
space H ∞ (D d , ω) by: H ∞ (D d , ω) = {f : D d → C| f (z) is analytic in D d , and ||f || < +∞},(2.2)
where the norm of f is given by:
||f || = sup z∈D d f (z) ω(z) . (2.3)
Consequentially, for ω(z) that decays double exponentially, the functions in H ∞ (D d , ω) decay double exponentially as well. Let us consider the N (= 2n+1)-point trapezoidal rule for the interval (−∞, +∞):
+∞ −∞ f (x) dx ≈ h +n k=−n f (k h),(2.4)
where the mesh size h is suitably chosen for a given positive integer n. For the trapezoidal rule, let E T N,h (H ∞ (D d , ω)) denote the error norm in H ∞ (D d , ω):
E T N,h (H ∞ (D d , ω)) = sup ||f ||≤1 +∞ −∞ f (x) dx − h +n k=−n f (k h) . (2.5)
Let B(D d ), originally introduced in [40], denote the family of all functions f analytic in D d such that:
N 1 (f, D d ) = ∂D d |f (z)| dz < +∞. (2.6)
Theorem 2.1 (Sugihara [44]). Suppose that the function ω(z) satisfies the following three conditions:
1. ω(z) ∈ B(D d ); 2. ω(z) does not vanish at any point in D d and takes real values on the real axis; 3. the decay rate of ω(z) on the real axis is specified by:
α 1 exp (−(β|x| ρ )) ≤ |ω(x)| ≤ α 2 exp (−(β|x| ρ )) , x ∈ R, (2.7)
where α 1 , α 2 , β > 0 and ρ ≥ 1. Then:
E T N,h (H ∞ (D d , ω)) ≤ C d,ω exp −(πdβN ) ρ ρ+1
, (2.8) where N = 2n + 1, the mesh size h is chosen optimally as:
h = (2πd) 1 ρ+1 (βn) − ρ ρ+1 ,(2.
9)
and C d,ω is a constant depending on d and ω. Theorem 2.2 (Sugihara [44]). Suppose that the function ω(z) satisfies the following three conditions:
1. ω(z) ∈ B(D d ); 2. ω(z) does not vanish at any point in D d and takes real values on the real axis; 3. the decay rate of ω(z) on the real axis is specified by:
α 1 exp(−β 1 e γ|x| ) ≤ |ω(x)| ≤ α 2 exp(−β 2 e γ|x| ), x ∈ R, (2.10)
where α 1 , α 2 , β 1 , β 2 , γ > 0. Then:
E T N,h (H ∞ (D d , ω)) ≤ C d,ω exp − πdγN log(πdγN/β 2 )
, (2.11) where N = 2n + 1, the mesh size h is chosen optimally as:
h = log(2πdγn/β 2 ) γn , (2.12)
and C d,ω is a constant depending on d and ω.
Since the trapezoidal rule is equivalent to the integration of the Sinc expansion of a function [41], the entire process of analyzing the convergence rates with different endpoint decay can also be useful for the Sinc expansion of a function, with subtle differences that arise in the mesh size and convergence rates. Let us consider the N (= 2n + 1)-point Sinc approximation of a function on the real line:
f (x) ≈ +n j=−n f (j h)S(j, h)(x),(2.13)
where S(j, h)(x) is the so-called Sinc function:
S(j, h)(x) = sin[π(x/h − j)] π(x/h − j) ,(2.14)
and where the step size h is suitably chosen for a given positive integer n. From l'Hôpital's rule, it can easily be seen that the Sinc functions are mutually orthogonal at the so-called Sinc points x k = k h:
S(j, h)(k h) = δ k,j ,(2.15)
where δ k,j is the Kronecker delta [1]. For the Sinc approximation, let E Sinc
N,h (H ∞ (D d , ω)) denote the error norm in H ∞ (D d , ω): E Sinc N,h (H ∞ (D d , ω)) = sup ||f ||≤1 sup x∈R f (x) − +n j=−n f (j h) S(j, h)(x) .
(2.16) Theorem 2.3 (Sugihara [46]). Suppose that the function ω(z) satisfies the following three conditions:
1. ω(z) ∈ B(D d ); 2. ω(z) does not vanish at any point in D d and takes real values on the real axis; 3. the decay rate of ω(z) on the real axis is specified by:
α 1 exp (−(β|x| ρ )) ≤ |ω(x)| ≤ α 2 exp (−(β|x| ρ )) , x ∈ R, (2.17)
where α 1 , α 2 , β > 0 and ρ ≥ 1. Then:
E Sinc N,h (H ∞ (D d , ω)) ≤ C d,ω N 1 ρ+1 exp − πdβN 2 ρ ρ+1
, (2.18) where N = 2n + 1, the mesh size h is chosen optimally as:
h = (πd) 1 ρ+1 (βn) − ρ ρ+1 ,(2.19)
and C d,ω is a constant depending on d and ω. Theorem 2.4 (Sugihara [46]). Suppose that the function ω(z) satisfies the following three conditions:
1. ω(z) ∈ B(D d ); 2. ω(z) does not vanish at any point in D d and takes real values on the real axis; 3. the decay rate of ω(z) on the real axis is specified by: 20) where α 1 , α 2 , β 1 , β 2 , γ > 0.
α 1 exp(−β 1 e γ|x| ) ≤ |ω(x)| ≤ α 2 exp(−β 2 e γ|x| ), x ∈ R,(2.
Then:
E Sinc N,h (H ∞ (D d , ω)) ≤ C d,ω exp − πdγN 2 log(πdγN/(2β 2 ))
, (2.21) where N = 2n + 1, the mesh size h is chosen optimally as:
h = log(πdγn/β 2 ) γn , (2.22)
and C d,ω is a constant depending on d and ω.
Last but not least, there is the nonexistence theorem, which provides a fundamental bound for the proposed optimization approach.
Theorem 2.5 (Sugihara [44]). There exists no function ω(z) that satisfies the following three conditions:
1. ω(z) ∈ B(D d ); 2. ω(z)
does not vanish at any point in D d and takes real values on the real axis; 3. the decay rate on the real axis of ω(z) is specified as:
ω(x) = O exp(−βe γ|x| ) , as |x| → ∞, (2.23)
where β > 0, and dγ > π/2.
3. Maximizing the convergence rates. From the previous theorems 2.2 and 2.4 on the convergence rates of the trapezoidal rule with a prescribed decay at the endpoints and the nonexistence theorem 2.5 of analytic functions with double exponential decay in too wide a strip, we may ask the following question. How can we use a conformal map φ to maximize the convergence rate of the trapezoidal rule:
∞ −∞ f (φ(t))φ (t) dt ≈ h +n k=−n f (φ(k h))φ (k h), (3.1)
or the Sinc approximation:
f (x) ≈ +n j=−n f (φ(j h))S(j, h)(φ −1 (x)),(3.2)
despite the singularities of f ∈ C which limit its domain of analyticity? To formulate this problem mathematically, let Φ ad be the admissible space of all functions φ satisfying the conditions of theorems 2.2 and 2.4:
Φ ad = φ : f (φ(·))φ (·) ∈ H ∞ (D d , ω) for some d > 0,
and for some ω such that:
1. ω(z) ∈ B(D d ); 2. ω(z) does not vanish at any point in D d
and takes real values on the real axis;
3. α 1 exp −β 1 e γ|x| ≤ |ω(x)| ≤ α 2 exp −β 2 e γ|x| , x ∈ R, where α 1 , α 2 , β 1 , β 2 , γ > 0.
We wish to find the φ ∈ Φ ad so that the convergence rates are maximized:
argmax φ∈Φ ad πdγN log(πdγN/β 2 ) Convergence Theorem 2.2 subject to dγ ≤ π 2 Nonexistence Theorem 2.5 argmax φ∈Φ ad πdγN 2 log(πdγN/(2β 2 )) Convergence Theorem 2.4 subject to dγ ≤ π 2
Nonexistence Theorem 2.5
As infinite-dimensional optimization problems for φ, these are challenging problems. However, the convergence rates of theorems 2.2 and 2.4 are asymptotic ones and therefore it is of equivalent interest to investigate the asymptotic solutions to the problem. Consider the asymptotic problems:
πdγN log(πdγN/β 2 ) = πdγN log N + log(πdγ/β 2 ) , ∼ πdγN log N , as N → ∞, (3.3) πdγN 2 log(πdγN/(2β 2 )) = πdγN 2 log N + 2 log(πdγ/(2β 2 )) , ∼ πdγN 2 log N , as N → ∞. (3.4)
Then, the linear appearance of dγ leads directly to the following result. Theorem 3.1. Let Φ as,ad = {Φ ad : dγ = π/2} be the asymptotically admissible subspace of the admissible space Φ ad . Then for every φ as ∈ Φ as,ad :
E T N,h (H ∞ (D d , ω)) ≤ C d,ω exp − π 2 N 2 log(π 2 N/2β 2 )
, (3.5) where N = 2n + 1, the mesh size h is chosen optimally as:
h = log(π 2 n/β 2 ) γn , (3.6)
and C d,ω is a constant depending on d and ω. This same φ as ensures that:
E Sinc N,h (H ∞ (D d , ω)) ≤ C d,ω exp − π 2 N 4 log(π 2 N/4β 2 ) ,(3.
7)
where N = 2n + 1, the mesh size h is chosen optimally as:
h = log(π 2 n/2β 2 ) γn , (3.8)
and C d,ω is a constant depending on d and ω.
The implication of such a theorem is that suitable mappings φ can be found which maximize the convergence rates by neutering the terrible effects of singularities near the approximation interval. In this section, we find such mappings by starting with the observation that in all of the maps in Table 2.1 for the finite, semi-infinite, and infinite canonical domains, an elementary map is composed with the sinh map. Therefore, it seems as though studying the sinh map, or some modification thereof, will be the best place to start.
Let f have a finite number of singularities located at the points {δ k ± i k } n k=1 . The four maps in Table 2.1 can be written as the composition of:
ψ(z) = tanh(z), ψ −1 (z) = tanh −1 (z), (3.9) ψ(z) = sinh(z), ψ −1 (z) = sinh −1 (z), (3.10) ψ(z) = log(e z + 1), ψ −1 (z) = log(e z − 1), (3.11) ψ(z) = exp(z), ψ −1 (z) = log(z). (3.12)
with the π 2 sinh function. In any of these cases, let a finite number of singularities of f be transformed as
{δ k ± i˜ k } n k=1 as the ordered set of {ψ −1 (δ k ± i k )} n k=1 , wherẽ δ 1 <δ 2 < · · · <δ n .
The sinh function is a conformal map from the strip D π 2 to the entire complex plane with two branch cuts emanating outward from the points ±i. It is actually the most rudimentary Schwarz-Christoffel formula mapping from the strip D π 2 to the entire complex plane [18] with those two aforementioned branches. Let g map the strip D π 2 to the polygonally bounded region P having vertices {w k } n k=1 = {δ 1 + i˜ 1 , . . . ,δ n + i˜ n } and interior angles {πα k } n k=1 . Let also π 2 α ± be the divergence angles at the left and right ends of the strip D π 2 . Then the function:
g(z) = A + C z e (α−−α+)ζ n k=1 [sinh(ζ − z k )] α k −1 dζ,(3.13)
where z k = g(w k ) and for some A and C maps the interior of the top half of the strip D π 2 to the interior of the polygon P . The solution of the constants A, C, and {z k } n k=1 is known as the Schwarz-Christoffel parameter problem. Figure 3.1 shows an example of the Schwarz-Christoffel map for the polygonal restrictions on C due to the possible singularities of f . The Schwarz-Christoffel map is actually the exact solution of the problem of maximizing the convergence rates, as it maps points on the top and bottom of the strip D π 2 to the singularities. However, the entire process is computationally intensive. Firstly, the nonlinear system of equations of the Schwarz-Christoffel parameter problem needs to be solved, and secondly, the map is defined as an integral. The parameter problem can be prohibitive to solve, requiring thousands of integrations of the map function. Also, the integral only has an analytical expression for a polygon with one finite vertex, and this gives the sinh map. The Schwarz-Christoffel Toolbox in MATLAB [8,56,57] is used to solve for the maps in Figure 3.1, and provides a precision of approximately 10 −8 for a computation time on the order of one minute. In Figure 3.1 and subsequent figures, the plots show the mapping of lines with constant imaginary values between −iπ/2 and +iπ/2 via the conformal map from the strip, then the composition of this conformal map with one of the maps ψ(z) of (3.9)-(3.12). Were it only for the difficulties posed by the Schwarz-Christoffel parameter problem, this approach may have some promise. However, the major problem is that even after the parameter problem is solved, the map itself is defined as an integral and requires a large computational effort compared to the following proposed approach. with singularities located on the boundary, in (b) the resulting Schwarz-Christoffel map, and in (c) a tanh map of the Schwarz-Christoffel map. In all three cases, the crosses track the singularities δ 1 ± i 1 = −1/2 ± i and δ 2 ± i 2 = 1/2 ± i/2. For the sake of comparison, an integral with these singularities is treated in Example 4.1.
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Fortunately, due to the framework of the double exponential transformation, we can make a polynomial adjustment to the sinh map while still retaining a variable transformation φ which induces double exponential decay. For any real values of the n + 1 parameters {u k } n k=0 , the function:
h(t) = u 0 sinh(t) + n j=1 u j t j−1 , u 0 > 0, (3.14)
still grows single exponentially. Therefore, the composition ψ(h(t)) for any ψ in (3.9)-(3.12) still induces a double exponential variable transformation. The benefit of choosing such functions is that we now have sufficient parameters which we can use to ensure the pre-images of the singularities {δ k ± i˜ k } n k=1 reside on the top and bottom edges of the strip D π 2 , respectively. This is done by solving the system of equations:
h(x k + iπ/2) =δ k + i˜ k , for k = 1, . . . , n. (3.15)
This is a system of n complex equations for the 2n + 1 unknowns {u k } n k=0 and the x-coordinates of the pre-images of the singularities {x k } n k=1 . Since there is one more unknown than equations, we are able to maximize the value of u 0 , which is proportional to β 2 in every case. Summing all n equations of (3.15) leads to the nonlinear program:
maximize u 0 = n k=1 ˜ k − n j=1 u j (x k + iπ/2) j−1 n k=1 cosh(x k ) , subject to h(x k + iπ/2) =δ k + i˜ k , for k = 1, . . . , n.
(3.16)
Because the maximization condition is obtained by summing the constraint equations, we have one additional degree of freedom in the program (3.16). In order to save from premature convergence, we impose ad hoc the condition:
x 1 = 0, for n = 1, |x 1 + x n | ≤x, for n ≥ 2,(3.17)
wherex is a parameter which ensures the singularities stay reasonably close to the origin. In all our examples, we setx = 20 which is sufficient. This nonlinear program is in close analogy to the Schwarz-Christoffel parameter problem. However, this method has many advantages over the Schwarz-Christoffel formula. Firstly, the map h(t) is defined in terms of elementary functions and not as an integral. Secondly, the accuracy of the values of {u k } n k=0 does not need to equal the accuracy required of the map, allowing the map (3.14) to be evaluated in arbitrary precision. One disadvantage of this method is that the map is an approximate solution to the original problem. Therefore, while the strip width will indeed be 2 d = π, we can expect a smaller than optimal β 2 . Nevertheless, given that β 2 only has a secondary effect on the convergence rates, according to theorem 3.1, this is a small price to pay to obtain a solution method that emulates the Schwarz-Christoffel formula while being amenable to arbitrary precision calculations.
A nonlinear program without any a priori information on the solution requires an iterative method for solving the parameter problem (3.16). An iterative method also requires a close initial guess to converge to the solution. To obtain an initial guess, we let¯ be the smallest of {˜ k } n k=1 andδ be theδ k of the same index. Then the nonlinear program with the singularities {δ + i˜ k } n k=1 is exactly solved by:
h(t) =¯ sinh t +δ. (3.18)
A homotopy H (t) is then constructed between the solution with singularities {δ + i˜ k } n k=1 at t = 0 and the solution with singularities {δ k + i˜ k } n k=1 at t = 1. The interval t ∈ [0, 1] is discretized, and the nonlinear program is solved with singularities that vary linearly between the two problems and initial guesses from the solution of the previous iterate. In practice, the locations of a function's singularities may not be known in advance. This can result from either incomplete theoretical information, or a non-local nature of singularities, such as branch cuts or "numerically singular" terms such as the error function, which while entire, is unbounded off the real axis [14]. In [55], an adaptive approach is taken to approximating the nearby singularities, whereby the interpolatory Chebyshev-Padé approximants are constructed, and approximants' poles are taken as the loci of the singularities of the underlying function. The map is then modified to exclude these points, and the iteration of this process is the adaptive algorithm.
Because we are working with Sinc approximations, and Sinc points, we modify their algorithm to make efficient use of the information we have at hand, i.e. the Sinc sampling of the function.
Definition 3.2. Let x k = k h be the Sinc points and let f (x k ) be the N (= 2n+1) Sinc sampling of f . Then for r + s ≤ 2n, the Sinc-Padé approximants {r/s} f (x) are given by:
{r/s} f (x) = r i=0 p i x i 1 + s j=1 q j x j ,(3.
19)
where the r + s + 1 coefficients solve the system: . For the Chebyshev-Padé approximants, the inverse cosine distribution of sample points leads to a stable linear system and the degrees of the numerator and denominator can add to equate the number of collocation points. For the Sinc-Padé approximants, double exponential growth of the sample points renders the system highly ill-conditioned. Therefore, these indices must be decoupled from n and the function must only be sampled near the centre. Our adaptive algorithm is based on the following principles:
r i=0 p i x i k − f (x k ) s j=1 q j x j k = f (x k ),(3.
1. Sinc-Padé approximants are useful only when the Sinc approximation obtains some degree of accuracy, 2. Sinc-Padé approximants are useful for r, s = O(log n) as n → ∞. The first principle follows from observations of our numerical experiments, and we found that a relative error of approximately 10 −3 in the Sinc approximation allows for a useful Sinc-Padé approximant. The second principle follows from the observation that we need not identify many singularities to remove, and that even at a logarithmic increase, the sample points tend to infinity at a single exponential rate, implying that they will ultimately cover the real line. These principles form the basis of the following algorithm.
Examples.
In this section, we will use the proposed nonlinear program (3.16) to maximize the convergence rate of the double exponential transformation. We compare the results of the trapezoidal rule with single, double, and optimized double exponential variable transformations on three integrals using arbitrary precision arithmetic. On a fourth integral, we use the adaptive algorithm 3.3 to approximate nearby singularities.
4.1. Example: endpoint and complex singularities. We wish to evaluate the integral:
1 −1 exp ( 2 1 + (x − δ 1 ) 2 ) −1 log(1 − x) ( 2 2 + (x − δ 2 ) 2 ) √ 1 + x dx = −2.04645 . . . ,(4.1)
for the values δ 1 +i 1 = −1/2+i and δ 2 +i 2 = 1/2+i/2. This integral has two different endpoint singularities and two pairs of complex conjugate singularities of different types near the integration axis.
Single Double Optimized Double φ(t) tanh(t/2) tanh π 2 sinh(t) tanh(h(t)) ρ or γ 1 1 1 β or β 2 1/2 π/4 0.06956 d 1.10715 0.34695 π/2
In addition, the optimized transformation is given by: In Figure 4.2 (a), the integrand of (4.1) is shown, and in Figure 4.2 (b), the logarithm of the relative errors of the trapezoidal rule of order n with single, double, and optimized double exponential variable transformations are plotted. The increase in convergence rate using the optimized variable transformation is a significant increase in efficiency over the double exponential transformation.
h(t) ≈
Example
: eight different complex conjugate singularities. We wish to evaluate the integral:
+∞ −∞ exp 10( 2 1 + (x − δ 1 ) 2 ) −1 cos 10( 2 2 + (x − δ 2 ) 2 ) −1 ( 2 3 + (x − δ 3 ) 2 ) 2 4 + (x − δ 4 ) 2 dx = 15.01336 . . . , (4.3)
for the values δ 1 +i 1 = −2+i, δ 2 +i 2 = −1+i/2, δ 3 +i 3 = 1+i/4, and δ 4 +i 4 = 2+i.
Single
Double Optimized Double φ(t) sinh(t) sinh π 2 sinh(t) sinh(h(t)) ρ or γ 1 1 1 β or β 2 2 π/2 5.7715 × 10 −6 d 0.35260 0.22640 π/2
In addition, the optimized transformation is given by:
h(t) ≈ 5.7715 ×
Example: for Goursat's infinite integral.
We wish to evaluate the integral:
+∞ 0 x dx 1 + x 6 sinh 2 x = 0.50368 . . . , (4.5)
which is evaluated in [16,33] as part of a high precision numerical evaluation of Goursat's infinite integral. While there are an infinite number of poles in the complex plane due to the sinh function, a four-parameter solution h(t) can be found using the nearest poles, while excluding the remainder. This shows the incredible versatility of the proposed optimization approach, because the same optimal asymptotic convergence rate is obtained in the complicated situation of an infinite number of singularities while not leading to a more complicated solution process. Table 4.3 summarizes the variable transformations used and the parameters in the theorems 2.1 and 2.2. For the sake of comparison, we use the same double exponential transformation used in [16]. In addition, the optimized transformation is given by: In Figure 4.6 (a), the integrand of (4.5) is shown, and in Figure 4.6 (b), the logarithm of the relative errors of the trapezoidal rule of order n with single, double, and optimized double exponential variable transformations are plotted. The increase in convergence rate using the optimized variable transformation is a significant increase in efficiency over the double exponential transformation.
h(t) ≈ 0.26725 sinh(t) + 0.30707 + 0.20337 t − 0.031966 t 2 .
In [16], the authors obtained the relative error of 10 −72 with 480 function evaluations, and claim this to be a nearly minimal number. However, as can be seen in Figure 4.6 (b), the optimized double exponential transformation can obtain the same relative error with only approximately 140 function evaluations. Neither of these results, however, compares to the million-digit algorithm of [33] using the Hilbert transform to construct a conjugate function, thereby removing all the singularities.
Example: adaptive optimization via Sinc-Padé approximants.
In this example, we will show the performance of the adaptive optimized method using the Sinc-Padé approximants to approximately locate the singularities. We wish to evaluate the integral:
∞ 0 x dx 2 1 + (x − δ 1 ) 2 ( 2 2 + (x − δ 2 ) 2 )( 2 3 + (x − δ 3 ) 2 ) = 12.55613 . . . ,(4.7)
for the values δ 1 + i 1 = 1 + i, δ 2 + i 2 = 2 + i/2, and δ 3 + i 3 = 3 + i/3.
Single
Double Table 4.5 shows the evolution of the six nearest roots of the Sinc-Padé approximants to the integration contour. The degrees of the Sinc-Padé approximants increase as r = log 2 (n) − 2 and s = log 2 (n) + 2. Table 4.5 Evolution of the six nearest roots of the Sinc-Padé approximants.
Optimized Double φ(t) exp(t) exp π 2 sinh(t) exp(h(t)) ρ or γ 1 1 1 β or β 2 2 π/2 9.4353 × 10 −3 d 0.11066 0.05762 π/2n δ 1 ± i 1 δ 2 ± i 2 δ 3 ± i 3 2 5 −0.58097 ± 1.2106i 2.0717 ± 0.28089i 3.0298 ± 0.45170i 2 6 −0.18822 ± 1.3571i 2.0008 ± 0.49777i 3.0004 ± 0.33311i 2 7 0.14091 ± 1.3982i 1.9963 ± 0.48734i 3.0009 ± 0.33279i 2 8 0.41762 ± 1.3767i 2.0498 ± 0.39481i 3.0022 ± 0.35279i
Exact 1.0000 ± 1.0000i 2.0000 ± 0.50000i 3.0000 ± 0.33333i Table 4.6 shows the evolution of the adaptive map. The coefficients of the optimized map are also shown for comparison. Figure 4.7 shows the three stages of the adaptive double exponential map. In Figure 4.8 (a), the integrand of (4.7) is shown, and in Figure 4.8 (b), the logarithm of the relative errors of the trapezoidal rule of order n with single, double, and optimized double exponential variable transformations are plotted. The increase in convergence rate using the optimized variable transformation is a significant increase in efficiency over the double exponential transformation.
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Applications.
Nonlinear waves.
For internal waves in stratified fluids of great depth, the Benjamin-Ono equation [5,32] is a nonlinear partial integro-differential equation involving the Hilbert transform. While the KdV equation has soliton solutions that decay exponentially on the real line, the soliton solutions to the homogeneous Benjamin-Ono equation decay algebraically. The Hilbert transform on the real line is defined as [20]:
Hy(x) = 1 π − +∞ −∞ y(s) s − x ds,(5.1)
where the dash in the integral sign denotes the Cauchy principal value. The forced Benjamin-Ono equation can then be written as:
y t + yy x + Hy xx = g(x − c t), x ∈ R, t ≥ 0, lim x→±∞ y(x, t) = 0. (5.2)
for some g with wave speed c ∈ R. If we let y(x, t) = y(x − c t), then we may consider the traveling wave solutions. Inserting such a substitution into (5.2) and integrating, we obtain: The traveling wave is then obtained by solving this equation for t = 0.
−c y + yy + Hy = g(x − c t),(5.
The forced solutions to this equation have many properties that can be deduced from f . For example, if f decays algebraically on the real line, the method of dominant balance shows that:
y(x) ∼ −c −1 f (x)
, as x → ±∞, (5.5) and will therefore behave similarly. As well, complex singularities in f can potentially be found in y.
To continue, it is clear we require an approximation for the Hilbert transform. In [42,43], Stenger derives a Sinc-based approximation for the Hilbert transform. We closely follow his development which is for a general variable transformation, and show how the optimized conformal map improves the convergence rate.
By making the invertible variable transformation s = φ(t) : R → R in (5.1), we obtain:
Hy(x) = 1 π − +∞ −∞ y(φ(t))φ (t) φ(t) − x dt. (5.6)
The integrand multiplied by t − φ −1 (x) has but a removable singularity at t = φ −1 (x). Therefore, we may approximate with a Sinc basis:
y(φ(t))φ (t) φ(t) − x (t − φ −1 (x)) ≈ +n j=−n y(φ(jh))φ (jh) φ(jh) − x (jh − φ −1 (x))S(j, h)(t). (5.7)
Using the Hilbert transform [20]:
H sin x x = 1 π − +∞ −∞ sin s s(s − x) ds = cos x − 1 x ,(5.8)
and dividing by the linear factor t − φ −1 (x) and integrating each basis function, the approximation (5.7) leads directly to:
Hy(x) ≈ h π +n j=−n y(φ(jh))φ (jh) cos π( φ −1 (x) h − j) − 1 x − φ(jh) . (5.9)
Using the Sinc-based approximation for the Hilbert transform, the solution of the forced Benjamin-Ono equation can be obtained by approximating the solution y(x) in a Sinc basis:
y(x) ≈ y n (x; φ) = +n j=−n y j S(j, h)(φ −1 (x)),(5.y(x) = m i=1 2 i ((x − δ i ) 2 + 2 i ) ,(5.12)
for the values m = 3, δ 1 + i 1 = −1 + 0.3i, δ 2 + i 2 = 0 + 0.1i, and δ 3 + i 3 = 1 + 0.2i. This allows for an exact comparison with an analytic expression. In addition, the wave speed c = 1 is used.
Single
Double Optimized Double φ(t) sinh(t) sinh(π/2 sinh(t)) sinh(h(t)) ρ or γ 1 1 1 β or β 2 1/2 π/4 5.7257 × 10 −8 d 0.10017 0.06381 π/2
In addition, the optimized transformation is given by:
h(t) ≈ 1.1451 × 10 −7 sinh(t) + 0.04531 + 0.06359 t − 1.2134 × 10 −4 t 2 .
(5.13)
In Figure 5.1, we approximate the relative error:
sup x∈R y(x) − y n (x; φ) y(x) ,(5.14)
by computing the maximum of the difference and quotient at 101 equally spaced points in the interval [−5, 5]. The inverse optimized map φ −1 is conveniently computed via Newton iteration, as the map and its first derivative are already required in the system of collocated equations. The increase in convergence rate using the optimized variable transformation is a significant increase in efficiency over the double exponential transformation.
Multidimensional integrals.
There are many applications in physics that warrant the evaluation of m-dimensional integrals. Examples we are interested in include: magnetic susceptibility integrals in the Ising theory of solid-state physics [2], which form terms in a series that represents the dependence of magnetic susceptibility on temperature; and, box integrals [3], which are essentially m-dimensional expectations of the s th power of the distance in a unit hypercube | r| s r∈[0,1] m . Such integrals have applications in probability theory, and in potential theory such as "jellium" potentials. After making substantial analytic advances in the theory of box integrals in [4], the authors acknowledge that some of the analytical techniques they used would not apply to other m-dimensional expectations, and posit that e −κ| r| r∈[0,1] m should remain extremely difficult to evaluate in any general way. In this section, we construct such a general way to calculate these integrals. Explicitly:
e −γ(m, a) = a m 1 0 x m−1 e −a x dx = (m − 1)! − e −a m−1 j=0 m − 1 j j! a j−m+1 ,(5.16)
we obtain:
e −κ| r| r∈[0,1] m = m 2 m−1 [−1,1] m−1 dx 1 · · · dx m−1 (m − 1)! κ m (x 2 1 + · · · + x 2 m−1 + 1) m/2 − m−1 j=0 m − 1 j j!e −κ(x 2 1 +···+x 2 m−1 +1) 1/2 κ j+1 (x 2 1 + · · · + x 2 m−1 + 1) (j+1)/2 . (5.17)
multidimensional integrals are a challenge for univariate numerical integration methods because of the curse of dimensionality, whereby the dimension m increases the number of sample points of the one-dimensional case N = 2n + 1 geometrically as O(N m ), reaching the limits of modern computational power quite quickly. Nevertheless, positive results may be obtained for lower dimensions, especially for extremely high accuracy, for which even tens of digits qualifies in this setting. We compare and contrast the trapezoidal rule on (5.15) for the single, double, and optimized double exponential transformations. These transformations are summarized in Table 5.2. Of course, m − 1 transformations {φ (t 1 , . . . , t m−1 )} m−1 =1 will be required to induce decay at all the boundaries of the hypercube in (5.15).
Single
Double Optimized Double
φ (t 1 , . . . , t m−1 ) tanh(t /2) tanh(π/2 sinh(t )) tanh tan −1 −1 j=1 φ 2 j + 1 sinh(t )
ρ or γ 1 1 1 β or β 2 1 π/2 π/4 d π/2 π/6 π/2
Note that while the th optimized double exponential transformation calls all previous ones through the term −1 j=1 φ 2 j + 1, this comes at no extra cost because it is extracted during the construction of the term x 2 1 + · · · + x 2 m−1 + 1, which occurs in (5.17).
In Figure 5.2, the logarithm of the relative errors of the trapezoidal rule of order n with single, double, and optimized double exponential variable transformations are plotted. The increase in convergence rate using the optimized variable transformation is a significant increase in efficiency over the double exponential transformation. The dimension of the so-called box integrals of [3] was reduced completely to one in every case by using an integral representation of the integrand | r| s , which allowed the integrals over r 1 , r 2 , etc. . . to be separated and written as the m th power of the error function. The authors postulated the exponential expectation to be a challenge because no integral representation was known to reduce the dimension of the problem.
However, using the Bessel representation for K −1/2 (κ| r|) from [12, §8.432 7.]: e −κ| r| = 2 κ| r| π K −1/2 (κ| r|) = κ 2π Because these are such challenging integrals, we include rounded approximate values in Table 5.3 for the sake of reproducibility. The one-dimensional integral (5.20) allows for the calculation of the high precision results, and it is also used for the calculation of the relative error in Figure 5.2. 6. Numerical Discussion. The numerical experiments are all programmed in Julia [6], calling at times GNU's MPFR library for arbitrary precision arithmetic, OpenBLAS for solving the linear systems and Ipopt [58] for solving the nonlinear program (3.16). As can be seen in the figures showing the relative errors, the maximization of the convergence rates provides a significant improvement for the double exponential variable transformations. With an equal number of function evaluations, the optimized double exponential formulas provide approximately 2.5-4 times as many correct digits. The conformal maps achieve this by locating singularity pre-images on the boundary of the strip ∂D π 2 . In Examples 4.1-4.4, one integral is treated on each of the canonical domains: in Example 4.1, two different endpoint and two pairs of different near-contour singularities are treated; in Example 4.2, four pairs of different near-contour singularities are treated; in Example 4.3, an infinite array of singularities is treated; and, in Example 4.4, the adaptive optimized method is shown to successfully approximate the loci of the three pairs of near-contour singularities.
In every case, the nonlinear program (3.16) still does not ensure analyticity in the strip D π 2 . In Examples 4.1, 4.2 and 4.4, the compositions ψ(h(t)) actually cross the branches of the square root functions, and in Example 4.3, there are even more poles than are shown in Figure 4.5. Yet still, a significant increase in convergence is observed. It is quite clear that some singular effects are numerically more damaging than others. Example 4.4 shows the use of the Sinc-Padé approximants for the adaptive optimized algorithm 3.3. While in Figure 4.8, the Sinc-Padé approximants serve as relatively poor approximations of the integrand, their ability to approximate singular points is acceptable. In Table 4.5, the Sinc-Padé approximants obtain 2-3 correct digits for the first order poles δ 2 ± i 2 and δ 3 ± i 3 , but struggle to obtaining an accurate location of the branch points δ 1 ± i 1 . This is entirely related to the rational limitations of the Sinc-Padé approximants, and suggests that approximating essential singularities, for example, surpasses the capabilities of the Padé methods.
While the nonlinear program (3.16) is successful in the current endeavours, further research in other conformal maps would be fruitful. For example, it is still unclear how to maximize the convergence rate in cases of countably infinite singularities, such as:
+∞ −∞ tanh x x(1 + x 2 )
dx. (6.1)
A single exponential transformation such as π 2 sinh creates an array of singularities along ± iπ 2 . Therefore, any further composition will inevitably cause the limits of this array to approach the real axis without bound. The problem has been discussed in [31,51], and the result is almost the same convergence property as a single exponential transformation.
Nevertheless, the applications show how the conformal maps can be used to obtain substantial increases in accuracy. For the forced Benjamin-Ono equation for nonlinear waves, only the optimized double exponential transformation is able to provide any accuracy that resembles the solution even for a large basis. And for the multidimensional integrals, the gain in accuracy is also significant.
Figure 3 . 1 .
31In (a) the plot of the strip D π 2
Figure 3
3
Figure 3. 2 .
2In (a) the exact solution H (0), in (b) the solution H (1/2), and in (c) the desired solution H (1). An integral with these singularities is treated in Example 4.2.
Algorithm 3. 3 .
3Set n = 1; while |Relative Error| ≥ 10 −3 do Double n and naïvely compute the n th double exponential approximation; end; while |Relative Error| ≥ do Compute the poles of the Sinc-Padé approximant; Solve the nonlinear program (3.16) for h(t); Double n and compute the n th adapted optimized approximation; end.
Figure 4 .
41 shows the three stages of the optimized double exponential map.
Figure 4 . 1 .
41In (a) the plot of the strip D π 2 with singularities located on the boundary, in (b) the optimized map h(·), and in (c) the optimized DE map. In all three cases, the crosses track the singularities.
Figure 4 . 2 .
42In (a) the plot of the integrand of (4.1) and in (b) the performance of the trapezoidal rule with single, double, and optimized double exponential variable transformations.
10 −6 sinh(t) + 0.25431 + 0.14936 t − 4.5433 × 10 −3 t 2 + 9.9880 × 10 −5 t 3 .
Figure 4 .
43 shows the three stages of the optimized double exponential map. InFigure 4.4 (a), the integrand of (4.3) is shown, and inFigure 4.4 (b), the logarithm of the relative errors of the trapezoidal rule of order n with single, double, and optimized double exponential variable transformations are plotted. The increase in convergence rate using the optimized variable transformation is a significant increase in efficiency over the double exponential transformation.
Figure 4 . 3 .
43In (a) the plot of the strip D π 2 with singularities located on the boundary, in (b) the optimized map h(·), and in (c) the optimized DE map. In all three cases, the crosses track the singularities.
Figure 4 . 4 .
44In (a) the plot of the integrand of (4.3) and in (b) the performance of the trapezoidal rule with single, double, and optimized double exponential variable transformations.
Figure 4 .
45 shows the three stages of the optimized double exponential map.
Figure 4 . 5 .
45In (a) the plot of the strip D π 2 with singularities located on the boundary, in (b) the optimized map h(·), and in (c) the optimized DE map. In all three cases, the crosses track the singularities.
Figure 4 . 6 .
46In (a) the plot of the integrand of (4.5) and in (b) the performance of the trapezoidal rule with single, double, and optimized double exponential variable transformations.
Figure 4 . 7 .
47In (a) the plot of the strip D π 2 with singularities located on the boundary, in (b) the adaptive map h(·), and in (c) the optimized DE map. In all three cases, the crosses track the singularities and the squares track the roots of the Sinc-Padé approximant for n = 2 8 .
+
Hy = f (x − c t) =
Figure 4 . 8 .
48In (a) the plot of the integrand of (4.7) and the Sinc-Padé approximant for n = 28 and in (b) the performance of the trapezoidal rule with single, double, optimized double, and adaptive optimized double exponential variable transformations.
the system of nonlinear equations obtained by collocating at the Sinc points x k = φ(kh):− c y n (k h; φ) + y 2 n (k h, φ) + Hy n (k h, φ) = f (k h), k = −n, . . . , n,(5.11)for the 2n + 1 unknowns {y j } |j|≤n by Newton iteration. For the purposes of illustration, we consider the functions f (x) which yield the solutions:
Figure 5 . 1 .
51In (a) the solution of (5.2) along with the three Sinc approximations for n = 2 5 and the scaled forcing function and in (b) the performance of the Sinc approximation with single, double, and optimized double exponential variable transformations.
Figure 5 . 2 .
52The performance of the trapezoidal rule with single, double, and optimized double exponential variable transformations. In all figures, κ = 1, and in (a) m = 2, in (b) m = 3, and in (c) m = 4.
Table 2 .1
2Variable transformations φ(t) for endpoint decay.
Table 4 .
41 summarizes the variable transformations
Table 4 .1
4Transformations and parameters for (4.1).
Table 4 .
42 summarizes the variable transformations used and the parameters in the
theorems 2.1 and 2.2.
Table 4 . 2
42Transformations and parameters for (4.3).
Table 4 .3
4Transformations and parameters for (4.5).Single
Double
Optimized Double
φ(t)
log(e t + 1)
exp(0.22t − 0.017e −t )
log(e h(t) + 1)
ρ or γ
1
0.22
1
β or β 2
2
2
0.26725
d
1.13615
1.58223
π/2
Table 4 .
44
Table 4 .
44
Transformations and parameters for (4.7).
Table 4 .6
4Evolution of the coefficients of the adaptive map.n
u 0
u 1
u 2
u 3
2 5
3.1344 × 10 −3
0.88233
0.072018
−1.6222 × 10 −3
2 6
9.1841 × 10 −3
0.95544
0.073207
−7.1021 × 10 −3
2 7
8.6135 × 10 −3
0.95359
0.072918
−6.7917 × 10 −3
2 8
6.1605 × 10 −3
0.93730
0.071916
−4.7927 × 10 −3
Optimized
9.4353 × 10 −3
0.93351
0.084087
−9.9846 × 10 −3
Table 5 .
51 summarizes the variable transformations and the
parameters used.
Table 5 .
51
Transformations and parameters for (5.2).
dr 1 · · · dr m . (5.15)Using the same dimensional-reduction technique in[3], and the incomplete gamma function [12, §8.350 1.]:κ| r|
r∈[0,1] m =
[0,1] m
e −κ(r 2
1 +···+r 2
m ) 1/2
Table 5 .2
5Transformations and parameters for (5.17).
Table 5 .3
5Numerical Evaluation of (5.15) using (5.20).
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|
[] |
[
"Comparative analysis of predictive methods for early assessment of compliance with continuous positive airway pressure therapy",
"Comparative analysis of predictive methods for early assessment of compliance with continuous positive airway pressure therapy"
] |
[
"Xavier Rafael-Palou \nEurecat Centre Tecnòlogic de Catalunya\neHealt Unit, Carrrer Bilbao, 7208005BarcelonaSpain\n\nDepartment of Information and Communication Technologies\nBCN Medtech\nUniversitat Pompeu Fabra\nBarcelonaSpain\n",
"Cecilia Turino ",
"Alexander Steblin \nEurecat Centre Tecnòlogic de Catalunya\neHealt Unit, Carrrer Bilbao, 7208005BarcelonaSpain\n",
"Manuel Sánchez-De-La-Torre ",
"Ferran Barbé ",
"Eloisa Vargiu \nEurecat Centre Tecnòlogic de Catalunya\neHealt Unit, Carrrer Bilbao, 7208005BarcelonaSpain\n"
] |
[
"Eurecat Centre Tecnòlogic de Catalunya\neHealt Unit, Carrrer Bilbao, 7208005BarcelonaSpain",
"Department of Information and Communication Technologies\nBCN Medtech\nUniversitat Pompeu Fabra\nBarcelonaSpain",
"Eurecat Centre Tecnòlogic de Catalunya\neHealt Unit, Carrrer Bilbao, 7208005BarcelonaSpain",
"Eurecat Centre Tecnòlogic de Catalunya\neHealt Unit, Carrrer Bilbao, 7208005BarcelonaSpain"
] |
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Background: Patients suffering obstructive sleep apnea are mainly treated with continuous positive airway pressure (CPAP). Although it is a highly effective treatment, compliance with this therapy is problematic to achieve with serious consequences for the patients' health. Unfortunately, there is a clear lack of clinical analytical tools to support the early prediction of compliant patients. Methods: This work intends to take a further step in this direction by building compliance classifiers with CPAP therapy at three different moments of the patient follow-up, before the therapy starts (baseline) and at months 1 and 3 after the baseline.Results:Results of the clinical trial shows that month 3 was the time-point with the most accurate classifier reaching an f1-score of 87% and 84% in cross-validation and test. At month 1, performances were almost as high as in month 3 with 82% and 84% of f1-score. At baseline, where no information of patients' CPAP use was given yet, the best classifier achieved 73% and 76% of f1-score in cross-validation and test set respectively. Subsequent analyzes carried out with the best classifiers of each time point revealed baseline factors (i.e. headaches, psychological symptoms, arterial hypertension and EuroQol visual analog scale) closely related to the prediction of compliance independently of the time-point. In addition, among the variables taken only during the follow-up of the patients, Epworth and the average nighttime hours were the most important to predict compliance with CPAP. Conclusions: Best classifiers reported high performances after one month of treatment, being the third month when significant differences were achieved with respect to the baseline. Four baseline variables were reported relevant for the prediction of compliance with CPAP at each time-point. Two characteristics more were also highlighted for the prediction of compliance at months 1 and 3. Trial registration: ClinicalTrials.gov Identifier, NCT03116958. Retrospectively registered on 17 April 2017.
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10.1186/s12911-018-0657-z
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5aee1441bf55cf52b28e9f82af4ab4815eae4454
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Comparative analysis of predictive methods for early assessment of compliance with continuous positive airway pressure therapy
Xavier Rafael-Palou
Eurecat Centre Tecnòlogic de Catalunya
eHealt Unit, Carrrer Bilbao, 7208005BarcelonaSpain
Department of Information and Communication Technologies
BCN Medtech
Universitat Pompeu Fabra
BarcelonaSpain
Cecilia Turino
Alexander Steblin
Eurecat Centre Tecnòlogic de Catalunya
eHealt Unit, Carrrer Bilbao, 7208005BarcelonaSpain
Manuel Sánchez-De-La-Torre
Ferran Barbé
Eloisa Vargiu
Eurecat Centre Tecnòlogic de Catalunya
eHealt Unit, Carrrer Bilbao, 7208005BarcelonaSpain
Comparative analysis of predictive methods for early assessment of compliance with continuous positive airway pressure therapy
10.1186/s12911-018-0657-zRafael-Palou et al. BMC Medical Informatics and Decision Making (2018) 18:81 RESEARCH ARTICLE Open Access *Correspondence: Full list of author information is available at the end of the articleObtrusive sleep apneaContinuous positive airway pressurePredictive methodsMachine learning
Background: Patients suffering obstructive sleep apnea are mainly treated with continuous positive airway pressure (CPAP). Although it is a highly effective treatment, compliance with this therapy is problematic to achieve with serious consequences for the patients' health. Unfortunately, there is a clear lack of clinical analytical tools to support the early prediction of compliant patients. Methods: This work intends to take a further step in this direction by building compliance classifiers with CPAP therapy at three different moments of the patient follow-up, before the therapy starts (baseline) and at months 1 and 3 after the baseline.Results:Results of the clinical trial shows that month 3 was the time-point with the most accurate classifier reaching an f1-score of 87% and 84% in cross-validation and test. At month 1, performances were almost as high as in month 3 with 82% and 84% of f1-score. At baseline, where no information of patients' CPAP use was given yet, the best classifier achieved 73% and 76% of f1-score in cross-validation and test set respectively. Subsequent analyzes carried out with the best classifiers of each time point revealed baseline factors (i.e. headaches, psychological symptoms, arterial hypertension and EuroQol visual analog scale) closely related to the prediction of compliance independently of the time-point. In addition, among the variables taken only during the follow-up of the patients, Epworth and the average nighttime hours were the most important to predict compliance with CPAP. Conclusions: Best classifiers reported high performances after one month of treatment, being the third month when significant differences were achieved with respect to the baseline. Four baseline variables were reported relevant for the prediction of compliance with CPAP at each time-point. Two characteristics more were also highlighted for the prediction of compliance at months 1 and 3. Trial registration: ClinicalTrials.gov Identifier, NCT03116958. Retrospectively registered on 17 April 2017.
Background
Obstructive sleep apnea (OSA) [1] is defined as repeated episodes of shallow or paused breathing during sleep, despite the effort to breathe. This syndrome is caused by complete or partial obstructions of the upper airway leading to daytime sleepiness and impaired cardiopulmonary function. Gold standard treatment of OSA involves the use of a device that administers continuous positive pressure (CPAP) in the respiratory tract of patients [2]. Patients should wear a mask, usually nasal [3,4], to receive the air from the CPAP device. Doctors are responsible for adjusting the air pressure provided by the device, preventing the throat from collapsing or damaging its tissue [5].
Several studies aimed at objectively assess the effects of using CPAP show that this kind of intervention is highly effective in improving symptoms, such as daytime sleepiness, morbidity, and mortality rates related to cardiovascular diseases [6,7]. Although it is highly effective in minimizing OSA symptoms, up to 36% of patients do not use or even discontinue CPAP [8,9].
Different factors have been found that influence the adherence to CPAP treatment, although they have not been reported consistently. In [10] patient demographic features, such as age, sex, and marital status were found to be weak predictors of CPAP adherence, as well as apneahypopnea index (AHI), oxygen desaturation index (ODI), or daytime sleepiness in [11][12][13]. In contrast, an improvement of ODI and deep sleep in patients during the CPAP titration increased the chances of complying with CPAP in [14]. Also, the adherence measured few days after CPAP initiation was shown to be a good predictor of long-term compliance in [15].
Despite these efforts, the reasons that lead to compliance with therapy remain an open field of research. In fact, knowing in more detail the initial factors that determine compliance with CPAP could help reduce dropout rates and the application of personalized treatments to improve patient compliance. Unfortunately, there is a clear lack of clinical analytical tools to support the early prediction of compliant patients.
When statistically analyzing the possible factors that might determine CPAP compliance, we found common complexities that compromise the predictive capacity and robustness of the findings. The limited number of participants, the large number of clinical variables and the quality of collected data are just to name a few. To overcome these limitations the use of machine learning (ML) might be an alternative solution to the aforementioned problems. Despite the numerous ML applications in the medical domain, (e.g. disease diagnosis [16,17]), compliance with therapy is usually constrained to the medication adherence problem [18,19]. Recent ML algorithms (e.g. support vector machines [20] and artificial neural networks [21]) often provide highly accurate predictive models. However, such models lack transparency and therefore their interpretation is difficult [22]. As a consequence, other classification algorithms in the medical field are still preferred (e.g. logistic regressions [23] or decision trees [24]).
The goal of the study presented in this paper is twofold. On the one hand, to provide a comparative analysis of predictive methods of CPAP compliance built using machine learning techniques in different stages of treatment. On the other hand, to define the most important factors associated with CPAP compliance identified by the best predictive methods obtained in the different stages of therapy up to month 6.
Methods
Participants
Fifty-one adult patients (> 18 years), diagnosed with OSA (15 or more apneas/hypopneas per hour in an overnight sleep study) and requiring CPAP treatment were recruited at Hospital Arnau de Vilanova (Lleida, Spain). Patients with impaired lung function (overlap syndrome, obesity hypoventilation, and restrictive disorders), severe heart failure, psychiatric disorders, periodic leg movements, pregnancy, other dyssomnias or parasomnias, and/or a history of previous CPAP treatment were excluded. The study was approved by the hospital ethics committee (Approval number: CEIC-1283). All recruited patients signed an informed consent form.
Of the 51 patients originally included in the study, 3 were excluded due to malfunction of the CPAP machine, 5 did not attend the last visit at the sleep unit and 1 patient died during the study.
The final sample consisted of 42 patients (29 males and 13 females) with a mean age of 56.93+/-12.58 yrs. Their BMI was 33.83+/-6.46 and their number of apnea or hypopneas per hour of sleep (apnea/hypopnea index or AHI) was 53.13+/-20.72 events/h. In our sample, 60% (25) of all patients were active workers and 33% (14) were retired. The sample also had 62% of nonsmokers (26) and 57% of non-alcoholic consumers (24). In terms of CPAP device use, the patients scored an average of nightly hours of use of 5.44+/-1.74 at month-1, 5.33+/-1.90 at month-3, 5.07+/-2.10 at month-6.
Datasets
The study variables from the 42 patients were manually collected by lung specialists along four visits at month-0 (baseline or T0), at month-1 (T1), at month-3 (T3), and at month-6 (T6). During the first visit (T0) clinicians gathered 77 features organized in five categories: clinical history (e.g. depression, anxiety, arterial hypertension (HTA), cardiopathy, neurological disease, respiratory disease), symptoms (e.g. irritability, apathy, depression, insomnia), co-morbidities (e.g. diabetes, obesity, dyslipidemia), therapies (e.g. beta blockers, diuretics), sleeping test (e.g. sleeping time, AHI, percentage of the night spent with oxygen saturation < 90% or CT 90) and basal information (e.g. size, weight, BMI, tas, tad, oxygen saturation). In the second visit (T1), after the patients had the CPAP machine at home during one consecutive month, 16 new features related to monitoring were collected (e.g. nightly average use, abandon or adverse effects of the treatment, such as dry mouth, allergies, and cutaneous irritations). At the third month (T3), the same number of features as in T1 were gathered together with 5 more: size, weight, BMI, removed drugs, and added drugs). At month-6 (T6), although some other variables were collected, for the purpose of this study only the average use of nightly hours was considered. Eventually, three datasets (D0, D1, and D3) with an incremental number of features (i.e. D1 features = D0 features + features collected at T1) were created with 77, 93, and 114 features, respectively. The full list of variables is described in Additional file 1: Table S1 of the supplementary material.
In this study, we addressed CPAP compliant users as those who had more than 4 h per night on average during the first 6 months of treatment. Therefore, all samples from each dataset (ds) were labeled using the collected information about nightly hours/use on average of the CPAP device at the end of the month-6. In so doing, 24 (57%) patients were labeled as "compliant", class "1", as they correctly followed the CPAP therapy prescription (more than 4h nightly on average). On the contrary, 18 (43%) patients did not achieve the prescribed treatment (minus or equal than 4 h nightly on average) and they were labeled as "non-compliant", class "0".
Data description
Datasets D0, D1, and D3 collected at time points T0, T1, and T3, respectively, were statistically analyzed for a better understanding of the sample. In this task, the Mann-Whitney U test was used to evaluate the statistical significance of quantitative variables with CPAP compliance and Chi-square tests for qualitative characteristics. Previously, the categorical features were converted into numeric to achieve a homogeneous data type sample. Variables with only two categories were directly mapped into binary values. Variables with more than two categories, given their underlying incremental meaning, were mapped into unsigned ordinal values.
Data cleansing
We carried out a set of empirical tasks, supervised by the lung specialists, to reduce possible noise and redundancy in the datasets. First, given the existence of null values in the datasets, an imputation process was carried out consisting of computing the mode for the categorical characteristics and the mean for the numerical characteristics.
Subsequently, the distributions of the categorical characteristics were analyzed, which revealed variables with few individuals by category. The features with a number of individuals less than a given threshold were removed from the study to avoid the noise they might introduce when building the predictive models. To catch up possible information redundancy in the datasets, we computed the mutual information (MI) [25] score for the categorical variables in a pair-wise manner. From each pair, we kept the variable more statistically significant with the dependent variable using Chi-square test. Among the numerical features, we applied a correlation analysis to detect highly redundant features. Given the existence of non-normally distributed numerical features, we used the Spearman correlation method on all numerical variables in a pair-wise manner. Empirically we set-up a threshold for the correlation scores above of which one feature of the pair was removed (i.e. the feature with the highest P-value). P-values lower than 0.05 were considered statistically significant.
Classification framework
All preprocessed datasets presented common particularities, such as a small number of samples, the presence of missing values, class unbalance and high multidimensionality feature space. To cope with these complexities we designed a classification framework flexible enough to enable the execution of heterogeneous pipelines or sequence of configurable machine learning steps. In particular, the pipelines were composed of three mandatory steps (i.e. imputation, variance filtering and data standardization), two optional steps (i.e. feature selection and feature sampling), and two more final steps (i.e. classifier training and evaluation). In total 80 pipelines were configured from 4 feature selection methods, 5 classifier algorithms, 2 sampling strategies, and 2 evaluation metrics.
The result of running (i.e. training or building) a pipeline (Pipe i ) on a dataset (D j ) with parameters (params i ) is a predictive model or classifier (M i,j ) with its associated predictive performance (Perf i,j ). Figure 1 sketches the scheme with the input, output, and the different steps that configure the pipelines for compliance with CPAP therapy.
The first step of a pipeline is the imputation of null values. To do this, given the small proportion of null values in the datasets, a simple strategy was proposed to replace the null values with their most frequent value (for categorical characteristics) and with the mean value (for numerical characteristics).
The second step consists of a simple filter method to eliminate features with zero variance, that is, to eliminate these characteristics that have the same value in all the samples and that do not provide any additional information to the dataset. Since the data come from different sources, the next step is to standardize them. This step consisted of homogenizing all features to zero mean and variance one. This transformation step is crucial for the construction of many classification algorithms since it allows them to compare features without harming their performance or execution time [26].
Feature selection (fs) was introduced in the pipelines given a large number of features compared with the number of samples for each dataset (p > n). This type of methods aims to reduce over-fitting by improving model performance and generalization, to provide faster and more cost-effective models and simplify models making them easier to interpret [27]. Feature selection methods are usually divided into three categories: filter, wrapper, and embedded [28]. Filter methods, in general, examine features individually with respect to the class, wrapper methods use a predictive model to generate subsets of features evaluated according to their predictive power, and embedded methods search for an optimal subset of features during the training of the prediction model. In this study, we used one method for each of the different feature selection strategies. For the filter-based strategy, we defined a simple method (Combine_fs) that makes a ranking of the features by their statistical significance with the class (i.e. applying ANOVA or chi-squared tests according to the data type of the characteristics). Then, this method returns the subset of features through a configurable threshold. For the wrapper strategy, we proposed the recursive feature elimination (RFE_RF_fs) method [29] configured with a random forest to provide the importance of the features. The embedded strategy was in charge of the application of the least absolute shrinkage and selection operator (Lasso_fs) [30]. To this purpose, we used a logistic regression model configured with the L1 norm. It was also considered the possibility of not using any method of feature selection. The next step is data sampling (sm). In particular, we proposed the use of the synthetic minority over-sampling technique (Smote) [31]. This technique consists of creating synthetic samples (i.e. detecting similar instances and performing small perturbations in their values) of the under-represented class samples instead of creating copies, as the over-sampling method would do. In particular, we used this technique to balance the minority class with the same number of instances of the majority class. The main idea behind this method is to avoid the bias produced by many standard classifier learning algorithms towards the class with a larger number of instances. As in the feature selection pipeline step, we also considered the option of not using data sampling in the experiments.
Regarding the training and evaluation stage, we selected several classification algorithms (cls) to deal with various classification strategies (i.e. linear, non-linear, distancebased, and tree-based). In fact, the provision of different classification strategies is especially appropriate in complex datasets when the distribution of data is not clear. As already mentioned, the interpretability of the resulting predictive models is also a desired condition. Therefore, we opted for logistic regression (LR) [23], k-nearest neighbor (k-NN) [32], and random forest (RF) [33] for the subset of interpretable classification algorithms (referred as "descriptive", hereinafter). In contrast, we chose support vector machines (SVM) [20] and artificial neural networks (NN) [21] for the subset of algorithms with less interpretative capacity but with a potential greater discriminatory capacity (referred as "non-descriptive", hereinafter).
Evaluation setup
In order to ensure adequate performance evaluation, the available data were shuffled, stratified, and randomly divided into train (29 rows, 12 non-compliant, and 17 compliant) and test (13 rows, 6 non-compliant, and 7 compliant) sets with a ratio of 70/30. Therefore, the training set partitions of the three datasets (D0, D1, and D3) contained the same individuals. The same rule applies for the test set.
Test sets remained untouched until the end of the process. Training sets were used for 10-fold cross-validation to enable proper model tuning and evaluation. This technique is particularly suitable when the sample size is small. Indeed, as suggested in [34], the entire sequence of processes that composed each pipeline was wrapped-up within the cross-validation technique in order to reduce the possibility of biased results. Thus, training data were shuffled and randomly split into stratified train-validation sets (20 rows, 8 non-compliant, and 12 compliant) and stratified test-validation sets (9 rows, 4 non-compliant, and 5 compliant) following a ratio of 70/30. Then, for each of the configured pipelines (i.e. 80 pipelines), we created as many experiments as combinations of values for the different hyper-parameters defined for each method of the pipeline (Table 1).
We performed 10-fold cross-validation for all experiments in each pipeline. This process was repeated for each of the proposed learning metrics (i.e. f1-weighted and precision-weighted). The learning metric, f1-weighted (f 1), was selected since it is a suitable measure for unbalanced datasets. This metric combines the precision and recall metrics weighted by the number of samples per each class. The other selected metric, precision-weighted (prec), tends to prefer classifiers with less incorrect compliant predicted patients, which indeed they are the most harmful cases to avoid. This metric computes the ratio of correctly classified cases (i.e. compliant patients) among all positive classified cases weighted by the number of samples per each class. As a result of this 10-fold cross-validation process, we reported for each experiment the average and standard deviation performance of the learning metric which the pipeline was configured with. A greedy-search strategy was applied to select the best experiment, i.e. best pipeline parameterization. Additionally, with the intention to avoid the possible bias introduced in this process to the validation dataset [35] and to reduce the chances of having obtained similar folds with repeated individuals, we evaluated the best parameterization of each pipeline (i.e. pipelines with the appropriated values for their hyper-parameters) using a final outer stratified 10fold cross-validation on the training data (i.e. with learning metric=f1-weighted, ratio=70/30 for cv-train and cvtest). As a result of this process we reported the final cross-validation performance (f1-score) of the pipelines. This score was provided by the f1-weighted metric since although having a high precision is desirable, a high recall (rec) is also needed especially for health institutions since it reduces false negatives and thus non-necessary clinical interventions and additional costs.
This whole process was repeated 80 times for all pipelines of each dataset. The best pipelines of each dataset were identified by ranking the cross-validation performances (i.e. f1-score) reported by each pipeline. The best pipelines of each dataset were compared together in order to find statistically significant differences. We did the same among the best descriptive and non-descriptive pipelines. To do this, we used a 10-fold cross-validated paired t-test.
To complete the reporting of this analysis, we computed on the test set and for each of the best pipelines of each dataset a comprehensive set of scores (i.e. f1, precision, recall, AUC, and confusion matrix) to enable a better understanding of the results. Also, we reported their ROC and learning curves.
Feature importances
We reported the most important features from the best descriptive pipelines. Let us note that, in this study, we only used those configured with random forest and logistic regression. for each dataset. To do this, we performed a ranking of features using a stability score [36]. This score measures how "stable" the features of a predictive model are. For that reason, we build a pipeline n times using n random subsets of fixed size s.
To compute this score, we created n = 100 randomly stratified partitions from the s = 70% of the entire dataset. For each data partition, we trained the best pipelines and keep a record of the selected features of the classifier and their weights (or feature importances for RF classifier). With this information, we computed the number of times any feature was selected (i.e. stability score) and its normalized absolute average weight (or importance). Since one classifier might report all features as relevant (i.e. non-zero), a threshold (t > 0.4) in the weights was empirically defined to make usable the stability score.
Results
Data cleansing
A descriptive analysis of the initial datasets was carried out and summarized in Tables S2, S3 and S4 of the Additional file 1. Subsequently, the data cleansing process was conducted under the supervision of the lung specialists. From this process, we found out that (11/27/42) features had null values with ratios between 2.3% and 12% from the total number of rows. After a null imputation, we found (14/7/10) variables with underrepresented categories (<= 10% of rows per category). We also detected 4 pairs of categorical features (i.e. no active, anti-depressives, ADO, and memory disorders) in the D0 with MI scores above 50%. From the correlation analysis applied on the numerical variables, we found 4 highly (> 80%) redundant features in D0 (i.e. abdomen and hip circumference, weight and CT90%) and 4 features in D3 (i.e. size_3, weight_3, bmi_3 and total_use_hours_3). After the validation by the specialists of setting aside the aforementioned features, we obtained the final datasets for the classification analysis. These were composed of (54/63/70) variables, respectively.
Classification analysis
We evaluated 76 out of 80 initially configured pipelines for each dataset. In particular, we rule out pipelines which had same classification algorithm (i.e. random forest) for feature selection and classification given their initial poor contribution to the experimental results and the long runtime required to complete their evaluation.
Best pipelines (p0, p1, p3) for D0, D1, and D3, respectively, achieved 0.73+/-0.18, 0.82+/-0.06, 0.87+/-0.15 of f1-score in cross-validation and 0.76, 0.84, 0.84 in test set ( Table 2). These pipelines were configured with precisionweighted metric and SVM algorithm for the D0 dataset; with Smote sampling, f1-weighted metric, and an SVM for the D1 dataset, and with Lasso feature selection, Smote sampling, precision-weighted metric and an RF for the D3 dataset.
To visually support these values, Figs. 2, 3, and 4 show the area under the receiver operating characteristic (ROC) curves and Figs. 5, 6, and 7 show the learning curves with the effects of increasing the size of the training set in their performances.
We also analyzed which was the contribution in classification performance of the different techniques configured in the pipelines (i.e. sampling strategy, feature selection, learning metric, and classification algorithm). In particular, the average in the performance (i.e. f1score) of all pipelines using the sampling strategy was (0.59+/-0.07,0.61+/-0.09,0.75+/-0.09) for each dataset. Let us recall that Smote [31] was the selected sampling technique to balance the training data from each fold of cross-validation by oversampling the minority class to the number of samples of the majority class (i.e. from 8 non-compliant/12 compliant to 12/12). On the contrary, not using any sampling strategy was (0.58+/-0.07,0.62+/-0.08,0.75+/-0.08). Concerning the average performance reached among the pipelines configured with the best feature selection methods they scored (0.59+/-0.03,0.63+/-0.07,0.77+/-0.08). On the other side, the average in performance reached by the pipelines without using any feature selection gave as result (0.66+/-0.07,0.68+/-0.11,0.77+/-0.09), respectively. Focusing on the evaluation metric with which the pipelines were configured, the pipelines with f1-weighted metric achieved an average in the performance of (0.58+/-0.07,0.62+/-0.09,0.76+/-0.09) while the pipelines configured with precision-weighted obtained an average performance of (0.59 + / − 0.07, 0.61 + / − 0.09, 0.75 + / − 0.09). Regarding the type of classification algorithm (Fig. 8), the best pipelines reported performances in cross-validation between 0.59+/-0.21 (using k-NN) and 0.73+/-0.18 (using SVM) of f1-score in D0. In D1 the best pipelines reported performances between 0.61+/-0.12 (using k-NN) and 0.82+/-0.06 (with SVM). In D3 the best pipelines reported performances between 0.75+/-0.07 (using k-NN) and 0.87+/-0.15 (with RF). Table 3 summarizes these differences of performance among the configured pipelines.
Regarding the comparison of performance between descriptive and non-descriptive pipelines, the best descriptive pipelines obtained f1-scores of 0.69 +/-0. 15 Table 4.
To complete the analysis, we extracted the most important features used by the best descriptive pipelines. In total (25/28/20) features were reported for each dataset after setting up a minimum threshold of 0.4 in the feature weights. Top-10 features of D0 and D1 reported stability scores above 0.6. In D3 only 2 features were above 0.4 of stability score. Figures 9, 10, and 11 provides a visual ranking of the features of each of the dataset ordered by the stability score.
Discussion
CPAP compliance has been demonstrated to reduce cardiovascular risk and symptoms in patients with sleep apnea. Therefore, an automatic support to the early identification of patients with poor compliance with CPAP therapy would allow personalized treatments and better management of hospital resources. Consequently, in this article we proposed the creation of classifiers that predict compliance with CPAP therapy in patients with OSA in the early stages of treatment.
One of the main challenges of this study has been to build predictive models from small data [37]. In this scenario, models tend to over-fit training and even validation data, which means they are not able to generalize well for previously unseen data. To overcome this problem, first, under the supervision of pulmonary specialists, we performed an initial process of data cleansing to eliminate the non-relevant and noisy characteristics of the dataset. Secondly, we set up several machine learning experiments (e.g. using data sampling, feature selection or one of the different proposed machine learning algorithms) to obtain those with the best generalization performance for each dataset. To do this, we proposed the use of machine learning pipelines to carry out all the steps of the learning experiment together for each of the folds of the crossvalidation. In this way, we avoided the knowledge leakage from the training to the evaluation stage. Thirdly, we used a nested cross-validation [38] to train, adjust, and evaluate the models. This mechanism reduces the chances of reporting over-fitted models since the validation scores are reported through a second level of cross-validation, once the models are tuned on the first level of crossvalidation. Lastly, we provided a thorough evaluation of the best models to understand their performance. The results of this evaluation are presented in Table 2 and show that the performance of the best predictive models, both for validation and test datasets, are high (i.e. greater than 0.7 for D0 and higher than 0.8 for D1 and D3 of precision, recall, and f1 scores) and close to each other (i.e. with only 0.02 and 0.03 difference in f1) for each of the analyzed datasets. These values reflect that current models are reliable and generalize well in data partitions that were not used for training. In addition, as can be seen in the Figs. 5, 6, and 7, validation curves reach the highest values when all training data are available to build the classifiers. However, as these are still a bit far from the training curves but the tendency is to get together, getting more data for training could be beneficial to obtain more accurate models (especially for datasets D1 and D3).
Going deep into the classification results obtained at three different moments of CPAP treatment, they show that dataset D0, collected before the start of treatment, is the most complex to learn compared to D1 and D3, collected in months 1 and 3 of the patient's therapy, Fig. 7 Learning curves of the best pipeline for dataset D3 respectively. Nevertheless, a considerable average f1-score of 0.73 +/-0.18 was achieved in cross-validation and 0.76 in the test set. As shown in Table 5, an important performance increase of 0.09 (p = 0.12) was reached in D1 with respect to D0. In D3, we get the best classification performance with a significant increase of 0.14 in f1-score (p = 0.024) with respect to D0. This same trend occurred in the test set where the best pipelines of D1 and D3 reported performances of 0.18 above the achieved in D0. The difference in f1-score between D3 and D1 did not prove to be significant (p = 0.30). These results seem to confirm that follow-up measurements help to increase baseline prediction performance. Indeed, the closer to the CPAP compliance cut-off we are the more confident the classifier is (i.e. performance in D0 is lower than performance in D1 and lower than D3). In addition, patterns of CPAP adherence appear early, in our case at month 1, since it is when the greatest increase in performance is achieved. This same finding was confirmed by other studies [15,39].
During the evaluation step, we noticed that the use of sampling, feature selection or a particular learning metric were not as substantial as expected in any of the datasets (Table 3). To be more specific, the maximum increase in performance, regardless of the method used, was between 0.02 and 0.04 of f1-score in cross-validation. Probably this confined contribution was due to the initial preprocessing and the fact that the data were not severely unbalanced. Indeed, in D0 and D1 the use of feature selection compromised the performances with a maximum decrease of 0.13. In contrast, important increments of performance in all datasets were produced depending on the classification algorithm used (i.e. 0.14, 0.21, and 0.20). In D0 and D1 best pipelines were using an SVM. This result was not surprising because this algorithm has been already found suitable for problems with few samples and with a high number of features [40], being able to build complex non-linear decision boundaries. In D3, the best pipeline was configured with an RF, although the one with SVM also provided a high score. RF algorithm is also suitable for difficult problems and especially indicated for handling categorical features [33]. The other non-descriptive classifier (i.e. NN) reported competent performances in all three datasets, especially in D3, where it exceeded the results reported by the best pipeline configured with an SVM. However, the k-NN algorithm reported the worst scores of the three datasets. This is partly because it does not usually work well with a large number of features [41].
Focusing on the differences of performance between the best descriptive and non-descriptive pipelines, those were always lower than 0.1 and not significant (p = 0.14) for D0 but significant in D1 (p = 0.02). In contrast, the best performance in D3 was achieved through a descriptive classifier although in cross-validation the difference in performance with the best non-descriptive pipeline (0.02+/-0.16 of f1-score) proved to be non-significant (p = 0.69).
Regarding the relevant factors related to the CPAP compliance prediction, four baseline features were found common in each of the best pipelines for the different datasets collected at time-points (T0, T1, and T3). Those were headaches, psychological symptoms (i.e. irritability, apathy, and depression), arterial hypertension and the visual analog scale (as part of EuroQol questionnaire).
From these four features, the headache was the most stable feature (i.e. with the highest stability score) at T0 and T1. In the baseline, all these characteristics were found significant with respect to the CPAP compliance but the latter (p = 0.079). In particular, compliant patients were more likely to not having headache (85%, 23 out of 27) nor psychological symptoms (67%, 18 out of 27), having arterial hypertension (74%, 20 out of 27) and worst visual analog scale score (9.15+/-1.02 on mean difference with respect to non-compliant patients). To the best of our knowledge, these features together have not previously been reported as relevant to predict patient compliance with CPAP therapy at either month 0, 1, and 3. In the literature, having morning headache was also found significant in a randomized control trial of OSA patients [42]. On the contrary, psychological factors did not show prediction capability in [43][44][45] but how patients were challenging difficult situations (active versus passive) [46]. In [47], authors highlighted the positive effect of CPAP treatment on blood pressure in patients with resistant hypertension. The visual analog scale, used as a generic method for measuring the quality of life, was reported useful to track treatment-induced changes in [48,49].
Different studies [50][51][52] have shown an improvement in snoring, gastro-esophageal reflux and oxygen saturation with CPAP treatment. In our case, they were found among the characteristics with the highest stability scores for the best pipelines of month-0 and month-1. However, only oxygen saturation (p < 0.001) predicted good compliance with CPAP.
Two of the features collected at months 1 and 3 (i.e. average hours of nightly CPAP use and Epworth) were found as the most important predictive features in these time-points. These features were significant regarding CPAP compliance. Interestingly, from months 1 to 3 the average of nightly hours of use for compliant users increased (from 5.9+/-1.51 to 6.17+/-1.29) while in non-compliant users decreased (from 4.4+/-1.75 to 3.56+/-1.76). On the contrary, the average of Epworth for compliant users decreased from 5.48+/-3.63 to 4.64+/-3.07, while for non-compliant increased from 7.33+/-3.7 to 8.46+/-4.16). Early measurements of the average hours of nightly CPAP use were already reported as predictive of CPAP compliance in [53,54]. Epworth was also reported as a relevant predictor of compliance in [15,[54][55][56]. The limitations of this study can be summarized in the following two points. First, although a common cutoff was selected for the definition of CPAP compliance, changes in this threshold might cause different performances as well as variations in the rank of the feature importance reported in this work, thus further explorations are required in this regard. Second, despite the positive scores obtained by the predictive models at the different time-points, the small number of individuals in the sample makes it highly recommendable to validate the obtained results with new data.
Finally, let us point out that the work proposed in this paper is part of the myOSA project (RTC-2014-3138-1), aimed at developing new ICT tools to support the OSA treatment. Under the umbrella of myOSA, we created an IoT system [57] that remotely monitors the patient's CPAP devices to provide indicators of progress such as early compliance, adherence level, as well as personalized recommendations to empower the patients. Moreover, we are currently investigating how to apply these predictive methods also in the context of the H2020 project CONNECARE (ID: 689802), in which we focus on patient's monitoring with the final goal of providing selfmanagement features to people in needs, such as chronic patients [58].
Conclusions
To the best of our knowledge, this article is the first attempt to analyze and compare the compliance with the CPAP therapy of patients with OSA at different points of the treatment by building classifiers. Three time-points were established to perform the analysis (i.e. before the treatment starts, at month 1, and at month 3). To build and evaluate the classifiers a flexible framework was designed relying on machine learning pipelines. High performances were reached yet after one month of treatment, being the third month when significant differences in performances were achieved with respect to the baseline. Four baseline variables were found relevant for the prediction of compliance with CPAP at each time-point. Two characteristics more, collected during the follow-up, were also highlighted for the prediction of compliance at months 1 and 3. Further tasks are devised to extend the present study, including the collection of new patients and exploring other CPAP compliance cut-offs, in order to validate the findings and reported performances. This work aims to take a step forward towards the creation of new tools to allow early and accurate detection of patients struggling to follow the CPAP treatment and thus enable personalized patient interventions that would lead to improving their quality of life.
Additional file
Fig. 1
1Pipeline steps designed for building classifiers for compliance with the CPAP therapy
= [1e−5, 0.00001, 0.0001, 0.001, 0.01,0.1,1,3,5,10] hidden_layer_sizes = [(30,), (50,), (70,), (100,), (150,),(30,30),(50,50),(70,70),(100,100),(30,30,30),(50,50,50),(70,70,70)]
, 0.75 +/-0.15 and 0.87 +/-0.15 in cross-validation and 0.76, 0.84, 0.84 in test set, while the best non-descriptive pipelines obtained scores of 0.73 +/-0.18, 0.82 +/-0.06 and 0.84 +/-0.08 in cross-validation and 0.76, 0.84, 0.84 in test set. Further details about these pipelines can be found in
Fig. 3
3ROC curves for cross-validation and test of the best pipeline for dataset D1
Fig. 4
4ROC curves for cross-validation and test of the best pipeline for dataset D3
Fig. 5
5Learning curves of the best pipeline for dataset D0Fig. 6Learning curves of the best pipeline for dataset D1
Fig. 8
8Best pipelines results in cross-validation and test at different time-points
Avg Max Avg Max Avg Max Sampling Smote vs None 0.01 -0.02 -0.01 0.04 0.0 0.02 Feature selection Best vs None -0.07 -0.07 -0.05 -0.13 0.0 0.02 Metrics f1 vs prec 0.01 0.02 0.01 0.04 0.01 -0.01 Classifier algorithm Best vs Worst 0.09 0.14 0.14 0.21 0.19 0.20
Fig. 9
9Stability scores and feature weights of the best pipeline for dataset D0
Fig. 10
10Stability scores and feature weights of the best pipeline for dataset D1Fig. 11Stability scores and feature weights of the best pipeline for dataset D3
Additional file 1 :
1Tables with the description of all the variables of the D0, D1 and D3 datasets. Tables with the results of the descriptive analysis of the datasets. (DOCX 944 kb) Abbreviations ADO: Oral antidiabetics; AHI: Apnea-hypopnea index; AUC: Area under the curve; CONNECARE: personalized connected care for complex chronic patients; CPAP: Continuous positive pressure; IoT: Internet of things; ICT: Information and communications technology; k-NN: k-nearest neighbor; LASSO: Least absolute shrinkage and selection operator; LR: Logistic regression; MI: Mutual information; ML: Machine learning; myOSA: A telemedicine system for management and treatment of patients with obstructive sleep apnea/hypopnea syndrome NN: Neural network; ODI: Oxygen desaturation index; OSA: Obstructive sleep apnea; RFE: Recursive feature elimination; ROC: receiver operating characteristic; SMOTE: Synthetic minority over-sampling technique; SVM: Support vector machine
Table 1
1Pipeline parameters tested using grid-search and 10-fold CVPipeline step
Parameter options
Combine_fs
percentile = [5, 10, 20, 30, 40, 50]
Lasso_fs
estimator = Logistic Regression
penalty = "l1"
C =[ 5, 10, 20, 30, 40, 50]
RFE_RF_fs
class_weight = 'balanced'
n_estimators = 100
step = [0,1 ]
n_features_to_select = [0.4,0.6,0.8]
Smote_fs
n_neighbors = [3,4,5]
ratio='auto'
kind='regular'
k-NN
n_neighbors = [1,3,5,7,9,11]
weights = ['uniform', 'distance']
LR
C = [0.00001, 0.0001, 0.0005, 0.001, 0.005,
0.01, 0.05, 0.1, 0.5, 1, 5, 10, 15, 30]
class_weight = [None, 'balanced']
penalty = ['l1', 'l2']
RF
n_estimators = [100,150,200,250,500]
criterion = ['entropy','gini']
max_depth = ['None',4,6]
class_weight = [None, 'balanced']
SVM
C = [0.01,0.1,0.5,1,5,10,15,30,50]
gamma = [0.0001,0.001,0.01, 0.1,1,5]
Table 2
2Performances of the best pipelines in each datasetid
ds
sm
fs
metric
cls
params
p0
D0
none
none
precision_weighted
SVM
[0.001, balanced, 30]
p1
D1
Smote
none
f1_weighted
SVM
[0.001, None, 4, 15]
p3
D3
Smote
Lasso_fs
precision_weighted
RF
[1, 250, gini, 4, None, None]
id
cv_prec
cv_rec
cv_f1
test_prec
test_rec
test_f1
p0
0.78+/-0.2
0.74+/-0.17
0.73+/-0.18
0.77
0.85
0.76
p1
0.84+/-0.06
0.82+/-0.05
0.82+/-0.06
0.85
0.88
0.84
p3
0.89+/-0.14
0.88+/-0.14
0.87+/-0.15
0.85
0.88
0.84
Table 3
3Performance difference of f1 cross-validation along the different datasets achieved among pipelines configured with different techniques
Table 4
4Best descriptive and non-descriptive pipelines by datasetds
sm
fs
metric cls
params
D0 none
none
prec
SVM [0.001, balanced, 30]
D0 Smote none
prec
LR
[None, 15, 4, l2]
D1 Smote none
f1
SVM [0.001, None, 4, 15]
D1 none
none
f1
LR
[None, 5, l2]
D3 Smote Lasso_fs prec
RF
[1, 250, gini, 4, None, None]
D3 none
none
f1
LR
[None, 0.5, l1]
Table 5
5Performance comparison between best pipelines for each datasetPipelines
Difference (cv_f1)
Statistic
p_value
p0 vs p1
0.09 +/-0.15
-1.71
0.1201
p0 vs p3
0.14 +/-0.18
-2.70
0.0241
p1 vs p3
0.05 +/-0.14
-1.0931
0.3027
AcknowledgementsThe authors wish to acknowledge all the members involved in the myOSA project for their enthusiasm, support and dedication. In addition, we also want to acknowledge the partial support by the Industrial Doctorates Program (AGAUR).Authors' contributionsAll the authors have been involved in drafting the manuscript and revising it. They all contributed in the conception and design of the analysis presented in this paper. XRP performed the analysis of the datasets and worked on the model definition and evaluation. CT gave clinical support in all the phases of the model definition and creation. AS contributed to data collection, labeling and pre-processing. MST and FB supervised the study from the clinical perspective. EV scientifically supervised the study. Finally, all the authors given final approval of the version to be published and agreed to be accountable for all aspects of the work.Ethics approval and consent to participateEthics approval was obtained from the Human Research Ethics Committee of University Hospital of Arnau de Vilanova (Lleida) with the reference number 10/2014, in accordance with the Declaration of Helsinki, and written informed consent to participate was obtained from each patient included in the investigation.Consent for publicationNot applicable.Competing interestsThe authors declare that they have no competing interests.
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Self-reported habitual snoring and risk of cardiovascular disease and all-cause mortality. D Li, D Liu, X Wang, D He, Atherosclerosis. 2351Li D, Liu D, Wang X, He D. Self-reported habitual snoring and risk of cardiovascular disease and all-cause mortality. Atherosclerosis. 2014;235(1):189-95.
Mortality in obstructive sleep apnea-hypopnea patients treated with positive airway pressure. F Campos-Rodriguez, N Pena-Grinan, N Reyes-Nunez, I De La Cruz-Moron, J Perez-Ronchel, F De La Vega-Gallardo, A Fernandez-Palacin, CHEST J. 1282Campos-Rodriguez F, Pena-Grinan N, Reyes-Nunez N, De la Cruz-Moron I, Perez-Ronchel J, De la Vega-Gallardo F, Fernandez-Palacin A. Mortality in obstructive sleep apnea-hypopnea patients treated with positive airway pressure. CHEST J. 2005;128(2):624-33.
Predictors of long-term adherence to continuous positive airway pressure therapy in patients with obstructive sleep apnea and cardiovascular disease in the save study. C L Chai-Coetzer, Y-M Luo, N A Antic, X-L Zhang, B-Y Chen, Q-Y He, E Heeley, S-G Huang, C Anderson, N-S Zhong, Sleep. 3612Chai-Coetzer CL, Luo Y-M, Antic NA, Zhang X-L, Chen B-Y, He Q-Y, Heeley E, Huang S-G, Anderson C, Zhong N-S, et al. Predictors of long-term adherence to continuous positive airway pressure therapy in patients with obstructive sleep apnea and cardiovascular disease in the save study. Sleep. 2013;36(12):1929-37.
Identifying poor compliance with cpap in obstructive sleep apnoea: a simple prediction equation using data after a two week trial. D Ghosh, V Allgar, M Elliott, Respir Med. 1076Ghosh D, Allgar V, Elliott M. Identifying poor compliance with cpap in obstructive sleep apnoea: a simple prediction equation using data after a two week trial. Respir Med. 2013;107(6):936-42.
Continuous positive airway pressure for sleep apnoea/hypopnoea syndrome: usefulness of a 2 week trial to identify factors associated with long term use. G Popescu, M Latham, V Allgar, M Elliott, Thorax. 569Popescu G, Latham M, Allgar V, Elliott M. Continuous positive airway pressure for sleep apnoea/hypopnoea syndrome: usefulness of a 2 week trial to identify factors associated with long term use. Thorax. 2001;56(9): 727-33.
Relationship between hours of cpap use and achieving normal levels of sleepiness and daily functioning. T E Weaver, G Maislin, D F Dinges, T Bloxham, C F George, H Greenberg, G Kader, M Mahowald, J Younger, A I Pack, Sleep. 306Weaver TE, Maislin G, Dinges DF, Bloxham T, George CF, Greenberg H, Kader G, Mahowald M, Younger J, Pack AI. Relationship between hours of cpap use and achieving normal levels of sleepiness and daily functioning. Sleep. 2007;30(6):711-9.
Towards an intelligent monitoring system for patients with obstructive sleep apnea. X Rafael-Palou, E Vargiu, C Turino, A Steblin, M Sánchez-De-La-Torre, F Barbé, ICST Trans Ambient Syst. 41Rafael-Palou X, Vargiu E, Turino C, Steblin A, Sánchez-de-la-Torre M, Barbé F. Towards an intelligent monitoring system for patients with obstructive sleep apnea. ICST Trans Ambient Syst. 2017;4:1.
Patient empowerment and case management in connecare. E Vargiu, J M Fernández, F Miralles, S Nakar, V Weijers, H Meetsma, S Mariani, M Mamei, F Zambonelli, F Michel, F Matthes, J Kelly, J Eaglesham, R Kaye, International Conference on Integrated Care. UtrechtVargiu E, Fernández JM, Miralles F, Nakar S, Weijers V, Meetsma H, Mariani S, Mamei M, Zambonelli F, Michel F, Matthes F, Kelly J, Eaglesham J, Kaye R. Patient empowerment and case management in connecare. In: International Conference on Integrated Care, Utrecht, May 23-25, 2018; 2018.
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[
"CONFORMAL CONTRACTIONS AND LOWER BOUNDS ON THE DENSITY OF HARMONIC MEASURE",
"CONFORMAL CONTRACTIONS AND LOWER BOUNDS ON THE DENSITY OF HARMONIC MEASURE"
] |
[
"Leonid V Kovalev "
] |
[] |
[] |
We give a concrete sufficient condition for a simply-connected domain to be the image of the unit disk under a nonexpansive conformal map. This class of domains is also characterized by having sufficiently dense harmonic measure. The relation with the harmonic measure provides a natural higher-dimensional analogue of this problem, which is also addressed.2010 Mathematics Subject Classification. Primary 30C62; Secondary 31A15, 31B05.
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10.1007/s11118-016-9586-6
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[
"https://arxiv.org/pdf/1601.02711v2.pdf"
] | 119,306,796 |
1601.02711
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c82359344782936cdf06ee61392e52edbb382b1f
|
CONFORMAL CONTRACTIONS AND LOWER BOUNDS ON THE DENSITY OF HARMONIC MEASURE
29 Jun 2016
Leonid V Kovalev
CONFORMAL CONTRACTIONS AND LOWER BOUNDS ON THE DENSITY OF HARMONIC MEASURE
29 Jun 2016
We give a concrete sufficient condition for a simply-connected domain to be the image of the unit disk under a nonexpansive conformal map. This class of domains is also characterized by having sufficiently dense harmonic measure. The relation with the harmonic measure provides a natural higher-dimensional analogue of this problem, which is also addressed.2010 Mathematics Subject Classification. Primary 30C62; Secondary 31A15, 31B05.
Introduction
The images of the unit disk D under conformal maps f with the normalization f (0) = 0 = f ′ (0) − 1 have long been understood and characterized in terms of their Green's function, capacity of the complement, and so on (e.g., the books [5] and [8] expose this circle of ideas). This paper studies the effect of a uniform bound on the derivative of a conformal map: namely, |f ′ (z)| ≤ 1 for all z ∈ D. This condition can be equivalently stated as |f (z) − f (w)| ≤ |z − w| for all z, w ∈ D; such f may be called a conformal contraction. Under the normalization f (0) = 0, it follows that the image f (D) must be contained in D. However, not every subdomain of the unit disk is its image under a conformal contraction.
Let us consider a convex domain Ω ⊂ C that contains 0 and has C 1,1smooth boundary. With such a domain we associate three radii:
• outer radius R O is the smallest radius of a disk centered at 0 and containing Ω; • inner radius R I is the largest radius of a disk centered at 0 and contained in Ω; • curvature radius R C is the minimal radius of curvature of ∂Ω. It is the largest radius R such that Ω can be written as a union of open disks of radius R.
Note that R O ≥ R I and R O ≥ R C , while there is no general relation between R I and R C .
Theorem 1.1.
Let Ω ⊂ C be a convex domain that contains 0 and has C 1,1 -smooth boundary. If the radii R O , R I , and R C satisfy
(1.1) (R O − R C ) log R I − log R C R I − R C + 1 2 log R C ≤ 0
then Ω = f (D) for some conformal map f such that f (0) = 0 and sup|f ′ | ≤ 1. (When R I = R C , the difference quotient is understood as 1/R I .)
We will also consider the harmonic measure of domain Ω with respect to 0, denoted ω Ω (·, 0). In the context of Theorem 1.1, of particular interest is the Radon-Nikodym derivative of ω Ω (·, 0) with respect to arclength, which will be called the density of harmonic measure.
The images of D under conformal contractions fixing 0 are precisely those domains Ω for which the density of ω Ω (·, 0) is at least 1/(2π) everywhere on the boundary. This follows immediately from the conformal invariance of harmonic measure and the fact that its density on the boundary of the unit disk is 1/(2π). Thus, Theorem 1.1 gives a sufficient condition for Ω to have harmonic measure with such a lower density bound. Theorem 1.1 was prompted by a question of J. E. Tener [7] which arose in the following context. When f is a conformal map of D into itself with f (0) = 0, the composition with f is a contraction on the Hardy space H 2 (D), see [3,Corollary 3.7]. By the conformal invariance of harmonic measure, this implies that the restriction operator R : H 2 (D) → L 2 (∂Ω, ω Ω (·, 0)) is a contraction. A lower bound on the density of ω Ω (·, 0) then allows one to estimate the norm of the restriction operator R : H 2 (D) → L 2 (∂Ω) where L 2 is taken with respect to arclength.
Concerning the structure of condition (1.1) it should be noted that the term
(R O − R C ) log R I − log R C R I − R C
is scale-invariant, while the second term, 1 2 log R C , tends to −∞ as the domain is scaled down. Thus, for any convex domain Ω of class C 1,1 Theorem 1.1 gives an explicit factor λ > 0 such that the scaled-down domain λΩ is the image of D under a conformal contraction. This can be compared to the classical Kellogg-Warschawski theorem [6, Theorem 3.5] which asserts that the conformal map of the disk onto a Dini-smooth Jordan domain Ω has a uniformly continuous derivative. The latter also implies that λΩ is the image of D under a conformal contraction for sufficiently small λ > 0. However, in contrast to Theorem 1.1, one does not have an explicit suitable value of λ in this case.
Examples illustrating and motivating the condition (1.1) are given in §2.
The higher-dimensional version of Theorem 1.1 is stated in terms of the harmonic measure, since there is no longer a rich supply of conformal maps. The desired property of Ω in this case is having the density of ω Ω (·, 0) at least 1/σ n−1 , where σ n−1 is the surface area of the unit sphere. The quantity 1/σ n−1 is the density of the harmonic measure of the unit ball with respect to its center. We will also use the notation
a + = max(a, 0), φ(a, b) = log a − log b a − b , φ(a, a) = 1/a. Theorem 1.2.
Let Ω ⊂ R n , n > 2, be a convex domain that contains 0 and has C 1,1 -smooth boundary. If the radii R O , R I , and R C satisfy
(1.2) R C R n−2 I e n(R O −R C −R I /2) + φ(R I /2,R C ) ≤ 2 n−2 − 1 2 n−1 (n − 2)
then the density of the harmonic measure of Ω with respect to 0 is bounded below by σ −1 n−1 .
The exponential term in (1.2) is scale invariant, while the factor R C R n−2 I makes sure that the left hand side of (1.2) tends to 0 as the domain is scaled down.
Examples and counterexamples
The sufficient conditions of Theorems 1.1 and 1.2 are not necessary; however, they are reasonably precise. For example, in the special case R O = R I = R C = R the hypothesis of Theorem 1.1 is that R ≤ 1, which is both necessary and sufficient in this case. The higher-dimensional estimate is less accurate: the inequality (1.2) simplifies to R ≤ 1 2 ((2 n−2 − 1)/(n − 2)) 1/(n−1) , where the right hand side is less than 1 but converges to 1 as n → ∞.
To justify the presence of three radii R O , R I , R C in Theorems 1.1 and 1.2, let us note that constraining just two of them would not be sufficient for the conclusion. Indeed, a convex polygon has zero density of harmonic measure at the vertices. Slightly rounding the corners, one obtains a domain that fails the conclusion of the theorem, which only the curvature radius detects. To show the necessity of R I , let Ω be the disk of radius 1 centered at the point 1 − ǫ; the density of ω Ω (·, 0) is small on most of the boundary. Finally, letting Ω be the convex hull of the union of two disks such as D(0, 1)∪D(n, 1) shows that the presence of R O is also necessary.
Preliminaries: hyperbolic and quasihyperbolic metrics
The hyperbolic metric on the unit disk D is
ρ D (z, w) = inf γ |dζ| 1 − |ζ| 2
where the infimum is taken over all rectifiable curves γ connecting z and w. In particular,
(3.1) ρ D (z, 0) = |z| 0 dt 1 − t 2 = 1 2 log 1 + |z| 1 − |z| .
On other simply-connected domains the hyperbolic metric can be defined by its conformal invariance property: ρ Ω (f (z), f (w)) = ρ D (z, w) if f is a conformal map of D onto Ω. In particular, for a disk Ω = D(a, R) we have
(3.2) ρ D(a,R) (z, a) = 1 2 log R + |z − a| R − |z − a| .
As a consequence of the Schwarz-Pick lemma, the hyperbolic metric is monotone with respect to domain: if G and Ω are two simply-connected domains and z, w ∈ G ⊂ Ω, then
(3.3) ρ G (z, w) ≥ ρ Ω (z, w).
The quasihyperbolic metric ρ * Ω is defined by
ρ * Ω (z, w) = inf γ |dζ| dist(ζ, ∂Ω) .
It is not conformally invariant, but is comparable to ρ Ω for every simplyconnected domain: When the domain Ω is rescaled by the map z → (1 − ǫ)z, the left side of (1.1) decreases. Therefore, we may assume that strict inequality holds in (1.1).
(3.4) 1 4 ρ * Ω (z, w) ≤ ρ Ω (z, w) ≤ ρ * Ω (z, w).
Let f be a conformal map of the unit disk D onto Ω, normalized by f (0) = 0. By the Kellogg-Warschawski theorem [6,Theorem 3.5], f ′ has a continuous extension to D, and for ζ ∈ ∂D we have
(4.1) lim z→ζ, z∈D f (z) − f (ζ) z − ζ = f ′ (ζ).
By the maximum principle, it suffices to show |f ′ | ≤ 1 on ∂D. By (4.1) it suffices to show that
(4.2) lim |z|ր1 dist(f (z), ∂Ω) 1 − |z| ≤ 1.
Fix z ∈ D and let d = dist(f (z), ∂Ω). Since the small values of d are of interest, we may assume d < R C . Our plan is to estimate ρ Ω (0, f (z)) from above, which will yield
(4.3) |z| < 1 − d
for sufficiently small d, thus proving (4.2). Choose a point w ∈ ∂Ω such that |f (z) − w| = d. By the definition of R C , there is a disk D = D(a, R C ) that has w on its boundary and is contained in Ω. Observe that f (z) lies on the radius of this disk connecting a to w, and therefore |f (z) − a| = R C − d. By (3.3) and (3.2), (4.4) ρ
Ω (f (z), a) ≤ ρ D(a,R C ) (f (z), a) = 1 2 log 2R C − d d ≤ 1 2 log 2R C d .
To estimate ρ Ω (a, 0) we use the comparison with ρ * Ω stated in (3.4). Since Ω contains D(0, R I ) and D(a, R C ), the convexity of Ω implies
(4.5) dist(ta, ∂Ω) ≥ tR C + (1 − t)R I , 0 ≤ t ≤ 1.
Integration along the line segment from 0 to a yields (4.6) ρ * Ω (a, 0) ≤ |a|
1 0 dt tR C + (1 − t)R I = |a|φ(R I , R C ).
Since D(a, R C ) ⊂ Ω ⊂ D(0, R O ), we have |a| ≤ R O − R C . In conclusion,
(4.7) ρ Ω (a, 0) ≤ (R O − R C )φ(R I , R C ).
Suppose that (4.3) fails, that is, |z| ≥ 1−d. From the conformal invariance of hyperbolic metric, (4.8) ρ
Ω (f (z), 0) = 1 2 log 1 + |z| 1 − |z| ≥ 1 2 log 2 − d d .
Combining (4.4), (4.7), and (4.8) we obtain
1 2 log 2 − d d ≤ 1 2 log 2R C d + (R O − R C )φ(R I , R C ), hence (4.9) 1 2 log 1 − d 2 ≤ 1 2 log R C + (R O − R C )φ(R I , R C ).
Since the right hand side of (4.9) is negative, the inequality implies a lower bound on d. Therefore, (4.3) hold provided that d is sufficiently small. This proves Theorem 1.1.
5.
Higher dimensions: proof of Theorem 1.2
In this section Ω is a convex domain in R n , n > 2, and 0 ∈ Ω. The density of ω Ω (·, 0) with respect to the surface measure of ∂Ω is related to Green's function g Ω by
ω Ω (E, 0) = E ∂g Ω ∂n .
Here the derivative is taken along the interior normal, and g Ω is Green's function with pole at 0, normalized by g Ω (x, 0) = 1 (n−2)σ n−1 |x| 2−n + O(1) as x → 0.
Thus, to prove that the density of harmonic measure is no less than σ −1 n−1 , it suffices to show that
(5.1) g Ω (x) ≥ 1 + o(1) σ n−1 dist(x, ∂Ω), x → ∂Ω.
To this end we use a lower bound for g Ω in terms of the quasihyperbolic metric. An estimate of this kind is given in Section 1.2 of [1], namely g Ω (x, 0) ≥ exp(−Aρ * Ω (x, 0)) with unspecified A. But we need a more explicit bound, since the presence of A in the exponent does not allow one to conclude with (5.1).
Fix x ∈ Ω and let d = dist(x, ∂Ω), assuming d < R C . Choose a point w ∈ ∂Ω such that |x − w| = d. By the definition of R C , there is a ball B(a, R C ) that has w on its boundary and is contained in Ω. Since |x − a| = R C − d, Harnack's inequality [2, Theorem 1.4.1] yields
(5.2) g Ω (x) ≥ (R C − |x − a|)R n−2 C (R C + |x − a|) n−1 g Ω (a) = d R n−2 C (2R C − d) n−1 g Ω (a).
Since B(0, R I ) ⊂ Ω, it follows that the restriction of g Ω to B(0, R I ) is minorized by Green's function of this ball: specifically,
(5.3) g Ω (x) ≥ 1 (n − 2)σ n−1 (|x| 2−n − R 2−n I ), |x| < R I .
In particular, at the point a ′ = R I 2 a |a| we have
(5.4) g Ω (a ′ ) ≥ 2 n−2 − 1 (n − 2)σ n−1 R 2−n I .
As a corollary of Harnack's inequality [2, Corollary 1.4.2], the gradient of a positive harmonic function on B(a, r) satisfies |∇u(a)| ≤ (n/r)u(a). Therefore,
|∇ log g Ω (x)| ≤ n/ dist(x, ∂Ω ′ )
where Ω ′ = Ω \ {0}. This implies (5.5) | log g Ω (x) − log g Ω (y)| ≤ nρ * Ω ′ (x, y). Case 1: |a| ≥ R I /2. Since |a ′ | = R I /2 ≤ |a| ≤ R O − R C and a ′ is a scalar multiple of a, it follows that |a − a ′ | ≤ (R O − R C − R I /2). Observe that the domain Ω ′ contains the balls B(a ′ , R I /2) and B(a, R C ), as well as their convex hull. Integration similar to (4.6) yields ρ * Ω ′ (a, a ′ ) ≤ |a − a ′ |φ(R I /2, R C ) ≤ (R O − R C − R I /2)φ(R I /2, R C ). Using (5.5) we obtain g Ω (a) ≥ 2 n−2 − 1 (n − 2)σ n−1 R 2−n I e −n(R O −R C −R I /2)φ(R I /2,R C ) .
Case 2: |a| < R I /2. Instead of using a ′ , we have which by virtue of (5.2) implies (5.6)
σ n−1 g Ω (x) d ≥ R n−2 C (2R C − d) n−1 2 n−2 − 1 n − 2 R 2−n I e −n(R O −R C −R I /2) + φ(R I /2,R C ) .
As d → 0, the right hand side of (5.6) converges to 2 n−2 − 1 2 n−1 (n − 2) 1 R C R n−2 I e −n(R O −R C −R I /2) + φ(R I /2,R C ) ≥ 1.
This proves (5.1) and concludes the proof of Theorem 1.2.
Figure 1 . 1 Figure 1
111A domain in Theorem 1.presents a concrete example of a domain that satisfies (1.1), namely a rounded triangle with R O = 0.6, R I = 0.5, and R C = 0.4. A domain satisfying (1.2) could have n = 3, R O = 3/4, and R I = R C = 1/2.
e
−n(R O −R C −R I /2) + φ(R I /2,R C )
E-mail address: [email protected]
Acknowledgement. I would like to thank the referee for carefully reading the paper and suggesting several improvements.
Intrinsic ultracontractivity via capacitary width. H Aikawa, Rev. Mat. Iberoam. 313H. Aikawa, Intrinsic ultracontractivity via capacitary width, Rev. Mat. Iberoam., 31, no. 3, 1041-1106 (2015).
D H Armitage, S J Gardiner, Classical potential theory. Springer-VerlagD. H. Armitage, S. J. Gardiner, Classical potential theory. Springer-Verlag (2001)
Composition operators on spaces of analytic functions. C C Cowen, B D Maccluer, CRC PressBoca Raton, FLC. C. Cowen, B. D. MacCluer, Composition operators on spaces of analytic functions. CRC Press, Boca Raton, FL, (1995)
Harmonic measure. J B Garnett, D E Marshall, Cambridge University PressJ. B. Garnett, D. E. Marshall, Harmonic measure. Cambridge University Press (2005)
Geometric theory of functions of a complex variable. G M Goluzin, American Mathematical Society. G. M. Goluzin, Geometric theory of functions of a complex variable. American Math- ematical Society, Providence, R.I. (1969)
Boundary behaviour of conformal maps. Ch, Pommerenke, Springer-VerlagBerlinCh. Pommerenke, Boundary behaviour of conformal maps. Springer-Verlag, Berlin (1992)
Bounds on the derivative of a Riemann map. J E Tener, J. E. Tener, Bounds on the derivative of a Riemann map, MathOverflow. http://mathoverflow.net/q/165037
Potential theory in modern function theory. M Tsuji, Maruzen Co., LtdTokyoM. Tsuji, Potential theory in modern function theory. Maruzen Co., Ltd., Tokyo (1959)
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[] |
[
"Composite fermion model for entanglement spectrum of fractional quantum Hall states",
"Composite fermion model for entanglement spectrum of fractional quantum Hall states"
] |
[
"Simon C Davenport \nT.C.M Group\nCavendish Laboratory\nJ.J. Thomson AvenueCB3 0HECambridgeUnited Kingdom\n",
"Iván D Rodríguez \nMax-Planck-Institut für Quantenoptik\n85748GarchingGermany\n",
"J K Slingerland \nDepartment of Mathematical Physics\nNational University of Ireland\nMaynoothIreland\n\nDublin Institute for Advanced Studies\nSchool of Theoretical Physics\n10 Burlington RdDublinIreland\n",
"Steven H Simon \nRudolf Peierls Centre for Theoretical Physics\n1 Keble RoadOX1 3NPOxfordUK\n"
] |
[
"T.C.M Group\nCavendish Laboratory\nJ.J. Thomson AvenueCB3 0HECambridgeUnited Kingdom",
"Max-Planck-Institut für Quantenoptik\n85748GarchingGermany",
"Department of Mathematical Physics\nNational University of Ireland\nMaynoothIreland",
"Dublin Institute for Advanced Studies\nSchool of Theoretical Physics\n10 Burlington RdDublinIreland",
"Rudolf Peierls Centre for Theoretical Physics\n1 Keble RoadOX1 3NPOxfordUK"
] |
[] |
We show that the entanglement spectrum associated with a certain class of strongly correlated many-body states -the wave functions proposed by Laughlin and Jain to describe the fractional quantum Hall effect -can be very well described in terms of a simple model of non-interacting (or weakly interacting) composite fermions. arXiv:1506.06741v1 [cond-mat.str-el]
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10.1103/physrevb.92.115155
|
[
"https://arxiv.org/pdf/1506.06741v1.pdf"
] | 118,600,307 |
1506.06741
|
1571a5bcfefe2e4c2a83eccad175ebc12f1880e2
|
Composite fermion model for entanglement spectrum of fractional quantum Hall states
22 Jun 2015
Simon C Davenport
T.C.M Group
Cavendish Laboratory
J.J. Thomson AvenueCB3 0HECambridgeUnited Kingdom
Iván D Rodríguez
Max-Planck-Institut für Quantenoptik
85748GarchingGermany
J K Slingerland
Department of Mathematical Physics
National University of Ireland
MaynoothIreland
Dublin Institute for Advanced Studies
School of Theoretical Physics
10 Burlington RdDublinIreland
Steven H Simon
Rudolf Peierls Centre for Theoretical Physics
1 Keble RoadOX1 3NPOxfordUK
Composite fermion model for entanglement spectrum of fractional quantum Hall states
22 Jun 2015(Dated: August 9, 2016)
We show that the entanglement spectrum associated with a certain class of strongly correlated many-body states -the wave functions proposed by Laughlin and Jain to describe the fractional quantum Hall effect -can be very well described in terms of a simple model of non-interacting (or weakly interacting) composite fermions. arXiv:1506.06741v1 [cond-mat.str-el]
Understanding of strongly correlated many-body quantum states has been significantly enhanced in recent years by the introduction of the entanglement spectrum (ES), i.e. the spectrum of eigenvalues of the reduced density matrix of a subsystem. 1 The scrutiny of the ES has been particularly fruitful in the study of fractional quantum Hall (FQH) wave functions -the archetype for topological phases of matter resulting from strong interaction effects -where this analysis often reveals a special underlying entanglement structure useful for identifying and classifying different topological phases. [1][2][3][4][5][6] For FQH states in particular it has been established that there is a deep connection between the entanglement structure of bulk wave functions (as seen in their ES [7][8][9] ) and the structure of edge excitations. [10][11][12][13] Central to this connection so far have been arguments based on the common conformal field theory description of the bulk and edge physics, which is known to apply for the class of wave functions that can be constructed in terms of expansion functions known as "conformal blocks". The ES for this class of what we shall refer to as "simple" FQH states has been extensively studied using the machinery of conformal field theory. 4,5,9 Largely separate from these developments, it has been well documented that the composite fermion model of the FQH effect can successfully account for many experimentally observed features of the effect, particularly in the lowest Landau level. 14 It does so by positing that they are due to the integer quantum Hall (IQH) effect of non-interacting (or weakly interacting) quasiparticles known as composite fermions. While composite fermion theory presents an appealingly simple physical picture to explain the FQH effect, it has proved to be very challenging to relate the many-body composite fermion wave functions to constructs in conformal field theory. While such constructs do exist, 15-18 they are somewhat complicated, and this has so far limited the effectiveness of conformal field theory machinery in describing the ES of the Jain states. In this sense the Jain states are not "simple" FQH states.
In this work we present a very different approach to constructing entanglement spectra that bypasses the po-tential difficulty associated with explicitly writing wave functions in terms of conformal blocks. We show that the low-lying (highest weight) part of the ES of the Jain FQH states can be accurately described by a modified ES of non-interacting (or weakly interacting) composite fermions in filled Landau levels (in other words a modified ES of the IQH states [19][20][21] ). To demonstrate the effectiveness of our method, we shall focus on presenting results for two fundamental examples, namely the Laughlin state at filling factor ν = 1/2 for bosons (a simple FQH state that was previously studied using conformal field theory methods 9 ) and the Jain state at ν = 2/3 for bosons (a non-simple FQH state). (The method applies equally well to fermionic FQH states.) Our results for the Laughlin case are in good agreement with the approach based on conformal field theory, but the real advantage of our method becomes evident for the Jain states, where the conformal field theory approach appears much more complicated, and has therefore not been worked out.
In the spirit of the original composite fermion model, 14 we start with the ES for the IQH states, 20,21 treating it in a general framework for determining the ES of noninteracting systems. 19 We shall briefly review its derivation here. For simplicity we shall consider spinless particles. Also, to remove any additional complications due to edge physics we consider a system without physical edges. A standard technique to achieve this is to solve the problem on the surface of sphere. 22 Here we have an integer n Landau level problem (where the filling factor is ν = n) for N particles on a sphere of radius √ Q (in units of magnetic length) that encloses a fictitious magnetic monopole of strength 2Q = (N − n 2 )/n. There is rotational symmetry about the z-axis, which leads to the single-particle orbitals of the problem being labelled by the z-component of angular momentum m = −(2Q + σ)/2, −(2Q + σ)/2 + 1, . . . , (2Q + σ)/2 in addition to a Landau level index σ = 0, 1, . . . , n − 1. The single-particle orbitals are φ m,σ (r), where r lies on the surface of the sphere (technically, φ m,σ (r) are monopole harmonics. 23 The monopole harmonics also depend explicitly on Q, but for simplicity we suppress this dependence in our notation).
In order to connect to existing calculations we shall focus on the entanglement spectrum for wave functions in real space (as opposed to e.g. momentum space), which leads to the so-called real-space entanglement spectrum (RSES). [4][5][6] We shall consider cuts along lines of latitude, so that that rotational invariance about the z-axis is preserved and the z-component of angular momentum L z remains a good quantum number. Once the system is cut, L z is bipartitioned as L z = L A z + L B z . Additionally, the total particle number is bipartitioned as
N = N A + N B .
For the full fermionic N -particle system, we have a basis for the Hilbert space that consists of Slater determinants built from the orbitals φ m,σ . For the A and B subsystems, the restrictions of these orbitals to the regions A and B still form a complete single-particle basis, although the restricted orbitals must be renormalized to account for the fact that only part of the single-particle weight is in each subsystem. Once restricted to a subsystem, the orbitals φ m1,σ1 and φ m2,σ2 are orthogonal whenever m 1 = m 2 (because the cut is made to preserve L z ), but are generally not orthogonal whenever σ 1 = σ 2 (though they remain linearly independent in this case). The Fock space of the A and B subspaces is spanned by Slater determinants in terms of these restricted orbitals.
We can now write the Schmidt decomposition of the state |ψ as
|ψ = i e −ξi/2 ψ A i ⊗ ψ B i (1)
where ψ A i and ψ B i belong to the Fock spaces for the A and B subsystems that we have just described. The nonzero Schmidt coefficients, e −ξi/2 , are conveniently written in terms of the entanglement energies, ξ i . The index i labels the vectors in the Schmidt basis for the A system (at least those with non-zero Schmidt coefficients). Since N A and L A z are conserved by the cut, we can choose the Schmidt basis to consist of eigenstates of N A and L A z . We can then write ξ N A ,L A z ,i for the entanglement energy of the i th Schmidt state with given N A and L A z . In fact, with any Schmidt state with N A particles and given L A z , we can associate an N A -tuple of single par-
ticle momenta m = (m 1 , . . . , m N A ) with L A z = p m p and m 1 < m 2 < . . . < m N A .
If there is only a single Landau level involved, the Schmidt state is just the unique Slater determinant in the Fock space of system A labelled by m and we can replace the label i above by m. With multiple Landau levels, there will be a number of Schmidt states associated with each m. These states are superpositions of the Slater determinants with this m and different choices of the numbers of particles in each Landau level.
One can find exact expressions for the entanglement energies and Schmidt states in the case of a noninteracting Landau level problem, 20 , using an argument proposed by Peschel. 19 If one constructs the correlation matrix, (2) then the eigenvalues, λ m,σ , of C are related to the entanglement energies by
C m1,σ1,m2,σ2 = A φ * m1,σ1 (r)φ m2,σ2 (r) dr = δ m1,m2 A φ * m1,σ1 (r)φ m1,σ2 (r) drξ i = m,σ o m,σ,i m,σ + constant,(3)
where the single-particle entanglement energy function, m,σ , is defined by
m,σ = log 1 − λ m,σ λ m,σ .(4)
Note that now σ labels the eigenstates of the correlation matrix at given m, rather than a Landau level. The full Schmidt states are Slater determinants built from these eigenstates and o m,σ,i denotes the occupation number of the correlation matrix eigenstates labelled by (m, σ) in the Schmidt state with label i. Examples of the RSES for the ν = 1 and ν = 2 IQH states calculated using this method are included in Fig. 1a and Fig. 2a for later comparison with our results for the FQH case.
We now come to the statement of our main result: For the class of FQH states proposed by Laughlin and Jain to describe the FQH effect in the lowest Landau level at filling factors ν = n /(2n + 1) [for fermions] and ν = n /(n + 1) [for bosons] with n integer, the low-lying ξ in the associated RSES can be accurately described by the same model as the IQH states (Eq. 3), but a different single-particle entanglement energy function (Eq. 4). In addition, when multiple effective Landau levels are present in the Jain wave functions, one also needs to take into account a simple exchange-entanglement-energy term that we shall shortly describe with an example. In this connection the Landau level index σ for the IQH wave functions becomes the effective Landau level index of the Jain states, σ = 0, 1, . . . , n − 1, the magnetic field described by the monopole strength Q is replaced with an effective magnetic field and an effective monopole strength Q = (N − n 2 )/n and the occupation basis of single-particle Landau level orbitals now becomes an occupation basis of so-called composite fermion orbitals in the effective Landau levels, whose single-particle eigenstates are labelled by m = −(2Q +σ )/2, −(2Q +σ )/2+ 1, . . . , (2Q + σ )/2.
It is useful to expand m,σ from Eq. 4 as a power series in m about the mid-point m = 0, as it turns out that we need to keep only very few terms in this expansion for a very accurate description of m,σ . For example, in the case of a single Landau level (ν = n = 1 for spinless particles) or a single effective Landau level (n = 1 and replacing m with m in the expansion) m,σ ≈ a 0 + a 1 m + a 2 m 2 + a 3 m 3 + . . . , where, for the IQH case, a 0 , a 1 etc. are functions that can be expressed in terms of the eigenvalues of the matrix C and its derivatives (note that they depend on the system size, via their dependence on the monopole strength Q). For 2 filled Landau levels (ν = n = 2 for spinless particles) or two effective Landau levels (n = 2) the expansion becomes instead m,σ ≈ a 0,0 + a 0,1 m + a 0,2 m 2 + a 0,3 m 3 + . . .
+ [σ − 1/2] a 1,0 + a 1,1 m + a 1,2 m 2 + . . . .(6)
In our approach for the FQH case the coefficients a j,k now play the role of fitting parameters in a model given by truncating the single particle energy function at a low degree in m . Their values are determined by fitting this model to the RSES generated numerically from a micro-scopic construction of the FQH state using other methods. Further details of the fitting procedure are given in Appendix A. To obtain an appropriate numerical spectrum to fit to for large system sizes we use the method proposed in Ref. 25 (a method that works more efficiently for bosonic rather than fermionic states).
To illustrate how our approach works we present two examples of applications to bosonic FQH states, where we choose symmetrical, equal-sized regions A and B (i.e. a cut along the equator of the sphere) and N A = N B = N/2. 24 The method also applies for more general cuts.
As our first example, we consider at the RSES of the Laughlin state of bosons at filling factor ν = 1/2 (this state also occurs in the Jain series for n = 1). The RSES for ν = 1/2 is accurately described by the singleparticle energy function in Eq. 5, written in terms of the composite fermion angular momentum labels m and truncating the expansion at order m 3 . In this truncated expansion, the coefficients a 1 , a 2 and a 3 are free fitting parameters. It is not necessary to include a constant term (a 0 ) in this fit since its value can be fixed by normalizing the spectrum (see Appendix A). An example fit to the numerically-calculated RSES of the ν = 1/2 state is shown in Fig. 1b. Fitting parameter data is included in Appendix B. We find that by far the most dominant contribution comes from the a 1 term, indicating that the spectrum is almost linear. In Fig. 1a we also show the RSES of the IQH state for ν = 1 for comparison.
The Laughlin state falls into the class of simple FQH states and its RSES has been studied previously using conformal field theory techniques. In particular Ref. 9 describes in detail a method to model the Laughlin RSES in terms of the energy levels of an entanglement Hamiltonian that is given by writing down every allowable conformal field theory operator (in this case chiral boson operators, their derivatives and powers) order-by-order in their scaling dimension (with terms at lower scaling dimension providing more relevant contributions). These operators come with unknown coefficients that can be fitted to the numerically-calculated RSES using a similar procedure to that described in Appendix A. In Ref. 9 it was shown that the RSES of the ν = 1/2 state can be very accurately described by truncating at scaling dimension 2, leading to an entanglement Hamiltonian containing 3 terms (and therefore 3 unknown fitting parameters). We have checked that for the same data set as used here (Fig. 1b) the fit of this truncated entanglement Hamiltonian to the RSES is in excellent agreement with our non-interacting model (in fact, the quality of the fit as defined in Appendix A is typically improved by an order of magnitude compared to our approach). One could in principle reproduce our model directly (with the inclusion of additional interaction terms) by fermionizing Dubail-Read-Rezayi's entanglement Hamiltonian. The fact that our result agrees closely with the numericallycalculated RSES implies that any additional interaction terms must only be small corrections, which justifies the assumptions made by the composite fermion approach. 2 2 2 2 2 2 2 6 2 2 2 2 6 2 2 2 2 2 2 2 2 6 2 2 2 2 6 2 2 2 2 2 2 2 . Fit of the single-particle entanglement energy model (Eq. 8) to the ν = 2/3 Jain state calculated numerically for the same cut and system size using the method in Ref. 25. Fitting parameter data is included in Appendix B. σ tot. labels the number of particles in the upper effective Landau level for each ξ. Entanglement energies ξ and angular momentum labels L A z are relative to the lowest lying state ξ0 in the lowest angular momentum sector L A 0 i.e. ∆ξ = ξ − ξ0 and ∆L A z = L A 0 − L A z . Number labels embedded in the plots indicate the degeneracy of ξ where ambiguous. 24 Our second example is the RSES of the Jain state of bosons at filling factor ν = 2/3 (this state occurs in the Jain series for n = 2 for bosons). Our technique was not able to improve upon conformal field theory in treating simple FQH wave functions, but where it really excels is in its treatment of the Jain states. In this case we now use the single-particle energy from the ν = 2 IQH state (Eq. 6), but we augment it to take into account an exchange-like interaction (treated at the level of "mean-field theory") for fermions in different Landau levels. This term can alternatively be thought of as a "charging energy". Our ansatz to describe the RSES of the ν = 2/3 Jain state is
ξ N A ,L A z ,i = m,σ n m,σ m,σ + c (∆N ) 2 ,(7)
with the single-particle energy function given by
m ,σ ≈ a 0,1 m + [σ − 1/2] {a 1,0 + a 1,1 m } ,(8)
and the exchange-interaction, or charging energy term, given by
∆N = m ,σ n m ,σ [σ − 1/2].(9)
This ansatz requires 4 fitting parameters a 0,1 , a 1,0 , a 1,1 and c (once again, a constant term a 0,0 is not fitted because it can be fixed by normalization).
In Fig. 2b we show an example fit to the RSES of the 2/3 state for a large system of N = 48 particles. Fitting parameter data is included in Appendix B. We observe that the most dominant contribution arises from the coefficient a 1,0 , which can be thought of as an entanglement "cyclotron energy" term. We also find that the coefficient of the exchange interaction, c, is positive (so due to this additional term, the branches in the RSES for ν = 2/3 are further apart than they would have been otherwise). Note that there are degeneracies in the fitted entanglement energy eigenvalues, but these can be lifted by truncating the entanglement energy expansion at higher order in m . For comparison, in Fig. 2a we plot the RSES of the ν = 2 IQH state.
To summarize, we have described how a RSES of noninteracting composite fermions can be constructed to accurately approximate the RSES for certain strongly correlated FQH states. Key to this construction is the observation that the single-particle entanglement energy function underlying the description of the RSES for the IQH states can be simply modified in order to describe the RSES of FQH states, treating the many-body RSES within a non-interacting approximation and allowing for very basic mean-field theory like exchange energy corrections in the multi-Landau level case. We note that this description of the RSES closely parallels the description of the real energy spectrum of the edge of a Hall droplet in terms of composite fermions (see e.g. Ref. 26).
The quality of this approximation could be improved by allowing for higher order corrections to the underlying entanglement energy function (at the cost of needing more fitting parameters). These corrections can be thought of as allowing for additional inter-composite fermion interactions at the level of "mean field theory". Alternatively, weak interaction corrections can be added to the model perturbatively, working in the framework of degenerate perturbation theory as applied to the entanglement Hamiltonian. However, we find that such corrections only provide a marginal improvement to the fit.
online). (a). Exact RSES for the ν = 1 state for N = 50 particles, with equal-size A and B regions and NA = 25. Embedded number labels indicate the degeneracy of ξ (up to ∆L A z = 8). 24 (b). Fit of the single-particle entanglement energy model (Eq. 5 in terms of the composite fermion angular momentum m and truncated at order m 3 ) to the ES of the ν = 1/2 Laughlin state calculated numerically for the same cut and system size using the method in Ref.25. Fitting parameter data is included in Appendix B. Entanglement energies ξ and angular momentum labels L A z are relative to the lowest lying state ξ0 in the lowest angular momentum sector L A 0 i.e. ∆ξ = ξ − ξ0 and ∆L A z = L A 0 − L A z .
FIG. 2 .
2(color online). (a). Exact RSES for the ν = 2 state for N = 48 particles, with equal-size A and B regions and NA = 24. (b)
Acknowledgements: We thank S. Sondhi for helpful discussions. SCD was supported by EPSRC grant EP/J017639/1. IDR was supported by EU project SIQS. JKS was supported by Science Foundation Ireland Principal Investigator Award 12/IA/1697. SHS was supported by EPSRC grants EP/I032487/1 and EP/I031014/1. We acknowledge use of the Hydra computer cluster at the Rudolf Peierls Centre for Theoretical Physics. Statement of compliance with EPSRC policy framework on research data: This publication reports theoretical work that does not require supporting research data.Appendix A: Fitting algorithmIn this appendix we shall briefly describe our procedure for evaluating the goodness of fit of the RSES to the single-particle models proposed in this work.The problem is a many-parameter optimization where we aim to minimize a weighted sum of squared differences between corresponding ξ in the lowest lying part of the RSES and single-particle energy spectrum for a given set of fitting parameters {a j,k }. By corresponding ξ we mean that, for comparison, we normalize both the model spectrum and the numerically calculated spectrum such that the lowest lying ξ in the lowest ∆L A z sector is set to zero. Then we order the set of ξ in the model spectrum (call them ξ model ) and the lowest lying part of the RSES (call them ξ RSES ) by their ∆L A z values, then for each sector we sort in order of increasing ξ RSES or ξ model value and finally we take the sum of squared difference between the lowest ξ RSES and lowest ξ model , the difference between the next lowest ξ RSES and next lowest ξ model and so on. This sum can also be weighted in various ways, for instance if we want to give increased importance to matching up states with the lowest values of ξ. In general, therefore, we aim to minimize a fitting function of the formwhere {a j,k } denotes the set of free parameters in the model, and the index i appearing in the sum denotes the ith value of ξ model , ξ RSES within the ordered set (ordered in the sense described above). The factor W i assigns an optional weight to each term in the sum. In addition, we only include entanglement spectrum eigenvalues below a certain specified cut-off sector ∆L A cut-off , i.e. W i = 0 if it refers a state with ∆L A z > ∆L A cut-off . In our application we always include a factor in W i that divides out the number of states in each sector (i.e. for every ξ i EH labelled by the same ∆L A z we divide by a factor of N ∆L A z that counts the total number of ξ model with that ∆L A z ). The reason for doing this is because otherwise the fitting function would assign overwhelmingly more weight to fitting the higher sectors (because the counting of states in the RSES grows superpolynomially). Once the N ∆L A z factor is divided out, each sector counts for the same total weight in Eq. A1, i.e. i with ∆L A z W i = 1 for all ∆L A z . Another aspect to consider is that, in the RSES, linearly smaller ξ correspond to exponentially greater coefficients in the Schmidt decomposition. In this sense, it 6 should be more important physically to match up the lower lying ξ. For this reason, we might also consider including a factor exp(−ξ i RSES ) in our weight function. This addition works well for the Laughlin case, however the Jain case is more complicated due to the presence of the branch structure in the RSES. Consequently, for studying the Laughlin state we use the fitting function(A2) whereas for fitting the Jain state we find that for fitting all branches with equal weight it is better to useThe quality of a given fitting model (e.g. a given set of fitting parameters {a j,k }) can be assessed by the minimal value of the fitting function obtained for that model. The lower the minimal value of the fitting function, the better the quality of the fit.In order to solve the minimization problem we use the Powell method provided by Scientific Python (SciPy) version 0.11 and above, which involves a sequential 1D minimization of each fitting parameter. The Powell method is found to be numerically stable for this problem. One also has to take steps to avoid finding the local rather than the global minimum, which we achieve by running the procedure a large number of times with different random starting parameters.Appendix B: Fitting parameter dataIn this appendix we tabulate the values of the fitting parameters a j,k obtained in the single-particle energy fits shown inFig. 1 and Fig. 2. We also include the corresponding values of the fitting function R({a j,k }) defined in Appendix A to assess the relative quality of those fits.Example fitting parameter values and R Laughlin (Eq. A2) values in the single-particle model proposed in this work for the Laughlin ν = 1/2 state. Corresponding results plotted inFig. 1Example fitting parameter values and R Jain (Eq. A3) values in the single-particle model proposed in this work for the Jain ν = 2/3 state. Corresponding results plotted inFig. 2. Smaller values of R Jain indicate a better quality of fit:
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Note that due to our choice of equal-sized regions A and B, degeneracies occur in the RSES of the integer quantum Hall states. These degeneracies can be associated with a particle-hole symmetry in the occupations of the singleparticle orbitals in the Schmidt decomposition, Eq. 1. Such particle-hole degeneracy is no longer present once interactions are introduced as can be seen in e.g. Fig. 1bNote that due to our choice of equal-sized regions A and B, degeneracies occur in the RSES of the integer quantum Hall states. These degeneracies can be associated with a particle-hole symmetry in the occupations of the single- particle orbitals in the Schmidt decomposition, Eq. 1. Such particle-hole degeneracy is no longer present once interac- tions are introduced as can be seen in e.g. Fig. 1b.
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[] |
[
"A new properties of varieties of Leibnitz algebras",
"A new properties of varieties of Leibnitz algebras"
] |
[
"A V Shvetsova ",
"T V Skoraya "
] |
[] |
[] |
The paper is devoted to the study of the new properties of varieties of Leibnitz algebras. The characteristic of base field Φ assumed to be zero. All undefined concepts can be found in[3]. The article presented two new results. The first result belongs to the second author and contains a proof of the sufficient conditions for the finiteness of the colength varieties of Leibnitz algebras. The second belongs to the first author. In it are found a basis of identities and a basis of the space of multilinear elements of variety V 3 of Leibnitz algebras.
|
10.18500/1816-9791-2013-13-4-124-129
|
[
"https://arxiv.org/pdf/1405.4603v1.pdf"
] | 119,261,283 |
1405.4603
|
615380beac4f7a94f26d99259f111bc057f78d8c
|
A new properties of varieties of Leibnitz algebras
19 May 2014 MSC2010 17D99
A V Shvetsova
T V Skoraya
A new properties of varieties of Leibnitz algebras
19 May 2014 MSC2010 17D99
The paper is devoted to the study of the new properties of varieties of Leibnitz algebras. The characteristic of base field Φ assumed to be zero. All undefined concepts can be found in[3]. The article presented two new results. The first result belongs to the second author and contains a proof of the sufficient conditions for the finiteness of the colength varieties of Leibnitz algebras. The second belongs to the first author. In it are found a basis of identities and a basis of the space of multilinear elements of variety V 3 of Leibnitz algebras.
Introduction
A linear algebra with bilinear multiplication, which is satisfies to the Leibnitz identity (xy)z ≡ (xz)y + x(yz), is called a Leibnitz algebra. Perhaps for the first time this concept was discussed in the article [2] as a generalization of Lie algebras. The Leibnitz identity allows any element expressed as a linear combination of elements in which the brackets are arranged from left to right. Therefore further agree omit brackets in left-normed products, i.e. (((ab)c) . . . d) = abc . . . d. A variety V of linear algebras over a field Φ is a set of algebras over this field that satisfy a fixed set of identities. Note, that the system of identities can be given implicitly. In this case the variety V is usually defined generating algebra given constructively.
Let F (X, V) be a relatively free algebra of variety V with countable set of free generators X = {x 1 , x 2 , . . . }. Consider the space of the multilinear elements of algebra F (X, V). This space we will denote P n (V) and call multilinear part of variety V. On this space naturally introduce the action of permutations that we can consider it as a ΦS n -module, where S n is a symmetric group. Since the field Φ has zero characteristic, then the space P n (V) is the direct sum of irreducible submodules. Denote by χ λ the character of the irreducible representations of the symmetric group, which corresponds to the partition λ of the number n. Then the character of module P n (V) is expressed by the formula
(1) χ(P n (V)) = λ⊢n m λ χ λ ,
where m λ are the multiplicities of irreducible submodules in this sum. An important numerical characteristic of variety V of linear algebras is the colength l n (V), which is defined as the number of terms in the decomposition of character in the sum of irreducible characters:
(2) l n (V) = λ⊢n m λ .
We say that the colength V of variety is finite if there exists a constant C independent of n such that for any n is true the inequality l n (V) ≤ C. The right multiplication operator, for example, on element z, we denote by Z, assuming that xz = xZ. This designation allows the element xy...y n to write in the form xY n . Recall that the standard polynomial of degree n has the form: St n (x 1 , x 2 , . . . , x n ) = q∈Sn (−1) q x q(1) x q(2) . . . x q(n) , where the summation is carried out by elements of the symmetric group, and (−1) q is equal to +1 or −1 depending on the parity of permutation q. Agree variables in standard polynomial denote with special characters above (below, wave and so on). For example the standard polynomial of degree n in the variables x 1 , x 2 , . . . , x n we will write as follows: St n = x 1 x 2 . . . x n . It is clear that the standard polynomial is skew symmetric. Variables in different skew symmetric sets will be denoted by different symbols, for example:
q∈Sn,p∈Sm (−1) q (−1) p x q(1) x q(2) . . . x q(n) y p(1) y p(2) . . . y p(m) = = x 1 x 2 . . . x n y 1 y 2 . . . y m .
Sufficient condition for the finiteness colength of varieties of Leibnitz algebras
Previously, in the article [7] are identified the necessary conditions for the finiteness colength of varieties of Leibnitz algebras. Further, we consider it sufficient conditions. Following article [6] denote the variety of all Leibnitz algebras (Lie algebras), that satisfy the identity (
x 1 x 2 )(x 3 x 4 ) . . . (x 2s+1 x 2s+2 ) ≡ 0, by N s A (relatively N s A)
. Let in addition V 1 = N 2 A is a variety of all Lie algebras, commutator of which is nilpotent of class not more then two, and V 1 is a variety of Leibnitz algebras defined by the identity
x 1 (x 2 x 3 )(x 4 x 5 ) ≡ 0.
Theorem 1. Let V be a subvariety of variety N s A, which for any natural numbers k, m, k ≤ m, and α 1 , . . . , α k ∈ K satisfies the identity
(3) xY k zY m−k ≡ k i=1 α i xY k−i zY m−k+i .
Then the variety V has the final colength.
Proof. Because identity (3) is not satisfied in the varieties V 1 and V 1 , from conditions of the theorem follows that V 1 , V 1 ⊂ V ⊂ N s A. Then by theorem 1 of article [4], there exists a constant C independent of n for such in the sum (1) is true the condition (n − λ 1 ) < C. In this case in the sum (2) the number of the non-zero terms is bounded by a constant independent of n. Thus, to prove the result, it suffices to establish that all multiplicities m λ are bounded by a constant, which also is independent of n.
The article [5] is proved that the multiplicity m λ (V) is equal to the number of linearly independent polyhomogeneous elements of special form. We will show that the dimension of the space of polyhomogeneous elements is bounded by a constant independent of n, which will complete the proof of the theorem.
Consider λ, for which m λ = 0. For such partition is true the condition (n − λ 1 ) < C and to it correspond monomials of the form
g s = Y α 1 x i 1 Y α 2 x i 2 Y α 3 . . . Y αs x is Y α s+1 ,
where s < C. Denote by Q λ 1 the space generated by elements g s . We prove that the number of linearly independent monomials g s bounded by a constant. The proof is by induction on the number s of generators x i and lexicographic order on lines of the form (α 1 , α 2 , . . . , α s+1 ).
Consider the case s = 1. Then generating monomials of the space Q λ 1 have the form: Y α 1 x 1 Y α 2 . If for these elements are true the conditions α 1 ≥ m and α 2 ≥ m, then by the identity (3) they can be represented as a linear combination of the elements, in which α 1 < m. Thus any monomial will be expressed through such monomials in which either only α 1 ≥ m or α 2 ≥ m. The number of such monomials is bounded by the constant 2m independent of n.
In the general case the space Q λ 1 will generate by elements, in which only one α i is not less then tm. Note, that the general number of such elements is bounded by a constant independent of n.
Let i will be a smallest index for which α i ≥ tm. consider the corresponding element:
g s = yY α 1 x t 1 Y α 2 . . . Y α i x t i Y αi+1 . . . Y αs x ts Y α s+1 .
If α i+1 ≥ tm, then the identity (3) allows to bring the element g s to a linear combination of words, that are lexicographically less.lexicographically smaller. If α i+1 < tm, then modulo words, lexicographically smaller, the element g s can be written as
Y α 1 x t 1 . . . Y α i −α i+1 −1 (y(y . . . (yx t i ) . . . )) α i+1 +1 · · (y(y . . . (yx t i+1 ) . . . )) α i+1 Y α i+2 . . . Y αs x ts Y α s+1 .
The Leibnitz identity allows to bring the last element to the sum of terms, that are lexicographically smaller, and term
Y α 1 x t 1 . . . Y α i −α i+1 −1 X ′ Y α i+2 . . . Y αs x ts Y α s+1 , where x ′ = (y(y . . . (yx t i ) . . . )) α i+1 +1 (y(y . . . (yx t i+1 ) . . . )) α i+1 .
We will contain element with fewer generators x ir covered by the induction assumption. The theorem is proved.
The basis of multilinear part of variety V 3 of Leibnitz algebras
The variety V 3 of Leibnitz algebras is an equivalent to the well-known variety V 3 of Lie algebras. Previously, in the article [1] the growth of this variety was designated, and in the article [8] -its multiplicities and colength.
Let T = Φ[t] be a ring of polynomial in the variable t. Consider threedimensional Heisenberg algebra H with the besis {a, b, c} and multiplication ba = −ab = c, the product of the remaining basis elements is zero. Well known and easy to verify that the algebra H is nilpotent of the class two Lie algebra. Transform the polynomial ring T in the right module of algebra H, in which the basis elements of algebra H act on the right on the polynomial f from T follows:
f a = f ′ , f b = tf, f c = f, where f ′ is a partial derivative of a polynomial f in the variable t.
Consider the direct sum of vector spaces H and T with multiplication by the rule:
(x + f )(y + g) = xy + f y, where x, y are from H; f, g are from T . Denote it by the symbol H. Direct verification shows that H is an algebra of Leibnitz. The algebra H is the Leibnitz algebra, satisfies to the identity x(y(zt)) ≡ 0 and generates the variety V 3 of Leibnitz algebras.
Lemma. The variety V 3 satisfies to the identities: (4) x(y(zt)) ≡ 0,
(5) x 0 Ax 1 Bx 2 Cx 3 Dx 4 ≡ 0, (6) x 0 (x 1 x 4 )(x 2 x 3 ) ≡ x 0 (x 1 x 2 )(x 3 x 4 ) + x 0 (x 1 x 3 )(x 2 x 4 ),
where A, B, C, D are some words from generators. Proof. The truth of identities (4) and (5) verified by arbitrary replacement generators by elements of algebra H and was showed in the paper [1]. Consider the following special form of the second identity:
x 0 x 1 x 2 x 3 x 4 ≡ 0.
Presenting it as a sum and using the identity xyz − xzy ≡ x(yz), we obtain:
2x 0 (x 1 x 2 )(x 3 x 4 ) − 2x 0 (x 1 x 3 )(x 2 x 4 ) + 2x 0 (x 1 x 4 )(x 2 x 3 ) ≡ 0.
Dividing this identity by 2 and moving the second term to the right, we obtain the identity (6). The lemma is proved.
Theorem 2. The set elements of form
θ(i, i 1 , ..., i m , j 1 , ..., j m ) = x i (x i 1 x j 1 )(x i 2 x j 2 )...(x im x jm )x k 1 x k 2 ...x k n−2m−1 ,
where i s < j s , s = 1, 2, ..., m, i 1 < i 2 < ... < i m , j 1 < j 2 < ... < j m , k 1 < k 2 < ... < k − 2m − 1, is a basis of the space P n ( V 3 ).
Proof. Consider an arbitrary element of the space P n ( V 3 ). Using corollary xy(zt) ≡ x(zt)y from the Leibnitz identity and identity (4), move the all pairs as far right as possible.
We order the elements obtained using the lexicographic ordering of lines (k 1 , k 2 , . . . , k n−2m−1 ). Let the considering element has a form:
x i (x i 1 x j 1 )(x i 2 x j 2 )...(x im x jm )x k 1 ...x ks x k s+1 ...x k n−2m−1
and k s > k s+1 . Using the identity xyz ≡ xzy + x(yz) we can write this element as a sum
x i (x i 1 x j 1 )(x i 2 x j 2 )...(x im x jm )x k 1 ...x k s+1 x ks ...x k−2m−1 + +x i (x i 1 x j 1 )(x i 2 x j 2 )...(x im x jm )x k 1 ...x k s−1 (x ks x k s+1 )x k s+2 ...x k−2m−1 ,
where the first term is lexicographically less, than parent element, and the second term has fewer number of single elements. Applying the same method to the resulting term, we eventually present our original element as a sum of terms, in that k 1 < k 2 < ... < k n−2m−1 .
Consider an arbitrary element, in which indexes of single elements are ordered. We choose the two lowest index in the considered element and redenote them through 1 ′ and 2 ′ relatively. We introduce the lexicographic order on lines (j 1 , j 2 , . . . , j m ). Using also the induction on the number of brackets, we will prove, that all received elements can be represented as a linear combination of elements θ(i, i 1 , ..., i m , j 1 , ..., j m ). The corollary x(yz) ≡ −x(zy) of Leibnitz identity allows to order the indexes of elements in couples, and the identity xy(zt) ≡ x(zt)y allows to order the brackets by the indexes of first elements. According to these identities, the element can be written either in the form
x i (x 1 ′ x 2 ′ )(x i 1 x j 1 )...(x i m−1 x j m−1 )x k 1 x k 2 .
..x k n−2m−1 , either in the form
x i (x 1 ′ x j 1 )(x 2 ′ x j 2 )...(x i m−2 x jm )x k 1 x k 2 .
..x k n−2m−1 . In the first case we can consider the ordering on m−1 brackets that runs by induction. In the second case we apply the identity (6) and obtain:
x i (x 1 ′ x 2 ′ )(x j 2 x j 1 )...(x i m−2 x jm )x k 1 x k 2 ...x k n−2m−1 + x i (x 1 ′ x j 2 )(x 2 ′ x j 1 )...(x i m−2 x jm )x k 1 x k 2 .
..x k n−2m−1 , where for the first term can be again apply the induction hypothesis, and the second term is lexicographically less. Therefore, any element of the space P n ( V 3 ) can be written as a linear combination of elements θ(i, i 1 , ..., i m , j 1 , ..., j m ) modulo Id( V 3 ).
We now prove, that the elements θ(i, i 1 , ..., i m , j 1 , ..., j m ) are linearly independent modulo Id( V 3 ). Consider the linear combination of these elements:
(i,i 1 ,...,im.j 1 ,...,jm) α(i, i 1 , ..., i m , j 1 , ..., j m )θ(i, i 1 , ..., i m , j 1 , ..., j m ) = 0
and show that all coefficient α(i, i 1 , ..., i m , j 1 , ..., j m ) are zero. Assume the contrary.Choose an element θ(i * , i * 1 , ..., i * m , j * 1 , ..., j * m ) with non-zero coefficient α(i * , i * 1 , ..., i * m , j * 1 , ..., j * m ) such that the number of commutators m in it is the least and the index j * 1 of element in the second position in the first bracket is the largest. Since each element is uniquely determined by the number m, by the element x i and by the sample (i, i 1 , ..., i m , j 1 , ..., j m ), then in the selected element θ(i * , i * 1 , ..., i * m , j * 1 , ..., j * m ) these rates are fixed. We replace its generators on the basis elements of algebra H as follows:.., m, the rest generators we replace on element c. After this substitution all elements, which are different from the chosen, will be equal to zero: if the element x i * will be replace on the basis element of Heisenberg algebra, than this element will be zero (because xf ≡ 0 for any x from H); if the element will have more then m commutators, then it will be also zero (since the element c from the center of algebra fall into the commutator); a similar situation arises, if the element will have m commutators but the sample will be different from the fixed. Indeed, there are two kinds of elements containing m commutators at a fixed sample (i * , i * 1 , ..., i * m , j * 1 , ..., j * m ): these are the elements that contain in the second position and the first bracket x j * 1 , and elements that contain in the second position and the first bracket x i * s , where i * s < j * 1 (s = 2, ..., m). All elements of the second kind are zero, as by described substitution the first bracket will be equal to (aa). In one of brackets of the elements of the first type fall generators x i * s and x i * t . As a result of described substitution this bracket also resets the element. Thus we obtained, that if f = 0, then (i,i 1 ,...,im,j 1 ,...,jm) α(i, i 1 , ..., i m , j 1 , ..., j m )θ(i, i 1 , ..., i m , j 1 , ..., j m ) = 0. Consequently, contrary to the assumption α(i * , i * 1 , ..., i * m , j * 1 , ..., j * m ) is zero. The theorem is proved.Note that in the proof of the theorem we used only the Leibnitz identity and corollaries from it, the identity x(y(zt)) ≡ 0, and lastly the identity (3). Consequently, any identity that runs in the variety V 3 , is a corollary from these identities. Hence we obtain the following assertion.Corollary. The identities(4)and(6)form a basis of identities of variety V 3 .The authors thank S.P. Mishchenko for useful advice and attention to this work.
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